Nonlinear trident in the high-energy limit: Nonlocality, Coulomb field, and resummations

We study nonlinear trident in laser pulses in the high-energy limit, where the initial electron experiences, in its rest frame, an electromagnetic field strength above Schwinger’s critical field. At lower energies the dominant contribution comes from the “two-step” part, but in the high-energy limit the dominant contribution comes instead from the one-step term. We obtain new approximations that explain the relation between the high-energy limit of trident and pair production by a Coulomb field, as well as the role of the Weizsäcker-Williams approximation and why it does not agree with the high-$\chi$ limit of the locally-constant-field approximation. We also show that the next-to-leading order in the large-$a_0$ expansion is, in the high-energy limit, nonlocal and is numerically very important even for quite large $a_0$. We show that the small-$a_0$ perturbation series has a finite radius of convergence, but using Padé-conformal methods we obtain resummations that go beyond the radius of convergence and have a large numerical overlap with the large-$a_0$ approximation. We use Borel-Padé-conformal methods to resum the small-$\chi$ expansion and obtain a high precision up to very large $\chi$. We also use newer resummation methods based on hypergeometric/Meijer-G and confluent hypergeometric functions.

Figure 1

Illustration of the fact that in the high-energy limit all the field dependence comes from the pair-production step. Double lines represent fermions dressed by the background field and the single line is an electron without interaction with the field. The wiggly line is a photon that is not part of the background field.

Figure 2

Comparison between the exact, the leading order (28), and the leading order plus the next-to-leading order correction (2). The field is monochromatic with a circular polarization, $\mathbf{f}=\{\sin \varphi ,\cos \varphi \}$. The first plot shows the part proportional to $\ln 2b_0$ and the second plot the rest. In both cases a factor of ${\alpha }^2\mathcal{T}$ has been factored out, where $\mathcal{T}$ is a volume factor. The nonlocal correction is clearly important here.

Figure 3

Same as 2 but with a Lorentzian ($n=1$ in (46) instead of a flat-top envelope. Since $\mathrm{LO}\sim a_0$ and $\mathrm{NLO}\sim \sqrt{a_0}$ (apart from log terms), the absolute difference increases with $a_0$, so NLO is even more important for this pulse shape. For the constant part the NNLO scales as $a_0^0$.

Figure 4

Exact evaluation of (23) (black lines) compared to the perturbative sums including 1 and up to 15 of the first terms in the $a_0^2$ expansion.

Figure 5

The black lines are exact evaluation of (23) and the other lines show the Padé approximants with $N=M=1,2,\dots ,6$.

Figure 6

The black dashed lines are exact evaluation of (23) and the other lines show the Padé approximant with $N=M=6$ and the large-$a_0$ expansion in (48).

Figure 7

The black lines are exact evaluation of (23) and the other lines show: the perturbative sum in the conformal variable $z$ up to $z^{30}$; the Padé approximant of the conformal series with $N=M=10$ ($N=M=14$) for the log part (constant part); and the large-$a_0$ expansion in (48).

Figure 8

The exponent and the prefactor for $\mu \to \infty$, low-energy and a sinusoidal field. The probability is given by $\mathbb{P}={\alpha }^2{E}^{2}G{e}^{F/E}$, where $E$ is the field strength. “LO,” “NLO,” “NNLO,” and “${\mathrm{N}}^{13}\mathrm{LO}$” are obtained by expanding in $1/a_0$. “${\mathrm{N}}^{13}\mathrm{LO}$” includes terms up to $1/a_0^{26}$, and $[6/6]$ Padé is the $N=M=6$ Padé approximant for the perturbative series in $1/a_0^2$.

Figure 9

${\text{Borel}}_i$ gives the Padé-Borel resummation with a diagonal Padé approximant with $N=M=i$. LO is the leading small-$\chi$ approximation (80), and NLO, NNLO and NNNLO are obtained by including the first couple of terms in the direct sum of the perturbation series (84). The “large-${\chi }_{\gamma }$” line shows (81). The black lines show the exact result.

Figure 10

Same as Fig. 9 but for the two-step part of trident.