Thermal corrections to the gluon magnetic Debye mass
Alejandro Ayala, Jorge David Castaño-Yepes, C. A. Dominguez, S. Hernandez-Ortiz, L. A. Hernandez, M. Loewe, D. Manreza Paret, R. Zamora
aa r X i v : . [ h e p - ph ] J a n Gluon polarization tensor in a hot and strongly magnetizedmedium: Thermal corrections to the gluon magnetic Debye mass
Alejandro Ayala , , Jorge David Casta˜no-Yepes , C. A. Dominguez ,S. Hern´andez-Ortiz , L. A. Hern´andez , , M. Loewe , , , D. Manreza Paret and R. Zamora , Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,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 F´ısica, Pontificia UniversidadCat´olica de Chile, Casilla 306, Santiago 22, Chile. Centro Cient´ıfico-Tecnol´ogico de Valpara´ıso CCTVAL,Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valapara´ıso, Chile. Facultad de F´ısica, Universidad de La Habana,San Lazaro y L, La Habana, Cuba. Instituto de Ciencias B´asicas, Universidad Diego Portales, Casilla 298-V, Santiago, Chile. Centro de Investigaci´on y Desarrollo en Ciencias Aeroespaciales (CIDCA),Fuerza A´erea de Chile, Santiago, Chile. (Dated: January 14, 2020) bstract We compute the gluon polarization tensor in a thermo-magnetic environment in the strong mag-netic field limit at zero and high temperature. The magnetic field effects are introduced usingSchwinger’s proper time method. Thermal effects are computed in the HTL approximation. Atzero temperature, we reproduce the well-known result whereby for a non-vanishing quark mass, thepolarization tensor reduces to the parallel structure and its coefficient develops an imaginary partcorresponding to the threshold for quark-antiquark pair production. This coefficient is infraredfinite and simplifies considerably when the quark mass vanishes. Keeping always the field strengthas the largest energy scale, in the high temperature regime we analyze two complementary hierar-chies of scales: q ≪ m f ≪ T and m f ≪ q ≪ T . In the latter, we show that the polarizationtensor is infrared finite as m f goes to zero. In the former, we discuss the thermal corrections tothe magnetic Debye mass. Keywords: gluon polarization tensor, magnetic fields, finite temperature . INTRODUCTION The properties of strongly interacting matter immersed in a magnetized medium havebeen the subject of intense research over the last years. The motivation for this activitystems from several fronts: On the one hand, lattice QCD (LQCD) [1] has shown that fortemperatures above the chiral restoration pseudo-critical temperature, the quark-antiquarkcondensate decreases and that this temperature itself also decreases, both as functions ofthe field intensity. This result, dubbed inverse magnetic catalysis (IMC), has sparked a largenumber of explanations [2–13]. On the other hand, it has been argued that intense magneticfields can be produced in peripheral heavy-ion collisions. Possible signatures of the presenceof such fields in the interaction region can be the chiral magnetic effect [14] or the enhancedproduction of prompt photons [15–18]. Moreover, magnetic fields can have an impact onthe properties of compact astrophysical objects, such as neutron stars [19].The dispersive properties for gluons propagating in a magnetized medium are encodedin the gluon polarization tensor. For QED, this tensor has been computed and extensivelystudied both at zero and finite temperature [20–28]. In particular, Refs. [23–25] study thecase of intense magnetic fields, where the lowest Landau Level (LLL) approximation can beused. Reference [26] works the one-loop zero temperature case to all orders in the magneticfield and finds a general expression in terms of an integral over proper time parameters. Noattempt to provide analytical results is made. In Ref. [27] the polarization tensor is computedboth at finite temperature and field strength. The findings are applied to study magneticfield effects on the Debye screening. Reference [29] expresses the one-loop polarizationtensor as a sum over Landau levels and evaluates it using numerical methods. An analyticalapproach to the sum over Landau levels at zero temperature has been recently carried outin Ref. [30]. Magnetic corrections to the QCD equation of state in the hard thermal loop(HTL) and the LLL approximations, applied to the description of heavy-ion collisions havebeen considered in Refs. [31, 32]. Analytic results can be obtained in several limits of interestsuch as the HTL and LLL by considering different hierarchies for the fermion mass, thermaland magnetic scales [25, 33].Nevertheless, in a thermo-magnetic medium, the gluon polarization tensor depends, inaddition to the temperature T and magnetic field strength | eB | , also on the square of itsmomentum components as well as on the fermion mass. The breaking of boost and rota-3 q−k qk FIG. 1. One-loop diagram representing the gluon polarization tensor. tional invariance introduced by thermal and magnetic effects, makes the polarization tensorto depend separately on the longitudinal and