Quasinormal modes of magnetic black branes at finite 't Hooft coupling

The aim of this work is to extend the knowledge about Quasinormal Modes (QNMs) and the equilibration of strongly coupled systems, specifically of a quark gluon plasma (which we consider to be in a strong magnetic background field) by using the duality between $\mathcal{N}=4$ Super Yang-Mills (SYM) theory and type IIb Super Gravity (SUGRA) and including higher derivative corrections. The behaviour of the equilibrating system can be seen as the response of the system to tiny excitations. A quark gluon plasma in a strong magnetic background field, as produced for very short times during an actual heavy ion collision, is described holographically by certain metric solutions to $5\text{D}$ Einstein-Maxwell-(Chern-Simons) theory, which can be obtained from type IIb SUGRA. We are going to compute higher derivative corrections to this metric and consider $\alpha'^3$ corrections to tensor-quasinormal modes in this background geometry. We find indications for a strong influence of the magnetic background field on the equilibration behaviour also and especially when we include higher derivative corrections.


Introduction
The formalism to qualitatively describe the early, far from equilibrium dynamics of the QCD phase of high energy density (for which the term quark gluon plasma (QGP) will be used even in the non-thermalized state) generated during heavy ion collisions at RHIC or LHC is one of the most prominent applications of gauge/gravity duality, more specifically of the duality between N = 4 super Yang-Mills theory (SYM) in 4 dimensions and supergravity (SUGRA) on AdS 5 × S 5 known as the AdS/CFT duality. In the weak limit of this duality the gauge group rank N of the boundary theory is taken to infinity, while the 't Hooft coupling λ is held fixed during the N → ∞ limit and afterwards is taken to infinity as well. This limit of the holographic duality allows for the description of far from equilibrium dynamics at strong coupling.
After having determined certain observables within the AdS/CFT duality an interesting next question would be how their higher derivative or α -corrections behave and how large they are. Computing finite 't Hooft coupling corrections is notoriously messy and involved, but necessary, if ones wishes to leave the unrealistic λ → ∞ limit.
Within a formalism that helps to describe QGPs far from equilibrium a natural aspect that should be analysed is how and how fast such a system equilibrates. At late times this question breaks down to the analysis of quasinormal modes (QNMs), fluctuations around the equilibrium state. The inverse of the absolute value of the imaginary part of QNM frequencies, which correspond to the poles of the propagator of such a fluctuation, is proportional to the equilibration time. Thus, the QNM with the smallest absolute imaginary part determines the time the system needs to equilibrate. The real part gives information about the energy of the mode, i.e. the frequency of the fluctuation.
Motivated by the work of [4], we are going to consider higher derivative corrections to the magnetic black brane metric and to tensor QNMs in a coupling corrected magnetic black brane background. The propagator of these perturbations h xy is dual to the two-point function of the xy component of the boundary stress energy tensor. Our numerical analysis gave a well converging result for the lowest α -corrected QNM frequeny, which is the most interesting regarding the equilibration of a QGP in a strong background field. The numerical errors of the α -corrections to the following QNMs were too large to give results, whose precision exceeds their rough size.
On the one hand we want to study how the late time behaviour of the QGP changes, if we consider it to be in a strong magnetic field, as produced for a very a short time during actual heavy ion collisions, and include higher α corrections, to leave the λ → ∞ limit. On the other hand this analysis also has a more abstract application: So far, we don't have a satisfying dual theory, that describes QCD. The most prominent AdS/CFT duality allows us to non-perturbatively compute quantities in a conformal field theory, with N → ∞ and λ = g 2 YM N → ∞. Whereas QCD has a finite coupling, a finite N = 3 and is not conformally invariant. Apart from (bottom-up-) modeling, one should try everything that is feasible on the gravity side, to bring the dual field theory closer to QCD in a top-down fashion.
This includes the computation of finite coupling corrections, 1/N corrections, breaking the scale invariance e.g. by introducing a magnetic background field, where the metric ansatz describing this setting can be deduced from a solution to 10D SUGRA, or several of the above simultaneously.
In the limit λ → ∞ the holographic description of a QGP in a magnetic background field was realized in [4] by considering a Einstein-Maxwell-Chern-Simons theory. That this setting describes the physical properties of the real SU (3) QGP at least qualitatively was shown in [11]. In this work we will show and also need that the ansatz chosen in [4] can be derived from a specific solution to SUGRA living in 10 dimensions (see [16]). This allows us to determine α 3 -corrections first to the metric of a magnetic black brane, where the magnetic background field back-reacts to the metric, and afterwards to QNM fluctuations around this specific solution. We are going to give a mathematical proof of a prescription, which was found in [5], to handle higher derivative correction to the five form F 5 in the presence of gauge fields for the specific case of a constant background field. The higher derivative corrections to the QNM frequencies and the metric will be computed numerically using pseudo-spectral methods.

