Transition Form-Factor for $η_Q$ at NNLO in the strong coupling $α_s$ and with all-order $v^2$ resummation

TL;DR

This work delivers the first NNLO QCD corrections to the transition form-factor for $\\gamma^*\\gamma^* \\to \\eta_Q$ in the double-off-shell regime, while resumming a class of relativistic $v^2$ corrections to all orders within NRQCD. The authors derive the NRQCD factorization for the $^{1}S_0$ state, extract short-distance coefficients, and compute two-loop amplitudes with two off-shell photons, using numerical master integrals and careful renormalization. They present three phenomenological applications in charmonium: single-space-like and double-space-like form-factor ratios compared to BaBar data, and the decay width $\\Gamma(\\eta_c \\to \\gamma\\gamma)$, including a fit to PDG values for $|R_{\\eta_c}(0)|^2$. The results show sizable negative NNLO corrections and demonstrate the importance of both relativistic and QCD effects, with prospects for future experimental tests and combined fits of nonperturbative inputs and relativistic parameters.

Abstract

In this work, we discuss both relativistic and perturbative QCD corrections to the transition form-factor ${\cal F}_{η_Q}{(t_1,t_2)}$ for the process $γ^*(q_1) γ^*(q_2) \leftrightarrow η_Q(P)$ with dependencies on the normalised photon virtualities $t_1=q_1^2/m_Q^2$ and $t_2=q_2^2/m_Q^2$, where $m_Q$ is the heavy quark mass. We resum a class of relativistic corrections to all orders in the relativistic parameter $\langle v^2 \rangle_{η_Q}$. In addition, we include perturbative QCD corrections up to Next-to-Next-to-Leading Order (NNLO) in the strong coupling constant $α_s$. This involves the computation of two-loop amplitudes with two off-shell photons. We explore three different phenomenological applications of our transition form-factors for the charmonium case. We first study the ratio $\vert {\cal F}_{η_c}{(t_1,0)} \vert/ \vert {\cal F}_{η_c}{(0,0)} \vert$ for the single space-like photon case and compare our results with the existing $η_c$ measurements from the BaBar collaboration. Secondly, we consider observables for the case of double space-like photons and discuss the impact of the different corrections. The NNLO corrections for this case are presented here for the first time in the literature. Finally, we revisit the decay width $Γ[η_c \rightarrow γγ]$ and compare it with the existing PDG value.

Figures (9)

• Figure 1: (a) Plot of $g(t_1, t_2)$ (see eq. (\ref{['eq:g_t1t2']})) as a function of $t_1$ and $t_2$ and (b) plot of $g(t_1, 0)$ and $g_n(t_1, 0)$ (see eq. (\ref{['eq:g_n_t1t2']})) with $n=\{1,2,3\}$ as a function of $t_1$ only. We set $\langle v^2 \rangle_{\eta_c} = 0.20 \pm 0.07$ as default parameter for the charmonium case, that will be discussed later in Section \ref{['Sec:Applications']}, and show the uncertainties of $g$ only.
• Figure 2: Two-loop diagrams for the form-factor $\gamma^* \gamma^* \leftrightarrow {^1S_0}$ with (a) regular contributions, (b) light-by-light contributions and (c) vacuum polarisation contributions.
• Figure 3: NLO and NNLO correction factors as a function of $t_1$ and $t_2$ in the double space-like region. We plot in (a) the real part of $K_{1}(t_1, t_2)$, while in (b) and (c) we plot the real and imaginary parts of $K_{2}(t_1, t_2)$ respectively. For $K_2(t_1, t_2)$, we set $l_{\mu_R}=0$, $l_{\mu_{\Lambda}}=0$, $n_h=1$, $n_l=3$ and $\tilde{n}_l=3/2$ relevant for the charmonium case.
• Figure 4: Plots of $\left\vert 1 + \Delta(t_1, t_2) \right\vert$ (see eq. (\ref{['eq:deltalabelp']})) at LO, NLO and NNLO for (a) $\eta_c$ and (b) $\eta_b$ as a function of $t_1$ and at three different values of $t_2=\{0,-5,-10\}$. We set $l_{\mu_R}=\log{4}$ with $\mu_R^2 = 4m_Q^2$, and $l_{\mu_{\Lambda}}=0$. While we use $m_c=1.5 \text{ GeV}$ for the $\eta_c$ case, we set $m_b=4.8 \text{ GeV}$ for $\eta_b$ and adjust the number of light flavours (see footnote \ref{['foot:bottomflavnum']}).
• Figure 5: Transition form-factor ratio $R{(Q_1^2)}$ with impact of (a) relativistic corrections, (b) QCD corrections, (c) combined relativistic and QCD corrections.