perpendicular components of the gluon mo-mentum. The competition between these different energy scales produces a rich structurethat can be better grasped if one resorts to analytic approximations that arise when consid-ering given hierarchies of these energy scales. In this work we take on this task and considerthe behavior of the gluon polarization tensor in vacuum and at high temperature for thecase when the field is strong. The paper is organized as follows. In Sec. II, we compute thegluon polarization tensor in the presence of a strong, uniform and constant magnetic fieldat zero and high temperature. For the latter case, working in the HTL approximation, westudy two different hierarchies of scales: q ≪ m f ≪ T and the the complementary case m f ≪ q ≪ T , writing for each case the most general tensor structure and discussing itsproperties. We compute the magnetic modifications to the Debye mass, paying attentionto the cases when longitudinal and the perpendicular components of the gluon momentumvanish at different rates.In Sec. III, we summarize and discuss our results. We reserve for the appendices thecalculation details for each of the regimes where the gluon polarization tensor is computed. II. THERMO-MAGNETIC GLUON POLARIZATION TENSOR
We proceed to compute the gluon polarization tensor at one-loop order in the presence ofa magnetic field, both in vacuum and in a thermal bath. In both cases we consider that thelargest of all energy scales is the field strength. As we proceed to show, for the vacuum case,there is no need to establish a hierarchy of scales between the gluon momentum squared q and the fermion mass squared m f . However, for the thermal case, care has to be taken for the4ierarchy between q and m f . Therefore, for the thermal case, we perform the computationin the two different regimes, namely q ≪ m f ≪ T and m f ≪ q ≪ T .In general, the one-loop contribution to the gluon polarization tensor, depicted in Fig. 1,is given by Π µνab = − Z d k (2 π ) Tr { igt b γ ν iS F ( k ) igt a γ µ iS F ( k − q ) } . (1)The factor 1 / g is the coupling constant, S F ( k ) is thequark propagator and t a,b are the generators of the color group. Since the quark anti-quarkpair in the loop interact with the magnetic field, the quark propagator is modified from itsvacuum expression and is written, omitting a trivial color factor, as S F ( x, x ′ ) = Φ( x, x ′ ) Z d k (2 π ) e − ik · ( x − x ′ ) S ( k ) , (2)where Φ( x, x ′ ) is called the Schwinger phase factor . The latter accounts for the loss ofLorentz invariance in the presence of the magnetic field. For the present calculation, thephase factor can be gauged away [12, 34] and we need to just work with the translationallyinvariant part of the fermion propagator. The latter is given by iS ( k ) = Z ∞ ds cos( q f Bs ) e is ( k k − k ⊥ tan( qf Bs ) qf Bs − m f ) × n [cos( q f Bs ) + γ γ sin( q f Bs )]( m f + /k k ) − /k ⊥ cos( q f Bs ) (cid:9) , (3)where q f is the the quark electric charge. Hereafter, we use the following notation for theparallel and perpendicular (with respect to the magnetic field) pieces of the scalar productof two four-vectors a µ and b µ ( a · b ) k = a b − a b , ( a · b ) ⊥ = a b + a b , (4)such that a · b = ( a · b ) k − ( a · b ) ⊥ . (5)Gauge invariance requires that the gluon polarization tensor be transverse. However, thebreaking of Lorentz symmetry makes this tensor to split into three transverse structures,5uch that the gluon polarization tensor can be written, omitting a trivial factor δ ab comingfrom the color trace in Eq. (1), as [21] (see also Refs. [24, 30, 35])Π µν = P k Π µν k + P ⊥ Π µν ⊥ + P Π µν , (6)where Π µν k = g µν k − q µ k q ν k q k , Π µν ⊥ = g µν ⊥ − q µ ⊥ q ν ⊥ q ⊥ , Π µν = g µν − q µ q ν q − (Π µν k + Π µν ⊥ ) . (7)Notice that the three tensor structures in Eq. (7) are orthogonal to each other, hence,their coefficients in Eq. (6) can be expressed as P k = Π k µν Π µν ,P ⊥ = Π ⊥ µν Π µν ,P = Π µν Π µν . (8)We now proceed to compute each of the coefficients in Eq. (8) in the strong field limit. A. Vacuum case, strong field approximation
We now proceed to calculate the polarization tensor in the strong field limit, namely | eB | ≫ q , m f . The (photon) polarization tensor in this limit has been computed inRef. [23] using Ritus’ method. Here we work instead using Schwinger’s proper time method.To implement this approximation, we work in the lowest Landau level (LLL) limit of thequark propagator, given by [28] iS LLL ( k ) = 2 ie − k ⊥| qf B | /k k + m f k k − m f O ± , (9)where O ± = 12 [1 ± iγ γ sign( q f B )] . (10)Figure 2 shows the diagrams contributing to the calculation. These represent one and theother possible electric charge flow direction within the loop, which, in the presence of the6 a) (b) FIG. 2. One-loop diagrams for the gluon polarization tensor in the strong field limit, using theLLL. magnetic field have both to be accounted for. Using Eq. (9), into the Eq. (1), the explicitexpression for diagram ( a ) in Fig. (2) is given by i Π µνa = − g X f Z d k ⊥ (2 π ) e − k ⊥| qf B | e − ( q ⊥− k ⊥ )2 | qf B | × Z d k k (2 π ) Tr[ γ ν ( m f − /k k ) O + γ µ ( m f − ( /k − /q ) k ) O + ][ k k − m f ][( k − q ) k − m f ] . (11)The contribution from diagram ( b ) in Fig. 2 is obtained by replacing O + → O − . Thepolarization tensor is obtained by adding these two contributions, resulting in the expression i Π µνa + i Π µνb = − g X f Z d k ⊥ (2 π ) e − k ⊥| qf B | e − ( q ⊥− k ⊥ )2 | qf B | × Z d k k (2 π ) k k − m f ][( k − q ) k − m f ] ×{ Tr[ γ ν ( m f − /k k ) O + γ µ ( m f − ( /k − /q ) k ) O + ]+Tr[ γ ν ( m f − /k k ) O − γ µ ( m f − ( /k − /q ) k ) O − ] } . (12)The explicit expressions for the traces are given byTr[ γ ν ( m f − /k k ) O ± γ µ ( m f − ( /k − /q ) k ) O ± ] = m f Tr[ γ ν O ± γ µ O ± ] + Tr[ γ ν /k k O ± γ µ ( /k − /q ) k O ± ]= m f Tr[ γ ν O ± γ µ k ] + Tr[ γ ν /k k O ± γ µ k ( /k − /q ) k ] . (13)7ubstituting Eqs. (10) and (13) into Eq. (12), we obtain i Π µνa + i Π µνb = − g X f Z d k ⊥ (2 π ) e − k ⊥| qf B | e − ( q ⊥− k ⊥ )2 | qf B | × Z d k k (2 π ) k k − m f ][( k − q ) k − m f ] × (cid:8) m f Tr[ γ ν ( O + + O − ) γ µ k ] + Tr[ γ ν /k k ( O + + O − ) γ µ k ( /k − /q ) k ] (cid:9) = − g X f Z d k ⊥ (2 π ) e − k ⊥| qf B | e − ( q ⊥− k ⊥ )2 | qf B | × Z d k k (2 π ) k k − m f ][( k − q ) k − m f ] × (cid:8) m f Tr[ γ ν γ µ k ] + Tr[ γ ν /k k γ µ k ( /k − /q ) k ] (cid:9) . (14)After integrating over the transverse components of the four-momentum, the expression forthe polarization tensor becomes i Π µνa + i Π µνb = − g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | × Z d k k (2 π ) k k − m f ][( k − q ) k − m f ] × h ( m f − k k · ( k − q ) k ) g µν k + k µ k ( k − q ) ν k + k ν k ( k − q ) µ k i . (15)In order to compute the two dimensional integral over the parallel components, we useFeynman’s parametrization. Thus, the denominator in Eq. (15) is written as1[ k k − m f ][( k − q ) k − m f ] = Z dx [ l k − ∆] , (16)with l k = k k − (1 − x ) q k and ∆ = m f − x (1 − x ) q k . Shifting the integration variable k k → l k in Eq. (15), we obtain i Π µνa + i Π µνb = − g X f (cid:16) | q f B | π (cid:17) e − q ⊥ | qf B | Z dx Z d l k (2 π ) l k − ∆] × { l µ k l ν k − g µν k l k − x (1 − x ) q µ k q ν k + g µν k ( m f + x (1 − x ) q k ) } , (17)where in the integrand we have already discarded linear terms in l k , since they give avanishing contribution. In order to compute the momentum integrals, we recall the following8ell known relations Z d d l k (2 π ) d l k − ∆] n = i ( − n (4 π ) d/ Γ( n − d/ n ) ! n − d/ , Z d d l k (2 π ) d l k [ l k − ∆] n = i ( − n − (4 π ) d/ d n − d/ − n ) ! n − d/ − , Z d d l k (2 π ) d l µ l ν [ l k − ∆] n = i ( − n − (4 π ) d/ g µν n − d/ − n ) ! n − d/ − . (18)Using these into Eq. (17), we get i Π µνa + i Π µνb = − g X f (cid:16) | q f B | π (cid:17) e − q ⊥ | qf B | − i (4 π ) d/ Z dx ( g µν k Γ(1 − d/ ! − d/ − g µν k d − d/ ! − d/ + 2 x (1 − x ) q µ k q ν k Γ(2 − d/ ! − d/ − g µν k ( m f + x (1 − x ) q k )Γ(2 − d/ ! − d/ ) . (19)Equation (19) is apparently divergent when taking the limit d →
2. To show that this isnot the case, we first combine the terms proportional to the tensor structure g µν k , namely,the first, second and fourth terms in Eq. (19), to obtain g µν k " Γ(1 − d/ ! − d/ − d − d/ ! − d/ − ( m f + x (1 − x ) q k )Γ(2 − d/ ! − d/ = g µν k " (1 − d/ − d/ − ( m f + x (1 − x ) q k )Γ(2 − d/ ! − d/ = g µν k " ( m f − x (1 − x ) q k ) − ( m f + x (1 − x ) q k ) Γ(2 − d/ ! − d/ = − g µν k q k Γ(2 − d/ ! − d/ (20)9ubstituting Eq. (20) into Eq. (19), we get i Π µνa + i Π µνb = − g X f (cid:16) | q f B | π (cid:17) e − q ⊥ | qf B | i (4 π ) d/ Γ(2 − d/ Z dx ! − d/ × n g µν k [ − m f + x (1 − x ) q k + m f + x (1 − x ) q k ] − x (1 − x ) q µ k q ν k o = − g X f (cid:16) | q f B | π (cid:17) e − q ⊥ | qf B | i (4 π ) d/ Γ(2 − d/ q k ( g µν k − q µ k q ν k q k ) × Z dx x (1 − x ) ! − d/ . (21)Notice that Eq. (21) is now free of divergences when taking the limit d →
2. We thus obtain i Π µνa + i Π µνb = ig X f (cid:16) | q f B | π (cid:17) e − q ⊥ | qf B | q k ( g µν k − q µ k q ν k q k ) Z dx − x (1 − x )∆ , (22)from where it is seen that the emerging tensor structure is equal to Π µν k . Therefore, in thestrong field limit, the coefficients P ⊥ and P turn out to be equal to zero.The integral over the x variable can be expressed in terms of the function I ( y k ) = y k Z dx x (1 − x ) − x (1 − x ) y k , (23)where y k ≡ q k /m f . For the case m f = 0, I = 1 and the gluon polarization tensor is given by i Π µν (cid:12)(cid:12) m f =0 = ( i Π µνa + i Π µνb ) (cid:12)(cid:12) m f =0 = ig ( g µν k − q µ k q ν k q k ) X f (cid:16) | q f B | π (cid:17) e − q ⊥ | qf B | . (24)This result coincides (albeit for the case of the photon polarization tensor) with the onefound in Ref. [23] (see also Refs. [24, 30]). Figure 3 shows the behavior of the function I ( y k )for the general case when m f = 0. Notice that this function develops an imaginary part for y k ≥
4, corresponding to the threshold for quark-antiquark production.
