IC/83/17
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
ON THE FATE OF SUPERHEAVY MAGNETIC MONOPOLES IN A NEUTRON STAR
V.A. Kuzmin
and INTERNATIONAL ATOMIC ENERGY V.A. Rubakov AGENCY
UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1982M1RAMARE-TRIESTE
1. The observation [l,2] that grand unified magnetic monopoies [3] can international Atomic Energy Agency catalyze the baryon decay with a strong interaction rate has recently raised and a growing interest. This effect can be observed in deep underground [1,^,5] United Nations Educational Scientific and Cultural Organization and deep underwater [U] experiments, an evidence for its existence in the sun can be provided by solar neutrino detectors [1,6], so the eosmological and INTEREATIONAL CENTRE FOR THEORETICAL PHYSICS astrophysical limits on the monopole flux are of particular importance. While the eosmological limit on the flux of monopoles catalyzing the baryon decay is rather weak [5], much stronger limits have been claimed to follow from the con- siderations of neutron stars [7-103. The neutron "decays" catalyzed by mono- poles with the cross-section [1,2] a = a Jv where o = 0(l GeV~ ) and v is a relative neutron-monopole velocity, tend to heat a neutron star. The ON THE FATE OF 5UPERHEAVY MAGNETIC MONOPOLES IH A HEuTROB STAR * observational limit on the X-ray luminosity of neutron stars [ll], Y 31 -1 L . < 10 erg s , implies the following limit on the number of monopoles • V.A. Kuzmin and V.A. Rubakov in any neutron star, N , and on the cross-section parameter o [10]* International Centre for Theoretical Physics, Trieste, Italy, and Institute for Nuclear Research of the Academy of Sciences of the USSR, 0 11* Moscow, USSR.** < 10 (1) 0.1 fm
If all monopoles whenever captured by the star were present inside the star
ABSTRACT up to now, Eq.(l) would lead to very strong constraints on the flux of relic monopoles [7-10] which would make their experimental observation almost impossible. We propose two possible scenarios of the behaviour of superheavy magnetic monopoles in a neutron star, in which the monopole-antimonopole We believe, however, that the question of the survival of magnetic annihilation rate is sufficiently large to prevent the enormous heating monopoles in a neutron star is by no means clear. In fact, the monopole- of a neutron star due to the monopole induced neutron decays. We find antimonopole annihilation rate crucially depends on the particular properties that the galactic monopcle flux of order 10~1<5 cm"2 s"1 ster"1 can be of matter inside the star core, which are still unknown. The main purpose compatible with the observational limit on the X-ray luminosity of of the present paper is to propose two possible scenarios of the behaviour neutron stars. of superheavy (rn^^jlO GeV) monopoles in the neutron star; in both of them the rocnopole-antimonopole annihilation is sufficiently effective to make M
W om s consistent with Eq.(l) even for the galactic monopole flux J(O' 10 ster , which is close to experimentally observable values [12,M as well as to other astrophysical and eosmological limits [13-15]. The general feature MIRAMAHE - TRIESTE of both scenarios is that at the first stage monopoles exhaust, magnetic February 1983 fields deep inside the star (but not near the star surface) and then con- centrate either near the centre (for neutron stars with non-superconducting interiors), or on the boundary of the central superconductive region, if it exists. In the first case the considerable annihilation rate is simply due * To be submitted for publication. to the small size of the region occupied by monopoles, while in the second ** Permanent address. case the annihilation is enhanced by the presence of the superconductor.
