-
The LHCb collaboration found a very narrow peak structure named
T^+_{cc} in theD^0D^0\pi^+ invariant mass spectrum in thepp \to X + D^0D^0\pi^+ process [1]. The mass parameters obtained from a generic constant-width Breit-Wigner fit were listed as\delta m_{\rm BW} = -273 \pm 61 \pm 5^{+11}_{-14}\; \mathrm{keV} ,\,\,\, \Gamma_{\rm BW} = 410 \pm 165 \pm 43^{+18}_{-38}\; \mathrm{keV}\,,
where
\delta m_{\rm BW} defines the mass shift with respect to theD^{*+}D^0 threshold. Later, it was suggested thatT^+_{cc} could more possibly be an isoscalar state with spin-parity quantum numbersJ^P = 1^+ [2], and in a more complicated model, the pole mass and width were extracted as\delta m_{\rm pole} = -360 \pm 40^{+4}_{-0} \; \mathrm{keV}\ ,\,\,\, \Gamma_{\rm pole} = 48\pm 2^{+0}_{-14}\; \mathrm{keV}\,.
The constituent of
T^+_{cc} iscc\bar{u}\bar{d} and there is no annihilated quark pair, similar toX_1(2900) (ud\bar{s}\bar{c} ) [3, 4].This experimental observation has stimulated numerous theoretical discussions. First of all, there have been some dynamical lattice QCD simulations about double charmed tetraquarks, although they have not provided a definite conclusion on the existence of the
T^+_{cc} state [5, 6]. Recently, based on (2+1)-flavor lattice QCD simulations, theD^*D system was studied more carefully. It was verified that there is a loosely-bound state near the threshold (–10 keV) [7]. Many phenomenological studies have also been conducted. A theoretical prediction indicated that there may exist acc\bar{u}\bar{d} tetraquark withJ^P = 1^+ near theD^{*+}D^0 threshold [8, 9]. In addition, the heavy meson chiral effective field theory (HMChEFT), which considers contact and one pion exchange (OPE) interaction, was used. The preferred conclusion of the analyses is that theT^+_{cc} state is a molecule ofD^{*+}D^0 andD^{*0}D^+ [10, 11]. The effect of the triangle diagram singularity was also evaluated withD^*D\pi interactions. It was found that the contribution is very weak compared with that of the tree diagram, which suggests thatT^+_{cc} is not generated from the triangle singularity [12]. The pole parameters ofT^+_{cc} extracted from a simple K-matrix amplitude were also studied and it was found thatT^+_{cc} may originate from aD^{*+}D^0 virtual state [13]. The extended chiral Lagrangian with K-matrix unitarity approach was also applied, and it was suggested that vector meson exchanges could play a crucial role in forming theT^+_{cc} bound state ofD^*D [14].In this work, to determine whether
T^+_{cc} is just a loosely-bound s-wave molecule ofD^*D or it contains thecc\bar{u}\bar{d} ingredient, different approaches are used. First, we adopt an approach similar to that of Ref. [14], in which a coupled channel unitarity approach (D^{*+}D^0 andD^{*0}D^+ ) with the interaction stemming from the extended hidden local gauge lagrangian was also applied. In Ref. [14], the authors only considered the vector exchanging diagram contributions and there was no fitting of the lineshape data. In this study, more complete interactions, including pseudoscalar, vector exchanges, andD^*D contact terms, are introduced, and a combined fit ofDD\pi ,D\pi , andDD channels is made. It indicates that the ρ vector meson exchange couplings really make non-negligible contributions in generating theT^+_{cc} resonance