Quasiclassical approach to synergic synchrotron–Cherenkov radiation in polarized vacuum

In any realistic set-up the vacuum refractive index n = 1 + δn is extremely close to unity. For example, to formally fulfill the Cherenkov condition in the polarized vacuum for low-energy photons emitted by electrons, χ = 70 is enough. Thus, one can choose γ = 7 × 10⁴ and B/B_S = 1 × 10⁻³. As seen from section 3.1, this value of the magnetic field lead to the refractive index such that δn ≈ 2 × 10⁻¹⁰.

In the classical electrodynamics the key feature of the emission process is the synchronism between the emitting current and the emitted wave, as demonstrated in appendix A. The synchronism is the only thing which can be influenced by the vacuum polarization with such small δn. Thus, to take into account the refractive index in the classical electrodynamics, one should use exact relation between the photon frequency and the wave vector k = nω/c only in the phase of (A.11).

In the quantum theory, similarly to the classical one, the refractive index should be taken into account first in the phase of the corresponding quasiclassical formulas (see equation (1.2) in [33]). However, an extra phase term ∝ω² − c²k² appears in the quasiclassical formula [32, 34] which though can be neglected in the case of refractive index of polarized vacuum (see appendix C; this term probably should be taken into account for electrons with χ ≳ 10³). Thus, in the polarized vacuum the energy emitted per unit frequency interval and per unit solid angle is determined by (C.9), where the refractive index is present only in the exponential function

$$\begin{equation}\mathrm{exp}\left[\mathrm{i}{\omega }^{\prime }\left(t-\mathbf{n}\boldsymbol{\rho }\right)\right],\end{equation} \tag{ 2 }$$

with n = ck/ω, |n| = n the refractive index, k and ω the wave vector and the cyclic frequency of the emitted photon, ω' = ωɛ/ɛ', ɛ the electron energy, ɛ' = ɛ − ℏω and ρ = r/c the normalized electron position.

Equation (C.9) differs from the classical one (A.11) by two quantum features. First, an additional 'spin' term appears in (C.9) (the last term, which originates from the spin flips [32, 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 (C.9), including the spin term, contain the same exponential function. In local constant field approximation the phase of the exponential function can be expanded using the Taylor series as in appendix B, exp[iω'(t − nρ)] ≈ exp[iϕ(t)] with

$$\begin{equation}\phi \left(t\right)=2\pi \left[\frac{\varsigma t}{{\tau }_{{\Vert}}}+{\left(\frac{t}{{\tau }_{\perp }}\right)}^{3}\right],\end{equation} \tag{ 3 }$$

$$\begin{equation}\varsigma =\mathrm{sgn}\left({\theta }^{2}+1/{\gamma }^{2}-2\delta n\right)\end{equation} \tag{ 4 }$$

(note that if the local constant field approximation is not valid, the refractive index should be used not only for the emitted photons, but also for the laser field [37]). Therefore, 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

$$\begin{equation}{\tau }_{{\Vert}}=\frac{4\pi }{{\omega }^{\prime }\vert {\theta }^{2}+1/{\gamma }^{2}-2\delta n\vert },\end{equation} \tag{ 5 }$$

$$\begin{equation}{\tau }_{\perp }={\left(\frac{12\pi {\gamma }^{2}}{{\omega }^{\prime }{\omega }_{B}^{2}}\right)}^{1/3},\end{equation} \tag{ 6 }$$

with θ the angle between the particle velocity at t = 0 and the wave vector k, ω_B = eB/mc the cyclotron frequency and B the magnetic field strength. Therefore the timescales differs from the timescales of the classical synchrotron spectrum given by (B.7) and (B.8).

