V. A. Gusseinov

Abstract: One of the basic directions of modern physics of elementary particles is the study of interactions of high-energy particles in intense external fields. The successes of cosmology and astrophysics confirm existence of strong gravitational fields near to black holes and strong magnetic fields in neutron stars - pulsars. The value of magnetic fields in the magnetosphere of pulsars reaches the value H³ H₀ , where H₀= m_e²c³/ eh = 4.41´ 10¹³ Gs is the characteristic quantumelectrodynamic intensity of a field.

Investigations of influence of strong external fields on interactions of particles of high energy became possibly in laboratory conditions last years.

The intensities of the effective external fields inside of the crystals are much more stronger than the intensities of the fields received by traditional methods. Thus when passing the ultrarelativistic particles through crystals the strong fields inside of the crystals exert essential influence to processes of interactions of elementary particles [ 1] .

The intensities of the fields achieved in laser radiation (H~ 10^7¸ 10⁸ Gs) less than the critical field H₀. Despite of it, if the particles of high energy participate in a process, in the system of readout connected to the particle, the magnetic field is increased in E/mc² in comparison with a field in laboratory system.

The study of processes of interactions of elementary particles in a magnetic field represents the special interest. Unlike an electrical field the magnetic field does not do work on a particle. As the forse of Lorents is always perpendicular trajectories of particles. For this reason the vacuum is stable concerning a magnetic field , i.e. even at H³ H₀ the magnetic field does not lead to appearance of pairs from the vacuum. From this point of view the check of the various theories represents the interest in case of extreme fields.

In a magnetic field at the expense of quantum effects the new channels of reactions even arise at the energy below threshold energy of the free case. Here the process is considered in a constant uniform magnetic field. It is possible precisely to calculate the cross section of this process. In the work [ 2] the annihilation of neutrino pair into electron-positron pair has been considered in a crossed field. If we have dealings with and pair or and pair neutral currents only take part in these processes. If we have dealings with and pair both neutral and charged currents take part in the process. Unlike the process charged currents only take part in the process .

Investigation of the asymptotics of the cross section of the process in various limiting cases represents the interest.

In the free case the considered process has the threshold

(1) ,

i.e. kinematically it is permitted in the area

(2) ,

where k m = 2q₁q₂/ mm ² , m_e is the mass of an electron, mm is the mass of a muon, q₁ and q₂ are 4-momenta of the muon neutrino and the electron neutrino, accordingly.

We research the influence of an external field on the process in the area of a comparatively low energy, when 4 - fermions approach of the standard model of Weinberg-Salam-Glashow is applicable . According to the Feynman rules the amplitude of the process in the considered approach of the standard model of an electroweak interaction is

(3) ,

where , , are the Dirac matrices, u(q₁) is the bispinor of the muon neutrino, is the bispinor of the electron antineutrino, is the current of the charged leptons:

(4) .

y _e is an exact wave function of the charged lepton l=m ^-,e⁺ in a constant external field ; q=q₁+q₂; L is the normalizing length.

As a constant external field we choose a constant uniform magnetic

field. We take the exact wave function of a charged lepton in cylindrical coordinates [ 3] . The gauge of the potential Am of the constant external field H­ ­ Oz is chosen as :

(5) Am = ( 0 , 0, xH, 0) .

(6) ,

where f = ( n, s, p_z, z ) and f ¢ = ( n¢ , s¢ , p_z¢ , z ¢ ) are the sets of quantum numbers, determining states of an electron and a muon in an external magnetic field [ 3] . n is the principal quantum number determining a value of a transverse momentum ; s is a radial quantum number; p_z determines a projection of a momentum to a direction of a magnetic field H; z is the spin number which determines the state with orientation along ( z = + 1) or against ( z = - 1) the orientation of the field. e ₁, e ₂, w , w ¢ are the energies of the muon, the positron, the muon neutrino and the electron antineutrino, accordingly. E = w + w ¢ is the total energy of the neutrinos. The obvious form of the currents ja has been shown in the work [ 4] .

We shall consider the case when the muon neutrino flies along the axis Ox and the electron neutrino flies under an arbitrary angle to the external field. Let the muon and the positron are produced in the ground state , i.e. n= 0, n¢ = 0. In order the muon and the positron not to be produced in the first exited state the following condition should be carried out:

(7) .

After summaring up on all quantum numbers of the final charged particles we receive the following expression for the cross section

(8) ,

where

(9) x,

When the intensity of the external magnetic field is comparatively strong the argument x of the exponential is x<<1. In this case the exponential can be replaced by its first main member, i.e. e^-x» 1 . Then the cross section of the process will accept the form

(10) .

