Gluon polarization tensor in a magnetized medium: Analytic approach to the sum over Landau levels

Alejandro Ayala, 2 Jorge David Castaño-Yepes, ∗ M. Loewe, 2, 4 and Enrique Muñoz 5 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, CdMx 04510, Mexico. Centre for Theoretical and Mathematical Physics, and Department of Physics, University of Cape Town, Rondebosch 7700, South Africa. Instituto de F́ısica, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile. Centro Cient́ıfico-Tecnológico de Valparáıso CCTVAL, Universidad Técnica Federico Santa Maŕıa, Casilla 110-V, Valaparáıso, Chile Research Center for Nanotechnology and Advanced Materials CIEN-UC, Pontificia Universidad Católica de Chile, Santiago, Chile.


I. INTRODUCTION
The production of hot and dense strongly interacting matter in heavy-ion reactions at high energies, constitutes a driving force for the formulation of novel approaches to study QCD subject to extreme conditions. For semi-central collisions, these conditions include the presence of strong, albeit short-lived, magnetic fields. Many theoretical efforts concentrate on describing these conditions considering that the temperature is the largest of the energy scales [1][2][3][4]. However, it has also been realized that the imprints of these strong fields [5,6], if any, should be searched for studying probes produced during the very early stages of the collision, where the system is not yet equilibrated and the largest of the energy scales is instead the magnetic field itself. Possible imprints include an enhanced prompt photon production and/or the chiral magnetic effect [7][8][9][10][11][12].
The early stages of a heavy-ion reaction are also characterized by the presence of a large number of low momentum gluons which are thought to give rise to the saturation phenomenon described by the Glasma [13]. When a magnetic field is present, gluon dynamics can also be affected. A deeper understanding of gluon properties within a magnetized medium is crucial to describe the evolution of observables coming from these early stages.
The gluon dispersive properties in a magnetized medium are encoded in the gluon polarization tensor Π µν . In a perturbative approach, deviations from its vac-uum properties come from the coupling of the magnetic field to virtual quarks. The quark propagator can be represented in terms of a sum over Landau levels. When the field is strong, calculations often resort to the approximation where these quarks occupy the lowest Landau level (LLL), which simplifies considerably the treatment [14][15][16]. Nevertheless, when the field is not as intense, it is important to perform a sum over Landau levels to capture effects that may be missing from expressions restricted to the LLL, in particular, the emergence of tensor polarization structures other than the parallel one that make up the full polarization tensor. This kind of calculations have been performed at one-loop level for the photon polarization tensor [17] in the context of the vacuum birefringence in strong magnetic fields, where the authors resort to a numerical treatment for the infinite sum over Landau levels. However, in order to gain a deeper insight, an analytical approach for the infinite sum over Landau levels is desirable. In this work, we undertake such task and present an analytic method to perform the sum over all Landau levels for the coefficients of the tensor structure that make up the gluon polarization tensor in the presence of a magnetic field of arbitrary intensity. The vacuum contribution is obtained in the limit when B → 0. We show that by this procedure one obtains the usual fermion contribution to the vacuum polarization tensor, together with a second term that is shown to vanish, given the properties of its coefficient under scaling transformations. Applying the same argument to the full, magnetic field-dependent polarization tensor, it is possible to isolate the physical tensor structures and their coefficients, thus getting rid of spurious terms. We then proceed to carefully subtract the vacuum pieces to remove ultraviolet divergences. The procedure ensures that the remaining, magnetic field dependent contributions are finite. In order to test the validity of the expressions thus obtained, we study the weak and strong magnetic field limits. The work is organized as follows: In Sec. II, we write the one-loop expression for the gluon polarization tensor in the presence of a constant external magnetic field. We chose the tensor basis to express the polarization tensor and outline the calculation to carry out the product of fermion propagators and the corresponding sums over Landau levels. We show that after the sum is made, there appear two spurious, non-transverse terms. These are shown to vanish, as in the vacuum case, from the properties of their coefficients under scaling transformations. In Sec. III we study the strong and in Sec. IV the weak field limits and show that the obtained expressions coincide with well known results. We summarize and discuss our results in Sec. V and leave for the appendices the calculation details.

