Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 10.5

Spinors — Schwartz M.D. z-lib.org

Physics — QFT · supersymmetry

Problem

Supersymmetry. (a) Show that the Lagrangian L=∂μϕ∗∂μϕ+χ†iσˉ∂χ+F∗F+mϕF+i2mχTσ2χ+h.c.(10.143)\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi + \chi^\dagger i \bar{\sigma}\partial\chi + F^* F + m\phi F + \frac{i}{2}m\chi^T \sigma^2 \chi + \text{h.c.} \tag{10.143} is invariant under δϕ=−iϵTσ2χ,δχ=ϵF+σμ∂μϕσ2ϵ∗,δF=−iϵ†σˉμ∂μχ,(10.144)\begin{aligned} \delta\phi &= -i\epsilon^T \sigma^2 \chi, \tag{10.144}\\ \delta\chi &= \epsilon F + \sigma^\mu \partial_\mu \phi \sigma^2 \epsilon^*, \\ \delta F &= -i\epsilon^\dagger \bar{\sigma}^\mu \partial_\mu \chi, \end{aligned} where ϵ\epsilon is an infinitesimal spinor, χ\chi is a spinor, and FF and ϕ\phi are scalars. All spinors anticommute. σ2\sigma^2 is the second Pauli spin matrix. (b) The field FF is an auxiliary field, since it has no kinetic term. A useful trick for dealing with auxiliary fields is to solve their equations of motion exactly and plug the result back into the Lagrangian. This is called integrating out a field. Integrate out FF to show that ϕ\phi and χ\chi have the same mass. (c) Auxiliary fields such as FF act like Lagrange multipliers. One reason to keep the auxiliary fields in the Lagrangian is because they make symmetry transformations exact at the level of the Lagrangian. After the field has been integrated out, the symmetries are only guaranteed to hold if you use the equations of motion. Still using δϕ=iϵTσ2χ\delta\phi = i\epsilon^T \sigma^2 \chi, what is the transformation of χ\chi that makes the Lagrangian in (b) invariant, if you are allowed to use the equations of motion?

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