Russian Physics Journal, Vol. 38, No. 7, 1995

G E N E R A T I O N O F M A S S I V E N E U T R I N O S BY V A R I A B L E

F I E L D S W I T H A L L O W A N C E F O R M I X I N G AND R A D I A T I V E

E F F E C T S

V. V. Skobelev UDC 539.1.01

The probabilities of the processes F --, ~i~j3" and FB --, ~,i~j of generation of massive neutrinos by a variable

electromagnetic field, due to the contribution of four-particle peaks in the one-loop approximation with

allowance for mixing, are obtained. A comparative analysis is made of the quantities considered here and of

generation mechanisms discussed earlier (F --, Pi~j) in the case of vacuum neutrino synchrotron emission.

INTRODUCTION

An invariant formulation of processes of generation of the set {f} of multiparticle states by a variable electromag-

netic field F within the framework of quantum electrodynamics (f = 3') was given in [1]. In particular, photon emission,

through the channels F --, 33" and FB --, 2% by the variable electric field of a classical charge moving in a circle in a magnetic

field B has been considered [2, 3] (vacuum three- and two-photon synchrotron emission), as well as vacuum emission in

collisions of coulomb centers [4]. These mechanisms of photon generation are negligible, of course, against the background

of the ordinary emission by charges moving with acceleration. The ratio of the intensities of vacuum and classical synchrotron emission by electrons, for example, is

S2~/Scl -- 10-2ot3X6 '

S3v/Scl .~. 10-5cx4X8 ' (1)

where

X = (B/Bo)(E/me)

is the standard invariant parameter; B 0 = m2/eo = 4.41-1013 G; o~ = 1/137; E and m e are the electron's energy and mass.

As shown in [5], the situation is different in processes of neutrino generation F ~ vi~, j, where Pi are eigenstates with

a certain mass in models with mixing. That is, in [5] we formulated the conditions under which the structure of the sources

of the variable field (protons, nuclei) is not treated, and the vacuum emission mechanism is the main one, which virtually

always occurs in the approximation being used. For i ~ j , however, the three-particle peak is suppressed by the factor

(mJmw) 2, which is due to the unitary nature of the mixing matrix in the lepton sector (the GIM mechanism [6]), determined by the equation

vL = E Uljvj, (2) J

where the sum is taken over all mass states, while ut are lepton population states.

Moscow State Pedagogical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 48- 52, July, 1995. Original article submitted November 30, 1994.

1064-8887/95/3807-0695512.50 ©1995 Plenum Publishing Corporation 695

It thus makes sense to consider radiative processes of the type FB ---, vi~, j, F ---, viT, j7, due to the contribution of four-

particle peaks, in which that suppression mechanism is absent. In [7], for example, we demonstrated the possibility that two-

photon decay v i ~ vj'y"y of massive neutrinos dominates over one-photon decay v i ---, Vgq,, which may have consequences on a

cosmological scale. Below we use an effective interaction Lagrangian of the type [7] (for simplicity we use the notation i --, 1, j --, 2)

L=iv~ (Cp+ 7SCs)v2F~f'F~f~+~ (C's + T s C'p)v2F~'f'F~f~+h. c ,

2

(3)

The structure constants in the one-loop approximation of the standard model are

c; =c;=o, {c,} * .Cp 2 4 ] / 2 ~ (m, +_ m,~) ~.~ U, tU2,

. l=e , :x . . . /TL~

(4)

where m 1,2 are the neutrino masses.

1. THE FB - , t,l~ 2 PROCESS IN A MAGNETIC FIELD

To analyze the generation of massive neutrinos by the variable electromagnetic field F of an ultrarelativistic classical

charge Q moving in a magnetic field B with a frequency o~ with allowance for a one-time interaction in a loop with the magnetic

field, in Eq. (3) we must set F ---, F ~ B and retain cross terms F ® B . The further transformations are fairly analogous to those

in [3, 5], and for the total probability of the emission of a pair of neutrinos per unit time we can obtain

~ (1 - ;~); [~ (1 - - ; -b 2 - 2 ~ ~ (1 - ~ ' ) X

( r )

1

X (m~ + m~) + (m~ -- m~)'] 1/2 j ' dcos 0 {[~2 (1 - - 7 ) - - m 21 X

- 1

X [Cs ~ v 2 sin 20J; = + C~, cos ~ OJ, ~ ] + [~' (1 - - ~=) - - (m I - - m2) 2] X

X [C'p2vasin2OJ;'+C 2 ° s cos 20Y~]},

(5)

where the argument of the Bessel functions and the domain of integration are defined by the equations

argJ~ = vv~-sinO,

r = {v2(1--~2) >t m2},

(6)

0 is the angle between the directions of the magnetic field and the vector r of total neutrino momentum, v is the velocity of

the charge, and

= - - , ~ = - , t n l , 2 = - - , m = m ~ + m ~ . tO /CO to

Because it is impossible to analyze the general equation (5), we consider characteristic special cases.

a) rh >> 1. When the stronger condition

m > > (7)

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is satisfied, the integrals can be taken analytically by complete analogy with [5], and after transformations we obtain

~v - 3_~,~VTc,mQ., ,'n~ t ~ \too )

+ 3 - ~ C ~ + 4 m l r n z C ~ + 2 C3 1 E \ Q, 9 m '

\ E / J

(8)

In the Weinberg-Salam model with mlm 2 ~ 0 and when (7) is satisfied, in particular, we have

