arXiv subjects

S. P. Klevansky

Publications and source records attributed to S. P. Klevansky.

At least 19 recordsLinked to original sources

arXiv2026

Emergence of kaonium as a sharp resonance in photon-photon to meson-meson cross-sections

We calculate the binding energies of the hypothetical mesonic atom, $K^+ K^-$ (kaonium), using the $K^+ K^- \to K^+ K^-$ elastic scattering amplitude. Our findings are in line with previously reported results, which involve solving an eigenvalue equation of the Kudryavtsev-Popov type. Using chiral perturbation theory, we show that kaonium manifests itself as a sharp resonance around 992 MeV accompanying $f_0 (980)$ or $a_0 (980)$ in cross-sections for processes $γγ\to π^0 π^0$ or $γγ\to π^0 η$. The latter process is particularly striking: the peak at the kaonium resonance energy is highly pronounced, with the ratio of the cross-sections $σ(γγ\to π^0 η) / σ(γγ\to π^0 π^0) \approx 9$. Due to the short lifetime of kaonium ($\sim 10^{-18}$ s) and its small decay width ($\sim 0.4$ keV), direct detection of this exotic atom poses a significant challenge and requires high experimental resolution. However, we show that once the formation of kaonium is considered in the cross-section, a better fit to the available experimental data is obtained.

hep-ph↗

arXiv2025

Amplitude determination for $M M \to M M$, $M = π, K$ and cross-sections for $γγ\to π^+ π^-, π^0 π^0, π^0 η$ in a chiral model

Dai and Pennington have performed a comprehensive analysis of essentially all pion and kaon pair production data from two-photon collisions below 1.5 GeV, including all high statistics results from Belle, as well as the older data from Mark II at SLAC, CELLO at DESY, and Crystal Ball at SLAC. Imposing the basic constraints required by analyticity, unitarity, and crossing symmetry and making use of Low's low-energy theorem for QED, they were able to extract the final-state, strong-interaction scattering amplitudes for the intermediate $ππ\to ππ$ and $ππ\to K \overline{K}$ reactions in a model-independent fashion. In addition, they provided good fits to the respective $γγ\to ππ$ cross-sections that are known in the low-energy sector in the restricted angular range, $| \cos θ| < 0.6 - 0.8$. Using the parameters obtained in this fashion, these authors constructed the $γγ\to ππ$ cross-sections integrated over the full angular range. In the present work, we use a version of chiral perturbation theory developed by Oller and Oset to evaluate the final-state, strong-interaction amplitudes theoretically, and we compare our low-energy QCD-based results with the amplitudes extracted by Dai and Pennington. We also calculate the $γγ\to ππ$ cross-sections (integrated over the full angular range) and compare them with those obtained by Dai and Pennington. These calculations give a more detailed insight into the fit of chiral perturbation theory, not just to the measured $γγ\to ππ$ cross-sections, as is usually presented, but rather to a higher level of detail through the available analysis of the experimental data for the underlying final-state, strong-interaction, meson-meson scattering amplitudes $ππ\to ππ$ and $ππ\to K \overline{K}$ themselves. The fits appear to be sensible.

hep-ph↗

arXiv2024

Three perspectives on entropy dynamics in a non-Hermitian two-state system

A comparative study of entropy dynamics as an indicator of physical behavior in an open two-state system with balanced gain and loss is presented. We distinguish the perspective taken in utilizing the conventional framework of Hermitian-adjoint states from an approach that is based on biorthogonal-adjoint states and a third case based on an isospectral mapping. In this it is demonstrated that their differences are rooted in the treatment of the environmental coupling mode. For unbroken $\mathcal{PT}$ symmetry of the system, a notable characteristic feature of the perspective taken is the presence or absence of purity oscillations, with an associated entropy revival. The description of the system is then continued from its $\mathcal{PT}$-symmetric pseudo-Hermitian phase into the regime of spontaneously broken symmetry, in the latter two approaches through a non-analytic operator-based continuation, yielding a Lindblad master equation based on the $\mathcal{PT}$ charge operator $\mathcal{C}$. This phase transition indicates a general connection between the pseudo-Hermitian closed-system and the Lindbladian open-system formalism through a spontaneous breakdown of the underlying physical reflection symmetry.

