Schwinger Formula Research Articles

Pulsed excitation of a quantum oscillator: A model accounting for damping

Abstract This article proposes a model that accounts for damping of a quantum oscillator (QO) during pulsed excitation. Our model is based on the Schwinger formula, which calculates oscillator’s excitation probability through the energy of an associated classical damped oscillator. We utilize this model to describe the influence of damping on temporal and spectral dependences of QO excitation, induced by electromagnetic pulses with exponential and double exponential envelopes. The oscillator excitation is analyzed in terms of transition probability between stationary states after pulse termination. Here, we present an analytical description of these dependences, along with numerical results. Specifically, we derive analytical expressions that depict the saturation effect during pulsed excitation, taking into account the damping of a QO. The evolution of the temporal dependence of the excitation probability with a change in the damping constant is numerically traced. We demonstrate that the number of maxima in this dependence is determined by the values of pulse parameters and the damping constant.

Dyon production from near-extremal Kerr–Newman–(anti-)de Sitter black holes

Using the enhanced symmetry in the near-horizon region of the near-extremal dyonic Kerr–Newman (KN) black hole in the (A)dS space, we find the exact solutions for dyonic charged scalar field in terms of the hypergeometric function and explicitly compute the Schwinger effect for the emission of electric and/or magnetic charges. The emission formula confirms a universal factorization of the Schwinger formula in the AdS_2 and another Schwinger formula in the two-dimensional Rindler space determined by the effective temperature and the Hawking temperature with the chemical potentials of electric and/or magnetic charges and the angular momentum. The emission of the same species of charges from the KN black hole is enhanced in the AdS boundary while it is suppressed in the dS boundary. In addition, the dragging of particles in the KN black hole diminishes the emission of charges in both AdS and dS spaces. The AdS geometry of near-horizon gives the Breitenloher–Freedman (BF) bound, within which the stability of dyonic KN black holes is guaranteed against both the emission of charges and Hawking radiation.

On the Temperature Dependence of the String-Breaking Distance in QCD

The temperature dependence of the QCD string-breaking distance is evaluated in terms of the string tension and the rate of production of light mesons in the chromo-electric field of a flux tube. As a function of the meson mass, the mentioned rate can be falling off either as a Gaussian, as suggested by the Schwinger formula, or as an exponential, which is the case in the London limit of the dual superconductor. We find an excellent agreement of the so-evaluated temperature dependence of the string-breaking distance with the respective lattice data, for the case of the meson-production rate corresponding to the London limit.

Effective field theory of magnons: Chiral magnets and the Schwinger mechanism

We develop the effective field theoretical descriptions of spin systems in the presence of symmetry-breaking effects: the magnetic field, single-ion anisotropy, and Dzyaloshinskii-Moriya interaction. Starting from the lattice description of spin systems, we show that the symmetry-breaking terms corresponding to the above effects can be incorporated into the effective field theory as a combination of a background (or spurious) $\SO(3)$ gauge field and a scalar field in the symmetric tensor representation, which are eventually fixed at their physical values. We use the effective field theory to investigate mode spectra of inhomogeneous ground states, with focusing on one-dimensionally inhomogeneous states, such as helical and spiral states. Although the helical and spiral ground states share a common feature of supporting the gapless Nambu-Goldstone modes associated with the translational symmetry breaking, they have qualitatively different dispersion relations: isotropic in the helical phase while anisotropic in the spiral phase. We also discuss the magnon production induced by an inhomogeneous magnetic field, and find a formula akin to the Schwinger formula. Our formula for the magnon production gives a finite rate for antiferromagnets, and a vanishing rate for ferromagnets, whereas that for ferrimagnets interpolates between the two cases.

