{"id":"83b3f027-8330-4239-bbd4-1b7d3bc318c8","arxiv_id":"1509.07167","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A C++ implementation of the Loop-Tree Duality computes one-loop scalar and tensor integrals up to pentagon and hexagon level, using contour deformation for singular regions.","lead":"This doctoral thesis develops and numerically implements the Loop-Tree Duality, which rewrites one-loop Feynman integrals as sums of tree-level single-cut integrals. It reports a C++ program that computes scalar and tensor one-loop diagrams up to pentagon and hexagon level and checks selected results against LoopTools.","discovery_kind":"unclear","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The only guard against the contour deformation enclosing an unwanted pole is the heuristic λ̄ rescaling of Eq. (7.23), derived from a single on-shell-energy pole; overlapping ellipsoids and the kji,0→0 limit are not covered by that argument.","rationale":"The reader's weakest_assumption points at the contour deformation, which is the right place to look, but frames it as 'hand-chosen parameters'. That framing is slightly off target: λ and A in Eq. (7.18) control accuracy and efficiency, not correctness, and the paper's own λ-scan in §7.6.1–7.6.2 shows the integral is stable across a range of them. The genuinely load-bearing question is narrower: whether the deformed contour stays on the correct side of every dual pole for arbitrary kinematics. The paper's construction is more defensible than the reader allows. Condition (a) of §7.5.2 holds for the full summed deformation, as I verified above, and condition (b) follows from the exponential suppression. The residual soft spot is condition (c), guarded only by the heuristic λ̄ rescaling of Eq. (7.23), which is a one-pole-at-a-time analysis. That is a real gap, but it is bounded: a crossed pole would produce a residue-sized discrepancy against LoopTools, and the paper does run that comparison. So the empirical check is the correct one, and the concern lands only if the comparison fails at some sampled point. The supplied manuscript truncates the numerical tables, so I cannot confirm or refute the agreement here. The paper also acknowledges several limitations that should not be re-raised as new findings: §7.5.2 footnote 1 states that soft and collinear singularities are out of scope because 'we are only dealing with finite diagrams'; §5.2.1 states 'Although no formal proof exists' for the IBP reduction to single poles; and the code is not released. These are honest disclosures, not hidden assumptions. The thesis is a compilation of published work ([1], [2]) plus a new implementation, and the published chapters are supported by their citations. On the central claim, the method is coherent and the validation strategy is appropriate; the verdict CONDITIONAL is the right level, and my read does not move it. I would only add that the abstract's 'for the first time' is consistent with the group's own earlier proceedings [5,6], which are self-cited and do not undercut the claim. One editorial note, not load-bearing: Table 8.4 in the hexagon section repeats the caption of Table 8.3 ('Examples for pentagons involving numerators of tensor rank two and three'), which looks like a copy-paste slip and should be corrected before the results are read as hexagon-level evidence.","tokens_in":66788,"tokens_out":9211,"duration_ms":218706,"concrete_test":"Take a box or pentagon configuration in which two ellipsoid singularities belonging to the same deformation group overlap in loop three-momentum space (for instance a planar box with two time-like separated external momenta), and tune one kji,0 close to zero. Evaluate the dual integral three times: with the published λ=−0.5, A=10⁶; with λ halved; and with λ doubled. If all three agree with LoopTools within the quoted Cuhre/VEGAS error, the deformation is robust at that point and the concern does not land. If the results differ by a residue-sized amount, or if the λ-scan shows a systematic drift, the λ̄ guard of Eq. (7.23) is incomplete and the validation should be restricted to the kinematics actually tested.