Emission of linearly polarized photons in a strongly coupled magnetized plasma from the gauge/gravity correspondence
Daniel ?vila, Tonantzin Monroy, Francisco Nettel, Leonardo Patiño
aa r X i v : . [ h e p - t h ] J a n Emission of linearly polarized photons in a QGP from the gauge/gravitycorrespondence
Daniel ´Avila, ∗ Tonantzin Monroy, ‡ Francisco Nettel, § and Leonardo Pati˜no k Departamento de F´ısica, Fac. de CienciasUniversidad Nacional Aut´onoma de M´exico,A. P. 50-542, M´exico D.F. 04510, M´exico (Dated: January 25, 2021)In this letter we use holographic methods to show that the photons emitted by a strongly cou-pled quark-gluon plasma are linearly polarized when an intense external magnetic field is present.This happens regardless of the photon four-momentum, except for the degenerate case in whichit propagates parallel to the magnetic field. The holographic model we use is a five-dimensionalconsistent truncation of ten-dimensional type IIB supergravity, which features the backreaction of aconstant magnetic field and a scalar field, dual to a single-trace scalar operator of scaling dimension2. In terms of the geometry of a collision experiment, our result can be stated by saying that anyphoton produced there has to be in its only polarization state parallel to the reaction plane, whichis perpendicular to the background magnetic field.
Thermal photons produced in the quark-gluon plasma(QGP) are very appealing probes to obtain informationabout the first stages of its evolution, as they are practi-cally not scattered by the plasma [1–3]. It was previouslysuggested that the QGP has global quark spin polariza-tion in non-central heavy-ion collisions [4, 5], and latterit was shown that this in turn leads to the polarization ofthe emitted photons [6], either direct [7] or virtual [8–10].It has also been established that high energy collisionsproduce an intense magnetic field that points perpendic-ularly to the reaction plane [11–15], thus understandingits effects becomes relevant to properly analyze experi-mental observations. In particular, in [16] it was pro-posed that this magnetic field could lead to the quarkspin polarization, and in turn induce a polarization onthe emitted photons.The gauge/gravity correspondence [17] has been ex-tensively used to explore some of the general dynamicalproperties of the QGP produced at high energy p-p orheavy-ion collisions, as it exists in a strongly coupledstate. Although the exact gravitational dual of QCD isnot known, what has been done is to consider theoriessimilar to QCD, such as N = 4 Super Yang-Mills (SYM)at finite temperature, and modify them to bring them asclose to QCD as possible. This modifications have beenextended to include spatial anisotropies [18–20] and, inparticular, the presence of a very intense external mag-netic field [21–24].The emission of photons by the QGP has been ana-lyzed using different configurations in the gauge/gravitycorrespondence. Starting with SYM N = 4 at finite tem-perature in [25], many developments have been consid-ered to improve the modelling of the experimental con-text, such as a non-vanishing quark mass [26] or chem-ical potential [27–29], spatial anisotropies [30], and anexternal magnetic field [31–33]. The contribution thata magnetic field turned on over the D / ¯ D U (1) from the SO (6) symmetry of the compact space and changing itto a gauge symmetry. Thus, from the dual gauge theoryperspective, the magnetic field in this case couples to theconserved current associated with a U (1) subgroup of the SU (4) R − symmetry. The importance of this model re-sides in the fact that it allows the introduction of massiveflavor degrees of freedom by means of the embedding ofD7-branes which naturally wrap a 3-cycle of the compactsector of the ten-dimensional geometry [23, 24]. However,here we will work with the unflavored 5-dimensional trun-cation, as this is enough to show the polarization