QGP universality in magnetic field?

We use top-down holographic models to study the thermal equation of state of strongly coupled quark-gluon plasma in external magnetic field. We identify different conformal and non-conformal theories within consistent truncations of ${\cal N}=8$ gauged supergravity in five dimensions (including STU models, gauged ${\cal N}=2^*$ theory) and show that the ratio of the transverse to the longitudinal pressure $P_T/P_L$ as a function of $T/\sqrt{B}$ can be collapsed to a 'universal' curve for a wide range of the adjoint hypermultiplet masses $m$. We stress that this does not imply any hidden universality in magnetoresponse, as other observables do not exhibit any universality. Instead, the observed collapse in $P_T/P_L$ is simply due to a strong dependence of the equation of state on the (freely adjustable) renormalization scale: in other words, it is simply a fitting artifact. Remarkably, we do uncover a different universality in ${\cal N}=2^*$ gauge theory in the external magnetic field: we show that magnetized ${\cal N}=2^*$ plasma has a critical point at $T_{crit}/\sqrt{B}$ which value varies by $2\%$ (or less) as $m/\sqrt{B}\in [0,\infty)$. At criticality, and for large values of $m/\sqrt{B}$, the effective central charge of the theory scales as $\propto \sqrt{B}/m$.


Introduction and summary
In [1] the authors used the recent lattice QCD equation of state (EOS) data in the presence of a background magnetic field [2,3], and the holographic EOS results 1 for the strongly coupled N = 4 SU(N) maximally supersymmetric Yang-Mills (SYM) to argue for the universal magnetoresponse. While N = 4 SYM is conformal, the scale invariance is explicitly broken by the background magnetic field B and its thermal equilibrium stress-energy tensor is logarithmically sensitive to the choice of the renormalization scale. It was shown in [1] that both the QCD and the N = 4 data (with optimally adjusted renormalization scale) for the pressure anisotropy R, i.e., defined as a ratio of the transverse P T to the longitudinal P L pressure, collapse onto a single universal curve as a function of T / √ B, at least for T / √ B 0.2 or correspondingly for R 0.5, see Fig. 6 of [1]. The authors do mention that the 'universality' is somewhat fragile: besides the obvious fact that large-N N = 4 SYM is not QCD (leading to inherent ambiguities as to how precisely one would match the renormalization schemes in both theories -hence the authors opted for the freelyadjustable renormalization scale in SYM), one observes the universality in R, but not in other thermodynamic quantities (e.g., P T /E -the ratio of the transverse pressure to the energy density).
So, is there a universal magnetoresponse? In this paper we address this question in a controlled setting: specifically, we consider holographic models of gauge theory/string theory correspondence [5,6] where all the four-dimensional strongly coupled gauge theories discussed have the same ultraviolet fixed point -N = 4 SYM. We discuss two classes of theories: • conformal gauge theories corresponding to different consistent truncations of N = 8 gauged supergravity in five dimensions 2 [7]; • non-conformal N = 2 * gauge theory (N = 4 SYM with a mass term for the N = 2 hypermultiplet) [7][8][9] (PW).
In the former case, the anisotropic thermal equilibrium states are characterized by the temperature T , the background magnetic field B and the renormalization scale µ; in the latter case, we have additionally a hypermultiplet mass scale m.
Before we present results, we characterize more precisely the models studied.
CFT diag : N = 4 SYM has a global SU(4) R-symmetry. In this model magnetic field is turned on for the diagonal U(1) of the R-symmetry. This is the model of [1], see also [4]. See section 2.1 for the technical details.
CFT ST U : Holographic duals of N = 4 SYM with U(1) 3 ⊂ SU(4) global symmetry are known as STU-models [10,11]. In this conformal theory the background magnetic field is turned on for one of the U(1)'s. This model is a consistent truncation of N = 8 fivedimensional gauged supergravity with two scalar fields dual to two dimension ∆ = 2 operators. As we show in section 2.2, in the presence of the background magnetic field these operators will develop thermal expectation values.
nCFT m : As we show in section 2.3, within consistent truncation of N = 8 fivedimensional gauged supergravity presented in [7], it is possible to identify a holographic dual to N = 2 * gauge theory with a single U(1) global symmetry. In this model the background magnetic field is turned on in this U (1). The label m ∈ (0, +∞) denotes the hypermultiplet mass of the N = 2 * gauge theory.
CFT P W,m=0 : This conformal gauge theory is a limiting case of the nonconformal nCFT m model: Its bulk gravitational dual contains two scalar fields dual to dimension ∆ = 2 and ∆ = 3 operators of the N = 2 * gauge theory. As we show in section 2.3.1, in the presence of the background magnetic field these operators will develop thermal expectation values.
