PMC Physics A

, 3:3 | Cite as

B meson decays

  • Marina Artuso
  • Elisabetta Barberio
  • Sheldon Stone
Open Access
Review

Abstract

We discuss the most important Physics thus far extracted from studies of B meson decays. Measurements of the four CP violating angles accessible in B decay are reviewed as well as direct CP violation. A detailed discussion of the measurements of the CKM elements V cb and V ub from semileptonic decays is given, and the differences between resulting values using inclusive decays versus exclusive decays is discussed. Measurements of "rare" decays are also reviewed. We point out where CP violating and rare decays could lead to observations of physics beyond that of the Standard Model in future experiments. If such physics is found by directly observation of new particles, e.g. in LHC experiments, B decays can play a decisive role in interpreting the nature of these particles.

PACS Codes: 13.25.Hw, 14.40.Nd, 14.65.Fy

1 Introduction

The forces of nature generally reveal their properties by how they act on matter. On the most fundamental material scale that we are aware of matter is formed from fermions. These take two forms, leptonic matter and quark matter. The former do not have any strong interactions, that is they lack a property called "color charge", which allow quarks to bind together either mesonically or baryonically. New and therefore as yet unknown forces could effect both leptons and quarks. Here, we concentrate on how such forces effect quarks, especially the b quark.

Light matter consists mostly of u, d and s quarks. In 1963 Cabibbo showed that weak interactions of mesons and baryons containing s quarks were suppressed with respect to those without s quarks by an amount tan θ C , where the "Cabibbo" angle θ C must be determined experimentally [1]. The s was further shown to have an important and at that time a mystifying role, by the discovery of CP violation in K L 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdUealnaaDaaaleaacqWGmbataeaacqaIWaamaaaaaa@2E52@ Open image in new window decays in 1964 [2, 3]. (In CP, C stands for charge conservation invariance and P for parity invariance. The operation of C takes a particle to anti-particle state and the operation of P changes left to right. [4, 5, 6].) When the c quark was discovered in Nov. 1974 [7, 8] (though its existence was speculated earlier [9, 10, 11]), it became clear that θ C was the mixing angle between two quark doublets, (c, s) and (u, d). However, it is not possible to generate a CP violating phase with only two quark doublets.

This was recognized even before the discovery of the charm quark by Kobayashi and Maskawa, who postulated the existence of yet another quark doublet (b, t) [12], in work for which they were awarded the Nobel Prize in 2008. While the t quark is the heaviest, having a mass of 173 GeV, they are difficult to produce and decay before they can form a hadron, thus excluding many otherwise possible studies. Much interesting work has been done with the s and c quarks, but in this article we discuss the physics of the b quark, which turns out to be the most interesting of the six quarks to study.

1.1 How B's Fit Into the Standard Model

First we will discuss how particles formed with b-quarks fit into current paradigm of particle physics, the "Standard Model" (SM) [13, 14, 15]. The SM has at its basis the gauge group SU(3)xSU(2)xU(1). The first term corresponds to the strong interaction and SU(3) describes the octet of colored gluons which are the strong force carriers of quantum chromodynamics. SU(2)xU(1) describes the weak interaction and is the product of weak isospin and hypercharge. We speak of the fundamental objects being spin-1/2 quarks and leptons and the force carriers generally spin-1 objects. The spin-0 Higgs boson, yet to be discovered, is necessary for generating mass (to view a cartoon of the mechanism of Higgs mass generation, see [16]).

Particles containing b quarks can be B0, B-, B s , or B c mesons, depending on whether the light anti-quark that it pairs with is d ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdsgaKzaaraaaaa@2C60@ Open image in new window, u ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdwha1zaaraaaaa@2C82@ Open image in new window, s ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdohaZzaaraaaaa@2C7E@ Open image in new window, or c ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdogaJzaaraaaaa@2C5E@ Open image in new window or a baryon containing two other quarks. Mesons containing b and b ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkgaIzaaraaaaa@2C5C@ Open image in new window quarks are also interesting especially for studies of quantum chromodynamics, but will not be discussed further in this article, as we will concentrate on weak decays and discuss strong interactions as an important and necessary complication that must be understood in many cases to extract information on fundamental b quark couplings.

The quarks come in three repetitions called generations, as do the leptons. The first generation is d u, the second s c and the third b t. In the second and third generations the charge +2/3 quark is heavier than the charge -1/3; the first generation has two very light quarks on the order of a few MeV with the d thought to be a bit heavier. (Isospin invariance is related to the equality of u and d quark masses. The PDG [17] gives the u quark mass between 1.5–3.3 MeV and the d mass between 3.5–6.0 MeV, where the large range indicates the considerable uncertainties.) Decays usually proceed within generations, so the c decays predominantly to the s quark via the quark level process cW+ s, though some decays do go to the first generation as cW+ d. The ratio of these amplitudes approximate the Cabibbo angle discussed earlier.

The mixing matrix proposed by Kobayshi and Maskawa [12] parameterizes the mixing between the mass eigenstates and weak eigenstates as couplings between the charge +2/3 and -1/3 quarks. We use here the Wolfenstein approximation [18] good to order λ4:
V C K M = ( V u d V u s V u b V c d V c s V c b V t d V t s V t d ) = ( 1 λ 2 / 2 λ A λ 3 ( ρ i η ) λ 1 λ 2 / 2 λ 4 ( 1 + 4 A 2 ) / 8 A λ 2 A λ 3 ( 1 ρ i η ) A λ 2 + A λ 4 ( 1 / 2 ( ρ + i η ) ) 1 A 2 λ 4 / 2 ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakqaaeeqaaiabdAfawnaaBaaaleaacqWGdbWqcqWGlbWscqWGnbqtaeqaaOGaeyypa0ZaaeWaaeaafaqabeWadaaabaGaemOvay1aaSbaaSqaaiabdwha1jabdsgaKbqabaaakeaacqWGwbGvdaWgaaWcbaGaemyDauNaem4CamhabeaaaOqaaiabdAfawnaaBaaaleaacqWG1bqDcqWGIbGyaeqaaaGcbaGaemOvay1aaSbaaSqaaiabdogaJjabdsgaKbqabaaakeaacqWGwbGvdaWgaaWcbaGaem4yamMaem4CamhabeaaaOqaaiabdAfawnaaBaaaleaacqWGJbWycqWGIbGyaeqaaaGcbaGaemOvay1aaSbaaSqaaiabdsha0jabdsgaKbqabaaakeaacqWGwbGvdaWgaaWcbaGaemiDaqNaem4CamhabeaaaOqaaiabdAfawnaaBaaaleaacqWG0baDcqWGKbazaeqaaaaaaOGaayjkaiaawMcaaaqaaiabg2da9maabmaabaqbaeqabmWaaaqaaiabigdaXiabgkHiTiabeU7aSnaaCaaaleqabaGaeGOmaidaaOGaei4la8IaeGOmaidabaGaeq4UdWgabaGaemyqaeKaeq4UdW2aaWbaaSqabeaacqaIZaWmaaGccqGGOaakcqaHbpGCcqGHsislcqWGPbqAcqaH3oaAcqGGPaqkaeaacqGHsislcqaH7oaBaeaacqaIXaqmcqGHsislcqaH7oaBdaahaaWcbeqaaiabikdaYaaakiabc+caViabikdaYiabgkHiTiabeU7aSnaaCaaaleqabaGaeGinaqdaaOGaeiikaGIaeGymaeJaey4kaSIaeGinaqJaemyqae0aaWbaaSqabeaacqaIYaGmaaGccqGGPaqkcqGGVaWlcqaI4aaoaeaacqWGbbqqcqaH7oaBdaahaaWcbeqaaiabikdaYaaaaOqaaiabdgeabjabeU7aSnaaCaaaleqabaGaeG4mamdaaOGaeiikaGIaeGymaeJaeyOeI0IaeqyWdiNaeyOeI0IaemyAaKMaeq4TdGMaeiykaKcabaGaeyOeI0IaemyqaeKaeq4UdW2aaWbaaSqabeaacqaIYaGmaaGccqGHRaWkcqWGbbqqcqaH7oaBdaahaaWcbeqaaiabisda0aaakiabcIcaOiabigdaXiabc+caViabikdaYiabgkHiTiabcIcaOiabeg8aYjabgUcaRiabdMgaPjabeE7aOjabcMcaPiabcMcaPaqaaiabigdaXiabgkHiTiabdgeabnaaCaaaleqabaGaeGOmaidaaOGaeq4UdW2aaWbaaSqabeaacqaI0aanaaGccqGGVaWlcqaIYaGmaaaacaGLOaGaayzkaaGaeiOla4caaaa@B440@ Open image in new window
(1)
In the Standard Model A, λ, ρ, and η are fundamental constants of nature like G, or α EM ; η multiplies the imaginary i and is responsible for all Standard Model CP violation. We know λ = 0.226, A ~0.8 and we have constraints on ρ and η. Often the variables ρ ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbeg8aYzaaraaaaa@2CCF@ Open image in new window and η ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbeE7aOzaaraaaaa@2CBB@ Open image in new window are used where
ρ ¯ + i η ¯ = ( ρ + i η ) ( 1 λ 2 / 2 ) = V u d V u b * V c d V c b * , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacuaHbpGCgaqeaiabgUcaRiabdMgaPjqbeE7aOzaaraGaeyypa0JaeiikaGIaeqyWdiNaey4kaSIaemyAaKMaeq4TdGMaeiykaKIaeiikaGIaeGymaeJaeyOeI0Iaeq4UdW2aaWbaaSqabeaacqaIYaGmaaGccqGGVaWlcqaIYaGmcqGGPaqkcqGH9aqpcqGHsisljuaGdaWcaaqaaiabdAfawnaaBaaabaGaemyDauNaemizaqgabeaacqWGwbGvdaqhaaqaaiabdwha1jabdkgaIbqaaiabcQcaQaaaaeaacqWGwbGvdaWgaaqaaiabdogaJjabdsgaKbqabaGaemOvay1aa0baaeaacqWGJbWycqWGIbGyaeaacqGGQaGkaaaaaOGaeiilaWcaaa@5621@ Open image in new window
(2)

where the definition in terms of the CKM matrix elements is correct to all orders in λ [19] (see also [20]).

Applying unitarity constraints allows us to construct the six independent triangles shown in Figure 1. Another basis for the CKM matrix are four angles labeled as χ, χ' and any two of α, β and γ since α + β + γ = π [21, 22]. (These angles are also shown in Figure 1.) CP violation measurements make use of these angles. (The Belle collaboration defines ϕ2α, ϕ1β, and ϕ3γ.)
Figure 1

The 6 CKM triangles resulting from applying unitarity constraints to the indicated row and column. The CP violating angles are also shown.

B meson decays can occur through various processes. Some decay diagrams with intermediate charged vector bosons are shown in Figure 2. The simple spectator diagram shown Figure 2(a) has by far the largest rate. Semileptonic decays proceed through this diagram, and allow us to measure the CKM couplings V cb and V ub by considering only the hadronic uncertainties due to the spectator quark. The color suppressed diagram Figure 2(b) exists only for hadronic decays. It can occur only when the colors of the quarks from the virtual W- decay match those of the initial B meson. Since this happens only 1/3 of the time in amplitude, the rate is down by almost an order of magnitude from the spectator decays. The annihilation Figure 2(c) describes the important decay B-τ- ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window and will be discussed in detail later. The W exchange Figure 2(d) diagram is small. The box diagram Figure 2(e) is the source of mixing in the B system. An analogous diagram exists also for the B s meson. Finally the Penguin diagram Figure 2(f) is an example of one of several loop diagrams leading to "rare decays" that will be discussed in detail in a subsequent section.
Figure 2

Some B decay diagrams.

1.1.1 Dark Matter

"Dark Matter" was first shown to exist by Zwicky studying rotation curves of galaxies [23]. The motion could only be explained if there was massive cloud of matter that was not luminous. We still do not know what composes this dark matter, though hopes are it will be discovered at the LHC. An even more mysterious phenomena called "Dark Energy" may also have a connection to particle physics experiments [24], perhaps via "Extra Dimensions" [25].

1.1.2 Baryogenesis

When the Universe began with the Big Bang, there was an equal amount of matter and antimatter. Now we have mostly matter. How did it happen? Sakharov gave three necessary conditions: Baryon ( MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ Open image in new window) number violation, departure from thermal equilibrium, and C and CP violation [26]. (The operation of Charge Conjugation (C) takes particle to anti-particle and Parity (P) takes a vector r MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkhaYzaalaaaaa@2C76@ Open image in new window to - r MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkhaYzaalaaaaa@2C76@ Open image in new window.)

These criteria are all satisfied by the Standard Model. MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ Open image in new window is violated in Electroweak theory at high temperature, though baryon minus lepton number is conserved; in addition we need quantum tunneling, which is powerfully suppressed at the low temperatures that we now have. Non-thermal equilibrium is provided by the electroweak phase transition. C and CP are violated by weak interactions. However the violation is too small. The ratio of the number of baryons to the entropy in the observed part of the Universe needs to be ~5 × 10-11, while the SM provides many orders of magnitude less. Therefore, there must be new physics [27].

1.1.3 The Hierarchy Problem

Our worry is why the Planck scale at ~1019 GeV is so much higher than the scale at which we expect to find the Higgs Boson, ~100 GeV. As Lisa Randall said [28] "The gist of it is that the universe seems to have two entirely different mass scales, and we don't understand why they are so different. There's what's called the Planck scale, which is associated with gravitational interactions. It's a huge mass scale, but because gravitational forces are proportional to one over the mass squared, that means gravity is a very weak interaction. In units of GeV, which is how we measure masses, the Planck scale is 1019 GeV. Then there's the electroweak scale, which sets the masses for the W and Z bosons. These are particles that are similar to the photons of electromagnetism and which we have observed and studied well. They have a mass of about 100 GeV. So the hierarchy problem, in its simplest manifestation, is how can you have these particles be so light when the other scale is so big." We expect the explanation lies in physics beyond the Standard Model [29].

1.2 BDecays as Probes for New Physics

When we make measurements on B decays we observe the contributions of SM processes as well as any other processes that may be due to physics beyond the SM or New Physics (NP). Other diagrams would appear adding to those in Figure 2 with new intermediate particles. Thus, when it is declared by those who say that there isn't any evidence of NP in B decays, we have to be very careful that we have not absorbed such new evidence into what we declare to be SM physics. There are several approaches that can be followed.

One approach is to simply predict the decay rate of a single process in the SM with known couplings and compare to the measurements. The classical case here is bs γ and we will discuss this and other specific examples later. Another approach is make different measurements of the CKM parameters in different ways and see if they agree. This is normally done by measuring both angles and sides of the CKM triangle, but other quantities can also be used. This is the approach used by the CKM fitter [19] (see also [20]) and UT fit groups [30]. In yet a third approach, the exact same quantity can be measured in several ways, even if cannot be predicted in the SM. An example here is measuring the CP violating angle β using B0J/ψ K S decays that proceed through the diagram in Figure 2(b), at least in the SM, and another process that uses the "Penguin" diagram in Figure 2(f), e.g. B0ϕ K S .

The punch line is that if new, more massive particles exist in a mass range accessible to the LHC then they MUST contribute to rare and CP violating B decays! Even if measurements are precise enough only to limit the size of these effects, the properties of these new particles will be much better understood. This is the raison d'être for the further study of B decays.

2 Measurements of mixing and CP violation

2.1 Neutral BMeson Mixing

Neutral B mesons can transform into their anti-particles before they decay. The diagrams for this process are shown in Figure 3 for the B d . There is a similar diagram for the B s . Although u, c and t quark exchanges are all shown, the t quark plays a dominant role mainly due to its mass, as the amplitude of this process is proportional to the mass of the exchanged fermion.
Figure 3

The two diagrams for B d mixing.

Under the weak interactions the eigenstates of flavor degenerate in pure QCD can mix. Let the quantum mechanical basis vectors be {|1⟩, |2⟩} ≡ {|B0⟩, | B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window⟩}. The Hamiltonian is then
H = M i 2 Γ = ( M M 12 M 12 * M ) i 2 ( Γ Γ 12 Γ 12 * Γ ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@59C5@ Open image in new window
(3)
The Schrödinger equation is
i d d t ( | B 0 ( t ) | B ¯ 0 ( t ) ) = H ( | B 0 ( t ) | B ¯ 0 ( t ) ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6200@ Open image in new window
(4)
Diagonalizing we have
Δ m = m B H m B L = 2 | M 12 | MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGTbqBcqGH9aqpcqWGTbqBdaWgaaWcbaGaemOqai0aaSbaaWqaaiabdIeaibqabaaaleqaaOGaeyOeI0IaemyBa02aaSbaaSqaaiabdkeacnaaBaaameaacqWGmbataeqaaaWcbeaakiabg2da9iabikdaYiabcYha8jabd2eannaaBaaaleaacqaIXaqmcqaIYaGmaeqaaOGaeiiFaWhaaa@3F3B@ Open image in new window
(5)

ΔΓ = Γ L - Γ H = 2|Γ12|cos ϕ

where H refers to the heavier and L the lighter of the two weak eigenstates, and ϕ = arg(-M1212). We expect that ΔΓ is very small for B0 mesons but should be significant for B s mesons.

B d mixing was first discovered by the ARGUS experiment [31] (There was a previous measurement by UA1 indicating mixing for a mixture of B d 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdkeacnaaDaaaleaacqWGKbazaeaacqaIWaamaaaaaa@2E70@ Open image in new window and B s 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdkeacnaaDaaaleaacqWGZbWCaeaacqaIWaamaaaaaa@2E8E@ Open image in new window[32]). At the time it was quite a surprise, since the top-quark mass, m t , was thought to be in the 30 GeV range. Since b-flavored hadrons are produced in pairs, it is possible to observe a mixed event by having both B's decay semileptonically. Thus the number of events where both Bs decay into leptons of the same sign is indicative of mixing. We can define R as the ratio of events with same-sign leptons to the sum of same-sign plus opposite-sign dilepton events. This is related to the mixing probability. The OPAL data for R are shown in Figure 4[33].
Figure 4

The ratio, R, of same-sign to total dilepton events as a function of proper decay time, for selected B D *+ X - ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window events. The jet charge in the opposite hemisphere is used to determine the sign correlation. The curve is the result of a fit to the mixing parameter. The.

Data from many experiments has been combined by "The Heavy Flavor Averaging Group," (HFAG) to obtain an average value Δm d = (0.507 ± 0.004) × 1012 ps-1 [34].

The probability of mixing is related to the CKM matrix elements as [3, 35]
x d Δ m Γ = G F 2 6 π 2 B B d f B 2 m B τ B | V t b * V t d | 2 m t 2 F ( m t 2 M W 2 ) η Q C D , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6BC5@ Open image in new window
(7)

where B B is a parameter related to the probability of the d and b ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkgaIzaaraaaaa@2C5C@ Open image in new window quarks forming a hadron and must be estimated theoretically, F is a known function which increases approximately as m t 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWG0baDaeaacqaIYaGmaaaaaa@2EEA@ Open image in new window, and η QCD is a QCD correction, with value about 0.8. By far the largest uncertainty arises from the decay constant, f B .

In principle f B can be measured. The decay rate of the annihilation process B- → ℓ- ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window is proportional to the product of f B 2 | V u b | 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAgaMnaaDaaaleaacqWGcbGqaeaacqaIYaGmaaGccqGG8baFcqWGwbGvdaWgaaWcbaGaemyDauNaemOyaigabeaakiabcYha8naaCaaaleqabaGaeGOmaidaaaaa@36CC@ Open image in new window. Of course there is a substantial uncertainty associated with |V ub |. The experimental evidence for B-τ- ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window is discussed in Section 5.1, and substantial uncertainty exists in the branching ratio measurement. Thus, we need to rely on theory for a value of f B .