Temporal Klein Model for Particle-Pair Creation

This work considers the creation of electron-positron pairs from intense electric fields in vacuum, for arbitrary temporal field variations. These processes can be useful to study quantum vacuum effects with ultra-intense lasers. We use the quantized Dirac field to explore the temporal Klein model. This model is based on a vector potential discontinuity in time, in contrast with the traditional model based on a scalar potential discontinuity in space. We also extend the model by introducing a finite time-scale for potential variations. This allows us to study the transition from a singular electric field spike, with infinitesimal duration, to the opposite case of a static field where the Schwinger formula would apply. The present results are intrinsically non-perturbative. Explicit expressions for pair-creation as a function of the potential time-scales are derived. This work explores the spacetime symmetry associated with pair creation in vacuum: the space symmetry breaking of the old Klein paradox model, in contrast with the time symmetry breaking of the temporal Klein model.

(F, G) -summed form of the QED effective action

We conjecture that the proper-time series expansion of the one-loop effective Lagrangian of quantum electrodynamics can be summed in all terms containing the field-strength invariants $\mathcal{F}=\frac{1}{4}{F}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}}(x)$, $\mathcal{G}=\frac{1}{4}{\stackrel{\texttildelow{}}{F}}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}}(x)$, including those also possessing derivatives of the electromagnetic field strength. This partial resummation is exactly encapsulated in a factor with the same form as the Heisenberg-Euler Lagrangian density, except that now the electric and magnetic fields can depend arbitrarily on spacetime coordinates. We provide strong evidence for this conjecture, which is proved to sixth order in the proper time. Furthermore, and as a byproduct, we generate some solvable electromagnetic backgrounds. We also discuss the implications for a generalization of the Schwinger formula for pair production induced by nonconstant electric fields. Finally, we briefly outline the extension of these results in the presence of gravity.

Axial Ward Identity and the Schwinger Mechanism: Applications to the Real-Time Chiral Magnetic Effect and Condensates.

We elucidate chirality production under parity breaking constant electromagnetic fields, with which we clarify qualitative differences in and out of equilibrium. For a strong magnetic field the pair production from the Schwinger mechanism increments the chirality. The pair production rate is exponentially suppressed with mass according to the Schwinger formula, while the mass dependence of chirality production in the axial Ward identity appears in the pseudoscalar term only. We demonstrate that, in a real-time formulation with in and out states, the axial Ward identity with an in-in expectation value leads to a chirality production rate consistent with the Schwinger formula, while the axial anomaly with an in-out expectation value is canceled by the pseudoscalar condensate for any mass. We illuminate that such an in- and out-state formulation clarifies subtleties in the chiral magnetic effect in and out of equilibrium, and we discuss further applications to real-time condensates.

Invited Article: Refined analysis of synchrotron radiation for NIST’s SURF III facility

We have developed a new method for the exact calculation of synchrotron radiation for the National Institute of Standards and Technology Synchrotron Ultraviolet Radiation Facility, SURF III. Instead of using the Schwinger formula, which is only an approximation, we develop formulae based on Graf's addition theorem for Bessel functions and accurate asymptotic expansions for Hankel functions and Bessel functions. By measuring the radiation intensity profile at two distances from the storage ring, we also confirm an apparent vertical emittance that is consistent with the vertical betatron oscillations that are intentionally introduced to extend beam lifetime by spreading the electron beam spatially. Finally, we determine how much diffraction by beamline apertures enhances the spectral irradiance at an integrating sphere entrance port at the end station. This should eliminate small but treatable components of the uncertainty budget that one should consider when using SURF III or similar synchrotrons as standard, calculable sources of ultraviolet and other radiation.

Quark and gluon production from a boost-invariantly expanding color electric field

Particle production from an expanding classical color electromagnetic field is extensively studied, motivated by the early stage dynamics of ultra-relativistic heavy ion collisions. We develop a formalism at one-loop order to compute the particle spectra by canonically quantizing quark, gluon and ghost fluctuations under the presence of such an expanding classical color background field; the canonical quantization is done in the $\tau$-$\eta$ coordinates in order to take into account manifestly the expanding geometry. As a demonstration, we model the expanding classical color background field by a boost-invariantly expanding homogeneous color electric field with lifetime $T$, for which we obtain analytically the quark and gluon production spectra by solving the equations of motion of QCD non-perturbatively with respect to the color electric field. In this paper we study (i) finite lifetime effect which is found to modify significantly the particle spectra from those expected from the Schwinger formula; (ii) the difference between the quark and gluon production; and (iii) the quark mass dependence of the production spectra. Implications of these results to ultra-relativistic heavy ion collisions are also discussed.

