Non-perturbative signatures of non-linear Compton scattering

The probabilities of various elementary laser - photon - electron/positron interactions display in selected phase space and parameter regions typical non-perturbative dependencies such as $\propto {\cal P} \exp\{- a E_{crit} /E\}$, where ${\cal P}$ is a pre-exponential factor, $E_{crit}$ denotes the critical Sauter-Schwinger field strength, and $E$ characterizes the (laser) field strength. While the Schwinger process with $a = a_S \equiv \pi$ and the non-linear Breit-Wheeler process in the tunneling regime with $a = a_{n \ell BW} \equiv 4 m / 3 \omega'$ (with $\omega'$ the probe photon energy and $m$ the electron/positron mass) are famous results, we point here out that also the non-linear Compton scattering exhibits a similar behavior when focusing on high harmonics. Using a suitable cut-off $c>0$, the factor $a$ becomes $a = a_{n \ell C} \equiv \frac23 c m /(p_0 + \sqrt{p_0^2 -m^2)}$. This opens the avenue towards a new signature of the boiling point of the vacuum even for field strengths $E$ below $E_{crit}$ by employing a high electron beam-energy $p_0$ to counter balance the large ratio $E_{crit} / E$ by a small factor $a$ to achieve $E / a \to E_{crit}$. In the weak-field regime, the cut-off facilitates a threshold leading to multi-photon signatures showing up in the total cross section at sub-threshold energies.


I. INTRODUCTION
The Schwinger process signals the instability of the vacuum against particle (pair) creation in an external field. The pair (e + e − ) production rate ∝ exp{−aE crit /E}, a = π [1-3], in a spatio-temporally homogeneous electric field of strength E is exceedingly small due to the large value of the (critical) Sauter-Schwinger field strength E crit = 1.3 × 10 18 V/m and therefore escaped a direct experimental verification until now. Much hope was therefore put on the progressing laser technology which however delivers even at present and near-future "ultrahigh intensities" far too low field strengths [4,5]. Many efforts on the theory side attempted to find field configurations which enhance the Schwinger type pair production. To cite a few entries of the fairly extended literature, which documents the ongoing enormous interest in that topic, we mention dynamical assistance [6][7][8][9][10][11][12][13][14][15][16][17][18] and double assistance effects [19,20] and multi-beam configurations [21] and their embedding into optimization procedures [22,23]. In essence, these attempts envisage a reduction of the factor a in the above exponent, which is in general a complicated function of the external parameters. Despite such a "practical goal", these investigations aim at understanding the QED as a pillar of the Standard Model in the non-perturbative, high-intensity regime. Given the seminal meaning of the Schwinger process as paradigm for related processes, e.g. particle production in cosmology [24] and at black hole horizons as Hawking radiation [25], up to the disputed Unruh radiation [26][27][28][29], various authors considered analog processes, e.g. in condensed matter physics [30,31] and in wave guides [32] etc., which display also the monomonial, genuinly non-perturbative dependence on an external field parameter.
Still within QED, one can search for more easily accessible processes which have the prototypical non-perturbative dependence ∝ exp{−aE crit /E}. For instance, the LUXE collaboration [33][34][35] envisages to exploit the non-linear Breit-Wheeler process which is known to behave as ∝ exp{−a n BW E crit /E} in the tunneling regime with a n BW = 4m/3ω , where ω is the energy of a probe photon traversing a strong laser pulse. LUXE is planed as next-generation follow-up of the seminal SLAC experiment E-144 [36], which operated in the multi-photon regime, by "Measuring the Boiling Point of the Vacuum of Quantum Electrodynamics" [33] via the non-linear Breit-Wheeler process since ω m reduces the exponential suppression, i.e. it makes the above quantity a n BW small when using probe photon energies ω much larger than the electron mass m, thus compensating the large value of E crit /E at presently attainable facilities. Note furthermore that the trident process shows also an exponential behavior under certain conditions [37,38], as originally elaborated in [39,40].
Here, we point out that the non-linear Compton process has a similar non-perturbative exponential field strength dependence under certain side conditions. The key is the suppression of the low harmonics which facilitate the Thomson limit and display a polynomial dependence. What is then left is the otherwise exponentially suppressed contribution. The analogy to the non-linear Breit-Wheeler process is not surprising since it is the crossing channel of the non-linear Compton process in the Furry picture. The crucial difference is in the final-state phase spaces. This is most clearly evident in the perturbative, weak-field limit, where the Breit-Wheeler process is a threshold process, while the Compton process without side conditions has no threshold (see [41,42] for the physical regions in the Mandelstam plane). We introduce here as side condition a cut-off which is related to exit channel kinematics. This in fact enforces the exponential behavior.
Our brief note is organized as follows. In section II, we outline the definition of a Lorentz invariant cut-off in the non-linear Compton scattering. In section III, the restriction of the physically accessible regions in the Mandelstam plane is discussed. The cut-off facilitates a clear signature of multi-photon effects in the total cross section in the weak-field regime (section IV). The moderately strong-field regime is considered in section V, where we compare the exact numerical results with some approximation formula to evidence the exponential dependence of the cross section. The discussion section VI contains a comparison with laser pulses and outlines of how the cut-off is realized by photon observables in the exit channel.
We summarize in section VII.

