Two-loop HTL-resummed thermodynamics for $\mathcal{N}=4$ supersymmetric Yang-Mills theory

$\mathcal{N}=4$ supersymmetric Yang-Mills theory (SYM) is the most famous example of a conformal field theory (CFT) in four dimensions, and is often taken as a model for hot QCD in the large number of colors $N_{c}$ and strong ’t Hooft coupling $\lambda=g^{2}N_{c}$ limits. The strong coupling behavior of the free energy has been computed using the anti-de Sitter space/CFT (AdS/CFT) correspondence Gubser:1998nz , with the result being

where $\mathcal{F}_{\textrm{ideal}}=-d_{A}\pi^{2}T^{4}/6$ is the ideal or Stefan-Boltzmann limit of the free energy and $\mathcal{S}_{\textrm{ideal}}=2d_{A}\pi^{2}T^{3}/3$, with $d_{A}=N_{c}^{2}-1$ being the dimension of the adjoint representation.

In the weak-couping limit the $\mathcal{N}=4$ SYM free energy has been calculated through order $\lambda^{3/2}$ giving Fotopoulos:1998es ; Kim:1999sg ; VazquezMozo:1999ic

Note that, since the beta function of the $\mathcal{N}=4$ SYM theory is zero, the coupling constant does not run and is independent of the temperature. As a result, we can vary the coupling between the two limits at each temperature.

One expects these two series to describe their respective asymptotic limits correctly, however, the radius of convergence of each of these series is unknown and, therefore, it is unclear to what degree each of these can trusted away from their respective limits. In Fig. 1 we plot the scaled entropy density resulting from Eqs. (1) and (2) as a function of $\lambda$ along with a $R_{[4,4]}$ Padé approximant constructed from these results Kim:1999sg ; Blaizot:2006tk .¹¹1Although a Padé approximant might provide a convenient interpolation between the weak- and strong-coupling limits their construction is in no sense systematic. In particular, the resulting expressions are incomplete since we know that, at least at weak coupling, the series will contain logarithms of the coupling constant beyond ${\mathcal{O}}(\lambda^{3/2})$. From this Figure, we can see that the two successive weak coupling approximations are only close to one another below $\lambda\sim 0.1$ and rapidly diverge beyond $\lambda\sim 1$. In the strong coupling limit, only the first two terms in the series are known. As can be seen from Fig. 1 the strong coupling result diverges quickly below $\lambda\sim 10$. The question then becomes, how can we systematically extend these two results into the intermediate coupling region $\lambda\sim 1-10$. In this paper, we present progress towards this goal in the weak-coupling limit using hard-thermal-loop (HTL) perturbation theory. Our study is complementary to the earlier work of Blaizot, Iancu, and Rebhan in which they applied HTL resummation using an approximately self-consistent scheme Blaizot:2006tk . The key difference from this earlier work is that our result is manifestly gauge invariant and based on the systematically improvable HTL perturbation theory (HTLpt) framework Andersen:1999fw ; Andersen:1999sf ; Andersen:1999va .

The goal of our work is to improve the convergence of the successive weak-coupling approximations. One promising approach is to use a variational framework in which the free energy $\mathcal{F}$ is expressed as the variational minimum of the thermodynamic potential $\Omega(T,\lambda;m^{2})$ that depends on one or more variational parameters that we denote collectively by ${\bf a}$

For example, the $\Phi$-derivable approximation is a widely used variational method in which the propagator is used as an infinite set of variational parameters Luttinger:1960ua ; Baym:1962sx . The $\Phi$-derivable thermodynamic potential $\Omega$ is given by the $2$-particle-irreducible (2PI) effective action, which is the sum of all diagrams that are $2$-particle-irreducible with respect to the complete propagator Cornwall:1974vz . This method is difficult to apply to relativistic field theories except for the case where the self-energy is momentum-independent. Despite this, there still have some progress in applications to quantum chromodynamics (QCD) and electrodynamics (QED) Braaten:2001en ; Braaten:2001vr ; vanHees:2001ik ; vanHees:2002bv ; Blaizot:2003br ; Andersen:2004re . Historically, the $\Phi$-derivable approximation was first applied to QCD by Freedman and McLerran Freedman:1976ub , who demonstrated that the thermodynamic potential $\Omega$ is gauge dependent beyond a given order in the coupling constant. The gauge parameter dependence appears at the same order in $\alpha_{s}$ as the series truncation when evaluated off the stationary point and at twice the order in $\alpha_{s}$ when evaluated at the stationary point Arrizabalaga:2002hn ; Blaizot:1999ip ; Andersen:2004re . Despite this issue, this method had been used as the starting point for approximately self-consistent HTL resummation of the entropy Blaizot:1999ap ; Blaizot:2000fc and the pressure Peshier:2000hx .