CFT P W,m=∞ : This conformal gauge theory is a limiting case of the nonconformal nCFT m model: Its holographic dual can be obtained from the N = 8 five dimensional gauged supergravity of [7] using the "near horizon limit" of [12] 3 , followed by the uplift to six dimensions -the resulting holographic dual is Romans F (4) gauged supergravity in six dimensions [15,16] 4 . The six dimensional gravitational bulk contains a single scalar, dual to dimension ∆ = 3 operator of the effective CFT 5 . There is no conformal anomaly in odd dimensions. Furthermore, there is no invariant dimension-five operator that can be constructed only with the magnetic field strength -as a result, the anisotropic stress-energy tensor of CFT P W,m=∞ plasma is traceless, and is free from renormalization scheme ambiguities. Details on the CFT P W,m=∞ model are presented in section 2.3.2. The renormalization scheme-independence of CFT P W,m=∞ is a welcome feature: we will use the pressure anisotropy (1.1) of the theory as a benchmark to compare with the other conformal and non-conformal models.
And now the results. There is no universal magnetoresponse. Qualitatively, among conformal/non-conformal models we observe three different IR regimes (i.e., when T / √ B is small): In CFT diag it is possible to reach deep IR, i.e., the T / √ B → 0 limit. For T / √ B 0.1 3 See appendix D of [13] for details of the isotropic (no magnetic field) thermal states of N = 2 * plasma in the limit m/T → ∞. The first hint that N = 2 * plasma in the infinite mass limit is an effective five dimensional CFT appeared in [14]. 4 See [17] for a recent discussion.
the thermodynamics is BTZ-like with the entropy density 5 [4] s → Both in CFT P W,m=0 and CFT P W,m=∞ (and in fact in all nCFT m models) there is a terminal critical temperature T crit which separates thermodynamically stable and unstable phases of the anisotropic plasma. Remarkably, this T crit is universally determined by the magnetic field B, (almost) independently 6 of the mass parameter m of nCFT m : i.e., the variation of T crit / √ B with mass about its mean value is 2% or less, see Fig. 7 (left panel). We leave the extensive study of this critical point to future work, and only point out that the specific heat at constant B at criticality has a critical exponent 7 where F is the free energy density, see Fig. 6.
The CFT ST U model in the IR is different from the other ones. We obtained reliable numerical results in this model for T / √ B 0.06: we neither observe the critical point as in the CFT P W,m=0 and CFT P W,m=∞ models, nor the BTZ-like behavior (1.2) as in the CFT diag model, see Fig. 3 (left panel).
In Fig. 1 we present the pressure anisotropy parameter R (1.1) for the conformal theories: CFT diag (black curves), CFT ST U (blue curves), CFT P W,m=0 (green curves) and CFT P W,m=∞ (red curves) as a function of 8 T / √ B. R is renormalization scheme 5 We independently reproduce this result. 6 A very weak dependence on the mass parameter has been also observed for the equilibration rates in N = 2 * isotropic plasma in [18]. 7 The critical point with the same mean-field exponent α has been observed in isotropic thermodynamics of N = 2 * plasma with different masses for the bosonic and fermionic components of the hypermultiplet [19]. 8 We use the same normalization of the magnetic field in holographic models as in [1].  Figure 1: Anisotropy parameter R = P T /P L for conformal models CFT diag (black curves), CFT ST U (blue curves), CFT P W,m=0 (green curves) and CFT P W,m=∞ (red curves) as a function of T / √ B. R CFT P W,m=∞ is renormalization scheme independent; for the other models there is a strong dependence on the renormalization scale δ = ln B µ 2 : different panels represent different choices for δ; all the models in the same panel have the same value of δ, leading to identical high-temperature asymptotics, T / √ B ≫ 1.
independent in the CFT P W,m=∞ model, while in the former three conformal models it is sensitive to  Figure 2: Renormalization scale δ is adjusted separately for the CFT diag , CFT ST U and CFT P W,m=0 models (see (1.7)) to ensure that in all these models the pressure anisotropy R = 0.5 occurs for the same value of T √ B as in the CFT P W,m=∞ model (see (1.6)). This matching point is highlighted with the dashed brown lines.
in the IR to be "to the left" of the scheme-independent (red) curve R CFT P W,m=∞ (top panels and the bottom left panel); or "to the right" of the scheme-independent (red) curve R CFT P W,m=∞ (the bottom right panel).
In Fig. 1 we kept δ the same for the conformal models CFT diag , CFT ST U and CFT P W,m=0 . This is very reasonable given that one can match δ across all the models by comparing the UV, i.e., T / √ B ≫ 1 thermodynamics (see appendix B) -there are no other scales besides T and B, and thus by dimensional analysis 9 , If we give up on maintaining the same renormalization scale for all the conformal models, it is easy to 'collapse' all the curves for the pressure anisotropy, see Fig. 2. We will not perform sophisticated fits as in [1], and instead, adjusting δ independently for 9 The asymptotic AdS 5 radius L always scales out from the final formulas. In a nutshell, this is what was done in [1] to claim a universal magnetoresponse for R 0.5. Rather, we interpret the collapse in Fig. 2 as nothing but a fitting artifact, possible due to a strong dependence of the anisotropy parameter R on the renormalization scale.
