{"id":"39bb4fbf-525e-4667-842d-049a7e37250d","arxiv_id":"2507.03145","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A finite-time QFT treatment removes the t-channel singularity that appears when an unstable particle scatters off another particle.","lead":"This paper studies scattering when the incoming particle is unstable, where standard quantum field theory predicts an unphysical infinite cross-section at certain angles. It presents a finite-time formalism in which the divergence disappears, because the unstable particle decays during the measurement window.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (13) as printed lacks the overall damping factor from I1,I2, so for Γ_C>0 the |k_A| integral contains terms growing as exp[Γ/2(1/γ(k_A)-1/γ(p_C))T], contradicting the claimed smooth vanishing as T→∞.","rationale":"I read the paper in good faith as a phenomenological cure for the t-channel singularity by including the finite lifetime of C in the external legs. The reader's CONDITIONAL verdict is reasonable; my stress-test focuses on the internal consistency of the central analytic formula. If Eq. (13) is correct as printed, it contains exponentially growing terms for generic off-shell |k_A|, so the cross-section would diverge with T, which is the opposite of the paper's central assertion. The likely resolution is a missing overall damping factor from the product of I1 and I2 in Appendix A.2, since each of I1 and I2 carries an explicit decay factor and the product gives e^{-ΓT/4(1/γ_p+1/γ_k)}. Without that factor, the imaginary parts of P1,P2 in the exponentials produce the growth. The paper's own figures (Fig. 4b,c) show decay, so the authors presumably used the corrected expression; the displayed Eqs. (13)-(14) would then be misprinted. This is a concrete, fixable issue, but it is load-bearing because the no-divergence claim is explicitly based on Eqs. (12)-(14). I therefore recommend CONDITIONAL: the manuscript should be corrected with the explicit damping factor, or a derivation demonstrating that it is already included, before the central claim is accepted. I do not see a fundamental objection to the finite-time idea if the corrected formula indeed vanishes as T→∞; the reader's concern about the ad hoc nature of the damping is real but secondary to this algebraic check.","tokens_in":10006,"tokens_out":28983,"duration_ms":325806,"concrete_test":"Evaluate Eqs. (11)-(14) numerically for the parameters of Fig. 2 (m_A=0.1, m_B=0.5, m_C=1, p_C=0.4, Γ_C=0.1) at a fixed angle, e.g. θ=0.5 rad, for T=10, 50, 100, 200, integrating over |k_A| from 0 to about 5. If dσ/dΩ increases with T instead of decreasing to zero, Eq. (13) is missing the damping factor. Then insert the factor e^{-ΓT/(4γ_p)-ΓT/(4γ_k)} into the definition of F (or into Eq. 13) and repeat; if the corrected result decays to zero, the authors' intended physics is recovered and the printed formula is simply misstated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The no-divergence claim rests on Eq. (11) with F defined by Eqs. (12)-(14). In Eq. (13), the exponentials e^{-i(P1-P2)T/2} and e^{i(P1-P2)T/2} are written with P1,P2 complex as in Eqs. (15)-(16). Let I1=Γ/(2γ(p_C)) and I2=Γ/(2γ(k_A)). For the off-shell final momenta integrated in Eq. (11), generically I1≠I2. The first term behaves as exp[-iRe(P1-P2)T/2 + (I2-I1)T/2] for I2>I1, and the third term grows for I2<I1. Thus for any |k_A|≠|p_C|, one of these exponentials grows without bound as T increases, and the integral over |k_A| in Eq. (11) includes a finite neighborhood of such momenta. Consequently F, and hence dσ/dΩ, would diverge as T→∞ for Γ_C>0, directly contradicting the paper's central claim that the cross-section vanishes smoothly. The likely cause is that Eq. (13) omits the overall factor e^{-ΓT/(4γ_p)-ΓT/(4γ_k)} that appears in the product I1 I2 in Appendix A.2; with that factor, the growing real parts cancel and all terms decay. The displayed formula therefore appears to be missing this damping factor, or the derivation of Eqs. (13)-(14) is incomplete. This is a concrete, checkable algebraic issue, not merely a matter of interpretation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper addresses the Peierls t-channel singularity in two-body scattering when an external particle C is unstable. The authors propose a finite-time QFT framework in which the free-field phases of the incoming