# Z(3)-symmetric effective theory for Su(3) Yang-Mills theory at high temperature

A. Vuorinen1  and Laurence G. Yaffe2
Department of Physics, University of Washington, Seattle, WA 98195–1560
11
22
###### Abstract:

A three-dimensional effective theory for high temperature gauge theory, which maintains the center symmetry of the full theory, is constructed. Such a invariant effective theory should be applicable to a wider temperature range than the usual effective theory, known as EQCD, which fails to respect the center symmetry. This center-symmetric effective theory can reproduce domain wall and phase transition properties that are not accessible in EQCD. After identifying a convenient class of invariant effective theories, we constrain the coefficients of the various terms in the Lagrangian using leading-order matching to EQCD at high temperature, plus matching of domain wall properties in the full theory. We sketch the expected structure of the phase diagram of the effective theory and briefly discuss the prospects of numerical simulations and the addition of quarks.

Thermal Field Theory, QCD
preprint:

## 1 Introduction

The thermodynamics of high-temperature QCD with various numbers of massless quarks has been widely studied using the method of dimensional reduction, in which one exploits the decoupling of the non-static degrees of freedom to build an effective three-dimensional theory known as electrostatic QCD (EQCD) . This approach has been used to derive the perturbative expansion of the QCD pressure through order , and to compute various gluonic and mesonic correlation lengths [2, 3, 4]. It has also led to improved understanding of the non-perturbative dynamics on the “ultra-soft” energy scale . This approach is applicable at asymptotically high temperatures where the running coupling is small. But closer to the confinement/deconfinement transition, at temperatures of a few times , even non-perturbative simulations of EQCD have failed to produce satisfactory results [6, 7]. There have been attempts to build effective theories for the Wilson line near the phase transition [8, 9, 10, 11], but none of these theories (which often come with very complicated Lagrangians) provide any connection to the perturbative regime at .

Our goal is to formulate a simple three-dimensional effective theory for hot QCD that can reproduce equilibrium properties at temperatures ranging from asymptotically large down to . For simplicity, most of our discussion will focus on the case of pure Yang-Mills theory, although the addition of dynamical quarks will be briefly discussed in the last Section. The construction of our theory relies on a separation between the inverse correlation length and the lowest non-zero Matsubara frequency, . In Yang-Mills theory, there is a parametrically large separation between these scales at very high temperatures, and a modest separation near .333 It is natural to distinguish the correlation lengths of time-reflection even and odd operators. The Debye screening length is the longest correlation length of time-reflection odd operators , such as the imaginary part of the trace of the Wilson line. The longest correlation length of time-reflection even operators, which we will denote by , is longer than the Debye length. In Yang-Mills theory at asymptotically high temperatures, approaches while is approximately as . (This is an estimate based on the corresponding result, plus the fact that should scale nearly linearly with .) Correlators of purely magnetic operators fall off like at large distance, while correlators of the real part of the trace of the Wilson line initially decrease like but eventually fall more slowly as . The correlator of the imaginary part of the trace of the Wilson line initially decreases like but switches to at large distance. As the temperature decreases, the ratio , as deduced from the Wilson line correlator, rises to about 0.5 at [14, 15] and then falls to about 0.1 just above . The ratio on the other hand rises to near unity at . Because the confinement/deconfinement phase transition in Yang-Mills theory is only weakly first order , this ratio should decrease, similar to the behavior of , closer to the phase transition, but we are unaware of any numerical simulations confirming this. Near , correlations of purely magnetic operators appear to fall exponentially with a length scale near but should eventually switch to the longer scale seen in the Wilson line correlator .

In pure Yang-Mills theory, the confinement/deconfinement phase transition is a symmetry-breaking phase transition. The theory has a global symmetry corresponding to invariance of the action under gauge transformations which are periodic in the temporal direction only up to a twist belonging to the center of the gauge group. This symmetry is spontaneously broken in the deconfined phase where the Wilson line operator (which is an order parameter for this symmetry) acquires a non-zero expectation value. Close to the phase transition region, fluctuations in the Wilson line become increasingly important and the invariance of the Wilson line probability distribution is an essential feature of the theory. Even in full QCD with dynamical quarks it has been argued that the symmetry, although explicitly broken by the presence of light quarks, may nevertheless play an important role in the dynamics near .

