Production of the $D_{s0}(2317)$ and $D_{s1}(2460)$ by kaon-induced reactions on a proton target

We investigate the possibility to study the charmed-strange mesons $D_{s0}(2317)$ and $D_{s1}(2460)$ by kaon-induced reactions on a proton target in an effective Lagrangian approach. The production process is described by the $t$-channel $D^{0}$ and $D^{*0}$ exchanges, respectively. Our theoretical approach is based on the chiral unitary theory where the $D_{s0}(2317)$ and $D_{s1}(2460)$ resonances are dynamically generated. Within the coupling constants of the $D_{s0}(2317)$ to $KD$ and $D_{s1}(2460)$ to $KD^{*}$ channels obtained from the chiral unitary theory, the total and differential cross sections of the $K^{-}p\to{}\Lambda_cD_{s0}(2317)$ and $K^{-}p\to{}\Lambda_cD_{s1}(2460)$ are evaluated. The $\bar{K}p$ initial state interaction mediated by Pomeron and Reggeon exchanges is also included, which reduces the production cross sections of the $D_{s0}(2317)$ and $D_{s1}(2460)$. If measured in future experiments, the predicted total cross sections and specific features of the angular distributions can be used to test the (molecular) nature of the $D_{s0}(2317)$ and $D_{s1}(2460)$.


I. INTRODUCTION
In recent years, many charmed-strange mesons have been observed [1]. Among them, the D s0 (2317) and D s1 (2460) are two peculiar states(we abbreviate it as D * s0 and D * s1 hereafter), since their masses are about 160 MeV and 70 MeV, respectively, below the quark model predicted values [2]. The charmed-strange meson D * s0 was first observed by the BaBar collaboration as a narrow peak in the D s π invariant mass distribution [3]. The state was shortly confirmed by CLEO [4] and Belle [5,6] collaborations. Nowadays it is well established in the PDG [1] with quantum numbers I(J P ) = 0(0 + ). The D * s1 was also observed in the CLEO experiment [4] in the D * s π channel and BaBar [7][8][9] also found a signal in that region. Nowadays it is also well established in the PDG [1] with quantum numbers I(J P ) = 0(1 + ). The mass and width of the D * s0 and D * s1 states reported by the above collaborations [3][4][5][6][7][8][9] are consistent with each other, i.e., The large disagreement between the quark model expectations [2] and the experimental measurements [3][4][5][6][7][8][9] made it difficult to assign these two states as conventional charmedstrange mesons. Since the masses of the D * s0 and D * s1 are about 40 MeV below the DK and D * K thresholds, respectively, many studies proposed that the D * s0 and D * s1 are S -wave DK and D * K molecular states. The studies in the Bethe-Salpeter approach [10] and potential model [11] showed indeed that the D * s0 could be a DK hadronic molecule. In Ref. [12], the D * s0 and D * s1 were considered as kaonic molecules bound by strong short range attraction. Assuming that the D * s0 and D * s1 are DK and D * K molecular states, the strong and radiative decays of the D * s0 and D * s1 were studied by several groups [13][14][15][16]. The production of the D * s0 and D * s1 from the nonleptonic B decay were calculated in Ref. [17], in which the D * s0 and D * s1 were also considered as hadronic molecules of DK and D * K, respectively. In the chiral unitary approach [18][19][20][21][22][23], the D * s0 and D * s1 can be dynamically generated from the DK/D * K and coupled channel interactions.
In addition to the interpretation of the D * s0 and D * s1 as DK and D * K molecules, the possibility to assign them as a conventional open charmed meson was also discussed in many different approaches, such as the relativistic quark model [24], the chiral perturbation theory [25], the quark pair-creation model [26,27], and the QCD sum rules [28][29][30][31][32]. On the other hand, the large-N c expansion indicated that the D * s0 could be a tetraquark meson [33]. The tetraquark interpretation was also proposed to understand the mass and decay behavior of the D * s0 [34]. We note that the QCD sum rules also supported the idea that the D * s0 does not seem to be a standard quarkantiquark meson [35].
The present knowledge about the D * s0 and D * s1 was obtained from the e + e − collision [3][4][5][6][7][8][9]. Thus, it will be helpful to understand the nature of the D * s0 and D * s1 if we can observe them in other production processes. High-energy kaon beams are available at OKA@U-70 [36] and SPS@CERN [37], which provide another alternative to study D * s0 and D * s1 . The kaon beam at J-PARC can also be upgraded to the energy region required in charmed-strange meson productions [38]. Therefore, it is interesting to study the D * s0 and D * s1 productions in the K − p → Λ c D * s0 and K − p → Λ c D * s1 reactions. Since there exist plenty of experimental information about the K p elastic interaction in the energy region relevant to the D * s0 and D * s1 production [3][4][5][6][7][8][9], the effect from the K p initial state interaction (ISI) can be taken into account in order to make a more reliable prediction. This paper is organized as follows. In Sec. II, we will present the theoretical formalism. In Sec. III, the numerical result of the kaon-induced D * s0 and D * s1 production on a proton target will be given, followed by discussions and conclusions in the last section.

