Status of two-loop automation in OpenLoops

Max F. Zoller; Natalie Sch\"ar

REVIEW 3 major objections 4 minor 20 references

Status of two-loop automation in OpenLoops

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read OpenLoops now builds fully renormalised two-loop QED and QCD amplitudes.

desk verdict Honest status report from the OpenLoops group: the two-loop renormalisation code is running and passes UV pole-cancellation checks, but finite parts and IR rational terms are explicitly still on the to-do list. read the letter →

arxiv 2412.14856 v1 pith:EL3BDJMB submitted 2024-12-19 hep-ph

Natalie Schär , Max F. Zoller This is my paper · ORCID

classification hep-ph

keywords two-loopamplitudesOpenLoopsrationalcountertermsdimensionalregularizationtensorintegralreductionNNLOpredictionsQEDandQCDcorrectionsrenormalisation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper reports the status of an effort to automate two-loop amplitude calculations in the OpenLoops framework. Its central claim is that the framework now contains a complete implementation of the renormalisation procedure for two-loop QED and QCD corrections to the Standard Model, combining two-loop amplitudes with the proper one-loop and tree-level counterterm insertions. The UV poles have been shown to cancel in several off-shell two- and three-point amplitudes. This is a step toward automated NNLO predictions for collider processes.

What carries the argument

The central mechanism is the decomposition of amplitudes according to equation (6), separating loop integrals with four-dimensional numerators from $(D-4)$-dimensional remainders, and the reconstruction of those remainders through process-independent rational counterterms combined with the usual UV counterterms. The four-dimensional pieces are built by a recursive algorithm that multiplies universal loop-segment building blocks into tensor coefficients, and the resulting two-loop tensor integrals are reduced to master integrals using projector techniques plus integration-by-parts identities.

What would settle it

Compute the finite part of an on-shell two-loop QED or QCD vertex function with the implemented pipeline and compare it with an independent analytic result; any mismatch shows the rational counterterms miss some (D-4)-dimensional effects. A cheaper test is to run the four-point QCD vertex function, which the paper says is still pending, and look for a residual UV pole.

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Extended reading notes

Core claim

The paper's central claim is that, following the master formula in equation (6), every renormalised two-loop amplitude can be assembled from the unrenormalised two-loop contribution plus three counterterm contributions: a one-loop amplitude with one-loop counterterm insertions, a tree-level amplitude with a single two-loop counterterm, and a tree-level amplitude with double one-loop counterterm insertions. Each counterterm includes not only the standard UV counterterm but also a rational term that restores the effects of the $(D-4)$-dimensional numerator parts on UV poles. The implemented pipeline has verified the cancellation of UV poles for several two- and three-point off-shell QED and QCD vertex functions.

Load-bearing premise

The whole scheme assumes that universal rational counterterms capture every way the extra dimensional components of loop numerators affect ultraviolet poles; so far only the pole cancellations in a few off-shell functions have been checked, not the finite leftovers.

Editorial extensions

If this is right

If the central claim is correct, the same counterterm machinery can generate the double-virtual contribution for any QED or QCD correction to a Standard Model process, once the tensor-integral reduction is extended beyond simple topologies.
The verified UV pole cancellation means the renormalisation bookkeeping, including the rational counterterms, is consistent for off-shell vertex functions.
Because the rational counterterms are model-dependent but process-independent, the implementation can be ported to other renormalisable theories without rederiving the recursion machinery.
The paper's stated next step, treating $(D-4)$-dimensional numerator parts interacting with infrared poles, is the difference between off-shell validation and physical on-shell observables.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the authors leave implicit is that if the rational-counterterm reconstruction works for UV poles, the same principle might eventually package infrared-singular $(D-4)$-dimensional effects into universal counterterms, potentially simplifying NNLO subtraction schemes.
The finite parts of the QED and QCD vertex functions are said to be under computation; comparing those finite parts against known analytic results would convert the current UV-pole check into a full validation of the whole rational-term construction.
The paper notes the projector/IBP reduction becomes impractical for high tensor ranks and masses, so the practical reach of the tool will likely be set by the new reduction method being developed, not by the renormalisation procedure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a progress report, and an honest one. For anyone tracking numerical NNLO automation, the news is that OpenLoops now has a working implementation of the two-loop renormalisation procedure including the rational counterterms, and that implementation cancels the 1/eps poles in a handful of two- and three-point QED and QCD vertex functions. That is a real milestone for their program, even though it is exactly the kind of check that catches gross errors, not the kind that confirms the finite parts.

