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Fermi National Accelerator Laboratory 



FERMI LAB- Pub- 8 9 /I 5 6 -A 
July 3, 1989 



Pair Production of Helicity-Flipped 

Neutrinos in Supernovae "^ 



Armando Perez 

NASA/Fermilab Astrophysics Center 

Fermi National Accelerator Laboratory 

Box 500, Batavia, II. 60510-0500 USA 

and 

Departamento de Fi'sica Teorica 

Universidad de Valencia 

46100 Burjassot (Valencia) Spain 

and 

Raj Gandhi 

NIKHEF 

P.O. Box 41882 

1009DB Amsterdam 

The Netherlands. 

Abstract 

We calculate the emissivity for the pair production of helicity-flipped neutrinos, 
in a way that can be used in supernova calculations. We also present some simple 
estimates which show that such process can act as an efficient energy-loss mechanism 
in the shocked supernova core, and we use this fact to extract neutrino mass limits 
from 5A'^1987^ neutrino observations. 



M89-25789 



iler.i national icc.leratcr let.) 16 F ^^^ 

G3/90 02 19942 



Operated by Universities Research Association Inc. under contract with the United States Department of Energy 



1 Introduction 

The observation of neutrinos from SN1987A [1,2], in fair agreement with predictions 
from supernova models, has been used by several authors to bound the properties and 
interactions of various exotic and non- exotic particles [3,4,5,6,7,8]. For those particles 
which act as an efficient energy loss mechanism for the supernova, the simplest constraint 
on their masses and couplings can be derived from the fact that the totzd energy carried 
away by them cannot be greater than the available energy of the star. The detected 
neutrino flux has substantiated several features of supernova theory which are now ac- 
cepted as being standard [9,10]. First, the total emitted energy is 2 — 4 x 10^^ ergs . 
Secondly, it is emitted in the form of neutrino- antineutrino paurs of all species (with 
roughly equal amounts carried by each) formed inside the core via the Z° exchange ^ 
process e'^ + e~ —* u + P. Finally, these neutrinos are trapped in the core and undergo 
slow thermal diffusion for several seconds until they reach the neutrinosphere, where they 
are released in large quantities. 

If neutrinos are massive Dirac particles, then it is possible to produce neutrino- an- 
tineutrino pairs via the above process such that one of them is non-interacting, i.e either 
a positive helicity neutrino or a negative helicity antineutrino, which does not undergo 
difFussion but leaves the core much faster than its trapped partner^. As will be shown, this 
process can lead to significant energy loss on a much shorter timescale from the shocked 
core, depending on how massive the neutrinos are. Our main aim in this paper is to pro- 
vide suitable expressions for the emissivity of this process in a form which can be easily 
incorporated into realistic supernova models to evaluate the energy lost in the form of 
these flipped neutrinos. To illustrate that the process can have significant consequences, 

we also derive, using general considerations, approximate expressions for these energy 

^Electron type neutrinos are also produced via the W exchange channel. 

'We confine our discussion to the Standard Model, [11,12,13] minimally extended to include Dirac 
masses for the neutrinos. A right-handed Majorana neutrino interacts in a manner similar (but not 
identical) to a right-handed Dirac anti-neutrino and hence cannot provide an avenue for rapid cooling. 



losses. The emissivity of the process is proportional to the mass squared of the neutrino 
and the seventh power of the core temprature. We show that, using these expressions 
, it is possible to reliably exclude neutrino masses in the range iMeV — lOOMeV. The 
above range of 1 — 100 MeV is obtained using low core temperatures [21], and hence is 
on the pessimistic side. If higher temperatures are used, as is the case in some supernova 
models [22] then it is possible to exclude all neutrino masses between 100 KeV - 100 MeV 
using the considerations discussed below. As we will discuss, these mass limits would be 
improved by performing full supernova calculations. 

This paper is organized as follows. In Section 2, we analyze the production amplitudes 
for helicity- flipped neutrinos. In Section 3, we calculate the emissivities associated to 
this processes, in a way that can be used in supernova calculations, and we make some 
estimations about mass limits. ^. We end by summarizing our most relevant conclusions. 

2 Pair production of helicity-flipped neutrinos 

In this section we obtain the cross-section for the Z° exchange process 

e+(Pi) + e-(p2) -> Pi(fci, Ai) + t^iiki, As) (2.1) 

Here the ki and pi are the particle momenta and A is the neutrino helicity. We recall that 
within the context of the standard theory, a neutrino interaction eigenstate (i/£, or ur) 
is a superposition of helicity (A) eigenstates i/±, where A = a.p = ±1. For a relativistic 
particle, this translates into the statement that a ul is predominantly in the A = — 1 state 
and a ur is predominantly in the A = +1 state, with small admixtures of the opposite 
helicity, of order m/Ei,. In particular, we are interested in the case where the final state 
neutrino has positive helicity, and is 'almost' non-interacting. 



