Renormalization in Minkowski space-time

I. Steib; J. Polonyi; S. Nagy

REVIEW 4 major objections 5 minor 1 cited by

Renormalization in Minkowski space-time

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Renormalization group flows computed directly in Minkowski space-time become complex, acquire new crossover scales, and cannot be recovered by Wick rotation from Euclidean flows.

desk verdict Worth a serious look for real-time FRG, but the headline claim that quasi-particle poles make Wick rotation singular is not established—it rests on a scheme-dependent pole in a sharp-cutoff beta function. read the letter →

arxiv 1908.11311 v2 pith:ZNWE4Q3D submitted 2019-08-29 hep-th

I. Steib , S. Nagy , J. Polonyi This is my paper · ORCID

classification hep-th

keywords renormalizationgroupMinkowskispace-timescalarfieldtheoryWickrotationmass-shellsingularitiescomplexcouplingsfunctionalquasi-particles

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

In a four-dimensional scalar field theory with $\varphi^4$ interaction, renormalization group flows are normally computed in Euclidean space-time and then Wick-rotated back to real time. This paper argues that this strategy breaks down in Minkowski space-time: when the subtraction point is placed in the quasi-particle domain, where real propagating excitations live, internal loop lines can go on their mass shell, and the running couplings become complex. The paper shows this concretely in both the multiplicative renormalization group scheme and the functional renormalization group scheme, where the numerical trajectories display complex couplings, non-monotonic behaviour, and new crossover scales, and where no trajectory in the studied truncation extends to arbitrarily high cutoff without losing convergence of the path integral. Because the quasi-particle poles make the Wick rotation singular, the authors conclude that real-time renormalization must be formulated in Minkowski space-time from the start. If they are right, Euclidean-space calculations can miss features that only appear in real time, such as the finite lifetime of quasi-particles encoded in the imaginary parts of the running parameters.

What carries the argument

The central object is the functional renormalization-group evolution equation (25), $\partial_\tau S = -(ik/2)\mathrm{Tr}\ln(\delta^2 S/\delta\varphi\delta\varphi)$, derived by blocking from cutoff $k$ to $k-\Delta k$ with the saddle point of the blocked field set to zero. The paper projects this equation onto a second-order gradient-expansion ansatz for the blocked action, uses a sharp non-relativistic cutoff $P_k = \{(p_0, p) : |p_0| < \Omega_k,\ |p| < k\}$, and extracts running parameters by evaluating the blocked action at a subtraction point built from a homogeneous field plus an on-shell plane wave, with $\omega_s$ chosen just above the quasi-particle mass. The mechanism that carries the argument is the mass-shell singularity: each loop integral acquires an imaginary part when an internal line can go on shell, making the $\beta$-function coefficients complex and generating new crossover scales.

What would settle it

Compute the same functional renormalization group flow in a less restrictive truncation, or keep the non-trivial saddle points in the blocking step: if a trajectory then extends to $k \to \infty$ without flipping the sign of $\mathrm{Im}\,g_2$ or $\mathrm{Im}\,g_4$, or if the two crossover scales in $\mathrm{Im}\,g_4$ disappear, the paper's central conclusions would be artifacts of the approximation. A cleaner kinematic check is to place the subtraction point just below versus just above the two-particle threshold and verify that the $\beta$ functions stay real below it and become complex only above it, as the mass-shell mechanism predicts.

Watch

Extended reading notes

Core claim

The paper's central discovery is that renormalization-group trajectories in Minkowski space-time are intrinsically complex and are not analytically continuable from Euclidean ones. In the multiplicative scheme, $\beta$ functions pick up imaginary parts as soon as loop-internal propagators can go on shell: the coefficient of $g^2$ in the $\beta$ function of $g$ develops an imaginary part for $\omega_s > 2m$, and the mass and wavefunction renormalization become complex for $\omega_s > 3m$. In the functional scheme, integrating the one-loop exact evolution equation in a second-order gradient expansion gives complex running couplings $g_2$, $g_4$, and $z_t$, with a fast change of $\mathrm{Im}\,g_4$ between two characteristic scales that the authors associate with the decreasing lifetime of two-particle resonances. No trajectory was found that extends to $k \to \infty$ without the imaginary part of $g_2$ or $g_4$ flipping sign, which would make the path integral diverge. The authors conclude that the quasi-particle poles make the Wick rotation singular, so real-time physics is inaccessible from Euclidean space-time by analytic continuation.

