Nonlinear Compton scattering (NCS) is the scattering of an electron with coherent photons, i.e., e + nγ_L → e + γ. It is a strong-field quantum electrodynamic (SFQED) process and is expected to be one of the dominant processes in ultra-high-intensity laser (UIL) fields. With recent progress in laser technology,1–15 SFQED processes16–31 have attracted wide research interest. The availability of 1–10 PW facilities15 and new designs of high-energy colliders32–35 have also encouraged investigations into new effects of NCS, such as interference effects,36 nonperturbative SFQED,37 quantum quenching,38 peak broadening,39 and electron trapping.40,41

The analytic theory of single NCS was established long ago.42–44 However, NCS is a high-probability quantum process, and electrons usually emit multiple times in UIL pulses, and therefore most investigations of NCS so far have relied on numerical simulations or numerical solution of integral equations,45 which are comparatively time-consuming and expensive tasks.

Peaks in spectra of particle beams are basic features. They are taken as strong evidence for new acceleration mechanisms in experiments and simulations46–53 or possible clues for dark matter annihilation and decay.54 An important constraint on high-energy colliders is that the radiation loss of the colliding beams at the interaction point should be within 20%.32–35 Searching for peaks in the invariant mass spectrum is one of the main approaches for finding new particles, and the spectrum of invariant mass recoiling against observable particles has the prominent advantage of being independent of specific decay modes in the detection of Higgs bosons and the search for dark matter particles.55–58

In this work, analytic formulas for the effects of multiple NCS that are similar to the formulas in the theory of multiple bremsstrahlung are developed. Based on these formulas, a new pure quantum effect on electron dynamics in strong fields called quantum peak splitting is identified: when electrons propagate in ultra-strong fields and the average radiation per electron is ∼5.1–9, the electron peak splits into two. This finding has a direct impact on electron dynamics in strong fields. It unveils a new stage of quantum radiation reaction and puts new constraints on future colliders. This new effect should also be important for the analysis of particle acceleration, and the detection of new particles, dark matter, and Higgs bosons, since the it can create fake peaks.

In ultra-strong fields, electron dynamics is mainly influenced by quantum emission of radiation and the reaction back on itself. NCS and quantum synchrotron radiation (QSR) are very important radiation processes. NCS is the dominant process16–23 and the major form of electron radiation when laser intensities reach levels of ≳10^21–23 W/cm² or higher, which are already achievable today.1,15 QSR of electrons and positrons in the strong fields generated by opposite bunches at the interaction points of colliders59–62 is called beamstrahlung.32–35,63

NCS and QSR have short coherence intervals in ultra-strong fields and are therefore usually taken as instantaneous and local processes, i.e., the local constant field approximation (LCFA) is adopted. χ≡e−(Fμνpν)2/m3 is the key parameter of NCS and QSR,16–20,64 where F is the electromagnetic field, p is the electron four-momentum, e is the electron charge, and m is the electron mass. The differential probability of single NCS42–44 isdWNCSdδdt=απ3m2p01−δ+11−δK2/32δ3χ(1−δ)−∫2δ/3χ(1−δ)∞dyK1/3(y),δ∈(0,1),(1)and that of single QSR is64dWQSRdtdδ=3α2πmqχqγ1−δδ×κ(y)+y33χq22(1−δ)K2/3(y),(2)where α is the fine structure constant, m_q and γ are the mass and Lorentz factor of the charge, χ_q = γB/B_q is the χ parameter in a pure magnetic field, B is the magnetic field, and Bq=mq2/q is the critical magnetic field. δ = k′k/pk for NCS and δ=k0′/p0 for QSR. p₀ is the electron energy before radiation, and k0′ is the emitted photon energy. For NCS of ultra-relativistic electrons, χ ≈ γ(1 − cos θ)a₀k₀/m, where θ is the angle between electron and laser, a₀ ≡ eE_L/mω₀ is the normalized laser amplitude, and E_L and ω₀ are the laser electric field and frequency. The function κ(y) is given byκ(y)=y∫y∞K5/3(x)dx,(3)where K_ν(x) is the modified Bessel function of the νth order andy=2δ3χq(1−δ).(4)

The differential probabilities of both NCS42–44 and QSR64–66 scale approximately asdWdδdt∝δ−2/3e−δ/δ0(5)when δ ≪ 1 and χ or χ_q ≪ 1,44,64 where δ=k0′/p0 is the radiation loss rate and δ₀ ≈ χ or χ_q when χ or χ_q ≪ 1.

