Collider Physics within the Standard Model pp 97157  Cite as
The Theory of Electroweak Interactions
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Abstract
In this chapter, we summarize the structure of the standard EW theory [Some recent textbooks are listed in Langacker (The standard model and beyond, CRC, Boca Raton, FL, 2010; Paschos, Electroweak theory (Cambridge University Press, Cambridge, 2007); Becchi and Ridolfi, An introduction to relativistic processes and the standard model of electroweak interactions, Springer, Berlin, 2006; Horejsi, Fundamentals of electroweak theory, Karolinum, Prague, 2002; Barbieri, Lectures on the electroweak interactions, Publications of the Scuola Normale Superiore, Pisa, 2007). See also Altarelli (The standard model of electroweak interactions. In: LandoltBoernstein I 21A: Elementary Particles, vol. 3, Springer, Berlin, 2008) and Quigg (Annu Rev Nucl Part Sci 59:505, 2009).] and specify the couplings of the intermediate vector bosons W^{±} and Z and those of the Higgs particle with the fermions and among themselves, as dictated by the gauge symmetry plus the observed matter content and the requirement of renormalizability. We discuss the realization of spontaneous symmetry breaking and the Higgs mechanism. We then review the phenomenological implications of the EW theory for collider physics, that is, we leave aside the classic low energy processes that are well described by the “old” weak interaction theory (see, for example, Commins, Weak interactions, McGraw Hill, New York, 1973; Okun, Leptons and quarks, North Holland, Amsterdam, 1982; Bailin, Weak interactions, 2nd edn. Hilger, Bristol, 1982; Georgi, Weak and modern particle theory, Benjamin, Menlo Park, CA, 1984).
Keywords
Higgs Boson Gauge Boson Higgs Mass Radiative Correction Higgs Sector3.1 Introduction
In this chapter, we summarize the structure of the standard EW theory^{1} and specify the couplings of the intermediate vector bosons W^{±} and Z and those of the Higgs particle with the fermions and among themselves, as dictated by the gauge symmetry plus the observed matter content and the requirement of renormalizability. We discuss the realization of spontaneous symmetry breaking and the Higgs mechanism. We then review the phenomenological implications of the EW theory for collider physics, that is, we leave aside the classic low energy processes that are well described by the “old” weak interaction theory (see, for example, [148]).
3.2 The Gauge Sector
As discussed in Sect. 1.5, the standard EW theory is a chiral theory, in the sense that ψ_{L} and ψ_{R} behave differently under the gauge group (so that parity and charge conjugation nonconservation are made possible in principle). Thus, mass terms for fermions (of the form \(\bar{\psi }_{\mathrm{L}}\psi _{\mathrm{R}} +\mathrm{ h.c.}\)) are forbidden in the symmetric limit. In the following, ψ_{L, R} are column vectors, including all fermion types in the theory that span generic reducible representations of SU(2) ⊗ U(1).
3.3 Couplings of Gauge Bosons to Fermions
3.4 Gauge Boson SelfInteractions
3.5 The Higgs Sector
We now turn to the Higgs sector of the EW Lagrangian [243]. Until recently, this simplest realization of the EW symmetry breaking was a pure conjecture. But in July 2012 the ATLAS and CMS Collaborations at the CERN LHC announced [2, 135] the discovery of a particle with mass m_{H} ∼ 126 GeV that looks very much like the long sought Higgs particle. More precise measurements of its couplings and the proof that its spin is zero are necessary before the identification with the SM Higgs boson can be completely established. But the following description of the Higgs sector of the SM can now be read with this striking development in mind.
If only one Higgs doublet is present, the change of basis that makes \(\mathcal{M}\) diagonal will at the same time diagonalize the fermion–Higgs Yukawa couplings. Thus, in this case, no flavourchanging neutral Higgs vertices are present. This is not true, in general, when there are several Higgs doublets. But one Higgs doublet for each electric charge sector, i.e., one doublet coupled only to utype quarks, one doublet to dtype quarks, one doublet to charged leptons, and possibly one for neutrino Dirac masses, would also be acceptable, because the mass matrices of fermions with different charges are diagonalized separately. For several Higgs doublets in a given charge sector, it is also possible to generate CP violation by complex phases in the Higgs couplings. In the presence of six quark flavours, this CP violation mechanism is not necessary. In fact, at the moment, the simplest model with only one Higgs doublet could be adequate for describing all observed phenomena.
3.6 The CKM Matrix and Flavour Physics
For N generations of quarks, V is a N × N unitary matrix that depends on N^{2} real numbers (N^{2} complex entries with N^{2} unitarity constraints). However, the 2N phases of up and downtype quarks are not observable. Note that an overall phase drops away from the expression of the current in (3.64), so that only 2N − 1 phases can affect V. In total, V depends on N^{2} − 2N + 1 = (N − 1)^{2} real physical parameters. Similar counting gives N(N − 1)∕2 as the number of independent parameters in an orthogonal N × N matrix. This implies that in V we have N(N − 1)∕2 mixing angles and (N − 1)^{2} − N(N − 1)∕2 = (N − 1)(N − 2)∕2 phases: for N = 2, one mixing angle (the Cabibbo angle θ_{C}) and no phases, for N = 3 three angles (θ_{12}, θ_{13}, and θ_{23}) and one phase φ, and so on.
In the SM, the nonvanishing of the \(\bar{\eta }\) parameter [related to the phase φ in (3.66) and (3.67)] is the only source of CP violation in the quark sector (we shall see that new sources of CP violation very likely arise from the neutrino sector). Unitarity of the CKM matrix V implies relations of the form ∑_{ a }V_{ ba }V_{ ca }^{∗} = δ_{ bc }.
For B mixing the dominant contribution is from the t quark. In this case, the partonic dominance is more realistic and the GIM factor O(m_{ t }^{2}∕m_{ W }^{2}) is actually larger than 1. More recently D mixing has also been observed [53]. In the corresponding box diagrams, downtype quarks are involved. But starting from \(D \sim c\bar{u}\), the b quark contribution is strongly suppressed by the CKM angles, given that V_{ cb }V_{ ub }^{∗} ∼ O(λ_{ C }^{5}). The masses of the d and s quarks are too small for a partonic evaluation of the box diagram, and nonperturbative terms cannot be neglected. This makes a theoretical evaluation of mixing and CP violation effects for D mesons problematic.
