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REVIEW 3 major objections 8 minor 117 references

Multi-partonic interactions, iterated discontinuities and the virtuality expansion in deep inelastic scattering

T0 review · 3 major / 8 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read At leading virtuality, multi-partonic DIS cross-section building blocks equal iterated discontinuities of Feynman-type embedding integrals; the NLO structure functions match the parton model up to a scheme change.

desk verdict A serious, internally consistent KLN-based model of multi-partonic DIS whose central doubled optical theorem is shown in examples, not proven in general, and whose physical relevance rests on an explicitly modeled uniform-density ansatz. read the letter →

arxiv 2506.16489 v1 pith:H6MFI576 submitted 2025-06-19 hep-ph hep-th

Zeno Capatti , Lucien Huber , Michael Ruf This is my paper · ORCID
classification hep-phhep-th
keywords multi-partonicinteractionsdeepinelasticscatteringKLNtheoremvirtualityexpansiongraphembeddingsiterateddiscontinuitiesreverseunitaritypartonmodel
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

This paper introduces a perturbative model of multi-partonic interactions in deep inelastic scattering. The proton is described as a uniform mixture of massless partons, and the leading term in an expansion in the invariant mass $\Lambda^2$ of the initial-state cluster is studied. The paper's central claim is that each infrared-finite class of interference diagrams, organized by "embeddings" of vacuum graphs, can be computed as a double discontinuity of a Feynman-type integral, a result the authors call a doubled optical theorem. If correct, single- and multi-parton initial states are treated on the same footing with building blocks that are separately infrared finite and computable by standard cut-integral and Reverse Unitarity techniques. At next-to-leading order the resulting structure functions agree with the parton model after a scheme change, with the same Altarelli-Parisi scale evolution in $\Lambda^2/Q^2$.

What carries the argument

The load-bearing object is an "embedding": a vacuum Feynman graph together with a puncture whose position is encoded by winding numbers of the graph's cycles. An embedding groups all cuts that can be moved around the puncture into one equivalence class, and the sum of the corresponding interference diagrams is separately infrared finite by the KLN theorem. The second essential element is the simplest clustering criterion: the initial-state partons are taken to be a uniform density matrix constrained only by total momentum, so that in the $\Lambda^2\to0$ limit the density collapses to a function of the proton momentum $p$ alone. The identity carrying the argument is the doubled optical theorem, eq. (136), together with the expansion-by-regions form $I_\Gamma=(p^2)^{-1}H+(-p^2)^{-1-\epsilon}C+O((p^2)^0)$; applying $\operatorname{disc}_{p^2}\operatorname{disc}_{1-x}$ to that expression reproduces the sum of cut diagrams, with the hard and collinear coefficients $H$ and $C$ carrying the KLN cancellations.

What would settle it

Perform the same model's next-to-leading-order Drell-Yan calculation: the paper's equivalence claim requires that the same scheme redefinition that converts multi-parton DIS into parton-model DIS also converts multi-parton Drell-Yan into parton-model Drell-Yan; a difference in the two conversions would make the model experimentally distinguishable from the parton model.

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Extended reading notes

Core claim

The central discovery is that at leading virtuality the partonic tensor of an embedding $\Gamma$ satisfies $$W_\Gamma(p,q)=\frac{1}{\operatorname{av}(q)}\,\operatorname{disc}_{$p^{2}$}\,\operatorname{disc}_{1-x}\,I_\Gamma($p^{2}$,$q^{2}$,1-x)+O(($p^{2}$)^0).$$ Interference diagrams that cancel infrared singularities through the KLN theorem are therefore obtained not one by one but as the iterated discontinuity of the embedding's loop integral, generalising the optical theorem and Cutkosky's cutting rules. The identity is checked in detail on triangle, box, and pentagon embeddings by direct Reverse Unitarity evaluation of the cut integrals. Using it, the paper computes next-to-leading order DIS structure functions: the longitudinal structure functions coincide with the parton model, the $F_2$ scale dependence is driven by the usual splitting kernels, and the finite pieces differ only by a scheme change, with genuinely new contributions such as $F_{2qq}$ appearing.

Load-bearing premise

The construction rests on modeling the proton as a uniform mixture of massless partons constrained only by total momentum, with no correlations among initial-state partons, together with the omission of multi-winding diagrams; if a real proton state has non-trivial initial-state correlations, or if those diagrams contribute, the claimed equivalence with the parton model is not guaranteed.

