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A limit on the solar monopole abundance
Volume 133B, number 6
PHYSICS LETTERS
29 December 1983
A LIMIT ON THE SOLAR MONOPOLE ABUNDANCE J. ARAFUNE
Institute for Cosmic Ray Research, University of Tokyo, Tanashi, Tokyo 188, Japan and M. FUKUGITA
Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan Received 2 August 1983 Revised manuscript received 28 September 1983
We discuss the solar neutrino flux which would arise from monopole catalysed proton decay processes. In particular we found that the solar neutrino experiment with 37C1leads to an upper limit on the flux of ve with E v ~ 35 MeV from p t~÷ + anything, followed by ta+ ~ Veb#e+, givingthe strong constraint n M < (1 monopole/4 × 101~ gr) • (ao/10 -27 cm2) -1, with oR = ¢r0/3-2 the cross section of monopole catalysis in hydrogen.
The existence of magnetic monopoles [1 ] ,1 is one of the most important predictions of grand unified theories. While the theoretical estimation of the local or cosmic abundance of the monopole is difficult at present, experimental efforts have been made to detect them not only in cosmic rays [3] but also in some materials [4] where one may expect monopoles being trapped. Recently Rubakov [5,6] and Callan [7] proposed the interesting process that proton decay is catalysed by a monopole passing nearby with a large cross section. This process would provide us with a powerful method to detect the monopole or at least to set an upper bound on the monopole abundance. A stringent b o u n d on the monopole flux was given from the X-ray excess limit for neutron stars [8], which is several orders of magnitude stronger ,2 than that obtained by Parker from the survival of galactic magnetic fields [10]. There is also an attempt to set an upper limit on the local monopole flux from the subterranian heat flow, giving also a strong limit on the local flux [11 ]. *1 -F-ora review see ref. [2]. ,2 If the interior of a neutron star is a normal conductor, this bound could be loosened when one takes account of the annihilationof a mono'pole and an antimonopole [9 ]. 380
We shall point out that this will almost disappear when we consider carefully the behaviour of monopoles in matter. In this paper we consider the Rubakov effect for monopoles in the sun. We shall show that we can obtain a strong upper bound on the solar abundance of monopoles from the limit on the neutrino flux from the sun. We pay particular attention to the peculiar velocity dependence of the cross section of the Rubakov process in matter, recently remarked by the present authors [12]. For convenience let us briefly repeat here the main results of ref. [12]. The system of a monopole with a minimum magnetic charge and a nucleus with charge Ze and spin s receives a centrifugal potential L2/(2A • mNr2), w i t h L 2 ~>Z[½ - (1 +k)s] for nuclei w i t h Z ~> 2s. Here A • m N is the mass of the nucleus, and Ze(1 + k)s/(A • raN) is the magnetic moment of the nucleus. This potential leads to a velocity-dependent factor (/3//30)2v for the Rubakov effect. The index takes v = - 51 or -½ + (L 2 + ~)1/2 according as (L 2 + ~-) is negative or positive, and/30 = 1/(A4/3mNro) with r 0 = 1.2 fm. This gives a suppression factor for even-even (therefore spinless) nuclei and an enhancement factor for protons. The cross section should have another 1/13 0.031-9163/83/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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factor because the Rubakov process is an exothermic reaction. As a result the Rubakov process cross section behaves as o R ~ 1//32 for a proton in agreement with the recent suggestion by Marciano and Muzinich [13] and by Rubakov and Serebryakov [6]. For e v e n - e v e n nuclei like 4He, 160. or 56Fe the cross section behaves as o R ~/3 -2+(2Z+1)i/2 . This velocity-dependent factor, therefore, causes a strong suppression of the Rubakov process for slowly-moving magnetic monopoles in ordinary matter. Another suppression factor also arises from a strong repulsive force against a slowly-moving monopole due to the Zeeman effect and diamagnetic effect o f the atomic electrons. This repulsive potential is estimated to be about 16 eV even for helium [14] and a monopole with velocity/3 ~ 10 - 4 can hardly touch the helium nucleus. For heavier atoms such a repulsive force is quite strong [15]. In the presence o f these repulsive forces the Rubakov process is greatly suppressed in the earth, and it is most unlikely to obtain any strong limit on the abundance o f monopoles from the subterranian heat flow. An exceptional case is a monopole trapped in the sun, which mostly consists of hydrogen. We expect that the thermal motion o f hydrogen will cause an overlap of a proton with a monopole and therefore the Rubakov process to occur, because we do not expect any suppression for the m o n o p o l e - h y d r o g e n system. In this paper we are only interested in the present abundance of monopoles in the sun. We do not ask whether monopoles are trapped by the time of the formation o f the sun or trapped during 4.5 × 109 yr after the formation. Since we can not separate the latter abundance, we do not discuss a limit on the monopole flux. The frequency o f monopole catalysed proton decay in the sun will be given b y
