Neutrino-matter interactions in the context of core-collapse supernovae

P E R G A M O N

Progress in Particle and

Nuclear Physics Progress in Particle and Nuclear Physics 46 (2001) 59--71

http://www.elsevier.nl/locate/npe

Neutrino-Matter Interactions in the Context of Core-Collapse Supernovae

A. B U R R O W S

Department of Astronomy. The University of Arizona, Tuscon, AZ, USA

A b s t r a c t

Neutrino processes determine the viability of the neutrino-driven mechanism of core-collapse supernova explosions. Here, I list and discuss some of the various neutrino scattering, absorption, and production processes that collectively determine the outcome of core collapse, the creation of neutron stars, the ejection of the stellar mantle, and the cooling of protoneutron stars. These are the essential inputs into the radiation/hydrodynamics codes with which astrophysicists simulate the supernova phenomenon. Hence, understanding these processes is a crucial prerequisite to the further advancement of supernova theory.

1 In troduc t ion

One of the great insights (discoveries) of the 20th Century was that most of the elements of nature are created by nuclear processes in stars. Supernova explosions are one major means by which these elements are injected into the intersteelar medium and, hence, into subsequent generations of stars. Therefore, supernovae are central to the chemical evolution and progressive enrichment of the universe. Core-collapse supernova explosions signal the death of a massive star and are some of the most majestic and awe-inspiring events in the cosmos. However, to fully understand the role of supernovae in the grand synthesis of creation, one must have a firm handle on the nuclear data. In particular, understanding the origin of iron-peak, r-process, and rp-process elements hinges upon improvements in our knowledge of the properties of exotic nuclei, some of which are far from the valley of beta stability.

The mechanism of core-collapse supernovae seems to depend upon the transfer of energy from the core to the mantle of the inner regions of a massive star after it becomes unstable to collapse. Neutrinos seem to be the mediators of this energy transfer. In order to understand this coupling and the role of neutrinos in supernova explosions, one needs to master the particulars of the neutrino- matter scattering, production, and absorption rates. Since recently there has been some progress in understanding the associated microphysics, it seems fitting to summarize the neutrino-matter cross sections and the production rates of neutrinos in the core-collapse context. To this end, I have assembled here a short precis of many of the relevant processes and physics. This contribution does not attempt to explain the hydrodynamics of supernova explosions, but does try to present the relevant neutrino processes that play a role. For the former, the reader is referred to Burrows, Hayes, and Fryxell[1] and Burrows[2].

In §2, I present a physical derivation of stimulated absorption (the Fermionic correlate to stimulated emission) and then in §3 I present the basic cross sections. Also in §3 are discussions of dynamic structure factors in neutrino-nucleon scattering and of fully corrected neutrino-nucleus scattering. In §4, I discuss e+e - annihilation spectra and rates and in §5 I summarize the v~ annihilation equations, taken from the paper by Janka[3]. I provide in §6 a derivation of the single and pair neutrino rates and spectra for nucleon-nucleon bremsstrahlung, a process that can compete with pair annihilation as

0146-6410/01/$ - see front matter © 2001 Published by Elsevier Science BV. PII: S0146-6410(01)00108-9

60 A. Burrows/Prog. Part. Nucl. Phys. 46 (2001) 59-71

a source for v,, ~u, v~, and P~ neutrinos. Some of this discussion can be found in Burrows et al. [4], but much is new. A mastery of these neutrino-matter processes is a prerequisite for progress in supernova theory and is provided here in that spirit.

2 S t i m u l a t e d A b s o r p t i o n

The concept of stimulated emission for photons is well understood and studied, but the corresponding concept of stimulated absorption for neutrinos is not so well appreciated. This may be because its simple origin in Fermi blocking and the Pauli exclusion principle ih the context of net emission is not often explained. The net emission of a neutrino is simply the difference between the emissivity and the absorption of the medium:

J~ = ~ - ~ o I v . (1 )

All absorption processes involving fermions will be inhibited by Pauli blocking due to final-state oc- cupancy. Hence, ~v in eq. (1) includes a blocking term, (1 - ~-v)[5]. ~'v is the invariant distribution function for the neutrino, whether or not it is in chemical equilibrium.