For light fermion exchange in the loop, the GIM suppression is also effective in \(\vert \Delta F\vert = 1\) amplitudes. For example, analogous leptonic transitions like μ → eγ or τ → μγ also exist, but in the SM are extremely small and out of reach for experiments, because the tiny neutrino masses enter into the GIM suppression factor. But new physics effects could well make these rare lepton flavourviolating processes accessible to experiment. In fact, the present limits already pose stringent constraints on models of new physics. Of particular importance is the recent bound obtained by the MEG Collaboration at SIN, near Zurich, Switzerland, on the branching ratio for μ → eγ, viz., \(B(\upmu \rightarrow e\upgamma )\lesssim 5.7 \times 10^{13}\) at 90% [16].
Among the exclusive processes of the b → s type, much interest is at present devoted to the channel B → K^{∗}μ^{+}μ^{−} [4, 106]. The differential decay distribution depends on three angles and on the μ^{+}μ^{−} invariant mass squared q^{2}. In general 12 + 12 form factors enter into the decay distribution (12 in B decay and 12 in the CP conjugated \(\bar{B}\) decay), and many observables can be defined. By suitable angular foldings and CP averages, the number of form factors is reduced. A sophisticated theoretical analysis allows one to identify and study a number of quantities that can be measured and are “clean”, i.e., largely independent of hadronic form factor ambiguities [106]. For those observables most of the results agree with the SM predictions (based on a Wilson operator expansion in powers of 1∕m_{ W } and 1∕m_{ b }, with coefficients depending on α_{s}), but a few discrepancies are observed. The significance, taking into account the number of observables studied and the theoretical ambiguities (especially in the estimate of 1∕m_{ b } corrections), is not compelling, but a substantial activity is under way on both the experimental and the theoretical side (see, for example, [248]). Watch this space!
In conclusion, the CKM theory of quark mixing and CP violation has been precisely tested in the last decade and turns out to be very successful. The expected deviations from new physics at the EW scale have not yet appeared. The constraints on new physics from flavour phenomenology are extremely demanding: when adding higher dimensional effective operators to the SM, the flavour constraints generically lead to powers of very large suppression scales Λ in the denominators of the corresponding coefficients. In fact, in the SM, as we have discussed in this section, there are very powerful protections against flavourchanging neutral currents and CP violation effects, in particular through the smallness of quark mixing angles. In this respect the SM is very special and, as a consequence, if there is new physics, it must be highly nongeneric in order to satisfy the present flavour constraints.
Only by requiring new physics to share the SM set of protections can one reduce the scale Λ down to O(1) TeV. For example, the class of models with minimal flavour violation (MFV) [152], where the SM Yukawa couplings are the only flavour symmetry breaking terms also beyond the SM, have been much studied and represent a sort of extreme baseline. Alternative, less minimal models that are currently under study are based on a suitably broken U(3)^{3} or U(2)^{3} flavour symmetry (the cube refers to the Q_{L} = u_{L}, d_{L} doublet and the two u_{R} and d_{R} singlets, while U(3) or U(2) mix the three or the first two generations) [81].
3.7 Neutrino Mass and Mixing
In the minimal version of the SM, the righthanded neutrinos ν_{ iR}, which have no gauge interactions, are not present at all. With no ν_{R}, no Dirac mass is possible for neutrinos. If lepton number conservation is also imposed, then no Majorana mass is allowed either, and as a consequence, all neutrinos are massless. But at present, from neutrino oscillation experiments, we know that at least two out of the three known neutrinos have nonvanishing masses (for reviews, see, for example, [36]): the two masssquared differences measured from solar (\(\Delta m_{12}^{2}\)) and atmospheric oscillations (\(\Delta m_{23}^{2}\)) are given by \(\Delta m_{12}^{2} \sim 8 \times 10^{5}\) eV^{2} and \(\Delta m_{23}^{2} \sim 2.5 \times 10^{3}\) eV^{2} [200, 201, 229].

Laboratory experiments, e.g., tritium β decay near the end point, which gives \(m_{\upnu }\lesssim 2\) eV [307].

Absence of visible neutrinoless double β decay (0νββ). From Ge^{76}, it has been shown that \(\vert m_{ee}\vert \lesssim 0.2\)–0.4 eV [21]. The range is from nuclear matrix element ambiguities and m_{ ee } is a combination of neutrino masses (for a review, see, for example, [373]). This result strongly disfavours, in a modelindependent way, the claimed observation of 0νββ decay in Ge^{76} decays [267]. From Xe^{136}, one obtains the combined result \(\vert m_{ee}\vert \lesssim 0.12\)–0.25 eV [69].

Cosmological observations [175]. After the recent release of the Planck data, the quoted bounds for Σ m_{ν}, the sum of (quasi)stable neutrino masses, span a range, depending on the data set included and the cosmological priors, like \(\varSigma m_{\upnu }\lesssim 0.98\) or \(\lesssim 0.32\) or \(\lesssim 0.23\) [18] (assuming three degenerate neutrinos, these numbers have to be divided by 3 in order to obtain the limit on individual neutrino masses).
If ν_{ iR} are added to the minimal model and lepton number is imposed by hand, then neutrino masses would in general appear as Dirac masses, generated by the Higgs mechanism, as for any other fermion. But for Dirac neutrinos, to explain the extreme smallness of neutrino masses, one should allow for very small Yukawa couplings. However, we stress that, in the SM, baryon B and lepton L number conservation, which are not guaranteed by gauge symmetries (although this is the case for the electric charge Q), are understood as “accidental” symmetries. In fact the SM Lagrangian should contain all terms allowed by gauge symmetry and renormalizability, but the most general renormalizable Lagrangian (i.e., with operator dimension d ≤ 4), built from the SM fields, compatible with the SM gauge symmetry, in the absence of ν_{ iR}, is automatically B and L conserving. (However, nonperturbative instanton effects break the conservation of B + L while preserving B − L, as discussed in Sect. 3.8.)
