On the shape of a rapid hadron in QCD

TL;DR

This paper investigates how the size and internal momentum distribution of a rapid hadron in QCD evolve with energy, revealing a rapid increase in transverse momenta and invariant mass of quark-antiquark pairs at high energies, with implications for deep inelastic scattering and pp collisions.

Contribution

It generalizes the dipole model with $k_t$ factorization to analytically evaluate the energy dependence of quark dipole interactions and invariant mass in high-energy QCD processes, extending prior approximations.

Findings

The invariant mass squared of the q$ar q$ pair increases with energy as a power law or exponential, depending on photon polarization.

The average transverse momentum squared of quarks grows rapidly with energy, indicating a shrinking transverse size of the hadron.

The shape of the final hadron state in DIS becomes biconcave at high energies, affecting the understanding of hadron structure.

Abstract

We visualize the fundamental property of pQCD: the smaller size of the colorless quark-gluon configurations leads to a more rapid increase of its interaction with energy. Within the frame of dipole model we use the $k_t$ factorization theorem to generalize the DGLAP approximation and/or leading $\ln(x_0/x)$ approximation and evaluate the interaction of quark dipole with a target. In the limit of fixed $Q^2$ and $x\to 0$ we found the increase with energy of transverse momenta of quark(antiquark) within q$\bar q$ pair produced by strongly virtual photon. The average $p^2_t$ is evaluated analytically within the double logarithmic approximation. We demonstrate that the invariant mass$^2$ of the q$\bar q$ pair increases with the energy as $M^2_0(x_0/x)^{\lambda}$, where $\lambda\sim 0.4\alpha_sN_c/\pi$ for transverse photons, and as $\sim M^2_0 \exp{0.17[(4\alpha_sN_c/\pi)\log(x_0/x)]^{1/2}}$ for longitudinal photons, where $M^2_0 \approx 0.7Q^2$ at the energies of the order $s_0\sim 10^4$ GeV$^2$ ($x_0\sim 10^{-2}$). The magnitude of the effect depends strongly on the small $x$ behavior of the gluon distribution. Similar pattern of the energy dependence of $M^2$ is found in the LO DGLAP approximation generalized to account for $k_t$ factorization. We discuss the impact of the found phenomenon on the dependence of the coherence length on the initial energy and demonstrate that the shape of final hadron state in DIS has biconcave form instead of pancake. Some implications of the found phenomena for the hard processes in pp collisions are discussed.