To further see that there is no universal physics, we can compare renormalization scheme-independent anisotropic thermodynamic quantities of the models: the entropy densities, see Fig. 3. The color coding is as before: CFT diag (black curves), CFT ST U (blue curves), CFT P W,m=0 (green curves) and CFT P W,m=∞ (red curves). We plot the entropy densities relative to the entropy density of the UV fixed point at the corre- sponding temperature (see eq. (D.13) for the CFT P W,m=∞ model in [13]): ( In nCFT m models it is equally easy to 'collapse' the data for the pressure anisotropy. In these models we have an additional scale m -the mass of the N = 2 hypermultiplet. In the absence of the magnetic field, i.e., for isotropic N = 2 * plasma, the thermodynamics is renormalization scheme-independent 10 [20]. Once we turn on the 10 Scheme-dependence arises once we split the masses of the fermionic and bosonic components of the N = 2 * hypermultiplet [20]. color coding is as in Fig. 4, except that we collected more data 11 in addition to (1. Indeed, the (lower) thermodynamically unstable branch has a negative specific heat (left panel); approaching the critical temperature from above we observe the divergence in the specific heat, both for the stable and the unstable branches. To extract a critical exponent α, defined as we plot (right panel) the dimensionless quantity c 2 B /s 2 as a function of T / √ B. Both the stable (solid) and the unstable (dashed) curves approach zero, signaling the divergence 11 To have a better characterization of the critical points. of the specific heat at the critical temperature (vertical dashed brown line), with a finite slope -this implies that the critical exponent is There is a remarkable universality of the critical points in nCFT m and conformal CFT P W,m=0 and CFT P W,m=∞ models. In Fig. 7 (left panel) we present the results for the critical temperature as a function of m/ √ 2B in nCFT m models (points). The horizontal dashed lines indicate the location of the critical points for the CFT P W,m=0 (green) and CFT P W,m=∞ (red) conformal models. In the right panel the dots represent the relative entropy, (1.14) One can understand the origin of the asymptote (1.13) from the fact that nCFT m models in the large m limit should resemble the conformal model CFT P W,m=∞ ; thus, we expect that γ ∞ ≈ γ CFT P W,m=∞ . Indeed, where we extracted numerically the value of s crit s U V for the CFT P W,m=∞ conformal model, used (1.8) to analytically compute the second factor in the first line, and substituted the numerical value for T crit / √ B of the CFT P W,m=∞ model in the second line.
We now outline the rest of the paper, containing technical details necessary to obtain the results reported above. In section 2 we introduce the holographic theory of [7] and explain how the various models discussed here arise as consistent truncations of the latter: CFT diag in section 2.1, CFT ST U in section 2.2, and nCFT m in section 2.3.
The conformal models CFT P W,m=0 and CFT P W,m=∞ are special limits of the nCFT m model and are discussed in sections 2.3.1 and 2.3.2 correspondingly. Holographic renormalization is by now a standard technique [21], and we only present the results for the boundary gauge theory observables. Our work is heavily numerical. It is thus important to validate the numerical results in the limits where perturbative computations (analytical or numerical) are available. We have performed such validations in appendix B, i.e., when T √ B ≫ 1. We did not want to overburden the reader with details, and so we did not present the checks of the agreement of the numerical parameters (e.g., as in (2.23)) with the corresponding perturbative counterparts -but we have performed such checks in all models. There are further important constraints on the numerically obtained energy density, pressure, entropy, etc., of the anisotropic plasma: the first law of the thermodynamics dE = T ds (at constant magnetic field and the mass parameter, if available), and the thermodynamic relation between the free energy density and the longitudinal pressure F = −P L . The latter relation can be proved (see appendix A) at the level of the equations of motion, borrowing the holographic arguments of [22] used to establish the universality of the shear viscosity to the entropy density in the holographic plasma models. Still, as the first law of thermodynamics, it provides an important consistency check on the numerical data -we verified these constraints in all the models, both perturbatively in the high-temperature limit, to inclusive, see appendix B, and for finite values of B/ √ T , see appendix Conce again, we present only partial results of the full checks.
Our paper is a step in broadening the class of strongly coupled magnetized gauge theory plasmas (both conformal and massive) amenable to controlled holographic analysis. We focused on the equation of state, extending the work of [1]. The next step is to analyze the magneto-transport in these models, in particular the magneto-transport at criticality.