Fits to neutrino oscillation data from [229] (free fluxes, including short baseline reactor data)
\(\Delta m_{\mathrm{sun}}^{2}\ (10^{5}\ \mathrm{eV}^{2})\)  7. 45_{−0. 16}^{+0. 19}  
\(\Delta m_{\mathrm{atm}}^{2}\ (10^{3}\ \mathrm{eV}^{2})\)  2. 417 ± 0. 013 ( − 2. 410 ± 0. 062)  
sin^{2}θ_{12}  0. 306 ± 0. 012  
sin^{2}θ_{23}  \(0.446 \pm 0.007\bigoplus 0.587_{0.037}^{+0.032}\)  
sin^{2}θ_{13}  0. 0229_{−0. 0019}^{+0. 0020}  
δ_{CP} (^{∘})  265_{−61}^{+56} 
In conclusion, neutrino masses are believed to be small because neutrinos are Majorana particles with masses inversely proportional to the large scale M of energy where L nonconservation is induced. This corresponds to an important enlargement of the original minimal SM, where no ν_{R} was included and L conservation was imposed by hand (but this ansatz would be totally unsatisfactory because L conservation is true “accidentally” only at the renormalizable level, but is violated by nonrenormalizable terms like the Weinberg operator and by instanton effects). Actually, L and B nonconservation are necessary if we want to explain baryogenesis and we have Grand Unified Theories (GUTs) in mind. It is interesting that the observed magnitudes of the masssquared splittings of neutrinos are well compatible with a scale M remarkably close to the GUT scale, where L nonconservation is indeed naturally expected. In fact, for \(m_{\upnu } \approx \sqrt{\Delta m_{\mathrm{atm } }^{2}} \approx 0.05\) eV (see Table 3.1) and m_{ν} ≈ m_{D}^{2}∕M with m_{D} ≈ v ≈ 200 GeV, we find M ≈ 10^{15} GeV which indeed is an impressive indication for M_{GUT}.
In the previous section, we discussed flavour mixing for quarks. But clearly, given that nonvanishing neutrino masses have been established, a similar mixing matrix is also introduced in the leptonic sector. We assume in the following that there are only two distinct neutrino oscillation frequencies, the atmospheric and the solar frequencies (both of them now also confirmed by experiments where neutrinos are generated on the Earth like K2K, KamLAND, and MINOS). At present the bulk of neutrino oscillation data are well reproduced in terms of three light neutrino species. However, some (so far not compelling) evidence for additional “sterile” neutrino species (i.e., not coupled to the weak interactions, as demanded by the LEP limit on the number of “active” neutrinos) are present in some data. We discuss here 3neutrino mixing, which is in any case a good approximate framework to discuss neutrino oscillations, while for possible sterile neutrinos we refer to the comprehensive review in [8].
Neutrino mixing is important because it could in principle provide new clues for the understanding of the flavour problem. Even more so since neutrino mixing angles show a pattern that is completely different from that of quark mixing: for quarks all mixing angles are small, while for neutrinos two angles are large (one is still compatible with the maximal value) and only the third one is small. In reality, it is frustrating that there has been no real illumination of the problem of flavour. Models can reproduce the data on neutrino mixing in a wide range of dynamical setups that goes from anarchy to discrete flavour symmetries (for reviews and references see, for example, [35, 37, 50, 51, 52, 264]), but we have not yet been able to single out a unique and convincing baseline for the understanding of fermion masses and mixings. Despite many interesting ideas and the formulation of many elegant models, the mysteries of the flavour structure of the three generations of fermions have not yet been unveiled.
3.8 Quantization and Renormalization of the Electroweak Theory
The formalism of the R_{ ξ } gauges is also very useful in proving that spontaneously broken gauge theories are renormalizable. In fact, the nonsingular behaviour of propagators at large momenta is very suggestive of the result. Nevertheless, it is not at all a simple matter to prove this statement. The fundamental theorem that a gauge theory with spontaneous symmetry breaking and the Higgs mechanism is in general renormalizable was proven by ’t Hooft and Veltman [278, 358].
For a chiral theory like the SM an additional complication arises from the existence of chiral anomalies. But this problem is avoided in the SM because the quantum numbers of the quarks and leptons in each generation imply a remarkable (and, from the point of view of the SM, mysterious) cancellation of the anomaly, as originally observed in [109]. In quantum field theory, one encounters an anomaly when a symmetry of the classical Lagrangian is broken by the process of quantization, regularization, and renormalization of the theory. Of direct relevance for the EW theory is the Adler–Bell–Jackiw (ABJ) chiral anomaly [19]. The classical Lagrangian of a theory with massless fermions is invariant under U(1) chiral transformations \(\psi {\prime} =\mathrm{ e}^{\mathrm{i}\gamma _{5}\theta }\psi\) (see also Sect. 2.2.3). The associated axial Noether current is conserved at the classical level. But at the quantum level, chiral symmetry is broken due to the ABJ anomaly and the current is not conserved. The chiral breaking is produced by a clash between chiral symmetry, gauge invariance, and the regularization procedure.
An important implication of chiral anomalies together with the topological properties of the vacuum in nonAbelian gauge theories is that the conservation of the charges associated with baryon (B) and lepton (L) numbers is broken by the anomaly [336], so that B and L conservation are actually violated in the standard electroweak theory (but B − L remains conserved). B and L are conserved to all orders in the perturbative expansion, but the violation occurs via nonperturbative instanton effects [87] [The amplitude is proportional to the typical nonperturbative factor exp(−c∕g^{2}), with c a constant and g the SU(2) gauge coupling.] The corresponding effect is totally negligible at zero temperature T, but becomes relevant at temperatures close to the electroweak symmetry breaking scale, precisely at T ∼ O(TeV). The nonconservation of B + L and the conservation of B − L near the weak scale plays a role in the theory of baryogenesis that aims quantitatively at explaining the observed matter–antimatter asymmetry in the Universe (for reviews and references, see, for example, [115]).
