Search for $C=+$ charmonium and XYZ states in $e^+e^-\to \gamma+ H$ at BESIII

Within the framework of nonrelativistic quantum chromodynamics, we study the production of $C=+$ charmonium states $H$ in $e^+e^-\to \gamma~+~H$ at BESIII with $H=\eta_c(nS)$ (n=1, 2, 3, and 4), $\chi_{cJ}(nP)$ (n=1, 2, and 3), and $^1D_2(nD)$ (n=1 and 2). The radiative and relativistic corrections are calculated to next-to-leading order for $S$ and $P$ wave states. We then argue that the search for $C=+$ $XYZ$ states such as $X(3872)$, $X(3940)$, $X(4160)$, and $X(4350)$ in $e^+e^-\to \gamma~+~H$ at BESIII may help clarify the nature of these states. BESIII can search $XYZ$ states through two body process $e^+e^-\to \gamma H$, where $H$ decay to $J/\psi \pi^+\pi^-$, $J/\psi \phi$, or $D \bar D$. This result may be useful in identifying the nature of $C=+$ $XYZ$ states. For completeness, the production of $C=+$ charmonium in $e^+e^-\to \gamma +~H$ at B factories is also discussed.

We calculate the production of C = + charmonium at e + e − annihilation at BESIII to test the nature of C = + XY Z states. Our paper is organized as follows. The calculation framework is given in Sec. 2. The numerical results of the cross-sections of C = + charmonium are discussed in Sec. 3. A discussion of X(3872) and other C = + XY Z states is given in Sec. 4. The summary is given in Sec. 5.

The frame of the calculation
In the NRQCD factorization framework, we can express the amplitude in the rest frame of H as [28,30,31] A(e − (k 1 )e + (k 2 ) → H cc ( 2S+1 L J )(2p 1 ) + γ) = LzSz s 1 s 2 jk where 3j;3k | 1 = δ jk / √ N c , s 1 ; s 2 | SS z is the color Clebsch-Gordan coefficient for cc pairs projecting out appropriate bound states, and s 1 ; is the quark level scattering amplitude. In the rest frame of H, q = (0, q), and p 1 = ( m 2 c + q 2 , 0, 0, 0). Φ H cc ( q) is the cc component wave function of hadron H in momentum space. For v 2 = q 2 /m 2 c ≪ 1 [50], we can expand Eq.(2.1) with v 2 : Here A(q) = A e − (k 1 )e + (k 2 ) → c s 1 j (p 1 + q) +c s 2 k (p 1 − q) + γ(k) . We consider the Fourier transform between the momentum space and position space as: [50,94], Here Z H cc is the possibility of cc component in hadron H. R cc (0) is the radial Schrodinger wave function at the origin. R l cc (0) is the derivative of the radial Schrodinger wave function at the origin is also written as long-distance matrix elements (LDMEs) as discussed in Ref. [94]. For example, We calculated the relativistic corrections for the S wave and P wave states and obtain two LDMEs for η c , four LDMEs for χ cJ , and one LDMEs for 1 D 2 states. To simplify the discussion of the numerical result, we assumed that Then there is only one LDME for S wave, P wave, and D wave respectively. More details can be found in Ref. [94]. The relativistic correction K factor is where r = 4m 2 c /s. − rv 2 1−r is the relativistic correction of the phase space. If we select r → 0, the K v 2 factor is consistent with the K factor at large p T in Ref. [94].
We can obtain a similar amplitude for the DD component in the molecule model. We can estimate the off-resonance amplitude of e + e − → H + γ from the DD component. The parton-level amplitudes may be compared with the hadron-level amplitudes: with the S wave l = 0 and P wave l = 1 for the binding energies of cc and DD are several hundreds of MeV and several MeV, respectively. If Z H cc ∼ Z H DD , we can consider the cc contributions only. In the numerical calculation, we consider the charm quark mass as half of the hadron mass consistent with the physics phase space. With a large charm quark mass, the wave functions at the origin are identified as the Cornell potential result in Ref. [96]. The sellected parameters are as follows: The wave functions at origin for higher states are estimated as In the numerical result, "σ LO " is the LO cross-section, "σ v 2 " is the cross-section including the LO and the relativistic correction, "σ αs " is the cross-section including the LO and the radiative correction, and "σ αs,v 2 " is the cross-section including the LO, the relativistic correction, and the radiative correction. In addition, "LO" is the LO cross-section, "RC" is the relativistic correction, "QCD" is the radiative correction, and "Total" is the cross-section including the LO, the relativistic correction, and the radiative correction.
For the LO, the cross-section is O(α 0 s v 0 ). As α s = 0.23 ± 0.03 and v 2 = 0.23 ± 0.03 are reasonable estimates, we can estimate that the uncertainty of the numerical result from α s and v 2 is < 10%.

