Srednicki — Quantum Field Theory · Problem 49.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · feynman-rules-majorana-fields-selectron

Problem

Let

\Psi

be a Dirac field (representing the electron and positron),

X

be a Majorana field (representing the photino, the hypothetical supersymmetric partner of the photon, with mass

m_{\tilde{\gamma}}

), and

E_{\mathrm{L}}

and

E_{\mathrm{R}}

be two different complex scalar fields (representing the two selectrons, the hypothetical supersymmetric partners of the left-handed electron and the right-handed electron, with masses

M_{\mathrm{L}}

and

M_{\mathrm{R}}

; note that the subscripts L and R are just part of their names, and do not signify anything about their Lorentz transformation properties). They interact via

\mathcal{L}_1 = \sqrt{2}\,e\, E_{\mathrm{L}}^\dagger \overline{X} P_{\mathrm{L}} \Psi + \sqrt{2}\,e\, E_{\mathrm{R}}^\dagger \overline{X} P_{\mathrm{R}} \Psi + \mathrm{h.c.} \;, \tag{49.9}

where

\alpha = e^2/4\pi \simeq 1/137

is the fine-structure constant, and

P_{\mathrm{L,R}} = \frac{1}{2}(1\mp\gamma_5)

. a) Write down the hermitian conjugate term explicitly. b) Find the tree-level scattering amplitude for

e^+e^- \to \tilde{\gamma}\tilde{\gamma}

. Hint: there are four contributing diagrams, two each in the

t

and

u

channels, with exchange of either

E_{\mathrm{L}}

E_{\mathrm{R}}

. c) Compute the spin-averaged differential cross section for this process in the case that

m_e

(the electron mass) can be neglected, and

|t|, |u| \ll M_{\mathrm{L}} = M_{\mathrm{R}}

. Express it as a function of

s

and the center-of-mass scattering angle

\theta

Related theory

Color Ordered Feynman Rules And N 1 Component Fields

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