Compton Scattering Sum Rules for Massive Vector Bosons

3.1 Decomposition of the Polarized Amplitude
The decomposition for arbitrary j, where j > 1 /2, is
T (ν) = W † [ε · ε ∗ f 0 (ν) (3.4)
+ ν
j∑
n ∈ {N+ 1 2 } f 2n (ν) [S · ε ∗ , S · ε] (S · q) 2n−1
j∑
]
+ ν 2 f 2n (ν) {S · ε ∗ , S · ε} (S · q) 2n−2 W,
n ∈ N
where the f i are the e.m. structure functions. The spin vector S can be constructed
via its relation to the Clebsch-Gordan coefficients [Sch07]. To accomplish this, it is
helpful to construct the (2j + 1) × (2j + 1) polarization matrices
( )
C
(S)
σ
= √ j(j + 1)C(1σ, jλ; jλ ′ ), (3.5)
λ ′ +j+1,λ+j+1
where
C(j 1 m 1 , j 2 m 2 ; jm) ≡ 〈j 1 j 2 m 1 m 2 |j 1 j 2 jm〉 (3.6)
are the Clebsch-Gordan coefficients. The indices run as σ = (−1, 1) and λ = (−s, s).
The components of the spin vector are given by
S 1 = 1 √
2
(C +1 − C −1 ) ,
S 2 =
S 3 = C 0 .
i √
2
(C +1 + C −1 ) ,
(3.7a)
(3.7b)
(3.7c)
It can be easily confirmed that these matrices satisfy the spin algebra su(2), [S k , S l ] =
iε klm S m . They also fulfill the additional properties of a spin operator,
S 2 = j(j + 1) and (S 3 ) λ ′ λ = λ δ λ ′ λ. (3.8)
We choose the 3-axis as the direction of propagation for the photons. The photon
momentum is q = νê 3 . Since we are using circular polarized photons with respect to
the 3-direction, the polarization vectors are defined as
ε = − 1 √
2
(ê 1 + iê 2 ) , (3.9)
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