Neutrino interaction with strongly magnetized electron-positron plasma

Journal of Experimental and Theoretical Physics, Vol. 91, No. 4, 2000, pp. 748–760.Translated from Zhurnal Éksperimental’no

œ

i Teoretichesko

œ

Fiziki, Vol. 118, No. 4, 2000, pp. 863–876.Original Russian Text Copyright © 2000 by Kuznetsov, Mikheev.

PLASMA,GASES

Neutrino Interaction with Strongly Magnetized Electron–Positron Plasma

A. V. Kuznetsov* and N. V. Mikheev**Yaroslavl State University, Yaroslavl, 150000 Russia

*e-mail: [email protected]**e-mail: [email protected]

Received April 27, 2000

Abstract—A study is made of a complete set of neutrino–electron processes in a magnetized plasma. It is

shown that processes involving neutrinos in the initial and final states and ν νe–e+ havekinematic amplification in the ultrarelativistic limit. Relatively simple expressions are obtained for the proba-bility and average neutrino energy–momentum loss which are convenient for quantitative analysis. It isobserved that the total contribution of νe processes did not depend on the chemical potential of the magnetizedelectron–positron plasma. © 2000 MAIK “Nauka/Interperiodica”.

νe+− νe+−

1. INTRODUCTION

Over the last few decades one of the most rapidlydeveloping physical sciences has been cosmoparticlephysics which lies at the junction between the physicsof elementary particles, astrophysics, and cosmology[1, 2]. The most important stimulus to its developmentwas understanding the important role of quantum pro-cesses in the dynamics of astrophysical objects and alsoin the early universe. However, the extreme physicalconditions existing inside these objects and, specifi-cally, the presence of a hot dense plasma and strongelectromagnetic fields should have a strong influenceon quantum processes. As a result, studies of the inter-actions of elementary particles in an external activemedium have attracted ongoing interest.

So far, essentially one-dimensional problems havebeen solved in astrophysical calculations of processessuch as supernova explosions, and analyses of the influ-ence of the active medium on quantum processes haveonly contained the plasma contribution. However, seri-ous arguments have been put forward to suggest thatthe physics of supernovas is considerably more com-plex. In particular, we need to allow for rotation of theshell and also for the possible existence of a strongmagnetic field, with these two phenomena being inter-related. In fact, the magnetic field generated during thecollapse of a supernova nucleus may reach the critical

Schwinger value Be = /e . 4.41 × 103 G.1 The pres-ence of rotation may increase the magnetic field by anadditional factor of 103–104 [3].

In astrophysical phenomena such as stellar collapse,the absence of strong magnetic fields is an exotic rather

1 We use the natural system of units c = " = 1. Everywhere in thearticle e > 0 is the elementary charge.

me2

1063-7761/00/9104- $20.00 © 20748

than a typical case. It is appropriate to discuss the fol-lowing set of questions.

(1) Which can be considered to be the more exoticobject: a star possessing a magnetic field or a star with-out such a field? As far as we know from astrodynam-ics, a star without a magnetic field should be taken asan exotic rather than a typical case. In exactly the sameway the presence of a primary magnetic field may beconsidered natural for a presupernova. As we know, aprimary magnetic field of 100 G leads to the generationof a field on the scale of 1012–1013 G during the collapseprocess as a result of the conservation of magnetic flux.

(2) Which can be considered to be the more typicalcase: a star possessing rotation or a star without rota-tion? Evidently a star without rotation appears to be themore exotic object.

(3) Which type of collapse looks more exotic: com-pression without or with an angular velocity gradient?Since the velocities at the edge of a compressible astro-physical object may reach relativistic scales, compres-sion with differential rotation, i.e., with an angularvelocity gradient, seems more probable.

All these factors are required to achieve the Bisno-vatyi-Kogan scenario for the rotational explosion of asupernova [3]. The main component of this scenario isthat the initially poloidal magnetic field lines of a fieldof 1012–1013 G are twisted and compacted as a result ofthe angular velocity gradient to form an almost toroidalfield of 1015–1017 G.

We stress that this field is in fact a very densemedium having the mass density

(1.1)ρ B2

8π------ . 0.4 1010

g

cm3--------- B

1016 G----------------

2

,×=

000 MAIK “Nauka/Interperiodica”

NEUTRINO INTERACTION WITH STRONGLY MAGNETIZED ELECTRON–POSITRON PLASMA 749

which is comparable with the characteristic mass densityof the shell of an exploding supernova, 1010–1012 g/cm3.Thus, in detailed studies of astrophysical processessuch as supernova collapse it is absolutely essential totake into account the influence of the complex activemedium including the plasma and the magnetic field.

We know that neutrino physics plays a decisive rolein astrophysical cataclysms such as supernova explo-sions and coalescence of neutron stars, and also in theearly universe. Consequently, studies of neutrino inter-actions and in particular neutrino–electron processes inan external active medium are of considerable interest.At the same time, an investigation of neutrino processesunder such extreme physical conditions is interestingfrom the conceptual viewpoint since it affects funda-mental problems of quantum field theory.

The first studies of neutrino–electron processes inan external electromagnetic field were devoted to the“synchrotron” radiation of neutrino pairs, e eν[4] and neutrino creation of electron–positron pairs,ν νe–e+ [5]. The analysis was made under condi-tions of a so-called relatively weak magnetic field whenthe initial particle energy is the dominant parameter,E2 @ eB; in fact, this limit corresponds to the crossedfield approximation. In our studies [6, 7] the ν νe–e+

process was investigated for arbitrary values of themagnetic field and in particular in the strong field limitE2 @ eB when an electron and a positron can only becreated in states corresponding to the Landau groundlevel. A canonical neutrino–electron interaction chan-nel, νe– νe– scattering, was investigated in [8]under conditions of a degenerate electron plasma tak-ing into account the influence of a relatively weak mag-netic field.

In the present study we show that a correct analysisof the neutrino propagation process in a hot denseplasma in the presence of a strong magnetic fieldrequires us to consider the complete set of neutrino–elec-

tron processes. Specifically, in addition to the

scattering reactions which also take place in theabsence of a field, and the ν νe–e+ pair creationprocess which is only possible in a magnetic field, wealso need to take into account the “exotic” processwhen a neutrino captures an electron–positron pairfrom the plasma: νe–e+ ν. This process is onlyallowed when both a magnetic field and a plasma arepresent. Then only the probability of the processsummed over all initial states of the plasma electronsand positrons is physically meaningful. The probability

of the scattering channels is defined sim-ilarly as the sum over all e– or e+ initial states. The totalprobability of neutrino interaction with an electron–positron plasma in a magnetic field is made up of theprobabilities of these processes.

