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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 anexplicit 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 onlyquadratic correction, with both quadratic and logarithmic corrections and withonly logarithmic correction. We solve the dilaton field and dilation potentialfrom the Einstein equation, and investigate the corresponding beta function inthe G{\"u}rsoy -Kiritsis-Nitti (GKN) framework. Our studies show that in thecase with only quadratic correction, a negative $\sigma$ or theAndreev-Zakharov model is favored to fit the heavy quark potential and toproduce the QCD beta-function at 2-loop level, however, the dilaton potentialis unbounded in infrared regime. One interesting observing for the case ofpositive $\sigma$, or the soft-wall ${\rm AdS}_5$ model is that thecorresponding beta-function exists an infrared fixed point. In the case withonly logarithmic correction, the heavy quark Cornell potential can be fittedvery well, the corresponding beta-function agrees with the QCD beta-function at2-loop level reasonably well, and the dilaton potential is bounded from belowin infrared. At the end, we propose a more compact model which has onlylogarithmic correction in the deformed warp factor and has less freeparameters.

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