{"id":"d809d693-685b-40ca-ad12-5d0ecaeeca48","arxiv_id":"2412.21054","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A dimension-changing transformation computes one-loop and fixed-branch Feynman integrals to high orders in the dimensional regulator by integrating auxiliary-mass integrals over the mass parameter, implemented in an open-source C++ package.","lead":"A new transformation, the dimension-changing transformation, computes one-loop Feynman integrals and fixed-branch integrals to arbitrarily high orders in the dimensional regulator epsilon. The method makes high-precision collider predictions much cheaper and comes with an open-source C++ package.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The DCT contour Eq. (5)/(10) is valid only if no fourth-quadrant η-poles exist; the paper asserts this without proof for general kinematics, complex masses, or FBIs, so the central identity may silently omit contributions.","rationale":"The reader's weakest_assumption identifies exactly the same point, and I agree: the entire one-loop and FBI algorithm hinges on Eq. (10), and the contour deformation is the only step not derived. The derivation of Eq. (4) from the loop-momentum split is solid, and the beta-function algebra in Eq. (9) is standard provided convergence is handled by analytic continuation. But moving the contour from the positive real axis to the negative imaginary axis is a global analytic statement; the paper's Fig. 1 merely asserts the absence of fourth-quadrant singularities. For physical kinematics with Feynman iδ this is plausible and consistent with AMFlow, but the paper explicitly claims complex masses and general FBIs, for which no sufficient condition is stated. This is a correctness risk, not a disagreement with consensus: if the condition fails only for unphysical inputs, the fix is to state and enforce the condition; if it can fail for physical inputs, the method is unsound as written. The published examples are not informative because they all sit in the safe region. The absence of independent validation (all comparisons are against the same group's AMFlow, and FBIs rest on unpublished [40]) increases the cost of a silent failure. I therefore keep the reader's CONDITIONAL verdict: the central idea is plausible and the numerics are impressive, but the contour condition and the fixed-branch derivation should be settled before full acceptance.","tokens_in":9069,"tokens_out":11917,"duration_ms":131231,"concrete_test":"Choose a one-loop scalar two-point function with mass m and external invariant s such that the Landau branch point η = s/4 - m^2 lies in the fourth quadrant (e.g., s real with s > 4 Re(m^2) and Im(m^2) > 0, or Im(s) < 0 with Re(s) > 4 Re(m^2)). Evaluate the left side of Eq. (5) by direct numerical integration of the Feynman-parameter representation with the same iδ prescription, and the right side with the DCT package. If the two disagree, the no-fourth-quadrant condition is necessary and must be proven and enforced; if they agree, repeat the scan for the FBI example of Fig. 5 with complex X, checking all solutions of F(X,y)+η U(y)=0 over the simplex for fourth-quadrant η.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section II.A derives Eq. (4) as an integral over positive real η, then replaces it by Eq. (5) on the negative imaginary axis, stating that this requires I^{D0}_ν(η) to have no poles in the fourth quadrant. No proof or sufficient condition is given. For FBIs, Eq. (10) has the same structure, with singularities at η = -F(X,y)/U(y), where F is the second Symanzik polynomial and X are branch parameters; nothing in Section II.B or III shows that these singularities cannot enter the fourth quadrant for allowed complex masses or complex X. If one does, the deformed contour differs from the original integral by the residue of the crossed singularity, and the package would return a wrong result without warning. The examples in Section IV use real X and Feynman-prescription kinematics, so they cannot expose this. The fixed-branch differential equations also rely on the unpublished companion paper [40], but that is secondary: the contour assumption is load-bearing even for one-loop integrals.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9184,"tokens_out":6797,"duration_ms":64738,"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":[{"comment":"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.","section":"II.A (Eq. (5)) and II.B (Eq. (10))"},{"comment":"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.","section":"III, Eq. (11)"},{"comment":"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.","section":"Table II"}],"minor_comments":[{"comment":"The sentence beginning 'since ∆ − δ is fixed' should have 'Since' capitalized at the start of the sentence.","section":"III, after Eq. (10)"},{"comment":"The word 'diferential' should be spelled 'differential' in the description of solving the η differential equations.","section":"IV, first example"},{"comment":"Reference [40] is listed with a placeholder arXiv number 'arXiv:2412.xxxxx'; this should be updated once the companion paper is available.","section":"References"},{"comment":"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.","section":"Fig. 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The central DCT identity is derived independently and is not fitted to the target results, which is a strength. The main concern for publication is the unproved analytic-continuation assumption in Section II; the other issues (unpublished reference and precision table) are fixable. The open-source package and numerical benchmarks are useful, but the authors should be asked to supply the missing proof or a precise sufficient condition, and to make the companion derivation available."