The genuinely new content is the implementation itself: the tensor coefficient recursion was described in refs [11,12], but this paper records that it is now coded for all diagram topologies and combined with the counterterm insertions of Eqs. (26)-(27). Section 4 also notes a practical subtlety worth remembering: Collier and OneLoop do not provide the O(eps) parts needed when the 1/eps pole of a UV counterterm multiplies the one-loop integral, so they had to use their own tensor reduction for those pieces. That detail will save someone time.

The soft spots are visible, and the paper points to them itself. Validation is only UV pole cancellation; finite parts are not yet compared with the literature, and the four-point QCD check is ongoing. The rational terms that would reconstruct the (D-4)-dimensional numerator parts interacting with IR poles are not implemented; Section 4 says that interplay is under investigation. So the stress-test concern is real, but it is not a hidden flaw: the authors state the limitation twice. I would hold them to that framing when citing the work.

The tensor integral reduction in Section 3 is deliberately limited to massless two- and three-point topologies plus a few four-point ones, so the validation covers a small slice of what a full two-loop tool needs. Again, this is disclosed.

If this were a full journal submission rather than a Loops-and-Legs proceedings write-up, the right referee report would be: sound as far as it goes, but do not claim full two-loop automation until the finite parts match the literature and the IR rational terms are in place. As a status report it deserves a serious read by anyone in the NNLO automation area, and it should be citable for 'OpenLoops has passed UV pole cancellation at two loops.' I would not reject it; I would publish it as proceedings and keep the caveat prominently in mind.

Referee Report

3 major / 4 minor

Summary. Schär and Zoller report the status of two-loop automation in OpenLoops. They decompose D-dimensional two-loop amplitudes into Feynman integrals with four-dimensional numerators plus (D-4)-dimensional remainders, the latter to be reconstructed through universal rational counterterms. The paper describes the recursive construction of tensor coefficients for irreducible and reducible two-loop diagrams (Section 2), a first projector/IBP-based reduction of two-loop tensor integrals for simple massless topologies (Section 3), and the implementation in OpenLoops of a two-loop renormalisation procedure with UV rational counterterms (Section 4). Validation so far consists of checking the cancellation of UV poles in Eq. (6) for several two- and three-point QED and QCD vertex functions; finite-part comparisons and the four-point QCD check are stated as ongoing, and the interplay of (D-4)-dimensional numerator parts with IR poles is not yet implemented.

Significance. If the framework delivers correct finite parts, it would be a notable step toward automated NNLO predictions, and the process-independent decomposition in Eqs. (7)-(9) together with the explicit master formulas in Eqs. (26)-(27) are valuable building blocks. The paper has no fitted free parameters, it presents a nontrivial self-consistency check (UV pole cancellation), and it is honest about many of its limitations. However, the current significance is conditional: the rational counterterms are not yet validated on finite parts, and the strength of the claims in the abstract and in Section 4 exceeds what the reported evidence supports.

major comments (3)

[Abstract and Section 4, Eq. (6)] The abstract says that the renormalisation procedure and the reconstruction of (D-4)-dimensional numerator parts through two-loop rational counterterms 'has been implemented and validated', and Section 4 states that the combination to 'fully renormalised D-dimensional amplitudes' is implemented. The only validation reported there, however, is the cancellation of UV poles in Eq. (6) for a few two- and three-point QED/QCD vertex functions; the finite parts are explicitly said to be still to be compared with the literature, and the interplay of (D-4)-dimensional numerator parts with IR poles is 'still under investigation'. Because a missing or incorrect rational term can shift finite parts while leaving pole cancellation intact, this evidence validates pole cancellation only, not the full renormalised amplitude. Please either report a finite-part check for at least one amplitude or replace 'validated' by 'partially validated (UV pole cancellation only)' throughout the abstract and Section 4.
[Section 2, final paragraph] The sentence 'The algorithms for all categories of two-loop diagrams are fully implemented and validated for QED and QCD corrections to the Standard Model' is stronger than what Section 4 documents, where validation is restricted to UV pole cancellation and the four-point QCD vertex-function check is still ongoing. Please state the exact validation criterion used for the tensor-coefficient algorithms in Section 2, or move the word 'validated' to the discussion in Section 4 where its domain is defined.
[Sections 3 and 4] The in-house reduction described in Section 3 (projectors plus IBP with FIRE, stored as Fortran libraries for simple topologies) is used both to build the two-loop amplitudes and to perform the pole-cancellation test of Section 4. Because the same reduction chain appears on both sides of Eq. (6), a systematic error in that reduction could in principle cancel between the loop terms and the counterterms. A comparison of the finite parts of the QED and QCD three-point functions with known analytical results would provide the independent check needed to underwrite the validation claim; the paper states this comparison is ongoing, but until it is reported the claim 'validated' should be qualified.

