Decay widthes of $^3 P_J$ charmonium to $DD,DD^*,D^*D^*$ and corresponding mass shifts of $^3 P_J$ charmonium

In this work, we calculate the amplitudes of the processes $c\bar c({^3P_J}) \rightarrow DD,DD^*, D^*D^* \rightarrow c\bar c({^3P_J})$ in the leading order of the nonrelativistic expansion. The imaginary parts of the amplitudes are corresponding to the branch decay widthes of the charmonium $c\bar c({^3P_J}) \rightarrow DD,DD^*, D^*D^*$ and the real parts are corresponding to the mass shifts of the charmonium $c\bar c({^3P_J})$ due to these decay channels. After absorbing the polynomial contributions which are pure real and include the UV divergences, the ratios between the branch decay widthes and the corresponding mass shifts are only dependent on the center-of-mass energy. We find the decay widthes and the mass shifts of the $^3P_2$ states are exact zero in the leading order. The ratios between the branch decay widthes and the mass shifts for the $^3P_0, {^3P_1}$ states are larger than 5 when the center-of-mass energy is above the $DD,DD^*, D^*D^*$ threshold. The dependence of the mass shifts on the center-of-mass energy is nontrivial especially when the center-of-mass energy is below the threshold. The analytic results can be extended to the $b$ quark sector directly.

of Belle [4], CDF [5], D0 [6], BABAR [7], Cleo-C [8], LHCb [9], BES [10], and CMS [11]. These charmonium-like states cannot be well understood in the traditional quark model and their masses usually lie above the open charm threshold where some new decay modes are opened. In the previous study [12], we studied the mass shifts of 1 S 0 and 3 P J heavy quarkonia due to the transition qq → 2g → qq. Physically, when the masses of the states lie above the threshold of D or D * pairs, the transitions cc to these mesons' pairs are opened. It is natural that these opened channels not only result in the visible branch decay widthes but also give contributions to the mass shifts of the corresponding charmonium. When the masses of the charmonium lie about the threshold of the meson pairs, one can expect that the nonrelativistic expansion is available, which means that one can take the mesons D, D * like the heavy quark in the nonrelativistic QCD to construct the effective nonrelativistic interactions order by order. In this work, we follow this spirit to calculate the amplitudes of cc( 3 P J ) → DD, DD * , D * D * → cc( 3 P J ) in the leading order of non-relativistic expansion. The imaginary parts of the results are corresponding to the branch decay widthes which can be used to determine the effective coupling constants. Furthermore, if these annihilation interactions are much smaller than the binding interaction, then the real parts can be used to estimate the corresponding mass shifts.
We organize the paper as follow. In Sec. II we describe the basic frame to calculate the amplitudes of cc( 3 P J ) → DD, DD * , D * D * →c( 3 P J ) in the leading order of nonrelativistic expansion, in Sec. III we give the analytic results for the amplitudes in the leading order of nonrelativistic expansion, in Sec. IV, we present some numerical results to show some properties in detail.