'Current experimental bounds on neutrino masses from particle accelerators are ttv, < 18eV, m^ < 
0.25MeV, rov^ < 35JkfeV [14]. More stringent bounds on the masses of stable neutrinos have been derived 
from big bang cosmology [15,16,17]. The neutrinos we consider here would reasonably be expected to 
decay via mixing and other modes. Bounds from cosmology and astrophysics on the masses and lifetimes 
of unstable neutrinos have been derived in [18,19,20]. 



In what follows below, the subscipts 1 and 2 denote antiparticle and particle respec- 
tively. The amplitude for ( 2.1) is given by: 

A = %ii(*2, A07''(l - iXki, Ax) • viPih^^iCv - C^-t'Hp^) (2.2) 

The u and v are the usual Dirac spinors, and use has been made of the fact that, when 
the processes (2.1) occur in the core of the collapsed star, the center of mass energies are 
at most 1 GeV , hence the amplitude may be written in its effective four-fermion form. 
The electron and positron helicity indexes have been supressed since they will be averaged 
over. We then write 



(where the spin averaging factors have been explicitly shown) with 



N'^'^ = irr[(^ + m)(l + 7' ^2)^(1 - l'){M - m)(l + 7' Mh^^i^ " 7')] (2.4) 

and 

E"'' = Tr[{jl2 + M)Y{Cv - Cx7')(yi - M)7''(Cv - C^7')] (2.5) 

Here s-i and S2 are the spin four-vectors associated with the anti-neutrino and neutrino 
respectively, while m and M are the neutrino and electron masses. These spin vectors 
satisfy the Lorentz invariant conditions 

Si 'Si - -1; Si'ki = 0; (2.6) 

and for a relativistic neutrino the additional constraint 

Si\\\iki fort = 1,2 (2.7) 

holds, where 

fc'' = {E^, \k\i) (2.8) 



with k being a unit vector along the three-momentum of the neutrino . 
We now introduce two four-vectors associated with the neutrino pair: 

K^ = k>^ + ms'l ■ K^ = k^ - ms^i (2.9) 

In conjunction with the properties in ( 2.6) and ( 2.7), these will allow us to write the am- 
plitude squared for the process under consideration in a compact and physically revealing 
form. As a first step towards this, we note that the spin vector may be expressed as 

s^ = Xm-''{\k\,Ej) (2.10) 

Using this and ( 2.9), we see that one may write 

Ki = r]i{lJi)- Ki = r)2ih-^2h (2.11) 

with 

Vi = E^ + (El - m'y/' ;m = E^- {El - m'fl' (2.12) 

Note that for m << E^ we have: 

7/1 « 2i;^ ; 7/2 « — (2.13) 

We can now evaluate the traces and the contraction N'^^E^^, in a straightforward way 
to obtain: 

iV^-i;^, = 16{Cv + CaY{pi-Ki){p2-K2) + 
ie{Cv-CAy{pi-K2){p2-K,) + 
16{Cl ~ C\)M^{Ki ' K2) (2.14) 

Here Cv, Ca are the usual weak vertex factors. From this expression and equations 
( 2.11) and ( 2.13) above one sees that the amplitude vanishes for massless neutrinos , 
as it should . Further, the expression ( 2.14) is akin to the usual weak pair production 



amplitude with the replacement Ki —y ki. Finally, the flip and non-flip cross-sections are 
related by more than just a simple factor of m'/4£?^, since the ?;,• carry a sign affixed 
to the three-momentum, which in general depends on the nature of the final state (i.e 
whether it is a particle or anti-particle) and its helicity eigenvalue [5] . 

In the next section we proceed to evcduate the emissivity for this process using ( 2.14). 

