STELLAR ENERGY-LOSS AND ANOMALOUS CONTRIBUTIONS TO GAMMA-GAMMA-]NU-NU

PHYSICAL REVIEW D VOLUME 36, NUMBER 10 15NOVEMBER 1987

Stellar energy loss and anomalous contributions to yy: ^vv

A. A. Natale, V.Pleitez, and A. Tacla

Instituto de Fssica Teorica, Universidade Estadua/ Paulista, Rua Pamplona 145, 01405 SaoPaulo, Sdo Paulo, Brazil

(Received 18 May 1987)

We compute the stellar energy-loss rate for several anomalous contributions to

yy~vv

and for massive neutrinos. Except for the cases where the mass scale ofthese new contributions is small, or the neutrino masses are large, these reactions are not important as a mechanism for stellar ener- gy loss.

Many years ago Pontecorvo and Chiu and Morrison' suggested that the process

yy~vv

might play an impor- tant role as a mechanism for stellar cooling. Gell-Mann subsequently showed that this process is forbidden in a local (V

—

_{A) theory.} However, it can occur at the one- loop level which has been computed by Levine for the intermediate-boson (V

—

A) theory, and the stellar energy-loss rate through

yy~vv

was found to be small- er than the rates for competing process (pair annihilation

e+e ~vv

and photoneutrinos ye

—

_+evv). This result is not modified when the cross section

of

the above process is computed in the Weinberg-Salam theory as was shown by Dicus.

The conditions for which the yy

~

^{vv reaction is}

suppressed are weakened

if

we have more exotic cou- plings between photons and neutrinos, one

of

the pho- tons is virtual, or the neutrinos are massive. Many ex- tensions

of

the standard model lead naturally to new in- teractions between photons and neutrinos, and there is not any fundamental principle implying that neutrinos should be massless. The existence

of

massive neutrinos have such far-reaching consequences that the determina- tion and understanding

of

neutrino masses are major is- sues for particle physics. Under this point

of

view it is interesting to investigate whether new contributions to yy

~vv

are astrophysically or cosmologically relevant.

New interactions between photons and neutrinos _may originate in different contexts (for instance, composite models, models with right-handed neutrinos,

etc.

). Here, without committing to a specific model we will compute the cross sections and energy-loss rates

of

new contribu- tions to

yy~vv.

^We compare them to the standard- model contribution, verifying that for some specific con- ditions they may be larger than this one. The eff'ective Lagrangians that we will consider are

+v

1

A (CL+L3

+L+CR +pl 4g

)e p(a

F"')F,

gf

⁴

or=

m_e

6 s

240

3 2

(15P+20P — _23P'),

_(2b)

~rrr

=4~an

^cos2

0~

sin

0~

Mg

2 m ² ₁ GeV

A

2

o._iv

=&

s A

2

1 GeV

A

2

(2c)

(2d) (le) where ^%' are neutrino fields,

F"

the electromagnetic field strength tensor,

Z"

the weak neutral _field, and A is the mass scale

of

the new interactions. Most

of

these have been studied in the case

of

charged-fermion fields.

has already been considered as a source

of

stellar energy loss through plasmon decay

(I ~vv),

and its coupling is constrained experimentally ^' as well as astrophysical-

6,⁸

The effective Lagrangians (la) and (lb) give rise to the diagrams

of

Fig. 1(a), Lagrangian

(lc)

to the diagram

of

Fig. 1(b), where the

Z

-neutrino coupling is the standard-model one, and (ld) and (le) have the contribu- tion depicted ⁱⁿ

Fig.

1(c).

To

compute the cross sections we will assume that the neutrinos have Dirac masses.

We obtain the following results:

[3P —

—,

'P' —

₍₁

— _P

)

X),

me

0^o-

+F~,

1 s

~v=

A

2 m ₁ GeV

A

2

(2e)

, ^{0 y„B.}

^VF~

2A

Z"[(Bg„)F +F ' (Bg„)],

2A

where

1 Ac

64m 1 GeV

2

(cm

),

44F„, F"',

1

4A (ld)

P = 1—

^4m

1+P

1—

36 3278

1987

The American Physical Society

36

BRIEF

REPORTS ₃₂₇₉

cos²ew

m,

Q»,

— —

_47TcXK

sin Hw

'6

'4 ^- -4

5T₉⁵

Jrrr(il»

w

(4c)

Qrv

=K

A

n'T9'Jrv(il»

8

Qv=K

i) T9

Jv(il),

A

(4d)

(b)

(c)

where

J;(i))=

^—,',

J

^dx

f

^dy

^{f (x,}

^y,

^q)fV;(y),

FIG.

