Soft thermal contributions to 3-loop gauge coupling

We analyze 3-loop contributions to the gauge coupling felt by ultrasoft ("magnetostatic") modes in hot Yang-Mills theory. So-called soft/hard terms, originating from dimension-six operators within the soft effective theory, are shown to cancel 1097/1098 of the IR divergence found in a recent determination of the hard 3-loop contribution to the soft gauge coupling. The remaining 1/1098 originates from ultrasoft/hard contributions, induced by dimension-six operators in the ultrasoft effective theory. Soft 3-loop contributions are likewise computed, and are found to be IR divergent, rendering the ultrasoft gauge coupling non-perturbative at relative order O(alphas^{3/2}). We elaborate on the implications of these findings for effective theory studies of physical observables in thermal QCD.


Introduction
Dimensionally reduced ("3d") thermal effective theories, originally conceived for studying thermodynamics and phase transitions in non-Abelian gauge theories [1][2][3], and still used for that purpose in the context of weak interactions (cf. e.g. refs. [4,5] for recent work and references), have been reinvigorated in another context some time ago. Indeed, quite remarkably, they also turn out to determine soft contributions to real-time lightcone observables [6]. As examples, they can be used for estimating the so-called transverse collision kernel related to jet quenching in a hot QCD plasma [7,8]; soft parts of the photon and dilepton production rates from a QCD plasma [9,10]; and the interaction rate experienced by neutrinos in an electroweak plasma [11]. Following standard terminology, we refer to the "soft" effective theory as EQCD, whereas the "ultrasoft" theory containing only the magnetostatic modes is called MQCD (cf. e.g. refs. [12][13][14][15]). The latter has been argued to give e.g. the leading non-perturbative contribution to jet quenching [16].
In the QCD context it is known, however, that EQCD fails to describe the full theory close to the phase transition or crossover temperature (T c ). This is obvious when light quarks are present: EQCD contains only gluonic degrees of freedom, and displays no remnant of the flavour symmetries that underlie the chiral transition. For pure-glue theory, the reason for the breakdown is more subtle. Even though the center symmetry that drives the transition in the imaginary-time formulation [17] is not explicit in EQCD, remnants of it are generated dynamically [18]. However the dynamical re-generation is incomplete, and a 3d lattice study in which soft EQCD dynamics was treated non-perturbatively did not achieve satisfactory agreement with thermodynamic functions obtained from full 4d lattice simulations [19].
One purpose of this paper is to demonstrate analytically that power-suppressed dimensionsix operators, truncated from the super-renormalizable EQCD description, play an essential role in soft and ultrasoft observables, and are therefore a likely culprit for EQCD's failure close to T c . More concretely, we determine the MQCD gauge coupling in terms of the EQCD gauge coupling and mass parameter up to 3-loop level, including the 1-and 2-loop contributions of all dimension-six operators; the result is contained in eqs. (3.13), (3.14) and (4.4).
Our presentation is organized as follows. After reviewing the form of EQCD and re-deriving the coefficients of its dimension-six operators in sec. 2, we determine overlapping soft/hard and ultrasoft/hard contributions to the ultrasoft gauge coupling in sec. 3. In terms of fourdimensional Yang-Mills we go up to 3-loop level; this implies 2-loop level in effects originating from dimension-six operators, which are themselves generated by 1-loop diagrams. A 3-loop computation of soft effects, as well as of overlapping ultrasoft/soft contributions, is presented in sec. 4, whereas conclusions are collected in sec. 5. Spacetime and colour tensors, tensorlike 1-loop sum-integrals, Feynman rules related to dimension-six operators, d-dimensional vacuum integrals, and some lengthier results, are collected in five appendices, respectively.

