EPJ manuscript No.
(will be inserted by the editor)
Plasmaneutrino spectrum
A. Odrzywo lek
M. Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland
November 15, 2018
Abstract. Spectrum of the neutrinos produced in the massive photon and longitudal plasmon decay process
has been computed with four levels of approximation for the dispersion relations. Some analytical formulae
in limiting cases are derived. Interesting conclusions related to previous calculations of the energy loss in
stars are presented. High energy tail of the neutrino spectrum is shown to be proportional to exp(-E/kT),
where E is the neutrino energy and kT is the temperature of the plasma.
1 Introduction & Motivation
Thermal neutrino loses from plasma are very important
for stellar astrophysics [1,2]. Plasmon decay is one of the
three main reactions. Extensive calculations for these pro-
cesses were done by group of Itoh [3,4,5,6,7,8,9,10,11].
Other influential article include [12,13,14,15,16,17,18,19,
20]. Meanwhile, our abilities to detect neutrinos has grown
by many orders of magnitude, beginning with 1.4 tonne
experiment of Reines&Cowan [21] up to the biggest exist-
ing now 50 kt Super-Kamiokande detector [22]. Recently,
”GADZOOKS!” upgrade to Super-Kamiokande proposed
by Beacom&Vagins [23] attract attention of both experi-
mental and theoretical physicists. At last one new source
of the astrophysical antineutrinos is guaranteed with this
upgrade, namely Diffuse Supernova Neutrino Background
[25,24]. Pre-supernova stars will be available to observa-
tions out to ∼2 kiloparsecs [25]. This technique is the
only extensible to megaton scale [25]. Memphys, Hyper-
Kamiokande and UNO (Mt-scale water Cherenkov detec-
tors cf. e.g. [26]) proposals now seriously consider to add
GdCl3 to the one of the tanks with typical three-tank
design [27]. Recently, the discussion on the geoneutrino
detection [28], increased attention to the deep underwa-
ter neutrino observatories [29] with target mass 5-10 Mt
[25] and even bigger [30]. It seems that (anti)neutrino as-
tronomy is on our doorstep, but numerous astrophysical
sources of the ν’s still are not analyzed from the detection
point of view.
Detection of the solar [31,32,33,34] and supernova neu-
trinos [35,36,37,38] was accompanied and followed with
extensive set of detailed calculations (see e.g. [39,40,41,
42,43,44] and references therein as a representatives of
this broad subject) of the neutrino spectrum. On the con-
trary, very little is known about spectral neutrino emission
from other astrophysical objects. Usually, some analytical
representation of the spectrum is used, based on earlier
experience and numerical simulations, cf. e.g. [45]. While
this approach is justified for supernovae, where neutrinos
are trapped, other astrophysical objects are transparent
to neutrinos, and spectrum can be computed with an ar-
bitrary precision. Our goal is to compute neutrino spec-
tra as exact as possible and fill this gap. Plasmaneutrino
process dominates dense, degenerate objects like red giant
cores [46], cooling white dwarfs [47] including Ia supernova
progenitors before so-called ,,smoldering” phase [48]. It is
also important secondary cooling process in e.g. neutron
star crusts [49] and massive stars [50]. Unfortunately, ther-
mal neutrino loses usually are calculated using methods
completely erasing almost any information related to the
neutrino energy Eν and directionality as well. This infor-
mation is not required to compute total energy Q radiated
as neutrinos per unit volume and time. From experimen-
tal point of view, however, it is extremely important if
given amount of energy is radiated as e.g. numerous keV
neutrinos or one 10 MeV neutrino. In the first case we
are unable to detect (using available techniques) any tran-
sient neutrino source regardless of the total luminosity and
proximity of the object. In the second case we can detect
astrophysical neutrino sources if they are strong and not
too far away using advanced detector which is big enough.
Few of the research articles in this area attempt to es-
timate average neutrino energy [16,17,52,53] computing
additionally reaction rate R. Strangely, they presented fig-
ures and formulae for Q/R instead of 1
Q/R. This gives
false picture of real situation, as former expression gives
〈Eν + Eν̄〉. Obviously, we detect neutrinos not ν-ν̄ pairs.
Q/R do not give average neutrino energy, as in general
neutrino and antineutrino spectra are different. As we will
see only for longitudal plasmon decay neutrinos energies
of neutrinos and antineutrinos are equal. However, dif-
ference in all situations where thermal neutrino loses are
important is numerically small and formula:
〈Eν〉 ≃
is still a ”working” estimate.
http://arxiv.org/abs/0704.1222v1
2 A. Odrzywo lek: Plasmaneutrino spectrum
Mean neutrino energy is useful in the purpose of quali-
tative discussion of the detection prospects/methods. Quan-
titative discussion require knowledge of spectrum shape
(differential emissivity dR/dEν). High energy tail is par-
ticularly important from an experimental detection point
of view. Detection of the lowest energy neutrinos is ex-
tremely challenging due to numerous background signal
noise sources e.g. 14C decay for Eν < 200 keV [51]. Rele-
vant calculations for the spectrum of the medium energy
〈Eν〉 ∼ 1MeV neutrinos emitted from thermal processes
has become available recently [52,53,54]. Purpose of this
article is to develop accurate methods and discuss various
theoretical and practical (important for detection) aspects
of the neutrino spectra from astrophysical plasma process.
This could help experimental physicists to discuss possi-
ble realistic approach to detect astrophysical sources of
the neutrinos in the future.
2 Plasmaneutrino spectrum
2.1 Properties of plasmons
Emissivity and the spectrum shape from the plasmon de-
cay is strongly affected by the dispersion relation for trans-
verse plasmons (massive in-medium photons) and longi-
tudal plasmons. In contrast to transverse plasmons, with
vacuum dispersion relation ω(k) = k, longitudal plasmons
exist only in the plasma. Dispersion relation, by the def-
inition is a function ω(k) where ~ω is the energy of the
(quasi)particle and ~k is the momentum. Issues related to
particular handling of these functions are discussed clearly
in the article of Braaten and Segel [15]. We will repeat here
the most important features of the plasmons.
For both types, plasmon energy for momentum k = 0
is equal to ω0. Value ω0 ≡ ω(0) is refereed to as plasma
frequency and can be computed from:
ω20 =
(f1 + f2) dp (2)
where v = p/E, E =
p2 + m2e (~ = c = 1 units are
used), me ≃ 0.511 MeV and fine structure constant is α =
1/137.036 [55]. Functions f1 and f2 are the Fermi-Dirac
distributions for electrons and positrons, respectively:
e(E−µ)/kT + 1
, f2 =
e(E+µ)/kT + 1
. (3)
Quantity µ is the electron chemical potential (including
the rest mass). Other important parameters include first
relativistic correction ω1:
ω21 =
v2 − v4
(f1 + f2) dp (4)
maximum longitudal plasmon momentum (energy) kmax:
k2max ≡ ω
max =
1 − v
1 + v
(f1 + f2) dp
Table 1. Plasma properties for typical massive star during Si
burning. All values in MeV.
kT µ ω0 ω1 mt ωmax ωA
0.32 1.33 0.074 0.070 0.086 0.133 0.002
and asymptotic transverse plasmon mass mt:
m2t =
(f1 + f2) dp. (6)
Value mt is often referred to as thermal photon mass. We
also define parameter v∗:
interpreted as typical velocity of the electrons in the plasma
[15]. Axial polarization coefficient is:
(f1 − f2) dp. (8)
Value of the ωA is a measure of the difference between
neutrino and antineutrino spectra. Set of numerical values
used to display sample result is presented in Table 1.
Values ω0, ωmax,mt define sub-area of the ω-k plane
where dispersion relations for photons ωt(k) and longitu-
dal plasmons ωl(k) are found:
max (k, ω0) ≤ ωl(k) ≤ ωmax, 0 ≤ k ≤ kmax (9a)
k2 + ω20 ≤ ωt(k) ≤
k2 + m2t , 0 ≤ k ≤ ∞ (9b)
Dispersion relations are solution to the equations [15]:
k2 = Πl (ωl(k), k) (10a)
k2 = ωt(k)
2 −Πt (ωt(k), k) (10b)
where longitudal and transverse polarization functions are
given as an integrals:
ωl + vk
ωl − vk
ω2l − k
ω2l − v
(f1+f2) dp.
(11a)
ω2t − k
ωt + vk
ωt − vk
(f1+f2) dp.
(11b)
Typical example of the exact plasmon dispersion re-
lations (dash-dotted) is presented in Fig. 1. As solving
eqns. (10a, 10b) with (11) is computationally intensive,
three levels of approximation for dispersion relations are
widely used:
1. zero-order analytical approximations
2. first order relativistic corrections
3. Braaten&Segel approximation
A. Odrzywo lek: Plasmaneutrino spectrum 3
0.05 0.1 0.15
k @MeVD
Longitudal
0.05 0.1 0.15
k @MeVD
Transverse
Fig. 1. Longitudal and transverse plasmon dispersion relation ωl,t(k) for plasma parameters from Table 1. Exact result (dot-
sahed) is very close to the Braaten & Segel approximation (solid). Zero-order (dotted) and first order (dashed) approximations
are very poor, especially for londitudal mode (left).
