Towards pair production in the non-perturbative regime

The creation of matter purely from the interaction of two light quanta in vacuum is one of the most fundamental predictions [1] in modern physics to elude experimental investigation to date. The opportunity afforded by the advent of ultra-intense lasers capable of reaching intensities of 10²¹ W cm⁻² and beyond [2, 3] and simultaneously driving laser-wakefield accelerators (LWFA) capable of reaching multi-GeV energies [4–6] over cm-distance scales has brought the challenge of testing vacuum pair creation—once thought insurmountable—into reach for mid-scale facilities. First concrete designs and proposals were made for the GEMINI laser system at the Rutherford Laboratory, UK [7] with several groups discussing aspects and possible geometries of such experiments since [8–12].

The creation of matter–antimatter pairs from the quantum vacuum occurs either via the Schwinger [13, 14] or the Breit–Wheeler (BW) mechanism [1] or combinations thereof [15, 16]. The Schwinger mechanism requires a field strong enough and capable to perform the work equivalent to the rest mass energy of the particle to create a particle pair over its reduced Compton wavelength ${\mathcal{E}}_{\mathrm{S}}={m}_{\mathrm{e}}{c}^{2}/(e{\lambda }_{c})=1.3\times 1{0}^{18}\enspace \mathrm{V}\enspace {\mathrm{m}}^{-1}$ corresponding to a laser intensity of I ≈ 10²⁹ W cm⁻². Pair production through the BW reaction [1] in its simplest form requires the collision of two energetic photons:

$$\begin{equation}\gamma +{\gamma }^{\prime }\to {\mathrm{e}}^{+}+{\mathrm{e}}^{-}.\end{equation} \tag{ 1 }$$

In the BW process, the energies of the participating photons with E = ℏω must be such that the total collision energy is sufficient to produce the rest mass energy of the electron–positron pair. The physics of the BW pair production process is governed by the two Lorentz and gauge invariant parameters: a₀ and χ_γ . The normalised laser vector potential, which is given in rationalized natural units (c = ℏ = ₀ = 1) by ${a}_{0}=e\enspace \mathcal{E}/({m}_{\mathrm{e}}{\omega }_{0})$, with the laser electric field amplitude $\mathcal{E}$, serves as the inverse Keldysh parameter of the problem.

The quantum nonlinearity (or quantum efficiency) parameter χ of a (massive) particle is the ratio of the laser field strength in the particle's rest frame to the Schwinger field ${\mathcal{E}}_{\mathrm{S}}$. For the collision of an electron with an intense field at an angle θ it can be expressed as ${\chi }_{\mathrm{e}}=(1-{\beta }_{\mathrm{e}}\enspace \mathrm{cos}\enspace \theta ){\gamma }_{\mathrm{e}}\mathcal{E}/{\mathcal{E}}_{\mathrm{S}}$. More pertinently, for our considerations for the collision of a γ-ray photon of frequency ω_γ with an intense laser beam at the angle θ the quantum nonlinearity parameter reads

$$\begin{equation}{\chi }_{\gamma }={a}_{0}\enspace \frac{{\omega }_{\gamma }{\omega }_{0}(1-\mathrm{cos}\enspace \theta )}{{m}_{\mathrm{e}}^{2}}.\end{equation} \tag{ 2 }$$

In analogy to atomic ionisation phenomena [17], pair creation experiments can be divided into three regimes, distinguished by the values of a₀ and χ_γ :

• (a)  
  The linear regime (a₀ ≪ 1 and χ_γ ≪ 1), first considered by Breit and Wheeler [1], where a binary collision of two photons with frequencies ω₀, ω_γ provides sufficient energy to produce an electron–positron pair. In the centre-of-momentum (COM) frame [18] the threshold condition can be written as ${\omega }_{0}\enspace {\omega }_{\gamma }\geqslant {m}_{\mathrm{e}}^{2}$. This is analogous to atomic photo-ionisation by a photon exceeding the binding energy of the bound electron.
• (b)  
  The multi-photon regime (a₀ ⪅ 1, χ_γ < 1), where N ⩾ N₀ photons from the intense field collectively contribute to meeting the pair-production threshold

  $$\begin{equation}\gamma +{N}_{\text{laser}}\enspace {\omega }_{0}\to {\mathrm{e}}^{+}+{\mathrm{e}}^{-}.\end{equation} \tag{ 3 }$$

