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Nonzero impact parameter turns Laguerre-Gaussian collisions into a controllable probe of vortex structure via total transverse momentum.

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2026-07-13 15:00 UTC pith:TTZNQK2O

load-bearing objection Clean analytic baseline for LG–LG kinematics at nonzero impact parameter; the closed forms and the re-framing of b as a dial are the real additions.

arxiv 2604.00575 v2 pith:TTZNQK2O submitted 2026-04-01 hep-ph quant-ph

Universal features of high-energy scattering of Laguerre-Gaussian states

Yaoqi Yang , Igor P. Ivanov This is my paper
classification hep-ph quant-ph
keywords Laguerre-Gaussian wave packetsvortex statesorbital angular momentumimpact parametertransverse momentum distributionparaxial approximationimpulse approximationhigh-energy scattering
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

High-energy vortex particles carry orbital angular momentum that ordinary plane-wave scattering never sees. When two paraxial Laguerre-Gaussian wave packets collide, the total final transverse momentum P_perp is no longer fixed; its distribution encodes how the packets were prepared. This paper isolates the purely kinematic piece of that distribution, called W0, and gives closed analytic forms for any orbital numbers, packet sizes, and impact parameter b. At nonzero b the formulas predict three concrete, process-independent effects: a conditional transverse-momentum imbalance, high-contrast non-radial interference fringes for opposite-sign orbital numbers, and the splitting of a single phase vortex into two well-separated vortices whose winding numbers equal the initial orbital numbers. The authors argue that b, long treated as a nuisance, is in fact the knob that makes these features visible and controllable, and that existing electron-microscope technology is already sufficient to observe them.

Core claim

Under the impulse and paraxial approximations the normalized weight W0 that multiplies the ordinary plane-wave cross section admits exact closed forms. For same-sign orbital numbers and nonzero impact parameter b these forms factor into two phase vortices of winding numbers ℓ1 and ℓ2 sitting at momentum-space locations set by b/σ²; for opposite-sign numbers they produce high-contrast, non-radial interference controlled by the same b.

What carries the argument

The normalized transverse-momentum density W0 = |I0⊥|^{2} / ∫|I0⊥|^{2} d^{2}P⊥, obtained by evaluating the transverse overlap integral of two Laguerre-Gaussian packets at impact parameter b and shown to be independent of the scattering amplitude whenever that amplitude is smooth.

Load-bearing premise

The calculation assumes the wave packets do not spread appreciably during the brief collision, so the time-dependent widths can be replaced by their values at the instant of closest approach.

What would settle it

Measure the total transverse-momentum map of two 300 keV vortex electrons colliding at a controlled impact parameter of order one nanometer; the appearance (or absence) of two distinct zeros whose separation scales with b/σ² would confirm or refute the predicted vortex-splitting formula.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper re-analyzes high-energy 2 o2 scattering of paraxial Laguerre-Gaussian (LG) wave packets at nonzero impact parameter b, focusing on the normalized transverse-momentum weight W0 that multiplies the plane-wave cross section. Under the impulse approximation and the assumption that the plane-wave amplitude M is smooth, the authors derive closed analytic expressions for the transverse integral I0 op (Appendix B; Eqs. 26–31, B20, B30). These expressions exhibit three universal kinematic features at b eq0: (i) a conditional transverse-momentum imbalance, (ii) high-contrast non-radial interference fringes for opposite-sign OAM, and (iii) splitting of a single phase vortex into two singularities of winding numbers ℓ1 and ℓ2. The work deliberately isolates process-independent kinematics, leaving process-specific dynamics for future papers, and argues that a controllable b is a useful experimental probe rather than a nuisance.

Significance. If the analytic forms and their geometric interpretation hold, the paper supplies a clean, process-independent baseline against which future vortex-scattering experiments and process-specific calculations can be compared. The closed expressions (Gaussian and associated Laguerre polynomials) are parameter-free once the LG parameters and b are fixed, recover known zero-b and LG–Gaussian limits, and make falsifiable predictions (vortex locations at Ai op, “wifi” interference patterns, conditional ⟨P op⟩). The explicit elevation of nonzero impact parameter from nuisance to diagnostic tool is a useful conceptual shift for the field. Strengths include fully analytic results, transparent geometric reading of the zeros, and a realistic experimental feasibility estimate with present-day electron-microscope technology.

