Dynamically generated 0^+ heavy mesons in a heavy chiral unitary approach

In terms of the heavy chiral Lagrangian and the unitarized coupled-channel scattering amplitude, interaction between the heavy meson and the light pseudoscalar meson is studied. By looking for the pole of scattering matrix on an appropriate Riemann sheet, a $DK$ bound state $D_{s0}^*$ with the mass of $2.312\pm0.041$ GeV is found. This state can be associated as the narrow $D_{sJ}^*(2317)$ state found recently. In the same way, a $B{\bar K}$ bound state $B_{s0}^*$ is found, and its mass of $5.725\pm0.039$ GeV is predicted. The spectra of $D_0^*$ and $B_0^*$ with $I=1/2$ are further investigated. One broad and one narrow states are predicted in both charm and bottom sectors. The coupling constants and decay widths of the predicted states are also calculated.

On the other hand, it has been shown that the light scalar mesons σ, f 0 (980), a 0 (980) and κ can dynamically be generated through the S wave interaction between Goldstone bosons in the chiral unitary approach (ChUA) [40,41,42,43,44,45,46]. In such an approach, the amplitudes from the chiral perturbation theory (ChPT) are usually adopted as the kernels of the factorized coupled-channel Bethe-Salpeter (BS) equations. In this procedure, a Lagrangian in a specific expanded order, where the symmetries of ChPT should be preserved, is chosen at the beginning, and then the higher order corrections to the amplitudes are resummed with the symmetries kept up to the order of the expansion considered. Namely, what the unitary CHPT does in the successive step is re-summing a string of infinite loop diagrams while the the symmetries of ChPT are held [47,48,49]. Moreover, ChUA has been applied to study the S wave interaction between the lower lying vector meson and the Goldstone boson, and most of the known axial-vector mesons can also be generated dynamically [50]. Based on the valuable achievements mentioned above, extending ChUA to the heavy-light meson sector to study the S wave interaction between the heavy pseudoscalar meson and the Goldstone boson, and consequently the structures of possible heavy scalar mesons, would be extremely meaningful. In fact, similar work, called χ-BS(3) approach, has been done [26,27]. In such an approach, heavy-light meson resonances and open-charm meson resonances were predicted through checking speed plots together with the real and imaginary parts of the reduced scattering amplitudes. In our opinion, studying the poles on the appropriate Riemann sheet of the scattering amplitude would be a powerful procedure to reveal the properties of the generated states in a more accurate way. In this paper, the S wave interaction between the heavy meson and the light pseudoscalar meson is studied by using the extended chiral unitary approach, called heavy chiral unitary approach. The poles that associate with the experimentally observed narrow D * sJ (2317) and broad D * 0 in the I = 0, S = 1 and I = 1 2 , S = 0 channels, where I and S denote the isospin and the strangeness, respectively, are searched. The corresponding coupling constants and decay widths are also discussed.

II. COUPLED-CHANNEL HEAVY CHIRAL UNITARY APPROACH
In order to describe the interaction between the Goldstone boson and the heavy pseudoscalar boson, we employ a leading order heavy chiral Lagrangian [51,52,53] where f π = 92.4 MeV is the pion decay constant, P represents the charmed mesons (cū, cd, cs), namely (D 0 , D + , D + s ), and Φ denotes the octet Goldstone bosons and can be written in the form of 3 × 3 matrix This Lagrangian is equivalent to the SU(4) extrapolation of the ordinary meson meson chiral Lagrangian, eliminating the exchanges of heavy vector mesons in the equivalent picture of vector meson exchange [54]. Obviously, the similar investigation in the bottom sector can be carried out by replacing P in Eq. (1) with the anti-bottom mesons (bū, bd, bs), namely We are interested in the heavy mesons in the I = 0, S = 1 and I = 1 2 , S = 0 channels that can be specified by their own isospins, respectively. In terms of Eq. (1), the amplitudes can easily be obtained by where i and j represent the initial state and the final state, respectively. In the I = 0 case, i (j) can be 1 and 2 which represent the coupled DK and D s η channels in the charmed sector, respectively, and BK and B s η channels in the bottom sector, respectively. In the I = 1 2 case, i (j) can take 1, 2 and 3 which denote the coupled Dπ, Dη and D sK channels in the charmed sector, respectively, and Bπ, Bη and B s K channels in the bottom sector, respectively. The coefficients C I ij are listed in Table I.
