{"id":"94b27c45-d977-4dd1-ae48-a2c45ab479b6","arxiv_id":"2506.09142","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A spectral renormalisation group computation extracts the anomalous dimension eta ~ 0.1 for 2+1-dimensional phi^4 theory in the scaling regime, within a truncated approximation.","lead":"The authors compute real-time spectral functions for a scalar field theory near a critical point using a renormalisation group method, and extract a critical exponent. The work is a step toward applying real-time methods to the QCD critical endpoint and transport phenomena.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reported η≈0.1 is likely dominated by the s-channel-only, φ^4-truncated four-point function rather than by the spectral CS method; the paper's own LPA' benchmark shows the φ^4 truncation alone shifts η by ~0.08.","rationale":"The reader's weakest-assumption pinpoints the same truncation; I agree. I chose this over the extrapolation issue because the paper's own LPA' benchmark makes the truncation effect quantifiable: it is large enough to account for the entire deviation from the Ising value. The consistency among the three extractions is weak evidence because ρ4 is derived from the propagator and the paper itself demotes it to a lower bound. The proposed full-potential run is the minimal extension that would settle whether the ~0.1 value is a property of the spectral method or of the missing higher-order couplings and channels. The extrapolation fragility in Appendix D is a secondary concern; I include a cheaper stability check in the concrete test rather than making it the primary objection. I recommend no change to the reader's CONDITIONAL verdict, since the paper is honest about the limitation and the concern is exactly what the condition should require.","tokens_in":21314,"tokens_out":6348,"duration_ms":68810,"concrete_test":"Run the spectral flow with the full effective potential (Appendix H 1), re-extracting η from ρ, ρ4, and Zφ; if η drops from ~0.095 toward 0.0802 (LPA' full-potential) or 0.036 (Ising), the reported exponents stem from the φ^4/s-channel truncation. As a cheaper cross-check, include the t-channel constant contribution described in Appendix H 2 and test whether the UV boundary of the scaling window shifts as predicted there. In parallel, rerun the current truncation one more decade down in k/λφ (to 10^-8) and compare the k→0 intercepts in Figure 7 with the quoted errors; if the intercept shifts by more than the asymmetric errors, the extrapolation is under-resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative results (32a)-(35) are extracted from a truncation whose dominant error is not controlled. The four-point function is resummed only in the s-channel, Eqs. (23)-(24), and the effective potential is truncated at order φ^4, V^(n>2)_eff ≡ 0 (Section II B 3). In the flow of the inverse propagator, Eq. (26), the vertex enters only through Dtad^dyn in the configuration Γ(4)(p,q,-q,-p); as the authors note in Appendix H 2, in this configuration the s- and u-channels contribute equally, so dropping t- and u-channel momentum dependence removes a quantitatively important part of the scattering kernel. The paper itself admits in Section III A 1 that eliminating the intrinsic scale requires 'the feedback of non-trivial momentum dependencies of the four-point function as well as a more sophisticated initial condition.' Thus the present truncation is not sufficient for a uniform scaling limit, and the reported η values are likely truncation-dependent. The apparent consistency between ηρ, ηρ4/2, and η is not an independent validation, because ρ4 is computed from the same propagator spectral function and inherits its scaling; the authors themselves label the four-point result a cross-check and lower bound. The sensitivity is quantified by the paper's own LPA' fixed-point benchmark (Appendix E): changing only the effective-potential truncation from V^(n>2)=0 to the full potential moves η from 0.1600 to 0.0802, a shift of ~0.08 that is comparable to the full gap between the reported 0.095 and the Ising value 0.036. Without a computation that includes the full effective potential or the t-channel, the claim that the spectral CS method captures universal scaling rests on an untested cancellation of omitted contributions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a real-time spectral functional Callan-Symanzik framework for three-dimensional scalar phi^4 theory and uses it to compute the single-particle spectral function and the s-channel four-point spectral function in and outside the scaling regime. Working in the symmetric phase with an on-shell renormalisation condition, the authors close the flow of the inverse propagator by an s-channel bubble-resummed four-point