The standard high temperature effective theory, EQCD, cannot provide a smooth interpolation between asymptopia and . This is evident even without considering numerical evidence, as the construction of EQCD violates one of the central tenets of effective field theory — it does not respect all the symmetries of the underlying theory. EQCD breaks the center symmetry explicitly, even in the absence of quarks. Its leading order Lagrangian is obtained by expanding the one-loop effective potential of the Wilson line  around one of its three degenerate minima. Effects of the explicit breaking are clearly visible in the phase diagram of the effective theory .444 Even though it has been argued that a “partial dynamical restoration” of the symmetry takes place in EQCD , this theory does not have three distinct, degenerate, and physically equivalent equilibrium states which would signal a spontaneously broken symmetry. These aspects will be discussed in more detail in Sec. 5.

An effective theory that preserves the symmetry can capture the dynamics of thermal fluctuations (above ) which create bubbles sampling all three degenerate equilibrium states, and thus should have a phase structure similar to that of the full four-dimensional theory. In particular, its phase diagram should contain a quadruple point, where all three deconfined phases coexist with the confining phase, thereby extending the description of the theory into a temperature range inaccessible to EQCD.

Our guide during the process of constructing a better effective theory for QCD is the requirement that the new theory provide a smooth link between asymptotically high temperatures, where perturbation theory is valid, and , where a symmetry restoring phase transition occurs. At high temperatures, its predictions must reproduce those of EQCD (within its domain of applicability). But a invariant effective theory should also reproduce the thermodynamics of domain walls which interpolate between different equilibrium states related by the symmetry .

We begin in Section 2 with a brief review of the relevant aspects of the thermodynamics of Yang-Mills theory, including dimensional reduction and the role of the Wilson line as an order parameter for the confinement/deconfinement phase transition. Section 3 presents the general form of our effective theory. We perform lowest order matching to ordinary EQCD at high temperatures in Section 4, and also compute the semiclassical domain wall profile in our effective theory. The expected phase diagram of our theory is discussed in Section 5, and compared to that of EQCD. Possible future directions, including the addition of quarks, are briefly discussed in the concluding Section 6. Some details of the analysis of Section 3 are relegated to the appendices.

## 2 Thermodynamics of Su(3) Yang-Mills

### 2.1 Z(3) center symmetry

The functional integral representation for the partition function of Yang-Mills theory involves an integral over all gauge fields which are periodic in (Euclidean) time with period . In addition to invariance under gauge transformations which are periodic in time, the action and the space of field configurations are invariant under gauge transformations which are only periodic in time up to a “twist” belonging to the center of the gauge group,

 Aμ(x)→s(x)(Aμ(x)+i∂μ)s(x)†, s(x)∈SU(3), (1a) s(x+β^et)=zs(x), z∈Z(3). (1b)

Under such a transformation, the trace of the Wilson line (or Polyakov loop)

 trΩ(x) ≡ tr{Pexp[i∫β0dτA0(τ,x)]}, (2)

transforms in the fundamental representation of ,

 trΩ(x) → ztrΩ(x). (3)

The magnitude of the expectation value of may be interpreted physically as the exponential of the change in free energy due to the addition of an infinitely heavy fundamental representation test quark at position ,

 |⟨trΩ(x)⟩| = e−βΔFq(x). (4)

This expectation value vanishes in the confining low temperature phase, where the free energy cost to introduce a single test quark is infinite, but the expectation value is non-zero in the deconfined high temperature phase, where the free energy cost of a test quark is finite. Hence, the center symmetry is unbroken in the confining phase, but is spontaneously broken in the deconfined phase, and the expectation value of is an order parameter for the center symmetry.555 Unlike more typical symmetry breaking phase transitions, the low-temperature (confining) phase is the disordered phase, while the high-temperature (deconfined) phase is the ordered phase.

Symmetry considerations , as well as lattice simulations , show that four-dimensional SU(3) Yang-Mills theory belongs to the same universality class as the three-dimensional three-state Potts model, and undergoes a weak first order phase transition at the critical temperature. At the phase transition, the theory has a quadruple point with four co-existing equilibrium phases (the three spontaneously broken deconfined states, related by transformations, plus the confining phase). No correlation lengths diverge as .

The transition from the deconfined plasma above to the confining glueball phase below may occur through two mechanisms: bubble nucleation or complete wetting. In the former, bubbles of the confining phase are formed via thermal fluctuations inside the (supercooled) deconfined matter just below , and as the temperature is lowered these bubbles grow, eventually filling all space. Alternatively, complete wetting takes place if one begins with domain walls separating deconfined phases with differing “magnetization” (i.e., differing phases for ). Such a domain wall may be viewed as a thin film of the confining phase residing between the two deconfined phases. As the temperature is lowered below , this film widens and splits the “deconfined-deconfined” domain wall into two “confined-deconfined” domain walls which repel, leading to the expansion of the confining phase in the direction perpendicular to the original domain wall.

