We consider the non-linear classical field theory which results from adding to the Maxwell's Lagrangian the contributions from the weak-field Euler-Heisenberg Lagrangian and a non-uniform part which involves derivatives of the electric and magnetic fields. We focus on the electrostatic case where the magnetic field is set to zero, and we derive the modified Gauss law, resulting in a higher order differential equation. This equation gives the electric field produced by stationary charges in the higher order non-linear electrodynamics. Specializing for the case of a point charge, we investigate the solutions of the modified Gauss law and calculate the correction to the Coulomb law.
Contrary to what was previously believed, two-loop radiative corrections to the $g$-factor of an electron bound in a hydrogen-like ion at $\mathcal{O}\left(α^2 (Zα)^5\right)$ exhibit logarithmic enhancement. This previously unknown contribution is due to a long-distance light-by-light scattering amplitude. Taking an effective field theory approach, and using the Euler-Heisenberg Lagrangian, we find $Δg = -\left( \fracαπ \right)^2\left(Zα\right)^5 \frac{56 π}{135}\ln Zα$.
I summarize what is known about the Euler-Heisenberg Lagrangian and its multiloop corrections for scalar and spinor QED, in various types of constant fields, and in various dimensions. Particular attention is given to the asymptotic properties of the weak-field expansion of the Lagrangian, which via Borel summation is related to Schwinger pair-creation, and the status of the "exponentiation conjecture" for the imaginary part of the Euler-Heisenberg to all loop orders.
This is the first part of lectures about effective field theories. Decoupling of heavy-particle loops is considered (heavy leptons in QED, heavy quarks in QCD).
On this 75th anniversary of the publication of the Heisenberg-Euler paper on the full non-perturbative one-loop effective action for quantum electrodynamics I review their paper and discuss some of the impact it has had on quantum field theory.
We clarify a discrepancy between two previous calculations of the two-loop QED Euler-Heisenberg Lagrangian, both performed in proper-time regularization, by calculating this quantity in dimensional regularization.
I present a pedagogical review of Heisenberg-Euler effective Lagrangians, beginning with the original work of Heisenberg and Euler, and Weisskopf, for the one loop effective action of quantum electrodynamics in a constant electromagnetic background field, and then summarizing some of the important applications and generalizations to inhomogeneous background fields, nonabelian backgrounds, and higher loop effective Lagrangians.
We examine the possibility of travelling wave solutions within the nonlinear Euler-Heisenberg electrodynamics. Since this theory resembles in its form the electrodynamics in matter, it is a priori not clear if there exist travelling wave solutions with a new dispersion relation for $ω(k)$ or if the Euler-Heisenberg theory stringently imposes $ω=k$ for any arbitrary ansatz $\mathbf{E}(ξ)$ and $\mathbf{B}(ξ)$ with $ξ\equiv \mathbf{k}\cdot\mathbf{r} -ωt$. We show that the latter scheme applies for the Euler-Heisenberg theory, but point out the possibility of new solutions with $ω\neq k$ if we go …
Considering the effective Euler-Heisenberg Lagrangian, i.e., non-linear photon-photon interactions, we study the circular polarization of electromagnetic radiation based on the time-evolution of Stokes parameters. To the leading order, we solve the Quantum Boltzmann Equation for the density matrix describing an ensemble of photons in the space of energy-momentum and polarization states, and calculate the intensity of circular polarizations. Applying these results to a linear polarized thermal radiation, we calculate the circular polarization intensity, and discuss its possible relevance to the…
The Heisenberg-Euler Lagrangian is not only a topic of fundamental interest, but also has a rich variety of diverse applications in astrophysics, nonlinear optics and elementary particle physics etc. We discuss the series representation of this Lagrangian and a few of its applications in this study. [In an appendix, we discuss issues related to the renormalization - and the renormalization-group invariance - of the Heisenberg-Euler Lagrangian and its two-loop generalization.]
