Scheme dependence of two-loop HTLpt-resummed $\text{SYM}_{4,4}$ thermodynamics

The resummed thermodynamics of ${\cal N}=4$ supersymmetric Yang-Mills theory in four space-time dimensions ($\text{SYM}_{4,4}$) has been calculated previously to two loop order within hard thermal loop perturbation theory (HTLpt) using the canonical dimensional regularization (DRG) scheme. Herein, we revisit this calculation using the regularization by dimensional reduction (RDR) scheme. Since the RDR scheme manifestly preserves supersymmetry it is the preferred scheme, however, it is important to assess if and by how much the resummed perturbative results depend on the regularization scheme used. Comparing predictions for the scaled entropy obtained using the DRG and RDR schemes we find that for $\lambda \lesssim 6$ they are numerically very similar. We then compare the results obtained in both schemes with the strict perturbative result, which is accurate up to order $\lambda^2$, and a generalized Pad\'{e} approximant constructed from the known large-$N_c$ weak- and strong-coupling expansions. Comparing the strict perturbative expansion of the two-loop HTLpt result with the perturbative expansion to order $\lambda^2$, we find that both the DRG and RDR HTLpt calculations result in the same scheme-independent predictions for the coefficients at order $\lambda$, $\lambda^{3/2}$, and $\lambda^2 \log\lambda$, however, at order $\lambda^2$ there is a residual regularization scheme dependence.


