Finite temperature quark gluon vertex with a magnetic field in the hard thermal loop approximation

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(1)PHYSICAL REVIEW D 91, 016007 (2015). Finite temperature quark-gluon vertex with a magnetic field in the hard thermal loop approximation Alejandro Ayala,1,5,* J. J. Cobos-Martínez,2 M. Loewe,3,5 María Elena Tejeda-Yeomans,4,1 and R. Zamora3,1 1. Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, México Distrito Federal 04510, Mexico 2 Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Ciudad Universitaria, Morelia, Michoacán 58040, Mexico 3 Instituto de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile 4 Departamento de Física, Universidad de Sonora, Boulevard Luis Encinas y Rosales, Colonia Centro, Hermosillo, Sonora 83000, Mexico 5 Centre for Theoretical and Mathematical Physics, and Department of Physics, University of Cape Town, Rondebosch 7700, South Africa (Received 24 October 2014; published 22 January 2015) We compute the thermomagnetic correction to the quark-gluon vertex in the presence of a weak magnetic field within the hard thermal loop approximation. The vertex satisfies a QED-like Ward identity with the quark self-energy. The only vertex components that get modified are the longitudinal ones. The calculation provides a first principles result for the quark anomalous magnetic moment at high temperature in a weak magnetic field. We extract the effective thermomagnetic quark-gluon coupling and show that this decreases as a function of the field strength. The result supports the idea that the properties of the effective quark-gluon coupling in the presence of a magnetic field are an important ingredient for understanding the inverse magnetic catalysis phenomenon. DOI: 10.1103/PhysRevD.91.016007. PACS numbers: 11.10.Wx, 25.75.Nq, 98.62.En, 12.38.Cy. I. INTRODUCTION The properties of strongly interacting matter under the influence of magnetic fields have been the subject of intense research over the last years. Lattice QCD results indicate that the transition temperature with 2 þ 1 quark flavors decreases with increasing magnetic field strength [1–3]. This behavior has been dubbed inverse magnetic catalysis. Possible explanations include a fermion paramagnetic contribution to the pressure with a sufficiently large magnetization, driven by confinement [4], a backreaction of the Polyakov loop [5,6], magnetic inhibition due to neutral meson fluctuations in a strong magnetic field [7], high baryon density effects [8], a decreasing magnetic field and temperature dependent coupling inspired by the QCD running of the coupling with energy, in the NambuJona-Lasinio model [9,10], the importance of gluons during the phase transition [11] and quark antiscreening from the effect of the anomalous magnetic moment of quarks in strong [12] and weak [13] magnetic fields. The phenomenon is not observed when only mean field approaches to describe the thermal environment are used [14–19], or when calculations beyond mean field do not include magnetic effects on the coupling constants [20]. In two recent studies [21,22] we have shown that the decrease of the coupling constant with increasing field *. Corresponding author. ayala@nucleares.unam.mx. 1550-7998=2015=91(1)=016007(9). strength can be obtained within a perturbative calculation in the Abelian Higgs model and the linear sigma model, where charged fields are subject to the effect of a constant magnetic field. This behavior introduces a dependence of the boson masses on the magnetic field and a decrease of the critical temperature for chiral symmetry breaking/ restoration. In order to establish whether a similar behavior takes place in QCD, the first step is the knowledge of the finite temperature and magnetic field dependence of the coupling constant. The purely magnetic field dependence of this coupling has been looked at in Ref. [12] for the case of a strong magnetic field. The authors find an anisotropic behavior and a decrease of the parallel part of the coupling with the field strength which in turn produces antiscreening and thus a decrease of the critical temperature for chiral symmetry restoration. However the strong field case is limited to extreme scenarios. For instance, in peripheral heavy ion collisions, though the initial intensity of the magnetic fields can be quite strong both at RHIC and the LHC, the field strength is a fast decreasing function of time [23–25]. By the time the gluons and quarks thermalize, the temperature becomes the largest of the energy scales. It seems that for most of the evolution of such systems a calculation that considers the strength of the magnetic field to be smaller than the square of the temperature is more appropriate. The field theoretical treatment of systems involving massless bosons, such as gluons, at finite temperature in the presence of magnetic fields is plagued with subtleties.. 016007-1. © 2015 American Physical Society.

