Supersymmetry. Theory, Experiment, and Cosmology

Spontaneous breaking of symmetry 379

F is bounded and reaches a maximum for some element gˆ(x). Using δg = iαagta, we have

ˆ ≡ −1 α

Hence φ(x) gˆ (x)φ(x) is orthogonal to the directions θ φ0 in field space: it parametrizes the (in general) massive excitations.

Since any element of G can be written as eiγαθα eiβiti and since gˆ(x) is in fact a

which generalizes (A.70): the (dim G − dim H) fields γα(x) are the Goldstone bosons. We note [82] that they transform nonlinearly under G: γα → γα (x) defined by

A.2.4 Spontaneous breaking of a local symmetry. Higgs mechanism

The conclusions that we have reached in the preceding section, especially the Goldstone theorem, are drastically changed when we turn to local transformations. Let us illustrate this on the case of a complex scalar field minimally coupled to an abelian gauge symmetry, a theory known as scalar electrodynamics. The Lagrangian reads

with Dµφ = ∂µφ − igqAµφ and V (φ†φ) given by (A.61) with a = −m2. The scalar field kinetic term thus yields a term quadratic in the vector field:

Let us pause for a while to discuss massive vector bosons. The free field action is

Gauge invariance is only recovered in the limit of vanishing mass M . The corresponding equation of motion is the Proca equation:

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

Acting with a derivative ∂ν and using the antisymmetry of F µν , we infer from the Proca equation the Lorentz condition ∂ν Aν = 0. Thus the massive vector field has 4 − 1 = 3 degrees of freedom: more precisely two transverse (as the massless photon) and one longitudinal. Using the Lorentz condition, we may finally write the Proca equation

which shows that, indeed, M is the mass of the gauge field.

Returning to our case of interest, we deduce that, once the gauge symmetry is

There seems, however, to be a discrepancy when we count the number of degrees of freedom. Using the gauge invariance built in the Lagrangian, we count two scalar degrees of freedom and two vector (transverse) degrees of freedom. However, once one translates the scalar field around its vev φ0 = 0, one seems to identify:

•a massless Goldstone boson ϕ2 and a real scalar ϕ1 of mass 2m2 (as in Section A.2.1);

•a vector field of mass MA, with three degrees of freedom (two transverse and one longitudinal).

It turns out that making the symmetry local has changed one of the scalar fields into a spurious degree of freedom. This is most easily seen by using the nonlinear parametrization (A.70) of the scalar degrees of freedom. We see that, in the local case, one can gauge away the field γ(x) by performing a local gauge transformation:

with the choice θ(x) = γ(x)/q. In other words, because the symmetry is local, the would be Goldstone boson is now a gauge artifact. It reappears in the theory as the longitudinal component of the massive vector particle. This is the famous Higgs mechanism.

The choice (A.98) corresponds to a gauge choice known as the unitary gauge: with this choice of gauge, all fields are physical and the S-matrix is unitary, but gauge symmetry is no longer apparent. One often prefers covariant gauges which retain some of the gauge invariance (through the dependence on some gauge parameters, which should drop from physical results) but include the spurious boson.

We may generalize this analysis to nonabelian gauge symmetries and will illustrate it on the example of SU (2) which provides the basis for the Standard Model. We

introduce a complex scalar field Φ(x) = φ1(x) which transforms as a doublet under

φ2(x)

SU (2) (cf. (A.37))

The Standard Model of electroweak interactions 381

Then the full Lagrangian reads:

where Fµνa is given in (A.52), and the potential is given by

The ground state is reached for Φ†0Φ0 = m2/(2λ). We choose to orient it along the second component of the doublet:

Then, studying as before the quadratic terms AaµAaµ coming from the term DµΦ†DµΦ, one concludes that all three gauge fields Aaµ, a = 1, 2, 3, acquire a mass

through the Higgs mechanism. This requires three longitudinal degrees of freedom.

Since V (Φ†Φ) = V ([v + η]2/2), the three fields γa(x) are the would-be Goldstone bosons. But a gauge transformation (A.99) with αa(x) = γa(x) shows that they are spurious degrees of freedom.

