Shedding light on charmonium

We investigate E1 radiative transitions within charmonium in a relativistic approach based on light-front QCD. In quantum field theory, two sets of processes are pure E1: $\chi_{c0} \to J/\psi \gamma$ ($\psi\to \chi_{c0}\gamma$) and $h_c \to \eta_c\gamma$ ($\eta_c' \to h_c\gamma$), both involving the $P$-wave charmonia. We compute the E1 radiative decay widths as well as the corresponding transition form factors of various processes including those involving $2P$ states. These observables provide an access to the microscopic structures of the $P$-wave charmonium. We show that our parameter-free predictions are in excellent agreement with the experimental measurements as well as lattice simulations whenever available.


INTRODUCTION
The discoveries of charmonium-like states, e.g., χ c1 (3872) and χ c0 (3915), have sparked renewed interests in the charmonium structure [1][2][3].The proximity of their masses to the DD threshold leads to the speculation that at least some of them may be meson molecules [4].On the other hand, their quantum numbers are consistent with conventional cc quark model and their masses are also in the vicinity of the 2P charmonia in various quark model predictions [5].Furthermore, decay patterns from the quark models can encapsulate various coupled channel effects [6][7][8][9][10][11][12][13][14][15][16][17][18][19].In any case, the investigation of the microscopic structures of charmonium offers new insights into the nature of the strong force, which, after 50 years of QCD, remains one of the biggest puzzles in physics [20].
The radiative transitions provide a clean probe with variable resolutions to the microscopic composition of the system [21][22][23][24][25][26][27][28][29][30][31][32].Furthermore, these transitions are sensitive to relativistic effects, which underlines some recent discrepancies between NRQCD and the experimental measurements [33][34][35][36].For example, the two-photon decay width in NRQCD converges poorly and deviates from the experimental measurements up to 7σ in NNLO [35].A possible explanation is that charmonium is an intrinsically relativistic system.And the relativistic effects are stronger for the excited states.Therefore, a systematic investigation of the radiative transitions for both the ground-state P -wave charmonia and their excitations is required to obtain a complete picture of these charmonium-like states [33,37,38].
In our previous works, parameter-free predictions are made for the two-photon widths [38] and the M1 widths [41], as well as the associated transition form factors (TFFs).Our results are based on light-front wave functions (LFWFs [42,43]) from basis light-front quantization (BLFQ [44]).This approach is a natural framework to tackle hadrons as relativistic many-body bound states in the nonperturbative regime [45].For the application FIG. 1. Schematic view of the pure E1 transitions (yellow arrows) within charmonium.Other radiative transitions, e.g., M1, M2, and E2, are not shown.The masses obtained in BLFQ along with similar relativistic approaches (CST [39] and DSE/BSE [40]) are shown for comparison (see Sec. 5.4 of Ref. [20] and therein).Figure is adapted from Ref. [20].
to charmonium, two parameters, the charm quark mass m c and the basis scale κ, were fit to the charmonium mass spectrum [45] (see also Fig. 1).Then the obtained LFWFs are used to make parameter-free predictions to hadronic observables, e.g., decay constants [45], as well as partonic observables, e.g., parton distribution functions [46][47][48][49][50].All of these results were shown to be in reasonable agreement with the experimental measurements whenever available.
We focus here on the E1 transitions (Fig. 1) between Pwave scalar charmonia χ c0 (0 ++ ) and vector charmonia ψ (1 −− ), as well as between P -wave axial vector h c (1 +− ) and pseudoscalars (0 −+ ), which are relevant for unraveling the relativistic structure of P -wave charmonium.We assume the states are pure cc's.Therefore, any significant deviation from the experimental measurements implies a deviation from the conventional cc picture.
The E1 transition has a similar helicity structure SVV with the scalar meson two-photon transition [38].Since
the structures of the photon and vector charmonia are well established, these two processes can be used to constrain the structure of the scalar charmonia at different scales.Fig. 2 combines the E1 widths Γ S→V γ (or Γ V →Sγ ) and the two-photon width Γ S→γγ for scalar charmonia χ c0 (1P ) and χ c0 (2P ), as obtained in BLFQ [38].Processes for 1P scalar χ c0 have been measured by several experiments and compiled by PDG [51].Our results are in good agreement with the PDG values, which provide a basis for making predictions for the 2P state χ ′ c0 .Experimentally, the only available E1 data come from Belle for χ c0 (3915), a prime candidate for the 2P scalar.Last year, Belle collaboration discovered a resonance with the mass 3.922 GeV, which can be identified as χ c0 (3915) or χ c2 (3930).Belle also measured the prod- which is shown as a red curve with a band in Fig. 2 [52].Our predicted E1 and diphoton widths are consistent with this result.

