We have seen in Problem 30 that the four-Fermi interaction in good approxi-mation can be written in terms of the exchange of a heavy vector particle. In lowest order we have resp. the diagrams in Figures 19.1(a) and 19.1(b). The first diagram comes from a four-fermion interaction term that can be written in terms of the product of two currents JµJµ, where Jµ =�γµ�. Here each fermion line typically carries its own flavour index, which was suppressed for simplicity. Figure 19.1(b) can be seen to effectively correspond to
−˜Jµ(−k)
gµν−kµkν/M2 k2−M2+iε
˜Jν(k). (19.1)
At values of the exchanged momentum k2�M2, one will not see a difference between these two processes, provided the coupling constant for the four-Fermi interactions [Figure 19.1(a)] is chosen suitably (see Problem 30). This is because for small k2, the propagator can be replaced by gµν/M2, which indeed converts eq. (19.1) to JµJµ/M2. It shows that the four-Fermi coupling constant is proportional to M−2, such that its weakness is explained by the heavy mass of the vector particle that mediates the interactions. Examples of four-Fermi interactions occur in the theory ofβ-decay, for example, the decay of a neutron into a proton, an electron and an antineutrino. In that case the current also contains aγ5(Problem 40).
It turns out that the four-Fermi theory cannot be renormalised. Its quantum corrections give rise to an infinite number of divergent terms that cannot be reabsorbed in a redefinition of a Lagrangian with a finite number of interac-tions. With the interaction resolved at higher energies by the exchange of a massive vector particle, the situation is considerably better. But it becomes crucial for the currents in question to be conserved, such that the kµkν part in the propagator has no effect. It would give rise to violations of unitarity in the scattering matrix at high energies [theσ field defined in eq. (16.15) has the wrong sign for its kinetic part]. To enforce current conservation, we typ-ically use gauge invariance. But gauge invariance would protect the vector particle from having a mass. The big puzzle therefore was how to describe a massive vector particle that is nevertheless associated to the vector potential of a gauge field.
133
134 A Course in Field Theory
(a)
M
(b) FIGURE 19.1
Lowest order diagrams.
The answer can be found in the theory of superconductivity, which prevents magnetic field lines from penetrating in a superconducting sample. If there is, however, penetration in the form of a quantised flux tube, the magnetic field decays exponentially outside the flux tube. This would indicate a mass term for the electromagnetic field within the superconductor. The Landau–
Ginzburg theory that gives an effective description of this phenomenon [the microscopic description being given by the Bardeen–Cooper–Schrieffer (BCS) theory of Cooper pairs] precisely coincides with scalar quantum electrody-namics.
L= −14FµνFµν+ Dµϕ∗
Dµϕ−κϕ∗ϕ−14λ ϕ∗ϕ2
. (19.2)
In the Landau–Ginzburg theory,ϕdescribes the Cooper pairs. It is also called the order parameter of the BCS theory. In usual scalar quantum electrody-namics, we would putκ = m2, where m is the mass of the charged scalar field. But in the Landau–Ginzburg theory of superconductivity, it happens to be the case thatκis negative. In that case the potential V(ϕ) for the scalar field has the shape of a Mexican hat, Figure 19.2.
The minimum of the potential is no longer atϕ =0, but atϕ∗ϕ = −2κ/λ, and is independent of the phase ofϕ. To find the physical excitations of this theory, we have to expand around the minimum. With a global phase rotation we can choose the point to expand around to be real,
ϕ0=
−2κ/λ. (19.3)
But this immediately implies that the quadratic terms in the gauge field give rise to a mass term for the photon field
|Dµϕ0|2=e2ϕ02AµAµ≡ 12M2AµAµ, M=2e
−κ/λ. (19.4) Furthermore, from the degeneracy of the minimum of the potential it follows that the fluctuation along that minimum [the phase inϕ = ϕ0exp(iχ)] has no mass (this is related to the famous Goldstone theorem, which states that if choosing a minimum of the potential would break the symmetry, called spon-taneous symmetry breaking, there is always a massless particle). However, this phaseχ is precisely related to the gauge invariance and can be rotated away by a gauge transformation. On the one handχcorresponds to a mass-less excitation; on the other hand it is the unphysical longitudinal component of the gauge field. But the photon becomes massive and has to develop an
The Higgs Mechanism 135 V
Im (ϕ)
Re (ϕ)
FIGURE 19.2
The Mexican hat potential V(ϕ).
additional physical polarisation, which is precisely the longitudinal compo-nent. In a prosaic way one states that the massless excitation (called a would-be Goldstone boson) was ‘eaten’ by the longitudinal component of the photon, which in the process got a mass (‘got fat’).
This means we have four massive degrees of freedom, three for the massive vector particle and one for the absolute value of the complex scalar field (its mass is determined by the quadratic part of the potential in the radial direction atϕ =ϕ0). This is exactly the same number as for ordinary scalar electrodynamics whereκ > 0, because in that case the massless photon has only two degrees of freedom, whereas the complex scalar field represents two massive real scalar fields. It looks, however, like there is a discontinuity in the description of these degrees of freedom when approachingκ = 0. But the interpretation of the phase of the complex field as a longitudinal component of the vector field is simply a matter of choosing a particular gauge. To count the number of degrees of freedom, we implicitly made two different gauge choices
κ >0 : ∂µAµ=0, Lorentz gauge,
κ <0 : Imϕ =0, Unitary gauge. (19.5)
136 A Course in Field Theory There is a gauge, called the ’t Hooft gauge, that interpolates between these two gauges
F≡∂µAµ−2ieξϕ0Imϕ =0, ’t Hooft gauge. (19.6) Rather than adding to the Lagrangian the gauge-fixing termLgf = −12α(∂µ Aµ)2, one addsLgf= −12αF2. Atξ =0 this corresponds to the Lorentz gauge, and atξ = ∞to the unitary gauge. For the choice ’t Hooft made (ξ =1/α), the terms that mix (ϕ −ϕ0) and Aµ at quadratic order disappear and one easily reads off the masses. Gauge fixing will be discussed in the next chapter, where it will be shown how extra unphysical degrees of freedom appear in the path integral, so as to cancel the unphysical components of the gauge and scalar fields. The scalar field, whose interactions give the gauge field a mass, is called the Higgs field. Problems 34 and 35 discuss the Higgs mechanism in detail for the Georgi–Glashow model, which is a non-Abelian gauge theory with gauge group SO(3), coupled to an SO(3) vector of scalar fieldsϕa.
DOI: 10.1201/b15364-20