{"id":"1ca8007a-c8c8-449e-9b4d-f5b5c1143d27","arxiv_id":"2502.05645","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A first interacting 4D Carrollian Yukawa theory is constructed, showing ultralocal fermion-scalar interactions and one-loop beta functions with mostly Gaussian fixed points.","lead":"This paper quantizes Carrollian Dirac fermions, particles that live in a spacetime where the speed of light is zero, and couples them to a scalar field in a toy Yukawa model. It finds that the interaction becomes a point-like, time-dependent delta potential and computes how the coupling constants change with energy scale.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Carrollian non-Gaussian fixed point in Table II is an artefact of an unjustified spatial-volume regulator and is retracted by the paper's own Appendix A, so the RG claim is unsupported.","rationale":"The reader correctly identifies the regulator dependence of K3 and the Appendix A retraction as the weakest assumption. I agree. The reason this is load-bearing is that the paper's advertised advance over previous Carrollian scalar work is the one-loop RG analysis and the fixed-point comparison in Table II; the abstract explicitly mentions renormalization and fixed-point analysis. If the non-Gaussian fixed point is an artefact of an arbitrary 'a*Lambda' replacement for an infinite spatial-volume integral, then the main quantitative result is not established. The concern is not disagreement with the Carrollian programme or with the possibility of quantizing ultralocal fermions; it is an internal-consistency problem. The main-text beta functions omit field-strength renormalization and use p = 0, while Appendix A supplies the omitted terms and reaches a different conclusion. No formal verification or reproducible code is provided, so there is no independent check of the algebra. The quantization and the tree-level potential (66) appear sound and would survive, but the RG claim needs reworking. A conditional verdict is appropriate; I would not reject the paper outright because the core quantization and CPT results are likely useful.","tokens_in":25807,"tokens_out":11346,"duration_ms":116758,"concrete_test":"Recompute the Carrollian one-loop beta functions using a finite-box regularization of the spatial integral, e.g. K3 = (2 pi)^3 N / V, and including the wavefunction renormalization terms delta_phi and delta_psi from Appendix A. Solve beta_i = 0 without imposing M^2 << Lambda^2 and compare the resulting fixed points with Eq. (102). If the non-Gaussian fixed point disappears, or if M*^2 and lambda* change when the same calculation is repeated with K3 = (4 pi / 3) Lambda_s^3, then the Table II claim is regulator-dependent and should be withdrawn.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central RG claim—that Carrollian Yukawa theory flows to the non-Gaussian fixed point (102), M*^2 = -Lambda^2/3, lambda* = 128 pi^4 Lambda^3 / (9 K3)—depends on treating the divergent spatial integral K3 = integral d^3k as a finite constant a*Lambda. This is not a standard Wilsonian shell integration: Carrollian propagators carry no spatial momentum, so every one-loop diagram in Eqs. (96)-(100) is multiplied by the same K3, which is an extensive volume factor rather than a UV momentum-shell contribution. The beta functions (101) and the fixed point therefore depend on the arbitrary choice K3 = a*Lambda; with K3 ~ (4 pi / 3) Lambda^3 or with a finite-box regularization, the fixed-point value and even its existence change. Independently, the fixed point itself has M*^2 = -Lambda^2/3, which violates the M^2 << Lambda^2 assumption used to derive the simplified beta functions, and Appendix A explicitly concludes that in this regime 'only Gaussian fixed points' remain. The main text's Table II is thus internally retracted. The free-field quantization and the tree-level ultralocal potential in Eq. (66) are not affected, but the one-loop renormalization and fixed-point structure claimed in the abstract and in Section V.A.2 are not supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the electric Carrollian limit of Dirac fermions in 3+1 dimensions. It constructs the free Carrollian Dirac action, canonically quantizes the field, computes the Feynman propagator, and analyzes charge conjugation, parity, and time reversal for the relevant bilinears. It then couples the fermions to a Carrollian scalar through a Yukawa interaction and obtains the tree-level scattering amplitude, which