Searching for doubly charmed tetraquark candidates T cc and T cc ¯ s in B c decays

In this work, we propose to search for the exotic doubly charmed meson T + cc and its analog T + cc ¯ s in B + c decays, which provide a good environment for the formation of the exotic state containing double charm quarks. Within the molecular scheme, the production of T + cc and T + cc ¯ s through various rescattering processes with different intermediate states are investigated. For the moderate values of model parameters, the branching ratios of B + c decaying into T + cc ¯ D 0 , T + cc ¯ D ∗ 0 , T + cc ¯ s ¯ D 0 and T + cc ¯ s ¯ D ∗ 0 are estimated to be of the order of 10 − 7 , 10 − 5 , 10 − 6 and 10 − 4 , respectively, which may be tested by future experiments.


I. INTRODUCTION
The LHCb collaboration recently reported the observation of a narrow doubly charmed tetraquark candidate T + cc in the prompt production of proton-proton collisions [1,2].This is the first observation of the exotic state containing double charm quarks.The T + cc is observed in the D 0 D 0 π + mass spectrum, and has mass of approximately 3875 MeV, which is just below the D * + D 0 threshold.Using the Breit-Wigner parametrization, the location of the resonance peak relative to the D * + D 0 threshold δ m and the width Γ are determined to be respectively.
The long-lived T + cc particle has the quark content ccū d and the spin-parity quantum number J P = 1 + .The flavor quantum number is absolutely exotic.In fact, the tetraquark state with double heavy quarks has been studied for many years.There are two popular theoretical pictures describing the underlying structures of such tetraquark states, i.e., the compact tetraquark picture and hadronic molecule picture.In the compact tetraquark picture, the doubly heavy state is usually thought to be composed of the compact diquark and anti-diquark , while in the hadronic molecule picture, it is composed of a pair of heavy mesons [26][27][28][29][30][31][32][33][34][35][36][37].In Refs.[38][39][40][41], the compact tetraquark and hadronic molecule pictures are taken into account simultaneously.According to the LHCb measurements, the rather closeness of the T + cc mass to the D * + D 0 threshold strongly favors the molecular explanation concerning the T + cc nature [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57].This is similar to the case of the famous X(3872) state, which is widely supposed to be a weakly bound hadronic molecule composed of . The boxes and dots represent the weak and strong vertices, respectively.c is treated as a spectator.In the naive factorization approach the effective Hamiltonian governing the process reads where (q 1 q 2 ) V −A ≡ q1 γ µ (1 − γ 5 )q 2 , G F is the Fermi constant, V cb and V cd are the CKM matrix elements, and c 1 and c 2 are the perturbatively calculable Wilson coefficients.
Neglecting the contributions from the nonfactorizable, colorsuppressed and annihilation terms, the decay amplitude of B + c → M cc D ( * )+ can be factorized as where a 1 = c 1 + c 2 /N c , and N c is the number of colors.The factorized amplitude can be expressed in terms of the form factors of the transition B + c → M cc and the decay constant of D ( * )+ .The form factors of B c decaying into the lower charmonia have been well investigated in the literature .To estimate the rescattering amplitudes, in this work we employ the relevant numerical results of Refs.[89,90], where the form factors are calculated by means of the covariant light-front approach.The form factors of B c → J/ψ and η c induced by the vector and axial-vector currents are defined by where P = p 0 + k 1 , q = p 0 − k 1 , and ε J is the polarization vector of J/ψ.The B c decaying to the P -wave charmonia χ c0 , χ c1 and h c form factors are defined by where the state A represents the axial vector meson χ c1 or h c .The second matrix element in the right-hand side of Eq. ( 3) is parameterized as One of the strong vertices in the rescattering diagrams of Figs. 1 and 2 involves a charmonium and a pair of open charm mesons.Taking into account the heavy quark spin symmetry, the effective Lagrangian describing these couplings is given by [79,115,116] with where is the charge conjugate field of H 1a , and a is the light flavor index.The Sand P -wave charmonia are collected into the fields J and χ µ , respectively.The pseudoscalar and vector charmed mesons are collected into the fields H 1 and H 2 .All of the heavy fields in the above equations contain a factor √ m with m the corresponding heavy meson mass.The • • • in Eqs. ( 12) and ( 13) means the trace over Dirac matrices.According to the effective Lagrangian L ψ and L χ , we can obtain the corresponding vertex function A(M cc → D( * )0 D ( * )0 ) in the rescattering diagram.The relevant vertex functions are collected in Appendix A. The coupling constants g ψ and g χ can be estimated by invoking the vector meson dominance arguments [78,79].The results are with f ψ and f χc0 the decay constants of J/ψ and χ c0 , respectively.
If we assume that the T + cc is a pure isoscalar hadronic molecule, its wave function can be written as The vertex functions in the rescattering diagrams for the Taking into account the isospin symmetry, the coupling constant g 1 should be equal to g 2 .However, the thresholds of D * + D 0 and D * 0 D + are located above m T + cc around 0.3 and 1.7 MeV, respectively, and in the molecular scheme the coupling of the loosely bound state T + cc with its component is relatively sensitive to the binding energy.Therefore we adopt the similar prescription used in Ref. [50] to account for the difference between g 1 and g 2 .The coupling constants take the form where where k 1 , k 2 and k 3 (m 1 , m 2 and m 3 ) correspond to the momenta (mass) of intermediate mesons M cc , D ( * )+ and D ( * )0 , respectively.The sum over polarization of intermediate state in Eq. ( 26) is implicit.For the intermediate spin-1 state, the sum over polarization reads pol 3 ) is introduced in Eq. ( 26), which takes the form This form factor is supposed to parameterize the off-shell effects of the intermediate state and to kill the ultraviolet divergence in the loop integrals.We employ a dipole form factor rather than a monopole form factor because the latter cannot kill the divergence appearing in some rescattering amplitudes.The loop integral is performed by employing the program package LoopTools [117].

