{"id":"8f27d047-7d19-4ea2-bb08-5a76d36ba257","arxiv_id":"hep-ph/0605001","paper_version":4,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Lagrangian-level zero-bin subtraction removes double counting between soft, ultrasoft, and collinear fields, converting spurious infrared singularities into ultraviolet divergences that renormalize cleanly.","lead":"Effective field theories that split one particle into several momentum-region fields can count the same low-energy physics twice. This paper gives a systematic subtraction rule for that overlap and shows it removes long-standing singularities in factorization formulas for particle decays.","discovery_kind":"new_method","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The core zero-bin method is well supported, but the exclusive SCET II factorization (Eqs. 146, 156) rests on an unproven decoupling of rapidity and invariant-mass renormalization, evidenced only by one one-loop scalar diagram and tree-level matching.","rationale":"The reader's weakest_assumption identifies exactly the right load-bearing condition: the independence of rapidity and invariant-mass renormalization in SCET II. I agree with the reader's verdict of CONDITIONAL and with the reasoning that the core zero-bin method is durable while the exclusive half is a program with a promising start. The paper's own hedging supports this: Section I explicitly declines a complete derivation of exclusive rapidity factorization; footnote 23 states the authors did not independently verify the tree-level matching of Ref. [23] used in Eq. (146); the one-loop SCET II matching in Eqs. (116) and (119) is computed for a single diagram and described as illustrative; and Eq. (156) is followed by 'A more detailed study of Eqs. (156) will be reported on in a future publication.' None of these are hidden, and the abstract's 'gives evidence for a factorization in rapidity space' is appropriately calibrated. The concern I would add is narrower than the reader's: the specific mechanism by which the decoupling could fail is that two-loop rapidity divergences might not be local in the endpoint-moment sense, which would invalidate the ø-distribution as defined in Eq. (129) and force non-local structures into Eqs. (146) and (156). That is a concrete, checkable failure mode, and the two-loop scalar computation is the natural test. I do not see a comparable load-bearing concern in the NRQCD or SCET I halves: the IR-to-UV conversion is explicit, the pinch-singularity removal is clean, and the regulator-independence is checked across three different regulator choices. The paper also has independent support in the form of the two-loop static-potential check mentioned in Section V B, which is a genuine non-trivial verification of the method beyond one loop. I would not raise the scheme-dependence of the zero-bin subtraction as a concern, because the paper explicitly acknowledges it ('unique up to possible finite scheme dependent contributions') and argues correctly that it only moves finite pieces between matching and matrix elements. I would also not raise the endpoint-behavior assumption on φπ (that φπ(0)=0 and φ'_π(0) exists) as a separate concern, because the paper states it as an assumption and the ø-distribution is defined for that class; it is a restriction on applicability, not an internal inconsistency. The verdict should remain CONDITIONAL, with the exclusive factorization claims read as evidence for a program rather than a completed derivation.","tokens_in":69218,"tokens_out":6227,"duration_ms":116001,"concrete_test":"Compute the O(αs²) correction to the scalar B→γℓν toy example of Fig. 15 by adding one more scalar-gluon exchange, and extract the rapidity divergence of the two-loop jet function J(ℓ+,p-,μ+,μ-). Check (i) whether the rapidity divergence is still a sum of endpoint moments φ(0), φ'(0) (locality), and (ii) whether μ+μ- = μ² still holds with the same counterterm structure. If a non-local rapidity divergence appears — a convolution over the full x range not reducible to endpoint moments — the decoupling assumption fails and the ø-distribution construction of Eq. (129) is incomplete. A cheaper proxy: repeat the one-loop computation of Eq. (119) with the rapidity regulator exponent in Eq. (103) changed from ε to 2ε; if the finite part shifts by a term not absorbable into C_ct(μ-), the construction is not regulator-independent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central method claim — Eq. (17), the IR-to-UV conversion in Eqs. (24), (33), (59), (68), and the removal of pinch singularities in Eq. (30) — is supported by explicit one-loop computations in NRQCD and SCET I, by a two-loop static-potential check, and by the regulator-independence checks of Sections VI A, VI B, and Appendix B. That half holds up. The load-bearing soft spot is the exclusive SCET II half. Section I states directly: 'we will make the simplifying assumption that the renormalization