The electromagnetic decays of $X(3823)$ as the $\psi_2(1^{3}D_{2})$ state and its radial excited states

We study the electromagnetic (EM) decays of $X(3823)$ as the $\psi_2(1^{3}D_{2})$ state by using the relativistic Bethe-Salpeter method. Our results are $\Gamma[X(3823)\rightarrow\chi_{_{c0}}\gamma]=1.2$ keV, $\Gamma[X(3823)\rightarrow\chi_{_{c1}}\gamma]=265$ keV, $\Gamma[X(3823)\rightarrow\chi_{_{c2}}\gamma]=57$ keV and $\Gamma[X(3823)\rightarrow\eta_{_c}\gamma]=1.3$ keV. The ratio ${\cal B}[X(3823)\rightarrow\chi_{_{c2}}\gamma]/{\cal B}[X(3823)\rightarrow\chi_{_{c1}}\gamma]=0.22$, agrees with the experimental data. Similarly, the EM decay widths of $\psi_{_2}(n^{3}D_{_2})$, $n=2,3$, are predicted, and we find the dominant decays channels are $\psi_{_2}(n^{3}D_{_2})\rightarrow\chi_{_{c1}}(nP)\gamma$, where $n=1,2,3$. The wave function include different partial waves, which means the relativistic effects are considered. We also study the contributions of different partial waves.


I. INTRODUCTION
The bound state of charm and anti-charm quarks (charmonium) is significant in our knowledge of quantum chromodynamics (QCD). It is a double-heavy meson, but not heavy enough that its relativistic corrections are still large [1]. Then the charmonium is crucial to test the validity of phenomenological models, such as the quark potential model, which already foresee a rich and meaningful quarkonium spectra [2]. More charmonia and charmoniumlike states have been discovered experimentally in the last decade, such as the X(3872) [3], X(3915) [4], χ c2 (3930) [5], ψ(4160) [6], Y (4260) [7], Z c (3900) [8] and Z cs (3985) [9], and these new states have stimulated great interests of studies, more details can be found in the review papers [10][11][12][13].
Recently, a new bound state X(3823) has been observed, which is considered to be a good candidate for spin triplet D wave charmonium ψ 2 (1 3 D 2 ). The Belle Collaboration first observed X(3823) in the B → χ c1 γK decay with a statistical significance of 3.8σ [14].
Most existing theoretical predictions of the X(3823) EM decay are provided by nonrelativistic methods. However, we have found the relativistic corrections are large for charmonia, especially for the higher excited states [1,32], so it is necessary to study the properties of X(3823) with different methods especially relativistic one. The Bethe-Salpeter (BS) equation is a relativistic dynamic equation used to describe bound state [33]. Salpeter equation [34] is its instantaneous version which is suitable for the heavy meson, especially the doubleheavy meson. We have solved the complete Salpeter equations for different states, see Refs. [35,36] as examples, and we have improved this method to calculate the transition amplitude [37] with relativistic wave function as input, where the transition formula is also relativistic.
Using this improved BS method, we can get relatively accurate theoretical results, which agree well with the experimental data [38][39][40].
So in this paper, the X(3823) as ψ 2 (1 3 D 2 ) state is studied by the improved BS method, we will focus on the EM decay processes of X(3823). Besides the dominant channels ψ 2 (1 3 D 2 ) → χ c1 γ and ψ 2 (1 3 D 2 ) → χ c2 γ, the radiative decays χ c0 γ and η c γ, whose studies are lacking in the literature, are also calculated. We also provide the results of ψ 2 (n 3 D 2 ) → χ cJ (mP )γ, ψ 2 (n 3 D 2 ) → η c (mS)γ and ψ 2 (n 3 D 2 ) → χ c2 (mF )γ, with n = 2, 3 and m = 1, 2, 3. Where χ c2 (mF ) is the F wave dominant 2 ++ state, mixed with sizable P and D partial waves [41]. This paper is organized as follows. In Sec II, we show theoretical method to calculate the transition matrix amplitude and the form factors as well as the relativistic wave functions of initial and final states. In Sec III, we give the results and compare them with other theoretical predictions and experimental data. Finally, we give the discussion and conclusion.

