Bound-neutrino sphere and spontaneous neutrino-pair creation in cold neutron stars

VOLUME 64, NUMBER 2 PHYSICAL REVIEW LETTERS 8 JANUARY 1990

Bound-Neutrino Sphere and Spontaneous Neutrino-Pair Creation in Cold Neutron Stars

Abraham LoebThe Institute for Adt anced Study, Princetoni, Yew Jerse& 08540

(Received 21 August 1989)

It is shown that neutrinos (massless or massive), produced with kinetic energies below —50 eU in asupernova, have bound orbits in the remnant neutron star. The binding is mediated by a radial weak-interaction force, caused by a gradient in the collective weak potential of the neutrons in the star. Thisforce is also able to create spontaneously neutrino-antineutrino pairs. If the bound-neutrino sphere is notfully degenerate at low momenta, a cold neutron star will shine continuously antineutrinos with energies

50 eV, as a result of the density gradient in it. In principle, these eA'ects can also be realized at small-er (e.g. , solid) densities for sufficiently low neutrino energies.

4'„f(0)n =1+ =1+

mnpo

J2KGF p E2mn, Po

where GF is the Fermi constant, f(0) is the forwardneutrino-neutron scattering amplitude, m„ is the neutronmass, and p„ is the mass density of neutrons. For Dirac

PACS numbers: 97.60.3d, 14.60.Gh, 42.20.Cc, 67.90.+z

It is usually assumed that massless particles, such asphotons or neutrinos, cannot stay bound to stars, andthat their escape is only delayed by scattering processes. '

This Letter will present an exception to this generalview.

In the neutron liquid of a cold neutron star, there is amacroscopic number of neutrons in a de Broglie-wavelength scale of neutrinos with energies of severaltens of eV. Consequently, the collective eff'ect of theneutrons (or quarks) on a neutrino is coherent and canbe averaged over the latter wavelength scale. This eff'ect

can be expressed in terms of a smooth weak-interactionpotential, or equivalently an index of refraction, that de-pends on the local mass density. The existence of this lo-

cal potential leads naturally, in an inhomogeneous medi-

um, to the formation of a weak-interaction force actingon each neutrino. It is important to stress that this forceis not a result of random scattering processes that delaythe diffusion of neutrinos from the star, but rather re-sults from the coherent weak interaction of the nuclearmatter with neutrinos. The neutron star has the same"weak charge" for all neutrino flavors because it con-tains many more neutrons than protons and electrons.The resulting force is directed along the radial direction,just like the collective gravitational force produced bythe baryons. While gravity cannot prevent massless neu-

trinos from escaping, this radial weak force can bindthem if their kinetic energies are smaller than the poten-tial depth. This capacity is naturally understood in

terms of the large refractive index of the neutron liquid,which can bend the low-momenta neutrino rays.

When its value is close to unity, the index of refractionfor low-energy neutrinos in an unpolarized medium,composed mostly of nonrelativistic neutrons, is givenb '-4

neutrinos,

K=+—1

2

1/2-

1+Ip 2

(2)

where the upper sign is for neutrinos with helicity, —I;the lower sign is for antineutrinos with helicity, I; andl = ~ 1. In the case of Majorana neutrinos,lim~~~ „,„K=0. For the sake of concreteness, this dis-cussion will refer only to Dirac particles with a diagonalmass matrix, where Im, I, -,„, is the correspondingmass of each flavor. We denote by po—:F. —

pyg, theneutrino momentum in vacuum at an energy F, and by

p =n po its corresponding eff'ective momentum in

matter (neglecting general-relativistic eII'ects). In Eq.(I) we use the units h =c =ktt =1, and ignore the imagi-nary part of n, since the inelastic scattering rates arenegligible on the short dynamical time scales of boundneutrinos. In the framework of weak interactions, thisequation was rigorously shown to be valid ' ' for

~n —

1~

&&1. The index of refraction is conventionallyintroduced into the spatial phase of the neutrino wavefunction, @(x)=npo x. The mass density in the neutronstar varies spatially on scales that are much larger thanthe de Broglie wavelength of the neutrinos under con-sideration, i.e., ~Vn/np ~