The problems encountered when applying the $\Phi$-derivable approximation to gauge theories motivated the use of alternative variational approximations. One such alternative, which in its simplest form involves a single variational parameter $m$, has been called optimized perturbation theory Stevenson:1981vj , variational perturbation theory kleinert2009path ; Sisakian:1994nn , or the linear $\delta$ expansion Duncan:1988hw ; Duncan:1992ba . This strategy has been successfully used for the thermodynamics of the massless $\phi^{4}$ field theory up to four-loop order using “screened perturbation theory” Karsch:1997gj ; Andersen:2000yj ; Andersen:2001ez ; Andersen:2008bz , and spontaneously broken field theories at finite temperature Chiku:1998kd ; Pinto:1999py ; Chiku:2000eu ; Kneur:2010yv . One impediment to applying such ideas to gauge theories was that one cannot simply introduce a scalar mass for the gluon without breaking gauge invariance. The solution to this problem was introduced in Refs. Andersen:1999sf and Andersen:1999va in which it was demonstrated that one could generalize the linear delta expansion by adding and subtracting the full gauge-invariant HTL effective Lagrangian Braaten:1991gm ; Mrowczynski:2004kv .

The resulting scheme was called hard-thermal-loop perturbation theory (HTLpt)  Andersen:1999sf ; Andersen:1999va . The HTLpt method has been used to improve the convergence of weak coupling calculations of the free energy in QED Andersen:2009tw and QCD up to three-loop order at finite temperature and chemical potential Andersen:2009tc ; Andersen:2010ct ; Andersen:2010wu ; Andersen:2011sf ; Andersen:2011ug ; Haque:2013sja ; Haque:2014rua . When confronted with finite temperature and chemical potential lattice QCD data the HTLpt resummation scheme works remarkably well down to temperatures on the order of 200-300 MeV where the QCD coupling constant is on the order of $g_{s}\sim 2$ Haque:2013sja ; Haque:2014rua ; Ghiglieri:2020dpq . The method successfully describes all thermodynamic variables including second- and fourth-order quark susceptibilities. Herein, we will take the first steps in applying this method to ${\mathcal{N}}=4$ SYM in the hope that a similar improvement in convergence can be achieved in this theory at intermediate couplings. We will calculate the one- and two-loop HTLpt-resummed thermodynamic potential Additionally, in ${\mathcal{N}}=4$ SYM using the same method as was used to obtain the one- and two-loop QCD results in Refs. Andersen:1999sf ; Andersen:1999va ; Andersen:2002ey ; Andersen:2003zk . Importantly, in these papers it was demonstrated that it was possible to renormalize the resummed thermodynamic potential at two-loop order using only known vacuum and mass counterterms. Herein we will demonstrate the same occurs in $\mathcal{N}=4$ SYM. In this theory the NLO contributions include scalar and scalar-gluon, scalar-quark interactions compared to the QCD calculation, however these are relatively straightforward to include. In $\mathcal{N}=4$ SYM instead of having only gluon and quark thermal masses, $m_{D}$ and $m_{q}$, respectively, we will also have a thermal mass for the scalar particles, $M_{D}$. Our results at one- and two-loop order are infinite series in $\lambda$ which, when trucated at ${\mathcal{O}}(\lambda^{3/2})$, reproduce the weak-coupling results obtained previously in Refs. Fotopoulos:1998es ; Kim:1999sg ; VazquezMozo:1999ic . In order to make the calculation tractable, we expand the HTLpt scalarized sum-integrals in a power series in the three mass parameters $M_{D}$, $m_{D}$, and $m_{q}$ such that it includes terms that would naively contribute throughx ${\mathcal{O}}(\lambda^{5/2})$. Our final results indicate that, in $\mathcal{N}=4$ SYM, NLO HTLpt provides a good approximation for the scaled entropy for couplings in the range $0\leq\lambda\lesssim 2$.

