High-Energy Vacuum Birefringence and Dichroism in an Ultrastrong Laser Field

TL;DR

This paper explores the theoretical prediction and experimental feasibility of observing vacuum birefringence and dichroism caused by quantum electrodynamics effects in an ultrastrong laser field, using high-energy probe photons.

Contribution

It derives the influence of a polarized probe beam on vacuum birefringence and dichroism in a strong laser field and proposes an experimental scheme with circularly polarized photons for high-confidence detection.

Findings

Feasibility of measuring vacuum birefringence at 5σ confidence within days using 10 PW lasers.

Demonstrates that dichroism and anomalous dispersion effects are detectable in vacuum at these facilities.

Proposes using circularly polarized high-energy photons to improve measurement sensitivity.

Abstract

A long-standing prediction of quantum electrodynamics, yet to be experimentally observed, is the interaction between real photons in vacuum. As a consequence of this interaction, the vacuum is expected to become birefringent and dichroic if a strong laser field polarizes its virtual particle--antiparticle dipoles. Here, we derive how a generally polarized probe photon beam is influenced by both vacuum birefringence and dichroism in a strong linearly polarized plane-wave laser field. Furthermore, we consider an experimental scheme to measure these effects in the nonperturbative high-energy regime, where the Euler-Heisenberg approximation breaks down. By employing circularly polarized high-energy probe photons, as opposed to the conventionally considered linearly polarized ones, the feasibility of quantitatively confirming the prediction of nonlinear QED for vacuum birefringence at the $5\sigma$ confidence level on the time scale of a few days is demonstrated for upcoming 10 PW laser systems. Finally, dichroism and anomalous dispersion in vacuum are shown to be accessible at these facilities.

Tables

Table 1. Table 1: Duration of the experiment τ 𝜏 \tau at different facilities ( χ = 0.25 𝜒 0.25 \chi=0.25 ). S 0 subscript 𝑆 0 S_{0} and S 1 subscript 𝑆 1 S_{1} , ⟨ R B ⟩ delimited-⟨⟩ subscript 𝑅 B \langle{R_{\textrm{B}}}\rangle , and N \indices ∗ γ B N\indices*{{}_{\gamma}^{\textrm{B}}} follow from Eq. ( 11 ), Eq. ( 14 ) and Eq. ( 15 ), respectively ( S ( 0 ) = { 1 , 0 , − 1 , 0 } superscript 𝑆 0 1 0 1 0 S^{(0)}=\{1,0,-1,0\} ; 5 ​ σ 5 𝜎 5\sigma confidence level, i.e., n = 5 𝑛 5 n=5 ). Note that the pair production probability in the strong laser field is much smaller than the conversion efficiency in the detector [ ( 1 − S 0 ) ≪ η = 10 − 2 much-less-than 1 subscript 𝑆 0 𝜂 superscript 10 2 (1-S_{0})\ll\eta=10^{-2} ].

	$1-S_0$	$S_1$	$⟨R_{\text{B}}⟩$	$N\text{indices}\ast {}_{\gamma }^{\text{B}}$	$\tau$
Apollon	$1.9\times {10}^{-5}$	0.06	$3.4\times {10}^{-3}$	$3.0\times {10}^8$	45 d
ELI-NP	$3.1\times {10}^{-5}$	0.09	$5.6\times {10}^{-3}$	$1.1\times {10}^8$	10 d
ELI-BL	$6.3\times {10}^{-5}$	0.18	$1.1\times {10}^{-2}$	$2.6\times {10}^7$	11 h
LINAC-L	$3.8\times {10}^{-6}$	0.01	$6.8\times {10}^{-4}$	$7.4\times {10}^9$	2 d

