{"id":"96063538-c33b-4a56-89c1-96fa7a5c9f93","arxiv_id":"2505.02520","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Hidden zeros and smooth splitting in Tr phi^3, NLSM, YMS, and the special Galileon are shown to follow from on-shell recursion under improved UV falloff, yielding generalized, triple, and 4d helicity zeros.","lead":"This paper traces the recently discovered hidden zeros and smooth splitting of scattering amplitudes to a contour-integral argument, provided the shifted amplitude has no pole at infinity. It also produces generalized and four-dimensional splitting and zero formulas, including new helicity zeros.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Improved UV falloff for YMS and special Galileon is unproven and partly circular; the recursion proof stands only if that premise holds at all multiplicities.","rationale":"The reader's weakest_assumption and my concern coincide: the improved UV falloff for YMS and special Galileon. I agree with the CONDITIONAL verdict. The Tr phi^3 and NLSM derivations are supported by the g-vector/delta-shift arguments and explicit examples; for YMS the paper admits circularity, and for Galileon only 6- and 8-point checks exist. A concrete higher-point falloff check would settle the matter. No ad hominem and no manufactured issue: the concern is internal to the proof structure.","tokens_in":30332,"tokens_out":1224,"duration_ms":14564,"concrete_test":"Numerically evaluate a 10-point YMS scalar amplitude on scalar split kinematics Z(X_B) with c* != 0, apply the (X_B,c*)-shift, and fit the leading z power at large z from explicit expressions; if it is not z^{-2}, the proof's premise fails for YMS at n=10. For special Galileon, perform the same check at 10 points (and ideally 12 points) on split kinematics; z^{-2} would support the conjecture, while z^0 or worse would refute it.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central claim is that splitting formulas follow from contour integration plus improved UV falloff. The recursive proof in Section 3.1 is valid provided the shifted amplitude has no residue at infinity on the relevant split kinematics. For Tr phi^3 this is proven via g-vector shifts (Section 2.3), and for NLSM it is derived by commuting g-vector and delta shifts, so those two theories are in good shape modulo the cited result from [43]. The fragile premise is the claimed z^{-2} falloff for YMS and special Galileon on scalar split kinematics. Section 2.2 states explicitly that for YMS this is only conjectured, and that assuming the splitting formula (2.9) makes the scaling follow, creating a circularity if used as input to the recursive proof. For special Galileon, the z^{-2} split-kinematics falloff is verified only at 6 and 8 points, with no proof at higher multiplicity. If at some higher n the Galileon or YMS amplitude fails to fall as z^{-2}, the contour-integral proof of the splitting theorem collapses for that theory, because the bonus relation (3.4) requires the sum of certain residues to vanish. The load-bearing concern is therefore not the analyticity/factorization input but the unproven improved UV behavior for YMS and Galileon.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a new family of complex kinematic shifts, the (Xij, ckl)-shifts, and uses contour integration in the style of BCFW to argue that the recently discovered 'hidden zeros' and 'smooth splitting' properties of tree amplitudes in Tr phi^3, NLSM, YMS, and the special Galileon follow from standard factorization plus improved large-z behavior. For Tr phi^3 and NLSM the improved UV scaling is argued from the g-vector/surfaceology framework; for YMS it is explicitly left as a conjecture, and for the special Galileon it is checked only at six and eight points. The paper also derives higher-order and triple-splitting formulas, discusses their kinematics in the mesh, and analyzes the realization of hidden zeros and new 'helicity zeros' in four dimensions via BCFW recursion.","tokens_in":30608,"tokens_out":4347,"duration_ms":57598,"significance":"If the improved-UV premises were established at all multiplicities, the paper would provide a unified and conceptual explanation of hidden zeros and smooth splitting across four different theories, with a clear recursive mechanism that also generates new splitting formulas, including triple-splitting formulas and generalized all-multiplicity expressions. The Tr phi^3 and NLSM results are the strongest part: there the argument is genuinely recursive and anchored in the cited g-vector/surfaceology results, so the proof is largely complete modulo external references. The paper is also honest about