Quasiclassical approach to synergic synchrotron-Cherenkov radiation in polarized vacuum

The photon emission by an ultrarelativistic charged particle in extremely strong magnetic field is analyzed, with vacuum polarization and photon recoil taken into account. The vacuum polarization is treated phenomenologically via refractive index. The photon emission occurs in the synergic (cooperative) synchrotron-Cherenkov process [J. Schwinger, W. Tsai and T. Erber, Annals of Physics, 96 303 (1976)] which is similar to the synchrotron emission rather than to the Cherenkov one. For electrons, the effect of the vacuum polarization on the emission spectrum is not evident even beyond the probable onset of non-perturbative quantum electrodynamics (QED). However, the effect of the vacuum polarization on the emission spectrum can be observable for muons already at $\gamma B / B_S \approx 30$, with $\gamma$ the muon Lorentz factor, $B$ the magnetic field strength and $B_S$ the critical QED field. Nevertheless, vacuum polarization leads to only 10% enhancement of the maximum of the radiation spectrum.


Introduction
Quantum electrodynamics (QED) predicts nonlinear dielectric properties of the vacuum in strong magnetic field caused by virtual electron-positron pairs. The Kramers-Kronig relations connect vacuum refractive index with pair photoproduction probability, and the latter have been studied in strong crystalline fields [1] and in laser field [2,3]. Still direct experimental evidence of vacuum refractive index is absent, and many set-ups have been proposed to detect and measure it, e.g. x-ray diffraction on a double-slit formed by two counterpropagating intense laser pulses [4,5,6], or changes in polarization of x or gamma photons due to vacuum birefrigence in strong laser field [7,8,9,10]. The idea behind these proposals is not only to measure vacuum refractive index but to test QED in a not-yet-investigated region of extreme laser fields. Moreover, investigation of vacuum polarization becomes important in the light of Ritus-Narozhny conjecture of perturbative QED breakdown at certain conditions [11,12,13,14,15].
The fields of intensity 10 23 −10 24 W cm −2 is expected in near future thanks to facilities such as ELI-NP [16], ELI-beamlines [17], Apollon [18], Vulcan 2020 [19], XCELS [20] and others. Therefore, the field of the order of 10 −3 × B S will be available which results the vacuum refractive index n such that δn = n − 1 ∼ 10 −10 for photons with energy 1 GeV [21], with B S = m 2 c 3 /eh the Sauter-Schwinger critical QED field [22], m and e > 0 the electron mass and charge magnitude, respectively, c the speed of light andh the reduced Plank constant. Despite such small value of δn, the Lorentz factor γ ∼ 10 5 , available for electrons nowadays, is enough to reach the speed of a charged particle greater than the phase speed of the photons, hence the Cherenkov emission may occur. Such estimates drives the recent interest to Cherenkov emission in the polarized vacuum [23,24,25]. However, the results of these papers should be reconsidered because of simplified approach used there. A charged particle in the strong field inevitably moves along a curved trajectory that prevents plain Cherenkov radiation. The trajectory curvature determines the radiation formation length and is crucial for the emission process. Furthermore, there are unified emission process [26], and it is not possible to distinguish "Cherenkov" and "Compton" emission mechanisms in the considered situation, as Refs. [23,24] does. Ref. [25], although considers Cherenkov emission and nonlinear Compton scattering as a single process, uses expression for the emitted energy and for the formation length [equations (16) and (17) wherein] as if the particle moves along the straight line and emits photons in a plain Cherenkov process. At the same time, earlier works on the considered topic contain not only qualitative estimates, but expressions for the spectrum and for the photon emission probability.
In 1966 Erber was the first who pointed out the possibility of Cherenkov radiation in polarized vacuum [27]. He used expression for pair photoproduction in a constant magnetic field and dispersion relation to compute the real part of the refractive index, following work of Toll in 1952, see reference [28] and references therein. In 1969 Ritus considered possibility of Cherenkov radiation in a constant crossed electromagnetic field [29], using photon Green's function obtained year before by Narozhny. Thus, there is no need in two laser pulses which create regions with pure magnetic field, and it is enough to use single laser pulse to induce vacuum polarization. Then, in 1976 Schwinger, Tsai and Erber with QED mass operator method obtained the general expression for the spectrum of the photon emission by a charged particle which moves both in a constant magnetic field and in a medium with n = 1 [26]. They pointed out that "there is actually only a single emission act, synergic synchrotron-erenkov radiation, for which a correspondence with either erenkov emission or synchrotron radiation can be established only in the respective limits of vanishing field or matter density", and that "the practical import of this synergism is that the radiation depends sensitively on both positive and negative values of n − 1". They demonstrated [26,30] that depending on parameters, both amplification and suppression (quenching) of the photon emission may occur. Finally, synergic synchrotron-Cherenkov radiation in gases was observed in the experiment [31], which results agree well with the analysis which treats Cherenkov and synchrotron radiation as limiting manifestations of a unified process.
