# Publications by

Dr. Felix Karbstein

## 2022

**All-optical quantum vacuum signals in two-beam collisions**

**106**, 116005 (2022)

**Abstract:** We study the collision of two optical laser pulses in a pump-probe setup using beams with circular and elliptic cross section and estimate the number of discernible signal photons induced by quantum vacuum nonlinearities. In this analysis we study strategies to optimize the quantum vacuum signal discernible from the background of the driving lasers. One of the main results is that the collision of two maximally focused lasers does not lead to the best discernible signal. Instead, widening the focus typically improves the signal to background separation in the far field. For petawatt class lasers, an optimal choice of the focus waist yields several discernible photons per shot in contrast to no discernible signal for tight focusing. Further enhancement is possible by using an elliptical waist.

**Direct Accessibility of the Fundamental Constants Governing Light-by-Light Scattering**

**129**, 061802 (2022)

**Abstract:** Quantum field theory predicts that the vacuum exhibits a nonlinear response to strong electromagnetic fields. This fundamental tenet has remained experimentally challenging and is yet to be tested in the laboratory. We present proof of concept and detailed theoretical analysis of an experimental setup for precision measurements of the quantum vacuum signal generated by the collision of a brilliant x-ray probe with a high-intensity pump laser. The signal features components polarized parallel and perpendicularly to the incident x-ray probe. Our proof-of-concept measurements show that the background can be efficiently suppressed by many orders of magnitude which should not only facilitate a detection of the perpendicularly polarized component of the nonlinear vacuum response, but even make the parallel polarized component experimentally accessible for the first time. Remarkably, the angular separation of the signal from the intense x-ray probe enables precision measurements even in the presence of pump fluctuations and alignment jitter. This provides direct access to the low-energy constants governing light-by-light scattering.

**Quantum vacuum processes in the extremely intense light of relativistic plasma mirror sources**

**24**, 065005 (2022)

**Abstract:** The advent of petawatt-class laser systems allows generating electromagnetic fields of unprecedented strength in a controlled environment, driving increasingly more efforts to probe yet unobserved processes through their interaction with the quantum vacuum. Still, the lowest intensity scale governing these effects lies orders of magnitude beyond foreseen capabilities, so that such endeavor is expected to remain extremely challenging. In recent years, however, plasma mirrors have emerged as a promising bridge across this gap, by enabling the conversion of intense infrared laser pulses into coherently focused Doppler harmonic beams lying in the X-UV range. In this work, we present predictions on the quantum vacuum signatures produced when such beams are focused to intensities between 1024 and 1028 W cm−2, specifically photon–photon scattering and electron–positron pair creation. These signatures are computed via the stimulated vacuum formalism, combined with a model of perfectly focused beam built from PIC-generated harmonics spectra, and implemented on state-of-the-art massively parallel numerical tools. In view of identifying experimentally favorable configurations, we also consider the coupling of the focused harmonic beam with an auxiliary optical beam, and provide comparison with other established schemes. Our results show that a single coherently focused harmonic beam can produce as much scattered photons as two infrared pulses in head-on collision, and confirm that the coupling of the harmonic beam to an auxiliary beam gives rise to significant levels of inelastic scattering, and hence holds the potential to strongly improve the attainable signal to noise ratios in experiments.

**Expanding Nuclear Physics Horizons with the Gamma Factory**

**534 (3)**, 2100284 (2022)

**Abstract:** Abstract The Gamma Factory (GF) is an ambitious proposal, currently explored within the CERN Physics Beyond Colliders program, for a source of photons with energies up to ≈400 MeV and photon fluxes (up to ≈1017 photons s-1) exceeding those of the currently available gamma sources by orders of magnitude. The high-energy (secondary) photons are produced via resonant scattering of the primary laser photons by highly relativistic partially-stripped ions circulating in the accelerator. The secondary photons are emitted in a narrow cone and the energy of the beam can be monochromatized, down to 10-3...10-6 level, via collimation, at the expense of the photon flux. This paper surveys the new opportunities that may be afforded by the GF in nuclear physics and related fields.

**Vacuum Birefringence at the Gamma Factory**

**534 (3)**, 2100137 (2022)

**Abstract:** Abstract The perspectives of studying vacuum birefringence at the Gamma Factory are explored. To this end, the parameter regime which can be reliably analyzed resorting to the leading contribution to the Heisenberg?Euler effective Lagrangian is assessed in detail. It is explicitly shown that?contrary to naive expectations?this approach allows for the accurate theoretical study of quantum vacuum signatures up to fairly large photon energies. The big advantage of this parameter regime is the possibility of studying the phenomenon in experimentally realistic, manifestly inhomogeneous pump and probe field configurations. Thereafter, two specific scenarios giving rise to a vacuum birefringence effect for traversing gamma probe photons are analyzed. In the first scenario the birefringence phenomenon is induced by a quasi-constant static magnetic field. In the second case it is driven by a counter-propagating high-intensity laser field.

**Towards perfectly linearly polarized x-rays**

**4**, 013220 (2022)

**Abstract:** In recent years, high-precision x-ray polarimeters have become a key method for the investigation of fundamental physical questions from solid-state physics to quantum optics. Here, we report on the verification of a polarization purity of better than 8×10−11 at an x-ray free-electron laser, which implies a suppression of the incoming photons to the noise level in the crossed polarizer setting. This purity provides exceptional sensitivity to tiny polarization changes and offers intriguing perspectives for fundamental tests of quantum electrodynamics.