Within the framework of the world-line formalism we write down in detail a two-loop Euler-Heisenberg type action for gluon loops in Yang-Mills theory and discuss its divergence structure. We exactly perform all the world-line moduli integrals at two loops by inserting a mass parameter, and then extract divergent coefficients to be renormalized.
We show that the two-loop Euler-Heisenberg effective Lagrangian for scalar QED in a constant Euclidean self-dual background has a simple explicit closed form expression in terms of the digamma function. This result leads to a simple analysis of the weak- and strong-field expansions, the two-loop scalar QED beta function, and the analytic continuation properties of the effective Lagrangian and its imaginary part.
We modify action in an external electromagnetic field to include effects of virtual axion-like particle (ALP) excitations. A measurable addition to QED-Euler-Heisenberg-Schwinger (EHS) action is obtained and incorporated into experimental constraints placed on ALP mass and coupling to two photons. The regime of these constraints in which the ALP vacuum effect surpasses the EHS effect is characterized. We show that probing of the virtual vacuum effect offers an alternative method in search for physics related to ALPs.
Vacuum polarization, a key prediction of quantum theory, can cause a variety of intriguing phenomena that can be triggered by high-intensity laser pulses. The Heisenberg-Euler theory of the quantum vacuum supplements Maxwell's theory of electromagnetism with nonlinear photon-photon interactions mediated by vacuum fluctuations. This work presents a numerical solver for the leading weak-field Heisenberg-Euler corrections. The present code implementation reaches an accuracy of order thirteen in the numerical scheme and takes into account up to six-photon interactions. Since theoretical approaches…
We calculate the multi-photon decay widths of the Higgs boson from an effective Lagrangian for a system of electromagnetic and Higgs fields. We utilize a low-energy theorem to connect the above Lagrangian to the Heisenberg-Euler effective Lagrangian induced by charged particles that gain mass from interactions with the Higgs boson. In the standard model only the W^\pm gauge bosons and the top quark are relevant, and we compute their contributions to the effective couplings and the multi-photon decay widths of the Higgs boson.
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.
The Euler-Heisenberg effective action in a self-dual background is remarkably simple at two-loop. This simplicity is due to the inter-relationship between self-duality, helicity and supersymmetry. Applications include two-loop helicity amplitudes, beta-functions and nonperturbative effects. The two-loop Euler-Heisenberg effective Lagrangian for QED in a self-dual background field is naturally expressed in terms of one-loop quantities. This mirrors similar behavior recently found in two-loop amplitudes in N=4 SUSY Yang-Mills theory.
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 $0\leq d\leq 4$ 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 vacu…
In high energy heavy ion collisions as well as in astrophysical objects like magnetars extreme magnetic field strengths are reached. Thus, there exists a need to calculate divers QED processes to all orders in the magnetic field. We calculate the vacuum polarization graph in second order of the electric field and all orders of the magnetic field resulting in a generalization of the Euler-Heisenberg Lagrangian. We perform the calculation in the effective Lagrangian approach of J. Schwinger as well as using modified Feynman rules. We find that both approaches give the same results provided that …
We study the divergence of large-order perturbation theory in the worldline expression for the two-loop Euler-Heisenberg QED effective Lagrangian in a constant magnetic field. The leading rate of divergence is identical, up to an overall factor, to that of the one-loop case. From this we deduce, using Borel summation techniques, that the leading behaviour of the imaginary part of the two-loop effective Lagrangian for a constant E field, giving the pair-production rate, is proportional to the one-loop result. This also serves as a test of the mass renormalization, and confirms the earlier analy…
In this paper, we examine the CP-violating term of the Euler-Heisenberg action. We focus in the aspects related with the generation of such a term from a QED-like model in terms of the effective action approach. In particular, we show that the generation of the CP-violating term is closely related with both of vector and axial fermionic bilinears. Although, these anomalous models are not a "viable" extension of QED, we argue that the CP-violating term in the photon sector is obtained only from this class of models, and not from any fundamental field theory.
