Electromagnetic Interactions and Chirality Flip of Neutrinos in a Thermal Background

C LAPP-TH-329/91

T. Altherr

Laboratoire d'Annecy-le- Vieux de Physique des Particules IN2P8-CNRS, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France

and K. Kainulainen

NORDITA, Blegdamsvej 17, DK-UOO, Copenhagen 0, Denmark

Abstract We investigate the finite temperature and density effects on the electromagnetic properties of neutrinos. For a Standard Model neutrino we find a left-handed effective charge and a chirality flipping term associated with the longitudinal photon exchange. We find that in the supernova the chirality flipping rate is two orders of magnitude larger than the previous estimates using the vacuum magnetic moment. 1 Introduction

The properties of neutrinos in a thermal background can differ very much from the ones in the vacuum. A famous example is given by the MSW [1] solution to the solar neutrino problem. Another explanation of the observed neutrino deficit from the sun is based on the chirality flip due to an hypothetical large magnetic moment. This could also account for the time variations observed in the 37Cl-data [2]. In the zero temperature field theory the chirality flipping rate of a neutrino state is related to the magnetic moment of the neutrino. In the Standard Model the magnetic moment is generated at the one loop level and is extremely small because the only mass scales in the problem are mu and Mw (the internal lepton mass does not intervene at the leading order) and hence one must have fi{v) « mu/M\V. In fact [3] "<"<> " W ^3-2 * ir"» (SvO • <"> where ^B = e/2me is the electron Bohr magneton. Due to the smallness of fi(ve) the chirality flipping rate is negligible for most physical applications in the Standard Model at zero temperature.

In the finite temperature and density there will be new natural scales, namely the temperature T and the chemical potentials Ji,. The dimensional argument that leads to the smallness of IM(VF) in the vacuum case is then not applicable and one can expect to find significantly enhanced chirality flipping rate if the neutrino is immersed in a heat bath having large temperature and/or large chemical potential. Such extreme conditions prevailed in the early Universe, and can even at present be found inside the superdense hot cores of the exploding supernovae. In a plasma, a neutrino will also acquire an effective charge [4]. In this paper we calculate the electromagnetic interaction Lagrangian for a Standard Model neutrino in a thermal background. In the soft photon momentum limit we find a left-handed charge renormalization term and a new flipping term which is not related to the magnetic moment but instead is connected with a longitudinal photon exchange. For supernova, we estimate the flipping rate and find that it is two orders of magnitude larger than the naive prediction.

2 Neutrino-photon coupling in matter

Because neutrinos are neutral particles they have no coupling to photons in the Standard Model at tree level. However, such a coupling is generated already at the one loop level. The relevant diagrams for the finite temperature and/or density vertex corrections are represented in fig.l and e.g. the contribution coming from the W-graph can be written down as fj (2.1) where L = ^(I - 7,5) as usual. The scalar part of the propagators is factorized in the f{p,p', fc)-term. Using the real time formalism [5,6] one can prove that the physical part of the vertex associated with the matter effect is 2 2TT \S((p - k) - m?e)NF{p - k) , ,. where the W is at rest. Indeed, for most physical applications, both temperatures and chemical potentials are much smaller than the vector gauge boson masses. In this case the Z-graph gives a similar contribution. The statistical weight TVf is the Fermi-Dirac distribution

Because of this thermal factor the finite temperature and/or finite density corrections are always ultraviolet finite. Hence, contrary to the vacuum case we only need to consider the diagrams represented in fig.l. The integration over k in eq.(2.1) can be worked out analytically in two important limits, namely when the exchanged momentum q = p—p' is much smaller than the momenta of the thermalized electrons (of the order of /2 or T) or, the opposite situation which turns out to be negligible as all results become inversely proportional to q2. Also, as we will see in the next section, the flipping rate is enhanced for small q (soft) exchanged photons as compared to the large q (hard) exchange. In the soft limit we find the matter induced vertex function to be [7]

- ieA^ = ierfL - i\}mvu^\ (2.4) where the first piece is the induced left-handed charge coupling and the second piece is a chirality flipping term. We have defined

and X1 = .A ^?*. (M)

with 2 A = 2\/2GF(1 + 4 sin Sn-) (2.6) and the finite temperature/density integrals