Editorial extensions

If this is right

  • Every infrared-finite multi-parton contribution at leading virtuality can be computed by taking discontinuities of Feynman-type integrals, so standard integration-by-parts, differential-equation, and Reverse Unitarity technology applies.
  • NLO DIS structure functions match the parton model up to a scheme change: the longitudinal structure functions are identical, and the $Q^2/\Lambda^2$ evolution of $F_{2q}$ and $F_{2g}$ is governed by the same Altarelli-Parisi kernels.
  • Class-three and class-four embeddings introduce branch cuts at $x=1/2$ and new objects such as the double-quark structure function $F_{2qq}$, which are integrable, suppressed by $1/(2N_c)$, and carry no $\Lambda^2$ logarithms.
  • The simplest clustering criterion generalises to Drell-Yan, where the cross-section becomes a sum of iterated discontinuities (one per incoming proton) and is finite after virtuality integration, with leading behaviour proportional to $\log(\Lambda_1^2/m_\star^2)\log(\Lambda_2^2/m_\star^2)$.
  • At NNLO, an explicit two-loop example shows that double, single, and on-shell discontinuities cancel their infrared poles after virtuality integration, leaving a finite expression in terms of powers of $\log(Q^2/\Lambda^2)$.

Reading between the lines

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

  • Editorial inference: because the scheme-change equivalence is established for only one process, the paper's own Drell-Yan roadmap makes it falsifiable; if the same redefinition does not convert the multi-parton Drell-Yan result into the parton-model one, the two models are experimentally distinguishable.
  • Editorial inference: adopting simplest clustering as an effective definition rather than a derived fact, the doubled optical theorem gives a direct route to $\Lambda^2/Q^2$ power corrections in terms of the $p^2$ derivative of the initial-state density; precision Drell-Yan data could constrain those corrections.
  • Editorial inference: the distinctive $\log|2x-1|$ term in $F_{2qq}$, though suppressed by $1/(2N_c)$, is a clean kinematic signature that could be searched for in low-virtuality DIS measurements to test the model against the parton model.
  • Editorial inference: the mechanism points toward a general rule that one discontinuity corresponds to each clustered external state; if that pattern holds, any momentum-conservation-only clustering prescription would define an optical theorem for multi-jet observables, but the paper does not prove this generality.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 8 minor

Summary. The paper proposes a perturbative model for multi-partonic initial states in deep inelastic scattering. Initial-state partons are clustered into a proton-like object using momentum conservation alone, with a uniform Fock-space density and a cutoff Lambda^2 on the cluster virtuality. The authors organize the resulting diagrammatic contributions by graph embeddings, i.e. vacuum graphs endowed with a puncture and winding numbers, and show that at leading virtuality the contribution of each embedding is an iterated double discontinuity of the corresponding embedding integral (eq. (136)). The framework is applied to NLO DIS structure functions; the authors find the same Altarelli-Parisi scale evolution as the parton model, identical longitudinal structure functions, and finite pieces that differ from the parton model. They also present a two-loop example and a first extension to Drell-Yan. The paper contains extensive cross-checks, including the Ward identity, reproduction of parton-model results when MP diagrams are removed, embedding-by-embedding infrared cancellations, integrability tests, and supplementary code and embedding data.

Significance. If the doubled optical theorem in eq. (136) holds at the claimed level of generality, it is a substantial technical result: infrared-finite multi-parton building blocks would be computable by standard cut-integral and reverse-unitarity methods, and the model provides explicit, falsifiable NLO predictions. The paper is also valuable for introducing a systematic graph-theoretic classification of KLN-related embeddings and for releasing code and data. The central caveat is that the theorem and the parton-model equivalence rest on modeling assumptions, in particular the uniform initial-state density and the truncation of higher-winding embeddings, whose scope is not fully established. The strengths of the paper are real, but the strongest claims need to be delimited more carefully.

