3.3 Quantum chromodynamics
3.3.4 Functional integrals for gauge fields
Since the Yang-Mills Lagrangian is invariant under local gauge transformations, some de-grees of freedom of the gauge field are unphysical because they can be adjusted arbitrarily by gauge transformations. This means that the functional integral over a gauge field must be defined carefully and that subtle aspects of this construction can introduce new ingredients into the quantum theory [7].
To quantize pure gauge theory, we have to define the functional integral Z =
Z
[DA(x)]eiSYM[A]= Z
[DA(x)] exp
i Z
d4x
− 1
4g2(Fµνi (x))2
, (3.64) where the measure is taken to be invariant under local gauge transformations. For both Abelian and non-Abelian gauge fields, the Lagrangian remains unchanged along the direc-tions in the space of field configuradirec-tions which correspond to local gauge transformadirec-tions.
Hence, the functional integral is badly defined because we are redundantly integrating over a continuous infinity of physically equivalent field configurations. A method for dealing with this redundancy was invented by Faddeev and Popov and works in the following way:
The integration over physically equivalent gauge configurations is constrained by imposing a gauge fixing condition, Gi(A) = 0, at each spacetime point (iis a color index). To this end, the identity
1 = ∆FP(A) Z
[Dh(x)]δ Gi
Ah
, (3.65)
which is basically the definition of the Faddeev-Popov determinant ∆FP, is inserted into the path integral [10]. HereAh denotes the gauge field which is obtained from the fieldA through the local gauge transformation (3.58) withh(x) = exp (iα(x)) = exp (iαj(x)tj),
Ahµ(x) =eiα(x)Aµ(x)e−iα(x)−i
∂µeiα(x)
e−iα(x), (3.66) whose infinitesimal form is given by
Aαµ(x) =Aµ(x) +i[α(x), Aµ(x)] +∂µα(x), (3.67) (Aα)iµ(x) =Aiµ(x) +fijkαk(x)Ajµ(x) +∂µαi(x) =Aiµ+ (Dµα)i, (3.68) where Dµ is a covariant derivative acting on a field in the adjoint representation of the gauge group. In the adjoint representation, the matrix elements of the generators are given by the structure constants, (tadjj )ik =ifijk, cf. Sec. 2.5, and therefore
∂µ−iAjµtadjj
ik =δik∂µ+fijkAjµ. (3.69) The integration measure is given by a product of Haar measures at each spacetime point,
[Dh(x)] =Y
x
dµ(h(x)), (3.70)
where dµ(h) is the invariant Haar measure of SU(N), cf. Sec. 2.5. Similarly, the delta functional in Eq. (3.65) is given by a product of delta functions at each spacetime point.
Since dµ(h) = dµ(hh0), the Faddeev-Popov determinant is invariant under local gauge transformations,
Inserting the identity (3.65) into the functional integral (3.64) and interchanging the order of integration (over hand A) leads to
Z =
We can now change the integration variable from A toAh−1 and use that SYM In this expression, the group integrationR
[Dh(x)] has been factored out and may therefore be ignored. It only leads to an (infinite) normalization constant which is fortunately irrelevant for the computation of correlation functions. The correct expression for Z is therefore given by [10]
Z = Z
[DA(x)]eiSYM[A]∆FP(A)δ(Gi(A)). (3.75) Let us consider, e.g., the generalized Lorentz gauge condition [7]
Gi(A(x)) =∂µAiµ(x)−ωi(x) (3.76) with an arbitrary function ωi(x). Equation (3.75) holds for anyωi(x), so it remains valid if we replace the right hand side by any normalized linear combination involving different functions ωi(x). We can therefore integrate over ωi(x) with a Gaussian weight function and obtain (up to an irrelevant normalization factor)
Z =
To compute the Faddeev-Popov determinant, it is sufficient to consider the infinitesimal gauge transformation (3.67). In analogy to the properties of delta functions of a finite number of integration variables,
36 3.3 Quantum chromodynamics
the Faddeev-Popov determinant is given by the functional determinant [10]
∆FP(A) = det
δGi(Aα(x)) δαj(y)
G=0
. (3.79)
The generalized Lorentz gauge condition (3.76) leads to δGi(Aα(x))
δαl(y) =∂µ δil∂µ+fijlAjµ(x)
δ(x−y) =∂µ(Dµ)ilδ(x−y). (3.80) For an Abelian gauge theory, such as QED, ∆FP(A) is independent of A (the structure constants are zero) and the Faddeev-Popov determinant can be treated as just another overall normalization factor. In the non-Abelian case however, the functional determinant contributes new terms to the Lagrangian. The determinant can be represented as a func-tional integral over new anticommuting fields (called Faddeev-Popov ghosts) belonging to the adjoint representation [7]
det (∂µDµ) = Z
[D¯c(x)][Dc(x)]eiRd4xLghost[c(x),¯c(x)] (3.81) with Lagrangian
Lghost[c(x),c(x)] = ¯¯ ci(x) −δik∂2−fijk∂µAjµ(x) ck(x)
= ¯ci(x) −δik∂2−fijk
∂µAjµ(x)
+Ajµ(x)∂µ
ck(x). (3.82) Although the scalar ghost fields are treated as anticommuting, the violation of the spin-statistics connection is acceptable because they are not associated with physical particles.