II. NON-LINEAR COMPTON SCATTERING WITH CUT-OFF
We consider here a monochromatic laser field in plane wave approximation for circular polarization. The non-linear Compton (n C ) cross section with cut-off c reads where F n (z n ) = −4J n (z n ) 2 + 2 + for c ≤ y n and F n = 0 elsewhere. The Lorentz and gauge invariant quantity a 0 is the classical non-linearity parameter charactering solely the laser beam, and α stands for the finestructure constant. The arguments of the Bessel functions J n read explicitly z n (x, y n , a 0 ) = 2na 0 1 yn , where the two invariants x = k · k /k · p and y n = 2n k·p m 2 * with 0 ≤ x ≤ y n enter. For c = 0, one recovers the text book formulas, e.g. in [41,42], where the effective mass m 2 * = m 2 (1 + a 2 0 ) and the (quasi-) momentum balance as well as the relation to asymptotic four-momenta (p/p for in/out-electrons and k/k for in/out-photons) are discussed in detail.
The only but decisive difference is the introduction of the cut-off c in (1) which pushes the lower limit of the x integration to higher values, i.e. it is aimed at suppressing the lower harmonics.

III. KINEMATICS IN THE MANDELSTAM PLANE
The meaning of the cut-off c can be visualized in a covariant manner by inspecting the Mandelstam plane. Defining the invariants s n = (q + nk) 2 , t n = (k − nk) 2 , u n = (q − nk) 2 for harmonics n = 1, 2, 3 · · · , the physical regions I -III in scaled triangular coordinateŝ s = s/m 2 * ,t = t/m 2 * ,û = u/m 2 * withŝ +t +û = 2 refer to processes related by crossing symmetry on amplitude level: I (red area in Fig. 1) for n C process, e − + nγ → e − + γ or q + nk = q + k with quasi-momenta q and q , II (upper gray area) for non-linear Breit-Wheeler (n BW ) pair production, γ + nγ → e + + e − or k + nk = q e + + q e − , and III (left gray area) as mirror of I, e.g. e + + nγ → e + + γ . In I, the harmonicsŝ n = const are parallel lines (in blue in Fig. 1), limited byt = 0 (on-axis forward scattering, where x = 0) and by the hyperbolaŝû = 1 (on-axis backscattering), i.e. the physical interval of each harmonic is given by 0 ≤t ≤ 2 −ŝ n −ŝ −1 n , which is another way of expressing the above quoted restriction 0 ≤ x ≤ y n . The scaled invariant-energy squared of the first harmonic isŝ 1 = 1 + ∆ŝ (measured from the bullet at the top of I in direction of theŝ coordinate, indicated by the arrow, as shown for the other coordinates too) and the spacing of adjacent harmonics is ∆ŝ =ŝ n+1 −ŝ n = 2k · p/m 2 * . Considering an optical laser (we use the frequency ω = 1 eV as representative value) colliding head-on with an electron beam, as available (i) in HZDR (40 MeV [43]) or planned (ii) at ELI (600 MeV [44]) and (iii) at LUXE (17.5 GeV [34]) for instance, one has (i) ∆ŝ ≈ 6.5 × 10 −3 /(1 + a 2 0 ), (ii) 9.6 × 10 −2 /(1 + a 2 0 ) and (iii) 2.8 × 10 −1 /(1 + a 2 0 ) in the red region displayed in Fig Harmonics of n C are parallel to (and may coincide with) the blue lines in region I, which become restricted to the dark-red region below the boundary x = 1 (in yellow) and above the hyperbolaŝû = 1 (in green) when imposing the cut-off c = 1 which facilitates the threshold at coordinatesŝ = 2,t = − 1 2 ,û = 1 2 . In region II, the harmonics of n BW are parallel to (and may coincide with) the horizontal green lines. narrow parallel lines representing the harmonics, we depict only a few representative proxies of them atŝ = 3 2 , 2, 5 2 , 3 etc. as blue lines. In contrast to the perturbative, weak-field limits of the linear processes, n = 1, a 0 → 0, the physical regions I -III of the non-linear processes are mapped out by the discrete harmonics n = 1 · · · ∞.
