Dispersion relations for fermions at high temperature and in a background
magnetic field are calculated in two different ways. First from a
straightforward one-loop calculation where, in the weak field limit, we find an
expression closely related to the standard dispersion relations in the absence
of the magnetic field. Secondly, we derive the dispersion relations directly
from the Hard Thermal Loop effective action, which allows for an exact solution
(i.e. to all orders in the external field), up to the last numerical integrals.Comment: Latex+epsf with uuencoded ps-figure

Elmfors, Per

English

arXiv

Dispersion relations for fermions at high temperature and in a background magnetic field are calculated in two different ways. First from a straightforward one-loop calculation where, in the weak field limit, we find an expression closely related to the standard dispersion relations in the absence of the magnetic field. Secondly, we derive the dispersion relations directly from the Hard Thermal Loop effective action, which allows for an exact solution (i.e. to all orders in the external field), up to the last numerical integrals.Dispersion relations for fermions at high temperature and in a background magnetic field are calculated in two different ways. First from a straightforward one-loop calculation where, in the weak field limit, we find an expression closely related to the standard dispersion relations in the absence of the magnetic field. Secondly, we derive the dispersion relations directly from the Hard Thermal Loop effective action, which allows for an exact solution (i.e. to all orders in the external field), up to the last numerical integrals