compared with the other two interactions. In this scheme, there exists a bound state near theD^{*+}D^0 threshold, which suggests thatT^+_{cc} may be aD^*D molecule. Furthermore, the Flatté-like parametrization is also examined. Through a combined fit on three-body and two-body invariant mass spectrum, we find that the result is the same based on the pole counting rule (PCR) and spectral density function sum rule calculation [15−20]: there is only one pole nearD^*D threshold and the corresponding\mathcal{Z} \simeq 1 . We also attempt to judge the compositeness ofT^{+}_{cc} by comparing its production (pp\to T^{+}_{cc} + X ) with different theoretical estimations. However, it is difficult to make a clear judgement using this approach.This paper is organized as follows: Sec. I is the introduction, in Sec. II, the K-matrix unitarization approach using an effective Lagrangian in s-wave approximation is introduced and its numerical fit is shown. In Sec. III, other frameworks are employed to analyze the compositeness of
T^+_{cc} . Finally, in Sec. IV, a brief conclusion on the structure ofT^+_{cc} is drawn. -
We start off from a
S U(4) invariant effective Lagrangian, and then modify the relevant parameters to only preserve theS U(2) symmetry later, as in [21]. Here, we list the relevant coupling terms,\begin{aligned}[b] \mathcal{L} =& \mathcal{L}_0 - {\rm i} g\mathrm{Tr}([P,\partial_\mu P]V^\mu)+ {\rm i} g\mathrm{Tr}([V^\nu, \partial_\mu V_\nu]V^\mu)\\ & -\frac{g^2}{2}\mathrm{Tr}([P,V_\mu]^2)+\frac{g^2}{4}\mathrm{Tr}([V_\mu,V_\nu]^2)\ , \end{aligned}
(1) where
\mathcal{L}_0 is the free Lagrangian for the pseudoscalar and vector mesons. P and V denote, respectively, the properly normalized pseudoscalar and vector meson matrices\begin{array}{*{20}{l}} P = \begin{pmatrix} \dfrac{\eta}{\sqrt{3}}+\dfrac{\eta'}{\sqrt{6}}+\dfrac{\pi^0}{\sqrt{2}}&\pi^+&K^+&\bar{D}^0\\ \pi^-&\dfrac{\eta}{\sqrt{3}}+\dfrac{\eta'}{\sqrt{6}}-\dfrac{\pi^0}{\sqrt{2}}&K^0&D^-\\ K^-&\bar{K}^0&-\dfrac{\eta}{\sqrt{3}}+\sqrt{\dfrac{2}{3}}\eta'&D^-_s\\ D^0&D^+&D^+_s&\eta_c \end{pmatrix}\ , \end{array} (2) \begin{array}{*{20}{l}} V = \begin{pmatrix} \dfrac{\omega}{\sqrt{2}}+\dfrac{\rho^0}{\sqrt{2}}&\rho^+&K^{*+}&\bar{D}^{*0}\\ \rho^-&\dfrac{\omega}{\sqrt{2}}-\dfrac{\rho^0}{\sqrt{2}}&K^{*0}&D^{*-}\\ K^{*-}&\bar{K}^{*0}&\phi&D^{*-}_s\\ D^{*0}&D^{*+}&D^{*+}_s&J/\psi \end{pmatrix}\ . \end{array}
(3) In the later discussions, all the coupling constants denoted as g in the vertices of the PPV, VVV, and PPVV types in Eqs. (2) and (3) could be different for the vertices of different isospin multiplets, such that only the
S U(2) isospin symmetry is retained. Hereafter, we only consider the vertices relevant to our discussions. We adopt previous theoretical works [18, 21, 22] to estimate these coupling constants because the interaction vertices are similar to theirs with only normalization constant differences, which can be tracked with a careful analysis.From Eq. (1), we can obtain the contact, t, and u channel diagrams of the
D^{*+}D^0\to D^{*0}D^+ process. We list their amplitudes successively. First, for the contact diagrams in Fig. 1,\begin{array}{*{20}{l}} {\rm i} M^{c}_{ij} = {\rm i}\, g_{D^*DD^*D}^2, \end{array}
(4) where