function and a phi^4-truncated effective potential. From the scaling of rho(lambda), rho4(lambda), and the k-dependence of the on-shell wave function they extract eta_rho = 0.101(+0.004/-0.028), eta_rho4/2 = 0.077(+0.002/-0.003), and eta = 0.095(9), and compare these with Euclidean LPA' benchmarks and with other real-time truncations. The paper is explicit that the phi^4 truncation is responsible for much of the distance from the conformal-bootstrap value eta ~ 0.036 and that improvements such as a full effective potential and t/u-channel momentum dependence are left to future work.","tokens_in":21639,"tokens_out":5388,"duration_ms":52539,"significance":"If the results are taken at face value, the paper provides a useful demonstration that a manifestly Lorentz-invariant, causal spectral fRG scheme can reach a scaling regime and produce power-law spectral functions with a nontrivial anomalous dimension. The technical execution has notable strengths: the momentum integrals in the tadpole and fish diagrams are performed analytically with explicit spectral representations (Appendix G), the extrapolation procedures are documented in detail (Appendix D), and the authors benchmark their truncation against LPA' fixed-point computations in different potential truncations (Appendix E). However, the significance is limited by the fact that the central quantitative output, eta ~ 0.1, is obtained in a truncation whose own benchmark shows a ~0.08 sensitivity to the effective-potential truncation alone; the method-specific validation would require an estimate of that systematic error or an improved truncation.","major_comments":[{"comment":"The uncertainties quoted in (32a), (32b), and (35) are extrapolation uncertainties only; no truncation error is included. The paper's own LPA' benchmark shows that changing only the effective-potential truncation from V^(n>2)_eff = 0 to the full potential moves eta from 0.1600 to 0.0802 (Table I and Table II, Appendix E), a shift of about 0.08 that is the same order as the gap between the three extractions and as the deviation from the Ising value. In this situation the statement that the three extractions are 'consistent with each other' (Section III A 1) supports only consistency within a single truncation, not a controlled extraction of the critical exponent. I would ask the authors to quote a systematic truncation-error band, or to reframe the quantitative claims as a demonstration of the real-time framework within a specified truncation rather than as an extraction of eta.","section":"Section III A and Appendix E"},{"comment":"The s-channel-only bubble resummation of Gamma^(4) omits the u-channel contribution that is equally important in the configuration Gamma^(4)(p,q,-q,-p) entering the tadpole diagram, as well as the constant t-channel part; the authors themselves note in Appendix H 2 that these channels shift the intrinsic scale and are needed for a uniform scaling limit. Because the flow equation (26) receives its entire momentum-dependent quantum correction through this vertex, the reported eta values may reflect the missing channels rather than the spectral CS method. A concrete test would be to include the constant t-channel contribution via the effective coupling described in Appendix H 2, or to show numerically how eta_rho changes when the u-channel is added; without such a test the central quantitative claim remains truncation-dominated.","section":"Eqs. (23)-(26) and Appendix H 2"},{"comment":"The four-point spectral function rho4 is computed from the propagator spectral function through the fish diagram (25), so eta_rho4 is not an independent determination of eta. The paper acknowledges this by calling (32b) 'a consistency check and a lower bound,' but the subsequent sentence that the agreement 'further validates the computation' overstates the evidential value: the agreement mainly confirms that the fish-diagram construction is implemented consistently, not that the underlying truncation is accurate. This should be reworded so that the logical status of the cross-check is not inflated.","section":"Section III A 1 and Appendix B"}],"minor_comments":[{"comment":"The choice of the polynomial order (Nmax = 2 for eta_rho, Nmax = 5 for eta_rho4) is based on chi^2_red and overfitting behaviour, but the final lambda -> 0 extrapolation in Figure 7a is estimated from only the last three points and a family of fit functions chosen 'roughly'; this procedure should be described more quantitatively, for example by reporting the fit ranges and the spread of the three extrapolations.","section":"Appendix D 1 and Figure 7a"},{"comment":"The value eta = 0.095(9) is quoted with an uncertainty, but the text does not explain how the error 0.009 is obtained; since this is one of the three