The ratio of B s to B d mixing frequency, however, provides a better situation in terms of reducing model dependent errors. Using Eq. 7 for B d mixing and an analogous relation for B s mixing and then dividing them results in
x d x s = B B B B s f B 2 f B s 2 m B m B s τ B τ B s | V t b * V t d | 2 | V t b * V t s | 2 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6AA5@ Open image in new window
(8)
The CKM terms are
| V t b * V t d | 2 = A λ 3 | ( 1 ρ i η ) | 2 = A λ 3 ( ρ 1 ) 2 + η 2  and | V t b * V t s | 2 = A λ 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@713E@ Open image in new window
(9)

ignoring the higher order term in V ts . Solving the ratio of the two above equations for ( 1 λ ) | V t d | | V t s | MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaabmaajuaGbaWaaSaaaeaacqaIXaqmaeaacqaH7oaBaaaakiaawIcacaGLPaaajuaGdaWcaaqaaiabcYha8jabdAfawnaaBaaabaGaemiDaqNaemizaqgabeaacqGG8baFaeaacqGG8baFcqWGwbGvdaWgaaqaaiabdsha0jabdohaZbqabaGaeiiFaWhaaaaa@3EB8@ Open image in new window gives a circle centered at (1,0) in the ρ - η plane whose radius depends on x d /x s . The theoretical errors are now due to the difference between having a light quark versus a strange quark in the B meson, called SU(3) splitting.

For many years experiments at the Z0 using e+ e- colliders at both LEP and the SLC had set lower limits on B s 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdkeacnaaDaaaleaacqWGZbWCaeaacqaIWaamaaaaaa@2E8E@ Open image in new window mixing [17]. In 2006 B s mixing was measured by the CDF collaboration [36, 37]. The D0 collaboration had previously presented both a lower and an upper limit [38].

We now discuss the CDF measurement. The probability, P MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaGqaaiab=bfaqbaa@2C25@ Open image in new window(t) for a B s to oscillate into a B ¯ s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaSbaaSqaaiabdohaZbqabaaaaa@2DB7@ Open image in new window is given as
P ( t ) ( B s B ¯ s ) = 1 2 Γ s e Γ s t [ 1 + cos ( Δ m s t ) ] , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaaieaacqWFqbaucqGGOaakcqWG0baDcqGGPaqkcqGGOaakcqWGcbGqdaWgaaWcbaGaem4CamhabeaakiabgkziUkqbdkeaczaaraWaaSbaaSqaaiabdohaZbqabaGccqGGPaqkcqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabikdaYaaakiabfo5ahnaaBaaaleaacqWGZbWCaeqaaOGaemyzau2aaWbaaSqabeaacqGHsislcqqHtoWrdaWgaaadbaGaem4Camhabeaaliabdsha0baakiabcUfaBjabigdaXiabgUcaRiGbcogaJjabc+gaVjabcohaZjabcIcaOiabfs5aejabd2gaTnaaBaaaleaacqWGZbWCaeqaaOGaemiDaqNaeiykaKIaeiyxa0LaeiilaWcaaa@563D@ Open image in new window
(10)

where t is the proper time.

An amplitude A for each test frequency ω, is defined as [39]
P ( t ) = 1 2 Γ s e Γ s t [ 1 + A cos ( ω t ) ] . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaaieaacqWFqbaucqGGOaakcqWG0baDcqGGPaqkcqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabikdaYaaakiabfo5ahnaaBaaaleaacqWGZbWCaeqaaOGaemyzau2aaWbaaSqabeaacqGHsislcqqHtoWrdaWgaaadbaGaem4Camhabeaaliabdsha0baakiabcUfaBjabigdaXiabgUcaRiabdgeabjGbcogaJjabc+gaVjabcohaZjabcIcaOiabeM8a3jabdsha0jabcMcaPiabc2faDjabc6caUaaa@4B90@ Open image in new window
(11)
For each frequency the expected result is either zero for no mixing, or one for mixing. No other value is physical, although measurement errors admit other values. Figure 5 shows the CDF results.
Figure 5

The measured amplitude values and uncertainties versus the B s B ¯ s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaSbaaSqaaiabdohaZbqabaaaaa@2DB7@ Open image in new window oscillation frequency Δ m s for a combination of semileptonic and hadronic decay modes from CDF.

At Δm s = 17.75 ps-1, the observed amplitude A = 1.21 ± 0.20 (stat.) is consistent with unity, indicating that the data are compatible with B s B ¯ s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaSbaaSqaaiabdohaZbqabaaaaa@2DB7@ Open image in new window oscillations with that frequency, while the amplitude is inconsistent with zero: A/σ A = 6.05, where σ A is the statistical uncertainty on A, the ratio has negligible systematic uncertainties. (This then is called a "6σ effect.) The measured value is
Δ m s = 17.77 ± 0.010 ± 0.07  ps 1 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGTbqBdaWgaaWcbaGaem4Camhabeaakiabg2da9iabigdaXiabiEda3iabc6caUiabiodaZiabigdaXmaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIXaqmcqaI4aaoaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIZaWmcqaIZaWmaaGccqGHXcqScqaIWaamcqGGUaGlcqaIWaamcqaI3aWncqqGGaaicqqGWbaCcqqGZbWCdaahaaWcbeqaaiabgkHiTiabigdaXaaakiabcYcaSaaa@4A37@ Open image in new window
(12)

where the first error is statistical and the second systematic.

In order to translate the mixing measurements to constraints on the CKM parameters ρ and η, we need to use theoretical values for the ratios B B B B s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaKqbaoaalaaabaGaemOqai0aaSbaaeaacqWGcbGqaeqaaaqaaiabdkeacnaaBaaabaGaemOqai0aaSbaaeaacqWGZbWCaeqaaaqabaaaaaaa@319B@ Open image in new window and f B f B s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaKqbaoaalaaabaGaemOzay2aaSbaaeaacqWGcbGqaeqaaaqaaiabdAgaMnaaBaaabaGaemOqai0aaSbaaeaacqWGZbWCaeqaaaqabaaaaaaa@322B@ Open image in new window in Eq. 8. It is interesting that in practice it is not possible to measure either of these numbers directly. It is usually assumed, however, that f B + MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAgaMnaaBaaaleaacqWGcbGqdaahaaadbeqaaiabgUcaRaaaaSqabaaaaa@2EA0@ Open image in new window, which could in principle be measured via the process B-τ- ν is the same as f B . There is no way to measure f B s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAgaMnaaBaaaleaacqWGcbGqdaWgaaadbaGaem4CamhabeaaaSqabaaaaa@2F2C@ Open image in new window. The charm system, on the other hand, provides us with both D+ → ℓ+ ν and D s + MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdseaenaaDaaaleaacqWGZbWCaeaacqGHRaWkaaaaaa@2E86@ Open image in new window → ℓ+ ν, and both rates have been measured. They also have been calculated in unquenched lattice quantum electrodynamics (QCD).

The combined efforts of the HPQCD and UKQCD collaborations predict f D + MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAgaMnaaBaaaleaacqWGebardaahaaadbeqaaiabgUcaRaaaaSqabaaaaa@2EA4@ Open image in new window = (207 ± 4) MeV [40], while the CLEO measurement is in astonishing agreement: (205.8 ± 8.5 ± 2.5) MeV [41]. Furthermore, the measurements of f D s + MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAgaMnaaBaaaleaacqWGebardaqhaaadbaGaem4CamhabaGaey4kaScaaaWcbeaaaaa@3013@ Open image in new window are not in such good agreement with the Follana et al calculation of (241 ± 3) MeV. The average of CLEO and Belle results as determined by Rosner and Stone is (273 ± 10) MeV [42]. (For updated results see [43, 44].) The discrepancy is at the 3 standard deviation level and not yet statistically significant, though it bears watching.

Unfortunately, the group that has calculated f D + MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAgaMnaaBaaaleaacqWGebardaahaaadbeqaaiabgUcaRaaaaSqabaaaaa@2EA4@ Open image in new window is not the same group that has determined f B + MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAgaMnaaBaaaleaacqWGcbGqdaahaaadbeqaaiabgUcaRaaaaSqabaaaaa@2EA0@ Open image in new window. The theoretical calculations used for the B B terms and the f B terms are summarized by Tantalo [45]. He suggests using values of f B s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAgaMnaaBaaaleaacqWGcbGqdaWgaaadbaGaem4CamhabeaaaSqabaaaaa@2F2C@ Open image in new window = (268 ± 17 ± 20) MeV, f B s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdAgaMnaaBaaaleaacqWGcbGqdaWgaaadbaGaem4CamhabeaaaSqabaaaaa@2F2C@ Open image in new window/f B = 1.20 ± 0.02 ± 0.05, B B s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdkeacnaaBaaaleaacqWGcbGqdaWgaaadbaGaem4CamhabeaaaSqabaaaaa@2EE4@ Open image in new window = 0.84 ± 0.03 ± 0.055 and B B = 0.83 ± 0.01 ± 0.06. These numbers allow us to measure the length of one side of the CKM triangle using Eq. 8. Use of this measurement will be discussed in more detail in Section 6.

2.2 CP Violation in the BSystem

We have two quantum mechanical operators: Charge Conjugation, C, and Parity, P. When applied to a particle wavefunction C changes particle to antiparticle and vice-versa. Applying P to a wavefunction ψ (r) we have P ψ (r) = ψ (-r). The P operator can be thought of changing the natural coordinate system from right-handed to left-handed. If nature was blind to handedness, then P would be always conserved. By applying the P operator twice we end up with P2ψ (r) = ψ (r), so the eigenvalues of P are ± 1. Therefore wave-functions, or particles represented by such wave-functions, have either intrinsic positive parity +1 (right-handed) or -1 (left-handed).

Weak interactions, characterized by a combination of vector minus axial-vector currents, are known to be left-handed. Therefore, handedness matters and its well known that Parity is maximally violated in weak decays [3]. Since C changes left-handed particles to right-handed anti-particles, the product CP symmetry could have been preserved, but nature decided otherwise. Different particle transitions involve the different CP violating angles shown in Figure 1. Measuring these independently allows comparisons with measurements of the sides of the triangle and any differences in constraints in ρ and η can be due to the presence of new physics.

Consider the case of a process Bf that goes via two amplitudes, A MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=bq8bbaa@366B@ Open image in new window and MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ Open image in new window each of which has a strong part e. g. s A MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdohaZnaaBaaaleaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFaeFqaeqaaaaa@3806@ Open image in new window and a weak part w A MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdEha3naaBaaaleaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFaeFqaeqaaaaa@380E@ Open image in new window. Then we have
Γ ( B f ) = ( | A | e i ( s A + w A ) + | | e i ( s + w ) ) 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqqHtoWrcqGGOaakcqWGcbGqcqGHsgIRcqWGMbGzcqGGPaqkcqGH9aqpdaqadaqaaiabcYha8nrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=bq8bjabcYha8jabdwgaLnaaCaaaleqabaGaemyAaKMaeiikaGIaem4Cam3aaSbaaWqaaiab=bq8bbqabaWccqGHRaWkcqWG3bWDdaWgaaadbaGae8haXheabeaaliabcMcaPaaakiabgUcaRiabcYha8jab=XsicjabcYha8jabdwgaLnaaCaaaleqabaGaemyAaKMaeiikaGIaem4Cam3aaSbaaWqaaiab=XsicbqabaWccqGHRaWkcqWG3bWDdaWgaaadbaGae8hlHieabeaaliabcMcaPaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaaaa@600B@ Open image in new window
(13)
Γ ( B ¯ f ¯ ) = ( | A | e i ( s A w A ) + | | e i ( s w ) ) 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqqHtoWrcqGGOaakcuWGcbGqgaqeaiabgkziUkqbdAgaMzaaraGaeiykaKIaeyypa0ZaaeWaaeaacqGG8baFt0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFaeFqcqGG8baFcqWGLbqzdaahaaWcbeqaaiabdMgaPjabcIcaOiabdohaZnaaBaaameaacqWFaeFqaeqaaSGaeyOeI0Iaem4DaC3aaSbaaWqaaiab=bq8bbqabaWccqGGPaqkaaGccqGHRaWkcqGG8baFcqWFSeIqcqGG8baFcqWGLbqzdaahaaWcbeqaaiabdMgaPjabcIcaOiabdohaZnaaBaaameaacqWFSeIqaeqaaSGaeyOeI0Iaem4DaC3aaSbaaWqaaiab=XsicbqabaWccqGGPaqkaaaakiaawIcacaGLPaaadaahaaWcbeqaaiabikdaYaaaaaa@6051@ Open image in new window
(14)
Γ ( B f ) Γ ( B ¯ f ¯ ) = 2 | A | sin ( s A s ) sin ( w A w ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqqHtoWrcqGGOaakcqWGcbGqcqGHsgIRcqWGMbGzcqGGPaqkcqGHsislcqqHtoWrcqGGOaakcuWGcbGqgaqeaiabgkziUkqbdAgaMzaaraGaeiykaKIaeyypa0JaeGOmaiJaeiiFaW3enfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8haXhKae8hlHiKaeiiFaWNagi4CamNaeiyAaKMaeiOBa4MaeiikaGIaem4Cam3aaSbaaSqaaiab=bq8bbqabaGccqGHsislcqWGZbWCdaWgaaWcbaGae8hlHieabeaakiabcMcaPiGbcohaZjabcMgaPjabc6gaUjabcIcaOiabdEha3naaBaaaleaacqWFaeFqaeqaaOGaeyOeI0Iaem4DaC3aaSbaaSqaaiab=XsicbqabaGccqGGPaqkcqGGUaGlaaa@667D@ Open image in new window
(15)

Any two amplitudes will do, though its better that they be of approximately equal size. Thus charged B decays can exhibit CP violation as well as neutral B decays. In some cases, we will see that it is possible to guarantee that | sin ( s A s ) | MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabcYha8jGbcohaZjabcMgaPjabc6gaUjabcIcaOiabdohaZnaaBaaaleaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFaeFqaeqaaOGaeyOeI0Iaem4Cam3aaSbaaSqaaiab=XsicbqabaGccqGGPaqkcqGG8baFaaa@4496@ Open image in new window is unity, so we can get information on the weak phases. In the case of neutral B decays, mixing serves as the second amplitude.

2.2.1 Formalism of CP Violation in Neutral BDecays

Consider the operations of Charge Conjugation, C, and Parity, P:
C | B ( p ) = | B ¯ ( p ) , C | B ¯ ( p ) = | B ( p ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaem4qamKaeiiFaWNaemOqaiKaeiikaGIafmiCaaNbaSaacqGGPaqkcqGHQms8cqGH9aqpcqGG8baFcuWGcbGqgaqeaiabcIcaOiqbdchaWzaalaGaeiykaKIaeyOkJeVaeiilaWcabaGaem4qamKaeiiFaWNafmOqaiKbaebacqGGOaakcuWGWbaCgaWcaiabcMcaPiabgQYiXlabg2da9iabcYha8jabdkeacjabcIcaOiqbdchaWzaalaGaeiykaKIaeyOkJepaaaaa@4DAD@ Open image in new window
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P | B ( p ) = | B ( p ) , P | B ¯ ( p ) = | B ¯ ( p ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaemiuaaLaeiiFaWNaemOqaiKaeiikaGIafmiCaaNbaSaacqGGPaqkcqGHQms8cqGH9aqpcqGHsislcqGG8baFcqWGcbGqcqGGOaakcqGHsislcuWGWbaCgaWcaiabcMcaPiabgQYiXlabcYcaSaqaaiabdcfaqjabcYha8jqbdkeaczaaraGaeiikaGIafmiCaaNbaSaacqGGPaqkcqGHQms8cqGH9aqpcqGHsislcqGG8baFcuWGcbGqgaqeaiabcIcaOiabgkHiTiqbdchaWzaalaGaeiykaKIaeyOkJepaaaaa@5195@ Open image in new window
(17)
C P | B ( p ) = | B ¯ ( p ) , C P | B ¯ ( p ) = | B ( p ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaem4qamKaemiuaaLaeiiFaWNaemOqaiKaeiikaGIafmiCaaNbaSaacqGGPaqkcqGHQms8cqGH9aqpcqGHsislcqGG8baFcuWGcbGqgaqeaiabcIcaOiabgkHiTiqbdchaWzaalaGaeiykaKIaeyOkJeVaeiilaWcabaGaem4qamKaemiuaaLaeiiFaWNafmOqaiKbaebacqGGOaakcuWGWbaCgaWcaiabcMcaPiabgQYiXlabg2da9iabgkHiTiabcYha8jabdkeacjabcIcaOiabgkHiTiqbdchaWzaalaGaeiykaKIaeyOkJeVaeiOla4caaaaa@5497@ Open image in new window
(18)
For neutral mesons we can construct the CP eigenstates
| B 1 0 = 1 2 ( | B 0 | B ¯ 0 ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqGG8baFcqWGcbGqdaqhaaWcbaGaeGymaedabaGaeGimaadaaOGaeyOkJeVaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaadaGcaaqaaiabikdaYaqabaaaaOWaaeWaaeaacqGG8baFcqWGcbGqdaahaaWcbeqaaiabicdaWaaakiabgQYiXlabgkHiTiabcYha8jqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaGccqGHQms8aiaawIcacaGLPaaacqGGSaalaaa@42C8@ Open image in new window
(19)
| B 2 0 = 1 2 ( | B 0 + | B ¯ 0 ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqGG8baFcqWGcbGqdaqhaaWcbaGaeGOmaidabaGaeGimaadaaOGaeyOkJeVaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaadaGcaaqaaiabikdaYaqabaaaaOWaaeWaaeaacqGG8baFcqWGcbGqdaahaaWcbeqaaiabicdaWaaakiabgQYiXlabgUcaRiabcYha8jqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaGccqGHQms8aiaawIcacaGLPaaacqGGSaalaaa@42BF@ Open image in new window
(20)
where
C P | B 1 0 = | B 1 0 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqadaaabaGaem4qamKaemiuaaLaeiiFaWNaemOqai0aa0baaSqaaiabigdaXaqaaiabicdaWaaakiabgQYiXdqaaiabg2da9aqaaiabcYha8jabdkeacnaaDaaaleaacqaIXaqmaeaacqaIWaamaaGccqGHQms8cqGGSaalaaaaaa@3B5B@ Open image in new window
(21)
C P | B 2 0 = | B 2 0 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqadaaabaGaem4qamKaemiuaaLaeiiFaWNaemOqai0aa0baaSqaaiabikdaYaqaaiabicdaWaaakiabgQYiXdqaaiabg2da9aqaaiabgkHiTiabcYha8jabdkeacnaaDaaaleaacqaIYaGmaeaacqaIWaamaaGccqGHQms8cqGGUaGlaaaaaa@3C50@ Open image in new window
(22)
Since B0 and B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window can mix, the mass eigenstates are a superposition of a|B0⟩ + b| B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window⟩ which obey the Schrodinger equation
i d d t ( a b ) = H ( a b ) = ( M i 2 Γ ) ( a b ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGPbqAjuaGdaWcaaqaaiabdsgaKbqaaiabdsgaKjabdsha0baakmaabmaabaqbaeqabiqaaaqaaiabdggaHbqaaiabdkgaIbaaaiaawIcacaGLPaaacqGH9aqpt0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaqadaqaauaabeqaceaaaeaacqWGHbqyaeaacqWGIbGyaaaacaGLOaGaayzkaaGaeyypa0ZaaeWaaeaacqWGnbqtcqGHsisljuaGdaWcaaqaaiabdMgaPbqaaiabikdaYaaakiabfo5ahbGaayjkaiaawMcaamaabmaabaqbaeqabiqaaaqaaiabdggaHbqaaiabdkgaIbaaaiaawIcacaGLPaaacqGGUaGlaaa@5292@ Open image in new window
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If CP is not conserved then the eigenvectors, the mass eigenstates |B L ⟩ and |B H ⟩, are not the CP eigenstates but are
| B L = p | B 0 + q | B ¯ 0 , | B H = p | B 0 q | B ¯ 0 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaeiiFaWNaemOqai0aaSbaaSqaaiabdYeambqabaGccqGHQms8cqGH9aqpcqWGWbaCcqGG8baFcqWGcbGqdaahaaWcbeqaaiabicdaWaaakiabgQYiXlabgUcaRiabdghaXjabcYha8jqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaGccqGHQms8cqGGSaalaeaacqGG8baFcqWGcbGqdaWgaaWcbaGaemisaGeabeaakiabgQYiXlabg2da9iabdchaWjabcYha8jabdkeacnaaCaaaleqabaGaeGimaadaaOGaeyOkJeVaeyOeI0IaemyCaeNaeiiFaWNafmOqaiKbaebadaahaaWcbeqaaiabicdaWaaakiabgQYiXlabcYcaSaaaaaa@571A@ Open image in new window
(24)
where
p = 1 2 1 + { B 1 + | { B | 2 , q = 1 2 1 { B 1 + | { B | 2 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@5802@ Open image in new window
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CP is violated if ϵ B ≠ 0, which occurs if |q/p| ≠ 1.

