{"id":"cd94a560-8e88-4a44-9803-a3143516465d","arxiv_id":"2411.17591","paper_version":7,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new soft hierarchy makes soft theorems for inflationary perturbations independent of off-shell cubic vertices and fixes superfluid and scaling superfluid amplitudes up to the sound speed.","lead":"This paper derives soft theorems for scattering amplitudes in boost-breaking effective field theories, showing that the theorems can be made independent of arbitrary cubic interactions by imposing a specific order of limits. It applies these theorems to bootstrap the amplitudes of superfluid and scaling superfluid effective field theories, matching previous Hamiltonian analyses.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim depends on a nonstandard soft hierarchy (q<<ε with ε fixed) whose equivalence to the conventional amplitude soft limit is asserted, not proven; if the limits do not commute, cubic-vertex independence fails.","rationale":"The Reader's weakest_assumption identifies the soft hierarchy (3.51) and the order of limits as the key unresolved point, and my read of the paper converges on the same issue. The paper is otherwise internally consistent: the Ward-Takahashi route is explicit, the worked examples through five points match the Hamiltonian analysis, and the iε prescription is carefully specified. However, the central claim—that soft theorems 'depend solely on on-shell data' and are cubic-vertex independent—only follows if one accepts that the hierarchy q<<ε, with ε removed after the soft limit, defines the physical soft limit. The paper's justification via phase-space integration is plausible but not a proof; Section 7's explicit O(p1^2) scaling of the exchange-diagram sum is manifestly a statement within that hierarchy. Since the conventional soft limit used elsewhere in the literature removes iε before (or as part of) defining the amplitude, the two limits may well disagree. This does not invalidate the paper's internal mathematics, but it means the central claim is conditional on a specific, nonstandard definition. My proposed concrete test—computing the same four-point observable under both orders of limits—would settle whether the concern actually lands. If the limits agree, the concern is moot; if they disagree, the paper's scope must be narrowed. The Reader already rendered CONDITIONAL, and my stress-test supports that verdict, so no change is needed.","tokens_in":32567,"tokens_out":6542,"duration_ms":65103,"concrete_test":"Compute the four-point amplitude for the superfluid EFT with both cubic vertices ˙π^3 and ˙π(∂π)^2, using the iε-prescribed propagator 1/(2s_{1a}+2iεE_{1a}). Take the soft limit p1→0 in two ways: (i) the paper's hierarchy—expand in p1 with ε fixed, then take ε→0; (ii) the conventional order—set ε→0 in the propagator first and then take p1→0 (or integrate against a test function to define the distributional limit). Compare the limits of (Ã_{E1}+Ã_{-E1})/2 and its derivative with the RHS of Eq. (5.8). If the two orders give different results, the soft theorem is not a statement about the conventional amplitude soft limit and the central claim requires qualification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—that soft theorems for on-shell amplitudes are independent of unconstrained off-shell cubic vertices—is derived under the soft hierarchy (3.51): q << p0_a - E_a << ε << p_a, with ε kept fixed while the soft momentum goes to zero and ε→0 taken only at the end. This is not the conventional definition of a soft limit of an on-shell amplitude, where the Feynman iε is an infinitesimal regulator that is removed when the amplitude is defined as a distribution. The paper argues the hierarchy corresponds to phase-space integration of cross-sections, but that is an argument, not a proof. If the limits do not commute, the cubic-independence claim and the all-orders statement apply only to this particular regulated limit, not to the standard amplitude soft limit used in [29,32]. The derivation itself shows this in Eq. (7.4): the O(p1^2) enhanced soft scaling of the sum of soft-cubic exchange diagrams is obtained by expanding with ε fixed; without the hierarchy the poles at s_{1,a}=0 do not cancel in the same way, and the soft theorem would contain explicit cubic-vertex dependence. Thus the load-bearing assumption is an order of limits that is asserted rather than established to agree with the conventional soft limit. Whether this is a legitimate redefinition of 'soft limit' or a limitation of scope is the central unresolved question.