Another interesting result of Erber et al. is that the Cherenkov condition for electrons v > c/n (with v the electron velocity) is not enough for spectrum of the synchrotron-Cherenkov radiation in polarized vacuum to be different from purely synchrotron spectrum. The sufficient condition for this occurs extremely strict {see equations (8.8e) and (8.11) in Ref. [30]}: which for B/B S ∼ 10 −3 yields enormous energy mc 2 γ ∼ 100 TeV. Here B is the magnetic field strength, γ the electron Lorentz factor, and the electron velocity v is assumed to be perpendicular to the magnetic field. Erber then suppose that the condition (1) indicates that higher order QED corrections besides vacuum polarization should be also taken into account, i.e. QED is no longer a perturbative theory. Indeed the threshold χ value (1) is even far beyond the conjectured value of the perturbative QED breakdown [11,29] The aim of the current paper is manifold. First, the physical picture of photon emission by ultrarelativistic particles is recalled and applied to the synchrotron emission in a medium with δn 1, within the classical theory (section 2). A special attention is paid to the synchronism between the emitting particle and the emitted wave. Second, the general quantum formulas for spectral and angular distribution of the emitted photons in synergic synchrotron-Cherenkov process are obtained (section 3), for that quasiclassical theory of Baier and Katkov [32] is used. This allows to take into account photon recoil neglected in Refs. [26,30]. Third, in section 4 the onset of Cherenkov corrections to the synchrotron spectrum is found. Following the proposal of Erber [30], synchrotron-Cherenkov emission by particles heavier than electrons is considered in details in section 4.2. It is shown that the onset of Cherenkov corrections to the synchrotron spectrum for muons occurs at much lower value of χ than that for the electrons, due to enlarged formation length and weakened photon recoil. Section 5 is the conclusion.

Photon emission by ultrarelativistic particle in classical theory
Calculation of the radiation of ultrarelativistic charged particle for n = 1 can be found in many textbooks, e.g. in [33]. However, these calculations are often difficult to tailor to the case n = 1. Here the general formulas for angular and spectral distributions of the emitted energy are recalled and applied to the synchrotron radiation. Despite n = 1 is used in this section, the approach used here allows obvious generalization to the case n = 1, if δn 1.

General formulas
For the sake of simplicity one can consider emission of electromagnetic waves by a current density j inside a virtual superconductive rectangular box (resonator or cavity) of size L x ×L y ×L z . The emitted field can be decomposed by complex resonator modes with well-known sine-cosine spatial and exp(−iω s t) temporal structure: where s is the generalized mode number and ω s is the mode cyclic frequency, E and B are the electric and magnetic field, respectively. The modes can be chosen orthogonal, with the following normalization: where the symbol * means complex conjecture. Hence the energy of the emitted field is and |C s | 2 can be interpreted as the emission probability of the photon of mode s. To find C s , one can start from Maxwell's equations: with ρ and j are the charge and the current density, respectively. Let the current j emits during t ∈ (t 1 , t 2 ), and j = 0, ρ = 0 for t < t 1 and t > t 2 . Thus, the decomposition (2) is valid for t > t 2 . One can multiply Eq. (5) on B * s , and subtract it from Eq. (6) multiplied on E * s . Then the result can be integrated over the space and time that yields with V a volume of the virtual box and S its boundary. The cavity can be chosen big enough such that E = B = 0 at the boundary, in this case the right-hand side of equation (8) is zero. Furthermore, E = B = 0 at t = t 1 , hence from equation (8), taking into account equations (2) and (3), one gets Equation (9) has clear physical meaning. Being multiplied byhω s C * s , it expresses the equality between the energy emitted into the mode s, and work of the current j over the one-half field of the emitted mode. This work peaks if there is a synchronism between the current and the field of the mode. Note also that formula (9) is similar to one for the amplitude of an oscillator driven by an external force.
For an ultrarelativistic electron, which emits mostly in the forward direction, the computation of C s can be further simplified. First, the current of the electron is with r(t) the electron position. Second, each of the complex modes is formed by eight complex plane waves ∝ exp(−iω s t + ik s r) (except a few modes with wave vector parallel to the box boundaries). This yields eight terms in the integral over t in equation (9). It can be noted that one of the terms oscillates much slower than the others which hence can be dropped {e.g., if k x ≈ ω s /c and x(t) ≈ ct, then exp[iωt − ik x x(t)] cannot be dropped, whereas exp[iωt + ik x x(t)] can be}. Let the remaining term corresponds to a wave with the polarization direction e s (with e 2 s = 1). The amplitude of this remaining wave, a s , can be found from the normalization (3): the wave energy ishω s /8 hence a s = (2πhω s /V ) 1/2 . Therefore, An ultrarelativistic particle emits photons in a narrow cone around the direction of the particle velocity. Thus the energy radiated in a certain direction can be readily computed from the energy of the modes. The density of the modes which has a planewave component in some certain unit solid angle and unit frequency interval can be found from the boundary conditions for the virtual superconducting box. From this, the full emitted energy can be expressed using the energy radiated per unit frequency interval and per unit solid angle: with e i (i = 1, 2) the polarization directions. Equation (12) is very useful in estimating the radiation timescales and the radiation formation length, that discussed for the sinchrotron radiation in the next section, and for the synchrotron-Cherenkov radiation in section 3.2.

