{"id":"495c7f09-6c26-4287-95eb-415dc7afd4a2","arxiv_id":"2506.24073","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A method for converting Minkowski-regime Feynman parameter integrals into sums of real, positive integrands with complex prefactors, eliminating contour deformation and speeding up numerical evaluation.","lead":"A new method rewrites hard Feynman integrals in the physical Minkowski region as sums of real, positive integrands with simple complex prefactors, removing the need for contour deformation. It speeds up existing numerical evaluations by up to four orders of magnitude in some multi-loop examples.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The split-at-F=0 identity and the bijectivity of the algebraic transformations for massive examples are asserted rather than proven; failure of either would break the central claim.","rationale":"The reader's verdict of CONDITIONAL is appropriate: the paper is a proof-of-principle with explicit constructions and numerical validations, but the central decomposition identity and the bijectivity/positivity of the transformations for the massive examples are not proven in generality. I agree with the reader that the unproven coverage and measure-preservation of the transformations is a key weak assumption, and I add that the even more basic question—whether splitting at F=0 and replacing the iδ prescription by a phase factor yields the exact original integral—is also asserted rather than derived. This concern is not an accusation of error; the examples strongly suggest the method works, and the numerical checks against known results provide independent support. However, without a general proof or a formalization, the method remains conditional. A concrete test on the 1-loop off-shell box at non-integer ε would directly probe the load-bearing assumption, because it isolates the iδ-to-phase reduction from the more complicated algebraic transformations. If that test passes, the remaining concern is the generality of the algorithm, which the paper itself acknowledges by limiting the claim to 'selected examples' and by noting that a general solution requires cylindrical algebraic decomposition. Given the paper's own scoping, the reader's CONDITIONAL verdict should stand unchanged; this stress-test identifies a specific verification that would strengthen the case for a future ACCEPT.","tokens_in":45001,"tokens_out":6451,"duration_ms":88375,"concrete_test":"Independently re-derive Eq. (3.1) for the 1-loop box with one off-shell leg (Section 4.1.2) by starting from the original (F−iδ) integral, splitting the real integration domain at F=0, and applying the paper's transformations, keeping ε explicit. Then evaluate both sides with high-precision numerical integration at a non-integer value, e.g. ε=0.3, and compare results. If the sum of the split integrals with the phase factor does not match the known analytic continuation of the box integral, the representation fails for non-integer exponents; if it matches, the identity still needs a proof for general integrals, but this check would settle whether the iδ-to-phase reduction is valid in the simplest nontrivial case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central representation Eq. (3.1) claims that the original Minkowski-regime integral equals a sum of integrals over F>0 and F<0 regions, with the iδ prescription reduced to a phase factor. This is not a trivial identity: splitting the real domain exactly at F=0 changes the integration contour from one that avoids the singular hypersurface to integrals that approach it as a boundary. For non-integer (ν−LD/2), the original integral picks up a branch determined by iδ, whereas each split integral is a separately singular real integral. The paper asserts the equality and verifies it numerically for specific examples, but provides no contour-deformation argument that boundary terms at F=0 vanish or cancel. For the massive examples (Sections 4.2.4 and 4.2.5), the transformations involve square roots and the regions are described geometrically; the paper states that the five positive banana integrands are obtained by 'a similar analysis' without writing them out, and the positivity of factors such as R3, R4, R5 in Eq. (4.129) is asserted, not demonstrated. If either the decomposition identity or the bijectivity/positivity of these transformations fails for some integral within the claimed class, the method would produce incorrect results without any contour deformation to flag the problem. The reader's weakest assumption identifies the same structural gap; I agree that this is the load-bearing point, but emphasize that it applies equally to the fundamental split identity, not only to the coordinate maps.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a method for rewriting dimensionally regulated Feynman parameter integrals in the Minkowski regime as a sum of integrals over regions where the second Symanzik polynomial F is strictly positive or strictly negative, multiplied by complex prefactors. The authors present an algorithmic procedure for a class of integrals they call univariate bisectable, and demonstrate it on massless examples up to three loops, including non-planar boxes and the crown graph. They then extend the construction by case-specific geometric/algebraic transformations to massive examples, including the 1-loop triangle, the 2-loop elliptic sunrise, and the 3-loop hyperelliptic banana. Numerical benchmarks with pySecDec show substantial speedups and, in some cases, convergence where contour deformation fails. The central claim is that the resulting integrands are manifestly non-negative and that the iδ prescription is fully encoded in an overall phase