and outgoing C states are replaced by exponentially damped phases, Eqs. (9)-(10), with damping rate Gamma_C/(2 gamma_C). The main result is a finite-time cross-section formula Eq. (11), with F defined by Eqs. (12)-(14). The paper claims that the t-channel singularity is absent for any time interval T, that the cross-section vanishes smoothly as T->infinity for Gamma_C>0, and that the Gamma_C=0 limit reproduces the standard QFT result. Applications to femtoscopy and to the treatment of long-lived weakly decaying particles as stable during strong interactions are discussed.","tokens_in":10383,"tokens_out":14498,"duration_ms":156679,"significance":"The proposed solution, if correct, would be a simple and useful phenomenological resolution of a long-standing problem: it gives analytic finite-time expressions, uses a width Gamma_C computed from the coupling rather than fitted, and passes the Gamma_C=0 consistency check. The extension to treating weakly decaying pions and similar states as stable in strong processes is a valuable corollary. However, the central no-divergence claim currently rests on an algebraic expression that is inconsistent with the appendix, so the significance can only be assessed after the correction.","major_comments":[{"comment":"The displayed expressions for I_F^up and I_F^down omit the overall damping factor e^{-Gamma T/(4 gamma(p_C))-Gamma T/(4 gamma(k_A))} that is present in the product I_1 I_2 of Eqs. (A10)-(A11). Since P_1 and P_2 in Eqs. (15)-(16) are complex, the exponentials e^{i(P_1-P_2)T/2} and e^{-i(P_1-P_2)T/2} carry real parts +(I_1-I_2)T/2 and -(I_1-I_2)T/2, with I_1=Gamma/(2 gamma(p_C)) and I_2=Gamma/(2 gamma(k_A)). For every |k_A| different from |p_C| one of these grows exponentially as T->infinity, and because the integration over |k_A| in Eq. (11) includes a continuum of such momenta, the differential cross-section as printed diverges for Gamma_C>0. This directly contradicts the claimed smooth vanishing and the absence of divergences in the infinite-time limit. With the missing factor restored, the growing exponentials become e^{-I_1 T} or e^{-I_2 T} and the conclusion is plausible; as it stands, however, the central result is not supported by the displayed equations.","section":"Sec. III, Eqs. (13)-(14) together with Appendix A.2"},{"comment":"The damping of the external C states is introduced as a phenomenological rule rather than derived from the Lagrangian or from a consistent treatment of unstable asymptotic states. Consequently, the disappearance of the singularity is, to a large extent, built into the ansatz: any sufficiently strong damping of the external-state overlap will suppress the on-shell t-channel exchange. The paper acknowledges this at a general level, but it should be explicit that the result is a property of the modified finite-time dynamics, not a theorem of the underlying QFT. The authors should also state the regime in which the time-dilation approximation Gamma/(2 gamma) is quantitatively controlled and, ideally, test the robustness of the conclusion by varying the damping profile.","section":"Sec. III, Eqs. (9)-(10)"}],"minor_comments":[{"comment":"The heading contains the typo 'Breit-Winger'; it should read 'Breit-Wigner'.","section":"Appendix A.1"},{"comment":"The notation is inconsistent: Eqs. (9)-(10) use 'Gamma_c' while the rest of the paper uses 'Gamma_C'; please unify.","section":"Sec. III"},{"comment":"The phrase 'at-channel singularity' is missing a hyphen and should read 't-channel singularity'.","section":"Abstract and Sec. II"},{"comment":"The text states 'Present work x10 for visibility' but the caption does not; the rescaling should be stated in the caption itself.","section":"Fig. 4 caption"},{"comment":"The discussion around Eq. (20) is heuristic; the claim that the three-body singularity reappears as T->infinity is not derived in the paper and should be explicitly labeled as a qualitative argument.","section":"Sec. IV, Eq. (20)"}],"recommendation":"major_revision","confidential_remarks":"The missing damping factor in Eqs. (13)-(14) is a serious algebraic error in the central formula, but it appears straightforwardly repairable from Eqs. (A10)-(A11), so I do not recommend rejection. The phenomenological status of the damping ansatz is the main conceptual limitation and should be addressed in the revision. The paper is within the scope of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely useful: it takes the old t-channel singularity problem for scattering off unstable states and shows how a finite-time treatment with damped external-state phases gives finite cross-sections at any T, with a clean Γ_C→0 limit back to standard QFT. The survey of earlier fixes (beam-size, B-width, Ginzburg complex mass, three-body embedding) is careful, and the authors correctly show why none is fully satisfactory. The derivation of Γ from the coupling is a plus. For someone working in femtoscopy or on the legitimacy of treating long-lived pions as stable, this is a relevant read.