Weak Effective Hamiltonians and Flavor-Changing Processes

A weak effective Hamiltonian separates short-distance electroweak and QCD physics into CKM factors and Wilson coefficients, and long-distance physics into renormalized operator matrix elements. Matching fixes the coefficients at a heavy scale, RG evolution resums logarithms between scales, threshold matching changes the active theory, and only the complete contraction $C_i(\mu)\langle Q_i(\mu)\rangle$ is basis, scheme, and scale independent. Flavor-changing neutral currents arise first beyond tree level and inherit GIM cancellations from CKM unitarity.

Required background. Quark mixing and the CKM matrix supplies charged-current factors and unitarity; dual evolution of operators and Wilson coefficients supplies the RG construction. Helpful background. Evanescent operators, finite renormalization, and RG closure supplies the dimensional-regularization scheme interface.

Matching a charged-current decay

Use the site convention $\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3$ and define $P_L=(1-\gamma_5)/2$. As a bounded example, consider the quark transition $b\to c\bar u d$ at energies well below $M_W$. A convenient color basis is

$$Q_1=(\bar c_i\gamma_\mu P_Lb_j) (\bar d_j\gamma^\mu P_Lu_i),$$ $$Q_2=(\bar c_i\gamma_\mu P_Lb_i) (\bar d_j\gamma^\mu P_Lu_j),$$

with color indices $i,j$. The effective Hamiltonian is normalized as

$$\mathcal H_{\rm eff}^{b\to c\bar ud} =\frac{4G_F}{\sqrt2}V_{cb}V_{ud}^* \left[C_1(\mu)Q_1(\mu)+C_2(\mu)Q_2(\mu)\right] +\text{h.c.}$$

The factor $4$ accompanies the use of $P_L$ currents; a basis written with $\gamma_\mu(1-\gamma_5)$ absorbs it into the operators. At tree level, color-singlet $W$ exchange gives

$$C_2(\mu_W)=1,\qquad C_1(\mu_W)=0$$

in this labeling, plus corrections in $\alpha_s,\alpha$, and external momenta over $M_W$. The matching equation is an equality of renormalized on-shell or suitably infrared-regulated Green functions in the full and effective theories:

$$\mathcal A_{\rm full} =\mathcal A_{\rm EFT} +O\!\left(\frac{p^2}{M_W^2}\right).$$

The same infrared regulator, external-state convention, and perturbative order must be used on both sides so infrared terms cancel in the difference. The resulting $C_i$ are independent of the later hadronic external state. This current-current example and its QCD corrections are developed in Schwartz 2014, §§ 31.3.1–31.3.3, pp. 657–666.

Operator mixing and leading-log evolution

Declare

$$Q^{\rm bare}=Z(\mu)Q(\mu), \qquad \gamma(\alpha_s)=Z^{-1}\mu\frac{dZ}{d\mu}.$$

Holding the bare operators fixed gives

$$\mu\frac{dQ}{d\mu}=-\gamma Q.$$

Since $\mathcal H_{\rm eff}=C^{\mathsf T}Q$, scale independence requires the dual equation

$$\mu\frac{dC}{d\mu}=\gamma^{\mathsf T}C.$$

Indeed,

$$\mu\frac{d}{d\mu}(C^{\mathsf T}Q) =(\gamma^{\mathsf T}C)^{\mathsf T}Q-C^{\mathsf T}\gamma Q=0.$$

At one loop in QCD, the current-current anomalous-dimension matrix in the displayed $(Q_1,Q_2)$ basis is

$$\gamma^{(0)} = \begin{pmatrix} -6/N_c&6\\ 6&-6/N_c \end{pmatrix}.$$

For $N_c=3$, the combinations

$$Q_\pm=\frac12(Q_2\pm Q_1), \qquad C_\pm=C_2\pm C_1$$

diagonalize the evolution with

$$\gamma_+^{(0)}=4,\qquad \gamma_-^{(0)}=-8.$$

Using

$$\mu\frac{d\alpha_s}{d\mu} =-\frac{\beta_0}{2\pi}\alpha_s^2+\cdots, \qquad \beta_0=11-\frac23n_f,$$

the leading-log solution within an interval of fixed active flavor number is

$$C_\pm(\mu) =\left[\frac{\alpha_s(M)}{\alpha_s(\mu)}\right]^{ \gamma_\pm^{(0)}/(2\beta_0)} C_\pm(M).$$

For evolution downward from $M$ to $\mu$, $\alpha_s(\mu)>\alpha_s(M)$: the $+$ coefficient decreases and the $-$ coefficient grows. This limiting behavior checks both the anomalous-dimension sign and the ratio of couplings. Crossing a heavy-quark threshold requires a new $n_f$, coupling matching, and a finite operator/coefficient matching matrix; simply continuing one power law through every threshold is inconsistent. The general OPE, RG, scheme, and evanescent-operator framework is given in Buchalla, Buras, and Lautenbacher 1996, §§ III.B–III.F.

