Connection between the string tension, light-meson Regge trajectories, and the QCD scale parameter.

Abstract

We compare the radial excitation spectrum of an {ital S}-wave solitonlike system containing an infinitely heavy quark and a massless scalar antiquark ({ital Q{bar q}}), as described in a renormalization-group-improved effective-action model (EAM) of QCD (log and log-log models), to the spectrum of an analogous system obeying a Klein-Gordon equation with a Lorentz-scalar linear potential {ital V}({ital r})={sigma}{ital r}. We show that the two systems have the same energy spectrum for high radial excitation numbers {ital N}, which enables us to establish a connection between the QCD scale {Lambda}{sub M{bar S}} (where M{bar S} denotes the modified minimal-subtraction scheme), and the effective string tension'' {sigma}. We find {sigma}=(1.475{Lambda}{sub M{bar S}}){sup 2} ((1.912{Lambda}{sub M{bar S}}){sup 2}) in the log (log-log) model. Moreover, we find for our solitonlike states that the ratio of the total energy of the system to the rms radius is, for {ital N}{much gt}1,{ital U}{sub {ital N}}/{l angle}{ital r}{sup 2}{r angle}{sub {ital N}}{sup 1/2}= {radical}3 Q{ital E}{sub vac}{congruent}{sigma}, where {ital Q}= {radical}4/3 and {ital E}{sub vac} is the vacuum color-electric field in the EAM. This is an additional indication that the light quark in the EAM experiences to a very good approximation an effective scalar potential {ital V}({italmore » r})={sigma}{ital r}. We then introduce a commonly used two-body Klein-Gordon equation that allows us to smoothly interpolate (for a given string tension {sigma}) between the {ital Q{bar q}} regime, where we found the connection between {Lambda}{sub M{bar S}} and {sigma}, and the {ital q{bar q}} regime (massless quark and antiquark), where we can make contact with the measured Regge slope {alpha}{prime}. We obtain in this way a novel connection between {alpha}{prime} and {sigma}, {alpha}{prime}=1/(8{sigma}).« less