Srednicki — Quantum Field Theory · Problem 63.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · vertex-function-form-factors

Problem

The most general possible form of

\overline{u}'V^\mu(p',p)u

is a linear combination of

\gamma^\mu

p^\mu

, and

p'^\mu

sandwiched between

\overline{u}'

and

u

, with coefficients that depend on

q^2

. (The only other possibility is to include terms with

\gamma_5

, but

\gamma_5

does not appear in the tree-level propagators or vertex, and so it cannot be generated in any Feynman diagram; this is a consequence of parity conservation.) Thus we can write

\overline{u}_{s'}(\mathbf{p}')V^\mu(p',p)u_s(\mathbf{p}) = e\overline{u}'[A(q^2)\gamma^\mu + B(q^2)(p'+p)^\mu + C(q^2)(p'-p)^\mu]u . \tag{63.23}

a) Use gauge invariance to show that

q_\mu \overline{u}'V^\mu(p',p)u = 0

, and determine the consequences for

A

B

, and

C

. b) Express

F_1

and

F_2

in terms of

A

B

, and

C

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