II. GLUON POLARIZATION TENSOR
We start from the one-loop contribution to the gluon polarization tensor, which is depicted in Fig. 1 and is given explicitly by where C.C. refers to the charge conjugate contribution, that is, the contribution where the flow of charge within the loop is in the opposite direction and g is the strong coupling. S(k) is the quark propagator and t a,b are the generators of the color group in the fundamental representation. The fermion propagator in the presence of a magnetic field B = Bẑ can be written in terms of a sum over Landau levels as [18,19]: where m f and q f are the quark mass and electric charge, respectively, and In Eq. (3), L α n (x) are the generalized Laguerre polynomials, with the index n labeling the n-th Landau level, and Also, we follow the convention whereby the square of the four-momentum p µ , expressed in terms of the square of its parallel and perpendicular (with respect to the magnetic field direction) components, is given by Computing Eqs. (1) and (2), after performing the sum over all Landau levels, the gluon polarization tensor can be written in terms of four tensor structures, given by where on the right-hand side, we have omitted a factor δ ab coming from using the relation Tr(t a t b ) = δ ab /2, and correspondingly, for notation simplicity, removed the color indices on the left-hand side. Here (x 1 , x 2 ) ∈ (0, ∞) are Schwinger parameters, with d 2 x = dx 1 dx 2 and For calculation details, see Appendix A.

A. Tensor Basis
The gluon polarization tensor should be represented by a symmetric tensor under the exchange of its Lorentz indices. It can be constructed out of the external products of the independent vectors describing the propagation of a gluon with momentum p µ in the presence of a magnetic field whose direction is specified by a fourvector b µ , in addition to the metric tensor g µν . Without loss of generality, we can choose a reference frame where the magnetic field points along theẑ axis. Due to the presence of this Lorentz invariance-breaking vector, it is convenient to split the metric itself into parallel and perpendicular (with respect to the magnetic field direction) components, that is where and We thus see that the most general symmetric tensor can be constructed out of combinations of the four possible independent tensors However, notice that in QCD, Π µν must satisfy the generalized Ward-Takahashi identity namely, the transversality condition Therefore, since Eq. (12), implies a relation between the coefficients of the tensors to express Π µν , only three transverse tensors turn out to be independent. A convenient basis to express the polarization tensor is such that the independent tensors are chosen each to be transverse, in such a way that Eq. (12) be satisfied already as This choice has the advantage that the basis can be used to express the polarization tensor either in QCD or in QED. In the present work, we chose the orthonormal basis Therefore, we can use this basis to express Eqs. (7) (see Appendix B) as where and Notice that, contrary to expectations, Eq. (17) contains also terms proportional to the tensors g µν and g µν ⊥ . In order to show that Π µν is made out only of combinations of transverse tensors, we need to prove that the coefficients A 1 and A 2 vanish. This is shown in Appendix C. For the time being, let us only emphasize that, had we simply projected out Eq. (6) onto the basis given by Eqs. (18)-(20), the spurious terms would have induced non-physical contributions that, given their complexity, could obscure the numerical evaluation of the physical coefficients [17,20,22].