Equations

$${\mathcal{L}}_{\text{EM}}=-\mathcal{F}+\frac{\alpha }{90\pi E_{\text{cr}}^2}(4{\mathcal{F}}^2+7{\mathcal{G}}^2)+\cdots ,$$

$$\chi =\frac{\sqrt{-(f^{\mu \nu }{q}_{\nu }{)}^2}}{E_{\text{cr}}m}\approx 0.5741\frac{\omega }{\mathrm{GeV}}\sqrt{\frac{I}{10^{22}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2}}$$

$$\xi =\frac{|e|\sqrt{-a^2}}{m}\approx 0.7495\frac{\mathrm{eV}}{{\omega }_{\text{L}}}\sqrt{\frac{I}{10^{18}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2}}.$$

$${\Phi }_q^{\mu }(x)={ϵ}^{\mu }(kx){\mathrm{e}}^{-iqx},{ϵ}^{\mu }(kx)=\sum _{i=1,2}c\text{indices}\ast {}_i(kx){\Lambda }_i^{\mu },$$

$${ϵ}^{\mu }(kx\to -\infty )={ϵ}^{(0)\mu }=\sum _{i=1,2}c\text{indices}\ast {}_i^{(0)}{\Lambda }_i^{\mu },$$

$$c\text{indices}\ast {}_i(kx)=c\text{indices}\ast {}_i^{(0)}\exp [i{\varphi }_i(kx)-{\lambda }_i(kx)],$$

$$\left[\begin{array}{c}{\varphi }_i(kx)\\ {\lambda }_i(kx)\end{array}\right]=-\frac{1}{2kq}\int {\text{displaylimits}}_{-\infty }^{kx}\mathrm{d}\varphi \left[\begin{array}{c}\mathrm{Re}\left[p_i(\varphi ,\chi )\right]\\ \mathrm{Im}\left[p_i(\varphi ,\chi )\right]\end{array}\right],$$

$$\left[\begin{array}{c}{p}_1(kx,\chi )\\ p_2(kx,\chi )\end{array}\right]=\frac{\alpha m^2}{3\pi }\int {\text{displaylimits}}_{-1}^{+1}\mathrm{d}v\left[\begin{array}{c}(w-1)\\ (w+2)\end{array}\right]\frac{f^{\prime }(u)}{u},$$

$${\varrho }^{(0)\mu \nu }$$

$${\varrho }^{\mu \nu }$$

$${\rho }^{(0)}=\frac{1}{2}\left(S_0^{(0)}\mathrm{I}+{\mathbf{S}}^{(0)}\mathbf{\sigma }\right),\rho =\frac{1}{2}\left(S_0\mathrm{I}+\mathbf{S}\mathbf{\sigma }\right)$$

$$\begin{array}{rl}\left(\begin{array}{c}{S}_0\\ S_3\end{array}\right)& ={\mathrm{e}}^{-({\lambda }_1+{\lambda }_2)}\left(\begin{array}{cc}\cosh \delta \lambda & \sinh \delta \lambda \\ \sinh \delta \lambda & \cosh \delta \lambda \end{array}\right)\left(\begin{array}{c}{S}_0^{(0)}\\ S_3^{(0)}\end{array}\right),\\ \left(\begin{array}{c}{S}_1\\ S_2\end{array}\right)& ={\mathrm{e}}^{-({\lambda }_1+{\lambda }_2)}\left(\begin{array}{cc}\cos \delta \varphi & -\sin \delta \varphi \\ \sin \delta \varphi & \cos \delta \varphi \end{array}\right)\left(\begin{array}{c}{S}_1^{(0)}\\ S_2^{(0)}\end{array}\right).\end{array}$$

$$\mathrm{d}\sigma_{\textrm{pp}}=\frac{\mathrm{d}\varphi}{2\pi}\big{\{}S_{0}\sigma_{0}+[S_{1}\sin(2\varphi)+S_{3}\cos(2\varphi)]\sigma_{1}\big{\}},$$

$$\begin{array}{rl}{R}_{\text{B}}& =\frac{(N_{\pi /4}+N_{5\pi /4})-(N_{3\pi /4}+N_{7\pi /4})}{(N_{\pi /4}+N_{5\pi /4})+(N_{3\pi /4}+N_{7\pi /4})},\\ R_{\text{D}}& =\frac{(N_0+N_{\pi })-(N_{\pi /2}+N_{3\pi /2})}{(N_0+N_{\pi })+(N_{\pi /2}+N_{3\pi /2})},\end{array}$$