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 2.2 and 3 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

$$\begin{equation}{\omega }_{\mathrm{c}}=\frac{\varepsilon {\omega }_{\mathrm{c}}^{\prime }}{\varepsilon +\hslash {\omega }_{\mathrm{c}}^{\prime }},\end{equation} \tag{ 7 }$$

$$\begin{equation}{\omega }_{\mathrm{c}}^{\prime }=\frac{3{\omega }_{B}{\gamma }^{2}}{\vert 1-2\delta n{\gamma }^{2}{\vert }^{3/2}}.\end{equation} \tag{ 8 }$$

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_Bγ², and the spin term is negligible. In this case (if additionally δn = 0), the maximum of d²I/dω dΩ 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), (8) yields ℏω'_c ≫ ɛ, 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: ℏω_c ≈ ɛ[1 − 1/(3χ)]. Thus ω_c is very close to the upper spectrum bound ɛ/ℏ. 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 3). As seen from (5) and (6), 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.

The radiation formation length is the length of the electron path which contributes most to the integrals in (A.11) and (C.9). 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:

$$\begin{equation}\mathcal{I}\left({t}_{a},{t}_{b}\right)={\int }_{{t}_{a}}^{{t}_{b}}f\left(t\right)\mathrm{sin}\left[\phi \left(t\right)\right]\mathrm{d}t,\end{equation} \tag{ 9 }$$

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:

$$\begin{align}\hfill \mathcal{I}\left({t}_{0},{t}_{2}\right)& ={\int }_{{t}_{0}}^{{t}_{2}}f\left(t\right)\mathrm{sin}\left[\phi \left(t\right)\right]\mathrm{d}t\hfill \\ \hfill & \approx 2fT{\vert }_{\left({t}_{0}+{t}_{1}\right)/2}-2fT{\vert }_{\left({t}_{1}+{t}_{2}\right)/2}\approx -{\int }_{{t}_{0}}^{{t}_{2}}\left(\frac{\mathrm{d}}{\mathrm{d}t}\enspace fT\right)\mathrm{d}t.\hfill \end{align} \tag{ 10 }$$

Here t₀, t₁, t₂ are the time instants which correspond to ϕ = 0, π, 2π, respectively. The continuous function T(t) is approximately equal to the local period of function sin[ϕ(t)]. Obviously, the estimate (10) 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 $\mathcal{I}\left(-\infty ,\infty \right)$ can be estimated as follows:

$$\begin{equation}\mathcal{I}\left(t,\infty \right)\sim f\left(t\right)T\left(t\right),\end{equation} \tag{ 11 }$$

where we assume that lim_t→∞ fT = 0.

The local oscillation period for the phase (3) far from the saddle points is

$$\begin{equation}T\left(t\right)\approx 2\pi {\left(\frac{\mathrm{d}\phi }{\mathrm{d}t}\right)}^{-1}={\left(\frac{\varsigma }{{\tau }_{{\Vert}}}+\frac{3{t}^{2}}{{\tau }_{\perp }^{3}}\right)}^{-1}.\end{equation} \tag{ 12 }$$

The integrals in (C.9) for βe_i from (B.3) and (B.4) 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 1(a)), and for t ≫ τ_⊥ one gets $\mathcal{I}\left(t,\infty \right)\propto 1/t$. Hence the radiation formation time is t_rf ∼ τ_⊥ in this case. For high frequencies [τ_⊥ ≳ τ_∥, ς = +1, see figure 1(b)] the integrand contribution decays, $\mathcal{I}\left(t,\infty \right)\propto 1/t$, only if t ≫ t_s with

$$\begin{equation}{t}_{s}=\sqrt{\frac{{\tau }_{\perp }^{3}}{3{\tau }_{{\Vert}}}}\end{equation} \tag{ 13 }$$

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_∥.