3.9 QED Tests: Lepton Anomalous Magnetic Moments
3.10 Large Radiative Corrections to Electroweak Processes
Since the SM theory is renormalizable, higher order perturbative corrections can be reliably computed. Radiative corrections are very important for precision EW tests. The SM inherits all the successes of the old V − A theory of charged currents and QED. Modern tests have focussed on neutral current processes, the W mass, and the measurement of triple gauge vertices. For Z physics and the W mass, the stateoftheart computation of radiative corrections include the complete oneloop diagrams and selected dominant multiloop corrections. In addition, some resummation techniques are also implemented, like Dyson resummation of vacuum polarization functions and important renormalization group improvements for large QED and QCD logarithms. We now discuss in more detail sets of large radiative corrections which are particularly significant (for reviews of radiative corrections for LEP1 physics, see, for example, [47], and for a more pedagogical description of LEP physics, see [338]).
Among the oneloop EW radiative corrections, a remarkable class of contributions are those terms that increase quadratically with the top mass. The sensitivity of radiative corrections to m_{ t } arises from the existence of these terms. The quadratic dependence on m_{ t } (and on other possible widely broken isospin multiplets from new physics) arises because, in spontaneously broken gauge theories, heavy virtual particles do not decouple. On the contrary, in QED or QCD, the running of α and α_{s} at a scale Q is not affected by heavy quarks with mass M ≫ Q. According to an intuitive decoupling theorem [60], diagrams with heavy virtual particles of mass M can be ignored at Q ≪ M, provided that the couplings do not grow with M and that the theory with no heavy particles is still renormalizable. In the spontaneously broken EW gauge theories, both requirements are violated.
First, one important difference with respect to unbroken gauge theories is in the longitudinal modes of weak gauge bosons. These modes are generated by the Higgs mechanism, and their couplings grow with masses (as is also the case for the physical Higgs couplings). Second, the theory without the top quark is no longer renormalizable since the gauge symmetry is broken because the (t, b) doublet would not be complete (also the chiral anomaly would not be completely cancelled). With the observed value of m_{ t }, the quantitative importance of the terms of order \(G_{\mathrm{F}}m_{t}^{2}/4\pi ^{2}\sqrt{2}\) is substantial but not dominant (they are enhanced by a factor m_{ t }^{2}∕m_{ W }^{2} ∼ 5 with respect to ordinary terms). Both the large logarithms and the G_{F}m_{ t }^{2} terms have a simple structure and are to a large extent universal, i.e., common to a wide class of processes. In particular, the G_{F}m_{ t }^{2} terms appear in vacuum polarization diagrams which are universal (virtual loops inserted in gauge boson internal lines are independent of the nature of the vertices on each side of the propagator) and in the \(Z \rightarrow b\bar{b}\) vertex which is not. This vertex is specifically sensitive to the top quark which, being the partner of the b quark in a doublet, runs in the loop. Instead, all types of heavy particles could in principle contribute to vacuum polarization diagrams. The study of universal vacuum polarization contributions, also called “oblique” corrections, and of top enhanced terms is important for an understanding of the pattern of radiative corrections. More generally, the important consequence of nondecoupling is that precision tests of the electroweak theory may a priori be sensitive to new physics, even if the new particles are too heavy for their direct production, but a posteriori no signal of deviation has clearly emerged.
While radiative corrections are quite sensitive to the top mass, they are unfortunately much less dependent on the Higgs mass. In fact, the dependence of oneloop diagrams on m_{H} is only logarithmic, viz., ∼ G_{F}m_{ W }^{2}log(m_{ H }^{2}∕m_{ W }^{2}). Quadratic terms ∼ G_{ F }^{2}m_{ H }^{2} only appear at twoloop level [356] and are too small to be detectable. The difference with the top case is that the splitting m_{ t }^{2} − m_{ b }^{2} is a direct breaking of the gauge symmetry that already affects the 1loop corrections, while the Higgs couplings are “custodial” SU(2) symmetric in lowest order.
3.11 Electroweak Precision Tests
3.12 Results of the SM Analysis of Precision Tests
We now extend the discussion of the SM fit of the data. One can think of different types of fit, depending on which experimental results are included or which answers one wants to obtain. For example, in Table 3.2 we present in column 1 a fit of all Z pole data plus m_{ W } and Γ_{ W } (this is interesting as it shows the value of m_{ t } obtained indirectly from radiative corrections, to be compared with the value of m_{ t } measured in production experiments), in column 2, a fit of all Z pole data plus m_{ t } (here it is m_{ W } which is indirectly determined), and finally, in column 3, a fit of all the data listed in Fig. 3.15 (which is the most relevant fit for constraining m_{H}).
Standard Model fits of electroweak data [350]
Fit  1  2  3  

Measurements  m_{ W }, Γ_{ W }  m _{ t }  m_{ t }, m_{ W }, Γ_{ W }  
m_{ t } (GeV)  178. 1_{−7. 8}^{+10. 9}  173. 2 ± 0. 9  173. 26 ± 0. 89  
m_{H} (GeV)  148_{−81}^{+237}  122_{−41}^{+59}  94_{−24}^{+29}  
log [m_{H}(GeV)]  2. 17 ± +0. 38  2. 09 ± 0. 17  1. 97 ± 0. 12  
α_{s}(m_{ Z })  0. 1190 ± 0. 0028  0. 1191 ± 0. 0027  0. 1185 ± 0. 0026  
m_{ W } (MeV)  80381 ± 13  80363 ± 20  80377 ± 12 
Thus the whole picture of a perturbative theory with a fundamental Higgs is well supported by the data on radiative corrections. It is important that there is a clear indication for a particularly light Higgs: at 95% confidence level \(m_{\mathrm{H}}\lesssim 152\) GeV (which becomes \(m_{\mathrm{H}}\lesssim 171\) GeV, including the input from the LEP2 direct search result). This was quite encouraging for the LHC search for the Higgs particle. More generally, if the Higgs couplings are removed from the Lagrangian, the resulting theory is nonrenormalizable. A cutoff Λ must be introduced. In the quantum corrections, logm_{H} is then replaced by logΛ plus a constant. The precise determination of the associated finite terms would be lost (that is, the value of the mass in the denominator in the argument of the logarithm). A heavy Higgs would need some unfortunate accident: the finite terms, different in the new theory from those of the SM, should by chance compensate for the heavy Higgs in a few key parameters of the radiative corrections (mainly ε_{1} and ε_{3}, see, for example, [48]). Alternatively, additional new physics, for example in the form of effective contact terms added to the minimal SM Lagrangian, should accidentally do the compensation, which again needs some sort of conspiracy.
