NNLO EW Fermionic Corrections

• NNLO electroweak fermionic corrections are order α² contributions with closed fermion loops that refine theoretical predictions for processes like ZH production and Drell–Yan.
• Semi-numerical two-loop techniques combine Feynman parameters, dispersion relations, and Passarino–Veltman reduction to achieve stable numerical integration and precision.
• Robust UV and IR subtraction along with strategic renormalization schemes significantly reduce theoretical uncertainties, aligning predictions with experimental targets.

Next-to-next-to-leading order (NNLO) electroweak fermionic corrections refer to the contributions at order $\mathcal{O}(\alpha^2)$ in the Standard Model (SM) perturbative expansion to scattering amplitudes and cross sections, involving Feynman diagrams with at least one closed fermion loop in the electroweak sector. These corrections are a crucial component in achieving the precision required for theoretical predictions at future lepton colliders and for benchmark processes such as Drell–Yan production and Higgsstrahlung ($e^+e^- \to ZH$). The evaluation and numerical stability of these corrections require advanced semi-numerical two-loop techniques, control of ultraviolet (UV) and infrared (IR) divergences, and detailed renormalization-scheme studies.

1. Perturbative Expansion and Classification of Corrections

The perturbative expansion of the unpolarized cross section for processes such as $e^+e^- \to ZH$ or Drell–Yan ($e^+e^- \to \ell^+\ell^- / q\bar{q}$) is organized as

$$\sigma = \sigma^{(0)} + \alpha \sigma^{(1)} + \alpha^2 \sigma^{(2)} + \ldots$$

where

• $\sigma^{(0)}$ is the tree-level (Born) cross section,
• $\sigma^{(1)}$ is the full one-loop (next-to-leading order, NLO) electroweak correction (composed of fermionic and bosonic contributions),
• $\sigma^{(2)}$ includes the NNLO contributions.

At the NNLO, “fermionic corrections” are defined as those with at least one closed fermion loop in the two-loop amplitude. These can be further subdivided according to the number of closed loops ($N_f = 1, 2$). For instance, in $e^+e^- \to ZH$, the decomposition is

$$\sigma^{(2)}_\text{fermionic} = \sigma^{(2)}_{N_f=2} + \sigma^{(2)}_{N_f=1}.$$

Typical topologies include reducible diagrams (self-energy insertions), two-loop vertices, box diagrams, and triangle sub-loops, all characterized by at least one (possibly two) closed fermion (lepton or quark) loop(s) (Freitas et al., 2022, Freitas et al., 2023, Freitas et al., 17 Dec 2025).

2. Semi-Numerical Two-Loop Evaluation Techniques

The computation of genuine two-loop electroweak diagrams with fermion loops is performed using semi-numerical strategies that combine analytical and numerical integration. The key steps are:

• Introduction of Feynman parameters for one sub-loop (e.g., $q_2$-loop).
• Use of dispersion relations for the same sub-loop leading to an integral over a spectral parameter $\sigma$,
• Application of Passarino–Veltman reduction for the remaining sub-loop (e.g., $q_1$), and
• Final numerical integration over the relevant Feynman and dispersion parameters.

For a generic two-loop vertex diagram, one encounters integrals of the form: $$\mathcal{I} = \int \frac{d^Dq_2}{i\pi^2} \frac{d^Dq_1}{i\pi^2} \frac{\mathcal{N}(q_1,q_2)}{\prod_j [(q_k+p_j)^2 - m_j^2]}$$ which are reduced, after a Feynman-parameter shift and dispersion for the $q_2$-loop, to a well-controlled integral over a finite region. The method systematically isolates UV subdivergences and global UV divergences through analytic subtraction terms. Tensor reduction and vacuum-integral subtraction further yield fully finite quantities that are then computed numerically with 3–4 digit precision per diagram; in cases of strong cancellations, quadruple precision is employed (Freitas et al., 2022, Freitas et al., 17 Dec 2025).

3. UV and IR Subtraction and Renormalization Schemes

Two-loop fermionic corrections feature nested UV divergences, as well as potential soft-photon IR singularities, necessitating a robust subtraction strategy. Three levels of subtraction are implemented:

• Sub-loop UV subtraction: Achieved by dropping external momenta in the propagators of the divergent sub-loop and treating the divergence analytically, e.g., using $B_0$ or $A_0$ master integrals.
• Global UV subtraction: Remaining divergences are canceled by vacuum diagrams (with all external momenta set to zero).
• Infrared subtraction: IR divergences due to soft internal photons are removed by process-specific eikonal counterterms following the Yennie–Frautschi–Suura scheme, ensuring the two-loop amplitude is both UV- and IR-finite (Freitas et al., 17 Dec 2025).