II. THEORETICAL FORMALISM
The tree level Feynman diagrams for the K − p → Λ c D * s0 and K − p → Λ c D * s1 reactions are depicted in Fig. 1, where the t−channel D and D * exchange are considered. In this work, the contributions from s− and u− channels are ignored, because the s− and u−channels, which involves the creation of an additional ss quark pair creation in the kaoninduced production, are usually strongly suppressed, Hence, the K − p → Λ c D * s0 and K − p → Λ c D * s1 reactions should be dominated by the Born terms through the t-channel D and D * exchanges, which makes the background very small.
We also show the definition of the kinematics (p 1 , p 2 , p 3 ,and p4) used in the calculation.

A. Lagrangians
To compute the diagrams shown in Fig. 1, we need the effective Lagrangian densities for the relevant interaction vertices. For the Λ c pD and Λ c pD * vertices, we adopt the commonly employed Lagrangian densities as follows [39], The coupling constants g Λ c pD = −13.98 and g Λ c pD * =-5.20 are determined from the SU(4) invariant Lagrangians [40] in terms of g πNN =13.45 and g ρNN =6.0.
In addition to the Λ c pD and Λ c pD * vertices, we also need the information on the KDD * s and KD * D * s1 vertices. As mentioned in the chiral unitary approach of Refs. [18][19][20][21][22][23], the D * s and D * s1 resonance are identified as s−wave meson-meson molecule that include bigKD andKD * component, respectively. We can write down the KDD * s and KD * D * s1 vertices of Fig. 1 as where the coupling of the D * s0 to D 0 K − , g KDD * s0 , is obtained from the coupling constant of the D * s0 to the DK channel in isospin I = 0, found to be g KDD * s = 10.21 in Ref. [18], multiplied by the appropriate Clebsch-Gordan (CG) coefficient, Similarly to Ref. [18], we rely on the chiral unitary approach [19] to obtain the coupling constant g K − D * 0 D * − s1 = 9.82/ √ 2. When evaluating the scattering amplitudes of the K − p → Λ c D * s and K − p → Λ c D * s1 reactions, we need to include form factors because hadrons are not point-like particles. We adopt here a common scheme used in many previous works [41,42], for the t-channel D ( * ) meson exchange. Here the q D ( * ) and M D ( * ) are the four momentum and the mass of the exchanged D ( * ) meson, respectively. In this model, the Λ D ( * ) is the hard cutoff and it can be directly related to the hadron size. Empirically, the cutoff parameter Λ D ( * ) should be at least a few hundred MeV larger than the D ( * ) mass. Hence, we chose GeV as used in previous works [43][44][45] for other reactions. The parameter α is usually close to unitary, and in this work a variation of the cutoff will be made to show the sensitivity of the results on the cutoff.

