Thermal free energy of large Nf QED in 2+1 dimensions from weak to strong coupling

In 2+1 dimensions, QED becomes exactly solvable for all values of the fermion charge $e$ in the limit of many fermions $N_f\gg 1$. We present results for the free energy density at finite temperature $T$ to next-to-leading-order in large $N_f$. In the naive large $N_f$ limit, we uncover an apparently UV-divergent contribution to the vacuum energy at order ${\cal O}(e^6 N_f^3)$, which we argue to become a finite contribution of order ${\cal O}(N_f^4 e^6)$ when resumming formally higher-order $1/N_f$ contributions. We find the finite-temperature free energy to be well-behaved for all values of the dimensionless coupling $e^2N_f/T$, and to be bounded by the free energy of $N_f$ free fermions and non-interacting QED3, respectively. We invite follow-up studies from finite-temperature lattice gauge theory at large but fixed $N_f$ to test our results in the regime $e^2N_f/T\gg 1$.


I. INTRODUCTION
The conjectured duality between strongly coupled gauge theories and classical gravity in one higher dimension has been an extremely successful tool to effectively calculate properties of large N gauge theories at strong coupling and finite temperature [1][2][3][4].
Unfortunately, while generally expected to be correct, there is no formal proof of the conjecture. Furthermore, only certain gauge theories have known gravity duals, and this list does not include gauge theories that are realized in nature such as QED or QCD.
Finally, while gauge-gravity duality allows calculations in a regime where the coupling of the field theory is effectively infinite, the gravity dual is just as hard (or harder) to solve than the original field theory for intermediate values of the coupling, which are often physically relevant.
This provides the motivation to revisit and generalize existing tools to solve quantum field theories (and specifically gauge theories realized in nature) at finite temperature for arbitrary (weak or strong) values of the coupling. At first glance, this project seems to be dead on arrival: if techniques existed to, say, solve QCD non-perturbatively, using gaugegravity dual results for N = 4 super-Yang-Mills theory as a proxy for QCD would not have been needed. Surprisingly, however, a number of large N quantum field theories can be solved at finite temperature for all values of the coupling, including scalar field theories [5][6][7], Wess-Zumino models [7] and Gross-Neveu models, albeit in two spatial dimensions (2+1d).
In 3+1 dimensions, divergences requiring a renormalization program spoil much of the beauty of the exact (and sometimes analytic) results found in 2+1d. This typically leads to the large N 3+1-dimensional theories exhibiting a Landau pole, as is the case for scalar theories [8] and four-dimensional QED [9,10]. While the theories are still useful in the effective theory sense, cut-off effects near the Landau pole imply that in 3+1 dimensions, the strong-coupling limit of these theories is ambiguous.
For this reason, we are led to consider QED in 2+1 dimensions ("QED3") at finite temperature in the limit of many fermions N f 1, which is free of a Landau pole, and hence is unambiguously defined for any value of the coupling (cf. Refs. [11][12][13]). Because the theory does not exhibit any logarithmic divergences at leading and next-to-leading order in large N f , in the massless fermion case QED3 is essentially a finite quantum field theory, and there are no logarithmic scale dependencies in the coupling. This implies that the free energy f ∝ T 3 of QED3 scales as the third power of the temperature, with a coefficient that is only dependent on the (dimensionless) coupling T . In this work, we determine the ratio f /T 3 in QED3 non-perturbatively for all (weak to strong) values of the dimensionless coupling e 2 N f T to NLO at large N f using well-established field theory techniques. Our results thus generalize studies of QED3 at T = 0 [12,13] to arbitrary temperature, and may be useful as a reference for lattice gauge theory studies [14][15][16][17][18][19], dualities found for "cousins" of QED in 2+1 dimensions [20][21][22], conformal QED3 studies [23,24], as well as condensed matter systems [25].

