The QED four-photon amplitudes off-shell. II. (English) Zbl 1530.81139
Summary: This is the second one of a series of four papers devoted to a first calculation of the scalar and spinor QED four-photon amplitudes completely off-shell. We use the worldline formalism which provides a gauge-invariant decomposition for these amplitudes as well as compact integral representations. It also makes it straightforward to integrate out any given photon leg in the low-energy limit, and in the present sequel we do this with two of the four photons. For the special case where the two unrestricted photon momenta are equal and opposite the information on these amplitudes is also contained in the constant-field vacuum polarisation tensors, which provides a check on our results. Although these amplitudes are finite, for possible use as higher-loop building blocks we evaluate all integrals in dimensional regularisation. As an example, we use them to construct the two-loop vacuum polarisation tensors in the low-energy approximation, rederive from those the two-loop \(\beta\)-function coefficients and analyse their anatomy with respect to the gauge-invariant decomposition. As an application to an external-field problem, we provide a streamlined calculation of the Delbrück scattering amplitudes in the low-energy limit. All calculations are done in parallel for scalar and spinor QED.
For Part I, see [the authors, ibid. 991, Article ID 116216, 36 p. (2023; Zbl 1529.81108)].
For Part I, see [the authors, ibid. 991, Article ID 116216, 36 p. (2023; Zbl 1529.81108)].
MSC:
| 81V10 | Electromagnetic interaction; quantum electrodynamics |
| 81R25 | Spinor and twistor methods applied to problems in quantum theory |
| 81V80 | Quantum optics |
| 20C12 | Integral representations of infinite groups |
| 70F10 | \(n\)-body problems |
| 55P35 | Loop spaces |
| 81T33 | Dimensional compactification in quantum field theory |
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
| 05C51 | Graph designs and isomorphic decomposition |
| 81U05 | \(2\)-body potential quantum scattering theory |
Citations:
Zbl 1529.81108Software:
MathematicaReferences:
| [1] | Ahmadiniaz, N.; Lopez-Arcos, C.; Lopez-Lopez, M. A.; Schubert, C., The QED four – photon amplitudes off-shell: part 1 |
| [2] | Batalin, I. A.; Shabad, A. E., Photon green function in a stationary homogeneous field of the most general form, Zh. Eksp. Teor. Fiz., 60, 894-900 (1971) |
| [3] | Baier, V. N.; Katkov, V. M.; Strakhovenko, V. M., An operator approach to quantum electrodynamics in external field: mass operator, Zh. Eksp. Teor. Fiz., 67, 453 (1974) |
| [4] | Baier, V. N.; Katkov, V. M.; Strakhovenko, V. M., An operator approach to quantum electrodynamics in external field 2. Electron loops, Zh. Eksp. Teor. Fiz., 68, 405 (1975) |
| [5] | Tsai, W.-y.; Erber, T., The propagation of photons in homogeneous magnetic fields: index of refraction, Phys. Rev. D, 12, 1132 (1975) |
| [6] | Urrutia, L. F., Leptonic and radiative meson decays: a phenomenologically broken symmetry approach, Phys. Rev. D, 18, 819 (1978) |
| [7] | Schubert, C., Vacuum polarization tensors in constant electromagnetic fields. Part 1, Nucl. Phys. B, 585, 407-428 (2000) · Zbl 0971.81182 |
| [8] | Dittrich, W.; Gies, H., Probing the quantum vacuum. Perturbative effective action approach in quantum electrodynamics and its application, vol. 166 (2000) |
| [9] | Karbstein, F.; Shaisultanov, R., Photon propagation in slowly varying inhomogeneous electromagnetic fields, Phys. Rev. D, 91, 8, Article 085027 pp. (2015) |
| [10] | Schmidt, M. G.; Schubert, C., Multiloop calculations in the string inspired formalism: the Single spinor loop in QED, Phys. Rev. D, 53, 2150-2159 (1996) |
| [11] | Jost, R.; Luttinger, J. M., Vacuumpolarisation und \(e^4\)-Ladungsrenormalisation für Elektronen, Helv. Phys. Acta I-II, 2150-2159 (1950) · Zbl 0036.26901 |
| [12] | Z. Bialynicka-Birula, Fourth Order Renormalization Constants in Quantum Electrodynamics, Bull. Acad. Polon. Sci., Ser. Sci., Math., Astron. Phys. 3. · Zbl 0109.21703 |
| [13] | Johnson, K.; Willey, R.; Baker, M., Vacuum polarization in quantum electrodynamics, Phys. Rev., 163, 1699-1715 (1967) |
| [14] | Broadhurst, D. J.; Delbourgo, R.; Kreimer, D., Unknotting the polarized vacuum of quenched QED, Phys. Lett. B, 366, 421-428 (1996) |
| [15] | Grauel, B., β function QED to two loops - traditionally and with Corolla polynomial (2014), Humboldt University: Humboldt University Berlin, MSc thesis |
| [16] | Schubert, C., Perturbative quantum field theory in the string-inspired formalism, Phys. Rep., 355, 2, 73-234 (2001) · Zbl 0988.81108 |
