3.5 Quantum field theory on a lattice
3.5.5 Wilson loops and confinement
Y
(x,µ)∈L/
dµ(Uµ(x))
f[U]e−βS[U] (3.156) is a function of the link variables corresponding to the links (x, µ)∈L,
U|L={Uµ(x)|(x, µ)∈L}. (3.157) Sincef and S are both invariant under the local gauge transformation (3.153), we obtain
F[U|L] = Z
Y
(x,µ)/∈L
dµ(Uµ(x))
f[U0]e−βS[U0]. (3.158) We can now change the integration variables fromU toU0. Due to the invariance of the Haar measure, dµ(U) =dµ(U0), we find that
F[U|L] =F[U0|L] =F[ ˜U] (3.159) is a constant [11]. This finally results in
hfi= 1
ZF[ ˜U] = 1 Z
Z
Y
(x,µ)∈L/
dµ(Uµ(x))
f[ ¯U]e−βS[ ¯U], (3.160) where
U¯µ(x) =
(Uµ(x) for (x, µ)∈/ L ,
U˜µ(x) for (x, µ)∈L . (3.161) This means that in calculations of expectation values of gauge-invariant functions, one may fix the link variables on L to arbitrary prescribed values. One example of such a gauge fixing is the temporal gauge, where on an infinite lattice one choosesU4(x) =1 for allx. A complete gauge fixing is achieved by fixing the link variables on a maximal set of links without closed loops [11].
3.5.5 Wilson loops and confinement
Let us consider first the Euclidean formulation of continuum gauge theories and a rectan-gular curve CR,T with sides of lengths R and T in Euclidean spacetime (R is associated with space, T with time). One can show that the energy of the gauge field in the pres-ence of two static color sources (a quark and an antiquark), separated by a distanceR, is related to the large T behavior of the Wilson loop (in the fundamental representation of the gauge group) corresponding to the curveCR,T,
WCR,T =P ei
H
CR,TdxµAµ
, (3.162)
cf. Sec. 3.4. The static quark-antiquark potential V(R) (including self-energy contribu-tions) is obtained from [11]
TrWCR,T
T→∞
∝ e−T V(R) (3.163)
so that
V(R) =− lim
T→∞
1 T log
TrWCR,T
. (3.164)
The expectation value is computed with respect to the Euclidean Yang-Mills action, cf. Sec. 3.3.4,
TrWCR,T
= 1 ZE
Z
[DA(x)] TrWCR,Te−SEYM[A]. (3.165) The coefficient
σ= lim
R→∞
1
RV(R) =− lim
R,T→∞
1 RT log
TrWCR,T
(3.166)
is called the string tension [11]. If the limitsR→ ∞and T → ∞lead to a finite, non-zero result for the string tension σ, the static quark-antiquark potential asymptotically rises linearly with the separation R,
V(R)R→∞∝ σR , (3.167)
which results in a constant force between widely separated color sources. This behavior is called static quark confinement [11]. In this case, large Wilson loops obey the area law
TrWCR,TR,T→∞
∝ e−σA, (3.168)
where A = RT is the area of the rectangular spacetime curve that defines the Wilson loop. It is customarily assumed that quarks are confined if an area law holds for loops of large area in pure gauge theory [17]. This suspected relation is referred to as the Wilson loop criterion, which is often used to distinguish different phases, with and without quark confinement. Consequently, the associated string tension is used as an order parameter in many numerical and analytical investigations of lattice gauge theories [11].
Within the continuum formulation, the path integral has only a formal meaning. To define it rigorously, we have to use the lattice versions of the above equations. The lattice version of the Wilson loop defined in Sec. 3.4 is given by an ordered product of link variables Uµ(x) around a closed contour on the spacetime lattice. A contour C on the lattice can be specified by its initial point x and by thendirections µi of the links which form the contour (µi may be both positive or negative). If the contour is closed, we have Pn
i=1µi = 0. The Wilson loop associated to the closed contour is defined by
WC =Uµn(x+µ1+µ2+. . . µn−1)· · ·Uµ2(x+µ1)Uµ1(x). (3.169) The smallest closed contour (with non-zero area) is given by an elementary square of the lattice, the associated Wilson loop is the plaquette variable defined in Eq. (3.141), which is used to construct the gauge field action.