0.05 Ω0 0.1 Ωmax
k @MeVD
Longitudal plasmon
0.2 0.4 0.6 0.8 1 1.2
k @MeVD
In-medium photon
Fig. 2. Longitudal and transverse plasmon mass. Dotted lines on the right panel show asymptotic transverse mass. Line
dashing the same as in Fig. 1.
2.1.1 Approximations for longitudal plasmons
For longitudal plasmons, the simplest zero-order approach
used in early calculations of Adams et al. [13] and more
recently in [53] for photoneutrino process is to put simply:
ω(k) = ω0 (12)
where ω0 is the plasma frequency (2). Maximum plasmon
energy ωmax = ω0 in this approximation. Zero-order ap-
proximation is valid only for non-relativistic regime, and
leads to large errors of the total emissivity [12].
First relativistic correction to (12) has been introduced
by Beaudet et al. [12]. Dispersion relation ωl(k) is given
in an implicit form:
ω2l = ω
, (13)
with maximum plasmon energy equal to:
ω(1)max =
ω20 +
ω21 (14)
This approximation, however, do not introduce really se-
rious improvement (Figs. 1, 2 (left) & 4). Breaking point
was publication of the Braaten&Segel approximation [15].
4 A. Odrzywo lek: Plasmaneutrino spectrum
Using simple analytical equation:
k2 = 3
ωl + v∗k
ωl − v∗k
where v∗ is defined in (7) one is able to get almost exact
dispersion relation, cf. Figs. 1 & 2, left panels. Solution to
the eq. (15) exist in the range 1 < k < kBSmax, where, in this
approximation, maximum longitudal plasmon momentum
ωBSmax
1 + v∗
1 − v∗
what gives value slightly different than exact value (Fig. 2,
left), but required for consistency of the approximation.
2.1.2 Approximations for transverse plasmons
For photons in vacuum dispersion relation is ωt = k. Zero
order approximation for in-medium photons is:
ω2t = ω
0 + k
2, k ≪ ω0 (17a)
valid for small k and:
ω2t = m
t + k
2, k ≫ ω0 (17b)
valid for very large k. Formulae (17a) and (17b) provide
lower and upper limit for realistic ωt(k), respectively (cf. Fig. 1,
right panel, dotted). First order relativistic corrections
lead to the formula:
ω2t = ω
0 + k
with asymptotic photon mass:
ω20 + ω
1/5 (19)
Finally, Braaten&Segel approximation leads to:
ω2t = k
2 + ω20
ω2t − v
2ωt v∗ k
ωt + v∗k
ωt − v∗k
Asymptotic photon mass mBSt derived from (20) is:
1 − v2
1 + v∗
1 − v∗
This is slightly smaller (left panel of Fig. 2, dashed) than
exact value (solid line).
All four relations are presented in Fig. 1. Differences
are clearly visible, but they are much less pronounced
for transverse than for longitudal plasmons. Inspection of
Fig. 2 reveals however, that in the large momentum regime
asymptotic behavior is correct only for exact integral re-
lations (10b) and may be easily reproduced using (17b)
with mt from (6).
Let us recapitulate main conclusions. Braaten&Segel
approximation provide reasonable approximation, as non-
linear equations (15) and (20) are easily solved using e.g.
νe,µ,τ
ν̄e,µ,τ
Fig. 3. Fenmann diagrams for plasmon decay.
bisection method. Zero and first-order approximations (12,
17a, 17b) with limiting values (9) provide starting points
and ranges. Approximation has been tested by [56] and
is considered as the best available [20]. Errors for part
of the kT -µ plane where plasmaneutrino process is not
dominant may be as large as 5% [56]. At present, these
inaccuracies are irrelevant for any practical application,
and Braaten&Segel approximation is recommended for all
purposes.
2.2 Plasmon decay rate
In the Standard Model of electroweak interactions, mas-
sive in-medium photons and longitudal plasmons may de-
cay into neutrino-antineutrino pairs:
γ∗ → νx + ν̄x. (22)
In the first-order calculations two Feynmann diagrams
(Fig. 3) contribute to decay rate [15,52].
For the decay of the longitudal plasmon squared ma-
trix element is:
M2l =
ω2l − k
2K ·Q1 K ·Q2
2k · q1 k · q2
−Q1 ·Q2
(23a)
where K = (ω,k) is four momentum of the plasmon.
Q1 = (E1,q1) and Q2 = (E2,q2) is four-momentum of
the neutrino and antineutrino, respectively.
Squared matrix element for decay of the massive pho-
ton is:
M2t =
C2V Π
t + C
E1E2 −
k · q1 k · q2
+ 2CV CAΠtΠA
E1 k · q2 − E2 k · q1
(23b)
where Πt is defined in (11b) and axial polarization func-
tion ΠA reads:
ω2t −k
ωt + vk
ωt − vk
ω2t − k
ω2t − v
(f1−f2) dp
A. Odrzywo lek: Plasmaneutrino spectrum 5
Table 2. Relative weight of the M2t (23b) terms for e and µ, τ
neutrinos.
Flavor Vector Axial Mixed
(CV ω
+CAωA)
(CV ω
+CAωA)
2CV CAω
(CV ω
+CAωA)
electron 0.74 0.02 0.24
mu/tau 0.07 0.39 0.54
Fermi constant is GF /(~c)
3 = 1.16637(1)× 10−5 GeV−2
[55] and, in standard model of electroweak interactions,
vector and axial coupling constants are:
CeV =
+ 2 sin2 θW , C
V = −
+ 2 sin2 θW , C
A = −
for electron and µ, τ neutrinos, respectively. The Weinberg
angle is sin2 θW = 0.23122(15) [55].
Terms containing CA (so-called axial contribution) in
(23b) are frequently treated separately [52] or removed at
all [3]. In calculations concentrated on the total emissiv-
ity this is justified as anti-symmetric term multiplied by
CV CA do not contribute at all and term C
A × . . . is sup-
pressed relative to the term beginning with C2V × . . . by
four orders of magnitude [3]. However, if one attempts to
compute neutrino energy spectrum all three terms should
be added together, as mixed V-A ,,channel” alone leads
to negative emission probability for some neutrino energy
range (Fig. 6), what is physically unacceptable. These
terms remains numerically small but only for electron neu-
trinos. For µ and τ neutrino spectra axial part contributes
at ∼ 1% level due to very small value C
V = −0.0376
while still CA = −0.5. ”Mixed” term leads to significant
differences between νµ,τ and ν̄µ,τ spectra, cf. Fig. 6. Rel-
ative contributions of the three transverse ”channels” for
electron and µ, τ are presented in Table 2.
In general, all the terms in the squared matrix element
(23b) should be added. We have only two different spectra:
longitudal and transverse one.
Particle production rate from plasma in thermal equi-
librium is:
(2π)5
Zi fγ∗ δ
4(K−Q1−Q2) M
where i = l for longitudal mode and i = t for transverse
mode. Bose-Einstein distribution for plasmons fγ∗ is:
fγ∗ =
eωt,l/kT − 1
. (28)
and residue factors Zt,l are expressed by polarization func-
tions Πt,l (11b, 11a):
Z−1t = 1 −
Z−1l = −
. (30)
For massive photons gt = 2 and for longitudal plasmon
gl = 1.
Differential rates1 has been derived for the first time
in [52]. Here, we present result in the form valid for both
types of plasmons, ready for calculations using any avail-
able form of dispersion relation:
dE1 dE2
i fγ∗ Ji S (31)
where i = l or i = t. Product S of the unit step functions
Θ in (31) restrict result to the kinematically allowed area:
S = Θ(4E1E2−m
i )Θ(E1+E2−ω0)Θ(ωmax−E1−E2) (32)
Four-momenta in the squared matrix element are:
Q1 = (E1, 0, 0, E1)
Q1 = (E2, E2 sin θ, 0, E2 cos θ)
K = (E1 + E2, E2 sin θ, 0, E1 + E2 cos θ)
m2i = K ·K = (E1 + E2)
2 − k′2
cos θ =
− E21 − E
2E1E2
k′ = ω−1l,t (E1 + E2)
ωi = E1 + E2
where ω−1i denotes function inverse to the dispersion re-
lation. Jacobian Ji arising from Dirac delta integration in
(27) is:
J−1i =
. (33)
Residue factors Zi are given in (30) and (29). Maximum
energy ωmax in (32) for longitudal plasmons must be in the
agreement with particular approximation used for ωl(k):
ω0, (14) or (16) for zero-order (12), first-order (13) or
Braaten&Segel (15) approximation, respectively. For trans-
verse plasmons ωmax → ∞ and last Θ function in (32) has
no effect and may be omitted.
2.3 Longitudal neutrino spectrum
2.3.1 Analytical approximation
We begin with general remark on the spectrum. Note, that
eq. (31) is symmetric for longitudal mode under change
E1,2 → E2,1 because (23a) is symmetric with respect to
exchange Q1,2 → Q2,1. Resulting energy spectrum is thus
identical for neutrinos and antineutrinos. This is not true
for transverse plasmons with axial contribution included,
cf. Sect. 2.4.