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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 1(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:

$$\begin{equation}\phi =2\pi \left(-\frac{t}{{\tau }_{{\Vert}}}+\frac{{t}^{3}}{{\tau }_{\perp }^{3}}\right)\approx \frac{6\pi {t}_{s}}{{\tau }_{\perp }^{3}}{\left(t-{t}_{s}\right)}^{2}+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t},\end{equation} \tag{ 14 }$$

where the term $2\pi {\left(t-{t}_{s}\right)}^{3}/{\tau }_{\perp }^{3}$ is neglected at the right-hand-side. Then the width of the bump at t_s can be found,

$$\begin{equation}T\left({t}_{s}\right)\sim {\left(\frac{{\tau }_{\perp }^{3}}{{t}_{s}}\right)}^{1/2}\sim {\tau }_{\perp }^{3/4}{\tau }_{{\Vert}}^{1/4},\end{equation} \tag{ 15 }$$

Hence, the cubic term is actually small: ${\left(t-{t}_{s}\right)}^{3}/{\tau }_{\perp }^{3}\sim {T}^{3}\left({t}_{s}\right)/{\tau }_{\perp }^{3}\sim {\left({\tau }_{{\Vert}}/{\tau }_{\perp }\right)}^{3/4}\ll 1$. The parabolic phase dependency leads to quite fast convergence of the integrals, e.g. fT ∼ 1/|t − t_s| for T(t_s) ≪ |t − t_s| ≪ t_s.

An important consequence 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, ${t}_{\mathrm{r}\mathrm{f}}\sim T\left({t}_{s}\right)\sim {\tau }_{\perp }^{3/4}{\tau }_{{\Vert}}^{1/4}\ll {\tau }_{\perp }$.

Summing up the above, the radiation formation length for synchrotron–Cherenkov radiation is

$$\begin{equation}{t}_{\mathrm{r}\mathrm{f}}/{\tau }_{\perp }\sim \begin{cases}1\qquad \quad \hfill & \mathrm{f}\mathrm{o}\mathrm{r}\quad {\tau }_{\perp }\lesssim {\tau }_{{\Vert}},\hfill \\ {\left({\tau }_{\perp }/{\tau }_{{\Vert}}\right)}^{1/2}\qquad \quad \hfill & \mathrm{f}\mathrm{o}\mathrm{r}\quad {\tau }_{\perp }\gtrsim {\tau }_{{\Vert}},\varsigma =1,\hfill \\ {\left({\tau }_{{\Vert}}/{\tau }_{\perp }\right)}^{1/4}\qquad \quad \hfill & \mathrm{f}\mathrm{o}\mathrm{r}\quad {\tau }_{\perp }\gtrsim {\tau }_{{\Vert}},\varsigma =-1.\hfill \end{cases}\end{equation} \tag{ 16 }$$

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 1(b)]. The emission probability becomes negligible in this case. Taking this into account, one notes from (16) 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/γ², one has ς = −1 and τ_∥ ≪ τ_⊥ [see figure 1(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 2, where I₀, I₊ and I₋ are the emitted energy for n = 1, n = 1 + 0.1/γ² and n = 1 − 0.1/γ², respectively. Figures 2(a)–(c) show frequency and angular distribution, whereas figure 2(d) depicts the energy emitted per unit frequency interval. Distributions d²I/dω dθ for figures 2(a)–(c) are computed numerically as described in section 2.3, for γ = 1 × 10⁵ and B/B_S = 3 × 10⁻⁵ (hence χ = 3). Lines in figure 2(d) are computed by summing up these distributions along the θ axis, except the black dotted line which corresponds to the analytical expression for the pure synchrotron emission (n ≡ 1) found in the framework of BKS theory [32, 36]. Note that for such artificial δn the phase term ∝ω² − c²k² (not included in the computations) is not very small (see appendix C, $\hat{r}\approx 0.2$ for ℏω ≈ ɛ), thus figure 2 is rather qualitative.

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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 figures 1(b) and (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 2, where the critical frequency is ℏω_c = 0.9 × ɛ. For the photons of ω ∼ ω_c the radiation formation length can be estimated with (16) and (6) as cτ_rf ≲ cτ_⊥ ∼ c/ω_B ∼ 10⁻² μm. Therefore, as expected, the curvature radius R ∼ γc/ω_B and the scale of the laser wavelengths ∼1 μm are much larger than the radiation formation length.

In the low-frequency region (ω ≪ ω_c hence τ_⊥ ≪ τ_∥) the synchrotron 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 2. If one increases the parameter χ, the critical frequency tends to ɛ/ℏ, and the region where the effect of the refractive index is noticeable becomes narrow and pinned to ɛ/ℏ.