The article is constructed as follows. In Section 2 wedescribe the concept of a strongly magnetized electron–

ν

νe+−

νe+−

νe+− νe+−

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHY

positron plasma and justify the feasibility of thesephysical processes occurring under the physical condi-tions during stellar collapse. In Section 3 we obtain thetotal amplitude of the neutrino–electron processesunder conditions of a strongly magnetized electron–positron plasma in the local ννee interaction limitderived from the standard model of electroweak inter-actions. A kinematic analysis is made and it is shownthat those neutrino–electron processes in which a neu-

trino is present in the initial and final states:

and ν νe–e+ exhibit kinematic amplification.Section 4 is devoted to a detailed description of the pro-cedure for calculating the probability of the ν νe–e+

process in a strongly magnetized electron–positronplasma. In Section 5 we give the probabilities of otherνe processes and obtain the total probability of neutrinointeraction with a magnetized e–e+ plasma. In Section 6we calculate the average energy and momentum lossesof a neutrino propagating through a magnetizedplasma. In Section 7 we calculate the characteristics ofthe integral action of a neutrino on a magnetizedplasma: the volume density of the energy transferred

from the neutrino to the plasma per unit time, , andthe volume density of the force acting on the plasmafrom the neutrino. An analysis is made of possibleastrophysical manifestations of neutrino–electron pro-cesses under these extreme physical conditions. It isshown that these processes may be important for adetailed description of the evolution of astrophysicalobjects.

2. WHAT WE UNDERSTAND BY STRONGLY MAGNETIZED e–e+ PLASMA

Here we are talking of conditions where, among allthe physical parameters characterizing an electron–positron plasma, the field parameter is the dominantone. These conditions can be characterized simply bythe relationship: eB @ µ2, T2, where µ is the chemicalpotential of the electrons and T is the plasma tempera-ture. In order to find a better substantiated relationshipwe compare the energy densities of the magnetic fieldB2/8π and the electron–positron plasma.

As we know, a magnetic field changes the statisticalproperties of an electron–positron gas [9]. Taking intoaccount degeneracy of the transverse momentum, thedependences of the concentration and energy density ofan electron–positron gas on the chemical potential andtemperature are described by the following sums overLandau levels:

(2.1)

νe+−

νe+−

%

n ne

– ne

+–=

= eB

2π2-------- p Φ p µ T, ,( ) Φ p µ T,–,( )–[ ] ,d

0

∞

∫

SICS Vol. 91 No. 4 2000

750 KUZNETSOV, MIKHEEV

(2.2)

(2.3)

Here we used the approximation of an ultrarelativisticelectron–positron gas since astrophysical processes arecharacterized by fairly high neutrino and plasma elec-tron energies E @ me. Thus, we shall neglect the elec-tron mass wherever this causes no misunderstandings.

In a strong field and specifically, when the condition

– µ @ T is satisfied, in practice only the Landauground level is filled. From (2.1) and (2.2) we thenobtain

(2.4)

(2.5)

Thus, a more exact condition that the electron–positronplasma is strongly magnetized may be written in theform

(2.6)

Selecting values of the physical parameters typical of asupernova shell as scales in the relationship (2.6), werewrite this in the form

(2.7)

where

(2.8)

ρ is the total plasma density in the shell, and Ye is theratio of the number of electrons to the number of bary-ons. It can be seen that the plasma magnetization con-dition is definitely satisfied.

% %e

– %e

++=

= eB

2π2-------- p p Φ p µ T, ,( ) Φ p µ T,–,( )+[ ] ,d

0

∞

∫

Φ p µ T, ,( )p µ–

T------------

exp 1+1–

=

+ 2p2 2keB+ µ–

T--------------------------------------

exp 1+1–

.k 1=

∞

∑

eB

neBµ2π2----------,=

% eBµ2

4π2------------

eBT2

12------------.+=

B2

8π------ @

π2n2

eB----------

eBT2

12------------.+

0.8 1032× B32

@ 1.7 1030×ρ12

2 Y0.12

B3---------------- 1.1 1027B3T5

2 erg

cm3---------

× ,+

B3B

103Be

--------------, ρ12ρ

1012 g/cm3--------------------------,= =

Y0.1

Ye

0.1-------, T5

T5 MeV-----------------,= =

JOURNAL OF EXPERIMENTAL

3. NEUTRINO–ELECTRON PROCESSESIN A STRONGLY MAGNETIZED PLASMA

3.1. Total Amplitude

Calculations will be made for the relatively small

momentum transfers |q2| ! , where mW is the Wboson mass. An analysis shows that when studying theprocesses in a magnetized plasma we need to add the

conditions @ eB and @ eBT, eBµ. Weak neu-trino–electron interaction can then be described in thelocal limit by the effective Lagrangian

(3.1)

where gV = ±1/2 + 2sin2θW, gA = ±1/2. Here the uppersigns refer to an electron neutrino (ν = νe) whenexchange with both Z and W bosons contributes to theprocess and the lower signs corresponds to muon andtau-neutrinos (ν = νµ, ντ) when only exchange with a Zboson contributes to the Lagrangian (3.1).

The Lagrangian (3.1) is written assuming masslessneutrinos and consequently the absence of mixing inthe lepton sector. A generalization to the case of mass-possessing neutrinos with mixing can be found, e.g., in[10]. However, it should be noted that the kinematics ofcharged particles is a magnetic field is such that allinteraction processes of high-energy neutrinos withelectrons become insensitive to lepton mixing and arepossible even in the massless neutrino limit. Thisimplies that the flavor of an ultrarelativistic neutrino isconserved in these processes in a magnetic field to

within terms of the order / regardless of the lep-ton mixing angles which makes the question of neu-trino mixing irrelevant.

The total amplitude for neutrino–electron processesis obtained directly from the Lagrangian (3.1) whereknown solutions of the Dirac equation in a magneticfield must be used for an electron and positron. As weknow (see, e.g., [11]), these solutions for the Landauground level may be expressed in the following form[the magnetic field is directed along the z-axis and thevector potential is taken in the form A = (0, xB, 0)]:

(i) for an electron of energy ω and “momentum” ky, kz2

(3.2)

2 Here kz is the kinetic momentum along the z-axis, and ky is thegeneralized momentum which determines the position of the cen-ter of a Gaussian packet along the x axis, and x0 = –ky/eB, see(3.4).

mW2

mW2 mW

3

+GF

2------- eγα gV gAγ5+( )e[ ] νγα 1 γ5+( )ν[ ] ,=

mν2 Eν

2

ψkeB( )1/4

π2ωLyLz( )1/2------------------------------------=

× i ωt kyy kzz––( )–[ ] ξ2

2-----–

u k ||( ),expexp

AND THEORETICAL PHYSICS Vol. 91 No. 4 2000


(ii) for a positron of energy ω' and “momentum”,

(3.3)

where k||( ) is the electron (positron) energy–momen-tum vector in the Minkowski (0,3) plane. Here and sub-sequently Lx, Ly, and Lz are auxiliary parameters whichdetermine the normalization volume V = LxLyLz, ω =

, ω' = , m is the electron mass, andξ and ξ' are dimensionless coordinates which describethe motion along the x-axis:

(3.4)

The bispinor amplitudes have the form

(3.5)

where Ψ = corresponds to a state whose spin is

directed opposite to the field. It is interesting to notethat the bispinor amplitudes (3.5) are exactly the sameas the solutions of the free Dirac equation for an elec-tron and positron having momenta directed along thez-axis. This separation of bispinor amplitudes which donot depend on the spatial coordinate x is only typical ofthe Landau ground level.