**Large N external-field quantum electrodynamics**

**01**, 057 (2022)

**Abstract:** We advocate the study of external-field quantum electrodynamics with N charged particle flavors. Our main focus is on the Heisenberg-Euler effective action for this theory in the large N limit which receives contributions from all loop orders. The contributions beyond one loop stem from one-particle reducible diagrams. We show that specifically in constant electromagnetic fields the latter are generated by the one-loop Heisenberg-Euler effective Lagrangian. Hence, in this case the large N Heisenberg-Euler effective action can be determined explicitly at any desired loop order. We demonstrate that further analytical insights are possible for electric-and magnetic-like field configurations characterized by the vanishing of one of the secular invariants of the electromagnetic field and work out the all-orders strong field limit of the theory.

## 2021

**X-ray vacuum diffraction at finite spatiotemporal offset**

**104**, 076015 (2021)

**Abstract:** We study the nonlinear QED signature of x-ray vacuum diffraction in the head-on collision of optical high-intensity and x-ray free-electron laser pulses at finite spatiotemporal offsets between the laser foci. The high-intensity laser driven scattering of signal photons outside the forward cone of the x-ray probe constitutes a prospective experimental signature of quantum vacuum nonlinearity. Resorting to a simplified phenomenological ad hoc model, it was recently argued that the angular distribution of the signal in the far-field is sensitive to the wavefront curvature of the probe beam in the interaction region with the high-intensity pump. In this work, we model both the pump and probe fields as pulsed paraxial Gaussian beams and reanalyze this effect from first principles. We focus on vacuum diffraction both as an individual signature of quantum vacuum nonlinearity and as a potential means to improve the signal-to-background separation in vacuum birefringence experiments.

**Derivative corrections to the Heisenberg-Euler effective action**

**09**, 070 (2021)

**Abstract:** We show that the leading derivative corrections to the Heisenberg-Euler effective action can be determined efficiently from the vacuum polarization tensor evaluated in a homogeneous constant background field. After deriving the explicit parameter-integral representation for the leading derivative corrections in generic electromagnetic fields at one loop, we specialize to the cases of magnetic- and electric-like field configurations characterized by the vanishing of one of the secular invariants of the electromagnetic field. In these cases, closed-form results and the associated all-orders weak-and strong-field expansions can be worked out. One immediate application is the leading derivative correction to the renowned Schwinger-formula describing the decay of the quantum vacuum via electron-positron pair production in slowly-varying electric fields.

**Vacuum birefringence at x-ray free-electron lasers**

**23**, 095001 (2021)

**Abstract:** We study the perspectives of measuring the phenomenon of vacuum birefringence predicted by quantum electrodynamics using an x-ray free-electron laser (XFEL) alone. We devise an experimental scheme allowing two consecutive XFEL pulses to collide under a finite angle, and thus act as both pump and probe field for the effect. The signature of vacuum birefringence is encoded in polarization-flipped signal photons to be detected with high-purity x-ray polarimetry. Our findings for idealized scenarios underline that the discovery potential of solely XFEL-based setups can be comparable to those involving optical high-intensity lasers. For currently achievable scenarios, we identify several key details of the x-ray optical ingredients that exert a strong influence on the magnitude of the desired signatures.

**Vacuum birefringence and diffraction at an x-ray free-electron laser: From analytical estimates to optimal parameters**

**104**, 013006 (2021)

**Abstract:** We study vacuum birefringence and x-ray photon scattering in the head-on collision of x-ray free electron and high-intensity laser pulses. Resorting to analytical approximations for the numbers of attainable signal photons, we analyze the behavior of the phenomenon under the variation of various experimental key-parameters and provide new analytical scalings. Our optimized approximations allow for quantitatively accurate results on the one-percent level. We in particular demonstrate that an appropriate choice of the x-ray focus and pulse duration can significantly improve the signal for given laser parameters, using the experimental parameters to be available at the Helmholtz International Beamline for Extreme Fields at the European XFEL as example. Our results are essential for the identification of the optimal choice of parameters in a discovery experiment of vacuum birefringence at the high-intensity frontier.

**Quantum vacuum signatures in multicolor laser pulse collisions**

**103**, 076009 (2021)

**Abstract:** Quantum vacuum fluctuations give rise to effective nonlinear interactions between electromagnetic fields. A prominent signature of quantum vacuum nonlinearities driven by macroscopic fields are signal photons differing in characteristic properties such as frequency, propagation direction and polarization from the driving fields. We devise a strategy for the efficient tracing of the various vacuum-fluctuation-mediated interaction processes in order to identify the most prospective signal photon channels. As an example, we study the collision of up to four optical laser pulses and pay attention to sum and difference frequency generation. We demonstrate how this information can be used to enhance the signal photon yield in laser pulse collisions for a given total laser energy.

## 2020

**Enhancing quantum vacuum signatures with tailored laser beams**

**101**, 113002 (2020)

**Abstract:** We demonstrate that tailored laser beams provide a powerful means to make quantum vacuum signatures in strong electromagnetic fields accessible in experiment. Typical scenarios aiming at the detection of quantum vacuum nonlinearities at the high-intensity frontier envision the collision of focused laser pulses. The effective interaction of the driving fields mediated by vacuum fluctuations gives rise to signal photons encoding the signature of quantum vacuum nonlinearity. Isolating a small number of signal photons from the large background of the driving laser photons poses a major experimental challenge. The main idea of the present work is to modify the far-field properties of a driving laser beam to exhibit a fieldfree hole in its center, thereby allowing for an essentially backgroundfree measurement of the signal scattered in the forward direction. Our explicit construction makes use of a peculiar far-field/focus duality.

