{"id":"02a4e545-6eca-4a2c-83ef-667937df68e2","arxiv_id":"1908.08033","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Tree-level amplitudes in a web of effective field theories are uniquely fixed by locality plus novel single-hard UV scaling constraints, with unitarity emerging in many cases.","lead":"This paper shows that demanding only locality and certain high-energy scaling behaviors can uniquely determine the tree-level scattering amplitudes of a family of effective field theories. It also introduces a new 'single hard limit' constraint and finds evidence that unitarity may not need to be assumed from the start.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The uniqueness program depends on single-hard scaling laws verified only through n=10 (NLSM) and n=7 (YM); an all-multiplicity proof or counterexample is the decisive missing piece.","rationale":"The reader's weakest_assumption pinpoints the finite-multiplicity verification of the single-hard scaling. I agree. The paper is honest about this: Sec. III.E states no all-multiplicity proof is available for YM/NLSM SHS, and Sec. III.D makes the same statement for the NLSM two-particle shift. These scaling laws are not peripheral; they are the constraints imposed on the ansätze in Claims 2–5. If a scaling law fails at higher n, the corresponding claim is false, not merely unproven. The sGal Claim 4 additionally imports Conjecture 1, also finite-n. The internal proofs, where given (e.g., bi-adjoint induction in App. A), appear coherent but inherit this external assumption. I therefore see no reason to change the CONDITIONAL verdict: the paper's conditional theorems are valuable and the finite-n verifications are real evidence, but the headline claim 'UV scaling is sufficient' is not established at all multiplicities. A concrete higher-n check of the SHS (or an analytic proof via CHY/double-copy) is the natural next step.","tokens_in":27664,"tokens_out":15881,"duration_ms":148466,"concrete_test":"Compute the 11-point and 12-point NLSM ordered amplitudes via the CHY formula (or the Carrasco–Mafra–Schlotterer numerator method) and numerically test the single-hard limit: fix a hard particle p, choose a p-favoring basis δ(i,[j,k]) that is compatible with the ordering and one that is incompatible, and extract the leading power of z in A_n(z p, ...). Check for O(z^0) (compatible) and O(z^1) (incompatible). Independently verify the two-particle BCFW shift scaling for n=11. If any case grows faster, Claims 3–5 fail; if all match, run the same test for the special Galileon at n=7 to test Conjecture 1. This would not replace an analytic proof, but it would either produce a counterexample or increase confidence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Claims 2–5 impose the single-hard scaling laws of Sec. III.E as defining constraints on arbitrary-n local ansätze. Yet Sec. III.E explicitly states: 'In the single-hard scaling case we do not have all multiplicity Yang-Mills or NLSM single-hard scaling arguments available, and so explicitly verified Yang-Mills through n = 7 and NLSM through n = 10.' The analogous two-particle-shift scaling for NLSM is likewise verified only through 10 points (Sec. III.D). Thus the load-bearing premise is not the internal logic of the induction (which is plausible) but the existence of a uniform all-n scaling law. If at n>10 an NLSM ordered amplitude grew as O(z^2) for an incompatible δ, or failed the compatible O(z^0) bound, the constraint set would not contain the true amplitude, so the statement that UV scalings 'fix' the theory would be false. The sGal claim (Claim 4) has the additional dependence on Conjecture 1, itself only verified through n=7. These are not mere technicalities: the paper's central claim is that UV scaling is sufficient, and that claim is only as strong as the scaling laws it assumes.