i,j = 1,2 refer to theD^{*+}D^0 andD^{*0}D^+ channels, respectively. The couplingg_{D^*DD^*D} was estimated when studyingX(3872) [18], that is,g (=g_{D^*DD^*D}) \simeq 16 1 .The t channel diagrams include vector meson (
J/\psi or ρ, ω) exchanges [14], as shown in Fig. 2.2 We also neglect the momentum dependence in the denominator of the propagator near the threshold. Here, the estimates about the coupling constants from the PPV and VVV vertices are as follows: Fori, j = 1,1 or2,2 , the coupling constantg(= g_{J/\psi D^{(*)}D^{(*)}})\simeq 7.7 , and fori, j = 1,2 or2,1 ,g(= g_{\rho D^{(*)}D^{(*)}}) \simeq 3.9 , which are obtained from the vector meson dominance (VMD) assumption [21]. The t-channel amplitudes are hence written down as follows:\begin{array}{*{20}{l}} {\rm i} M^t_{ij} = {\rm i} D_{ij} (p_1 +p_3)\cdot(p_2 +p_4)\; \epsilon(p_1)\cdot\epsilon^*(p_3), \end{array}
(5) where
\begin{array}{*{20}{l}} D_{ij} = \begin{pmatrix} \dfrac{g^2_{J/\psi D^{(*)}D^{(*)}}}{M^2_{J/\psi}}&\dfrac{g^2_{\rho D^{(*)}D^{(*)}}}{m^2_\rho}\\ \dfrac{g^2_{\rho D^{(*)}D^{(*)}}}{m^2_\rho}&\dfrac{g^2_{J/\psi D^{(*)}D^{(*)}}}{M^2_{J/\psi}}\\ \end{pmatrix}_{ij}. \end{array}
(6) The third type are u channel diagrams with π exchanges as shown in Fig. 3, and the amplitudes are
{\rm i} M^u_{ij} = {\rm i} E_{ij} g_{\pi DD^*}^2\frac{\epsilon(p_1)\cdot (p_1-2p_4)\; \epsilon^*(p_3)\cdot (p_3-2p_2)}{(p_1-p_4)^2-m_\pi^2},
(7) where
\begin{array}{*{20}{l}} E_{ij} = \begin{pmatrix} -1&\dfrac{1}{2}\\ \dfrac{1}{2}&-1\\ \end{pmatrix}_{ij}. \end{array}
(8) The coupling strength
g_{\pi DD^*} can be restricted by the decay processD^{*+} \to D^0 \pi^+ , and we take the valueg(=g_{\pi DD^*}) \simeq 8.4 [22]. u-channel diagrams with pseudo-scalar meson\eta_c exchanges also exist. They are not important at this energy range as it is tested numerically, so we neglect these diagrams. As for the amplitudes corresponding to Fig. 3, the u-channel π exchange is somewhat special because it is possible to exchange one on-shell π meson. After partial wave projection, there exists, in tree level amplitudes, an additional cut in the energy region above theD^{*+}D^0 threshold. Here, this singularity will affect the unitarity. To remedy this, we adopt the approximate relationm_{D^*} = m_{D} + m_{\pi} to keep the unitarity threshold away from the singularity, as in Refs. [10, 11]. At last, we obtain the total coupled channel amplitudes\begin{array}{*{20}{l}} M_{ij} = M^c_{ij} + M^t_{ij} +M^u_{ij} \ . \end{array}
(9) Furthermore, with the assumption that the full amplitude is mainly contributed by the s-wave amplitude and the d-wave amplitude can be neglected, we consider the s-wave amplitude, which can be unitarized by the relation
\begin{array}{*{20}{l}} {\bf{T}}^{-1} = {\bf{K}}^{-1} - {\bf{g}}(s), \end{array}
(10) where
{\bf{T}} is the unitarized s-wave scattering T matrix,{\bf{K}} is the two-channel s-wave scattering amplitude matrix fromM_{ij} [23], and{\bf{g}}(s)\equiv \mathrm{diag}\{g_i(s)\} . In our normalization convention\begin{aligned}[b]& g_i(s;M_i,m_i) \\=& -16\pi^2 {\rm i}\int \frac{{\rm d}^4q}{(2\pi)^4} \frac{1}{(q^2-M_i^2+ {\rm i}\epsilon)((P-q)^2-m_i^2+ {\rm i}\epsilon)},\,\, (s = P^2) \end{aligned}
(11) where
M_i is the vector meson mass andm_i is the pseudoscalar meson mass in the i-th channel. The expression ofg_i(s) in Eq. (11) is renormalized using the standard\overline{\mathrm{MS}} scheme, which introduces an explicit renormalization scale (μ) dependence. In our fit, we select the same μ parameter in the two channels.To obtain a finite width for the