**Visualization of relativistic laser pulses in underdense plasma**

**23**, 032801 (2020)

**Abstract:** We present experimental evidence of relativistic electron-cyclotron resonances (RECRs) in the vicinity of the relativistically intense pump laser of a laser wakefield accelerator (LWFA). The effects of the RECRs are visualized by imaging the driven plasma wave with a few-cycle, optical probe in transverse geometry. The probe experiences strong, spectrally dependent and relativistically modified birefringence in the vicinity of the pump that arises due to the plasma electrons’ relativistic motion in the pump’s electromagnetic fields. The spectral birefringence is strongly dependent on the local magnetic field distribution of the pump laser. Analysis and comparison to both 2D and 3D particle-in-cell simulations confirm the origin of the RECR effect and its appearance in experimental and simulated shadowgrams of the laser-plasma interaction. The RECR effect is relevant for any relativistic, magnetized plasma and in the case of LWFA could provide a nondestructive, in situ diagnostic for tracking the evolution of the pump’s intensity distribution with propagation through tenuous plasma.

**Probing Vacuum Polarization Effects with High-Intensity Lasers**

**3**, 39 (2020)

**Abstract:** These notes provide a pedagogical introduction to the theoretical study of vacuum polarization effects in strong electromagnetic fields as provided by state-of-the-art high-intensity lasers. Quantum vacuum fluctuations give rise to effective couplings between electromagnetic fields, thereby supplementing Maxwell’s linear theory of classical electrodynamics with nonlinearities. Resorting to a simplified laser pulse model, allowing for explicit analytical insights, we demonstrate how to efficiently analyze all-optical signatures of these effective interactions in high-intensity laser experiments. Moreover, we highlight several key features relevant for the accurate planning and quantitative theoretical analysis of quantum vacuum nonlinearities in the collision of high-intensity laser pulses.

## 2019

**X-ray photon scattering at a focused high-intensity laser pulse**

**100**, 033002 (2019)

**Abstract:** We study x-ray photon scattering in the head-on collision of an XFEL pulse and a focused high-intensity laser pulse, described as paraxial Laguerre-Gaussian beam of arbitrary mode composition. For adequately chosen relative orientations of the polarization vectors of the colliding laser fields, this gives rise to a vacuum birefringence effect manifesting itself in polarization flipped signal photons. Throughout this article the XFEL is assumed to be mildly focused to a waist larger than that of the high-intensity laser beam. As previously demonstrated for the special case of a fundamental paraxial Gaussian beam, this scenario is generically accompanied by a scattering phenomenon of x-ray energy signal photons outside the forward cone of the XFEL beam, potentially assisting the detection of the effect in experiment. Here, we study the fate of the x-ray scattering signal under exemplary deformations of the transverse focus profile of the high-intensity pump.

**Boosting Quantum Vacuum Signatures by Coherent Harmonic Focusing**

**123**, 091802 (2019)

**Abstract:** We show that coherent harmonic focusing provides an efficient mechanism to boost all-optical signatures of quantum vacuum nonlinearity in the collision of high-intensity laser fields, thereby offering a promising route to their first experimental detection. Assuming two laser pulses of given parameters at our disposal, we demonstrate a substantial increase of the number of signal photons measurable in experiments where one of the pulses undergoes coherent harmonic focusing before it collides with the fundamental-frequency pulse. Imposing a quantitative criterion to discern the signal photons from the background of the driving laser photons and accounting for the finite purity of polarization filtering, we find that signal photons arising from inelastic scattering processes constitute a promising signature. By contrast, quasielastic contributions which are conventionally assumed to form the most prospective signal remain background dominated. Our findings may result in a paradigm shift concerning which photonic signatures of quantum vacuum nonlinearity are accessible in experiment.

**All-Loop Result for the Strong Magnetic Field Limit of the Heisenberg-Euler Effective Lagrangian**

**122**, 211602 (2019)

**Abstract:** We provide an explicit expression for the strong magnetic field limit of the Heisenberg-Euler effective Lagrangian for both scalar and spinor quantum electrodynamics. To this end, we show that the strong magnetic field behavior is fully determined by one-particle reducible contributions discovered only recently. The latter can efficiently be constructed in an essentially algebraic procedure from lower-order one-particle reducible diagrams. Remarkably, the leading strong magnetic field behavior of the all-loop Heisenberg-Euler effective Lagrangian only requires input from the one-loop Lagrangian. Our result revises previous findings based exclusively on one-particle irreducible contributions. In addition, we briefly discuss the strong electric field limit and comment on external field QED in the large N limit.

**Photon-Photon Scattering at the High-Intensity Frontier: Paraxial Beams**

**1206**, 012016 (2019)

**Abstract:** Our goal is to study optical signatures of quantum vacuum nonlinearities in strong macroscopic electromagnetic fields provided by high-intensity laser beams. The vacuum emission scheme is perfectly suited for this task as it naturally distinguishes between incident laser beams, described as classical electromagnetic fields driving the effect, and emitted signal photons encoding the signature of quantum vacuum nonlinearity. Using the Heisenberg-Euler effective action, our approach allows for a reliable study of photonic signatures of QED vacuum nonlinearity in the parameter regimes accessible by all-optical high-intensity laser experiments. To this end, we employ an efficient, flexible numerical algorithm, which allows for a detailed study of the signal photons emerging in the collision of focused paraxial high-intensity laser pulses. Due to the high accuracy of our numerical solutions we predict the total number of signal photons, but also have full access to the signal photons’ characteristics, including their spectrum, propagation directions and polarizations. We discuss setups offering an excellent background-to-noise ratio, thus providing an important step towards the experimental verification of quantum vacuum nonlinearities.