3 Calculation of emissivity 

We now proceed to calculate the emissivity associated with the pair production of helicity- 
flipped neutrinos. This is given by 

^'" ^ V (^(^ f'-iEe-)feAEe.)<PuP2) (3.1) 

In the last equation Qm is the emissivity, pi (pi) are the e~ (e+) momenta. Eg- and Ee+ 
are their energies, and f^- (/«+) are their respective equilibrium Fermi-Dirac distribution 
functions, which are assumed to be 



^'-(^'-)- l-.exp(i;.--.,-)/T (^-2) 



1 -|-exp(£e+ + f^e-)/T 

Here, fie- is the chemical potential for the electrons and T is the temperature (we take 
Kb = 1 for the Boltzmann constant) 
In eq. (3.1), e(pi,p2) stands for 

e(Pi,P2) = j -^-^ klvd(x{piP2 -^ kiki) (3.4) 

where v is the relative e"*" — e~ velocity, and da(pip2 —* ^1^2) is the differential cross 
section for the process described in (2.1) (later on, we will also include the process e'^e" -+ 



ULi^L, the cross section being the same that for e+e" — > vri^r). Since the typical energies 
which are involved are in the order of magnitude of ~ lOOM eF, electrons will be treated 
as relativistic. Therefore, we have 



Qm = r-rT' r dxidx2xlxl /e-(a;i)/e+(i2) / d{cos 6) 
27r* ^0 J-i 



KPuPi) 



(3.5) 



Where Xi = Eg-fT, x-i = Eg-t- /T and 9 is the angle formed by pi and p2. 
We have performed the integrals in (3.4). After a lenghty (cdthough strightforward) 
calculation, we obtzun 



c(Pi,P2) = -^ — ^PuP2) 



(3.6) 



The function fl{pi,P2) is given by 

n{puP2) = E {{CI + C\) [Ci(q) + C2{a)^'^{l - cos Of + C3(a)(l - cos <?)] + 



2CvCAC,{a) ^'' ^^'^ {1 - cos ^)| 



(3.7) 



where E = E^- + E^^ , a = |pi + P2I IE and 



^ , , [(a"» - Q< - 4a3 + 3a' + 4a - 3)//(a) - 2a^ + 8a* - Aa^ - 8a' + 6a] . ^, 
Ca(a) = — (3.8) 



C2{cc) = 



[(3a* - 4a3 + 2a' - 4a + 3)//(a) - &a^ + 8a' - 6a] 



2a5 



(3.9) 



Cz{a) = 



[(a« - 3a* + a' + 4a - 3)/j(a) + 2a^ + 4a* - 8a' + 6a] 



2a' 



(3.10) 



C,{a) = 



[(3a* - 4a3 + 2a' ~ 4a + 3)/j(a) - Aa^ - 6a^ + 8a' - 6a] 



(3.11) 



In the last equations, //(a) = log(i3^). The integral in cos 6 can be performed numer- 
ically, for different vzdues of Ee- and £«+. We realized that, for our purposes, the result 
of this integral can be approximated by the simple expression 



/ <f(cos^) 



^{Pl,P2) 



= Ai(C^ + Ci)(xi + xj) - XiCvCAixi - X2) (3.12) 



T 

Here, Ai and Aj are two numerical constants to be determined later. By inserting (3.6) 
and (3.12) in eq. (3.5) one can get a final expression in terms of the relativistic Fermi 
functions Fn{'T]), defined as 

n! yo 1 + exp(x — 77) 
here, 7/ is the electron degenerancy parameter (jj = fig-fT). We compared the resulting 
expression with the exact numerical integral, as given in (3.5). We then found that 
Aj ~ 2.5 and Aj c:; 8.0 is a good choice for 77 < 10, whereas Ai ~ 3.5 and A2 c^ 9.0 seems 
more appropiate when 77 > 10. In that way, we obtain the final expressions 

Qm = ^-^m'rHir,) (3.14) 

with the following approximations for ^(7;) : 

H{ri) = 2.5{Cl + Cl)A+{ri) - 8CvCAA-{ri) (3.15) 

(77 < 10) 



H{r}) = 3.5{Cl + Cl)A+{v) - QCvC^A-irj) (3.16) 



(77 > 10) 
where 



AHv) = F,{r])F,{-v) ± F,{T,)F^{-r,) (3.17) 

In eq. (3.14) we introduced a factor of 2 to account for the combined emissivity of the 
two processes e^e~ — > I'x.ia and e+e" — v vrur 

Substituing for the values of the constants, one obtains 



^"^ - = 5.281 X 10^\m/lMeVy{T/10MeVy H{t,) (3.18) 



erg.cTn~^3 



In Fig. 1 we show the two approximations given in (3.15) and (3.16) (dashed line and 
dot-dashed line, respectively) for H{t]). We also have plotted the values that result from 
a direct numerical integration of (3.5). The error in using this approximate expressions is 
only a few percent. 