Reviewing magnetic black branes in the λ → ∞ limit
In this chapter we give a review of calculations and results of [4] and present the computations in a way, that makes it more intuitive to extend them to the finite λ case.
The action in five dimensions, which is the starting point of the λ = ∞ calculations of [4] reads As shown by the authors of [10] the five sphere metric components depend on the radial coordinate of the AdS-space, if we consider α corrections. This will, of course, stay true, when we include a strong magnetic background field with back-reaction on the geometry. Therefore it is advisable to return to the 10-dimensional type IIB SUGRA action, from which (2.1) can be derived by integrating out the five sphere coordinates. 2) The metric ansatz for a constant magnetic background field with field strength tensor F xy = br 2 h = −F yx = const. is given by where with A y = r 2 h xb, A µ = 0 for other directions and u = r 2 h r 2 . The five-sphere S 5 is described by the coordinates y 1 , . . . , y 5 with µ 1 = sin(y 1 ), µ 2 = cos(y 1 ) sin(y 2 ), µ 2 = cos(y 1 ) cos(y 2 ), φ 1 = y 3 , φ 2 = y 4 , φ 3 = y 5 . (2.5) We have chosen the xy-direction of the field strength tensor to be r 2 h b, such that b coincides with the corresponding magnetic field strength parameter chosen in [4]. In the following we are going to set r h = 1, which corresponds to a rescaling of the coordinates. Reintroducing r h in the final differential equations for e.g. tensor fluctuations by ω 2 →ω = ω 2r h and q 2 →q = q 2r h , where ω and q are the frequency and the momentum of the mode corresponds to a rescaling to get the original form of the metric (2.4), (2.3). The relation between b and the physical magnetic field is given by [4] where the constant v can be computed from the near boundary metric.
The self dual solution to the EoMs for the five form components and with F 2 = dA. Here and henceforth we call F el 5 the electric part of the five form and its Hodge dual F mag 5 = * F el 5 the magnetic part. 1 In the λ = ∞ case the action (2.1) is the result of this setup in 10 dimensions with L(u) = 1. The factor 1 L(u) in front of the second term in (2.8) was not omitted, although L(u) = 1 in this order in α , since later on we will need an expression for F 5 for which for arbitrary L(u) and (2.8) does the job. The Einstein equations, or equivalently the differential equations obtained by varying action (2.1) with respect to U ,Ũ , W , L and V are given by 0 =b 2 L(u) 12 where we already inserted (2.17). The ansatz to solve these can be written as As said above we have for λ = ∞ that L(u) = 1, which can be seen from the form of the solution below. Furthermore we use the freedom to set u 0 = 0, in order to obtain a blackening factor and set v 0 = w 0 = 0, which can be achieved by rescaling. As pointed out by [4] u 1 is linked to the temperature of the system. In practical calculations we can set u 1 = 2 to give a Schwartzschild black hole for b → 0, which together with our metric ansatz (2.4) links the temperature to the horizon radius r h . Solving this system of differential equations near the horizon gives The next order term in this expansion is given in the Appendix 5.2.
Setting l 0 = 1 gives the same expansion as in [4], with l i = 0 for all i > 0. What we are after is a solution in order O(γ 0 ) with minimal error on a sufficiently large u-interval we expand U 0 (u), V 0 (u) and W 0 (u) in 1 − u and solve the resulting equations order by order up to order 260 in (1 − u).

Higher derivative corrections
In the following we will include 't Hooft coupling corrections in our calculations. We start again from the action in 10 dimensions, but now with α 3 -correction terms determined in [1].
These terms can be schematically written as where we have ignored a factor containing the exponential of the dilaton field, which is 0 for λ → ∞, and written the contractions between the tensors C and T , which will be defined below, as products. The quantity γ is defined as γ = ζ(3)λ − 3 2 8 and is thus proportional to α 3 .
Correction terms to the type IIb SUGRA action of order α and α 2 vanish. The action we work with in the following can be written as C abcd is the Weyl tensor of the ten dimensional manifold and T is given by The relation between b and B deduced from the trace anomaly of the stress energy tensor might get finite coupling corrections, too. Since our focus is on how QNMs behave for large magnetic background fields, without the need to prioritize a precise value for B, we will carry out the calculation including coupling corrections also with the choice b = 5 4 , while stressing that this only approximately corresponds to the λ → ∞ result B ≈ 34.5T 2 .
with antisymmetrized indices a, b, c and d, e, f and symmetrized with respect to the interchange of (a, b, c) ↔ (d, e, f ) [1]. Here F + is the self dual part of the F 5 ansatz or F + = 1 2 (1 + * )F 5 (working with Lorentzian signature ensures that this part of F 5 exists). Using the notation in [1] we write and The higher derivative corrected EoM for the five form is given by which yields where we set δW δF 5 := 2κ δS γ 10 δF 5 . (3.11)