Using the Lagrangian (3.1) and the wave functions(3.2), (3.3), and (3.5), we write the S-matrix element ofthe process ν νe–e+ in the following form (theamplitudes of the other neutrino–electron processes arethen obtained by crossing transformations):

(3.6)

where q = p – p' is the change in the four-vector of theneutrino momentum, q⊥ is the projection of the vector q

ky' kz'

ψk'eB( )1/4

π2ω'LyLz( )1/2-------------------------------------=

× i ω't ky' y kz' z––( )–[ ] ξ '2

2------–

u k ||'( ),expexp

k ||'

m2 kz2+ m2 kz'

2+

ξ eB xky

eB------+

,=

ξ' eB xky'

eB------–

.=

u k ||( )1

ω m+------------------

ω m+( )ΨkzΨ–

,=

u k ||'( )1

ω' m–-------------------

ω' m–( )Ψkz'Ψ–

,=

01

S iGF

2------- 2π( )3

2EV2E'V2ωLyLz2ω'LyLz

--------------------------------------------------------------------=

× δ ω ω' q0–+( )δ ky ky' qy–+( )δ kz kz' qz–+( )

×q⊥

2

4eB----------–

iqx ky ky'–( )2eB

---------------------------– u k||( ) j gV gAγ5+( )u k ||'–( )[ ] ,exp


on the plane perpendicular to the vector B, = +

, ja = γα(1 + γ5)ν(p) is the Fourier transform ofthe current of left neutrinos. Note that in this approxi-mation where the field is the largest energy parameter

of the problem, the exponential factor exp(– ) inthe amplitude (3.6) differs little from unity and may beomitted. Direct calculations using the bispinors (3.5)and taking into account the conservation laws in (3.6)give

(3.7)

where ϕαβ = Fαβ/B, = εαβµνϕµν are the dimension-

less tensor and the dual tensor of the magnetic field, and

= – = ( ). Inside the parentheses the ten-sor indices are contracted systematically, for example

( ) = jα qλ.

Expression (3.7) and thus the total amplitude (3.6)for an arbitrary neutrino–electron process contain sup-pression associated with the relative smallness of theelectron mass. This suppression is not random andreflects the angular momentum conservation law. Forexample, in the ν e–e+ process the total spin of aneutrino–antineutrino pair in the center-of-inertia sys-tem is one whereas the total spin of an electron–positron pair in the Landau ground level is zero. Conse-quently, the amplitude of the process would be zero formassless particle and contain suppression in the relativ-istic limit under study. However, an analysis shows thatwhen integration is performed over the phase volume

kinematic regions exist where ~ m and this suppres-sion disappears for some neutrino–electron processes.

3.2. Kinematic Analysis

All neutrino–electron processes determined by theLagrangian (3.1) can be divided into two groups.

1. Processes in which a neutrino is present in the initial

and final states: , ν νe–e+, νe–e+ νand similar antineutrino processes.

2. Processes involving creation or absorption of aneutrino–antineutrino pair e–e+ ν , ν e–e+,

e eν , and eν e. It can be seen from (3.7) that the square of the

amplitude of each neutrino–electron process contains

the factor m2/ . However, the value of = – differs fundamentally for processes of the first and sec-ond types. For processes with a neutrino–antineutrino

q⊥2

qx2

qy2 ν p'( )

q⊥2 /4eB

u k ||( ) j gV gAγ5+( )u k ||'–( )[ ]

= 2m

q||2

---------qz

qz

------- gV jϕq( ) gA jϕ ϕq( )+[ ] ,

ϕαβ12---

q||2 q0

2 qz2 qϕ ϕq

jϕ ϕq ϕαβϕβλ

ν

q||2

νe+− νe+−

ν νν ν

q||2 q||

2 q02 qz

2

SICS Vol. 91 No. 4 2000


pair we have q = p + p' (p and p' are the four-momentaof a neutrino and an antineutrino, respectively), and

consequently q2 > 0. Since = + , where both

terms are positive, the value of can only be small

when both q2 and are small which is only possiblein a small region of phase space. This implies that

almost everywhere in phase space ~ E ~ T @ mwhich leads to reduction of the probability by a factorm2/T2 ! 1.

At the same time, we have q = p – p' for processesinvolving neutrinos in the initial and final states and

consequently q2 < 0 and the value of may be smallover a fairly wide region of phase space. Calculationsconfirm that kinematic amplification is achieved forthese processes, leading to the disappearance of the fac-tor m2/T2 in the probabilities.

Hence, neutrino interaction with a strongly magne-tized electron–positron plasma is determined by theprocesses νe– νe–, νe+ νe+, ν νe–e+, andνe–e+ ν. Figure 1 shows kinematic regions inmomentum space of a finite neutrino for the processesdescribed above in a convenient frame of referencewhere the momentum of the initial neutrino is perpen-dicular to the magnetic field. The main contribution tothe probability comes from regions near the parabola

= 0 where this kinematic amplification takes place.

q||2 q2 q⊥

2

q||2

q⊥2

q||2

q||2

q||2

I

II

III

0.5

–1.0

–0.5

0

0.5

1.0

II

1.0 1.5 p'⊥ /E

p'z/E

Fig. 1. Kinematic regions in momentum space of a finalneutrino: I—for the ν νe–e+ pair creation process; II—for the νe– νe–, νe+ νe+ scattering channels;III—for the νe–e+ ν pair capture process. The lines

correspond to = 0,q||2


4. PROBABILITY OF THE ν νe–e+ PROCESS

We express the probability of neutrino creation of ane–e+ pair per unit time in the form

(4.1)

where 7 is the total interaction time, dΓ is an elementof the particle phase volume,

(4.2)

