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Z(N) interface tension in a hot SU(N) gauge theory

arXiv:hep-ph/9205231v1 24 May 1992BNL–February, 1992Z(N) interface tension in a hot SU(N) gauge theoryTanmoy BhattacharyaService de Physique Th´eoriqueCEN–Saclay91191 Gif–sur–Yvette Cedex, FranceAndreas GockschDepartment of PhysicsBrookhaven National LaboratoryUpton, New York 11973, USAChris Korthals AltesCentre Physique Th´eorique au CNRSSection 2, B.P. 907 LuminyF 13288 Marseille, FranceRobert D. PisarskiDepartment of PhysicsBrookhaven National LaboratoryUpton, New York 11973, USAAbstractThe interface tension between Z(N) vacua in a hot SU(N) gaugetheory (without dynamical fermions) is computed at next to leading orderin weak coupling.

The Z(N) interface tension is related to the instantonof an effective action, which includes both classical and quantum terms;a general technique for treating consistently the saddle points of sucheffective actions is developed. Loop integrals which arise in the calculationare evaluated by means of zeta function techniques.

As a byproduct, upto two loop order we find that the stable vacuum is always equivalent tothe trivial one, and so respects charge conjugation symmetry.

I.IntroductionIn the absence of dynamical fermions, SU(N) gauge theories possess a globalZ(N) symmetry associated with the center of the gauge group [1].Confinementimplies that at zero temperature the vacuum is Z(N) symmetric, but at nonzerotemperature there is a phase transition to a deconfining phase, where the Z(N)symmetry is spontaneously broken [2]. In the deconfined phase, a system of infiniteextent falls into one of the N degenerate vacua, but in a finite volume bubbles ofdifferent vacua form [3], separated by domain walls.

The action of these domain wallsis proportional to the interface tension, and so controls the dynamics of large Z(N)bubbles.We previously discussed how to compute the interface tension α in terms of thetemperature T and the coupling constant g [4]. In weak coupling,α ≃4(N −1)π23√3NT 3g 1 −(15.27853...) g2N16π2!.

(1.1)The term at leading order, T 3/g, is the result of [4]; the principal result of this paperis the computation of α at next to leading order, gT 3. In this expression the couplingg represents a running coupling constant at a temperature T; the exact relationshipof this g to the bare coupling constant is given in (4.11).The physical applications of our results is at best indirect, for it only applies toa world devoid of dynamical quarks.

It can be compared to numerical simulations oflattice gauge theories [3]; indeed, it was these measurements of α which led us to askif it is computable in weak coupling.Nevertheless, the methods and techniques which we have developed to compute theinterface tension are, we believe, of general interest. The calculation of the interfacetension reduces to an instanton problem in an effective theory.

This effective theoryis one dimensional [5, 6], as it describes the profile of the interface in the directionperpendicular to the domain wall. Remember how a standard instanton calculationproceeds [6]: the instanton is the solution of the classical equations of motion, withthe action for the domain wall proportional to 1/g2.

The corrections to the classicalaction are given by expanding the classical action in a background instanton field.Corrections at next to leading order are given by integrating over terms of quadraticorder, which gives corrections to the action of order one.Contrast this with the interface tension in (1.1). The action of the domain wall is1

α times the transverse volume. By its mass dimensionality, α is proportional to T 3,while the constants, and the factors of N, result from the detailed path the Z(N)interface takes in the space of SU(N) gauge fields.What is surprising is that αstarts out as 1/g in weak coupling, for if the Z(N) interface were a solution to theclassical equations of motion, it would start instead as 1/g2.

This is because theeffective action (in one dimension) which controls the Z(N) interface is the sum ofthe classical action plus a quantum term, obtained by by integrating out fluctuationsat one loop order. In the effective action the classical piece acts like a kinetic term,and the quantum piece like a potential.

The Z(N) instanton is a stationary pointonly of the full effective action, and this balance between classical and quantum termstransforms the usual factor of 1/g2 into a 1/g.It is then unclear how to compute corrections to α beyond leading order. Surelyone cannot compute blindly: if the effective action is expanded in fluctuations abouta background instanton field, and these fluctuations integrated out, how is doublecounting avoided?

That is, how to differentiate between the quantum fluctuationswhich generate the effective action in the first place, from the quantum fluctuationswhich are properly included in the expansion about the instanton?In this paper we solve this problem for the Z(N) interface in a manner applicableto arbitrary effective actions. The method generalizes what is known as the “con-strained” effective potential [7].

Here we use it to reduce the four dimensional gaugetheory to an effective scalar theory in one dimension. The method is trivial in design:a delta function constraint for new degrees of freedom is inserted into the functionalintegral.

The original degrees of freedom are then integrated out, producing an effec-tive theory for the new field. There is obviously no problem with double counting,since extra degrees of freedom are introduced in the first place.For the interface tension, this method shows that the leading corrections are oforder g2 times that at leading order.

These effects are due entirely to corrections ingoing from the four dimensional theory to the effective, one dimensional theory. Theyenter both for the kinetic and potential terms in the effective action.

For example,the corrections to the kinetic term transform the bare into a renormalized couplingconstant.The order parameter which distinguishes different Z(N) vacua is the Wilson lineat nonzero temperature [2]. It is convienient to parametrize a vacuum expectationvalue for the Wilson line by giving the gauge field A0 a nonzero value; the effective2

action for a Z(N) instanton is then related to the free energy in such a backgroundfield. The Z(N) instanton is slowly varying in space, so that previous results in aconstant A0 field [8 – 14], especially those by Belyaev and Eletsky [11], Enqvist andKajantie [12], and Belyaev [14], can be used.

The technical problem of computingthe free energy in a background A0 field is done most easily by using zeta functiontechniques [15].Much of the interest in considering a hot theory with A0 ̸= 0 concerns the pos-sibility of the vacuum spontaneously generating an expectation value for A0, and sofor the Wilson line [9–14]. If this happens for N ≥3, it implies that the vacuum atnonzero temperature spontaneously breaks charge conjugation symmetry.

For theo-ries without dynamical fermions, following Belyaev [14] we do not find evidence forthe spontaneous breaking of charge conjugation symmetry at two loop order: up tothe usual Z(N) rotations, the stable vacuum is the trivial state, with A0 = 0 and theWilson line equal to one.The outline of the paper is as follows. In sec.

II we review the calculations of[4]. Sec.

III outlines the general calculation of the interface tension. The calculationof the interface tension at next to leading order in Feynman gauge is given in sec.IV.

Sec. V considers the computation in arbitrary covariant gauges.

There are twoappendices. In appendix A we prove that the path chosen for the Z(N) instanton hasminimal action for N = 3 and ∞.

Appendix B summarizes various integrals neededin a constant background A0 field.II.Z(N) interface at leading orderWe begin by rederiving the results for the Z(N) interface tension at leading order[4]. This is not mere repetition, for here we use a more natural basis for the generatorsof SU(N) matrices than before.

