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

Abstract

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.

Thermal Schwinger pair production at arbitrary coupling

“…Further it gives an all-orders correction to the known weak coupling results. This will be of interest to current and future experimental studies of Schwinger pair production (see for example [62] for a discussion), as well as to multi-loop and asymptotic analyses of the QED perturbation series [36][37][38].…”

Propagator from nonperturbative worldline dynamics

“…Using the ansatz for the PDF given by (30), it is straightforward to obtain an estimate of the expectation value in terms of the fit parameters, 6 The contribution from the uncertainties given in eq. (35) for the α and β parameters are smaller than the width of the plot line in Figure 7 (right).…”

“…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.…”

Asymptotic Behaviour of the QED Perturbation Series

“…Here I will summarize the state of a long-term effort [7][8][9][10][11][12][13][14][15] to get information on the large-order behaviour of the QED perturbation series using the Euler-Heisenberg Lagrangian and its higher-loop radiative corrections.…”