II. THE THEORETICAL CALCULATIONS
In order to avoid tediousness, we will not introduce the BS equation and Salpeter equation, interested reader can find them in Refs. [33,34] or our previous paper, for example, [35].

A. Transition Amplitude
Take the EM decay X(3823) → χ cJ γ as an example, we show how to use our method to calculate the transition amplitude, which can be written as where ǫ 0 , ǫ 1 and ǫ 2 are the polarization vectors (tensor) of the photon, initial and final mesons, respectively. P , P f and k are the momenta of initial meson, final meson and photon, respectively. Invariant amplitude M ξ consists of two parts, corresponding to the two subgraphs in Figure 1, where photons are emitted from quark and anti-quark, respectively. The amplitude can be written as where χ P (q), χ P f (q f ) are the relativistic BS wave functions for X(3823) and χ cJ , respectively.
q and q f are the internal relative momenta of the initial and final mesons, respectively. p 1 , p 2 , p ′ 1 and p ′ 2 are the momenta of quark and anti-quark in the initial and final mesons, respectively. Q 1 and Q 2 are the electric charges (in unit of e) of quark and anti-quark, respectively. S 1 , S 2 are the propagators for quark and anti-quark.
Since instead of BS equation, the Salpeter equation is solved, where we have used the instantaneous approximation, we need to make the same approximation to the invariant amplitude. Here we only show the amplitude formula we used, interested reader can find the details in Ref. [37]. The amplitude has the following form M + ω 1 + ω 2 ∼ 2M. So the contribution of ϕ ++ is much larger than others. Therefore, to simplify the calculation, the decay amplitude in Eq.(3) can be written as We will compare the decay widths given by Eq. (3) Using Eq.(5), where we integrate internal q ⊥ over the initial and final state wave functions, then obtain the amplitude described using form factors.
(1) For the channel X(3823) → η c ( 1 S 0 )γ, there are two form factors h 1 and h 2 , (2) For X(3823) → χ c0 ( 3 P 0 )γ, there is only one form factor t 1 , where ǫ µ f is the polarization vector of χ c1 . (4) For X(3823) → χ c2 ( 3 P 2 )γ or X(3823) → χ c2 ( 3 F 2 )γ, the amplitude is more complicated, which can be represented by eight form factors g i , where ǫ f,µν is the polarization tensor of χ c2 ( 3 P 2 ), and we have used some abbreviations, If the final state is 3 F 2 state, the definitions of the form factors are same as those for 3 P 2 state. Since the expressions of h i , s i , t i and g i are complex and long, their specific expressions are not given here, we put their detailed description in Appendix B.
The thing to note here is that most of these form factors are not independent. Due to the Ward identity (P ξ − P f,ξ )M ξ i = 0 (i = 1, 2, 3, 4), they are linked by the following constrain conditions: Other form factors such as t 1 , g 5 and g 6 are independent and have no such constraints.
Then, the amplitude square for the EM decay of X(3823) is where, ε (γ) ξ is the polarization vector of the final state photon γ, J is the total angular momentum of the initial state. For the X(3823) → η c ( 1 S 0 )γ decay channel, we have For X(3823) → χ c1 ( 3 P 1 )γ and X(3823) → χ c2 ( 3 P 2 )γ, the modulus square of amplitudes is more complex, and for brevity they are placed in Appendix C for the reader's reference.
Finally, the two-body decay width formulation can be written as where,