The time dependence of the mass eigenstates is
| B L ( t ) = e Γ L t / 2 e i m L t / 2 | B L ( 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqGG8baFcqWGcbGqdaWgaaWcbaGaemitaWeabeaakiabcIcaOiabdsha0jabcMcaPiabgQYiXlabg2da9iabdwgaLnaaCaaaleqabaGaeyOeI0Iaeu4KdC0aaSbaaWqaaiabdYeambqabaWcdaWcgaqaaiabdsha0bqaaiabikdaYaaaaaGccqWGLbqzdaahaaWcbeqaaiabdMgaPjabd2gaTnaaBaaameaacqWGmbataeqaaSWaaSGbaeaacqWG0baDaeaacqaIYaGmaaaaaOGaeiiFaWNaemOqai0aaSbaaSqaaiabdYeambqabaGccqGGOaakcqaIWaamcqGGPaqkcqGHQms8aaa@4C46@ Open image in new window
(26)
| B H ( t ) = e Γ H t / 2 e i m H t / 2 | B H ( 0 ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqGG8baFcqWGcbGqdaWgaaWcbaGaemisaGeabeaakiabcIcaOiabdsha0jabcMcaPiabgQYiXlabg2da9iabdwgaLnaaCaaaleqabaGaeyOeI0Iaeu4KdC0aaSbaaWqaaiabdIeaibqabaWcdaWcgaqaaiabdsha0bqaaiabikdaYaaaaaGccqWGLbqzdaahaaWcbeqaaiabdMgaPjabd2gaTnaaBaaameaacqWGibasaeqaaSWaaSGbaeaacqWG0baDaeaacqaIYaGmaaaaaOGaeiiFaWNaemOqai0aaSbaaSqaaiabdIeaibqabaGccqGGOaakcqaIWaamcqGGPaqkcqGHQms8cqGGSaalaaa@4D06@ Open image in new window
(27)
leading to the time evolution of the flavor eigenstates as
| B 0 ( t ) = e ( i m + Γ 2 ) t ( cos Δ m t 2 | B 0 ( 0 ) + i q p sin Δ m t 2 | B ¯ 0 ( 0 ) ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6B9C@ Open image in new window
(28)
| B ¯ 0 ( t ) = e ( i m + Γ 2 ) t ( i P q sin Δ m t 2 | B 0 ( 0 ) + cos Δ m t 2 | B ¯ 0 ( 0 ) ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6C54@ Open image in new window
(29)

where m = (m L + m H )/2, Δm = m H - m L and Γ = Γ L ≈ Γ H , and t is the decay time in the B0 rest frame, the so-called "proper time". Note that the probability of a B0 decay as a function of t is given by ⟨B0 (t)|B0(t)⟩*, and is a pure exponential, et/2, in the absence of CP violation.

2.2.2 CP Violation for BVia the Interference of Mixing and Decay

Here we choose a final state f which is accessible to both B0 and B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window decays [3]. The second amplitude necessary for interference is provided by mixing. Figure 6 shows the decay into f either directly or indirectly via mixing. It is necessary only that f be accessible directly from either state; however if f is a CP eigenstate the situation is far simpler. For CP eigenstates
Figure 6

Two interfering ways for a B 0 to decay into a final state f.

CP|f CP ⟩ = ± |f CP ⟩.

It is useful to define the amplitudes
A = f C P | H | B 0 , A ¯ = f C P | H | B ¯ 0 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaemyqaeKaeyypa0JaeyykJeUaemOzay2aaSbaaSqaaiabdoeadjabdcfaqbqabaGccqGG8baFt0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecscqGG8baFcqWGcbGqdaahaaWcbeqaaiabicdaWaaakiabgQYiXlabcYcaSaqaaiqbdgeabzaaraGaeyypa0JaeyykJeUaemOzay2aaSbaaSqaaiabdoeadjabdcfaqbqabaGccqGG8baFcqWFlecscqGG8baFcuWGcbGqgaqeamaaCaaaleqabaGaeGimaadaaOGaeyOkJeVaeiOla4caaaaa@5518@ Open image in new window
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If | A ¯ A | MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaaemaajuaGbaWaaSaaaeaacuWGbbqqgaqeaaqaaiabdgeabbaaaOGaay5bSlaawIa7aaaa@30EF@ Open image in new window ≠ 1, then we have "direct" CP violation in the decay amplitude, which we will discuss in detail later. Here CP can be violated by having
λ = q p A ¯ A 1 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqaH7oaBcqGH9aqpjuaGdaWcaaqaaiabdghaXbqaaiabdchaWbaakiabgwSixNqbaoaalaaabaGafmyqaeKbaebaaeaacqWGbbqqaaGccqGHGjsUcqaIXaqmcqGGSaalaaa@3943@ Open image in new window
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which requires only that λ acquire a non-zero phase, i.e. |λ| could be unity and CP violation can occur.

Other useful variables, that are independent of any phase convention are
ϕ 12 = arg ( M 12 Γ 12 ) ,  and Im Γ 12 M 12 = 1 | q / p | 4 1 + | q / p | 4 A S L ( t ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6ABB@ Open image in new window
(33)
The first quantity can be related to CKM angles, while the second can be measured by the "semileptonic asymmetry," or for that matter in any flavor specific decay [46]:
A S L ( t ) = Γ [ B ¯ 0 ( t ) X ] Γ [ B 0 ( t ) + X ] Γ [ B ¯ 0 ( t ) X ] + Γ [ B 0 ( t ) + X ] = Δ Γ Δ M tan ϕ 12 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGbbqqdaWgaaWcbaGaem4uamLaemitaWeabeaakiabcIcaOiabdsha0jabcMcaPiabg2da9KqbaoaalaaabaGaeu4KdCKaei4waSLafmOqaiKbaebadaahaaqabeaacqaIWaamaaGaeiikaGIaemiDaqNaeiykaKIaeyOKH4QaeS4eHW2aaWbaaeqabaGaeyOeI0caaiabdIfayjabc2faDjabgkHiTiabfo5ahjabcUfaBjabdkeacnaaCaaabeqaaiabicdaWaaacqGGOaakcqWG0baDcqGGPaqkcqGHsgIRcqWItecBdaahaaqabeaacqGHRaWkaaGaemiwaGLaeiyxa0fabaGaeu4KdCKaei4waSLafmOqaiKbaebadaahaaqabeaacqaIWaamaaGaeiikaGIaemiDaqNaeiykaKIaeyOKH4QaeS4eHW2aaWbaaeqabaGaeyOeI0caaiabdIfayjabc2faDjabgUcaRiabfo5ahjabcUfaBjabdkeacnaaCaaabeqaaiabicdaWaaacqGGOaakcqWG0baDcqGGPaqkcqGHsgIRcqWItecBdaahaaqabeaacqGHRaWkaaGaemiwaGLaeiyxa0faaOGaeyypa0tcfa4aaSaaaeaacqqHuoarcqqHtoWraeaacqqHuoarcqWGnbqtaaGccyGG0baDcqGGHbqycqGGUbGBcqaHvpGzdaWgaaWcbaGaeGymaeJaeGOmaidabeaakiabcYcaSaaa@7EBE@ Open image in new window
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for either B0 or for B s mesons, separately.

A comment on neutral B production at e+ e- colliders is in order. At the ϒ(4S) resonance there is coherent production of B0 B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window pairs. This puts the B's in a C = -1 state. In hadron colliders, or at e+ e- machines operating at the Z0, the B's are produced incoherently. For the rest of this article we will assume incoherent production except where explicitly noted.

The asymmetry, in the case of CP eigenstates, is defined as
a f C P = Γ ( B 0 ( t ) f C P ) Γ ( B ¯ 0 ( t ) f C P ) Γ ( B 0 ( t ) f C P ) + Γ ( B ¯ 0 ( t ) f C P ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6B59@ Open image in new window
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which for |q/p| = 1 gives
a f C P = ( 1 | λ | 2 ) cos ( Δ m t ) 2 Im λ sin ( Δ m t ) 1 + | λ | 2 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGHbqydaWgaaWcbaGaemOzay2aaSbaaWqaaiabdoeadjabdcfaqbqabaaaleqaaOGaeyypa0tcfa4aaSaaaeaadaqadaqaaiabigdaXiabgkHiTiabcYha8jabeU7aSjabcYha8naaCaaabeqaaiabikdaYaaaaiaawIcacaGLPaaacyGGJbWycqGGVbWBcqGGZbWCcqGGOaakcqqHuoarcqWGTbqBcqWG0baDcqGGPaqkcqGHsislcqaIYaGmcyGGjbqscqGGTbqBcqaH7oaBcyGGZbWCcqGGPbqAcqGGUbGBcqGGOaakcqqHuoarcqWGTbqBcqWG0baDcqGGPaqkaeaacqaIXaqmcqGHRaWkcqGG8baFcqaH7oaBcqGG8baFdaahaaqabeaacqaIYaGmaaaaaOGaeiOla4caaa@5D31@ Open image in new window
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For the cases where there is only one decay amplitude A, |λ| equals 1, and we have
a f C P = Im λ sin ( Δ m t ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGHbqydaWgaaWcbaGaemOzay2aaSbaaWqaaiabdoeadjabdcfaqbqabaaaleqaaOGaeyypa0JaeyOeI0IagiysaKKaeiyBa0Maeq4UdWMagi4CamNaeiyAaKMaeiOBa4MaeiikaGIaeuiLdqKaemyBa0MaemiDaqNaeiykaKIaeiOla4caaa@40BF@ Open image in new window
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Only the factor -Imλ contains information about the level of CP violation, the sine term is determined by B mixing. In fact, the time integrated asymmetry is given by
a f C P = x 1 + x 2 Im λ = 0.48 Im λ . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGHbqydaWgaaWcbaGaemOzay2aaSbaaWqaaiabdoeadjabdcfaqbqabaaaleqaaOGaeyypa0JaeyOeI0scfa4aaSaaaeaacqWG4baEaeaacqaIXaqmcqGHRaWkcqWG4baEdaahaaqabeaacqaIYaGmaaaaaOGagiysaKKaeiyBa0Maeq4UdWMaeyypa0JaeyOeI0IaeGimaaJaeiOla4IaeGinaqJaeGioaGJagiysaKKaeiyBa0Maeq4UdWMaeiOla4caaa@4710@ Open image in new window
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This is quite lucky for the study of B d mesons, as the maximum size of the coefficient for any x is -0.5, close to the measured value of x d .

Let us now find out how Imλ relates to the CKM parameters. Recall λ = q p A ¯ A MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeU7aSjabg2da9KqbaoaalaaabaGaemyCaehabaGaemiCaahaaOGaeyyXICDcfa4aaSaaaeaacuWGbbqqgaqeaaqaaiabdgeabbaaaaa@3643@ Open image in new window. The first term is the part that comes from mixing:
q p = ( V t b * V t d ) 2 | V t b V t d | 2 = ( 1 ρ i η ) 2 ( 1 ρ + i η ) ( 1 ρ i η ) = e 2 i β  and MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@715C@ Open image in new window
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Im q p = 2 ( 1 ρ ) η ( 1 ρ ) 2 + η 2 = sin ( 2 β ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaajuaGcyGGjbqscqGGTbqBdaWcaaqaaiabdghaXbqaaiabdchaWbaacqGH9aqpcqGHsisldaWcaaqaaiabikdaYiabcIcaOiabigdaXiabgkHiTiabeg8aYjabcMcaPiabeE7aObqaaiabcIcaOiabigdaXiabgkHiTiabeg8aYjabcMcaPmaaCaaabeqaaiabikdaYaaacqGHRaWkcqaH3oaAdaahaaqabeaacqaIYaGmaaaaaiabg2da9iGbcohaZjabcMgaPjabc6gaUjabcIcaOiabikdaYiabek7aIjabcMcaPiabc6caUaaa@4E98@ Open image in new window
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2.2.3 Measurements of sin(2β)

To evaluate the decay part we need to consider specific final states. For example, consider the final state J/ψ K s . The decay diagram is shown in Figure 7. In this case we do not get a phase from the decay part because
Figure 7

The Feynman diagram for the decay B0 → J/ψK0.

A ¯ A = ( V c b V c s * ) 2 | V c b V c s | 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiqbdgeabzaaraaabaGaemyqaeeaaOGaeyypa0tcfa4aaSaaaeaadaqadaqaaiabdAfawnaaBaaabaGaem4yamMaemOyaigabeaacqWGwbGvdaqhaaqaaiabdogaJjabdohaZbqaaiabcQcaQaaaaiaawIcacaGLPaaadaahaaqabeaacqaIYaGmaaaabaGaeiiFaWNaemOvay1aaSbaaeaacqWGJbWycqWGIbGyaeqaaiabdAfawnaaBaaabaGaem4yamMaem4CamhabeaacqGG8baFdaahaaqabeaacqaIYaGmaaaaaaaa@466A@ Open image in new window
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is real to order 1/λ4.

In this case the final state is a state of negative CP, i.e. CP|J/ψ K s ⟩ = -|J/ψ K s ⟩. This introduces an additional minus sign in the result for Imλ. Before finishing discussion of this final state we need to consider in more detail the presence of the K s in the final state. Since neutral kaons can mix, we pick up another mixing phase (similar diagrams as for B0, see Figure 3). This term creates a phase given by
( q p ) K = ( V c d * V c s ) 2 | V c d V c s | 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaadaqadaqcfayaamaalaaabaGaemyCaehabaGaemiCaahaaaGccaGLOaGaayzkaaWaaSbaaSqaaiabdUealbqabaGccqGH9aqpjuaGdaWcaaqaamaabmaabaGaemOvay1aa0baaeaacqWGJbWycqWGKbazaeaacqGGQaGkaaGaemOvay1aaSbaaeaacqWGJbWycqWGZbWCaeqaaaGaayjkaiaawMcaamaaCaaabeqaaiabikdaYaaaaeaacqGG8baFcqWGwbGvdaWgaaqaaiabdogaJjabdsgaKbqabaGaemOvay1aaSbaaeaacqWGJbWycqWGZbWCaeqaaiabcYha8naaCaaabeqaaiabikdaYaaaaaGaeiilaWcaaa@4AD6@ Open image in new window
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which is real to order λ4. It necessary to include this term, however, since there are other formulations of the CKM matrix than Wolfenstein, which have the phase in a different location. It is important that the physics predictions not depend on the CKM convention. (Here we do not include CP violation in the neutral kaon since it is much smaller than what is expected in the B decay.)

In summary, for the case of f = J/ψ K s , Imλ = - sin(2β). The angle β is the best measured CP violating angle measured in B meson decays. The process used is B0J/ψ K S , although there is some data used with the ψ (2S) or with K L (where the phase is + sin(2β)).

Although it is normally thought that only one decay amplitude contributes here, in fact one can look for the presence of another source of CP violation, presumably in the decay amplitude, by not assuming |λ| equals one in Eq. 36. Then the time dependence of the decay rate is given by
a f C P = 2 Im λ 1 + | λ | 2 sin ( Δ m t ) 1 | λ | 2 1 + | λ | 2 cos ( Δ m t ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@63EA@ Open image in new window
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Thus a f C P MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdggaHnaaBaaaleaacqWGMbGzdaWgaaadbaGaem4qamKaemiuaafabeaaaSqabaaaaa@3033@ Open image in new window (t) has both sin(Δmt) and cos(Δmt) terms, and the coefficients of these terms can be measured. Let us assign the labels
S = 2 Im λ 1 + | λ | 2 , C = 1 | λ | 2 1 + | λ | 2 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NeXpLaeyypa0tcfa4aaSaaaeaacqaIYaGmcyGGjbqscqGGTbqBcqaH7oaBaeaacqaIXaqmcqGHRaWkcqGG8baFcqaH7oaBcqGG8baFdaahaaqabeaacqaIYaGmaaaaaOGaeiilaWcabaGae8NaXpKaeyypa0tcfa4aaSaaaeaacqaIXaqmcqGHsislcqGG8baFcqaH7oaBcqGG8baFdaahaaqabeaacqaIYaGmaaaabaGaeGymaeJaey4kaSIaeiiFaWNaeq4UdWMaeiiFaW3aaWbaaeqabaGaeGOmaidaaaaakiabc6caUaaaaaa@58D3@ Open image in new window
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(Note that the sign of the S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jr8tbaa@368F@ Open image in new window term changes depending on the CP of the final state, but not the C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ Open image in new window term.)

The most precise measurements of sin(2β) have been made by the BaBar and Belle experiments [47, 48]. These measurements are made at the ϒ(4S) resonance using e+ e- → ϒ(4S) → B0 B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window, and with one of the neutral B's decaying into J/ψ K0. Because the ϒ(4S) is a definite quantum state with C = -1, the formulae given above have to modified somewhat. One important change is the definition of t. The time used here is the difference between the decay time of the J/ψ K0 and the other B0, also called the "tagging" B because we need to determine its flavor, whether B0 or B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window, in order to make the asymmetry measurement.

While we will not discuss flavor tagging in general, it is an important part of CP violation measurements. At the ϒ(4S) once such very useful tag is that of high momentum lepton as a b-quark, part of a B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window meson decays semileptonically into an ℓ-, while a b ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkgaIzaaraaaaa@2C5C@ Open image in new window-quark decays into an e+. Other flavor tagging observables include the charge of kaons and fast pions. Modern experiments combine these in a "neural network," to maximize their tagging efficiencies [49] (see also [50]). The BaBar data are shown in Figure 8, Belle has similar results.
Figure 8

(a) The number of J/ψK s candidates in the signal region with either a B0 tag ( N B 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaabmaabaGaemOta40aaSbaaSqaaiabdkeacnaaCaaameqabaGaeGimaadaaaWcbeaaaOGaayjkaiaawMcaaaaa@300F@ Open image in new window , or a B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window tag ( N B ¯ 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaabmaabaGaemOta40aaSbaaSqaamaanaaabaGaemOqaieaamaaCaaameqabaGaeGimaadaaaWcbeaaaOGaayjkaiaawMcaaaaa@3020@ Open image in new window as a function of Δ t. (b) The raw asymmetry, ( N B 0 N B ¯ 0 ) / ( N B 0 + N B ¯ 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaabmaabaGaemOta40aaSbaaSqaaiabdkeacnaaCaaameqabaGaeGimaadaaaWcbeaakiabgkHiTiabd6eaonaaBaaaleaadaqdaaqaaiabdkeacbaadaahaaadbeqaaiabicdaWaaaaSqabaaakiaawIcacaGLPaaacqGGVaWldaqadaqaaiabd6eaonaaBaaaleaacqWGcbGqdaahaaadbeqaaiabicdaWaaaaSqabaGccqGHRaWkcqWGobGtdaWgaaWcbaWaa0aaaeaacqWGcbGqaaWaaWbaaWqabeaacqaIWaamaaaaleqaaaGccaGLOaGaayzkaaaaaa@3F1C@ Open image in new window. The solid (dashed) curves represent the fit projections as functions of Δt for both B0 and B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window tags. The shaded regions represent the estimated background contributions.