**The Vacuum Emission Picture Beyond Paraxial Approximation**

**1206**, 012017 (2019)

**Abstract:** Optical signatures of the effective nonlinear couplings among electromagnetic fields in the quantum vacuum can be conveniently described in terms of stimulated photon emission processes induced by strong classical, space-time dependent electromagnetic fields. Recent studies have adopted this approach to study collisions of Gaussian laser pulses in paraxial approximation. The present study extends these investigations beyond the paraxial approximation by using an efficient numerical solver for the classical input fields. This new numerical code allows for a consistent theoretical description of optical signatures of QED vacuum nonlinearities in generic electromagnetic fields governed by Maxwell’s equations in the vacuum, such as manifestly non-paraxial laser pulses. Our code is based on a locally constant field approximation of the Heisenberg-Euler effective Lagrangian. As this approximation is applicable for essentially all optical high-intensity laser experiments, our code is capable of calculating signal photon emission amplitudes in completely generic input field configurations, limited only by numerical cost.

**All-optical signatures of quantum vacuum nonlinearities in generic laser fields**

**99**, 016006 (2019)

**Abstract:** All-optical experiments at the high-intensity frontier offer a promising route to unprecedented precision tests of quantum electrodynamics in strong macroscopic electromagnetic fields. So far, most theoretical studies of all-optical signatures of quantum vacuum nonlinearity are based on simplifying approximations of the beam profiles and pulse shapes of the driving laser fields. Since precision tests require accurate quantitative theoretical predictions, we introduce an efficient numerical tool facilitating the quantitative theoretical study of all-optical signatures of quantum vacuum nonlinearity in generic laser fields. Our approach is based on the vacuum emission picture, and makes use of the fact that the dynamics of the driving laser fields are to an excellent approximation governed by classical Maxwell theory in vacuum. In combination with a Maxwell solver, which self-consistently propagates any given laser field configuration, this allows for accurate theoretical predictions of photonic signatures of vacuum nonlinearity in high-intensity laser experiments from first principles. We employ our method to simulate photonic signatures of quantum vacuum nonlinearity in laser pulse collisions involving a few-cycle pulse, and show that the angular and spectral distributions of the emitted signal photons deviate from those of the driving laser beams.

## 2018

**Determination of ΛMSbar (nf=2) and analytic parametrization of the static quark-antiquark potential**

**98**, 114506 (2018)

**Abstract:** While lattice QCD allows for reliable results at small momentum transfers (large quark separations), perturbative QCD is restricted to large momentum transfers (small quark separations). The latter is determined up to a reference momentum scale Λ, which is to be provided from outside, e.g., from experiment or lattice QCD simulations. In this article, we extract ΛMSbar for QCD with nf=2 dynamical quark flavors by matching the perturbative static quark-antiquark potential in momentum space to lattice results in the intermediate momentum regime, where both approaches are expected to be applicable. In a second step, we combine the lattice and the perturbative results to provide a complete analytic parametrization of the static quark-antiquark potential in position space up to the string breaking scale. As an exemplary phenomenological application of our all-distances potential, we compute the bottomonium spectrum in the static limit.

**High magnetic fields for fundamental physics**

**765-766**, 1 (2018)

**Abstract:** Various fundamental-physics experiments such as measurement of the magnetic birefringence of the vacuum, searches for ultralight dark-matter particles (e.g., axions), and precision spectroscopy of complex systems (including exotic atoms containing antimatter constituents) are enabled by high-field magnets. We give an overview of current and future experiments and discuss the state-of-the-art DC- and pulsed-magnet technologies and prospects for future developments.

**Vacuum birefringence in the head-on collision of x-ray free-electron laser and optical high-intensity laser pulses**

**98**, 056010 (2018)

**Abstract:** The focus of this article is on providing compact analytical expressions for the differential number of polarization-flipped signal photons constituting the signal of vacuum birefringence in the head-on collision of x-ray free electron (XFEL) and optical high-intensity laser pulses. Our results allow for unprecedented insights into the scaling of the effect with the waists and pulse durations of both laser beams, the Rayleigh range of the high-intensity beam, as well as transverse and longitudinal offsets. They account for the decay of the differential number of signal photons in the far field as a function of the azimuthal angle measured relative to the beam axis of the probe beam in the forward direction, typically neglected by conventional approximations. Moreover, they even allow us to extract an analytical expression for the angular divergence of the perpendicularly polarized signal photons. We expect our formulas to be very useful for the planning and optimization of experimental scenarios aiming at the detection of vacuum birefringence in XFEL/high-intensity laser setups, such as the one put forward at the Helmholtz International Beamline for Extreme Fields at the European XFEL.

**Photon-photon scattering at the high-intensity frontier**

**97**, 076002 (2018)

**Abstract:** The tremendous progress in high-intensity laser technology and the establishment of dedicated high-field laboratories in recent years have paved the way towards a first observation of quantum vacuum nonlinearities at the high-intensity frontier. We advocate a particularly prospective scenario, where three synchronized high-intensity laser pulses are brought into collision, giving rise to signal photons, whose frequency and propagation direction differ from the driving laser pulses, thus providing various means to achieve an excellent signal to background separation. Based on the theoretical concept of vacuum emission, we employ an efficient numerical algorithm which allows us to model the collision of focused high-intensity laser pulses in unprecedented detail. We provide accurate predictions for the numbers of signal photons accessible in experiment. Our study is the first to predict the precise angular spread of the signal photons, and paves the way for a first verification of quantum vacuum nonlinearity in a well-controlled laboratory experiment at one of the many high-intensity laser facilities currently coming online.

