Gauge Theories and the Standard Model

  • Guido Altarelli
  • James Wells
Open Access
Part of the Lecture Notes in Physics book series (LNP, volume 937)


A possible goal of fundamental physics is to reduce all natural phenomena to a set of basic laws and theories which, at least in principle, can quantitatively reproduce and predict experimental observations. At the microscopic level all the phenomenology of matter and radiation, including molecular, atomic, nuclear, and subnuclear physics, can be understood in terms of three classes of fundamental interactions: strong, electromagnetic, and weak interactions. For all material bodies on the Earth and in all geological, astrophysical, and cosmological phenomena, a fourth interaction, the gravitational force, plays a dominant role, but this remains negligible in atomic and nuclear physics. In atoms, the electrons are bound to nuclei by electromagnetic forces, and the properties of electron clouds explain the complex phenomenology of atoms and molecules. Light is a particular vibration of electric and magnetic fields (an electromagnetic wave). Strong interactions bind the protons and neutrons together in nuclei, being so strongly attractive at short distances that they prevail over the electric repulsion due to the like charges of protons. Protons and neutrons, in turn, are composites of three quarks held together by strong interactions occur between quarks and gluons (hence these particles are called “hadrons” from the Greek word for “strong”). The weak interactions are responsible for the beta radioactivity that makes some nuclei unstable, as well as the nuclear reactions that produce the enormous energy radiated by the stars, and in particular by our Sun. The weak interactions also cause the disintegration of the neutron, the charged pions, and the lightest hadronic particles with strangeness, charm, and beauty (which are “flavour” quantum numbers), as well as the decay of the top quark and the heavy charged leptons (the muon μ and the tau τ). In addition, all observed neutrino interactions are due to these weak forces.


Gauge Theory Gauge Transformation Gauge Boson Gauge Field Lagrangian Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1.1 An Overview of the Fundamental Interactions

A possible goal of fundamental physics is to reduce all natural phenomena to a set of basic laws and theories which, at least in principle, can quantitatively reproduce and predict experimental observations. At the microscopic level all the phenomenology of matter and radiation, including molecular, atomic, nuclear, and subnuclear physics, can be understood in terms of three classes of fundamental interactions: strong, electromagnetic, and weak interactions. For all material bodies on the Earth and in all geological, astrophysical, and cosmological phenomena, a fourth interaction, the gravitational force, plays a dominant role, but this remains negligible in atomic and nuclear physics. In atoms, the electrons are bound to nuclei by electromagnetic forces, and the properties of electron clouds explain the complex phenomenology of atoms and molecules. Light is a particular vibration of electric and magnetic fields (an electromagnetic wave). Strong interactions bind the protons and neutrons together in nuclei, being so strongly attractive at short distances that they prevail over the electric repulsion due to the like charges of protons. Protons and neutrons, in turn, are composites of three quarks held together by strong interactions occur between quarks and gluons (hence these particles are called “hadrons” from the Greek word for “strong”). The weak interactions are responsible for the beta radioactivity that makes some nuclei unstable, as well as the nuclear reactions that produce the enormous energy radiated by the stars, and in particular by our Sun. The weak interactions also cause the disintegration of the neutron, the charged pions, and the lightest hadronic particles with strangeness, charm, and beauty (which are “flavour” quantum numbers), as well as the decay of the top quark and the heavy charged leptons (the muon μ and the tau τ). In addition, all observed neutrino interactions are due to these weak forces.

All these interactions (with the possible exception of gravity) are described within the framework of quantum mechanics and relativity, more precisely by a local relativistic quantum field theory. To each particle, treated as pointlike, is associated a field with suitable (depending on the particle spin) transformation properties under the Lorentz group (the relativistic spacetime coordinate transformations). It is remarkable that the description of all these particle interactions is based on a common principle: “gauge” invariance. A “gauge” symmetry is invariance under transformations that rotate the basic internal degrees of freedom, but with rotation angles that depend on the spacetime point. At the classical level, gauge invariance is a property of the Maxwell equations of electrodynamics, and it is in this context that the notion and the name of gauge invariance were introduced. The prototype of all quantum gauge field theories, with a single gauged charge, is quantum electrodynamics (QED), developed in the years from 1926 until about 1950, which is indeed the quantum version of Maxwell’s theory. Theories with gauge symmetry in four spacetime dimensions are renormalizable and are completely determined given the symmetry group and the representations of the interacting fields. The whole set of strong, electromagnetic, and weak interactions is described by a gauge theory with 12 gauged non-commuting charges. This is called the “Standard Model” of particle interactions (SM). Actually, only a subgroup of the SM symmetry is directly reflected in the spectrum of physical states. A part of the electroweak symmetry is hidden by the Higgs mechanism for spontaneous symmetry breaking of the gauge symmetry.

The theory of general relativity is a classical description of gravity (in the sense that it is non-quantum mechanical). It goes beyond the static approximation described by Newton’s law and includes dynamical phenomena like, for example, gravitational waves. The problem of formulating a quantum theory of gravitational interactions is one of the central challenges of contemporary theoretical physics. But quantum effects in gravity only become important for energy concentrations in spacetime which are not in practice accessible to experimentation in the laboratory. Thus the search for the correct theory can only be done by a purely speculative approach. All attempts at a description of quantum gravity in terms of a well defined and computable local field theory along similar lines to those used for the SM have so far failed to lead to a satisfactory framework. Rather, at present, the most complete and plausible description of quantum gravity is a theory formulated in terms of non-pointlike basic objects, the so-called “strings”, extended over much shorter distances than those experimentally accessible and which live in a spacetime with 10 or 11 dimensions. The additional dimensions beyond the familiar 4 are, typically, compactified, which means that they are curled up with a curvature radius of the order of the string dimensions. Present string theory is an all-comprehensive framework that suggests a unified description of all interactions including gravity, in which the SM would be only a low energy or large distance approximation.

A fundamental principle of quantum mechanics, the Heisenberg uncertainty principle, implies that, when studying particles with spatial dimensions of order \(\Delta x\) or interactions taking place at distances of order \(\Delta x\), one needs as a probe a beam of particles (typically produced by an accelerator) with impulse \(p\gtrsim \hslash /\Delta x\), where is the reduced Planck constant ( = h∕2π). Accelerators presently in operation, like the Large Hadron Collider (LHC) at CERN near Geneva, allow us to study collisions between two particles with total center of mass energy up to \(2E \sim 2pc\lesssim 7\)–14 TeV. These machines can, in principle, study physics down to distances \(\Delta x\gtrsim 10^{-18}\) cm. Thus, on the basis of results from experiments at existing accelerators, we can indeed confirm that, down to distances of that order of magnitude, electrons, quarks, and all the fundamental SM particles do not show an appreciable internal structure, and look elementary and pointlike. We certainly expect quantum effects in gravity to become important at distances \(\Delta x \leq 10^{-33}\) cm, corresponding to energies up to E ∼ MPlanckc2 ∼ 1019 GeV, where MPlanck is the Planck mass, related to Newton’s gravitational constant by GN = ℏ cMPlanck2. At such short distances the particles that so far appeared as pointlike may well reveal an extended structure, as would strings, and they may be described by a more detailed theoretical framework for which the local quantum field theory description of the SM would be just a low energy/large distance limit.

From the first few moments of the Universe, just after the Big Bang, the temperature of the cosmic background gradually went down, starting from kT ∼ MPlanckc2, where k = 8. 617 × 10−5 eV K−1 is the Boltzmann constant, down to the present situation where T ∼ 2. 725 K. Then all stages of high energy physics from string theory, which is a purely speculative framework, down to the SM phenomenology, which is directly accessible to experiment and well tested, are essential for the reconstruction of the evolution of the Universe starting from the Big Bang. This is the basis for the ever increasing connection between high energy physics and cosmology.

1.2 The Architecture of the Standard Model

The Standard Model (SM) is a gauge field theory based on the symmetry group \(SU(3)\bigotimes SU(2)\bigotimes U(1)\). The transformations of the group act on the basic fields. This group has 8 + 3 + 1 = 12 generators with a nontrivial commutator algebra (if all generators commute, the gauge theory is said to be “Abelian”, while the SM is a “non-Abelian” gauge theory). \(SU(2)\bigotimes U(1)\) describes the electroweak (EW) interactions [225, 316, 359] and the electric charge Q, the generator of the QED gauge group U(1) Q , is the sum of T3, one of the SU(2) generators and of Y∕2, where Y is the U(1) generator: Q = T3 + Y∕2. SU(3) is the “colour” group of the theory of strong interactions (quantum chromodynamics QCD [215, 234, 360]).

In a gauge theory,1 associated with each generator T is a vector boson (also called a gauge boson) with the same quantum numbers as T, and if the gauge symmetry is unbroken, this boson is of vanishing mass. These vector bosons (i.e., of spin 1) act as mediators of the corresponding interactions. For example, in QED the vector boson associated with the generator Q is the photon γ. The interaction between two charged particles in QED, for example two electrons, is mediated by the exchange of one (or occasionally more than one) photon emitted by one electron and reabsorbed by the other. Similarly, in the SM there are 8 gluons associated with the SU(3) colour generators, while for \(SU(2)\bigotimes U(1)\) there are four gauge bosons W+, W, Z0, and γ. Of these, only the gluons and the photon γ are massless, because the symmetry induced by the other three generators is actually spontaneously broken. The masses of W+, W, and Z0 are very large indeed on the scale of elementary particles, with values m W  ∼ 80. 4 GeV and m Z  ∼ 91. 2 GeV, whence they are as heavy as atoms of intermediate size, like rubidium and molybdenum, respectively.

In the electroweak theory, the breaking of the symmetry is of a particular type, referred to as spontaneous symmetry breaking. In this case, charges and currents are as dictated by the symmetry, but the fundamental state of minimum energy, the vacuum, is not unique and there is a continuum of degenerate states that all respect the symmetry (in the sense that the whole vacuum orbit is spanned by applying the symmetry transformations). The symmetry breaking is due to the fact that the system (with infinite volume and an infinite number of degrees of freedom) is found in one particular vacuum state, and this choice, which for the SM occurred in the first instants of the life of the Universe, means that the symmetry is violated in the spectrum of states. In a gauge theory like the SM, the spontaneous symmetry breaking is realized by the Higgs mechanism [189, 236, 243, 261] (described in detail in Sect. 1.7): there are a number of scalar (i.e., zero spin) Higgs bosons with a potential that produces an orbit of degenerate vacuum states. One or more of these scalar Higgs particles must necessarily be present in the spectrum of physical states with masses very close to the range so far explored. The Higgs particle has now been found at the LHC with mH ∼ 126 GeV [341, 345], thus making a big step towards completing the experimental verification of the SM. The Higgs boson acts as the mediator of a new class of interactions which, at the tree level, are coupled in proportion to the particle masses and thus have a very different strength for, say, an electron and a top quark.

The fermionic matter fields of the SM are quarks and leptons (all of spin 1/2). Each type of quark is a colour triplet (i.e., each quark flavour comes in three colours) and also carries electroweak charges, in particular electric charges + 2∕3 for up-type quarks and − 1∕3 for down-type quarks. So quarks are subject to all SM interactions. Leptons are colourless and thus do not interact strongly (they are not hadrons) but have electroweak charges, in particular electric charges − 1 for charged leptons (e, μ and τ) and charge 0 for neutrinos (ν e , νμ and ντ). Quarks and leptons are grouped in 3 “families” or “generations” with equal quantum numbers but different masses. At present we do not have an explanation for this triple repetition of fermion families:
$$\displaystyle{ \left [\,\begin{array}{*{10}c} u\,&\,u\,&\,u\,&\,\upnu _{e} \\ d\,&\,d\,&\,d\,&\,e\\ \end{array} \,\right ]\;,\qquad \left [\,\begin{array}{*{10}c} c\,&\,c\,&\,c\,&\,\upnu _{\upmu }\\ s\, &\,s\, &\,s\, & \,\upmu \\ \end{array} \,\right ]\;,\qquad \left [\,\begin{array}{*{10}c} t\,&\,t\,&\,t\,&\,\upnu _{\uptau } \\ b\,&\,b\,&\,b\,&\,\uptau \\ \end{array} \,\right ]\;. }$$
The QCD sector of the SM (see Chap.  2) has a simple structure but a very rich dynamical content, including the observed complex spectroscopy with a large number of hadrons. The most prominent properties of QCD are asymptotic freedom and confinement. In field theory, the effective coupling of a given interaction vertex is modified by the interaction. As a result, the measured intensity of the force depends on the square Q2 of the four-momentum Q transferred among the participants. In QCD the relevant coupling parameter that appears in physical processes is αs = es2∕4π, where es is the coupling constant of the basic interaction vertices of quarks and gluons: qqg or \(ggg\ \big[\) see (1.28)–(1.31)\(\big]\).

Asymptotic freedom means that the effective coupling becomes a function of Q2, and in fact αs(Q2) decreases for increasing Q2 and vanishes asymptotically. Thus, the QCD interaction becomes very weak in processes with large Q2, called hard processes or deep inelastic processes (i.e., with a final state distribution of momenta and a particle content very different than those in the initial state). One can prove that in four spacetime dimensions all pure gauge theories based on a non-commuting symmetry group are asymptotically free, and conversely. The effective coupling decreases very slowly at large momenta, going as the reciprocal logarithm of Q2, i.e., αs(Q2) = 1∕blog(Q2Λ2), where b is a known constant and Λ is an energy of order a few hundred MeV. Since in quantum mechanics large momenta imply short wavelengths, the result is that at short distances (or Q > Λ) the potential between two colour charges is similar to the Coulomb potential, i.e., proportional to αs(r)∕r, with an effective colour charge which is small at short distances.

In contrast, the interaction strength becomes large at large distances or small transferred momenta, of order Q < Λ. In fact, all observed hadrons are tightly bound composite states of quarks (baryons are made of qqq and mesons of \(q\bar{q}\)), with compensating colour charges so that they are overall neutral in colour. In fact, the property of confinement is the impossibility of separating colour charges, like individual quarks and gluons or any other coloured state. This is because in QCD the interaction potential between colour charges increases linearly in r at long distances. When we try to separate a quark and an antiquark that form a colour neutral meson, the interaction energy grows until pairs of quarks and antiquarks are created from the vacuum. New neutral mesons then coalesce and are observed in the final state, instead of free quarks. For example, consider the process \(e^{+}e^{-}\rightarrow q\bar{q}\) at large center-of-mass energies. The final state quark and antiquark have high energies, so they move apart very fast. But the colour confinement forces create new pairs between them. What is observed is two back-to-back jets of colourless hadrons with a number of slow pions that make the exact separation of the two jets impossible. In some cases, a third, well separated jet of hadrons is also observed: these events correspond to the radiation of an energetic gluon from the parent quark–antiquark pair.

In the EW sector, the SM (see Chap.  3) inherits the phenomenological successes of the old (VA) ⊗ (VA) four-fermion low-energy description of weak interactions, and provides a well-defined and consistent theoretical framework that includes weak interactions and quantum electrodynamics in a unified picture. The weak interactions derive their name from their strength. At low energy, the strength of the effective four-fermion interaction of charged currents is determined by the Fermi coupling constant GF. For example, the effective interaction for muon decay is given by
$$\displaystyle{ \mathcal{L}_{\mathrm{eff}} = \frac{G_{\mathrm{F}}} {\sqrt{2}}\big[\bar{\nu }_{\mu }\gamma _{\alpha }(1 -\gamma _{5})\mu \big]\big[\bar{e}\gamma ^{\alpha }(1 -\gamma _{5})\nu _{e}\big]\;, }$$
with [307]
$$\displaystyle{ G_{\mathrm{F}} = 1.166\,378\,7(6) \times 10^{-5}\ \mathrm{GeV}^{-2}\;. }$$
In natural units  = c = 1, GF (which we most often use in this work) has dimensions of (mass)−2. As a result, the strength of weak interactions at low energy is characterized by GFE2, where E is the energy scale for a given process (E ≈ mμ for muon decay). Since
$$\displaystyle{ G_{\mathrm{F}}E^{2} = G_{\mathrm{ F}}m_{\mathrm{p}}^{2}(E/m_{\mathrm{ p}})^{2} \approx 10^{-5}(E/m_{\mathrm{ p}})^{2}\;, }$$
where mp is the proton mass, the weak interactions are indeed weak at low energies (up to energies of order a few tens of GeV). Effective four-fermion couplings for neutral current interactions have comparable intensity and energy behaviour. The quadratic increase with energy cannot continue for ever, because it would lead to a violation of unitarity. In fact, at high energies, propagator effects can no longer be neglected, and the current–current interaction is resolved into current–W gauge boson vertices connected by a W propagator. The strength of the weak interactions at high energies is then measured by g W , the W–μ–νμ coupling, or even better, by α W  = g W 2∕4π, analogous to the fine-structure constant α of QED (in Chap.  3, g W is simply denoted by g or g2). In the standard EW theory, we have
$$\displaystyle{ \alpha _{W} = \sqrt{2}G_{\mathrm{F}}m_{W}^{2}/\pi \approx 1/30\;. }$$
That is, at high energies the weak interactions are no longer so weak.
The range r W of weak interactions is very short: it was only with the experimental discovery of the W and Z gauge bosons that it could be demonstrated that r W is non-vanishing. Now we know that
$$\displaystyle{ r_{W} = \frac{\hslash } {m_{W}c} \approx 2.5 \times 10^{-16}\ \mathrm{cm}\;, }$$
corresponding to m W  ≈ 80. 4 GeV. This very high value for the W (or the Z) mass makes a drastic difference, compared with the massless photon and the infinite range of the QED force. The direct experimental limit on the photon mass is [307] m γ  < 10−18 eV. Thus, on the one hand, there is very good evidence that the photon is massless, and on the other, the weak bosons are very heavy. A unified theory of EW interactions has to face this striking difference.

Another apparent obstacle in the way of EW unification is the chiral structure of weak interactions: in the massless limit for fermions, only left-handed quarks and leptons (and right-handed antiquarks and antileptons) are coupled to W particles. This clearly implies parity and charge-conjugation violation in weak interactions.

The universality of weak interactions and the algebraic properties of the electromagnetic and weak currents [conservation of vector currents (CVC), partial conservation of axial currents (PCAC), the algebra of currents, etc.] were crucial in pointing to the symmetric role of electromagnetism and weak interactions at a more fundamental level. The old Cabibbo universality [120] for the weak charged current, viz.,
$$\displaystyle\begin{array}{rcl} J_{\alpha }^{\mathrm{weak}}& =& \bar{\nu }_{\mu }\gamma _{\alpha }(1 -\gamma _{ 5})\mu +\bar{\nu } _{e}\gamma _{\alpha }(1 -\gamma _{5})e +\cos \theta _{\mathrm{c}}\bar{u}\gamma _{\alpha }(1 -\gamma _{5})d \\ & & +\sin \theta _{\mathrm{c}}\bar{u}\gamma _{\alpha }(1 -\gamma _{5})s + \cdots \;, {}\end{array}$$
suitably extended, is naturally implied by the standard EW theory. In this theory the weak gauge bosons couple to all particles with couplings that are proportional to their weak charges, in the same way as the photon couples to all particles in proportion to their electric charges. In (1.7), d  = dcosθc + ssinθc is the weak isospin partner of u in a doublet. The (u, d ) doublet has the same couplings as the (ν e , ) and (νμ, μ) doublets.

Another crucial feature is that the charged weak interactions are the only known interactions that can change flavour: charged leptons into neutrinos or up-type quarks into down-type quarks. On the other hand, there are no flavour-changing neutral currents at tree level. This is a remarkable property of the weak neutral current, which is explained by the introduction of the Glashow–Iliopoulos–Maiani (GIM) mechanism [226] and led to the successful prediction of charm.

The natural suppression of flavour-changing neutral currents, the separate conservation of e, μ, and τ leptonic flavours that is only broken by the small neutrino masses, the mechanism of CP violation through the phase in the quark-mixing matrix [269], are all crucial features of the SM. Many examples of new physics tend to break the selection rules of the standard theory. Thus the experimental study of rare flavour-changing transitions is an important window on possible new physics.

The SM is a renormalizable field theory, which means that the ultraviolet divergences that appear in loop diagrams can be eliminated by a suitable redefinition of the parameters already appearing in the bare Lagrangian: masses, couplings, and field normalizations. As will be discussed later, a necessary condition for a theory to be renormalizable is that only operator vertices of dimension not greater than 4 (that is m4, where m is some mass scale) appear in the Lagrangian density \(\mathcal{L}\) (itself of dimension 4, because the action S is given by the integral of \(\mathcal{L}\) over d4x and is dimensionless in natural units such that  = c = 1). Once this condition is added to the specification of a gauge group and of the matter field content, the gauge theory Lagrangian density is completely specified. We shall see the precise rules for writing down the Lagrangian of a gauge theory in the next section.

1.3 The Formalism of Gauge Theories

In this section we summarize the definition and the structure of a Yang–Mills gauge theory [371]. We will list here the general rules for constructing such a theory. Then these results will be applied to the SM.