Synchrotron emission and the timescales
The key feature of the photon emission by an ultrarelativistic particle is the synchronism between the particle and the emitted wave, as seen from equations (9), (11) and (12). The phase of the exponent in these equations in the case n = 1 (hence k = ω/c), varies slowly in vicinity of the point where the angle between v and k is minimal. For the sake of simplicity we assume that there is only one such point, and it is in the origin of the local coordinates (figure 1), and the particle is in the origin at t = 0. In the synchrotron approximation, or local-constant-field approximation, the particle trajectory is described locally like a circular orbit fully determined by the local curvature radius R and the Lorentz factor γ: x z e₁ e₂ θ k x y e₁ e₂ k R Figure 1. Local coordinates used in the computations. For a given wave number k the origin is the point on the electron trajectory (thick blue line) where k is perpendicular to the normal vector of the trajectory. Thus, the x axis is tangent to the trajectory, the y axis is parallel to the normal vector (hence the xy plane is the osculating plane), and the z axis is chosen by the right-hand rule. The polarization vector e 1 is chosen to be on the y axis, and e 2 to be perpendicular to e 1 and k.
Then, the pre-exponential functions in the integrand of equation (12) can be written as follows: where the phase φ contains only linear and cubic terms: Here τ and τ ⊥ are the timescales of dephasing between the electron and the emitted wave caused by the longitudinal (along the x axis) and transverse (along the y axis) electron motion, respectively: where the effective magnetic field strength B is introduced for convenience such that v = ω B R/γ, and is the cyclotron frequency in this field. Note that τ depends on v (hence on γ) and does not depend on R, whereas τ ⊥ depends on R ∝ γ/ω B and does not depend separately on γ.
Well-known equations (14.78) and (14.83) from the textbook [33] which describe angular and spectral distribution of the synchrotron photons can be easily got from equations (12) and (15)-(18). The key feature of the synchrotron spectrum is the critical frequency [33] The energy emitted per unit frequency interval per unit solid angle, d 2 I/dωdΩ, has maximum at θ = 0 and ω ≈ 0.42 × ω c . If ω ω c or θ 1/γ, the emitted energy sharply tends to zero, that can be explained with τ and τ ⊥ . The critical frequency corresponds to τ ⊥ /τ = 3/(4π) 2/3 ≈ 0.56. If ω increases beyond ω c , or if ω ∼ ω c and θ increases beyond 1/γ, then τ becomes smaller than τ ⊥ . Hence, the exponent (17)-(18) oscillates strongly hence the synchrotron integrals tend to zero. As shown in the next sections, the presence of the refractive index, spin contribution and photon recoil change the basic equation for the photon emission probability. However, the emission probability is still governed by the synchronism between the emitted wave and the electron, hence, by τ and τ ⊥ , though equations for them should be corrected.

Quasiclassical theory of the synchrotron-Cherenkov radiation
In order to take into account the refractive index n = 1+δn (which is assumed close to unity, |δn| 1) in the classical formula (12), one should not change anything, except the relation between the photon frequency and the wave vector in the phase, The mode structure, the energy of the modes and their normalization can be taken unchanged in the case |δn| 1. This situation replicates in QED. If one follows the Baier-Katkov quasiclassical derivation of the spectral and angular distribution of the synchrotron photons [32], he/she finds that the presence of the refractive index changes nothing in it except the phase in the exponential. However, to isolate the phase one should reorganize the quasiclassical formula [32] {see also, for example, equation (6) in the supplementary material of Ref. [34]}: One can note that d dt Hence, integrating by parts one gets where the product n × [n × β] is rewritten using βe 1 and βe 2 . Now, to take into account the refractive index n, one should set |n|≡ |ck/ω|= n in the exponential functions in equation (30). Equation (30) differs from the classical one (12) by two quantum features. First, an additional "spin" term appears in (30) {the last term, which originates from the spin flips [35,36]}. Second, the radiation recoil arises, which is reflected, first of all, in the fact that ω is substituted with a higher frequency ω in the exponential. Hence, if the photon energy is about the electron energy, the synchronism is strongly affected by the recoil. Particularly, the recoil effect squeezes the photon spectrum such that it is limited by the energy of the electron.
All the terms in equation (30), including the spin term, contain the same phase This means that, as for the classical synchrotron emission, the spectrum of the synchrotron-Cherenkov emission at the given frequency ω is governed by the only two timescales which though differs from the timescales of the classical synchrotron spectrum given by equations (19) and (20). The refractive index brings a novel effect: the sign of the linear term can be changed, i.e. if Cherenkov condition is fulfilled, βn > 1, then at least for θ ≈ 0 one has ς = −1. In the subsequent sections 3.2 and 4 it is demonstrated that in the case of ς < 0 the emission spectrum can differ dramatically from the synchrotron one. In the remaining part of this section the effect of the radiation recoil is considered in detail, because of its importance both for the case ς = +1 and ς = −1.