main numerical results, the derivation of both the central value and the error should be specified.","section":"Section III A 2, Eq. (35)"},{"comment":"The Bethe-Salpeter kernel is taken as classical and the authors cite [1,3,6,11], but a one-sentence justification of why this is the leading non-trivial kernel in the present truncation would help the reader.","section":"Section II B 2, Eq. (23)"},{"comment":"The sentence 'The exponent in (29a) is the critical exponent eta ~ 0.036 of the three-dimensional Ising model, if no approximation is applied' is clear, but it could be sharpened to distinguish the exact theory value from the value obtained in the present truncation, which is the relevant comparison for the reader.","section":"Section III A 1, below Eq. (29a)"},{"comment":"The convention Im(arctanh(x > 1)) = +pi/2 is stated, but the sign convention should be checked explicitly against the retarded limit p0 = -i(omega + i0+) to avoid ambiguity for readers who use the opposite arctanh branch.","section":"Appendix G 1, Eq. (G1c)"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its truncation, but the central numerical claim is not yet robust: the quoted errors are extrapolation errors, while the paper's own benchmarks show a truncation sensitivity of order 0.08 in eta. I would send back for major revision rather than reject, since the spectral CS framework itself is promising and the missing systematic error estimate or an improved truncation appears achievable within the scope of the project."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a careful, honest proof-of-principle for extracting critical exponents from real-time spectral functions, but the headline η≈0.1 is a truncation-dependent estimate, not a quantitative result.\n\nThe genuinely new piece is the first spectral-fRG computation that pushes the single-particle and four-point spectral functions of 3D φ^4 into the scaling regime and extracts η from their power laws. The flow equations are derived cleanly, the momentum integrals are done analytically, and the paper gives explicit error estimates. The three extractions — from ρ, ρ4, and the k-scaling of Z_φ — agree within errors, which is a fair internal consistency check. The authors also clearly flag that the ρ4 extraction is a cross-check, not independent, since it inherits its scaling from the propagator.\n\nThe soft spots are real and, in large part, self-admitted. The four-point function is resummed only in the s-channel, and in the configuration that enters the flow s and u contribute equally, so the s-only vertex likely misses a quantitatively important part of the momentum dependence. The effective potential is truncated at order φ^4, and the authors' own LPA' benchmark shows that lifting that potential truncation moves η by about 0.08 in that scheme — comparable to the gap between the reported 0.095 and the Ising value 0.036. That makes it plausible that the reported values are dominated by truncation rather than by the spectral CS method itself. The k→0 and λ→0 extrapolations look fragile: in Figure 7a only the last three points show the onset of convergence, and the error bars on η_ρ blow up. No code or data are shipped, so the numerical claims are not independently checkable.\n\nNone of this kills the paper. The authors are up front about the limitations and explicitly frame the computation as a first step. The comparison with spectral DSE in comparable truncations is reasonable, and the reference list is not padded with irrelevant work. For people working on real-time fRG for QCD near a critical endpoint, this is a useful proof-of-concept: the infrastructure works, and the path to improvement is clearly laid out.\n\nIf I were the editor, I would send it to a serious referee. The right verdict is probably 'accept after major revision' with a request to include the full-potential computation or a quantitative bound on the truncation error, and to release the code. It is not a paper to desk-reject.","headline":"A careful proof-of-principle for real-time critical exponents, but the extracted η≈0.1 is truncation-dependent and not a quantitative result.","tokens_in":22225,"tokens_out":4994,"would_cite":false,"duration_ms":47113,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Real-time spectral flows in 2+1D phi^4 theory yield a critical exponent near 0.10, consistent across three independent extractions.","keywords":["spectral functions","Callan-Symanzik equation","functional renormalisation group","critical exponents","scalar phi^4 theory","Ising universality class","Wilson-Fisher fixed point","real-time dynamics"],"falsifier":"Run the same spectral flow with the full effective potential and with t- and u-channel contributions included, the extension sketched in Appendix H: if the