An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 3.5

The Dirac Field — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · supersymmetry-field-theory

Problem

Supersymmetry. It is possible to write field theories with continuous symmetries linking fermions and bosons; such transformations are called *supersymmetries*. (a) The simplest example of a supersymmetric field theory is the theory of a free complex boson and a free Weyl fermion, written in the form L=∂μϕ∗∂μϕ+χ†iσˉ⋅∂χ+F∗F.\mathcal{L} = \partial_\mu\phi^*\partial^\mu\phi + \chi^\dagger i\bar{\sigma}\cdot\partial\chi + F^*F. Here FF is an auxiliary complex scalar field whose field equation is F=0F = 0. Show that this Lagrangian is invariant (up to a total divergence) under the infinitesimal transformation δϕ=−iϵTσ2χ,\delta\phi = -i\epsilon^T\sigma^2\chi, δχ=ϵF+σ⋅∂ϕ σ2ϵ∗,\delta\chi = \epsilon F + \sigma\cdot\partial\phi\,\sigma^2\epsilon^*, δF=−iϵ†σˉ⋅∂χ,\delta F = -i\epsilon^\dagger\bar{\sigma}\cdot\partial\chi, where the parameter ϵa\epsilon_a is a 2-component spinor of Grassmann numbers. (b) Show that the term ΔL=[mϕF+12imχTσ2χ]+(complex conjugate)\Delta\mathcal{L} = \left[m\phi F + \frac{1}{2}im\chi^T\sigma^2\chi\right] + \text{(complex conjugate)} is also left invariant by the transformations of part (a). Add this term to the Lagrangian and find the field equation for FF. Show that, when FF is eliminated by this equation, the resulting Lagrangian gives equal masses to the boson and the fermion.

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