{"id":"3e6d2a2e-ac87-47ec-be81-4aeb01713f6e","arxiv_id":"2505.21631","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the 3d NJL model at large charge, the scaling dimension has a convergent small-q series and an asymptotic large-q series whose nonperturbative corrections are worldline-instanton terms e^{-α√q}.","lead":"This paper derives both a convergent small-charge series and an asymptotic large-charge series for the conformal dimension of charged operators in the 3d Nambu-Jona-Lasinio model, and identifies exponentially small instanton corrections in the large-charge limit. It extends the large-charge resurgence program from bosonic O(N) models to a fermionic theory, suggesting a geometric worldline-instanton picture.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq (4.28) quotes a purely imaginary nonperturbative correction to the real scaling dimension; the Stokes-discontinuity vs actual-correction distinction is never resolved, so the central exponential prediction is ambiguous.","rationale":"The reader's weakest assumption correctly identifies the zeta-function regularization and the saddle identification as foundations, but the most immediately checkable load-bearing gap is the reality of Eq (4.28). In resurgence theory, Stokes discontinuities are generically imaginary while the physical observable is real; quoting one without the other is not a complete prediction. The paper's own remark that the reality condition fixes the Stokes parameter shows awareness, but the subsequent results are written as imaginary exponentials and are never converted into a real correction. This does not invalidate the small-q convergent series or the asymptotic large-q expansion themselves, but it makes the advertised exponential corrections, which are the paper's central claim, ambiguous rather than testable. A concrete check would settle whether the real physical correction matches Eq (4.28), and the verdict should remain conditional pending that check.","tokens_in":20142,"tokens_out":5268,"duration_ms":67603,"concrete_test":"Construct the real transseries explicitly: evaluate the integral representation (4.2) for the zeta function at large rΦ0 without separating Dawson's function into perturbative and exponential pieces, or compute the median resummation of the formal large-q expansion generated by Eq (4.9), including both branches of the Stokes discontinuity and both signs of k. Extract the coefficient of exp(-2π sqrt(κ0^{-1}-κ0^{-3})|k| sqrt(q)) in the real part of Δ/(2N) and compare it with Eq (4.28). If the real part differs from the real part of Eq (4.28), or if the relative phase of the cos and sin terms changes, the central exponential prediction is incorrect as stated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central result for the nonperturbative correction to Δ(Q)/(2N), Eq (4.28), is purely imaginary: the prefactor is -i times a real combination of cos and sin. Since Δ(Q)/(2N) is a real observable, Eq (4.28) cannot be the physical exponential correction itself. At most it is the Stokes discontinuity of the transseries, or one of the two half-discontinuities entering the Dawson-function expansion (4.9). The paper states that the reality condition fixes the Stokes parameter, but it never constructs the real physical combination from the k and -k contributions and from the two sides of the Stokes line. This is not cosmetic: taking the real part, taking a half-sum, or combining k and -k with different relative weights changes the numerical prefactor and the relative weight of the cos and sin terms in Eq (4.28). The worldline comparison in Section 5 and Appendix B is made against this imaginary expression, not against a real observable, so the claimed exponential corrections are not defined as a prediction for the conformal dimension.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the three-dimensional Nambu–Jona–Lasinio model at its UV fixed point in the double-scaling limit Q,N→∞ with q=Q/(2N) fixed. The grand potential is computed from the zeta-function-regularized one-loop determinant on R×S^2, the gap equation is solved in both small-q and large-q regimes, and the scaling dimension Δ(Q)/(2N) is obtained from a Legendre transform. The authors claim that the small-q series is convergent with radius |q|<0.35(3), that the large-q series is asymptotic with (2n)! growth, and that the leading nonperturbative corrections take the explicit form of the transseries in Eq. (4.28), which they interpret geometrically as worldline instantons wrapping geodesics on S^2.","tokens_in":20333,"tokens_out":4440,"duration_ms":50610,"significance":"If the main claims hold, the paper would provide the first fermionic example of a full large-charge transseries with explicit exponential corrections, extending the bosonic