(2) ALEJANDRO AYALA et al.. PHYSICAL REVIEW D 91, 016007 (2015). It is well known that unless a careful treatment is implemented, infrared divergences, associated with the effective dimensional reduction of the momentum integrals, appear. The divergence comes when accounting for the separation of the energy levels into transverse and longitudinal directions (with respect to the magnetic field direction). The former are given in terms of discrete Landau levels. Thus, the longitudinal mode alone no longer can tame the divergence of the Bose-Einstein distribution. This misbehavior can be overcome by a proper treatment of the physics involved when magnetic fields are introduced. For instance, it has recently been shown that it is possible to find the appropriate condensation conditions by accounting for the plasma screening effects [26]. Here we show that a simple prescription where the fermion mass acts as the infrared regulator allows one to obtain the leading behavior of the QCD coupling for weak magnetic fields at high temperature, that is, in the hard thermal loop (HTL) approximation. As a first step, we compute the thermomagnetic corrections to the quark-gluon vertex in the weak field approximation. We use this calculation to compute the thermomagnetic dependence of the QCD coupling. To include the magnetic field effects we use Schwinger’s proper time method. We should point out that the weak field approximation means that one considers the field strength to be smaller than the square of the temperature but does not imply a hierarchy with respect to other scales in the problem such as the fermion mass. The work is organized as follows: In Sec. II we recall how the magnetic field effects are included when charged virtual fermions are present. We set up the calculation of the Feynman diagrams involved. In Sec. III we work in the weak field approximation and find the thermomagnetic behavior of the quark-gluon vertex. We show that the vertex thus found satisfies a QED-like Ward identity with the fermion self-energy computed within the same approximation. In Sec. IV we use this result to study the thermomagnetic dependence of the coupling and show that this is a decreasing function of the magnetic field. We finally summarize and conclude in Sec. V. II. CHARGED FERMION PROPAGATOR IN A MEDIUM The presence of a constant magnetic field breaks Lorentz invariance and leads to a charged fermion propagator which is a function of the separate transverse and longitudinal momentum components (with respect to the field direction). Considering the case of a magnetic field pointing ~ ¼ Bẑ, the vector potential, along the ẑ direction, namely B in the so called symmetric gauge, is Aμ ðxÞ ¼. B ð0; −x2 ; x1 ; 0Þ: 2. ð1Þ. The fermion propagator in coordinate space can no longer be written as a simple Fourier transform of a momentum propagator but instead it is written as [27] Z. d4 p −ip·ðx−x0 Þ e SðkÞ; ð2πÞ4. ð2Þ. 1 dξμ Aμ þ Fμν ðξ − x0 Þν 2. ð3Þ. Sðx; x0 Þ ¼ Φðx; x0 Þ where Z Φðx; x0 Þ ¼ exp iq. x0. x. is called the phase factor and q is the absolute value of the fermion’s charge, in units of the electron charge. SðkÞ is given by Z ∞ 2 2 tanðqBsÞ 2 ds SðkÞ ¼ −i eisðk∥ −k⊥ qBs −m Þ 0 cosðqBsÞ k⊥ × ½cosðqBsÞ þ γ 1 γ 2 sinðqBsÞðm þ k∥ Þ − ; cosðqBsÞ ð4Þ where m is the quark mass and we use the definitions for the parallel and perpendicular components of the scalar product of two vectors aμ and bμ given by ða · bÞ∥ ¼ a0 b0 − a3 b3 ; ða · bÞ⊥ ¼ a1 b1 þ a2 b2 :. ð5Þ. Figure 1 shows the Feynman diagrams contributing to the quark-gluon vertex. Diagram (a) corresponds to a QED-like contribution whereas diagram (b) corresponds to a pure QCD contribution. The computation of these diagrams requires using the fermion propagator given by Eq. (2), which involves the phase factor in Eq. (3). Notice that the phase factor does not depend on the chosen path. Taking a straight line path parametrized by ξμ ¼ x0μ þ tðx − x0 Þμ ;. ð6Þ. with t ∈ ½0; 1, and using that Fμν is antisymmetric, we can check that the phase factor becomes Z 1 0 0 μ Φðx; x Þ ¼ exp ð7Þ dtAμ ðx − x Þ : 0. For Aμ given by Eq. (1) the phase factor does not vanish. This factor can however be made to vanish by choosing an appropriate gauge transformation Aμ ðξÞ → A0μ ðξÞ ¼ Aμ þ with. 016007-2. ∂ ΛðξÞ; ∂ξμ. ð8Þ.