B. Numerical analysis
In this subsection, we give the results from explicit calculations of the rescattering amplitudes.First we list the input parameters used in the numerical calculation.In the weak decay amplitude A(B + c → M cc D ( * )+ ), the central values of the CKM matrix elements |V cb | = 0.0408 and |V cd | = 0.221 reported by the Particle Data Group (PDG) [58] are adopted, and the combination of Wilson coefficients a 1 is set to be 1.14 [112].For the relevant decay constants, we use the following values: f D = 0.209 GeV [58], f D * = 0.245 GeV [118], f ψ = 0.416 GeV [58] and f χc0 = 0.510 GeV [78].Concerning the relevant particle mass, the PDG 2022 central values are used [58].
The numerical results of B c → η c , J/ψ, χ c0,c1 and h c form factors in Table II of Ref. [89] and Table I of Ref. [90] are employed.In our calculation of the loop integral, as an approximation we do not take into account the q 2 -dependence of the form factors.The values of these weak decay form factors are fixed at q 2 = m 2 D ( * )+ , i.e., an on-shell approximation of the weak vertex function is adopted.
There is still a free parameter, i.e., the cutoff energy Λ, in the dipole form factor Eq. ( 27).Its explicit value should be determined from the experimental data.As an theoretical prediction, the empirical value of Λ is usually set to be larger than the mass of the exchanged particle, and it also depends on the formalism of the form factor.For instance, in Ref. [78], one obtains Λ ≈ 2.7 GeV to roughly fit the experimental data of Br(B − → K − χ c0 ), where the form factor is the monopole type and the exchanged particles are D and D * .This cut-off energy is just around the typical values of the mass of the radially excited states of D ( * ) mesons.Taking into account the uncertainty induced by the cutoff Λ, we plot the branching ratio of B + c → T + cc D( * )0 via the rescattering processes as a function of Λ.The numerical results are displayed in Figs. 3  and 4, where we consider a relatively larger range 2.5-5 GeV for Λ.The uncertainties from the T + cc mass are also taken into account.
The thresholds of final states T + cc D0 and T + cc D * 0 are around 5739.7 and 5881.7 MeV, respectively, which are not far from the B + c mass 6274.5 MeV.Therefore the higher partial-wave amplitudes of the two channels are expected to be highly suppressed by the limited phase space.From Figs. 3 and 4, we can see that the two branching ratios increase monotonically with Λ increasing.Within the cutoff range 2.

III. PRODUCTION OF T
Inspired by the observation of T + cc , one may guess some other analogs can also exist, such as the doubly heavy states with the strange quark.In this work, we are interested in a T + cc analog named T + ccs .We assume T + ccs is a hadronic molecule composed of D * + s D 0 /D + s D * 0 , and the wave function is defined as The T + ccs mass relative to the lower D * 0 D + s threshold δ m is defined as The relation between T + cc and T + ccs is similar to that between Z c (3900) and Z cs (3985), which are widely supposed to be the hadronic molecules composed of D D * /D * D and D s D * /D * s D. In Ref. [37], the doubly heavy systems composed of a pair of heavy mesons have been systematically studied in a quasi-potential Bethe-Salpeter equation approach.The authors predict that the bound state with J P = 1 + can be found from the D * s D-D s D * coupled channel interactions, i.e., the hadronic molecule T + ccs we defined here may exist.We are mainly interested in the production of the T + ccs state in this work.In the B + c decays, the production mechanism of T + ccs is rather similar with that of T + cc .Replacing the D ( * )+ in Figs. 1 and 2  In the rescattering diagrams, the effective Hamiltonian gov-erning the weak process with |V cs | = 0.975 [58].The decay constants in the factorized amplitudes are taken as f Ds = 0.247 GeV [58] and f D * s = 0.272 GeV [118].Notice that is a Cabibbofavored process compared with B + c → M cc D ( * )+ .Therefore the branching ratio of B + c → T + ccs D( * )0 via the rescattering processes is also expected to be larger.
Following the rather similar procedures as that in Section II, we calculate the rescattering contributions and the numerical results are presented in Figs.7 and  .This is a sizable branching ratio.If the T + ccs state truly exists, it is very likely that we can find it in the B c decays.
The two lines in Fig. 7 has a crossing point.This is because that the T + ccs D * 0 D + s coupling is sensitive to the δ m when δ m is smaller.Employing the similar coupling definitions as those in Eqs.(23) and ( 24), The T + ccs D * + s D 0 coupling g 1 and T + ccs D * 0 D + s coupling g 2 are determined to be (g 1 , g 2 ) (7.1 GeV, 3.4 GeV) when δ m = 100 keV, and (g 1 , g 2 ) (7.8 GeV, 6.0 GeV) when δ m = 1 MeV.As a result the interferences among rescattering diagrams for the two cases behave differently.The δ m = 100 keV line is not simply scaled by the δ m = 1 MeV line.