of rapidity space effects and invariant mass effects can be carried out independently.' Everything in Section VII depends on this: the μ± scales, the ø-distribution of Eq. (129), and the finite ρ–π and B→π formulas of Eqs. (146) and (156). The evidence offered is (a) a single one-loop scalar diagram in which μ+μ- = μ² emerges (Eqs. 98–101, 109–111, 115–119), and (b) tree-level matching with ø-distributions. The paper does not show that at two loops the rapidity divergences remain local — i.e. proportional to endpoint moments φ(0), φ'(0) — and cancelable by the same counterterm structure of Eqs. (113), (116), (126), (131). If rapidity and invariant-mass renormalization mix at two loops, the ø-distribution is not the correct finite object and Eqs. (146) and (156) would require additional non-local structures. The authors flag this ('We will not give a complete derivation of an exclusive rapidity factorization formula here'), so it is an acknowledged limitation rather than a hidden error; but it is the load-bearing one for the exclusive claims, and it is why CONDITIONAL rather than ACCEPT is the right verdict.","agreement_with_reader":"agree"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"This manuscript proposes and develops the zero-bin subtraction for effective field theories that use several distinct fields to cover different infrared momentum regions of the same particle. The central prescription is Eq. (17): a sum over nonzero labels is converted into a full momentum integral minus subtraction terms obtained by expanding the integrand in the scaling of the vanishing label, with the subtractions organized iteratively over single, double, and higher zero-bins. The paper applies this to NRQCD (soft and ultrasoft gluons, soft and potential quarks), to SCET I (collinear and ultrasoft modes), and to SCET II (soft and collinear modes separated in rapidity). In NRQCD the subtraction is shown to remove the pinch singularity of the soft box graph (Eq. 30), to convert the infrared divergence of the box and crossed box into an ultraviolet one (Eqs. 24, 33), and thereby to reproduce the pull-up mechanism from the Lagrangian rather than by hand. In SCET I the same conversion is exhibited for b to s gamma with a hard cutoff (Section VI A), with an offshellness regulator in dimensional regularization (Section VI B), and for two-jet production (Section VI C); the zero-bin is also implemented at tree level and in phase-space integrals for gamma* to q qbar g (Section VI D). In SCET II the authors argue that avoiding double counting in the rapidity variable zeta_p requires a rapidity regulator, introduce scales mu+ and mu- with mu^2 = mu+ mu-, define a plus-type ø-distribu­","tokens_in":69501,"tokens_out":6588,"duration_ms":103749,"significance":"If the method holds, it is a genuine conceptual contribution rather than an incremental calculation. Three things stand out. First, the pull-up mechanism in NRQCD, previously implemented by hand as a necessary ingredient of the mode decomposition, is derived here from the effective Lagrangian, which is a real gain in understanding. Second, the paper makes the infrared-to-ultraviolet conversion of divergences systematic and, importantly, regulator independent; the parallel treatments of b to s gamma with a hard cutoff (Section VI A), with an offshellness plus dimensional regularization (Section VI B), and the three-mass and offshell variants collected in Appendix B are exactly the kind of cross-check that makes such a claim credible, and I was able to verify the arithmetic of Eqs. (53)-(55) and (58)-(60). Third, the ø-distribution of Eq. (129) is a concrete, explicitly defined object with displayed finite parts, not a formal gesture, and the same construction is carried through to the rho-pi form factor (Eq. 146) and the soft B to pi form factor (Eq. 156). The paper is also candid about scope: the abstract says the analysis 'gives evidence for' rapidity factorization, and Section I,","major_comments":[{"comment":"The paper presents Eq. (17) as a prescription valid at any order, but the only evidence beyond one loop is stated rather than shown: 'One can check explicitly that the subtractions remove the pinch singularities in the double box' and 'We have checked that the subtracted three gluon exchange graphs give the correct contribution to the two-loop static potential [61] in Feynman gauge.' Neither calculation is displayed. Since the all-orders validity of the subtraction is part of what the paper is claiming, please either add a short appendix showing the double-box subtraction explicitly (the single- and double-pinch structure, in the spirit of Eqs. (27)-(30)), or state plainly in the text that the higher-order behavior is argued by construction and checked only in the cases shown.","section":"Section V B, text following Eq. (33)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sentence 'Due to the ø-distributions the result