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REVIEW 2 major objections 5 minor 39 references

Scattering off unstable states

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2507.03145 v2 pith:6OCNGJA5 submitted 2025-07-03 hep-ph hep-thnucl-th

Francesco Giacosa , Vanamali Shastry This is my paper
classification hep-phhep-thnucl-th
keywords t-channelsingularityunstableparticlesfinite-timequantumfieldtheorydecayingexternalstatesscatteringcross-sectionfemtoscopylong-livedresonancesrelativisticwidth
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $T\to\infty$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Sec. III, Eqs. (13)-(14) together with Appendix A.2] 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.
  2. [Sec. III, Eqs. (9)-(10)] 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.
minor comments (5)
  1. [Appendix A.1] The heading contains the typo 'Breit-Winger'; it should read 'Breit-Wigner'.
  2. [Sec. III] The notation is inconsistent: Eqs. (9)-(10) use 'Gamma_c' while the rest of the paper uses 'Gamma_C'; please unify.
  3. [Abstract and Sec. II] The phrase 'at-channel singularity' is missing a hyphen and should read 't-channel singularity'.
  4. [Fig. 4 caption] The text states 'Present work x10 for visibility' but the caption does not; the rescaling should be stated in the caption itself.
  5. [Sec. IV, Eq. (20)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the finite-time damping is a physical input with Γ_C computed from the Lagrangian, the Γ_C=0 limit reproduces the standard QFT singularity, and the self-citations are not load-bearing.

full rationale

The central result (finite-time cross-section with no t-channel singularity) is derived by inserting a Breit-Wigner-style complex phase for the external unstable C states (Eqs. 9-10). This is an explicit physical ansatz, not a parameter fitted to the target. Γ_C is the tree-level Lagrangian decay width (Eq. 2), and the Γ_C→0 limit returns the standard QFT amplitude and singularity (Eq. 17), an independent consistency check. The displacement of the energy denominators P1,P2 off the real axis is a mathematical consequence of that ansatz, not an equation defined in terms of the cross-section it 'predicts'. The self-citations (Refs. [24,25] for the moving-particle decay law) support a standard approximation and are not the sole justification; the Breit-Wigner propagator is independently cited. The displayed Eq. (13) may have an algebraic issue regarding an overall damping factor (as noted by a skeptic), but that is a correctness concern, not circularity. No fitted input is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. Hence no circular step is established.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new particles or forces are invented; Gamow states are adopted from the literature. The central input is the ad hoc exponential damping of the external C, plus standard perturbation theory and the Breit-Wigner approximation. No parameters are fitted to data: in the toy model Γ_C is computed from g and the masses, and T is a physical measurement window.

assumptions (5)
  • standard math Standard second-order Dyson perturbation theory with the interaction L=gABC is valid for finite-time scattering.
    Equation (8) defines S^(2) over [-T/2,T/2]; the whole cross-section derivation builds on this.
  • ad hoc to paper The unstable C field modes are damped as e^{-Γ_C t/(2γ_C(p))}, so initial and final C states are Gamow-like states.
    Equations (9)-(10) and (A2) insert the lifetime by hand; this damping is what removes the singularity, but it is not derived from the Lagrangian.
  • domain assumption The unstable propagator has the relativistic Breit-Wigner form with width Γ_C.
    Appendix A.1, Eq. (A1); used for the position-space propagator and the cross-section; acknowledged as an approximation to be improved.
  • domain assumption The decay law of a moving unstable particle is the exponential e^{-Γ_C t/γ_C}, i.e., time dilation holds exactly.
    Used in Eqs. (9)-(10); the text notes it is 'typically an excellent approximation' with small deviations, citing Refs. [24,25].
  • domain assumption The ratio in Eq. (11) defines the finite-time differential cross-section, including integration over final momentum magnitude at fixed angle.
    Equation (11) with F from (12); this is the observable whose finiteness is claimed, but it is not the usual fixed-energy cross-section.

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Pith. "Pith review of Scattering off unstable states." pith.science (2026). https://pith.science/paper/6OCNGJA5

@misc{pith2026250703145,
  author       = {Pith},
  title        = {Pith review of: Scattering off unstable states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6OCNGJA5}},
  note         = {Machine review of arXiv:2507.03145}
}
abstract

Unstable states that live long enough may appear as in(out)going particles in scattering experiments. Yet, the standard QFT approach strictly applies only to fully stable asymptotic states. This is evident when scattering involving unstable particles develops a $t$-channel singularity at specific angles. We employ a finite-time formalism leading to analytic results without the singularity (even in the infinite-time limit), thus solving the problem at a phenomenological level. In turn, the approach also justifies treating long-lived particles, like weakly decaying pions, as stable during strong interactions.

Figures

Figures reproduced from arXiv: 2507.03145 by the authors.