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates to what extent tree-level amplitudes in the 'web of theories' (Yang-Mills, gravity, bi-adjoint phi^3, NLSM, DBI-VA, Galileon and special Galileon) can be fixed by locality, unitarity, mass dimension, and UV scaling. The authors introduce a 'single hard scaling' (SHS) limit and use it alongside two-particle BCFW-type shifts. Section IV proves (Claim 1) that flavor-ordered NLSM amplitudes are uniquely determined by locality, mass dimension 2, cyclic invariance, and the BCJ relations, via reduction to the Adler zero. In Section V they claim that bi-adjoint amplitudes are fixed by locality and SHS (Claim 2, proven by induction in Appendix VIII.A); NLSM by locality, unitarity, and SHS/2S (Claim 3); sGal by locality, unitarity, mass dimension, SHS, and an unproven Conjecture 1 on the Galileon vertex (Claim 4); and BI by locality, unitarity, mass dimension, and two-particle shift scaling (Claim 5). They also provide low-multiplicity numerical and analytic checks suggesting that unitarity may be replaced by UV constraints.","tokens_in":27968,"tokens_out":8517,"duration_ms":82427,"significance":"If the proofs were complete, the paper would establish UV scaling as an on-shell principle as powerful as the IR Adler-zero soft bootstrap for a wide class of effective field theories, and would unify the double-copy web under a single UV constraint. The proof of Claim 1 is elegant and the explicit low-multiplicity checks (NLSM through 8/10 points, YM through 7, BI through 6, sGal through 6) are valuable evidence. The bi-adjoint induction in Appendix VIII.A is explicit and goes beyond mere verification. The double-copy argument in Sec. V.E gives a structural explanation for the SHS of composite theories, assuming the building-block scalings. However, the significance is currently conditional: the main uniqueness theorems rely on scaling laws whose all-multiplicity validity is unproven.","major_comments":[{"comment":"The load-bearing assumption is that the SHS laws (66)–(74) and the 2S law (60)–(63) hold at arbitrary multiplicity. The text explicitly states that no all-multiplicity proof is available and that YM was checked through n=7 and NLSM through n=10 (Sec. III.E), with NLSM 2S scaling checked only through 10 points (Sec. III.D). Claims 2–5 impose these laws as defining constraints on arbitrary-n local ansätze; if a higher-multiplicity amplitude violated the assumed O(z^k) behavior for some δ, the constraints would not contain the true amplitude and the claimed uniqueness would fail. The claims should either be accompanied by an all-n proof of the scaling laws, or be restated as conditional results with the verification order made explicit.","section":"Sec. III.E; Sec. V.A–V.D"},{"comment":"The sGal uniqueness theorem is not self-contained: it assumes Conjecture 1, that the Galileon vertex is uniquely fixed by mass dimension [2n-2], SHS O(z^4), and 2S O(z^2). The paper verifies Conjecture 1 only through n=7 and provides no proof. Since the proof of Claim 4 uses Conjecture 1 both to identify any allowed contact term as the Galileon vertex and to exclude it via its SHS behavior, the claim is not established beyond n=7. The finite verification range should be stated explicitly in the theorem, and the proof of Conjecture 1 or a precise conditional statement is needed.","section":"Sec. V.C; Claim 4; Conjecture 1"},{"comment":"The proof of Claim 5 depends on the lemma that no polynomial of mass dimension [n] can have double-soft O(τ^3) scaling. Appendix VIII.B does not actually prove this lemma; after tensorizing, it asserts that no cancellations can occur 'in arbitrarily high dimensions, which is obvious.' That assertion is not demonstrated. A complete linear-algebra argument is needed, since this lemma is what rules out the higher double-soft orders in the BI contact-term analysis. As written, the BI uniqueness theorem is not fully proven. Similarly, the conclusion after Eq. (103) that verifying through τ^2 is sufficient because of 'arguments of the type given in ref. [14]' is a citation rather than a proof for all n.","section":"Sec. V.D; Appendix VIII.B"}],"minor_comments":[{"comment":"The symbol σ denotes both the soft-scaling exponent in Eq. (50) and permutation labels throughout the paper; a different symbol would remove ambiguity.","section":"Sec. III.C"},{"comment":"The word 'theor ies' in the header should read 'theories'.","section":"First page header"},{"comment":"The induction excluding NLSM contact terms is terse; expanding the step from 'C_{n+1} scales as O(τ)' to 'ruled out by Adler-zero uniqueness' would make the argument easier to follow.","section":"Sec. V.B"},{"comment":"The double-copy scaling derivation assumes the NLSM and YM SHS/2S scalings; the text should