Compton Scattering Sum Rules for Massive Vector Bosons

2.6 Electromagnetic Moments and Natural Values
which is related to the charge current J(q) by
ρ M (q) ≡ J(q) = ie √ τ
= 2ie √ τ
2j
∑
l=1
l odd
√
(l + 1)(−τ) (l−1) /2 4π l!
2l + 1 (2l − 1)!! G Ml(Q 2 )Y l0 (0) (2.44)
[
G M1 (Q 2 ) − 4 5 τG M3(Q 2 ) + . . .
]
(2.45)
The electromagnetic moments are defined as the low-energy constants of the multipole,
or Sachs form factors at zero momentum transfer, i.e. Q 2 = 0. The l th electric moment
Q l is thus given by
Q l = e M l (l!) 2
while the l th magnetic moment is defined as
2 l G El (0), (2.46)
µ l = e M l (l!) 2
2 l−1 G Ml(0). (2.47)
For j = 1, the most general electromagnetic interaction current is
[
J µ (1) = −W α(p ∗ ′ , λ ′ ) g αβ P µ F 1 (Q 2 ) + (g µβ q α − g µα q β )F 2 (Q 2 )
− qα q β ]
2M P µ F 2 3 (Q 2 ) W β (p, λ).
(2.48)
The interaction is thus described in terms of the independent covariant vertex structures
− g αβ P µ ,
g µβ q α − g µα q β ,
q α q β
2M 2 P µ .
and
(2.49)
Fixing λ = λ ′ = +1, we can obtain the relation between the Sachs form factors and
the form factors F i corresponding to these structures. For the electric moment, the
charge density evaluates to
(
J(1) 0 = 2p 0 F1 (Q 2 ) + τ(F 1 (Q 2 ) − F 2 (Q 2 ) + (1 − τ)F 3 (Q 2 )) sin 2 θ ) (2.50)
with θ the scattering angle, while for the magnetic part, we get
∇ · (J
(1) × q ) = i √ τ2p 0 F 2 (Q 2 )2 √ 4π3Y 10 (0). (2.51)
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