Pair production with neutrinos in an intense background magnetic field
Pair production with neutrinos in an intense background magnetic field
Duane A. Dicus,1 Wayne W. Repko,2 and Todd M. Tinsley3
Department of Physics, University of Texas at Austin, Austin, Texas 78712
Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824
Physics and Astronomy Department, Rice University, Houston, Texas 77005
(Dated: July 7, 2021)
We present a detailed calculation of the electron-positron production rate using neutrinos in an
intense background magnetic field. The computation is done for the process ν → νeē (where ν
can be νe, νµ, or ντ ) within the framework of the Standard Model. Results are given for various
combinations of Landau-levels over a range of possible incoming neutrino energies and magnetic
field strengths.
PACS numbers: 13.15.+g, 12.15.-y, 13.40.Ks
I. INTRODUCTION
Neutrino interactions are of great importance in astrophysics because of their capacity to serve as mediators for
the transport and loss of energy. Their low mass and weak couplings make neutrinos ideal candidates for this role.
Therefore, the rates of neutrino interactions are integral in the evolution of all stars, particularly the collapse and
subsequent explosion of supernovae, where the overwhelming majority of gravitational energy lost is radiated away in
the form of neutrinos.
Neutrinos have held a prominent place in models of stellar collapse ever since Gamow and Schoenberg suggested
their role in 1941 [1]. While supernova models have progressed a great deal in the last 65 years, the precise mechanism
for explosion is still uncertain. A common feature, however, among all models is the sensitivity to neutrino transport.
Neutrino processes once thought to be negligible now become relevant, and this has inspired many authors to calculate
rates for neutrino interactions beyond that of the fundamental “Urca” processes
p e → n νe
n → p e ν̄e .
Recent examples include neutrino-electron scattering, neutrino-nucleus inelastic scattering, and electron-positron pair
annihilation [2, 3]. Furthermore, the large magnetic field strengths associated with supernovae (1012–1017 G) are
likely to cause significant changes in the behavior of neutrino transport.
While the the electromagnetic field does not couple to the Standard Model neutrino, it does affect neutrino physics
by altering the behavior of any charged particles, real or virtual, with which the neutrino may interact. A number
of authors have considered such effects on Urca-type processes [4, 5, 6, 7] and on neutrino absorption by nucleons
(and its reversed processes) [8, 9, 10]. Furthermore, Bhattacharya and Pal have prepared a very nice review of other
processes involving neutrinos that are affected by the presence of a magnetic field [11].
The problem of interest in this work is the production of electron-positron pairs with neutrinos in an intense
magnetic field
ν → ν e ē . (1)
Normally this process is kinematically forbidden, but the presence of the magnetic field changes the energy balance
of the process, thereby permitting the interaction.
Stimulation of this process with high-intensity laser fields has been shown to have an unacceptably low rate of
production [12], but such an interaction could have important consequences in astrophysical phenomena where large
magnetic field strengths exist. The process would most likely serve to transfer energy in core-collapse supernovae [13].
However, Gvozdev et al. have proposed that its role in magnetars could even help to explain observed gamma-ray
∗Electronic address: dicus@physics.utexas.edu
†Electronic address: repko@pa.msu.edu
‡Electronic address: tinsley@rice.edu
http://arxiv.org/abs/0704.1695v1
mailto:dicus@physics.utexas.edu
mailto:repko@pa.msu.edu
mailto:tinsley@rice.edu
(a)Neutral current reaction
(b)Charged current reaction
FIG. 1: Possible diagrams considered for the process ν → νeē. Both diagrams contribute for electron-type neutrinos, but
only the neutral current reaction (FIG. 1(a)) contributes for νµ and ντ .
bursts [14]. The interest in this reaction has led to a previous treatment in the literature [15], but those authors
present results for two special limiting cases: (1) when the generalized magnetic field strength eB is greater than the
square of the initial neutrino energy E2, and (2) when the square of the initial neutrino energy E2 is much greater than
the generalized magnetic field strength eB. In both cases the incoming neutrino energy E is much greater than the
electron’s rest energy me. In this paper we present a more complete calculation of the production rate as mediated by
the neutral and the charged-current processes (FIG. 1). We present the results of the calculation for varying Landau
levels, neutrino energies, and magnetic field strengths. A comparison with the approximate method is also discussed.
II. FIELD OPERATOR SOLUTIONS
As we have pointed out in section I, the standard model neutrino can only be affected by the electromagnetic field
through its interactions with charged particles. This means that for the process ν → νeē the Dirac field solution for
the final state electron and positron must change relative to their free-field solutions. The magnetic field will also
change the form of theW -boson’s field solution which can mediate the process when electron neutrinos are considered.
However, in our analysis we take the limit that the momentum transfer for this reaction is much less than the mass
of the W -boson (Q2 ≪ m2W ) and ignore any effects the magnetic field may have on this charged boson. Thus, in
this section we review the results of our derivation of the Dirac field operator solutions for the electron and positron.
We closely follow the conventions used by Bhattacharya and Pal and refer the reader to their work [9] for a detailed
derivation. The reader who is familiar with these solutions may wish to begin with section III where we calculate the
production rate.
We choose our magnetic field to lie along the positive z-axis
~B = B0k̂ (2)
which allows us some freedom in the choice of vector potential A(x). We make the choice
Aµ(x) = (0,−yB, 0, 0) (3)
both for its simplicity and its agreement with the choice found in reference [9]. This choice in vector potential leads us
to assume that all of the y space-time coordinate dependence is within the spinors. The absence of any y dependence
in, for instance, the phase leads us to define a notation such that
y−µ = (t, x, 0, z) (4)
~Vy− = (Vx, 0, Vz) , (5)
where ~V is any 3-vector.