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives soft theorems for on-shell scattering amplitudes from non-linearly realised spacetime symmetries arising in the flat-space and decoupling limits of the EFT of inflation. The derivation follows the Noether-current route: a Ward-Takahashi identity is LSZ-reduced, and a new 'soft hierarchy' q << eps << p is imposed, with eps kept fixed while the soft momentum goes to zero and eps tending to zero only at the end. The central claim is that, under this hierarchy, the soft theorems are independent of unconstrained off-shell cubic vertices, depend only on on-shell data, and hold to all orders in perturbation theory. The paper works out polynomial shift symmetries, the non-linear boost symmetry of the superfluid EFT, and the combined boost-dilatation symmetry of the scaling superfluid, and uses the theorems to bootstrap Wilson coefficients up to five points, matching a Hamiltonian analysis.","tokens_in":32836,"tokens_out":9084,"duration_ms":87346,"significance":"If the central claim is correct, this is a significant step: it would remove the long-standing obstruction that soft theorems for boost-breaking amplitudes require explicit subtraction of soft-cubic-vertex contributions, and it would turn the soft theorems into a systematic bootstrap for superfluid and inflationary EFTs. The paper's strengths are the explicit tree-level checks, the careful treatment of the minimal basis and energy-momentum delta functions, the concrete cancellation mechanism in Eq. (7.4), and the explicit agreement with Hamiltonian Wilson coefficients at five points. The main unresolved issue is the status of the nonstandard order of limits, which is load-bearing for the all-orders and cubic-independence claims.","major_comments":[{"comment":"The central claim is derived under the soft hierarchy q << eps << p, with eps kept fixed while q goes to zero and eps going to zero only at the end. The paper argues that this corresponds to phase-space integration of cross-sections, but it does not prove that this regulated limit equals the conventional soft limit of the on-shell S-matrix, where the Feynman i eps is removed when the amplitude is defined as a distribution. Eq. (7.4) makes the issue concrete: the O(p1^2) enhanced scaling of the sum of soft-cubic exchange diagrams is obtained by expanding with eps fixed, and without this order of limits the poles at s_{1,a}=0 do not cancel in the same way. Since the abstract's all-orders cubic-independence statement depends on this order of limits, the manuscript must either prove that the two limits commute or explicitly state that the theorem refers to this regulated soft limit rather than to the conventional amplitude soft limit.","section":"Sec. 3.5, Eq. (3.51); Sec. 7, Eq. (7.4)"},{"comment":"The reduction of the quadratic-current contribution G(2) to a regular term determined only by free-field data and symmetry variation, and the vanishing G(m)=0 for m>2, are asserted in general but demonstrated only for the specific boost current in Section 5. The text states that the factorization 'holds for all loop orders' and cites Ref. [38], but no loop-level derivation of the pole structure or of the soft limit is provided. Thus the 'all orders in perturbation theory' claim in the abstract and in Eqs. (3.45)-(3.49) is not backed by the derivation as written. Either supply a general argument, including at least one explicit loop-level check, or restrict the claim to tree level.","section":"Sec. 3.2, Eqs. (3.28)-(3.40)"},{"comment":"The RHS of the Ward-Takahashi identity is set to zero by applying the i eps shift to the LSZ pole at p0_a = E_{p_a+q} - q0 and then taking p0_a -> E_a with eps fixed. This step is not derived, and it is not the same as the treatment in Section 2, where the same kind of regulated pole was expanded and produced the nontrivial boost and rotation constraints. As written, the argument seems capable of eliminating the external-leg variation for any symmetry, so the reader cannot tell which contributions are being discarded. A derivation showing that the combination of the soft limit and the eps-fixed prescription makes these terms vanish without also killing the constraints of Section 2 is needed.","section":"Sec. 3.3, Eqs. (3.42)-(3.44)"},{"comment":"The enhanced Adler zero for the collection of soft-cubic exchange diagrams relies on the specific energy-dependent imaginary