We use an interpolation functionfin(δ)≈Aαm2p0χδ2/3⁡exp−δχ(6)to test this approximation. The comparisons in Fig. 1 show that this approximation agrees well with Eqs. (1) and (2) in the region where δ ≪ 1 and χ ≪ 1. Note that this region covers the main part of f_in and the parameter space that existing and near-future experiments can reach (χ ≲ 0.3). The unphysical part of this approximation is less than 0.6%∫1∞fin(x)dx/∫0∞fin(x)dx when χ ≲ 0.3.

We will restrict our discussion of multiple NCS to the weak radiation-dominated regime (WRDR), where χ ≪ 1 and a₀ ≪ χγ. The former ensures that most emissions only consume a small fraction of the electron energy and the latter ensures that the influence of the Lorentz force on electrons is much weaker compared with that of radiation. The WRDR is important because experiments on NCS so far67,68 and the beamstrahlung of existing colliders fall in this regime. In the WRDR, pair production is suppressed, electrons lose only a small fraction of energy through most emissions, and the influence of the electromagnetic force is much weaker than that of NCS, and hence electron dynamics is dominated by weak emissions that can be treated as perturbations.

We will show that the effects of multiple NCS and QSR in the WRDR can be described analytically. Consider a physical quantity x (spin, energy, angle, etc.) of an electron beam propagating in ultra-strong fields in the WRDR. The distribution function F(x) normalized by electron number is approximatelyF(x)=P0f0(x)+∑i=1∞Pifi(x),(7)where P_i is the probability of emitting i times (∑P_i = 1) and f_i are the normalized spectra of electrons radiating i times (∫f_i(x)dx = 1).

P_i has an analytic approximation, which is possible because the emission rate in the WRDR is almost unmodified by quantum emissions. The radiation rate of NCS,dWNCSdt=∫01dδdWdδdt≈1.44αχmγℏ≈0.01(1−cosθ)a0ω0,(8)is approximately proportional to a₀38,44 and approximately independent of γ in the WRDR, where r is the average number of emissions per electron. Similarly, in the WRDR, the emission rate of QSR,64–66dWQSRdt≈5αBm23Bqℏ,(9)is also approximately proportional to the external field and insensitive to γ.61 Since there is almost no modification of the emission rate by NCS and QSR, P_i is approximately determined by the fields. According to probability theory,71 the number of emissions i approximately follows a Poisson distribution, i.e.,Pi(r)≈rii!e−r(i≥0).(10)This is similar to the distribution of multiple bremsstrahlung.69,70 As shown in Fig. 2(a), this simple approximation agrees very well with the simulation results. The distribution peaks at i = [r], where [r] denotes the integer part of r, and the width of the distribution is of the scale of r.

We develop an analytic approximation for f_i(x) in the comparatively simple case that x is the electron energy. In the WRDR, radiation reaction can be taken as a perturbation to the electron energy, and hencefi(η)≈∫dη′fi−1(η′)R̄(η′−η)(i≥1),(11)where η = E/E₀ is the electron energy E normalized by the primary energy E₀, andR̄(δ)=∫dtdWdδdt∫dδ∫dtdWdδdt(12)is the normalized average differential radiation probability for a electron with p₀ = E₀ to emit a photon with k₀ = δE₀.

