Logarithmic correction in the deformed ${\rm AdS}_5$ model to produce the heavy quark potential and QCD beta function

We stude the \textit{holographic} QCD model which contains a quadratic term $ -\sigma z^2$ and a logarithmic term $-c_0\log[(z_{IR}-z)/z_{IR}]$ with an explicit infrared cut-off $z_{IR}$ in the deformed ${\rm AdS}_5$ warp factor. We investigate the heavy quark potential for three cases, i.e, with only quadratic correction, with both quadratic and logarithmic corrections and with only logarithmic correction. We solve the dilaton field and dilation potential from the Einstein equation, and investigate the corresponding beta function in the G{\"u}rsoy -Kiritsis-Nitti (GKN) framework. Our studies show that in the case with only quadratic correction, a negative $\sigma$ or the Andreev-Zakharov model is favored to fit the heavy quark potential and to produce the QCD beta-function at 2-loop level, however, the dilaton potential is unbounded in infrared regime. One interesting observing for the case of positive $\sigma$, or the soft-wall ${\rm AdS}_5$ model is that the corresponding beta-function exists an infrared fixed point. In the case with only logarithmic correction, the heavy quark Cornell potential can be fitted very well, the corresponding beta-function agrees with the QCD beta-function at 2-loop level reasonably well, and the dilaton potential is bounded from below in infrared. At the end, we propose a more compact model which has only logarithmic correction in the deformed warp factor and has less free parameters.