Searching for $X_{0}(2900)$ and $X_{1}(2900)$ through the kaon induced reactions

Abstract

Within an effective Lagrangian approach, a theoretical study of the $K^{+}p \rightarrow \Sigma _{c}^{++}X_{0}(2900)$ and $K^{+}p \rightarrow \Sigma _{c}^{++}X_{1}(2900)$ reactions is carried out to search for the exotic states $X_{0}(2900)$ and $X_{1}(2900)$. The production process is described by the $t-$channel $D^{-}$ meson exchange. The numerical results show that the total cross sections may reach up to about 0.5 nb and 25 nb for the productions of $X_0(2900)$ and $X_1(2900)$, respectively. The predicted differential cross sections are sensitive to the scattering angles and have strong forward enhancements. The cross sections for the $K^{+}n \rightarrow \Lambda _{c}^{+}X_{0,1}$ are also calculated and found to be larger than that in $K^{+}p$ collisions by more than one order of magnitude. The results are helpful to the possible experimental measurements with kaon beam.

1 Introduction

In the past two decades, amounts of new hadronic states were discovered experimentally in the B-meson decay processes. The explorations for these new particles have broaden our understanding on the nature of hadrons. However, a substantial part of these particles cannot easily find a position in the predicted conventional hadron spectrum. People speculate these states might be the missing exotic hadron states expected from QCD. In the year 2020, LHCb reported two exotic states $X_0(2900)$ and $X_1(2900)$ by reconstructing the $D^-K^+$ invariant mass in the $B^+\rightarrow D^+D^-K^+$ decay channel with model-depend [1] and model-independent [2] assumptions. The masses and widths of $X_{0,1}(2900)$ were determined to be

$$\begin{aligned}{} & {} X_0(2900): M=2866 \pm 7 \pm 2 \text {MeV}, \\{} & {} \quad \Gamma =57 \pm 12 \pm 4 \text {MeV};\\{} & {} X_1(2900): M=2904 \pm 5 \pm 1 \text {MeV}, \\{} & {} \quad \Gamma =110 \pm 11 \pm 4 \text {MeV}. \end{aligned}$$

And the spin parity quantum numbers of $X_0(2900)$ and $X_1(2900)$ are fitted to be $0^{+}$ and $1^{-}$, respectively. Up to now, knowledge is still sparse on $X_{0,1}(2900)$. Appearance of the $D^-$ and $K^+$ mesons in the final state implies that both of $X_{0,1}(2900)$ can strongly couple to $D^-K^+$. If interpreted as resonances, they should be constituted of at least four quarks $(ud\bar{s}\bar{c})$ and regarded as the first observation of exotic hadrons with open flavour and without heavy quark-antiquark pairs. Once comfirmed, it should help to deepen our understanding of the QCD confinement.

Before the release of the LHCb announcement, several explorations have been carried out to discussed the existence of hadrons containing a charm and a strange quark through a coupled channel unitary approach using the extended local hidden gauge model [3], chromomagnetic interaction model [4] and color-magnetic interaction model [5], in which a lowest-lying state at a mass around 2.85 GeV was predicted.

Subsequently, the discovery of $X_{0,1}(2900)$ has attracted a great deal of attention and various identifications are proposed to explain the nature of $X_{0,1}(2900)$. One of the mainstream views is that $X_{0,1}(2900)$ can be treated as diquark-antidiquark states or hadronic molecules. With the QCD sum rule method [6, 7] and a phenomenological model [8], the authors regarded $X_0(2900)$ as the ground-state scalar tetraquark $sc\bar{u}\bar{d}$. While an assumption was made that $X_0(2900)$ and $X_1(2900)$ may be the radial excitation and orbitally excitation of $ud\bar{s}\bar{c}$ tetraquark state, respectively [9]. The authors modeled the $X_0(2900)$ [10] and $X_1(2900)$ [11, 12] as an exotic scalar and vector state built of $ud\bar{s}\bar{c}$ and got the mass and width consistent with the LHCb experiment. However, the authors pointed that the compact $ud\bar{s}\bar{c}$ tetraquark in $0^+$ state disfavors $X_0(2900)$ in an extended relativized quark model [13]. As for the molecular picture, the $X_0(2900)$ and $X_1(2900)$ are assigned as $\bar{D}^{*}K^{*}$ molecular states in overwhelming majority of investigations [14,15,16,17,18,19,20,21,22,23]. In Refs. [24,25,26], the authors investigated $X_1(2900)$ as $\bar{D}_1 K$ molecular state. However, there exists different opinion that does not support the $X_1(2900)$ as molecular state [15, 17, 27]. Although the $X_{0,1}(2900)$ can be well described with compact tetraquark and molecular pictures, other interpretations such as hadronic rescattering effects still cannot be ruled out [28,29,30]. More discussions can be found in Ref. [31].