B. Thermo-magnetic polarization tensor in the HTL and LLL approximations
We now proceed to calculate the polarization tensor in the strong field limit within theHTL approximation. We consider the case where | eB | ≫ T . At finite temperature, dueto the presence of the vector u µ that defines the medium’s reference frame, the tensor10 - - - - - - y || I ( y || ) Im [ I ( y || )] Re [ I ( y || )] FIG. 3. Behavior of I as a function of y k . Shown are the real (black) and imaginary (red) parts. Thisfunction presents a discontinuity at y k = 4 that corresponds to the threshold for quark-antiquarkproduction. g µν − q µ q ν /q splits into (three dimensional) transverse Π µνT and longitudinal Π µνL structures,such that − g µν + q µ q ν q = Π µνL + Π µνT , (25)with Π µνT = − g µν + q ~q ( q µ u ν + u µ q ν ) − ~q ( q µ q ν + q u µ u ν ) , Π µνL = − q ~q ( q µ u ν + u µ q ν ) + 1 ~q h ( q ) q q µ q ν + q u µ u ν i , (26)where, in the medium’s reference frame, u µ = (1 , , , µν k , given by the first of Eqs. (7), span the most general expressionfor the polarization tensor for the case | q f B | ≫ T . Indeed, recall that when the gluon splitsinto a virtual quark-antiquark pair, and this occupies the same Landau level (the LLL in thiscase), the pair’s transverse momentum vanishes, because the quark moves in the oppositedirection than the antiquark around the field lines. From momentum conservation and fora finite gluon momentum, the motion of the virtual pair can therefore only happen alongthe direction of the external magnetic field. Since the quark/antiquark motion is thrust bythe chromo-electric field, only a polarization vector having a component along the externalmagnetic field can push the motion of the virtual pair [22]. The only polarization vector,11nd thus tensor, having a non-vanishing projection along the magnetic field is the parallelstructure Π µν k . In fact, from this argument, it can be expected that the transverse (to themagnetic field) polarization structure Π µν ⊥ will be absent from the tensor basis. Also, whenworking in the limit | eB | ≫ T , neither the virtual quark nor the antiquark have enoughthermal energy to transit between Landau levels. Thus the motion of the virtual pair in theLLL would keep being along the external field.Concentrating only on the temperature dependent part of the polarization tensor e Π µν ,we can write e Π µν = e P L Π µνL + e P T Π µνT . (27)To find the coefficients e P L and e P T , we need to compute the corresponding projections onto e Π µν . This gives Π Lµν e Π µν = e P L Π Tµν e Π µν = 2 e P T . (28)In order to compute the contractions on the left-hand side of Eq. (28), we follow aprocedure similar to the one that lead to Eq. (15). This implies using that, in the strongfield limit, transverse and parallel structures factorize, and also that temperature effects areobtained, in the Matsubara formalism, from the time-like component of the integration four-vector. Therefore all temperature effects are comprised to the parallel pieces of the integrals.Explicitly, within the Matsubara formalism, we transform the integrals to Euclidean spaceby means of a Wick rotation, namely Z d k (2 π ) f ( k ) → iT ∞ X n = −∞ Z d k (2 π ) f ( i ˜ ω n , ~k ) , (29)where the integral over the time-like component of the fermion momentum has been dis-cretized and we introduced the fermion Matsubara frequencies ˜ ω n = (2 n + 1) πT , k = i ˜ ω n and q = iq = ω . In order to obtain the coefficients, we need to compute the sum overMatsubara frequencies and the integral over k . Notice that after performing the sum, onegets two terms: one corresponding to the, T -independent magnetized vacuum and anothercorresponding to the thermo-magnetic contribution. The magnetized vacuum term turnsout, of course, to be equal to Eq. (22) and therefore it comes entirely as the coefficient ofΠ µν k . This is explicitly shown in A. The temperature contribution is obtained from the terms12roportional to the Fermi-Dirac distributions. For these terms, we work in two limits: (1)the external momentum q is taken as the softest energy scale, thus we keep the quark massfinite and (2) the quark mass is the smallest energy scale and thus we keep the externalmomentum q finite.
1. Case where q ≪ m f ≪ T When the external momentum q is considered as the smallest energy scale, in the HTLapproximation, we can neglect q and k · q in each of the numerators. Thus, the explicithierarchy of scales for this calculation is q ≪ m f ≪ T . Performing the integral over k ,and after analytical continuation back to Minkowski space, we get e P L = g π X f | q f B | e − q ⊥ | qf B | " q ( q ⊥ + 2 q ) ~q q − ln (cid:16) π T m f (cid:17) − γ E ! (30)and e P T = g π X f | q f B | e − q ⊥ | qf B | " q ⊥ ~q ln (cid:16) π T m f (cid:17) − γ E ! . (31)The explicit calculations to obtain Eqs. (30) and (31) are shown in B. Notice that knowledgeof the complete expression is important when discussing their evolution properties under therenormalization group [36] and for the study of the thermo-magnetic effects on the Debyemass [27].
2. Case where m f ≪ q ≪ T Another way to compute the polarization tensor in the HTL approximation is to takeinto account the complementary hierarchy of scales m f ≪ q ≪ T . After the integral over k is performed and the analytical continuation to Minkowski space is done, we obtain e P L = g π X f | q f B | e − q ⊥ | qf B | " q + q + q q q q ~q q k !(cid:16) q T (cid:17) , (32)and e P T = − g π X f | q f B | e − q ⊥ | qf B | " q ⊥ q q k ~q q T (cid:17) . (33)13he explicit calculations to obtain Eqs. (32) and (33) are shown in C. For this hierarchyof scales, we have already considered the massless limit m f → q becomes thesmallest energy scale. Also, notice that the coefficients P T , P L vanish when we take the limit q →