Load-bearing premise

The derivation of the functional flow equation assumes that in each blocking step the dominant field configuration is the trivial zero configuration; if a non-zero configuration is important, the numerically found crossover scales and the absence of infinite-cutoff trajectories could be artifacts of that assumption.

Editorial extensions

If this is right

Euclidean functional renormalization group studies cannot be continued to real-time quantities by a simple Wick rotation once the subtraction point sits in the quasi-particle domain.
Minkowski-space running parameters must be treated as complex, doubling the number of real couplings and allowing non-monotonic trajectories and extra crossover scales absent in Euclidean theories.
The rapid change of $\mathrm{Im}\,g_4$ between two characteristic scales points to a regime of fast-decreasing two-particle resonance lifetime, a genuinely real-time feature.
Within the studied truncation, no flow reaches $k \to \infty$ without flipping the sign of $\mathrm{Im}\,g_2$ or $\mathrm{Im}\,g_4$, so the theory may require a finite cutoff or a more complete real-time construction such as the Closed Time Path formalism.
The non-relativistic cutoff needed for practical calculations breaks boost invariance explicitly, making the boost-symmetry problem an unavoidable part of non-perturbative Minkowski-space renormalization group schemes.

Reading between the lines

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

Editorial inference: the same mass-shell mechanism should make complex flows appear in other theories with long-lived particles, such as Yukawa or gauge theories, with similar extra crossover scales.
Editorial inference: a testable extension is to extract spectral functions from the complex Minkowski propagators; the imaginary parts of running masses and couplings should appear as resonance widths, and the two crossover scales should be visible as two-particle thresholds.
Editorial inference: analytic continuation around, rather than through, the quasi-particle poles may recover some Euclidean predictions, but cannot reproduce the full real-time flow; this could be checked by constructing such contours explicitly.
Editorial inference: the absence of infinite-cutoff trajectories may be a truncation artifact; retaining non-local saddle-point contributions could either restore such trajectories or strengthen the cutoff dependence, and the distinction is testable numerically.

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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

The genuinely new thing here is the direct construction of RG schemes in Minkowski space for phi^4, avoiding the usual Euclidean-then-continue route. The explicit loop integrals in Sec. III show cleanly that on-shell intermediate states put imaginary parts into the running parameters via the optical theorem, and the emphasis on subtraction-point choice is well argued. The functional RG setup with a non-relativistic cutoff and the numerical flows that find crossover scales—sign flips of Im(g2), the Landau-like peak in |g4|, and the absence of infinite-cutoff trajectories—are a real extension of the literature. The authors also deserve credit for flagging their own limitations: they call the results preliminary, admit the cutoff breaks boost invariance, note the saddle-point assumption, and list open questions.

Where the paper oversells is the big claim, repeated in the abstract and intro, that the quasi-particle poles make the Wick rotation singular and the results are inaccessible from Euclidean space. The evidence is a pole in the beta function at omega_s^2 = omega_k^2 in the simplified flow, computed with a sharp, non-relativistic cutoff. That shows the naive Wick rotation fails for this scheme; it does not show a fundamental obstruction. In standard QFT, Minkowski Green functions are boundary values of Euclidean ones, and threshold branch cuts give imaginary parts without making analytic continuation impossible if the i-epsilon contour is handled correctly. So the strong wording should be softened unless the singularity survives in a smooth, covariant regulator—which is not shown.

Two other soft spots, both acknowledged but real. The full functional RG equations are omitted as "too lengthy to record," which hurts reproducibility; the numerics cannot be reconstructed from the text. And the non-relativistic cutoff, while justified for technical reasons, makes the quantitative claims—like the location of the new crossover scales—hard to trust until the boost issue is addressed or at least better controlled.

The complexification of the parameters is solid, standard physics. The inaccessibility claim is not. For someone working in real-time FRG, quark-meson models, or nonequilibrium dynamics, this is a useful and honest paper that raises the right questions, but it needs a referee who will push on the analytic-continuation argument and ask for the omitted derivations.

Send it to peer review. It is a serious contribution, not a desk reject, though it needs substantial revision before the qualitative claims are fully accepted.