Similarly to the classical synchrotron emission, the critical frequency ω c can be introduced such that for it τ ⊥ and τ are of the same order for θ = 0, namely for ω = ω c one has τ ⊥ /τ = 3/(4π) 2/3 ≈ 0.56.
If the quantum parameter is small, χ = γB/B S 1, quantum formulas tend to classical ones, i.e. radiation recoil is negligible, ω c ≈ ω c ≈ 3ω B γ 2 , and the spin term is negligible. In this case (if additionally δn = 0), the maximum of d 2 I/dωdΩ is in the point θ = 0 and ω ≈ 0.42ω c that reveals the physical meaning of ω c in this case.
In the quantum limit, χ 1 (and for δn = 0), equation (36) yieldshω c ε, that looks non-physical if one neglects the effect of radiation recoil and sets ω c = ω c . Actually, the radiation recoil changes significantly the critical frequency, and ω c differs significantly from ω c :hω c ≈ ε[1−1/(3χ)]. Thus ω c is very close to the upper spectrum bound ε/h. Therefore, in the quantum limit almost for all frequencies one has τ ⊥ τ , and the term t/τ can be neglected in the phase of the exponential.
For the refractive index of the polarized vacuum the Cherenkov condition is fulfilled only in the quantum case, i.e. nβ > 1 can be reached only if χ 1 (this is discussed in section 1 and especially in section 4). As seen from equations (33) and (34), the refractive index affects τ only, hence only the linear term in the phase. However, in the quantum case this term is negligible for almost all frequencies in the synchrotron spectrum. Therefore, in order to modify the synchrotron spectrum, the refractive index should be large enough not only to change τ , but make it much lower than in the case of δn = 0. That is why the Cherenkov condition is far from being enough to change the synchrotron spectrum.

Radiation formation length
The radiation formation length is the length of the electron path which contributes most to the integrals in equations (12) and (30). Obviously, the radiation formation length depends on the frequency of the emitted wave, although for synchrotron emission often ω ∼ ω c is assumed.
The radiation formation time (i.e. the radiation formation length divided by c) can be estimated by consideration of the following integral: with f (t) and φ(t) slowly varying functions and sin[φ(t)] contains many oscillation periods on the interval [t a , t b ]. The contribution of a single oscillation period can be estimated as follows: Here t 0 , t 1 , t 2 are time instants which correspond to φ = 0, π, 2π, respectively. Continuous function T (t) is approximately equal to the local period of function sin[φ(t)]. Obviously, the estimate (38) for I(t a , t b ) can be extended to an arbitrary integer number of periods between t a and t b . In this case the integral can be estimated as the difference between the integral of the first "bump" (the first half-period) and the last one, whereas the intermediate "bumps" do not contribute. Finally, the integral which determines the convergence speed of I(−∞, ∞) can be estimated as follows: where we assume that lim t→∞ f T = 0. Function cos[φ(t )] for the phase given by equation (18), namely φ(t ) = 2π(at + t 3 ) with t = t/τ ⊥ and a = ςτ ⊥ /τ . Proper values of a are given in the lower-right corners.
The local oscillation period for the phase (31) far from the saddle points is The integrals in equation (30) contain f (t) = 1 and f (t) = t, both lead to the same radiation formation time, thus for simplicity we set f (t) = t from here on. For synchrotron emission of low frequencies (τ ⊥ τ , ς = +1) the leading "bump" of the integrand has width about τ ⊥ [see figure 2(a)], and for t τ ⊥ one gets I(t, ∞) ∝ 1/t. Hence the radiation formation time is t rf ∼ τ ⊥ in this case. For high frequencies [τ ⊥ τ , ς = +1, see figure 2(b)] the integrand contribution decays, a point where the linear and the cubic terms in the phase yield the same oscillation periods. Thus, here the radiation formation time is t rf = t s τ ⊥ τ . A special consideration needed for the Cherenkov branch of the synchrotron-Cherenkov emission (ς = −1). If τ ⊥ τ , the sign of the linear term is unimportant and t rf ∼ τ ⊥ . However, in the case τ ⊥ τ the integrand changes significantly [see figures 2(c) and (d)]: most contribution to the synchrotron-Cherenkov integrals comes from the regions around two saddle points t = ±t s . If τ τ ⊥ , the phase near a saddle point (say, t = +t s ) can be approximated with a parabolic dependency: where the term 2π(t − t s ) 3 /τ 3 ⊥ is neglected at the right-hand-side. Then the width of the bump at t s can be found, Hence, the cubic term is actually small 1. The parabolic phase dependency leads to quite fast convergence of the integrals, e.g.
