Renormalization group optimized $\lambda \phi^4$ pressure at next-to-next-to-leading order

We investigate the renormalization group optimized perturbation theory (RGOPT) at the next-to-next-to-leading order (NNLO) for the thermal scalar field theory. From comparing three thus available successive RGOPT orders we illustrate the efficient resummation and very good apparent convergence properties of the method. In particular the remnant renormalization scale dependence of thermodynamical quantities is drastically improved as compared to both standard perturbative expansions and other related resummation methods, such as the screened perturbation theory. Our present results thus constitute a useful first NNLO illustration in view of NNLO applications of this approach to the more involved thermal QCD.


I. INTRODUCTION
For thermodynamical quantities at equilibrium, and a weakly coupled theory, one could hope at first that a perturbative expansion would give reliable results. In contrast, as is well-known, for a massless theory infrared divergences spoil a naive perturbation theory (PT) approach in thermal field theories. Even though those infrared divergences can be efficiently resummed (see e.g. [1][2][3] for reviews), it leads to nonanalytical terms in the coupling, that happen to give poorly convergent successive orders, even when pushed to the highest perturbative order available, furthermore with increasingly sizeable remnant scale dependence. This situation is notoriously illustrated for asymptotically free thermal QCD, where however lattice simulations (LS) offer a powerful genuinely nonperturbative alternative bypassing those issues. At least so far LS has been very successful in the description of the nonperturbative physics of the QCD phase transitions at finite temperatures and near vanishing or small baryonic densities [4]. Nevertheless, the famous numerical sign problem [5] at finite density (equivalently at finite chemical potential), prevents LS to successfully describe compressed baryonic matter at sufficiently high densities and to explore a large part of the QCD phase diagram. More generally despite the very success of LS it is still highly desirable to explore more analytical improvements of thermal PT. Accordingly many efforts have been devoted in the past to overcome the generically observed issues of poor PT convergence. Apart from the most important case of QCD, the above mentioned behavior is generic for any thermal quantum field theory, and typically the scalar λφ 4 interaction is often used as a simpler model to study new alternative approaches. Although the λφ 4 model is not asymptotically free, it shares some important features with thermal QCD. Typically the dynamical generation of a thermal screening mass m D ∼ √ λT impacts the relevant expansion of thermodynamical quantities such as the pressure, that exhibit weak expansion terms λ (2p+1)/2 , p ≥ 1.
Various approximations attempting to more efficiently resum thermal perturbative expansions have been developed and refined over the years, typically the screened perturbation theory (SPT) [6,7], the nonperturbative renormalization group (NPRG) [8] approach, the two-particle irreducible (2PI) formalism [9][10][11], or other approaches [12]. In particular for the λφ 4 model, SPT essentially redefines the weak expansion about a quasiparticle mass, avoiding in this way infrared divergences from the start. It has been investigated up to three-loop [13][14][15] and even four-loop orders [16]. SPT may also be viewed as a particular case, in the thermal context, of the so-called optimized perturbation theory (OPT) 1 , in which more generally the (thermal or nonthermal) perturbative expansion is redefined about an unphysical test mass parameter, fixed by a variational prescription, that provides a resummation of perturbative expansion. The generalization of SPT tailored to treat the much more involved thermal gauge theories [20], the Hard Thermal Loop perturbation theory (HTLpt) [21], has been pushed to three-loop order [22,23]. The three-loop results [23] agreement with LS is quite remarkable down to about twice the critical temperature, for a renormalization scale choice ∼ 2πT . However, both SPT and HTLpt exhibit a very sizeable remnant scale dependence at NNLO, that definitely call for further improvement. More recently, OPT at vanishing temperatures and densities was extended to the so-called renormalization group optimized perturbation theory (RGOPT) [24,25]. The basic novelty is that it restores perturbative RG invariance at all stages of calculations, in particular when fixing the variational mass parameter, by solving the (mass) optimization prescription consistently with the RG equation. At vanishing temperatures and densities it has given precise first principle determinations [25] of the basic QCD scale (Λ MS ) or related coupling α S , and of the quark condensate [26,27]. The RGOPT was extended at finite temperatures for the scalar λφ 4 in [28,29] and for the non-linear sigma model (NLSM) in [30], showing how it substantially reduces the generic scale dependence and convergence problem of thermal perturbation theories at increasing perturbative orders. More recently the RGOPT in the quark sector contribution to the QCD pressure was investigated at NLO, for finite densities and vanishing temperatures [31], and at finite temperature and density [32,33], leading to drastic improvements with respect to both perturbative QCD and HTLpt, specially at nonzero temperature. In the present work, as a first step to investigate the RGOPT in thermal theories beyond NLO, we explore the three-loop order (NNLO) for the technically simpler scalar λφ 4 model pressure. We investigate the stability and convergence properties of the method, also assessing the remnant renormalization scale dependence improvement as compared to standard PT, and to SPT.