I. INTRODUCTION
Supersymmetric field theories have generated a great deal of interest in the past decades [1][2][3][4][5]. Such field theories are invariant under supersymmetry transformations in which bosonic and fermionic degrees of freedom are transformed into one another. Although there is currently no experimental evidence that such theories are realized in nature, they have recently proven to be useful due to the ability to employ the conjectured (and strongly evidenced) holographic duality between strongly-coupled conformal field theory (CFT) in the large-N c limit and weakly-coupled gravity in five-dimensional anti-de Sitter space (AdS) [6]. This conjectured mathematical equivalence is called the AdS/CFT correspondence and, in the context of thermodynamics, has been used to calculate the strong-coupling limit of N = 4 supersymmetric field theory in four space-time dimensions (SYM 4,4 ) [7]. In this paper, we focus on resummed perturbative calculations of thermodynamics in SYM 4,4 which complement such calculations.
One of the motivations for the calculation presented herein is to test methods that have been applied in the context of finite temperature and density quantum chromodynamics (QCD). In particular, we would like to apply perturbative reorganizations which have been used to improve the convergence of the successive perturbative approximations to the QCD thermodynamic potential . These studies have demonstrated that it is possible to obtain excellent agreement with continuum extrapolated lattice calculations of QCD thermodynamics for T 250 − 300 MeV using such methods. Unlike SYM 4,4 , QCD is a confining theory at low temperature, however, at high temperature there are many similari-ties between SYM 4,4 and QCD. This stems from the fact that (a) QCD is asymptotically free and (b) the two theories are similar in the weak coupling limit. Comparing perturbative SYM 4,4 and QCD one finds that (1) the forms of gluonic and fermionic collective modes are the same, and the scalar collective modes are those of a massive relativistic particle; (2) the transport coefficients, such as the shear viscosity η, which are dominated by the Coulomb-like interactions, are quite similar [30]; and (3) the energy loss and momentum broadening of highly energetic test particles are also alike [31]. These studies indicate that the key difference between perturbative finite-temperature QCD and SYM 4,4 is the number and type of degrees of freedom, with SYM 4,4 having, in addition to the adjoint gauge field, four adjoint Majorana fermions and six adjoint scalars.
Unlike QCD, however, SYM 4,4 is ultraviolet finite due to its supersymmetric nature and has a vanishing β function. Due to this, the 't Hooft coupling λ = g 2 N c in SYM 4,4 does not run and is independent of the temperature. In the weak-coupling limit, the thermodynamics of SYM 4,4 has been calculated through order λ 2 with the result being [32][33][34][35][36][37] where F ideal = −d A π 2 T 4 /6 is the ideal or Stefan-Boltzmann limit of the free energy and S ideal = 2d A π 2 T 3 /3, with d A = N 2 c −1 being the dimension of the arXiv:2201.03510v2 [hep-th] 17 Apr 2022 adjoint representation and ζ(z) being the Riemann zeta function. The ratios of the free energy and entropy density to their corresponding ideal limit are the same owing to the fact that the 't Hooft coupling is temperature independent. Note, importantly that the weak-coupling expression (1) is valid for all N c . In the strong-coupling limit, the behavior of the SYM 4,4 free energy has been computed using the AdS/CFT correspondence. Most information is known about the large-N c limit where one has [7] One issue that must be faced when using strict weak-or strong-coupling expansions is that they might have poor convergence as additional orders are included in the expansion or may not converge at all. This is a known issue with the weak-coupling expansion of QCD thermodynamics, in which case the series seems to converge only for T 10 5 GeV. This poor convergence of strict perturbation theory motivated research into methods for reorganizing the weak-coupling expansion in order to improve its convergence as one goes to higher loop order. In the context of QCD, the methods used have included the Φ-derivable method [38][39][40][41] and the hard-thermal-loop perturbation theory (HTLpt) reorganization [8,10,12].
Despite its appeal, a fundamental issue with the Φderivable method is that it is not manifestly gauge invariant, with gauge parameter dependence appearing at the same order in λ as the series truncation when evaluated off the stationary point and at twice the order in λ when evaluated at the stationary point [9,42,43]. The HTLpt approach, on the contrary, is manifestly gauge invariant due to the fact that the HTL effective action used as the starting point is gauge invariant by construction. HTLpt has been used to improve the convergence of weak coupling calculations of the free energy in scalar field theories [44][45][46], QED [47], and QCD up to threeloop order at finite temperature and chemical potential [19-23, 26, 27]. Based on its success in QCD applications, in Ref. [48] we applied this method to SYM 4,4 , however, in this prior work canonical dimensional regularization (DRG) [49][50][51] was used to regulate divergences generated during the calculation in the same manner as was done in QCD.
One issue with this prior work is that the use of canonical DRG breaks supersymmetry because in DRG the size of the bosonic representation depends on the dimensional regulation parameter . In this paper we address this issue by using a regularization scheme called regularization by dimensional reduction (RDR). The RDR method was introduced by Siegel [52] and is a modified version of dimensional regularization [53,54] which manifestly preserves gauge invariance, unitarity, and supersymmetry [55,56]. In Refs. [36,37] this method was used to compute the λ 2 and λ 2 log λ coefficients in Eq. (1). Herein we will compute the order λ, λ 3/2 , and λ 2 log λ coefficients using two-loop HTLpt and demonstrate that these coefficients are scheme independent. We will additionally demonstrate that the order λ 2 coefficient is regularization-scheme dependent; however, this is somewhat expected, since this coefficient is beyond the strict perturbative accuracy of a two-loop calculation. Importantly, we find that the coefficient of λ 2 log λ is exactly the same as obtained in the prior two-loop DRG HTLpt calculation [48], resummation using the Arnold-Zhai method [36], and resummation using effective field theory methods [37]. This firmly establishes the existence of logarithms in the weak-coupling expansion of SYM 4,4 thermodynamics and provides confidence in the computed coefficient.
Note that since the high-order terms in the expansion of the free energy are unknown, it not possible to determine if the finite temperature perturbative series for the free energy has a finite radius of convergence. We note, however, that even if the perturbative series has zero radius of convergence (an asymptotic series), it is possible to apply variational perturbation theory methods such as HTLpt to improve the convergence of successive loop approximations. In cases where all orders expansions of quantities are known, e.g., the ground state energy of a zero temperature anharmonic oscillator [57,58], it has been shown that by using variational perturbation theory one can even self-consistently obtain the strong coupling limit coefficients from a divergent weak-coupling expansion [59,60].
In the case of SYM 4,4 , our recent calculations through order λ 2 [36,37] suggest that the perturbative expansion of SYM 4,4 thermodynamics has a finite and potentially large radius of convergence. The resummed results obtained in this paper show that HTLpt can be used to improve the convergence of successive approximations to the SYM 4,4 free energy. As will we demonstrate, the twoloop HTLpt resummed result, although having strict perturbative accuracy of order λ 3/2 , reproduces the order λ 2 result to within 2% for λ 2 (see Fig. 2