&&1. Thus, the trajectory of aneutrino wave packet can be described in the context ofgeometrical optics, where a ray is defined as the curveorthogonal to the geometrical wavefront: @(x)=const.Then, similarly to the photon optics, the ray trajectoryin a medium with a weakly inhomogeneous index of re-fraction is described by the equation (d/ds)(ndr/ds)=&n, where r is the position vector at a given pointalong the ray and s is the length of the ray at this posi-tion, as measured from a fixed point on the ray. Inspherical symmetry, this equation conserves the effective"angular momentum" of the neutrino, npod, where d isthe perpendicular distance from the center to the tangentof the ray at a given position. Accordingly, geometricalconsiderations provide a simple solution for the neutrinoray trajectory in a spherical star, with n =n(r) of an ar-

1990 The American Physical Society 115


bitrary magnitude,

=—[[n(r)r]' b—'i "'-.dO b

(3)

Here r=(r, 0) =re, is written in terms of the polar coor-dinates in the plane defined by the neutrino angularmomentum, and b =const has the dimension of length.The trajectory generally bends towards higher values ofn(r); i.e. , neutrinos (0 & K) will be attracted towards thecenter of the star. For example, neutrinos that are pro-duced where [8(nr)/Br] i, =0, with an angular momen-tum ppb, will stay bound to the neutron star in a closedorbit, sampling a circle on the constant-index-of-refraction surface at the radius r. In spherical symme-

try, all the bound neutrinos will conserve their adiabaticinvariants and may escape from the star only as a resultof rare inelastic collisions. In the more general case, thestochasticity of their orbits depends on the spatial andtemporal details of the neutron density distribution.

Equation (3) is equivalent to the classical equation ofmotion of a neutrino in a spherically symmetric poten-tial V(r) obeying

p =(F. —V) —m, =n po.

According to the weak-interaction Hamiltonian, this po-tential should depend (up to a factor of —

1 (K~ 1)only on the neutron density. Thus, it can be derivedfrom Eqs. (1) and (4) with in —

1i «1,

V(r)—:—KVo(r)—:—KGF

Jam„,p„(r)

38K is 3eV.10"g/cm',

(5)

This expression is of general validity and can also be de-

rived directly from the effective weak-interaction La-grangian for the unpolarized neutron medium,

L = (GF/242) y„y, (1+ys) y, ty„y'y„. The correspond-

ing classical equation of motion for a neutrino or an an-

tineutrino isa(KV, ) .~ =F—= —VV= e, , (6)

dt 8rwhere K = + 1 for massless particles with I = 1. Theneutrino behaves analogously to an electron with a nega-tive charge —K in an electric potential Vp. In the caseof antineutrinos, —K is positive and the central force Fbecomes repulsive, as one would expect from the analogy

i

to positrons. In principle, a mixed beam of neutrinos andantineutrinos can be separated by this force. If(p +m, )' —m„& —V, a neutrino will stay bound tothe star, consistently with Eq. (3). Consequently, theneutrinos that are produced thermally in a type-II super-nova with kinetic energies below their local potentialdepth i Vi -50 eV will stay bound to the remnant neu-

tron star, captured in its weak-interaction potential well

(gravity changes these values only by &10%). The an-tineutrinos observed in terrestrial detectors are accelerat-ed by this force and gain the small amount of —50 eV in

their kinetic energies. The weak potential at their emis-sion surface causes a net delay of

dr = (20.6 sec)100 kpc 0. 1 MeV

—3

10 eV

Vp

10 eV

between the arrival times of neutrinos and antineutrinoswith the same initial momentum, from a supernova at adistance L. In addition, because of the excess emissionof v, during the neutronization process, the emergent su-

pernova neutrinos will deposit a net fraction of about10%x Vo/(10 MeV) —0.5&10 of their total momen-tum in pushing the neutrons outwards. The amount ofenergy exchanged (—10 ergs) is, however, too small toaffect the supernova hydrodynamics.