A helpful prescription and its mathematical proof
In this section we claim and proof the validity of the following prescription, which will facilitate our calculation noticeably. It is equivalent to strictly applying the variational principle, treating both the four form components and the metric as independent fields and solve the resulting system of highly coupled, finite coupling corrected differential equations simultaneously including the back-reaction of a strong background field: Solve the equation of motion for F 5 in the lowest order in α for a strong background field, such that it depends on the metric components of the ansatz made in (2.3, 2.4) (which we allow to be of order O(γ)) and choose the L(u)-factor of the components of the electric part of the five form in such a way that Now replace the F 2 5 term in the action with 2 times (F mag ) 2 and insert F 5 as given in (2.8, 2.9), which depends on metric components, that still have to be determined, into the higher derivative part of the action. The resulting action only depends on the absolute value of the z-component of the magnetic background field b and the metric, whose solution in order O(γ) will be determined by solving the system of differential equations obtained by varying this effective action with respect to g µν . 3 We justify this claim with the following proof, where we work with the metric ansatz proof. Let us first focus on the tuzy 3 is the starting point of generalizing the following proof to arbitrary gauge fields.
the system of differential equations it appears in, derived from (3.9), is given by where the right hand side has to be equal to the corresponding directions of In order O(x 0 ) there are no other contributions from C 4 to the right hand side of the diagram.
From diagram (3.14) we can derive that modulo terms, which are independent of u, the following equations hold  proof. This claim follows by carefully inspecting the magnetic part F mag 5 = * F el 5 of F 5 given in (2.8) and (2.9) and by using the self duality of this five form.
for all directions abcde.
proof. Let {X i } i∈I be equal to the set {g µν } µν∈{1,...,10} and let X 0 = X. Then we have where we made use of the sum convention.
proof. The claim follows immediately with Lemma 3.4.

Theorem 3.6. The prescription given in the introduction of this section is valid.
proof. Due to Lemma 3.1 and due to the fact that the effective action for the metric is not allowed to depend on x, because of gauge invariance, the theorem 3.6 holds, if we can show that for any given direction abcde, for which the electric part of the five form F 5 is non-zero, the expression given by −γ(3.21)| g→g is the same as for X ∈ {g µν } µν∈{1,...,10} and g being the solution for the metric with back-reaction and without higher derivative corrections. The claim now follows immediately by applying Lemma 3.5 and Lemma 3.1, since comment 3.3 implies We also can extend the prescription to include tensor fluctuations.  proof. Since ∂ ∂h xy |g| g xx g yy − (g xy ) 2 = 0 (3.26) the Lemma follows immediately.
The proof of the validity of the extension of the prescription is now entirely analogous to the one presented for theorem 3.6.

