Prediction of new Tcc states of D∗D∗ and Ds∗ D∗ molecular nature

Prediction of new Tcc states of DDand D sDmolecular nature L. R. Dai ,1,2,* R. Molina,2,†and E. Oset2,‡ 1School of Science, Huzhou University, Huzhou, 313000 Zhejiang, China 2Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC Institutos de Investigación de Paterna, Aptdo.22085, 46071 Valencia, Spain (Received 5 November 2021; accepted 19 January 2022; published 31 January 2022) We extend the theoretical framework used to describe the Tcc state as a molecular state of DDand make predictions for the DDand D sDsystems, finding that they lead to bound states only in the JP¼1þ channel. Using input needed to describe the Tcc state, basically one parameter to regularize the loops of the Bethe-Salpeter equation, we find bound states with bindings of the order of MeVand similar widths for the DDsystem, while the D sDsystem develops a strong cusp around the threshold. DOI: 10.1103/PhysRevD.105.016029 I. INTRODUCTION The discovery of the mesonic Tcc state with two open charm quarks [1–4] has brought a new example of an exotic meson state, challenging the standard nature of mesons as q¯ qstates. The reaction of the theory community has been fast and many articles have been written proposing different interpretations about the nature of the state [5–27]. Rates in production reactions had been calculated prior to the Tcc discovery [28]. The relevant analysis carried out in [29] by means of a unitary amplitude, with explicit consideration of the experimental resolution, has shown that the width of the Tcc state is much smaller than the one extracted from the raw data in [4], and in line with the width obtained in [8]. Similar conclusions are obtained in [30]. The proximity of the peak position in [29] to the DDthreshold strongly supports the DDnature of this state, something assumed in most of the works done on the Tcc state. In [8] the state was studied within a unitary coupled channel approach with the DþD0and D0Dþchannels and the interaction was obtained from the exchange of vector mesons in a straight extrapolation of the local hidden gauge approach [31–34] to the charm sector [35]. The only parameter in [8] was a regulator in the meson meson loop function of the Bethe-Salpeter equation, which was tuned to obtain the mass of the state at the right position, then the width and the D0D0πþmass distribution were obtained in good agreement with experiments, with a width of around 43 KeV. Given the fact that heavy quark spin symmetry [36–38] allows us to relate the Dand Dsectors, it is tempting to extend the results of [8] to the DDsystem to make predictions on this state. The task is rendered easier because this system and the D sDwere studied before the recent experimental finding on the DDsystem [4] in [39]. Indeed, in [39] it was found that the DDsystem in isospin I¼0, and JP¼1þand the D sDsystem in I¼ 1=2;JP¼1þhad an attractive potential, strong enough to support a bound state, and predictions were done with a binding of around 35 MeV. The DDwith I¼1and D sD ssystems were also investigated in [39] and the interaction was found repulsive, so we do not study these systems here. The predictions for the binding are tied to the regulator of the meson meson loop function, and right now we have experimental information from [4,29] to fix it, such that more accurate predictions can be done. On the other hand, in [39] the width of the states was obtained from the pseudoscalar-pseudoscalar decay channel, which in the present case is forbidden by spin-parity conservation. Yet, there is another source of decay which is the pseudoscalar-vector channel, which we will evaluate here. This decay channel was investigated in the study of the D¯ Ksystem [40] which produced a bound state in [39] and was shown in [40] to be suited to reproduce the properties of the X0ð2866Þstate recently observed by the LHCb Collaboration in [41]. The width