Scalar Spectral Function Research Articles

Scalar spectral functions from the spectral functional renormalization group

We compute spectral functions in a scalar ϕ4-theory in three spacetime dimensions via the spectral functional renormalization group. This approach allows for the direct, manifestly Lorentz covariant computation of correlation functions in Minkowski spacetime, including a physical on-shell renormalization. We present numerical results for the spectral functions of the two- and four-point correlation functions for different values of the coupling parameter. These results agree very well with those obtained from another functional real-time approach, the spectral Dyson-Schwinger equation. Published by the American Physical Society 2024

Inhomogeneous mean-field approach to collective excitations near the superfluid–Mott glass transition

We develop an inhomogeneous quantum mean-field approach to the behavior of collective excitations across the superfluid–Mott glass quantum phase transition in two dimensions, complementing recent quantum Monte Carlo simulations (Puschmann et al. 2020). In quadratic (Gaussian) approximation, the Goldstone (phase) and Higgs (amplitude) modes completely decouple. Each is described by a disordered Bogoliubov Hamiltonian which can be solved by an inhomogeneous multi-mode Bogoliubov transformation. We find that the Higgs mode is spatially localized in both phases. The corresponding scalar spectral function shows a broad peak that is noncritical in the sense that its peak frequency does not soften but remains nonzero across the quantum phase transition. In contrast, the lowest-energy Goldstone mode delocalizes in the superfluid phase, leading to a zero-frequency spectral peak. We compare these findings to the results of the quantum Monte Carlo simulations. We also relate them to general results on the localization of bosonic excitations, and we discuss the limits and generality of our approach.

Spectral Functions for the Vector-Valued Fourier Transform

A scalar distribution function σ(s) is called a spectral function for the Fourier transform φ^(s)=∫Reitsφ(t)dt (with respect to an interval I⊂R) if for each function φ∈L2(R) with support in I the Parseval identity ∫Rφ^s2dσ(s)=∫Rφt2dt holds. We show that in the case I=R there exists a unique spectral function σ(s)=(1/2π)s, in which case the above Parseval identity turns into the classical one. On the contrary, in the case of a finite interval I=(0,b), there exist infinitely many spectral functions (with respect to I). We introduce also the concept of the matrix-valued spectral function σ(s) (with respect to a system of intervals {I1,I2,…,In}) for the vector-valued Fourier transform of a vector-function φ(t)={φ1(t),φ2(t),…,φn(t)}∈L2(I,Cn), such that support of φj lies in Ij. The main result is a parametrization of all matrix (in particular scalar) spectral functions σ(s) for various systems of intervals {I1,I2,…,In}.

Nucleon form factors in dispersively improved chiral effective field theory: Scalar form factor

We propose a method for calculating the nucleon form factors (FFs) of $G$-parity-even operators by combining Chiral Effective Field Theory ($\chi$EFT) and dispersion analysis. The FFs are expressed as dispersive integrals over the two-pion cut at $t > 4 M_\pi^2$. The spectral functions are obtained from the elastic unitarity condition and expressed as products of the complex $\pi\pi \rightarrow N\bar N$ partial-wave amplitudes and the timelike pion FF. $\chi$EFT is used to calculate the ratio of the partial-wave amplitudes and the pion FF, which is real and free of $\pi\pi$ rescattering in the $t$-channel ($N/D$ method). The rescattering effects are then incorporated by multiplying with the squared modulus of the empirical pion FF. The procedure results in a marked improvement compared to conventional $\chi$EFT calculations of the spectral functions. We apply the method to the nucleon scalar FF and compute the scalar spectral function, the scalar radius, the $t$-dependent FF, and the Cheng-Dashen discrepancy. Higher-order chiral corrections are estimated through the $\pi N$ low-energy constants. Results are in excellent agreement with dispersion-theoretical calculations. We elaborate several other interesting aspects of our method. The results show proper scaling behavior in the large-$N_c$ limit of QCD because the $\chi$EFT calculation includes $N$ and $\Delta$ intermediate states. The squared modulus of the timelike pion FF required by our method can be extracted from Lattice QCD calculations of vacuum correlation functions of the operator at large Euclidean distances. Our method can be applied to the nucleon FFs of other operators of interest, such as the isovector-vector current, the energy-momentum tensor, and twist-2 QCD operators (moments of generalized parton distributions).