below).
Finally, we emphasize that the scheme dependence discussed herein is related to whether or not one uses a supersymmetry-preserving regularization scheme. This is a fundamental symmetry requirement and, hence, the RDR scheme is better suited to this problem. In the context of QED and QCD there exists a more general scheme dependence which stems from the choice of the method of regularization. The standard scheme for renormalizing QCD is the MS (modified minimal subtraction) scheme [61], which is related to the MS (minimal subtraction) scheme [62,63] through a rescaling of the MS scale by (e γ E /4π) where γ E is the Euler-Mascheroni constant. As a result of this relationship, one can connect quantities computed in the two schemes in a straightforward manner using the running coupling itself [61]. We note importantly, however, that physical observables such as scattering rates and the free energy are scheme independent [61,64,65]. In strict perturbation theory, renormalization scheme invariance of observables can be established order-by-order in the coupling, however, the parameters in the theory which are not directly observable, such as the running coupling constant, are in general scheme dependent. When computing observables the scheme-dependence of the parameters is compensated for by the scheme-dependence of the coefficients in the perturbative expansion thereby ensuring renormalization group invariance [61,64,65].
In SYM 4,4 an analogous scheme dependence does not occur because the theory is conformal and the coupling does not run. As a result, the perturbative coefficients are renormalization group invariants and hence, scheme independent in the QED/QCD sense. This is evidenced by Eq. (1) which is scale invariant through the perturbative accuracy determined. Finally, we note that HTLpt goes beyond strict perturbation theory through an all orders resummation in the soft sector. As a result, a residual scale dependence can remain in HTLpt when truncating at finite loop order; however, as one extends the HTLpt calculation to higher loop order, the scale dependence is systematically pushed to higher orders in the 't Hooft coupling, resulting in mathematically unique predictions for the fully determined perturbative coefficients.
The structure of our paper is as follows. We begin with a brief introduction to the basics of SYM 4,4 in Sec. II. In section III, we present a summary of HTLpt applied to SYM 4,4 . We list the terms which change in the high temperature expansion when going from DRG to RDR in Sec. IV. Based on these results, we present the complete expressions for the leading-(LO) and next-to-leading order (NLO) thermodynamic potentials in the RDR scheme in Sec. V. In Sec. VI, we present our numerical results for the RDR HTLpt-resummed NLO SYM 4,4 scaled thermodynamic functions and compare to our previous results obtained using the DRG scheme. We also compare to strict perturbative expression for the scaled thermodynamic functions through order λ 2 which were obtained using the RDR scheme and to a generalized Padé approximant based on this result and the corresponding result in the large-N c strong coupling limit. In Sec. VII we present our conclusions and an outlook for the future.
Notation: We use lower-case letters for Minkowski space four-vectors, e.g., p, and upper-case letters for Euclidean space four-vectors, e.g., P . We use the mostly minus convention for the metric.

