Supersymmetry. Theory, Experiment, and Cosmology

390 A review of the Standard Model and of various notions of quantum field theory

We now turn to the fermion sector. We will only consider here a single family of quarks and leptons: e, νe, u, and d. We have identified in the previous section the quantum numbers of these fields. Restoring the color SU (3) degrees of freedom, we thus define

The minimal coupling to gauge fields is obtained from a standard action of the

symmetry because the left and right chiralities transform di erently.

This chiral nature of the Standard Model is one of its key properties: fermion masses as well as gauge boson masses only appear once the symmetry is spontaneously broken. This is especially important when one considers the Standard Model as a low energy e ective theory of an underlying fundamental theory with a very high mass scale Λ. If the Standard Model was vectorlike, i.e. if left and right chiralities were transforming the same way, then masses of order Λ would be allowed. Because the Standard Model is chiral, only masses of order v are allowed.

We note also that the only dimensionful parameter (mass scale) in the Standard Model is the scale m in the scalar potential (A.127): it is this scale which fixes the scale of electroweak symmetry breaking v in (A.128). It is also this scale which fixes, at tree-level, the Higgs mass. This can easily be seen by considering the scalar potential

This establishes the key rˆole of the Higgs mass parameter in the Standard Model. As discussed in Chapter 1, this is one of the most sensitive issues when one tries to go beyond the Standard Model.

The Standard Model of electroweak interactions 391

Fermion mass terms arise from the coupling of fermions to the scalar sector. One may check that the following renormalizable couplings, known as Yukawa couplings, are the most general couplings allowed by the SU (2) × U (1)Y local gauge symmetry:

The case of the neutrino mass is complicated by the possible presence of a Majorana mass term which is not constrained by the SU (2) × U (1)Y symmetry since NR is an electroweak singlet:

As discussed in Chapter 1, this leads in a fairly straightforward way to the seesaw mechanism for neutrino masses9.

It remains for us to check that the low energy limit of the Standard Model is indeed the Fermi theory described by the current–current interaction (A.110). This will provide some important information on the mass scales involved.

A.3.3 Low energy e ective theory

Let us identify the currents coupled to the di erent gauge bosons of the Standard Model. We have already done this in (A.115): it su ces to add the coupling to the U (1)Y gauge boson Bµ. Thus

9Alternatively, if one does not introduce NR at all, one obtains a nonvanishing Majorana neutrino mass by allowing the following nonrenormalizable coupling:

ˆ

LM = λν ψcψ φφ + h.c. (A.143)

M

This term is interpreted as arising in the context of an e ective theory valid at energies smaller than the scale M of a more fundamental theory.

392 A review of the Standard Model and of various notions of quantum field theory

Expressing the W mass in terms of the Higgs vacuum expectation value as in

Thus the electroweak symmetry breaking scale is completely fixed by the low energy Fermi theory. We have seen earlier that this scale is related to the mass parameter in the Higgs potential, and thus to the Higgs mass. We conclude from (A.138) that, if the Higgs is light compared to the scale υ, the symmetry breaking (Higgs) sector is weakly coupled (λ < 1), whereas it becomes strongly coupled for Higgs masses much larger than υ, i.e. in the TeV range.

The Z0 exchange also leads to low energy e ective four-fermion interaction. One

with coupling constant g/ cos θW . The exchange of Z0 at low energy thus leads to the e ective “neutral current” interaction:

where we have included a symmetry factor of 1/2 (absent in the charged current e ective interaction because Jµ+ = Jµ−). Such an interaction scales like the Fermi

The Standard Model of electroweak interactions 393

constant GF but was much more di cult to detect because it does not correspond to a change in the nature of the quarks (as in the beta decay). It was discovered at CERN in 1973, following the prediction of the Standard Model.

A.3.4 Flavor structure

The Standard Model has a very simple family structure which leads to specific predictions, such as the quasi-absence of flavor changing neutral currents, which are di cult to reproduce in the context of its extensions. We will thus first review the main results.

In the Standard Model, we are in presence of three families which are replicas (from the point of view of gauge quantum numbers) of the set (νe, e, u, d). We will thus denote the fermions of each three family as

(νi, ei, ui, di) i = 1 (νe, e, u, d) i = 2 (νµ, µ, c, s) i = 3 (ντ , τ, t, b)

where color indices have been suppressed on the quark fields. We call the index i a family index. If we include the three families, then the Yukawa couplings have a matrix structure in family space: all couplings in

L ¯ ¯ ¯ ¯

Y = Λ ij ψ i ΦeRj + Λνij ψ i ΦNRj + Λdij ψqi ΦdRj + Λuij ψqi ΦuRj + h.c. (A.154)

are allowed by the gauge symmetries of the Standard Model (since these are horizontal,

i.e. do not depend on the family of the quark or lepton). In (A.154), ψ = νLi and

i eLi uLi .

dLi

The Yukawa matrices Λ , Λd, and Λu are generic complex 3 × 3 matrices. In all generality, a complex matrix has a polar decomposition, i.e. it can be written as a product of a hermitian matrix H and a unitary matrix U (see Exercise 3)

The matrix H, being hermitian , can be diagonalized with a unitary matrix VL: H = VLDVL†. Then the general complex matrix Λ is diagonalized with the help of two unitary matrices VL and VR ≡ U †VL

Using this general result, one may then write for example the electron mass term which arises from (A.154) as

where we have suppressed some of the summed family indices. Thus, charge eigenstates, i.e. fields belonging to definite representations of the gauge group such as (u, d)

394 A review of the Standard Model and of various notions of quantum field theory

or (ντ , τ ), should not be confused with mass eigenstates i.e. fields which are observed in the detectors:

Charged and neutral currents should be expressed in terms of the observable fields, that is in terms of the mass eigenstates. This induces some new structure. For example, the charged current reads

Hence the quark charged current coupled to the W + is not diagonal. It involves the mixing matrix

known as the Cabibbo–Kobayashi–Maskawa matrix. And similarly for the leptons

where VMNS is the Maki–Nakagawa–Sakata [280] matrix.