The electron and positron distribution functions

(4.3)

allow for the presence of a plasma; here µ and T are thechemical potential and temperature of the electron–positron gas. To be general, we also allowed for thepossible presence of a quasiequilibrium neutrino gasdescribed by the distribution function . In general,the question of the accuracy of the description of thestate of a neutrino gas under conditions of stellar col-lapse or another astrophysical process using an equilib-rium distribution function and also the determination ofthis function is a complex astrophysical problem (see,e.g., [12]). Quite clearly, the approximation of an equi-librium neutrino Fermi gas using the distribution func-tion

(4.4)

where µν and Tν are the chemical potential and the tem-perature of the neutrino gas, should give satisfactoryresults inside the neutrinosphere. Outside the neutrino-sphere, where an outgoing neutrino flux forms and theneutrino momenta become asymmetric, a factorizationof the local distribution is usually assumed

(4.5)

where the energy distribution is assumed to be approx-imately equilibrium, the function Φ(ϑ , R) determinesthe neutrino angular distribution, ϑ = cosα, α is theangle between the neutrino momentum and the radialdirection in the star, and R is the distance from the cen-ter of the star. An analysis shows [12] that in the vicin-ity of the neutrinosphere the function Φ(ϑ , R) differsnegligibly from unity. In order to calculate the proba-

W ν νe–e+( )

= 17----- S 2 Γ

e– 1 f

e––( )d Γ

e+ 1 f

e+–( )d Γν' 1 f ν'–( ),d∫

dΓe

–

d2kLyLz

2π( )2-------------------, dΓ

e+

d2k'LyLz

2π( )2--------------------,= =

dΓν'd3 p'V

2π( )3--------------.=

fe

–

1ω µ–( )/T[ ]exp 1+

------------------------------------------------,=

fe

+

1ω' µ+( )/T[ ]exp 1+

-------------------------------------------------,=

f ν'

f νeq( ) 1

E µν–( )/Tν[ ]exp 1+----------------------------------------------------,=

f νΦ ϑ R,( )

E µν–( )/Tν[ ]exp 1+----------------------------------------------------,=



bility we shall use the neutrino distribution function inthe form (4.4), neglecting the asymmetry. Later in Sec-tion 7 when analyzing possible astrophysical manifes-tations of these neutrino–electron processes, we shallalso allow for asymmetry in the distribution function(4.5) for the initial and final neutrinos.

Substituting (3.6) into (4.1) and integrating usingδ-functions with respect to d2k' [where, as is usually thecase δ3(0) = 7LyLz/(2π)3], we obtain

(4.6)

where we need to substitute ω' = , =qz – kz. It is easy to see that the expression in the inte-grand in (4.6) does not depend on ky and consequentlyintegration over ky essentially determines the degree ofdegeneracy of an electron having a given energy [seefootnote to Eq. (3.2)]

(4.7)

Integrating over the electron momentum in (4.6)taking into account (4.7) we obtain the probability ofthe ν νe–e+ process in the form of the followingintegral over the final neutrino momentum:

(4.8)

In this expression the electron and positron energies ωand ω' appearing in the distribution function are

determined by the conservation law ω + ω' – q0 = 0 andare given by

(4.9)

Expression (4.8) is a generalization of Eq. (3.2) fromour study [7], where we investigated the neutrino–elec-tron process ν νe–e+ in a high-intensity purelymagnetic field, to the case where electron–positron andneutrino gases are present.

WGF

2

32 2π( )4E------------------------ 1

Lx

----- p'3dE'

--------- 1 f ν'–( )ky kzdd

ω2---------------δ ω ω' q0–+( )∫=

× 1 fe

––( ) 1 fe

+–( ) u k ||( ) j gV gAγ5+( )u k ||'–( )2,

m2 qz kz–( )2+ kz'

NE

Ly

2π------ kyd∫

eBLy

2π------------ x0d

Lx/2–

Lx/2

∫eBLxLy

2π------------------.= = =

WGF

2 eBm2

64π4E-------------------- p'3d

E'---------θ q0 qz

2 4m2+–( )∫=

× 1

q||2( )3/2

q||2 4m2–( )1/2

---------------------------------------------- gV jϕq( ) gA jϕ ϕq( )+ 2

× 1 f ν'–( ) 1 fe

––( ) 1 fe

+–( ) qz qz–( )+[ ] .

fe+−

ω 12--- q0 qz 1 4m2

q||2

---------–+

,=

ω'12--- q0 qz 1 4m2

q||2

---------––

.=


Further integration over the final neutrino momen-tum can be conveniently performed in a referenceframe where the initial neutrino momentum is perpen-dicular to the magnetic field pz = 0. For the case of apurely magnetic field we could convert to this systemwithout any loss of generality by performing a Lorentztransformation parallel to the field. In fact, we can seethat, besides the statistical Fermi factors, the value ofEW determined from Eq. (4.8) only contains invariantswith respect to this transformation (including the signof the argument of the θ function). However, we nowhave an isolated reference frame, the plasma rest sys-tem, in which the distribution functions (4.3) and (4.5)are formulated. In order to convert to a system wherepz = 0 we express these functions in a relativisticallyinvariant form:

(4.10)

Here we introduce the four-vector of the plasma veloc-ity v α, v2 = 1 which in its rest system is v α = (1, 0) andthe distribution functions (4.10) are exactly the same asthe functions (4.3) and (4.5). In the system pz = 0 wehave

where θ is the angle between the vectors of the initialneutrino momentum and the magnetic field induction inthe plasma rest system.

In Eq. (4.8) it is convenient to convert the dimen-sionless cylindrical coordinates in the space of the finalneutrino momentum vector p':

Here E⊥ is the energy of the initial neutrino in the sys-tem pz = 0 which is related to its energy E in the plasmarest system by E⊥ = Esinθ. In terms of the variables ρ,ζ, expression (4.8) is rewritten in the form

fe

–

kv( ) µ–T

--------------------- exp 1+

1–

,=

fe

+

k'v( ) µ+T

---------------------- exp 1+

1–

,=

f ν'p'v( ) µν–

Tν-------------------------

exp 1+1–

.=

v α v 0 0 0 v z, , ,( ), v 0 1/ θ,sin= =

v z θ/ θ,sincos–=

ρpx'

2py'

2+

E⊥------------------------, φtan

py'

px'-----, ζ

pz'

E⊥------.= = =

EWGF

2 m2eBE⊥2

4π3---------------------------=

× ρρ ζd

β ρ2 ζ2+ 1 2 ρ2 ζ2+ ρ2+–( )2

-----------------------------------------------------------------------------

ζm–

ζm

∫d

0

1 λ–

∫

SICS Vol. 91 No. 4 2000


(4.11)

where we need to substitute in the distribution func-tions (4.10)

and also introduce the notation

Note that the expression in the integrand in (4.11)exhibits a gain which completely compensates for thereduction by the smallness of the electron mass. Themain contribution then comes from the region near theupper limits of the integrals over ρ, ζ corresponding to

the values ~ m. Converting to the new integrationvariables β and x = E⊥ (1 – ρ2)/4Tsinθ in Eq. (4.11) and

isolating the leading contribution ~ /m2, we trans-form the expression for the probability to the form