$$⟨R_{\text{B}}⟩=\frac{\sin (2\beta )}{2\beta }\frac{{\sigma }_1}{{\sigma }_0}\frac{S_1}{S_0},⟨R_{\text{D}}⟩=\frac{\sin (2\beta )}{2\beta }\frac{{\sigma }_1}{{\sigma }_0}\frac{S_3}{S_0}.$$

$$N\text{indices}\ast {}_{\gamma }^{\text{B}}=\frac{\pi n^2}{4\eta \beta S_0⟨R_{\text{B}}{⟩}^2},N\text{indices}\ast {}_{\gamma }^{\text{D}}=\frac{\pi n^2}{4\eta \beta S_0⟨R_{\text{D}}{⟩}^2}$$

Full text

In the realm of classical electrodynamics, the electromagnetic field experiences no self-interaction in vacuum Landau and Lifshitz (1987). According to quantum electrodynamics (QED), however, a finite photon-photon coupling is induced by the presence of virtual charged particles in the vacuum Berestetskii et al. (1982). For low-frequency electromagnetic fields $F^{\mu\nu}$, such vacuum polarization effects are described by the Euler-Heisenberg Lagrangian density Dunne (2012); Dittrich and Gies (2000); Schwinger (1951); Heisenberg and Euler (1936). Below the QED critical field $E_{\text{cr}}={m^{2}}/{\left|e\right|}\approx 1.3\times 10^{18}\,{\mathrm{V}}/{\mathrm{m}}$, low-frequency vacuum polarization effects are suppressed King and Heinzl (2016); Battesti and Rizzo (2013); Di Piazza et al. (2012); Marklund and Shukla (2006); Ritus (1985); Mitter (1975) and the density is given by

[TABLE]

where $\mathcal{F}={F_{\mu\nu}F^{\mu\nu}}/{4}$ and $\mathcal{G}={\tilde{F}_{\mu\nu}F^{\mu\nu}}/{4}$ are the electromagnetic field invariants 111Here, $F^{\mu\nu}$ denotes the electromagnetic field tensor, $\tilde{F}^{\mu\nu}={\epsilon^{\mu\nu\sigma\tau}F_{\sigma\tau}}/{2}$ is its dual, and $\epsilon^{\mu\nu\sigma\tau}$ is the totally antisymmetric pseudotensor. Heaviside and natural units are used ($\epsilon_{0}=\hbar=c=1$), i.e., the fine-structure constant is given by $\alpha={e^{2}}/{(4\pi)}\approx{1}/{137}$; and the metric tensor $g^{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$ is employed ($m$ and $e<0$ denote the electron mass and charge, respectively)..

The Euler-Heisenberg Lagrangian predicts that the vacuum resembles a birefringent medium Bialynicka-Birula and Bialynicki-Birula (1970); Baier and Breitenlohner (1967); Klein and Nigam (1964); Toll (1952). The smallness of the QED prediction for the light-by-light scattering cross section in the low-energy regime opens up the possibility to search for physics beyond the Standard Model, e.g., axionlike or minicharged particles and paraphotons, by measuring optical vacuum polarization effects Villalba-Chávez et al. (2016); Villalba-Chávez and Di Piazza (2013); Tommasini et al. (2009); Abel et al. (2008); Gies et al. (2006), see also Jaeckel and Spannowsky (2016); Jaeckel and Ringwald (2010).

Recent astronomical observations hint at the existence of vacuum birefringence Mignani et al. (2017) (see also the remarks in Capparelli et al. (2017); Turolla et al. (2017)). However, a direct laboratory-based verification of this fundamental property of the vacuum is still missing. Laboratory experiments like BFRT Cameron et al. (1993), BMV Cadène et al. (2014), PVLAS Della Valle et al. (2016), and Q&A S.-J. Chen et al. (2007) have so far employed magnetic fields to polarize the vacuum and optical photons to probe it, though without reaching the required sensitivity.