We can derive stimulated absorption using Fermi's Golden rule. For example, the net collision term for the process, v~n ++ e-p, is:

[ da!i~ f d3pn d3pv d~fie ( )

x E(v~n ++ e-p) (2r) 4 54(pv, + Pn - Pp - Pe) , (2)

where p is a four-vector and

~ ( ~ , ~ ~ e - p ) = . r ~ o . r , ( 1 - . r~ ) (1 - . r , ) - . r , . r p ( 1 - . r , , ) ( 1 - . r v , ) . (3)

The final-state blocking terms in eq. (3) are manifest, in particular that for the v~ neutrino. Algebraic manipulations convert :~(u~n ++ e-p) in eq. (3) into:

E(uen ~, e-p) = .T'n(1 - .T'~)(1 - J:p) [ 7 - : - ~ p , (1 - Sv,) - S~, L 1 -s' .

where .T'v~ = [e ( e " ' - ( ' - ' a ) ) # + 1] -1 (5)

is an equilibrium distribution function for the v~ neutrino and it has been assumed that only the electron, proton, and neutron are in thermal equilibrium. Note that in 9t~'o there is no explicit reference to a neutrino chemical potential, though of course in beta equilibrium it is equal to #e - / i . There is no need to construct or refer to a neutrino chemical potential in neutrino transfer.

We see that eq. (4) natural ly leads to:

j .~ , = ~___L_~ (By - I~) = ~ ( B v - Zv). (6) 1 - J = -

If neutrinos were bosons, we would have found a (1 + bye) in the denominator, but the form of eq. (6) in which Iv is manifestly driven to B~, the equilibrium intensity, would have been retained. From eqs. (4) and (6), we see that the stimulated absorption correction to ~ is 1/(1 - ~'~). By writing the collision term in the form of eq. (6), with ~a corrected for stimulated absorption, we have a net source term that clearly drives Iv to equilibrium. The timescale is 1/c~*~. Though the derivation of the stimulated absorption correction we have provided here is for the Pen ~Pr e-p process, this correction is quite general and applies to all neutrino absorption opacities.

A. Burrows/Prog. Part. Nucl. Phys. 46 (2001) 59-71 61

Kirchhoff's Law, expressing detailed balance, is:

~o = ~ / B ~ or ~; = ~'~IB~, (7)

where fl'~ is not corrected for final-state neutrino blocking. Furthermore, the net emissivity can be written as the sum of its spon taneous and induced c o m p o n e n t s :

'~ ~ + 1 ' (8)

where + or - is used for bosons or fermions, respectively.

3 N e u t r i n o C r o s s S e c t i o n s

Neutrino-matter cross sections, both for scattering and for absorption, play the central role in neutrino transport. The major processes are the super-allowed charged-current absorptions of ve and ~ neutrinos on free nucleons, neutral-current scattering off of free nucleons, alpha particles, and nuclei[6], neutrino-electron/positron scattering, neutrino-nucleus absorption, neutrino-neutrino scattering, neutrino-antineutrino absorption, and the inverses of various neutrino production processes such as nucleon-nucleon bremsstrahlung and the modified URCA process (v~ + n + n --~ e- + p + n). Com- pared with photon-matter interactions, neutrino- matter interactions are relatively simple functions of incident neutrino energy. Resonances play little or no role and continuum processes dominate. Nice summaries of the various neutrino cross sections of relevance in supernova theory are given in Tubbs and Schramm[7] and in Bruenn[5]. In particular, Bruenn[5] discusses in detail neutrino-electron scattering and neutrino-antineutrino processes using the full energy redistribution formalism. He also provides a serviceable approximation to the neutrino-nucleus absorption cross section[8, 9, 10]. Recall that for a neutrino energy of ~10 MeV the ratio of the charged-current cross section to the re-electron scattering cross section is ,,~100. However, neutrino-electron scattering does play a role, along with neutrino- nucleon scattering and nucleon-nucleon bremsstrahlung, in the energy equilibration of emergent u, neutrinos[Ill.

Below, we list and discuss many of the cross sections one needs in detailed supernova calculations. In §3.5, we provide some straightforward formulae that can be used to properly handle inelastic scattering off of nucleons. These formulae include in a self-consistent way final-state nucleon blocking. In addition, a variety of useful facts and equations are presented to clarify and summarize the major neutrino- matter interactions. In sections 4, 5, and 6 we discuss aspects of e+e - annihilation, v~ annihilation, and nucleon-nucelon bremsstrahlung. Collectively, these sections encapsulate the microphysics most relevant to neutrino atmospheres and core-collapse supernovae.