say explicitly that this does not supply the missing all-multiplicity proof of those scalings.","section":"Sec. V.E"},{"comment":"The list of quantities fixed by UV conditions mixes proven claims, conditional claims, and finite-order checks (Galileon vertex through n=7, sGal through n=6, BI through n=6, DBI-VA through n=8); adding the verification order to each bullet would avoid overstating the results.","section":"Sec. VI"},{"comment":"The 'or better' clauses make the stated scaling axioms non-sharp; the authors might specify the minimal set of inequalities actually used in the induction.","section":"Eqs. (69)–(71)"}],"recommendation":"major_revision","confidential_remarks":"To the editor: this is a well-written paper and the Claim 1 proof, the bi-adjoint induction, and the low-multiplicity evidence are valuable. My main concern is the gap between the unconditional language ('fixed uniquely') and the conditional status of the theorems. I would be comfortable with major revision if the authors either prove the all-order scaling laws and Conjecture 1, or carefully restate every theorem as conditional on those assumptions with the finite verification orders documented in the theorem statements. There do not appear to be fatal internal inconsistencies in the conditional arguments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: genuinely new kinematic probe (single hard scaling) plus a set of uniqueness proofs that carry the bootstrap program into EFT territory. The NLSM-from-BCJ result (Claim 1) is the cleanest new result and looks solid conditional on the Adler-zero uniqueness it cites. The bi-adjoint soft-theorem induction in Appendix VIII.A is explicit and, as far as I can tell, valid. The double-copy inheritance arguments for the scaling of BI and sGal are neat and plausible.\n\nThe main soft spot is exactly what the stress-test note identifies: the SHS scaling laws are load-bearing, and they are verified only through n=7 (YM) and n=10 (NLSM). The paper says so itself. The authors impose these scalings as defining constraints on arbitrary-n ansätze, so if a higher-n amplitude violates the assumed O(z^k) behavior the uniqueness claims would not follow. This is not a hidden flaw; it is an acknowledged gap, but it is a real one. The same applies to Conjecture 1 for the Galileon vertex, verified only through n=7 and then used to get sGal. The 'unitarity emerges' evidence is also only low-multiplicity, and the authors are appropriately cautious there.\n\nProportionately: this is a solid paper, not a revolution. It extends an existing program and introduces one new tool plus several clean results. The gaps are clearly stated, and the paper is worth serious referee time. A referee should push on the all-multiplicity scaling question and ask whether the inductions can be tightened, but I would not desk-reject it. I would bring it to our reading group and would cite it if I worked in this area.","headline":"A solid extension of the on-shell bootstrap program with a genuinely new probe (single hard scaling); the proofs are clean where proven, but the acknowledged all-multiplicity gap is the main thing a referee should press on.","tokens_in":28425,"tokens_out":1610,"would_cite":true,"duration_ms":15784,"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":"Locality plus UV scaling fixes tree amplitudes across theories","keywords":["scattering amplitudes","nonlinear sigma model","Born-Infeld","special Galileon","double copy","BCFW shift","single hard scaling","effective field theory"],"falsifier":"Compute the NLSM tree amplitude at eleven or twelve points, expand it in a $p$-favoring basis with a non-compatible $\\delta$, and check whether the single-hard limit grows at most as $O(z^1)$; a single amplitude growing as $z^2$ would break Claim 3. Alternatively, run the same ansatz bootstrap at ten points and search for a local quartic object with mass dimension two, the claimed $O(z^0)/O(z^1)$ scalings, and unitarity that is not the NLSM amplitude.","tokens_in":27472,"feed_emoji":"⚛️","tokens_out":6873,"duration_ms":61778,"temperature":0.7,"pith_summary":"This paper argues that the tree-level scattering amplitudes of a family of effective field theories—the non-linear sigma model, bi-adjoint scalar, Born-Infeld, and special Galileon—are uniquely fixed by their