factor.","tokens_in":45268,"tokens_out":5598,"duration_ms":65998,"significance":"If the central claims hold, this would be a practically useful alternative to contour deformation for numerical and semi-numerical evaluation of Feynman integrals in physical kinematics. The massless resolved integrands are explicit and are validated against independent analytic results, most notably Ref. [102] for the 2-loop non-planar boxes, and the numerical benchmarks show gains of several orders of magnitude in difficult kinematic regions. The construction is parameter-free: no fitted parameters enter, and the resolved integrands are derived rather than inferred. The paper also makes a conceptual contribution by tying the iδ prescription to a simple prefactor and by showing that elliptic and hyperelliptic integrals can be brought to positive-integrand form. However, the proof-of-principle status is evident: the general validity of the split identity, the bijectivity of the transformations, and the claimed positivity of several massive integrands remain asserted rather than rigorously established.","major_comments":[{"comment":"The central decomposition identity is asserted rather than derived. Splitting the real integration domain at the F=0 hypersurface changes the contour from one that avoids the singular surface to integrals that approach it as a boundary. For non-integer ν−LD/2, each split integral is separately singular on F=0, and the equality requires a precise prescription for these boundary singularities and a proof that no boundary terms survive. The paper verifies Eq. (3.1) for individual examples, but the method is claimed for a broad class of integrals. Please provide a contour-deformation or analytic-continuation argument establishing Eq. (3.1), or explicitly restrict the claim to cases where the identity is proven, stating the conditions on ν, D, and F under which the split integrals are well defined.","section":"Section 3.1, Eq. (3.1)"},{"comment":"Algorithm 1 returns success as soon as Reduce produces a set of the form (3.5) or (3.6) for a single variable xi. It does not check that the F<0 set has only the component described by that interval, that the transformation yi or y'i maps the original domain onto the new domain exactly once, or that the F>0 complement is covered by the companion transformation. For a quadratic F, the F<0 set can have two components and a single interval check is insufficient. The examples are verified case by case, but the stated scope of 'univariate bisectable integrals' needs a precise sufficient condition on F and sR under which the returned transformations are bijections and the Jacobians keep all integrands positive. Without this, the algorithm is a heuristic that succeeds on the examples shown.","section":"Section 3.2, Algorithm 1"},{"comment":"The six-region decomposition of the 3-loop banana is summarized schematically: the negative integrand is stated, but the five positive integrands are not written out, and the text says 'A similar analysis is carried out.' Since the central claim for the hyperelliptic case is that all resulting integrands are manifestly non-negative over the integration domain, the absence of the explicit positive integrands prevents verification of positivity and of the covering property. Please provide the complete set of transformed integrands (or a supplementary file) for all six regions. Similarly, in Section 4.2.4 the positivity of the factors R1–R5 in Eq. (4.129) is asserted but not demonstrated; please include an argument or a systematic check over the full parameter range.","section":"Section 4.2.5, Eq. (4.140)"},{"comment":"The crown-graph decomposition is stated as a sum of twelve integrals, but the positive resolutions of integrals B and D are deliberately omitted ('For brevity, we do not state all of the positive resolutions'). Given that this example is used to demonstrate that the method works when contour deformation fails, the complete set of resolved integrands should be available, at least as an ancillary file, so that Eq. (4.84) can be checked term by term.","section":"Section 4.1.6, Eq. (4.84)"}],"minor_comments":[{"comment":"The notation 'lim δ/∫hortrightarrow0+' appears repeatedly (e.g. Eq. (2.1), Eq. (3.2)) and should be typeset as δ→0+; this appears to be an OCR or conversion artifact that should be corrected in the final version.","section":"Throughout"},{"comment":"The text contains garbled expressions such as '2 /∫hortrightarrow2 scattering' and 'resulting in6'; these should be cleaned up to '2→2 scattering' and 'resulting in' respectively.","section":"Section 1 and Section 4.1"},{"comment":"The expression for f(x1) in Eq. (4.128) is extremely unwieldy and the chosen branch of the square root is not discussed. Introducing auxiliary variables and stating the branch selection would substantially aid readability and verifiability.","section":"Section 4.2.4, Eq. (4.128)"},{"comment":"The benchmark discussion references many figures (Figures 18–28) and tables; please ensure that all figures are present with captions in the final submission, since the extracted text does not contain them.","section":"Section 5"},{"comment":"The connection to cylindrical algebraic decomposition is mentioned but not developed. A short description of how a CAD-based resolution would certify coverage and positivity would strengthen the claim that the method is generalizable beyond the examples.","section":"Section 3.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is a credible and useful proof of principle, and the numerical results appear solid for the examples shown. The main risk is the unproven split identity in Eq. (3.1) and the lack of explicit positivity/covering checks for the massive examples. I do not think this requires rejection, because the identity is plausible and can likely be proven for the stated class, but it is load-bearing and must be addressed before publication. The manuscript fits the scope of JHEP."