<p>We use the recently proposed supersymmetric expansion algorithm to obtain a complete analytical solution to the Schrödinger equation with the Cornell potential. We find that the energy levels <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>E</mi><mrow><mi>n</mi><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></math> depend on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>n</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>L</mi><mn>2</mn></msup><mo>=</mo><mi>l</mi><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></math>. For a given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>, the energy <i>decreases</i> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>l</mi></math> and the radial probabilities have the Coulomb shape but their peaks are shifted toward smaller radius. We study the heavy quarkonium structure on the light of these results, showing that the measured <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>b</mi><mo stretchy="false">¯</mo></mover><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>c</mi></math> meson masses follow the inverted spectrum pattern predicted by the Cornell potential. Details of the structure of heavy quarkonium like the mean inverse radius and mean squared velocity for the different quarkonium configurations can be obtained from our solution. These details point to significant relativistic corrections for all the configurations of real heavy quarkonium. We calculate relativistic corrections using perturbation theory finding an expansion in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>α</mi><mi>s</mi><mn>2</mn></msubsup></math> for the heavy quarkonium masses. The mass hierarchies in the fine splittings can be qualitatively understood from this expansion. The quantitative analysis of the Bohr-like levels and of the fine splittings in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>l</mi><mo>=</mo><mn>0</mn></math> sector allow us to make well-defined predictions for the masses of some of the missing heavy quarkonium states, to identify the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ψ</mi><mo stretchy="false">(</mo><mn>4040</mn><mo stretchy="false">)</mo></math> as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>3</mn><mn>3</mn></msup><msub><mi>S</mi><mn>1</mn></msub></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>c</mi></math> state and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ψ</mi><mo stretchy="false">(</mo><mn>3842</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ψ</mi><mo stretchy="false">(</mo><mn>3823</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ψ</mi><mo stretchy="false">(</mo><mn>3770</mn><mo stretchy="false">)</mo></math> as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>3</mn><mn>3</mn></msup><msub><mi>D</mi><mn>3</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>3</mn><mn>3</mn></msup><msub><mi>D</mi><mn>2</mn></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>3</mn><mn>3</mn></msup><msub><mi>D</mi><mn>1</mn></msub></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>c</mi></math> states, respectively.</p>

Mexico

M. Napsuciale

S. Rodríguez

A.E. Villanueva-Gutiérrez

Complete analytical solution to the Cornell potential and heavy quarkonium structure

The well-known Cornell quark-antiquark potential in momentum space contains singularities both in its one-gluon-exchange (OGE) and linear confining parts, which prevents a direct use of the convenient Nyström method to solve the corresponding bound-state integral equation for the meson masses. While it has been known for a long time how the Coulomb-type singularity in the OGE potential can be treated with a subtraction technique, only very complicated methods have been developed to deal with the stronger singularity in the linear potential. In this work, we present a simple subtraction method to remove this singularity from the kernel, such that the Nyström method becomes applicable. Derivatives of the wave function, that appear as a result of the subtraction, are represented by means of interpolating functions, for which we found Lagrange polynomials to be very efficient. Test calculations show excellent agreement with exactly known energy eigenvalues. By increasing the number of integration points and the order of the Lagrange interpolation polynomials, extremely high accuracy can be achieved. This method can also be extended to relativistic Bethe-Salpeter type equations with singular kernels.

Portugal

Alfred Stadler

Elmar P. Biernat

Vasco Valverde

Simple high-accuracy method for solving bound-state equations with the Cornell potential in momentum space

In this paper, we mainly predict the rates of <math><mi>M</mi><mn>1</mn></math> radiative and spin-nonflip <math><mi>π</mi><mi>π</mi></math> transitions of the <math><msub><mi>B</mi><mi>c</mi></msub></math>-meson under the nonrelativistic Cornell potential model with a screening potential effect. We employ the numerical wave function to determine the <math><mi>M</mi><mn>1</mn></math> radiative transition widths of <math><msub><mi>B</mi><mi>c</mi></msub></math> excited states and utilize the Kuang-Yan proposed method for the spin-nonflip <math><mi>π</mi><mi>π</mi></math> transitions among <math><msub><mi>B</mi><mi>c</mi></msub></math> states. Our theoretical results are valuable for studying the <math><mi>M</mi><mn>1</mn></math> radiative and spin-nonflip <math><mi>π</mi><mi>π</mi></math> transition processes of <math><msub><mi>B</mi><mi>c</mi></msub></math> states in experiments.