\n\nBut the central formula as printed has a problem. Eq. (13) drops the real damping factor that appears in the product I1 I2 in Appendix A.2 (the e^{-ΓT/(4γ_p)-ΓT/(4γ_k)} multiplying everything). Without that factor, the exponentials in Eq. (13) contain terms that grow like exp(±ΓT/4(1/γ_k-1/γ_p)) for off-shell momenta, so the integral over |k_A| would not vanish smoothly as T→∞; it would diverge. The appendix has the factor, so I suspect a typo, but as written Eqs. (11)-(14) do not support the paper's main claim of no divergence for all T. The authors need to fix the display and, ideally, show explicitly how the damping cancels the growing parts.\n\nBeyond that, the damping insertion itself (Eqs. 9-10) is an ansatz, not derived from a proper LSZ-type formulation with unstable states. The paper acknowledges this and frames it as phenomenological, so it is a mild concern. The finite-time cross-section is also a nonstandard observable; its meaning depends on the chosen T, which is physically reasonable for femtoscopy but needs a clearer operational definition.\n\nOverall: the idea is good, the review of prior work is solid, and the Γ_C=0 limit is a strong consistency check. But the missing damping factor is a load-bearing algebraic issue that must be corrected. I would send this to a serious referee (PRD or EPJC level) with a request to verify the algebra and the numerical plots. If the factor is restored, the result stands as a useful phenomenological cure.","headline":"A useful finite-time cure for the t-channel singularity, but the printed central formula drops a real damping factor and would not vanish in the T→∞ limit as claimed.","tokens_in":10870,"tokens_out":4350,"would_cite":true,"duration_ms":45237,"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":"Scattering an unstable particle off another state produces no t-channel singularity when the unstable state's lifetime is built into the finite-time amplitude; the cross-section stays finite for any interval and vanishes smoothly as the…","keywords":["t-channel singularity","unstable particles","finite-time quantum field theory","decaying external states","scattering cross-section","femtoscopy","long-lived resonances","relativistic width"],"falsifier":"Evaluate the second-order $CA\\to CA$ amplitude with the unstable particle represented by its full spectral integral instead of the damped phases of Eqs. (9)--(10), and take $T\\to\\infty$: if a $t=m_B^2$ enhancement survives for any positive width $\\Gamma_C$, the central claim is wrong. A femtoscopic measurement of $K_S\\pi$ or $\\phi K$ correlations at the singular kinematics, with source size varied over a large range, would provide the experimental counterpart.","tokens_in":9804,"feed_emoji":"⚛️","tokens_out":8666,"duration_ms":94410,"temperature":0.7,"pith_summary":"The paper addresses a longstanding puzzle: scattering experiments routinely involve unstable particles in the initial or final state, but the standard quantum field theory S-matrix is built for stable asymptotic states, and for processes such as $CA\\to CA$ with $C$ unstable it develops a t-channel singularity that makes the cross-section infinite at a specific angle. The authors propose a finite-time QFT formalism in which the unstable particle's wave is given exponentially damped phases over the interval $T$ between production and detection. This yields an analytic expression for the differential cross-section with no t-channel pole for any $T$, and as $T\\to\\infty$ the cross-section vanishes smoothly because the unstable state decays before interacting. In the limit where the width $\\Gamma_C$ is zero, the standard QFT result is recovered, which serves as a consistency check. The practical consequence would be that long-lived unstable