In other words, there are different and controversial interpretations of the structures of $X_0(2900)$ and $X_1(2900)$ in literatures. That means our understanding of the nature of the $X_{0,1}(2900)$ is sitll incomplete. More detailed investigations are required with larger data samples or different decay modes. One notices that $X_0(2900)$ and $X_1(2900)$ are observed in B meson decay. Searching these two states in other production channels should implement complementary measurements and help to further understand these two states. In fact, several investigations have been carried out. For example, productions of the $X_{0,1}(2900)$ in the $\Lambda _b$ and $\Xi _b$ decays were suggested in Ref. [32]. The authors in Ref. [33] investigated the hadronic effects on the $X_{0,1}(2900)$ in heavy-ion collisions. Besides the two processes, one may expect the productions of the $X_{0,1}(2900)$ in the kaon induced reactions due to the strong coupling between the $X_{0,1}(2900)$ and $D^-K^+$. The reaction of kaon and proton is an ideal process to explore the new hadronic states [34,35,36,37,38]. Thus in the present work, we explore the productions of the $X_{0,1}(2900)$ in the processes $K^+ p \rightarrow \Sigma _c^{++} X_{0,1}$ and $K^+ n \rightarrow \Lambda _c^{+} X_{0,1}$ with $t-$channel $D^-$ exchange, where the effective Lagrangian approach is adopted. Along the way, the cross sections and angular momentum distributions are discussed.

The paper is organized as follows. After the Introduction, we explore the production mechanisms of the $X_{0,1}(2900)$ which are induced by the kaon in Sect. 2. In Sect. 3, the numerical results are given and discussed. This paper ends with a summary.

2 Production relevant to $X_0$ and $X_1$ induced by a kaon

The tree level Feynman diagrams for the productions of $X_0$ and $X_1$ are depicted in Fig. 1, where only the t-channel $D^-$ exchange is considered. In the present work, the contributions from the u and s channels are ignored because they are strongly suppressed due to the reason that an additional $s\bar{s}$ quark pair creation is created. Hence, the $K^+ p \rightarrow X \Sigma _c^{++}$ reaction should be dominated by the Born term through the t-channel. As in the above case, the $X_{0,1}$ can also be produced in the t-channel $K^+ n \rightarrow X \Lambda _{c}^{+}$, which can be depicted with the replacements of $p\rightarrow n$ and $\Sigma _{c}^{++}\rightarrow \Lambda _{c}^{+}$ in Fig. 1.

Fig. 1

The diagram responsible for the production of $X_0$ and $X_1$ in $K^+ p$ collisions, where the definitions of the kinematics are also shown

2.1 Lagrangians

To compute the processes as shown in Fig. 1, the following effective Lagrangian densities are introduced as [39,40,41,42,43,44]

$$\begin{aligned} \mathcal {L}_{DN\Lambda _c}= & {} ig_{DN\Lambda _c}\bar{N} \gamma _5 \bar{D}\Lambda _c + \mathrm {H.c.}, \end{aligned}$$

(1)

$$\begin{aligned} \mathcal {L}_{DN\Sigma _c}= & {} ig_{DN\Sigma _c}\bar{N} \gamma _5 D\varvec{\tau }\cdot \varvec{\Sigma _c} + \mathrm {H.c.}, \end{aligned}$$