(4.12)

× gV2 gA

2+( ) 1 ρ2+( ) ρ2 ζ2+ 2ρ2–[ ]{

– 2gVgA 1 ρ2–( )ζ } 1 f ν'–( )

× 1 fe

––( ) 1 fe

+–( )σ +1=

1 fe

––( ) 1 fe

+–( )σ 1–=

+[ ] ,

kv( )E⊥

2 θsin--------------=

× 1 ρ2 ζ2+–( ) 1 σβ θcos+( ) ζ θcos σβ+( )–[ ] ,

k'v( )E⊥

2 θsin--------------=

× 1 ρ2 ζ2+–( ) 1 σβ θcos–( ) ζ θcos σβ–( )–[ ] ,

p'v( )E⊥

θsin----------- ρ2 ζ2+ ζ θcos+( ),=

β 1 4m2

q||2

---------– 1 λ2

1 2 ρ2 ζ2+ ρ2+–( )2

---------------------------------------------------– ,= =

λ 2mE⊥-------, ζm

12--- 1 ρ2 λ2–+( )2

4ρ2– .= =

q||2

E⊥2

EWGF

2 eBE⊥2 T2 θsin

2

2π3----------------------------------------=

× x xd

0

ετ /4

∫ dβgV gA+( )2

1 ε– 2x 1 u+( )/τ ην+ +[ ]exp+----------------------------------------------------------------------------

0

1

∫× f β u η, ,( )f β– u η–, ,( ) f β u η–, ,( )f β– u η, ,( )+[ ]

+gV gA–( )2

1 ε– 2x 1 u–( )/τ ην+ +[ ]exp+----------------------------------------------------------------------------


where

Integrating (4.12) with respect to the variable βusing the relationship

(4.13)

where a = x(1 + u) and converting to the plasma restsystem, we finally obtain

(4.14)

where ε = E/Tν. The dependence of the probability(4.14) on the electron–positron gas concentration n =

– is defined in terms of its chemical potential

[see (2.4)]. Note that the formula for the probability(4.14) holds for hot (µ ! T) and cold (µ @ T) plasmas.For low-density electron–positron and neutrino gases(T, µ, Tν, µν 0), Eq. (4.14) reproduces our result[6, 7] for the probability of the process ν νe–e+ inthe strong magnetic field limit eB @ E2sin2θ without aplasma:

(4.15)

× f β u– η, ,( )f β– u– η–, ,( ) f β u– η–, ,( )f β– u– η, ,( )+[ ]

,

η µT---, ην

µν

Tν-----, ε

E⊥

Tν θsin-----------------,= = =

u θ, τcos Tν/T ,= =

f β u η, ,( )1

1 x 1 β+( )– 1 u+( ) η+[ ]exp+------------------------------------------------------------------------.=

βf β u η, ,( )f β– u η–, ,( )d

0

1

∫

= 1

a 1 e 2a––( )------------------------ 1 e 2a– η++

1 eη+------------------------- 1 ea η++

1 e a– η++-----------------------

,ln

W ν νe–e+( )GF

2 eBT2E

4π3------------------------ gV gA+( )2 1 u–( )2

=

× ξd

1 e ξ––( ) 1 ε– ξ /τ ην+ +( )exp+[ ]-----------------------------------------------------------------------------------

0

ετ 1 u+( )/2

∫

× ξcosh ηcosh+1 ηcosh+

------------------------------------ gV gA–( )2 1 u+( )2+ln

× ξd

1 e ξ––( ) 1 ε– ξ /τ ην+ +( )exp+[ ]-----------------------------------------------------------------------------------

0

ετ 1 u–( )/2

∫


------------------------------------

,ln

ne

– ne

+

WB

GF2 gV

2 gA2+( )

16π3-----------------------------eBE3 θsin

4.=



In the absence of a neutrino gas (Tν, µν 0) theexpression for the probability (4.14) for a hot electron–positron plasma (T ∞) has the form WB/4 as weindicated in [7] since the statistical factors for an elec-tron and positron in this limit are 1/2.

5. TOTAL PROBABILITY OF NEUTRINO INTERACTION WITH A MAGNETIZED

ELECTRON–POSITRON PLASMA

As we noted in the Introduction, the influence of the

scattering and νe–e+ ν pair capturechannels on the neutrino propagation process in aplasma should be taken into account in terms of theprobabilities summed over initial electron and/orpositron states. Thus, the probabilities of scatteringprocesses should be defined as

(5.1)

Similarly for the pair capture process

(5.2)

It can be seen from Fig. 1 that the scattering and paircapture processes correspond to infinite kinematicregions since the initial electrons and positrons can for-mally have any energy. Convergence of the integrals isensured by the distribution functions.

The expressions (5.1) and (5.2) are integrated by thesame scheme as that described above for the ν νe–e+

pair creation process. An important factor for the inte-gration will be that the energy imparted from the neu-trino to the active medium q0 = E – E ' is not positive-definite. For the probability (per unit time) of neutrinoscattering on magnetized plasma electrons we have

(5.3)

νe+− νe+−

W νe+− νe+−( )

= 17----- S 2 Γ

e+− fe+−d Γ

e+−' 1 fe+−'–( )d Γν' 1 f ν'–( ).d∫

W νe–e+ ν( )

= 17----- S 2 Γ

e– f

e–d Γ

e+ f

e+d Γν' 1 f ν'–( ).d∫

W νe– νe–( )GF

2 eBT2E

4π3------------------------ gV gA+( )2 1 u–( )2

=

× ξd

1 e ξ––( ) 1 ε– ξ /τ ην+ +( )exp+[ ]-----------------------------------------------------------------------------------

0

ετ 1 u+( )/2

∫

× 1 eη+

1 e ξ– η++----------------------- gV gA–( )2 1 u+( )2+ln

× ξd

1 e ξ––( ) 1 ε– ξ /τ ην+ +( )exp+[ ]-----------------------------------------------------------------------------------

0

ετ 1 u–( )/2

∫


Taking into account the distribution functions (4.10),the probability of scattering at positrons is obtainedfrom (5.3) by substituting η –η. For the pair cap-ture channel we have

(5.4)

As we have already noted, only the total probabilityof neutrino interaction with an electron–positronplasma is physically meaningful:

(5.5)

It was found that this quantity has a substantially sim-pler form:

(5.6)

where

(5.7)

and significantly, the dependence on the chemicalpotential µ of the electron–positron gas, which waspresent in the probabilities of the various processes,was cancelled in the total probability. At present, aphysical cause of this reduction is unknown. Possiblysome property involving the completeness of this set ofprocesses in relation to electrons is manifest.