The average value as determined by the Heavy Flavor Averaging Group is sin(2β) = 0.671 ± 0.024, where the dominant part of the error is statistical. No evidence is found for a non-zero C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ Open image in new window term with the measured average given as 0.005 ± 0.020.

Determining the sine of any angle gives a four-fold ambiguity in the angle. The decay mode B0J/ψ K*0, K*0K s π0 offers a way of measuring cos 2β and resolving the ambiguities.

This is a subtle analysis that we will not go into detail on [51, 52]. The result is that β = 42.14 1.83 + 1.88 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabek7aIjabg2da9iabisda0iabikdaYiabc6caUiabigdaXiabisda0maaDaaaleaacqGHsislcqaIXaqmcqGGUaGlcqaI4aaocqaIZaWmaeaacqGHRaWkcqaIXaqmcqGGUaGlcqaI4aaocqaI4aaoaaaaaa@3BE2@ Open image in new window degrees.

2.2.4 Measurements of α

The next state to discuss is the CP + eigenstate fπ+ π-. The simple spectator decay diagram is shown in Figure 9(a). For the moment we will assume that this is the only diagram, though the Penguin diagram shown in Figure 9(b) could also contribute; its presence can be inferred because it would induce a non-zero value for C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ Open image in new window, the coefficient of the cosine term in Eq. 43. For this bu u ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdwha1zaaraaaaa@2C82@ Open image in new windowd process we have
Figure 9

Tree (a) and Penguin (b) processes for neutral B decay into either π + π - or ρ + ρ - .

A ¯ A = ( V u d * V u b ) 2 | V u d V u b | 2 = ( ρ i η ) 2 ( ρ i η ) ( ρ + i η ) = e 2 i γ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiqbdgeabzaaraaabaGaemyqaeeaaOGaeyypa0tcfa4aaSaaaeaadaqadaqaaiabdAfawnaaDaaabaGaemyDauNaemizaqgabaGaeiOkaOcaaiabdAfawnaaBaaabaGaemyDauNaemOyaigabeaaaiaawIcacaGLPaaadaahaaqabeaacqaIYaGmaaaabaGaeiiFaWNaemOvay1aaSbaaeaacqWG1bqDcqWGKbazaeqaaiabdAfawnaaBaaabaGaemyDauNaemOyaigabeaacqGG8baFdaahaaqabeaacqaIYaGmaaaaaOGaeyypa0tcfa4aaSaaaeaacqGGOaakcqaHbpGCcqGHsislcqWGPbqAcqaH3oaAcqGGPaqkdaahaaqabeaacqaIYaGmaaaabaGaeiikaGIaeqyWdiNaeyOeI0IaemyAaKMaeq4TdGMaeiykaKIaeiikaGIaeqyWdiNaey4kaSIaemyAaKMaeq4TdGMaeiykaKcaaOGaeyypa0Jaemyzau2aaWbaaSqabeaacqGHsislcqaIYaGmcqWGPbqAcqaHZoWzaaaaaa@6718@ Open image in new window
(45)

and

Im (λ) = Im (e-2i βe-2i γ) = Im (e2i α) = - sin (2α

Time dependent CP violation measurements have been made by both the BaBar and Belle collaborations [53, 54]. Both groups find a non-zero value of both C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ Open image in new window and S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jr8tbaa@368F@ Open image in new window, though their values are not in particularly good agreement. The HFAG average is shown in Figure 10 along with the experimental results. The value of C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ Open image in new window clearly is not zero, thus demonstrating direct CP violation. (Historically, this was an important observation because it showed that CP violation could not be due to some kind of "superweak" model ala Wolfenstein [55].)
Figure 10

Coefficients of the sine term S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jr8tbaa@368F@ Open image in new window and the cosine term C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ Open image in new windowin time dependent CP violation for neutral B decay into π+π-, showing BaBar and Belle results, and the HFAG average. Contours are shown at 60.7% confidence level.

The non-zero value of C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ Open image in new window shows the presence of at least two amplitudes, presumably tree and penguin, in addition to the mixing amplitude, making extraction of α difficult. All is not lost, however. Gronau and London [56] showed that α could be extracted without theoretical error by doing a full isotopic spin analysis of the π π final state. The required measurements include: C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ Open image in new window, S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jr8tbaa@368F@ Open image in new window, MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ Open image in new window(B+π+π0), MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ Open image in new window (B0π0π0), and MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ Open image in new window ( B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new windowπ0π0). The last two items require a flavored tagged analysis that has not yet been done and whose prospects are bleak. Grossman and Quinn have showed, however, that an upper limit on Γ (B0π0π0)/Γ(B+π+π0) can be used to limit the penguin shift to α [57, 58, 59]. Unfortunately, current data show a relative large MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ Open image in new window(B0π0π0) = (1.55 ± 0.19) × 10-6 rate, compared with MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ Open image in new window(B0π+ π-) = (5.16 ± 0.22) × 10-6, implying a large penguin contribution [17, 34], and the limit is very weak.

Use of the ρ+ ρ- final state is in principle quite similar to π+ π-, with some important caveats. First of all, it is a vector-vector final state and therefore could have both CP + and CP- components. It is however possible, doing a full angular analysis to measure the CP violating phase separately in each of these two amplitudes. The best method for this is in the "transversity" basis and will be discussed later [60] in Section 2.2.6. It is possible, however, for one polarization component to be dominant and then the angular analysis might not be necessary. In fact the longitudinal polarization is dominant in this case. BaBar measures the fraction as 0.992 ± 0.024 0.013 + 0.026 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabiMda5iabiMda5iabikdaYiabgglaXkabicdaWiabc6caUiabicdaWiabikdaYiabisda0maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqaIZaWmaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaI2aGnaaaaaa@419D@ Open image in new window[61], and Belle measures it as 0.941 0.040 + 0.034 ± 0.030 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabiMda5iabisda0iabigdaXmaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaI0aancqaIWaamaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIWaamcqaIZaWmcqaI0aanaaGccqGHXcqScqaIWaamcqGGUaGlcqaIWaamcqaIZaWmcqaIWaamaaa@4193@ Open image in new window[62]. Thus we can treat this final state without worrying about the angular analysis and just apply a correction for the amount of transverse amplitude.

In addition, ρ mesons are wide, so non-B backgrounds could be a problem and even if the proper B is reconstructed, there are non-resonant and possible a1 π contributions.

Furthermore, it has been pointed out that the large ρ width could lead to the violation of the isospin constraints and this effect should be investigated [63]. The relevant branching ratios are given in Table 1.
Table 1

Branching ratios Bρ ρ modes in units of 10-6

Mode

BaBar

Belle

Average [17, 34]

ρ + ρ -

25.5 ± 2.1 3.9 + 3.6 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabikdaYiabiwda1iabc6caUiabiwda1iabgglaXkabikdaYiabc6caUiabigdaXmaaDaaaleaacqGHsislcqaIZaWmcqGGUaGlcqaI5aqoaeaacqGHRaWkcqaIZaWmcqGGUaGlcqaI2aGnaaaaaa@3B17@ Open image in new window

22.8 ± 3.8 2.6 + 2.3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabikdaYiabikdaYiabc6caUiabiIda4iabgglaXkabiodaZiabc6caUiabiIda4maaDaaaleaacqGHsislcqaIYaGmcqGGUaGlcqaI2aGnaeaacqGHRaWkcqaIYaGmcqGGUaGlcqaIZaWmaaaaaa@3B17@ Open image in new window

24.2 ± 3.1

ρ + ρ 0

16.8 ± 2.2 ± 2.3

31.7 ± 7.1 6.7 + 3.8 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabiodaZiabigdaXiabc6caUiabiEda3iabgglaXkabiEda3iabc6caUiabigdaXmaaDaaaleaacqGHsislcqaI2aGncqGGUaGlcqaI3aWnaeaacqGHRaWkcqaIZaWmcqGGUaGlcqaI4aaoaaaaaa@3B25@ Open image in new window

18 ± 4

ρ 0 ρ 0

0.92 ± 0.32 ± 0.14

0.4 0.2 + 0.4 ± 0.3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabisda0maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIYaGmaeaacqGHRaWkcqaIWaamcqGGUaGlcqaI0aanaaGccqGHXcqScqaIWaamcqGGUaGlcqaIZaWmaaa@3A05@ Open image in new window

0.74 0.27 + 0.30 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabicdaWiabc6caUiabiEda3iabisda0maaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaIYaGmcqaI3aWnaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIZaWmcqaIWaamaaaaaa@382B@ Open image in new window

Nevertheless, the small branching ratio for ρ0 ρ0, if indeed it has been observed at all, shows that the effects of the Penguin diagram on the extracted value of sin(2α) are small, and this may indeed be a good way to extract a value of α. The time dependent decay rates separately for B0ρ+ ρ- and B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new windowρ+ ρ- and their difference are shown in Figure 11 from BaBar. In is interesting to compare these results with those in Figure 8. We see that the measured asymmetry in the ρ+ ρ- decay more or less cancels that from B0 - B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window mixing inferring that sin(2α) is close to zero.
Figure 11

The number of ρ+ ρ- candidates in the signal region with either a B0 tag (a) ( N B 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaabmaabaGaemOta40aaSbaaSqaaiabdkeacnaaCaaameqabaGaeGimaadaaaWcbeaaaOGaayjkaiaawMcaaaaa@300F@ Open image in new window, or a B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new windowtag (b) ( N B ¯ 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaabmaabaGaemOta40aaSbaaSqaamaanaaabaGaemOqaieaamaaCaaameqabaGaeGimaadaaaWcbeaaaOGaayjkaiaawMcaaaaa@3020@ Open image in new windowas a function of Δ t. (c) The raw asymmetry, ( N B 0 N B ¯ 0 ) / ( N B 0 + N B ¯ 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaabmaabaGaemOta40aaSbaaSqaaiabdkeacnaaCaaameqabaGaeGimaadaaaWcbeaakiabgkHiTiabd6eaonaaBaaaleaadaqdaaqaaiabdkeacbaadaahaaadbeqaaiabicdaWaaaaSqabaaakiaawIcacaGLPaaacqGGVaWldaqadaqaaiabd6eaonaaBaaaleaacqWGcbGqdaahaaadbeqaaiabicdaWaaaaSqabaGccqGHRaWkcqWGobGtdaWgaaWcbaWaa0aaaeaacqWGcbGqaaWaaWbaaWqabeaacqaIWaamaaaaleqaaaGccaGLOaGaayzkaaaaaa@3F1C@ Open image in new window. The dashed curves represent the estimated backgrounds.

Results from the time dependent CP violation analysis from both Belle and BaBar are shown in Figure 12.
Figure 12

Coefficients of the sine term S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jr8tbaa@368F@ Open image in new windowand the cosine term C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dbaa@366F@ Open image in new windowin time dependent CP violation for neutral B decay into ρ+ρ- showing BaBar and Belle results, and the HFAG average. Contours are shown at 60.7% confidence level.

The data can be averaged and α determined by using the isospin analysis and the rates listed in Table 1. Unfortunately the precision of the data leads only to constraints. These have been determined by both the CKM fitter group [64] and the UT fit group [30]. These groups disagree in some cases. The CKM fitter group use a frequentist statistical approach, while UT fit uses a Bayseian approach. The basic difference is the that the Bayesian approach the theoretical errors are taken as having a Gaussian distribution. Here we show in Figure 13 the results from CKM fitter.
Figure 13

The probability density, defined as 1 – CL (confidence level) for the angle α as determined by measurements of B ρ ρ.

The final state π+ π- π0 can also be used to extract α. Snyder and Quinn proposed that a Dalitz plot analysis of Bρ ππ+ π- π0 can be used to unravel both α and the relative penguin-tree phases [65]. The Dalitz plot for simulated events unaffected by detector acceptance is shown in Figure 14.
Figure 14

Dalitz plot for Monte-Carlo generated B 0 ρ π π + π - π 0 decays. The decays have been simulated without any detector effect and the amplitudes for ρ+π-, ρ-π+ and ρ0π0 have all been generated with equal magnitudes in order to have destructive interferences where the ρ bands overlap. The main overlap regions between the ρ bands are indicated by the hatched areas. Adapted from [66].

The task at hand is to do a time dependent analysis of the magnitude of the decay amplitudes and phases. The analyses of both collaborations allow for ρ(1450) and ρ(1700) contributions in addition to the ρ(770). There are a total of 26 free parameters in the fit. The statistics for such an analysis are not overwhelming: BaBar has about 2100 signal events [66], while Belle has about 1000 [67].

The results for the confidence levels of α found by both collaborations are shown in Figure 15. Note that there is also a mirror solution at α + 180°. The Belle collaboration uses their measured decay rates for B+ρ+ π0 and B+ρ0 π+ coupled with isospin relations [68, 69] to help constrain α.
Figure 15

The experimental confidence levels for α as determined separately by the Belle collaboration (left) and the BaBar collaboration (right). The dotted curve for Belle shows the result without using constraints from charged B to ρ π final states.

The LHCb experiment expects to be able to significantly improve on the determination of α using the ρ π mode. They expect 14,000 events in for an integrated luminosity of 2 fb-1, with a signal to background ratio greater than 1 [70]. It is expected that this amount of data can be accumulated in one to two years of normal LHC operation.

Combining all the data, both the CKM fitter and UT fit groups derive a the confidence level plot for α shown in Figure 16. There is a clear disagreement between the two groups, UT fit preferring a solution in the vicinity of 160°, and CKM fitter a value closer to 90°. The CKM fitter group believes that this is due to the UT fit group's use of Bayesian statistics which they criticize [71].
Figure 16

The experimental confidence levels for α as determined separately by the CKM fitter and UT fit groups.

2.2.5 The Angle γ

The angle γ = arg [ V u d V u b * V c d V c b * ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeo7aNjabg2da9iGbcggaHjabckhaYjabcEgaNnaadmaabaGaeyOeI0scfa4aaSaaaeaacqWGwbGvdaWgaaqaaiabdwha1jabdsgaKbqabaGaemOvay1aa0baaeaacqWG1bqDcqWGIbGyaeaacqGGQaGkaaaabaGaemOvay1aaSbaaeaacqWGJbWycqWGKbazaeqaaiabdAfawnaaDaaabaGaem4yamMaemOyaigabaGaeiOkaOcaaaaaaOGaay5waiaaw2faaaaa@470B@ Open image in new window. It can be viewed as the phase of the |V ub | matrix element, with respect to |V cb |. Interference measurements are required to determine phases. We can use neutral or even charged B decays to measure γ, and there are several ways to do this without using any theoretical assumptions, as is the case for α and β. The first relies on the color suppressed tree diagrams shown in Figure 17, and the second on using the interference in B s mixing. At first glance, the diagrams in Figure 17 don't appear to have any interfering components. However, if we insist that the final state is common to D0 and D ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D3B@ Open image in new window, then we have two diagrams which differ in that one has a bc amplitude and the other a bu amplitude, the relative weak phase is then γ. Explicitly the amplitudes are defined as
Figure 17

Two diagrams for a charged B decay into a neutral D meson and a charged kaon.

A ( B D 0 K ) A B A ( B D ¯ 0 K ) A B r B e i ( δ B γ ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@56F0@ Open image in new window
(47)

where r B reflects the amplitude suppression for the bu mode and δ B is the relative phase. We have not yet used identical final states for D0 and D ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D3B@ Open image in new window decays, yet we can see that there will be an ambiguity in our result between δ B and γ that could be resolved by using different D0 decay modes.

There are several suggestions as to different D0 decay modes to use. In the original paper on this topic Gronau and Wyler [72, 73] propose using CP eigenstates for the D0 decay, such as K+ K-, π+ π- etc.., combining with charge specific decays and comparing B- with B+ decays. In the latter, the sign of the strong phase is flipped with respect to the weak phase. In fact modes such as D*0 or K*- can also be used. (When using D*0 there is a difference in δ B of π between γ D0 and π0 D0 decay modes [74].)

It is convenient to define the follow variables:
A C P ± = [ Γ ( B D C P ± 0 ( * ) K ( * ) ) Γ ( B + D C P ± 0 ( * ) K + ( * ) ) ] [ Γ ( B D C P ± 0 ( * ) K ( * ) ) + Γ ( B + D C P ± 0 ( * ) K + ( * ) ) ] , R C P ± = [ Γ ( B D C P ± 0 ( * ) K ( * ) ) Γ ( B + D C P ± 0 ( * ) K + ( * ) ) ] [ Γ ( B D 0 ( * ) K ( * ) ) + Γ ( B + D ¯ 0 ( * ) K + ( * ) ) ] / 2 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@E377@ Open image in new window
(48)
These are related to the variables defined in Eq. 47 as
R C P ± = 1 + r B 2 ± 2 r cos δ B cos γ , A C P ± = ± 2 r sin δ B sin γ / R C P ± . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@675D@ Open image in new window
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Measurements have been made by the BaBar, Belle and CDF collaborations [75, 76, 77, 78, 79]. These data, however, are not statistically powerful enough by themselves to give a useful measurement of γ. Atwood, Dunietz and Soni suggested using double-Cabibbo suppressed decays as an alternative means of generating the interference between decay amplitudes [80]. For example the final state B-D0 K- can proceed via the tree level diagram in Figure 2(a) when the W- u ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdwha1zaaraaaaa@2C82@ Open image in new windows. Then if we use the doubly-Cabibbo suppressed decay D0K+ π-, we get a final state that interferes with the D ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D3B@ Open image in new windowK- final state with the D ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D3B@ Open image in new windowK+ π-, which is Cabibbo allowed. The advantage of this method, besides extending the number of useful final states, is that both amplitudes are closer to being equal than in the above method, which can generate larger interferences. Any doubly-Cabibbo suppressed decay can be used. Similar equations exist relating the measured parameters to γ and the strong phase shift.

Measurements have been made mostly using the K± π final state [81, 82]. Again, these attempts do not yet produce accurate results.

Thus far, the best method for measuring γ uses the three-body final state K S π+ π-. Since the final state is accessible by both D0 and D ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D3B@ Open image in new window decays interference results. By its very nature a three-body state is complicated, and consists of several resonant and non-resonant parts. It is typically analyzed by examining the Dalitz plot where two of the possible three sub-masses (squared) are plotted versus one another [83]. In this way, the phase space is described as a uniform density and thus any resonant structure is clearly visible. For our particular case a practical method was suggested by Giri et al [84], where the decay is analyzed in separate regions of the Dalitz plot. Results from this analysis have been reported by BaBar [85] (they also include the mode D0K S K+ K-) and Belle [86]. BaBar finds γ = (76 ± 22 ± 5 ± 5)°, and Belle finds γ = ( 76 13 + 12 ± 4 ± 5 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeo7aNjabg2da9iabcIcaOiabiEda3iabiAda2maaDaaaleaacqGHsislcqaIXaqmcqaIZaWmaeaacqGHRaWkcqaIXaqmcqaIYaGmaaGccqGHXcqScqaI0aancqGHXcqScqaI1aqncqGGPaqkdaahaaWcbeqaaiablIHiVbaaaaa@3E49@ Open image in new window. In both cases the first error is statistical, the second systematic, and the third refers to uncertainties caused by the Dalitz plot model. (There is also a mirror solution at ± 180°.)

It has been shown [87, 88] that measurement of the amplitude magnitudes and phases found in the decays of ψ (3770) → (CP ± Tag)(K S π+ π-) D provide useful information that help the narrow model error. The CLEO collaboration is working on such an analysis, and preliminary results have been reported [89]. (CLEO also is working on incorporating other D0 decay modes.)