(2)

$$\begin{aligned} \mathcal {L}_{KDX_0}= & {} - g_{KDX_0} \bar{X}_{0}KD + \mathrm {H.c.}, \end{aligned}$$

(3)

$$\begin{aligned} \mathcal {L}_{KDX_1}= & {} i g_{KDX_1} \bar{X}_1^\mu (K\partial _\mu D - \partial _\mu KD) + \mathrm {H.c.}, \end{aligned}$$

(4)

where N, D and K represent the isospin doublets for necleon, the pseudoscalar D and K mesons, respectively. $\Lambda _c$ and $\Sigma _c$ denote the $\Lambda _c(2286)$ and $\Sigma _c(2455)$ isospin triplet, respectively. In the following formulae, the abbreviations $g_{\Lambda _c}\equiv g_{DN\Lambda _c}$, $g_{\Sigma _c}\equiv g_{DN\Sigma _c}$, $g_{0}\equiv g_{KDX_0}$ and $g_{1}\equiv g_{KDX_1}$ are implemented. The coupling constants $g_{\Lambda _c}$ and $g_{\Sigma _c}$ can be determined from the SU(4) invariant Lagrangians [45]. In the present work, $g_{\Lambda _c}=-13.98$ and $g_{\Sigma _c}=-2.69$ are adopted as obtained in Ref. [41] in terms of $g_{\pi NN}=13.45$. The coupling constants $g_{0}$ and $g_{1}$ will be discussed later.

It should be noted that the above effective Lagrangians depend on the quantum numbers rather than the internal components of $X_{0,1}$. The information on the internal structure is coded in the coupling constants $g_0$ and $g_1$, which can be inputted from different models assigned to the $X_{0,1}$. In addition, the experimental information can be used to determine $g_0$ and $g_1$ by comparing with our results. Thus the coupling constants $g_0$ and $g_1$ can be related to the corresponding decay widths

$$\begin{aligned} \Gamma _{X_0 \rightarrow K^+D^-}= & {} \frac{g_0^2 |\varvec{k}_K|}{8\pi m_X^2} \end{aligned}$$

(5)

$$\begin{aligned} \Gamma _{X_1 \rightarrow K^+D^-}= & {} \frac{g_1^2 |\varvec{k}_K|}{6\pi m_X^2}f^2(m_X,m_K,m_D), \end{aligned}$$

(6)

where $m_X$, $m_K$ and $m_D$ are the masses of $X_{0,1}$, $K^+$ meson and $D^-$ meson, respectively. $\varvec{k}_K$ is the three momentum of the $K^+$ meson in the $X_{0,1}$ rest frame. The function f is defined as $f(a,b,c)=\sqrt{a^4+b^4+c^4-2(a^2b^2+b^2c^2+c^2a^2)}/(2a)$. Unfortunately, the partial decay width of the $X_{0,1} \rightarrow D^- K^+$ processes are still unknown from the LHCb measurements. However, it is expected that the branching fraction of the $K^+D^-$ mode is large since it is the only two-body strong decay mode observed. Here, we make an approximation that the partial decay widths are the same as the total width of $X_{0,1}$ resonances. Then the two coupling constants are computed to be $g_0=4.07~\text {GeV}$ and $g_1=6.53$. We also notice that partial decay width $\Gamma _{X_1 \rightarrow K^+D^-}$ was obtained through the QCD sum rules and the diquark antidiquark assignment [12]. The obtained value of $g_1$ is twice as ours when a same decay width is adopted. These results will be utilized in the following calculations.