II. BASIC FORMULA
When the mass of the charmonium is about 2m D or 2m D * with m D,D * being the masses of the D, D * mesons, the three-momenta of the c quarks and the mesons in the decay channels cc( 3 P J ) → DD, DD * , D * D * are much smaller than c quarks' mass m c or m D,D * . In this case, one can take m c ≈ m D ≈ m D * as the large scale comparing with Λ QCD and expand the interaction on the small variables | ⇀ q|/m c with ⇀ q the three-momenta of the c quarks and the mesons. This nonrelativistic expansion is similar with the spirit of NRQCD where the contact four point interactions are introduced. Different from NRQCD, now there is no hard gluon in the decay channels cc( 3 P J ) → DD, DD * , D * D * , but only nonrelativistic heavy quarks and heavy mesons. This means that there are only contact interactions between the c quarks and the D, D * mesons. In the leading order of | ⇀ q|/m c , naively the most general interactions with C, P, T invariance can be written as follows: where ψ, φ D , A µ D * are the fields of the c quark, the D meson, and the D * meson, respectively. Here we do not assume that there is spin asymmetry between the D and D * mesons since the dynamics of the light quarks insider the D and D * mesons may break the spin symmetry strongly. This means that the couplings g a,b,c are independent.
By these interactions, the Feynman diagrams for the amplitudes of cc( 3 P J ) → DD, DD * , D * D * → cc( 3 P J ) in the leading order are showed in Fig. 1 Similar with any effective theory, usually the contract interactions are needed to absorb the UV divergence in the loop diagrams. To absorb the UV divergence in Fig. 1(a, b, c), the following contact interactions are needed: where the higher orders of the interactions are also kept. We want to point out that we just write down such contact interactions here to show the exact cancellation of the UV divergence and the polynomial contributions. In the practical calculation, one can get the same final results even without knowing the form of the contact interactions. The Feynmann diagram for the contribution due to these contact interactions is showed in Fig. 1(d).
In the center of mass frame, we choose the four external momenta as follows: For simplicity we define P def = ( √ s, 0, 0, 0) and use the instantaneous approximation for q i,f which means that we assume q i = (0, q i ) and q f = (0, q f ), where we use the bold formatting to refer to the three momentum here and in the following.
To project the cc pairs to the 3 P J states we use the project matrices in the on-shell case [13][14][15] which are defined as follows: where the Clebsch-Gordan coefficients are the standard ones as in Ref. [14], and the Dirac spinors are normalized as u + u = ν + ν = 1, whose definitions are expressed as with Finally the project matrices can be written as where and N i,f are the normalized global factors which can be expressed as follows in the nonrelativistic In principle the form of the project matrix for a bounded cc pair should be deduced from the Bethe-Salpeter wave funciton or similar Lorentz covariant matrix element, while in the ultra nonrelativistic limit the above expressions are expected to be correct.
In the leading order of nonrelativistic expansion, the structure of a meson H( 3 P J ) can be expressed as follow: where N c = 3 and φ(|p|) is the wave function of H( 3 P J ) in the momentum space which is defined with the normalization condition Combining the structure of H( 3 P J ) and the project matrices, the expression for the amplitudes in the leading order can be expressed as where the index (X) refers to (a, b, c, d) which are corresponding to the contributions from the diagrams (a), (b), (c) and (d) showed in Fig. 1, respectively. G (X) ( 3 P J ) are expressed as with where is the color factor, and and the propagators of the pseudoscalar S and the vector D µν are defined as In the practical calculation, the package FeynCalc [16] is used to do the trace in the d dimension.
The packages FIESTA [17] and PackageX [18] are independently used to do the loop integration for double check. After the loop integrations, G (X) (s i , s f ) can be expressed in the following form: where C (X) i can be expressed as To get the coefficients G (X) ( 3 P J ), usually the sums of the spins and the integrations of the angles are calculated independently to simplify the expressions [19]. In our calculation, we directly calculate the sums of the spins and the integrations of the angles together after getting the expres- in . This method is more efficient and has been used in our previous work [12]. The relevant expressions are listed in the Appendix.

III. THE ENERGY SHIFT OF 3 P J IN THE LEADING ORDER
We expand G (X) ( 3 P J ) on |q i |, |q f | to order 1 as following forms: Here we want to emphasis that the contributions G (d) ( 3 P J ) are used to absorb the UV divergences in G (a+b+c) ( 3 P J ) and give no contributions to the decay widthes of 3 P J states. The finite parts of the contributions G (d) ( 3 P J ) are arbitrary. Actually, they not only absorb the UV divergences but also absorb the polynomial contributions in G (a+b+c) ( 3 P J ). This situation is a little different from the results in the cc( 3 P J ) → 2g → cc( 3 P J ) cases where there are no any contact interactions in the original QCD interaction. The important point is that these absorptions are universal and independent on the processes, and we discuss the details in the following subsection.
A. The energy shift of 3 P 0 state After the loop integration, the sum of the spins, the integration of the angles, and the Taylor expansion, we get the following results in the 3 P 0 channel.
where c

