Astroparticle Physics

ELSEVIER Astroparticle Physics 4 ( 1095) I W- I94

Magnetic moment of neutrinos in the statistical background

Samina S. Masood Ilc;r,urtmertt of Phystcs. C)uatd-r-Axm Cinrt rrsity. Islamahad 45320, Pakistan

Received 24 January lYY5: revised manuscript received 6 June 199s

Abstract

The magnetic moment of neutrinos is found to have nonzero corrections from the background heat bath if we deal with the massive Dirac neutrinos. These statistical corrections are calculated in different extensions of the standard model with the right handed neutrinos. WC expect that these statistical corrections may be useful in understanding the physics of superdense media such as supernovae.

1. Introduction

The left-handed massless neutrinos are present in the standard electroweak model of Glashow, Salam and Weinberg. There is no fundamental reason [I] to avoid the right-handed neutrinos which could have paired with vL through the Higgs mechanism. Even the grand unified theories can accomodate the massive neutrinos with two degrees of freedom so that the leptons and quarks can be put in the same multiplet. These massive neutrinos are, of course, somewhat different in nature and are expected to be helpful in solving some astrophysical and cosmological issues [2]. In this context, the neutrino flavour oscillation (MSW effect [3]) and the spin oscillations (i.e. the suggestion of OVV [4]) were studied in detail, in particular, to resolve the solar neutrino problem (SNP). The role of the neutrino electromag- netic moments in the stellar energy loss rate is also found to be very interesting [5]. However, the MSW effect and the OVV proposals were not able to resolve the solar neutrino problem. The cosmological limit [6] on the magnetic moment of neutrinos could not be achieved from the standard electroweak model and attempts were made to extend the standard model in different directions. The minimal extension of the standard model (MESM) was obviously the Salam-Weinberg model with right-handed neutrinos. Among the other models, the left-right symmetric model [l] and the standard model with scalar Higgs as a singlet [71 or as a doublet 181 (i.e. the Japanese model) are the interesting ones. The addition of more particles in a model leads to an increase in the number of ad hoc parameters, therefore, we restrict our self to the MESM and the Japanese model.

It is now well known that the electromagnetic properties [91 of the particles change in the statistical background. The electric charge, the mass, and the wavefunction of particles are modified [lo] at finite temperature and density (FTD). The massless gauge bosons can acquire the dynamically generated mass [II] (see also Landsman and Weert in Ref. [IO]) due to the plasma screening effect. Hence the refractive indices, the magnetic moments, and the refractive energies are expressible [9] as functions of tempera-

0927~h505/95,/$09.50 C 1995 Elsevier Science B.V. All rights KS~ITW~ SSDI 0927-hSOS(Yc)OOO.1 1 -.i

-f 5?-- ‘:

Z”

3 9 v

r 9

(a) (b)

Fig. I. Lowest ordrr hgr;m~ contributing to tlx magnetic moment of neutrino in MESM. (a) Bubble diagram, (b) tadpole diagram.

ture T and chemical potential p. The magnetic moments of the charged fermions can be obtained from tree level diagrams because the charges directly couple with the magnetic field. The magnetic moment of neutrinos is a perturbative effect because neutrinos can couple with the external magnetic field through charged leptons. These charged leptons induce some magnetic moment in the neutrinos. Since the magnetic moment of charged leptons is increased [ 121 in the statistical background, the same is expected for neutrinos as well. The massless neutrinos with one degree of freedom, however, do not show any magnetic moment. Even the Majorana type massive neutrinos have zero magnetic moment. The massive Dirac neutrinos, however? exhibit magnetic moment at the one loop level. The dominant contribution up to the order (Y can be obtained from Fig. 1 which are only possible in models having right-handed neutrinos. The magnetic moment of neutrinos is calculated [l] from Fig. la, whereas the contribution of Fig. lb vanishes in vacuum. Olivo, Nieves, and Pal have shown in Ref. [12] that the FTD corrections to the magnetic moment of neutrinos, a!, vanish at low temperature and thus those of Fig. la because both diagrams are linked through a Fierz type transformation. In an earlier work [9], we have mentioned that the contribution of Fig. lb is zero in a heat bath and hence that of Fig. la even at high temperatures. Later, we noticed that if we calculate both the diagrams separately, in detail, they do not come out to be equal to each other. The vanishing contribution of the tadpole diagram can be understood from the closed fermion loop because the virtual charged fermion and antifermion induce equal and opposite recession to the neutrino in the magnetic field and are cancelled. On the other hand, Fig. la gives a nonzero value of the magnetic moment.

In this paper we estimate the statistical background correction from the heat bath in the MESM and the Japanese model. The next section comprises the explicit calculations of the magnetic moment of neutrino at FTD. The results obtained in Section 2 and some of their applications are discussed in Section 3.

