Collinear factorization in wide-angle hadron pair production in $e^{+}{e}^{-}$ annihilation

We compute the inclusive unpolarized dihadron production cross section in the far from back-to-back region of $e^{+}{e}^{-}$ annihilation in leading order pQCD using existing fragmentation function fits and standard collinear factorization, focusing on the large transverse momentum region where transverse momentum is comparable to the hard scale (the center-of-mass energy). We compare with standard transverse-momentum-dependent (TMD) fragmentation function-based predictions intended for the small transverse momentum region with the aim of testing the expectation that the two types of calculation roughly coincide at intermediate transverse momentum. We find significant tension, within the intermediate transverse momentum region, between calculations done with existing nonperturbative TMD fragmentation functions and collinear factorization calculations if the center-of-mass energy is not extremely large. We argue that $e^{+}{e}^{-}$ measurements are ideal for resolving this tension and exploring the large-to-small transverse momentum transition, given the typically larger hard scales ($\gtrsim 10\mathrm{GeV}$) of the process as compared with similar scenarios that arise in semi-inclusive deep inelastic scattering and fixed-target Drell-Yan measurements.

Figure 1

(a) The photon frame. The $x$ and $z$ axes have been aligned with the spatial components of $X^{\mu }$ and $Z^{\mu }$ from Eq. (3). The blue plane is the $e^{+}{e}^{-}$ plane. (b) The hadron frame, with the hadrons exactly back to back. See text for further explanation.

Figure 2

(a) The general diagrammatic structure contributing to Eq. (1) at large $q_{\mathrm{T}}$ and at LO in ${\alpha }_s$. The outgoing partonic lines are dotted to indicate that generally they can be of any type. In the region of interest for this paper, their momenta deviate by wide angles from the back-to-back orientation for the dihadron pair. $H$ represents the hard part of the interaction and the $C_{A,B,C}$ are the collinear subgraphs [6]. (b) The $O({\alpha }_s)$ partonic contribution to the square-modulus amplitude in the factorization of (a).

Figure 3

Partonic channels that contribute at order ${\alpha }_s$. A detailed explanation is in Sec. 3a.

Figure 4

(Left) LO collinear factorization predictions for the inclusive $e^{+}{e}^{-}$ to dihadron cross section (Sec. 3 and Appendix pp2), for $Q=12$, 50 GeV. The red band shows the range covered by switching the renormalization group scale between $\mu =Q$ (lower edge) and $q_{\mathrm{T}}$ (upper edge). The blue band is the calculation performed using TMD ffs, and the band shows the range covered by the values of the nonperturbative parameters discussed in Sec. 5. (Right) Correlation between partonic momentum fractions ${\zeta }_{A,B}$ for various values of $q_{\mathrm{T}}/q_{\mathrm{T}}^{\mathrm{Max}}$.

Figure 5

The lowest order collinear factorization calculation from Sec. 3 compared with ${\pi }^{+}/{\pi }^{-}$ pair production simulated by PYTHIA-8 with default settings for different ranges of $z_{A,B}$ and for increasing values of $Q$, starting with $Q=12\mathrm{GeV}$. Both the fixed order calculation and the simulation are averaged in the $z_{A,B}$ bins. The uncertainty on the bands is purely statistical.