The effect of the refractive index becomes more dramatic if the Cherenkov condition is fulfilled, δn > 1/(2γ²). Such case for δn = 1/γ² is shown in figure 3 for γ = 1 × 10⁵ and B/B_S = 3 × 10⁻⁵ (hence χ = 3 and ℏω_c = 0.9 ×ɛ). Note that for such artificial δn the phase term ∝ω² − c²k² which is not included in the computations cannot be neglected (see appendix C, $\hat{r}\approx 2$ for ℏω ≈ ɛ), thus figure 3 demonstrates the spectrum qualitatively. 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 (3) 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 3(a) and solid red curve in figure 3(c)] and δn > 0 [upper half of figure 3(a) and dashed blue curve in figure 3(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.

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Points A, B, C and D in figures 3(a) and (b) exactly correspond to a ≡ ςτ_⊥/τ_∥ used in figures 1(a)–(d), respectively. The fine structure in the radiation distribution at high frequencies is seen in figure 3(b), which shows in details the rectangular region of figure 3(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 1(d).

Figure 3 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 the refractive index influences 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 3, whereas the next section, section 2.3, is devoted to details of numerical computation of the spectrum.

Similarly to the transition from (A.10) and (A.11), one can find from d²I/dω dΩ [see (C.9)] the photon emission probability summed up for both polarizations, $W={\sum }_{i}\vert {C}_{{e}_{i}}{\vert }^{2}$, which is more convenient for numerical simulations:

$$\begin{equation}W=\frac{{e}^{2}{c}^{2}\pi {{\omega }^{\prime }}^{2}}{4\hslash {\omega }^{3}V}\left\{\frac{\left({\varepsilon }^{2}+{{\varepsilon }^{\prime }}^{2}\right)}{2{\varepsilon }^{2}}\sum _{e}{\left\vert \int \mathrm{d}t\enspace \boldsymbol{\beta }e\enspace \mathrm{exp}\left[\mathrm{i}{\omega }^{\prime }\left(t-\mathbf{n}\boldsymbol{\rho }\right)\right]\right\vert }^{2}+\frac{1}{2}{\left(\frac{\hslash \omega m{c}^{2}}{{\varepsilon }^{2}}\right)}^{2}{\left\vert \int \mathrm{d}t\enspace \mathrm{exp}\left[\mathrm{i}{\omega }^{\prime }\left(t-\mathbf{n}\boldsymbol{\rho }\right)\right]\right\vert }^{2}\right\}.\end{equation} \tag{ 17 }$$

The integrals in (17) can be found analytically in some cases if Taylor series for the phase (3) and for the other factors are used. Also, analytical results of [26] can be considered. However, for the sake of simplicity, expression (17) is used here in the numerical computations. Equation (17) is implemented in the open-source code jE [38] where a circle particle trajectory is used. As for jE code of version 1.0.0, for the integrals in (17) the trapezoidal rule of integration with a fixed time step is implemented. The time step is computed as one-half of the minimal oscillation period reached on the integration interval [−t_b, t_b] (or two integration intervals in the case ς = −1, τ_⊥ ≫ τ_∥, see previous subsection). As it is shown above, numerical error caused by finiteness of the integration interval decreases quite slowly, $\mathcal{I}\left({t}_{b},\infty \right)\propto 1/{t}_{b}$. 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, $T\left({t}_{b}\right)\sim 1/\dot {\phi }\left({t}_{b}\right)\propto {t}_{b}^{-2}$. 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:

$$\begin{equation}{\int }_{-\infty }^{\infty }f\left(t\right)\mathrm{sin}\left[\phi \left(t\right)\right]\mathrm{d}t\approx {\int }_{-{t}_{b}}^{{t}_{b}}g\left(t\right)f\left(t\right)\mathrm{sin}\left[\phi \left(t\right)\right]\mathrm{d}t.\end{equation} \tag{ 18 }$$

