L int = 1. V MSW =2 1/2 G F ρ e. (2) V W (r) =2 1/2 G F ρ ν (r), (5)

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1 Electrostatic corrections to the Mikheyev-Smirnov-Wolfenstein potential of a neutrino in the sun C. J. Horowitz Nuclear Theory Center and Dept. of Physics, Indiana University, Bloomington, IN 705 (December, 00) A neutrino in matter polarizes the medium via the weak interaction and acquires a small effective electric charge. This can interact with stellar electric fields. Stars have electric fields to help support the ions against gravity. This gives an electrostatic correction that increases the MSW potential of a neutrino in the sun by about a factor of five. This can significantly effect neutrino oscillations. A neutrino in matter acquires a small flavor dependent potential because of its weak interactions with electrons []. This can dramatically change neutrino oscillations via the MSW effect [,]. We start from the interaction Lagrangian, L int = G ψ(γ / F µ c v γ µ γ 5 c a )ψ νγ µ ( γ 5 )ν. () Here G F is the Fermi constant, ψ is an electron, proton, or neutron field and ν is the neutrino field. The coupling constants c v and c a are collected in Table I. For an electron neutrino both neutral Z 0 and charged W ± bosons contribute. However for mu or tau neutrinos only the Z 0 contributes. The expectation value of the interaction Hamiltonian from Eq. () gives rise to a potential. The difference of this potential for ν e and ν µ neutrinos is the MSW potential V MSW, see Fig. (a), V MSW = / G F ρ e. () Here ρ e is the number density of electrons. An MSW resonance occurs when the density is such that V MSW compensates for differences in neutrino masses and can lead to large neutrino flavor conversion, V MSW resonance = m E cos θ. () Here m = m m for neutrino states of mass m and m and mixing angle θ and E is the neutrino energy. Resonances are possible even for small values of V MSW. At the center of the sun, with a density near 50 g/cm, V MSW (r =0) ev. () This small potential can be very important for neutrino oscillations. We emphasize, the interpretation of all solar neutrino experiments, many terrestrial ν oscillation experiments, and the inferred neutrino masses and mixing angles, all depend on the form of V MSW in Eq. (). The MSW potential V MSW, that Wolfenstein described as the neutrino forward scattering amplitude times the density [], is equal to the self-energy of a neutrino in the system. This is a complex many-body quantity. Can there be many-body corrections to V MSW? Many works have considered corrections from magnetic fields, see for example []. In this paper we calculate the electrostatic correction from the effective electric charge of a neutrino in matter. This interacts with the electric field in the sun. I. NEUTRINO EFFECTIVE CHARGE A neutrino polarizes the medium via the weak interactions and acquires a very small effective electric charge e ν [ 6]. We present a simple Debye Hückel derivation [7] of this charge for a neutrino in the solar plasma. This derivation shows that e ν is closely related to V MSW.We model the sun as a classical ideal gas of electrons and ions. We neglect small corrections from coulomb interactions and electron degeneracy. If a neutrino has an interaction V MSW then an electron has an interaction V W, V W (r) = / G F ρ ν (r), (5) where the neutrino density ρ ν is normalized to one ( d rρ ν (r) = ). Note that this is actually the interaction with a ν e minus the interaction with a ν µ or ν τ. This will lead to an electron density, δρ e (r)+ρ e = ρ e exp( V W (r)/kt ). (6) Here ρ e is the unperturbed electron density and T is the temperature. In thermal equilibrium, the density is related by a Boltzmann factor to the potential energy. Expanding Eq. (6) to first order in V W (a truly excellent approximation) yields the polarization correction to the electron density δρ e (r), δρ e (r) = V MSW kt ρ ν(r). (7) Integrating this over all space and multiplying by the electron charge e e = e yields the effective charge e ν for the neutrino, e ν = e V MSW kt. (8)