II. THE BASICS OF THE SYM4,4 THEORY
In SYM 4,4 all fields belong to the adjoint representation of the SU (N c ) gauge group. The definition of gauge field is the same as QCD and A µ can be expanded as A µ = A a µ t a , with real coefficients A a µ , and Hermitian color generators t a in the adjoint representation which satisfy where a, b = 1, · · · , N 2 c − 1 and the group structure constants f abc are real and completely antisymmetric.
In addition to the gauge field and Majorana spinors, there are six independent real scalar fields which are represented by a multiplet where X p and Y q are Hermitian, with p,q = 1, 2, 3. X p and Y q denote scalar and pseudoscalar fields, respectively. We will use a capital Latin index A to denote components of the vector Φ. Therefore Φ A , X p , and Y q can be expanded as Φ A = Φ a A t a , with A = 1, · · · , 6 or alternatively, X p = X a p t a and Y q = Y a q t a . The Minkowski-space Lagrangian density for SYM 4,4 can be expressed as where µ, ν = 0, 1, 2, 3 and α p and β q are 4 × 4 matrices that satisfy The matrices α and β satisfy α p ik α p kj = −3δ ij and β q ij β p ji = −4δ pq , with δ ii = 4 for the four Majorana fermions and δ pp = 3 for each set of three scalars.
To quantize the theory, gauge-fixing and ghost terms should be added to the Lagrangian density. In general covariant gauge, their forms are the same as in QCD, with ξ being the gauge parameter.
The HTLpt reorganization of SYM 4,4 can be obtained in the same manner as in QCD by introducing an expansion parameter δ, treating it as a formal expansion parameter, expanding around δ = 0 to a fixed order, and then setting δ = 1 in the end. The HTL reorganized Lagrangian density for SYM 4,4 can be written as The HTL improvement term L HTL SYM4,4 is where y µ = (1,ŷ) is a light-like four vector, · · · ŷ represents an average over the direction ofŷ defined in App. A, the index j ∈ {1 . . . 4} labels the four Majorana fermions, and the index A ∈ {1 . . . 6} labels the six independent real-valued scalars. The parameters m D and M D are the electric screening masses for the gauge field and the adjoint scalar fields, respectively. The parameter m q is the induced finite temperature quark mass. We note that, in general, the gluon mass in RDR is not equal to −g µν Π µν , where Π µν is the gluonic self energy and care must be taken due to this. This stems from the necessity of employing integration by parts during the calculation of the gluon self energy.
In the RDR scheme, all momentum-space integrals will be evaluated in d = 4 − 2 dimensions while the size of the gauge field and fermionic field representations will be taken to be integer valued, and correspond to D = 4 dimensional fields. Since, as we will demonstrate, all ultraviolet divergences generated in the calculation of HTLthermodynamics are canceled by the HTLpt counterterms independently of the value of the gluon, quark, and scalar mass parameters, we need to only consider the leading-order contributions in the momentum-space regulation parameter .
The HTLpt reorganization of QCD and SYM 4,4 generates new ultraviolet (UV) divergences compared to the vacuum Lagrangian. In QCD, these divergences can be eliminated using the counterterm Lagrangian ∆L HTL QCD and the thermodynamic potential at the two-loop level can be renormalized by using a simple counterterm Lagrangian ∆L HTL QCD which includes vacuum energy and mass counterterms [16,17]. Although not proven at arbitrary loop order, it has been explicitly demonstrated that one can renormalize the HTLpt thermodynamic potential through three-loop order using only vacuum, gluon thermal mass, quark thermal mass, and gauge coupling constant counterterms [22,27]. The same method can be used in SYM 4,4 .
We find that in SYM 4,4 the vacuum counterterm ∆ 0 E 0 , which is the leading order counterterm in the δ expansion of the vacuum energy E 0 , can be obtained by calculating the free energy to leading order in δ, and the next-to-leading-order contribution ∆ 1 E 0 can be obtained by expanding ∆E 0 to linear order in δ. As a result, we find that in the RDR scheme the counterterm ∆E 0 has the form Comparing to the DRG HTLpt result obtained in Ref. [48], one sees that Eq. (11) is precisely the same as obtained therein.
To calculate the NLO HTLpt-improved free energy one needs to expand the partition function to order δ. Correspondingly, to cancel the remaining UV divergences we need the counterterms ∆E 0 , ∆m 2 D , ∆m 2 q , and ∆M 2 D to order δ. In order to remove all UV divergences which appear at two-loop level, the mass counterterms required are Once again we find that the mass counterterms are precisely the same as the ones necessary to remove the HTLpt divergences in DRG HTLpt [48].
To calculate HTLpt-improved physical observables in SYM 4,4 we use the same method as in QCD, namely expanding the path-integral in powers of δ, truncating at some specified order, and then setting δ = 1. The results for physical observables will depend on m D , M D , and m q which are functions of T and λ. These parameters are fixed in HTLpt by minimizing the free energy. If we use Ω RDR N (T, λ, m D , M D , m q , δ) to represent the thermodynamic potential expanded to N -th order in δ, then the corresponding variational prescription is These three equations are called the gap equations. The free energy can be obtained by evaluating the thermodynamic potential at the solution to the gap equations, and other thermodynamic functions can be obtained from the free energy and its derivatives with respect to T . In the following section, we calculate the thermodynamic potential through order λ 5/2 , which can be expressed as an expansion in powers of m D /T , m q /T , and M D /T . Instead of repeating all results from our DRG calculation, we list all terms which change when going from DRG to RDR. As mentioned above, at order δ, all divergences in the two-loop thermodynamic potential can be removed by the HTLpt vacuum and mass counterterms which are scheme independent.