3 . 1 ve + n ~ e - + p:

The cross section per baryon for ve neutrino absorption on free neutrons is larger than that for any other process. Given the large abundance of free neutrons in protoneutron star atmospheres, this process is central to v~ neutrino transport. A convenient reference neutrino cross section is ~o, given by

4G2(m~c2)2 ~ 1.705 × 10 -44 c m 2 . (9)

The total v~ - n absorption cross section is then given by

where gA is the axial-vector coupling constant (~ -1.26), Anp = m, ,c 2 - mpc 2 = 1.29332 MeV, and for a collision in which the electron gets all of the kinetic energy c~- = s~, + Anp. W M is the correction for


weak magnetism and recoil[12] and is approximately equal to (1 + 1.1:,Jm~c2). At e~, = 20 MeV, this correction is only ,,~ 2.5%. We include it here for symmetry's sake, since the corresponding correction (W~) for D~ neutrino absorption on protons is (1 - 7.1:~,/m~c2), which at 20 MeV is a large -15%. To calculate ~c;, a ~,~ must be multiplied by the stimulated absorption correction, 1/(1 - br'o), and final-state blocking by the electrons and the protons g la eq. (4) must be included.

3 . 2 ~e + P - - + e + + n :

The total De - p absorption cross section is given by

G~ =ao \ "m7 , I- ~°' k : U : : o , ) J w ~ , (11)

where :~+ = ee, - A= v and W~ is the weak magnetism/recoil correction given in §3.1. Note that * a- ~ must also W~ is as large as many other corrections and should not be ignored. To calculate ~ , ~,p

be corrected for stimulated absorption and final-state blocking. However, the sign of #~ - / 2 in the stimulated absorption correction for ~ neutrinos is flipped, as is the sign of/~e in the positron blocking term. Hence, as a consequence of the severe electron lepton asymmetry in core-collapse supernovae, both coefficients are very close to one. Note that the De + p --+ e + + n process dominates the supernova neutrino signal in proton-rich underground neutrino telescopes on Earth, such as Super Kami~l~nde, LVD, and MACRO, a fact that emphasizes the interesting complementarities between emission at the supernova and detection in Cerenkov and scintillation facilities.

3 . 3 v~ +p --+ vi +p:

The total vi - p scattering cross section for all neutrino species is:

ap = ~- \~ec2 ] 4sin4 0w -- 2sin2 0w + , (12)

where 0w is the Weinberg angle and sin 20w ~- 0.23. In terms of C{: = 1/2 + 2 sin 20w and C~ = 1/2, eq. (12) becomes[13]:

Go ( h2 go [(c~ - 1) 2 + ace(c; - 1) 2] (la) °" = ¥ ~--7/

The differential cross section is:

e-g = (1.+ ~ , . ) , (14)

where l 2 2 I 2

~ = (cv - :) - gA(CA - 1) (15)

Note that ($p, and 5~ below, are negative (~p ~ -0.2 and ~n ~ -0.1) and, hence, that these processes are backward-peaked.

The transport cross section is simply

(Yo ( ~2 Ev t r [(C~: - 1) 2 + 5g2A(CrA - - 1) 2] (16) ~, = -g \.~---y/

3 . 4 v/ + n --+ u~ + n"

The total vi - n scattering cross section is:

(17)

A. Burrows / Prog. Part. Nucl. Phys. 46 (2001) 59-71

The corresponding differential cross section is:

dan d~

where

The transport cross section is

63

~ (1 + (in#) (18) 4~r

( i~ = 1 - g____~ (19) 1 + 3g l "

= " (20)

t~ is greater than ai increases the The fact that 5p and (in are negative and, as a consequence, that a i neutrino-matter energy coupling rate for a given neutrino flux in the semi-transparent region.

3 .5 D y n a m i c S t r u c t u r e F a c t o r s f o r N e u t r i n o - N u c l e o n I n t e r a c t i o n s

Recent explorations into the effects of many-body correlations on neutrino-matter opacities at high densities have revealed that for densities above 1014 gm cm -3 both the charged-current and the neutral- current interaction rates are decreased by a factor of perhaps 2 to 3, depending on the density and the equation of state[14, 15, 16, 17]. Furthermore, it has been shown that the rate of energy transfer due to neutral-current scattering off of nucleons exceeds that due to v~electron scattering[18]. Previously, it had been assumed that neutrino-nucleon scattering was elastic[19]. However, these recent reappraisals reveal that the product of the underestimated energy transfer per neutrino-nucleon scattering with cross section exceeds the corresponding quantity for neutrino-electron scattering. Since ue and Pe neutrinos participate in super-allowed charged-current absorptions on nucleons, neutrino-nucleon scattering has little effect on their rate of equilibration. However, such scattering would seem to be important for v~ and u~ equilibration. Since the many-body correlation suppressions appear only above neutrinosphere densities (~ 1011 - 1013 gm cm-3), it is only the kinematic effect, and not the interaction effect, that need be considered when studying the emergent spectra. Without interactions, the relevant dynamical structure factor, S(q, w), for neutrino-nucleon scattering is simply

S(q, w) = 2 / ~ - ~ - ( I p l ) ( 1 dap - ~-(Ip + qlll2=(i(w + ep - ep+q), (21)

where JC(Ip] ) is the nucleon Fermi-Dirac distribution function, ep is the nucleon energy, w is the energy transfer to the medium, and q is the momentum transfer. The magnitude of q is related to w and El, the incident neutrino energy, through the neutrino scattering angle, 8, by the expression,

q = [E~ + (El - w) 2 - 2El(E1 - w) cos0] 1/2 . (22)

In the elastic limit and ignoring final-state nucleon blocking, S(q, w) = 27~(i(w)nn, the expected result, where nn is the nucleon's number density.

