The setting sun diagram with complex external momenta

David Dudal; Duifje Maria van Egmond; Gastao Krein

REVIEW 4 major objections 7 minor 40 references

The setting sun diagram with complex external momenta

T0 review · 4 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper argues that inserting a complex external momentum $p^2$ directly into the two-dimensional setting sun Feynman integral yields a result inconsistent with the spectral representation, and that one must instead integrate at real…

desk verdict A small, honest counterexample showing that naive complex p^2 in the standard +iε Minkowski integral fails, but the paper overreaches when it turns that into a blanket rule. read the letter →

arxiv 2507.05921 v1 pith:F4FCSWCU submitted 2025-07-08 hep-th hep-lathep-ph

David Dudal , Duifje Maria van Egmond , Gastao Krein This is my paper

classification hep-thhep-lathep-ph

keywords settingsundiagramanalyticcontinuationcomplexexternalmomentaEuclidean-MinkowskicorrespondenceWickrotationspectralrepresentationcontourintegration

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 asks whether the standard identity $S_E(p^2)=S_M(-p^2)$ between the Euclidean and Minkowski versions of the two-dimensional setting sun diagram survives when the external momentum $p^2$ is allowed to be complex before the loop integral is done. It finds, by explicit contour integration and by direct Wick rotation, that it does not: for $p^2$ in the upper or lower half-plane the Minkowski integral evaluates to $S_M(-p^2)=\frac{1}{2}S_E(p^2)-\frac{i}{4}\frac{\sqrt{p^2}}{\sqrt{4m^2+p^2}}$, which is not defined on the negative real axis and is not analytically connected to the real-axis result $S_M(-p^2)=S_E(p^2)$. The authors conclude that the correct procedure is to integrate first at real $p^2$ and only then analytically continue to complex values, matching the spectral representation. This is relevant because Green's functions with complex momenta are needed in nonperturbative studies of quarks and gluons, and the failure shows that a priori complex external momenta cannot be inserted naively into Feynman integrals.

What carries the argument

The object carrying the argument is the two-dimensional setting sun integral with two equal real masses, evaluated by closing the internal energy contour in the complex plane. The key identities are the residue formula Eq. (7), the branch relations Eq. (10) between $\sqrt{-p^2}$ and $\sqrt{p^2}$, and the spectral representation Eq. (3) used as the benchmark. The mechanism producing the discrepancy is the motion of the integrand poles $g_3$ and $g_4$ across the real $k_0$ axis as $p^2$ moves off the real axis; each crossing costs a residue, so the upper and lower half-plane results differ by exactly the term shown in Eq. (14). The alternative tilted contour of Eq. (18), which sends $k_0$ to $\pm\infty(1+i\epsilon)$ with $\epsilon$ growing, keeps the contour away from all moving poles and restores the standard relation.

What would settle it

Take $p^2=4m^2e^{i\pi/3}$, evaluate the $k_0$ integral in Eq. (9) with a contour that never crosses the moving poles, and compare with $S_E(p^2)$; if they agree, the claimed failure of direct complex continuation is wrong.

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Extended reading notes

Core claim

Starting from the Euclidean representation Eq. (5), with a fixed branch $\sqrt{p^2}$ having positive real part, the $k_2$ integral is done by residues and yields $S_E(p^2)=-\frac{i}{\pi}\frac{\arctan(i\sqrt{p^2}/\sqrt{4m^2+p^2})}{\sqrt{p^2}\sqrt{4m^2+p^2}}$, which agrees with the spectral representation over the whole cut plane. The same manipulation applied to the Minkowski integral Eq. (9), with the relation between $\sqrt{-p^2}$ and $\sqrt{p^2}$ given by Eq. (10), gives different results depending on which half-plane contains $p^2$: for $p^2\in\mathbb{C}^+\cup\mathbb{C}^-$ one obtains Eq. (14), $S_M(-p^2)=\frac{1}{2}S_E(p^2)-\frac{i}{4}\frac{\sqrt{p^2}}{\sqrt{4m^2+p^2}}$, which is neither well-defined on $\mathbb{R}^-$ nor analytically connected to the real-axis identity. The extra term is traced to a pole crossing the real $k_0$ axis as $p^2$ moves into the complex plane; the difference between approaching the negative real axis from above and below is exactly the residue contribution. A tilted contour proposed in Eq. (18), with $k_0\to\pm\infty(1+i\epsilon)$ interpreted as an actually growing imaginary part, restores $S_M(-p^2)=S_E(p^2)$ for all complex $p^2$, but the paper notes caveats about the meaning of $\infty\cdot\epsilon$ and about complex masses.