1 Double differential rate d2Ri/dEd cos θ has an identical
form as (31) but now four momenta cannot be given explicitly,
unless simple analytical approximation for ωi(k) is used. Ana-
lytical approximations for the specrum shape are derived this
6 A. Odrzywo lek: Plasmaneutrino spectrum
Using zero-order dispersion relation for longitudal plas-
mons (12) we are able to express spectrum by the elemen-
tary functions. Longitudal residue factor Zt is now:
Z0l = 1, (34)
and Jacobian Jl resulting from the integration of the Dirac
delta function is:
J0l = 1. (35)
Now, differential rate d2R/dEd cos θ (cf. (31) and foot-
note 1) becomes much more simple and integral over d cos θ
can be evaluated analytically. Finally, we get the longitu-
dal spectrum:
≡ λ(E) =
1260 π4 α ~3 c9
f(E/ω0)
eω0/kT − 1
where normalized spectrum is:
f(x) =
4x(x− 1)(8x4 − 16x3 + 2x2 + 6x− 3)
+ 3(1 − 2x)2 ln(1 − 2x)2
Let us note that f is undefined at x = 1/2; use limit
instead:
x→1/2
f(x) = 105/32.
Function f(x) is symmetric with respect to point x = 1/2,
where f has a maximum value (Fig. 4, dotted line).
In this limit, correct for non-relativistic, non-degenerate
plasma, average neutrino and antineutrino energy is 〈E〉 =
ω0/2 and maximum ν energy is ω0.
Inspection of Fig. 4 reveals little difference between
analytical result (36) and result obtained with first-order
relativistic corrections to the dispersion relation (13).
2.3.2 Numerical results
Simple formula (36) significantly underestimates flux and
the maximum neutrino energy, equal to ωmax rather than
ω0. Therefore we have used Braaten & Segel approxima-
tion for longitudal plasmon dispersion relation.
To derive spectrum we will use form of differential rate
(31) provided by [52]. In the Braaten&Segel approxima-
tion:
ZBSl =
ω2l − k
2(ω2l − v
3ω20 − ω
l + v
JBSl =
1 − βl
βBSl =
ωl + v∗k
ωl − v∗k
ω2l v∗
k2(ω2l − v
Spectrum is computed as an integral of (31) over dE2.
Example result is presented in Fig. 4. Integration of the
function in Fig. 4 over neutrino energy gives result in well
agreement with both (30) from [15] and (54) from [52].
Ω0�2 Ω0 Ωmax
@MeVD
Fig. 4. Longitudal plasmon approximate analytical (36) neu-
trino spectrum (dotted), with first-order correction used by
BPS [12] (dashed), and spectrum computed using [15] disper-
sion relation (solid). Plasma properties according to Table 1.
2.4 Transverse plasmon decay spectrum
2.4.1 Analytical approximation
Derivation of massive in-medium photon decay spectrum
closely follows previous subsection. Semi-analytical for-
mula can be derived for dispersion relations (17). For dis-
persion relation (17b) transverse residue factor Zt is:
Z0t = 1, (38)
polarization function Πt is equal to:
Π0t = m
t , (39)
and Jacobian resulting from integration of the Dirac delta
function Jt is:
J0t =
E1 + E2
. (40)
Approximate spectrum, neglecting differences between
neutrinos and antineutrinos, is given by the following in-
tegral:
λ(E) =
64 π4α
P (cos θ, E/mt) d cos θ
2E(1−cos θ)
where rational function P (ct, x) is:
1 + 2(ct− 1)2(2x2 − 1)x2
x(ct− 1)2[1 − 2ct(ct− 1)x2 + 2(ct− 1)2)x4]
A. Odrzywo lek: Plasmaneutrino spectrum 7
Ω0 Ωmax kT 0.5
@MeVD
Fig. 5. Transverse plasmaneutrino spectrum computed from
[15] approximation (solid) with upper (17b) and lower (17a)
limits for the dispersion relation (dotted). First-order rela-
tivistic correction leads to the spectrum shown as dashed line.
Plasma parameters as in Fig. 4.
Result presented in Fig. 5 show that spectrum (41)
obtained with dispersion relation (17b) agree well in both
low and high neutrino energy part with spectrum obtained
from Braaten&Segel approximation for dispersion rela-
tions. Dispersion relation (17a) produces much larger er-
ror, and spectrum nowhere agree with correct result. This
fact is not a big surprise: as was pointed out by Braaten
[16] dispersion relation is crucial. Therefore, all previous
results, including seminal BPS work [12], could be eas-
ily improved just by the trivial replacement ω0 → mt.
Moreover, closely related photoneutrino process also has
been computed [12,3,17,14] with simplified dispersion re-
lation (17a) with ω0. One exception is work of Esposito
et. al. [57]. It remains unclear however, which result is
better, as accurate dispersion relations have never been
used within photoneutrino process context. For plasma-
neutrino, Eq. (17b) is much better approximation than
(17a), especially if one put mt from exact formula (6).
High energy tail of the spectrum also will be exact in this
case.
As formula (41) agree perfectly with the tail of the
spectrum, we may use it to derive very useful analyti-
cal expression. Leaving only leading terms of the rational
function (42)
P (ct, x) ∼ x−1(1 − ct)−2
one is able to compute integral (41) analytically:
λ(E) ≃
64 π4α
eaκ/2 − 1
where κ = 2x + (2x)−1, x = E/mt, a = mt/kT . Interest-
ingly, spectrum (43) is invariant under transformation:
E ′E = m2t/4
and all results obtained for high energy tail of the spec-
trum immediately may be transformed for low-energy ap-
proximation. The asymptotic behavior of (43) for E ≫ kT
is of main interest:
λ(Eν) = A kT m
t exp
where for electron neutrinos :
= 2.115 × 1030 [MeV−8cm−3s−1]
and mt, kT are in MeV. For µ, τ neutrinos just replace A
with A (C
Formula (44) gives also quite reasonable estimates of
the total emissivity Qt and mean neutrino energies 〈Eν〉:
Qt = A kT
3m6t (45a)
〈Eν〉 = kT (45b)
For a comparison, Braaten & Segel [15] derived exact for-
mulae in the high temperature limit kT ≫ ω0:
QBSt =
V ζ(3)
12π4α
kT 3m6t = 0.8A kT
3m6t (46a)
〈EBSν 〉 =
6ζ(3)
kT = 0.73 kT (46b)
Formulae above agree with ∼25% error in the leading co-
efficients.
2.4.2 Numerical results
Calculation of the spectrum in the framework of Braaten&Segel
approximation requires residue factor, polarization func-
tion [15] (transverse&axial) and Jacobian [52]:
ZBSt =
2ω2t (ω
t − v
3ω20ω
t + (ω
t + k
2)(ω2t − v
k2) − 2ω2t (ω
t − k
ΠBSt =
ω2t − v
ωt + v∗k
ωt − v∗k
, (48)
ΠBSA = ωA k
ω2t − k
ω2t − v
3ω20 − 2 (ω
t − k
, (49)
JBSt =
E1 + E2
1 − βBSt
βBSt =
ωt + v∗k
ωt − v∗k
Example spectrum, computed as an integral of (31)
over dE2 is shown in Fig. 5.
8 A. Odrzywo lek: Plasmaneutrino spectrum
Ω0 Ωmax 0.5
@MeVD
Fig. 6. Spectrum of the muon neutrinos (dotted) and antineu-
trinos (dashed) from transverse plasmon decay. Contributions
to the spectra from so-called mixed ,,vector-axial channel” pro-
duces significant differences. For electron flavor, contribution
from ”mixed channel” lead to unimportant differences. For
both flavors contribution from ”axial channel” remains rela-
tively small: 10−4 for νe and 10
−2 for νµ. Overall contribution
to the total emissivity from µ, τ flavors is suppressed relatively
to electron flavor by a factor (C
/CeV )
≃ 3.3 × 10−3.
3 Summary
Main new results presented in the article are analytical
formulae for neutrino spectra (36, 41) and exact analyti-
cal formula (44) for the high energy tail of the transverse
spectrum. The latter is of main interest from the detection
of astrophysical sources point of view: recently available
detection techniques are unable to detect keV plasmaneu-
trinos emitted with typical energies 〈Eν〉 ∼ ω0/2 (Fig. 4,
5), where ω0 is the plasma frequency (2). Tail behavior
of the transverse spectrum quickly ”decouple” from ω0
dominated maximum area, and becomes dominated by
temperature-dependent term exp (−Eν/kT ). Calculation
of the events in the detector is then straightforward, as de-
tector threshold in the realistic experiment will be above
maximum area. This approach is much more reliable com-
pared to the typical practice, where an average neutrino
energy is used as a parameter in an arbitrary analytical
formula.