Figure 1
Figure 1. Using the standard QFT (sQFT) approach for asymptotic states and neglecting the fact that C is not stable, the differential and the total cross-section reads in the center-of-mass (c.o.m.) frame [3]: dσ dΩ = 1 16E2 in |iM|2 (2π) 2 , σ = Z dΩ dσ dΩ = Z π 0 dθ dσ dΩ 2π sin θ , (4) where Ein = ωA(pc) + ωB(pc) is the total initial energy with ωA,C (p) = q p2 + m2 A,C being the energy of A(C). Quite interestingly, if cer… view at source ↗
Figure 2
Figure 2. FIG. 2: Summary of plots for sQFT approach: (a) Minimum and maximum values of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: One of the 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Overview of cross-section behavior. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Works this paper leans on

39 extracted references · 22 canonical work pages

  1. [1]

    [7] and further elaborated in Ref

    A solution proposed in Ref. [7] and further elaborated in Ref. [8] takes into account the finite size of the beam, which cures the divergences. Following the heuristic discussion above, if the flux ofChas a limited section, the number of incident particles per unit time is finite and the cross-section cannot diverge. Interestingly, the reduction of the cr...

    work page

  2. [2]

    In the case of N ∗πscattering, when the neutron is considered as the exchanged particle, the infinity is cured by the fact that the neutron has a nonzero width

    The exchanged particleBhas a finite width because it is not really stable:m 2 B →m 2 B −im BΓB. In the case of N ∗πscattering, when the neutron is considered as the exchanged particle, the infinity is cured by the fact that the neutron has a nonzero width. Yet, since it is extremely small, the cross-section is very large. Moreover, this solution does not ...

    work page

  3. [3]

    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

    The particleCis considered rightly as unstable. 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. This simple change removes the divergence and generates ‘bumps’ in the cross-section

    work page

  4. [4]

    This amounts to considering the scattering process AAB→AABcontainingAC→ACas a subprocess [12–15]

    One may consider only stable states in the initial/final states. This amounts to considering the scattering process AAB→AABcontainingAC→ACas a subprocess [12–15]. Alternative formalisms such as localizedS-matrix theory have also been proposed to cure the divergence [16, 17]. None of the above proposals fully resolves the issue. That is, in the first and s...

    work page

  5. [5]

    T reatment of an unstable field The propagator of an unstable particleCin the relativistic Breit-Winger approximation reads ∆C(E,k)≡∆ C(k) = i k2 −m 2 C +im CΓC +iε ≃ i 2ωC(k) " 1 E−ω C(k) +i ΓC 2γC (k) − 1 E+ω C(k)−i ΓC 2γC (k) # . (A1) This propagator is obtained by the fieldC(x) that includes its finite decay width: C(x) = Z d3k (2π)3 1p 2ωC(k) cke−ik·...

    work page

  6. [6]

    Derivation of the cross section for finiteT Here we explicitly derive the differential cross section formula given in Eq. 11. The quantityF[T,|k A|, θ] takes the form: F[T,|k A|, θ] = 1 T 2π Z +∞ −∞ dq0 (iMt)I 1(T, q0)I2(T, q0) +s-channel 2 . (A9) Theθdependence is contained in the Feynman amplitudeiMof Eq. (3). The functionsI 1,2(T, q0) read: I1(T, q0) =...

    work page

  7. [7]

    Navas et al

    S. Navas et al. (Particle Data Group), Phys. Rev. D110, 030001 (2024)

    2024

  8. [8]

    Fabbietti, V

    L. Fabbietti, V. Mantovani Sarti, and O. Vazquez Doce, Ann. Rev. Nucl. Part. Sci.71, 377 (2021), arXiv:2012.09806 [nucl-ex]

    arXiv 2021

Show all 39 references
  1. [9]

    M. E. Peskin and D. V. Schroeder, An Introduction to quantum field theory (Addison-Wesley, Reading, USA, 1995)

    1995

  2. [10]

    R. F. Peierls, Phys. Rev. Lett.6, 641 (1961)

    1961

  3. [11]

    Grzadkowski, M

    B. Grzadkowski, M. Iglicki, and S. Mr´ owczy´ nski, Nucl. Phys. B984, 115967 (2022), arXiv:2108.01757 [hep-ph]

    2022 arXiv

  4. [12]

    The scattering angleθis chosen as the angle between⃗ p C and ⃗kA, in agreement with thet-channel

    Strictly speaking, the exchange channel is of theu-type, but the literature deals witht-channel singularity, see below. The scattering angleθis chosen as the angle between⃗ p C and ⃗kA, in agreement with thet-channel

  5. [13]

    Melnikov and V

    K. Melnikov and V. G. Serbo, Nucl. Phys. B483, 67 (1997), [Erratum: Nucl.Phys.B 662, 409 (2003)], arXiv:hep- ph/9601290

    1997

  6. [14]

    Dams and R

    C. Dams and R. Kleiss, Eur. Phys. J. C29, 11 (2003), arXiv:hep-ph/0212301

    2003 arXiv

  7. [15]

    A. E. Blinov et al., Phys. Lett. B113, 423 (1982)

    1982

  8. [16]

    Iglicki, JHEP06, 006, arXiv:2212.00561 [hep-ph]

    M. Iglicki, JHEP06, 006, arXiv:2212.00561 [hep-ph]

    arXiv

  9. [17]