Super-renormalizable part
The super-renormalizable truncation of the dimensionally reduced "electrostatic" QCD, called EQCD, is defined by the action (2.1) is an adjoint scalar, X abcd is defined in eq. (A.6), Latin indices take values i, j ∈ {1, ..., d}, we have in mind d ≡ 3 − 2ǫ, and repeated indices are summed over. We employ a convention in which the fields A a i and A a 0 have the same dimensionality as in four-dimensional Yang-Mills theory. Then explicit factors of 1/T and T appear in configuration and momentum space integration measures, respectively, where T is the temperature.
Focussing on pure SU(N c ) gauge theory, 1 i.e. suppressing contributions proportional to the number of fermion flavours (N f ), the parameters appearing in eq. (2.1) have the expressions is the bare coupling of the original four-dimensional theory, µ is the scale parameter introduced in the context of dimensional regularization, and g 2 ≡ 4πα s is the renormalized coupling. By Σ ′ P we denote a sum-integral over P , with the prime indicating that the Matsubara zero mode is omitted. A 1-loop re-derivation of eqs. (  . (2.9) The starting point for our analysis is the 3-loop determination of g 2 E from four-dimensional Yang-Mills theory [22,23]. It is helpful to display the result in the form of a background field effective action [26]. After gauge coupling and wave function renormalization through vacuum counterterms, refs. [22,23] found an expression containing a logarithmic (1/ǫ) divergence, (2.12) Here ζ n ≡ ζ(n) andμ 2 ≡ 4πµ 2 e −γ E . The renormalized gauge coupling is given by g 2 ER = g 2 /Z B , and the corresponding counterterm by δg 2 E = −g 2 µ 2ǫ δZ B + O(g 10 ). We stress that eqs. (2.11) and (2.12) are gauge independent [27].
An essential technical goal of our investigation is to demonstrate how the divergence in eq. (2.12) is cancelled by overlapping soft/hard and ultrasoft/hard contributions, originating from dimension-six operators within EQCD and MQCD, respectively.
At this point we would like to clarify why such logarithmic divergences (which are "universal", i.e. present in any regularization scheme) originate first at 3-loop level. In three dimensions, 1-loop graphs may contain power divergences but no logarithmic divergences. Logarithmic divergences first originate at 2-loop level. However, within the super-renormalizable truncation of EQCD, they lead to the counterterms in eq. (2.9), i.e. the gauge coupling is finite. Divergences affecting the gauge coupling can only emerge when dimension-six operators are added to EQCD. Given that dimension-six operators are themselves generated by 1-loop diagrams, the divergences correspond to the 3-loop level in terms of the fundamental theory. In sec. 3, where effects originating from integrating out the hard scale are considered, 3-loop level corresponds to the relative accuracy O(g 6 ), whereas in sec. 4, where effects originating from integrating out the soft scale are at focus, the expansion parameter is ∼ g, and the 3-loop effects are of relative magnitude O(g 3 ).

Dimension-six operators
The dimension-six operators that can be added to eq. (2.1) were determined in ref. [28]. We represent the operators as matrices in the adjoint representation. Letting Greek indices take values µ ∈ {0, ..., d}, computing the coefficients at 1-loop level, and choosing to rephrase the gauge coupling as the same g E as appears inside F a ij and D ab i , the dimension-six action can be written as The colour trace refers to the adjoint representation: The value of the sum-integral over P evaluates to (2.14) The values of c i were given for d = 3 in ref. [28]. We need to generalize the expressions to d dimensions, because some of the operators lead to divergent loop integrals at the second stage of our analysis (cf. sec. 3). Beyond leading order, the coefficients are also functions of g 2 , but these contributions are of higher order than the effects that we are interested in. As mentioned in sec. 2.1, we are also suppressing effects proportional to N f . As a first step, it may be realized that the operator basis in eq. (2.13) is redundant: it can be verified that Therefore a simultaneous change of the coefficients (c new i ≡ c i + δc i , i = 4, ..., 7) has no physical effect, provided that δc 4 = δc 5 = −2δc 6 = 2δc 7 . (2.16) In particular, we could tune c 7 to zero as was done in ref.