× 1 eη+

1 e ξ– η++-----------------------ln

+ gV gA+( )2 1 u–( )2 gV gA–( )2 1 u+( )2+[ ]

× ξd

eξ 1–( ) 1 ε– ξ /τ– ην+( )exp+[ ]-------------------------------------------------------------------------------- 1 eη+

1 e ξ– η++-----------------------ln

0

∞

∫

.

W νe–e+ ν( )GF

2 eBT2E

4π3------------------------=

× gV gA+( )2 1 u–( )2 gV gA–( )2 1 u+( )2+[ ]

× ξd

eξ 1–( ) 1 ε– ξ /τ– ην+( )exp+[ ]--------------------------------------------------------------------------------

0

∞

∫


------------------------------------.ln

W ν ν( ) W ν νe–e+( ) W νe–e+ ν( )+=

+ W νe– νe–( ) W νe+ νe+( ).+

W ν ν( )GF

2 eBT2E

4π3------------------------ gV gA+( )2 1 u–( )2

=

× F1ετ 1 u+( )

2-----------------------

F1 ∞–( )–

---+ gA gA; u u––( )

,

Fk z( )ξk ξd

1 e ξ––( ) 1 ε– ην ξ /τ+ +( )exp+[ ]-----------------------------------------------------------------------------------,

0

z

∫=

SICS Vol. 91 No. 4 2000


For a sparse neutrino gas the probability (5.6) isexpressed in terms of the dilogarithm Li2(x):

(5.8)

where Lin(x) is an nth-order polylogarithm:

(5.9)

The relative contributions of the plasma and themagnetic field to the process of neutrino interactionwith the active medium are illustrated in Fig. 2 whichgives the ratio of the probabilities of neutrino interac-tion with a magnetized plasma and a pure magneticfield Rw = WB + pl/WB for the angle θ = π/2 as a functionof the ratio of the neutrino energy to the plasma temper-ature. It can be seen that as the temperature increases,the interaction probability increases.

W ν ν( )GF

2 eBT2E

4π3------------------------ gV

2 gA2+( )E2 θsin

4

4T2-------------------

=

+ gV gA+( )2 1 θcos–( )2

× Li2 1 E 1 θcos+( )2T

-----------------------------– exp–

+ gV gA–( )2 1 θcos+( )2Li2 1 E 1 θcos–( )

2T----------------------------–

exp–

+π2

3----- gV

2 gA2+( ) 1 θcos

2+( ) 4gVgA θcos–[ ]

,

Lin x( ) xk

kn----

k 1=

∞

∑ .=

01

Rw

E/T2 4 6 8 10 12

2

3

4

5

6

Fig. 2. Ratio of the probabilities of neutrino interaction witha magnetized plasma and a pure magnetic field, Rw =WB + pl/WB for θ = π/2 as a function of the ratio of the neu-trino energy to the plasma temperature.


The probability (5.6) determines the partial contri-bution of these processes to the opacity for neutrinopropagation in a medium. An estimate of the mean freepath associated with neutrino–electron processes gives

(5.10)

This should be compared with the neutrino mean freepath as a result of interaction with nucleons, which is ofthe order of a kilometer at density ρ ~ 1012 g/cm3. Atfirst glance the influence of neutrino–electron reactionson the neutrino propagation process is negligible. How-ever, the mean free path does not exhaust the neutrinophysics in the medium. Other important quantities inastrophysical applications are the neutrino energy andmomentum losses. Of particular importance is theasymmetry of the neutrino momentum loss caused bythe influence of an external magnetic field. Manyattempts have been made to calculate these asymme-tries caused by neutrino–nucleon processes associatedwith the problem of the high proper velocities of pul-sars (see [13] and the references therein). As we shallshow, despite the relatively low probability of neu-trino–electron processes, their contribution to theasymmetry may be comparable to the contributions ofneutrino–nucleon processes.

6. AVERAGE LOSSES OF NEUTRINO ENERGY AND MOMENTUM

In studies of these neutrino–electron interactions ina magnetic field and/or plasma [5, 8], the analysis hasusually been confined to calculation of the probabilitiesand cross sections of processes. As we have noted, notonly the probabilities of the processes are of practicalinterest for astrophysics but also the average loss ofneutrino energy and momentum in the medium3 whichis determined by the four-vector

(6.1)

where E and p are the energy and momentum of the ini-tial neutrino, q is the difference between the momentaof the initial and final neutrinos, q = p – p', and dW isthe total differential probability of all the processesspecified in (5.5). The zeroth component Q0 is associ-ated with the average energy lost by a single neutrinoper unit time and the spatial components Q are associ-ated with the loss of neutrino momentum per unit time.

For a purely magnetic field the four-vector of thelosses Qα was calculated in our studies [6,7]. In thiscase, the losses are caused by the only possible processin the absence of plasma, pair creation during motion ofa neutrino in a strong magnetic field ν νe–e+. In the

3 In general a neutrino can lose and acquire energy and momentumso that we shall subsequently understand “loss” of energy andmomentum in the algebraic sense.

λ e1W----- . 170 km

103Be

B--------------

5 MeVT

-----------------

3

.=

Qα EdEdt------- dp

dt------,

– E qα W ,d∫= =



strong magnetic field limit for the zeroth and z-compo-nents of the vector Qα we obtained (the field is directedalong z)

(6.2)

It can be seen from Eq. (6.2) in particular that even foran isotropic neutrino momentum distribution the aver-age momentum loss will be nonzero (proportional togVgA) because of parity nonconservation in weak inter-action. As we showed in [6,7] in fields of ~103Be theintegral asymmetry of the neutrino emission caused bythe component Qz and determined by the expression

A = / could reach the scale of ~1% requiredto explain the observed intrinsic pulsar velocities [14]as a result of the ν νe–e+ process only.

In the presence of a magnetized plasma our calcula-tions yield the following result for the same compo-nents of the loss four-vector:

(6.3)

where the function F2(z) was determined in expression(5.7), and the plus or minus signs correspond to thezeroth and z components. Our result for the loss four-vector obtained for the case of a purely magnetic field(6.2) is reproduced from Eq. (6.3) in the low-densityplasma limit (T, Tµ, µν 0).