It means that with growth of intensity of an external field the cross section of the process increases linearly. For the head-on neutrino collisions the answer is simple:

(11)

It is derived that with growth of the total energy of the initial neutrinos the cross section decreases on a quadratic law. It means that with growth of the initial total energy producing of the muon and the positron in the exited states becomes possibly, i.e. producing of these particles in the ground state becomes more difficult.

Now let us discuss the case when the muon neutrino flies along the axis Ox and electron neutrino flies against the axis Ox. We consider that the charged leptons have large transverse momenta in the magnetic field H<< Hm = mm ² /e, i.e. the motions of the charged leptons are quasi-classical. (We have to note that the signature of the metrics is (+ - - - ) and we use the system of units where

h =c=1). In this case the principal quantum numbers of the charged leptons are n >>1, n´ >>1.

Let us enter the spectral variable

(12) ,

the kinematic parameter

(13)

and the field parameter

(14) c m ,

where is the tensor of the external field.

The spectral distribution of the cross section is

(15) ,

where

;

is the derivative of the Aire function,

It represents the interest to investigate the cross section of the process at the very large and small values of the parameter c m .

Let us consider the cross section of this process at small c m << 1. There are two opportunities here:

1. k m > k ₀ - the region which is above the threshold;
2. k m < k ₀ - the region which is below the threshold.

when c m << 1 and k m >> k ₀ .

(16) .

The magnetic field gives the very small contribution to the cross section. In this case if to take into account the spin member, the contribution of polarization effects to the cross section should be more essential than the contribution of the only magnetic effects [ 5] .

The asymptotics of the cross section has the following form in the region which is below the threshold of the free case, i.e. when c m << 1 and k m << k ₀

(17) .

In spite of the fact that the process in the free case in the region k m <k ₀ does not go, the section in this case is distinct from zero but it is exponentially small.

From here follows that in a magnetic field for threshold processes the new channels of reactions even below threshold of a free case open.

At last we receive the following asymptotics of the cross section when c m >> 1

(18) ,

Let us consider the condition of applicability of 4-fermions approach used above. The modulus of the square of the transmitted 4-momentum has to be relatively small: |q²|_eff << m_w² , where m_w is the mass of W-boson. Using the formula (13) and the fact we can obtain the following condition

(19) » 5.88´ 10⁵ .

We can write the formula (11) obtained above in relativistic-invariant form.Using

(20)

and

(21)

we can write the formula (11) as follows

(22) ,

where

(23)

And now we shall estimate the opportunity of observation the effects of the external field. The influence of an external field on the given process is determined with parameter

(24) .

The field effects become essential at h ³ 1.

If we put w = 20 GeV (muon neutrinos used in the experiments on inverse muon decay [ 6] ), H=10⁸ Gs (impulse magnetic fields, effective internal fields of monocrystals [ 7] ) then we obtain the estimate h » 10^-9 . With w =100 MeV and the other parameters used above, we obtain h » 6´ 10^-5.

These numerical estimations show that the field effects are substantial in that case when we have dealings with the low-energy neutrinos. It allows to register the low-energy neutrinos.

Thus the strong magnetic field substantially changes the cross section of the considered process having the threshold in absence of the field and results in characteristic quantum effects that can be used for check of the theory of electroweak interactions in extreme external fields and in astrophysics.

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

The cross section of the process øòin an external constant uniform magnetic field is calculated in the framework of the Weinberg-Salam-Glashow model. The asymptotic behaviour of the cross section as a function of the kinematic and field parameters is investigated. The possibility of the new reaction channel in the region under the threshold of the free process is shown. The numerical estimations show that the field effects become especially substantial in that case when we have dealings with the low-energy neutrinos. This opens a new possibility for experimental registration of the low-energy neutrinos.

References

[1] Bayer V. N., Katkov V. M. and Strakhovenko V. M., Uspekhi Fiz. Nauk, 159 (1989) 455.

[2]Lyulka V. A., Yad. Fiz., 42 (1985) 1211.

[3] Sokolov A. A. and Ternov I. M., Relativistic electron (Nauka, Moscow) 1983, chapter 4.

[4] Borisov A. V. and Gusseinov V. A., Yad. Fiz., 57 (1994) 496.

[5] Borisov A. V., Gusseinov V. A. and Pavlova O.S., Yad. Fiz., 61 (1998) 103.

[6] Vilain P. et al., Preprint CERN- PPE / 96- 01.Geneva. 1996.

[7] Bayer V. N., Katkov V. M. and Strakhovenko V. M., Results of a science and engineering. In ser. Beams of the charged particles and solid (VINITI, Moscow), 4 (1992) p. 57.

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