The strongest electromagnetic fields of macroscopic extent are nowadays produced by lasers. However, even the intensities $I\sim 10^{23}\,{\mathrm{W}}/{\mathrm{cm^{2}}}$ envisaged for future $10\,\mathrm{PW}$-class optical lasers Danson et al. (2015); Jeong and Lee (2014) are still well below the critical intensity $I_{\text{cr}}=E^{2}_{\text{cr}}\approx 4.6\times 10^{29}\,{\mathrm{W}}/{\mathrm{cm^{2}}}$. Therefore, the leading-order correction given in Eq. (1) is sufficient to describe low-frequency vacuum polarization effects. Recently, various setups have been considered to measure them Shakeri et al. (2017); H.-P. Schlenvoigt et al. (2016); Karbstein and Sundqvist (2016); Zavattini et al. (2016); Tennant (2016); Gies et al. (2015); Fillion-Gourdeau et al. (2015); Karbstein and Shaisultanov (2015); Hu and Huang (2014); Mohammadi et al. (2014); Monden and Kodama (2012); King and Keitel (2012); Kryuchkyan and Hatsagortsyan (2011); Homma et al. (2011); King et al. (2010); Tommasini et al. (2008); Lundström et al. (2006); Di Piazza et al. (2006); Heinzl et al. (2006), but all suggested experiments will remain challenging in the foreseeable future.

As the light-by-light scattering cross section attains its maximum at the pair-production threshold Berestetskii et al. (1982), it is natural to consider high-energy photons to probe vacuum birefringence King and Elkina (2016); Ilderton and Marklund (2016); Nakamiya and Homma (2017); Dinu et al. (2014); Wistisen and Uggerhøj (2013); Cantatore et al. (1991). A photon four-momentum $q^{\mu}$ ($q^{0}=\omega$, $q^{2}=0$) allows us to construct a third invariant, the quantum nonlinearity parameter (see Berestetskii et al. (1982), § 101)

[TABLE]

[for a plane-wave background field with amplitude $f^{\mu\nu}$ and phase-dependent pulse shape $\psi^{\prime}(\phi)$, i.e., $F^{\mu\nu}=f^{\mu\nu}\psi^{\prime}(\phi)$, details are given below; the last relation in Eq. (2) assumes a head-on collision]. As gamma photons with energies $\omega\gtrsim 1\,\mathrm{GeV}$ are obtainable from Compton backscattering Muramatsu et al. (2014); Weller et al. (2009); Fukuda et al. (2003); Ginzburg et al. (1984); Berestetskii et al. (1982), the regime $\chi\sim 1$ is attainable in future laser-based vacuum birefringence experiments.

In the nonperturbative regime $\chi\gtrsim 1$ the Euler-Heisenberg approximation is no longer applicable, as it neglects the contribution of the probe photon momentum which flows in the electron-positron loop (see Fig. 1a). Instead, the polarization operator in the background field must be employed (see Fig. 1b). For low-energy photons, both objects in Fig. 1 are related by functional derivatives Bialynicka-Birula and Bialynicki-Birula (1970). The regime $\chi\gtrsim 1$ is qualitatively different from the one where the Euler-Heisenberg approximation is valid, in particular, due to the following two reasons: 1) electron-positron photoproduction becomes sizable, and thus, the vacuum acquires dichroic properties; 2) the vacuum exhibits anomalous dispersion Heinzl and Ilderton (2009); Dinu et al. (2014); Ritus (1985); Baier et al. (1976); Becker and Mitter (1975).

