Implementing the exact kinematical constraint in the saturation formalism

We revisit the issue of the large negative next-to-leading-order (NLO) cross section for single inclusive hadron production in $\mathrm{p}A$ 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.

Figure 1

A typical real diagram at NLO.

Figure 2

The comparison between $\frac{Q_s^2}{S_{\perp }}{L}_q(k_{\perp })$ and ${Q_s}^{2}F(k_{\perp })$ with $S_{\perp }$ factored out. (One can also simply set $Q_s=1\mathrm{GeV}$.) 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. (10) with mathematica, while the red circles represent the evaluation of Eq. (14). The golden diamonds correspond to the numerical results obtain from Eq. (14) by using our solo code programmed with $\mathrm{C}++$. The asymptotic $k_{\perp }$ behavior 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 the RHIC) beam. Some results published by ALICE [15] and ATLAS [21] 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 [9] with the center-of-mass energy of $\sqrt{s_{NN}}=200\mathrm{GeV}$ 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 $\mathrm{LO}+\mathrm{NLO}$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 10 to $50{\mathrm{GeV}}^2$.

Figure 5

Comparison of STAR data [10] with $\sqrt{s_{NN}}=200\mathrm{GeV}$ at $y=4$ with results from solo for the GBW and rcBK models. The color scheme is the same as in Fig. 4, and again, the error band comes from ${\mu }^2=10$ and $50{\mathrm{GeV}}^2$. 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 $\sqrt{s_{NN}}{e}^{-y}$.

Figure 6

Comparison of ATLAS forward-rapidity data [21] with the center-of-mass energy of $\sqrt{s_{NN}}=5.02\mathrm{TeV}$ at $y=1.75$ with solo results for the GBW and rcBK models. Again, the color scheme is the same as in Fig. 4. Here the error band shows plots for ${\mu }^2=10{\mathrm{GeV}}^2$ and ${\mu }^2=100{\mathrm{GeV}}^2$. Since the numerical data for these measurements are not published, we have extracted the ATLAS points from Fig. 6 of Ref. [21]. The extraction procedure introduces uncertainties comparable to the size of the points.

Figure 7

Comparison of mid-rapidity data at $\sqrt{s_{NN}}=5.02\mathrm{TeV}$ at $y=0$ from ALICE [15] and ATLAS [21] with solo results for the GBW and rcBK models. Both data sets correspond to the same kinematic parameters. Again, the color scheme is the same as in Fig. 4. These results display the breakdown of the dilute-dense factorization approach when the separation between $x_p$ and $x_g$ is not sufficiently large. $\frac{x_g}{x_p}$ is roughly on the order of 1.