B. Vacuum Polarization Tensor
As one can expect, the gluon polarization tensor contains divergences which come from the vacuum contribution. In order to proceed to isolate these contributions we notice that two possible vacua can be defined: • A vacuum where p µ = 0 and B = 0, corresponding to a situation where particles and magnetic field appear simultaneously.
• A vacuum with B = 0 and p µ = 0, representing a situation where the external field is turned on with pre-existing gluons with four-momentum p µ .
The first choice is ambiguous, given that the energy scales associated to the magnetic field and the transverse momentum appear within the combination p 2 ⊥ / |q f B|, and thus, p 2 ⊥ and B cannot be set to zero simultaneously. Therefore, we chose to extract the vacuum working in the situation described by the second case. The vacuum contribution is thus given by Notice that Eq. (23) contains a term that does not simply vanish under contraction with p µ , namely, the term proportional to g µν . In order to show that the coefficient of this term vanishes, we follow the argument in Ref. [21]. We introduce the scaling transformation for the Schwinger parameters in such a way that x i → λz i , where λ is a real parameter. Under this transformation, the coefficient of the term proportional to g µν becomes It is easy to show that the integral I can also be written as If we now scale back z i → x i /λ we observe that the integral becomes λ-independent and thus its derivative with respect to λ vanishes, namely Therefore, the vacuum polarization tensor becomes Notice that Eq. (27) can also be written as where P µν 0 , P µν and P µν ⊥ are given by Eqs. (14)- (16). A similar argument is valid for a non-vanishing magnetic field. This means that the coefficients A 1 and A 2 , in Eqs. (21) and (22), respectively, do not contribute to Π µν , since they vanish. The systematic evaluation of these terms is shown in Appendix C. Thus, the full polarization tensor with the desired physical properties is given by where Π , Π ⊥ and Π 0 are given by Eqs. (18), (19) and (20), respectively. To cancel the vacuum piece, we subtract from Eq. (29) the contribution from Eq. (27). Therefore, the finite, magnetic field-dependent part of the gluon polarization tensor is explicitly given by where and we have used the symmetry of the integral under the exchange x 1 ↔ x 2 . In order to check the validity of the above expression, we proceed to study its limits in the strong and weak magnetic field cases.

III. STRONG FIELD LIMIT
In order to study the strong field limit, let us first introduce the dimensionless variables and the new variables s and y related to y 1 and y 2 by so that Eq. (29) becomes Note that in the strong field limit  For the kinematical region such that y(1 − y)ρ 2 < 1, the integration over s can be performed, yielding which coincides with the result obtained in Refs. [14][15][16] where the gluon polarization tensor is computed by considering only the contribution from the LLL. Figure 2 shows the real and imaginary parts of I(ρ 2 ). Notice the discontinuity at the threshold value ρ = 4 or equivalently at p 2 = 4m 2 f . Notice also that Eq. (30) implies the existence of an infinite sequence of momentum thresholds when the external gluon momentum becomes resonant with twice the quark/antiquark magnetic mass, whose square is defined as m 2 (B)f = m 2 f + 2n |q f B|. The threshold corresponds to the value of the longitudinal momentum squared for the creation of a quark-antiquark pair, each particle having a magnetic mass corresponding to the given Landau level.
These thresholds can be obtained from our calculation by concentrating on the conditions where the hyperbolic functions become divergent. For these purposes let us examine the term proportional to coth(Bs) in Eq. (33): Using that we can write so that, the dominant term in Eq. (37) is given by where I(x) is defined in Eq. (36) and In this way, the resonant behavior of the thresholds is explicit: the gluon polarization tensor has divergences when its momentum reaches the value p 2 = 4m 2 (Bn)f , where n labels each of the Landau levels. In other words, the creation of quark-antiquark pairs is allowed when the gluon momentum is large enough to generate not only the inertial mass of the pair but rather the magnetic mass, induced by the magnetized medium. Figure 3 shows several thresholds of the function J(ρ 2 ) in a broad range of ρ 2 for a maximum value of n, n max = 100. The same argument is valid for all terms in Eq. (33) given that its dominant contribution is given by a power of the series in Eq. (39).

IV. WEAK FIELD LIMIT
Let us study the case where the field satisfies the hierarchy of energy scales |eB| < m 2 f . We call this the weak field limit. For this purpose, we can perform a power series of Eq. (33) around B = 0 to obtain where the vacuum contribution of Eq. (28) can be identified as Subtracting this contribution, we are left with the Bdependent part. The integrations over s and y can be performed analytically, so that The coefficientsΠ ,Π ⊥ andΠ 0 consist of real and imaginary parts. The imaginary parts can be obtained from the corresponding real parts from the Kramers-Kronig relations. With the notation we have where P is the Principal Value. Examples of these coefficients as functions of ρ 2 , for various values of ρ 2 ⊥ are shown in Fig. 4.