The paper is organized as follows. In Sec. II we briefly review the standard thermal perturbative pressure of the λφ 4 model up to NNLO, to set our conventions and basic expressions that serve as a starting point for our construction. In Sec. III we recall the main ingredients of the RGOPT construction and some previous NLO results from [28,29]. Our main new NNLO results are derived and illustrated in Sec IV, while finally some conclusions and outlook are given in Sec. V. Some additional technical ingredients can be found in three appendixes.

II. REVIEW OF STANDARD (MASSIVE) THERMAL PERTURBATIVE EXPANSION
We consider the Lagrangian for one neutral scalar field with a quartic interaction, where a generic mass term m is arbitrary at this stage. We first recall the result [2,13] for the two-loop free energy (equivalently minus the pressure) including a mass term, corresponding to the first two graphs 2 in Fig.1, where in the imaginary time formalism P 2 = ω 2 n + p 2 , ω n = 2πT n (n = 0, 1, · · ·) represents the bosonic Matsubara frequencies. The sum-integral in Eq.(2.2) is defined as usual as the sum over Matsubara frequencies times remaining integration over three-momentum, using dimensional regularization and the MS renormalization scheme: 3) The three-loop contributions involves the basic last two graphs in Fig. 1 and read (2.4) We recall that the renormalization is most easily performed as follows. First, applying mutiplicative renormalization to the (bare) coupling and mass in expressions above, as where Z λ , Z m are the standard coupling and mass counterterms for the massive λφ 4 model, given for completeness respectively in Eqs.(B5),(B6) in Appendix B. Those are expanded for the relevant three-loop order case here to perturbative order λ 2 . The remaining divergences are then removed by an additive renormalization from the vacuum energy counterterms [13], formally represented as F 2l,ct 0 , F 3l,ct 0 in the above expressions, and also given for completeness in Eq.(B7) in Appendix B.
Once the mass, coupling, and vacuum energy counterterms have been accounted to cancel the original divergences, one obtains the (MS-scheme) renormalized free energy. The one-and two-loop contributions read [2,13]: where L ≡ ln(µ 2 /m 2 ) and we explicitly separated the thermal and non-thermal contributions. Here and in all related renormalized expressions below, µ stands for the arbitrary renormalization scale introduced by dimensional regularization in the MS-scheme, Eq.(2.3), and λ ≡ λ(µ). Note carefully that E 0 in Eq.(2.6) represents a finite vacuum energy term, to be speficied below, that plays a crucial role in our approach as will be reexamined in Sec. II B. The standard (dimensionless) thermal integrals appearing in Eq.(2.6) and below are given by where t = p/T and x = m/T . Different integrals can be easily related by employing derivatives such as Also, a high-T expansion [2] such as is often a rather good approximation as long as x < ∼ 1, i.e., m T .
Next the NNLO contribution involves the additional genuine massive three-loop second integral in Eq.(2.4), first calculated in [34]. After algebra, the complete three-loop contribution can be expressed as [13] and it involves two irreducible, respectively two-loop K 2 (m/T ) and three-loop K 3 (m/T ) integrals, given explicitly in ref. [34], reproduced for selfcontainedness in Eqs.(C1),(C4) in Appendix C.
To complete this subsection on basic thermal perturbative expressions, for latter easier reference and comparison purpose we also recall the expression of the (massless) PT pressure [1,35,36] up to NNLO: where P 0 = (π 2 /90)T 4 is the ideal bosonic gas pressure.