effectcaused by the magnetic field.To calculate the photon production in the magnetizedQGP at hand, we first establish the description in thegauge theory side. We consider a 4-dimensional N = 4super Yang-Mills theory over Minkowski spacetime, withgauge group SU ( N c ) at large N c and ’tHooft coupling λ = g YM2 N c . In this theory, the matter fields are inthe adjoint representation of the gauge group, thus, theso-called quarks are massless. The produced photons aremodeled by adding a U (1) kinetic term to the SYM actionthat couples to the electromagnetic current associated toa U (1) subgroup of the global SU (4) R -symmetry groupof the theory. Therefore, the action adopts the form of a SU ( N c ) × U (1) gauge theory S = S SU ( N c ) − Z d x (cid:0) F − eA µ J EM µ (cid:1) , (1)where F is the field strength of the U (1) component,comprising the background magnetic field which sourcesthe anisotropy, e stands for the electric charge, and theelectromagnetic current J EM µ is given by J EM µ = ¯Ψ γ µ Ψ + i ∗ ( D µ Φ) − i D µ Φ ∗ ) ∗ Φ , (2)with Ψ and Φ generically representing the fermionic andscalar matter content of the SU ( N c ) gauge theory, re-spectively. There is also an implicit sum over the fla-vor indices, and the operator D µ = D µ − ieA µ actingupon the scalar fields is the covariant derivative for the su ( N c ) × u (1)-connection.The electromagnetic coupling α EM = e / π is smallcompared to ’tHooft coupling λ = g YM2 N c (at large N c ),so, even if the two-point correlation function necessaryto compute photon production has to be calculated non-perturbatively in the SU ( N c ) theory that involves λ , itis enough to determine it to leading order in α EM and ig-nore terms of order O ( α ). In view of the above, onlythe gravitational dual of the SU ( N c ) gauge theory is nec-essary, and there is no need to extend the gauge/gravitycorrespondence to include the full SU ( N c ) × U (1) group.Assuming that the QGP is in thermal equilibriumwhen the production of the thermal photons takes place,the rate of emitted photons with null wave four-vector k µ = ( k , ~k ) and polarization ǫ µs ( ~k ) can be calculated as[25, 26] d Γ s d~k = e (2 π ) | ~k | n B ( k ) ǫ µs ( ~k ) ǫ νs ( ~k ) χ µν ( k ) (cid:12)(cid:12)(cid:12)(cid:12) k =0 , (3)where n B ( k ) is the Bose-Einstein distribution for thephoton energy and the spectral density χ µν ( k ) = − G R µν ( k )] is given in terms of the two-point retardedcorrelation function for the electromagnetic current (2) G R µν ( k ) = − i Z d xe − ik · x Θ( t ) h (cid:2) J EM µ ( x ) , J EM ν (0) (cid:3) i , (4)with the expectation value taken in the thermal equilib-rium state.The spatial polarization four-vectors ǫ µs are orthog-onal to the null wave four-vector with respect to the Minkowski metric, thus satisfying ǫ is k j δ ij = 0, and canalso be chosen to satisfy ǫ i ǫ j δ ij = 0. If we also fix ourcoordinate system such that the background magneticfield is directed along the z -direction, i.e. F = B d x ∧ d y ,there would be a rotational symmetry on the reactionplane ( xy -plane), allowing us to conveniently set the wavefour-vector to lie in the xz -plane, and to denote by ϑ theangle that ~k forms with the background magnetic field.For these choices, the wave and polarization four-vectorstake the form k = k ( ∂ t + sin ϑ∂ x + cos ϑ∂ z ) , (5)and ǫ = ∂ y , ǫ = cos ϑ ∂ x − sin ϑ ∂ z , (6)respectively, and the rate of emitted photons (3) can bedecomposed into d Γ d~k ∝ χ yy , (7)for the polarization state ǫ and d Γ d~k ∝ cos ϑχ xx − ϑ sin ϑχ xz + sin ϑχ zz (8)for ǫ .To set up the gravitational theory dual to thegauge theory we are considering, we start from a five-dimensional gauged supergravity theory obtained fromthe S reduction of ten-dimensional type IIB supergrav-ity and a further consistent truncation to N = 2 super-gravity theory [34]. For the purposes of this letter, it isuseful to express the relevant five-dimensional action interms of an orthonormal frame { e a } , a = 0 , . . . ,