f= f NMORVrelPHNAd3X,
(1)
where N M is the monopole number density, a R the cross section o f the Rubakov process for the m o n o p o l e proton system, Ore1 the relative velocity, PH the hydrogen weight density, and N A is the Avogadro number, ~ 6 × 1023. The space integration is to be done over the interior o f the sun. The contribution from helium is negligible because o f the small cross section due to the suppression factor discussed above. We take Ore1 as the thermal velocity of ionized hydrogen,
29 December 1983
Ore1 = c~3 = (2T/mN) I/2, with T t h e temperature. We note that a monopole once trapped by the sun will be thermalised within a short period, t ~ 103(C/VM)s, with vM the initial velocity of the monopole, as is easily estimated from the energy loss o f a monopole [ 16], - d E / d x ~ 30" (2E/mmonopole)l/2 GeV/gr/cm 2. Putting OR = 00//32 ,
(2)
we obtain
f = fNMNACaO (PH/fl) d3x
•
(3)
While we do not know the distribution NM(X ) in the sun, we see, from the knowledge o f the properties of the solar interior [17], that PHI~3 depends only weakly on the distance from the centre o f the sun: PH//3 = 3.3 × 104 gr/cm 3
at the centre r = 0 ,
pH//3 = 1.3 X 104 gr/cm 3
atMr/Me<~0.5, orr/r® <<,0.3. (4)
(Here M o and r o are the mass and the radius o f the sun, respectively.) Therefore the integral f i s not sensitive to the distribution o f the monopole in the sun. Taking PH//3~> 1.3 X 104 gr/cm 3, and integrating over Mr/M ®<~0.5, we obtain f~> 2.3 X 1011 × (o0/10 _27 cm 2)
X
fNmd3X
decays/s.
(5)
Because of the symmetry breaking pattern o f the SU(5) theory [ 18], a direct decay o f a proton to an antineutrino, p ~ n + + if, is forbidden in monopole catalysis [19]. There are yet various indirect processes which would cause a neutrino and/or antineutrino flux: Proton decay will result in an electron neutrino flux, coming mainly from the beta decay of #+ or K 0, /a+ ~ v e + e + + ~ ,
or
K0~v e+e ++Tr-.
(6)
The #+ is produced either directly in proton decay or indirectly through the decay ,3 o f n +. It is estimated ~-3 If monopoles are concentrated near the centre of the sun, the nuclear absorption (mainly due to helium) reduces the ve flux by a factor 1.9. In this case, however, p/f3 [see eq. (4)] is larger by a factor of 2.5, which compensates the reduction due to the nuclear absorption effect, and we will recover our bound eq. (10). If monopoles are away from the centre, say Mr/M~ ~ 0.5, the reduction of ve due to ~r+ absorption is only 15%. 381
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that most o f the/~+ decays after it is stopped, and therefore the mean energy of such a neutrino is around 35 MeV. This neutrino would cause an increase in the neutrino capture rate in the solar neutrino experiment. In a typical SU(5) model we expect the probability BTr+ to have such rr+ is ~ 0 . 5 [20] ,4. (The branching ratio B,r+ of catalysed proton decay, however, could be different from that of free proton decay. Therefore we show explicitly the dependence of our result on the branching ratio.) We estimate the capture cross section of the process ve + 37C1 + e + 3 7 A ,
(7)
for a neutrino with E ~ 35 MeV by taking the lowest four levels of 37A as the final states. The capture cross section for each final state I' is [22] ¢7i,= 36.8 X s o (frl/2 )w2G(Z, We),
29 December 1983
n M ~< (1 monopole/4 X 1012 gr)- (10 -27 cm2/o0)
× (0.5IBm.+).
(10)
Detection of such a neutrino with E v ~ 35 MeV may be possible using underground facilities for the proton decay experiment; a search for the excess o f this neutrino will be important, since it would be the only way to see the effect of monopoles in the sun when the abundance is so small as in eq. (10). If we assume SU(5) grand unified theory, proton decay is also accompanied by an energetic neutrino or antineutrino with E ~> 200 MeV through the twostep process p + monopole -->/J+ + K ° + m o n o p o l e ,
I
(8)
--+ KL -+ Ve + e + zr.