major comments (3)
  1. [§II.B, eqs. (20)–(22)] The collapse of the general diagonal density to the uniform density is asserted, not derived. The text moves from f(p1,...,pn) to a function of p alone via the statement that infrared safety 'by the usual degeneracy arguments' forces this collapse. The KLN theorem constrains cancellations among degenerate states; it does not fix the relative weights of different initial-state configurations. If f retains dependence on the individual momentum fractions, the sum over interference diagrams in eq. (90) is weighted by f, and the identification with the unweighted double discontinuity of a single embedding integral in eq. (136) is not guaranteed. This collapse is load-bearing for the doubled optical theorem and for the NLO comparison with the parton model. Please either provide an explicit derivation of the collapse or state it as a defining axiom of the model, and test its necessity, for example by computing an NLO embedding with a non-uniform f and showing what breaks in eq. (136).
  2. [§III.B.3, §VI.A.b, §VI.B.h] The exclusion of higher-winding embeddings is a truncation, not a derived property. The paper itself describes the |r_e| <= 1 condition as the least physically motivated constraint, and §VI.B.h states that the full KLN sum over windings is infinite and requires an additional truncation or resummation principle. In addition, the leading-order matching to the parton model requires the extra condition f_q(1/q) = q^alpha f_q(1), which is not fixed by the model. Consequently the NLO structure functions and the claimed scheme-change equivalence to the parton model are established only for the particular finite subset Embdis. The abstract and Section VI state the equivalence more strongly than what is proven. Please either give a physical principle that selects Embdis, bound the contribution of higher-winding embeddings at NLO, or rephrase the equivalence claim as applying only to the truncated set.
  3. [§V.D, eq. (136)] Equation (136) is presented as a general doubled optical theorem, but the evidence is an explicit match for NLO one-loop embeddings (triangle, box, pentagon) and one two-loop example. The derivation relies on the leading-virtuality analytic forms of eqs. (119)–(120) and the reverse-unitarity reduction of eqs. (132)–(135); these are checked, not proven, at arbitrary order. Furthermore, the text itself notes that the discontinuity and the virtuality expansion do not commute, so eq. (136) holds only order-by-order in the virtuality expansion. Please state the theorem with its actual domain (verified at NLO, conjectural beyond), or supply a proof, and make the order-by-order qualification part of the statement of eq. (136).
minor comments (8)
  1. [§II.B, eq. (22)] In eq. (22) the density is written as f(p), while the surrounding text says f does not depend on p^2 but can depend on the momentum fraction; please clarify the notation by writing f(ξ) or specifying the p-dependence after integration over the phase space.
  2. [§III.B.3] The sentence 'By using Z_n we bound the possible winding numbers to be less than n' uses an integer n that is not defined; it presumably depends on the edge or cut set, but this should be stated explicitly.
  3. [§VI.B.a] The validation tests a)–d) are listed but not documented in detail; a short table or a description of how the embedding-by-embedding cancellation and the integrability checks were performed would make these claims reproducible.
  4. [§II.D.c, eq. (50)] The claim that equality of the longitudinal structure functions is necessary and sufficient for a scheme change is stated without proof; please provide a derivation or a reference, since the sufficiency direction is not immediate for two structure functions that both depend on the PDF transformation.
  5. [§VII.B] There are several typos in this section, including 're-routing okk' and 'compeletely' in §II.B; a careful proofread is needed.
  6. [§III.B.3.d] The code is described as hosted on GitHub 'at this link', but no URL is given in the text; please include the actual repository address.
  7. [Acknowledgments] The acknowledgments thank 'the work of M.H.' but the author list contains M. Ruf; this should be corrected to M.R.
  8. [Appendix C] The embedding tables are very hard to read in the text version; since the .m data files are provided as supplementary material, it would help to keep a few illustrative rows in the paper and move the full tables to the supplementary material.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NLO matching, scale dependence, and iterated-discontinuity identities are derived and externally checked, not fitted or defined into existence.

