Spin asymmetries arising in neutrino-

lepton processes in a magnetic field

and their macroscopic appearance

Vali Huseynova,b, Rasmiyya Gasimovaa, Nurida

Akbarovaa and Billura Hajiyevaa

a) Department of General and Theoretical Physics,

Nakhchivan State University, AZ 7000, Nakhchivan,

Azerbaijan; b) Laboratory of Physical Research,

Nakhchivan Division of Azerbaijan National Academy

of Sciences , AZ 7000, Nakhchivan, Azerbaijan

E-mail : vgusseinov@yahoo.com

Considered processes

,l le e ,l le e

l l l l ,l e l e

l ee l

(1)

(2)

(3)

(4)

(5)

Motivation●The presence of a strong magnetic field leads

on the one hand to the anisotropy and asymmetry

in the heating of the stellar matter on the other

hand to the anisotropy and asymmetry of the

subsequent explosion of the outer layers of the

collapsing stellar core. To clarify the reasons of the

anisotropy and the asymmetry arising in astrophysical

phenomena connected with neutrino-lepton

processes and to show the macroscopic appearance

of these asymmetries it is important to investigate the

dependence of the differential cross section of the

considered processes in a magnetic field on the spin

variables of the charged leptons.

● Analyses of polarization effects in some

neutrino-lepton processes in a magnetic field

enable to use polarized electrons as a target in

neutrino (antineutrino) detectors.

● The polarization asymmetry in some neutrino-

lepton processes provide a sensitive tool for

distinguishing neutrino flavor of the incoming

neutrinos (antineutrinos).

Some related published papers

[1] Benesh C. J. and Horowitz C. J. astro-ph/ 9708033

[2] Bezchastnov V. G. and Haensel P. 1996 Phys.

Rev. D 54 3706.

[3] Guseinov V. A. 2000 J. Phys. G 26 1313.

[4] Kuznetsov A. V. and Mikheev N. V. 1997 Phys. Lett.

B 394 123.

[5] Borisov A. V., Nanaa M. K. and Ternov I.M. 1993

Vestnik Moskovskogo universiteta.Fizika. Astronomiia,

48(2) 15.

The main purposes

● to present analytic formulae for the rates

(differential cross sections, differential probabilities)

of the considered neutrino-lepton processes in a

magnetic field with allowance for the longitudinal

and transverse polarizations of the charged leptons;

● to analyze polarization effects;

● to show possible applications of the obtained

results.

Assumptions

● Neutrino (antineutrino) energies, transverse

momenta and the Landau energy levels of the

charged leptons, the strength of a magnetic

field, the temperature of matter are arbitrary.

● Polar and azimuthal angles of the incident

(scattered) neutrinos and antineutrinos are

arbitrary.

0;m signature ; 1.c

We use the standard Weirberg-Salam-Glashow

(WSG) electroweak interaction theory.When the

momentum transferred is relatively small,

2 2 2,w zq m m ( wm is the W -boson mass,

zm is the Z

-b

-boson mass), the four-fermion approximation of the

WSG standard model can be used. The gauge of a

4-potential is , , ,A o o xH o and an external magnetic field vector Is directed along the axis

H��������������

Oz

2 i fM A E E

is the matrix element of the considered processes.

2

1 2

2 2 Fi f y y y y z z z z

y z

i n ni L

GA p p k k p p k k

VL L

e e u k u k J

The amplitude and the matrix element

of the processes

is the amplitude of the considered processes.

FG the Fermi constant;

V the normalization volume;

,y zL L the normalization lengths;

k k 4-momentum of the incident (scattered)

neutrino and/or antineutrino with the energy ;

y zk k and y zk k y z components of the 4- momenta

of the incident (scattered) neutrino and/or antineutrino;

, 1 the sign of energy for the incident (scattered)

neutrino and/or antineutrino;

y zp p and y zp p y z component of the 4-momenta

of the initial (final) charged lepton and initial (final) charged

antilepton with the energy .E E

1 1 the sign of energy for the charged lepton

(the charged antilepton)

0,1...n and 0,1...n enumerate the Landau energy

levels of the charged leptons (the charged antileptons)

2; ,tan y xq q ,q k k 2 ,x y yq h p p

,u k u k the Dirac bispinors

J -the transition amplitude of the 4-current for the

considered prosesses. The structure of the components of J

depends on the kind of polarization of the charged

leptons (the charged antileptons).

,h eH

-the Dirac matrices

5 5 0 1 2 31 2;L i

Longitudinal polarization of charged leptons

0, 1, 2,3

V A

V A i

g s g s fJ i

g s g s f

0,3 1 1 1, 1 2 2 ,n n nnf B B I B B I

1 1 2 , 1 1 2 1, ,i i

n n n nf B B e I B B e I

2 1 2 , 1 1 2 1, ,i i

n n n nf iB B e I iB B e I

1 1 2 2

1

4s A A A A

1 2 1 2

1

4s A A A A

20.5 2sin ,V wg 0.5Ag

for

e e

-processes,20.5 2sin ,V wg 0.5Ag

for

ee -processes.