in Eq. (146) is finite, independent of assumptions about the non-perturbative distribution functions' is stronger than what the preceding analysis establishes. The finiteness rests on the ø-distribution definition of Eq. (129) and on the counterterm operators of Eqs. (126) and (131), which are derived at one loop and at tree level, together with the Section I assumption that rapidity and invariant-mass renormalization can be carried out independently. The authors do acknowledge that assumption in Section I and state that they will not give a complete derivation of an exclusive rapidity factorization formula, so this is not a new objection; the request is that the qualification be carried into the Section VII D and VII E statements themselves, so that the finiteness claim in Eqs. (146) and (156) is explicitly conditional on that assumption and on the order shown.","section":"Section VII D, Eq. (146); Section VII E, Eq. (156)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The cancellation of the mu- dependence between the ø-distribution in Eq. (129) and the finite part of the counterterm coefficient C_ct(mu-) in Eq. (128) is asserted by reference to [91]. Because the entire mu+ mu- = mu^2 structure of Section VII, and hence the claimed rapidity factorization, depends on this cancellation holding order by order in alpha_s, please state explicitly at which order it is established here and identify the result in [91] that supplies it, or, if it is demonstrated in this paper, point to the equation.","section":"Section VII C, Eqs. (128)-(129)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The footnote states 'Note that we have not independently verified the calculations in Ref. [23].' Since Eq. (146) is presented as a result of this paper, please indicate which terms are new here and which are taken over from Ref. [23], so that the reader can attribute the matching coefficients correctly.","section":"Section VII D, footnote 23","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The iterative definition of the subtractions over U_i, U_ij, U_ijk is clear in the surrounding prose but not in the equation itself. A forward pointer to the explicit two-label example in Eqs. (27)-(29), where the double subtractions I_box4 and I_box5 are added back, would make Eq. (17) readable on its own.","section":"Section IV, Eq. (17)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The subtraction series in J_1 and J_2 runs over k up to alpha-2. Please state explicitly that for alpha < 2 the sum is empty and the subtraction vanishes, which is currently only implicit in the surrounding discussion.","section":"Section V C, Eq. (35)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The zero-bin restrictions on x1 and x2 in Eq. (74) are the substance of this section, but the region boundaries x1 = 1, x2 = 1, and x1 + x2 = 1 are not marked on Fig. 11, and the caption notes that the soft regions are 'slightly exaggerated for visibility.' Please draw the boundaries explicitly so the restrictions can be read off the figure.","section":"Section VI D, Fig. 11 and Eq. (74)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The caption contains a garbled clause: 'The n to this entire set one must add the graphs with the current insertion on the other quark line.' Please repair the sentence.","section":"Fig. 16 caption","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The conclusion that soft-collinear messenger modes are not needed differs from Refs. [75, 77, 80], and the argument given is that the mu+ and mu- parameters are not infrared sensitive. A single sentence stating what operational difference would distinguish the two pictures, for example a specific observable or a specific coefficient that would differ, would sharpen the comparison considerably.","section":"Section VII A, messenger modes","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is long and its two halves are of unequal maturity: the NRQCD and SCET I material is complete and carefully cross-checked, while Section VII is programmatic and the authors say so. The abstract is calibrated accordingly ('gives evidence for'), which I take as good faith. If the journal wishes, Section VII D-E could be shortened or flagged as a longer paper in its own right, but I would not require it; the method half stands on its own and is the part likely to be cited."