Lagrangian formulation for neutrino plasma interactions

Transcription

1 PHYSICS OF PLASMAS VOLUME 6, NUMBER 4 APRIL 1999 Lagrangian formulation for neutrino plasma interactions Alain J. Brizard and Jonathan S. Wurtele Department of Physics, University of California and Lawrence Berkeley National Laboratory, Berkeley, California Received 30 November 1998; accepted 15 December 1998 A Lagrangian formalism is used to derive coupled nonlinear equations for collective interactions between an intense neutrino flux and a relativistic cold plasma fluid with multiple particle species. In order to focus on the self-consistent collective treatment of neutrino plasma interactions, quantum effects are ignored throughout and the spinless neutrinos are represented by a complex-valued Klein Gordon scalar field. Through the application of Noether s method, the conservation laws for energy, momentum and wave action are derived explicitly. The transfer of energy, momentum and wave action between the neutrinos and the electromagnetic-plasma is discussed in the context of astrophysical applications e.g., type II supernova explosions American Institute of Physics. S X I. INTRODUCTION In a series of papers Refs. 1 9, Bingham et al. recently introduced a set of nonlinear coupled equations to study the collective interaction between an intense electron neutrino flux and a dense cold plasma. Such interactions involving the weak force may occur during a type II supernova explosion when most of the energy released during gravitational collapse is thought to go to neutrinos. Indeed, during the initial phase of a type II supernova explosion, 10,12 electron capture reactions (e p n e ) in the core produce a large outward electron neutrino flux, which can then interact with the outer layers of the collapsing star. Some typical supernova neutrino approximate parameters 10,12 are J released in approximately 1 second yielding a neutrino flux of W/cm 2 at a distance of 100 km from the center of the core. Approximately 1% of the neutrino energy and momentum needs to be transferred to the surrounding plasma in order to generate the observed supernova explosion. In addition, the intense neutrino flux is expected to produce significant plasma heating through resonant 3-wave interactions of the type neutrino neutrino plasma. The investigation of these transfer processes thus requires exact conservation laws for energy, momentum and wave action. The investigation of neutrino plasma interactions in the present work is based partially on the assumptions used by Bingham et al. 1 9 First, a classical description for the plasma and the neutrinos is adopted i.e., quantum and general relativistic effects are ignored entirely. Furthermore, as pointed out in Ref. 3, a classical wave description for the neutrinos is used to emphasize the ponderomotive i.e., collective nature of the neutrino plasma interactions. Next, the background plasma is treated as a cold, unmagnetized multi-species fluid. The relevance of these assumptions can be checked by comparing when appropriate the predictions of the present classical neutrino plasma model and those of neutrino plasma models constructed from finite-temperature quantum field theory QFT. Work presented in Ref. 13, for example, considers the propagation of neutrinos in a magnetized plasma. We point out, however, that QFT models usually do not consider self-consistent neutrino plasma interactions, in so far as they ignore the action of the neutrinos on the background plasma, whereas the emphasis of the present work is to study the effects of an intense neutrino flux on a background plasma. Our purpose in this paper is to derive coupled nonlinear equations describing the interaction between an intense neutrino flux with a dense plasma from a variational principle. The Lagrangian formalism is ubiquitous in quantum field theory, 14 where it serves as a guiding principle for deriving correct field equations. In the present work, we use this formalism to correctly and systematically derive a set of coupled nonlinear equations for neutrino plasma interactions which possess the important conservation laws of energy, momentum and wave action. These exact conservation laws are derived from the Lagrangian density through the Noether method. The Lagrangian formalism presented here builds on the Lagrangian formulation recently introduced for laser plasma interactions. 