Interaction between neutrinos and nonstationary plasmas

__ _i!J

cm

11 December 1995 PHYSICS

ELSEVIER

LETTERS

A

Physics Letters A 209 (1995) 78-82

Interaction between neutrinos and nonstationary plasmas J.T. Mendoqa

a, R. Bingham b, P.K. Shukla ‘, J.M. Dawson d, V.N. Tsytovich

e

a GOLP, Centro de Electrodiniimica. Instituto Superior Tecnico, 1096 Lisbon Codex. Portugal b Rutherford Appleton Laboratory, Chilton. Didcot, Oxon, OX11 OQX, UK ’ Institut fir Theoretische Physik, Ruhr-Universitiit Bochum. D-44780 Bochum, Germany ’ Physics Department, University of California Los Angeles, Los Angeles, CA 90024.1547, USA ’ General Physics Institute. Russian Academy of Sciences, I1 7924 Moscow, Russian Federation

Received 19 July 1995; accepted for publication 11 October 1995 Communicated by M. Porkolab

Abstract The collective interaction of electron neutrinos with a dense non-stationary plasma is described here. This effect can eventually lead to a significant loss of energy of the neutrinos emitted by a collapsing supernova. The present effect is an exact analogue of the photon acceleration in an ionization front. We present here a classical estimate of the neutrino energy losses and also give the corresponding quantum description.

1. Introduction The emission of neutrinos by a star, and in particular by an exploding supernova, is an important and not completely solved problem in astrophysics. The collisional losses between the neutrinos and the fermions in the dense core of the star [ 1I are not high enough to avoid gravitational collapse and one has to identify new and more efficient mechanisms by which the emitted neutrinos could lose part of their energy to the star. Very recently, a non-linear interaction of the neutrino field with plasma oscillations was proposed [2], which could lead to significant neutrino damping. This effect is a resonant process, which can be seen as an equivalent to the well known stimulated Raman scattering of electromagnetic waves in a plasma

131.

In this paper we propose a new mechanism of neutrino-plasma interaction by which neutrino energy losses can also be explained. In contrast with the previous Raman scattering process we consider here a non-resonant mechanism which is the exact analog of the photon acceleration occurring in time varying plasmas [4,5]. Due to its intrinsic non-resonant character this neutrino acceleration (or deceleration) process will affect every neutrino emitted by a star during its collapse. A simple energy loss estimate can be obtained with the aid of a classical approach which is the analogue of the ray tracing theory for photon acceleration [S]. Such an estimate will be presented in Section 2. But, of course, these simple calculations have to be validated by a more reliable quantum description, which will be given in Section 3. Finally, in Section 4 we state the conclusions.

0375.9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00788-l

J.T. Mendonp

2. Classical

estimate

of neutrino

et al. / Physics Letters A 209 f 1995) 78-82

losses

and

It is well known that, in the presence of matter, the neutrino masses are changed due to the weakcurrent interaction [6,7]. In particular, the charged current will couple the electron neutrinos with the plasma electrons existing in the dense core of a star. In order to describe such a coupling we can use the dispersion relation of a neutrino in the presence of a plasma, relating its momentum p with its energy E [7l, ( E - V)’ - p*c* - m2c4 = 0,

79

(1)

dk _=-dt

aw (6)

ar .

During the collapse of a supernova, the neutrinos emitted by the dense core of the star will interact with the surface of the star, which is moving with collapsing velocity u,. In this case, we can write n,(r,

t) = n,(r-

U,f).

(7)

We can now go from the time-dependent Hamiltonian w(r, k, t) to an autonomous Hamiltonian fl(r, k) by using a simple canonical transformation, q=r-v,t,

(8)

where m is the (eventually existing) neutrino rest mass and V is an equivalent potential energy such that

and

V= CGn,.

This canonical transformation can be formally generated by the following generating function,

(2)

Here n, is the electron number density and G is the Fermi constant of the weak interaction. Because this coupling constant is a small quantity, we can assume that even for very high densities the equivalent potential energy is always much less than the total neutrino energy V +Z pc - E. It will be useful to rewrite the dispersion relation (1) in terms of the neutrino frequency w and the wavevector k, such that E = h w and p = fik. Thus, w=/m+;.

(3)

Clearly, in an inhomogeneous and nonstationary plasma, the potential V will vary in space and time. This means that a neutrino moving in a space and time varying plasma will have a variable eigenfrequency w = dr, k, r> satisfying the local approximate dispersion relation w(r,

k, t) = /m+

d+,(r,

t). (4)

The canonical equations of motion, or equivalently, the ray tracing equations for the neutrino field, will be given by dr dt=ak

k’ = k.

