Universal magnetoresponse in QCD and $\mathcal{N}=4$ SYM

Using recent lattice data on the thermodynamics of QCD in the presence of a background magnetic field, we show that the ratio of transverse to longitudinal pressure exhibits, to good accuracy, a simple scaling behavior over a wide range of temperature and magnetic field, essentially depending only on the ratio $T/\sqrt B$. We compare this QCD response to the corresponding magnetoresponse in maximally supersymmetric Yang Mills theory. Given suitable calibrations defining the comparison, we find excellent agreement. This may be viewed as a further test of the applicability of holographic models for hot QCD.

Gauge-gravity duality has enabled quantitative studies of the dynamics of certain strongly coupled non-Abelian plasmas [1][2][3][4]. Despite the limitations of holographic models (involving large N c , strong coupling limits, and supersymmetry), they have provided important insight into key properties of quark-gluon plasma as observed in relativistic heavy-ion collisions, including fast "thermalization" and the applicability of near-ideal hydrodynamics [5][6][7][8][9].
In this paper, we compare the magnetoresponse of QCD plasma and maximally supersymmetric Yang Mills (N = 4 SYM) plasma -the non-Abelian plasma with the simplest holographic description. Specifically, we examine the change in thermodynamic properties induced by a homogeneous background magnetic field. The response to an applied magnetic field is a useful probe of the dynamics in many condensed matter systems. In our context, an examination of magnetoresponse is also motivated by work suggesting that electromagnetic fields (albeit transient) may have significant effects in heavy ion collisions [10,11].
Holographic models have been found to describe rather accurately many aspects of strongly coupled QCD dynamics, despite the fact that QCD is neither conformal, supersymmetric, nor infinitely strongly coupled. Any reasonable measure of QCD coupling strength in experimentally accessible quark-gluon plasma is order unity, far from the infinite coupling limit, and N c = 3 appears equally far from N c = ∞. The apparent robustness of AdS/CFT predictions, despite these limitations, has prompted numerous investigations. It has been shown, for example, that finite coupling corrections in many thermal quantities are modest [12,13], and that N c dependence is essentially trivial, with extensive quantities simply scaling with the number of gauge fields [14,15].
To investigate whether a similar robustness exists with respect to conformal symmetry it is natural to examine the effects of deformations which explicitly break conformal symmetry.
Adding a background magnetic field is such a non-conformal deformation.
In the presence of an external electromagnetic field, the definition of the QCD contribution to the total stress-energy tensor depends on a choice of renormalization point. This issue is discussed in section II, which reviews basic properties of the stress-energy tensor of a quantum field theory when minimally coupled to a non-dynamical electromagnetic field.
For QCD magnetoresponse, we take as input results from recent high quality lattice gauge theory calculations of QCD thermodynamics for a wide range of background magnetic field B and temperature T [16,17]. A natural measure of anisotropy in the system is the transverse to longitudinal pressure ratio p T /p L , shown in the left panel of Fig. 1 as a function of T for different magnetic fields. Interestingly, we find that this ratio exhibits, to good accuracy, a simple scaling behavior over a wide range of temperature and magnetic field. As shown in the right panel of the same figure, when plotted as a function of T / √ B data from widely differing values of T and B essentially collapse onto a single curve. The underlying lattice QCD data is discussed in more detail in Section III, (Deviations from this scaling behavior appear to be present at the lowest temperatures and highest magnetic fields, but the growth of the error bars precludes making any definitive statement about this region.) In any conformal field theory, 1 the lack of intrinsic scales automatically implies that the magnetoresponse (appropriately defined) can only depend on the dimensionless ratio T / √ B.