2 Again this is the effective charge of a ν e minus the effective charge of a ν µ or ν τ. Alternatively, one can calculate e ν by simply evaluating the diagram in Fig. (b). This yields, densities creates an electric field that helps support the ions. This field also keeps the electrons in the star. Thus, there is an electric field inside the plasma with a strength related to the gravitational field. e ν = e / G F Π 00 (q 0 =0,q 0), (9) Here Π 00 is the electron polarization insertion in the medium at finite density. See for example ref. [5] at zero temperature or ref. [8] at finite temperature. This shows that the effective charge depends on the frequency q 0 and momentum transfer q of the electromagnetic field. Here we are interested in a static electric field. This corresponds to the limit q 0 =0andthenq 0. In this limit Π 00 (q 0 =0,q 0) = ρ e /kt, for a nondegenerate system, which reproduces Eq. (8). Note that this nonzero Π 00 causes electric fields to be screened in a plasma. For a degenerate system, for example in a core collapse supernova, Π 00 (q 0 =0,q 0) = k F (kf + m e )/ /π, with k F the Fermi momentum and m e the electron mass. Therefore Eq. (8) is replaced by, e ν = e / G F k F (k F + m e) / /π. (0) This can be large, e ν 0 8 efork F 00 MeV, in a supernova. At the center of the sun, with kt.6 kev, e ν from Eq. (8) is, e ν (r =0) e. () We note that e ν is positive. The repulsive weak interaction induces a small reduction in the electron density near the neutrino. This reduction is equivalent to a small positive charge. Thus a neutrino in matter acts as a tiny fraction of an electron hole. This charge allows a neutrino to interact with electric fields. However most electric fields are screened in the plasma. There is a basic Catch- that limits the role of the effective charge of a neutrino in matter. If an electromagnetic field, such as the magnetic field, is not screened, it can penetrate the plasma. However, if it is not screened, there will be no polarization cloud for the neutrino to couple to. Therefore the neutrino does not develop a transverse effective charge for coupling to magnetic fields. Alternatively, the electric field is screened by the plasma. In this case the screening cloud can couple via the weak interaction and the neutrino does develop an effective charge. However, because of the screening, it is difficult for an electric field to penetrate the plasma. Therefore there may be no electric field for the neutrino to interact with. Stars may get around this Catch- by using a gravitational field to induce an electric field. The gravitational field acts mostly on the positive ions because they are much heavier than the electrons. This causes the ions to sag slightly until the difference in ion and electron charge II. THE ELECTRIC FIELD IN STELLAR PLASMAS An electron in a stellar plasma will feel a buoyancy force F B from the gradient in the pressure. Hydrostatic equilibrium insures that this force is equal to the gravitational force on material of average density. We model the sun as an ideal gas of electrons, protons, and He and 6 O ions. The mass fraction of protons is X, He is Y and metals (all ions with charge greater than that we assume to be 6 O) have mass fraction Z = X Y. Note that metals make only a very small contribution. The average mass per particle of bulk material in the sun is <M>=6m p /( + 0X Z), with m p the proton mass. Therefore F B at a position r in the sun is, F B = GM(r) r <M>, () and is directed radially outward. Here M(r) = π r 0 r dr ρ m (r ) is the enclosed mass with ρ m the mass density. This force is balanced by an electric field of strength E, E = e F GM(r) <M> B = er, () Integrating Eq. () yields the electrostatic potential φ in the sun, φ(r) = R r GM(r ) <M> er dr. () We assume this vanishes at the surface φ(r ) = 0. Inside the sun, φ is positive, since it supports positive ions against gravity. At the center of the sun we calculate, φ(r =0) 500 V. (5) The electric energy eφ is the same order of magnitude as the gravitational energy per nucleon. Furthermore, the virial theorem insures that this is the same order of magnitude as the thermal energy kt. Indeed at r =0, kt 60 ev, (6) which is similar to e times Eq. (5). A neutrino in the sun will feel an additional electromagnetic potential V E that is the product of its effective charge Eq. (8) times the electrostatic potential Eq. (), V E (r) =e ν φ(r), (7)