Consider a Lagrangian density \(\mathcal{L}[\phi,\partial _{\mu }\phi ]\) which is invariant under a D dimensional continuous group Γ of transformations:
$$\displaystyle{ \phi ^{{\prime}}(x) = U(\theta ^{A})\phi (x)\quad \quad (A = 1,2,\ldots,D)\;, }$$
$$\displaystyle{ U(\theta ^{A}) =\exp \bigg[\mathrm{i}g\sum _{ A}\theta ^{A}T^{A}\bigg] \sim 1 + \mathrm{i}g\sum _{ A}\theta ^{A}T^{A} + \cdots \;. }$$
The quantities θ A are numerical parameters, like angles in the particular case of a rotation group in some internal space. The approximate expression on the right is valid for θ A infinitesimal. Then, g is the coupling constant and T A are the generators of the group Γ of transformations (1.8) in the (in general reducible) representation of the fields ϕ. Here we restrict ourselves to the case of internal symmetries, so the T A are matrices that are independent of the spacetime coordinates, and the arguments of the fields ϕ and ϕ in (1.8) are the same.
If U is unitary, then the generators T A are Hermitian, but this need not be the case in general (although it is true for the SM). Similarly, if U is a group of matrices with unit determinant, then the traces of the T A vanish, i.e., tr(T A ) = 0. In general, the generators satisfy the commutation relations
$$\displaystyle{ [T^{A},T^{B}] = \mathrm{i}C_{ ABC}T^{C}\;. }$$
For A, B, C, , up or down indices make no difference, i.e., T A  = T A , etc. The structure constants C ABC are completely antisymmetric in their indices, as can be easily seen. Recall that if all generators commute, the gauge theory is said to be “Abelian” (in this case all the structure constants C ABC vanish), while the SM is a “non-Abelian” gauge theory.
We choose to normalize the generators T A in such a way that, for the lowest dimensional non-trivial representation of the group Γ (we use t A to denote the generators in this particular representation), we have
$$\displaystyle{ \mathrm{tr}\big(t^{A}t^{B}\big) = \frac{1} {2}\delta ^{AB}\;. }$$
A normalization convention is needed to fix the normalization of the coupling g and the structure constants C ABC . In the following, for each quantity f A , we define
$$\displaystyle{ \mathbf{f} =\sum _{A}T^{A}f^{A}\;. }$$
For example, we can rewrite (1.9) in the form
$$\displaystyle{ U(\theta ^{A}) =\exp (\mathrm{i}g\boldsymbol{\theta }) \sim 1 + \mathrm{i}g\boldsymbol{\theta } + \cdots \;. }$$
If we now make the parameters θ A depend on the spacetime coordinates, whence θ A  = θ A (x μ ), then \(\mathcal{L}[\phi,\partial _{\mu }\phi ]\) is in general no longer invariant under the gauge transformations U[θ A (x μ )], because of the derivative terms. Indeed, we then have μ ϕ  =  μ () ≠ U∂ μ ϕ. Gauge invariance is recovered if the ordinary derivative is replaced by the covariant derivative
$$\displaystyle{ D_{\mu } = \partial _{\mu } + ig\mathbf{V}_{\mu }\;, }$$
where V μ A are a set of D gauge vector fields (in one-to-one correspondence with the group generators), with the transformation law
$$\displaystyle{ \mathbf{V}_{\mu }^{{\prime}} = U\mathbf{V}_{\mu }U^{-1} - \frac{1} {\mathrm{i}g}(\partial _{\mu }U)U^{-1}. }$$
For constant θ A , V reduces to a tensor of the adjoint (or regular) representation of the group:
$$\displaystyle{ \mathbf{V}_{\mu }^{{\prime}} = U\mathbf{V}_{\mu }U^{-1} \approx \mathbf{V}_{\mu } + \mathrm{i}g[\boldsymbol{\theta },\mathbf{V}_{\mu }] + \cdots \;, }$$
which implies that
$$\displaystyle{ V _{\mu }^{{\prime}C} = V _{\mu }^{C} - gC_{ ABC}\theta ^{A}V _{\mu }^{B} + \cdots \;, }$$
where repeated indices are summed over.
As a consequence of (1.14) and (1.15), D μ ϕ has the same transformation properties as ϕ :
$$\displaystyle{ (D_{\mu }\phi )^{{\prime}} = U(D_{\mu }\phi )\;. }$$
In fact,
$$\displaystyle\begin{array}{rcl} (D_{\mu }\phi )^{{\prime}}& =& (\partial _{\mu } + ig\mathbf{V}^{{\prime}}_{ \mu })\phi ^{{\prime}} \\ & =& (\partial _{\mu }U)\phi + U\partial _{\mu }\phi + igU\mathbf{V}_{\mu }\phi - (\partial _{\mu }U)\phi = U(D_{\mu }\phi )\;.{}\end{array}$$
Thus \(\mathcal{L}[\phi,D_{\mu }\phi ]\) is indeed invariant under gauge transformations. But at this stage the gauge fields V μ A appear as external fields that do not propagate. In order to construct a gauge invariant kinetic energy term for the gauge fields V μ A , we consider
$$\displaystyle{ [D_{\mu },D_{\nu }]\phi = \mathrm{i}g\big\{\partial _{\mu }\mathbf{V}_{\nu } - \partial _{\nu }\mathbf{V}_{\mu } + \mathrm{i}g[\mathbf{V}_{\mu },\mathbf{V}_{\nu }]\big\}\phi \equiv \mathrm{i}g\mathbf{F}_{\mu \nu }\phi \;, }$$
which is equivalent to
$$\displaystyle{ F_{\mu \nu }^{A} = \partial _{\mu }V _{\nu }^{A} - \partial _{\nu }V _{\mu }^{A} - gC_{ ABC}V _{\mu }^{B}V _{\nu }^{C}\;. }$$
From (1.8), (1.18), and (1.20), it follows that the transformation properties of F μ ν A are those of a tensor of the adjoint representation:
$$\displaystyle{ \mathbf{F}_{\mu \nu }^{{\prime}} = U\mathbf{F}_{\mu \nu }U^{-1}\;. }$$
The complete Yang–Mills Lagrangian, which is invariant under gauge transformations, can be written in the form
$$\displaystyle{ \mathcal{L}_{\mathrm{YM}} = -\frac{1} {2}Tr\mathbf{F}_{\mu \nu }\mathbf{F}^{\mu \nu } + \mathcal{L}[\phi,D_{\mu }\phi ] = -\frac{1} {4}\sum _{A}F_{\mu \nu }^{A}F^{A\mu \nu } + \mathcal{L}[\phi,D_{\mu }\phi ]\;. }$$
Note that the kinetic energy term is an operator of dimension 4. Thus if \(\mathcal{L}\) is renormalizable, so also is \(\mathcal{L}_{\mathrm{YM}}\). If we give up renormalizability, then more gauge invariant higher dimensional terms could be added. It is already clear at this stage that no mass term for gauge bosons of the form m2V μ V μ is allowed by gauge invariance.

1.4 Application to QED and QCD

For an Abelian theory like QED, the gauge transformation reduces to U[θ(x)] = exp[ieQθ(x)], where Q is the charge generator (for more commuting generators, one simply has a product of similar factors). According to (1.15), the associated gauge field (the photon) transforms as
$$\displaystyle{ V _{\mu }^{{\prime}} = V _{\mu } - \partial _{\mu }\theta (x)\;, }$$
and the familiar gauge transformation is recovered, with addition of a 4-gradient of a scalar function. The QED Lagrangian density is given by
$$\displaystyle{ \mathcal{L} = -\frac{1} {4}F^{\mu \nu }F_{\mu \nu } +\sum _{\psi }\bar{\psi }(\mathrm{i}D/ - m_{\psi })\psi \;. }$$
Here D∕ = D μ γ μ , where γ μ are the Dirac matrices and the covariant derivative is given in terms of the photon field A μ and the charge operator Q by
$$\displaystyle{ D_{\mu } = \partial _{\mu } + \mathrm{i}eA_{\mu }Q }$$
$$\displaystyle{ F_{\mu \nu } = \partial _{\mu }A_{\nu } - \partial _{\nu }A_{\mu }\;. }$$
Note that in QED one usually takes e to be the particle, so that Q = −1 and the covariant derivative is D μ  =  μ −ieA μ when acting on the electron field. In the Abelian case, the F μ ν tensor is linear in the gauge field V μ , so that in the absence of matter fields the theory is free. On the other hand, in the non-Abelian case, the F μ ν A tensor contains both linear and quadratic terms in V μ A , so the theory is non-trivial even in the absence of matter fields.
According to the formalism of the previous section, the statement that QCD is a renormalizable gauge theory based on the group SU(3) with colour triplet quark matter fields fixes the QCD Lagrangian density to be
$$\displaystyle{ \mathcal{L} = -\frac{1} {4}\sum _{A=1}^{8}F^{A\mu \nu }F_{\mu \nu }^{A} +\sum _{ j=1}^{n_{\mathrm{f}} }\bar{q}_{j}(iD/ - m_{j})q_{j}\;. }$$
Here q j are the quark fields with nf different flavours and mass m j , and D μ is the covariant derivative of the form
$$\displaystyle{ D_{\mu } = \partial _{\mu } + \mathrm{i}e_{\mathrm{s}}\mathbf{g}_{\boldsymbol{\mu }}\;, }$$
with gauge coupling es. Later, in analogy with QED, we will mostly use
$$\displaystyle{ \alpha _{\mathrm{s}} = \frac{e_{\mathrm{s}}^{2}} {4\pi } \;. }$$
In addition, \(\mathbf{g}_{\boldsymbol{\mu }} =\sum _{A}t^{A}g_{\mu }^{A}\), where g μ A , A = 1, , 8, are the gluon fields and t A are the SU(3) group generators in the triplet representation of the quarks (i.e., t A are 3 × 3 matrices acting on q). The generators obey the commutation relations [t A , t B ] = iC ABC t C , where C ABC are the completely antisymmetric structure constants of SU(3). The normalizations of C ABC and es are specified by those of the generators t A , i.e., \(\mathrm{Tr}[t^{A}t^{B}] =\delta ^{AB}/2\ \big[\) see (1.11)\(\big]\). Finally, we have
$$\displaystyle{ F_{\mu \nu }^{A} = \partial _{\mu }g_{\nu }^{A} - \partial _{\nu }g_{\mu }^{A} - e_{\mathrm{ s}}C_{ABC}g_{\mu }^{B}g_{\nu }^{C}\;. }$$
Chapter  2 is devoted to a detailed description of QCD as the theory of strong interactions. The physical vertices in QCD include the gluon–quark–antiquark vertex, analogous to the QED photon–fermion–antifermion coupling, but also the 3-gluon and 4-gluon vertices, of order es and es2 respectively, which have no analogue in an Abelian theory like QED. In QED the photon is coupled to all electrically charged particles, but is itself neutral. In QCD the gluons are coloured, hence self-coupled. This is reflected by the fact that, in QED, F μ ν is linear in the gauge field, so that the term F μ ν 2 in the Lagrangian is a pure kinetic term, while in QCD, F μ ν A is quadratic in the gauge field, so that in F μ ν A2, we find cubic and quartic vertices beyond the kinetic term. It is also instructive to consider a scalar version of QED:
$$\displaystyle{ \mathcal{L} = -\frac{1} {4}F^{\mu \nu }F_{\mu \nu } + (D_{\mu }\phi )^{\dag }(D^{\mu }\phi ) - m^{2}(\phi ^{\dag }\phi )\;. }$$
For Q = 1, we have
$$\displaystyle{ (D_{\mu }\phi )^{\dag }(D^{\mu }\phi ) = (\partial _{\mu }\phi )^{\dag }(\partial ^{\mu }\phi ) + \mathrm{i}eA_{\mu }\big[(\partial ^{\mu }\phi )^{\dag }\phi -\phi ^{\dag }(\partial ^{\mu }\phi )\big] + e^{2}A_{\mu }A^{\mu }\phi ^{\dag }\phi \;. }$$
We see that for a charged boson in QED, given that the kinetic term for bosons is quadratic in the derivative, there is a gauge–gauge–scalar–scalar vertex of order e2. We understand that in QCD the 3-gluon vertex is there because the gluon is coloured, and the 4-gluon vertex because the gluon is a boson.

1.5 Chirality

We recall here the notion of chirality and related issues which are crucial for the formulation of the EW Theory. The fermion fields can be described through their right-handed (RH) (chirality + 1) and left-handed (LH) (chirality − 1) components:
$$\displaystyle{ \psi _{\mathrm{L,R}} = [(1 \mp \gamma _{5})/2]\psi \;,\quad \bar{\psi }_{\mathrm{L,R}} =\bar{\psi } [(1 \pm \gamma _{5})/2]\;, }$$
where γ5 and the other Dirac matrices are defined as in the book by Bjorken and Drell [102]. In particular, γ52 = 1, γ5 = γ5. Note that (1.34) implies
$$\displaystyle{\bar{\psi }_{\mathrm{L}} =\psi _{ \mathrm{L}}^{\dag }\gamma _{ 0} =\psi ^{\dag }[(1 -\gamma _{ 5})/2]\gamma _{0} =\bar{\psi }\gamma _{0}[(1 -\gamma _{5})/2]\gamma _{0} =\bar{\psi } [(1 +\gamma _{5})/2]\;.}$$
The matrices P± = (1 ±γ5)∕2 are projectors. They satisfy the relations P±P± = P±, P±P = 0, P+ + P = 1. They project onto fermions of definite chirality. For massless particles, chirality coincides with helicity. For massive particles, a chirality + 1 state only coincides with a + 1 helicity state up to terms suppressed by powers of mE.
The 16 linearly independent Dirac matrices (Γ) can be divided into γ5-even (ΓE) and γ5-odd (ΓO) according to whether they commute or anticommute with γ5. For the γ5-even, we have
$$\displaystyle{ \bar{\psi }\varGamma _{\mathrm{E}}\psi =\bar{\psi } _{\mathrm{L}}\varGamma _{\mathrm{E}}\psi _{\mathrm{R}} +\bar{\psi } _{\mathrm{R}}\varGamma _{\mathrm{E}}\psi _{\mathrm{L}}\quad \quad (\varGamma _{\mathrm{E}} \equiv 1,\mathrm{i}\gamma _{5},\sigma _{\mu \nu })\;, }$$
whilst for the γ5-odd,
$$\displaystyle{ \bar{\psi }\varGamma _{\mathrm{O}}\psi =\bar{\psi } _{\mathrm{L}}\varGamma _{\mathrm{O}}\psi _{\mathrm{L}} +\bar{\psi } _{\mathrm{R}}\varGamma _{\mathrm{O}}\psi _{\mathrm{R}}\quad \quad (\varGamma _{\mathrm{O}} \equiv \gamma _{\mu },\gamma _{\mu }\gamma _{5}). }$$

We see that in a gauge Lagrangian, fermion kinetic terms and interactions of gauge bosons with vector and axial vector fermion currents all conserve chirality, while fermion mass terms flip chirality. For example, in QED, if an electron emits a photon, the electron chirality is unchanged. In the ultrarelativistic limit, when the electron mass can be neglected, chirality and helicity are approximately the same and we can state that the helicity of the electron is unchanged by the photon emission. In a massless gauge theory, the LH and the RH fermion components are uncoupled and can be transformed separately. If in a gauge theory the LH and RH components transform as different representations of the gauge group, one speaks of a chiral gauge theory, while if they have the same gauge transformations, one has a vector gauge theory. Thus, QED and QCD are vector gauge theories because, for each given fermion, ψL and ψR have the same electric charge and the same colour. Instead, the standard EW theory is a chiral theory, in the sense that ψL and ψR behave differently under the gauge group (so that parity and charge conjugation non-conservation are made possible in principle). Thus, mass terms for fermions (of the form \(\bar{\psi }_{\mathrm{L}}\psi _{\mathrm{R}}\) + h.c.) are forbidden in the EW gauge-symmetric limit. In particular, in the Minimal Standard Model (MSM), i.e., the model that only includes all observed particles plus a single Higgs doublet, all ψL are SU(2) doublets, while all ψR are singlets.

1.6 Quantization of a Gauge Theory

The Lagrangian density \(\mathcal{L}_{\mathrm{YM}}\) in (1.23) fully describes the theory at the classical level. The formulation of the theory at the quantum level requires us to specify procedures of quantization, regularization and, finally, renormalization. To start with, the formulation of Feynman rules is not straightforward. A first problem, common to all gauge theories, including the Abelian case of QED, can be realized by observing that the free equations of motion for V μ A , as obtained from (1.21) and (1.23), are given by
$$\displaystyle{ \big(\partial ^{2}g_{\mu \nu } - \partial _{\mu }\partial _{\nu }\big)V ^{A\nu } = 0\;. }$$
Normally the propagator of the gauge field should be determined by the inverse of the operator 2g μ ν μ ν . However, it has no inverse, being a projector over the transverse gauge vector states. This difficulty is removed by fixing a particular gauge. If one chooses a covariant gauge condition μ V μ A  = 0, then a gauge fixing term of the form
$$\displaystyle{ \Delta \mathcal{L}_{\mathrm{GF}} = -\frac{1} {2\lambda }\sum _{A}\vert \partial ^{\mu }V _{\mu }^{A}\vert ^{2} }$$
has to be added to the Lagrangian (1∕λ acts as a Lagrangian multiplier). The free equations of motion are then modified as follows:
$$\displaystyle{ \big[\partial ^{2}g_{\mu \nu } - (1 - 1/\lambda )\partial _{\mu }\partial _{\nu }\big]V ^{A\nu } = 0\;. }$$
This operator now has an inverse whose Fourier transform is given by
$$\displaystyle{ D_{\mu \nu }^{AB}(q) = \frac{\mathrm{i}} {q^{2} + \mathrm{i}\epsilon }\left [-g_{\mu \nu } + (1-\lambda ) \frac{q_{\mu }q_{\nu }} {q^{2} + \mathrm{i}\epsilon }\right ]\delta ^{AB}\;, }$$
which is the propagator in this class of gauges. The parameter λ can take any value and it disappears from the final expression of any gauge invariant, physical quantity. Commonly used particular cases are λ = 1 (Feynman gauge) and λ = 0 (Landau gauge).
While in an Abelian theory the gauge fixing term is all that is needed for a correct quantization, in a non-Abelian theory the formulation of complete Feynman rules involves a further subtlety. This is formally taken into account by introducing a set of D fictitious ghost fields that must be included as internal lines in closed loops (Faddeev–Popov ghosts [197]). Given that gauge fields connected by a gauge transformation describe the same physics, there are clearly fewer physical degrees of freedom than gauge field components. Ghosts appear, in the form of a transformation Jacobian in the functional integral, in the process of elimination of the redundant variables associated with fields on the same gauge orbit [14]. By performing some path integral acrobatics, the correct ghost contributions can be translated into an additional term in the Lagrangian density. For each choice of the gauge fixing term, the ghost Lagrangian is obtained by considering the effect of an infinitesimal gauge transformation V μ ′ C  = V μ C gC ABC θ A V μ B μ θ C on the gauge fixing condition. For μ V μ C  = 0, one obtains
$$\displaystyle{ \partial ^{\mu }V _{\mu }^{{\prime}C} = \partial ^{\mu }V _{\mu }^{C} - gC_{ ABC}\partial ^{\mu }(\theta ^{A}V _{\mu }^{B}) - \partial ^{2}\theta ^{C} = -\big[\partial ^{2}\delta _{ AC} + gC_{ABC}V _{\mu }^{B}\partial ^{\mu }\big]\theta ^{A}\;, }$$
where the gauge condition μ V μ C  = 0 has been taken into account in the last step. The ghost Lagrangian is then given by
$$\displaystyle{ \Delta \mathcal{L}_{\mathrm{Ghost}} =\bar{\eta } ^{C}\big[\partial ^{2}\delta _{ AC} + gC_{ABC}V _{\mu }^{B}\partial ^{\mu }\big]\eta ^{A}\;, }$$
where η A is the ghost field (one for each index A) which has to be treated as a scalar field, except that a factor − 1 has to be included for each closed loop, as for fermion fields.
Starting from non-covariant gauges, one can construct ghost-free gauges. An example, also important in other respects, is provided by the set of “axial” gauges n μ V μ A  = 0, where n μ is a fixed reference 4-vector (actually, for n μ spacelike, one has an axial gauge proper, for n2 = 0, one speaks of a light-like gauge, and for n μ timelike, one has a Coulomb or temporal gauge). The gauge fixing term is of the form
$$\displaystyle{ \Delta \mathcal{L}_{\mathrm{GF}} = -\frac{1} {2\lambda }\sum _{A}\vert n^{\mu }V _{\mu }^{A}\vert ^{2}\;. }$$
With a procedure that can be found in QED textbooks [102], the corresponding propagator in Fourier space is found to be
$$\displaystyle{ D_{\mu \nu }^{AB}(q) = \frac{\mathrm{i}} {q^{2} + \mathrm{i}\epsilon }\left [-g_{\mu \nu } + \frac{n_{\mu }q_{+}n_{\nu }q_{\mu }} {(nq)} - \frac{n^{2}q_{\mu }q_{\nu }} {(nq)^{2}}\right ]\delta ^{AB}\;. }$$
In this case there are no ghost interactions because n μ V μ ′ A , obtained by a gauge transformation from n μ V μ A , contains no gauge fields, once the gauge condition n μ V μ A  = 0 has been taken into account. Thus the ghosts are decoupled and can be ignored.