Adding the 5-point vertex permits for us to fix the combinations in eq. (2.17) as (2.19) In addition the 5-point vertex shows the presence of so-called evanescent operators whose coefficients vanish for d = 3, The coefficient c 10 is also evanescent and can be determined from the 6-point vertex; we find c 10 = (5 − d)(3 − d)(d − 1) 2 /180 but this does not contribute to any of our results. For d = 3 eqs. (2.18)-(2.20) agree with ref. [28]. (Expressions for a general d were derived in ref. [29], but unfortunately a rather different notation was employed.)

Details on the determination of dimension-six coefficients
In this section we provide some details on the determination of the coefficients listed in eqs. (2.18)- (2.20). The derivation of eq. (2.13) is most conveniently formulated with the background field method [26], and as a reminder the gauge potentials are denoted by B a µ . The object computed is the background field effective action, Γ EQCD [B], whereby the vertices are automatically symmetrized in the appropriate way. After a field redefinition, viz.
. We choose to work directly in momentum space, with the background fields denoted by B a µ (q). The momenta q have spatial components only: Specific tensors are defined for showing the dependence of the vertices on spacetime and colour indices; these are summarized in appendix A. The structure naturally emerging from the computation is one in which there are Lorentz-invariant structures (δ µν etc.) and additional terms that only appear for the zero components of the gauge potentials; the latter are identified through the tensors T µν ≡ δ µ0 δ ν0 etc. Results for various 1-loop sum-integrals in this basis are given in appendix B.
Computing the 2-point and 3-point vertices in the background field gauge, we obtain the 1-loop correction where summations and integrations are implied, and T q δ(q) ≡ 1. Expanding in 1/P 2 ∼ 1/(πT ) 2 , the 2-point vertex reads where c 1 and c 2 have the values in eq. (2.18). The term proportional to Σ ′ P 1 P 2 yields the parameter m 2 E in eq. (2.2), whereas the terms proportional to Σ ′ P 1 P 4 yield wave function corrections. The existence of a term Σ ′ P T µν q 2 P 4 indicates that temporal and spatial components of the gauge potentials need to be normalized differently. For the 3-point vertex a similar computation leads to where c 3 and c 4 − c 5 have the values shown in eq. (2.18). 4 The terms proportional to Σ ′ P 1 P 4 can be partly accounted for by wave function corrections; the remainder yields the effective gauge coupling of eq. (2.3). The same result for g 2 E is obtained both from a purely spatial vertex (∼ q ρ δ µi δ νi ) and from a vertex mixing two A a 0 's and one A a i (∼ q ρ T µν ). The 4-point vertex can similarly be written as The notations X {ab}{cd} and X [ab] [cd] are defined in appendix A. The term proportional to Σ ′ P T µναβ P 4 yields λ E in eq. (2.4), whereas the other terms proportional to Σ ′ P 1 P 4 correspond to wave function corrections and g 2 E . The dimension-six part of the 4-point vertex is rather complicated (it is shown in appendix C) and we have not used it for determining c i 's.
Proceeding finally to the 5-point vertex, we find no contribution ∼ Σ ′ P 1 P 4 . The contribution of the dimension-six operators from eq. (2.13) can be written as We have computed the corresponding Feynman diagrams, shown in fig. 1. Making use of momentum conservation and appropriate symmetrizations, and identifying g 2 ), we obtain precisely the same structure from Feynman diagrams. There are 20 independent terms that permit for a crosscheck of eq. (2.18) and, most importantly, for a unique determination of the combinations appearing in eqs. (2.19) and (2.20).

Overlapping soft/hard and ultrasoft/hard contributions
In EQCD, the gauge field components A a 0 have turned into massive adjoint scalar fields when the non-zero Matsubara modes were integrated out (cf. eq. (2.1)). Our goal now is to integrate out the massive A a 0 , and thereby construct the MQCD action. Its super-renormalizable part has the form of the spatial part of eq. (2.1). We denote it by even though F a ij now contains a different gauge coupling than eq. (2.1): The main goal of this section is to determine the contributions to g 2 M that originate from the dimension-six operators in eq. (2.13). These are termed soft/hard (secs. 3.1 and 3.2) and ultrasoft/hard (sec. 3.3) contributions.