quant-ph↗

arXiv2023

Dyson-Schwinger equations in zero dimensions and polynomial approximations

The Dyson-Schwinger (DS) equations for a quantum field theory in $D$-dimensional space-time are an infinite sequence of coupled integro-differential equations that are satisfied exactly by the Green's functions of the field theory. This sequence of equations is underdetermined because if the infinite sequence of DS equations is truncated to a finite sequence, there are always more Green's functions than equations. An approach to this problem is to close the finite system by setting the highest Green's function(s) to zero. One can examine the accuracy of this procedure in $D=0$ because in this special case the DS equations are just a sequence of coupled polynomial equations whose roots are the Green's functions. For the closed system one can calculate the roots and compare them with the exact values of the Green's functions. This procedure raises a general mathematical question: When do the roots of a sequence of polynomial approximants to a function converge to the exact roots of that function? Some roots of the polynomial approximants may (i) converge to the exact roots of the function, or (ii) approach the exact roots at first and then veer away, or (iii) converge to limiting values that are unequal to the exact roots. In this study five field-theory models in $D=0$ are examined, Hermitian $ϕ^4$ and $ϕ^6$ theories and non-Hermitian $iϕ^3$, $-ϕ^4$, and $-i ϕ^5$ theories. In all cases the sequences of roots converge to limits that differ by a few percent from the exact answers. Sophisticated asymptotic techniques are devised that increase the accuracy to one part in $10^7$. Part of this work appears in abbreviated form in Phys.~Rev.~Lett.~{\bf 130}, 101602 (2023).

math-ph↗

arXiv2023

Underdetermined Dyson-Schwinger equations

This paper examines the effectiveness of the Dyson-Schwinger (DS) equations as a calculational tool in quantum field theory. The DS equations are an infinite sequence of coupled equations that are satisfied exactly by the connected Green's functions $G_n$ of the field theory. These equations link lower to higher Green's functions and, if they are truncated, the resulting finite system of equations is underdetermined. The simplest way to solve the underdetermined system is to set all higher Green's function(s) to zero and then to solve the resulting determined system for the first few Green's functions. The $G_1$ or $G_2$ so obtained can be compared with exact results in solvable models to see if the accuracy improves for high-order truncations. Five $D=0$ models are studied: Hermitian $ϕ^4$ and $ϕ^6$ and non-Hermitian $iϕ^3$, $-ϕ^4$, and $iϕ^5$ theories. The truncated DS equations give a sequence of approximants that converge slowly to a limiting value but this limiting value always {\it differs} from the exact value by a few percent. More sophisticated truncation schemes based on mean-field-like approximations do not fix this formidable calculational problem.

math-ph↗

arXiv2022

Thermodynamic properties of non-Hermitian Nambu--Jona-Lasinio models

We investigate the impact of non-Hermiticity on the thermodynamic properties of interacting fermions by examining bilinear extensions to the $3+1$ dimensional $SU(2)$-symmetric Nambu--Jona-Lasinio (NJL) model of quantum chromodynamics at finite temperature and chemical potential. The system is modified through the anti-$PT$-symmetric pseudoscalar bilinear $\barψγ_5 ψ$ and the $PT$-symmetric pseudovector bilinear $iB_ν\,\barψγ_5γ^νψ$, introduced with a coupling $g$. Beyond the possibility of dynamical fermion mass generation at finite temperature and chemical potential, our findings establish model-dependent changes in the position of the chiral phase transition and the critical end-point. These are tunable with respect to $g$ in the former case, and both $g$ and $|B|/B_0$ in the latter case, for both lightlike and spacelike fields. Moreover, the behavior of the quark number, entropy, pressure, and energy densities signal a potential fermion or antifermion excess compared to the standard NJL model, due to the pseudoscalar and pseudovector extension respectively. In both cases regions with negative interaction measure $I = ε-3p$ are found. Future indications of such behaviors in strongly interacting fermion systems, for example in the context of neutron star physics, may point toward the presence of non-Hermitian contributions. These trends provide a first indication of curious potential mechanisms for producing non-Hermitian baryon asymmetry. In addition, the formalism described in this study is expected to apply more generally to other Hamiltonians with four-fermion interactions and thus the effects of the non-Hermitian bilinears are likely to be generic.