D. Decay Widths in Non-relativistic Approximation
Although this article presents a relativistic calculation, we like to give the decay width in the non-relativistic approximation, since the later has simplified formula and may help to see the problem clearly. Using the non-relativistic wave functions in Eqs. (12)(13)(14)(15)(16), we obtain the radiative decay widths of X(3823).
For the X(3823) → η c ( 1 S 0 )γ decay channel, we have where E γ is the energy of emitted photon, q ≡ | q |, θ is the angle between q and P f . In non-relativistic limit, since ω c = m c , wave functions F 1 and A f 1 are related to the original radial wave functions directly, and correspond to the two diagrams in Fig.1, where the photons emitted by quark and anti-quark, respectively.
Then it can be seen that we have already consider the recoil effect in the transition.
In the above equation of the decay width, the representations of the radial wave functions and Therefore, it can be seen from the formula of decay width that this is a M 1 magnetic radiative transition, and a subscript M 1 is marked.
where, subscript M 2 denote the M 2 magnetic radiative transitions. Normalization condition where When giving the upper representation, the normalization condition for the 1 ++ state, where Here, The non-relativistic expression of decay widths in Eqs. (28,29,30,34) can be further simplified. Since in radiative decay, compared with initial meson mass M, the recoil momentum | P f | = E γ ≡ r is usually a small quantity, for example, in the radiative decays of X(3823) to η c , χ c0 , χ c1 and χ c2 , the recoil momenta are 0.746 MeV, 0.386 MeV, 0.298 MeV and 0.256 MeV, respectively. Then the wave functions, for example, can be expanded in a dimensionless quantity r M cos θ. If the first four terms of Taylor expansion are retained, then we have 3 . Then after integrating the angle θ, the lowest order contribution in decay width Γ 1 for

So only even power of r exists in
where we can see that the leading order 2A f 1 does not contribute, which is consistent with the non-relativistic results in Refs. [27,47].
For X(3823) → χ c1 ( 3 P 0 )γ, we find that the contribution of E 1 transition expanded to all orders is zero, which also confirms the results in Refs. [27,47]. Further, only the M 2 transition has contribution, and the lowest order result is The decay widths of X(3823) → χ c1 ( 3 P 1 )γ and X(3823) → χ c2 ( 3 P 2 )γ can be simplified as where we retain the lowest order contribution of the E 1 transition and the lowest cross term between E 1 and M 2 .
From the simplified expression of non-relativistic decay widths, it can be seen that, sition has zero contribution, then its contrition comes from the M 2 transition. While the main contributions of X(3823) → χ c1 ( 3 P 1 )γ and X(3823) → χ c2 ( 3 P 2 )γ come from the E 1 transition, so we conclude that the decay widths of X(3823) → χ c1 ( 3 P 1 )γ and X(3823) →

A. Masses
In our calculation, some model-dependent parameters have been used, for example, the mass of the charm quark is fixed at m c = 1.62 GeV [1]. Since V 0 in the kernel originates from QCD non-perturbative effects, its value is to account the states with J P C , so we fix it by fitting the masses of the ground states. Thus the parameter V 0 vary with J P C . And we vary the free parameter V 0 [43] to fit the mass of the ground state. For example, M3 D 2 (1D) = 3.823 GeV [44] is actually not our prediction, but an input, while those of the first and second radial excited states are our predictions, For other charmonia, we have calculated the mass spectrum in Ref. [43]. For example, the masses of some highly excited states are predicted as, It can be seen from Ref. [43], most of our predictions about the mass spectrum consist well with experimental data, especially the case of bottomonium. However, there are still some states whose theoretical masses are different from the experimental data. For example, our prediction of M χ c1 (2P ) = 3.929 GeV [43], while the data is M X(3872) = 3.872 GeV, another is the mass of η c (2S), our prediction 3.576 GeV is lower than data 3.636 GeV. To see the difference in decays, for these two states, we use the theoretical mass as well as the experimental data to calculate the decay width, and give two groups of results.