Both the CKM fitter and UT fit groups have formed liklihoods for γ based on the measured results, as shown in Figure 18. In this case CKM fitter and UT fit agree on the general shape of the liklihood curve. The CKM fitter plot shows a small disagreement between the Daltiz method and a combination of the other two methods.
Figure 18

The experimental confidence levels for γ as determined separately by the CKM fitter and UT fit groups.

2.2.6 The Angle χ

The angle χ shown in Figure 1 is the phase that appears in the box diagram for B s mixing, similar to the diagram for B0 mixing shown in Figure 3, but with the d quark replaced by an s quark. The analogous mode to B0J/ψ K s in the B s system is B s J/ψ η. The Feynman diagrams are shown in Figure 19. This is very similar to measuring β so χ is often called β s . We also have a relation between ϕ s defined in Eq. 33, ϕ s = -2χ.
Figure 19

The Feynman diagram for the decay B s J / ψ η or ϕ.

Since there are usually two photons present in the η decay, experiments at hadron colliders, which can perform time-dependent studies of B s mesons, preferentially use the J/ψ ϕ final state. This, unfortunately, introduces another complexity into the problem; as the B s is spinless the total angular momentum of the final state particles must be zero. For a decay into a vector and scalar, such as J/ψ η, this forces the vector J/ψ to be fully longitudinally polarized with respect to the decay axis. For a vector-vector final state both angular momentum state vectors are either longitudinal (L), both are transverse with linear polarization vectors parallel (||) or they are perpendicular (⊥) to one another [90]. Another way of viewing this is that a spin-0 B decay into two massive vector mesons can form CP even states with L = 0 or 2, and a CP odd state with L = 1. The relative populations in the two CP states are determined by strong interactions dynamics, but to study the weak phase here we are not particularly interested in the actual amount, unless of course one state dominated. We do not expect this to be the case however, since the SU(3) related decay B0J/ψ K*0, K*0K+ π- has a substantial components of both CP states; the PDG quotes gives the longitudinal fraction as (80 ± 8 ± 5)% [17].

The even and odd CP components can be disentangled by measuring the appropriate angular quantities of each event. Following Dighe et al [91], we can decompose the decay amplitude for a B s as
A ( B s J / ψ ϕ ) = A 0 ( m ϕ ) / E ϕ e J / ψ * L A | | e J / ψ * T / 2 i A e ϕ * p ^ / 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqWGbbqqcqGGOaakcqWGcbGqdaWgaaWcbaGaem4CamhabeaakiabgkziUkabdQeakjabc+caViabeI8a5jabew9aMjabcMcaPiabg2da9iabdgeabnaaBaaaleaacqaIWaamaeqaaOGaeiikaGIaemyBa02aaSbaaSqaaiabew9aMbqabaGccqGGPaqkcqGGVaWlcqWGfbqrdaWgaaWcbaGaeqy1dygabeaaruGqLXgBGC0B0HwAJbYu0rgicXwyJTgaiqGakiab=vgaLnaaDaaaleaacqWGkbGscqGGVaWlcqaHipqEaeaacqGGQaGkcqWGmbataaGccqGHsislcqWGbbqqdaWgaaWcbaGaeiiFaWNaeiiFaWhabeaakiab=vgaLnaaDaaaleaacqWGkbGscqGGVaWlcqaHipqEaeaacqGGQaGkcqWGubavaaGccqGGVaWldaGcaaqaaiabikdaYaWcbeaakiabgkHiTiabdMgaPjabdgeabnaaBaaaleaacqGHLkIxaeqaaOGae8xzau2aa0baaSqaaiabew9aMbqaaiabcQcaQaaakiabgwSixlqbhchaWzaajaGaei4la8YaaOaaaeaacqaIYaGmaSqabaGccqGGSaalaaa@70DE@ Open image in new window
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where ϵJ/ψand ϵ ϕ are polarization 3-vectors in the J/ψ rest frame, p ^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbhchaWzaajaaaaa@2C74@ Open image in new window is a unit vector giving the direction of the ϕ momentum in the J/ψ rest frame, and E ϕ is the energy of the ϕ in the J/ψ rest frame. We note that the corresponding amplitude for the B ¯ s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaSbaaSqaaiabdohaZbqabaaaaa@2DB7@ Open image in new window decay are A ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdgeabzaaraWaaSbaaSqaaiabicdaWaqabaaaaa@2D34@ Open image in new window = A0, A ¯ | | MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdgeabzaaraWaaSbaaSqaaiabcYha8jabcYha8bqabaaaaa@2F46@ Open image in new window = A||, and A ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdgeabzaaraWaaSbaaSqaaiabgwQiEbqabaaaaa@2DF7@ Open image in new window = -A. The amplitudes are normalized so that

dΓ(B s J/ψ ϕ)/dt = |A0|2 + |A|||2 + |A|2.

The ϕ meson direction in the J/ψ rest frame defines the x ^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdIha4zaajaaaaa@2C80@ Open image in new window direction. The z ^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdQha6zaajaaaaa@2C84@ Open image in new window direction is perpendicular to the decay plane of the K+ K- system, where p y (K+) ≥ 0. The decay direction of the ℓ+ in the J/ψ rest frame is described by the angles (θ, ϕ). The angle ψ is that formed by the K+ direction with the x ^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdIha4zaajaaaaa@2C80@ Open image in new window-axis in the ϕ rest frame. Figure 20 shows the angles.
Figure 20

Pictorial description of the decay angles. On the left θ and ϕ defined in the J/Ψ rest frame and on the right Ψ defined in the ϕ rest frame. (From T. Kuhr [235].).

The decay width can be written as
d 4 Γ [ B s ( + ) J / ψ ( K + K ) ϕ ] d cos θ d ϕ d cos ψ d t = 9 32 π [ 2 | A 0 | 2 cos 2 ψ ( 1 sin 2 θ cos 2 ϕ ) + sin 2 ψ { | A | | | 2 ( 1 sin 2 θ sin 2 ϕ ) + | A | 2 sin 2 θ Im ( A | | * A ) sin 2 θ sin ϕ } + 1 2 sin 2 ψ { Re ( A 0 * A | | ) sin 2 θ sin 2 ϕ + Im ( A 0 * A ) sin 2 θ cos ϕ } ] . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=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@E9A9@ Open image in new window
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The decay rate for B ¯ s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaSbaaSqaaiabdohaZbqabaaaaa@2DB7@ Open image in new window can be found by replacing A in the above expression with -A.

Another complexity arises from the expectation that the width difference ΔΓ s s ≈ 15%. This complicates the time dependent rate equations. For convenience, setting ρ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbeg8aYzaalaaaaa@2CC9@ Open image in new window ≡ (cos θ, ϕ, cosψ), we have for the decay width for B s :
d 4 P ( t , ρ ) d t d ρ | A 0 | 2 T + f 1 ( ρ ) + | A | | | 2 T + f 2 ( ρ ) + | A | 2 T f 3 ( ρ ) + | A | | | | A | U + f 4 ( ρ ) + | A 0 | | A | | | cos ( δ | | ) T + f 5 ( ρ ) + | A 0 | | A | V + f 6 ( ρ ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaafaqaaeabdaaaaKqbagaadaWcaaqaaiabdsgaKnaaCaaabeqaaiabisda0aaacqWGqbaucqGGOaakcqWG0baDcqGGSaalcuaHbpGCgaWcaiabcMcaPaqaaiabdsgaKjabdsha0jabdsgaKjqbeg8aYzaalaaaaaGcbaGaeyyhIulabaGaeiiFaWNaemyqae0aaSbaaSqaaiabicdaWaqabaGccqGG8baFdaahaaWcbeqaaiabikdaYaaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaakiab=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rr8vnaaBaaaleaacqGHRaWkaeqaaOGaemOzay2aaSbaaSqaaiabisda0aqabaGccqGGOaakcuaHbpGCgaWcaiabcMcaPaqaaaqaaiabgUcaRaqaaiabcYha8jabdgeabnaaBaaaleaacqaIWaamaeqaaOGaeiiFaWNaeiiFaWNaemyqae0aaSbaaSqaaiabcYha8jabcYha8bqabaGccqGG8baFcyGGJbWycqGGVbWBcqGGZbWCcqGGOaakcqaH0oazdaWgaaWcbaGaeiiFaWNaeiiFaWhabeaakiabcMcaPiab=nr8unaaBaaaleaacqGHRaWkaeqaaOGaemOzay2aaSbaaSqaaiabiwda1aqabaGccqGGOaakcuaHbpGCgaWcaiabcMcaPaqaaaqaaiabgUcaRaqaaiabcYha8jabdgeabnaaBaaaleaacqaIWaamaeqaaOGaeiiFaWNaeiiFaWNaemyqae0aaSbaaSqaaiabgwQiEbqabaGccqGG8baFcqWFveVvdaWgaaWcbaGaey4kaScabeaakiabdAgaMnaaBaaaleaacqaI2aGnaeqaaOGaeiikaGIafqyWdiNbaSaacqGGPaqkcqGGSaalaaaaaa@C624@ Open image in new window
(53)
where
T ± = [ ( 1 ± cos ( 2 χ ) ) e Γ L t + ( 1 cos ( 2 χ ) ) e Γ H t ] / 2 , U ± = ± e Γ t × [ sin ( δ δ | | ) cos ( Δ m s t ) cos ( δ δ | | ) cos ( 2 χ ) sin ( Δ m s t ) ± cos ( δ δ | | ) sin ( 2 χ ) sinh ( Δ Γ t / 2 ) ] , V ± = ± e Γ t × [ sin ( δ ) cos ( Δ m s t ) cos ( δ ) cos ( 2 χ ) sin ( Δ m s t ) ± cos ( δ ) sin ( 2 χ ) sinh ( Δ Γ t / 2 ) ] . f 1 ( ρ ) = 2 cos 2 ψ ( 1 sin 2 θ cos 2 ϕ ) , f 2 ( ρ ) = sin 2 ψ ( 1 sin 2 θ sin 2 ϕ ) , f 3 ( ρ ) = sin 2 ψ sin 2 θ , f 4 ( ρ ) = sin 2 ψ sin ( 2 θ ) sin ϕ , f 5 ( ρ ) = sin ( 2 ψ ) sin 2 θ sin ( 2 ϕ ) / 2 , f 6 ( ρ ) = sin ( 2 ψ ) sin ( 2 θ ) cos ( 2 ϕ ) / 2 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@011F@ Open image in new window

The quantities δ and δ|| are the strong phases of A and A|| relative to A0, respectively [92]. The expression for B ¯ s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaSbaaSqaaiabdohaZbqabaaaaa@2DB7@ Open image in new window mesons can be found by substituting U + MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=rr8vnaaBaaaleaacqGHRaWkaeqaaaaa@37A1@ Open image in new window U MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=rr8vnaaBaaaleaacqGHsislaeqaaaaa@37AC@ Open image in new window and V + MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=vr8wnaaBaaaleaacqGHRaWkaeqaaaaa@37A3@ Open image in new window V MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=vr8wnaaBaaaleaacqGHsislaeqaaaaa@37AE@ Open image in new window.

The most interesting quantities to be extracted from the data are χ and ΔΓ. There are many experimental challenges: the angular and lifetime distributions must be corrected for experimental acceptances; flavor tagging efficiencies and dilutions must be evaluated; backgrounds must be measured. Both the CDF [93] and D0 [94] experiments have done this complicated analysis. Updated results as of this writing are summarized by the CKM fitter derived limits shown in Figure 21.
Figure 21

Constraints at 68% confidence level in the ( ϕ s , ΔΓ s ) plane. Overlaid are the constraints from the CDF and D0 measurements, the constraint from Δ Γ s = cos ϕ s Δ Γ s S M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabfs5aejabfo5ahnaaBaaaleaacqWGZbWCaeqaaOGaeyypa0Jagi4yamMaei4Ba8Maei4CamNaeqy1dy2aaSbaaSqaaiabdohaZbqabaGccqqHuoarcqqHtoWrdaqhaaWcbaGaem4CamhabaGaem4uamLaemyta0eaaaaa@3EBD@ Open image in new window, the constraint from the flavor specific B s lifetime [34], and the overall combination. The SM prediction is also given. From [95, 96].

The Standard Model allowed region is a very thin vertical band centered near zero at ϕ s of -0.036 ± 0.002 (shown in red). (Recall ϕ s ≡ -2χ ≡ -2β s .) The region labeled "all" (green) shows the allowed region at 68% confidence level. Although the fit uses several input components besides the CP asymmetry measurements in B s J/ψ ϕ, including use of measured total widths introduced via the constraint equation Δ Γ s = cos ϕ s Δ Γ s S M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabfs5aejabfo5ahnaaBaaaleaacqWGZbWCaeqaaOGaeyypa0Jagi4yamMaei4Ba8Maei4CamNaeqy1dy2aaSbaaSqaaiabdohaZbqabaGccqqHuoarcqqHtoWrdaqhaaWcbaGaem4CamhabaGaem4uamLaemyta0eaaaaa@3EBD@ Open image in new window, it is the measurement of ϕ s that dominates the fit result. We see that there is a discrepancy that may be as large as 2.7 standard deviations [95, 96]. While this is as not yet significant, it is very tantalizing. The LHCb experiment plans to vastly improve this measurement [97].

It has been pointed out, however, that there is likely an S-wave K+ K- contribution in the region of the ϕ that contributes 5–10% of the event rate as estimated using D s + MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdseaenaaDaaaleaacqWGZbWCaeaacqGHRaWkaaaaaa@2E86@ Open image in new window decays [98]. For example, analysis of the analogous channel B0J/ψ K*0 reveals about 8% S-wave in the K π system under the K*0, and BaBar has used this to extract a value for cos 2β thus removing an ambiguity in β [99]. The S-wave amplitude and phase needs to be added to Eq. 53. Note that the errors will increase due to the addition of another amplitude and phase. The S-wave can manifest itself as a π+ π-, so it is suggested that the decay B s J/ψ f0 (980) be used to measure χ; here the f0π+ π- [98]; the estimate is that the useful f0 rate would be about 20% of the ϕ rate, but the J/ψ f0 is a CP eigenstate, so an an angular analysis is unnecessary, and these events may provide a determination of χ with an error comparable to that using J/ψ ϕ.

2.2.7 Measurements of Direct CP Violation in BK πDecays

Time integrated asymmetries of B mesons produced at the ϒ(4S) resonance can only be due to direct CP asymmetry, as the mixing generated asymmetry must integrate to zero due to the fact that the initial state has J PC = 1--. The first evidence for such direct CP violation at the greater than four standard deviation level in the K π± final state was given by BaBar [100]. The latest BaBar result is [101]
A K π + = Γ ( B ¯ 0 K π + ) Γ ( B 0 K + π ) Γ ( B ¯ 0 K π + ) + Γ ( B 0 K + π ) = 0.107 ± 0.016 0.004 + 0.006 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@87DD@ Open image in new window
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showing a large statistical significance.

This result was confirmed by the Belle collaboration, but Belle also measured the isospin conjugate mode. Consider the two-body decays of B mesons into a kaon and a pion, shown in Figure 22. For netural and charged decays, it can proceed via a tree level diagram (a) or a Penguin diagram (b). There are two additional decay diagrams allowed for the B-, the color-suppressed tree level diagram (c) and the elusive "Electroweak" Penguin diagram in (d). (So named because of the intermediate γ or Z boson.) Since it is expected that diagrams (c) and (d) are small, the direct CP violating asymmetries in both charged and neutral modes should be the same. Yet Belle observed [102]
Figure 22

Processes for B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window→ K -π+ and B- → K- π0, (a) via tree and (b) Penguin diagrams, and B- → K- π0(c) via color-suppressed tree and (d)"Electroweak" Penguin diagrams.

A K π 0 = Γ ( B K π 0 ) Γ ( B + K + π 0 ) Γ ( B K π 0 ) + Γ ( B + K + π 0 ) = 0.07 ± 0.016 ± 0.01. MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@8072@ Open image in new window
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The Belle data are shown in Figure 23.
Figure 23

The beam constrained mass distributions from Belle in units of GeV for the four different indicated final states. The dotted curves indicate peaking backgrounds, the dot-dashed curves the signal and the solid curves the sum. The differences in event numbers between the charge conjugate modes are apparent.

The difference between A K π + MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=bq8bnaaBaaaleaacqWGlbWsdaahaaadbeqaaiabgkHiTaaaliabec8aWnaaCaaameqabaGaey4kaScaaaWcbeaaaaa@3BB6@ Open image in new window and A K π 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=bq8bnaaBaaaleaacqWGlbWsdaahaaadbeqaaiabloHiTbaaliabec8aWnaaCaaameqabaGaeGimaadaaaWcbeaaaaa@3C08@ Open image in new window is not naively expected in the Standard Model and Belle suggests that this may be a sign of New Physics. Peskin commented on this possibility [103].

2.3 Conclusions from CPViolation Measurements

All CP violation in the quark sector is proportional to the parameter η in Eq. 1. In fact all CP asymmetries are proportional to the "Jarlskog Invariant", J = A2 λ6 η, which represents the equal area of all the CKM triangles [104]. Since we know the value of these three numbers, we do know the amount of CP violation we can expect, even without making the measurements. We also can estimate the amount of CP violation necessary using cosmology. To reproduce the observed baryon to entropy ratio requires many orders of magnitude more CP violation than thus far found in heavy quark decays [105]. Thus we believe there are new sources of CP violation that have not yet been found.

3 The CKM parameter |Vcb|

There are two experimental methods to determine |V cb |: the exclusive method, where |V cb | is extracted by studying the exclusive B ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraaaaa@2C1C@ Open image in new windowD(*)- ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window decay process; and the inclusive method, which uses the semileptonic decay width of b-hadron decays. In both methods, the extraction of |V cb | is systematics limited and the dominant errors are from theory. The inclusive and exclusive determinations of |V cb | rely on different theoretical calculations of the hadronic matrix element needed to extract it from measured quantities, and make use of different techniques which, to a large extent, have uncorrelated experimental uncertainties. Thus, the comparison between inclusive and exclusive decays allows us to test our understanding of hadronic effects in semileptonic decays. The latest determinations differ by more than 2σ, with the inclusive method having a stated error half of the size of the exclusive one.

3.0.1 Beauty Quark Mass Definitions

Due to confinement and the non-perturbative aspect of the strong interaction, the concept of the quark masses cannot be tied to an intuitive picture of the rest mass of a particle, as for leptons. Rather, quark masses must be considered as couplings of the SM Lagrangian that have to be determined from processes that depend on them. As such the b-quark mass (m b ) is a scheme-dependent, renormalised quantity.

In principle, any renormalisation scheme or definition of quark masses is possible. In the framework of QCD perturbation theory the difference between two mass schemes can be determined as a series in powers of α s . Therefore, higher-order terms in the perturbative expansion of a quantity that depends on quark masses are affected by the particular scheme employed. There are schemes that are more appropriate and more convenient for some purposes than others. Here we examine the main quark mass definitions commonly used in the description of B decays.

Pole mass: The pole mass definition is gauge-invariant and infrared-safe [106, 107, 108] to all orders in perturbation theory and has been used as the standard mass definition of many perturbative computations in the past. By construction, it is directly related to the concept of the mass of a free quark. The presence of a renormalon ambiguity [109, 110] makes the numerical value of the pole mass an order-dependent quantity, leading to large perturbative corrections for Heavy Quark Effective Theory (HQET) parameters (see below for a discussion of HQET). These shortcomings are avoided by using quark mass definitions that reduce the infrared sensitivity by removing the ΛQCD renormalon of the pole mass. Such quark mass definitions are generically called "short-distance" masses.

