Spin Polarized Photons from Axially Charged Plasma at Weak Coupling: Complete Leading Order
RRBRC-1158
Spin Polarized Photons from Axially ChargedPlasma at Weak Coupling: Complete Leading Order
Kiminad A. Mamo ∗ and Ho-Ung Yee , † Department of Physics, University of Illinois, Chicago, Illinois 60607 RIKEN-BNL Research Center, Brookhaven National Laboratory,Upton, New York 11973-5000
Abstract
In the presence of (approximately conserved) axial charge in the QCD plasma atfinite temperature, the emitted photons are spin-aligned, which is a unique P- andCP-odd signature of axial charge in the photon emission observables. We computethis “P-odd photon emission rate” in weak coupling regime at high temperaturelimit to complete leading order in the QCD coupling constant: the leading log aswell as the constant under the log. As in the P-even total emission rate in theliterature, the computation of P-odd emission rate at leading order consists of threeparts: 1) Compton and Pair Annihilation processes with hard momentum exchange,2) soft t- and u-channel contributions with Hard Thermal Loop re-summation, 3)Landau-Pomeranchuk-Migdal (LPM) re-summation of collinear Bremstrahlung andPair Annihilation. We present analytical and numerical evaluations of these contri-butions to our P-odd photon emission rate observable. ∗ e-mail: [email protected] † e-mail: [email protected] a r X i v : . [ h e p - ph ] A p r Introduction
Possible fluctuation of axial charge in QCD plasma through topological color field config-urations, either from initial color glass fields [1] or from thermal sphaleron transitions, isone of the fundamental aspects of QCD dynamics. Axial charge is both P- and CP-odd,and this distinct symmetry entails several interesting and unique phenomena associated toit, such as Chiral Magnetic Effect [2, 3, 4]. In Ref.[5] we explored and classified possible P-and CP-odd observables in photon and di-lepton emission rates, and found that the P- andCP-odd signals can be encoded in spin asymmetries of emitted photons and di-leptons.See Ref.[6] for a study of spin polarization of photons from a rotating plasma. Denotingthe photon emission rate with fixed photon helicity h = ± ± , the unique P- and CP-odd photon observable is ∗ A ± γ ≡ Γ + − Γ − Γ + + Γ − . (1.1)For di-leptons, let Γ s ,s be the rate with fixed helicities ( s , s ) = (cid:0) ± , ± (cid:1) of a leptonand anti-lepton pair respectively, and the P- and CP-odd observable is given by A ± l ¯ l ≡ Γ + , + − Γ − , − Γ + , + + Γ − , − , (1.2)whereas the total di-lepton emission rate (at a given momentum bin) isΓ l ¯ l = Γ + , + + Γ + , − + Γ − , + + Γ − , − . (1.3)As these observables share the same P- and CP-odd parities with the axial charge,their signals naturally arise from the axial charge of the QCD plasma. QCD is a P- andCP-even theory, and the axial charge can only exist as temporal and local fluctuations.The relaxation rate of axial charge via sphaleron transitions in a deconfined QCD plasmaat weak coupling is given by [7, 8] τ − R = (2 N F ) Γ sph χT ∼ α s log(1 /α s ) T , (1.4)where Γ sph is the sphaleron rate and χ is the charge susceptibility. The effect of smallquark mass m q to the relaxation rate is expected to be ∼ α s m q /T [9]. On the other hand,the photon and di-lepton emission rates for hard momenta comparable to T are d Γ /d k ∼ ∗ The Γ ± can be the differential rates in momentum space. EM α s log(1 /α s ) T at leading order. We will assume in our work that α s , α s ( m q /T ) (cid:28) α EM α s at sufficiently high temperature, so that the axial charge, once created by initialconditions or fluctuations, stays long enough to justify our computation of the above P-and CP-odd observables at weak coupling in the presence of an approximately constantvalue of axial chemical potential in the massless chiral limit. In this work, we will presentthe computation of A ± γ for photons with hard momenta at complete leading order in α s ,and postpone a computation of di-lepton observable A ± l ¯ l to a future study.In heavy-ion experiments, since the axial charge fluctuation averages to zero over manyevents, our observables should be measured either on the event-by-event basis, or one canlook at the average of the squared. If the latter is chosen, one needs to take care ofpossible background fluctuations as well.In Ref.[5], we derived explicit expressions relating the axial chemical potential to ourP- and CP-odd observables (1.1), (1.2). Letting the momentum direction of a photon bealong ˆ x , and defining G R ± ≡ ( G R ± iG R ) (rotational invariance dictates that G R = G R and G R = − G R ) where G Rij is the retarded correlation function of electromagnetic currentin momentum space † G Rij ( k ) = ( − i ) (cid:90) d x e − ikx θ ( x ) (cid:104) [ J i ( x ) , J j (0)] (cid:105) , (1.5)we found d Γ ± d k = e (2 π ) ω n B ( ω )( − (cid:2) ( (cid:15) µ ± ) ∗ (cid:15) ν ± G Rµν (cid:3) = e (2 π ) ω n B ( ω )( − G R ± , (1.6)for the emission rates with spin aligned polarization vectors (cid:15) µ ± = 1 √ , , ± i, . (1.7)Note that their sum is simply the total photon emission rate that has been computed inliterature. The difference that appears in our observable A ± γ is given by d Γ odd d k ≡ d Γ + d k − d Γ − d k = e (2 π ) ω n B ( ω )( − G R . (1.8)We will refer d Γ odd /d k simply as “P-odd photon emission rate” in the following. † Our definition of currents does not include an explicit factor of e in front, that is, they are “number”currents. G R ( k ) when k = | k | ˆ x arises from the P-odd part of the retarded correla-tion functions. Rotational invariance and Ward identity allow us to have a unique P-oddstructure in addition to the usual P-even part, G Rij ( k ) ∼ iσ χ ( k ) (cid:15) ijl k l , (1.9)which is in fact responsible for the Chiral Magnetic Effect at finite frequency-momentum k of the external magnetic field [10, 11], J = σ χ ( k ) e B ( k ) . (1.10)Since Re G R ( k ) = − Im σ χ ( k ), the P-odd emission rate d Γ odd /d k measures the imaginarypart of chiral magnetic conductivity σ χ ( k ) at light-like momenta. For small values of axialchemical potential, the chiral magnetic conductivity, and hence the P-odd emission rate,is proportional to the axial chemical potential. In our present study, although our resultsand expressions are in full dependency on axial chemical potential beyond linear order,we will present our numerical results only for linear dependency.Note that the Chiral Magnetic Effect at zero momentum limit that has been shownto be universal, lim k → σ χ ( k ) = N c π (cid:32)(cid:88) F Q F (cid:33) µ A ≡ σ , (1.11) does not contribute to the imaginary part of σ χ ( k ), and the P-odd photon emission rateis insensitive to this topological result. The imaginary part of σ χ ( k ) is a dynamics drivenquantity, and is highly sensitive to microscopic content and interactions of the theory.For example, its small frequency limit at zero spatial momentum was recently computedin Ref.[12] at leading log order in the QCD coupling α s = g / (4 π ) to findIm σ χ ( ω, ) = − ξ QCD5 ω + O ( ω ) , ξ QCD5 = − . g log(1 /g ) µ A T , (1.12)which appears in the first time-derivative correction to the Chiral Magnetic Effect as J = σ e B + ξ QCD5 e d B dt + · · · . (1.13)The computation of ξ shares many common features with that of the ordinary electricconductivity (which also has ∼ / ( g log(1 /g )) behavior), and is sensitive to the sameQCD dynamics that the electric conductivity is subject to. Nonetheless it relies on theexistence of axial chemical potential, dictated by P- and CP-odd parities.3n the next section 2, we will formalize the dynamical nature of Im σ χ ( k ) by introduc-ing the concept of “P-odd spectral density”, which naturally appears in the fluctuation-dissipation relation of P-odd part of current correlation functions. The section 3 presentsthe main steps and results of our computation of the P-odd photon emission rate d Γ odd /d k at complete leading order in α s . We summarize and discuss our results in section 4. One can formalize the dynamical nature of the imaginary part of chiral magnetic con-ductivity by the concept of “P-odd spectral density”, first introduced in Ref.[12] (seeAppendix 1 of that reference). We choose to discuss it in real-time Schwinger-Keldyshformalism, where we have two time contours joined at future infinity, one is going for-ward in time (labeled as contour 1) and the other is going backward (contour 2). Initialthermal density matrix is realized by attaching an imaginary time thermal contour at thebeginning time (at past infinity). By placing operators in suitable positions in the twocontours, one can generate all kinds of time orderings for correlation functions. In termsof “ra”-variables defined by O r = 12 ( O + O ) , O a = O − O , (2.14)our starting point is the thermal relation for the current-current correlation functions G rrij ( k ) = (cid:18)
12 + n B ( k ) (cid:19) (cid:0) G raij ( k ) − G arij ( k ) (cid:1) . (2.15)The retarded Green’s function is given in this notation by G Rij ( k ) = − iG raij ( k ) , (2.16)and by hermiticity of the current operator, the retarded Green’s function should be real-valued in coordinate space. This requires to have ( G Rij ( k )) ∗ = G Rij ( − k ) in momentumspace, or equivalently ( G raij ( k )) ∗ = − G raij ( − k ) . (2.17)On the other hand, by definition, G raij ( x ) = G arji ( − x ), so that in momentum space we have G arij ( k ) = G raji ( − k ) = − ( G raji ( k )) ∗ , (2.18)where the last equality comes from (2.17). 4n the relation (2.15), the left-hand side means the fluctuation amplitude, and theright-hand side, besides the statistical factor, represents the spectral density G rrij ( k ) = (cid:18)