B. Perturbatively RG-invariant massive free energy
At this stage an important feature is that the (massive) free energy is lacking RG invariance. Namely, applying to Eq.(2.6), with E 0 = 0, the standard RG operator, with β(λ), γ m (λ) given in Eqs.(B1),(B2), yields a remnant contribution of leading (one-loop) order: −(m 4 /2) ln µ, for arbitrary m. Indeed the latter term is not compensated by lowest orders terms from β(λ) or γ m (λ) in Eq.(2.13), those being at least of next order O(λm 4 ). This is a manifestation that perturbative RG invariance generally occurs from cancellations between terms from the RG equation at order λ k and the explicit µ dependence at the next order λ k+1 . Nevertheless perturbative RG invariance can easily be restored by adding the finite vacuum energy term E 0 to the action, although this term is usually ignored, minimally set to zero in the (thermal) literature [13,21,22]. Following Refs. [28,29] the easiest way to construct a (perturbatively) RG-invariant renormalized vacuum energy is to determine E 0 order by order as a perturbative series from the reminder of Eq. (2.13) applied on the non RG-invariant finite part of Eq. (2.6): (2.14) Accordingly E 0 has the form where the coefficients s k can be determined at successive orders from knowing the (single) powers of ln µ at order k + 1 (or equivalently the single poles in 1/ of the unrenormalized expression) [25]. This procedure leaves non RG-invariant remnant terms of higher orders, that may be treated similarly once higher order terms are considered. Explicitly, we obtain [28] and similarly for the next orders presently relevant, 480 π 2 0.01399 , The explicit RG coefficients in intermediate expressions emphasizes the general form of these results, while the s 3 expression is specific to the N = 1 λφ 4 model. We stress that the previous construction, being only dependent on the renormalization procedure, does not depend on temperature-dependent contributions: at arbitrary perturbative orders the s k coefficients are determined from the T = 0 contributions only. Indeed as Eq.(2.14) suggests, its RHS precisely defines the vacuum energy anomalous dimension, that has been calculated even to five-loop order for the general O(N ) scalar model [37]. Our independent results for the s k are fully consistent with [37]. A subtlety is that according to Eq. (2.15), s k is strictly required for perturbative RG invariance at order λ k , but contributes at order λ k−1 . So at order λ k one may choose minimally to include only s 0 , · · · s k , or more completely include also s k+1 , incorporating in this way higher order RG dependence within the resulting expression.

III. RGOPT λφ 4 PRESSURE AT LO AND NLO
In this section we briefly recall the RGOPT construction, as investigated for the λφ 4 model in [28,29] at LO and NLO, before to extend our approach at the technically more involved NNLO. We also underline some important features that were not plainly discussed in [28]. After restoring in a first stage perturbative RG invariance of the massive free energy, leading to the crucial additional term in Eq.(2.15), one performs on the resulting complete expression the variational modification, according to where m is from now an arbitrary variational mass, and the crucial role of the exponent a will be specified just next. One then reexpands Eq.(3.1) at successive orders, δ k , at the same order as the original perturbative expression, and set δ → 1 afterwards. This leaves a remnant m dependence at any order k, that may be conveniently fixed by a stationarity prescription [19], thus determining a dressed massm(λ) with a "nonperturbative" (all order) λ-dependence. In OPT [17] or similarly SPT [7] applications, the linear δ-expansion has been mostly used, i.e. assuming a = 1/2 in Eq.(3.1), that corresponds to the "add and subtract a mass" intuitive prescription, one mass being treated as an interaction term. Yet it was pointed out that the rather drastic modification implied by Eq.(3.1) is generally not compatible with RG invariance [28]: in contrast a can be uniquely fixed [25,28] by (re)imposing RG invariance now for the variationally modified perturbative expansion. Once combined with Eq.(3.2), the RG Eq.(2.13) takes the massless form that at leading RG order uniquely fixes [25,28] simply in terms of the universal (renormalization scheme independent) first order RG coefficients 3 . At higher orders, Eq.