A study of the resonances K0*(8OO) and K0*(1430)

We study the scalar kaonic states K0* (800) and K0* (1430) by using a relativistic QFT Lagrangian in which only a single kaonic field corresponding to the well-established scalar state K0* (1430) is considered and in which both derivative and non-derivative interaction terms are taken into account. Even if the scalar spectral function shows a unique peak close to 1.4 GeV, we find two poles in the complex plane: 1.413 ± 0.002 — i(0.127 ± 0.003) GeV, which is related to the seed quark-antiquark state K0* (1430), and 0.746 ± 0.019 — i(262 ± 0.014) GeV, which is an additional companion pole related to K0* (800). As a further investigation for increasing Nc confirms, K0* (800) emerges as a dynamically generated four-quark object as a consequence of pion-kaon loops.

Large-$N_{\rm c}$ Pole Trajectories of the Vector Kaon $K^{\ast }(892)$ and of the Scalar Kaons $K_{0}^{\ast }(800)$ and $K_{0}^{\ast }(1430)$

We study the spectral functions, the poles and their trajectories for increasing $N_{c}$ of the vector kaon state $K^{\ast}(892),$ characterized by $I(J^{P})=\frac{1}{2}(1^{-})$, and of the scalar kaons $K_{0}^{\ast}(800)$ and $K_{0}^{\ast}(1430),$ characterized by $I(J^{P})=\frac{1}{2}(0^{+})$. To this end, we use relativistic QFT's Lagrangians with both derivative and non-derivative terms. In the vector kaonic sector the spectral function is well approximated by a Breit-Wigner function: there is one single peak and, correspondingly, a single pole in the complex plane. On the contrary, in the scalar sector, although the Lagrangian contains only one scalar kaonic field, we find two poles, one corresponding to a standard quark-antiquark ,,seed\textquotedblright state $K_{0}^{\ast}(1430),$ and one to a \textquotedblleft companion\textquotedblright dynamically generate pole $K_{0}^{\ast}(800)$. The latter does not correspond to any peak in the scalar kaonic spectral function, but only to an enhancement in the low-energy regime.

Eigenfunction expansions associated with the one‐dimensional Schroedinger operator

AbstractWe consider the one‐dimensional Schroedinger operator H on L2(−∞, ∞) associated with where q(r) : R → R is locally integrable, λ ∈ R is a spectral parameter, and the differential expression is in Weyl's limit point cases at both endpoints. Starting from the well known Weyl‐Kodaira formulation of the eigenfunction expansion associated with H [1], an alternative formulation can be derived in such a way that the principal features contributing to it, namely a scalar spectral density function, the location and multiplicity of the spectrum, and the spectral types of the generalised eigenfunctions, can be explicitly exhibited in the resulting expansion. The proof is based on classical work of Kac [2] and the theory of subordinacy [3], [4]. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Spectral function of the Higgs mode in4−εdimensions

We investigate the amplitude (Higgs) mode of the relativistic O(N) model in the vicinity of the Wilson-Fisher quantum critical point in D=4-epsilon spacetime dimensions. We compute the universal part of the scalar spectral function near the transition, to leading non-trivial order in the ordered phase, and to next to leading order in both the disordered phase and the quantum critical regime. We find that, in the disordered phase, the spectral function has a threshold behavior with no Higgs-like peak, whereas in the ordered phase, the Higgs mode appears as a well defined resonance. The pole associated with this resonance is purely real in the D->4 limit, evolving smoothly with dimensionality to become purely imaginary at D=3 in the N->infty limit. Our results complement previous studies of the scalar spectral function, and demonstrate that the resonance found in these studies can indeed be directly identified with the Higgs mode.

