Pionic and radiative transitions from T + c ¯ s 0 ( 2900 ) to D + s 1 ( 2460 ) as a probe of the structure of D + s 1 ( 2460 )

In this work, we evaluated the widths of the pionic and radiative transitions from the T + c ¯ s 0 ( 2900 ) to the D + s 1 ( 2460 ) in the D + s 1 ( 2460 ) molecular frame and the D + s 1 ( 2460 ) charmed-strange meson frame. Our estimations demonstrate that the transition widths in the D + s 1 ( 2460 ) molecular frame are much larger than those in the the D + s 1 ( 2460 ) charmed-strange meson frame. Speciﬁcally, the ratio of the widths of (cid:2)( T + c ¯ s 0 ( 2900 ) → D + s 1 π 0 ) and (cid:2)( T + c ¯ s 0 ( 2900 ) → D + ( 0 ) K 0 ( + ) ) is estimated to be around 0.1 in the D + s 1 ( 2460 ) charmed-strange meson frame, whereas the lower limit of this ratio is 0.67 in the D + s 1 ( 2460 ) molecular frame. Thus, the aforementioned ratio could be employed as a tool for testing the nature of the D + s 1 ( 2460 ) .

Recently, the LHCb Collaboration reported two new structures T and respectively.
2 Hadronic molecular structure of T + cs0 (2900) and D + s1 (2460) In the molecular scheme, the T + cs0 (2900) and D + s1 (2460) could be considered as S-wave molecular states composed of D * K * and D * K , respectively.Here, we employ the effective Lagrangian approach to describe the coupling of the molecular states with their components, and the effective Lagrangians related to T + cs0 (2900) and D + s1 (2460) are, respectively, where ω i j = m i /(m i + m j ) is the kinematical parameter.The T cs0 (y 2 ) and D s1 (y 2 ) are the correlation function for T + cs0 (2900) and D + s1 (2460), respectively, which are introduced to describe the molecular inner structure.The Fourier transformation of the correlation function is, In principle, the correlation function in momentum space should decrease sharply enough to avoid the divergence in the ultraviolet region.Here, we employ the correlation function in the Gaussian form [48][49][50][51], which is, where P E is the Jacobi momentum in the Euclidean space, and M is a model parameter to depict the distribution of components in the molecule.
For the coupling constants g T cs0 and g D s1 in Eq. ( 3), they could be determined by the Weinberg's compositeness condition, which means that the possibility of finding the molecular in a bare elementary state is set equal to zero [52][53][54][55][56], i.e., with T cs0 to be the derivative of mass operator of the T cs0 .While for D + s1 (2460), the mass operator μν D s1 could be divided into the transverse part D s1 and the longitudinal part L D s1 , which is, where respectively.
3 Pionic and radiative transitions from In the present work, the initial state T + cs0 (2900) is considered as a D * K * molecule.Subsequently, the pionic and radiative transitions from T + cs0 (2900) to D + s1 (2460) could occur through two possible subprocesses.The first one is via the subprocess K * → K π/γ and the K and D * couple to the D + s1 (2460), which is shown in Figs. 3, 4. The second one is through the subprocess D * → Dπ/γ and the D and K * couple to the D + s1 (2460), where the exchanged meson is D meson.The mass of D meson is much greater than the one of K meson, thus, the contributions from the second subprocess should be suppressed.In addition, in the kaon exchange diagram, the final D s1 (2460) couples to DK , and the threshold of DK is close to the mass of the D s1 (2460), thus in the triangle diagram, all the involved internal particles are almost on-shell, which will enhance the loop integral.On the contrary, in the D meson exchange diagram, the final D s1 (2460) couples to DK * , while the threshold of DK * is far above the mass of D s1 (2460), thus, the involve internal particles are off-shell, which further suppress the contributions from the D meson exchange diagrams.Thus in the present estimation, we only consider the diagrams in Figs. 3, 4.
In the present calculations, the diagrams in Figs. 3, 4 are evaluated in the hadronic level.The interactions of the involved particles are depicted by effective Lagrangian.The effective Lagrangian depicts the subprocess K * → K π could be constructed by SU(3) symmetry interaction  [12,27,47,[57][58][59][60], which is while the one for K * → K γ is, where α are the field-strength tensors.According to the decay width of K * K π [61], the coupling constant g K * K π is estimated to be 3.12.Additionally, we utilize the coupling constants g K * 0 K 0 γ = −1.27and g K * + K + γ = 0.83, which are estimated from the corresponding partial width of K * +/0 → K +/0 γ [59-62]. where The amplitudes corresponding to Fig. 3b can be obtained by M a through replacing the masses of the involve meson and the relevant coupling constants with the corresponding values, which is, Then, the total amplitude of T + cs0 (2900 In the similar way, one can obtain the amplitudes for T + cs0 (2900) → D + s1 (2460)γ corresponding to the diagrams in Fig. 4, which are, then, the total amplitude of T + cs0 (2900 here, we introduce a monopole form factor to depict the inner configuration and off-shell effect of the exchanging mesons, After performing the loop integral, we find the amplitude of T + cs0 (2900) → D + s1 (2460)γ can be simplified as, (17) which satisfies the principle of gauge invariance for the photon field.
In our previous work [47], we have estimated the pionic transition from T + cs0 (2900) to D + s1 (2460) in the D + s1 (2460) charmed-strange meson frame.Thus, in this subsection, we only present the amplitude for T + cs0 (2900) → D + s1 (2460)γ .The amplitudes corresponding to the diagrams in Fig. 4 are, The total amplitude for T + cs0 (2900 Similar to Eq. ( 15), the above amplitude also satisfies the principle of gauge invariance for the photon field.Additionally, in the above amplitudes, a phenomenological form factor in monopole form is introduced, which can describe the inner structure and off-shell effect of the exchanging mesons.The concrete form of the form factor is, with to be a model parameter, which should be of the order of unity.In the present estimations, the loop integrals in the mass operators and the amplitudes can be estimated by Schwinger parameterizations [73].This parameterization scheme is more convenient to handle the four-momentum integrals with the correlation functions in the Gaussian form.