hep-ph↗

arXiv2021

Towards perturbative renormalization of $ϕ^2(iϕ)^\varepsilon$ quantum field theory

In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Green's functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=\frac{1}{2}(\nablaϕ)^2+\frac{1}{2}ϕ^2(iϕ)^\varepsilon$ in $D$-dimensional Euclidean spacetime, where $ϕ$ is a pseudoscalar field. In this earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed as perturbation series in powers of $\varepsilon$ and were calculated to first order in $\varepsilon$. (The parameter $\varepsilon$ is a measure of the nonlinearity of the interaction rather than a coupling constant.) This paper extends these perturbative calculations to the Euclidean Lagrangian $L= \frac{1}{2}(\nablaϕ)^2+\frac{1}{2}μ^2ϕ^2+\frac{1}{2} gμ_0^2ϕ^2\big(iμ_0^{1-D/2}ϕ\big)^\varepsilon-ivϕ$, which now includes renormalization counterterms that are linear and quadratic in the field $ϕ$. The parameter $g$ is a dimensionless coupling strength and $μ_0$ is a scaling factor having dimensions of mass. Expressions are given for the one-, two, and three-point Green's functions, and the renormalized mass, to higher-order in powers of $\varepsilon$ in $D$ dimensions ($0\leq D\leq2$). Renormalization is performed perturbatively to second order in $\varepsilon$ and the structure of the Green's functions is analyzed in the limit $D\to 2$. A sum of the most divergent terms is performed to {\it all} orders in $\varepsilon$. Like the Cheng-Wu summation of leading logarithms in electrodynamics, it is found here that leading logarithmic divergences combine to become mildly algebraic in form. Future work that must be done to complete the perturbative renormalization procedure is discussed.

hep-th↗

arXiv2021

Fermion and meson mass generation in non-Hermitian Nambu--Jona-Lasinio models

We investigate the effects of non-Hermiticity on interacting fermionic systems. We do this by including non-Hermitian bilinear terms into the 3+1 dimensional Nambu--Jona-Lasinio (NJL) model. Two possible bilinear modifications give rise to $\mathcal{PT}$ symmetric theories; this happens when the standard NJL model is extended either by a pseudovector background field $ig \barψγ_5 B_μγ^μψ$ or by an antisymmetric-tensor background field $g \barψF_{μν}γ^μγ^νψ$. The three remaining bilinears are {\it anti}-$\mathcal{PT}$-symmetric in nature, $ig \barψB_μγ^μψ, ig\barψγ_5 ψ$ and $ig\barψ{1}ψ$, so that the Hamiltonian then has no overall symmetry. The pseudovector $ig \barψγ_5 B_μγ^μψ$ and the vector $ig \barψB_μγ^μψ$ combinations, are, in addition, chirally symmetric. Thus, within this framework we are able to examine the effects that the various combinations of non-Hermiticity, $\mathcal{PT}$ symmetry, chiral symmetry and the two-body interactions of the NJL model have on the existence and dynamical generation of a real effective fermion mass (a feature which is absent in the corresponding modified massless free Dirac models) as well as on the masses of the composite particles, the pseudoscalar and scalar mesonic modes ($π$ and $σ$ mesons). Our findings demonstrate that $\mathcal{PT}$ symmetry is not necessary for real fermion mass solutions to exist, rather the two-body interactions of the NJL model supersede the non-Hermitian bilinear effects. The effects of chiral symmetry are evident most clearly in the meson modes, the pseudoscalar of which will always be Goldstone in nature if the system is chirally symmetric. Second solutions of the mesonic equations are also discussed.

hep-ph↗

arXiv2020

Non-Hermitian extension of the Nambu--Jona-Lasinio model in 3+1 and 1+1 dimensions