In order to evaluate the macroscopic properties of thebound neutrinos, one should specify their distributionfunction. In a supernova the weak binding energies,given by Eq. (5), are much smaller than the chemicalpotential and temperature of the degenerate neutrinosduring the collapse. ' Accordingly, the neutrino distri-bution function, p„,, is almost fully degenerate for kineticenergies below

iV

i in the neutron liquid, i.e., for all thebound species P,, (p) = 1. Ignoring nonadiabaticchanges in the neutron-star potential and neglecting dis-sipative processes, the bound neutrinos will preserve theirintegrals of motion and have an equilibrium number den-

goo ~ Jr'0 d 3

e...= g J J (p'+m, ') 'i'P, (p) P, d'x-e, . ' ' ' (2z)'

3GFp (r)r dr .

8am„(7)

sity given by

~ V0

n., (r) = J P,, (p)4zcp dp(2') =V (r0)/6z

The upper limit in this integral (that represents the ma-

jority of the occupied phase-space volume) was taken ac-cording to the relativistic case, assuming that m, ((Vp,based on the existing bounds on stable light neutrinomasses. '" Because of the strong dependence of n, on

the local mass density in the star, most of the bound neu-

trinos surround the center with n, —0.02(50 eV) =3x 10' cm . The total energy stored in the threespecies of bound (mostly left-handed relativistic) neutri-nos depends strongly on the density profile in the star,

116


In a crude approximation, e„,—0.02(50 eV) (10 km)=3&10 ergs. Thus, the bound neutrino sphere addsabut 30 kg to the neutron star mass. In total, the energystored in bound neutrinos is small and may have only aminor eAect on the late cooling stages of the star. '

It is interesting to note that the neutron star also has acloud of unbound particles in it, containing the cosmo-logical background neutrinos. If the neutrinos are mas-sive, their cosmological density is enhanced appreciablyby their gravitational interaction with the galaxy. Nev-ertheless, because of their relatively low temperature anddensity they will not be significant inside the neutronstar, compared with the bound-neutrino sphere there.For example, in the case of massless neutrinos, theVlasov equation and Eq. (6) yield a cosmological distri-bution function of unbound neutrinos

P,, (p) = (I —V,/p) '

exp([p —Vo]/T, ) + 1

f«p ~ Vo, which provides a number density,

n,, =) P,, (p)d'p/(2z) '=0.09T,'

inside the neutron star, where T, =1.9 K is the predictedneutrino background temperature. The local energydensity of the unbound neutrinos is 3n, Vo while they fol-low almost radial orbits, because T, &( Vo. Sincen, /n, =5.4(T,/Vo) -2x10 ', the existence of thiscloud can be ignored near the center of the star.

The distribution function of the bound neutrinos will

evolve slowly as a result of inelastic collisions with theleptons and hadrons in the star. The v„v„, and v, 's will

increase their energy mostly by scattering on the neu-

trons, electrons, and muons. The evaporation rate ofbound neutrinos is dictated by the inelastic collision fre-quencies, being limited by their final-state degeneracyblocking (I —P, ) in each collision. On average, a rela-tivisitic neutrino can increase its local energy (E —V)only by the fraction ' —T„/m„«1 in each collision with

a neutron at a temperature T„&&Vo. Therefore, theheating will start being eAective first for the highest-

energy neutrinos, and then continue at lower energies asthese high-energy states will be evacuated. By takingthe standard weak-interaction cross sections and acharacteristic central composition, the minimal timescales for evaporation (without neutrino degeneracyblocking) are found to be of the order of (10 yr)[(p +m, ) ' /(1 eV)], at p„= 10' g/cm and T„= 10 eV. In reality, the details of the evaporation pro-cess for the three diAerent neutrino species depend onthe density structure and the late temperature history(down to T„—50 eV after —10 yr) of the neutron-starinterior. " As a result of this slow evaporation, the de-pleted distribution function may be populated again byvarious mechanisms, up to a new equilibrium stage, ob-tained when the population rate balances the evaporationrate. This equilibrium state will correspond to some spa-tial and temporal averages of the star temperature, be-

cause of the large mean free path of the bound neutrinos.In the case of an extensive evaporation of the initial

neutrino sphere in old neutron stars, the weak field ap-pearing in Eq. (6) is capable of creating spontaneouslyneutrino-antineutrino pairs with I = 1 [in a similar way,

any weakly interacting particles that are reflected by theneutron star from outside (e.g. , the cosmological an-

tineutrino background or galactic halo particles) with

IE+m] & V, can induce pair creation according toKlein's paradox' ]. The produced antineutrinos at a ra-dius r will be accelerated outwards up to a kinetic energyof