|t − t s | t s . An important consecuence of the estimations above is that τ T (t s ) τ ⊥ t s in the case ς = −1 and τ τ ⊥ . This means that the saddle points t = ±t s are far away from each other, and there is almost random phase shift between the integrals around these points. Thus, instead of coherent sum of the integrals one should sum resulting probabilities which are computed separately for t = +t s and t = −t s points. This also means that the radiation formation length in this case should be estimated not as the distance between these points, but as the width of the leading bumps, Summing up the above, the radiation formation length for synchrotron-Cherenkov radiation is for τ ⊥ τ , Besides the radiation formation length, the synchrotron-Cherenkov integrals depend on the ratio τ ⊥ /τ itself. For instance, one can note that the integrals decay exponentially with the increase of τ ⊥ /τ , if τ ⊥ τ and ς = +1 [see figure 2(b)]. The emission probability becomes negligible in this case. Taking this into account, one notes from equation (44) that in all the noticeable cases the radiation formation length is less or about τ ⊥ , which do not depend on the refractive index. Note also that in the regime of seemingly dominance of Cherenkov radiation, δn 1/γ 2 , one has ς = −1 and τ τ ⊥ [see figure 2(d)], and the radiation formation time is small, t rf τ ⊥ . This differs significantly from the case of plain Cherenkov radiation, where the radiation formation length can be extremely large.
The presence of the refractive index with δn = 0 can however significantly influence the emission probability. The effect of the refractive index in the case ς = +1 is shown in figure 3, where I 0 , I + and I − are the emitted energy for n = 1, n = 1 + 0.1/γ 2 and n = 1 − 0.1/γ 2 , respectively.  [32,36].
In the high-frequency region, ω ω c , the timescales relates as τ ⊥ τ in the absence of δn. Thus, as the presence of the refractive index with δn > 0 leads to the increase of τ , it also leads to the substantial increase of the emission probability [compare figure 2(b) and figure 2(a)]. In the opposite case of refractive index with δn < 0, the timescale τ decreases that quenches the emission probability. The described picture is confirmed well by figure 3, where the critical frequency is hω c = 0.9 × ε.
In the low-frequency region (ω ω c hence τ ⊥ τ ) the sinchrotron integrals do not depend on τ which the only depend on the refractive index. Therefore, the presence of the refractive index with δn = 0 does not change the low-frequency   spectrum, as seen in figure 3. If one increases the parameter χ, the critical frequency tends to ε/h, and the region where the effect of the refractive index is noticeable becomes narrow and pinned to ε/h. The effect of the refractive index becomes more dramatic if the Cherenkov condition is fulfilled, δn > 1/(2γ 2 ). Such case for δn = 1/γ 2 is shown in figure 4 for γ = 1 × 10 5 and b = 3 × 10 −5 (hence χ = 3 andhω c = 0.9 × ε). For the chosen refractive index and θ = 0, τ remains the same as for the case δn = 0, but the sign of the linear term in the phase (31) changes, ς = −1. As before, the low-frequency part of the spectrum remains the same in the both cases, δn = 0 [lower half of figure 4(a) and solid red curve in figure 4(c)] and δn > 0 [upper half of figure 4(a) and dashed blue curve in figure 4(c)]. However, in the case δn > 0 the high-frequency part of the spectrum is extremely enhanced, such that the overall emitted energy is more than an order of magnitude greater than in the case δn = 0. Points A, B, C and D in figures 4(a, b) exactly correspond to a ≡ ςτ ⊥ /τ used in figures 2 (a), (b), (c) and (d), respectively. The fine structure in the radiation distribution at high frequencies is seen in figure 4(b), which shows in details the rectangular region of figure 4(a) marked with black solid line (note the different color scale in these figures). This fine structure emerges due to the interference of the contributions yielded by the two bumps seen in figure 2(d). Figure 4 looks quite encouraging, however, the refractive index of the polarized vacuum depend on the photon frequency, and have both δn > 0 and δn < 0 parts. The latter corresponds to high photon energies. This, together with the fact that the high-frequency region (where refractive index influence the radiation spectrum) becomes extremely narrow in the case χ 1, makes almost impossible to reveal the effect of vacuum polarization on the synchrotron radiation, at least for electrons (or positrons). The radiation spectrum for electrons is discussed in details in section 4, whereas the next section, section 3.3, is devoted to details of numerical computation of the spectrum.

Numerical implementation
Similarly to the transition from equation (11) to equation (12), one can find from d 2 I/dωdΩ [equation (30)] the photon emission probability summed up for both polarizations, W = e |C e | 2 , which is more convenient for numerical simulations: (or two integration intervals in the case ς = −1, τ ⊥ τ , see previous subsection). As it is shown above, numerical error caused by finitness of the integration interval decreases quite slowly, I(t b , ∞) ∝ 1/t b . Hence, to obtain proper accuracy one should choose t b much greater than t rf , say t b ≈ 100 t rf for accuracy of about 1%. At the same time the oscillation period hence the timestep sharply decreases with time, Thus the resulting number of nodes (time steps) yielding proper accuracy becomes extremely large. However, the computation of the integrals can be performed on a much smaller interval, if the artificial attenuation g is added: Here t b should be just several times larger than t rf (in the code t b ≈ 3t rf is chosen), and the function g should fade smoothly near the boundaries of the integration interval from 1 to 0. Equation (46) can be easily proven by estimating the difference of its left-hand-side and its right-hand-side with formula (39): (1 − g)f T ≈ 0 at the point where g(t) just starts to fade as well as at t b . In the code the following attenuation function is chosen: which together with the given timestep and the integration interval yields error less than 3% in comparison with the integrals without g computed on a extremely wide integration interval by a different numerical method, at least for ςτ /τ ⊥ ∈ [−48, 0.8].
