General Formula for Impulse Losses in the Process of Emission of Neutrino Pairs by Electrons in a Magnetic Field

Бюллетень науки и практики / Bulletin of Science and Practice

UDC 539.123                                        

An impulse is transmitted to the stellar conditions by the emission of neutrino pairs by electrons in the process of neutrino synchrotron radiation of electrons in a magnetic field. Impulse transmitted to the scope of the conditions in a momentum unit (impulse losses) is generally calculated by the formula [1–5]:

dP = 1 jn dn i f i n dn, ( i - f, )' M r q dt V _i         f

Here in the dn and dnf scope element of the phase, respectively, the number of initial and final cases, the distribution functions of the particles in f — initial state, the distribution functions of the particles in f, — final state, the momentum unit value of the modulus of \Mif | /3 — process matrix element, 3 — at the moment of the total interaction, in q — reaction it is an impulse transmitted to the conditions by a neutrino. The formulae for the impulse losses transmitted to a single scope of the stellar condition at a momentum unit due to the emission of neutrino pairs by electrons in the process of neutrino synchrotron radiation of electrons in a magnetic field are correct.

dP = 1 faf.^ (1 - h Ж (1 — fv   (1 — fv)

dt V□

Ly     L(3)

dn_e = —dPv —dp_z eyz

2n2

dn'e = Ly-dP'yLT-dP'z 2n2

,      Vdk'

' = ( П

,      Vdk:

dn~ =

^{( 2 n )}

By including here, we obtain the general expression for the impulse losses transmitted to a single scope of the stellar conditions at a momentum unit by the emission of neutrino pairs by electrons in the process of neutrino synchrotron radiation of electrons in a magnetic field.

dP = Gp  LL zd Qt ( 1 - f X¹ - f v ) f e ( 1 - f e W E' + ”' + ® - E ) *

dt   (2п) ^^ V x Xp'y + qy - Py k(pZ + qz - Pz dpydpzdp'ydp'zdkdk'

In (7) and (1) formulae

f. = f e ( E ) =   E - 1                                            (8) e T + 1

Fermi-Dirac distribution of electrons in the initial state, Fermi-Dirac distribution of electrons in f_e, = f_e ‘ ( E' ) final state, chemical potential of p_e electrons, temperature of T — electron gas

} =      1                                                  (9) J V    J V ( ® )     ^ - p e T + 1

Fermi-Dirac distribution of neutrinos, chemical potential of p _v — neutrinos, — temperature of

T V — neutrino gas: 1                                                (10) Jv = Jv   = ^+p e T +1 are the Fermi-Dirac distribution of antineutrinos. We use it in the unit system which is h = c = kB = 1. Here kB is the Bolasman constant. The Q quantity in formula (7) is determined as follows:

Й ®

Q = N αβ J ^α J ^{∗ β}

N αβ = k α k β ′+ k α ′ k β - g αβ ( kk ′ ) + i ε αβµν k ^µ k ′ ^ν                              (12)

Here g _αβ is a metric tensor and ε is an antisymmetric tensor. ( α , β , µ , ν = 0,1,2,3 ) .

The following general expression is true for the R quantity for the arbitrary kinematics of the motion of neutrinos and antineutrons:

Q ωω x x y y z z         ωω x x y y z z 1 +(ωω′-k k′ +k k′ -k k′)× xx   yy   zz             xx   y y   zz           xx   y y   zz

× J ²   + ( ωω ′ - k k ′ - k k ′ + k k ′ )( J ³ ) - 2 ( ω k ′ + k ω ′ ) Re J ⁰ J ¹ - 2 ( ω k ′ + k ω ′ ) Re J ⁰ J ² -

- 2 ( ω k ′ + k ω ′ ) J ⁰ J ³ + 2( k k ′ + k k ′ )Re J ¹ J ^{2 ∗} + 2( k k ′ + k k ′ )Re J ¹ J ³ + z   z           xy   yx            xz   zx

+ 2( k_yk_z ′+ k_zk ′ _y )Re J ² J ³ - 2( k_yk_z ′- k_zk ′ _y )Im J ⁰ J ¹ - 2( k_zk_x ′- k_xk_z ′ )Im J ⁰ J ² +

+ 2 ( ω ′ k - ω k ′ ) Im J ^{1 ∗} J ² - 2( ω ′ k - ω k ′ )Im J ² J ³ + 2( ω ′ k - ω k ′ )Im J ¹ J ³                 (13)

Taking into account that p determines x coordinate of moving orbit of quantum number electron

py x=x =-

0    eH and this changes in this coordinate part

L

- ^x ≤ x = x

_≤ L_x

It is possible to carry out integral for dpy in (7) expression.

∫dp =eHL yx

Integrals for dp ^′ _y and dp ^′ _z are implemented by means of delta functions.

∫ F(...)δ(p′y +qy - py)dp′y = F(...)

∫F(...)δ(p′z +qz -pz)dp′z = F(...)

Indicating the polar angle of antineutrinos (neutrinos) with ϑ(ϑ′) and the azimuthal angle with α(α′) in the spherical coordinate system, the components of the impulses of antineutrinos (neutrinos) can be written as follows, k = ωsinϑcosα,

k = ωsinϑsin α, k = ωcosϑ, k′ = ω′ sinϑ′ cosα′, k′ = ω′ sinϑ′ sinα′,(23)

k′ = ω′ cosϑ′

Here, non-dashed quantities refer to antineutrinos, and dashed quantities refer to neutrinos. Taking (19-24) expressions into account, it is possible to write (13) expression as:

Q = ωω′Q0

The structure of the quantity depends on the polarization of the electrons in the initial and final cases, and we will give it later.

dk = a2 dadQ

dk ′ = ω ′ ² d ω ′ d Ω′

Considering the angle elements of the body in the formula (7), the formula is obtained for the impulse losses transmitted to a single scope of the stellar conditions simultaneously due to the emission of neutrino pairs by electrons in the process of neutrino synchrotron radiation of electrons in a magnetic field:

dp dt

^GF ² ( 2 π ) ⁷

eH f Qo a a’2 2 q(1 - f )(1 - f~)f (1 - f X E' + a'+ a - E )dpzdada'dQdQ’

As we view the effects of polarization, we do not average the spins of the electrons in the initial (final) state.

It is possible to write delta function simplifying the energy conservation as:

δ ( E ′+ ω ′+ ω - E ) = ∑

i

^Ei^Ei ^′

E_i ′ p_zi - E_i p ′ _zi

δ(pz - pzi)

Here E , E ′ satisfies energy conservation while the law of conservation of the third component of the impulse is satisfied by ( p = p ′ + k + k ′ ) . In this case, the impulse losses transmitted by electrons to a unit scope of the stellar conditions simultaneously due to the emission of neutrino pairs by electrons in the process of neutrino synchrotron radiation in a magnetic field are determined by the following formula:

P       ²

—=—^H X I,., i '  , Qa a q(i - v )(1 - f~)f. (I - л dada'dndn’ dt    (2π)7       i  Ei′ pzi - Ei p′zi

Conclusion

During neutrino synchrotron radiation, these gases will be stimulated by the transmission of impulses as a result of the emission of neutrino pairs. However, a gas consisting of polarized electrons in the direction of the magnetic field and spins composed of polarized electrons in the opposite direction of the magnetic field would receive a different impulse due to the asymmetric transmission of the impulse.