The subtleties of how to derive the effective actionare deferred until sec. III.We work in euclidean spacetime at a temperature T, so the euclidean time τvaries from 0 to β = 1/T.

In the spatial directions the system is a long tube, oflength L in the z direction, of length Lt in the two remaining spatial directions, ⃗xt,with L ≫Lt ≫β. The volume in the directions transverse to z is Vtr = βL2t.A Z(N) interface is constructed by assuming that the system is in one Z(N) phaseat one end of the tube, z = 0, and in another Z(N) phase at the other end, for z = L.3

This forces a Z(N) interface along the z direction, with the action of the interfaceequal to the interface tension, α, times the transverse volume, Vtr. As a practicalmatter, while L ≫Lt, in the end both are taken to infinity.The Z(N) symmetry is determined by the the trace of the Wilson line in thefundamental representation,tr Ω(A) =1N tr P exp igZ β0 A0(x)dτ!!

;(2.1)P refers to path ordering. Let A0 have the value,Acl0 (x) = 2πTgcN q tN ,(2.2)where tN is the diagonal matrixtN = c1N−100−(N −1),c =1q2N(N −1).

(2.3)In the presence of this A0 field the Wilson line equalstr Ω(Acl) = e2πiq/N1 −1N1 −e−2πiq. (2.4)The trivial vacuum is A0 = q = 0, with tr Ω= 1; Z(N) transforms of the trivialvacuum occur for q = j, with tr Ω= e2πij/N; j is an integer between 1 and (N −1).Thus the simplest Z(N) interface is constructed by promoting the parameter q in(2.2) to a function of z, satisfying q(0) = 0 and q(L) = 1, so that tr Ω= 1 at z = 0,and tr Ω= e2πi/N at z = L.In order to proceed we need a useful parametrization for the remaining generatorsof SU(N) .

The diagonal generators are chosen similarly to (2.3). For example,t(N−1) =1q2(N −1)(N −2)1N−2...−(N −2)0. .

.00. (2.5)Altogether there are N −1 diagonal generators, with ti from i = 2 to N. In SU(2)t2 is proportional to the Pauli matrix σ3, while in SU(3) t3 is proportional to theGell–Mann matrix λ8.4

For the off–diagonal generators we follow Belyaev and Eletsky [11] and use a ladderbasis. For example,t+N,1 =1√2.

. .01. .

.00......,(2.6)t−N,1 =1√2......00. . .10. .

.. (2.7)All elements not indicated vanish.

Each diagonal generator ti has 2(i −1) laddergenerators associated with it: t±i,j, with j running from j = 1 to i −1. For a laddergenerator only one element is nonzero,(t+i,j)mn =1√2 δin δjm , (t−i,j)mn =1√2 δim δjn .

(2.8)We are working in the fundamental representation, so the matrix indices m and nvary from 1 to N.These generators form an orthogonal set, with the diagonal generators normalizedastr(titi′) = 12 δii′ ,(2.9)and the offdiagonal generators astr(t+i,jt−i′,j′) = 12 δii′δjj′,tr(t+i,jt+i′,j′) = tr(t−i,jt−i′,j′) = 0 . (2.10)Observe that the metric in the ladder basis is diagonal in the i and j indices, butoff-diagonal in the ± indices.The advantage of the ladder basis is the simplicity of the commutation relations.For the generator tN, since from (2.3) it involves the unit matrix in the first N −1components, 1N−1, the only nontrivial commutator of tN is with its associated laddergenerators, t±N,j.

This commutator is just a constant times the same ladder generator:[tN, t±N,j] = ± cN t±N,j . (2.11)This relation is familiar from SU(2), from where up to overall constants, t2 is σ3, andt±2,1 are the matrices σ±.5

The Wilson line in the adjoint representation can be computed using (2.11). Thisis defined asΩabadj(A) =2N2 −1 ta P exp igZ β0 [A0(x), ]!tb!,(2.12)where “a” and “b” refer to the (N2 −1) adjoint indices, and [A0, ] denotes the adjointoperator, [A0, ]X = [A0, X].

For the constant A0 field of (2.2), the only nontrivialelements of the adjoint Wilson line are those involving the ladder operators t±N,j, andso its trace istr Ωadj(Acl) = 1 −2N + 1 (1 −cos(2πq)) . (2.13)The adjoint Wilson line is unaffected by the Z(N) symmetry: for the Z(N) degeneratevacua, where q is an integer, tr Ωadj = 1.The classical action isScl(A) =Z β0 dτZd3x 12 trG2µν,(2.14)where Gµν = ∂µAν −∂νAµ −ig[Aµ, Aν] is the field strength tensor.

For the interfaceproblem the field q of (2.2) is assumed to be a function only of z, the length alongthe tube. For this field the classical action reduces toScl(Acl) = Vtr4π2T 2g2N(N −1)Zdz dqdz!2.

(2.15)Using the classical action, a solution to the equations of motion is q(z) = z/L. There isno true interface, since the action vanishes like 1/L as L →∞.

But this is misleading,for classically there is no sign of the Z(N) symmetry, as all values of q degenerate.This classical degeneracy is lifted by quantum effects [8]. This is shown by calcu-lating the action in the presence of the background field in (2.2).

For the time beingwe assume that q is independent of z, and concentrate on the q–dependent termswhich lift the degeneracy in q. With Aµ = Aclµ + Aquµ , in background field gauge [16]the gauge fixing and ghost terms areSgf(A, η) =Z β0 dτZd3x 1ξ trDclµ Aquµ2 + ¯η−Dclµ Dµη!,(2.16)where Dµ = ∂µ −ig[Aµ, ] is the covariant derivative in the adjoint representation,Dclµ = ∂µ −ig[Aclµ, ], and η is the ghost field.6

For a constant field q it is especially easy to expand the full action, Scl + Sgf, inquadratic order in the fluctuations Aqu, and then integrate them out:Squ1 (Acl) = 12 tr ln −D2clδµν + 1 −1ξ!Dclµ Dclν!−tr ln−D2cl. (2.17)The subscript on Squ1indicates the quantum action at one loop order.The firstterm on the right hand side is from the integration over the gauge fields, the secondfrom that over the ghosts.

Note that because Aclµ is assumed to be independent ofspacetime, there are no terms in the inverse gauge propagator proportional to Gclµν.This quantum action is independent of the gauge fixing parameter, ξ. To see this,note that the derivative of Squ1with respect of ξ−1 is∂Squ1 (Acl)∂ξ−1= 12 tr −Dclµ Dclν δµν−D2cl+ (1 −ξ) Dcl,µDcl,ν(−D2cl)2!!.

(2.18)That is, the derivative is −DµclDνcl times the gauge propagator in the background field.Normally this propagator is difficult to compute because the covariant derivativedoesn’t commute with itself. For a constant background field, however, it does, andso the ordering of the Dclµ ’s is inconsequential.