Technical details
The starting point for the holographic analysis is the effective action of [7]: where the F (J) are the field strengths of the U(1) gauge fields, A (J) , and P is the scalar potential. We introduced The scalar potential, P, is given in terms of a superpotential In what follows we set gauged supergravity coupling g = 1, this corresponds to setting the asymptotic AdS 5 radius to L = 2. The five dimensional gravitational constant G 5 is related to the rank of the supersymmetric N = 4 SU(N) UV fixed point as The holographic dual to the CFT diag conformal model is a consistent truncation of (2.1) with leading to where we used the normalization of the bulk U(1) to be consistent with [1].
This model has been extensively studied in [1,4] and we do not review it here.

CFT ST U
The holographic dual to the CFT ST U is a special case of the STU model [10,11,23], a consistent truncation of the effective action (2.1) with and the scalar potential We would like to keep a single bulk gauge field, so we can set two of them to zero and work with the remaining one. The symmetries of the action allow us to choose whichever gauge field we want. To see this, notice that the action (2.9) is invariant µν = 0 for the gauge fields and with the scalar field redefinitions ρ → ν 1/2 ρ −1/2 and ν → ν 1/2 ρ 1/2 . Thus, we arrive to the holographic dual of CFT ST U as where once again we used the normalization of the remaining gauge field as in [1].
Solutions to the gravitational theory (2.11) representing magnetic black branes dual to anisotropic magnetized CFT ST U plasma correspond to the following background ansatz 12 : where all the metric warp factors c i as well as the bulk scalars ρ and ν are functions of the radial coordinate r, where r 0 is a location of a regular Schwarzschild horizon, and r → +∞ is the asymptotic and denoting we obtain the following system of ODEs (in a radial coordinate x, ′ = d dx ):

16)
12 Note that we fixed the radial coordinate r with the choice of the metric warp factor in front of (2.20) Notice that r 0 is completely scaled out from all the equations of motion. Eqs. (2.16)-(2.20) have to be solved subject to the following asymptotics: in the UV, i.e., as x → 0 + , in the IR, i.e., as y ≡ 1 − x → 0 + , which is the correct number of parameters necessary to provide a solution to a system of three second order and two first order equations, 3 × 2 + 2 × 1 = 8. The parameters n 1 and r 1 correspond to the expectation value of two dimension ∆ = 2 operators of the boundary CFT ST U ; the other two parameters, a 1,2 and a 2,2 , determine the expectation value of its stress-energy tensor. Using the standard holographic renormalization we find: for the components of the boundary stress-energy tensor, and for the entropy density and the temperature. Note that, as in N = 4 SYM [1], (2.30) • Anisotropy introduced by the external magnetic field results in P T = P L . From the elementary anisotropic thermodynamics (see [1] for a recent review), the free energy density of the system F is given by We emphasize that holographic renormalization (even anisotropic one) naturally enforces (2.27) (see [25] for one of the first demonstrations), but not (2.31). In appendix A we present a holographic proof 13  Technical details presented here are enough to generate the CFT ST U model plots reported in section 1. 13 The proof follows the same steps as in the first proof of the universality of the shear viscosity to the entropy density in holography [22].
14 Additionally, as in the nCFT m model with m/ √ 2B = 1 (see appendix C), we checked both relations for finite b.