In this Letter, we put forward an experimental scheme to measure high-energy vacuum birefringence and dichroism in the nontrivial regime $\chi\gtrsim 1$. It is based on Compton backscattering to produce polarized gamma photons and exploits pair production in matter to determine the polarization state of the probe photon after it has interacted with a linearly polarized strong laser pulse. By analyzing the consecutive stages of this type of experiment, we show that for vacuum birefringence, the required measurement time is reduced by two orders of magnitude if a circularly polarized probe photon beam is employed (hitherto, only linearly polarized probe gamma photons have been considered for setups similar to ours King and Elkina (2016); Ilderton and Marklund (2016); Nakamiya and Homma (2017); Dinu et al. (2014),222So far, the case of circularly polarized photons has only been discussed for proposals to measure vacuum birefringence in magnetic fields at $\chi\ll 1$ Wistisen and Uggerhøj (2013); Cantatore et al. (1991).).

Assuming conservative experimental parameters, we demonstrate that with this type of setup and the observables we introduce [see Eq. (13)], the quantitative verification of the strong-field QED prediction for vacuum birefringence and dichroism is feasible with an average statistical significance of $5\sigma$ on the time scale of a few days at upcoming $10\,\mathrm{PW}$ laser facilities.

In the following, we consider a linearly polarized plane-wave laser pulse, described by the four-potential $A^{\mu}(kx)=a^{\mu}\psi(kx)$. Here, $x^{\mu}$ is the position four-vector, $k^{\mu}$ is a characteristic laser photon four-momentum ($k^{0}=\omega_{\textrm{L}}$, $k^{2}=0$), $a^{\mu}$ characterizes the amplitude of the field ($a^{2}<0$, $ka=0$, $f^{\mu\nu}=k^{\mu}a^{\nu}-k^{\nu}a^{\mu}$), and $\psi(kx)$ defines its pulse shape ($|\psi(kx)|,\;|\psi^{\prime}(kx)|\lesssim 1$; a prime denotes the derivative of a function with respect to its argument).

A gauge- and Lorentz-invariant measure of the laser field strength is the classical intensity parameter Ritus (1985)

[TABLE]

Here, we focus on high-intensity optical lasers ($I\gtrsim 10^{20}\,{\mathrm{W}}/{\mathrm{cm^{2}}}$, $\omega_{\textrm{L}}\sim 1\,\mathrm{eV}$), i.e., the regime $\xi\gg 1$.

Inside a plane-wave background field an incoming external photon line (see Fig. 2) in a Feynman diagram corresponds (up to normalization) to the function $\Phi_{q}^{\mu}(x)$, which is a solution of the Dyson equation Meuren et al. (2015a); Berestetskii et al. (1982) with initial condition $\Phi_{q}^{\mu}(x)\to\Phi_{q}^{(0)\mu}(x)=\epsilon^{(0)\mu}\mathrm{e}^{-iqx}$ as $kx\to-\infty$ ($\epsilon^{(0)}\epsilon^{(0)*}=-1$, $q\epsilon^{(0)}=0$). After applying the local constant field approximation (valid if $\xi\gg 1$) and following Meuren et al. (2015a), we find that to leading order, $\Phi_{q}^{\mu}(x)$ is given by (see also Villalba-Chávez et al. (2016); Dinu et al. (2014); Baier et al. (1976); Becker and Mitter (1975))

[TABLE]

where

[TABLE]

and $\Lambda_{1}^{\mu}={f^{\mu\nu}q_{\nu}}/{\sqrt{qf^{2}q}}$, $\Lambda_{2}^{\mu}=-{\tilde{f}^{\mu\nu}q_{\nu}}/{\sqrt{qf^{2}q}}$ ($q\Lambda_{i}=k\Lambda_{i}=0$, $\Lambda_{i}\Lambda_{j}=-\delta_{ij}$; note that $\Lambda_{2}^{\mu}$ is actually a pseudo four-vector) Meuren et al. (2015b, a, 2013). The coefficients $c\indices*{{}_{i}}(kx)$ and $c\indices*{{}_{i}^{(0)}}$ are connected via

[TABLE]

where

[TABLE]

[we refer to $\phi_{i}=\phi_{i}(kx\to\infty)$ as phase shifts and to $\lambda_{i}=\lambda_{i}(kx\to\infty)$ as decay parameters] with

[TABLE]

$w=4/(1-v^{2})$, $u=[w/\chi(kx)]^{2/3}$, $\chi(kx)=\chi|\psi^{\prime}(kx)|$, and $f(u)=\pi\left[\mathrm{Gi}(u)+i\mathrm{Ai}(u)\right]$ Olver et al. (2010); Ritus (1985).