Then (2.18) reduces to∂Squ1 (Acl)∂ξ−1= ξ2 tr(1) . (2.19)Thus the variation of the quantum action with respect to ξ is a constant independentof the background field, which can be dropped.Adopting Feynman gauge, ξ = 1, the commutation relations of the ladder basisreduce the color trace in Squ1to an abelian problem.

As Dcl0 is the adjoint covariantderivative, it is independent of the background field unless it acts upon the laddermatrices t±N,j:Dcl0 t±N,j = (∂0 ∓2πTqi) t±N,j ≡D0±t±N,j . (2.20)With the euclidean four momentum equal to (k0,⃗k), at nonzero temperature k0 =2πnT for integral n, and the covariant derivative becomesiD0± →k0± ≡2πT(n ± q) .

(2.21)The sum over n implicit in the trace includes both positive and negative values, withthe sum over k0−equal to that for k0+. Hence the quantum action reduces toSqu1 (Acl) = 2(N −1) tr ln(k0+)2 + k2,(2.22)7

k2 = ⃗k2.This result is typical of loop effects in a constant background field. For the degreesof freedom along the ladder operators t±N,j, the propagators are as in zero backgroundfield, except that k0 is shifted by a constant amount, to k0±.

The propagators for theremaining degrees of freedom are unaffected by the background field.From (2.22), at one loop order the q–dependence of the free energy reduces to(N −1) copies of that for SU(2). To isolate the q–dependence in Squ1 , consider itsderivative with respect to q:∂Squ1 (Acl)∂q= 4(N −1)(2πT) tr k0+(k0+)2 + k2!.

(2.23)The integral is most easily done by integrating first over the spatial momenta:(Vtr L) T+∞Xn=−∞Zd3k(2π)3 k0+(k0+)2 + k2!= −(Vtr L) πT 3+∞Xn=−∞(n + q)|n + q| . (2.24)This sum, while formally divergent, is interpreted using zeta function regularization[15].

The zeta function ζ(p, q) is defined asζ(p, q) =+∞Xn=01(n + q)p . (2.25)Hencetr k0+(k0+)2 + ⃗k2!= −(Vtr L) πT 3 (ζ(−2, q) −ζ(−2, 1 −q)) .

(2.26)Usingζ(−2, q) = −112ddqq2(1 −q)2,(2.27)integration of (2.23) givesSqu1 (Acl) = Vtr4π2T 43(N −1)Zdz q2(1 −q)2 . (2.28)From the nature of the sum in (2.24), Squ1is periodic in q, and is invariant undershifts of q →q + l for any integer l. Thus in (2.28) q is defined modulo one.

Also, aq–independent constant in (2.28) was dropped; this constant is just the free energyof an ideal gas of N2 −1 gluons at a temperature T.As promised, the classical degeneracy in q is lifted by quantum effects: the minimaof the theory are now at integral values of q = j, where tr Ω= exp(2πij/N).8

In (2.28) the length in the z direction, L, is replaced by the integral over z. Ofcourse for a constant field this substitution doesn’t matter, but consider the interfaceproblem. Introducing the dimensionless coordinate z′,z′ =sN3 gT z ,(2.29)the sum of the classical and quantum actions becomesScl(Acl) + Squ1 (Acl) = Vtr4π2(N −1)√3NT 3gZdz′ dqdz′!2+ q2(1 −q)2.

(2.30)We can view minimization of this effective action as a problem in mechanics, withthe coordinate z′ as the “time”. The classical action contributes the kinetic energy,(dq/dz′)2, while the quantum action produces a standard double well potential, q2(1−q)2.

For any solution to the equations of motion the energy, E = (dq/dz′)2−q2(1−q)2is conserved, dE/dz′ = 0. With the Z(N) interface we want a solution which obeysthe boundary conditions q(0) = 0, q(L′) = 1, as L′ = (qN/3) gT L →∞[5,6].

Bythe boundary conditions the instanton has zero energy, E = 0. Consequently, for theZ(N) instantonZdz′ dqdz′!2+ q2(1 −q)2= 2Z 10dq q(1 −q) = 13 .

(2.31)If the total action of the interface is the transverse volume Vtr times the interfacetension α, then at leading order [4]α = 4π2(N −1)3√3NT 3g . (2.32)Two assumptions must be justified.

The first is why Squ1can be computed for aconstant field q, and then applied to the Z(N) instanton, where q is clearly a functionof z. The reason can be seen from the definition of z′ in (2.29).

As there is no lengthscale in the rescaled action, in terms of z′ the instanton’s width is of order one. Forthe original coordinate, z, this implies that the Z(N) instanton is “fat”, with a widthof order 1/(gT).

In weak coupling this is much larger than the natural length scale ina gas of massless, nearly ideal gluons, which is 1/T. Thus while the Z(N) instantonfield is large in magnitude, Acl ∼T/g, it varies slowly in space, and at leading orderthis variation can be neglected.

Corrections to this approximation do enter beyondleading order.9

The second assumption is whether in the space of SU(N) gauge fields the pathchosen is of minimal action. The Z(N) instanton interpolates between tr Ω= 1 andtr Ω= exp(2πi/N).

By a global gauge rotation Ωcan be chosen as a diagonal matrix,involving the (N −1) diagonal generators of SU(N) , the ti’s. The quantum actionfor a general (constant) field can be computed directly [8].

We chose the simplestpath possible — straight along the tN direction — but it is not obvious that otherpaths, which wander offinto the direction of the other (N −2) ti’s, might not havelower action. (Our path is at least a local minimum.) For SU(2) there is only onepath possible.

In appendix A we show that for N = 3 and N = ∞, the path alongtN is minimal. On this basis we conjecture that this remains true for all 4 ≤N < ∞.III.General analysis of the Z(N) interfaceThe partition function of an SU(N) gauge theory isZ =Z[dAµ(x)] [dη(x)] e−Scl(A)−Sgf (A,η) ,(3.1)where Scl and Sgf are the classical and gauge fixing actions of (2.14) and (2.16).

Thegauge field Aµ = Aclµ + Aquµ ; for the time being the choice of Aclµ is left open. Thecoordinate of four dimensional spacetime is x = (τ, ⃗xt, z)In order to reduce this four dimensional theory to an effective theory in one di-mension we introduce the fieldq(z) =1VtrZ β0 dτZd2xtgcN2πT Acl0 (x) .

(3.2)This choice is obviously motivated by the definition of Acl0 ; unlike the scalar field q of(2.2), q is a matrix valued field in the adjoint representation of SU(N) . The onlysubtlety in the introduction of q(z) is our insistence on defining it not just as theaverage of Acl0 over time, but over the spatially transverse directions as well.