IV. SUMMARY
In this work, we study the production of the doubly charmed state T + cc and its analog T + ccs in B c decays, which provide a good environment for the formation of the exotic meson containing double charm quarks.The T + cc or T + ccs is produced from a charmonium and a charmed meson reascattering via exchanging another charmed meson.The contributions from various rescatterings with different intermediate states are taken into account.The calculation of rescattering amplitudes is performed under the ansatz that T + cc and T + ccs are weakly bound hadronic molecules.For the moderate cutoff energy, the branching ratios Br We should also mention that even if T + cc , T + ccs and some other analogs are not hadronic molecules, the production mechanism of the doubly charm mesons proposed in this paper still works.Correspondingly, the couplings between the tetraquark states and open charm mesons need to be modified.
The predicted relatively sizable branching ratios suggest that in future experiments one may search for the T + cc and its analogs in the B c decay processes discussed here.Besides, investigating the different production mechanisms of these exotic doubly charmed states is also crucial in revealing their underlying structures.
Here we give the vertex functions A(M cc (k 1 ) → D( * )0 (p 1 )D ( * )0 (k 3 )) corresponding to a charmonium coupling with a pair of open charm mesons:

FIG. 2 :
FIG. 2: Rescattering diagrams contributing to B + c → T + cc D * 0 .The boxes and dots represent the weak and strong vertices, respectively.
) with f D and f D * the decay constants of D and D * , respectively.
and µ 1 (µ 2 ) is the reduced mass of D * + D 0 (D * 0 D + ).The decay amplitude of B + c → T + cc D( * )0 via one of the rescattering diagrams in Figs. 1 and 2 can be expressed in a general form as follows

5 - 5
GeV, the branching ratio of B + c → T + cc D0 is of the order of 10 −7 , while that of B + c → T + cc D * 0 increases from O(10 −6 ) to O(10 −5 ) with Λ increasing.The branching ratio of B + c → T + cc D * 0 is much larger than that of B + c → T + cc D0 .This is because that the S-wave decay is allowed in B + c → T + cc D * 0 but forbidden in B + c → T + cc D0 to conserve the angular momentum.For the moderate cutoff Λ around 3 GeV, Br(B + c → T + cc D * 0 ) is of the order of 10 −5 , and we can expect it may be detectable in future experiments.

7 ]FIG. 3 :
FIG. 3: Λ-dependence of the branching ratio of B + c → T + cc D0 via the rescattering processes in Fig. 1.The band is obtained by taking into account the uncertainties of the δm.

5 ]FIG. 4 :
FIG. 4: Λ-dependence of the branching ratio of B + c → T + cc D * 0 via the rescattering processes in Fig. 2. The band is obtained by taking into account the uncertainties of the δm.
with D ( * )+ s , the rescattering diagrams contributing to the B + c → T + ccs D0 and B + c → T + ccs D * 0 are illustrated in Figs. 5 and 6, respectively.

8 .
We choose two typical values δ m = 100 keV and δ m = 1 MeV for a weakly bound hadronic molecule to estimate the couplings between T + ccs and its components.For the B + c → T + ccs D0 process, varying the Λ from 2.5 to 5 GeV, the branching ratio increases from O(10 −6 ) to O(10 −5 ) for both δ m =100 keV and 1 MeV.For the moderate cutoff Λ around 3 GeV, Br(B + c → T + ccs D0 ) is of the order of 10 −6 .Compared with B + c → T + ccs D0 , the B + c → T + ccs D * 0 process has a larger branching ratio.The argument is similar as that in Section II.For both of the two δ m values, Br(B + c → T + ccs D * 0 ) increases from O(10 −4 ) to O(10 −3 ) with Λ increasing.For the moderate cutoff Λ around 3 GeV, Br(B + c → T + ccs D * 0 ) is of the order of 10 −4
(B + c → T + cc D0 ) and Br(B + c → T + cc D * 0 ) are estimated to be of the order of 10 −7 and 10 −5 , respectively.The T + ccs production in the B + c decay is a Cabibbo-favored process.For the moderate cutoff energy Br(B + c → T + ccs D0 ) and Br(B + c → T + ccs D * 0 ) are estimated to be of the order of 10 −6 and 10 −4 , respectively.