Advances in QED with intense background fields

Abstract

Upcoming and planned experiments combining increasingly intense lasers and energetic particle beams will access new regimes of nonlinear, relativistic, quantum effects. This improved experimental capability has driven substantial progress in QED in intense background fields. We review here the advances made during the last decade, with a focus on theory and phenomenology. As ever higher intensities are reached, it becomes necessary to consider processes at higher orders in both the number of scattered particles and the number of loops, and to account for non-perturbative physics — for example the Schwinger effect — with extreme intensities requiring resummation of the loop expansion. In addition to increased intensity, experiments will reach higher accuracy, and these improvements are being matched by developments in theory such as in approximation frameworks, the description of finite-size effects, and the range of physical phenomena analysed. Topics on which there has been substantial progress include: radiation reaction, spin and polarisation, nonlinear quantum vacuum effects and connections to other fields including physics beyond the Standard Model.

1. Introduction

In high-intensity laser pulses, electrons can be accelerated to relativistic velocities over a single laser wavelength. Such lasers, made possible by chirped pulse amplification, for which the 2018 Nobel prize was awarded, have great potential not only for applications in the sciences, industry and medicine, but also as a tool to probe fundamental quantum physics. Pulses of light in which the photon density surpasses one photon per Compton wavelength cubed are now routinely produced at modern laser facilities. They provide a means of experimental investigation complementary to accelerator searches for probing the "intensity frontier" of the Standard Model.

Laser light is well described by a coherent state, in which the interaction of laser photons with charged matter adds coherently. In high intensity laser pulses, the charge-field coupling becomes large enough that the perturbative hierarchy is disrupted and the interaction between the charge and the laser must be accounted for to all orders in perturbation theory, or non-perturbatively. This is an example of "non-perturbativity at weak coupling"; while the fine structure constant remains small, the photon density in a laser scales as the square of the dimensionless intensity parameter divided by the fine structure constant, and that intensity parameter nowadays easily exceeds unity, so that the effective charge-field coupling — the square root of the fine structure constant times the photon density — is of the order of the intensity parameter and is much greater than one.

This situation is in contrast to existing high-precision tests of quantum electrodynamics, where electromagnetic fields are low intensity and calculations can be performed perturbatively. For example, the electron anomalous magnetic moment and fine-structure constant have been measured to agree with theory up to fifth order in the fine structure constant, or better than one part in a billion. Other recent high-profile tests of QED in ultra-peripheral heavy ion collisions, where light-by-light scattering, by ATLAS and CMS, and linear Breit-Wheeler pair-creation, by STAR, have been measured for the first time, are found to be consistent with perturbative QED calculations.

An intense, or strong, electromagnetic field can be regarded as a coherent state of high occupation number or, by the correspondence principle, an essentially classical field. For high occupation number, one can also neglect back-reaction on the field and consider it as fixed. Again, this approximation eventually breaks down, as will be discussed. The appropriate theory for studying, for example, high-intensity laser-matter interactions is therefore quantum field theory in an external, or background, field — sometimes the term "strong-field QED" is used.

This is a subject as old as quantum field theory itself, and within it lies, to illustrate, the topic of Schwinger pair-production from an external field, also called the Sauter-Schwinger effect. This is perhaps the most familiar, and "prototype", example of a non-perturbative quantum field theory effect which can admit an analytic treatment. Significant progress has been made in understanding the theory and phenomenology of the Schwinger effect in more realistic backgrounds, with a particular focus on models of colliding laser pulses, which offer one route toward eventual experimental measurement. Despite this progress, there remain many unanswered questions about the time-resolved Schwinger effect: when are the pairs produced, or "become real", what can one say about the behaviour of the system at non-asymptotic times, and what can be measured? These are questions which go to the heart of quantum mechanics and which arise also in tunneling ionisation and in cosmological scenarios.

Much of the work on which this review builds began in the 1960s, shortly after, and inspired by, the invention of the laser itself. In most of these early papers, laser fields were modelled as monochromatic plane waves, or their low frequency limit, constant crossed fields. This setup allowed for analytic progress in the calculation of scattering observables while, crucially, treating the strong background exactly. However, the model neglects the finite duration of a real laser pulse as well as the structure transverse to its propagation direction, focusing effects, which go hand in hand with shorter pulses and higher intensities. As such, much of the work in subsequent decades has concentrated on the incorporation of more realistic structure in the modelling of laser pulses — "finite size effects" — in tandem with experimental developments.