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 ap…
We derive an analytic expression for one-loop effective action of QCD+QED at zero and finite temperatures by using the Schwinger's proper time method. The result is a nonlinear effective action not only for electromagnetic and chromo-electromagnetic fields but also the Polyakov loop, and thus reproduces the Euler-Heisenberg action in QED, QCD, and QED+QCD, and also the Weiss potential for the Polyakov loop at finite temperature. As applications of this "Euler-Heisenberg-Weiss" action in QCD+QED, we investigate quark pair productions induced by QCD+QED fields at zero temperature and the Polyako…
We develop integration-by-parts rules for Feynman diagrams involving massive scalar propagators in a constant background electromagnetic field, and use these to show that there is a simple diagrammatic interpretation of mass renormalization in the two-loop scalar QED Heisenberg-Euler effective action for a general constant background field. This explains why the square of a one-loop term appears in the renormalized two-loop Heisenberg-Euler effective action. No integrals need be evaluated, and the explicit form of the background field propagators is not needed. This dramatically simplifies the…
We analyze the structure of the imaginary part of the two-loop Euler-Heisenberg QED effective Lagrangian for a constant self-dual background. The novel feature of the two-loop result, compared to one-loop, is that the prefactor of each exponential (instanton) term in the imaginary part has itself an asymptotic expansion. We also perform a high-precision test of Borel summation techniques applied to the weak-field expansion, and find that the Borel dispersion relations reproduce the full prefactor of the leading imaginary contribution.
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…
We calculate the exact width for a dark photon decaying to three photons at one loop order for dark photon masses m' below the e+e- production threshold of 2m_e. We find substantial deviations from previous results derived from the lowest order Euler-Heisenberg effective Lagrangian in the range m_e < m' < 2m_e, where higher order terms in the derivative expansion are nonnegligible. This mass range is precisely where the three photon decay takes place on cosmologically relevant timescales. Our improved analysis opens a window for dark photons in the range 850 keV < m' < 2m_e, 10^-5 < epsilon < …
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 Lagrangia…
The main goal of this paper is a direct diagrammatic evaluation of the effective four-photon Lagrangian of the Euler-Heisenberg type for the quantum electrodynamics of massive charged vector bosons. This QED model is naturally embedded in the standard electroweak theory, and we have carried out the corresponding one-loop calculation in the unitary gauge. As far as we know, such a work has not been published before since usually the $R$-gauge techniques are preferred for the computation of loop Feynman diagrams. We have recovered the result obtained many years ago by Vanyashin and Terent'ev, wh…
In this work, we study the radiative generation of the Lorentz-violating Euler-Heisenberg action, in the weak field approximation. For this, we first consider a nonperturbative calculation in the coefficient $c_{μν}$, however, by assuming rotational invariance. Within this approach, we also obtain the results of the amplitude for the photon triple splitting. In the following, we take into account the perturbative approach, where $c_{μν}$ is treated as a insertion in the propagator and a new vertex. The partial results are in fact an expansion up to first order in $c_{μν}$ of the nonperturbativ…
Spontaneous pair production for spin-$1/2$ and spin-$0$ particles is explored in a quantitative manner for a static $\tanh$-Sauter potential step (SS), evaluating the imaginary part of the effective action. We provide finite-valued per unit-surface results, including the exact sharp-edge Klein paradox (KP) limit, which is the upper bound to pair production. At the vacuum instability threshold the spin-$0$ particle production can surpass that for the spin-$1/2$ rate. Presenting the effect of two opposite sign Sauter potential steps creating a well we show that spin-$0$ pair production, contrary…
We show that, for both scalar and spinor QED, the two-loop Euler-Heisenberg effective Lagrangian for a constant Euclidean self-dual background has an extremely simple closed-form expression in terms of the digamma function. Moreover, the scalar and spinor QED effective Lagrangians are very similar to one another. These results are dramatic simplifications compared to the results for other backgrounds. We apply them to a calculation of the low energy limits of the two-loop massive N-photon `all +' helicity amplitudes. The simplicity of our results can be related to the connection between self-d…