A. Electron field operator
Solving the Dirac equation for our choice of vector potential (Eq. 3) results in the following electron field operator
ψe(x) =
d2~py−
(2π)2
En +me
us(~py−, n, y) e
−ip·y− âse ~py−,n + v
s(~py−, n, y) e
+ip·y− b̂
e ~py−,n
, (6)
where the creation and annihilation operators obey the following anti-commutation relations
âse ~py−,n, â
e ~p′
b̂se ~py−,n, b̂
e ~p′
= (2π)2 δss
δnn′ δ
2(~py− − ~py−′) . (7)
In Eq. 6 we sum over all possible spins s and all Landau levels n where En is the energy of fermion occupying the
nth Landau level
p2z +m
e + 2neB , n ≥ 0 . (8)
The Dirac bi-spinors are
u+(~py−, n, y) =
In−1(ξ−)
En+me
In−1(ξ−)
En+me
In(ξ−)
, u−(~py−, n, y) =
In(ξ−)
En+me
In−1(ξ−)
En+me
In(ξ−)
, (9a)
v+(~py−, n, y) =
En+me
In−1(ξ+)√
En+me
In(ξ+)
In−1(ξ+)
, v−(~py−, n, y) =
En+me
In−1(ξ+)
En+me
In(ξ+)
In(ξ+)
. (9b)
The Im(ξ) are functions of the Hermite polynomials
Im(ξ) =
2/2Hm(ξ) (10)
where the dimensionless parameter ξ is defined by
eB y ± px√
. (11)
Recall that the Hermite polynomials Hm(ξ) are only defined for nonnegative values of m. Therefore, we must define
I−1(ξ) = 0. This means that the electron in the lowest Landau energy level (n = 0) cannot exist in spin-up state and
the positron in the lowest Landau energy level cannot exist in the spin-down state.
The normalization in Eq. (10) has been chosen such that the functions Im(ξ) obey the following delta-function
representation [16, p. 86]
δ(y − y′) = δ(ξ − ξ
|∂y/∂ξ|
eB δ(ξ − ξ′)
2n n!
2/2Hn(ξ) e
−ξ′2/2Hn(ξ
δ(y − y′) =
In(ξ)In(ξ
′) . (12)
For convenience we choose to normalize our 1-particle states in a “box” with dimensions LxLyLz = V such that
the states are defined as
|e〉 = |~p1y−, n1, s1〉 =
e ~p1y−,n1
|0〉 (13a)
|ē〉 = |~p2y−, n2, s2〉 =
e ~p2y−,n2
|0〉 , (13b)
and the completeness relation for the states is
d2~py−
(2π)2
LxLz |~py−, n, s〉 〈~py−, n, s| . (14)
B. Spin sums
In order to evaluate the production rate for our process, we must derive the completeness relations for summations
over the spin of the fermions. For a detailed calculation of the rules see reference [10]. The results of the calculation
are as follows
s=+,−
us(~py−, n, y
′)ūs(~py−, n, y) = (2(En +me))
me(1− σ3)+ 6p‖+ 6q‖γ5
−)In(ξ−)
me(1 + σ
3)+ 6p‖− 6q‖γ5
In−1(ξ
−)In−1(ξ−)
γ1 + iγ2
In−1(ξ
−)In(ξ−)
γ1 − iγ2
−)In−1(ξ−)
(15a)
vs(~py−, n, y)v̄
s(~py−, n, y
′) = (2(En +me))
−me(1− σ3)+ 6p‖+ 6q‖γ5
In(ξ+)In(ξ
−me(1 + σ3)+ 6p‖− 6q‖γ5
In−1(ξ+)In−1(ξ
γ1 + iγ2
In−1(ξ+)In(ξ
γ1 − iγ2
In(ξ+)In−1(ξ
, (15b)
where
‖ = (E, 0, 0, pz) (16)
‖ = (pz, 0, 0, E) . (17)
The above results have been derived using the standard “Bjorken and Drell” representation for the γ-matrices [17]
, γi =
−σi 0
. (18)
C. Neutrino field operator
Having no charge, the neutrino’s field operator solution ψν(x) is not modified due to the magnetic field. We present
it here for easy reference
ψν(x) =
(2π)3
us(p) e−ip·x âsν + v
s(p) e+ip·x b̂s †ν
, (19)
where the creation and annihilation operators obey the conventional anticommutation relations
âsν , â
b̂sν , b̂
= (2π)3 δss
δ3(~p− ~p ′) . (20)
The neutrino bi-spinors follow the standard spin sum rules
us(p)ūs(p) =
vs(p)v̄s(p) = 6p , (21)
where we take the Standard Model neutrino mass to be zero.
With “box” normalization the 1-particle states for the neutrino are
|ν〉 = |~p, s〉 = 1√
ν ~p |0〉 , (22)
satisfying the completeness relation
(2π)3
V |~p, s〉 〈~p, s| . (23)
III. THE PRODUCTION RATE
The quantity of interest for the process ν → νeē in a background magnetic field is the rate at which the electron-
positron pairs are produced Γ. The production rate is defined as the probability per unit time for creation of pairs
Γ = lim
. (24)
where T is the timescale on which the process is normalized. We begin by finding the probability P of our reaction
n1,n2=0
d3~p′
(2π)3
d2~p1y−
(2π)2
d2~p2y−
(2π)2
s,s′,s1,s2
~p′, s′; ~p1y−, n1, s1; ~p2y−, n2, s2
~p, s
. (25)
In Eq. 25 quantities with the index 1 correspond to the electron, those with index 2 to the positron, the primed
quantities to the final neutrino, and the unprimed quantities correspond to the initial neutrino.