M S ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaanaaabaGaemyta0Kaem4uamfaaaaa@2D5A@ Open image in new windowmass: The most common short-distance mass definition is the M S ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaanaaabaGaemyta0Kaem4uamfaaaaa@2D5A@ Open image in new window mass m ¯ b MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbd2gaTzaaraWaaSbaaSqaaiabdkgaIbqabaaaaa@2DEB@ Open image in new window (μ) [111, 112, 113], where the scale μ is typically chosen to be the order of the characteristic energy scale of the process. In the M S ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaanaaabaGaemyta0Kaem4uamfaaaaa@2D5A@ Open image in new window scheme the subtracted divergencies do not contain any infrared sensitive terms and the M S ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaanaaabaGaemyta0Kaem4uamfaaaaa@2D5A@ Open image in new window mass is only sensitive to scales larger than m b . The M S ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaanaaabaGaemyta0Kaem4uamfaaaaa@2D5A@ Open image in new window mass arises naturally in processes where the b-quark is far off-shell, but it is less adequate when the b-quark has non-relativistic energies.

Kinetic mass: The shortcomings of the pole and the M S ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaanaaabaGaemyta0Kaem4uamfaaaaa@2D5A@ Open image in new window mass in describing non-relativistic b-quarks can be resolved by so-called threshold masses [114], that are free of an ambiguity of order ΛQCD and are defined through subtractions that contain universal contributions for the dynamics of non-relativistic quarks. Since the subtractions are not unique, an arbitrary number of threshold masses can be constructed. The kinetic mass is defined as [115, 116]:
m b , kin ( μ kin ) = m b , pole [ Λ ¯ ( μ kin ) ] pert [ μ π 2 ( μ kin ) 2 m b , kin ( μ kin ) ] pert + , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@7E35@ Open image in new window
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where μkin is the nominal kinetic mass renormalisation scale. For μkin → 0 the kinetic mass reduces to the pole mass.

• 1S mass: The kinetic mass depends on an explicit subtraction scale to remove the universal infrared sensitive contributions associated with the non-relativistic b-quark dynamics. The 1S mass [117, 118] achieves the same task without a factorisation scale, since it is directly related to a physical quantity. The b-quark 1S mass is defined as half of the perturbative contribution to the mass of the ϒ(S1) in the limit m b m b vm b v2 ≫ ΛQCD.

A list of b-quark mass determinations, converted into the M S ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaanaaabaGaemyta0Kaem4uamfaaaaa@2D5A@ Open image in new window mass scheme is shown in Table 2.
Table 2

List of m b determinations converted into the M S ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaanaaabaGaemyta0Kaem4uamfaaaaa@2D5A@ Open image in new window mass scheme.

m b (m b ) GeV

Method

4.243 ± 0.042

From BX s γ and BX c ν fit [238]

4.25 ± 0.02 ± 0.11

Lattice UKQCD [239]

4.346 ± 0.07

ϒ(1S) NNNLO [240]

4.164 ± 0.025

low-moment sum rules NNNLO [241]

4.17 ± 0.05

ϒ sum rules NNLO [242]

3.1 Determination Based on Exclusive Semileptonic BDecays

The exclusive |V cb | determination is obtained by studying the decays BD* ℓν and BDν, where ℓ denotes either an electron or a muon. The exclusive measurements of a single hadronic final state, e.g. the ground state D or D*, restrict the dynamics of the process. The remaining degrees of freedom, usually connected to different helicity states of the charmed hadron, can be expressed in terms of form factors, depending on the invariant mass of the lepton-ν pair, q2. The shapes of those form factors are unknown but can be measured. However, the overall normalization of these functions needs to be determined from theoretical calculations.

Isgur and Wise formulated a theoretical breakthrough in the late 1980's. They found that in the limit of an infinitely heavy quark masses QCD possess additional flavor and spin symmetries. They showed that since the heavier b and c quarks have masses much heavier than the scale of the QCD coupling constant, they are heavy enough to posses this symmetry but that corrections for the fact that the quark mass was not infinite had to be made. They showed there was a systematic method of making these corrections by expanding a series in terms of the inverse quark mass [119]. This theory is known as Heavy Quark Effective Theory (HQET).

When studying b quark decays into c quarks, it is convenient to view the process in four-velocity transfer (w) space as opposed to four-momentum transfer space, because at maximum four-velocity transfer, where w equals one, the form-factor in lowest order is unity, i.e. the b transforms directly into a c quark without any velocity change. The value of |V cb | can be extracted by studying the decay rate for the process B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new windowD* + ℓ- ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window as a function of the recoil kinematics of the D*+ meson. Specifically, HQET predicts that:
d Γ ( B D v ) d w = G F 2 | V c b | 2 48 π 3 K ( w ) F ( w ) 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabdsgaKjabfo5ahnaabmaabaGaemOqaiKaeyOKH4Qaemiraq0aaWbaaeqabaGaey4fIOcaaiabloriSjabdAha2bGaayjkaiaawMcaaaqaaiabdsgaKjabdEha3baakiabg2da9KqbaoaalaaabaGaem4raC0aa0baaeaacqWGgbGraeaacqaIYaGmaaWaaqWaaeaacqWGwbGvdaWgaaqaaiabdogaJjabdkgaIbqabaaacaGLhWUaayjcSdWaaWbaaeqabaGaeGOmaidaaaqaaiabisda0iabiIda4iabec8aWnaaCaaabeqaaiabiodaZaaaaaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGccqWFke=sdaqadaqaaiabdEha3bGaayjkaiaawMcaaiab=ftignaabmaabaGaem4DaChacaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmaaGccqGGSaalaaa@6057@ Open image in new window
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where w is the product of the four-velocities of the D*+ and the B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new window mesons, related to q2. K MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NcXVeaaa@3B6E@ Open image in new window(w) is a known phase space factor and the form factor F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window(w) is generally expressed as the product of a normalization factor and a shape function described by three form factors, constrained by dispersion relations [120]:
F ( w ) 2 = | h A 1 ( w ) | 2 ( 1 r ) 2 [ ( w r ) 2 ( 1 R 2 ( w ) ) 2 + 2 ( 1 2 w r r 2 ) ( 1 w 1 w + 1 R 2 2 ( w ) ) ] , h A 1 ( w ) h A 1 ( 1 ) = 1 8 ρ 2 z + ( 53 ρ 2 15 ) z 2 ( 231 ρ 2 91 ) z 3 , R 1 ( w ) = R 1 ( 1 ) 0.12 ( w 1 ) + 0.05 ( w 1 ) 2 , R 2 ( w ) = R 2 ( 1 ) 0.11 ( w 1 ) 0.06 ( w 1 ) 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakqaaeeqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=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@F0A7@ Open image in new window
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where r = m D / m B , z = w + 1 2 w + 1 + 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdkhaYjabg2da9iabd2gaTnaaBaaaleaacqWGebardaahaaadbeqaaiabgEHiQaaaaSqabaGccqGGVaWlcqWGTbqBdaWgaaWcbaGaemOqaieabeaakiabcYcaSiabdQha6jabg2da9KqbaoaalaaabaWaaOaaaeaacqWG3bWDcqGHRaWkcqaIXaqmaeqaaiabgkHiTmaakaaabaGaeGOmaidabeaaaeaadaGcaaqaaiabdEha3jabgUcaRiabigdaXaqabaGaey4kaSYaaOaaaeaacqaIYaGmaeqaaaaaaaa@434E@ Open image in new window; The linear slope of the form-factor is given by the parameter ρ2, and must be determined from the data. In the infinite mass limit, F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window(w = 1) = h A 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdIgaOnaaBaaaleaacqWGbbqqdaWgaaadbaGaeGymaedabeaaaSqabaaaaa@2EAF@ Open image in new window (w = 1) = 1; for finite quark masses, non-perturbative effects can be expressed in powers of 1/m Q . There are several different corrections to the infinite mass value F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window(1) = 1:
F ( 1 ) = η QED η A [ 1 + δ 1 / m Q 2 + ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqGacqWFXeIrdaqadaqaaiabigdaXaGaayjkaiaawMcaaiabg2da9iabeE7aOnaaBaaaleaacqqGrbqucqqGfbqrcqqGebaraeqaaOGaeeiiaaIaeq4TdG2aaSbaaSqaaiabdgeabbqabaGcdaWadaqaaiabigdaXiabgUcaRiabes7aKnaaBaaaleaacqaIXaqmcqGGVaWlcqWGTbqBdaqhaaadbaGaemyuaefabaGaeGOmaidaaaWcbeaakiabgUcaRiablAcilbGaay5waiaaw2faaaaa@4EC2@ Open image in new window
(59)

Note that the first term in the non-perturbative expansion in powers of 1/m Q vanishes [121]. QED corrections up to leading logarithmic order give η A = 0.960 ± 0.007. Different estimates of the 1/ m Q 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGrbquaeaacqaIYaGmaaaaaa@2EA4@ Open image in new window corrections, involving terms proportional to 1/ m b 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIYaGmaaaaaa@2EC6@ Open image in new window, 1/ m c 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGJbWyaeaacqaIYaGmaaaaaa@2EC8@ Open image in new window and 1/(m b m c ), have been performed in a quark model with QCD sum rules, and, more recently, with an HQET based lattice gauge calculation. The best estimate comes from lattice QCD, h A 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdIgaOnaaBaaaleaacqWGbbqqdaWgaaadbaGaeGymaedabeaaaSqabaaaaa@2EAF@ Open image in new window (1) = F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window (1) = 0.921 ± 0.013 ± 0.020 [122]. This result does not include a 0.7% QED correction.

Since the phase-space factor K MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NcXVeaaa@3B6E@ Open image in new window(w) tends to zero as w → 1, the decay rate vanishes and the accuracy of the |V cb | value extracted with this method depends upon experimental and theoretical uncertainties in the extrapolation. Experiments determine the product | F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window(1)·V cb |2 by fitting the measured dΓ/dw distribution.

This decay has been analyzed by CLEO, BaBar and Belle using B mesons from the ϒ(4S) decay, and by ALEPH, DELPHI, and OPAL at the Z0 center of mass energy. Experiments that exploit the ϒ(4S) have the advantage that w resolution is quite good. However, they suffer from lower statistics near w = 1 in the decay BD*+ν due to the lower reconstruction efficiency of the slow π±. On the other hand, the decay BD*0ν is not affected by this problem [123]. In addition, kinematic constraints enable these experiments to identify the final state including D* without large contamination from the poorly known portion of semileptonic decays with a hadronic system recoiling against the lepton-ν pair with masses higher than the D and D*, commonly identified as 'D**'. B-factories and CLEO fit for the signal and background components in the distribution of the cosine of the angle between the direction of the B and the direction of the D*ℓ system. At LEP, B mesons are produced with a large variable momentum (about 30 GeV on average), giving a relatively poor w resolution and limited physics background rejection capabilities. By contrast, LEP experiments benefit from an efficiency that is only mildly dependent upon w.

LEP experiments extracted |V cb | by performing a two-parameter fit, for F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window(1)|V cb | and the slope ρ2. The first measurements of both ratio R1 and R2, and ρ2, were made by the CLEO collaboration [124]. Belle [125] and BaBar [126] improved upon these measurements using the B ¯ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraWaaWbaaSqabeaacqaIWaamaaaaaa@2D37@ Open image in new windowD*+ e- ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window e decay. They determined R1, R2, and ρ2 using an unbinned maximum likelihood fit to the full decay distribution. BaBar and CLEO results are combined to give: R1 = 1.396 ± 0.060 ± 0.035 ± 0.027, R2 = 0.885 ± 0.040 ± 0.022 ± 0.013, and ρ2 = 1.145 ± 0.059 ± 0.030 ± 0.035. The stated uncertainties are the statistical from the data and systematic uncertainty, respectively.

Table 3 summarizes the available data. Values of F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window(1)|V cb | from different experiments can be combined if they are extracted using the same F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window(w) parametrization. All measurements included in the exclusive |V cb | world average relying on the above form factor ratios R1 and R2. Using the F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window(1)|V cb | world average in Table 3 and h A 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdIgaOnaaBaaaleaacqWGbbqqdaWgaaadbaGaeGymaedabeaaaSqabaaaaa@2EAF@ Open image in new window (1) = 0.921 ± 0.013 ± 0.020 [122], we find
Table 3

Experimental results based on BD*ν after the correction to common inputs and the world average as obtained by HFAG [243].

experiment

F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window(1)|V cb | (×10-3)

ρ 2

Corrstat

Branching fraction(%)

ALEPH [132]

31.6 ± 1.8 ± 1.3

0.50 ± 0.20 ± 0.09

94%

5.44 ± 0.25 ± 0.20

BaBar [244]

33.9 ± 0.3 ± 1.1

1.18 ± 0.05 ± 0.03

27%

4.53 ± 0.04 ± 0.13

BaBar [245]

34.9 ± 0.8 ± 1.4

1.11 ± 0.06 ± 0.08

90%

5.40 ± 0.16 ± 0.25

BaBar [246]

35.7 ± 0.2 ± 1.2

1.20 ± 0.02 ± 0.07

38%

-

BELLE [125]

34.7 ± 0.2 ± 1.0

1.16 ± 0.04 ± 0.03

91%

BELLE [247]

34.7 ± 0.2 ± 1.0

1.16 ± 0.04 ± 0.03

91%

4.75 ± 0.25 ± 0.19

CLEO [248]

41.3 ± 1.3 ± 1.8

1.37 ± 0.08 ± 0.18

91%

6.02 ± 0.19 ± 0.20

DELPHI(excl) [132]

36.3 ± 1.8 ± 1.9

1.04 ± 0.14 ± 0.15

89%

5.53 ± 0.19 ± 0.34

DELPHI(part rec) [132]

35.8 ± 1.4 ± 2.3

1.18 ± 0.13 ± 0.25

94%

5.00 ± 0.15 ± 0.18

OPAL(excl) [132]

36.9 ± 1.6 ± 1.5

1.24 ± 0.20 ± 0.14

77%

5.17 ± 0.20 ± 0.38

OPAL(par rec) [132]

37.6 ± 1.2 ± 2.4

1.14 ± 0.13 ± 0.27

94%

5.63 ± 0.27 ± 0.43

world average

35.4 ± 0.5

1.16 ± 0.05

20%

5.05 ± 0.10

All numbers are corrected to use R1 and R2 from a global fit to CLEO and BaBar data. The newest BELLE result is not included in the world average.

† – Not used in the World Average

|V cb | = (38.2 ± 0.5 exp ± 1.0 theo ) × 10-3,

where the dominant error is theoretical, and it will be difficult to improve upon.

The study of the decay BDν poses new challenges from the experimental point of view. The differential decay rate for BDν can be expressed as [127, 128]
d Γ D d w ( B D v ) = G F 2 | V c b | 2 48 π 3 K D ( w ) G ( w ) 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabdsgaKjabfo5ahnaaBaaabaGaemiraqeabeaaaeaacqWGKbazcqWG3bWDaaGaeiikaGIccqWGcbGqcqGHsgIRcqWGebarcqWItecBcqWG2bGDjuaGcqGGPaqkkiabg2da9KqbaoaalaaabaGaem4raC0aa0baaeaacqWGgbGraeaacqaIYaGmaaWaaqWaaeaacqWGwbGvdaWgaaqaaiabdogaJjabdkgaIbqabaaacaGLhWUaayjcSdWaaWbaaeqabaGaeGOmaidaaaqaaiabisda0iabiIda4iabec8aWnaaCaaabeqaaiabiodaZaaaaaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGccqWFke=sdaWgaaWcbaGae83aXteabeaakiabcIcaOiabdEha3jabcMcaPiab=zq8hjabcIcaOiabdEha3jabcMcaPmaaCaaaleqabaGaeGOmaidaaOGaeiilaWcaaa@6424@ Open image in new window
(60)
where K D MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=Pq8lnaaBaaaleaacqWFdepraeqaaaaa@386A@ Open image in new window (w) is the phase space factor and the form factor G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=zq8hbaa@3679@ Open image in new window (w) is generally expressed as the product of a normalization factor G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=zq8hbaa@3679@ Open image in new window (1) and a function, g D (w), constrained by dispersion relations [120]. The strategy to extract G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=zq8hbaa@3679@ Open image in new window (1)|Vcb| is identical to that used for the BD⋆ℓν decay. However, dΓ D /dw is more heavily suppressed near w = 1 than d Γ D * / d w MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdsgaKjabfo5ahnaaBaaaleaacqWGebarcqGGQaGkaeqaaOGaei4la8IaemizaqMaem4DaChaaa@3381@ Open image in new window due to the helicity mismatch between initial and final states. Moreover, this channel is much more challenging to isolate from the dominant background BD⋆ℓν as well as from fake D-ℓ combinations. Table 4 shows the results of two-dimensional fits to |V cb | G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=zq8hbaa@3679@ Open image in new window(1) and ρ2 for different experiments and the world average.
Table 4

Experimental results from BDν using corrections to the common inputs and world averages.

experiment

G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=zq8hbaa@3679@ Open image in new window(1)|Vcb|(×10-3)

ρ D 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeg8aYnaaDaaaleaacqWGebaraeaacqaIYaGmaaaaaa@2EE7@ Open image in new window

Corrstat

Branching fraction(%)

ALEPH [132]

38.1 ± 11.8 ± 6.1

0.91 ± 0.98 ± 0.36

98%

2.25 ± 0.18 ± 0.36

BaBar [249]

42.3 ± 1.9 ± 1.0

1.20 ± 0.09 ± 0.04

95%

-

BaBar [250]

43.8 ± 0.8 ± 2.1

1.22 ± 0.04 ± 0.06

63%

-

BaBar [251]

-

-

-

2.20 ± 0.11 ± 0.12

BELLE [252]

40.7 ± 4.4 ± 5.1

1.12 ± 0.22 ± 0.14

96%

2.09 ± 0.12 ± 0.39

CLEO [253]

44.5 ± 5.9 ± 3.4

1.27 ± 0.25 ± 0.14

95%

2.10 ± 0.13 ± 0.15

World average

42.4 ± 0.7 ± 1.4

1.19 ± 0.04 ± 0.04

96%

2.16 ± 0.12

In the limit of infinite quark masses, G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=zq8hbaa@3679@ Open image in new window(w = 1) coincides with the Isgur-Wise function [129]. In this case there is no suppression of 1/m Q corrections and QCD effects on G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=zq8hbaa@3679@ Open image in new window(1) are calculated with less accuracy than F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window(1). Corrections to this prediction have recently been calculated with improved precision, based on unquenched lattice QCD [130], specifically G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=zq8hbaa@3679@ Open image in new window(1) = 1.074 ± 0.018 ± 0.016. Using this result we get

|V cb | = (39.5 ± 1.4 exp ± 0.9 theo ) × 10-3

consistent with the value extracted from BD⋆ℓν decay, but with an experimental uncertainty about twice as large.

BaBar recently has also studied the differential decay widths for the decays B-D0ν and B-D0ν to extract the ratio G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=zq8hbaa@3679@ Open image in new window(1)/ F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYlPi=FI8qipeYlI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdc9as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceiGae8xmHyeaaa@3AC2@ Open image in new window(1) = 1.23 ± 0.09 [131], compatible with the lattice theory prediction of 1.16 ± 0.04.

3.2 BD**ℓνDecays

It is important to understand the composition of the inclusive B semileptonic decay rate in terms of exclusive final states for use in semileptonic B decay analyses. The BD(*)ν decays are well measured, but a sizeable fraction of semileptonic B decay are to D**ℓν. The D** resonances have larger masses than the D* and not well studied. (D** refers to the resonant states, D 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdseaenaaDaaaleaacqaIWaamaeaacqGHxiIkaaaaaa@2E12@ Open image in new window, D 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdseaenaaDaaaleaacqaIXaqmaeaacqGHxiIkaaaaaa@2E14@ Open image in new window, D 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdseaenaaDaaaleaacqaIYaGmaeaacqGHxiIkaaaaaa@2E16@ Open image in new window and D1.) The measurements of these branching fractions require a good understanding of the different D(*) systems, which can be either resonant or non-resonant, each with characteristic decay properties.