of the Tcc state is tied to the D→πDdecay [29], and results in about 40 KeV–50 KeV. On the contrary, here the DDand D sDsystems will decay into a pseudoscalar-vector system which has a much larger phase space for decay than the D→πD. Hence, we can already guess that we shall have a much larger width, yet, still reasonably small, as one can induce from the results of [40]. *[email protected]n †[email protected].es ‡[email protected].es Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 105, 016029 (2022) 2470-0010=2022=105(1)=016029(12) 016029-1 Published by the American Physical Society The DDsystem has been relatively unexplored from the molecular point of view. It is studied in [42] using meson exchange for the interaction and in [43] using the exchange of vector mesons guided by the local hidden gauge approach. Only the diagonal-interaction and no decay channels are considered in [43], which aims at providing guidelines on possible states at the qualitative level. The DDsystem in I¼0and JP¼1þappears in [43] as a candidate for a bound state, and in [30,44] it also appears as a bound state using arguments of heavy quark spin symmetry and the data of [4].In[42] a state of DðÞDðÞ nature, without distinguishing between DDand DD, is also found to be a good candidate for a bound state in I¼0and JP¼1þ. Similarly, a possible ðDðÞDðÞÞsstate in JP¼1þis also reported in [42]. The widths are not considered in this latter work. On the other hand there are many works dealing with mesons with two heavy quarks as tetraquark states, using quark models, sum rules or lattice QCD. The QQq¯ q systems were studied already in [45] indicating an increased attraction on some channels as the ratio of M=m (heavy to light quark masses) increases, something corroborated in [46–48]. Calculations of binding energies of doubly heavy tetraquarks were done in [49–51] using different quark models, in [52–61] using QCD sum rules, lattice QCD calculation [62–70], and other models [71–73]. The predictions of the different models ranged from about 250 MeV above to 250 MeV below the meson-meson threshold, indicating the difficulties to make accurate predictions with these methods. In the present work, with the background from the molecular studies discussed above and the valuable information from the Tcc state, we retake the task of studying the DDand D sDsystems, paying attention to the decay channels, with the purpose of making a precise determination of the mass and widths of the bound states emerging from the interaction of these systems. II. FORMALISM A. Direct interaction The interaction of DDand D sDis studied in [39].Itis based on the extrapolation of the local hidden-gauge approach [31–34] to the charm sector. The local hiddengauge approach was first used in [74,75] to study the interaction between vector mesons in the SUð3Þsector. It contains a contact term and the exchange of vector mesons which requires a three-vector vertex. In [74,75] this interaction was shown to produce bound states or resonances which could be associated to existing states. Extrapolated to the charm sector, it predicted the pentaquark states with hidden charm and hidden charm and strangeness [35,76] which were found later by the LHCb Collaboration [77–79]. The basic ingredients to calculate the potential between two vectors are the Lagrangians LðcÞ¼g2 2hVμVνVμVν−VνVμVμVνi;ð1Þ with g¼MV 2fðMV¼800 MeV;f ¼93 MeVÞ, and Vμthe q¯ qmatrix written in terms of vector mesons Vμ¼ 0 B B B B B @ ω ffiffi2 pþρ0 ffiffi2 pρþKþ ¯ D0 ρ−ω ffiffi2 p−ρ0 ffiffi2 pK0D− K−¯ K0ϕD− s D0Dþ Dþ sJ=ψ 1 C C C C C Aμ ;ð2Þ and LVVV ¼ighðVμ∂νVμ−∂νVμVμÞVνi:ð3Þ LðcÞis a contact term and LVVV stands for the three-vector vertex. By means of it, one generates an interaction between vectors exchanging vector mesons. The mechanisms for the interaction are depicted in Fig. 1. In Table XVI of [39] it was shown that the contact term for DDin I¼0gives no contribution. By