-£- 2. Our first scenario applies to the neutron stars with normal (non- d Hv, (5) superconducting) interiors. As has been argued in Bef.8, a neutron star dt 'M°M is likely to contain no mcnopoles at the mon •• , " i*r- format "on. Just after this moment, it begins to capture the relic monopoles with the rate [7-10] where n^ is the monopole density. From (5) we find that the characteristic time of the exhaustion of the magnetic field is independent of H (cf. [IT])
and is equal to t = a/8ifgMn,, , The restoration of the magnetic field due dt = F = (2) to the magnetic flux diffusion occurs with the characteristic time 2 i 33 6 h where ag ~ 1O '" °
= magnetic field in this region H(t) (cf. (3)). So, we can rewrite Eq.(5) as dx a v,, (1*) follows:
11 where vM is the monopole velocity and a ~ 10 GeV/cm (although the second dR(t) (6) dt process can give somewhat larger contribution to a). Therefore, the monopole RS(t) velocity is related to the magnetic field as follows, v = g^ H/a, This where Ft is the number of monopoles inside the star. Therefore, leads to the estimate of the dissipation of the energy density of the magnetic the monopolV*>e collapse ends up at t field per unit time, coll
-h- -3- IC/S3/17 1. The observation [1,2] that grand unified magnetic monopoles [3] can International Atomic: F.nergy Agency catalyze the baryon decay with a strong interaction rate has recently raised and a growing interest. This effect can be observed in deep underground [1,1+ ,5] Dili ted Nations id'.-.rational Scientific and Cultural Organization and deep underwater [k] experiments, an evidence for its existence in the sun can be provided by solar neutrino detectors [1,6], so the cosmological and IK&HHATIOKAL CENTRE FOR THEORETICAL PHYSICS astrophysical limits on the monopole flux are of particular importance. While the cosmological limit on the flux of monopoles catalyzing the "bafyon decay is rather weak [5], much stronger limits have been claimed to follow from the con- siderations of neutron stars [7-10]. The neutron "decays" catalyzed by mono- poles with the cross-section [1,2] 0 = a /v where a = 0(1 QeV ) and v
OM THE FATE OF SUPEEHEAVY MAGNETIC MONOPOLES IN A UEUTROH STAR * is a relative neutron-monopole velocity, tend to heat a neutron star. The observational limit on the X-ray luminosity of neutron stars [11], Y 31 -1 V.A. Kuzmin and V.A. Rubakov L . < 10 erg s , implies the following limit on the number of monopoles in any neutron star, N , and on the cross-section parameter 0 [lO]" International Centre for Theoretical Physics, Trieste, Italy, and Institute for nuclear Research of the Academy of Sciences of the USSR, lli Moscow, USSR.** 10 (1) 0.1 fm
If all monopoles whenever captured by the star were present inside the star
ABSTRACT up to now, Eq. (l) would lead to very strong constraints on the flux of relic monopoles [7-10] which would make their experimental observation almost impossible. We propose two possible scenarios of the behaviour of superheavy magnetic monopoles in a neutron star, in which the monopole-antimonopole We believe, however, that the question of the survival of magnetic annihilation rate is sufficiently large to prevent the enormous heating monopoles in a neutron star is by no means clear. In fact, the monopole- of a neutron star due to the monopole induced neutron decays. We find antiraonopole annihilation rate crucially depends on the particular properties that the galactic monopole flux of order 10"16 cm"2 s"1 ster"1 can he of matter inside the star core, which are still unknown. The main purpose compatible with the observational limit on the X-ray luminosity of of the present paper is to propose two possible scenarios of the behaviour neutron stars. of superheavy (n^ ^,10 G-eV) monopoles in the neutron star; In both of them the mcnopole-antimonopole annihilation is sufficiently effective to make H 16 2 1 consistent with Eq.(l) even for the galactic monopole flux J—*" 10~ cm s~
ster , which is close to experimentally observable values [12,1*] as well as to other astrophysical and cosmological limits [13-15]. The general feature MIRAMARE - TRIESTE of both scenarios is that at the first stage taonopoles exhaust magnetic February 1983 fields deep inside the star (but not near the star surface) and then con- centrate either near the centre (for neutron stars with non-superconducting interiors), or on the boundary of the central superconductive region, if it
* To be submitted for publication. exists. In the first case the considerable annihilation rate is simply due ** Permanent address. to the small size of the region occupied by monopoles, while in the second case the annihilation is enhanced by the presence of the superconductor.