O(N) analysis of [19] and supporting the conjecture that gapped fixed-charge systems have e^{-α√q} corrections. The computation is parameter-free in the sense that κ0 is fixed by the transcendental equation κ0 tanh(κ0)=1 and no data are fitted. The small-q convergence radius and the explicit form of the leading nonperturbative correction are concrete, falsifiable predictions. However, two load-bearing issues currently prevent the central results from being accepted as stated: an inconsistency in the leading large-q coefficient between Section 2 and Section 4, and the fact that Eq. (4.28) is purely imaginary despite being presented as a correction to a real observable.","major_comments":[{"comment":"The leading large-q coefficient of Δ(Q)/(2N) is stated in two incompatible forms. Eq. (2.23) and the abstract give (2/3)κ0^{3/2} q^{3/2} + (1/6)κ0^{1/2} q^{1/2} + …, while Eq. (4.19) gives (2/3)(q/κ0)^{3/2} + (1/6)(q/κ0)^{1/2} + …. These differ by a factor κ0^3 ≈ 1.727 in the leading term. Since the entire large-q expansion, the optimal truncation order, and the comparison with the worldline result depend on this coefficient, the contradiction must be resolved and the correct expression identified.","section":"Section 2, Eq. (2.23); Section 4, Eq. (4.19); Abstract Eq. (1.5)"},{"comment":"The central nonperturbative prediction is not a prediction for a real observable as written. Eq. (4.28) states a contribution to Δ(Q)/(2N) that is explicitly proportional to -i times a real combination of cos and sin, while Δ(Q)/(2N) is real. At most Eq. (4.28) can be the Stokes discontinuity of the transseries, or one of the two half-discontinuities entering the Dawson-function expansion (4.9). The text after Eq. (4.9) says that the reality condition fixes the Stokes parameter, but it never constructs the real physical combination from the k and -k contributions or from the two sides of the Stokes line. This is not cosmetic: taking the real part, taking a half-sum, or combining k and -k with different relative weights changes the numerical prefactor and the relative weight of the cos and sin terms. The comparisons in Section 5 and Appendix B are made against this imaginary expression, so the claimed exponential correction is not yet defined as a physical correction to the conformal dimension.","section":"Section 4, Eq. (4.28)"},{"comment":"The geometric interpretation is incomplete for the full nonperturbative correction. The heat-kernel exponential contribution in Eq. (4.20) contains both kπr cos(2πkμr) and tμ sin(2πkμr) terms, but the worldline determinant computation in Appendix B.3 reproduces only the cos term through the factor cos(2πkμr) in Eq. (B.27). Section 5 explicitly states that the sin term is a two-loop correction that is not reproduced, that the detailed construction of the action and saddle is lacking, and footnote 3 says that omitted counterterms cannot be neglected in the physical limit at fixed μt/r. Therefore, the claim that the nonperturbative corrections are controlled by worldline instantons on S^2 geodesics is not established for the complete expression that enters Eq. (4.28).","section":"Section 5 and Appendix B"}],"minor_comments":[{"comment":"The argument of the Bessel function in Eq. (4.23) is written as 2πrΦ0|k|r, which contains an extra factor of r compared with the natural dimensionless combination 2πrΦ0|k| used in Eq. (4.24) and in the exponent below it; please correct this typo.","section":"Eq. (4.23)"},{"comment":"The quantities R1 and R2 in Figure 4 are used for Richardson transforms but are never defined; please define them in the caption or in the text.","section":"Section 3.2, Figure 4"},{"comment":"The abstract says the exponential corrections 'relate' the convergent small-q expansion to the asymptotic large-q behavior, but the paper does not exhibit a full transseries that connects the two regimes beyond the leading exponential correction; please clarify what relation is actually established.","section":"Abstract and Section 4"},{"comment":"The small-q convergence radius is established only numerically through ratio tests, Richardson transforms, and Darboux analysis of the first twenty coefficients, not by a proof; the text should state this limitation explicitly in the summary of the claim.","section":"Section 3.2"},{"comment":"The concluding paragraph correctly acknowledges that the worldline interpretation is suggestive rather than derived; this is commendable, but it should be reflected in the abstract's phrasing, which currently states the geometric interpretation more definitively.","section":"Conclusion, Section 6"}],"recommendation":"major_revision","confidential_remarks":"The two leading issues — the κ0^{3/2} vs κ0^{-3/2} inconsistency and the imaginary Eq. (4.28) — are load-bearing and must be fixed before the paper can be considered for publication. If the authors can identify the correct leading coefficient and convert Eq. (4.28) into a well-defined real (or explicitly Stokes-discontinuity) statement, the paper would be a valuable contribution. Otherwise the central exponential prediction remains ambiguous."