Massive Dirac neutrinos in a background electromagnetic field

15 June 1995

PHYSICS

ELSEVIER

LETTERS

B

Physics Letters B 352 (1995) 346-356

Massive Dirac neutrinos in a background electromagnetic field Alexander V. Kurilin Department

ofPhysics,

Moscow State Open Pedagogical Institute, Verkhnyaya Radishcheuskaya Received 29 November

1994; revised manuscript Editor: P.V. Landshoff

16-18, Moscow 109004, Russia

received 3 April 1995

Abstract

We investigate the propagation of a massive Dirac neutrino ve in a background electromagnetic field F”“. In particular we have obtained exact solutions for the Dirac-Schwinger equation with the self-energy operator of the neutrino which involves the effective electron and W-boson propagators allowing for interactions of these particles with an intense electromagnetic background. In the framework of the approach described above, we analyze the dispersion relation for the neutrino and calculate the effects induced by the external field. For example, in the massless case we obtain the following shift of the pole in the neutrino propagator: p@pP + (gze2/6~*~~)[ln(M,/m,) + ~]ppFp,F”“pu = 0. The external field impact on the dipole magnetic moment pLyof the neutrino is also considered. We find that under certain conditions the magnitude of the static magnetic moment p,, = (3$?/16n’)G, em, can be enhanced by the electromagnetic field. However this effect is very small: p,/p,, = 1.016148. We also demonstrate the possibility of non-perturbative contributions being non-linear in F ph to influence sp in precession phenomena. Some applications of these results to astrophysics and cosmology are discussed.

1. Introduction

The properties of the neutrino and its interactions play one of the central roles in modern particle physics, astrophysics and cosmology. The neutrino experiments used to appear as the best indicators for unusual phenomena which require principally new approaches for adequate theoretical interpretation. Indeed, twenty years ago this was the case when neutrino physics led to the discovery of neutral currents, providing thus the first confirmation of the standard electroweak model. And nowadays there are experimental results concerning neutrinos that cannot be easily interpreted within the framework of the minimal standard theoretical conception. Here we can mention the famous “solar neutrino puzzle” dealing with the discrepancy between the predictions of the standard solar model [l] and the rates of neutrinos detected in different experiments [2-51. Although it seems difficult to decide whether the deficit of neutrinos is due either to the basic properties of this particle or to the shortcomings of the standard solar model, now there are cogent arguments in favor of particle physics rather than the astrophysical solution [6]. There are also serious reasons to think about new physics in view of recent 0370.2693/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0370-2693(95)00447-5

A. V. Kurilin / Physics Letters B 352 (1995) 346-356

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theory contradicting experiments in which the ratio u,/v, of neutrinos produced by cosmic ray interactions in the atmosphere were measured [7,8]. From the aforesaid it is clear that present experimental data strongly evidence in favor of different extensions of the minimal standard electroweak model which rely on the occurrence of neutrino oscillations between different flavor states. All such approaches face the main question of neutrino physics, the question of whether the neutrino has a nonzero mass. There are many models for the neutrino mass (see e.g. [9] and references therein), all of which have good and bad features. Perhaps, the most natural and economical one is the possibility to generate a Dirac mass for the neutrino by the vacuum expectation value of the neutral component of the Higgs scalar doublet. It is very attractive that in this approach the neutrinos are treated exactly like other fermions. The main shortcoming of the above procedure is that it is hard to understand why the neutrinos are so light in comparison with the other particles. However, this question may be considered in connection with the general problem of explaining why the fermion masses range over at least five orders of magnitude, and there is hope that it can be settled in future superstring inspired models [lo]. The problem of the neutrino mass touches upon a subject concerning the electromagnetic interactions of this particle. The major motivation for considering neutrino interactions with electromagnetic fields is the possibility to obtain an explanation of the solar neutrino puzzle by magnetic transitions of the left handed neutrino ve,_ into a sterile right handed component v/a [ll]. However, this phenomenon requires an unnaturally large magnetic moment of the neutrino, II,, = (3-10) X 10-‘lpa, whereas in the standard electroweak model with a Dirac mass one expects py = 3 X 10-‘9(m,/l eV)pa [12] (here ha is the Borh magneton). In this connection it is very interesting to investigate alternative mechanisms for electromagnetic interactions of the neutrino being able to contribute to the effect pointed to above. From this viewpoint it is worth noticing that besides the dipole magnetic moment, the electromagnetic current of the massive Dirac neutrino contains other form factors [12,13] which could also affect the propagation of this particle in an intense electromagnetic background. In particular the Dirac neutrino possesses an anapole (toroidal) moment that is nonvanishing even in the massless case. Moreover in strong electromagnetic fields additional form factors of the neutrino can arise due to the non-perturbative effects being non-linear in the field strength F,,. As an example we would like to mention the field induced dipole electric moment of the neutrino which could reveal itself in constant parallel (h, # 0 see below (5)) electric and magnetic fields [14] or in the ellipsoidal field of a plane electromagnetic wave [15]. Recently the question of neutrino interactions in an electromagnetic environment has gained more attention in view of the mechanism of resonant neutrino conversion in twisting magnetic fields [16]. It has been realized that the effect of the neutrino spin precession can be considerably amplified due to the presence of a magnetic field rotating in a plane perpendicular to the neutrino momentum. In this paper we investigate the dispersion relation for the massive Dirac neutrino and its propagation in an intense electromagnetic background. The related topics were discussed in [17] where the case of Majorana neutrinos was also considered. However in those papers only effects linear with respect to external field Fph were analyzed. Here we intend to take into account the non-perturbative contributions and to study their influence on the spin precession phenomena.

