Compton Scattering Sum Rules for Massive Vector Bosons

2.5 Feynman Rules for L Eff
• W µν transforms covariantly (c.f. section 1.2), hence the quadratic term in the
field strength tensor is invariant;
• F µν is invariant by construction;
• ( W ∗ µW µ) ′
= e −iα(x) W ∗ µe iα(x) W µ = W ∗ µW µ since W and e ±iα(x) commute;
• ( W ∗ µνW µ) ′
= e −iα(x) W ∗ µνe iα(x) W µ = W ∗ µνW µ .
As all appearing terms are gauge invariant, so are linear combinations. It follows that
L Eff is invariant under U(1) gauge transformations.
2.5 Feynman Rules for L Eff
In order to describe interactions with the newly-constructed effective Lagrangian L Eff ,
we need to derive the corresponding Feynman rules. In this theory, two types of
vertices appear: the 3-point vertex γW W and the 4-point contact vertex γγW W . In
order to recover the correct Feynman rules, the states have to be contracted with the
fields coming from the Lagrangian, or more precise, the action (c.f. section 1.3). For a
quick overview over the resulting Feynman rules, see App. A.1.
A Taylor expansion of the S-Matrix yields the interaction part
∫
S = 〈f|T exp (i
= ✘ 〈f|1|i〉 ✘ ✘ ✘✿0
dxL Eff )|i〉 (2.26)
∫
+ i d 4 x〈f|L Eff |i〉
which corresponds to a Feynman rule describing the transition from the initial state |i〉
to the final state |f〉 ≠ |i〉. The contraction is evaluated using the plane-wave expansions
which can be found in App. A.2.1. There, we have also derived the contraction relations.
In the following we will neglect the integration ∫ d 4 x; however, it is still implied in
this symbolic notation.
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