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arxiv: 2606.26247 · v1 · pith:ARJCM7INnew · submitted 2026-06-24 · ✦ hep-ph

Double-real corrections to color singlet decay in a parton-shower inspired scheme

John M. Campbell , Stefan H\"oche , Max Knobbe This is my paper

Pith reviewed 2026-06-26 01:44 UTC · model grok-4.3

classification ✦ hep-ph
keywords NNLO QCDinfrared subtractioncolor singlet decaydouble-real correctionsparton shower inspirede+e- annihilationlocal counterterms
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0 comments X

The pith

Local subtraction with scalar radiators renders double-real corrections finite for color-singlet decays at NNLO.

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

The paper presents a subtraction scheme for next-to-next-to-leading order QCD calculations of color singlet particle decays. It constructs counterterms from scalar radiators and splitting functions to handle infrared singularities in double-real emissions. Overlapping singularities are resolved through partial fractioning, with kinematics mapped iteratively like next-to-leading order. The method is tested on the double-real remainder for electron-positron to quark-antiquark, showing finiteness in unresolved limits. Phase-space integrals of the counterterms are computed analytically and numerically.

Core claim

The authors introduce a local infrared subtraction method for NNLO QCD in color singlet decays. Counterterms are based on scalar radiators and pure splitting functions. Singularities are disentangled by partial fractioning and kinematics correspond to iterated NLO mappings. They verify that the double-real remainder for e+e- to q qbar is finite in single and double unresolved limits. The phase-space integrals of scalar counterterms are computed in the back-to-back configuration.

What carries the argument

The local infrared subtraction method based on scalar radiators and splitting functions with partial fractioning for overlapping singularities.

If this is right

  • The subtracted double-real contribution can be integrated numerically with Monte Carlo methods.
  • Phase space integrals of the counterterms can be evaluated both analytically and numerically in back-to-back kinematics.
  • The scheme allows handling of overlapping singularities in multipole radiation patterns for color singlet processes.
  • Finiteness holds in both single and double unresolved limits for the tested decay.

Where Pith is reading between the lines

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

  • The scheme may extend to processes with initial-state radiation or more complex final states.
  • It could facilitate consistent matching between fixed-order NNLO results and parton showers.
  • Numerical convergence properties might be tested in applications beyond the back-to-back configuration.

Load-bearing premise

The counterterms from scalar radiators and splitting functions, after partial fractioning, cancel all infrared singularities in the double-real emissions for color singlet decays.

What would settle it

A numerical evaluation showing that the subtracted double-real remainder diverges in the double unresolved limit for the e+e- to q qbar process would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.26247 by John M. Campbell, Max Knobbe, Stefan H\"oche.

Figure 1
Figure 1. Figure 1: FIG. 1. Left: Scaling behavior of the subtracted [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Differential jet rates in the Durham algorithm at [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Sketch of the kinematics mapping in the identified particle subtraction algorithm of [64]. See the main text for details. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Sketch of the two-emission phase space mapping in parametrization (a). [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
read the original abstract

We introduce a local infrared subtraction method for next-to-next-to-leading order QCD calculations in color singlet decays, with counterterms based on scalar radiators and pure splitting functions. Overlapping singularities in the multipole radiation pattern are disentangled by partial fractioning, and the kinematics mapping corresponds to iterated next-to-leading order kinematics. We verify that the double-real remainder to $e^+e^-\to\;q\bar{q}$ is rendered finite in the single and double unresolved limits and investigate the numerical convergence of the Monte-Carlo integral. We compute the phase-space integrals of the scalar counterterms in the back-to-back configuration, both analytically and with the help of numerical techniques based on sector decomposition.

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 / 2 minor

Summary. The manuscript introduces a local infrared subtraction method for NNLO QCD calculations in color singlet decays, with counterterms constructed from scalar radiators and pure splitting functions. Overlapping singularities are handled via partial fractioning, and the kinematics mapping uses iterated NLO mappings. The central result is a verification that the double-real remainder to e+e−→q q¯ is finite in the single and double unresolved limits, together with analytic and numerical (sector-decomposition) evaluation of the scalar counterterm phase-space integrals in the back-to-back configuration and Monte-Carlo checks of numerical convergence.

Significance. If the cancellation holds, the scheme supplies a parton-shower-inspired local subtraction for double-real emissions in color-singlet processes, complementing existing dipole and antenna methods. Explicit analytic results for the counterterm integrals and the reported numerical convergence tests constitute concrete strengths that can be directly used or cross-checked by other groups.

minor comments (2)
  1. The description of the partial-fractioning procedure would benefit from an explicit example of how the overlapping double-unresolved terms are decomposed before the kinematics mapping is applied.
  2. A short table comparing the analytic expressions for the scalar counterterm integrals with the numerical sector-decomposition results would improve readability of the verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs a local subtraction scheme from standard splitting functions and scalar radiators, applies partial fractioning and iterated NLO mappings, then verifies finiteness of the double-real remainder for e+e- -> q qbar via direct analytic phase-space integrals and Monte-Carlo checks. No equation reduces a claimed prediction to a fitted input by construction, no load-bearing self-citation chain is invoked, and the finiteness result is externally checkable against the matrix elements. This matches the default non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the method assumes standard QCD infrared factorization but introduces no explicit free parameters or new entities in the provided text.

axioms (1)
  • domain assumption Infrared singularities in double-real emissions for color singlet decays can be subtracted locally using scalar radiators and pure splitting functions after partial fractioning.
    This is the foundational premise stated in the abstract for constructing the counterterms.