In order to illustrate the relationship between thecontributions of the plasma and the magnetic field tothe four-vector of the neutrino energy and momentumlosses in an active medium we shall consider the sim-pler situation of a low-density neutrino gas and rewriteEq. (6.3) for the angle θ = π/2 in the following form:

(6.4)

It can be seen from a comparison of (6.4) with Eq. (6.2)for θ = π/2 that the function ^(E/T) is the ratio of thecomponents of the loss vector in a magnetized plasma

Q0 z,B( ) GF

2 eBE5 θsin4

48π3---------------------------------=

× gV2 gA

2 2gVgA θcos+ + gV2 gA

2+( ) θcos 2gVgA+,{ } .

p∑ p∑

Q0 z,GF

2 eBT3E2

4π3-------------------------- gV gA+( )2 1 u–( )2

=

× F2ετ 1 u+( )

2-----------------------

F2 ∞–( )–

---± gA gA; u u––( )

,

Q0 z, θ π2---=

GF2 eBE5

48π3-------------------- gV

2 gA2+ 2gVgA,( )^ E

T---

,=

^ x( ) 16x--- 1 e x/2––( )ln+=

–24

x2------Li2 e x/2–( ) 48

x3------Li3 e x/2–( ).–


and in a purely magnetic field. Figure 3 gives a graph ofthe function (E/T). It can be seen that at E = E0 . 3.4Tthere is a unique “window of transparency” when aneutrino does not exchange energy and momentumwith a magnetized plasma. The negative values of thefunction ^(E/T) at lower energies imply that the neu-trino captures energy from the plasma and acquiresmomentum in the opposite direction to the magneticfield. At energies higher than E0 the neutrino impartsenergy to the plasma and also momentum in the direc-tion of the field. This may have extremely interestingastrophysical consequences.

7. INTEGRAL ACTION OF NEUTRINOSON A MAGNETIZED PLASMA

As an illustration of the application of our results toastrophysical conditions we estimate the volume

energy density lost by neutrinos per unit time andthe component ^z (parallel to the field) of the volumedensity of the force acting on the plasma from neu-trinos

(7.1)

where dnν is the initial neutrino density:

(7.1)

Here the angular distribution of the initial neutrinos istaken into account in the function Φ(ϑ , R), ϑ = cosα,

%

% ^z,( ) nν1E---Q0 z, ,d∫=

dnνd3 p

2π( )3------------- Φ ϑ R,( )

E µν–( )/Tν[ ] 1+exp----------------------------------------------------.=

0–8

2 4 6 8

–6

–4

–2

0

2

E/T

^

Fig. 3. The function ^(E/T) introduced in (6.4) and deter-mining the dependence of the components of the four-vectorof the neutrino energy and momentum losses in a magne-tized plasma on the ratio of the neutrino energy to theplasma temperature.

SICS Vol. 91 No. 4 2000


α is the angle between the neutrino momentum and theradial direction in the star, and R is the distance fromthe center of the star. At the same time, the similar func-tion Φ(ϑ ', R) should be introduced in the statistical fac-tor 1 – when integrating over the momenta of thefinal neutrino. In a supernova shell the neutrino angulardistribution is close to isotropic [12] so that in theexpansion of the function Φ in terms of ϑ , we can con-fine ourselves to the lowest Legendre polynomials andthis function can be uniquely expressed in terms of theaverage values ⟨ϑ⟩ and ⟨ϑ 2⟩ (which depend on R) as fol-lows:

(7.2)

Neutrinos leaving the central region of a star at hightemperature enter the peripheral region where a strongmagnetic field is generated and the temperature of theelectron–positron gas is lower. In this case the spectraltemperatures for different types of neutrino differ [12, 15]:

(7.3)

The action of a neutrino on a plasma leads to the estab-

lishment of thermal equilibrium = 0. When analyz-ing this equilibrium we need to take into account the

contributions to made by all processes of neutrinointeraction with the medium. As we have noted, theprobability of the β processes νe + n e– + p is sub-stantially higher than that for neutrino–electron pro-cesses so that these dominate in the energy balance.The energy transferred per unit time per unit plasmavolume as a result of these processes involving onlyelectron neutrinos may be expressed in the form

(β) . @( – T)/T.

From this it follows that as a result of neutrino heatingthe plasma temperature should be very close to thespectral temperature of the electron neutrinos (T .

). However, the contribution to made by othertypes of neutrino whose spectral temperatures exceed

, has the result that the plasma temperature is

slightly higher (T * ). It is therefore meaningful to

make separate estimates of the contributions to ( , z)made by neutrino–electron processes involving νe andall other neutrinos and antineutrinos.

We stress that the appearance of the force density z

in expression (7.1) is caused by interference betweenthe vector and axial-vector coupling in the effectiveLagrangian (3.1) and is a macroscopic manifestation ofparity nonconservation in weak interactions. At firstglance, the main contribution to ^z should be made by

f ν'

Φ ϑ R,( ) . 1 3 P1 ϑ( )⟨ ⟩ P1 ϑ( ) 5 P2 ϑ( )⟨ ⟩ P2 ϑ( ).+ +

Tνe . 4 MeV, Tνe

. 5 MeV,

Tνµ τ, . Tνµ τ,

. 8 MeV.

%tot

%tot

% Tνe

Tνe%

Tνe

Tνe

%


electron neutrinos since gV(νe) @ gV(ν

µ

,

τ

). However, aswe shall show below, the main contributions are madeby

µ

and

τ

neutrinos and antineutrinos (as a result of theconservation of CP parity neutrinos and antineutrinospush the plasma in the same direction). This is becausein the vicinity of the

ν

e

neutrinosphere the spectral tem-peratures of the other types of neutrinos differ substan-tially from the plasma temperature

T

.

.

7.1. Processes Involving Electron Neutrinos

We obtained the following expression for the vol-ume density of the neutrino energy losses and the forcedensity (7.1):

(7.4)

where

τ

e

= /

T

. This formula is written assuming asmall deviation from thermal equilibrium between theneutrino gas and the electron–positron plasma (( τ e – 1) ! 1) and thus comparatively weak asymmetry of the neu-trino distribution (

⟨ϑ

2

⟩

– 1/3)

!

1).

A numerical estimate gives

(7.5)

7.2. Processes Involving

,

ν

µ

,

τ

,

In this case

T

ν

/

T

– 1 cannot be considered to be asmall parameter. However, the relative contribution ofthe asymmetry of the neutrino distribution is small [12]and can be neglected.

