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

2. 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~10 4 km) collapses in a proto-neutron star (R~10 2 km) in a fraction of second. Only ~1% of the energy available from the collapse (  10 53 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 masses and mixings (for a review see [ 3 ]). The SN ’s give us the unique possibility to probe both the “solar” and the “atmospheric”  m 2 ’s with the same “neutrino beam”. In fact, when the neutrinos move from the neutrinosphere (the start of the free streaming) to the surface of the star, the potential V(x)=  2G F Y e  (x) varies from ~10 -2 eV 2 /MeV to 0, thus crossing zones with V(x)~  m 2 atm /2E (“higher” transition) and V(x)~  m 2 ⊙ /2E (“lower” transition). Moreover, SN neutrinos are sensitive to values of the mixing matrix element U 13 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 e  e survival probability P ee 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. II [ 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. XX International Conference on Neutrino Physics and Astrophysics In the case of two family oscillations e    = , we label the mass eigenstates ( 1, 2 ) so that 1 is the lightest (  m 2 =m 2 2  m 1 2 >0) and parameterize the mixing matrix U(  ) as follows: with . We also define the vacuum wavenumber k=  m 2 /2E. III In the case of three family oscillations, we label the mass eigenstates ( 1, 2, 3 ) as in figure. We define  m 2  m 2 2  m 1 2 (>0 by definition) and m 2  m 3 2  m 1,2 2 >0 (<0) for a direct (inverse) hierarchy. From the phenomenology of solar, atmospheric, and reactor neutrino oscillations we argue that  m 2   3  10 -3 eV 2 and  m 2  7  10 -4 eV 2, so that the hierarchical hypothesis  m 2 <<  m 2  is satisfied. The associated wavenumber are k H =  m 2  /2E and k L =  m 2 /2E. The relevant mixing matrix elements are parameterized in terms of the two mixing angles ( ,  )  (  13,  12 ): The two family oscillation scenario is recovered in the limit  0 (pure 1  2 transitions) or  0 (pure 1  3 transitions). Notation m2m2 m2m2 m 2 (   m 2 ) 1 2 1 2 3 3 “inverse” hierarchy “direct” hierarchy

4. XX International Conference on Neutrino Physics and Astrophysics IV Neutrino potential In matter, the flavor dynamics depends on the potential where Y e is the electron/nucleon fraction and  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  10 5 km is the solar radius, and V 0 =1.5  10 -8 eV 2 /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. XX International Conference on Neutrino Physics and Astrophysics V Crossing Probability With 2 generations, the relevant quantity in calculating the survival probability P ee is the crossing probability, i.e., the probability that the heavier mass eigenstate in matter flips into the lighter: P c =P( 2 m  1 m ). 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 x p : The common choice for the point x p is the resonance point, i.e., the point where the mixing angle in matter - defined as sin2  m (x)=k/k m (x)  sin2 , where k m (x)=[V 2 (x)+2kV(x)cos  +k 2 ] ½ is the neutrino wavenumber in matter - is maximal: cos  m (x r )=0  V(x r )=k  cos . This choice is not adequate for large , and in particular for for  /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 [k m (x)] -1 d  m (x)/dx is maximum. This point corresponds to the flex point of the function cos  m (x): [d 2 cos  m (x)/dx 2 ] x=x MVA =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+  (n,  )], where   (n,  )   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  (n,  ) and take the point x p defined as V(x p )=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. XX International Conference on Neutrino Physics and Astrophysics VI Blue (solid) line: our analytical recipe (using the MVA prescription) Red (dotted) line: direct (numerical) solution of the MSW equation

7. XX International Conference on Neutrino Physics and Astrophysics VII 2 transitions The figure shows the isolines of constant P ee 2 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 P c 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  /2 . The isolines for antineutrinos are just the mirror images around the line tan 2  =1. At the start of neutrinosphere we have V(x 0 )  k for all the values of the  m 2 /E of phenomenological interest, so we have  m (x 0 )  e. The calculation of the 2 survival probability P ee 2 can be factorized as follows: final rotation to the e state initial  m (= e ) state  m   m transition