The introduction of a suitable regularization method that preserves gauge invariance is essential for the definition and the calculation of loop diagrams and for the renormalization programme of the theory. The method that is currently adopted is dimensional regularization [334], which consists in the formulation of the theory in n dimensions. All loop integrals have an analytic expression that is actually valid also for non-integer values of n. Writing the results for n = 4 −ε the loops are ultraviolet finite for ε > 0 and the divergences reappear in the form of poles at ε = 0.

1.7 Spontaneous Symmetry Breaking in Gauge Theories

The gauge symmetry of the SM was difficult to discover because it is well hidden in nature. The only observed gauge boson that is massless is the photon. The gluons are presumed massless but cannot be directly observed because of confinement, and the W and Z weak bosons carry a heavy mass. Indeed a major difficulty in unifying the weak and electromagnetic interactions was the fact that electromagnetic interactions have infinite range (m γ  = 0), whilst the weak forces have a very short range, owing to m W, Z  ≠ 0. The solution to this problem lies in the concept of spontaneous symmetry breaking, which was borrowed from condensed matter physics.

Consider a ferromagnet at zero magnetic field in the Landau–Ginzburg approximation. The free energy in terms of the temperature T and the magnetization M can be written as
$$\displaystyle{ F(\mathbf{M},T) \simeq F_{0}(T) + \frac{1} {2}\mu ^{2}(T)\mathbf{M}^{2} + \frac{1} {4}\lambda (T)(\mathbf{M}^{2})^{2} + \cdots \;. }$$
This is an expansion which is valid at small magnetization. The neglect of terms of higher order in \(\mbox{ $\mathbf{M}$}^{2}\) is the analogue in this context of the renormalizability criterion. Furthermore, λ(T) > 0 is assumed for stability, and F is invariant under rotations, i.e., all directions of M in space are equivalent. The minimum condition for F reads
$$\displaystyle{ \partial F/\partial M_{i} = 0\;,\quad \big[\mu ^{2}(T) +\lambda (T)\mathbf{M}^{2}\big]\mathbf{M} = 0\;. }$$
There are two cases, shown in Fig. 1.1. If \(\mu ^{2}\gtrsim 0\), then the only solution is M = 0, there is no magnetization, and the rotation symmetry is respected. In this case the lowest energy state (in a quantum theory the vacuum) is unique and invariant under rotations. If μ2 < 0, then another solution appears, which is
$$\displaystyle{ \vert \mathbf{M}_{0}\vert ^{2} = -\mu ^{2}/\lambda \;. }$$
In this case there is a continuous orbit of lowest energy states, all with the same value of | M | , but different orientations. A particular direction chosen by the vector M0 leads to a breaking of the rotation symmetry.
Fig. 1.1

The potential V = μ2M2∕2 +λ(M2)2∕4 for positive (a ) or negative μ2 (b ) (for simplicity, M is a 2-dimensional vector). The small sphere indicates a possible choice for the direction of M

For a piece of iron we can imagine bringing it to high temperature and letting it melt in an external magnetic field B. The presence of B is an explicit breaking of the rotational symmetry and it induces a nonzero magnetization M along its direction. Now we lower the temperature while keeping B fixed. Both λ and μ2 depend on the temperature. With lowering T, μ2 goes from positive to negative values. The critical temperature Tcrit (Curie temperature) is where μ2(T) changes sign, i.e., μ2(Tcrit) = 0. For pure iron, Tcrit is below the melting temperature. So at T = Tcrit iron is a solid. Below Tcrit we remove the magnetic field. In a solid the mobility of the magnetic domains is limited and a non-vanishing M0 remains. The form of the free energy is again rotationally invariant as in (1.45). But now the system allows a minimum energy state with non-vanishing M in the direction of B. As a consequence the symmetry is broken by this choice of one particular vacuum state out of a continuum of them.

We now prove the Goldstone theorem [228]. It states that when spontaneous symmetry breaking takes place, there is always a zero-mass mode in the spectrum. In a classical context this can be proven as follows. Consider a Lagrangian
$$\displaystyle{ \mathcal{L} = \frac{1} {2}\vert \partial _{\mu }\phi \vert ^{2} - V (\phi )\;. }$$
The potential V (ϕ) can be kept generic at this stage, but in the following we will be mostly interested in a renormalizable potential of the form
$$\displaystyle{ V (\phi ) = -\frac{1} {2}\mu ^{2}\phi ^{2} + \frac{1} {4}\lambda \phi ^{4}\;, }$$
with no more than quartic terms. Here by ϕ we mean a column vector with real components ϕ i (1 = 1, 2, , N) (complex fields can always be decomposed into a pair of real fields), so that, for example, ϕ2 =  i ϕ i 2. This particular potential is symmetric under an N × N orthogonal matrix rotation ϕ  = , where O is an SO(N) transformation. For simplicity, we have omitted odd powers of ϕ, which means that we have assumed an extra discrete symmetry under ϕ ↔ −ϕ. Note that, for positive μ2, the mass term in the potential has the “wrong” sign: according to the previous discussion this is the condition for the existence of a non-unique lowest energy state. Further, we only assume here that the potential is symmetric under the infinitesimal transformations
$$\displaystyle{ \phi \rightarrow \phi ^{{\prime}} =\phi +\updelta \phi \;,\quad \updelta \phi _{ i} = \mathrm{i}\updelta \theta ^{A}t_{ ij}^{\,A}\phi _{ j}\;, }$$
where δθ A are infinitesimal parameters and t ij A are the matrices that represent the symmetry group on the representation carried by the fields ϕ i (a sum over A is understood). The minimum condition on V that identifies the equilibrium position (or the vacuum state in quantum field theory language) is
$$\displaystyle{ \frac{\partial V } {\partial \phi _{i}} (\phi _{i} =\phi _{ i}^{0}) = 0\;. }$$
The symmetry of V implies that
$$\displaystyle{ \updelta V = \frac{\partial V } {\partial \phi _{i}} \updelta \phi _{i} = \mathrm{i}\updelta \theta ^{A}\frac{\partial V } {\partial \phi _{i}} t_{ij}^{\,A}\phi _{ j} = 0\;. }$$
By taking a second derivative at the minimum ϕ i  = ϕ i 0 of both sides of the previous equation, we obtain that, for each A,
$$\displaystyle{ \frac{\partial ^{2}V } {\partial \phi _{k}\partial \phi _{i}}(\phi _{i} =\phi _{ i}^{0})t_{ ij}^{\,A}\phi _{ j}^{0} + \frac{\partial V } {\partial \phi _{i}} (\phi _{i} =\phi _{ i}^{0})t_{ ik}^{\,A} = 0. }$$
The second term vanishes owing to the minimum condition (1.51). We then find
$$\displaystyle{ \frac{\partial ^{2}V } {\partial \phi _{k}\partial \phi _{i}}(\phi _{i} =\phi _{ i}^{0})t_{ ij}^{\,A}\phi _{ j}^{0} = 0\;. }$$
The second derivatives M ki 2 = (2V∂ ϕ k ∂ ϕ i )(ϕ i  = ϕ i 0) define the squared mass matrix. Thus the above equation in matrix notation can be written as
$$\displaystyle{ M^{2}t^{A}\phi ^{0} = 0\;. }$$
In the case of no spontaneous symmetry breaking, the ground state is unique, and all symmetry transformations leave it invariant, so that, for all A, t A ϕ0 = 0. On the other hand, if for some values of A the vectors (t A ϕ0) are non-vanishing, i.e., there is some generator that shifts the ground state into some other state with the same energy (whence the vacuum is not unique), then each t A ϕ0 ≠ 0 is an eigenstate of the squared mass matrix with zero eigenvalue. Therefore, a massless mode is associated with each broken generator. The charges of the massless modes (their quantum numbers in quantum language) differ from those of the vacuum (usually all taken as zero) by the values of the t A charges: one says that the massless modes have the same quantum numbers as the broken generators, i.e., those that do not annihilate the vacuum.

The previous proof of the Goldstone theorem has been given for the classical case. In the quantum case, the classical potential corresponds to the tree level approximation of the quantum potential. Higher order diagrams with loops introduce quantum corrections. The functional integral formulation of quantum field theory [14, 250] is the most appropriate framework to define and compute, in a loop expansion, the quantum potential which specifies the vacuum properties of the quantum theory in exactly the way described above. If the theory is weakly coupled, e.g., if λ is small, the tree level expression for the potential is not too far from the truth, and the classical situation is a good approximation. We shall see that this is the situation that occurs in the electroweak theory with a moderately light Higgs particle (see Sect.  3.5).

We note that for a quantum system with a finite number of degrees of freedom, for example, one described by the Schrödinger equation, there are no degenerate vacua: the vacuum is always unique. For example, in the one-dimensional Schrödinger problem with a potential
$$\displaystyle{ V (x) = -\frac{\mu ^{2}} {2}x^{2} + \frac{\lambda } {4}x^{4}\;, }$$
there are two degenerate minima at x = ±x0 = (μ2λ)1∕2, which we denote by | + 〉 and | −〉. But the potential is not diagonal in this basis: the off-diagonal matrix elements
$$\displaystyle{ \langle +\vert V \vert -\rangle =\langle -\vert V \vert +\rangle \sim \exp (-khd) =\delta }$$
are different from zero due to the non-vanishing amplitude for a tunnel effect between the two vacua given in (1.57), proportional to the exponential of minus the product of the distance d between the vacua and the height h of the barrier, with k a constant (see Fig. 1.2). After diagonalization the eigenvectors are \((\vert +\rangle +\vert -\rangle )/\sqrt{2}\) and \((\vert +\rangle -\vert -\rangle )/\sqrt{2}\), with different energies (the difference being proportional to δ). Suppose now that we have a sum of n equal terms in the potential, i.e., V =  i V (x i ). Then the transition amplitude would be proportional to δ n and would vanish for infinite n: the probability that all degrees of freedom together jump over the barrier vanishes. In this example there is a discrete number of minimum points. The case of a continuum of minima is obtained, still in the Schrödinger context, if we take
Fig. 1.2

A Schrödinger potential V (x) analogous to the Higgs potential

$$\displaystyle{ V = \frac{1} {2}\mu ^{2}\mathbf{r}^{2} + \frac{1} {4}\lambda (\mathbf{r}^{2})^{2}\;, }$$
with r = (x, y, z). The ground state is also unique in this case: it is given by a state with total orbital angular momentum zero, i.e., an s-wave state, whose wave function only depends on | r | , i.e., it is independent of all angles. This is a superposition of all directions with the same weight, analogous to what happened in the discrete case. But again, if we replace a single vector r, with a vector field M(x), that is, a different vector at each point in space, the amplitude to go from a minimum state in one direction to another in a different direction goes to zero in the limit of infinite volume. Put simply, the vectors at all points in space have a vanishingly small amplitude to make a common rotation, all together at the same time. In the infinite volume limit, all vacua along each direction have the same energy, and spontaneous symmetry breaking can occur.
A massless Goldstone boson corresponds to a long range force. Unless the massless particles are confined, as for the gluons in QCD, these long range forces would be easily detectable. Thus, in the construction of the EW theory, we cannot accept physical massless scalar particles. Fortunately, when spontaneous symmetry breaking takes place in a gauge theory, the massless Goldstone modes exist, but they are unphysical and disappear from the spectrum. In fact, each of them becomes the third helicity state of a gauge boson that takes mass. This is the Higgs mechanism [189, 236, 243, 261] (it should be called the Englert–Brout–Higgs mechanism, because of the simultaneous paper by Englert and Brout). Consider, for example, the simplest Higgs model described by the Lagrangian [243, 261]
$$\displaystyle{ \mathcal{L} = -\frac{1} {4}F_{\mu \nu }^{2} +\big \vert (\partial _{\mu } + \mathrm{i}eA_{\mu }Q)\phi \big\vert ^{2} +\mu ^{2}\phi ^{{\ast}}\phi - \frac{\lambda } {2}(\phi ^{{\ast}}\phi )^{2}\;. }$$
Note the “wrong” sign in front of the mass term for the scalar field ϕ, which is necessary for the spontaneous symmetry breaking to take place. The above Lagrangian is invariant under the U(1) gauge symmetry
$$\displaystyle{ A_{\mu } \rightarrow A_{\mu }^{{\prime}} = A_{\mu } - \partial _{\mu }\theta (x)\;,\qquad \phi \rightarrow \phi ^{{\prime}} = \mathrm{exp}\big[\mathrm{i}eQ\theta (x)\big]\phi \;. }$$
For the U(1) charge Q, we take  = −ϕ, as in QED, where the particle is e. Let ϕ0 = v ≠ 0, with v real, be the ground state that minimizes the potential and induces the spontaneous symmetry breaking. In our case v is given by v2 = μ2λ. Exploiting gauge invariance, we make the change of variables
$$\displaystyle\begin{array}{rcl} \phi (x)& \rightarrow & \left [v + \frac{h(x)} {\sqrt{2}} \right ]\mathrm{exp}\left [-\mathrm{i} \frac{\zeta (x)} {v\sqrt{2}}\right ], \\ A_{\mu }(x)& \rightarrow & A_{\mu } - \partial _{\mu } \frac{\zeta (x)} {ev\sqrt{2}}. {}\end{array}$$
Then the position of the minimum at ϕ0 = v corresponds to h = 0, and the Lagrangian becomes
$$\displaystyle{ \mathcal{L} = -\frac{1} {4}F_{\mu \nu }^{2} + e^{2}v^{2}A_{\mu }^{2} + \frac{1} {2}e^{2}h^{2}A_{\mu }^{2} + \sqrt{2}e^{2}hvA_{\mu }^{2} + \mathcal{L}(h)\;. }$$
The field ζ(x) is the would-be Goldstone boson, as can be seen by considering only the ϕ terms in the Lagrangian, i.e., setting A μ  = 0 in (1.59). In fact, in this limit the kinetic term μ ζ ∂ μ ζ remains but with no ζ2 mass term. Instead, in the gauge case of (1.59), after changing variables in the Lagrangian, the field ζ(x) completely disappears (not even the kinetic term remains), whilst the mass term e2v2A μ 2 for A μ is now present: the gauge boson mass is \(M = \sqrt{2}ev\). The field h describes the massive Higgs particle. Leaving a constant term aside, the last term in (1.62) is now
$$\displaystyle{ \mathcal{L}(h) = \frac{1} {2}\partial _{\mu }h\partial ^{\mu }h - h^{2}\mu ^{2} + \cdots \;, }$$
where the dots stand for cubic and quartic terms in h. We see that the h mass term has the “right” sign, due to the combination of the quadratic tems in h which, after the shift, arise from the quadratic and quartic terms in ϕ. The h mass is given by m h 2 = 2μ2.
The Higgs mechanism is realized in well-known physical situations. It was actually discovered in condensed matter physics by Anderson [58]. For a superconductor in the Landau–Ginzburg approximation, the free energy can be written as
$$\displaystyle{ F = F_{0} + \frac{1} {2}\mathbf{B}^{2} + \frac{1} {4m}\big\vert (\boldsymbol{\nabla }- 2\mathrm{i}e\mathbf{A})\phi \big\vert ^{2} -\alpha \vert \phi \vert ^{2} +\beta \vert \phi \vert ^{4}\;. }$$
Here B is the magnetic field, | ϕ | 2 is the Cooper pair (ee) density, and 2e and 2m are the charge and mass of the Cooper pair. The “wrong” sign of α leads to ϕ ≠ 0 at the minimum. This is precisely the non-relativistic analogue of the Higgs model of the previous example. The Higgs mechanism implies the absence of propagation of massless phonons (states with dispersion relation ω = kv, with constant v). Moreover, the mass term for A is manifested by the exponential decrease of B inside the superconductor (Meissner effect). However, in condensed matter examples, the Higgs field is not elementary, but rather a condensate of elementary fields (like for the Cooper pairs).

1.8 Quantization of Spontaneously Broken Gauge Theories: R ξ Gauges

In Sect. 1.6 we discussed the problems arising in the quantization of a gauge theory and in the formulation of the correct Feynman rules (gauge fixing terms, ghosts, etc.). Here we give a concise account of the corresponding results for spontaneously broken gauge theories. In particular, we describe the R ξ gauge formalism [14, 207, 250]: in this formalism the interplay of transverse and longitudinal gauge boson degrees of freedom is made explicit and their combination leads to the cancellation of the gauge parameter ξ from physical quantities. We work out in detail an Abelian example that will be easy to generalize later to the non-Abelian case.

We go back to the Abelian model of (1.59) (with Q = −1). In the treatment presented there, the would-be Goldstone boson ζ(x) was completely eliminated from the Lagrangian by a nonlinear field transformation formally identical to a gauge transformation corresponding to the U(1) symmetry of the Lagrangian. In that description, in the new variables, we eventually obtain a theory with only physical fields: a massive gauge boson A μ with mass \(M = \sqrt{2}ev\) and a Higgs particle h with mass \(m_{h} = \sqrt{2}\mu\). This is called a “unitary” gauge, because only physical fields appear. But if we work out the propagator of the massive gauge boson, viz.,
$$\displaystyle{ \mathrm{i}D_{\mu \nu }(k) = -\mathrm{i} \frac{g_{\mu \nu } - k_{\mu }k_{\nu }/M^{2}} {k^{2} - M^{2} + \mathrm{i}\epsilon }\;, }$$
we find that it has a bad ultraviolet behaviour due to the second term in the numerator. This choice does not prove to be the most convenient for a discussion of the ultraviolet behaviour of the theory. Alternatively, one can go to a different formulation where the would-be Goldstone boson remains in the Lagrangian, but the complication of keeping spurious degrees of freedom is compensated by having all propagators with good ultraviolet behaviour (“renormalizable” gauges). To this end we replace the nonlinear transformation for ϕ in (1.61) by its linear equivalent (after all, perturbation theory deals with small oscillations around the minimum):
$$\displaystyle{ \phi (x) \rightarrow \left [v + \frac{h(x)} {\sqrt{2}} \right ]\mathrm{exp}\left [-\mathrm{i} \frac{\zeta (x)} {v\sqrt{2}}\right ] \sim \left [v + \frac{h(x)} {\sqrt{2}} -\mathrm{i}\frac{\zeta (x)} {\sqrt{2}}\right ]\;. }$$
Here we have only applied a shift by the amount v and separated the real and imaginary components of the resulting field with vanishing vacuum expectation value. If we leave A μ as it is and simply replace the linearized expression for ϕ, we obtain the following quadratic terms (those important for propagators):
$$\displaystyle\begin{array}{rcl} \mathcal{L}_{\mathrm{quad}}& =& -\frac{1} {4}\sum _{A}F_{\mu \nu }^{A}F^{A\mu \nu } + \frac{1} {2}M^{2}A_{\mu }A^{\mu } \\ & & +\frac{1} {2}(\partial _{\mu }\zeta )^{2} + MA_{\mu }\partial ^{\mu }\zeta + \frac{1} {2}(\partial _{\mu }h)^{2} - h^{2}\mu ^{2}\;.{}\end{array}$$
The mixing term between A μ and μ ζ does not allow us to write diagonal mass matrices directly. But this mixing term can be eliminated by an appropriate modification of the covariant gauge fixing term given in (1.38) for the unbroken theory. We now take
$$\displaystyle{ \Delta \mathcal{L}_{\mathrm{GF}} = -\frac{1} {2\xi }(\partial ^{\mu }A_{\mu } -\xi M\zeta )^{2}\;. }$$
By adding \(\Delta \mathcal{L}_{\mathrm{GF}}\) to the quadratic terms in (1.67), the mixing term cancels (apart from a total derivative that can be omitted) and we have
$$\displaystyle\begin{array}{rcl} \mathcal{L}_{\mathrm{quad}}& =& -\frac{1} {4}\sum _{A}F_{\mu \nu }^{A}F^{A\mu \nu } + \frac{1} {2}M^{2}A_{\mu }A^{\mu } -\frac{1} {2\xi }(\partial ^{\mu }A_{\mu })^{2} \\ & & +\frac{1} {2}(\partial _{\mu }\zeta )^{2} - \frac{\xi } {2}M^{2}\zeta ^{2} + \frac{1} {2}(\partial _{\mu }h)^{2} - h^{2}\mu ^{2}\;.{}\end{array}$$
We see that the ζ field appears with a mass \(\sqrt{\xi }M\) and its propagator is
$$\displaystyle{ \mathrm{i}D_{\zeta } = \frac{\mathrm{i}} {k^{2} -\xi M^{2} + \mathrm{i}\epsilon }\;. }$$
The propagators of the Higgs field h and of gauge field A μ are
$$\displaystyle{ \mathrm{i}D_{h} = \frac{\mathrm{i}} {k^{2} - 2\mu ^{2} + \mathrm{i}\epsilon }\;, }$$
$$\displaystyle{ \mathrm{i}D_{\mu \nu }(k) = \frac{-\mathrm{i}} {k^{2} - M^{2} + \mathrm{i}\epsilon }\left [g_{\mu \nu } - (1-\xi ) \frac{k_{\mu }k_{\nu }} {k^{2} -\xi M^{2}}\right ]\;. }$$
As anticipated, all propagators have good behaviour at large k2. This class of gauges are called “R ξ gauges” [207]. Note that for ξ = 1 we have a sort of generalization of the Feynman gauge with a Goldstone boson of mass M and a gauge propagator:
$$\displaystyle{ \mathrm{i}D_{\mu \nu }(k) = \frac{-\mathrm{i}g_{\mu \nu }} {k^{2} - M^{2} + \mathrm{i}\epsilon }\;. }$$
Furthermore, for ξ →  the unitary gauge description is recovered, since the would-be Goldstone propagator vanishes and the gauge propagator reproduces that of the unitary gauge in (1.65). All ξ dependence present in individual Feynman diagrams, including the unphysical singularities of the ζ and A μ propagators at k2 = ξ M2, must cancel in the sum of all contributions to any physical quantity.
An additional complication is that a Faddeev–Popov ghost is also present in R ξ gauges (while it is absent in an unbroken Abelian gauge theory). In fact, under an infinitesimal gauge transformation with parameter θ(x), we have the transformations
$$\displaystyle\begin{array}{rcl} A_{\mu }& \rightarrow & A_{\mu } - \partial _{\mu }\theta, {}\\ \phi & \rightarrow & (1 -\mathrm{i}e\theta )\left [v + \frac{h(x)} {\sqrt{2}} -\mathrm{i}\frac{\zeta (x)} {\sqrt{2}}\right ], {}\\ \end{array}$$
so that
$$\displaystyle{ \updelta A_{\mu } = -\partial _{\mu }\theta \;,\quad \updelta h = -e\zeta \theta \;,\quad \updelta \zeta = e\theta \sqrt{2}\left (v + \frac{h} {\sqrt{2}}\right )\;. }$$
The gauge fixing condition μ A μ ξ M ζ = 0 undergoes the variation
$$\displaystyle{ \partial _{\mu }A^{\mu } -\xi M\zeta \; \rightarrow \; \partial _{\mu }A^{\mu } -\xi M\zeta -\left [\partial ^{2} +\xi M^{2}\left (1 + \frac{h} {v\sqrt{2}}\right )\right ]\theta \;, }$$
where we have used \(M = \sqrt{2}ev\). From this, recalling the discussion in Sect. 1.6, we see that the ghost is not coupled to the gauge boson (as usual for an Abelian gauge theory), but has a coupling to the Higgs field h. The ghost Lagrangian is
$$\displaystyle{ \varDelta \mathcal{L}_{\mathrm{Ghost}} =\bar{\eta } \left [\partial ^{2} +\xi M^{2}\left (1 + \frac{h} {v\sqrt{2}}\right )\right ]\eta \;. }$$
The ghost mass is seen to be \(m_{\mathrm{gh}} = \sqrt{\xi }M\) and its propagator is
$$\displaystyle{ \mathrm{i}D_{\mathrm{gh}} = \frac{\mathrm{i}} {k^{2} -\xi M^{2} + \mathrm{i}\epsilon }\;. }$$
The detailed Feynman rules follow for all the basic vertices involving the gauge boson, the Higgs, the would-be Goldstone boson, and the ghost, and are easily derived, with some algebra, from the total Lagrangian including the gauge fixing and ghost additions. The generalization to the non-Abelian case is in principle straightforward, with some formal complications involving the projectors over the space of the would-be Goldstone bosons and over the orthogonal space of the Higgs particles. But for each gauge boson that takes mass M a , we still have a corresponding would-be Goldstone boson and a ghost with mass \(\sqrt{\xi }M_{a}\). The Feynman diagrams, both for the Abelian and the non-Abelian case, are listed explicitly, for example, in the textbook by Cheng and Li [250].