To get more insight on some properties of the solution of Eq.(3.7), one may conveniently use the high-temperature expansion of the relevant J n (x), from Eq.(2.9) with x ≡ m/T 4 . Accordingly Eq. (3.7) simplifies to a quadratic equation for x, with a unique physical (x > 0) solution: x =m Perturbatively at high temperature, the variationally determined mass Eq.(3.9) has the form of a screening mass,m 2 ∼ O(λT 2 ), but note that this variational mass parameter is unrelated to the physical Debye mass [36] definition. The corresponding one-loop RGOPT pressure, from Eq.(3.5), reads where α ≡ b 0 λ. Note in particular that Eq.(3.11) contains the nonanalytic term ∼ λ 3/2 , originating from the bosonic zero mode resummation, but here readily obtained from RG properties. Expanding at higher orders Eq.(3.9) it is easily seen that it entails nonanalytic terms λ (2p+1)/2 , p ≥ 1 at all orders. Incidentally it is worth mentioning that the higher orders beyond Eq.(3.11) obtained from Eq.(3.9) correctly reproduce all orders of the O(N ) scalar model large N -results (e.g. Eq. (5.8) of [38]) (once including higher order O(x 6 ) terms, not given in Eq.(2.9)), as can be checked upon identifying the correct large-N b 0 = 1/(16π 2 ) value [38]. Accordingly although the LO RGOPT is essentially built on the very first one-loop graph of Fig.1 augmented by the optimized RG construction as above described, it happens to correctly resum the whole set of 'foam' graphs as illustrated in Fig.2.

B. NLO RGOPT
The NLO (two-loop) O(δ 1 ) contribution to the free energy, for δ = 1, takes a rather compact form in terms of Σ R in Eq. (3.6): where the subscript on the integral term means to taking its corresponding finite (renormalized) expression. Eq.(2.15) gives explicitly The exact two-loop OPT and RG Eqs. (3.2) and (3.3) can be written compactly as [28] , and we introduced the reduced (dimensionless) self-energy, with, from Eq. (3.6), At this stage at NLO in principle we could use three different possible prescriptions to obtain a thermally dressed massm(λ, T ) as function of the coupling, as investigated in details in [28]: Either the OPT Eq.(3.14), or alternatively the (massless) RG Eq.(3.15), or else the full RG, Eq.(2.13). The latter is not an independent equation, being a linear combination of Eq.(3.14) and (3.15): We speculate that if one could calculate to all orders, the solutions of those different prescriptions would presumably converge towards a unique, nonperturbatively dressed massm(g, T ) (as it happens [24] in the large-N limit of the O(N ) Gross-Neveu model, where the original perturbative series is known to all orders). But due to the inherent perturbative truncations those prescriptions give formally different solutions, thus providing useful variants of the method. The resulting NLO solutions form/T and P/P 0 , once reexpanded, are perturbatively consistent with Eqs. (3.9), (3.10) for the first two order terms, but contain modifications at higher orders [28]. However, for rather large g(2πT ) 1 reference coupling and µ = 4πT , the OPT Eq.(3.14) no longer gives a real solution: in fact, the possible occurence of nonreal solutions at higher orders is a recurrent burden of such variational approach, quite generically expected from the nonlinear m dependence if requiring to solvem(λ, T ) exactly. Alternatively using Eq.(3.15) to determinem(λ, T ) gives an unphysical solution at NLO [28], being driven towards the NLO UV fixed point at λ = −b 0 /b 1 (an artefact of the scalar model two-loop beta function approximation due to b 1 < 0). These features lead us to rather consider the third option at NLO, taking the full RG Eq. (2.13), explicitly Eq.(3.18) at NLO, that happens to give real solutions at least in a much larger range of coupling values 6 .
