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Stimulated scattering of massless fermionic neutrinos by plasmas
19 July 1999
Physics Letters A 258 Ž1999. 141–144 www.elsevier.nlrlocaterphysleta
Stimulated scattering of massless fermionic neutrinos by plasmas A. Serbeto, J.A. Souza Instituto de Fisica, UniÕersidade Federal Fluminense Gragoata, 24.210-340, Niteroi, Rio de Janeiro, Brazil Received 30 April 1998; received in revised form 24 March 1999; accepted 6 May 1999 Communicated by M. Porkolab
Abstract The Weyl equation has been used to study the interaction of a neutrino flux with a plasma in a stellar medium. The resulting collective stimulated scattering of the neutrino by the plasma is described by a cubic dispersion relation, which is solved numerically and analytically which gives a maximum growth rate structurally identical to growth rate obtained by Bingham et al. wPhys. Lett. A 193 Ž1994. 279x by using a Klein–Gordon like equation model, in which the neutrinos are considered as bosons, to describe the linear regime of the collective interaction. q 1999 Elsevier Science B.V. All rights reserved. PACS: 95.30.Qd; 95.30.Cq; 97.60.Bw; 97.60.Gb
In three very interesting recent papers, Bingham et al. w1,2x and Shukla et al. w3x have analyzed the collective interactions between neutrinos and plasma in a background stellar medium. The applications of their results seem very promising, specially in the field of supernovae dynamics. The above authors, however, following an early suggestion by Bethe w4x, described the neutrino field by a scalar function C , which obeys a Klein–Gordon like equation Žbosonic description. and assumed the neutrino mass to be different from zero in order to explain the possibility of the collective stimulated scattering of the neutrinos by plasma, which are coupled to each other due to the weak Fermi interaction and the ponderomotive force due to the neutrino flux. A question naturally arises: what modifications will appear in the results from Refs. w1–3x if the fermionic nature of the neutrino is taken into account? In a trial to get an answer, we have studied the neutrino-plasma interac-
tions but now describing the neutrino field by a massless Dirac equation ŽWeyl equation.. In this case, as is well known, the states of positive and negative helicities become completely uncoupled, and the left handed Žnegative helicity. state corresponds to the neutrino. The solutions of the Dirac equation with mass equal to zero
gˆ m EmC s 0
Ž 1.
are, in principle, four component spinors. Nevertheless, the separation of the states according to their helicities referred to above implies the neutrino being really a two component object. The extra degrees of freedom are removed by imposing the chiral constraint w5x
Ž 1ˆ q igˆ 5 . C s 0 ,
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 3 5 0 - 3
Ž 2.
A. Serbeto, J.A. Souzar Physics Letters A 258 (1999) 141–144
142
here g mˆ are the Dirac matrices, and gˆ 5 s igˆ 0gˆ 1gˆ 2gˆ 3. Thus, if C is originally of the form
a C s a1 , 2
ž /
Ž 3.
where a 1 and a 2 are two components objects, the constraint Ž2. implies a 1 s a 2 if we use the Dirac gˆ matrices in the Bjorken–Drell representation w6x. With all these in mind, let us write Weyl equation assuming a ‘minimal’ coupling to the weak field, like the one assumed in Refs. w1–3x. Using the trivector notation, we have for the neutrino dynamics the following equation: Vˆ
" EC
aˆ P pˆC s i
c Et
q c
C,
Ž 4.
where Vˆ s '2 G F n e , with G F being the weak coupling constant ŽFermi constant. and n e the electron plasma density. For one-dimensional interaction and using coordinate representation, Eq. Ž4. reduces to
EC yi" a xˆ P
Ex
'2
" EC si
c Et
q c
G F n eC ,
Ž 5.
scripts o,1, and l correspond to pump, scattered, and electron plasma wave Žfor neutrinos and Langmuir plasmons., respectively. Considering these matching conditions and supposing that C s Co q C 1 and n e s n p q d n, where Co and C 1 stand for the incident and scattered neutrino spinors, respectively, we have from Eq. Ž5.
ECo Et EC 1 Et
0 0 1 0
0 1 0 0
1 0 , 0 0
0
0
EC 1 Ex
y i GpCo s i Gp
dn
y i GpC 1 s i Gp
np
C1 ,
d n) np
Ž 8a .
