Frame dependence of transition form factors in light-front dynamics

We calculate the transition form factor between vector and pseudoscalar quarkonia in both the timelike and the spacelike region using light-front dynamics. We investigate the frame dependence of the form factors for heavy quarkonia with light-front wave functions calculated from the valence Fock sector. This dependence could serve as a measure for the Lorentz symmetry violation arising from the Fock-space truncation. We suggest using frames with minimal longitudinal momentum transfer for calculations in the valence Fock sector, namely, the Drell-Yan frame for the spacelike region and a specific longitudinal frame for the timelike region; at $q^2=0$ these two frames give the same result. We also use the transition form factor in the timelike region to investigate the electromagnetic Dalitz decay ${\psi }_A\to {\psi }_B{l}^{+}{l}^{-}$ ($l=e$, $\mu$) and predict the effective mass spectrum of the lepton pair.

Figure 1

The two dominant contributions to the transition ${\psi }_A\to {\psi }_B{\gamma }^{(*)}$ on the light front: (a) the parton-number-conserving term, and (b) the parton-number-nonconserving term. Diagrams are light-front time ordered. Light-front time flows from the left to the right.

Figure 2

Visualization of the Lorentz invariant momentum transfer squared $q^2$ as a function of $z$ and ${\overrightarrow{\mathrm{\Delta }}}_{\perp }$ at $\arg {\overrightarrow{\mathrm{\Delta }}}_{\perp }=0,\pi$. (a) Regional plot of $q^2$. The timelike region ($q^2>0$) is the orange oval shape, bounded by ${\mathrm{\Delta }}_{\mathrm{node}}=(m_A^2-m_B^2)/2m_A$ and $z_{\mathrm{node}}=1-m_B^2/m_A^2$. The spacelike region ($q^2<0$) is in light gray. Contour lines of $q^2$ are indicated with thin dashed curves. The maximal value $q_{\max }^2=(m_A-m_B{)}^2$ occurs at $(z_{\mathrm{turn}}=1-m_B/m_A,{\mathrm{\Delta }}_{\perp }=0)$. (b) Three-dimensional plot of $q^2$ showing a convex shape in the $(z,{\mathrm{\Delta }}_{\perp })$ representation. The blue flat plane is the reference plane of $q^2=0$. In each figure, the Drell-Yan frame is shown as a thick solid line, and the longitudinal I and II frames are shown as thick dotted and thick dashed lines, respectively.

Figure 3

The valence light-front wave functions of mesons as they contribute [see Eq. (5)] to the convolution in the transition $J/\psi \to {\eta }_c+{\gamma }^{(*)}$ at $q^2=-3{\mathrm{GeV}}^2$ in different frames. According to Eq. (5), in this $2\to 2$ parton-number-conserving term, the initial state wave function of $J/\psi$ would appear shifted and stretched to overlap with the final state wave function of ${\eta }_c$, when plotted on the $(x,{\overrightarrow{k}}_{\perp })$ space. Shown in (a), the wave function of $J/\psi$ is shaped differently at different $(z,{\overrightarrow{\mathrm{\Delta }}}_{\perp })$, i.e., in different frames. The longitudinal dimension is preserved most in the Drell-Yan frame where $z=0$. At larger $z$, the information in the longitudinal region is reduced, and the transverse shift becomes smaller. The largest $z$ is achieved when ${\mathrm{\Delta }}_{\perp }=0$ in the longitudinal frame, in this case, $z=0.45$. Plotted in (b) is the wave function of ${\eta }_c$. All light-front wave functions that we employ are calculated by Ref. [28] and here we only plot the dominant spin components for the purpose of illustration.

Figure 4

The transition form factor of the transition $\mathcal{V}(nS)\to \mathcal{P}(nS)\gamma$ of charmonia (blue curves/shades) and bottomonia (red curves/shades), calculated with light-front wave functions at $N_{\max }=L_{\max }=32$ basis truncation. Meson masses are taken from experimental data [1] in defining the frames according to Eq. (4). The solid curves represent the Drell-Yan frame while the other curves represent the longitudinal I (dotted lines) and II (dashed lines) frames. The shaded areas represent the results from all other frames. The left panel shows the transition form factor at a larger scale of $q^2$, and the right panel focuses on the small $q^2$ region.

Figure 5

The transition form factor of the transition ${\psi }_A(2S)\to {\psi }_B(1S)\gamma$ (${\psi }_A,{\psi }_B=\mathcal{V},\mathcal{P}$ or $\mathcal{P},\mathcal{V}$) of charmonia (blue curves/shades) and bottomonia (red curves/shades), calculated with light-front wave functions at $N_{\max }=L_{\max }=32$ basis truncation. Meson masses are taken from experimental data [1] for defining the frames according to Eq. (4). The solid curves represent the Drell-Yan frame while the other curves represent the longitudinal I (dotted lines) and II (dashed lines) frames. The shaded areas represent the results from all other frames. The left panel shows the transition form factor at a larger scale of $q^2$, and the right panel focuses on the small $q^2$ region.

Figure 6

The transition form factors for charmonia (left panels) and bottomonia (right panels) with different basis truncations. Meson masses are taken from experimental data [1] in defining the frames according to Eq. (4). The solid curves represent the Drell-Yan frame while the other curves represent the longitudinal I (dotted lines) and II (dashed lines) frames. The shaded areas represent the results from all other frames.

Figure 7

The effective mass spectrum of the lepton pairs in the Dalitz decays for charmonia (left panels) and bottomonia (right panels). The dashed and solid curves represent the longitudinal I and II frames, respectively. The shaded areas represent the results from all other frames. $\mathrm{\Delta }{m}^2=(m_A-m_B{)}^2$ is the square of the mass difference between the initial and the final mesons. Meson masses are taken from experimental data [1].