(11)
where
We estimate the probability of this process as
s O = 1.91 X 10 -46 cm 2 ,
Bhig h -_ x B r ( M + p + M + tl + + K 0)" h " Br(K L + ve err)" r
(f71/2){-I = ((l)~, + (gA/gv) 2 (O)2 , ) / 2 ,
>~2 × 10 - 6 × [Br(M + p - + M +/x + + K0)/0.05] , (12) where h represents the hindrance factor o f the decay in flight of K L through the absorption or regeneration of K 0 in hydrogen in the sun [h "~ 0.5% at the centre of the sun (O ~ 160 gr/cm3), h cc O-1 ], and r is the fraction of high energy neutrinos with E >~ 200 MeV (r ~ 4.5%). (We note that in the absence o f the monopole Br(p -+/I+K 0) ~ 5% in a typical calculation o f the SU(5) model [20, 21].) The flux o f such a neutrino or antineutrino at the e a r t h Ihig h iS then/high = Bhighf/47rR2. These neutrinos should be detected as background events in the underground experiments. It will be quite safe to say that the excess event rate of such a neutrino interaction is less than 5 events/ (100 t o n . yr) [24]. Taking the cross section for the v - N charged current interaction as 0.5 X 10-38(E/ GeV) cm 2, we find
and
G(Z, We) = cPeF(Z, We)/ZTraZI¢e , with F the Coulomb correction factor, and w e = We/ me e2. Here We is the total energy o f the final electron. The nuclear matrix elements ( l ~I, and (o) 2, are given in ref. [22]. The dominant contribution comes from the fourth level, JP =23 + a n d T = ~ - a t E ~ 5 M e V , i t s matrix elements being (/)2 = 3 and (o) 2 = ~-. 1 Taking also account o f the other three low lying levels, we obtain the total capture cross section Oca p = 4.2 X 105 X s O = 8.1 X 10 -41 cm 2 .
(9)
This is regarded as a lower bound on the capture cross section, since only the lowest four levels of 37 A are taken into account. The capture rate in the solar neutrino experiment is given by IvOcap, where I v = B l o w f / 47rR 2 is the flux o f a neutrino from ~+ at the earth with R the distance between the earth and the sun. We take the upper bound for an excess neutrino cap, ture rate to be less than 1 SNU [23]. Then we are led to an upper bound on the solar monopole abundance as , 4 For a review see ref. [21 ].
382
f<4X
1033/s.
(13)
Eqs. (5) and (8) thus lead to an upper bound for the total number of monopoles in the sun (we concentrate on monopoles in the relatively inner region o f the sun, say M r <~0.5 M®),
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fNMd3X ~< 1.7 X 1022 • (10 -27 cm2/ao) X [0.05/Br(M + p - + M + # + + K0)] ,
Mr
(14)
or dividing by the mass 0.5 M®, we obtain the bound on the density o f the monopole: n M ~< (1 monopole/0.6 X 1011 gr)" (10 -27 cm2/a0) X [0.05/Br(M + p -+ M + # + + K0)] .
(15)
The underground proton decay experiments [25,26] currently being carried out will further improve this bound. The above considerations depend on the model for the decay modes o f the proton. Even if proton decay is not accompanied b y either a neutrino or an antineutrino in some particular models, we can still obtain some constraints on the monopole abundance from the observed solar neutrino flux: proton decay due to monopoles also affects the heat flow o f the sun. I f 1% o f the solar luminosity is to be attributed to monopole catalysed nucleon decay, the energy generation by the ordinary p - p chain should need a contribution 1% less, or a temperature near the centre 0.25% less. The neutrino production from B 8, which is proportional to T 22-5 , would then be reduced due to this temperature decrease by 6%, or 7% using the result b y Bahcall et al. [27], ~ In ¢(B8)/O In L , = 7.2. In this respect we expect that the solar luminosity should not be attributed to the Rubakov process b y more than 10%, namely, mNc2f<.-..0.1 X I o in order not to disturb the theoretical calculation [27] o f the thermonuclear neutrino flux b y more than a factor 2. This gives also an upper bound on the solar abundance of the monopole n M ~< (1 monopole/109gr) • (10 -27 c m 2 / o 0 ) .
