IC/94/60 INTERNAL REPORT 1. INTRODUCTION (Limited distribution) The investigation of processes with participation of supersynunetric (sec [ 1,2]) in external fields is one of the intriguing branches of the modern element ary International Atomic Energy Agency [3-13]. In [10,11] we have discussed 's interactions with supersymmetric and particles in the presence of external electromagnetic fields. Four types of three-particle United Nations Educational Scientific and Cultural Organisation processes l—>w + v, J->i + 7, w —t I + ir, I —• 1 + 7 (1) INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS (here / is a charged lepton, w is wino, v is sneutrino, /is a charged slepton, 7 is photino) have been studied in external constant electromagnetic field. The effective Lagrangians of these processes are given by INFLUENCE OF SUPERSYMMETRIC PARTICLE PAIR EMISSION BY ON EVOLUTION OF STAR £1 =

9\ = \72sin0,,,1 (2) Gennady G. Likhachev and the exact solutions of the corresponding wave equations accounting for external Department of Theoretical Physics, Physics Faculty, Moscow State University, field are used for the wave functions of charged particles (see [14]). Leninskie Gory, 119899 Moscow, Russian Federation The first two of these processes (1) describe the lepton decays into the pair of supersymmetric particles. Such processes are forbidden in vacuum and could proceed and only in the case of relativistic energy of the initial lepton and in the presence of strong external electromagnetic field. These extreme physical conditions can exist in astro- Alexander I. Studtnikin physical objects (for example in neutron star [15,16] ). Therefore the neutron star can International Centre for Theoretical Physics, Trieste, Italy be regarded as a natural laboratory for the investigation of supersymmetric particles. Indeed very strong magnetic fields of the order 1012 — 10I4G are believed to exist and 16 18 Department of Theoretical Physics, Physics Faculty, Moscow State University, near neutron star [15,17] and even much stronger magnetic fields (10 — 10 G) may Leninskie Gory, 119899 Moscow, Russian Federation. exist in the interion of neutron star as theoretical assumptions [16,18]. Such intense magnetic fields should be great importance in inducing new mechanisms of cooling of neutron stars due to hypothetical particles emission (see, for example [19,20]). In this ABSTRACT paper we should like to discuss new possible mechanisms of energy losses of a neutron star due to processes of emission of supersymmetric particles by : The processes of emission of a pair of supersymmetric particles (wino and sneutrino or selectron and photino) by electron in strong magnetic field of neutron star are dis- w + V, + 7, (3) cussed. The contributions to the energy losses of the neutron star due to these processes that are possible in the presence of a magnetic field. are calculated. The comparison of these new proposed mechanisms of energy losses with It is supposed that sneutrinos and photinos created in the crust of neutron star the standard mechanism is presented. escape freely from the interior of the star and must carry away a certain amount of energy. Therefore these new proposed processes of emission of supersymmetric particles may give certain contribution to energy losses of neutron star [11,13]. Note that another possible mechanism of energy losses of a neutron star due to the familon emission was MIRAMARE TRIESTE discussed in [19,20], see also [21]. April 1«J94 Using the formula (2) for the lagrangians we obtain the following expressions for probabilities of processes (3) (see Refs.[10,ll]):

(4) *

where i\ = kg , «2 = e2, V{ = n — n'-, d = sinOdBdtp. We suppose that the supersyrametric particles are emitted in the crust of neutron For the coefficients Ai,Bi,d we obtain star. In accordance with the modern astrophysical theories the crust of neutron star may be described as a crystal lattice of atomic nuclears plunged into electron gas. M 3 The electrons of this gas are characterized by electron number density nt ~ 10 cm~ and a temperature T of about 10*A' [22,23]. At these conditions for electron density, temperature and star magnetic field the electron gas is extremely relativistic and highly degenerated. In Refs.[10,ll] we have shown that the probabilities of processes (3) may not be small, rnd moreover, their values may reach the values of probabilities of processes of the in the case of large energies of particles and/or in the case of strong magnetic fields. There are two different cases for possible values of energy of electron and strength f<-)_(+) or of external field. The first case corresponds to the quasiclassical approach when the energies of electrons are large, the fields are not very strong (B < Ba) and the quantum numbers of all particles are large (n > 1, n\ 3> 1 , n is the quantum number of electron, nj are the quantum numbers of final charged particles). This case is studied in Section 2. In Section 3 we investigate the opposite situation when the energies of particles are not so large, the magnetic fields are very strong [B > B ) and the quantum numbers s/Zyn a € j_ sin

