Unintegrated parton distributions and prompt photon hadroproduction

TL;DR

This work develops a formalism to construct unintegrated parton distributions $f_a(x, k_t^2, \mu^2)$ from conventional integrated PDFs, incorporating Sudakov suppression and coherence effects. It introduces the parton-parton luminosity ${\cal L}_{ab}(x_1, x_2, q_t)$ and shows how incoming partonic transverse momentum $q_t$ enters prompt photon hadroproduction cross sections under kt-factorization, exploring both BFKL and double-log limits. Applying the framework to $pp$ and $p\bar{p}$ collisions, the authors find that including $q_t$ primarily boosts cross sections while leaving the $p_{t}^{\gamma}$ spectrum shape largely unchanged, with the effect sensitive to the scale at which PDFs are sampled. The results validate the kt-factorization approach for hard hadronic processes and outline how full unintegrated PDFs and higher-order corrections are needed for precision phenomenology, especially across different energy regimes.

Abstract

We introduce a general expression which enables the parton distribution, unintegrated over the parton transverse momentum, to be obtained from the conventional parton densities. We use the formalism to study the effects of the transverse momentum q_t of the incoming partonic system on the calculation of the transverse momentum spectra of prompt photons produced in high energy pp and p\bar{p} collisions. For the purposes of illustration, we use the double logarithm approximation. For large q_t we calculate the effect directly from the perturbative formalism, whereas for small q_t we bound the effect using two extreme hypotheses. In both q_t domains we find that the shapes of the prompt photon spectra are not significantly modified, although the cross sections are enhanced.

Figures (5)

• Figure 1: A schematic diagram describing both the subprocesses $gq \rightarrow \gamma q$ and $\bar{q}q \rightarrow \gamma g$, in which the hard momentum transfer squared is either $\hat{t}$ or $\hat{t}^\prime$ respectively.
• Figure 2: The parton luminosity function ${\cal L}_{gq} (x_1, x_2, q_t)$ of (\ref{['eq:a6']}) as a function of the incoming partonic $q_t$ for $p_{t \gamma} = 4$ GeV at the UA6 energy of $\sqrt{s} = 24.3$ GeV and for $p_{t \gamma} = 20$ GeV at the Tevatron energy of $\sqrt{s} = 1800$ GeV. In each case we take $x_1 = x_2 = x_T = 2p_{t \gamma}/\sqrt{s}$. The hard scale $\mu$ is taken to be $p_{t \gamma}$.
• Figure 3: The effect of partonic $q_t$ arising from the non-perturbative $(q_t < q_0)$, perturbative $(q_t > q_0)$, and total contributions to the $p_{t \gamma}$ spectrum of prompt photon production in $pp$ collisions at $\sqrt{s} = 24.3$ GeV and in $p\bar{p}$ collisions at 1800 GeV. Two extreme hypotheses are used for the non-perturbative contribution, but the predictions are very similar (and in fact are indistinguishable on the $\sqrt{s} = 1800$ GeV plot). We see that the non-perturbative contribution dominates at $\sqrt{s} = 24.3$ GeV, whereas the perturbative dominates at $\sqrt{s} = 1800$ GeV. The hard scale $\mu$ is taken to be $p_{t \gamma}$ and $q_0^2 = 1.25$ GeV$^2$.
• Figure 4: The scale dependence of the predictions for production of prompt photons in $pp$ collisions at $\sqrt{s} = 24.3$ GeV, $p$Be collisions at $\sqrt{s} = 38.8$ GeV and $p\bar{p}$ collisions at $\sqrt{s} = 1.8$ TeV shown together with UA6 UA6, E706 E706 and CDF FNAL data respectively. The continuous curves are the predictions with the incoming partonic transverse momentum $q_t$ included, whereas the dashed curves correspond to the unsmeared $(q_t = 0)$ results in which the integrated partons are sampled at the hard scale $\mu$. In each case the upper curve corresponds to the scale $\mu = p_{t \gamma}/2$, while the lower corresponds to $\mu = p_{t \gamma}$.
• Figure 5: A clearer comparison of the data also plotted in Fig. \ref{['fig4']} with our theoretical calculations at the scale $\mu = p_{t \gamma}$; we use $x_{T}\equiv 2p_{t\gamma}/\sqrt{s}$ as the transverse momentum variable.