Thereason for this will become apparent shortly.The field q(z) is introduced into the functional integral by a delta function con-straint:Z =Z[dAµ(x)] [dη(x)] [dλ(z)][dq(z)] e−Scl(A)−Sgf (A,η)−Scon(λ,q,A) ,(3.3)Scon(λ, q, A) =Zdz 2i tr λ(z) q(z) −1VtrZ β0 dτZd2xtgcN2πT Acl0 (x)!!. (3.4)10

The constraint field λ(z) is introduced to enforce the definition of q(z) in (3.2), andso is also an adjoint matrix. With the overall factor of i in Scon, the original contourof integration for the constraint field λ(z) is along the real axis.As an aside, note that q(z) does not transform in a simple fashion under gaugetransformations.

For an infinitesimal gauge transformation ω, where Aµ →∂µω −ig[Aµ, ω],q(z) →1VtrZ β0 dτZd2xtgcN2πT (−ig)[A0(x), ω(x)] ,(3.5)assuming that ω(x) = ω(τ, ⃗xt, z) is periodic in τ.Nevertheless, no further termsbesides those in (3.3) are required in the measure of the functional integral: gaugefixing for Aµ takes care of that.The effective action Seff(q), is defined as the integral over all fields, exceptingq(z):Z =Z[dq(z)] e−Seff(q) . (3.6)How the effective action is computed in practice depends upon the problem at hand.But by introducing q(z) as an extra field into the functional integral, clearly there isno confusion possible about double counting degrees of freedom.The calculation of Seff(q) is straightforward at one loop order [7], and so ourdiscussion is brief.

The field q(z) is held fixed, while the gauge field and the constraintfield λ are seperated into classical plus quantum terms: Aµ = Aclµ + Aquµ and λ =λcl + λqu. Expanding the action of (3.3) to quadratic order,Scl(A) + Sgf(A, η) + Scon(λ, q, A) ≈S0 + S1 + S2 .

(3.7)The leading term is the sum of the classical action plus the constraint:S0 = Scl(Acl) + Scon(λcl, q, Acl) . (3.8)The linear terms determine the equations of motion:S1 = Scon(λqu, q, Acl) + 2 iZd4x trλclδν0 + iDclµ GclµνAquν .

(3.9)The constraint term, Scon(λqu, q, Acl), is linear in λqu, and so determines Acl(x). Theobvious choice is to take Acl(x) to be a function only of z,Acl(z) = 2πTgcN q(z) .

(3.10)11

The constraint term in S0 then vanishes, Scon(λcl, q, Acl) = 0.The second term in S1 modifies the equations of motion for the gauge field: theconstraint term acts as a source for the gauge field, proportional to λcl. As Acl isdetermined, letλcl(z) = −iDclµ Gclµ0(z) ,(3.11)from which S1 = 0.

(As Acl(x) is assumed to depend only upon z, the other com-ponents of the equations of motion for the gauge field are automatically satisfied. )At the stationary point λcl is purely imaginary, while the fluctuations λqu remainreal.

This shift in the contour for the constraint field λ — by an imaginary amount,keeping it parallel to the real axis — is standard.The quadratic terms in the action areS2 = −gcN2πTVtrZd4x(2i) tr (λqu(z) Aqu0 (x))+Zd4xZd4y trAquµ ∆cl,−1µνAquν+ ¯η−D2clη. (3.12)The first term in S2, involving λqu, is special to the constraint action.

Since λqu onlyenters into S2 linearly, it can be integrated out. This introduces a constraint for theintegration over Aqu0 (x):1VtrZ β0 dτZd2xt Aqu0 (τ, ⃗xt, z) = 0 .

(3.13)This constraint is completely innocuous. Remember that for the two spatial directions⃗xt, each length Lt is taken to infinity.

Integration is over all modes of λqu(z) whichobey (3.13), but the only modes which don’t are those constant both in τ and ⃗xt —in momentum space, modes with k0 = ⃗kt = 0. If the length in the ⃗xt directions areinfinite, the corresponding momenta ⃗kt take on all continuous values, and those with⃗kt = 0 have zero measure, and can be ignored.Being able to drop the constraint is an important point.

Suppose the constraintfield is defined not as an integral over τ and ⃗xt, but just as an integral over τ: thenq is a field in three, instead of one, dimension. Going through the same proceedureas above, at quadratic order integration is over all fields constant in time.

But thesemodes have nonzero measure in the functional integral. At finite temperature, themomentum k0 = 2πnT for integral n, and the constant modes, with n = 0, are ofcountable extent.12

For situations in which the constraint doesn’t matter — that is, where the effectivefields are of zero measure in the space of the original fields — the constraint methodsgive the same result as for the usual effective potential [7]. The constraint field λclplays the role of the external source, while the exchange of Acl for q mimics preciselythe process of Legendre transformation.

The effective potential is the energy of thevacuum in the presence of the external source, and so is properly minimized.The meaning of the effective action when the effective fields are of nonzero measurein the space of the original fields is unclear. We dwell on this point because of suchan analysis by Oleszczuk and Polonyi [10], who introduce an effective field in threedimensions, as the integral of A0 with respect to time.

Integrating out modes withk0 ̸= 0, they find a potential different from that of (2.28). The potential in (2.28) is aconstant times q2(1−q)2 = q2−2q3+q4.

The potential for the three dimensional fieldof [10] is just q2 +q4: the term −2q3 is missing, as that arises from the k0 = 0 mode ofthe integral. More generally, with an effective three dimensional theory, the manifestZ(N) symmetry of the original theory is broken by the separation into modes withzero and nonzero k0.

The loss of the Z(N) symmetry seems a grievous price to pay.Returning to (3.12), the remaining terms are standard in a background field ex-pansion. In background field gauge, (2.16), the inverse gauge field propagator is∆cl,−1µν= −D2cl δµν + Dclν Dclµ −1ξ Dclµ Dclν + ig[Gclµν, ],= −D2cl δµν + 1 −1ξ!Dclµ Dclν + 2ig[Gclµν, ] .

(3.14)After integrating out λqu, Aqu, and η,Seff(q) = Scl(Acl) + Squ1 (Acl) ,(3.15)where Acl is related by q(z) by (3.10), andSqu1 (Acl) = 12 tr log∆cl,−1µν−tr log−D2cl. (3.16)For the Z(N) interface letq(z) = q(z) tN ,(3.17)and then repeat the analysis of sec.

II. Taking the field of the Z(N) instanton asslowly varying, to leading order in g2 the effective action for q(z), Seff(q), is givenby (2.30).13

At next to leading order corrections arise from two sources. Viewing the action of(2.30) as a type of quantum mechanics, these terms can be understood as correctionsto the potential and kinetic terms.

As the potential was first generated by the freeenergy at one loop order, so corrections to this potential are produced by the freeenergy at two loop order.These two loop effects are g2 times those at one looporder. Like the calculation at leading order in sec.

