Anomalous triple gauge couplings in the light of dimension-8 operators in $W^+W^-$

Abstract: We compare the size of dimension-8 effects on $W<sup>+W<sup>-$ production at the LHC arising from the $q\bar q$ and $γγ$ initial states. In particular, we consider bosonic operators, which contribute to the anomalous triple and quadruple gauge couplings. The relevant dimension-6 and dimension-8 operators are matched to the anomalous triple gauge couplings that contribute to the $q\bar q$ channel, allowing for the resummation of large logarithmic contributions arising in the presence of a jet-veto through the program MCFM-RE. For the $γγ$ channel, which receives contributions from both triple and quadruple gauge couplings, such resummation can be performed through the program MadGraph, by simply setting the factorisation scale to the jet-veto scale. We find that, when neglecting fermionic dimension-8 operators, the $γγ$ channel has a dominant bosonic dimension-8 contribution and this channel can be used to understand the range of validity of the Effective Field Theory. With this in mind, and carefully considering theoretical and experimental uncertainties, we provide constraints on higher dimensional operators using current and future data.

• The paper demonstrates that photon-induced dimension-8 contact terms can dominate at high invariant masses, challenging the EFT truncation at dimension-6.
• It uses SMEFT analysis combining q̄q and γγ channels with NLL jet veto resummation to predict anomalous TGC contributions.
• Numerical results highlight that rigorous EFT validity checks and optimal factorization scale choices are essential for reliable constraints on Wilson coefficients.

Anomalous Triple Gauge Couplings and the Role of Dimension-8 Operators in $W^+W^-$ Production

Overview and Motivation

The production of $W^+W^-$ pairs at the LHC is a critical probe for electroweak gauge structure, sensitive to deviations from Standard Model (SM) gauge couplings due to Beyond Standard Model (BSM) physics. Adopting the framework of the Standard Model Effective Field Theory (SMEFT), this analysis focuses on the interplay between dimension-6 and dimension-8 bosonic operators—specifically, their impact on anomalous triple (aTGC) and quadruple gauge couplings in both $q\bar q$ (quark-antiquark initiated) and $\gamma\gamma$ (photon-photon initiated) channels.

Key motivations include:

• Disentangling the relative importance of dimension-8 effects—especially for operators unique to $\gamma\gamma$, such as direct $\gamma\gamma WW$ contact terms absent at lower operator dimension.
• Assessing the practical validity of the EFT expansion by quantifying the regimes where dimension-8 contributions become sizable relative to dimension-6, potentially invalidating a truncated SMEFT description.
• Providing NLL-accurate predictions for aTGCs in the presence of jet-vetoes, enabled by implementing operator-matching in MCFM-RE and appropriately treating the factorization scale for MadGraph $\gamma\gamma$ simulations.

SMEFT Operator Structure and Mapping to Physical Processes

The EFT analysis identifies four CP-even bosonic dimension-6 operators and twenty dimension-8 operators relevant for diboson production. Dimension-6 operators, in the Warsaw basis, include $\mathcal{O}_{HW}$, $\mathcal{O}_{HB}$, $\mathcal{O}_{HWB}$, and $W^+W^-$0. Of these, all but $W^+W^-$1 contribute to both $W^+W^-$2 and $W^+W^-$3 channels, while $W^+W^-$4 exclusively induces $W^+W^-$5 contributions.

At dimension-8, only a subset directly modifies the $W^+W^-$6 channel; most notably, many induce $W^+W^-$7 four-point vertices that are absent in the SM and dimension-6. This leads to energy-scaling enhancements for $W^+W^-$8 initiated effects at high invariant mass.

The mappings from EFT coefficients to anomalous TGCs and QGCs, following standardized conventions (Lagrangian formalisms in MCFM-RE), enable direct implementation in event generators for both precision SM and BSM predictions. The paper provides explicit conversion formulae, distinguishing unique Lorentz structures for certain operators, notably $W^+W^-$9, which produces a hermitian (rather than imaginary) modification to the TGC vertex.

Impact of Jet Vetoes and Factorization Scale in $q\bar q$0 Initiated Processes

Jets from QCD radiation play a major role in background suppression for $q\bar q$1 production, making jet vetoes an experimental necessity. However, such vetoes introduce large logarithmic corrections in QCD, especially for $q\bar q$2 and $q\bar q$3 initial states, necessitating resummation (at NLL/NNLL) in theoretical predictions. The $q\bar q$4 channel, being colorless, does not radiate gluons and is affected mainly through its PDF evolution. The optimal factorization scale ($q\bar q$5) for $q\bar q$6 is set to the jet-veto scale, efficiently resumming all collinear emissions below this threshold.

Figure 2: Pictorial representation of the PDF factorization and radiation structure in event generator predictions, with hard-process scale set to $q\bar q$7 in the presence of a jet veto.

Comparison of LO and NLO predictions in the $q\bar q$8 channel demonstrates significantly better perturbative convergence when $q\bar q$9 is taken at the jet-veto threshold (see Figure 3). This method robustly incorporates jet-veto effects into photon-induced processes.

Figure 4: Convergence between LO and NLO for the SM photon-induced contribution at HL-LHC (14 TeV), highlighting the importance of setting $\gamma\gamma$0 for improved stability in the presence of jet vetoes.

Suppression effects are channel-dependent: while the gluon-induced contributions are most strongly suppressed by the jet veto (due to largest color charge), the photon channel also experiences noticeable suppression when appropriate account of PDF evolution and factorization is included.

Figure 1: Jet-veto effects on different channels: gluon (red), quark (black), and photon (yellow), for relevant operators; BSM and SM suppression compared.

