Process-independent strong running coupling

We unify two widely different approaches to understanding the infrared behavior of quantum chromodynamics (QCD), one essentially phenomenological, based on data, and the other computational, realized via quantum field equations in the continuum theory. Using the latter, we explain and calculate a process-independent running coupling for QCD, a new type of effective charge that is an analogue of the Gell-Mann–Low effective coupling in quantum electrodynamics. The result is almost identical to the process-dependent effective charge defined via the Bjorken sum rule, which provides one of the most basic constraints on our knowledge of nucleon spin structure. This reveals the Bjorken sum to be a near direct means by which to gain empirical insight into QCD’s Gell-Mann–Low effective charge.

Figure 1

Solid (blue) curve, complete effective charge in Eq. (7); and dot-dashed (black) curve, Taylor-scheme effective charge, i.e. computed in the absence of crucial pieces of the gluon-ghost vacuum polarisation [$LF\equiv 0$ in Eq. (1c)]. The k-axis scale is linear to the left of the vertical line and logarithmic otherwise, an artifice which enables us to show saturation of the effective charge.

Figure 2

Solid (blue) curve: predicted process-independent RGI running coupling ${\widehat{\alpha }}_{\mathrm{PI}}(k^2)$, Eq. (7). The shaded (blue) band bracketing this curve combines a 95% confidence-level window based on existing lattice-QCD results for the gluon two-point function with an error of 10% in the continuum extraction of the RGI product $LF$ in Eqs. (1). World data on ${\alpha }_{g_1}$ [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81]. The shaded (yellow) band on $k>1\mathrm{GeV}$ represents ${\alpha }_{g_1}$ obtained from the Bjorken sum by using QCD evolution [84, 85, 86] to extrapolate high-$k^2$ data into the depicted region, following Refs. [56, 57]; and, for additional context, the dashed (red) curve is the light-front holographic model of ${\alpha }_{g_1}$ canvassed in Ref. [46].