[Paper Review] Light and Heavy Mesons in The Complex Mass Scheme

М. Н. Сергеенко|arXiv (Cornell University)|Sep 21, 2019

Cold Atom Physics and Bose-Einstein Condensates2 references4 citations

TL;DR

This paper proposes a relativistic two-body framework using a position-dependent mass model and complex-mass interpolation to describe light and heavy $Q\bar{q}$ mesons and their resonances. By applying a modified funnel-type potential with $\alpha_{\rm S}(r)$-dependent coupling and solving a relativistic wave equation, it derives an interpolating complex-mass formula that simultaneously predicts both the real (mass) and imaginary (width) parts of meson resonances with high accuracy across $\rho$ and $B^*$ states.

ABSTRACT

Mesons containing light and heavy quarks are studied. Interaction of quarks is described by the funnel-type potential with the distant dependent strong coupling, $α_§(r)$. Free particle hypothesis for the bound state is developed: quark and antiquark move as free particles in of the bound system. Relativistic two-body wave equation with position dependent particle masses is used to describe the flavored $Qq$ systems. Solution of the equation for the system in the form of a~standing wave is given. Interpolating complex-mass formula for two exact asymptotic eigenmass expressions is obtained. Mass spectra for some leading-state flavored mesons are calculated.

Motivation & Objective

• To develop a unified relativistic quantum mechanical description of light and heavy quarkonia and their resonances.
• To address the challenge of describing both nonrelativistic heavy quarkonia and ultra-relativistic light quark mesons within a single framework.
• To incorporate both mass and width (resonance width) of mesons in a single analytic formula using complex-mass interpolation.
• To validate the model against experimental data for $\rho$ and $B^*$ mesons and their excited states.
• To demonstrate that quark masses are not constant but depend on spatial position in the bound state, consistent with relativistic kinematics.

Proposed method

• Formulates a relativistic two-body wave equation with position-dependent quark masses based on the correspondence principle and relativistic kinematics.
• Uses a modified funnel-type Lorentz-scalar potential $V(r) = -\frac{4}{3}\frac{\alpha_{\rm S}(r)}{r} + \sigma r$, with $\alpha_{\rm S}(r)$ depending on distance.
• Derives two exact asymptotic solutions: one for the nonrelativistic limit (heavy quarks) and one for the ultra-relativistic limit (light quarks).
• Applies an interpolation procedure between these two solutions to construct a complex-mass formula for the invariant mass squared: $\mathcal{M}_{\rm N}^2 = (m_1 + m_2)^2(1 - v_{\rm N}^2) \pm 2i m_+ m_- v_{\rm N} + \mathsf{M}_1^2|_L$.
• Solves the wave equation in three regions (inside, middle, outside the classically allowed region) using elementary functions, with phase shifts determined by turning points.
• Normalizes the wave function using $C_n = \sqrt{2|p_n| / [\pi(n + \frac{1}{2}) + 1]}$, and determines eigenmomenta from the quantization condition.

Experimental results

Research questions

• RQ1 Can a single relativistic two-body framework describe both nonrelativistic heavy quarkonia and ultra-relativistic light quark mesons?
• RQ2 How can the complex-mass formula be derived from exact asymptotic solutions of the relativistic wave equation?
• RQ3 What is the role of position-dependent quark masses in describing meson spectra and resonance widths?
• RQ4 Can the model simultaneously predict both the real mass and total width of meson resonances with high accuracy?
• RQ5 How does the effective mass of the light quark evolve with increasing heavy quark mass in $Q\bar{q}$ systems?

Key findings

• The complex-mass formula (14) successfully interpolates between nonrelativistic and ultra-relativistic limits, yielding both the real mass and total width of mesons.
• For the $\rho(1S)$ state, the theoretical mass is 776 MeV, matching the experimental value exactly.
• The $B^*(1S)$ state is predicted at 5325 MeV, in perfect agreement with the experimental value.
• The model predicts the $\rho(2S)$ resonance at 1688 MeV, close to the experimental value of 1720 MeV.
• The effective mass of the $d$ quark decreases from 119 MeV to 25 MeV as the heavy quark mass increases, reflecting reduced spatial separation.
• The universal string tension $\sigma = 140$ MeV$^2$ and gluon mass $m_g = 416$ MeV are consistent across $\rho$ and $B^*$ states, supporting their fundamental role in hadron physics.