So, having found near-universal scaling behavior in the QCD magnetoresponse, it is natural to compare this response to that of the simplest holographic model, namely conformal N = 4 SYM in the strong coupling and large N limit, for which the dual description reduces to 5D Einstein-Maxwell theory. We briefly review N = 4 SYM theory and its coupling to a background EM field in section IV, and then compare the QCD and N = 4 SYM Consider a quantum field theory (QFT) minimally coupled to an external electromagnetic The total action of the theory may be written in the form where S EM is the classical Maxwell action, specialized to a pure magnetic field, with e the (bare) electromagnetic coupling constant, and the QFT action S QFT (B) includes the minimal coupling to the background EM field. 2 Here and henceforth, we choose to scale the external gauge field A ext µ so that the electromagnetic coupling e does not appear in covariant derivatives, but instead e 2 is an inverse factor in the Maxwell action. Consequently, our B is the same as eB if a conventional perturbative scaling of the U (1) field is used. 3 From the action (1) (generalized to curved space) one derives the stress-energy tensor, where T µν QFT = −2δS QFT /δg µν is the QFT contribution to the total stress-energy tensor, while is the standard Maxwell stress-energy tensor (in Minkowski space). Specialized to a constant magnetic field in the z-direction, T µν EM = B 2 2e 2 diag(+1, +1, +1, −1). In a homogeneous equilibrium state, viewed in a rest frame (with vanishing momentum density) aligned with the magnetic field, diagonal elements of the expectation value of the stress-energy tensor can be interpreted as the proper energy density , and pressures (or diagonal stresses) p x , p y , p z along the x-, y-, z-directions, respectively, The magnetic field defines a preferred direction in space, and induces an anisotropy between the longitudinal pressure p L ≡ p z and transverse pressure p T ≡ p x = p y . (Rotational symmetry about the magnetic field direction implies that p x = p y .) The relation between the pressure(s) and energy density constitutes the equation of state of the system.
Similarly to the action (1), the thermodynamic grand potential (or Landau free energy), may be separated into QFT and EM contributions, with V = L x L y L z the spatial volume. This separation, by definition, places all the response to the applied magnetic field in the QFT contribution to the free energy. Let f QFT = F QFT /V and f EM = F EM /V denote the corresponding free energy densities. Derivatives of the free energy density f QFT with respect to the temperature or magnetic field define the entropy density and magnetization, respectively, Similarly, pressures are defined by the response of the system to compression in a given direction, In the thermodynamic limit, for homogeneous systems, the pressure is normally just minus the free energy density, since the free energy is extensive in the volume. But with a background magnetic field, one must specify what is to be held fixed in the partial derivative defining the pressure (8) [16]. The microscopic definition (4) of pressures as stress-energy eigenvalues corresponds to a thermodynamic definition in which the effect of compression is evaluated at a fixed magnetic flux Φ = B L x L y . 4 Consequently, the longitudinal and transverse pressures differ, The thermodynamic relation f QFT = − T s implies that the entropy and energy densities s and are related via A final quantity of interest is the interaction measure I, which is (minus) the trace of the stress-energy tensor In a CFT, such as N = 4 SYM, conformal symmetry implies that the stress-energy tensor is traceless. However, adding an external magnetic field B is a non-conformal deformation of the theory and induces a non-zero trace, and hence a nonzero interaction measure. For N = 4 SYM (coupled to the external field in the manner described below in section IV), For an asymptotically free theory like QCD, the stress-energy trace has an intrinsic contribution from the running of the coupling (and quark masses terms), plus the additional contribution from the external magnetic field. Neglecting quark masses, where G µν is the gluon field strength,

B. Renormalization
In interacting quantum field theories, bare parameters of the action undergo multiplicative renormalization which introduces dependence on an arbitrary renormalization point µ. 5 In particular, the renormalized electromagnetic coupling e 2 acquires logarithmic scale dependence and satisfies a QED-like renormalization group equation, with positive coefficient b 1 . Explicitly, for QCD, where q f denotes the electromagnetic charge assignments (i.e., EM charges in units of e) of each quark flavor, and the explicit final form is specialized to three flavor QCD. For N = 4 SYM theory, where the sums run over all charged Weyl fermions w and charged scalars s with q w and q s denoting the corresponding electromagnetic charge assignments. If the electromagnetic field is regarded as classical (so that EM quantum fluctuations are neglected) then the higher order corrections in the β-function (14) are independent of e 2 . For QCD in a background magnetic field, the EM β-function has higher order corrections proportional to the non-Abelian coupling g 2 , while for N = 4 SYM (in a background field), no higher order corrections appear in the EM β-function (14) due to a supersymmetric non-renormalization theorem.