minor comments (5)
  1. The smoothness assumption that lets M be replaced by M0 (Eq. 12 and surrounding text) is stated clearly but could be flagged more prominently in the abstract or introduction as the precise domain of “universality,” so that readers do not over-apply W0 near resonances or the forward peak.
  2. Figures 3–8 use shade intensity without a color bar or absolute scale. Adding a normalized color bar (or stating that only relative contrast matters) would improve readability.
  3. A short sentence comparing the LG radial-oscillation pattern with the Bessel-beam case (already discussed in §III D) to the earlier numerical study of Zhao (Ref. [49]) would help situate the analytic advance.
  4. Notation for the auxiliary vectors A1 op, A2 op (Eq. 29) is introduced after their first appearance in the geometric discussion; moving the definition slightly earlier would aid the reader.
  5. Occasional typographical slips (e.g., “wefocus,” “theformersource,” missing spaces after commas in a few places) should be cleaned in proof.

Circularity Check

0 steps flagged

No significant circularity: W0 is obtained by direct integration of the LG Ansatz under stated approximations; analytic forms and plots are pure consequences of those integrals.

full rationale

The paper's central results are the closed-form expressions for the normalized transverse weight W0 (Eqs. 26–31, B6, B20, B30) obtained by evaluating the overlap integral I0⊥ of two paraxial LG wave packets at impact parameter b. Appendix B performs standard Gaussian and associated-Laguerre integrals after the impulse approximation factorizes I into IL·I⊥; no free parameters are fitted to data, and the numerical density plots are direct illustrations of those expressions. The smoothness assumption that replaces M by M0 is stated explicitly and is conventional for non-resonant kinematics; it is not used to force the kinematic features of W0. Self-citations ([39, 48, 52] etc.) supply the standard wave-packet scattering formalism and the impulse-approximation condition (A7), which are re-derived or used as background rather than as load-bearing uniqueness theorems. The claimed effects (vortex splitting, non-radial interference, conditional momentum imbalance) follow by construction from the analytic forms once b eq0 is allowed; they are not circular re-labelings of external data or of prior fitted results. The derivation is therefore self-contained against its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The central claims rest on standard QFT wave-packet scattering plus three domain approximations (paraxial LG principal modes, impulse approximation, smoothness of M) that are stated explicitly and are conventional for the subfield. No free parameters are fitted to data; the numerical values of σ and b are purely illustrative. No new physical entities are postulated.

axioms (5)
  • domain assumption Paraxial approximation for the LG wave packets (transverse momenta ≪ longitudinal momenta; spin–OAM coupling neglected).
    Stated in Sec. I D and used throughout the factorization of I into IL·I⊥; standard for all experimentally realized high-energy vortex states to date.
  • domain assumption Impulse approximation: wave-packet spreading is negligible during the collision duration (condition (A7)).
    Appendix A; required to replace time-dependent σi(t) by their collision-time values and thereby obtain closed transverse integrals.
  • domain assumption Plane-wave scattering amplitude M is sufficiently smooth that it may be replaced by its value at the average momenta (M≈M0) when extracting the universal weight W0.
    Sec. II B–C; excludes narrow resonances and the extreme forward region of Møller scattering, which are deferred to future work.
  • domain assumption Final-state particles are detected as ordinary plane waves with conventional pixelized detectors; only the initial states are LG packets.
    Sec. I D; matches near-term experimental capability and defines the observable as the P⊥ distribution of the two-particle final state.
  • standard math Standard Lorentz-invariant normalization of momentum-space wave packets and the luminosity function constructed from the space-time overlap of the two packets.
    Eqs. (4)–(10); taken from the established Kotkin–Serbo–Schiller / Karlovets–Serbo formalism.

pith-pipeline@v1.1.0-grok45 · 34115 in / 2638 out tokens · 25430 ms · 2026-07-13T15:00:12.676837+00:00 · methodology