(16)
This abundance will disturb the solar magnetic field b y more than 10 - 6 G on the surface of the sun. Although eq. (16) is less stringent than eq. (10) and (15), it is still significant compared with the bounds obtained so far: (i) n M < 1/(2 X 107 gr) for the sun considering the decay of the solar magnetic fields [28], (ii) n M < 1/(5 X 109 gr) for the earth
29 December 1983
considering the terrestrial magnetic field [28], (iii) n M < 1.7 × 10-4/gr for the moon obtained from a monopole search in lunar material [29], (iv) n M < 5 × 10-5/gr [30] for the earth considering monop o l e - a n t i m o n o p o l e annihilation. It seems unlikely that the abundance of monopoles in the planets such as the earth and Jupiter or in meteorites exceeds much o f that in the sun. Therefore we may regard our limit on the monopole abundance as that for the entire solar system. It is interesting to note that the Rubakov process together with the monopole abundance just allowed b y the bound eq. (10), might give rise to the excess heat of the jovian planets , s , which has long been a puzzle [32]. *s For a review see ref. [31 ].
References [1 ] G. 't Hooft, Nucl. Phys. B79 (1974) 276; A.M. Polyakov, Zh. Eksp. Teor. Fiz. Pis'ma 20 (1974) 430 [JETP Lett. 20 (1974) 194]. [2] D. Olive,Phys. Rep. 49 (1979) 165. [3] B. Cabrera, Phys. Rev. Lett. 48 (1982) 1378; T. Mashimo et al., Phys. Lett. 128B (1983) 327; E.N. Alexeyev et al., Lett. Nuovo Cimento 35 (1982) 413; J. Bartelt et al., Phys. Rev. Lett. 50 (1983) 655; F. Kajino et al., University of Tokyo ICRR preprint (1983), and references therein. [4] E. Goto, J. Phys. Soc. Japan 13 (1958) 1413; B. Cabrera and W.P. Trouwer, Found. Phys. 13 (1983) 195, and references therein. [5 ] V.A. Rubakov, Institute of Nuclear Research (Moscow) preprint P-0211 (1981); Zli. Eksp. Teor. Fiz. Pis'ma 33 (1981) 658 [JETP Lett. 33 (1981) 644]; Nucl. Phys. B203 (1982) 311. [6] V.A. Rubakov and M.S. Serebryakov, Nucl. Phys. B218 (1983) 240. [7] C.G. CaUan Jr., Phys. Rev. D25 (1982) 2141; D26 (1982) 2058. [8] E.W. Kolb, S.A. Colgate and J.A. Harvey, Phys. Rev. Lett. 49 (1982) 1373; S. Dimopoulos, J. Preskill and F. Wilczek, Harvard University preprint HUTP-82/A047. [9] V.A. Kuzmin and V.A. Rubakov, Phys. Lett. 125B (1983) 372. [10] E.N. Parker, Astrophys. J. 160 (1970) 383; M.S. Turner, E.N. Parker, and T.J. Bogdan, Phys. Rev. D26 (1982) 1296. [11] M.S. Turner, Nature 302 (1983) 804. [12] J. Arafune and M. Fukugita, Phys. Rev. Lett. 50 (1983) 1901. 383
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[13] W.J. Marciano and I.J. Muzinich, Phys. Rev. Lett. 50 (1983) 1035. [14] S.D. Drell, N.M. KroU, M.T. Mueller, S.I. Parke and M.A. Ruderman, Phys. Rev. Lett. 50 (1983) 644. [15] W.V.R. Malkus, Phys. Rev. 83 (1951) 899. [16] S.P. Ahlen and K. Kinoshita, Phys. Rev. D26 (1982) 2347. [ 17] D.D. Clayton, Principles of stellar evolution and nucleosynthesis (McGraw-Hill, New York, 1968). [18] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438; H. Georgi, H.R. Quinn and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451. [19] J. Ellis, D.V. Nanopoulos and K.A. Olive, Phys. Lett. 116B (1982) 127. [20] G. Kane and G. Karl, Phys. Rev. D22 (1980) 2808. [21 ] P. Langacker, Phys. Rep. 72 (1981) 185. [22] J.N. Bahcali, Phys. Rev. 135 (1964) B137. [23] R. Davis Jr. et al., Phys. Rev. Lett. 20 (1968) 1205; J.K. Rowley et al., BNL report 27190 (1980). [24] M.R. Krishnaswamy et al., Phys. Lett. 115B (1982) 349, and references therein; PRAMANA (Indian Academy of Science) 19 (1982) 525.
