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 : [email protected]
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.