Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 119
teracting with a hot nuclear medium, even though these issues are of
paramount importance for a proper quantitative understanding of SN
physics where the interaction of neutrinos with the medium dominates
the thermal and dynamical evolution. My discussion can only be a
starting point for future work that may actually yield some answers to
the questions raised.
4.2 Axionic Bremsstrahlung Process
4.2.1 Matrix Element for NN → NNa
The simplest neutral-current process of the kind to be discussed in this
chapter is bremsstrahlung emission of axions or other pseudoscalars
(Fig. 4.1) because the single-particle axion phase space is particularly
simple. It will turn out that the result thus derived can be applied
to neutrino processes almost without modification. The interaction
Hamiltonian with nucleons is of the form
H int = − C N
2f a
ψ N γ µ γ 5 ψ N ∂ µ ϕ, (4.1)
where f a is an energy scale (the Peccei-Quinn scale for axions), C N with
N = n or p is a dimensionless, model-dependent coupling constant of
order unity, the ψ N are the proton and nucleon Dirac fields, and ϕ is
the axion field or any other pseudoscalar Nambu-Goldstone boson.
Consider a single species of nonrelativistic nucleons interacting by a
one-pion exchange (OPE) potential. The spin-summed squared matrix
element is (Brinkmann and Turner 1988; Raffelt and Seckel 1995)
⎡
∑
|M| 2 = 16 (4π)3 απα 2 (
a k
⎣
2 ) 2 (
l 2 ) 2
+
spins
3m 2 N k 2 + m 2 π l 2 + m 2 π
+ k2 l 2 − 3 (k · l) 2 ]
. (4.2)
(k 2 + m 2 π)(l 2 + m 2 π)
Here, α a ≡ (C N m N /f a ) 2 /4π and α π ≡ (f 2m N /m π ) 2 /4π ≈ 15 with f ≈
1 are the axion-nucleon and pion-nucleon “fine-structure constants,”
respectively. Further, k = p 2 −p 4 and l = p 2 −p 3 with p i the momenta
of the nucleons N i as in Fig. 4.1.
In a thermal medium k 2 ≈ 3m N T so that [k 2 /(k 2 + m 2 π)] 2 varies
between 0.86 for T = 80 MeV and 0.37 for T = 10 MeV. Therefore,
neglecting the pion mass causes only a moderate error in a SN core
(Brinkmann and Turner 1988; Burrows, Ressell, and Turner 1990).