In the simplest case of a constant external field, since the most emissions fall in the δ ≪ 1 region and R(δ) is approximately proportional to δ^−2/3e^−/ in the WRDR,R(δ)≈RA(δ)=χ−1/3δ−2/3Γ(1/3)e−δ/χ.(13)Then, for a quasi-monoenergetic electron beam, according to Eq. (11), an analytic approximationfi,χ(η)≈fi,χA(η)=χ−i/3Γ(i/3)(1−η)i/3−1e−(1−η)/χ(14)for i ≥ 1 is obtained, which is similar to the expression in the case of multiple bremsstrahlung69,70 if screening effects are significant. This analytic approximation fiA shows that f_i increases monotonically and peaks at η = 1 when i = 1, 2, 3. When i ≥ 4, fiA vanishes at η = 1 and η = 0 and peaks at η = 1 − (χ/3)(i − 3). The peak jumps with a step of Δη = χ/3 as i increases. These features predicted by fiA are confirmed by the functions f_i(η) obtained by numerical integration of Eq. (11) and shown in Fig. 2(b). The unphysical part of f_i(η < 0) obtained with Eq. (11) is within 0.5% when χ < 0.10 and i ≤ 10, and the unphysical part of F(η) is less than 1% when r < 10 and χ < 0.10.

In Sec III, we developed analytic formulas for the electron spectrum after stochastic emissions of radiation in the WRDR. The analytic approximation is very simple when the electron bunch has a narrow primary energy divergence.

We compare the results of the analytic formulas for multiple NCS with those of Monte Carlo simulations. We employed the program used in Refs. 72 and 73 to carry out the simulations. In these simulations, emissions of radiation were included as instantaneous and local processes. Between emissions, the classical equations of motion described the laser Lorentz force and particle propagation. The laser fields of the tightly focused laser used in the simulations were those given in Ref. 74. The 0.8 μm laser lasted for τ₀ = 2λ/c = 30 fs, its peak intensity was I0=2.74(a0/λ0[μm])2×1018W/cm2=5×1020W/cm2, and the laser waist was 3.6 μm. N = 10⁵ electrons were primarily distributed uniformly in a D = 0.8 μm sphere. The electron beam and the laser pulse were perpendicular to each other, and they were both 15 μm from the focus before the simulations, which lasted for 100 fs, with a time step of 1.67 × 10⁻² fs. The electromagnetic force between the electrons was ignored, since the it was four to five orders of magnitude weaker than the laser Lorentz force. Pair production was ignored, since the its probability is strongly suppressed when χ < 1.67,68 Other SFQED effects and processes24–31 are also negligible in the WRDR, owing to their low probabilities; a detailed method to estimate the significance of other SFQED processes can be found in the supplementary information of Ref. 73. Figures 3(a)–3(c) show that the analytic formula agrees well with the simulations when χ₀ ≪ 1. Even in the χ₀ = 0.5 case [Fig. 3(d)], the agreement is not bad.

With the analytic approximations of P_i and f_i in Eqs. (10) and (14), we can give a complete picture of the electron spectrum deformation caused by quantum radiation reaction induced by NCS and QSR in the WRDR. When r is small, most electrons only emit a few times, f₀ to f₃ dominate the spectrum, and the spectral peak stays at η = 1, as shown by the simulation results in Fig. 4(a). We employed the programs used in Refs. 72 and 73 to carry out the simulations. In these simulations, NCS and pair production were included, although the latter is negligible because the probability of pair production is strongly suppressed in χ < 1 regions.67,68 The classical equations of motion described the laser Lorentz force and electron propagation between emissions. The electromagnetic forces between the charges were four to five orders of magnitude weaker than the laser Lorentz force, and therefore were also ignored. Note that i = 0 electrons, i.e., quantum quenching of radiation, were believed to suspend the peak shift,38 but actually all the electrons radiating i = 0, …, 3 times contribute.

When r has moderate values, both i = 0, …, 3 and i ≥ 4 electrons are important. The functions f_i≥4 strongly overlap with each other, and hence the i ≥ 4 electrons together form a peak, i.e., the spectral peak splits into two, as shown in Fig. 4(b). With further increase in r, the contributions of i ≤ 3 electrons and thus also the peak at η = 1 decay rapidly and eventually disappear when r ∼ 9, as shown in Fig. 4(c).

