Field-Induced Axion Luminosity of Photon Gas via a-interaction N.V. Mikheev, A.Ya. Parkhomenko and L.A. Vassilevskaya Yaroslavl State (Demidov) Univer

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1 Field-Induced Axion Luminosity of Photon Gs vi -Interction N.V. Mikheev, A.Y. Prkhomenko nd L.A. Vssilevsky Yroslvl Stte (Demidov) University, Sovietsky 14, Yroslvl , Russi Abstrct The interction of pseudosclr prticle with two photons in n externl electromgnetic eld is used to study the photon decy! nd the photon colescence! where is pseudosclr prticle ssocited with Peccei-Quinn U (1) symmetry. A strong ctlyzing inuence of the externl eld on these processes reduces to tht the eld removes the suppression ssocited with the smllness of the xion mss. The eld-induced xion emission by photon gs is nlyzed s one more possible source of energy losses by strophysicl objects. Introduction Axion is pseudosclr prticle which ws introduced for solving the CP problem in QCD [1, 2]. This prticle ppers fter the brekdown of the Peccei-Quinn chirl U(1) symmetry nd crries smll mss [3, 4] 10 5 ev < m < 10 2 ev (1) becuse the underlying symmetry is not exct t low energies. Now we cn consider xions or ny other pseudosclr mssless or low-mss bosons s nturl consequence of certin extensions of the stndrd model. Axions couple to photons ccording to the Lgrngin [3]: L = g 4 (F ~ F ) (2) 1

2 u HH H H H u_ ^_ ^_ ^ Hu _ ^ ^ ^ Figure 1: The eective vertex of the interction of pseudosclr prticle with two photons. with strength g = f (3) where f is the energy scle of the symmetry breking nd is modeldependent fctor of order unity. All existing xion models contin the interction of n xion with chrge fermions (usul or exotic) which utomticlly leds to n electromgnetic coupling of the form Eq. (2) becuse of the tringle fermion loop mplitude shown on Fig. 1. The two-photon-xion interction vertex llows for the xion rditive decy! [3], for the Primko conversion $ in the presence of electric or mgnetic elds [5] s well s for the photon decy T! L [6,7]ndcolescence L T! [7] in plsm. The lst two processes re kinemticlly possible becuse of the dispersion reltions of electromgnetic excittions in plsm which dier signicntly from the vcuum dispersion [3]. Pseudosclrs re of gret importnce in n ppliction to strophysics s n dditionl source of str energy losses becuse of the very wek interction with mtter. In strophysicl objects one hs to tke into ccount the inuence of both components of the ctive medium, plsm nd n externl electromgnetic eld, on processes inside. A sitution is lso possible when the eld component domintes nd one cn consider the xion processes in n externl eld only. Note tht n rbitrry reltively smooth eld in which reltivistic prticle propgtes is well described by the constnt crossed eld limit (E? B, E = B). In this cse the dynmic prmeter 2 = e 2 (qffq)=m 6, where m is mss of n interctive prticle (rel or, possibly, virtul) with n electric chrge e, is the only eld invrint. In supernov explosion region outside the neutrinosphere of order of hundred kilometers with rther rreed plsm with the temperture 2

3 (q) ' f & u u Q Q f $ % (q) Figure 2: The digrm describing the polriztion opertor of photon. Double lines correspond to fermion propgtors in n externl electromgnetic eld. of order of MeV nd strong mgnetic eld of order of the Schwinger vlue B e ' 4: G could exist. Under the supernov conditions considered the limit of smll vlues of the dynmic prmeter is relized in the processes of the xion emission by photon gs. In this tlk we discuss the forbidden in vcuum xion processes { the photon decy! nd the photon colescence! in n externl electromgnetic eld using the eective -interction [] nd estimte their possible inuence on the supernov cooling. Dispersions of Photon nd Axion in Externl Electromgnetic Field In clcultion of probbilities nd luminosities of the xion emission processes in n externl eld one hs to tke into ccount the non-trivil kinemtics of interctive prticles. The kinemtics depends substntilly on the prticle dispersion reltions which cn chnge in the presence of the electromgnetic eld becuse this eld plys the role of n nisotropic medium. In this section the inuence of the externl electromgnetic crossed eld on the photon [9] nd xion [10] dispersions is considered. The polriztion opertor of photon in n externl eld cn be obtined from the two-point fermion loop digrm shown on Fig. 2. The double solid lines denote the exct fermion propgtors in n externl electromgnetic 3

