Daniele Montanino Università degli Studi di Lecce & Sezione INFN, Via Arnesano, 73100, Lecce, Italy [email protected] Analytic Treatment of

Daniele Montanino Università degli Studi di Lecce & Sezione

INFN, Via Arnesano, 73100, Lecce, Italy

[email protected]

Analytic Treatment of Neutrino Analytic Treatment of Neutrino Oscillations in SupernovaeOscillations 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

AbstractWe 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


IntroductionA 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~130 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” m2’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)=2GFYe(x) varies

from ~10-2 eV2/MeV to 0, thus crossing zones with V(x)~m2atm/2E (“higher” transition) and

V(x)~m2⊙/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 ee

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.

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).


In the case of two family oscillations e=, we label the mass eigenstates (1, 2) so that 1 is the lightest (m2=m2

2m12>0) and parameterize the mixing matrix U() as

follows:

cossin

sincos)(U

with . We also define the vacuum wavenumber k=m2/2E.

III

In the case of three family oscillations, we label the mass eigenstates (1, 2, 3) as in figure. We define m2m2

2m12 (>0

by definition) and m2m32m1,2

2>0 (<0) for a direct (inverse)

hierarchy. From the phenomenology of solar, atmospheric, and reactor neutrino oscillations we argue that m2310-3

eV2 and m2710-4 eV2, so that the hierarchical hypothesis m2<<m2 is satisfied. The associated wavenumber are kH=m

2/2E and kL=m2/2E. The relevant mixing matrix

elements are parameterized in terms of the two mixing angles (, )(13, 12): )sin ,sincos ,cos(cos)U ,U ,U( 222222

3e22e

21e

The two family oscillation scenario is recovered in the limit 0 (pure 12 transitions) or 0 (pure 13 transitions).

Notation

m2

m2

m2(m2)

1

2

1

2

3

3

“inverse”

hierarchy

“direct” hierarch

y


IV

Neutrino potentialIn matter, the flavor dynamics depends on the potential

where Ye 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):

3e

28

eF cm/g)x( )x(Y

MeVeV

1057.7)x( )x(YG2)x(V

n

0 Rx

V)x(V

where n=3, R⊙=6.96105 km is the solar radius,

and V0=1.510-8 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).


V

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(2

m1m). A widely used formula is the “double exponential” [6]:

1)rk2exp(1)cosrk2exp(

),k(P2

c

where r is a scale factor, i.e., the inverse of the logarithmic derivate of the potential V(x) in the crossing point xp: 1

xx pdx

)x(dV)x(V

1r

The common choice for the point xp is the resonance point, i.e., the point where the mixing

angle in matter - defined as sin2m(x)=k/km(x)sin2, where km(x)=[V2(x)+2kV(x)cos +k2]½ is

the neutrino wavenumber in matter - is maximal: cosm(xr)=0 V(xr)=kcos. 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 [km(x)]-1dm(x)/dx is maximum. This point corresponds to

the flex point of the function cosm(x): [d2cosm(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+(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 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).


VI

Blue (solid) line: our analytical recipe (using the MVA

prescription)

Red (dotted) line: direct (numerical) solution of the MSW

equation


VII

2 transitions

The figure shows the isolines of constant Pee2

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 (12), and then to take /2. The isolines for antineutrinos are just the mirror images around the line tan2=1.

At the start of neutrinosphere we have V(x0)k

for all the values of the m2/E of phenomenological interest, so we have m(x0)e. The calculation of the 2 survival

probability Pee2 can be factorized as follows:final rotation to the e

state

final rotation to the e

stateinitial m(=e)

state

initial m(=e)

statemm

transition

mm

transition

2c

2c

cc

cc222ee

sin)P1(cosP

1

0

P1P

PP1sin,cosP


VIII

3 transitions

Using the hierarchical hypothesis m2<<m2, it is possible to factorize the dynamics in the 2 “high” and “low” subsystems [7]:

0

1

P1P0

PP10

001

100

0P1P

0PP1

U,U,U P

HH

HHLL

LL23e

22e

21e

3ee

where =1 (=0) if the initial state is the heavier (lighter) mass eigenstate in matter. In the following we consider the phenomenological input sin2=Ue3

2few %, so that we can safely take

V(x)cos2V(x) within the uncertainties on the SN density profile. With this assumption, the Pee

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).


state


state

mm (“higher”)

transition

mm (“higher”)

transitionmm (“lower”)

transition

mm (“lower”)

transition initial m(=e)

state

initial m(=e)

state

k)x(V

2

c dx)x(dV

)x(V1

r , 1)rk2exp(

1)cosrk2exp(),k(P

and PH±=Pc

±(kH,) and PL±=Pc

±(kL,) as the “higher” and “lower” transition probabilities

respectively, we have:)P1(UP)]P1(UPU[P H

23eHL

22eL

21e

3ee

(PH±, PL

±)(,) [(PH±, PL

±)(,)] for neutrinos [antineutrinos] and direct

hierarchy;

(PH±, PL

±)(,) [(PH±, PL

±)(,)] for neutrinos [antineutrinos] and inverse

hierarchy.

where:


IX


in the (m2, tan2) plane, assuming m2=310-3

eV2, tan2=210-5, 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/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” (Ue1

2, Ue22, Ue3

2)

in the calculation of the Pee3 with (Pe1, Pe2,

Pe3), where Pei=P(ie) along the neutrino

path in the Earth. Within our phenomenological assumptions, it is:

]sin ,Pcos ),P1([cos)P ,P ,P( 2E

2E

23e2e1e

where PEPE(kL,)=P2(2e). 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].

Black (solid) line: P3ee survival probability, no Earth effect

Red (dotted) line: P3ee survival probability, with Earth effect (8500km,

mantel)


X

Non-standard interactions

f f

Gf

h.c.2

GHeff

ff

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 Gf 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:

F

edudue

eF G

)x(Y/)G2G(GGG)x( with (x), )x(YG2)x()x(

V

Here we neglect possible variations of Ye along x, so that the are assumed constant.

Moreover, the are assumed to be small: <<1. In particular, we assume few10-2,

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 Pc.

2

ec2

ec2

ee sin)],k(P1[cos),k(PP


XI

Black (solid) line: P3ee survival probability, no Earth effect

Red (dotted) line: P3ee 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 Pee3 survival probability can be

written as [2]: )P~

1(UP~

)]P~

1(UP~

U[P H23eHL

22eL

21e

3ee

where:

),k(PP~

),,k(PP~

LLLLHHHH

sin2cos2sin2

cos)sincos(

cossin

eeL

eeHand:

Here is the 23 mixing angle.


in the (m2, tan2) plane, for E=15 MeV, m2=310-3 eV2 and for two representative values of tan2(+H) (here we assume that

tan2 is small, so that Ue121Ue2

2cos2 and

Ue320). Solid line: no Earth matter effect.

Dotted (red) line: 8500 km path in the Mantle. In particular, we have PEPE(kL,m+L),),

where the shift in is performed only in the calculation of the mixing angle m in matter.


XII

Discussion and conclusions

We have described a simple and accurate analytical prescription for the calculation of the 2 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 3 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 GF) 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).