full rationale

The paper's central construction is explicitly model-based rather than derived from QCD: the density matrix in eq. (12) is introduced as 'the simplest model for the clustering of partons', and the reduction of a general f(p1,...,pn) to f(p) in eq. (22) is asserted from infrared safety and 'usual degeneracy arguments'. Even if this inference is under-derived, it is an assumption about the input state, not an output defined in terms of the structure functions or the doubled optical theorem. The main identity (136) is obtained by comparing two independent computations of the same NLO embeddings: the iterated-discontinuity evaluation of sect. VB and the Reverse Unitarity cut calculation of sect. VC. The agreement is checked explicitly for the triangle, box, and pentagon examples, and no parton-model result is substituted into the definition of WGamma. The NLO structure functions are not fitted: the Lambda^2/Q^2 logarithms emerge from integrating the (p^2)^(-1-epsilon) collinear term, the longitudinal structure functions agree exactly with the parton model, and the F2 finite parts differ, which is the opposite of what would happen if the parton-model answer were being projected onto the model. The pipeline is validated against the independent NLO parton-model calculation of ref. [93]. The LO higher-winding relation f_q(1/q)=q^alpha f_q(1) is explicitly introduced as one possible constraint 'to retrieve the parton model result', not as a predicted consequence; this is under-determination of the model rather than circularity. The truncation of higher-winding embeddings is admitted to be 'the least physically motivated constraint', but this limits the scope of the theorem rather than making the derivation feed on itself. Self-citations to Local Unitarity works are used as computational technology, not as the justification of the doubled optical theorem or of the matching to the parton model. No step reduces by construction to its own inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central claim depends on a uniform density matrix for the initial-state parton cluster, the KLN theorem, a leading-virtuality truncation, and a specific finite set of embeddings with higher-winding diagrams excluded. The only numerical freedom explicitly used to match the parton model is the exponent alpha in the higher-winding LO sum, while Lambda^2 is an arbitrary scale analogous to a factorization scale.

free parameters (2)
  • Clustering scale Lambda^2 = Not fitted; arbitrary input scale
    Introduced in eqs. (5), (18) and (19) as the upper bound on the invariant mass of the initial-state parton cluster. Structure functions depend on it logarithmically, and it plays the role of a factorization scale that is not determined by the model.
  • Exponent alpha in higher-winding PDF matching = Not fitted; chosen to match parton model
    In sect. VIa, the LO higher-winding sum is made equal to the parton model by imposing f_q(1/q) = q^alpha f_q(1). The value of alpha is not predicted by the model and is introduced ad hoc to recover the standard result.
assumptions (5)
  • domain assumption KLN theorem applies to the summed multi-partonic initial and final states
    Used throughout to assert IR finiteness of each embedding class and to motivate the uniform initial-state density; assumes massless QCD degeneracy sums are sufficient.
  • ad hoc to paper Uniform initial-state density with simplest clustering
    Eqs. (12), (20) and (22): the proton is a uniform Fock mixture constrained by momentum conservation, and infrared safety is claimed to force f(p) dependence only. This is the core model assumption.
  • domain assumption Leading-virtuality truncation is physically meaningful
    All results are leading order in Lambda^2/Q^2 via eqs. (36)-(43). The physical significance of this truncation is assumed, and the authors note that discontinuity and virtuality expansion do not commute.
  • ad hoc to paper Exclusion of higher-winding embeddings
    The |r_e| <= 1 filter in sect. IIIB3 is admitted to be the least physically motivated constraint. All NLO numbers depend on this choice of embedding subset.
  • domain assumption Embedding analytic structure is captured by p^2, 1-x and 1-2x
    Eqs. (119) and (120) assert that hard and collinear coefficients have this analytic form; this is the basis for the proof of the doubled optical theorem.
invented entities (1)
  • Vacuum graph embedding (G,w) with a puncture
    purpose: Mathematical classification of infrared-finite classes of interference diagrams and the object whose double discontinuity gives the cross-section.
    No falsifiable observable outside the paper is associated with this construct; its validity is judged by the usefulness of the resulting cross-sections.

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Cite this review

Pith. "Pith review of Multi-partonic interactions, iterated discontinuities and the virtuality expansion in deep inelastic scattering." pith.science (2026). https://pith.science/paper/H6MFI576

@misc{pith2026250616489,
  author       = {Pith},
  title        = {Pith review of: Multi-partonic interactions, iterated discontinuities and the virtuality expansion in deep inelastic scattering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H6MFI576}},
  note         = {Machine review of arXiv:2506.16489}
}
read the original abstract

We introduce a perturbative model that accounts for the contribution of multi-partonic interactions to collider observables. A key feature of this multi-parton model is that cross sections are organised in terms of building blocks that are separately infrared-finite in virtue of the KLN theorem. We find compact expressions for these building blocks in terms of double discontinuities of Feynman-type integrals obtained by endowing a vacuum graph with additional topological information. The framework is applied to the computation of next-to-leading order structure functions in deep inelastic scattering. We comment on the scale dependence of these structure functions and discuss the possibility of relating them by a scheme change to those computed in the parton model. Finally we lay down next steps for analogous computations for deep inelastic scattering at NNLO and Drell-Yan at NLO.

