Coupled-Channel-Induced $S-D$ mixing of Charmonia and Possible Assignments for $Y(4260)$ and $Y(4360)$

We calculate the $S-D$ mixing of the $J^{PC}=1^{--}$ charmonium states induced by the coupled-channel effects. The mass shifts, the open-charm decay widths and the di-electron decay widths are evaluated. We find that it is possible to assign $Y(4260)$ and $Y(4360)$ as the mixtures of $4S$ and $3D$ states.


A. Coupled-Channel Dynamics
In naive quark models, mesons are bound states of a quark and an anti-quark bounded by a QCD-inspired potential. The masses and wave functions can be obtained by solving the eigenvalue problem formally expressed as: where the M 0 and |ψ n are usually referred as the bare mass and the bare state. The Hamilton H 0 only includes the interaction described by the potential binding the quark-anti-quark pair. In coupled-channel formulism, the Hilbert space under consideration is enlarged to include the continuum states which the bare states can decay into. We are focusing on vector charmona decaying into two-body open-charm channels, and in this situation, the Hamilton can be formally written as [23]: where H BC is the free Hamilton (by "free", we mean that the interactions between the two mesons are neglected) for the continuum states with two particles B and C: where The quark-pair creation Hamilton H QPC induces the decays. With the presence of the non-diagonal elements in the Hamilton, the physical states become mixtures of all bare states: |ψ ′ n = i a ni |ψ i + BC dk c n,BC |B, C; P B , P C , where dk denotes an integration over all three-momenta of B and C. The problem now turns into the eigenvalue problem [20,21]: where Π mn (M ) = BC dk m|H † QPC |B, C; P B , P C B, C; P B , P C |H QPC |n M − E BC + iǫ .
Above the open-charm threshold, Π mn develops an imaginary part, so in general, the equation allows complex value solutions. In this case, the physical mass and the width of a physical state |ψ ′ n are related to the corresponding eigenvalue M n as M phys.,n = ReM n , For real M n 's, of cause, M n is just the physical mass of the corresponding state, and these M n 's should satisfy the condition M n < 2M D , where 2M D denotes the open-charm threshold.
In vector charmonium sector, the mass of nD state is closest to the (n + 1)S state. So it is expected that these two states should mix each other the most, which is verified in Ref. [22] and taken as granted by many authors [19,27]. In this work, we follow these researches and only take into account the mixing between the nD and the (n + 1)S states, which means Π nD,(n+1)S and Π (n+1)S,nD are the only non-vanishing non-diagonal elements. With this simplification, Eq. (6) decomposes into several sectors. For 1S sector, we have For (n + 1)S − nD sector, we have where the principle quantum numbers in front of S and D are omitted. Since we are focusing on the S − D mixing in this work, and the continuum components in the physical states are irrelevant, we drop off these components in the physical states as Refs. [22,27] did. So a mixed physical state is expressed as with coefficients a S and a D satisfying |a S | 2 + |a D | 2 = 1.

B. Instantaneous Bethe-Salpeter Equation and the 3 P0 Model
Now the remaining problem is to solve the naive quark model to obtain the bare masses and the wave functions, and to calculate the open-charm decay amplitudes B, C; P B , P C |H QPC |n . To this end, we make use of the instantaneous Bethe-Salpeter equation and the 3 P 0 model. The instantaneous BS equation, also known as the Salpeter equation, is a well developed relativistic two-body bound state equation, and is very suitable to apply on the heavy quarkonium system. The instantaneous BS wave function ϕ(q µ P ⊥ ) may be decomposed into positive and negative energy parts: ϕ = ϕ ++ + ϕ −− , where q P ⊥ is the perpendicular part of relative momentum q. For any momentum l µ , we have where P µ is the meson's momentum. The Salpeter equation then takes the form as coupled equations for ϕ ++ and ϕ −− [28]: where with j = 1 for quark and j = 2 for anti-quark. p 1 is the quark momentum and p 2 is the anti-quark momentum.