I3 = — 2 K0 = ^-Jd"k6(k -ml)NF(k)~. (2.7)

These results are valid either for a hard but almost light-like photon (q2 —» 0) for which ei = (q-u)Xf, or for a soft photon (in which case g2 cannot be neglected in front of (q-u)2). For an ultra-relativist!c and ultra-degenerate electron gas, the above integrals reduce to (in the plasma rest-frame)

ft X i- * + « (2.8) qo-Q The charge term in eq. (2.4) leads to the well-known phenomenon of plasmon decay- in stars [8j. Typical value of the induced charge inside the supernova core is ei/e ~ 4 2 12 2 8 x HT (//,./10OMeV) and in the early Universe eL/e ~ 1.4 x 10~ (T/IMeV) . It should be noted that because this charge is purely left-handed it does not have any new significant physical consequences. The existence of the flipping term is a unique property of the heat bath. It is due to the presence of the longitudinal photons which can couple to the internal electron without changing its helicity, so that the chirality flip can occur in the outer leg. Let us recall that in vacuum, the magnetic moment appears only in the ultra-violet sector, i.e., there is no magnetic moment in a contact theory. Furthermore, the photon is purely transverse which has the consequence that the magnetic moment term flips both chirality and helicity (at the leading order). It should be noted that while the vertex function (2.4) cannot change the helicity of an incoming particle, in most applications the incoming neutrino states are left-chiral wave packets rather than helicity eigenstates [9], whence the flipping term will have physical consequences.

3 Chirality flipping rate

Recently there has been lots of progress concerning the calculation of scattering rates at finite temperature and/or finite density. The main difference with the vacuum theory is the possibility of having incoming particles from the heat bath. For soft momenta, a partial resummation of multiple scatterings is necessary as was shown by Braaten and Pisarski [10] in a general context and in certain interesting examples [11,12]. The emission rate of a given particle can be obtained through the use of cutting rules [13]. With the effective coupling given in eq.(2.4) the emission rate of a right-handed neutrino produced by a thermal bath of left-handed neutrinos is given by 2ET(p) = ml f

x Yi, "(P. *)£7s( fo- À + Tn11)^Ru(P, s), (3.1) 3 where the cut propagators for the neutrino are

Sï(p -q) = (9(q0 - Po) - NF(p - ç)) 2*6((p - qf - ml) (3.2) and for the photon the contraction with the tensor v?u" leaves intact only the longitudinal part

( ^) n-L(q). (3.3)

The photon being soft, we use the effective propagator given in [12] where nfj(q) is the usual Bose-Einstein distribution function and T?L is the longitudinal part of the photon polarization tensor. The delta function term in eq.(3.4) corresponds to a quasi particle propagation and the second one is associated with Landau damping. Then, eq.(3.1) can be written as (in the plasma rest-frame)

l 2 2 2ET(p) = - -(e\muY J ^ (qj3 - q K3) ^(^(p - q). (3.5)

It is easy to see that since the internal neutrino is on the mass shell, the quasiparticle interaction process is actually forbidden and only the Landau-damping part contributes to the nipping rate. Let us now estimate the flipping rate (3.5) in the case of supernovae. Inside the nascent neutron star the left-handed neutrinos are trapped and thermalized. Electrons and neutrinos are highly degenerate so that typical values for their chemical potentials are p,e ~ 300 MeV and //„, ^ 200 MeV. The soft momentum scale is given by the plasmon frequency uj{) = ejlf/TTy/Z ~ 20 MeV. As in [12] the main contribution of the rate comes from the "pole-cut" term, that is with the term associated with Landau damping, a soft space-like photon interacting with the neutrino. In the soft region we can estimate the emission rate with the energy E as

since in the integral of eq. (3.5) one gets ~ U>Q from the phase space and l/u>o from the propagator. C is a remaining dimensionless number, which we have numerically found to be ~ 2. For hard photon one would get ~ p,*T from the phase space (taking Pauli blocking into account) and «»//*« from the propagator leading to a smaller contribution to the flipping rate. Using eq.(3.6) one can obtain the total energy release in form of right-handed states for the SN1987A event. Taking the number of neutrinos in the core region to be Nv ~ 56 0.05 x Nnucl ~ 10 and E ~ \ftVf we obtain

(^)2 (3.7)

which, for the allowed range of mVc < 20 eV [14], is still much smaller than the total 52 observed [15] luminosity of Q0*, ~ 2 x 10 erg/s. However, the energy release given by eq.(3.7) is by 2 orders of magnitude larger than the naive estimation Q ~ 4 x 1038erg obtained using the vacuum magnetic moment of eq.(l.l) [16].