F,(r,

(9)

k’, t) =k’.(r-v,t).

The new Hamiltonian

O(q, k’) = w(q)

(10) can be written as

+ 2

+i’+,(q)

-0,

-

k’.

(11)

The new canonical equations for the neutrino motion can now be solved for an arbitrary plasma density profile in a way similar to that used for the ray tracing studies of photon acceleration in the presence of an ionization front [5]. But we can use the invariance of the new Hamiltonian O(q, k’) in order to derive the total energy loss of the neutrinos crossing the moving boundary of the star, without having to explicitly integrate the canonical equations. Let us assume a neutrino initially created in the dense core of the star, with an energy E, = /i w. in the presence of an electron plasma density n,a. The associated momentum state of the neutrino p,, = fik, will have to obey the local dispersion relation (1) which, by assuming the rest mass to be zero (m = 0). can simply be written as

aw (5)

( 12)

80

J.T. MendonGa et al./ Physics Letters A 209 (1995) 78-82

With these two values, constant of motion,

w,, and k,, we can define the

(13) where we have used & = v,/c. This same neutrino when coming out of the star (neo = 0) will have a final energy Ef = h of such that n=

or(l

+ p,>.

(14)

The total energy variation of the neutrino AE = w. - wr will be obtained by equating these two equations. The result is

(15) Eq. (15) can be used to estimate the total energy lost by the neutrinos during the process of gravitational collapse of the star. It is important to stress that this energy is independent of the specific density profile of the star surface and of the neutrino energy. It only depends on the electron density at the core and on the velocity of the collapsing star boundary.

3. The quantum

a -iii-

(16)

ar

and

1

a2

m2c2

ax2

c2

at2

A2

(

+(x7

Making tion

t>

I

t).

= ~“(x-v’t);JI(x,

(1%

now use of the old d’Alembert

transforma-

P-4 (21)

5=x-v,t, 7)=x+v,t and performing that

a Fourier

transformation

in 77 such

(22) we obtain

In order to get simple but physically relevant solutions, let us first assume that V( 5) is a slowly varying function. We can then use a WKB solution in 5 such that t,$( 6) = kro exp i ‘PC 6’) d-5’ (1 1 This leads to the following valid locally in 5, (1 -P,‘)[P=(&)

E-+ifii.

(17)

The result is [2] (h=c=V= - n&4

a2

------

description

Up to now we have only considered the neutrino motion in a purely classical description. It is, however, quite straightforward to derive a wave equation for the neutrino field from the dispersion relation (1) by using p-+

plasma density perturbation moving with a constant velocity u,. If the perturbation is uniform in the plane perpendicular to us, or in other words, if we have a well defined moving front, we can choose the x coordinate parallel to o, and reduce the above wave equation to

-

fi2a2/at2)g=

2ifiVg,

(18)

where + is the normalized wave function associated with the neutrino field. But, as we saw in the previous section, we are mainly interested in the case of a

+4=1

dispersion

+2(1

(24) relation that is

+P,2)P(5)9

~&V(t)[4-P(f)l.

+(mc/hj2=

(25)

The relation between the parameters q and P(5) with the local frequency w( 5 > and the wavevector k( 5) can easily be obtained from their own definition,

45) = const

2q=k(6) - -

US

J.T. Mendonp

et al./ Physics Letters A 209 (19951 78-82

and

(27)

2P(5) =k(O +45)/u,.

Using the invariance of 4 we can now calculate the energy shift AE of a neutrino after crossing the potential perturbation located around 5 = 0. Actually, if we have an initial neutrino state (0”’ k,) inside the star at t+ --oo and a different neutrino state (wr, k,) outside the star at 5 + m, we can write, for a given value q = q. and for L’, anti-parallel to the wavevectors, -

2 qd’s

= k”U, +

w.

=

kp, +

Of.

(28)

But we know that k, and k, are related to the frequencies w and wt by the dispersion relation (1) for V( 5 > = &C&./h = V, and V( 5) = 0, respectively. We can derive from this a neutrino energy shift 6.~) identical to that given by the clasAE=h(w,sical theory of Section 2. In this way, we not only confirm the qualitative result of the classical theory but also find a relation between the invariant Hamiltonian L! and the new parameter q: L! = - 2qu,. In addition, this quantum model is also able to predict the existence of neutrinos reflected by the potential discontinuity around .$ = 0 and propagating inwards in a state (w,, k,). In this case, the previous equation is replaced by - 2 qoo, = koLl, + w. = - kp, + w, .