T_{cc}^+ state below theD^*D threshold, we need to consider the finite width of theD^* state. This is accomplished by performing a convolution of theg_i(s) functions with the mass distribution of theD^* states [24]:S\left(s_V;M_V,\Gamma_V\right) \equiv -\frac{1}{\pi} \operatorname{Im}\left\{\dfrac{1}{s_V-M_V^2+ {\rm i} M_V \Gamma_V}\right\}
(12) such that
\tilde{g}_i(s;M_i,m_i)=\mathcal{C} \int_{s_{V \min}}^{s_{V \max}} {\rm d} s_V g_i(s;\sqrt{s_V},m_i) S\left(s_V;M_i,\Gamma_{i}\right) ,
(13) where
\mathcal{C} is a normalization factor. The main contribution to this integration is from the region around the unstable masss_V\sim M_V^2 , so we can introduce a cutoffs_{V \min} ands_{V \max} . For example, for\tilde{g}_1 , it is integrated from(m_{D_0}+ m_{\pi^+})^2 to(m_{D^{*+}}+2\Gamma_{D^{*+}})^2 , whereas for\tilde{g}_2 , it is integrated from(m_{D_0}+m_{\pi^0})^2 to(m_{D^{*0}}+2\Gamma_{D^{*0}})^2 . Here, in principle,\Gamma_i is s-dependent as in Ref. [14]. However, we approximate the decay widths as constants because we only focus on a small region near theD^{*+}D^0 threshold, and we also verified that it makes little difference if the s dependence is included in the numerical calculations. The constant decay widths suggested by PDG [25] read\begin{array}{*{20}{l}} \Gamma_{D^{*+}} = 83.4\; \mathrm{keV},\; \; \Gamma_{D^{*0}} = 55.3\; \mathrm{keV}. \end{array}
(14) To fit the final state three-body invariant mass spectrum of
D^0D^0\pi^+ inpp \to X + D^0D^0\pi^+ , the final-state interaction (FSI) [20] betweenD^{*+}D^0 and/orD^{*0}D^+ needs to be considered as above, before contemplating theD^{*+}\to D^0+\pi^+ decay. The amplitude for theD^{*+} D^0 final state reads\begin{array}{*{20}{l}} \mathcal{F}_{D^{*+} D^{0}}(s)=\alpha_{1}\; T_{11}+\alpha_{2}\; T_{21}\ , \end{array}
(15) where
\alpha_1 ,\alpha_2 are smooth real polynomials parametrizing the amplitude of producingD^{*+}D^0 andD^{*0}D^+ , respectively, and as the energy region of interest is very small, we treat them as constant parameters near the thresholds ofD^{*+}D^0 andD^{*0}D^+ . Finally, the decay ofT^+_{cc}\to D^0D^0\pi^+ can therefore be expressed as in Fig. 4, and the final s-wave scattering amplitude is written as3 \begin{aligned}[b] t = &\mathcal{F}_{D^{*+}D^0}\left[ \frac{ {\epsilon}\cdot[({p}_1-{p}_2)+\dfrac{m_{\pi^+}^2-m_{D^0}^2}{m_{D^{*+}}^2}({p}_1+{p}_2) ]}{M_{12}^2-m_{D^{*+}}^2+ {\rm i} M_{12}\Gamma_{D^{*+}}(M_{12})}\right.\\ &\left.+\frac{ {\epsilon}\cdot [({p}_3-{p}_2)+\dfrac{m_{\pi^+}^2-m_{D^0}^2}{m_{D^{*+}}^2}({p}_3+{p}_2) ]}{M_{23}^2-m_{D^{*+}}^2+ {\rm i} M_{23}\Gamma_{D^{*+}}(M_{23})} \right]. \end{aligned}
(16) Here,
M_{12} andM_{23} are Dalitz kinematic variables of the final three-body state. The corresponding definition iss_{ij}=M_{ij}^2 = (p_i + p_j)^2 , and{\epsilon}={\epsilon}(P) corresponds to the polarization vector ofT^+_{cc} ,P=\left(p_{1}+p_{2}+p_{3}\right) , andP^{2}=s . These invariants have the relationM^2_{12} + M^2_{13} + M^2_{23} = P^2 + p_1^2+ p_2^2+ p_3^2 . Finally, the decay width ofT_{cc}^+ is given by{\rm d}\Gamma(\sqrt{s}) = \frac{\mathcal{N}}{2} \frac{32}{\pi}\frac{1}{s^{3/2}} |t|^2 {\rm d} s_{12} {\rm d} s_{23}.