12 + n B ( k ) (cid:19) ρ ij ( k ) , ρ ij ( k ) ≡ G raij ( k ) − G arij ( k ) . (2.19)The relation (2.18) gives us ρ ij ( k ) = G raij ( k ) + ( G raji ( k )) ∗ , (2.20)so that the spectral density is twice of the hermitian part of G raij ( k ) in terms of spatial i, j indices. In a P-even ensemble, rotational invariance dictates that G raij ( k ) be proportionalto δ ij or k i k j , and hence be symmetric with respect to i, j . The resulting spectral densityfrom this should then be real-valued by (2.20).In a P-odd ensemble, such as with axial chemical potential, rotational invariance allowsus to have a purely imaginary and anti-symmetric (and hence hermitian) spectral density, ρ ij ( k ) ∼ ρ odd ( k ) i(cid:15) ijl k l , (2.21)with a real valued function ρ odd ( k ). From (1.9), we have ρ odd ( k ) = − σ χ ( k ), that is,the P-odd spectral density is in fact the imaginary part of chiral magnetic conductivity.We see that the imaginary part of chiral magnetic conductivity governs P-odd thermalfluctuations of currents, while the topological real part at zero momentum limit (1.11)does not contribute to thermal fluctuations. This gives some intuition why Im σ χ ( k ) issubject to microscopic real-time dynamics of the theory.From (2.17), and (2.20), we have ρ odd ( − k ) = − ρ odd ( k ) . (2.22)Rotational invariance dictates that ρ odd ( k ) be a function of | k | , so ρ odd ( ω, | k | ) is an oddfunction on ω , similarly to P-even spectral densities. In small frequency, zero momentumlimit we expect to have ρ odd ( ω, ) ∼ ξ ω + · · · , ω → , (2.23)where the hydrodynamic transport coefficient ξ has the meaning of (1.13). As the signof ξ depends both on the chirality and the axial chemical potential, there seems to be noconcept of positivity constraint on it, contrary to electric conductivity. However, explicitcomputations indicate that the “relative” sign between σ (defined in (1.13)) and ξ is5lways negative, reminiscent of magnetic induction [12]. We are not yet aware of anyformal proof on this.Our P-odd photon emission rate is related to the P-odd spectral density via (1.8) by d Γ odd d k = − e (2 π ) n B ( ω ) ρ odd ( ω, k ) (cid:12)(cid:12) ω = | k | , (2.24)which explains that the P-odd photon emission rate, while it is P- and CP-odd, is adynamics driven observable. In this section, we compute the P-odd photon emission rate at complete leading order inQCD coupling α s , d Γ odd d k ≡ d Γ + d k − d Γ − d k ∼ α EM α s (log(1 /α s ) + c ) , (3.25)with an (approximate) axial chemical potential µ A in the chiral limit of QCD.A massless Dirac quark consists of a pair of left- and right-handed Weyl fermions. Atleading order in α s , the QCD interaction between them gives a higher order correction tothe photon emission rate, and hence we can treat them independently. This will be clearin the Feynman diagrams we compute in the following. The only effect of having the otherchiral Weyl fermion appears in the value of Debye mass m D in the gluon Hard ThermalLoop self-energy which enters the Landau-Pomeranchuk-Migdal (LPM) resummation ofcollinear Bremstrahlung and pair-annihilation that we compute in subsection 3.4. Wetherefore present our computational details only for the right-handed Weyl fermion withits chemical potential µ = µ A . The other left-handed Weyl fermion then has µ = − µ A ,and the total contribution to our P-odd photon emission rate is simply twice of thatfrom the right-handed Weyl fermion, up to the above mentioned modification of m D . Weassume our Dirac quark has a electromagnetic charge Q = +1, and the full result for twoflavor QCD is simply Q u + Q d = 59 , (3.26)times of the result for Q = +1 (where again m D has to include two flavor contributions).6e briefly summarize our notation and convention for a right-handed Weyl fermiontheory. Our metric convention is η = ( − , + , + , +). Let us define σ µ = ( , σ ) , ¯ σ µ = ( , − σ ) , (3.27)which satisfy σ µ ¯ σ ν + ¯ σ µ σ ν = − η µν . (3.28)The equation ( p · σ )( p · ¯ σ ) = − p = ( p ) − | p | , (3.29)and the following trace formula will be useful,Tr( σ µ ¯ σ ν σ α ¯ σ β ) = 2( η µν η αβ + η µβ η να − η µα η νβ + i(cid:15) µναβ ) . (3.30)The right-handed Weyl fermion action with QCD coupling g is L = iψ † σ µ ( ∂ µ − igt a A aµ ) ψ , (3.31)Upon quantization, we have ψ ( x ) = (cid:90) d p (cid:112) | p | (cid:16) u ( p ) a p e − i | p | t + i p · x + v ( p ) b †− p e i | p | t + i p · x (cid:17) , (3.32)where particle and antiparticle spinors are defined by( − σ · ˆ p ) u ( p ) = 0 , ( + σ · ˆ p ) v ( p ) = 0 , ˆ p ≡ p | p | , (3.33)with normalization u ( p ) u † ( p ) = − p · ¯ σ , v ( p ) v † ( p ) = − p · σ , p µ = ( | p | , p ) . (3.34)Note also that v ( − p ) v † ( − p ) = − p · ¯ σ . It will be convenient to define spin projectionoperators to quark/anti-quark states P s ( p ) ≡
12 ( + s ˆ p · σ ) = − s p s · ¯ σ | p | , p s ≡ ( s | p | , p ) , s = ± , (3.35)in terms of which the (bare) real-time propagators in “r/a” basis are S ra ( p ) = i p · ¯ σp (cid:12)(cid:12)(cid:12)(cid:12) p → p + i(cid:15) = (cid:88) s = ± ip − s | p | + i(cid:15) P s ( p ) ,S ar ( p ) = (cid:88) s = ± ip − s | p | − i(cid:15) P s ( p ) ,S rr ( p ) = (cid:18) − n + ( p ) (cid:19) ( S ra ( p ) − S ar ( p )) = (cid:18) − n + ( p ) (cid:19) ρ F ( p ) , (3.36)7here n ± ( p ) = 1 / ( e β ( p ∓ µ ) + 1) and the (bare) fermionic spectral density is ρ F ( p ) = (2 π ) (cid:88) s = ± δ ( p − s | p | ) P s ( p ) . (3.37)The Feynman rules are as usual, for example, for incoming (out-going) quark of mo-mentum p , we have u ( p ) ( u † ( p )), and for the incoming (out-going) antiquark of momen-tum p , we have v † ( − p ) ( v ( − p )). We remind ourselves of the rules for polarization statesas it is important to get the correct sign for our P-odd photon emission rate. For out-goingphoton of polarization (cid:15) µ , we attach ( (cid:15) µ ) ∗ contracted with the photon vertex ieσ µ in thediagram. The same is true for gluons. For incoming gluon of polarization ˜ (cid:15) µ , we attach ˜ (cid:15) µ contracted with the gluon vertex igt a σ µ . Finally, with these normalizations, the naturalmomentum integration measure is (cid:90) d p (2 π ) | p | . (3.38) The leading order rate consists of three distinct contributions: 1) Compton and Pair An-nihilation with hard (that is, comparable to T ) momentum exchanges, 2) Soft (that is,much less than T ) t-channel exchange contribution with IR divergence regulated by HardThermal Loop (HTL) re-summation of exchanged fermion line, and 3) collinear Brem-strahlung and pair-annihilation contributions induced by multiple scatterings with softthermal gluons, referred to as Landau-Pomeranchuk-Migdal (LPM) effect. The leadinglog result in α s is produced by 1) and 2), and the matching of the two logarithms from1) and 2) to have the cut-off dependence removed is an important consistency check forthe computation. We will see that this happens for our result.Our methods of computation for the above three contributions closely follow the well-known ones in literature [13, 14, 15, 16], and we apply them to our case of P-odd emissionrate, modulo a few subtleties. The complexity of numerical evaluation is somewhat heavierthan the P-even total emission rate.In this subsection, we describe hard Compton and Pair Annihilation rate computa-tions. Let the final photon momentum be k . For Pair Annihilation we label the momentaof incoming quark and antiquark pair by p and p (cid:48) respectively, and let k (cid:48) be the momen-tum of out-going gluon of polarization ˜ (cid:15) µ and color a . There are two Feynman diagrams8igure 1: Pair Annihilation diagrams with hard momentum exchanges.as in Figure 1 with the total amplitude given as M pair ( (cid:15) ± ) = − iegv † ( − p (cid:48) ) (cid:20) t a σ ν ( p − k ) · ¯ σ ( p − k ) σ µ + σ µ ( k − p (cid:48) ) · ¯ σ ( k − p (cid:48) ) t a σ ν (cid:21) u ( p )( (cid:15) ± µ ) ∗ (˜ (cid:15) ν ) ∗ , (3.39)where (cid:15) µ ± are the spin polarized photon states. Summing over colors in the squaredamplitude produces a simple color factor (cid:88) a tr( t a t a ) = C ( R ) d R = 12 ( N c − , (3.40)for the fundamental representation of SU ( N c ). The summation over gluon polarizationcan be replaced by (cid:88) ˜ (cid:15) (˜ (cid:15) ν ) ∗ ˜ (cid:15) ν (cid:48) → η νν (cid:48) , (3.41)thanks to Ward identities. Since our P-odd photon emission rate is the difference betweenthe rates with (cid:15) + and (cid:15) − , what we need is the difference |M pair ( (cid:15) + ) | − |M pair ( (cid:15) − ) | ≡ |M pair | , (3.42)and the Pair Annihilation contribution to the P-odd photon emission rate is written as(2 π ) ω d Γ odd d k = (cid:90) d p (2 π ) | p | (cid:90) d p (cid:48) (2 π ) | p (cid:48) | (cid:90) d k (cid:48) (2 π ) | k (cid:48) | (2 π ) δ ( p + p (cid:48) − k − k (cid:48) ) × |M pair | n + ( | p | ) n − ( | p (cid:48) | )(1 + n B ( | k (cid:48) | )) . (3.43)The computation of P-odd amplitude |M pair | is algebraically complicated, althoughconceptually straightforward. Using (3.34) and (3.41), and the polarization vectors (cid:15) µ ± = 1 √ , , ± i, , (3.44)9fter choosing k = | k | ˆ x , it reduces to computing traces of 8 σ matrices. After someamount of efforts, we obtain a compact expression |M pair | = C ( R ) d R · e g ( t − u ) (cid:32) t + 1 u − (cid:18) p ⊥ t − p (cid:48)⊥ u (cid:19) (cid:33) , (3.45)where t ≡ ( p − k ) , u ≡ ( k − p (cid:48) ) , and p ⊥ is the component of p perpendicular to thephoton momentum k .The momentum integration in the emission rate (3.43) with the above P-odd amplitudepossesses logarithmic IR divergences near t ∼ u ∼
0, corresponding to soft fermionexchanges. From the diagrams in Figure 1, it is clearly seen that the u ∼ t ∼ p ←→ p (cid:48) , t ←→ u , n + ( | p | ) ←→ n − ( | p (cid:48) | ) , (3.46)and we can replace singular ∼ /u terms in the amplitude with ∼ /t terms, so that theIR divergence appears in the new expression only around t ∼
0. Explicitly, we can havea replacement |M pair | n + ( | p | ) n − ( | p (cid:48) | )(1 + n B ( | k (cid:48) | )) −→ C ( R ) d R · e g (cid:18) − ut − t − u ) (cid:18) p ⊥ t − p ⊥ · p (cid:48)⊥ tu (cid:19)(cid:19) × ( n + ( | p | ) n − ( | p (cid:48) | ) − n − ( | p | ) n + ( | p (cid:48) | ))(1 + n B ( | k (cid:48) | )) . (3.47)The integral with the above new expression has an additional advantage besides theabsence of IR divergence near u ∼
