# A Virtual Tribute to Quark Confinement and the Hadron Spectrum 2021

Aug 2 – 6, 2021
online
Europe/Brussels timezone

## Hadrons as QCD Bound States

Aug 3, 2021, 9:20 AM
20m
online

#### online

Parallel contribution A: Vacuum structure and confinement

### Speaker

Paul Hoyer (University of Helsinki)

### Description

Quarkonia are well described by the Schrödinger equation, with a linear confining potential that agrees with lattice QCD. For QED atoms the classical $-α/r$ potential is determined by Gauss’ law. Taking the similarity with QED at face value the confinement scale of QCD should be given by a boundary condition on Gauss’ law.

Temporal gauge $(A^0=0)$ is well suited for bound states defined at equal time of the constituents. It preserves rotational invariance, allowing eigenstates of $J^2$ and $J^z$ (in the rest frame). Canonical quantization is straightforward since $\vec A$ (unlike $A^0$) has a conjugate field. In temporal gauge Gauss’ law is implemented as a constraint, not as an operator equation. It determines the value of $A_L$ for each physical state, and thus also the classical potential.

In QCD the classical color octet gluon field vanishes for color singlet states (whereas there is a dipole electric field for $e^+e^-$ states in QED). However, each color component $A$ of a $q_A\bar{q}_A$ state does have a longitudinal gluon field. There is a homogeneous solution of Gauss’ constraint in QCD which gives a spatially constant field energy density, thus preserving translational and rotational invariance for (globally) color singlet states. This leads to an instantaneous confining potential for each color compoent of the state. The potential is linear for $q\bar q$, and confining also for $qqq$, $q\bar qg$ and $gg$ states. The confinement scale is given by the magnitude of the universal field energy density.

The confining potential is of $O(α_s^0)$ and determines the strong binding of hadrons. The remaining contributions (gluon exchange, Fock states with transverse gluons and $q\bar q$ sea quarks) may be included perturbatively.

This approach to gauge theory bound states is described in arXiv 2101.06721.

### Primary author

Paul Hoyer (University of Helsinki)