~V(r) ~, and escape from the star; while the produced

neutrinos will be attracted to the center and accumulatearound it in bound orbits. In principle, the emission

spectrum can be used as a probe of the internal densitystructure of a neutron star. The energy for this processis provided by very small adjustments in the neutron

density gradient that is gravitationally supported in thestar. Clearly, a significant evaporation of the star energycannot take place because of baryon number conserva-

tion. Before any structural backreaction eAect becomessignificant this pair-creation process will be stronglyself-inhibited by the bound neutrinos it generates. Let us

first consider the most eA'ective pair-creation conditionsof massive neutrinos, obtained when there are no bound

neutrinos in the star. In this situation, the productionrate of massive Dirac neutrino pairs (with I = 1 ) per unit

volume per unit time, w, can be evaluated in analogywith electron-pair production in a homogeneous electricfield, ' yielding

F 21w=, g g, exp

eprJ 1 J

jism,

i.e., for the neutrino mass range m, ((0.05 eV. For neu-

trino masses larger than 10 eV the above rate is al-

ready negligible; thus the weak field appearing in it canbe considered as locally homogeneous. At typicalneutron-star conditions,

~F

~

—5 eV/km, yielding

w —(10 cm -'sec ')+exp[ —m, , /(2x10 eV) ],for 10 eV ~m, The field will produce mostly the

lightest species in this range. If evacuated of its neutri-

nos, a neutron star with an eAective emission volume of——, n(10 km), will shine initially with a luminosity of

—[3 x 10 ' ergs/sec] g exp[ —m, /(2 x 10 ~ eV) 2]

in antineutrinos with energies 10 eV«F =~

V~

~ 50eV, because of the neutron density gradient in it. Theemission process will eliminate itself gradually by filling

where F is given by Eqs. (5) and (6), with'~

It'~

= l.This nonperturbative expression assumes homogeneity ofthe local weak-interaction force, and

~F ~/m, &&1. The

homogeneity assumption is valid if'

)~F/F'

I&& ( ( F ( /m, '. ) ' 'm „

117


up to degeneracy the bound neutrino sea at low momenta(if not already filled up from the supernova). Conse-quently, the above rate will be inhibited by the Pauliprinciple. In this situation, the virtual neutrinos musttunnel through a potential barrier that has a magnitudeof about the lowest available neutrino energy state, in-

stead of 2m„ in order for new pairs to be materialized. '

Therefore, even if the neutrino is massless it will acquirewithin a very short time scale (510 km=3X10 ' sec)an effective mass larger than (~ F ~/tr)

' in this process.Following the exponential dependence in the above esti-mates, this pair-production mechanism will not be ableto populate significantly the neutrino sea above —10eV, on a Hubble time scale. Nevertheless, most of thethermal evaporation of the bound neutrinos takes placeat much higher energies. Thus, when the initial neutrinosphere is depleted by inelastic collisions, it will be popu-lated mainly by thermal processes, such as the URCA orpionic reactions and the nucleon or neutrino pair brems-strahlung. '

In conclusion, we have shown that low-energy neutri-nos have bound orbits in cold neutron stars. The weakbinding force may decay spontaneously (in a nonpertur-bative process) into neutrino pairs, if their effective massis sufficiently small. This second eAect adds the weak in-

teraction to the known examples of spontaneous paircreation in electrodynamics, ' gravitation, ' and quan-turn chromodynamics. Clearly, the eAects mentionedabove can also be realized at other conditions, e.g. , in

other stars, near black holes, or in cosmology, but typi-cally with a smaller magnitude. In particular, neutrinoswith p ~ 10 [rrt„/(10 eV)] '/ eV can be confined insidesolids. ' In order to detect the pair-creation eAect atsolid densities (where potential gradients similar to thosementioned above can be obtained only over A boun-daries), one may use time-dependent (e.g. , rotating)structures. Although difficult to measure, any experi-mental evidence for the existence of these phenomenacan be used to probe, at relatively low energies, some ofthe fundamental and as yet unresolved questions in

weak-interaction theory.The author thanks John Bahcall, Murph Goldberger,

Andy Gold, Lars Hernquist, Ofer Lahav, Glenn Stark-man, Leo Stodolsky, Martin Weinberg, and Ed Wittenfor fruitful discussions. This work was supported in partby the NSF, Grant No. PHY-86-20266.