The cut-off c = 1 in (1) restricts the region I to the dark-red area, limited by a section of the hyperbolaŝû = 1 and the line x ≡ k · k /k · p =t n /(1 −ŝ n −t n ) ≥ c. This excludes the low harmonicsŝ n < 2 and restricts the admissiblet intervals of the harmonicsŝ n ≥ 2 to are in the admissible region. In such a way, a non-trivial threshold is introduced, depicted by the blue bullet at the tip of the dark-red area at coordinatesŝ = 2,t = − 1 2 andû = 1 2 . Imagine now that we keep the laser frequency ω = | k| but lower the electron energy p 0 , i.e. the values ofŝ n would become gradually smaller. Then, a certain number of harmonics drop out the admissible area as they pass the threshold by moving to the left-above: less and less harmonics contribute to the n C process by (i) imposing a threshold by the cut-off c > 0 and/or (ii) diminishingŝ 1 (and all otherŝ n ).  [45] (complementary approaches to multi-photon effects are considered in [46]). Thus, the channel closing effect is exactly analog to subthreshold n BW pair production in region II [47]. There, the thresholdt = 4 (depicted as bullet at bottom of the green top parabolaŝû = 1 in Fig. 1) limits the physically admissible region: only harmonics witht n ≥ 4 contribute. The notion "sub-threshold" meanst n=1 < 4.
Similar to the n C process, we have displayed in Fig. 1 only two possible proxies (horizontal green lines) of two harmonics of n BW in the region II. Note hat, in considering n BW pair production per se, one changes usually the coordinate namest n →ŝ n etc. according to the crossing symmetry relations [42].
V. NON-PERTURBATIVE REGIME, a 0 1 After enforcing a non-trivial threshold in n C process by the cut-off c > 0, one expects a further similarity to the n BW in the region a 0 > 1 despite different phase spaces. As shown originally in [48][49][50], in the tunneling regime a 0 1/ √ κ 1, the n BW pair creation rate scales as ∝ κ exp{−8/3κ}, where κ = a 0 k · k /m 2 (here, k and k are the in four-momenta of the laser and probe photons). In head-on collisions, κ = 2 ω m E E crit since a 0 = m ω E E crit . That yields the Schwinger type dependence ∝ exp{−a n BW E crit /E} with a n BW = 4 3 m ω . The large ratio E crit /E can be compensated by a small ratio m/ω , thus making the pair creation rate accessible in present day experiments by using hard probe photons with ω m, in contrast to the plain Schwinger rate, even with assistance effects. As emphasized in [33], such a Schwinger type rate of n BW is found numerically already for a 0 1 and κ 1.
Quite in contrast to n BW , the n C cross section without cut-off displays a polynomial dependence on the invariant Ritus variable 1 χ ≡ a 0 k·p/m 2 = a 0 (s 1 −m 2 * )/2m 2 [49]. However, imposing the cut-off c > 0, thus suppressing the low harmonics in (1) by a threshold, turns the behavior to an exponential one. In fact, evaluating (1) numerically, one obtains the solid curves in Fig. 3 for c = 1 (left panel) and 2 (right panel). Since at 1/χ < 1 the curves display an a 0 dependence, we have employed scaling factors. Without the latter ones, the curves at 1/χ > 1 are nearly perfectly on top of each other, i.e. independent of a 0 . To quantify the 1 The Ritus variable χ is a measure of the field strength E/E crit in the rest frame of the electron; χ encodes the energy of the laser + electron beams as well as the laser intensity. The high-energy limit and the high-intensity limit do not commute albeit they yield both a high-χ asymptotic [51,52]. 1/χ dependence we depict for a comparison the dashed curves based on where f (c) = (5 + 7c + 5c 2 )/(1 + c) 3 and erfc stands for the complementary error function.
One avenue to (3) is to start with (1) in the limit a 0 → ∞ with side condition (1 − z 2 n /n 2 )a 2 0 = const and then to convert the sum via the Euler-Maclaurin formula into an integral, F ∞ (χ, c) = − 4 3π ∞ c dx f (x)x −2/3 Ai (z(x)) with z(x) = (x/χ) 2/3 . Under the condition c χ, the derivative of the Airy function, Ai , can be replaced by its asymptotic representation and the integral can be executed upon a shift of the variable x and a suitable Taylor expansion.
Surprisingly, the small-χ leading-order term ∝ exp{−2c/3χ} in (3) numerically approximates (1) fairly well in the non-asymptotic region, a 0 1 and χ < 1, irrespectively of the assumptions made in the sketched derivation. As a consequence, the n C cross section also displays a Schwinger type dependence σ(c > 0) ∝ exp{−a n C E crit /E} for suitable values of the cut-off c > 0, in general with a n C (c, a 0 , s 1 ). That is, the paradigmatic transmonomial behavior [53] is provided not only for pair creation but shows up also in high-harmonics Compton scattering on the level of "total" cross section, which actually means integration over a fraction of the out-phase space.

