The role of the massive photon decay via intermediate states of electron-electron-holes and proton-proton-holes into neutrino-anti-neutrino pairs in the course of neutron star cooling is investigated. These reactions may ba operative in hot neutron stars in the region of proton pairing. The corresponding contribution to the neutrino emissivity is calculated. It varies with the temperature as T3/2e−mγ/T for T < m γ, where mγ is an effective photon mass in superconducting matter. Estimates show that this process appears as strong cooling channel of neutron stars at temperatures T ≈ ( 10^9 - 10^10) K

Kolomeitsev, E. E.

Kämpfer, Burkhard

Voskresensky, D. N.

English

Qucosa – Hemholtz-Zentrum Dresden-Rossendorf

FORSCHUNGSZENTRUM ROSSENDORF e.v. FZR Decernber 1995 Preprint D. N. Voskresensky, E. E. Kolomeitsev and B. Kämpfer The role of the massive photon decay channel for the neutrino cooling of neutron stars Fctrschungszentrum Rossendorf e.V. Postfach 51 01 19 - D-01314 Dresden Bundesrepublik Deutschland Telefon (0351) 260 3258 Telefax (0351) 260 3709 E-Mail kaempferafz-rossendorf. de The role of the massive photon decay channel for the neutrino cooling of neutron stars IlrIoscow Engineering Physical Insti t U te, Kashirskoe shosse 31, 1 I5409 Moscow, Russia Institut für Kern und Hadronenphysik, Forschungszentrum Rossendorf e. V., FF 51 01 19, D-01 31 4 Dresden, Germany Institut für Theoretische Physik, T U  Dresden, Mommsenstr. 13, D-01 062 Dresden, Germany Abstract The r d e  of the massive photon decay via intermediate states of electron-electron- holes and proton-proton-holes into neutrino-anti-neutrino pairs in the Course of neutron star cooling is investigated. These reactions may be operative in hot neutron stars in the region of proton paiiring. The corresponding contribution to the neutrino emissivity is calculated. It varies with the temperature as ~ ~ / ~ e - ~ . r l ~  for T < m„ where m, is an effective photon mass in superconducting matter. Estimates show that this process appears as strong cooling channel of neutron stars at temperatures I" E (10' - 10'') I(. PACS number(s): 21.65.+f, 95.85.R, 97.60.1 key words: nuclear matter, neutron stars, neutrino cooling 'e-mail: voskresn@theor.rnephi.msk.su; voskreOrzri6f.gsi.de 'e-mail: kolomei@fi-rossendorf.de 3e-mail: kaempferOfz-rossendorf.de 1. The EIESTEI'S, EXOSAT and ROSAT observatories have measured surface temperatures of certain neutron stars and put upper limits on the surface temperatures of some other neutron stars ( cf. [l, 2, 31 and further references therein). The data for the supernova remnants in 3C58, Crab, RCW103 are related to rather slow cooling, while the data for Vela, PSR2334+61, PSR0656+14 and Geminga point to an essentially more rapid cooling. In the standard scenario of the neutron star cooling the most important channel belongs to the modified Urca process n n + n p  e- ü with an emissivity &FM calculate, e.g., by Friman and hIaxwell [4]. In ref. [4] the nucleon - nucleon interaction is treated within a model with free one-pion exchange (plus slight modifications). In a system with nucleon pairing this emissivity is suppressed by the factor exp(- (An + Ap)/T) [2], where An and Ap are the respective neutron and proton gaps determined by &(T) = Ai(0) (TcTi -T) T;' @(T,,; - T), (here @(X) is the step- function; i = {p, n), and Tc+ is the corresponding critical temperature for nucleon pairing). At temperatures T « Tc,p, Tc,n the cooling is determined by the photon radiation from the neutron star surface. We suggest here that the decay processes of massive photons (7,) via the electron-electron- hole (ee-l) and the proton-proton-hole (PP-') intermediate states to neutrino - antineutrino pairs, y„ + e e-I + pp-I -+ vivi, i = {e, V, T),  might be operative in hot neutron stars in the region of proton pairing T < Tc,p. These processes are determined by the diagrams Fat vertices in the nucleon diagrams (1) include the nucleon-nucleon correlations. In a Fermi system with pairing, besides the graphs (I) ,  tkere are also diagrams witk ansma- lous Green's functions of protons [5 ] .  