(3) FINITE TEMPERATURE QUARK-GLUON VERTEX WITH A …. PHYSICAL REVIEW D 91, 016007 (2015). (a). (b). FIG. 1. Feynman diagrams contributing to the thermomagnetic dependence of the quark-gluon vertex. Diagram (a) corresponds to a QED-like contribution whereas diagram (b) corresponds to a pure QCD contribution.. B 0 ðx ξ − x01 ξ2 Þ; 2 2 1 ∂ B ΛðξÞ ¼ ð0; x02 ; −x01 ; 0Þ: ∂ξμ 2 ΛðξÞ ¼. SðKÞ ¼ ð9Þ. Therefore, when a single fermion propagator is involved, we can gauge away the phase factor and work with the momentum space representation of the propagator. The situation is similar when two fermion propagators stemming from a common vertex are involved. Choosing straight line paths, the product of the phase factors becomes Z Z x00 x Φðx; x0 ÞΦðx00 ; xÞ ¼ exp iq dξμ Aμ þ dξμ Aμ x x0 Z 00 x ¼ exp iq dξμ Aμ x0 Z 1 00 0 μ ¼ exp ð10Þ dtAμ ðx − x Þ ;. m − K∥ m−K − iγ 1 γ 2 2 ðqBÞ: 2 2 K þm ðK þ m2 Þ2. Using the propagator in Eq. (11) and extracting a factor gta common to the bare and purely thermal contributions to the vertex, the magnetic field dependent part of diagram (a) in Fig. 1 is expressed as ðaÞ δΓμ. ¼. −ig2 ðCF. − CA =2ÞðqBÞT. d3 k ð2πÞ3. ~ 2 − KÞ × γ ν ½γ 1 γ 2 K ∥ γ μ K ΔðP ~ 1 − KÞγ ν þ Kγ μ γ 1 γ 2 K ∥ ΔðP ~ 2 − KÞΔðP ~ 1 − KÞ; × ΔðKÞΔðP. ð12Þ. where in the spirit of the HTL approximation, we have ignored terms proportional to m in the numerator and 1 ; þ k2 þ m2 1 ΔðKÞ ≡ 2 ; ωn þ k2 ~ ΔðKÞ ≡. ω~ 2n. ð13Þ. with ω~ n ¼ ð2n þ 1ÞπT and ωn ¼ 2nπT the fermion and boson Matsubara frequencies. CF , CA are the factors corresponding to the fundamental and adjoint representations of the SUðNÞ Casimir operators N2 − 1 ; 2N CA ¼ N;. CF ¼. III. QCD VERTEX AT FINITE TEMPERATURE WITH A WEAK MAGNETIC FIELD To compute the leading magnetic field dependence of the vertex at high temperature, we work in the weak field limit of the momentum representation of the fermion propagator [28]. We work in Euclidean space which is suited for calculations at finite temperature in the imaginary-time formalism of finite temperature field theory. The fermion propagator up to OðqBÞ is written as. XZ n. 0. where in the last step we have also chosen a straight line path connecting x0 and x00 . We can now employ a gauge transformation similar to Eq. (8) to gauge away the phase factor. Therefore, for the computation of diagrams (a) and (b) in Fig. 1, we can just work with the momentum representation of the fermion propagators since the phase factors do not contribute. The situation would have been nontrivial in case the computation had required a three fermion propagator closed loop [28].. ð11Þ. ð14Þ. respectively. Hereafter, capital letters are used to refer to four-momenta in Euclidean space with components ~ ¼ ð−ω; kÞ, ~ with ω either a Matsubara fermion K μ ¼ ðk4 ; kÞ or boson frequency. In the same manner, the magnetic field dependent part of diagram (b) in Fig. 1 is expressed as. 016007-3.

(4) ALEJANDRO AYALA et al. ðbÞ. δΓμ ¼ −2ig2. CA ðqBÞT 2. XZ n. PHYSICAL REVIEW D 91, 016007 (2015) 3. dk ð2πÞ3. × ½−Kγ 1 γ 2 K ∥ γ μ þ 2γ ν γ 1 γ 2 K ∥ γ ν K μ − γ μ γ 1 γ 2 K ∥ K 2 ~ ΔðP1 − KÞΔðP2 − KÞ: × ΔðKÞ. ð15Þ. The explicit factor 2 on the right-hand side of Eq. (15) accounts for the two possible fermion channels. These two channels are already accounted for in Eq. (12) since the magnetic field insertion on each quark internal line is thereby included at the order we are considering. The contribution from the two channels in each Feynman diagram is illustrated in Fig. 2. Recall that in Euclidean space it is convenient to work with the set of Dirac gamma matrices γ μ , μ ¼ 1; …; 4, with γ 4 ¼ iγ 0. ð16Þ. representing