The neutral current scattering rate off of either neutrons or protons is[14],

d2r dwd cos 0 - (4r2)-lG~v(E1 - w)2[1 - 9rv(E1 - W)]ZNC,

where

and

zNo = [(1 + c o s e ) V + (3 - cose)A]S(q, )

(23)

(24)

S(q,~) = 2ImII(°)(1 - e-Z~) -1. (25)

V and A are the applicable vector and axial-vector coupling terms (see §3.3 and §3.4) and/~ -- 1/kT. The free polarization function, II (°), contains the full kinematics of the scattering, as well as blocking due to the final-state nucleon, and the relevant imaginary part of H (°) is given by:

m 2 [ l+e-Q~ +#" ] ImYI(°)(q, w) = 2-~qB log |-1 ~ - e - ~ | ' (26)


where

Q~ = : ~ + , (27)

# is the nucleon chemical potential, and m is the nucleon mass. The dynamical structure factor, S(q, w), contains all of the information necessary to handle angular and energy redistribution due to scattering. The corresponding term on the r ight-hand-side of the transport equation is:

27T-3 2 / 3 r t I 8[~'~] = ( ) Gw d p~2:NC {[1 -- ~'~(E1)]~"~(E1 - w)e - ~ - 3%(E1)[1 - 3c'~(E1 - w)]}, (28)

where p" is the final state neutrino momentum. In the non-degenerate nucleon limit, eq. (26) can be expanded to lowest order in Q2 to obtain,

using eq. (25), an approximation to the dynamical structure factor:

S(q,w) = n(27rm/3)l/2 e -@~, (29) q

where n is the nucleon number density. This says that for a given momentum transfer the dynamical structure factor is approximately a Gaussian in w.

For charged-current absorption process, ~ + n --~ e- + p, ImH(°)(q,w) is given by a similar expression:

m 2 1 + e-Q~ + ~ - ImH (°) (q, w) = ~ log[1 + e_q~_+0u,_0w ] . (30)

Eq. (30) inserted into eq. (25) with a (1 - e-~(~+P)), as is appropriate for the charged-current process, substi tuted for (1 - e - '~ ) , results in an expression that is a bit more general than the one employed to date by most practitioners, i.e., S = ( X , - Xp)/(1 - e-MT). In the non-degenerate nucleon limit, the structure factor for the charged-current process can be approximated by eq. (29) with n = n~. Note that for the structure factor of a charged-current interaction one must distinguish between the ini t ia l- and the final-state nucleons and, hence, between their chemical potentials. To obtain the structure factor for the Pe absorption process, one simply permutes #~ and #p in eq. (30) and substitutes -/2 for

in the (1 - e -0(~+~)) term.

3 .6 vi + A -+ vi + A:

In the post-bounce phase, nuclei exist in the unshocked region and temporarily in a cap of nuclei just exterior to the nucleon/nucleus phase transition near 0.5xnuclear density. However, at the high entropies in shocked protoneutron star atmospheres there are no nuclei. There are alpha particles, but their fractional abundances are generally low, growing to interesting levels due to reassociation of free nucleons just interior to the shock only at late times. Hence, v~ - A processes are of secondary importance after bounce, except at the' highest neutrino energies for which the neutrinosphere is exterior to the shock. Nevertheless, for completeness and because it is important during infall and collapse, we include here a discussion of v~ - A scattering, with its various corrections.

The differential v~ - A neutral-current scattering cross section may be expressed as:

daA ao ( e v ~2 A 2 { W g F F + C L O S } 2 (8 ,o~) (1+#) (31) d~ 64r \ ~ c 2 /

where 2Z

]/V = 1 - --~-(1 - 2 sin2 Ow), (32)

Z is the atomic number, A is the atomic weight, and (Sion) is the ion-ion correlation function, deter- mined mostly by the Coulomb interaction between the nuclei during infall[20].