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[25 ] J. Vandervelde et al., Proc. Neutrino '81 (Maui, Hawaii, 1981), Vol. 1, p. 205; R.M. Bionta et al., Phys. Rev. Lett. 51 (1983) 27. [26] T. Suda et al., Proc. Neutrino '81 (Maui, Hawaii, 1981), Vol. 1, p. 224. [27] J.B. Bahcall, N.A. Bahcall and R.K. Ulrich, Astrophys. J. 156 (1969) 559; see also J.N. Bahcall et al., Rev. Mod. Phys. 54 (1982) 767. [28] S. Dimopoulos, S.L. Glashow, E.M. Purcell and F. Wilczek, Nature 298 (1982) 824. [29] R.R. Ross, P.H. Eberhard, L.W. Alvarez and R.D. Watt, Phys. Rev. D8 (1973)698, and references therein. [30] R.A. Carrigan, Nature 288 (1980) 348. [31] D.J. Stevenson, Ann. Rev. Earth Planet. Sci. 10 (1982) 257; see also W.B. Hubbard, Rev. Geophys. Space Phys. 18 (1980) 1. [32] J. Arafune, M. Fukugita and S. Yanagita, Kyoto University preprint (1983).
PHYSICS LETTERS
29 December 1983
A LIMIT ON THE SOLAR MONOPOLE ABUNDANCE J. ARAFUNE
Institute for Cosmic Ray Research, University of Tokyo, Tanashi, Tokyo 188, Japan and M. FUKUGITA
Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan Received 2 August 1983 Revised manuscript received 28 September 1983
We discuss the solar neutrino flux which would arise from monopole catalysed proton decay processes. In particular we found that the solar neutrino experiment with 37C1leads to an upper limit on the flux of ve with E v ~ 35 MeV from p t~÷ + anything, followed by ta+ ~ Veb#e+, givingthe strong constraint n M < (1 monopole/4 × 101~ gr) • (ao/10 -27 cm2) -1, with oR = ¢r0/3-2 the cross section of monopole catalysis in hydrogen.
The existence of magnetic monopoles [1 ] ,1 is one of the most important predictions of grand unified theories. While the theoretical estimation of the local or cosmic abundance of the monopole is difficult at present, experimental efforts have been made to detect them not only in cosmic rays [3] but also in some materials [4] where one may expect monopoles being trapped. Recently Rubakov [5,6] and Callan [7] proposed the interesting process that proton decay is catalysed by a monopole passing nearby with a large cross section. This process would provide us with a powerful method to detect the monopole or at least to set an upper bound on the monopole abundance. A stringent b o u n d on the monopole flux was given from the X-ray excess limit for neutron stars [8], which is several orders of magnitude stronger ,2 than that obtained by Parker from the survival of galactic magnetic fields [10]. There is also an attempt to set an upper limit on the local monopole flux from the subterranian heat flow, giving also a strong limit on the local flux [11 ]. *1 -F-ora review see ref. [2]. ,2 If the interior of a neutron star is a normal conductor, this bound could be loosened when one takes account of the annihilationof a mono'pole and an antimonopole [9 ]. 380
We shall point out that this will almost disappear when we consider carefully the behaviour of monopoles in matter. In this paper we consider the Rubakov effect for monopoles in the sun. We shall show that we can obtain a strong upper bound on the solar abundance of monopoles from the limit on the neutrino flux from the sun. We pay particular attention to the peculiar velocity dependence of the cross section of the Rubakov process in matter, recently remarked by the present authors [12]. For convenience let us briefly repeat here the main results of ref. [12]. The system of a monopole with a minimum magnetic charge and a nucleus with charge Ze and spin s receives a centrifugal potential L2/(2A • mNr2), w i t h L 2 ~>Z[½ - (1 +k)s] for nuclei w i t h Z ~> 2s. Here A • m N is the mass of the nucleus, and Ze(1 + k)s/(A • raN) is the magnetic moment of the nucleus. This potential leads to a velocity-dependent factor (/3//30)2v for the Rubakov effect. The index takes v = - 51 or -½ + (L 2 + ~)1/2 according as (L 2 + ~-) is negative or positive, and/30 = 1/(A4/3mNro) with r 0 = 1.2 fm. This gives a suppression factor for even-even (therefore spinless) nuclei and an enhancement factor for protons. The cross section should have another 1/13 0.031-9163/83/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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factor because the Rubakov process is an exothermic reaction. As a result the Rubakov process cross section behaves as o R ~ 1//32 for a proton in agreement with the recent suggestion by Marciano and Muzinich [13] and by Rubakov and Serebryakov [6]. For e v e n - e v e n nuclei like 4He, 160. or 56Fe the cross section behaves as o R ~/3 -2+(2Z+1)i/2 . This velocity-dependent factor, therefore, causes a strong suppression of the Rubakov process for slowly-moving magnetic monopoles in ordinary