(17) The factor
\dfrac{1}{2} in the above equation results from averaging the two integrals ofD^0 in the final state. To fit the experimental data, the normalization factor\mathcal{N} should be introduced. As for the two FSI parameters,\alpha_1 can be absorbed in the coefficients\mathcal{N} . Thus,\alpha_1 = 1 is fixed and there remains one free parameter\alpha_2 . Besides, to obtain the yields for theD^0D^0\pi^+ invariant mass spectrum, the resolution function needs to be convoluted\mathrm{Yields}(l)=\int_{l-2 \sigma}^{l+2 \sigma} {\rm d} l^{\prime} \frac{1}{\sqrt{2 \pi} \sigma} \Gamma\left(l^{\prime}\right) \mathrm{e} ^{-\frac{\left(l^{\prime}-l\right)^2}{2 \sigma^2}},
(18) where
\sigma = 1.05 \times 263\;\; \mathrm{keV} [1]. At last, invariant mass distributions for the selected two-body state (particles 2 and 3, for example) can also be derived as the following function:\frac{{\rm d} Br}{{\rm d} M_{23}} = \mathcal{N}' \int_{m^2_{D^0D^0\pi^+}}^{m_{\max}^2} {\rm d} s \int {\rm d} s_{12} |t(s,s_{12},s_{23})|^2 ,
(19) where
\mathcal{N}' is another normalization constant, andM_{23} is the invariant mass of particles 2 and 3. TheT^+_{cc} energy is integrated from the initial energym_{D^0D^0\pi^+} to a cutoffm_{\max} .4 Data obtained from the LHCb collaboration about three-body final states
D^0D^0\pi^+ [1] and two-body invariant mass distributionsD^0\pi^+ ,D^0D^0 , andD^+D^0 [2] are used to make a combined fit. The normalization\mathcal{N} ,\mathcal{N'} , FSI parameter\alpha_2 , and renormalization scale μ are regarded as free parameters to be fitted, and all coupling constants found in the literature are regarded as fixed parameters.The fit result is presented in Fig. 5 and Table 1. It is found that the fit result is very sensitive to the parameter μ. That occurs because the peak (
T^+_{cc} state) is too narrow, considering that the unit of μ is GeV but the signal range is in MeV. The discussion above seems to suggest that the fit result prefers a particular choice of parameter μ. In [14], a specific fixed\mu=1.5 GeV was also taken in their analysis, which is similar to our result.Figure 5. (color online)
D^0D^0\pi^+ final state invariant mass spectrum. The vertical purple dash line indicates theD^{*+}D^0 threshold and the green one corresponds to theD^{*0}D^+ threshold. Data obtained from Ref. [1].The pole location on the s-plane is also studied. If
D^{*} is taken as a stable particle, thenT_{cc}^+ appears as a bound state pole located at\sqrt{s} = 3.8746 , that is, approximately500 keV below theD^{*+}D^0 threshold (\sqrt{s} = 3.8751 ). As there is no accompanying virtual pole nearby, we conclude that, according to the pole counting rule,T^+_{cc} is a pure molecule composed ofDD^* . However, due to the instability ofD^* , theD^{*}D channel opens at the energy somewhat smaller thanm_{D^*} + m_D and the decay ofT_{cc}^+ takes place [14].Furthermore, invariant mass distributions for any two of three final state particles are also taken into consideration. The fit result is shown in Fig. 6. As for the
D^0\pi^+ state, which comes fromD^{*+}D^0 , we takem_{\max} = 3.8751 GeV. As forD^0D^0 states, we take the samem_{\max} = 3.8751 GeV (\mathcal{N}' = \mathcal{N}_{DD} here). TheD^+D^0 final state, which comes from theD^+D^0\pi^0 final state, is different. SinceD^+D^0 state may come from two channels,D^{*+}D^0 andD^{*0}D^+ , they need to be considered altogether aided by isospin symmetry. Since the threshold of the second channel is higher, we takem_{\rm max} = 3.8766\;\;\mathrm{GeV} , and on account of a symmetry factor{1}/{2} in the channel includingD^0D^0 , the normalization constant here is doubled (\mathcal{N}' = 2\mathcal{N}_{DD} ). The fitting results are plotted in Fig. 7. Both invariant mass spectra (D^0D^0 andD^+D^0 ) have an incoherent background component, parametrized as a product of the two-body phase space function\Phi_{DD} and a linear function. ForD^+D^0 from channelT^+_{cc} \to D^{*0}D^+\to D^+D^0\pi^0/D^+D^0\gamma , because the decay channelD^{*0}\to D^0\gamma accounts for 35% of the totalD^{*0} decay width, this incoherent background contribution is non-negligible and needs to be counted specially. Here, we take this estimation from Ref. [2] directly.Figure 6. (color online)
D^0\pi^+ invariant mass spectrum from the three body final stateD^0D^0+X (Data from Ref. [2]).Figure 7. (color online)
D^0D^0 (D^+D^0 ) invariant mass spectrum from the three body final stateD^0D^0+X (D^+D^0 + X ) [2], where the vertical dashed line indicates theD^{0}D^{0} (D^+D^0 ) threshold.Table 1. Parameters.