4, forwhich the dependence on the metric is explicitly shownby the Hodge dual operator ⋆ associated to it S grav = 116 πG Z (cid:20) R ab ∧ ⋆ ( e a ∧ e b ) + 4 L X i =1 X − i ⋆ − X I =1
12 d ϕ I ∧ ⋆ d ϕ I − X i =1 X i − F i ∧ ⋆ F i − F ∧ F ∧ A (cid:21) , (9)where G is the five-dimensional gravitational constant, L − is proportional to a negative cosmological constant, R ab is the curvature two-form, ϕ I , I = 1 ,
2, are two scalarfields, F i = d A i , i = 1 , ,
3, are the field intensities of the U (1) × U (1) × U (1) sector (gauged field action) and X i = e − a Ii ϕ I , a Ii = ( a i , a i ) and ϕ I = ( ϕ , ϕ )(10)where we are using Einstein summation convention overthe a, b, . . . and I, J, . . . indices. Notice that we are usingcalligraphic letters for the U (1) fields to distinguish themfrom those of the gauge theory side (cf. equation (1)).The Einstein field equation from this action is (cid:20) R ab ∧ e c ∧ e d + 13! L X i =1 X i − e a ∧ e b ∧ e c ∧ e d (cid:21) ε abcde = τ e , (11)where ε abcde is the Levi-Civita pseudo-tensor and τ e isthe energy-momentum 4-form given by the following ex-pression τ e = X I =1
112 (d ϕ I ) a ε abcde d ϕ I ∧ e b ∧ e c ∧ e d − X I =1 (d ϕ I ) e ⋆ d ϕ I + 12 X i =1 X i − (cid:20) F iab ε abcde F i ∧ e c ∧ e d − F ide ⋆ F i ∧ e d (cid:21) . (12)The field equations for the scalar fields ϕ I are (cid:3) ϕ I + 2 L X i =1 X i − a Ii − X i =1 X i − a Ii F i = 0 , (13)while for the Maxwell fields F i d( X i − ⋆ F i ) + F j ∧ F k = 0 , (14)where i = j = k .As can be noticed, the last expression in the action (9)is independent of the metric degrees of freedom, i.e. atopological term.A further truncation is imposed considering the follow-ing specific choice for the gauge and scalar fields ϕ = 2 ϕ = 2 √ ϕ , A = 0 , A = A = √ A , (15)while a I = (cid:18) √ , √ (cid:19) , a I = (cid:18) √ , −√ (cid:19) , a I = (cid:18) − √ , (cid:19) . (16)Einstein equation reduces in this case to (cid:20) R ab ∧ e c ∧ e d + 13! L (cid:0) X + 2 X − (cid:1) e a ∧ e b ∧ e c ∧ e d (cid:21) ε abcde = τ e , (17) with X = e √ ϕ and the energy-momentum 4-form is τ e = 112 (d ϕ ) a ε abcde d ϕ ∧ e b ∧ e c ∧ e d −
12 (d ϕ ) e ⋆ d ϕ + 12 X − F ab ε abcde F ∧ e c ∧ e d − X − F de ⋆ F ∧ e d . (18)For the scalar field ϕ (cid:3) ϕ + 4 L r (cid:0) X − X − (cid:1) + r X − F = 0 , (19)while the equations (14) yield the sourceless Maxwellequations d( X − ⋆ F ) = 0 , (20)and a topological condition on the Maxwell field intensity F ∧ F = 0 . (21)It will be shown that this last equation entails the linearpolarization effect that the QGP produces on the emittedphotons. Equations (17)-(20) can be obtained from theeffective action for the gravitational configuration S eff = 116 πG Z (cid:20) R ab ∧ ⋆ ( e a ∧ e b ) + 4 L ( X + 2 X − ) ⋆ −
12 d ϕ ∧ ⋆ d ϕ − X − F ∧ ⋆ F (cid:21) , (22)while equation (21) must be imposed as a further con-straint on the system. In the following we will take L = 1without loss of generality.In order to holographically describe the gauge theorydiscussed above, we need to look for solutions of theequations of motion coming from (22) that have a finitetemperature, a magnetic field, and that are asymptoti-cally AdS in the boundary. Such family of solutions wasfound in [22], where the thermodynamics of the unfla-vored plasma was studied extensively. The metric of anymember of the family of solutions can be written as ds = dr U ( r ) − U ( r ) dt + V ( r )( dx + dz )+ W ( r ) dz , (23)while the gauge field is taken to be a constant magneticfield pointing in the z -direction and the scalar field is afunction of r alone F = B dx ∧ dy, ϕ = ϕ ( r ) . (24)The metric asymptotes AdS at the boundary located at r = ∞ , while it features an event horizon at r = r h wherethe function U ( r ) vanishes. Note that the constant mag-netic field automatically satisfices the constraint (21) andthe Maxwell equations for the metric (23). The scalarfield ϕ is dual to a single-trace scalar operator of dimen-sion ∆ = 2, and its presence has very important conse-quences on the thermodynamics of the plasma [22–24].The only known analytical solution to the equationsof motion is for B = 0, while any other member of thefamily of solutions needs to be computed numerically.Although the general procedure is described in detail in[22], for the purpose of this letter the explicit numericalsolutions are unnecessary. The polarization of the emit-ted photons is an analytical result.According to the gauge/gravity correspondence, thecorrelation function (4) can be obtained from a pertu-bative calculation in the gravitational dual theory [35–38]. We need to consider perturbations around the back-ground solutions (23) and (24) g mn = g BG mn + ǫ h mn (25) F = F BG + ǫ d A , ϕ = ϕ BG + ǫ φ, (26)where ǫ is a small auxiliary parameter introduced to keeptrack of the order of the perturbations. The gauge field A in the gravitational side evaluated at the boundary, A b , is dual to the source of the gauge field A , i.e. to theelectromagnetic current J EM in the gauge theory side,as long as we work in the gauge A r = 0.In order to obtain the correlation function (4) we mustcompute the on-shell action (22) and take the secondvariation with respect to A b G R µν ( k ) = δ S b eff δ A bµ δ A bν . (27)To follow this procedure, and given that the boundary isperpendicular to r , we further impose the gauge h mr =0 for the metric perturbations (see [32] for a discussionregarding this gauge choice). As the equations for theperturbations are coupled, in order to properly take thisvariation we need to resort to the methods described in[32, 39]. We must find in the action (22) those termswith second order derivatives with respect to r , integrateby parts to obtain the boundary term, and evaluate thelatter on-shell to second order in ǫ . Schematically weobtain S b eff ∝ Z d x ( O ( AA ′ ) + O ( φφ ′ ) + O ( h ) + O ( hh ′ )) , (28)where the prime denotes differentiation with respect to r ,the limit r → ∞ is meant to be taken and zeroth, first and higher than second order terms in the perturbation fieldsare not written. Following the prescription in [32, 39],we see that the O ( φφ ′ ), O ( h ) and O ( hh ′ ) terms do notcontribute to the second variation with respect to A b .Thus the only relevant part for this calculation is S b eff = − πG Z d x p − γ BG U ( r ) / γ BG µν X − A µ A ′ ν , (29)where γ BG µν is the background boundary metric givenby ds bdry = − U ( r ) dt + V ( r )( dx + dy ) + W ( r ) dz . (30)The only other terms that could have contributed tothe second variation are of the form O ( h ′ h ′ ), O ( φ ′ φ ′ ), O ( φ ′ h ′ ), O ( A h ′ ), O ( A φ ′ ), O ( A ′ h ′ ) and O ( A ′ φ ′ ), butnone of these appear in the action.To compute the differential photon production (3) thenext step is to solve the equations of motion obtainedfrom (17)-(20) for the perturbations to linear order in ǫ .Given that these equations are highly non-linear, we needto resort to numerical methods to find the solutions and adetailed study of the photon production as a function ofthe photon momentum will be presented in a forthcomingpaper. To show that the precise linear polarization of theemitted photons can be described analytically, let us turnour attention to the constraint that (21) imposes on theirpropagation. From (21) and (26) we observe that up tolinear order in ǫ F ∧ F = F BG ∧ F BG + 2 F BG ∧ d A . (31)Given the orientation of the background magnetic field(23), the first term on the right hand side of (31) is zero,hence the constraint is relevant only at the order of theperturbation A . Expressing the gauge field in the samecoordinate basis of (23) and imposing the gauge A r = 0we can write A = A µ ( x ν , r ) d x µ , (32)which components can be decomposed as A µ ( x ν , r ) = Z d k (2 π ) e − ik · x ˜ A µ ( k, r ) . (33)Thus, for an arbitrary