part in the propagator 1/[2(s_{1,a}+i eps E_{1a})]; replacing it by 1/[2(s_{1,a}+i eps)] breaks the cancellation. Since the standard Feynman prescription is usually stated with a momentum-independent i eps, the paper should explain why this particular energy-dependent form is the correct physical regulator for the on-shell amplitudes considered here, and how this choice is compatible with the conventional definition of the S-matrix.","section":"Sec. 7, Eqs. (7.5)-(7.7)"}],"minor_comments":[{"comment":"The four-vector notation for the energy-flipped state is inconsistent: Eq. (1.17) and Fig. 3 set p' = (-E_p, p), while the text below Eq. (3.18) defines q' = (E_q, -q). Please make the sign convention uniform throughout.","section":"Eq. (1.17) and Fig. 3 vs. text near Eq. (3.18)"},{"comment":"There are typos in the summary: 'flat space and decoupling limts' should be 'limits', and 'superluid' should be 'superfluid'.","section":"Sec. 1, Summary of results"},{"comment":"The notation g^H_{m,n} for Hamiltonian Wilson coefficients is used before the subscript convention is explained; a short definition of m and n (e.g. total derivative order and number of spatial derivatives) would help the reader.","section":"Sec. 5, Eqs. (5.26)-(5.31)"},{"comment":"The two-point 'amplitude' with the dimensionless factor delta(0) is unusual; the replacement delta(E2-E3) -> delta(0)/E2 deserves a comment on why this normalization is consistent with the standard LSZ reduction for two-point functions.","section":"Sec. 6, around Eq. (6.21)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a hep-th/amplitudes journal, and the worked examples are explicit and internally consistent. The main risk is the order-of-limits issue: the soft hierarchy is asserted to reproduce the physical soft limit, but the manuscript does not prove this, and the abstract overstates all-orders validity relative to the tree-level checks. If the author can establish commutativity of the limits, or carefully restrict the claims, the paper could become publishable. The derivation also leans heavily on the author's prior paper [31] for the tower structure; the editor may wish to verify that [31] is available and published, since the present paper's novelty partly depends on it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read Du's paper. Bottom line: the cubic-vertex independence is real, but only under a non-standard order of limits, and the paper hasn't shown that order is the physical one. The derivation is careful and the worked examples hold together; up to five points, the bootstrap matches the Hamiltonian coefficients, which is a strong check. The enhanced soft scaling of the summed soft-cubic exchange diagrams (Eq. 7.4) is the clearest part of the paper.\n\nWhat's new: the soft hierarchy q << ε, keeping ε fixed while q goes to zero and only taking ε to zero afterward, and the resulting claim that unconstrained cubic vertices never enter the soft theorem. That's absent from [29,31,32]; [31] covered field-independent symmetries, and the other two either subtract cubic contributions or leave them explicit. So this is a genuine step inside the author's own program.\n\nThe soft spot is the one the stress-test flagged. This hierarchy is not the conventional soft limit of an on-shell amplitude, where the Feynman iε is removed before you take kinematics. The paper argues the hierarchy is natural because phase-space integration keeps ε fixed, but that's an argument, not a proof. Eq. (7.4) shows the cancellation of poles at s_{1,a}=0 explicitly uses ε fixed; if ε→0 first, the leading term is O(E1^2/ε) and does not vanish in the soft limit—the cubic vertices re-enter. So the central claim is conditional on the limits commuting, and that commutativity is asserted, not established.\n\nTwo smaller concerns. The all-orders statement is not backed by a loop-level derivation; the argument that higher-order currents J_(m) vanish is schematic. And the claim that the regular part of the quadratic current is determined by free EOM and symmetry is demonstrated in the superfluid example, not proved for the generic theorem. The selective iε prescription (only on indeterminate terms) is also a choice, though the paper is transparent about it.