2. Massive neutrino self-energy operator Propagation of neutrinos in an intense electromagnetic field is governed by the Dirac-Schwinger which involves the self-energy operator 2(x, x’):

(ir*j-mv)JI(x)

-/

d4X’ 2(x,

x’)+(x’)

=O.

equation

(1)

348

A. V. Ku&in /Physics

Letters B 352 (199.5) 346-356

Fig. 1. Feynman diagrams contributing to the neutrino self-energy in an intense electromagnetic field. Double lines represent the effective particle propagators in the non-trivial background.

In the standard model of electroweak interactions the neutrino second-order self-energy can be determined through the calculation of the diagrams depicted in Figs. la-lc. However in the nonlinear ‘t Hooft-Feynman gauge [18] the contribution of the graph (lb) is suppressed by the factor (m,/M,)’ which is due to the small value of the gauge scalar (4 *> Yukawa coupling h, = (g/ fiXm,/~,). So let us confine ourselves to the estimation of the diagram (la) yielding the expression S( X, x’) = $g’r”(l+

y5)yP(1

+ r’)D,,(

x, x’).

(2)

Now we wish to take into account the effects of neutrino interactions with the background electromagnetic field. This can be done effectively by substituting the Green functions of the electron and W-boson which adequately describe propagation of these particles in an intense non-perturbative field F,,. The given approach being widely known in quantum field theory as the Furry perturbative picture [19] has successfully proved its efficiency in earlier investigations (for a review see [20]). The corresponding equations for the W-boson and the electron propagators follow from the lagrangian of the standard model: [(a~+ieAfi)2+M;]Q& r

ry”a, + ey”A,

-nQ,]S(x,

x, x’) +2ieF,“D&, x’) = syx-x’).

x’) =gag64(x-x’),

(3) (4)

Except in a few special cases described in [20], it is very difficult to solve Eqs. (3), (4) for a generic electromagnetic field. However there is a good approximation which allows one to analyze in detail the domain of relatively weak external fields. The main idea lies in the fact that exact solutions of Eqs. (3), (4) being Lorentz-covariant must include the invariant parameters

A.V. Kurilin / Physics Letters B 352 (1995) 346-356

349

which fix the scale of the background electromagnetic fields (here flLcLA = ie’LAUBFUSis a dual field strength tensor). Then for weak fields one can neglect the corrections being proportional to the small quantities h,, h, and put the condition h, = h, = 0. This conception is known in the literature as the crossed field approximation representing the semiclassical limit for the propagators [21]. In a constant homogeneous crossed field Eqs. (3), (4) can be easily solved employing the Fock-Schwinger proper time formalism [22], and one obtains -km:

x2

- i-g-

-

2 izXPF,,FABXp

X”

X ( gaB + 2eTFap + 2e2r2F,,Fi),

(7)

and @(x, x’) is a gauge factor which is determined by the integral along a straight line where XcL=x~-xrc from the point xp to the point XL: @(x, x’) = exp ie xA,(z) (i x’

dr”

.

(8)

I

Inserting propagators (61, (7) into Eq. (2) and performing the Fourier transformation ;S( p, p’) = / d4x / d4x’ exp( ipx - ip’x’)Z( x, x’) = (27r)4S4( p -p’) Z( p),

(9)

after some tedious but not difficult calculations, we get the following expression for the neutrino self-energy: W)

=&(p)

+-Up,

F).