II, for the two loop potential thebackground field can be taken as constant. Secondly, at next to leading order it isnecessary to account for the spatial variation of the Z(N) instanton.

For this onlythe free energy at one loop order is required, expanding to leading order in (dq/dz)2;thus these terms correct the kinetic term in the effective action. With our definitionof q, the classical action is proportional to 1/g2 times (dq/dz)2, (2.15); the terms fromthe free energy are of order one times (dq/dz)2, and so are smaller by g2.Ultimately, corrections are small because the Z(N) instanton is fat: the ratio of itssize to the thermal wavelength is of order 1/g.

Hence an expansion in the derivativesof the instanton field is automatically an expansion in g2. This is what makes theproblem tractable.Both of these effects are due to the effects of fluctuations in four dimensions asthey generate the effective, one dimensional action Seff(q).

The functional integralover q in (3.6), however, is still treated classically. When do fluctuations in q(z)enter?Remember that the quantity of physical interest [5] is the partition function, Z,for a system with the appropriate boundary conditions to enforce a domain wall inthe spacetime tube:Z = c e−αVtr .

(3.18)While we have concentrated on the interface tension, α, of course the prefactor “c”is also of significance.Integration over fluctuations in four dimensions generate the effective theory in onedimension, Seff, and so determine α. Fluctuations in Seff, though, do not contributeto α, only to the prefactor.

This is simply because Seff itself is proportional to Vtr,and so the integral over the effective, one dimensional fields cannot generate a constanttimes Vtr in the exponent, but merely powers of Vtr in the prefactor [5]. The prefactorfor a Z(N) domain wall is given by the integral over Seff at one loop order; however,it is necessary to compute Seff for a general path in group space, (a.1), instead ofthe “classical” path of (3.17).

This we defer.14

Consequently, we confess that the machinery of the effective action Seff(q) devel-oped in this section is not essential for what follows. We discussed it at such lengthin order to ensure that there are no problems of principle, and to emphasize thegenerality of the method.IV.Z(N) interface at next to leading order: Feynman gaugeIn this section we compute the leading corrections to the interface tension inbackground field Feynman gauge, ξ = 1; in the next section, for arbitrary ξ. Whilethe methods are the same for all ξ, technically the calculations are simpler in Feynmangauge, and so in this section we discuss our methods in some detail.

In sec. V theresults are merely summarized, in order to emphasize the physical interpretation ofthe gauge dependence which arises in the effective action.As discussed following (3.17), at next to leading order there are two pieces neededfor the interface tension.

The first is the effective potential in a constant backgroundAcl0 field to two loop order. This was calculated for SU(3) by Belyeav and Eletsky [11]and by Enqvist and Kajantie [12].

We have independently computed the potentialfor the field of (2.2) at arbitrary N, but given previous calculations, are content hereto just establish the N–dependence at two loop order.At one loop order the N dependence of Squ1 (Acl) is obvious. The background fieldenters only through adjoint covariant derivatives; from (2.11), tN only has nontriv-ial commutators with the generators t±N,j.

Thus the only q–dependence is from thefree energy of the 2(N −1) fields for these ladder operators, and at one loop orderSqu1 (Acl) ∼(N −1), (2.22).The diagrams which enter at two loop order involve either two three–gluon verticesor one four–gluon vertex.Both types of diagrams involve a product of structureconstants. The only diagrams that depend nontrivially upon the background fieldare those in which two lines are along the ladder operators t±N,j.

Denoting the q–dependent terms in the free energy at two loop order as Squ2 (Acl), after writing eachstructure constant as a traceSqu2 (Acl) ∼N−1Xj=1Xatrta[t+N,j, t−N,j]2 = (N −1)Xatrta[t+N,(N−1), t−N,(N−1)]2 . (4.1)The sum is over all generators ta with nonzero trace.

The last expression follows by15

noting that from the form of tN in (2.3), each value of “j” contributes equally to(4.1). Thus the complete sum over “j” is (N −1) times that for any single term, suchas j = (N −1).

The commutator for t±N,(N−1) is:[t+N,(N−1), t−N,(N−1)] = 12....... . .−10.

. .01.

(4.2)All elements not indicated vanish. From (4.2) the only terms which contribute to(4.1) are if ta is one of two diagonal generators, tN or tN−1, (2.3) and (2.5).

It isthen easy to show that the sum over “a” in (4.1) is proportional to N, so that in allSqu2 (Acl) ∼N(N −1).Knowing the N–dependence, the general result can be read offfrom that for N = 3[11, 12]. For the constant field of (2.2), the sum of the the free energies at one looporder, Squ1 (Acl) (2.28), and at two loop order in Feynman gauge, Squ2,ξ=1(Acl), isSqu1 (Acl) + Squ2,ξ=1(Acl)= Vtr4π2T 43(N −1)Zdz q2(1 −q)2 + g2N16π2!

3 q2(1 −q)2 −2 q(1 −q)!. (4.3)The second piece required for the interface tension arises from the free energy atone loop order for a background field which varies in z.

This corrects the kineticterm in the effective action, as a function of the background field, q, times (dq/dz)2:once one factor of (dq/dz)2 is extracted, the remaining factors of q can be taken asconstant. At one loop order the quantum action Squ1 (Acl) is given by (3.16), with theinverse gluon propagator of (3.14), setting ξ = 1 in Feynman gauge.

To calculatethis some, although not all, of the tricks of sec. II can be used.

For example, from(2.17)-(2.19), for a constant field the one loop quantum action is independent of ξ.This is no longer true for a spatially varying field.The ladder basis of sec. II can be used to simplify the color algebra.

DefiningG+0z = −G+z0 = dq/dz, else zero, and D2+ = (D+0 )2 + ∂2i , with D+0 as in (2.20), forarbitrary fields q(z) the one loop quantum action reduces toSqu1,ξ=1(Acl) = (N −1)tr log−D2+ δµν + 4πT i G+µν−2 tr log−D2+. (4.4)One kinetic term arises by expanding to quadratic order in G+µν:Squ1a,ξ=1(Acl) = −16π2T 2 (N −1) dqdz!2tr 1((k0+)2 + k2)2!,(4.5)16

with k0+ as in (2.21).Having extracted this, the remaining kinetic term arises from the expansion ofSqu1b,ξ=1(Acl) = 2(N −1)1 −ǫ2tr log(−D2+) . (4.6)The factor of 1 −ǫ/2 appears because the theory is regularized in 4 −ǫ dimensions,and in covariant gauges the number of gluons equals the dimensionality.Calculating the momentum dependence for such a one loop action is a standardproblem: see, for example, the treatment of Iliopoulos, Itzykson, and Martin [17].

Thecomputations of [17], however, are in coordinate space, which for most problems israther awkward. Instead, it is much simpler to perform the calculations in momentumspace.