| [17] | Schmidt, M. G.; Schubert, C., Worldline Green functions for multiloop diagrams, Phys. Lett. B, 331, 69-76 (1994) |
| [18] | Schubert, C., The Structure of the Bern-Kosower integrand for the N gluon amplitude, Eur. Phys. J. C, 5, 693-699 (1998) |
| [19] | Ahmadiniaz, N.; Schubert, C.; Villanueva, V. M., String-inspired representations of photon/gluon amplitudes, J. High Energy Phys., 01, Article 132 pp. (2013) |
| [20] | Shaisultanov, R., On the string-inspired approach to QED in external field, Phys. Lett. B, 378, 1, 354-356 (1996) |
| [21] | Reuter, M.; Schmidt, M. G.; Schubert, C., Constant external fields in gauge theory and the spin 0, 1/2, 1 path integrals, Ann. Phys., 259, 2, 313-365 (1997) · Zbl 0988.81523 |
| [22] | Ahmadiniaz, N.; Edwards, J. P.; Ilderton, A., Reducible contributions to quantum electrodynamics in external fields, J. High Energy Phys., 05, Article 038 pp. (2019) · Zbl 1416.81209 |
| [23] | Ahmadiniaz, N.; Bastianelli, F.; Corradini, O.; Edwards, J. P.; Schubert, C., One-particle reducible contribution to the one-loop spinor propagator in a constant field, Nucl. Phys. B, 924, 377-386 (2017) · Zbl 1373.81401 |
| [24] | Ilderton, A.; Torgrimsson, G., Worldline approach to helicity flip in plane waves, Phys. Rev. D, 93, 8, Article 085006 pp. (2016) |
| [25] | Edwards, J. P.; Schubert, C., N-photon amplitudes in a plane-wave background, Phys. Lett. B, 822, Article 136696 pp. (2021) · Zbl 07417992 |
| [26] | Schubert, C.; Shaisultanov, R., Master formulas for photon amplitudes in a combined constant and plane-wave background field · Zbl 1528.81178 |
| [27] | Meitner, L.; Kösters, H., Über die streuung kurzwelliger γ-strahlen, Z. Phys., 84, 3-4, 137-144 (1933) |
| [28] | Bethe, H.; Rohrlich, F., Small angle scattering of light by a coulomb field, Phys. Rev., 86, 1, 10 (1952) · Zbl 0046.43814 |
| [29] | Jarlskog, G.; Jönsson, L.; Prünster, S.; Schulz, H.; Willutzki, H.; Winter, G., Measurement of delbrück scattering and observation of photon splitting at high energies, Phys. Rev. D, 8, 11, 3813 (1973) |
| [30] | Cheng, H.; Wu, T. T., High-energy collision processes in quantum electrodynamics. i, Phys. Rev., 182, 5, 1852 (1969) |
| [31] | Cheng, H.; Wu, T. T., High-energy collision processes in quantum electrodynamics. ii, Phys. Rev., 182, 5, 1868 (1969) |
| [32] | Cheng, H.; Wu, T. T., High-energy collision processes in quantum electrodynamics. iii, Phys. Rev., 182, 5, 1873 (1969) |
| [33] | Cheng, H.; Wu, T. T., High-energy collision processes in quantum electrodynamics. iv, Phys. Rev., 182, 5, 1899 (1969) |
| [34] | Mil’shtein, A.; Strakhovenko, V., Quasiclassical approach to the high-energy delbrück scattering, Phys. Lett. A, 95, 3-4, 135-138 (1983) |
| [35] | Mil’shtein, A.; Strakhovenko, M., Coherent scattering of high energy photons in a coulomb field, Zh. Èksp. Teor. Fiz., 85, 14-23 (1983) |
| [36] | Schumacher, M.; Borchert, I.; Smend, F.; Rullhusen, P., Delbrück scattering of 2.75 MeV photons by lead, Phys. Lett. B, 59, 2, 134-136 (1975) |
| [37] | Milstein, A.; Schumacher, M., Present status of delbrück scattering, Phys. Rep., 243, 4, 183-214 (1994) |
| [38] | Schumacher, M., Delbrück scattering, Radiat. Phys. Chem., 56, 1, 101-111 (1999) |
| [39] | Edwards, J. P.; Mata, C. M.; Müller, U.; Schubert, C., New techniques for worldline integration, SIGMA, 17, Article 065 pp. (2021) · Zbl 1508.81840 |
| [40] | Itzykson, C.; Zuber, J., Quantum Field Theory, International Series in Pure and Applied Physics (1980), McGraw-Hill: McGraw-Hill New York · Zbl 0453.05035 |
| [41] | Schmidt, M. G.; Schubert, C., Multiloop calculations in the string-inspired formalism: the single spinor loop in QED, Phys. Rev. D, 53, 2150-2159 (1996) |
| [42] | Fliegner, D.; Reuter, M.; Schmidt, M.; Schubert, C., The two loop Euler-Heisenberg Lagrangian in dimensional renormalization, Theor. Math. Phys., 113, 1442-1451 (1997) |
| [43] | Costantini, V.; De Tollis, B.; Pistoni, G., Nonlinear effects in quantum electrodynamics, Nuovo Cimento A, 2, 3, 733-787 (1971) |
| [44] | Cvitanovic, P., Asymptotic estimates and gauge invariance, Nucl. Phys. B, 127, 176-188 (1977) |
| [45] | Huet, I.; Rausch de Traubenberg, M.; Schubert, C., Asymptotic behaviour of the QED perturbation series, Adv. High Energy Phys., 2017, Article 6214341 pp. (2017) · Zbl 1404.81298 |
| [46] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series (1965), Dover Publications · Zbl 0515.33001 |
| [47] | Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL, 2017. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