IV. HIGH-TEMPERATURE EXPANSION USING THE RDR SCHEME
In this section, we list only the contributions which change when going from the DRG to RDR scheme. Since the Feynman rules and diagrams needed are the same as in Ref. [48], the high-temperature forms are the same as in this reference up to the dimension of the gluon tensor and metric tensor. The labels on each Feynman diagram contribution below correspond to the diagrams shown in Ref. [48].
A. One-loop sum-integrals in the RDR scheme Based on the above knowledge, at one-loop order, only the hard contribution from the gluonic one-loop free energy and two-loop HTL counterterm are modified in the RDR scheme. The remaining contributions come from fermions and scalars and remain unchanged compared to the DRG HTLpt result.

RDR modified gluon hard contributions
The form of the hard contribution to the one-loop gluon free energy expanded to second order in m 2 D has been given in [16], and can be written in the RDR scheme as where T P is given in Eq. (A14), which enters into the HTL gluon self-energy, and Setting D = 4, using Eq. (15), and the formulas in App. B, (14) reduces to The form of the hard contribution to the two-loop gluon HTL counterterm has been given in [16], and can be expressed as Setting D = 4, using Eq. (15), and the formulas in App. B, (18) reduces to We find, as can be expected on general grounds, that the only difference between the result of the one-loop gluonic contributions in the RDR scheme and the DRG scheme is the finite contribution which is proportional to m 4 D .

B. Two-loop sum-integrals in the RDR scheme
Considering the two-loop contributions in the RDR scheme, the soft-soft (ss) contribution from all diagrams and the F (hh) 3qs contribution are unchanged. All other contributions are modified due to the dimension of the gluon tensor and metric tensor changing from 4 − 2 to 4.

Contributions from the (hh) region
The form of the (hh) contributions to the two-loop gluonic free energy expanded to first order in m 2 D has been given in [16], and can be written in the RDR scheme as Setting D = 4, using Eq. (15), and the formulas in App. B, (20) reduces to The form of the (hh) contribution to F 3qg and F 4qg expanded to first order in m 2 D can be obtained from Ref. [17], and in the RDR scheme become where f n is defined as Setting D = 4, using Eqs. (15) and (23), and formulas in App. B, (22) reduces to The form of the (hh) contributions to F 4s , F 3gs and Setting D = 4, using Eq. (15), and the formulas in

Contributions from the (hs) region
The form of the (hs) contribution to the two-loop gluon free energy expanded to first order in m 2 D was given in [16] and in the RDR scheme can be written as where Setting D = 4, using Eqs. (15) and (28), and the formulas in App. B, (27) reduces to The form of the (hs) contribution to F 3qg and F 4qg expanded to first order in m 2 D can be obtained from Ref. [17], and in the RDR scheme can be expressed as The (hs) contribution to F 4s , F 3gs , and F 4gs were presented in [48] and in the RDR scheme can be expressed as Setting D = 4, using Eq. (15), and the formulas in App. B, (32) reduces to Finally, the (hs) contribution to F 3qs was first presented in [48] and in the RDR scheme is Setting D = 4, using Eqs. (15), (23), and (28)

V. NLO HTL THERMODYNAMIC POTENTIAL IN THE RDR SCHEME
In this section, we will combine all contributions and counterterms to obtain the LO and NLO HTLpt-resummed SYM 4,4 thermodynamic potential Ω RDR (T, λ, m D , M D , m q , δ = 1). In the following subsections, we list the various terms contributing and compare them to the corresponding results obtained in the DRG scheme.