matter. Another suppression factor also arises from a strong repulsive force against a slowly-moving monopole due to the Zeeman effect and diamagnetic effect o f the atomic electrons. This repulsive potential is estimated to be about 16 eV even for helium [14] and a monopole with velocity/3 ~ 10 - 4 can hardly touch the helium nucleus. For heavier atoms such a repulsive force is quite strong [15]. In the presence o f these repulsive forces the Rubakov process is greatly suppressed in the earth, and it is most unlikely to obtain any strong limit on the abundance o f monopoles from the subterranian heat flow. An exceptional case is a monopole trapped in the sun, which mostly consists of hydrogen. We expect that the thermal motion o f hydrogen will cause an overlap of a proton with a monopole and therefore the Rubakov process to occur, because we do not expect any suppression for the m o n o p o l e - h y d r o g e n system. In this paper we are only interested in the present abundance of monopoles in the sun. We do not ask whether monopoles are trapped by the time of the formation o f the sun or trapped during 4.5 × 109 yr after the formation. Since we can not separate the latter abundance, we do not discuss a limit on the monopole flux. The frequency o f monopole catalysed proton decay in the sun will be given b y
f= f NMORVrelPHNAd3X,
(1)
where N M is the monopole number density, a R the cross section o f the Rubakov process for the m o n o p o l e proton system, Ore1 the relative velocity, PH the hydrogen weight density, and N A is the Avogadro number, ~ 6 × 1023. The space integration is to be done over the interior o f the sun. The contribution from helium is negligible because o f the small cross section due to the suppression factor discussed above. We take Ore1 as the thermal velocity of ionized hydrogen,
29 December 1983
Ore1 = c~3 = (2T/mN) I/2, with T t h e temperature. We note that a monopole once trapped by the sun will be thermalised within a short period, t ~ 103(C/VM)s, with vM the initial velocity of the monopole, as is easily estimated from the energy loss o f a monopole [ 16], - d E / d x ~ 30" (2E/mmonopole)l/2 GeV/gr/cm 2. Putting OR = 00//32 ,
(2)
we obtain
f = fNMNACaO (PH/fl) d3x
•
(3)
While we do not know the distribution NM(X ) in the sun, we see, from the knowledge o f the properties of the solar interior [17], that PHI~3 depends only weakly on the distance from the centre o f the sun: PH//3 = 3.3 × 104 gr/cm 3
at the centre r = 0 ,
pH//3 = 1.3 X 104 gr/cm 3
atMr/Me<~0.5, orr/r® <<,0.3. (4)
(Here M o and r o are the mass and the radius o f the sun, respectively.) Therefore the integral f i s not sensitive to the distribution o f the monopole in the sun. Taking PH//3~> 1.3 X 104 gr/cm 3, and integrating over Mr/M ®<~0.5, we obtain f~> 2.3 X 1011 × (o0/10 _27 cm 2)
X
fNmd3X
decays/s.
(5)
Because of the symmetry breaking pattern o f the SU(5) theory [ 18], a direct decay o f a proton to an antineutrino, p ~ n + + if, is forbidden in monopole catalysis [19]. There are yet various indirect processes which would cause a neutrino and/or antineutrino flux: Proton decay will result in an electron neutrino flux, coming mainly from the beta decay of #+ or K 0, /a+ ~ v e + e + + ~ ,
or
K0~v e+e ++Tr-.
(6)
The #+ is produced either directly in proton decay or indirectly through the decay ,3 o f n +. It is estimated ~-3 If monopoles are concentrated near the centre of the sun, the nuclear absorption (mainly due to helium) reduces the ve flux by a factor 1.9. In this case, however, p/f3 [see eq. (4)] is larger by a factor of 2.5, which compensates the reduction due to the nuclear absorption effect, and we will recover our bound eq. (10). If monopoles are away from the centre, say Mr/M~ ~ 0.5, the reduction of ve due to ~r+ absorption is only 15%. 381
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that most o f the/~+ decays after it is stopped, and therefore the mean energy of such a neutrino is around 35 MeV. This neutrino would cause an increase in the neutrino capture rate in the solar neutrino experiment. In a typical SU(5) model we expect the probability BTr+ to have such rr+ is ~ 0 . 5 [20] ,4. (The branching ratio B,r+ of catalysed proton decay, however, could be different from that of free proton decay. Therefore we show explicitly the dependence of our result on the branching ratio.) We estimate the capture cross section of the process ve + 37C1 + e + 3 7 A ,
(7)
for a neutrino with E ~ 35 MeV by taking the lowest four levels of 37A as the final states. The capture cross section for each final state I' is [22] ¢7i,= 36.8 X s o (frl/2 )w2G(Z, We),
29 December 1983
n M ~< (1 monopole/4 X 1012 gr)- (10 -27 cm2/o0)
× (0.5IBm.+).
