Implementing the exact kinematical constraint in the saturation formalism

TL;DR

The study tackles the problematic negative NLO cross section in forward hadron production within the small-$x$ saturation formalism by implementing the exact kinematical constraint in the dipole framework. This yields two additional NLO corrections, $L_q(k_ot)$ and $L_g(k_ot)$, whose high-$k_ot$ tails scale as $\sim 1/k_ot^4$, offsetting the negativity and extending the formalism’s applicability to larger transverse momenta. Numerically, the improved implementation in SOLO with GBW and rcBK dipole amplitudes yields excellent agreement with forward RHIC and LHC data, particularly in the forward region, while exposing limitations at mid-rapidity that require matching to collinear factorization. Overall, the work provides a more robust, quantitative tool for testing gluon saturation effects at high energy and sets the stage for systematic comparisons across kinematic regimes.

Abstract

We revisit the issue of the large negative next-to-leading order (NLO) cross section for single inclusive hadron production in $pA$ collisions in the saturation formalism. By implementing the exact kinematical constraint in the modified dipole splitting functions, two additional positive NLO correction terms are obtained. In the asymptotic large $k_\perp$ limit, we analytically show that these two terms become as large as the negative NLO contributions found in our previous calculation. Furthermore, the numerical results demonstrate that the applicable regime of the saturation formalism can be extended to a larger $k_\perp$ window, where the exact matching between the saturation formalism (in the asymptotic $k_\perp$ regime) and the collinear factorization calculations will have to be performed separately. In addition, after significantly improving the numerical accuracy of the NLO correction, we obtain excellent agreement with the LHC and RHIC data for forward hadron productions.

Figures (7)

• Figure 1: A typical real diagram at NLO
• Figure 2: The comparison between $\frac{Q_s^2}{S_\perp}L_q (k_\perp)$ and $Q_s^2F(k_\perp)$ with $S_\perp$ factored out. (One can also simply set $Q_s=1\, \textrm{GeV}$.) Here we have employed three different numerical methods to evaluate $L_q (k_\perp)$. The blue dots indicate the direct numerical evaluation of $L_q(k_\perp)$ as in Eq. (\ref{['lq1']}) with Mathematica, while the red circles represent the evaluation of Eq. (\ref{['lq2']}). The golden diamonds correspond to the numerical results obtain from Eq. (\ref{['lq2']}) by using our SOLO code programmed with $C++$. The asymptotic $k_\perp$ behaviour of $L_q (k_\perp)$ is indicated by the green dashed line. The numerical uncertainties are very small as shown in the right plot.
• Figure 3: The orientation of rapidities used by SOLO and throughout this paper. Positive (or forward) rapidity is always in the direction of the proton (or deuteron, for RHIC) beam. Some results published by ALICE ALICE:2012mj and ATLAS AlexanderMilovonbehalfoftheATLAS:2014rta use the opposite orientation. In this paper we always use $y$ to represent the rapidity in the center-of-mass frame.
• Figure 4: Comparisons of BRAHMS data Arsene:2004ux with the center-of-mass energy of $\sqrt{s_{NN}}=200GeV$ per nucleon at rapidity $y=2.2, 3.2$ with our results. As illustrated above, the crosshatch fill shows LO results, the grid fill indicates LO+NLO results, and the solid fill corresponds to our new results which include the NLO corrections from $L_q$ and $L_g$ due to the kinematical constraint. The error band is obtained by changing $\mu^2$ from $10GeV^2$ to $50GeV^2$.
• Figure 5: Comparison of STAR data Adams:2003qm with $\sqrt{s_{NN}}=200GeV$ at $y=4$ with results from SOLO for the GBW and rcBK models. The color scheme is the same as in figure \ref{['g']}, and again, the error band comes from $\mu^2 = 10GeV^2$ and $50GeV^2$. We do not see the negative total cross section because the cutoff momentum above which the cross section becomes negative is larger than the $p_\perp$ of the available data, and in fact larger than the kinematic limit $\sqrt{s_{NN}} e^{-y}$.