Referee Report

4 major / 5 minor

Summary. This paper develops renormalization group schemes for a four-dimensional scalar φ^4 theory directly in Minkowski space-time. Using multiplicative renormalization with renormalized and bare perturbation expansions, it shows that the running parameters become complex when the subtraction point lies in the quasi-particle (on-shell) regime, with imaginary parts arising from mass-shell contributions. It then constructs a functional RG flow using a sharp non-relativistic momentum-space cutoff and a gradient expansion truncated to second order, and presents numerical trajectories. The authors claim that the quasi-particle poles make the Wick rotation singular, that the choice of subtraction point is more important in Minkowski than in Euclidean space, and that the complexified flow generates new crossover scales. They also report that within their truncation no trajectory extends to arbitrarily high cutoff without a sign flip in g2 or g4.

Significance. If substantiated, the central claim would be significant: it would show that Minkowski-space RG flows are not simply Wick-rotated Euclidean flows and that real-time methods are necessary for on-shell physics. The paper's treatment of the complex running parameters is supported by explicit two-loop integrals and the optical theorem, and the authors are commendably explicit about the limitations of their truncations. However, the strongest version of the claim—the inaccessibility by Wick rotation—is not proven, and several load-bearing statements rest on unshown equations or unvalidated extrapolations. The paper is a useful exploratory contribution, but it requires substantial revision before it can be accepted.

major comments (4)

[Section III.C and Eq. (32)] The claim, repeated in the abstract and in Section V, that the quasi-particle poles make the Wick rotation singular and make the results inaccessible from Euclidean space-time by Wick rotation is not established by the evidence presented. The finite imaginary parts in the two-loop beta functions and the singular term 1/[2(ω_s^2 − ω_k^2)] in Eq. (32) are computed with the sharp non-relativistic cutoff P_k and with a subtraction point on the quasi-particle shell. Threshold branch cuts in Minkowski Green functions are, in the standard analytic structure, boundary values of Euclidean functions, and the iε prescription is designed to navigate around them. The paper demonstrates that this particular sharp-cutoff beta function has a pole when the subtraction energy equals the cutoff-shell energy, but it does not show that a Euclidean RG flow cannot be continued around the threshold. Because beta functions and threshold singularities are scheme-dependent, the 'inaccessible by Wick rotation' conclusion requires either a smooth/covariant regulator or a direct analytic-continuation comparison; neither is provided.
[Section IV.B and IV.C] The numerical flows in Figs. 4 and 5 are the main functional-RG results, but the full beta functions from which they are obtained are not given; after Eq. (31) the authors state they are 'too lengthy to record here.' This makes the results irreproducible and prevents a full assessment of truncation errors. Moreover, Eq. (25) is derived under the assumption φ0 = 0, and the paper itself notes in Section IV.A(a) that non-trivial saddle points generate non-local contributions omitted by the gradient expansion. The conclusion in Section IV.C that no trajectory extends to k → ∞ without flipping the sign of g2,i or g4,i could therefore be an artifact of the vanishing-saddle-point assumption. Please provide the explicit projected flow equations, or at least an independent check of the numerical integration, and a discussion of the sensitivity to non-trivial saddle points.
[Section III.B and III.D] The statement that the theory is asymptotically free because complex beta-function coefficients turn the Landau pole into a Landau peak is not supported. The one-loop beta functions (12)-(13) are derived in the renormalized perturbation expansion, whose validity is explicitly limited to g_B(Λ0) Λ^2 ≪ m_B^2(Λ0) in the text following Eq. (15). Equation (14) is then extrapolated to arbitrarily large Λ to infer the absence of a Landau pole and the UV decay of |g_B|. This extrapolation requires control over the scheme dependence of the complex beta-function coefficients and over higher-order terms, neither of which is provided. The claim should either be derived within a controlled approximation (e.g., a resummation or a proper functional-RG analysis) or removed.
[Section IV.A(g) and V] The sharp energy-momentum cutoff P_k = { (p0,p) : |p0| < Ω_k, |p| < k } is not boost invariant, and the paper's remedy is to project the blocked action back to the relativistic form (26) after each step. The paper acknowledges that this is an open problem. Since the numerical flows in Section IV.C are obtained with this cutoff, the claimed crossover scales and the sign-flip barriers could be artifacts of the non-relativistic regulator. Please provide evidence of insensitivity to the shape of the cutoff and to the choice of Ω_k, or at least quantify the symmetry-breaking effects.

minor comments (5)