The spin coefficients1 2

1 1 ;lmAE

1 2

2 1 ;lmAE

1 2

1 2 21 ;z

l

pB

E m

1 2

2 2 21 .z

l

pB

E m

1 1 corresponds to right-hand (left-hand) helicity.

Transverse polarization of charged leptons

00 2 4 4 0 2 3 3 1, 1nn n nJ g s g s B B I g s g s B B I

11 3 3 4 , 1 1 3

3 4 1,

iV A n n V A

in n

J g s g s B B e I g s g s

B B e I

21 3 3 4 , 1 1 3

3 4 1,

iV A n n V A

in n

J i g s g s B B e I i g s g s

B B e I

32 0 4 4 2 0 3 3 1, 1V A nn V A n nJ g s g s B B I g s g s B B I

0,2 3 3 4 4

1,

4s A A A A

1,3 3 4 3 4

1.

4s A A A A

1 2

3 1 ;zpAE

1 2

4 1 ;zpAE

1 2

3 2 21 ;l

z

mB

E p

1 2

4 2 21 l

z

mB

E p

1 1 corresponds to the charged lepton

spin oriented parallel (antiparallel) to the magnetic

field H.

Rate of the processes

2

11i i f f

i f

Mdn f dn f

V

22

y zi y z

dp dpdn L L

-the phase-spase element of a

particle

2,

2

y zf y z

dp dpdn L L

1

1E Tf E e the Fermi-Dirac distribution function

1

1E Tf E e

the total interaction time

22 2

2

22 F

y z

y y y y z z z z

GME E

V L L

p p k k p p k k R

N k k k k g kk i k k

R N J J

for transverse polarization case

2 2 2 21 1 4 2 3 3 3 4 2 4 1 5 22 2R d I I I I d I I

2 23 6 4 7 3 8 3 4 4 9 1 22 4d I I I I d I I

2 2 2 25 10 4 11 3 12 3 4 6 4 1 5 22 2d I I I I d I I

7 13 1 3 14 1 4 15 2 3 16 2 42d I I I I I I I I

8 17 1 3 18 1 4 19 2 3 20 2 42d I I I I I I I I

9 17 1 3 18 1 4 19 2 3 20 2 42d I I I I I I I I

10 13 1 3 14 1 4 15 2 3 16 2 42d I I I I I I I I

R

1

11 2 1 1

4z z z zp p p p

g gE E E E

2

11 2 1 1

4z z zp p pp

g gE E E E

1 2 1 22 23

11 1 1

4z z z zp p p p

g gE E E E

1 2 1 22 2

4 2 2

11 1 1 2 1 1

8z z z z z zp p p p p p

g g gE E E E E E

1 2 1 22 2

5 2 2

11 1 1 2 1 1

8z z z z z zp p p p p p

g g gE E E E E E

1 2 1 22 2

6 2 2

11 1 1 1

4z zp p

gE E

1 2 1 22 2

7 2 2

11 1 1 1

4z zp p

gE E

1 2 1 22 2

1 2 1 22 28 2 2

12 1 1 2 1 1

8z z z zp p p p

g gE E E E

1 2 1 22 2

1 2 1 22 29 2 2

11 1 1 1 1

8z z z zp p p p

g gE E E E

10

12 1 1 1

8z z z zp p p p

g gE E E E

11

12 1 1 1

8z z z zp p p p

g gE E E E

1 2 1 22 2

1 2 1 22 212 2 2

12 1 1 1 1

8z z z zp p p p

g gE E E E

1 2 1 2 1 22 2 2

1 2213 2 2 2

11 2 1 1 1

8z z z zp p p p

g g gE E E E

1 2 1 2 1 22 2 2

1 2214 2 2 2

11 1 2 1 1 1

8z z z zp p p

g g gE E E E

1 2 1 2 1 2 1 22 2 2 2

15 2 2 2

11 1 2 1 1 1

8z z z zp p p p

g g gE E E E

1 2 1 2 1 22 2 2

1 2216 2 2 2

11 1 2 1 1 1

8z z z zp p p

g g gE E E E

1 2 1 2 1 22 2 2

1 2217 2 2 2

11 1 2 1 1 1

8z z z z zp p p p p

g g gE E E E E

1 2 1 2 1 22 2 2

1 2218 2 2 2

11 1 2 1 1 1

8z z z z zp p p p p

g g gE E E E E

1 2 1 2 1 22 2 2

1 2219 2 2 2

11 1 2 1 1 1

8z z z z zp p p p p

g g gE E E E E

1 2 1 2 1 22 2 2

1 2220 2 2 2

11 1 2 1 1 1

8z z z z zp p p p p

g g gE E E E E

2 2 ,Ag g g Ag g g

1,2 1 cos cos ,d

3 sin sin cos ,d

4 sin sin cos 2 ,d

5,6 cos cos ,d

7,8 sin cos sin cos ,d

9,10 cos sin cos cos sin cos ,d

the polar angle of the incident (scattered) neutrino and/or antineutrino

the azimuthal angle of the incident(scattered) neutrino and/or antineutrino

1 , 1,n nI I 2 1, ,n nI I 3 1, 1,n nI I

4 ,nnI I

1 2

2!

!n n x n n

nn n

nI x x e L x

n

the Laguerre function

n nnL x the associated Laguerre polynomial

2 2

2x yq q

xh

2 2 2 22 21 1 4 2 3 3 3 4 2 4 1 5 2

R d r I r I r I I d r I r I

2 2 2 43 6 4 7 3 8 3 4 4 9 1 2d r I r I r I I d r I I

2 2 2 22 25 510 4 11 3 6 4 1 2d r I r I d r I r I

72 12 1 4 13 2 4 14 1 3 15 2 3d r I I r I I r I I r I I