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 (18) can be easily proven by estimating the difference of its left-hand-side and its right-hand-side with formula (11): (1 − g)fT ≈ 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:

$$\begin{equation}g\left(t\right)=\frac{1}{4}\left\{1-\mathrm{tanh}\left[8\left(t/{t}_{b}-0.7\right)\right]\right\}\left\{1+\mathrm{tanh}\left[8\left(t/{t}_{b}+0.7\right)\right]\right\},\end{equation} \tag{ 19 }$$

which together with the given time step and the integration interval yields error less than 3% in comparison with the integrals without g computed on an 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 (17). 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, d²I/dω dΩ, computed numerically, is compared with analytical results, namely with (14.83) from [39]. 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 d²I/dω dΩ is already more than 500 times lower than the maximal value of d²I/dω dΩ, thus although the error becomes greater with the increase of ω and θ, the whole value of d²I/dω dΩ can be neglected there. Furthermore, a well-known asymptotic behavior of the full radiated energy is also tested (namely $cI/2\pi R\approx \left(1-55\sqrt{3}\chi /16+48{\chi }^{2}\right){P}_{\mathrm{c}\mathrm{l}}$ at χ ≪ 1 with P_cl = 2e⁴B²γ²/3 m²c² and cI/2πR ≈ 0.37 × e²m²c⁴χ^2/3/ℏ² at χ ≫ 1, see [32, 36]).

The radiation spectrum calculated numerically and analytically for n = 1 and χ = 3 is shown in figure 2(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 [38].

One can ask for the conditions necessary to modify the well known synchrotron spectrum because of 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 [21, 27, 29] and references therein]:

$$\begin{equation}n\left(\varkappa \right)=1+\frac{\alpha }{4\pi }{\left(\frac{B}{{B}_{S}}\right)}^{2}N\left(\varkappa \right),\end{equation} \tag{ 20 }$$

with N(ϰ) is presented in figure 4(a), ϰ = (ℏω/mc²)(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:

$$\begin{equation}N\left(\varkappa \right)=\begin{cases}14/45\qquad \quad \hfill & \mathrm{f}\mathrm{o}\mathrm{r}\enspace \varkappa \ll 1\hfill \\ -0.278{\times}{\varkappa }^{-4/3}\qquad \quad \hfill & \mathrm{f}\mathrm{o}\mathrm{r}\enspace \varkappa \gg 1\hfill \end{cases}\end{equation} \tag{ 21 }$$

where ϰ = (ℏω/mc²)(B/B_S). As seen from figure 4, δn = n − 1 is positive for ϰ ≲ 15 and negative for ϰ ≳ 15.

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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 (3). At the same time the linear term of the phase influences the integrals in (C.9) 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 (5)–(7), the necessary condition of the spectrum change at a given photon frequency ω is the following:

If condition (22) holds, then τ_∥ changes noticeably, and if (23) is holds too (with ω'_c computed either with or without vacuum polarization taken into account), then the spectrum changes. Here ω' = ωɛ/(ɛ − ℏω), and ω'_c is determined by (8) for a given frequency ω (note that δn depends on ω).

It should be noted that the conditions (22) and (23) are very weak: if δn is, say, 10% of 1/γ² 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/γ², one has ω' ≈ 0.1 × ω'_c(ω) for ℏω ≈ 0.5ɛ, however, the changes in the spectrum are evident for this frequency, as seen in figure 2(d).

For low-energy photons (ϰ ≪ 1) equation (22) can be rewritten using the electron χ parameter only:

$$\begin{equation}\chi \gtrsim {\left(\frac{90\pi }{7\alpha }\right)}^{1/2}\approx 70.\end{equation} \tag{ 24 }$$

However, condition (23) in this case much harder to fulfill: it also can be written in terms of χ and yields for ϰ = 1

$$\begin{equation}\alpha {\chi }^{2/3}\gtrsim \frac{{3}^{2/3}45\pi }{7}\approx 42,\end{equation} \tag{ 25 }$$

hence χ ≳ 4 × 10⁵. One can note that the result (25) 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 (21) 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 (22) yields χ ≳ 80 ϰ^2/3, and (23) 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 × 10⁵ 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 (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.

