Reducible contributions to quantum electrodynamics in external fields

Abstract

We consider one-particle reducible (1PR) contributions to QED and scalar QED processes in external fields, at one-loop and two-loop order. We investigate three cases in detail: constant crossed fields, constant magnetic fields, and plane waves. We find that 1PR tadpole contributions in plane waves and constant crossed fields are non-zero, but contribute only divergences to be renormalised away. In constant magnetic fields, on the other hand, tadpole contributions give physical corrections to processes at one loop and beyond. Our calculations are exact in the external fields and we give strong and weak field expansions in the magnetic case. arXiv:1901.09416v2 [hep-th] 11 May 2019making such instances of great theoretical and phenomenological interest. This is possible if the field configuration is simple, or highly symmetric.This area of field theory was pioneered by Euler and Heisenberg who, taking a constant electromagnetic background, calculated the one-loop effective Lagrangian for QED [1] (see the calculations of Schwinger and Weisskopf [2, 3] for the corresponding calculations in scalar QED, and [4] for a review of these results). As is well known, one physical consequence revealed by the Euler-Heisenberg Lagrangian (EHL) is the instability of the vacuum to the application of strong electric fields, which leads to particle / anti-particle pair creation (the Schwinger mechanism). 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. These have been used to study low energy photon amplitudes [24,25] and the structure of the quantum vacuum, see [26] for a recent review. Aside from this, the particle propagator can also be constructed exactly (non-perturbatively) in the presence of constant fields, plane waves, and other symmetric fields, allowing the calculation of a variety of electron-seeded and photon-seeded processes, see [26][27][28][29] for reviews.Recently, however, it was found that historical calculations had overlooked the possibility of one particle reducible (1PR) contributions to processes in constant background fields [30][31][32]. These contributions involve a tadpole, displayed in figure 1, attached somewhere in the corresponding Feynman diagram describing the process. The tadpole is linear in the exchanged (off-shell) photon momentum, k µ , and momentum conservation implies that it can be ...

Quasiclassical representation of the Volkov propagator and the tadpole diagram in a plane wave

“…Thus, we conclude that the contribution of the tadpole to an arbitrary process can be absorbed into a renormalization of the external plane-wave field in agreement with Ref. [75].…”

“…Also, it shows that only linear terms in the plane wave electromagnetic field contribute to the tadpole (see also Ref. [75]). Now, one notices that the second term in the integrand in Eq.…”

“…It was found in Ref. [75] by using the worldline formalism that the contribution of the tadpole is linear in the external plane wave and that for this reason it can be renormalized out. Below, we show that for the plane-wave case the computation of the tadpole is significantly simplified by starting from the general definition of the vacuum four-current and from its relation with the Volkov propagator.…”

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

“…The forthcoming second part will focus on on-shell amplitudes and cross sections involving open fermion lines, and in the third part we will add an external constant field (partial results of the third part have already been published in [109,110]).…”

Large N external-field quantum electrodynamics

“…section 3 below. Note, however, that all physical tadpole contributions vanish for the special cases of constant crossed and plane-wave fields [12,18].…”