\n\nTake-home: this is a serious, clearly written paper that checks out internally. But the strongest conclusions—no off-shell cubic vertices, all orders—hold for a particular regulated soft limit, not yet for the standard amplitude soft limit. That should be the referee's main question. I'd send it to review; it deserves a serious referee. I wouldn't cite it for the all-orders claim until the hierarchy question is settled.","headline":"A careful derivation of cubic-independent soft theorems whose central claim rests on a non-standard soft limit; the equivalence to the conventional limit is argued, not proven.","tokens_in":33343,"tokens_out":5707,"would_cite":false,"duration_ms":52817,"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":"Soft theorems for boost-breaking amplitudes become independent of all off-shell cubic vertices under a prescribed ordering of limits.","keywords":["soft theorems","Adler zeros","boost-breaking amplitudes","EFTs of inflation","superfluid EFT","scaling superfluid","soft hierarchy","Ward-Takahashi identity"],"falsifier":"Compute the four-point amplitude of the $\\dot\\pi^3$ vertex, form $(\\tilde{A}_{E_1}+\\tilde{A}_{-E_1})$ with the $i\\varepsilon$ shift applied only to the collinear pole $s_{1,a}$, send $p_1\\to0$ with $\\varepsilon$ fixed, and then let $\\varepsilon\\to0$; if any term of order $p_1^0$ or $p_1^1$ survives without matching the right-hand side, the enhanced Adler zero is absent. A complementary check is to compare this regulated soft limit with the standard simultaneous limit $\\varepsilon\\to0$, $p\\to0$ in a one-loop soft amplitude in a boost-breaking theory: any difference would show the hierarchy changes the physics rather than merely ordering two equivalent limits.","tokens_in":32307,"feed_emoji":"🎯","tokens_out":8910,"duration_ms":76248,"temperature":0.7,"pith_summary":"This paper claims that soft theorems for on-shell scattering amplitudes in boost-breaking theories do not depend on unconstrained off-shell interactions, including cubic vertices, once one adopts the soft hierarchy $q \\ll \\varepsilon \\ll p_a$: the soft momentum is taken to zero while the $i\\varepsilon$ regulator stays fixed, and $\\varepsilon \\to 0$ is taken only at the end. The derivation runs through the Ward-Takahashi identity and LSZ reduction, and shows that with this ordering the left-hand side is fixed by the free equation of motion and the linear part of the symmetry transformation alone. The exchange diagrams whose soft leg attaches to a cubic vertex are collectively indeterminate in the soft limit but acquire an enhanced soft scaling and drop out, so no explicit subtraction of cubic vertices is needed. The paper applies the resulting theorems to the superfluid and scaling superfluid EFTs that arise from the flat-space, decoupling limit of the EFT of inflation, and shows they fix the Wilson coefficients up to known degrees of freedom, matching a Hamiltonian analysis through five points. If the claim is right, the soft theorems for inflationary perturbations are universal, all-orders statements that depend solely on on-shell data.","feed_headline":"Cubic vertices vanish from soft theorems under one ordering rule","feed_subtitle":"With a q << ε << p ordering of limits, inflationary soft theorems depend only on on-shell data.","key_machinery":"The central mechanism is the soft hierarchy $q \\ll \\varepsilon \\ll p_a$: the soft momentum $q$ tends to zero while the $i\\varepsilon$ regulator that tames the asymptotic time integrals is kept fixed, and $\\varepsilon \\to 0$ is taken only after the soft and on-shell limits. This ordering makes the front factor $q^\\mu$ from the Ward-Takahashi identity soft enough to kill the collinear poles produced by cubic vertices, so only the linear current $J^\\mu_{(1)}$ and the regular part of the quadratic current survive. The energy-flip combinations $\\tilde{A}_{E_p}+\\tilde{A}_{-E_p}$ and $(\\tilde{A}_{E_p}-\\tilde{A}_{-E_p})/(2E_p)$ carry the soft information, and the momentum derivatives acting on the energy-momentum delta functions generate the tower structure that matches the inverse Higgs constraints of the symmetry algebra. The $i\\varepsilon$ shift is applied exclusively to terms that are divergent or indeterminate in the soft limit, which the paper argues is required for field-redefinition invariance.","core_discovery":"The central result is a generic soft theorem for non-linearly realised space-time symmetries. For a spatial polynomial shift $\\delta^{(0)}\\pi = b_{i_1\\ldots i_N} x^{i_1}\\cdots x^{i_N}$, the theorem takes the form $\\lim_{p\\to 0} \\partial_{p^{i_1}}\\cdots\\partial_{p^{i_N}}[(\\tilde{A}^{n+1}_{E_p}+\\tilde{A}^{n+1}_{-E_p})/2] = -\\sum_a O_L(p_a,\\partial_{p_a}) \\tilde{A}^n$, with all