Partial Wave Decomposition Research Articles

Positivity of the hypergeometric Veneziano amplitude

Recently, an infinite family of one-parameter generalizations of the Veneziano amplitude were bootstrapped using as input assumptions an integer mass spectrum, crossing symmetry, high-energy boundedness, and exchange of finite spins. This new result was dubbed the hypergeometric Veneziano amplitude, with a real-valued deformation parameter r. For concreteness we work in a setup where the lowest-mass state is a tachyon of mass m02=−1 and using the partial-wave decomposition and the positivity of said decomposition’s coefficients we are able to bound the deformation parameter to r≥0 and, also, to obtain an upper bound on the number of spacetime dimensions D≤26, which is the critical dimension of bosonic string theory. Published by the American Physical Society 2024

Progress in the Partial-Wave Analysis Methods at COMPASS

Abstract. We study the excitation spectrum of light and strange mesons in diffractive scattering. We identify different hadron resonances through partial wave analysis, which inherently relies on analysis models. Besides statistical uncertainties, the model dependence of the analysis introduces dominant sys tematic uncertainties. We discuss several of their sources for the π−π−π+ and K0S K¯ final states and present methods to reduce them. We have developed a new approach exploiting a-priori knowledge of signal continuity over adjacent final-state-mass bins to stably fit a large pool of partial-waves to our data, al lowing a clean identification of very small signals in our large data sets. For two-body final states of scalar particles, such as K0S K¯, mathematical ambigui ties in the partial-wave decomposition lead to the same intensity distribution for different combinations of amplitude values. We will discuss these ambiguities and present solutions to resolve or at least reduce the number of possible so lutions. Resolving these issues will allow for a complementary analysis of the aJ-like resonance sector in these two final states.

Relativistic Faddeev 3D equations for three-body bound states without two-body t-matrices

Abstract This paper explores a novel revision of the Faddeev equation for three-body (3B) bound states, as initially proposed in Ref. [J. Golak, K. Topolnicki, R. Skibiński, W. Glöckle, H. Kamada, A. Nogga, Few Body Syst. 54, 2427 (2013)]. This innovative approach, referred to as t-matrix-free in this paper, directly incorporates two-body (2B) interactions and completely avoids the 2B transition matrices. We extend this formalism to relativistic 3B bound states using a three-dimensional (3D) approach without using partial wave decomposition. To validate the proposed formulation, we perform a numerical study using spin-independent Malfliet–Tjon and Yamaguchi interactions. Our results demonstrate that the relativistic t-matrix-free Faddeev equation, which directly implements boosted interactions, accurately reproduces the 3B mass eigenvalues obtained from the conventional form of the Faddeev equation, referred to as t-matrix-dependent in this paper, with boosted 2B t-matrices. Moreover, the proposed formulation provides a simpler alternative to the standard approach, avoiding the computational complexity of calculating boosted 2B t-matrices and leading to significant computational time savings.

Electron scattering on finite-temperature quantum screened potential

Elastic scattering of electrons on a finite-temperature quantum screened potential is studied by using the partial wave decomposition of scattering wavefunction. The influence of plasma temperature on the total and l = 0, 1, and 2 partial wave cross sections is systematically analyzed in the present work. It is found that when the plasma density is fixed, the screened strength does not change monotonically with temperature, which leads to different conditions for forming shape resonance in the scattering cross sections for the different temperature domains. The scattering cross sections under the environments of low temperature limit are compared with those for zero-temperature quantum plasmas.

Spinning partial waves for scattering amplitudes in d dimensions

Partial wave decomposition is one of the main tools within the modern S-matrix studies. We present a method to compute partial waves for 2 → 2 scattering of spinning particles in arbitrary spacetime dimension. We identify partial waves as matrix elements of the rotation group with definite covariance properties under a subgroup. This allows to use a variety of techniques from harmonic analysis in order to construct a novel algebra of weight-shifting operators. All spinning partial waves are generated by the action of these operators on a set of known scalar seeds. The text is accompanied by a Mathematica notebook to automatically generate partial waves. These results pave the way to a systematic studies of spinning S-matrix bootstrap and positivity bounds.

Coupled-mode separation of multiply scattered wavefield components in two-dimensional waveguides.