Coupling constants
Since the T + cs0 (2900) has not been observed yet, in the present estimation, we take the same resonance parameters as those of T 0 cs0 (2900) for easy comparison with our previous work in Ref. [28].Besides the coupling constants discussed in the above section, the coupling constants related to the molecular states are also needed to estimate the relevant decay widths.In the present work, we estimate the coupling constants g D s1 and g T cs0 according to the compositeness conditions as shown in Eq. ( 6).Here, T cs0 and D s1 are phenomenological model parameters, which should be of order 1 GeV.Considering both K and K * are S-wave strange mesons and the similarity between D + s1 (2460) and T + cs0 (2900), we take T cs0 = D s1 = M for simplify.In Ref. [47], our estimations indicate that the total width of T 0 cs0 (2900) could be well reproduced with M < 1.6, thus, in the present work, we vary the parameter M from 0.5 to 1.6 GeV, and the M dependences of g D s1 and g T cs0 are presented in Fig. 5.In the considered parameter range, one can find the coupling constants g D s1 and g T cs0 are weakly dependent on the model parameter M .Particularly, with the parameter M increase from 0.5 to 1.2 GeV, the coupling constants g T cs0 and g D s1 decrease from 4.98 to 3.84 GeV and from 15.08 to 10.51 GeV, respectively.