**Photon polarization tensor in circularly polarized Hermite- and Laguerre-Gaussian beams**

**33**, 1850044 (2018)

**Abstract:** We derive analytical expressions for the photon polarization tensor in circularly polarized Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams, complementing the corresponding results for linearly polarized beams obtained recently. As they are based upon a locally constant field approximation of the one-loop Heisenberg–Euler effective Lagrangian for quantum electrodynamics (QED) in constant fields, our results are generically limited to slowly varying electromagnetic fields, varying on spatial (temporal) scales much larger than the Compton wavelength (time) of the electron.

**All-optical signatures of strong-field QED in the vacuum emission picture**

**97**, 036022 (2018)

**Abstract:** We study all-optical signatures of the effective nonlinear couplings among electromagnetic fields in the quantum vacuum, using the collision of two focused high-intensity laser pulses as an example. The experimental signatures of quantum vacuum nonlinearities are encoded in signal photons, whose kinematic and polarization properties differ from the photons constituting the macroscopic laser fields. We implement an efficient numerical algorithm allowing for the theoretical investigation of such signatures in realistic field configurations accessible in experiment. This algorithm is based on a vacuum emission scheme and can readily be adapted to the collision of more laser beams or further involved field configurations. We solve the case of two colliding pulses in full 3+1-dimensional spacetime and identify experimental geometries and parameter regimes with improved signal-to-noise ratios.

## 2017

**Photon polarization tensor in pulsed Hermite- and Laguerre-Gaussian beams**

**96**, 116004 (2017)

**Abstract:** In this article, we provide analytical expressions for the photon polarization tensor in pulsed Hermite- and Laguerre-Gaussian laser beams. Our results are based on a locally constant field approximation of the one-loop Heisenberg-Euler effective Lagrangian for quantum electrodynamics. Hence, by construction they are limited to slowly varying electromagnetic fields, varying on spatial and temporal scales significantly larger than the Compton wavelength/time of the electron. The latter criterion is fulfilled by all laser beams currently available in the laboratory. Our findings will, e.g., be relevant for the study of vacuum birefringence experienced by probe photons brought into collision with a high-intensity laser pulse which can be represented as a superposition of either Hermite- or Laguerre-Gaussian modes.

**Tadpole diagrams in constant electromagnetic fields**

**2017**, 75 (2017)

**Abstract:** We show how all possible one-particle reducible tadpole diagrams in constant electromagnetic fields can be constructed from one-particle irreducible constant-field diagrams. The construction procedure is essentially algebraic and involves differentiations of the latter class of diagrams with respect to the field strength tensor and contractions with derivatives of the one-particle irreducible part of the Heisenberg-Euler effective Lagrangian in constant fields. Specific examples include the two-loop addendum to the Heisenberg-Euler effective action as well as a novel one-loop correction to the charged particle propagator in constant electromagnetic fields discovered recently. As an additional example, the approach devised in the present article is adopted to derive the tadpole contribution to the two-loop photon polarization tensor in constant fields for the first time.

**Heisenberg-Euler effective action in slowly varying electric field inhomogeneities of Lorentzian shape**

**95**, 076015 (2017)

**Abstract:** We use a locally constant field approximation (LCFA) to study the one-loop Heisenberg-Euler effective action in a particular class of slowly varying inhomogeneous electric fields of Lorentzian shape with 0≤d<4 inhomogeneous directions. We show that, for these fields, the LCFA of the Heisenberg-Euler effective action can be represented in terms of a single parameter integral, with the constant field effective Lagrangian with rescaled argument as integration kernel. The imaginary part of the Heisenberg-Euler effective action contains information about the instability of the quantum vacuum towards the formation of a state with real electrons and positrons. Here, in particular, we focus on the dependence of the instantaneous vacuum decay rate on the dimension d of the field inhomogeneity. Specifically, for weak fields, we find an overall parametric suppression of the effect with (E0/Ecr)^(d/2), where E0 is the peak field strength of the inhomogeneity and Ecr the critical electric field strength.

**An addendum to the Heisenberg-Euler effective action beyond one loop**

**03**, 108 (2017)

**Abstract:** We study the effective interactions of external electromagnetic fields induced by fluctuations of virtual particles in the vacuum of quantum electrodynamics. Our main focus is on these interactions at two-loop order. We discuss in detail the emergence of the renowned Heisenberg-Euler effective action from the underlying microscopic theory of quantum electrodynamics, emphasizing its distinction from a standard one-particle irreducible effective action. In our explicit calculations we limit ourselves to constant and slowly varying external fields, allowing us to adopt a locally constant field approximation. One of our main findings is that at two-loop order there is a finite one-particle reducible contribution to the Heisenberg-Euler effective action in constant fields, which was previously assumed to vanish. In addition to their conceptual significance, our results are relevant for high-precision probes of quantum vacuum nonlinearity in strong electromagnetic fields.

**Photon Propagation in Slowly Varying Electromagnetic Fields**

**59**, 1 (2017)

**Abstract:** Effective theory of soft photons in slowly varying electromagnetic background fields is studied at one-loop order in QED. This is of relevance for the study of all-optical signatures of quantum vacuum nonlinearity in realistic electromagnetic background fields as provided by high-intensity lasers. The central result derived in this article is a new analytical expression for the photon polarization tensor in two linearly polarized counterpropagating pulsed Gaussian laser beams. Treating the peak field strengths of both laser beams as free parameters, this field configuration can be considered as interpolating between the limiting cases of a purely right- or left-moving laser beam (if one of the peak field strengths is set equal to zero) and the standing-wave type scenario with two counter-propagating beams of equal strength.