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the first numerical LTD implementation, validated against LoopTools. For that to hold, the deformed contour must (a) carry the correct i0 sign at every dual pole, (b) vanish at infinity, and (c) not enclose any other pole. Conditions (a) and (b) are argued in §7.5.2: Eq. (7.16) requires κ·qi<0, and the exponential in Eq. (7.18) localizes the deformation. I checked that (a) survives the full sum over pairs, not just term by term: v_ij·qi = |qi| + qj·qi/|qj| ≥ 0, so with λij<0 every term contributes κ·qi ≤ 0. The paper does not spell this out, but the step is correct. Condition (c) rests entirely on Eq. (7.23), which solves q(+)i,0(𝓁+iλ̄κ)=0 for one on-shell energy at a time and caps λ̄ at (1/2)√Y or √(X²−Y/4). That is a single-pole analysis. It does not address configurations where two ellipsoid singularities of the same deformation group overlap in loop three-momentum space, so that the summed deformation of Eq. (7.18) at one pole is dominated by a different pair than the one being guarded. Nor does it address the limit kji,0→0, where the dual prescription −i0 kji,0 degenerates and the required deformation direction becomes ambiguous. If the contour crosses a dual pole, the result differs from LoopTools by a residue, so the empirical comparison is the right test. But the supplied text truncates the tables in §7.7.2–§8.5, so the agreement cannot be verified here, and §7.6 concedes 'very good results for most of the examples', implying some points fail.","agreement_with_reader":"partial"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"This is a doctoral thesis (IFIC, Universitat de València, June 2015) on the Loop-Tree Duality (LTD). After a Standard Model and numerical-integration preamble (Ch. 2), it reviews the derivation of the one-loop duality relation from the residue theorem and contrasts it with the Feynman Tree Theorem (Ch. 3), recasts the relation in a notation that extends to two loops (Ch. 4), treats propagators raised to higher powers by reducing them with integration-by-parts identities before applying the single-pole duality theorem (Ch. 5), analyses the cancellation of singularities among dual integrands and sketches a phase-space mapping to real corrections (Ch. 6), and then presents what the abstract calls the first numerical implementation of LTD: a C++ program built on the Cuba library that evaluates dual integrals with a three-dimensional contour deformation, applied to finite scalar integrals up to the pentagon (Ch. 7) and to tensor integrals up to the hexagon (Ch. 8), with reference values from LoopTools 2.10. The load-bearing claim is the numerical one, stated at abstract:s7; the theoretical chapters largely restate the author's published work [1, 48, 49].","tokens_in":67311,"tokens_out":7112,"duration_ms":125669,"significance":"If the numerical claim holds, the thesis is a useful proof of concept: it shows that dual integrals, which are tree-level objects over a modified phase space, can actually be integrated numerically for multi-leg one-loop configurations, which is the prerequisite for the programme of combining virtual and real corrections in a single Monte Carlo generator. Several things deserve explicit credit. The validation is against LoopTools, an independent implementation, so the central numerical check is not circular. The integrand-level analysis of Ch. 6, in particular the demonstration in Eqs. (6.11)-(6.17) that singularities from intersecting forward hyperboloids cancel between dual contributions, is a genuine structural result and is what makes the dual representation numerically attractive in the first place. The construction of the deformation in Eq. (7.18), with a vector part that orients the shift away from the ellipsoid and a Gaussian suppression that localizes it, is concrete and implementable rather than schematic. The thesis is also candid about scope: it states that only finite diagrams are treated and that the reduction of multiple poles to single poles is not proved in general","major_comments":[{"comment":"The safety argument for the contour deformation is a single-pole argument, and the numerical claim rests on it. Eq. (7.23) solves q^{(+)}_i0(l + i*lambda_bar*kappa) = 0 for one on-shell energy at a time and caps lambda_bar accordingly. It does not address two configurations that the deformation of Eq. (7.18) will actually encounter. First, when two ellipsoid singularities of the same deformation group overlap in loop three-momentum space, the summed kappa at the location of one pole is dominated by a different pair than the one being guarded, so the cap computed for pair (i,j) need not protect the pole of pair (i,k). Second, the limit k_ji,0 -> 0, where the dual prescription -i0*k_ji,0 degenerates and the required deformation direction becomes ill-defined. A concrete test is available and cheap: for the scans of §7.7.3-§7.7.4, plot the relative deviation from LoopTools against (a) the number of overlapping ellipsoids in the group and (b) min|k_ji,0| over the contributing pairs. If the residuals show no structure in