Load-bearing premise

The load-bearing premise is that the Minkowski Feynman integral with a priori complex external momentum is defined by the standard real-axis contour with a single $+i\epsilon$ prescription; if the physically correct definition is the alternative tilted contour discussed in Section V, the contradiction disappears.

Editorial extensions

If this is right

For real $p^2$ in $(-4m^2,0)$, the relation $S_M(-p^2)=S_E(p^2)$ is safe; the pathologies appear once $p^2$ is moved into the complex half-planes.
A naive Wick rotation performed with complex external momentum picks up the residue of the pole that crosses the contour, and the missing piece in Eq. (14) is exactly that residue.
The alternative tilted integration contour Eq. (18), if accepted, gives a unique result $S_M(-p^2)=S_E(p^2)$ across the whole complex plane, but its definition relies on an interpretation of $\epsilon$ that grows with $k_0$.
The safe route recommended by the paper is to compute the integral at real external momentum and then use the spectral representation to reach complex $p^2$.
With complex masses the spectral representation itself breaks down, so the ambiguity becomes worse; the paper flags this as a problem for future work.

Reading between the lines

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

One direct test of the paper's reading: for a fixed complex $p^2$, numerically integrate Eq. (2) along the real $k_0$ axis with a contour deformation that never crosses the moving poles; if the result matches Eq. (14), the failure is a genuine property of the fixed-$\epsilon$ definition, while agreement with $S_E(p^2)$ would show the discrepancy is an artifact of the residue bookkeeping.
The same pole-crossing mechanism should appear in any one-loop two-point function whenever the external momentum is complex enough to push an internal pole onto the integration contour; the sunset is a minimal example of a general phenomenon.
If the tilted-contour prescription of Eq. (18) is accepted as the correct Minkowski definition, then the usual $+i\epsilon$ propagator prescription is not unique for complex external momenta; different deformations at infinity would define different Green's functions, which would then have to be fixed by physical requirements such as causality or spectral positivity.
A natural extension is to repeat the calculation with a complex mass; the paper notes that the spectral density would lose its positivity or even its definition, so the benchmark itself would have to be replaced by another principle.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Net take: this is a small but real result. The authors show explicitly, for the d=2 setting sun with real mass, that if you define the Minkowski integral via the real-k0-axis + iε contour of Eq. (2) and insert a priori complex p^2, you do not get SM(-p^2) = SE(p^2). The residue computation is coherent, and the Kallen-Lehmann representation is a fair external benchmark; the discrepancy term in Eq. (14) is a genuine counterexample to naive continuation. That is the paper's contribution, and it is new in this explicit form. The authors are also honest about the alternative contour in Section V, which restores the spectral relation, and about the ambiguities of that contour for complex masses.

Where I part company with them is the conclusion. Section VI says one should not start with a complex value of p^2 inside the integral. That is too strong. The failure is a property of one particular contour definition, Eq. (2)/(9), not of "the Minkowski integral" in general. Their own Eq. (18), following Eichmann et al., gives SM(-p^2)=SE(p^2) for all complex p^2, and they do not show that Eq. (2) is the physically correct prescription for complex momenta. So the correct statement is: within that fixed +iε real-axis prescription, direct complex p^2 fails; other prescriptions exist. The paper should soften the blanket guideline or justify why Eq. (2) is preferred.

Minor soft spots: the direct contour evaluation is restricted to Im(sqrt(p^2)) > -m, so the full complex plane is handled by continuation, not by the explicit residue sum. There are also reversed equation references in Section III (Eq. (1) and (2) swapped) and some figure cross-reference slips. These are cosmetic but should be cleaned up. The citation pattern is fair; the key alternative [29] is engaged rather than dismissed.

If you work on analytic continuation of propagators in models with complex poles, this is a useful, citable caution. It is not a new framework, and the generalization to complex masses is left open. The calculation is self-contained and honest, and the conclusion is mostly right once narrowed. I would send it to a referee for a short-notice paper; with modest revisions it is publishable. I would not desk reject it.

