{"id":"935c5aa3-f525-49f3-8fcf-69aa8af4d130","arxiv_id":"2607.09550","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Under minimal assumptions, the large-charge bootstrap in 3d U(1) CFTs requires a Goldstone Regge trajectory with sound speed squared equal to 1/2; extra trajectories decouple from current and stress tensor under mild extra assumptions.","lead":"Crossing symmetry in three-dimensional CFTs with a U(1) charge forces at least one family of operators whose energies match the Goldstone mode of a conformal superfluid. The result tightens when the stress tensor is used as a probe and shows that extra light modes, if present, can only talk to scalars at this order.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the finite-N assumption as the strongest modelling restriction, but that assumption is not load-bearing for the existence claim that constitutes the paper’s strongest result. The existence proof in §5.1 uses only positivity of the TT spectral weights, the vanishing of the n=1 (TT,δδ) moment, and the strictly positive spin-1 TT normalization; finite N enters only when one proceeds to uniqueness and to the residual SS recurrence under the additional assumptions. The parity-even restriction is likewise flagged and excludes conformal solids by design. Because the central existence argument is self-contained and free of internal gaps, the reader’s ACCEPT verdict stands.","tokens_in":39895,"tokens_out":546,"duration_ms":5007,"concrete_test":"Independently recompute the n=1 (TT,δδ) algebraic equation from the spectral sum (3.40) and the contact-singularity formula (4.22)–(4.27) without using the reduced independent set of §4.4; confirm that the right-hand side is identically zero and that every term on the left is non-negative, so that the existence of a Goldstone trajectory follows solely from positivity plus the spin-1 normalization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the TT equations under the five minimal assumptions force at least one trajectory with ω_ℓ^{2} = J_ℓ^{2} = ℓ(ℓ+1)/2. The argument is local to the TT sector: the n=1 (TT,δδ) moment (eq. 5.3) is a sum of non-negative terms that vanishes only if every T-coupled state satisfies ω^{2} = J^{2} or has vanishing λ_T; the spin-1 TT equation then forbids the all-vanishing case because P_n^(TT)(1) = 1 + ∑ T_{1,i} > 0. Finite N is used only later for uniqueness and for the residual SS recurrence; it is not required for the existence statement. The contact-singularity and macroscopic-limit inputs that produce the algebraic moments are standard and are checked against the EFT in Appendix F. No internal inconsistency or hidden gap in this chain is apparent.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the large-charge bootstrap for three-dimensional CFTs with a global U(1) symmetry, using four-point functions of two heavy charge-Q scalars with light probes chosen from a charged scalar, the conserved current J^a, and the stress tensor T^{ab}. After constructing the H-basis tensor structures and writing the s-channel expansions to next-to-leading order in 1/Q (restricted to parity-even exchanges), the authors convert crossing plus the macroscopic-limit singularity structure into algebraic moment equations on the spectral data. Under five minimal assumptions (unique scalar ground state per charge, existence of the macroscopic limit, NLO content of the s-channel, finitely many Regge trajectories, and parity-even exchanges only), the TT sector forces every trajectory that couples to T to satisfy ω_ℓ^{2} = J_ℓ^{2} = ℓ(ℓ+1)/2, and consistency of the spin-1 equations requires that at least one such trajectory exist. With two further assumptions (no non-Goldstone trajectory reaches zero energy at ℓ=0, and non-degeneracy of primaries at fixed spin), the Goldstone trajectory is unique and all non-Goldstone trajectories decouple from J and T, contributing only to the scalar-scalar channel.","tokens_in":40111,"tokens_out":877,"duration_ms":8692,"significance":"The central result is a genuine bootstrap derivation that the standard conformal-superfluid Goldstone trajectory must appear in the finite-energy, parity-even sector of any 3d large-charge CFT satisfying the stated assumptions. The argument is local to the TT positivity sum (eq. 5.3) and does not insert the Goldstone dispersion by hand; the macroscopic-limit and contact-singularity inputs that produce the algebraic moments are standard and are checked against the EFT in Appendix F. The paper also supplies a complete independent set of crossing equations for the six correlator sectors, two independent proofs of current decoupling (Appendix H), and a clean separation between minimal and additional assumptions. Within the finite-trajectory, parity-even framework the result substantially strengthens earlier scalar and current analyses and brings the large-charge bootstrap close to the expected EFT structure (universal Goldstone plus