II. SETUP
Let us consider QED with N f massless fermions defined by the Lagrangian where A µ is the photon gauge field, F µν = ∂ µ A ν − ∂ ν A µ is the photon field strength tensor, like direction compactified on a circle with radius β = T −1 (see e.g. [26]). The resulting D-dimensional Euclidean action is given by where A µ , ψ a are the Euclidean versions of the gauge field and the fermion, respectively, the Euclidean γ-matrices. Note that while the gauge field obeys periodic boundary conditions in the time-like direction, the fermions require anti-periodic boundary conditions. Gauge invariance of S E implies that there are gauge configurations A µ along which S E does not change. The existence of these "flat directions" implies that the QED partition function, defined as Z = DAe −S E , is ill-defined, because integration along the flat di-rections leads to divergences 1 . In order to make sense of the theory in the non-compact formulation, it is necessary to break gauge invariance. This is customarily done using the Faddeev-Popov formalism by introducing the ghost fieldsc, c, such that for instance in the class of covariant gauges the gauge-fixed Euclidean action becomes [26] where the anti-commuting ghosts fulfill periodic boundary conditions just like the bosonic gauge field. The partition function defined from the gauge-fixed action (3) is well-defined, and hence (3) will be used as the definition of QED in the following. While not gauge invariant, the action (3) is invariant under BRST transformations where ζ is an anti-commuting space-time independent parameter such that {ζ, c} = {ζ,c} = Taylor series by suitably rewriting S E = S 0 + S I , for instance with where the same term was added and subtracted in (3). Using S 0 instead of (3) with e = 0 as the reference action allows one to non-perturbatively resum an infinite number of Feynman diagrams ("Dyson series"). Nevertheless, it is important to maintain BRST invariance of S 0 in order to avoid introducing gauge-dependent artifacts. One finds that BRST invariance of S 0 requires ∂ µ Π µν = 0, which is a condition that we will check a posteriori. 1 Note that this is different when choosing a compact formulation of the Lagrangian by trading the gauge field A µ with a compact link variable U = e iAµ .

Photon Self-Energy
As in Refs. [27,28], the quantity Π µν is fixed by calculating the full connected photon two-point function, which in the limit N f → ∞ becomes or, taking into account the extra minus sign arising from the fermion loop, Note that here G µν (x), ∆(x) denote fully dressed propagators, but to leading order in large N f we can take the fermion propagator ∆(x) to be free. It is easiest to express the ∆(x) by going to Fourier space where where {ω n } = πT (2n + 1) with n ∈ Z are the fermionic Matsubara frequencies, µ is the renormalization scale parameter and we use dimensional regularization with > 0. With these conventions, the free fermion propagator becomes which leads to the photon self-energy given by The trace is readily evaluated using the properties of γ-matrices, finding In Fourier space, the photon self-energy thus becomes Let us first calculate the zero-temperature (vacuum) part of Π, which is given by Shifting the integration variable K → K + (1 − x)P , the momentum integration is straight- There are no logarithmic divergences in dimensional regularization, and one can take the limit → 0, finding [12] Π T =0,D=3 At finite temperature, Lorentz covariance is broken through the presence of a local matter rest frame. This implies that Π µν may be decomposed into the most general tensor structure that can be built out of δ µν , P µ and the rest frame vector n µ = (1, 0). The corresponding decomposition is standard in quantum field theory (cf. Ref. [29]) and we use the complete and orthogonal tensor basis spanned by to evaluate the structure functions for Π µν = Π A A µν + Π B B µν + Π C C µν + Π D D µν . Herẽ n µ ≡ n ν (A µν + B µν ). Evaluating P µ Π µν from (11) one finds which implies Π C = Π D = 0 and confirms that BRST invariance is satisfied for the action (5). The structure functions Π A , Π B may be found by considering the components The corresponding thermal sums may be evaluated using standard finite-temperature field theory methods [30], and the finite temperature parts are given for instance in the appendix of Ref. [31]: where Ref ). For D = 3, the remaining angular integration may be carried out to find