A. Leading order
By combining Eq. (17) and contributions which are the same as in DRG scheme from Ref. [48], our final result for the one-loop free energy is wherem D ,M D ,m q andμ are dimensionless variables, which are defined aŝ Comparing to the one-loop free energy Ω 1-loop obtained in the DRG scheme, the only difference are the finite contributions which are proportional to m 4 D . Since the RDR divergences are the same as in the DRG scheme, the corresponding leading order vacuum energy counterterm ∆ 0 E 0 is also the same. After adding ∆ 0 E 0 to (36), our final result for the LO renormalized thermodynamic potential in the RDR scheme is B. Next-to-leading order By combining Eqs. (20), (22), (26), (27), (30), (33), (35), and contributions which are the same as in DRG scheme from Ref. [48], and multiplying by λd A , we obtain the final result for the two-loop HTLpt thermodynamic potential Comparing to the two-loop DRG result Ω 2-loop , we find that there are differences in the finite contributions proportional to m 2 The NLO RDR HTLpt counterterm contribution is obtained from the sum of Eqs. (18) and contributions which are the same as in DRG scheme from Ref. [48], giving Comparing to the DRG scheme result for Ω HTL , the only difference is the finite contribution proportional to m 4 D . We find that the form of ∆ 1 E 0 is the same as in the DRG scheme. Since the divergent terms in Eq. (39) are the same as the ones in the DRG scheme, we find the same result for ∆Ω RDR NLO as in the DRG scheme. By adding the one-loop (38), and two-loop (39) HTLpt thermodynamic potentials, the HTLpt gluon and quark counterterms, and the HTLpt vacuum and mass renormalizations, we obtain the final expression for the NLO HTL thermodynamic potential in SYM 4,4 theory in the RDR scheme Note that the final expression is free from singularities and valid for all N c . Comparing to the result obtained using the DRG scheme in Ref. [48], we find differences in terms without logarithms, which are proportional to m 4 D , λm 2 q , λm 2 D , λM 2 D , λm 3 D and λM 3 D . These terms all contribute to the coefficient of λ 2 in the strict perturbative limit.

C. The strict perturbative limit
We next consider the strict perturbative limit through order λ 2 . To obtain this limit we evaluate the NLO HTLpt thermodynamic potential with LO perturbative gluon, scalar, and quark masses. Expressed numerically, the NLO HTLpt result in the DRG scheme obtained in Ref. [48] is We find the same dependence on the renormalization scaleμ and all terms are same except the term proportional to λ 2 without logarithms. We note that the coefficient of λ 2 log λ is the same using both schemes, showing that it is scheme independent. As further a comparison, the perturbative free energy computed to three-loop order, keeping all terms through order λ 2 was obtained in Refs. [36,37]. Expressed numerically the full result is Comparing Eqs. (42) and (43) to (44), we find all terms are the same except the terms proportional to λ 2 . This is expected since both the RDR and DRG scheme HTLpt calculations were two-loop calculations and hence cannot reproduce the full λ 2 coefficient. At two-loop order one could consider fixingμ as to reproduce the full order λ 2 coefficient. This occurs whenμ 0.133554 andμ 0.143428, in the DRG and RDR schemes, respectively.

