Euler-Heisenberg Lagrangians and asymptotic analysis in 1+1 QED. Part I: two-loop.

*(English)*Zbl 1294.81301Summary: We continue an effort to obtain information on the QED perturbation series at high loop orders, and particularly on the issue of large cancellations inside gauge invariant classes of graphs, using the example of the \(l\) – loop \(N\) – photon amplitudes in the limit of large photons numbers and low photon energies. As was previously shown, high-order information on these amplitudes can be obtained from a nonperturbative formula, due to Affleck et al., for the imaginary part of the QED effective Lagrangian in a constant field. The procedure uses Borel analysis and leads, under some plausible assumptions, to a number of nontrivial predictions already at the three-loop level. Their direct verification would require a calculation of this ‘Euler-Heisenberg Lagrangian’ at three-loops, which seems presently out of reach. Motivated by previous work by Dunne and Krasnansky on Euler-Heisenberg Lagrangians in various dimensions, in the present work we initiate a new line of attack on this problem by deriving and proving the analogous predictions in the simpler setting of 1+1 dimensional QED. In the first part of this series, we obtain a generalization of the formula of Affleck et al. to this case, and show that, for both Scalar and Spinor QED, it correctly predicts the leading asymptotic behaviour of the weak field expansion coefficients of the two loop Euler-Heisenberg Lagrangians.

##### MSC:

81V10 | Electromagnetic interaction; quantum electrodynamics |

81T16 | Nonperturbative methods of renormalization applied to problems in quantum field theory |

81T18 | Feynman diagrams |

81V80 | Quantum optics |

70S05 | Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems |

##### References:

[1] | J.C. Le Guillou and J. Zinn-Justin, Large Order Behavior Of Perturbation Theory, North-Holland, (1990). |

[2] | Stevenson, PM, Optimized perturbation theory, Phys. Rev., D 23, 2916, (1981) |

[3] | Stevenson, PM, Optimization and the ultimate convergence of QCD perturbation theory, Nucl. Phys., B 231, 65, (1984) |

[4] | Fischer, J., The use of power expansions in quantum field theory, Int. J. Mod. Phys., A 12, 3625, (1997) |

[5] | G.V. Dunne, Perturbative-nonperturbative connection in quantum mechanics and field theory, hep-th/0207046 [SPIRES]. |

[6] | Badger, S.; Bjerrum-Bohr, NEJ; Vanhove, P., Simplicity in the structure of QED and gravity amplitudes, JHEP, 02, 038, (2009) |

[7] | Badger, SD; Henn, JM, Compact QED tree-level amplitudes from dressed BCFW recursion relations, Phys. Lett., B 692, 143, (2010) |

[8] | Cvitanovic, P.; Kinoshita, T., Sixth order magnetic moment of the electron, Phys. Rev., D 10, 4007, (1974) |

[9] | Cvitanovic, P., Asymptotic estimates and gauge invariance, Nucl. Phys., B 127, 176, (1977) |

[10] | http://www.nbi.dk/∼predrag/papers/finitness.html. |

[11] | Dunne, GV; Schubert, C., Two-loop self-dual Euler-Heisenberg Lagrangians. I: real part and helicity amplitudes, JHEP, 08, 053, (2002) |

[12] | Dunne, GV; Schubert, C., Two-loop self-dual Euler-Heisenberg Lagrangians. II: imaginary part and Borel analysis, JHEP, 06, 042, (2002) |

[13] | Martin, LC; Schubert, C.; Villanueva Sandoval, VM, On the low-energy limit of the QED N-photon amplitudes, Nucl. Phys., B 668, 335, (2003) |

[14] | Dunne, GV; Schubert, C., Multiloop information from the QED effective Lagrangian, J. Phys. Conf. Ser., 37, 59, (2006) |

[15] | W. Heisenberg and H. Euler, Consequences of Dirac’s theory of positrons, Z. Phys.98 (1936) 714 [physics/0605038]. |

[16] | C. Itzykson and J. Zuber, Quantum Field Theory, McGraw-Hill, (1985). |

[17] | G.V. Dunne, Heisenberg-Euler effective Lagrangians: Basics and extensions, Ian Kogan Memorial Collection, From Fields to Strings: Circumnavigating Theoretical Physics. volume I, M. Shifman et al.eds., hep-th/0406216 [SPIRES]. |

[18] | L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [SPIRES]. |

[19] | Weisskopf, V., The electrodynamics of the vacuum based on the quantum theory of the electron, K. Dan. Vidensk. Selsk. Mat. Fy. Medd., 14, 1, (1936) |

[20] | Schwinger, JS, On gauge invariance and vacuum polarization, Phys. Rev., 82, 664, (1951) |

[21] | Sauter, F., Über das verhalten eines elektrons im homogenen elektrischen feld nach der relativistischen theorie diracs, Zeit. f. Phys., 69, 742, (1931) |

