A. Zee, Quantum Field Theory in a Nutshell · Problem VIII.3.5

Field Theory Redux

Physics — QFT

Problem

Imagine a mythical (and presumably impossible) race of physicists who only understand physics at energies less than the electron mass

m_e

. They manage to write down the effective field theory for the one particle they know, the photon,

\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{m_e^4}\{a(F_{\mu\nu}F^{\mu\nu})^2 + b(F_{\mu\nu}\tilde{F}^{\mu\nu})^2\} + \cdots

with

\tilde{F}^{\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}

the dual field strength as usual and

a

and

b

two dimensionless constants presumably of order unity. (a) Show that

\mathcal{L}

respects charge conjugation (

A \to -A

in this context), parity, and time reversal. (and of course gauge invariance.) (b) Draw the Feynman diagrams that give rise to the two dimension 8 terms shown. The coefficients

a

and

b

were calculated by Euler and Kockel in 1935 and by Heisenberg and Euler in 1936, quite a feat since they did not know about Feynman diagrams and any of the modern quantum field theory set up. (c) Explain why dimensional 6 terms are absent in

\mathcal{L}

. [Hint: One possible term is

\partial_\lambda F_{\mu\nu} \partial^\lambda F^{\mu\nu}

.] (d) Our mythical physicists do not know about the electron, but they are getting excited. They are going to start doing photon-photon scattering experiments with a machine called LPC that could produce photons with energy greater than

m_e

. Discuss what they will see. Apply unitarity and the Cutkosky rules.

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