(10)
Detection of such a neutrino with E v ~ 35 MeV may be possible using underground facilities for the proton decay experiment; a search for the excess o f this neutrino will be important, since it would be the only way to see the effect of monopoles in the sun when the abundance is so small as in eq. (10). If we assume SU(5) grand unified theory, proton decay is also accompanied by an energetic neutrino or antineutrino with E ~> 200 MeV through the twostep process p + monopole -->/J+ + K ° + m o n o p o l e ,
I
(8)
--+ KL -+ Ve + e + zr.
(11)
where
We estimate the probability of this process as
s O = 1.91 X 10 -46 cm 2 ,
Bhig h -_ x B r ( M + p + M + tl + + K 0)" h " Br(K L + ve err)" r
(f71/2){-I = ((l)~, + (gA/gv) 2 (O)2 , ) / 2 ,
>~2 × 10 - 6 × [Br(M + p - + M +/x + + K0)/0.05] , (12) where h represents the hindrance factor o f the decay in flight of K L through the absorption or regeneration of K 0 in hydrogen in the sun [h "~ 0.5% at the centre of the sun (O ~ 160 gr/cm3), h cc O-1 ], and r is the fraction of high energy neutrinos with E >~ 200 MeV (r ~ 4.5%). (We note that in the absence o f the monopole Br(p -+/I+K 0) ~ 5% in a typical calculation o f the SU(5) model [20, 21].) The flux o f such a neutrino or antineutrino at the e a r t h Ihig h iS then/high = Bhighf/47rR2. These neutrinos should be detected as background events in the underground experiments. It will be quite safe to say that the excess event rate of such a neutrino interaction is less than 5 events/ (100 t o n . yr) [24]. Taking the cross section for the v - N charged current interaction as 0.5 X 10-38(E/ GeV) cm 2, we find
and
G(Z, We) = cPeF(Z, We)/ZTraZI¢e , with F the Coulomb correction factor, and w e = We/ me e2. Here We is the total energy o f the final electron. The nuclear matrix elements ( l ~I, and (o) 2, are given in ref. [22]. The dominant contribution comes from the fourth level, JP =23 + a n d T = ~ - a t E ~ 5 M e V , i t s matrix elements being (/)2 = 3 and (o) 2 = ~-. 1 Taking also account o f the other three low lying levels, we obtain the total capture cross section Oca p = 4.2 X 105 X s O = 8.1 X 10 -41 cm 2 .
(9)
This is regarded as a lower bound on the capture cross section, since only the lowest four levels of 37 A are taken into account. The capture rate in the solar neutrino experiment is given by IvOcap, where I v = B l o w f / 47rR 2 is the flux o f a neutrino from ~+ at the earth with R the distance between the earth and the sun. We take the upper bound for an excess neutrino cap, ture rate to be less than 1 SNU [23]. Then we are led to an upper bound on the solar monopole abundance as , 4 For a review see ref. [21 ].
382
f<4X
1033/s.
(13)
Eqs. (5) and (8) thus lead to an upper bound for the total number of monopoles in the sun (we concentrate on monopoles in the relatively inner region o f the sun, say M r <~0.5 M®),
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fNMd3X ~< 1.7 X 1022 • (10 -27 cm2/ao) X [0.05/Br(M + p - + M + # + + K0)] ,
Mr
(14)
or dividing by the mass 0.5 M®, we obtain the bound on the density o f the monopole: n M ~< (1 monopole/0.6 X 1011 gr)" (10 -27 cm2/a0) X [0.05/Br(M + p -+ M + # + + K0)] .