Analytical formulae for the spectrum are shown to be
a poor approximation of the realistic situation, especially
for longitudal plasmons (Fig. 4). This is in the agreement
with general remarks on the dispersion relations presented
by Braaten [16]. On the contrary, Braaten & Segel [15] ap-
proximation is shown to be a very good approach not only
for the total emissivities, but also for the spectrum. Excep-
tion is the tail of the massive photon decay neutrino spec-
Ω0Ωmax 1.0 10
@MeVD
Fig. 7. Typical spectra from the plasma process. Dotted
line is a longitudal and dashed transverse spectrum. Only
∼ exp(−Eν/kT ) tail of the transverse spectrum (solid line)
contributes to (possibly) detectable signal. Plasma properties
according to Table 1.
trum: Braaten & Segel [15] formulae lead to underestimate
of the thermal photon mass while the formula (44) gives
exact result. Numerical difference between mt from (6)
and (21) is however small [15]. Calculating of the emissiv-
ities by the spectrum integration seems much longer route
compared to typical methods, but we are given much more
insight into process details. For example, we obtain exact
formula for the tail for free this way. Interesting surprise
revealed in the course of our calculations is importance
of the high-momentum behavior of the massive photon.
While mathematically identical to simplest approach used
in the early calculations, formula (17b) gives much better
approximation for the total emissivity than (17a).
This work was supported by grant of Polish Ministry of Edu-
cation and Science (former Ministry of Scientific Research and
Information Technology, now Ministry of Science and Higher
Education) No. 1 P03D 005 28.
References
1. D. Arnett, Supernovae and nucleosynthesis (Princeton
University Press, 1996)
2. G.S. Bisnovatyi-Kogan, Stellar physics. Vol.1: Fundamen-
tal concepts and stellar equilibrium (Springer, 2001)
3. H. Munakata, Y. Kohyama, N. Itoh, Astrophys. J 296, 197
(1985)
4. H. Munakata, Y. Kohyama, N. Itoh, Astrophys. J 304, 580
(1986)
A. Odrzywo lek: Plasmaneutrino spectrum 9
5. Y. Kohyama, N. Itoh, H. Munakata, Astrophys. J 310, 815
(1986)
6. N. Itoh, T. Adachi, M. Nakagawa, Y. Kohyama, H. Mu-
nakata, Astrophys. J 339, 354 (1989)
7. N. Itoh, T. Adachi, M. Nakagawa, Y. Kohyama, H. Mu-
nakata, Astrophys. J 360, 741 (1990)
8. N. Itoh, H. Mutoh, A. Hikita, Y. Kohyama, Astrophys. J
395, 622 (1992)
9. Y. Kohyama, N. Itoh, A. Obama, H. Mutoh, Astrophys. J
415, 267 (1993)
10. Y. Kohyama, N. Itoh, A. Obama, H. Hayashi, Astrophys.
J 431, 761 (1994)
11. N. Itoh, H. Hayashi, A. Nishikawa, Y. Kohyama, Astro-
phys. Js 102, 411 (1996)
12. G. Beaudet, V. Petrosian, E.E. Salpeter, Astrophys. J 150,
979 (1967)
13. J.B. Adams, M.A. Ruderman, C.H. Woo, Physical Review
129, 1383 (1963)
14. D.A. Dicus, Phys. Rev. D 6, 941 (1972)
15. E. Braaten, D. Segel, Phys. Rev. D 48(4), 1478 (1993)
16. E. Braaten, Phys. Rev. Lett. 66(13), 1655 (1991)
17. P.J. Schinder, D.N. Schramm, P.J. Wiita, S.H. Margolis,
D.L. Tubbs, Astrophys. J 313, 531 (1987)
18. S.I. Blinnikov, M.A. Rudzskij, Astron. Zh. 66, 730 (1989)
19. S.I. Blinnikov, M.A. Rudzskii, Sov. Astron. 33, 377 (1989)
20. M. Haft, G. Raffelt, A. Weiss, Astrophys. J 425, 222 (1994)
21. F. Reines, C.L. Cowan, Phys. Rev. 113(1), 273 (1959)
22. http://www-sk.icrr.u-tokyo.ac.jp/sk/index-e.html
23. J.F. Beacom, M.R. Vagins, Phys. Rev. Lett. 93(17),
171101 (2004)
24. J. F. Beacom and L. E. Strigari, Phys. Rev. C 73, 035807
(2006), M. Wurm et. al., Phys. Rev. D 75, 023007 (2007)
25. http://sn1987a-20th.physics.uci.edu/
26. G.L. Fogli, E. Lisi, A. Mirizzi and D. Montanino, JCAP
0504, 002 (2005)
27. http://neutrino.phys.washington.edu/nnn06/
28. J. G. Learned, S. T. Dye and S. Pakvasa, ”Neutrino Geo-
physics Conference Introduction”, Earth, Moon, and Plan-
ets 99 (2006) 1
29. http://www.phys.hawaii.edu/~sdye/hano.html
30. J. G. Learned, ”White paper on Gigaton Array”,
www.phys.hawaii.edu/~jgl/post/gigaton_array.pdf
31. R. Davis, Jr. Phys. Rev. Lett. 12, 303 (1964)
J. N. Bahcall and R. Davis, Jr. Science 191, 264-267 (1976)
32. GALLEX-Collaboration: P. Anselmann et al. Physics Let-
ters B 357(1-2) (1995) 237-247
W. Hampel et al. Physics Letters B 388(2) (1996) 384-396
N. Bahcall, B. T. Cleveland, R. Davis et.al. Phys. Rev.
Lett. 40, 1351-1354 (1978)
33. The SNO Collaboration, Phys.Rev.Lett. 87 (2001) 071301
34. S. Hirata et al., Phys. Rev. Lett. 65, 1297, 1301 (1990); 66,
9 (1991); Phys. Rev. D44, 2241 (1991).
35. Hirata, K. S. et al. (Kamiokande), Phys. Rev. D38 (1988)
448-458; Phys. Rev. Lett. 58 (1987) 1490-1493.
36. Bionta, R. M. et al. (IMB), Phys. Rev. Lett. 58 (1987)
1494.
37. Galeotti, P. et al., Helv. Phys. Acta 60 (1987) 619-628.
38. Alekseev, E. N., Alekseeva, L. N., Volchenko, V. I.,
Krivosheina, I. V., JETP Lett. 45 (1987) 589-592.
Pisma Zh. Eksp. Teor. Fiz. 45, 461-464 (1987)
Chudakov, A. E., Elensky, Ya. S., Mikheev, S. P., JETP
Lett. 46 (1987) 373-377.
Pisma Zh. Eksp. Teor. Fiz. 46, 297 (1987).
Alekseev, E. N., Alekseeva, L. N., Krivosheina, I. V.,
Volchenko, V. I., Phys. Lett. B205 (1988) 209-214.
39. J. N. Bahcall and M. H. Pinsonneault, Rev. Mod. Phys.
64, 885 (1992)
J. N. Bahcall and R. N. Ulrich, Rev. Mod. Phys. 60, 297
(1988)
S. Turck-Chieze and I. Lopes, Astrophys. J. 408, 347 (1993)
40. H.-Th. Janka et. al. astro-ph/0612072
41. A. Burrows, Nature, 403, 727 (2000)
42. J. Blondin, A. Mezzacappa, Nature 445, 58-60 (4 January
2007)
43. K. Kotake, S. Yamada, and K. Sato Phys. Rev. D, 68,
044023, (2003)
44. Bethe, H. A., Rev. Mod. Phys. 62 (1990) 801-866.
45. J. A. Pons, A. W. Steiner, M. Prakash, and J. M. Lattimer,
Phys. Rev. Lett. 86 (2001) 5223
46. G. Raffelt & A. Weiss, Astron. Astrophys. 264 (1992) 536-
47. L. G. Althaus, E. Garcia-Berro, J. Isern, A. H. Corsico,
A&A 441, 689-694 (2005)
48. W. Hillebrandt and J. C. Niemeyer, Annual Review of As-
tronomy and Astrophysics 38 (2000) 191-230
49. Yakovlev, D. G. and Kaminker, A. D. and Gnedin, O. Y.
and Haensel, P., Physics Reports 354 1 (2001)
50. S. E. Woosley, A. Heger, & T. A. Weaver, RMP 74 (2002)
51. S. Schnert et al. (BOREXINO Collaboration),
physics/0408032 [Nucl. Instrum. Meth. A (to be pub-
lished)].
52. S. Ratkovic, S.I. Dutta, M. Prakash, Phys. Rev. D 67(12),
123002 ( 21) (2003),
53. S.I. Dutta, S. Ratković, M. Prakash, Phys. Rev. D 69(2),
023005 (2004)
54. M. Misiaszek, A. Odrzywolek, M. Kutschera, Phys. Rev.
D 74(4), 043006 (2006)
55. W.M. Yao, C. Amsler, D. Asner, R. Barnett,
J. Beringer, P. Burchat, C. Carone, C. Caso, O. Dahl,
G. D’Ambrosio et al., Journal of Physics G 33 (2006) 1,
http://pdg.lbl.gov
56. N. Itoh, A. Nishikawa, Y. Kohyama, Astrophys. J 470,
1015 (1996)
57. S. Esposito, G. Mangano, G. Miele, I. Picardi, O. Pisanti,
Nuclear Physics B 658, 217 (2003)
http://arxiv.org/abs/astro-ph/0612072
http://arxiv.org/abs/physics/0408032
	Introduction & Motivation
	Plasmaneutrino spectrum
	Summary

  Spectrum of the neutrinos produced in the massive photon and longitudal
plasmon decay process has been computed with four levels of approximation for
the dispersion relations. Some analytical formulae in limiting cases are
derived. Interesting conclusions related to previous calculations of the energy
loss in stars are presented. High energy tail of the neutrino spectrum is shown
to be proportional to exp(-E/kT), where E is the neutrino energy and kT is the
temperature of the plasma.