15 Applications such as magneticfield generation through self-consistent neutrino plasma interactions will be discussed in a future publication. II. VARIATIONAL STRUCTURE OF THE NEUTRINO PLASMA EQUATIONS A. Lagrangian density for neutrino plasma interactions The investigation of collective interactions between electron neutrinos and an unmagnetized plasma fluid with multiple particle species is modeled by the interaction Lagrangian density: 16 L int g n n c c, 1a obtained by taking the classical fluid limit of the QFT Lagrangian density. 17 Here, the summation is over particle species (e for electrons, n for neutrons, p for pro X/99/6(4)/1323/6/$ American Institute of Physics

2 1324 Phys. Plasmas, Vol. 6, No. 4, April 1999 A. J. Brizard and J. S. Wurtele tons and i for ions other than protons, n and n v are the plasma density and plasma density flux for particle species, respectively, while n and are the neutrino density and neutrino flux, respectively. The parameter g in 1a denotes the coupling between the neutrinos and particle species ; each coupling coefficient is calculated by QFT methods. 13,18 Taking into account the charged and neutral weak currents for each particle species, we use for e,p,n: g e 2G F, g p 0, 1b g n G F /2, where G F ( ev cm 3 ) is the Fermi weakinteraction coupling constant; since ions are composed of Z protons, N neutrons and ZI electrons, the parameter g i is approximately 2(ZI)NG F /2. We note that the work of Bingham et al. 1 9 considers the case of a nonrelativistic plasma background only i.e., their work assumes that the neutrinos interact with a background plasma at rest, for which the dominant term in the interaction Lagrangian density 1a is g n n. Here, using 1a the neutrino plasma interactions are treated covariantly. We now turn to the definitions of the neutrino density n and the neutrino flux appearing in 1a. In the present work, the classical neutrino field is described in terms of a complex-valued scalar field. 11 In the absence of interactions with the plasma, is assumed to satisfy the unperturbed Klein Gordon equation: 2 t c , 2 where m c 2 / is a frequency associated with the neutrino mass which we allow to be finite. In the Klein Gordon representation, 19 the neutrino density n and the neutrino flux are expressed in terms of as n i* t t *, 3a ic 2 **. With these definitions note that 2 has units of ev 1 cm 3 ), the unperturbed Klein Gordon equation 2 yields the unperturbed neutrino continuity equation, t n 0. 3b The Lagrangian density for the perturbed Klein Gordon equation is thus obtained by combining the Lagrangian density for the unperturbed Klein Gordon equation 2 with 1a: L 2 t 2 c g n n v c 2, 4 where n and are defined in 3a. The Lagrangian formulation for the cold multi-fluid plasma and the electromagnetic field is given in terms of the densities n and the fluid velocity v for each particle species, and the electric and magnetic fields E and B expressed in terms of the potentials and A as E c 1 t A and BA). In the absence of neutrino plasma interactions, the Lagrangian density for the electromagneticplasma EMP equations is 15 L EMP 1 8 E2 B 2 m c 2 n 1 1 q n v c A, 5 where each cold fluid species is treated fully relativistically, with (1v 2 /c 2 ) 1/2, and q denotes the electric charge of particle species. By combining the Lagrangian densities 4 and 5 we arrive at the Lagrangian density for the neutrino plasma interactions: L 1 8 E2 B 2 2 t 2 c m c 2 n 1 1 n v c, 6a where the generalized scalar potential and the generalized vector potential are defined, respectively, as q g n, 6b q Ag /c. Note how the electromagnetic potentials (,A) and the neutrino potentials (n, ) combine in 6b to form the generalized electromagnetic neutrino potentials (, ). In the Lagrangian density 6a, the first two terms represent the electromagnetic and neutrino fields, respectively, while the remaining two terms inside the summation over species represent, respectively, a free relativistic fluid and its interaction with the electromagnetic neutrino fields. B. Variational principle The variational principle d 3 xdtl0 requires an expression for L in terms of, A,, * and the virtual fluid displacement. The constrained variations of n and v are expressed in terms of as 15 and n n 7a v t v v. The expressions 7a ensure that the continuity equation for each particle species, n n t v 0, 7b is satisfied; note that 7b also ensures that the Lagrangian density 6a is invariant with respect to an electromagnetic gauge transformation see 17c below. After some manipulations, the variation of the Lagrangian density 6a yields