0: from the new structure of distribution functionfactor, the fact that the result is an odd function on the chemical potential µ is manifest.For Compton scatterings, let us first consider the Compton scattering with incomingquark of momentum p and incoming gluon of momentum p (cid:48) . The momentum of out-goingquark will then be k (cid:48) . The kinematics is identical to the Pair Annihilation case with thesame definitions of t ≡ ( p − k ) , u ≡ ( k − p (cid:48) ) and s ≡ ( k + k (cid:48) ) . Note that t + u + s = 0 . (3.48)There are two Feynman diagrams as in Figure 2 with the amplitude M Comptonquark ( (cid:15) ± ) = − iegu † ( k (cid:48) ) (cid:20) σ ν ( p − k ) · ¯ σ ( p − k ) σ µ + σ µ ( k + k (cid:48) ) · ¯ σ ( k + k (cid:48) ) σ ν (cid:21) u ( p )( (cid:15) ± µ ) ∗ ˜ (cid:15) ν , (3.49)10igure 2: Compton scattering diagrams with hard momentum exchanges.where we omit color generators as it produces the same C ( R ) d R factor in the final result.The P-odd amplitude square is then computed after some amount of algebra as |M Comptonquark | ≡ |M Comptonquark ( (cid:15) + ) | − |M Comptonquark ( (cid:15) − ) | = C ( R ) d R · e g ( s − t ) (cid:32) t + 1 s − (cid:18) p ⊥ t + k (cid:48)⊥ s (cid:19) (cid:33) . (3.50)The P-odd emission rate with this Compton amplitude for quarks is given by(2 π ) ω d Γ odd d k = (cid:90) d p (2 π ) | p | (cid:90) d p (cid:48) (2 π ) | p (cid:48) | (cid:90) d k (cid:48) (2 π ) | k (cid:48) | (2 π ) δ ( p + p (cid:48) − k − k (cid:48) ) × |M Comptonquark | n + ( | p | )(1 − n + ( | k (cid:48) | )) n B ( | p (cid:48) | ) . (3.51)There arises a logarithmic divergence near t ∼ M Comptonantiquark ( (cid:15) ± ) = − iegv † ( − p ) (cid:20) σ µ ( k − p ) · ¯ σ ( k − p ) σ ν + σ ν ( − k − k (cid:48) ) · ¯ σ ( k + k (cid:48) ) σ µ (cid:21) v ( − k (cid:48) )( (cid:15) ± µ ) ∗ ˜ (cid:15) ν . (3.52)Besides to this sign flip compared to the quark Compton contribution, the distributionfunction n + in (3.51) has to be replaced by n − for anti-quarks, so the final Compton rate11s given as(2 π ) ω d Γ odd d k = (cid:90) d p (2 π ) | p | (cid:90) d p (cid:48) (2 π ) | p (cid:48) | (cid:90) d k (cid:48) (2 π ) | k (cid:48) | (2 π ) δ ( p + p (cid:48) − k − k (cid:48) ) × |M Comptonquark | ( n + ( | p | )(1 − n + ( | k (cid:48) | )) − n − ( | p | )(1 − n − ( | k (cid:48) | ))) n B ( | p (cid:48) | ) . (3.53)The fact the the result is an odd function on the chemical potential is also apparent here.To perform the phase space integrations in (3.43) and (3.53) with P-odd amplitudes(3.45) and (3.50), we follow the technique nicely introduced and explained in Refs.[17, 18].The idea is to introduce auxiliary energy variable q corresponding to either t-channel en-ergy transfer (“t-channel parametrization” according to Ref.[18]), or s-channel energytransfer (”s-channel parametrization”). Its essential role is to trade the angular integra-tion, coming from the energy δ -function, for a scalar integration of q . The price to payis a somewhat complicated, but manageable integration domain. The choice between t-channel and s-channel parametrizations is simply for convenience: t-channel parametriza-tion is convenient for terms with 1 /t , and vice versa for s-parametrization.We will give a brief summary on these parametrizations that one can also find in theoriginal Refs.[17, 18]. Let us focus on the common phase space integration measure in(3.43) and (3.53), (cid:90) d p (2 π ) | p | (cid:90) d p (cid:48) (2 π ) | p (cid:48) | (cid:90) d k (cid:48) (2 π ) | k (cid:48) | (2 π ) δ ( p + p (cid:48) − k − k (cid:48) ) . (3.54)For t-channel parametrization, we perform d k (cid:48) integration, and shift the integrationvariable p to q ≡ p − k to obtain ‡ (cid:90) d q (2 π ) | q + k | (cid:90) d p (cid:48) (2 π ) | p (cid:48) | | q + p (cid:48) | (2 π ) δ ( | q + k | + | p (cid:48) | − | k | − | q + p (cid:48) | ) . (3.55)We then introduce a variable q to write the energy δ function as δ ( | q + k | + | p (cid:48) | − | k | − | q + p (cid:48) | ) = (cid:90) + ∞−∞ dq δ (cid:0) | q + k | − | k | − q (cid:1) δ (cid:0) q + | p (cid:48) | − | q + p (cid:48) | (cid:1) , (3.56)where the meaning of Q ≡ ( q , q ) as the t-channel exchange momentum is obvious. ‡ When we perform k (cid:48) integration or shift the integration variable to q , we should of course keep trackof their effects in the amplitude and distribution function parts. We will present final summary on theseparts as well. q , ˆ x (cid:48) , ˆ x ) form an orthonormalbasis rotated by θ qk , and p (cid:48)(cid:48) is a projection of p (cid:48) onto the ( ˆ x (cid:48) , ˆ x ) plane.The next step is to express the energy δ -functions in terms of angle variables. Denotingthe angle between q and k as θ , we have δ (cid:0) | q + k | − | k | − q (cid:1) = | k | + q | q || k | δ (cos θ − cos θ qk ) , (3.57)where cos θ qk = ( q ) − | q | + 2 | k | q | q || k | . (3.58)There appears constraints on ( q , | q | ) simply from the requirement that | cos θ qk | ≤ θ (cid:48) between q and p (cid:48) we have δ (cid:0) q + | p (cid:48) | − | q + p (cid:48) | (cid:1) = | p (cid:48) | + q | p (cid:48) || q | δ (cos θ (cid:48) − cos θ p (cid:48) q ) , (3.59)with cos θ p (cid:48) q = ( q ) − | q | + 2 | p (cid:48) | q | p (cid:48) || q | . (3.60)Using these, one can perform the angular integrals of cos θ from d q and cos θ (cid:48) from d p (cid:48) ,localizing cos θ and cos θ (cid:48) to the values cos θ qk and cos θ p (cid:48) q . Since we need to compute p ⊥ = q ⊥ and p (cid:48)⊥ that appear in the P-odd amplitudes, it is convenient to fix the photonmomentum direction to be along ˆ x , and using the overall rotational symmetry in ( x , x )-plane, we can align q to be in ( x , x ) plane. See Figure 3 for the illustration. Thisalignment will produce a trivial (2 π ) azimuthal integration factor in the integral of d q .13igure 4: The integration domain of ( q , | q | ) (shaded blue). The domain for | p (cid:48) | is | p (cid:48) | > ( | q | − q ) /
2. The soft region A (shaded red) is responsible for leading log IRdivergence, and the region B produces the energy logarithm that is described in thefollowing.Note that the azimuthal angle φ of p (cid:48) with respect to q as defined in Figure 3 still has tobe integrated explicitly. From the geometry in Figure 3, we have q ⊥ = ( | q | sin θ qk , , (3.61)in ( x , x ) plane, and the p (cid:48) in ( x , x , x )-basis is given as p (cid:48) = | p (cid:48) | cos θ qk θ qk − sin θ qk θ qk sin θ p (cid:48) q cos φ sin θ p (cid:48) q sin φ cos θ p (cid:48) q = | p (cid:48) | cos θ qk sin θ p (cid:48) q cos φ + sin θ qk cos θ p (cid:48) q sin θ p (cid:48) q sin φ − sin θ qk sin θ p (cid:48) q cos φ + cos θ qk cos θ p (cid:48) q , (3.62)which will be used in computing the P-odd amplitudes (3.45) and (3.50). Finally, theintegration domain for ( q , | q | , | p (cid:48) | ) is depicted in Figure 4.From all these, the phase space integration in the t-channel parametrization becomes (cid:90) d p (2 π ) | p | (cid:90) d p (cid:48) (2 π ) | p (cid:48) | (cid:90) d k (cid:48) (2 π ) | k (cid:48) | (2 π ) δ ( p + p (cid:48) − k − k (cid:48) )= 18(2 π ) | k | (cid:90) ∞ d | q | (cid:90) | q | max( −| q | , | q |− | k | ) dq (cid:90) ∞ | q |− q d | p (cid:48) | (cid:90) π dφ . (3.63)For the amplitudes, we need to express various quantities in terms of integration variablesand the angles θ kq and θ p (cid:48) q . The following expressions can be derived from (3.61) and143.62) and the previous definitions: t = − ( q ) + | q | , u = 2 | k || p (cid:48) | (1 + sin θ qk sin θ p (cid:48) q cos φ − cos θ qk cos θ p (cid:48) q ) , q ⊥ = | q | sin θ qk , q ⊥ · p (cid:48)⊥ = | q || p (cid:48) | (cid:0) sin θ qk cos θ qk sin θ p (cid:48) q cos φ + sin θ qk cos θ p (cid:48) q (cid:1) ,s = − t − u , p ⊥ = q ⊥ , k (cid:48)⊥ = q ⊥ + p (cid:48)⊥ , (3.64)where θ kq and θ p (cid:48) q are given by (3.58) and (3.60). Finally, for the arguments that enterthe distribution functions, we have | p | = q + | k | , | k (cid:48) | = q + | p (cid:48) | . (3.65)The above data are enough, at least numerically, to compute the phase space integrationsin (3.43) and (3.53) to obtain our P-odd emission rate from the hard Compton and PairAnnihilation processes. This t-channel parametrization is not efficient for the terms of ∼ /s or ∼ /s type, for which we use s-channel parametrization.The geometry of s-channel parametrization is similar, so we simply summarize it. Thephase space measure becomes (cid:90) d p (2 π ) | p | (cid:90) d p (cid:48) (2 π ) | p (cid:48) | (cid:90) d k (cid:48) (2 π ) | k (cid:48) | (2 π ) δ ( p + p (cid:48) − k − k (cid:48) )= 18(2 π ) | k | (cid:90) ∞| k | dq (cid:90) q | | k |− q | d | q | (cid:90) q | q | q −| q | d | p | (cid:90) π dφ , (3.66)and we have s = − ( q ) + | q | , t = 2 | k || p | (1 + sin θ qk sin θ pq cos φ − cos θ qk cos θ pq ) , q ⊥ = | q | sin θ qk , p ⊥ · q ⊥ = | p || q | (cid:0) sin θ qk cos θ qk sin θ pq cos φ + sin θ qk cos θ pq (cid:1) , k (cid:48)⊥ = q ⊥ , (3.67)where cos θ qk = | q | − ( q ) + 2 q | k | | q || k | , cos θ pq = | q | − ( q ) + 2 q | p | | q || p | , (3.68)and finally, we have to replace | p (cid:48) | → q − | p | , | k (cid:48) | → q − | k | , (3.69)in the arguments of distribution functions.The φ integrations in both t-channel and s-channel methods are at most of the type (cid:90) π dφ A + B cos φC + D cos φ , (3.70)15hich can be done analytically. The rest parts of the integration have to be done nu-merically, but we can identify the leading log parts of log(1 /α s ) and log( ω/T ) for ω (cid:29) T analytically (recall ω = | k | ), which we now describe. The Pair Annihilation contribution (3.43) with (3.45) has a logarithmic IR divergence near t ∼
