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Daniele Montanino Università degli Studi di Lecce & Sezione INFN,

I. XX International Conference on Neutrino Physics and Astrophysics. Analytic Treatment of Neutrino Oscillations in Supernovae. Daniele Montanino Università degli Studi di Lecce & Sezione INFN, Via Arnesano, 73100, Lecce, Italy daniele.montanino@le.infn.it. Abstract

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Daniele Montanino Università degli Studi di Lecce & Sezione INFN,

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  1. I XX International Conference on Neutrino Physics and Astrophysics Analytic Treatment of Neutrino Oscillations in Supernovae Daniele Montanino Università degli Studi di Lecce & Sezione INFN, Via Arnesano, 73100, Lecce, Italy daniele.montanino@le.infn.it Abstract We present a simple analytical prescription for the calculation of the neutrino transition probability in supernovae. We generalize the results in the most general case of  three-neutrino flavor transition and in presence of non standard flavor changing and flavor diagonal interactions Based on [1] G.L. Fogli, E.Lisi, D. M., and A. Palazzo, Supernova neutrino oscillations: A simple analytic approach, Phys. Rev. D65, 073008 (2002) (hep-ph/0111199); [2] G.L. Fogli, E. Lisi, A. Mirizzi, and D. M., Revisiting nonstandard interaction effects on supernova neutrino flavor oscillations, hep-ph/0202269, submitted to PRD.

  2. II XX International Conference on Neutrino Physics and Astrophysics Introduction A star with a mass M>8M⊙ terminates its life in a dramatic way: the iron core (R~104km) collapses in a proto-neutron star (R~102 km) in a fraction of second. Only ~1% of the energy available from the collapse (1053 erg) becomes “visible”, leading to a spectacular explosion (type II Supernova, SNII). The remaining ~99% of the energy is emitted in ~10 seconds after the collapse in the form of neutrinos and antineutrinos of all flavors, with energy E~130 MeV. A SNII is thus one of the most intense sources of neutrinos in the Universe. The detection of galactic SN neutrinos may shed light not only to the mechanism of the SN explosion but also to the neutrino properties, in particular n masses and mixings (for a review see [3]). The SN n’s give us the unique possibility to probe both the “solar” and the “atmospheric” Dm2’s with the same “neutrino beam”. In fact, when the neutrinos move from the neutrinosphere (the start of the n free streaming) to the surface of the star, the n potential V(x)=2GFYer(x) varies from ~10-2 eV2/MeV to 0, thus crossing zones with V(x)~Dm2atm/2E (“higher” transition) and V(x)~Dm2⊙/2E (“lower” transition). Moreover, SN neutrinos are sensitive to values of the mixing matrix element U13 up to 10-3, beyond of the range of the current and planned terrestrial experiments. For this reason, a general treatment for the calculation of the relevant nene survival probability Pee is highly desirable. We propose a simple unified approach, based on the condition of the maximum violation of adiabaticity (discussed in [4] in the context of solar neutrinos) valid for all the values of the oscillations parameters of phenomenological interest. We then extend the method to include also small effects due to non-standard flavor changing and flavor diagonal interactions. [3] G. Raffelt, Stars as Laboratories for Fundamental Physics, (Chicago U. Press, Chicago, 1996). [4] E. Lisi, A. Marrone, D. M., A. Palazzo, and S.T. Petcov, Phys. Rev. D63, 093002 (2001).

  3. III XX International Conference on Neutrino Physics and Astrophysics Notation In the case of two family oscillations nena (a=m, t), we label the mass eigenstates (n1, n2) so that n1 is the lightest (Dm2=m22-m12>0) and parameterize the mixing matrix U(q) as follows: “direct” hierarchy n2 n3 n1 dm2 with q[0,p/2]. We also define the vacuum wavenumber k=Dm2/2E. In the case of three family oscillations, we label the mass eigenstates (n1, n2, n3) as in figure. We define dm2m22-m12 (>0 by definition) and m2m32-m1,22>0 (<0) for a direct (inverse) hierarchy. From the phenomenology of solar, atmospheric, and reactor neutrino oscillations we argue that m2310-3 eV2 and dm2710-4 eV2, so that the hierarchical hypothesis dm2<<m2 is satisfied. The associated wavenumber are kH=m2/2E and kL=dm2/2E. The relevant mixing matrix elements are parameterized in terms of the two mixing angles (, w)(q13, q12): m2(>>dm2) dm2 n2 n3 n1 “inverse” hierarchy The two family oscillation scenario is recovered in the limit 0 (pure n1n2 transitions) or w0 (pure n1n3 transitions).