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read the original abstract

Vortex states of photons, electrons, and other particles are wave packets that carry intrinsic orbital angular momentum (OAM) and exhibit other features unavailable for plane waves. Collisions of high-energy vortex states can become a promising tool for nuclear and particle physics, once experimental challenges are overcome. An extensive literature exists on scattering processes involving vortex states; however, most works rely on assumptions that will be challenging to achieve in experiment. In this work, we initiate a systematic re-analysis of vortex-state scattering processes using paraxial Laguerre-Gaussian (LG) wave packets colliding at a non-zero impact parameter $b$. Since the total final transverse momentum $P_\perp$ is no longer fixed, we focus on how the differential cross section depends on $P_\perp$. We emphasize that non-trivial $P_\perp$-dependent features can originate either from the shape of the LG wave packets or from the dynamics of the scattering process under interest. Here, we focus on the former source and explore in detail these universal kinematic features, while the study of process-specific modifications, along with the novel insights they may bring, is delegated to a future work. Interestingly, the non-zero impact parameter $b$ plays a key role in many $P_\perp$-dependent effects, making it a useful probe of vortex states, not a nuisance factor as often assumed.

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Works this paper leans on

77 extracted references · 40 linked inside Pith

  1. [1]

    instrumental

    examined kinematic features that emerge in the scattering of two Laguerre-Gaussian wave packets. However, that work had limitations: it did not address specific processes, the calculations were performed numerically, and the dependence on the impact parameter was not discussed. Although the above works set the stage for future research and provide interes...

  2. [2]

    =i(2π) 4δ(4)(k1 +k 2 −k ′ 1 −k ′

  3. [3]

    Here, the invariant amplitudeMdepends on all four momenta,M(k 1,k 2;k ′ 1,k ′ 2)

    Mp 16E1E2E′ 1E′ 2 ,(1) where we tacitly assume the usual PW normalization of unit flux. Here, the invariant amplitudeMdepends on all four momenta,M(k 1,k 2;k ′ 1,k ′ 2). The plane wave scattering cross section can then be written as dσPW = (2π)4δ(4)(k1 +k 2 −k ′ 1 −k ′ 2)|M|2 4IPW · d3k′ 1 (2π)32E′ 1 d3k′ 2 (2π)32E′ 2 .(2) whereI PW = p (k1 ·k 2)2 −m 4 is...

  4. [4]

    The latter form of the cross section is especially useful for the small-angle scattering, when|k′ 1⊥| ≪ |k ′ 1|andk ′ 1z >0

    Working in the center of mass frame and performing the necessary final phase space integration, we obtain the well-known angular distribution dσPW = |M|2 64π2E2 0 dΩ1 = |M|2 64π2E2 0 d2k′ 1⊥ |k′ 1|k ′ 1z .(3) Here,Ω 1 refers to the solid angle of the first final particle; the momentum of the second final particle isk′ 2 =−k ′ 1. The latter form of the cro...

  5. [5]

    In the exact, non-paraxial treatment,Lcan be expressed in terms of Wigner’s functions for the two colliding wave packets [39]

    = 1 L ·(2π) 8|I|2 d3k′ 1 (2π)32E′ 1 d3k′ 2 (2π)32E′ 2 .(8) Here, instead of the PW scattering amplitudeM, we deal with the wave packet scattering amplitude I= Z d3k1 (2π)32E1 d3k2 (2π)32E2 ϕ1(k1)ϕ 2(k2)δ (4)(k1 +k 2 −k ′ 1 −k ′ 2)· M.(9) The quantityLrepresents the luminosity function for collision of two wave packets [2, 39]. In the exact, non-paraxial t...

  6. [6]

    non-scattered

    The transverse part of the total final state momentumk′ 1⊥ +k ′ 2⊥ =P ⊥ is non-zero and lies in a region of the order of1/σ i⊥. Since the four-dimensional delta-function that encodes the energy-momentum conservation is absorbed in the defini- tion of the integralIin Eq. (9), it no longer appears in the differential cross section given in Eq. (8). This imp...

    2013

  7. [7]

    We consider free-space collisions of two paraxial LG wave packets in their principal modes. The key parameters are the OAM projectionsℓi and transverse localization parametersσi⊥ of the two wave packets, as well as the controllable impact parameterb⊥ between the two vortex axes. 17

  8. [8]

    We describe the final state particles as plane waves with momentak′ 1 andk ′ 2, to be detected with conventional pixelized detectors with a suitable resolution

  9. [9]

    We followed the standard procedure for wave packet scattering developed in [2] and refined recently in [39]

    Calculations are done under the impulse approximation, in which the spreading of the wave packets during the collision event is neglected. We followed the standard procedure for wave packet scattering developed in [2] and refined recently in [39]. The usage of the well-defined LG wave packets eliminates all the artefacts originating from the unnormalized ...