Domain walls separating rotated deconfined phases are topologically stable, non-perturbative objects which play an important role in the dynamics of quarkless QCD at temperatures just above . At sufficiently high temperatures, their properties may be computed using semi-classical techniques . In hot QCD, the effective potential for the Wilson line only arises at one loop order . As a result, the Debye mass (which is determined by the curvature of this potential at its minimum) is of order , while the energy density at the top of the barrier separating minima is (not order , as one might have expected for a semiclassical object). The width of a domain wall is determined by the Debye screening length. So high temperature domain walls have an energy density over a distance of order , yielding a tension which scales as . This parametric dependence must be reproduced by any effective theory that claims to capture the symmetry breaking physics of hot Yang-Mills theory.

### 2.2 Dimensional reduction and EQCD

Due to the compact (Euclidean) temporal direction in a finite-temperature field theory, all four-dimensional fields may be decomposed into Fourier sums running over Matsubara frequencies. Each Fourier component acts as a three-dimensional field, with a mass (for bosonic fields) given by an integer multiple of . At sufficiently high temperatures, all non-static modes act like heavy fields and decouple from dynamics on length scales large compared to . The remaining static modes are the relevant degrees of freedom for physics on longer distance scales, yielding an effective three dimensional theory. To lowest order, the form of the effective theory may be found by integrating out the heavy fields explicitly . At higher orders, it is more efficient to start with the most general form of the Lagrangian and adjust its parameters by demanding that the effective theory reproduce a minimal set of physical quantities which are computable in both the effective and underlying theories [1, 20].

In the case of Yang-Mills theory at high temperature, the degrees of freedom of the effective theory correspond to the static components of the gluon field . Integrating out non-static fluctuations generates an order mass for the temporal component , due to Debye screening. In addition, self-interaction terms are generated for the field. The resulting Lagrangian of EQCD reads666 This form of the effective theory is valid for and . With four or more colors, there is an additional term. But for both and , the two fourth-order terms are not independent, as .

 L\scriptsize EQCD=g−23{12trF2ij+tr[(DiA0)2]+m2Etr(A20)+λEtr(A40)}+δL\scriptsize E, (5)

with the three-dimensional gauge coupling and an adjoint representation covariant derivative. The final term stands for higher dimension operators whose effects, on length scales large compared to , are subleading and only affect the thermodynamics at order . The operators in will be irrelevant for our purposes.

A straightforward way to obtain the form of the above Lagrangian, plus the leading-order values of its parameters, is to consider the one-loop effective potential for the Wilson line in the full theory. The resulting potential  has three degenerate minima related by transformations belonging to the center of . Choosing to work around the minimum where the Wilson line expectation value is real and positive (and subtracting from the potential an contribution originating from fluctuations in the static gauge field, which are not to be integrated out in the present case), one obtains Eq. (5) as the resulting effective potential for together with the lowest-order values for the effective theory parameters,

 m2E=g(T)2T2,λE=34g(T)2/π2. (6)

To lowest order, the mass parameter is the same as the physical Debye mass .

This derivation also reveals the major shortcoming of the effective theory: when constructing the EQCD Lagrangian, one has chosen to work in the neighborhood of , corresponding to a real, positive Wilson line, and thus loses the invariance of the original theory. The other minima correspond to eigenvalues of near , and are outside the domain of validity of the effective theory. At asymptotically high temperatures this is of little consequence for most physical quantities, as the probability of thermal fluctuations crossing the barriers separating different minima is exponentially small [with an exponent scaling as ]. But fluctuations between the different minima become crucial for the dynamics closer to , and hence a non--invariant effective theory cannot properly describe physics near the confinement/deconfinement phase transition.

## 3 Z(3) invariant effective theory

### 3.1 Degrees of freedom

To build a three-dimensional invariant effective theory, the minimal degrees of freedom are the spatial gauge field and the Wilson line . But because is a unitary matrix, a theory with polynomial interactions for the Wilson line will not be perturbatively renormalizable. For an effective theory which is only intended to be valid below a UV cutoff, this is not a fundamental problem. However, it is a serious practical nuisance. It complicates perturbative matching to the underlying fundamental theory, as well as matching between lattice and continuum regulated versions of the effective theory.

