Proc. Neutrino Astrophysics - MPP Theory Group

So far the damping rate corresponds to the zero field case, but the generalization to a
background magnetic field is straightforward after identifying the different factors in Eq. (3).
The δ-functions come from the imaginary part of the usual electron and neutrino propagators
(or rather from the thermal part of the real-time propagators). We now have to use the
electron propagator in an external field instead, and here it is convenient to use Schwinger’s
formulation
i(�P + me)
P 2 − m 2 e + iǫ →
� ∞
ds
0
eieBsσz
cos(eBs) exp
�
is(P 2 tan(eBs)
� +
�
�P� + e−ieBsσz
cos(eBs) �P⊥
�
+ me
eBs P 2 ⊥ − m2 e
�
+ iǫ)
81
, (5)
where a·b � = a0bb −azbz and a·b⊥ = −axbx −ayby. This expression is quite difficult to use in
general but the complicated parts, cos(eBs) and tan(eBs)/eBs, have no linear dependence
on eB and can therefore be approximated by 1 for weak fields. The s-integral can then be
performed and after taking the imaginary part we end up with the replacement rule
(�P + me)δ(P 2 − m 2 e ) → (�P � + me)eBσzsign(P 2 − m 2 e )δ((P2 − m 2 e )2 − (eB) 2 ) . (6)
To obtain this rule I have kept the full eB dependence in the exponentials in Eq. (5) since
they are necessary for the IR convergence of the energy integrals. In the final result, after
performing all integrals, the damping rate has a very clear linear dependence on eB for
weak fields.
To compute the damping from absorption and creation of neutrinos in URCA processes
one can use the same formalism but with neutron and proton propagators in the loop. In the
present case absorption is dominted by ν + n → p + e and creation by the inverse process.
The nucleons are also essentially nonrelativistic which simplifies the matrix elements. On the
other hand, since the general expression comes out for free and the trace is easily evaluated
with a symbolic algebraic program, we might as well keep the full relativistic expression and
all possible channels when we perform the numerical integrals.
Results
The physical conditions in a supernova vary considerably during the short time of the explosion.
I have as an illustration evaluated the damping asymmetry for two more or less typical
conditions. The first case has a high electron chemical potential (µe = 200 MeV) and neutrino
energy q0 = 100 MeV, and the other one has µe = 50 MeV and q0 = 30 MeV. In both cases
I used T = 10 MeV and Ye = 0.3. Local thermal equilibrium was assumed and the proton
chemical potential was determined from charge neutrality. These two cases correspond to
the densities 2×10 14 g/cm 3 and 4×10 12 g/cm 3 . For simplicity the incoming neutrino has its
momentum parallel to the magnetic field (the linear term in eB is of course antisymmetric
in qz). Writing the damping factor as Γ = Γ (0) + eB Γ (1) , I found the relative asymmetry for
URCA and ν-e processes to be (B14 is the field strength in units of 10 14 Gauss)
�
�
�eB
Γ
�
�
(1)
URCA
Γ (0)
�
�
�
�
�
URCA
≃ 3.6×10−6 B14 ,
�
�
�
�
eB Γ
�
�
(1)
ν−e
Γ (0)
�
�
�
�
�
ν−e � ≃ 9.8×10−7 B14 , (7)