Pure C = + charmonium states
We can estimate the cross-sections for pure C = + charmonium states H in e + e − → γ + H at BESIII with H = η c (nS) (n=1, 2, 3, and 4), χ cJ (nP ) (n=1, 2, and 3), and 1 D 2 (nD) (n=1 and 2). The mass of the lower states can be found in Ref. [24], and the mass of the higher states is selected from Ref. [17]. Σ e e ΓΗ c2 nD fb The cross-section of e + e − → η c + γ as a function of √ s is shown in Fig.1. The crosssections of e + e − → η c2 (1D, 2D) + γ as a function of √ s are shown in Fig.2. The numerical results for nS with n = 1, 2, 3, 4 and nD with n = 1, 2 are listed in Table 2. We determined Table 2. The cross-sections of e + e − → H + γ for η c (nS) with n = 1, 2, 3, 4 and η c2 (nD) for n = 1, 2 charmonium states in fb. The labels LO, RC, QCD and Total are defined near the end of Section 2. The mass of η c (3S), η c (4S), η c2 (1D), and η c2 (2D) are selected from Ref. [17]. The other mass can be found in Ref. [ that the radiative and relativistic corrections are negative and large for η c (nS), respectively. The LO cross-sections for η c2 (1D, 2D) is very small at BESIII; hence, the high order corrections are ignored.
The cross-sections of e + e − → χ cJ + γ as a function of √ s are shown in Fig.3, Fig.4, and  Table 3, Table 4, and Table 5 for J = 0, 1, 2, respectively. We determined that the QCD corrections are large but negative and the relativistic corrections are large and positive. Hence, many P wave states can be searched at BESIII. The NRQCD requires that the energy of photon at the center of the mass frame of e + e − be larger than Λ QCD ∼ 300 MeV ∼ m c v 2 . Although this process is a QED process, the prediction is not reliable and only a reference value if this requirement is not satisfied. If we replace photon with gluon, the soft photon contributions correspond to the long-distance color octet contributions [31,50]. Σ e e Χ c0 Γ fb Figure 3. The cross-sections of e + e − → χ c0 + γ as a function of √ s in fb. The cross-section "σ LO ", "σ v 2 ", "σ αs ", and "σ αs,v 2 " are defined near the end of Section 2. Table 3. The cross-sections of e + e − → χ c0 (nP )+γ with n = 1, 2, 3 in fb. The labels LO, RC, QCD and Total are defined near the end of Section 2. The χ c0 (2P ) is considreed as X(3915)(X(3945)/Y (3940)) [1,33]. The mass of χ c0 (3P ) are selected from Ref. [17]. The other mass can be found in Ref. [ Σ e e Χ c1 Γ fb Figure 4. The cross-sections of e + e − → χ c1 + γ as a function of √ s in fb. The cross-section "σ LO ", "σ v 2 ", "σ αs ", and "σ αs,v 2 " are defined near the end of Section 2.
To clarify the nature of X(3872), we also give the numerical calculation of e + e − → Σ e e X 3872 Γ J ΨΠΠ Γ Figure 6. The cross-sections of e + e − → χ c2 + γ as a function of √ s in fb. The cross-section "σ LO ", "σ v 2 ", "σ αs ", and "σ αs,v 2 " are defined near the end of Section 2. The uncertainty bind of σ αs,v 2 is from the uncertainty of k = 0.018 ± 0.04. The cross-sections as a function of √ s is shown in Fig.6. Many 1 −− states with M H < 5 GeV are also observed. We can predict the cross-sections from continuous contributions at this point, and the result is listed in Table 6. We ignore the 1 −− resonances contributions here. We emphasize that if we select √ s = 4.009GeV, the energy of photon E γ = 134 MeV and smaller than Λ QCD ∼ m c v 2 ∼ 300 MeV. Hence, NRQCD cannot accurately predict the cross-sections with a soft photon with √ s = 4.009GeV [50]. If √ s = 4.160GeV, the energy of photon is E γ = 270MeV. Although this process is a QED process, the prediction is not reliable and only a reference value [31]. We determined that the NRQCD prediction of the continuous contributions can be compared with the BESIII data of the cross-sections of e + e − → γX(3872) [46,47] in Eq.(1.1). When we only considered the continuum production, the resonance contributions can be estimated as that: We take into account only one resonance here and ignore continuum and other resonances here.
If we ignore the interference between one resonance and continuum and other resonances, the gamma energy dependence of the Γ[Res → γX], and DD contributions of decay of Res → γX, we can estimate the resonance contributions. With X(3872) considered as 2P states, the largest decay widths are ψ(4040) and ψ(4160), which are considered as the mixing of ψ(3S) and ψ(2D) [97,98]. The Γ[Res → γX] for other states will be less than 1 keV [98],   [43]. η c and χ c0 are recoiled with J/ψ, but χ c1 and χ c2 are missed [43]. The theoretical predictions are consistent with the experimental data [61,69,99,100]. So there should be large η c (nS) and χ c0 (nP ) component in X(3940) and X(4160), respectively. The mass of η c (3S) and χ c0 (3P ) are predicted as 3994 MeV and 4130 MeV respectively [17]. Compared with Table 2 and Table  3, we can found that the cross-sections of η c (3S) is small even negative at √ s < 5 GeV. But χ c0 (3P ) is large. The cross-sections as a function of √ s is shown in Fig 7. Here Z X cc ≤ 1 is the possibility of η c (3S) and χ c0 (3P ) component in X(3940) and X(4160) respectively. The BESIII collaboration can search X(3940) and X(4160) in the process e + e − → γ + X(DD). The result may be useful in identifying the nature of X(3940) and X(4160).