III. PARTITION FUNCTION FOR QED3
The partition function for QED3 is given by with S 0 , S I given in Eqns. (5). To leading and next-to-leading order in large N f , S 0 already resums all the relevant "daisy-type" diagram contributions, such that contributions from S I only appear at order O(N −1 f ), which we neglect. Hence the free energy density for QED3 to NLO in large N f is given by with where we used the fact that all the path integrals are Gaussian in momentum space. Here G −1 µν (K) is the inverse photon propagator in momentum space, which from the expression given in S 0 takes the form using the projectors given in (15). The determinant in f photon is given by the product of the eigenvalues of G −1 µν , which are the factors multiplying the orthogonal projectors above. Therefore, the O N 0 f contribution to the free energy is given by where we used that ln ξ = 0 in dimensional regularization. The photon polarization contributions Π A,B consist of a zero-temperature piece and a finite-temperature contribution given in (14), (20) above, which for D = 3 become and where the integration momenta have been scaled by the temperature. Particular care must be taken when evaluating the thermal contributions in the static limit p 0 → 0, finding One recognizes the 2+1 dimensional Debye mass in the zero momentum limit of Π B − Π V .
The fermion contribution and the remaining ghost contribution are easy to evaluate: where only the matter contribution of the thermal sums give non-vanishing contributions.
For the photons, we note that where for large K the asymptotic form of Π A,B − Π V ∝ 1 K 2 means that the second term in (30) is both IR-and UV-safe, and thus can be handled numerically.
The remaining term is given by where n B (x) = 1 e x/T −1 . The thermal contribution may be rewritten by deforming the contour to run along the Minkowski axis rather than the Euclidean axis because the integrand only has a branch cut, but no singularities anywhere on the principal Riemann sheet [9]. Taking the limit → 0, this leads to The contribution may be further simplified as which is readily evaluated numerically. Alternatively, one may investigate the weak coupling ( e 2 N f T 1) and strong coupling (

A. Apparently divergent vacuum energy at four-loop order
Finally, let us discuss the vacuum contribution which vanishes identically for both e 2 N f = 0, ∞. However, at face value f V,2 includes a logarithmic divergence for any finite value of e 2 N f . This divergence arises at four-loop order in a perturbative expansion, which can be seen by expanding (35) in powers of Π V ∝ The appearance of the e 6 coefficient is similar to the g 6 infrared divergence encountered for non-abelian gauge theories, also known as the "Linde problem" [32]. However, we believe these issues are unrelated because for the case of QED3, the apparent divergence is in the ultraviolet, not in the infrared. The naively UV divergent contribution to f V,2 may be calculated by considering (13) and γ E is Euler's constant. Subsequently integrating w.r.t. c we find Further inspection reveals that the problem lies with the naive N f → ∞ limit. It is possible to consider further corrections to the photon polarization tensor which are formally suppressed by powers of N f , for instance at the two-loop level, cf. Ref. [33]. One two-loop contribution (which by itself is not gauge invariant) originates from a non-vanishing fermion self-energy, modifying the fermion propagator as To leading order in large N f , Σ(K) may be calculated by using the resummed photon propagator to find which in turn suggests that similar contributions of higher order in N f may be non-perturbatively resummed to give [12] 1 with a calculable constant c 0 . Including the self-energy correction into the evaluation for the photon polarization tensor (11) then suggests the modification While these modifications do not modify most of the results for the free energy discussed above at the O(N 0 f ) level, there are two notable exceptions. First, consider the contribution f V,2 in light of these non-perturbative resummations of formally sub-leading 1/N f corrections. Expanding (35) in powers of Π V as before, but with N f π 2 one finds that the four-loop perturbative expression is finite in dimensional regularization because 8 N f π 2 takes over the role of . Hence we find Thus the result of including the naively sub-leading terms in the 1/N f expansion is that the apparent UV divergence of the free energy gets turned into a finite contribution to order O(N f ). Therefore, after resummation, the vacuum free energy is no longer renormalizationscale dependent, but there is a non-vanishing and finite cosmological constant contribution at order O(e 6 N 4 f ).

