Three-loop Euler-Heisenberg Lagrangian in 1+1 QED. Part I. Single fermion-loop part

Abstract

We study the three-loop Euler-Heisenberg Lagrangian in spinor quantum electrodynamics in 1+1 dimensions. In this first part we calculate the one-fermion-loop contribution, applying both standard Feynman diagrams and the worldline formalism which leads to two different representations in terms of fourfold Schwinger-parameter integrals. Unlike the diagram calculation, the worldline approach allows one to combine the planar and the non-planar contributions to the Lagrangian. Our main interest is in the asymptotic behaviour of the weak-field expansion coefficients of this Lagrangian, for which a nonperturbative prediction has been obtained in previous work using worldline instantons and Borel analysis. We develop algorithms for the calculation of the weak-field expansions coefficients that, in principle, allow their calculation to arbitrary order. Here for the nonplanar contribution we make essential use of the polynomial invariants of the dihedral group D 4 in Schwinger parameter space to keep the expressions manageable. As expected on general grounds, the coefficients are of the form r 1 + r 2 ζ 3 with rational numbers r 1 , r 2 . We compute the first two coefficients analytically, and four more by numerical integration. arXiv:1812.08380v1 [hep-th]

Reducible contributions to quantum electrodynamics in external fields

“…Related results now exist for the effective action at two-and three loops [14,15] in a constant background, (anti-)self-dual backgrounds [16][17][18] and at one-loop order for various non-constant backgrounds such as Sauter pulses [19][20][21] and a pulsed Hermite and Laguerre-Gaussian laser beam [22]. See also [23] for the full mass range analysis of the QED effective action for a nontrivial background with some special symmetry.…”

“…This effect has recently received renewed attention [5][6][7][8][9] due to the prospects of investigating pair creation using future laser facilities. For the status of current and future laser facilities, making study of these backgrounds of great experimental interest for the coming years see, for example, the information at [10-13]).Related results now exist for the effective action at two-and three loops [14,15] in a constant background, (anti-)self-dual backgrounds [16][17][18] and at one-loop order for various non-constant backgrounds such as Sauter pulses [19][20][21] and a pulsed Hermite and Laguerre-Gaussian laser beam [22]. See also [23] for the full mass range analysis of the QED effective action for a nontrivial background with some special symmetry.…”

Worldline master formulas for the dressed electron propagator. Part I. Off-shell amplitudes

“…This does not come unexpected, since it had been noted already in [101] that the ξ = −1 gauge in two dimensions has the property of removing the divergence of the one-loop fermion propagator (which in two dimensions is an IR one). More recently, this property has turned out to be extremely useful for multiloop calculations in the Schwinger model [102]. It will be interesting to see whether further simplification can be achieved by one of the generalisations (8.19), (8.22).…”

Propagator from nonperturbative worldline dynamics

“…This is, because subclasses of different topologies can be combined into a single worldline expression, cf. [9,29].…”

“…Obtaining nonperturbative results from such all-order expressions is nevertheless challenging, since their evaluation also requires a nonperturbative way to perform the necessary renormalization, i.e., the fixing of physical parameters. In fact, a whole research program has been initiated to cross-check the result for the Schwinger pair production rate evaluated from the all-order worldline expression by semiclassical instanton methods [10] with explicit higher-loop calculations [30,31,32,33,29], the latter requiring an explicit treatment of the mass renormalization. If semiclassical methods turned out to be reliable also at higher loop order, many studies on Schwinger pair production using worldline instanton methods [20,34,35,36,37,38,39,40] could be generalized beyond perturbative loop expansions.…”