  The minimum number of photons N₀ required for the reaction to occur is determined by the threshold for the multi-photon channels $(N{\omega }_{0}){\omega }_{\gamma }\geqslant {m}_{\star }^{2}$, where ${m}_{\star }={m}_{\mathrm{e}}\sqrt{1+{a}_{0}^{2}/2}$ is the intensity-dependent effective mass of the leptons [15, 19]. In the multi-photon regime, the N-photon channels exhibit a perturbative scaling with the Nth power of the laser intensity.
• (c)  
  The non-perturbative, quasi-static strong-field regime a₀ ≫ 1, where typically a very large number of laser photons are required to overcome the threshold. In this regime, the strong laser field can be considered to be 'deforming' the mass-gap between the negative and positive continuum. This is analogous to the suppression of the atomic potentials where strong-field ionization proceeds by either tunnelling of the wavefunction through (or propagation over) the potential barrier. In this regime, the pair production rate depends strongly on the value of the quantum parameter χ_γ . For χ_γ ≪ 1 the pair production rate scales as P ∼ χ_γ  exp(−8/3χ_γ ), with an exponential suppression typical for tunnelling processes [18–20]. Contrary, if χ_γ ≫ 1 the rate scales as $P\sim {\chi }_{\gamma }^{2/3}$, which could be considered the equivalent of the over-the-barrier regime.

An experimental investigation of the linear regime (a) has been precluded to date by the small cross-section (about 150 mb) and the lack of photon beams with the required energy and luminosity. Consequently, the linear BW process is still considered as the most difficult fundamental QED process to be observed experimentally on Earth.

Despite the importance of this reaction in cosmology and its relevance to γ-ray colliders it has yet to be demonstrated experimentally in the absence of charged particles in the interaction region and therefore free of all competing processes. First experimental evidence of the non-linear BW process was obtained in 1996 in an experiment in which pairs were observed in laser-electron beam interactions. In this experiment, performed at SLAC, 47 GeV electron bunches collided with an intense (I = 5 × 10¹⁷ W cm⁻², a₀ = 0.4, λ = 527 nm) laser pulse [21–23]. The interaction resulted in the production of Compton γ-rays with photon energies up to 29.2 GeV. At this very high γ-energy, conservation of energy demands at least 5 laser photons to take part in the interaction, placing this experiment into the multi-photon regime (b).

Figure 1 shows a plot of the different interaction regimes as a function of γ-ray energy (E_γ = ℏω_γ ) and laser strength a₀. The experiment detailed here assumes e-beam parameters of 2.5 GeV electrons and a₀ ≈ 66, placing it into the unexplored quasi-static regime of strong-field pair production, case (c).

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In the following we present the detailed design case developed by the Research Collaboration 'Probing the Quantum Vacuum' [24] for the ATLAS laser facility. We show various aspects of the current design and highlight that single lepton detection above background is possible at experimentally accessible rates in the current configuration. We highlight the trade-off between signal rates and the background which affects the signal-to-noise ratio (SNR) in this experimental configuration, which optimise for relatively thin converter foils and intensities below the theoretical maximum achievable. We also present full shielding simulation with GEANT4 that minimises the production Bethe–Heitler pairs from apertures etc and demonstrate detector simulations showing an SNR of ≈10 for our design point.

The challenge of investigating the BW phenomenon in a laboratory has always been recognised as formidable and engaged physicists for decades. The first proposal to perform an 'all-optical' experiment based on CPA laser technology [7, 25] and LWFA accelerators [5, 26, 27] was centred on the Astra-Gemini laser [7], which demonstrated the key components of the current experimental design—in particular the analyser magnet system [28], LWFA beams with 2.5 GeV energy from [29] and successful timing overlap of an LWFA driving laser and the collider laser beams [30].