In the code the integration method described above is used in the function which computes photon emission probability with equation (45). A number of tests is implemented for this function. For instance, in the classical limit (χ 1) energy radiated per unit frequency interval per unit solid angle, d 2 I/dωdΩ, computed numerically, is compared with analytical results, namely with equation (14.83) from [33]. This test shows accuracy of the code better than 0.5% for |θ|≤ 1/γ and ω ≤ 1.6×ω c . At this point the value of d 2 I/dωdΩ is already more than 500 times lower than the maximal value of d 2 I/dωdΩ, thus although the error becomes greater with the increase of ω and θ, the whole value of d 2 I/dωdΩ can be neglected there. Furthermore, a well-known asymptotic behavior of the full radiated energy is also tested {namely cI/2πR ≈ (1 − 55 √ 3χ/16 + 48χ 2 )P cl at χ 1 with P cl = 2e 4 B 2 γ 2 /3m 2 c 2 and cI/2πR ≈ 0.37 × e 2 m 2 c 4 χ 2/3 /h 2 at χ 1, see [32,36]}. The radiation spectrum calculated numerically and analytically for n = 1 and χ = 3 is shown in figure 3(d), where the result of jE code is shown with solid red line and the analytical result with dotted black line. A number of tests also is written in order to demonstrate that the mass of the emitting particle, the spin term and the refractive index are treated correctly [37].

Synchrotron-Cherenkov radiation of electrons
One can ask for the conditions necessary to modify the well known synchrotron spectrum because of the vacuum polarization. To answer, one first needs to discuss an expression for the vacuum index of refraction in a strong field. The refractive index depends on the photon polarization. For example, in a constant magnetic field δn for low-energy photons is about twice greater for the polarization perpendicular to the magnetic field, in comparison with the polarization parallel to the magnetic field. However, the most of the synchrotron photons are polarized perpendicularly to the magnetic field, and the following expression for the real part of the vacuum refractive index can be used [see [27,29,21] and references therein]: with N (κ) is presented in figure 5(a), κ = (hω/mc 2 )(B/B S ) is the photon analogue of the χ parameter, and B the (effective) magnetic field. The asymptotics of N (κ) are given by: where κ = (hω/mc 2 )(B/B S ). As seen from figure 5, δn = n − 1 is positive for κ 15 and negative for κ 15.
As described in the previous sections, the refractive index influences only the timescale τ in which the electron becomes out of phase with the wave due to the difference of its velocity along the wave vector k and the wave phase velocity. Hence the vacuum polarization influences only the linear term in the phase (31). At the same time the linear term of the phase influences the integrals in equation (30) only if the timescale τ is less or about τ ⊥ . In the timescale τ ⊥ the electron becomes out of the phase with the wave due to the trajectory curvature (because the curvature affects the electron velocity along the wave vector). Summing up, and taking into account equations (33), (34) and (35), the necessary condition of the spectrum change at a given photon frequency ω is the following: If condition (50) holds, then τ changes noticeably, and if (51) is holds too (with ω c computed either with or without vacuum polarization taken into account), then the spectrum changes. Here ω = ωε/(ε −hω), and ω c is determined by equation (36) for a given frequency ω (note that δn depends on ω). It should be noted that the conditions (50) and (51) are very weak: if δn is, say, 10% of 1/γ 2 it nevertheless can lead to sizable changes in the spectrum. Also, if ω is 10% of ω c (ω), the spectrum changes noticeably. For instance, for χ = 3 and δn = 0.1/γ 2 , one has ω ≈ 0.1 × ω c (ω) forhω ≈ 0.5ε, however, the changes in the spectrum are evident for this frequency, as seen in figure 3(d).
For low-energy photons (κ 1) equation (50) can be rewritten using the electron χ parameter only: However, condition (51) in this case much harder to fulfill: it also can be written in terms of χ and yields for κ = 1 hence χ 4 × 10 5 . One can note that the result (53) is far beyond the conjectured threshold of the perturbative QED breakdown [11], αχ 2/3 1. Therefore, the BKS approach used here and the expression for the refractive index (49) are hardly valid if αχ 2/3 1 and even more so for αχ 2/3 42. Thus the considered theory predicts no change in the synchrotron spectrum in the region of the perturbative QED.
For high-energy photons (κ 1) equation (50) yields χ 80 κ 2/3 , and equation (51) yields αχ 2/3 20 κ(χ − κ)/χ. To fit the latter to the region of perturbative QED applicability, one can try χ − κ χ, however, the former condition in this case yields χ 1/3 80 hence χ 5 × 10 5 which is again beyond the region of perturbative QED. Therefore, the evidence of vacuum polarization in synchrotron spectrum for high-energy photons is also unreachable. The estimates above are in agreement with the results of reference [30] [see equations (8.8e) and (8.11) therein], which, however, do not take radiation recoil into account and do not discuss the region of the perturbative QED applicability.

