Cherenkov Radiation from the Quantum Vacuum

PHYSICAL REVIEW LETTERS 122, 161601 (2019)

Editors' Suggestion

‡ † Alexander J. Macleod, Adam Noble,* and Dino A. Jaroszynski SUPA Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom (Received 17 September 2018; revised manuscript received 1 March 2019; published 24 April 2019)

A charged particle moving through a medium emits Cherenkov radiation when its velocity exceeds the phase velocity of light in that medium. Under the influence of a strong electromagnetic field, quantum fluctuations can become polarized, imbuing the vacuum with an effective anisotropic refractive index and allowing the possibility of Cherenkov radiation from the quantum vacuum. We analyze the properties of this vacuum Cherenkov radiation in strong laser pulses and the magnetic field around a pulsar, finding regimes in which it is the dominant radiation mechanism. This radiation process may be relevant to the excess signals of high energy photons in astrophysical observations.

DOI: 10.1103/PhysRevLett.122.161601

Quantum electrodynamics (QED) is one of the most lead to the emission of radiation due to the buildup of wave successful and well tested theories in physics. An early fronts propagating from the particle, producing the well prediction of QED is the presence of virtual particle- known “Cherenkov cone” of radiation behind the particle. antiparticle pairs which fluctuate in and out of existence Since vacuum fluctuations can also reduce the phase in the quantum vacuum. It has been known since the seminal velocity of light (see, for example, [14]), the same argument work of Euler and Heisenberg [1] (see also [2]) that a strong implies that high-energy particles traveling through strong electromagnetic field can polarize these vacuum fluctua- electromagnetic fields should emit Cherenkov radiation, tions. This, in turn, can mediate an indirect interaction in addition to the usual synchrotron radiation caused by between a probe photon and the strong field such that the acceleration in the field. First steps towards analyzing this photon propagates as it would in a dielectric medium (for effect were taken by Erber [15], who used the principles of extensive reviews, see [3–5] and references therein). The QED to obtain semiquantitative predictions for the radia- Euler-Heisenberg theory is the result of integrating out the tion emitted by an electron in a strong magnetic field. This fermion degrees of freedom in the QED path integral, was followed by Ritus [16], who derived the analogous producing a nonlinear effective theory in which the photon process for an electron in constant crossed fields from interacts directly with the strong field. Other nonlinear the effective photon mass. Subsequently, Dremin [17] theories of electrodynamics have been proposed, most made more quantitative estimates for the Cherenkov notably the Born-Infeld theory [6], which, in its original radiation produced by particles crossing a laser pulse, form, was an attempt to resolve the electron self-energy while Marklund et al. [18] explored the possibility of problem before the advent of QED. More recently, it has Cherenkov radiation from a particle in a photon gas. found a resurgence of interest due to its emergence in the low In this Letter, we provide a unified description of vacuum energy limit of some string theories [7,8]. Cherenkov radiation in nonlinear electrodynamics, appli- It is well known that a charged particle moving through a cable to arbitrary field configurations. To illustrate the material medium can emit Cherenkov radiation [9,10]. The approach, we analyze the effect in the context of both first theoretical work to explain these results was presented upcoming laser facilities (e.g., the Extreme Light by Frank and Tamm [11] (though earlier work by Heaviside Infrastructure (ELI) [19]) and astrophysical sources of [12] and Sommerfeld [13] considered similar effects). This strong fields. In the latter, we find regimes in which the effect occurs because, in a medium with refractive index n, Cherenkov radiation dominates over other radiation processes, highlighting a new and, as yet, unexplored mecha- the phase velocity of light is reduced, vp ¼ c=n,soa particle traveling through the medium with velocity V>v nism for generating gamma rays, which we suggest should p be further investigated in the context of the observed excess will outrun any electromagnetic waves it emits. This can signals of astrophysical high energy photons [20–23]. Lorentz invariance of the vacuum requires nonlinear theories of electrodynamics to be constructed from Published by the American Physical Society under the terms of Lagrangians, LðX; YÞ, depending only on the two electro- the Creative Commons Attribution 4.0 International license. 1 μν 1 ˜ μν magnetic invariants, X ¼ − 4 F Fμν and Y ¼ − 4 F Fμν, Further distribution of this work must maintain attribution to μν ˜ μν the author(s) and the published article’s title, journal citation, where F and F are the electromagnetic field and its and DOI. Funded by SCOAP3. dual, and repeated indices imply summation. Given the