) now becomes the QCD-inspired potential between quark and anti-quark. In our model, we use the Cornell potential with screening effect. The Cornell potential without screening effect can be written as: The screening effect is introduced by modify the scalar part V s to be [15,29]: Now the potential at sufficient long range, i.e. for r ≫ 1 α , is suppressed (becomes more and more flat), while for r ≪ 1 α , it retains the linear form. So 1/α sets the distance where the screening effect becomes important. Transforming it into momentum space, we obtain: where q is the three-momentum of q µ P ⊥ , i.e., q µ P ⊥ = (0, q) in the meson's rest frame. A constant V 0 has been added into V s to adjust the ground state energy as usual. An infrared cut is introduced in the vector part of the potential to avoid the infrared divergence, which we set it equaling to α because the results are insensitive to this approximation [29]. α s (q) = 12π 33−2N f 1 log(a+q 2 /Λ 2 QCD ) is the QCD running coupling constant; the constants λ, α, a, V 0 and Λ QCD are the parameters characterizing the potential.
The method of solving the full Salpeter equation is given in Ref. [30]. After solving the Salpeter equation, we obtain the bare mass spectrum and the wave functions. Then we can use them to calculate the open-charm decay amplitudes. The A → BC open-charm decay is a typical OZI-allowed strong decay process, which in essence is a non-perturbative QCD problem. Due to our pure knowledge of QCD in its non-perturbative region, such processes are usually evaluated using models. Among others, the 3 P 0 model is a widely accepted one [31,32]. This model assumes the decay takes place via creating a quark anti-quark pair from the vacuum baring the quantum number 3 P 0 , so it is also known as the quark pair creation model. The 3 P 0 model is a non-relativistic model, and majority of works using this model stick on its non-relativistic form. On the other hand, the Salpeter wave function is a relativistic wave function containing the Dirac spinor of quark/anti-quark. So incorporating the Salpeter wave function into 3 P 0 model requires the relativistic extension of the original 3 P 0 model, which has been done in Ref. [33]. In this work we employ the formula derived in that reference and write the amplitude for the A → BC open-charm decay as where quantities referred to A, B and C are labeled with the subscript/superscript A, B and C respectively. g = 2m q γ with m q being the the mass of the created quark q or anti-quarkq and γ being a universal constance characterizing the strength of the decay. Negative-energy contributions have been neglected due to their smallness comparing to positive-energy contributions. For J P = 1 − meson, the wave function ϕ ++ can be decomposed into two parts: the S-wave part and the D-wave part, i.e., where R nl is the radial wave function with principle number n and orbital angular momentum l. Y lm is the spherical harmonic function. p. This expression clearly shows the L − S coupling inside the meson, and it is the same as the corresponding nonrelativistic wave function except that: a) the non-relativistic spinor is replaced with the Dirac spinor; b) the radial wave function is the solution of the Salpeter equation rather than the non-relativistic Schrödinger equation.
Inserting the S-wave part or the D-wave part of the wave function into Eq. (19), one arrives at the familiar form of the decay amplitude of the 3 P 0 model: where ψ nlm = √ 2M 0 4πR nl Y lm and ω qP = p 2 q + m 2 q . The decay width is then given by where f A→BC is introduced to make connection with the convention which is widely used in literature (such as Ref. [34]). pol means summation over polarizations. f A→BC is related to the matrix element as Given these decay amplitudes, whenever Π mn develops imaginary part, Π mn can be calculated as [23] ImΠ where E th BC represents the threshold energy of BC channel. θ(x) is the step function. The expressions of ImΠ mn for each decay channels are given in Appendix A.