V. RESULTS, DISCUSSION AND CONCLUSIONS
The results of this work can be used to study birefringence of the gluon polarization in a magnetized medium. Recall that birefringence is the optical property exhibited by a material whose refractive index depends on the polarization and propagation direction of light. Recall that in solid-state, crystals with non-cubic lattice symmetry show birefringence, with calcite being a typical and historical example. The simplest type of birefringence corresponds to the so-called uniaxial type, where a single direction governs the optical anysotropy while all the other directions orthogonal to it are optically equivalent. Thus, rotations of the crystal with respect to this axis leave the optical response invariant. On the other hand, a material that is otherwise optically isotropic, can manifest birefringence under the presence of external agents, such as strain and, more importantly, an external magnetic field. This last case is often called Faraday effect [23]. An analogous situation is studied in the context of high-energy physics, particularly in QED under the presence of static magnetic or electric fields, where the index of refraction depends on the photon polarization state.
Despite the absence of an underlying discrete symmetry as in crystalline materials, the presence of these static fields is often sufficient to induce optical birefringence under certain conditions, which in this context is called vacuum birefringence. This effect has been extensively studied theoretically [17,24]. Moreover, recent experimental evidence for this phenomenon has been provided from astronomical observations of neutron stars, where intense magnetic fields are present [25].
In QED, the microscopic mechanism behind the effect are the vacuum fluctuations due to the spontaneous emergence of virtual electron-positron pairs that act as dipoles, in analogy with dielectric crystals. In the absence of external fields, Lorentz invariance ensures an isotropic optical response. However, when a static electric or magnetic field is present, Lorentz invariance is broken and an anisotropic optical response is triggered. In particular, when a magnetic field is responsible for the effect, the virtual fermion pair exists in general in a combination of Landau levels.
In this work, we show that vacuum birefringence arises also for gluons in QCD, where the virtual fermionantifermion pairs correspond to quark-antiquark pairs that play the same role as electron-positron pairs in QED. Just as in QED, the QCD version of the phenomenon necessarily implies the existence of an infinite sequence of momentum thresholds, that correspond to the condition where the external gluon momentum is resonant with the magnetic mass of a pair occupying a given Landau level, which are successively occupied by the pair of virtual quarks participating in the process.
We have presented a method to compute the oneloop magnetic correction to the gluon polarization tensor starting from the Landau-level representation of the quark propagator in the presence of an external magnetic field. We have shown that the general expression contains the vacuum contribution that can be isolated from the zero-field limit for finite gluon momentum. An important observation is that, the general tensor structure for the gluon polarization contains two spurious terms that do not satisfy the transversality properties. We have shown that, in analogy with the case in vacuum, these terms have vanishing coefficients and thus do not contribute to the polarization tensor, as expected. In order to check the validity of the expressions thus found, we have studied the strong and weak field limits and shown that we reproduce well established results. The results of this work can be used to study the conditions for gluons to equilibrate with a magnetized medium, for example during the early stages of a relativistic heavy-ion collision. This is work in progress and it will be reported elsewhere. Let us begin from the general expression of the gluon polarization tensor of Eq. (1): The trace in the above expression involves two fermion propagator factors, each given by Eqs.
(2)-(3). This product produces nine terms, that are explicitly given by In order to perform the sum over Landau levels, we write the denominators introducing Schwinger parameters such that We start with the expression given by Eq. (A2) and By using the generating function of the Laguerre Polynomials, given by and therefore Putting all together t µν 1 = −8g 2 with and J µν 1 = d 2 k (2π) 2 e α(k )x1 e β(k )x2 k · p + m 2 f − k 2 g µν + 2k µ k ν − k µ p ν − k ν p µ .
The transverse integral I 1 is performed by making the shift which turns the integral into a simple Gaussian form. It is straightforward to prove that For the parallel integral J µν 1 , the appropriate shift is and by performing a rotation to Euclidean space, the integral becomes of a Gaussian form in the variable l 2 E = l 2 4 + l 2 3 , and thus Collecting terms × exp − tanh(|q f B| x 1 ) tanh(|q f B| x 2 ) tanh(|q f B| x 1 ) + tanh(|q f B| x 2 )