China

Zhi-bin Gao

Yan-yue Fan

Hao Chen

Cheng-Qun Pang

<math><mrow><mi>M</mi><mn>1</mn></mrow></math> radiative and spin-nonflip <math><mi>π</mi><mi>π</mi></math> transitions of <math><msub><mi>B</mi><mi>c</mi></msub></math> states in the Cornell potential model

The Cornell potential can be derived from a recently proposed non-local extension of Abelian electrodynamics. Non-locality can be alternatively described by an extended charge distributions in Maxwell electrodynamics. We state that in these models the energy momentum tensor necessarily requires the presence of the interaction term between the field and the charge itself. We show that this extended form of energy momentum tensor leads to an exact solution of the Einstein equations describing a charged AdS black hole. We refer to it as the “Cornell black hole”(CBH). Identifying the effective cosmological constant with the pressure of Van der Waals fluid, we study the gas-liquid phase transition and determine the critical parameters.

Italy

Euro Spallucci

Anais Smailagic

The Cornell black hole

We make a critical analysis on the free parameters of the Cornell potential  $$-\frac{4\alpha _{s}}{3r}+br+c$$  <math> <mrow> <mo>-</mo> <mfrac> <mrow> <mn>4</mn> <msub> <mi>α</mi> <mi>s</mi> </msub> </mrow> <mrow> <mn>3</mn> <mi>r</mi> </mrow> </mfrac> <mo>+</mo> <mi>b</mi> <mi>r</mi> <mo>+</mo> <mi>c</mi> </mrow> </math>   and provide a parameterisation space for the strong coupling constant  $$\alpha _{s}$$  <math> <msub> <mi>α</mi> <mi>s</mi> </msub> </math>   and the constant shift ‘c’ for choosing linear part as perturbation in the potential model. In the analysis of heavy-light mesons  $$(D,D_{s},B,B_{s}\ \text {and}\ B_{c})$$  <math> <mrow> <mo>(</mo> <mi>D</mi> <mo>,</mo> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>,</mo> <mi>B</mi> <mo>,</mo> <msub> <mi>B</mi> <mi>s</mi> </msub> <mspace width="4pt"></mspace> <mtext>and</mtext> <mspace width="4pt"></mspace> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> </math>  , we have found a wide range of values for the coupling constant i.e  $$0.20 \le \alpha _{s}\le 0.64$$  <math> <mrow> <mn>0.20</mn> <mo>≤</mo> <msub> <mi>α</mi> <mi>s</mi> </msub> <mo>≤</mo> <mn>0.64</mn> </mrow> </math>   with  $$-1.2 \le c \le -0.66$$  <math> <mrow> <mo>-</mo> <mn>1.2</mn> <mo>≤</mo> <mi>c</mi> <mo>≤</mo> <mo>-</mo> <mn>0.66</mn> </mrow> </math>   which can be used to treat the confining part as perturbation.

India

Krishna Pathak

Satyadeep Bhattacharya

Tapashi Das

Parameterisation space for Cornell potential in a QCD potential model

In the present work, we have studied the differential scattering cross-section for ground states of charmonium and bottomonium in the frame work of the medium-modified form of quark-antiquark potential and Born approximation using the nonrelativistic quantum chromodynamics approach. To reach this end, quasiparticle (QP) Debye mass depending upon baryonic chemical potential (<math id="M1"><msub><mrow><mi>μ</mi></mrow><mrow><mi>b</mi></mrow></msub></math>) and temperature has been employed, and hence the variation of differential scattering cross-section with baryonic chemical potential and temperature at fixed value of the scattering angle (<math id="M2"><mi>θ</mi></math>=<math id="M3"><mn>9</mn><msup><mrow><mn>0</mn></mrow><mrow><mi>o</mi></mrow></msup></math>) has been studied. The variation of differential scattering cross-section with scattering angle <math id="M4"><mi>θ</mi></math> (in degree) at fixed temperature and baryonic chemical potential has also been studied. We have also studied the effect of impact parameter and transverse momentum on differential scattering cross-section at <math id="M5"><mi>θ</mi><mo>=</mo><msup><mrow><mn>90</mn></mrow><mrow><mi>o</mi></mrow></msup></math>.