the medium’s rest frame and the direction of the magnetic field, respectively. Note that in the HTL approximation P1 and P2 are small and can be taken to be of the same order. Thus, to extract the leading temperature behavior let us write ~ 2 − KÞ þ Kγ μ γ 1 γ 2 K ∥ ΔðP ~ 1 − KÞ ½γ 1 γ 2 K ∥ γ μ K ΔðP ~ 1 − KÞ ≃ ½γ 1 γ 2 K ∥ γ μ K þ Kγ μ γ 1 γ 2 K ∥ ΔðP. in the integrand of Eq. (12). With the use of Eqs. (19) and (21) we proceed to work out the gamma-matrix structure in Eq. (12) and obtain ðaÞ. ~ μ ðP1 ; P2 Þ; δΓμ ¼ −2iγ 5 g2 ðCF − CA =2ÞðqBÞG. XZ n. fγ μ ; γ ν g ¼ −2δμν :. ð17Þ. ~ 2 − KÞ: ~ 2 ðP1 − KÞΔðP × ΔðKÞΔ ð18Þ. γ 1 γ 2 K ∥ ¼ γ 5 ½ðK · bÞu − ðK · uÞb;. ð19Þ. ðbÞ. δΓμ ¼ 2iγ 5 g2. Gμ ðP1 ; P2 Þ ¼ 2T. XZ n. bμ ¼ ð0; 0; 0; 1Þ;. CA ðqBÞGμ ðP1 ; P2 Þ; 2. d3 k fðK · bÞKuμ − ðK · uÞKbμ ð2πÞ3. þ ½ðK · bÞu − ðK · uÞbK μ g. ð20Þ. ~ 2 ðKÞΔðP1 − KÞΔðP2 − KÞ: ×Δ. ð25Þ. Note that the gamma-matrix and vector structure of Eqs. (23) and (25) is the same. Also note that Eq. (23) is obtained from Eq. (25) by changing a boson line into a fermion line, which is accomplished by replacing the BoseEinstein distribution fðEÞ with minus a Fermi-Dirac ~ distribution −fðEÞ and therefore this amounts for an overall change of sign [29], namely. (c). ~ μ ðP1 ; P2 Þ ¼ −Gμ ðP1 ; P2 Þ: G (b). ð24Þ. where. where we have introduced the four-vectors uμ ¼ ð1; 0; 0; 0Þ;. ð23Þ. In the same fashion we work out the gamma-matrix structure in Eq. (15) and obtain. which anticommutes with the rest of the gamma matrices. We use this set of matrices and its properties to write. (a). d3 k fðK · bÞKuμ − ðK · uÞKbμ ð2πÞ3. þ ½ðK · bÞu − ðK · uÞbK μ g. Also, γ 5 is defined by γ5 ¼ γ4γ1γ2 γ3 ;. ð22Þ. where ~ μ ðP1 ; P2 Þ ¼ 2T G. satisfying the algebra. ð21Þ. (d). FIG. 2. Explicit Feynman diagrams accounting for the two fermion channels contributing to the quark-gluon vertex. The expansion of the fermion propagator at OðqBÞ can be represented by the insertion of a photon line attached from an external source to each internal fermion line.. ð26Þ. Therefore, adding the contributions from the two Feynman diagrams in Fig. 1 we get. 016007-4. ðaÞ. ðbÞ. δΓμ ¼ δΓμ þ δΓμ. ¼ 2iγ 5 g2 CF ðqBÞGμ ðP1 ; P2 Þ:. ð27Þ.

(5) FINITE TEMPERATURE QUARK-GLUON VERTEX WITH A …. Gμ ðP1 ; P2 Þ can be computed from the tensor J αi ðα ¼ 1; …4; i ¼ 3; 4Þ given by J αi ¼ T. XZ n. d3 k ~ 2 ðKÞΔðP1 − KÞΔðP2 − KÞ; KαKi Δ ð2πÞ3 ð28Þ. which in turn requires one to compute the frequency sums X ~ 2 ðKÞΔðP1 − KÞΔðP2 − KÞ Y~ 0 ¼ T Δ n. X ∂ ~ T ¼ − ΔðKÞΔðP 1 − KÞΔðP2 − KÞ ∂m2 n ∂ X~ 0 ; ≡ − ∂m2 X ~ 2 ðKÞΔðP1 − KÞΔðP2 − KÞ Y~ 1 ¼ T ωn Δ. PHYSICAL REVIEW D 91, 016007 (2015). calculation of X~ 0 for those terms. We make the approximation where fðE1 Þ ≃ fðE2 Þ ≃ fðEÞ, namely,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that the Bose-Einstein distribution depends of E ¼ k2 þ m2 and thus on the quark mass. This approximation allows one to find the leading temperature behavior for m → 0 since it amounts to keeping the quark mass as an infrared ~ i · k̂, i ¼ 1; 2, we get regulator. Also, using that Ei ≃ k − p ~ 1 ½fðEÞ þ fðEÞ 1 ~ X0 ≃ − 2 E 8k ~ 1 · k̂Þðiω2 þ p ~ 2 · k̂Þ ðiω1 þ p 1 þ ; ð31Þ ~ 1 · k̂Þðiω2 − p ~ 2 · k̂Þ ðiω1 − p where in the denominator of the first fraction we have set E1 ¼ E2 ¼ k. In a similar fashion. n. X ∂ ~ T ωn ΔðKÞΔðP ¼ − 1 − KÞΔðP2 − KÞ ∂m2 n ∂ X~ 1 ; ≡ − ∂m2 X ~ 2 ðKÞΔðP1 − KÞΔðP2 − KÞ Y~ 2 ¼ T ω2n Δ n. X ∂ ~ T ω2n ΔðKÞΔðP ¼ − 1 − KÞΔðP2 − KÞ ∂m2 n ∂ X~ 2 ; ð29Þ ≡ − ∂m2. ~ i ½fðEÞ þ fðEÞ 1 X~ 1 ≃ − 8k E ~ 1 · k̂Þðiω2 þ p ~ 2 · k̂Þ ðiω1 þ p 1 − ; ~ 1 · k̂Þðiω2 − p ~ 2 · k̂Þ ðiω1 − p ~ 1 ½fðEÞ þ fðEÞ 1 ~ X2 ≃ 8 E ~ 1 · k̂Þðiω2 þ p ~ 2 · k̂Þ ðiω1 þ p 1 þ : ð32Þ ~ 1 · k̂Þðiω2 − p ~ 2 · k̂Þ ðiω1 − p Using Eqs. (31) and (32) into Eqs. (29) and (28), we get. where X~ 0 , X~ 1 , X~ 2 are in turn given by. Z ∞ 1 ∂ dxx2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 2 2 8π ∂y 0 x2 þ y2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 þ f x2 þ y2 × f~ Z dΩ K̂ α K̂ i ; × 4π ðP1 · K̂ÞðP2 · K̂Þ. J αi ¼ −. X ss1 s2 1 iðω 8EE E − ω Þ − s1 E1 þ s2 E2 1 2 1 2 s;s1 ;s2 ~ ~ 1 − fðsEÞ þ fðs1 E1 Þ 1 − fðsEÞ þ fðs2 E2 Þ ; × − iω1 − sE − s1 E1 iω2 − sE − s2 E2 X s1 s2 E 1 X~ 1 ¼ i 8EE1 E2 iðω1 − ω2 Þ − s1 E1 þ s2 E2 s;s1 ;s2 ~ ~ 1 − fðsEÞ þ fðs1 E1 Þ 1 − fðsEÞ þ fðs2 E2 Þ ; × − iω1 − sE − s1 E1 iω2 − sE − s2 E2 X ss1 s2 E2 1 X~ 2 ¼ iðω 8EE E − ω Þ − s1 E1 þ s2 E2 1 2 1 2 s;s1 ;s2 ~ ~ 1 − fðsEÞ þ fðs1 E1 Þ 1 − fðsEÞ þ fðs2 E2 Þ : × − iω1 − sE − s1 E1 iω2 − sE − s2 E2 X~ 0 ¼ −. ð33Þ. where we defined x ¼ k=T, y ¼ m=T, K̂ ¼ ð−i; k̂Þ, ~ 1 Þ, and P2 ¼ ð−ω2 ; p ~ 2 Þ. The integrals P1 ¼ ð−ω1 ; p over x can be expressed in terms of the well-known functions [30]. hn ðyÞ ¼ f n ðyÞ ¼. ð30Þ The leading temperature behavior is obtained from the terms with s ¼ −s1 ¼ −s2 . Let us consider in detail the. 1 ΓðnÞ 1 ΓðnÞ. Z. ∞. dxxn−1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ; 2 þy2 2 2 x x þy e −1. ∞. dxxn−1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ; 2 þy2 2 2 x x þy e þ1. 0. Z. 0. which satisfy the differential equations. 016007-5. ð34Þ.

(6) ALEJANDRO AYALA et al.. PHYSICAL REVIEW D 91, 016007 (2015). ∂hnþ1 h ¼ − n−1 ; 2 2n ∂y ∂f nþ1 f n−1 ¼− ; 2n ∂y2. In the spirit of the HTL approximation we ignore loose factors of P1 , P2 in the numerator. Noting that ð35Þ. therefore J αi ¼ −. 1 ½h1 ðyÞ þ f 1 ðyÞ 16π 2. Z. dΩ K̂ α K̂ i : 4π ðP1 · K̂ÞðP2 · K̂Þ. 1 1 ðP1 − P2 Þ · K ≃ ðP2 − KÞ2 − ðP1 − KÞ2 2 2 1 −1 1 ¼ Δ ðP2 − KÞ − Δ−1 ðP1 − KÞ; 2 2 we get. ð36Þ Using the high temperature expansions for h1 ðyÞ and f 1 ðyÞ [31] π 1 y 1 h1 ðyÞ ¼ þ ln þ γE þ ; 2y 2 4π 2 1 y 1 f 1 ðyÞ ¼ − ln − γE þ ; ð37Þ 2 π 2. ðP1 − P2 Þ · GðP1 ; P2 Þ X Z d3 k fðK · bÞu − ðK · uÞbg ¼T ð2πÞ3 n ~ 2 ðKÞ − ΔðP2 − KÞΔ ~ 2 ðKÞ: × ½ΔðP1 − KÞΔ. J αi. ðP1 − P2 Þ · δΓðP1 ; P2 Þ ¼ ΣðP1 Þ − ΣðP2 Þ; ð38Þ. ΣðPÞ ¼. 2. δΓμ ðP1 ; P2 Þ ¼ 4iγ 5 g CF M ðT; m; qBÞ Z dΩ 1 × 4π ðP1 · K̂ÞðP2 · K̂Þ þ ½ðK̂ · bÞu − ðK̂ · uÞbK̂ μ g;. 2iγ 5 g2 CF ðqBÞT. XZ n. d3 k ~ 2 ðKÞ ΔðP − KÞΔ ð2πÞ3. × fðK · bÞu − ðK · uÞbg:. × fðK̂ · bÞK̂uμ − ðK̂ · uÞK̂bμ ð39Þ. where we have defined the function M 2 ðT; m; qBÞ as qB πT 2 M ðT; m; qBÞ ¼ : ð40Þ lnð2Þ − 2m 16π 2 It is no surprise that the thermomagnetic correction to the quark-gluon vertex is proportional to γ 5 since the magnetic field is odd under parity conjugation. Furthermore, it is important to note that the vertex thus found satisfies a QEDlike Ward identity. The simplest way to see this is to look at Eq. (27) with Gμ ðP1 ; P2 Þ given by Eq. (25). Contracting this function with ðP1 − P2 Þμ we get. ð44Þ. where, as can be computed from the diagram in Fig. 3, the quark self-energy in the presence of a weak magnetic field in the HTL approximation is given by. Using Eqs. (38) and (25) into Eq. (27) we obtain 2. ð43Þ. Therefore, the expression in Eq. (27) satisfies the QED-like Ward identity, in the HTL approximation, given by. and keeping the leading terms, we get Z 1 πT dΩ K̂ α K̂ i lnð2Þ − : ¼ 2 2m 4π ðP1 · K̂ÞðP2 · K̂Þ 16π. ð42Þ. ð45Þ. Using the same approach that lead to finding the expression for the vertex, we get the explicit expression for the selfenergy, given by 2. 2. ΣðPÞ ¼ 2iγ 5 g CF M ðT; m; qBÞ. Z. dΩ ½ðK̂ · bÞu − ðK̂ · uÞb : 4π ðP · K̂Þ ð46Þ. This remarkable result shows that even in the presence of the magnetic field and provided the temperature is the largest of the energy scales, the thermomagnetic correction to the quark-gluon vertex is gauge invariant.. ðP1 − P2 Þ · GðP1 ; P2 Þ X Z d3 k fðK · bÞKðP1 − P2 Þ · u ¼ 2T ð2πÞ3 n − ðK · uÞKðP1 − P2 Þ · b þ ½ðK · bÞu − ðK · uÞb ~2. × ðP1 − P2 Þ · KgΔ ðKÞΔðP1 − KÞΔðP2 − KÞ:. ð41Þ. FIG. 3. Feynman diagram for the the quark self-energy. The internal quark line represents the quark propagator in the presence of the magnetic field in the weak field limit.. 016007-6.