nCFT m
There is a simple consistent truncation of the effective action (2.1) to that of the PW action [8], supplemented with a single bulk U(1) gauge field. Indeed, setting where P P W is the Pilch-Warner scalar potential of the gauged supergravity:

(2.35)
We use the same holographic background ansatz, the same radial coordinate x, as for the CFT ST U model (2.12)-(2.15); except that now we have the bulk scalar fields α and .

(2.40)
As in the CFT ST U model, r 0 is completely scaled out from all the equations of motion.
Eqs. (2.36)-(2.40) have to be solved subject to the following asymptotics: in the UV, i.e., as x → 0 + , (2.41) in the IR, i.e., as y ≡ 1 − x → 0 + , (2.42) The non-normalizable coefficients α 1,1 (of the dimension ∆ = 2 operator) and χ 0 (of the dimension ∆ = 3 operator) are related to the masses of the bosonic and the fermionic components of the hypermultiplet of N = 2 * gauge theory. When both masses are the same (see [20]) Furthermore, carefully matching to the extremal PW solution [8,9] (following the same procedure as in [20]) we find B m 2 = 2b where m is the hypermultiplet mass. We find it convenient to use for the components of the boundary stress-energy tensor, for the expectation values of the relevant operators, and for the entropy density and the temperature. Note that, as expected [26], (2.52) In appendix C we have verified FT and TR in the nCFT m model with m/ √ 2B = 1 numerically.
Technical details presented here are enough to generate nCFT m model plots reported in section 1.

CFT P W,m=0
The CFT P W,m=0 model is a special case of the nCFT m model when the hypermultiplet mass m is set to zero. This necessitates setting the non-normalizable coefficients α 1,1 and χ 0 to zero =⇒ η = 0 in (2.45). From (2.40) it is clear that this m = 0 limit is consistent with we also present O(B 4 /T 8 ) results for R CFT P W,m=0 and confirm that the renormalization scheme choice of κ as in (2.28) leads to (2.55)

