arxiv: 2604.16599 · v1 · submitted 2026-04-17 · ✦ hep-ph

Recognition: unknown

The effect of the two-loop SMEFT RGEs at future colliders

Luca Mantani , Pablo Olgoso , Alejo N. Rossia

Authors on Pith no claims yet

Pith reviewed 2026-05-10 07:53 UTC · model grok-4.3

classification ✦ hep-ph

keywords SMEFTtwo-loop RGEsWilson coefficientsHL-LHCFCC-eefour-quark operatorstop YukawaHiggs-gluon operators

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The pith

Two-loop RGE contributions produce non-negligible shifts in SMEFT fits for four-quark, top Yukawa and Higgs-gluon operators at future colliders.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the two-loop renormalization group equations for SMEFT Wilson coefficients lead to observable changes in how these coefficients evolve from high scales down to collider energies. A sympathetic reader would care because these higher-order effects modify the predicted signals of new physics in precision measurements planned for the High-Luminosity LHC and the FCC-ee. The authors solve the equations with full numerical integration, compare the resulting evolution matrices to the one-loop case, and perform both individual and global fits at linear and quadratic order for 61 coefficients. They identify non-negligible impacts in particular for four-quark operators, the top Yukawa, and Higgs-gluon interactions, while top-down fits to scalar and fermion extensions of the Standard Model show percent-level changes in the sensitivity to their couplings.

Core claim

By solving the two-loop SMEFT RGEs numerically, the authors show that two-loop contributions alter mixing patterns and break zeroes that exist at one-loop order. In bottom-up phenomenological studies at HL-LHC and FCC-ee, this produces non-negligible effects for four-quark, top Yukawa and Higgs-gluon operators. In top-down fits to all scalar and fermion extensions of the Granada dictionary matched onto SMEFT at one-loop level, including couplings that enter only at one loop, the sensitivity to the model couplings changes at the percent level.

What carries the argument

The full numerical integration of the two-loop renormalization group equations for the Wilson coefficients of dimension-six SMEFT operators, which yields an evolution matrix that includes additional mixing terms absent at one loop.

If this is right

Two-loop RGEs break some of the zeroes in the one-loop mixing matrix, allowing new operator mixing channels.
Global and individual fits at HL-LHC and FCC-ee shift for four-quark operators, the top Yukawa, and Higgs-gluon interactions when two-loop running is used.
Quadratic-level fits exhibit different behavior once two-loop contributions are retained.
Sensitivity to the couplings of scalar and fermion UV extensions changes by a few percent when one-loop matching is combined with two-loop running.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Current one-loop-only SMEFT analyses for other processes may require re-evaluation once high-precision data arrive.
The breaking of one-loop zeroes opens new ways to constrain previously decoupled operators through running alone.
Extending the top-down study to two-loop matching of the same UV models would likely enlarge the observed corrections in some cases.

Load-bearing premise

Higher-order corrections to the matching of UV models onto SMEFT remain negligible compared with the two-loop RGE effects themselves.

What would settle it

A direct measurement at HL-LHC or FCC-ee showing that the extracted bounds on four-quark, top Yukawa or Higgs-gluon Wilson coefficients remain unchanged at the percent level when two-loop RGE evolution is included versus one-loop only.

Figures

Figures reproduced from arXiv: 2604.16599 by Alejo N. Rossia, Luca Mantani, Pablo Olgoso.

