### Speaker

Prof.
Paul Hoyer
(University of Helsinki)

### Description

A perturbative expansion for bound states starts with a lowest order approximation that has a non-perturbative wave function, with all powers in the coupling $\alpha$.
The distinction between lowest and higher orders is blurred since contributions of finite order in $\alpha$ may be assigned to either one. A criterion beyond powers of $\alpha$ is needed to fully define a perturbative expansion for bound states.
I consider the criterion that the gauge field should satisfy the classical field equations at lowest order. This corresponds to lowest order in $\hbar$, but all orders in $\alpha$, and ensures that the expansion starts from a configuration with stationary action. Poincare invariance as well as unitarity is expected to hold at each order in the fundamental constant $\hbar$.
Higher order corrections are defined by an expression for the S-matrix similar to the Interaction Picture, but with in- and out-states that are eigenstates of a Hamiltonian that includes the classical gauge field. The formal derivation and higher order terms of this "Potential Picture" remain to be studied.
When applied to non-relativistic QED atoms the classical criterion leads to lowest order atoms described by the Schrödinger equation. The Schrödinger equation cannot be derived from QED based only on a power expansion in $\alpha$, for reasons stated in the beginning.
The QCD scale $\Lambda_{QCD}$ can arise through a boundary condition on the classical gluon field equations. The homogeneous solution for the classical gluon field that satisfies basic physical requirements appears to be unique, up to an overall scale. It is of O($\alpha_s^0$) and leads to a strictly linear potential for mesons and a related one for baryons.
Meson wave functions may be found iteratively, corresponding to an expansion in $1/N_c$. At lowest order in $1/N_c$ there is no string breaking and the states lie on linear Regge trajectories for vanishing quark mass $m$. The overlap of these states determine decays and meson loop corrections, as required by unitarity at higher orders in $1/N_c$.
There are massless solutions which allow to include the effect of spontaneous chiral symmetry breaking when $m=0$. The massless $0^{++}$ sigma state can mix with the perturbative vacuum without breaking Poincare invariance. A chiral transformation of such a condensate creates massless pions. For a small quark mass $m \neq 0$ the $0^{-+}$ pion gains a mass $M \propto \sqrt{m}$ and is annihilated by the axial vector current as expected for a Goldstone boson.

### Primary author

Prof.
Paul Hoyer
(University of Helsinki)