Therefore, we take as our goal the construction of an effective theory whose short distance behavior is as benign as possible. To do so, we will replace the unitary matrix by an unconstrained complex matrix . This should be viewed as analogous to the relation between non-linear and linear sigma models (or between an Ising model and a double-well field theory). One may view the complex field as the result of applying a block-spin renormalization group transformation which averages over a small spatial region, producing a result which is no longer unitary.

An arbitrary complex matrix may always be decomposed into a product of unitary and hermitian matrices, so we may write with . One may regard the matrix as containing unphysical degrees of freedom (not present in the underlying Yang-Mills theory) which we have chosen to “integrate in” in order to construct an effective theory with good short distance behavior. We will arrange for these extra degrees of freedom to be “heavy” — to have masses large compared to the inverse correlation length.

We require that the effective theory Lagrangian be invariant under three dimensional gauge transformations,

 Z(x)→s(x)Z(x)s(x)†,A(x)→s(x)(A(x)+i∇)s(x)†, (7)

for , as well as global phase rotations,

 Z(x) →e2πin/3Z(x), (8)

for integer .

Instead of using the polar decomposition of mentioned above, it will be more convenient for our purposes to decompose this field into a traceless part and its trace (times a unit matrix). Therefore, we define

 L(x) ≡ trZ(x), (9) and M(x) ≡ Z(x)−13trZ(x)to0.0pt11, (10) so that Z(x) = M(x)+13L(x)to0.0pt11. (11)

### 3.2 Effective Lagrangian

The Lagrange density for the effective theory will have standard derivative terms plus a potential for the complex scalar ,

 L = g−23{12trF2ij+tr(DiZ†DiZ)+V(Z)}, (12)

with and .777 The freedom to make multiplicative rescalings of is used to fix the coefficient of in the Lagrangian (12). One could also add a term, as this preserves all the required symmetries. But this additional term may be eliminated with a field redefinition. The invariant potential will consist of two pieces,

 V(Z)=V0(Z)+g23V1(Z), (13)

with

 V0(Z) (14) and V1(Z) =~c1tr[M†M]+~c2(tr[M3]+tr[(M†)3])+~c3tr[(M†M)2]. (15)

We will refer to as the “hard potential” and as the “soft potential”. This terminology is appropriate at high temperatures where is small, in which case the effective masses arising from will be large compared to those arising from .

The hard potential is composed of the three lowest-dimension operators which (in addition to being invariant) are invariant under the transformation

 Z(x)→AZ(x)B, (16)

for arbitrary . Note that the free kinetic term is also invariant under this transformation. A short exercise, presented in Appendix A, shows that has extrema when for any element of and a real root of . The coefficient must be positive for stability. We will require that and . These conditions ensure that the global minimum corresponds to the largest positive real root which we will write, for later convenience, as with

 v≡34(−c2+√c22−8c1c3c3). (17)

Therefore, is minimized when equals an arbitrary matrix times .888 Note that sending the coefficients to infinity, while holding their ratios fixed, would cause to act like a delta-function constraint forcing the matrix to equal an element (up to a scale factor). The resulting effective theory would be a gauged non-linear sigma model for the Wilson line . But this limit would return us to a non-renormalizable effective theory, which we wish to avoid.

The coupling constants in the soft potential will be required to satisfy the inequalities and . As shown in Appendix A, these conditions ensure that is strictly positive for non-zero . Hence, the soft potential is minimized when , or equivalently when is proportional to the identity matrix. The overall factor of multiplying the soft potential is designed to mimic the fact, noted earlier, that the effective potential for the Wilson line in high temperature Yang-Mills theory only arises at one loop order.

We have only included in the soft potential the three lowest dimension invariant terms which can be built just from the traceless field . There are more invariant terms, with the same dimensions, which we could have added — for example, terms including factors of the trace . However, if is restricted to the manifold of minima of the hard potential , and so is proportional to an matrix, then there are only three independent operators which are gauge invariant, invariant, and at most fourth order in the field. The three terms we have included in the potential (15) are a convenient choice for these three independent terms. As shown in the next section, these terms will be sufficient for matching with the EQCD Lagrangian.999 Had we chosen to work with a gauge group or higher, we would need to include a quartic double trace term in Eq. (15) in order to reproduce the EQCD Lagrangian. This is unnecessary for , due to the identity mentioned in footnote 6.