(7) FINITE TEMPERATURE QUARK-GLUON VERTEX WITH A …. IV. THERMOMAGNETIC QCD COUPLING In order to look at the thermomagnetic dependence of the quark-gluon coupling let us look explicitly at the quantities Z Jαi ðP1 ; P2 Þ ≡. dΩ K̂ α K̂ i ; 4π ðP1 · K̂ÞðP2 · K̂Þ. 1 1 2 iω1 p2 þ iω2 p1 Z 1 p1 p2 × dx þ iω1 þ p1 x iω2 − p2 x −1 1 1 ¼− 2 iω1 p2 þ iω2 p1 iω1 þ p1 iω2 þ p2 þ ln : × ln iω1 − p1 iω2 − p2. J44 ðP1 ; P2 Þ ¼ −. Furthermore, let us look now at the static limit, that is, where the quarks are almost at rest, namely p → 0, then 1 : p20. ð50Þ. Now consider J33 ðP1 ; P2 Þ in the same momenta configuration 1 1 2 iω1 p2 þ iω2 p1 Z 1 p1 p2 2 þ × dxx iω1 þ p1 x iω2 − p2 x −1 1 ¼− iω1 p2 þ iω2 p1 iω1 iω1 iω1 þ p1 1− × ln p1 2p1 iω1 − p1 iω2 iω2 iω2 þ p2 þ 1− : ð51Þ ln p2 2p2 iω2 − p2. J33 ðP1 ; P2 Þ ¼. ð52Þ. In the limit where p → 0, p→0. J 33 ⟶. 1 : 3p20. ð53Þ. For the same choice of momenta, the rest of the components of Jαi vanish, which means that only the longitudinal components of the thermomagnetic vertex are modified. Using Eqs. (47), (49), and (53) into Eq. (39), the explicit longitudinal components are given by 2 4g2 CF M 2 ðT; m; qBÞ~γ ∥ Σ3 ; 3p20. ~ ∥ ðp0 Þ ¼ δΓ. We now take the analytic continuation to Minkowski space iω1;2 → p01;02 [K̂ → ð−1; k̂Þ] and consider the scenario where p01 ¼ p02 ≡ p0 and p1 ¼ p2 ≡ p, then we get 1 p0 þ p J44 → J00 ¼ ln : ð49Þ 2p0 p p0 − p. p→0. J33. 1 p0 p0 þ p ¼− 2 1− ln : p0 − p 2p p. . ð48Þ. J00 ⟶. After analytical continuation to Minkowski space and in the same scenario where p01 ¼ p02 ≡ p0 and p1 ¼ p2 ≡ p, we get. ð47Þ. appearing on the right-hand side of Eq. (39). For the sake of simplicity, let us choose a configuration where the ~ 1 and p ~ 2 make a relative angle θ12 ¼ π. This momenta p configuration corresponds for instance to a thermal gluon decaying into a quark-antiquark pair in the center of mass system and is therefore general enough. Consider first J44 ðP1 ; P2 Þ. PHYSICAL REVIEW D 91, 016007 (2015). ð54Þ. where ~γ ∥ ¼ ðγ 0 ; 0; 0; −γ 3 Þ and we have rearranged the gamma matrices to introduce the spin operator in the ẑ direction Σ3 ¼ iγ 1 γ 2 i ¼ ½γ 1 ; γ 2 : 2. ð55Þ. That the vertex correction in Eq. (54) is proportional to the third component of the spin operator is of course natural since the first order magnetic correction to the vertex is in turn proportional to the spin interaction with the magnetic field, which we chose to point along the third spatial direction. The correction thus corresponds to the quark anomalous magnetic moment at high temperature in a weak magnetic field. Note also that Eq. (54) depends on the scales p0 and m. p0 is the typical energy of a quark in the medium and therefore the simplest choice for this scale is to take it as the temperature. The quark mass represents the infrared scale and it is therefore natural to take it as the thermal quark mass. We thus set p0 ¼ T; 1 m2 ¼ m2f ¼ g2 T 2 CF : 8. ð56Þ. As is well known, the purely thermal correction to the quark-gluon vertex is given by [29]. 016007-7. δΓtherm ðP1 ; P2 Þ μ. ¼. −m2f. Z. K̂ μˆK dΩ : 4π ðP1 · K̂ÞðP2 · K̂Þ. ð57Þ.