Considering more realistic descriptions of laser fields, even by introducing only a finite pulse duration, has allowed for progress in many areas. For example, at the time of the E144 experiment on pair production in the collision of a laser and a high-energy electron beam, there was no complete theory of the process being investigated, namely "nonlinear trident pair production", as even the tree level amplitude had not been calculated exactly in strong backgrounds. This illustrates the extreme complexity of amplitudes in strong fields, even for low numbers of scattered particles — four in the case of trident. This situation has now changed; by going beyond the simplest models, and doing so from a quantum field theory perspective, we now have a fairly complete understanding of the nonlinear trident process.

Indeed, it has been found in many cases that the simple constant and monochromatic backgrounds originally studied can often obscure the physics rather than exemplify it. Going beyond this has allowed the resolution of long-standing problems, such as the nature of the much discussed "effective mass" in a laser pulse, and the refinement of models and approximation schemes needed to plan and analyse experiments in the high intensity regime, the advent of which has been a significant driving factor for the field. Predictions for signals of vacuum birefringence have, for example, been made more precise and refined in preparation for upcoming experiments which aim to measure this subtle consequence of light-by-light scattering.

Experimental developments have also renewed interest in the behaviour of QED in extremely strong fields, far beyond what we can realise today. This has led to a great deal of activity surrounding the Ritus-Narozhny conjecture, which suggests that at sufficiently high intensities, not only is the charge-field coupling large, but the fine structure constant itself becomes enhanced by intensity effects. The implication is that QED becomes "fully non-perturbative" in such a regime, requiring all loop orders to be resummed in order to yield reliable results. This conjecture, as well as the application of resurgence in quantum field theory, has inspired new interest in the behaviour of higher loop processes in strong fields, and their resummation.

This review has been written now because of both technological progress and increasing research activity. QED in intense fields has also begun to attract attention from the high energy physics community, and experiments colliding conventionally-accelerated electron beams with intense laser pulses will be performed in E320 at SLAC and LUXE at DESY in the near future. At the same time, a new generation of laser facilities have recently come online (CoReLS), are being commissioned (ELI), are being built (SEL) or are in the process of consultation (MP3). Therefore, a review of recent theory developments is timely for experts and newcomers alike.

1.1. Experimental landscape

QED in intense background fields can be tested in a number of ways. We give here an overview of the region of parameter space that has been, and will be, probed in experiment. This section also introduces some standard parameters for quantifying total particle yields that will be used throughout the review.

The intensity of a background field is often quantified using the dimensionless and gauge-invariant "classical nonlinearity parameter", also called the intensity parameter despite being proportional to the square root of intensity, and written in the literature as xi, or a-zero, or occasionally eta. This parameter occurs naturally in scattering calculations as the dimensionless charge-field coupling, and can be written in a more physical way as the electron charge times the electric field strength times the Compton wavelength, divided by Planck's reduced constant times the background photon frequency. In this form, it is the work done by the background, over the Compton wavelength of the electron, in units of the background photon energy. The probability of leading-order perturbative calculations of tree-level processes is proportional to its square, and so that square can be understood as an approximate measure of the number of photons interacting with an electron. It can be defined for a plane wave in a manifestly gauge-invariant way as the square root of the squared charge times the phase-cycle average of the squared contraction of the probe momentum with the classical field strength tensor, divided by the squared product of the electron mass with the contraction of the background wavevector and the probe momentum.

The energy of the collision between probe and plane wave background can be quantified using a dimensionless linear quantum parameter — in the literature also b-zero — which for a plane-wave background is Planck's reduced constant times the contraction of the background wavevector with the probe momentum, divided by the squared mass times the fourth power of the speed of light. That parameter is therefore equal to the laser frequency in the rest frame of an accelerated charge. If the particle is a photon, it is half the centre of mass energy in a collision with a single laser photon; the threshold for linear Breit-Wheeler pair creation is that it reach two.

The "quantum nonlinear parameter" — occasionally written upsilon or eta in the literature — is given by Equation 1 in the source: the electron charge times Planck's reduced constant times the square root of minus the squared contraction of the probe momentum with the field strength tensor, divided by the cube of the mass times the fourth power of the speed of light. It applies to a general background field and can be interpreted in many ways: for an electron or positron, as the work done by the background over the Compton wavelength in the particle's rest frame, in units of the particle's rest energy; as the ratio of the electric field to the Schwinger limit — the squared mass times the cube of the speed of light, divided by the charge times Planck's reduced constant — in the rest frame of an accelerated charge; as the proper acceleration; or as the worldline curvature of a particle moving under the Lorentz force, times the Compton wavelength. Therefore when this parameter reaches order one, quantum nonlinear processes such as pair creation should become probable. These three parameters are related in a plane wave background: the quantum nonlinear parameter is the product of the classical intensity parameter and the linear quantum parameter.