X(4350)
X(4350) are found in γγ → H → φJ/ψ at B factories [45]. And J P C is 0 ++ or 2 ++ . So there should be large χ c0 (nP ) or χ c2 (nP ) component in X(4350). In Ref. [17], The mass of χ c2 (3P ) is 4208 MeV. Ignore more detail of the mass, we considered it as χ c0 (nS) or χ c2 (nP ), the wave function at origin are estimated as The cross-sections of e + e − → X(4350) + γ as a function of √ s is show in Fig.8. Here Z X cc is the possibility of χ c0 (nP ) or χ c2 (nP ) component in X(4350). The cross-section for χ c2 (nP ) is larger than χ c0 (nP ) by a factor of 6. The result may be useful in identifying the nature of X(4350).

Summary and discussion
While BESIII and Belle have collected a large amount of data, some final states may be searched by the experimentalists. We can estimate the possible event number at BESIII and Belle. The possible event number is Σ X 4350 Γ Z c c x fb Figure 8. The cross-sections of e + e − → X(4350) + γ as a function of √ s in fb. The cross-section "σ LO ", "σ v 2 ", "σ αs ", and "σ αs,v 2 " are defined near the end of Section 2. And Z X cc is the possibility of χ c0 (nP ) or χ c2 (nP ) component in X(4350).