1. (a) Diagrams contributing to

yy~vv

coming out from the effective interactions (1a) and (1b); (b) the same for the interaction (1c);(c) the same for (1d) and (le).

Wr(y⁾

=

_(2y

+

^10y

+

³⁾

9

+

^—,

^'(1 —

_6y

_)In(y++y — 1),

and m is the neutrino mass. In

(la)

we have replaced

(A '₎ by (

f

^{/2m, ), where}

f

^is the neutrino magnetic moment in units

of

electron Bohr magnetons, and in (le) we assumed, for sinlplieity, Cz

— _{— 0}

_{and CL}

— ^— _1.

The rate at which the energy

of

the photons (in thermal equilibrium) is converted into neutrino pairs is given by

IVrr(y⁾

= _{(192y +490y} —

_72Sy

+

^10y

+

¹⁵⁾

+

^—,',

ln(y+ +y' ^— 1),

2

W»i(y ⁾

=

_(4Sy

—

_Sy⁴

—

_1oy

—

₁₅₎

12

—

—,

'ln(y+ +y ^— 1), (2mfic).

4c CO)

exp

—

1

672

exp

d

krd

k2(cor+co2) ~v„r~o'(s)

2

—

1

Wrv(y)=

₄

(16y —

_24y

_+2y +3)

and

cu&

— —

_m

_x,

_C02

— —

m y 2m

kT

f ^(x,

^y,

^{rI)—: x+y}

(3) where co& and cu2 are the photon energies, k; ^are their momenta, ~v„r~ is the relative velocity factor, and the cross sections

[a(s)]

given by (2a)—_(2e) can be expressed in terms

of

co&, co2, and

8,

^where

8

is the angle between the photons.

For

each

of

the cross sections (2a)—_(2e),after perform- ing the angular integrations, and defining the variables

—

—,

'ln(y+V

_y

— 1),

~v(y

⁾

=

_{~rrr (y})

E =

(kTO)' (erg/cm

sec),

m (rric)

T9= To=

10

K

To

These equations, (4a)

—

_{(4e), can} be integrated analyti- cally with the approximation 2m c

&kT

(or, as we did too, numerically), obtaining

Q,

=

',

Kf

kTo ^—T9

I (4)I (5g'(4)g(5),

(5a) mec

r 8

~kT,

Qrr

— —

_~5K

_T9' I (6)I (7)g(6)g(7),

(5b)

we obtain

exp 'QX

2 exp

2 cos Ow kTo ⁴ kTo

Q,

»

— —

64rraK

sin2ew

Ac'

mwc2

2 m

4

Q,

=Kf ' _{g'T9'Jr (rl),}

m, ^(4a) X

T, "I (5)I (6)g(5)g(6),

Q„=K

8

n'T9'Jn(n»

(4b)

kTo '6

Qrv

=16K

_T9

"I (5)l (6)g(5)g(6),

Ac (5d)

3280 BRIEFREPORTS 36

1Q30

QsM=2X10

^'

^T9'

(erg/cm sec)

.

(6)

20 1Q C3

(D 10

C3 10

10 9

10 10¹² 10¹³

Qv

=

16I(.

kTQ 6

Ac

2

T9

"1 (5)1 (6)g(5)g(6)

^.

10 11

10 10

TEMP.

(K)

FIG.

2. Results of the numerical integration of Eqs.

(4a)—_(4e)for m

=10

eV.

As noticed before, the reaction yy

—

_{+vv can be im-}

portant only at cosmological temperatures. The interac- tion (ld) gives an energy output some orders

of

magni- tude larger than the standard-model one. All

of

these contributions are smaller than the one

of e+e ~v~

at the same temperature.

In some

of

the cases we have studied there are no reasons to have a small value

of

_Q, other than the small cross section due to the suppression

of

many powers in

(1/A),

and we have been conservative when we adopted

A=1

TeV. A smaller value

of

A, as ^is possible in some composite models, would cause a substantial modifica- tion

of

our results. We shall also have another consider- able enhancement in the emitted energy

if

we have larger

neutrino masses' [seeEqs. (5c) and (Se)].