Renormalization can be performed in the on-shell (OS) mass scheme for all fields, with two main input definitions for $\alpha$:

• The $\alpha(0)$-scheme, using the Thomson limit value of $\alpha$, and requiring inclusion of $\Delta\alpha = 1 - \alpha(m_Z)/\alpha(0)$,
• The $G_\mu$ scheme, using Fermi constant $G_\mu$ from muon decay, absorbing leading universal corrections ($\Delta r$) in the input parameters (Freitas et al., 2023).

4. Numerical Results and Polarization Dependence

At $\sqrt{s}=240$ GeV for $e^+e^- \to ZH$, the numerical evaluation in the $\alpha(0)$ scheme finds: $\begin{array}{lcl} \text{LO} && 222.958\,\text{fb} \ \text{NLO (total)} && 229.893\,\text{fb} \ \text{NNLO (fermionic)} && 231.546\,\text{fb} \ \text{– %%%%27%%%%} && +1.881\,\text{fb} \ \text{– %%%%28%%%%} && -0.226\,\text{fb} \ \end{array}$ resulting in a net shift at NNLO of $+1.655$ fb or approximately $+0.72\%$ relative to the NLO result (Freitas et al., 2022).

For polarized beams:

• $e^+_L e^-_R$: NNLO correction increases cross-section by $+0.76\%$,
• $e^+_R e^-_L$: NNLO correction decreases cross-section by $-0.04\%$ (Freitas et al., 2023).

In neutral-current Drell–Yan-like processes ($e^+e^- \to \mu^+\mu^-$ at $\sqrt{s}=240\,$GeV): $$\begin{array}{lcl} \sigma_{LO} &=& 1.797\,\text{pb} \ \sigma_{LO+NLO} &=& 1.990\,\text{pb} \ \sigma_{LO+NLO+NNLO} &=& 2.010\,\text{pb} \ \end{array}$$ thus the NNLO fermionic correction is $+1\%$ at this energy. For quark final states, effects are $+0.5\%-1\%$ (Freitas et al., 17 Dec 2025). Modifications of differential distributions (e.g., $d\sigma/d\cos\theta$) are of similar relative size, and the shape shows mild forward–backward asymmetries.

5. Renormalization-Scheme Dependence and Theory Uncertainties

The inclusion of NNLO fermionic corrections leads to a notable reduction of renormalization-scheme dependence. For $e^+e^- \to ZH$, the scheme difference between $\alpha(0)$ and $G_\mu$:

• At LO: $\sim7.3\%$,
• At NLO: $\sim1\%$,
• At NNLO (fermionic): $\sim0.5\%$,
• With mixed QCD–EW and NNLO fermionic: $\sim0.05\%$ (Freitas et al., 2023).

Estimates of missing bosonic NNLO two-loop corrections indicate that the associated theory error is below $0.3\%$, dominated by the square of the NLO bosonic amplitude. Thus, the full fermionic NNLO contribution is essential to reach sub-percent-level precision.

6. Phenomenological Impact and Future Directions

Given experimental precision targets of $0.4\%$ (FCC-ee), $0.5\%$ (CEPC), and $1.2\%$ (ILC) for the $ZH$ cross section, the inclusion of NNLO electroweak fermionic corrections is indispensable to match the anticipated measurement uncertainty (Freitas et al., 2022, Freitas et al., 2023). For Drell–Yan and $e^+e^- \to f\bar{f}$ processes, these corrections are similarly essential for FCC-ee, ILC, CEPC physics programs and high-luminosity LHC analyses requiring per-mille accuracy (Freitas et al., 17 Dec 2025).

A plausible implication is that the methods and computational tools developed (dispersive semi-numerical approaches, robust subtraction frameworks) will form the cornerstone for future calculations of complete NNLO electroweak corrections, including the outstanding bosonic two-loop pieces and real emission. This will further reduce theory uncertainties and enable sensitivity to new-physics deviations at future colliders. Moreover, polarization-dependent effects at the per-mille to percent level may provide additional handles to disentangle possible new-physics contributions in electroweak observables (Freitas et al., 2023).