high-energy (ultraviolet) scaling behavior once locality is assumed. To make this precise, the authors introduce a single-hard limit, sending one external momentum to infinity in a carefully chosen basis of kinematic invariants, and they determine the required scaling powers that select each theory. They prove that NLSM amplitudes follow from locality, unitarity, mass dimension two, and two UV scaling conditions, and they give evidence that unitarity can be dropped and instead emerge from locality plus UV constraints. Because these theories are linked by double-copy relations, the UV rules can be inherited from the building blocks, tying the constraints together across the web. If correct, the result puts effective field theories on the same on-shell footing as Yang-Mills and gravity, where IR and UV information each suffice to rebuild the amplitudes.","feed_headline":"Locality plus UV scaling fixes tree amplitudes across theories","feed_subtitle":"A new single-hard limit shows unitarity can emerge as a bonus, not an input, for NLSM, BI and the special Galileon.","key_machinery":"The central object is the single hard scaling (SHS), a one-line deformation $p\\to zp$ of a single external momentum, evaluated in a $p$-favoring basis of momentum invariants $\\delta(i,[j,k])$ so that momentum conservation cannot obscure the growth in $z$. The growth rate $O(z^k)$ is enhanced, meaning smaller than naive power counting, only for special theories, and the compatibility of $\\delta$ with the color or flavor ordering decides which scaling applies. The two-particle BCFW shift is the other UV probe, and the SHS is effectively half of it. These scalings carry the argument by forcing contact terms to vanish or to cancel against factorization channels, and the double-copy/KLT structure lets the scaling of composite theories such as Born-Infeld and the special Galileon be read off from the scalings of their factors, Yang-Mills and NLSM.","core_discovery":"On the paper's own terms, the central discovery is that prescribed large-momentum scaling is a defining, not merely diagnostic, property of special amplitudes. Claim 3 states that NLSM amplitudes are uniquely fixed by locality, unitarity, and mass dimension two together with: single-hard scaling $A_n(\\sigma)\\sim O(z^0)$ when the chosen basis $\\delta(i,[j,k])$ is compatible with the ordering $\\sigma$, and $O(z^1)$ otherwise; and two-particle-shift scaling $O(z^1)$ for adjacent shifted legs and $O(z^0)$ for non-adjacent legs. The same logic fixes doubly-ordered bi-adjoint amplitudes from locality plus $O(z^{-3})/O(z^{-2})$ single-hard scaling, Born-Infeld from locality, unitarity, mass dimension $n$, and $O(z^0)$ BCFW scaling, and the special Galileon from locality, unitarity, and $O(z^3)$ single-hard scaling, with the last relying on an unproven uniqueness conjecture for the Galileon vertex. The paper further proves that NLSM amplitudes are fixed by locality, cyclic invariance, mass dimension two, and the BCJ color-kinematic relation, without assuming unitarity. Throughout, the authors present explicit checks that unitarity may be a consequence rather than an assumption: local ansätze subjected to UV scaling alone get fixed, with factorization emerging.","pith_inferences":["If the scaling laws are proven at all multiplicities, the bootstrap becomes an algorithm: enumerate local ansätze, impose the single-hard scalings, and solve linear systems, with no Lagrangian needed.","Because the single-hard limit is half of a BCFW shift, two-particle-shift scaling can never be worse than single-hard scaling; this relation may be the seed of an inductive all-multiplicity proof for NLSM and Yang-Mills.","The apparent IR/UV equivalence for massless amplitudes hints that soft and hard limits are dual descriptions of the same on-shell data; probing this duality in celestial or Mellin-transformed amplitudes could be a fruitful test.","The unproven Galileon-vertex conjecture is the weakest link; extending the polynomial check beyond seven points, or proving the uniqueness via determinant identities, would complete the special-Galileon argument."],"forward_implications":["NLSM amplitudes need not be defined by the Adler zero; the same physics follows from locality plus two UV scaling conditions, so IR and UV descriptions become