In order to extend the above result from a single photon to a photon beam (which is, in general, not in a pure polarization state), we introduce the following density tensors, which describe the initial ($\varrho^{(0)\mu\nu}$) and the final ($\varrho^{\mu\nu}$) polarization state of the beam Berestetskii et al. (1982); Meuren et al. (2016); Blum (2012)

[TABLE]

Here, $w\indices{{}_{a}}$ represents the probability to find a photon with polarization four-vector $\epsilon\indices*{{}_{a}^{(0)\mu}}$ ($\epsilon\indices*{{}_{a}^{\mu}}$) in the initial (final) beam.

Using the identity matrix $\mathrm{I}$ and the Pauli matrices $\boldsymbol{\sigma}=\left(\sigma_{1},\sigma_{2},\sigma_{3}\right)$ Berestetskii et al. (1982), we expand the initial ($\rho\indices*{{}_{ij}^{(0)}}$) and the final ($\rho\indices*{{}_{ij}}$) polarization density matrices as Berestetskii et al. (1982); Meuren et al. (2016); Blum (2012)

[TABLE]

[$\mathrm{Tr}(\rho^{(0)})=S_{0}^{(0)}$, $\mathrm{Tr}(\rho)=S_{0}$; $S_{0}\leq S_{0}^{(0)}$, in general, as the photons can decay in the strong background field].

The real Stokes parameters $S^{(0)}=\{S_{0}^{(0)},\boldsymbol{S}^{(0)}\}$ [$\boldsymbol{S}^{(0)}=(S_{1}^{(0)},S_{2}^{(0)},S_{3}^{(0)})$] and $S=\{S_{0},\boldsymbol{S}\}$ [$\boldsymbol{S}=\left(S_{1},S_{2},S_{3}\right)$] completely characterize the initial (final) polarization state of the beam Blum (2012); Born and Wolf (1999). Therefore, the following relations describe any possible vacuum birefringence and/or dichroism experiment [see Eqs. (4), (6), (9), and (10)]

[TABLE]

Here, $\delta\phi=\phi_{2}-\phi_{1}$ is related to vacuum birefringence and $\delta\lambda=\lambda_{2}-\lambda_{1}$ to vacuum dichroism.

In the following, we discuss possible high-energy vacuum birefringence and/or dichroism experiments (see Fig. 3a) at the Apollon facility (F1/F2 laser) Papadopoulos et al. (2016), ELI-NP (two $10\,\mathrm{PW}$ lasers) Negoita et al. (2016); Turcu et al. (2016), and ELI-Beamlines (ELI-BL; L3/L4 laser) Rus et al. (2013). At each facility, a 10 PW laser is employed to polarize the vacuum and the second laser is utilized to produce electron bunches via laser wakefield acceleration Leemans et al. (2014); Wang et al. (2013). We also consider a possible experiment (denoted as LINAC-L) at a conventional electron accelerator, e.g., the European XFEL eur , FACET-II fac , or SACLA Yabashi et al. (2015), combined with a high-repetition (10 Hz) 1 PW laser. The parameters of the considered facilities are summarized in the Supplemental Material 333See Supplemental Material for technical details, which includes Refs. Akhiezer and Berestetskii (1969); Le Garrec et al. (2014); eli ; Tsai (1974); Riley et al. (2006); James (2006); Ku (1966).

We assume that $N_{e}=10^{8}$ monoenergetic few-$\mathrm{GeV}$ electrons are used in one experimental cycle for the generation of probe gamma photons via Compton backscattering.