B. Wave functions
We consider X(3823) as the 2 −− ground state ψ 2 (1 3 D 2 ). From the Eq. state has the property f 1 ≃ f 2 , this is correct, since in a non-relativistic limit f 1 = f 2 .
We also show the numerical results of the radial wave functions for excited states ψ 2 (2 3 D 2 ) and ψ 2 (3 3 D 2 ) in Fig. 2. In general, from the number of nodes of the wave function, we can tell whether the state is a ground state or an excited one. For example, the radial wave function of the ground state has no node, while that of the first excited state has one node and the second excited state has two nodes, etc..
For the S wave states η c (nS) and the P wave states χ c0 (nP ), χ c1 (nP ) and χ c2 (nP ), we have shown the 1S, 2S, 1P and 2P wave functions in previous paper [43], but since the theoretical masses of 2 1 S 0 and 2 3 P 1 states are a little different from data, which make the wave functions a little difference from the old ones in Ref. [43], we like to show the wave functions for all the excited S and P states one more time in this paper. In Fig. 3 From the above results, it can be seen that the contributions of the positive energy wave functions ϕ ++ to the decay width are dominant, and the contributions of other terms are about 2.4% and 3.0% for the two channels. Therefore, in the following calculation, for simplicity, the formula Eq.(5) of decay amplitude is adopted.
We can see the dominant decay channel is X(3823) → χ c1 (1P )γ, and its decay width is much larger than others.
For comparison, we show our results and other model predictions [20,[25][26][27][28][29] in Table   I. Where, RE represents a relativistic method, NR the non-relativistic method, GI is the relativistic Godfrey-Isgur model, RV and RS represent the relativistic method using vector and scalar potential, respectively, while RV S the mixture of them. In our results, the value in parentheses is calculated using the experimental mass. It can be seen that the decay width is insensitive to the mass of particle. We can also see that our results of X(3823) → χ {c1, c2} (1P )γ are close to those of relativistic method RE in Refs. [20,25] and relativistic GI model in Ref. [27].
In Table I, we also show the ratio of the decay rate X(3823) → χ c2 γ to that of X(3823) → χ c1 γ, our result is This result and all other theoretical predictions in Table I are within the range of current experimental value 0.28 +0.14 −0.11 ± 0.02 [31]. The consistence shows that this ratio cancels some model dependent uncertainties, and it is more reflective of the true value. So we also give the ratio The result is within the experimental limit < 0.24 detected by BESIII [31]. The channel of X(3823) → χ c0 γ was also calculated in Ref [45], and they gave a decay width of 1.42 keV, witch is a little bigger than ours, while their ratio Our predictions for the EM decay widths of the excited state ψ 2 (2D) and other theoretical results are shown in Table II. The dominant decay channel is ψ 2 (2D) → χ c1 (2P )γ, which is close to those of the relativistic GI model in Ref. [27] and non-relativistic potential model NR 1 in Ref. [29].
If instead of using the theoretical mass of χ c1 (2P ), the experimental value is used, then the decay width for ψ 2 (2D) → χ c1 (2P ) becomes to 230 keV, see the value in parenthesis in Table II. Combined with the result of ψ 2 (2D) → η c (2S) in Table II, two groups of values are also given, we confirm the previous conclusion that the radiative electromagnetic decay width is not very sensitive to the mass.
The channels ψ 2 (2D) → χ c1 (1P )γ and ψ 2 (2D) → χ c2 (2P )γ also have sizable contributions, so we also calculate their decay ratios to the channel ψ 2 (2D) → χ c1 (2P )γ, and list and ψ 2 (2D) → η c γ. [27] [45] [29] ours  Table II. We can see that, unlike the case of ψ 2 (1D), the ratios of ψ 2 (2D) are much different from model to model. The reason may due to the relativistic corrections being not included or fully considered, because in previous paper [41], we have pointed out that higher excited states have much larger relativistic corrections than those of lower excited and ground states. This conclusion has been confirmed in the weak transition process [32].
The predictions for the EM decay width of excited state ψ 2 (3D) are shown in Table III.