particles can legitimately be treated as stable in stronger interactions, as in pion-pion or nucleon-nucleon scattering.","feed_headline":"Unstable-particle scattering loses its t-channel singularity","feed_subtitle":"Damped unstable-state phases make the cross-section finite and drive it smoothly to zero in the infinite-time limit.","key_machinery":"The machinery is a finite-time S-matrix in which the external legs of the unstable field are treated as decaying states whose time evolution carries a decay factor $\\Gamma_C/(2\\gamma_C(p))$ in the exponent. Concretely, the free phase of the unstable particle is modified as in Eqs. (9)--(10), with the width-to-energy ratio representing the lifetime of a moving particle in the relativistic width-corrected propagator approximation. This damping makes the energy integrals in the second-order amplitude converge and shifts the poles $P_1$ and $P_2$ of Eqs. (15)--(16) off the real axis, so that no kinematical point $t=m_B^2$ can be reached. The cross-section formula (11) with $F$ from Eqs. (12)--(14) is then analytic in the scattering angle for all $T$, and the standard QFT result is recovered in the $\\Gamma_C=0$, $T\\to\\infty$ limit.","core_discovery":"The central claim is that replacing the plane-wave phases of the incoming and outgoing unstable particle $C$ in the second-order S-matrix element by exponentially damped phases, $e^{-i\\omega_C(p_C)(t_1+T/2) - \\Gamma_C (t_1+T/2)/(2\\gamma_C(p_C))}$ and the analogous expression for the final state, removes the t-channel singularity of the $CA\\to CA$ cross-section for every finite time interval $T$. The resulting amplitudes $I_F^{(up)}$ and $I_F^{(down)}$ have only off-shell poles $P_1, P_2$ controlled by $\\Gamma_C$, so the divergent $t=m_B^2$ pole of the standard amplitude never appears. In the infinite-time limit the cross-section tends to zero rather than to the standard QFT value, because the unstable particle has decayed; only for strictly stable $C$ ($\\Gamma_C=0$) does the standard $\\delta(E_{in}-E_f)$ result emerge. The authors emphasize that previous cures, such as finite beam size, an effective width for the exchanged stable particle, or the ad hoc prescription $m_C^2\\to m_C^2 - i m_C\\Gamma_C$, are incomplete, whereas this approach is analytic and free of divergences even as $T\\to\\infty$.","pith_inferences":["A natural extension not pursued here: applying the same damped-phase machinery to $\\mu^+\\mu^-$ or $W^+e^-$ scattering would give concrete finite-time predictions that next-generation femtoscopy could test.","If the exact nonexponential decay of a resonant state replaces the exponential approximation, the finite-time amplitude (11) acquires corrections that grow with $T$; experiments with long-lived unstable beams could search for such deviations.","The same $T$-dependent damping offers a way to regulate t-channel singularities in thermal or in-medium calculations, where the width of an unstable hadron depends on temperature and density, without giving the stable exchanged particle a width."],"forward_implications":["The t-channel singularity of two-body scattering with unstable initial or final states is removed for any time interval $T$, including $T\\to\\infty$, for any finite width $\\Gamma_C>0$.","The finite-time cross-section vanishes smoothly as $T\\gg 1/\\Gamma_C$, so an unstable particle that lives much shorter than the experimental time interval effectively does not scatter.","When $\\Gamma_C=0$ the standard QFT cross-section with the energy-conserving delta function is recovered, validating the formalism against textbook results.","Long-lived unstable particles such as pions, kaons, and muons can be treated as stable for strong or electromagnetic subprocesses, because the corresponding large-$T$ limit applies to those interactions.","The framework gives a formal reason that subtracting sequential pairwise scattering, as done in relativistic three-body formalisms, is not the right cure for the two-body t-channel problem, since $C\\to AB\\to C$ is not a sequence of asymptotic-state scatterings."],"supporting_citations":[{"why":"Supplies the standard QFT cross-section formula that the paper must reproduce in the $\\Gamma_C=0$ limit.","marker":"[3]"},{"why":"Introduced the t-channel singularity in $\\pi N^*$ scattering via exchange of a stable nucleon, the problem being solved.","marker":"[4]"},{"why":"Lists