0, or when ( q , | q | ) (cid:28) | k | , | p (cid:48) | in the t-channel parametrization. The same is true forthe Compton rate (3.53) with (3.50). These divergences are regulated by including HTLself-energy [19] in the t-channel fermion propagator, which screens the fermion exchangefor soft momenta ( q , | q | ) (cid:46) gT (“soft region”). When ( q , | q | ) (cid:29) gT (“hard region”),the HTL correction is sub-leading in α s and what we have in the above as hard Comptonand Pair Annihilation contributions give the leading order result.A practical way to organize the leading order contributions from both regions is tointroduce an intermediate scale gT (cid:28) q ∗ (cid:28) T [20], which serves as a t-channel IR cutofffor the above hard Compton and Pair Annihilation rates in the hard region, and as at-channel UV cutoff for the same rates in the soft region with now the HTL self-energyincluded in the fermion propagator. The latter soft region will be described in the nextsubsection 3.2. The two logs of log q ∗ from both regions have to match to produce a finalresult independent of q ∗ : after identifying log q ∗ from each region, we neglect q ∗ /T and( gT ) /q ∗ corrections in the rest parts of the two regions by sending q ∗ → q ∗ → ∞ in the soft region. The resulting (numerical) constant is the leadingorder constant under the log.Let us identify the leading log from the hard region in this subsection. The t-channelparametrization is most efficient for this purpose. The q ∗ is introduced as an IR cutoff of d | q | -integral in (3.63) § :(2 π ) ω d Γ oddhard d k = 18(2 π ) | k | (cid:90) ∞ q ∗ d | q | (cid:90) | q | max( −| q | , | q |− | k | ) dq (cid:90) ∞ | q |− q d | p (cid:48) | (cid:90) π dφ I , (3.71)where I is the sum of the integrands in (3.47) and (3.53) from the Compton and Pair § This meaning of q ∗ has to be identical to the one in the soft region computation in subsection 3.2. I = C ( R ) d R · e g (cid:18) − ut − t − u ) (cid:18) q ⊥ t − q ⊥ · p (cid:48)⊥ tu (cid:19)(cid:19) × ( n + ( q + | k | ) n − ( | p (cid:48) | ) − n − ( q + | k | ) n + ( | p (cid:48) | ))(1 + n B ( q + | p (cid:48) | ))+ C ( R ) d R · e g ( s − t ) (cid:32) t + 1 s − (cid:18) q ⊥ t + ( q ⊥ + p (cid:48)⊥ ) s (cid:19) (cid:33) × (cid:0) n + ( q + | k | )(1 − n + ( q + | p (cid:48) | )) − n − ( q + | k | )(1 − n − ( q + | p (cid:48) | )) (cid:1) n B ( | p (cid:48) | ) , (3.72)with the use of expressions in (3.64) and (3.65) for the t-channel parametrization.From the distribution functions, | p (cid:48) | integral is dominated by | p (cid:48) | ∼ T . The logdivergence appears in small ( q , | q | ) (cid:28) | k | , | p (cid:48) | ∼ T since we assume hard photons T (cid:46) | k | .Figure 4 shows this region (region A ). In this case, from (3.58) and (3.60), we havecos θ qk ≈ cos θ p (cid:48) q ≈ q | q | , (3.73)and the leading behavior in A comes from the terms of ( u, s ) /t or ( u, s ) q ⊥ /t types, whichgives after some algebra, I ∼ C ( R ) d R · e g | k || p (cid:48) || q | (1 + cos φ ) × ( n + ( | k | ) n − ( | p (cid:48) | )(1 + n B ( | p (cid:48) | )) + n + ( | k | ) n B ( | p (cid:48) | )(1 − n + ( | p (cid:48) | )) − ( n + ↔ n − ))= C ( R ) d R · e g | k || p (cid:48) || q | (1 + cos φ ) × ( n + ( | k | ) n − (0) − n − ( | k | ) n + (0)) ( n + ( | p (cid:48) | ) + n − ( | p (cid:48) | ) + 2 n B ( | p (cid:48) | )) , (3.74)where in the last line, we use an interesting identity n ∓ ( | p (cid:48) | )(1 + n B ( | p (cid:48) | )) + n B ( | p (cid:48) | )(1 − n ± ( | p (cid:48) | )) = n ∓ (0) ( n + ( | p (cid:48) | ) + n − ( | p (cid:48) | ) + 2 n B ( | p (cid:48) | )) . (3.75)We then have a leading log behavior(2 π ) ω d Γ oddhard d k ∼ C ( R ) d R · e g (2 π ) ( n + ( | k | ) n − (0) − n − ( | k | ) n + (0)) × (cid:90) ∼ Tq ∗ d | q | | q | (cid:90) | q |−| q | dq (cid:90) ∞ d | p (cid:48) | | p (cid:48) | ( n + ( | p (cid:48) | ) + n − ( | p (cid:48) | ) + 2 n B ( | p (cid:48) | )) ∼ C ( R ) d R · e g (2 π ) (cid:0) π T + µ (cid:1) ( n + ( | k | ) n − (0) − n − ( | k | ) n + (0)) log ( T /q ∗ )= d R e (2 π ) m f ( n + ( | k | ) n − (0) − n − ( | k | ) n + (0)) log ( T /q ∗ ) , (3.76)17here we use (cid:90) ∞ d | p (cid:48) | | p (cid:48) | ( n + ( | p (cid:48) | ) + n − ( | p (cid:48) | ) + 2 n B ( | p (cid:48) | )) = 12 (cid:0) π T + µ (cid:1) , (3.77)and in the last line we write the result in terms of the asymptotic fermion thermal mass m f = C ( R ) g (cid:18) T + µ π (cid:19) . (3.78)We will check that the leading log from the hard Compton and Pair Annihilation givenin (3.76) nicely matches to the soft region result with HTL re-summation in the nextsubsection.For an ultra-hard photon energy ω = | k | (cid:29) T , there appears a logarithmic rise oflog( ω/T ) in the energy dependence of the leading order constant under the log. We closethis subsection by identifying this “energy logarithm”. For this aim, it is convenient towork with the light cone variables q ± ≡ | q | ± q , (3.79)with the measure change d | q | dq = 2 dq + dq − . The energy logarithm appears in the domainwhere q − (cid:46) | p (cid:48) | ∼ T (cid:28) q + (cid:28) | k | = ω , (3.80)which is also indicated in Figure 4 (region B ). In this case, we havecos θ qk ≈ q | q | ≈ , cos θ p (cid:48) q = − q + q − + 2 q | p (cid:48) | | p (cid:48) || q | ≈ − q − | p (cid:48) | , (3.81)and the leading behavior in A arises again from the same ( u, s ) /t or ( u, s ) q ⊥ /t terms,with I ∼ C ( R ) d R · e g | k | q + ( n + ( | k | ) ( n − ( | p (cid:48) | ) + n B ( | p (cid:48) | )) − ( n + ↔ n − )) , (3.82)so that we have(2 π ) ω d Γ oddhard d k ∼ C ( R ) d R e g (2 π ) (cid:90) | k |∼ T dq + q + (cid:90) ∞ dq − (cid:90) ∞ q − d | p (cid:48) |× ( n + ( | k | ) ( n − ( | p (cid:48) | ) + n B ( | p (cid:48) | )) − ( n + ↔ n − ))= C ( R ) d R e g (2 π ) log( | k | /T ) (cid:18) n + ( | k | ) (cid:90) ∞ dq − q − (cid:0) n − ( q − ) + n B ( q − ) (cid:1) − ( n + ↔ n − ) (cid:19) , (3.83)18here in the first line, we can safely let the upper cutoff of q − be infinity, due to thepresence of effective cutoff by the distribution functions (more precisely, the cutoff isgiven by ∼ q + (cid:29) T ).The integrals that appear in the above (cid:90) ∞ dq − q − (cid:0) n ∓ ( q − ) + n B ( q − ) (cid:1) = T (cid:0) π − (cid:0) − e ∓ µ/T (cid:1)(cid:1) , (3.84)are not simple polynomials in T and µ , contrary to the case of leading log in coupling(3.76). In this subsection, we compute the soft t-channel contributions from Compton and PairAnnihilation processes, whose IR divergence is regulated by re-summing fermion HTLself-energy in the fermion exchange line. Following the original treatment in Refs.[13, 14],we compute this directly in terms of 1-loop current-current correlation functions thatenter the emission rate formula (1.6) or (1.8), with one internal fermion line being soft,and hence HTL re-summed, corresponding to soft t-channel exchange. The emission ratewritten in (1.6) is given by suitable imaginary part of the correlation functions, and byapplying the cutting-rule, it is easy to see that the result should be equivalent to that fromcomputing Feynman diagrams of only t-channel Compton and Pair Annihilation processes(with the HTL re-summed propagator) that we described in the previous subsection.We compute the following with the soft t-channel momentum with an UV cutoff q ∗ ,(2 π ) ω d Γ( (cid:15) ± ) d k = e n B ( ω )( −
2) Im (cid:2) ( (cid:15) µ ± ) ∗ (cid:15) ν ± G Rµν ( k ) (cid:3) = e n B ( ω ) 2 Re (cid:2) ( (cid:15) µ ± ) ∗ (cid:15) ν ± G raµν ( k ) (cid:3) . (3.85)Since ( (cid:15) µ ± ) ∗ (cid:15) ν ± is a hermitian matrix in terms of µ, ν indices, the emission rate picks uponly the hermitian part of G raµν ( k ). There are two real-time Feynman diagrams for G raµν ( k )depicted in Figure 5, which gives G raµν ( k ) = ( − d R (cid:90) d p (2 π ) tr [ σ ν S rr ( p ) σ µ S ra ( p + k ) + σ ν S ar ( p ) σ µ S rr ( p + k )] , (3.86)where d R is the dimension of color representation. Recall the thermal relation S rr ( p ) = (cid:18) − n + ( p ) (cid:19) ( S ra ( p ) − S ar ( p )) ≡ (cid:18) − n + ( p ) (cid:19) ρ F ( p ) , (3.87)19igure 5: Two real-time Feynman diagrams for G raµν ( k ) in the “ra”-basis.and by the reality property S ar ( p ) † = − S ra ( p ), S rr ( p ) and ρ F ( p ) are hermitian matricesin terms of 2 component spinor indices. Using the same relations and the hermiticity of σ µ , it is easy to find the hermitian part of G raµν ( k ) as (we denote ω ≡ k = | k | ) G raµν ( k ) + ( G raνµ ( k )) ∗ = d R (cid:90) d p (2 π ) (cid:0) n + ( p ) − n + ( p + ω ) (cid:1) tr [ σ ν ρ F ( p ) σ µ ρ F ( p + k )] . (3.88)The emission rate is given solely by (fermion) spectral density ρ F , which conforms to theexpectation from cutting rules.Bare fermion spectral density is easy to read off from (3.36) or (3.37): ρ bare F ( p ) = (2 π ) (cid:88) s = ± δ ( p − s | p | ) P s ( p ) , (3.89)with the projection operators we repeat here for convenience, P s ( p ) = 12 ( + s ˆ p ) = − s ¯ σ · p s | p | , p µs ≡ ( s | p | , p ) . (3.90)In general, fermion spectral density in a Weyl fermion theory including HTL self-energyis written as (see Appendix 2 of Ref.[12]), ρ HTL F ( p ) = (cid:88) s ρ HTL s ( p ) P s ( p ) , ρ HTL s ( p ) = − (cid:20) p − s | p | + Σ R, HTL s ( p ) (cid:21) , (3.91)where the HTL self-energy is given byΣ R, HTL s ( p ) = − m f | p | (cid:18) s + (cid:18) − s p | p | (cid:19) log (cid:18) p + | p | + i(cid:15)p − | p | + i(cid:15) (cid:19)(cid:19) , (3.92)20ith the asymptotic fermion thermal mass that is introduced before in (3.78), m f = C ( R ) g (cid:18) T + µ π (cid:19) . (3.93)Inserting (3.88) into (3.85), choosing the direction of k = | k | ˆ x explicitly and comput-ing the σ -matrix traces using (3.30), we end up to an expression for our P-odd emissionrate as (2 π ) ω d Γ odd d k = d R e n B ( ω ) (cid:90) d p (2 π ) (cid:0) n + ( p ) − n + ( p + ω ) (cid:1) × (cid:88) s,t ρ s ( p ) ρ t ( p + k ) (cid:18) t ( p + | k | ) | p + k | − s p | p | (cid:19) , (3.94)where ρ s,t in the above can be either bare or HTL, depending on whether the momentumargument is hard or soft. We should consider the region of p where one of the twomomenta, p or p + k , is soft, corresponding to soft t- or u-channel processes.It would be convenient to combine the two soft regions into one, say soft p region.That is, for soft p + k region, let us change the variable p → − p − k , so that in the newvariable, p is soft. Under this transform, we have n + ( p ) − n + ( p + ω ) → n − ( p ) − n − ( p + ω ) , ρ s ( p ) → ρ − s ( p + k ) , ρ t ( p + k ) → ρ − t ( p ) , (3.95)and relabeling − t → s and − s → t , we arrive at the precisely the same form as in (3.97),with the replacement (cid:0) n + ( p ) − n + ( p + ω ) (cid:1) → − (cid:0) n − ( p ) − n − ( p + ω ) (cid:1) , (3.96)therefore, we can study only the soft p region of the following expression(2 π ) ω d Γ oddsoft d k = d R e n B ( ω ) (cid:90) d p (2 π ) (cid:0) n + ( p ) − n + ( p + ω ) − ( n + ↔ n − ) (cid:1) × (cid:88) s,t ρ HTL s ( p ) ρ bare t ( p + k ) (cid:18) t ( p + | k | ) | p + k | − s p | p | (cid:19) , (3.97)where we explicitly indicated the HTL (bare) spectral density for soft (hard) p ( p + k ). Anadditional bonus is that the result is manifestly an odd function in the chemical potential.This is reminiscent of what happens in our previous computation of hard Compton andPair Annihilation processes. 