the conjectural status of the YMS scaling, which is a strength. The 4d helicity-zero claim is new, concrete, and falsifiable, and the BCFW-based reasoning is a promising route even if the general inductive proof is only sketched. The significance is tempered by the fact that the headline claim for YMS and the special Galileon rests on unproven, and for YMS partly circular, UV-scaling assumptions.","major_comments":[{"comment":"The recursive proof of the YMS splitting formula is not self-contained as written. The z^-2 scaling on scalar splitting kinematics is introduced as an empirical conjecture, and the text explicitly notes that 'If one assumes the splitting formula (2.9) then this scaling follows; to avoid a circular argument it would be preferable to have an independent understanding of this fact.' But in Section 3.1 the bonus relation (3.4) is precisely the z^-2 scaling that is used to derive the splitting formula (2.9). Thus, for YMS the argument proves only the conditional statement: (2.9) implies improved UV behavior, which implies (2.9). An independent derivation of the z^-2 falloff for (Xeo, c*)-shifts, or an alternative argument that avoids the circularity, is needed before the YMS section can support the paper's stated claim.","section":"2.2 (YMS bullet); 3.1, Eq. (3.4)"},{"comment":"The special Galileon splitting claim rests on the same type of unproven premise. The improved z^-2 falloff on split kinematics is verified only for six- and eight-point amplitudes, and the accompanying KLT discussion explicitly does not establish the higher-multiplicity behavior, because it would require cancellations among NLSM amplitudes with different orderings that are not demonstrated. Since the contour argument in Section 3.1 and the conclusion that the special Galileon 'also splits' depend on this falloff at all n, higher-point checks or a proof are required. As it stands, the Galileon result is a conjecture supported by two examples.","section":"3.3, Eqs. (3.36)-(3.37)"},{"comment":"The all-multiplicity higher-order splitting formula (3.23) is stated as a general result, but the derivation in the text is limited to the 10-point examples (3.18)-(3.22), and the step from examples to the general formula is described only by 'the pattern is clear'. The recursive procedure of Section 3.1 could plausibly supply an inductive proof, but no such induction is written down. As presented, the higher-order splitting formulas beyond the explicitly derived examples are conjectures, not consequences of the contour argument.","section":"3.2.1, Eq. (3.23)"},{"comment":"The claim that BCFW proves helicity zeros in all helicity sectors of YM and gravity is supported by one explicit 8-point YMS example and a 6-point gluon example. The general statement is asserted by saying that the argument 'always follows in the same manner', but the full induction requires a precise specification of the shift, the treatment of all BCFW term topologies, and the base cases for every N^kMHV sector and for gravity. Please either provide the general inductive argument or state the all-sector statement as a conjecture. This matters because the abstract advertises the helicity zeros as proven in all sectors of YM and gravity.","section":"4.3, Fig. 9 and Eq. (4.22)"}],"minor_comments":[{"comment":"There is an apparent mismatch in the text: Eq. (4.15) is labeled with Zspin(X35), while the following sentence says 'where Zspin(X24) sets...'. Please correct the labels so that the zero condition matches the displayed kinematics.","section":"4.2, around Eq. (4.15)"},{"comment":"In the sentence 'Unlike YMS the direction of the row...', the intended comparison appears to be with NLSM rather than with YMS itself; please clarify the wording.","section":"3.2.3, paragraph after Eq. (3.27)"},{"comment":"The notation c126 in the example (3.34) is not defined; please state that it denotes the Mandelstam invariant (p1+p2+p6)^2 or introduce a general definition for multi-index c-variables.","section":"3.3, after Eq. (3.33)"},{"comment":"The indices in the generic splitting formula (2.9)-(2.10) are used before the ranges of i,j,k,l are fully specified in the paragraph; a reader unfamiliar with the mesh would benefit from an explicit display of the ranges accompanying the formula.","section":"2.1, after Eq. (2.9)"},{"comment":"The proof that the (Xij, ckl)-shift is a g-vector shift is constructive but somewhat compressed; explicitly stating the direction vector t in the basis {(Xij)} for a generic rectangle would make Section 2.3 easier to check.","section":"2.3, g-vector shift derivation"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is interesting and likely publishable in JHEP after revision. The strongest parts are the Tr phi^3 and NLSM recursive derivations and the new 4d helicity-zero observations. My main concern is that the YMS and special Galileon sections advertise results whose central UV-scaling input is conjectural, and in the YMS case explicitly circular. I would ask the authors either to prove the scaling at all multiplicities, or to carefully rephrase the abstract, introduction, and conclusions so that those results are presented as conditional/conjectural. The authors' own transparency in Section 2.2 is to their credit; the revision should match the framing of the paper to that level of honesty."