For numerical estimates we can conveniently

express the values and

^

z

(7.1) in the followingform:

(7.6)

Tνe

% ^z,( )νe

GF2 eBT7

3π5-------------------- gV

2 gA2+ 2gVgA,( )=

× τe 1–( ) x3 xd

ex 1–-------------

0

∞

∫ y3 yd

1 ex– y– ην+

+( ) 1 ey ην–

+( )--------------------------------------------------------------

0

∞

∫

+278------ ϑ 2⟨ ⟩ 1

3---–

x3 xd

ex 1–-------------

0

∞

∫ y3 3y x–( ) yd

x y+( )2 1 ey ην–

+( )---------------------------------------------

0

∞

∫

,

Tνe

% ^z,( )νe . 2.0 1030 erg

cm2 s------------- 0.57 1020

dyne

cm3-----------×,×

× B

1016 G----------------

T4 MeV-----------------

7

× ην( ) τe 1–( ) 0.53 ϑ 2⟨ ⟩ 13---–

+ .exp

νe νµ τ,

%

% ^,( )νi . ! gV2 gA

2+ 2gVgA,( )ϕ η i( )ψ τ i( ),



where

(7.7)

(7.8)

(7.9)

Formulas (7.5)–(7.10) demonstrate in particular thatthe action of each individual neutrino fraction on anelectron–positron plasma goes to zero when thermody-namic equilibrium is established between this fractionand the plasma τi = 1, ⟨ϑ⟩ = 0, ⟨ϑ 2⟩ = 1/3.

We show that the main contribution to the neutrinoaction on the plasma is made by µ and τ neutrinos andantineutrinos. In fact the function ψ(τi) (7.10) increasesrapidly as the difference between the spectral tempera-ture of the neutrinos and the plasma temperatureincreases. For example, at temperatures (7.4) we haveψ(1.25) . 0.824 for electron antineutrinos and ψ(2) .38.47 for µ and τ neutrinos and antineutrinos. This fac-tor leads to compensation for the smallness of the con-stant gV(νµ, τ) and makes the contribution νµ, τ, notonly comparable with the contribution of the electronneutrinos and antineutrinos but also dominant.

As we have noted, the contribution of neutrino–electron processes to the energy action of a neutrino onthe plasma is small compared with the contribution ofβ processes and leads to a small departure from equilib-rium between electron neutrinos and the plasma so thatthe total contribution of β processes and all νe pro-

cesses to the value of is zero.

For the force action of a neutrino on the plasma par-allel to the magnetic field described by ^z in Eqs. (7.5)–(7.10) the total contribution of all types of neutrino isgiven by

(7.10)

Here we assumed for estimates that the chemical poten-tials of the neutrinos are zero [15]. Note that the value

!12GF

2 eBT7

π5--------------------------=

= B

1016 Gs------------------

T4 MeV-----------------

7

1.6 1030 erg

cm3 s-------------×

0.55 1020 dyne

cm3-----------× ,

ϕ η i( )η i

4

24------

π2η i2

12----------- 7π4

360--------- Li4 η i–( )exp–[ ] ,+ + +=

ϕ 0( )7π4

720--------- . 0.947,=

ψ τ i( )τ i

7

6---- y2 yd

eτ i y 1–

---------------- τ i 1–( )y[ ] 1–exp{ } ,

0

∞

∫=

ψ τ i( ) τ i 1→ . τ i 1.–

νµ τ,

%

^z . 3.6 1020 dyne

cm2----------- B

1016 G----------------

T4 MeV-----------------

7

× .


(7.11) was independent of the chemical potential of anelectron–positron plasma.

The force density (7.11) should be compared withthe recent result for a similar force caused by β-pro-cesses [16]. Under the same physical conditions ourvalue of the force as a result of neutrino–electron pro-cesses is of the same order of magnitude and, which isparticularly important, of the same sign as the result of[16]. Thus, the role of neutrino–electron processes in ahigh-intensity magnetic field may be significant inaddition to the contribution of β processes.

Note that the force density (7.11) is five orders ofmagnitude lower than the density of the gravitationalforce and thus negligibly influences the radial dynam-ics of the supernova shell. However, when a toroidalmagnetic field [3] is generated in the shell, the force(7.11) directed along the field can fairly rapidly (withintimes of the order of a second4 lead to substantial redis-tribution of the tangential plasma velocities. Then intwo toroids in which the magnetic field has oppositedirections, the tangential neutrino acceleration of theplasma will have different signs relative to the rota-tional motion of the plasma. This effect can then lead tosubstantial redistribution of the magnetic field lines,concentrating them predominantly in one of the tor-oids. This leads to considerable asymmetry of the mag-netic field energy in the two hemispheres and may beresponsible for the asymmetric explosion of the super-nova [17] which could explain the phenomenon of highproper pulsar velocities [14]. In our view it is interest-ing to model the mechanism for toroidal magnetic fieldgeneration taking into account the neutrino force actionon the plasma both via neutrino–nucleon and neutrino–electron processes.

8. CONCLUSIONS

As we know, in existing systems for numerical mod-eling of astrophysical cataclysms such as supernovaexplosions and coalescing of neutron stars, where thephysical conditions being studied can be achieved inprinciple, the neutrino–electron interaction effectsstudied by us were neglected. However, in detailedanalyses of these astrophysical processes it may beimportant to allow for the influence of an activemedium such as a magnetized e–e+ plasma on quantumprocesses involving neutrinos.

In the present study we have investigated the entirerange of neutrino–electron processes in a magnetized

plasma. In addition to canonical scatter-ing and e–e+ annihilation reactions we havealso considered exotic processes of synchrotron radia-tion and pair absorption, e e , and also neutrinoradiation and absorption of an electron–positron pair bya neutrino ν νe–e+. We have shown that among this

4 We know that the cooling stage of a supernova shell, known as theKelvin–Helmholz stage, lasts for around 10 s.

νe+− νe+−

νν

νν

SICS Vol. 91 No. 4 2000


entire range processes involving neutrino pair creationand absorption are kinematically suppressed for thecase of relatively high neutrino energies Eν @ me andhot dense plasmas T, µ @ me. The total probability of allprocesses including neutrinos in the initial and finalstates does not have this suppression. In addition, weobserved that the total probability of these processesand also the average neutrino energy and momentumlosses do not depend on the chemical potential of thee−e+ plasma whereas the contributions of the variousprocesses do contain this dependence. This is a new andunexpected result.

We assume that these results will be useful for adetailed analysis of the dynamics of supernova explo-sion.

ACKNOWLEDGMENTSWe are grateful to G.G. Raffelt, V.A. Rubakov, and

V.B. Semikov and also all participants at the interna-tional symposium on “Strong Magnetic Fields in Neu-trino Astrophysics” (Yaroslavl, 1999) for stimulatingdiscussions.

This work was partly financed by the Russian Foun-dation for Basic Research (project no. 98-02-16694).

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Vol. 91

No. 4

2000

Documents

Neutrino interaction with strongly magnetized electron-positron plasma