For a waveguide that is invariant in one of the horizontal directions, this paper presents a mathematically exact partial-wave decomposition of the wavefield for assessment of multiple scattering by horizontally displaced medium anomalies. The decomposition is based on discrete coupled-mode theory and combination of reflection/transmission matrices. In particular, there is no high-frequency ray approximation. Full details are presented for the scalar case with plane-wave incidence from below. An application of interest concerns ground motion induced by seismic waves, which may be severely amplified by local medium anomalies such as alluvial valleys. Global optimization techniques are used to design an artificial medium termination at depth for a normal-mode representation of the field. A derivation of a horizontal source array to produce an incident plane wave gives, as a by-product, an extension of a previous Fourier-transform relation involving Bessel functions. Like purely numerical methods, such as finite differences and finite elements, the method can handle all kinds of (two-dimensional) anomaly shapes.

Towards the non-perturbative cosmological bootstrap

We study quantum field theory on a de Sitter spacetime dSd+1 background. Our main tool is the Hilbert space decomposition in irreducible unitary representations of its isometry group SO(d + 1, 1). As the first application of the Hilbert space formalism, we recover the Källen-Lehmann spectral decomposition of the scalar bulk two-point function. In the process, we exhibit a relation between poles in the corresponding spectral densities and the boundary CFT data. Moreover, we derive an inversion formula for the spectral density through analytical continuation from the sphere and use it to find the spectral decompisiton for a few examples. Next, we study the conformal partial wave decomposition of the four-point functions of boundary operators. These correlation functions are very similar to the ones of standard conformal field theory, but have different positivity proper- ties that follow from unitarity in de Sitter. We conclude by proposing a non-perturbative conformal bootstrap approach to the study of these late-time four-point functions, and we illustrate our proposal with a concrete example for QFT in dS2.

Role of bound states and resonances in scalar QFT at nonzero temperature

We study the thermal properties of quantum field theories (QFT) with three-leg interaction vertices $g{\ensuremath{\varphi}}^{3}$ and $gS{\ensuremath{\varphi}}^{2}$ ($\ensuremath{\varphi}$ and $S$ being scalar fields), which constitute the relativistic counterpart of the Yukawa potential. We follow a nonperturbative unitarized one-loop resummed technique for which the theory is unitary and well defined for a large range of values of the coupling constant $g$. Using the partial wave decomposition of two-body scattering we calculate the phase shifts, whose derivatives are used to infer the pressure of the system at nonzero temperature by using the so-called phase shift formalism. A $\ensuremath{\varphi}\ensuremath{\varphi}$ bound state is formed when the coupling $g$ is larger than a certain critical value. As the main outcomes of this work, we estimate the influence of particle interaction on the pressure (both without and with the bound state), and we demonstrate that the latter is always continuous as a function of the coupling constant $g$ (no sudden jumps occur when the bound state forms), and we show that the contribution of the bound state to the pressure does not count as one state in the thermal gas, since a cancellation with the residual $\ensuremath{\varphi}\ensuremath{\varphi}$ interaction occurs. The amount of this cancellation depends on the details of the model and its parameters and a variety of possible scenarios is presented. We also show how the overall effect of the interaction, including eventual resonances and bound states, can be formally described by a unique expression that makes use of the phase shift continued below the threshold.

Scattering of e± by C2H6 Molecule over a Wide Range of Energy: A Theoretical Investigation.

The present work reports the theoretical investigation of the scattering of electrons and positrons by the ethane (C2H6) molecule over the energy range 1 eV-1 MeV. The investigation was carried out by taking into account the screening correction arising from a semiclassical analysis of the atomic geometrical overlapping of the scattering observables calculated in the independent atom approximation. The study is presented through the calculations of a broad spectrum of observable quantities, namely differential, integrated elastic, momentum transfer, viscosity, inelastic, grand total, and total ionization cross-sections and the Sherman functions. A comparative study was carried out between scattering observables for electron impact with those for positron impact to exhibit the similarity and dissimilarity arising out of the difference of the collisions of impinging projectiles with the target. Partial-wave decomposition of the scattering states within the Dirac relativistic framework employing a free-atom complex optical model potential was used to calculate the corresponding observable quantities of the constituent atoms. The results, calculated using our recipe, were compared with the experimental and theoretical works available in the literature. The Sherman function for a e±-C2H6 scattering system is presented for the first time in the literature. The addition of the screening correction to the independent atom approximation method was found to substantially reduce the scattering cross-sections, particularly at forward angles for lower incident energies.