We begin with a brief summary of HTLpt for $\mathcal{N}=4$ SYM in Sec. 2. In Sec. 3, we give the expressions for the one- and two-loop diagrams contributing to the SYM thermodynamic potential. In Sec. 4, we reduce these diagrams to scalar sum-integrals. As mentioned in the prior paragraph, since it would be intractable to calculate the resulting sum-integrals analytically, in Sec. 5 we expand these expressions by treating $m_{D}$, $M_{D}$ and $m_{q}$ as $\mathcal{O}(\lambda^{1/2})$ and expanding the integrals in powers of $m_{D}/T$, $M_{D}/T$ and $m_{q}/T$, keeping all terms up to $\mathcal{O}(\lambda^{5/2})$. In Sec. 6, we combine the results obtained in Sec. 5 to obtain the complete expressions for the leading- (LO) and next-to-leading order (NLO) thermodynamic potentials. In Sec. 7, we present our numerical results for the HTLpt-resummed LO and NLO scaled thermodynamic functions in $\mathcal{N}=4$ SYM and compare to prior results in the literature. For details concerning the transformation to Euclidean space and the sum-integrals necessary we refer the reader to the appendices of Refs. Andersen:2002ey ; Andersen:2003zk .

In $\mathcal{N}=4$ SYM theory all fields belong to the adjoint representation of the $SU(N_{c})$ gauge group. For the fermionic fields, a massless two-component Weyl fermion $\psi$ in four dimensions can be converted to four-component Majorana fermions Quevedo:2010ui ; bertolini2015lectures ; Yamada:2006rx ; DHoker:1999yni ; Kovacs:1999fx

where $\alpha=1,2$ and the Weyl spinors satisfy $\bar{\psi}^{\dot{\alpha}}\equiv[\psi^{\alpha}]^{\dagger}$. The conjugate spinor $\bar{\psi}$ is not independent, but is related to $\psi$ via the Majorana condition $\psi=C\bar{\psi}$, where $C=\begin{pmatrix}\begin{smallmatrix}\epsilon_{\alpha\beta}&0\\ 0&\epsilon^{\dot{\alpha}\dot{\beta}}\end{smallmatrix}\end{pmatrix}$ is the charge conjugation operator with $\epsilon_{02}=-\epsilon_{11}\equiv-1$. In the following, we will use the indices $i,j=1,2,3,4$ to enumerate the Majorana fermions and use $\psi_{i}$ to denote each bispinor.

The definition of gauge field is the same as QCD, and $A_{\mu}$ can be expanded as $A_{\mu}=A_{\mu}^{a}t^{a}$, with real coefficients $A_{\mu}^{a}$, and Hermitian color generators $t^{a}$ in the fundamental representation that satisfy

where $a,b=1,\cdots,N_{c}^{2}-1$, the structure constants $f_{abc}$ are real and completely antisymmetric. The fermionic fields can similarly be expanded in the basis of color generators as $\psi_{i}=\psi_{i}^{a}t^{a}$. The coefficients $\psi_{i}^{a}$ are four-component Grassmann-valued spinors.

There are six independent real scalar fields which are represented by a multiplet

where $X_{\texttt{p}}$ and $Y_{\texttt{q}}$ hermitian, with ${\texttt{p,q}}=1,2,3$. $X_{\texttt{p}}$ and $Y_{\texttt{q}}$ denote scalars and pseudoscalar fields, respectively. We will use a capital Latin index $A$ to denote components of vector $\Phi$. Therefore $\Phi_{A}$, $X_{\texttt{p}}$, and $Y_{\texttt{q}}$ can be expanded as $\Phi_{A}=\Phi_{A}^{a}t^{a}$, with $A=1,\cdots,6$, and $X_{\texttt{p}}=X_{\texttt{p}}^{a}t^{a}$, $Y_{\texttt{q}}=Y_{\texttt{q}}^{a}t^{a}$.

The Lagrangian density that generates the perturbative expansion for $\mathcal{N}=4$ SYM theory in Minkowski-space can be expressed as

where the field strength tensor is $G_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ig[A_{\mu},A_{\nu}]$, and $D_{\nu}=\partial_{\nu}-ig[A_{\nu},\cdot]$ is the covariant derivation in the adjoint representation. $\alpha^{\texttt{p}}$ and $\beta^{\texttt{q}}$ are $4\times 4$ matrices that satisfy

and their explicit form can be given as

where $\sigma_{i}$ with $i\in\{1,2,3\}$ are the $2\times 2$ Pauli matrices. And $\alpha$ and $\beta$ satisfies $\alpha_{ik}^{\texttt{p}}\alpha_{kj}^{\texttt{p}}=-3\delta_{ij}$ and $\beta_{ij}^{\texttt{q}}\beta_{ji}^{\texttt{p}}=-4\delta^{\texttt{pq}}$, with $\delta_{ii}=4$ for four Majorana fermions and $\delta^{\texttt{pp}}=3$ for three scalars.