AdS-Schwarzschild black hole solution
In this chapter we present a way to compute higher derivative corrections to the AdS-Schwarzschild black hole solution, so in the case b = 0, on an interval u = r 2 The interval boundaries l and k have to be chosen sufficiently close to 0 and 1. The following procedure can be generalized to the case of a non-vanishing background field with back-reaction on the geometry. In that case we cannot hope to be able to determine the higher derivative corrections to the metric analytically. Even a near boundary and a near horizon analysis of the higher derivative correction terms to the differential equations of the metric with back-reaction of a strong magnetic background field turns out to be extremely difficult. We motivate the computational strategy we are going to apply to determine these corrections to the metric numerically by performing an analogous calculation in the case b = 0 and show that it delivers the same results (with very small errors) as the analytical solutions first derived in [10].
Our metric ansatz is of the form ( Let now L 10 be the action defined in (2.2) with F el 5 = − 4 L(u) 5 AdS . In addition we define where the contributions of the T -tensors to the EoM vanish in the case of absent background fields b = 0. We have to solve the differential equations We choose the ansätze Only the X 0 (u) parts are entering the terms we can see that L W 10 (X) is regular at the horizon for X(u) ∈ {Ũ (u), V (u), L(u)}, whereas for X(u) = U (u) it has a pole of at maximum first order at u = 1. 4 In the following our aim is to determine the terms L W 10 (X). Our strategy will be to calculate the terms on the rescaled Gauss-Lobatto grid for the u-coordinate with l = 0.1 and k = 0.99, such that for u ∈ [l, k] we have The functions U 0 (u),Ũ 0 (u), V 0 (u), W 0 (u) for a fixed value b = 5 4 were determined numerically in section 2, in such a way, that the numerical error is negligible on the interval [l, k] on which we have defined our Gauss-Lobatto grid (3.33). Since we consider the case b = 0 in this section we perform this calculation with U 0 (u),Ũ 0 (u), V 0 (u), W 0 (u) chosen such that (2.4) is the Schwarzschild black hole metric. The higher derivative corrections will be determined with the help of spectral methods by expanding the ansätze in the following way  The maximal absolute value of the relative error, which again appears for R X = R L , is now 7.3 × 10 −9 .

Calculating higher derivative corrections to the magnetic black brane metric
In this chapter we are going to generalize techniques derived previously to determine an approximation of higher derivative corrections to the metric computed in section 2. First of all we have to use the theorem derived in section 3.1. We apply the prescription from there to simplify our calculation. Following this theorem we define the five form F 5 in the following way: Starting with the fluctuation free electric part and its Hodge dual, we get g tt 10 g uu 10 g xx 10 g yy 3 10 g zz 10 dy 1 ∧ dy 2 ∧ dy ∧ dy 4 ∧ dy 5 + g tt 10 g uu 10 g xx 10 g yy 4 10 g zz 10 dy 1 ∧ dy 2 ∧ dy 3 ∧ dy ∧ dy 5 + g tt 10 g uu 10 g xx 10 g yy 5 10 g zz 10 dy 1 ∧ dy 2 ∧ dy 3 ∧ dy 4 ∧ dy . (3.47) The electric components of the five form including the gauge field A y = bx is explicitly given by while its Hodge dual simplifies to * (F el 5 ) 1 = − 2b √ 3 L(u) 4 det(g S 5 ) sin(y 1 ) cos(y 1 )g y 1 y 1 10 (g y 3 y 3 10 − sin(y 2 ) 2 g y 4 y 3 10 − cos(y 1 ) 2 g y 5 y 3 10 )dx ∧ dy ∧ dy 2 ∧ dy 5 ∧ dy 4 + cos(y 1 ) 2 sin(y 2 ) cos(y 2 )× g y 2 y 2 10 (g y 4 y 4 10 − g y 5 y 4 10 )dx ∧ dy ∧ dy 1 ∧ dy 5 ∧ dy 3 − sin(y 1 ) cos(y 1 )g y 1 y 1
Each part of the higher derivative terms, which are schematically written above, will con- We define L 10 to be (2.2) with F 2 5 being replaced by 2 (F 5 ) mag 2 . As before we consider the system of differential equations  different values for k in the vicinity of 1 6 to ensure that the numerical error we commit, due 5 When we compute the variation, we are allowed to assume that the metric components abbreviated with X ∈ {L, U,Ũ , W, V } do not depend on x, since terms of the form ∂ ∂∂xX L W 10 , ∂ ∂∂ 2 x X L W 10 must vanish, exactly as in the case O(γ 0 ). Otherwise the EoM for the gauge field Ay = bx would get mass terms. In addition Ay = bx is also a solution to the higher derivative corrected EoM for gauge fields. 6 Divergences of several terms in the non-simplified version of (3.52), which cancel analytically, if we would expand them around the horizon, but not numerically due to finite machine precision, make it also impossible to choose k = 1. where both the minimal and the maximal value for T γ are taken in the case l = 0.14, the maximal l-value of our analysis. Finally let us consider the function where T γ max/min (l) is the maximal/minimal value for T γ we obtained for a certain l. The results are displayed in figure (2).
In figure (3) we display the results for the correction factor to the temperature obtained by calculations on intervals [0.1, k], we extrapolated the resulting coupling corrections to the metric to u = 1.