yields an ultralocal, time-dependent delta-function potential, Eq. (66). The rest of the paper applies Wilsonian renormalization at one loop: the authors derive beta functions (101), identify a non-Gaussian fixed point (102), analyze its stability via the matrix (104), and summarize the fixed-point structure in Table II. An appendix presents a more complete one-loop calculation and reaches a different conclusion about the Carrollian fixed points.","tokens_in":26089,"tokens_out":13716,"duration_ms":145057,"significance":"The free-field quantization of the electric Carrollian Dirac fermion and the derivation of the tree-level ultralocal Yukawa potential are clean, self-contained, and potentially useful: they provide an explicit fermionic example in Carrollian field theory, connect the Carrollian limit to time-dependent point-like interactions relevant in condensed matter and quantum mechanics, and the discrete symmetry analysis is systematic. The paper is also honest in presenting limitations in Appendix A. However, the renormalization-group analysis, which is advertised in the abstract and Section V, is not supported as written: the non-Gaussian fixed point in the main text is contradicted by the paper's own Appendix A, and the one-loop beta functions depend on an ad hoc regularization of the divergent spatial momentum integral K3. The fixed-point claim should therefore be corrected or substantially qualified before the paper can be accepted.","major_comments":[{"comment":"The non-Gaussian fixed point M*^2 = -Λ^2/3, λ* = 128π^4Λ^3/(9K3) presented in the main text is internally contradicted by Appendix A. The beta functions (101) were derived assuming M^2 << Λ^2, but the fixed point violates this condition. Appendix A explicitly states that the non-Gaussian fixed points for M^2 are O(Λ^2), that this 'invalidates the condition M^2 << Λ^2', and that 'Consequently, we are left with only Gaussian fixed points.' Table II and the discussion around Eq. (102) therefore do not represent a consistent result of the paper as a whole; this contradiction must be resolved before the renormalization claim can be accepted.","section":"Section V.A.2, Eq. (102) and Table II"},{"comment":"The one-loop results depend on the divergent spatial volume K3 = ∫d^3k, which is treated as a finite constant. Since the Carrollian propagators depend only on frequency, K3 factors out of every loop integral and is not a Wilsonian momentum-shell contribution. The replacement K3 = aΛ used in Appendix A is an ad hoc regularization, and the fixed point value λ* in Eq. (102) changes with the choice of K3 (for example K3 ~ (4π/3)Λ^3 versus K3 = aΛ); the appendix's own conclusion is that only Gaussian fixed points survive in the regime of validity. The beta functions (101) and the stability analysis (104) are therefore regulator-dependent as stated, and a physical justification for K3, or a regulator-independent statement of the results, is required.","section":"Section V.A.2, Eqs. (96)-(101) and Appendix A"},{"comment":"The main text derives the beta functions (101) after setting external momenta to zero and neglecting field-strength renormalization, referring to Appendix A for details. The full one-loop beta functions in Appendix A contain additional contributions from field-strength renormalization, and the appendix finds that the non-Gaussian Carrollian fixed points lie outside the M^2 << Λ^2 regime used in the derivation. The relationship between the truncated beta functions (101) and the more complete expressions (A9)-(A11) is not explained, and it is not shown that the fixed point (102) is robust under the inclusion of the omitted terms. The authors should either justify the truncation or present the fixed-point structure only in the regime where the approximations are controlled.","section":"Section V.A.2 and Appendix A"}],"minor_comments":[{"comment":"The tree-level potential V(t,x) = -i g^2 e^{-iM|t|} δ^3(x)/(2M) is complex, while the interaction term -g φ Ψbar Ψ in the Lagrangian (62) is Hermitian. The authors should clarify whether this is an effective or non-Hermitian potential and how the imaginary coefficient is to be interpreted physically.","section":"Eq. (66)"},{"comment":"The operator ordering in the expression for the Carroll boost charge C^i is not normal ordered; using the anticommutation relation (29) may introduce a divergent c-number term. The authors should specify the normal-ordering prescription or verify the commutator [P^i, C^j] = iδ^{ij}H with the ordering as written.","section":"Eq. (34)"},{"comment":"Reference [7] appears to contain an incomplete author name ('J., E. Have' should presumably be 'J. Hartong, E. Have'), and a few equation numbers are missing or inconsistently referenced; a final proofreading pass would be helpful.