(8) ALEJANDRO AYALA et al.. PHYSICAL REVIEW D 91, 016007 (2015). In order to extract the effective modification to the coupling constant in one of the longitudinal directions, let us look at the contribution proportional to γ 0 from Eq. (57). For the same working momenta configuration and using Eqs. (49) and (50) δΓtherm ðp0 Þ ¼ − 0. m2f p20. γ0:. ð58Þ. Using Eqs. (54) and (58), the effective thermomagnetic modification to the quark-gluon coupling, extracted from the effective longitudinal vertex, is given by. geff. m2f 8 2 2 g CF M ðT; mf ; qBÞ ; ¼g 1− 2 þ 3T 2 T. ð59Þ. where we have used a spin configuration with eigenvalue 1 for Σ3. Figure 4 shows the behavior of geff normalized to gtherm , where m2f gtherm ≡ g 1 − 2 ; T. ð60Þ. for αs ¼ g2 =4π ¼ 0.2; 0.3 as a function of the scaled variable b ¼ qB=T 2 . Plotted as a function of b and for our choice of momentum scales, the function geff =gtherm is temperature independent. Note that the effective thermomagnetic coupling geff decreases as a function of the magnetic field. The decrease becomes more significant for larger αs and for the considered values of αs it becomes. 1.00 0.95. geff gtherm. 0.90 0.85 0.80. s. 0.2. s. 0.3. 0.75 0.70. 0.0. 0.2. 0.4. 0.6. 0.8. 1.0. qB T 2. FIG. 4 (color online). The effective thermomagnetic coupling geff normalized to the purely thermal coupling gtherm as a function of the field strength scaled by the squared of the temperature, for αs ¼ 0.2, 0.3. Note that geff decreases down to about 15%–25% for the largest strength of the magnetic field within the weak field limit with respect to the purely thermal correction and that the decrease is faster for larger αs.. about 15%–25% smaller than the purely thermal correction for qB ∼ T 2 ∼ 1. V. SUMMARY AND CONCLUSIONS In this work we have computed the thermomagnetic corrections to the quark-gluon vertex for a weak magnetic field and in the HTL approximation. We have shown that this vertex satisfies a QED-like Ward identity with the quark self-energy. This result hints to the gauge independence of the calculation. The thermomagnetic correction is proportional to the spin component in the direction of the magnetic field and affects only the longitudinal components of the quark-gluon vertex. It thus corresponds to the quark anomalous magnetic moment at high temperature in a weak magnetic field. From the quark-gluon vertex we have extracted the behavior of the magnetic field dependence of the QCD coupling. We have shown that the coupling decreases as the field strength increases. For analytic simplicity, the explicit calculation has been performed for a momentum configuration that accounts for the conditions prevailing in a quarkgluon medium at high temperature, namely back-to-back slow moving quarks whose energy is of the order of the temperature and with the infrared scale of the order of the thermal particle’s mass. The chosen values for these scales provide a good representation of the coupling constant’s strength within the plasma conditions. We stress that the weak field approximation means that the field strength is smaller than the square of the temperature but does not require a hierarchy with respect to other scales in the problem such as the fermion thermal mass. Note that for situations where the temperature is not the largest of the energy scales, as for the largest magnetic field strengths achieved in peripheral heavy-ion collisions, or for compact astrophysical objects, a calculation such as the present one cannot provide conclusions about the behavior of the coupling constant nor for its effect on the critical temperature. Nevertheless, it is important to note that lattice QCD simulations for the critical temperature are performed for magnetic field intensities starting from qB ¼ 0 (see e.g. Fig. 9 in Ref. [1]). The results of these simulations show that the critical temperature is a decreasing function of the field intensity all the way to qB ¼ 1 GeV2 . Also, one should note that it is likely that the effects on the quarkgluon plasma in peripheral heavy-ion collisions take place when the field intensity is already smaller than the square of the temperature, since the field strength is a rapidly decreasing function of time. It is thus important to explore in QCD the regime where qB is small compared to T 2 as a necessary bridge to further studies on the behavior of the coupling constant for larger field strengths. In this context, the development of the framework to study the large field behavior of the QCD vertex, consistent with gauge invariance, is also a relevant issue that will be explored elsewhere.. 016007-8.