We conclude that the renormalizability of non-Abelian gauge theories, also in the presence of spontaneous symmetry breaking, was proven in the fundamental work by t’Hooft and Veltman [358], and is discussed in detail in [278].


  1. 1.

    Much of the material in this chapter is a revision and update of [32].


  1. 1.
    Aad, G., et al., [ATLAS Collaboration]: Eur. Phys. J. C 72, 2039 (2012). ArXiv:1203.4211; and talk by Giordani, M.P., [ATLAS Collaboration], presented at ICHEP 2012, Melbourne (2012)Google Scholar
  2. 2.
    Aad, G., et al., [ATLAS Collab.]: Phys. Lett. B 716, 1 (2012). ArXiv:1207.7214Google Scholar
  3. 3.
    Aad, G., et al., ATLAS and CMS Collaborations: JHEP 1608, 045 (2016). arXiv:1606.02266 [hep-ex]Google Scholar
  4. 4.
    Aaij, R., et al., LHCb Collaboration: (2013). ArXiv:1304.6325; (2013). ArXiv:1308.1707; Chatrchyan, S., et al., CMS Collaboration: (2013). ArXiv:1308.3409Google Scholar
  5. 5.
    Aaij, R., et al., LHCb Collaboration: Phys. Rev. Lett. 111, 101805 (2013). ArXiv:1307.5024Google Scholar
  6. 6.
    Aaltonen, T., et al., [CDF Collaboration]: Phys. Rev. D 83, 112003 (2011). ArXiv:1101.0034; CDF note 10807Google Scholar
  7. 7.
    Aaltonen, T., et al., [CDF Collaboration]: (2013). ArXiv:1306.2357Google Scholar
  8. 8.
    Abazajian, K.N., et al.: (2012). ArXiv:1204.5379Google Scholar
  9. 9.
    Abazov, V.M., et al., [D0 Collaboration]: Phys. Rev. D 84, 112005 (2011). ArXiv:1107.4995Google Scholar
  10. 10.
    Abbate, R., Fickinger, M., Hoang, A.H., Mateu, V., Stewart, I.W.: Phys. Rev. D 83, 074021 (2011). ArXiv:1006.3080Google Scholar
  11. 11.
    Abbate, R., Fickinger, M., Hoang, A.H., Mateu, V., Stewart, I.W.: Phys. Rev. D 86, 094002 (2012). ArXiv:1204.5746Google Scholar
  12. 12.
    Abe, K., et al., T2K Collaboration: Phys. Rev. Lett. 107, 041801 (2011). ArXiv:1106.2822Google Scholar
  13. 13.
    Abe, Y., et al., DOUBLE-CHOOZ Collaboration: (2011). ArXiv:1112.6353Google Scholar
  14. 14.
    Abers, E.S., Lee, B.W.: Phys. Rep. 9, 1 (1973)ADSCrossRefGoogle Scholar
  15. 15.
    Abulencia, A., et al., CDF Collaboration: Phys. Rev. D 74, 072005 (2006); Phys. Rev. D 74, 072006 (2006); Abazov, V.M., et al., the D0 Collaboration: Phys. Rev. D 74, 112004 (2006)Google Scholar
  16. 16.
    Adam, J., et al., MEG Collaboration: (2013). ArXiv:1303.0754Google Scholar
  17. 17.
    Adamson, P., et al., MINOS Collaboration: Phys. Rev. Lett. 107, 181802 (2011). ArXiv:1108.0015Google Scholar
  18. 18.
    Ade, P.A.R., et al., Planck Collaboration: (2013). ArXiv:1303.5076Google Scholar
  19. 19.
    Adler, S.L.: Phys. Rev. 177, 24261 (969); Adler, S.L., Bardeen, W.A.: Phys. Rev. 182, 1517 (1969); Bell, J.S., Jackiw, R.: Nuovo Cim. A 60, 47 (1969); Bardeen, W.A.: Phys. Rev. 184, 1848 (1969)Google Scholar
  20. 20.
    Aglietti, U., Bonciani, R., Degrassi, G., Vicini, A.: Phys. Lett. B 595, 432 (2004); Degrassi, G., Maltoni, F.: Phys. Lett. B 600, 255 (2004)Google Scholar
  21. 21.
    Agostini, M., et al.: GERDA Collaboration, Phys. Rev. Lett. 111, 122503 (2013)ADSCrossRefGoogle Scholar
  22. 22.
    Ahmed, M.A., Ross, G.G.: Nucl. Phys. B 111, 441 (1976)ADSCrossRefGoogle Scholar
  23. 23.
    Ahn, J.K., et al., RENO Collaboration: (2012). ArXiv:1204.0626Google Scholar
  24. 24.
    Aidala, C.A., Bass, S.D., Hasch, D., Mallot, G.K.: (2012). ArXiv:1209.2803; Burkardt, M., Miller, C.A., Nowak, W.D.: Rep. Prog. Phys. 73, 016201 (2010)Google Scholar
  25. 25.
    Alekhin, S., Djouadi, A., Moch, S.: Phys. Lett. B 716, 214 (2012). ArXiv:1207.0980Google Scholar
  26. 26.
    Alekhin, S., Blumlein, J., Moch, S.: ArXiv:1202.2281 [hep-ph]Google Scholar
  27. 27.
    Alekhin, S., Djouadi, A., Moch, S.: (2012). ArXiv:1207.0980; see also Beneke, M., Falgari, P., Klein, S., Schwinn, C.: (2011). ArXiv:1112.4606Google Scholar
  28. 28.
    Altarelli, G.: Proceedings of the International Conference on High Energy Physics, Geneva, vol. 2, p. 727 (1979)Google Scholar
  29. 29.
    Altarelli, G.: Phys. Rep. 81, 1 (1982)ADSCrossRefGoogle Scholar
  30. 30.
    Altarelli, G.: Annu. Rev. Nucl. Part. Sci. 39, 357 (1989)ADSCrossRefGoogle Scholar
  31. 31.
    Altarelli, G.: In: Zichichi, A. (ed.) Proceedings of the E. Majorana Summer School, Erice. Plenum, New York (1995); Beneke, M., Braun, V.M.: In: Shifman, M. (ed.) Handbook of QCD, vol. 3, p. 1719. World Scientific, Singapore (2001)Google Scholar
  32. 32.
    Altarelli, G.: Gauge theories and the standard model. In: Landolt-Boernstein I 21A: Elementary Particles, vol. 2. Springer, Berlin (2008)Google Scholar
  33. 33.
    Altarelli, G., QCD: The theory of strong interactions. In: Landolt-Boernstein I 21A: Elementary Particles, vol. 4. Springer, Berlin (2008)Google Scholar
  34. 34.
    Altarelli, G.: The standard model of electroweak interactions. In: Landolt-Boernstein I 21A: Elementary Particles, vol. 3. Springer, Berlin (2008)Google Scholar
  35. 35.
    Altarelli, G.: (2012). ArXiv:1210.3467Google Scholar
  36. 36.
    Altarelli, G., Feruglio, F.: New J. Phys. 6, 106 (2004). hep-ph/0405048; Mohapatra, R.N., Smirnov, A.Y.: Annu. Rev. Nucl. Part. Sci. 56, 569 (2006). hep-ph/0603118; Grimus, W., PoS P2GC, 001 (2006). hep-ph/0612311; Gonzalez-Garcia, M.C., Maltoni, M.: Phys. Rep. 460, 1 (2008). ArXiv:0704.1800Google Scholar
  37. 37.
    Altarelli, G., Feruglio, F.: Rev. Mod. Phys. 82, 2701–2729 (2010). ArXiv:1002.0211Google Scholar
  38. 38.
    Altarelli, G., Isidori, G.: Phys. Lett. B 337, 141 (1994); Casas, J.A., Espinosa, J.R., Quirós, M.: Phys. Lett. B 342, 171 (1995); Casas, J.A., et al.: Nucl. Phys. B 436, 3 (1995); B 439, 466 (1995); Carena, M., Wagner, C.E.M.: Nucl. Phys. B 452, 45 (1995)Google Scholar
  39. 39.
    Altarelli, G., Martinelli, G.: Phys. Lett. B 76, 89 (1978)ADSCrossRefGoogle Scholar
  40. 40.
    Altarelli, G., Parisi, G.: Nucl. Phys. B 126, 298 (1977)ADSCrossRefGoogle Scholar
  41. 41.
    Altarelli, G., Ross, G.G.: Phys. Lett. B 212, 391 (1988); Efremov, A.V., Terayev, O.V.: In: Proceedings of the Czech Hadron Symposium, p. 302 (1988); Carlitz, R.D., Collins, J.C., Mueller, A.H.: Phys. Lett. B 214, 229 (1988); Altarelli, G., Lampe, B.: Z. Phys. C 47, 315 (1990)Google Scholar
  42. 42.
    Altarelli, G., Ellis, R.K., Martinelli, G.: Nucl. Phys. B 143, 521 (1978); Altarelli, G., Ellis, R.K., Martinelli, G.: Nucl. Phys. B 146, 544 (1978); Nucl. Phys. B 157, 461 (1979)Google Scholar
  43. 43.
    Altarelli, G., Ellis, R.K., Greco, M., Martinelli, G.: Nucl. Phys. B 246, 12 (1984); Altarelli, G., Ellis, R.K., Martinelli, G.: Z. Phys. C 27, 617 (1985)Google Scholar
  44. 44.
    Altarelli, G., Mele, B., Pitolli, F.: Nucl. Phys. B 287, 205 (1987)ADSCrossRefGoogle Scholar
  45. 45.
    Altarelli, G., Nason, P., Ridolfi, G.: Z. Phys. C 68, 257 (1995)ADSCrossRefGoogle Scholar
  46. 46.
    Altarelli, G., Sjostrand, T., Zwirner, F.: Physics at LEP2, CERN Report 96-01 (1996). LEP2Google Scholar
  47. 47.
    Altarelli, G., Kleiss, R., Verzegnassi, C. (eds.): Z Physics at LEP1 (CERN 89-08, Geneva, 1989), vols. 1–3; Precision Calculations for the Z Resonance, ed. by Bardin, D., Hollik, W., Passarino, G., CERN Rep 95-03 (1995); Vysotskii, M.I., Novikov, V.A., Okun, L.B., Rozanov, A.N.: hep-ph/9606253 or Phys. Usp. 39, 503–538 (1996)Google Scholar
  48. 48.
    Altarelli, G., Barbieri, R., Caravaglios, F.: Int. J. Mod. Phys. A 13, 1031 (1998) and references thereinGoogle Scholar
  49. 49.
    Altarelli, G., Ball, R., Forte, S.: Nucl. Phys. B 799, 199 (2008). ArXiv:0802.0032Google Scholar
  50. 50.
    Altarelli, G., Feruglio, F., Merlo, L.: (2012). ArXiv:1205.5133Google Scholar
  51. 51.
    Altarelli, G., Feruglio, F., Merlo, L., Stamou, E.: (2012). ArXiv:1205.4670Google Scholar
  52. 52.
    Altarelli, G., Feruglio, F., Masina, I., Merlo, L.: (2012). ArXiv:1207.0587Google Scholar
  53. 53.
    Amhis, Y., et al., the Heavy Flavor Averaging Group: (2012). ArXiv:1207.1158; Aaij, R., et al., LHCb Collaboration: (2012). ArXiv:1211.1230Google Scholar
  54. 54.
    An, F.P., et al., DAYA-BAY Collaboration: (2012). ArXiv:1203.1669Google Scholar
  55. 55.
    Anastasiou, C., Dixon, L.J., Melnikov, K., Petriello, F.: Phys. Rev. D 69, 094008 (2004). hep-ph/0312266Google Scholar
  56. 56.
    Anastasiou, C., Melnikov, K., Petriello, F.: Nucl. Phys. B 724, 197 (2005); Ravindran, V., Smith, J., van Neerven, W.L.: hep-ph/0608308Google Scholar
  57. 57.
    Anastasiou, C., Melnikov, K., Petriello, F.: Phys. Rev. D 72, 097302 (2005). hep-ph/0509014Google Scholar
  58. 58.
    Anderson, P.W.: Phys. Rev. 112, 1900 (1958); Phys. Rev. 130, 439 (1963)ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    Aoyama, T., Hayakawa, M., Kinoshita, T., Nio, N.: (2012). ArXiv:1205.5368; ArXiv:1205.5370Google Scholar
  60. 60.
    Appelquist, T., Carazzone, J.: Phys. Rev. D 11, 2856 (1975)ADSCrossRefGoogle Scholar
  61. 61.
    Arkani-Hamed, N., Dimopoulos, S.: JHEP 0506, 073 (2005). hep-th/0405159; Arkani-Hamed, N., et al.: Nucl. Phys. B 709, 3 (2005). hep-ph/0409232; Giudice, G., Romanino, A.: Nucl. Phys. B 699, 65 (2004); erratum ibid. B 706, 65 (2005); hep-ph/0406088; Arkani-Hamed, N., Dimopoulos, S., Kachru, S.: hep-ph/0501082; Giudice, G., Rattazzi, R.: Nucl. Phys. B 757, 19 (2006). hep-ph/0606105ADSCrossRefGoogle Scholar
  62. 62.
    Arkani-Hamed, N., Dimopoulos, S., Dvali, G.: Phys. Lett. B 429, 263 (1998). hep-ph/9803315; Antoniadis, I., Arkani-Hamed, N., Dimopoulos, S., Dvali, G.R.: Phys. Lett. B 436, 257 (1998). hep-ph/9804398; Arkani-Hamed, N., Hall, L.J., Smith, D.R., Weiner, N.: Phys. Rev. D 62 (2000). hep-ph/9912453; Giudice, G.F., Rattazzi, R., Wells, J.D.: Nucl. Phys. B 544, 3 (1999). hep-ph/9811291; Han, T., Lykken, J.D., Zhang, R.-J.: Phys. Rev. D 59, 105006 (1999). hep-ph/9811350; Hewett, J.L.: Phys. Rev. Lett. 82, 4765 (1999)Google Scholar
  63. 63.
    Arkani-Hamed, N., et al.: JHEP 0208, 021 (2002). hep-ph/0206020; Arkani-Hamed, N., Cohen, A., Katz, E., Nelson, A.: JHEP 0207, 034 (2002). hep-ph/0206021; Schmaltz, M., Tucker-Smith, D.: Annu. Rev. Nucl. Part. Sci. 55, 229 (2005). hep-ph/0502182; Perelstein, M.: Prog. Part. Nucl. Phys. 58, 247 (2007). hep-ph/0512128; Perelstein, M., Peskin, M.E., Pierce, A.: Phys. Rev. D 69, 075002 (2004). hep-ph/0310039ADSCrossRefGoogle Scholar
  64. 64.
    Arnold, P., Kauffman, R.: Nucl. Phys. B 349, 381 (1991); Ladinsky, G.A., Yuan, C.P.: Phys. Rev. D 50, 4239 (1994); Ellis, R.K., Veseli, S.: Nucl. Phys. B 511, 649 (1998)Google Scholar
  65. 65.
    Ashman, J., et al., EMC Collaboration: Phys. Lett. B 206, 364 (1988)ADSGoogle Scholar
  66. 66.
    Atlas Collaboration: Coupling properties of the new Higgs-like boson observed with the ATLAS detector at the LHC. CERN Council Open Symposium: European Strategy for Particle Physics, Krakow, Poland, 10–12 September 2012. ATLAS-CONF-20122127 (2012)Google Scholar
  67. 67.
    ATLAS Collaboration: Eur. Phys. J. C 71, 1846 (2011). ArXiv:1109.6833; CMS Collaboration: JHEP 04, 084 (2012). ArXiv:1202.4617Google Scholar
  68. 68.
    ATLAS Collaboration ATLAS-CONF-2013-034: Phys. Lett. B 726, 88 (2013). ArXiv:1307.1427; ATLAS Collaboration ATLAS-CONF-2013-034: Phys. Lett. 726, 120 (2013). ArXiv:1307.1432; CMS Collaboration: PAS HIG-13-005; Landsberg, G.: Proceedings of EPS-HEP Conference, Stockholm (2013). ArXiv:1310.5705Google Scholar
  69. 69.
    Auger, M., et al., (EXO Collaboration): Phys. Rev. Lett. 109, 032505 (2012); Gando, A., et al., KamLAND-Zen Collaboration: ArXiv:1211.3863Google Scholar
  70. 70.
    Awramik, M., Czakon, M.: Phys. Rev. Lett. 89, 241801 (2002). hep-ph/0208113; Onishchenko, A., Veretin, O.: Phys. Lett. B 551, 111 (2003). hep-ph/0209010; Awramik, M., Czakon, M., Onishchenko, A., Veretin, O.: Phys. Rev. D 68, 053004 (2003). hep-ph/0209084; Awramik, M., Czakon, M.: Phys. Lett. B 568, 48 (2003). hep-ph/0305248; Freitas, A., Hollik, W., Walter, W., Weiglein, G.: Phys. Lett. B 495, 338 (2000); erratum ibid. B 570, 260 (2003). hep-ph/0007091; Freitas, A., Hollik, W., Walter, W., Weiglein, G.: Nucl. Phys. B 632, 189 (2002). erratum ibid. B 666, 305 (2003). hep-ph/0202131Google Scholar
  71. 71.
    Awramik, M., Czakon, M., Freitas, A., Weiglein, G.: Phys. Rev. Lett. 93, 201805 (2004). hep-ph/0407317; Awramik, M., Czakon, M., Freitas, A.: (2006). hep-ph/0608099; Phys. Lett. B 642, 563 (2006). hep-ph/0605339; Hollik, W., Meier, U., Uccirati, S.: Nucl. Phys. B 731, 213 (2005). hep-ph/0507158; Nucl. Phys. B 765, 154 (2007). hep-ph/0610312ADSCrossRefGoogle Scholar
  72. 72.
    Azatov, A., Galloway, J.: (2006). ArXiv:1212.1380; Hollik, W., Meler, U., Uccirati, S.: Nucl. Phys. B 731, 213 (2005); Nucl. Phys. B 765, 154 (2007)Google Scholar
  73. 73.
    Baak, M., et al., Gfitter group: (2012). ArXiv:1209.2716Google Scholar
  74. 74.
    Baikov, P.A., Chetyrkin, K.G., Kuhn, J.H.: Phys. Rev. Lett. 101, 012002 (2008). ArXiv:0801.1821; Phys. Rev. Lett. 104, 132004 (2010). ArXiv:1001.3606; Baikov, A., Chetyrkin, K.G., Kuhn, J.H., Rittinger, J.: ArXiv:1210.3594ADSCrossRefGoogle Scholar
  75. 75.
    Baker, C.A., et al.: Phys. Rev. Lett. 97, 131801 (2006). hep-ex/0602020v3ADSCrossRefGoogle Scholar
  76. 76.
    Ball, R.D., et al.: Phys. Lett. B 707, 66 (2012). ArXiv:1110.2483Google Scholar
  77. 77.
    Banfi, A., Dasgupta, M.: JHEP 0401, 027 (2004); Idilbi, A., Ji, X.-d.: Phys. Rev. D 72, 054016 (2005). hep-ph/0501006; Bolzoni, P., Forte, S., Ridolfi, G.: Nucl. Phys. B 731, 85 (2005); Delenda, Y., Appleby, R., Dasgupta, M., Banfi, A.: JHEP 0612, 044 (2006); Bolzoni, P.: Phys. Lett. B 643, 325 (2006); Mert, S., Aybat, Dixon, L.J., Sterman, G.: Phys. Rev. D 74, 074004 (2006); Dokshitzer, Yu.L., Marchesini, G.: JHEP 0601, 007 (2006); Laenen, E., Magnea, L.: Phys. Lett. B 632, 270 (2006); Lee, C., Sterman, G.: Phys. Rev. D 75, 014022 (2007); Becher, T., Neubert, M.: Phys. Rev. Lett. 97, 082001 (2006). hep-ph/0605050; de Florian, D., Vogelsang, W.: ArXiv:0704.1677; Chay, J., Kim, C.: Phys. Rev. D 75, 016003 (2007). hep-ph/0511066; Becher, T., Neubert, M., Pecjak, B.D.: JHEP 0701, 076 (2007). ArXiv:0607228; Abbate, R., Forte, S., Ridolfi, G.: ArXiv:0707.2452; Becher, T., Neubert, M., Xu, G.: JHEP 0807, 030 (2008). ArXiv:0710.0680; Ahrens, V., Becher, T., Neubert, M., Yang, L.L.: Phys. Rev. D 79, 033013 (2009); ArXiv:0808.3008; Bonvini, M., Forte, S., Ridolfi, G.: Nucl. Phys. B 808, 347 (2009). ArXiv:0807.3830; Nucl. Phys. B 847, 93 (2011). ArXiv:1009.5691; Bonvini, M., Ghezzi, M., Ridolfi, G.: Nucl. Phys. B 861, 337 (2012); ArXiv:1201.6364Google Scholar