3 so that the total potential is, Using Eq. (8), this is (r) =V E (r)+v MSW (r). (8) (r) =( eφ(r) kt +)V MSW(r). (9) Evaluating eφ/kt at r = 0 with Eqs. (5,6) yields, (0).9 V MSW (0). (0) We evaluate φ(r) ande ν using solar model profiles of X(r), Y (r), kt, andρ m (r) from Ref. [9]. We do not expect our results to be very sensitive to the solar model. Figure shows φ(r), e ν, (r) and /V MSW. We find that /V MSW is 5 over most of the core, but drops quickly near the surface. Note that our calculation may not be accurate very near the surface of the sun. However, we expect to be very small there. We find an increase in by about a factor of 5 compared to V MSW. This could have large effects on solar neutrino oscillations and on the m and mixing angles one might extract from experiment. Before considering oscillations, we discuss another way to derive this effect that may clarify its origin. In the medium there is mixing between the Z 0 and photon propagators as described by the polarization diagram in Fig. (c). Note that in addition to Z 0 exchange there is also a similar exchange diagram with a W ±. In the vacuum, this polarization vanishes as qµ 0. However, in the medium Π 00 (q 0 =0,q 0) does not vanish. This leads to mixing of order G F Π 00 between the Coulomb potential and a weak potential. Therefore, along with the Coulomb potential of Eq. () there will always be a weak potential φ W of order / G F Π 00 eφ = e ν φ. This weak potential is present in the sun at all times, even without neutrinos. A neutrino couples to φ W with a weak charge of order one. This reproduces V E. III. EFFECTS OF V TOT ON NEUTRINO OSCILLATIONS We now discuss possible implications of. () One should redo calculations of neutrino oscillations using the new larger potential and search for new neutrino parameters to try and reproduce all of the solar neutrino experiments. () The larger potential will allow neutrinos with a larger m > 0 ev to still undergo an MSW resonance in the sun. () Perhaps is large enough to allow matter enhanced oscillations with the atmospheric m 0 ev to fit the solar neutrino data? This scenario could free up a m for the LSND neutrino oscillation signal []. () We suggest that matter effects will depend on more than just the electron density. Matter effects in the earth can be different from matter effects in the sun, even at the same density. Therefore, one should reanalyze daynight differences in solar neutrino signals. (5) One should test by comparing detailed solar neutrino experiments with terrestrial experiments such as Kamland [] to see if a common set of neutrino oscillation parameters can explain both results. (6) We expect to be much larger in core collapse supernovae because of the high densities, see Eq. (0), and very large gravitational fields. This could strongly impact neutrino oscillations. (7) For oscillations of active to sterile neutrinos there will be a similar large correction. This will involve a neutrino effective charge that has contributions from polarization of the ions by Z 0 exchange in addition to polarization of the electrons. (8) It may be possible to reproduce this effect in terrestrial experiments using laboratory electric fields. One could consider either neutrino oscillations or perhaps a similar effect will be present for electric field influenced matter enhanced oscillations of neutral K mesons. We note that the solar electric field Eq. (5) is relatively modest and that e ν in the sun is significantly reduced by the sun s very high temperature. The effective charge in degenerate matter is