Neutrino propagation in magnetized media

PROCEEDtNGS SlJPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 70 (1999)267-269

Neutrino propagation

in magnetized

media

D. Grasso’” ‘Departament E-46100

de F&a

Burjassot,

Teorica,

Valencia,

Universitat

de Valencia

SPAIN

After a short presentation of the general techniques used to determine neutrino potentials in a magnetized medium I will discuss some applications to MSW resonant oscillations. I will also consider the relevance of our results for the pulsar velocity problem.

2.

1. Introduction The effect of media on the propagation mentary

particles

branch

of particle

Among

many important

search

subject

theoretical cillations solution

is nowadays physics

and astrophysics

applications

one of most

study

of MSW

[2] including

of ele-

a well established

was the

resonant

os-

their possible role for the

of the solar-neutrino

induced effects are expected matic in the environment

problem.

Medium

to be even more dra-

of collapsed stars and in

the early Universe where temperature

and matter

density can be much larger than in the Sun. In

such

another role:

extreme

physical

environments,

parameter

this is the magnetic

however,

ofte_n play a crucial field B.

Indeed,

mag-

netic fields as large as B N 1012 Gauss have been observed on the surface of pulsars and fields as large as 10 l6 Gauss are expected

to be present in

the core of neutron

and supernovas

(SN’s).

stars

(NS’s)

Several models also predict

strong

mag-

netic fields to be present in the early Universe. As I will discuss, such a strong fields can modify in several ways the properties a consequence,

of a medium and, as

those of the particles

through this medium. magnetic field induced

propagating

Only recently, however, effects have been started

to receive a wide attention (see e.g. [3-51). In this contribution I will concentrate my attention on neutrinos collapsing, stars.

propagating

in collapsed,

or

of the

neutrino

The main tool to study neutrino a medium,

as in the vacuum,

potentials propagation

in

is the Dirac equa-

tion. In the Fourier space this reads

of this re-

remarkable neutrino

[l].

Computation

(a,?

- m, - E(T, B, R)) where

Here C(T B, pi), and pi the chemical species,

&,(A)

of the i-th particle induced

contribution

den in the neutrino

(1)

T is the temperature

potential

is only the medium

being the vacuum

= 0 .

physical

self-energy,

to C already vacuum

hid-

mass m,.

Two kind of Feynman diagrams contribute to C(T, B, pi). These are the bubble and the tadpole diagrams

(see Fig.

ning in the internal to be intended

1 in ref.[5]).

The fermions run-

lines of these diagrams

as real (not virtual)

particles

have be-

longing to the heat-bath.

For temperatures

densities

in NS’s we can safely

as those present

population

and

neglect

any thermal

so that

W and 2 in ours diagrams

of gauge-bosons, are the usual

virtual

ones. Thermal

of muons and

populations

tauons are also negligible,

so that only electrons,

positrons

, proton and neutrons give a relevant contribution to C(T, B, pi). If we neglect (see below)

nucleon

polarizations,

the ambient

netic field can affect C only through the electron

propagator.

mag-

its effect on

A very powerful tool to

study this effect is given by the finite-temperature electron propagator in the presence of an external magnetic field. This propagator has been derived in refs.

[6] and applied for the first time to neu-

trino physics in [5]. The complete expression of this propagator is quite long and I do not report it here. The interested reader can find it in refs. 0920-5632/98/$19.00 0 1998 Elsevier Science B.V. PI1 SO920-5632(98)00436-S

All rights reserved

D. Grasso/Nuclear Physics B (Pmt. Suppl.) 70 (1999) 267-269

268

[5,6]. Using this propagator it is possible to evaluate the contributions of the bubble and tadpole diagrams to C(T, B, pi) [5]. Inserting such a result into the neutrino dispersion relation det (8,~~

- my - X(2’, B, p;))

= 0

(2)

we can then derive the matter induced potential of neutrinos propagating through an electrically neutral plasma I+,)

=

+

&F

-?+N.+2N;

c N,Li- ;Np

V&)