Nature of the X(6900) in partial wave decomposition of J/ψJ/ψ scattering

In this letter, we perform partial wave decomposition on coupled-channel scattering amplitudes, $J/\psi J/\psi$-$J/\psi \psi(2S)$-$J/\psi \psi(3770)$, to study the resonance appears in these processes. Effective Lagrangians are used to describe the interactions of four charmed vector mesons, and the scattering amplitudes are calculated up to the next-to-leading order. Partial wave projections are performed, and unitarization is implemented by Pad\'e approximation. Then we fit the amplitudes to the $J/\psi J/\psi$ invariant mass spectra measured by LHCb and determine the unknown couplings. The pole parameters of the $X(6900)$ are extracted as $M=6861.0^{+6.3}_{-8.8}$~MeV and $\Gamma=129.0^{+5.6}_{-3.4}$~MeV. Our analysis implies that its quantum number prefers to be $0^{++}$. The pole counting rule and phase shifts show that it is a normal Breit-Wigner resonance and, hence, should be a compact tetraquark.

Two-body binding energy calculation using homogeneous and inhomogeneous Lippmann-Schwinger equations within three-dimensional approach

Bound state equations are mostly solved in partial-wave truncated basis. In partial-wave method, we need to consider many partial waves to reach the accurate results. In this paper, we avoid partial-wave decompositions and directly work with vector variables to solve homogeneous and inhomogeneous Lippmann-Schwinger equations. We study binding energies of deuteron and atomic hydrogen using three-dimensional approach.

Scattering of relativistic spinless particles within the Feshbach-Villars formalism

We consider the relativistic quantum scattering of spinless particles in one-spatial dimension using the Feshbach-Villars formalism. We construct the general form of the scattering matrix, for symmetric and non-symmetric potentials, based on the symmetry properties of the Feshbach-Villars equation. Then, since in one dimension there are only two partial waves associated with even and odd parities, we show, in a simple and comprehensive way, how to describe this transmission-reflection problem using partial-wave decomposition. As an illustration, we also discuss the special case of scattering by a symmetric square-well potential.

Combined molecular and atomic potentials for elastic cross sections of electrons scattering off diatomic molecules at intermediate energies

A simple model is proposed to compute electron-diatomic molecule elastic differential cross sections at intermediate energies within the framework of an analytical local optical potential. In a spherical harmonic expansion of the molecular potential we treat the isotropic term with the partial-wave decomposition and apply the independent atom model (IAM) on all higher orders. This model is seen to properly converge to the IAM at high energies, while bringing significant improvement at lower energies. We compare the results with a well-tested program called ELSEPA, tailored for high-energy electron-atom scattering and address its further extension to molecules with the method proposed. The simplicity of the calculations and the encouraging agreement in shape with experimental data could promote attractiveness among plasma physics simulations in need of coherent and well-resolved differential cross sections.

Partial wave analysis for the in-hadron condensate

In-hadron condensates, defined as the scalar form factors at zero-momentum transfer, are investigated for flavor-symmetric mesons in pseudoscalar and vector channels under the rainbow-ladder (RL) truncation within the Dyson-Schwinger equations framework. We confirm the efficiency of the in-hadron condensates in describing the effects of dynamical chiral symmetry breaking from both global and structural perspectives by comparing the meson masses, the dimensionless in-hadron condensates, and the partial wave decompositions of in-hadron condensates as functions of current-quark mass. From partial wave analysis, $\ensuremath{\pi}(1300)$ is inferred as a radial excitation dominated by $s$ waves, while $\ensuremath{\rho}(1450)$ needs further studies beyond the RL truncation. This work provides a new insight into the studies of hadron properties with partial wave analysis for the in-hadron condensates.

J-Matrix time propagation of atomic hydrogen in attosecond fields

The J-Matrix approach for scattering is extended to the time-dependent Schrödinger equation (TDSE) for one electron atoms in external few cycle attosecond fields. To this purpose, the wave function is expanded in square integrable (L^2) Sturmian functions and an equation system for the transition amplitudes is established. Outside the interaction zone, boundary conditions are imposed at the border in the L^2 function space. These boundary conditions correspond to outgoing waves (Siegert states) and minimize reflections at the L^2 boundary grid. Outgoing wave behaviour in the asymptotic region is achieved by employing Pollaczek functions. The method enables the treatment of light - atom interactions within arbitrary external fields. Using a partial wave decomposition, the coupled differential equation system is solved by a Runge-Kutta method. As a proof of the method ionization processes of atomic hydrogen in half and few cycle attosecond fields are examined. The electron energy spectrum is calculated and the numerical implementation will be presented. Different forms of the interaction operator are considered and the convergence behaviour is discussed. Results are compared to other studies which use independent approaches like finite difference methods. Remarkable agreement is achieved even with strong field strengths of the electromagnetic field. It is demonstrated that expanding in L^2 functions and imposing boundary conditions at the limit in the L^2 function space can be an advantageous alternative to conventional propagation methods using complex absorbing potentials or complex scaling.