Dynamics and conductivity near quantum criticality

Relativistic O(N) field theories are studied near the quantum critical point in two space dimensions. We compute dynamical correlations by large scale Monte Carlo simulations and numerical analytic continuation. In the ordered side, the scalar spectral function exhibits a universal peak at the Higgs mass. For N=3 and 4 we confirm its \omega^3 rise at low frequency. On the disordered side, the spectral function exhibits a sharp gap. For N=2, the dynamical conductivity rises above a threshold at the Higgs mass (density gap), in the superfluid (Mott insulator) phase. For charged bosons, (Josephson arrays) the power law rise above the Higgs mass, increases from two to four. Approximate charge-vortex duality is reflected in the ratio of imaginary conductivities on either side of the transition. We determine the critical conductivity to be \sigma^*_c = 0.3 (\pm 0.1) 4e^2/h. In the appendices, we describe a generalization of the worm algorithm to N>2, and also a singular value decomposition error analysis for the numerical analytic continuation.

Charm mass effects in bulk channel correlations

The bulk viscosity of thermalized QCD matter at temperatures above a few hundred MeV could be significantly influenced by charm quarks because their contribution arises four perturbative orders before purely gluonic effects. In an attempt to clarify the challenges of a lattice study, we determine the relevant imaginary-time correlator (of massive scalar densities) up to NLO in perturbation theory, and compare with existing data. We find discrepancies much larger than in the vector channel; this may hint, apart from the importance of taking a continuum limit, to larger non-perturbative effects in the scalar channel. We also recall how a transport peak related to the scalar density spectral function encodes non-perturbative information concerning the charm quark chemical equilibration rate close to equilibrium.

Essentials of the σ resonance in meson–meson scattering and spectral functions

We study the roles of resonances in semi-local duality from meson-meson scattering and in the scalar and pseudoscalar spectral function sum rules. The scattering amplitudes and spectral functions are first systematically calculated within chiral perturbation theory (χPT) with explicit η1 field in addition to the pseudo-Goldstone octet π,K,η8 and then unitarized through the N/D approach. In this note, we emphasize the important roles of the σ or f0(500) resonance in these quantities.

Interesting features of scalar and pseudoscalar spectral function sum rules within aU(3) chiral framework

We compute the scalar and pseudoscalar spectral functions by using their form factors calculated within the unitarized U(3) chiral perturbation theory (�PT). The extrapolation to larger values of NC and the spectral function sum rules are discussed as well. Some interesting features of the scalar spectral functions are revealed.

On the localization of the spectrum for quasi-selfadjoint extensions of a Carleman operator

In the present work, using a formula describing all scalar spectral functions of a Carleman operator $A$ of defect indices $( 1,1) $ in the Hilbert space $L^{2}( X,\mu ) $ that we obtained in a previous paper, we derive certain results concerning the localization of the spectrum of quasi-selfadjoint extensions of the operator $A$.

Holographic approach to finite temperature QCD: The case of scalar glueballs and scalar mesons

We study scalar glueballs and scalar mesons at $T \neq 0$ in the soft wall holographic QCD model. We find that, using the Anti-de Sitter-Black Hole metric for all values of the temperature, the masses of the hadronic states decrease and the widths become broader when T increases, and there are temperatures for which the states disappear from the scalar glueball and scalar meson spectral functions. However, the values of the temperatures in correspondence of which such phenomena occur are low, of the order of 40-60 MeV. A consistent holographic description of in-medium effects on hadron properties should include the Hawking-Page transition, which separates the phase with the Anti-de Sitter metric at small temperatures from the phase with Anti-de Sitter-Black Hole metric at high temperatures.

Sturm-Liouville Operators with Singularities and Generalized Nevanlinna Functions

The Titchmarsh–Weyl function, which was introduced in Fulton (Math Nachr 281(10):1418–1475, 2008) for the Sturm-Liouville equation with a hydrogen-like potential on (0, ∞), is shown to belong to a generalized Nevanlinna class $${\bf N_\kappa}$$ . As a consequence, also in the case of two singular endpoints for the Fourier transformation defined by means of Frobenius solutions there exists a scalar spectral function. This spectral function is given explicitly for potentials of the form $${\dfrac{q_0}{x^2}+\dfrac{q_1}{x},\,-\dfrac 14\le q_0 < \infty}$$ .

