performing the Euclidean momentum integration. The general result in d-dimensional Euclidean spacetime is given by (cf. App. A)
(DE)jkµν(x−y) =δjkΓ d2 −1
In the previous section, we have introduced the covariant derivative to achieve invariance under local gauge transformations. This derivative can be regarded as a consequence of the concept of parallel transport (inspired from the theory of general relativity). If we allow for local transformations of the fields ψ with spacetime dependent h(x), fields at different spacetime points are measured with respect to different basis systems in the internal SU(N) symmetry space. The gauge field in the covariant derivative has to be introduced in order to compare ψ(x +dx) with the value ψ(x) would have if it were transported fromx tox+dx, keeping the axes (in the internal space) fixed.
3.4.1 Wilson lines
Consider now a pathx(s) in four-dimensional spacetime, which continuously connects two pointsx0=x(0) andx1 =x(1). We associate to the spacetime curve the so-calledWilson line, which is a curve-dependent element of the gauge group SU(N) and is defined as [7]
W(x1, x0) =P with ˙xµ(s) = dxdsµ. In this definition, the symbol P denotes the path-ordering operator, which is defined analogously to the time-ordering operator introduced in Sec. 3.1. This means that the Wilson line is defined as the power-series expansion of the exponential with the integrands ordered in such a way that higher values ofsstand to the left. For the (n+ 1)-th term in the expansion, we get nnon-commuting factorsAµ1(x(s1)),Aµ2(x(s2)),. . . , Aµn(x(sn)) (the si’s are the integration variables parametrizing the curve). These can be ordered in n! ways, which leads to the cancellation of the factorial that we get from the exponential series, i.e., consider instead W(x(t), x0) with 0 ≤t≤ 1 and assume that it is a continuous function of the parameter t, we see from Eq. (3.93) that this Wilson line fulfills the differential equation
d
dtW(x(t), x0) =ix˙µ(t)Aµ(x(t))W(x(t), x0), (3.94) which can be rewritten as
˙
xµ(t)DµW(x(t), x0) = 0. (3.95)
By changing the parametrization of the integration domain (restricted to s1 ≥s2≥. . .≥ we find a similar differential equation for W(x1, x(t)),
W(x1, x(t))
The covariant backward derivative D←µ has the property that the local gauge transforma-tions (3.57) and (3.58) result in
ψ(x)¯ ←
∂µ+iAµ(x)
→ψ(x)¯ ←
∂µ+iAµ(x)
h−1(x). (3.98) To determine the behavior of W under local gauge transformations, we transform the gauge field according to Eq. (3.58). It is obvious that Eqs. (3.95) and (3.97) have to hold also for the transformed variables, i.e.,
˙
xµ(t)Dµ0W0(x(t), x0) = 0 and W0(x1, x(t))
←
D0µx˙µ(t) = 0, (3.99) where the covariant derivativesD0µ and
←
D0µ are built from the transformed field A0µ. The transformation law forW then follows from the transformation of the covariant derivatives (cf. Eqs. (3.59) and (3.98)),
W(x1, x0)→W0(x1, x0) =h(x1)W(x1, x0)h−1(x0). (3.100) This implies that
W(x1, x0)ψ(x0)→W0(x1, x0)ψ0(x0) =h(x1)W(x1, x0)ψ(x0), (3.101) which is just the transformation law of a field at the spacetime point x1. Obviously, the combination ¯ψ(x1)W(x1, x0)ψ(x0) is invariant under local gauge transformations. The Wilson lineW(x1, x0) compensates for the different transformations at different spacetime pointsx1andx0 (corresponding to a local change of basis in the internal symmetry space).
It can be viewed as a parallel transporter for the fieldψ along the curvex(s), allowing for the comparison of fields at different points in spacetime by converting the transformation law at point x0 to that atx1.
3.4.2 Closed Wilson lines: Wilson loops
If x(s) describes a closed curve C in spacetime, i.e., x(0) =x(1), the Wilson line is called a Wilson loop6,
6Note that in the literature, the term “Wilson loop” often refers to the trace of the Wilson line around a closed path. Here, we usually use the term “Wilson loop” for the untraced matrix. We often refer to the size of the underlying spacetime curveCsimply as “the size of the loop”.