Finally, the fit parameters are listed in Table 1.
-
In this section, the production of
T^+_{cc} in some other methods is also analyzed to determine its compositeness. First of all, a single channel Flatté-like parametrization is used, where we do not distinguishD^{*+}D^0 andD^{*0}D^+ anymore. As in the previous calculation in Sec. II, this process is regarded as a cascade decay, as shown in Fig. 8. The propagator ofT^+_{cc} is approximated by Flatté formula. The later propagator ofD^* selected here is a simple Breit–Wigner amplitude form because the energy of this process is near the threshold ofD^*D and its range is sufficiently small enough. Besides, the momentum dependent polynomial in the numerator is also normalized by a constant factor\mathcal{N} for convenience. Numerical calculations indicate that these approximations make little difference.The total s-wave approximation amplitude about the process
T^+_{cc} \to D^0D^0\pi^+ can be written as\begin{aligned}[b] t= & \frac{1}{s-M^2+{\rm i} M\left(\hat g \rho(s)\right)}\\ &\times\left(\frac{1}{M_{12}^2-m_{D^{*+}}^2+{\rm i} M_{12} \Gamma_{D^{*+}}}+\frac{1}{M_{23}^2-m_{D^{*+}}^2+{\rm i} M_{23} \Gamma_{D^{*+}}}\right), \end{aligned}
(20) where
\hat g presents the coupling strength withD^*D . The doubling of the kinetic variables of theD^* propagator is due to the indistinguishability ofD\pi in the three-body final state, and the symmetry factor\dfrac{1}{2} is absorbed by the total normalization\mathcal{N} . By this parametrization, we make the energy resolution convolution as that before and fit the three-body decay width and two-body invariant mass spectra at the same time using the previous Eqs. (17) and (19). It is worth pointing out that under normal conditions, it will form a divergent peak because of the zero partial decay width. However, if we regardD^* as an unstable particle, in other words, if the amplitude can develop an imaginary part when the energy has not yet reached theD^*D threshold, the peak is not divergent anymore. We can use the same stratagem as Eq. (13) to treat ρ in Eq. (20), or more simply take the valuem_{D^*} in ρ with an imaginary part\Gamma_{D^*} . This selection does not affect the result except for the goodness of fit. Here, we take the latter scenario. The results of the combined fit are shown in Fig. 9.We list the corresponding parameters in Table 2, and the pole structure of the Flatté amplitude is displayed in Fig. 10.
\chi^2/d.o.f \hat g M 0.81 keV0.075\pm 0.015 3874.1\pm 0.2 MeVTable 2. Fit parameter.