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 [40], 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

$$\begin{equation}{\omega }_{\mathrm{c},\mu }=3{\omega }_{B}{\gamma }^{2}m/{m}_{\mu },\end{equation} \tag{ 26 }$$

that gives the ratio of the photon energy to the muon energy: $\hslash {\omega }_{\mathrm{c},\mu }/{\varepsilon }_{\mu }=3\chi {\left(m/{m}_{\mu }\right)}^{2}$ (with χ = γB/B_S the same as for electrons). Therefore, this ratio is small, ℏω_c,μ/ɛ_μ ≪ 1, up to χ ∼ 10⁴, 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 (22) and (23).

As a starting point, one can consider ω ∼ ω_c that ensures fulfillment of condition (28). By virtue of (5), the difference between τ_∥ with δn ≠ 0 taken into account, and τ_∥ with δn = 0 (τ_∥,0), is determined by 2γ²δn, if this quantity is small:

$$\begin{equation}\frac{{\tau }_{{\Vert}}-{\tau }_{{\Vert},0}}{{\tau }_{{\Vert}}}\approx 2{\gamma }^{2}\delta n,\end{equation} \tag{ 29 }$$

where θ = 0 is assumed. One can note that this quantity depends on χ and ϰ only [see (20)]. 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, χ² = ϰm_μ/m. Thus 2γ²δn can be expressed in terms of ϰ only, and 2γ²δn as function of ϰ is plotted as dashed green line in figure 4(a). It is seen from figure 4(a) that 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 vacuum polarization, that agrees well with the numerical results. Figures 4(b) and (c) demonstrate the radiation spectrum for χ = 30 and χ = 60, respectively, computed with jE code (value B/B_S = 0.01 is used, however, the shapes of the resulting spectra almost do not depend on it). At χ = 30 the spectrum maximum becomes 10% higher thanks to 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 (27) and (28), one can consider χ ≳ 70 that ensures the fulfillment of the first of them. Then, similarly to the previous section, the low-energy part of the spectrum (ϰ ≲ 1) can be considered. In this case condition (28) yields

$$\begin{equation}\alpha {\chi }^{2/3}\gtrsim 42{\times}{\left(\frac{m}{{m}_{\mu }}\right)}^{2/3}\approx 1\end{equation} \tag{ 30 }$$

[compare this with (25)]. 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 5 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 5(a) the ratio

$$\begin{equation}J=\left.\left[\frac{{\mathrm{d}}^{2}I}{\mathrm{d}\omega \enspace \mathrm{d}\theta }\right]/{\left[\frac{{\mathrm{d}}^{2}I}{\mathrm{d}\omega \enspace \mathrm{d}\theta }\right]}_{\delta n=0}\right.,\end{equation} \tag{ 31 }$$

computed for θ = 0 and value of ω providing ϰ = 2, is shown with green solid line. According with the estimate (30), value of J differs noticeable from unity for χ ∼ 10³. 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 [see (26)] that

$$\begin{equation}\frac{\hslash {\omega }_{\mathrm{c},\mu }}{m{c}^{2}}\frac{B}{{B}_{S}}=\frac{3{\chi }^{2}m}{{m}_{\mu }}.\end{equation} \tag{ 32 }$$

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 ∼ 10³ the timescales τ_⊥ and τ_∥ becomes of the same order thanks to 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 3(b). In figure 5(a) this is seen as the oscillations of J at high values of χ.

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Figure 5(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 × 10⁴). Note that even such high value of χ the phase term ∝(ω² − c²k²) which is neglected here (see appendix C) is small, $\hat{r}\lesssim 2{\times}1{0}^{-3}$. Figure 5(c) shows the energy spectra which correspond to the distributions in figure 5(b). Although the difference between dashed and solid curves in figure 5(c) is dramatic, it is hardly fit as possible experimental evidence of 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 frequencies 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 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 mc²γ. 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 [11–15] in future experiments.