21rom ρ bare t ( p + k ) = (2 π ) δ ( p + | k | − t | p + k | ) , (3.98)and since p is soft while ( ω = | k | , k ) is hard, we see that only t = 1 contributes. Thetotal integrand has a rotational symmetry on ( x , x )-plane, so the azimuthal integralof p around k will trivially give (2 π ). The polar integration can be done by the sametechnique we use in (3.58): for p (cid:28) k , we can write the integral measure including theenergy δ -function as (cid:90) d p (2 π ) (2 π ) δ ( p + | k | − | p + k | ) = 1(2 π ) (cid:90) ∞ d | p || | p | (cid:90) | p |−| p | dp (cid:18) p | k | (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) p →| p | cos θ pk , (3.99)where cos θ pk = ( p ) − | p | + 2 p | k | | p || k | . (3.100)Using this, our P-odd rate (3.97) from soft region is compactly written as(2 π ) ω d Γ oddsoft d k = d R e (2 π ) n B ( ω ) (cid:90) q ∗ d | p || | p | (cid:90) | p |−| p | dp (cid:18) p | k | (cid:19) × (cid:0) n + ( p ) − n + ( p + ω ) − ( n + ↔ n − ) (cid:1) × (cid:88) s ρ HTL s ( p , | p | ) (cid:18) | p | cos θ pk + | k | p + | k | − s cos θ pk (cid:19) , (3.101)where we introduce the UV cutoff q ∗ for the t-channel momentum integral of | p | to regulatethe logarithmic diveregence. The meaning of q ∗ here is identical to that used in the hardCompton and Pair Annihilation rates in the previous subsection, which is important toget the correct leading order constant under the log.Since the cutoff is q ∗ (cid:28) T (cid:46) | k | (while q ∗ (cid:29) m f ∼ gT ), we have a further simplifica-tion at leading order tocos θ pk ≈ p | p | , (cid:18) | p | cos θ pk + | k | p + | k | − s cos θ pk (cid:19) ≈ − s p | p | , (3.102)22nd we arrive at(2 π ) ω d Γ oddsoft d k ≈ d R e (2 π ) n B ( ω ) ( n + (0) − n + ( ω ) − ( n + ↔ n − )) × (cid:90) q ∗ d | p || | p | (cid:90) | p |−| p | dp (cid:88) s ρ HTL s ( p , | p | ) (cid:18) − s p | p | (cid:19) = d R e (2 π ) ( n + ( ω ) n − (0) − n − ( ω ) n + (0)) × (cid:90) q ∗ d | p || | p | (cid:90) | p |−| p | dp (cid:88) s ρ HTL s ( p , | p | ) (cid:18) − s p | p | (cid:19) , (3.103)where in the last line, we use an interesting identity n B ( ω )( n ± (0) − n ± ( ω )) = n ± ( ω ) n ∓ (0) . (3.104)As it happens, the remaining integral is something that has been already computed inliterature: the same integral appears in the P-even total emission rate. In fact, a similarmanipulation in our language produces the usual P-even total emission rate from softt-channel region at leading order as(2 π ) ω d Γ totalsoft d k ≈ d R e (2 π ) ( n + ( ω ) n − (0) + n − ( ω ) n + (0)) × (cid:90) q ∗ d | p || | p | (cid:90) | p |−| p | dp (cid:88) s ρ HTL s ( p , | p | ) (cid:18) − s p | p | (cid:19) , (3.105)and matching to the known results in Refs.[13, 16] when µ = 0, we have at leading order (cid:90) q ∗ d | p || | p | (cid:90) | p |−| p | dp (cid:88) s ρ HTL s ( p , | p | ) (cid:18) − s p | p | (cid:19) = (2 π ) m f (log( q ∗ /m f ) − . (3.106)Using this in (3.103) we finally have the leading order expression for our P-odd emissionrate as(2 π ) ω d Γ oddsoft d k ≈ d R e (2 π ) m f ( n + ( ω ) n − (0) − n − ( ω ) n + (0)) (log( q ∗ /m f ) − . (3.107)Nonetheless, it is instructive to see how the leading log arises from the above integral,using the sum rules for the fermion spectral densities ρ HTL s . The leading log comes from the23egion m f (cid:28) | p | (cid:28) q ∗ , and in this case, we have sum rules (see, for example, Refs.[21, 22]) (cid:90) | p |−| p | dp ρ HTL s ( p , | p | ) = π m f | p | (cid:32) log (cid:32) | p | m f (cid:33) − (cid:33) , (cid:90) | p |−| p | dp p ρ HTL s ( p , | p | ) = s π m f | p | (cid:32) log (cid:32) | p | m f (cid:33) − (cid:33) , (3.108)which gives(2 π ) ω d Γ oddsoft d k ≈ d R e (2 π ) m f ( n + ( ω ) n − (0) − n − ( ω ) n + (0)) (cid:90) q ∗ m f d | p | | p | = d R e (2 π ) m f ( n + ( ω ) n − (0) − n − ( ω ) n + (0)) log( q ∗ /m f ) . (3.109)Looking at the leading log from the hard Compton and Pair Annihilation processes (3.76),(2 π ) ω d Γ oddhard d k ≈ d R e (2 π ) m f ( n + ( | k | ) n − (0) − n − ( | k | ) n + (0)) log ( T /q ∗ ) , (3.110)we see that the log( q ∗ ) nicely cancels in their sum, which is an important consistencycheck of our computation. Looking at the leading log expressions for both P-even case (3.105) and the P-odd emissionrate (3.107),(2 π ) ω d Γ totalsoft d k ≈ d R e (2 π ) m f ( n + ( ω ) n − (0) + n − ( ω ) n + (0)) log( q ∗ /m f ) , (2 π ) ω d Γ oddsoft d k ≈ d R e (2 π ) m f ( n + ( ω ) n − (0) − n − ( ω ) n + (0)) log( q ∗ /m f ) , (3.111)and recalling that they are given in terms of spin polarized emission rates asΓ total = Γ( (cid:15) + ) + Γ( (cid:15) − ) , Γ odd = Γ( (cid:15) + ) − Γ( (cid:15) − ) , (3.112)we find that the leading log spin polarized emission rates are given, after matching thelogarithmic dependence on q ∗ with the hard rate, as(2 π ) ω d Γ( (cid:15) ± ) d k (cid:12)(cid:12)(cid:12)(cid:12) Leading Log = d R e (2 π ) m f n ± ( ω ) n ∓ (0) log( T /m f ) , (3.113)which can be physically understood as follows.24igure 6: Leading log contributions from soft t- or u-channel exchanges: a hard fermionmaking conversion to a collinear photon. The blob represents Hard Thermal Loop (HTL)re-summed propagator.Recall that the leading log comes from the soft t-channel fermion exchange, and thet-channel momentum is space-like as can be seen in the integral in (3.103); we have p < | p | . The spectral density in this kinematics is non-zero due to Landau damping thatis captured by HTL self-energy, and represents thermally excited (fermionic) fluctuationsof soft momentum that are present in the finite temperature plasma. The leading logprocess can be understood as a process of a hard fermion making conversion into a collinearphoton after being annihilated by a soft fermion of momentum gT , as in the Figure 6. Atleading order, this gT momentum can be taken as zero.For definite spin helicity of the final photon in Γ( (cid:15) ± ), the conservation of angularmomentum dictates that the incoming hard fermion which is collinear to the photonshould have a spin ± / ± / ± / − /
2. This means thatthe leading log rate of Γ( (cid:15) + ) (for photons of spin helicity +1) can appear only from theincoming particle of helicity +1 /
2, while an incoming anti-particle of helicity − / (cid:15) + ). Since the incoming particle can annihilate only with a soft anti-particle, the rate Γ( (cid:15) + ) should be proportional to n + ( ω ) n − (0), where the first factor is thenumber density of incoming particle and the second is the number density of annihilatinganti-particle of zero (soft) momentum. See Figure 7. The precisely same logic tells usthat the leading log rate of Γ( (cid:15) − ) should be proportional to n − ( ω ) n + (0). This argument25igure 7: Angular momentum conservation in leading log spin polarized emission rates.nicely explains the result in (3.113). The overall m f is nothing but the strength of thefermionic spectral density in soft momentum range that arises from the HTL self-energy:the same self-energy also gives arise to the asymptotic thermal mass m f . In this section, we compute collinear Bremstrahlung and Pair Annihilation contributionsto the P-odd photon emission rate that are induced by multiple scatterings with softthermal gluons in the plasma [15]. The incoming quark or anti-quark of a hard momentumexperiences soft transverse kicks by thermal gluons of momenta ∼ gT , becoming off-shell by small amount g T , during which a nearly collinear photon is emitted, or quark-antiquark pair annihilates to a collinear photon. The rate of these soft scatterings is well-known to be ∼ g T (which causes the damping rate of ∼ g T ). The scattering gluons aregenuine thermal effects: their momenta are space like and the non-zero spectral densityin this kinematics arises only due to the Landau damping. Since the life time of theintermediate states dictated by small virtuality g T is of 1 / ( g T ), which is comparableto the scattering rate, one has to sum over all multiple scatterings to get the correctleading order result, coined as the LPM re-summation [15]. These contributions add tothe leading order constant under the log. The effect of re-summation typically gives asuppression compared to the single scattering contribution.In diagrammatic language, the LPM re-summation corresponds to summing over allladder diagrams of the type depicted in Figure 8 for the retarded (or “ra”) current-26igure 8: Ladder diagrams to be summed over to get the correct leading order LPMcontribution to (our P-odd) photon emission rate.current correlation functions that enter the photon emission rate formula [15]. The reasonwhy these multiple ladder diagrams are not suppressed by higher powers in couplingconstant is the presence of collinear “pinch” singularities arising from nearly on-shellfermion propagators: the momentum transfer by exchanged gluon lines are soft, and eachpair of fermion propagators, one from the upper line and the other from the lower line,are nearly on-shell and have an IR pinch singularity when the internal momentum isnearly collinear to the external photon momentum (the detail will become clear in thefollowing). This singularity is regulated by soft transverse component of the fermionmomentum, p ⊥ ∼ g T , induced by soft kicks from thermal gluons. Then, one has to alsoinclude in the propagators the fermion thermal mass m f ∼ g T and the leading orderdamping rate ζ ∼ g T which are of the same order as p ⊥ .Since the exchanged gluons have soft momenta for leading order contributions, weneed to re-sum gluonic HTL self-energy in their propagators. To get a Bose-Einsteinenhancement n B ( q ) ∼ T /q ∼ /g in the exchanged gluon lines, the gluon propagatorsneed to be of the rr -type in the “ra”-basis of Schwinger-Keldysh formalism: only thesediagrams give leading order contributions in g . Imposing this requirement and the max-imal number of pinch singularities (that arise from a pair of S ra and S ar propagators),there are essentially two types of ladder diagrams to be summed over in the “ra”-basisas depicted in Figure 9. Defining the re-summed“rr”-type fermion-current vertex Λ i ( p, k )which has two r-type fermions legs, the re-summed G raij ( k ) current-current correlationfunction is written as G raij ( k ) = ( − d R (cid:90) d p (2 π ) tr (cid:2) S ra ( p + k ) σ j S rr ( p )Λ i ( p, k ) + S rr ( p + k ) σ j S ar ( p )Λ i ( p, k ) (cid:3) . (3.114)Since the pinch singularity appears from a pair of S ra and S ar , and using the thermalrelation S rr ( p ) = (1 / − n + ( p ))( S ra ( p ) − S ar ( p )), the singular part of G raij ( k ) is given by27igure 9: Two types of real-time ladder diagrams for leading order LPM contributions.The shaded part represents the re-summed rr-type current vertex Λ i ( p, k ). The rr-typegluon lines are the HTL re-summed ones.