The schematic layout of the experiment is as shown in figure 2 and follows the same basic arrangement first introduced in the GEMINI campaign [7, 28]. Two intense, PW-class laser beams are required to drive the experiment, a weakly focussed LWFA beam and the tightly focussed high-intensity beam. In the case of our experiment the ATLAS laser at CALA produces a single beam with the energy limited to 45 J (of the available 60 J) in 30 fs and consequently a power of 1.5 PW with a central wavelength of λ = 800 nm. This beam is then split spatially into a central part containing 9.5 J for the ultra-intense beam and ring-shaped beam containing about 30 J (after splitting losses) to drive the multi-GeV laser-wakefield accelerator.

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The weakly focussed 'LWFA-laser' beam accelerates electrons to multi-GeV energies using the LWFA method [27]. The electrons then interact with a high-Z converter foil to produce high-energy bremsstrahlung γ-ray photons. The electrons are deflected by the first magnet to an electron spectrometer, while the bremsstrahlung γ-rays interact with the tightly focussed 'collider' laser at the interaction point (IP). A key consideration is minimising the distance D from the LWFA stage to the IP to increase the γ-photon flux at the IP. The installation into a multi-purpose facility favours the use of permanent magnets with field strengths of approximately 1 T.

At the IP, the interaction between the γ-photons and the collider laser produces BW pairs that propagate towards the analyser magnet system consisting of a separator and collimator magnet located after a radiation shield wall. This allows the intense γ-ray beam to be separated from the pairs, which are detected on a single event sensitive detector array consisting of Cherenkov calorimeters and a pixelated LYSO array. Calibration of the detector system will be performed by placing a thin foil source of Bethe–Heitler pairs in the IP.

2.1. Pair creation probability and optimal intensity

The primary experimental parameters of laser intensity and γ-ray energy are set by using the available laser energy to achieve high pair creation probability in the IP. We base our experimental design on the previously published theoretical work considering the non-linear BW interaction [19, 31], and note the significant body of further work e.g. [8, 32–41].

Assuming that a γ-ray photon of energy E_γ collides with a linearly polarized laser beam of intensity I_laser, photon energy ℏω₀ = 1.55 eV, the probability rate of pair creation dP/dt of the nonlinear BW process is given, in rationalized natural units and employing the locally constant field approximation, by [19]

$$\begin{equation}\frac{\text{d}P}{\text{d}t}=-\frac{\alpha {m}_{\mathrm{e}}^{2}{\chi }_{\gamma }}{16{\omega }_{\gamma }}{\int }_{{z}_{0}}^{\infty } \text{d}z\enspace \frac{8u+1}{\sqrt{z}\sqrt{{u}^{3}(u-1)}}\enspace {\mathrm{A}\mathrm{i}}^{\prime }(z),\end{equation} \tag{ 4 }$$

with ${z}_{0}={(4/{\chi }_{\gamma })}^{2/3}$ and $u={(z/{z}_{0})}^{3/2}$. Figure 3(a) shows the pair conversion probability per cycle of the laser pulse, i.e., the rate in (4) integrated over one laser cycle, as a function of the laser intensity. The exponential suppression of the pair production probability predicted by theory for a non-perturbative process below threshold is clearly visible at low intensities. At high intensities (for values of χ > 1) the effective non-linearity q of the pair production probability P(a₀) ∝ I^q reduces significantly and the pair production probability per cycle exceeds 10% per incident γ-photon requiring depletion of the γ-photons to be taken into account.

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The optimal intensity to maximise the pair yield for fixed laser power is determined by the effective non-linearity parameter q for this experimental configuration. Under the assumption of fixed laser power and a large γ-spot area A_γ > A_laser the number of γ-photons interacting with the intense field increases linearly with focal spot area as N_γ ∝ A_laser, while the intensity decreases linearly with $I\propto {A}_{\text{laser}}^{-1}$.