CFT P W,m=∞
The holographic dual to the CFT P W,m=∞ model can be obtained as a particular decoupling limit χ → ∞ of the effective action (2.34). As emphasized originally in [12], 15 It is interesting to investigate whether this Z 2 symmetry can be spontaneously broken, and if so, what is the role of the magnetic field. This, however, is outside the scope of the current paper.
the supersymmetric vacuum, and the isotropic thermal equilibrium states of the theory [13,14] are locally that of the 4 + 1 dimensional conformal plasma. We derive the 5 + 1 dimensional holographic effective action S CFT P W,m=∞ (trivially) generalizing the arguments of [12].
Solutions to the gravitational theory (2.63) representing magnetic branes dual to anisotropic magnetized CFT P W,m=∞ plasma correspond to the following background ansatz: where all the metric warp factors c i as well as the bulk scalar φ 1 are functions of the radial coordinate r. The rescaled, i.e.,ˆcoordinates, are related to PW coordinates x µ and the KK direction x 6 as follows (compare with (2.59)): It is convenient to fix the radial coordinate r and redefine the metric warp factor, the bulk scalar, and the magnetic field as (2.66) The radial coordinate r changes where r 0 is a location of a regular Schwarzschild horizon, and r → +∞ is the asymptotic AdS 6 boundary 17 . The bulk scalar field p is dual to a dimension ∆ = 3 of the effective 16 The identification is as follows: A i = 0, B = 0, X = e −φ1 , m = 1 4 and g 2 = 1 2 . 17 AdS 6 of radius L AdS6 = 3 3/4 2 1/2 is a solution with r 0 = 0,b = 0 and a 1 = a 2 = a 4 ≡ 1 and five-dimensional boundary conformal theory. Introducing a radial coordinate x as in (2.14) we obtain the following system of ODEs (in a radial coordinate x, ′ = d dx ): have to be solved subject to the following asymptotics: in the UV, i.e., as x → 0 + , (2.72) in the IR, i.e., as y ≡ 1 − x → 0 + , In total, givenb, the asymptotic expansions are specified by 6 parameters: {a 1,5 , a 2,5 , p 3 , a 1,h,0 , a 2,h,0 , p h,0 } , (2.74) which is the correct number of parameters necessary to provide a solution to a system of two second order and two first order equations, 2 × 2 + 2 × 1 = 6. The parameter p 3 corresponds to the expectation value of a dimension ∆ = 3 operator of the boundary theory; the other two parameters, a 1,5 and a 2,5 , determine the expectation value of its stress-energy tensor. Using the standard holographic renormalization we find: for the components of the boundary stress-energy tensor, and for the entropy density and the temperature. Note that, There is no renormalization scheme dependence in (2.75), and the trace of the stressenergy tensor vanishes -there is no invariant dimension-five operator that can be constructed only with the magnetic field strength. The (holographic) free energy density is given by the standard relation (2.27). In (2.75)-(2.76) we used the subscript [5] to indicate that the thermodynamic quantities are measured from the perspective of the effective five-dimensional boundary conformal theory; to convert to the fourdimensional perspective, we need to account for (2.65), see also [13], E, P T , P L = E [5] , P [5]T , P [5]L × 2k 3 As for the other models discussed in this paper, the first law of thermodynamics dE = T ds (at fixed magnetic field) and the thermodynamic relation F = −P L lead to constraints on the numerically obtained parameter set (2.74) (here ′ = d db ): A Proof of −P L = E − sT in holographic magnetized plasma The proof follows the argument for the universality of the shear viscosity to the entropy density in holographic plasma [22].
Consider a holographic dual to a four dimensional 18 gauge theory in an external magnetic field. We are going to assume that the magnetic field is along the z-direction, as in (2.12). We take the (dimensionally reduced -again, this can be relaxed) holographic background geometry to be At extremality (whether or not the extremal solution is singular or not within the truncation is irrelevant), the Poincare symmetry of the background geometry guarantees where R µν is the Ricci tensor in the orthonormal frame. Clearly, an analogous condition must be satisfied for the full gravitational stress tensor of the matter supporting the 18 Generalization to other dimensions is straightforward.
Explicitly evaluating the ratio of the const in (A.4) in the UV (r → ∞) and IR (r → r horizon ) we recover for each of the models we study.
We should emphasize that the condition (A.2) can be explicitly verified using the equations of motion in each model studied. The point of the argument above (as the related one in [22]) is that this relation is true based on the symmetries of the problem alone.

B Conformal models in the limit
In holographic models, supersymmetry at extremality typically guarantees that equilibrium isotropic thermodynamics is renormalization scheme independent (compare the N = 2 * model with the same masses for the bosonic and the fermionic components 20]). This is not the case for the holographic magnetized gauge theory plasma in four space-time dimensions, e.g., see [1] for N = 4 SYM. In this appendix we discuss the high temperature anisotropic equilibrium thermodynamics of the conformal (supersymmetric in vacuum) models. For the (locally) four dimensional models ( CFT diag , CFT ST U and CFT P W,m=0 ) matching high-temperature equations of state is a natural way to relate renormalization schemes in various theories. In the CFT P W,m=∞ model, which is locally five dimensional, magnetized thermodynamics is scheme independent.

B.2 CFT P W,m=0
The high temperature expansion of the Z 2 symmetric, i.e., χ ≡ 0 phase, of anisotropic CFT P W,m=0 plasma thermodynamics corresponds to the perturbative expansion in b.
In what follows we study anisotropic thermodynamics to order O(b 4 ) inclusive. Introducing we find at order n = 1: and at order n = 2 (we will not need α (2) ): , to the fact that the value a 2,2,0, (2) in the CFT P W,m=0 dual is "sourced" by a single dimension ∆ = 2 operator (the scalar field α in the holographic dual), while the value a 2,2, (2) in the CFT ST U model is "sourced" by two dimension ∆ = 2 operators (the scalar fields ρ and ν in the holographic dual).

B.3 CFT P W,m=∞
The high temperature expansion corresponds to the perturbative expansion inb. In