Introduction & Motivation
Thermal neutrino loses from plasma are very important
for stellar astrophysics [1,2]. Plasmon decay is one of the
three main reactions. Extensive calculations for these pro-
cesses were done by group of Itoh [3,4,5,6,7,8,9,10,11].
Other influential article include [12,13,14,15,16,17,18,19,
20]. Meanwhile, our abilities to detect neutrinos has grown
by many orders of magnitude, beginning with 1.4 tonne
experiment of Reines&Cowan [21] up to the biggest exist-
ing now 50 kt Super-Kamiokande detector [22]. Recently,
”GADZOOKS!” upgrade to Super-Kamiokande proposed
by Beacom&Vagins [23] attract attention of both experi-
mental and theoretical physicists. At last one new source
of the astrophysical antineutrinos is guaranteed with this
upgrade, namely Diffuse Supernova Neutrino Background
[25,24]. Pre-supernova stars will be available to observa-
tions out to ∼2 kiloparsecs [25]. This technique is the
only extensible to megaton scale [25]. Memphys, Hyper-
Kamiokande and UNO (Mt-scale water Cherenkov detec-
tors cf. e.g. [26]) proposals now seriously consider to add
GdCl3 to the one of the tanks with typical three-tank
design [27]. Recently, the discussion on the geoneutrino
detection [28], increased attention to the deep underwa-
ter neutrino observatories [29] with target mass 5-10 Mt
[25] and even bigger [30]. It seems that (anti)neutrino as-
tronomy is on our doorstep, but numerous astrophysical
sources of the ν’s still are not analyzed from the detection
point of view.
Detection of the solar [31,32,33,34] and supernova neu-
trinos [35,36,37,38] was accompanied and followed with
extensive set of detailed calculations (see e.g. [39,40,41,
42,43,44] and references therein as a representatives of
this broad subject) of the neutrino spectrum. On the con-
trary, very little is known about spectral neutrino emission
from other astrophysical objects. Usually, some analytical
representation of the spectrum is used, based on earlier
experience and numerical simulations, cf. e.g. [45]. While
this approach is justified for supernovae, where neutrinos
are trapped, other astrophysical objects are transparent
to neutrinos, and spectrum can be computed with an ar-
bitrary precision. Our goal is to compute neutrino spec-
tra as exact as possible and fill this gap. Plasmaneutrino
process dominates dense, degenerate objects like red giant
cores [46], cooling white dwarfs [47] including Ia supernova
progenitors before so-called ,,smoldering” phase [48]. It is
also important secondary cooling process in e.g. neutron
star crusts [49] and massive stars [50]. Unfortunately, ther-
mal neutrino loses usually are calculated using methods
completely erasing almost any information related to the
neutrino energy Eν and directionality as well. This infor-
mation is not required to compute total energy Q radiated
as neutrinos per unit volume and time. From experimen-
tal point of view, however, it is extremely important if
given amount of energy is radiated as e.g. numerous keV
neutrinos or one 10 MeV neutrino. In the first case we
are unable to detect (using available techniques) any tran-
sient neutrino source regardless of the total luminosity and
proximity of the object. In the second case we can detect
astrophysical neutrino sources if they are strong and not
too far away using advanced detector which is big enough.
Few of the research articles in this area attempt to es-
timate average neutrino energy [16,17,52,53] computing
additionally reaction rate R. Strangely, they presented fig-
ures and formulae for Q/R instead of 1
Q/R. This gives
false picture of real situation, as former expression gives
〈Eν + Eν̄〉. Obviously, we detect neutrinos not ν-ν̄ pairs.
Q/R do not give average neutrino energy, as in general
neutrino and antineutrino spectra are different. As we will
see only for longitudal plasmon decay neutrinos energies
of neutrinos and antineutrinos are equal. However, dif-
ference in all situations where thermal neutrino loses are
important is numerically small and formula:
〈Eν〉 ≃
is still a ”working” estimate.
http://arxiv.org/abs/0704.1222v1
2 A. Odrzywo lek: Plasmaneutrino spectrum
Mean neutrino energy is useful in the purpose of quali-
tative discussion of the detection prospects/methods. Quan-
titative discussion require knowledge of spectrum shape
(differential emissivity dR/dEν). High energy tail is par-
ticularly important from an experimental detection point
of view. Detection of the lowest energy neutrinos is ex-
tremely challenging due to numerous background signal
noise sources e.g. 14C decay for Eν < 200 keV [51]. Rele-
vant calculations for the spectrum of the medium energy
〈Eν〉 ∼ 1MeV neutrinos emitted from thermal processes
has become available recently [52,53,54]. Purpose of this
article is to develop accurate methods and discuss various
theoretical and practical (important for detection) aspects
of the neutrino spectra from astrophysical plasma process.
This could help experimental physicists to discuss possi-
ble realistic approach to detect astrophysical sources of
the neutrinos in the future.
2 Plasmaneutrino spectrum
2.1 Properties of plasmons
Emissivity and the spectrum shape from the plasmon de-
cay is strongly affected by the dispersion relation for trans-
verse plasmons (massive in-medium photons) and longi-
tudal plasmons. In contrast to transverse plasmons, with
vacuum dispersion relation ω(k) = k, longitudal plasmons
exist only in the plasma. Dispersion relation, by the def-
inition is a function ω(k) where ~ω is the energy of the
(quasi)particle and ~k is the momentum. Issues related to
particular handling of these functions are discussed clearly
in the article of Braaten and Segel [15]. We will repeat here
the most important features of the plasmons.
For both types, plasmon energy for momentum k = 0
is equal to ω0. Value ω0 ≡ ω(0) is refereed to as plasma
frequency and can be computed from:
ω20 =
(f1 + f2) dp (2)
where v = p/E, E =
p2 + m2e (~ = c = 1 units are
used), me ≃ 0.511 MeV and fine structure constant is α =
1/137.036 [55]. Functions f1 and f2 are the Fermi-Dirac
distributions for electrons and positrons, respectively:
e(E−µ)/kT + 1
, f2 =
e(E+µ)/kT + 1
. (3)
Quantity µ is the electron chemical potential (including
the rest mass). Other important parameters include first
relativistic correction ω1:
ω21 =
v2 − v4
(f1 + f2) dp (4)
maximum longitudal plasmon momentum (energy) kmax:
k2max ≡ ω
max =
1 − v
1 + v
(f1 + f2) dp
Table 1. Plasma properties for typical massive star during Si
burning. All values in MeV.
kT µ ω0 ω1 mt ωmax ωA
0.32 1.33 0.074 0.070 0.086 0.133 0.002
and asymptotic transverse plasmon mass mt:
m2t =
(f1 + f2) dp. (6)
Value mt is often referred to as thermal photon mass. We
also define parameter v∗:
interpreted as typical velocity of the electrons in the plasma
[15]. Axial polarization coefficient is:
(f1 − f2) dp. (8)
Value of the ωA is a measure of the difference between
neutrino and antineutrino spectra. Set of numerical values
used to display sample result is presented in Table 1.
Values ω0, ωmax,mt define sub-area of the ω-k plane
where dispersion relations for photons ωt(k) and longitu-
dal plasmons ωl(k) are found:
max (k, ω0) ≤ ωl(k) ≤ ωmax, 0 ≤ k ≤ kmax (9a)
k2 + ω20 ≤ ωt(k) ≤
k2 + m2t , 0 ≤ k ≤ ∞ (9b)
Dispersion relations are solution to the equations [15]:
k2 = Πl (ωl(k), k) (10a)
k2 = ωt(k)
2 −Πt (ωt(k), k) (10b)
where longitudal and transverse polarization functions are
given as an integrals:
ωl + vk
ωl − vk
ω2l − k
ω2l − v
(f1+f2) dp.
(11a)
ω2t − k
ωt + vk
ωt − vk
(f1+f2) dp.
(11b)
Typical example of the exact plasmon dispersion re-
lations (dash-dotted) is presented in Fig. 1. As solving
eqns. (10a, 10b) with (11) is computationally intensive,
three levels of approximation for dispersion relations are
widely used:
1. zero-order analytical approximations
2. first order relativistic corrections
3. Braaten&Segel approximation
A. Odrzywo lek: Plasmaneutrino spectrum 3
0.05 0.1 0.15
k @MeVD
Longitudal
0.05 0.1 0.15
k @MeVD
Transverse
Fig. 1. Longitudal and transverse plasmon dispersion relation ωl,t(k) for plasma parameters from Table 1. Exact result (dot-
sahed) is very close to the Braaten & Segel approximation (solid). Zero-order (dotted) and first order (dashed) approximations
are very poor, especially for londitudal mode (left).