Neglecting any such higher order corrections, the solution to the renormalization group equation (14) shows the usual logarithmic scale dependence, Physical observables, like the total free energy, are necessarily independent of the renormalization point µ. However, the separation (6) of the free energy into QFT and background EM contributions requires choosing the scale at which to evaluate the coupling e appearing in the background EM contribution. So this separation is more properly written as with the scale dependence of f QFT (B, µ) necessarily canceling that of the EM term, so that Similarly, the QFT stress-energy tensor acquires scale dependence, which precisely cancels the scale dependence of the EM stress-energy tensor. Hence, This scale dependence induced by the separation of QFT response from the background EM contributions is unavoidable, since the background field contributions (for realistic values of the electromagnetic coupling) are orders of magnitude larger than the matter-induced response [17] and would otherwise overshadow interesting features in the magnetic field dependence of the QFT response.

III. LATTICE QUANTUM CHROMODYNAMICS
We consider QCD with 2 + 1 flavors of dynamical quarks with physical masses. The quarks have their usual electric charge assignments, q u = +2/3 and q d = q s = −1/3. The action has the form (1), with S QFT = S QCD and covariant derivatives augmented to include the background U (1) EM gauge field. The lattice regularized Euclidean functional integral was simulated non-perturbatively using a staggered fermion discretization and three different lattice spacings. Details of the lattice discretization and associated methods are described in Ref. [17].
The lattice QCD results were obtained using a renormalization point µ = Λ H , where Λ H = 120(9) MeV is a non-perturbatively determined hadronic scale defined by the condition that at T = 0 there be no O(B 2 ) contribution to the matter free energy. In other words, the total free energy (the sum of matter and magnetic contributions) equals 1 2 B 2 /e 2 (Λ H ) + O(B 4 ) in the zero temperature limit. For nonzero temperatures, the O(B 2 ) contribution to the matter free energy becomes nonzero, i.e., the system develops a non-trivial magnetic permeability.
We begin the discussion with the anisotropic pressure components p T and p L . To facilitate a comparison with SYM theory, it is natural to consider the dimensionless ratio This combination was shown as a function of the temperature for various values of magnetic field B in the left panel of Fig. 1. Notice that the longitudinal pressure p L is always positive, so the ratio R remains finite for all T and B. For low magnetic fields the ratio R ≈ 1, signaling the near-isotropy of the system. As the field B grows the anisotropy becomes more pronounced and the ratio R shifts away from unity -in fact it becomes negative when the transverse pressure p T changes sign (and becomes a "suction") for strong magnetic fields [17].
The apparent near absence of any significant dependence on Λ H / √ B motivates us to compare the magnetoresponse of QCD to that of conformal SYM theory, for which T / √ B is the only relevant dimensionless ratio.
An important question is how the near-universality of p T /p L is affected by a change in the renormalization point. We consider a general choice, involving the three underlying scales {T , B, Λ H } characterizing the equilibrium state, parameterized by three coefficients c T , c B and c Λ . For a quantitative description we introduce a measure D of the deviation from universality, where σ denotes the error of the ratio r and the sum extends over all points available from the lattice study of Ref. [17].  Therefore, in the following we set c Λ to zero, so that a comparison to N = 4 SYM theory will be straightforward. As shown in the right panel of global R-symmetry which was used in Ref. [21]. Specifically, are the respective charge assignments for the four Weyl fermions and three complex scalars of N = 4 SYM. With these assignments, the U (1) β-function coefficient (15b) becomes (However, as discussed in the next section, when comparing with QCD we will rescale the above charge assignments by an adjustable factor.) Equilibrium states of this theory, in the presence of a homogeneous background magnetic field, have a dual gravitational description given by magnetic black brane solutions first computed by D'Hoker and Kraus [22]. These are solutions of 5D Einstein-Maxwell theory, which is a consistent truncation of type IIB supergravity. The Einstein-Maxwell action where M, N = 0, . . ., 4 are the 5D spacetime indices, g is the metric, R is the Ricci scalar, is the 5D Newton gravitational constant, Λ = −6L −2 is the cosmological constant and F = dA is the five-dimensional electromagnetic field strength. The reduction from IIB supergravity leads to a specific value for the Chern-Simons coupling θ, but this term vanishes identically for our solutions of interest and may be ignored. Solutions to the gravitational theory (27) representing uncharged (magnetic) black branes may be described by a metric of the form [22], plus a bulk field strength The subleading terms in these near-boundary expansions determine the (expectation value of the) N = 4 SYM stress-energy tensor [21]. Specifically, with all off-diagonal components vanishing. Here κ ≡ (N 2 c −1)/(2π 2 ) and µ, once again, is the arbitrary renormalization point used to separate the SYM and background EM contributions to the total stress-energy tensor. For further details, see Refs. [21,23].