We note that in analogy with eq. (2.13), S MQCD also has a dimension-six part, δS MQCD . It is given in eq. (3.16) and discussed in more detail in sec. 3.3.
In order to determine g 2 M , we once again make use of the background field effective action, Γ MQCD [B]. In particular, we consider its quadratic part, where δZ B collects any possible divergences.
In the background field gauge, Γ is gauge invariant in terms of B [26]. Consequently the 3-point and 4-point vertices are fully determined by eq. (3.2). After a subsequent field redefinition, this implies that Z B determines the gauge coupling of MQCD: Here δg 2 E is from eq. (2.8). The following discussion is carried out in terms of Z B and δZ B . When the field A a 0 is integrated out and one vertex from eq. (2.13) is included, we expect to find terms of the types where # (6) may contain logarithms. The corresponding effects are of O(g 5 ) and O(g 6 ) in terms of the original QCD coupling. The latter effect is comparable to eq. (2.12). Before proceeding let us explain why we consider "2-loop soft × 1-loop hard" contributions, i.e. 2-loop graphs with one insertion of dimension-six operators, but not "1-loop soft × 2-loop hard" ones. In terms of Z B defined in eq. (3.2), "1-loop hard" gives a factor ∼ g 2 /T 2 , "1-loop soft" gives a factor ∼ g 2 T m ER ∼ g 3 T 2 , and "2-loop soft" gives a factor ∼ (g 2 T ) 2 ∼ g 4 T 2 . The overall effects of these orders are ∼ g 5 , g 6 , cf. eq. (3.4). In contrast "2-loop hard" would give dimension-six operators proportional to ∼ g 4 /T 2 . The overall effect from "1-loop soft × 2-loop hard" would therefore be ∼ g 7 , i.e. of higher order than our computation. The same applies to dimension-eight operators, whose coefficients are ∼ g 2 /T 4 and who get a further suppression factor < ∼ g 2 T m 3 ER ∼ g 5 T 4 from soft effects.
Computing the graphs in fig. 2 in dimensional regularization and expanding in q 2 /m 2 E , all of them can be related to a single 1-loop tadpole integral, denoted by Inserting the values of the coefficients in terms of the c i 's from appendix C, the terms proportional to m 2 E δ ij and q i q j drop out as required by gauge invariance, and we are left with The corresponding contribution to Z B is shown on the first row of eq. (3.13).   (the notation is as in fig. 3).

2-loop results with dimension-six operators
6-point Chapman vertex also contribute. The general expressions for these, parametrized through the coefficients κ 1 , ..., κ 10 , λ 1 , ..., λ 10 and χ 1 , ..., χ 16 , are given in eqs. (C. 19) and (C.21), respectively, and the corresponding diagrams are shown in fig. 6. In order to display the result, we introduce a 2-loop "sunset" integral, Then δΓ (2) where C 1 , C 2 , C 3 are given in appendix E in terms of the coefficients η 1 , ..., χ 16 . Inserting the values of the coefficients from appendix C, we find that C 1 and C 3 and terms proportional to α in C 2 cancel. The remaining contribution reads , (3.12) where in the last step we made use of eqs. (2.18)-(2.20). We note that the evanescent operators parametrized by c 8 and c 9 do not play a role for d ≈ 3, because the coefficients with which they contribute in eq. (3.11) themselves vanish for d → 3. Setting d = 3 − 2ǫ, inserting eqs. (2.14), (3.7) and (3.9), and going over to renormalized parameters, we obtain (3.14) Remarkably, setting g 2 ER = g 2 (1 + O(g 2 )), the divergence in eq. (3.14) cancels 1097/1098 of the coefficient of 1/ǫ in eq. (2.12). The remaining 1/1098 can be expressed as where in the round brackets we have isolated the master integral in eq. (2.14).