either variable, the heuristic is empirically supported even without a proof; if they do, the deformation needs a group-level rather than pair-level safety criterion.","section":"§7.5.2-§7.6, Eqs. (7.18) and (7.23)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There is no quantitative accuracy statement anywhere in the numerical chapters. Agreement with LoopTools is displayed as overlay plots (red curve versus blue points) and as a handful of tabulated phase-space points, but no relative deviation per point, no stated tolerance for what counts as agreement, and no separation of the integration error reported by Cuba from the error introduced by the deformation. For a thesis whose headline claim is a working numerical implementation, the accuracy of that implementation is load-bearing and should be stated as a number. §7.6 concedes that lambda = -0.5 and A = 10^6 'produce very good results for most of the examples', which implies some points do not; those points and their failure mode should be identified rather than absorbed into a scan. Please add, for each of the triangle, box, pentagon and hexagon families, a table of relative deviation against LoopTools with the Cuba error estimate alongside, and state the tolerance within which the method is claimed to work.","section":"§7.7 and §8.2-§8.5, Tables 7.1-8.4 and Figs. 7.13-8.3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The abstract claims a program 'that calculates one-loop scattering diagrams', but what Chapters 7 and 8 actually evaluate are finite scalar and tensor integrals with up to six external legs. No scattering amplitude with gauge vertices, spinors or external polarisation vectors is computed anywhere in the thesis, and §7.5.2 states in a footnote that only finite diagrams are treated, so no ultraviolet- or infrared-divergent configuration is exercised. The gap matters because the motivation offered in Ch. 1 and Ch. 6 is the combination of virtual and real corrections, which is precisely the regime the implementation does not yet touch. The claim should be calibrated to 'one-loop scalar and tensor integrals' in the abstract, with the restriction to finite configurations stated there rather than only in a footnote. Relatedly, the priority claim 'for the first time' at abstract:s7 is not supported by any discussion of prior or concurrent numerical LTD work; a short paragraph placing this implementation against other efforts would make the claim assessable.","section":"Abstract (abstract:s7) versus Chapters 7-8","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The central artefact of the thesis is a computer program, and it is not made available, nor are the per-point inputs needed to reproduce a single number. Appendix C collects the momenta of the example points and scans, which is helpful, but the deformation parameters actually used for each point are not tabulated, and §7.6.2 says the width sometimes has to be chosen differently from the default. As it stands a reader cannot reproduce even one entry of Table 7.3 or Table 8.2. At minimum, tabulate (lambda_ij, A_ij, lambda_bar) alongside each quoted result, and state whether the code can be obtained on request. This is not a demand for a public release; it is a demand that the reported numbers be traceable to stated inputs.","section":"§7.7.1 and Appendix C","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The radial mapping is typeset as r = x/(x-1), which for x in [0,1] runs from 0 to -infinity and has the wrong sign for the intended [0,1] -> [0,infinity) map; it should presumably read r = x/(1-x). The Jacobian for the two-step change of variables in Eqs. (7.8)-(7.9) is stated to be included but never written out; giving it explicitly would let a reader check the normalisation against a known integral.","section":"Eq. (7.9)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The piecewise definition of lambda_bar is discontinuous at X_i^2 = Y_i: the middle branch gives sqrt(X^2 - Y/4) -> sqrt(3Y/4) as X^2 -> Y from below, while the third branch gives 1. The two branches agree at X^2 = Y/2 but not at X^2 = Y. Either the boundary should be adjusted or the jump should be acknowledged, since a discontinuous safety factor is awkward to reason about when the deformation is summed over a group.","section":"Eq. (7.23)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The requirement kappa·q_i < 0 is stated but never checked against the summed deformation of Eq. (7.18). The check is one line and it works: with v_ij = q_i/|q_i| + q_j/|q_j| one has v_ij·q_i = |q_i| + q_j·q_i/|q_j| >= 0, so every term with lambda_ij < 0 contributes