-2- 2,,?- •=•-• O'rc first scenario applies to the neutron stars with normal (non- = (5) dt VM - superconducting) interiors. As has beer, argued in Ref.8, a neutron star is likely to contain no monopoles at the maiiK . ... "' s r—r.iMon. Just after this moment, it begins to capture the relic monopoles with the rate [7-10] where i\ is the monopole density. From (5) we find that the characteristic time of the exhaustion of the magnetic field is independent of H (cf. [17]) and is equal to t = a/Bngji , The restoration of the magnetic field du dii.it) fV -,2 A t A ltTrcl R dt esc (2) to the magnetic flux diffusion occurs with the characteristic time D "' l g Q> where <*„ ~ 1033 (lO6 VO^ s""~1 [l8] is the electric conductivity of the "2 -3 core matter and T is the neutron star temperature. Note that the observational where v^ '"-10 ± 10 is the monopole velocity in the Galaxy, FL ~ 10 Km 6 v 0 1 s limits [11] imply Tig ^ 10 °K. is the star radius and esc~ - * i the escape velocity at the star surface. Near the star surface, the magnetic force aX (g^ is the monopole magnetic The number of monopoles inside the star steadily increases, and when charge and H^. ~10 Q is the magnetic field on the surface) is much weaker t becomes of order of t the magnetic fields become weak and the radius exh D than the gravitational one, so the monopoles proceed to the star core. If of the region occupied by monopoles becomes smaller. Since, for a given there were no magnetic fields inside the core, the monopoles would fall down 3 2 to the star centre. However, a neutron star initially has rather large number of monopoles, the exhaustion time is proportional to R , while t -3- IC/B3/17 1, The observation [l,2j that grand unified magnetic monopoles [3] can Tnte^iiational Atomic tinergy Agency catalyze the baryon decay with a strong interaction rate has recently raised and a growing interest. This effect can be observed in deep underground [l,lt,5] United Nations Educational Scientific? and Cultural Organization and deep underwater [It] experiments, an evidence for its existence in the sun can be provided by solar neutrino detectors [1,6], so the cosmological and IHTERHATIQt.'AL CENTRE FOR THEORETICAL PHYSICS astrophysical limits on the monopole flux are Of particular importance. While the cosmological limit on the flux of monopoles catalyzing the baryon decay is rather weak [5], much stronger limits have been claimed to follow from the con- siderations of neutron stars [7-10]. The neutron "decays" catalyzed by mono- poles with the cross-section [1,2] a = 0 /v where a = 0(l GeV ) and v ON THE FATE OF SUPEHHEAVTf MAGNETIC MONOPOLES IH A NSUTRON STAH * is a relative neutron-monopole velocity, tend to heat a. neutron star. The observational limit on the X-ray luminosity of neutron stars [ll], T 31 -1 L . < 10 erg s , implies the following limit on the number of monopoles V.A. Kuamin and V.A. Rubakov in any neutron star, N , and on the cross-section parameter 0 [10]' International Centre for Theoretical Physics, Trieste, Italy, M 0 and Institute for nuclear Research of the Academy of Sciences of the USSR, Moscow, USSR.** NH < 10 (1) 0.1 fm If all monopoles whenever captured by the star were present inside the star ABSTRACT up to now, Ecj.(l) would lead to very strong constraints on the flux of relic monopoles [7-10] which would make their experimental observation almost impossible. We propose two possible scenarios of the behaviour of superheavy magnetic monopoles in a neutron star, in which the monopole-antimonopole tfe believe, however, that the question of the survival of magnetic annihilation rate is sufficiently large to prevent the enonnous heating monopoles in a neutron star is by no means clear. In fact, the monopole- of a neutron star due to the monopole Induced neutron decays. We find antiraonopole annihilation rate crucially depends on the particular properties that the galactic monopole flux of order 10~ cm "~ s ater can he of matter inside the star core, which are still unknown. The main purpose compatible with the observational limit on the X-ray luminosity of of the present paper is to propose two possible scenarios of the behaviour neutron stars. of superheavy (m ^,10 GeV) monopoles in the neutron star; in both of them the mcnopole-antimonopole annihilation is sufficiently effective to make M -l6 2 1 consistent with Eq.