(9) FINITE TEMPERATURE QUARK-GLUON VERTEX WITH A …. PHYSICAL REVIEW D 91, 016007 (2015). The result supports the idea [21,22] that the decreasing of the coupling constant may be one of the important ingredients to understanding the inverse magnetic catalysis obtained in lattice QCD. It remains to include this result within a calculation to allow the extraction of the magnetic field dependence of the critical temperature for chiral symmetry restoration/deconfinement transition in QCD. This is work in progress and will be reported elsewhere.. A. A. acknowledges useful conversations with G. Krein. J. J. C.-M. acknowledges support from a CONACyTMéxico postdoctoral grant with Grant No. 290807UMSNH. M. E. T.-Y. acknowledges support from the CONACyT-Mexico sabbatical Grant No. 232946. Support for this work has been received in part from CONACyTMéxico under Grant No. 128534 and FONDECYT under Grants No. 1130056 and No. 1120770. R. Z. acknowledges support from CONICYT under Grant No. 21110295.. [1] G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer, and K. K. Szabo, J. High Energy Phys. 02 (2012) 044. [2] G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, and A. Schafer, Phys. Rev. D 86, 071502 (2012). [3] G. S. Bali, F. Bruckmann, G. Endrodi, S. D. Katz, and A. Schafer, J. High Energy Phys. 08 (2014) 177. [4] E. S. Fraga and L. F. Palhares, Phys. Rev. D 86, 016008 (2012); E. S. Fraga, J. Noronha, and L. F. Palhares, Phys. Rev. D 87, 114014 (2013). [5] F. Bruckmann, G. Endrodi, and T. G. Kovacs, J. High Energy Phys. 04 (2013) 112. [6] M. Ferreira, P. Costa, D. P. Menezes, C. Providencia, and N. N. Scoccola, Phys. Rev. D 89, 016002 (2014). [7] K. Fukushima and Y. Hidaka, Phys. Rev. Lett. 110, 031601 (2013). [8] J. O. Andersen and A. Tranberg, J. High Energy Phys. 08 (2012) 002; J. O. Andersen, W. R. Naylor, and A. Tranberg, J. High Energy Phys. 04 (2014) 187. [9] R. L. S. Farias, K. P. Gomes, G. Krein, and M. B. Pinto, Phys. Rev. C 90, 025203 (2014). [10] M. Ferreira, P. Costa, O. Lourenço, T. Frederico, and C. Providência, Phys. Rev. D 89, 116011 (2014). [11] G. S. Bali, F. Bruckmann, G. Endrodi, F. Gruber, and A. Schaefer, J. High Energy Phys. 04 (2013) 130. [12] E. J. Ferrer, V. de la Incera, and X. J. Wen, arXiv:1407.3503. [13] Sh. Fayazbakhsh and N. Sadooghi, Phys. Rev. D 90, 105030 (2014). [14] J. O.Andersen,W. R.Naylor,andA.Tranberg,arXiv:1410.5247. [15] E. S. Fraga and A. J. Mizher, Phys. Rev. D 78, 025016 (2008).. [16] M. Loewe, C. Villavicencio, and R. Zamora, Phys. Rev. D 89, 016004 (2014). [17] N. O. Agasian and S. M. Fedorov, Phys. Lett. B 663, 445 (2008). [18] A. J. Mizher, M. N. Chernodub, and E. S. Fraga, Phys. Rev. D 82, 105016 (2010). [19] E. S. Fraga, B. W. Mintz, and J. Schaffner-Bielich, Phys. Lett. B 731, 154 (2014). [20] A. Ayala, L. A. Hernández, A. J. Mizher, J. C. Rojas, and C. Villavicencio, Phys. Rev. D 89, 116017 (2014). [21] A. Ayala, M. Loewe, A. J. Mizher, and R. Zamora, Phys. Rev. D 90, 036001 (2014). [22] A. Ayala, M. Loewe, and R. Zamora, Phys. Rev. D 91, 016002 (2015). [23] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl. Phys. A803, 227 (2008). [24] V. Skokov, A. Y. Illarionov, and V. Toneev, Int. J. Mod. Phys. A 24, 5925 (2009). [25] A. Bzdak and V. Skokov, Phys. Lett. B 710, 171 (2012). [26] A. Ayala, M. Loewe, J. C. Rojas, and C. Villavicencio, Phys. Rev. D 86, 076006 (2012). [27] J. Schwinger, Phys. Rev. 82, 664 (1951). [28] T.-K. Chyi, C.-W. Hwang, W. F. Kao, G. L. Lin, K.-W. Ng, and J.-J. Tseng, Phys. Rev. D 62, 105014 (2000). [29] M. Le Bellac, Thermal Field Theory (Cambridge University Press, Cambridge, England, 1996). [30] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974). [31] J. I. Kapusta, Finite-Temperature Field Theory (Cambridge University Press, Cambridge, England, 1989).. ACKNOWLEDGMENTS. 016007-9.

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