Numerical Results and Relative Importance of Higher-Dimensional Operators

Dimension-6 Effects

Interference and squared contributions of dimension-6 operators can be sizable in both $\gamma\gamma$1 and $\gamma\gamma$2 channels. Notably, for some operators (e.g., $\gamma\gamma$3), the photon-initiated interference and squared terms can reach up to 60% and 40% (respectively) of the $\gamma\gamma$4 contribution at high $\gamma\gamma$5, far exceeding naive expectations based on SM cross-section ratios.

(Figure 6), (Figure 7)

Figure 3: Comparisons of dimension-6 interference and squared contributions for $\gamma\gamma$6 (left) and $\gamma\gamma$7 (right) channels at $\gamma\gamma$8 TeV, showing competitiveness of the photon channel for certain operators.

Dimension-8 Effects and EFT Validity

Although dimension-8 $\gamma\gamma$9 contributions are parametrically suppressed, the dimension-8 $\gamma\gamma$0-induced operators, particularly contact terms, show a rapid energy scaling. At high invariant masses, the squared contribution from these photon-induced operators can become comparable to or even exceed the dimension-6 $\gamma\gamma$1 squared contributions.

(Figure 8), (Figure 9)

Figure 5: SM interference and squared contributions for bosonic dimension-8 operators in $\gamma\gamma$2 and $\gamma\gamma$3 at $\gamma\gamma$4 TeV; photon-induced contact terms dominate at high $\gamma\gamma$5.

The truncation of EFT at dimension-6 becomes questionable if dimension-8 effects reach similar magnitudes. The analysis quantifies a bin-wise minimum allowed value of $\gamma\gamma$6 ($\gamma\gamma$7) required to ensure a valid EFT expansion in each region of $\gamma\gamma$8.

Figure 6: Minimum EFT validity scale $\gamma\gamma$9 as a function of $\gamma\gamma WW$0 for $\gamma\gamma WW$1 and $\gamma\gamma WW$2 channels, demonstrating the region where dimension-6 dominance is preserved.

Constraints, Sensitivity Studies, and Uncertainties

Current Experimental Constraints

Using ATLAS 2019 data, the study derives exclusion contours on the Wilson coefficients of dimension-6 operators, carefully respecting EFT validity by restricting fits to kinematic regions where dimension-8 effects remain subdominant. The strongest constraints are set on $\gamma\gamma WW$3 and $\gamma\gamma WW$4, while $\gamma\gamma WW$5 and especially $\gamma\gamma WW$6 are much weaker (the latter being only relevant for the subleading $\gamma\gamma WW$7 channel).

Figure 7: 2D sensitivity contours for $\gamma\gamma WW$8 at $\gamma\gamma WW$9 TeV using ATLAS 2019 $\gamma\gamma$0 data with jet veto.

HL-LHC Projections and the Role of Jet Veto

HL-LHC projections highlight that, after imposition of EFT validity, constraints improve only mildly due to large theoretical (QCD and EW) and systematic uncertainties, especially at higher $\gamma\gamma$1. Jet-veto effects do not significantly improve sensitivity for these channels, unlike in $\gamma\gamma$2-initiated processes.

(Figure 12), (Figure 13)

Figure 14: Projected sensitivity at HL-LHC, including interference and squared contributions, systematically analyzed as functions of $\gamma\gamma$3.

Theoretical and Experimental Uncertainties

Electroweak Sudakov logs and missing QCD-EW mixed terms introduce significant uncertainties in high-$\gamma\gamma$4 tails. The difference between multiplicative and additive schemes for combining QCD and EW corrections is used as an estimate, with errors at the $\gamma\gamma$5 level depending on bin and jet-veto presence.

(Figure 15), (Figure 16)

Figure 17: Comparison of predictions with/without EW corrections, and breakdown of QCD and QCD-EW uncertainty contributions across $\gamma\gamma$6 bins.

Systematic uncertainties remain competitive or dominant below $\gamma\gamma$7 TeV in $\gamma\gamma$8. The constraints on Wilson coefficients are thus determined by a complex interplay of available fitting bins, theoretical errors, and the requirements for EFT reliability.

Implications and Future Directions

The analysis demonstrates:

• Neglecting photon-induced dimension-8 effects leads to underestimation of EFT breakdown; in several kinematic regions, $\gamma\gamma$9 contact terms set the lowest valid scale for a reliable SMEFT expansion even though the SM $\mathcal{O}_{HW}$0 background is subdominant.
• ATGC fits for $\mathcal{O}_{HW}$1 production must include both $\mathcal{O}_{HW}$2 and $\mathcal{O}_{HW}$3 channels at dimension-6 and dimension-8 to avoid biased or overly optimistic constraints.
• The need for precision in EW corrections (including Sudakov resummation) is acute as sensitivity at HL-LHC will be ultimately systematics-limited in much of the accessible phase space.
• Validity-driven bin selection is essential: global fits using high-energy bins must explicitly verify the parametric suppression of higher-order EFT terms if SMEFT truncation is to be justified.
• The complementarity of multiple diboson channels and the inclusion of additional SMEFT vertices in event generators are clear avenues for improving global new-physics searches.

Conclusion

This work rigorously quantifies the contributions of bosonic dimension-6 and dimension-8 SMEFT operators to $\mathcal{O}_{HW}$4 production at the LHC, with a comprehensive treatment of photon-fusion effects, jet-veto, and theoretical uncertainties. It provides a blueprint for robust SMEFT analyses, emphasizing the nuanced impact of dimension-8 contact operators and arguing for their systematic inclusion in future ATGC studies. The results underscore the importance of EFT validity checks and motivate enhanced theoretical and experimental precision—particularly for setups aiming to leverage high-mass tails at the HL-LHC.