(l) even for the galactic monopole flux j(JO'*' 10~ cm a ster , which is close to experimentally observable values [12,1+] as well as to other astrophysical and cosmological limits [13-15], The general feature MIRAMARE - TRIESTE of both scenarios is that at the first stage monapoles exhaust, magnetic February 1983 fields deep inside the star (but not near the star surface) and then con- centrate either near the centre (for neutron stars with non-superconducting interiors), or on trie boundary of the central superconductive region, if it * To be submitted for publication. exists. In the first case the considerable annihilation rate is simply due ** Permanent address. to the small size of the region occupied by monopoles, while in the second case the annihilation is enhanced by the presence of the superconductor. -2- 1O33 (10 °K/Tj2 S"1 [18] is the electric conductivity of the ~2 -3 where v^ '•"10 ~ 10 is the monopole velocity in the Galaxy, JL ~ 10 Kin core matter and T is the neutron star temperature. Note that the observational is the star radius and v ~0,U is the escape velocity at the star surface. limits [11] imply Tg -S 10 °K. Hear the star surface, the magnetic force gH (g^ is the monopole magnetic The number of monopoles inside the star steadily increases, and when charge and Hg »vlO G is the magnetic field on the surface) is much weaker t •becomes of order of t , the magnetic fields become weak and the radius exh D than the gravitational one, so the monopoles proceed to the star core. If of the region occupied by monopoles becomes smaller. Since, for a given there were no magnetic fields inside the core, the monopoles would fall down 3 2 number of monopoles, the exhaustion time is proportional to E , while t <* R , to the star centre. However, a neutron star initially has rather large 12 both the processes of the exhaustion of the magnetic field and the monopole magnetic fields inside the core, HQ *W0 G [l6], which are expected to be Q t we find the collapse to the star centre are catastrophic. From t h ""* -D 3 random [16]. Therefore, the monopoles are initially distributed over the a RQ/2!+iTg^as. 10 rlt 3 critical number of monopoles33 ,1 ^ ^ ^^ region of radius RQl at which the magnetic force roughly equals the R ~ 100 m and •*„ ~ 1O s" .) U b gravitational one This argument is not, however, completely correct. Indeed, for H = uCrlt the exhaustion time much exceeds the age of the Universe. There- • (3) crit 0 fore, the exhaustion process is very slow when HM = MM ' Actually, the above consideration shows that the restoration of the magnetic field due to ,,16 -3 (R ^,100 m for OeV), where 0.2 fm is the neutron density the flux diffusion is negligible, once the number of monopoles exceeds N^ of the core. Note also that the magnetic field far outside the region occupied by the mono- poles is not affected by the processes inside this region because of the The crucial point of our first scenario is that the monapoles exhaust enormous time of the flux diffusion from the periphery of the star (cf. [l8]). the magnetic fields (cf. [ll4,17]). The energy losses of the (non-relativistic) Therefore there is no conflict between our scenario and the calculations (see, monopoles inside the star originate mainly from the electromagnetic inter- e.g. [19]) of the magnetic field near the star surface. actions with remaining electrons [8] and from the baryon-number violating interactions [°1; in both cases To estimate the actual time of the collapse of the monopoles, we note that the radius of the region occupied by them, R(t), is proportional to the magnetic field in this region H(t) (cf. (3)). So, we can rewrite Eq.