The tree level amplitudes can be projected to the S wave by using and and the on-shell condition for the Mandelstam variables, s + t + u = 4 i=1 m 2 i , is applied. In ChUA, under the on-shell approximation, the full scattering amplitude can be converted into an algebraic BS equation [40] where V is a matrix whose elements are the S wave projections of the tree diagram amplitudes and G is a diagonal matrix with the element being a two-meson loop integral where p 1 and p 2 are the four-momenta of the two initial particles, respectively, and m 1 and m 2 are the masses of the particles appearing in the loop. It was shown that the scattering matrix derived in such a way satisfies the unitary relation [41,42,46].
The loop integral can usually be calculated in the center-of-mass frame by using a threemomentum cut-off parameter q max [40]. However, in this method, an artificial singularity of the loop function might be produced [46], and the applicability of the method is limited.
The better way to remove the singularity of the loop integral is using the dispersion relation where a subtraction constant is employed. Then, the analytic expression of G ii (s) can be expressed by [42] G ii (s) = 1 where a(µ) is the subtraction constant, µ denotes the regularization scale, This result is independent of µ, because the change in G ii , caused by a variation of µ, is cancelled by the corresponding change of the subtraction constant a(µ).

III. POLES ON APPROPRIATE RIEMANN SHEETS
The physical states are closely associated with the poles of the scattering amplitude on the appropriate Riemann sheet of the energy plane. For instance, considering only one channel, a bound state is associated with a pole below the threshold value in the real axis of the energy plane, and the three-momentum of the scattered meson in the center of mass frame of the two mesons system can be written as p cm = i|p cm |. A resonance should be related with a pole on the second Riemann sheet, namely, Imp cm < 0. In the coupled channel case, the situation is somewhat complicated. Detailed relation can be found in Ref. [55].
Before searching for poles of the scattering amplitude, the range of subtraction constant values in the dispersion relation method should firstly be estimated. It can be done by comparing the calculated value of loop integration in the dispersion relation method with the one obtained in the cut-off method, although there might be an artificial singularity problem in the cut-off method [46]. The cut-off momentum can approximately be chosen as The corresponding values of a(µ) and q max are tabulated in Table II. With the estimated a(µ) value, the full scattering amplitude can be calculated.   Table III, respectively. These poles are apparently associated with the DK bound state and the BK bound state, respectively.
Due to the existence of thes quark, these bound states should be scalar heavy mesons, namely D * s0 andB * s0 , respectively. More specifically, when a(m D ) = −0.630, corresponding to q max = 0.8 GeV, the mass of the DK state, namely D * s0 , is about 2317 MeV, which is almost the same as the measured value of D * sJ (2317). Taking into account the uncertainty of subtraction constant, the mass of the D * s0 (0, 1) state in our model is 2.312 ± 0.041 GeV. Also due to the uncertainty of a(m B ), the predicted mass of the BK bound state, namely B * s0 (0, 1) state, is 5.725 ± 0.039 GeV. This mass is consistent with the mass predicted in Refs. [8,33], but larger than that in Refs. [16,26]. For comparison, we list the mass of B * s0 predicted in different models in Table IV. In the I = 1 2 , S = 0 case, the poles are located on nonphysical Riemann sheets. Usually, if Imp cm is negative for all the channels open for a certain energy, the width obtained would correspond more closely with the physical one. We search for poles in this particular sheet.