2 8 16 1 4 17 1 3 18 2 4 19 2 3d r I I r I I r I I r I I

2 9 16 1 4 17 1 3 18 2 4 19 2 3d r I I r I I r I I r I I

10 12 1 4 13 2 4 14 1 3 15 2 32d r I I r I I r I I r I I

for longitudinal polarization caseR

Analyses of the cross section of the process e e

2

4, 0

132F

zn n

G eHddp E E f f R

d d

1 22 2/ ,z ep E E

When the neutrinos scatter on the electrons with right-

hand circular polarization , we have 1

2 230, 0, 1 4 1 1 1RR g I

where 1 22 2/z ep E m

This expression shows that, if , and the

neutrinos scatter on the electrons with right-hand circular

polarization, the final electrons can only have right-hand

circular polarization. When , and

the differential cross section only depends on and it is

not sensitive to the neutrino flavor.

0 0

0 0 ,1

When the neutrinos scatter on the electrons with left-

hand circular polarization , we have ,1

2 240, 0, 1 4 1 1 1LR g I

0

The obtained expression shows that, if , and

the neutrinos scatter on the electrons with left-hand

circular polarization the final electrons can only

have left-hand circular polarization. When , and

the differential cross section depends on and

therefore it is sensitive to the neutrino flavor. When

the differential cross section is different for and

. This result enables to come to the conclusion that

initial electrons with left-hand circular polarization in

neutrino-electron scattering in a magnetic

0 0

,1

0 0

1 Lg

1 e

field can be used as a polarized target in neutrino

detectors. The polarized electron targets in neutrino-

electron scattering in a magnetic field can also be used

for distinguishing neutrino flavor.

When , and the neutrinos scatter on the

electrons with right-hand left-hand circular

polarization, we have the following expressions for :

R

24

2 11141,, IgR R

and

23

2 11141,, IgR L

The last two expressions show that, if and

the neutrinos scatter on the electrons with right-hand

(left-hand) circular polarization, the final electrons can

only have right-hand (left-hand) circular polarization.

When and the differential cross

sections only depends on and it is not sensitive to the

neutrino flavor. When , and the differential cross

section depends on and therefore it is sensitive to the

neutrino flavor. In this case the differential cross section

is different for and .

,

, 1,

e

So, spin asymmetry in neutrino-electron scattering in a

magnetic field enables to use electrons with left-hand circular

polarization as polarized electron targets for distinguishing

neutrino flavor and for detection of neutrinos.

The analyses of the obtained expressions for the cross

section show that electrons with right-hand circular

polarization are heated by , and equally. However, the

electrons with left-hand circular polarization in neutrino-

electron scattering in a magnetic field are heated by and

unequally. This fact leads to the asymmetry in the

heating of the stellar matter.

e

e

0,n 0,zp

In the case of the transverse polarizations of initial and

final electrons when the initial electrons are on the

lowest Landau level, i.e. we have1

2 240, 0 4 1 1 Re

R v g I

Analyses of the cross section of the

process

When the antineutrinos scatter on the electrons with

right-hand circular polarization , we have

e e

1

2 230, 0, 1 4 1 1 1LR g I

This expression shows that, if , and the

antineutrinos scatter on the electrons with right-hand

circular polarization, the final electrons can only have

right-hand circular polarization.