(5) as dx follows: where v is the monopole velocity and a ~ 10 GeV/cm (although the second dRttj. (6) dt process can give somewhat larger contribution to a). Therefore, the monopole velocity is related to the magnetic field as follows, v = g^ H/a, This where H,(t) = Ft is the number of monopoles inside the star. Therefore, leads to the estimate of the dissipation of the energy density of the magnetic the monopole collapse ends up at t .. = field per unit time, -3- i. Our second scenario can be valid if interiors of neutron stars are At this time, the number of monopoles inside the star is superconductive. The monopole behaviour in the superconductive neutron stars 1/2 has been discussed by Harvey [10] under the assumption that the superconductivity U .. = F-t (T) arises due to the proton-proton pa:ring,and it has been found that the limits coll coll 9 s,. on the monopole flux [7-9] remained unaltered. However, the density of protons -i Note that for R -> 100 m and for F 10 s which is consistent with in a star core is much less than that of neutrons, while proton-neutron inter- -I -f 5 1 ~\ M - ""12, which is less than actions are as strong as protcn-proton ones, so it seems likely that a Cooper Joo **1 0 cm s ster , Eq.(9) gives coll the allowed value (1). pair would consist of one proton and one neutron, instead of two protons. The th1 e As discussed in Ref.10, monopoles in the neutron star are likely to question oi ^actual nature of the superconductivity under the neutron star he described by their own temperature T —' 100 MeV arising due to the conditions is interesting by itself; here we simply assume that protons are superconductive due to the formation of (pn) pairs (electrons in the neutron catalysis of neutron decays. Therefore, the collaped raonopoles spread over star interior are likely to be non-superconductive ([21]), and that the super- the region of radius R. = (3T M^j/VfinLn m ) at which their gravitational conductivity is of the second order (cf. [22]). energy is equal to I,, [10]. There they would eventually annihilate; however, the initial annihilation rate turns out to be small (see below). The phase transition from normal to superconducting state leads to the squeezing of the initial magnetic field H n/10 G into flux tubes of Now we turn to the monopoles captured liy the star after the collapse the size equal to the penetration length 1 r~> 10 cm [2£] each carrying the of the monopole cluster (i.e. later than t _„ after the star formation). magnetic flux corresponding to that of one Dirac charge monopole (in the case e coll of proton-proton pairing the tubes would carry two such units of flux. Due The magnetic field of the star tends to be restored due to the magnetic flux to the enormous conductivity of (normal) electrons, the characteristic time diffusion. However, as is clear from the above considerations, it is quickly of the expelling of the magnetic fields out of the superconducting region much exhausted by incoming monopoles and does not prevent them to fall down to exceeds the age of the Universe [22], so we can neglect this process. Except for the star centre. In fact, one can show that at any moment the number of the magnetic fields squeezed into the flux tubes, there are also initial (more monopoles present outside the star centre is much less than that allowed by or less random) magnetic fields near the boundary of the superconducting region, Ed.(l). which are directed along this boundary [10], The number of monopoles inside the central region of radius R^ is determined by their annihilation rate and the incoming flux F, According A magnetic monopole coming from the neighbouring space looses its to Refs.[20,10] the annihilation cross-section is kinetic energy in the non-superconductive outer region of the neutron star and approaches the boundary of the superconductor with the velocity determined by l6 n"35 10 'lOO , (8) the gravitational attraction and by the energy losses (k). Entering the super- conductor it forms its own flux,tube. If the monopole mass is less than IT s> •? This yields the equilibrium number of monopoles 2.10 GeV, the magnetic force turning it back to the boundary, l+irgT/TrX , exceeds the gravitational attraction to the centre, so the monopoles are con- 10 centrated on the boundary. In what follows we consider the case m <: 10 GeV. 100 M 10 s-1 (9) Monopoles follow magnetic lines near the boundary and eventually drop down to a flux tube. Since the flux tube carries the magnetic flux : where v = »/3TM/2mM' is the average monopole velocity. From Eqs.