contrast, the vector exchange term gives null contribution for spin J¼0, 2 but produces an attractive potential in JP¼1þ, VDD→DD¼1 4g22 m2 J=ψþ1 m2 ω − 3 m2 ρ ×fðp1þp4Þ·ðp2þp3Þ þðp1þp3Þ·ðp2þp4Þg:ð4Þ In Table XVII of [39] it is shown that for I¼1the interaction for J¼0, 2 is repulsive and null for J¼1. The situation is similar for the D sD ssystem, which has I¼0, with repulsive interaction in J¼0, 2 and null interaction for J¼1(see Table XIX of [39]). On the contrary, as shown in Table XVIII of [39], the D sDsystem, which has I¼1 2, gives attraction for J¼1and repulsion for J¼0,2. (a) (b) FIG. 1. Terms for the VV interaction: (a) contact term, (b) vector exchange. L. R. DAI, R. MOLINA, and E. OSET PHYS. REV. D 105, 016029 (2022) 016029-2 We are thus left with two candidates for bound states DDwith I¼0,JP¼1þand D sDwith I¼1 2,JP¼1þ. The interaction for this latter case is given by VD sD→D sD¼−g2ðp1þp4Þ·ðp2þp3Þ m2 K þg2ðp1þp3Þ·ðp2þp4Þ mJ=ψ2 :ð5Þ Note that ðp1þp3Þ·ðp2þp4Þprojected in s-wave can be written as [80] 1 2f3s−ðM2 1þM2 2þM2 3þM2 4Þ− 1 sðM2 1−M2 2ÞðM2 3−M2 4Þg: ð6Þ In [39] the T-matrix was obtained from these potentials using the Bethe-Salpeter equation T¼½1−VG−1V; ð7Þ with Gthe intermediate vector-vector (VV) loop function which was regularized by means of a cutoff and also dimensional regularization. On the other hand, the pseudoscalar-pseudoscalar (PP) decay channels were considered for all the states studied there, but the VV states with 1þcannot decay to PP if we want to conserve spin and parity. Here we consider instead the decay into vectorpseudoscalar ðVPÞchannels which will give a width to the bound states that we find. B. Vector-pseudoscalar decay channels 1. DD→DD decay We take the I¼0DDstate. With the isospin doublets ðDþ;−D0Þand ðDþ;−D0Þ, the I¼0state is given by jDD;I ¼0i¼− 1 ffiffiffi 2 pjDþD0−D0Dþi:ð8Þ This system can decay into DþD0or D0Dþand we shall take into account these decays by means of the imaginary part of the box diagrams of Fig. 2. The diagrams shown in Fig. 2have all the same structure and only the isospin coefficients are different. Note that intermediate Dsor D s states are not possible while having two open charm quarks in the two-meson system. Taking the first diagram as reference and the coupling of π0to DþDþ as 1, by means of Clebsch-Gordan coefficients we find the weight FIG. 2. Box diagrams according for DD;I ¼0decay into DþD0and D0Dþ. PREDICTION OF NEW TCC STATES OF DD…PHYS. REV. D 105, 016029 (2022) 016029-3 −1for π0D0D0, and ffiffiffi 2 pfor πþor π−coupling to DþD0 (consistent with our phase convention πþ¼−j11i). The total weight of the diagrams is 1 4ð1þ2þ2þ4þ4þ2þ2þ1Þ¼18 4¼9 2: We have to consider in addition the diagrams where the pseudoscalar meson is on the upper line of the diagram and the vector in the lower line, which are depicted in Fig. 3.If we take the second diagram of Fig. 3as reference (with the exchange of two π0), the rest of them are included as before and altogether we have a weight 9 2of the second diagram of Fig. 3. The set of diagrams must be completed exchanging the vectors Dðp3Þ↔Dðp4Þin the final state, given the identity of the two Din the final state (in the isospin formalism). Then the diagrams of Fig. 2give rise to the diagrams of Fig. 4. We observe now that the third diagram of Fig. 4(the one with two π0exchange) is equivalent to the first diagram of Fig. 2, except that p3;ϵ3↔p4;ϵ4(ϵiis the polarization vector of particle i) are exchanged and there is a relative (−1) sign. The same happens when we exchange the final states of Fig. 3, which we do not depict. Altogether, the sum of the 32 diagrams can be calculated as shown in Fig. 5. The evaluation of these diagrams requires now the use of two new vertices, the ordinary VPP coupling and the anomalous VVP coupling given by the Lagrangians LVPP ¼−igh½P; ∂μPVμi;ð9Þ LVVP ¼G0 ffiffiffi 2 pϵμναβh∂μVν∂αVβPi:ð10Þ LVPP appears in the local hidden-gauge approach and LVVP