A. The scattering matrix
The scattering matrix
~p′, s′; ~p1y−, n1, s1; ~p2y−, n2, s2
~p, s
naturally depends on the flavor of the neutrino. While the process involving the electron neutrino can advance through
either the charged (W ) or neutral (Z) current, the muon (or tau) neutrino can only proceed through the latter. For
this reason we will break the scattering matrix into a neutral component
~p′, s′; ~p1y−, n1, s1; ~p2y−, n2, s2
~p, s
(27a)
and a charged component
~p′, s′; ~p1y−, n1, s1; ~p2y−, n2, s2
~p, s
, (27b)
where the scattering operators are defined by the Standard Model Lagrangian as
ŜZ =
23 cos2 θW sin
d4xψe(x)γ
geV − geAγ5
ψe(x)Zµ(x)
d4x′ ψνl(x
1− γ5
ψνl(x
′)Zσ(x
′) (28a)
ŜW =
23 sin2 θW
d4xψe(x)γ
1− γ5
ψνe(x)W
µ (x)
d4x′ ψνe(x
1− γ5
′)W+σ (x
′) , (28b)
and θW is the weak-mixing angle, νl indicates a neutrino of any flavor, νe refers to a electron neutrino, and the vector
and axial vector couplings for the electron are
geV = −
+ 2 sin2 θW (29a)
geA = −
. (29b)
In our analysis we will be using incoming neutrino energies that are well below the rest energies of the Z and W
bosons. Therefore, we can safely make the 4-fermion effective coupling approximation to the Z and W propagators
〈0 |T (Zµ(x)Zσ(x′))| 0〉 → δ4(x− x′)
(30a)
W−µ (x)W
→ δ4(x− x′)
. (30b)
After making this approximation our expressions for the scattering operators simplify to
ŜZ =
d4xψe(x)γ
geV − geAγ5
ψe(x)ψνl(x)γµ
1− γ5
ψνl(x) (31a)
ŜW =
d4xψe(x)γ
1− γ5
ψνe(x)ψνe(x)γµ
1− γ5
ψe(x) , (31b)
where GF /
2 = e2/(8 sin2 θW m
W ), and we have made use of the fact that cos
2 θW = m
After substituting of the scattering operators (Eqs. (31)) into the expressions for the components of the scattering
matrix (Eqs. (27)), we can use our results from sections IIA and IIC to write the components in the form of
SZ/W =
i(2π)3 δ3
py− − p′y− − py−,1 − py−,2
Lx Lz V
MZ/W , (32)
where
(En1 +me)(En2 +me)
EE′En1En2
(p′)γµ
1− γ5
us(p)
dy ei(py−p
y)y ūs1 (~p1y−, n1, y) γ
geV − geAγ5
vs2 (~p2y−, n2, y) (33a)
(En1 +me)(En2 +me)
EE′En1En2
(p′)γµ
1− γ5
vs2 (~p2y−, n2, y)
dy ei(py−p
y)y ūs1 (~p1y−, n1, y) γ
1− γ5
us(p) . (33b)
The reversal of sign on Eq. (33b) relative to Eq. (33a) is from the anticommutation of the field operators. The
scattering amplitude for the charged component MW can be transformed into the form of the neutral component
MZ by making use of a Fierz rearrangement formula
ū1γµ
1− γ5
u2 ū3γ
1− γ5
u4 = −ū1γµ
1− γ5
u4 ū3γ
1− γ5
u2 , (34)
such that
(En1 +me)(En2 +me)
EE′En1En2
(p′)γµ
1− γ5
us(p)
dy ei(py−p
y)y ūs1 (~p1y−, n1, y) γ
1− γ5
vs2 (~p2y−, n2, y) . (35)
With the rearrangement of MW in Eq. (35), we can now express the scattering amplitude in terms of the type of
incoming neutrino. The muon neutrino can only proceed through exchange of a Z-boson, so its scattering amplitude
is just that of MZ
Mνµ = MZ
Mνµ =
(En1 +me)(En2 +me)
EE′En1En2
(p′)γµ
1− γ5
us(p)
dy ei(py−p
y)y ūs1 (~p1y−, n1, y) γ
G−V − γ
vs2 (~p2y−, n2, y) . (36)
The scattering matrix for a tau neutrino, and the subsequent decay rate, is exactly the same as the muon neutrino.
We will keep the notation as νµ for simplicity.
The electron neutrino has both a Z-boson exchange component and an W -boson exchange component. Therefore
we must add the amplitudes to find its scattering amplitude
Mνe = MZ +MW
Mνe =
(En1 +me)(En2 +me)
EE′En1En2
(p′)γµ
1− γ5
us(p)
dy ei(py−p
)y ūs1 (~p1y−, n1, y)γ
G+V − γ
vs2 (~p2y−, n2, y) . (37)
Note that the scattering amplitudes for electron (Eq. 37) and non-electron neutrinos (Eq. 36) depend on a generalized
vector coupling GV defined by
G±V = 1± 4 sin
2 θW . (38)
We see that the scattering amplitudes for an incoming electron neutrino versus an incoming muon neutrino differ
only in the value of the generalized vector coupling and an overall sign. And the overall sign will be rendered
meaningless once the amplitude is squared. Therefore, we choose to make no distinction between the two processes,
other than keeping the generalized vector coupling as G±V , until we discuss the results in section IV.