III. RESULTS AND DISCUSSIONS
Previous section gives the formula for calculating the charmonium spectrum and the coupled-channel effects. To obtain the bare masses and the bare states of vector charmonia, we use the following parameters: The value of α in this work is comparable to the corresponding parameter in Ref. [15]. While all the other parameters lie in reasonable ranges, the parameter Λ QCD is a bit larger than its typical value. At two-loop order in the M S scheme, Λ  [35]. Since in our model Λ QCD is in essence a model parameter, we do not require it strictly binding to Λ (4) MS . So our value of Λ QCD is still acceptable. The results of bare masses are listed in Table I. With the obtained wave functions, the coupled-channel effects are calculated. The decay strength parameter of 3 P 0 model is fitted to be γ = 0.43, and we also set the strength parameter of creating ss to be γ s = γ √ 3 as usual [36,37]. For consistency, the wave functions of D ( * ) and D  Table I. One can see that from the ψ ′ (1S) through ψ ′ (3S), the masses are comparable with the experimental data, which justifies our model calculations. (ψ ′ (2D) is one exception, which is also a problem of another screened potential model calculation [15].) We find that the masses of the two physical states ψ ′ (4S) and ψ ′ (3D) are roughly comparable with the masses of Y (4260) and Y (4360) respectively.
In Table II, we give the coefficients a S and a D as in Eq. (12) for different states. We also extracted the mixing angles defined as in |ψ ′ (S) = cos θ|S + sin θ|D , |ψ ′ (D) = − sin θ|S + cos θ|D . In extracting the mixing angles, we neglect the phase of the complex number a S/D and set the sign equal to the sign of its real part. The magnitude of mixing angles for 2S − 1D states in our model are smaller than 10 • , the result from Ref. [20]. The difference may due to different model settings in these two works. The magnitude of mixing angles for 3S − 2D states are larger than those for 2S − 1D states as expected. For 4S − 3D states, the mixing angles are around −6 • ∼ −5 • .
To provide more details, we plot the real and imaginary parts of Π mn in Figs. 1-3. The real parts of diagonal elements roughly reflect the mass shifts at given energy scale. The imaginary part of diagonal elements roughly reflect half-widths at given energy scale. And non-diagonal elements may reflect how much the S − D mixing is at given energy scale. One can find that the non-diagonal elements are in general smaller than the diagonal ones. Actually, the diagonal elements ImΠ BC SS and ImΠ BC DD for each channel BC always ≤ 0, so the contributions to ImΠ SS and ImΠ DD add up. But the non-diagonal elements ImΠ BC SD for different channels may have different signs, so ImΠ SD is generally smaller than ImΠ SS and ImΠ DD , and oscillates around zero as E varies. This in turn results in ReΠ SD < ReΠ SS and ReΠ DD in general. The diagonal elements ImΠ SS and ImΠ DD also oscillate but without changing signs. The oscillation behavior of the diagonal elements of Π is a reflection of the node structures of initial states, so this behavior becomes more frequent (against E) for higher excitation states. In general, oscillations make Π and relevant observables sensitive to the energy scale E. In view of this, we calculate the decay widths of physical states at their real mass scales, i.e., the experimental masses. The decay widths of physical states are given in Table III open-charm decays. So for these states, Γ theo. tot. should be comparable to the observed width of the corresponding particle. From Table III, one can see that our results are consistent with the experimental data. For ψ ′ (4S) and ψ ′ (3D), it is interesting to find that their Γ theo.
tot. are notably smaller than those of ψ ′ (3S) and ψ ′ (2D). Especially Γ DD of ψ ′ (4S) is less than 1 MeV. If we use the experimental data of total widths, we can find the branching ratios for these channels: The small branching ratios of Y (4260) and Y (4360) in these channels may be the reason why we haven't seen them in open-charm channels. The reason of the smallness of decay widths of ψ ′ (4S) and ψ ′ (3D) are the oscillation behavior of decay amplitudes and the mixing between S and D wave components. The node structures of initial states make the decay amplitude of tot.