Leinson et al. [21] have investigated the electron polarization correction, CLOS, and find that

Z (1 + 4 sin s Ow CLos = -~ i ~ : ~ ) ' (33)

where the Debye radius is

I rh2c (34) rD = 4c~pFEF '

k s = ]p _ p, iS = 2(e~/c)2(1 _ #), PF and EF are the electron Fermi momentum and energy, and ~ is the fine-structure constant (-~ 137-a). Note that ro ~ lOh/pF in the ultra-relativistic limit (PF > > m~c). The CLOS term is important only for low neutrino energies, generally below ~ 5 MeV.

Following Tubbs and Schramm[7] and Burrows et al. [22], the form factor term, CFF, in eq. (31) can be approximated by:

OFF -~- e - y ( 1 - # ) / 2 , (35)

where

Y t~- - f f -~ s t T-5-6 s '

and (rs) 1/2 is the rms radius of the nucleus. 7FF differs from 1 for large A and e,, when the de Broglie wavelength of the neutrino is smaller than the nuclear radius.

When (Sion) = CFF = CLOS = 1, we have simple coherent Freedman scattering[6]. The physics of the polarization, ion-ion correlation, and form factor corrections to coherent scattering is interesting in its own right, but has little effect on supernovae[23].

3 . 7 vi + e - -+ ui + e - :

The differential and total scattering cross sections for ui - e- scattering are[24]:

dse- = 2 m~c 2 - ~ , /

and

where

oe- = 5OoA, ~ +

(36)

(37)

, ( 1 ) A i = ~ A i + ~ B i ,

A~ = (Cv + CA) s ,

and

Bi = (Cv - CA) s •

Cv = 1/2 + 2sin29w for u~ and Pe neutrinos, Cv = - 1 / 2 + 2sins 0w for u~, u¢, uN, and Or neutrinos, CA = +1/2 for ue, Pu, and ~T neutrinos and CA = --1/2 for ~e, vu, and u¢ neutrinos. Neutrino- electron cross sections, in particular those for "uu" neutrinos, are small compared to those for u-nucleon scattering (see §3.4 and §3.3). Generally, neutrino cross sections depend upon the square of the center- of-momentum energy of the neutrino, s, but neutrin0-electron cross sections are linear, not quadratic, in cv~ because the small mass of the electron makes s ..~ 2~v~ee-. Furthermore, due to the partial electron degeneracy of protoneutron star atmospheres, neutrino-electron cross sections are diminished by final-state electron blocking.


4 e + e - A n n i h i l a t i o n

Ignoring phase space blocking of neutrinos in the final state and taking the relativistic limit (rn~ ~ 0), the total electron-positron annihilation rate into neutr ino-ant ineutr ino pairs can be written in terms of the electron and positron phase space densities[25/:

4 3 + e,-e,+) dee- de,+ , (38)

where Ki = (1/187r4)cao(C~ + C~). Again, Cv = 1/2 + 2 sin 2 Ow for electron types, Cv = - 1 / 2 + 2 sin ~ Ow for mu and tau types, and C~ = (1/2) 2. Rewriting eq. (38) in terms of the Fermi integral F~(r/), we obtain:

~m,c:] \ h c / [F4(r/,)F3(-V,) + F4(-r/,)F3(rT,)] , (39)

where r/, ~ # e / k T and fO c'z X n F, (~7) ~ - - d x . (40) e x-r1 + 1

Integrating eq. (38), we obtain

Q,,~, _ 9.7615 x 1024 r kT ]~ [M--e--e-e-~] f(7/e) ergscm-3s -1 , (41)

where F,(~o)Fz(-~,) + F4(-~,)Fz(~,) (42)

f(r/~) = 2F4(0)F3(0)

For uuP u and uCp¢ production combined,

[ k T ] 9 Qv.,.~.,. ~- 4.1724 x 10 ~4 ~ f(rl.) ergscm-3s -1. (43)

One can easily derive the spectrum of the total radiated neutrino energy (eT) by inserting a delta function ( f 5 ( e T - e~- --e~+)deT) into eq. (38). RecaLl that the total energy of the neutrinos in the final state is equal to the sum of the electron and positron energies in the initial state. Integrating first over ee+ to annihilate the delta function and then over c~- to leave a function of er , one obtains:

dQ = Ki eT(ZT -- ee- ) g,-Ye-[Ze-].T'e+[~T -- ~e-] de,- (44) de--~ ~ ~ / Jo The numerical evalution of eq. (44) is straightforward. The average of ~T is equal to:

leT) = (Fa(rle) + F4(-~?e)'~T (45) ',F~(~) F~( -w)J '

which near r/~ "~ 0 is ,,* 8T and for ~7~ > > 1 is ~ 4T(1 + 77,/5). However, while the total energy loss rate (eq. 41) and the spectrum of E T pose no great mathematical

problems, the production spectrum of an individual neutrino is not so easily reduced to a simple integral or to an analytic expression. This is due primarily to the awkward integration of the angular phase space terms, while subject to the momentum conservation delta function, and to the explicit dependence of the matrix elements on the electron/neutrino angles. From Dicus[25], averaging over initial states and summing over final states, the matrix element for the e+e - -+ ~ process in the m~ = 0 limit is:

! E IMI ~ = 16G2[(Cv + CA) 2p" qv P'" q~' + (Cv - CA)2p • qv P ' ' qp] (46) 4 ~

A. Burrows / Prog. Part. Nucl. Phys. 46 (2001) 59-71 67

where p and /¢ are the four-momenta of the electron and positron, respectively, and q. and qo are the four-momenta of the neutrino and ant(neutrino, respectively. Using the formalism of Bruenn[5] and Fermi's Golden ru-le, expanding the production kernel in the traditional truncated Legendre series, performing the trivial angular integrals, taking the non-trivial angular integrals from Bruenn[5], and ignoring final-state neutrino blocking, we obtain for the single-neutrino source spectrum due to e+e - annihilation:

dQ 87r 2 fo °° dcp c~ Og(g~, co) (47)

3

dc----~ = (27rhc) 6 E~,

where

and

G2 f6~+¢~ cg(c . ,co) = T ~ 0 dE.- J:,-[~.-]~'.+[c~ + c o - c . - ] H 0 ( ~ , c a c . - ) , (48)

Ho(c.,~o,e~-) = (Cv + CA) 2 J~(ev, ea, c~-) + (Cv - CA) 2 Jdt(¢~,ep, ee-) . (49) The J0s in eq. (49) come from the more obdurate angular integrals required by the dot products in eq. (46) and the momentum delta function and have the symmetry:

Jro (C,,, co, c~- ) -- Jto' (C~, e,,, e~- ) . (50)

From eqs. (47) and (49), we see that the differences between the spectra of the ue and % neutrinos flow solely from their correspondingly different values of (Cv + CA) 2 and (Cv - CA) 2. One can use 4-point Gauss-Legendre integration to calculate eq. (48) and 16-point Gauss-Laguerre integration to calculate eq. (47).

At small r/e, the e+e - annihilation spectra and total energy loss rates for the ~ and ~e neutrinos are similar, as are the average emitted t% and ~ neutrino energies. However, as r/e increases, both the total energy radiated in ~e neutrinos and the average ~e energy start to lag the corresponding quantities for the ve neutrinos. This is true despite the fact that the total number of Ue and ~e neutrinos radiated is the same. If final-state blocking is ignored, (e i ) /T is a function of r/e alone, becoming linear with r/e at high r/e and one half of eq. (45) (~ 4.0) at low r/~. Note also that (~ , , ) /T and (co , ) /T are closer to one another than are (c , , ) /T and (co,)/T. The individual production spectra vary in peak strength, in peak energy, and in low-energy shape, but they are quite similar on the high-energy tail. Due to the parity-violating matrix element for the e+e - --r up process and the fact that r]~ is positive, the ant(neutrino spectra of all species are softer than the neutrino spectra. The pair sums of the integrals under these curves are given by eqs. (41) and (43). For rle = 0, 50% of the pair energy emission of electron types is in Pe neutrinos, but at r/e = 10 only 42% of this total energy is in p~ neutrinos. However, at r/~ = 10, the p~ neutrinos still constitute 48.5% of the t,,/5~ pair emission. These differences reflect differences in the corresponding coupling constants Cv and CA.

5 uiPi A n n i h i l a t i o n

In the limit of high temperatures and ignoring electron phase space blocking, the v~Pi annihilation rate into e+e - pairs can be written[3]:

Q~,,a,=4Ki?r 4 f / ~'J~,,Jo,(e,~, +ca,)de~,,dei,, , (51)

where Jv is the zeroth moment of the radiation field, ~, is the neutrino energy, Ki is defined as before (i.e., Ki : (1/187ra)cao(C 2 + CA)), and

3 [1 - 2(#~,,)(#oi) + p~,,po, + ~(1 - p,,)(1 - Pa,)] , (52) • ' (( ,~,), ( ,~,),p~,,pa,) =

where the flux factor (#.,) = H./J~, and the Eddington factor p . = (#~,) = P~./J,,. Eq. (51) can be rewritten in terms of the invariant distribution functions ~'~:

( 1 ) 2 ( 1 ) ~ ' f f 9 ' . T ' , . ~ , . ( ~ : ~ +c~ ,c~ . )dc , . d~ , , . (53) Q~.,a,=Ki ~ ~ . . , , , , , ,


Note that when the radiation field is isotropic (if' = 1) and when r/~ = 0 the total rate for e+e - annihilation given in eq. (38) equals that for u/Pi annihilation given in eq. (53), as expected.