Contribution from dimension-six operators in MQCD
As already alluded to below eq. (3.1), there are dimension-six operators also in MQCD. These originate from the purely spatial part of eq. (2.13), and also from 1-loop effects within EQCD, as will be discussed in sec. 4. The corresponding action can be written as 6 where (recalling g 2 M = g 2 E (1 + O(g))) the hard contribution is δC i = Σ ′ P c i /P 6 . The dimension-six operators in eq. (3.16) give a contribution to physical observables determined by MQCD, such as the spatial string tension or "magnetostatic" screening masses. Given that MQCD is a confining theory, these effects cannot be computed analytically. We would like to know, however, whether the MQCD dynamics can give an ultraviolet (UV) divergent contribution, compensating against the term in eq. (3.15).
In order to determine the UV divergence, we employ a trick similar to that in ref. [31]. All infrared (IR) contributions are "shielded" by employing the propagators where c a ,c b are ghost fields, α is a gauge parameter, and m G ≡ g 2 M T /π is a fictitious mass. Once again, we compute a background field effective action, now denoted by Γ IR [B] given that the most IR fluctuations have been accounted for. We extract from it a 2-point function like in eq. (3.2). The technical implementation follows that in secs. 3.1 and 3.2. 6 There are many alternative representations, for instance tr where we denoted the dual field strength by F a i ≡ ǫ ijk 2 F a jk and defined ǫ ijk ǫ lmn ≡ δ il (δ jm δ kn − δ jn δ km ) + δ im (δ jn δ kl − δ jl δ kn ) + δ in (δ jl δ km − δ jm δ kl ).
Most contributions that we find are α-dependent and void of physical significance. For instance, the 1-loop result has a structure similar to eq. (3.6) but with m E → m G : This result is finite and proportional to m G and vanishes when we send m G → 0. However, at 2-loop order a non-trivial and gauge-independent result emerges. Writing the contribution from Chapman vertices in a form reminiscent of eq. (3.10), we get The function H 3 is the three-mass variant of eq. (3.9), cf. eq. (D.10), and has the same UV divergence, viz.
For ǫ → 0, D 1,3 are of O(ǫ) and yield no divergence, whereas D 2 has a finite α-independent part: Substituting C 3 → Σ ′ P c 3 /P 6 , inserting c 3 from eq. (2.18), and setting g 2 M = g 2 µ 2ǫ (1+O(g)), yields a gauge-independent UV divergence and logarithmic part: Comparing with eq. (3.15), the divergence exactly cancels. Therefore we have now established our main technical goal, demonstrating that the IR-divergence in eq. (2.12) is fully cancelled by soft/hard and ultrasoft/hard contributions from dimension-six operators.

Soft and overlapping ultrasoft/soft contributions
In sec. 3 we considered the soft/hard contributions to the MQCD effective action, cf. eq. (3.4). However, there are other contributions to Z B , namely those associated with the purely "soft" contributions from the scale m E . In order to distinguish these from the effects considered in sec. 3, we denote them by Z B . For this section, we can take the super-renormalizable truncation in eq. (2.1) as a starting point, and m E as the only scale being integrated out.

Direct soft terms up to 3-loop level
Up to 2-loop level, the value of Z B was determined in ref. [32] (the dependence on scalar couplings was added in ref. [20]): 7 We now turn to the 3-loop contribution. The determination of Z B is a rather straightforward exercise in computer-algebraic methods for loop integrals. The Feynman diagrams were generated with QGRAF [33]. After expanding in the external momentum and projecting onto the transverse and longitudinal polarizations, we have to deal with vacuum-like master integrals. The subsequent simplifications, making use of renamings of integration variables and integration-by-parts (IBP) identities [34,35], have been programmed in FORM [36]. The values of the 3-loop master integrals can be found in refs. [31,37] and are given in eqs. (D.12) and (D.13). As a crosscheck, we have carried out two independent computations, whose results coincide perfectly. Our final "bare" expression reads 8 δ Γ (2)  The 1/ǫ-divergences in eq. (4.2) could a priori have an IR or UV origin. To find out, we have carried out the same computation by shielding all masses like in eq. (3.17), but with m G → m E . Then only the divergence proportional to 4(κ 2 − 4λ) remains. This indicates that the divergence not containing scalar self-couplings is purely of IR origin.