  10. [10]

    instrumental

    Repeating the point made in [43, 44], we argued that the main novelty of vortex-vortex scattering comes not from thek′ 1-dependence but from theP-dependence ofdσ(k ′ 1,P), in particular from itsP⊥-distribution. Integratingdσ(k ′ 1,P)over all total momenta will result in a cross section nearly identical to the plane wave cross sectiondσ(k′ 1⊥). Hence the f...

  11. [11]

    The transverse integral given in Eq

    Zero impact parameter:b= 0 We start with case of the zero impact parameter. The transverse integral given in Eq. (22) becomes I0⊥ = Z d2k1⊥ d2k2⊥ δ(2)(k1⊥ +k 2⊥ −P ⊥) (σ1⊥k1⊥)|ℓ1| (σ2⊥k2⊥)|ℓ2| ei(ℓ1φ1+ℓ2φ2) ·exp − k2 1⊥σ2 1⊥ 2 − k2 2⊥σ2 2⊥ 2 .(B1) The two-dimensional delta function can be recast in the form the a coordinate space integral (2π)2δ(2)(k1⊥ +k...

  12. [12]

    helicity

    Non-zero impact parameter:b̸= 0, same-signℓ 1, ℓ2 For non-zerob ⊥, switching to the coordinate space integration can also be done in the same way, with the result shown in Eq. (23). However, it does not bring us any simplification. Thus, we start again from the definition Eq. (22) and follow a different strategy. First, we perform thek2⊥ integration: I0⊥ ...

  13. [13]

    For definiteness, support ℓ1 <0andℓ 2 >0

    Non-zero impact parameter:b̸= 0, opposite-signℓ 1, ℓ2 Ifℓ 1 andℓ 2 are of the opposite signs, the above procedure needs to be slightly modified. For definiteness, support ℓ1 <0andℓ 2 >0. The expression (B9) still holds, withℓ1 → |ℓ1|but the exponent functionS(t1, t2)now differs from Eq. (B10) by usinge− instead ofe + in thet 1 term. This bears consequence...

  14. [14]

    An Introduction to quantum field theory,

    M. E. Peskin and D. V. Schroeder, “An Introduction to quantum field theory,” Addison-Wesley, 1995. 23

    1995

  15. [15]

    G. L. Kotkin, V. G. Serbo and A. Schiller, Int. J. Mod. Phys. A07, 4707–4745 (1992)

    1992

  16. [16]

    I. P. Ivanov, Prog. Part. Nucl. Phys.127, 103987 (2022) [arXiv:2205.00412 [hep-ph]]

    Pith/arXiv arXiv 2022

  17. [17]

    Allen, M

    L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw and J. P. Woerdman, Phys. Rev. A45, 8185-8189 (1992)

    1992

  18. [18]

    B. A. Knyazev, V. G. Serbo, Phys. Uspekhi61, 449 (2018)

    2018

  19. [19]

    Forbes, M

    A. Forbes, M. de Oliveira, M. R. Dennis, Nature Photonics15, 4, 253 (2021)

    2021

  20. [20]

    B. K. Das, C. Granados, and M. F. Ciappina, Advances in Physics: X11, no. 1, 2608076 (2026)

    2026

  21. [21]

    K. Y. Bliokh, S. Savel’ev and F. Nori, Phys. Rev. Lett.99, 190404 (2007) [arXiv:0706.2486 [quant-ph]]

    Pith/arXiv arXiv 2007

  22. [22]

    Uchida and A

    M. Uchida and A. Tonomura, Nature464, 737 (2010)

    2010

  23. [23]

    Verbeeck, H

    J. Verbeeck, H. Tian, and P. Schlattschneider, Nature467, 301 (2010)

    2010

  24. [24]

    B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, Science331, 192 (2011)

    2011

  25. [25]

    Nori,et al.Phys

    K.Y.Bliokh, I.P.Ivanov, G.Guzzinati, L.Clark, R.VanBoxem, A.Béché, R.Juchtmans, M.A.Alonso, P.Schattschneider and F. Nori,et al.Phys. Rept.690, 1 (2017) [arXiv:1703.06879 [quant-ph]]

    Pith/arXiv arXiv 2017

  26. [26]

    S. M. Lloyd, M. Babiker, G. Thirunavukkarasu and J. Yuan, Rev. Mod. Phys.89, 035004 (2017)

    2017

  27. [27]