This paper presents a non-Hermitian PT-symmetric extension of the Nambu--Jona-Lasinio (NJL) model of quantum chromodynamics in 3+1 and 1+1 dimensions. In 3+1 dimensions, the SU(2)-symmetric NJL Hamiltonian $H_{\textrm{NJL}} = \barψ(-i γ^k \partial_k + m_0) ψ- G [ (\barψψ)^2 + (\barψi γ_5 \vecτ ψ)^2 ]$ is extended by the non-Hermitian, PT- and chiral-symmetric bilinear term $ig\barψγ_5 B_μ γ^μ ψ$; in 1+1 dimensions, where $H_{\textrm{NJL}}$ is a form of the Gross-Neveu model, it is extended by the non-Hermitian PT-symmetric but chiral symmetry breaking term $g \barψγ_5 ψ$. In each case, the gap equation is derived and the effects of the non-Hermitian terms on the generated mass are studied. We have several findings: in previous calculations for the free Dirac equation modified to include non-Hermitian bilinear terms, contrary to expectation, no real mass spectrum can be obtained in the chiral limit; in these cases a nonzero bare fermion mass is essential for the realization of PT symmetry in the unbroken regime. Here, in the NJL model, in which four-point interactions are present, we {\it do} find real values for the mass spectrum also in the limit of vanishing bare masses in both 3+1 and 1+1 dimensions, at least for certain specific values of the non-Hermitian couplings $g$. Thus, the four-point interaction overrides the effects leading to PT symmetry-breaking for these parameter values. Further, we find that in both cases, in 3+1 and in 1+1 dimensions, the inclusion of a non-Hermitian bilinear term can contribute to the generated mass. In both models, this contribution can be tuned to be small; we thus fix the fermion mass to its value when $m_0=0$ in the absence of the non-Hermitian term, and then determine the value of the coupling required so as to generate a bare fermion mass.

hep-ph↗

arXiv2019

Continuous quantum phase transition in the fermionic mass solutions of the Nambu-Jona-Lasinio model

Recently quantum simulators have been constructed to investigate experimentally the most prominent theoretical four-point many-body system described by the Hubbard model. By varying the coupling strength of the four-point interaction in relation to the kinetic term, one can analyze the phase structure of the model. This intriguing fact leads us to investigate whether similar Hamiltonians with four-point interactions can also be studied as a function of their four-point coupling strength. In this paper, we reexamine the Nambu-Jona-Lasinio model, regarding it generally beyond the context of quantum chromodynamics. Essentially, it is a model in which particle-antiparticle pairing leads to a BCS-like condensate, with the result that chiral symmetry is broken dynamically in the strong-coupling regime. To study the behavior of the system, it is necessary to move from this regime to a hypothetical regime of weak coupling, altering the coupling strength of the interaction arbitrarily. In order to do this, the gap equation must be regarded as complex and its Riemann surface structure must be known. We do this and obtain a continuous quantum phase transition characterized by the development of a complex order parameter (the dynamically generated mass) from the second sheet of the Riemann surface, as we move into the weak-coupling regime. The power-law behavior of the order parameter in the vicinity of the phase transition point is demonstrated to be independent of the choice of the regularization scheme with the critical exponent as $β\approx 0.55$. At the same time, the isovector pseudoscalar modes retain their feature as Goldstone modes and still have zero mass, while the isoscalar scalar meson follows the behavior of the order parameter and gains a width. Energetically, this mode is not favored over the normal, uncondensed mode but would have to be accessed through an excitation process.

