PROCEEDtNGS SlJPPLEMENTS

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

Neutrino propagation in magnetized media D. Grasso’”

‘Departament de F&a Teorica, Universitat de Valencia E-46100 Burjassot, 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.

1. Introduction

The effect of media on the propagation of ele- mentary particles is nowadays a well established branch of particle physics and astrophysics [l]. Among many important applications of this re- search subject one of most remarkable was the theoretical study of MSW neutrino resonant os- cillations [2] including their possible role for the solution of the solar-neutrino problem. Medium induced effects are expected to be even more dra- matic in the environment of collapsed stars and in the early Universe where temperature and matter density can be much larger than in the Sun.

In such extreme environments, however, another physical parameter ofte_n play a crucial role: this is the magnetic 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 stars (NS’s) and supernovas (SN’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 of a medium and, as a consequence, those of the particles propagating through this medium. Only recently, however, magnetic field induced effects have been started to receive a wide attention (see e.g. [3-51).

In this contribution I will concentrate my at- tention on neutrinos propagating in collapsed, or collapsing, stars.

2. Computation of the neutrino potentials

The main tool to study neutrino propagation in a medium, as in the vacuum, is the Dirac equa- tion. In the Fourier space this reads

(a,? - m, - E(T, B, R)) &,(A) = 0 . (1)

Here C(T B, pi), where T is the temperature and pi the chemical potential of the i-th particle species, is only the medium induced self-energy, being the vacuum contribution to C already hid- den in the neutrino physical vacuum mass m,.

Two kind of Feynman diagrams contribute to C(T, B, pi). These are the bubble and the tadpole diagrams (see Fig. 1 in ref.[5]). The fermions run- ning in the internal lines of these diagrams have to be intended as real (not virtual) particles be- longing to the heat-bath. For temperatures and densities as those present in NS’s we can safely neglect any thermal population of gauge-bosons, so that W and 2 in ours diagrams are the usual virtual ones. Thermal populations of muons and 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 mag- netic field can affect C only through its effect on the electron propagator. 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.

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

[5,6]. Using this propagator it is possible to eval- uate the contributions of the bubble and tadpole diagrams to C(T, B, pi) [5]. Inserting such a re- sult 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 cos.4

1 (3)

i=p,7

V&) = &‘GF -+ + 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

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 de- pendent 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

terms in the neutrino potentials. Although nu- cleon anomalous magnetic moments are much smaller than the electrons’, it is however possible that a strong degeneration of the electron gas sup- press 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 electron- positron gas inducing an enlargement of the po- tential to which the V, is submitted. Besides that, the magnetic field induces a polarization of par- ticles 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 sec- tion to study MSW-type neutrino resonant oscil- lations. 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 (8)

where Am2 _ m2 YP,Z - m2 . Since in NS’s and SN’s the charge asymmet$ of neutrinos is usu- ally 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~ N

2E [ e (1, $cos+) - ;N+‘)

Am2 -cos28=fiGF N 2E

[ c (+COS#J) - fN+O)

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

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

for which mvs > my._. We then see from (10) that u~,~ ++ us resonant oscillations are only pos- sible if m,, < mVe * T. , >

4. Consequences for pulsar velocities

One of the most challenging problems in mod- ern astrophysics is to find a consistent explana- tion 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 reso- nance sphere lays between the tauon and electron neutrino spheres. In this case, in fact, the distor- tion of the resonance sphere would induce a tem- perature 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% asymme- try in the escaping neutrino total momentum is required. Using (8) we find the predicted asym- metry to be [12]

Ak 1 hN, --_-_x k 9 hT

(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 so- lution of the pulsar velocities puzzle may be in- voked. Such a solution is related with an asym-

metric neutrino transport in SN’s due to the an- gular 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.

G. Raffelt, Stars as a Laboratory for Fun- damental Physics, The University of Chicago Press, 1996. L. Wolfenstein, Phys. Rev. D1’7 2369 (1978); S.P. Mikheyev and A.Y. Smirnov, Nuovo Ci- me&o QC, 17 (1986). J.C. D’Olivo, J.F. Nieves and P.B. Pal, Phys. Rev. D40 3679 (1989). S. Esposito and G. Capone, Z. Phys. C70, 55 (1996); J.C. D’Olivo, J.F. Nieves and P.B. Pal, Phys. Lett. B383 87 (1996). P. Elmfors, D.Grasso and G. Raffelt, Nucl. Phys. B479 3 (1996). K.W. Mak, Phys. Rev. D49 6939 (1994); P. Elmfors, D. Persson and B.-S. Skagerstam, AShpart. Phys. 2 299 (1994). H. Nunokawa, V.B. Semikoz, A.Y. Smirnov and J.W.F. Valle, Nucl. Phys. B501 17 (1997). E.Kh. Akhmedov, A. Lanza and D.W.Sciama, Phys. Rev. D56 6117 (1997). A.G. Lyne and D.R. Lorimer, Nature 369 127 (1994).

10. A. Kusenko and G. Segrk, Phys. Rev. Lett. 77 4872 (1996).

11. A. Kusenko and G. Segre, Phys. Lett. B396 197 (1997).

12. Y.Z. Qian, Phys. Rev. Lett. 79 2750 (1997). 13. A. Vilenkin, Astrophys. J. 451 700 (1995). 14. P. Elmfors, D. Grass0 and P. Ullio, work in

progress.