Standard Model and Beyond · Problem 2.30

Lie Groups, Lie Algebras, and Symmetries

Physics — Particle Physics · electron-vertex-non-relativistic-edm

Problem

Consider the non-relativistic limit of the matrix element (2.356) for an

e^-

in a static external field. (a) Compute the limit to linear order in the momenta. Hint: use the explicit forms for the spinors in the Pauli-Dirac representation. It simplifies the calculation to rewrite

\bar{u}_2\Gamma^\mu u_1

using the Gordon decomposition. (b) Suppose that

\Gamma^\mu(p_2,p_1)

in (2.352) contained a term

\frac{i\sigma^{\mu\nu}}{2m}q_\nu\gamma^5 G_2(q^2)

. This violates

P

and

T

but in principle could be generated by a new interaction. Show how

G_2(0)

is related to the electric dipole moment

\vec{d}_e

of the electron, which is defined by the non-relativistic interaction

H_{EDM} = -\vec{d}_e\cdot\vec{E}(\vec{x})

, where

\vec{E}

is an external electric field. Note that the Hermiticity condition (2.354) requires that

G_2

is pure imaginary.

Related theory

Dirac Equation Relativistic Electron

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