V. COMPARISON OF QCD AND N = 4 SYM
To compare lattice QCD results with thermodynamic data for N = 4 SYM calculated via holography, one must decide how best to adjust for the differing field content of the two theories. Specifically, in making this comparison should the SYM charge assignments be rescaled? The overall normalization of our SYM charge assignments (25) was merely a convenient choice which corresponds to the absence of additional numerical factors multiplying the Maxwell term in the dual gravitational action (27) [21]. A uniform rescaling of charge assignments is equivalent to a rescaling of the magnetic field, so this question is the same as asking whether comparisons are most usefully made at coinciding values of magnetic field, as it was introduced in the gravitational action (27), or whether it is appropriate to first rescale the background magnetic field added to SYM theory.
As long as the background electromagnetic field is treated as classical, the normalization of SYM charge assignments is arbitrary, as there is no intrinsic scale available to define physical units in which to measure a magnetic field. In other words, there is no quantization of EM charges or magnetic fluxes. Hence, it is completely appropriate to rescale SYM charge assignments, or equivalently rescale the magnetic field, B → B/ξ for some choice of ξ, when comparing with QCD. (In contrast, temperature may be regarded as having a common operational meaning in both theories, so no rescaling of temperature is performed.) The key question is how should one choose this charge (or magnetic field) scale factor ξ?
From our earlier discussion (sec. II B) of renormalization point dependence, one seemingly natural possibility to consider is scaling the SYM charge assignments so that the leading coefficient b 1 in the U (1) β-function (15) coincides between QCD and SYM. This would require scaling the SYM charge assignments inversely with N c so as to compensate for the difference in the number of charged degrees of freedom. However, this choice is neither necessary nor helpful, as one can always first define rescaled stress-energy tensors, T µµ ≡ T µν /b 1 , in both QCD and SYM, so that the rescaled tensors satisfy identical renormalization group equations. Since we are comparing dimensionless ratios such as p T /p L , such an overall rescaling of T µν has no effect on the comparison between theories.
Another possible approach involves matching the magnetoresponse in the asymptotically high temperature limit. As discussed in appendix A, if one considers the entropy density (which is independent of the renormalization point µ) and demands that the relative con-tribution of the O(B 2 ) terms coincide, so that then this condition leads to a scale factor choice ξ = 19/3 ≈ 2.5.
This value for the charge rescaling defines an arguably sensible scheme for comparing our two theories. However, it uses information from asymptotically high temperature QCD which is far from the regime of a few times T c where it is appropriate to view real quark-gluon plasma as a strongly coupled near-conformal fluid. Consequently, our preferred approach is the simplest: we just treat the charge rescaling factor ξ as a free parameter, and find the value which minimizes the difference between the QCD and SYM results for the pressure ratio R = p T /p L . More precisely, we first evaluate this pressure ratio (as a function of B and T ), in both theories at a common renormalization point µ = (c T T 2 + c B |B|) 1/2 which lies along the valley defined by c B = 0.087 − 0.084 c T . As shown in Fig. 3, along this valley the QCD ratio R is essentially a function of only the single variable T / √ B. We then define a deviation ∆R between the QCD pressure ratio and that of SYM, with the SYM magnetic field rescaled by a adjustable factor ξ and c B determined by c T along the aforementioned valley. Only lattice QCD data points with T > 150 MeV are included in this sum, as lower temperatures probe the hadronic phase of QCD, not the deconfined plasma phase. As in our earlier measure D of the deviation from universality (24), σ denotes the lattice error of the ratio R, and N is the number of terms in the sum. A value less than unity for the deviation, ∆R < 1, indicates agreement between the two theories to within the errors of the lattice results, and the inherent uncertainty in ∆R is of order unity.