can be found from [81,82], where G0is given by G0¼3g0 4π2f;g 0¼−GVmρ ffiffiffi 2 pf2; GV¼55 MeV;f¼93 MeV: In addition to Vμof Eq. (2) we now need the matrix Pfor the pseudoscalar mesons given by FIG. 3. Box diagrams according for the DD;I ¼0decay into D0Dþ and DþD0. L. R. DAI, R. MOLINA, and E. OSET PHYS. REV. D 105, 016029 (2022) 016029-4 P¼ 0 B B B B B @ η ffiffi3 pþη0 ffiffi6 pþπ0 ffiffi2 pπþKþ¯ D0 π−η ffiffi3 pþη0 ffiffi6 p−π0 ffiffi2 pK0D− K−¯ K0−η ffiffi3 pþffiffi2 3 qη0D− s D0DþDþ sηc 1 C C C C C A ; ð11Þ where the standard η;η0mixing of [83] has been used. We evaluate the amplitudes at the DDthreshhold since we expect small binding energies. We have then ϵ0¼0for all the external vector mesons and we take it also for the propagating vectors in the loop given the large mass of the particles. We get the following vertices: (1) D0π0→D0, −it ¼−2igqϵðD0Þ1 ffiffi2 p, (2) D0→π0D0, −it ¼−2igqϵðD0Þ1 ffiffi2 p, (3) Dþ →π0Dþ, −it ¼−iG0 ffiffi2 pϵijkEðDþ extÞϵiðDþ extÞqjϵkðintÞ1 ffiffi2 p, (4) Dþπ0→Dþ, −it ¼−iG0 ffiffi2 pϵijkEðDþ extÞϵiðDþ extÞqjϵkðintÞ1 ffiffi2 p, where all vector and tensor components are contravariant (even if we write them as lower indices) and EðDÞstands for the energy of the D. The indices “ext”or “int”stand for the external or internal vectors of the diagrams. Taking into account that X pole ϵkðintÞϵk0ðintÞ¼δkk0; FIG. 4. Diagrams obtained from Fig. 2exchanges Dðp3Þ↔Dðp4Þin the final state. FIG. 5. Diagrams to be calculated with their respective weights. PREDICTION OF NEW TCC STATES OF DD…PHYS. REV. D 105, 016029 (2022) 016029-5 the product of all four vertices gives ðffiffiffi 2 pgÞ2G0 22 Eð1ÞEð3Þfϵið1Þϵlð2Þϵið3Þϵmð4Þq2qlqm −ϵjð1Þϵlð2Þϵið3Þϵmð4Þqiqjqlqmg;ð12Þ where the indices 1,2,3,4 refer to the particles on the order of Fig. 5. Let us note that at threshold all the propagators in the loop depend only on q2which allows us to write Zd3qfðq2Þqlqm¼Zd3qfðq2Þ1 3q2δlm; Zd3qfðq2Þqiqjqlqm¼Zd3qfðq2Þ1 15 q4ðδijδlm þδilδjm þδimδjlÞ:ð13Þ The second combination in Eq. (12) gives rise to the product of polarization vectors in the order of 1,2,3,4 ϵjϵlϵjϵlþϵjϵiϵiϵjþϵjϵjϵiϵi;ð14Þ and using the projectors into the spin states of J¼1,2,3, Pð0Þ,Pð1Þ,Pð2Þfrom [40,74] we have ϵjϵjϵiϵi¼3Pð0Þ; ϵjϵlϵjϵl¼Pð0ÞþPð1ÞþPð2Þ; ϵjϵiϵiϵj¼Pð0Þ−Pð1ÞþPð2Þ:ð15Þ Hence, the combination of Eq. (14) gives Pð0ÞþPð1ÞþPð2ÞþPð0Þ−Pð1ÞþPð2Þþ3Pð0Þ; and we see that this term does not contribute to our state with JP¼1þ. The first term of Eq. (12) gives rise to the combination ϵiϵlϵiϵl¼Pð0ÞþPð1ÞþPð2Þ; One can see that the diagrams of Fig. 3give rise to the same combination, and those of Fig. 5and the equivalent to Fig. 3exchanging p3;ϵ3↔p4;ϵ4give the same contribution except for a minus sign and the exchange of ϵ3↔ϵ4. Hence we get the combination now of −ϵiϵlϵlϵi¼−ðPð0Þ−Pð1ÞþPð2ÞÞ; and we see that they give the same contribution to Pð1Þas the other diagrams. Altogether we find now for the JP¼1þ state the contribution for the four diagrams of Fig. 5, keeping the positive-energy part of the propagators of the heavy particles −it ¼49 2 1 3Zd4q ð2πÞ4 1 2EDðqÞ i p0 1−q0−EDðqÞþiϵ 1 2EDðqÞ ×i p0 2þq0−EDðqÞþiϵ i q2−m2 πþiϵ ×i ðp2−p4þqÞ2−m2 πþiϵq4:ð16Þ The pion propagator cannot be placed on shell and if we are interested in the imaginary part only the DDintermediate particles can be put on shell. We can perform the q0 analytically and then use Im 1 xþiϵ¼−iπδðxÞ(alternatively one can use Cutkowsky rules) and we find ImVbox ¼−61 8π 1 ffiffiffi s pq5E2 Dðffiffiffi 2 pgÞ2G0 22 ×1 ðp0 2−EDðqÞÞ2−q2−m2 π2 F4ðqÞFHQ;ð17Þ where q¼λ1=2ðs; m2 D;m 2 DÞ 2ffiffiffi s p;E D¼ffiffiffi s p 2; where we have added the form factor FðqÞused in [39,40] and the heavy quark correcting factor FHQ to correct the VPP vertex for heavy particles, as discussed in [84] FðqÞ¼eððq0Þ2−q2Þ=Λ2ð18Þ with q0¼p0 1−EDðqÞ; and FHQ ¼mD mK2 2. D sD→D sD+D sDdecay We take the Dþ sDþ state and consider the decay into Dþ sDþand Dþ sDþ. The diagrams that we must