Several important experimental tests of QED in intense background fields have been performed by colliding proton beams with amorphous media and oriented crystals, where the high energy of the proton beams and the strong static inter-planar crystalline fields combine to give a quantum nonlinearity parameter of order 0.1 to 7. The NA63 experiment collides electrons and protons with energies of order 100 gigaelectronvolts, provided by CERN's Super Proton Synchrotron, with fixed targets of different proton numbers. It has had widespread success in measuring strong-field QED effects in the crystal's background field, which can vary, depending on the transverse momentum variation, from undulator-like to synchrotron-like. In the last decade, NA63 has measured the quantum suppression of synchrotron radiation in the range 0.05 to 0.7 of the quantum nonlinearity parameter, measured the Landau-Pomeranchuk-Migdal effect of radiation suppression due to multiple Compton scattering within the photon formation length, measured radiation reaction in the classical limit, verifying the Landau-Lifshitz equation, and has also observed quantum effects.

Although they have seen much success reaching intensity parameters as large as of order one hundred, a possible future limitation of using oriented crystals is the maximum intensity parameter that can be produced. In contrast, intensities reachable at the next generation of multi-petawatt lasers can in principle soon exceed an intensity parameter of order one hundred, and at multi-petawatt lasers could exceed order one thousand. For example, if a laser of wavelength 800 nanometres is focussed with linear polarisation to an intensity of ten to the twenty-third watts per square centimetre, the intensity parameter corresponds to about 150. Therefore, in the near future, lasers will push further into the region where the intensity parameter is much greater than one, allowing the quantum nonlinear region, where the quantum nonlinearity parameter is of order one, to be probed with lasers for the first time.

(Sections 2 to 11 — the technical body of the review — are omitted for length; the complete text is at the source.)

12. Conclusions and open questions

We have reviewed work in the theory and phenomenology of strong-field QED over the past decade. We conclude by summarising progress made, identifying important lessons learnt, and outlining some open problems and possible avenues of future research, for each of the topics covered in this review.

Light-by-light scattering

In the first part of Section 7 we focused on the Heisenberg-Euler effective Lagrangian, highlighted the importance of one-particle reducible contributions, studied its strong field limit and demonstrated that a perturbative loop expansion of it breaks down for exponentially large fields. It would certainly be very interesting and important to advance the study of higher loop orders of the Heisenberg-Euler Lagrangian from lower to three-plus-one space-time dimensions, particularly by using resummation and resurgence techniques. Another topical research direction is to put forward strategies to go beyond the loop expansion and so obtain insights into the manifestly non-perturbative parameter regime, where such an expansion no longer makes sense.

An important lesson learnt in this context is that a larger Feynman diagram containing a sub-diagram which, considered as an isolated object, vanishes identically because of an overall momentum conserving delta function, can still yield a finite result. Therefore, one should never set such a superficially vanishing contribution to a larger diagram to zero from the outset, but rather keep the full expression and only apply simplifications on the level of the final expression to be calculated. This subtlety was precisely the reason why, for example, one-particle reducible contributions to the Heisenberg-Euler Lagrangian were previously erroneously assumed to vanish.

Moreover, we detailed that analytical results for the photon polarization tensor are presently only available in homogeneous constant and generic plane-wave backgrounds. Remarkably, even in a constant field the one-particle irreducible contribution to the photon-polarization tensor at two loops and arbitrary momentum transfer has not been evaluated to date. Together with the one-particle reducible contribution already determined, this would constitute the full result for the two-loop photon polarization tensor. Apart from this, it would be worthwhile to go beyond constant and plane wave backgrounds and evaluate the one-loop photon polarization tensor in a localized inhomogeneous field. This would for instance allow for the study of vacuum diffraction and quantum reflection phenomena generic to inhomogeneous fields from first principles without needing to employ a slowly varying field approximation.

Finally, to actively assist the discovery of light-by-light scattering phenomena in high-intensity laser experiments, in the upcoming years it will definitely be very important to refine various theoretical estimations of prospective photonic quantum vacuum signals to account for the full details of the actual discovery experiment. This requires an accurate modelling of the precise field configurations available in experiment as well as accounting for any real-world complications such as the inevitable presence of shot-to-shot fluctuations and jitter. The combination of the vacuum emission picture with a Maxwell solver, as well as complementary approaches based on a direct numerical solution of the non-linear wave equation in three-plus-one dimensions, should allow tackling this challenge.