[22] | Dunne, GV; Schubert, C., Two-loop Euler-Heisenberg QED pair-production rate, Nucl. Phys., B 564, 591, (2000) |

[23] | Ritus, VI, The Lagrange function of an intensive electromagnetic field and quantum electrodynamics at small distances, Zh. Eksp. Teor. Fiz, 69, 1517, (1975) |

[24] | Ritus, VI; Ginzburg, VI (ed.), The Lagrangian function of an intense electromagnetic field and quantum electrodynamics at short distances, (1987), New York U.S.A |

[25] | W. Dittrich and M. Reuter, Effective Lagrangians in Quantum Electrodynamics, Springer (1985). |

[26] | Reuter, M.; Schmidt, MG; Schubert, C., Constant external fields in gauge theory and the spin 0, 1/2, 1 path integrals, Annals Phys., 259, 313, (1997) |

[27] | Fliegner, D.; Reuter, M.; Schmidt, MG; Schubert, C., Two-loop Euler-Heisenberg Lagrangian in dimensional regularization, Theor. Math. Phys., 113, 1442, (1997) |

[28] | Körs, B.; Schmidt, MG, The effective two-loop Euler-Heisenberg action for scalar and spinor QED in a general constant background field, Eur. Phys. J., C 6, 175, (1999) |

[29] | Schubert, C., Perturbative quantum field theory in the string-inspired formalism, Phys. Rept., 355, 73, (2001) |

[30] | Ritus, VI, Connection between strong-field quantum electrodynamics with short-distance quantum electrodynamics, Zh. Eksp. Teor. Fiz, 73, 807, (1977) |

[31] | Dunne, GV; Huet, A.; Rivera, D.; Schubert, C., Closed-form weak-field expansion of two-loop Euler-Heisenberg Lagrangians, JHEP, 11, 013, (2006) |

[32] | Lebedev, SL; Ritus, VI, Virial representation of the imaginary part of the Lagrange function of the electromagnetic field, Sov. Phys. JETP, 59, 237, (1984) |

[33] | Duff, MJ; Isham, CJ, Self-duality, helicity, and supersymmetry: the scattering of light by light, Phys. Lett., 86B, 157, (1979) |

[34] | Affleck, IK; Alvarez, O.; Manton, NS, Pair production at strong coupling in weak external fields, Nucl. Phys., B 197, 509, (1982) |

[35] | Feynman, RP, Mathematical formulation of the quantum theory of electromagnetic interaction, Phys. Rev., 80, 440, (1950) |

[36] | Blau, SK; Visser, M.; Wipf, A., Analytical results for the effective action, Int. J. Mod. Phys., A 6, 5409, (1991) |

[37] | Gavrilov, SP; Gitman, DM, Vacuum instability in external fields, Phys. Rev., D 53, 7162, (1996) |

[38] | Gavrilov, SP; Gitman, DM; Goncalves, AE, QED in external field with space-time uniform invariants: exact solutions, J. Math. Phys., 39, 3547, (1998) |

[39] | Lin, Q-g, Electron positron pair creation in vacuum by an electromagnetic field in 3+1 and lower dimensions, J. Phys., G 25, 17, (1999) |

[40] | Dunne, GV; Krasnansky, M., ’background field integration-by-parts’ and the connection between one-loop and two-loop Heisenberg-Euler effective actions, JHEP, 04, 020, (2006) |

[41] | Krasnansky, M., Two-loop vacuum diagrams in background field and Heisenberg-Euler effective action in different dimensions, Int. J. Mod. Phys., A 23, 5201, (2008) |

[42] | Loskutov, YM; Lysov, BA; Skobelev, VV, Field asymptotic behavior of a polarization operator, Theor. Math. Phys., 53, 1252, (1982) |

[43] | Dunne, GV; Schubert, C., Worldline instantons and pair production in inhomogeneous fields, Phys. Rev., D 72, 105004, (2005) |

[44] | I. Huet and C. Schubert, Euler-Heisenberg lagrangians and asymptotic analysis in 1+1 QED, part 2: Three-loop, in preparation. |

[45] | I. Huet, D.G.C. McKeon and C. Schubert, Three-loop Euler-Heisenberg Lagrangian and asymptotic analysis in 1+1 QED, Proceedings of the Ninth Nonference on Quantum Field Theory Under The Influence Of External Conditions (QFEXT ’09), K.A. Milton and M. Bordag eds., World Scientific (2010), [arXiv:0911.0227]. |

[46] | Feynman, RP, An operator calculus having applications in quantum electrodynamics, Phys. Rev., 84, 108, (1951) |

[47] | Huet, A., New relations between spinor and scalar one-loop effective Lagrangians in constant background fields, Int. J. Mod. Phys., A 25, 4055, (2010) |

[48] | G.V. Dunne and C. Schubert, Bernoulli Number Identities from Quantum Field Theory, math/0406610 [SPIRES]. |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.