We can envisage two possible sources for the IR divergence. One is related to ultrasoft contributions of the same type as in sec. 3.3; these are analyzed in sec. 4.2. The other is related to the mass parameter m 2 E . It is well known that the physical Debye mass, defined as a screening mass related to a "heavy-light" state, is non-perturbative starting at next-toleading order [38,39]. Our m 2 E is not such a physical mass but rather a Lagrangian parameter. Nevertheless, m 2 E can still be considered IR sensitive at O(g 4 ER T 2 ). Indeed, if we compute the 2-point function of A a 0 at zero momentum, and shield all masses like in eq. (3.17), we find the UV divergence cancelled by the mass counterterm in eq. (2.9). In contrast, if we compute the , where the integrals are given in eq. (D.1). 8 The full d-dimensional form is given in appendix E, cf. eqs. (E.4)-(E.12).
2-point function without IR-shielding, we find an additional 1/ǫ-divergence proportional to g 4 ER T 2 , which depends on the gauge parameter α. This is an IR divergence, i.e. ∼ g 4 ER T 2 /ǫ IR . If we naively insert an ambiguity of this type into the 1-loop term in eq. (4.1) and re-expand up to 3-loop order, the result is On the non-perturbative level, 1/ǫ IR would turn into a multiple of ln(c m G /m ER ), where c is a non-perturbative constant and the scale m G was defined around eq. (3.17). Keeping in mind this expectation, we renormalize eq. (4.2) by employing the proper mass counterterm from eq. (2.9). The UV divergences proportional to κ 2 − 4λ duly cancel, and we find the 3-loop result (4.4)

Contribution from dimension-six operators in MQCD
Parallelling sec. 3.3, let us finally consider contributions from ultrasoft effects to the gauge coupling, in the presence of dimension-six operators in MQCD. The action has the form in eq. (3.16), with the coefficients now completed to include the soft contribution: (4.5) The spatial integral appearing is related to that in eq. (3.5) as shown by eq. (D.1), Including the overall prefactor from eq. (3.16) and the integral from eq. (4.6), the new contributions to the coefficients of the dimension-six operators are ∼ g 2 M T /m 3 E at 1-loop level. Including a fictitious IR-regulator like in eq. (3.17), the 1-loop contribution from these operators to Z B comes with a factor ∼ g 2 M T m G and vanishes for m G → 0, whereas the 2loop contribution comes with a factor ∼ g 4 M T 2 and can yield a contribution ∼ g 6 M T 3 /m 3 E ∼ O(g 3 ) to Z B . 2-loop contributions to the coefficients of dimension-six operators would be ∼ g 4 M T 2 /m 4 E and therefore lead to effects suppressed by ∼ O(g 4 ). Dimension-eight operators, whose coefficients are ∼ g 2 M T /m 5 E , lead to effects suppressed by ∼ g 10 M T 5 /m 5 E ∼ O(g 5 ). According to eq. (2.24), the value ofc 1 can be inferred from the 2-point and that ofc 3 from the 3-point vertex of the background field effective action. To be sure that no operators got overlooked, we have also determined them from the 5-point vertex, cf. the spatial part of eq. (2.27), which leads to several independent crosschecks (the diagrams are shown in fig. 7). We find that the results are related in a curious way to the d-dependence of c 1 and c 3 in eq. (2.18): 9,10c Inserting these values into eq. (3.20), and substituting g 2 M = g 2 ER µ 2ǫ (1 + O(g)), we find a gauge-independent UV divergence and logarithmic part: This implies that the counterterm needed in MQCD reads δ Z 45ǫ . Obviously, eq. (4.8) does not match the divergence in eq. (4.4). In other words, if we subtract the part needed to serve as δ Z (3) B from eq. (4.4), an IR divergence remains. In terms of the coefficient β introduced in eq. (4.3), it amounts to β = − 13 15 . Let us stress that we have verified the gauge independence of this result. Therefore we are left to speculate that a non-perturbative mass ambiguity of the type discussed around eq. (4.3) prohibits a purely perturbative determination of Z