0. This shows that in this case, the matter contributions are all free of any infrareddivergences.
3. Magnetic corrections to the gluon Debye mass
If we now keep the fermion mass finite and work with the premises spelled out in subsec-tion II B 1, we can consider the matter contribution of Eqs. (30) and (31) to the dispersionrelation. For this purpose, we need to take the limit when the three-momentum goes to zero.Nevertheless, notice that the result may be different depending on whether the parallel orperpendicular momentum component, with respect to the magnetic field, is taken first tozero. This behaviour is due to the breaking of the spatial isotropy and is the analog to thepurely thermal case, where the limits when either q or | ~q | goes first to zero, do not commute,due to the loss of Lorentz (boost) invariance. In the present case, the limits when either q or q ⊥ go first to zero in general do not commute either. Let us explore the case when q ⊥ and q goes to zero at different rates writing q = a q ⊥ . Thus, the matter contribution inEqs. (30) and (31) can be written as e P L = g π X f | q f B | e − q ⊥ | qf B | " q ( q ⊥ + 2 aq ⊥ )(1 + a ) q ⊥ ( q − (1 + a ) q ⊥ ) − × ln (cid:16) π T m f (cid:17) − γ E ! = g π X f | q f B | e − q ⊥ | qf B | " q (1 + 2 a )(1 + a )( q − (1 + a ) q ⊥ ) − × ln (cid:16) π T m f (cid:17) − γ E ! . (34) e P T = g π X f | q f B | e − q ⊥ | qf B | " q ⊥ (1 + a ) q ⊥ ln (cid:16) π T m f (cid:17) − γ E ! = g π X f | q f B | e − q ⊥ | qf B | "
11 + a ln (cid:16) π T m f (cid:17) − γ E ! . (35)14otice that the behavior of e P T and e P L depend on a , giving rise to possibly differentmasses of the corresponding excitations when q ⊥ and q go to zero at different rates. Thus,let us use Eqs. (34) and (35) to study the Debye mass, defined as the solution of[ q − e P L,T ( q , q ⊥ , q )] | ~q =0 = 0 , (36)for q = m D , in the different cases. • q ⊥ → q finite:In this case, Eqs. (34) and (35) become e P L = g π X f | q f B | q q k ! ln (cid:16) π T m f (cid:17) − γ E !e P T = 0 , (37)which shows that transverse modes are not screened. If we now take q → m D ) L = g π X f | q f B | ln (cid:16) π T m f (cid:17) − γ E ! . (38) • q → q ⊥ finite:In this other case, Eqs. (34) and (35) become e P L = g π X f | q f B | (cid:18) q q − q ⊥ − (cid:19) ln (cid:16) π T m f (cid:17) − γ E !e P T = g π X f | q f B | ln (cid:16) π T m f (cid:17) − γ E ! . (39)If we now take q ⊥ → e P L is equal to zero and this time it is the transverse modeswhich develop a Debye mass given by( m D ) T = g π X f | q f B | ln (cid:16) π T m f (cid:17) − γ E ! , (40)whereas the longitudinal one is not screened. Notice that the right-hand side of thisexpression coincides with the Debye mass for the longitudinal mode in the previouscase given by Eq. (38). 15 q , q ⊥ → e P L = e P T = g π X f | q f B | ln (cid:16) π T m f (cid:17) − γ E ! , (41)and both the longitudinal and transverse modes develop a Debye mass given by( m D ) L = ( m D ) T = g π X f | q f B | ln (cid:16) π T m f (cid:17) − γ E ! . (42)Notice that when q and q ⊥ vanish at the same rate, both modes are screened and theDebye mass of the longitudinal mode is equal to that of the transverse mode.For the three cases( m D ) L + ( m D ) T = g π X f | q f B | ln (cid:16) π T m f (cid:17) − γ E ! . (43)We emphasize that the thermo-magnetic contribution to the polarization tensor can beexpressed using only the orthogonal tensors Π µνL and Π µνT , given by Eqs. (26). Indeed, onecould have thought that the tensor Π µν k could also develop a thermo-magnetic coefficient.However, notice that since the projections Π µνL Π k µν and Π µνT Π k µν are non-vanishing, one canalways express Π µν k as a linear combination of Π µνL and Π µνT . Therefore it is sufficient to obtainthe thermo-magnetic dependence of the coefficients of these latter, as has been expressed inEq. (28). In physical terms, this means that the matter corrections to the gluon dispersionproperties can be calculated over a background of a magnetized vacuum. III. SUMMARY AND DISCUSSION
In this work we have computed the gluon polarization tensor in a thermo-magneticmedium. The computation has been performed including the magnetic field effects bymeans of Schwinger’s proper time method. Although the vacuum polarization tensor forgauge fields has been previously studied in several other works (see for example Refs. [20–23, 26, 27]), here we have analytically studied in detail the strong field limit at zero andhigh temperature. The latter case has been implemented within the HTL approximation.For the T = 0 case, we have computed the coefficients of the tensor structures describingthe polarization tensor including the case where the quark mass is non-vanishing. We have16hown that in the LLL, the only surviving structure corresponds to the tensor Π µν k . Thecoefficient is given by an expression that develops an imaginary part, corresponding to thethreshold for quark-antiquark pair production, appearing when the parallel gluon momentumsquared is such that q k = 4 m f . For the case where the quark mass vanishes, this coefficientis real and equal to 1. To understand why it is that q k dictates the threshold behaviour,we have noticed that when the gluon splits into a virtual quark-antiquark pair, in the sameLandau level, the LLL in this case, the pair’s total transverse momentum vanishes. Thishappens because the quark moves in the opposite direction than the antiquark around thefield lines. From momentum conservation the motion of the virtual pair can only happenalong the direction of the external magnetic field and thus the kinematics is dictated by q k . This also helps understanding why the parallel tensor structure is the only one thatsurvives. Indeed, since the quark/antiquark motion is thrust by the chromo-electric field,only a polarization vector having a component along the external magnetic field can pushthe motion of the virtual pair. The only polarization vector, and thus tensor, having anon-vanishing projection along the magnetic field is the parallel structure [22].For large T , we have also implemented the calculation in the LLL approximation. Wehave explicitly separated the magnetized vacuum and the thermo-magnetic contributions.This is particularly useful for a renormalization group analysis of the gluon polarizationtensor, suited to extract the behavior of the strong coupling in a thermo-magnetic envi-ronment [36]. We have worked within the hierarchy of scales | eB | ≫ T to make sure thatthermal fluctuations do not induce transitions between higher Landau levels. In this approx-imation, temperature and magnetic field effects factorize, due to the dimensional reductionin the LLL. This factorization is not possible when T is larger than | eB | and to describesuch case, one would need to include in the calculation the contribution from other Landaulevels.We analyzed the coefficients of the two tensor structures that appear at finite temperatureand magnetic field and considered the cases q ≪ m f ≪ T and m f ≪ q ≪ T . We showedthat both cases are free from infrared divergences. In particular, we have obtained thethermo-magnetic behavior of the Debye mass from the gluon polarization tensor in the HTLand LLL approximations. However, since the magnetic field breaks rotational invariance,the result depends separately on the momentum components q and q ⊥ . Therefore one needsto consider distinct limits when either q and q ⊥ go to zero at different rates. We distinguish17hree different cases: When q ⊥ ( q ) goes first to zero, only the longitudinal (transverse) modedevelops a Debye mass. However, when both q ⊥ and q go to zero at the same rate, bothmodes develop the same Debye mass. Consequences of these different screening patternswill be explored in the future and reported elsewhere. ACKNOWLEDGMENTS
This work was supported by Consejo Nacional de Ciencia y Tecnolog´ıa grant number256494, by UNAM-DGAPA-PAPIIT grant number IG100219, by University of Cape Town,by Fondecyt (Chile) grant numbers 1170107, 1190192, Conicyt/PIA/Basal (Chile) grantnumber FB0821. R. Z. would like to thank support from CONICYT FONDECYT Iniciaci´onunder grant number 11160234. D. M. acknowledges support from a PAPIIT-DGAPA-UNAMfellowship.