2. TEMPERATURE DEPENDENCE OF LUMINOSITY In these formulas we use the following denotations: OF NEUTRON STAR

(±) ±) /, = F44 ±F1],/< = F33 ± F,,,Ft> = Now using our recent results (see formulas (12), (13) in [10] and formula (3) in [11]) for the probabilities of processes I —> wv and / —> l-f in magnetic field and taking *,j = 1,2,3,4; i = 1,2, into account the distribution functions of the particles we investigate the temperature dependence of the luminosities of neutron star. Averaging over initial states of particles and summing over final ones and supposing Q]^ = 1 in formula (3) from [11] we can obtain the expressions for the <7i = 1 + 55' """' , (T2 = 1 + 55— COS 9, probabilities (i = 1 for the process e —> wit and i — 1 for the process e -+ e~f):

du, 12 {y )Al (1 + (5) ex ^7 ^ * * ' ~ (+) _. Pa Pi .2 We use the following denotations: KS-' = 1 -

/3 /3 M = «, , A2 = i4 , B, = 1 + m ~ A,, B2 = 1 - + A2,

6 B Here m is the mass of electron, m\ is the mas: of wino, m'^ is the mass of selectron, mj «.= ^, X = ± is the mass of sneutrino, mj is the mass of ph< tino, e is the energy of electron, t\ is the energy of wino (< = 1) or selectron (i = 2), p is the momentum of electron, pj_ is the '= /° perpendicular to the field B component of the momentum, kg = e — e\ is the energy of = / £*. sneutrino (t = 1) or photino (i = 2), k, is the momentum of sneutrino (i = 1) or photino Jo (j = 2), 5, 5', S are the spin states of electron, wino and photino correspondingly, n denotes the Landau levels. particles are relativistic and degenerated. Then taking into account that tin: selectrou chemical potential /jj 's negative and that the electron chemical potential fi is equal to its Fermi energy at zero temperature we can obtain the following approximations for Here \ is the field parameter, *(y) is the Eiry function. the expressions of Jx and J2 in(10): Let us use a constant uniform magnetic field Bwa model for the field of neutron star (the reference frame can be chosen so that B \\ OZ). ~ T, ~ 2pF. (11) Note that now we suppose quantum numbers of all particles are large (n > 1 and n[ > 1). Such a case corresponds to the quasiclassical approach (the electrons are Tile integration over u can be performed analogously to the calculations of the probabilities of processes (3) and the integration method depends on the argument y of relativistic (px > ™) and external fields are not very strong (B < BB = ^-)). see The rate of the energy losses per unit volume from the crust of the neutron star the Eiry functions and field parameter \ ( for details Refs.[10,ll]). e v& e K ar e due to the processes of emission of the supersymmetric particles (3) is characterized by In the case when \ < ('/1 + ^i)\/^ ^ l" °f V l g and luminosities are the integration of the probabilities (4): exponential small values. In the opposite case, \ ~%> (rji f A;)3'2 (this case corresponds to vpry large energies of electrons and small fields B < Bo} the argument of Eiry functions is small. After (6) integration over u and 0 , taking into account that £j_ — PF sin 0 and x — ^rr we can obtain from (10) for sneutrino and photino luminosities of neutron star (in ordinary where k^ — E - e\ is the energy of sneutrino (i = 1) or photino (t = 2). The functions units) : / and /{ are the distribution functions of electron and wino described by Fermi-Dirac enj statistics: , y P,= -T v ' 1V" ' (12) - • (7) where A'i = 2SkT, K'i = 16yyc. /; = (8) The comparibwin of the obtained results for the luminosities of the neutron star due to processes e —> wv and c —* C7 gives : function /j is the distribution function of selectron described by Bose-Einstein statistics: (13) gs) ikT'

(9) B 38 J For the temperature T = 10 A' and th« electron density ne ~ 10 cm~ it follows that ^=3x 10s. Hi are the chemical potentials. The luminosity of neutron star due to process e —• tbv is proportional to tempe- Using formulas (4),(5),(6) and after integration over momentum p (accounting for rature and the luminosity of neutron star due to process e —> £7 has no temperature the fact that in the case of degenerated electron gas the main contribution to the integral dependence in the main term. This difference in the luminosities can be explained due (6) is given by electrons with momentum within the narrow range near the Fermi value to properties of final particles : stJectrons are the , and winos are the pF ) we get: and y must obey the Pauli principle. As it follows from Eq.(12) in the case of degen- erated and relativistic electron gas (T

