THE R ξ GAUGES

, (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· · ·A^M, 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 bosonsA^M+1· · ·A^N.

4.4 THE R

GAUGES

The unitary gauge for a spontaneously broken theory, introduced inSection 4.3, is extremely useful for identifying the physical states of the theory, but it is not very convenient for explicit calculations, especially when higher-order loop corrections are involved, because it is very singular. It is useful to work instead in a new class of gauges called the R_ξ gauges (Fujikawa et al., 1972; Weinberg, 1973c,d; Lee and Zinn-Justin, 1973), which are less singular, though the particle content is less obvious. They are therefore better behaved in higher-order calculations and for proving the renormalizability of spontaneously broken gauge theories.

Consider a gauge theory with n Hermitian scalars φ_a, a = 1· · ·n, arranged in a col-umn vector φ, andm fermion fields represented by a column vector ψ. The most general renormalizable Lagrangian density with a conserved fermion number is then

L=−1

4F_µνⁱ F^iµν +1

2(D^µφ)(Dµφ)−V(φ) + ¯ψ(iD6 −m0)ψ+ ¯ψΓ^aψφ_a, (4.51) whereF_µνⁱ , i= 1· · ·N, are the field strength tensors for the gauge groupG,D_µand6Dare the covariant derivatives for the scalars and fermions, andV is the scalar potential containing gauge-invariant terms up to order φ⁴. There may be present a fermion bare mass term described by the m×mmatrixm0 =m0LPL+m0RPR, withm0L=m^†_0R, if it is allowed by the symmetries of the theory. Similarly, there will in general be Yukawa interactions between the fermions and scalars described by m×mmatrices Γ^a = Γ^a_LP_L+ Γ^a_RP_R, with Γ^a_L= Γ^a†_R.

Just as in Section 4.3, we define a column vector ν = h0|φ|0iof VEVs, where ν_a = 0 and some (Lⁱν)a 6= 0 for a=p+ 1· · ·n, where n−p=N−M is the number of broken generators. In an arbitrary gauge one can writeφ=ν+φ⁰, whereφ⁰ is the shifted field with h0|φ⁰|0i= 0,

φ=ν+φ⁰=







ν1+φ⁰₁ ... ν_n+φ⁰_n





. (4.52)

We saw in (4.45) that φ⁰_a, a= 1· · ·p, represent physical scalars, while φ⁰_a, a=p+ 1· · ·n, are associated with the Goldstone degrees of freedom. The latter disappear in the unitary gauge, which can be defined by the condition

hLⁱν|φ⁰i= 0, i= 1· · ·N. (4.53) In a general gauge, φ⁰ will include the unphysical Goldstone bosons. Similar to(4.38) we can rewrite (4.51) in terms of the shifted fields

L=−1

4F_µνⁱ F^iµν+1

2(D^µφ⁰)(Dµφ⁰) +M_ij²

2 A^µiA^j_µ−ighν|Lⁱ∂^µφ⁰iAⁱ_µ+g²hν|LⁱL^jφ⁰iAⁱ_µA^jµ

−V(ν+φ⁰) + ¯ψ(iD6 −m₀+ Γ^aν_a)ψ+ ¯ψΓ^aψφ⁰_a,

(4.54)

whereL≡Lφ=−L^T are the representation matrices for the Hermitian scalars. The terms in the first line represent the gauge and Higgs kinetic energies and gauge interactions, just as in the non-spontaneously broken case (Figure 4.1). Both the physical and the Goldstone boson states are included. The first term in the second line is the induced gauge boson mass term. It is of the same form as in the unitary gauge, with M² given in (4.48). The next will be cancelled by terms added toL to fix the gauge, and the third is the induced cubic interaction, leading to theφ⁰_bAⁱ_µA^j_ν vertex

ig²g_µνν_a(LⁱL^j+L^jLⁱ)ab (4.55) shown inFigure 4.2. In the last line,V becomes the scalar potential forφ⁰, including mass terms for the physical states and scalar self-interactions. The n×n dimensional mass-squared matrix ˆµ² is given by (3.169), just as for the case of a global symmetry. It has p (generally) nonzero eigenvalues corresponding to the physical scalars, and n−p zero

eigenvalues corresponding to the Goldstone bosons. The next term in (4.54) includes the fermion mass matrix

m=m₀−Γ^aν_a≡m_LP_L+m_RP_R, m_L=m^†_R, (4.56) which may in general have both bare (m0) and spontaneously-generated pieces. m may involveγ⁵’s unlessm_L=m_R. Techniques for “diagonalizing”mto obtain the fermion mass eigenvalues and eigenvectors are described in the standard model context in Section 8.2.2.

The Yukawa interactions involving the shifted fields are given by the last term in (4.54).

TheR_ξ gauges are defined by the condition

Fⁱ(A(x), φ(x), ξ) =αⁱ(x), (4.57)

where

Fⁱ≡p ξ

∂^µAⁱ_µ+ig

ξhν|Lⁱφ⁰i

, (4.58)

ξ is a real parameter which runs from 0 to ∞ and specifies the gauge, and αⁱ(x) is a number that depends onxonly.αⁱ is averaged with an exponential weighting factor in the quantization procedure, so its precise value is unimportant. The limitξ→0 corresponds to the unitary gauge.

The quantization procedure in theR_ξ gauges introduces additional terms into the effec-tive Lagrangian density that modify the structure of the vector and scalar propagators. The Goldstone bosons have not been removed from the Lagrangian, and occur in internal lines in Feynman diagrams. They do not occur as external states (except in connection with the equivalence theorem, introduced inSection 8.5.1). The quantization also introduces terms that can be represented by a set of N ghost fields η_i, i = 1,· · ·, N. These are fictitious particles that do not correspond to physical states but do circulate in internal loops. They are needed to ensure unitarity and renormalizability, and can be treated like complex scalar fields except that they obey Fermi statistics. The ghost vertices are given by the effective interaction

L^ghost=g(∂^µη^†_i)c_ijkη_jA^k_µ+g²

ξ η_i^†η_jhν|LⁱL^jφ⁰i (4.59) and are shown inFigure 4.2.There is a factor of−1 for each closed ghost loop.η^†_i andη_iare to be viewed as independent anticommutingc-numbers, not necessarily related by Hermitian or complex conjugation. The notation emphasizes that there is an effective propagator iD^G(x−x⁰) =h0|T[η(x), η^†(x⁰)]|0i. Clear discussions and derivations of the formulae may be found in (Weinberg, 1995; Pokorski, 2000).

The momentum space propagator for the gauge bosons in an arbitraryRξ gauge is iD^µν_V (k) =−i

g^µν−k^µk^ν(1−1/ξ) k²−M²/ξ

1

k²−M², (4.60)

which is an N ×N matrix in the space of gauge indices.⁵ For practical calculations it is often convenient to rewrite this (within the subspace of broken generators) as

iD^µν_V (k) =−i

g^µν−k^µk^ν M²

1

k²−M² − i M²

k^µk^ν

k²−M²/ξ. (4.61) Theξ dependent piece, which will ultimately cancel otherξ dependent terms in the Higgs

5There is an implicit +iin each propagator, e.g., 1/(k²−M²)→1/(k²−M²+i).

b

Figure 4.2

Left: Induced three-point vertex between one scalar and two gauge fields,