exchangeable.","If unitarity really emerges from locality and UV scaling, factorization is not an independent axiom for these theories, suggesting S-matrix formulations where locality and scaling are primary.","The double copy becomes a tool for predicting UV scaling: scaling powers of a composite theory are sums of the scaling powers of its factors, so constraints can be organized across the web rather than case by case.","Higher-derivative NLSM corrections are strongly constrained: with unitarity and BCJ relations, the $O(p^\\kappa)$ contact terms are unique through $\\kappa=10$ at six points, with the first ambiguity at $\\kappa=12$.","The same single-hard-scaling bootstrap can likely be applied to other effective theories or supersymmetric variants; the paper already checks DBI-VA photon and fermion sectors through eight points."],"supporting_citations":[{"why":"Establishes that locality and unitarity of Yang-Mills and gravity amplitudes follow from singularities and gauge invariance; it provides the template for uniqueness from on-shell principles.","marker":"[1]"},{"why":"Proves uniqueness from gauge invariance and the Adler zero; used in proving the NLSM amplitude-relation claim.","marker":"[2]"},{"why":"Shows soft theorems and infrared behavior suffice to constrain scattering amplitudes; it is the IR counterpart the paper complements.","marker":"[14]"},{"why":"Argues Yang-Mills and gravity tree amplitudes follow from locality and improved BCFW scaling; this is the method the paper extends to effective field theories.","marker":"[26]"},{"why":"Introduces BCFW recursion and the two-particle shift used as a UV probe throughout the paper.","marker":"[15]"},{"why":"Provides the improved UV scaling of Yang-Mills and gravity and links it to spin symmetry; supplies scaling facts used in comparisons.","marker":"[25]"},{"why":"Introduces color-kinematic duality and the amplitude relations central to the double-copy web and to the NLSM claim.","marker":"[27]"},{"why":"Gives the abelian Z-theory representation of NLSM amplitudes, connecting NLSM to the KLT and double-copy structure.","marker":"[79]"},{"why":"Shows vector effective field theories such as Born-Infeld are fixed by infrared soft limits; this is the result the paper recovers from UV behavior.","marker":"[83]"},{"why":"Identifies the special Galileon via a hidden symmetry; it defines the object whose amplitude is constrained in Claim 4.","marker":"[88]"}],"fun_headline_variants":["UV scaling fixes tree amplitudes, unitarity emerges","Hard-scaling constraints pin down amplitudes across theories","Single hard limit defines amplitudes, unitarity is bonus","Locality plus UV scaling fixes amplitudes, unitarity emerges"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the single-hard scaling laws imposed as constraints hold for every multiplicity; the paper verifies Yang-Mills through seven points and NLSM through ten points but has no all-multiplicity proof, and the special Galileon argument additionally depends on the unproven uniqueness of the Galileon vertex under its UV scalings.","fun_headline_variants_meta":{"raw":{"variants":["UV scaling fixes tree amplitudes, unitarity emerges","Hard-scaling constraints pin down amplitudes across theories","Single hard limit defines amplitudes, unitarity is bonus","Locality plus UV scaling fixes amplitudes, unitarity emerges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001197,"raw_usage":{"total_tokens":5032,"prompt_tokens":1137,"completion_tokens":3895,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":753,"completion_tokens_details":{"reasoning_tokens":3833}},"tokens_in":753,"tokens_out":3895,"duration_ms":29889,"temperature":1.0,"reasoning_tokens":3833,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:51:21.542482+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the NLSM tree amplitude at eleven or twelve points, expand it in a $p$-favoring basis with a non-compatible $\\delta$, and check whether the single-hard limit grows at most as $O(z^1)$; a single amplitude growing as $z^2$ would break Claim 3. Alternatively, run the same ansatz bootstrap at ten points and search for a local quartic object with mass dimension two, the claimed $O(z^0)/O(z^1)$ scalings, and unitarity that is not the NLSM amplitude.","supporting_citations":[],"review_version":1}