The soft potential lifts the continuous degeneracy at the minimum of , leaving three discrete minima related by the symmetry. If equals times an element of the center of , then it will simultaneously minimize both the hard and soft potentials, and hence will trivially minimize the combined potential .

The theory defined by Eqs. (12)–(15) is a superrenormalizable three-dimensional field theory. The only renormalizations required (in addition to vacuum energy subtraction) are one and two loop adjustments of the coefficients and of the quadratic terms.

To carry out perturbative matching, as discussed in the next section, we will need the expansion of the Lagrangian (12) about the minima of the potential. In the vicinity of the minimum at , we may decompose as

 Z=e2πin/3{13vto0.0pt11+g3[1√6(ϕ+iχ)to0.0pt11+(h+ia)]}, (18)

where and are real numbers, while and are traceless Hermitian matrices. We will also rescale the three-dimensional gauge field to switch to conventional perturbative normalization of the gauge field. Note that the fluctuation fields , , and , as well as the gauge field , will now have canonical dimension , as appropriate for a three dimensional theory. Expressed in terms of these fluctuation fields, the effective theory Lagrangian has the form

 L=Vmin +12trF2ij+12[(∂iϕ)2+m2ϕϕ2]+12[(∂iχ)2+m2χχ2] +tr[(Dih)2+m2hh2]+tr[(Dia)2]+Vint(ϕ,χ,h,a), (19)

where and is the value of at its minimum.

The interaction terms in are suppressed by one or more powers of , and come from the soft potential plus the parts of the hard potential which involve more than two fluctuation fields. These terms are displayed explicitly in Appendix B. The leading order masses induced by the hard potential are

 m2ϕ=−2c1−13vc2,m2χ=−vc2,m2h=−2c1−43vc2, (20)

with given in Eq. (17). The field stays massless at this lowest order. Note that , so this mass is not independent of the other two. The relations between the couplings in and the tree level masses (and ) may be inverted to give

 c1=16(m2χ−3m2ϕ),c2=−m2χ/v,c3=34(m2χ+3m2ϕ)/v2. (21)

The tree-level heavy masses and , plus the value of , should be regarded as arbitrary parameters for now, subject to the condition that . This is equivalent to the inequality which ensures that the global minima of the hard potential lie at . The value of the potential (divided by ) at these minima is .

## 4 Parameter matching at high temperature

### 4.1 Reduction to EQCD

We now want to deduce the constraints on the coefficients in the soft potential which must be satisfied if the effective theory is to reproduce the long distance dynamics of Yang-Mills theory at asymptotically high temperature. In this regime, fluctuations on the scale of the temperature [as well as on the Debye scale ] are weakly coupled, so perturbative matching techniques may be used. The most efficient way to carry this out is to start with our invariant effective theory and integrate out the heavy fluctuation fields , and . This will produce a non- invariant effective theory only involving the light field (and the three dimensional gauge field), which may be directly compared to EQCD.

To lowest non-trivial order, the result of integrating out the heavy fields is a ‘‘light’’ effective theory with Lagrange density101010 We omitted the unit operator from the original effective theory Lagrangian (12), as well as from (22), for simplicity of presentation. The coefficient of the unit operator gives the short distance contribution to the free energy density (divided by ). To match this contribution to the free energy, the coefficient of the unit operator in our effective theory must differ from the corresponding coefficient in EQCD by , up to higher order corrections suppressed by powers of divided by a heavy mass.

 Llight = 12trF2ij+tr[(Dia)2+m2aa2+~λa4]. (22)

Except for the trivial rescaling of the gauge field, and renaming , this has the same form, and identical derivative terms, as the EQCD Lagrangian (5).

The effective light mass , and the quartic coefficient , can have contributions arising solely from the hard potential , as well as having contributions which involve the soft potential . The lowest order contributions generated by the hard potential are illustrated in Fig. 1. Using the coefficients of the various terms in the expansion of about its minima, as given in Eq. (57), one may easily evaluate these diagrams explicitly. However, this is not necessary — the diagrams shown in Fig. 1, at zero external momentum, sum to zero. This is a consequence of the invariance (16) which the effective theory would have in the absence of the soft potential and the static gauge field. An infinitesimal transformation acts as a shift on the light field . Integrating out the heavy fields, without the soft potential, must produce a result which respects this invariance, and thus can only involve derivative interactions for the light field .

