Stars as Laboratories for Fundamental Physics - MPP Theory Group

136 Chapter 4
4.6 Structure Functions
4.6.1 Formal Definition
To treat axion and neutrino pair emission and absorption on the same
footing it became useful in Sect. 4.3.1 to define the quantity S µν which
was the nuclear part of the squared matrix element of nucleon-nucleon
collisions, integrated over the nucleon phase space. The structure function
thus obtained embodied the medium properties relevant for different
processes without involving the radiation phase space. Clearly, this
method is not limited to the bremsstrahlung process. For example, one
could include the interaction of nucleons with thermal pions or a pion
condensate, three-nucleon collisions, and so forth. Whatever the details
of the medium physics, in the end one will arrive at some function
S µν (ω, k) of the energy and momentum transfer which embodies all of
its properties. High-density properties of the medium should also appear
in the structure function so that multiple-scattering modifications
can be consistently applied to all relevant processes such as neutrino
scattering, pair emission, axion emission, and others.
A formal definition of the structure function without reference to
specific processes begins with a neutral-current interaction Hamiltonian
H int = (g V V µ + g A A µ ) J µ , (4.37)
where g V and g A are (usually dimensionful) coupling constants. The
radiation or “probe” is characterized by a current J µ which for axions
is ∂ µ ϕ, for neutrino interactions ψ ν γ µ (1 − γ 5 )ψ ν , and for photons the
electromagnetic vector potential. The medium is represented by the
vector and axial-vector currents V µ and A µ . If the probe couples only
to one species N of nucleons, V µ = ψ N γ µ ψ N and A µ = ψ N γ µ γ 5 ψ N .
Next, one imagines that the medium properties are experimentally
investigated with a neutrino beam with fixed momentum k 1 which is
directed at a bulk sample of the medium, and the distribution of finalstate
momenta and energies are measured. The transition probability
W (k 1 , k 2 ) is proportional to (gV 2 S µν
V +gAS 2 µν
A +g V g A S µν
V A)N µν where N µν
was defined in Eq. (4.17). A standard perturbative expansion (Sect. 9.3)
yields for the dynamical structure functions
S µν
V (ω, k) = 1 ∫ +∞
n B −∞
dt e iωt ⟨ V µ (t, k)V ν (0, −k) ⟩ , (4.38)
and analogous expressions for S µν
A in terms of ⟨A µ A ν ⟩ and for S µν
V A
involving ⟨V µ A ν + A µ V ν ⟩. The expectation values are to be taken with
respect to a thermal ensemble of medium states.