(2) and (9) equal to that of a monopole, this leads to th>> disappearance of the tube M • ^ 16 -P -1 -1 we realize that the galactic monopole flux of order j^^-JO cm s ster (see Pig.l), [In this respect the superconductor with (pn)-pairing drastically is allowed within our first scenario provided v f*- 10 '" (so that F^3 s ). differs from that with (pp)-pairing.] The oagnetic field inside the Luce p 1 c H = 6g;.j/^ "-"3,10 Gj accelerates the monopole up to the velocity v , . = H g /a~0.3, so the monopole destroys the whole tube in 1 1 For F = 10 a" , TH = 100 MeV and = 10 GeV we find = lO""" *, which 1 -k B is still consistent with (1). t -- Ra / v, w . % ~ 10 - I* is straightforward to estimate the time of d 13 Mt t J exhaustion of the magnetic fields inside the superconducting region. The The lifetime of a confined pair shown in Fig.2d is very short. In- 9 2 31 initial number of the flux tubes is roughly M. ~ I+TTIOL,/2TTA I 10 while t b b t deed, one can show that the volume of the bubble is typically less than 9 the rate of the destruction of the flux tubes is dN /dt = - NM(t)/td, where 10 fur; using Eq.(lO) we obtain the lifetime of order 10 e. Therefore, as before, KM(t) = Ft. Therefore, all flux tubes disappear at only N = FT ~»10 pairs are present in the superconductive region at each t. = (N, t , at this time the number of monopoles in the neutron star moment, so they produce no effect on the star. 1 /? 1 lit is NM(t0) (F Mttd) For F = 10 s" we have ^tQ) ~ 10 , which is consistent with Eq.(l). It is worth noting that the destruction of the flux k. We have discussed two possible scenarios of the neutron star tubes inside the superconducting region has no effect on the magnetic fields evolution in the presence of a considerable galactic monopole flux, which near the star surface, since the superconductive interior is surrounded by make the flux jm -J 10 cm s ster and the neutron "decay" parameter the thick thrust of the non-superconductive matter [19] (presumably with 2 0 ~ 0.1 fm consistent with the observational limit on the X-ray luminosity superfluid neutrons [23,19]), in which the electric conductivity is sufficiently of the neutron stars. None of these scenarios gets in conflict with the high to prevent the magnetic field diffusion into the inner region (see observed or theoretically well established properties of the neutron stars, above). such as their density, radius or surface magnetic field. We concludetnat tlle Once the magnetic fields inside (and thus near the boundary of) the reliable derivation of the limits on the monopole flux and/or "decay" superconductive region are exhausted, monopoles spend all time at the parameter requires better knowledge of particular features of the neutron star boundary and eventually annihilate. The annihilation is enhanced by the interiors, and that the observation of the superheavy monopoles catalyzing following mechanism. Once the distance between a monopole and an anti- the baryon decay is much less unlikely than it has been thought previously. monopole becomes less than X , the magnetic force preventing them to fall On the other hand, in both scenarios the number of monopoles inside a star down to the star centre disappears, and the pair penetrates into the super- turns out to be close to the allowed value (l), so that the discovery of the conductor (Figs.2a-e). There the monopole and antimonopole are confined X-ray temperature of the neutron stars (including old ones!) at the level inside a small bubble (Fig,2d) and then annihilate. The whole process is close to the present limit would provide some evidence of the existence of possible only if the monopole-antimonopoie relative velocity v is small the