However, their contribution to the correspsnding matrix elements is as small as Ap/CFP « 1 for T < T,, « cpp (cFp is the protsn Fermi energy Within this accuracy one can drop the ano~nalous diagrams a ~ i d  use for protsns in Green's functions for the normal Ferrni liquid. The contribution of the massive photon decay via the electrsn-efectron-hsle intermediate states (i.e., the first diagram in (1)) has been calculated by several authors references) for the case of the electron gas in white dwarfs and neutrsn star ezusts. 111 an electron plasma the photon acquires an effective in-medium maas which is equal to the electrow plasrna frequency WPI 2 2 e (31/"p)1/3, where e is the electron charge, and p, denotes the electron density (tve employ units with 6 = c = 1). Therefore, the contribution to the emissivity of the mentioned process is suppressed by the factor exp(-wpr/T). In white dwarfs and neutron star crusts the electron density is not too large and the process is effective. In neutron star interiors the electron density is equal to the proton density pp, due to the electroneutrality, which is where po 2 0.17 fm-3 stands for the nuclear Saturation density, and we used the values [4] of the neutron and proton Fermi momenta, p ~ ,  = 340(p/po) MeV and p ~ ,  = 85(p/po)2/3 MeV. Shus, at densities typical for the neutron star interiors, the value of the electron plasma frequency is large, i.e., wpr(p - po) - 8 MeV, and at temperatures T < wpl the process -I, -+ e e-l -+ viüi is strongly suppressed. Oppositely, in a superconducting medium, formed in neutron stars at T < Tc,„ photons acquire the effective in-medium mass due to the Higgs - Meissner effect [5] 4 and obey the dispersion relation w = Jz2  + m:, where w and k are respectively the frequency and the momentum of the photon, and mp stands for the effective in-medium proton mass. The quantity P;(T) = pp(T,,, - T)/T, ,  denotes the paired proton density. Supposing m;(po) = 0.8mN (with mN as the free nucleon mass) with eqs. (2,3) we estimate m,(p = po, T)  2 1.6 J(T,, - i')/T,,, MeV < wpl(p PO) - 8 MeV. Due to the rather small effective photon mass in superconducting neutron star matter at T < T,, « wpl one may expect a corresponding increase of the contribution of the processes (1) to the neutrino emissivity. We shall calculate the contribution of these processes to the neutrino emissivity E; and compare the results with the emissivity of the modified Urca process E:" [4] and with the photon emissivity E; from the star surface. We show that the processes (1) may play an important r6le in the Course of neutron star cooling at temperatures T = (10" 1O1O) K. 2. The matrix element of the diagrams (1) for the ith neutrino species (i = {U„ V„ U,}) reads where 1 and I is the in-medium electron (proton) Green's function, and n j ( p )  = O ( p ~ j  - p) .  E; is the cor- responding polarization four-vector of the massive photon with three polarization states in ~u~erconducting matter. The factor I', takes into account the ~iucleon-nucleon correlations in the photon vertex. The quantity G = 1. l'i . 10-5 GeV-2 is ithe Fermi constant of the weak in- teraction. Above, 1, denotes the neutrino weak current. The electron aad proton weak currents are determined by where CF) = CF) = 1 + 4sin2 Ow 2 1.92 and c(V.') = c $ ~ )  = C$) = 1 - 4sin2 Bw E 0.08. Bw stands for the Weinberg angle, and c p )  = -C?J~') = 1. The proton coupling is corrected by the nucleon-nucleon correlations, i.e., by factors rcPp and -fPp [7]. By integrating in eq. (5) over the energy variable, we obtain for the i-th neutrino species -4 - k k p  = u2 - i2. The four-velocity of tlie medium where j P  = (k u)kP - upk2, kP = (W, k), k P u P  is introduced for the sake of a covariant notation. The transverse (rt), longitudinal (71) and axial (r5)  components of the tensors in eq. (7) render Here we note that the contribution of the axial component TS to the resulting neutrino emissivity is small ( T ~ / T ~  ~ : T ~ / W ~ T [  m,/m> for protons and .- (m,m,/p:.e) In(p~./rn,) for electrons), so that i t  will be dropped below. The squared matrix element (4) Tor a certain neutrino species, summed over the lepton Spins, and averaged over the three photon polarizations, may be cast into the following form 4 where (k- - q1,z) = ww1,2 - (kqi ,~) ,  and w1,2 and (1;,2 denote the frequencies and the momenta of the neutrino and anti-neutrino. We have also used that Tr{l'lu) = 8[qfq; + q;q; - g'U(ql . q2) - . 4-i e~~~~ q l~q~p] .  