Nevertheless, we can treat these excitations as additional particles in the computation of Feynman diagrams.
The final Lagrangian, including quark fields and all of the effects of the Faddeev-Popov gauge fixing procedure, is given by [7]
L=− 1
4g2 Fµνi 2
− 1
2ξ ∂µAiµ2
+ ¯ψ(i /D−m)ψ+ ¯ci −∂µDµik
ck. (3.83) Correlation functions can be computed in perturbation theory by splitting the Lagrangian into terms which are quadratic in the fields on one side and into interaction terms which contain more than two field variables on the other side. If we rescale the gauge fields by A → gA (and change the parameter ξ → g2ξ), all the terms in the Lagrangian that are not quadratic in the fields are proportional to the coupling constantg.
In perturbation theory, one first computes the “free” propagators, which are obtained from the quadratic terms in the Lagrangian just as in a free theory (this corresponds to setting g = 0). Expanding exp(iR
d4xL) in powers of g, these propagators can be used to take the effects of the interaction terms into account in a systematic way (assuming that the coupling is small), and correlation functions of the full theory are obtained in a series in g (which needs to be renormalized, cf. Sec. 3.5.6). The Lagrangian resulting from the gauge-fixing procedure is in fact only meaningful in perturbation theory since the gauge-fixing condition does not always have a unique solution (in the space of gauge fields that are equivalent up to gauge transformations). This ambiguity, first noted by Gribov, is related to the property that there exist gauge fields for which the operator ∂µDµ has vanishing eigenvalues, cf. Ref. [14].
Computing correlation functions in perturbation theory is somehow an unnatural act since it involves splitting the highly symmetric Lagrangian into two parts. The alter-native proposed by Wilson is to violate Lorentz invariance rather than gauge invariance
and to study gauge field theories on a discrete lattice of spacetime points, cf. Sec. 3.5.
Lattice gauge theory provides the only known regularization scheme which is entirely non-perturbative.
The quadratic terms in the gauge field A are diagonal in the internal color space;
therefore the free gauge-field propagator in SU(N) gauge theory is basically the same as in the Abelian U(1) case (QED). The quadratic action can be written as (after rescaling the fields)
In complete analogy to scalar field theory (cf. Sec. 3.2), the free gauge-field propagator Djkνσ(x−y) = is found by inverting the differential operator in the Lagrangian. By Fourier transforming, we find that a solution of
δij gµν∂2− 1−ξ−1 where theiε-term in the denominator arises exactly as in the scalar case [7]. The Faddeev-Popov method guarantees that correlation functions of gauge-invariant operators are in-dependent of the value of ξ. The choice ξ = 0 is called Landau gauge, Feynman gauge corresponds to setting ξ= 1.
In a similar way, the quadratic terms of the ghost Lagrangian lead to the free ghost-field propagator [7]
The Faddeev-Popov gauge fixing procedure can be carried over to Euclidean Yang-Mills theory without complication. In this case, the starting point is the Lagrangian LE = 4g12 Fµνi 2
, whereFµν is the Euclidean field strength tensor. The Euclidean action is obtained by integrating over the Euclidean spacetime, SE = R
d4xELE, cf. Sec. 3.2.3.
After fixing the gauge, correlation functions are obtained from the generating functional ZE[J] =
The free gauge-field propagator in Euclidean spacetime is obtained from the quadratic terms in the Lagrangian and is found to be
(DE)jkνσ(x−y) =δjk where the integral is over Euclidean momenta (and the fields have again been rescaled by A → gA). A position space representation of the propagator can be obtained by