Partial-wave decomposition of the one-electron properties of the LiH molecule computed with explicitly correlated basis sets

Two one-electron properties of the LiH molecule in its ground state at the internuclear distance of 3.015 bohr, namely the number of electrons and their kinetic energy, are decomposed into contributions due to projections (described by the quantum number m) of individual angular momenta onto the molecular axis. For sufficiently accurate electronic wavefunctions obtained with explicitly correlated Gaussian basis sets, the results of these partial-wave decompositions are found to obey the asymptotics valid at that stem from the off-diagonal cusp in the one-electron reduced density matrix. It turns out that the kinetic energy may be accurately extrapolated to the complete-basis-set limit, even if only the contributions pertaining to small values of are available.

Ladder resummation of spin 1/2 fermion many-body systems with arbitrary partial-wave content

We resum the ladder diagrams for the calculation of the energy density E of a spin 1/2 fermion many-body system in terms of arbitrary vacuum two-body scattering amplitudes. The partial-wave decomposition of the in-medium two-body scattering amplitudes is developed, and the expression for calculating E in a partial-wave amplitude expansion is also given. The case of contact interactions is completely solved for any content in partial waves and it is shown to provide renormalized results, expressed directly in terms of scattering data parameters, within cutoff regularization in a wide class of schemes. The S- and P-wave interactions are considered up to including the first three-terms in the effective-range expansion, paying special attention to the parametric region around the unitary limit.

Relativistic nucleon\u2013nucleon potentials in a spin-dependent three-dimensional approach

The matrix elements of relativistic nucleon–nucleon (NN) potentials are calculated directly from the nonrelativistic potentials as a function of relative NN momentum vectors, without a partial wave decomposition. To this aim, the quadratic operator relation between the relativistic and nonrelativistic NN potentials is formulated in momentum-helicity basis states. It leads to a single integral equation for the two-nucleon (2N) spin-singlet state, and four coupled integral equations for two-nucleon spin-triplet states, which are solved by an iterative method. Our numerical analysis indicates that the relativistic NN potential obtained using CD-Bonn potential reproduces the deuteron binding energy and neutron-proton elastic scattering differential and total cross-sections with high accuracy.

Scattering amplitudes for monopoles: pairwise little group and pairwise helicity

On-shell methods are particularly suited for exploring the scattering of electrically and magnetically charged objects, for which there is no local and Lorentz invariant Lagrangian description. In this paper we show how to construct a Lorentz-invariant S-matrix for the scattering of electrically and magnetically charged particles, without ever having to refer to a Dirac string. A key ingredient is a revision of our fundamental understanding of multi-particle representations of the Poincaré group. Surprisingly, the asymptotic states for electric-magnetic scattering transform with an additional little group phase, associated with pairs of electrically and magnetically charged particles. The corresponding “pairwise helicity” is identified with the quantized “cross product” of charges, e1g2− e2g1, for every charge-monopole pair, and represents the extra angular momentum stored in the asymptotic electromagnetic field. We define a new kind of pairwise spinor-helicity variable, which serves as an additional building block for electric-magnetic scattering amplitudes. We then construct the most general 3-point S-matrix elements, as well as the full partial wave decomposition for the 2 → 2 fermion-monopole S-matrix. In particular, we derive the famous helicity flip in the lowest partial wave as a simple consequence of a generalized spin-helicity selection rule, as well as the full angular dependence for the higher partial waves. Our construction provides a significant new achievement for the on-shell program, succeeding where the Lagrangian description has so far failed.

Conformal partial waves in momentum space

The decomposition of 4-point correlation functions into conformal partial waves is a central tool in the study of conformal field theory. We compute these partial waves for scalar operators in Minkowski momentum space, and find a closed-form result valid in arbitrary space-time dimension d \geq 3d≥3 (including non-integer dd). Each conformal partial wave is expressed as a sum over ordinary spin partial waves, and the coefficients of this sum factorize into a product of vertex functions that only depend on the conformal data of the incoming, respectively outgoing operators. As a simple example, we apply this conformal partial wave decomposition to the scalar box integral in d = 4d=4 dimensions.