In the exponentially suppressed regime, characterised by a high effective non-linearity q, it is clearly desirable to maximise the laser intensity. However, in the limit of high intensities where q < 1 the pair yield N_pairs ∼ N_γ P increases sub-linearly and larger laser spots at the optimum intensity rather than the highest intensity result in maximum pair yield. The ideal intensity can be seen to be in the range of 0.5...3 × 10²² W cm⁻² in figure 4 for the accessible electron-beam energy range of 1...5 GeV. We choose to base our design on an electron beam energy of 2.5 GeV and a peak intensity close to 10²² W cm⁻², where q ≈ 1. The collision angle between the γ-ray beam and the laser pulse was chosen to be 162° which does not significantly reduce the pair yield as compared to the theoretical optimum of 180° (see figure 3(b), a detailed explanation for the chosen angle is given in the following section 3.1). The potential gains in pair yield from higher electron beam energies would be offset by the increase in the magnet length and therefore D assuming a constant γ-ray divergence set by the electron beam.

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The expected pair spectrum is then determined by the γ-ray spectrum. For a monoenergetic γ-ray beam, the energy spectra for the pairs are symmetrical and centred at E_γ /2 as is to be expected from the kinematics. For our parameters, the most efficient approach to generate a γ-beam is via bremsstrahlung in a high-Z foil (details see below) placed shortly after the LWFA target as seen in figure 2, which results in broad spectra with the highest photon numbers produced at low energies. This leads to asymmetrical experimental pair spectra shifted to lower energies. Figure 5 shows the positron energy spectrum broadening and shifting for different values of the normalized vector potential a₀.

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2.2. Laser wakefield accelerator and γ-ray beam production

The LWFA stage is designed to be capable of producing multi-GeV electron beams. The LWFA drive beam will contain an energy of up to 30 J, focussed with an f/40 parabolic mirror. Experiments with similar laser power have previously been used to generate electron beams of up to 7.8 GeV [42]. Using a conservative estimate the setup is thus expected to allow stable generation of electron beams with an endpoint energy of 2.5 GeV. While these electron beam energies are substantially below the ultimate limit of what is achievable with 30 J in the LWFA beam [4] using external guiding, the simplicity of a self-guided accelerator [29] is preferred for this initial experiment.

First tests with the ATLAS laser were performed at a reduced laser energy of ≈5 J, using an f/30 focussing geometry and a 20 mm-long slit nozzle with a plateau electron density of ≈10¹⁸ cm⁻³. Electrons were injected via ionization-induced process in a gas mixture containing 96% hydrogen and 4% nitrogen and routinely reached a peak energy of around 1 GeV. A typical electron spectrum with an FWHM divergence in the peak of θ_e = 0.48 mrad can be seen in figure 6. The observed beam energies follow approximately the scalings for matched acceleration in the bubble regime where the maximum energy gain is given by [43]

$$\begin{equation}{\Delta}{E}_{\mathrm{max}}\enspace (\mathrm{G}\mathrm{e}\mathrm{V})\approx 1.7\cdot {(P[100\enspace \mathrm{T}\mathrm{W}])}^{1/3}\cdot \enspace {({n}_{\mathrm{e}}[1{0}^{18}\enspace {\mathrm{c}\mathrm{m}}^{-3}])}^{-2/3}\cdot {(\lambda [800\enspace \mathrm{n}\mathrm{m}])}^{-4/3},\end{equation} \tag{ 5 }$$

which evaluates to ΔE_max > 2.5 GeV for a driving laser power of P_L > 300 TW at λ = 800 nm and a plasma density n_e < 10¹⁸ cm⁻³. It is therefore expected that operation at the design energy of the experiment (30 J, P_L ≈ 1 PW) and at matching density will lead to the desired peak energies above 2.5 GeV.

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The LWFA stage is followed by a thin tape target, to separate the remaining laser light of LWFA beam from the remaining set-up by forming a highly reflective plasma mirror [44], while negligibly perturbing the electron beam.

The conversion of the electron beam to γ-rays takes place in a high-Z tungsten target for high efficiency. Tungsten has among the highest radiation yield of readily available target materials (368 MeV cm² g⁻¹ for interaction with a 2.5 GeV electron).