Conclusions
The main technical ingredient of this investigation was the analysis carried out in sec. 3. We considered dimension-six operators induced by integrating out the "hard" momenta ∼ πT from thermal QCD [28]. Specifically, we computed at 1-loop and 2-loop levels the influence of 9 To our knowledge these values were first obtained for d = 3 by P. Giovannangeli (unpublished, 2005), along lines that have recently been documented in ref. [40]. 10 We note in passing that even though thec i contribution in eq. these operators on the gauge coupling felt by ultrasoft (magnetostatic) modes. Remarkably, including UV divergences originating both from "soft" loops at the Debye scale m E ∼ gT and "ultrasoft" loops at the non-perturbative scale ∼ g 2 T /π, we observed an exact cancellation of the IR divergence found in a 3-loop determination of the EQCD gauge coupling (cf. eq. (2.12)) [22,23]. This represents a nice crosscheck of the effective theory setup as a whole.
As a second technical ingredient, discussed in sec. 4, we considered the "soft" contributions to the ultrasoft gauge coupling. We determined direct 3-loop effects (cf. eq. (4.4)) and compared them with overlapping ultrasoft/soft contributions originating from dimensionsix operators induced by integrating out the soft momenta ∼ m E (cf. eq. (4.8)). This time only a partial cancellation of soft IR divergences against ultrasoft/soft UV divergences was observed. As a culprit, we speculate that a non-perturbative ambiguity of the soft scale within EQCD sets an upper bound on the accuracy with which effects depending on m E can be determined within perturbation theory. This may be surprising insofar as no such problem was met in 3-loop or 4-loop studies of the EQCD vacuum energy density [15,31]. However, the present quantity is different, being not directly a physical observable but rather an effective Lagrangian parameter (the MQCD gauge coupling g 2 M ). On a more general level, the main conclusions that we draw are as follows: (i) Even if the colour-electric scale m E ∼ gT is formally larger than the colour-magnetic scale ∼ g 2 T /π, it does play an essential role in the IR dynamics. Concretely, in terms of the IR divergence found by integrating out the hard scale ∼ πT , the colour-electric scale is 1097 times more important than the colour-magnetic scale (cf. eq. (3.14)).
(ii) Dimension-six operators need to be included in EQCD if good precision is required.
Indeed, as we have demonstrated analytically (cf. point (i)), they do influence the IR dynamics of the system. This is a possible reason for why the super-renormalizable truncation of EQCD fails close to T c even in pure Yang-Mills theory [19].
(iii) Apart from the indications in point (i) that the scale m E is important, we also find trouble if we try to integrate it out. The reason could be that EQCD is a confining theory, and that physics at the scale m 2 E should in general be affected by non-perturbative ambiguities of O(g 4 T 2 /π 2 ). Once m E is integrated out, some remnant of these ambiguities may remain, if the parameters of MQCD are determined up to the corresponding relative precision. It would be interesting to find a way to determine the leading nonperturbative contribution to g 2 M through lattice methods, even if this requires the simultaneous inclusion of the 1/m 3 E -suppressed MQCD dimension-six operators in eq. (3.16).
As a second ingredient, we report the full d-dimensional version of eq. (4.2). The result can be expressed as δ Γ (2) where the pure gauge contributions are parametrized by