To the list of precision tests of the SM, one should add the results on low energy tests obtained from neutrino and antineutrino deep inelastic scattering (NuTeV [353]), parity violation in Cs atoms (APV [274]), and the recent measurement of the parityviolating asymmetry in Moller scattering [354]. When these experimental results are compared with the SM predictions, the agreement is good except for the NuTeV result, which differs by three standard deviations. The NuTeV measurement is quoted as a measurement of sin^{2}θ_{W} = 1 − m_{ W }^{2}∕m_{ Z }^{2} from the ratio of neutral to charged current deep inelastic crosssections from ν_{μ} and \(\bar{\upnu }_{\upmu }\) using the Fermilab beams. But it has been argued, and it is now generally accepted, that the NuTeV anomaly probably simply arises from an underestimation of the theoretical uncertainty in the QCD analysis needed to extract sin^{2}θ_{W}. In fact, the lowest order QCD parton formalism upon which the analysis has been based is too crude to match the experimental accuracy.
Even taking this spread into account, it is clear that the implications for m_{H} are significantly different. One might imagine that some new physics effect could be hidden in the \(Zb\bar{b}\) vertex. For instance, for the top quark mass there could be other nondecoupling effects from new heavy states or a mixing of the b quark with some other heavy quark. However, it is well known that this discrepancy is not easily explained in terms of any new physics effect in the \(Zb\bar{b}\) vertex. A rather large change with respect to the SM of the b quark righthanded coupling to the Z is needed in order to reproduce the measured discrepancy (in fact, a ∼ 30% change in the righthanded coupling), an effect too large to be a loop effect, but which could be produced at the tree level, e.g., by mixing of the b quark with a new heavy vectorlike quark [140], or some mixing of the Z with ad hoc heavy states [170]. But then this effect should normally also appear in the direct measurement of A_{ b } performed at SLD using the left–right polarized b asymmetry, even within the moderate accuracy of this result. The measurements of neither A_{ b } at SLD nor R_{ b } confirm the need for such a large effect (recently a numerical calculation of NLO corrections to R_{ b } [204] appeared at first to indicate a rather large result, but in the end the full correction turned out to be rather small). Alternatively, the observed discrepancy could simply be due to a large statistical fluctuation or an unknown experimental problem. As a consequence of this problem, the ambiguity in the measured value of sin^{2}θ_{eff} is in practice greater than the nominal error, reported in (3.114), obtained from averaging all the existing determinations, and the interpretation of precision tests is less sharp than it would otherwise be.
In conclusion, the experimental information on the Higgs sector, obtained from EW precision tests at LEP1 and 2 and the Tevatron can be summarized as follows. First, the relation M_{ W }^{2} = M_{ Z }^{2}cos^{2}θ_{W} in (3.52), modified by small, computable radiative corrections, has been demonstrated experimentally. This relation means that the effective Higgs (be it fundamental or composite) is indeed a weak isospin doublet. The direct lower limit \(m_{\mathrm{H}}\gtrsim 114.5\) GeV (at 95% confidence level) was obtained from searches at LEP2. When compared to the data on precision EW tests, the radiative corrections computed in the SM lead to a clear indication of a light Higgs, not too far from the direct LEP2 lower bound. The upper limit for m_{H} in the SM from the EW tests depends on the value of the top quark mass m_{ t }. The CDF and D0 combined value after Run II is at present m_{ t } = 173. 2 ± 0. 9 GeV [350]. As a consequence, the limit on m_{H} from the LEP and Tevatron measurements is rather stringent [350]: m_{H} < 171 GeV (at 95% confidence level, after including the information from the 114.5 GeV direct bound).
3.13 The Search for the SM Higgs
The Higgs problem is really central in particle physics today. On the one hand, the experimental verification of the Standard Model (SM) cannot be considered complete until the structure of the Higgs sector has been established by experiment. On the other hand, the Higgs is also related to most of the major problems of particle physics, like the flavour problem and the hierarchy problem, the latter strongly suggesting the need for new physics near the weak scale (something that so far has not been found). In its turn, the discovery of new physics could throw light on the nature of dark matter. It was already clear before the LHC that some sort of Higgs mechanism is at work. The W or the Z with longitudinal polarization that we observe are not present in an unbroken gauge theory (massless spin1 particles, like the photon, are transversely polarized): the longitudinal degrees of freedom for the W or the Z are borrowed from the Higgs sector and hence provide evidence for it.
Furthermore, it has been precisely established at LEP that the gauge symmetry is unbroken in the vertices of the theory: all currents and charges are indeed symmetric. Yet there is obvious evidence that the symmetry is instead badly broken in the masses. Not only do the W and the Z have large masses, but the large splitting of, for example, the t–b doublet shows that even a global weak SU(2) is not at all respected by the fermion spectrum. This is a clear signal of spontaneous symmetry breaking and the implementation of spontaneous symmetry breaking in a gauge theory is via the Higgs mechanism.
The big questions are about the nature and the properties of the Higgs particle(s). The search for the Higgs boson and for possible new physics that could accompany it was the main goal of the LHC from the start. On the Higgs the LHC should answer the following questions: do some Higgs particles exist? And if so, which ones: a single doublet, more doublets, additional singlets? SM Higgs or SUSY Higgses? Fundamental or composite (of fermions, of WW, or other)? PseudoGoldstone bosons of an enlarged symmetry? A manifestation of large extra dimensions (fifth component of a gauge boson, an effect of orbifolding or of boundary conditions, or other)? Or some combination of the above, or something so far unthought of? By now we have a candidate Higgs boson that really looks like the simplest realization of the Higgs mechanism, as described by the minimal SM Higgs. In the following we first consider the a priori expectations for the Higgs sector and then the profile of the Higgs candidate discovered at the LHC.
