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Efficient Computation of One-Loop Feynman Integrals and Fixed-Branch Integrals to High Orders in $\epsilon$

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The dimension-changing transformation rewrites one-loop Feynman integrals and fixed-branch integrals as a single contour integral over an auxiliary mass, so that solving differential equations once in one dimension yields the full…

desk verdict The DCT trick is a real efficiency win for high-order one-loop epsilon expansions, but the unproved contour condition needs scrutiny before the method is used outside the paper's safe zone. read the letter →

arxiv 2412.21054 v1 pith:C2JEK2Y3 submitted 2024-12-30 hep-ph

Rui-Jun Huang , Dong-Shan Jian , Yan-Qing Ma , Dao-Ming Mu , Wen-Hao Wu This is my paper · ORCID
classification hep-ph MSC 81T1881T1581-08 PACS 11.15.Bt
keywords dimension-changingtransformationone-loopFeynmanintegralsfixed-branchepsilonexpansionauxiliarymassflowmastercontourdeformationdimensionalregularization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a dimension-changing transformation (DCT) that relates a one-loop Feynman integral or a fixed-branch integral (a Feynman-parameter integral representing a branch of a multi-loop diagram) in any spacetime dimension to the same family of integrals carrying an auxiliary mass in a different dimension, through a single contour integral over that mass. Because the dimensional regulator $\epsilon$ enters the contour integral only through the factor $\eta^{\delta-1}/\Gamma(\delta)$, the $\epsilon$-expansion can be extracted by expanding that factor and integrating the auxiliary-mass series term by term, so the differential equations in the auxiliary mass need to be solved only once, at one nonsingular dimension. The transformation is implemented in an open-source C++ package and demonstrated on a five-point and a six-point one-loop family and a two-loop six-point fixed-branch family, producing Laurent expansions to $O(\epsilon^{10})$ with 14-15 digit agreement against existing results. If correct, the method removes the need to solve differential equations at many $\epsilon$ values for one-loop and fixed-branch integrals, which is the main computational bottleneck for high-precision predictions.

What carries the argument

The central object is the dimension-changing transformation, Eq. (10), a contour-integral identity expressing an integral in $D$ dimensions as an integral over the auxiliary mass $\eta$ of the same integral in $D_0=D-2\delta$ dimensions, with the contour taken along the negative imaginary axis. The work it does is to separate the $\epsilon$-dependence from the hard part of the calculation: after the auxiliary-mass integrals are expanded in power series around $\eta=0$, $\eta=\infty$, and regular points, each term is integrated against $\eta^{\delta-1}$ analytically, and $\Gamma(\delta)$ and $\eta^{\delta-1}$ are expanded in $\epsilon$ before integration. The identity is obtained by splitting the loop momentum into $D_0$ parallel and $D-D_0$ orthogonal components, integrating out the orthogonal angular variables, and then deforming the radial $\eta$ integration path from the positive real axis to the negative imaginary axis. The paper also gives the auxiliary-mass differential equation used to generate the series expansions, with boundary conditions at $\eta\to\infty$.

What would settle it

Set up a one-loop integral with a Breit-Wigner propagator mass $m^2 = M^2 - iM\Gamma$ and physical scattering kinematics, compute the auxiliary-mass integral along the negative imaginary axis, and scan the fourth quadrant numerically for poles; if a pole is found, the DCT contour integral in Eq. (10) will not reproduce the original integral, and comparing DCT with a direct numerical integration at that point would reveal the discrepancy.

Watch

Extended reading notes

Core claim

The central claim is the identity in Eq. (10): $$I^\Delta_{\vec\nu}(X) = \frac{1}{\Gamma(\delta)} \int_{-i\infty}^{-i0^+} d\eta\, \$eta^{{\delta-1}}$\, I^{\$\Delta$-\delta}_{\vec\nu}(X,\eta),$$ where $I^{\Delta-\delta}_{\vec\nu}(X,\eta)$ is the fixed-branch integral with auxiliary mass $\eta$ in a dimension $D_0=2(\Delta-\delta)$, and $I^\Delta_{\vec\nu}(X)$ is the same integral in a different dimension $D=2\Delta$ after removing the auxiliary mass. The same relation reduces to a one-loop Feynman integral when there is a single branch. The authors argue that, because the $\epsilon$-dependence sits in the prefactor and in $\eta^{\delta-1}$, one can solve the auxiliary-mass differential equations once at a conveniently chosen non-integer dimension and then produce the full Laurent series in the dimensional regulator at very low cost. Their numerical examples confirm that this one-solution-then-transform route reproduces known results to 14-15 digits, including cases with vanishing Gram determinants.

Load-bearing premise

The whole derivation rests on the claim that the auxiliary-mass integral $I^{D_0}(\eta)$ has no poles in the fourth quadrant of the $\eta$-plane, so the contour may be rotated from the positive real axis to the negative imaginary axis; the paper states this condition but never proves it for arbitrary kinematics, complex masses, or general fixed-branch integrals.