F. Contributions of different partial waves
In a previous work [41], we point out that, in a complete relativistic method, the relativistic wave function for a J P state is not a pure wave. This conclusion is also valid for the charmonium. For the X(3823) as the 2 −− state ψ 2 (1 3 D 2 ), besides the main D wave, it also includes a small part of F wave; for the η c (1S), it is dominated by S wave with a small amount of P partial wave, while for the χ c0 (1P ) state, as a P wave dominant state, it includes a small component of S wave, etc, see the details in Sec.II.B.
In this subsection, we study the contributions of different partial waves of the initial and finial mesons to the decay width. The results are shown in Table IV∼ IX, where 'complete' means the complete or whole wave function is used, 'S wave' means only the S partial wave 0.14 0.14 ∼ 0 has contribution and other partial waves are deleted. From these tables, we can see that in all the decays, the main contribution of 2 −− state ψ 2 comes from its dominant partial wave, namely D wave, which is also its non-relativistic term, and its relativistic correction term, namely F partial wave, has a relatively small contribution. Table IV shows the case of ψ 2 (1D) → η c (1S)γ. We know that η c (1S) is a S-wave dominant state, which only contains a small amount of P partial wave. But from Table IV, we can see that the contribution of D wave → S wave transition is suppressed, indicates that the major contribution of this decay process is due to relativistic effect (dominant by D wave → P wave transition).    suppressed. And in the non-relativistic limit, the contribution of all these P wave terms is zero. So for the EM decay ψ 2 (1D) → χ c0 (1P )γ, the contribution of S wave which provides the relativistic correction is greater than the that of P wave. Table VI show the result of ψ 2 (1D) → χ c1 (1P )γ. We can see that, the main contribution of the final state come from the dominant P partial wave which provides the non-relativistic result, and the relativistic correction (D partial wave in 1 ++ state) contribute very small.
The form factors for this decay are shown in Appendix, but they are very complicated, we will not discuss the details.
Tables VII, VIII and IX show the results of ψ 2 (1D) → χ c2 (1P )γ, ψ 2 (2D) → χ c2 (1P )γ and ψ 2 (2D) → χ c2 (1F )γ, respectively, where three final mesons are all 2 ++ states. The first two are 1P wave dominant states combined with small D and F partial waves, the third one is 1F wave dominant state but combined with sizable P and D partial waves [41]. Tables VII and VIII show us that compared with the dominant P wave, the contributions of D and F partial waves in 1P dominant final state are small, and the nodal structure in the wave function of ψ 2 (2D) results in the smaller decay width of ψ 2 (2D) → χ c2 (1P )γ compared with Table IX, we can see that besides the large contribution of F wave in the 1F dominant state, the contribution of D partial wave is also large, but those of P wave are suppressed.

G. Discussion and Conclusion
In a previous paper [23], we have estimated the annihilation decay (including ggg and ggγ final states) width of X(3823), which is about 9.8 keV. From Eichten's work [25], we can get the decay width Γ[ψ 2 ( 3 D 2 ) → J/ψππ] ≈ 45 keV. So the total decay width of X(3823) can be estimated as, Therefore, the process X(3823) → χ c1 γ whose partial width is estimated as 265 keV, is the dominant decay channel of X(3823). The detection of this channel in experiment is crucial to confirm X(3823) being the state ψ 2 ( 3 D 2 ).
We find for ψ 2 (n 3 D 2 ), the dominant EM decay channel is ψ 2 (n 3 D 2 ) → χ c1 (nP )γ. Our where F = F (q 2 ⊥ , q 2 f ⊥ ), and we have used the following abbreviations The coefficient f ij are calculated as where θ is the angle between q and P f , we have defined q ≡ | q | and r ≡ | P f |.

Appendix B: Form factors
Here, we will give the detailed expression of the form factors in the corresponding decay channel. For the decay channel X(3823) → η c ( 1 S 0 )γ, the form factors h 1 and h 2 are where α f = α 1 = α 2 = 0.5.