A.3 The Standard Model of electroweak interactions

From the very beginning, electrodynamics has inspired the theory of weak interactions. In electrodynamics, a vector current of matter fields couples to the photon. Electromagnetic interactions through photon exchange is thus described by a current–current interaction with a photon propagator to account for the action at distance. In nuclear beta decay, which we write here at the level of nucleons n → p + e− + νe, the range seemed to be zero or very small and [156, 157] proposed to keep a current–current structure but to make it a local one (in order to account for the zero range):

where the letters p, n, . . . denote the corresponding spinor fields (proton, neutron, . . .) and GF is the Fermi constant:

where mp is the proton mass.

The Standard Model of electroweak interactions 383

Fig. A.3 Tree-level Fermi interaction for nuclear beta decay.

sections have become so large that they are no longer compatible with the unitarity of the S-matrix (which expresses the conservation of probability). At such an energy the fundamental theory must be modified.

Since this type of reasoning will be repeated for gravity (another nonrenormalizable theory where the coupling, Newton’s constant GN , has also mass dimension −2), let us sketch the argument more precisely.

If we consider the tree-level process depicted in Fig. A.3, then, on purely dimensional grounds, we may deduce that the corresponding cross-section σ (given by the square of

(expressed through the unitarity of the S-matrix) imposes that σ E−2. Thus at energies E G−F 1/2, the theory is no longer compatible with first principles and must be replaced by a more complete theory (which involves new fields whose exchange modifies the high energy behavior of the cross-section). This is the so-called unitarity limit that we consider in more details in Section 1.2.1 of Chapter 1.

This behavior of the cross-sections for tree-level processes may be related here to

the nonrenormalizable character of the theory. If we compute the one-loop (second order) amplitude given in Fig. A.4, we obtain a contribution of order G2F ∞ EdE (to be compared with the tree-level GF as in Fig. A.3) which is infinite if we assume that the theory remains valid up to arbitrarily large energies, i.e. to arbitrarily small distances. The renormalization procedure may take care of a finite number of such infinities but, in the case considered, new infinities appear at each order of perturbation theory and thus perturbative renormalization cannot help.

These considerations show that, because the coupling GF of the weak theory has mass dimension −2, the theory at energy E is characterized by the dimensionless combination5 GF E2. When E reaches a scale of order G−F 1/2 300 GeV, the dimensionless combination is of order one and the low energy weakly coupled theory is no longer valid: it must be replaced by a more complete fundamental theory6.

Since the problem occurs at short distance (i.e. when the two vertices of the graph of Fig. A.4 become arbitrarily close), an obvious way of curing the problem is to replace

integral over the energy.

6The precise value of scale may be found by imposing the unitarity bound on varied processes. The best limit thus obtained is 630 GeV.

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

Fig. A.4 One-loop contribution to beta decay.

the contact interaction between the four fermions in the Fermi interaction by a vector particle exchange as in Fig. A.5, much in the spirit of quantum electrodynamics. This (charged) vector particle was introduced long before the days of the Standard Model and called the intermediate vector boson. It had to be be massive in order to account for the short range interaction. The above arguments showed that its mass was expected to be smaller than a few hundred GeVs. Once it was realized that a nonvanishing mass for a vector boson was compatible with gauge symmetry (although spontaneously broken), it was natural to try to fit weak interactions into a gauge theory. We will now see that the minimal nontrivial possibility leads to the Standard Model.

A.3.1 Identifying the gauge structure and quantum numbers

We will try here to fit the current–current theory of weak interactions just described with the structure of a gauge theory. In other words, pursuing the analogy with QED which motivated Fermi, we try to identify the intermediate vector boson W ± as a gauge field. Our guiding principle will be here minimality: there is no a priori reason that the theory of weak interactions should be the minimal one but, as we will see, it turns out to be the case.

From the graph of Fig. A.5 (and a similar one for muon beta decay), we infer that the couplings of the intermediate vector boson to the quarks u and d and the leptons e, νe, µ, νµ must be of the form

Since a charged field is complex, we need to introduce two real gauge fields; minimality leads to consider an abelian symmetry U (1) × U (1), with a global symmetry between the two U (1), or a nonabelian gauge symmetry SU (2). The first choice is very close to QED. Since the physics of weak interactions seems quite di erent from electrodynamics, we pursue the second possibility.