  4. IV XX International Conference on Neutrino Physics and Astrophysics Neutrino potential In matter, the n flavor dynamics depends on the potential where Ye is the electron/nucleon fraction and r is the matter density. In figure, the dotted line shows the potential profile above the neutrinosphere as function of the radius x for a typical Supernova progenitor [5]. The solid line shows a power-law approximation of the potential V(x): where n=3, R⊙=6.96105 km is the solar radius, and V0=1.510-8 eV2/MeV. For definiteness, we will use both the realistic and the power-law profiles in this figure to illustrate our method. However, our main results are applicable to a generic SN density profile. [5] from T. Shigeyama and K. Nomoto, Astophys.J. 360, 242 (1990).

  5. V XX International Conference on Neutrino Physics and Astrophysics Crossing Probability With 2 generations, the relevant quantity in calculating the survival probability Pee is the crossing probability, i.e., the probability that the heavier mass eigenstate in matter flips into the lighter: Pc=P(n2mn1m). A widely used formula is the “double exponential” [6]: where r is a scale factor, i.e., the inverse of the logarithmic derivate of the potential V(x) in the crossing point xp: The common choice for the point xp is the resonance point, i.e., the point where the mixing angle in matter - defined as sin2qm(x)=k/km(x)sin2q, where km(x)=[V2(x)+2kV(x)cosq+k2]½ is the neutrino wavenumber in matter - is maximal: cosqm(xr)=0  V(xr)=kcosq. This choice is not adequate for large q, and in particular for for qp/4, were the resonance point is not defined. A better choice is the so called point of maximum violation of adiabaticity (MVA), defined as the point where the adiabaticity parameter [km(x)]-1dqm(x)/dx is maximum. This point corresponds to the flex point of the function cosqm(x): [d2cosqm(x)/dx2]x=xMVA=0 [4]. For a power-law profile (which is a good approximation for Supernovae) V(x)x-n, the MVA point is defined by the equation V(x)=k[1+F(n,q)], where F(n,q)0.1 for n=3±1. The uncertainty on the value of V(x) is ~few %. For this reason, we can safely neglect the function F(n,q) and take the point xp defined as V(xp)=k as the effective MVA point. This simple recipe can be extended also to the realistic potential profile. [6] S.T. Petcov, Phys. Lett. B200, 373, (1988).

  6. VI XX International Conference on Neutrino Physics and Astrophysics Blue (solid) line: our analytical recipe (using the MVA prescription) Red (dotted) line: direct (numerical) solution of the MSW equation

  7. VII XX International Conference on Neutrino Physics and Astrophysics 2n transitions At the start of neutrinosphere we have V(x0)>>k for all the values of the Dm2/E of phenomenological interest, so we have n2m(x0)ne. The calculation of the 2n survival probability Pee2n can be factorized as follows: final rotation to the ne state n2mn1m transition initial n2m(=ne) state The figure shows the isolines of constant Pee2n in the mass-mixing plane for a representative value of the neutrino energy (E=15 MeV) both for the realistic SN potential (solid line) and the its power law approximation (dotted line). The crossing probability Pc is calculated with our analytical recipe. For antineutrinos, V(x) -V(x). By conventionally keeping V>0, this is equivalent to swap the mass labels (12), and then to take qp/2-q. The isolines for antineutrinos are just the mirror images around the line tan2q=1.

  8. VIII XX International Conference on Neutrino Physics and Astrophysics 3n transitions Using the hierarchical hypothesis dm2<<m2, it is possible to factorize the dynamics in the 2n “high” and “low” subsystems [7]: n2mn1m (“lower”) transition n3mn2m (“higher”) transition initial nm(=ne) state final rotation to the ne state where a=1 (a=0) if the initial state is the heavier (lighter) mass eigenstate in matter. In the following we consider the phenomenological input sin2=Ue32few %, so that we can safely take V(x)cos2V(x) within the uncertainties on the SN density profile. With this assumption, the Pee3n survival probability can be simply calculated from the 2n case as follows (see [1] for details). Defining and PH±=Pc±(kH,) and PL±=Pc±(kL,w) as the “higher” and “lower” transition probabilities respectively, we have: (PH±, PL±)(+,+) [(PH±, PL±)(-,-)] for neutrinos [antineutrinos] and direct hierarchy; (PH±, PL±)(+,-) [(PH±, PL±)(-,+)] for neutrinos [antineutrinos] and inverse hierarchy. where: [7] see e.g., T.K. Kuo, and J. Pantaleone, Rev. Mod. Phys. 61, 937 (1989).