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arXiv:hep-ph/9510314v1  17 Oct 1995CERN-TH/95-274hep-ph/9510314Dispersion Relations from theHard Thermal Loop Effective Actionin a Magnetic FieldPer ElmforsTH-Division, CERN, CH-1211 Geneva 23, SwitzerlandEmail: elmfors@cern.chAbstractDispersion relations for fermions at high temperature and in a background magneticfield are calculated in two different ways. First from a straightforward one-loop cal-culation where, in the weak field limit, we find an expression closely related to thestandard dispersion relations in the absence of the magnetic field. Secondly, we derivethe dispersion relations directly from the Hard Thermal Loop effective action, whichallows for an exact solution (i.e. to all orders in the external field), up to the lastnumerical integrals.CERN-TH/95-274October 1995Contribution to The 4th International Workshop on Thermal Field Theories and their Applications,Dalian, China, 6 – 12 August, 1995.1 IntroductionWe know that thermal effects on dispersion relations are of extreme importance at hightemperature, where the whole concept of propagating particles is drastically changed [1, 2].Since these effects are large and influence the mass shell conditions, it is also importantto include them consistently to all orders and not only as first-order corrections. This isconsistent to leading order in (eT )2. Both fermions and gauge bosons develop new branchesand the zero-temperature masses become less important. New decay channels can open upbecause of the effective thermal masses of the propagating excitations. It is also interestingto study how strong external fields influence the physics of these effective modes. We haveconsidered two ways of calculating the dispersion relations for fermions at high temperaturein a constant background magnetic field. One method is to simply do the one-loop calcula-tion at finite temperature, using exact propagators in the external field [3]. The other is tostart from the already resummed effective action for Hard Thermal Loops [4, 5] (HTLs). Itis an effective action that generates all HTLs at tree level and which contains the externalfields to all orders.2 One-loop self-energyIn Ref. [3] the fermionic dispersion relation at finite temperature in a constant magneticfield was calculated using the electron propagator in the Furry picture and Schwinger’sproper-time method. In the high-temperature limit it turns out that Schwinger’s proper-time formulation [6] of the exact propagator is the easiest to use for the loop calculation.The thermal propagator can be constructed as [7]iSvac(p)− fF (p0)(iSvac(p)− iS∗vac(p)), (1)where for a magnetic field parallel to the z-direction1iSvac(p) =∫ ∞0dseieBsσzcos eBsexp[is(p2‖ −tan eBseBsp2⊥ −m2 + iε)]×{γp‖ − e−ieBsσzcos eBsγp⊥ +m}. (2)After performing the loop integrals in Σ(x′, x) = 〈x′|Σˆ|x〉 it is possible to extract the gauge-invariant operator Σˆ(p0, pz,Π⊥), where Π⊥ = (px − eAx, py − eAy), in a gauge with zeroz-component. (We shall use the gauge Aµ = (0, 0,−B x, 0).) Then we take the weak fieldlimit, but keep eB exactly wherever it is added linearly to the canonical momentum likeΠ2⊥−eBσz . The reason being that Π2⊥−eBσz = Π/2⊥, so eB can be reabsorbed in Π/2⊥, which1The sign convention here, that the particle has a positive charge e, differs from Ref. [3].1need not be small. When the final expression for the self-energy is used in the effectiveDirac equation, we obtain[Π/−m− Σˆ(p0, pz,Π⊥)]Ψ =[s(p0,Π2)γ0p0 − r(p0,Π2)γzpz − r(p0,Π2 − eBσz)Π/⊥ −m]Ψ = 0 , (3)where Π2 = Π2⊥ + p2z ands(p0,Π2) =(1− M22p0 |Π| ln∣∣∣∣∣p0 + |Π|p0 − |Π|∣∣∣∣∣), (4)r(p0,Π2) =(1 +M2Π2(1− p02 |Π| ln∣∣∣∣∣p0 + |Π|p0 − |Π|∣∣∣∣∣)). (5)The temperature dependence enters only through the thermal mass M = e2T 2/8. It isalmost possible to guess the expression in Eq. (3) from the standard expression for the HTLDirac equation [1, 2]. The usual momentum pµ should be replaced with the gauge-invariantmomentum Πµ, but there is an ambiguity in replacing p2 by Π2 or by Π/ Π/ . The correct wayfollows from the calculations in Ref. [3].In the massless limit (i.e. zero vacuum mass) the dispersion relation divides into a left-and a right-handed part. With a wave function Ψ = (L,R)T in the chiral representationthe two dispersion relations becomeL : (Esn + pzrn)(Esn−1 − pzrn−1)− 2eBnr2n− 12= 0 , (6)R : (Esn − pzrn)(Esn−1 + pzrn−1)− 2eBnr2n− 12= 0 , (7)where sn = s(E, p2z+eB(2n+1)) and similarly for rn. These relations are valid for all n ≥ 1,but in the lowest Landau level there is only one non-zero component for each of L and R,so the dispersion relations reduce toL : Es(E, p2z + eB) + pzr(E, p2z + eB) = 0 , (8)R : Es(E, p2z + eB)− pzr(E, p2z + eB) = 0 . (9)The numerical solution in the lowest Landau level is shown in Fig. 1 for the right-handed par-ticle. There are eight different branches corresponding to the states (L/R)× (particle/hole)× (positive/negative energy), for