World-line instantons and the Schwinger effect as a Wentzel–Kramers–Brillouin exact path integral

A detailed study of the semiclassical expansion of the world line path integral for a charged relativistic particle in a constant external electric field is presented. We show that the Schwinger formula for charged particle pair production is reproduced exactly by the semiclassical expansion around classical instanton solutions when the leading order of fluctuations is taken into account. We prove that all corrections to this leading approximation vanish and that the WKB approximation to the world line path integral is exact.

One-loop effective action and Schwinger effect in (anti-) de Sitter space

We study the Schwinger mechanism by a uniform electric field in ${\rm dS}_2$ and ${\rm AdS}_2$ and the curvature effect on the Schwinger effect, and further propose a thermal interpretation of the Schwinger formula in terms of the Gibbons-Hawking temperature and the Unruh temperature for an accelerating charge in ${\rm dS}_2$ and an analogous expression in ${\rm AdS}_2$. The exact one-loop effective action is found in the proper-time integral in each space, which is determined by the effective mass, the Maxwell scalar, and the scalar curvature, and whose pole structure gives the imaginary part of the effective action and the exact pair-production rate. The exact pair-production rate is also given the thermal interpretation.

Thermalization of anisotropic quark–gluon plasma produced by decays of color flux tubes

Kinetic equations are used to study thermalization of the anisotropic quark-gluon plasma produced by decays of color flux tubes possibly created at the very early stages of relativistic heavy-ion collisions. The decay rates of the initial color tubes are given by the Schwinger formula, while the collision terms are taken in the relaxation-time approximation. By connecting the relaxation time with viscosity we analyze production and thermalization processes in the plasma characterized by different values of the ratio of the shear viscosity to entropy density.

Finite pulse effects one+e−pair creation from strong electric fields

We investigate electron-positron pair creation from the vacuum in a pulsed electric background field. Employing the Sauter-type pulsed field $E(t)=E_0 {\rm sech}^2 (t/\tau)$ with height $E_0$ and width $\tau$, we demonstrate explicitly the interplay between the nonperturbative and perturbative aspects of pair creation in the background field. We analytically compute the number of produced pairs from the vacuum in the Sauter-type field, and the result reproduces Schwinger's nonperturbative formula in the long pulse limit (the constant field limit), while in the short pulse limit it coincides with the leading-order perturbative result. We show that two dimensionless parameters $\nu = |eE_0| \tau^2$ and $\gamma = |eE_0| \tau /m_e$ characterize the importance of multiple interactions with the fields and the transition from the perturbative to the nonperturbative regime. We also find that pair creation is enhanced compared to Schwinger's formula when the field strength is relativity weak $|eE_0|/m_e^2 \lesssim 1$ and the pulse duration is relatively short $m_e \tau \lesssim 1$, and reveal that the enhancement is predominantly described by the lowest order perturbation with a single photon.

On Schwinger's formula for pair production

We present some comments on Schwingers's calculation of electron–positron production in a prescribed constant electric field. The range of validity of 2 Im ℒ(1)(E) is discussed thoroughly and limiting cases are provided.

Equilibration of anisotropic quark-gluon plasma produced by decays of color flux tubes

A set of kinetic equations is used to study equilibration of the anisotropic quark-gluon plasma produced by decays of color flux tubes possibly created at the very early stages of ultra-relativistic heavy-ion collisions. The decay rates of the initial color fields are given by the Schwinger formula, and the collision terms are treated in the relaxation-time approximation. By connecting the relaxation time with viscosity we are able to study production and thermalization processes in the plasma characterized by different values of the ratio of the shear viscosity to entropy density, eta/s. For the lowest (KSS) value of this ratio, 4 pi eta/s = 1, and realistic initial conditions for the fields, the system approaches the viscous-hydrodynamics regime within 1-2 fm/c. On the other hand, for larger values of the viscosity, 4 pi eta/s greater or equal to 3, the collisions in the plasma become inefficient to destroy collective phenomena which manifest themselves as oscillations of different plasma parameters. The presence of such oscillations brings in differences between the kinetic and hydrodynamic descriptions, which suggest that the viscous-hydrodynamics approach after 1-2 fm/c is not complete if 4 pi eta/s is greater or equal to 3 and should be extended to include dissipative phenomena connected with color conductivity.

