The $T_{c\bar{s}}(2900)$ as a threshold effect from the interaction of the $D^*K^*$, $D^*_s\rho$ channels

We look at the mass distribution of the $D_s^+ \pi^-$ in the $B^0 \to \bar{D}^0 D_s^+ \pi^-$ decay, where a peak has been observed in the region of the $D^*_s \rho$, $D^* K^*$ thresholds. By creating these two channels together with a $\bar{D}^0$ in $B^0$ decay and letting them interact as coupled channels, we obtain a structure around their thresholds, short of producing a bound state, which leads to a peak in the $D_s^+ \pi^-$ mass distribution in the $B^0 \to \bar{D}^0 D_s^+ \pi^-$ decay. We conclude that the interaction between the $D^*K^*$ and $D^*_s\rho$ is essential to produce the cusp structure that we associate to the recently seen $T_{c\bar{s}}(2900)$, and that its experimental width is mainly due to the decay width of the $\rho$ meson. The peak obtained together with a smooth background reproduces fairly well the experimental mass distribution observed in the $B_0 \to \bar{D}^0 D_s^+ \pi^-$ decay.


Introduction
The D * K * system was investigated in [1] and three states were found corresponding to I = 0; J P = 0 + , 1 + and 2 + .The 2 + state was identified with the D * s2 (2573) state, and served to set the scale for the regularization of the loops, allowing predictions in the other sectors.There, the I = 1 interaction of the D * K * and D * s ρ channels was also studied and, a cusp was found for J = 0 and J = 1 around the D * s ρ threshold.Recently, the LHCb Collaboration has observed an state in the D + s π − , D + s π + mass distributions in the B 0 → D0 D + s π − and B + → D − D + s π + decays, respectively, at 2900 MeV [2,3].Indeed, the state branded as T c s(2900) with J P = 0 + , as seen in D + s π − and D + s π + , exhibits an I = 1 character and it has also been associated with J P = 0 + .On the other hand, 2900 MeV is just the threshold of the D * K * channel.Thus, one is finding a I = 1 J P = 0 + state in the threshold of D * K * (the D * s ρ is only 14 MeV below neglecting the ρ width), which could correspond to the cusp found in [1].
In the present work we look again at the interaction of D * K * and D * s ρ channels, taking into account the K * and ρ widths and also the decay of the states found into the D s π channel where it has been observed, comparing our results with the recent experimental findings.

Formalism
In Ref. [2] a peak is found in the D s π invariant mass in the B 0 → D0 D + s π − and B + → D − D + s π + decays.In order to have a b quark rather than a b quark, we look at the reaction We evaluate the scattering matrix using the Bethe-Salpeter equation in the with G the diagonal loop function for the intermediate mesons and V the transition potential.However, the state is observed in D s π, hence, the mechanism by means of which the reaction proceeds is given in Fig. 2. The amplitude for the process of Fig. 2 is given by, where a is a normalization constant that we do not evaluate, unnecessary to show the shape of the πD s mass distribution in the B0 decay, and M inv is the invariant mass distribution of the D s π final state.The vertex function Ṽ corresponding to the triangle loop of Fig. 3 can be easily evaluated.We assume the resonance to be in J = 0, and that the vectors have small momenta with respect to their masses.Then, the structure of the triangle diagram of Fig. 3 is given by The loop function Ṽ is naturally regularized with a cutoff q max , the same one used to regularize the D * K * and D * s ρ loops when studying their interactions.The equivalent q max used in [1] was 1100 MeV.We find, which shows the different cuts of the loop diagram when pairs of the internal particles of the loop are placed on-shell.Then, we consider that the transition amplitude for B0 → D 0 D − s π + is given by a constant background (considering the dominance of s-wave in the coupling of the bottom meson to the pseudoscalars), see Fig. 1 (right), together with the scattering amplitude of the diagram in Fig. 2, which accounts for the interaction of the coupled channels.It reads as Therefore, the mass distribution of πD − s in the B0 decay is given by, where

Results
We take into account the decay widths of the vector mesons K * and ρ by means the convolution of the G function in Eq. ( 1) [4].The result for the T matrix in I = 1; J = 0 is shown in Fig. 4. The cusp obtained for J = 0 has become wider.The position of the cusp is similar, it shows up slightly above the D * K * threshold and around 2920 MeV, with a width coming basically from the decay of the ρ into ππ.We have also obtained visible peaks in the scattering amplitudes for J = 1 and 2 [4].
Finally, we show the result of the invariant mass distribution of the decay B0 → D − s D 0 π + , Eq. ( 6), in comparison with the LHCb experimental data [2,3] in Fig. 5 (left).In Eq. ( 6), we adjusted the constants a and b to reproduce well the experimental data around the T c s(2900) resonance, and we obtain a = 2.1 × 10 3 and b = −1.45× 10 3 .As can be seen, our model describes well the experimental data.A peak is obtained around the threshold of the D * K * channel.Since these results were obtained fixing the subtraction constant to obtain the T cs (2900), this also supports the molecular picture of this state as D * K * of [5].Thus, our model strongly supports the T c s(2900) as a cusp structure originated by the non-diagonal interaction D * K * → D * s ρ, with a width mainly due to the decay of the ρ meson into ππ Finally, it is interesting to give a band of errors by changing the background, we do this to show the sensitivity of the results to this background.We have done this, keeping the value of a, needed to get the strength of the peak of the distribution, by varying the parameter b of the background by 5% (up and down).This is shown in Fig. 5 (right).The band obtained overlaps with the errors of the data.

Figure 1 .Figure 2 .
Figure 1.Left: B0 decay to D * − s c d with hadronization of the c d pair to produce D * − s D 0 ρ + .Right: B0 decay into D − s D 0 π + (contribution to the background).

Figure 3 .
Figure 3. Triangle diagram accounting for the R → π Ds decay of the R resonance of I = 1 generated with the ρ Ds , D * K * coupled channels.