A complete peak splitting process is shown in Fig. 4(d), and the contributions from i ≤ 3 and i ≥ 4 electrons are shown separately in Figs. 4(e) and 4(f). These results clearly confirm the composition of the two split peaks: one is formed by electrons emitting zero to three times and the other by electrons emitting ≥4 times, and the origin of the quantum peak splitting is the discreteness of NCS/QSR.

The normalized spectra of emitted photons when the electron beam has propagated D = 1, 2, and 4 μm shown in Figs. 4(a)–4(c) are almost the same. This verifies the assumption we adopted to obtain the analytic formulas (10)–(14), namely, that the differential radiation probability of successive NCS is insensitive to NCS emissions in the WRDR.

This new picture of electron dynamics in strong fields corrects the conventional picture and makes it self-consistent. The conventional picture is based on two longstanding, widely accepted, but incompatible concepts about electron dynamics in strong fields. On the one hand, since the radiation decreases the electron energy, it naturally can shift the electron peak. On the other hand, quantum quenching of radiation loss can lock the electron peak at the primary energy.32,33,38 So, the conventional diagram shown in Fig. 5(a) needs a transition point. However, a peak locked by quantum quenching of radiation loss cannot shift anyway, and therefore such a transition point cannot exist. With quantum peak splitting, a new stage is inserted between the peak locking and shifting stages, as shown in Fig. 5(b), and the picture of electron dynamics in strong fields becomes self-consistent.

Such peak splitting is a pure quantum effect induced by the discreteness of quantum emissions of radiation. In classical theory,75,76 continuous and deterministic radiation reaction shifts the peak continuously and nonlinearly rather than splitting it, as shown in Fig. 4(d).

A feature of quantum peak splitting is that the derivative of the spectrum,∂ηF(η)≈−χ∑i=1∞ri[(i/3−1)ξi/3−2−ξi/3−1]i!Γ(i/3)e−re−ξ,(15)has a zero point between η = 0 and η = 1, where ξ = (1 − η)/χ. Since the this is a series in η whose coefficients are determined by r, the threshold for quantum peak splitting is determined by the value of r. We obtain r = 5.35 by solving the equation numerically. In our simulations, the splitting of the peak is slightly earlier: it usually happens around r = 5.1. Hence, the average number of emissions r controls the quantum peak splitting process.

Then what is the interval between split peaks? The P_i distribution is approximately symmetric around i = [r] when r is near [r] + 1/2. Hence, the peak formed by i ≥ 4 electrons is located near η = 1 − (χ/3)(r − 3.5), and the interval between the split peaks isΔηpeak≈χ3(r−3.5).(16)The splitting interval is compared with simulations in Fig. 4(d): it grows linearly with r as long as Δη_peak is not too large, i.e., ≤0.5. The initial position of the i ≥ 4 peak when it splits from the i = 0, …, 3 peak is then at η ≈ 1 − χ/2. An additional condition for quantum peak splitting is that this splitting should not be flooded by the primary beam energy spread Δη₀, i.e., Δη₀ < Δη_peak.

In colliders, quantum radiation reaction can also split the peaks of the luminosity spectrum, i.e., the center-of-mass energy distribution at the interaction point, and this is due to quantum beamstrahlung. In most future e⁺e⁻ collider designs, the two bunches are symmetric, and the beamstrahlung parameter χe−=χe+=χ. When χ ≪ 1, the beamstrahlung energy loss ΔE is small compared with the primary energy E₀, and then ηcm≡s/2E0≈1−(ΔEe−+ΔEe+)/2E0. In this case, the luminosity spectrumF(ηcm)≈∑i,i′=0∞Pi(nγ,e−)Pi′(nγ,e+)×∫ηcm1dηcm′fi,χ/2A(ηcm′)fi′,χ/2A(1−ηcm′+ηcm),(17)where nγ,e−/e+ is the average emission number of an e⁻/e⁺ bunch. Consideringfi,χA(η)≈∫η1dη′fi′,χA(η)fi−i′,χA(1+η−η′)(18)andPk(nγ,e−+nγ,e+)=∑k′=0kPk′(nγ,e−)Pk−k′(nγ,e+),(19)the luminosity spectrum has a simple approximationF(ηcm)≈∑k=0∞Pk(2nγ)fi,χ/2A(ηcm).(20)Hence, when nγ,e−+nγ,e+ fall between 5.1 and 9, i.e., nγ=nγ,e−=nγ,e+ fall between 2.55 and 4.5, the peak in the luminosity spectrum will split.