Augmentations to the Euler-Heisenberg Lagrangian (QED one-loop effective action in homogeneous electromagnetic fields) under a constant background axial gauge are examined. Two special configurations admit an exact eigendecomposition, and hence effective action as a spectral sum, of the augmented Dirac operator: one with a magnetic field with chiral chemical potential, and the other with an electric field with spatial axial gauge, which resembles an emergent vorticity. An enhancement to Schwinger pair production is found for the latter, which is more fully analyzed using the worldline instanto…
A generalized Heisenberg-Euler formula is given for an Abelian gauge theory having vector as well as axial vector couplings to a massive fermion. So, the formula is applicable to a parity-violating theory. The gauge group is chosen to be $U(1)$. The formula is quite similar to that in quantum electrodynamics, but there is a complexity in which one factor (related to spin) is expressed in terms of the expectation value. The expectation value is evaluated by the contraction with the one-dimensional propagator in a given background field. The formula affords a basis to the vacuum magnetic birefri…
We continue an effort to obtain information on the QED perturbation series at high loop orders, and particularly on the issue of large cancellations inside gauge invariant classes of graphs, using the example of the l - loop N - photon amplitudes in the limit of large photons numbers and low photon energies. As was previously shown, high-order information on these amplitudes can be obtained from a nonperturbative formula, due to Affleck et al., for the imaginary part of the QED effective lagrangian in a constant field. The procedure uses Borel analysis and leads, under some plausible assumptio…
We discuss quantum electrodynamics emerging in the vacua with anisotropic scaling. Systems with anisotropic scaling were suggested by Horava in relation to the quantum theory of gravity. In such vacua the space and time are not equivalent, and moreover they obey different scaling laws, called the anisotropic scaling. Such anisotropic scaling takes place for fermions in bilayer graphene, where if one neglects the trigonal warping effects the massless Dirac fermions have quadratic dispersion. This results in the anisotropic quantum electrodynamics, in which electric and magnetic fields obey diff…
We consider both massive Euler-Heisenberg-like and Euler-Heisenberg-like Electrodynamics in the approximation of the strong-field limit. Our analysis shows that massive Euler-Heisenberg-type Electrodynamics displays the vacuum birefringence phenomenon. Afterwards, we calculate the lowest-order modifications to the interaction energy for both classes of Electrodynamics, within the framework of the gauge-invariant path-dependent variables formalism. Interestingly enough, for massive Euler-Heisenbeg-like electrodynamics (Whichmann-Kroll) we obtain a new long-range ($1/r^3$- type) correction, apar…
In this paper we calculate the non-perturbative Euler-Heisenberg Lagrangian for massless QED in a strong magnetic field $H$, where the breaking of the chiral symmetry is dynamically catalyzed by the external magnetic field via the formation of an electro-positron condensate. This chiral condensate leads to the generation of dynamical parameters that have to be found as solutions of non-perturbative Schwinger-Dyson equations. Since the electron-positron pairing mechanism leading to the breaking of the chiral symmetry is mainly dominated by the contributions from the infrared region of momenta m…
We show that the one-loop Euler-Heisenberg QED effective Lagrangian in a constant background field acquires a very different non-perturbative trans-series structure at two-loop and higher-loop order in the fine structure constant. Beyond one-loop, virtual particles interact, causing fluctuations about the instantons, whereby the simple poles of the one-loop Borel transform become branch points. We illustrate this in detail at two-loop order using Ritus's seminal result for the renormalized two-loop effective Lagrangian as an exact double-integral representation, and propose a possible new appr…
I reexamine the phenomena of the chromomagnetic gluon condensation in Yang-Mills theory. The extension of the Heisenberg-Euler Lagrangian to the Yang-Mills theory allows to calculate the effective action, the energy-momentum tensor and demonstrate that the energy density curve crosses the zero energy level of the perturbative vacuum state at nonzero angle and continuously enters to the negative energy density region. At the crossing point and further down the effective coupling constant is small and demonstrate that the true vacuum state of the Yang-Mills theory is below the perturbative vacuu…