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a direct contradiction between Section V.A.2 and Appendix A on the existence of a non-Gaussian Carrollian fixed point. If the appendix is the corrected analysis, the abstract, Table II, and the related discussion in Section V.A.2 need to be revised as a unit. The free-field quantization and tree-level ultralocal potential are sound and could form the basis of a publishable paper even if the fixed-point claim is removed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First: the free Carrollian Dirac quantization and the tree-level ultralocal potential are solid enough to be useful. The paper's real new content is the electric-limit Dirac field, its CPT table, and the Yukawa coupling whose tree-level amplitude becomes -i g^2/(2M) e^{-iM|t|} delta^3(x). That is a legitimate extension and the derivation is clean. For someone working in Carrollian QFT or ultralocal toy models, that half is worth reading.\n\nThe one-loop renormalization section is the soft spot, and the stress-test note is right. The beta functions (101) are all multiplied by K3 = integral d^3k, the total spatial volume, not by a loop-momentum shell. Treating it as a*Lambda is an arbitrary choice; the fixed point values and even their existence depend on it. More seriously, the fixed point in Table II, M*^2 = -Lambda^2/3, violates the M^2 << Lambda^2 assumption used in the same derivation, and Appendix A says so explicitly: in that regime only Gaussian fixed points remain. That is an internal retraction of the central RG claim. I don't think this is fatal to the quantization part, but it is a load-bearing flaw for the abstract's claims about renormalization and fixed points.\n\nThere are smaller things: the Wilsonian procedure is only sketched, the distinction between shell integration and volume regularization is never confronted, and the comparison with relativistic beta functions in (83) is standard but not the issue. The citation pattern looks fine; the Carrollian references are appropriate, and the self-citations to the scalar RG work are used as benchmarks, not as inputs. The paper is honest about its own limitations, which I appreciate.\n\nWho this is for: people doing Carrollian QFT and ultralocal models will get value from the quantization and the tree-level potential. The RG section should not be taken at face value.\n\nRecommendation: I would not desk-reject it, because the quantization half deserves a referee. I'd tell the referees to focus on whether the RG section can be repaired or should be cut. If I were the editor, I'd send it to review and expect heavy revision.","headline":"The quantization half is solid and worth a referee; the one-loop fixed-point claim is withdrawn by the paper's own Appendix A and should not survive in current form.","tokens_in":26595,"tokens_out":1794,"would_cite":false,"duration_ms":19034,"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":"Taking the electric Carrollian limit of Yukawa theory yields quantized Dirac fermions whose interaction is a time-dependent Dirac-delta potential, and one-loop renormalization flow that is either unstable or Gaussian.","keywords":["Carrollian symmetry","ultralocal field theory","Dirac fermion quantization","Yukawa theory","Wilsonian renormalization group","renormalization fixed points","CPT transformations","flat-band systems"],"falsifier":"Compute the one-loop Carrollian $\\beta$ functions keeping the field-strength renormalization terms and without imposing $M^2 \\ll \\Lambda^2$; if the non-Gaussian fixed point at $M_*^2 = -\\Lambda^2/3$ shifts or disappears, the reported fixed-point structure is an artifact of the regulator. A complementary check is to repeat the Wilsonian analysis with a dimensionally regularized spatial integral instead of $K_3 = a\\Lambda$; genuine regulator independence would keep the fixed points fixed, while scheme dependence would expose the assumption.","tokens_in":25613,"feed_emoji":"⚛️","tokens_out":9683,"duration_ms":86909,"temperature":0.7,"pith_summary":"Carrollian field theories are the $c \\to 0$ contraction of relativistic theories: the light cone closes, particles cannot move through space, and only