Transverse momentum distribution of Upsilon production in hadronic collisions

TL;DR

The paper develops a QCD-based framework to predict the transverse-momentum distribution of Upsilon states in hadronic collisions across the full $p_T$ range. It combines a two-step factorization with Collins-Soper-Sterman resummation in impact-parameter space to sum initial-state gluon-shower logarithms, and fixes large-$p_T$ behavior with ${\cal O}(\alpha_s^3)$ perturbation theory, incorporating a nonperturbative transition ${\cal F}(Q^2)$ to model Υ formation. The authors fit a small set of parameters to Tevatron data, demonstrating that the low-$p_T$ region is dominated by perturbative small-$b$ contributions and achieving good agreement with data for all Υ(1S–3S) states. This work provides a robust, QCD-grounded method for quarkonium p_T spectra and outlines clear paths for improvement via higher-order corrections and applications to higher-energy colliders.

Abstract

We calculate the transverse momentum p_T distribution for production of the Upsilon states in hadronic reactions. For small $p_T (\leq M_Υ)$, we resum to all orders in the strong coupling alpha_s the process-independent large logarithmic contributions that arise from initial-state gluon showers. We demonstrate that the p_T distribution at low p_T is dominated by the region of small impact parameter b and that it may be computed reliably in perturbation theory. We express the cross section at large p_T by the alpha_s^3 lowest-order non-vanishing perturbative contribution. Our results are consistent with data from the Fermilab Tevatron collider.

Figures (7)

• Figure 1: Hadronic production of an $\Upsilon$ via an intermediate heavy quark pair $b$ and $\bar{b}$.
• Figure 2: Diagram that illustrates multiple gluon radiation from an initial-state parton.
• Figure 3: The $b$-space distributions for $\Upsilon$ production: (a) $gg$ channel, and (b) the sum of all$q\bar{q}$ channels. Note that the magnitude of the $gg$ distribution has been scaled by a factor of 20. The functions are evaluated at rapidity $y = 0$ and are integrated over the mass range $2m_b < Q < 2M_B$.
• Figure 4: Inclusive transverse momentum distribution of the $\Upsilon(1S)$ in $p \bar{p}$ interactions at $\sqrt S = 1.8$ TeV. The solid, dashed, and dotted lines in the region of large $p_T$ are obtained from fixed-order ${\cal O} (\alpha_s^3)$ perturbative QCD for three different values of the scale $\mu$; solid for $\mu = 0.5\mu_0$, dashed for $\mu = \mu_0$, and dotted for $\mu = 2\mu_0$; with $\mu_0 = \sqrt{Q^2+p_T^2}$. The dot-dashed line in the region of small $p_T$ is the resummed prediction.
• Figure 5: Calculated differential cross sections times leptonic branching fractions $B$, evaluated at $y = 0$, as functions of transverse momentum for hadronic production of (a) $\Upsilon(1S)$, (b) $\Upsilon(2S)$, and (c) $\Upsilon(3S)$, along with CDF data cdf2Acosta:2001gv at $\sqrt S = 1.8$ TeV. The dashed, solid, and dotted lines show the result of our full calculation for $b_{max}=0.3$, 0.5, and 0.7 GeV$^{-1}$, respectively. The 1995 CDF cross sections are multiplied by $C_n = 0.88$.