Consequently, the light mass and the quartic coupling in the Lagrangian (22) can arise only from contributions involving the non- invariant soft potential . To lowest order, one may simply read off their values from the soft terms in the shifted potential (57), giving

 m2a=~c1g23,~λ=~c3g43. (23)

Demanding that these coefficients coincide with the corresponding terms in the EQCD Lagrangian (5) (after identifying ) fixes

 ~c1 = T+O(g23), (24) and ~c3 = 34π2T+O(g23T2), (25)

while is left undetermined.

Integrating out the heavy fields beyond leading order will, of course, produce corrections to the coefficients (23), suppressed by additional powers of divided by a heavy mass, as well as induce additional higher dimension operators in the light effective Lagrangian (22). The first new operators, involving six powers of the light field , will have a coefficient of order .

### 4.2 Domain wall properties

Having ensured that our theory reproduces the perturbative physics of Yang-Mills theory at very high temperature, we are left with two undetermined parameters, and (in addition to the deliberately introduced heavy masses and ). As we will show, these two parameters may be fixed by demanding that our effective theory reproduce the tension and width of domain walls separating different minima.

Consider a domain wall lying in the - plane, which interpolates between the state with at and at . To find the domain wall profile and evaluate its tension, we need to minimize the free energy for configurations of the scalar field which depend on a single coordinate and satisfy the boundary conditions

 L(z=−∞) = v,L(z=∞)=e2πi/3v, (26a) M(z=−∞) = 0,M(z=∞)=0. (26b)

To evaluate the (leading order) domain wall tension in the high temperature or weak coupling limit it will be necessary to include:

1. The contribution of the hard potential, .

3. The contribution of the soft potential, .

4. The one-loop contribution from fluctuations in the gauge and scalar fields.

Our task is greatly simplified by the hierarchical structure of the potential. Because of their overall factors of , one might expect the first two contributions to be most important. However, a translationally non-invariant domain wall configuration can minimize , everywhere in space, provided equals times some (spatially varying) matrix. And the gradient energy will be comparable to the energy due to the soft potential, and not parametrically larger, if the width of the domain wall is inversely proportional to . Because the soft potential is multiplied by an explicit factor of in the Lagrangian (12), its contribution is comparable to the effects due to fluctuations in the fields. Hence, the one-loop contribution cannot be neglected.111111 This parallels the computation of the leading order domain wall tension in the full theory .

Using gauge invariance, we may diagonalize and write it as

 Z(z)=diag(f1(z),f2(z),f3(z)), (27)

with

 f1(z) ≡ v3e2πi3[α(z)+3β(z)], (28a) f2(z) ≡ v3e2πi3[α(z)−3β(z)], (28b) f3(z) ≡ v3e−4πi3α(z). (28c)

Here and are real functions of that satisfy the boundary conditions

 α(z=−∞) = 0,α(z=∞)=1, (29a) β(z=−∞) = 0,β(z=∞)=0. (29b)

Note that shifting by any integer multiple of leaves invariant (up to a permutation of its diagonal elements).

It will be convenient to introduce a dimensionless rescaled coordinate and Wilson line expectation value via

 ¯z≡g3√Tz,¯v≡vT. (30)

The gradient and soft potential contributions to the domain wall free energy (per unit area) then become

with , etc., and

 U1(α,β) = 14π2{3−cos(4πβ)−2cos(2πα)cos(2πβ)} + ¯v~c218π2{6+cos(4πα)−3cos(4πβ)+2cos(2πα)[cos(6πβ)−3cos(2πβ)]} + (¯v12π2)2{9+cos(8πβ)+2cos(4πβ)[cos(4πα)−2]−8cos(2πα)cos(2πβ)}.

Fluctuations of the fields in the effective theory, at one-loop order, produce functional determinants depending on the background domain wall profile . The width of the domain wall will be parametrically large compared to , and therefore one may use a gradient expansion in the evaluation of the resulting functional determinants. It is sufficient to keep just the first term, leading to121212 The evaluation of this one-loop contribution is similar to the calculation in Ref. . One may neglect the soft potential . Transverse fluctuations of acquire an effective mass due to the background value of the scalar field, and yield the above functional determinant. If one uses the gauge fixing term and lets , with the background value of the scalar field, then the longitudinal part of mixes with the traceless anti-Hermitian part of (which receives no tree-level mass from the hard potential ). The resulting functional determinant is independent of , and equals one with dimensional regularization. Fluctuations of the trace and traceless Hermitian parts of are also independent of , and produce an irrelevant constant (given in footnote 10).