three extractions continue to agree but the common $\\eta$ stays near 0.1 instead of moving toward the Ising value $\\eta = 0.03631(3)$, the claim that the current truncation resolves the universal scaling would be falsified.","tokens_in":21087,"feed_emoji":"⚛️","tokens_out":6689,"duration_ms":66337,"temperature":0.7,"pith_summary":"This paper tries to show that real-time (Minkowski-space) correlation functions of 2+1-dimensional $\\phi^4$ theory can be computed through the critical regime using the spectral Callan-Symanzik equation, and that the scaling of those spectral functions yields the Ising critical exponent $\\eta$. The authors compute the single-particle spectral function and the s-channel spectral function of the four-point vertex, both in and outside the scaling window, and extract $\\eta$ in three independent ways. The three extractions, $\\eta_\\rho = 0.101^{+0.004}_{-0.028}$, $\\eta_{\\rho_4}/2 = 0.077^{+0.002}_{-0.003}$, and $\\eta = 0.095(9)$ from the pole flow of $Z_\\phi$, are consistent with each other and with comparable real-time truncations in the literature. The point of the exercise is that a real-time renormalisation group flow that preserves causality can reach critical scaling, which matters for future applications to strongly correlated systems such as QCD near a critical end point.","feed_headline":"Real-time flows extract phi^4 critical exponent ~0.10","feed_subtitle":"Three independent spectral-function extractions agree with each other and with prior real-time truncations.","key_machinery":"The load-bearing mechanism is the spectral functional Callan-Symanzik equation, a mass-flow renormalisation group equation whose regulator is the mass parameter itself, $m_\\phi^2 = Z_\\phi k^2$, so that the cutoff scale is exactly the on-shell pole mass. Combined with the Källén-Lehmann representation, the flow lives on real frequencies and preserves manifest Lorentz invariance and causality. The system is closed by an inhomogeneous Bethe-Salpeter equation for the four-point function, which amounts to an s-channel bubble resummation, $\\Gamma^{(4)}(p) = \\lambda_\\phi / (1 + (\\lambda_\\phi/2) D_{\\mathrm{fish}}(p))$, and by an effective potential with vanishing couplings beyond $\\phi^4$. The momentum integrals in the tadpole and fish diagrams are done analytically, leaving one- and two-dimensional spectral integrals whose only non-perturbative input is the spectral functions themselves.","core_discovery":"The central claim is that the spectral Callan-Symanzik equation, closed with an s-channel bubble-resummed four-point function and an effective potential truncated at order $\\phi^4$, already captures the universal scaling of the 2+1D $\\phi^4$ theory in the symmetric phase. On this truncation the propagator spectral function develops a power-law window $\\rho(\\lambda) \\propto \\lambda^{-2+\\eta}$ and the s-channel vertex spectral function $\\rho_4(\\lambda) \\propto \\lambda^{1-2\\eta}$, with the plateau of the sliding exponents allowing a numerical extraction of $\\eta$. The authors report that the extracted exponents from the propagator, from the vertex, and from the on-shell wave-function flow agree within errors, and compare favourably with other real-time computations at a comparable truncation level, while remaining above the precise Ising value $\\eta = 0.03631(3)$, a deviation the paper attributes to the $\\phi^4$ truncation of the effective potential.","pith_inferences":["Editorial inference: since the paper itself attributes much of the gap between $\\eta \\approx 0.10$ and the Ising value $0.036$ to the $V_{\\mathrm{eff}}^{(n>2)} = 0$ truncation, a natural next test is to couple the spectral propagator flow to the full effective potential; if the exponent then moves toward $0.036$ while the three methods stay mutually consistent, that would confirm the truncation, n","Editorial inference: the s-channel-only spectral machinery could be tested at the upper boundary of the scaling window; the paper notes that this boundary is set by the classical coupling, so replacing the $\\phi^4$ initial condition by a running effective coupling, as sketched in its Appendix H, is a concrete extension that would sharpen $\\eta_\\rho$ and turn $\\eta_{\\rho_4}$ from a lower bound into","Editorial inference: a cross-check by computing the same quantities in the broken phase, where the fixed-point field value is nonzero and convergence is expected to be faster, would test whether the symmetric-phase extraction underestimates the reliability of the truncation; agreement between the two phases would strengthen the universal-scaling claim."],"forward_implications":["The scaling regime sets in at pole masses below about one percent of the classical coupling, $m_\\phi/\\lambda_\\phi \\lesssim 10^{-2}$, meaning