0.05 Ω0 0.1 Ωmax
k @MeVD
Longitudal plasmon
0.2 0.4 0.6 0.8 1 1.2
k @MeVD
In-medium photon
Fig. 2. Longitudal and transverse plasmon mass. Dotted lines on the right panel show asymptotic transverse mass. Line
dashing the same as in Fig. 1.
2.1.1 Approximations for longitudal plasmons
For longitudal plasmons, the simplest zero-order approach
used in early calculations of Adams et al. [13] and more
recently in [53] for photoneutrino process is to put simply:
ω(k) = ω0 (12)
where ω0 is the plasma frequency (2). Maximum plasmon
energy ωmax = ω0 in this approximation. Zero-order ap-
proximation is valid only for non-relativistic regime, and
leads to large errors of the total emissivity [12].
First relativistic correction to (12) has been introduced
by Beaudet et al. [12]. Dispersion relation ωl(k) is given
in an implicit form:
ω2l = ω
, (13)
with maximum plasmon energy equal to:
ω(1)max =
ω20 +
ω21 (14)
This approximation, however, do not introduce really se-
rious improvement (Figs. 1, 2 (left) & 4). Breaking point
was publication of the Braaten&Segel approximation [15].
4 A. Odrzywo lek: Plasmaneutrino spectrum
Using simple analytical equation:
k2 = 3
ωl + v∗k
ωl − v∗k
where v∗ is defined in (7) one is able to get almost exact
dispersion relation, cf. Figs. 1 & 2, left panels. Solution to
the eq. (15) exist in the range 1 < k < kBSmax, where, in this
approximation, maximum longitudal plasmon momentum
ωBSmax
1 + v∗
1 − v∗
what gives value slightly different than exact value (Fig. 2,
left), but required for consistency of the approximation.
2.1.2 Approximations for transverse plasmons
For photons in vacuum dispersion relation is ωt = k. Zero
order approximation for in-medium photons is:
ω2t = ω
0 + k
2, k ≪ ω0 (17a)
valid for small k and:
ω2t = m
t + k
2, k ≫ ω0 (17b)
valid for very large k. Formulae (17a) and (17b) provide
lower and upper limit for realistic ωt(k), respectively (cf. Fig. 1,
right panel, dotted). First order relativistic corrections
lead to the formula:
ω2t = ω
0 + k
with asymptotic photon mass:
ω20 + ω
1/5 (19)
Finally, Braaten&Segel approximation leads to:
ω2t = k
2 + ω20
ω2t − v
2ωt v∗ k
ωt + v∗k
ωt − v∗k
Asymptotic photon mass mBSt derived from (20) is:
1 − v2
1 + v∗
1 − v∗
This is slightly smaller (left panel of Fig. 2, dashed) than
exact value (solid line).
All four relations are presented in Fig. 1. Differences
are clearly visible, but they are much less pronounced
for transverse than for longitudal plasmons. Inspection of
Fig. 2 reveals however, that in the large momentum regime
asymptotic behavior is correct only for exact integral re-
lations (10b) and may be easily reproduced using (17b)
with mt from (6).
Let us recapitulate main conclusions. Braaten&Segel
approximation provide reasonable approximation, as non-
linear equations (15) and (20) are easily solved using e.g.
νe,µ,τ
ν̄e,µ,τ
Fig. 3. Fenmann diagrams for plasmon decay.
bisection method. Zero and first-order approximations (12,
17a, 17b) with limiting values (9) provide starting points
and ranges. Approximation has been tested by [56] and
is considered as the best available [20]. Errors for part
of the kT -µ plane where plasmaneutrino process is not
dominant may be as large as 5% [56]. At present, these
inaccuracies are irrelevant for any practical application,
and Braaten&Segel approximation is recommended for all
purposes.
2.2 Plasmon decay rate
In the Standard Model of electroweak interactions, mas-
sive in-medium photons and longitudal plasmons may de-
cay into neutrino-antineutrino pairs:
γ∗ → νx + ν̄x. (22)
In the first-order calculations two Feynmann diagrams
(Fig. 3) contribute to decay rate [15,52].
For the decay of the longitudal plasmon squared ma-
trix element is:
M2l =
ω2l − k
2K ·Q1 K ·Q2
2k · q1 k · q2
−Q1 ·Q2
(23a)
where K = (ω,k) is four momentum of the plasmon.
Q1 = (E1,q1) and Q2 = (E2,q2) is four-momentum of
the neutrino and antineutrino, respectively.
Squared matrix element for decay of the massive pho-
ton is:
M2t =
C2V Π
t + C
E1E2 −
k · q1 k · q2
+ 2CV CAΠtΠA
E1 k · q2 − E2 k · q1
(23b)
where Πt is defined in (11b) and axial polarization func-
tion ΠA reads:
ω2t −k
ωt + vk
ωt − vk
ω2t − k
ω2t − v
(f1−f2) dp
A. Odrzywo lek: Plasmaneutrino spectrum 5
Table 2. Relative weight of the M2t (23b) terms for e and µ, τ
neutrinos.
Flavor Vector Axial Mixed
(CV ω
+CAωA)
(CV ω
+CAωA)
2CV CAω
(CV ω
+CAωA)
electron 0.74 0.02 0.24
mu/tau 0.07 0.39 0.54
Fermi constant is GF /(~c)
3 = 1.16637(1)× 10−5 GeV−2
[55] and, in standard model of electroweak interactions,
vector and axial coupling constants are:
CeV =
+ 2 sin2 θW , C
V = −
+ 2 sin2 θW , C
A = −
for electron and µ, τ neutrinos, respectively. The Weinberg
angle is sin2 θW = 0.23122(15) [55].
Terms containing CA (so-called axial contribution) in
(23b) are frequently treated separately [52] or removed at
all [3]. In calculations concentrated on the total emissiv-
ity this is justified as anti-symmetric term multiplied by
CV CA do not contribute at all and term C
A × . . . is sup-
pressed relative to the term beginning with C2V × . . . by
four orders of magnitude [3]. However, if one attempts to
compute neutrino energy spectrum all three terms should
be added together, as mixed V-A ,,channel” alone leads
to negative emission probability for some neutrino energy
range (Fig. 6), what is physically unacceptable. These
terms remains numerically small but only for electron neu-
trinos. For µ and τ neutrino spectra axial part contributes
at ∼ 1% level due to very small value C
V = −0.0376
while still CA = −0.5. ”Mixed” term leads to significant
differences between νµ,τ and ν̄µ,τ spectra, cf. Fig. 6. Rel-
ative contributions of the three transverse ”channels” for
electron and µ, τ are presented in Table 2.
In general, all the terms in the squared matrix element
(23b) should be added. We have only two different spectra:
longitudal and transverse one.
Particle production rate from plasma in thermal equi-
librium is:
(2π)5
Zi fγ∗ δ
4(K−Q1−Q2) M
where i = l for longitudal mode and i = t for transverse
mode. Bose-Einstein distribution for plasmons fγ∗ is:
fγ∗ =
eωt,l/kT − 1
. (28)
and residue factors Zt,l are expressed by polarization func-
tions Πt,l (11b, 11a):
Z−1t = 1 −
Z−1l = −
. (30)
For massive photons gt = 2 and for longitudal plasmon
gl = 1.
Differential rates1 has been derived for the first time
in [52]. Here, we present result in the form valid for both
types of plasmons, ready for calculations using any avail-
able form of dispersion relation:
dE1 dE2
i fγ∗ Ji S (31)
where i = l or i = t. Product S of the unit step functions
Θ in (31) restrict result to the kinematically allowed area:
S = Θ(4E1E2−m
i )Θ(E1+E2−ω0)Θ(ωmax−E1−E2) (32)
Four-momenta in the squared matrix element are:
Q1 = (E1, 0, 0, E1)
Q1 = (E2, E2 sin θ, 0, E2 cos θ)
K = (E1 + E2, E2 sin θ, 0, E1 + E2 cos θ)
m2i = K ·K = (E1 + E2)
2 − k′2
cos θ =
− E21 − E
2E1E2
k′ = ω−1l,t (E1 + E2)
ωi = E1 + E2
where ω−1i denotes function inverse to the dispersion re-
lation. Jacobian Ji arising from Dirac delta integration in
(27) is:
J−1i =
. (33)
Residue factors Zi are given in (30) and (29). Maximum
energy ωmax in (32) for longitudal plasmons must be in the
agreement with particular approximation used for ωl(k):
ω0, (14) or (16) for zero-order (12), first-order (13) or
Braaten&Segel (15) approximation, respectively. For trans-
verse plasmons ωmax → ∞ and last Θ function in (32) has
no effect and may be omitted.
2.3 Longitudal neutrino spectrum
2.3.1 Analytical approximation
We begin with general remark on the spectrum. Note, that
eq. (31) is symmetric for longitudal mode under change
E1,2 → E2,1 because (23a) is symmetric with respect to
exchange Q1,2 → Q2,1. Resulting energy spectrum is thus
identical for neutrinos and antineutrinos. This is not true
for transverse plasmons with axial contribution included,
cf. Sect. 2.4.
