Expanding running coupling effects in the hard Pomeron

TL;DR

This work develops a controlled perturbative framework for high-energy QCD in the hard Pomeron regime by exploiting a small-$b$ limit with fixed coupling, effectively suppressing nonperturbative tunneling to the Pomeron. The authors define a perturbative gluon Green's function via a precise double-$\gamma$ spectral formalism and implement a systematic $b$-expansion to capture diffusion and running-coupling corrections to the hard Pomeron exponent, including analytic (Airy-diffusion) and numerical results. They derive explicit expressions for the perturbative Green's function, analyze the diffusion parameter $z$ and the convergence radius $z_c$, and demonstrate the approach's viability through numerical extractions of expansion coefficients and effective exponents. The results offer a principled way to quantify perturbative high-energy evolution and to delineate the perturbative domain from nonperturbative tunneling effects in small-$x$ QCD phenomenology. The methodology holds potential for improved predictions in high-energy scattering where running-coupling and diffusion corrections are important.

Abstract

We study QCD hard processes at scales of order k^2 > Lambda^2 in the limit in which the beta-function coefficient - b is taken to be small, but alphas(k) is kept fixed. The (nonperturbative) Pomeron is exponentially suppressed in this limit, making it possible to define purely perturbative high-energy Green's functions. The hard Pomeron exponent acquires diffusion and running coupling corrections which can be expanded in the b parameter and turn out to be dependent on the effective coupling b alphas^2 Y. We provide a general setup for this b-expansion and we calculate the first few terms both analytically and numerically.

Figures (10)

• Figure 1: Shape of the potential $V(x)$ in Schroedinger-like equation from the diffusion model with running coupling, Eq. (\ref{['schroedinger']}). Solid line: leading term $-\omega_s(t(x))$; dotted line: leading term + quantum corrections $-\frac{5}{36}\frac{b^2}{x^2}$. The dashed line illustrates the boundary $x(\bar{t})$ at which one has specified the regularisation of ${\bar{\alpha}}_s(t)$.
• Figure 2: Schematic representation of the states in an arbitrary potential well. The continuum, which corresponds to Re $\omega<0$, is the perturbative part; $\omega_\mathrm{I\!P}$ is the lowest lying bound state corresponding to the Pomeron; $\omega_b$ denotes the subleading (discrete or continuum) bound states.
• Figure 3: Plot of the linear potential $b\omega t$ with freezing below $\bar{t}$. $\chi_m$ has here the interpretation of the energy in the Schroedinger-like problem. Dashed lines correspond to the principle value regularisation of the running coupling.
• Figure 4: Spectrum of $\omega$ values with a gap and the corresponding contour for the perturbative Green's function.
• Figure 5: Plot of the $g(\bar{p})$ function (\ref{['dzeta']}) in the case of collinear model $\chi(\gamma) = \frac{1}{\gamma(1-\gamma)}$.