## 2016

**Probing vacuum birefringence using x-ray free electron and optical high-intensity lasers**

**94**, 013004 (2016)

**Abstract:** Vacuum birefringence is one of the most striking predictions of strong field quantum electrodynamics: Probe photons traversing a strong field region can indirectly sense the applied “pump” electromagnetic field via quantum fluctuations of virtual charged particles which couple to both pump and probe fields. This coupling is sensitive to the field alignment and can effectively result in two different indices of refraction for the probe photon polarization modes giving rise to a birefringence phenomenon. In this article, we perform a dedicated theoretical analysis of the proposed discovery experiment of vacuum birefringence at an x-ray free electron laser/optical high-intensity laser facility. Describing both pump and probe laser pulses realistically in terms of their macroscopic electromagnetic fields, we go beyond previous analyses by accounting for various effects not considered before in this context. Our study facilitates stringent quantitative predictions and optimizations of the signal in an actual experiment.

**Photon merging and splitting in electromagnetic field inhomogeneities**

**93**, 085034 (2016)

**Abstract:** We investigate photon merging and splitting processes in inhomogeneous, slowly varying electromagnetic fields. Our study is based on the three-photon polarization tensor following from the Heisenberg-Euler effective action. We put special emphasis on deviations from the well-known constant field results, also revisiting the selection rules for these processes. In the context of high-intensity laser facilities, we analytically determine compact expressions for the number of merged/split photons as obtained in the focal spots of intense laser beams. For the parameter range of typical petawatt class laser systems as pump and probe, we provide estimates for the numbers of signal photons attainable in an actual experiment. The combination of frequency upshifting, polarization dependence and scattering off the inhomogeneities renders photon merging an ideal signature for the experimental exploration of nonlinear quantum vacuum properties.

**Crystalline ground states in Polyakov-loop extended Nambu-Jona-Lasinio models**

**93**, 014032 (2016)

**Abstract:** Nambu–Jona-Lasinio-type models have been used extensively to study the dynamics of the theory of the strong interaction at finite temperature and quark chemical potential on a phenomenological level. In addition to these studies, which are often performed under the assumption that the ground state of the theory is homogeneous, searches for the existence of crystalline phases associated with inhomogeneous ground states have attracted a lot of interest in recent years. In this work, we study the Polyakov-loop extended Nambu–Jona-Lasinio model using two prominent parametrizations and find that the existence of a crystalline phase is stable against a variation of the parametrization of the underlying Polyakov loop potential.

**High Field Physics and QED Experiments at ELI-NP**

**68**, S145 (2016)

**Abstract:** ELI-NP facility will enable for the first time the use of two 10 PW laser beams for quantum electrodynamics (QED) experiments. The first beam will accelerate electrons to relativistic energies. The second beam will subject relativistic electrons to the strong electromagnetic field generating QED processes: intense gamma ray radiation and electron-positron pair formation. The laser beams will be focused to intensities above 10^21 W/cm^2 and reaching 10^22–10^23 W/cm^2 for the first time. We propose to use this capability to investigate new physical phenomena at the interfaces of plasma, nuclear and particle physics at ELI-NP. This High Power Laser System Technical Design Report (HPLS-TDR2) presents the experimental area E6 at ELI-NP for investigating high field physics and quantum electrodynamics and the production of electron-positron-pairs and of energetic gamma-rays. The scientific community submitted 12 commissioning runs for E6 interaction chamber with two 10 PW laser beams and one proposal for the CETAL interaction chamber with 1 PW laser. The proposals are representative of the international high field physics community being written by 48 authors from 14 European and US organizations. The proposals are classified according to the science area investigated into: Radiation Reaction Physics: Classical and Quantum; Compton and Thomson Scattering Physics: Linear and Non Linear Regimes; QED in Vacuum; Atoms in Extreme Fields. Two pump-probe colliding 10 PW laser beams are proposed for the E6 interaction chamber. The focused pump laser beam accelerates the electrons to relativistic energies. The accelerated electron bunches interact with the very high electro-magnetic field of the focused probe laser beam. We propose two main types of experiments with: (a) gas targets in which the pump laser-beam is focused by a long focal length mirror and drives a wakefield in which the electron bunch is accelerated to multi-GeV energies and then exposed to the EM field of the probe laser which is tightly focused; (b) solid targets in which both the pump and probe laser beams are focused on the solid target, one accelerating the electrons in the solid and the other, delayed, providing the high electric field to which the relativistic electrons are subjected. We propose four main focusing configurations for the pump and probe laser beams, two for each type of target: counter-propagating 10 PW focused laser beams and the two 10 PW laser beams focused in the same direction. For solid targets we propose an additional configuration with plasma-mirror on the pump laser beam: the plasma mirror placed between the focusing mirror and target. It is proposed that the 10 PW laser beams will have polarization control and focus control by means of adaptive optics. Initially only one 10 PW may have polarization control and adaptive optics. In order to accommodate the two laser beams and diagnostics the proposed interaction chamber is quasi-octagonal with a diameter of 4.5 m. A large electron-spectrometer is proposed for multi-GeV electrons. Other diagnostics are requested for: gamma-rays, electrons and positrons, protons and ions, plasma characterization, transmitted and reflected laser beam. Targets will be provided by the ELI-NP Target Laboratory or purchased. The E6 experiments and diagnostics will benefit from the ELI-NP Electronics Laboratory, the Workshop and the Optics Laboratory. In order to ensure radiation-protection, a large beam-dump is planned for both multi-GeV electrons and multi-100 MeV protons.