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is the first systematic fermionic large-charge resurgence analysis in the NJL model, and the small-q convergent series computation is a genuinely new piece of work. I would not desk-reject it. But before the central exponential result can be cited, two things need fixing.\n\nWhat is new and good: the all-orders small-q expansion for Delta/(2N) with a numerical convergence radius around 0.35, supported by Padé and Darboux analysis; the large-q asymptotic expansion with factorial growth; and the explicit transseries-style heat kernel with a worldline-instanton interpretation. The application to the NJL model is new, and the authors are honest that the framework and the e^{-alpha sqrt{q}} conjecture come from the bosonic O(N) work in [19] and from [14]. There are no fitted parameters: kappa0 is fixed by a transcendental equation. The worldline construction in Section 5 and Appendix B is explicitly labeled incomplete, which is the right call.\n\nSoft spots. First, the leading large-q coefficient is stated in two incompatible forms: Eq. (2.23) and Eq. (1.5) have 2/3 kappa0^{3/2} q^{3/2} + ..., while Eq. (4.19) has 2/3 (q/kappa0)^{3/2} + .... One of these is a typo, but it is the first term of the central expansion and must be corrected.\n\nSecond, and more substantive: Eq. (4.28) is purely imaginary. A scaling dimension is real, so an imaginary term cannot be the physical nonperturbative correction. At most it is the Stokes discontinuity, or one side of it. The paper says the reality condition fixes the Stokes parameter, but it never constructs the real physical combination from the k and -k contributions and from the two sides of the Stokes line. This is not cosmetic: the numerical prefactor and the relative weight of the cos and sin terms change depending on how the real part is taken. The worldline comparison in Section 5 and Appendix B is matched against this imaginary expression, so it does not define a prediction for Delta. This is a load-bearing gap in the paper's central claim.\n\nThird, the convergence radius is established numerically, not analytically, and the connection to a massless radial mode is plausible but left to future work. The authors say so, and that is fine.\n\nWho is this for: people working on large-charge CFT and resurgence. The machinery is detailed and mostly self-consistent, and the e^{-alpha sqrt{q}} structure is likely robust even if the prefactor is not. I would send this to a serious referee with the expectation of major revision, and I would bring it to a reading group to argue about the Stokes issue.","headline":"First systematic fermionic large-charge resurgence analysis for the NJL model, with a solid small-q convergent expansion; the claimed exponential correction is at best a Stokes discontinuity, not a real prediction.","tokens_in":20862,"tokens_out":3891,"would_cite":true,"duration_ms":43655,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In the 3d NJL fixed point, with $q=Q/(2N)$ fixed, the scaling dimension is a small-$q$ convergent series and a large-$q$ transseries whose exponential corrections are $S^2$ worldline instantons.","keywords":["large charge expansion","Nambu-Jona-Lasinio model","double-scaling limit","resurgence","transseries","worldline instantons","conformal field theory","zeta-function regularization"],"falsifier":"The claim would be settled by evaluating the exact zeta function (2.15) numerically at fixed $q$ and comparing against the truncated transseries (perturbative terms plus the Eq (4.28) exponentials): a mismatch beyond estimated higher-order corrections, or an independent computation of $\\Delta(Q)/(2N)$ that fails to reproduce the radius $|q|\\approx0.35(3)$ and the $e^{-\\alpha\\sqrt{q}}$ corrections, would falsify it.","tokens_in":19943,"feed_emoji":"⚛️","tokens_out":17880,"duration_ms":144003,"temperature":0.7,"pith_summary":"This paper studies the three-dimensional Nambu–Jona–Lasinio model at its ultraviolet fixed point in a double-scaling limit: charge $Q$ and flavor number $N$ both go to infinity with the ratio $q=Q/(2N)$ held fixed. The authors claim that the conformal dimension of the lowest charged operator, $\\Delta(Q)/(2N)$, is governed by two complementary expansions: a convergent power series in $q$ at small $q$, with radius $|q|<0.35(3)$, and an asymptotic series in inverse powers of $q$ at large $q$ whose coefficients grow like $(2n)!