2.2 Feynman amplitudes

With the Lagrangians given above, the Born amplitudes corresponding to the diagrams in Fig. 1 can be expressed as

$$\begin{aligned} \mathcal {M}_{0}= & {} g_0\overline{u}(p_2) \mathcal {C}_{a} u(p_1)\mathcal {P}_{D}^{F}F^2(q^2), \end{aligned}$$

(7)

$$\begin{aligned} \mathcal {M}_{1}= & {} g_1\overline{u}(p_2) \mathcal {C}_{a} u(p_1)(q^\mu -k_1^\mu )\mathcal {P}_{D}^{F}\varepsilon _\mu ^{*}F^2(q^2) \end{aligned}$$

(8)

with $\mathcal {C}_{a}=\sqrt{2}g_{\Sigma _c}\gamma _5$ and $\mathcal {P}_{D}^{F} = 1/(q^2-m_D^2)$ the Feynman propagator of the D meson. To take into account the finite extension of the relevent hadrons, we also introduce the form factor in the amplitude with the monopole form

$$\begin{aligned} F(m_{q}^{2},q^2)=\frac{\Lambda ^2-m_{q}^2}{\Lambda ^2-q^2}, \end{aligned}$$

(9)

where $m_{q}$ is the mass of the exchanged meson. The cutoff in the form factor can be parametrized as $\Lambda =m_{q}+\alpha \Lambda _{QCD}$ with $\Lambda _{QCD}=220$ MeV. The parameter $\alpha$ will be discussed in further detail below.

Finally, the differential cross section in the center of mass (c.m.) frame is written as

$$\begin{aligned} \frac{\text {d}\sigma }{\text {d}\cos \theta } = \frac{1}{32\pi s}\frac{|\varvec{k}_2|}{|\varvec{k}_1|}\left( \frac{1}{2}\sum _\lambda |\mathcal {M}|^2\right) , \end{aligned}$$

(10)

with $s=(k_1+p_1)^2$ and $\theta$ the angle of the outgoing $X_0/X_1$ meson relative to the kaon beam direction in the c.m. frame. $\varvec{k}_1$ and $\varvec{k}_2$ are the three-momentum of the initial kaon and the final $X_0/X_1$, respectively.

3 Numerical results

With the formalism and ingredients given above, the total cross sections versus the c. m. energy W for $K^{+}p \rightarrow \Sigma _{c}^{++}X_{0,1}$ are calculated. The theoretical numerical results are presented in Fig. 2. In these calculations, the parameter $\alpha$ in the parameterized cutoff is still a free parameter. It has been discussed in some literatures. For instance, by fitting the experimental data, a value of $\alpha =2.2$ was extracted for the exchanged particles $D^{*}$ and D [46]. However, $\alpha$ is expected to be of order unity and it depends both on the exchanged particle and the external particles involved in the strong interaction vertex. Since we do not calculate $\alpha$ from first-principles, in the present work we just employ a typical value of $\alpha =1.5$ lying between the unity and 2.2 to study the production rate.

Fig. 2

The energy dependence of the total cross section for the productions of the $X_0$ and $X_1$ in $K^+ p$ collisions with $\alpha =1.5 \pm 0.5$ GeV. The red solid and violet dashed lines are for the productions of $X_0$ and $X_1$, respectively. The bands stand for the error bar of the cutoff $\alpha$

In Fig. 2, it is easy to find that the line shape of the cross section of $K^{+}p \rightarrow \Sigma _{c}^{++}X_{0}$ goes very rapidly near the threshold and has a maximum 0.5 nb around $W=6.3$ GeV. It indicates that $W=6.3$ GeV is the best energy window to the search for $X_{0}$ via kaon induced reaction. The cross section for $K^{+}p \rightarrow \Sigma _{c}^{++}X_{1}$ reaction is also presented. The evolutionary trend of the line shape is similar to that of $X_{0}$ production. The total cross section reaches a maximum of about 25 nb at $W=6$ GeV. It’s obvious that the total cross section for $K^{+}p \rightarrow \Sigma _{c}^{++}X_{1}$ is about fifty times larger than that of $X_{1}$.

Besides the cross sections with fixed $\alpha$, the effect of different $\alpha$ on the cross section is also studied with a range of $\alpha =1.0-2.0$. The corresponding cross sections with these different $\alpha$ values are depicted as the bands in Fig. 2. The lower and upper edges of the band correspond to $\alpha =1.0$ and $\alpha =2.0$, respectively. That means the cross section increases with the increasing of the $\alpha$ value in the presented region.