Simulating fermion production in1+1dimensional QED

We investigate fermion-antifermion production in $1+1$ dimensional QED using real-time lattice techniques. In this nonperturbative approach the full quantum dynamics of fermions is included, while the gauge field dynamics can be accurately represented by classical-statistical simulations for relevant field strengths. We compute the nonequilibrium time evolution of gauge-invariant correlation functions, implementing ``low-cost'' Wilson fermions. Introducing a lattice generalization of the Dirac-Heisenberg-Wigner function, we recover the Schwinger formula in $1+1$ dimensions in the limit of a static background field. We discuss the decay of the field due to the backreaction of the created fermion-antifermion pairs and apply the approach to strongly inhomogeneous gauge fields. The latter allows us to discuss the striking phenomenon of a linear rising potential building up between produced fermion bunches after the initial electric pulse ceased.

Holographic Schwinger Effect

We study tunneling pair creation of W bosons by an external electric field on the Coulomb branch of N=4 supersymmetric Yang-Mills theory. We use AdS/CFT holography to find a generalization of Schwinger's formula for the pair production rate to the strong coupling, planar limit which includes the exchange of virtual massless particles to all orders. We find that the pair creation formula has an upper critical electric field beyond which the process is no longer exponentially suppressed. The value of the critical field is identical to that which occurs in the Born-Infeld action of probe D3-branes in the AdS(5)×S(5) background, where AdS(5) and S(5) are 5-dimensional anti-de Sitter space and the 5-sphere, respectively.

Pairs emission in a uniform background field: an algebraic approach

A fully algebraic general approach is developed to treat the pairs emission and absorption in the presence of some uniform external background field. In particular, it is shown that the pairs production and annihilation operators, together with the pairs number operator, do actually fulfil the SU(2) functional Lie algebra. As an example of application, the celebrated Schwinger formula is consistently and nicely recovered, within this novel approach, for a Dirac spinor field in the presence of a constant and homogeneous electric field in four spacetime dimensions.

Nonperturbative enhancement of heavy quark-pair production in a strong SU(2) color field

Non-perturbative charm and bottom quark-pair production is investigated in the early stage of heavy ion collisions. Following our earlier works, the time-dependent study is based on a kinetic description of fermion-pair production in strong non-Abelian fields. We introduce time-dependent chromo-electric external field with a pulse-like time evolution, which simulates the overlap of two colliding heavy ions. The calculations is performed in a SU(2) color model with finite current quark masses. Yields of heavy quark-pairs are compared to the ones of light and strange quark-pairs. We show that the small inverse duration time of the field pulse determines the efficiency of the quark-pair production. Thus we do not see the expected suppression for heavy quark production, as follows from the Schwinger formula for constant field, but rather an enhanced heavy quark production at ultrarelativistic energies. We convert pulse duration time-dependent results into collisional energy dependence and introduce flavour-dependent energy string tensions, which can be used in string based model calculations at RHIC and LHC energies.

On Schwinger pair creation in gravity and in closed superstring theory

We investigate the Schwinger pair creation process in the context of gravitational models with the back reaction of the electric field included in the geometry. The background is also an exact solution of type II superstring theory, where the electric field arises by Kaluza-Klein reduction. We obtain a closed formula for the pair creation rate that incorporates the gravitational back reaction. At weak fields it has the same structure as the general Schwinger formula, albeit pairs are produced by a combination of Schwinger and Unruh effect, the latter due to the presence of a Rindler horizon. In four spacetime dimensions, the rate becomes constant at strong electric fields. For states with mass of Kaluza-Klein origin, the rate has a power-like dependence in the electric field, rather than the familiar (non-perturbative) exponential dependence. We also reproduce the same formula from the string partition function for winding string states. Finally, we comment on the generalization to excited string states.