Vineet Kumar Agotiya

Siddhartha Solanki

Manohar Lal

Study of Differential Scattering Cross-Section Using Yukawa Term of Medium-Modified Cornell Potential

CP-violating contributions to Higgs-fermion couplings are absent in the standard model of particle physics (SM), but are motivated by models of electroweak baryogenesis. Here, we employ the framework of the SM effective theory (SMEFT) to parameterise deviations from SM Yukawa couplings. We present the leading contributions of the relevant operators to the fermionic electric dipole moments (EDMs). We obtain constraints on the SMEFT Wilson coefficients from the combination of LHC data and experimental bounds on the electron, neutron, and mercury EDMs. We perform, for the first time, a combined fit to LHC and EDM data allowing the presence of CP-violating contributions from several fermion species simultaneously. Among other results, we find non-trivial correlations between EDM and LHC constraints even in the multi-parameter scans, for instance, when floating the CP-even and CP-odd couplings to all third-generation fermions.

United States

Joachim Brod

Jonathan Cornell

Germany

Dimitrios Skodras

Emmanuel Stamou

Global constraints on Yukawa operators in the standard model effective theory

We assess the status of a wide class of WIMP dark matter (DM) models in light of the latest experimental results using the global fitting framework GAMBIT. We perform a global analysis of effective field theory (EFT) operators describing the interactions between a gauge-singlet Dirac fermion and the Standard Model quarks, the gluons and the photon. In this bottom-up approach, we simultaneously vary the coefficients of 14 such operators up to dimension 7, along with the DM mass, the scale of new physics and several nuisance parameters. Our likelihood functions include the latest data from Planck, direct and indirect detection experiments, and the LHC. For DM masses below 100 GeV, we find that it is impossible to satisfy all constraints simultaneously while maintaining EFT validity at LHC energies. For new physics scales around 1 TeV, our results are influenced by several small excesses in the LHC data and depend on the prescription that we adopt to ensure EFT validity. Furthermore, we find large regions of viable parameter space where the EFT is valid and the relic density can be reproduced, implying that WIMPs can still account for the DM of the universe while being consistent with the latest data.

Australia

Peter Athron

Canada

Neal Kozar

Csaba Balázs

Belgium

Ankit Beniwal

United Kingdom

Sanjay Bloor

Norway

Torsten Bringmann

Christopher Chang

Ben Farmer

Andrew Fowlie

Tomás Gonzalo

Will Handley

Felix Kahlhoefer

Anders Kvellestad

CERN

France

Farvah Mahmoudi

Markus Prim

Are Raklev

Sweden

Janina Renk

Andre Scaffidi

Pat Scott

Patrick Stöcker

Aaron Vincent

Martin White

Sebastian Wild

Jure Zupan

Thermal WIMPs and the scale of new physics: global fits of Dirac dark matter effective field theories

In this paper, we study the finite temperature-dependent Schrödinger equation by using the Nikiforov-Uvarov method. We consider the sum of the Cornell, inverse quadratic, and harmonic-type potentials as the potential part of the radial Schrödinger equation. Analytical expressions for the energy eigenvalues and the radial wave function are presented. Application of the results for the heavy quarkonia and <math id="M1"><msub><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow></msub></math> meson masses are in good agreement with the current experimental data except for zero angular momentum quantum numbers. Numerical results for the temperature dependence indicates a different behaviour for different quantum numbers. Temperature-dependent results are in agreement with some QCD sum rule results from the ground states.

Azerbaijan

A.I. Ahmadov

K.H. Abasova

M. Sh. Orucova

Bound State Solution Schrödinger Equation for Extended Cornell Potential at Finite Temperature

We study the scattering of   $J/\psi$   -   $J/\psi$   mesons using quadratic and Cornell potentials in our tetraquark (   $J/\psi$  ) system. The system’s wavefunction in the restricted gluonic basis, which is written by utilizing the adiabatic approximation and Hamiltonian, is used via a quark potential model. The resonating group technique is used to obtain the integral equations, which are solved to obtain the unknown inter-cluster dependence of the total wavefunction of our tetraquark system. T-Matrix elements are calculated from the solutions, and eventually, the scattering cross sections are obtained using the two potentials. We compare these cross sections and find that the magnitudes of scattering cross sections of quadratic potential are higher than the Cornell potential.