3. The emissivity of the processes (1) is given by Substituting eq. (11) into eq. (12), we obtain eventually after some integrations where rr = 7 and Some numerically small terms are dropped in eq. (14). The integral I can be calculated analytically in two limiting cases a « 1 and cu » 1, Thus, cornbining eqs. (SJ3-151, tve. obtain an estimate for the emissivity of the reactions (1) (we present here the result for m, > T an$ for three neutrino species): Here the quantity T9 stands for the temperature measured in 10' E(. The unity in squared brackets of eq. (16)  corresponds to the electron-electron-hole diagram, whereas the factor 17 is related to the proton-proton-hole (first term in eq. (17))  and the interference diagrams (second term in eq. (17)) .  The emissivity eq. (16) varies with the temperature as exp(-m-, /T),  whereas the emis- sivity of the modified Urca process varies as T 8 e x p ( - ( A ,  + An)/T)  in the region of proton (LI, # 0 )  and neutron (An # 0 )  pairing. Hence, one can expect that the process y, -+ v6 will dominate at comparatively low temperatures, when Ap(T) + & ( T )  - m,(T) > 0 and T < T,,p. 4. In order to perform quantitative estimates we need the values of the nucleon-nucleon correlation factors i1;„ and I?,. According to ref. [7] one has where fnp  E -0.75 and fnn E 1.25 stand for the constants in the theory of finite Fermi systems [8, 71; C;' = m n p ~ , / 7 ( - ~  is the density of states at the Fermi surface; Am is the neutron- neutron-hole loop, 4 for values of W » I k lpFn/m; of interest, and I'-'( f„) = 1 - 2 fnnCoAm. We note that the second terrn in eq. (18) is not proportional to a small factor CL-', because the nucleon - nucleon correlations also allow for the radiation of v6-pairs frsm the nn-' losp. Numerical estimates of the ratio R, are as follows: for cr » 1 R, 2 1.6 fsr p = po, rnE(p0) 2 0.8mn and B, E 2.1 for p = 2po, m;(2po) 2 0.7mn; for cu « 1 R, E 1  and csrrelation effects are negligible. The factor r, is approximately unity, since the correction is proportional to a small proton-proton-hole loop factor ( / I p p )  being suppressed at comparatively small protsn densities. With these estimates we observe that the main contributisn to the neutrino emissivity is given by the electron-electron-hole processes. 5. The ratio of the emissivity E; (16)  to the emissivity &yhf of thc modified Urca process yields For further estimates we need the values of the neutron and proton gaps, which unfortunately are essentially model dependent. E.g., the evaluation in ref. [9] yields &(0) 2 8.4T,, Y 0.6 MeV, T,,, = 0.07 MeV for 3P2 neutron pairing at p = po, and A,(0) = 1.76T,,p = 3 MeV, T,, 1.7 MeV for 1s proton pairing, while ref. [10] uses A,(O) 2 2.1 MeV, T , ,  2 0.25 MeV and A,(0) z 0.7 MeV, T , ,  E 0.4 MeV for p = po. Employing these estimates of the zero- temperature gaps, their temperature dependence and the photon effective mass, we obtain from eq. (20) the temperature dependence of the ratio RF,i. In order to  find the lower temperature limit, at which the processes ;im -t v F  are still operative, we need to compare the value E: with the value of the photon ernissivity from the neutron star surface, E; = 3aT,4/R, where a is the Stefan-Boltzmann constant, T, denotes the surface temperature of the star and R stands for the star radius. By employing a relation [11] between the surface and interior temperatures, we obtain &-' 9. - 0 . 7  m~ ( 7 ) I 2  ( 3 T ) (:)'I3 R, = 2% 1.2.10 J' MeV I$- -  Es, 2 m, [l +I], where the star radius and the mass are supposed to be respectively 10 km and 1.4-M@, with n/r, as solar mass, and is some averaged value of the density in the neutron star interior. The ratios RFIM and R, are plotted as a function of the temperature in Fig. 1 for the both mentioned above parameter choices. We See that the processes (1) are operative in the temperature range 1-10' K5 T K' 8.10' K for the parameter choice [9] and 1-10' KK' T 5 4-10" for the parameters of ref. [10]. 6 .  