optional light fields that couple only to scalars).","major_comments":[],"minor_comments":[{"comment":"The abstract and Introduction state that the Goldstone trajectory has sound speed c_s^{2} = 1/2. It would help the reader if this were written once as ω_ℓ^{2} = ℓ(ℓ+1)/2 = J_ℓ^{2} with the explicit identification of c_s, rather than only in the prose of Section 1.","section":null},{"comment":"In Section 4.3 the values of β for the various H-functions are listed without a short derivation. A one-sentence reminder that each explicit power of η^a can raise the singularity by at most one inverse power of distance would make the list self-contained.","section":null},{"comment":"Appendix G reprints the full (redundant) set of crossing equations after the independent subset has already been given in Section 4.4. A brief pointer that Appendix G is for completeness only would avoid the impression of duplication.","section":null},{"comment":"The notation for the residual polynomials R_n(w) and the elementary symmetric polynomials Q_k(w) in Section 5.2.2 is introduced without a forward reference to the standard finite-trajectory recurrence of Refs. [9,10]; a short citation would help.","section":null},{"comment":"A few typographical inconsistencies appear (e.g., occasional missing spaces around = in the displayed equations of Appendix B, and the arXiv identifier in the header). These are purely cosmetic.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a natural and carefully executed sequel to the authors’ earlier current-probe paper. The finite-N assumption is the only real limitation of scope, but it is stated clearly and the existence claim does not rely on it. I see no reason to delay publication."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new result is simple and useful: once you put the stress tensor into the large-charge bootstrap (parity-even exchanges only), the TT spectral sum at the first moment is a sum of non-negative terms that vanishes only if every T-coupled state has ω^{2} = J^{2} or zero OPE coefficient. The spin-1 TT equation then forbids the all-zero case, so at least one Goldstone trajectory with c_s^{2} = 1/2 must exist. That is forced by positivity, not inserted by hand.\n\nThey do the technical work carefully. The H-basis reduction, the conversion of crossing plus contact singularities into algebraic moment equations, the conservation constraints, and the two independent proofs that non-Goldstone trajectories decouple from the current under the extra assumptions are all written out. Appendices are complete enough that you can check the chain without reverse-engineering. Relative to their own scalar+current paper and to Jafferis-Mukhametzhanov-Zhiboedov, the necessity from TT, uniqueness under non-degeneracy, and the residual SS-only freedom for extra trajectories are new.\n\nSoft spots are the ones they flag. Finite number of Regge trajectories is load-bearing for uniqueness and for the residual recurrence, and it excludes Fermi liquids by construction. Parity-even restriction excludes conformal solids (transverse phonons). Macroscopic-limit saturation of the singularity is an input, not a derivation. None of these break the central TT existence claim under the five minimal assumptions; they just mark the domain. The stress-test note is right that finite N is not needed for existence itself.\n\nThis is for people already working large-charge EFT or spinning bootstrap. It organizes which phases are allowed inside the finite-trajectory, parity-even sector and shows how much is fixed once T is included. Math and citation pattern look solid; self-cites are to the direct predecessors.\n\nI would send it to referees. Worth engaging if you care about the large-charge program.","headline":"Clean TT-channel argument that forces the superfluid Goldstone under stated assumptions; finite-N and parity-even are the real scope limits, not hidden gaps.","tokens_in":40701,"tokens_out":511,"would_cite":true,"duration_ms":7880,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Crossing with spinning probes forces the Goldstone dispersion of the conformal superfluid in large-charge 3d CFTs.","keywords":["large charge","conformal bootstrap","conformal superfluid","Regge trajectories","stress tensor","U(1) current","3d CFT"],"falsifier":"Exhibit a unitary 3d CFT with a global U(1) that satisfies the macroscopic limit and unique scalar ground states per charge, yet whose lowest large-charge spectrum contains no parity-even Regge trajectory with ω_ℓ^{2} = ℓ(ℓ+1)/2, or whose TT four-point function is inconsistent with that dispersion.","tokens_in":40806,"feed_emoji":"⚛","tokens_out":991,"duration_ms":10093,"temperature":0.7,"pith_summary":"The paper asks how much of the large-charge physics of three-dimensional CFTs with a U(1) symmetry is fixed by crossing, unitarity, and the existence of the current and stress tensor. Using four-point functions of two heavy charge