of order ev MSW /E F where E F is the Fermi energy. Therefore it may only require an electric field in the medium that yields a potential difference of order E F, perhaps a few volts, to significantly influence oscillations. If feasible, a terrestrial experiment might be useful for studying neutrino properties, testing symmetries, or preparing neutrino beams. The following is both preliminary and speculative. We assume 5V MSW in the sun and V MSW in the earth. If the solar mixing angle θ is reasonably large and the neutrinos undergo an adiabatic transition in the sun, the survival probability will be, P νe ν e sin θ. () This is independent of neutrino energy and m.therefore the electrostatic correction V E may only shift the location of the MSW resonance in the sun to lower densities without changing P νe ν e. The constraint on m may come from the low threshold Ga experiments that see a larger fraction of the standard solar model flux than the Cl experiment. This may require that very low energy neutrinos have a resonance density larger than the central density of the sun. Presently, the LMA solution including the SNO data is near m ev []. Perhaps with V E, m will be about five times larger, m.5 0 ev. Note that constraints on m from day/ night asymmetries and from experiments with reactor anti-neutrinos should not be modified by V E. In conclusion, a neutrino in matter polarizes the medium via the weak interaction and acquires a small

4 effective electric charge. Inside stellar plasmas there is an electric field that helps support the ions against gravity. Therefore, there will be an electrostatic correction that increases the MSW potential for a neutrino in the sun by about a factor of five. This may significantly impact neutrino oscillations. We thank Wick Haxton and Brian Serot for useful discussions and Steven Girvin, Malcolm Macfarlane, Jorge Piekarewicz and Mike Snow for helpful comments on the manuscript. This work is supported in part by DOE grant DE-FG0-87ER065. Here g a.60, sin θ w 0.5. [] L. Wolfenstein, Phys. Rev. D 7 (978) 69; Phys. Rev. D 0 (979) 6. [] S. P. Mikheyev, and A. Yu. Smirnov, Sov. J. Nuc. Phys. (986) 9; JETP 6 (986) ; Nuovo Cimento, 9C (986) 7. [] A. Kusenko and G. Segre, Phys. Rev. Lett. 77 (996) 87. [] V. N. Oraevsky, V. B. Semikoz, and Ya A. Smordinsky, JETP Lett. (986) 709; L. B. Leinson, V. N. Oraevsky and V. B. Semikoz, Phys. Lett. B 09 (988) 80; V. N. OraevskyandV. N. Ursov, Phys. Lett. B09 (988) 8. [5] C. J. Horowitz and K. Wehrberger, Phys Rev. Lett. 66 (99) 7; C. J. Horowitz, Phys. Rev. Lett. 69 (99) 67. [6] A. I. Rez, V. B. Semikoz, hep-ph/0059. [7] P. Debye and E. Hückel, Physik. Z. (9) 85. See for example A. L. Fetter and J. D. Walecka, The Quantum Theory of Many-Particle Systems, McGraw Hill, NY 97, p79. [8] C. J. Horowitz and K. Wehrberger, Phys. Lett. B 66 (99) 6. Note that Eqs (7) and () of this reference should be divided by e q 0/T. [9] John N. Bahcall, M. H. Pinsonneault, Sarbani Basu, Ap J. 555 (00) 990. [0] Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 8 (998) 56. [] LSND Collaboration, C. Athanassopoulos et al., Phys. Rev. C5 (996) 685; Phys. Rev. C58 (998) 89. [] V. Barger, D. Marfatia and B. P. Wood, Phys. Lett. B98 (00) 5. [] Q. R. Ahmad et al, Phys Rev. Lett. 89 (00) 00. Table I Coupling Constants Interaction c v c a ν e e +sin θ w ν µ e, ν τ e sin θ w ν p sin θ w g a / ν n g a / FIG.. Feynman diagrams for (a) the MSW potential V MSW, (b) the neutrino effective charge e ν, and (c) the mixing between weak and electromagnetic interactions induced by the in medium polarization. The loop represents the electrons at finite density, the heavy dot is the weak interaction, and the X is the coulomb potential. / V MSW 6 5 φ e ν /V MSW r/r sun FIG.. Ratio of /V MSW (solid curve) versus radius in units of the solar radius R. Also shown is the electric potential φ in kv (dotted curve), the neutrino effective charge in units of 0 5 e (dashed curve), and in units of 0 ev (dot-dashed curve).

5 6 5 / V MSW φ /V MSW e ν r/r sun