Russian Physics Journal, Vol. 37, No. 9, 1994

THE EMISSION OF MASSIVE N E U T R I N O S BY RAPIDLY

M O V I N G C H A R G E S

V. V. Skobelev and I. F. Botsiev UDC 539.1.01

In a mixed model, the process of the generation of a pair of massive neutrinos pitPj by means of the electromagnetic field F of a classical charge moving with a constant acceleration w in its own frame of

reference is examined. In the low-energy approximation a general formula is obtained for the probability of emission with analysis of the important special case mi, j < < co. Possible astrophysical applications are discussed.

In a number of publications [1-5] one of the authors has developed a computation method, the essence of which consists

in a universal invariant of the representation of the probability of emission by an alternating classical field F (electromagnetic

or gravitational) of some set of particles {f} of a concrete field model based on the corresponding effective Lagrangian of the

interaction

L = L(F, {[}). (1)

Thus, in [4] the emission of massless neutrino pairs by an alternating electromagnetic field induced during the

accelerated motion of classical Coulomb centers was examined. The effective Lagrangian used, in satisfying the requirements

of invariance, as is now clear, in fact contained an extra degree of the small energy parameter in comparison with the effective

Lagrangian of the now accepted combined Gleshau-Weinberg-Salama (GWS) model. Moreover, according to current ideas it

is necessary to take into account two additional important circumstances. First, neutrinos are above all massive particles, as

is indicated both by investigations of the uj-decay spectrum in laboratory experiments and by arguments of an astrophysical

character associated with the problem of the latent mass of the universe. Second, neutrinos participating in weak-electrical

interactions may not be the fundamental states of the uj with a particular mass, but rather associated with the latter through the

unitary transformation

v~ -- 2]U~jvj, j (2)

where U is a mixing matrix. Thus, a more general formulation of the problem is to determine the probability of emission by

an alternating electromagnetic field F, taken into account in the first Born approximation, of a pair of massive neutrinos of the

i sort (with a mass mi) and of the j sort (with a mass mj). The corresponding effective low-energy Lagrangian has the form [6]

L~j = [~'~ (aq + 0ij "r ~) ~°~ ~Fj] F,:, + ~. c., (3)

where ,Iqj are the operators of the Dirac neutrino field; aij, bij are structural constants, the value of which depends on the details

of the excitation of the vacuum states in the given model. For example, in the single-loop approximation of the GWS model,

their form is established by the relation (i ¢ j)

3eO , {bl~f} = -T-64---~2xu (mj-:-mi) ~,~ U~ Ujz( m' I s. (4)

~=e.~... \m~v /

Moscow State Open Pedagogical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 89-92, September, 1994. Original article submitted July 28, 1993.

880 1064-8887/94/3709-0880512.50 © 1995 Plenum Publishing Corporation

where m I are the masses of the charged leptons and G is the Fermi constant, and the summation is carried out over the lepton

generations. For the sake of simplicity we will assume that

i-+1, j-+2, aiF+a, bu--~b, (5)

regarding the generation process F ~ Vl.5 2.

Omitting the details of the transformations, we present the result for the total probability of emission of a pair of Dirac neutrinos:

W - - 2 (~ d ~ s¢ (2~):' j ,~ Fo~ (~) F~," ( - x ) A ( ~ ) , (6)

A (~-) = [~" - -2~ ' (m l + m ~ ) + (m ~, - - r a ~ ) q , 2 ×

X (a ~ + b 2) 6zT [ ~ + z'2 (m~ + m~) - - 2 (m? - - m~) 21 + m, m,2 (a z - - b ' ) . (6a)

Here F~(K) is the Fourier form of the field tensor and r is the overall impulse of the pair, while the area of integration is de-

termined by the condition K 2 > (m I + m2) 2. As a result of the low-energy character of the approximation, it is assumed that

the effective value of K contributing to the integral is much less than the mass of the lightest of the particles of the vacuum

of the model. For example, in the single-loop approximation of the GWS model and the majority of the other models, it should

be the case that K2eff < < m2e.