0031-9007=19=122(16)=161601(6) 161601-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 122, 161601 (2019) success of Maxwell’s theory, we consider only leading encodes the effect of the photon mass operator in order corrections, i.e., Lagrangians of the form QED [15,16]. To interpret these solutions as Cherenkov radiation, we 2 2 L ¼ X þ λþX þ λ−Y ; ð1Þ must relate them to the source of the radiation—i.e., the charged particle. The analogous problem in a material where the constants λ determine the specific theory. For medium has been well studied [11] and leads to the well- Euler-Heisenberg, known expressions for the emission angle θC relative to the particle’s velocity and the power radiated per unit 1 1 α 1 ω λþ ¼ λ− ¼ ; ð2Þ frequency dP=d 4 7 90π E2 S 2 vp dP e 2 cos θC ¼ ; ¼ ωsin θC: ð7Þ where α ≃ 1=137 is the fine-structure constant and ES ¼ jβj dω 4π 2 18 me=e ≃ 1.3 × 10 V=m is the Schwinger field [2] (we work throughout in units where c ¼ ℏ ¼ 1). In the Born- These are calculated for Cherenkov radiation in a homo- geneous, isotropic medium (ICR). Each of the expressions Infeld theory, the (unknown) constants coincide, λþ ¼ λ− [6]. Although our results are readily extendible to more (7) are relatively simple in their structure, and it can be general Lagrangians, (1), (2) remains a good approximation clearly seen that the key parameters are the phase velocity vp, and the particle velocity β. The definition of the for field strengths approaching ES and so is sufficient for our purposes. Cherenkov angle ensures that no Cherenkov radiation is ˜ μν β β ≡ jβj The field equations following from (1) are ∂μF ¼ 0 observed for anisotropy of the background field implies μν the phase velocity itself depends on the direction of radiation field f , and linearizing in the latter, yields ˆ emission, vp ¼ vpðkÞ. This can be accounted for using (6) ˜ μν μναβ ∂μf ¼ 0; ∂μðχ fαβÞ¼0; ð3Þ 1 2θ ¼ ð1 þ 2λ F F λ ˆμ ˆν Þ ð Þ cos C β2 λμ νkk ; 8 with the constitutive tensor which is valid in any (slowly varying) background field. μναβ μα νβ μβ να χ ¼ð1 þ 2λþXÞðg g − g g Þ Note that, in theories exhibiting birefringence, we have − 2λ F μνF αβ − 2λ F˜ μνF˜ αβ ð Þ two Cherenkov cones, corresponding to the different phase þ − : 4 velocities of the two polarizations. It is clear that the Cherenkov power formula in (7) cannot μν X ¼ − 1 F μνF g is the metric tensor, and we define 4 μν and be adopted directly into the nonlinear theories as, in similarly for Y. θ general, C depends on the azimuthal angle, and we must, The system (3), (4) has been well studied (e.g., [24,25]). instead, determine the differential power emitted per unit Neglecting derivatives of the background [26], and defining frequency per unit azimuthal angle, d2P=dωdϕ. We follow φ ¼ μ the phase kμx , the radiation field can be expressed the approach taken by Altschul to describe Cherenkov ¼ð − Þ iφ as fμν kμaν kνaμ e , where the polarization aμ and radiation in Lorentz-violating vacua [27]. Although the wave vector kμ are determined algebraically from physical basis of such theories is the reverse of nonlinear electrodynamics (where Lorentz invariance is strictly pre- μναβ χ kνkβaα ¼ 0: ð5Þ served), the linearization treats the background field as an external structure, and (3) is formally equivalent to CPT- μ μ ν μ ˜ μ ν This has solutions aþ ¼ F νkþ, a− ¼ F νk− [24], with even Lorentz-violating electrodynamics. The key observa- the wave vectors obeying the dispersion relations tion is that Cherenkov modes corresponding to different 2 λ μ ν μ k ≃ 2λF λμF νkk, where only the leading order wave vectors k propagate independently and, hence, behavior in λ has been included. Evidently, for behave as waves propagating in an isotropic medium with 2 2 2 ˆ λþ ≠ λ−, we have birefringence. Using k ¼ðω − jkj Þ¼ scalar refractive index n ¼ 1=vpðkÞ. ICR is linearly polar- ð 2 − 1Þjkj2 ˆ ˆ ˆ vp , the dispersion relations yield the phase ized in the plane ðβ; kÞ, and orthogonal to k, i.e., in the velocity v ˆ ˆ ˆ p direction ϵ0 ¼ðβ − cos θCkÞ= sin θC (throughout, spatial vectors with carets are unit normalized). In the nonlinear 2 ≃ 1 þ 2λ F F λ ˆμ ˆν ð Þ μ vp λμ νkk; 6 case, there are two independent polarization modes, aþ and μ a−, the spatial parts of which do not, in general, coincide ˆμ μ where k ¼ k=jkj is the direction 4-vector of the with ϵˆ0. As such, only the projection of ICR along these radiation. For Euler-Heisenberg, the last term in (6) directions will propagate