D. Gap equations
To go beyond strict perturbation theory we need to fix the mass parameters in a non-perturbative manner. Using the NLO HTLpt thermodynamic potential, the gluon, scalar, and quark masses can be fixed using the variational method described previously. The gluon, scalar, and quark mass parameters m D , M D , and m q can be determined by requiring the derivative of Ω RDR NLO with respect to each parameter is zero. The first equation in Eq. (45) giveŝ The second equation giveŝ Finally, the third equation giveŝ We find that, similar to the DRG scheme,m q in Eqs. (47) and (48) can be written in terms ofM D and m D by using (46). By numerically solving these three equations simultaneously, we obtain the gap equation solutions,m gap q (λ,μ),m gap D (λ,μ), andM gap D (λ,μ). In Fig. 1, we compare numerical solution of quark, gluon, and scalar gap equations in the RDR scheme to the result obtained using the DRG scheme. In these three panels, the results are obtained using the central value of the renormalization scale,μ = 1, and are scaled by their corresponding leading-order weak-coupling limits. The red line is the result obtained using the RDR scheme, and the black dotted line is the result obtained using the DRG scheme. As one can see from Fig. 1, the results obtained in these two schemes are numerically similar to each other. At the central value µ = 2πT , for the quark mass, the RDR result is slightly higher than the DRG result when λ → 0; 1 for the gluon mass, the RDR result is a little lower than the DRG result at intermediate λ; for the scalar mass, the two results are nearly identical.

VI. THERMODYNAMIC FUNCTIONS IN THE RDR SCHEME
The NLO HTLpt-resummed free energy in the RDR scheme can be obtained by evaluating the NLO thermodynamic potential (41) at the solution of the gap equations (45) The pressure, entropy density, and energy density can be obtained using In SYM 4,4 these three quantities satisfy since λ is temperature independent. 1 The gap equation solution for mq does not approach its perturbative limit when λ approaches to zero in both schemes.

A. Numerical results
In Fig. 2, we present a comparison of our NLO HTLpt result in the RDR and DRG schemes with the next-toleading approximation (NLA) result of Blaizot, Iancu, Kraemmer, and Rebhan (BIKR) [71]. The green, blue, and cyan dashed/dotted lines represent the strict perturbation theory result given in Eq. (1), truncated at order λ, λ 3/2 , λ 2 (including the logarithmic term), respectively. The orange dashed line is the strong coupling result given  1) and (2) [37]. The black and blue lines with a shaded bands are the NLO HTLpt result in the RDR and DRG schemes, respectively. The bands correspond to variation ofμ in the range 1/2 ≤μ ≤ 2 and the solid lines correspond toμ = 1. Finally, the red line with a shaded band is the NLO BIKR result with the bands and line corresponding to scale variation and central value. As Fig. 2 demonstrates, the DRG and RDR schemes result in similar predictions for λ 6, however, both are below the generalized Padé which is constructed using the full order λ 2 and λ 2 log λ coefficients. Since the DRG scheme breaks supersymmetry, the RDR result should be taken as the correct NLO HTLpt result. In comparison with the BIKR result, which made use of the DRG scheme, we see that it is also in agreement to within a few percent for λ 6.
In Fig. 3, we present a similar comparison, but for smaller λ in order to see the differences more clearly. The various strict perturbative results in this figure are the same as in Fig. 2. As can be seen from this figure, there is excellent agreement between the BIKR result and RDR NLO HTLpt in this coupling range. The generalized Padé approximant, however, only overlaps with both resummed results for λ 0.5.