  78. 78.
    Banfi, A., Salam, G.P., Zanderighi, G.: JHEP 0503, 073 (2005)ADSCrossRefGoogle Scholar
  79. 79.
    Barbieri, R.: (2013). ArXiv:1309.3473Google Scholar
  80. 80.
    Barbieri, R., Strumia, A.: (2000). hep-ph/0007265Google Scholar
  81. 81.
    Barbieri, R., et al.: Eur. Phys. J. C 71, 1725 (2011). ArXiv:1105.2296; JHEP 1207, 181 (2012). ArXiv:1203.4218; ArXiv:1206.1327; ArXiv:1211.5085; Crivellin, A., Hofer, L., Nierste, U.: PoS EPS-HEP2011, 145 (2011); Buras, A.J., Girrbach, J.: JHEP 1301, 007 (2013). ArXiv:1206.3878Google Scholar
  82. 82.
    Bardeen, W.A., Buras, A.J., Duke, D.W., Muta, T.: Phys. Rev. D 18, 3998 (1978)ADSCrossRefGoogle Scholar
  83. 83.
    Bauer, D., (for the CDF and D0 Collaborations): Nucl. Phys. B (Proc. Suppl.) 156, 226 (2006)Google Scholar
  84. 84.
    Bauer, C.W., Fleming, S., Luke, M.E.: Phys. Rev. D 63, 014006 (2000). hep-ph/0005275; Bauer, C.W., et al.: Phys. Rev. D 63, 114020 (2001). hep-ph/0011336; Bauer, C.W., Stewart, I.W.: Phys. Lett. B 516, 134 (2001). hep-ph/0107001; Bauer, C.W., Pirjol, D., Stewart, I.W.: Phys. Rev. D 65, 054022 (2002). hep-ph/0109045;Google Scholar
  85. 85.
    Bazavov, A., et al.: Phys. Rev. D 85, 054503 (2012). ArXiv:1111.1710Google Scholar
  86. 86.
    Becher, T., Neubert, M.: Phys. Rev. Lett. 98, 022003 (2007). hep-ph 0610067; see also Misiak, M., et al.: Phys. Rev. Lett. 98, 022002 (2007). hep-ph/0609232ADSCrossRefGoogle Scholar
  87. 87.
    Belavin, A., Polyakov, A., Shvarts, A., Tyupkin, Y.: Phys. Lett. B 59, 85 (1975)ADSMathSciNetCrossRefGoogle Scholar
  88. 88.
    Benayoun, M., David, P., Del Buono, L., Jegerlehner, F.: (2012). ArXiv:1210.7184; Davier, M., Malaescu, B.: (2013). ArXiv:1306.6374Google Scholar
  89. 89.
    Beneke, M., Boito, D., Jamin, M.: (2012). ArXiv:1210.8038Google Scholar
  90. 90.
    Beneke, M., Jamin, M., JHEP 0809, 044 (2008). ArXiv:0806.3156; Davier, M., et al., Eur. Phys. J. C 56, 305 (2008). ArXiv:0803.0979; Maltman, K., Yavin, T.: Phys. Rev. D 78, 094020 (2008). ArXiv:0807.0650; Narison, S.: Phys. Lett. B 673, 30 (2009). ArXiv:0901.3823; Caprini, I., Fischer, J., Eur. Phys. J. C 64, 35 (2009); ArXiv:0906.5211; Phys. Rev. D 84, 054019 (2011). ArXiv:1106.5336S; Abbas, G., Ananthanarayan, B., Caprini, I., Fischer, J.: Phys. Rev. D 87, 014008 (2013). ArXiv:1211.4316; Menke, S.: ArXiv:0904.1796; Pich, A.: ArXiv:1107.1123; Magradze, B.A.: ArXiv:1112.5958; Abbas, G., et al.: ArXiv:1202.2672; Boito, D., et al.: Phys. Rev. D 84, 113006 (2011). ArXiv:1110.1127; Boito, D., et al.: ArXiv:1203.3146Google Scholar
  91. 91.
    Berends, F.A., Giele, W.: Nucl. Phys. B 294, 700 (1987); Mangano, M.L., Parke, S.J., Xu, Z.: Nucl. Phys. B 298, 653 (1988)Google Scholar
  92. 92.
    Berger, C., et al.: Phys. Rev. D 78, 036003 (2008). ArXiv:0803.4180; ArXiv:0912.4927; Bern, Z., et al.: ArXiv:1210.6684Google Scholar
  93. 93.
    Berger, C.F., et al.: Phys. Rev. D 82, 074002 (2010). ArXiv:1004.1659Google Scholar
  94. 94.
    Berger, C.F., et al.: Phys. Rev. Lett. 106, 092001 (2011). ArXiv:1009.2338; Ita, H., et al.: Phys. Rev. D 85, 031501 (2012). ArXiv:1108.2229ADSCrossRefGoogle Scholar
  95. 95.
    Bern, Z., Dixon, L.J., Kosower, D.A.: Annu. Rev. Nucl. Part. Sci. 46, 109 (1996). hep-ph/9602280 and references thereinADSCrossRefGoogle Scholar
  96. 96.
    Bern, Z., Forde, D., Kosower, D.A., Mastrolia, P.: Phys. Rev. D 72, 025006 (2005)ADSMathSciNetCrossRefGoogle Scholar
  97. 97.
    Bern, Z., et al., Phys. Rev. D 88, 014025 (2013). ArXiv:1304.1253Google Scholar
  98. 98.
    Bertolini, S., Di Luzio, L., Malinsky, M.: (2012). ArXiv:1205.5637 and references thereinGoogle Scholar
  99. 99.
    Bethke, S.: (2009). ArXiv:0908.1135Google Scholar
  100. 100.
    Bevilacqua, G., Czakon, M., Papadopoulos, C.G., Worek, M.: Phys. Rev. Lett. 104, 162002 (2010). ArXiv:1002.4009; Phys. Rev. D 84, 114017 (2011); ArXiv:1108.2851ADSCrossRefGoogle Scholar
  101. 101.
    Bilenky, S.M., Hosek, J., Petcov, S.T.: Phys. Lett. B 94, 495 (1980); Schechter, J., Valle, J.W.F.: Phys. Rev. D 22, 2227 (1980); Doi, M., et al.: Phys. Lett. B 102, 323 (1981)Google Scholar
  102. 102.
    Bjorken, J.D., Drell, S.: Relativistic Quantum Mechanics/Fields, vols. I, II. McGraw-Hill, New York (1965)zbMATHGoogle Scholar
  103. 103.
    Bloch, F., Nordsieck, H.: Phys. Rev. 52, 54 (1937)ADSCrossRefGoogle Scholar
  104. 104.
    Blumlein, J., Bottcher, H.: Nucl. Phys. B 841, 205–230 (2010). ArXiv:1005.3113Google Scholar
  105. 105.
    Blumlein, J., Bottcher, H., Guffanti, A.: Nucl. Phys. B 774, 182 (2007). ArXiv:0607200 [hep-ph]Google Scholar
  106. 106.
    Bobeth, C., Hiller, G., van Dyk, D.: JHEP 1107, 067 (2011). ArXiv:1105.0376; Beaujean, F., Bobeth, C., van Dyk, D., Wacker, C.: JHEP 1208, 030 (2012). ArXiv:1205.1838; Bobeth, C., Hiller, G., van Dyk, D.: Phys. Rev. D 87, 034016 (2013). ArXiv:1212.2321; Descotes-Genon, S., Matias, J., Virto, J.: Phys. Rev. D 88, 074002 (2013). ArXiv:1307.5683 and references thereinADSCrossRefGoogle Scholar
  107. 107.
    Bona, M., [UTFit Collaboration]: UPdates from the UTfit on the Unitarity Triangle analysis. In: ICHEP 2016, Chicago (2016)Google Scholar
  108. 108.
    Bonciani, R., Catani, S., Mangano, M.L., Nason, P.: Nucl. Phys. B 529, 424 (1998) and references thereinGoogle Scholar
  109. 109.
    Bouchiat, C., Iliopoulos, J., Meyer, P.: Phys. Lett. B 38, 519 (1972)ADSCrossRefGoogle Scholar
  110. 110.
    Bozzi, G., Catani, S., de Florian, D., Grazzini, M.: Phys. Lett. B 564, 65 (2003); Nucl. Phys. B 737, 73 (2006); Kulesza, A., Sterman, G., Vogelsang, W.: Phys. Rev. D 69, 014012 (2004)Google Scholar
  111. 111.
    Branchina, V., Messina, E.: (2013). ArXiv:1307.5193Google Scholar
  112. 112.
    Brandt, R., Preparata, G.: Nucl. Phys. B 27, 541 (1971)ADSCrossRefGoogle Scholar
  113. 113.
    Bredenstein, A., Denner, A., Dittmaier, S., Pozzorini, S.: Phys. Rev. Lett. 103, 012002 (2009). ArXiv:0905.0110; JHEP 1003, 021 (2010). ArXiv:1001.4006; Bevilacqua, G., et al.: JHEP 0909, 109 (2009). ArXiv:0907.4723ADSCrossRefGoogle Scholar
  114. 114.
    Britto, R., Cachazo, F., Feng, B.: Nucl. Phys. B 715, 499 (2005). hep-th/0412308; Britto, R., Cachazo, F., Feng, B., Witten, E.: Phys. Rev. Lett. 94, 181602 (2005). hep-th/0501052Google Scholar
  115. 115.
    Buchmuller, W., Peccei, R.D., Yanagida, T.: Annu. Rev. Nucl. Part. Sci. 55, 311 (2005). hep-ph 0502169; Fong, C.S., Nardi, E., Riotto, A.: Adv. High Energy Phys. 2012, 158303 (2012). ArXiv:1301.3062; Buchmuller, W.: ArXiv:1212.3554ADSCrossRefGoogle Scholar
  116. 116.
    Buras, A.J.: Lect. Notes Phys. 558, 65 (2000). hep-ph/9901409Google Scholar
  117. 117.
    Buras, A.J., Girrbach, J., Guadagnoli, D., Isidori, G.: Eur. Phys. J. C 72, 2172 (2012). ArXiv:1208.0934; Buras, A.J., Fleischer, R., Girrbach, J., Knegjens, R., JHEP 07, 077 (2013). ArXiv:1303.3820; see also Amhis, Y., et al., (Heavy Flavor Averaging Group): ArXiv:1207.1158Google Scholar
  118. 118.
    Buttazzo, D., Degrassi, G., Giardino, P.P., Giudice, G.F., Sala, F., Salvio, A., Strumia, A.: JHEP 1312, 089 (2013). doi:10.1007; JHEP 12, 089 (2013). arXiv:1307.3536 [hep-ph]ADSCrossRefGoogle Scholar
  119. 119.
    Butterworth, J.M., Dissertori, G., Salam, G.P.: Annu. Rev. Nucl. Part. Sci. 62, 387 (2012). doi:10.1146/annurev-nucl-102711-094913, arXiv:1202.0583 [hep-ex]ADSCrossRefGoogle Scholar
  120. 120.
    Cabibbo, N.: Phys. Rev. Lett. 10, 531 (1963)ADSCrossRefGoogle Scholar
  121. 121.
    Cabibbo, N.: Phys. Rev. Lett. 10, 531 (1963); Kobayashi, M., Maskawa, T.: Prog. Theor. Phys. 49, 652 (1973)ADSCrossRefGoogle Scholar
  122. 122.
    Cabibbo, N.: Phys. Lett. B 72, 333 (1978)ADSCrossRefGoogle Scholar
  123. 123.
    Cabibbo, N., Maiani, L., Parisi, G., Petronzio, R.: Nucl. Phys. B 158, 295 (1979)ADSCrossRefGoogle Scholar
  124. 124.
    Cacciari, M., Greco, M.: Nucl. Phys. B 421, 530 (1994); Cacciari, M., Greco, M., Nason, P., JHEP 9805, 007 (1998)Google Scholar
  125. 125.
    Cacciari, M., et al., JHEP 0404, 068 (2004); Cacciari, M., et al., JHEP 0407, 033 (2004)ADSCrossRefGoogle Scholar
  126. 126.
    Cacciari, M., Salam, G.P., Soyez, G.: JHEP 0804, 005 (2008). ArXiv:0802.1188ADSCrossRefGoogle Scholar
  127. 127.
    Cacciari, M., et al.: Phys. Lett. B 710, 612, (2012). ArXiv:1111.5869Google Scholar
  128. 128.
    Cachazo, F., Svrcek, P.: PoS RTN2005:004 (2005). hep-th/0504194; Dixon, L.J.: In: Proceedings of the EPS International Europhysics Conference on High Energy Physics, Lisbon, Portugal, 2005, PoS HEP2005:405 (2006). hep-ph/0512111 and references thereinGoogle Scholar
  129. 129.
    Campbell, J.M., Ellis, R.K.: JHEP 1207, 052 (2012). ArXiv:1204.5678ADSCrossRefGoogle Scholar
  130. 130.
    Campbell, J.M., Huston, J.W., Stirling, W.J.: Rep. Prog. Phys. 70, 89 (2007)ADSCrossRefGoogle Scholar
  131. 131.
    Caswell, W.E.: Phys. Rev. Lett. 33, 244 (1974); Jones, D.R.T.: Nucl. Phys. B 75, 531 (1974); Egorian, E., Tarasov, O.V.: Theor. Math. Phys. 41, 863 (1979). [Teor. Mat. Fiz. 41 (1979) 26]ADSCrossRefGoogle Scholar
  132. 132.
    Catani, S., Dokshitzer, Y.L., Seymour, M.H., Webber, B.R.: Nucl. Phys. B 406, 187 (1993) and references therein; Ellis, S.D., Soper, D.E.: Phys. Rev. D 48, 3160 (1993). hep-ph/9305266Google Scholar
  133. 133.
    Catani, S., de Florian, D., Grazzini, M.: JHEP 0105, 025 (2001); JHEP 0201, 015 (2002); Harlander, R.V.: Phys. Lett. B 492, 74 (2000); Ravindran, V., Smith, J., van Neerven, W.L.: Nucl. Phys. B 704, 332 (2005); Harlander, R.V., Kilgore, W.B.: Phys. Rev. D 64, 013015 (2001); Harlander, R.V., Kilgore, W.B.: Phys. Rev. Lett. 88, 201801 (2002); Anastasiou, C., Melnikov, K.: Nucl. Phys. B 646, 220 (2002); Ravindran, V., Smith, J., van Neerven, W.L.: Nucl. Phys. B 665, 325 (2003)ADSCrossRefGoogle Scholar
  134. 134.
    Catani, S., de Florian, D., Grazzini, M., Nason, P.: JHEP 0307, 028 (2003); Moch, S., Vogt, A.: Phys. Lett. B 631, 48 (2005); Ravindran, V.: Nucl. Phys. B 752, 173 (2006) (and references therein)ADSCrossRefGoogle Scholar
  135. 135.
    Chatrchyan, S., et al., [CMS Collab.]: Phys. Lett. B 716, 30 (2012). ArXiv:1207:7235Google Scholar
  136. 136.
    Chatrchyan, S., et al., CMS Collaboration: Phys. Rev. Lett. 111, 101804 (2013). ArXiv:1307.5025Google Scholar
  137. 137.
    Cheng, T.-P., Li, L.-F.: Gauge Theory of Elementary Particle Physics. Oxford University Press, Oxford (1988); Bailin, D., Love, A.: Introduction to Gauge Field Theory, Revised edition. CRC, Boca Raton, FL (1993); Bardin, D.Y., Passarino, G.: The Standard Model in the Making: Precision Study of the Electroweak Interactions. Clarendon, Oxford (1999); Aitchison, I.J.R., Hey, A.J.G.: Gauge Theories in Particle Physics. Taylor and Francis, London (2003)Google Scholar
  138. 138.
    Chetyrkin, K.G., Kataev, A.L., Tkachev, F.V.: Phys. Lett. B 85, 277 (1979); Dine, M., Sapirstein, J.: Phys. Rev. Lett. 43, 688 (1979); Celmaster, W., Gonsalves, R.J.: Phys. Rev. Lett. 44, 560 (1979); Phys. Rev. D 21, 3772 (1980)Google Scholar
  139. 139.
    Chetyrkin, K., Kniehl, B., Steinhauser, M.: Phys. Rev. Lett. 79, 353 (1997)ADSCrossRefGoogle Scholar
  140. 140.
    Choudhury, D., Tait, T.M.P., Wagner, C.E.M.: Phys. Rev. D 65, 053002 (2002). hep-ph/0109097Google Scholar
  141. 141.
    Ciafaloni, M., Colferai, D., Salam, G.P., Stasto, A.M.: Phys. Lett. B 587, 87 (2004); see also Salam, G.P.: hep-ph/0501097; White, C.D., Thorne, R.S.: hep-ph/0611204Google Scholar
  142. 142.
    Ciuchini, M., Franco, E., Mishima, S., Silvestrini, L.: (2013). ArXiv:1306.4644Google Scholar
  143. 143.
    CMS Collaboration: CMS PAS B2G-12-012 (2012)Google Scholar
  144. 144.
    CMS Collaboration: (2011). CMS-PAS-TOP-11-030; and talk by Chwalek, T., [CMS Collaboration]: Presented at ICHEP 2012, Melbourne (2012)Google Scholar
  145. 145.
    Coleman, S., Gross, D.: Phys. Rev. Lett. 31, 851 (1973)ADSCrossRefGoogle Scholar
  146. 146.
    Collins, J.C., Soper, D.E., Sterman, G.: Nucl. Phys. B 261, 104 (1985); a review by the same authors can be found in Perturbative QCD, edited by Mueller, A.H., World Scientific, Singapore (1989); Bodwin, G.T.: Phys. Rev. D 31, 2616 (1985); erratum D 34, 3932 (1986); Nucl. Phys. B 308, 833 (1988); Phys. Lett. B 438, 184 (1998). hep-ph/9806234Google Scholar
  147. 147.
    Combridge, B.L., Kripfganz, J., Ranft, J.: Phys. Lett. B 70, 234 (1977)ADSCrossRefGoogle Scholar
  148. 148.
    Commins, E.D.: Weak Interactions. McGraw Hill, New York (1973); Okun, L.V.: Leptons and Quarks. North Holland, Amsterdam (1982); Bailin, D.: Weak Interactions, 2nd edn. Hilger, Bristol (1982); Georgi, H.M.: Weak and Modern Particle Theory. Benjamin, Menlo Park, CA (1984)Google Scholar
  149. 149.
    Cornwall, J.M., Levin, D.N., Tiktopoulos, G.: Phys. Rev. D 10, 1145 (1974); Vayonakis, C.E.: Lett. Nuovo Cim. 17, 383 (1976); Lee, B.W., Quigg, C., Thacker, H.: Phys. Rev. D 16, 1519 (1977); Chanowitz, M.S., Gaillard, M.K.: Nucl. Phys. B 261, 379 (1985)Google Scholar
  150. 150.
    Czakon, M., Fiedler, P., Mitov, A., Rojo, A.: (2013). ArXiv:1305.3892; See also: Baernreuther, P., Czakon, M., Mitov, A.: Phys. Rev. Lett. 109, 132001 (2012). ArXiv:1204.5201; Czakon, M., Mitov, A.: JHEP 1212, 054 (2012). ArXiv:1207.0236; JHEP 1301, 080 (2013). ArXiv:1210.6832; Czakon, M., Fiedler, P., Mitov, A.: Phys. Rev. Lett. 110, 252004 (2013). ArXiv:1303.6254Google Scholar
