N E U T R I N O P A I R E M I S S I O N B Y A M A G N E T I Z E D

S T E L L A R P L A S M A

NORMANJ. MORGENSTERN H O R I N G and CHARLES ACQUISTA Dept. o f Physics and Cryogenics Center, Stevens Institute of

Technology, Hoboken, N.J., U.S.A.

(Received 20 June, 1973)

Abstract. The decay of a plasmon into two neutrinos in the presence of an intense magnetic field has been studied by Canuto et al. (1970). They suggest that one of the principal longitudinal plasmon modes, which occurs only in magnetized plasmas, would cause certain magnetic stars to cool more rapidly than their unmagnetized counterparts. We show here that this mechanism is inoperative since the plasmon mode involved cannot be excited in the direction parallel to the magnetic field as consid- ered by Canuto et al. Moreover, for coc/o)p ~ 1, we show that the other principal longitudinal plasmon mode considered earlier by Adams et al. (1963) (which is largely independent of the magnetic field) dominates the plasmon-neutrino decay cooling of magnetic stars.

I. Introduction

Adams et al, (1963) have shown that plasma photons (plasmons) in a star can decay into neutrino-antineutrino pairs which can escape from the star without further inter- action, thus providing a mechanism for energy loss from the stellar interior, cooling

the star. This decay mechanism is active by virtue of the fact that plasma photons behave as if they have a nonzero rest mass m7 = hOOp/C 2 (in contrast to photons in vacuum

whose vanishing rest mass makes the decay into neutrinos impossible because of energy-momentum conservation). Canuto et al. (1970a, b) have extended the con- sideration of this decay mechanism to include the effects of an intense magnetic field such as occurs in some white dwarfs. The magnetic field splits the principal local

longitudinal plasmon mode spectrum into two branches, and one of these plasmon modes lies near the cyclotron frequency co c for propagation nearly parallel to the magnetic field (limiting to o)~ for strictly parallel propagation). I t is this plasmon mode (which exists only in a magnetic field) that Canuto et al. (1970b) suggest to be re- sponsible for enhanced plasmon-neutrino energy loss. However, we show here that it is impossible to excite this plasmon mode in the limit of parallel propagation con- sidered by Canuto et al. Moreover, for nonparallel propagation, we show that excita- tion of this plasmon mode is extremely unlikely as long as c%/~%~ I. Consequently, the energy loss associated with the decay of this plasmon mode into neutrinos is negligibly small.

2. Plasmon Decay into Neutrinos in a Magnetic Field

The decay process under consideration involves the transition of a longitudinal plasma photon (plasmon) into a virtual electron-positron pair by the electromagnetic inter-

Astrophysics and Space Science 26 (1974) 38%390. All Rights Reserved Copyright �9 1974 by D. Reidel Publishing Company, Dordrecht-Holland

388 NORMAN J, MORGENSTERN HORING AND CHARLES ACQUISTA

action, followed by the transition of the virtual electron-positron pair into a neutrino- antineutrino pair by the weak ('four Fermi') interaction, as indicated in the accom- panying Feynman diagram:

e +

The decay rate for this process has been evaluated by Adams et al. (1963), and an account of it is presented in the review article of Chiu and Zaidi (1967) whose notation we follow. It involves the evaluation of the S-matrix element for the transition from longitudinal plasmon to neutrino-antineutrino pair to lowest order in the electro- magnetic and weak interactions jointly. The longitudinal vector potential describing the initial longitudinal plasma photon is given by Chiu and Zaidi (Equation 14.30b) as

4 (v + (,4)* (1)

where d is the frequency dependent longitudinal dielectric function of the plasma, whose zeros define the frequencies of the longitudinal ptasmons. It should furthermore be noted that 0e~/c~e) appearing in the denominator of the normalization amplitude in (1) is identical with the inverse of the natural excitation amplitude Z = (~et/&o) - 1 eval- uated at the plasmon frequency determined by ez=0. The occurrence of the natural excitation amplitude Z in this context should not be surprising since Z measures the relative importance of a given plasma mode in response to excitation as determined by the internal dynamics of the plasma (see for example Horing, 1965). Finally the decay rate for the process under consideration is given by Chiu and Zaidi (Equation 17.26) as

6 ~ ( k . k ) 2

z [ l = o h _ 6 ~ V e 2 0st/o~ (2)

It is once again apparent that the behavior of the plasmon-neutrino decay rate (and consequently the energy loss rate) is governed by the natural excitation amplitude Z = ( & ~ / ~ ) - 1.

