Electroweak Renormalization and Input Schemes

Beyond tree level, $g,g',v,\lambda$, the weak angle, and the vector-boson masses cannot all be chosen independently. A calculation must select a complete input set, define masses and tadpoles, solve for the dependent parameters, and use the corresponding counterterms everywhere. Different consistent schemes predict the same physical observable up to terms beyond the retained order; mixing their finite parts generally double counts or omits corrections.

Required background. The Fermi limit of weak interactions supplies the tree relation between $G_F$ and $v$. Renormalized perturbation theory supplies fields, counterterms, and order-by-order ultraviolet cancellation.

Helpful background. Renormalization conditions, schemes, and finite parts compares subtraction prescriptions. Unstable-particle observables and resonance approximations supplies the pole definitions required for $W$, $Z$, and Higgs states.

From bare parameters to an input map

With $Q=T_3+Y$, $H\sim(\mathbf1,\mathbf2)_{1/2}$, and

$$D_\mu H= \left(\partial_\mu-igT^aW_\mu^a-ig'\frac12B_\mu\right)H,$$

the bare Lagrangian may be parameterized by $g_0,g'_0,\mu_0^2,\lambda_0$ together with the Yukawa and QCD sectors. Perturbation theory introduces, for example,

$$\begin{aligned} g_0&=g+\delta g,& g'_0&=g'+\delta g',& v_0&=v+\delta v,\\ M_{V,0}^2&=M_V^2+\delta M_V^2,& V_{0\mu}&=\left(1+\frac12\delta Z_V\right)V_\mu,& t_0&=t+\delta t . \end{aligned}$$

Here $t$ is the coefficient of the Higgs one-point function. The ultraviolet parts are fixed by finiteness, but finite parts require renormalization conditions. Denner gives the complete one-loop on-shell organization, including field mixing and electroweak counterterms, in Denner 1993, §§3–4, pp. 334–364.

An input scheme is an invertible map

$$\{\text{measured or declared inputs}\} \longrightarrow \{g,g',v,\lambda,Y_f,\ldots\}_{\mathcal S}$$

at a specified perturbative order. The map must be applied before evaluating dependent vertices. Three common organizations illustrate the choices.

Scheme family	Representative inputs	Tree map	Essential higher-order statement
On-shell electromagnetic	$\alpha(0),M_W,M_Z$	$e^2=4\pi\alpha(0)$, $s_{\rm OS}^2=1-M_W^2/M_Z^2$, $g=e/s_{\rm OS}$, $v=2M_W/g$	define mass poles, electric charge, residues, and mixing counterterms consistently
$G_F$-based	$\alpha(0),G_F,M_Z$ or another complete set	solve the $G_F$ relation together with $s_{\rm OS}^2=1-M_W^2/M_Z^2$	include the finite conversion conventionally summarized by $\Delta r$
Short-distance	running $\widehat g(\mu),\widehat g'(\mu)$, or equivalent $\widehat\alpha(\mu),\widehat s^2(\mu)$, plus masses or $G_F$	$\widehat s^2=\widehat g'^2/(\widehat g^2+\widehat g'^2)$	state subtraction scheme, scale, active fields, thresholds, and pole conversion

The table gives families, not interchangeable rows. For example, taking both $M_W$ and $G_F$ as independent inputs while also imposing their tree relation overconstrains the same parameter set unless an additional quantity is being fitted or tested.

On-shell angle and the Fermi-constant conversion

In an on-shell mass convention,

$$s_{\rm OS}^2=1-\frac{M_W^2}{M_Z^2}, \qquad c_{\rm OS}^2=\frac{M_W^2}{M_Z^2}.$$

Differentiation fixes the weak-angle counterterm:

$$\frac{\delta s_{\rm OS}^2}{s_{\rm OS}^2} =\frac{c_{\rm OS}^2}{s_{\rm OS}^2} \left( \frac{\delta M_Z^2}{M_Z^2} -\frac{\delta M_W^2}{M_W^2} \right).$$

It is not an independent counterterm in this scheme. In a short-distance scheme, by contrast, the running angle is defined through the renormalized couplings and receives its own subtraction prescription.

Muon decay supplies a second useful map. With the convention used here,

$$\frac{G_F}{\sqrt2} =\frac{\pi\alpha(0)} {2s_{\rm OS}^2M_W^2} \left(1+\Delta r\right)$$

through one loop. Some references instead define the exact rearrangement with $1/(1-\Delta r)$; the two agree through first order but differ in which higher-order terms are implicitly resummed. A calculation must state the convention. When $\{\alpha(0),G_F,M_Z\}$ are inputs, this equation and $s_{\rm OS}^2=1-M_W^2/M_Z^2$ determine $M_W$ order by order. One must not then reinsert an independently fitted $M_W$ without redefining the input problem.