Approximating higher derivative corrections to tensor QNMs without magnetic background field
Let us now turn to fluctuations of the metric of a coupling corrected AdS-Schwartzschild black hole. Quasinormal modes can be thought of as tiny perturbations of the geometry, which can be separated according to their transformation behaviour, respectively with the help of symmetry arguments. They are dual to quasiparticles on the field theory side and encode the response of the system to excitations around the equilibrium. We will consider tensor, or spin-2-fluctuations h x,y in the x, y-plane with momentum in z direction. In this section we are going to approximate higher derivative corrections to these tensor QNMs without considering background magnetic fields. The coupling corrections to spin-2-QNMs in this setup were first computed in [2]. Our aim is to reproduce these results by applying a technique, which can be extended to derive coupling corrections to tensor QNMs of the coupling corrected magnetic black brane geometry, which we now know on an interval [l, k] ⊂ [0, 1]. We consider the linearized differential equations obtained by varying the higher derivative corrected action with respect to fluctuations h xy dxdy of the background geometry. These EoM were first derived in [13] and are given in the Appendix (5.1). The characteristic exponents of the differential equation (5.1) are given by ± iω 2 , such that where φ(u) is regular at the horizon and the exponent of (1 − u) − iω 2 was chosen to correspond to infalling wave solutions. Hereω is defined asω = ω 2πT to be consistent with the convention in [2]. In the case of b = 5 4 we will use the conventionω = ω πT ,ω = ω r h to be consistent with [4]. Considering the grid (3.33) again, we define the discrete differentiation matrix A(M ) as where c j is the j-th Chebyshev cardinal function corresponding to the j-th grid point u j . An alternative and numerically more convenient definition of A(M ) is given in [17]. Expanding with v = (a i ) i∈{0,...,M } and This allows us to write (3.61) as a generalized Eigenvalue problem The idea is to solve (3.64) forω exactly in γ with .  [2]. We plotted the first order coefficients of the γ-expansion of the first QNM frequency.
Applying this method with 25 M corrupts the results noticeably. For 10 < M < 20 we obtain good agreement with the already known results. Since the aim is to give a numerical approximation to the higher derivative corrections of tensor QNMs in the presence of a strong magnetic background field, that backreacts on the coupling corrected geometry, we take this result as motivation to apply this technique in the case b = 5 4 .

Approximating higher derivative corrections to the first tensor QNM in the presence of a strong magnetic background field
The way we have chosen our background field together with considering fluctuations ensures that the linearized differential equations for h x,y decouple from those of other fluctuations.
Our aim is to determine γ-corrections to the results in [4]. As before the calculation is done for the case q = 0. We already have found the metric, respectively the functionsŨ , U, V, W, L up to order O(γ) for the parameter b = 5 4 in the previous sections. The following metric ansatz describes tensor fluctuations of this geometry.
dy dy 3 sin(y 1 ) 2 + dy 4 cos(y 1 ) 2 sin(y 2 ) 2 + dy 5 cos(y 1 ) 2 cos(y 2 ) 2 L(u) 2 dy 2 1 + cos(y 1 ) 2 dy 2 2 + sin(y 1 ) 2 dy 2 3 + cos(y 1 ) 2 sin(y 2 ) 2 dy 4 + cos(y 1 ) 2 cos(y 2 ) 2 dy 2 5 + h x,y (u, t)dxdy. (3.67) Our strategy is very similar to the one of the previous chapters. We choose the same grids as before and evaluate the functions  . Together with the Fourier transformed version we can write (3.72) A straightforward calculation shows that the rest of the action can be written as 8 Here and in the following we will use the conventionω = ω πT andω = ω r h . Since we have to consider solutions that are infalling at the horizon we set again  The λ → ∞ limit coincides with the findings in [4]. The correction γ(−1.6 + 2.7i)10 4 to the lowest QNM in the case of very strong magnetic background field is, similar to the higher derivative correction to the temperature, one order of magnitude larger than in the case b = 0. This is not surprising, but it raises the question, whether it makes sense, to evaluate this coupling corrected first QNM at values for the 't Hooft coupling that would correspond to a more realistic QCD limit λ ≈ 11, which is obtained by naively choosing behaves similarly to the analogous quantity in the case b = 0.