kappa·q_i <= 0. Including that line would close the first condition properly rather than leaving it to the reader.","section":"§7.5.2, Eq. (7.16)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The text states that the O(lambda^{-1}) terms also cancel in the sum 'although not shown for simplicity'. Since the point of the section is that the dual integrand is finite in the triple-collinear limit, the subleading cancellation is part of the claim and should either be shown or explicitly stated to have been verified numerically.","section":"§6.2, Eq. (6.15)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The statement that multiple poles can always be reduced to single poles is explicitly conjectural: 'Although no formal proof exists, in all cases studied so far it has been possible... The generality of this result seems plausible [82].' The authors acknowledge this, and the conclusion of §5.3 should be softened to match, since as written it reads as a general prescription rather than a pattern observed on the topologies of Fig. 5.2.","section":"§5.2.2 and §5.3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Table 8.4 carries the caption 'Examples for pentagons involving numerators of tensor rank two and three' but is placed in §8.5, Tensor Hexagons, and its caption is identical to that of Table 8.3. One of the two captions is wrong; given the section, Table 8.4 should refer to hexagons.","section":"Tables 8.3 and 8.4","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The figure numbering has gaps: 7.11, 7.14 and 7.19 are absent, and Figs. 7.15 and 7.20 carry the same caption ('Singularities of a box graph in loop-momentum space and singularity scheme for comparison'). Renumbering and differentiating the captions would help the reader follow §7.7.3.","section":"List of Figures","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Please check the placement of the (4 pi)^{2-epsilon} factor in the definition of c_Gamma; in the standard convention it sits in the denominator. The subsequent comparison with the phase-space volume factor in Eq. (3.50) depends on getting this right.","section":"Eq. (3.45)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There is a consistent layer of typographical errors that should be cleaned up: 'equivalentely' and 'previosly' (Ch. 1), 'strengthend' and 'diagarams' (§2, §2.1.1), 'inportant' and 'basisc' (Ch. 3), 'progam' (§2.3.1), 'remormalization' (§6.3), 'Minkowsky' for Minkowski (Ch. 3), and 'Teoria Cu´ antica de Campos' for 'Teoría Cuántica de Campos' in the Resumen. The inconsistent use of 'divergencies' and 'divergences' should also be unified.","section":"Chapters 1-3, typography","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The Standard Model review in §2.1-§2.2 is generic and largely disconnected from the thesis's contribution, whereas §2.3 on numerical integration is directly relevant to Ch. 7. Trimming the former and expanding the latter, for instance with a sentence on why a deterministic cubature rule is competitive in three dimensions, would serve the thesis better.","section":"§2.1-§2.2","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision"},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"The thing worth your time here is Chapter 7 and 8: an actual C++ program that evaluates dual integrals numerically, with a three-dimensional contour deformation, checked against LoopTools for scalar integrals through the pentagon and tensor integrals through the hexagon. The theory chapters are recaps of the author's published papers [1,2,48,49] and are presented as such; the new content is the implementation and the deformation. That is a genuine enabling step for the LTD program, and the thesis is honest about which parts are borrowed and which are new. The singularity classification in §7.3–7.4 — hyperboloid pairs cancel among dual contributions, ellipsoids need deformation, coupled contributions must share the same deformation — is clean and useful, and the grouping strategy in Fig. 7.4 is the kind of practical detail that saves the next person months. I checked the sign condition κ·qi<0 in §7.5.2; it survives the full sum over pairs, not just term by term, though the thesis doesn't spell that out. The stress-test concern about the deformation is fair and lands. The only guard against the contour enclosing an unwanted pole is the λ̄ rescaling of Eq. (7.23), which solves for one on-shell energy at a time. Overlapping ellipsoids within a deformation group, and the kji,0→0 limit where the dual prescription degenerates, are not covered by that argument. §7.6 concedes the chosen