Optimising the bremsstrahlung conversion is a critical part of the optimising the signal and controlling the background. As is well known the overall bremsstrahlung yield optimises for target lengths close to the material radiation length, L_rad = 3.5 mm in the case of tungsten. The bremsstrahlung photon spectrum produced by a relativistic charged particle with energy E₀ passing through a thin converter target of high-Z material with radiation length equal to L_rad and thickness L produces a radiation spectrum given by the approximated equation [45]:

$$\begin{equation}\frac{\text{d}{N}_{\gamma }}{\text{d}{E}_{\gamma }}=\frac{L}{{E}_{\gamma }{L}_{\mathrm{r}\mathrm{a}\mathrm{d}}}\left[\frac{4}{3}-\frac{4}{3}\frac{{E}_{\gamma }}{{E}_{0}}+{\left(\frac{{E}_{\gamma }}{{E}_{0}}\right)}^{2}\right].\end{equation} \tag{ 6 }$$

In figure 7(a) we see how the Bremsstrahlung spectrum evolves for increasing converter thickness L. The rapid reduction of the pair production probability with decreasing γ energy places the emphasis on optimising the flux of high energy γ-photons with energy E_γ > 0.9E₀. The softening of the spectrum due to electron energy loss and scattering of the high energy γ-rays shifts the optimum to lengths of L < L_rad. Note the negligible increases in yield above 0.9E₀ = 2.25 GeV as the target thickness is increased from 1.0 mm to 2.5 mm.

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Multiple scattering of the electrons with increasing L also increases the FWHM angular distribution of the emitted photons. As shown in figure 7(b), the angular distribution begins to exceed the limit for a collimated beam reducing the gains in flux at the IP, with 90% of the maximum γ-flux at the IP reached L ≈ 0.4L_rad. Figure 7(c) shows the hard γ-flux for different tungsten converter target thicknesses at the IP. Note that although the overall Bremsstrahlung yield increases the hard γ-flux saturates due to increased divergence and softening of the spectrum with increased converter thickness. The simulated γ-flux distribution for different converter targets is shown in figure 8 with the increasing divergence and stochastic nature of the photon distribution visible.

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3.1. Photon–photon interaction point

We can now estimate the expected pair yield N_pairs for the experiment per primary electron bunch. In doing so we proceed by averaging the pair production probability per cycle P over the top 10% bremsstrahlung spectrum obtaining the pair yield per incident primary electron from the LWFA stage,

$$\begin{equation}{N}_{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s}}={{\Phi}}_{\text{gamma}}\cdot {A}_{\text{laser}}\cdot {\theta }_{\text{div}}\cdot {N}_{\text{cycles}}\cdot P.\end{equation} \tag{ 7 }$$

where Φ_gamma is the gamma-photon flux provided in figure 8 for different target thickness, A_laser is the collider laser focal spot area, N_cycles is the number of laser cycles and P is the average pair production probability per cycle as provided in figure 3(a). Note that the photon flux Φ_γ in figure 8 was calculated for collimated electron beam with zero source size.

The constant factor θ_div represents the 'divergence loss' of the bremsstrahlung flux, which in our setup is dominated by the divergence of the electron beam θ_e and, which is given by ${\theta }_{\text{div}}={\theta }_{\gamma }^{2}/({\theta }_{\gamma }^{2}+{\theta }_{\mathrm{e}}^{2})\approx 0.14$, where θ_γ is the divergence of the bremsstrahlung photons for collimated electrons.

While the pair conversion rate per γ photon is high, reaching over 50% at the centre of the laser focus, the overlap of the laser spot with the γ-beam is small therefore reducing the number of gamma-laser interactions.

Based on a charge of 10 pC per shot we find that our design point can achieve a maximum number of created positrons of 0.22 per laser shot. Figure 9 presents the pair yield expected per hour for different converter target thicknesses for 0.1 Hz repetition rate. Therefore, at our design point (electron energy of E₀ = 2.5 GeV with divergence θ_e = 0.5 mrad, 50 μm tungsten converter foil and collider laser intensity of 9.5 × 10²¹ W cm⁻²), the pair yield is expected to be of the order of 80 pairs per hour. Much higher pair production rates of >1000  pairs/h are possible with thicker targets and higher bunch charges, which could realistically reach 100 pC per shot.