Expression (6) can also be represented through the currents j generating the field F, since

F=~ (tQ F ~ ( - - to) = - - 32r. 2 1 j.~ ( z ) j ~ ( _ ~c). (7) K"

Here we will consider the case of the generation of neutrino pairs during the one-dimensional (along the 1 axis) motion

of a classical charge q with constant acceleration w in its own frame of reference (this corresponds to the invariant formulation

of the problem). Three-photon emission in this situation was examined previously in [3], where a type of Fourier form of the

current was presented. Using this result, we find

,~3 [~j2 ,,j K,l r.#.)

where K 1 is the MacDonald function, and the total probability is reduced to the time interval (0, co), when the velocity varies

from zero, thereafter tending to the speed of light (in [3] determining the intensity of the emission).

It is expedient to introduce the dimensionless variables

/¢o /¢1 l • = - - , ~, = - - , ~ , = - - (x~ + tc~) ~/2 - ( 9 ) LU /~,'o Ko

and the cylindrical angle ,p of the vector K with the area of integration

(ml,2 = ml,2/w.

(9a)

After the transformations we obtain a probability in the energy interval (rh, rE)

~ X W -

(l--m2/v~) U2 ~E

. " 2 V K'~ (z'/2) X o 7~,(1 _,~)],,2 (10)

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X (a2 _}_ b2) __~z _k_ m, m~ (a~ _ b~ ) _ Rl~..}_ z m~~ .+ m] t n A l + z l n A 2 ;

rn~ -m~

z='~2(1-- '~),R=z ~-2z(m~ +,~,~)+(m~--m]) 2,

A 1 ~-

I I

~t Az -- m

I [ ( m , ~ +, 'Tn~) z - - ( m ~ - - ,~.~)= - (Tn, ~ - ,~,~) R ' 2 I ,

2 t n I /7 l 2 Z

1 [z - ( r ~ + m D + R~:2I, m, > m~. 2m, me

(lOa)

The total energy of the emitted pairs in the interval in question is obtained by adding the factor vco in the integrand;

as a result of the low-energy character of the approximation it is necessary, generally speaking, to assume that v E < < me~CO.

Explicit computation of the integrals in expression (10) is impossible, and so, because of the assumed smallness of the

mass of the neutrinos, we will consider the special case r~ 1,2 < < 1 (m 1 = m 2 = 0). Then from (10) we obtain

A I V = 4 - - ~ - - q 2 w 2 ( a ~ + b ' ) 3 d ' q ( 1 j ' (11) 3=2 - - , ~ ) cb. "~3 K ? (z~/2). 0 0

After substitution of the variable and partial integration, the integral reduces to the single

'~E

4 q,.w~ x 2 y 12 ~W 3= z (a ~ + b e) • x '~ arcsin - - - -

0

which is a complex function of u e. Since, however, the acceleration m e ~ 105 krn/sec 2, for reasonable values of CO < < m e,

remaining within the bounds of the low-energy approximation, it is possible to set u E ~ co (the additional contribution is

vanishingly small). The corresponding integral is tabular [7] (6.586), and the total probability of emission of a neutrino pair

is equal to

V~ = 4 (a2.4_ b~)q2 tez. (13) 9r~

It is possible analogously to obtain the total energy of a neutrino emission with a pair energy < m e

2 (14) A E ~ ~' - - m e,

where the value of W is determined by the preceding formula. Assuming for the evaluations q --, e in (13) and

U~t U it -+

in expression (4), we obtain within the framework of the GWS model

A E ~

Let us try to apply this result to obtain cosmological evaluations. For a value of the Hubble constant of -102

Km. s e c - 1 - M p c - 1 the characteristic acceleration is - 10-12 Kin. sec-2, and for an average concentration of several protons

per M 3 and for rnv ~ 1 eV (which will give a too-large result in comparison with the precise (10)), the neutrino component

density obtained from (15) is negligibly small in comparison with the critical density and therefore does not solve the problem of bidden mass.

The role of the process in question in the mechanism of energy loss in gravitational collapse might be more interesting, since acceleration in the fundamental frame of reference will be significantly greater here (by an order of magnitude co -

TpR, where 0 and R are the initial density and the dimensions of the collapsing sphere and y is the gravitational constant),

However, the corresponding evaluations are too indefinite from the point of view both of the ideal of the approximation used

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(the motion of noninteracting charges with w = const) and of the strong dependence on the details of the collapse, the

composition of the object etc.

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