2. Calculation of the magnetic moment

The lowest order (dominant) contribution to the magnetic moment of neutrino can be estimated from Fig. 1. We calculate Fig. la in detail here. Fig. lb is already calculated in Refs. [9], [12]. The QED perturbations in the electromagnetic media induce the magnetic moment which gives rise to sizeable corrections from the background heat bath. We work in the covariant real-time formalism and use the hot and dense fermion propagators given as [ 131

191

with

in the usual notation with ,C3 being the inverse temperature and O(po) the step function. The matrix element corresponding to Fig. la can be written as

x - I ?;riS{( p? - k)’ - mf} ,2 ( p (j7, -k)‘-mf t I-k)+Pl++P:

I (2)

in the standard notation. The propagator of W-boson is approximated as l/m& with k’ CC m& where m, is its mass. The magnetic moment of the neutrino then comes out to be

GFmf ,Ci=---m “C 4$ l’i

where the neutrino mass is taken in electronvolts and tin denominator is approximately equal to the mass of W. The

I

In( 1 + e I !!I 1 !A w ) ( 1 + c cm il 10 ) .

G(m,B. P) = I*-n-I f c

( - l)‘I

Pr:,,l:, (rzpp)‘-’ c

llrn,li

t(m,/3. p) =

m,p, l-&-h ’ 1 (3) is the Bohr magneton. The mass A4 in the tilde functions in Eq. (2) are given by

T>F,

cosh(nPp), T</J, (da)

T>I-L,

T<p. (4b)

T>P.,

(4c) I, em”“‘@ cosh(nPp), T>p.

These tilde functions are discussed in detail in Ref. [9]. and the magnetic moment can be evaluated using them in different limits of T and F giving

(5a)

for T>m,>p.

(5b)

192 S.S. Masood / Astroparticle Physics 4 (1995) 189-194

Fig. 2. Diagrams contributmg to the magnetic moment of neutrino in Japanese model where the lepton flavour symmetry is broken.

(5c)

in the classical limit, i.e. T < m, +C p. For low values of T and p the magnetic moment corrections from the heat bath vanishes because the fermion background effect is vanishingly small in this case.

The Japanese model was mainly constructed to get the magnetic moment of neutrino high enough to meet the cosmological bound. The couplings in this model are expressed in terms of the Higgs mass and are the additional parameters. The diagrams contributing to the Japanese model are given in Fig. 2. The non-zero corrections to the magnetic moment of the neutrino in this model can be obained from Fig. 2a giving

(6)

whereas Fig. 2b vanishes. I and I’ in Eq. (6) correspond to lepton flavours which can either be same or different because their conservation is violated in this model. mb is the Higgs mass and the coupling constants are estimated [8] in the units of the Higgs mass as follows

fe,h,,/m~ 5 2.8 x IOph GeV’, Pa)

f6,Th,,/mi 2 0.8 x lWh GeV’.

The FTD corrections to the magnetic moment in this model also depend on the tilde functions giving

for F <m, < T, and

for T < m, < y whereas,

Ei( -x) = -IX? dt. 1

S.S. Musood /Asfropurtdr~ Physm 4 (19951 1X9-194 193

3. Results and discussion

The magnetic moment of a neutrino arises as a perturbative effect in weak interactions. The neutrino, being a neutral particle,cannot couple with the magnetic field directly. It is, therefore,the charged lepton which induces a magnetic moment to the neutrino. In the MESM, due to the lepton flavour conservation, only the lepton of the same flavour induces some magnetic moment to the neutrino whereas in the Japanese model all flavours of charged leptons can induce some recession to the one type of neutrino and can attain higher value. It is already calculated that the magnetic moment of electron increases at FTD so it should obviously induce more recession to the neutrino. Therefore, the FTD corrections to the magnetic moment of neutrino is not expected to vanish. We know that the electromagnetic properties of media change at FTD and the propagation of the particles in media is affected by the perturbative corrections to the self energies of the particles. Since the statistical effects appear in the form of tilde functions which depend on rnlP and p/m, so the temperature and chemical potential can be written in the units of the mass of charged leptons m,. Therefore, in Eq. (2) for heavy lepton masses, the significant corrections can only be achieved at very high temperatures (e.g., T - lOI K). It can be interesting for T sufficiently greater than m, (i.e., around 2 GeV) which is at very high temperature and is possible in the early universe only. However, in the Japanese model due to the breaking of flavour symmetry the lighter lepton terms can give dominant contribution (though small) such that uf, is around 5 X lo-l5 at T - m,. This is so because the LZ~- in Eq. (5) only depend on b(m/3, II) which is negligibly small if /.L < T. The b function attains integral values only if T - m,. It is also worth mentioning that the diagonal couplings in the Japanese model are those of the standard model so the dominant contribution is achieved by the off-diagonal couplings. It so happens because the Higgs doublet blocks the flavour symmetry of leptons and therefore the heavy neutrinos get sizeable corrections from the light leptons. On the other hand, the same order of corrections is obtained for a!? when T - m7 which is around 2 GeV. Thus the Japanese model at FTD does not change the vacuum results much. Moreover, the tadpole term need not be vanishing in this model.