As one can see from eq. (3.14-17) the emissivity is highly dependent on the tem- 
perature and electron degenerancy in the supernova core. Therefore, in order to get an 
appropiate neutrino mass limit, one should incorporate our expressions in a realistic su- 
pernova calculation. However, in order to motivate such a calculation, we present some 
simple estimates. The highest temperatures in a supernova collapse are reached in the 
shocked, outer core, during the first few seconds of the neutron star cooling. In this re- 
gion, the temperature amounts to several tens of MeV, and 7/ w (see ref [21]. For this 
model, the peak temperature is higher than 35MeF). 

In Fig. 2, we show our estimates for the total luminosity of the shocked core in the 
form of heli city-flipped neutrinos, for various temperatures (we assumed a characteristic 
core radius R = 10 Km). As can be seen, for modest temperatures T = 35Afey [21] 
one can exclude neutrino masses larger than about IMeV, using arguments identical to 
those in ref. 5. However, for supernova models with higher temperatures T ~ 70MeV 
[22,23], the corresponding mass limit is about lOOKeV (we are convinced that this limits 
can be improved by performing realistic calculations, since the neutrino luminosity would 

8 



be integrated over a few seconds). 

We now address the question of whether very massive neutrinos can be ruled out 
from the above considerations. For these neutrinos, the re-flipping process into standard 
neutrinos can proceed via scattering with targets such as electrons, neutrons and protons 
in the core [5]. In fact, for neutrino masses larger than a few MeV, the mean free path 
associated with re-flipping becomes comparable to the core radius, and hence one could 
claim that flipped neutrinos will be trapped, rather than freely escaping. However, they 
will be trapped only temporarily, because neutrinos will continue to be flipped. The 
important criterion is when transport is more effectively done by flipped neutrinos rather 
than unflipped ones. Hence, the relevant quantity is the difussion time-scale associated 
with re-flipping 

7r*c 
with n being the number density of the targets, and <r/up the cross section for re- flipping 
(which is proportional to the neutrino square mass). If tdiff is of the order of 1 sec. or 
so, helicity-flipped neutrinos become efl'ectively trapped, and no longer act as an energy 
loss source. For standard values in (3.19), this only happens when the neutrino mass is 
m > IQOMeV [5]. Therefore, neutrinos which are more massive than this limit can not 
be excluded by heli city- flipping processes (however we note that such high masses for fi 
and r neutrinos are anyways ruled out by accelerator limits and hence reflipping is not a 
relevant concern here). 

4 Conclusions 

We have calculated accurate expressions for the emissivity due to pair production of 
helicity-flipped neutrinos with a Dirac mass. We showed that this process can act as an 
efficient energy-loss mechanism in the shocked core of a supernova. Therefore, this can 



9 



be used to put limits on the neutrino mass, by using the data of the detected neutrinos 
from SN1987A. 

Because of the high temperature-dependence of the corresponding emissivity, these 
mass limits are better in the case of supernova models with large core temperatures 
[22,23]. If these models are reliable, all Dirac neutrino masses in the range lOOA'cF- 
lOOMeV can be ruled out by our simple estimates. A full supernova calculation should 
certanly lower the lOOiiTeV limit (probably within a factor of 3 or so), since the neutrino 
luminosity would be integrated over several seconds. In this way, mass limits which are 
comparable to the ones obtained from scattering helicity- flipping processes [5,7] can be 
reproduced by an independent mechanism. 

If one addopts models with much lower core temperatures [21], the corresponding 
(pessimistic) mass limit is higher in about a factor of 10. However, even in this case, the 
extreme sensitivity of the supernova explosion to neutrino properties (specially for the 
so-called delayed explosion scenario [22,24]) could lead to a much more stringent limit. 

Acknowledgments 

This work was supported in part by the DOE and by NASA at Fermilab (grant Nr. 
NAGW-1340). RG would like to thank Edward Kolb and the Astrophysics Group at 
Fermilab for hospitality, where this work was begun, and AP is grateful to Leo v.d. 
Horn and the Cheaf Group at Nikhef, where it was completed. We also would like to 
acknowledge useful conversations with Jim Lattimer, E. Kolb and M. S. Turner. 



10 



References 

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11 



[22] J. Wilson in Numerical Astrophysics, ed J. Centrella et al, (Jones and Bartlett, 
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12 



Figure Caption 
Fig. 1.- The function H{t]) (solid line), as given by a numerical integration of (3.5). 
Also shown, the two approximations corresponding to eq. (3.15) (dashed line) and (3.16) 
(dot-dashed line). 

Fig. 2.- Totad emissivity for helicity-flipped neutrinos in the core, as a function of the 
neutrino mass, for three different core temperatures. A typical core radius R = lOKm 
has been assumed. 



13 



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