There are two poles in either charmed sector or bottom sector. The width of the lower pole is broad and the width of the higher one is narrow. The obtained poles are listed in Table   V. In either the charmed or bottom sector, the lower pole is located on the second Riemann sheet (Imp cm1 < 0, Imp cm2 > 0, Imp cm3 > 0, where p cmi denotes the momentum of one of the interacting mesons in the i-th channel in the center of mass system). This pole should be associated with a Dπ (Bπ) resonance in the charmed (bottom) sector. Consequently, this state should easily decay into Dπ (Bπ) in the charmed (bottom) sector. The higher pole in either charmed or bottom sector is found on the third Riemann sheet (Imp cm1 < 0, Imp cm2 < 0, Imp cm3 > 0) when a(µ) corresponds to q max = 0.6 GeV or 0.8 GeV, or on the second Riemann sheet when a(µ) corresponds to q max = 1.0 GeV. The pole should be associated with an unstable D sK (B s K) bound state in the charmed (bottom) sector due to its narrow width. It should be mentioned that the situation for the higher pole in the later case, namely a(µ) corresponding to q max = 1.0 GeV, is somewhat complicated.
Besides a pole on the second Riemann sheet, pole II = 2.429 − i0.002 GeV shown in Table   V, there is a shadow pole, pole III = 2.397 − i0.043 GeV, on the third Riemann sheet.
Note that Re(pole II ) > m D + m η and Re(pole III ) < m D + m η . A sketch plot for the paths of these two poles to the physical region in the energy plane is shown in Fig. 1.
From this cartoon, one sees that pole II corresponds more closely with the physical one.
Therefore, we choose pole II = 2.429 − i0.002 GeV as the result. Similar complexity appears at pole III = 2.488 − i0.039 GeV in Table V, due to the existence of pole V = 2.048 − i0.020 GeV. With the same reason, we disregard pole V . Considering the deviations of the data caused by the uncertainty of a(µ) Table V, we predict the mass and the width of the broad D * 0 ( 1 2 , 0) state as 2.097 ± 0.018 GeV and 0.213 ± 0.080 GeV, respectively, and the mass and the width of the narrow D * 0 ( 1 2 , 0) state as 2.448 ± 0.030 GeV and 0.051 ± 0.047 GeV, respectively. In the same way, we forecast the mass and the width of the broad B * 0 ( 1 2 , 0) state as 5.536 ± 0.029 GeV and 0.234 ± 0.086 GeV, respectively, and the mass and the width of the narrow B * 0 ( 1 2 , 0) state as 5.842 ± 0.022 GeV and 0.035 ± 0.019 GeV, respectively.
Recalling the predictions in Refs. [26,27], we noticed that by checking the reduced scattering amplitude curves in the speed plot, the authors in Ref. [26] found a broad state with mass of 2138 MeV and a narrow states with mass of 2413 MeV in the charmed sector, and by further adjusting free parameters in the next-to-leading order to reproduce the D * s0 (2317) state with mass of 2317±3 MeV and the D * 0 state with mass of 2308±60 MeV and width of 276±99 MeV given in Ref. [37], the authors in Ref. [27]  MeV in the bottom sector. It seems that our predicted D * 0 ( 1 2 , 0) states are consistent with those in Ref. [27], although they still deviate from the experimental data [37,38]. It should be mentioned that because of the large uncertainty in the data analysis and existence of the predicted higher narrow state just around the D * 2 (2460) region, the present model could not be disregarded rudely.

IV. COUPLING CONSTANTS AND DECAY WIDTHS
The decay properties of predicted states are studied by making the Laurent expansion of the amplitude around the pole [56] where g i and g j are coupling constants of the generated state to the i-th and j-th channels.
g i g j can be obtained by calculating the residue of the pole [42] g i g j = lim In the case where a(µ) corresponds to q max = 0.8 GeV, we calculate the residues of the poles, and consequently the coupling constants. The resultant coupling constants for the D * s0 and B * s0 (D * 0 and B * 0 ) states are tabulated in Table VI (VII). From these tables, one sees that the coupling constants again are consistent with the results in the pole analysis.