'S. A. Shapiro and S. A. Teukolsky, 8lack Holes, 8 hite

Dwarfs and iVeutron Stars (Wiley, New York, 1983), pp.306-334.

~M. Lax, Rev. Mod. Phys. 23, 287 (1951).3P. Langacker„J. P. Leveille, and J. Sheiman, Phys. Rev. D

27, 1228 (1983).4P. F. Smith, Nuovo Cimento A 83, 263 (1984).~N. Cabbibo and L. Maiani, Phys. Lett. 114B, 115 (1982).6L. Wolfenstein, Phys. Rev. D 17, 2369 (1978); R. R. Lewis,

Phys. Rev. D 21, 663 (1980).M. Born and E. Wolf, Principles of Optics (Pergamon,

New York, 1975), pp. 121-123~

8H. Goldstein, Classical Mechanics (Addison-Wesley, Cali-fornia, 1980), pp. 87 and 484-492.

For simplicity, this discussion ignores a possible rotation ofthe neutron star that may introduce an azimuthal neutroncurrent in Eq. (1) and a nonradial force in Eq. (6).

' A. Burrows and J. M. Lattimer, Astrophys. J. 307, 178(1986); R. Mayle, J. R. Wilson, and D. N. Schramm, Astro-phys. J. 318, 288 (1987).

' 'F. Boehm and P. Vogel, Physics of Massive iVeutrinos(Cambridge Univ. Press, Cambridge, 1987), pp. 193-195.

' W. D. Arnett, J. N. Bahcall, R. P. Kirshner, and S. E.Woosley, Ann. Rev. Astron. Astrophys. 27, 629 (1989).

' The first-order Doppler effect, in the small thermal veloci-ties of the neutrons, vanishes because of the isotropy of thisthermal motion. Apart from the neutrino degeneracy blocking,this heating mechanism resembles qualitatively the well-known

process of soft-photon Comptonization by hot nonrelativisticelectrons.

' See, e.g. , S. Tsuruta, Max Planck Institute Report No.MPA 183, 1985 (to be published).

'sW. Greiner, B. Miiller, and J. Rafelsky, Quantum Electrodynamics of Strong Fields (Springer-Verlag, Berlin, 1985), pp.121 and 291.

' J. Schwinger, Phys. Rev. 82, 664 (1951);C. Itzykson and J.B. Zuber, Quantum Field Theory (McGraw-Hill, New York,1980), p. 195.

' In the limit of a homogeneous weak force over all space,~ po/E ~

1 in Eq. (2), and mostly left-handed neutrinos andright-handed antineutrinos (l = 1 particles) will be createdspontaneously.

' This inhibition would clearly not hold if a low-mass weaklyinteracting boson had existed.

' S. Hawking, Nature (London) 248, 30 (1974); Commun.Math. Phys. 43, 199 (1975).

A. Casher, H. Neuberger, and S. Nussinov, Phys. Rev. D20, 179 (1979); N. K. Glendenning and T. Matsui, Phys. Rev.D 28, 2890 (1983); K. Sumiyoshi, K. Kusaka, T. Kamio, andT. Kajino, Phys. Lett. B 225, 10 (1989).

2' For atoms with mass and atomic numbers, A and Z, Eq. (2)should be multiplied by 1

—2Z/(A —Z) for electron neutrinos,because of the charged-current contribution.

See, E. Brez in and C. I tzykson, Phys. Rev. D 2, 1 191{1970),for the electromagnetic analog.

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Bound-neutrino sphere and spontaneous neutrino-pair creation in cold neutron stars