Yukawa Theory

Yukawa theory couples a scalar field to a fermion bilinear. It shows how spin-zero exchange generates forces and how a scalar can decay into a particle–antiparticle pair. Its simplicity makes it useful for learning amplitudes and renormalization without the gauge constraints of QED.

Scalar–fermion interaction

For one real scalar $\phi$ and one Dirac field $\psi$ in $3+1$ dimensions, take

$$\begin{aligned} \mathcal L={}&\frac12(\partial\phi)^2-V(\phi) +\bar\psi(i\gamma^\mu\partial_\mu-m)\psi\\ &-y\phi\bar\psi\psi . \end{aligned}$$

Expand about a chosen stable vacuum with $\langle\phi\rangle=0$ and $V''(0)=M^2>0$. The coupling $y$ is real and dimensionless. A bounded potential and the counterterms allowed by the actual symmetries complete the model; specifying only the Yukawa vertex is insufficient at loop level. A fermion mass and a scalar Yukawa term do not generally preserve $\phi\mapsto-\phi$ by itself.

A decay threshold and its spin dependence

When $M>2m$, the scalar can decay into $\psi\bar\psi$. For a scalar at rest, the tree-level amplitude is proportional to $y\bar u(p)v(p')$. Summing the two outgoing spin labels gives

$$\sum_{\mathrm{spins}}\lvert\mathcal M\rvert^2 =y^2\operatorname{tr}\bigl[(p\!\!\!/+m)(p'\!\!\!/-m)\bigr] =2y^2(M^2-4m^2).$$

The two-body phase-space factor supplies another square root, yielding

$$\Gamma_{\phi\to\psi\bar\psi} =\frac{y^2M}{8\pi} \left(1-\frac{4m^2}{M^2}\right)^{3/2}.$$

This is for one Dirac species, no additional color multiplicity, and leading order in $y$. Below threshold the on-shell two-body decay closes. The $3/2$ power is characteristic of this scalar bilinear; a pseudoscalar coupling has a different threshold law. The spin sums and phase space follow Srednicki 2007, §§11 and 45–48.

Effective interactions and limits

If the scalar is heavy and external momenta are small compared with $M$, its equation of motion gives $\phi\simeq-y\bar\psi\psi/M^2$. Substitution produces the leading local term

$$\Delta\mathcal L_{\mathrm{eff}} =\frac{y^2}{2M^2}(\bar\psi\psi)^2,$$

with derivative corrections suppressed by further powers of momentum over $M$. This is an effective interaction, not an exact replacement near the scalar pole.

Pole masses, decay rates, and scattering cross sections are the useful observables. Weak-coupling calculations require small loop corrections and controlled logarithms. A scalar expectation value shifts the fermion mass, but neither that shift nor the tree decay calculation establishes the model’s full vacuum stability.

Develop the calculation

Fermion and Yukawa Tree Amplitudes gives the detailed amplitude conventions. Compare Dirac Field for the free limit and Gross–Neveu Model for a lower-dimensional four-fermion theory with different ultraviolet behavior.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§11, 45–48, and 51–52. DOI.