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper should be on your radar if you compute one-loop integrals to high orders in epsilon. The dimension-changing transformation (DCT) is a genuine workhorse improvement: it lets you solve the auxiliary-mass differential equations once at a convenient dimension, then extract any epsilon order from a single contour integral. The derivation is clean—split the loop momentum, integrate out the orthogonal components, and you get Eq. (4) without free parameters. The implementation is open source and the examples (pentagon with multiple masses, massless hexagon at vanishing Gram, and a two-loop FBI) show high numerical precision and real speedups over both LoopTools and an in-house AMFlow.\n\nThe central soft spot is the contour deformation. The authors replace the positive-real integration by the negative imaginary axis and state that this requires no fourth-quadrant poles. They do not prove that condition, and it is not obvious for complex masses or for FBIs with complex branch parameters. If a pole is lurking there, the method returns a number that differs from the true integral by the residue, and the package gives no warning. For the kinematics in the paper (real masses, Feynman prescription) the condition presumably holds—the validation against AMFlow supports that—but a user going beyond those cases could be misled. This is a load-bearing caveat, not a fatal flaw.\n\nThe second issue is the unapologetic reliance on the companion paper [40] for the FBI differential equations. That paper is unpublished and its arXiv number is a placeholder. For the one-loop results this does not matter, but the FBI claims cannot be fully assessed yet. A referee should insist that [40] be available before the FBI section is taken at face value.\n\nValidation is against AMFlow, from the same group, so the agreement is not independent. It would be more convincing to see a cross-check with an unrelated package for at least one case.\n\nOverall: the core identity is sound, the method is useful, and the paper is worth refereeing. My recommendation is to send it to review, and in the report ask for a sharper statement (or a numerical check) of the no-pole condition, and for the companion paper to be posted.","headline":"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.","tokens_in":9790,"tokens_out":2515,"would_cite":true,"duration_ms":26267,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81T18","81T15","81-08"],"pacs":["11.15.Bt"],"model":"deepseek-v4-flash","headline":"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…","keywords":["dimension-changing transformation","one-loop Feynman integrals","fixed-branch integrals","epsilon expansion","auxiliary mass flow","master integrals","contour deformation","dimensional regularization"],"falsifier":"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.","tokens_in":8807,"feed_emoji":"🧮","tokens_out":10132,"duration_ms":91745,"temperature":0.7,"pith_summary":"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.","feed_headline":"One contour integral yields all epsilon orders of one-loop integrals","feed_subtitle":"Solve the auxiliary-mass differential equations once, then recycle one solution into the full epsilon expansion.","key_machinery":"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 $\\eta\\to\\infty$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the auxiliary-mass regularization and the differential-equation flow used to obtain the series expansions of the auxiliary-mass integral.","marker":"[36]"},{"why":"Introduces fixed-branch integrals and their eta-differential equation, the objects whose dimension-changing relation is derived in this paper.","marker":"[40]"},{"why":"Provides the reference results used to validate the method's numerical output in the examples.","marker":"[39]"},{"why":"Supplies the massless six-point one-loop benchmark family computed to high order in epsilon in the examples.","marker":"[34]"},{"why":"Supplies the five-point multi-scale topology and physical kinematics used as the main efficiency benchmark.","marker":"[35]"}],"fun_headline_variants":["One auxiliary-mass solution yields all ε orders of loop integrals","DCT: solve once, transform to any dimension, get every ε order","Solve once, recycle: DCT yields all ε orders of one-loop integrals","One contour transform yields full ε series for loop integrals","All ε orders from one solution: DCT for one-loop integrals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["One auxiliary-mass solution yields all ε orders of loop integrals","DCT: solve once, transform to any dimension, get every ε order","Solve once, recycle: DCT yields all ε orders of one-loop integrals","One contour transform yields full ε series for loop integrals","All ε orders from one solution: DCT for one-loop integrals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001201,"raw_usage":{"total_tokens":4933,"prompt_tokens":910,"completion_tokens":4023,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":3947}},"tokens_in":526,"tokens_out":4023,"duration_ms":28449,"temperature":1.0,"reasoning_tokens":3947,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:03:26.197196+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Huang, R.-J","cited_arxiv_id":null,"evidence_quote":"Introduces fixed-branch integrals and their eta-differential equation, the objects whose dimension-changing relation is derived in this paper."}],"review_version":1}