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Schwinger pair production in spacetime fields: Moiré patterns, Aharonov-Bohm phases, and Sturm-Liouville eigenvalues

Gianluca Degli Esposti1,2,* and Greger Torgrimsson3,†

  • *Contact author: g.degli-esposti@hzdr.de
  • †Contact author: greger.torgrimsson@umu.se
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Phys. Rev. D 112, 016026 – Published 30 July, 2025

DOI: https://doi.org/10.1103/3143-w4w3

Abstract

We use a worldline-instanton formalism to study the momentum spectrum of Schwinger pair production in spacetime fields with multiple stationary points. We show that the interference structure changes fundamentally when going from purely time-dependent to space-time-dependent fields. For example, it was known that two time-dependent pulses give interference if they are antiparallel, i.e., Ez(t)−Ez(t−Δt), but here we show that two spacetime pulses will typically give interference if they instead are parallel, i.e., Ez(t,z)+Ez(t−Δt,z−Δz). We take into account the fact that the momenta of the electron, pz, and of the positron, pz′, are independent for Ez(t,z) [it would be pz+pz′=0 for E(t)], and find a type of fields which give moiré patterns in the pz−pz′ plane. Depending on the separation of the two pulses, we also find an Aharonov-Bohm phase. We also study complex momentum saddle points in order to obtain the integrated probability from the spectrum. Finally, we calculate an asymptotic expansion for the eigenvalues of the Sturm-Liouville equation that corresponds to the saddle-point approximation of the worldline path integral, use that expansion to compute the product of the eigenvalues, and compare this with the result obtained with the Gelfand-Yaglom method.

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References (43)