The energy of a static quark-antiquark pair measured in lattice units is related to the expectation value of a rectangular Wilson loop on the lattice. Let ˆR be the number of links in the space direction and ˆT the number of links in the time direction, then the expression analogous to Eq. (3.164) is [12]
Vˆ( ˆR) =− lim
Tˆ→∞
1 TˆlogD
TrWCˆ
R,Tˆ
E
. (3.170)
50 3.5 Quantum field theory on a lattice
This relation allows for the computation of the static interquark potential on the lattice using numerical methods [11]. To determine the physical potentialV(R), one has to take the appropriate continuum limit of the lattice version (3.170), cf. Sec. 3.5.6.
When ˆV( ˆR) is determined from Monte Carlo simulations where both ˆR and ˆT are not very large, the potential is not expected to be of the form ˆV = ˆσR. In this case, self-energyˆ contributions, which are proportional to the perimeter ˆR+ ˆT of the curve, will compete with the area term in
D
TrWCˆ
R,Tˆ
E
=e−ˆσRˆTˆ−ˆα(R+ ˆˆ T)+ ˆβ. (3.171) The string tension ˆσ (in lattice units) can be extracted from so-called Creutz ratios,
γ( ˆR,Tˆ) =−log D
TrWCR,ˆTˆ
E D
TrWCR−1,ˆ T−1ˆ
E D
TrWCR,ˆT−1ˆ
E D
TrWCR−1,ˆ Tˆ
E (3.172)
where the perimeter terms are eliminated [12].
In two spacetime dimensions, the partition function factorizes and one finds that an exact area law holds for all couplings and all (unitary) gauge groups, cf. Sec. 4.
In a four-dimensional spacetime, however, the situation is more complicated. A lin-early rising quark-antiquark potential (up to distances where dynamical fermions might screen the interaction) cannot be generated in perturbation theory. Since there is no way to calculate the potential analytically, one has to rely on numerical methods. However, ignoring vacuum polarization effects (in the absence of dynamical fermions) analytic re-sults can be obtained in the strong coupling regime. In analogy to the high temperature expansion in statistical mechanics, one expands the exponential e−βS[U] in the partition function in powers of the inverse coupling β. In the leading order of this approximation, one finds that QCD confines quarks since the expectation value of the trace of the Wilson loop exhibits an area law behavior [12]. However, the same result is obtained in the strong coupling approximation for the Abelian U(1) gauge theory, where it is known that the potential is given by the Coulomb law and where we do not expect confinement in the continuum. It can be shown that the lattice U(1) gauge theory possesses a weak coupling Coulomb phase, and it has been confirmed in numerical simulations that the strong cou-pling regime is separated from the weak coucou-pling regime by a phase transition. Therefore, it is necessary to check if the confining property of QCD persists into the small coupling regime [12]. Although it has not (yet) been possible to analytically relate the strong cou-pling limit of QCD to perturbative weak coucou-pling results, the hope (based on numerical evidence) is that no such phase transition exists for non-Abelian SU(N) gauge theories and confinement survives in the weak coupling regime [18].
In the strong coupling limit, the flux lines connecting the quark-antiquark pair are squeezed into narrow tubes (strings) along the shortest path joining the pair. This string is not allowed to fluctuate forg→ ∞,β →0. Fluctuations may, however, destroy confine-ment when one studies the continuum limit. In two spacetime dimensions, the persistence of confinement in the continuum limit is not surprising since in one space dimension there is no way the string can fluctuate. In real QCD, however, there is no a priori reason why confinement could not be lost in the continuum limitg→0,β→ ∞ [12]. Up to now, this question can only be answered with the help of numerical lattice simulations. Although the linear rising potential in pure non-Abelian SU(N) gauge theories could be established as a numerical fact, it is not yet known how the theory results in the formation of flux tubes with constant energy density [18].