  151. 151.
    Czarnecki, A., Krause, B., Marciano, W.J.: Phys. Rev. Lett. 76, 3267 (1996); Knecht, M., Peris, S., Perrottet, M., De, E., Rafael, J.: High Energy Phys. 11, 003 (2002); Czarnecki, A., Marciano, W.J., Vainshtein, A.: Phys. Rev. D 67, 073006 (2003)ADSCrossRefGoogle Scholar
  152. 152.
    D’Ambrosio, G., Giudice, G.F., Isidori, G., Strumia, A.: Nucl. Phys. B 645, 155 (2002). hep-ph/0207036; Buras, A.J.: Acta Phys. Polon. B 34, 5615 (2003). hep-ph/0310208Google Scholar
  153. 153.
    Dasgupta, M., Salam, G.P.: Phys. Lett. B 512, 323 (2001); Dasgupta, M., Salam, G.P.: J. Phys. G 30, R143 (2004); Banfi, A., Corcella, G., Dasgupta, M., Delenda, Y., Salam, G.P., Zanderighi, G.: (2005). hep-ph/0508096; Forshaw, J.R., Kyrieleis, A., Seymour, M.H.: JHEP 0608, 059 (2006)Google Scholar
  154. 154.
    Dasgupta, M., Salam, G.P., J. Phys. G. 30, R143 (2004). hep-ph/0312283; Biebel, O.: Phys. Rep. 3450, 165 (2001); Kluth, S.: Rep. Prog. Phys. 69, 1771 (2006). hep-ex/0603011ADSCrossRefGoogle Scholar
  155. 155.
    Davier, M., Hoecker, A., Malaescu, B., Zhang, Z.: Eur. Phys. J. C 71, 1515 (2011). ArXiv:1010.4180Google Scholar
  156. 156.
    Dawson, S.: Nucl. Phys. B 359, 283 (1991); Djouadi, A., Spira, M., Zerwas, P.M.: Phys. Lett. B 264, 440 (1991); Spira, M., Djouadi, A., Graudenz, D., Zerwas, P.: Nucl. Phys. B 453, 17 (1995)Google Scholar
  157. 157.
    de Boer, W., Sander, C.: Phys. Lett. B 585, 276 (2004)ADSCrossRefGoogle Scholar
  158. 158.
    de Florian, D., Grazzini, M., Kunszt, Z.: Phys. Rev. Lett. 82, 5209 (1999); Ravindran, V., Smith, J., Van Neerven, W.L.: Nucl. Phys. B 634, 247 (2002); Glosser, C.J., Schmidt, C.R.: JHEP 0212, 016 (2002)ADSCrossRefGoogle Scholar
  159. 159.
    de Florian, D., Sassot, R., Stratmann, M., Vogelsang, W.: Phys. Rev. D 80, 034030 (2009). ArXiv:0904.3821; ArXiv:1112.0904Google Scholar
  160. 160.
    Degrassi, G., et al.: JHEP 1208, 098 (2012). ArXiv:1205.6497ADSCrossRefGoogle Scholar
  161. 161.
    Denner, A., Dittmaier, S., Kallweit, S., Pozzorini, S.: Phys. Rev. Lett. 106, 052001 (2011). ArXiv:1012.3975; Bevilacqua, G., et al.: JHEP 1102, 083 (2011). ArXiv:1012.4230ADSCrossRefGoogle Scholar
  162. 162.
    DeTar, C.E., Heller, U.M.: Eur. Phys. J. A. 41, 405 (2009). ArXiv:0905.2949ADSCrossRefGoogle Scholar
  163. 163.
    Dinsdale, M., Ternick, M., Weinzierl, S.: JHEP 0603, 056 (2006). hep-ph/0602204ADSCrossRefGoogle Scholar
  164. 164.
    Dissertori, G., et al.: JHEP 0908, 036 (2009). ArXiv:0906.3436ADSCrossRefGoogle Scholar
  165. 165.
    Dittmaier, S., et al.: Handbook of LHC Higgs Cross Sections (2011). ArXiv:1101.0593, (2012). ArXiv:1201.3084Google Scholar
  166. 166.
    Dixon, L.: Talk at ICHEP 2012, Melbourne (2012)Google Scholar
  167. 167.
    Dixon, L.J., Glover, E.W.N., Khoze, V.V.: JHEP 0412, 015 (2004); Badger, S.D., Glover, E.W.N., Khoze, V.V.: JHEP 0503, 023 (2005); Badger, S.D., Glover, E.W.N., Risager, K.: Acta Phys. Polon. B 38, 2273–2278 (2007)Google Scholar
  168. 168.
    Djouadi, A.: Eur. Phys. J. C 73, 2498 (2013) doi:10.1140/epjc/s10052-013-2498-3. arXiv:1208.3436 [hep-ph]Google Scholar
  169. 169.
    Djouadi, A.: (2005). hep-ph/0503172; (2012). ArXiv:1203.4199Google Scholar
  170. 170.
    Djouadi, A., Moreau, G., Richard, F.: (2006). hep-ph/0610173Google Scholar
  171. 171.
    Dokshitzer, Yu.L.: Sov. Phys. JETP 46, 641 (1977)ADSGoogle Scholar
  172. 172.
    Dokshitzer, Yu.: J. Phys. G 17, 1572 (1991); Brown, N., Stirling, W.J.: Z. Phys. C 53, 629 (1992)ADSGoogle Scholar
  173. 173.
    Dokshitzer, Y.L., Khoze, V.A.: Basics of Perturbative QCD. Frontieres, Gif-sur-Yvette (1991); Muta, T.: Foundation of Quantum Chromodynamics: An Introduction to Perturbative Methods in Gauge Theories, 3rd edn. World Scientific, Singapore (2009); Yndurain, F.J.: The Theory of Quark and Gluon Interactions, 4th edn. Springer, Berlin (2006); Altarelli, G.: The Development of Perturbative QCD. World Scientific, Singapore (1994); Greiner, W., Schramm, S., Stein, E.: Quantum Chromodynamics, 3rd edn. Springer, Berlin (2007); Shifman, M. (eds.): At the Frontier of Particle Physics: Handbook of QCD, vols. 1–4. World Scientific, Singapore (2001); Ellis, R.K., Stirling, W.J., Webber, B.R.: QCD and Collider Physics. Cambridge Monographs. Cambridge University Press, Cambridge (2003); Dissertori, G., Knowles, I., Schmelling, M.: Quantum Chromodynamics: High Energy Experiments and Theory. Oxford University Press, Oxford (2003); Narison, S.: QCD as a Theory of Hadrons: From Partons to Confinement. Cambridge Monographs. Cambridge University Press, Cambridge (2007); Collins, J.C.: Foundations of Perturbative QCD. Cambridge Monographs. Cambridge University Press, Cambridge (2011)Google Scholar
  174. 174.
    Dokshitzer, Y.L., Leder, G.D., Moretti, S., Webber, B.R.: JHEP 9708, 001 (1997). hep-ph/9707323; Wobisch, M., Wengler, T.: hep-ph/9907280ADSCrossRefGoogle Scholar
  175. 175.
    Dolgov, A.D.: Phys. Rep. 370, 333 (2002). hep-ph/0202122; Lesgourgues, J., Pastor, S.: Phys. Rep. 429, 307–379 (2006); ArXiv:astro-ph/0603494; ArXiv:1212.6154Google Scholar
  176. 176.
    Donoghue, J.F., Golowich, E., Holstein, B.: Dynamics of the Standard Model. Cambridge University Press, Cambridge (1992)zbMATHCrossRefGoogle Scholar
  177. 177.
    Douglas, M.R.: (2006). hep-th/0602266; (2012). ArXiv:1204.6626Google Scholar
  178. 178.
    Eichten, E., Hill, B.: Phys. Lett. B 234, 511 (1990); Georgi, H.: Phys. Lett. B 240, 447 (1990); Grinstein, B.: Nucl. Phys. B 339, 253 (1990); Mannel, T., Roberts, W., Ryzak, Z.: Nucl. Phys. B 368, 204 (1992)Google Scholar
  179. 179.
    Ellis, J., Hwang, D.S.: JHEP 1209, 071 (2012). ArXiv:1202.6660; Ellis, J., Hwang, D.S., Sanz, V., You, T.: ArXiv:1208.6002; ArXiv:1210.5229ADSCrossRefGoogle Scholar
  180. 180.
    Ellis, R.K., Sexton, J.C.: Nucl. Phys. B 269, 445 (1986); Aversa, F., Chiappetta, P., Greco, M., Guillet, J.P.: Nucl. Phys. B 327, 105 (1989); Phys. Rev. Lett. 65, 401 (1990); Ellis, S.D., Kunszt, Z., Soper, D.E.: Phys. Rev. Lett. 64, 2121 (1990); Kunszt, Z., Soper, D.E.: Phys. Rev. D 46, 192 (1992); Giele, W.T., Glover, E.W.N., Kosower, D.A.: Nucl. Phys. B 403, 633 (1993)Google Scholar
  181. 181.
    Ellis, J., Gaillard, J.M., Nanopoulos, D.V.: Nucl. Phys. B 106, 292 (1976); Bjorken, J.D.: SLAC Report 198 (1976)Google Scholar
  182. 182.
    Ellis, J., Gaillard, M., Nanopoulos, D.: Nucl. Phys. B 106, 292 (1976)ADSCrossRefGoogle Scholar
  183. 183.
    Ellis, R.K., Martinelli, G., Petronzio, R.: Nucl. Phys. B 211, 106 (1983); Arnold, P., Reno, M.H.: Nucl. Phys. B 319, 37 (1989); erratum B 330, 284 (1990); Arnold, P., Ellis, R.K., Reno, M.H.: Phys. Rev. D 40, 912 (1989); Gonsalves, R., Pawlowski, J., Wai, C.F.: Phys. Rev. D 40, 2245 (1989)Google Scholar
  184. 184.
    Ellis, R.K., Giele, W.T., Zanderighi, G.: Phys. Rev. D 72, 054018 (2005); erratum D 74, 079902 (2006); Campbell, J.M., Ellis, R.K., Zanderighi, G.: JHEP 0610, 028 (2006); Anastasiou, C., et al.: JHEP 0701, 082 (2007); Davatz, G., et al.: JHEP 0607, 037 (2006); Catani, S., Grazzini, M.: Phys. Rev. Lett. 98, 222002 (2007); Anastasiou, C., Dissertori, G., Stoeckli, F.: ArXiv:0707.2373; Balázs, C., Berger, E.L., Nadolsky, P.M., Yuan, C.-P.: hep-ph/0702003. ArXiv:0704.0001; Ciccolini, M., Denner, A., Dittmaier, S.: ArXiv:0707.0381; Furlan, E.: JHEP 1110, 115 (2011). ArXiv:1106.4024; Anastasiou, C., et al.: ArXiv:1202.3638; Bonvini, M., Forte, S., Ridolfi, G.: Phys. Rev. Lett. 109, 102002 (2012). ArXiv:1204.5473; Grazzini, M., Sargsyan, H.: JHEP (2013). ArXiv:1306.4581Google Scholar
  185. 185.
    Ellis, S.D., et al.: Prog. Part. Nucl. Phys. 60, 484 (2008). ArXiv:0712.2447ADSCrossRefGoogle Scholar
  186. 186.
    Ellis, R.K., Giele, W.T., Kunszt, Z., Melnikov, K.: Nucl. Phys. B 822, 270 (2009). ArXiv:0806.3467Google Scholar
  187. 187.
    Ellis, R.K., Melnikov, K., Zanderighi, G.: Phys. Rev. D 80, 094002 (2009). ArXiv:0906;1445; JHEP 0904, 077 (2009). ArXiv:0901.4101; Melnikov, K., Zanderighi, G.: Phys. Rev. D 81, 074025 (2010). ArXiv:0910.3671; Berger, F., et al.: Phys. Rev. Lett. 102, 222001 (2009). ArXiv:0902.2760; Berger, C.F., et al.: Phys. Rev. D 80, 074036 (2009). ArXiv:0907.1984Google Scholar
  188. 188.
    Ellis, R.K., Kunszt, Z., Melnikov, K., Zanderighi, G.: Phys. Rep. 518, 141 (2012). ArXiv:1105.4319ADSMathSciNetCrossRefGoogle Scholar
  189. 189.
    Englert, F., Brout, R.: Phys. Rev. Lett. 13, 321 (1964); Higgs, P.W.: Phys. Lett. 12, 132 (1964); Phys. Rev. Lett. 13, 508 (1964)ADSMathSciNetCrossRefGoogle Scholar
  190. 190.
    Ermolaev, B.I., Greco, M., Troyan, S.I.: (2009). ArXiv:0905.2841 and references thereinGoogle Scholar
  191. 191.
    Fadin, V.S., Lipatov, L.N.: Phys. Lett. B 429, 127 (1998); B 429, 127 (1998); Fadin, V.S., et al.: Phys. Lett. B 359, 181 (1995); B 387, 593 (1996); Nucl. Phys. B 406, 259 (1993); Phys. Rev. D 50, 5893 (1994); Phys. Lett. B 389, 737 (1996); Nucl. Phys. B 477, 767 (1996); Phys. Lett. B 415, 97 (1997); B 422, 287 (1998); Camici, G., Ciafaloni, M.: Phys. Lett. B 412, 396 (1997); Phys. Lett. B 430, 349 (1998); del Duca, V., Phys. Rev. D 54, 989 (1996); D 54, 4474 (1996); del Duca, V., Schmidt, C.R.: Phys. Rev. D 57, 4069 (1998); Bern, Z., del Duca, V., Schmidt, C.R.: Phys. Lett. B 445, 168 (1998)Google Scholar
  192. 192.
    Fahri, E.: Phys. Rev. Lett. 39, 1587 (1977)ADSCrossRefGoogle Scholar
  193. 193.
    Faisst, M., Kuhn, J.H., Seidensticker, T., Veretin, O.: Nucl. Phys. B 665, 649 (2003). hep-ph/0302275; Chetyrkin, K.G., Faisst, M., Kuhn, J.H., Maierhofer, P., Sturm, C.: Phys. Rev. Lett. 97, 102003 (2006). hep-ph/0605201; Boughezal, R., Czakon, M.: Nucl. Phys. B 755, 221 (2006). hep-ph/0606232Google Scholar
  194. 194.
    Falkowski, A., Riva, F., Urbano, A.: JHEP 1311, 111 (2013). arXiv:1303.1812 [hep-ph]ADSCrossRefGoogle Scholar
  195. 195.
    Farina, M., Pappadopulo, D., Strumia, A.: (2013). ArXiv:1303.7244Google Scholar
  196. 196.
    Fayet, P.: Nucl. Phys. B 90, 104 (1975); Ellwanger, U., Hugonie, C., Teixeira, A.M.: Phys. Rep. 496, 1 (2010). ArXiv:0910.1785; Maniatis, M.: Int. J. Mod. Phys. A 25, 3505 (2010); Hall, L.J., Pinner, D., Ruderman, J.T.: JHEP 1204, 131 (2012). ArXiv:1112.2703; Agashe, K., Cui, Y., Franceschini, R.: JHEP 1302, 031 (2013). ArXiv:1209.2115; Athron, P., Binjonaid, M., King, S.F.: ArXiv:1302.5291; Barbieri, R., et al.: ArXiv:1307.4937Google Scholar
  197. 197.
    Feynman, R.: Acta Phys. Pol. 24, 697 (1963); De Witt, B.: Phys. Rev. 162, 1195, 1239 (1967); Faddeev, L.D., Popov, V.N.: Phys. Lett. B 25, 29 (1967)Google Scholar
  198. 198.
    Floratos, E.G., Ross, D.A., Sachrajda, C.T.: Nucl. Phys. B 129, 66 (1977); B 139, 545 (1978); B 152, 493 (1979); Gonzales-Arroyo, A., Lopez, C., Yndurain, F.J.: Nucl. Phys. B 153, 161 (1979); Curci, G., Furmanski, W., Petronzio, R.: Nucl. Phys. B 175, 27 (1980); Furmanski, W., Petronzio, R.: Phys. Lett. B 97, 438 (1980); Floratos, E.G., Lacaze, R., Kounnas, C.: Phys. Lett. B 99, 89, 285 (1981); Herrod, R.T., Wada, S.: Z. Phys. C 9, 351 (1981)Google Scholar
  199. 199.
    Fodor, Z., Katz, S.D.: arXiv:0908.3341; Acta Phys. Polon. B 42, 2791 (2009)Google Scholar
  200. 200.
    Fogli, G., et al.: (2012). ArXiv:1205.5254Google Scholar
  201. 201.
    Forero, D., Tortola, M., Valle, J.: (2012). ArXiv:1205.4018Google Scholar
  202. 202.
    Forte, S., Ridolfi, G.: Nucl. Phys. B 650, 229 (2003); Moch, S., Vermaseren, J.A.M., Vogt, A.: Nucl. Phys. B 726, 317 (2005). hep-ph/0506288; Becher, T., Neubert, M., Pecjak, B.D.: JHEP 0701, 076 (2007). hep-ph/0607228Google Scholar
  203. 203.
    Frederix, R., et al.: Phys. Lett. B 701, 427 (2011). ArXiv:1104.5613Google Scholar
  204. 204.
    Freitas, A., Huang, Y.-C.: JHEP 1208, 050 (2012); erratum ibid 1305, 074 (2013); 1310, 044 (2013). ArXiv:1205.0299Google Scholar
  205. 205.
    Frieman, J., Turner, M., Huterer, D.: Annu. Rev. Astron. Astrophys. 46, 385 (2008). ArXiv:0803.0982ADSCrossRefGoogle Scholar
  206. 206.
    Frixione, S., Webber, B.R.: JHEP 06, 029 (2002). hep-ph/0204244; hep-ph/0612272 and references thereinADSCrossRefGoogle Scholar
  207. 207.
    Fujikawa, K., Lee, B.W., Sanda, A.: Phys. Rev. D 6, 2923 (1972); Yao, Y.P.: Phys. Rev. D 7, 1647 (1973)Google Scholar
  208. 208.
    Fukaya, H., et al.: Phys. Rev. D 83, 074501 (2011). ArXiv:1012.4052Google Scholar
  209. 209.
    Gaillard, M.K., Lee, B.W.: Phys. Rev. Lett. 33, 108 (1974); Altarelli, G., Maiani, L.: Phys. Lett. B 52, 351 (1974)ADSCrossRefGoogle Scholar
  210. 210.
    Gambino, P.: Int. J. Mod. Phys. A 19, 808 (2004). hep-ph/0311257Google Scholar
  211. 211.
    Gardi, E., Korchemsky, G.P., Ross, D.A., Tafat, S.: Nucl. Phys. B 636, 385 (2002). hep-ph/0203161; Gardi, E., Roberts, R.G.: Nucl. Phys. B 653, 227 (2003). hep-ph/0210429Google Scholar
  212. 212.
    Gasser, J.: Nucl. Phys. Proc. Suppl. 86, 257 (2000). hep-ph/9912548ADSCrossRefGoogle Scholar
  213. 213.
    Gehrmann-De Ridder, A., Gehrmann, T., Glover, E.W.N., Heinrich, G.: Phys. Rev. Lett. 99, 132002 (2007). ArXiv:0707.1285; JHEP 0711, 058 (2007). ArXiv:0710.0346; Phys. Rev. Lett. 100, 172001 (2008). ArXiv:0802.0813; JHEP 0712, 094 (2007). ArXiv:0711.4711ADSCrossRefGoogle Scholar
  214. 214.
    Gell-Mann, M.: Phys. Lett. 8, 214 (1964); Zweig, G.: CERN TH 401 and 412 (1964); Greenberg, O.W.: Phys. Rev. Lett. 13, 598 (1964)ADSCrossRefGoogle Scholar
  215. 215.
    Gell-Mann, M.: Acta Phys. Austriaca Suppl. IX, 733 (1972); Fritzsch, H., Gell-Mann, M.: In: Proceedings of XVI International Conference on High Energy Physics, Chicago-Batavia (1972); Fritzsch, H., Gell-Mann, M., Leutwyler, H.: Phys. Lett. B 47, 365 (1973)Google Scholar
  216. 216.
    Georgi, H., Glashow, S., Machacek, M., Nanopoulos, D.: Phys. Rev. Lett. 40, 692 (1978)ADSCrossRefGoogle Scholar
  217. 217.