Referee Report

4 major / 7 minor

Summary. The paper examines whether the standard Euclidean-Minkowski relation SE(p^2)=SM(-p^2) holds when a complex value of p^2 is inserted directly into the loop integrals before integration. Working in d=2 for the setting sun diagram with real mass m, the authors first review the Källen-Lehmann representation, which predicts SE(p^2)=SM(-p^2) throughout the cut complex plane. They then evaluate the Minkowski integral SM(-p^2) using the real-k0-axis contour with a fixed +iε prescription and find, for p^2 in the upper or lower half-plane, the different result of Eq. (14), which is neither defined on the negative real axis nor analytically connected to the real-axis result. Section IV argues that the same discrepancy appears when a Wick rotation is attempted for complex p^2. Section V discusses an alternative Minkowski contour, Eq. (18), which restores SM(-p^2)=SE(p^2) for all complex p^2, but flags ambiguities in its ε∞ interpretation and in the presence of complex masses. The conclusion recommends not starting from a complex p^2 inside the integral, but rather computing for real p^2 and then analytically continuing.

Significance. The result, if it holds as stated, is a useful caution for the nonperturbative QCD community: analytic continuation of loop integrals from real to complex momenta is prescription-dependent, and the standard real-axis +iε contour may not reproduce the spectral-representation continuation when p^2 is complex from the outset. The manuscript is commendably self-contained: it benchmarks against the independent Källen-Lehmann representation, contains no fitted parameters or assumed target results, and explicitly presents the alternative formulation of Ref. [29] rather than hiding it. The scope is narrow (d=2, one diagram, real mass), but the paper is clearly positioned as a first study in a program. The main issue is that the central conclusion is broader than what the calculation actually establishes, and the key formula Eq. (14) needs to be displayed and checked more carefully.

major comments (4)

[Section VI (Conclusion), with Section V, Eq. (18)] The blanket conclusion that "one should not start with a complex value of p^2 inside the integral" is stronger than the calculation supports. In Section V the authors themselves show that the alternative Minkowski contour of Eq. (18) restores SM(-p^2)=SE(p^2) for all complex p^2 in this example, and they do not prove that the real-axis +iε contour of Eq. (2) is the physically correct definition of the Feynman integral for a priori complex external momenta. As stated, the paper establishes that the specific real-axis prescription leads to a different result, not that complex external momenta inside loop integrals are generally illegitimate; the conclusion should be restricted accordingly, or the correctness of Eq. (2) for complex p^2 must be argued explicitly.
[Section III, Eqs. (12)-(14)] The central contradiction rests on Eq. (14), but the displayed formula appears to have inconsistent mass dimension: in d=2, SE(p^2) has mass dimension -2, while the term (-i/4)√p^2/√(4m^2+p^2) as printed is dimensionless. Either the notation omits a factor 1/√p^2, or the result of the dk integration is misstated. The step from Eqs. (12)-(13) to Eq. (14) is also only described as "after performing the integral" with no intermediate expression; the authors should display the ∫dk residue integrals and their evaluation, including the branch choice for √p^2 and √(4m^2+p^2), so that the contradiction with the spectral representation can be verified independently.
[Section III, after Eq. (7)] The residue evaluation is restricted to Im√(p^2)>-m (and |Im p^2|≫ε), while Eq. (14) is presented as the result for all p^2∈C±. For Im√(p^2)<-m the authors state that "direct evaluation is not possible and the integral requires an analytic continuation," but no such continuation is supplied. The paper should either state the precise domain on which Eq. (14) has been established and explain how the claimed contradiction continues to hold outside that domain, or provide the analytic continuation.
[Section III, paragraph following Eq. (14)] The assertion that the expressions for p^2∈C+, p^2∈R-, and p^2∈C- are "not connected through analytical continuation" is asserted rather than demonstrated. Since Eq. (14) is defined on the two half-planes and the real-axis result holds on (-4m^2,0), the authors should compute the limits Im p^2→0± of Eq. (14) and show explicitly that no analytic function on the cut plane can interpolate between the two sides; the existence of branch cuts does not by itself rule out a continuation through the cut.

minor comments (7)