3 Phys. Plasmas, Vol. 6, No. 4, April 1999 A. J. Brizard and J. S. Wurtele 1325 L J t J L L EA L A 1 c t L E L B t L L v v L n v v L n 2Re L t v L t, L 8a where the generalized density J and the generalized density flux J are defined, and J A c L E J L 2Re L v t 8b L L v n v n I L E B AL 2 Re. L 8c These densities play a fundamental role in the Noether method; the derivation of conservation laws by this method is done in the next Section. When the variational principle d 3 xdtl0 is used, we note that only variations (,,A, ) which vanish at the integration boundaries are considered. Hence the first two terms on the right side of 8a have zero contributions when integrated over space time. Using 6a we now evaluate the partial derivatives of L which appear in 8a 8c. For partial derivatives with respect to (,A,E,B) we find L/ q n, L/A q n v, 9a L/EE/4, L/BB/4. For partial derivatives with respect to (*, t *, *)we find L/* 2 2 i t v, L/ t * 2 t i 2 ˆt, 9b L/ * 2 c 2 i 2 c 2 ˆ, v /c 2 where g n / is the weak-interaction plasma frequency for species. Lastly, for partial derivatives with respect to (n,v ) we find L/n m c v /c, L/v n m v /c. 9c The requirement that d 3 xdtl0 for arbitrary variations and A yield, respectively, the Maxwell equations: E4 q n, 10 t EcB4 q n v. 11 The requirement that d 3 xdtl0 for arbitrary virtual fluid displacements yields the cold, relativistic fluid equation of motion for species : m t v v 1 c t v c, 12a where we have made use of the continuity equation 7b. Note that the previous work of Bingham et al. 1 9 retains the neutrino ponderomotive term n only on the right side of 12a. Note also that the form of 12a is independent of the representation used for the neutrino field, i.e., this form follows immediately from the covariant expression for the interaction Lagrangian density 1a. Using the Klein Gordon representation 3a for the neutrinos, 12a becomes m t v v q E v c B 2ig d t * *d t, 12b where d t t v. Finally, the requirement that d 3 xdtl0 for arbitrary variations * and yield the perturbed Klein Gordon equations: 2 t c i d t, 13 2 t c *2i d t *, where the continuity equation 7b was used on the right side. Note that, in the presence of neutrino plasma interactions, the neutrino continuity equation 3c becomes or t n 2 d t 2, 14a t nˆ ˆ0, 14b where the neutrino density and neutrino flux are now nˆ i*ˆ t(ˆ t)* and ˆ ic 2 ( ˆ )** ˆ, with the operators ˆ t and ˆ defined in 9b. III. NOETHER METHOD AND CONSERVATION LAWS Since the neutrino plasma equations hold true for arbitrary variations,, A, ), the variation equation 8a becomes L J t J, 15

4 1326 Phys. Plasmas, Vol. 6, No. 4, April 1999 A. J. Brizard and J. S. Wurtele where explicit expressions for J and J are obtained by substituting 9a 9c into 8b and 8c, respectively: J A 4c E n p 2 2 Re*ˆ t, 16a where p m v /c is the generalized canonical momentum for species, and J n v p L I 1 4 EAB 2 2 c 2 Re* ˆ, 16b where L m c 2 (1 1 )( v /c) is the Lagrangian for a single particle of species moving in the combined electromagnetic neutrino fields. Equations such as 15 will henceforth be known as Noether equations. Note that the right side of 15 is invariant under the transformation, J J C and J J t C D, 17a where C and D are arbitrary vector fields. For example, consider an infinitesimal electromagnetic gauge transformation generated by the scalar field : c 1 t, A, and LL/A L/A. Then 16a and 16b become J 1 L ce 1 c L E 1 L c E, 17b J 1 L t c E L B 1 L c t E L B t 1 c L, B and the Noether equation 15 becomes 1 c t L A0, L L E 17c or the charge conservation equation t ( q n ) ( q