    Georgiou, G., Khoze, V.V.: JHEP 0405, 015 (2004); Wu, J.B., Zhu, C.J.: JHEP 0409, 063 (2004); Georgiou, G., Glover, E.W.N., Khoze, V.V.: JHEP 0407, 048 (2004)Google Scholar
  218. 218.
    Giardino, P.P., Kannike, K., Masina, I., Raidal, M., Strumia, A.: JHEP 1405, 046 (2014). arXiv:1303.3570 [hep-ph]ADSCrossRefGoogle Scholar
  219. 219.
    Gildener, E.: Phys. Rev. D 14, 1667 (1976); Gildener, E., Weinberg, S.: Phys. Rev. D 13, 3333 (1976); Maiani, L.: In: Davier, M., et al. (eds.) Proceedings of the Summer School on Particle Physics, Gif-sur-Yvette, 3–7 September 1979, IN2P3, Paris (1979); t’Hooft, G.: In: Proceedings of the 1979 Cargese Institute on Recent Developments in Gauge Theories, p. 135. Plenum, New York (1980); Veltman, M.: Acta Phys. Polon. B 12, 437 (1981); Witten, E.: Nucl. Phys. B 188, 513 (1981); Phys. Lett. B 105, 267 (1981)Google Scholar
  220. 220.
    Giudice, G.: In: Kane, G., Pierce, A.: Perspectives on LHC Physics. World Scientific, Singapore (2008). ArXiv:0801.2562Google Scholar
  221. 221.
    Giudice, G.: Proceedings of EPS-HEP Conference, Stockholm (2013). ArXiv:1307.7879Google Scholar
  222. 222.
    Giudice, G.F., Grojean, C., Pomarol, A., Rattazzi, R.: JHEP 0706, 045 (2007). hep-ph/0703164; Alonso, R., et al.: ArXiv:1212.3305; Contino, R., et al.: ArXiv:1303.3876; Pomarol, A., Riva, F.: ArXiv:1308.2803 and references thereinADSCrossRefGoogle Scholar
  223. 223.
    Giudice, G.F., Grojean, C., Pomarol, A., Rattazzi, R.: JHEP 0706, 045 (2007). hep-ph/0703164; Csaki, C., Falkowski, A., Weiler, A.: JHEP 0809, 008 (2008). ArXiv:0804.1954; Contino, R.: ArXiv:1005.4269; Barbieri, R., et al.: ArXiv:1211.5085; Keren-Zur, B., et al.: Nucl. Phys. B 867, 429 (2013). ArXiv:1205.5803ADSCrossRefGoogle Scholar
  224. 224.
    Giudice, G.F., Rattazzi, R., Strumia, A.: Phys. Lett. B 715, 142 (2012). ArXiv:1204.5465Google Scholar
  225. 225.
    Glashow, S.L.: Nucl. Phys. 22, 579 (1961)CrossRefGoogle Scholar
  226. 226.
    Glashow, S.L., Iliopoulos, J., Maiani, L.: Phys. Rev. 96, 1285 (1970)Google Scholar
  227. 227.
    Gleisberg, T., et al.: JHEP 0402, 056 (2004). hep-ph/0311263; Gleisberg, T., et al.: JHEP 0902, 007 (2009). ArXiv:0811.4622; Krauss, F., Kuhn, R., Soff, G.: JHEP 0202, 044 (2002). hep-ph/0109036; Gleisberg, T., Krauss, F.: Eur. Phys. J. C 53, 501 (2008). ArXiv:0709.2881ADSCrossRefGoogle Scholar
  228. 228.
    Goldstone, J.: Nuovo Cim. 19, 154 (1961); Goldstone, J., Salam, A., Weinberg, S.: Phys. Rev. 127, 965 (1962)Google Scholar
  229. 229.
    Gonzalez-Garcia, M.C., Maltoni, M., Salvado, J., Schwetz, T.: (2012). ArXiv:1209.3023Google Scholar
  230. 230.
    Gorishny, S.G., Kataev, A.L., Larin, S.A.: Phys. Lett. B 259, 144 (1991); Surguladze, L.R., Samuel, M.A.: Phys. Rev. Lett. 66, 560 (1991)Google Scholar
  231. 231.
    Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  232. 232.
    Greiner, N., Guffanti, A., Reiter, T., Reuter, J.: Phys. Rev. Lett. 107, 102002 (2011). ArXiv:1105.3624ADSCrossRefGoogle Scholar
  233. 233.
    Gribov, V.N., Lipatov, L.N.: Sov. J. Nucl. Phys. 15, 438 (1972)Google Scholar
  234. 234.
    Gross, D., Wilczek, F.: Phys. Rev. Lett. 30, 1343 (1973); Phys. Rev. D 8, 3633 (1973); Politzer, H.D.: Phys. Rev. Lett. 30, 1346 (1973)ADSCrossRefGoogle Scholar
  235. 235.
    Grunewald, M., Gurtu, A., Particle Data Group, Beringer, J., et al.: Phys. Rev. D 86, 010001 (2012)Google Scholar
  236. 236.
    Guralnik, G.S., Hagen, C.R., Kibble, T.W.B.: Phys. Rev. Lett. 13, 585 (1964)ADSCrossRefGoogle Scholar
  237. 237.
    H1 Collaboration: Eur. Phys. J. C 71, 1579 (2011). ArXiv:1012.4355Google Scholar
  238. 238.
    Haber, H.E., Kane, G., Dawson, S., Gunion, J.F.: The Higgs Hunter’s Guide. Westview, Boulder, CO (1990)Google Scholar
  239. 239.
    Hagiwara, K., Liao, R., Martin, A.D., Nomura, D., Teubner, T.: J. Phys. G 38, 085003 (2011)ADSCrossRefGoogle Scholar
  240. 240.
    Hamberg, R., van Neerven, W.L., Matsuura, T.: Nucl. Phys. B 359, 343 (1991); erratum ibid. B 644, 403–404 (2002); van Neerven, W.L., Zijlstra, E.B.: Nucl. Phys. B 382, 11 (1992)Google Scholar
  241. 241.
    Hambye, T., Riesselmann, K.: Phys. Rev. D 55, 7255 (1997)ADSCrossRefGoogle Scholar
  242. 242.
    Hanneke, D., Fogwell, S., Gabrielse, G.: Phys. Rev. Lett. 100, 120801 (2008); Hanneke, D., Fogwell Hoogerheide, S., Gabrielse, G.: Phys. Rev. A 83, 052122 (2011)ADSCrossRefGoogle Scholar
  243. 243.
    Higgs, P.W.: Phys. Rev. 145, 1156 (1966)ADSMathSciNetCrossRefGoogle Scholar
  244. 244.
    Hirai, M., Kumano, S.: Nucl. Phys. B 813, 106–122 (2009). ArXiv:0808.0413Google Scholar
  245. 245.
    Hoecker, A., Marciano, W.: The Muon Anomalous Magnetic Moment, in [307]Google Scholar
  246. 246.
    Holdom, R.: Phys. Rev. D 24, 1441 (1981); Phys. Lett. B 150, 301 (1985); Appelquist, T., Karabali, D., Wijewardhana, L.C.R.: Phys. Rev. Lett. 57, 957 (1986); Appelquist, T., Wijewardhana, L.C.R.: Phys. Rev. D 36, 568 (1987); Yamawaki, K., Bando, M., Matumoto, K.: Phys. Rev. Lett. 56, 1335 (1986); Akiba, T., Yanagida, T.: Phys. (1986); Hill, C.T.: Phys. Lett. B 345, 483 (1995); Lane, K.D., Mrenna, S.: Phys. Rev. D 67, 115011 (2003)Google Scholar
  247. 247.
    Hollik, W., Pagani, D.: Phys. Rev. D 84, 093003 (2011). ArXiv:1107.2606; Kuhn, J.H., Rodrigo, G.: JHEP 1201, 063 (2012). ArXiv:1109.6830; Manohar, A.V., Trott, M.: Phys. Lett. B 711, 313 (2012). ArXiv:1201.3926; Bernreuther, W., Si, Z.-G.: ArXiv:1205.6580Google Scholar
  248. 248.
    Hurth, T., Mahmoudi, F.: Nucl. Phys. B 865, 461 (2012). ArXiv:1207.0688; Altmannshofer, W., Straub, D.M.: ArXiv:1308.1501; Gauld, R., Goertz, F., Haisch, U.: ArXiv:1308.1959; ArXiv:1310.1082; Buras, A.J., Girrbach, J.: ArXiv:1309.2466Google Scholar
  249. 249.
    Isidori, G., Ridolfi, G., Strumia, A.: Nucl. Phys. B 609, 387 (2001)ADSCrossRefGoogle Scholar
  250. 250.
    Itzykson, C., Zuber, J.: Introduction to Quantum Field Theory. McGraw-Hill, New York, (1980); Cheng, T.P., Li, L.F.: Gauge Theory of Elementary Particle Physics. Oxford University Press, New York (1984); Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Perseus Books, Cambridge, MA (1995); Weinberg, S.: The Quantum Theory of Fields, vols. I, II. Cambridge University Press, Cambridge, MA (1996); Zee, A.: Quantum Field Theory in a Nutshell. Princeton University Press, Princeton, NJ (2003)Google Scholar
  251. 251.
    Jarlskog, C.: Phys. Rev. Lett. 55, 1039 (1985)ADSCrossRefGoogle Scholar
  252. 252.
    Jegerlehner, F., Nyffeler, A.: Phys. Rep. 477, 1 (2009). ArXiv:0902.3360sADSCrossRefGoogle Scholar
  253. 253.
    Jezabek, M., Kuhn, J.H.: Nucl. Phys. B 314, 1 (1989)ADSCrossRefGoogle Scholar
  254. 254.
    Jimenez-Delgado, P., Reya, E.: Phys. Rev. D 79, 074023 (2009). ArXiv:0810.4274Google Scholar
  255. 255.
    Kaczmarak, O., Karsch, F., Laermann, E., Lutgemeier, M.: Phys. Rev. D 62, 034021 (2000). ArXiv:hep-lat/9908010Google Scholar
  256. 256.
    Kaczmarek, O., Karsch, F., Zantow, F., Petreczky, P.: Phys. Rev. D 70, 074505 (2004); erratum ibid. D 72, 059903 (2005); hep-lat/0406036 and references thereinGoogle Scholar
  257. 257.
    Kannike, K., et al.: (2011). ArXiv:1111.2551Google Scholar
  258. 258.
    Kaplan, D.B., Georgi, H.: Phys. Lett. B 136, 183 (1984); Dimopoulos, S., Preskill, J.: Nucl. Phys. B 199, 206 (1982); Banks, T.: Nucl. Phys. B 243, 125 (1984); Kaplan, D.B., Georgi, H., Dimopoulos, S.: Phys. Lett. B 136, 187 (1984); Georgi, H., Kaplan, D.B., Galison, P.: Phys. Lett. B 143, 152 (1984); Georgi, H., Kaplan, D.B.: Phys. Lett. B 145, 216 (1984); Dugan, M.J., Georgi, H., Kaplan, D.B.: Nucl. Phys. B 254, 299 (1985)Google Scholar
  259. 259.
    Kettell, S.: Talk at NuFact 2013. IHEP, Beijing (2013)Google Scholar
  260. 260.
    Khachatryan, V., et al., CMS Collaboration: Eur. Phys. J. C 75(5), 212 (2015). doi:10.1140/epjc/s10052-015-3351-7. arXiv:1412.8662 [hep-ex]Google Scholar
  261. 261.
    Kibble, T.W.B.: Phys. Rev. 155, 1554 (1967)ADSCrossRefGoogle Scholar
  262. 262.
    Kim, J.E., AIP Conf. Proc. 1200, 83 (2010). ArXiv:0909.3908Google Scholar
  263. 263.
    Kim, J.E., Carosi, G.: Rev. Mod. Phys. 82, 557 (2010). ArXiv:0807.3125ADSCrossRefGoogle Scholar
  264. 264.
    King, S.F., Luhn, C.: (2013). ArXiv:1301.1340Google Scholar
  265. 265.
    Kinoshita, T.: J. Math. Phys. 3, 650 (1962); Lee, T.D., Nauenberg, M.: Phys. Rev. 133, 1549 (1964)ADSCrossRefGoogle Scholar
  266. 266.
    Kinoshita, T., Nio, N.: Phys. Rev. D 73, 013003 (2006)ADSCrossRefGoogle Scholar
  267. 267.
    Klapdor-Kleingrothaus, H.V., Krivosheina, I.V.: Mod. Phys. Lett. A 21, 1547 (2006)ADSCrossRefGoogle Scholar
  268. 268.
    Kluth, S.: Rep. Prog. Phys. 69, 1771 (2006)ADSCrossRefGoogle Scholar
  269. 269.
    Kobayashi, M., Maskawa, T.: Prog. Theor. Phys. 49, 652 (1973)ADSCrossRefGoogle Scholar
  270. 270.
    Kramer, M., Laenen, E., Spira, M.: Nucl. Phys. B 511, 523 (1998)ADSCrossRefGoogle Scholar
  271. 271.
    Kribs, G.D., Martin, A., Menon, A.: ArXiv:1305.1313; Krizka, K., Kumar, A., Morrissey, D.E.: ArXiv:1212.4856; Auzzi, R., Giveon, A., Gudnason, S.B., Shacham, T.: JHEP 1301, 169 (2013). ArXiv:1208.6263; Espinosa, J.R., Grojean, C., Sanz, V., Trott, M.: JHEP 1212, 077 (2012). ArXiv:1207.7355; Han, Z., Katz, A., Krohn, D., Reece, M.: JHEP 1208, 083 (2012). ArXiv:1205.5808; Lee, H.M., Sanzand, V., Trott, M.: JHEP 1205, 139 (2012). ArXiv:1204.0802; Bai, Y., Cheng, H.-C., Gallicchio, J., Gu, J.: JHEP 1207, 110 (2012). ArXiv:1203.4813; Allanach, B., Gripaios, B.: JHEP 1205, 062 (2012). ArXiv:1202.6616; Larsen, G., Nomura, Y., Roberts, H.L.: JHEP 1206, 032 (2012). ArXiv:1202.6339; Buchmueller, O., Marrouche, J.: ArXiv:1304.2185; Bi, X.-J., Yan, Q.-S., Yin, P.-F.: Phys. Rev. D 85, 035005 (2012). ArXiv:1111.2250; Papucci, M., Ruderman, J.T., Weiler, A.: ArXiv:1110.6926; Brust, C., Katz, A., Lawrence, S., Sundrum, R., JHEP 1203, 103 (2012). ArXiv:1110.6670; Arganda, E., Diaz-Cruz, J.L., Szynkman, A.: Eur. Phys. J. C 73, 2384 (2013). ArXiv:1211.0163; Phys. Lett. B 722, 100 (2013). ArXiv:1301.0708; Cao, J., Han, C., Wu, L., Yang, J.M., Zhang, Y.: JHEP 1211, 039 (2012). ArXiv:1206.3865; Hardy, E.: ArXiv:1306.1534; Baer, H., et al.: ArXiv:1306.3148; ArXiv:1306.4183; ArXiv:1310.4858Google Scholar
  272. 272.
    Kronfeld, A.S.: Annu. Rev. Nucl. Part. Sci. 62, 265 (2012). doi:10.1146/annurev-nucl-102711-094942. arXiv:1203.1204 [hep-lat]ADSCrossRefGoogle Scholar
  273. 273.
    Kubar-Andre, J., Paige, F.: Phys. Rev. D 19, 221 (1979)ADSCrossRefGoogle Scholar
  274. 274.
    Kuchiev, M.Yu., Flambaum, V.V.: (2003). hep-ph/0305053Google Scholar
  275. 275.
    Lane, K.: (2002). hep-ph/0202255; Chivukula, R.S.: (2000). hep-ph/0011264Google Scholar
  276. 276.
    Langacker, P.: The Standard Model and Beyond. CRC, Boca Raton, FL (2010); Paschos, E.A.: Electroweak Theory. Cambridge University Press, Cambridge, 2007; Becchi, C.M., Ridolfi, G.: An Introduction to Relativistic Processes and the Standard Model of Electroweak Interactions. Springer, Berlin (2006); Horejsi, J.: Fundamentals of Electroweak Theory. Karolinum, Prague (2002); Barbieri, R.: Lectures on the ElectroWeak Interactions. Publications of the Scuola Normale Superiore, Pisa (2007)Google Scholar
  277. 277.
    Leader, E., Sidorov, A.V., Stamenov, D.B.: Phys. Rev. D 82, 114018 (2010). ArXiv:1010.0574Google Scholar
  278. 278.
    Lee, B.W., Zinn-Justin, J.: Phys. Rev. D 5, 3121, 3137 (1972); 7, 1049 (1973)Google Scholar
  279. 279.
    Lee, B.W., Quigg, C., Thacker, H.B.: Phys. Rev. D 16, 1519 (1977)ADSCrossRefGoogle Scholar
  280. 280.
    Lee, B.W., Pakvasa, S., Shrock, R., Sugawara, H.: Phys. Rev. Lett. 38, 937 (1977); Lee, B.W., Shrock, R.: Phys. Rev. D 16, 1444 (1977)ADSCrossRefGoogle Scholar
  281. 281.
    LEPEWWG, figures included with permission. See also Schael, S., et al.: ALEPH and DELPHI and L3 and OPAL and LEP Electroweak Collaborations. Phys. Rep. 532, 119 (2013). doi:10.1016/j.physrep.2013.07.004 arXiv:1302.3415 [hep-ex]Google Scholar
  282. 282.
    LHCb Collaboration: (2013). ArXiv:1309.6534 and PAPER-2013-054Google Scholar
  283. 283.
    Lipatov, L.N.: Sov. J. Nucl. Phys. 20, 94 (1975)Google Scholar
  284. 284.
    Lipatov, L.N.: Sov. J. Nucl. Phys. 23, 338 (1976); Fadin, V.S., Kuraev, E.A., Lipatov, L.N.: Phys. Lett. B 60, 50 (1975); Sov. Phys. JETP 44, 443 (1976); 45, 199 (1977); Balitski, Y.Y., Lipatov, L.N.: Sov. J. Nucl. Phys. 28, 822 (1978)Google Scholar
  285. 285.
    Lombard, V.: Moriond QCD (2013). ArXiv:1305.3773Google Scholar
  286. 286.
    Maiani, L.: Proceedings of International Symposium on Lepton and Photon Interactions at High Energy, Hamburg (1977)Google Scholar
  287. 287.
    Martin, A.D., Stirling, W.J., Thorne, R.S., Watt, G.: Eur. Phys. J. C 64, 653 (2009). ArXiv:0905.3531Google Scholar
  288. 288.
    McNeile, C., et al., [HPQCD Collab.]: Phys. Rev. D 82, 034512 (2010) ArXiv:1004.4285; Davies, C.T.H., et al., [HPQCD Collab., UKQCD Collab., and MILC Collab.]: Phys. Rev. Lett. 92, 022001 (2004). ArXiv:0304004; Mason, Q., et al., [HPQCD and UKQCD Collaborations]: Phys. Rev. Lett. 95, 052002 (2005). hep-lat/0503005; Maltman, K., et al.: Phys. Rev. D 78, 114504 (2008). ArXiv:0807.2020; Aoki, S., et al., [PACS-CS Collab.]: JHEP 0910, 053 (2009). ArXiv:0906.3906; Shintani, E., et al., [JLQCD Collab.]: Phys. Rev. D 82, 074505 (2010). ArXiv:1002.0371; Blossier, B., et al., [ETM Collab.]: ArXiv:1201.5770Google Scholar
  289. 289.
    Melia, T., Melnikov, K., Rontsch, R., Zanderighi, G., JHEP 1012, 053 (2010). ArXiv:1007.5313; Phys. Rev. D 83, 114043 (2011). ArXiv:1104.2327; Greiner, N., et al.: Phys. Lett. B 713, 277 (2012). ArXiv:1202.6004; Denner, A., Hosekova, L., Kallweit, S. ArXiv:1209.2389; Jager, B., Zanderighi, G.: JHEP 1111, 055 (2011). ArXiv:1108.0864ADSCrossRefGoogle Scholar
  290. 290.
    Melnikov, K., Vainshtein, A.: Phys. Rev. D 70, 113006 (2004); Bijnens, J., Prades, J.: Mod. Phys. Lett. A 22, 767 (2007); Prades, J., de Rafael, E., Vainshtein, A.: In: Roberts, B.L., Marciano, W.J. (eds.) Lepton Dipole Moments, pp. 303–319. World Scientific, Singapore, (2009); Nyffeler, A.: Phys. Rev. D 79, 073012 (2009)Google Scholar