1 Double differential rate d2Ri/dEd cos θ has an identical
form as (31) but now four momenta cannot be given explicitly,
unless simple analytical approximation for ωi(k) is used. Ana-
lytical approximations for the specrum shape are derived this
6 A. Odrzywo lek: Plasmaneutrino spectrum
Using zero-order dispersion relation for longitudal plas-
mons (12) we are able to express spectrum by the elemen-
tary functions. Longitudal residue factor Zt is now:
Z0l = 1, (34)
and Jacobian Jl resulting from the integration of the Dirac
delta function is:
J0l = 1. (35)
Now, differential rate d2R/dEd cos θ (cf. (31) and foot-
note 1) becomes much more simple and integral over d cos θ
can be evaluated analytically. Finally, we get the longitu-
dal spectrum:
≡ λ(E) =
1260 π4 α ~3 c9
f(E/ω0)
eω0/kT − 1
where normalized spectrum is:
f(x) =
4x(x− 1)(8x4 − 16x3 + 2x2 + 6x− 3)
+ 3(1 − 2x)2 ln(1 − 2x)2
Let us note that f is undefined at x = 1/2; use limit
instead:
x→1/2
f(x) = 105/32.
Function f(x) is symmetric with respect to point x = 1/2,
where f has a maximum value (Fig. 4, dotted line).
In this limit, correct for non-relativistic, non-degenerate
plasma, average neutrino and antineutrino energy is 〈E〉 =
ω0/2 and maximum ν energy is ω0.
Inspection of Fig. 4 reveals little difference between
analytical result (36) and result obtained with first-order
relativistic corrections to the dispersion relation (13).
2.3.2 Numerical results
Simple formula (36) significantly underestimates flux and
the maximum neutrino energy, equal to ωmax rather than
ω0. Therefore we have used Braaten & Segel approxima-
tion for longitudal plasmon dispersion relation.
To derive spectrum we will use form of differential rate
(31) provided by [52]. In the Braaten&Segel approxima-
tion:
ZBSl =
ω2l − k
2(ω2l − v
3ω20 − ω
l + v
JBSl =
1 − βl
βBSl =
ωl + v∗k
ωl − v∗k
ω2l v∗
k2(ω2l − v
Spectrum is computed as an integral of (31) over dE2.
Example result is presented in Fig. 4. Integration of the
function in Fig. 4 over neutrino energy gives result in well
agreement with both (30) from [15] and (54) from [52].
Ω0�2 Ω0 Ωmax
@MeVD
Fig. 4. Longitudal plasmon approximate analytical (36) neu-
trino spectrum (dotted), with first-order correction used by
BPS [12] (dashed), and spectrum computed using [15] disper-
sion relation (solid). Plasma properties according to Table 1.
2.4 Transverse plasmon decay spectrum
2.4.1 Analytical approximation
Derivation of massive in-medium photon decay spectrum
closely follows previous subsection. Semi-analytical for-
mula can be derived for dispersion relations (17). For dis-
persion relation (17b) transverse residue factor Zt is:
Z0t = 1, (38)
polarization function Πt is equal to:
Π0t = m
t , (39)
and Jacobian resulting from integration of the Dirac delta
function Jt is:
J0t =
E1 + E2
. (40)
Approximate spectrum, neglecting differences between
neutrinos and antineutrinos, is given by the following in-
tegral:
λ(E) =
64 π4α
P (cos θ, E/mt) d cos θ
2E(1−cos θ)
where rational function P (ct, x) is:
1 + 2(ct− 1)2(2x2 − 1)x2
x(ct− 1)2[1 − 2ct(ct− 1)x2 + 2(ct− 1)2)x4]
A. Odrzywo lek: Plasmaneutrino spectrum 7
Ω0 Ωmax kT 0.5
@MeVD
Fig. 5. Transverse plasmaneutrino spectrum computed from
[15] approximation (solid) with upper (17b) and lower (17a)
limits for the dispersion relation (dotted). First-order rela-
tivistic correction leads to the spectrum shown as dashed line.
Plasma parameters as in Fig. 4.
Result presented in Fig. 5 show that spectrum (41)
obtained with dispersion relation (17b) agree well in both
low and high neutrino energy part with spectrum obtained
from Braaten&Segel approximation for dispersion rela-
tions. Dispersion relation (17a) produces much larger er-
ror, and spectrum nowhere agree with correct result. This
fact is not a big surprise: as was pointed out by Braaten
[16] dispersion relation is crucial. Therefore, all previous
results, including seminal BPS work [12], could be eas-
ily improved just by the trivial replacement ω0 → mt.
Moreover, closely related photoneutrino process also has
been computed [12,3,17,14] with simplified dispersion re-
lation (17a) with ω0. One exception is work of Esposito
et. al. [57]. It remains unclear however, which result is
better, as accurate dispersion relations have never been
used within photoneutrino process context. For plasma-
neutrino, Eq. (17b) is much better approximation than
(17a), especially if one put mt from exact formula (6).
High energy tail of the spectrum also will be exact in this
case.
As formula (41) agree perfectly with the tail of the
spectrum, we may use it to derive very useful analyti-
cal expression. Leaving only leading terms of the rational
function (42)
P (ct, x) ∼ x−1(1 − ct)−2
one is able to compute integral (41) analytically:
λ(E) ≃
64 π4α
eaκ/2 − 1
where κ = 2x + (2x)−1, x = E/mt, a = mt/kT . Interest-
ingly, spectrum (43) is invariant under transformation:
E ′E = m2t/4
and all results obtained for high energy tail of the spec-
trum immediately may be transformed for low-energy ap-
proximation. The asymptotic behavior of (43) for E ≫ kT
is of main interest:
λ(Eν) = A kT m
t exp
where for electron neutrinos :
= 2.115 × 1030 [MeV−8cm−3s−1]
and mt, kT are in MeV. For µ, τ neutrinos just replace A
with A (C
Formula (44) gives also quite reasonable estimates of
the total emissivity Qt and mean neutrino energies 〈Eν〉:
Qt = A kT
3m6t (45a)
〈Eν〉 = kT (45b)
For a comparison, Braaten & Segel [15] derived exact for-
mulae in the high temperature limit kT ≫ ω0:
QBSt =
V ζ(3)
12π4α
kT 3m6t = 0.8A kT
3m6t (46a)
〈EBSν 〉 =
6ζ(3)
kT = 0.73 kT (46b)
Formulae above agree with ∼25% error in the leading co-
efficients.
2.4.2 Numerical results
Calculation of the spectrum in the framework of Braaten&Segel
approximation requires residue factor, polarization func-
tion [15] (transverse&axial) and Jacobian [52]:
ZBSt =
2ω2t (ω
t − v
3ω20ω
t + (ω
t + k
2)(ω2t − v
k2) − 2ω2t (ω
t − k
ΠBSt =
ω2t − v
ωt + v∗k
ωt − v∗k
, (48)
ΠBSA = ωA k
ω2t − k
ω2t − v
3ω20 − 2 (ω
t − k
, (49)
JBSt =
E1 + E2
1 − βBSt
βBSt =
ωt + v∗k
ωt − v∗k
Example spectrum, computed as an integral of (31)
over dE2 is shown in Fig. 5.
8 A. Odrzywo lek: Plasmaneutrino spectrum
Ω0 Ωmax 0.5
@MeVD
Fig. 6. Spectrum of the muon neutrinos (dotted) and antineu-
trinos (dashed) from transverse plasmon decay. Contributions
to the spectra from so-called mixed ,,vector-axial channel” pro-
duces significant differences. For electron flavor, contribution
from ”mixed channel” lead to unimportant differences. For
both flavors contribution from ”axial channel” remains rela-
tively small: 10−4 for νe and 10
−2 for νµ. Overall contribution
to the total emissivity from µ, τ flavors is suppressed relatively
to electron flavor by a factor (C
/CeV )
≃ 3.3 × 10−3.
3 Summary
Main new results presented in the article are analytical
formulae for neutrino spectra (36, 41) and exact analyti-
cal formula (44) for the high energy tail of the transverse
spectrum. The latter is of main interest from the detection
of astrophysical sources point of view: recently available
detection techniques are unable to detect keV plasmaneu-
trinos emitted with typical energies 〈Eν〉 ∼ ω0/2 (Fig. 4,
5), where ω0 is the plasma frequency (2). Tail behavior
of the transverse spectrum quickly ”decouple” from ω0
dominated maximum area, and becomes dominated by
temperature-dependent term exp (−Eν/kT ). Calculation
of the events in the detector is then straightforward, as de-
tector threshold in the realistic experiment will be above
maximum area. This approach is much more reliable com-
pared to the typical practice, where an average neutrino
energy is used as a parameter in an arbitrary analytical
formula.