Scheme and basis cancellation

Individual Wilson coefficients are not observables. Under an invertible finite basis transformation

$$Q'=RQ, \qquad C'=R^{-\mathsf T}C,$$

one has exactly

$$C'^{\mathsf T}\langle Q'\rangle =C^{\mathsf T}\langle Q\rangle.$$

At next-to-leading order a scheme change commonly takes the form

$$Q'=\left(\mathbf1+\frac{\alpha_s}{4\pi}r\right)Q, \qquad C'=\left(\mathbf1-\frac{\alpha_s}{4\pi}r^{\mathsf T}\right)C +O(\alpha_s^2).$$

The contraction is unchanged through $O(\alpha_s)$. The finite matrix $r$ includes choices made for $\gamma_5$ and evanescent operators in $d=4-2\epsilon$. A four-dimensional Fierz identity can differ by an evanescent operator before renormalization; dropping that operator without the compensating finite transformation changes the NLO coefficient and matrix element separately.

Scale cancellation is likewise order by order:

$$\mu\frac{d}{d\mu} \sum_iC_i(\mu)\langle Q_i(\mu)\rangle =O(\text{first omitted order}).$$

Residual $\mu$ variation can diagnose missing perturbative orders only when coefficients and matrix elements are varied consistently in the same scheme. Varying $C_i$ while holding a scheme-converted matrix element fixed creates an artificial uncertainty.

The full decay-amplitude workflow

For a hadron $H$ decaying to a final state $f$,

$$\mathcal A(H\to f) =\langle f|\mathcal H_{\rm eff}|H\rangle =\frac{G_F}{\sqrt2} \sum_{p,i}\lambda_p\,C_i(\mu) \langle f|Q_i(\mu)|H\rangle +\mathcal A_{\rm nonlocal}.$$

Here the normalization of $Q_i$ determines whether an additional factor $4$ appears, $\lambda_p$ denotes CKM products, and $\mathcal A_{\rm nonlocal}$ collects time-ordered products or long-distance propagation not captured by the chosen local basis. The route from theory to a rate is

$$d\Gamma =\frac{1}{2M_H}|\mathcal A|^2\,d\Phi_f,$$

with spin sums, identical-particle factors, and phase-space conventions stated explicitly.

Stage	Output	Required record	Characteristic failure
electroweak matching	$C_i(\mu_W)$ and power remainder	input scheme, masses, gauge checks, operator normalization, perturbative order	importing a coefficient from a differently normalized basis
QCD/QED running	$C_i(\mu)$	anomalous dimension, transpose/sign convention, active flavors, thresholds	evolving coefficients without the mixed operator basis
hadronic calculation	$\langle f\rvert Q_i(\mu)\lvert H\rangle$	same scheme/scale, normalization, continuum/volume or model method, covariance	combining $\overline{\rm MS}$ coefficients with unconverted matrix elements
long-distance assembly	$\mathcal A_{\rm nonlocal}$ and strong phases	intermediate states, subtractions, analyticity, double-counting boundary	treating a short-distance coefficient as the whole amplitude
kinematics and inference	rate or angular/time distribution	phase space, radiative treatment, detector/fit covariance, source identity	reading a fitted CKM factor independently of theory inputs

Strong phases may come from on-shell rescattering or absorptive perturbative pieces; weak phases reside in rephasing-invariant combinations of CKM factors and any additional couplings. Their separation depends on a phase convention, while the complete amplitude and a $CP$ asymmetry do not.

Operator sectors are distinct

Sector	Representative structure	What it organizes
current–current	$(\bar q\gamma_\mu P_Lq)(\bar q\gamma^\mu P_Lq)$	tree-level charged-current decays plus QCD mixing
QCD penguin	four-quark sums with color-singlet/octet contractions	loop-generated $\Delta F=1$ hadronic transitions
electroweak penguin	charge-weighted four-fermion operators	photon/$Z$ short-distance effects
dipole	$m_q\bar q'\sigma_{\mu\nu}P_{R,L}q\,F^{\mu\nu}$ or $G^{a\mu\nu}$	radiative and chromomagnetic transitions
semileptonic	$(\bar q'\Gamma q)(\bar\ell\Gamma'\ell)$	rare or charged-current semileptonic amplitudes
$\Delta F=2$	$(\bar q'\gamma_\mu P_Lq)^2$ and extensions	dispersive neutral-meson mixing

These rows are categories, not one universal basis. Hermitian conjugates, flavor labels, mass factors, and factors of $e$, $g_s$, or $16\pi^2$ vary across conventions. Every quoted coefficient must travel with its exact operator definition.