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one if you care about hidden zeros or on-shell recursion. The new idea is the (Xij,ckl) shift and the contour argument: splitting formulas follow from standard factorization plus absence of a residue at infinity. That is a real conceptual upgrade over the empirical/geometric derivations in the earlier literature, and the paper is honest about what is proven and what is not.\n\nWhat is actually new and good: identifying the shift as a special g-vector shift gives a genuine proof for Tr phi^3, and the NLSM argument via the delta-shift is clean. The inductive contour proof in Section 3.1 is coherent, and the generalized splitting formulas, the triple-split limits, and the 4d helicity zeros go beyond prior work and are supported by explicit examples. The formulas look right, and the paper cites earlier empirical and geometric work fairly.\n\nWhere the soft spots are: the load-bearing UV falloff for YMS and special Galileon is not established. Section 2.2 admits for YMS that assuming the splitting formula makes the z^-2 scaling follow, which is circular if used as input to the recursion. For the Galileon, the z^-2 split-kinematics falloff is only checked at 6 and 8 points. If that scaling fails at higher multiplicity, the contour proof of splitting collapses for those theories. So I would not call the YMS or Galileon splitting theorems proven; they are well-motivated conjectures with strong example support. The 4d BCFW arguments are also sketched on representative helicity configurations rather than written out as a complete induction for all sectors, though the pattern is persuasive. Minor point: the all-multiplicity higher-order splitting formula (3.23) is plausible but mostly shown by example, not fully proven in the text.\n\nThe central equivalence—splitting as improved UV behavior—is convincing and the paper states its own limitations explicitly, which is a point in its favor. A serious editor should send this to referees; the right outcome is probably a revision where the YMS/Galileon scaling is either proven or clearly framed as conjecture with more evidence. I did not re-run the amplitude checks, but nothing I saw in the logic makes me doubt the Tr phi^3/NLSM core.","headline":"A genuinely new contour-based derivation of splitting/zeros via a new kinematic shift, with a solid Tr phi^3/NLSM core and an explicitly conjectural, partly circular YMS/special-Galileon edge that needs referee scrutiny.","tokens_in":31145,"tokens_out":1642,"would_cite":true,"duration_ms":23922,"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":"This paper shows that the recently discovered hidden zeros and smooth splitting of tree amplitudes in Tr $\\phi^3$, NLSM, YMS, and the special Galileon follow from a single contour-integration argument, once a specially chosen kinematic…","keywords":["hidden zeros","smooth splitting","on-shell recursion","kinematic mesh","g-vector shifts","scattering amplitudes","special Galileon","BCFW recursion"],"falsifier":"Compute a 10- or 12-point special Galileon or YMS tree amplitude on scalar splitting kinematics, apply an $(X_{eo}, c_{kl})$-shift, and inspect the large-$z$ behavior of the shifted function; a falloff worse than $z^{-2}$ (such as $z^{-1}$ or $z^0$) in any such example would falsify the improved-UV premise and collapse the recursive proof of splitting in that theory.","tokens_in":30096,"feed_emoji":"✂️","tokens_out":6600,"duration_ms":76332,"temperature":0.7,"pith_summary":"The paper's central claim is that 'smooth splitting'—the factorization of certain amplitudes into products of lower-point amplitudes when a small set of non-pole kinematic invariants is tuned to zero—and the associated 'hidden zeros' are not accidental algebraic miracles but consequences of a contour integration argument of the same type used in on-shell recursion. The key is a new linear shift of planar Mandelstam variables, the $(X_{ij}, c_{kl})$-shift, which