Analytical formulae for the spectrum are shown to be
a poor approximation of the realistic situation, especially
for longitudal plasmons (Fig. 4). This is in the agreement
with general remarks on the dispersion relations presented
by Braaten [16]. On the contrary, Braaten & Segel [15] ap-
proximation is shown to be a very good approach not only
for the total emissivities, but also for the spectrum. Excep-
tion is the tail of the massive photon decay neutrino spec-
Ω0Ωmax 1.0 10
@MeVD
Fig. 7. Typical spectra from the plasma process. Dotted
line is a longitudal and dashed transverse spectrum. Only
∼ exp(−Eν/kT ) tail of the transverse spectrum (solid line)
contributes to (possibly) detectable signal. Plasma properties
according to Table 1.
trum: Braaten & Segel [15] formulae lead to underestimate
of the thermal photon mass while the formula (44) gives
exact result. Numerical difference between mt from (6)
and (21) is however small [15]. Calculating of the emissiv-
ities by the spectrum integration seems much longer route
compared to typical methods, but we are given much more
insight into process details. For example, we obtain exact
formula for the tail for free this way. Interesting surprise
revealed in the course of our calculations is importance
of the high-momentum behavior of the massive photon.
While mathematically identical to simplest approach used
in the early calculations, formula (17b) gives much better
approximation for the total emissivity than (17a).
This work was supported by grant of Polish Ministry of Edu-
cation and Science (former Ministry of Scientific Research and
Information Technology, now Ministry of Science and Higher
Education) No. 1 P03D 005 28.
References
1. D. Arnett, Supernovae and nucleosynthesis (Princeton
University Press, 1996)
2. G.S. Bisnovatyi-Kogan, Stellar physics. Vol.1: Fundamen-
tal concepts and stellar equilibrium (Springer, 2001)
3. H. Munakata, Y. Kohyama, N. Itoh, Astrophys. J 296, 197
(1985)
4. H. Munakata, Y. Kohyama, N. Itoh, Astrophys. J 304, 580
(1986)
A. Odrzywo lek: Plasmaneutrino spectrum 9
5. Y. Kohyama, N. Itoh, H. Munakata, Astrophys. J 310, 815
(1986)
6. N. Itoh, T. Adachi, M. Nakagawa, Y. Kohyama, H. Mu-
nakata, Astrophys. J 339, 354 (1989)
7. N. Itoh, T. Adachi, M. Nakagawa, Y. Kohyama, H. Mu-
nakata, Astrophys. J 360, 741 (1990)
8. N. Itoh, H. Mutoh, A. Hikita, Y. Kohyama, Astrophys. J
395, 622 (1992)
9. Y. Kohyama, N. Itoh, A. Obama, H. Mutoh, Astrophys. J
415, 267 (1993)
10. Y. Kohyama, N. Itoh, A. Obama, H. Hayashi, Astrophys.
J 431, 761 (1994)
11. N. Itoh, H. Hayashi, A. Nishikawa, Y. Kohyama, Astro-
phys. Js 102, 411 (1996)
12. G. Beaudet, V. Petrosian, E.E. Salpeter, Astrophys. J 150,
979 (1967)
13. J.B. Adams, M.A. Ruderman, C.H. Woo, Physical Review
129, 1383 (1963)
14. D.A. Dicus, Phys. Rev. D 6, 941 (1972)
15. E. Braaten, D. Segel, Phys. Rev. D 48(4), 1478 (1993)
16. E. Braaten, Phys. Rev. Lett. 66(13), 1655 (1991)
17. P.J. Schinder, D.N. Schramm, P.J. Wiita, S.H. Margolis,
D.L. Tubbs, Astrophys. J 313, 531 (1987)
18. S.I. Blinnikov, M.A. Rudzskij, Astron. Zh. 66, 730 (1989)
19. S.I. Blinnikov, M.A. Rudzskii, Sov. Astron. 33, 377 (1989)
20. M. Haft, G. Raffelt, A. Weiss, Astrophys. J 425, 222 (1994)
21. F. Reines, C.L. Cowan, Phys. Rev. 113(1), 273 (1959)
22. http://www-sk.icrr.u-tokyo.ac.jp/sk/index-e.html
23. J.F. Beacom, M.R. Vagins, Phys. Rev. Lett. 93(17),
171101 (2004)
24. J. F. Beacom and L. E. Strigari, Phys. Rev. C 73, 035807
(2006), M. Wurm et. al., Phys. Rev. D 75, 023007 (2007)
25. http://sn1987a-20th.physics.uci.edu/
26. G.L. Fogli, E. Lisi, A. Mirizzi and D. Montanino, JCAP
0504, 002 (2005)
27. http://neutrino.phys.washington.edu/nnn06/
28. J. G. Learned, S. T. Dye and S. Pakvasa, ”Neutrino Geo-
physics Conference Introduction”, Earth, Moon, and Plan-
ets 99 (2006) 1
29. http://www.phys.hawaii.edu/~sdye/hano.html
30. J. G. Learned, ”White paper on Gigaton Array”,
www.phys.hawaii.edu/~jgl/post/gigaton_array.pdf
31. R. Davis, Jr. Phys. Rev. Lett. 12, 303 (1964)
J. N. Bahcall and R. Davis, Jr. Science 191, 264-267 (1976)
32. GALLEX-Collaboration: P. Anselmann et al. Physics Let-
ters B 357(1-2) (1995) 237-247
W. Hampel et al. Physics Letters B 388(2) (1996) 384-396
N. Bahcall, B. T. Cleveland, R. Davis et.al. Phys. Rev.
Lett. 40, 1351-1354 (1978)
33. The SNO Collaboration, Phys.Rev.Lett. 87 (2001) 071301
34. S. Hirata et al., Phys. Rev. Lett. 65, 1297, 1301 (1990); 66,
9 (1991); Phys. Rev. D44, 2241 (1991).
35. Hirata, K. S. et al. (Kamiokande), Phys. Rev. D38 (1988)
448-458; Phys. Rev. Lett. 58 (1987) 1490-1493.
36. Bionta, R. M. et al. (IMB), Phys. Rev. Lett. 58 (1987)
1494.
37. Galeotti, P. et al., Helv. Phys. Acta 60 (1987) 619-628.
38. Alekseev, E. N., Alekseeva, L. N., Volchenko, V. I.,
Krivosheina, I. V., JETP Lett. 45 (1987) 589-592.
Pisma Zh. Eksp. Teor. Fiz. 45, 461-464 (1987)
Chudakov, A. E., Elensky, Ya. S., Mikheev, S. P., JETP
Lett. 46 (1987) 373-377.
Pisma Zh. Eksp. Teor. Fiz. 46, 297 (1987).
Alekseev, E. N., Alekseeva, L. N., Krivosheina, I. V.,
Volchenko, V. I., Phys. Lett. B205 (1988) 209-214.
39. J. N. Bahcall and M. H. Pinsonneault, Rev. Mod. Phys.
64, 885 (1992)
J. N. Bahcall and R. N. Ulrich, Rev. Mod. Phys. 60, 297
(1988)
S. Turck-Chieze and I. Lopes, Astrophys. J. 408, 347 (1993)
40. H.-Th. Janka et. al. astro-ph/0612072
41. A. Burrows, Nature, 403, 727 (2000)
42. J. Blondin, A. Mezzacappa, Nature 445, 58-60 (4 January
2007)
43. K. Kotake, S. Yamada, and K. Sato Phys. Rev. D, 68,
044023, (2003)
44. Bethe, H. A., Rev. Mod. Phys. 62 (1990) 801-866.
45. J. A. Pons, A. W. Steiner, M. Prakash, and J. M. Lattimer,
Phys. Rev. Lett. 86 (2001) 5223
46. G. Raffelt & A. Weiss, Astron. Astrophys. 264 (1992) 536-
47. L. G. Althaus, E. Garcia-Berro, J. Isern, A. H. Corsico,
A&A 441, 689-694 (2005)
48. W. Hillebrandt and J. C. Niemeyer, Annual Review of As-
tronomy and Astrophysics 38 (2000) 191-230
49. Yakovlev, D. G. and Kaminker, A. D. and Gnedin, O. Y.
and Haensel, P., Physics Reports 354 1 (2001)
50. S. E. Woosley, A. Heger, & T. A. Weaver, RMP 74 (2002)
51. S. Schnert et al. (BOREXINO Collaboration),
physics/0408032 [Nucl. Instrum. Meth. A (to be pub-
lished)].
52. S. Ratkovic, S.I. Dutta, M. Prakash, Phys. Rev. D 67(12),
123002 ( 21) (2003),
53. S.I. Dutta, S. Ratković, M. Prakash, Phys. Rev. D 69(2),
023005 (2004)
54. M. Misiaszek, A. Odrzywolek, M. Kutschera, Phys. Rev.
D 74(4), 043006 (2006)
55. W.M. Yao, C. Amsler, D. Asner, R. Barnett,
J. Beringer, P. Burchat, C. Carone, C. Caso, O. Dahl,
G. D’Ambrosio et al., Journal of Physics G 33 (2006) 1,
http://pdg.lbl.gov
56. N. Itoh, A. Nishikawa, Y. Kohyama, Astrophys. J 470,
1015 (1996)
57. S. Esposito, G. Mangano, G. Miele, I. Picardi, O. Pisanti,
Nuclear Physics B 658, 217 (2003)
http://arxiv.org/abs/astro-ph/0612072
http://arxiv.org/abs/physics/0408032
	Introduction & Motivation
	Plasmaneutrino spectrum
	Summary