time evolution remains in the electric limit. This paper claims to provide the first quantized interacting example with Dirac fermions, obtained as the electric Carrollian limit of Yukawa theory. The fermions are canonically quantized, their charge-conjugation, parity, and time-reversal properties are worked out, and the tree-level Yukawa potential is exactly a spatial Dirac delta with a time-dependent factor, $V(t,x) = - i g^2 e^{-iM|t|}\\delta^3(x)/(2M)$. The paper then applies one-loop Wilsonian renormalization to the same model and finds that the fixed-point structure is inverted relative to the relativistic theory: with a $\\phi^4$ term the Carrollian theory has an unstable non-Gaussian fixed point, and without it only the Gaussian fixed point survives. The broader interest is that this connects Carrollian physics to well-studied point-like interactions in quantum optics and condensed matter and provides a field-theoretic origin for ultralocal interactions.","feed_headline":"Carrollian fermions interact via a time-dependent delta potential","feed_subtitle":"First quantized Carrollian Dirac fermions; one-loop flow lands on unstable or Gaussian fixed points.","key_machinery":"The load-bearing object is the electric Carrollian limit, implemented by the rescalings $\\tilde{\\Psi} = \\sqrt{c}\\,\\Psi$ and $\\tilde{m} = m/c$, which sends the Dirac equation to $i\\gamma^0\\dot{\\Psi} = m\\Psi$ and the Klein-Gordon equation to $\\ddot{\\phi} + M^2\\phi = 0$. Because no spatial derivatives survive, all propagators depend only on the frequency $w$, and the interaction vertex is local in space; this is what turns the Yukawa potential into a time-dependent Dirac delta. The Wilsonian machinery then scales only the frequency, $w' = bw$, leaving the spatial momenta untouched, which produces the unusual scaling dimensions $d_\\phi = -1/2$ and $d_\\psi = 0$ and the linear terms in the Carrollian $\\beta$ functions. The divergent spatial momentum integral $K_3 = \\int d^3k$ is regularized as $K_3 = a\\Lambda$, so every loop correction carries this factor and the fixed-point values depend on it.","core_discovery":"The core claim is that an interacting quantum field theory of Carrollian Dirac fermions exists and is described by the electric Carrollian limit of Yukawa theory. The authors start from the Carrollian Dirac Lagrangian $\\mathcal{L}_{CD} = \\bar{\\Psi} i\\gamma^0 \\dot{\\Psi} - m\\bar{\\Psi}\\Psi$, obtain the Feynman propagator $S_F(w) = i(w\\gamma^0 + m)/(w^2 - m^2 + i\\epsilon)$, and couple the fermions to a Carrollian scalar through $g\\phi\\bar{\\Psi}\\Psi$ plus a $\\lambda\\phi^4$ self-interaction. Computing the tree-level scattering amplitude gives $V(t,x) = - i g^2 e^{-iM|t|}\\delta^3(x)/(2M)$, an ultralocal, time-dependent point interaction. At one loop, Wilsonian renormalization with only the frequency rescaled gives $\\beta$ functions whose fixed points are summarized by Table II: in the Carrollian theory with $\\lambda \\neq 0$ there is a non-Gaussian fixed point $(M^2_* = -\\Lambda^2/3, m_* = 0, \\lambda_* = 128\\pi^4\\Lambda^3/(9K_3), g_* = 0)$ with a relevant direction, while deleting $\\lambda\\phi^4$ leaves only Gaussian fixed points; the relativistic theory shows the opposite pattern. The paper also shows that the allowed CPT-invariant Lagrangian terms are only the kinetic and mass terms, because $\\Psi^\\dagger\\Psi$ is odd under charge conjugation and $i\\bar{\\Psi}\\gamma^5\\Psi$ is odd under time reversal.","pith_inferences":["Editorial inference: if the spatial-momentum integral $K_3$ factorizes at all loop orders, the ultralocal interaction may be exact rather than a tree-level accident, and a lattice regularization of the spatial directions would provide a direct numerical test.","Editorial inference: the negative scalar mass-squared at the non-Gaussian fixed point hints at a Carrollian analogue of spontaneous symmetry breaking; adding a symmetry-breaking term could lift the instability and turn the fixed point into a physical critical point.","Editorial inference: because the free Carrollian Dirac propagator carries no spatial momentum, the model may be exactly solvable in the ultralocal sector, and the same Wilsonian treatment applied to magnetic Carrollian fermions (which keep spatial derivatives) could restore a richer fixed-point structure.","Editorial inference: since $\\Psi^\\dagger\\Psi$ is odd under charge conjugation, a Carrollian theory coupled to an electromagnetic-like field through this bilinear