lower-derivative towers vanishing; for time-dependent shifts an analogous combination $(\\tilde{A}^{n+1}_{E_p}-\\tilde{A}^{n+1}_{-E_p})/(2E_p)$ appears. The paper's key assertion is that $O_L$ and the right-hand side are determined only by the free theory and the linear part of the symmetry, never by unconstrained cubic vertices, provided the soft hierarchy is enforced. Specialising to the non-linear boost $\\delta_B\\pi = b_i[x_i + (x_i\\partial_t + t\\partial_i)\\pi]$ gives the superfluid soft theorems, and adding the non-linear dilatation gives the scaling superfluid theorems; in both cases the paper checks the constraints against explicit amplitudes and a Hamiltonian analysis up to five points. It also shows that the sum of exchange diagrams whose soft momentum is attached to a cubic vertex and which are singular in the collinear limit has an enhanced soft scaling $O(p_1^2)$, and that the $i\\varepsilon$ prescription restricted to divergent or indeterminate terms guarantees field-basis independence.","pith_inferences":["If the soft hierarchy is the physically correct definition of the soft limit, the same mechanism should carry over to unequal-time correlators and wavefunction coefficients through the flat-space residue relation; the paper gestures at this but does not prove it.","A diagrammatic selection rule suggests itself: in boost-breaking EFTs, soft emissions attached to cubic vertices through collinear poles can be dropped from the start of a soft bootstrap, which would simplify higher-point constructions.","A natural stress test the paper leaves open is non-linear dispersion relations such as the ghost condensate, where the free theory is not $E=|\\vec p|$; whether the hierarchy argument survives that change is not addressed.","The intermediate regulator dependence (terms such as $E_1^2/(i\\varepsilon)$) means the theorem is tied to a non-standard ordering of limits; if the conventional $\\varepsilon\\to0$ before $p\\to0$ limit is used instead, the claimed cubic-independence would need to be reconsidered."],"forward_implications":["Off-shell cubic vertices no longer need to be subtracted or assumed absent: the collection of exchange diagrams carrying a soft cubic vertex vanishes collectively with an enhanced $O(p_1^2)$ Adler zero.","For the superfluid EFT, the non-linear boost soft theorem fixes the Wilson coefficients up to one unconstrained coefficient per order in the field, agreeing with the Hamiltonian analysis through five points.","For the scaling superfluid, the additional non-linear dilatation soft theorem fixes even the three-point amplitude, leaving only the sound-speed parameter $c_s$ as a free input.","Field-basis independence follows from applying $i\\varepsilon$ only to terms that are divergent or indeterminate in the soft limit, so on-shell soft theorems are stable under field redefinitions within the minimal basis."],"supporting_citations":[{"why":"Supplies the minimal-basis and tower-structure technology and the earlier soft theorem for field-independent symmetries that this work extends to field-dependent non-linear symmetries.","marker":"[31]"},{"why":"Derives a boost soft theorem whose form depends on explicit cubic vertices, the baseline the paper claims to make cubic-independent.","marker":"[29]"},{"why":"Provides the soft phonon theorem framework and the energy-flip amplitude notation $\\tilde{A}_{E_p}\\pm\\tilde{A}_{-E_p}$ used throughout.","marker":"[32]"},{"why":"Shows correlator soft theorems are insensitive to exchange diagrams with soft cubic vertices, the correlator analogue that motivates the amplitude-level result.","marker":"[33]"},{"why":"Defines the effective field theory of inflation whose flat-space decoupling limit produces the boost-breaking superfluid theories studied here.","marker":"[19]"},{"why":"Establishes the wavefunction soft-limit structure with $\\tilde{A}_{E_p}\\pm\\tilde{A}_{-E_p}$ combinations that the paper's amplitude soft theorems mirror.","marker":"[34]"},{"why":"Defines the symmetric superfluid action and the scaling superfluid model whose Wilson coefficients are bootstrapped in Sections 5 and 6.","marker":"[35]"},{"why":"Gives the non-linear boost and dilatation transformations and the Hamiltonian Wilson coefficients used to check the soft-theorem constraints.","marker":"[36]"}],"fun_headline_variants":["Soft theorems ignore cubic vertices under one ordering","Ordering rule kills cubic vertex soft contributions","Soft limits depend only on on-shell data with q<ε