Two Schrödinger-like Equations for hadrons

Ruben Sandapen

doi:10.1051/epjconf/202225810009

Abstract

In this talk, based on [1, 2], I argue that the holographic Schrödinger Equation of (3 +1)-dim, conformal light-front QCD and the ’t Hooft Equation of (1+1)-dim, large Nc QCD, can be complementary to each other in providing a first approximation to hadron spectroscopy. Together, the two equations play a role in hadronic physics analogous that of the ordinary Schrödinger Equation in atomic physics.

Highlights

IntroductionIn (3 + 1)-dim light-front QCD, the internal dynamics of a quark-antiquark meson are governed by the Schrödinger-like equation: [3, 4]
In (3 + 1)-dim light-front QCD, the internal dynamics of a quark-antiquark meson are governed by the Schrödinger-like equation: [3, 4]− ∇2b⊥ + m2q + m2qx(1 − x) x 1 − x+ U(x, b⊥) Ψ(x, b⊥) = M2Ψ(x, b⊥), (1)where M is the meson mass and Ψ(x, b⊥) is the meson light-front wavefunction, with x being the light-front momentum fraction carried by the quark, and b⊥ the transverse distance between the quark and the antiquark
To use the ’t Hooft Equation for baryons and tetraquarks, we transform the antiquark into a diquark, followed by the transformation of the quark into an antidiquark

Summary

Introduction

In (3 + 1)-dim light-front QCD, the internal dynamics of a quark-antiquark meson are governed by the Schrödinger-like equation: [3, 4]. Schrödinger Equation, despite Eq (1) being fully relativistic and frame-independent. It remains that deriving U(x, b⊥) from first principles in QCD is an open question. Insights into its analytic form can be gained by introducing a new variable, ζ = x(1 − x)b⊥. The x-dependence of ζ means that, in this type of factorization, the so-called transverse and longitudinal dynamics do not decouple completely. The variable ζ plays a key role in the search of an analytical form for U⊥(ζ) [5,6,7]

The holographic Schrödinger Equation

The ’t Hooft Equation

Symmetry constraints

Computing the hadronic spectrum

Conclusions