Logarithmic EW corrections at two-loop

Abstract: We present the implementation of next-to-next-to-leading order (NNLO) electroweak (EW) virtual corrections at next-to-leading logarithmic (NLL) accuracy in the amplitude generator OpenLoops. The implementation covers the automated computation of processes involving massless fermions and transversely polarised vector bosons. For energies above the EW scale, logarithmic EW corrections are strongly enhanced in the tails of kinematic distributions of key LHC processes, reaching several tens of percent at NLO and several percent at NNLO. The two-loop implementation is validated against analytical results from the literature. We present phenomenological results for representative LHC processes and discuss the role of two-loop EW corrections in reducing theoretical uncertainties from missing higher-order contributions.

• The paper introduces an automated framework for computing NNLO two-loop electroweak corrections at NLL accuracy, crucial for high-energy SM processes.
• It employs pseudo-counterterm techniques in amplitude generators to factorize and resum LL and NLL contributions with effective two-point vertices.
• The study quantifies significant impacts on LHC observables, with corrections reaching -35% at NLO and up to +6% at NNLO, influencing precision measurements and new physics searches.

Two-Loop Logarithmic Electroweak Corrections: Formalism, Implementation, and Phenomenology

Motivation and Background

The paper "Logarithmic EW corrections at two-loop" (2604.14320) addresses the calculation and implementation of next-to-next-to-leading order (NNLO) electroweak (EW) virtual corrections at next-to-leading logarithmic (NLL) accuracy. The focus is on automated and process-independent evaluation of logarithmic EW corrections for high-energy Standard Model processes involving massless fermions and transversely polarized vector bosons, a regime where large logarithms $\log(Q^2/M^2)$ (with $Q \gg M_W, M_Z$) dominate radiative effects.

At leading order (LO), EW processes at large $Q$ show little sensitivity to Sudakov logarithms. However, next-to-leading order (NLO) and NNLO corrections are enhanced by powers of these logs, reaching several tens of percent at NLO and several percent at NNLO in the tails of LHC observables. These corrections critically influence the interpretation of precision measurements and backgrounds for new physics searches.

Logarithmic Structure of EW Corrections

Radiative corrections at high energy can be expanded in towers of LL and NLL terms: $$\left(\frac{\alpha}{4\pi}\right)^k \log^n\left(\frac{Q^2}{M^2}\right), \qquad 0 \leq n \leq 2k$$ with $n=2k$ as LL and $n=2k-1$ as NLL. Double logarithms stem from both UV (renormalization group) and IR (soft/collinear emission) origins, allowing for universal factorization and resummation of logarithmic corrections.

Resummation is achieved either via the IREE method, which treats $Q > M$ in a symmetric $SU(2)\otimes U(1)$ phase and $Q < M$ in unbroken $U(1)_{em}$, or via SCET, which organizes soft and collinear effects systematically. Both analytic (diagrammatic) and effective field theory approaches confirm exponentiation of LL and NLL terms.

Two-Loop Logarithmic Expansion: Factorization and Operators

The two-loop EW corrections are written as: $Q \gg M_W, M_Z$0 where $Q \gg M_W, M_Z$1 is angular-independent and $Q \gg M_W, M_Z$2 is angular-dependent.

The angular-independent NLL resummation generates Sudakov form factors for each external leg, exponentiating LL and NLL corrections. Angular-dependent NLL terms arise from wide-angle soft emission between external legs and require non-trivial operators in $Q \gg M_W, M_Z$3 space.

Implementation in Amplitude Generators: OpenLoops

The translation of analytic results into efficient computational tools is realized via pseudo-counterterm techniques, where the insertion of effective two-point vertices on external lines automates the evaluation of $Q \gg M_W, M_Z$4-flipped Born amplitudes needed for both LL and NLL corrections.

