A complete ${\cal O}(αα_s)$ calculation of the Photon + 1 Jet rate in $e^+e^-$ annihilation

TL;DR

This work delivers a complete O(α αs) calculation of the photon+1 jet rate in e+e− annihilation, extending phase-space slicing to handle double unresolved regions and employing a democratic clustering framework. It combines prompt-photon and fragmentation contributions, derives and factorizes the relevant singularities into the quark-to-photon fragmentation function D_{q→γ}, and provides an exact NLO evolution solution ensuring μ_F independence. By fitting D_{q→γ}^{np} to ALEPH data, the authors achieve improved agreement for the photon+1 jet rate and demonstrate the universality of the fragmentation function across observables, with clear implications for ISOLATED Photon phenomenology at colliders. The methodology and results offer a path toward NNLO-like precision in jet-related photon observables and furnish a data-driven input for photon production in hadron-collider environments such as the LHC.

Abstract

We present a complete calculation of the photon +~1 jet rate in $e^+e^-$ annihilation up to ${\cal O}(αα_{s})$. Although formally of next-to-leading order in perturbation theory, this calculation contains several ingredients appropriate to a next-to-next-to-leading order calculation of jet observables. In particular, we describe a generalization of the commonly used phase space slicing method to isolate the singularities present when more than one particle is unresolved. Within this approach, we analytically evaluate the singularities associated with the following double unresolved regions; triple collinear, soft/collinear and double single collinear configurations as well as those from the collinear limit of virtual graphs. By comparing the results of our calculation with the existing data on the photon +~1 jet rate from the ALEPH Collaboration at CERN, we make a next-to-leading order determination of the process-independent non-perturbative quark-to-photon fragmentation function $D_{q \to γ}(z,μ_{F})$ at ${\cal O}(αα_{s})$. As a first application of this measurement allied with our improved perturbative calculation, we determine the dependence of the isolated photon +~1 jet cross section in a democratic clustering approach on the jet resolution parameter $\ycut$ at next-to-leading order. The next-to-leading order corrections to this observable are moderate but improve the agreement between theoretical prediction and experimental data.

Figures (9)

• Figure 1: Parton level subprocesses contributing to the photon + 1 jet rate at ${\cal O}(\alpha\alpha_s)$.
• Figure 2: Different final state 'photon' + 1 jet topologies arising from the tree level $\gamma^* \to q \bar{q}\gamma g$ process. The 'photon' jet is moving to the left while the recoiling hadronic jet moves to the right. Square brackets denote theoretically unresolved particles, round brackets experimental clusters.
• Figure 3: Phase space decomposition of the real $\gamma^{\star}\to q\bar{q} \gamma g$ contributions. Note that the single and double unresolved regions where the photon clusters with the antiquark are not shown. For these regions, the necessary cuts are obtained by exchanging $q$ and $\bar{q}$. Altogether, there are five single unresolved and six double unresolved regions.
• Figure 4: Different final state 'photon' + 1 jet topologies arising from the $\gamma^* \to q \bar{q} (g)$ process with subsequent fragmentation of the quark into a photon. The 'photon' jet is moving to the left while the recoiling hadronic jet moves to the right. Square brackets denote theoretically unresolved particles, round brackets represent experimental clusters.
• Figure 5: Contributions of the individual terms ( (A),(B),(C),(D)) to the total cross section as function of $y_{\rm min}$ for $y_{\rm cut}=0.1$ and $z_{\rm cut}=0.7$. For clarity, only the next-to-leading order contributions are shown. Furthermore we take $\alpha e_{q}^2=2\pi$ and $\alpha_s C_F =2\pi$.