[Paper Review] Exclusive heavy vector meson production at next-to-leading order in the dipole picture

Heikki Mäntysaari, Jani Penttala|arXiv (Cornell University)|Apr 6, 2021

High-Energy Particle Collisions Research89 references70 citations

TL;DR

This paper presents the first next-to-leading order (NLO) calculation of exclusive heavy vector meson production in the dipole picture, including both QCD corrections (∼αs) and relativistic corrections (∼v²). It demonstrates that both corrections are numerically significant for J/ψ production, providing a precision tool for probing nonlinear dynamics in the Color Glass Condensate framework at small x.

ABSTRACT

We calculate exclusive production of a longitudinally polarized heavy vector meson at next-to-leading order in the dipole picture. The large quark mass allows us to separately include both the first QCD correction proportional to the coupling constant $\alpha_s$, and the first relativistic correction suppressed by the quark velocity $v^2$. Both of these corrections are found to be numerically important in $\mathrm{J}/\psi$ production. The results obtained are directly suitable for phenomenological calculations. We also demonstrate how vector meson production provides complementary information to structure function analyses when one extracts the initial condition for the energy evolution of the proton small-$x$ structure.

Motivation & Objective

• To extend exclusive heavy vector meson production calculations to next-to-leading order (NLO) in the dipole picture for improved precision.
• To include both the first QCD correction (∼αs) and the first relativistic correction (∼v²) in the calculation of J/ψ production.
• To provide a phenomenologically viable framework for studying nonlinear dynamics in the Color Glass Condensate (CGC) at small x.
• To demonstrate that exclusive vector meson production offers complementary information to structure function analyses for extracting initial conditions in small-x evolution.
• To develop a rigorous NLO framework for coherent, diffractive scattering at t = 0, focusing on impact parameter-integrated dipole amplitudes.

Proposed method

• Formulates the exclusive scattering amplitude at NLO using the dipole picture in the high-energy limit, with the photon and vector meson wave functions at NLO.
• Uses the Balitsky-Kovchegov (BK) equation to describe the energy evolution of the dipole amplitude N01(Y), with a non-perturbative initial condition fitted to HERA data.
• Incorporates NLO corrections to the photon and vector meson wave functions, including virtual corrections (δZ) and real gluon emission contributions.
• Evaluates the amplitude at t = 0, focusing on the impact parameter-integrated dipole amplitude to avoid modeling long-range confining effects.
• Derives explicit expressions for the NLO amplitude kernel, including special functions such as modified Bessel functions K0 and K1, and integrals I(f), I(g), and their imaginary parts.
• Solves the NLO amplitude using the wave function renormalization coefficient δZ and includes relativistic corrections via a nonrelativistic expansion of the quarkonium wave function up to order v².

Experimental results

Research questions

• RQ1 How do QCD corrections (∼αs) and relativistic corrections (∼v²) affect the cross section for exclusive J/ψ production at small x?
• RQ2 To what extent are these NLO corrections numerically significant in the context of J/ψ production at HERA and the LHC?
• RQ3 Can exclusive vector meson production at NLO provide complementary constraints on the initial condition of the small-x evolution compared to traditional structure function analyses?
• RQ4 What is the structure of the NLO amplitude kernel in the dipole picture, and how do the virtual and real emission contributions combine?
• RQ5 How do the NLO corrections modify the t-integrated cross section for longitudinally polarized photons?

Key findings

• Both the first QCD correction (∼αs) and the first relativistic correction (∼v²) are found to be numerically significant in J/ψ production, with v² ∼ αs for charm quarks.
• The NLO amplitude kernel includes contributions from virtual corrections (via δZ and K0(τ) terms), real gluon emission (via I(f), I(g)), and logarithmic terms in |x01| and mq.
• The calculation yields a finite, well-defined NLO amplitude at t = 0, with all infrared divergences regulated by a non-zero lower limit on the z2 integral (z2 > α).
• The relativistic correction to the wave function is derived up to order v² using a nonrelativistic expansion of the rest-frame wave function, with explicit expressions for the φq¯q wave function including ∇²φRF(0)/φRF(0) terms.
• The final NLO cross section is expressed in terms of special functions and integrals (e.g., K0, K1, Ii, I), all of which are finite and suitable for phenomenological implementation.
• The framework is directly applicable to phenomenological studies of nonlinear dynamics in the CGC, particularly in exclusive processes with heavy nuclei at future Electron-Ion Colliders.