B. The form of the production rate
Having determined the scattering matrix S and scattering amplitude M in section IIIA, we can now make series
of substitutions of those results to find the expression for the production rate Γ. We begin by substituting the form
of the scattering matrix (Eq. (32)) into the expression for the production rate (Eq. (24)
Γ = lim
= lim
T,V→∞
n1,n2=0
d3~p′
(2π)3
d2~p1y−
(2π)2
d2~p2y−
(2π)2
s,s′,s1,s2
~p′, s′; ~p1y−, n1, s1; ~p2y−, n2, s2
~p, s
= lim
T,V→∞
n1,n2=0
d3~p′
(2π)3
d2~p1y−
(2π)2
d2~p2y−
(2π)2
s,s′,s1,s2
i(2π)3 δ3
py− − p′y− − py−,1 − py−,2
Lx Lz V
Γ = lim
T,V→∞
(2πTV )−1
n1,n2=0
d3~p′
d2~p1y−
d2~p2y−
δ3(py− − p′y− − py−,1 − py−,2)
)2 |M|2 , (39)
where |M|2 is the square of the scattering amplitude after summing over spins
|M|2 =
s,s′,s1,s2
|M|2 . (40)
We can simplify the square of the 3-dimensional delta function by expressing one of the 3-dimensional delta functions
as a series of integrals over space-time coordinates
δ3(py− − p′y− − py−,1 − py−,2)
= δ3(py− − p′y− − py−,1 − py−,2)
(2π)3
ei(p−p
′−p1−p2)· y− . (41)
By using the remaining set of delta functions to reduce the exponential to unity, we can write the integrand in terms
of the dimensions of our normalization “box”
δ3(py− − p′y− − py−,1 − py−,2)
= δ3(py− − p′y− − py−,1 − py−,2)
(2π)3
δ3(py− − p′y− − py−,1 − py−,2)
= δ3(py− − p′y− − py−,1 − py−,2)
TLxLz
(2π)3
. (42)
With the above result for the square of the delta function, the production rate in Eq. (39) simplifies to
Γ = lim
n1,n2=0
d3~p′
d2~p1y−
d2~p2y− δ
3(py− − p′y− − py−,1 − py−,2)
(2π)4Ly
. (43)
The square of the scattering amplitude goes as the product of two traces
|M|2 =
s,s′,s1,s2
(En1 +me)(En2 +me)
EE′En1En2
ūs(p)γσ
1− γ5
(p′)ūs
(p′)γµ
1− γ5
us(p)
dy ei(py−p
dy′ e−i(py−p
s1,s2
v̄s2 (~p2y−, n2, y
′) γσ
G±V − γ
us1 (~p1y−, n1, y
′) ūs1 (~p1y−, n1, y)γ
G±V − γ
vs2 (~p2y−, n2, y)
|M|2 = G
(EE′En1En2)
dy ei(py−p
dy′ e−i(py−p
)y′Tr
1− γ5
6p′γµ
1− γ5
G±V − γ
me(1 − σ3)+ 6p1‖+ 6q1‖γ5
In1 (ξ
−,1)In1(ξ−,1)
2n1eB
γ1 + iγ2
In1−1(ξ
−,1)In1 (ξ−,1)
2n1eB
γ1 − iγ2
In1(ξ
−,1)In1−1(ξ−,1)
me(1 + σ
3)+ 6p1‖− 6q1‖γ5
In1−1(ξ
−,1)In1−1(ξ−,1)
G±V − γ
−me(1− σ3)+ 6p2‖+ 6q2‖γ5
In2(ξ
+,2)In2 (ξ+,2)
2n2eB
γ1 + iγ2
In2−1(ξ
+,2)In2 (ξ+,2)
2n2eB
γ1 − iγ2
In2(ξ
+,2)In2−1(ξ+,2)
−me(1 + σ3)+ 6p2‖− 6q2‖γ5
In2−1(ξ
+,2)In2−1(ξ+,2)
where we have used our result for the summations over spin from Eqs. (15) and (21).
The space-time dependence of Eq. (44) can be factored into terms like
In,m =
dy ei(py−p
y)y In(ξ−,1) Im(ξ+,2) (45)
I∗n,m =
dy′ e−i(py−p
)y′ In(ξ
−,1) Im(ξ
+,2) , (46)
where the In,m are functions of the momenta in the problem.
We have included a detailed calculation for the general form of In,m in appendix A, but we only present the result
In,m =
2/2 eiφ0 (ηx + iηy)
Lm−nn
, m ≥ n ≥ 0
2/2 eiφ0 (−ηx + iηy)n−m Ln−mm
, n ≥ m ≥ 0
where
p1x + p2x√
py − p′y√
(py − p′y)(p1 − p2)
η2 = η2x + η
y , (51)
and Lm−nn (η
2) are the associated Laguerre polynomials.
The full results of the traces and their subsequent contraction are nontrivial but have been included in appendix B. It
is important to note, however, that the only dependence on the x-components of the electron and positron momentum
is that which appears in Eq. (47) for In,m. Furthermore, we notice that all terms in the averaged square of the scattering
amplitude have factors that go as a product of In,m and I
n′,m′ . Therefore, the coefficient e
i φ0 in Eq. (47) will vanish
when this product is taken. The only remaining x-dependence of these two momenta appear as their sum in the
parameter ηx = (p1x + p2x)/
2eB. This helps to simplify the phase-space integral for our production rate (Eq. (43))
which is proportional to
Γ ∝ lim
dp2x . (52)
If we make a change of variable from the x-component of the positron momentum p2x to the parameter ηx, the
relationship in Eq. (52) is rewritten as
Γ ∝ lim
dp1 x
dηx . (53)
Because there is no longer any explicit dependence on the x-component of the electron’s momentum p1,x in the
averaged square of our scattering amplitude, we can simply evaluate the integral
dp1,x .
To evaluate this integral we must determine its limits. As discussed previously, we have elected to use “box” normal-
ization on our states. This means that our particle is confined to a large box with dimensions Lx, Ly, and Lz. The
careful reader will note that we have already taken the limit that these dimensions go
to infinity in some places, particularly in Eq. (A5), but it is imperative that we be cautious here, as we could naively
evaluate the integral over p1,x to be infinite.
Physically, the charged particles in our final state act as harmonic oscillators circling about the magnetic field lines.