4 eld. The polriztion opertor cn be presented s: = i 3X =1 () " () " () : (4) The set of the eigenvectors of coincides with the photon polriztions: " (1) = (qf) q " (2) = (q ~F ) q (5) (qffq) (qffq) " (3) = q2 (qff) (qffq)q q : q2 (qffq) 2 The rst two vectors " (1) nd " (2) describe the rel trnsverse photon polriztions. In dierence with plsm where besides the trnsverse photons the longitudinl excittion { \plsmon" { is ppered, in n externl electromgnetic eld the photon with the longitudinl polriztion " (3) is bsent. The eigenvlues () with =1 2 of the polriztion opertor (4) determine the dispersion reltions q 2 () =0of the trnsverse photon excittions. The nlysis of the polriztion opertor shows tht (), in generl, re complex () = 2 2i!. The rel prt 2 hs mening of the eldinduced \eective mss" squred nd is the probbility of the photon decy! e + e. 1 The photon dispersion curves in the externl crossed eld [9] re presented on Fig. 3 in dependence on the dynmic prmeter. The \eective msses" squred of the trnsverse photon polriztion being like in their qulittive behvior re dierent quntittively. The dierence in vlues of the \eective msses" squred of the photon eigenmodes mkes possible the photon decy! where is n rbitrry reltively light pseudosclr. The nlysis shows tht in the physiclly interesting region of the dynmic prmeter ( < 10 2 )thephoton \eective msses" squred re limited s: j 2 j < 10 kev: (6) The eld-induced contribution m to the smll xion mss m (1) cn be clculted s the rel prt of! e + e! trnsition mplitude vi 1 We will consider the contribution of n electron only s the most sensitive to the externl eld fermion. 4

5 B 2 m P Figure 3: The photon dispersion curves in the crossed eld. electron loop. As result m cn be estimted s [10]: m 2 m C 2 e 2=3 (7) where C e is the model-dependent fctor which determines the electron-xion coupling g e. This contribution is negligibly smll, nd herefter we willtke the xion s mssless prticle. Photon Decy! The eld-induced two-photon-xion vertex ws obtined by us erlier [] nd used to nlyze the xion rditive decy [11].As in the cse of! the min contribution to the mplitude of the photon decy! comes from the biliner on the externl eld terms of -vertex. 5

6 In the cse of the smll vlues of the dynmic prmeters the decy ofthe photon with the rst polriztion " (1) is llowed kinemticlly (see Fig. 3) due to the condition 2 1 > 2 2: (1)! (2) + : With the photon polriztion vectors (5) the mplitude of the decy is: M d ' 4C em 2 t (1 t) 2 1 J(t 1 1 ) () f J(t 1 1 ) 11 ' (1 2t) where t =! 2 =! 1 is the reltive energy of the nl photon, 1 is the dynmic prmeter of the decying photon. Tking ll the prticles s ultrreltivistic the probbility of the photon decy is: W (F ) d = 1 Z 1 dt jm d j 2 ' 4: C 2 e m 4 16! 1 3 f 2! 1: (9) 1 0 The decy probbility is proportionl to the eighth power of the dynmic prmeter 1 nd, hence, the eighth power of the externl eld strength. Photon Colescence! Both the photon decy! nd the photon colescence! give the contribution to the str energy losses due to the xion emission. The mplitude of the photon colescence my be esily produced from the xion rditive decy mplitude [11] by chnging ll the 4-moment of prticles on opposite ones [12]: M c ' 4C em 2 ( )[ 1 J( 1 2 )+ 2 J( 2 1 )] (10) f 1 J( 1 2 )+ 2 J( 2 1 ) 1 21 ' ( ) where 1 nd 2 re the dynmic prmeters of the initil photons. Becuse the nl stte is the one-prticle one the photon decy probbility hs the 6