Figures

Figures reproduced from arXiv: 2506.16489 by the authors.

Figure 1
Figure 1. In the above graph, the purple lines denote the deep inelastic scattering electrons, while the dotted [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗
Figure 2
Figure 2. The 5 base cut translations. Notice that all of them disconnect the graph, and bipartition the [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. An isomorphism invariant characterisation of an embedding of the sunrise graph. [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Embeddings may be classified in four different types. If their integrand has both a [PITH_FULL_IMAGE:figures/full_fig_p031_4.png]
Figure 5
Figure 5. Figure 5: Three example embeddings, with the purple line denoting the electron, and dotted line denoting [PITH_FULL_IMAGE:figures/full_fig_p032_5.png]
Figure 6
Figure 6. Figure 6: Interference diagrams belonging to the equivalence class identified by the embedding [PITH_FULL_IMAGE:figures/full_fig_p033_6.png]
Figure 7
Figure 7. Figure 7: Interference diagrams belonging to the equivalence class identified by the embedding thruple [PITH_FULL_IMAGE:figures/full_fig_p033_7.png]
Figure 8
Figure 8. Figure 8: Interference diagrams belonging to the equivalence class identified by the embedding thruple [PITH_FULL_IMAGE:figures/full_fig_p034_8.png]
Figure 9
Figure 9. Figure 9: After routing, we can build the integrand for the three embeddings [PITH_FULL_IMAGE:figures/full_fig_p034_9.png]
Figure 10
Figure 10. Figure 10: The three-point integral, including the additional propagator [PITH_FULL_IMAGE:figures/full_fig_p055_10.png]

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Reference graph

Works this paper leans on

117 extracted references · 25 canonical work pages

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    ground truth

    Partonic tensor as an iterated discontinuity Having detailed the analytic properties of embeddings in sec. VB and those of interference diagrams in sec. VC, we are ready to establish a relationship between the two. We have, by direct comparison of the NLO coefficients in eq. (126),(127)(128) and the Reverse Unitarity calculation: WΓ(p,q ) = discp2disc1−xI...

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    MP cross-section and the KLN theorem Recall that the partonic tensorWq(p,q ) is related to the partonic cross-section by eq. (27). In other words, in order to obtain an object that gives the full cross-section by simple contraction with the leptonic tensor, we first have to integrateWq(p,q ) over p2 taking all values from 0 to the infrared regulator Λ2: W...

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    Triangle diagram Let us start by discussing the case of the embedding in fig. 5a. The embedding reads IΓ1 = T (p,q,−p−q) p2(p +q)2 . (143) 4 In general, the finiteness statement also requires UV renormalisation. In the case of NLO DIS no UV renormalisation procedure is needed, since the electric charge does not receive QCD corrections. Furthermore, contra...

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    7 can be obtained as double discontinuities of the integrand associated to the embedding of fig

    Box diagram We will now show that the interference diagrams of fig. 7 can be obtained as double discontinuities of the integrand associated to the embedding of fig. 5b, which we identified with the box diagram of fig. 9b. The embedding is partial-fractioned into triangles: IΓ2 = B(p,q,−p,−q) p2 = T (p,q,−p−q) +T (−p,q,p −q) p2(p·q) . (152) Although the di...

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    higher- winding

    Pentagon diagram The most involved example we study relates to the embedding of fig. 5c. As we have seen, its integrand is the same as the pentagon of fig. 9c. Upon partial fractioning, the pentagon diagram may be expressed in terms of triangles: IΓ3 =− T (p,q,−p−q) p2q· (p +q) + T (q,p +q,−p− 2q) 4q· (p +q)(p 2 +q)2 + T (p 2,q,−p 2−q) +T (p 2, p 2 +q,−p−...

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    (200) Note that this expression is fully symmetric in its arguments

    = Z dDℓ1dDℓ2 (iπD/2)2 1 ℓ2 1ℓ2 2(ℓ1 +p1)2(ℓ2 +p2)2(ℓ1−ℓ2 +p1)2(ℓ2−ℓ1 +p2)2 . (200) Note that this expression is fully symmetric in its arguments. This integral has been studied frequently in the literature, for example in refs. [95, 96]. We will only compute in an expansion aboutΛ2 = 0, which means we can expand inp2. As for the NLO case we will need this...

