Threshold resummation at NNLL for single-particle production in hadronic collisions

We develop threshold resummation for single-particle inclusive cross sections in hadron-hadron collisions to the level of next-to-next-to-leading logarithm, up to full matching with two-loop hard functions. We define and calculate all one-loop soft functions for all partonic channels. This enables us to separate the hard and soft functions at one loop. Along with these results, the one-loop finite parts of jet functions are used to check that the full soft, collinear, and virtual corrections are reproduced to one loop for all partonic reactions. We exhibit these next-to-leading order (NLO) results explicitly. NLO expansions of the resummed cross section match the exact NLO results extremely well numerically, and two-loop expansions result in substantial corrections over many kinematic configurations. Explicit results are given in Mellin moment space, and a number of options for generating resummed cross sections are discussed.

Figure 1

NLO and NNLO $K$ factors for $pp\to {\pi }^{0}X$ for E706 kinematics with $\sqrt{s}=31.5\mathrm{GeV}$. All results are normalized to the LO cross section. We have chosen ${\mu }_F={\mu }_R=p_T$.

Figure 2

NLO and NNLO $K$ factors for $pp\to {\pi }^{0}X$ for RHIC kinematics with $\sqrt{s}=200\mathrm{GeV}$. All results are normalized to the LO cross section.

Figure 3

Same as Fig. 1, but for scales ${\mu }_F={\mu }_R=2p_T$.

Figure 4

Same as Fig. 2, but for scales ${\mu }_F={\mu }_R=2p_T$.

Figure 5

Upper left: ${\mu }_R$ dependence of the invariant cross section for E706 energy at $p_T=5\mathrm{GeV}$ for fixed ${\mu }_F=2p_T$. Upper right: ${\mu }_F$ dependence of the invariant cross section for E706 energy at $p_T=5\mathrm{GeV}$ for fixed ${\mu }_R=2p_T$. Lower left: scale dependence of the invariant cross section for E706 energy at $p_T=5\mathrm{GeV}$, setting ${\mu }_R={\mu }_F\equiv \mu$. Lower right: scale dependence of the invariant cross section for RHIC energy at $p_T=12\mathrm{GeV}$, setting ${\mu }_R={\mu }_F\equiv \mu$.