6 N u c l e o n - N u c l e o n Bremsstrahlung

A production process for neutrino/anti-neutrino pairs that has recently received attention in the supernova context is neutral-current nucleon-nucleon bremsstrahlung (nl +n2 -+ n~ +n4+ up). It importance in the cooling of old neutron stars, for which the nucleons are quite degenerate, has been recognized for years[26], but only in the last few years has it been studied for its potential importance in the quasi- degenerate to non-degenerate atmospheres of protoneutron stars and supernovae [27, 28, 29, 4, 11]. Neutron-neutron, proton-proton, and neutron-proton bremsstrahlung are all important, with the lat- ter the most important for symmetric matter. As a source of ue and P~ neutrinos, nucleon-nucleon bremsstrahlung can not compete with the charged-current capture processes. However, for a range of temperatures and densities realized in supernova cores, it may compete with e+e - annihilation as a source for u,, P~, u~, and Or neutrinos ("u~"s). The major obstacles to obtaining accurate estimates of the emissivity of this process are our poor knowledge of the nucleon-nucleon potential, of the degree of suitability of the Born Approximation, and of the magnitude of many-body effects[27, 30, 31]. Since the nucleons in protoneutron star atmospheres are not degenerate, we present here a calculation of the total and differential emissivities of this process in that Limit and assume a one-pion exchange (OPE) potential model to calculate the nuclear matrix element. For the corresponding calculation for arbitrary nucleon degeneracy, the reader is referred to Thompson, Burrows, and Horvath[ll]. The formalism we employ has been heavily influenced by those of Brinkman and Turner[31] and Hannestad and Raffelt[27], to which the reader is referred for details and further explanations.

Our focus is on obtaining a useful single-neutrino final-state emission (source) spectrum, as well as a final-state pair energy spectrum and the total emission rate. For this, we start with Fermi's Golden Rule for the total rate per neutrino species:

Qnb = (27r) 4 / t~__~r]14 (27r) 3jd~ia ~ ] (2r-~w~ d3~v (2~-~w;daq~ w ~s ].A/l]264(P).T'l.T'2(1 - .T'3) (1 - .T'4) , (54)

where 64(p) is four-momentum conservation delta function, w is the energy of the final-state neutrino pair, (w~,¢.) and (wa,q'o) are the energy and momentum of the neutrino and anti-neutrino, respectively, and lgl is the momentum of nucleon i. Final-state neutrino and anti-neutrino blocking have been dropped.

The necessary ingredients for the integration of eq. (54) are the matrix element for the interaction and a workable procedure for handling the phase space terms, constrained by the conservation laws. We follow Brinkmann and Turner[31] for both of these elements. In particular, we assume for the n + n --+ n + n + u~ process that the matrix element is:

6 4 G 2 , , , m ~4 2 r, k ,2 + . . Z ] M I 2 = - - ~ j / ,~) g~tt,~--~--i----~-~) (55)

s 4 k + rn~ "J w 2 022 '

where the 4 in the denominator accounts for the spin average for identical nucleons, G is the weak coupling constant, f (,,~ 1.0) is the pion-nueleon coupling constant, gA is the axial-vector coupling constant, the term in brackets is from the OPE propagator plus exchange and cross terms, k is the nucleon momemtum transfer, and rn~ is the pion mass. In eq. (55), we have dropped ~'~ •/~ terms from the weak part of the total matrix element. To further simplify the calculation, we set the "propagator" term equal to a constant ~, a number of order unity, and absorb into ~ all interaction ambiguities. The constant A in eq. (55) remains.

Inserting a f 5(w - aJ~ - w~)dw by the neutrino phase space terms times ww~,wo/w ~ and integrating over wo yields:

ca - ~ (2~)a2w, ' (27c)32w,~ ~ ~ so J0 w '

A. Burrows / Prog. Part. Nucl. Phys. 46 (2001) 59--71 69

where again w equals (w~ + wv). If we integrate over w~, we can derive the w spectrum. A further integration over w will result in the total volumetric energy emission rate. If we delay such an integration, after the nucleon phase space sector has been redueed to a function of w and if we multiply eq. (54) and/or eq. (56) by w~/w, an integration over w from w~ to infinity will leave the emission spectrum for the single final-state neutrino. This is of central use in mult i-energy group transport calculations and with this differential emissivity and Kirchhoff's Law (§2) we can derive an absorptive opacity.