The NLO pressure P/P 0 with exact T -dependence obtained from Eq. (3.18), as function of the reference coupling g ≡ λ(2πT )/24, is illustrated in Fig. 3, with scale dependence πT ≤ µ ≤ 4πT using the exact two-loop running Eq.(B8). It is compared with LO RGOPT, Eq.(3.10), and importantly also with the PT pressure, Eq.(2.12), respectively truncated at the lowest O(λ) ∼ g 2 and next O(λ 3/2 ) ∼ g 3 orders. We recall that the latter nonanalytic terms, obtained from resummation of a certain class of individually infrared divergent (massless) graphs, are largely responsible for the poorly convergent, oscillating behavior of successive perturbative orders, as illustrated. In contrast going from LO to NLO RGOPT appears very stable, despite the fact that both approximations also incorporate O(λ 3/2 ) ∼ g 3 contributions (as well as arbitrary higher order contributions as above explained). Moreover the reduction of the remnant scale dependence as compared to standard PT pressure is also sizeable, although a moderate residual scale dependence appears at NLO, visible on the figure for g > ∼ 0.6. A more accurate analysis [28] shows that at NLO the remnant scale dependence reappears first at perturbative order λ 3 . As above mentioned the exact scale invariance The LO and NLO RGOPT pressures in Fig. 3 trivially reproduce the Stefan-Boltzmann limit for λ → 0, see Eq.(3.10), but one can notice that beyond moderate coupling values they differ rather importantly from the PT pressure: this is more pronounced in such a plot, conventionally often used [1] to illustrate different approximations to P (λ), but where the φ 4 model is not fully specified by fixing a physical input scale µ = T 0 and corresponding λ(T 0 ) value. Note however that upon expressing our results in terms of a more physical mass scale, namely solving e.g. Eq. (3.2) forλ(m) with a resulting P (m/T ), and inserting the physical Debye screening mass m(λ) [36], correctly reproduces [28] the first two terms of the standard PT pressure Eq. (2.12). Yet the RGOPT pressure crucially differs from PT at higher orders, otherwise it would merely reproduce the same PT behavior and issues. Thus it is important to use the exact (all order)m(λ, T ) solution from Eq.(3.14) or (3.15), that corresponds to the results shown in Fig. 3. (In contrast, using low orders perturbative reexpansions of the RGOPTm(λ) solution leads to a behavior more similar to the PT pressure, showing large differences between successive orders and a larger scale dependence).
In Fig. 3 we also compare with the NLO SPT results elaborated in Ref. [13]. Note that all the relevant SPT expressions can be obtained consistently by 1) discarding the vacuum energy subtraction E 0 in Eq.(2.6); 2) taking a = 1/2 in Eq.(3.1), expanding the result to order δ, and setting δ → 1; and finally 3) calculating the variational mass gap, Eq. (3.2), that gives explicitly 7 see Eq.(3.6), to be solved self-consistently form SPT . Alternatively a simpler prescription was also used in Ref. [13], taking instead the (NLO) perturbative Debye screening mass [36]: therefore we illustrate these two SPT prescriptions in Fig. 3. As compared to the two lowest orders of standard PT shown, the SPT pressure is significantly more stable and with a better remnant scale dependence, that reflects its more elaborate resummation properties. The SPT pressure values obtained from the two prescriptions are quite close, but using the variational mass gap gives a much better remnant scale dependence than using the PT Debye mass.
The RGOPT remnant scale dependence at NLO is however significantly reduced in comparison: more precisely for the largest shown (rescaled) coupling g = 1, the relative P/P 0 variation for πT ≤ µ ≤ 4πT is 8%, 1.5%, 0.8%, and 0.4% respectively for the O(g 3 ) PT, SPT with screening PT mass, SPT with variational mass gap, and RGOPT.

IV. RGOPT λφ 4 PRESSURE AT NNLO
Coming back to the basic free energy in Eq.(2.2), if neglecting the subtraction terms in Eq.(2.15), the formal lack of RG invariance from unmatched m 4 ln µ terms remains relatively screened at one-and two-loop orders of thermal perturbative expansions for sufficiently small coupling, since perturbatively m 4 ∼ λ 2 . This can essentially explain why the remnant scale dependence of SPT remains quite moderate even at NLO, see Fig. 3, so that in comparison the NLO RGOPT improvement by merely a factor two is not spectacular. But conversely this can largely explain why a very sizeable scale dependence resurfaces for the NNLO SPT pressure [13], where the formally same order genuine three-loop λ 2 contributions are considered. In contrast the RGOPT scale dependence is expected to further improve at higher orders, at least formally: being built on perturbatively restored RG invariance of the free energy at order m 4 λ k for arbitrary m, the resulting mass gap exhibits a leading remnant scale dependence asm 2 ∼ λT 2 (1 + · · · + O(λ k ln µ)). Thus the dominant scale dependence in the free energy, coming from the leading term ∼ s 0 m 4 /λ, is expected to appear first only at O(λ k+1 ). Nevertheless this expected trend could be largely spoiled, either by large perturbative coefficients (generically expected to grow at higher orders), or by the well-known thermal PT issues due to infrared divergent bosonic zero modes. It is thus important to investigate more explicitly the outcome of our construction at NNLO, where standard thermal PT starts to badly behave, to delineate the RGOPT scale dependence improvement that can be actually obtained. Moreover, concerning the λφ 4 model, the peculiar sign alternating beta-function coefficients b i from one-to three-loop orders, see Eq.(B3), implies that considering the running coupling alone, at three-loop order it has a comparatively worse scale dependence than at two-loop order, as illustrated in Fig.4. This feature tends to partly counteract the benefits of our RG-improved construction, when comparing NLO with NNLO 8 . Applying the variational modification from Eqs.(3.1), (3.4) to the complete three-loop free energy, sum of Eq.(2.6) and Eq.(2.10), expanded consistently now to order δ 2 , and taking δ → 1, gives after algebra, The explicit expressions at NNLO for the RG and OPT Eqs.(3.3), (3.2) can be obtained straightforwardly from Eq.(4.1) after algebra. They are more involved than their NLO analogs in Eqs.(3.14),(3.15) and not particularly telling: some relevant expressions are given explicitly in Appendix A.