If this happened in a collider, its application and performance would be limited by such a double-peaked structure of the luminosity spectrum. This is especially severe for experiments based on measurements of invariant mass recoiling against observable particles,55–58 which have the prominent advantage of being independent of specific decay modes when probing particles that are hard to detect directly in processes such as Higgstrahlung e⁺e⁻ → Zh → μ⁺μ⁻h and processes producing dark matter e+e−→D̄D+X, where X can be a γ, a photon, a single jet, a Z boson, etc. In these experiments, peaks in the spectrum of invariant mass recoiling against observable particles are the main signals, and thus split peaks in the luminosity spectrum may introduce fake signals.

The conventional constraint on beamstrahlung in colliders is that the average energy loss should be within 10%–20%. Since the average energy loss per emission has an upper limit ∼25% even in χ ≫ 1 regions, this constraint is also usually expressed as the requirement that n_γ should not go far beyond 1, i.e., n_γ ≲ 1. With increasing demands on energy and luminosity, n_γ has increased rapidly, and has already reached 2.232–35 in collider designs. Quantum peak splitting brings in a new constraint of n_γ < 2.55.

Quantum peak splitting also provides a new mechanism for peak formation. New peaks are usually taken as strong evidence for new acceleration mechanisms46–53 or the decay of unknown particles.54 Quantum peak splitting provides a new possibility, especially in a strong-field background such as in ultra-strong laser experiments.

The above investigations have been limited to the WRDR where χ ≪ 1. A natural question is whether quantum peak splitting occurs when χ ≲ 1 or ≫1. Although a thorough understanding of this needs further careful research, we expect that quantum peak splitting will also exist in χ ≲ 1 regions. When χ ≲ 1, a large portion of electrons will lose a large ratio of their energies in a few emissions. Therefore, the split peaks will be located at η = 1 and η ∼ 0, the threshold for quantum peak splitting will be different from r = 5.1, and therefore the distance between split peaks can no longer be described by Eq. (16). When χ ≫ 1, pair production will become significant, and therefore the analytic description of the electron spectrum needs to incorporate it.

In conclusion, inspired by the theory of multiple bremsstrahlung, analytic formulas to describe the effects of multiple NCS and QSR on the electron spectrum have been developed. Using these formulas, the electron spectrum deformations induced by quantum radiation reaction in the weak radiation-dominated regime have been investigated, and a new pure quantum effect of multiple NCS/QSR called quantum peak splitting has been found. Most electron beams produced by accelerators have a single peak in the spectrum, but after propagating in ultra-strong fields, this peak can split owing to the quantum nature of the emissions (NCS/QSR). Electrons that radiate i = 0, …, 3 times form one peak, and electrons that radiate i ≥ 4 times form the other. The condition for quantum peak splitting in the WRDR is that electrons radiate r ∼ 5.1–9 times on average. As well as being a new phenomenon, quantum peak splitting also has an impact in three ways. First, newly formed peaks in charged particle spectra are usually attributed to particle acceleration or decay, and quantum peak splitting provides a third mechanism. Second, quantum peak splitting makes the picture of quantum radiation reaction self-consistent, since the it introduces a new splitting stage. Third, quantum peak splitting imposes a new constraint on future colliders that the electrons/positrons should radiate less than 2.55 times on average, otherwise the luminosity spectrum would have two peaks.

Acknowledgment. The authors would like to thank Bin Zhu, Dr. Gang Li, and Dr. Professor Wei Hong for fruitful discussions. This work is supported by the National Key Program for Science and Technology Research and Development (Grant No. 2018YFA0404804), the Science and Technology on Plasma Physics Laboratory, Laser Fusion Research Center Funds for Young Talents (Grant No. RCFPD2-2018-4), and the National Natural Science Foundation of China (Grant No. 11805181).