monopoles inside them. Moreover, the dependence of the temperature of enough, namely, if the time t = A/v during which the distance between them a neutron star on its age and other parameters (radius, mass, etc.) is is less than A is sufficient for the gravitational force to cause their predictable within both scenarios, so it might be possible (at least in penetration into the superconductor at the depth larger than X principle) to identify the heat source of the neutron stars with the neutron 3M2,) (t /2) > \ This determines the critical velocity, decays catalyzed by the grand unified monopoles. The velocity distribution of the monopoles on the boundary follows the two- dimensional Boltzmann law with the temperature T ~ 100 MeV (see above), so the surface density of the monopoles with the velocity below v is ACKNOWLEDGMENTS "n(vM < v ) = ^^.A /T )), wherwheree rf is a total surface density (for ^ -^10 GeV, mu^n << T>J- Since the "cross section" of the process of Fig. 2 is of order We are indebted to F.A. Bais, N.S, Craigie and C. Peterson for A (remeber that monopoles move along the two-dimensional surface) provided helpful discussions and to A. Mann and M.E. Shaposhnikov for discussions that v < v , the rate of the formation of the confined pairs is at the initial stage of this work. We ape grateful to W. Nahm who pointed dN /dt = H., "n Av, (m,,v~/T,,) , where H,, ~ ^irR^n is the total number of out Ref.10 to us. We are deeply indebted to Professor Abdus Salam, the p M u M U |vl ^ ^ International Atomic Energy Agency and UNESCO for hospitality at the monopoles on the boundary of the superconductor. We conclude that the International Centre for Theoretical Physics, Trieste. equilibrium number is (10) -f- \ KiffEBEHCEG [16] M.A. Ruderman and P.G. Sutherland, nature Phys. Sci. 2^6 (1973) 93. [ll] DM., Eitson,, Stand ford preprint SLAC-PUB-2977 (1982). [i] V.A. Rubakov, Pis'iaa ZhETF (19S"•.. • '=iS. [18] . G. Baym, C. Pethick and D. Pines, Nature 22^ (1969) 6lh. [2] V.A. Rubakov, Inst. Nucl. Res. preprint P-0211, Moscow (1981); [19] R.M. Manchester and J.H. Taylor, Pulsars (W.H. Freeman and Co., Nuel. Pnys. B203. (l9fe) 311; S. Francisco 1977). C.G. Callan, Phys. Rev. D2£ (1982) 211*1; D26_ (1982) 2050; V.A. Rubakov and M.S. Serebryakov, Proc. Seminar "Quarks-82", Sukhumi, [20] D.A. Dicus, D.H. Page and V.L. Teplitz, Phys. Rev. D26. (1982) 1306. USSR, May (1982) and to appear in Nuc.l. Phys. B. [21] V.L. Ginzburg, Usp. Fiz. Mauk 9J, (1969) 601. [3] G. 't Hooft, Mucl. Phys. BJ_9_ (1971*) 276; [22] G. Baym, C. Pethick and D. Pines, Nature 221*. (1969) 673. A.M. Polyakov, Pis'ma ZhETF 20 (197I1) U30. [23] V.L. Ginzburg and D.A. Kirzhnitz, Mature P2£ (1969) Itt8. V.A. Rubakov, B.E. Stern and l.M. Zheleznykh, Proe. Seminar "Quarks-82", Sukhumi, USSR, Hay (1982). [5] J. Ellis, D.V. Nanopoulos and K.A. Olive, Phys. Lett. ll6B (1982) 127. [6] S. Dimopoulos, S.L, Glashow, E.M, Purcell and F.A. Wilezek, Mature 298 C1982) 824. [7] E.W. Kolb, S.A. Colgate and J.A. Harvey, Phys. Rev. Lett. k9_ (1982) 13T3. [8] S. Dimopoulos, J. Preskill and F. Wilczek, Harvard preprint HUTP-82/AOltT (19S2). [9] F.A. Bais, J. Ellis, D.V. Nanopoulos and K.A. Olive, CERH preprint Bef. TH.3382 (1982). [10] J.A. Harvey, "Monopoles in neutron stars", Princeton preprint (1983). [ll] F.S. Cordova, K.O. Mason and J.E. Kelson, Astrophys. J. glj^ (1981) 609; G.A. Reichert, K.P. Mason, J.R. Thorstensen and S. Bowler, quoted in Ret.6. [12] E.N. Alexeev, M.M. Boliev, A.E. Chudatov, B.A. Makoev, S.P. Mikheev and Yu.M. Sten'kin, Lett. Nuovo Cimento 35_ (1982) Ul3; G. Giacomelli, Talk given at the Intern. Colloquium on Matter Non- Conservation (ICOMAN-83), Frascati , jEinuary 1983. [13] G.V. Domogatsky and l.M. Zheleznykh, Yad. Fiz. 10_ (1969) 1238, [lh] A.E. Chudakov, quoted in Ref.13; E.B. Parker, Astrophys. J. 163 (1971) 225. [15] Ya. B. Zel'dovich and M.Yu. Khlopov, Phys. Lett. 79B. (1978) 239; J. Preskill, Phys. Rev. Lett. 1*3. (1979) 1365; G. Lazarides, Q. Shafi and T,F. Walsh, Phys. Lett. 100E (1981) 20; M.S. Turner, E.M. Parker and T.J. Bogdan, Phys. Rev. D2§_ (1982) 1269- -10- -9- CURHfcHT ICTP PUBLICATIONS AI4D IKTEPJIAL REPORTS IC/82/150 I.M. RKDA and H. BANUERT - Glass forming ability and kinetics of INT.REP.