For a rectangular pulse with $N$ cycles {$\psi^{\prime}(kx)=\sin(kx)$ if $kx\in[-N\pi,N\pi]$ and $\psi^{\prime}(kx)=0$ otherwise}, the relative phase shift $\delta\phi$ depends only on $\chi$ and $\xi N$; it is plotted in Fig. 4. We conclude that $\left|\delta\phi\right|\lesssim 0.1$ for upcoming laser systems in the regime $0.1\lesssim\chi<1$, where a clean vacuum birefringence measurement is feasible as pair production is exponentially suppressed. Notably, the quantity $\delta\phi$ decreases with the increase of the probe photon energy for $\chi\gtrsim 2.5$, which characterizes the anomalous dispersion of the vacuum in this regime Heinzl and Ilderton (2009); Dinu et al. (2014); Ritus (1985); Baier et al. (1976); Becker and Mitter (1975).

For obtaining better estimates as those given in Fig. 4, in the following, we employ a Gaussian pulse envelope $\psi^{\prime}(kx)=\exp[-({kx}/{\Delta\phi})^{2}]\sin(kx)$, where $\Delta\phi$ is related to the duration of the pulse $\Delta t$ (FWHM of the intensity) via $\Delta\phi=\omega_{\textrm{L}}\Delta t/\sqrt{2\ln 2}$. This pulse collides with $N_{\gamma}=N_{e}\sigma_{\textrm{bs}}({I_{\textrm{bs}}}/{\omega_{\textrm{bs}}})\Delta t_{\textrm{bs}}$ gamma photons, where $\sigma_{\textrm{bs}}$ is the cross section of Compton scattering Berestetskii et al. (1982), and the index “bs” indicates the parameters characterizing the backscattering process. To obtain a high degree of polarization, we consider only photons which are scattered in the region $\theta\in(0,\theta_{\textrm{max}}\ll 1)$, where $\theta$ denotes the polar angle ($\theta=0$ corresponds to perfect backscattering) Muramatsu et al. (2014); Weller et al. (2009); Fukuda et al. (2003); Ginzburg et al. (1984); Berestetskii et al. (1982),Note (3).

Below, we employ $\Delta t_{\textrm{bs}}=\Delta t$, $\omega_{\textrm{bs}}=1.55\,\mathrm{eV}$, and $I_{\textrm{bs}}=4.3\times 10^{16}\,{\mathrm{W}}/{\mathrm{cm^{2}}}$ [considering linear Compton scattering is sufficient as $\xi_{\textrm{bs}}=0.1$ for this laser; see Eq. (3)].

One of the main experimental challenges is to analyze the final polarization state of the gamma photons. Here, we consider pair production in a screened Coulomb field of charge $Z|e|$ Hunter et al. (2014); Bernard (2013); Kelner et al. (1975); Olsen and Maximon (1959). The spin-summed pair production cross section is given by

[TABLE]

where $\varphi$ denotes the azimuth angle of the electron momentum in the transverse plane. For $\sigma_{0}$, $\sigma_{1}$, we use expressions exact in $Z\alpha$ and valid for ultrarelativistic particles Kelner et al. (1975); Olsen and Maximon (1959),Note (3). In the following, we assume a head-on collision [$q^{\mu}=\omega(1,0,0,1)$, $k^{\mu}=\omega_{\textrm{L}}(1,0,0,-1)$, $\Lambda_{1}^{\mu}=(0,1,0,0)$, $\Lambda_{2}^{\mu}=(0,0,1,0)$], and tungsten ($Z=74$) as conversion material.

As the pair-production cross section is only sensitive to linear polarization [$S_{1}$ and $S_{3}$, see Eq. (12)], we conclude from Eq. (11) that we need to utilize circularly polarized probe photons (e.g., $S^{(0)}=\{1,0,-1,0\}$) in order to obtain probabilities which depend on $\delta\phi$ [rather than $(\delta\phi)^{2}$] if $\left|\delta\phi\right|\ll 1$ (see also Wistisen and Uggerhøj (2013); Cantatore et al. (1991)). Therefore, inverting the standard scheme by using circularly instead of linearly polarized probe photons is highly beneficial in the regime $\left|\delta\phi\right|\lesssim 0.1$.