    Sarenac, M

    D. Sarenac, M. E. Henderson, H. Ekinci, C. W. Clark, D. G. Cory, L. Debeer-Schmitt, M. G. Huber, C. Kapahi, and D. A. Pushin, Sci. Adv.8, eadd2002 (2022) [arXiv:2205.06263 [physics.app-ph]]

    Pith/arXiv arXiv 2022

  28. [28]

    Luski, Y

    A. Luski, Y. Segev, R. David, O. Bitton, H. Nadler, A. R. Barnea, A. Gorlach, O. Cheshnovsky, I. Kaminer, E. Narevicius, Science373(6559) 1105 (2021)

    2021

  29. [29]

    I. P. Ivanov, Phys. Rev. D83, 093001 (2011) [arXiv:1101.5575 [hep-ph]]

    Pith/arXiv arXiv 2011

  30. [30]

    Sarenac, J

    D. Sarenac, J. Nsofini, I. Hincks, M. Arif, C. W. Clark, D. G. Cory, M. G. Huber and D. A. Pushin, New J. Phys.20, 103012 (2018) [arXiv:1803.02295 [quant-ph]]

    Pith/arXiv arXiv 2018

  31. [31]

    Z. P. Li, Y. Wang, Y. I. Salamin, M. Ababekri, F. Wan, Q. Zhao, K. Xue, Y. Tian and J. X. Li, Phys. Rev. Lett.135, no.13, 135001 (2025) [arXiv:2504.11113 [physics.optics]]

    Pith/arXiv arXiv 2025

  32. [32]

    A. Deur, S. J. Brodsky and G. F. De Téramond, Rept. Prog. Phys.82, 076201 (2019) [arXiv:1807.05250 [hep-ph]]

    arXiv 2019

  33. [33]

    X. Ji, F. Yuan and Y. Zhao, Nature Rev. Phys.3, no.1, 27 (2021) [arXiv:2009.01291 [hep-ph]]

    Pith/arXiv arXiv 2021

  34. [34]

    Carrara, F

    P. Carrara, F. Fortuna, R. Delaunay, J. Vila-Comamala, B. Rösner, C. David, S. Pizzini, C. Fourniols, L. Vila and M. Pancaldi,et al.[arXiv:2512.24191 [cond-mat.mtrl-sci]]

    arXiv

  35. [35]

    M. Wei, S. Chen, Y. Wang, P. L. He, X. Hu, M. Zhu, H. Xu, W. Zhou, J. Jia and X. Ge,et al.Phys. Rev. Lett.136, no.2, 025001 (2026) [arXiv:2503.18843 [physics.plasm-ph]]

    Pith/arXiv arXiv 2026

  36. [36]

    A. S. Dyatlov, V. V. Kobets, A. E. Levichev, M. V. Maksimov, D. A. Nikiforov, M. A. Nozdrin, K. Popov, K. A. Sibiryakova, K. E. Yunenko and D. V. Karlovets, [arXiv:2509.00732 [physics.acc-ph]]

    arXiv

  37. [37]

    F. An, D. Bai, H. Cai, S. Chen, X. Chen, H. Duyang, L. Gao, S. F. Ge, S. Ge and J. He,et al.Chin. Phys. Lett.42, no.11, 110102 (2025) [arXiv:2504.21050 [hep-ph]]

    arXiv 2025

  38. [38]

    Morgan and B

    J. Morgan and B. W. J. McNeil, Opt. Express30, no.17, 31171-31181 (2022) [arXiv:2205.01211 [physics.acc-ph]]

    Pith/arXiv arXiv 2022

  39. [39]

    Z. Zhou, Y. Kang, W. Wan and C. Feng, Phys. Rev. Lett.134, no.20, 205001 (2025) [arXiv:2409.16542 [physics.optics]]

    Pith/arXiv arXiv 2025

  40. [40]

    Q. Meng, Z. Huang, X. Liu, W. Ma, Z. Yang, L. Lu, A. J. Silenko, P. Zhang and L. Zou, Phys. Rev. Res.7, no.4, 043213 (2025) [arXiv:2506.18492 [physics.acc-ph]]

    Pith/arXiv arXiv 2025

  41. [41]

    A. Y. Murtazin, G. K. Sizykh, D. V. Grosman, U. G. Rybak, A. A. Shchepkin and D. V. Karlovets, Phys. Rev. D113, no.3, 036024 (2026) [arXiv:2512.09921 [hep-ph]]

    arXiv 2026

  42. [42]