The finite quantity $\Delta r$ contains self-energy, vertex, box, charge-renormalization, and related contributions appropriate to the inclusive muon-decay definition. It is not a universal multiplier for an arbitrary weak process.

Tadpoles and the renormalized vacuum

The tree statement $v^2=\mu^2/\lambda$ does not define a unique loop-level vacuum parameter. Two internally consistent organizations are common:

1. impose that the renormalized Higgs one-point function vanishes, fixing a tadpole counterterm;
2. keep the bare vacuum at the bare-potential minimum and retain explicit tadpole contributions in parameter counterterms.

Fleischer and Jegerlehner develop the second organization and show how it avoids attaching spurious gauge dependence to parameter definitions through a shifted vacuum Fleischer and Jegerlehner 1981, §§II–III, pp. 2004–2011. Physical amplitudes agree when all terms are translated consistently. Taking a mass counterterm from one prescription and omitting the tadpole graphs according to the other is not such a translation.

A useful calculation record therefore states

$$\left\{\text{input set},\ \text{subtraction scheme and scale},\ \text{tadpole prescription},\ \text{mass definition},\ \text{perturbative order}\right\}$$

before presenting dependent couplings.

Complex poles and unstable electroweak states

For an unstable field, the gauge-invariant resonance location is the complex pole

$$s_p=\mu_p^2-i\mu_p\gamma_p$$

on the appropriate analytically continued sheet. A real on-shell condition, a running-width line-shape parameter, and $(\mu_p,\gamma_p)$ need not coincide beyond leading order. Stuart shows how analyticity around the pole organizes gauge-invariant resonance parameters and residues Stuart 1991, pp. 113–119.

If pole masses enter $s_{\rm OS}^2=1-\mu_W^2/\mu_Z^2$, say so. If a line-shape convention is used instead, provide the conversion at the same order. The same rule applies to a Higgs mass used in $\lambda= m_h^2/(2v^2)$: inserting a pole mass into a counterterm derived for another convention leaves a finite mismatch.

Validation and failure modes

Ultraviolet check. Poles in the regulator must cancel after all counterterms and diagrams of the declared order are included. A finite answer obtained by dropping a divergent subset is not a renormalized prediction.

Gauge-parameter check. Physical pole locations and complete observables must be independent of the gauge-fixing parameters through the calculated order. Intermediate self-energies, tadpoles, and off-shell vertices need not be.

Input reconstruction check. Recompute every chosen input from the solved parameters. The residual must begin beyond the retained order.

Scheme-translation check. Convert both inputs and parameters before comparing two schemes. Their physical predictions should differ only by uncalculated higher orders; the residual difference can be an uncertainty diagnostic, not an exact error bar.

Double-counting check. Running of $\alpha$, a $G_F$-based normalization, and an explicit $\Delta r$ correction can contain overlapping vacuum-polarization or weak effects. Trace which finite terms have already been absorbed into the inputs.

Common pitfalls

Calling $s_W^2$ a unique observable. On-shell, short-distance, and process-effective weak angles are different renormalized quantities. Always attach the definition.

Changing one input inside a completed amplitude. Replacing $\alpha(0)$ by a running coupling without transforming counterterms and finite terms is not a scheme conversion.

Hiding a tadpole prescription. Gauge cancellations can fail even when every displayed self-energy looks finite. State how the vacuum and one-point function were renormalized.

Handoff

A prediction-ready electroweak parameter set passes

$$\left\{\mathcal S,\mu;\ I_{\mathcal S};\ \theta_{\mathcal S}(I_{\mathcal S});\ \delta\theta_{\mathcal S};\ \text{tadpoles};\ \{s_p,Z_p\};\ \text{order and residual scheme test}\right\},$$

where $I_{\mathcal S}$ is the independent input set and $\theta_{\mathcal S}$ the solved Lagrangian parameters. This is the required entry point for Higgs production, decay, and pole observables and for Precision Standard Model.

References

• Denner, Ansgar. “Techniques for the Calculation of Electroweak Radiative Corrections at the One-Loop Level and Results for W-Physics at LEP 200.” Fortschritte der Physik 41, no. 4 (1993): 307–420. DOI. Open PDF.
• Fleischer, J., and F. Jegerlehner. “Radiative Corrections to Higgs Decays in the Extended Weinberg–Salam Model.” Physical Review D 23, no. 9 (1981): 2001–2026. DOI.
• Stuart, Robin G. “Gauge Invariance, Analyticity and Physical Observables at the $Z^0$ Resonance.” Physics Letters B 262, no. 1 (1991): 113–119. DOI.