[Introduction, second paragraph] The word "whetnet" should be "whether."
[Section III, first paragraph] The text says "start from the Euclidean setting sun diagram ... as given in Eq. (2)" and "Minkowskian ... as given in Eq. (1)"; the equation numbers appear to be swapped, since Eq. (1) is the Euclidean integral and Eq. (2) is the Minkowski integral.
[Section III, paragraph after Eq. (14)] The sentence "for three different choices (p^2∈C+, p^2∈R- and p^2∈C+)" lists C+ twice and never lists C-; it should read "p^2∈C+, p^2∈R- and p^2∈C-."
[Figure III.1 caption] The caption refers to "The k0 integration path" but the Euclidean integral of Eq. (5) is over k2; the caption should say "k2 integration path."
[Section II, first sentence] The word "expressable" should be "expressible."
[Section III, last paragraph] The phrase "Schwartz reflection principle" should be "Schwarz reflection principle," and the statement "SM(p2)=SM(p2)" in the same paragraph should read "SM(p2)=SM(p2)*" (complex conjugation), as the notation for conjugation is introduced in footnote 2.
[Footnote 2] The sentence "√p2 = √p2 for all p2∈C/R−" is confusing because the two square roots are not visually distinguished; please clarify which one is the principal branch and which denotes complex conjugation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the direct contour integrations are benchmarked against the external Källén-Lehmann representation, and no fitted parameter or load-bearing self-citation enters the derivation.

full rationale

The paper's derivation is self-contained. Equations (8) and (14) are obtained by explicit contour integration of the displayed integrals (5) and (9), using the stated pole locations (6) and (11) and residue definitions (7); no parameter is fitted and no target relation is assumed in that computation. The comparison is made against the independent, standard Källén-Lehmann spectral representation (3), which is an external benchmark rather than an input tailored to the paper's conclusion. The self-references [34] and [37] supply only conventions and a square-root identity, not any load-bearing argument. The alternative contour (18), taken from Eichmann et al., is explicitly presented in Section V with its own caveats about ε∞ and complex masses; the paper's conclusion is therefore conditional on the definition (2), which is a substantive definitional question, not circular reasoning. The skeptic's concern that another contour restores SE=SM is a correctness or interpretive limitation, not a circularity, and the paper itself flags the ambiguity. Hence no circular step is present.

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

No free parameters or invented entities are introduced. The calculation uses a real mass m and standard complex analysis. The load-bearing domain assumptions are the fixed i epsilon integration path for Minkowski integrals and the Kallen-Lehmann spectral representation as benchmark.

assumptions (4)

domain assumption The Minkowski propagator for complex external momenta is defined by inserting p^2 into the real-axis integral Eq. (2) with a fixed +i epsilon prescription.
Section III evaluates Eq. (9) with the standard real k0 contour; if the alternative path Eq. (18) from [29] is chosen instead, the discrepancy disappears, so this axiom is load-bearing.
domain assumption The Kallen-Lehmann spectral representation with positive spectral density for real mass m is the correct benchmark for the Euclidean and Minkowski continuation.
Section II states the spectral form and uses it to conclude that the direct integral result is wrong; the paper relies on this standard QFT assumption.
standard math Contour deformation and the principal-value formula Eq. (15) are valid for the pole crossings encountered.
Section III uses Eq. (15) to account for poles crossing the real k0 axis; this is standard complex analysis but is essential to the derivation of Eqs. (12)-(14).
standard math The branch choice for sqrt(p^2) with Re(sqrt(p^2))>0 and cut on the negative real axis is legitimate and propagates to sqrt(-p^2) via Eq. (10).
Section III defines sqrt(p^2) and sqrt(-p^2); the signs in Eq. (14) depend on these branch choices.

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Cite this review

Pith. "Pith review of The setting sun diagram with complex external momenta." pith.science (2026). https://pith.science/paper/F4FCSWCU

@misc{pith2026250705921,
  author       = {Pith},
  title        = {Pith review of: The setting sun diagram with complex external momenta},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F4FCSWCU}},
  note         = {Machine review of arXiv:2507.05921}
}

abstract

We revisit the issue of analytically continuing Feynman integrals from Euclidean to Minkowski signature, allowing for generic complex momenta. Although this is well-known in terms of the K\"all\'{e}n-Lehmann representation, we consider potential alternative takes on the same problem and discuss how these are not necessarily equivalent to the K\"all\'en-Lehmann integral outcome. We present our analysis for a simple enough case -- the setting sun diagram in $d=2$ with a real mass -- but already with an eye out to the more general case with complex masses which will further complicate matters.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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Scattering amplitudes and contour deformations

G. Eichmann, P. Duarte, M. T. Pe˜ na, and A. Stadler, Phys. Rev. D 100, 094001 (2019), arXiv:1907.05402 [hep-ph]

work page Pith review arXiv 2019

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work page 1978

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work page Pith review arXiv 2004

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