40 3.4 Wilson loops
If the field ψ(x) is transported around the closed curve C, it accumulates a non-trivial SU(N) “phase factor” given by the Wilson loop matrix WC.
Under local gauge transformations, the Wilson loop transforms according to
WC(x0, x0)→WC0(x0, x0) =h(x0)WC(x0, x0)h−1(x0), (3.103) which means that the trace of the Wilson loop is a non-local, gauge-invariant observable,
Tr
WC0 (x0, x0) = Tr
h(x0)WC(x0, x0)h−1(x0) = Tr{WC(x0, x0)}. (3.104) For an infinitesimal transport along a straight line from x to x+dx, the Wilson line can be approximated by
W(x+dx, x)≈eiAµ(x)dxµ = 1 +iAµ(x)dxµ+. . . . (3.105) Requiring the transformation law W(x +dx, x) → W0(x+dx, x) = h(x+dx)W(x + dx, x)h−1(x) for the infinitesimal Wilson line results in
1 +iA0µ(x)dxµ
= (h(x) +∂νh(x)dxν) (1 +iAµ(x)dxµ)h−1(x) (3.106) and we recover the transformation law for the gauge field given in Eq. (3.58).
Let us now consider an infinitesimal Wilson loop associated to an infinitesimal square in spacetime with corners at x,x+ε, x+ε+δ, x+δ (with infinitesimal four vectors ε and δ). The parallel transport along the sides of the square, fromx tox+δ tox+δ+ε tox+εand back tox, is given by
W(x, x) =W(x, x+ε)W(x+ε, x+ε+δ)W(x+ε+δ, x+δ)W(x+δ, x)
≈e−iAµ(x)εµe−iAµ(x+ε)δµeiAµ(x+δ)εµeiAµ(x)δµ. (3.107) By making use of the relation
eεBeεC =eε(B+C)+ε
2
2[B,C]+O(ε3) (3.108)
and by expanding the fields around the pointx, we obtain
W(x, x)≈ei(∂µAν(x)−∂νAµ(x)−i[Aµ(x),Aν(x)])δµεν =eiFµν(x)dσµν (3.109) with the infinitesimal area element dσµν = δµεν. The field strength tensor Fµν can be interpreted as the curvature tensor of the internal symmetry space (in analogy to general relativity).
3.4.3 Divergences in perturbation theory
Expanding the path-ordered exponential in Eq. (3.102) in powers of Aµ leads to a per-turbative expansion for Wilson loops (rescaling the fields by A→ gA immediately leads to an expansion in powers of the coupling constant). We will now study the divergences occurring in calculations of expectation values of Wilson loops in perturbation theory. For simplicity, let us first consider the Abelian gauge group U(1). In the Abelian case, the fields commute and therefore path ordering becomes irrelevant. With
A= I
C
Aµ(x)dxµ (3.110)
we obtain, according to Wick’s theorem, which means that we have to compute only a single expectation value. It is instructive to perform this calculation for a circular curve in four-dimensional Euclidean spacetime with a sharp momentum cut-off in the Fourier representation of the gauge-field propagator, i.e., the integral (3.90) is restricted tok2<Λ2 (all quantities in this section are Euclidean; the subscript E is omitted). We parametrize the circle of radiusR by
xµ(s) =R(cos(2πs),sin(2πs),0,0), (3.112) which leads to
˙
xµ(s) ˙xµ(t) = (2πR)2cos(2π(s−t)). (3.113) In Feynman gauge (ξ = 1), the expectation value (3.111) is obtained from
A2 Integrating over the momentum components perpendicular to the plane of the circular curve (the components k3 and k4 in the parametrization (3.112)),
Z Changing the integration variable fromκtox=κ/Λ and using the integral representation of the Bessel function,
Making use of the asymptotic form of the Bessel functions for large arguments x ΛR1 , Jn(xΛR)≈
one finds that the expectation value is linearly divergent in the momentum cut-off Λ R1, A2
=c(ΛR) + subleading terms, (3.120)