4 Conclusion

We have calculated the electromagnetic vertex of neutrinos in a thermal background in the context of the Standard Model. We have found a left-handed effective charge and a chirality flipping term. We find that in supernova the chirality flipping rate induced by thermal effects is 102 times larger than what is obtained in previous calculations. This enhancement is still too small to give restrictions on the Standard Model, but it casts some doubt on the validity of the bound obtained for the magnetic moments in some exotic models. We expect such an enhancement for any model where the vertex includes a thermalized particle. It should be noted that the presence of longitudinal photons (forbidden in the vacuum) which induce chirality flipping is a model independent feature.

Let us finally mention that our calculation, applied on the sun, should be greatly modified as this system is nonrelativistic and therefore there is no reason to expect the same enhancement as in a supernova. We also found no significant enhancement in the case of the early Universe. Acknowledgements

We would like to thank Kari Enqvist for many discussions, and Keijo Kajantie and Victor Semikoz for helpful comments. One of us (KK) also thanks the Emil Aaltonen foundation for a grant. References •1; L. Wolfenstein, Phys. Rev. D17 (1978) 2369; S.P. Mikheyev and A. Yu. Smirnov, Sov. J. Nucl. Phys. 42 (1985) 913. [2] G. Fiorentini and G. Mezzorani, Phys. Lett. B253 (1991) 181. [3] B.W. Lee and R.E. Shrock, Phys. Rev. D16 (1977) 1444; K. Fujikawa and R.E. Shrock, Phys. Rev. Lett. 45 (1980) 963. [4] V.N. Oraevsky and V.B. Semikoz, Physica 142A (1987) 135. [5] For a review, see N.P. Landsman and Ch. G. Van Weert, Phys. Rep. 145 (1987) 141. [6] R. Kobes, Phys. Rev. D42 (1990) 562. [7] Electromagnetic properties of neutrinos have previously been studied in Ref.[4] and in J.C. D'Olivo, J.F. Nieves and P.B. Pal, Phys. Rev. D40 (1989) 3679. [8] J.B. Adams, M.A. Ruderman and C. Woo, Phys. Rev. 129 (1963) 1383; M.H. Zaidi, Nuovo Cimento 4OA (1965) 502. [9] K. Kainulainen, J. Maalampi and J.T. Peltoniemi, Nucl. Phys. B, to appear. [10) E. Braaten and R.D. Pisarski, Nucl. Phys. B337 (1990) 569 and Nucl. Phys. B339 (1990) 310. [11] E. Braaten, R.D. Pisarski and T. Chiang-Yuan, Phys. Rev. Lett. 64 (1990) 2242. [12] T. Altherr, Annals of Physics, to appear. [13] H.A. Weldon, Phys. Rev. D28 (1983) 2007; R.L. Kobes and G.W. Semenoff, Nucl. Phys. B260 (1985) 714 and Nucl. Phys. B272 (1986) 329. [14] T.J. Loredo and D.Q. Lamb, Ann. N.Y. Acad. Sci. 571 (1989) 601. [15] R.M. Bîonta et al., Phys. Rev. Lett. 58 (1987) 1494; CB. Bratton et al., Phys. Rev. D37 (1988) 3361; K.S. Hirata et al., Phys. Rev. Lett. 58 (1987) 1490 and Phys. Rev. D38 (1988) 448. [16] R. Barbieri and R. N. Mohapatra, Phys. Rev. Lett. 61 (1988) 27. Figure Captions

Fig.l The W and Z vertex diagrams for neutrinos. W

Fig. 1