λ=-0.5, A=10^6 give 'very good results for most of the examples', which implies some points don't. The empirical LoopTools comparison is the right test, but the tables are truncated in what I could read, so I can't confirm the agreement or the error budget. Two smaller things: the code isn't public, and §5.2.1 states outright that no formal proof exists that IBP always removes multiple poles — the author flags this himself, so it's a known gap, not a hidden one. Also, everything validated is finite; IR/UV cases are explicitly out of scope (footnote 1, §7.5.2). None of this is fatal. It's a thesis doing a first implementation, and it does it. Worth a referee, mainly to force a proper error budget and a public code release.","headline":"A real first numerical implementation of LTD, but the contour deformation rests on hand-tuned parameters and a single-pole safety argument rather than a proof.","tokens_in":67782,"tokens_out":914,"duration_ms":14509},"pith_extraction":{"msc":[],"pacs":["12.38.Bx","11.15.Bt","02.70.-c"],"model":"selfhosted/deepseek-v4.1-flash","headline":"A one-loop Feynman integral can be rewritten as a sum of tree-level cuts, and now evaluated numerically.","keywords":["loop-tree duality","Feynman integrals","one-loop scattering amplitudes","contour deformation","integration by parts","Monte Carlo integration","perturbative QCD","numerical methods"],"falsifier":"Take a kinematic point outside the tested set — for instance one sitting very close to a forward-backward threshold where two dual propagators nearly coincide — and evaluate it both with this program and with an independent established integrator such as SecDec or LoopTools; a discrepancy that persists above the quoted integration error and does not shrink when the deformation width and scaling are retuned would show that the deformation fails rather than that the integrator is noisy.","tokens_in":68224,"feed_emoji":"⚛️","tokens_out":3961,"duration_ms":74884,"temperature":0.7,"pith_summary":"This thesis sets out to show that a virtual one-loop Feynman integral can be traded for a sum of single-cut, tree-level objects integrated over a three-dimensional phase space, and — the part that is new here — that this rewriting can actually be run as a computer program for multi-leg kinematics. The route is the residue theorem applied to the loop-energy integral: each residue puts one internal line on shell and turns the remaining propagators into 'dual' propagators carrying a modified $i0$ prescription, giving $L^{(1)}=-~L^{(1)}$ with no double or higher cuts. The author then pushes the formalism outward in three directions: to two loops, to propagators raised to higher powers (double and triple poles, reduced away with integration-by-parts identities), and to a study of how singularities are shared and cancelled between dual contributions and between virtual and real corrections. The concrete deliverable is a C++ program built on the Cuba integrators that deforms the loop three-momentum into the complex plane to step around the surviving poles, and whose output is compared against LoopTools for scalar triangles, boxes and pentagons and for tensor numerators up to hexagon level. If the approach holds up, the practical payoff is that loop and real corrections stop being separate computational objects and could be handled inside one Monte Carlo event generator.","feed_headline":"Loop integrals rewritten as tree-level cuts, then computed","feed_subtitle":"A C++ program evaluates dual integrals with contour deformation and matches LoopTools up to the hexagon level.","key_machinery":"The load-bearing object is the dual propagator $G_D(q_i;q_j)=1/(q_j^2-m_j^2-i0~k_{ji,0})$, obtained by evaluating the other Feynman propagators at the residue of the cut line, with $k_{ji}=q_j-q_i$ fixed by external momenta alone. Its virtue is that it can be rewritten as $~L^{(1)}=-~L^{(1)}$-form $~L^{(1)}=-~L^{(1)}$, so that each dual integral is a genuine three-dimensional integral over the on-shell forward hyperboloid of one internal line, with the energy already fixed. Two singularity types follow from this form: the ellipsoid ($k_{ji}^2-(m_j+m_i)^2>0$, $k_{ji,0}<0$) and the hyperboloid ($k_{ji}^2-(m_j-m_i)^2<0$). The second piece of machinery is the contour deformation $~L^{(1)}=-~L^{(1)}$ with $i~L^{(1)}=-~L^{(1)}$, a sum over ellipsoids of a vector part pointing outward from the singularity surface times a Gaussian-like suppression $~L^{(1)}=-~L^{(1)}$, whose scale $~L^{(1)}=-~L^{(1)}$ and width $~L^{(1)}=-~L^{(1)}$ are tuned per contribution and then globally rescaled by a safety factor derived from requiring that