would break C and could produce distinct transport signatures in flat-band systems."],"forward_implications":["The tree-level potential $V(t,x) = -i g^2 e^{-iM|t|}\\delta^3(x)/(2M)$ gives Carrollian Yukawa theory as a field-theoretic origin for the time-dependent Dirac-delta potentials used in quantum tunnelling, quantum defects, and quantum optics.","Because the Carrollian beta functions contain linear terms in $g$ and $\\lambda$, the deformed ($\\lambda=0$) Carrollian theory has only irrelevant couplings and flows to the Gaussian fixed point, so it remains perturbatively trivial at all scales.","With $\\lambda\\phi^4$ included, the non-Gaussian Carrollian fixed point has stability eigenvalues $-3/2$, $-1$, $1-\\sqrt{10}$, and $1+\\sqrt{10}$, so exactly one direction is relevant and the fixed point is unstable.","In the relativistic theory the roles are reversed: the $\\lambda \\neq 0$ theory has only Gaussian fixed points, while the deformed theory has non-Gaussian points $m_* = \\pm\\Lambda$, $g_* = \\pm4\\pi$ with oscillatory unstable flow, as collected in Table II.","The Carrollian limit systematically converts marginal relativistic couplings into irrelevant couplings, which suggests that interacting Carrollian theories do not generate new universality classes at one loop."],"supporting_citations":[{"why":"Supplies the Wilsonian RG method and the Carrollian scalar fixed-point calculation that this paper extends to Yukawa theory.","marker":"[30]"},{"why":"Provides the classical electric and magnetic Carrollian Dirac actions and the field/mass rescalings used to quantize the fermions.","marker":"[83]"},{"why":"Supplies spinor normalization, canonical quantization, propagator, and CPT conventions used for the fermion sector.","marker":"[84]"},{"why":"Provides the first classical Carrollian Yukawa theory that the present paper promotes to the quantized toy model.","marker":"[27]"},{"why":"Establishes that electric Carrollian theories are an infinite collection of one-dimensional quantum-mechanical systems, grounding the ultralocal interaction interpretation.","marker":"[36]"},{"why":"Gives the Carrollian symmetry framework and free scalar quantization that the interacting construction builds on.","marker":"[1]"}],"fun_headline_variants":["First quantized Carrollian fermions get time-dependent delta force","Carrollian fermions interact via ultralocal time-dependent delta","First interacting Carrollian Dirac fermions: one-loop fixed points","Ultralocal Yukawa coupling for Carrollian fermions, first quantized","Time-dependent delta interaction emerges in Carrollian fermion QFT"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the divergent spatial momentum integral $K_3 = \\int d^3k$ can be treated as a finite constant times the cutoff, $K_3 = a\\Lambda$, and that the resulting cutoff-dependent $\\beta$ functions still describe the physics; this assumption is strained by the theory's own fixed point $M_*^2 = -\\Lambda^2/3$, which violates the $M^2 \\ll \\Lambda^2$ condition used in the calculation.","fun_headline_variants_meta":{"raw":{"variants":["First quantized Carrollian fermions get time-dependent delta force","Carrollian fermions interact via ultralocal time-dependent delta","First interacting Carrollian Dirac fermions: one-loop fixed points","Ultralocal Yukawa coupling for Carrollian fermions, first quantized","Time-dependent delta interaction emerges in Carrollian fermion QFT"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000509,"raw_usage":{"total_tokens":2534,"prompt_tokens":1059,"completion_tokens":1475,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":675,"completion_tokens_details":{"reasoning_tokens":1385}},"tokens_in":675,"tokens_out":1475,"duration_ms":9435,"temperature":1.0,"reasoning_tokens":1385,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T18:31:02.962280+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the one-loop Carrollian $\\beta$ functions keeping the field-strength renormalization terms and without imposing $M^2 \\ll \\Lambda^2$; if the non-Gaussian fixed point at $M_*^2 = -\\Lambda^2/3$ shifts or disappears, the reported fixed-point structure is an artifact of the regulator. A complementary check is to repeat the Wilsonian analysis with a dimensionally regularized spatial integral instead of $K_3 = a\\Lambda$; genuine regulator independence would keep the fixed points fixed, while scheme dependence would expose the assumption.","supporting_citations":[],"review_version":1}