VeB (15) (19) where dNn is the number of electrons in the volume V being in the states with quantum whore number n and momentum pj in the range dpj. Using the calculations of the probabilities of the processes of emission of supersym- . 4 l-f,'- metric particles in the presence of external magnetic field (see formulas (7) and (8) in [10]) we obtain for the energy losses rates due to these processes from one electron the and following expressions: p' ' =,

( 2 In order to receive the sneutrino find photino luminosities (P^V and P V) per unit J,° cos 9 volume of the crust of the neutron star we average the energy losses rates (19) over the density of electron states (IS). It is possible to show [19,30] that for the magnetic field 2 (16) B < Bt - -f- = (^) Bu only the terms with n » 1 contribute to the sum over n. This is why we replace the summation over n by the integration over pL and obtain

"5 ~ where m^ = mf — m2, m^ = {m\ -+- mj2 — m2, Here 8 is the angle between the wavevector k, of the sneutrino (z = 1) or photino (i = 2) and the field B in the frame s which moves along the field with velocity v\\ = f. In this frame the energy of sneutrino ^(<). (20) (i = 1) and photino (1 = 2) is given by The integral in (20) can be approximately evaluated using the paramctrization m iA Pi = *^} + i) where 0 < z < 1 and A, is also assumed to be a small quantity, A £i *0 = ' ~ m*'" • Finally we get for the sneutrino and photino luminosities (in ordinary units):

1 e\ = m n\\\B -ni+fl. numerical estimations,we can obtain from Eq.(24) that the masses of supersymmetric particles wino and select ron connect with field B : where (25)

For the case mj ~ m- (sneutrino and photino are massive) we find the analogous formulas: 2 (i) 2eBh 32v/2V/ P0 fiV i>/2 3 n C (ln A -J- 1 i -2/1 i «.2\^^ D^<.\5/2(J(>) I I*") The total sneutrino and photino luminosities per unit volume from the crust of the

neutron star can be obtained after summation over all possible n't in Eq.(21): and for the sum of these masses at identical conditions: r2.8£ i + ml zr m! (27) p(0 _ V" p(') (22) to««J / , n'; ' 13 Here m = 0.51 MtV is the electron mass, Bo = ra'/e ~ 4.41 x 10 G is the critical field and the B is expressed in Gauss. However, in the case of strong magnetic fields of the neutron star terms corre- So from Eqs.(24)-(27) we can obtain that in magnetic fields of the order B ~ 10)3 — sponding to low laying levels (small values of n\ ~ 1 or n\ = 0) give the most sufficient 1014G the processes of emission of fina! supersymmetric particles with masses of order contributions to the total luminosities, the contributions of terms with big values of n' t m-, rhi ~ 105 - 5 x 106eV can give the sufficient contributions in the energy losses of decrease when the field increases. neutron star, and at B ~ 10I6-1018G these estimations areof order 5x 107-5x 108eF. It worth noting that the constraints on the masses become more restrictive in the strong magnetic fields. For example the similar considerations lead to the result 4. THE COMPARISON BETWEEN DIFFERENT MECHANISMS that the role of emission of the heavy supersymmetric particles with masses of order OF ENERGY LOSSES OF NEUTRON STAR 50 GeV will be comparable with the role of neutrinos emission into the evolution of neutron star only at very strong magnetic fields (B ~ 1020G , see also Refs.[ll,13]). At Now suppose that the formula (21) accounts for all masses of supersymmetric par- less strong fields we have the domination of the standard mechanism of energy losses ticles exactly let us compare the obtained results (formula (21)) for the energy losses or the mechanisms of emission of the light particles,such as neutrinos or some other of neutron star due to emission of supersymmetric particles and the standard neutrino hypothetical particles such as familons, etc. (see, for example Refs.[16,19,20,25]). luminosity that was estimated in Refs.[16,25] as follows Now we can choose the real masses of supersymmetric particles (for example m j = ma, = 60 GeV, M'2 = me = 60 GeV,rh, = mj = 60 GeV, m2 = m=, = 20 GeV ) _0_y/3 r erg 1 (23) and estimate the fields at which the emission of new supersymmetric particles and the 108/ ' Lcm3 x J neutrino emission can be of order one. The comparison between P and P(l/) show that at field B < 1021G the neutrino 7 (for the case BpU* > 8 x 10 T,pm is the density). luminosity is larger then the new proposed mechanisms. This phenomenon can be Let us determine the constraints on the masses of supersymmetric particles and explained by the strong exponential dependence of P (due to term ( — ^) in exponent). the values of fields in which the new proposed mechanisms of energy losses can be The luminosities of P and P("' can be of one order at very strong field (B ~ comparable with the standard one due to the neutrino emission. 8.5 x 1021£? for process e —> ej and B ~ 1.2 x lO^G for process e —> w) and these Now we investigate the estimations of masses. Using Eq.(21) with n[ = 0 and luminosities can achieve the values of order 13 erg xcm"3xs"' and 16 ergxcm~3 xs'1 supposing that sneutrino and photino are light (so that m< <£ m\ and its values can be correspondingly. neglected) it is possible to get the approximate expressions for the rru, :