The ghost term $\mathcal{L}_{\textrm{gh}}$ depends on the choice of the gauge-fixing term $\mathcal{L}_{\textrm{gf}}$ and is the same as in QCD. Here we work in general covariant gauge, giving

with $\xi$ being the gauge parameter.

In general, perturbative expansion in powers of $g$ in quantum field theory generates ultraviolet divergences. The renormalizability of perturbation theory guarantees that all divergences in physical quantities can be removed by the renormalization of masses and coupling constants. The coupling constant in $\mathcal{N}=4$ SYM theory is denoted as $\lambda=g^{2}N_{c}$. Unlike QCD, $\mathcal{N}=4$ SYM theory does not run.

Similar to the case of QCD presented in Refs. Andersen:2002ey ; Andersen:2003zk , HTLpt is also a reorganziation of the perturbation series for the SYM theory, and can be defined by introducing an expansion parameter $\delta$. The HTL “shifted” Lagrangian density can be written as

The HTL improvement term is

where $y^{\mu}=(1,\hat{\textbf{y}})$ is a light-like four vector defined in App. A, $j\in\{1\ldots 4\}$ indexes the four Majorana fermions, $A\in\{1\ldots 6\}$ indexes the scalar degrees of freedom, and $\langle\cdots\rangle_{\hat{\textbf{y}}}$ represents the average over the direction of $\hat{\textbf{y}}$. The parameters $m_{D}$ and $M_{D}$ are the electric screening masses for the gauge field and the adjoint scalar field, respectively. The parameter $m_{q}$ can be seen as the induced finite temperature quark mass. We note that, if we set $\delta=1$, the Lagrangian above (11) reduces to the vacuum $\mathcal{N}=4$ SYM Lagrangian (7). HTLpt is defined by treating $\delta$ as a formal expansion parameter, expanding around $\delta=0$ to a fixed order, and then setting $\delta=1$. In the limit that this expansion is taken to all orders, one reproduces the QCD result by construction, however, the loop expansion is now shifted to be around the high-temperature minimum of the effective action, resulting in a reorganization of the perturbation series which has better convergence than the naive expansion loop expansion around the $T=0$ vacuum. In addition, this reorganization eliminates all infrared divergences associated with the electric sector of the theory.

The HTLpt reorganization generates new ultraviolet (UV) divergences and, due to the renormalizability of perturbation theory, the ultraviolet divergences are constrained to have a form that can be canceled by the counterterm Lagrangian $\Delta\mathcal{L}_{\textrm{HTL}}$. References Andersen:2002ey ; Andersen:2003zk demonstrated that at two-loop order the thermodynamic potential can be renormalized using a simple counterterm Lagrangian $\Delta\mathcal{L}_{\textrm{HTL}}$ containing vacuum and mass counterterms. Although the general structure of the ultraviolet divergences is unknown, it has been demonstrated that one can renormalize the next-to-leading order HTLpt thermodynamic potential through three-loop order using only vacuum, gluon thermal mass, quark thermal mass, and coupling constant counterterms Andersen:2011sf ; Haque:2014rua . In this paper, we demonstrate that the same method can be used for $\mathcal{N}=4$ SYM and we compute the vacuum and screening mass counterterms necessary.

We find that the vacuum counterterm $\Delta_{0}\mathcal{E}_{0}$, which is the leading order counterterm in the $\delta$ expansion of the vacuum energy $\mathcal{E}_{0}$, can be obtained by calculating the free energy to leading order in $\delta$. In Sec. 6.1, we show that $\Delta_{1}\mathcal{E}_{0}$ can be obtained by expanding $\Delta\mathcal{E}_{0}$ to linear order in $\delta$. As a result, the counterterm $\Delta\mathcal{E}_{0}$ has the form

To calculate the NLO free energy we need to expand to order $\delta$ and we will need the counterterms $\Delta\mathcal{E}_{0}$, $\Delta m_{D}^{2}$, $\Delta m_{q}^{2}$, and $\Delta M_{D}^{2}$ to order $\delta$ in order to cancel the UV divergences. We find that in order to remove the divergences to two-loop order, the mass counterterms should have the form

In the $\mathcal{N}=4$ SYM theory, we will use the same method as in QCD to calculate physical observables in HTLpt, namely expanding the path-integral in powers of $\delta$, truncating at some specified order, and then setting $\delta=1$. The results of the physical observables will depend on $m_{D}$, $M_{D}$, and $m_{q}$ for any truncation of the expansion in $\delta$, and some prescription is required to determine $m_{D}$, $M_{D}$, and $m_{q}$ as a function of $\lambda$. In this work, we will follow the two-loop HTLpt QCD prescription and determine them by minimizing the free energy. If we use $\Omega_{N}(T,\lambda,m_{D},M_{D},m_{q},\delta)$ to represent the thermodynamic potential expanded to $N$-th order in $\delta$, then our full variational prescription is

We will call Eqs. (2) the gap equations. The free energy is obtained by evaluating the thermodynamic potential at the solution to the gap equations. Other thermodynamic functions can then be obtained by taking appropriate derivatives of free energy with respect to $T$.