hep-ph↗

arXiv2019

Relativistic PT-symmetric fermionic theories in 1+1 and 3+1 dimensions

Relativistic PT-symmetric fermionic interacting systems are studied in 1+1 and 3+1 dimensions. The objective is to include non-Hermitian PT-symmetric interaction terms that give {\it real} spectra. Such interacting systems could describe new physics. The simplest non-Hermitian Lagrangian density is $L=L_0+L_{int}=\barψ(i\not\partial-m)ψ-g\barψγ^5ψ$. The associated relativistic Dirac equation is PT invariant in 1+1 dimensions and the associated Hamiltonian commutes with PT. However, the dispersion relation $p^2=m^2-g^2$ shows that the PT symmetry is broken in the chiral limit $m\to0$. For interactions $L_{int}=-g(\barψγ^5ψ)^N$ with N=2,3, if the associated Dirac equation is PT invariant, the dispersion relation gives complex energies as $m\to0$. Other models are studied in which x-dependent PT-symmetric potentials such as $ix^3$, $-x^4$, $iκ/x$, Hulthén, or periodic potentials are coupled to $ψ$ and the classical trajectories plane are examined. Some combinations of these potentials give a real spectrum. In 3+1 dimensions, the simplest system $L=L_0+L_{int}=\barψ(i\not\partial-m)ψ-g\barψγ^5ψ$ resembles the 1+1-dimensional case but the Dirac equation is not PT invariant because $T^2=-1$. This explains the appearance of complex eigenvalues as $m\to0$. Other Lorentz-invariant 2-point and 4-point interactions give non-Hermitian PT-symmetric terms in the Dirac equation. Only the axial vector and tensor Lagrangian interactions $L_{int}=-i\barψ\tilde B_μγ^5γ^μψ$ and $L_{int}=-i\barψT_{μν}σ^{μν}ψ$ fulfil both requirements of PT invariance of the associated Dirac equation and non-Hermiticity. Both models give complex spectra as $m\to0$. The effect on the spectrum of the additional constraint of selfadjointness of the Hamiltonian with respect to the PT inner product is investigated.

math-ph↗

arXiv2018

PT-symmetric quantum field theory in D dimensions

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^\varepsilon$. A surprising feature of this non-Hermitian Hamiltonian is that its eigenvalues are discrete, real, and positive when $\varepsilon\geq0$. This paper examines the corresponding quantum-field-theoretic Hamiltonian $H=\frac{1}{2}(\nablaϕ)^2+\frac{1}{2}ϕ^2(iϕ)^\varepsilon$ in $D$-dimensional spacetime, where $ϕ$ is a pseudoscalar field. It is shown how to calculate the Green's functions as series in powers of $\varepsilon$ directly from the Euclidean partition function. Exact finite expressions for the vacuum energy density, all of the connected $n$-point Green's functions, and the renormalized mass to order $\varepsilon$ are derived for $0\leq D<2$. For $D\geq2$ the one-point Green's function and the renormalized mass are divergent, but perturbative renormalization can be performed. The remarkable spectral properties of PT-symmetric quantum mechanics appear to persist in PT-symmetric quantum field theory.

hep-th↗

arXiv2018

Analytic eigenvalue structure of a coupled oscillator system beyond the ground state

By analytically continuing the eigenvalue problem of a system of two coupled harmonic oscillators in the complex coupling constant $g$, we have found a continuation structure through which the conventional ground state of the decoupled system is connected to three other lower {\it unconventional} ground states that describe the different combinations of the two constituent oscillators, taking all possible spectral phases of these oscillators into account. In this work we calculate the connecting structures for the higher excitation states of the system and argue that - in contrast to the four-fold Riemann surface identified for the ground state - the general structure is eight-fold instead. Furthermore we show that this structure in principle remains valid for equal oscillator frequencies as well and comment on the similarity of the connection structure to that of the single complex harmonic oscillator.

math-ph↗

arXiv2018

No-signaling principle and quantum brachistochrone problem in $PT$-symmetric fermionic two- and four-dimensional models

Fermionic systems differ from bosonic ones in several ways, in particular that the time-reversal operator $T$ is odd, $T^2=-1$. For $PT$-symmetric bosonic systems, the no-signaling principle and the quantum brachistochrone problem have been studied to some degree, both of them controversially. In this paper, we apply the basic methods proposed for bosonic systems to {\it fermionic} two- and four-dimensional $PT$-symmetric Hamiltonians, and obtain several surprising results: We find - in contrast to the bosonic case - that the no-signaling principle is upheld for two-dimensional fermionic Hamiltonians, however, the $PT$ symmetry is broken. In addition, we find that the time required for the evolution from a given initial state, the spin-up, to a given final state, the spin-down, is a constant, independent of the parameters of the Hamiltonian, under the eigenvalue constraint. That is, it cannot - as in the bosonic case - be optimized. We do, however, also find a dimensional dependence: four-dimensional $PT$-symmetric fermionic Hamiltonians considered here again uphold the no-signaling principle, but it is not essential that the $PT$ symmetry be broken. The symmetry is, however, broken if the measure of entanglement is conserved. In the four-dimensional systems, the evolution time between orthogonal states is dependent on the parameters of the Hamiltonian, with the conclusion that it again can be optimized, and approach zero under certain circumstances. However, if we require the conservation of entanglement, the transformation time between these two states becomes the same constant as found in the two-dimensional case, which coincides with the minimum time for such a transformation to take place in the Hermitian case.