4(4 +
0,poly = 256 π 3 (g 10 + g 11 s + g 12 s 2 ), An important property of the two contributions c (a,c) 0,poly is that they can be absorbed by the contact interactions L c 1 independently. These contact interactions are independent and give no contributions to the decay widthes of the charmonium. This means that their effects can be absorbed by the models which are used to calculate the energy spectrum and do not include the annihilation effects. Here we are only interested in the mass shifts due to the decay modes, then we only focus on the contributions including the imaginary parts due to the loop calculation and neglect the terms c (a,c) 0,poly . The choices of g 10,11,12 which can cancel all the polynomial contributions in c (a,c) 0 can be got directly.
From Eq. (20), one can easily get the imaginary parts as follows: Matching the amplitude with the corresponding amplitude in quantum mechanism with a perturbativel potential, one has Finally the decay widthes of 3 P 0 to DD and D * D * in the leading order are expressed as follows: where we have used the relation φ(p)p 2n+3 dp = (−1) n 2n + 3 4π R The corresponding mass shifts labeled as ∆m( 3 P 0 ) are expressed as where c In the 3 P 1 channel, we have the following results The polynomial terms are expressed as with At first glance, this property is very different from that in the 3 P 0 channel due to the nonzero values of c 1,−2 and c 1,−1 which seems is un-physical. While actually when taking the nonrelativistic approximation m D ≈ m D * , one has c 1;−2 , c 1;−2 ≈ 0, this means that there contributions are very small in the nonrelativistic approximation and can be neglected. The numerical calculations also shows such property and we neglect these two terms.
Similarly, the term c In the leading order, the decay width of 3 P 1 to DD * , are expressed as and the corresponding mass shift labeled as ∆m( 3 P 1 ) is expressed as where c 1,poly .
C. The energy shift of 3 P 2 state These results means that the decay widthes Γ( 3 P 2 → DD, DD * , D * D * ) are exact zero and there are no mass shifts for 3 P 2 states in the leading order. This result is a strong property which can be tested by the experiments and be used to judge whether a state is pure 3 P 2 heavy quarkonium or not.
Comparing our results with those results given by the 3 P 0 model in Ref. [20], one can find that both the two methods give the zero results for cc( 3 P 0 ) → DD and cc( 3 P 1 ) → DD * . But in Ref. [20], the contributions cc (  1 ] which are represented by the solid black curves are always negative. This means that after considering the annihilation effects, the masses of the 3 P 0,1 states move up and the masses of 3 P 2 states do not move. (2) When √ s is on the threshold of DD, DD * or D * D * the corresponding mass shifts are exact zero.
(3) When √ s is above the threshold, the mass shifts are much smaller than the corresponding decay widthes, the largest mass shift is about 1/5 of the corresponding decay width when √ s ≈ 4.5 GeV which is much larger than the threshold. This property gives a strong constrain on the mass shifts to all the 3 P 0,1 states. that the corresponding mass shifts are non-linear and can not be absorbed by some constants.
Experimentally, up to now there are still no definite results for the branch decay widthes Γ( 3 P 0,1 → DD, DD * , D * D * ) [21], this makes it difficult to determine the mass shifts certainly.
The experiments reported that the decay widthes Γ(X(3915), χ c2 (3930) → DD, DD * , D * D * ) are seen. By our calculation, we expect that the decay widthes Γ( 3 P 2 → DD, DD * , D * D * ) are zero in the leading order which suggests that the decay widthes Γ( 3 P 2 → DD, DD * , D * D * ) should be much smaller than Γ( 3 P 0 → DD, D * D * ) and Γ( 3 P 1 → DD * ). A relative larger decay widthes of a resonance to DD, DD * , D * D * suggest that it maybe is not a pure cc( 3 P J ) state. These properties are more reliable in the b quark part and can be tested by the further precise experiments.
Furthermore, the similar discussion can be extended to the S wave states and compared with the similar studies in Ref. [22]. In summary, the nonrelativistic asymptotic behavior of the transitions cc( 3 P J ) → DD, DD * , D * D * → cc( 3 P J ) with J = 0, 1, 2 are discussed. We find that the decay widthes Γ( 3 P 0 → DD * ), Γ( 3 P 1 → DD, D * D * ) and Γ( 3 P 2 → DD, DD * , D * D * ) are exact zero in the leading order of nonrelativistic expansion. For other channels, the ratios between the branch decay widthes and the mass shifts are larger than 5 when the center-of-mass energy is above the threshold. When below the threshold, the mass shifts are dependent on the center-of-mass energy nontrivially and can not be absorbed by a constant.

V. ACKNOWLEDGEMENTS
The author Hai-Qing Zhou would like to thank Zhi-Yong Zhou and Dian-Yong Chen for their helpful discussion. This work was supported by the National Natural Science Foundation of China

VI. APPENDIX: THE FIESTA INTEGRATIONS
We define the following functions to refer to the results after summing the spins and integrating the angles: where X are some functions dependent onq i ,q f , ǫ p (s i ), and ǫ * p (s f ) withq i,f def = q i,f /|q i,f |, P (J, X, n) are not dependent on J z . When J = 0, 1, 2 and n = 0, 1, we have P (J, ǫ p (s i ) · ǫ * p (s f ), 1) = 4π 3 , P (0, ǫ p (s i ) ·q i ǫ * p (s f ) ·q f , 0) = 4π, and others are zero.