Resumming finite λ corrections to the first tensor QNM in a strong magnetic background field
Finally we are going to consider resummed coupling corrections to the first tensor QNM.
Computing all higher derivative corrections of order α 4 to type IIb SUGRA or even higher orders dramatically exceeds current computational resources. However, there is a subset of higher derivative corrections in all orders α n to the QNM spectrum or to any other coupling corrected quantity computed within the AdS/CFT duality, that are already easily accessible, namely those that follow from the first order correction to the EoM of the corresponding field, in this case h xy . Resumming these higher order corrections analogously to [7] will allow us to decrease λ to almost arbitrarily small values without witnessing non-physical behaviour like an positive imaginary part of QNMs. Also the size of the resummed corrections is small compared to the λ → ∞ spectrum for a wide range of λ values. It should be added that this obviously covers only one of many possible resummation schemes [18] and that these partial resummations should be enjoyed with a grain of salt, as already pointed out in [7]. Their reliability at large γ is uncertain and they should not be understood as exact predictions but rather as rough estimates, which, if taken seriously, should be tested with other equivalent schemes. We postpone this additional analysis of this section to future work. Nonetheless the resummation we are going to present exhibits interesting features that we are going to discuss in the following.
We resum by truncating the EoM for h xy deduced in the previous section after the first order in γ and compute exactly in γ henceforth. In entire analogy to the calculation there, we apply spectral methods and write the task of finding the QNM spectrum as a generalized Eigenvalue problem, only that we are now interested in the resulting λ-curves of coupling correction resummed QNMs in the complex plane instead of their slope at λ → ∞. We display the results in the figures (7,8,9). The quantityω there is defined asω = ω πT . We find that for small values of λ both the imaginary and the real part of the first tensor QNM with q = 0 and b = 5 4 converge to a fixed value. In consistency with the λ → ∞ results these values are smaller than in the case b = 0. For b = 0 the imaginary part of the QNM converges to 0 for small λ (see figure (9)), whereas for b = 5 4 it converges to −2.5 as seen in figure (7). This is expected to happen, since without a background field and with very small 't Hooft coupling nothing drives the equilibration of the QGP and the thermalization time, that can be estimated from the negative inverse of the imaginary part of the lowest QNM, diverges.
The electromagnetic coupling doesn't approach zero for small values of λ (see e.g. (2.8) of [15] and corresponding footnote). Thus, in the case of a strong background field the QGP still equilibrates even if λ is sent to small values 11 , which is reflected by the comparison between 11 Any evaluation at λ → 0 is neither feasible nor meaningful in this context. When we call λ small, we mean λ 10, λ > 0. and q = 0 averaged over different grid sizes and interval sizes. The maximal deviation from those suggest a negligible error for large λ, an error of ≈ 1% for the imaginary part and ≈ 10% for the real part for λ → 11. Interestingly the average values as well as the curves for large M and large interval sizes [l, k] converge for γ8(11) 3/2 ζ(3) → 1, or λ → 11 to 2b(1 − i). The constant shape of the curve at small λ suggests that this is also the limit towards which the mode converges for λ 10.
the results displayed on the right hand side of figure (7) and figure (8). Out of caution it should be stressed that we treated only one possible channel. Therefore and because of the uncertain validity of partial resummations at small λ our results suggest and don't prove this statement.

Discussion
In this work we provided a proof of the prescription found in [5], regarding the higher derivative corrected five form in the presence of gauge fields, for the special case of a magnetic background field F = bdx ∧ dy. Using the higher derivative corrections to the type IIb SUGRA action [1] we computed the finite 't Hooft coupling corrected black brane metric, in which the strong background field back-reacts to the geometry. In this setting we found the 't Hooft coupling correction to the temperature (3.54) and computed the α 3 correction to the first tensor QNM (3.76). These correction terms turned out to be one order of magnitude larger than without a magnetic background field. The resummation of higher order corrections to this QNM frequency revealed an interesting pattern that reflects the intuitive expectation. For a vanishing background field and a vanishing 't Hooft coupling the imaginary part of the lowest (tensor) QNM frequency approaches 0, this suggests that the equilibration time diverges in this case. For a strong background field of b = 5 4 (which corresponds to B ≈ 34.5T 2 for λ → ∞) the imaginary part of the lowest QNMω(λ) converges to −2.5 for λ → 11, which is the value for the 't Hooft coupling that naively corresponds to the QCD limit. The (coupling correction resummed) QNM frequency itself approaches 2b(1 − i) for λ → 11. The form of the curve (7) suggests that this is also the limit for λ 10, indicating that the equilibration time of a QGP in a magnetic background field stays finite (and is of the same order of magnitude as in the λ → ∞ limit) even if the 't Hooft coupling becomes extremely small.
It should be added that there are many different ways to resum higher order corrections and that these resummations also should be taken with a grain of salt, when applied to compute quantities at large γ. They should be tested with other resummation schemes, otherwise the resummed results for small values of λ have to be understood as rough qualitative estimates at best.