$. They then identify the exponentially small corrections that connect the two regimes, of the form $e^{-\\alpha\\sqrt{q}}$ with explicit prefactor and phase given in Eq (4.28), and argue that these corrections are worldline-instanton contributions from particles winding around great circles of the $S^2$ in the cylinder geometry. If true, the large-charge expansion of this fermionic CFT is a complete transseries with a geometric interpretation, matching the pattern previously found for the bosonic O(N) model.","feed_headline":"Fermion large-charge expansion is a worldline-instanton transseries","feed_subtitle":"Fixed-ratio charge and flavor make the 3d NJL spectrum a transseries: convergent small q, instanton-driven large q.","key_machinery":"The argument is carried by the zeta-function-regularized one-loop grand potential, $S(\\mu,\\Phi_0)=-\\beta N\\,\\zeta(-1/2)$ with the gap equation $\\Phi_0\\,\\zeta(1/2)=0$, in which $\\Phi_0$ is the Hubbard–Stratonovich condensate (the fermion gap) and $\\mu$ the chemical potential. Around $\\mu=1/(2r)$ the small-$q$ regime is reached by a trinomial expansion of the zeta function whose convergence region $\\mathcal{C}=\\{r|\\tilde{\\mu}|<1/4,\\ r^2|\\Phi_0^2+\\tilde{\\mu}^2|<1/4\\}$ controls the radius of convergence. Around large $q$ the Mellin transform of the heat kernel with a Poisson resummation converts the $\\ell$-sum into a sum over winding numbers $k$; the asymptotic expansion of Dawson's function, with the Stokes parameter fixed by reality of the heat kernel, produces both the factorial growth $(2n)!$ and the exponential corrections. The dimensionless constant $\\kappa_0$ defined by $\\kappa_0\\tanh\\kappa_0=1$ fixes the large-$q$ relation $\\Phi_0=\\mu\\sqrt{\\kappa_0^2-1}$ and appears in the exponential coefficient $\\alpha=2\\pi\\sqrt{\\kappa_0^{-1}-\\kappa_0^{-3}}$. The geometric interpretation is provided by worldline instantons: geodesics on $S^2$ with action $S_I=(\\pi k r)^2$ and fluctuation determinant $\\cos(2\\pi k\\mu r)$ reproducing the nonperturbative terms.","core_discovery":"The central discovery is that at the 3d NJL fixed point, in the double-scaling limit $q=Q/(2N)$ fixed, the scaling dimension admits a convergent small-$q$ expansion with radius of convergence $|q|\\approx 0.35(3)$, whose leading singularity lies on the negative real axis with Darboux exponents $p=-1/2$ for the gap and $p=-3/2$ for the dimension, and a divergent asymptotic large-$q$ expansion whose coefficients grow like $(2n)!$. Using a Mellin-transform representation of the zeta-regulated determinant and a Poisson-resummed heat kernel, the paper extracts the nonperturbative part of the transseries: $\\Delta(Q)/(2N)$ receives corrections of the form $-i (\\kappa_0^2-1)^{1/4} q^{3/4} e^{-2\\pi \\sqrt{\\kappa_0^{-1}-\\kappa_0^{-3}}\\, |k| \\sqrt{q}} \\, (2\\pi \\kappa_0^{9/4}\\sqrt{|k|})^{-1} \\left[\\sqrt{\\kappa_0^2-1}\\,\\cos(2\\pi k\\sqrt{q}/\\kappa_0^{3/2}) + \\sin(2\\pi k\\sqrt{q}/\\kappa_0^{3/2})\\right] + \\ldots$, where $\\kappa_0\\approx 1.199678640257733\\ldots$ is the positive solution of $\\kappa_0\\tanh\\kappa_0=1$. These exponential corrections are the resurgence partners of the factorial divergence of the large-$q$ series, and the paper identifies their geometric origin: worldline instantons with action $S_I=(\\pi k r)^2$, i.e. fermions whose worldlines wind $k$ times around geodesics of the sphere. The reality of the effective action fixes the Stokes parameter, removing the ambiguity that normally plagues asymptotic expansions.","pith_inferences":["The paper leaves open the detailed first-quantized action at fixed $\\mu t/r$; a testable extension would be to construct the complexified saddle that reproduces the $\\sin(2\\pi k\\mu r)$ term in Eq (4.20), which the paper notes is not explained by the simple real-geodesic fluctuation determinant.","Because the canonical ensemble simplifies the singularity structure of the small-$q$ series, the same simplification may occur at large $q$; writing the worldline path integral directly in fixed-charge variables could produce a cleaner instanton action and settle which of the competing worldline formulations is correct.","The methods should transfer to the Gross–Neveu model at large charge, whose large-$N$ phase is a Fermi surface with a BCS instability; a similar Poisson-resummed heat-kernel analysis would predict whether its transseries has the same $e^{-\\alpha\\sqrt{q}}$ form with a different gap equation."],"forward_implications":["The small-$q$ expansion of $\\Delta(Q)/(2N)$ is convergent rather than asymptotic, so the lowest-charge spectrum in that regime is determined by a finite-radius power series, with radius estimated as $|q|<0.35(3)$.","The large-$q$ expansion is asymptotic with coefficients growing like $(2n)!