Decay widths
In addition to the parameter M introduced by the correlation functions for T + cs0 (2900) and D + s1 (2460), there are another parameter involved by the form factor in the amplitudes, which is also of the order of 1 GeV.Here, we take several typical values for , which are 1.6, 1.8 and 2.0 GeV, respectively [47].With the above preparations, we can evaluate the decay widths of T + cs0 (2900) → D + s1 (2460)π and T + cs0 (2900) → D + s1 (2460)γ in different frames, which are the D + s1 (2460) molecular frame and the D + s1 (2460) charmed strange meson frame, respectively.
In Fig. 6, we present the decay width of T + cs0 (2900) → D + s1 (2460)π 0 in the D + s1 (2460) molecular scenario and in the D + s1 (2460) charmed strange meson frame, respectively.From our estimation, one can find the transition width in the D + s1 (2460) molecular frame are much larger than that in the D + s1 (2460) charmed-strange meson frame.It should be noted that in the D s1 (2460) molecular frame, D s1 (2460) is composed of D * and K , thus, the coupling between D s1 (2460) and D * K is much stronger than that in the charmed strange meson frame.In addition, the kaon is considered as an ordinary meson in the D s1 (2460) molecular frame, and the coupling between D s1 (2460) and D * K is S wave, which is momentum independent.However, in the D s1 (2460) charmed strange meson frame, the kaon is considered as a chiral particle, thus, the D s1 D * K vertex is momentum dependent as shown in Eq. ( 18) due to the chiral symmetry, which further suppress the width of T cs (2900) → D s1 (2460)π in the charmed strange meson frame.
In Ref. [47], we have investigated the decay properties of T cs0 (2900) in the D * K * molecular frame.The decay widths of DK, D s π, D * s ρ, D s1 (2460)π, D s1 (2536)π and D * K π channels were estimated [47], where the D s1 (2460) is considered as a charmed strange meson.For simplify, we just quote the results of Ref. [47] in the following discussions.By comparing the estimated total width with the one reported by the LHCb Collaboration, we obtained the proper M ranges for different , which is collected in Table 1.In these parameter range, the estimated widths for D s1 (2460)π and DK channels are also listed.The ratio of D s1 π and DK is estimated to be, which is very weakly dependent on the model parameters and M .Moreover, one can investigate the decay properties of T + cs0 (2900) in the D + s1 (2460) molecular frame.Considering the isospin symmetry, the strong decay behaviors of T + cs0 (2900) and T 0 cs0 (2900) should be the same.When we investigate the decay properties of the T + cs0 (2900) in the D + s1 (2460) molecular frame, the partial widths of other decay channels without D + s1 (2460) should be the same as those in Ref. [28].Together with the width of T + cs0 (2900) → D + s1 (2460)π 0 estimated in the present work, we can roughly obtain the total width of T + cs0 (2900).Along the same line [47], one can determine the value of M by reproducing the width of T + cs0 (2900) → D + s1 π 0 , which are also listed in Table 1.From the table, one can find that the determined range of M in the D + s1 (2460) molecular frame is smaller than the one in the D + s1 (2460) charmed strange meson frame [47].In these model parameter, one can find that the partial widths of D s1 (2460)π and DK channels are in the same order, and the ratio of Ds1π and DK is estimated to be, By analyzing Eqs. ( 23) and ( 24), one can find the ratio of the widths of T + cs0 (2900) → D + s1 π 0 and T + cs0 (2900) → DK sig- In the present work, we investigate the pionic and radiative transitions from the T + cs0 (2900) to D + s1 (2460) in the D + s1 (2460) charmed-strange meson frame and the D + s1 (2460) molecular scenario, respectively.Our estimations indicate the ratio of the widths of T + cs0 (2900) → D + s1 π 0 and T + cs0 (2900) → DK are rather different in two different frame.Particularly, the ratio is estimated to be around 0.1 in the D + s1 (2460) charmed-strange frame, while the lower limit of this ratio is 0.67 in the molecular frame.Thus, we suggest that the ratio could be employed as a tool for testing the nature of the D + s1 (2460).Moreover, our estimation also find the radiative transition width estimated in the D + s1 (2460) molecular frame is much larger than the one estimated in the D + s1 (2460) charmed-strange meson frame.

Fig. 5
Fig.5 The coupling constants g T cs0 and g Ds1 depending on model parameter M , where T cs0 = Ds1 = M