B. Initial state interaction(ISI)
Following Ref. [46], the initial state interaction for the K − p → K − p reaction at high energies will be taken into account and the relevant Feynman diagram for the Initial state interaction(ISI) is shown in Fig. 2 When the center-of-mass energy √ s is large, the elastic K − p scattering amplitude is a sum of the following terms, where i = I P for Pomeron and f 2 , a 2 , ω, and ρ Reggeons. The energy scale s 0 = 1 GeV 2 . The coupling constants C KN i , the parameters of the Regge linear trajectories[α i (t) = α i (0) + α ′ i t], the signature factors(η i ), and the B i KN used in Ref. [46] provide a rather good description of the experimental data. The parameters determined in Ref. [46] are listed in Table. I.  [46]. First, we calculate the total cross section of the K − p → Λ + c D * − s and K − p → Λ + c D * − s1 reactions The corresponding unpolarized differential cross section reads where s = (p 1 + p 2 ) 2 , θ is the scattering angle of the outgoing meson relative to the beam direction, p 1cm and p 3cm are the K − and D * − s (D * − s1 ) three momenta in the center of mass frame, where λ(x, y, z) is Källen function with λ(x, y, z) = (x−y−z) 2 − 4yz. The m K − , m p , and m Λ c are the masses of the K − meson, proton, and Λ c , respectively. Here, we take m K − = 493.68 MeV, m p = 938.27 MeV, and m Λ c = 2286.46 MeV. Taking the ISI of the K − p system into account, the full amplitude for the process K − p → Λ + c D * − s0/s1 is a sum of the Born and ISI amplitudes. With the Lagrangians given in the previous section, the Born amplitude of the K − (p 1 )p(p 2 ) → Λ + c (p 4 )D * − s0/s1 (p 3 ) reaction can be obtained as, whereū(p 4 , s c ) and u(p 2 , s 2 ) are the Dirac spinors with s c (p 4 ) and s 2 (p 2 ) being the spins (the four-momenta) of the outgoing Λ c and the initial proton, respectively. The ǫ * ν (p 3 ) is the polarization vector of the D * s1 .
Following the strategy of Ref. [46], the ISI amplitude can be written as where k t is the momentum transfer in the K − p → K − p reaction.
With the formalism and ingredients given above, the total cross section versus the beam momentum of the K − p system for the K − p → Λ + c D * − s0 and K − p → Λ + c D * − s1 reactions are evaluated. The results for beam momentum P k − from the reaction threshold to 20.0 GeV are shown in Fig. 3. Because the cutoff can not be well determined, the results obtained with several cutoffs are also presented. It is worthy mentioning that the value of the cross section is very sensitive to the model parameter α when varying the cutoff parameter α from 1.0 to 2.0. This is because the model parameters we selected are very close to the masses of the exchanged particles. To make a reliable prediction for the cross section of the K − p → Λ + c D * − s0 and K − p → Λ + c D * − s1 reactions thus requires a good knowledge on the form factors. More and accurate experimental data can also be used to constrain the value of the cutoff parameter. The results in Fig. 3 also show that the total cross section increases sharply near the D * − s0/s1 Λ c threshold. At higher energies, the cross section increases continuously but relatively slowly compared with that near threshold. However, the total cross section decreases but very slowly for the D * s0 production in the K − p → Λ + c D * − s1 reaction when we change the beam energy P K − from 14.6 GeV to 20.0 GeV. With the increase of the cutoff, the total cross section increases. Comparing the cross section of the K − p → Λ + c D * − s0 reaction with that of the K − p → Λ + c D * − s1 reaction, we found that the line shapes of the cross section are very different. A possible explanation for this may be that KD interaction to form the D * s0 is stronger than that KD * interaction to form D * s1 due to the D * meson decays completely to the final state containing the D meson [1].
The results show that the total cross section for D * s0 production is bigger than that for D * s1 production. At a beam mo-mentum about 14.6 GeV and the parameter α = 1 the cross section is of the order of 10 nb, which is quite large for an experimentally observation of the D * s0 at current and future facilities. Our results suggest that it will takes high energy, at least above 14.6 GeV, to observe the production of D * s1 in the K − p → Λ + c D * − s1 reaction. To show the effect from the K − p ISI, we compare the cross sections obtained with and without ISI for the cutoff of α = 2.0 in Fig. 4 reactions, respectively. In Fig. 4, the dashed red lines are the pure Born amplitude contribution, while the solid black lines are the full results. It shows that the role of the ISI is to reduce the cross section by approximately 20%, in agreement with the conclusions drawn from Refs. [39,47,48] that the ISI for pp or pp reactions reduces the cross section.
In addition to the total cross section, we also compute the differential cross section for the K − p → Λ + c D * − s0 and K − p → Λ + c D * − s1 reactions as a function of the scattering angle of the outgoing meson relative to the beam direction at different beam energies,i.e,P K − =12.0,14.0,16.0, and 18.0 GeV. The theoretical results are shown in Fig. 5. We note that the differential cross section is the largest at the extreme forward angle and decreases with the increase of the scattering angle. This is because we have only considered the contributions from the t-channel D and D * exchanges. It should be pointed out that, if there are contributions from s− and u− channels, there will be a clear bump (or peak) in the total cross section which can be distinguished easily.

IV. SUMMARY
In this work, the production of the D * − s0 and D * − s1 resonances in the K − p → Λ + c D * − s0 and K − p → Λ + c D * − s1 reactions was studied in an effective Lagrangian approach. The production process is described by the t-channel D 0 and D * 0 meson ex-changes, respectively. The coupling constants of the D * s0 to KD and D * s1 to KD * are obtained from the chiral unitary the- ory [18,19]. where the D * − s0 and D * − s1 resonance are dynamically generated. The K − p initial state interaction(ISI) was included by Pomeron and Reggeon exchanges [46], which was shown to reduce the cross section by about 20%. the total and differential cross sections computed can be used to test the molecular picture of the D * s0 and D * s1 mesons in facilities such as OKA@U-70,SPS@CERN,and future J-PARC.
Finally, we would like to stress that, thanks to the absence of the s−channel, u−channel, and background contribution in the K − p → Λ + c D * − s and K − p → Λ + c D * − s1 reactions, future experimental data for these two reactions can be used to improve our knowledge of D * − s0 and D * − s1 properties, which are at present poorly known.