Similarly to the NLO, at this stage without examining further constraints one may a priori use any of the three possible (not independent) prescriptions to obtain the NNLO dressed optimized massm(λ, T ): namely taking the solution of the OPT Eq. (3.2), or the massless RG Eq. (3.3), or the full RG Eq.(2.13). One may further exploit the freedom to incorporate the highest order subtraction term s 3 of Eq.(2.17) or not, the latter being formally a three-loop contribution but depending on four-loop RG coefficients, thus not necessary for NNLO RG invariance. The latter flexibility happens to give a relatively simple handle to circumvent the annoyance of possibly nonreal NNLO solutions, that occur only at relatively large couplings in the λφ 4 model, quite similarly to what happens at NLO. The behavior of those solutions for the different prescriptions is detailed in Appendix A for completeness.
The outcome is, in order to maximize the range of coupling and scale πT ≤ µ ≤ 4πT values where real solutions are obtained, it is appropriate to minimally neglect s 3 if using the OPT Eq.(3.2), and incorporating s 3 = 0 when using the RG Eq. (3.3). We mention that at NNLO the full RG Eq.(2.13) gives no real solutions, at least for the relevant scale choice µ = 4πT that maximizes the values of g(µ) for a reference coupling g ≡ g(2πT ). (Actually one could recover real solutions only if truncating Eq.(2.13) maximally, keeping only O(g 2 ) terms, but one then loses the crucial perturbative-matching properties, so that the corresponding solutions have to be rejected). Some of these features could be intuitively expected: incorporating higher orders, either via s 3 = 0 or taking the more involved exact Eq.(2.13), renders the expression even more nonlinear in m, favoring the occurence of nonreal solutions. But for the massless RG Eq.(3.3), provided that it has real solutions, incorporating s 3 could be expected to give better results, since the subtraction coefficients in Eq.(2.17) entering the pressure are originating directly from the RG coefficients. Eq.(4.1) to give the physical pressure. Fig. 6 illustrates the corresponding pressures obtained from the two alternative OPT and RG mass prescriptions. As one can see, despite the quite different m(g) in Fig. 5 the corresponding OPT and RG pressures are very close and similar in shape, and have comparable very moderate scale dependence. This feature appears as a very good indication of the previously mentioned expected convergence at higher order of those a priori different prescriptions. Finally we illustrate our main results at successive LO, NLO, and NNLO in Fig. 7, compared with the PT pressure, Eq.(2.12), and SPT pressure at successive orders. The NNLO PT, with successive terms up to O(λ 2 ), has a substantially larger remnant scale dependence than the O(λ 3/2 ) PT in Fig. 3. Concerning the SPT, at NNLO we only show the results from using the mass gap Eq. (3.19), following the very same prescription as in [13], namely using also Eq. (3.19) at NNLO. (N.B. using the perturbative screeening mass instead, Eq. (3.20), gives a much larger scale dependence, that we do not illustrate). In contrast one can see that the RGOPT pressure is remarkably stable from comparing LO to NLO and NNLO, and has a very moderate remnant scale dependence at NNLO, almost invisible at the figure scale until relatively large g 0.8. Actually the improvement from NLO to NNLO becomes only moderate for relatively large coupling values 0.9 g 1, that we understand as the counter-effect from the worse scale dependence of the sole NNLO running coupling as above explained, see Fig. 4. Overall the scale dependence is drastically improved as compared to PT and SPT.