},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"The thing to take away: the zero-bin subtraction is the durable result here, and it is well supported. Eq. (17) is a clean rule — restrict labeled fields to nonzero labels, then convert the restricted sum to a full integral minus subtraction terms obtained by expanding the integrand in the scaling of the vanishing label. The worked examples back it: the crossed-box and box in Eqs. (24) and (33) show the IR-to-UV conversion explicitly, the pinch singularity is gone in Eq. (30), and the pull-up mechanism drops out of the Lagrangian instead of being put in by hand. The regulator-independence checks are the part I'd point a student at — Section VI A with a hard rapidity cutoff, Section VI B with dimensional regularization plus offshellness, and Appendix B with three unequal IR masses all land on the same answer. That is real work, not assertion. The two-loop static-potential check is a nice touch too. Anyone doing NRQCD or SCET I resummation should read Sections IV through VI carefully; the anomalous dimensions people had been extracting by treating IR poles as UV now have a proper justification. I'd cite this without hesitation for that half. The exclusive SCET II half is a different animal, and the paper is upfront about it. Section I says outright they will not give a complete derivation of an exclusive rapidity factorization formula, and that they assume rapidity and invariant-mass renormalization can be carried out independently. Footnote 23 says they did not independently verify the Ref. [23] matching that Eq. (146) inherits. The one-loop SCET II matching in Eqs. (116) and (119) is one scalar diagram, described as illustrative. Eq. (156) is followed by a promise of a future publication. So the stress-test framing of this as the load-bearing soft spot is right in substance but slightly off in tone — it is an acknowledged limitation, not something a reader has to dig for. Worth noting that the sentence classifier found no stated limitations in the abstract; that is wrong, the limitations are stated plainly in the body. The one thing I would actually want checked is whether the rapidity counterterm structure of Eqs. (113), (116), (126), (131) stays local at two loops. If it does not, the ø-distribution is not the right finite object and Eqs. (146) and (156) need more than they have. Nobody has shown that here. But the paper does not claim to. Read it for the method, treat Section VII as a promising direction with an honest caveat attached. Send it out.","headline":"The zero-bin method is the real contribution and it holds up; the exclusive SCET II half is explicitly a program, and the paper says so.","tokens_in":70255,"tokens_out":837,"would_cite":true,"duration_ms":15073,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["12.38.Bx","12.39.Hg","11.10.Gh","13.20.He"],"model":"selfhosted/deepseek-v4.1-flash","headline":"Effective field theories that split one particle into several momentum-region fields quietly count the same low-momentum configuration twice; the paper proposes a single subtraction rule that removes the overlap and makes the resulting loop","keywords":["zero-bin subtraction","soft-collinear effective theory","non-relativistic QCD","factorization","rapidity divergences","endpoint singularities","pull-up mechanism","effective field theory"],"falsifier":"A calculation in which rapidity and invariant-mass renormalization do not separate would settle it: for example, a two-loop computation of a mixed soft-collinear operator in SCET II showing that the $mu_pm$ dependence of the jet function fails to cancel that of the soft and collinear matrix elements when $mu_+ mu_- = mu^2$, or a regulator in which the zero-bin-subtracted collinear integral retains an infrared divergence not present in the full theory.","tokens_in":70474,"feed_emoji":"⚛️","tokens_out":3181,"duration_ms":64344,"temperature":0.7,"pith_summary":"Effective field theories often describe a single particle with several fields, each covering a different range of momentum: soft versus ultrasoft gluons in non-relativistic QCD, or collinear versus soft quarks in the soft-collinear effective theory used for energetic particles. The paper argues that this bookkeeping silently counts the same infrared configuration twice, in the corner where a field's large momentum label is taken to zero, and that the cure is a subtraction fixed once, at the level of the Lagrangian: labeled fields never carry a zero label, and every loop integral has the zero-label limit of its integrand subtracted. Working through explicit one-loop examples, the authors show that the subtraction removes unphysical pinch singularities from soft box graphs, derives rather than postulates the previously hand-imposed 'pull-up' of ultrasoft modes, turns infrared divergences into ultraviolet ones that local counterterms absorb, and makes the effective theory reproduce the full theory's infrared divergences whatever ultraviolet and infrared regulators are chosen. Applied to exclusive processes, the same mechanism converts apparently divergent convolutions of hard kernels with light-cone wavefunctions into finite integrals, at the price of introducing two rapidity scales whose product is the ordinary renormalization scale.","feed_headline":"Subtract the zero-momentum bin, cure factorization singularities","feed_subtitle":"One subtraction rule removes double counting between soft and collinear fields and makes divergent convolutions finite.","key_machinery":"The