n v )0 which follows from the continuity equation 7b for each species. We now derive the conservation laws of energy, momentum and wave action by inserting into 16a and 16b explicit expressions for the variations (,,A, ) and L. Within the Noether method, each symmetry of the action integral d 3 xdtl generates these explicit expressions. A. Energy conservation law We obtain the energy conservation from 15 through the Noether method as follows. First, we consider the following variations: tv, t t, t t, 18a At t ActE, Lt t L, where t tt represents a uniform translation in time. Substituting these expressions into 16a and 16b, wefind JtEL te/4, 18b JtS t te/4 ctb/4, where E 1 8 E2 B 2 m c 2 n 1 2 t 2 c a is the energy density and S ce 4 B m c 2 n 1v 2 2 c 2 Re t * 19b is the energy density flux. With these definitions, the energy conservation law is E t S0. 20 B. Momentum conservation law In a similar way the momentum conservation law follows from 15 through the Noether method. First, we consider the variations: x, x, x Ec 1 t A x, 21a Ax AxB A x, Lx L, where x xx represents a uniform translation in space. Substituting these expression into 16a and 16b, wefind JP x EA x/4c, 21b JTLI x t EA x/4c BA x/4, where P E 4c B22 Re * t m n v 22a is the momentum density and T m n v v 2 2 c 2 Re * 4E 1 2 B 2 I 2 EEBB 2 t 2 c I 22b

5 Phys. Plasmas, Vol. 6, No. 4, April 1999 A. J. Brizard and J. S. Wurtele 1327 is the momentum density flux. With these definitions, the momentum conservation law is P t T0. C. Wave-action conservation law 23 Our final conservation law is the wave-action conservation law. Although this law is normally derived within the context of linearized wave dynamics i.e., the interaction between linear waves of different types coexisting in a background medium 20, Hayes 21 has shown how to generalize the concept of wave-action conservation for nonlinear dissipationless systems. Following Hayes, 21 we first consider a family of solutions of the neutrino plasma equations parametrized by a real-valued angle and require that each field (n,v,,a,) be-periodic with period 2. Next, we consider the variations: A L A L, 24 where represents a constant phase shift and now represents a physical fluid displacement. Substituting these expressions into 16a and 16b and averaging over denoted by ), we obtain the generalized wave-action conservation law: A B0, 25 t where the wave-action density is defined in terms of A A 4c E n p 2 2 Re *ˆ t, 26a while the wave-action density flux is defined in terms of B n v p L I 1 4 E AB v v ṽ e i ṽ *e i, 28a e i *e i, into the relations 27a and linearize with respect to the small perturbation parameter, we obtain the familiar linear relations: ñ n and ṽ t v v, 28b where n and v characterize the background medium. This remark is helpful in interpreting the role played by the angle. In particular, it is used in 28b to decompose a realvalued field e.g., n ) into a complex-valued field e.g., ñ ). IV. TRANSFER OF ENERGY, MOMENTUM AND WAVE-ACTION As mentioned in the Introduction, an important aspect of neutrino plasma interaction during a type II supernova explosion concerns the transfer of energy, momentum and wave-action from the neutrinos to the dense surrounding plasma. These transfer processes can be described in terms of the conservation laws derived above as follows. First, we derive the energy-transfer equation. The electromagnetic plasma EMP energy density E EMP and energy density flux S EMP : E EMP 1 8 E2 B 2 m c 2 n 1, 29 S EMP c/4eb m c 2 n 1v satisfy, from 20, the energy-transfer equation, E EMP S t EMP g n v n 1 c 2 t. 30 We note that the right side of 30 is the neutrino plasma analog of the electromagnetic plasma energy-transfer term q n v E. The momentum-transfer equation is found, from 23, using the definitions of the EMP momentum density P EMP and the momentum density flux T EMP : 2 2 c 2 Re * ˆ. 26b The fact that the wave-action conservation equation 25 holds exactly relies on the following two relations: n n, 27a v t v v. With these definitions, we find t A B L, 27b and 25 follows as a result of the periodicity of the fields with respect to once 27b has been -averaged. Note that if we substitute the following expressions: n n ñ e i ñ *e i, P EMP 1/4cEB m n v, T EMP m n v v 1 4 E 2 B 2 I 2 EEBB, to be P EMP T t EMP g n n 1 v c 2 t c