the deformed on-shell energy not cross another pole.","core_discovery":"The central claim is a first: the Loop-Tree Duality has been implemented as a working numerical program that computes one-loop scattering diagrams. Concretely, the thesis derives the dual representation $L^{(1)}(p_1,...,p_N)=-~L^{(1)}(p_1,...,p_N)$ with $~L^{(1)}=-i$-normalised single-cut integrals $~L^{(1)}=-~L^{(1)}$ built from $~L^{(1)}=-~L^{(1)}$-type terms, then shows how to organise the numerical evaluation: map the infinite loop three-momentum onto the unit cube in spherical coordinates, classify each dual propagator's singularity as a forward-forward hyperboloid (which cancels between dual contributions) or a forward-backward ellipsoid (which must be dodged), group the dual contributions that share hyperboloids so they receive an identical deformation, and integrate with Cuhre or VEGAS. The program is validated point-by-point and by scans against LoopTools, including scans through thresholds and through the region where a deformation becomes mandatory and the integrand turns complex. Alongside this, the thesis extends the formalism itself: a Lorentz-invariant residue formula for double poles, an integration-by-parts strategy that removes multiple poles before duality is applied at two and three loops, and an argument that singularities from intersecting forward light-cones cancel at the integrand level while the survivors are confined to a finite region of loop momentum of the order of the external momenta — which is what makes a local map onto the real-correction phase space conceivable.","pith_inferences":["The natural next test is a process where the dual and real integrands are actually summed before integration rather than compared afterwards; the thesis argues the cancellation is local but does not demonstrate it numerically.","Since the deformation parameters are hand-tuned per contribution and then rescaled, an adaptive scheme that reads the singularity structure off the external momenta would remove the main source of fragility and is the obvious engineering step.","The tensor results up to hexagon level suggest the method is not blocked by numerator algebra, which matters because integration-by-parts reduction of multiple poles generates exactly such numerators.","The framework's real target — NLO cross sections with many external legs — would require the dual integrand to be built from full scattering amplitudes rather than scalar integrals, including gauge-dependent numerator terms the thesis explicitly sets aside."],"forward_implications":["Dual integrals are three-dimensional rather than four-dimensional, so the numerical cost of a one-loop virtual correction scales like a phase-space integral rather than a loop integral.","Because forward-forward hyperboloid singularities cancel between dual contributions at the integrand level, a large class of integrable singularities needs no contour deformation at all.","Multiple poles, unavoidable at two loops, can be removed by integration-by-parts reduction without reducing to a master-integral basis, leaving the single-pole duality relation applicable unchanged.","The surviving singularities are confined to loop momenta of the order of the external momenta, which is the precondition for mapping the virtual region onto the real-correction phase space and cancelling soft and collinear divergences locally.","A working dual-integral evaluator makes it plausible to combine loop and tree contributions inside a single Monte Carlo generator instead of subtracting them against each other."],"supporting_citations":[{"why":"Supplies the one-loop duality relation and its comparison with the Feynman Tree Theorem that the derivation in Chapter 3 follows.","marker":"[48]"},{"why":"Supplies the extension of the duality relation to multi-loop diagrams, on which the two-loop formula of Chapter 4 rests.","marker":"[49]"},{"why":"The author's own paper on the duality relation beyond simple poles, the basis for the double-pole residue and the integration-by-parts strategy.","marker":"[1]"},{"why":"The author's own paper on the singular behaviour of scattering amplitudes, which carries the analysis of singularity cancellation among dual contributions.","marker":"[2]"},{"why":"The Cuba library, providing the Cuhre and VEGAS integrators used for every numerical result reported.","marker":"[71]"},{"why":"LoopTools 2.10, the source of all reference values against which the dual-integral program is checked.","marker":"[98]"},{"why":"FIRE and Reduze 2, the automated reduction codes used to remove double and triple poles at two and three