Co ,
Ž 8b .
with d nrn p Ž d n ) rn p is the complex conjugate. being the electron plasma density perturbation normalized to the equilibrium plasma density n p Žmean density.. Following Ref. w2x the equation for the electron plasma density perturbation in the fluid model can be written as
½
E2 Et
2
E2 y Õt2
Ex
2
q v p2
dn
5
np
E2
Ž 9. Ž C1†Co q C1† P a x P Co . , where Gn s '2 G Fr4p m e " vn , Gp s Vor", and Vo s '2 G F n p . The right hand side of this equation is due s Gn
Ž 6.
and, according to the chiral constraint mentioned above f g Cs f g
Ex
q c a xˆ P
where in the Bjorken–Drell representation 0 0 aˆ x s 0 1
ECo
q c a xˆ P
Ž 7.
is the neutrino spinor Žnot for a single neutrino. normalized in such way that the neutrino energy density, Nn " vn , is W s ŽC † P C q C † P a x P C .r4p , where Nn is the neutrino density w1–3x. Since in this stimulated scattering process there is a feedback loop between the neutrino flux and the ripples of the electron plasma density, the seed of the collective instability due to the electron plasma wave and neutrino interaction must satisfy the energy momentum conservation matching conditions: " vn o s " vn 1 q " v l and "kn o s "kn 1 q "k l , where the sub-
E x2
to the ponderomotive force of the neutrinos acting on electron plasma w5x. Hence Eqs. Ž8. and Ž9. form a set of nonlinear basic equations to describe the dynamics of the stimulated scattering of neutrinos by plasma. In order to study the linear mechanism of the instability Žthe dispersion relation. we will consider a homogeneous pump amplitude ŽCo s constant. and take all space-time dependences to be proportional to expw iŽ kx y v t .x, where v and k are, respectively, the frequency and wave number associated to all dynamic quantities. Using Fourier transform in Eqs. Ž8. and Ž9. and combining the resulting equations we have the following matricial equation
Ž v l2 y v k2 . Ž vn 1 q Gp . Iˆy kn 1 c a xˆ C˜ 1 s k l2Gp Gn C˜o† P C˜ 1 C˜o q C˜ o† P a x P C˜ 1 C˜o ,
ž
/
ž
/
Ž 10 .
A. Serbeto, J.A. Souzar Physics Letters A 258 (1999) 141–144
143
where C˜o and C˜ 1 are the Fourier amplitudes of the pump and scattered neutrino spinors, respectively. Iˆ stands for 4 = 4 identity matrix and the matrix a xˆ is given by Eq. Ž6.. Using the representation given by Eq. Ž7. and the matching conditions the full dispersion relation can be written as
the parameters Gp and Gn , the normalized growth rate far from the threshold, in terms of the original physical parameters, is given by
Ž v l2 y v k2 .
where we have assumed the plasma wave phase velocity approximately equal to c. Here the normalized source term, S pn , of the instability is defined as
vn o y v l q Gp y Ž kn o y k l . c 2
s Gp Gn k l2 Co q Co† P a x P Co ,
Ž 11 .
where v k2 s v p2 q Õt2 k l2 and Õt is the electron thermal speed. In a realistic astrophysical scenario, due to the smallness of Fermi constant, G F , a weak coupling regime plays the main role in the interaction of neutrino fluxes with the stellar plasma. Hence the pump neutrino flux does not alter greatly the linear dispersion relation of the electron plasma wave,i.e, v l , v k q dv , where dv < v k . Then, substituting into Eq. Ž11. and neglecting second order terms in dv , the dispersion relation reduces to
d 2v , y
Gp Gn 2 vk
2
k l2 Co q Co† P a x P Co .
Ž 12 .
For dv s ig and assuming the resonant approximation of the neutrino dispersion relation, i.e., vn o y v k q Gp y Ž kn o y k l . c , 0, and from de definition of
'2
g vp
, 2
S pn s
s
S p1r2 n ,
Gp Gn vp
Ž 13 .
2
Co q Co† P Ž a xˆ P Co .
Vo2
Nn 2
" vp me c n p
.
Ž 14 .
As we can see, in the linear regime, the fermionic nature of the neutrino, given by the second term in Eq. Ž14., does not modify structurally the result for the growth rate pointed in Ref. w1x, in which the neutrinos are considered as bosons. The value of the Langmuir wave number Žplasmon linear momentum. which maximizes the growth rate can be obtained from the neutrino dispersion relation to yeld k l , v prc, which means that the maximum growth rate occurs when the plasma wave phase velocity is near to c. The numerical solution of Eq. Ž10. gives the
Fig. 1. Normalized growth rate,grv p , as a function of the source term,S pn .