A few remarks are in order: (i) Photons with electron plasma frequency wpl may also decay into neutrino pairs. How- ever, the corresponding contribution to the emissivity of the neutron star interior is negligible compared to that for the modified Urca process. (ii) The processes (1) may also occur in a charged-pion (or kaon) condensate state, however, they are suppressed due to the large value of the effective photon mass4 m, z d m  Y 6 MeV for the condensate field y, 2 O.lm, 2 14 MeV. (iii) Deriving the above used value erfilZ"I, one describes the nucleon-nucleon interaction es- sentially by the free one-pion exchange. However, in reality at p > (0.5 - l)po the total nucleon-nucleon interaction does not reduce to the free one-pion exchange because of the strong polarization of the medium, by which an essential part belongs to the in-medium pionic excita- tions [7, 12, 13, 143. Occurring in the intermediate states of the reaction, the in-medium pions 41n this estirnate the peculiarities of the coridensate with the non-vanishing rnomentum [12] are for simplicity ignored. can also decay into eü, or first into a nucleon-nucleon-hole, which then radiates ey, thereby substantially increasing the resulting emissivity. She other reaction channels [T ,  141 in the superfluid phase with paired nucleons n 4 n„;,vü and p t p„;,vV give rise to even a larger contribution to the emissivity than that of the modified Urca process. A b o ~ e  we compare the value EJ, with the value erbf just because the latter is used in the st.andard scenarios of cooling, while the mentioned in-medium processes are not yet included in computer simulation code. 7. In Summary, the processes .;/, t e e-I $ pp-I -+ vv might be operative in some temperature interval T ( I O ~ - ~ O ' ~ )  K, T < T,,„ and, together with other in-medium modified processes, it should be incorporated into computer simulations of neutron star cooling. Acknowledgments: We acknowledge V.M. Osadchiev for fruitful discussions. The research described in this publication was made possible in part by Grants K3W000 from the International Science Foundation and N3W300 from the International Science Foundation and the Riissian Government. B.K. and E.E.K. are supported by the BMBF grant 06DR666. E.E.K. acknowledges the Heisenberg- Landaii program for the possibility to participate in the school "Development in Nuclear Theory and Particle Physics"', Diibna, August 24 - September 8, 1995, where a part of this work has been performed. D.N.V. and E.E.K. are indebted to JINR Dubna for warm hospitality. References [l] S. Shapiro and S.X. Teukolsky, Black Holes, White Dwarfs and Neutron Stars: The  Physics of Compact Objects, chapter 10 (Wiley, 'J.Y., 1983). [2] D. Pines, R. Tamagaki and S. Tsuruta (eds.), Neutron Stars, (Addison-Weseley, N.Y., 1992). [3] H. Umeda, K. Nomoto, S. Tsuruta, T. Muto and T. Tatsumi, Ap. J .  431 (1994). [4] B. Friman and O.V. Maxwell, Ap. J. 232 (1979) 541; O.V. Maxwell, Ap. J. 231 (1979) 201. (51 E.M. Lifshiz and L.P. Pitaevsky, Statistical Physics, Part 2 (Pergamon Press, 1980). [6] J.B. Adams, M.A. Ruder~nan and C.-H. Woo, Phys. Rev. 129 (1963) 1383; D.A. Dicus, Phys. Rev. D6 (1972) 941; M. Soyeur and G.E. Brown, Nucl. Phys. A324 (1979) 464; V.N. Oraevskii, V.B. Semikoz and Ya. A. Smorodinsky, Part. and Nuclei 55 (1995) 312; J.C. D'Olivio, J.F. Nieves and P. Pal, Phys. Rev. D40 (1989) 3679. [7] D.N. Voskresensky and A.V. Senatorov, Yad. Fiz. 45 (1987) 657 (Sov. J .  Nucl. Phys. 45 (1987) 411). [8] A B .  Migdal, Theory of Finite Systems and Properties of Atomic Nuclei, (in Russian), (Nauka, Moscow , 1983). [9] T. Takatsuka, Prog. Theor. Phys. 48 (1972) 1517 [10] D. Pines and M.A. Alpar, Nature 316 (1985) 27. [ll] E.H. Gudmundsson, C.J. Pethick and R.I. Epstein, Ap. J .  272 (1983) 286. [12] A.B. bligdal, E.E. Saperstein, M.A. Troitsky and D.N. Voskresensky, Phys. Rep. 192 (1990) 179. [13] D.N. Voskresensky and A.V. Senatorov, Pis'ma V %hETF 40 (1984) 395 (JETP Lett. 40 (1984) 1212); D.N. Voskresensky and A.V. Senatorov, ZhETF 90 (1986) 1505 (JETP 63 (1986) 885). 1141 A.V. Senatorov and D.N. Voskresensky, Phys. Lett. B184 (1987) 119. Figure I: The temperature dependence of the ratios RF.+I and R, at nucleon density p = po. The solid curves correspond to the parameter choice of ref. 191, whereas the dashed curves depict results with parameters of ref. 1101. 

Contribution of the massive photon decay channel to the neutrino cooling of neutron stars

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Contribution of the massive photon decay channel to the neutrino cooling of neutron stars

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