consider are now depicted in Fig. 6. Since D sdoes not couple to D sπ0we see that the first and fourth diagrams to the left in Fig. 6are zero and so are all the diagrams of the right in Fig. 6. We repeat the calculations as done in the former subsection, omitting details and we obtain now L. R. DAI, R. MOLINA, and E. OSET PHYS. REV. D 105, 016029 (2022) 016029-6 ImVbox ¼− 1 3 1 8π 1 ffiffiffi s pð2gÞ2G0 ffiffiffi 2 p2 ðE1E3þE2E4Þ ×q51 ðp0 2−EDsðqÞÞ2−q2−m2 K2 F4ðqÞFHQ; ð19Þ with FðqÞgiven by Eq. (18) with q0¼p0 2−EDsðqÞ;q¼λ1=2ðs; m2 D;m 2 DsÞ 2ffiffiffi s p; p0 2¼sþm2 D−m2 D s 2ffiffiffi s p: We then solve the Bethe-Salpeter equation of Eq. (7) with V→VþiImVbox for the two cases DD,I¼0,JP¼1þand D sD,I¼1 2, JP¼1þand calculate the T-matrix. By plotting jTj2we find the mass of the state and its width which we report in the next section. III. RESULTS We will use the cutoff method to regularize the loops— something advised in [85]. The value of the cutoff, or subtraction constant in dimensional regularization, is normally the only free parameter in the theory, and this is the case here. One can take reasonable cutoffs from 450 MeV to 750 MeV and make reasonable predictions, but in the present case we can use the cutoff that was needed in [8] to get the experimental binding of the DDstate and rely upon it. We would then be making use of the findings of [86,87] encouraging the use of the same cutoff to respect rules of heavy quark symmetry, although there are limitations to the use of this symmetry in the case of two heavy quarks as we have here [88–90]. In Fig. 7we show the results for jTDD;DDj2as a function of ffiffiffi s pcalculated with a cutoff for Gin Eq. (7), qmax ¼750 MeV, and a value of Λof Eq. (18) of 1200 MeV, similar to what was used in [39], and also for Λ¼1400 MeV. We see that the results do not change FIG. 6. Diagrams for the decay of Dþ sDþ sinto Dþ sDþand Dþ sDþ. PREDICTION OF NEW TCC STATES OF DD…PHYS. REV. D 105, 016029 (2022) 016029-7 much by changing the value of Λand we get a bound state around 3960 MeV. It should be noted that we have two different scales to regularize the different loops. The first one, qmax, is used for the loops resulting from the iteration of the diagrams of Fig. 1, where in the t-channel we have the exchange of vector mesons, which are very off shell, to the point that, as usual, we neglect q2versus M2 V, and we obtain the G function for the loop where there are only two propagators. For the diagrams of Figs. 2,3,4, we have pion exchange in the t-channel. This pion is light (or kaon in Figs. 6) and we can no longer neglect q2versus m2 π, which forces us to make an exact calculation of the loop with four propagators. The range of qin the integration is also different now. In principle we only know that the scale of the form factors (or cutoffs) is of the order of 1 GeV, but to be more precise we use phenomenological information and take qmax to regularize the Gfunction from the study if the Tcc state in [8] and Λof the form factor to regularize the box diagram from the study of the D¯ Kmolecule X0ð2866Þ decaying to D¯ K, considered in [39,40] with a similar box diagram. Note that since we only evaluate the imaginary part of the box, the effect of the form factor is moderate as shown in Fig. 7. As we mentioned, the vector mesons exchanged in Fig. 1 (b) are very off shell. They are spacelike (q2is negative and small compared to M2 V) and hence they have no width.1 Hence, the approximation of 1=ðq2−M2 Vþiðq2Þ1=2Γðq2ÞÞ by 1=ð−M2 VÞ, as we have done, is a sensible one. We should note that this approximation is the one that produces the chiral Lagrangians from the exchange of vector mesons in the local hidden gauge approach in the SUð3Þ space [91,92]. In Fig. 8we show instead the results of jTDD;DDj2for a fixed value of Λ¼1200 MeV and different values of the cutoff. We can see that there is always a bound state. The binding energy depends on the