3.14 Theoretical Bounds on the SM Higgs Mass
It is well known that, as described in [241] and references therein, in the SM with only one Higgs doublet an upper bound on m_{H} (with mild dependence on m_{ t } and the QCD coupling α_{s}) is obtained from the requirement that the perturbative description of the theory remains valid up to a large energy scale Λ where the SM model breaks down and new physics appears. Similarly, a lower bound on m_{H} can be derived from the requirement of vacuum stability [38, 123, 323] (or, in milder form, a requirement of moderate instability, compatible with the lifetime of the Universe [160, 249]). The Higgs mass enters because it fixes the initial value of the quartic Higgs coupling λ in its running up to the large scale Λ. We now briefly recall the derivation of these limits.
In conclusion, for m_{ t } ∼ 173 GeV, only a small range of values for m_{H} is allowed, viz., 130 < m_{H} < ∼ 180 GeV, if the SM holds and the vacuum is absolutely stable up to an energy scale Λ ∼ M_{GUT} or M_{Planck}. For Higgs masses below this range, one can still have a domain where the SM is viable because the vacuum can be unstable, but with a lifetime longer than the age of the Universe [111, 118, 160]. We shall come back to this later (see Fig. 3.21).
3.15 SM Higgs Decays
3.16 The Higgs Discovery at the LHC
On 4 July 2012 at CERN, the ATLAS and CMS Collaborations [341, 345] announced the observation of a particle with mass around 126 GeV that, within the present accuracy, does indeed look like the SM Higgs boson. This is a great breakthrough which, by itself, already makes an adequate return for the LHC investment. With the Higgs discovery, the main building block for the experimental validation of the SM is now in place. The Higgs discovery is the last milestone in the long history (some 130 years) of the development of a field theory of fundamental interactions (apart from quantum gravity), starting with the Maxwell equations of classical electrodynamics, going through the great revolutions of relativity and quantum mechanics, then the formulation of quantum electrodynamics (QED) and the gradual buildup of the gauge part of the Standard Model, and finally completed with the tentative description of the electroweak (EW) symmetry breaking sector of the SM in terms of a simple formulation of the Englert–Brout–Higgs mechanism [189].
The other extremely important result from the LHC at 7 and 8 TeV centerofmass energy is that no new physics signals have been seen so far. This negative result is certainly less exciting than a positive discovery, but it is a crucial new input which, if confirmed in the future LHC runs at 13 and 14 TeV, will be instrumental in redirecting our perspective of the field. In this section we summarize the relevant data on the Higgs signal as they are known at present, while the analysis of the data from the 2012 LHC run is still in progress.
The Higgs particle has been observed by ATLAS and CMS in five channels γγ, ZZ^{∗}, WW^{∗}, \(b\bar{b}\), and τ^{+}τ^{−}. If we also include the Tevatron experiments, especially important for the \(b\bar{b}\) channel, the combined evidence is by now totally convincing. The ATLAS (CMS) combined values for the mass, in GeV∕c^{2}, are m_{H} = 125. 5 ± 0. 6 (m_{H} = 125. 7 ± 0. 4). This light Higgs is what one expects from a direct interpretation of EW precision tests [73, 142, 350]. The possibility of a “conspiracy” (the Higgs is heavy, but it falsely appears to be light because of confusing new physics effects) has been discarded: the EW precision tests of the SM tell the truth and in fact, consistently, no “conspirators”, namely no new particles, have been seen around.
In order to be sure that this is the SM Higgs boson, one must confirm that the spinparity is 0^{+} and that the couplings are as predicted by the theory. It is also essential to search for possible additional Higgs states, such as those predicted in supersymmetric extensions of the SM. As for the spin (see, for example, [179]), the existence of the H → γγ mode proves that the spin cannot be 1, and must be either 0 or 2, in the assumption of an swave decay. The \(b\bar{b}\) and τ^{+}τ^{−} modes are compatible with both possibilities. With large enough statistics the spinparity can be determined from the distributions of H → ZZ^{∗} → 4 leptons, or WW^{∗} → 4 leptons. Information can also be obtained from the HZ invariant mass distributions in the associated production [179]. The existing data already appear to strongly favour a J^{ P } = 0^{+} state against 0^{−}, 1^{+∕−}, or 2^{+} [68]. We do not expect surprises on the spinparity assignment because, if different, then all the Lagrangian vertices would be changed and the profile of the SM Higgs particle would be completely altered.
Within the somewhat limited present accuracy (October 2013), the measured Higgs couplings are in reasonable agreement (at about a 20% accuracy) with the sharp predictions of the SM. Great interest was excited by a hint of an enhanced Higgs signal in γγ, but if we put the ATLAS and CMS data together, the evidence appears now to have evaporated. All included, if the CERN particle is not the SM Higgs, it must be a very close relative! Still it would be really astonishing if the H couplings were exactly those of the minimal SM, meaning that no new physics distortions reach an appreciable level of contribution.
Actually, a more ambitious fit in terms of seven parameters has also been performed [194] with a common factor like a for couplings to WW and ZZ, three separate cfactors c_{ t }, c_{ b }, and c_{τ} for utype and dtype quarks and for charged leptons, and three parameters c_{ gg }, c_{γγ}, and c_{ Zγ} for additional gluon–gluon, γ–γ and Z–γ terms, respectively. In the SM a = c_{ t } = c_{ b } = c_{ τ } = 1 and c_{ gg } = c_{γγ} = c_{ Zγ} = 0. The present data allow a meaningful determination of all seven parameters which turns out to be in agreement with the SM [194]. For example, in the MSSM, at the tree level, a = sin(β −α), for fermions the u and dtype quark couplings are different: c_{ t } = cosα∕sinβ and c_{ b } = −sinα∕cosβ = c_{τ}. At the tree level (but radiative corrections are in many cases necessary for a realistic description), the α angle is related to the A, Z masses and to β by tan2α = tan2β(m_{ A }^{2} − m_{ Z }^{2})∕(m_{ A }^{2} + m_{ Z }^{2}). If c_{ t } is enhanced, c_{ b } is suppressed. In the limit of large m_{ A }, a = sin(β −α) → 1.