FORMALISM
The E1 amplitude of a scalar meson S decaying into a vector meson V plus a (virtual) photon is described by the hadronic matrix element (HME), where, J µ (x) is the current operator, Q c = 2/3 is the charge number, e = √ 4πα em is the electron charge, q = p ′ − p is the four-momentum of the photon, and ε µ λγ (q) is the polarization vector of the photon.Following Ref. [53], we parametrize the HME in terms of its Lorentz structures, where, Q 2 = −q 2 , and Ω(Q V .e µ λ (p) is the polarization vector of the vector meson.The form factors E 1 and C 1 defined here can be extracted from the transverse and longitudinal amplitudes, respectively, viz., In particular, the E1 radiative decay width is propor- Here, j i , M i are the initial state spin and mass, respectively, and M f is the final state mass.The vector meson decaying into a scalar plus a photon can be similarly expressed.The Lorentz structures of the HME between pseudoscalar 0 −+ and the C-odd axial vector 1 +− are identical to Eq. ( 2).
In a nonrelativistic quark model, the E1 transition is induced by the electric dipole interaction.The corresponding electric charge density on the light front is J + = J 0 +J 3 , where we adopt the light-front coordinates J + is also known as the "good current" in light-front dynamics as it is not contaminated by the spurious Lorentzsymmetry violating contributions [54][55][56].We further adopt the Drell-Yan frame q + = 0 which simplifies the expression dramatically [54,56].The relevant diagrams for the E1 process are shown in Fig. 3.The nonperturbative structures of the initial-and final-state mesons are encoded in the LFWFs, where, x = p + /P + is the longitudinal momentum fraction of the quark, and ⃗ k ⊥ = ⃗ p ⊥ − x ⃗ P ⊥ is the relative transverse momentum of the quark.The momenta of the quark and the antiquark are is the color index and N c = 3.The LFWF ψ (mj ) ss/h (x, ⃗ k ⊥ ) is frame-independent and only depends on the relative motion of the quark and antiquark.The ellipsis represents contributions beyond the valence Fock sector |cc⟩, which are shown to be small from previous investigations and will be neglected for the present work as well (cf.[57]).
Using the LFWFs, the TFF E1 can be represented as where, Q 2 = −q 2 = q 2 ⊥ , and we have adopted arg ⃗ q ⊥ = 0 for simplicity.The coupling constant E 1 (0) is associated with dipole transition between the transversely polarized vector meson and the scalar meson: (x, ⃗ r ⊥ )ψ ss/S (x, ⃗ r ⊥ ).(7) This expression resembles the nonrelativistic expression of the E1 transition.The TFF can also be extracted from the spatial current ⃗ J ⊥ [38,41].Since the E1 transition is induced by the electric dipole, we adopt the charge density operator J , which has a smooth nonrelativistic limit as shown by Eq. ( 7).
The TFF E 1 (Q 2 ) provides further resolution of the system.Alas, the TFFs of these processes are not currently available from the experiments.We thus compare our BLFQ results with recent lattice simulations [53,71,72,75].Fig. 5(a) compares the TFFs of the transition between χ c0 and J/ψ as predicted by BLFQ and several lattice calculations [71,72,75].Given the scattering of the lattice data from different groups, our BLFQ prediction is in reasonable agreement with these results, in particular at low Q 2 .Our approach also provides access to moderately high Q 2 , where the lattice simulations suffer from low statistics.
The transitions involving the C-odd axial vector h c are shown in Figs.5(g)-5(h).The TFF of the process h c → η c (1S) + γ is computed by Refs.[53,72] in lattice.Our results are in good agreement with Ref. [53] while deviating from [72].Note that our E1 width of this process is in better agreement with the experimental data [66][67][68][69][70].

SUMMARY
In this work, we investigate the E1 radiative transitions within the charmonium system using the basis light-front quantization approach.We derived the light-front wave function representation of the decay width as well as the transition form factors.These representations are exact as long as the wave functions are exactly known.The wave functions adopted in this work come from fitting to the charmonium spectrum.Therefore, we are able to make parameter-free predictions for the E1 transitions.The results, including the widths and the form factors, are in excellent agreement with the experimental measurements as well as lattice simulations whenever available.
We also compute the E1 widths and the corresponding transition form factors of χ c0 (3915) by treating it as the 2P cc state.The obtained results are consistent with the recent measurement from Belle [52].Further experimental measurements are required to discern the nature of this particle.
We note similar successes in describing the charmonium structures, viz., M1 transitions [41], two-photon transitions [38] as well as the decay constants [45] using the same set of light front wave functions.These applications provide confidence that the intrinsic structure of charmonium is accurately described by these BLFQ wave functions and lend support to the adopted phenomenological form of confinement [76].We envision that further applications of the charmonium LFWFs will help to resolve the non-perturbative dynamics of the strong interaction in high-energy processes, such as the gluon distributions, generalized parton distributions (GPDs), hadronic anomalous energy, etc., which are among the central goals of the forthcoming electron-ion colliders [77][78][79][80][81][82][83][84][85][86][87].