  291. 291.
    Minkowski, P.: Phys. Lett. B 67, 421 (1977); Yanagida, T.: In: Proceedings of the Workshop on Unified Theory and Baryon Number in the Universe, KEK (1979); Glashow, S.L.: In: Levy, M., et al. (eds.) Quarks and Leptons. Cargèse. Plenum, New York (1980); Gell-Mann, M., Ramond, P., Slansky, R.: In: Supergravity. Stony Brook, New York (1979); Mohapatra, R.N., Senjanovic, G.: Phys. Rev. Lett. 44, 912 (1980)Google Scholar
  292. 292.
    Moch, S., Vermaseren, J.A.M., Vogt, A.: Nucl. Phys. B 688, 101 (2004). hep-ph/0403192; Vogt, A., Moch, S., Vermaseren, J.A.M.: Nucl. Phys. B 691, 129 (2004). hep-ph/0404111; Vermaseren, J.A.M., Vogt, A., Moch, S.: Nucl. Phys. B 724, 3 (2005). hep-ph/0504242Google Scholar
  293. 293.
    Moch, S., Vermaseren, J.A.M., Vogt, A.: Phys. Lett. B 606, 123 (2005). hep-ph/0411112Google Scholar
  294. 294.
    Moretti, S., Lonnblad, L., Sjostrand, T.: JHEP 9808, 001 (1998). hep-ph/9804296ADSCrossRefGoogle Scholar
  295. 295.
    Mrazek, J., Wulzer, A.: Phys. Rev. D 81, 075006; Dissertori, G., et al., JHEP 1009, 019 (2010). ArXiv:1005.4414; Contino, R., Servant, G.: JHEP 06, 026 (2008). ArXiv:0801.1679; Vignaroli, N., JHEP 1207, 158 (2012). ArXiv:1204.0468; De Simone, A., et al.: ArXiv:1211.5663; Buchkremer, M., Cacciapaglia, G., Deandrea, A., Panizzi, L.: ArXiv:1305.4172; Grojean, C., Matsedonskyi, O., Panico, G.: ArXiv:1306.4655Google Scholar
  296. 296.
    Muller, B., Schukraft, J., Wyslouch, B.: Annu. Rev. Nucl. Part. Sci. 62, 361 (2012). doi:10.1146/annurev-nucl-102711-094910. arXiv:1202.3233 [hep-ex]ADSCrossRefGoogle Scholar
  297. 297.
    Muon g-2 Collab., Bennett, G.W., et al.: Phys. Rev. D 73, 072003 (2006); Roberts, B.L.: Chin. Phys. C 34, 741 (2010)Google Scholar
  298. 298.
    Narison, S.: QCD Spectral Sum Rules. World Scientific Lecture Notes in Physics, vol. 26, p. 1. World Scientific, Singapore (1989)Google Scholar
  299. 299.
    Nason, P.: JHEP 0411, 040 (2004). hep-ph/0409146; Frixione, S., Nason, P., Oleari, C.: JHEP 11, 070 (2007). ArXiv:0709.2092; Alioli, S., Nason, P., Oleari, C., Re, E.: JHEP 1006, 043 (2010). ArXiv:1002.2581ADSCrossRefGoogle Scholar
  300. 300.
    Nason, P., Dawson, S., Ellis, R.K.: Nucl. Phys. B 303, 607 (1988); Beenakker, W., Kuijf, H., van Neerven, W.L., Smith, J.: Phys. Rev. D 40, 54 (1989); Nason, P., Dawson, S., Ellis, R.K.: Nucl. Phys. B 327, 49 (1989); erratum ibid. B 335, 260 (1989); Mangano, M.L., Nason, P., Ridolfi, G.: Nucl. Phys. B 373, 295 (1992)Google Scholar
  301. 301.
    Neubert, M.: Phys. Rep. 245, 259 (1994). hep-ph/9306320; Manohar, A.V., Wise, M.B.: Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 10, 1 (2000)ADSCrossRefGoogle Scholar
  302. 302.
    Nilles, H.P.: Phys. Rep. C 110, 1 (1984); Haber, H.E., Kane, G.L.: Phys. Rep. C 117, 75 (1985); Barbieri, R.: Riv. Nuovo Cim. 11, 1 (1988); Martin, S.P.: hep-ph/9709356; Drees, M., Godbole, R., Roy, P.: Theory and Phenomenology of Sparticles. World Scientific, Singapore (2004); Aitchinson, I.: Supersymmetry in Particle Physics: An Elementary Introduction. Cambridge University Press, Cambridge (2007)Google Scholar
  303. 303.
    NNPDF Collaboration, Ball, R.D., et al.: Nucl. Phys. B 874, 36 (2013). ArXiv:1303.7236Google Scholar
  304. 304.
    Ossola, G., Papadopoulos, C.G., Pittau, R.: Nucl. Phys. B 763, 147 (2007). hep-ph/0609007; JHEP 0803, 042 (2008). ArXiv:0711.3596; JHEP 0805, 004 (2008). ArXiv:0802.1876Google Scholar
  305. 305.
    Parke, S.J., Taylor, T.R.: Phys. Rev. Lett. 56, 2459 (1986)ADSCrossRefGoogle Scholar
  306. 306.
    Particle Data group, Yao, W.-M., et al.: J. Phys. G 33, 1 (2006)Google Scholar
  307. 307.
    Particle Data Group, Beringer, J., et al.: Phys. Rev. D 86, 010001 (2012)Google Scholar
  308. 308.
    Peccei, R.D.: In: Jarlskog, C. (ed.) CP Violation. Advanced Series on Directions in High Energy Physics, p. 503. World Scientific, Singapore (1989)Google Scholar
  309. 309.
    Peccei, R.D., Quinn, H.R.: Phys. Rev. Lett. 38, 1440 (1977); Phys. Rev. D 16, 1791 (1977); Weinberg, S.: Phys. Rev. Lett. 40, 223 (1978); Wilczek, F.: Phys. Rev. Lett. 40, 279 (1978)ADSCrossRefGoogle Scholar
  310. 310.
    Peskin, M.E., Takeuchi, T.: Phys. Rev. Lett. 65, 964 (1990); Phys. Rev. D 46, 381 (1991)ADSCrossRefGoogle Scholar
  311. 311.
    Pich, A.: (2013). ArXiv:1303.2262Google Scholar
  312. 312.
    Pontecorvo, B., Sov. Phys. JETP 6, 429 (1957); Zh. Eksp. Teor. Fiz. 33, 549 (1957); Maki, Z., Nakagawa, M., Sakata, S.: Prog. Theor. Phys. 28, 870 (1962); Sov. Phys. JETP 26, 984 (1968); Zh. Eksp. Teor. Fiz. 53, 1717 (1968); Gribov, V.N., Pontecorvo, B.: Phys. Lett. B 28, 493 (1969)Google Scholar
  313. 313.
    Quigg, C.: Annu. Rev. Nucl. Part. Sci. 59, 505 (2009). ArXiv:0905.3187ADSCrossRefGoogle Scholar
  314. 314.
    Randall, L., Sundrum, R.: Phys. Rev. Lett. 83, 3370 (1999); 83, 4690 (1999); Goldberger, W.D., Wise, M.B.: Phys. Rev. Lett. 83, 4922 (1999)ADSMathSciNetCrossRefGoogle Scholar
  315. 315.
    Ross, G.G.: Grand Unified Theories. Benjamin, Reading, MA (1985); Mohapatra, R.N.: Unification and Supersymmetry. Springer, Berlin (1986); Raby, S.: in [307]; Masiero, A., Vempati, S.K., Vives, O.: ArXiv:0711.2903Google Scholar
  316. 316.
    Salam, A.: In: Svartholm, N. (ed.) Elementary Particle Theory, p. 367. Almquist and Wiksells, Stockholm (1969)Google Scholar
  317. 317.
    Salam, G.P.: Eur. Phys. J. C 67, 637 (2010). ArXiv:0906.1833Google Scholar
  318. 318.
    Schellekens, A.N.: (2013). ArXiv:1306.5083Google Scholar
  319. 319.
  320. 320.
    See, for example, Lindner, M.: Z. Phys. 31, 295 (1986); Hambye, T., Riesselmann, K.: Phys. Rev. D55, 7255 (1997). hep-ph/9610272Google Scholar
  321. 321.
    See, for example, Kamenik, J.F., Shu, J., Zupan, J.: (2011). ArXiv:1107.5257; Drobnak, J., et al.: (2012). ArXiv:1209.4872Google Scholar
  322. 322.
    Shaposhnikov, M.: (2007). ArXiv:0708.3550; Canetti, L., Drewes, M., Shaposhnikov, M.: (2012). ArXiv:1204.3902; Canetti, L., Drewes, M., Frossard, T., Shaposhnikov, M.: (2012). ArXiv:1208.4607Google Scholar
  323. 323.
    Sher, M.: Phys. Rep. 179, 273 (1989); Phys. Lett. B 317, 159 (1993)ADSCrossRefGoogle Scholar
  324. 324.
    Shifman, M.: (2013). ArXiv:1310.1966 and references thereinGoogle Scholar
  325. 325.
    Shifman, M., Vainshtein, A., Zakharov, V.: Nucl. Phys. B 147, 385 (1979)ADSCrossRefGoogle Scholar
  326. 326.
    Sissakian, A., Shevchenko, O., Ivanov, O.: Eur. Phys. J. C 65, 413 (2010). ArXiv:0908.3296Google Scholar
  327. 327.
    Stelzer-Chilton, O.: Hadron Collider Physics Symposium, Kyoto (2012); see also supplement to Phys. Rev. Lett. 112, 191802 (2014)ADSCrossRefGoogle Scholar
  328. 328.
    Stenzel, H.: JHEP 0507, 0132 (2005)Google Scholar
  329. 329.
    Sterman, G.: Nucl. Phys. B 281, 310 (1987); Catani, S., Trentadue, L.: Nucl. Phys. B 327, 323 (1989); Kidonakis, N., Sterman, G.: Nucl. Phys. B 505, 321 (1997)Google Scholar
  330. 330.
    Sudakov, V.V.: Sov. Phys. JETP 3, 75 (1956); Dokshitzer, Yu., Dyakonov, D., Troyan, S.: Phys. Lett. B 76, 290 (1978); Phys. Rep. 58, 269 (1980); Parisi, G., Petronzio, R.: Nucl. Phys. B 154, 427 (1979); Curci, G., Greco, M., Srivastava, Y.: Phys. Rev. Lett. 43 (1979); Nucl. Phys. B 159, 451 (1979); Collins, J., Soper, D.: Nucl. Phys. B 139, 381 (1981); B 194, 445 (1982); B 197, 446 (1982); Kodaira, J., Trentadue, L.: Phys. Lett. B 112, 66 (1982); B 123, 335 (1983); Collins, J., Soper, D., Sterman, G.: Nucl. Phys. B 250, 199 (1985); Davies, C., Webber, B., Stirling, J.: Nucl. Phys. B 256, 413 (1985)Google Scholar
  331. 331.
    Sundrum, R.: (2005). hep-th/0508134; Rattazzi, R.: (2006). hep-ph/ 0607055Google Scholar
  332. 332.
    Susskind, L.: Phys. Rev. D 20, 2619 (1979); Dimopoulos, S., Susskind, L.: Nucl. Phys. B 155, 237 (1979); Eichten, E., Lane, K.D., Phys. Lett. B 90, 125 (1980)Google Scholar
  333. 333.
    ’t Hooft, G., Veltman, M.: Nucl. Phys. B 44, 189 (1972); Bollini, C.G., Giambiagi, J.J.: Nuovo Cimento 12B 20 (1972); Ashmore, J.F.: Nuovo Cimento Lett. 4, 289 (1972); Cicuta, G.M., Montaldi, E.: Nuovo Cimento Lett. 4, 329 (1972)Google Scholar
  334. 334.
    ’t Hooft, G., Veltman, M.: Nucl. Phys. B 44, 189 (1972); Bollini, C.G., Giambiagi, J.J.: Phys. Lett. B 40, 566 (1972); Ashmore, J.F.: Nuovo Cim. Lett. 4, 289 (1972); Cicuta, G.M., Montaldi, E.: Nuovo Cim. Lett. 4, 329 (1972)Google Scholar
  335. 335.
    ’t Hooft, G.: Nucl. Phys. B 61, 455 (1973)Google Scholar
  336. 336.
    ’t Hooft, G.: Phys. Rev. Lett. 37, 8 (1976); Phys. Rev. D 14, 3432 (1976); erratum ibid. D 18, 2199 (1978)Google Scholar
  337. 337.
    Tarasov, O.V., Vladimirov, A.A., Zharkov, A.Yu.: Phys. Lett. B 93, 429 (1980)ADSCrossRefGoogle Scholar
  338. 338.
    Tenchini, R., Verzegnassi, C.: The Physics of the Z and W Bosons. World Scientific, Singapore (2008)Google Scholar
  339. 339.
    The ATLAS Collaboration: ATLAS-CONF-2012-024, 031,097,134 and 149 (2012)Google Scholar
  340. 340.
    The ATLAS Collaboration: ATLAS-CONF-2011-129; Guimares, J.: Talk at the ICHEP’12 Conference, Melbourne (2012)Google Scholar
  341. 341.
    The ATLAS Collaboration: Phys. Lett. B 716, 1 (2012). ArXiv:1207.7214 (2012)Google Scholar
  342. 342.
    The ATLAS Collaboration: ATL-PHYS-PUB-2011-01 (2011).
  343. 343.
    The ATLAS Collaboration: Phys. Rev. D 85, 012005 (2012). ArXiv:1108.6308; Phys. Lett. B 705, 415 (2011). ArXiv:1107.2381; the CMS Collaboration, Phys. Rev. D 85, 032002 (2012). ArXiv:1110.4973 Phys. Rev. D 65, 112003 (2002); Phys. Rev. D 70, 074008 (2004); the D0 Collaboration, Phys. Rev. Lett. 87, 251805 (2001); Phys. Rev. Lett. 84, 2786 (2000)Google Scholar
  344. 344.
    The CKM Fitter Group: Accessed 2013
  345. 345.
    The CMS Collaboration, Aad, G., et al.: Phys. Lett. B. (2012). ArXiv:1207.7235Google Scholar
  346. 346.
    The CMS Collaboration: JHEP 1305, 065 (2013). ArXiv:1302.0508; Eur. Phys. J. C 73, 2386 (2013). ArXiv:1301.5755; Phys. Lett. B 720, 83 (2013). ArXiv:1212.6682, top-12-006, 007, arXiv:1602.09024Google Scholar
  347. 347.
    The D0 collaboration: Phys. Rev. Lett. 84, 2792 (2000); Phys. Rev. D 61, 032004 (2000); see also the CDF Collaboration, Phys. Rev. Lett. 84, 845 (2000)Google Scholar
  348. 348.
    The FLAG Working Group, Colangelo, G., et al.: (2010). ArXiv:1011.4408Google Scholar
  349. 349.
    The H1, ZEUS Collaborations: JHEP 1001, 109 (2010). ArXiv:0911.0884Google Scholar
  350. 350.
    The LEP Electroweak Working Group: Accessed 2013
  351. 351.
    The LEP and SLD Collaborations: Phys. Rep. 427, 257 (2006). hep-ex/0509008Google Scholar
  352. 352.
    The LHCb and CMS Collaborations: LHCb-CONF-2013-012 (2013)Google Scholar
  353. 353.
    The NuTeV Collaboration, Zeller, G.P., et al.: Phys. Rev. Lett. 88, 091802 (2002)Google Scholar
  354. 354.
    The SLAC E158 Collaboration, Anthony, P.L., et al.: (2003). hep-ex/0312035, (2004). hep-ex/0403010Google Scholar
  355. 355.
    The Unitary Triangle Fit Group: Accessed 2013
  356. 356.
    van der Bij, J., Veltman, M.J.G.: Nucl. Phys. B 231, 205 (1984)ADSCrossRefGoogle Scholar
  357. 357.
    van Ritbergen, T., Vermaseren, J.A.M., Larin, S.A.: Phys. Lett. B 400, 379 (1997); see also Czakon, M.: Nucl. Phys. B 710, 485 (2005)Google Scholar
  358. 358.
    Veltman, M.: Nucl. Phys. B 21, 288 (1970); ’t Hooft, G.: Nucl. Phys. B 33, 173 (1971); 35, 167 (1971)Google Scholar
  359. 359.
    Weinberg, S.: Phys. Rev. Lett. 19, 1264 (1967)ADSCrossRefGoogle Scholar
  360. 360.
    Weinberg, S.: Phys. Rev. Lett. 31, 494 (1973)ADSCrossRefGoogle Scholar
  361. 361.
    Weinberg, S.: Phys. Rev. D 11, 3583 (1975)ADSCrossRefGoogle Scholar
  362. 362.
    Weinberg, S.: Phys. A 96, 327 (1979); Gasser, J., Leutwyler, H.: Ann. Phys. 158, 142 (1984); Nucl. Phys. B 250, 465 (1985); Bijnens, J., Ecker, G., Gasser, J.: The Second DAΦNE Physics Handbook (Frascati, 1995). hep-ph/9411232; Bijnens, J., Meissner, U.: hep-ph/9901381; Ecker, G.: hep-ph/9805500, hep-ph/0011026; Leutwyler, H.: hep-ph/9609465, hep-ph/0008124; Pich, A.: hep-ph/9806303; de Rafael, E.: hep-ph/9502254; Maiani, L., Pancheri, G., Paver, N.: The Second DAΦNE Physics Handbook (Frascati, 1995); Ecker, G.: Prog. Part. Nucl. Phys. 35, 1 (1995); Colangelo, G., Isidori, G.: hep-ph/0101264Google Scholar
  363. 363.
    Weinberg, S.: Phys. Rev. D 22, 1694 (1980)ADSCrossRefGoogle Scholar
  364. 364.
    Weinberg, S.: Phys. Rev. Lett. 59, 2607 (1987)ADSCrossRefGoogle Scholar
  365. 365.
    Wilson, K.: Phys. Rev. 179, 1499 (1969)ADSMathSciNetCrossRefGoogle Scholar
  366. 366.
    Wilson, K.G.: Phys. Rev. D 10, 2445 (1974)ADSCrossRefGoogle Scholar
  367. 367.
    Witten, E.: Nucl. Phys. B 156, 269 (1979); Veneziano, G.: Nucl. Phys. B 159, 213 (1979)Google Scholar
  368. 368.
    Witten, E.: Commun. Math. Phys. 252, 189 (2004). hep-th/0312171ADSCrossRefGoogle Scholar
  369. 369.
    Wobisch, M., Britzger, D., Kluge, T., Rabbertz, K., Stober, F., (the Fast NLO Collaboration): (2011). ArXiv:1109.1310Google Scholar
  370. 370.
    Wolfenstein, L.: Phys. Rev. Lett. 51, 1945 (1983)ADSCrossRefGoogle Scholar
  371. 371.
    Yang, C.N., Mills, R.: Phys. Rev. 96, 191 (1954)ADSCrossRefGoogle Scholar
  372. 372.
    Zijlstra, E.B., van Neerven, W.L.: Phys. Lett. B 273, 476 (1991); Phys. Lett. B 297, 377 (1992); Guillen, J.S., et al.: Nucl. Phys. B 353, 337 (1991)Google Scholar
  373. 373.
    Zuber, K.: Acta Phys. Polon. B 37, 1905–1921 (2006). nucl-ex/0610007Google Scholar

Copyright information

© The Author(s) 2017

This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Authors and Affiliations

  • Guido Altarelli
    • 1
  • James Wells
    • 2
  1. 1.CERNGenevaSwitzerland
  2. 2.Physics DepartmentUniversity of MichiganAnn ArborUSA

Personalised recommendations