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

We compare the radial excitation spectrum of an $S$-wave solitonlike system containing an infinitely heavy quark and a massless scalar antiquark ($Q\overline{q}$), 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 $V\left(r\right)=\sigma r$. We show that the two systems have the same energy spectrum for high radial excitation numbers $N$, which enables us to establish a connection between the QCD scale ${\Lambda }_{\overline{\mathrm{MS}}}$ (where $\overline{\mathrm{MS}}$ denotes the modified minimal-subtraction scheme), and the effective "string tension" $\sigma$. We find $\sigma ={\left(1.475\mathrm{}{\Lambda }_{\overline{\mathrm{MS}}}\right)}^2$ [${\left(1.912\mathrm{}{\Lambda }_{\overline{\mathrm{MS}}}\right)}^2$] 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 $N\gg 1$, $\frac{U_N}{{〈r^2〉}_N^{\frac{1}{2}}}=\sqrt{3}Q{E}_{\mathrm{vac}}\simeq \sigma$, where $Q=\sqrt{\frac{4}{3}}$ and $E_{\mathrm{vac}}$ 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 $V\left(r\right)=\sigma r$. 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 $Q\overline{q}$ regime, where we found the connection between ${\Lambda }_{\overline{\mathrm{MS}}}$ and $\sigma$, and the $q\overline{q}$ 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 $\sigma$, ${\alpha }^{\prime }=\frac{1}{\left(8\mathrm{}\sigma \right)}$. This allows us to estimate ${\Lambda }_{\overline{\mathrm{MS}}}$ in terms of the measured value ${\alpha }^{\prime }\approx {\left(1\mathrm{} \mathrm{GeV}\right)}^{-2\mathrm{}}$. We find ${\Lambda }_{\overline{\mathrm{MS}}}\approx 240$ MeV (185 MeV) in the log (log-log) model.