While they are free to slide about the lines along the z-axis, the particles are confined to circular orbits in the x and
y-directions no larger than the dimensions of the box. For a charged particle undergoing circular motion in a constant
magnetic field, the x-component of momentum is related to the y-position vector by
px = −eQBy (54)
where Q is the charge of the particle in units of the proton charge e = |e|. Therefore, the limits on p1,x are proportional
to the limits on the size of our box in the y-direction. The integral over the electron’s momentum in the x-direction is
∫ eBLy/2
−eBLy/2
dp1,x = eBLy , (55)
and the result helps to cancel the factor of Ly that already appears in the form of the production rate. We can now
safely take the limit that our box has infinite size, and the production rate now has the form
n1,n2=0
d3~p′
d~p1z
d~p2z
2eB δ3(py− − p′y− − py−,1 − py−,2)
eB |M|2
(2π)4
. (56)
IV. RESULTS
In our expression for the total production rate (Eq. (56)), one will notice is that there is a sum over all possible
values of the Landau levels. As a consequence of energy conservation, upper limits do exist for the summation over
the electron’s Landau level n1
E = E′ + En1 + En2
E ≥ En1 +me
E −me ≥
m2e + 2n1eB
E(E − 2me)
, (57)
and a similar one for the positron’s Landau level
E = E′ + En1 + En2
m2e + 2n1eB + En2
m2e + 2n1eB ≥
m2e + 2n2eB
m2e + 2n1eB
. (58)
These relationships help to constrain the extent of the summations. Physically, these constraints can be thought of
as limits on the size of the electron’s (or positron’s) effective mass, where the electron (or positron) occupying the nth
Landau level has an effective mass
m2e + 2neB (59)
and energy
p2z +m∗
2 . (60)
For low incoming neutrino energies and large magnetic field strengths (eB > m2e), the constraints put very tight
bounds on the limits of the summations. However, higher incoming energies and low magnetic field strengths impose
limits that still require a great deal of computation time. For instance, at threshold (E = 2me) there can exist only
one possible configuration of Landau levels (n1 = n2 = 0), while at an energy ten times that of threshold and a
magnetic field equal to the critical field (B = Bc = m
e/e = 4.414× 1013 G) there are nearly 7000 possible states. At
the same magnetic field but an energy that is 100 times that of threshold, there are almost 70 million states. However,
for incoming neutrino energies less than a certain value
E < me +
m2e + 2eB (61)
only the lowest Landau level is occupied, n1 , n2 = 0. And even at energies above, yet near, this value we expect that
production of electrons and positrons in the n1 , n2 = 0 level is still the dominant mode of production because it has
more phase space available.
Production rates at the 0, 0 Landau level are presented in FIG. 2 for both the electron and muon neutrinos. (All
of the results for muon-type neutrinos are valid for tau-type neutrinos.) One interesting feature of these results is
the flattening out of the rates at higher energies. The energy region at which this flattening begins increases with
increasing magnetic field strength, and it appears to be in the neighborhood of energies just above the limit set in
Eq. (61). At energies in this regime we expect that modes of production into other Landau levels are stimulated,
which helps to explain why the behavior of the 0, 0 production rates change above this area.
We should note that the results given in this work are all for an incoming neutrino traveling transversely to the
magnetic field. The rates are maximized in this case as can be seen in the example found in FIG. 3 for an initial
electron neutrino with energy Eνe = 20me in a magnetic field equal to the critical field B = Bc = m
For comparison purposes, the production rates for other combinations of Landau levels have been calculated. These
include the 1, 0 and 0, 1 cases (FIG. 4), the 20, 0 and 0, 20 cases (FIG. 5), and the 10, 10 case (FIG. 6). The first
noteworthy feature of these results is that the production rates are decreasing at higher Landau levels. Because the
energy required to create the pair goes as
Epair = En1 + En2
p21 z + 2n1eB +m
p22 z + 2n2eB +m
Epair ≥
2n1eB +m2e +
2n2eB +m2e ,
B/Bc = 1000
B/Bc = 100
B/Bc = 10
B/Bc = 1
B/Bc = 0.1
νe → νeeē
Eνe (MeV)
1000100101
10−10
10−15
10−20
10−25
10−30
(a)Incoming electron neutrino
B/Bc = 1000
B/Bc = 100
B/Bc = 10
B/Bc = 1
B/Bc = 0.1
νµ → νµeē
Eνµ (MeV)
1000100101
10−10
10−15
10−20
10−25
10−30
(b)Incoming muon (tau) neutrino
FIG. 2: Production rates for the n1 , n2 = 0 Landau levels where Γ is the rate of production, Eν is the energy of the incoming
neutrino, and the magnetic field is measured relative to the critical field Bc = 4.414× 10
13 G. All plots are for a neutrino that
is perpendicularly incident to the magnetic field.
the available phase space for the process should decrease in the order 0, 0 → 0, 1 → 0, 20 → 10, 10. And as can
be seen in FIGS. 2, 4, 5, and 6, the production rates fall off accordingly.
Another interesting feature of these results is the apparent preference for the creation of electrons in the highest of
the two Landau levels. That is, the rate of production is larger for the state n1 = i , n2 = 0 than for n1 = 0 , n2 = i
(FIGS. 4 and 5). This behavior is especially significant over the range of incoming neutrino energies near its threshold
value for creating pairs in the given states. Though the i, 0 production rate is larger and increases more quickly
in this “near-threshold” range than its 0, i counterpart, both curves plateau at higher energies, and their difference
approaches zero. This difference is presumably caused by the positron having to share the W ’s energy with the final
electron-type neutrino. This also explains why such an effect is not seen for muon and tau-type neutrinos that only
proceed through the neutral current reaction.
It was mentioned in section I that previous authors have considered this process under two limiting cases [15]. One
νµ → νµeē
νe → νeeē
Directional Dependence
sin2 θ
10.90.80.70.60.50.40.30.20.10
10−15
10−16
10−17
10−18
FIG. 3: The production rate’s dependance on the direction of the incoming neutrino.. The production rate is for the 0, 0 Landau
level with an electron of energy Eν = 20me traveling at an angle θ relative to a magnetic field of strength equal to the critical
field B = Bc. Data is included for both an incoming electron-type neutrino (solid line) and a muon-type neutrino (dashed line).
If we average over θ, then the average production rate is 1.38 × 10−16 cm−1 for electron-type neutrinos or 2.94 × 10−17 cm−1
for muon or tau-type.
is when the square of the energy of the initial-state neutrino and the magnetic field strength satisfy the conditions
Eν ≫ eB ≫ m2e. Under these conditions many possible Landau levels could be stimulated, offering a multitude of
production modes. Therefore, it would be inappropriate to compare their expression to our results for a specific set
of Landau levels. However, the second limiting case is for eB > E2ν ≫ m2e. This condition is slightly more restrictive
than our condition for the energies below which only the lowest energy Landau levels are occupied (Eq. (61)). In this
regime our results for the 0, 0 state are the total production rates, and we can compare our results to the expression
derived by the previous authors [15]
eB E3ν
E2/eB
, (62)
where we have taken the direction of the incoming neutrino to be perpendicular to the magnetic field’s direction.