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2 2 d P e 2 2 locally approximate the laser beam as a constant crossed ¼ jϵˆ0:ϵˆj ωsin θ : ð9Þ μν dωdϕ 8π2 C field. We will consider the background field F to represent a constant crossed field (i.e., X ¼ Y ¼ 0)of zˆ Here, ϵˆ are the (unit normalized) spatial components of strength E, with Poynting vector in the direction. We μ the polarization modes a (see Supplemental Material [28] consider the electron to be counterpropagating with respect for details). The derivation of (9) treats the particle’s orbit to the Poynting vector, as, in this configuration, the energy as rectilinear. Therefore, it is valid only for wavelengths transfer between background and electron will be greatest, that the particle can emit while turning a negligible angle. leading to the strongest effect. With the set up described, See Supplemental Material [28] for a demonstration that the phase velocity can be determined via (6), which leads to this includes almost all the radiation in the examples below. the simple expression for the Cherenkov angles As can be seen from (7) and (9), the Cherenkov spectrum 1 has an explicit linear dependence on the frequency ω. 2 θ ¼ ð Þ cos C ½β2 þ 2ð1 þ βÞ2λ 2 : 11 This means that the spectrum appears to diverge at high E frequencies. In a material medium, dispersive effects give In this case, the Cherenkov angles are independent of the θC an ω dependence, so that, at high frequencies, azimuthalpffiffiffiffiffiffiffiffiffiffiffiffi angle ϕ and yield the Cherenkov condition Cherenkov radiation is suppressed. In nonlinear electro- 2 2 2 dynamics, this is not the case, and we must assume a cutoff ðγ þ γ − 1Þ E > 1=2λ. Cherenkov radiation will frequency will arise from physics not captured in LðX; YÞ occur whenever this condition is satisfied; however, to (see [29] for an analogous discussion in the context of be observable, it must be non-negligible in comparison Lorentz-violating electrodynamics). In the case of QED, for with the emission of synchrotron radiation by electrons example, the Euler-Heisenberg Lagrangian must be sup- oscillating in the field. The synchrotron spectrum is [32] plemented by higher derivative terms at very high frequen- pffiffiffi Z 3 3 ω ∞ cies [3]. We could simply impose a cutoff directly on the dPsynch e E ¼ dxK5 3ðxÞ; ð12Þ ω ω π 3 ω = frequency . However, since frequency is not a Lorentz d mec c ω=ωc invariant, this would not be a physically meaningful condition. Instead, we assume (9) is valid for photons with where KνðxÞ is the order ν modified Bessel function of the 2 a small quantum nonlinearity parameter [16] second kind and ωc ¼ 3ðeE=mecÞγ . To compare the two