In the imaginary-time formalism, Minkoswski energies have discrete imaginary values $p_{0}=i(2\pi nT)$, and the integrals over Minkowski space should be replaced by Euclidean sum integrals. There are two ways to do this which have been discussed in Refs. Andersen:2002ey ; Andersen:2003zk . One is transforming the Feynman rules in Minkowski space given in App. A into the form in Euclidean space firstly, then calculating the free energy. The other way is using the Feynman rules in Minkowski space to get the forms of free energy, after reducing these forms, transforming it into the form in Euclidean space. Results from the two methods must be the same.

The HTL perturbative thermodynamic potential at next-to-leading order in $\mathcal{N}=4$ SYM can be expressed as

where $\Omega_{\textrm{LO}}$ is the leading order thermodynamic potential, ${\mathcal{O}}(\delta^{0})$, which includes the one-loop graphs shown in Fig. 2 and the LO vacuum renormalization counterterm. We first discuss the contributions at this order.

Since we can make use of prior QCD results, we only need to calculate $\mathcal{F}_{s}$, $\mathcal{F}_{sct}$, $\mathcal{F}_{4s},\mathcal{F}_{3gs}$, $\mathcal{F}_{4gs}$, $\mathcal{F}_{3qs}$, and the HTL counterterms contributing to Eq. (27). The first step to calculate the new SYM contributions in Figs. 2 and 3 is to reduce the sum of these diagrams to scalar sum-integrals. In Euclidean space, by substituting $p_{0}$ to $iP_{0}$ the scalar propagator can be written as

so its inverse is

The leading-order scalar contribution can be written as

The HTL scalar counterterm can be written as

We proceed to simplify the sum of formulas in Eq. (3.2) in general covariant gauge parameterized by $\xi$. Using Eqs. (32) and (111), we obtain

There are two terms which depend on $\xi$ in (36), however, using $P+R-Q=0$, one finds that they cancel each other, so that the sum of these contributions is gauge independent. Similarly, the results for $\mathcal{F}_{3g+4g+gh}$ and $\mathcal{F}_{3qg+4qg}$ are independent of gauge parameter $\xi$ as shown in prior QCD calculations. Therefore, we have verified explicitly that the NLO HTLpt resummed thermodynamic potential in $\mathcal{N}=4$ SYM is gauge independent.

Finally, we simplify Eq. (26) to

where in Euclidean space

with ${\mathcal{T}}_{P}$ is the angular average $\mathcal{T}^{00}(p,-p)$ in Euclidean space, can be expressed as

where the angular brackets denote an average over $c$ defined by

where $\omega(\epsilon)$ is given in (102).

Having reduced $\mathcal{F}_{s}$, $\mathcal{F}_{4s+3gs+4gs}$, $\mathcal{F}_{3qs}$, and the HTL counterterm $\mathcal{F}_{sct}$ to scalar sum-integrals, we will now evaluate these sum-integrals approximately by expanding them in powers of $m_{D}/T$, $m_{q}/T$, and $M_{D}/T$. We will keep terms that contribute through ${\mathcal{O}}(g^{5})$ if $m_{D},m_{q}$ and $M_{D}$ are taken to be of order $g$ at leading-order. Additionally, each of these terms can be divided into contributions from hard and soft momentum, so we will proceed to calculate their hard and soft contributions, respectively. In some cases, the results presented here were obtained in previous one- and two-loop QCD HTLpt papers Andersen:1999va ; Andersen:2002ey ; Andersen:2003zk . When converting the prior QCD graphs involving quarks, as mentioned previously, one has to take into account that the four SYM quarks are Majorana fermions. Here we list results for all contributions to the ${\mathcal{N}}=4$ SYM Feynman graphs and counterterms for completeness and ease of reference. In all cases, we use the integral and sum-integral formulas from Refs. Andersen:2002ey ; Andersen:2003zk to obtain explicit expressions.