Results of this comparison are shown in FIG. 7.
The results in FIG. 7 demonstrate the drawbacks of using the approximation in Eq. (62). While the expression is
very simple, it gives only reasonable agreement with the production rate at a magnetic field equal to 100 times that
of the critical field (B = 100Bc). Here it overestimates, at the very least, by a factor of two, and the inclusion of
higher order corrections makes no significant improvement. One reason for the disagreement at this field strength is
that there is only a very small range of energies that satisfy the condition eB > E2ν ≫ m2e. Therefore at higher field
strengths we should get better agreement, and we do. Closer inspection of FIG. 7 reveals that the differences are
less than a factor of three for neutrino energies in the range 2 MeV < Eν < 20 MeV, and the expression successfully
provides a good order of magnitude estimation. Though the estimate will improve at higher magnetic field strengths,
it begins to loose relevance as there are only a handful of known objects (namely magnetars) that can conceivably
possess fields as high as 1015 G. Even for these objects, fields stronger than 1015 G cause instability in the star and
the field begins to diminish [13].
Probing the limiting case Eν ≫
eB is imperative because our present work has already demonstrated nontrivial
deviation from approximate methods for realistic astrophysical magnetic field strengths and neutrino energies near
and below the value
eB. But, as was mentioned previously, the number of Landau level states which contribute to
the total production rate grows very rapidly in this higher energy regime, and we need to sum over these states. Future
work will attempt to do these sums by using an approximation routine that can interpolate between rates for known
sets of Landau levels. This will provide a flexible way to balance accuracy with computation time while determining
when the production rate deviates from its limiting behavior. The significance of these deviations will only be known
when a more complete understanding of the role that neutrino processes play in events such as supernova core-collapse
B/Bc = 1000
B/Bc = 100
B/Bc = 10
B/Bc = 1
B/Bc = 0.1
νe → νeeē
Eνe (MeV)
1000100101
10−10
10−15
10−20
10−25
10−30
(a)Incoming electron neutrino
B/Bc = 1000
B/Bc = 100
B/Bc = 10
B/Bc = 1
B/Bc = 0.1
νµ → νµeē
Eνµ (MeV)
1000100101
10−10
10−15
10−20
10−25
10−30
(b)Incoming muon neutrino
FIG. 4: Production rates for the n1 = 0 , n2 = 1 (solid) and n1 = 1 , n2 = 0 (dashed) Landau levels where Γ is the rate
of production, Eν is the energy of the incoming neutrino, and the magnetic field is measured relative to the critical field
Bc = 4.414 × 10
13 G.
and in the formation of the resulting neutron star. This work aims to improve that understanding.
Acknowledgments
It is our pleasure to thank Craig Wheeler for several discussions about supernovae and Palash Pal for helping us
to understand Ref. [11]. This work was supported in part by the U.S. Department of Energy under Grant No. DE-
B/Bc = 1000
B/Bc = 100
B/Bc = 10
B/Bc = 1
B/Bc = 0.1
νe → νeeē
Eνe (MeV)
104103102101100
10−10
10−12
10−14
10−16
10−18
10−20
10−22
10−24
10−26
(a)Incoming electron neutrino
B/Bc = 1000
B/Bc = 100
B/Bc = 10
B/Bc = 1
B/Bc = 0.1
νµ → νµeē
Eνµ (MeV)
104103102101100
10−10
10−12
10−14
10−16
10−18
10−20
10−22
10−24
10−26
10−28
(b)Incoming muon neutrino
FIG. 5: Production rates for the n1 = 0 , n2 = 20 (solid) and n1 = 20 , n2 = 0 (dashed) Landau levels where Γ is the rate
of production, Eν is the energy of the incoming neutrino, and the magnetic field is measured relative to the critical field
Bc = 4.414 × 10
13 G.
F603-93ER40757 and by the National Science Foundation under Grant PHY-0244789 and PHY-0555544.
[1] G. Gamow and M. Schoenberg, Physical Review 59, 539547 (1941).
[2] S. W. Bruenn and W. C. Haxton, Astrophysical Journal 376, 678 (1991).
[3] A. Mezzacappa and S. W. Bruenn, Astrophysical Journal 410, 740 (1993).
[4] O. F. Dorofeev, V. N. Rodionov, and I. M. Ternov, JETP Lett. 40, 917 (1984).
[5] D. A. Baiko and D. G. Yakovlev, Astron. Astrophys. 342, 192 (1999), astro-ph/9812071.
[6] A. A. Gvozdev and I. S. Ognev, JETP Lett. 69, 365 (1999), astro-ph/9909154.
[7] P. Arras and D. Lai, Phys. Rev. D60, 043001 (1999), astro-ph/9811371.
B/Bc = 1000
B/Bc = 100
B/Bc = 10
B/Bc = 1
B/Bc = 0.1
νe → νeeē
Eνe (MeV)
104103102101100
10−10
10−12
10−14
10−16
10−18
10−20
10−22
10−24
10−26
(a)Incoming electron neutrino
B/Bc = 1000
B/Bc = 100
B/Bc = 10
B/Bc = 1
B/Bc = 0.1
νµ → νµeē
Eνµ (MeV)
104103102101100
10−10
10−12
10−14
10−16
10−18
10−20
10−22
10−24
10−26
(b)Incoming muon neutrino
FIG. 6: Production rates for the n1 , n2 = 10 Landau levels where Γ is the rate of production, Eν is the energy of the incoming
neutrino, and the magnetic field is measured relative to the critical field Bc = 4.414 × 10
13 G.
[8] H. Duan and Y.-Z. Qian, Phys. Rev. D72, 023005 (2005), astro-ph/0506033.
[9] K. Bhattacharya and P. B. Pal, Pramana 62, 1041 (2004), hep-ph/0209053.
[10] K. Bhattacharya, Ph.D. thesis, Jadavpu University (2004), hep-ph/0407099.
[11] K. Bhattacharya and P. B. Pal, Proc. Ind. Natl. Sci. Acad. 70, 145 (2004), hep-ph/0212118.
