Nonperturbative QCD coupling and its $\beta$ function from light-front holography

The light-front holographic mapping of classical gravity in anti-de Sitter space, modified by a positive-sign dilaton background, leads to a nonperturbative effective coupling ${\alpha }_s^{\mathrm{AdS}}(Q^2)$. It agrees with hadron physics data extracted from different observables, such as the effective charge defined by the Bjorken sum rule, as well as with the predictions of models with built-in confinement and lattice simulations. It also displays a transition from perturbative to nonperturbative conformal regimes at a momentum scale $\sim 1\mathrm{GeV}$. The resulting $\beta$ function appears to capture the essential characteristics of the full $\beta$ function of QCD, thus giving further support to the application of the gauge/gravity duality to the confining dynamics of strongly coupled QCD. Commensurate scale relations relate observables to each other without scheme or scale ambiguity. In this paper we extrapolate these relations to the nonperturbative domain, thus extending the range of predictions based on ${\alpha }_s^{\mathrm{AdS}}(Q^2)$.

Figure 1

The effective coupling from LF holographic mapping for $\kappa =0.54\mathrm{GeV}$ is compared with effective QCD couplings extracted from different observables and lattice results. Details on the comparison with other effective charges are given in Ref. 39.

Figure 2

Holographic model prediction for the $\beta$ function compared to JLab and CCFR data, lattice simulations and results from the Bjorken sum rule.

Figure 3

How different schemes can lead to different values for the IR fixed point. The couplings are computed in the UV region. They freeze in the IR region. The interpolation between UV and IR is drawn freely and is meant to be illustrative, as are the various IR fixed point values. We note that the $V$ and $g_1$ schemes are numerically close.

Figure 4

Ratio ${\alpha }_V(Q)/{\alpha }_{g_1}(Q)$. The continuous line represents the domain where the CSR are computed at leading twist [Eq. (19)]. The dashed line is the extrapolation to the nonperturbative domain using the fixed point IR conformal behavior of QCD.

Figure 5

Holographic model predictions for ${\alpha }_s(r)$ in configuration space in the $g_1$ scheme (left panel) and $V$ scheme (right panel). The dashed lines are the holographic AdS results, the continuous lines correspond to the modified holographic results from Eq. (11) normalized, respectively, to ${\alpha }_{g_1}$ and ${\alpha }_V$ at $Q=0$. The lower bands (cross-hatched pattern) correspond to the JLab data and the higher (sparser pattern) to lattice results.

Figure 6

Contribution of the running coupling to the quark-antiquark Coulomb potential. The continuous line represents the result from light-front holography modified for pQCD effects using Eq. (11) and transformed to the $V$ scheme using the extrapolated CSR results shown in Fig. 4. The continuous band is the JLab results transformed to the $V$ scheme using the same CSR. The dashed line is the Cornell potential. The inserted figure zooms into the deep UV domain. PQCD results at two loops (Peter, Ref. 61) and three loops (Anzai et al., Ref. 62) are shown, respectively, by the cross-hatched band and the dot-dashed line. An arbitrary offset is applied to the Anzai et al. results.

Figure 7

Check of the CSR validity. Top panel: Comparison of ${\alpha }_V(Q^2)$ from the two-loop pQCD calculation of Ref. 61 and ${\alpha }_V(Q^2)$ obtained using the CSR with ${\alpha }_{g_1}$ as input. Bottom panel: Comparison of ${\alpha }_{g_1}(Q^2)$ computed using four different methods. The good agreement on top and bottom panels is a fundamental check of QCD.