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arxiv: 2505.10406 · v2 · pith:WGFRMZQ3new · submitted 2025-05-15 · ✦ hep-ph

One-loop amplitudes for tbar{t}j and tbar{t}γ productions at the LHC through mathcal{O}(ε²)

Souvik Bera , Colomba Brancaccio , Dhimiter Canko , Heribertus Bayu Hartanto This is my paper

Pith reviewed 2026-05-22 14:45 UTC · model grok-4.3

classification ✦ hep-ph
keywords one-loop amplitudeshelicity amplitudestop quark pair productionQCD correctionspentagon functionsdimensional regularizationLHC phenomenologyNNLO calculations
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The pith

Analytic expressions for one-loop QCD helicity amplitudes in top-pair production with jet or photon are given through O(ε²).

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

The paper establishes analytic expressions for the one-loop QCD helicity amplitudes in top-quark pair production associated with a jet or a photon at the LHC. These amplitudes are expanded through the second order in the dimensional regularization parameter epsilon. The results are needed to construct two-loop hard functions for next-to-next-to-leading order QCD calculations. A sympathetic reader would care because higher-order precision improves the accuracy of theoretical predictions for these important LHC processes. The amplitudes are built from linear combinations of pentagon functions with rational coefficients expressed in momentum-twistor variables.

Core claim

We present analytic expressions for the one-loop QCD helicity amplitudes contributing to top-quark pair production in association with a photon or a jet at the Large Hadron Collider, evaluated through O(ε²) in the dimensional regularisation parameter ε. These amplitudes are required to construct the two-loop hard functions that enter the NNLO QCD computation. The helicity amplitudes are expressed as linear combinations of algebraically independent components of the ε-expanded master integrals known as pentagon functions with the corresponding rational coefficients written in terms of momentum-twistor variables. Differential equations for the pentagon functions are derived and solved using a

What carries the argument

Pentagon functions, the algebraically independent components of the ε-expanded master integrals, whose rational coefficients are written in momentum-twistor variables and whose differential equations are solved numerically.

If this is right

  • The amplitudes allow construction of the two-loop hard functions required for NNLO QCD predictions of ttj and ttgamma production.
  • The expressions support numerical evaluations to the precision needed for LHC phenomenology using generalized power series methods.
  • Compact forms in momentum-twistor variables facilitate further analytic or numerical work on related processes.
  • These results reduce theoretical uncertainties in cross-section predictions for top-quark associated production at current colliders.

Where Pith is reading between the lines

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

  • Similar decompositions into pentagon functions could extend to other multi-particle processes with massive quarks at higher perturbative orders.
  • Integration of these amplitudes into parton-shower Monte Carlo programs would enable more precise event simulations for experimental analyses.
  • Cross-validation against existing numerical one-loop libraries at multiple phase-space points would strengthen in the results for practical use.

Load-bearing premise

The helicity amplitudes can be expressed as linear combinations of algebraically independent pentagon function components with rational coefficients in momentum-twistor variables.

What would settle it

An independent numerical evaluation of the amplitudes at a specific kinematic point that disagrees with the analytic expressions beyond expected precision would show the expressions are incorrect.

read the original abstract

We present analytic expressions for the one-loop QCD helicity amplitudes contributing to top-quark pair production in association with a photon or a jet at the Large Hadron Collider (LHC), evaluated through $\mathcal{O}(\epsilon^2)$ in the dimensional regularisation parameter, $\epsilon$. These amplitudes are required to construct the two-loop hard functions that enter the NNLO QCD computation. The helicity amplitudes are expressed as linear combinations of algebraically independent components of the $\epsilon$-expanded master integrals--known as pentagon function--with the corresponding rational coefficients written in terms of momentum-twistor variables. We derive differential equations for the pentagon functions and solve them numerically using the generalised power series expansion method.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript presents analytic expressions for the one-loop QCD helicity amplitudes for top-quark pair production in association with a jet or a photon at the LHC, expanded through O(ε²) in dimensional regularization. The amplitudes are reduced via IBP to a basis of algebraically independent pentagon functions, with rational prefactors expressed in momentum-twistor variables; differential equations for these functions are derived and solved numerically to the required order using the generalised power series expansion method. These results are intended as input for constructing two-loop hard functions in NNLO QCD calculations.

Significance. If correct, the explicit one-loop amplitudes to O(ε²) supply a necessary ingredient for NNLO QCD predictions of ttj and ttγ processes, which are phenomenologically relevant at the LHC for precision top-quark studies. The workflow follows established multi-loop techniques (IBP reduction, pentagon-function basis, momentum-twistor coefficients, and power-series DE solution), and the manuscript supplies the explicit coefficient expressions together with the DE system. This constitutes a concrete, reusable contribution that can be directly incorporated into higher-order phenomenology codes.

minor comments (3)
  1. [Abstract] Abstract: no explicit mention is made of numerical validation, error estimates, or cross-checks against known lower-order results or independent codes; adding a brief statement on these checks would strengthen reader confidence in the O(ε²) coefficients.
  2. [Section 3 (or wherever the basis is introduced)] The manuscript should specify the precise choice of pentagon-function basis (including any linear independence checks) and list the complete set of algebraically independent components used for each process.
  3. [Numerical results section] Figure or table presenting sample numerical values: include at least one benchmark point with comparison to a lower-order analytic result or an independent numerical integrator to illustrate the achieved precision.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for recommending minor revision. The work supplies the one-loop helicity amplitudes to O(ε²) expressed in a pentagon-function basis with momentum-twistor coefficients, together with the associated differential equations solved via generalised power series expansion, as a concrete input for NNLO hard functions.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper reduces one-loop helicity amplitudes for ttj and ttγ to a basis of algebraically independent pentagon functions via IBP reduction, expresses rational coefficients in momentum-twistor variables, derives the associated differential equations, and obtains the O(ε²) terms by numerical solution of those DEs using the generalised power-series method. This is a standard, externally validated workflow in the literature with no reduction of the final result to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The manuscript supplies the explicit coefficient expressions and DE system, rendering the derivation independent and falsifiable against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical frameworks of perturbative QCD and integral reduction; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond conventional dimensional regularization and master-integral reduction.

axioms (2)
  • standard math Dimensional regularization with parameter ε is used to regulate infrared and ultraviolet divergences
    Invoked throughout the abstract as the regularization scheme for the one-loop amplitudes.
  • domain assumption Feynman integrals for these processes reduce to a basis of pentagon functions whose ε-expansion components are algebraically independent
    Stated when the amplitudes are expressed as linear combinations of these components.

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    Relation between the paper passage and the cited Recognition theorem.

    The helicity amplitudes are expressed as linear combinations of algebraically independent components of the ε-expanded master integrals—known as pentagon functions—with the corresponding rational coefficients written in terms of momentum-twistor variables. We derive differential equations for the pentagon functions and solve them numerically using the generalised power series expansion method.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Tensor decomposition of $e^+e^-\to\pi^+\pi^-\gamma$ to higher orders in the dimensional regulator

    hep-ph 2026-04 unverdicted novelty 7.0

    First beyond-NLO tensor decomposition and higher-order analytic one-loop amplitudes for e+e- to pi+pi-gamma, paired with a fast numerical five-point integral evaluator.

  2. Double virtual QCD corrections to $t\bar{t}+$jet production at the LHC

    hep-ph 2025-11 unverdicted novelty 7.0

    Leading-colour two-loop virtual amplitudes for ttbar+jet are extracted analytically via finite-field evaluations and differential equations, then packaged in a C++ library with new numerical integration techniques.

Reference graph

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