In conclusion it really appears that the Higgs sector of the minimal SM, with good approximation, is realized in nature. Apparently, what was considered just as a toy model, a temporary addendum to the gauge part of the SM, presumably to be replaced by a more complex reality and likely to be accompanied by new physics, has now been experimentally established as the actual realization of the EW symmetry breaking (at least to a very good approximation). If the role of the newly discovered particle in the EW symmetry breaking is confirmed, it will be the only known example in physics of a fundamental, weakly coupled, scalar particle with vacuum expectation value (VEV). We know many composite types of Higgslike particles, like the Cooper pairs of superconductivity or the quark condensates that break the chiral symmetry of massless QCD, but the Higgs found at the LHC is the only possibly elementary one. This is a death blow not only to Higgsless models, to straightforward technicolor models, and to other unsophisticated strongly interacting Higgs sector models, but actually a threat to all models without fast enough decoupling, in the sense that, if new physics comes in a model with decoupling, the absence of new particles at the LHC helps to explain why large corrections to the H couplings are not observed.
3.17 Limitations of the Standard Model
No signal of new physics has been found, either by direct production of new particles at the LHC, or in the electroweak precision tests, or in flavour physics. Given the success of the SM, why are we not satisfied with this theory? Once the Higgs particle has been found, why don’t we declare particle physics closed? The reason is that there are both conceptual problems and phenomenological indications for physics beyond the SM. On the conceptual side the most obvious problems are that quantum gravity is not included in the SM and that the famous hierarchy (or naturalness or finetuning) problem remains open. Among the main phenomenological hints for new physics we can list coupling unification, dark matter, neutrino masses (discussed in Sect. 3.7), baryogenesis, and the cosmological vacuum energy. At accelerator experiments, the most plausible departure from the SM is the muon anomalous magnetic moment which, as discussed in Sect. 3.9, shows a deviation by about 3 σ, but some caution should be applied since a large fraction of the uncertainty is of theoretical origin, in particular that due to the hadronic contribution to light–light scattering [245].
Thus GUTs and the realm of quantum gravity set a very distant energy horizon that modern particle theory cannot ignore. Can the SM without new physics be valid up to such high energies? One can imagine that some obvious problems of the SM could be postponed to the more fundamental theory at the Planck mass. For example, the explanation of the three generations of fermions and the understanding of fermion masses and mixing angles can be postponed. But other problems must find their solution in the low energy theory. In particular, the structure of the SM could not naturally explain the relative smallness of the weak scale of mass, set by the Higgs mechanism at \(v \sim 1/\sqrt{G_{\mathrm{F}}} \sim 250\) GeV, where G_{F} is the Fermi coupling constant. This socalled hierarchy problem [219] is due to the instability of the SM with respect to quantum corrections. In fact, nobody can believe that the SM is the definitive, complete theory but, rather, we all believe it is only an effective low energy theory.
The hierarchy problem arises because the coefficient of \(\mathcal{L}_{2}\) is not suppressed by any symmetry. This term, which appears in the Higgs potential, fixes the scale of the Higgs VEV and of all related masses. Since empirically the Higgs mass is light, (and by naturalness, it should be of O(Λ), we would expect Λ, i.e., some form of new physics, to appear near the TeV scale. The hierarchy problem can be put in very practical terms (the “little hierarchy problem”): loop corrections to the Higgs mass squared are quadratic in the cutoff Λ, which can be interpreted as the scale of new physics.
It is important to note that, although the hierarchy problem is directly related to the quadratic divergences in the scalar sector of the SM, the problem can actually be formulated without any reference to divergences, directly in terms of renormalized quantities. After renormalization, the hierarchy problem is manifested by the quadratic sensitivity of μ^{2} to the physics at high energy scales. If there is a threshold at high energy, where some particles of mass M coupled to the Higgs sector can be produced and contribute in loops, then the renormalized running mass μ will evolve slowly (i.e., logarithmically according to the relevant beta functions [195]) up to M and there, as an effect of the matching conditions at the threshold, rapidly jump to become of order M (see, for example, [79]). In fact, in Fig. 3.26, we see that, under the assumption of no thresholds, the running Higgs mass m evolves slowly, starting from the observed low energy value, up to very high energies. In the presence of a threshold at M one needs a finetuning of order μ^{2}∕M^{2} in order to fix the running mass at low energy to the observed value.
Thus for naturalness either new thresholds appear endowed with a mechanism for the cancellation of the sensitivity or they had better not appear at all. But certainly there is the Planck mass, connected to the onset of quantum gravity, which sets an unavoidable threshold. One possible point of view is that there are no new thresholds up to M_{Planck} (at the price of giving up GUTs, among other things) but, miraculously, there is a hidden mechanism in quantum gravity that solves the finetuning problem related to the Planck mass [221, 322]. For this one would need to solve all phenomenological problems, like dark matter, baryogenesis, and so on, with physics below the EW scale. Possible ways to do so are discussed in [322]. This point of view is extreme, but allegedly not yet ruled out.

Supersymmetry [302]. In the limit of exact boson–fermion symmetry, quadratic bosonic divergences cancel so that only log divergences remain. However, exact SUSY is clearly unrealistic. For approximate SUSY (with soft breaking terms and Rparity conservation), which is the basis for most practical models, Λ^{2} is essentially replaced by the splitting of SUSY multiplets, Λ^{2} ∼ m_{SUSY}^{2} − m_{ord}^{2}, with m_{ord} the SM particle masses. In particular, the top loop is quenched by partial cancellation with stop exchange, so the stop cannot be too heavy. After the bounds from the LHC, the present emphasis is to build SUSY models where naturalness is restored not too far from the weak scale, but the related new physics is arranged in such a way that it would not have been visible so far. The simplest ingredients introduced in order to decrease the fine tuning are either the assumption of a split spectrum with heavy first two generations of squarks (for some recent work along this line see, for example, [271]) or the enlargement of the Higgs sector of the MSSM by adding a singlet Higgs field (see, for example, [196] on nexttominimal SUSY SM or NMSSM) or both.