Furthermore, according to the Flatté-like parametrization, it is natural to calculate the probability of finding an 'elementary' state in the continuous spectrum by the spectral density function [26]
\omega(E)=\frac{1}{2 \pi} \frac{\tilde{g} \sqrt{2 \tilde{M} E} \theta(E)+\tilde{\Gamma}_0}{\left|E-E_f+\dfrac{\mathrm{i}}{2} \tilde{g} \sqrt{2 \tilde{M} E} +\dfrac{\mathrm{i}}{2} \tilde{\Gamma}_0\right|^2},
(21) where
E = \sqrt{s} - m_{D^*D} ,E_f = M - m_{D^*D} ,\tilde{M} is the reduced mass ofD^*D , θ is the step function at threshold,\tilde{g} = 2 \hat{g} / m_{D^*D} is the dimensionless coupling constant of the concerned coupling mode, and\tilde{\Gamma}_0 is the constant partial width for the remaining couplings. By integrating it with a cut off (usually comparable to the total decay width\sim \Gamma ), the possibility of finding an 'elementary' state in the final state is\mathcal{Z}=\int_{E_{\min }}^{E_{\max }} \omega(E) \mathrm{d} E.
(22) Considering that no other channels are coupling with
T^+_{cc} under theD^*D threshold, the\tilde{\Gamma}_0 here should be set to zero. In this case, the integrated results in different sections are as follows.The result suggests that in a simple single channel Flatté-like parametrization framework,
T^+_{cc} is a pure molecular state. This is in agreement with the result of Ref. [27], obtained using an effective range expansion approximation. Our result is much more definite than that obtained in Ref. [2].Furthermore, the compositeness may also be discussed from the production rate of a particle. The cross section relation between the confined states
\Xi_{cc} (ccu/ccd ) [28] andT^+_{cc} can be obtained from the experiment. Since 2016, these two sets of experimental data are both collected under the same experimental conditions, such as transverse momentum truncationp_T and luminance9~\rm fb^{-1} . After taking account of the detection efficiency and branching fraction differences [29], there is an approximate relation\frac{\sigma(pp\to T^+_{cc})}{\sigma(pp\to \Xi_{cc})} \sim \frac{1}{3} \times \frac{1}{10}.
(23) If we agree that there exists a universal ratio between the
(Q/QQ)q and(Q/QQ)qq productivities in high energy collision [30], where Q represents a heavy quark and q is a light quark, we can obtain a factor1/3 , which means that catching two light quarks is always more difficult, that is,\sigma(pp\to \Xi_{cc}) \simeq 3 \sigma(pp\to (cc\bar{u}\bar{d})) . Thus, the ratio between the cross sections of the observedT^+_{cc} and the hypothetical tetraquark can be obtained as\frac{\sigma(pp\to T^+_{cc})}{\sigma(pp\to (cc\bar{u}\bar{d}))} \sim \frac{1}{10}.
(24) It is also possible to estimate the different orders of magnitude of the theoretical cross sections between the `elementary' and `molecular' pictures of
T^+_{cc} . Because theX(3872) resonance has analogous characteristics [31, 32] (e.g., binding energy and quark composition), some comparisons have been made about these orders of magnitude forX(3872) [33, 34]. If one can borrow the discussions here, it can be estimated that approximately forT^+_{cc} \frac{\sigma(pp\to (c\bar{u})(c\bar{d}))}{\sigma(pp\to (cc\bar{u}\bar{d}))} \sim \mathcal{O}(10^{-2})-\mathcal{O}(10^{-3}).
(25) A similar result was also obtained in [35]. By comparing Eq. (24) and Eq. (25), one can find that the production of
T^+_{cc} just falls in between two different cases. However, this analysis depends on some uncertain assumptions and is not quite reliable. In Ref. [36], the production cross section forT_{cc} as a molecule was estimated to be approximately an order of magnitude higher than that as a tetraquack, which creates more confusion. Thus, using the production argument cannot provide a clear conclusion on the nature ofT^+_{cc} . On the contrary, the analysis provided in this paper, for example in Table 3, clearly indicates the molecular nature ofT^+_{cc} .[M-\Gamma, M+\Gamma] [M-2\Gamma, M+2\Gamma] [M-3\Gamma, M+3\Gamma] \sim 0\sim 00.01 Table 3. Spectral density function integrating
\mathcal{Z} .
Some remarks on compositeness of {{\boldsymbol T}^{\bf +}_{\boldsymbol{cc}}}
- Received Date: 2023-10-26
- Available Online: 2024-04-15
Abstract: Recently, the LHCb experimental group found an exotic state