    I. F. Ginzburg, Nucl. Phys. B Proc. Suppl.51, 85 (1996), arXiv:hep-ph/9601272

    1996 arXiv

  10. [18]

    Goebel, Phys

    C. Goebel, Phys. Rev. Lett.13, 143 (1964)

    1964

  11. [19]

    M. Mai, B. Hu, M. Doring, A. Pilloni, and A. Szczepaniak, Eur. Phys. J. A53, 177 (2017), arXiv:1706.06118 [nucl-th]

    2017 arXiv

  12. [20]

    Jackura, C

    A. Jackura, C. Fern´ andez-Ram ´ ırez, V. Mathieu, M. Mikhasenko, J. Nys, A. Pilloni, K. Salda˜ na, N. Sherrill, and A. P. Szczepaniak (JPAC), Eur. Phys. J. C79, 56 (2019), arXiv:1809.10523 [hep-ph]

    2019 arXiv

  13. [21]

    Mikhasenko, Y

    M. Mikhasenko, Y. Wunderlich, A. Jackura, V. Mathieu, A. Pilloni, B. Ketzer, and A. P. Szczepaniak, JHEP08, 080, arXiv:1904.11894 [hep-ph]

    1904 arXiv

  14. [22]

    Nowakowski and A

    M. Nowakowski and A. Pilaftsis, Z. Phys. C60, 121 (1993), arXiv:hep-ph/9305321

    1993 arXiv

  15. [23]

    Karamitros and A

    D. Karamitros and A. Pilaftsis, Phys. Rev. D108, 036007 (2023), arXiv:2208.10425 [hep-th]

    2023 arXiv

  16. [24]

    [5] the width of the stateBis generated by medium effects

    In Ref. [5] the width of the stateBis generated by medium effects. The latter beingde factoalways present, thet-channel singularity is absent if the scattering occurs in the medium. In this respect, this type of solution is complete, but requires an external medium/field. Remo...

  17. [25]

    3), resulting in distinct sub-processes of the AB→ABtypes

    Namely, one may cutAAB→AABdiagram through an internalB(c.f.Fig. 3), resulting in distinct sub-processes of the AB→ABtypes. A full treatment of this case is left for the future

  18. [26]

    Collins, arXiv:1904.10923 [hep-ph] (2019)

    J. Collins, arXiv:1904.10923 [hep-ph] (2019)

    2019 arXiv

  19. [27]

    Bernardini, L

    C. Bernardini, L. Maiani, and M. Testa, Phys. Rev. Lett.71, 2687 (1993)

    1993

  20. [28]

    Blasone, F

    M. Blasone, F. Giacosa, L. Smaldone, and G. Torrieri, Eur. Phys. J. C83, 736 (2023), arXiv:2305.07107 [hep-ph]

    2023 arXiv

  21. [29]

    Blasone, F

    M. Blasone, F. Giacosa, L. Smaldone, and G. Torrieri, Eur. Phys. J. C85, 523 (2025), arXiv:2501.17111 [hep-ph]

    2025 arXiv

  22. [30]

    Giacosa, Acta Phys

    F. Giacosa, Acta Phys. Polon. B47, 2135 (2016), arXiv:1512.00232 [hep-ph]

    2016 arXiv

  23. [31]

    Gavassino and F

    L. Gavassino and F. Giacosa, Phys. Rev. A106, 042215 (2022), arXiv:2206.05125 [physics.gen-ph]

    2022 arXiv

  24. [32]

    Civitarese and M

    O. Civitarese and M. Gadella, Phys. Rept.396, 41 (2004)

    2004

  25. [33]

    de la Madrid and M

    R. de la Madrid and M. Gadella, Am. J. Phys.70, 626 (2002), arXiv:quant-ph/0201091

    2002 arXiv

  26. [34]

    Gegelia and S

    J. Gegelia and S. Scherer, Eur. Phys. J. A44, 425 (2010), arXiv:0910.4280 [hep-ph]

    2010 arXiv

  27. [36]

    V. S. Potapov and J. R. Taylor, Phys. Rev. A16(1977)

    1977

  28. [37]

    Rubin, R

    M. Rubin, R. Sugar, and G. Tiktopoulos, Phys. Rev.146, 1130 (1966)

    1966

  29. [38]

    Physically, for three particles to interact, all three worldlines must meet in a very small space–time region

    A universal definition of an analogous of the cross-section for three-body scattering is not feasible, since the specific invariant flux does not exist. Physically, for three particles to interact, all three worldlines must meet in a very small space–time region. The frequency...

  30. [39]

    M. T. Hansen and S. R. Sharpe, Phys. Rev. D90, 116003 (2014), arXiv:1408.5933 [hep-lat]

    2014 arXiv

  31. [40]

    Baeza-Ballesteros, J

    J. Baeza-Ballesteros, J. Bijnens, T. Husek, F. Romero-L´ opez, S. R. Sharpe, and M. Sj¨ o, JHEP05, 187, arXiv:2303.13206 [hep-ph]

    arXiv

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