A strongly interacting EW symmetrybreaking sector. The archetypal model of this class is technicolor, where the Higgs is a condensate of new fermions [332]. In these theories there is no fundamental scalar Higgs field, hence no quadratic divergences associated with the μ^{2} mass in the scalar potential. But this mechanism needs a very strong binding force, Λ_{TC} ∼ 10^{3}Λ_{QCD}. It is difficult to arrange for such a nearby strong force not to show up in precision tests. Hence, this class of models was abandoned after LEP, although some special classes of models have been devised a posteriori, like walking TC, topcolor assisted TC, etc. [246] (for reviews see, for example, [275]). But the simplest Higgs observed at the LHC has now eliminated another score of these models. Modern strongly interacting models, like little Higgs models [63] [in these models extra symmetries allow m_{h} ≠ 0 only at twoloop level, so that Λ can be as large as O(10 TeV)], or composite Higgs models [223, 258] (where nonperturbative dynamics modifies the linear realization of the gauge symmetry and the Higgs has both elementary and composite components) are more sophisticated. All models in this class share the idea that the Higgs is light because it is the pseudoGoldstone boson of an enlarged global symmetry of the theory, for example SO(5) broken down to SO(4). There is a gap between the mass of the Higgs (similar to a pion) and the scale f where new physics appears in the form of resonances (similar to the ρ, etc.). The ratio ξ = v^{2}∕f^{2} defines a degree of compositeness that interpolates between the SM at ξ = 0 up to technicolor at ξ = 1. Precision EW tests impose ξ < 0. 05–0.2. In these models the bad quadratic behaviour from the top loop is softened by the exchange of new vectorlike fermions with charge 2/3, or even with exotic charges like 5/3 (see, for example, [143, 295]).

Extra dimensions [62, 314] (for pedagogical introductions, see, for example, [331]). The idea is that M_{Planck} appears very large, or equivalently that gravity appears very weak, because we are fooled by hidden extra dimensions, so that either the real gravity scale is reduced down to a lower scale, even possibly down to O(1 TeV) or the intensity of gravity is redshifted away by an exponential warping factor [314]. This possibility is very exciting in itself and it is really remarkable that it is compatible with experiment. It provides a very rich framework with many different scenarios.

The anthropic evasion of the problem. The observed value of the cosmological constant Λ also poses a tremendous, unsolved naturalness problem [205]. Yet the value of Λ is close to the Weinberg upper bound for galaxy formation [364]. Possibly our Universe is just one of infinitely many bubbles (a multiverse) continuously created from the vacuum by quantum fluctuations. Different physics takes place in different universes according to the multitude of string theory solutions [177] ( ∼ 10^{500}). Perhaps we live in a very unlikely universe, but the only one that allows our existence [61, 220, 318]. Personally, I find the application of the anthropic principle to the SM hierarchy problem somewhat excessive. After all, one can find plenty of models that easily reduce the fine tuning from 10^{14} to 10^{2}: why make our universe so terribly unlikely? If we add, say, supersymmetry to the SM, does the universe become less fit for our existence? In the multiverse, there should be plenty of less finely tuned universes where more natural solutions are realized and which are still suitable for us to live in them. By comparison, the case of the cosmological constant is very different: the context is not as fully specified as the one for the SM (quantum gravity, string cosmology, branes in extra dimensions, wormholes through different universes, and so on). Further, while there are many natural extensions of the SM, so far there is no natural theory of the cosmological constant.
It is true that the data impose a substantial amount of apparent fine tuning, and our criterion of naturalness has certainly failed so far, so that we are now lacking a reliable argument to tell us where precisely the new physics threshold is located. On the other hand, many of us remain confident that some new physics will appear not too far from the weak scale.
While I remain skeptical I would like to sketch here one possibility of how the SM can be extended in agreement with the anthropic idea. If we completely ignore the finetuning problem and only want to reproduce, in a way compatible with GUTs, the most compelling data that demand new physics beyond the SM, a possible scenario is the following. The SM spectrum is completed by the recently discovered light Higgs and there is no other new physics in the LHC range (how sad!). In particular there is no SUSY in this model. At the GUT scale of M_{GUT} ≥ 10^{16} GeV, the unifying group is SO(10), broken at an intermediate scale, typically M_{int} ∼ 10^{10}–10^{12} down to a subgroup like the Pati–Salam group \(SU(4)\bigotimes SU(2)_{\mathrm{L}}\bigotimes SU(2)_{\mathrm{R}}\) or \(SU(3)\bigotimes U(1)\bigotimes SU(2)_{\mathrm{L}}\bigotimes SU(2)_{\mathrm{R}}\) [98]. Note that, in general, unification in SU(5) would not work because we need a group of rank larger than 4 to allow for (at least) twostep breaking: this is needed, in the absence of SUSY, to restore coupling unification and to avoid a too fast proton decay. An alternative is to assume some ad hoc intermediate threshold to modify the evolution towards unification [224].
The dark matter problem is one of the strongest pieces of evidence for new physics. In this model it should be solved by axions [262, 263, 309]. It must be said that axions have the problem that their mass has to be fixed ad hoc to reproduce the observed amount of dark matter. In this respect, the WIMP (weakly interacting massive particle) solution, like the neutralinos in SUSY models, is much more attractive. Lepton number violation, Majorana neutrinos, and the seesaw mechanism give rise to neutrino mass and mixing. Baryogenesis occurs through leptogenesis [115]. One should one day observe proton decay and neutrinoless beta decay. None of the alleged indications for new physics at colliders would survive (in particular, even the claimed muon g − 2 [297] discrepancy should be attributed, if not to an experimental problem, to an underestimate of the theoretical uncertainties, or otherwise to some specific addition to the above model [257]). This model is in line with the nonobservation of the decay μ → eγ at MEG [16], of the electric dipole moment of the neutron [75], etc. It is a very important challenge to experiment to falsify such a scenario by establishing firm evidence for new physics at the LHC or at some other “low energy” experiment.
In 2015 the LHC will restart at 13–14 TeV and in the following years should collect a much larger statistical sample than available at present at 7–8 TeV. From the above discussion it is clear that it is extremely important for the future of particle physics to know whether the extraordinary and unexpected success of the SM, including the Higgs sector, will continue, or whether clear signals of new physics will finally appear, as we very much hope.
Footnotes
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