( ω ≡ k = | k | ) G raij ( k ) ≈ d R (cid:90) d p (2 π ) (cid:0) n + ( p + ω ) − n + ( p ) (cid:1) tr (cid:2) S ra ( p + k ) σ j S ar ( p )Λ i ( p, k ) (cid:3) . (3.115)The re-summation of the vertex Λ i ( p, k ) is achieved by solving the Schwinger-Dyson equa-tion described in the Figure 10,Λ i ( p, k ) = σ i + ( ig ) C ( R ) (cid:90) d Q (2 π ) σ β S ar ( p + Q )Λ i ( p + Q, k ) S ra ( p + Q + k ) σ α G rrαβ ( Q ) , (3.116)where G rrαβ is the HTL re-summed gluon propagator. We will solve this integral equationand compute G raij ( k ) in leading collinear pinch singularity limit.The real-time fermion propagators, including the thermal mass and the leading orderdamping rate, are given as S ra ( p ) = (cid:88) s i P s ( p ) p − s (cid:113) | p | + m f + i ζ = − ( S ar ( p )) † , (3.117)where the damping rate is independent of momentum p and the species s at leading order ζ = C ( R ) g π log(1 /g ) T . (3.118)28igure 10: The real-time Schwinger-Dyson equation for the re-summed vertex Λ i ( p, k ).Let’s consider the pair of S ra ( p + k ) and S ar ( p ) in (3.115) to illustrate the pinch singularityand its leading order treatment. Looking at the expression S ra ( p + k ) S ar ( p ) = (cid:88) s,t i P s ( p + k ) (cid:16) p + | k | − s (cid:113) | p + k | + m f + i ζ (cid:17) i P t ( p ) (cid:16) p − t (cid:113) | p | + m f − i ζ (cid:17) , (3.119)the two poles in the complex p -plane, one in the upper half plane and the other in thelower half plane, p = −| k | + s (cid:113) | p + k | + m f − i ζ , p = t (cid:113) | p | + m f + i ζ , (3.120)may be close to each other with a distance of ∼ g T , if p is nearly collinear to k and p ⊥ ∼ gT . In computing p integral, we close the p integral contour, say, in the upperhalf plane, picking up the pole of p = t (cid:113) | p | + m f + iζ/
2, then the residue from theother pole is 1 | k | + t (cid:113) | p | + m f − s (cid:113) | p + k | + m f + iζ . (3.121)Let’s fix the direction of k to be along ˆ z = ˆ x direction, and write the ˆ z component ofmomentum p as p (cid:107) , and the perpendicular component as p ⊥ , so that we can expand upto order g T as (cid:113) | p | + m f ≈ | p (cid:107) | + p ⊥ + m f | p (cid:107) | , (cid:113) | p + k | + m f ≈ | p (cid:107) + | k || + p ⊥ + m f | p (cid:107) + | k || . (3.122)29he pinch singularity happens when the leading collinear terms in the denominator cancelwith each other, that is | k | + t | p (cid:107) | − s | p (cid:107) + | k || = 0, to result in ∼ g T in the denominatorwhich enhances the contribution. There are three physically distinct cases where thishappens:1) s = t = 1: in this case, | k | + | p (cid:107) |−| p (cid:107) + | k || = 0 is satisfied when p (cid:107) >
0. Consideringthe kinematics, one easily sees that this case corresponds to quark of momentum p + k emitting the collinear photon of momentum k by Bremstrahlung. The residue becomes p ⊥ + m f p (cid:107) − p ⊥ + m f p (cid:107) + | k | ) + iζ = | k | ( p ⊥ + m f )2 p (cid:107) ( p (cid:107) + | k | ) + iζ ≡ δE ( p ⊥ ) + iζ . (3.123)2) s = 1 , t = −
1: the condition | k |−| p (cid:107) |−| p (cid:107) + | k || = 0 is fulfilled when −| k | < p (cid:107) < p + k and an anti-quark of momentum − p . Considering signs of p (cid:107) and p (cid:107) + | k | , one finds thatthe residue has the precisely the same expression, δE + iζ with δE is defined as above.3) s = t = −
1: we have p (cid:107) < −| k | , which corresponds to Bremstrahlung of anti-quarkof momentum p + k . Again the residue has the precisely the same form as δE + iζ .Note that in all three cases, ( s, t ) are correlated with p (cid:107) in such a way that s ( p (cid:107) + | k | ) > tp (cid:107) >
0. Since we only care about the above pinch singularity enhancedcontributions, the ( s, t ) are uniquely chosen for each value of p (cid:107) as above, and we consideronly these terms in the following.In leading order treatment, the location of the pole can be approximated as p = t (cid:113) | p | + m f + iζ/ ≈ t | p (cid:107) | = p (cid:107) in all other places in the integral once the above residuesare correctly identified. In summary, we can replace the two poles in (3.119) by1 (cid:16) p + | k | − s (cid:113) | p + k | + m f + i ζ (cid:17) (cid:16) p − t (cid:113) | p | + m f − i ζ (cid:17) → (2 πi ) δ ( p − p (cid:107) ) δE + iζ , (3.124)and depending on the value of p (cid:107) ∈ [ −∞ , + ∞ ], the suitable ( s, t ) as described in theabove has to be chosen. For example, we have for (3.115), S ra ( p + k ) σ j S ar ( p ) → (cid:18) P + ( p + k ) σ j P + ( p )Θ( p (cid:107) ) + P + ( p + k ) σ j P − ( p )Θ( − p (cid:107) )Θ( p (cid:107) + | k | )+ P − ( p + k ) σ j P − ( p )Θ( − p (cid:107) − | k | ) (cid:19) − (2 πi ) δ ( p − p (cid:107) ) δE ( p ⊥ ) + iζ . (3.125)Since Q carried by exchange gluons is soft, we have an essentially same structure for S ar ( p + Q )Λ i ( p + Q, k ) S ra ( p + Q + k ) appearing in the integral equation for Λ i ( p, k ) in303.116), S ra ( p + Q )Λ i ( p + Q, k ) S ar ( p + Q + k ) → (cid:18) P + ( p + q )Λ i ( p + Q, k ) P + ( p + q + k )Θ( p (cid:107) )+ P − ( p + q )Λ i ( p + Q, k ) P + ( p + q + k )Θ( − p (cid:107) )Θ( p (cid:107) + | k | )+ P − ( p + q )Λ i ( p + Q, k ) P − ( p + q + k )Θ( − p (cid:107) − | k | ) (cid:19) − (2 πi ) δ ( q − q (cid:107) ) δE ( p ⊥ + q ⊥ ) + iζ , (3.126)the only difference of which are the argument p ⊥ + q ⊥ in δE instead of p ⊥ . In writingthe δ ( q − q (cid:107) ) factor, we used p = p (cid:107) that is imposed by (3.125) when we compute thecorrelation function G raij ( k ) by (3.115). We will solve the integral equation (3.116) for Λ i ,with the above replacement (3.126) that is enough for the leading order result.Looking at (3.115), (3.125), and (3.126), what we need are the projected vertices P s ( p + k ) σ j P t ( p ) ≡ Σ jst ( p , k ) P s ( p + k ) P t ( p ) , (3.127)and we define a vector function F i ( p ⊥ ) as (we ignore p (cid:107) and | k | arguments in F i as theyare common in all subsequent expressions) P t ( p )Λ i ( p, k ) (cid:12)(cid:12) p = p (cid:107) P s ( p + k ) ≡ ( δE ( p ⊥ ) + iζ ) F i ( p ⊥ ) P t ( p ) P s ( p + k ) . (3.128)Here, we emphasize again that the ( s, t ) are the choice depending on the value of p (cid:107) suitable for the pinch singularity that we discuss in the above. Note that Σ jst and F i arecomplex valued functions, not 2 × ω ≡ k = | k | ) G raij ( k ) = d R ( − i ) (cid:90) d p (2 π ) ( n + ( p + ω ) − n + ( p ))Σ jst ( p , k ) F i ( p ⊥ )tr ( P s ( p + k ) P t ( p )) × (2 π ) δ ( p − p (cid:107) ) ≈ d R ( − i ) (cid:90) d p (2 π ) ( n + ( p + ω ) − n + ( p ))Σ jst ( p , k ) F i ( p ⊥ )(2 π ) δ ( p − p (cid:107) ) , (3.129)where in the last line, we usetr ( P s ( p + k ) P t ( p )) = 12 (cid:16) st ˆ p · (cid:92) p + k (cid:17) ≈ , (3.130)to leading order in p ⊥ /p (cid:107) ∼ g and we use tp (cid:107) > s ( p (cid:107) + | k | ) > G raij ( k ) as(2 π ) ω d Γ odd d k = e n B ( ω )( − G ra ( k ) − G ra ( k )] , (3.131)given the choice of k = | k | ˆ x . Hence, we need only the transverse components of Σ jst and F i . A short computation from the definition (3.127) after taking the trace of the bothsides gives Σ jst ( p , k ) = s (cid:92) p + k j + t ˆ p j + ist (cid:15) jlm ˆ p l (cid:92) p + k m st ˆ p · (cid:92) p + k , (3.132)and the integral equation (3.116) after being contracted with P t ( p ) on the left and P s ( p + k ) on the right gives( δE ( p ⊥ ) + iζ ) F i ( p ⊥ ) = (cid:0) Σ ist ( p , k ) (cid:1) ∗ + g C ( R ) (cid:90) d Q (2 π ) F i ( p ⊥ + q ⊥ )ˆ v α ˆ v β G rrαβ ( Q )(2 πi ) δ ( q − q (cid:107) ) , (3.133)where in the integral kernel, we used an approximation P t ( p ) σ β P t ( p + q ) ≈ P t ( p ) σ β P t ( p ) = p βt / | p (cid:107) |P t ( p ) , (3.134)for soft Q , where p αt = ( | p | , t p ) ≈ ( | p (cid:107) | , , , tp (cid:107) ) at leading order, so that p αt / | p (cid:107) | is alight-like 4-velocity ˆ v α along the collinear vector t p . Considering the correlation between p (cid:107) and the sign of t that we describe before, we see that tp (cid:107) > v α = (1 , , , P s ( p + q + k ) σ α P s ( p + k ) sothat we have P t ( p ) σ β P t ( p + q ) P s ( p + q + k ) σ α P s ( p + k ) ≈ ˆ v α ˆ v β P t ( p ) P s ( p + k ) , (3.135)which has been used to arrive at our integral equation for F i in (3.133). Since F i ∼ /g and the both sides of (3.133) are of order ∼ g , this approximation is enough for the leadingorder computation.One subtle point is that the HTL gluon fluctuations in G rrαβ contains a P-odd spectraldensity ¶ which is anti-symmetric in α and β , which could potentially contribute to our P-odd photon emission rate, if we keep Q corrections in (3.134). We estimated them to findthat these corrections are higher order in g . The fluctuations contracted with light-likevector ˆ v α in (3.133), ˆ v α ˆ v β G rrαβ (which are the correlations along the Eikonalized light-likeWilson line) receive only the usual P-even longitudinal and transverse contributions. ¶ See the appendices in Ref.[12] for some of its sum rules in the HTL approximation.