  9. IX XX International Conference on Neutrino Physics and Astrophysics Black (solid) line:P3nee survival probability, no Earth effect Red (dotted) line: P3nee survival probability, with Earth effect (8500km, mantel) The figure shows the isolines of constant Pee3n in the (dm2, tan2w) plane, assuming m2=310-3 eV2, tan2=210-5, and E=15 MeV for both neutrinos and antineutrinos in the direct and inverse hierarchical scenarios. The density profile in the SN is assumed power-law. Solid line: no Earth matter effect. Dotted (red) line: 8500 km path in the Mantle (r=4.5 g/cm3 and Ye=0.5), of interest for the SN1987A phenomenology. The inclusion of the Earth matter effect is done by replacing the “final rotation” (Ue12, Ue22, Ue32) in the calculation of the Pee3n with (Pe1, Pe2, Pe3), where Pei=P(nine) along the neutrino path in the Earth. Within our phenomenological assumptions, it is: where PEPE(kL,w)=P2n(n2ne). A good approximation is to divide the interior of the Earth into two shells with different densities, the Core and the Mantle. In this case the PE can be calculated analytically [1].

  10. na nb Gfab f f X XX International Conference on Neutrino Physics and Astrophysics Non-standard interactions Several extension of the Standard Electroweak model (e.g., SUSY with broken R-parity) allow new four-fermions interactions with an effective flavor changing and flavor diagonal interaction hamiltonian of the kind: with (a, b) flavor indices and Gfab is the “strength” of the interaction. The net effect in ordinary matter is to provide to the standard n potential matrix V (x)=diag{V(x), 0, 0} an extra potential of the kind: Here we neglect possible variations of Ye along x, so that the eab are assumed constant. Moreover, the eab are assumed to be small: eab<<1. In particular, we assume eabfew10-2, compatible with the present phenomenology in the hypothesis of neutrino-fermion universality of the interactions. In the 2n case, in the hypothesis of the smallness of the eab, the potential matrix V+dV can be diagonalized through a matrix U(eea). In this way the MSW equation in matter can be formally cast in its standard form, modulo the replacement qq+eea. The details of calculation can be found in [2]. The final result is the following: i.e., the net effect is a shift of the angle q in the calculation of the crossing probability Pc.

  11. XI XX International Conference on Neutrino Physics and Astrophysics Black (solid) line:P3nee survival probability, no Earth effect Red (dotted) line: P3nee survival probability, with Earth effect (8500km, mantel) In the 3n case by factorizing the dynamics in the “high” and “low” subsystems, one obtains again that the Pee3n survival probability can be written as [2]: where: and: Here y is the q23 mixing angle. The figure shows the isolines of constant Pee3n in the (dm2, tan2w) plane, for E=15 MeV, m2=310-3 eV2 and for two representative values of tan2(+eH)(here we assume that tan2 is small, so that Ue121-Ue22cos2w and Ue320). Solid line: no Earth matter effect. Dotted (red) line: 8500 km path in the Mantle. In particular, we have PEPE(kL,wm(w+eL),w), where the shift in w is performed only in the calculation of the mixing angle wm in matter.

  12. XII XX International Conference on Neutrino Physics and Astrophysics Discussion and conclusions We have described a simple and accurate analytical prescription for the calculation of the 2n survival probability Pee inspired by the condition of maximum violation of the adiabaticity. The prescription holds in the whole oscillation parameter space and for a generic Supernova density profile. The analytical approach has been extended to cover 3n transitions with mass spectrum hierarchy, and to include Earth matter effects. We have found that, within the present phenomenology, if tan2>10-6 the survival probability is suppressed in the case of neutrinos (antineutrinos) with direct (inverse) hierarchy. This allow not only to probe small values of the mixing angle , but also to discriminate between the two mass spectrum hierarchies. Moreover, we have revisited the effects of nonstandard four-fermion interactions (with strength eabGF) on SN n oscillations. We have found that, as far as the transitions at high and low density are concerned, the main effects of the new interactions can be embedded through (positive or negative) shifts of the relevant mixing angles  and w, namely, +eH and ww+eL respectively. Barring the case of small w (disfavored by solar neutrino data), the main phenomenological implication of such results is a strict degeneracy between standard () and nonstandard (eH) effects on the high SN n transition. These results are complementary to studies of independent nonstandard effects that may occur at the neutrinosphere [8]. Acknowledgments This work was supported in part by the ItalianIstituto Nazionale di Fisica Nucleare (INFN) and Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR) under the project “Fisica Astroparticellare”. A copy of this presentation can be found at the following URL: http://www.ba.infn.it/~montan/Documents/nu2002.zip [8] H. Nunokawa et al., Phys. Rev. D 54, 4365 (1996); H. Nunokawa et al., Nucl. Phys. B 482, 481 (1996).

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