144
A. Serbeto, J.A. Souzar Physics Letters A 258 (1999) 141–144
maximum growth rate, grv p , for different values of the source term, S pn , in an universal way, as shown in Fig. 1. We note that the maximum growth rate exhibits a kind of band structure with null values for some intervals of S pn values, with all the maxima satisfying the analytical solution given by Eq. Ž13.. It should be pointed out that this band structure disappears for very large values of the source term, which corresponds to a non physically realistic scenario. In supernovae of Type II, for instance, 10 58 neutrinos of all flavors are emitted with average energy near 15 MeV w7x, which gives us a total energy in the vicinity of 3 = 10 53 erg, resulting in an energy density, at 300 km from the center of the star, near 2.7 = 10 30 erg cmy3 which corresponds to a neutrino density close to 10 35 cmy3 . For these parameters the normalized growth rate grv p ; 10y7 , if we consider the plasma density n p ; 10 40 cmy3 . In conclusion we have derived a set of nonlinear basic equations taking into account the fermionic nature of the neutrinos, in opposition to the bosonic description, in order to study the linear theory of the scattering of neutrinos by plasma in a stellar medium. We have shown that the fermionic and bosonic descriptions of the neutrino, at least in the linear regime, yield structurally the same growth rate as pointed out by Bingham et al. w1x. Hence, as we see from our results, we can treat the collective interaction of neutrinos with dense plasma, in a linear
regime, by either fermionic or bosonic descriptions. The model can also be extended to consider the forward scattering to analize the system in other regimes, such as the modulational instability. We should point out that our results can also be applied to the interaction of anti-neutrinos with plasma.
Acknowledgements This work was partially supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnologico ŽCNPq, The Brazilian Council of Scientific and Technologic Developments..
References w1x R. Bingham, J.M. Dawson, J.J. Su, H.A. Bethe, Phys. Lett. A 193 Ž1994. 279. w2x R. Bingham, H. A Bethe, J.M. Dawson, P.K. Shukla, J.J. Su Phys. Lett. A 220 Ž1996. 107. w3x P.K. Shukla, L. Stenflo, R. Bingham, H.A. Bethe, J.M. Dawson, T.T. Mendonca, Phys. Lett. A 230 Ž1997. 353. w4x H. Bethe, Phys. Rev. Lett 56 Ž1986. 1305. w5x L.O. Silva, R. Bingham, J.M. Dawson, W.B. Mori, to appear in Phys. Rev. E Ž1999. w6x J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics, Mc Graw Hill, 1964. w7x K. Hirata, Phys. Rev. Lett 58 Ž1987. 1490.
Physics Letters A 258 Ž1999. 141–144 www.elsevier.nlrlocaterphysleta
Stimulated scattering of massless fermionic neutrinos by plasmas A. Serbeto, J.A. Souza Instituto de Fisica, UniÕersidade Federal Fluminense Gragoata, 24.210-340, Niteroi, Rio de Janeiro, Brazil Received 30 April 1998; received in revised form 24 March 1999; accepted 6 May 1999 Communicated by M. Porkolab
Abstract The Weyl equation has been used to study the interaction of a neutrino flux with a plasma in a stellar medium. The resulting collective stimulated scattering of the neutrino by the plasma is described by a cubic dispersion relation, which is solved numerically and analytically which gives a maximum growth rate structurally identical to growth rate obtained by Bingham et al. wPhys. Lett. A 193 Ž1994. 279x by using a Klein–Gordon like equation model, in which the neutrinos are considered as bosons, to describe the linear regime of the collective interaction. q 1999 Elsevier Science B.V. All rights reserved. PACS: 95.30.Qd; 95.30.Cq; 97.60.Bw; 97.60.Gb
In three very interesting recent papers, Bingham et al. w1,2x and Shukla et al. w3x have analyzed the collective interactions between neutrinos and plasma in a background stellar medium. The applications of their results seem very promising, specially in the field of supernovae dynamics. The above authors, however, following an early suggestion by Bethe w4x, described the neutrino field by a scalar function C , which obeys a Klein–Gordon like equation Žbosonic description. and assumed the neutrino mass to be different from zero in order to explain the possibility of the collective stimulated scattering of the neutrinos by plasma, which are coupled to each other due to the weak Fermi interaction and the ponderomotive force due to the neutrino flux. A question naturally arises: what modifications will appear in the results from Refs. w1–3x if the fermionic nature of the neutrino is taken into account? In a trial to get an answer, we have studied the neutrino-plasma interac-
tions but now describing the neutrino field by a massless Dirac equation ŽWeyl equation.. In this case, as is well known, the states of positive and negative helicities become completely uncoupled, and the left handed Žnegative helicity. state corresponds to the neutrino. The solutions of the Dirac equation with mass equal to zero
gˆ m EmC s 0
Ž 1.
are, in principle, four component spinors. Nevertheless, the separation of the states according to their helicities referred to above implies the neutrino being really a two component object. The extra degrees of freedom are removed by imposing the chiral constraint w5x
Ž 1ˆ q igˆ 5 . C s 0 ,
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 3 5 0 - 3
Ž 2.