Synchrotron-Cherenkov radiation of muons
The effect of the vacuum polarization on the synchrotron spectrum can be enhanced if heavy charged particles are used instead of electrons. For definiteness, and because of the recent progress in their acceleration technique [38], muons are considered here. The advantage of using muons is a two-fold. First, their big mass, m µ ≈ 207 m, yields much greater curvature radius hence much greater timescale τ ⊥ than that for the electrons. For a given photon frequency this makes synchrotron spectrum much more sensible to the longitudinal synchronism between the particle and the emitted wave, i.e. to τ which, opposite to τ ⊥ , depends on the refractive index. Second, high mass makes the critical frequency significantly lower, hence a more sizable part of the spectrum lies in the low-frequency region κ 1, in which δn for the vacuum refractive index is maximal.
The classical critical frequency for muons is that gives the ratio of the photon energy to the muon energy:hω c,µ /ε µ = 3χ(m/m µ ) 2 (with χ = γB/B S the same as for electrons). Therefore, this ratio is small, hω c,µ /ε µ 1, up to χ ∼ 10 4 , and it is reasonable to neglect the radiation recoil for muons. In this case the conditions sufficient for the synchrotron spectrum to be noticeably modified due to vacuum polarization are the following: Obviously, these conditions can be derived similarly to equations (50) and (51). As a starting point, one can consider ω ∼ ω c that ensures fulfillment of condition (56). By virtue of equation (33), the difference between τ with δn = 0 taken into account, and τ with δn = 0 (τ ,0 ), is determined by 2γ 2 δn, if this quantity is small: where θ = 0 is assumed. One can note that this quantity depends on χ and κ only [see equation (48)]. For ω = ω c /3 (which corresponds to the maximum of the spectrum better than ω c itself) the parameters χ and κ become related, χ 2 = κm µ /m. Thus 2γ 2 δn can be expressed in terms of κ only, and 2γ 2 δn as function of κ is plotted as dashed green line in figure 5(a). It is seen from figure 5(a) that the vacuum polarization causes maximal change in τ of about 10% for κ ≈ 2 (χ ≈ 20). Further increase of the parameter χ (hence κ) leads to the decrease of the change of τ at this photon frequency.
The estimates above demonstrate that relatively small value of χ is enough to see the change in the synchrotron spectrum caused by the vacuum polarization, that agrees well with the numerical results. Figures 5(b, c) demonstrate the radiation spectrum for χ = 30 and χ = 60, respectively, computed with jE code (value B/B S = 0.01 is used, hovewer, the shapes of the resulting spectra almost do not depend on it). At χ = 30 the spectrum maximum becomes 10% higher thanks to the vacuum polarization, and at higher χ changes in the spectrum occurs at frequencies lower than the frequency of the spectrum maximum. At the same time, for photon energies corresponding to κ 15, the change in the refractive index becomes negative that causes slight quenching in the synchrotron spectrum.
Returning to the conditions (55) and (56), one can consider χ 70 that ensures the fulfillment of the first of them. Then, similarly to the previous section, the lowenergy part of the spectrum (κ 1) can be considered. In this case condition (56) yields [compare this with equation (53)]. Therefore, the change in the low-energy part of the synchrotron spectrum can be pronounced only near the conjectured breakdown threshold of the perturbative QED. Figure 6 demonstrates the low-energy part of the radiation spectrum of muons for B/B S = 0.01 (note that the subplots almost do not depend on this value). In figure 6(a) the ratio computed for θ = 0 and value of ω providing κ = 2, is shown with green solid line. According with the estimate (58), value of J differs noticeable from unity for χ ∼ 10 3 . The dashed brown line shows the ratio a = ςτ ⊥ /τ for the given frequency which  corresponds to κ = 2. It is seen from the expression for the classical critical frequency [equation (54)] that Hence, for χ 10 the frequency which provides κ = 2 is lower than the critical frequency. Thus, one expects τ ⊥ τ here, for n = 1. However, at χ ∼ 10 3 the timescales τ ⊥ and τ becomes of the same order thanks to the vacuum polarization. At even higher values of χ, τ becomes much smaller than τ ⊥ that together with ς = −1 leads to the interference patterns similar to that shown in figure 4(b). In figure 6(a) this is seen as the oscillations of J at high values of χ. Figure 6(b) shows the radiated energy per unit frequency and per unit angle θ for the vacuum refractive index taken into account (upper half) and δn = 0 (lower half), for χ = 800 (B/B S = 0.01 and γ = 8 × 10 4 ). Figure 6(c) shows the energy spectra which correspond to the distributions in figure 6(b). Although the difference between dashed and solid curves in figure 6(c) is dramatic, it is hardly fit as possible experimental evidence of the vacuum polarization. First, it rises only at high χ values, where some other high-order terms of QED can give even bigger contribution. Second, the change in the spectrum occurs only for frequences for which κ 10, which is much lower than for the critical frequency, hence this difference occurs for a small fraction of the photons. Therefore, photon emission by muons with χ ≈ 30 is still the most promising probe for the vacuum polarization effect in the radiation spectrum.