direction of propagation (5), theconstraint (21), up to the relevant order in A , reduces to B ( i k (cid:0) ˜ A z ( k, r ) + A t ( k, r ) cos ϑ (cid:1) d t ∧ d x ∧ d y ∧ d z − ˜ A ′ z ( k, r ) d x ∧ d y ∧ d z ∧ d r − ˜ A ′ t ( k, r ) d t ∧ d x ∧ d y ∧ d r ) = 0 . (34)Therefore, for a non-vanishing magnetic field, the com- ponents of A must satisfy the following relations˜ A z ( k, r ) − ˜ A t ( k, r ) cos ϑ = 0 , (35)˜ A ′ z ( k, r ) = 0 , ˜ A ′ t ( k, r ) = 0 . (36)In particular, (36) directly implies that both A ′ t and A ′ z vanish, and in turn, given the manner in which the dif-ferential equations couple the components of A for prop-agation in any direction other than z , this also imposesthat A ′ x = 0, reducing (29) to S b eff = − πG Z d xU ( r ) p W ( r ) X − A y A ′ y , (37)where (30) has been substituted.The lack of other terms in (37) readily shows that itssubstitution in (27) leads to δ S b eff δ A bz δ A bz = δ S b eff δ A bz δ A bx = δ S b eff δ A bx δ A bx = 0 , (38)and hence to the vanishing of the spectral densities χ zz , χ zx , and χ xx , that are the sole contributors to d Γ d~k in (8).We have just proved that once the back reaction to theinterplay of the background magnetic field and the elec-tromagnetic perturbations is accounted for, our modelpredicts that regardless of its energy, any photon pro-duced by the plasma at an angle ϑ different from zero,will be precisely polarized along the ǫ direction.The result here presented can be stated in terms ofthe geometry of a collision experiment by claiming thata photon produced in such an event propagating in anydirection, except that of the background magnetic field,has to be emitted in its only polarization state whichis parallel to the reaction plane, that is, the radiationis linearly polarized. This suggests that direct photonsfrom the QGP might share this property as a commongeneral feature obtained by means of the gauge/gravityduality.For photons propagating parallel to the backgroundmagnetic field no restrictions are imposed; both polariza-tion states, which are directed along the reaction plane,are present and contribute to the rate of photon pro-duction through the non-vanishing components d Γ d~k and d Γ d~k . This can be seen from the field equations (17)-(20)at first order in the perturbations, where setting ϑ = 0fixes k x = 0 through (5), and in turn implies that the re-striction that otherwise set A ′ x = 0 is not longer presentfor this direction of propagation. In this case the term A x A ′ x is present in (37) along with A y A ′ y symmetrically.Previous holographic studies [16, 32] showed that thepresence of the external magnetic field causes an increasein the production of photons with one polarization stateover the other. The strict linear polarization of the pho-tons in our model is due to the Chern-Simons term in theaction (9) in the presence of a non-vanishing magneticfield, as this term is the one that gives the constriction F ∧F = 0. Such term was not taken into consideration in[32], and while there is a Chern-Simons term in the actionin [16], their construction is such that the effect that the electromagnetic perturbations have on the embedding ofthe branes is not taken into account.Something interesting to note is that, strictly speaking,we showed that the current density in the gauge theory(2) is polarized through (4) in some way , as this is theoperator that is dual to the gravitational gauge field A .It is because the optical theorem that this in turn inducesa polarization on the emited photons. This is reminiscentof what was proposed in [4–6], although this is not a spinpolarization effect.It would be interesting to know if considering massivequarks modifies this effect in any form. As we mentionedbefore, the holographic model we use allows a simple de-scription of the embedding of flavor D7-branes [23, 24]. ACKNOWLEDGMENTS
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