radiation processes, we integrate the Cherenkov spectrum qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j j with respect to the azimuthal angle and consider the total χ ¼ e −F μλF ν ≲ 1 ð Þ γ 3 λkμkν ; 10 power per unit frequency me Z 2π 2 2 ω dPCher d Pþ d P− which can be solved for the maximum frequency, max. ¼ dϕ þ ; ð13Þ This is not strictly a cutoff, and Cherenkov radiation may dω 0 dωdϕ dωdϕ still occur for higher frequencies, but our results may not be ω where the contribution from each individual mode is reliable above max. In principle, the Lorentz factor γ should reduce as the determined by (9). particle loses energy to radiation. This could be accounted Future laser facilities such as ELI [19] are expected to ∼ 10−3 for by introducing a damping force F ¼ −P β, where P reach field strengths on the order of E ES × , with d C C γ ∼ 105 ≃50 is the integral of (9), and solving simultaneously for the access to electrons up to ( GeV). Thus, we motion of the particle and the radiation. In practice, consider this parameter regime in comparing the spectra from Cherenkov and synchrotron radiation in a constant however,qffiffiffiffiffiffiffiffiffiffiffiffiffi this is generally unnecessary, as, for crossed field. We also specialize to the Euler-Heisenberg γ ≫ 1 1 − 2 β ¼ 1 = vp, we can set in (8), (9). Lagrangian (2), as this arguably represents the best moti- With these considerations, we now have all the ingre- vated nonlinear extension to Maxwell electrodynamics. dients necessary to determine the Cherenkov radiation Figure 1 shows the calculated power per unit frequency due emitted by a particle moving in any given field configu- to each of the radiation processes as a function of the ration. To demonstrate more concretely the vacuum emitted photon energy ℏω. The black dashed line repre- ℏω ∼ ð 3 5Þ Cherenkov effect, we consider two examples of field sents the cutoff found from (10), max mec = configurations: a constant crossed field (representing a ð2eℏEÞ ≃ 0.25 GeV. Below this limit, synchrotron radia- laser pulse) and a constant magnetic field (representing the tion is always the dominant process. Thus, observing the field around a pulsar). Cherenkov effect appears unlikely for even future laser Advances in laser technology have begun to provide a facilities. This is primarily due to the limitation on the platform to study strong field effects experimentally, for ability to produce high energy electrons in the lab. For example, the recent results concerning radiation reaction γ ≫ 1, the Cherenkov spectrum becomes proportional to [30,31]. Many results pertaining to strong field physics E2, to leading order, and so, for a fixed field strength,