Editorial extensions

If this is right

  • High-order $\epsilon$ expansions of one-loop master integrals can be obtained by solving the auxiliary-mass differential equations at a single value of $\epsilon$, so the wall time grows only with the cost of the DCT integration, not with repeated differential-equation solves.
  • The same efficiency extends to fixed-branch integrals, which appear in the representation of multi-loop integrals; this opens a route to high-precision multi-loop predictions once the fixed-branch decomposition is known.
  • The method handles real and complex kinematic configurations, including Breit-Wigner propagators, provided the auxiliary-mass integral has no fourth-quadrant poles, so it applies to physical scattering regions rather than only Euclidean kinematics.
  • Exceptional cases such as vanishing Gram determinants are handled by the construction of the differential-equation system, and arbitrary precision is available through the package's support for high-precision complex arithmetic.

Reading between the lines

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

  • Editorial inference: the same dimension-changing trick could be applied to any master-integral family whose auxiliary-mass differential equations are already solved, converting a stock of existing solutions at one dimension into $\epsilon$-expansions at many dimensions without new differential-equation work.
  • Editorial inference: if the fourth-quadrant analyticity condition can be proven for broad classes of kinematics, DCT may become a general-purpose tool for evaluating Feynman integrals directly at complex or non-integer dimensions, not just at $D=4-2\epsilon$.
  • Editorial inference: the Laplace-transform variant sketched in the outlook suggests a family of dimension-changing identities parameterized by different integration weights; comparing their numerical convergence might yield even faster contour choices for specific 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

3 major / 4 minor

Summary. This paper introduces a dimension-changing transformation (DCT) that relates one-loop Feynman integrals and fixed-branch integrals (FBIs) with an auxiliary mass in D0 dimensions to the corresponding integrals without the auxiliary mass in D dimensions, via an integral over the auxiliary parameter η (Eqs. (4)-(5) and (10)). The authors propose to solve auxiliary-mass differential equations at a single non-singular dimension and then evaluate the DCT integral term by term, which yields the full ϵ expansion at low cost. The method is implemented in an open-source C++ package and demonstrated on a five-point one-loop topology, a massless six-point one-loop topology, and a two-loop three-branch FBI example, with comparisons to AMFlow and LoopTools.

Significance. The DCT identity is derived from first principles with no fitted parameters, and the manuscript provides an open-source implementation and numerical evidence of efficiency gains over AMFlow for high ϵ orders. If the contour-deformation assumption is justified, the method would be a useful tool for high-precision one-loop and FBI computations. However, the paper currently relies on an unpublished companion paper for the FBI differential equations, and the central analytic-continuation step is asserted rather than proved, so the significance is conditional on resolving these points.

major comments (3)
  1. [II.A (Eq. (5)) and II.B (Eq. (10))] The deformation of the integration contour from the positive real axis to the negative imaginary axis is stated to be valid provided the auxiliary-mass integral has no poles in the fourth quadrant, but no proof or sufficient condition is given. For one-loop integrals, the poles of I^{D0}_ν(η) in the η-plane are threshold singularities; for complex masses or general kinematic regions their location is not obvious. For FBIs, the integrand in Eq. (6) has singularities at η = -F(X,y), and nothing in Sections II.B or III shows that these singularities cannot enter the fourth quadrant for allowed complex branch parameters X or complex masses. If a fourth-quadrant pole is crossed, Eq. (5)/(10) differs from the original integral by the residue of that pole, and the package would return an incorrect result without warning. The numerical examples in Section IV all use real X and Feynman-prescription kinematics and therefore cannot expose this issue. Please provide a proof or a precise, checkable condition for the absence of fourth-quadrant poles, and add a numerical test with complex masses and/or complex branch parameters.
  2. [III, Eq. (11)] The differential equations for FBIs are quoted from Ref. [40], which is unpublished and cited with a placeholder arXiv number (arXiv:2412.xxxxx). Because the FBI differential equations are essential for the method, the manuscript is not self-contained and cannot be independently checked. The authors should either include a derivation of Eq. (11) and the construction of the matrix S and the constants C_b and z_α, or cite a publicly available version of [40], and should clearly state which results are inherited from [40] and which are new in this work.
  3. [Table II] Table II lists the precision of the hexagon calculation using 1000-digit boost::mpc as '-3 -5 -4 ...' for the ϵ orders. This is inconsistent with the definition precision = -log10(ε) with ε the relative error, since negative values would imply relative errors larger than one. If these are typographical errors (e.g., missing decimal points or exponents), please correct them; as written, the table does not support the claimed high-precision validation of the hexagon example.
minor comments (4)
  1. [III, after Eq. (10)] The sentence beginning 'since ∆ − δ is fixed' should have 'Since' capitalized at the start of the sentence.
  2. [IV, first example] The word 'diferential' should be spelled 'differential' in the description of solving the η differential equations.
  3. [References] Reference [40] is listed with a placeholder arXiv number 'arXiv:2412.xxxxx'; this should be updated once the companion paper is available.
  4. [Fig. 3 caption] The caption states 'Time spent for computing a pentagon family...' but the y-axis is labeled 't(ms)'; please define the symbol t explicitly in the caption.