I 1 ~¢7WS ~'~" 37/41/~ ~ C~B2(FlzlM72)32ex,~ __ ]//._~lTt(iTt o ~2 - - -Z ' P '

m9 2 \ E )

(9)

where the value of C s is determined by Eq. (4). b) t~ << 1. In contrast to vacuum synchrotron emission with the three-particle peak [5], in Eq. (5) one cannot simply

set ~ = 0 because the integral diverges logarithmically. The singularity is actually absent in the limit of zero neutrino mass, of course, since in any versions of electroweak models, the structure constants in (3) are proportional to it and the probability- is reduced to zero. In the limiting case under consideration, it is therefore sufficient to retain the "large logarithms" ln(1/ff0 and ln(E/ma), and the standard calculation procedure leads to the result

2O

3V~= B, Q2 ,.o3 (meEf {5 (C~" +C~)~n [1~_ (~_o) ~ ] +

, } + ~ (c~ + c~) in (1 'm) .

(10)

2 . R A D I A T I V E E M I S S I O N O F M A S S I V E N E U T R I N O S B Y A V A R I A B L E F I E L D

Omitting details of the cumbersome calculations, we give a general equation for the total probability of radiative emission of a pair of massive neutrinos 0,1~23,) by a variable electromagnetic field with a Fourier transform A(K) in the form

of an integral with respect to the transferred momentum:

4 f d*tc [A (u) A* (tc)] {C~- [-- L -F (m~ q- m]) L'] q- ve = (2#--- z (11)

+ 2CL rn~ m2 L'},

c~+ = c~ + c ; + c~ + c;;, c ~_ = c~ + c'i - c ~s - c;;,

L (m~ m,,Y ~ I 1 . (l la) - 61c: ~ ~ (3tt 4 ÷ 24tt 3 + 238-.'t 2 F 760tt ÷ 843) D I_

_ (1 l b) L" (mlm2)4113u3+2(9-+- l l ,~+)u ' - '~ - ' ) ( - -3+44:~++15~.~)u+

72u4 [~ , -

-]- 12 (-- 5 ÷ 6~,.+ d- 5~-~ + ~_)] D -- 6 [~+ u 3 + 3 (-- 2 ,'-- 2?.+ 4- ~ . ) tt ~ +

m~ m,_ to" (llc) X ln 4 - - - l u , IIn '~+(u-k-2) 54-+---~=! D)

, + 2 . 12 ' - p+ (u -k 2) -~ 4 - I P _ I D ] '

697

z~" - (m~ + m2): ',',"~ 5 .,v_:~ (l ld) , U , ? ~ , - ,

1711 777, 2 11l ~ ITl 2 I [ D - - [ u ( u + 4)1 ~.

In the synchrotron emission mechanism, the probability per unit time is obtained from (11) in the obvious way by

analogy with (5) and (6), while an analysis of the limit rh << 1, in contrast to Sec. 1, does not lead to the difficulties noted

above, and the result has the form

W _ 3~0 [/:~(2v,) 2 , , . , C?~. (12)

The other limiting case th >> 1 is characterized by an exponential cutoff factor, which coincides with the asymptotic representation (8).

3. DISCUSSION AND ANALYSIS OF RESULTS

We first note that the conclusion of [5] that there is a fundamentally new possibility for detecting the mass of a neutrino

is confirmed by the results obtained above, that is, at the average energy of ultrarelativistic neutrino pairs, which ranges from

mv(E/m Q) to co(E/ma) 3, the probability depends primarily on the neutrino's mass, which ranges from the exponentially small

value (8), (9) to (10), (12). As noted in [5], the very fact of the detection of neutrino radiation will establish an upper mass

limit mv < o:(E/ma) 2. Corresponding estimates of the possibilities for detecting the effect through the F --> Vl~ 2 channel have

been made in [5]. For this reason, it is interesting to compare the relative contributions of the channels discussed above and

the latter channel to the synchrotron emission mechanism in the region of possible detection rh << 1.

We first compare the probabilities of emission of neutrino pairs obtained in Secs. 1 and 2. Taking C~ = C; = 0 (i.e.,

in the standard model, for example) to simplify the notation, from (10) and (12) we find

W(F-+vl~z~) 2.10 -2 Q2 7_', (13) V¢" (FB --> "~1 ~)

i n ( I /m)

where

- (--~-B'/(-~-E t, BQ= mS Z=\BQ]\m ° ] IQI"

Since we always have ~ << 1 for reasonable values of E and B, the contribution of the F --> v1923, radiative process can be neglected in any case.

Now using Eq. (15) from [5], we estimate the relative contributions of the four- and three-particle peaks of the one-loop approximation to the synchrotron mechanism of neutrino generation:

W(FB~,~q)_ = 4 , 6 In(1/m) KN, W (F-+ ~1 ~2)

m~ (C~ + C~) (14) ;: (no)' \ me, eg (a s + b 2 )

while the form of the structure constants a and b has been given, for example, in [5]. The value of the expression (14) is

determined mainly by the factors K and N. For neutrino emission in the same mass states or in the absence of mixing, we have

N - 1° and by virtue of the smallness of K in existing and planned proton machines ( - 10-(9-12)), the contribution of the four-

particle peaks is negligible. And if the neutrino states differ because of mixing, then we have the estimate

N - (mw/ml) 4,

698

where m l is the mass of the heaviest charged lepton. For the three known lepton populations, we have I = r and N - 106 with

KN - 10 -3 in the best case. Thus, only upon a substantial change in the characteristics of proton storage rings with an increase in 2 by several

orders of magnitude might it become necessary to allow for the contribution of the four-particle peaks.

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