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a solid proof-of-principle that mostly delivers on its main promise. The genuinely new content is real: the univariate bisection algorithm and the explicit resolutions of the two-loop non-planar boxes, the crown graph, the massive triangle, the elliptic sunrise, and the hyperelliptic banana go well beyond the earlier PoS proceedings. The resolved integrands are written out for most cases and checked against known analytic results, including the non-planar two-loop box from Tausk and standard bubble/sunrise results. The numerical benchmarks are also convincing where they matter most: for BNP7 and the crown graph, contour deformation fails outright, while the resolved integrals converge to useful precision. That is a strong practical demonstration.\n\nThe soft spots are real but not fatal. The central identity Eq. (3.1) is asserted and verified case by case, but no general contour-deformation argument is given for why splitting at F = 0 and treating the negative region as a real integral with a phase factor is valid for non-integer regulators. The stress-test note is right that this is the load-bearing point. The coordinate maps in the massive examples involve square roots and the positivity of factors like R3, R4, R5 in the sunrise is asserted, not shown; for the three-loop banana, the positive integrands are not written out at all. The paper is honest about these gaps and points to cylindrical algebraic decomposition as a future general algorithm, which is fair for a proof-of-principle. A referee should push for a rigorous proof of the decomposition identity and for at least one fully explicit algebraic-geometry check on a non-trivial massive example.\n\nMinor issues: the headline speedups sometimes rely on skipping coefficients of spurious poles, and the pySecDec comparisons exclude prefactor parsing time. That makes the speedup factors optimistic, but the qualitative conclusion still holds because several contour-deformed runs never converged. The citation pattern looks clean; the self-citation to the authors' own proceedings is appropriate since this work substantially extends it.\n\nWho should read it: anyone doing numerical multi-loop integrals in physical regions, and people interested in positive representations of amplitudes. It is not a finished automated tool, but it is an important step in that direction.\n\nRecommendation: send it to peer review. The referee should require a proof or a much more careful statement of the decomposition identity, but the evidence is strong enough that this deserves referee time rather than a desk rejection.","headline":"A genuine proof-of-principle that Minkowski-regime Feynman parameter integrals can be rewritten as positive integrands without contour deformation, with the main open question being a general proof of the split identity.","tokens_in":150,"tokens_out":1831,"would_cite":true,"duration_ms":34351,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Feynman parameter integrals in the Minkowski regime can be rewritten as sums of positive integrands with complex prefactors, removing the need for contour deformation.","keywords":["Feynman integrals","Feynman parameter space","Minkowski regime","contour deformation","positive integrands","sector decomposition","univariate bisection","elliptic and hyperelliptic integrals"],"falsifier":"Numerically integrate the full resolved sum for the 3-loop banana, whose five positive-region integrands are asserted rather than written out, at a kinematic point such as $\\beta=0.8$, $m=2$, and compare with a contour-deformed reference evaluation; any mismatch beyond integration error, or an explicit check showing that a transformation double-covers or misses part of the original simplex, would disprove the bijectivity assumption.","tokens_in":44808,"feed_emoji":"⚛️","tokens_out":10106,"duration_ms":103823,"temperature":0.7,"pith_summary":"This paper tries to establish that dimensionally regulated Feynman parameter integrals in the Minkowski regime, where the second Symanzik polynomial $F$ changes sign inside the integration domain, can be rewritten as a sum of integrals with strictly non-negative integrands multiplied by complex prefactors. The central move is to map the hypersurface $F=0$ to the boundary of integration, so the causal $i\\delta$ prescription is absorbed into a single overall phase and contour deformation is never needed. The authors supply an algorithm for a class they call univariate bisectable integrals and construct resolutions by hand for massive, elliptic, and hyperelliptic examples up to three loops. If the representation is valid, direct numerical evaluation becomes faster and more stable, with reported speed-ups of up to four orders of magnitude in sector-decomposition benchmarks.","feed_headline":"Minkowski Feynman integrals become sums of positive integrands","feed_subtitle":"Mapping the F=0 surface to the integration boundary removes contour deformation and speeds up numerical evaluation.","key_machinery":"The central object is the Feynman parameter integral of Eq. (2.1), built from the Symanzik polynomials $U(x)$ and $F(x;s)$, and the decomposition identity Eq. (3.1) that splits the integration domain into regions of definite sign of $F$. For the algorithmic class, the load-bearing tool is univariate bisection (Algorithm 1): one iterates over Feynman parameters $x_i$, solves the inequality system $\\{F<0\\}\\cup\\{0