* formation of metplasses. IC/82/151 I.M. BEDA and A. WAGENDRI3TEL - Atomic transport in highly INT,REP.* disordered solids. IC/H2/152 M.Y, iiL-ASIIRY - On the kinetic theory of parametric resonance in relativistic plasma. 1C/32/153 LI XIMZHOU, WANG KELIN and SHAKO JIAHSU - The relation mR << 1 INT.REP.* in the preon model. S.M. PIETRUSZKO - /^jrp:;ous silicon prepared from silane-hydrogen INT.HEP.* mixture. IC/82/155 J. ZIELIKSKI - Semi-phenomenoiogical approach to the problem of IMT.BEP.* discontinuous change of valence. IC/82/156 H.B. GHASSIB and S. CHATTERJEE - On density fluctuations in dilute (a) He- He thin films. IC/82/15T G.B. CELMIHI, S. NUSEIMOV and T. YANAGIDA - Does nature like Hajnbu- Fig.l Goldstone bosons? IC/82/158 A.-6.F. OBADA and M.H. MAHRAN - Electric and magnetic dipole contributions INT.REP.* to a theory of radiation reaction field and atom self-energy: an operator reaction field, IC/82/159 R, KIMARAVADIVEL - Thermodynamic properties of liquid metals. INT.SEP.* IC/82/160 K.W. WOJCIECHOWSKI, P. PIERAtJEKI and J. MASECKI - An Instability in a hard-disc system in a narro« box. IC/82/161 J, CHELA FLORES - Abrupt onset of scaling violation. IC/S2/162 J. CHELA FLORES and V. VAHELA - Strong gravity: an approach to its 12 IC/82/163 S.A.H. ABOU STEIT - Additional evidence for the existence of the C INT.REP.* nucleus as three a-partlcles and not as Be + a. ic/82/i6it M. ROVERE, H.P. TOSI and H.H, MAHCH - Freezing theory of HbCl. (a) INT.REP.* IC/82/165 C.A. MAJID - Radial distribution analysis of amorphous carbon. INT.REP.• IC/82/166 P. BALLOBE, G. SENATORS and M.P. TOSI - Surface density profile and surface tension of the one-component classical plasma. IC/82/167 G. KAMIEHIARZ - Zero-temperature renormalization of the 2D transverse Ising model. IC/82/168 G. KAMIENIABZ, L. KOHALEWSKI ans W, PIECHOCKI - The spin S quantum INT.HEP.* Ising model at T ^ 0. ic/82/169 J, GORECKI and J. POPIELAWSKI - On the applicability of nearly free electron model for resistivity calculations in liquid metals. IC/82/1T0 K.F, WOJCIECHOWSKI - Trial charge density profiles at the metallic surface and uork function calculations. THESE PREPRINTS ARE AVAILABLE FROM THE PUBLICATIONS OFFICE, ICTP, P.O. Box 586, 1-31*100 TRIESTE, ITALY. * (Limited distribution). • -11- IC/82/171 H.B. GRASSIB and A.M. KHUDEIR - Tow irda a 'jr-nprehensive theory IC/82/197 FREM P. SRIVASTAVA - Gauge and non-gauge curvature tensor copies. INT,REP,* for He II: I - A zero-temperature hybrid -.pr-orioh. IC/82/198 R.P. HAZOUME - Orientational ordering in cholesterics and smectics, IC/82/172 J. CHELA FLORES and H.B. GHASSIB - Towards a esmpr-'^isive theor; IHT.REP.* IHT.REP.* for He II: - A temperature-depei.dcTu field-t'aeore-cic approach. IC/82/199 S. RAHDJBAR-DAEMI - On the uniqueness of the SU(2} instanton. IC/62/173 C.A. MAJID - Electrical, optical and o-s:;;- t j format ion behaviour : IC/82/200 M. TOMAK - On the photo-ionization of impurity centres in semi- IHT.REP.* of amorphous solids - An : r;tro lu.-tion •„<.. :'.,,: -^r'nls . conductors, IC/82/17lt H. PAK, C. PAlsAGIOTAKOPi'1!:; '/J ond •..;. 3HA:'1! - ''.;.;::-*:ic monopcl^ and baryons decay. -IC/82/201 0. BALCTOGLU - Interaction between ionization and gravity waves in the IHT.REP.* upper atmosphere. IC/82/175 M. OZER - Fermion mass ratios ana ;nw-er:ett;y .jr-ani: unfi'J-ition. IC/82/202 S.A. BARAII - Perturbation of an exact strong gravity solution. IC/82/176 S. CECOTTI and L. GIHARBELLG - Motes oi, J;. processes in IKT.RaP.* lattice (extended) supersymmeLries. IC/82/203 IC/82/177 D. RAJPOOT - 1-OV mass-scale parity ;-'_.;torat ion iv. expanded gau^e INT.REP,* H.II. BHATTARAI - Join decomposition of certain spaces. theories. M. CARMELI - Extension of the principle of minimal coupling to particles IC/82/20U IC/82/178 K. SAEED - Identification of r-^eosary and sufficient conditions J:- with magnetic moments. IHT.REP.* real non-negativeness