(IO)

The augend in Eq. (10) represents the renormalized vacuum contribution depending on the neutrino momentum pP only:

A( P) =

+(l -

y’>[ r”p,(&

-RI)

+ m,&] y

(11)

with the abbreviation R,=

R*W)

--

mtu’(l

8

-u)

’ du 81r2 II-J Miu+mZ,(l-u)-mZu(l-u)’

= &J-l

duu In 0

(12)

Miu+m2,(1-u)

-p’u(l-u)

Miu+m:(l-u)

-mEu(l-u)

(13)

whereas the addend corresponds to the field induced correction which has no divergences and can be written in the form

(14)

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A.V. Ku&n /Physics Letters B 352 (I 995) 346-356

It is important to stress that in addition to the apparent F FA-dependence in Eq. (14) there is also an implicit one through the scalar functions A$, N,, N. being defined in the following manner: ’ du u&(z),

&( P2,K) = -

$/,I

++(l,,

(16)

+.)(Z+i+)if(z).

(17)

du u(l

N2(P2,K) = --&‘duu(l

(15)

The Hardy-Stocks functions fi(z), f(z), f’(z) (see the Appendix) entering expressions (15)-(17) depend on the argument Z =

[

KU( 1 - U)] -2’3y,

where ~=u+(m,/M,)2(1-u)-(pZ/M;)u(l-u),

(18)

which involves the fundamental background field parameter K=ei%f;3/m.

(19)

In the semiclassical limit being considered when all other invariants (5) are disregarded, the parameter K is the only quantity which determines the magnitude of the effects induced by the background electromagnetic field. Let us now consider the momentum representation of the Dirac-Schwinger equation (1) which allows one to transform it into the following relation for spin components of the massive neutrino wavefunction: [V&-%-W)]+(P)

=O*

(20)

There are several ways to solve this linear equation and to find out the neutrino dispersion law in the electromagnetic background. The perturbative approach which has been employed in [14,17] is based on the calculation of the mass shift Am,, by using the non-perturbed wavefunction u(p) of the neutrino: p2-m~=2m,Am,=ii(p)~(p)u(p)~p~=,g,

(21)

which obeys the conditions (rPPF-m,)u(p)

=O,

U(p)u(p)

=2mV9

U(p)Vu(p)

=2PFL.

(22)

This method yields the following expression for the field induced correction to the neutrino mass: p’-m~=mZN,(mZ,

~)+[~rn~M~KiV~(rn~,

K)+(~-{,,)M~K~N,(~~,

K),

(23)

where the parameters [, , [,, = f 1 are connected with the polarization vector of the neutrino s P characterizing the spin direction:

In particular $.[, corresponds to the neutrino spin projection on the magnetic field while is,, is the parallel mean helicity. Eq. (23) was analyzed in detail in [14,17], but those results represent only the first perturbative correction to the neutrino dispersion. Here we intend to step further and to obtain exact solutions of the Dirac-Schwinger equation (20).

A.V. Kurilin / Physics Letters B 352 (1995) 346-356

3.51

3. Nonperturbative approach to the neutrino dispersion The starting point for the nonperturbative analysis is the self-consistency condition of the linear algebraic system (20) which ensures the existence of non-trivial solutions: det[y’“p,-m,-P(p)]

=O.

(25)

Inserting the neutrino self energy operator (lo)-(17)

into Eq. (25) we obtain the following dispersion relation:

(P2-m~)(l+Ro-RI-No)=m~(No+R,)+M,$.K2N2-5MWK

p2N:+M,$tc2N;,

(26)

which should be compared with the perturbative mass shift (23). We see that the expression (26) in contrast with (23) depends on only one parameter of neutrino polarization l= f 1 which substitutes the quantities 5, , Jr1 (24). From the series expansion of the radical in Eq. (26) one concludes that the above equations become approximately identical if the appropriate spin correlation holds: 5 = 5 I = - l,,. Moreover it is evident that the neutrino wavefunction in the background electromagnetic field $(p> differs from the Dirac spinor u(p) (22). To specify this difference we will seek exact solutions of Eq. (20) in the following way: HP)

= [YPP+mV(l+RO)

-%##0,S).

(27)

Substituting (27) in (20) we find that the function $(p, 4 ) must be the eigenvector of the definite spin matrix

B = rsrarP(C,~,

- CBpa),

B.~(P,

C)=~~{(CP)~-C~P~

.+(P,

5),

(28)

where C,=

&P’)NI

&(

+ &4(

W

(29)

F,,FA"p,)N2. W

The condition (28) may be met if we take the Dirac spinor u(p) on the mass-shell of E!q. (26) and write down the auxiliary function ~$(p, 5 ) in the following manner: 4(Pv S) = (1+ T’)4P)? where r5

(30)

represents the generalization of the usual chiral y5 matrix in an intense electromagnetic background.