zero-bin subtraction of Eq. (17): the identity that a sum over nonzero labels plus an integral over residual momenta equals a full momentum integral minus the scaling limit of the integrand in each vanishing-label region, applied recursively so that overlapping subtractions are added back. Its companion for exclusive observables is the $o$-distribution, a plus-like distribution that subtracts the first terms of the endpoint expansion of a light-cone wavefunction, so that a kernel such as $1/x^2$ acts finitely and leaves a $ln(mu_-)$-type term generated by operator renormalization in rapidity space.","core_discovery":"The central claim is a replacement rule for converting a restricted sum over momentum labels into a full integral: instead of $I = I_0$, one writes $I = I_0 - I_0^{sub}$, where $I_0^{sub}$ is obtained by expanding the integrand in the scaling of the vanishing label and integrating over all momentum space (the paper's Eq. (17), extended recursively to multiple simultaneous zero-bins). Because the subtracted integrand vanishes in the zero-bin region, no infrared configuration is counted by two different fields, and because the subtraction is integrated over all momenta it also generates new ultraviolet divergences, which counterterms remove. The authors verify the rule in NRQCD, where it eliminates the pinch singularity of the soft box graph and reproduces the anomalous dimensions previously obtained by the ad hoc pull-up procedure, and in SCET, where collinear loops lose their infrared $1/epsilon$ poles and gain ultraviolet ones, so that the sum of effective-theory graphs matches the full theory's infrared structure for both an on-shell cutoff regulator and an off-shell dimensional-regularization treatment. For exclusive processes the same subtraction is packaged into a distribution the authors call $o$-distribution, defined by subtracting the endpoint values and derivatives of the light-cone distribution from the hard kernel, leaving a finite convolution plus a logarithm of the large energy over a rapidity scale $mu_-$; the claim is that a consistent mode separation requires two such scales with $mu^2 = mu_+ mu_-$, and that with them the $rho$-$pi$ form factor at large $Q^2$ and the soft $B to pi$ form factor admit finite expressions in terms of individual $B$ and $pi$ distribution functions.","pith_inferences":["If the rule is correct, it supplies a general diagnostic for any multi-mode effective theory: whenever a loop integral in one sector produces an infrared divergence that the full theory does not have, the missing ingredient is likely a zero-bin subtraction rather than a new degree of freedom.","The rapidity scales $mu_pm$ behave like the Collins-Soper rapidity parameter of transverse-momentum factorization, which suggests the exclusive results here could be recast in that language and cross-checked against the known resummation for Drell-Yan and semi-inclusive deep inelastic scattering.","The claim that no messenger or soft-collinear mode is needed in SCET II is a concrete, checkable difference from the non-factorization arguments in the literature; a two-loop computation of a mixed soft-collinear operator would decide between them.","Because the subtraction is defined on the integrand, it should be implementable numerically with hard momentum cutoffs, which would allow the mode separation to be placed on a lattice rather than relying on dimensional regularization."],"forward_implications":["Anomalous dimensions of SCET currents for endpoint processes such as $B to X_s gamma$ and $B to X_u ell nu$, previously computed by treating parts of the collinear $1/epsilon$ poles as if they were ultraviolet, are justified: the zero-bin subtraction makes those divergences genuinely ultraviolet and absorbable by counterterms.","The pull-up mechanism in NRQCD and NRQED, needed for correlated running of soft and ultrasoft couplings and for the higher-order Lamb-shift and hyperfine logarithms, follows from the effective Lagrangian rather than being imposed by hand.","Effective-theory predictions become independent of the choice of ultraviolet and infrared regulator, since the subtracted integrand is finite in the overlap region and the leftover divergences are local.","Inclusive phase-space calculations such as $gamma^* to q bar q g$ avoid double counting at the edges of the Dalitz plot once the zero-bin is applied to real emission, not only to virtual loops.","Exclusive amplitudes acquire a rapidity factorization: pieces depend on two scales $mu_+$ and $mu_-$ with $mu_+ mu_- = mu^2$, and the $rho$-$pi$ form factor and the soft $B to pi$ form factor are written as finite convolutions over $B$ and $pi$ distribution functions."],"supporting_citations":[{"why":"Supplies the method of regions, the alternative partition of loop integrals into hard, soft, potential and ultrasoft contributions that the zero-bin procedure is contrasted with and partly