    Karlovets, D

    D. Karlovets, D. Grosman and I. Pavlov, Phys. Rev. Lett.136, no.8, 085002 (2026) [arXiv:2507.08763 [physics.acc-ph]]

    arXiv 2026

  43. [43]

    U. D. Jentschura and V. G. Serbo, Phys. Rev. Lett.106, no.1, 013001 (2011) [arXiv:1008.4788 [physics.acc-ph]]

    Pith/arXiv arXiv 2011

  44. [44]

    U. D. Jentschura and V. G. Serbo, Eur. Phys. J. C71, 1571 (2011) [arXiv:1101.1206 [physics.acc-ph]]

    Pith/arXiv arXiv 2011

  45. [45]

    I. P. Ivanov and V. G. Serbo, Phys. Rev. A84, 033804 (2011) [arXiv:1105.6244 [hep-ph]]

    Pith/arXiv arXiv 2011

  46. [46]

    D. V. Karlovets, S. S. Baturin, G. Geloni, G. K. Sizykh and V. G. Serbo, Eur. Phys. J. C82, no.11, 1008 (2022) [arXiv:2201.07997 [hep-ph]]

    Pith/arXiv arXiv 2022

  47. [47]

    D. V. Karlovets, S. S. Baturin, G. Geloni, G. K. Sizykh and V. G. Serbo, Eur. Phys. J. C83, no.5, 372 (2023) [arXiv:2203.12012 [hep-ph]]

    Pith/arXiv arXiv 2023

  48. [48]

    S. Lei, S. Liu, W. Wang, Z. Bu, B. Shen and L. Ji, Phys. Rev. D108, no.3, 036001 (2023) [arXiv:2303.10851 [hep-ph]]

    Pith/arXiv arXiv 2023

  49. [49]

    Ababekri, J

    M. Ababekri, J. L. Zhou, R. T. Guo, Y. Z. Ren, Y. H. Kou, Q. Zhao, Z. P. Li and J. X. Li, Phys. Rev. D110, no.7, 076024 (2024) [arXiv:2404.11952 [hep-ph]]

    Pith/arXiv arXiv 2024

  50. [50]

    I. P. Ivanov, Ukr. J. Phys.71, no. 2, 100 (2026)

    2026

  51. [51]

    Seipt, A

    D. Seipt, A. Surzhykov and S. Fritzsche, Phys. Rev. A90, no.1, 012118 (2014) [arXiv:1407.4329 [physics.optics]]

    Pith/arXiv arXiv 2014

  52. [52]

    D. V. Karlovets and V. G. Serbo, Phys. Rev. D101, no.7, 076009 (2020) [arXiv:2002.00101 [hep-ph]]

    Pith/arXiv arXiv 2020

  53. [53]

    Taira, T

    Y. Taira, T. Hayakawa and M. Katoh, Sci. Rep.7, 5018 (2017) [arXiv:1608.04894 [physics.acc-ph]]

    Pith/arXiv arXiv 2017

  54. [54]

    Y. Y. Chen, J. X. Li, K. Z. Hatsagortsyan and C. H. Keitel, Phys. Rev. Lett.121, no.7, 074801 (2018) [arXiv:1802.04748 [physics.plasm-ph]]

    Pith/arXiv arXiv 2018

  55. [55]

    Y. Liao, Q. Y. Wang and Y. Wu, [arXiv:2601.07530 [hep-ph]]

    arXiv

  56. [56]

    I. P. Ivanov, Phys. Rev. D85, 076001 (2012) [arXiv:1201.5040 [hep-ph]]

    Pith/arXiv arXiv 2012

  57. [57]

    I. P. Ivanov, D. Seipt, A. Surzhykov and S. Fritzsche, Phys. Rev. D94, no.7, 076001 (2016) [arXiv:1608.06551 [hep-ph]]

    Pith/arXiv arXiv 2016

  58. [58]

    D. V. Karlovets, EPL116, 31001 (2016) [arXiv:1608.08858 [hep-ph]]

    Pith/arXiv arXiv 2016

  59. [59]

    Karlovets, JHEP03, 049 (2017) [arXiv:1611.08302 [hep-ph]]