The Schwinger effect and spontaneous pair production

Our understanding of field inhomogeneities in the Schwinger effect is essentially limited to one-dimensional cases. Complete analytical frameworks have not yet been established to go beyond this, though there are attempts based on semi-classical methods and Furry picture expansions. Numerical simulations also become resource-heavy in the presence of multi-dimensional field inhomogeneities. Magnetic-field effects become important for multi-dimensional inhomogeneities, and can induce non-trivial phenomena such as Stern-Gerlach forces, spin-current generation, chiral-magnetic effects and anomaly-induced dynamical refringence, all of which require further investigation.

Field inhomogeneities may also significantly affect radiative corrections to the Schwinger effect, a topic in which there remains a great deal to be understood. Even for the simplest case of a constant electric field, for example, only the two-loop result is available. The exponentiation conjecture still remains an open question. It is also interesting that the holographic approach predicts an upper limit for electric fields in the super-critical regime, which is worthwhile to be tested with field-theoretical calculations. Photon emission due to radiative corrections is also an interesting subject, as it may provide characteristic experimental signatures such as high-harmonic generation, but still requires further theoretical investigation.

There is no established realtime framework for the Schwinger effect that goes beyond the mean-field approximation. Going beyond this is crucially important for the discussion of equilibration processes and possible QED cascades. It is predicted that there exist universal behaviours in far-from-equilibrium quantum processes initiated by overpopulated gauge fields, and it would be interesting to make connections between those processes and the Schwinger effect.

It is envisaged that available laser strengths may reach a few orders of magnitude below the Schwinger field of ten to the sixteenth volts per centimetre, corresponding to an intensity of ten to the twenty-ninth watts per square centimetre, in the near future. It is therefore timely to discuss possible experimental signatures; for laser experiments, it is important to continue investigation into enhancement effects including pulse shape optimisation and multiple colliding pulse setups.

Strong fields can also be realised in other systems, for example heavy-ion collisions, which may offer an opportunity to explore the regime of super strong fields beyond the Schwinger field. This implies that the Schwinger effect may be testable via electromagnetic probes in various collision geometries, but because the associated spacetime volume of the strong fields generated is very small, care must be taken to properly account for the impact of finite volume effects which affect the non-perturbativity of the pair creation process. This, along with quark and gluon production by strong colour flux tubes, remain open questions, but may be analysed in a quantitative manner using the developments reviewed here.

The Ritus-Narozhny conjecture

The Ritus-Narozhny conjecture applies to those backgrounds and parameter regimes where the locally constant field approximation holds. Therefore it is required to conceive of a scenario where the high quantum-nonlinearity region can be accessed experimentally, whilst that approximation remains valid. This is an active area of research and several suggestions are reviewed in Section 9.2.

However, on the theoretical side the main challenge is to specify what to resum, and to perform that resummation, in the region where the conjecture holds. In spite of a certain progress in understanding the nature of bubble chain corrections and the reasons for their enhancement in a constant crossed field, the overall understanding of the possible non-perturbative regime of QED, where the fine structure constant times the two-thirds power of the quantum nonlinearity parameter approaches one, has not yet been achieved. It is likely that the most important feature of this regime, yet to be accounted for properly, is the severe instability of photons and electrons in a strong constant crossed field, with respect to nonlinear Breit-Wheeler and nonlinear Compton scattering. A systematic route to future progress in understanding this regime, including a better understanding of the role of vertex corrections, might be based on a more rigorous selection of diagrams with potentially leading scaling for resummation, possibly starting with a general form of the Dyson-Schwinger equations.

Beyond QED

It would be intriguing to explore how more of the strong-field methods from QED could be brought to bear on heavy-ion physics and the colour glass condensate, where strong classical fields play a crucial role. Developing methods for going beyond the mean-field approximation is for example important for the thermalisation and hydrodynamisation of the quark-gluon plasma in heavy-ion collisions.

As gravitational wave astronomy becomes an ever more well-established field, new approaches will be needed for calculations of observables in gravity, calculations which are extremely challenging even classically. Motivated by this, recent years have seen the development of efficient methods for extracting the classical limits of observables from quantum scattering amplitudes without having to calculate the full amplitude itself. There is clearly scope for adapting such methods to strong-field QED. First-quantised and worldline approaches have very recently been applied to double copy. In gravity, such methods can greatly simplify the calculation of graviton scattering amplitudes compared to standard diagrammatic approaches; there remains a great deal of potential, and work to be done, in the extension of these methods to include background gravitational fields.