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discussion (0)

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Reference graph

Works this paper leans on

103 extracted references · 2 canonical work pages · 1 internal anchor

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    The NLO case We begin the discussion with the simple example of an NLO calculation. The scalar radiator for the production of a single on-shell gluon from a QCD multipole formed by massless on-shell particles is given by [41, 71] Sg({p};q 1) = X i,k ˆTi ˆTk Si;k(q1),whereS i;k(q1) =− 4pipk p2 i1p2 k1 . (9) Note that we have used current conservation to el...

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    The NNLO case – Abelian contributions References [41, 72] discussed the process-independent form of the two-gluon soft / scalar radiators, which can be determined from the squared two-gluon current. The abelian component of the scalar radiator is given by S(ab) gg ({p}, q1, q2) = 2 X i,k X l,m n ˆTa i ˆTa l , ˆTb k ˆTb m o S(ab) i,k;l,m(q1, q2) + 2 X i,k ...

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    The NNLO case – Nonabelian contributions The non-abelian scalar radiator was computed in Ref. [41]. As in the abelian case, one can use charge conservation to simplify the expressions considerably. We obtain S(nab) gg ({p}, q1, q2) = X i,l CA ˆTc i ˆTc l h S(ab) i,i;l (q1, q2) +S (ab) l,l;i (q1, q2) + (1−2δ il)S (ab) i;l (q1, q2)− S (nab) i;l (q1, q2) i ....

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    (30) As in the case of gluon emissions discussed above, it is possible to eliminate gauge-dependent terms with the help of color conservation

    The NNLO case – Emission of aq¯qpair The emission of a quark-antiquark pair from a pair of scalar radiators is described by the insertion operator [41, 72] Sq¯q({p};q 1, q2) = X i,k ˆTi ˆTk TR S(q¯q) i;k (q1, q2). (30) As in the case of gluon emissions discussed above, it is possible to eliminate gauge-dependent terms with the help of color conservation. ...

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    (14) S(q¯q) i;k (q1, q2) =− sik si12 sk12 4 s12 1−4 ϵik,f12 ϵ12,12 2 + 2 si12sk12 ,(32) 7 where ˜qµ 12 =q µ 12 +s 12/sik pµ ik

    The singularities are made explicit by using the notation of Eq. (14) S(q¯q) i;k (q1, q2) =− sik si12 sk12 4 s12 1−4 ϵik,f12 ϵ12,12 2 + 2 si12sk12 ,(32) 7 where ˜qµ 12 =q µ 12 +s 12/sik pµ ik. The mapping to the individual collinear sectors can be performed by an angular partial fractioning of the form of Eq. (10), leading to partitioned radiatorsS (q¯q),...

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    Polarization vectors for color monopoles It is well known that axial gauges simplify the computation of collinear splitting functions and related quantities [74– 82]. The polarization tensor of an axial gauge dµν(q, n) =−g µν + qµnν +q νnµ nq − n2 qµqν (nq)2 ,(37) satisfies the physical requirements for on-shell gluons, namelyd µ µ(p, n) =D−2 andp µdµν(p,...

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    Spin-correlated subtraction terms Consider now an NLO counterterm comprised of ane +e− →q¯qgunderlying Born process, and ag→q¯qor a g→ggsplitting function. In both cases, we will have a component of the splitting function proportional to the dyadic product of two scalar interaction terms of the formd µ ν(q12, n)(q1 −q 2)ν, which arise from the “decay” of ...

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    At NNLO, the technique is applied twice, with the successive assignment of the radiator performed according to the algorithms in Sec

    NLO kinematics Figure 3 shows the kinematics mapping we employ at NLO. At NNLO, the technique is applied twice, with the successive assignment of the radiator performed according to the algorithms in Sec. II C. The basic momentum 13 mapping was introduced in Sec. 5.6 of Ref. [64]. Beginning with Eq. (3), pµ i =z˜pµ i , N µ = ˜K µ + (1−z) ˜pµ i ,(A2) we ca...

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    NNLO kinematics The phase-space factorization at NNLO can be achieved in different ways. In a triple-collinear parametrization, the polar angle of both emitted partons is typically measured against the direction of ˜p µ i , and the longitudinal recoil is taken by partoni. We will begin by deriving theD-dimensional phase-space element in this case. A sketc...

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