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

Abstract

The solution of the Dirac equation in the presence of an arbitrary plane wave, corresponding to the so-called Volkov states, has provided an enormous insight in strong-field QED.In [Phys. Rev. A 103, 076011 (2021)] a new "fully quasiclassical" representation of the Volkov states has been found, which is equivalent to the one known in the literature but which more transparently shows the quasiclassical nature of the quantum dynamics of an electron in a plane-wave field. Here, we derive the corresponding expression of the propagator by constructing it using the fully quasiclassical form of the Volkov states. The found expression allows one, together with the fully quasiclassical expression of the Volkov states, to compute probabilities in strong-field QED in an intense plane wave by manipulating only 2-by-2 rather than 4-by-4 Dirac matrices as in the usual approach. Moreover, apart from the exponential functions featuring the classical action of an electron in a plane wave, the fully quasiclassical Volkov propagator only depends on the electron kinetic four-momentum in the plane wave, which is a gauge-invariant quantity. Finally, we also compute the tadpole diagram in a plane wave starting from the Volkov propagator and we show that, although it is divergent, its contribution can be always absorbed via a renormalization of the external field. I. INTRODUCTIONThe investigation of quantum electrodynamical processes occurring in the presence of intense laser fields has significantly improved our theoretical understanding of the strong-field regime of QED [1][2][3][4][5][6][7][8][9]. Strong-field QED signatures in the emission spectra of ultrarelativistic electrons colliding with an intense laser beam have also been observed in recent experiments [10,11]. In the strong-field regime of QED the leptons involved in the processes experience in their rest frames

Advances in QED with intense background fields

“…4.) These results for the spinor structure, together with the Volkov exponent being the classical Hamilton-Jacobi action, underline the fact that the Volkov wavefunctions are semiclassical-exact; for recent investigations of this statement, and alternative representations of the Volkov solutions designed to make it explicit, see [150].…”

“…The exact WKB method resolves the factorialdivergence problem via Borel resummation, which is achieved by computing a Laplace transformation of a Borel transform B[φ ± ]. The resulting Borel-summed φ ± is a well-defined function, and its asymptotic expansion coincides with the naïve WKB expansion (150). The Borel sum φ ± is, however, defined only locally and cannot be continued analytically to the whole complexified time space t ∈ R → w ∈ C without experiencing discontinuous jumps, which is the Stokes phenomenon.…”

“…An important lesson learnt in this context is that a larger Feynman diagram containing a sub-diagram which, considered as an isolated object, vanishes identically because of an overall momentum conserving delta function can still yield a finite result; cf. [110,433,509,150]. This can be illustrated by the following schematic example: while g(k) ∼ kδ(k) clearly vanishes, upon sewing it to another (finite) contribution h(k) ∼ k with a 'propagator' ∼ 1/k 2 which is non-regular at k = 0, and integrating over k, we obtain k g(k)f (k)/k 2 ∼ k δ(k) = 0.…”

“…For completeness, we note that subsequently analogous 1PR tadpole corrections have been evaluated for the charged particle propagator [503,504,505] and photon-graviton conversion [506] in a constant field at one loop, the lowenergy limit of the QED N -photon amplitudes at two loops [507], and arbitrary processes in constant fields [433]; see also [508]. Interestingly, all physical tadpole contributions vanish identically in constant crossed and plane-wave fields due to the special field and momentum structure of these backgrounds [110,509,150].…”

Advances in QED with intense background fields

“…This, together with the fact that the Volkov exponent is the classical Hamilton-Jacobi action, underlines the result that the Volkov wavefunctions are semiclassical-exact. For representations of the Volkov solutions designed to make this explicit, including investigation of other spin bases, see [158,159].…”