    D. Karlovets, JHEP03, 049 (2017) [arXiv:1611.08302 [hep-ph]]

    Pith/arXiv arXiv 2017

  60. [60]

    Korchagin, Phys

    N. Korchagin, Phys. Rev. D111, no.7, 076005 (2025) [arXiv:2403.08949 [hep-ph]]

    Pith/arXiv arXiv 2025

  61. [61]

    Karlovets, Phys

    D. Karlovets, Phys. Rev. A98, no.1, 012137 (2018) [arXiv:1803.10166 [quant-ph]]

    Pith/arXiv arXiv 2018

  62. [62]

    Zhao, Phys

    P. Zhao, Phys. Rev. D109, no.9, 096015 (2024) [arXiv:2312.00424 [hep-ph]]. 24

    Pith/arXiv arXiv 2024

  63. [63]

    I. P. Ivanov, B. Liu and P. Zhang, Phys. Rev. A105, no.1, 013522 (2022) [arXiv:2109.01323 [quant-ph]]

    Pith/arXiv arXiv 2022

  64. [64]

    Sheremet, A

    N. Sheremet, A. Chaikovskaia, D. Grosman and D. Karlovets, [arXiv:2404.11497 [physics.atom-ph]]

    Pith/arXiv arXiv

  65. [65]

    Liu and I

    B. Liu and I. P. Ivanov, Phys. Rev. A107, no.6, 6 (2023) [arXiv:2212.03624 [hep-ph]]

    Pith/arXiv arXiv 2023

  66. [66]

    Z. Li, S. Liu, B. Liu, L. Ji and I. P. Ivanov, Phys. Rev. Lett.133, no.26, 265001 (2024) [arXiv:2406.06795 [hep-ph]]

    Pith/arXiv arXiv 2024

  67. [67]

    S. Liu, B. Liu, I. P. Ivanov and L. Ji, Phys. Rev. A112, no.3, 032814 (2025) [arXiv:2503.08433 [hep-ph]]

    Pith/arXiv arXiv 2025

  68. [68]

    Verbeeck, P

    J. Verbeeck, P. Schattschneider, S. Lazar, M. Stöger-Pollach, S. Löffler, A. Steiger-Thirsfeld, G. Van Tendeloo, Applied Phys. Lett.99, no. 20, 203109 (2011)

    2011

  69. [69]

    S. M. Barnett and M. Berry, J. Opt.15, 125701 (2013)

    2013

  70. [70]

    Al-Drees, O

    W. Al-Drees, O. M. Aldossary, V. E. Lembessis, Phys. Rev. A113, no. 2, 023527 (2026)

    2026

  71. [71]

    Afanasev, C

    A. Afanasev, C. E. Carlson and A. Mukherjee, Phys. Rev. Res.3, no.2, 023097 (2021) [arXiv:2007.05816 [quant-ph]]

    Pith/arXiv arXiv 2021

  72. [72]

    Afanasev and C

    A. Afanasev and C. E. Carlson, Annalen Phys.534, no.3, 2100228 (2022) [arXiv:2105.07271 [hep-ph]]

    Pith/arXiv arXiv 2022

  73. [73]

    Z. Bu, L. Zhang, S. Liu, B. Shen, R. Li, I. P. Ivanov and L. Ji, Commun. Phys.9, no.1, 144 (2026) [arXiv:2502.14286 [physics.optics]]

    Pith/arXiv arXiv 2026

  74. [74]

    Gaudout, R

    S. Gaudout, R. Si-Ahmed, C. Debavelaere, M. Door, P. Cladé and S. Guellati-Khelifa, Phys. Rev. A112, no.6, L061301 (2025) [arXiv:2507.19157 [physics.atom-ph]]

    arXiv 2025

  75. [75]

    I. P. Ivanov, N. Korchagin, A. Pimikov and P. Zhang, Phys. Rev. D101, no.1, 016007 (2020) [arXiv:1911.09528 [hep-ph]]

    Pith/arXiv arXiv 2020

  76. [76]

    I. P. Ivanov, N. Korchagin, A. Pimikov and P. Zhang, Phys. Rev. Lett.124, no.19, 192001 (2020) [arXiv:1911.08423 [hep-ph]]

    Pith/arXiv arXiv 2020

  77. [77]

    I. S. Gradshteyn and I. M. Ryzhik, Academic Press, New York, 2014, ISBN 978-0-12-384933-5

    2014