Appendix A: Vacuum contribution in the HTL and LLL approximations
Here we show that the vacuum contribution, after the sum over Matsubara frequencies,is equal to the integral over k in Eq. (15). This means that the vacuum contribution at T, eB = 0 in the LLL and HTL approximations yields the same result as the case for T = 018n the presence of a magnetic field in the LLL approximation. We begin with Eq. (15) i Π µνa + i Π µνb = g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | × Z d k k (2 π ) k k − m f ][( k − q ) k − m f ] × h ( m f − k k · ( k − q ) k ) g µν k + k µ k ( k − q ) ν k + k ν k ( k − q ) µ k i = g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | × Z d k k (2 π ) k k − m f ][( k − q ) k − m f ] × h ( m f + k − k q + k q − k ) g µν k + 2 (cid:0) k − k ( q + q )+ k (2 k − q − q ) + k (cid:1)i = g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | × Z d k k (2 π ) k k − m f ][( k − q ) k − m f ] × h ( m f + k − k q ) g µν k + 2( k − k ( q + q ))+ k ( q g µν k + 2( k − q − q )) + k ( − g µν k + 2) i , (A1)Changing to Euclidean space by means of a Wick rotation and working in the imaginarytime formalism with q = iω and k = i e ω n , the polarization tensor becomes i Π µνa + i Π µνb = ig X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | T ∞ X n = −∞ Z dk (2 π ) × " ( m f + k − k q ) g µν k + 2( k − k ( q + iω ))[˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] − ( i e ω n )( iωg µν k + 2( k − q − iω ))[˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ]+ ( i e ω n ) ( − g µν k + 2)[˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] . (A2)19n order to compute the sum over Matsubara frequencies, we use the following expressions I = T ∞ X n = −∞ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ]= − X s ,s = ± s s E E h − ˜ f ( s E ) − ˜ f ( s E ) iω − s E − s E i I = T ∞ X n = −∞ i e ω n [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ]= − X s ,s = ± s s ( s E )4 E E h − ˜ f ( s E ) − ˜ f ( s E ) iω − s E − s E i I = T ∞ X n = −∞ ( i e ω n ) [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ]= − X s ,s = ± s s ( s E )( s E − iω )4 E E h − ˜ f ( s E ) − ˜ f ( s E ) iω − s E − s E i . (A3)We now substitute the vacuum pieces in Eqs. (A3) into Eq. (A2). This means that we onlyconsider the terms independent of the distribution functions ˜ f . We obtain i Π µνa + i Π µνb = − ig X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | × Z dk (2 π ) "(cid:2) ( m f + k − k q ) g µν k + 2( k − k ( q + iω )) (cid:3) × E E (cid:16) iω − E − E − iω + E + E (cid:17) − ( iωg µν k + 2( k − q − iω )) × E E E (cid:16) iω − E − E − iω + E + E (cid:17) + ( − g µν k + 2)4 E E − E E (cid:16) iω − E − E − iω + E + E (cid:17) + E ( iω ) (cid:16) iω − E − E + 1 iω + E + E (cid:17)! . (A4)20n the other hand, computing the integral over k in Eq. (15), one gets i Π µνa + i Π µνb = g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | × Z d k k (2 π ) k k − m f + iǫ ][( k − q ) k − m f + iǫ ] × h ( m f + k − k q ) g µν k + 2( k − k ( q + q ))+ k ( q g µν k + 2( k − q − q )) + k ( − g µν k + 2) i = g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | Z dk (2 π ) Z dk (2 π ) × h ( m f + k − k q ) g µν k + 2( k − k ( q + q )) × k ( q g µν k + 2( k − q − q )) + k ( − g µν k + 2) i × k − E + iǫ ][ k + E − iǫ ] × k − ( q + E ) + iǫ ][ k − ( q − E ) − iǫ ] , (A5)Integrating over k in the complex plane, where we take a closed path that encloses eitherthe upper or the lower half-plane, since, in either case, one always encloses two of the poles,the result is i Π µνa + i Π µνb = − ig X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | × Z dk (2 π ) "(cid:2) ( m f + k − k q ) g µν k + 2( k − k ( q + q )) (cid:3) × E E (cid:16) q − E − E − q + E + E (cid:17) − (cid:0) q g µν k + 2( k − q − q ) (cid:1) × E E E (cid:16) q − E − E − q + E + E (cid:17) + ( − g µν k + 2)4 E E − E E (cid:16) q − E − E − q + E + E (cid:17) + E q (cid:16) q − E − E + 1 q + E + E (cid:17)! . (A6)Comparing Eq. (A6) and Eq. (A4), we see that the result is the same upon the analyticalcontinuation iω → q . 21 ppendix B: Integrals for the polarization tensor in the LLL and HTL approxima-tions ( q ≪ m f ≪ T ) Here we show the explicit steps that lead to the results for the matter contributions inEqs. (30) and (31). First, we introduce temperature effects into the corresponding projec-tions onto e Π µν , Eq. (28), using the Matsubara formalism of thermal field theory, obtaining e P L = − g T ∞ X n = −∞ X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | Z dk (2 π ) × ( ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ) × (cid:26)h ω n ( − ˜ ω n ω − k q ) + 2˜ ω n ( ω n + q ) + 2 ω (˜ ω n + k + m f ) i ω~q + h ω n ω + k q + 2˜ ω n ωk q ) − ( ω n + q )(˜ ω n ω + k q + m f ) i ω ~q q + h ˜ ω n + 2˜ ω n ω − k + ˜ ω n ω + k q − m f i q ~q (cid:27) (B1)and e P T = − g T ∞ X n = −∞ X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | q ⊥ ~q Z dk (2 π ) × ( ( − ˜ ω n + k − m f − ˜ ω n ω − k q )[˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ) . (B2)We notice that since in the HTL approximation terms proportional to ˜ ω n and to k in thenumerators do not contribute, the calculation of Eqs. (B1) and (B2) involves only two kindsof sums over the Matsubara frequencies. These are explicitly given by [37] T ∞ X n = −∞ ω n − ω ) + ( k − q ) + m f ]= 12 E (1 − f ( E )) , (B3)and T ∞ X n = −∞ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ]= X s ,s = ± − s s E E h − ˜ f ( s E ) − ˜ f ( s E ) iω − s E − s E i , (B4)22ith E = k + m f , E = ( k − q ) + m f and ˜ f ( x ) = 1 / ( e x/T + 1). Using Eqs. (B3) and (B4)and continuing within the HTL approximation, we notice that the external momentum q is a soft energy scale compared with the temperature and quark mass, and therefore wecan neglect q and k · q in each of the numerators. We can therefore compute all the sumsappearing in Eqs. (B1) and (B2), ignoring the vacuum contributions, we obtain e P L = − g X f (cid:16) | q f B | π (cid:17) e − q ⊥ | qf B | × (Z dk f ( E ) E ! (cid:18) ω + ~q ~q q (cid:19) + Z dk − f ( E ) E ! (cid:18) q ω − ~q ~q q (cid:19)) (B5)and e P T = − g X f (cid:16) | q f B | π (cid:17) e − q ⊥ | qf B | q ⊥ ~q Z dk − f ( E ) E ! . (B6)We now turn to compute the matter contribution. For this we require an expression for Z ∞−∞ dk ˜ f ( E ) E = Z ∞−∞ dk k + m f ) / e √ ( k + m f ) /T + 1 . (B7)Using the general expression from Ref. [38] f n (˜ y ) = 1Γ( n ) Z ∞ dxx n − p x + ˜ y e √ x +˜ y + 1 , (B8)we identify Eq. (B7) as corresponding to the case with n = 1 and ˜ y = m f /T . In the limitwhere ˜ y is small, Eq. (B8) becomes f ( m f /T ) = −
12 ln (cid:16) m f πT (cid:17) − γ E + . . . , (B9)therefore we obtain Z ∞−∞ dk ˜ f ( E ) E = − (cid:16) ln (cid:16) m f πT (cid:17) + γ E (cid:17) . (B10)Using the above expression, we get e P L = − g π X f | q f B | e − q ⊥ | qf B | " − ω ( q ⊥ + 2 q ) ~q q − ln (cid:16) m f π T (cid:17) + 2 γ E ! (B11)and e P T = − g π X f | q f B | e − q ⊥ | qf B | " q ⊥ ~q ln (cid:16) m f π T (cid:17) + 2 γ E ! . (B12)Finally, to obtain Eqs. (30) and (31) we perform the analytical continuation iω → q , backto Minkowski space. 23 ppendix C: Integrals for the polarization tensor in the LLL and HTL approxima-tions ( m f ≪ q ≪ T ) We now compute the matter contribution in the same fashion as B but with the newhierarchy of scales m f ≪ q ≪ T . We show the explicit steps that lead to the results forthe matter contributions in Eqs. (32) and (33), where the internal momentum k is the largestenergy scale and the quark mass m f is the smallest one. First, we introduce temperatureeffects into the corresponding projections onto e Π µν , Eq. (28), using the Matsubara formalismof thermal field theory, obtaining e P L = − g T ∞ X n = −∞ X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | Z dk (2 π ) × ( ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ) × (cid:26)h ω n ( − ˜ ω n ω − k q ) + 2˜ ω n ( ω n + q ) + 2 ω (˜ ω n + k + m f ) i ω~q + h ω n ω + k q + 2˜ ω n ωk q ) − ( ω n + q )(˜ ω n ω + k q + m f ) i ω ~q q + h ˜ ω n + 2˜ ω n ω − k + ˜ ω n ω + k q − m f i q ~q (cid:27) (C1)and e P T = − g T ∞ X n = −∞ X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | q ⊥ ~q Z dk (2 π ) × ( ( − ˜ ω n + k − m f − ˜ ω n ω − k q )[˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ) . (C2)We notice that since in the HTL approximation terms proportional to m f , ˜ ω n , and k in thenumerators do not contribute, the calculation of Eqs. (C1) and (C2) involves only two kindsof sums over the Matsubara frequencies, T ∞ X n = −∞ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ]= X s ,s = ± − s s E E h − ˜ f ( s E ) − ˜ f ( s E ) iω − s E − s E i ≡ χ , (C3)24nd T ∞ X n = −∞ ˜ ω n [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ]= X s ,s = ± s s E E h − ˜ f ( s E ) − ˜ f ( s E ) iω − s E − s E i ≡ χ , (C4)with E = k + m f , E = ( k − q ) + m f and˜ f ( x ) = 1( e x/T + 1) . (C5)We observe that Eqs. (C3) and (C4) are related by χ = − E χ . Therefore, we should onlyfind an expression for the sum in Eq. (C4). In this approximation ( k > q > m f ) Eq. (C4) isgiven by T ∞ X n = −∞ ˜ ω n [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ]= − q q k h q e k /T T ( e k /T + 1) + 12 q e k /T ( e k /T − T ( e k /T + 1) + ... i , (C6)and thus, Eqs. (C1) and (C2) take the following form e P L = − g T ∞ X n = −∞ X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | Z dk (2 π ) × ( ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ) × (cid:26)h − ω n ω + 2 ω (˜ ω n + k ) i ω~q + h ω n ω + k q ) i ω ~q q + h ˜ ω n − k i q ~q (cid:27) (C7)and e P T = − g T ∞ X n = −∞ X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | q ⊥ ~q Z dk (2 π ) × ( − ˜ ω n + k [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ) . (C8)We can therefore compute all the sums appearing in Eqs. (C7) and (C8). Ignoring thevacuum contributions, we get e P L = g π X f | q f B | e − q ⊥ | qf B | " − ω + q − ω q q q ~q q k !