[Abstract and Introduction] There are several typos, including 'renormalization gro up' in the abstract and 'phyicis' in the Introduction; please proofread the manuscript carefully.
[Eq. (31)] The expression 'Θ( k − |p| < k )' appears garbled; the theta function should presumably enforce the momentum-shell condition, but the current notation is not well-defined.
[Section IV.C] The phrase 'The importance of gt, 2,r < 0' appears to contain a typo; the symbol should likely be zt,2,r, based on the surrounding discussion of the wave-function renormalization constant.
[Figure 1 and its caption] The caption describes dashed lines along the imaginary and real axes, but the figure itself is not displayed clearly in the text; please ensure the figure actually shows the claimed behavior and that the axes are labeled unambiguously.
[References] Reference [44] is incomplete as printed ('JHEP 05, 021 (2012)'), and reference [48] lacks the year; please complete all bibliographic entries.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Minkowski RG results are computed from explicit loop integrals and initial conditions, with only non-load-bearing self-citations.

full rationale

I find no circular step that makes a claimed prediction equivalent to an input by construction. The running parameters are computed from the explicitly displayed loop integrals (8)-(9) and the functional evolution equation (25), with stated initial conditions; no parameter is fitted to external data and then relabeled as a prediction. The complex character of m^2 and g in Minkowski space follows from the i factors and the on-shell intermediate-state cuts in the standard loop integrals, not from assuming the desired conclusion. The self-citations ([48], [49], and [52], [53]) concern the choice of a non-relativistic sharp cutoff and spinodal saddle-point background; they are technical scheme choices and do not carry the central claim that the quasi-particle subtraction point produces complex flows. The potentially singular denominator in Eq. (32) is tied to the subtraction-point projector (33), with c_s=3 chosen away from the pole; this is openly discussed scheme dependence, not a hidden fit. Whether the Wick-rotation statement is ultimately correct is a physics-correctness question, not a circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central observation of complex running parameters follows from standard unitarity and explicit loop integrals, not from a fitted model. The functional RG results rely on a truncation of the action, a vanishing saddle point, and a non-relativistic cutoff that breaks boost invariance. No new physical entities are introduced.

free parameters (3)

cs = 3
Dimensionless parameter in the subtraction point definition (Eq. 33); chosen to smooth out irregularity at the onset of the condensate, controls where the trajectory is evaluated.
Omega_k / omega_s ratio = 10^4
Energy cutoff scale in the non-relativistic regulator; chosen large but finite to regulate energy integrals (Sec. IV.C).
initial Im g2 = -0.001
Initial imaginary part of mass-squared chosen negative to ensure convergence of the path integral and used in both flows (Sec. IV.C).

assumptions (5)

domain assumption Optical theorem and unitarity make on-shell intermediate states produce complex vertex functions.
Used throughout to argue that running parameters become complex, e.g., Sec. II.B and Sec. III.D.
domain assumption The one-loop evolution equation (25) with saddle point phi0 = 0 is exact for an infinitesimal blocking step.
Section IV.A (i), where non-trivial saddle points are omitted and the gradient expansion is imposed.
domain assumption The gradient expansion truncated to O(delta^2) with a polynomial potential of degree 4 adequately represents the blocked action.
Section IV.A (ii); authors note the polynomial representation can only be justified for weakly coupled theories.
ad hoc to paper The path integral converges when Im g4 < 0, used as the only stability restriction.
Section IV.A remark (c); replaces the usual lower-bounded energy condition with a complex-parameter condition.
standard math Standard perturbative loop integrals and renormalization conditions in Minkowski space with sharp momentum cutoff and epsilon prescription are valid.
Section III, Eqs. (8) and (9), and the underlying BG/BPHZ-style renormalization framework cited in Refs. 5-7.

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Cite this review

Pith. "Pith review of Renormalization in Minkowski space-time." pith.science (2026). https://pith.science/paper/ZNWE4Q3D

@misc{pith2026190811311,
  author       = {Pith},
  title        = {Pith review of: Renormalization in Minkowski space-time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZNWE4Q3D}},
  note         = {Machine review of arXiv:1908.11311}
}

read the original abstract

The multiplicative and the functional renormalization group methods are applied for the four dimensional scalar theory in Minkowski space-time. It is argued that the appropriate choice of the subtraction point is more important in Minkowski than in Euclidean space-time. The parameters of the cutoff theory, defined by a subtraction point in the quasi-particle domain, are complex due to the mass-shell contributions and the renormalization group flow becomes much more involved than its Euclidean counterpart.

Figures

Figures reproduced from arXiv: 1908.11311 by the authors.

**Figure 2.** FIG. 2: Graphs of the two-loop self energy. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 4.** FIG. 4: The trajectories of the local potential. [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: FIG. 5: The trajectories of wave function renormalization c [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reviewed August 14, 2026 · model on record in the stance chip above.

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