We have seen that gauge theories do not allow elementary mass terms for gauge bosons, because they would break the gauge invariance and spoil the renormalizability. This ap-pears problematic for the weak interactions, which are short-ranged and require massive mediators.
Another potential problem, for theories with spontaneous symmetry breaking, is that they imply massless Goldstone bosons, but there are no known exactly-massless Goldstone bosons associated with the elementary particle interactions. Fortunately, whenGis a gauge symmetry the two problems of the unwanted Goldstone bosons and the unwanted massless gauge bosons can cure each other when the symmetry is spontaneously broken (Anderson, 1963). A particularly simple implementation is the Higgs mechanism, involving elementary spin-0 fields (Higgs, 1964, 1966; Englert and Brout, 1964; Guralnik et al., 1964). Instead of existing as a massless spin-0 particle, the degree of freedom carried by the Goldstone boson manifests itself as the longitudinal spin component of a gauge boson, which has in the process acquired a mass. (One says that the Goldstone boson has been “eaten.”) Remarkably, SSB via the Higgs mechanism preserves the renormalizability of the theory (’t Hooft, 1971a;
’t Hooft and Veltman, 1972; Lee and Zinn-Justin, 1972, 1973).
To illustrate this, consider aU(1) gauge theory with a single complex scalar fieldφ, with L=−1
4FµνFµν+ [(∂µ+igAµ)φ]†(∂µ+igAµ)φ−µ2φ†φ−λ(φ†φ)2 (4.36) as in (2.133) or (4.8). We studied the analogous problem of a global U(1) symmetry in Section 3.3.3. It was found that forµ2<0,φacquired a VEV,
h0|φ|0i= ν
√2, ν= r−µ2
λ . (4.37)
Expandingφ≡(ν+σ+iχ)/√
2 aroundν/√
2, there was one massive physical scalarη and one massless Goldstone bosonχ.
The minimization of the potential for a gauge symmetry is the same as for the global case, and (4.37) continues to hold. Rewriting (4.36) in terms of the new fields,
L=−1 to the kinetic energy terms for the gauge fields and forσandχthere are two new quadratic terms. The third term in (4.38) has the form of a mass term for the gauge field with mass gν. This mass can be interpreted as arising from the interaction of the gauge field with the condensate of Higgs fields. The fourth term in (4.38) is proportional to the gauge field times the derivative of the Goldstone boson fieldχ. To interpret it, we can recombine theχ and Aterms as
g2ν2 2 A0µ2
, A0µ≡Aµ+∂µχ
gν , (4.39)
suggesting that ∂µχ/gν is the longitudinal component of a now massive vector field A0µ. However, χstill enters the cubic and quartic terms in a way that is hard to interpret.
To see what is going on it is useful to use the Kibble reparametrization (Kibble, 1967), which makes the physical particle content manifest and was already introduced for the global case inSection 3.3.5. While working at the level of the classical field theory, one can define new fieldsη, ξ related toσandχ by a non-linear field redefinition:
φ=ν+σ+iχ
For smallσandχ one has approximately exp(iξ/ν)
so thatη∼σandξ∼χ. The fieldsη andξtherefore represent the massive and Goldstone scalars, respectively. In terms of the new fields,
L=−1 which is clearly the Lagrangian density for a massive vector field, including 3- and 4-point gauge and self interactions for the massive scalar η. The Goldstone boson has disappeared from the theory.
One can interpret (4.42) as the result of choosing a special gauge. Before quantizing one can make aU(1) gauge transformation3 as in (4.6), choosingβ(x) =−ξ(x)/ν. Then
Equation (4.42) then follows from (4.36) by gauge invariance, L(A, φ) = L(A0, φ0). This unitary gaugeis useful because it makes the physical particle content of the theory manifest.
However, it is not so useful for explicit calculations, especially in higher orders where it is rather singular.
The number of degrees of freedom was not changed by the Higgs mechanism. Before SSB there were two massless gauge degrees of freedom and two Hermitian scalars, while afterwards there are three gauge degrees of freedom and one massive real scalar.