We have checked that for the problem at hand, as well as for the scalar exampletreated in [17], the results agree.To work in momentum space, let q →q + δq in (4.6), and expand to quadraticorder in δq. The idea is to isolate the momentum dependence in the fluctuation, δq:tr log−D2+∼8π2T 2 12 tr 1−D2+(δq)2!+ tr D0+−D2+δqD0+−D2+δq!!.

(4.7)The momentum dependence only arises through the second term on the right handside, since the first term is a type of tadpole, independent of the momentum flowingthrough δq. The field q varies only in z, so its momentum is purely spatial.

If (0, ⃗p)is the external momenta, then, and (k0,⃗k) the loop momenta, (4.7) becomes8π2T 2 δq(⃗p)δq(−⃗p) tr (k0+)2((k0+)2 + k2)((k0+)2 + (⃗p −⃗k)2)!. (4.8)In this form it is trivial to expand to order p2.

Trading L (δq (p2) δq) forR dz (dq/dz)2,the second kinetic term isSqu1b,ξ=1(Acl)= 16π2T 2 (N −1) dqdz!2 1 −ǫ2tr −k2((k0+)2 + k2)3 +43 −ǫk4((k0+)2 + k2)4!. (4.9)At nonzero temperature dimensional continuation is carried out by changing thenumber of spatial dimensions to 3 −ǫ, which produces the factor of 4/(3 −ǫ) above.The integrals required are given in appendix B, (b.4), (b.6), and (b.7).

Withoutworrying about their detailed form, one feature is evident.Each integral is loga-rithmically divergent in four dimensions, so in 4 −ǫ dimensions, there are poles in17

1/ǫ. These are the standard terms which produce the renormalization of the couplingconstant at one loop order.

Thus it is instructive to combine the results for (4.5) and(4.9) with the classical action of (2.15) to findScl(Acl) + Squ1a,ξ=1(Acl) + Squ1b,ξ=1(Acl)= Vtr4π2T 2g2(T)N (N −1)Zdz dqdz!2 1 + 113g2N16π2ψ(q) + ψ(1 −q) + 111! ;(4.10)ψ(q) = d(log Γ(q))/dq is the digamma function.

The prefactor includes the runningcoupling constant at a temperature T, g2(T), which is related to the bare couplingconstant g2 as1g2(T) =1g2 1 −113g2N16π2 2ǫ + log µ2πT 2!+ ψ(1/2)! !,(4.11)with µ the renormalization mass scale.

The relationship between the bare and renor-malized coupling constants in (4.11) is arbitrary up to a constant; our choice is similarbut not identical to the modified minimal subtraction scheme, and is convenient atnonzero temperature. At high temperature, (4.11) exhibits the standard logarithmicfall offof the running coupling constant, g2(T), with the coefficient of 11N/3 appro-priate for the β–function of an SU(N) gauge theory at one loop order.

Notice thatthe q–dependence in (4.10), through the digamma functions of q and 1 −q, enterswith precisely the same coefficient as for the β–function, 11N/3.The effective action which governs the Z(N) instanton at next to leading orderis the sum of (4.3) and (4.11). The action is determined by the properties of thesolution at leading order.

As discussed following (2.30), the Z(N) instanton has zeroenergy, and soZdz′ dqdz′!2(ψ(q) + ψ(1 −q))= 2Z 10 dqq −12log Γ(q)Γ(1 −q)!∼−.995018 . .

. ;(4.12)the value of the integral was determined by numerical integration.

Hence at next toleading order, the interface tension α isα = 4(N −1)π23√3NT 3g(T) 1 −(15.2785...)g2(T)N16π2!,(4.13)which is the result quoted in (1.1). We have taken the liberty of writing the correctionsas proportional not just to the bare coupling constant, g2, but to the running couplingconstant, g2(T).18

V.Z(N) interface at next to leading order: general gaugesThe calculation of the interface tension in an arbitrary background field gauge issimilar to that for Feynman gauge. Nevertheless, the calculation illuminates somefeatures which are missed by working at fixed ξ.Including the terms at both one and two loop order, the free energy in the constantbackground field of (2.2) isSqu1 (Acl) + Squ2,ξ(Acl) = Vtr4π2T 43(N −1)Zdzq2(1 −q)2+ g2N16π2!

(7 −4ξ) q2(1 −q)2 −(3 −ξ) q(1 −q)!. (5.1)The two loop potential for general ξ was computed first by Enqvist and Kajantie [12];we agree with their result when ξ = 1, but not for ξ ̸= 1.The potential changes in a rather dramatic fashion in going from one to two looporder.

At one loop order the potential is just a standard double well, q2(1 −q)2. Attwo loop order the potential becomes ξ–dependent.

This includes a correction to thecoefficient of the double well potential, as well as a new term, proportional to q(1−q).This new term is peculiar, for it controls the behavior of the potential for small q. Ifξ < 3, the stable minima are not at q = 0 and q = 1, but at q0 ∼(3 −ξ)g2 and 1 −q0.On the other hand, if ξ > 3, the stable minima remain q = 0 and q = 1.Such a nonzero value of the stable minima would have profound consequences fora gauge theory at high temperature [9-14]. For N ≥3, the trace of the Wilson line inthe fundamental representation is a complex number.

Under charge conjugation (C)or time reversal (T) transformations, the trace of the Wilson line goes into its complexconjugate (up to global Z(N) transformations). Thus if the stable vacuum indeedhas q ̸= 0 (modulo 1), then the vacuum spontaneously breaks C and T symmetries,conserving CT.

While conceivable, it is unexpected to find C symmetry breakingarising spontaneously in a pure gauge theory.Of course a physical phenomenoncannot depend upon the choice of the gauge fixing parameter, while q0 changes withξ.Leaving these questions aside for the moment, in a general background gauge thekinetic terms in the effective action are, to one loop order,Scl(Acl) + Squ1a,ξ(Acl) + Squ1b,ξ(Acl)19

= Vtr4π2T 2g2(T)N (N−1)Zdz dqdz!2 1 + 113g2N16π2 ψ(q) + ψ(1 −q) + 7 −6ξ11!!. (5.2)The running coupling constant, g2(T), remains as in (4.11).In (5.2) the only ξ dependence is an overall constant, proportional to 7 −6ξ, andis independent of the background field q.

As discussed following (b.7) in appendix B,for each of the integrals which contribute to the kinetic term, the coefficient of thepole in 1/ǫ is always the same as for the digamma functions of q and 1 −q, which ishow the q–dependence arises. As is customary in background field calculations [16],the poles in 1/ǫ generate the β–function at one loop order, and is independent of ξ.Thus if the coefficient of the digamma functions is the same as for 1/ǫ, at one looporder it also must be independent of ξ.

The only remaining ξ–dependence possible isas a constant, which does appear.The effective action for arbitrary ξ is the sum of (5.1) and (5.2). It is easy to showthat while each term depends individually upon ξ, for any solution with zero energy,E = 0, the ξ–dependence cancels in the sum.