 FflucT = lndet(−∂2+[Z†,[Z,⋅]])/(Area) (33) = ∑ij∫dz∫d3k(2π)3ln(k2+|fi(z)−fj(z)|2) = −∫dz3π(|f1−f2|3+|f2−f3|3+|f3−f1|3) = g−13π2¯v2T(23√T)3∫∞−∞d¯zU2(α,β),

with

 U2(α,β)=−¯v3π3{∣∣sin(2πβ)∣∣3+∣∣sin(π(α−β))∣∣3+∣∣sin(π(α+β))∣∣3}. (34)

Combining the three contributions gives

 Fdw[α,β] ≡ Fgrad+Fsoft+Ffluc (35) = g−13(π¯vT)2(23√T)3∫∞−∞d¯z[(α′)2+3(β′)2+U1(α,β)+U2(α,β)].

This is the excess free energy density due to a domain wall (relative to the spatially constant equilibrium free energy density). Note that all three contributions are parametrically the same size, as anticipated. Minimizing this functional, subject to the boundary conditions (29), will yield the leading order domain wall tension.

The combined potential is plotted in Fig. 2. From the general form of the potential it is easy to see that for positive , and all values of , there is a local minimum in the -direction at . This suggests, and further inspection confirms, that one may simply set in the search for the minimum. This reduces the computation to the minimization of

 Fdw[α,0]=g−13(π¯vT)2(23√T)3∫∞−∞d¯z[(α′)2+˜U(α)], (36)

with

 ˜U(α)≡1π2sin2(πα)−2¯v3π3sin3(πα)+¯v2+4π2¯v~c29π4sin4(πα). (37)

Solutions to the equation of motion produced by varying the functional (36) conserve the “energy” , and the domain wall boundary conditions (29) imply that . The free energy functional (36) is invariant under the transformation , which is a combination of parity, charge conjugation, and a symmetry transformation. The minimal energy domain wall solution should have the same invariance, implying that . Hence, the appropriate solution to the zero energy equation of motion is given by

 ∫α(¯z)1/2dα˜U(α)−1/2=¯z, (38)

while the resulting domain wall tension is

 σ ≡ Fdw[α,0]=g−13(4π¯vT)2(13√T)3∫10dα˜U(α)1/2. (39)

We will define the width of the domain wall as the ratio of the first moment of the domain wall free energy density divided by the tension,131313 Alternative definitions of the domain wall width could, of course, be chosen, but make little difference. For example, defining the width from the second moment of the domain wall free energy density changes the resulting optimal parameters (44) by about 0.1% for and 2.6% for . or

 Δz ≡ (4π¯v)2(13T)3g23σ∫∞−∞d¯z|¯z|˜U(α(¯z)) (40) = 2(4π¯v)2(13T)3g23σ∫11/2dα1˜U(α1)1/2∫α11/2dα2˜U(α2)−1/2.

None of the integrals (38)–(40) are analytically tractable but evaluating them numerically, or integrating the equation of motion , for particular values of and is straightforward. The resulting tension and width may then be compared to the corresponding results in the underlying Yang-Mills theory. The (leading order) domain wall free energy density in high temperature Yang-Mills theory is 

 FYM(z) = π2T43cosh4(¯z(z)/2). (41)

The resulting domain wall tension is

 σ\scriptsize YM = ∫∞−∞dzFYM(z)=8π29T3g(T), (42)

and the (first moment) domain wall width equals

 (43)

A straightforward calculation shows that the effective theory reproduces the domain wall tension and width of the underlying Yang-Mills theory when

 ¯v = 3.005868,~c2=0.118914. (44)

[Note that this value for , almost equal to 3, makes the minima of the potential lie almost exactly at times an element of .] In Fig. 3 we plot for these values of the parameters, and in Fig. 4 we compare the resulting domain wall free energy density in the effective theory, given by

 Feff(z)=(4π¯vT2)2(13)3˜U(α(¯z(z))), (45)

with the full theory result (41). The agreement is rather remarkable.

### 4.3 Discussion

Perturbative matching with EQCD, plus comparison of domain wall properties, has determined (to leading order) all parameters of the effective theory [leading to Eqs. (24), (25), and (44)] except the two heavy masses and . These heavy masses are arbitrary, as long as they are parametrically of order (or ), and satisfy .

