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Positive Integrands from Feynman Integrals in the Minkowski Regime

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Feynman parameter integrals in the Minkowski regime can be rewritten as sums of positive integrands with complex prefactors, removing the need for contour deformation.

desk verdict 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. read the letter →

arxiv 2506.24073 v1 pith:IY4XX3KA submitted 2025-06-30 hep-ph hep-th

Stephen Jones , Anton Olsson , Thomas Stone This is my paper
classification hep-phhep-th
keywords FeynmanintegralsparameterspaceMinkowskiregimecontourdeformationpositiveintegrandssectordecompositionunivariatebisectionellipticandhyperelliptic
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 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.

What carries the argument

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 $\{F<0\}\cup\{0<x\}\cup s_R$ for $x_i$, and if the solution has the form $0<x_i<f(x_{\neq i})$ or $f(x_{\neq i})<x_i$, maps the $F=0$ hypersurface to a boundary using $y_i=x_i/(x_i+x_j f(x_{\neq i}))$ or $y_i=x_i+f(x_{\neq i})$. For massive integrals that are not univariate bisectable, the paper uses geometric visualisation of the $F=0$ surface and algebraic transformations involving square roots, as in the elliptic sunrise and banana resolutions.

What would settle it

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.

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Extended reading notes

Core claim

The paper claims that a Minkowski-regime Feynman integral can be decomposed according to Eq. (3.1): $$J(s)=\sum_{n_+} $J^{{+,n_+}}$(s)+\lim_{\delta\to0^+}(-1-i\delta)^{-(\nu-LD/2)}\sum_{n_-} $J^{{-,n_-}}$(s),$$ where the integrands of all $J^{+,n_+}$ and $J^{-,n_-}$ are real and non-negative on the integration domain, and the prefactor $(-1-i\delta)^{-(\nu-LD/2)}$ carries the entire analytic continuation. The $F=0$ variety is mapped to the integration boundary through rational or algebraic changes of variables, so that $F$ no longer vanishes in the interior of any region; singularities remain only on boundaries, where sector decomposition can handle them. The construction is demonstrated on massless boxes, pentagons, non-planar two- and three-loop boxes, massive bubbles and triangles, the two-loop elliptic sunrise, and the three-loop hyperelliptic banana, including examples where contour deformation fails outright.

Load-bearing premise

The representation only works if the coordinate changes that send each sign region of $F$ to the positive orthant are one-to-one, preserve volume, and cover the original integration domain exactly once; the paper verifies this example by example but supplies no general proof.

Editorial extensions

If this is right

  • Contour deformation can be removed for a broad class of parameter integrals, eliminating the complex Jacobian and the instability near pinch or endpoint singularities.
  • Analytic continuation becomes manifest: the branch structure of the integral is encoded entirely in $(-1-i\delta)^{-(\nu-LD/2)}$.
  • Integrals that defeat contour deformation, such as the three-loop crown graph and the two-loop non-planar seven-propagator box, become numerically accessible.
  • Resolved integrands work with sector decomposition, giving speed-ups of up to roughly four orders of magnitude, especially at extreme kinematics and small internal masses.
  • The same construction works for massive integrals with elliptic and hyperelliptic geometry, so the method is not limited to polylogarithmic cases.

Reading between the lines

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

  • If the decomposition is built for every integral in an amplitude, the analytic continuation of the whole scattering process becomes explicit, since each term carries only the phase $(-1-i\delta)^{-(\nu-LD/2)}$.
  • Because every integrand is non-negative, integration strategies designed for Euclidean integrals, such as positivity-constrained fits or neural-network surrogates, could plausibly be applied to Minkowski integrals; the paper notes the possibility but does not test it.
  • The bijectivity assumption is directly testable: for any asserted resolution, one can numerically compare the sum of resolved integrals against a contour-deformed reference, or check the Jacobian mapping for double coverage, without waiting for a general proof.
  • The practical limit of geometric visualisation beyond four propagators is not obviously fundamental; a cylindrical algebraic decomposition, which the authors identify as a general constructive route, could automate the resolution for higher-loop integrals.
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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

4 major / 5 minor

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.