each given value of n ≥ 1 and py. For n = 0 there areonly half as many states, since only one spin orientation is possible in the lowest Landaulevel. Figure 1 only shows the right-handed branch, and the left-handed branch is obtainedby reflection in pz. In fact, from Eqs. (6, 8) we find the combined symmetries (L ↔ R,pz ↔ −pz), (L ↔ R,E ↔ −E) and (E ↔ −E, pz ↔ −pz). In particular, right-handedparticles in the lowest Landau level can only propagate along the magnetic field (which2-2.0 -1.0 0.0 1.0 2.00.00.51.01.52.0pzEZhZpERhERpeB = 0.2n = 0Figure 1: Dispersion relation and spectral weight for the right-handed branch in the lowestLandau level. All dimensionful parameters are given in units of the thermal mass M.points in the positive z-direction), while the left-handed particles and the right-handedholes propagate against the field.The spectral weight presented in Fig. 1 is defined byZi(pz, n)−1 =ddω∣∣∣∣∣ω=Ei(pz ,n)(Tr[(D(ω, pz, n)γ0)−1])−1, (10)where D is the 4 × 4 matrix in Eq. (3) that remains after acting on an off-shell Landaulevel. For a given chirality, Zi(pz) is obviously very asymmetric in pz since only particlespropagate in one direction and the holes in the other.3 HTL effective action in a background fieldInstead of computing the one-loop self-energy as in Section 2, we can use the fact that theHTL effective action already contains the leading high-temperature contribution to all ordersin the gauge field. The equation of motion that follows from a variation of the fermionicfield should immediately give the dispersion relation. The HTL effective action for QED3can be written as [8]:LHTL = −14F 2 +34M2γFµα〈uαuβ(∂u)2〉F µβ+ Ψ(i∂/− eA/−m)Ψ−M2Ψγµ〈uµu ·Π〉Ψ , (11)where the average 〈·〉 is defined by〈f(u0, ~u)〉 =∫ dΩ4πf(1, ~u) , (12)where ~u is a spatial unit vector. The equation of motion for Ψ that follows is[Π/−m−M2γµ〈uµu · Π〉]Ψ = 0 . (13)Equation (13) is a non-local and non-linear differential equation, which is, in general, verydifficult to deal with. What makes this equation much less tractable than the thermal Diracequation in the absence of the B-field is that the average over ~u is difficult to performbecause [Πµ,Πν ] = −ieFµν 6= 0. Since the spatial symmetries of the system are unalteredby the thermal heat bath, we still expect the eigenfunctions to have the same spatial form asat zero temperature. In fact, after performing the u-integral in Eq. (13) the result can onlybe a function of the invariants Π2⊥, p20 and pz, and the γ-structure has to be proportional toγΠ⊥, γ0p0 and γzpz. We shall therefore compute the matrix elements〈Φκ′ |〈uµu · Π〉|Φκ〉 , (14)between the vacuum eigenstates〈x|Φκ〉 = exp[i(−p0t+pyy+pzz)]In;py(x) , (15)In;py(x) =(eBπ)1/4exp[−12eB(x− pyeB)2]× 1√n!Hn[√2eB(x− pyeB)], (16)where κ = {p0, n, py, pz} and Hn[x] are Hermite polynomials. These states form a completeset of functions in four dimensions when the energy is off shell. In the chiral representa-tion suitable spinors can be formed from Φκ as Ψκ = diag[Φκ,Φκ−1,Φκ,Φκ−1]χ where χ isan undetermined space-time-independent spinor. The vacuum Dirac operator in Eq. (13)obviously gives an eigenvalue when acting on Ψκ, but it is more difficult to determine theaction of the thermal part since Φκ cannot be an eigenfunction to u ·Π for all u. One way tocalculate the matrix element in Eq. (14) is to find a basis such that v ·Π|vp〉 = v · p|vp〉 andinsert a unit operator∫d4p|vp〉〈vp|. This unit operator is, of course, independent of v after4the p-integration, so in particular we can choose v = u and change the order of integrations.After computing the matrix elements in Eq. (14) we find indeed that they are diagonal in κfor u0 and uz, and have a mixing with the first subdiagonals for ux and uy. Define 〈u0,z,±〉by〈Φκ′ |〈u0,zu · Π〉|Φκ〉 = (2π)3δκ′,κ〈u0,z〉κ , (17)〈Φκ′|〈ux ± iuyu ·Π〉|Φκ〉 = (2π)3δκ′,κ∓1〈u±〉κ , (18)and κ∓1 = {p0, n∓1, py, pz}. These are exactly the components that occur naturally whenwe include the γ-matrices in the chiral representation. The calculation of 〈u0,z,±〉 is a bitlengthy but straightforward, and the result reads〈u0〉κ = 1n!√2π∫ ∞−∞dsH2n(s)e−s2/2× pz2p2 lnp0 + pzp0 − pz +Es√2eB2p2√p20 − p2arctans√2eB2√p20 − p2 , (19)〈uz〉κ = 1n!√2π∫ ∞−∞dsH2n(s)e−s2/2×{−pzp2+p0(2p2z − eBs2)4p4lnp0 + pzp0 − pz+pz(2E2 − p2)2p4s√2eB√p20 − p2arctans√2eB2√p20 − p2 , (20)〈u+〉κ = i√2πn!(n− 1)!∫ ∞−∞dsHn(s)Hn−1(s)e−s2/2×{s√2eB2p2− Espz√2eB2p4lnp0 + pzp0 − pz+2p2z(p20 − p2)− p20eBs22p4√p20 − p2arctans√2eB2√p20 − p2 , (21)〈u−〉κ = −〈u+〉κ+1 , (22)where p2 = p2z + eBs2/2. With these definitions the Dirac equation effectively reduces to a4×4 matrix in the spinor indices, since the other quantum numbers have been diagonalized.In the massless limit we take the determinant to find the dispersion relations. They factorize5again as in Eqs. (6, 8) and the equations for the right-handed component aren ≥ 1 :(p0 − pz −M2(〈u0〉κ − 〈uz〉κ))×(p0 + pz −M2(〈u0〉κ−1 + 〈uz〉κ−1))−(√2eBn− iM2〈u+〉κ)2= 0 ,n = 0 : p0 − pz −M2(〈u0〉κ − 〈uz〉κ) = 0 .