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the field, since any parallel component of β can be removed by a Lorentz transformation which does not alter the form of the background field. Taking the zˆ axis along the particle’s velocity, the polar angle of the emitted radiation and the Cherenkov angle coincide, θ ¼ θC. This immediately gives an azimuthal dependence to the Cherenkov angle

1 − 2λ 2 2 ϕ 2 θ ¼ B cos ð Þ cos C 2 2 2 ; 14 β þ 2λB sin ϕ

2 2 which gives the Cherenkov condition, γ B > 1=2λ. Again, we need to compare the Cherenkov and synchrotron spectra. In the case of the magnetic field, we use (12), with the substitution E → Bc=2 (the factor 2 arises because the constant crossed field has both magnetic and electric components, essentially doubling the contribution). We are FIG. 1. Radiated power from the interaction of an electron with considering high energy cosmic rays, which are predomi- γ ¼ 105 E ¼ E 10−3 and a crossed field with field strength S × , nantly protons, so we consider the two radiation processes due to: Synchrotron radiation (red line); total Cherenkov radi- → ation (blue, solid line); Cherenkov þ mode (blue, dashed line); for these [36]. This amounts to changing me mp in (12). Cherenkov − mode (blue, dotted line). The cutoff energy (black, However, factors of me appearing in the Cherenkov λ dashed line) is ℏωmax ≃ 0.25 GeV. spectrum (through the parameters ) and the cutoff are not changed: the nonlinear terms in the Lagrangian (1), (2) and the mass scale in the cutoff (10) are determined by increasing the energy of the particles has very little effect electron-positron fluctuations in the vacuum. The total on the Cherenkov spectrum. Conversely, the synchrotron power radiated per unit frequency is again determined spectrum becomes increasingly suppressed as γ increases by (13), with (9). for fixed E. For the field strength considered here, an For radiation from protons, we also need to compare (13) electron Lorentz factor γ ∼ 2.5 × 106, corresponding to an with the radiation of pions, which, subsequently, decay into energy of 1.3 TeV, would be required to have the con- photons. The spectrum for such radiation is given by [37] tributions from Cherenkov and synchrotron processes Z 2 ∞ approximately equal at the cutoff. There is also the concern dPπ pgffiffiffi −2 ¼ γ ω dxK1=3ðxÞ; ð15Þ that to reach these high field strengths in a real experiment, dω 3πc y strong focusing techniques are needed to compress the laser ¼ 2 ðℏω Þð Þð Þγ−2½1 þðℏω pulse, and this brings in a significant range of other effects where y 3 =me mp=me BS=B = 2 3=2 2 which would act to drown out the Cherenkov signal, or γmπÞ , mπ is the pion mass, and g ≃ 14ℏc is the deplete the electron energy sufficiently that, by the time it pion-proton coupling strength. reaches the peak intensity of the pulse, its energy has fallen The strongest magnetic fields observed are those pro- below the Cherenkov threshold [33]. It might be hoped that duced by rapidly rotating pulsars. These objects have protons offer a viable alternative, since their mass greatly characteristic attributes of mass and radius, which, with suppresses synchrotron radiation. However, the Cherenkov rotational period, determine the typical field strengths threshold corresponds to a proton energy of 33 TeV, well produced. There are two broad classes of pulsar, those beyond what can currently be produced. Therefore, the with a relatively longer rotational period which have 8 possibility of observing Cherenkov radiation in this context magnetic field strengths B ∼ 10 T, and rapidly rotating seems bleak. “millisecond pulsars” which have typical field strengths 4 Since the main obstacle to observing Cherenkov radi- B ∼ 10 T [38]. The cutoff energy found through (10) is ℏω ∼ ð 3 4Þ ð ℏ Þ ation is the availability of high energy particles, it is natural max mec = e B . This corresponds to 22.5 MeV for to turn our attention to astrophysics, where the only limit on B ¼ 108 T, or 225 GeV for B ¼ 104 T. Since we are the particle energy is the so-called Greisen-Zatsepin- interested in high energy gamma rays, we illustrate the Kuzmin limit, γ ≲ 1011 [34,35]. Astrophysical objects such results for millisecond pulsars. as pulsars have also been observed to generate magnetic Figure 2 shows the spectra for Cherenkov, synchrotron, fields up to and exceeding the Schwinger magnetic field and pion radiation for a proton moving perpendicularly 9 ¼ 104 γ ¼ 5 107 BS ¼ ES=c ≃ 4.4 × 10 T. This makes such a scenario to a magnetic field B T, for × (just above ideal for the study of nonlinear vacuum Cherenkov radi- the Cherenkov threshold) and γ ¼ 5 × 109. For clarity, ation. As such, we consider a constant magnetic field of we include only the total Cherenkov contribution. For strength B. We take the particle’s velocity perpendicular to γ ¼ 5 × 107, the Cherenkov radiation exceeds synchrotron