tBU 1/2 -In- (24) 5. CONCLUSION i« ^ Lcf-33 In this paper we have discussed new possible mechanisms of energy losses of neutron (l) (the new mechanisms of energy losses of neutron star can star due to emission of pair of supersymmetric particles by electron. Our analyses Demanding P0 ( g be of one order with one due to neutrinos emission), using the formulfl a (23) fof r T = have shown that in case of ordinary values of energies of particles and strength of 6 3 13 magnetic fields typical for neutron star the contributions of these mechanisms are small. 10 A", pm =s 9.1 x 10'°ff x cm.- and B > 10 G and taking a' = 1 and A ~ 0.1 in our

10 The emission of supersymmetric particles may give a considerable contribution and the REFERENCES luminosity of neutron star due to these processes may not be small, and its value may reach the value of luminosity of neutron star due to standard neutrino emission in the l.H. Haber and G.Kane, Phys. Rrp. 117 (1985) 75. 20 case of high energies of particles ex > 10 eV and/or in the case of strong magnetic 2.H. Nilles, Phys. Rep. 110 (1984) 1. 21 field (B> 10 G). 3.A.I. Studenikin, Preprint ITP-89-83E (1989) Kiev, USSR. 4.A.I. Studenikin, Sov. JETP. 07 (1990) 1407 Acknowledgments 5.P.A. Eminov, Sov. J. Nucl.Phys. 51 (1990) 542. 6.A.I. Studenikin and I.M. Ternov, Preprint MSU N13 (1990) Moscow,USSR. One of the authors (A.I.S.) would like to thank Professor Abdus Salam, the In- 7.V.Ch. Zhukovsky and P.A.Eminov, Sov. J. Nucl. Phys. 52 (1990) 1473. ternational Atomic Energy Agency and UNESCO for hospitality at the International 8.A.V. Kurilin, Phys. Lett. B. 249 (1990) 455. Centre for Theoretical Physics, Trieste. 9.E.V. Mersonova, A.I. Studenikin and I.M. Ternov, Vest. MSU, Phys., Astron. 32 (1991) 76. 10.G.G. Likhachev, A.I. Studenikin and I.M. Trrnov, Sov. J. Nucl. Phys. 53 (1991) 1614. G.G. Likhachev and A.I. Studeuikin, Dep. VINITI, 6374V90. ll.G.G. Likhachev and A.I. Studenikin, Sov. J. Nucl. Phys. 55 (1992) 150. 12.E.V. Arbuzova and A.I. Studenikin, ICTP, Trieste, Preprint No.IC/93/lC7 (1993). 13.G.G. Likhachev and A.I. Studenikin, ICTP, Trieste, Preprint No.IC/93/168 (1993). 14.A.A. Sokolov and I.M.Ternov, Relativistic Electron, Nauka, Moscow (1983). 15.V.M. Lipunov, Astrophysics of Neutron Stars, Nauka, Moscow (1987). 16.J.P. Landstreet, Phys. Rev. 153 (1967) 1372. 17.T. Gun and T. Ostriker, 221 (1969) 454. 1S.G.A. Shulman, Sov. Astron. J. 67 (1990) 334. 19.A.V. Averin, A.V. Dorisov ami A.I.St.udenikin, Sov, J, Nucl. Phys. 50 (1989) 1058. 20.A.V. Averin, A.V. Borisov and A.I- Studenikin, Phys. Lett. 231B (1989) 280. 21.G. Ruffett, Phys. Rep. 198 (1990) 1. 22.J.B. Zeldovitch and I.D. Novikov, Structure and Evolution of , Nauka, Moscow (1975). 23.S.L. Shapiro and S.A. Teukolsky, Black Holes, White Dwarfts and Neutron Stars, Wiley, New-York (1983). 24.A.D. Kaminker, K.P. Levenfish and D.G. Jakovlev, Sov. Lett, in Astron. ,1. 17 (1991) 1090. 25.V. Canuto, H.-Y. Chiu, C.K. Chou and L. Fassoi-Canuto, Phys. Rev. D2 (1970) 281.

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