Two-loop diagrams, after Ward identity cancellations, reduce to compact expressions involving products of one-loop Sudakov-type integrals and $Q \gg M_W, M_Z$5-function coefficients. This enables a factorized implementation with three classes of topologies—insertions on two, three, or four external legs—using pseudo-counterterms.

Validation against analytic results for representative processes confirms exact agreement at amplitude-squared level for all logarithmic orders, including angular-independent and angular-dependent terms.

Figure 1: Energy scan for $Q \gg M_W, M_Z$6 and $Q \gg M_W, M_Z$7, showing the decomposition and magnitude of two-loop EW corrections.

Numerical Results for LHC Observables

The authors provide a survey of NNLO EW corrections in key LHC processes, comparing one-loop and two-loop contributions for $Q \gg M_W, M_Z$8jets, $Q \gg M_W, M_Z$9jets, and $Q$0 production. Maximum EW corrections reach $Q$1 at NLO and up to $Q$2--$Q$3 at NNLO for transverse-momentum distributions in the TeV regime.

The analysis demonstrates a clear hierarchy in logarithmic terms: LL corrections dominate two-loop EW effects in inclusive transverse momentum distributions, while NLL corrections remain subleading (a factor five smaller). However, for observables violating LA (logarithmic approximation), such as invariant mass distributions with separated scales, angular-dependent NLL terms can overshoot the LL, leading to negative total two-loop corrections as large as $Q$4 in the tails.

Figure 2: Differential distributions in $Q$5 for $Q$6 jet processes, illustrating the relative size of one- and two-loop logarithmic EW corrections.

Figure 3: Two-loop EW corrections to di-jet invariant mass distributions, with breakdown of LL and NLL angular-dependent contributions.

Theoretical and Practical Implications

The process-independent and automated implementation of two-loop EW logarithms streamlines the inclusion of precision EW effects in collider phenomenology, significantly reducing theoretical uncertainties from missing higher-order terms. Accurate treatment of Sudakov logarithms is necessary for extracting signals in high-$Q$7 regions and for assessing backgrounds in new physics searches at current and future colliders.

For processes and observables where all kinematic invariants are of similar scale, the LA—and therefore the log hierarchy—is reliable. However, observables sensitive to forward or unbalanced configurations (e.g., large invariant mass with soft components) require careful treatment as angular-dependent NLL terms can violate the anticipated hierarchy.

Figure 4: Transverse momentum distributions for $Q$8, contrasting the dominance of LL in $Q$9 versus the compensation by NLL in $$\left(\frac{\alpha}{4\pi}\right)^k \log^n\left(\frac{Q^2}{M^2}\right), \qquad 0 \leq n \leq 2k$$0.

Future Directions

The presented implementation supports massless fermions and transversely polarized gauge bosons. Extending the framework to include longitudinal vector bosons, massive external fermions, and scalars will demand careful handling of symmetry breaking, Goldstone equivalence, and additional mixing effects. Further uncertainty quantification is advisable for exclusive LHC observables and QCD+EW matching.

Automation facilitates the study of two-loop EW Sudakov logarithms in exclusive Monte Carlo simulations. As energy scales at colliders rise, the relevance of two-loop EW effects will increase, making such systematic and general tools indispensable for both SM precision and BSM searches.

Conclusion

The paper provides a comprehensive framework for the inclusion of two-loop EW logarithmic corrections at NLL accuracy in amplitude generators. The analytic and computational approach is validated against the literature and enables practical evaluation of these corrections in a wide array of phenomenological applications. Two-loop EW corrections at LL+NLL are several percent in the high-$$\left(\frac{\alpha}{4\pi}\right)^k \log^n\left(\frac{Q^2}{M^2}\right), \qquad 0 \leq n \leq 2k$$1 regime, essential for precision studies at the LHC and future colliders. The methodology establishes a new standard for the systematic treatment of high-order EW effects in exclusive observables, with expected extensions covering longitudinal modes and massive particles in subsequent work.