Whatever our final goal, we need to reduce the nucleon phase space integrals and to do this we use the coordinates and approach of Brinkmann and 25arner[31]. We define new momenta: p+ = (p~ +p2)/2, p - = (pl - p2) /2 , P3~ = Pa - P+, and p4~ = P4 - P+, where nucleons 1 and 2 are in the initial state. Useful direction cosines are 71 = P+ " P - l iP+l iP- [ and % = p+ . paJ[p+Hp3c I. Defining it i = p ~ / 2 m T and using energy and momentum conservation, we can show that:

d3pld3p2 = 8d3p+dap_

w = 2 T ( u _ - u 3 ~ )

Ul,2 = u+ A- U_ "4- 2(U+U_)1/2,',/1

U3,4 = u+ -b U3c -4- 2(U+U3c)l/2%. (57)

In the non-degenerate limit, the 9v1~'2 (1 -9v3) (1 -9Ca) term reduces to e2Ye -2(u++'-), where y is the nucleon degeneracy factor. Using eq. (57), we see that the quanti ty (u+ + u_) is independent of both 71 and %. This is a great simplification and makes the angle integrations trivial. Annihi lat ing d3pa with

~ - u~. pairing the remaining the momentum delta function in eq. (54), noting that p2dp = 2roT a /2 1/2z. 2 ~' Z,

energy delta function with u_, and integrating u+ from 0 to co, we obtain:

28 × 3 × 5~ . o e - ~ ( ~ + x ~ / T ) l / 2 d x . dQ,~b (58)

The variable x over which we are integrating in eq. (58) is equal to 2uac. That integral is analytic and yields:

~o c¢ e -X(x 2 + x w / T ) l / 2 d x = ~e'TK1 (~), (59)

where K1 is the s tandard modified Bessel function of imaginary argument, related to the Hankel functions, and ~l = w / 2 T . Hence, the w spectrum is given by:

dQnb (x e - ~ / 2 T w b K t ( w / 2 T ) . (60) dw

It can easily be shown that (w) = 4.364T. Integrating eq. (58) over w and using the thermodynamic identity in the non-degenerate limit:

/ 2Ir \3/2 . e ~ = ~ - y ) n , / 2 , (61)

where n~ is the density of neutrons (in this case), we derive for the total neut ron-neut ron bremsstrahlung emissivity of a single neutrino pair:

30 2 T 5 5 3 1 Q~b --= 1.04 × 10 ~(X,~p14) (~---e-~) " e rgscm- s- , (62)

where p14 is the mass density in units of 1014 gm cm -3 and X~ is the neutron mass fraction. Interestingly, this is within 30% of the result in Suzuki[29], even though he has substituted, without much justification, (1 + w / 2 T ) for the integral in eq. (58). ([1 + (~r~7/2) 1/2] is a better approximation.) The proton- proton and neutron-proton processes can be handled similarly and the total bremsstrahlung rate is then obtained by subst i tut ing Xn 2 + X 2 + 2--SX3 nXp for X~ in eq. (62)[31]. At X , -- 0.7, Xp = 0.3, p = 1012 gm cm -3, and T = 10 MeV, and taking the ratio of augmented eq. (62) to eq. (43), we obtain the promising ratio of ~.. 5~. Setting the correction factor ~ equal to ~ 0.5127], we find that near and just deeper than the u, neutrinosphere, bremsstrahlung is larger than classical pair production.


If in eq. (56) we do not integrate over w,, but at the end of the calculation we integrate over w from o;~ to oo, after some manipulation we obtain the single neutrino emissivity spectrum:

! q, nb 3 c~ e - r l (~nb 3 oo e-2rl~l~ 2

3×~×7×m (~ 0.564), and for the second where r/~ = wv/2T, C is the normalization constant equal to expression we have used the integral representation of KI(U) and reversed the order of integration. In eq. (63), Q,b is the emissivity for the pair.

Eq. (63) is the approximate neutrino emission spectrum due to nucleon-nucleon bremsstrahlung. A useful fit to eq. (63), good to better than 3% over the full range of important values of r/v, is:

dQtnb ~ 0.234Qnb (w_y_v~2.4 -H~./T (64) dw~ T ~ T ] e .

7 Conclus ion

The processes that have been described above are the essential factors of the neutrino-driven supernova explosion mechanism. Coupling these with radiation-hydrodynamics codes, an equation of state, beta- decay and electron capture microphysics, and nuclear rates, one explores the viability of various scenarios for the explosion of the cores of massive stars. Multi-dimensional effects may play a role[l], but neutrinos are the key that will unlock the mystery of one of the most violent, yet ubiquitous, phenomena in this nuclear universe in which we live.

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Neutrino-matter interactions in the context of core-collapse supernovae