8. XX International Conference on Neutrino Physics and Astrophysics VIII 3 transitions Using the hierarchical hypothesis  m 2 <<  m 2 , it is possible to factorize the dynamics in the 2 “high” and “low” subsystems [ 7 ]: where  =1 (  =0) if the initial state is the heavier (lighter) mass eigenstate in matter. In the following we consider the phenomenological input sin 2  =U e3 2  few %, so that we can safely take V(x)  cos 2  V(x) within the uncertainties on the SN density profile. With this assumption, the P ee 3 survival probability can be simply calculated from the 2 case as follows (see [ 1 ] for details). Defining [ 7 ] see e.g., T.K. Kuo, and J. Pantaleone, Rev. Mod. Phys. 61, 937 (1989). final rotation to the e state  m   m (“higher”) transition  m   m (“lower”) transition initial m (= e ) state and P H ± =P c ± (k H,  ) and P L ± =P c ± (k L,  ) as the “higher” and “lower” transition probabilities respectively, we have: (P H ±, P L ± )  ( ,  ) [(P H ±, P L ± )  ( ,  )] for neutrinos [antineutrinos] and direct hierarchy; (P H ±, P L ± )  ( ,  ) [(P H ±, P L ± )  ( ,  )] for neutrinos [antineutrinos] and inverse hierarchy. where:

9. XX International Conference on Neutrino Physics and Astrophysics IX The figure shows the isolines of constant P ee 3 in the (  m 2, tan 2  ) plane, assuming m 2 =3  10 -3 eV 2, tan 2  =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 (  =4.5 g/cm 3 and Y e =0.5), of interest for the SN1987A phenomenology. The inclusion of the Earth matter effect is done by replacing the “final rotation” (U e1 2, U e2 2, U e3 2 ) in the calculation of the P ee 3 with (P  e1, P  e2, P  e3 ), where P  ei =P( i  e ) along the neutrino path in the Earth. Within our phenomenological assumptions, it is: where P E  P E (k L,  )=P 2 ( 2  e ). 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 P E can be calculated analytically [ 1 ]. Black (solid) line: P 3 ee survival probability, no Earth effect Red (dotted) line: P 3 ee survival probability, with Earth effect (8500km, mantel)

10. XX International Conference on Neutrino Physics and Astrophysics X Non-standard interactions   ff G f  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 ( ,  ) flavor indices and G f  is the “strength” of the interaction. The net effect in ordinary matter is to provide to the standard potential matrix V (x)=diag{V(x), 0, 0} an extra potential of the kind: Here we neglect possible variations of Y e along x, so that the   are assumed constant. Moreover, the   are assumed to be small:     << 1. In particular, we assume      few  10 -2, compatible with the present phenomenology in the hypothesis of neutrino-fermion universality of the interactions. In the 2 case, in the hypothesis of the smallness of the  , the potential matrix V +  V can be diagonalized through a matrix U(  e  ). In this way the MSW equation in matter can be formally cast in its standard form, modulo the replacement  +  e . 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  in the calculation of the crossing probability P c.

11. XX International Conference on Neutrino Physics and Astrophysics XI Black (solid) line: P 3 ee survival probability, no Earth effect Red (dotted) line: P 3 ee survival probability, with Earth effect (8500km, mantel) In the 3 case by factorizing the dynamics in the “high” and “low” subsystems, one obtains again that the P ee 3 survival probability can be written as [ 2 ]: where: and: Here  is the  23 mixing angle. The figure shows the isolines of constant P ee 3 in the (  m 2, tan 2  ) plane, for E=15 MeV, m 2 =3  10 -3 eV 2 and for two representative values of tan 2 (  +  H ) (here we assume that tan 2  is small, so that U e1 2  1  U e2 2  cos 2  and U e3 2  0). Solid line: no Earth matter effect. Dotted (red) line: 8500 km path in the Mantle. In particular, we have P E  P E (k L,  m  +  L ),  ), where the shift in  is performed only in the calculation of the mixing angle  m in matter.

12. XX International Conference on Neutrino Physics and Astrophysics XII Discussion and conclusions We have described a simple and accurate analytical prescription for the calculation of the 2 survival probability P ee 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 3 transitions with mass spectrum hierarchy, and to include Earth matter effects. We have found that, within the present phenomenology, if tan 2  >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   G F ) on SN 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 , namely,  +  H and  +  L respectively. Barring the case of small  (disfavored by solar neutrino data), the main phenomenological implication of such results is a strict degeneracy between standard (  ) and nonstandard (  H ) effects on the high SN 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 Italian Istituto 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).