From Eq. (11), we conclude that $S_{1}$ is sensitive to vacuum birefringence ($\delta\phi$), whereas $S_{3}$ depends on vacuum dichroism ($\delta\lambda$). To disentangle both effects, we introduce the following asymmetries:

[TABLE]

where $N_{\beta_{0}}$ denotes the number of pairs detected in the azimuth angle range $\varphi\in(\beta_{0}-\beta,\beta_{0}+\beta)$ of the transverse plane, with $\beta$ being specified below (see Fig. 3b). The expectation values of $R_{\textrm{B}}$ and $R_{\textrm{D}}$ are given by [see Eq. (12)]

[TABLE]

In order to detect vacuum birefringence (dichroism) at the $n\sigma$ confidence level on average, we require that the expectation value $\langle{R_{\textrm{B}}}\rangle$ ($\langle{R_{\textrm{D}}}\rangle$) differs from zero by $n$ standard deviations. Therefore, we obtain the following expressions for the number of required incoming gamma photons (see Supplemental Material Note (3)):

[TABLE]

[by minimizing $N\indices*{{}_{\gamma}^{\textrm{B}}}$ ($N\indices*{{}_{\gamma}^{\textrm{D}}}$), we find the optimal angle $\beta=\beta_{\textrm{opt}}\approx 0.58\approx 33^{\circ}$ for both observables]. Here, $\eta=n_{z}l\sigma_{0}$ denotes the photon to pair conversion efficiency ($n_{z}$ and $l$ are the number density and the thickness of the conversion material, respectively). The thickness of a conversion foil should be $\lesssim 1$ milliradiation length (mRL), otherwise multiple Coulomb scattering affects the measured angle Hunter et al. (2014); Kelner et al. (1975). Supposing that several conversion foils alternating with silicon detectors are cascaded Peitzmann (2013); Atwood et al. (2009); Tavani et al. (2003), we assume here $\eta=10^{-2}$ (i.e., an effective thickness of $\sim 10\,\mathrm{mRL}$).

To obtain a clean vacuum birefringence experiment without real electron-positron pair production, we consider the case $\chi=0.25$. The results for the four facilities under consideration are summarized in Table 1. As expected from Fig. 4, ELI-Beamlines is the most suitable facility for carrying out the measurement in this regime (the expected measurement time is less than one day).

As the number of required gamma photons $N\indices*{{}_{\gamma}^{\textrm{B}}}$ scales as $\langle{R_{\textrm{B}}}\rangle^{-2}$ [see Eq. (15)], the use of circularly polarized probe photons instead of linearly polarized ones reduces the measurement time by a factor $\approx 100$ ($\delta\phi\approx 0.1$, see Fig. 4).

Finally, we consider the case $\chi=2.5$ (attainable, e.g., at ELI-NP by utilizing $8.4\,\mathrm{GeV}$ electrons for backscattering; $\theta_{\textrm{max}}=7.6\times 10^{-6}$, $\sigma_{\textrm{bs}}=0.135r_{e}^{2}$, $\omega=1.4$ GeV, $\sigma_{1}/\sigma_{0}=0.077$; $r_{e}=\alpha/m=2.818\times 10^{-13}$ cm is the classical electron radius). In this regime, vacuum dichroism and anomalous dispersion come into play and the Euler-Heisenberg approximation breaks down completely (see Fig. 4), whereas the production of particles, heavier than electrons and positrons, and QCD corrections are still suppressed Bern et al. (2001). As the produced pairs radiate gamma photons, a discrimination of primary from secondary photons is necessary, e.g., via determination of the photon energy. For $S^{(0)}=\{1,0,-1,0\}$, we obtain that $S=\{0.18,0.11,-0.12,0.09\}$ at ELI-NP (see Fig. 5). Correspondingly, $\langle{R_{\textrm{B}}}\rangle=3.6\times 10^{-2}$ and $\langle{R_{\textrm{D}}}\rangle=3.0\times 10^{-2}$, implying a measurement time of 3-4 days [$5\sigma$ confidence level, see Eq. (15)].