0 0

When , and the differential cross section

depends on and it is sensitive to the antineutrino

flavor. When the differential cross section is

different for and . This result enables to come to

the conclusion that initial electrons with right-hand

circular polarization in antineutrino-electron scattering in

a magnetic field can be used as a polarized target in

antineutrino detectors. The polarized electron targets in

antineutrino-electron scattering in a magnetic field can

also be used for distinguishing antineutrino flavor.

0 0 ,1

Lg

,1

e

When the antineutrinos scatter on the electrons

with left-hand circular polarization we have ,1

2 240, 0, 1 4 1 1 1RR g I

The obtained expression shows that, if , and

the antineutrinos scatter on the electrons with left-hand

circular polarization the final electrons can only have

left-hand circular polarization. When , and the

differential cross section depends on and therefore

it is not sensitive to the antineutrino flavor.

0 0

,1

Rg

When , and the antineutrinos scatter on the

electrons with right-hand left-hand circular polarization,

we have the following expressions for :

R

2 24, , 1 4 1 1 1LR g I

and

2 23, , 1 4 1 1 1RR g I

The last two expressions show that, if , and the

antineutrinos scatter on the electrons with right-hand

(left-hand) circular polarization, the final electrons can

only have right-hand (left-hand) circular polarization.

, ,1

Rg

When and the differential cross

and it is not sensitive to the

sections only depends on

antineutrino flavor. , 1,

Lg

e .

When

,

and

the differential cross section depends on

.

and therefore

it is sensitive to the antineutrino flavor. In this case the

differential cross section is different for and

So, spin asymmetry in antineutrino-electron scattering

in a magnetic field enables to use electrons with right-

hand circular polarization as polarized electron targets for

distinguishing antineutrino flavor and for detection of

antineutrinos. It is derived from the obtained expressions

for the cross section that electrons with left-hand circular

polarization in antineutrino-electron scattering in a

magnetic field are heated by and equally.

However, the of the obtained expressions for

the cross section show that electrons with right-hand

circular polarization are heated by and

unequally.

,e

e

This fact leads to the asymmetry in the heating

of the stellar matter.

For antineutrino-electron scattering we have

2 240, 0 4 1 1 Le

R v g I

When the incident and scattered antineutrinos fly along

the magnetic field and the initial electrons are on the

lowest Landau level, the cross section of the

antineutrino-electron scattering is sensitive to the

incoming antineutrino flavor. Such electrons can be

used as polarized electron targets for distinguishing

antineutrino flavor and for detection of antineutrinos.

Analyses of the cross sections of the

processes

, ,l e l e l ll e e l e e

To obtain the general expressions for and for the cross

section of these processes (via -boson exchange)

we have to consider that

W

2 1sin 0, , 2

2w V A F wg g G G

The process l ee l is forbidden for electrons

with right-hand circular polarization 1 .

The final neutrino and charged lepton are emitted asymmetrically.

The ultra relativistic charged lepton and ultra relativistic positron in

the process are emitted at small angles with

respect to the direction of the high-energy-(anti) neutrino

momentum.

In the process the electrons and the

positrons with left (right) helicity are emitted

asymmetrically.

l e l e

l l e e

CONCLUSIONS

● Spin asymmetry in neutrino-electron scattering in a

magnetic field enables to use electrons with left-hand

circular polarization as polarized electron targets for

distinguishing neutrino flavor and for detection of

neutrinos.

● The electrons with right-hand circular polarization

in neutrino-electron scattering in a magnetic field are

heated by , and equally. e

However, the electrons with left-hand circular polarization in neutrino-electron

scattering in a magnetic field are heated by and unequally. This fact

leads to the asymmetry in the heating of the stellar matter.

● Spin asymmetry in antineutrino-electron scattering in a magnetic field enables

to use electrons with right-hand circular polarization as polarized electron

targets for

distinguishing antineutrino flavor and for detection of antineutrinos.

e

● When the incident and scattered antineutrinos fly

along the magnetic field and the initial electrons are on

the lowest Landau level, the cross section of the

antineutrino-electron scattering is

●The electrons with left-hand circular polarization in

antineutrino-electron scattering in a magnetic field are

heated by and equally. However, the

electrons with right-hand circular polarization are heated

by , and unequally that leads to the asymmetry

in the heating of the stellar matter.

e

e

sensitive to the incoming antineutrino flavor. Such electrons can

be used as polarized electron targets for distinguishing

antineutrino flavor and for detection of antineutrinos.

● The process is forbidden for electrons with right-

hand circular polarization (=1). The final neutrino and charged

lepton are emitted asymmetrically.

● In the process the electrons and the positrons

with left (right) helicity are emitted asymmetrically.

l ee l

l l e e

Acknowledgements We are very grateful to the Organizing Committee

of the 17th International Spin Physics Symposium

and Professor Kenichi Imai for the kind invitation, the

support to attend this symposium and warm

hospitality. We are also very thankful to the

participants of the SPIN2006 Symposium for useful

discussions.