On Convex Hull of Orthogonal Scalar Spectral Functions of a Carleman Operator

In this paper we describe the closed convex hull of orthogonal resolvents of an abstract symmetric operator of defect indices (1; 1), then we study the convex hull of orthogonal spectral functions of a Carleman operator in the Hilbert space L^2 (X; \mu) :

Heavy quarkonium in any channel in resummed hot QCD

We elaborate on the fact that quarkonium in hot QCD should not be thought of as a stationary bound state in a temperature-dependent real potential, but as a short-lived transient, with an exponentially decaying wave function. The reason is the existence of an imaginary part in the pertinent static potential, signalling the ``disappearance'', due to inelastic scatterings with hard particles in the plasma, of the off-shell gluons that bind the quarks together. By solving the corresponding Schr\odinger equation, we estimate numerically the near-threshold spectral functions in scalar, pseudoscalar, vector and axial vector channels, as a function of the temperature and of the heavy quark mass. In particular, we point out a subtlety in the determination of the scalar channel spectral function and, resolving it to the best of our understanding, suggest that at least in the bottomonium case, a resonance peak can be observed also in the scalar channel, even though it is strongly suppressed with respect to the peak in the vector channel. Finally, we plot the physical dilepton production rate, stressing that despite the eventual disappearance of the resonance peak from the corresponding spectral function, the quarkonium contribution to the dilepton rate becomes more pronounced with increasing temperature, because of the yield from free heavy quarks.

Bounds on the light quark masses: The scalar channel revisited

We evaluate lower bounds on the sum of the up and down quark masses. The bounds follow from the constraints provided by the dispersion relation obeyed by the two-point function of the scalar current density when combined with properties of the scalar spectral function at long-distances and perturbative QCD at short-distances. Our results point to values of mu+md somewhat higher than those reported recently in the literature using lattice QCD simulations.

ScalarKπform factor and light-quark masses

Recent experimental improvements on $K$-decay data allow for a precise extraction of the strangeness-changing scalar $K\ensuremath{\pi}$ form factor and the related strange scalar spectral function. On the basis of this scalar as well as the corresponding pseudoscalar spectral function, the strange quark mass is determined to be ${m}_{s}(2\text{ }\text{ }\mathrm{GeV})=92\ifmmode\pm\else\textpm\fi{}9\text{ }\text{ }\mathrm{MeV}$. Further taking into account chiral perturbation theory mass ratios, the light up and down quark masses turn out to be ${m}_{u}(2\text{ }\text{ }\mathrm{GeV})=2.7\ifmmode\pm\else\textpm\fi{}0.4\text{ }\text{ }\mathrm{MeV}$ as well as ${m}_{d}(2\text{ }\text{ }\mathrm{GeV})=4.8\ifmmode\pm\else\textpm\fi{}0.5\text{ }\text{ }\mathrm{MeV}$. As a by-product, we also find a value for the Cabibbo angle $|{V}_{us}|=0.2236(29)$ and the ratio of meson decay constants ${F}_{K}/{F}_{\ensuremath{\pi}}=1.203(16)$. Performing a global average of the strange mass by including extractions from other channels as well as lattice QCD results yields ${m}_{s}(2\text{ }\text{ }\mathrm{GeV})=94\ifmmode\pm\else\textpm\fi{}6\text{ }\text{ }\mathrm{MeV}$.

Electromagnetic volume scattering in the half-space: a moment-method analysis without Sommerfeld integrals or their approximations

Traditionally, in moment-method analyses of electromagnetic scattering, the elements of the impedance matrix are calculated as convolutions of the basis elements with the appropriate dyadic Green's function. However, for scattering in the half-space, the vertical and azimuthal copolar terms of the Green's function require evaluation of Sommerfeld integrals which are computationally burdensome. In this paper, it is shown that, in populating the impedance matrix for the half-space problem, evaluation of Sommerfeld integrals is, in fact, not necessary. For monochromatic excitation, the plane-wave expansion of the scattered field constitutes a Fourier transform, in the horizontal plane, of a vector spectral function. This vector function results from the convolution, in the vertical dimension, of the respective angular spectra of the Green's function and the equivalent current. On application of the moment method, through the Weyl identity, the impedance-matrix elements corresponding to the singular terms of the Green's function are convolutions in the horizontal plane of spherical potentials, and Fourier transforms of scalar spectral functions. These scalar functions are derived from the basis elements and, with a judicious choice of basis, they are well behaved and of compact support, and consequently their Fourier transforms can be computed as FFTs.