7 energy -function nd in the cse of ultrreltivistic interctive prticles is the following: W (F ) c = 2(! 1 +! 2 E )! 1! 2 E V ' 162 C 2 e m 4 441f 2 jm c j 2 (11) (! 1 +! 2 E )! 1! 2 E V ( ) 4 where V is the normlized volume. The probbility of the photon colescence hs the sme dependence of the externl eld strength s the photon decy probbility. Axion Luminosity of Photon Gs To illustrte possible strophysicl ppliction of the results obtined we clculte the contributions of the photon decy nd the photon colescence to the xion emissivity Q of photon gs, i.e. the rte of energy losses per unit volume. The decy Q (d) re: Q (d) = Z d 3 q 1 (2) 3! 1 n B (! 1 ) [10] nd colescence Q (c) Z 1 dt dw (F ) d dt B Be [12] xion emissivities (1 t) (1+n B (! 1 t)) (12) 0 ' 2:15 2 C 2 e m 7 T 11 5 f 2 m Q (c) = 1 Z Vd3 Z q 1 2V (2) n Vd3 q 2 B(! 3 1 ) (2) n B(! 3 2 )(! 1 +! 2 ) W (F ) c (13) ' 121:5 2 C 2 e m 7 T 11 B 5 f 2 m Be where n B (! i )istheplnck distribution function of photon with the energy! i t temperture T, B e =4: GistheSchwinger vlue. The fctor 1=2 in Eq. (13) tkes into ccount the equivlence of the initil photons. The xion emissivity due to the eective -vertex is determined by the photon colescence becuse the contribution of the photon decy is the correction of order of percent ccording to Eqs. (12) nd (13). The emissivities (12) nd (13) llow to estimte the contribution of the considered processes into the xion luminosity L (the energy losses by the 7

8 escping xions per unit time) in supernov explosion from region of order of hundred kilometers in size outside the neutrinosphere. In this region rther rreed plsm with the temperture of order of MeV nd mgnetic eld with the strength of order of G cn exist. Under these conditions the estimtion of the xion luminosity of photon gs is: 3 erg L ' 2 10 s ge T 1MeV 11 B G R 10 3 km 3 : (14) The comprison of Eq. (14) with the totl neutrino luminosity from the neutrinosphere L erg/s shows tht the contribution of the xion emission processes! nd! by the photon gs to the energy losses in supernov explosion is very smll. Conclusions In our tlk the eld-induced photon decy! nd colescence! re studied where is light pseudosclr prticle. For the pseudosclr prticle we considered the most widely discussed prticle, the xion, corresponding to the spontneous breking of the Peccei-Quinn symmetry. The forbidden in vcuum photon decy becomes kinemticlly possible becuse photons of dierent polriztions obtin dierent eld-induced \eective msses" squred. At the sme time n externl eld inuence on the xion mss is negligible. The processes! nd! could be of interest s n dditionl source of energy losses by strophysicl objects. We considered the cse of smll vlues of the dynmic prmeter which cn be relized, for exmple, in supernov explosion. The xion emission by the photon decy is the correction of order of percent to the emission by the photon colescence. While these processes nd their evlution re conceptully quite intruiging, the ctul energy-loss rte ppered to be rther smll in comprison with the neutrino luminosity in the conditions considered. Acknowledgements The work of N.V. Mikheev ws supported under grnt No. d9-11 by Interntionl Soros Science Eduction Progrm. This work ws prtilly sup-

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