$, so optimal truncation occurs at about $\\sqrt{q}$ terms and the remaining error is exponentially small.","The leading nonperturbative corrections take the explicit form $e^{-\\alpha\\sqrt{q}}$ with $\\alpha=2\\pi\\sqrt{\\kappa_0^{-1}-\\kappa_0^{-3}}\\,|k|$, phased by $\\cos(2\\pi k\\sqrt{q}/\\kappa_0^{3/2})$ and $\\sin(2\\pi k\\sqrt{q}/\\kappa_0^{3/2})$, making the large-charge spectrum a transseries rather than a plain divergent series.","The same exponential corrections are reproduced by a worldline-instanton calculation with action $S_I=(\\pi k r)^2$ and fermion-fluctuation determinant $\\cos(2\\pi k\\mu r)$, giving a geometric origin for the transseries.","The conjecture from the bosonic large-charge analysis, that nonperturbative corrections to gapped fixed-charge systems scale as $e^{-\\alpha\\sqrt{Q}}$, is confirmed for a fermionic model with a Cooper-pair condensate."],"supporting_citations":[{"why":"Supplies the state-operator correspondence $\\Delta(Q)=rF(Q)$ that turns the conformal dimension into a fixed-charge ground-state energy.","marker":"[1]"},{"why":"Establishes the double-scaling limit $Q\\to\\infty$, $N\\to\\infty$, $q=Q/(2N)$ fixed in the bosonic O(N) model, the method this paper extends to the NJL model.","marker":"[3]"},{"why":"Provides the earlier NJL double-scaling study that defines the setup, the Cooper-pair collective field, and the first large- and small-$q$ terms the paper extends.","marker":"[14]"},{"why":"Gives the bosonic large-charge resurgence analysis, the conjecture that nonperturbative corrections scale as $e^{-\\alpha\\sqrt{Q}}$, and the worldline-instanton interpretation the paper confirms for fermions.","marker":"[19]"},{"why":"Supplies the analytic-structure method connecting the radius of convergence to a gapless radial mode, used to characterize the small-$q$ singularities.","marker":"[20]"},{"why":"Provides the Darboux-theorem large-order analysis used to extract the singularity exponent from the series coefficients.","marker":"[26]"},{"why":"Sets up the worldline formalism for one-loop effective actions used in the geometric interpretation.","marker":"[30]"},{"why":"Proposes the unified worldline treatment of axial couplings used to reproduce the instanton contributions to the fermionic heat kernel.","marker":"[37]"}],"fun_headline_variants":["NJL large-charge transseries from worldline instantons","Resurgent NJL: geodesic instantons unify small and large charge","3d NJL: large-charge spectrum is a worldline-instanton transseries","Worldline instantons tame the NJL large-charge expansion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the zeta-function-regularized one-loop determinant $S(\\mu,\\Phi_0)=-\\beta N\\,\\zeta(-1/2)$, together with the gap equation $\\Phi_0\\,\\zeta(1/2)=0$, is the exact grand potential in the double-scaling limit at the NJL UV fixed point, and that requiring the heat kernel to be real uniquely fixes the Stokes parameter; if either premise fails, both the convergent small-$q$ series and the large-$q$ transseries would not describe the actual fixed point.","fun_headline_variants_meta":{"raw":{"variants":["NJL large-charge transseries from worldline instantons","Resurgent NJL: geodesic instantons unify small and large charge","3d NJL: large-charge spectrum is a worldline-instanton transseries","Worldline instantons tame the NJL large-charge expansion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000794,"raw_usage":{"total_tokens":3568,"prompt_tokens":1087,"completion_tokens":2481,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":703,"completion_tokens_details":{"reasoning_tokens":2400}},"tokens_in":703,"tokens_out":2481,"duration_ms":19242,"temperature":1.0,"reasoning_tokens":2400,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:26:58.855203+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The claim would be settled by evaluating the exact zeta function (2.15) numerically at fixed $q$ and comparing against the truncated transseries (perturbative terms plus the Eq (4.28) exponentials): a mismatch beyond estimated higher-order corrections, or an independent computation of $\\Delta(Q)/(2N)$ that fails to reproduce the radius $|q|\\approx0.35(3)$ and the $e^{-\\alpha\\sqrt{q}}$ corrections, would falsify it.","supporting_citations":[{"cited_title":"The analytic structure of the fixed charge expansion","cited_arxiv_id":"2202.13165","evidence_quote":"Supplies the analytic-structure method connecting the radius of convergence to a gapless radial mode, used to characterize the small-$q$ singularities."},{"cited_title":"A New Approach to Axial Vector Model Calculations","cited_arxiv_id":"hep-th/9807072","evidence_quote":"Proposes the unified worldline treatment of axial couplings used to reproduce the instanton contributions to the fermionic heat kernel."}],"review_version":1}