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Note that the calculated pair yields are based on a constant laser spot-size and a γ-ray divergence of 0.5 mrad set by the LWFA beam. Adding a collimating lens after the LWFA stage would allow the divergence of a 2.5 GeV bremsstrahlung beam to be reduced to ≈1/γ = 0.2 mrad and commensurately higher γ-ray flux and pair yield. Varying the laser intensity by maintaining constant laser pulse energy but increasing the laser spot-size results in a slower decay of the pair yield than the case considered in figure 9. The experimentally accessible intensity range is therefore greater than might be inferred from the figure and measurements in the range of 0.5 < χ_γ < 2 are feasible. The ability to vary the LWFA electron-beam energy will allow the impact of γ-ray energy on the cross-section to be evaluated experimentally in the range of 2...5 GeV.

The limitation of the design point to 50 μm thin targets and low charge beams may seem surprizing given the substantial reduction in pair yield by approximately two orders of magnitude and is explained by maintaining low backgrounds in a detector system with low granularity and tracking capability.

While single GeV leptons are easily detected with a variety of different approaches, the challenge is to ensure a sufficiently high SNR in the presence of 10⁸ GeV electrons and a similar number of γ-ray photons with energies above 10 MeV. Care needs to be taken to prevent the production of high energy leptons via Bethe–Heitler pair production and Compton scattering.

The detector system consists of a multi-channel lead glass Cherenkov calorimeter with LYSO screens. Both systems have ns-scale temporal resolution to suppress scattering (e.g. from the beam dump) that arrives at the detectors with significant time delay. This system is described in detail in [46].

For the entire range of pair rates considered in figure 9 the BW leptons will be detected as single events (no pile-up) per detector resolution element in both LYSO and Cherenkov detectors. Therefore each BW-event results in well defined signal on the detectors that does not depend on the converter thickness L or the beam charge.

By contrast, the backgrounds scale linearly with beam charge and increase non-linearly when converter thickness is taken into account. This behaviour can be understood by considering the bremsstrahlung emission previously presented. In figure 7, we see that the overall yield of bremsstrahlung photons increases linearly with converter thickness for the range of L of interest here, while the spectral component >2.2 GeV saturates for small values of L < 0.5. Additionally, we find that the ratio of peak γ-intensity to the wide-angle radiation decreases with converter length L as seen in figure 7(b). While the useful γ-conversion in the top 10% of the spectrum increases only by a factor of ≈2 the signal at close to 1 mrad increases 4 fold when the converter thickness reaches 2.5 mm. To optimise the SNR we therefore find thin converter foils with L ≪ L_rad to be ideal with our design choosing 50 μm W-foil (compared to 3.5 mm radiation length).

We conducted GEANT4 [47–49] simulations as shown in figure 10 taking into account the full experimental geometry. For a 50 μm converter foil our simulations indicate only a small number of background events after temporal gating of the detectors as shown in figure 11 and therefore a signal to noise ratio (SNR) of >10 for a 500 MeV lepton incident on the Cherenkov detectors. This reduces to SNR < 1 for the thicker target considered.

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In conclusion we have demonstrated a fully simulated design to measure the non-linear BW effect using an all-optical set-up. Ensuring a detectable and sufficiently high signal rate requires balancing a range of requirements within a large parameter space. An important consideration in optimising the experimental design is to maximise the SNR at detectable signal rates rather than maximising the signal rate itself. Counterintuitively, the trade-off between signal rate and SNR optimises for sufficiently thin bremsstrahlung converter foils at moderate charge (L ≪ L_rad) and an e-beam energy limited to 2.5 GeV—significantly below the achievable maximum. Similarly, rather than simply maximising the laser intensity, there is an e-beam energy dependent optimum in the mid-10²¹ W cm⁻² range. Under these conditions experimentally viable yields are predicted with very low background. Introduction of a plasma-lens to focus the electron beam can increase the signal rates by up to a further order of magnitude.

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG) under Project Nos. 416708866; 416699545 within the Research Unit FOR2783/1.

The data that support the findings of this study are available upon reasonable request from the authors.