There are four orbitally excited states with L = 1. They can be grouped in two pairs according to the value of the spin on the light system, j = L ± 1/2 (L = 1). States with j = 3/2 can have JP = 1+ and 2+. The 1+ state decays only through D*π, and the 2+ through D π or D*π. Parity and angular momentum conservation imply that in the 2+ the D* and π are in a D wave but allow both S and D waves in the 1+ state. However, if the heavy quark spin is assumed to decouple, conservation of j = 3/2 forbids S waves even in the 1+ state. A large D-wave component and the fact that the masses of these states are not far from threshold imply that the j = 3/2 states are narrow. These states have been observed with a typical width of 20 MeV/c2. On the contrary, j = 1/2 states can have JP = 0+and 1+, so they are expected to decay mainly through an S wave and manifest as broad resonances, with typical widths of several hundred MeV/c2.

The ALEPH [132], CLEO [133], DELPHI [132], and D0 [134] experiments have reported evidence of the narrow resonant states (D1 and D 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdseaenaaDaaaleaacqaIYaGmaeaacqGHxiIkaaaaaa@2E16@ Open image in new window) in semileptonic decays, whereas more recent measurements by the BaBar [135] and Belle [136] experiments provide semi-inclusive measurements to D(*)πν final states [137, 138].

The differences between the measured inclusive semileptonic branching fraction and the sum of all exclusive B semileptonic measurements for Belle, BaBar and World averages are given in Table 5 for B0 and B- decays. In the case where multiple measurements exist, only the most precise measurements have been used in the BaBar and Belle columns, i.e. no attempt at an average is made. In all cases the sum of the exclusive components does not saturate the B semileptonic rate.
Table 5

Inclusive versus sum of exclusive measured B semileptonic branching fractions (%).

MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ Open image in new window(%)

BaBar

Belle

World Average

 

B 0

B -

B 0

B -

B 0

B -

BDν

2.20 ± 0.16

2.30 ± 0.10

2.09 ± 0.16

-

2.16 ± 0.12

2.32 ± 0.09

BD*ℓν

4.53 ± 0.14

5.37 ± 0.21

4.42 ± 0.25

-

5.05 ± 0.10

5.66 ± 0.18

BD πν

0.42 ± 0.09

0.42 ± 0.07

0.43 ± 0.09

0.42 ± 0.06

0.43 ± 0.06

0.42 ± 0.05

BD* πν

0.48 ± 0.09

0.59 ± 0.06

0.57 ± 0.22

0.68 ± 0.11

0.49 ± 0.08

0.61 ± 0.05

Σ (Exc.)

7.63 ± 0.25

8.68 ± 0.25

7.51 ± 0.73

-

8.13 ± 0.19

9.01 ± 0.21

Inc.

10.14 ± 0.43

10.90 ± 0.47

10.46 ± 0.38

11.17 ± 0.38

10.33 ± 0.28

10.99 ± 0.28

Inc. – Σ (Exc.)

2.51 ± 0.50

2.22 ± 0.53

2.95 ± 0.82

-

2.20 ± 0.34

1.98 ± 0.35

All measured rates for the D** narrow states are in good agreement. Experimental results seem to point towards a larger rate for broader states. If it is due mainly to D 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaafaWaaSbaaSqaaiabigdaXaqabaaaaa@2D30@ Open image in new window and D 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdseaenaaDaaaleaacqaIWaamaeaacqGHxiIkaaaaaa@2E12@ Open image in new window decay channels, these results disagree with the prediction of QCD sum rules. However, Belle set an upper limit for the D 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdseaezaafaWaaSbaaSqaaiabigdaXaqabaaaaa@2D30@ Open image in new window channel below the rate measured by BaBar and DELPHI. More measurements need to be performed to elucidate this puzzle.

3.3 Determination Based on Inclusive Semileptonic BDecays

Inclusive determinations of |V cb | are obtained using combined fits to inclusive B decay distributions [139, 140]. These determinations are based on calculations of the semileptonic decay rate in the frameworks of the Operator Product Expansion (OPE) [141] and HQET [139, 142]. They predict the semileptonic decay rate in terms of |V cb |, the b-quark mass m b , and non-perturbative matrix elements. The spectator model decay rate is the leading term in a well-defined expansion controlled by the parameter ΛQCD/m b [142, 143, 144, 145, 146] with non-perturbative corrections arising to order 1/ m b 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIYaGmaaaaaa@2EC6@ Open image in new window. The key issue in this approach is the ability to separate perturbative and non-perturbative corrections (expressed in powers of α s ). Thus the accuracy of an inclusive determination of |V cb | is limited by our knowledge of the heavy quark nonperturbative matrix elements and the b quark mass. It also relies upon the assumption of local quark hadron duality. This is the statement that a hadronic matrix element can be related pointwise to a theoretical expression formulated in terms of quark and gluon variables.

Perturbative and non-perturbative corrections depend on the m b definition, i.e. the non-perturbative expansion scheme, as well as the non-perturbative matrix elements that enter the expansion. In order to determine these parameters, Heavy Quark Expansions (HQE) [139, 146, 147] express the semileptonic decay width ΓSL, moments of the lepton energy and hadron mass spectra in BX c ν decays in terms of the running kinetic quark masses m b kin MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqqGRbWAcqqGPbqAcqqGUbGBaaaaaa@31ED@ Open image in new window and m c kin MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGJbWyaeaacqqGRbWAcqqGPbqAcqqGUbGBaaaaaa@31EF@ Open image in new window as well as the b-quark mass m b 1 S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIXaqmcqqGtbWuaaaaaa@2FF1@ Open image in new window in the 1S expansion scheme. These schemes should ultimately yield consistent results for |V cb |. The precision of the b-quark mass is also important for |V ub |.

The shape of the lepton spectrum and of the hadronic mass spectrum provide constraints on the heavy quark expansion, which allows for the calculation of the properties of BX c ν transitions. So far, measurements of the hadronic mass distribution and the leptonic spectrum have been made by BaBar [148], Belle [149], CLEO [150, 151], DELPHI [152]. CDF [153] provides only the measurement of the hadronic mass spectrum with a lepton momentum cut of 0.6 GeV in the B rest frame.

The inclusive semileptonic width can be expressed as
Γ ( B ¯ X c ν ¯ ) = G F 2 m b 5 | V c b | 2 192 π 3 ( f ( ρ ) + k ( ρ ) μ π 2 2 m b 2 + g ( ρ ) μ G 2 2 m b 2 + d ( ρ ) ρ D 3 m b 3 + l ( ρ ) ρ L S 3 m b 3 + O ( m b 4 ) ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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5q8pjabcIcaOiabd2gaTnaaDaaaleaacqWGIbGyaeaacqGHsislcqaI0aanaaGccqGGPaqkcqGGPaqkcqGGSaalaaaaaa@A2EC@ Open image in new window
(61)
where ρ = m c 2 / m b 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeg8aYjabg2da9iabd2gaTnaaDaaaleaacqWGJbWyaeaacqaIYaGmaaGccqGGVaWlcqWGTbqBdaqhaaWcbaGaemOyaigabaGaeGOmaidaaaaa@364D@ Open image in new window, and μ π 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeY7aTnaaDaaaleaacqaHapaCaeaacqaIYaGmaaaaaa@2F89@ Open image in new window, μ G 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeY7aTnaaDaaaleaacqWGhbWraeaacqaIYaGmaaaaaa@2EE3@ Open image in new window, ρ D and ρ LS are non-perturbative matrix elements of local operators in HQET. To make use of equations such as Eq. 61, the values of the non-perturbative expansion parameters must be determined experimentally. Although some of these parameters are related to the mass splitting of pseudoscalar and vector meson states, most non-perturbative parameters are not so easily obtained. Measurements of the moments of different kinematic distributions in inclusive B-decays are used to gain access to these parameters. The first moment of a distribution is given by:
M 1 = M 1 ( x ) d x d Γ d x , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaadaaadaqaaiabd2eannaaBaaaleaacqaIXaqmaeqaaaGccaGLPmIaayPkJaGaeyypa0Zaa8qaaeaacqWGnbqtdaWgaaWcbaGaeGymaedabeaakmaabmaabaGafmiEaGNbaSaaaiaawIcacaGLPaaacqWGKbazcuWG4baEgaWcaKqbaoaalaaabaGaemizaqMaeu4KdCeabaGaemizaqMafmiEaGNbaSaaaaaaleqabeqdcqGHRiI8aOGaeiilaWcaaa@40C6@ Open image in new window
(62)
corresponding to the mean. Subsequent (central) moments are calculated around the first moment,
M n = ( x M 1 ) n , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaadaaadaqaaiabd2eannaaBaaaleaacqWGUbGBaeqaaaGccaGLPmIaayPkJaGaeyypa0ZaaaWaaeaadaqadaqaaiqbdIha4zaalaGaeyOeI0Iaemyta00aaSbaaSqaaiabigdaXaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiabd6gaUbaaaOGaayzkJiaawQYiaiabcYcaSaaa@3A80@ Open image in new window
(63)

corresponding to a distribution's width, kurtosis and so on.

A B ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraaaaa@2C1C@ Open image in new windowX c ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window decay observable calculated with the OPE is a double expansion in terms of the strong coupling α s (m b ) and the ratio ΛQCD/m b . Observables are typically calculated with the cuts used in the experimental determination for background suppression, and enhanced sensitivity such as the lepton energy cut. The spectral moments are defined as:
< E n E x m ( M x 2 ) > = 1 Γ 0 E ^ min d E d E x d M x d Γ d M X d E x d E E n E x m ( M x 2 ) = f 0 [ n , E ^ ] + f 1 [ n , E ^ ] Λ ¯ m b + i = 1 2 f i + 1 [ n , E ^ ] λ i m b 2 + O ( α s , Λ QCD 3 m b 3 ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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5q8pnaabmaabaGaeqySde2aaSbaaSqaaiabdohaZbqabaGccqGGSaaljuaGdaWcaaqaaiabfU5amnaaDaaabaGaeeyuaeLaee4qamKaeeiraqeabaGaeG4mamdaaaqaaiabd2gaTnaaDaaabaGaemOyaigabaGaeG4mamdaaaaaaOGaayjkaiaawMcaaiabcYcaSaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaeGOmaidaniabggHiLdaaaaaa@C865@ Open image in new window
(64)
where
Γ 0 = G F 2 m b 5 | V c b | 2 192 π 3 . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciGacaGaaeqabaqabeGadaaakeaacqqHtoWrdaWgaaWcbaGaeGimaadabeaakiabg2da9KqbaoaalaaabaGaem4raC0aa0baaeaacqWGgbGraeaacqaIYaGmaaGaemyBa02aa0baaeaacqWGIbGyaeaacqaI1aqnaaGaeiiFaWNaemOvay1aaSbaaeaacqWGJbWycqWGIbGyaeqaaiabcYha8naaCaaabeqaaiabikdaYaaaaeaacqaIXaqmcqaI5aqocqaIYaGmcqaHapaCdaahaaqabeaacqaIZaWmaaaaaOGaeiOla4caaa@4439@ Open image in new window
(65)

The measurement of |V cb | from inclusive decays requires that these decays be adequately described by the OPE formalism. The motivation of the moment approach is to exploit the degree of experimental and theoretical understanding of each moment and for different E ^ min MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdweafzaajaWaa0baaSqaaiabloriSbqaaiGbc2gaTjabcMgaPjabc6gaUbaaaaa@319A@ Open image in new window, directly examining the dependence of the various coefficient functions f i on these terms. The possibility of deviations from the OPE predictions due to quark hadron duality violations have been raised [154]. To compare the OPE predictions with data, one also has to define how uncertainties from 1/ m b 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIZaWmaaaaaa@2EC8@ Open image in new window corrections are estimated. These uncertainties are hard to quantify reliably, as the only information on the size of the matrix elements of the dimension six operators comes from dimensional analysis. The only method to check the reliability of |V cb | extraction from inclusive semileptonic decays is how well the OPE fits the data.

To compare with theoretical predictions, the moments are measured with a well defined cut on the lepton momentum in the B rest frame. The measured hadronic mass distribution and lepton energy spectrum are affected by a variety of experimental factors such as detector resolution, accessible phase space, radiative effects. It is particularly important to measure the largest fraction of the accessible phase space in order to reduce both theoretical and experimental uncertainties. Each experiment has focused on lowering the lepton energy cut.

The hadronic mass spectrum in BX c ν decays can be split into three contributions corresponding to D, D*, and D**, where D** here stands for any neutral charmed state, resonant or not, other than D and D*. Belle [149], BaBar [148] and CLEO [150] explored the moments of the hadronic mass spectrum M X 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2eannaaDaaaleaacqWGybawaeaacqaIYaGmaaaaaa@2E72@ Open image in new window as a function of the lepton momentum cuts. CLEO performs a fit for the contributions of signal and background to the full three-dimensional differential decay rate distribution as a function of the reconstructed quantities q2, M X 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2eannaaDaaaleaacqWGybawaeaacqaIYaGmaaaaaa@2E72@ Open image in new window, cos θ. Belle [155, 156] and BaBar [148] use a sample where one of the B mesons, produced in pairs from ϒ(4S) decays, is fully reconstructed and the signal side is tagged by a well identified lepton. The 4-momentum p X of the hadronic system X, recoiling against the lepton and neutrino, is determined by summing the 4-momenta of the remaining charged tracks and unmatched clusters. Belle reconstructs the full hadronic mass spectrum and measures the first, second central and second non-central moments of the unfolded M X 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2eannaaDaaaleaacqWGybawaeaacqaIYaGmaaaaaa@2E72@ Open image in new window spectrum in BX c ν, for lepton energy thresholds, Emin, varying from 0.7 to 1.9 GeV [149]. BaBar extracts the moments from the measured distributions using a calibration curve derived from Monte Carlo data, with a minimum momentum for the electron in the B meson rest frame of 0.9 GeV. The latest BaBar analysis [149] measures the first, second central and second non-central moments of the M X 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2eannaaDaaaleaacqWGybawaeaacqaIYaGmaaaaaa@2E72@ Open image in new window spectrum for Emin from 0.9 to 1.9 GeV. The main systematic errors originate from background estimation, unfolding and signal model dependence.

DELPHI follows a different approach in extracting the moments, measuring the invariant mass distribution of the D** component only and fixing the D and D* components. DELPHI measures the first moment with respect to the spin averaged mass of D and D*. At LEP b-quarks were created with an energy of approximately 30 GeV allowing the measurement of the hadronic mass moments without a cut on the lepton energy [152].

The shape of the lepton spectrum provides further constraints on the OPE. These measurements are sensitive to higher order OPE parameters and are considerably more precise experimentally. Moments of the lepton momentum with a cut p ≥ 1.0 GeV/c have been measured by the CLEO collaboration [157]. BaBar [148] extract up to the third moment of this distribution, using a low momentum cut of p ≥ 0.6 GeV/c. Both BaBar and CLEO use dilepton samples. The most recent measurement of the electron energy spectrum is from Belle [155, 156]. Events are selected by fully reconstructing one of the B mesons, produced in pairs from ϒ(4S) decays and it determines the true electron energy spectrum by unfolding [158] the measured spectrum in the B meson rest frame. Belle measures B0 and B+ weighted average partial branching fractions ( B X c ν ) E > E min MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=XsicjabcIcaOiabdkeacjabgkziUkabdIfaynaaBaaaleaacqWGJbWyaeqaaOGaeS4eHWMaeqyVd4MaeiykaKYaaSbaaSqaaiabdweafnaaBaaameaacqWItecBaeqaaSGaeyOpa4Jaemyrau0aaSbaaWqaaiGbc2gaTjabcMgaPjabc6gaUbqabaaaleqaaaaa@4938@ Open image in new window and the first four moments of the electron energy spectrum in BX c e ν, for Emin from 0.4 to 2.0 GeV [159]. All the lepton moment measurements are consistent with theory and with the moment of the hadronic and bs γ photon energy spectrum. Hadronic and lepton energy measurements are consistent within their errors. When compared with theory there is no sign of inconsistencies. The B-factories provide the most precise HQE parameter estimates.

3.3.1 HQE Parameters

Using the moment measurements described above, it is possible to determine the CKM matrix element |V cb | and HQE parameters by performing global fits in the kinetic and 1S b-quark mass schemes [160]. The photon energy moments in BX s γ decays [161] are also included in order to constrain the b-quark mass more precisely. Measurements that are not matched by theoretical predictions and those with high cutoff energies are excluded (i.e. semileptonic moments with Emin > 1.5 GeV and photon energy moments with Emin > 2 GeV). The results are preliminary.

The inclusive spectral moments of BX c ν decays have been derived in the 1S scheme up to ( B X c ν ) E > E min MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=XsicjabcIcaOiabdkeacjabgkziUkabdIfaynaaBaaaleaacqWGJbWyaeqaaOGaeS4eHWMaeqyVd4MaeiykaKYaaSbaaSqaaiabdweafnaaBaaameaacqWItecBaeqaaSGaeyOpa4Jaemyrau0aaSbaaWqaaiGbc2gaTjabcMgaPjabc6gaUbqabaaaleqaaaaa@4938@ Open image in new window(1/ m b 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIZaWmaaaaaa@2EC8@ Open image in new window) [139]. The theoretical expressions for the truncated moments are given in terms of HQE parameters with coefficients determined by theory, as functions of Emin. The non-perturbative corrections are parametrized in terms of the following non-perturbative parameters: Λ at O MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=5q8pbaa@3689@ Open image in new window(m b ), λ1 and λ2 at O MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=5q8pbaa@3689@ Open image in new window(1/ m b 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIYaGmaaaaaa@2EC6@ Open image in new window), and τ1, τ2, τ3, τ4, ρ1 and ρ2 at O MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=5q8pbaa@3689@ Open image in new window(1/ m b 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIZaWmaaaaaa@2EC8@ Open image in new window). In Table 6 one finds the following results for the fit parameters [160]. The first error is from the fit including experimental and theory errors, and the second error (on |V cb | only) is due to the uncertainty on the average B lifetime. If the same fit is performed to all measured moments of inclusive distributions in BX c ν and Bs γ decays, the |V cb | and m b values obtained are in Table 6. The fit results for |V cb | and m b 1 S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIXaqmcqqGtbWuaaaaaa@2FF1@ Open image in new window to BX c ν data only and BX c ν and BX s γ data combined are displayed in Fig. 24.
Table 6

Experimental results using corrections to the common inputs and world averages for the 1S scheme.

 

|V cb |(×10-3)

m b 1 S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIXaqmcqqGtbWuaaaaaa@2FF1@ Open image in new window (GeV)

λ1(GeV2)

χ2/n.d.f.

X c ν + X s γ

41.81 ± 0.34 fit ± 0.08 τ B MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabisda0iabigdaXiabc6caUiabiIda4iabigdaXiabgglaXkabicdaWiabc6caUiabiodaZiabisda0maaBaaaleaacqqGMbGzcqqGPbqAcqqG0baDaeqaaOGaeyySaeRaeGimaaJaeiOla4IaeGimaaJaeGioaGZaaSbaaSqaaiabes8a0naaBaaameaacqWGcbGqaeqaaaWcbeaaaaa@428C@ Open image in new window

4.700 ± 0.030

-0.315 ± 0.026

24.7/56

X c ν

42.03 ± 0.42 fit ± 0.08 τ B MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabisda0iabikdaYiabc6caUiabicdaWiabiodaZiabgglaXkabicdaWiabc6caUiabisda0iabikdaYmaaBaaaleaacqqGMbGzcqqGPbqAcqqG0baDaeqaaOGaeyySaeRaeGimaaJaeiOla4IaeGimaaJaeGioaGZaaSbaaSqaaiabes8a0naaBaaameaacqWGcbGqaeqaaaWcbeaaaaa@4280@ Open image in new window

4.656 ± 0.060

-0.343 ± 0.046

19.0/45

Figure 24

Fit results for | Vcb| and m b 1 S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIXaqmcqqGtbWuaaaaaa@2FF1@ Open image in new window to B → Xcℓ ν data only (larger blue ellipse) and B → Xcℓ ν and B → Xsγ data combined (smaller red ellipse) for the 1 S scheme. The ellipses are for Δχ2 = 1 [236].