(31) Except for the massless case (p2 = 0) when rs coincides with the ordinary y5 matrix, the left-handed neutrinos as well as the right-handed ones are no longer the eigen-vectors of the spin matrix B (28), so these states cannot be characterized by a definite energy and momentum. On the other hand, under the assumption mf = 0 from Eq. (26) it follows that p2 = 0 for the right-handed components, and p2 = 2 Mi K 2N2(1 + R, - R, -N,,-’ for the left-handed neutrinos. This conclusion implies that even in the massless case the way of neutrino propagation in the electromagnetic field substantially depends on its chirality. The mass-shell for neutrinos in the external background (26) involves the parameter K. When K K 6, = m,/M, the neutrino mass-shift is determined mainly by the contribution of the dipole magnetic moment ( CL,): p2 - rn: = ~m,M,KN,

= -21&(

In the opposite case when 8, +z with the effects of [-chirality: p2-m~=M~K2N2(1+~).

p’“F,,FA“pg) = -2&,MG(

K -sc 6, =

mv/M,

K/e).

(32)

the leading correction to the neutrino mass is connected

(33)

352

A. V. Kurilin / Physics Letters B 352 (1995) 346-356

In order to make these results more definite let us calculate the integrals (121, (13), (15)-(17) in the domain of relatively weak external fields (K e Se, p2 -=KMi). (34)

&,(P2,K) = -

&r’(l +o(K2)],

Nl(p2,

K)=

&(

K) = - j$[N&/m,)

P*,

-g[l+0(r”)J,

(37) + i +O(K*)].

Now we are able to make numerical estimations of the effects being analyzed. Assuming that the neutrino moves in a constant magnetic field directed along the z-axis (B = B,) one can find the neutrino dispersion explicitly:

PO=

d

P2+m2-(P2+P~)(1+5)12.rr211,14 g2e2B2 [ln(M,/m,)

+ i]

.

W

We see that the impact of the magnetic field becomes essential when the field strength is sufficiently high: B 2 102’( m,/pl)T.

(40)

Besides, it affects the neutrino motion only if there is a momentum component being perpendicular to the magnetic field direction: pL = p - (pB>B/B* # 0. Under th e above conditions it is possible to observe the neutrino refraction in a varying magnetic field. For relativistic neutrinos it is often useful to consider the refractive index IZ= ) p ) /p. being defined in analogy with optics (see e.g. [23]). Then from (39) it is evident that in a magnetic field the refractive index for right-handed neutrinos nt = 1 - (m,/po)2 differs from the one for the left-handed pattern,

nL=%

i

1-n g2e2B2

[ln(M,/m,)

+ f] sin2q -l’*,

W

(41)

I

implying thus birefringence for the opposite helicity states. Moreover, the refractive index for the left-handed neutrinos depends on the angle between the neutrino momentum and the magnetic field direction q = L(p, B). Depending on the spin projection, neutrinos of given momentum are split in energy by the amount AP,=P,(~=~)

-PO(~=

-l)

g2e2B2 = - 12~2~4

[ln( M,/m,)

+ i]

( p:

/PO)-

W

This effect being caused by the parity violation properties of weak interactions can influence the neutrino helicity flipping induced by the background magnetic field. In particular if ( Ape 1z+-2 /_L,, B then the spin precession is suppressed [24]. Inserting the neutrino magnetic dipole moment [121 3g2mye b% = cto =

64T2M;

7

(43)

353

A. V. Kurilin /Physics Letters B 352 (1995) 346-356

it is easy to find that this phenomenon could really take place when the magnetic field strength exceeds the lower limit given by the estimation (40).

4. Neutrino magnetic moment in a background field In the previous section the domain where the magnetically induced spin precession is suppressed by refractive phenomena has been evaluated. However we have ignored the possibility of an intense external field to influence the magnitude of the neutrino magnetic moment. This is the question we are going to investigate more carefully. It is well known that the self-mass and electromagnetic properties of many charged fermions such as the electron and other leptons are modified in a non-trivial background [20,21]. For example in quantum electrodynamics it is established that under certain conditions the electron anomalous magnetic moment becomes a function of the background field strength and the electron energy [25]. Similar types of changes are expected in electromagnetic form factors of the neutrino. In order to illustrate this effect let us analyze the magnetic dipole moment CL,,.Combining the results of Eqs. (231, (32) we find that in general the pV-dependence on the background field parameter K (19) is given by the function iV1(pz, K) (16): &(K)

=

Re &(m;,

-2

(44)

K).