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 (4)
  1. [Section 3.1, Eq. (3.1)] 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.
  2. [Section 3.2, Algorithm 1] 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.
  3. [Section 4.2.5, Eq. (4.140)] 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.
  4. [Section 4.1.6, Eq. (4.84)] 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.
minor comments (5)
  1. [Throughout] 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.
  2. [Section 1 and Section 4.1] 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.
  3. [Section 4.2.4, Eq. (4.128)] 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.
  4. [Section 5] 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.
  5. [Section 3.3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the split-at-F=0 identity is an explicit algebraic rewriting, and examples are validated against independent results.

full rationale

The central representation Eq. (3.1) is not fitted and does not reduce to its inputs by construction: it is an explicit pointwise decomposition of the Feynman parameter integral into regions F>0 and F<0, with the iδ prescription factored into (-1-iδ)^-(ν-LD/2) for negative regions. The paper supplies concrete transformations for each example and checks that they map F=0 to the boundary, cover the original domain, and produce non-negative integrands; these checks are case-by-case verifications, not definitions. The only self-citations (Refs. [23] and [65]) are cited for inspiration and for an earlier 1-loop box resolution that is re-derived in the text; they do not carry the central claim. Numerical results are cross-checked against independent analytic expressions (e.g., Ref. [102] for BNP7, and known bubble/triangle/sunrise results), so the output is externally falsifiable. Gaps such as the lack of a general bijectivity proof or the schematic treatment of the 3-loop banana positive regions are completeness and rigor concerns, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The method is parameter-free; its reliability rests on mathematical assumptions about coordinate changes, branch choices, and distributional integration rather than on fitted constants or new physical entities.

assumptions (4)
  • domain assumption The Feynman i-delta prescription can be represented by a global prefactor of the form (-1 - i delta)^(-(nu - L D / 2)) for regions where F < 0, with the remaining integrand defined using |F|.
    Used in Eq. (3.1) and throughout Section 4 to split the integral into sign-definite pieces; relies on the causal branch of the propagator.
  • domain assumption The sign regions of F can be transformed bijectively to the positive orthant so that F = 0 is mapped to the integration boundary, with no missed or double-counted regions.
    Stated as a requirement in Section 3.1 and Algorithm 1; verified only on a case-by-case basis, not proven in general.
  • standard math Dimensional regularization and sector decomposition can be applied to integrals whose only singularities lie on the integration boundary after resolution.
    Used to evaluate resolved integrands in Section 5; standard in the field but not re-derived here.
  • domain assumption Square-root and other algebraic transformations are admissible changes of variables that preserve the integral and the positivity of U in the massive examples.
    Introduced in Sections 4.2.4 and 4.2.5 with branch choices; no general proof is given that the chosen branches are globally consistent.

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Cite this review

Pith. "Pith review of Positive Integrands from Feynman Integrals in the Minkowski Regime." pith.science (2026). https://pith.science/paper/IY4XX3KA

@misc{pith2026250624073,
  author       = {Pith},
  title        = {Pith review of: Positive Integrands from Feynman Integrals in the Minkowski Regime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IY4XX3KA}},
  note         = {Machine review of arXiv:2506.24073}
}
read the original abstract

We present a method for rewriting dimensionally regulated Feynman parameter integrals in the Minkowski regime as a sum of real, positive integrands multiplied by complex prefactors. This representation eliminates the need for contour deformation, allowing for direct numerical or analytic evaluation of the integrals. We develop an algorithm to construct such representations for a broad class of integrals and demonstrate its generalisation through selected examples. Our approach is applied to integrals up to three loops, including cases with internal masses and off-shell external legs. The resulting expressions are suitable for evaluation using existing techniques, such as sector decomposition, where we observe performance gains of up to four orders of magnitude in certain cases.

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Forward citations

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Reference graph

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