quant-ph↗

arXiv2018

Two- and four-dimensional representations of the PT- and CPT-symmetric fermionic algebras

Fermionic systems differ from their bosonic counterparts, the main difference with regard to symmetry considerations being that $T^2=-1$ for fermionic systems. In PT-symmetric quantum mechanics an operator has both PT and CPT adjoints. Fermionic operators $η$, which are quadratically nilpotent ($η^2=0$), and algebras with PT and CPT adjoints can be constructed. These algebras obey different anticommutation relations: $ηη^{PT}+η^{PT}η=-1$, where $η^{PT}$ is the PT adjoint of $η$, and $ηη^{CPT}+η^{CPT}η=1$, where $η^{CPT}$ is the CPT adjoint of $η$. This paper presents matrix representations for the operator $η$ and its PT and CPT adjoints in two and four dimensions. A PT-symmetric second-quantized Hamiltonian modeled on quantum electrodynamics that describes a system of interacting fermions and bosons is constructed within this framework and is solved exactly.

quant-ph↗

arXiv2017

Behavior of eigenvalues in a region of broken-PT symmetry

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^\varepsilon$. When $\varepsilon\geq0$, the eigenvalues of this non-Hermitian Hamiltonian are discrete, real, and positive. This portion of parameter space is known as the region of unbroken PT symmetry. In the region of broken PT symmetry $\varepsilon<0$ only a finite number of eigenvalues are real and the remaining eigenvalues appear as complex-conjugate pairs. The region of unbroken PT symmetry has been studied but the region of broken PT symmetry has thus far been unexplored. This paper presents a detailed numerical and analytical examination of the behavior of the eigenvalues for $-4<\varepsilon<0$. In particular, it reports the discovery of an infinite-order exceptional point at $\varepsilon=-1$, a transition from a discrete spectrum to a partially continuous spectrum at $\varepsilon=-2$, a transition at the Coulomb value $\varepsilon=-3$, and the behavior of the eigenvalues as $\varepsilon$ approaches the conformal limit $\varepsilon=-4$.

math-ph↗

arXiv2017

Analytic structure of eigenvalues of coupled quantum systems

By analytically continuing the coupling constant $g$ of a coupled quantum theory, one can, at least in principle, arrive at a state whose energy is lower than the ground state of the theory. The idea is to begin with the uncoupled $g=0$ theory in its ground state, to analytically continue around an exceptional point (square-root singularity) in the complex-coupling-constant plane, and finally to return to the point $g=0$. In the course of this analytic continuation, the uncoupled theory ends up in an unconventional state whose energy is lower than the original ground state energy. However, it is unclear whether one can use this analytic continuation to extract energy from the conventional vacuum state; this process appears to be exothermic but one must do work to vary the coupling constant $g$.

math-ph↗

arXiv2016

Instantaneous modulations in time-varying complex optical potentials

We study the impact of a spatially homogeneous yet non-stationary dielectric permittivity on the dynamical and spectral properties of light. Our choice of potential is motivated by the interest in PT-symmetric systems as an extension of quantum mechanics. Because we consider a homogeneous and non-stationary medium, PT symmetry reduces to time-reversal symmetry in the presence of balanced gain and loss. We construct the instantaneous amplitude and angular frequency of waves within the framework of Maxwell's equations and demonstrate the modulation of light amplification and attenuation associated with the well-defined temporal domains of gain and loss, respectively. Moreover, we predict the splitting of extrema of the angular frequency modulation and demonstrate the associated shrinkage of the modulation period. Our theory can be extended for investigating similar time-dependent effects with matter and acoustic waves in PT-symmetric structures.

physics.optics↗