Spectral moments of BX c ν decays have been derived up to O MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=5q8pbaa@3689@ Open image in new window(1/ m b 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIZaWmaaaaaa@2EC8@ Open image in new window) in the kinetic scheme [146]. The theoretical expressions used in the fit contain improved calculations of the perturbative corrections to the lepton energy moments [116] and account for the Emin dependence of the perturbative corrections to the hadronic mass moments [162]. For the BX s γ moments, the (biased) OPE prediction and the bias correction have been calculated [147]. All these expressions depend on the b- and c-quark masses m b (μ) and m c (μ), the non-perturbative parameters, defined at the scale μ = 1 GeV: μ π 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeY7aTnaaDaaaleaacqaHapaCaeaacqaIYaGmaaaaaa@2F89@ Open image in new window (μ) and μ G 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeY7aTnaaDaaaleaacqWGhbWraeaacqaIYaGmaaaaaa@2EE3@ Open image in new window (μ) ( O MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=5q8pbaa@3689@ Open image in new window(1/ m b 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIYaGmaaaaaa@2EC6@ Open image in new window)), ρ ˜ D 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbeg8aYzaaiaWaa0baaSqaaiabdseaebqaaiabiodaZaaaaaa@2EF8@ Open image in new window(μ) and ρ L S 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeg8aYnaaDaaaleaacqWGmbatcqWGtbWuaeaacqaIZaWmaaaaaa@3028@ Open image in new window(μ) ( O MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabciab=5q8pbaa@3689@ Open image in new window(1/ m b 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaIZaWmaaaaaa@2EC8@ Open image in new window)), and α s . The CKM element |V cb | is a free parameter in the fit, related to the semileptonic width Γ (BX c ν) [142]. The fit results for various inputs and correlations are shown in Table 7.
Table 7

Experimental results using corrections to the common inputs and world averages for the kinetic scheme.

 

|V cb |(×10-3)

m b (GeV)

μ π 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeY7aTnaaDaaaleaacqaHapaCaeaacqaIYaGmaaaaaa@2F89@ Open image in new window(GeV2)

χ2/n.d.f.

X c ν + X s γ

41.67 ± 0.47 f i t ± 0.08 τ B ± 0.58 t h MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabisda0iabigdaXiabc6caUiabiAda2iabiEda3iabgglaXkabicdaWiabc6caUiabisda0iabiEda3maaBaaaleaacqWGMbGzcqWGPbqAcqWG0baDaeqaaOGaeyySaeRaeGimaaJaeiOla4IaeGimaaJaeGioaGZaaSbaaSqaaiabes8a0naaBaaameaacqWGcbGqaeqaaaWcbeaakiabgglaXkabicdaWiabc6caUiabiwda1iabiIda4maaBaaaleaacqWG0baDcqWGObaAaeqaaaaa@4B58@ Open image in new window

4.601 ± 0.034

0.440 ± 0.040

29.7/57

X c ν + X s γ

40.85 ± 0.68 f i t ± 0.08 τ B ± 0.57 t h MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabisda0iabicdaWiabc6caUiabiIda4iabiwda1iabgglaXkabicdaWiabc6caUiabiAda2iabiIda4maaBaaaleaacqWGMbGzcqWGPbqAcqWG0baDaeqaaOGaeyySaeRaeGimaaJaeiOla4IaeGimaaJaeGioaGZaaSbaaSqaaiabes8a0naaBaaameaacqWGcbGqaeqaaaWcbeaakiabgglaXkabicdaWiabc6caUiabiwda1iabiEda3maaBaaaleaacqWG0baDcqWGObaAaeqaaaaa@4B5A@ Open image in new window

4.605 ± 0.031

0.312 ± 0.060

54.2/46

X c ν

41.48 ± 0.47 f i t ± 0.08 τ B ± 0.58 t h MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabisda0iabigdaXiabc6caUiabisda0iabiIda4iabgglaXkabicdaWiabc6caUiabisda0iabiEda3maaBaaaleaacqWGMbGzcqWGPbqAcqWG0baDaeqaaOGaeyySaeRaeGimaaJaeiOla4IaeGimaaJaeGioaGZaaSbaaSqaaiabes8a0naaBaaameaacqWGcbGqaeqaaaWcbeaakiabgglaXkabicdaWiabc6caUiabiwda1iabiIda4maaBaaaleaacqWG0baDcqWGObaAaeqaaaaa@4B56@ Open image in new window

4.659 ± 0.049

0.428 ± 0.044

24.1/46

– New theoretical correlations are applied [163].

The first error is from the fit (experimental error, non-perturbative and bias corrections), and the second error is a 1.5% uncertainty from the theoretical expression for the semileptonic width [142].

All the measured moments of inclusive distributions in BX c ν and Bs γ decays are used in a fit to extract |V cb | and the b and c quark masses. The |V cb | and m b values obtained are listed in the Table 7. The default fit also gives m c = 1.16 ± 0.05 GeV, and μ G 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeY7aTnaaDaaaleaacqWGhbWraeaacqaIYaGmaaaaaa@2EE3@ Open image in new window = 0.27 ± 0.04 GeV2. The errors are experimental and theoretical (HQE and ΓSL) respectively. In this fit the following variations were considered when in the extraction of the HQ parameters, ± 20 MeV for the b and c quark masses, ± 20% for μ p 2 i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeY7aTnaaDaaaleaacqWGWbaCaeaacqaIYaGmaaGccqWGPbqAaaa@309A@ Open image in new window and μ G 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeY7aTnaaDaaaleaacqWGhbWraeaacqaIYaGmaaaaaa@2EE3@ Open image in new window, ± 30% for the 3 rd order non perturbative terms and α2 = 0.22 ± 0.04 for the perturbative corrections. The bias corrections uncertainties for Bs γ were varied by the full amount of their value.

There are open issues relating to the global fits. First of all, the χ2/n.d.f. are very small, pointing to an underestimate in the theoretical correlations. In a recent study [163], the theoretical correlations used in the fit were scrutinized, and new correlation coefficients were derived from the theory expressions using a "toy Monte Carlo" approach, showing that the theoretical correlations were largely underestimated. The result of this new fit is shown in Table 7. The second issue is related the size of the theoretical error. Recently the NNLO full two-loop calculations become available [164, 165, 166]. In the Kinetic scheme NNLO calculations include an estimate of the non-BLM terms and lead to a roughly 0.6 % reduction of the |V cb | value -0.25 × 10-3. In the 1S scheme the shift on |V cb | is of about -0.14 × 10-3 [167]. From the new power corrections at NLO we expect the chromo-magnetic corrections to be more important as the tree level corrections are more important, and a change of about 20–30% in the extracted value of μ π 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeY7aTnaaDaaaleaacqaHapaCaeaacqaIYaGmaaaaaa@2F89@ Open image in new window in the pole expansion (may be less with the other schemes) [168].

HQE has been carried out up to 1/ m b 4 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqaI0aanaaaaaa@2ECA@ Open image in new window and the effects are expected to be of the order δ(4)Γ/Γ ≈ 0.25% [169]. All these newly calculated results can be used to scrutinize the earlier error estimates. In the kinetic scheme, the full NNLO value for A pert = 0.919, which is in good agreement with A pert = Γ (BX c ν)/Γ (BX c ν) tree = 0.908 ± 0.009. In the 1S scheme the estimated uncertainty from the non-BLM two-loop is half of the BLM part, equivalent to 1.5% of the tree-level rate, more than 3 times the actual correction. The new results are not implemented in the global fits as they are available mainly in the pole-mass scheme.

3.4 Outlook

The error on the inclusive and exclusive determination of V cb is limited by theory. The two method have a 2σ disagreement, even when results in the same experiment are compared.

For exclusive determinations, the experimental determinations in BD*ℓν from different experiments are not in agreement. Improving the statistical error for the determination from BDν will help in elucidating the origin of this discrepancy.

In the inclusive method, errors of less than 2% are quoted. However, the latest theoretical results and the introduction of better correlation between theoretical error show shifts in the central value larger than the quoted fit error. The situation needs to be reevaluated when all the new calculations and corrections are implemented in the fit. Another puzzling result of the global fit to BX c ν moments only is the value of m b kin MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqqGRbWAcqqGPbqAcqqGUbGBaaaaaa@31ED@ Open image in new window, which seems to be in disagreement with other determinations, such as the ones in Table 2. (For a detailed comparison between m b kin MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2gaTnaaDaaaleaacqWGIbGyaeaacqqGRbWAcqqGPbqAcqqGUbGBaaaaaa@31ED@ Open image in new window and different mass determinations see A. Hoang, talk at Joint Workshop on |V ub | and |V cb | at the B-Factories Heidelberg, December 14–16, 2007.)

4 The CKM parameter |Vub|

The parameter |V ub | determines one of the sides of the unitarity triangle, and thus affects one of the crucial tests of the Yukawa sector of the Standard Model. Also in this case, there are two general methods to determine this parameter, using B meson semileptonic decays. The first approach relies on the determination of branching fractions and form factor determinations of exclusive semileptonic decays, such as Bπ ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window. The relationship between experimental measurements and |V ub | requires a theoretical prediction of the hadronic form factors governing these decays. The complementary approach relies on measurements of inclusive properties of B meson semileptonic decays. In this case, the hadronic matrix element is evaluated in the context of the OPE.

Both methods pose challenges to both experimenters and theorists. The branching fractions are small, and a substantial background induced by the dominant bc ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window needs to be suppressed or accounted for. The large data samples accumulated at the two b-factories, Belle and BaBar, have made possible the development of new experimental techniques that have reduced the experimental errors substantially. In parallel, theorists devoted considerable efforts to devise measurable quantities for which reliable predictions could be produced, and to improve the accuracy of the predictions through refinements in the calculation. Although the precision of the stated errors improved, so did the difference in central values between the inclusive and exclusive estimates of |V ub |, at least in most determinations. Possible interpretations of this discrepancy will be discussed.

4.1 Determinations Based on Exclusive BSemileptonic Decays

The decay Bπ ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window is the simplest to interpret, as it is affected by a single form factor. The differential decay width is given by
d Γ d q 2 = G F 2 | V u b | 2 192 π 3 m B 3 λ ( q 2 ) 3 / 2 | f + ( q 2 ) | 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@5BB7@ Open image in new window
(66)

where G F is the Fermi constant, λ ( q 2 ) = ( q 2 + m B 2 m π 2 ) 2 4 m B 2 m π 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeU7aSjabcIcaOiabdghaXnaaCaaaleqabaGaeGOmaidaaOGaeiykaKIaeyypa0JaeiikaGIaemyCae3aaWbaaSqabeaacqaIYaGmaaGccqGHRaWkcqWGTbqBdaqhaaWcbaGaemOqaieabaGaeGOmaidaaOGaeyOeI0IaemyBa02aa0baaSqaaiabec8aWbqaaiabikdaYaaakiabcMcaPmaaCaaaleqabaGaeGOmaidaaOGaeyOeI0IaeGinaqJaemyBa02aa0baaSqaaiabdkeacbqaaiabikdaYaaakiabd2gaTnaaDaaaleaacqaHapaCaeaacqaIYaGmaaaaaa@4AD2@ Open image in new window, and f+ (q2) is the relevant form factor.

The first experimental challenge is to reconstruct this exclusive channel without significant background from the dominant charm semileptonic decays, and the additional background component from other bu ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window transitions. The advantage of e+ e- B-factories is that the B decaying semileptonically originates from the decay e+ e- B B ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraaaaa@2C1C@ Open image in new window. Thus, if the companion B is fully reconstructed, the ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window 4-momentum can be inferred from the rest of the event.

CLEO pioneered this approach by reconstructing the ν from the missing energy (E miss ≡ 2E B - Σ i E i ), and momentum ( p m i s s i p i ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaabmaabaGafmiCaaNbaSaadaWgaaWcbaGaemyBa0MaemyAaKMaem4CamNaem4CamhabeaakiabggMi6oaaqababaGafmiCaaNbaSaadaWgaaWcbaGaemyAaKgabeaaaeaacqWGPbqAaeqaniabggHiLdaakiaawIcacaGLPaaaaaa@3BD5@ Open image in new window in the event; in these definitions the index i runs over the well reconstructed tracks and photons, and cuts are applied to enhance the probability that the only missing particle in the event is the ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window. This approach allows the application of a low momentum cut of 1.5 GeV on the lepton for Bπ ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window and 2 GeV for Bρ ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window. Averaging the results from these two exclusive channels, they obtain | V u b | = ( 3.3 ± 0.2 0.4 + 0.3 ± 0.7 ) × 10 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaaemaabaGaemOvay1aaSbaaSqaaiabdwha1jabdkgaIbqabaaakiaawEa7caGLiWoacqGH9aqpdaqadaqaaiabiodaZiabc6caUiabiodaZiabgglaXkabicdaWiabc6caUiabikdaYmaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaI0aanaeaacqGHRaWkcqaIWaamcqGGUaGlcqaIZaWmaaGccqGHXcqScqaIWaamcqGGUaGlcqaI3aWnaiaawIcacaGLPaaacqGHxdaTcqaIXaqmcqaIWaamdaahaaWcbeqaaiabgkHiTiabiodaZaaaaaa@4EA4@ Open image in new window. Using their relatively modest full data set (16 fb-1) at the center-of-mass energy of the ϒ(4S) and considering only the Bπ ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window channel, they get the branching fraction MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ Open image in new window (Bπ+ ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window) = (1.37 ± 0.15 ± 0.11) × 10-4, and | V u b | = ( 3.6 ± 0.4 ± 0.2 ± 0.4 + 0.6 ) × 10 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamaaemaabaGaemOvay1aaSbaaSqaaiabdwha1jabdkgaIbqabaaakiaawEa7caGLiWoacqGH9aqpdaqadaqaaiabiodaZiabc6caUiabiAda2iabgglaXkabicdaWiabc6caUiabisda0iabgglaXkabicdaWiabc6caUiabikdaYiabgglaXoaaDaaaleaacqGHsislcqaIWaamcqGGUaGlcqaI0aanaeaacqGHRaWkcqaIWaamcqGGUaGlcqaI2aGnaaaakiaawIcacaGLPaaacqGHxdaTcqaIXaqmcqaIWaamdaahaaWcbeqaaiabgkHiTiabiodaZaaaaaa@5098@ Open image in new window[170].

BaBar [171] uses a sample of 206 fb-1 to obtain MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Xsicbaa@35C8@ Open image in new window(Bπ+ ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window) = (1.46 ± 0.07 ± 0.08) × 10-4. Recently, the availability of very large data sets at Belle and BaBar have made possible tagged analyses, where semileptonic decays are studied in samples where the other B is fully reconstructed, thus defining the event kinematics even more precisely. The first implementation relies on the the partial reconstruction of exclusive BD(⋆)+ ν decay to tag the presence of a B B ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdkeaczaaraaaaa@2C1C@ Open image in new window event [172, 173].

Belle uses fully reconstructed hadronic tags, achieving the best kinematic constraints and thus the highest background suppression capabilities and the most precise determination of q2. Branching fractions obtained with this technique have a bigger statistical error because of the penalty introduced by the tag requirement, but the overall error is already comparable with the other methods. Table 8 summarizes the present status of the experimental information on this decay. Both the total branching fraction for Bπ ν ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbe27aUzaaraaaaa@2CC7@ Open image in new window and the partial branching fraction for q2 ≥ 16 GeV2 are shown. The latter partial branching fraction is useful to extract |V ub | using unquenched lattice calculations, as this is the only q2 interval where their calculation is reliable.
Table 8

Partial and total branching fractions, in units of 10-4, for the decay Bπ+ν.

Experiment

total

q2 > 16 GeV2

Method

CLEO [170]

1.37 ± 0.15 ± 0.11

0.41 ± 0.08 ± 0.04

Untagged Analysis

BaBar [171]

1.46 ± 0.07 ± 0.08

0.38 ± 0.04 ± 0.04

Untagged Analysis

Belle [172]

1.38 ± 0.19 ± 0.14 ± 0.03

0.36 ± 0.10 ± 0.04 ± 0.01

B→ D(*)ℓν Tag

Belle [254]

1.49 ± 0.26 ± 0.06

0.31 ± 0.12 ± 0.01

B0π-ℓ+ν (hadron tags)

Belle [254]

1.53 ± 0.20 ± 0.06

0.39 ± 0.12 ± 0.02

Combined B0 and B+ tags

BaBar [173]

1.33 ± 0.17 ± 0.11

0.46 ± 0.10 ± 0.06

Combined B0 and B+ tags

combined [243]

1.34 ± 0.06 ± 0.05

0.37 ± 0.03 ± 0.02

HFAG ICHEP08

– Preliminary results.

Whenever possible, B0 and B+ data are combined.

In order to interpret these results, we need theoretical predictions for the form factor f+ (q2). This problem can be split into two parts: the determination of the form factor normalization, f+ (0), and the functional form of the q2 dependence. Form factor predictions have been produced with quark models [174] and QCD sum rule calculations [175]. Lattice calculations provide evaluations of f+ (q2) at specific values of q2 or, equivalently, pion momenta (p π ).

Authors then fit these data points with a variety of shapes. Typically a dominant pole shape has been used in the literature. Nowadays more complex functional forms are preferred. Becirevic and Kaidalov (BK) [176] suggest using
f + ( q 2 ) | B K = c B ( 1 α ) ( 1 q 2 / M B 2 ) ( 1 α q 2 / M B 2 ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@603B@ Open image in new window
(67)
where c B M B 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2eannaaDaaaleaacqWGcbGqdaahaaadbeqaambvLH1qn1uy0Hws0fgBPngaryWyT1wAXadaiqaacqWFglYpaaaaleaacqaIYaGmaaaaaa@37E9@ Open image in new window is the residue of the form factor at q2 = M B 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2eannaaDaaaleaacqWGcbGqdaahaaadbeqaambvLH1qn1uy0Hws0fgBPngaryWyT1wAXadaiqaacqWFglYpaaaaleaacqaIYaGmaaaaaa@37E9@ Open image in new window, and M B 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabd2eannaaDaaaleaacqWGcbGqdaahaaadbeqaambvLH1qn1uy0Hws0fgBPngaryWyT1wAXadaiqaacqWFglYpaaaaleaacqaIYaGmaaaaaa@37E9@ Open image in new window/α is the squared mass of an effective 1- B⋆' excited state. Ball and Zwicky [175] propose
f + ( q 2 ) | B Z = r 1 1 q 2 / M B 2 + r 2 1 α q 2 / M B 2 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@5C24@ Open image in new window
(68)
where the parameters r1, r2, and α are fitted from available data. Lastly, parameterizations that allow the application of constraints derived from soft-collinear effective theory (SCET), and dispersion relations have been proposed by Boyd, Grinstein, and Lebedev [120], and later pursued also by Hill [177]. They define
f + ( t ) = 1 P ( t ) Φ ( t , t 0 ) 0 a k ( t 0 ) z ( t , t 0 ) k MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@531E@ Open image in new window
(69)
where t is (p B - p π )2, defined beyond the physical region, t± = (m B ± m π )2, and t0 is an expansion point. BGD use t0 = 0.65t-. The parameter α K allows the modeling of different functional forms, and the variable