=

(3)

cos.4

i=p,7

&‘GF

-+

1

+ 2 c

N;

i=e,p,r

-

N;+;N,OcoSQ . 1

In my notation Ni represents the charge density of the i-th particle species. N,” stands for the electron charge density in the lowest Landau level (LLL). Finally, 4 is the angle between neutrino momentum and the magnetic field vector. It is important to observe that both quantities N, and N,” are functions of T, pi and B’. Indeed, we have

terms in the neutrino potentials. Although nucleon anomalous magnetic moments are much smaller than the electrons’, it is however possible that a strong degeneration of the electron gas suppress electron polarization. If, at same time, the nucleons are non-degenerate it may then happen that nucleon polarization is not negligible [7,8]. A similar situation can actually be realized in the central regions of hot NS’s. To conclude this section we summarise that the effect of the magnetic field on the neutrino propagation is two-folds. Strong magnetic fields may modify the bulk properties of the electronpositron gas inducing an enlargement of the potential to which the V, is submitted. Besides that, the magnetic field induces a polarization of particles having a non-vanishing magnetic moment. We have shown that this latter effect gives rise to an angular dependence in the neutrino potential. 3. Resonant

oscillations

I will now apply the result of the previous section to study MSW-type neutrino resonant oscillations. The resonance condition for the MSW transition between u, and u,+~ is

(7) Substituting (3,4) in (7) we find (see also [7] for an independent derivation of the same result) ~cos28=~GsNe(1+Xeos~)=0

where 7~labels Landau levels and f,+r (n, p,, pe) are the Fermi-Dirac distributions of electrons and positrons. For a fixed electron chemical potential, both N, and N,” tend to increase almost linearly with B whenever eB >> p:,T2. Since, due to the double degeneracy of the n > 1 Landau level, only the LLL contributes to the spin-polarization of the electron-positron gas, it is possible [7] to rewrite the angular dependent terms in (3,4) in terms of a polarization parameter defined by A I -Nf/Ne. The reader should keep in mind that nucleon polarizations may also induce angular dependent

(8)

where Am2 _ m2YP,Z- m2 . Since in NS’s and SN’s the charge asymmet$ of neutrinos is usually negligible, I have ignored here the heath-bath neutrino contribution to the potential. Resonant MSW oscillations between active and sterile neutrinos can also be studied. As V(YS) = 0 we find the resonance conditions Am2 -cos20=fiG~ 2E

[ N e (1,

Am2 -cos28=fiGF 2E

[ N c (+COS#J)

$cos+)

- ;N+‘)

- fN+O)

respectively for u, H vs and vP,+ H vs oscillations. Here we assumed Am2 > 0 in those cases

D. Gra.wo/Nuclear Physics B (Proc. Suppl.) 70 (1999) 267-269

for which mvs > my._. that u~,~ ++ us resonant sible if m,, < mVe ,* >T. 4.

Consequences

We then see from (10) oscillations are only pos-

for pulsar

velocities

One of the most challenging problems in modern astrophysics is to find a consistent explanation for the high velocity of pulsars. Observations [9] shows that these velocities range from zero up to 900 km/s with a mean value of 450 f 50 km/s . It has been recently proposed that a solution of this puzzle may be provided by an interplay between neutrino resonant oscillations and SN’s magnetic fields [lO,ll]. This idea is based on the angular dependence of the resonance condition that we discussed in the previous section. In the case of ye - v, oscillation [lo] such a mechanism heavily bears on the assumption that the resonance sphere lays between the tauon and electron neutrino spheres. In this case, in fact, the distortion of the resonance sphere would induce a temperature anisotropy of the escaping tau-neutrinos produced by the oscillations, hence a recoil kick of the proto-neutron star. In order to give rise to the observed pulsar velocities an 1% asymmetry in the escaping neutrino total momentum is required. Using (8) we find the predicted asymmetry to be [12] Ak 1 hN, --_-_x k 9 hT

metric neutrino transport in SN’s due to the angular dependence of the neutrino mean free time in the presence of a magnetic field. This effect has been first considered in [13]. I point here that a more careful analysis of this effect may be performed through the determination of the imaginary part of the neutrino self-energies using thermal-field theory techniques. This approach would directly provide the thermalization rates of neutrinos as a function of the angle 4, hence the neutrino momentum asymmetry [14]. REFERENCES 1.

2.

3. 4.

5. 6.

(11)

where hN, f ]dlnN,/dr]-i and hT f Id ln T/dT 1-l are, respectively, the variation scale heights of T and N, at the resonance radius. The value of the ratio * ranges in the interval O.l- 1 the exact value depending on the adopted SN model. We then see from (11) that at least a N 10% electron polarization is required to achieve the desired anisotropy. Using (5,6) (see also [7]) this requirement translates into a minimal value of the dipolar component of the magnetic field strength which lies in the range 1015+101s Gauss. In conclusion, it is worthwhile to observe that, if such high values of magnetic field strength are generally present in SN’s, a more conventional solution of the pulsar velocities puzzle may be invoked. Such a solution is related with an asym-

269

10. 11. 12. 13. 14.

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