(29)

Assuming that k - w/ c, we see that Eq. (29) describes the well known relativistic mirror effect, namely

q =

w.

1 +P, 1 - P, .

81

but we also know that the number of reflected neutrinos N, is a fraction (V/E,>2 of the transmitted one, N,. We then have &(AE)M(AE),;.

(32)

n

We conclude from this qualitative discussion that the energy shift of the neutrinos crossing the moving boundary of a collapsing star can lead to a significant neutrino energy loss to the star. To complete this section, we now show how the coupling between an incident and a reflected neutrino state can be described. First of all we note that these two states correspond to the same value of the parameter q but to two different values of the parameter p, such that 2p,r,

=

&L’,

-

W”

=

2p,u, = - k,c, -.o,=

-

wg(

1 -

-Wr(l

p,)

(33)

+p,).

(34)

Using the approximate relativistic mirror relation between the two frequencies w,, and w, we obtain

(1 + PA’ Pr2Po

(1

_

(35)

p,)2.

Eq. (35) shows that the moving plasma boundary can couple two different values of p, that are compatible with the same value of q. We can obtain the coupling mode equations from the wave equation for #q if we assume a solution of the following form,

where tqj sense that

is the slowly

varying

amplitude,

in the

(30) (37)

For a non-relativistic collapsing star we have & -=z 1, which corresponds to a frequency-upshift by a reflection of (A W)T = p, w,,. Comparing this energy gain with the energy lost by the transmitted neutrinos, we obtain

(31)

Assuming that the mode I,!J~,is largely dominant and its amplitude is only slightly perturbed by the interaction, which is physically valid for a weak coupling, we can write the evolution equation for the mode $q2 as (38)

82

J.T. Mendonp

where the mode coupling

coefficient

et al./Physics

is determined

by

Xexp (jb, I

72)

dl’).

Eq. (39) shows that the coupling of an initial neutrino state 1 with a different state 2 is proportional to the velocity of the plasma front and to the Fourier component of V( 5 ) for p = p, - p2. Assuming an initial value of $q2( to> = 0, we obtain $2(

6)

=

qtt

5’)

d5’.

For a given plasma potential profile V(t), Eq. (40) could be used for a more precise estimation of the reflected neutrino state.

4. Conclusion In this paper, we have studied the coupling of a neutrino field with a non-stationary background plasma via collective weak interections between the plasma electrons and the neutrinos. The effects considered here are essentially non-resonant in the sense that they affect every neutrino state independently of their energy. We have presented an estimate of the neutrino energy losses using two different descriptions: one based on the classical Hamiltonian equations of motion and the other based on a Klein-Gordon wave equation. The quantum description confirmed the

Letters A 209 (1995) 78-82

estimates obtained with the classical model and demonstrated the existence of reflected neutrino modes. The mechanism proposed here could eventually explain the additional energy losses of the neutrino fluxes emitted by a collapsing star, and it also contributes to the clarification of a still open problem in astrophysics. The present theory could be extended, in a future work, in order to give a more precise estimation of the anomalous neutrino losses if consistently coupled with the theoretical models of star collapse.

Acknowledgement This work was supported in part by the Commission of the European Union (Brussels) through the Network on “Energetic Particles in Astrophysical and Space Plasmas” of the Human Capital and Mobility program, under contract No. CHRX-CT 94-0604.

References Ill H.A. Bethe, Rev. Mod. Phys. 62 (1990) 801. l21 R. Bingham, J.M. Dawson, J.J. Su and H.A. Bethe, Phys. Lett. A 193, (1994) 279. [31 P.K. Kaw, W.L. Kruer, C.S. Liu and K. Nishikawa, Advances in plasma physics, Vol. 6 (Wiley, New York, 1986) Part 1. [41 S.C. Wilks, J.M. Dawson, W.B. Mori, T. Katsouleas and M.E. Jones, Phys. Rev. Lett. 62, (1989) 2600. El J.T. Mendonca and L. Oliveira e Silva, Phys. Rev. E 49 (1994) 3529. lb1 L. Wolfenstein, Phys. Rev. D 17 (1978) 2369. [71 H.A. Bethe, Phys. Rev. Lett. 56 (1986) 1305.