[12] T. M. Tinsley, Phys. Rev. D71, 073010 (2005), hep-ph/0412014.
[13] C. Thompson and R. C. Duncan, Astrophysical Journal 408, 194 (1993).
[14] A. A. Gvozdev, A. V. Kuznetsov, N. V. Mikheev, and L. A. Vassilevskaya, Phys. Atom. Nucl. 61, 1031 (1998), hep-
ph/9710219.
[15] A. V. Kuznetsov and N. V. Mikheev, Phys. Lett. B394, 123 (1997), hep-ph/9612312.
[16] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, San Diego, CA, 1995), 4th ed.
[17] J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, International Series in Pure and Applied Physics (McGraw-
Hill, Inc., New York, 1964).
B/Bc = 1000
B/Bc = 100
νe → νeeē
103102101100
10−10
10−12
10−14
10−16
(a)Incoming electron neutrino
B/Bc = 1000
B/Bc = 100
νµ → νµeē
103102101100
10−10
10−11
10−12
10−13
10−14
10−15
10−16
10−17
10−18
(b)Incoming muon neutrino
FIG. 7: The total production rate (solid lines) and its approximation (dashed lines)[15] for energies and magnetic field satisfying
the condition E2ν < eB. Eν is the energy of the incoming neutrino, me is the mass of the electron, and the magnetic field is
measured relative to the critical field Bc = 4.414 × 10
13 G.
APPENDIX A: CALCULATION OF In,m
In section III B we discuss the fact that the squared scattering amplitude has coefficients that are integrals over the
space-time coordinate y
In,m =
dy ei(py−p
)y In(ξ−,1) Im(ξ+,2) . (A1)
In this appendix we will derive the result after integrating over y.
By defining new parameters
eB y (A2)
ζi = pix/
eB (A3)
ζ0 = (py − p′y)/
eB (A4)
and using the definition of ξ (Eq. (11)) we can make a change of variable from y to ζ and rewrite In,m as
In,m =
eiζ0ζ In(ζ − ζ1) Im(ζ + ζ2) (A5)
where the limits of integration are ±∞ because we have taken the limit of Ly as it approaches ∞. The In(ξ) in
Eq. (10) depend on the Hermite polynomials Hn(ξ), which can be represented as a contour integral in the following
way [16, Eq. (13.8)]
Hn(ξ) =
dt t−n−1 e−t
2+2tx . (A6)
Substituting this definition of the Hermite polynomial into Eq. (10) allows us to write the In,m as
In,m =
eiζ0 ζ
2n n!
e−(ζ−ζ1)
2/2Hn(ζ − ζ1)
e−(ζ+ζ2)
2/2Hm(ζ + ζ2)
2n+m n!m!π
)−1/2
dζ eiζ0 ζ e−(ζ−ζ1)
2/2Hn(ζ − ζ1)e−(ζ+ζ2)
2/2Hm(ζ + ζ2)
2n+m n!m!π
)−1/2
dζ eiζ0 ζ e−(ζ−ζ1)
2/2 e−(ζ+ζ2)
dt t−n−1 e−t
2+2t(ζ−ζ1) m!
ds s−m−1 e−s
2+2s(ζ+ζ2) . (A7)
Next, we isolate all of the ζ dependence, interchange the order of the integrals, and perform the integration over ζ
Int1 =
dζ exp
−ζ2 + ζ(ζ1 − ζ2 + iζ0 + 2t+ 2s)
π exp
(ζ1 − ζ2 + iζ0 + 2t+ 2s)2 /4
. (A8)
Substitution of this result back into Eq. (A7) gives
In,m =
2n+m n!m!
)−1/2
e−((ζ1+ζ2)
0)/4 eiζ0(ζ1−ζ2)/2
dt t−n−1 et(−ζ1−ζ2+iζ0)
ds s−m−1 es(ζ1+ζ2+iζ0) e2st .
If m ≥ n, then we can perform the integration over s first
Int2 =
ds s−m−1 es(ζ1+ζ2+iζ0) e2st .
es(ζ1+ζ2+iζ0+2t)
= (ζ1 + ζ2 + iζ0 + 2t)
m , (A10)
such that
In,m =
2n+m n!m!
)−1/2
(ζ1 + ζ2)
2 + ζ20
exp (iζ0(ζ1 − ζ2)/2)
(ζ1 + ζ2 + iζ0 + 2t)
exp (t(−ζ1 − ζ2 + iζ0)) . (A11)
The integration over t is made easier by making the following changes of variable
ζ1 + ζ2√
p1x + p2x√
(A12)
py − p′y√
(A13)
ζ0 (ζ1 − ζ2)
(py − p′y)(p1 − p2)
(A14)
η± = ηx ± iηy (A15)
η2 = η+ η− = η2x + η
y (A16)
u− η+
2 . (A17)
The integration over the variable t can now be written as
Int3 =
2 u)m
(u− η+) /
e((u−η
= 2(n+m)/2 eη
(u− η+)n+1
= 2(n+m)/2 eη
um e−uη
= 2(n+m)/2 eη
(η−)n−m
d(η2)
2)m e−η
= n! 2(n+m)/2 (η+)(m−n) Lm−nn (η
where we have used the Rodrigues’ representation for Laguerre polynomials [16, Eq. (13.47)]
Lkn(x) =
ex x−k