deforms the amplitude while leaving all invariants inside a chosen maximal rectangle of the kinematic mesh fixed. If the shifted amplitude has no pole at infinity, the residue theorem reconstructs the amplitude as a sum over factorization poles; all poles except two carry lower-point amplitudes that vanish by induction, leaving exactly the splitting formula. The paper proves the required UV falloff for Tr $\\phi^3$ and NLSM by identifying the shift as a g-vector shift from surfaceology, observes it for the special Galileon, and conjectures it for YMS. If the argument is right, the mysterious zero and splitting structure of these theories is a direct consequence of standard unitarity plus improved UV behavior, and the same mechanism yields new higher-order splitting formulae and new four-dimensional helicity zeros.","feed_headline":"One shift proves hidden zeros and splitting in four theories","feed_subtitle":"A contour integral derives smooth splitting in Tr phi^3, NLSM, YMS, and the special Galileon from UV behavior alone.","key_machinery":"The central object is the $(X_{ij}, c_{kl})$-shift, a linear deformation of the planar Mandelstam variables in two regions of the kinematic mesh by $\\mp z$ and $\\pm z$, defined relative to a maximal rectangle with bottom $X_B = X_{ij}$ and one relaxed plaquette $c_{kl}$. Its defining property is that all $c$-variables inside the rectangle—the ones set to zero in the hidden-zero locus—are unchanged by the shift, while the two special planar invariants $X_B$ and $X_T$ shift with opposite signs. This makes the contour integral around the shifted amplitude factorize: on every unwanted pole, one factor is a lower-point amplitude evaluated on zero kinematics and vanishes by induction; the only surviving residues at $z = X_B$ and $z = -X_T$ are related by a bonus relation when the amplitude falls as $z^{-2}$ at infinity. The identification of this shift with a g-vector shift, proven for Tr $\\phi^3$ and inherited by NLSM through the $\\delta$-shift, supplies the needed $z^{-2}$ falloff.","core_discovery":"The discovery is that hidden zeros and smooth splitting are equivalent to improved UV behavior of a carefully chosen deformation. For an amplitude in the kinematic mesh with a maximal rectangle bounded by $X_B$ and $X_T$, an $(X_{ij}, c_{kl})$-shift moves $X$-variables by $\\pm z$ so that the interior $c$-invariants defining the zero stay fixed. Deforming the contour in Cauchy's theorem and using factorization on each propagator, every residue except those at $z = X_B$ and $z = -X_T$ contains a lower-point amplitude evaluated on zero kinematics, which vanishes. The remaining two residues combine through a 'bonus relation' provided the shifted amplitude falls at least as $z^{-2}$; reading off the residue at $z = X_B$ reproduces the splitting formula with its kinematic remapping. Setting the relaxed $c_*$ to zero turns splitting into the hidden zero. Since the induction starts at four points, the splitting theorem is proven recursively to all multiplicity whenever the $z^{-2}$ falloff is available: for Tr $\\phi^3$ and NLSM the falloff follows from the identification of the shift as a g-vector shift, for YMS it is an explicitly flagged conjecture, and for the special Galileon it is checked at six and eight points.","pith_inferences":["The same contour argument suggests a search strategy for hidden zeros in other theories: any tree amplitude with a chosen maximal rectangle and $z^{-2}$ falloff under an $(X_{ij}, c_{kl})$-shift should exhibit splitting. The paper's checks that DBI and Einstein-Maxwell-Scalar amplitudes scale poorly are natural negative controls for this criterion.","The bonus relation between residues at $z = X_B$ and $z = -X_T$ is a UV constraint that may encode a hidden symmetry or soft theorem, analogous to how enhanced cancellations in gravity are captured by bonus relations; extracting that symmetry could explain why the falloff is far better than naive power counting.","The four-dimensional checkerboard zeros force non-adjacent pairs of external momenta to become proportional, which suggests the $d$-dimensional hidden-zero locus restricts to 4d as a multi-collinear limit; this may make the zeros visible as constraints in collinear factorization and useful as benchmark identities for numerical amplitude programs.","The helicity zeros, which vanish term by term in BCFW expansions, are new selection rules for 4d gluon and graviton amplitudes and appear to be the natural 4d avatar of the hidden-zero phenomenon, likely connecting to flattening limits of positive geometries built from BCFW terms."],"forward_implications":["For Tr $\\phi^3$ and NLSM, hidden zeros and near-zero splitting become theorems provable inductively from standard tree-level factorization plus the $z^{-2}$ behavior of g-vector shifts, without needing a positive-geometry argument.","Relaxing several zero conditions in the same row of the kinematic mesh produces closed higher-order splitting formulae, including limits in which the amplitude becomes a product of three lower-point amplitudes in both Tr $\\phi^3$ and NLSM.","If the conjectured $z^{-2}$ falloff for YMS scalar splitting kinematics is established from first principles, the same contour argument gives all-multiplicity smooth splitting in scalar channels of YMS.","The four-dimensional analysis identifies exactly which helicity configurations can support hidden zeros in YM and YMS, and proves a new class of helicity zeros that holds in all helicity sectors of Yang-Mills and gravity.","The equivalence between splitting and improved UV behavior provides a new organizing principle: any theory whose amplitudes satisfy the required large-$z$ falloff under an $(X_{ij}, c_{kl})$-shift automatically exhibits hidden zeros and smooth splitting."],"supporting_citations":[{"why":"Supplies the original hidden zeros and smooth splitting phenomena, the delta-shift relation between NLSM and Tr phi^3, and the scalar-scaffolding perspective.","marker":"[2]"},{"why":"Provides the BCFW contour-integration method and the criterion of good UV behavior that the paper adapts to its new shift.","marker":"[8, 9]"},{"why":"Introduces g-vector shifts and triangulations of the ABHY polytope, used to prove the z^-2 falloff of Tr phi^3 amplitudes.","marker":"[14, 15]"},{"why":"Gives the surfaceology description of amplitudes, which supplies the g-vector shift framework and the proof tools for Tr phi^3 and NLSM.","marker":"[35, 36, 37]"},{"why":"Studies large g-vector deformations of Tr phi^3 and proves commutativity of shifts, which the paper uses to transfer the z^-2 falloff to NLSM.","marker":"[43]"},{"why":"Establishes hidden zeros in the special Galileon and other theories via double-copy structures, providing the target phenomena the paper rederives.","marker":"[3, 4]"},{"why":"Reports universal splitting of tree-level string and particle amplitudes, including the special Galileon, which motivates the recursive proof.","marker":"[6, 7]"},{"why":"Provides the CHY representation and double-copy constructibility of the special Galileon, background for its hidden zeros and splitting.","marker":"[45]"},{"why":"Gives earlier BCFW-based proofs of hidden zeros in Tr phi^3 and states the equivalence between hidden zeros and enhanced UV scaling, which this paper extends.","marker":"[19, 46]"}],"fun_headline_variants":["On-shell shift ties splitting and zeros to UV falloff","New shift reveals hidden zeros via contour argument","One contour argument explains splitting and zeros","Recursive proof links zeros and smooth splitting"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument hinges on the shifted amplitude having no pole at infinity, falling as $z^{-2}$ on the chosen kinematics; this is proven for Tr $\\phi^3$ and NLSM, but for YMS it is only conjectured from examples, and for the special Galileon it is checked only at six and eight points.","fun_headline_variants_meta":{"raw":{"variants":["On-shell shift ties splitting and zeros to UV falloff","New shift reveals hidden zeros via contour argument","One contour argument explains splitting and zeros","Recursive proof links zeros and smooth splitting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000303,"raw_usage":{"total_tokens":1842,"prompt_tokens":1145,"completion_tokens":697,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":761,"completion_tokens_details":{"reasoning_tokens":640}},"tokens_in":761,"tokens_out":697,"duration_ms":8247,"temperature":1.0,"reasoning_tokens":640,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:48:49.133503+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute a 10- or 12-point special Galileon or YMS tree amplitude on scalar splitting kinematics, apply an $(X_{eo}, c_{kl})$-shift, and inspect the large-$z$ behavior of the shifted function; a falloff worse than $z^{-2}$ (such as $z^{-1}$ or $z^0$) in any such example would falsify the improved-UV premise and collapse the recursive proof of splitting in that theory.","supporting_citations":[{"cited_title":"Large deformations of Tr($\\Phi^3$) and the world at infinity","cited_arxiv_id":"2504.11253","evidence_quote":"Studies large g-vector deformations of Tr phi^3 and proves commutativity of shifts, which the paper uses to transfer the z^-2 falloff to NLSM."}],"review_version":1}