**The quantum vacuum in electromagnetic fields: From the Heisenberg-Euler effective action to vacuum birefringence**

**Abstract:** The focus of these lectures is on the quantum vacuum subjected to classical electromagnetic fields. To this end we explicitly derive the renowned Heisenberg-Euler effective action in constant electromagnetic fields in a rather pedagogical and easy to conceive way. As an application, we use it to study vacuum birefringence constituting one of the most promising optical signatures of quantum vacuum nonlinearity.

## 2015

**Vacuum birefringence in strong inhomogeneous electromagnetic fields**

**92**, 071301 (2015)

**Abstract:** Birefringence is one of the fascinating properties of the vacuum of quantum electrodynamics (QED) in strong electromagnetic fields. The scattering of linearly polarized incident probe photons into a perpendicularly polarized mode provides a distinct signature of the optical activity of the quantum vacuum and thus offers an excellent opportunity for a precision test of nonlinear QED. Precision tests require accurate predictions and thus a theoretical framework that is capable of taking the detailed experimental geometry into account. We derive analytical solutions for vacuum birefringence which include the spatio-temporal field structure of a strong optical pump laser field and an x-ray probe. We show that the angular distribution of the scattered photons depends strongly on the interaction geometry and find that scattering of the perpendicularly polarized scattered photons out of the cone of the incident probe x-ray beam is the key to making the phenomenon experimentally accessible with the current generation of FEL/high-field laser facilities.

**Search for inhomogeneous phases in fermionic models**

**91**, 116006 (2015)

**Abstract:** We revisit the Gross-Neveu model with N fermion flavors in 1+1 dimensions and compute its phase diagram at finite temperature and chemical potential in the large-N limit. To this end, we double the number of fermion degrees of freedom in a specific way which allows us to detect inhomogeneous phases in an efficient manner. We show analytically that this “fermion doubling trick” predicts correctly the position of the boundary between the chirally symmetric phase and the phase with broken chiral symmetry. Most importantly, we find that the emergence of an inhomogeneous ground state is predicted correctly. We critically analyze our approach based on this trick and discuss its applicability to other theories, such as fermionic models in higher dimensions, where it may be used to guide the search for inhomogeneous phases.

**Stimulated photon emission from the vacuum**

**91**, 113002 (2015)

**Abstract:** We study the effect of stimulated photon emission from the vacuum in strong space-time-dependent electromagnetic fields. We emphasize the viewpoint that the vacuum subjected to macroscopic electromagnetic fields with at least one nonzero electromagnetic field invariant, as, e.g., attainable by superimposing two laser beams, can represent a source term for outgoing photons. We believe that this view is particularly intuitive and allows for a straightforward and intuitive study of optical signatures of quantum vacuum nonlinearity in realistic experiments involving the collision of high-intensity laser pulses, and exemplify this view for the vacuum subjected to a strong standing electromagnetic wave as generated in the focal spot of two counterpropagating, linearly polarized, high-intensity laser pulses. Focusing on a comparably simple electromagnetic field profile, which should nevertheless capture the essential features of the electromagnetic fields generated in the focal spots of real high-intensity laser beams, we provide estimates for emission characteristics and the numbers of emitted photons attainable with present and near future high-intensity laser facilities.

**Quantum reflection of photons off spatio-temporal electromagnetic field inhomogeneities**

**17**, 043060 (2015)

**Abstract:** We reconsider the recently proposed nonlinear quantum electrodynamics effect of quantum reflection of photons off an inhomogeneous strong-field region. We present new results for strong fields varying both in space and time. While such configurations can give rise to new effects such as frequency mixing, estimated reflection rates based on previous one-dimensional studies are corroborated. On a conceptual level, we critically re-examine the validity regime of the conventional locally-constant-field approximation and identify kinematic configurations which can be treated reliably. Our results further underline the discovery potential of quantum reflection as a new signature of the nonlinearity of the quantum vacuum.

**Photon propagation in slowly varying inhomogeneous electromagnetic fields**

**91**, 085027 (2015)

**Abstract:** Starting from the Heisenberg-Euler effective Lagrangian, we determine the photon current and photon polarization tensor in inhomogeneous, slowly varying electromagnetic fields. To this end, we consider background field configurations varying in both space and time, paying special attention to the tensor structure. As a main result, we obtain compact analytical expressions for the photon polarization tensor in realistic Gaussian laser pulses, as generated in the focal spots of high-intensity lasers. These expressions are of utmost importance for the investigation of quantum vacuum nonlinearities in realistic high-intensity laser experiments.

## 2014

**LambdaMSbar (nf=2) from a momentum space analysis of the quark-antiquark static potential**

**09**, 114 (2014)

**Abstract:** We determine LambdaMSbar (nf=2) by fitting perturbative expressions for the quark-antiquark static potential to lattice results for QCD with nf=2 dynamical quark flavors. To this end we use the perturbative static potential at the presently best known accuracy, i.e. up to O(α_s^4), in momentum space. The lattice potential is computed on a fine lattice with a≈0.042fm in position space. To allow for a comparison and matching of both results, the lattice potential is transformed into momentum space by means of a discrete Fourier transform. The value of LambdaMSbar (nf=2) is extracted in momentum space. All sources of statistical and systematic errors are discussed. The uncertainty in the value of LambdaMSbar (nf=2) is found to be smaller than that obtained in a recent position space analysis of the static potential based on the same lattice data.