minor comments (4)

[Eq. (21)] The initial condition 'N-1 = 1 1' in Eq. (21) is garbled; it should presumably be an identity or 'N_-1 = 1', and the notation should be defined explicitly.
[Footnote 6] In footnote 6 the newly introduced counterterm is written as 'δZ1,γ' after 'while', but the equation and text use 'δZ~1,γ'; please correct the symbol.
[Section 1] The statement that 'D-dimensional quantities cannot be computed numerically in a direct way' is imprecise; many numerical tools evaluate D-dimensional integrals via analytic continuation or dimensional recurrence. What matters here is that OpenLoops' recursive construction requires four-dimensional numerators, and the text should say that.
[Section 4] The validation section would benefit from specifying the exact list of amplitudes (processes, number of points, kinematics) and from saying how many diagrams and topologies were exercised; 'several two and three-point amplitudes' is too vague for reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the implementation claims are supported by independent tools and the self-citations to prior rational-term derivations are not used to define the target result.

full rationale

The paper is a status report on implementing two-loop automation in OpenLoops, not a derivation of predictions from fitted inputs. The claimed new content is the automated construction of two-loop tensor coefficients (Section 2) and the implementation of the renormalisation formula (26) together with UV rational counterterms. The master formula (26) is taken from the same authors' earlier work [8], and the two-loop rational terms are taken from [9,10]; this is self-citation, but it is not circular because those papers contain parameter-free derivations with stated assumptions, and the present paper's UV-pole cancellation check (Section 4) is an implementation test rather than the construction of the target result. The in-house tensor-integral reduction tool of Section 3 relies on IBP reduction [15], FIRE [16,17], analytical master integrals [18], and FIESTA [19], providing independent checks for the massless two- and three-point topologies used in the validation. There are no fitted free parameters, and no quantity is defined in terms of the quantity it is used to predict. The main limitation is explicit in the paper: 'the calculation of the finite parts of all QED and QCD vertex functions, to be compared to the literature, are ongoing' and 'the interplay of (D-4)-dimensional numerator parts with IR poles is still under investigation.' These are completeness and correctness risks, not circularity: the UV-pole cancellation check alone does not validate finite parts, but the paper does not claim that it does. Therefore the derivation chain is self-contained to the extent claimed, and no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claims rest on standard tools (dimensional regularization, FIRE, FIESTA) and on the group's own prior formulas for two-loop rational counterterms. No free parameters are fitted to data. The strongest dependency is the unverified assumption that those rational counterterms correctly reconstruct all (D-4)-dimensional numerator effects in UV poles, which the ongoing validation is meant to test.

assumptions (3)

standard math Dimensional regularization in D = 4 - 2 epsilon is used throughout.
Standard regularization scheme in perturbative QFT, used implicitly in all integrals and counterterms.
domain assumption The two-loop rational counterterms from references [8]-[10] correctly capture the (D-4)-dimensional numerator effects in UV poles.
This is the central unproven input of the validation; the paper is currently testing it only through UV pole cancellation checks.
standard math FIRE and FIESTA correctly perform integration-by-parts reductions and numerical master integral evaluations.
Section 3 relies on FIRE [16,17] for IBP reduction and FIESTA [19] for numerical evaluation; these are established external tools.