V. CONCLUSIONS AND OUTLOOK
We have illustrated in the λφ 4 model up to NNLO how the RGOPT resummation of thermal perturbative expansions, consistently maintaining perturbative RG invariance, leads to drastically improved convergence and remnant renormalization scale dependence at successive orders, as compared with PT and with related thermal perturbative resummation approaches such as SPT. In particular RGOPT remains very stable from NLO to NNLO up to relatively large coupling values. We have compared two different prescriptions a priori available in our framework to defining a resummed dressed mass at a given order, that give very close and stable results. As could be intuitively expected, the prescription defining the dressed mass from the RG Eq. (3.3), thus more directly embedding RG properties, appears to give a slightly better remnant scale dependence than the more traditional variational mass prescription from Eq.(3.2) optimizing the pressure.
The RGOPT has been applied recently at NLO in thermal QCD [32,33], yet only in the quark sector, thus treating the gluon contributions apart purely perturbatively, due to some present technical limitations. Although such a relatively simple approximation shows a very good agreement with lattice simulations down to relatively low temperatures near the pseudo-transition, it is clearly an important next step to extend our approach to similarly treat the crucial gluon sector, largely responsible for the poorly convergent weak coupling expansion of thermal QCD, due to zero mode infrared divergences that start to show up at NNLO. Since the very stable NNLO RGOPT properties here demonstrated for the scalar model entails an appropriate resummation to all orders of infrared divergent bosonic zero modes, we anticipate similarly good properties to hold also in the QCD gluon sector, once the technical (computational) difficulties to readily adapt the RGOPT approach to the gluon sector will be overcome.
This appendix examines in some details the properties of the different possible prescriptions at NNLO. We first give the explicit expression of the NNLO RG Eq.(3.3), straightforward to obtain after algebra from Eq.(4.1), that takes the rather compact form: where with F 3l given in Eq.(2.10) and Σ R in Eq.(3.6). The OPT Eq.(3.2) at NNLO is similarly easily obtained from Eq.(4.1), but it gives a somewhat more lengthy expression that we thus refrain to give explictly. respectively, as function of m for two representative g ≡ λ/24 values and for the renormalization scale µ = 4πT , giving the largest coupling for a given g input, thus the most problematic case for sufficiently large g. For the simpler s 3 = 0 prescription, the RG equation has a unique perturbative-matching solution for 0 < g .75 (for µ = 4πT ), but above this maximal g value an inflection point appears that pushes the previous perturbative-matching solution to complex values, although being close to real values as can be seen in Fig. 8 (bottom). Note also that another RG solution appears at a higher mass value, but the latter is not matching standard perturbative behavior, so it has to be rejected. Rather similar features are obtained for µ = 2πT , but the disappearance of the real perturbativematching solution is delayed to higher g 0.95 values. In contrast for s 3 = 0 a similar behaviour is obtained but the disappearance of real perturbative-matching RG solution is delayed to substantially higher g 1 values for any µ ≤ 4πT : accordingly for 0 < g ≤ 1 and πT ≤ µ ≤ 4πT the perturbative-matching solution is real and unique. The OPT equation has a unique perturbative-matching solution for both s 3 = 0 or s 3 = 0, and we adopt the simpler minimal choice s 3 = 0 for our NNLO results. Notice that despite their very different expressions, the RG and OPT equations give m solutions that are rather close to each other. (Even when the RG solution is not real it is seen to be very close to the OPT real one). Finally note that the full RG Eq.(2.13) has no real solutions for 0 < g ≤ 1 for µ = 4πT : the latter is not illustrated on Fig. 8 but its shape looks quite similar to the RG one in the range where the latter gives complex solutions (also being close to give a real solution).
where L µ ≡ ln(µ/µ 0 ) and W (x) ≡ ln(W/x) is the Lambert implicit function. For the range of coupling values illustrated in our main figures, g ≡ λ/24 1, Eqs.(B8), (B9) give not much visible differences with a simpler perturbatively truncated expansion at order λ 3 : At three-loop order we used quite similarly an exact integral giving the running coupling λ(µ) as a more involved implicit function, numerically solved for µ as a function of µ 0 . For not too large coupling values there is not much visible differences with a more common perturbatively truncated running coupling: and E στ (p, q, r) = p 2 + m 2 + σ q 2 + m 2 + τ r 2 + m 2 . (C6) In the limit x ≡ m/T → 0 Eqs.(C1), (C4) can be expressed analytically as: For the numerical results we rather use the exact expressions Eqs.(C1), (C4).