Compton Scattering Sum Rules for Massive Vector Bosons

4.1 QFT: Yang-Mills Theory
angle [Wei72]. The Weinberg angle is a free parameter of the electroweak theory. In
consequence, it is valid to choose
θ W = 90 ◦ , or sin(θ W ) = 1, (4.7)
so that the two neutral bosons coincide with the fields,
|γ〉 = |W 0 〉 and |Z 0 〉 = |B 0 〉. (4.8)
The mass of the Z 0 is related to the W ± mass via M Z = M W/cos θ W , hence in this case
it diverges, M Z → ∞. Thus, the Z 0 decouples. Furthermore, g ′ → ∞, and from the
definition of the electric charge we find
e ≡
gg ′
√
g2 + g ′ 2
= g. (4.9)
We therefore obtain a Yang-Mills theory containing three bosons, i.e. two (massive)
bosons W and a (massless) photon. Note that the non-zero mass breaks the SU(2)
gauge symmetry down to U(1). One could introduce the mass without breaking
the symmetry explicitly, i.e. through the Higgs mechanism, see e.g. [BD65, PS95].
However, this point is not relevant to our forthcoming discussion. In what follows
we consider the electroweak theory for θ W = 90 ◦ and examine to which extent it
coincides with our previously constructed Lagrangian. It will be seen that the latter is
a truncated YM, as it lacks the boson self-interaction term.
4.1.2 Yang-Mills Theory
In sect. 1.2 we introduced SU(N) Yang-Mills theories. We will now concentrate on
the case N = 2. The generators of this group, in matrix notation, are
T a
ij = − i 2 (τ a) ij
(4.10)
and fulfill the algebra
[
T a , T b] = 1 4 [τ a, τ b ] = if abc T c . (4.11)
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