On the other hand, neutral currents, because they are flavor diagonal, remain diagonal when expressed in terms of mass eigenstates. For example, Jµ3 which appears in JµZ = Jµ3 − sin2 θW JµQ (see equation (A.152)), is written:

since the diagonalization matrices cancel (VLu†VLu = 1, . . .). Thus neutral current interactions proceed without any change of flavor (i.e. type of quarks). This is why neutral currents were so much more di cult to detect experimentally than charged currents (nuclear beta decay), although they are of the same strength (see (A.181) below).

The absence of flavor changing neutral currents extends to the one-loop level through a cancellation known as the GIM mechanism [193]. We illustrate it here on the simpler case of two families where the quark mixing matrix can be written (see

below) as the 2 × 2 orthogonal Cabibbo matrix

where θC is the Cabibbo angle (sin θC 0.2). Since charged currents are flavor changing, we should expect that the simultaneous exchange of a W + and a W − in a one-loop

The Standard Model of electroweak interactions 395

process (box diagram) yields a flavor-changing neutral current. Take for example two

(there are two other diagrams with the u quark internal line on the left replaced by a c quark).

In the limit of vanishing masses or equal masses, it is clear that the two diagrams have the same contribution apart from the couplings of the internal quark line on the right, which is cos θC sin θC for the first diagram and − sin θC cos θC for the second diagram. Thus they cancel and this flavor-changing process vanishes in the limit of equal masses: it is at most of the order of the quark mass di erences and thus small. A careful study of the size of the e ect allowed [171] to estimate the charm quark mass.

We close this presentation of the flavor sector of the Standard Model by a discussion of CP violation: it turns out that the only source of CP violation in the Standard Model is a single phase in the fermion sector.

requires that the mass matrix is real: M = M . Indeed, invariance under parity P

396 A review of the Standard Model and of various notions of quantum field theory

Thus, we will have a natural violation of the CP symmetry if there exists at least one physical phase in the fermion mass matrix or, if we are working with the mass eigenstates, in the Cabibbo–Kobayashi–Maskawa matrix [254]. By “physical” we mean a phase which cannot be set to zero by a redefinition of the fermion fields. Let us count the number of physical phases of VCKM in the general case of NF families of quarks:

would be a NF ×NF orthogonal matrix and would have NF (NF −1)/2 real parameters. The remaining (NF − 1)(NF − 2)/2 parameters are complex phases.

Thus, with NF = 2 families of quarks, there is no source of CP violation and the Cabibbo matrix (A.163) can be made real by redefining the phases of the quark fields. And NF = 3 corresponds to the minimum number of families necessary to have a CP violating phase in the Standard Model, as noted first by Kobayashi and Maskawa. This single phase is generically referred to as δCKM .

We may for example write the Cabibbo–Kobayashi–Maskawa matrix in the

may be represented geometrically in the complex plane and is called a unitarity triangle. It has become customary to rescale the length of one side, i.e. |VcdVcb| which is well-known, to 1 and to align it along the real axis. The unitarity triangle thus looks like Fig. A.7.

The angles α, β, γ are defined as:

Fig. 12.1 of Chapter 12 shows how the experimental limits obtained in 2005 constrain the unitarity triangle. The fact that all data is consistent with a small region with nonvanishing η shows that CP is violated and that its violation is consistent with a single origin: the phase of the CKM matrix.

10A global phase redefinition would have no e ect; hence the −1.

11We neglect terms of order λ4 and higher [59]; in the same spirit, we use in what follows the variables ρ¯ and η¯ which are precisely defined as ρ¯ ≡ ρ(1 − λ2/2) and η¯ ≡ η(1 − λ2/2).

Electroweak precision tests 397

Fig. A.7 Unitarity triangle

More quantitatively, one may show that CP is violated if and only if [238]

(JCKM λ6A2η in the Wolfenstein parametrization), one may write the condition (A.169) in the mass eigenstate basis as

(m2t − m2c )(m2t − m2u)(m2c − m2u)(m2b − m2s)(m2b − m2d)(m2s − m2d)JCKM = 0. (A.171)

A.4 Electroweak precision tests

The Standard Model has been tested at the CERN electron–positron collider LEP to a precision of the order of the per mil (‰). Such a precision allows us to go beyond the tree-level which has been described above and to test the theory at the quantum level. Indeed, if quantum fluctuations were not included, the Standard Model would be in disagreement with experimental data. We review in this section how these quantum corrections are included in order to confront the theory with experiment.

A.4.1 Principle of the analysis of the radiative corrections in the Standard Model

As soon as renormalization is involved, it is important to rest the discussion on observables: only observable quantities are free of the arbitrariness inherent to the renormalization procedure.

Let us consider for example the three parameters which, besides the Higgs mass,

describe the gauge sector of the Standard Model: the SU (2) gauge coupling g, the

√

hypercharge U (1) gauge coupling g /2, and the Higgs vacuum expectation value v/ 2. Since they are not directly measurable, one prefers to replace them by observable quantities:

•α, the fine structure constant;

•Gµ, the Fermi constant measured in µ decay;

•MZ , the Z boson mass.