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗

read the original abstract

The search for New Physics requires ever increasing precision from experimental and theoretical efforts. Within the Standard Model Effective Field Theory (SMEFT) framework, the latest achievement in this quest has been the complete computation of the two-loop Renormalisation Group Equations (RGEs) for the Wilson Coefficients of dimension-six operators. In this work, we solve the two-loop SMEFT RGEs with full numerical integration and compare the evolution matrix obtained at one and two loops to analyze how two-loop contributions alter mixing patterns and break zeroes present at one-loop order. Then, we perform a first comprehensive analysis of the impact of the two-loop RGEs in phenomenological studies at HL-LHC and FCC-ee. From a bottom-up perspective, we carry out individual and global fits at linear and quadratic level for a set of 61 Wilson coefficients and compare against the results obtained by including only one-loop RGE effects. We find non-negligible two-loop induced effects in some cases, in particular for four-quark, top Yukawa and Higgs-gluon operators. From a top-down perspective, we perform fits to all the scalar and fermion extensions of the Granada dictionary matched onto SMEFT at one-loop level, including for the first time the couplings that enter only at one loop, and find percent-level effects in the sensitivity to the couplings of some models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Two-loop RGEs shift some SMEFT fits at HL-LHC and FCC-ee by noticeable amounts, but the top-down UV matching section cannot cleanly isolate running from matching order.

read the letter

Two-loop SMEFT running produces non-negligible shifts for four-quark, top Yukawa, and Higgs-gluon operators in projected collider fits, and this paper is the first to run a full set of 61 Wilson coefficients through both one- and two-loop RGEs in global linear and quadratic fits plus top-down matching exercises. That is the core new result worth knowing. They numerically integrate the two-loop equations, compare the evolution matrices directly, and show how one-loop zeroes get filled. The bottom-up fits then quantify how those changes propagate into HL-LHC and FCC-ee sensitivities. This part is straightforward and gives concrete numbers that people doing precision phenomenology can check against their own codes. The top-down half matches all scalar and fermion UV extensions from the Granada dictionary at one loop, including the couplings that first appear at that order, then evolves them with two-loop RGEs and reports percent-level changes in sensitivity for some models. The execution looks careful on the numerical side for the running itself. The soft spot is exactly where the stress-test note flags it. For the operators that show the largest two-loop running effects, two-loop matching corrections are generically the same size as the one-loop matching terms already kept. Without estimating or including those O(1/16π²) pieces, the claimed percent-level shift in UV coupling sensitivity cannot be attributed to running alone. The abstract also gives no information on numerical stability, step-size checks, or validation against known one-loop limits, which leaves a small but real gap in reproducibility. This work is for SMEFT phenomenologists who run global fits or match UV models to future collider data. A reader updating projections or codes will get direct value from the bottom-up comparisons. It deserves a serious referee because the numerical RGE implementation is new and the collider question is timely, even though the top-down claims will need extra attention on matching order during review.

Referee Report

2 major / 2 minor

Summary. The paper numerically integrates the complete two-loop SMEFT RGEs for dimension-six operators and compares the resulting evolution matrices to their one-loop counterparts, identifying altered mixing patterns and broken zeros. It then performs bottom-up linear and quadratic fits to a set of 61 Wilson coefficients using projected HL-LHC and FCC-ee data, reporting non-negligible two-loop-induced shifts especially for four-quark, top-Yukawa and Higgs-gluon operators. In a top-down analysis, all scalar and fermion UV completions from the Granada dictionary are matched onto SMEFT at one loop (including one-loop-only couplings) and evolved with the two-loop RGEs, yielding percent-level changes in the extracted sensitivities for some models.

Significance. If the numerical results hold, the work demonstrates that two-loop running cannot be neglected in precision SMEFT studies at future colliders for selected operator classes, and supplies the first systematic top-down survey that incorporates one-loop-only matching coefficients. The full numerical integration and the breadth of the 61-coefficient fits constitute clear technical strengths.

major comments (2)