A. Serbeto, J.A. Souzar Physics Letters A 258 (1999) 141–144
142
here g mˆ are the Dirac matrices, and gˆ 5 s igˆ 0gˆ 1gˆ 2gˆ 3. Thus, if C is originally of the form
a C s a1 , 2
ž /
Ž 3.
where a 1 and a 2 are two components objects, the constraint Ž2. implies a 1 s a 2 if we use the Dirac gˆ matrices in the Bjorken–Drell representation w6x. With all these in mind, let us write Weyl equation assuming a ‘minimal’ coupling to the weak field, like the one assumed in Refs. w1–3x. Using the trivector notation, we have for the neutrino dynamics the following equation: Vˆ
" EC
aˆ P pˆC s i
c Et
q c
C,
Ž 4.
where Vˆ s '2 G F n e , with G F being the weak coupling constant ŽFermi constant. and n e the electron plasma density. For one-dimensional interaction and using coordinate representation, Eq. Ž4. reduces to
EC yi" a xˆ P
Ex
'2
" EC si
c Et
q c
G F n eC ,
Ž 5.
scripts o,1, and l correspond to pump, scattered, and electron plasma wave Žfor neutrinos and Langmuir plasmons., respectively. Considering these matching conditions and supposing that C s Co q C 1 and n e s n p q d n, where Co and C 1 stand for the incident and scattered neutrino spinors, respectively, we have from Eq. Ž5.
ECo Et EC 1 Et
0 0 1 0
0 1 0 0
1 0 , 0 0
0
0
EC 1 Ex
y i GpCo s i Gp
dn
y i GpC 1 s i Gp
np
C1 ,
d n) np
Ž 8a .
Co ,
Ž 8b .
with d nrn p Ž d n ) rn p is the complex conjugate. being the electron plasma density perturbation normalized to the equilibrium plasma density n p Žmean density.. Following Ref. w2x the equation for the electron plasma density perturbation in the fluid model can be written as
½
E2 Et
2
E2 y Õt2
Ex
2
q v p2
dn
5
np
E2
Ž 9. Ž C1†Co q C1† P a x P Co . , where Gn s '2 G Fr4p m e " vn , Gp s Vor", and Vo s '2 G F n p . The right hand side of this equation is due s Gn
Ž 6.
and, according to the chiral constraint mentioned above f g Cs f g
Ex
q c a xˆ P
where in the Bjorken–Drell representation 0 0 aˆ x s 0 1
ECo
q c a xˆ P
Ž 7.
is the neutrino spinor Žnot for a single neutrino. normalized in such way that the neutrino energy density, Nn " vn , is W s ŽC † P C q C † P a x P C .r4p , where Nn is the neutrino density w1–3x. Since in this stimulated scattering process there is a feedback loop between the neutrino flux and the ripples of the electron plasma density, the seed of the collective instability due to the electron plasma wave and neutrino interaction must satisfy the energy momentum conservation matching conditions: " vn o s " vn 1 q " v l and "kn o s "kn 1 q "k l , where the sub-
E x2
to the ponderomotive force of the neutrinos acting on electron plasma w5x. Hence Eqs. Ž8. and Ž9. form a set of nonlinear basic equations to describe the dynamics of the stimulated scattering of neutrinos by plasma. In order to study the linear mechanism of the instability Žthe dispersion relation. we will consider a homogeneous pump amplitude ŽCo s constant. and take all space-time dependences to be proportional to expw iŽ kx y v t .x, where v and k are, respectively, the frequency and wave number associated to all dynamic quantities. Using Fourier transform in Eqs. Ž8. and Ž9. and combining the resulting equations we have the following matricial equation
Ž v l2 y v k2 . Ž vn 1 q Gp . Iˆy kn 1 c a xˆ C˜ 1 s k l2Gp Gn C˜o† P C˜ 1 C˜o q C˜ o† P a x P C˜ 1 C˜o ,
ž
/
ž
/
Ž 10 .