B. Suppression of in-medium tensor contributions at strong coupling
The second instance where the formally sub-leading corrections (41) become important is in the numerical evaluation of the in-medium contribution in Eq. (30) near zero temperature. Specifically, without taking (41) into account, the temperature-dependence for the polarization tensor components (26) may be scaled out by taking P → TP , e 2 → Tê 2 . As a consequence, one would expect the in-medium contributions to Π A,B to have non-vanishing contributions to the free energy f A,B even in the zero temperature limit.
However, taking into account (41), our calculation with naive in-medium contributions can only be trusted in a regime where whereas for e 2 N f T → ∞, the O(N 0 f ) photon contribution to the free energy is given by f V in (31).

C. Numerical evaluation of thermal contribution
The thermal photon polarization tensor contribution to the free energy is handled fully numerically by directly evaluating Specifically, this is done by performing the sum over Matsubara frequencies and using Gauss-Legendre quadrature for the remaining integral as in Refs. [27,28]: where we used k i = tan x i π 2 + 0 + to compactify the infinite interval including a small regulator to avoid any IR divergences. Here x i , W i are the nodes and modified weights, respectively, defined by the roots of the Legendre polynomial of order N: In practice, because of the symmetries of the integrand, only nodes with n ≥ 0, x i ≥ 0 need to be summed over. Tabulated values for x i can be easily generated with high precision for N up to N 2000, but in practice N 200 seems sufficient to obtain percent level precision.
We note that, in practice, we find that f A ≤ 0 and f B ≥ 0 and of similar magnitude for all values of the coupling. The numerical code for obtaining f

(M )
A,B and f V,1 as well as tabulated numerical results are publicly available at [34].

IV. RESULTS AND DISCUSSION
The full free energy density for QED3 in the large N f limit is given by where f fermion and f ghost are given in Eq. (29). Here where f V,1 , f V,2 and the matter contributions f (M ) A,B are given in Eqns. (33), (37) and (45), respectively. As pointed out in section III A, f V,2 is UV-divergent in the naive large N f limit, with the expectation that this divergence gets turned into a finite O(N f ) contribution once higher order terms in 1/N f are resummed. Since this resummation is beyond the scope of the present work, we focus on the difference between vacuum and finite-temperature quantities where f V,2 drops out. In particular, we study the pressure (minus the free energy density) where we have normalized the pressure to the pressure of a free (non-interacting) bosonic degree of freedom. Note that the normalized pressure is 3N f and not 4N f because in three dimensions each fermionic degree of freedom contributes only 3 4 of a bosonic degree of freedom.
As discussed above, results for f V,1 , f For comparison, the normalized pressure using only the vacuum polarization tensor is also shown (dashed line). Horizontal axis has been compactified in order to show the full range e 2 N f T ∈ [0, ∞). Arrows indicate weak-coupling and infinite coupling limit, respectively. The question mark indicates that we do not trust our results using the full in-medium polarization tensor to be a good approximation at large, but fixed N f in this region. See text for details.
been reported in the case of QED4 in Refs. [9,10] (see Fig. 1). (Note that apparent nonmonotonic behavior shown in Fig. 1 is a result of the normalization used for plotting; the (un-normalized) pressure itself is always monotonically increasing with temperature as it should.) Eventually, the normalized pressure hits a maximum below 3N f + 1 and starts to decrease again for e 2 N f T ≥ 100, with the numerical evaluation of f

(M )
A,B becoming more challenging in this region.
Based on the arguments given in section III B, we suspect that for fixed, but large N f , the normalized pressure for e 2 N f T 16 may continue to decrease towards 3N f , departing from our calculation that is using the in-medium polarization tensor evaluated in the naive large N f limit. This is indicated by a question mark and the result using only the vacuum polarization tensor shown in Fig. 1. We would invite follow-up studies from lattice gauge