The magnetic field enters into the calculation of the energy loss rate only through the longitudinal dielectric function ez which is given in the local limit by

2 cos 2 0 2 sin 2 0 J = 1 top (Dp

ma m2 2, (3) - - ( ~ c

where 0 is the angle between the plasmon propagation direction and the magnetic

NEU'IRINO PAIR EMISSION BY A MAGNETIZED STELLAR PLASMA 389

field. This local result holds quantum mechanically as well as classically (see Horing, 1965). The local plasmon dispersion relation J---0 then yields two roots given by

1 2 = _ - 4copco~ cos 2 0 (4)

and it is apparent from this that the magnetic field splits the local plasmon spectrum into two branches, whose relative importance is determined by Z. For usual condi- tions in astrophysical magnetoplasmas, coo/cop ~ 1, and the plasmon roots co-+ are ap- proximately given by

,.~ z z sin20 __. z (5a) 0)2+ ----. COp + (De COP

(2)2 ~ 2 = co~ cosZ0. (5b)

From the definition of Z - 1 = ~e~/~co, we have

Z- 1 _ 2co~ COS 2 0 2coco~ sin e 0 + (6)

which must be evaluated at the frequency roots of the plasmon dispersion relation. For the case co~/co~ ~ 1 under consideration, we employ Equations (5a, b) for co+. The resulting excitation amplitudes for co+ are given by Z-+ as follows:

2 Z+ 1 ~ (7a)

COp

Z 2 ~ ~ 2co~ - 3 ( 7 b )

coc sin 2 0 cos 0"

3 . C o n c l u s i o n s

Canuto e t al . (1970b) have focused attention on the plasmon mode co_ (which is strongly dependent on the magnetic field and highly anisotropic) in the limit of pro- gation parallel to the magnetic field,

co- ~ coc as 0 ~ 0,

suggesting that this branch of the plasmon spectrum is responsible for enhanced ener- gy loss. However, it is clear from Equation (7b) that the natural excitation amplitude Z_ corresponding to co_ must vanish for parallel propagation. This is to say that it is impossible to excite c o in the case of parallel propagation, as one should expect since alI trace of c o vanishes from the structure of J as given by Equation (3) in the limit 0=0 . Since Z_ vanishes for 0 = 0 it is clear that the corresponding decay rate associated with co_ as given by Equation (2) must vanish in the limit 0---0, and the associated energy loss rate must likewise vanish. This result is in disagreement with the nonzero energy loss rate for co_ =coc at 0 = 0 obtained by Canuto e t al . (1970b, Equation 15), as well as being in disagreement with their further conclusions con- cerning the predominant importance of the co_ plasmon mode in enhancement of

390 NORMAN J. MORGENSTERN HORING AND CHARLES ACQUISTA

energy loss, since we find the mechanism involved to be inoperative due to the im- possibility of exciting the co_ mode at 0=0.

Considering arbitrary direction of propagation 0 relative to the magnetic field, one can evaluate the ratio of the excitation amplitudes Z_/Z+ corresponding to the plasmon modes co_ and co+ from Equations (7a, b) with the result

- - = sin 2 0 cos 0 ~ 1 (8) Z+ \cop/

for coc/cop~ 1. It is clear that even at arbitrary propagation angles the natural excita- tion amplitude for co_ is vanishingly small compared to that of m+ because of the smallness of coo/cop. Consequently, the decay rate and energy loss rate associated with o9_ are very small compared to those associated with co+; therefore the co+ mode dominates the plasmon-neutrino decay cooling of magnetic stars. It is readily seen from Equations (5a) and (7a) that co+ is the essentially isotropic plasmon mode corre- sponding to the zero field limit (and therefore independent of magnetic held). Indeed, this is the mode considered by Adams et al. (1963), and their analysis of plasmon- neutrino decay cooling thus remains accurate for magnetic stars as long as coJcop ~ 1, with negligibly small corrections arising from the ~o_ mode (for 0 r 0) and from the weak magnetic field dependence of the co+ mode.

Note added in proof. After this work was completed and submitted to the Journal, we were informed of related work by Chen, Ruderman, and Canuto which should appear soon. The principal conclusions of their work are in substantial agreement with our results.

References

Adams, J. B., Ruderman, M. A., and Woo, C. H.: 1963, Phys. Rev. 129, 1383. Canuto, V., Chiuderi, C., and Chou, C. K. : 1970a, Astrophys. Space Sci. 7, 407. Canuto, V., Chiuderi, C., and Chou, C. K. : 1970b, Astrophys. Space Sci. 9, 453. Chiu, H. Y. and Zaidi, M. H. : 1967, in C. DeWitt, E. Schatzman, and P. Veron (eds.), High Energy

Astrophysics, Vol. II, Gordon and Breach, N.Y. Horing, N. J. : 1965, Ann. Phys. 31, 1.