6 1328 Phys. Plasmas, Vol. 6, No. 4, April 1999 A. J. Brizard and J. S. Wurtele This transfer equation indicates that the neutrino flux can not only transfer momentum to the plasma but that it can also transfer momentum to the electromagnetic field thereby generating Poynting flux. Lastly, Eq. 25 can be used to find the wave-actiontransfer equation: t A EMP B EMP g n n v c 2, 33 where the EMP wave-action density A EMP and the waveaction-density flux B EMP are A EMP 1/4c A E n m v q A/c, 34 B EMP n v m v q A/cm c q v A/cI 1/4 E AB. The wave-action transfer equation 34 can be used in the context of the Manley Rowe relations for resonant threewave processes. The transfer equations 30, 32 and 34 imply that electromagnetic fields can be generated during the interaction between an intense neutrino flux and a background plasma. Here, collective ponderomotive neutrino forces drive currents and produce density perturbations in the background plasma which in turn act as sources for the electromagnetic field. The latter field can appear either in the form of electromagnetic radiation or in the form of a quasi-static magnetic field. Future work in this area will study these transfer processes more closely for realistic supernova scenarios. V. DISCUSSION A Lagrangian formulation of the coupling between an intense neutrino flux and a cold relativistic multi-fluid plasma has been used to derive equations of motion and the exact conservation laws of energy 20, momentum 23 and wave action 25. These equations differ from those more commonly used 1 9 in that additional neutrinoinduced ponderomotive terms appear in the cold plasma fluid equations of motion 12a. These new terms naturally arise from the covariant formulation of the neutrino plasma interaction Lagrangian density 1a. The extent to which the new terms change any previous work has not been examined. Straightforward extensions of this Lagrangian formulation to derive dynamical equations and conservation laws associated with more realistic physical situations will be made in a future publication. These extensions will include the problem of neutrino plasma interactions in magnetized plasmas represented either by a warm relativistic fluid description or by a relativistic kinetic description, a more realistic description of neutrino evolution, and the problem of magnetic-field generation. ACKNOWLEDGMENTS The authors thank G. Shvets for pointing out the work of T. Tajima and K. Shibata 16 on the Lagrangian formulation for the interaction between a single electron and single neutrino. This work was performed under Department of Energy Contract No. PDDEFG-03-95ER R. Bingham, J. M. Dawson, J. J. Su, and H. A. Bethe, Phys. Lett. A 193, J. T. Mendonça, R. Bingham, P. K. Shukla, J. M. Dawson, and V. N. Tsytovich, Phys. Lett. A 209, R. Bingham, H. A. Bethe, J. M. Dawson, P. K. Shukla, and J. J. Su, Phys. Lett. A 220, P. K. Shukla, L. Stenflo, R. Bingham, H. A. Bethe, J. M. Dawson, and J. T. Mendonça, Phys. Lett. A 224, R. Bingham, R. A. Cairns, J. M. Dawson, R. O. Dendy, C. N. Lashmore- Davies, and V. N. Tsytovich, Phys. Lett. A 232, P. K. Shukla, L. Stenflo, R. Bingham, H. A. Bethe, J. M. Dawson, and J. T. Mendonça, Phys. Lett. A 233, P. K. Shukla, L. Stenflo, R. Bingham, H. A. Bethe, J. M. Dawson, and J. T. Mendon, Phys. Plasmas 5, P. K. Shukla and L. Stenflo, Phys. Rev. E 57, P. K. Shukla, R. Bingham, J. T. Mendonça, and L. Stenflo, Phys. Plasmas 5, J. Cooperstein, Phys. Rep. 163, H. A. Bethe, Phys. Rev. Lett. 56, H. A. Bethe, Rev. Mod. Phys. 62, S. Esposito and G. Capone, Z. Phys. C 70, S. Weinberg, The Quantum Theory of Fields, Vol. I, Foundations Cambridge University Press, Cambridge, 1995, Chap A. J. Brizard, Phys. Plasmas 5, T. Tajima and K. Shibata, Plasma Astrophysics Addison-Wesley, Reading, MA, 1997, Sec L. Wolfenstein, Phys. Rev. D 17, D. Notzold and G. Raffelt, Nucl. Phys. B 307, P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I McGraw-Hill, New York, 1953, Chap G. B. Whitham, Linear and Nonlinear Waves Wiley, New York, W. D. Hayes, Proc. R. Soc. London, Ser. A 320,