xn+k e−x , n, k ≥ 0 . (A18)
With the result from Eq. (A18), we can now express the In,m as
In,m =
2/2 eiφ0 (ηx + i ηy)
Lm−nn (η
2) , m ≥ n ≥ 0 , (A19)
For the case when n > m we first integrate over t in Eq. (A9) and follow a similar procedure to find
In,m =
2/2 eiφ0 (−ηx + i ηy)n−m Ln−mm (η2) , n ≥ m ≥ 0 . (A20)
APPENDIX B: RESULT OF TRACE
We can express the trace result for the average of the squared scattering amplitude from from Eq. (44) as a sum of
terms
|M|2 = G
29EE′En1En2
Ai Ti , (B1)
where the coefficients Ai depend on the products of In,m and I
n′,m′ defined in Eq. (47) and presented in appendix A,
and the Ti are the parts that depend on the contraction of the traces in Eq. (44). The results are as follows:
A1 = In1,n2I
n1,n2
T1 = Tr
G±V − γ
m(1− σ3)+ 6p1‖+ 6q1‖γ5
G±V − γ
−m(1− σ3)+ 6p2‖+ 6q2‖γ5
T1 = −27
2 − 1
x + pyp
G±V + 1
(E − pz)(E′ − p′z)(En1 + p1 z)(En2 + p2 z)
G±V − 1
(E + pz)(E
′ + p′z)(En1 − p1 z)(En2 − p2 z) (B3)
A2 = In1,n2−1I
n1,n2
T2 = Tr
G±V − γ
m(1− σ3)+ 6p1‖+ 6q1‖γ5
G±V − γ
2n2eB)
γ1 + iγ2
T2 = −26
G±V + 1
2n2eB (px + i py)(E
′ − p′z)(En1 + p1 z)
G±V − 1
2n2eB (p
x + i p
y)(E + pz)(En1 − p1 z) (B5)
A3 = In1,n2I
n1,n2−1 = A
2 (B6)
T3 = T
2 (B7)
A4 = In1,n2−1I
n1,n2−1 (B8)
T4 = Tr
G±V − γ
m(1− σ3)+ 6p1‖+ 6q1‖γ5
G±V − γ
−m(1 + σ3)+ 6p2‖− 6q2‖γ5
T4 = 2
2 − 1
m2e(E + pz)(E
′ − p′z) + 26
G±V + 1
(E + pz)(E
′ − p′z)(En1 + p1 z)(En2 − p2 z)
G±V − 1
(E + pz)(E
′ − p′z)(En1 − p1 z)(En2 + p2 z) (B9)
A5 = In1,n2I
n1−1,n2 (B10)
T5 = Tr
G±V − γ
2n1eB)
γ1 + iγ2
G±V − γ
−m(1− σ3)+ 6p2‖+ 6q2‖γ5
T5 = 2
G±V + 1
2n1eB (p
x + i p
y)(E − pz)(En2 + p2 z)
G±V − 1
2n1eB (px + i py)(E
′ + p′z)(En2 − p2 z) (B11)
A6 = In1,n2−1I
n1−1,n2 (B12)
T6 = Tr
G±V − γ
2n1eB)
γ1 + iγ2
G±V − γ
2n2eB)
γ1 + iγ2
T6 = −26
G±V + 1
2n1eB
2n2eB (px + i py)(p
x + i p
G±V − 1
2n1eB
2n2eB (px + i py)(p
x + i p
y) (B13)
A7 = In1,n2I
n1−1,n2−1 (B14)
T7 = Tr
G±V − γ
2n1eB)
γ1 + iγ2
G±V − γ
2n2eB)
γ1 − iγ2
T7 = −26
G±V + 1
2n1eB
2n2eB (px − i py)(p′x + i p′y)
G±V − 1
2n1eB
2n2eB (px + i py)(p
x − i p′y) (B15)
A8 = In1,n2−1I
n1−1,n2−1 (B16)
T8 = Tr
G±V − γ
2n1eB)
γ1 + iγ2
G±V − γ
−m(1 + σ3)+ 6p2‖− 6q2‖γ5
T8 = 2
G±V + 1
2n1eB (p
x + i p
y)(E + pz)(En2 − p2 z)
G±V − 1
2n1eB (px + i py)(E
′ + p′z)(En2 + p2 z) (B17)
A9 = In1−1,n2I
n1,n2
= A∗5 (B18)
T9 = T
5 (B19)
A10 = In1−1,n2−1I
n1,n2
= A7 (B20)
T10 = T
7 (B21)
A11 = In1,n2I
n1,n2−1 = A
6 (B22)
T11 = T
6 (B23)
A12 = In1−1,n2−1I
n1,n2−1 = A
8 (B24)
T12 = T
8 (B25)
A13 = In1−1,n2I
n1−1,n2 (B26)
T13 = Tr
G±V − γ
m(1 + σ3)+ 6p1‖− 6q1‖γ5
G±V − γ
−m(1− σ3)+ 6p2‖+ 6q2‖γ5
T13 = 2
2 − 1
m2e(E − pz)(E′ + p′z) + 26
G±V + 1
(E − pz)(E′ + p′z)(En1 − p1 z)(En2 + p2 z)
G±V − 1
(E − pz)(E′ + p′z)(En1 + p1 z)(En2 − p2 z) (B27)
A14 = In1−1,n2−1I
n1−1,n2 (B28)
T14 = Tr
G±V − γ
m(1 + σ3)+ 6p1‖− 6q1‖γ5
G±V − γ
2n2eB)
γ1 + iγ2
T14 = −26
G±V + 1
2n2eB)(px + ipy)(E
′ + p′z)(En1 − p1 z)
G±V − 1
2n2eB)(p
x + ip
y)(E − pz)(En1 + p1 z) (B29)
A15 = In1−1,n2I
n1−1,n2−1 = A
14 (B30)
T15 = T
14 (B31)
A16 = In1−1,n2−1I
n1−1,n2−1 (B32)
T16 = Tr
G±V − γ
m(1 + σ3)+ 6p1‖− 6q1‖γ5
G±V − γ
−m(1 + σ3)+ 6p2‖− 6q2‖γ5
T16 = −27
2 − 1
m2e(pxp
x + pyp
y) + 2
G±V + 1
(E + pz)(E
′ + p′z)(En1 − p1 z)(En2 − p2 z)
G±V − 1
(E − pz)(E′ − p′z)(En1 + p1 z)(En2 + p2 z) . (B33)
	Introduction
	Field operator solutions
	Electron field operator
	Spin sums
	Neutrino field operator
	The production rate
	The scattering matrix
	The form of the production rate
	Results
	Acknowledgments
	References
	Appendices
	Calculation of In,m
	Result of trace