Circularity Check

0 steps flagged · score 1.0 of 10

DCT identity is a self-contained change of variables; no circular step found.

full rationale

The central claim is the dimension-changing identity, Eq. (4) for one-loop Feynman integrals and Eq. (10) for fixed-branch integrals. Both are derived directly in the text rather than assumed: Eq. (4) follows from splitting the loop momentum into parallel and transverse components, integrating over the transverse angular variables, and substituting l_perp^2 = eta; Eq. (10) follows from the Feynman-parameter representation by explicitly evaluating the integral over eta, giving Gamma(delta) I^{Delta+delta}_nu(X). The epsilon-dependence enters only through eta^{delta-1} and Gamma(delta), which are expanded analytically, while the auxiliary-mass input I^{Delta-delta}_nu(eta) is solved once at a fixed non-integer dimension D0 = 7/13. No parameter appearing in the examples is fitted to the target integrals, and the reported high-epsilon coefficients are not imposed by the input data by construction. The contour deformation from Eq. (4) to Eq. (5) is justified by the stated condition that I^{D0}_nu(eta) has no fourth-quadrant poles; this condition is asserted rather than proven for arbitrary complex kinematics and fixed-branch integrals, which is a rigor gap but not a circularity. The numerical inputs are obtained by solving differential equations cited from the authors' companion work [40] and following the AMFlow method [36,39]; these are dependencies with overlapping authorship, but they supply the integrand, not the claimed transformation. The DCT identity is independently cross-checked against directly computed AMFlow expansions in the examples. Under the review rules, self-citation and same-author validation are not by themselves circularity. Therefore no step in the derivation chain reduces to its own input by construction; the paper is assigned a low score reflecting only minor self-citation dependencies, not actual circular reasoning.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted. The computation relies on the analytic-continuation and contour assumptions summarized above, on the companion paper [40] for fixed-branch integrals, and on AMFlow as solver and validator. No new particles, forces, dimensions, or other invented entities are introduced; the auxiliary mass eta is the standard AMFlow device.

assumptions (5)
  • domain assumption Dimensional regularization admits analytic continuation so that the DCT relation holds for any values of D0 and D.
    Invoked immediately after Eq. (4) in Section II.A without proof. This is the basis for using a non-integer value D0 = 7/13 in the examples.
  • domain assumption The auxiliary-mass integral I^{D0}(eta) has no poles in the fourth quadrant for the kinematics of interest.
    Needed to rotate the DCT contour from the positive real axis to the negative imaginary axis in Eq. (5) and Eq. (10). It is stated as a condition but not proved for general configurations.
  • domain assumption The differential equation system for fixed-branch integrals, Eq. (11), and the associated Gram-matrix construction are correct as taken from Ref. [40].
    The companion paper [40] is unpublished and its arXiv identifier is a placeholder. For B = 1 it reduces to the AMFlow one-loop equation, but for general FBIs it is imported rather than derived here.
  • domain assumption Master integrals of fixed-branch integrals can always be chosen as corner integrals, so the differential-equation system closes with z0 and C not both vanishing.
    Stated in Section III without proof. It is used to guarantee a solvable closed system of differential equations for general fixed-branch integrals.
  • domain assumption AMFlow correctly solves the boundary-value problems for the auxiliary-mass integrals at the chosen dimension D0.
    AMFlow is the numerical engine used to produce the input integrals and also the validation baseline. It is a separate, established method, but it is from the same group and is taken as correct.

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Pith. "Pith review of Efficient Computation of One-Loop Feynman Integrals and Fixed-Branch Integrals to High Orders in $\epsilon$." pith.science (2026). https://pith.science/paper/C2JEK2Y3

@misc{pith2026241221054,
  author       = {Pith},
  title        = {Pith review of: Efficient Computation of One-Loop Feynman Integrals and Fixed-Branch Integrals to High Orders in $\epsilon$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C2JEK2Y3}},
  note         = {Machine review of arXiv:2412.21054}
}
abstract

We propose a novel method, called the dimension-changing transformation (DCT), to compute one-loop Feynman integrals and recently introduced fixed-branch integrals to arbitrary orders in $\epsilon$. The DCT relates one-loop Feynman integrals or fixed-branch integrals in one spacetime dimension to their corresponding quantities with auxiliary mass in any other dimension, making the expansion to high orders in $\epsilon$ highly efficient. We applied this method to several examples to demonstrate its validity and efficiency. The approach introduced in this work has been implemented in an open-source C++ package, available at \href{https://gitlab.com/multiloop-pku/dct}{https://gitlab.com/multiloop-pku/dct}.

Figures

Figures reproduced from arXiv: 2412.21054 by the authors.

Figure 1
Figure 1. FIG. 1. Different choices of integration paths for recover [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. A pentagon diagram in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Time spent for computing a pentagon family with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. A two-loop diagram with three branches, red lines [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Forward citations

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