operators with light scalar, current, and stress-tensor probes, and restricting to parity-even exchanged operators, the authors convert the next-to-leading large-charge crossing equations into algebraic moment constraints on Regge trajectories. Under five minimal assumptions, the stress-tensor channel alone forces at least one trajectory with the sound-speed-squared equals one-half dispersion of the standard conformal superfluid Goldstone mode. Extra non-degeneracy and zero-energy assumptions make that trajectory unique and confine every other light trajectory to the pure scalar channel. A sympathetic reader cares because this is a pure CFT derivation that the superfluid EFT spectrum is not an optional model but a necessary consequence of consistency once spinning universal operators are included.","feed_headline":"Bootstrap forces the superfluid Goldstone in large-charge CFTs","feed_subtitle":"Spinning probes and crossing alone require the sound-speed-squared equals one-half trajectory","key_machinery":"Algebraic moment equations obtained by integrating the τ-discontinuity of the H-basis coefficient functions against Gegenbauer polynomials near the t-channel contact singularity; these force the spectral densities of trajectories that couple to T to vanish unless ω^{2} equals the Casimir J^{2} = ℓ(ℓ+1)/2.","core_discovery":"Under the paper's five minimal assumptions, the tensor-tensor bootstrap equations require that every Regge trajectory coupling to the stress tensor satisfy ω_ℓ^{2} = ℓ(ℓ+1)/2 and that at least one such trajectory exist; that dispersion is exactly the standard Goldstone mode of the conformal superfluid. With the two additional assumptions of non-degeneracy among primaries and no non-Goldstone zero at spin zero, the Goldstone trajectory is unique and all other trajectories decouple from both the current and the stress tensor at this order, contributing only to the scalar-scalar channel.","pith_inferences":["Including parity-odd exchanges should recover the transverse-phonon trajectories of conformal solids as an independent bootstrap solution.","The same spinning-probe method may force analogous Goldstone-like dispersions in higher dimensions or for non-Abelian global symmetries.","A concrete microscopic check would be to extract the lowest large-charge spectrum of the O(2) Wilson-Fisher fixed point and verify the ω^{2} = J^{2} relation for the lightest parity-even tower."],"forward_implications":["Any large-charge phase that couples to the stress tensor at this order must contain at least the conformal-superfluid Goldstone mode.","Under the extra non-degeneracy and zero-energy assumptions, current and stress-tensor correlators are completely fixed by that single Goldstone trajectory.","All remaining freedom at this order sits in the pure scalar channel, corresponding to optional light fields that do not couple to J or T.","Conformal solids and Fermi liquids are excluded from the present system because they violate parity-even or finite-trajectory assumptions."],"fun_headline_variants":["Bootstrap requires Goldstone Regge trajectory in large-charge 3D CFTs","Tensor probes force ω_ℓ²=ℓ(ℓ+1)/2 Goldstone mode via bootstrap","Large-charge bootstrap demands superfluid sound-speed-1/2 trajectory","Spinning probes and crossing enforce Goldstone dispersion relation","Minimal assumptions force unique Goldstone trajectory in large-charge CFT"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"Only a finite number of Regge trajectories stay unsuppressed at the order considered; if infinitely many light trajectories remain, as expected for a Fermi liquid, the finite-order algebraic reduction fails.","fun_headline_variants_meta":{"raw":{"variants":["Bootstrap requires Goldstone Regge trajectory in large-charge 3D CFTs","Tensor probes force ω_ℓ²=ℓ(ℓ+1)/2 Goldstone mode via bootstrap","Large-charge bootstrap demands superfluid sound-speed-1/2 trajectory","Spinning probes and crossing enforce Goldstone dispersion relation","Minimal assumptions force unique Goldstone trajectory in large-charge CFT"]},"model":"grok-4.5","effort":"low","cost_usd":0.003744,"raw_usage":{"total_tokens":1100,"prompt_tokens":666,"num_sources_used":0,"completion_tokens":102,"cost_in_usd_ticks":37440000,"prompt_tokens_details":{"text_tokens":666,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":332,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":666,"tokens_out":102,"duration_ms":3492,"temperature":1.0,"reasoning_tokens":332,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T02:11:59.220152+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a unitary 3d CFT with a global U(1) that satisfies the macroscopic limit and unique scalar ground states per charge, yet whose lowest large-charge spectrum contains no parity-even Regge trajectory with ω_ℓ^{2} = ℓ(ℓ+1)/2, or whose TT four-point function is inconsistent with that dispersion.","supporting_citations":[],"review_version":1}