The Fermi Limit of Weak Interactions

At momentum transfer $|q^2|\ll m_W^2$, the charged-current $W$ propagator can be expanded and replaced by a local four-fermion interaction. With currents normalized using $P_L=(1-\gamma^5)/2$,

$$\mathcal L_{\rm Fermi} =-\frac{4G_F}{\sqrt2}J_+^\mu J_{-\mu}, \qquad \frac{G_F}{\sqrt2} =\frac{g^2}{8m_W^2} =\frac{1}{2v^2}$$

at tree level. The first relation defines the matching normalization; the last two are tree-level electroweak identities that receive radiative corrections.

Required background. Charged and neutral weak currents supplies $J_\pm^\mu$ and the $g/\sqrt2$ vertex normalization. Tree-level matching by classical elimination supplies the heavy-field elimination procedure.

Helpful background. Effective field theory as a controlled expansion supplies the power counting for omitted terms.

Expanding the charged-vector exchange

Start from

$$\mathcal L\supset m_W^2W_\mu^+W^{-\mu} +\frac{g}{\sqrt2} \left(W_\mu^+J_+^\mu+W_\mu^-J_-^\mu\right) +\text{kinetic terms}.$$

The scalar denominator in the tree propagator obeys

$$\frac{1}{q^2-m_W^2} =-\frac1{m_W^2} \left(1+\frac{q^2}{m_W^2} +O\!\left(\frac{q^4}{m_W^4}\right)\right).$$

Equivalently, solving the $W$ equation of motion to leading order and substituting it back gives

$$\mathcal L_{\rm eff}^{(6)} =-\frac{g^2}{2m_W^2}J_+^\mu J_{-\mu}.$$

Matching this coefficient to $-4G_F/\sqrt2$ yields

$$\frac{G_F}{\sqrt2}=\frac{g^2}{8m_W^2}.$$

Using $m_W=gv/2$ then gives $G_F/\sqrt2=1/(2v^2)$. This normalization and its application to muon decay are derived in Schwartz 2014, §29.4, pp. 602–604.

The longitudinal $q_\mu q_\nu$ part of the vector propagator either vanishes against conserved massless currents or combines with Goldstone exchange and fermion-mass terms as required by gauge invariance. Dropping it without checking the current is not a valid general matching rule.

Example: muon decay

For the leptonic pieces

$$J_+^\mu\supset \bar\nu_{\mu L}\gamma^\mu\mu_L +\bar\nu_{eL}\gamma^\mu e_L,$$

the cross term relevant to $\mu^-\to e^-\bar\nu_e\nu_\mu$ is

$$\mathcal L_{\mu{\rm decay}} =-\frac{4G_F}{\sqrt2} \left(\bar\nu_\mu\gamma^\mu P_L\mu\right) \left(\bar e\gamma_\mu P_L\nu_e\right) +\mathrm{h.c.}$$

Because each $(1-\gamma^5)$ current equals twice a $P_L$ current, the same expression is often written

$$\mathcal L_{\mu{\rm decay}} =-\frac{G_F}{\sqrt2} \left[\bar\nu_\mu\gamma^\mu(1-\gamma^5)\mu\right] \left[\bar e\gamma_\mu(1-\gamma^5)\nu_e\right] +\mathrm{h.c.}$$

The two formulas are identical. A factor-of-four discrepancy usually comes from combining the coefficient of one convention with the currents of the other.

For semileptonic processes the quark current contains the appropriate element of $V_{\rm CKM}$. The short-distance operator coefficient is still universal at this stage, while hadronic matrix elements and QCD running are separate ingredients.

Accuracy and domain of validity

The leading omitted propagator term is suppressed parametrically by

$$\frac{q^2}{m_W^2}.$$

It generates derivative dimension-eight operators, together with mass-suppressed terms associated with nonconserved currents. A controlled low-energy prediction therefore records the characteristic momentum transfer, the retained operator basis, and the perturbative order.

The local expansion fails near the $W$ pole, where $q^2-m_W^2$ is small and the full propagator, its width, and radiative corrections are essential. It also does not by itself account for

• electroweak loop corrections to the relation among $G_F,g,m_W$;
• QED radiation and infrared-safe measurement definitions;
• QCD evolution and hadronic matrix elements for quark currents;
• new light mediators that cannot be integrated out at $m_W$.

Beyond tree level one commonly uses $G_F$ as an input and packages the finite conversion to other electroweak parameters in a quantity such as $\Delta r$. That conversion must be performed in one declared renormalization scheme.

Checks and failure modes

Dimension check. $G_F$ has mass dimension $-2$, matching a dimension-six four-fermion operator.

Chirality check. Every minimal charged weak current contains $P_L$. A scalar, tensor, or right-handed current is a different operator and should receive its own Wilson coefficient.

Sign and normalization check. Compare a physical amplitude using the full $W$ propagator and its low-$q^2$ expansion. Do not infer the sign from the propagator denominator alone while omitting the two vertex factors.

Scale check. If the process samples a broad $q^2$ distribution, bound $q^2/m_W^2$ over the accepted phase space rather than at a single nominal point.

Common pitfalls

Using the Fermi theory at the resonance. The series is an expansion about $q^2/m_W^2=0$. It has no controlled truncation when the heavy propagator is nearly on shell.

Calling $v=(\sqrt2G_F)^{-1/2}$ exact. That equation is the tree matching relation or a convenient input definition. Loop calculations require the corresponding finite correction and tadpole convention.

Confusing universal matching with universal matrix elements. The same weak coefficient multiplies different currents, but QCD dynamics and external-state normalization can make their low-energy matrix elements very different.

Handoff

A reusable low-energy result passes

$$\left\{J_+^\mu J_{-\mu},\ -\frac{4G_F}{\sqrt2},\ \text{projector normalization},\ \mu_{\rm match},\ \frac{q^2}{m_W^2}\text{ bound},\ \text{operator and perturbative order}\right\}.$$

The radiative conversion of $G_F$ belongs to electroweak renormalization and input schemes. Flavor factors and their invariants belong to Quark Flavor and CP.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §29.4, pp. 602–604. DOI.