  1. F. Sauter, Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs, Z. Phys. 69, 742 (1931).
  2. J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).
  3. E. Brezin and C. Itzykson, Pair production in vacuum by an alternating field, Phys. Rev. D 2, 1191 (1970).
  4. V. S. Popov, Pair production in a variable and homogeneous electric field as an oscillator problem, Sov. Phys. JETP 35, 659 (1972).
  5. V. S. Popov, Imaginary-time method in quantum mechanics and field theory, Phys. At. Nucl. 68, 686 (2005).
  6. S. P. Kim and D. N. Page, Schwinger pair production via instantons in a strong electric field, Phys. Rev. D 65, 105002 (2002).
  7. S. P. Kim and D. N. Page, Improved approximations for fermion pair production in inhomogeneous electric fields, Phys. Rev. D 75, 045013 (2007).
  8. C. K. Dumlu and G. V. Dunne, The Stokes phenomenon and Schwinger vacuum pair production in time-dependent laser pulses, Phys. Rev. Lett. 104, 250402 (2010).
  9. C. K. Dumlu and G. V. Dunne, Interference effects in Schwinger vacuum pair production for time-dependent laser pulses, Phys. Rev. D 83, 065028 (2011).
  10. E. Strobel and S. S. Xue, Semiclassical pair production rate for time-dependent electrical fields with more than one component: WKB-approach and world-line instantons, Nucl. Phys. B886, 1153 (2014).
  11. C. Schubert, Perturbative quantum field theory in the string inspired formalism, Phys. Rep. 355, 73 (2001).
  12. J. P. Edwards and C. Schubert, Quantum mechanical path integrals in the first quantised approach to quantum field theory, arXiv:1912.10004.
  13. I. K. Affleck, O. Alvarez, and N. S. Manton, Pair production at strong coupling in weak external fields, Nucl. Phys. B197, 509 (1982).
  14. G. V. Dunne and C. Schubert, Worldline instantons and pair production in inhomogeneous fields, Phys. Rev. D 72, 105004 (2005).
  15. G. V. Dunne, Q. h. Wang, H. Gies, and C. Schubert, Worldline instantons. II. The fluctuation prefactor, Phys. Rev. D 73, 065028 (2006).
  16. C. K. Dumlu and G. V. Dunne, Complex worldline instantons and quantum interference in vacuum pair production, Phys. Rev. D 84, 125023 (2011).
  17. X. Bai-Song, M. Melike, and D. Sayipjamal, Electron-positron pair production in an elliptic polarized time varying field, Chin. Phys. Lett. 29, 021102 (2012).
  18. A. Ilderton, G. Torgrimsson, and J. Wårdh, Nonperturbative pair production in interpolating fields, Phys. Rev. D 92, 065001 (2015).
  19. C. Kohlfürst, N. Ahmadiniaz, J. Oertel, and R. Schützhold, Sauter-Schwinger effect for colliding laser pulses, Phys. Rev. Lett. 129, 241801 (2022).
  20. G. V. Dunne and Q. h. Wang, Multidimensional worldline instantons, Phys. Rev. D 74, 065015 (2006).
  21. C. Schneider, G. Torgrimsson, and R. Schützhold, Discrete worldline instantons, Phys. Rev. D 98, 085009 (2018).
  22. O. Gould and A. Rajantie, Thermal Schwinger pair production at arbitrary coupling, Phys. Rev. D 96, 076002 (2017).
  23. G. Degli Esposti and G. Torgrimsson, Worldline instantons for nonlinear Breit-Wheeler pair production and Compton scattering, Phys. Rev. D 105, 096036 (2022).
  24. G. Degli Esposti and G. Torgrimsson, Worldline instantons for the momentum spectrum of Schwinger pair production in spacetime dependent fields, Phys. Rev. D 107, 056019 (2023).
  25. G. Degli Esposti and G. Torgrimsson, Momentum spectrum of Schwinger pair production in four-dimensional e-dipole fields, Phys. Rev. D 109, 016013 (2024).
  26. F. Hebenstreit, R. Alkofer, G. V. Dunne, and H. Gies, Momentum signatures for Schwinger pair production in short laser pulses with sub-cycle structure, Phys. Rev. Lett. 102, 150404 (2009).
  27. C. K. Dumlu, Schwinger Vacuum pair production in chirped laser pulses, Phys. Rev. D 82, 045007 (2010).
  28. E. Akkermans and G. V. Dunne, Ramsey fringes and time-domain multiple-slit interference from vacuum, Phys. Rev. Lett. 108, 030401 (2012).
  29. F. Hebenstreit, R. Alkofer, and H. Gies, Particle self-bunching in the Schwinger effect in spacetime-dependent electric fields, Phys. Rev. Lett. 107, 180403 (2011).
  30. C. Kohlfürst, Electron-positron pair production in inhomogeneous electromagnetic fields, arXiv:1512.06082.
  31. M. Ababekri, B. S. Xie, and J. Zhang, Effects of finite spatial extent on Schwinger pair production, Phys. Rev. D 100, 016003 (2019).
  32. I. A. Aleksandrov and C. Kohlfürst, Pair production in temporally and spatially oscillating fields, Phys. Rev. D 101, 096009 (2020).
  33. C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).
  34. A. O. Barut and I. H. Duru, Pair production in an electric field in a time dependent gauge, Phys. Rev. D 41, 1312 (1990).
  35. G. Degli Esposti and G. Torgrimsson, Momentum spectrum of nonlinear Breit-Wheeler pair production in space-time fields, Phys. Rev. D 110, 076017 (2024).
  36. Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115, 485 (1959).
  37. W. Ehrenberg and R. E. Siday, The refractive index in electron optics and the principles of dynamics, Proc. Phys. Soc. London Sect. B 62, 8 (1949).
  38. A. Ilderton, Coherent quantum enhancement of pair production in the null domain, Phys. Rev. D 101, 016006 (2020).
  39. I. Aniceto, G. Basar, and R. Schiappa, A primer on resurgent transseries and their asymptotics, Phys. Rep. 809, 1 (2019).
  40. R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, New York 1953), Vol. 1.
  41. R. Carlson, Large eigenvalues and trace formulas for matrix Sturm-Liouville problems, SIAM J. Math. Anal. 30, 949 (1999).
  42. M. Morse, The Calculus of Variations in the Large, AMS Colloquium Series XVIII, (AMS, New York, 1934).
  43. S. Levit and U. Smilansky, A new approach to Gaussian path integrals and the evaluation of the semiclassical propagator, Ann. Phys. (N.Y.) 103, 198 (1977).

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Phys. Rev. D 112, 016026– Published 30 July, 2025
  • Received 30 March 2025
  • Accepted 8 July 2025
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