The angular dependence of the differential cross section of unpolarized light-by-light scattering summed over final polarizations is the same in any low-energy effective theory of quantum electrodynamics and also in Born-Infeld electrodynamics. In this paper we derive general expressions for polarization-dependent low-energy scattering amplitudes, including a hypothetical parity-violating situation. These are evaluated for quantum electrodynamics with charged scalar or spinor particles, which give strikingly different polarization effects. Ordinary quantum electrodynamics is found to exhibit r…
Taking into account the Euler-Heisenberg effective Lagrangian of one-loop nonperturbative quantum electrodynamics (QED) contributions, we formulate the Einstein-Euler-Heisenberg theory and study the solutions of nonrotating black holes with electric and magnetic charges in spherical geometry. In the limit of strong and weak electromagnetic fields of black holes, we calculate the black hole horizon radius, area, and total energy up to the leading order of QED corrections and discuss the black hole irreducible mass, entropy, and maximally extractable energy as well as the Christodoulou-Ruffini m…
We investigate the Landau-level structures encoded in the famous Heisenberg-Euler (HE) effective action in constant electromagnetic fields. We first discuss the HE effective actions for scalar and spinor QED, and then extend it to the QCD analogue in the covariantly constant chromo-electromagnetic fields. We identify all the Landau levels and the Zeeman energies starting out from the proper-time representations at the one-loop order, and derive the vacuum persistence probability for the Schwinger mechanism in the summation form over independent contributions of the all-order Landau levels. We …
We study the three-loop Euler-Heisenberg Lagrangian in spinor quantum electrodynamics in 1+1 dimensions. In this first part we calculate the one-fermion-loop contribution, applying both standard Feynman diagrams and the worldline formalism which leads to two different representations in terms of fourfold Schwinger-parameter integrals. Unlike the diagram calculation, the worldline approach allows one to combine the planar and the non-planar contributions to the Lagrangian. Our main interest is in the asymptotic behaviour of the weak-field expansion coefficients of this Lagrangian, for which a n…
The low energy effective field model for the multilayer graphene (at ABC stacking) is considered. We calculate the effective action in the presence of constant external magnetic field $B$ (normal to the graphene sheet). We also calculate the first two corrections to this effective action caused by the in-plane electric field $E$ at $E/B \ll 1$ and discuss the magnetoelectric effect. In addition, we calculate the imaginary part of the effective action in the presence of constant electric field $E$ and the lowest order correction to it due to the magnetic field ($B/E \ll 1$).
We analyze vacuum instability of strongly coupled gauge theories in a constant electric field using AdS/CFT correspondence. The model is the N=2 1-flavor supersymmetric large N_c QCD in the strong 't Hooft coupling limit. We calculate the Euler-Heisenberg effective Lagrangian L(E), which encodes the nonlinear response and the quantum decay rate of the vacuum in a background electric field E, from the complex D-brane action in AdS/CFT. We find that the decay rate given by Im L(E) becomes nonzero above a critical electric field set by the confining force between quarks. A large-E expansion of Im…
Particle production in strong electromagnetic fields is a recurring theme in solid state physics, heavy ion collisions, early universe cosmology and formal quantum field theory. In this paper we discuss the Dirac equation in a background of parallel electric and magnetic fields. We review the Schwinger particle production rate, clarify the emergence of the chiral anomaly equation and compute the induced current of charged fermions. We distinguish the contributions from non-perturbative particle production, from the running of the gauge coupling constant and from non-linearities in the effectiv…
The nonlinear Euler-Heisenberg interaction bends light toward an electric charge. The bending angle and trajectory of light in a Coulombic field are computed in geometric optics.
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 demonst…