One more interesting prospect of QED study should be noted regarding the photon emission by muons. Let muons and electrons are of the same Lorentz factor γ. For the electrons the energy of the emitted photons in the regime of χ 1 is limited due to the recoil effect by mc 2 γ. Despite higher curvature radius, for the muons the photon energy can be much higher, because the recoil effect for them is negligible. Definitely, one can reach ακ 2/3 ∼ 1 for photons emitted by the muons already at χ ∼ 300, for which αχ 2/3 ∼ 0.3. This potentially opens perspectives to reach the non-perturbative QED [11,12,13,14,15] in future experiments.

Conclusion
The general formula which describes the photon emission by an ultrarelativistic electron in a strong magnetic field can be found in the framework of the quasiclassical theory of Baier and Katkov [32]. Baier-Katkov formula can be extended to the case of a constant non-unity refractive index n, |n − 1| 1 [see equation (30)]. From this, one can find photon emission probability that generalizes both the synchrotron and the Cherenkov emission, and takes into account photon recoil and spin flips. The obtained expression clearly shows that the emission probability is not the sum of the synchrotron emission probability and the Cherenkov emission probability. Hence, the photon emission occurs in the synergic (cooperative) synchrotron-Cherenkov radiation process.
The electron motion along its curved trajectory prevents the pure Cherenkov radiation. The trajectory curvature determines the radiation formation time for the synchrotron-Cherenkov radiation [see equation (44)], which is much shorter than that for the pure Cherenkov radiation (which can be extremely large in the case of the Cherenkov synchronism, nβ = 1). Furthermore, the radiation formation time for the synchrotron-Cherenkov radiation is less or about of that for the pure synchrotron emission.
The photon emission probability is determined not only by the radiation formation time, and the probability can be either greater or less for the synchrotron-Cherenkov radiation (n = 1) than that for the synchrotron one (n = 1). The radiation spectrum is sensible to δn = n − 1 in the both cases, δn > 0 and δn < 0, however, the changes in the spectrum occur first for frequencies which are higher than the critical frequency ω c [see equation (35)]. If the Cherenkov condition holds, v > c/n, the overall emitted energy can be much higher than in the case n = 1. For numerical simulations of the synchrotron-Cherenkov spectrum the open source code jE is implemented [37], and the numerical results are in a good agreement with the analytical predictions.
One can use formulas for the refractive index of vacuum polarized by a strong external magnetic field {see reference [28] and references therein}, in order to find how the synchrotron spectrum is modified due to vacuum polarization. The estimates and numerical simulations demonstrate that in the framework of the considered model the changes in the spectrum emitted by electrons becomes noticeable far beyond the Cherenkov threshold v = c/n and even far beyond the conjectured breakdown of the perturbative QED αχ 2/3 ∼ 1. The cause of this is that the vacuum refractive index depend on the photon frequency, and for the photon energies greater or abouthω c (for which the spectrum modification is expected first) δn is negative and |δn| is very small. Moreover, the critical frequency for electrons with χ 1 is very close to the electron energy.
Muons have much larger curvature radius of the trajectory in a strong field than the electrons, if the muons and the electrons are of the same Lorentz factor. This makes the radiation spectrum of the muons much more sensible to the refractive index than that of the electrons. Opposite to the electrons with χ 1 for which the critical frequency always yields (hω c /mc 2 )(B/B S ) 1, for the muons this is not the case. E.g., for χ = 30 the maximum of the spectrum corresponds to κ = (hω/mc 2 )(B/B S ) ≈ 2 which is favourable for the vacuum refractive index.
The radiation spectrum is enhanced up to 10% in this case thanks to the vacuum polarization [see figure 5(b)]. From the point of view of possible experiments, the muons with χ ≈ 30 probable are the most promising tool to probe the influence of the vacuum polarization on the synchrotron spectrum.
Regarding possible experiments using laser pulses and muon accelerators, a simple head-on collision geometry with single laser pulse can be considered. The expression for the vacuum refractive index for the emitted photons in this case is quite close to that for a constant magnetic field [29,21]. However, the results of this paper can not be applied directly to the laser field. First, as the most of the emitted photons have κ 1, the pair photoproduction and the photon emission by the secondary electrons and positrons should be taken into account. Second, χ = 30 will be reached rather at a 0 = eE 0 /mcω L ∼ m µ /m = 207 (with E 0 and ω L the electric field amplitude and the photon frequency of the laser pulse, respectively), e.g. for a 0 = 800 and γ = 2×10 4 (for hω L = 1 eV). For such values of a 0 the local constant field approximation applied here should be used with caution [39]. This constraint becomes even more pronounced for protons, for which the dipole-Cherenkov radiation should be considered rather than the synchrotron-Cherenkov radiation. Third, in the linearly polarized laser the refractive index is not uniform (however, it probably can be considered uniform on a scale of the radiation formation length). Therefore, the realistic proposal for probing vacuum polarization with synchrotron emission of heavy charged particles in laser fields needs further investigations.