32s is well-known [15], the integral equation is further simplified due to the fact thatthe integral on the right in (3.133) without F i is identical to the leading order dampingrate ζ , ζ = g C ( R ) (cid:90) d Q (2 π ) ˆ v α ˆ v β G rrαβ ( Q )(2 π ) δ ( q − q (cid:107) ) , (3.136)so that we can move iζ F i ( p ⊥ ) term in the left to the right to arrive at δE ( p ⊥ ) F i ( p ⊥ ) = (cid:0) Σ ist ( p , k ) (cid:1) ∗ (3.137)+ g C ( R ) (cid:90) d Q (2 π ) (cid:0) F i ( p ⊥ + q ⊥ ) − F i ( p ⊥ ) (cid:1) ˆ v α ˆ v β G rrαβ ( Q )(2 πi ) δ ( q − q (cid:107) ) . This form has a good infrared behavior so that only the well-controlled soft scale Q ∼ gT contributes at leading order, while the magnetic scale of g T gives a finite, sub-leadingcontributions.Finally, for soft Q we replace G rrαβ ( Q ) = (cid:18)
12 + n B ( q ) (cid:19) ρ gluon αβ ( Q ) ≈ Tq ρ gluon αβ ( Q ) , (3.138)for leading order, where ρ gluon αβ is the gluon spectral density in HTL approximation, andthe amazing sum rule in Ref.[23] gives the integral over ( q , q (cid:107) ) as T (cid:90) dq dq (cid:107) (2 π ) ˆ v α ˆ v β q ρ gluon αβ ( Q )(2 π ) δ ( q − q (cid:107) ) = T m D q ⊥ ( q ⊥ + m D ) , (3.139)where m D = g (cid:18) T µ π (cid:19) ( T A + N F T R ) = g (cid:18) T µ π (cid:19) ( N c + N F / , (3.140)is the Debye mass for N F Dirac quarks in fundamental representation, so that the integralequation for F i ( p ⊥ ) is finally recast to δE ( p ⊥ ) F i ( p ⊥ ) = (cid:0) Σ ist ( p , k ) (cid:1) ∗ + i (cid:90) d q ⊥ (2 π ) C ( q ⊥ ) (cid:0) F i ( p ⊥ + q ⊥ ) − F i ( p ⊥ ) (cid:1) , (3.141)with C ( q ⊥ ) = g C ( R ) T m D q ⊥ ( q ⊥ + m D ) . (3.142)Since we need only the transverse parts of (3.141) and (3.129) for G raij ( k ), we expandΣ ist ( p , k ) given in (3.132) to linear order in p ⊥ /p (cid:107) ∼ g , which is enough for leading order,Σ ist ( p , k ) ≈ (cid:18) p (cid:107) + 1 p (cid:107) + | k | (cid:19) p i ⊥ + i (cid:18) p (cid:107) − p (cid:107) + | k | (cid:19) (cid:15) il ⊥ p l ⊥ = 2 p (cid:107) + | k | p (cid:107) ( p (cid:107) + | k | ) p i ⊥ + i | k | p (cid:107) ( p (cid:107) + | k | ) (cid:15) il ⊥ p l ⊥ , (3.143)33here we used the fact that tp (cid:107) > s ( p (cid:107) + | k | ) >
0, and (cid:15) ⊥ = − (cid:15) ⊥ = 1. We use thisexpansion in both (3.129) and (3.141). From (3.141), we see that the solution for F i ( p ⊥ )is given by F i ( p ⊥ ) = 2 p (cid:107) + | k | p (cid:107) ( p (cid:107) + | k | ) f i ⊥ ( p ⊥ ) − i | k | p (cid:107) ( p (cid:107) + | k | ) (cid:15) il ⊥ f l ⊥ ( p ⊥ ) , (3.144)where f i ⊥ ( p ⊥ ) is the solution of the integral equation δE ( p ⊥ ) f i ⊥ ( p ⊥ ) = p i ⊥ + i (cid:90) d q ⊥ (2 π ) C ( q ⊥ ) (cid:0) f i ⊥ ( p ⊥ + q ⊥ ) − f i ⊥ ( p ⊥ ) (cid:1) . (3.145)This equation for f i ⊥ ( p ⊥ ) is identical to the integral equation obtained by Arnold-Moore-Yaffe in Ref.[15], with the identification f i ⊥ ( p ⊥ ) = − i (cid:0) f i AMY ( p ⊥ ) (cid:1) ∗ , (3.146)so that the techniques of solving this integral equation that are known in literature canbe utilized to find our object F i ( p ⊥ ). Using this expression for F i and (3.129) for G raij ( k ),we obtain after short manipulations, G ra − G ra ( k ) = − d R (cid:90) dp (cid:107) d p ⊥ (2 π ) ( n + ( p (cid:107) + ω ) − n + ( p (cid:107) )) | k | (2 p (cid:107) + | k | ) p (cid:107) ( p (cid:107) + | k | ) ( p ⊥ · f ⊥ ) , (3.147)and using an interesting identity n B ( ω ) (cid:0) n + ( p (cid:107) + ω ) − n + ( p (cid:107) ) (cid:1) = − n + ( p (cid:107) + ω ) (cid:0) − n + ( p (cid:107) ) (cid:1) , (3.148)we finally arrive at an expression for our P-odd photon emission rate in terms of thesolution f ⊥ ( p ⊥ ) of the integral equation (3.145) (recall ω = | k | ),(2 π ) ω d Γ oddLPM d k = e d R (cid:90) dp (cid:107) d p ⊥ (2 π ) n + ( p (cid:107) + ω ) (cid:0) − n + ( p (cid:107) ) (cid:1) ω (2 p (cid:107) + ω ) p (cid:107) ( p (cid:107) + ω ) ( − p ⊥ · f ⊥ )] . (3.149)This is the main outcome of this section. Our numerical evaluation is based on thisexpression with the integral equation (3.145), where δE is given in (3.123) (see also(3.152)).Although it is not manifestly obvious that the above expression is an odd functionin (axial) chemical potential µ that enters the distribution function n + , one way to seethis is to first observe that the factor n + ( p (cid:107) + ω ) (cid:0) − n + ( p (cid:107) ) (cid:1) is easily recognized as thestatistical factor for the collinear Bremstrahlung process of a fermion of momentum p + k k , provided that p (cid:107) >
0. In the case p (cid:107) < −| k | , usingthe identity n + ( p (cid:107) + ω ) (cid:0) − n + ( p (cid:107) ) (cid:1) = n − ( − p (cid:107) ) (cid:0) − n − ( − p (cid:107) − ω ) (cid:1) (3.150)we see that the process is in fact the Bremstrahlung of anti-fermion of momentum − p emitting a photon of momentum k . It is more convenient to change the integrationvariable in this case to p (cid:107) → − (˜ p (cid:107) + ω ) so that we have ˜ p (cid:107) > n − (˜ p (cid:107) + ω ) (cid:0) − n − (˜ p (cid:107) ) (cid:1) , (3.151)which makes the interpretation clearer. From the expression for δE in (3.123), we have δE = ω ( p ⊥ + m f )2 p (cid:107) ( p (cid:107) + ω ) = ω ( p ⊥ + m f )2˜ p (cid:107) (˜ p (cid:107) + ω ) , (3.152)so that the integral equation (3.145) and hence the solution f ⊥ ( p ⊥ ) is invariant under thischange of variable, but the integral kernel in our P-odd emission rate in (3.149) changessign under this transformation as ω (2 p (cid:107) + ω ) p (cid:107) ( p (cid:107) + ω ) → − ω (2˜ p (cid:107) + ω )˜ p (cid:107) (˜ p (cid:107) + ω ) , (3.153)so that the net sign of the contribution from anti-fermion Bremstrahlung is opposite tothe one from fermion Bremstrahlung. This is expected since fermion and anti-fermionfrom our right-handed Weyl fermion field have opposite chirality, so their contributionsto Γ odd should be opposite. From the above, if we sum over p (cid:107) > p (cid:107) > p (cid:107) as p (cid:107) ), we see that the final result is proportional to n + ( p (cid:107) + ω ) (cid:0) − n + ( p (cid:107) ) (cid:1) − n − ( p (cid:107) + ω ) (cid:0) − n − ( p (cid:107) ) (cid:1) , (3.154)which is indeed an odd function on the (axial) chemical potential µ . More generally, bythe change of variable from p (cid:107) to ˜ p (cid:107) for the entire range of p (cid:107) , we can simply replace thestatistical factor in our main formula (3.149) with the average n + ( p (cid:107) + ω ) (cid:0) − n + ( p (cid:107) ) (cid:1) → (cid:0) n + ( p (cid:107) + ω ) (cid:0) − n + ( p (cid:107) ) (cid:1) − n − ( p (cid:107) + ω ) (cid:0) − n − ( p (cid:107) ) (cid:1)(cid:1) , (3.155)so that the LPM contribution to our P-odd emission rate, (3.149), is now manifestly anodd function in µ .Following Ref.[24], the integral equation (3.145) can be transformed to the one in thetransverse 2-dimensional coordinate space b , which takes a form ω ( −∇ b + m f )2 p (cid:107) ( p (cid:107) + ω ) f i ⊥ ( b ) = − i ∇ i b δ (2) ( b ) + i C ( b ) f i ⊥ ( b ) , (3.156)35here f i ⊥ ( b ) = (cid:90) d p ⊥ (2 π ) e i b · p ⊥ f i ⊥ ( p ⊥ ) , (3.157)and C ( b ) ≡ (cid:90) d q ⊥ (2 π ) C ( q ⊥ ) (cid:0) e − i b · q ⊥ − (cid:1) = − g C ( R ) T π ( K ( | b | m D ) + γ E + log( | b | m D / . (3.158)From rotational symmetry, one can write f ⊥ ( b ) = b f ( b ) , b ≡ | b | , (3.159)in terms of a scalar function f ( b ) which satisfies the following second order differentialequation ω p (cid:107) ( p (cid:107) + ω ) (cid:18) − ∂ b − b ∂ b + m f (cid:19) f ( b ) = i C ( b ) f ( b ) , (3.160)with the boundary conditions f ( b →