loops.","marker":"[79, 80]"},{"why":"Supplies the conditions a valid contour deformation must satisfy, which the three-dimensional deformation in Chapter 7 is built to meet.","marker":"[87]"}],"fun_headline_variants":["A working numerical code for loop-tree duality at one loop","Dual integrals checked against LoopTools through hexagon scans","Loop momentum mapped to a cube, ellipsoid poles dodged","Double poles removed by integration by parts before duality","Dual singularities cancel; survivors stay near external momenta"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The numerical agreement rests on a contour deformation whose shape and scale ($~L^{(1)}=-~L^{(1)}$) were chosen by hand and checked only on the momentum configurations that were tested, so the claim that the program computes one-loop diagrams generally depends on that deformation avoiding every pole and leaving the integral unchanged for kinematics nobody sampled; separately, the author acknowledges that no formal proof exists that integration-by-parts reduction always eliminates multiple poles, only that it did so in every case studied.","fun_headline_variants_meta":{"raw":{"variants":["A working numerical code for loop-tree duality at one loop","Dual integrals checked against LoopTools through hexagon scans","Loop momentum mapped to a cube, ellipsoid poles dodged","Double poles removed by integration by parts before duality","Dual singularities cancel; survivors stay near external momenta"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":2307,"prompt_tokens":1077,"completion_tokens":1230,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":1161}},"tokens_in":1077,"tokens_out":1230,"duration_ms":26572,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":1161,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-14T18:26:07.609417+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"pipeline_version":"pith-pipeline@v1.5.0-archive-selfhost","input_kind":null,"input_chars":null,"input_total_chars":null,"passes_missing":null,"published_standing":null,"published_confidence":null,"objection_audit":null,"run_record":null,"basis":{"kind":"not_recorded","source":null,"chars_given":null,"chars_total":null},"receipt":null,"falsifier":"Take a kinematic point outside the tested set — for instance one sitting very close to a forward-backward threshold where two dual propagators nearly coincide — and evaluate it both with this program and with an independent established integrator such as SecDec or LoopTools; a discrepancy that persists above the quoted integration error and does not shrink when the deformation width and scaling are retuned would show that the deformation fails rather than that the integrator is noisy.","supporting_citations":[{"cited_title":"An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC","cited_arxiv_id":"hep-ph/0703012","evidence_quote":"Supplies the one-loop duality relation and its comparison with the Feynman Tree Theorem that the derivation in Chapter 3 follows."},{"cited_title":"From loops to trees by-passing Feynman's theorem","cited_arxiv_id":"0804.3170","evidence_quote":"Supplies the extension of the duality relation to multi-loop diagrams, on which the two-loop formula of Chapter 4 rests."},{"cited_title":"write newline","cited_arxiv_id":null,"evidence_quote":"The author's own paper on the duality relation beyond simple poles, the basis for the double-pole residue and the integration-by-parts strategy."},{"cited_title":"Tree-Loop Duality Relation beyond simple poles","cited_arxiv_id":"1211.5048","evidence_quote":"The author's own paper on the singular behaviour of scattering amplitudes, which carries the analysis of singularity cancellation among dual contributions."},{"cited_title":"Introduction to Monte Carlo methods","cited_arxiv_id":"hep-ph/0006269","evidence_quote":"The Cuba library, providing the Cuhre and VEGAS integrators used for every numerical result reported."},{"cited_title":"Wolfram Research, `` Mathematica '', Wolfram Research, Inc., version 8.0 ed., 2010","cited_arxiv_id":null,"evidence_quote":"LoopTools 2.10, the source of all reference values against which the dual-integral program is checked."},{"cited_title":"Choosing integration points for QCD calculations by numerical integration","cited_arxiv_id":"hep-ph/0103262","evidence_quote":"Supplies the conditions a valid contour deformation must satisfy, which the three-dimensional deformation in Chapter 7 is built to meet."}],"lane_accuracy":{"status":"not yet measured","lane":{"model":"selfhosted/deepseek-v4.1-flash","input_kind":null},"url":"https://pith.science/accuracy"},"review_version":1}