Compton Scattering Sum Rules for Massive Vector Bosons

2.2 Charged Proca Fields
2.2 Charged Proca Fields
In order to describe a massive vector particle with electric charge, we need two real
spinor fields χ (1) and χ (2) which are described by the Lagrangian
( )
L P χ
(1)
µ , χ (2)
µ , ∂ µ χ (1)
ν , ∂ µ χ (2)
ν
(2.8)
= L 1 + L 2
2∑
( 1
=
4 χ(i) µνχ (i)µν + 1 )
2 M 2 χ (i)
µ χ (i)µ
i=1
where χ (i)
µν := ∂ µ χ (i)
ν − ∂ ν χ (i)
µ .
This can be described equivalently by introducing the complex fields
W µ = √ 1 ( )
χ
(1)
µ + iχ (2)
µ
2
and Wµ ∗ = √ 1 ( )
χ
(1)
µ − iχ (2)
µ
2
(2.9)
which obviously fulfill the Proca equation. One can thus find a Lagrangian
L ′ = L ′ ( W µ , W ∗ µ, ∂ µ W ν , ∂ µ W ∗ ν
)
(2.10)
which is equivalent to the real field Lagrangian L ( χ (1)
µ , χ (2)
µ , ∂ µ χ (1)
ν , ∂ µ χ (2) )
ν .
Proof.
L P = L 1 + L 2
2∑
(
= − 1 4 χ(i) µνχ (i)µν + 1 )
2 M 2 χ (i)
µ χ (i)µ
i=1
= − 1 2 (∂ µW ∗ ν ) (∂ µ W ν ) + 1 2 (∂ µW ∗ ν ) (∂ ν W µ )
(2.11)
+ 1 ( )
∂ν Wµ
∗ (∂ µ W ν ) − 1 ( )
∂ν Wµ
∗ (∂ ν W µ )
2
2
+ M 2
4 (W · W + 2W · W ∗ + W ∗ · W ∗ − W · W + 2W · W ∗ − W ∗ · W ∗ )
= − 1 2 ˜W ∗ µν˜W µν + M 2 W ∗ µW µ .
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