Radiative return at NLOPS accuracy

TL;DR

We address radiative return measurements of the pion form factor, essential for data-driven hadronic vacuum polarization in $(g-2)_\mu$, by delivering the first full NLO corrections matched to a Parton Shower for $e^+e^-\to X^+X^-\gamma$ with $X=\pi,\mu$. The method implements exact NLO SV corrections together with LL-resummed multi-photon emission, including ISR, FSR and IFI, and non-perturbative inputs through $\Lambda(Q^2)=\{\alpha(Q^2),F_\pi(Q^2)\}$, all integrated in an updated BabaYaga@NLO framework with multi-channel phase-space sampling and a CKKW-like clustering for hard photons. Validation against Phokhara, McMule, Recola and Alpha shows per-mille accuracy for the leptonic channel and reliable control of the hadronic channel, including exclusive three-photon configurations. The study reveals that higher-order (beyond NLO) corrections can reach the percent level in realistic radiative-return setups, underscoring the need for NLOPS modelling to achieve sub-percent precision in the extraction of the pion form factor and the HVP contribution to $a_\mu$ at flavour factories.

Abstract

The radiative return, together with the energy scan, is the method used at flavour factories to measure the pion form factor, which is a crucial input for the data-driven dispersive computation of the leading-order hadronic contribution to the muon anomalous magnetic moment. We consider the radiative hadronic and leptonic channels of main experimental interest, namely the processes $e^+e^-\to X^+X^-γ$, with $X = \{π\, , μ\}$. For such processes, we compute the exact next-to-leading order (NLO) corrections matched to a Parton Shower (PS) to describe exclusive multiple photon emission. All sources of radiative corrections from initial-state and final-state radiation, as well as their interference, are considered according to QED for $e^+e^-\toμ^+μ^-γ$ and QED$\oplus$F$\times$sQED (Factorised scalar QED) for $e^+e^-\toπ^+π^-γ$. We describe in detail the novel features of our PS approach to compute the fixed-order corrections in association with higher-order contributions to $2\to3$ processes, with a hard photon in the final state. We present validation tests and comparisons with NLO predictions available in the literature to cross-check various ingredients of our formulation. We also show numerical results at NLOPS accuracy according to realistic event selection criteria for precision measurements at flavour factories. Our calculation is implemented in an updated version of the Monte Carlo event generator BabaYaga@NLO, which can be used for fully exclusive simulations and data analysis in radiative return experiments.

Figures (18)

• Figure 1: Tree-level diagrams contributing to the $e^+e^-\to X^+X^-\gamma$ process: the first two are ISR diagrams, while the latter three represent FSR. The first four contribute to the signatures with $X=\left\{\mu,\pi\right\}$, while the last one contributes to the pion channel only. The shaded blob represents the pion form factor.
• Figure 2: Examples of virtual diagrams where the photon is emitted by the initial state, divided by classes.
• Figure 3: Examples of classes of diagrams for the $2\to2+2\gamma$ process. The first diagram accounts for ISR photon emission, the central diagram exemplifies the set of diagrams in which one photon is emitted from the initial current and one from the final one. The rightmost diagram represents the double FSR emission.
• Figure 4: Comparison of the exact NLO calculation implemented in BabaYaga@NLO with those of independent programs. Left panel: $\mu^+\mu^-\gamma$ channel, compared with McMule and Phokhara. Right panel: $\pi^+\pi^-\gamma$ channel, compared with Phokhara. Both panels correspond to setup (a) of Tab. \ref{['tab:scenarios']}, and show the differential cross section as a function of the tagged photon energy.
• Figure 5: Comparison of the exact NLO calculation implemented in BabaYaga@NLO with Phokhara for the $\pi^+\pi^-\gamma$ channel. Left panel: differential cross section as a function of $\theta^{+}$ in setup (a). Right panel: differential cross section as a function of $M_{\pi\pi}$ in setup (d) in the range $M_{\pi\pi}\in \left[2\,m_\pi,2 \,\, \rm GeV\right]$. Event selection provided in Tab. \ref{['tab:scenarios']}.