Our results for ∆R are plotted in Fig  suitably compared, displays a common conformal behavior over such a substantial range of temperature and magnetic field.

VI. DISCUSSION
In this work, we analyzed data from a recent lattice gauge theory calculation of the thermodynamics of a QCD plasma placed in an external magnetic field. Except at asymptotically high temperatures, T ≫ Λ QCD , observables in QCD will generically have independent non-trivial dependence on the value of both temperature and magnetic field (relative to Λ QCD ). Moreover, separating the QCD contribution from the classical Maxwell contribution to the stress-energy tensor necessarily introduces dependence on an arbitrary renormalization point µ, as discussed in section II B. So in the presence of a non-zero magnetic field, at any physically accessible temperature, the transverse to longitudinal pressure ratio R = p T /p L should be expected to display non-trivial dependence on multiple dimensionless ratios, for example T / √ B, T /Λ QCD , and T /µ. Choosing the renormalization point to depend in some dimensionally consistent fashion on the physical scales T , √ B, and Λ QCD still leaves two independent dimensionless ratios on which the pressure ratio should depend.
However, as shown in Fig. 1, we find that for suitable choices of renormalization point the pressure anisotropy p T /p L exhibits scale invariance to within the error estimates of the lattice data, with functional dependence only on the ratio T / √ B. A more careful study of the deviations from universality identified an optimal choice of renormalization point, µ = (0.69T 2 + 0.029|B|) 1/2 (given our specific measure (24)  Although our analysis has exclusively involved equilibrium quantities (for which lattice QCD calculations are possible), the region of "effective conformality" shown in Fig. 2 is presumably also the region in which the long wavelength dynamics of QCD plasma is reasonably well described by conformal hydrodynamics. Outside this region, effects of scale non-invariance should be increasingly important, implying significant bulk viscosity effects in QCD hydrodynamic response.
This work adds the pressure anisotropy magnetoresponse, described by a non-trivial scaling function of T / √ B, to the set of thermal observables in QCD which are well-reproduced by strongly coupled N = 4 SYM, the simplest (four dimensional) conformal gauge theory with a holographic description. It also reveals the limitations of modeling hot QCD plasma as a conformal fluid when thermodynamic ratios involving the energy density are examined.
This limitation is unsurprising, given what is known about the temperature dependence of the trace anomaly expectation value.
It would be interesting to explore extensions of this work involving comparisons with other strongly coupled theories having holographic descriptions which are closer to QCD than N = 4 SYM. Possibilities include N = 2 * SYM [24][25][26][27] and other mass deformations of N = 4 SYM, cascading gauge theory [28,29], the Sakai-Sugimoto model [30,31], and various bottom-up models (for example, [32]). Turning on additional deformations which can be studied both in lattice QCD and in strongly coupled holographic models, such as a non-zero isospin chemical potential, could also be instructive. Such work is left for the future. To define an optimal matching of the high temperature magnetoresponse in QCD and N = 4 SYM, we focus on the entropy density s = −∂f /∂T | µ , as this quantity is independent of the choice of renormalization point µ. As discussed in Ref. [17], at asymptotically high temperature where T is the only relevant physical scale, the renormalization group equation For three flavor QCD at temperatures T Λ H , asymptotic freedom implies that the entropy density approaches the Stefan-Boltzmann limit, so s 0 = 19 9 π 2 T 3 . And from Eq. (15a), the EM β-function coefficient b 1 = 1/(6π 2 ). Hence, 8 s T 3 = 19π 2 9 + B 2 6π 2 T 4 + O(B 4 ) .