(15)
The underground proton decay experiments [25,26] currently being carried out will further improve this bound. The above considerations depend on the model for the decay modes o f the proton. Even if proton decay is not accompanied b y either a neutrino or an antineutrino in some particular models, we can still obtain some constraints on the monopole abundance from the observed solar neutrino flux: proton decay due to monopoles also affects the heat flow o f the sun. I f 1% o f the solar luminosity is to be attributed to monopole catalysed nucleon decay, the energy generation by the ordinary p - p chain should need a contribution 1% less, or a temperature near the centre 0.25% less. The neutrino production from B 8, which is proportional to T 22-5 , would then be reduced due to this temperature decrease by 6%, or 7% using the result b y Bahcall et al. [27], ~ In ¢(B8)/O In L , = 7.2. In this respect we expect that the solar luminosity should not be attributed to the Rubakov process b y more than 10%, namely, mNc2f<.-..0.1 X I o in order not to disturb the theoretical calculation [27] o f the thermonuclear neutrino flux b y more than a factor 2. This gives also an upper bound on the solar abundance of the monopole n M ~< (1 monopole/109gr) • (10 -27 c m 2 / o 0 ) .
(16)
This abundance will disturb the solar magnetic field b y more than 10 - 6 G on the surface of the sun. Although eq. (16) is less stringent than eq. (10) and (15), it is still significant compared with the bounds obtained so far: (i) n M < 1/(2 X 107 gr) for the sun considering the decay of the solar magnetic fields [28], (ii) n M < 1/(5 X 109 gr) for the earth
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considering the terrestrial magnetic field [28], (iii) n M < 1.7 × 10-4/gr for the moon obtained from a monopole search in lunar material [29], (iv) n M < 5 × 10-5/gr [30] for the earth considering monop o l e - a n t i m o n o p o l e annihilation. It seems unlikely that the abundance of monopoles in the planets such as the earth and Jupiter or in meteorites exceeds much o f that in the sun. Therefore we may regard our limit on the monopole abundance as that for the entire solar system. It is interesting to note that the Rubakov process together with the monopole abundance just allowed b y the bound eq. (10), might give rise to the excess heat of the jovian planets , s , which has long been a puzzle [32]. *s For a review see ref. [31 ].
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[13] W.J. Marciano and I.J. Muzinich, Phys. Rev. Lett. 50 (1983) 1035. [14] S.D. Drell, N.M. KroU, M.T. Mueller, S.I. Parke and M.A. Ruderman, Phys. Rev. Lett. 50 (1983) 644. [15] W.V.R. Malkus, Phys. Rev. 83 (1951) 899. [16] S.P. Ahlen and K. Kinoshita, Phys. Rev. D26 (1982) 2347. [ 17] D.D. Clayton, Principles of stellar evolution and nucleosynthesis (McGraw-Hill, New York, 1968). [18] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438; H. Georgi, H.R. Quinn and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451. [19] J. Ellis, D.V. Nanopoulos and K.A. Olive, Phys. Lett. 116B (1982) 127. [20] G. Kane and G. Karl, Phys. Rev. D22 (1980) 2808. [21 ] P. Langacker, Phys. Rep. 72 (1981) 185. [22] J.N. Bahcali, Phys. Rev. 135 (1964) B137. [23] R. Davis Jr. et al., Phys. Rev. Lett. 20 (1968) 1205; J.K. Rowley et al., BNL report 27190 (1980). [24] M.R. Krishnaswamy et al., Phys. Lett. 115B (1982) 349, and references therein; PRAMANA (Indian Academy of Science) 19 (1982) 525.
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[25 ] J. Vandervelde et al., Proc. Neutrino '81 (Maui, Hawaii, 1981), Vol. 1, p. 205; R.M. Bionta et al., Phys. Rev. Lett. 51 (1983) 27. [26] T. Suda et al., Proc. Neutrino '81 (Maui, Hawaii, 1981), Vol. 1, p. 224. [27] J.B. Bahcall, N.A. Bahcall and R.K. Ulrich, Astrophys. J. 156 (1969) 559; see also J.N. Bahcall et al., Rev. Mod. Phys. 54 (1982) 767. [28] S. Dimopoulos, S.L. Glashow, E.M. Purcell and F. Wilczek, Nature 298 (1982) 824. [29] R.R. Ross, P.H. Eberhard, L.W. Alvarez and R.D. Watt, Phys. Rev. D8 (1973)698, and references therein. [30] R.A. Carrigan, Nature 288 (1980) 348. [31] D.J. Stevenson, Ann. Rev. Earth Planet. Sci. 10 (1982) 257; see also W.B. Hubbard, Rev. Geophys. Space Phys. 18 (1980) 1. [32] J. Arafune, M. Fukugita and S. Yanagita, Kyoto University preprint (1983).