Pseudoscalar Yukawa interaction ($ig\bar{\psi }{\gamma }^5\psi \varphi$)

A pseudoscalar Yukawa interaction couples a Dirac field bilinear containing the chirality matrix to a spin-zero field. In the standard mostly-minus gamma matrix convention with ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and the usual Dirac adjoint, $ig\bar{\psi }{\gamma }^5\psi \varphi$ is Hermitian for real $g$ and real $\varphi$. If $\varphi$ is a pseudoscalar, the interaction also preserves parity symmetry in quantum field theory; if $\varphi$ is parity even, it breaks that symmetry. Moving the factor $i$ into the definition of ${\gamma }^5$ changes the displayed vertex coefficient but not the interaction. Each vertex has one bosonic leg and two oppositely oriented fermion legs, and preserves Dirac fermion number conservation.

For an unbroken parity symmetry in quantum field theory with $\varphi$ a pseudoscalar, pure-scalar local counterterms of mass dimension at most four are even in $\varphi$. A four-vertex fermion loop requires a quartic counterterm, even when the initial Lagrangian density shows only a Yukawa interaction. The one-point and three-point pure-boson vertices vanish at all orders: their available external four-momenta cannot form a parity-odd Lorentz scalar.

With the usual Dirac adjoint, Hermitian ${\gamma }^5$ and $\{{\gamma }^0,{\gamma }^5\}=0$, taking the adjoint gives ${\psi }^{†}{\gamma }^5{\gamma }^0\psi =-{\psi }^{†}{\gamma }^0{\gamma }^5\psi$. Consequently a real pseudoscalar with coupling $-g\varphi \bar{\psi }{\gamma }^5\psi$ has a Hermitian interaction only if $g^{\ast }=-g$. Writing $g=i{g}_P$ with real $g_P$ gives the familiar interaction $-i{g}_P\varphi \bar{\psi }{\gamma }^5\psi$. Formal Euler-Lagrange field equations obtained with independent spinor and adjoint variables can still be computed for the original symbolic coupling; it must not silently be identified with a real physical coupling.