Γex. [ each channel oscillate, which can be reflected by ImΠ BC SSorDD in each channel. Fig. 4 shows that the decay amplitudes (actually, the square of decay amplitudes) approach zero at particular energies. ψ(3D) → D * D * dose not reach zero but is close to zero at some energy scales. From Fig. 4, we can see that in the range E ≃ [4.2, 4.4] GeV, where Y (4260) and Y (4360) lie in, the decay amplitude of each channel, except ψ(4S) → D * D * , has one trough, and this exception has one crest. So, except ψ(4S) → D * D * process, ImΠ BC SSorDD of all channels at the mass scales of Y (4260) and Y (4360) have good chance to be small. One may wonder that the locations of the zeros of these decay amplitudes may vary as model parameters vary. But as long as we require a reasonable spectrum to be able to accommodate Y (4260) and Y (4360), the model parameters can not vary too far, and the locations of zeros are not sensitive to small variations in parameters. The S − D mixing then affect the final results in the following ways. For the case when the S wave amplitude and the D wave amplitude are not comparable to each other, for example in D * D * channel the S wave amplitude is larger than the D wave amplitude, the decay width of the mixed state dominated by S wave is depressed by mixing with the smaller D wave amplitude and vice versa. For the case when the S wave amplitude is comparable to the D wave amplitude, the decay widths of mixed states either enhanced by the constructive interference or depressed by the destructive interference. Y (4260) → DD is an example of the constructive interference case and Y (4260) → DD * +c.c. is an example of the destructive interference case. In Ref. [18], the authors noticed that the S wave amplitude and the D wave amplitude of Y (4260) → DD decay are of opposite signs, but since the mixing angle is negative, the two amplitudes actually interfere constructively.
Finally, we indicate that from the di-electron decay widths of vector charmonia, one may expect larger mixing angles than the coupled-channel-induced mixing angles presented in this work. To show this we calculate the Γ ee for each physical states, and the results are shown in Table IV. In these calculations, the QCD correction factor 1 − 16 3 αs π with α s = 0.3 is included. On the other hand, by fitting the experimental data of Γ ee under the assumption of Eqs. (26) and (27), we obtain θ 2S−1D = −11.5 • and θ 3S−2D = −30.7 • . This may imply that there are some unrevealed   26) and (27) against |θ| in Fig. 5. We find that the decay widths of Y (4360) → DD and Y (4260) → DD * + c.c. decrease as |θ| increases as expected, because the S wave and D wave amplitudes interfere destructively in these two channels. The summation of these channels Γ tot. for both the resonances decrease as |θ| becomes larger, which means they would be harder to be observed in these open-charm decay channels.

IV. CONCLUSION
In this work, we studied the coupled-channel effects in J P C = 1 −− charmonium states up to ψ(3D). The mass shifts are found to be from tens MeV up to 100 MeV. We focused on the mixing between (n + 1)S and nD states induced by the coupled-channel effects. The mixing angles are extracted. We find that the mixing in 3S − 2D sector is larger than those in 2S − 1D sector and in 4S − 3D sector. The two-body open-charm decay widths and the di-electron decay widths are calculated. Most of the widths and the masses are consistent with the corresponding experimental data for states from J/ψ to ψ(4160). The calculations are performed using the instantaneous BS equation with a screened Cornell potential and the 3 P 0 model which has been reexpressed in the form suitable for the Salpeter wave functions.
Based on these calculations, we discussed the possibility of assigning the resonant state Y (4260) as the mixture of 4S − 3D with lower mass and Y (4360) as the mixture of 4S − 3D with higher mass. The predicted masses of these two states are 4285 MeV and 4319 MeV respectively. The branching ratios of Y (4260) → DD and Y (4360) → DD are found to be 0.758% and 8.10% respectively. Such small branching ratios may be the reason of non-observation of these two states in DD channels. Combining the predictions on mass and on the open-charm decays and comparing them to the experimental observations, although not perfect, we find it is still possible for Y (4260) and Y (4360) to have conventional charmonium interpretations. Of course, the present study does not exclude other possible exotic interpretations. So, we expect further studies on these resonances. where R A/B/C is the radial wave function of corresponding particle. α B/C 1,2 is a partition parameter in defining relative momentum of quark and anti-quark of the corresponding meson. We define α 1 = m1 m1+m2 and α 2 = m2 m1+m2 , where m 1 is the quark's constituent mass and m 2 is the anti-quark's constituent mass.