VII. CONCLUSIONS
In this work, we obtained the LO and NLO HTLpt resummed thermodynamic potential in SYM 4,4 using the RDR regularization scheme. We then analytically and numerically compared the RDR result to the result obtained in the DRG scheme. From the analytical perspective, we found that differences emerged only in finite contributions, while the divergences were unaffected. This causes only a small change compared to DRG HTLpt result, which can be seen from the figures presented in Sec. VI A. Importantly, we found that, when expanded in strict perturbation theory, both schemes gave the same results for the coefficients of all terms contributing at orders λ, λ 3/2 , and λ 2 log λ, while at order λ 2 we found a weak residual regularization scheme dependence.
The results reported herein provide an important cross check of the coefficient of the order λ 2 log λ term in the free energy, independently confirming our prior result for this coefficient, which was obtained in Refs. [36,37]. The differences at order λ 2 are not yet conclusive, however, since in both the RDR and DRG schemes we have only performed a two-loop HTLpt calculation and a three-loop calculation is needed to properly fix this coefficient. Such a calculation was performed in strict perturbation theory in Refs. [36,37]. It would be interesting to extend the RDR HTLpt calculation to three-loop order in order to see how well it performs at large coupling, however, we first plan to complete the computation of the order λ 5/2 coefficient using effective field theory methods since this calculation is somewhat more manageable. n µ specifies the thermal rest frame and in the local rest frame of the plasma given by n = (1, 0).
We begin by noting that, since all ultraviolet divergences which are generated from the HTLpt reorganization are all canceled by systematically derivable counterterms, we only need to consider the leading order contributions to the gluon, quark, and scalar masses where ∆ T and ∆ L are the transverse and longitudinal propagators, respectively, and defined by , and ∆ X (p) is defined as 3. HTLpt quark self energy, propagator, and counterterm The HTLpt quark self energy is where we have suppressed the trivial Kronecker deltas and Similar to the gluon polarization tensor, the angular average in T µ can be written as As was the case with the gluonic self energy, the quark self energy (A20) is the same as in QCD [17] up to the definition of m q , and taking into account the four Majorana fermions which are indexed by i. This will also occur in the HTLpt quark propagator, quark-gluon three vertex, and quark-gluon four vertex in SYM 4,4 [48].
The HTLpt Feynman rule for the quark propagator is where i, j index the four independent Majorana fermions and where A µ (p) = (A 0 (p), A s (p)p) with The insertion of an HTL quark counterterm into a quark propagator corresponds to iδ ab δ ij Σ(p) .
(A26) 4. HTLpt scalar self energy, propagator, and counterterm The HTL scalar self energy can be expressed as and the corresponding Feynman rule for the scalar propagator is where ∆ s (p) = 1 (A29) The insertion of an HTL scalar counterterm into a scalar propagator corresponds to −iδ ab δ AB P AA aa (p) . (A30)
The rank-one tensorT µ is only defined for p + r − q = 0 and is even under the permutation of q and r. It satisfies The quark-gluon vertex therefore satisfies the Ward identity

HTLpt four scalar vertex
The four-scalar vertex is It satisfies

HTLpt scalar-gluon four vertex
The scalar-gluon four vertex is independent of the direction of the momentum, and can be expressed as It satisfies δ ac δ bd δ AB Γ µν,AB abcde (p, q, r, s) = (2ig 2 )(6N c d A )g µν . (A43)

HTLpt quark-scalar vertex
There are two types of quark-scalar vertices, the quarkscalar vertex and the quark-pseudoscalar vertex. This is due to the fact that there are two types of interactions between quarks and scalars (X p ), and between quarks and pseudoscalars (Y q ) which are different. The quarkscalar vertex, with incoming scalar momentum p, incoming quark momentum r and outgoing quark momentum q, and their corresponding colors indexed by a, c, b, can be written as Γ p abc,ij (p, q, r) = −igf abc α p ij . (A44) The quark-pseudoscalar vertex can be written as Γ q abc,ij (p, q, r) = gf abc β q ij γ 5 . (A45)

Appendix B: Integrals required in the HTLpt calculation
The one-loop sum-integrals required for the SYM 4,4 NLO HTLpt calculation were presented in Appendix B of Refs. [16,17]. We list them here for completeness. 1 + 2 + 2 ζ (−1) ζ(−1) , (B1) The two-loop sum-integrals required can be split into two types. The first type are related to bosonic momentum integrations, {P Q} p 2 P 2 Q 2 R 2 q 2 = In a similar manner, the two-loop sum-integrals involving T P can be split into two types. The first type are related to the bosonic momentum integrations, P Q T R P 2 Q 2 r 2 =