A. Serbeto, J.A. Souzar Physics Letters A 258 (1999) 141–144
143
where C˜o and C˜ 1 are the Fourier amplitudes of the pump and scattered neutrino spinors, respectively. Iˆ stands for 4 = 4 identity matrix and the matrix a xˆ is given by Eq. Ž6.. Using the representation given by Eq. Ž7. and the matching conditions the full dispersion relation can be written as
the parameters Gp and Gn , the normalized growth rate far from the threshold, in terms of the original physical parameters, is given by
Ž v l2 y v k2 .
where we have assumed the plasma wave phase velocity approximately equal to c. Here the normalized source term, S pn , of the instability is defined as
vn o y v l q Gp y Ž kn o y k l . c 2
s Gp Gn k l2 Co q Co† P a x P Co ,
Ž 11 .
where v k2 s v p2 q Õt2 k l2 and Õt is the electron thermal speed. In a realistic astrophysical scenario, due to the smallness of Fermi constant, G F , a weak coupling regime plays the main role in the interaction of neutrino fluxes with the stellar plasma. Hence the pump neutrino flux does not alter greatly the linear dispersion relation of the electron plasma wave,i.e, v l , v k q dv , where dv < v k . Then, substituting into Eq. Ž11. and neglecting second order terms in dv , the dispersion relation reduces to
d 2v , y
Gp Gn 2 vk
2
k l2 Co q Co† P a x P Co .
Ž 12 .
For dv s ig and assuming the resonant approximation of the neutrino dispersion relation, i.e., vn o y v k q Gp y Ž kn o y k l . c , 0, and from de definition of
'2
g vp
, 2
S pn s
s
S p1r2 n ,
Gp Gn vp
Ž 13 .
2
Co q Co† P Ž a xˆ P Co .
Vo2
Nn 2
" vp me c n p
.
Ž 14 .
As we can see, in the linear regime, the fermionic nature of the neutrino, given by the second term in Eq. Ž14., does not modify structurally the result for the growth rate pointed in Ref. w1x, in which the neutrinos are considered as bosons. The value of the Langmuir wave number Žplasmon linear momentum. which maximizes the growth rate can be obtained from the neutrino dispersion relation to yeld k l , v prc, which means that the maximum growth rate occurs when the plasma wave phase velocity is near to c. The numerical solution of Eq. Ž10. gives the
Fig. 1. Normalized growth rate,grv p , as a function of the source term,S pn .
144
A. Serbeto, J.A. Souzar Physics Letters A 258 (1999) 141–144
maximum growth rate, grv p , for different values of the source term, S pn , in an universal way, as shown in Fig. 1. We note that the maximum growth rate exhibits a kind of band structure with null values for some intervals of S pn values, with all the maxima satisfying the analytical solution given by Eq. Ž13.. It should be pointed out that this band structure disappears for very large values of the source term, which corresponds to a non physically realistic scenario. In supernovae of Type II, for instance, 10 58 neutrinos of all flavors are emitted with average energy near 15 MeV w7x, which gives us a total energy in the vicinity of 3 = 10 53 erg, resulting in an energy density, at 300 km from the center of the star, near 2.7 = 10 30 erg cmy3 which corresponds to a neutrino density close to 10 35 cmy3 . For these parameters the normalized growth rate grv p ; 10y7 , if we consider the plasma density n p ; 10 40 cmy3 . In conclusion we have derived a set of nonlinear basic equations taking into account the fermionic nature of the neutrinos, in opposition to the bosonic description, in order to study the linear theory of the scattering of neutrinos by plasma in a stellar medium. We have shown that the fermionic and bosonic descriptions of the neutrino, at least in the linear regime, yield structurally the same growth rate as pointed out by Bingham et al. w1x. Hence, as we see from our results, we can treat the collective interaction of neutrinos with dense plasma, in a linear
regime, by either fermionic or bosonic descriptions. The model can also be extended to consider the forward scattering to analize the system in other regimes, such as the modulational instability. We should point out that our results can also be applied to the interaction of anti-neutrinos with plasma.
Acknowledgements This work was partially supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnologico ŽCNPq, The Brazilian Council of Scientific and Technologic Developments..
References w1x R. Bingham, J.M. Dawson, J.J. Su, H.A. Bethe, Phys. Lett. A 193 Ž1994. 279. w2x R. Bingham, H. A Bethe, J.M. Dawson, P.K. Shukla, J.J. Su Phys. Lett. A 220 Ž1996. 107. w3x P.K. Shukla, L. Stenflo, R. Bingham, H.A. Bethe, J.M. Dawson, T.T. Mendonca, Phys. Lett. A 230 Ž1997. 353. w4x H. Bethe, Phys. Rev. Lett 56 Ž1986. 1305. w5x L.O. Silva, R. Bingham, J.M. Dawson, W.B. Mori, to appear in Phys. Rev. E Ž1999. w6x J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics, Mc Graw Hill, 1964. w7x K. Hirata, Phys. Rev. Lett 58 Ž1987. 1490.