the power-law window is narrow in this truncation.","In that window the propagator spectral function follows $\\rho(\\lambda) \\propto \\lambda^{-2+\\eta}$ and the s-channel vertex spectral function follows $\\rho_4(\\lambda) \\propto \\lambda^{1-2\\eta}$.","The three extractions, $\\eta_\\rho = 0.101^{+0.004}_{-0.028}$, $\\eta_{\\rho_4}/2 = 0.077^{+0.002}_{-0.003}$, and $\\eta = 0.095(9)$ from the flow of $Z_\\phi$, are mutually consistent within errors.","The results agree with equally truncated real-time Dyson-Schwinger ($\\eta \\approx 0.11$) and Keldysh-fRG ($\\eta = 0.0988$) computations, and the deviation from the conformal bootstrap value $0.03631(3)$ is largely attributed by the paper to the $\\phi^4$ truncation of the effective potential.","Because the method preserves Lorentz invariance and causality, the setup is positioned to be transferred to the mesonic sector of QCD to study real-time physics near a potential critical end point."],"supporting_citations":[{"why":"Supplies the renormalised functional Callan-Symanzik equation with manifest finiteness, the foundational flow used throughout.","marker":"[1]"},{"why":"Introduces the spectral Callan-Symanzik regulator that preserves Lorentz invariance and causality.","marker":"[2]"},{"why":"Provides the previous spectral $\\phi^4$ implementation and the on-shell renormalisation conditions used here.","marker":"[3]"},{"why":"Gives the comparable spectral Bethe-Salpeter/Dyson-Schwinger computation in the broken phase with $\\eta \\approx 0.11$, used as a real-time benchmark.","marker":"[11]"},{"why":"Provides the derivative-expansion fRG value $\\eta = 0.0361(3)$ used as a quantitative reference.","marker":"[13]"},{"why":"Provides the conformal bootstrap value $\\eta = 0.03631(3)$ used as the precise benchmark for the Ising universality class.","marker":"[17]"},{"why":"Provides a Keldysh-fRG real-time result $\\eta = 0.0988$ used as a consistency benchmark.","marker":"[38]"},{"why":"Outlines the scheme with full effective potential and momentum-dependent dressings that the paper points to for systematic improvement.","marker":"[56]"}],"fun_headline_variants":["Three spectral probes agree on phi^4 exponent","Spectral Callan-Symanzik yields critical scaling","Phi^4 eta from spectral functions ~0.10","Real-time spectral flows capture phi^4 scaling","Spectral functions fix phi^4 critical exponent"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The computation stands or falls on the assumption that keeping only the s-channel momentum dependence of the four-point function and cutting the effective potential at order $\\phi^4$ still captures the universal scaling of the propagator; if the omitted t- and u-channel momentum dependence or higher-order couplings dominate in the scaling regime, the extracted $\\eta$ reflects the truncation rather than the physics.","fun_headline_variants_meta":{"raw":{"variants":["Three spectral probes agree on phi^4 exponent","Spectral Callan-Symanzik yields critical scaling","Phi^4 eta from spectral functions ~0.10","Real-time spectral flows capture phi^4 scaling","Spectral functions fix phi^4 critical exponent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000833,"raw_usage":{"total_tokens":3568,"prompt_tokens":812,"completion_tokens":2756,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":428,"completion_tokens_details":{"reasoning_tokens":2681}},"tokens_in":428,"tokens_out":2756,"duration_ms":22076,"temperature":1.0,"reasoning_tokens":2681,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:55:57.543951+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same spectral flow with the full effective potential and with t- and u-channel contributions included, the extension sketched in Appendix H: if the three extractions continue to agree but the common $\\eta$ stays near 0.1 instead of moving toward the Ising value $\\eta = 0.03631(3)$, the claim that the current truncation resolves the universal scaling would be falsified.","supporting_citations":[{"cited_title":"In the deep IR, the finite numerical precision leads to small, numeri- cal deviations from the sum rule, which, if not corrected, can build up to destabilize the flow","cited_arxiv_id":null,"evidence_quote":"Gives the comparable spectral Bethe-Salpeter/Dyson-Schwinger computation in the broken phase with $\\eta \\approx 0.11$, used as a real-time benchmark."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the derivative-expansion fRG value $\\eta = 0.0361(3)$ used as a quantitative reference."},{"cited_title":"Dynamic universality class of Model C from the functional renormalization group","cited_arxiv_id":"1307.1700","evidence_quote":"Provides a Keldysh-fRG real-time result $\\eta = 0.0988$ used as a consistency benchmark."}],"review_version":1}