Compton Scattering Sum Rules for Massive Vector Bosons

3.2 Scattering Kinematics
A generalized parametrization of the 4-momenta is
p = (E, p), p ′ = (E ′ , p ′ ),
q = (ω, q), q ′ = (ω ′ , q ′ ),
(3.20)
where E (′) and ω (′) are the initial-(final-)state target particle and photon energies in a
given frame, respectively. In all frames, the energy-momentum conservation holds:
p µ + q µ = p ′µ + q ′µ . (3.21)
Due to the on-shell condition for external particles, the Mandelstam variables are
constrained by the relation
s + t + u = ∑ i
m 2 i = 2M 2 (3.22)
In general, the differential cross section for unpolarized 2 → 2 scattering is defined as
dσ(s) =
1
8π 2 λ 1 2 (s, m 2 a, m 2 b ) ∫ d 3 p 1
2E 1
d 3 p 2
2E 2
δ (4) (p a + p b − p 1 − p 2 ) |M fi | 2 . (3.23)
Here, |M fi | 2 is the averaged sum over all spin states of the matrix element,
|M fi | 2 := 1 ∑
|M fi | 2 . (3.24)
4j
s i ,r i
The cross section contains the kinematic triangle function, which is defined as
λ ( s, m 2 a, m 2 b) :=
( (√m
2
a +
√ ) (
)
m 2 2
b
s − (√ √ )
)
m 2 a − m 2 2
b
, (3.25)
The relativistically covariant integral over the final state momenta is the N-body phase
space integral which has the form
∫
R 2 (s; m 2 1, m 2 2) := d 4 p 1 d 4 p 2 δ(p 2 1 − m 2 1)δ(p 2 2 − m 2 2)δ (4) (p a + p b − p 1 − p 2 ). (3.26)
After integrating over p 2 and identifying p a ≡ p, p b ≡ q, p 1 ≡ p ′ as above, we obtain
R 2 (s) =
∫ d 3 p
2E δ(s + p′2 − 2(p + q) · p ′ ) (3.27)
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