As the Z(N) instanton has zero energy,the value of the interface tension for ξ ̸= 1 is equal to that for ξ = 1, (4.13).The cancellation of ξ–dependence is a necessary check on the consistency of ourmethod, but by itself is rather unsatisfactory. To understand this better we followBelyaev [14].

In SU(2) Belyaev showed that the apparent ξ–dependence in the po-tential for a constant A0 field can be understood as a renormalization of the Wilsonline. We now generalize his results to arbitrary SU(N) , and show that they explainthe ξ–dependence of both the potential and kinetic terms in the effective action.The point is that while the vacuum expectation value of the trace of the Wilsonline is a gauge invariant quantity, the fields which we have been using to parametrizethe Wilson line — Acl0 , and so q — are not.

For instance, from (3.5), our effective fieldtransforms in a nonlocal manner under infinitesimal gauge transformations. At treelevel this doesn’t matter, but it does at one loop order and beyond, as the Wilsonline undergoes both infinite and finite renormalizations.

To compute these, defineA0(x) = Acl0 + Aqu0 (x) ,(5.3)with Acl0 related to q as in (2.2).In general the Wilson line is a function of thespatial position, and so we consider the trace of the Wilson line, averaged over space.20

Expanding in powers of Aqu,Z d2xtL2tZ dzL ⟨tr Ω(A)⟩= tr Ω(Acl) + Ω1 + Ω2 + . .

. .

(5.4)The first term on the right hand side, tr Ω(Acl), is the value in the classical backgroundfield, (2.4). The term linear in Aqu, Ω1, can be written asΩ1 = igZdzLZ β0 dτZ d2xtL2ttrAqu0 (τ, ⃗xt, z) Ω(Acl).

(5.5)Due to the constraint imposed upon the quantum fluctuations in (3.13), this termvanishes.The term quadratic in quantum fluctuations is nontrivial. Consider first its valuein zero background field, Acl = 0:Ω2 = −g2(N2 −1)2βZd3k(2π)312 k2 .

(5.6)This is a standard renormalization of the Wilson line, a type of “wave function”renormalization, proportional to β, the length in euclidean time.In dimensionalregularization this vanishes identically.However these infinite terms are regularized, they are independent of the back-ground field. In addition, there are finite terms which depend upon Acl.

Using thecommutation relations of (2.11), for an arbitrary constant “y”eytN t±N,j e−ytN = e± ycN t±N,j . (5.7)Using this relation, and remembering the path ordering required for the Wilson line,the terms dependent on the background field areΩ2 = −g2NN−1Xj=1Z β0 dτZ τ ′0dτ ′ Aqu0,j+(τ) Aqu0,j−(τ ′) e−2πiq(τ−τ ′) trt+N,jt−N,j Ω(Acl)+ Aqu0,j−(τ) Aqu0,j+(τ ′) e2πiq(τ−τ ′) trt−N,jt+N,j Ω(Acl).

(5.8)Aqu0,j± is the component of Aqu0 in the direction of t±N,j. The τ integrals are evaluatedusing the background field propagator, as in (2.18).

After doing the color trace, inmomentum spaceΩ2 = ig2β(N −1)2Ne2πiq/N 1 −e−2πiqtr 1k0+∆00(k0+, k)!. (5.9)21

∆00(k0, k) is the usual covariant gauge propagator,∆00(k0, k) =1(k0)2 + k2 −(1 −ξ)(k0)2((k0)2 + k2)2 ,(5.10)with the only dependence on the background field in (5.9) through the shifted mo-mentum k0+, (2.21).How then to interpret these corrections at one loop order to the classical valueof the Wilson line? By the constraint imposed upon Aqu, the linear term in (5.5)vanishes, and so they cannot absorbed in Aqu.

We then introduce “renormalized”fields for A0 and q asAren0= 2πTgcN qren tN,qren = q + δq ,(5.11)and require that the spatial average of the vacuum expectation value of the Wilsonline be given by Aren:Z d2xtL2tZ dzL ⟨tr Ω(A)⟩= tr Ω(Aren) . (5.12)Expanding the right hand side to linear order in δq, δq is proportional to Ω2.

Theintegrals required for (5.9) are (b.9) and (b.10) of appendix B, and giveqren = q + g2N16π2 (3 −ξ)q −12. (5.13)This relation is valid for 0 < q < 1.

For SU(2), it agrees with the result of Belyaev[14]. Note that the renormalization from q to qren is entirely a matter of a finite shift.The gauge dependent actions, (5.1) and (5.2), can be trivially rewritten in termsof the renormalized fields.

The potential term becomesSqu1 (Aren) + Squ2 (Aren) = Vtr4π2T 43(N −1)Zdz 1 −5 g2N16π2!q2(1 −q)2 ,(5.14)while for the kinetic terms,Scl(Aren) + Squ1a(Aren) + Squ1b (Aren)= Vtr4π2T 2g2(T)N (N −1)Zdz dqdz!2 1 + 113g2N16π2 (ψ(q) + ψ(1 −q) + 1)!. (5.15)22

With the effective action of (5.14) and (5.15), the corrections to the interface are ofcourse unchanged, equal to (4.11).Once written in terms of qren, all of the gauge dependence found previously inthe kinetic and potential terms cancels. Further, the effect of two loop terms in therenormalized potential is just to change the coefficient of the one loop terms: thenew terms found previously at two loop order, proportional to q(1 −q), have all beenabsorbed by qren.Hence the apparent instability of the perturbative vacuum at two loop ordermerely results from a classical parametrization of the renormalized Wilson line.

Aftercorrecting for loop effects, the stable vacuum is the trivial one (plus Z(N) transformsthereof) and is C symmetric.It seems unlikely that the cancellations found at two loop order are mere coinci-dence. We conclude with a conjecture: that the stable vacuum of hot gauge theories —both with and without fermions — is symmetric under charge conjugation to arbitraryloop order.The research of A.G. and R.D.P.

was supported in part by the U.S. Departmentof Energy under contract DE–AC02–76–CH0016.Appendix A:Proof of minimal action for N = 3, ∞In this appendix we prove that for N = 3 and ∞, the path chosen for the Z(N)instanton is of minimal action.By a global gauge rotation a constant background field Acl can be chosen to be adiagonal matrix. Thus the most general constant field for the Acl of (3.10) isq =NXi=2qi ti .

(a.1)The ti are the N −1 diagonal generators of SU(N) , as in (2.3) and (2.5). For clarity,in (a.1) the field q is relabeled as qN.Promoting each qi to be a function of z, for this ansatz the classical action becomesa sum over N −2 independent kinetic termsScl(Acl) = Vtr4π2T 2g2N(N −1)ZdzNXi=2 dqidz!2.

(a.2)23

For the Z(N) interface the boundary conditions required areqi(0) = qi(L) = 0,i = 2 . .