W

When the external electromagnetic field is absent (K + 0) the dynamic value p,(~) coincides with the well known static one ~~(0) = CL,,.However with augmentation of the parameter K the dynamic neutrino magnetic moment ~JK) at first slowly increases reaching the maximum value and then diminishes monotonously to zero. Asymptotic evaluations of the integral (16) yield the following piecemeal continuous approximation:

= (3~%)/(20~)

r4i(3~)-2’3,

for

Kc<

a,,

for

K x-

1,

which evidences that the absolute maximum of the ratio pU,(~)/pV(0) is reached when the parameter to %ll,X= fi exp(y, - +) = 0.110051 and

(45)

K

is equal

PV( %I,, /CL,(O) = 1+ 4 exp(2y, - !$) = 1.016148.

5. Conclusions and discussion In this paper we have presented a general discussion of the neutrino dispersion and of its magnetic moment in the presence of a background electromagnetic field. The dispersion properties of neutrinos are very important because they govern the plane-wave propagation of these particles in such an unusual environment. Our analysis has shown that if the magnetic field strength is sufficiently high (40) then there is a birefringence of neutrino waves containing different polarization states. This effect may be of crucial significance for a proper theoretical interpretation of the neutrino signals coming from the Universe. The birefringence of the neutrinos with opposite helicity states inevitably influences the magnetically induced phenomenon of left-right neutrino oscillations. Thus under certain conditions neutrino refractive effects could impede the spin precession and freeze neutrinos in their initial helicity states.

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Letters B 352 (1995) 346-356

LWO

”

I).,0

001

020

0.1s

Fig. 2. The anomalous magnetic moment of the neutrino depicted as a function of the background field parameter K (19).

On the other hand, we have found that the anomalous magnetic dipole moment of a massive Dirac neutrino is really a dynamic quantity which depends on the background field parameter K (see Fig. 2). So there is a region where the static magnitude /.L”(43) is enhanced due to the strong field impact which intensifies the neutrino spin flipping. Unfortunately, for a constant homogeneous magnetic field the observed enhancement is rather small, but we suppose that there could be a somewhat resonant amplifying in a varying electromagnetic background. Besides, we have not taken into account the Z”-tadpole graph (Fig. lc) which could contribute to the neutrino magnetic moment as well. In any case we think that here additional theoretical analysis is needed. It is also worth noticing that recently there has been a great interest in the problem of neutrino interactions with a thermal background [23,24,26,27]. In particular similar calculations for neutrino dispersion and for the electromagnetic form factors at finite temperature and density have been carried out [28]. These studies demonstrated that in a hot and dense medium the Dirac neutrinos acquire an induced electric charge and dipole magnetic moment which could influence the propagation of these particles in such an unusual background. Comparing those results with ours we see that the external electromagnetic field can be treated to some extent as a birefringent medium where different neutrino states experience different refractive indices. However, the electromagnetic characteristics of neutrinos moving in an intense electromagnetic field essentially differ from the ones in a heat bath. Thus, the induced magnetic moment in a plasma which interacts only with the longitudinal component (B,,) of the external electromagnetic field cannot flip the neutrino helicity, in contrast with the field induced corrections to the neutrino magnetic moment. Besides, it is the transversal component (B I > of the magnetic field that governs the neutrino dispersion in the electromagnetic background. From this viewpoint it seems very promising to consider the simultaneous impact of the heat bath and background field effects which is practically important for the study of neutrino oscillations in the early Universe [29].

Appendix We use the natural system of units h = c = 1, with y-matrices corresponding to the metric tensor IrA g = diag( + 1, - 1, - 1, - 1). The following notations are employed throughout the text: ppqF=poqo

-pq,

y5 = -iy”y’y2y3,

r(i)

= 2.6789.. .,

yE = 0.5772.. . .

In Section 2 we have introduced special mathematical functions [30], which can be written in the following way:

f(z) = r[Gi(z) +i A(Z)],

f’(z) =

df(z) dzy

fl(z)

=

lrndt[f(t)

-l/t],

A. V. Kurilin /Physics Letters B 352 (H95) 346-356

where t cos(zt+t3/3),

Gi(z) = iimdt

sin(zt+t3/3).

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/Physics

Letters B 352 (1995) 346-356

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