Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 305 / 1

Write $m=|m|e^{i\theta }$. Parity reverses the spatial coordinates and must exchange the two chiral Weyl spinors. One convenient choice isThe identities ${\sigma }^0={\bar{\sigma }}^0$ and ${\sigma }^i=-{\bar{\sigma }}^i$ exchange the two kinetic terms after $\mathbf{x}\mapsto -\mathbf{x}$. The mass bilinear $m{\bar{\psi }}_R{\psi }_L$ maps to its complex conjugate $m^{\ast }{\bar{\psi }}_L{\psi }_R$, so the two mass terms are exchanged and the action is invariant. More generally, independent unit phases $a,b$ may multiply the two transformations provided $b^{\ast }a=m^{\ast }/m$.

Charge conjugation reverses the gauge charge, soUsing the antisymmetric spinor metric $ϵ=i{\sigma }^2$, a compatible action on the two Weyl fields isComplex conjugation reverses the sign of $i$ and of the gauge representation, while $A_{\mu }\mapsto -A_{\mu }$ restores the original gauge covariant derivative. The identities ${\sigma }^2({\sigma }^{\mu }{)}^{\ast }{\sigma }^2={\bar{\sigma }}^{\mu }$ and ${\sigma }^2({\bar{\sigma }}^{\mu }{)}^{\ast }{\sigma }^2={\sigma }^{\mu }$ then exchange the two equations of motion. Unit phases can be inserted in the two spinor transformations without changing the conclusion, subject to the mass-term relation.

Take the most general parity action compatible with the kinetic and mass terms as ${\psi }_L\mapsto a{\psi }_R$, ${\psi }_R\mapsto b{\psi }_L$, where $|a|=|b|=1$ and $b^{\ast }a=m^{\ast }/m$, and let $\varphi \mapsto c_P\varphi$ with $|c_P|=1$. The two holomorphic Yukawa interactions are exchanged whenup to the common sign convention for antisymmetric spinor contraction. The freely chosen intrinsic phases satisfy these equations exactly when

For charge conjugation, write ${\psi }_L\mapsto a_{C}i{\sigma }^2{\psi }_R^{\ast }$, ${\psi }_R\mapsto b_{C}i{\sigma }^2{\psi }_L^{\ast }$ and $\varphi \mapsto c_C{\varphi }^{\ast }$. It exchanges each Yukawa term with the Hermitian conjugate of the opposite-chirality term. The coefficient equations have the form ${\lambda }_2^{\ast }={\lambda }_1{c}_C{a}_C^2$ and ${\lambda }_1^{\ast }={\lambda }_2{c}_C{b}_C^2$; the available intrinsic phases again solve them exactly whenFor conventional choices of all intrinsic phases these statements reduce to phase-sensitive relations such as ${\lambda }_2={\lambda }_1$ under parity and ${\lambda }_2={\lambda }_1^{\ast }$ under charge conjugation; those phases are conventions, while equality of magnitudes is invariant.

In four spacetime dimensions a Weyl spinor has mass dimension $3/2$, so each chiral current $J_L^{\mu }={\bar{\psi }}_L{\bar{\sigma }}^{\mu }{\psi }_L$ or $J_R^{\mu }={\bar{\psi }}_R{\sigma }^{\mu }{\psi }_R$ has dimension three. Since the interaction is $(g{J}_L+g^{\prime }{J}_R{)}^2$, dimensional homogeneity of the Lagrangian givesEquivalently, the expanded four-fermion interaction has coefficients $g^2,g{g}^{\prime },g^{\prime 2}$ of mass dimension minus two.

Parity exchanges $J_L$ and $J_R$. The square is invariant precisely when the two same-chirality coefficients agree, $g^2=g^{\prime 2}$, while the mixed coefficient is already symmetric. Therefore