ABSTRACT
The cross section for the production of ultrarelativistic electron-positron pair in muon neutrino and antineutrino annihilation in an external electromagnetic field is calculated. The asymptotic behavior of the cross section as a function of the kinematical and field parameters is studied. It is shown that the new channel of the reaction arises at the expense of quantum effects. For the strong magnetic field case the cross section of the process does not depend on the masses of the charged leptons. The contribution of the weak external field to the cross section of the process is very small. It is shown that the considered process can be one of the possible sources of cosmic electrons and positrons of high energy.

The study of high-energy particles in intense external fields is one of the basic directions of modern particle physics. The intensity of the magnetic field in the magnetosphere of neutron stars - pulsars reaches the value H³H₀, where  Gs [ 1] is the characteristic quantum electrodynamic (Schwinger) intensity of a field. Investigations of influences of strong external fields on interactions of particles of high energy have become possibly in laboratory conditions in recent years.

When passing the ultrarelativistic particles through the monocrystals the strong fields inside of the monocrystals [ 2] exert essential influence on processes of interactions of elementary particles.

The purpose of this work is to investigate the influence of an external electromagnetic field to the process

. (1)

Production of electron-positron pair in neutrino-antineutrino annihilation is a purely leptonic process. Its cross section can be calculated precisely. As we have dealings with  and  pair, neutral currents only take part in the process. Production of an electron-positron pair in a neutrino-antineutrino annihilation has been considered in the limiting case of a crossed electromagnetic field (EH=E^2-H²=0) in [ 3] . It is well-known that in the case of ultrarelativistic particles a constant crossed electromagnetic field simulates an arbitrary constant electromagnetic field of a comparatively small strength F<<H₀ [ 4] . Here F is the strenght of the constant field (electrical or magnetic) produced by the crossed electromagnetic field. In [ 3] the general expression for the cross section of the process has been obtained as the function of the kinematical parameter

(2)

and the field parameter

, (3)

where q_{1, q2} are 4-momenta of a muon neutrino and antineutrino, which we consider massless, m_e is the mass of an electron or a positron, Fmnis the tensor of the external electromagnetic field. In general, it is difficult to integrate the expression received for the cross section of the process. But the investigation of the cross section of the process at the very large and small values of the field parameter represents the special interest for the concrete applications, especially in astrophysics.

Taking into account that the general expression for the cross section of the considered process has already been derived (see [ 3] and the references there) we do not revise the calculations. Therefore, let us write the expression for the cross section of the process:

, (4)

where

, (5)

, (6)

, (7)

, (8)

, (9)

wand w¢ are the energies of the muon neutrino and antineutrino, accordingly. G_w is the Fermi constant, u is the spectral variable [ 3,5] . Here

and . (10)

For  -scattering

; , (11)

where  is the Weinberg angle and  »0,23 [6].

F (z) is the Airy function

(12)

and it depends on the argument

. (13)

F¢ (z) is the derivative of the Airy function on the argument z and .

And now let us consider the asymptotic behavior of the cross section of the process at the various values of the parameter c .

When c>> 1 the region |z|<<1 gives the main contribution to the integral (5). In this case we use the following values of the functions F¢ (z) and F₁(z):

,

. (14)

In consequence, we can write the asymptotic expression for the cross section of the process in a strong field case

. (15)

The main contribution to the cross section of the process is given by the term which is proportional to :

. (16)

Let us suppose, for instance, the neutrino flies along the axis Ox  , the antineutrino flies against the axis Ox and we have chosen a uniform constant magnetic field H||Oz. Using these facts, we can rewrite the parameters (2) and (3) as follows

, (17)

, (18)

where . We use the pseudoeuclidean metric with the signature (+ - - - ) and the system of units where . If we put (17) and (18) in the expression (16), then we can be convinced of independence of the cross section of the process from the mass of the particles in the strong magnetic field (compare with the corresponding results of [ 7,8] ).

When c<< 1 and k>> 1, the following asymptotic expansions of the Airy functions are true [ 9] :

,

(19)

,

where  is the Dirac delta function and A is the parameter

. (20)

Using (19) we obtain the asymptotics for the cross section in the weak external field case

. (21)

The term proportional to c² determines the contribution of the external field. In an external field the cross section of the process is substantially distinguished from the cross section of the free process even in the case of a comparatively weak field (compare with pion decay in an external field [ 4] ). In (21) the terms which do not depend on c correspond to the cross section of the free process (see [ 10] and compare with [ 3] ).

It should be noted that if to take into account the polarization effects which are determined with the term proportional to F (z), then the contribution of the polarization effects will be more substantial than the own contribution of the external field.

Now let us consider the region where c<< 1 and k<< 1. The free process is forbidden in the region k<< 1. In this case the argument z of the Airy functions is much greater than one. To calculate of the cross section of the process we can use the following known asymptotic formulae of the Airy functions (z>>1)

,

(22)

.

To integrate on the spectral variable u we use the method of steepest descent. The expansion of the argument z(u) in the vicinity of the extremal point u₀ is as follows

, (23)

where .

In consequence, we obtain the asymptotic expression for the cross section of the process when c<< 1 and k<< 1

. (24)

In spite of the fact that the process (1) is forbidden in the region k<< 1 but the cross section of this process in an external field in that region is distinct from zero. The cross section is exponentially small. This situation is typical for the threshold processes [ 4, 7, 11] . Decreasing of the cross section of the process on exponential law at c® 0 shows impossibility of producing electron-positron pair in muon neutrino and antineutrino annihilation in the classical limit. Thus, arising of the new channel of reaction occurs at the expense of quantum effects.

Now we can estimate the opportunity of observing of the effects of the external field. The influence of the external field on the considered process is determined with the parameter (compare with [ 7] , [ 8] , [ 11] )

, (25)

where  is the threshold of the considered process in absence of the external field.

The field effects become significant for h³ 1.

If we put H=10⁸ Gs [ 2] and w¢=10 MeV (reactor or solar neutrinos), then we obtain .

The influence of the external field will be strong in the very narrow region of the value of the kinematical parameter  :  . For the strong influence region we can write

. (26)

For w¢ =1 MeV we obtain that w»0.26 MeV which is in the boundary of the detectable region. To get into the strong influence region the parameter  has to be trimmed precisely . From (26), it follows that whas to be chosen for given w¢. But it is impossible. Because we can not change w. Therefore, it is impossible to observe the effects of an external field in laboratory conditions. On the other hand, because of the low density of the neutrino bunch it is difficult to observe the effects of an external field in laboratory conditions.

And now let us discuss the astrophysical applications of the results obtained. Nowadays various processes of inelastic scattering of cosmic neutrinos (antineutrinos) of ultrahigh energy on low energy relic antineutrinos (neutrinos) in the Milky Way Galaxy are considered as a possible source of cosmic rays (see [ 12] and references there). We can estimate the energy (temperature) of the relic radiation assuming that relic neutrinos are massless:

.

Our results are applicable in the region where

.

Then we obtain the following condition for the field parameter c

For the mean-galactic magnetic field (H~10^-6 Gs) we obtain . For the magnetic field of neutron stars the field parameter c is

.

The cross section of the process  in a magnetic field increases considerably compared with the cross section of the free process at the expense of the factor  (see (16)):

.

Thus, the process  can be one of the possible sources of cosmic electrons and positrons of high energy.

I would like to thank the Professors I.G. Jafarov, V.Ch.Zhukovskii and A.V.Borisov for many helpful discussions on this work.
 
 

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