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Cite this review

Pith. "Pith review of Status of two-loop automation in OpenLoops." pith.science (2026). https://pith.science/paper/EL3BDJMB

@misc{pith2026241214856,
  author       = {Pith},
  title        = {Pith review of: Status of two-loop automation in OpenLoops},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EL3BDJMB}},
  note         = {Machine review of arXiv:2412.14856}
}

abstract

The calculation of hard scattering amplitudes up to NLO is automated in numerical tools, such as OpenLoops. The LHC and future experiments, however, demand high-precision predictions at NNLO and beyond for a wide range of particle processes. Hence, the development of a fully automated tool for numerical NNLO calculations is an important goal. In order to perform a numerical calculation, we decompose $D$-dimensional two-loop amplitudes into Feynman integrals with four-dimensional numerators and $(D-4)$-dimensional remainders, which contribute to the finite result through the interaction with the poles of Feynman integrals and are reconstructed during the subtraction procedure for these poles from universal rational terms. The integrals with four-dimensional numerators are further decomposed into loop momentum tensor integrals and tensor coefficients. We present the status of OpenLoops with respect to these building blocks. The algorithm for the construction of the tensor coefficients is implemented for QED and QCD corrections to the SM in a fully automated way. Recently, the renormalisation procedure and the reconstruction of the interplay of $(D-4)$-dimensional numerator parts with UV poles through two-loop rational counterterms has been implemented and validated using an in-house library for the reduction of simple tensor integrals.

Figures

Figures reproduced from arXiv: 2412.14856 by the authors.

**Figure 1.** Categorisation of two-loop diagrams into irreducible (Irred) and reducible (Red) ones. The latter are further split into two subcategories, where two one-loop subdiagrams are either connected to a tree structure 𝑃 through two vertices (Red2) or are attached to each other through a common quartic vertex (Red1). The blue blobs denote subtrees connected to internal and external lines. In general V0, V1 can be quartic v… view at source ↗

**Figure 2.** Categories of one-loop diagrams with counterterm insertions. The master formula for two-loop diagrams of type Irred requires up to three contributions of type O1a and O1b. Two-loop Red1 diagrams each require up to two O1b contributions, Red2 diagrams each require up to two contributions of type O2. all UV-divergent one-loop subdiagrams 𝛾 ⊂ Γ and M1,Γ/𝛾 the one-loop amplitude resulting from contracting 𝛾 to a vertex … view at source ↗

**Figure 3.** Categories of tree-level diagrams with counterterm insertions. The master formula for two-loop diagrams of type Irred and Red1 with a global UV divergence require a contribution of type T1, while two-loop diagrams of type Red2 require a contribution of type T2 if both subdiagrams are UV divergent. in the previous sections. As a first step, we checked the cancellation of the UV poles in (6) for several two and three-… view at source ↗

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Reference graph

Works this paper leans on

20 extracted references · 5 canonical work pages

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Cascioli, P

F. Cascioli, P. Maierhöfer and S. Pozzorini,Scattering Amplitudes with Open Loops, Phys.Rev.Lett.108 (2012) 111601 [1111.5206]

arXiv 2012

[2] [2]

Buccioni, S

F. Buccioni, S. Pozzorini and M. Zoller,On-the-fly reduction of open loops,Eur. Phys. J. C 78 (2018) 70 [1710.11452]

arXiv 2018

[3] [3]

Buccioni, J.-N

F. Buccioni, J.-N. Lang, J. M. Lindert, P. Maierhöfer, S. Pozzorini, H. Zhang et al., OpenLoops 2,Eur. Phys. J. C79 (2019) 866 [1907.13071]. 9 Status of two-loop automation in OpenLoops M. F. Zoller

arXiv 2019

[4] [4]

Ossola, C

G. Ossola, C. G. Papadopoulos and R. Pittau,On the Rational Terms of the one-loop amplitudes, JHEP05(2008) 004 [0802.1876]

arXiv 2008

[5] [5]

Draggiotis, M

P. Draggiotis, M. V. Garzelli, C. G. Papadopoulos and R. Pittau,Feynman Rules for the Rational Part of the QCD 1-loop amplitudes, JHEP04(2009) 072 [0903.0356]

arXiv 2009

[6] [6]

M. V. Garzelli, I. Malamos and R. Pittau,Feynman rules for the rational part of the Electroweak 1-loop amplitudes, JHEP01(2010) 040 [0910.3130]

arXiv 2010

[7] [7]

Pittau,Primary Feynman rules to calculate the epsilon-dimensional integrand of any 1-loop amplitude, JHEP02(2012) 029 [1111.4965]

R. Pittau,Primary Feynman rules to calculate the epsilon-dimensional integrand of any 1-loop amplitude, JHEP02(2012) 029 [1111.4965]

arXiv 2012

[8] [8]

Pozzorini, H

S. Pozzorini, H. Zhang and M. F. Zoller,Rational Terms of UV Origin at Two Loops, JHEP 05 (2020) 077 [2001.11388]

arXiv 2020