[top-down perspective section] Top-down analysis (abstract and corresponding results section): the reported percent-level shifts in UV-model sensitivities are obtained after one-loop matching followed by two-loop RGE evolution. For the four-quark, top-Yukawa and Higgs-gluon operators where two-loop RGE effects are stated to be non-negligible, two-loop matching corrections are generically O(1/16π²) and of the same order as the one-loop matching terms already included. No estimate or bound on these higher-order matching contributions is provided, so the claimed impact cannot be cleanly attributed to running alone.
[RGE integration and evolution-matrix comparison] Numerical RGE integration section: the manuscript states that the two-loop RGEs are solved by full numerical integration, yet supplies no information on the integrator, step-size control, convergence tests, or validation against known one-loop analytic limits and benchmark points. Given that the central phenomenological claims rest on the difference between one- and two-loop evolution matrices, such technical details are required to assess numerical reliability.

minor comments (2)

[abstract] The abstract and introduction should explicitly state the number of UV models considered and the precise set of 61 Wilson coefficients retained in the global fits.
[figures showing evolution matrices] Figure captions for the evolution-matrix comparisons should indicate the renormalization scale at which the matrices are evaluated and the reference scale used for the initial conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have revised the manuscript to incorporate the suggestions where appropriate.

read point-by-point responses

Referee: [top-down perspective section] Top-down analysis (abstract and corresponding results section): the reported percent-level shifts in UV-model sensitivities are obtained after one-loop matching followed by two-loop RGE evolution. For the four-quark, top-Yukawa and Higgs-gluon operators where two-loop RGE effects are stated to be non-negligible, two-loop matching corrections are generically O(1/16π²) and of the same order as the one-loop matching terms already included. No estimate or bound on these higher-order matching contributions is provided, so the claimed impact cannot be cleanly attributed to running alone.

Authors: We agree that two-loop matching corrections are of the same perturbative order as the one-loop matching terms we include. Our top-down analysis is explicitly constructed to isolate the effect of two-loop RGE evolution by using identical one-loop matching coefficients for both the one-loop and two-loop running comparisons. This isolates the running contribution to the observed percent-level shifts in sensitivities. We have added a clarifying paragraph in the revised top-down section stating the scope of the study and noting that a complete NLO treatment would require two-loop matching for each UV model, which is a substantial separate computation beyond the present work. revision: partial
Referee: [RGE integration and evolution-matrix comparison] Numerical RGE integration section: the manuscript states that the two-loop RGEs are solved by full numerical integration, yet supplies no information on the integrator, step-size control, convergence tests, or validation against known one-loop analytic limits and benchmark points. Given that the central phenomenological claims rest on the difference between one- and two-loop evolution matrices, such technical details are required to assess numerical reliability.

Authors: We thank the referee for this important observation. In the revised manuscript we have added a dedicated paragraph in the numerical RGE integration section with the requested technical details. The two-loop SMEFT RGEs are integrated using an adaptive fourth-order Runge-Kutta integrator with relative and absolute tolerances set to 10^{-10}. Step-size control is performed dynamically. Validation consists of recovering the exact analytic one-loop evolution matrices when the two-loop beta functions are switched off, for multiple benchmark points including cases with vanishing two-loop mixing. Convergence was verified by tightening the tolerances by an order of magnitude, with the resulting evolution matrices stable to better than 0.1 percent, which is sufficient for the precision of our collider projections. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper solves the two-loop SMEFT RGEs via independent numerical integration (external computation) and performs explicit bottom-up and top-down fits to projected collider data. Top-down matching of UV models is done at one-loop order as an explicit assumption, with results compared to one-loop RGE evolution; no step reduces by construction to a fitted parameter renamed as prediction, self-definition, or load-bearing self-citation. The central claims rest on direct numerical comparisons and data fits that remain falsifiable against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard SMEFT assumptions and numerical methods for solving differential equations.

axioms (2)

domain assumption Validity of SMEFT with dimension-six operators up to collider energy scales
Assumes higher-dimensional operators are not needed for the relevant processes.
domain assumption One-loop matching of UV models to SMEFT is adequate for RGE impact studies
Top-down fits neglect higher-loop matching corrections when evaluating two-loop RGE effects.

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discussion (0)

Forward citations

Cited by 1 Pith paper

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