A. Bandwidth effects
While (1) is for monochromatic laser beams with the four-potential A(φ) = g(φ) [ a 1 cos φ + a 2 sin φ] of the e.m. field with invariant phase φ and obeying g(φ) = 1, a 1 a 2 = 0, a 1 2 = a 2 2 , one has to check whether laser pulses are well approximated when focusing on total cross sections. In Fig. 4, the cross section as a function ofs is exhibited as in Fig. 2, however, for short and ultra-short pulses. The calculations are based on equation (33) in [46] with the replacement 0 → c in the lower limit of the u integral. The pulse shape envelope is here especially g(φ) = 1/ cosh(φ/N π), where N characterizes the number of oscillations of the field. This envelope g(φ) does have neither an extended flat-top section nor narrow ramping sections. The former property makes it distinctive to a near-monochromatic beam with very broad flat-top envelope. The related bandwidth effects smoothen the step like shape of the total cross sections, as known from n BW [45]. In particular, for the ultrashort pulse, the strong bandwidth effect overwrites the multi-photon effects; the cross section is stark enhanced in the sub-threshold region. We conclude from this particular example that (1) provides a useful expression for sufficiently long pulses with N > 10, i.e. a pulse duration of > 30 fs for optical laser pulses.

B. Imposing the cut-off
The cut-off c > 0 in (1) looks quite innocent, but in practice it may become challenging.
For on-axis backscattering, these relations evidence that one has to reject events with too low values of ν or select sufficiently high harmonics to realize the request x ≥ c, see left panel of Fig. 5. The meaning of these curves is that the realization of x ≥ c requires in general ν (n, Θ ; a 0 , ν, ζ) ≥ ν (x, Θ ; ζ) as a function of Θ , where (4) determines the ν independent function ν (x, Θ ; ζ). These relations are exhibited in the right panel of Fig. 5.
In the preferred backward direction Θ → π, the curves ν (x, Θ ; ζ) are nearly flat, with the benefit that only an energy-resolved measurement is necessary to select the wanted range x ≥ c. At smaller angles Θ , i.e. going further to the right, beyond the region displayed in the right panel of Fig. 5, the curves ν (x, Θ ; ζ) bend up, which would require also an angular-resolved measurement. However, the contributions of the very high harmonics are exceedingly small in that phase space region and can be neglected.
In addition to this purely kinematic relations one has to account for the dead cone effect which is special for the dynamics in circularly polarized lasers according to (1): Ignoring for the moment being the cut-off, the harmonics n > 1 are (multiply) peaked within the interval 0 < x < y n and drop smoothly towards zero at the boundaries x → 0 and x → y n .
This translates into peaked structures at certain values of Θ 0 within the interval 0 · · · π when transforming the variable x to the laboratory-related angle Θ . For relativistic electrons, i.e. e ζ 1, e.g. for the LUXE type kinematics, one finds Θ 0 π, with Θ 0 slightly dropping with increasing harmonic number as long as n < 10 and saturating for higher harmonics, as the widening dead cones does too. For the given kinematics we find from (1) the dead cone location at angles Θ > π − 7.7 × 10 −6 and the maximum of the distributions dF/d cos Θ at π − 1.8 × 10 −5 for n = 2 and at π − 4.3 × 10 −5 for n > 10.

VII. SUMMARY
In summary we point out that the non-linear Compton process obeys a field strength dependence ∝ P exp{−a n C E crit /E}, similar to the Schwinger process of "vacuum break down", when imposing a suitable cut-off c which suppresses the low harmonics. We focus on the slope coefficient a n C = 2 3 cm/(p 0 + p 2 0 − m 2 ) by a comparison with some approximation formula which displays a dependence ∝ exp{−2c/3χ} already in the non-asymptotic region. Albeit the Compton process does obviously not have such a tunneling regime as the pair production processes, its formal similarity with the non-linear Breit-Wheeler process provides evidence [49] for selected differential contributions with an exponential field dependence. The here introduced cut-off acts as a threshold and enforces a large gap between inand out-Zel'dovich levels; it makes the otherwise hidden exponential contributions visible in the "total" cross section, which actually refers to a fraction of the out-phase space. This opens another avenue towards a measurement of the boiling point of the vacuum, complementary to plans of the LUXE collaboration [33]. While for LUXE a high-energy photon beam is vital, our approach requires either a moderately high-energy (p 0 ) electron beam and the selection of very high harmonics or a high-energy electron beam and the selection of moderately high harmonics. The experimental challenge is anyway the isolation of the high harmonics characterized by the out-photon kinematics.
The present considerations apply to a plane-wave, mono-chromatic laser beam, i.e. a very long pulse duration, with circular polarization. Obvious extensions should take into account general laser polarizations and further bandwidth effects due to finite-duration pulses and their detailed temporal structures. Planned follow-up work is devoted to energy-and angular-differential spectra and suitable realizations of the crucial cut-off implementation in non-perfect head-on collisions.