(23)In general there are imaginary parts in the functions 〈u0,z,±〉κ, which make the spectral-1.0 -0.5 0.0 0.5 1.00.81.01.21.41.6   pzEHTLWeak fieldeB = 0.0eB = 0.2n = 0Figure 2: Comparison of the dispersion relation from the HTL effective action and the weakfield approximation. All dimensionful parameters are given in units of the thermal mass M.functions more complicated. The imaginary parts of Eqs. (19) to (22) are determined bythe analytic continuation p0 → p0 + iǫ for positive p0; this amounts to the replacement1√p20 − p2arctans√2eB2√p20 − p2→ − 12√p2 − p20lns√2eB + 2√p2 − p20s√2eB − 2√p2 − p20− iπ√p2 − p20, (24)6for p2 > p20. It is anyway useful to solve Eq. (23), ignoring the imaginary part, since the realpart indicates where the spectral functions are peaked, at least if the imaginary parts aresmall enough. This can conveniently be done numerically as all the integrals in Eqs. (19)to (22) are well convergent. The result for the lowest Landau level in a weak magnetic field(eB = 0.2M2) is shown in Fig. 2, and it agrees rather well with the weak-field result fromEq. (8).-1.0 -0.5 0.0 0.5 1.00.00.51.01.52.0    pzEeB = 0.5eB = 1.0eB = 2.0eB = 10.0Figure 3: Dispersion relation for the right-handed branch in the lowest Landau level (n = 0).As the B-field increases thermal effects become less important and the dispersion approaches thelight cone, which is indicated by solid lines. All dimensionful parameters are given in units of thethermal mass M.It is, of course, more interesting to see what happens at larger field strength, whichcannot be treated by Eq. (8). The dispersion relations for several field strengths are shownin Fig. 3. Apart from the changes in the particle branch there is a new branch comingfrom the light cone, which eventually joins the hole branch and disappears. This newbranch can be understood mathematically, from Eq. (23), by studying it close to the lightcone. In the absence of the B-field, the hole branch exists because the logarithm in Eq. (4)becomes dominant close to the light cone, and the sign is such that it allows for a positiveenergy solution to the part of that Dirac equation which normally only gives the antiparticle7solution. In the present case, there is a compensation from the new terms in Eq. (23), whichchange the behaviour close to the light cone and a new branch can exist. As the B-fieldincreases the hole branch and its partner become less extended, and the particle branchapproaches a massless mode. This is physically very reasonable since for very strong fieldstrengths the thermal effects should disappear.At this point it is worth discussing the approximations involved. The HTL effectiveaction is derived under the assumption that the temperature is much larger than the externalmomenta. Here, the momentum is at least Π ∼ √eB, which should be kept smaller than T .But, if we consider the coupling constant e to be very small this approximation should bevalid even for eB ∼M2. On the other hand, when Π≫ T the vacuum part is dominant andthe dispersion relation is still approximately valid. The other approximation was to neglectthe imaginary part. This approximation becomes worse for increasing eB and cannot bemotivated for large eB. A new massless excitation, such as the new hole partner, could haveimportant physical consequences only if its spectral weight is non-negligible. This remainsto be studied and it can only be done correctly using the full spectral function.AcknowledgementsI would like to thank David Persson and Bo-Sture Skagerstam for collaboration on Ref. [3],on which this contribution to a large extent is based.References[1] V. V. Klimov, Collective excitations in a hot quark gluon plasma, Sov. Phys. JETP 55(1982) 199.[2] H. A. Weldon, Effective fermion masses of order gT in high temperature gauge theorieswith exact chiral invariance, Phys. Rev. D26 (1982) 2789; Dynamical holes in a quark-gluon plasma, Phys. Rev. D40 (1989) 2410.[3] P. Elmfors, D. Persson and B.-S. Skagerstam, Thermal fermionic dispersion relationsin a magnetic field, preprint CERN-TH/95-243, hep-ph/9509418.[4] E. Braaten and R. D. Pisarski, Soft amplitudes in hot gauge theories: a general analysis,Nucl. Phys. B337 (1990) 569.[5] J. Frenkel and J. C. Taylor, High temperature limit of thermal QCD, Nucl. Phys. B334(1990) 199; J. C. Taylor and S. M. H. Wong, The effective action of hard thermal loopsin QCD, Nucl. Phys. B346 (1990) 115.8[6] J. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664and Particles, sources and fields, Vol. 3 (Addison-Wesley Pub. Co., 1988).[7] W–y. Tsai, Modified electron propagator function in strong magnetic fields, Phys. Rev.D10 (1974) 1342.[8] E. Braaten, Effective theory for plasmas at all temperatures and densities, Can. J.Phys. 71 (1993) 215.9

Dispersion relations from the Hard Thermal Loop effective action in a magnetic field

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