**Laser photon merging in an electromagnetic field inhomogeneity**

**90**, 033007 (2014)

**Abstract:** We study the effect of laser photon merging, or equivalently high harmonic generation, in the quantum vacuum subject to inhomogeneous electromagnetic fields. Such a process is facilitated by the effective nonlinear couplings arising from charged particle-antiparticle fluctuations in the quantum vacuum subject to strong electromagnetic fields. We derive explicit results for general kinematic and polarization configurations involving optical photons. Concentrating on merged photons in reflected channels which are preferable in experiments for reasons of noise suppression, we demonstrate that photon merging is typically dominated by the competing nonlinear process of quantum reflection, though appropriate polarization and signal filtering could specifically search for the merging process. As a byproduct, we devise a novel systematic expansion of the photon polarization tensor in plane wave fields.

**Quark-antiquark static energy from a restricted Fourier transform**

**04**, 144 (2014)

**Abstract:** We provide a fully analytical determination of the perturbative quark-antiquark static energy in position space as defined by a restricted Fourier transformation from momentum to position space. Such a determination is complicated by the fact that the static energy genuinely decomposes into a strictly perturbative part (made up of contributions ∼α_s^n, with n∈ℕ) which is conventionally evaluated in momentum space, and a so-called ultrasoft part (including terms ∼α_s^{n+m}ln^m(α_s), with n≥3 and m∈ℕ) which, conversely, is naturally evaluated in position space. Our approach facilitates the explicit determination of the static energy in position space at the accuracy with which the perturbative potential in momentum space is known, i.e., presently up to order α_s^4.

## 2013

**Photon polarization tensor in a homogeneous magnetic or electric field**

**88**, 085033 (2013)

**Abstract:** We revisit the photon polarization tensor in a homogeneous external magnetic or electric field. The starting point of our considerations is the momentum space representation of the one-loop photon polarization tensor in the presence of a homogeneous electromagnetic field, known in terms of a double parameter integral. Our focus is on explicit analytical insights for both on- and off-the-light-cone dynamics in a wide range of well-specified physical parameter regimes, ranging from the perturbative to the manifestly nonperturbative strong field regime. The basic ideas underlying well-established approximations to the photon polarization tensor are carefully examined and critically reviewed. In particular, we systematically keep track of all contributions, both the ones to be neglected and those to be taken into account explicitly, to all orders. This allows us to study their ranges of applicability in a much more systematic and rigorous way. We point out the limitations of such approximations and manage to go beyond at several instances.

**Quantum Reflection as a New Signature of Quantum Vacuum Nonlinearity**

**15**, 083002 (2013)

**Abstract:** We show that photons subject to a spatially inhomogeneous electromagnetic field can experience quantum reflection. Based on this observation, we propose quantum reflection as a novel means to probe the nonlinearity of the quantum vacuum in the presence of strong electromagnetic fields.

**Symmetries of the three-heavy-quark system and the color-singlet static energy at next-to-next-to-leading logarithmic order**

**87**, 074014 (2013)

**Abstract:** We study the symmetries of the three-heavy-quark system under exchange of the quark fields within the effective field theory framework of potential nonrelativistic QCD. The symmetries constrain the form of the matching coefficients in the effective theory. We then focus on the color-singlet sector and determine the so far unknown leading ultrasoft contribution to the static potential, which is of order α_(s)^(4)lnμ, and consequently to the static energy, which is of order α_(s)^(4)lnα_(s). Finally, in the case of an equilateral geometry, we solve the renormalization group equations and resum the leading ultrasoft logarithms for the static potential of three quarks in a color singlet, octet and decuplet representation.

**Magnetically amplified light-shining-through-walls via virtual minicharged particles**

**87**, 025022 (2013)

**Abstract:** We show that magnetic fields have the potential to significantly enhance a recently proposed light-shining-through-walls scenario in quantum-field theories with photons coupling to minicharged particles. Suggesting a dedicated laboratory experiment, we demonstrate that this particular tunneling scenario could provide access to a parameter regime competitive with the currently best direct laboratory limits on minicharged fermions below the meV regime. With present day technology, such an experiment has the potential to even overcome the best model-independent cosmological bounds on minicharged fermions with masses below O(10^(-4)) eV.

## 2012

**Magnetically Amplified Tunneling of the Third Kind as a Probe of Minicharged Particles**

**109**, 131802 (2012)

**Abstract:** We show that magnetic fields significantly enhance a new tunneling mechanism in quantum field theories with photons coupling to fermionic minicharged particles (MCPs). We propose a dedicated laboratory experiment of the light-shining-through-walls type that can explore a parameter regime comparable to and even beyond the best model-independent cosmological bounds. With present-day technology, such an experiment is particularly sensitive to MCPs with masses in and below the meV regime as suggested by new-physics extensions of the standard model.

**Lambda_MS from the static potential for QCD with n_f=2 dynamical quark flavors**

**025**, 1 (2012)

**Abstract:** We determine Lambda((MS)over-bar) for QCD with n_f = 2 dynamical quark flavors by fitting the QQ̅ static potential known analytically in the perturbative regime up to terms of O(α^4_s) and ~ α^4_s ln(α_s) to corresponding results obtained from lattice simulations. This has become possible, due to recent advances in both perturbative calculations, namely the determination and publication of the last missing contribution to the QQ̅ static potential at O(α^4_s), and lattice simulations with n_f = 2 dynamical quark flavors performed at the rather fine lattice spacing of a ≈ 0.042 fm. Imposing conservative error estimates we obtain Lambda((MS)over-bar) = 315(30) MeV.