(cid:16) q T (cid:17) (C9)25nd e P T = − g π X f | q f B | e − q ⊥ | qf B | " q ⊥ q q k ~q q T (cid:17) . (C10)Finally, to obtain Eqs. (32) and (33), we perform the analytical continuation iω → q ,back to Minkowski space.For consistency, we compute the coefficients e P L and e P T in an alternatively way. First,in Euclidean space, we add and subtract the term k + m f in the numerators, such that theintegrands look like e P L = − g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | × i T X n Z dk π ( (˜ ω n + k + m f )2 ( iω ) ~q [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] − k − m f ) ( iω ) ~q [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] − (˜ ω n + k + m f )(( iω ) + q ) ( iω ) ~q q [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ]+ [ k (( iω ) + q ) + ( iω ) m f ] iω ) ~q q [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] − (˜ ω n + k + m f ) q ~q [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ]+ 2( k + m f ) q ~q [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ) (C11)and e P T = − g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | × i T X n Z dk π ( (˜ ω n + k + m f ) q ⊥ ~q [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ]+ (2 k − k q ) q ⊥ ~q [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ) . (C12)We simplify Eqs. (C11) and (C12), and also ignore the terms propotional to m f , thus we26btain e P L = − g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | × i T X n Z dk π ( ( iω ) ~q [(˜ ω n − ω ) + ( k − q ) + m f ] − k
23 ( iω ) ~q [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] − (( iω ) + q ) ( iω ) ~q q [(˜ ω n − ω ) + ( k − q ) + m f ]+ k (( iω ) + q ) iω ) ~q q [˜ ω n + k + m f ][(˜ ω − ω ) + ( k − q ) + m f ] − q ~q [(˜ ω n − ω ) + ( k − q ) + m f ]+ 2 k q ~q [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ) (C13)and e P T = − g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | × i T X n Z dk π ( q ⊥ ~q [(˜ ω n − ω ) + ( k − q ) + m f ]+ (2 k − k q ) q ⊥ ~q [˜ ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ) . (C14)From Eqs. (C13) and (C14), we identify two kinds of sums over the Matsubara frequencieswhich are equal to Eqs. (B3) and (B4), where the notation does not change. Since, we workwith the hierarchy of energy scales m f ≪ q ≪ T , the matter terms from the sums can beexpressed in the following way T X n ω n − ω ) + ( k − q ) + m f ] ≈ − p ( k − q ) ˜ f (cid:0) ( k − q ) (cid:1) (C15)27nd T X n ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ≈ k p ( k − q ) ((cid:0) ˜ f ( k ) + ˜ f ( p ( k − q ) ) (cid:1)(cid:16) iω − k − p ( k − q ) − iω + k + p ( k − q ) (cid:17) + (cid:0) ˜ f ( k ) − ˜ f ( p ( k − q ) ) (cid:1)(cid:18) iω + k − p ( k − q ) − iω − k + p ( k − q ) (cid:19) , (C16)where in Eqs. (C15) and (C16), we have ignored the smallest energy scale, namely, thefermion mass. We now need to consider that the external momentum q is softer than thetemperature, therefore we neglect subdominant terms and perform a Taylor expansion in ˜ f around q = 0 to get T X n ω n − ω ) + ( k − q ) + m f ] ≈ − k ˜ f ( k ) (C17)and T X n ω n + k + m f ][(˜ ω n − ω ) + ( k − q ) + m f ] ≈ − k × ( k ˜ f ( k ) − q ( iω ) − q "(cid:16) q T (cid:17) e k /T ˜ f ( k ) + (cid:16) q k T (cid:17) e k /T (cid:18) k T ( e k /T −
1) + ( e k /T + 1) (cid:19) ˜ f ( k ) . (C18)Substituting Eqs. (C17) and (C18) into Eqs. (C13) and (C14), the pure matter contribution28o the coefficients becomes e P L = − g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | Z dk π ( − iω ) ~q (cid:16) k (cid:17) ˜ f ( k ) + 2( iω ) ~q (cid:16) k (cid:17) ˜ f ( k ) − ( iω ) ~q h(cid:16) q T (cid:17) e k /T ˜ f ( k ) + (cid:16) q k T (cid:17) e k /T (cid:18) k T ( e k /T −
1) + ( e k /T + 1) (cid:19) ˜ f ( k ) i + ( iω ) ~q q (( iω ) + q ) (cid:16) k (cid:17) ˜ f ( k ) − ( iω ) ~q q (( iω ) + q ) (cid:16) k (cid:17) ˜ f ( k )+ ( iω ) ~q q (( iω ) + q ) h(cid:16) q T (cid:17) e k /T ˜ f ( k ) + (cid:16) q k T (cid:17) e k /T (cid:18) k T ( e k /T −
1) + ( e k /T + 1) (cid:19) ˜ f ( k ) i + q ~q (cid:16) k (cid:17) ˜ f ( k ) − q ~q (cid:16) k (cid:17) ˜ f ( k )+ q ~q h(cid:16) q T (cid:17) e k /T ˜ f ( k ) + (cid:16) q k T (cid:17) e k /T (cid:18) k T ( e k /T −
1) + ( e k /T + 1) (cid:19) ˜ f ( k ) i) = g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | (cid:16) q ~q q k (cid:17)(cid:20) ( iω ) (cid:18) q q + 2 (cid:19) + q (cid:21)(cid:16) q πT (cid:17) (C19)and e P T = g X f (cid:16) π | q f B | π (cid:17) e − q ⊥ | qf B | Z dk π ( q ⊥ ~q (cid:16) k (cid:17) ˜ f ( k ) − q ⊥ ~q (cid:16) k (cid:17) ˜ f ( k )+ q q ⊥ ~q (( iω ) − q ) h(cid:16) q T (cid:17) e k /T ˜ f ( k ) + (cid:16) q k T (cid:17) e k /T (cid:18) k T ( e k /T −
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