0) = − i p (cid:107) ( p (cid:107) + ω ) πωb + O ( b ) , f ( b → ∞ ) = 0 . (3.161)In terms of the scalar function f ( b ) which can be easily solved from the above differentialequation, the p ⊥ integral in our P-odd emission rate (3.149) takes a simple form (cid:90) d p ⊥ (2 π ) ( − p ⊥ · f ⊥ ( p ⊥ )] = ( − − i ) ∇ b · f ⊥ ( b )] (cid:12)(cid:12)(cid:12)(cid:12) b → = 2 Re f (0) , (3.162)so that the final expression for the LPM contribution to the P-odd photon emission ratebecomes(2 π ) ω d Γ oddLPM d k = e d R (cid:90) + ∞−∞ dp (cid:107) π (cid:0) n + ( p (cid:107) + ω ) (cid:0) − n + ( p (cid:107) ) (cid:1) − n − ( p (cid:107) + ω ) (cid:0) − n − ( p (cid:107) ) (cid:1)(cid:1) × ω (2 p (cid:107) + ω ) p (cid:107) ( p (cid:107) + ω ) Re f (0) . (3.163)This is what we practically use for numerical evaluations, and the computation reducesto solving the second order differential equation (3.160) with the boundary conditions(3.161). In summary, the leading order P-odd photon emission rate for a single species of right-handed Weyl fermion is a sum of the three contributions: 1) hard Compton and Pair36nnihilation rate given by (in t-channel parametrization) the equation (3.71) with (3.72)where one has to use (3.64), 2) soft t- and u-channel contributions given in (3.107), 3)the LPM re-summed collinear Bremstrahlung and Pair Annihilation contribution givenin (3.163) with (3.160) and (3.161). For a theory with N F Dirac fermions with an axialchemical potential µ A , one has to multiply the above results by a factor2 (cid:32)(cid:88) F Q F (cid:33) , (4.164)with a replacement µ → µ A in the distribution functions, where Q F are electromagneticcharges of flavor F in units of e . Recall also that the Debye mass m D = g (cid:18) T µ π (cid:19) ( N c + N F / , (4.165)has to be adjusted according to the number of flavors N F .We choose to present our result in a way similar to the existing literature. Define A ( ω ) ≡ α EM (cid:32)(cid:88) F Q F (cid:33) d R m f, (0) ω n f ( ω ) , (4.166)where n f ( ω ) is the Fermi-Dirac distribution with zero chemical potential and m f, (0) ≡ C ( R ) g T / m f = C ( R ) g (cid:18) T + µ A π (cid:19) . (4.167)The hard Compton and Pair Annihilation rate is then written as(2 π ) d Γ oddhard d k = A ( ω ) 2(2 π ) Tω n f ( ω ) (cid:90) ∞ q ∗ d | q | T (cid:90) | q | max( −| q | , | q |− | k | ) dq T (cid:90) ∞ | q |− q d | p (cid:48) | T (cid:90) π dφ ¯ I , (4.168)where¯ I = (cid:18) − ut − t − u ) (cid:18) q ⊥ t − q ⊥ · p (cid:48)⊥ tu (cid:19)(cid:19) × ( n + ( q + | k | ) n − ( | p (cid:48) | ) − n − ( q + | k | ) n + ( | p (cid:48) | ))(1 + n B ( q + | p (cid:48) | ))+ ( s − t ) (cid:32) t + 1 s − (cid:18) q ⊥ t + ( q ⊥ + p (cid:48)⊥ ) s (cid:19) (cid:33) × (cid:0) n + ( q + | k | )(1 − n + ( q + | p (cid:48) | )) − n − ( q + | k | )(1 − n − ( q + | p (cid:48) | )) (cid:1) n B ( | p (cid:48) | ) . (4.169)37ote that what is multiplied to A ( ω ) is a dimensionless function on ω/T (recall | k | = ω ),and the phase space integral as well as the integrand ¯ I is in terms of dimensionlessvariables | q | /T , etc. The soft t- and u-channel contribution is written as(2 π ) d Γ oddsoft d k = A ( ω ) m f m f, (0) n f ( ω ) ( n + ( ω ) n − (0) − n − ( ω ) n + (0)) (log( q ∗ /m f ) − . (4.170)Finally, the LPM contribution is(2 π ) d Γ oddLPM d k = A ( ω ) 1 n f ( ω ) (cid:90) + ∞−∞ d ¯ p (cid:107) (cid:0) n + ( p (cid:107) + ω ) (cid:0) − n + ( p (cid:107) ) (cid:1) − n − ( p (cid:107) + ω ) (cid:0) − n − ( p (cid:107) ) (cid:1)(cid:1) × ¯ ω (2¯ p (cid:107) + ¯ ω )¯ p (cid:107) (¯ p (cid:107) + ¯ ω ) Re ¯ f (0) , (4.171)where ¯ p (cid:107) ≡ p (cid:107) /T and ¯ ω ≡ ω/T , and ¯ f (¯ b ) is the solution of the differential equation¯ ω p (cid:107) (¯ p (cid:107) + ¯ ω ) (cid:18) − ∂ b − b ∂ ¯ b + m f m D (cid:19) ¯ f (¯ b ) = − i π m f, (0) m D (cid:0) K (¯ b ) + γ E + log(¯ b/ (cid:1) ¯ f (¯ b ) , (4.172)with the boundary conditions¯ f (¯ b →
0) = − i ¯ p (cid:107) (¯ p (cid:107) + ¯ ω ) π ¯ ω ¯ b m D m f, (0) , ¯ f (¯ b → ∞ ) = 0 . (4.173)The final result can be recast to the form(2 π ) d Γ oddLO d k = A ( ω ) (cid:0) C oddLog ( ω/T ) log ( T /m f ) + C odd2 ↔ ( ω/T ) + C oddLPM ( ω/T ) (cid:1) , (4.174)with the dimensionless functions C oddLog , C odd2 ↔ , C oddLPM , where C oddLog = m f m f, (0) n f ( ω ) ( n + ( ω ) n − (0) − n − ( ω ) n + (0)) ,C odd2 ↔ = lim q ∗ → (cid:18) π ) Tω n f ( ω ) (cid:90) ∞ q ∗ d | q | T (cid:90) | q | max( −| q | , | q |− | k | ) dq T (cid:90) ∞ | q |− q d | p (cid:48) | T (cid:90) π dφ ¯ I− C oddLog ( ω/T ) (log( T /q ∗ ) + 1 − log 2) (cid:19) ,C oddLPM = 1 n f ( ω ) (cid:90) + ∞−∞ d ¯ p (cid:107) (cid:0) n + ( p (cid:107) + ω ) (cid:0) − n + ( p (cid:107) ) (cid:1) − n − ( p (cid:107) + ω ) (cid:0) − n − ( p (cid:107) ) (cid:1)(cid:1) × ¯ ω (2¯ p (cid:107) + ¯ ω )¯ p (cid:107) (¯ p (cid:107) + ¯ ω ) Re ¯ f (0) . (4.175)38ote that we have not extracted out the energy logarithm given in (3.83), but one couldchoose to do so to redefine C odd2 ↔ .The above result is valid for full dependence in the axial chemical potential µ A , butwe will present our numerical evaluations only for its linear dependency by expanding thedimensionless functions C oddLog , C odd2 ↔ , C oddLPM in linear order in µ A /T . In this case, m f canbe identified with m f, (0) and one can also neglect µ A in the Debye mass m D . Writing thislinear expansion as(2 π ) d Γ oddLO d k ≈ A ( ω ) (cid:16) C odd , (1)Log ( ω/T ) log ( T /m f ) + C odd , (1)2 ↔ ( ω/T ) + C odd , (1)LPM ( ω/T ) (cid:17) µ A T + O ( µ A ) , (4.176)we have C odd , (1)Log = 12 (1 − n f ( ω )) , (4.177)while the other two functions, C odd , (1)2 ↔ , C odd , (1)LPM , have to be evaluated numerically. Thenumerical evaluation involves three dimensional integrals and solving second order differ-ential equation, and can be performed with a reasonable precision using Mathematica.We present our numerical results in Figure 11 for the range 0 . < ω/T <
3. We see thatthe LPM contributions to the constant under the log is 2-3 times bigger than the onefrom 2 ↔ ↔ total = Γ( (cid:15) + ) + Γ( (cid:15) − ) , Γ odd = Γ( (cid:15) + ) − Γ( (cid:15) − ) , (4.178)we get(2 π ) d Γ totalLO d k ≈ A ( ω ) (cid:16) log ( T /m f ) + C total , (0)2 ↔ ( ω/T ) + C total , (0)LPM ( ω/T ) (cid:17) + O ( µ A ) , (4.179)where C total , (0)2 ↔ ( ω/T ) = 12 ln (cid:18) ωT (cid:19) + 0 . Tω − . . e − . ω/T , . < ωT , (4.180) C total , (0)LPM ( ω/T ) = 2 (cid:34) .
316 ln(12 .
18 +
T /ω )( ω/T ) / + 0 . ω/T (cid:112) ω/ (16 . T ) (cid:35) , . < ωT < , (4.181)39 ↔ ( ) C LPModd, ( ) ω T C odd, ( ) ( ω / T ) Figure 11: Numerical results for C odd , (1)2 ↔ ( ω/T ), C odd , (1)LPM ( ω/T ) for N F = 2 QCD.which is nothing but AMY’s result for µ A = 0 [16].Therefore, the circular polarization asymmetry A ± γ = Γ odd Γ total ≈ .
03 for ω/T = 2, α s = 0 .
2, and µ A /T = 0 . A ± γ ≈ .
01 that we found in [5] using AdS/CFT correspondence.
Acknowledgment
We thank Meseret Demise, Jacopo Ghiglieri, J.-F. Paquet, Rob Pisarski, MatthewSiebert, and Derek Teaney for helpful discussions.
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