content_image/370502.jpg content_image/370503.jpg content_image/370504.jpg
to consistently extend the AdS coupling to the UV domain, consistent with pQCD. 

## VIII. HOLOGRAPHIC COUPLING IN CONFIGURATION SPACE 

In order to obtain modifications to the instantaneous Coulomb potential in configuration space V ( r ) = − C$_{F}$ α$_{V}$ ( r ) /r from the running coupling, one must transform the coupling defined by the static quark potential V ( q ) = − 4 πC$_{F}$ α$_{V}$ ( q ) /q 2 in the nonrelativistic limit and extract the coefficient of 1 /r to define the coupling α$_{V}$ ( r ) in the V scheme. The couplings are related by the Fourier transform [60] 

From (9) we find the expression 

<img src='content_image/370502.jpg'>

<img src='content_image/370503.jpg'>

where C = α$_{V}$ ( Q = 0) = α$_{V}$ ( r → ∞ ) since erf(x → + ∞ ) = 1. We have written explicitly the normalization at Q = 0 in the V scheme since it is not expected to be equal to the normalization in the g$_{1}$ scheme for the reasons discussed in Sec. VII. 

The couplings in the V and g$_{1}$ schemes are related at leading twist by the CSR: [10] 

<img src='content_image/370504.jpg'>

with Q ∗ = 1 . 18 Q , Q ∗∗ = 2 . 73 Q , and we set Q ∗∗∗ = Q $^{∗∗}$. We have verified that this relation numerically holds at least down to Q 2 = 0 . 6 GeV $^{2}$, as shown in the figure in the Appendix (Fig. 7). In order to transform α$_{g}$$_{1}$ ( Q $^{2}$) into α$_{V}$ ( Q $^{2}$) over the full Q 2 range, we extrapolate the CSR to the nonperturbative domain. For guidance, we use the fact that QCD is near conformal at very small Q ; thus, the ratio α$_{V}$ /α$_{g}$$_{1}$ is Q independent. A model for the ratio α$_{V}$ ( Q ) /α$_{g}$$_{1}$ ( Q ) is shown in Fig. 4. We apply this ratio to α AdS Modified,g $_{1}$( Q ), Eq. (11), and then Fourier transform the result using Eq. (17) to obtain α AdS $_{Modified,V}$( r ). We find C ≃ 2 . 2. 

## A. Comparison of V and g$_{1}$ Results 

The right panel of Fig. 5 displays α AdS V ( r ) (dashed line) and α$_{V}$ ( r ) obtained with the same procedure but applied to the JLab data (lower cross-hatched band). Also shown for 