Now let us consider the non-abelian case with an arbitrary scalar sector. Just as in the general discussion of the Nambu-Goldstone theorem, it is more convenient to choose a Hermitian basis for the scalars for formal manipulations, though a complex basis may be simpler for concrete calculations. We assume there arenHermitian scalar fields φa, which can be arranged in a column vectorφ= (φ1· · ·φn)T with VEVν =h0|φ|0i. As discussed in Section 3.3.6, we assume thatMof the generators are not broken,Liν= 0,i= 1· · ·M, while the remainingN−M are broken, i.e.,Liν 6= 0, i=M+1· · ·N. Then, for a global symmetry we expect that there will beN−M massless Goldstone bosons andp=n−(N−M) massive physical scalar particles. According to (3.173) the Goldstone bosons are linear combinations of the original Hermitian fields, corresponding to the directions of the massless eigenvectors iLiν, i =M + 1· · ·N. These span an N −M dimensional vector space, but need not be orthogonal. (The i is because the Li are imaginary). We can therefore label then scalars so that the subspace spanned by theiLiν. There are no VEV’s in those directions: we are working in a Hermitian basis, which implies Li = −LiT = −Li∗ and therefore that hν|Liνi = 0.
The p fields σi will be associated with the massive scalars. Not all of the νi, i = 1· · ·p, are necessarily non-zero. The second form is a polar reparametrization. Only the broken generators are included in the exponent, and the ξ are the Goldstone boson fields. Unlike (4.40) we have not attempted to normalize theξi by factors of ν, as it is not needed for a gauge symmetry in unitary gauge.4
The Goldstone fields may then be removed by going to the unitary gauge, i.e., the Kibble transformation, just as for the U(1) example. Gauge invariance ensures thatL will be unchanged in form under the gauge transformations in (4.2) and (4.24). In particular, chooseβi(x) =−ξi(x), i=M+1· · ·N, andβi(x) = 0 otherwise. Then,L(A, φ) =L(A0, φ0), whereφ0 =ν+η.A0 is given generally by (4.24), but for small ξi is
A0iµ=Aiµ+cijkξjAkµ+1
g∂µξi, (4.46)
which contains the longitudinal term∂µξi. The relevant term for our present considerations is the gauge covariant kinetic term for the Hermitian scalars. This becomes, in terms of the
4For a global symmetry theξfields would have a non-canonical kinetic energy matrix, and field redefi-nitions similar to those inSection 2.14to restore the correct diagonal form 12P
i(∂µξ)2would be required.
new variables, 1
2DµφDµφ→ 1
2(ν+η)T(←−∂µ−ig ~Aµ·L)(~ −→∂µ+ig ~Aµ·~L)(ν+η)
= 1
2Mij2AiµAjµ+1
2DµηDµη+g2 νTLiLjη
AiµAjµ,
(4.47)
where the primes have been dropped for notational simplicity. The Goldstone fieldsξ have disappeared from the theory, reemerging as the longitudinal gauge field components, and the gauge bosons corresponding to the broken generators have acquired mass, described by the first term in the second line. The second term is the normal gauge covariant kinetic energy for the physical η fields, which include the three and four point gauge interactions, and the last is an induced cubic gauge interaction.
The gauge boson mass matrix in (4.47) is Mij2 =Mji2 =g2νTLiLjν
=g2hν|LiLj|νi=g2hLiν|Ljνi=g2
n
X
a=p+1
(Liν)∗a(Ljν)a, (4.48) where the inner product is defined by hx|yi =Px∗aya. The induced cubic term in (4.47) may be similarly rewritten as
g2 νTLiLjη
AiµAjµ=g2hLiν|LjηiAiµAjµ=g2hν|LiLjηiAiµAjµ
=g2
n
X
a=p+1
(Liν)∗a(Ljη)aAiµAjµ. (4.49) It is easy to prove, using the explicit form in (4.48) and the fact that the representation matricesLi are antisymmetric and purely imaginary, that M2 is real, symmetric, and has non-negative eigenvalues. From (3.174),M2 must have the block-diagonal form
M2=
0 0 0 M2
, (4.50)
where the upper diagonal block is M ×M dimensional and M2 is (N −M)×(N−M) dimensional. There are therefore M massless gauge bosons A1· · ·AM, corresponding to the unbroken generators, and N−M massive gauge bosons, corresponding to theN−M non-zero eigenvalues ofM2. TheN−M Goldstone bosons have been eaten to become the longitudinal modes ofN−M massive gauge bosonsAM+1· · ·AN.