Compton Scattering Sum Rules for Massive Vector Bosons

3.4 Optical Theorem
3.4 Optical Theorem
The optical theorem relates the absorptive part of the amplitude, Abs T , to the total
photoabsorption cross section,
σ(ν) = 4π ν
Abs T (ν) =
e2
2Mν ε∗ µε ν Abs T µν . (3.51)
A derivation of the optical theorem can be found in [Pan98]. Using the decompositions
in sect. 3.1, we obtain for the polarized forward Compton scattering amplitude, via
eq. (3.15) and (3.18a), the optical theorems for j = 1 /2 and j = 1, respectively:
Optical Theorem for Spin 1 /2
Imf 0 (ν) = ν (
8π
(
Imf 1 (ν) = 1
8π
σ 1
2
σ 1
2
(ν) + σ 3
2
)
(ν)
)
(ν) − σ 3 (ν)
2
Optical Theorem for Massive Vector Bosons
, (3.52a)
. (3.52b)
Imf 0 (ν) = ν
8π (σ +1(ν) − σ 0 (ν) + σ −1 (ν)), (3.53a)
} {{ }
=:σ T (ν)
Imf 1 (ν) = 1
8π (σ −1(ν) − σ +1 (ν)), (3.53b)
} {{ }
=:∆σ(ν)
Imf 2 (ν) = 1
16πν (2σ 0(ν) − (σ −1 (ν) + σ +1 (ν))) . (3.53c)
} {{ }
=:σ Q (ν)
3.5 Derivation of Forward Dispersion Relations
We have now laid the foundation to be able to derive sum rules for the electromagnetic
moments from our theory: We have expanded our general Compton scattering amplitude
in terms of the target helicity λ (c.f. section 3.1, eqs. (3.4) and (3.18a)). We
have then constructed a gauge invariant phenomenological effective Lagrangian L Eff
to describe our interactions (section 2.3) and have derived Feynman rules from it
(section 2.5). Based on this, we have derived the low-energy theorems (LETs) for the
structure functions from L Eff , including a new quadrupole LET.
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