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    Double discontinuity For the double discontinuity we directly evaluate the required cuts in the limit of smallp2 using the method of regions. In this way we can obtain expressions that are exact in the dimensional regulator 56 thus allowing to perform the integration inp2 and the extraction of the distributional part in(p +q)2. When expanding inϵ we find ...

  8. [8]

    For this we can specifyp2 1 = 0 and p2 2 > 0, p2 3 < 0 and then obtain the two cases needed for the problem at hand by settingp1 =p,p 2 = p +q and p1 =p +q,p 2 =p respectively

    Single discontinuity We need the integral with a single discontinuity and a on-shell leg. For this we can specifyp2 1 = 0 and p2 2 > 0, p2 3 < 0 and then obtain the two cases needed for the problem at hand by settingp1 =p,p 2 = p +q and p1 =p +q,p 2 =p respectively. We proceed by computing univariate differential equations inz =p2 2/p2

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    in the limitp2→ 0

    The boundary conditions can be fixed by the method of regions e.g. in the limitp2→ 0. From this we can obtain the discontinuity in a form, e.g. for case 2 F4(0,p 2 2,p 2

  2. [10]

    = (−p2 3/µ2)−2ϵf0(p2 2/p2

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    + (−p2 3/µ2)−ϵ(p2 2/µ2)−ϵf1(p2 2/p2

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    (203) Wherefi are analytic atp2 = 0

    + (p2 2/µ2)−2ϵf2(p2 2/p2 3). (203) Wherefi are analytic atp2 = 0. From this expression, it is straightforward to obtain the discontinuity. In order to perform thep2 integration as required for case 2, we expand the functionsfi in small arguments. Keeping the leading order all ...

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    vertical particle

    On-shell integral Finally, we also need the integral with two on-shell legs. Given that this is the hard region of the integral we can obtain it from Eq. (203) C4 =δ((p +q)2)F4(0, 0,q 2) =δ((p +q)2)(−q2/µ2)−2ϵf0(0). (205) Interestingly, the contribution from the double discont...

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    simplest clustering

    In summary, including only the leading-virtuality diagram labellings, and dividing by 2 to average overP1↔P2 symmetry, the cross-section can be written as σ =2δ(x1x2s−m2 ⋆) Z dΠHH ˜δ(m2 1)˜δ(m2 2)Re[T (P1,P 2,−P1−P2)] +˜δ(m2 2) discm2 1 T (P1,P 2,−P1−P2) m2 1 + ˜δ(m2 1) discm2...

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    VC, the Reverse Unitarity calculation takes the virtual bubbleB(p2), the cut bubble /B(p2) and the doubly-cut triangle /T (p2 1,p 2 2,p 2

    Cut masters As mentioned in sect. VC, the Reverse Unitarity calculation takes the virtual bubbleB(p2), the cut bubble /B(p2) and the doubly-cut triangle /T (p2 1,p 2 2,p 2

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    as inputs. The virtual bubble is obtained straight-forwardly by Feynman parametrisation (2π)dB(p2) = Z ddk 1 k2(p−k)2 = iπ d 2 Γ d−2 2 2 Γ 4−d 2 Γ (d− 2) (−p2) d−4 2 63 = iπ2−ϵΓ (1−ϵ)2 Γ (ϵ) Γ (2− 2ϵ) (−p2)−ϵ (A1) where we setd = 4−2ϵ. The cut bubble and the doubly cut triangl...

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    Calculation of Th The hard functionTh required in the region expansion of sect. VB is simply the two-mass triangle, which IBP-reduces to bubbles: Th(p,q ) = (d− 3)[B(q)−B(p +q)] (d− 4)p· (p +q) . (A4) Using the result for the bubble of eq. (A1): (2π)dTh(p,q ) = iπd/2 Γ(1−ϵ)2Γ(...

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    Calculation of Tc The collinear function Tc required in the region expansion of sect. VB is computed in light-cone coordinates. It reads: (2π)d(−p2 1)−ϵTc(p1,p 2) = Z ddk 1 k2(k−p1)2(2k+(n+·p2) +p2 2). (A6) Where we have used light-cone variables,k =k+ˆn+ +k−ˆn− +k⊥, ˆn±·k⊥ = ...

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