GIM suppression of neutral flavor change

Tree-level neutral currents are diagonal because the same unitary rotation appears on both sides of the current. At loop level, a typical flavor-changing amplitude contains

$$\mathcal A_{q\to q'} \propto \sum_{i=u,c,t} V_{iq'}^*V_{iq}\, F\!\left(\frac{m_i^2}{M_W^2}\right), \qquad q'\ne q.$$

Column unitarity gives

$$\sum_iV_{iq'}^*V_{iq}=0.$$

Therefore, for any reference flavor $r$,

$$\mathcal A_{q\to q'} \propto \sum_iV_{iq'}^*V_{iq} \left[ F\!\left(\frac{m_i^2}{M_W^2}\right) -F\!\left(\frac{m_r^2}{M_W^2}\right) \right].$$

The amplitude vanishes when the internal masses are degenerate and is controlled by mass splittings otherwise. This is the GIM mechanism. Omitting one internal flavor before using unitarity destroys the cancellation, can leave gauge-dependent terms, and produces the wrong heavy-mass limit.

Independent checks and failure modes

• Dimensions: four-fermion $Q_i$ have dimension six; $G_F$ has dimension $-2$; $C_i$ and CKM products are dimensionless in the displayed normalization.
• Tree matching: setting $\alpha_s\to0$ and $\mu=\mu_W$ must recover $C_2=1,C_1=0$ for the declared basis.
• RG sign: differentiate the leading-log solution and reproduce $\mu\,dC_\pm/d\mu=(\alpha_s/4\pi)\gamma_\pm^{(0)}C_\pm$.
• Scheme round trip: apply $Q'=RQ$ and $C'=R^{-\mathsf T}C$, then recover the same amplitude before comparing numerical coefficients.
• Threshold limit: matching matrices approach the identity at tree level, while the active-flavor beta function changes at the threshold.
• GIM limit: set all internal up-type masses equal; every off-diagonal neutral-current loop amplitude must vanish by CKM unitarity.
• Strong phases: direct $CP$ asymmetry vanishes if either the relative weak phase or relative strong phase is zero, even if individual amplitude terms are complex.
• Evidence boundary: current decay averages, anomalies, fitted coefficients, and lattice numbers require dated covariance and scheme records and are intentionally absent here.

Common pitfalls

Comparing coefficients with the same name but different operators. Factors of $4$, color ordering, quark masses, couplings, and $16\pi^2$ are often absorbed differently. Compare complete $C_iQ_i$ terms.

Running through a threshold without matching. The beta function, operator basis, and Wilson coefficients belong to an EFT with a specified active field set. Change all three together.

Calling a loop coefficient an FCNC observable. A physical rate also needs CKM factors, matrix elements, nonlocal contributions, phase space, and correlated uncertainties.

Informal self-check

Starting from $Q^{\rm bare}=ZQ$ and $\mathcal H=C^{\mathsf T}Q$, derive the coefficient RGE and show that the Hamiltonian is scale independent.

Answer

Differentiating the bare relation gives $\mu\,dQ/d\mu=-Z^{-1}(\mu\,dZ/d\mu)Q=-\gamma Q$. Then $0=\mu\,d(C^{\mathsf T}Q)/d\mu=(\mu\,dC/d\mu)^{\mathsf T}Q-C^{\mathsf T}\gamma Q$, so $\mu\,dC/d\mu=\gamma^{\mathsf T}C$. The cancellation holds for the complete basis and through the perturbative order at which $\gamma$, matching, and matrix elements have been computed.

Handoffs

• Send the $\Delta F=2$ Hamiltonian, coefficient scheme, and hadronic matrix element to neutral-meson propagation.
• Send invariant CKM products and separated weak/strong phases to the unitarity-triangle and $CP$ route.
• Send generic matching, evanescent, and higher-order RG questions back to dual operator/coefficient evolution.
• Send local-current hadronic inputs to Hadron Form Factors and Current Structure or to their versioned nonperturbative calculation, retaining the same scheme and covariance.

References

• Buchalla, Gerhard, Andrzej J. Buras, and Markus E. Lautenbacher. “Weak Decays Beyond Leading Logarithms.” Reviews of Modern Physics 68 (1996): 1125–1244. DOI · Open PDF
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014, § 31.3, pp. 657–666. DOI