replaces.","marker":"[13]"},{"why":"Source of the pull-up mechanism between soft and ultrasoft fields, which the paper claims to derive rather than impose by hand.","marker":"[14, 15]"},{"why":"Provides the label-plus-residual momentum decomposition of NRQCD fields and the velocity renormalization group that the zero-bin constraint modifies.","marker":"[8]"},{"why":"Supplies the SCET Lagrangians and the decoupling field redefinition used in every SCET computation in the paper.","marker":"[4]"},{"why":"Source of the heavy-to-light SCET current and of the anomalous-dimension computation whose infrared/ultraviolet interpretation the zero-bin subtraction is meant to correct.","marker":"[2]"},{"why":"The Collins-Soper rapidity-parameter factorization for fragmentation, the precedent the paper invokes for rapidity-dependent factorization in exclusive processes.","marker":"[33, 34]"},{"why":"The analysis of endpoint divergences in SCET that concluded factorization fails; the paper's finite convolutions are offered as the counterexample.","marker":"[78]"},{"why":"The non-factorization and messenger-mode arguments, and the operator basis for the soft $B to pi$ form factor that the paper reuses while dropping the messenger operator.","marker":"[77, 80]"},{"why":"The tree-level matching computation for the $rho$-$pi$ form factor that the paper re-expresses with $o$-distributions to obtain a finite result.","marker":"[23]"}],"fun_headline_variants":["Zero-bin subtraction removes double counting between modes","Zero-bin rule targets pinch singularities in effective theories","One subtraction links soft and collinear factorization","Rapidity scales and o-distributions for exclusive decays","Evidence for rapidity factorization in exclusive processes"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The exclusive-process results assume, as the paper states in its introduction, that renormalization of rapidity-space effects and of invariant-mass effects can be carried out independently, so that the $mu_pm$ scales and the $o$-distribution can be defined without mixing.","fun_headline_variants_meta":{"raw":{"variants":["Zero-bin subtraction removes double counting between modes","Zero-bin rule targets pinch singularities in effective theories","One subtraction links soft and collinear factorization","Rapidity scales and o-distributions for exclusive decays","Evidence for rapidity factorization in exclusive processes"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":1777,"prompt_tokens":1088,"completion_tokens":689,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":630}},"tokens_in":1088,"tokens_out":689,"duration_ms":12620,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":630,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-29T15:55:08.387076+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"falsifier":"A calculation in which rapidity and invariant-mass renormalization do not separate would settle it: for example, a two-loop computation of a mixed soft-collinear operator in SCET II showing that the $mu_pm$ dependence of the jet function fails to cancel that of the soft and collinear matrix elements when $mu_+ mu_- = mu^2$, or a regulator in which the zero-bin-subtracted collinear integral retains an infrared divergence not present in the full theory.","supporting_citations":[{"cited_title":"Asymptotic expansion of Feynman integrals near threshold","cited_arxiv_id":"hep-ph/9711391","evidence_quote":"Supplies the method of regions, the alternative partition of loop integrals into hard, soft, potential and ultrasoft contributions that the zero-bin procedure is contrasted with and partly replaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the label-plus-residual momentum decomposition of NRQCD fields and the velocity renormalization group that the zero-bin constraint modifies."},{"cited_title":"Soft-Collinear Factorization in Effective Field Theory","cited_arxiv_id":"hep-ph/0109045","evidence_quote":"Supplies the SCET Lagrangians and the decoupling field redefinition used in every SCET computation in the paper."},{"cited_title":"An effective field theory for collinear and soft gluons: heavy to light decays","cited_arxiv_id":"hep-ph/0011336","evidence_quote":"Source of the heavy-to-light SCET current and of the anomalous-dimension computation whose infrared/ultraviolet interpretation the zero-bin subtraction is meant to correct."},{"cited_title":"Factorization of heavy-to-light form factors in soft-collinear effective theory","cited_arxiv_id":"hep-ph/0311335","evidence_quote":"The analysis of endpoint divergences in SCET that concluded factorization fails; the paper's finite convolutions are offered as the counterexample."},{"cited_title":"Phys. Rept. 112, 173","cited_arxiv_id":null,"evidence_quote":"The tree-level matching computation for the $rho$-$pi$ form factor that the paper re-expresses with $o$-distributions to obtain a finite result."}],"review_version":1}