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predicted by these theories in regions of strong fields—may provide an alternate radiation mechanism for very high energy particles. We considered two examples of background field with relevance to future experimental or observational campaigns, and determined the possibility of observing Cherenkov radiation in each case. When the background field is a constant crossed field (approximating a laser pulse), the availability of high energy particles appears to put observation of Cherenkov radiation out of reach. In contrast, astrophysics provides environments in which the vacuum Cherenkov effect may be observed due to the presence of very high energy cosmic rays and strong magnetic fields. We have demonstrated regimes in which radiation due to the nonlinear Cherenkov effect dominates over radiation produced through synchrotron and pion emission, generating very high energy photons. A notable excess of gamma rays with energies in the GeV–TeV range has been observed in various astrophysical contexts, and the vacuum Cherenkov process could provide an alternate explanation for their origin, not previously considered in the literature. We would like to thank other members of the ALPHA-X Collaboration for useful discussions. This work was sup- ported by the UK EPSRC (Grant No. EP/N028694/1), the European Union H2020 Research and Innovation Programme LASERLAB EUROPE (654148) and a University of Strathclyde Doctoral Training Programme Studentship. All of the results can be fully reproduced FIG. 2. Power radiated via synchrotron, pion, and Cherenkov using the methods described in the Letter. emission, by protons in a magnetic field B ¼ 104 T, with Lorentz factor γ ¼ 5 × 107 (upper panel) and γ ¼ 5 × 109 (lower panel). * The cutoff energy is ℏωmax ≃ 225 GeV. [email protected] † [email protected] ‡ Present address: Centre for Mathematical Sciences, Uni- emission for photon energies above 8.5 GeV but remains versity of Plymouth, PL4 8AA, United Kingdom. ℏω γ ¼ 5 109 [1] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). below the pion emission up to max.For × , however, Cherenkov radiation is, by far, the dominant [2] J. Schwinger, Phys. Rev. 82, 664 (1951). 78 emission channel for photon energies from 54 MeV up to [3] M. Marklund and P. K. Shukla, Rev. Mod. Phys. , 591 the cutoff. So for the highest energy proton cosmic rays, the (2006). [4] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. highest energy radiation is completely dominated by the Keitel, Rev. Mod. Phys. 84, 1177 (2012). Cherenkov process. [5] R. Battesti and C. Rizzo, Rep. Prog. Phys. 76, 016401 There is currently a debate within the astrophysics (2013). community concerning the origin of observed excesses [6] M. Born and L. Infeld, Proc. R. Soc. Ser. A 144, 425 (1934). of high energy photons found in recent data. For example, [7] E. Fradkin and A. Tseytlin, Phys. Lett. B 163, 123 (1985). observations of intense gamma rays from the Galactic [8] F. Abalos, F. Carrasco, E. Goulart, and O. Reula, Phys. Rev. Center [20] have prompted a range of possible explan- D 92, 084024 (2015). ations, such as dark matter annihilation [22] and unresolved [9] P. A. Cherenkov, C. R. Ac. Sci. U.S.S.R. 8, 451 (1934). 8 pulsar sources [23]. The Cherenkov process detailed in this [10] S. Vavilov, C. R. Ac. Sci. U.S.S.R. , 457 (1934). 14 Letter provides a new, and so far unexplored, gamma-ray [11] I. Tamm and I. Frank, C. R. Ac. Sci. U.S.S.R. , 109 (1937). production mechanism, which we believe warrants further [12] O. Heaviside, The Electrician 22, 147 (1888). study in this context. [13] A. Sommerfeld, Knkl. Acad. Wetensch. 7, 345 (1904). To summarize, in this Letter, we have provided a [14] S. P. Flood and D. A. Burton, Europhys. Lett. 100, 60005 comprehensive, quantitative study of the Cherenkov effect (2012). in nonlinear theories of vacuum electrodynamics. This [15] T. Erber, Rev. Mod. Phys. 38, 626 (1966). effect—expected due to the reduced phase velocity of light [16] V. I. Ritus, J. Sov. Laser Res. 6, 497 (1985).

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