In the (0, 1) channel, the coupling of D * s0 (B * s0 ) to the D s η (B s η) channel is weaker than that to the DK (BK) channel. This is because the D * s0 (B * s0 ) state is the DK (BK) bound state. In the ( 1 2 , 0) channel, the coupling of the lower broad D * 0 (B * 0 ) state to the Dπ (Bπ) channel is stronger than the coupling of the higher narrow one to the Dπ (Bπ) channel; the coupling of the lower state to the Dπ (Bπ) channel is stronger than that to the D sK (B s K)   The decay widths of generated states are further evaluated. We first study the states in the (0, 1) channel. The D * s0 state cannot decay into either DK or D s η, because the mass of the state is lower than the threshold of the DK channel. Moreover, the D * + s0 (2317) → D + s π 0 decay violates the isospin symmetry. Thus, the decay width of D * + s0 (2317) should be very small. This decay can only occur through π 0 -η mixing. According to Dashen's theorem [58], the π 0 -η transition matrix should be t πη = π 0 |H|η = −0.003 GeV, (12) and the decay width reads where M is the mass of the initial state, g 2 represents the coupling of D * s0 (2317) to Dη, and p cm denotes the three-momentum in the center of mass frame and can be written as Then, the partial decay width of the D * + s0 (2317) → D + s π 0 process can be obtained as This value is compatible with that in Ref. [35,36]. Similarly, the partial decay width of the isospin violated decay B * 0 s0 (5729) → B 0 s π 0 can be evaluated as Γ(B * 0 s0 (5729) → B 0 s π 0 ) = 7.92 keV.
We then study the states in the ( 1 2 , 0) channel. For the higher state, two strong decay channels are opened. The fraction ratio of the decay widths for these two decay channels can be calculated by utilizing the coupling constants given in Table VII. Let Γ 1 and Γ 2 denote the partial decay widths with the final states being D(B)π and D(B)η, respectively.
The ratio Γ 1 /(Γ 1 + Γ 2 ) can be written by For higher D * 0 and B * 0 states, we have It is shown that in the bottom sector, the higher narrow state is easier to decay into Bπ than into Bη, but in the charmed sector, the higher narrow state can decay into Dπ and Dη in almost the same weight.

V. CONCLUSION
Based on the heavy chiral unitary approach, the S wave interaction between the pseudoscalar heavy meson and the Goldstone boson is studied. By calculating full scattering amplitudes via an algebraic BS equation, the poles on some appropriate Riemann sheets are found. These poles can be associated with bound states or resonances. With a reasonably estimated single parameter a(µ) in the loop integration, a pole on the real axis on the first Riemann sheet, which is associated with the bound state, in the two-coupled-channel calculation in the (0, 1) channel is found. Because the mass of the pole in the charmed sector is about 2.312 ± 0.041 GeV, this state should be a 0 + DK bound state and can be regarded as the recently observed D * sJ (2317). Meanwhile, a 0 + state B * s0 , which should be a BK bound state, is predicted. Its mass is about 5.725 ± 0.039 GeV. In the I = 1 2 , S = 0 case, three-coupled-channel calculations are performed in both charmed and bottom sectors. In the charm sector, a broad pole structure, which is associated with a resonance, is found at about (2.097 ± 0.018 − i0.107 ± 0.040) GeV. Besides, a narrow pole structure, which can be interpreted as a quasi-bound state of D sK , at about (2.448 ± 0.030 − i0.026 ± 0.024) GeV is also found. In the bottom sector, one broad and one narrow poles are found at about They are consistent with the results in the pole structure analysis. In the (0, 1) channel, the width of the isospin violated decays D * + s0 (2317) → D + s π 0 and B * 0 s0 (5729) → B 0 s π 0 are calculated. They are about 8.69 and 1.54 keV, respectively. Finally in the ( 1 2 , 0) channel, the decay ratio Γ 1 /(Γ 1 + Γ 2 ) for the higher narrow state is also estimated.