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Remarks on the two-dimensional power correction in the soft wall model

We present a direct derivation of the two-point correlation function of the vector current in the soft wall model by using the AdS/CFT dictionary. The resulting correlator is exactly the same as the one previously obtained from dispersion relation with the same spectral function as in this model. The coefficient $C_2$ of the two-dimensional power correction is found to be $C_2=-c/2$ with $c$ the slope of the Regge trajectory, rather than $C_2=-c/3$ derived from the strategy of first quantized string theory. Taking the slope of the $ρ$ trajectory $c\approx0.9{GeV}^2$ as input, we then get $C_2\approx-0.45{GeV}^2$. The gluon condensate is found to be $<α_sG^2>\approx0.064{GeV}^4$, which is almost identical to the QCD sum rule estimation. By comparing these two equivalent derivation of the correlator of scalar glueball operator, we demonstrate that the two-dimensional correction can't be eliminated by including the non-leading solution in the bulk-to-boundary propagator, as was done in \cite{Colangelo2}. In other words, the two-dimensional correction does exist in the scalar glueball case. Also it is manifest by using the dispersion relation that the minus sign of gluon condensate and violation of the low energy theorem are related to the subtraction scheme.