It may seem surprising that domain wall properties can be used to fix parameters of the effective theory which were left undetermined by the perturbative matching to EQCD. Correct matching with EQCD should imply that our effective theory reproduces all physics of hot Yang-Mills theory on energy scales small compared to . Fields in the domain wall only vary on the Debye screening length scale which, for asymptotically high temperatures, is parametrically large compared to . So why didn’t matching to EQCD yield an effective theory that automatically reproduces the correct domain wall structure, without any further tuning of parameters?

The resolution of this puzzle involves the realization that the inverse domain wall width is not the only relevant energy scale in a domain wall. As seen in the fluctuation contribution (33) to the tension, near the center of the domain wall off-diagonal components of the static gauge field receive effective masses which are order in the effective theory, and order in the underlying Yang-Mills theory.141414 An alternative way to see this is to consider the electric field contribution to the Yang-Mills action, . A background value of effectively shifts the Matsubara frequencies from to minus the eigenvalues of (in the adjoint representation). Near the center of a domain wall, the eigenvalues of (in a gauge where is static) are of order — since this corresponds to an order one change in the phase of the Wilson line . Consequently, near the center of a domain wall there is no separation of scales between the effective frequencies of the static and non-static modes. Hence, the value of , which determines the magnitude of the scalar field in our effective theory, is a physically relevant scale in the domain wall. Equivalently, the excess free energy density (45) near the center of the domain wall is , and thus comparable to the equilibrium free energy density due to fluctuations on the scale of .

The symmetry structure of the effective theory guarantees that it will have domain walls interpolating between different minima, and the matching to EQCD guarantees that these solutions will have the correct exponential decay, proportional to , far from the domain wall. However, the precise value of the domain wall tension, and its width, are sensitive to physics on the scale of . This is why reproducing these observables in the effective theory requires tuning of parameters beyond the perturbative matching to EQCD. If one were to demand that the effective theory exactly reproduce all moments of the domain wall energy density (41), then it would be necessary to add an infinite number of additional higher dimension operators to the effective theory, with appropriately tuned coefficients. But from Fig. 4, it is evident that near perfect agreement is achieved, without introducing any higher dimension operators, just by suitably adjusting those low energy parameters ( and ) of our superrenormalizable effective theory which were left undetermined by the matching to EQCD.

To conclude this section, let us briefly review the different sources of corrections to the Lagrangian and parameters of the effective theory. These will become important if one aims at next-to-leading-order (NLO) accuracy in the determination of various physical quantities through the effective theory, such as bulk equilibrium thermodynamics or the domain wall tension and width.

Integrating out the heavy fields, with vertices originating from the hard potential , will produce corrections to the scalar and gauge field kinetic terms. At first non-trivial order, the former involves determining the leading momentum dependence of the two-point graphs of Fig. 1b, and will yield an order correction to the normalization of the scalar kinetic term in the Lagrangian of Eq. (22).151515 Terms involving powers of the external momentum, which correspond to higher derivative terms in the Lagrangian, will be proportional to inverse powers of the heavy masses and are thus suppressed by additional factors of when evaluated on momentum scales of order which are relevant in the effective theory. These corrections amount to multiplicative wave function renormalizations for the fields in question.

Integrating out heavy fields at NLO will also generate corrections to the values of the parameters in the light theory (22), whose matching to EQCD will require relative adjustments to the couplings in the soft potential . For the parameters and , one-loop contributions involving one soft vertex will produce and corrections, respectively. For bulk thermodynamics, the NLO correction to is more important than the leading order value of .

Because the invariance of is not respected by the soft potential , or by the coupling of the scalar fields to the gauge field, integrating out the heavy fields may also produce entirely new operators involving and of dimension three or higher. The first such new operators to appear include and , and are suppressed by .161616 For , one factor of comes from the necessary inclusion of one soft vertex from . For , the multiplicative coupling constant is of order , but the covariant derivatives contribute additional factors of — either explicitly or via spatial derivatives applied to scale physics. The same is true for other higher dimension operators involving derivatives.

Ensuring that the domain wall properties of our theory agree with corresponding results in full Yang-Mills theory beyond leading order would, in addition to the above, require extending the derivative expansion of the one-loop fluctuation contribution to next-to-leading order, as well as evaluating the two-loop effective potential for . The analogous NLO calculation in the full Yang-Mills theory is discussed in more detail in Ref. . The required analysis in the effective theory should be very similar.

## 5 Phase diagrams

### 5.1 Eqcd

The lack of center symmetry in EQC
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