In "new physics" particle searches, weak magnetic fields are often combined with high-finesse cavities to achieve a long interaction length and hence increase the production probability by increasing the product of field and length. The equivalent quantity in the interaction of a probe with a weak laser pulse is the intensity parameter times the frequency times the length. Using strong fields introduces nonlinearities, which compared to this perturbative scaling tend to decrease the probability. It remains an open question as to how other properties of intense fields phenomenology might be used to enhance signals of new physics.

Closing summary

The topics reviewed here lie at the confluence of high-power laser, particle, and plasma physics, and of theory, simulation, and experiment. As such they are relevant to the intensity frontier of particle physics, to future colliders such as the International Linear Collider and the Compact Linear Collider through strong field effects at the interaction point, and also to matter exposed to extreme electromagnetic fields, such as in exotic astrophysical objects, as tested using high intensity lasers.

The past decade has seen huge progress in the understanding of the "traditional" processes of strong-field QED, such as nonlinear Compton and Breit-Wheeler, the Schwinger effect, and vacuum birefringence, while access to higher-order processes, such as trident and double nonlinear Compton, has been made possible by a combination of novel methods and appropriate approximations. More recently, calculations have been pushed to higher loop orders, and in some cases to all orders, motivated by results on the very high intensity behaviour of QED in strong fields, and by results on resurgence in quantum field theory. Connections to non-Abelian gauge theories and gravity have also begun to be explored, and there is scope here for a fruitful exchange of ideas and methods.

This review has concentrated on the last decade of progress. At the beginning of this period, the state-of-the-art experimental result for testing QED in intense laser fields was still dominated by the landmark E144 experiment from the mid-1990s. In the last decade, several multi-petawatt lasers have come or are coming online, that will be able to reach intensities of the order of ten to the twenty-third watts per square centimetre and higher; the E144 experiment operated with a peak intensity of half of ten to the eighteenth watts per square centimetre. The increase in the number of papers published in the last ten years has likely been driven in part by recent experiments that have probed the edge of the nonlinear quantum regime and in part by upcoming experiments that will use more energetic probes and higher field strengths. These future experiments will be able to probe the nonlinear quantum regime in depth, and do so at a higher precision. It is a reasonable expectation that ten years from now, there will be papers that feature comparisons of experimental data with the theoretical models and predictions reviewed here.

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A. Fedotov, National Research Nuclear University MEPhI, Moscow; A. Ilderton, Higgs Centre, University of Edinburgh; F. Karbstein, Helmholtz-Institut Jena, GSI Darmstadt and Friedrich-Schiller-Universität Jena; B. King, Centre for Mathematical Sciences, University of Plymouth; D. Seipt, Helmholtz-Institut Jena and GSI Darmstadt; H. Taya, RIKEN iTHEMS; G. Torgrimsson, Helmholtz-Zentrum Dresden-Rossendorf and Umeå University. Published as Physics Reports 1010, pages 1 to 138 (2023), doi.org/10.1016/j.physrep.2023.01.003, open access under CC BY 4.0; accepted manuscript at arXiv 2203.00019.

(A 138-page review, reproduced here only in part: the abstract, the introduction, the experimental-landscape parameters, and the conclusions. Sections 2 to 11 are omitted for length and the complete text is at the source. Running heads, page numbers, figures and reference-number markers have been dropped; display equations are rendered in words and inequalities and Greek symbols written out; the eleven hundred references are at the source.)

(On this site: Schwinger’s own 1951 paper, the origin of the critical field this whole subject is measured against, is at /library/stm-6e110e2441. The long experimental hunt for vacuum birefringence — the effect Section 7 is about — runs through the PVLAS papers at /library/stm-a5446902ec, /library/stm-e58fdfd01c and /library/stm-08601764f6. For the vacuum measured at the opposite, gentle end of the field scale: the Casimir force at /library/stm-208d347532, direct sampling of vacuum field fluctuations at /library/stm-6095b3deaf and the separation of vacuum fluctuations from source radiation at /library/stm-d38d7e2394; Milonni’s survey of the quantum vacuum is at /library/stm-d0a2779af6. The cosmological accounting of the same vacuum energy is at /library/stm-568da33759 and /library/stm-bea1f2cb9c.)