Fig. 3

The differential cross section $\text {d}\sigma /\text {d}\cos {\theta }$ of the $X_0$ production in $K^{+}p$ collisions at different center-of-mass energies with $\alpha =1.5$ GeV

Fig. 4

The differential cross section $\text {d}\sigma /\text {d}\cos {\theta }$ of the $X_1$ production in $K^{+}p$ collisions at different center-of-mass energies with $\alpha =1.5$ GeV

In Figs. 3 and 4, the differential cross sections of the reactions $K^{+}p \rightarrow \Sigma _{c}^{++}X_{0}$ and $K^{+}p \rightarrow \Sigma _{c}^{++}X_{1}$ are also predicted, where $\alpha =1.5$ is adopted. It is found that the differential cross sections are very sensitive to the $\theta$ angle. It’s obvious that with the increase of energy, the strong forward-scattering enhancement becomes more and more apparent. Thus the experimental measurement at forward angles is highly recommended.

Fig. 5

The energy dependence of the total cross section for the productions of the $X_0$ and $X_1$ in $K^+ n$ collisions with $\alpha =1.5$ GeV

Besides the proton target, the neutron and deuteron targets should be alternative with kaon beam to study the productions of $X_{0}$ and $X_{1}$. In the present work, we also explore the productions of $X_{0,1}$ through the process $K^{+}n \rightarrow \Lambda _{c}^{+}X_{0,1}$ with $t-$channel $D^{-}$ exchange. The relative amplitudes can be directly obtained from Eqs. (7) and (8) with the replacement of $\mathcal {C}_{a}$ by $\mathcal {C}_{b}=g_{\Lambda _c}\gamma _5$. The numerical results are shown in Fig. 5 with the fixed $\alpha =1.5$. It’s obvious that the cross section of $X_{0} (X_{1})$ in $K^{+}n$ reaction is about one order of magnitude larger than that in $K^{+}p$. The main reason is that the coupling of $DN\Lambda _{c}$ is stronger than that of $DN\Sigma _{c}$.

4 Summary

In this work, we study the productions of the exotic mesons $X_{0}(2900)$ and $X_{1}(2900)$ through the kaon induced reactions $K^{+}p \rightarrow \Sigma _{c}^{++}X_{0}$ and $K^{+}p \rightarrow \Sigma _{c}^{++}X_{1}$. With the assumption for the decay widths, the total cross sections are estimated to reach about 0.5 nb and 25 nb for the productions of $X_{0}$ and $X_{1}$, respectively. Correspondingly, the best energy windows for searching these two states are $W=6.3$ GeV and $W=6.0$ GeV, respectively. Hence, an experimental study of $X_{0}$ and $X_{1}$ via $K^{+}p$ reaction is suggested. Moreover, a further analysis indicates that the differential cross section is quite sensitive to the $\theta$ angle and gives a considerable contribution at forward angles. These results will provide an important basis for studying the production mechanisms of $X_{0}$ and $X_{1}$ in the future.

Meanwhile, the productions of $X_{0,1}$ with kaon beam and neutron target are also studied. The numerical results indicate that the cross sections reach up to nearly 8 and 400 nb for the production of $X_{0}$ and $X_{1}$, respectively. Thus it may be more recommended to search $X_{0,1}$ through $K^{+}n$ scattering reaction. Our results are helpful to the possible experimental research on $X_{0}$ and $X_{1}$ at the kaon beam facilities such as the K10 Beam Line at J-PARC [47], OKA at U-70 [34, 48], CERN SPS [49], etc.

In addition, one notes that the spin partners of the $X_0(2900)$, e.g., the $J^P=1^+$ and $J^P=2^+$ S-wave hadronic molecules were predicted in theories [14, 15, 19, 25]. Some authors also suggest to search $Z^{++}$, which is a doubly charged exotic state with inner structure of $K^+D^+(cu\bar{s}\bar{d})$ predicted in Ref. [50, 51]. Thus, it is an interesting topic to study these predicted states via kaon induced reactions in the future.