Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 137
In addition, one frequently uses the static structure functions which
are functions of the momentum transfer alone. For example,
S µν
V (k) =
∫ +∞
−∞
dω
2π Sµν V (ω, k) = 1
n B
⟨
V µ (k)V ν (−k) ⟩ . (4.39)
It was used that ∫ +∞
−∞ dω eiωt = 2πδ(t) so that ∫ dt in Eq. (4.38) is
trivially done and yields the operators at equal times. In Eq. (4.39)
V (k) is V (t, k) at an arbitrary time, for example t = 0. It only matters
that both V (k) and V (−k) are taken at equal times.
In an isotropic medium the tensorial composition of the dynamical
structure function can be obtained only from the energy-momentum
transfer K and the four-velocity U of the medium; U = (1, 0, 0, 0) in its
rest frame. The general form of the vector term is (Kirzhnits, Losyakov,
and Chechin 1990)
S µν
V = S 1,V U µ U ν + S 2,V (U µ U ν − g µν )
+ S 3,V K µ K ν + S 4,V (K µ U ν + U µ K ν ). (4.40)
An analogous expression pertains to S µν
A
while the mixed term is
S µν
V A = i S V A ϵ µναβ U α K β (4.41)
because of its transformation properties under parity. The functions
S l,V and S l,A (l = 1, . . . , 4), and S V A depend on medium properties
and on the Lorentz scalars K 2 and U · K that can be constructed from
U and K; the third possibility U 2 = 1 is a constant. Instead of K 2 and
U · K one may use the energy and momentum transfer ω and k = |k|
measured in the medium rest frame.
The structure functions are defined for both positive and negative
energy transfers because the medium can both give or take energy from
a probe. Taking axion emission and absorption as an example, the rate
of change of the occupation number of an axion field mode k is given by
( ) 2
CA n [ B
∂ t f k =
(fk + 1) S µν
A (−ω, k) − f k S µν
A (ω, k) ] K µ K ν .
2f a 2ω
(4.42)
If axions are trapped and reach thermal equilibrium, ∂ t f k = 0 and f k =
(e ω/T − 1) −1 , a Bose-Einstein distribution. This implies the detailedbalance
condition
S µν
A (ω, k) = S µν
A (−ω, k) e ω/T . (4.43)
Recall that a positive energy transfer is energy given to the medium.