the affected processes such as $W^+e^-$, $Z^0e^-$, and $\\mu^+\\mu^-$ scattering and discusses the in-medium width cure that the paper argues is incomplete.","marker":"[5]"},{"why":"Proposed the prescription $m_C^2 \\to m_C^2 - i m_C\\Gamma_C$ that the paper compares with and replaces by a finite-time treatment.","marker":"[11]"},{"why":"Argues that the standard asymptotic-state construction is not valid for unstable particles, motivating the finite-time approach.","marker":"[20]"},{"why":"Provides one of the finite-time QFT formulations used to build the finite-time S-matrix.","marker":"[21]"},{"why":"Establishes the time-dilated exponential decay law for a moving unstable particle that underlies the phase replacements in Eqs. (9)--(10).","marker":"[24]"},{"why":"Quantifies deviations from the exponential decay law and shows they are negligible for the finite-time amplitude.","marker":"[25]"},{"why":"Gives the decaying-state interpretation of the damped external states used for the unstable particle.","marker":"[27]"}],"fun_headline_variants":["Finite-time fix kills t-channel pole in unstable scattering","Unstable scattering: no singularity even in infinite-time limit","Damped phases erase t-channel singularity for unstable states","Unstable-particle scattering stays finite via damped phases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole result depends on the claim that an unstable particle's wave function decays as a simple exponential, with its lifetime stretched by the relativistic time-dilation factor, over the whole time between production and interaction; if real moving unstable particles do not behave this way, the divergence-free cross-section does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Finite-time fix kills t-channel pole in unstable scattering","Unstable scattering: no singularity even in infinite-time limit","Damped phases erase t-channel singularity for unstable states","Unstable-particle scattering stays finite via damped phases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000147,"raw_usage":{"total_tokens":1158,"prompt_tokens":887,"completion_tokens":271,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":503,"completion_tokens_details":{"reasoning_tokens":206}},"tokens_in":503,"tokens_out":271,"duration_ms":3876,"temperature":1.0,"reasoning_tokens":206,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:17:57.356782+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the second-order $CA\\to CA$ amplitude with the unstable particle represented by its full spectral integral instead of the damped phases of Eqs. (9)--(10), and take $T\\to\\infty$: if a $t=m_B^2$ enhancement survives for any positive width $\\Gamma_C$, the central claim is wrong. A femtoscopic measurement of $K_S\\pi$ or $\\phi K$ correlations at the singular kinematics, with source size varied over a large range, would provide the experimental counterpart.","supporting_citations":[{"cited_title":"This point of view is based on the solution proposed by Ginzburg [11] and anticipated by Peierls [4]:m 2 C →m 2 C −im CΓC","cited_arxiv_id":null,"evidence_quote":"Supplies the standard QFT cross-section formula that the paper must reproduce in the $\\Gamma_C=0$ limit."},{"cited_title":"This amounts to considering the scattering process AAB→AABcontainingAC→ACas a subprocess [12–15]","cited_arxiv_id":null,"evidence_quote":"Introduced the t-channel singularity in $\\pi N^*$ scattering via exchange of a stable nucleon, the problem being solved."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Lists the affected processes such as $W^+e^-$, $Z^0e^-$, and $\\mu^+\\mu^-$ scattering and discusses the in-medium width cure that the paper argues is incomplete."},{"cited_title":"[5] the width of the stateBis generated by medium effects","cited_arxiv_id":null,"evidence_quote":"Establishes the time-dilated exponential decay law for a moving unstable particle that underlies the phase replacements in Eqs. (9)--(10)."},{"cited_title":"3), resulting in distinct sub-processes of the AB→ABtypes","cited_arxiv_id":null,"evidence_quote":"Quantifies deviations from the exponential decay law and shows they are negligible for the finite-time amplitude."},{"cited_title":"Bernardini, L","cited_arxiv_id":null,"evidence_quote":"Gives the decaying-state interpretation of the damped external states used for the unstable particle."}],"review_version":1}