. (N −1);qN(0) = 0,qN(L) = 1 .

(a.3)We wish to show that the path with q2(z) = . .

. qN−1(z) = 0, and qN(z) = q(z)as before, is the path of minimal action.

To demonstrate this we need the potentialgenerated by fluctuations at one loop order. We do so in two special cases where theanalysis is elementary.N=3: There are two independent fields, q2 and q3.

The potential term isSqu1 (Acl) = Vtr4π2T 43Zdz Vtot(q2, q3) ,Vtot(q2, q3) = V(q2) + V(q2/2 + q3) + V(−q2/2 + q3) ,(a.4)whereV(q) = [q]2(1 −[q])2,[q] = |q|mod 1 . (a.5)Because the V(q) is a function of [q], the absolute value modulo one, it is not quitethe simple polynomial form it first appears to be.

(This restriction could be ignoredbefore, since the Z(N) instanton only involves q : 0 →1. )The effective action is the sum of (a.3) and (a.4).

It is not difficult to see whyfor the boundary conditions of (a.3), the path with q2(z) = 0 is of minimal action.Considering the problem as classical mechanics in two dimensions, from energy con-servation the action for any solution to the equations of motion is proprotional toZdsqVtot(q2, q3) ,(a.6)where ds is the arc length in the space of q2 and q3. It can be shown that for anyfixed value of q3, the potential is minimized for q2 = 0: Vtot(q2, q3) ≥Vtot(0, q3).

Giventhe boundary conditions, the path chosen is clearly of minimal length. As both thearc length and the potential are bounded by the path with q2 = 0, by (a.6), so is theaction.N = ∞: The large N limit is taken by holding eg2 ≡g2N fixed as N →∞; we workin weak coupling, for small eg.

Then the interface tension is of order N, α ∼N/eg,while the Z(N) instanton remains fat, with a width of order 1/(egT).24

Besides qN, there are of order N qi’s which can contribute. Assume first that thereare a finite fraction of the qi’s for which qi(z) ̸= 0.

From (a.3), since each field hasa kinetic term proportional to N, if order N fields contribute, the sum of the kineticterms for all fields is of order N2. Similarly, the potential term is also of order N2.Because both the kinetic and potential terms are positive definite, any solution haspositive action, equal to a pure number times N2.

This is N times the action of thepath chosen and so is not minimal.Hence we can assume that at large N, only a finite number of the qi’s contribute.For simplicity, assume that they are just qN and qN−1. The generators for tN, (2.3),and tN−1, (2.5), simplify greatly at large N, reducing to just one diagonal element.

(Essentially, the terms which enforce tracelessness can be ignored, as a correction in1/N.) In this case it is easy to work out the full potential term, which is proportionaltoVtot(qN−1, qN) = (N −2)V(qN) + (N −2)V(qN−1) + V(qN−1 −qN)∼N (V(qN) + V(qN−1)) .

(a.7)The last term is the leading term at large N; like the kinetic energy, it is of orderN. But notice that at large N, in the potential the coupling between qN and qN−1has dropped out.

Thus even though the potential for qN−1 is nontrivial, because it ispositive definite, the path of minimal action which satisfies qN−1(0) = qN−1(L) = 0 isjust qN−1(z) = 0.This argument generalizes immediately to any finite number of qi’s: the term oforder N in the potential is a sum over decoupled potentials, and the path of minimalaction excites only one field, qN.Appendix B:Integrals in a constant A0 fieldIn this appendix we catalog the integrals required for the computation of quantumactions in the constant background field of (2.2).The trace is defined astr = V T+∞Xn=−∞Zd3−ǫk(2π)3−ǫ . (b.1)Remember that the system is of length Lt in the ⃗xt directions, and L in the z direction.At a temperature T, β = 1/T, and the momentum k0 = 2πnT for integral n. The25

volume of spacetime V = βL2tL. Dimensional continuation in 3 −ǫ dimensions isused.As discussed following (2.22), after using the ladder basis to reduce the coloralgebra, the propagators in the background field of (2.2) are given by their valuesin zero field, except for the replacement of k0 →k0+ = 2πT(n + q).

Following theexample of (2.22)–(2.28), the integrals are most easily computed by zeta functiontechniques [15]. First, the spatial integrals over d3−ǫk are done using the standardformulas of dimensional regularization.

This leaves a sum over n, which is evaluatedin terms of zeta functions. For instance, the simplest integral which arises in theexpansion of the kinetic term at one loop order istr1((k0+)2 + k2)2 =V16π21 + ǫ2 ψ(1/2) −log(πT 2)+∞Xn=−∞1|n + q|1+ǫ .

(b.2)Using the definition of the zeta function, (2.25), and then expanding to order ǫ,+∞Xn=−∞1|n + q|1+ǫ = ζ(1+ǫ, q) + ζ(1+ǫ, 1−q) ∼2ǫ −(ψ(q) + ψ(1 −q)) , (b.3)with ψ(q) = d(log Γ(q))/dq the digamma function. Hence in alltr1((k0+)2 + k2)2 =V16π2 2ǫ′ −(ψ(q) + ψ(1 −q)),(b.4)where2ǫ′ = 2ǫ + ψ(1/2) −log(πT 2) .

(b.5)Similarly,tr(k0+)2((k0+)2 + k2)3 =V64π2 2ǫ′ −(ψ(q) + ψ(1 −q)) + 2,(b.6)tr(k0+)4((k0+)2 + k2)4 =V128π2 2ǫ′ −(ψ(q) + ψ(1 −q)) + 83,(b.7)These integrals are all those required to compute the one loop corrections to thekinetic term in sec.’s IV and V. The factors of the renormalization group scale µwhich enter into the running coupling constant, (4.11), arise by taking g2 →g2µǫ inthe classical action, and expanding in ǫ.Notice that for the three integrals of (b.4), (b.6), and (b.7), the dependence on thebackground field q, through the digamma functions, always has the same coefficient26

as the pole in 1/ǫ. This is because in all of these integrals, q enters through the zetafunctions ζ(1 + ǫ, q) + ζ(1 + ǫ, 1 −q).

Expanding about small ǫ, (b.3), generates acommon factor of 1/ǫ and digamma functions in each case, so that up to their overallnormalization, these integrals only differ by a constant.Other integrals required for the potential at two loop order, and the renormaliza-tion of the Wilson line at one loop order, include (2.22), (2.26), and the following.These are all finite integrals, so it is safe to set ǫ = 0.tr1(k0+)2 + k2 = V T 22q2 −q + 16,(b.8)tr1k0+ ((k0+)2 + k2) = V T2πq −12,(b.9)trk0+((k0+)2 + k2)2 = −V T4πq −12. (b.10)The last two integrals, (b.9) and (b.10), are singular for q = 0 or 1, and as writtenare defined over 0 < q < 1.

Properly defined, they are discontinuous and vanish forq = 0 or 1.27

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