This picture becomes clearer if we look at Eqs. (4). As a typical example, we calculate at$rnve - 10 eV) in different limits of T and p in both the models and we find that the temperature corrections to the magnetic moment of neutrino in general are not very important because it does not enhance the magnetic moment value high enough to resolve SNP or meet the cosmological bound whereas the density corrections may some time serve the purpose in MESM. We expect that the statistical corrections in MESM in superdense media can probably help to get the required amount of emission energy of neutrinos in supernovae. It is, therefore, clear from this analysis that neither the MSW effect nor the OW suggestion is workable to solve the SNP or help to understand the cosmological issues. Thus one needs to think about other more sophisticated models like supersymmetric models or incorporate the dark matter effects to resolve these issues. If dark matter exists it experiences the MSW effect also [14]. It has already been noticed 1151 that the neutrino in SUSY models can attain a magnetic moment value

Table 1

Comparison of the statistical corrections to the magnetic moment of electron neutrino in MESM and the Japanese model. expressed in units of pa.

T P u/?, (MESM) a!<, (Japanese)

me. 0 lo- I7 ,020

m, 0 10 Ih lo-lx

m, 0 IO ‘I 10 IS 0 1?1(, 0 /,I u I:: Ih

0

0 112 j IO ‘-

194 S S Musood /A,woywtdc~ Physrcs 4 (1995) IX9-I94

high enough to touch the cosmological bound. i.c., 10” /~a. Hence SUSY probably may prove to be a more successful theory.

References

[l] See for example. R.N. Mohapatra and P.B. Pal, Maswe Neutrinob in Physics and Astrophysics (World Scientific, Singapore. 1991).

[2] Good reviews on the subject can be tound in T.K. Kuo and J. Pantaleone. Rev. Mod. Phys. 61 (lY8Y) 937; S.M. Bilenkey and S.T. Petcov, Rev. Mod. Phys. 5Y (1987) 671.

[3] S.P. Mikheyev and A.Yu Smirnov. Nuovo Cimento UC (IYXh) 17:

L. Wolfenstein, Phys. Rev. D 17 (lY78) 2369. [4] L.B. Okun. M.B. Voloshyn and M.I. Vystosky. Yad. Fit. Yl (1986) 751 (Soviet .I. Nucl. Phys. 4h (1986) 446). [5] B.K. Keimov. Phys. Lett. B 274 (1992) 477.

[6] J.A. Morgan. Phys. Lett. B 102 (1981) 247. [7] KS Babu and V.S. Mathur, Phys. Lett. B 1Yh (IYX7) 21X:

A. Zee. Phys. Lett. B IhI (lY753 141. [X] M. Fukujita and S. Yazaki, Phys. Rev. D3h (1987) 3817:

M. Fukujita and T. Yanagida. Phys. Rev. Lett. 58 (1987) IX117 [Y] S.S. Masood, Phys. Rev. D 48 (1093) 3250:

D. Notzold and G. Raffelt. Nucl. Phys. B 307 ( 1988) 324. [lO] See for example. N.P. Landsman and Ch.G. Weert. Phy\. Rep. I45 (IYX7l 141:

S.S. Masood, Phys. Rev. D 44 (1091) 3943.

[ 1 l] K. Ahmed and S.S. Masood. Ann. Phys. (NY) 707 (IOY I ) JO: S.S. Masood, Phys. Rev. D 47 (1993) 648: S.S. Masood and M.Q. Haseeb, Gluon polarization at fimte tcmperaturc and density, Astropart. Phys. 3 (1995) 405.

[12] J.C. Olive, J.F. Nieces and P.B. Pal. Phys. Rev. D 40 (1YXY) 3679 and references therein. [13] E.J. Levinson and D. Boal. Phys. Rev. D 31 (19X5) 3280. [l4] J. Ellis, R. Flares, and S.S. Masood, Phys. Lett. B 7Y4 (lYY1) 170.

1151 R. Barbier. M.M. Guzro. A. Mahieri and D. Tomahinl, Phy\. 1,211. B 252 (IYYO) 251; K.S. Babu and R.N. Mohapatra. Phys. Rev. Lett. 63 I 19YO) 1705