Comparing the energy output

of py~vv

^{with the}

plasmon decay one (1

~vv)

(Ref. 13), we see that the first one wins from the second ⁱⁿ the temperature range

of

10 —

10' K

and densities

(ply,

^{) up to 10}

^g/cm,

^for

larger densities (10—10 g/cm ) the plasmon decay takes over up to temperatures

— 10" _K.

The photons are quite sensible to the plasma effect at large densities;

introducing these effects into the computation

of

_Q for

yy~vv,

the temperature dependence will be modified and we expect that it may lead to larger neutrino lumi- nosities. We are currently studying this possibility.

(5e)

Our results are shown in

Fig.

2, where we used m

„=

^{10 eV,}

f ⁼

^2.

⁰

^X10

"

_{which comes out from a}

superstring-inspired model, ^'

A=1

TeV, sin

0~ — — 0.

226, and

M~ — —

_{83 GeV. We have drawn in}

_Fig.

_{2 the}

energy-loss rate for

yy~vv of

the standard model, _QsM (Ref.

11):

We wish to thank

J. C.

Montero for helping us in the numerical calculations. We are indebted to Conselho Nacional de Desenvolvimento Cientifico e Tecnologico and Financiadora de Estudos e Projectos, Brazil (A.N.), Coordenaqao de Aperfeiqoamento de Pessoal do Ensino Superior Brazil (A.

T.

) for their financial support. This work was also partially supported by the Universidade Estadual Paulista-Instituto de Fiisica Teorica contract.

IB.M. Pontecorvo, Zh. Eksp. Tear. Fiz. 36, 1615(1959)[Sov.

Phys. JETP 9, ¹¹⁴⁸(1959)]; H. Y. Chiu and P.Morrison, Phys. Rev. Lett.5, 573(1960).

2M. Gell-Mann, Phys. Rev.Lett. 6,70(1961).

3M.

J.

Levine, Nuovo Cimento 48A, 67(1966).

4D.A.Dicus, Phys. Rev. D 6,941(1972).

5See, for example, S.

F.

King and S.

R.

Sharpe, Nucl. Phys.

B253, ¹ (1985), for interaction (1b); Y. Tomozawa, Phys.

Lett. 139B,455 (1984); L.

F.

^Abbott and E. Farhi, ibid.

1018,

69 (1981); Nucl. Phys. B189, 547 (1981);and Z.

Ryzak, ibid. B289,301 (1987),for (1c);S.D.Drell and S.

J.

Parke, Phys. Rev. Lett. 53, 1993 (1984);D.A.Dicus and

X.

Tata, Phys. Lett. 155B, ¹⁰³(1985);and D. A. Dicus, Phys.

Rev. D31,2999 (1985),for (1e). Interaction (¹d) must be re- garded with caution because ifit is applied to the charged- leptonic sector it will give a too large muon and electron anomalous magnetic moment [see

F.

del Aguila, A.Mendez, and R.Pascual, Phys. Lett. 104B, 431 (1984)].

J.

Bernstein, M. Ruderman, and G.Feinberg, Phys. Rev. 132, 1227(1963).

7J.

E.

Kim, V.S.^Mathur, and S.Okubo, Phys. Rev. D 9, 3050 (1974).

P. Sutherland et al., Phys. Rev. D 13, 2700 (1976);D. A.

Dicus and

E.

W. Kolb,ibid. 15, 977 (1977).

9We do not have the anomalous-magnetic-moment contribu- tion (1a) for Majorana masses; however, most of our con- clusions will not change ifthe masses are ofone or another type.

J.

A.Grifols and

J.

Sola, Phys. Lett. B182,53(1987).

IIEquation (6)is Levine's result (Ref.3} corrected for M~

=83

GeV. In the case ofmassive neutrinos the standard-model contribution has been calculated _by A. Halprin, Phys. Rev.

D 11,147(1975);for small neutrino masses it does not devi- ate substantially from the result ofEq.(6).

There are models allowing neutrino masses even near to the experimental upper limits; see G. B.Gelmini and

J.

W.

F.

Valle, Phys. Lett. 142B, 181 (1984); S. L. Glashow, Phys.

Lett. B187,307(1987).

We used the calculations ofG. Beaudet, V. Petrossian, and

E.

Salpeter, Astrophys.

J.

150, 979 (1967}.