cutoff value and for values of qmax of the order of 450 MeV, as needed in [8] to get the Tcc state, we also obtain a bound state very close to the DDthreshold. We observe a curious phenomena which is that the width becomes smaller as we get closer to the threshold, in spite of the fact that the phase space for decay increases with increasing energy. To understand this feature we recall that including the box diagram in our approach, adding it to the potential from vector exchange and solving the Bethe-Salpeter equation, is an effective way to include the DDchannel together with DD, or the D sD,DsD channels together with the D sDchannel. Then one must recall a well-known fact, based on the Weinberg compositeness condition [93–95], that the coupling squared of the bound state to the hadron-hadron component in a single channel goes as the square root of the binding energy. As we go closer to the DDthreshold the coupling of the resonance to this channel becomes smaller. What is less known is that in the case of coupled channels, if we approach one threshold, all the couplings to the different channels that couple to the one of that threshold also go to zero [95,96]. Then the width of the DDstate obtained will be proportional to the square of the coupling of that state to DDand will go to zero as we approach the DD threshold. In Fig. 9we show the results for the D sDcase. We show the results of jTD sD;D sDj2for qmax ¼750 MeV and two values of the parameter Λ. As we can see, we get a bound state and the width does not change much with the value FIG. 8. Squared amplitude jTDD→DDj2with Λ¼ 1200 MeV. The vertical line indicates the DDthreshold at 4017.1 MeV. FIG. 7. Squared amplitude jTDD→DDj2with qmax ¼ 750 MeV. 1One should use a full propagator for the exchanged vector including its width. However, unitarity of the amplitudes requires that the propagator (equivalent to a Breit-Wigner form) is 1=ðq2−M2 Vþiðq2Þ1=2Γðq2ÞÞ, with the width as a function of q2, but for q2≤0,Γðq2Þ¼0. L. R. DAI, R. MOLINA, and E. OSET PHYS. REV. D 105, 016029 (2022) 016029-8 of Λ. In Fig. 10 we show the results of jTD sD;D sDj2for Λ¼1200 MeV and three values of qmax. We see a similar trend as before, but the bindings are smaller as a consequence of the smaller strength of the potential. In Figs. 11 and 12 we show the enlarged picture of the states in Figs. 8and 10 for qmax ¼450 MeV, together with the results for qmax ¼420 MeV (the value taken in [8] to get the Tcc state). For the D sDsystem we do not get bound states in these latter cases, and instead we find pronounced cusps at the D sDthreshold. This is a consequence of the weaker potential Vin the case of D sDcompared to DD, Eqs. (4) and (5) (see for instance Tables XVI and XVIII of [39] for numerical values). We summarize this information in Table I, providing the mass and width of the states. Values of the binding around 0.5 MeV–1.5 MeV for the DDsystem are also obtained in [30,44] using arguments of heavy quark spin symmetry and the data of [4]. The width is not calculated there. The work of [30] explores the possibility of an I¼1state, not fully ruled out by the data, but our theoretical framework excludes such a state. We think these are sensible predictions that should encourage the search of these states at LHCb. We can compare the results obtained here with those in Ref. [39] for the DDand D sDsystems. Since these systems were investigated theoretically for the first time from the molecular perspective, one could only use a general regulator for the loops which were evaluated with dimensional regularization and a typical subtraction constant was used. Here we have experimental constraints from the mass of the Tcc and this allows us to be more precise in the predictions. The masses 4015 MeV and 4122 MeV for DDand D sD, respectively, obtained here were found in Ref. [39] as 3969 MeV and 4101 MeV, respectively. Both systems were found as bound systems, but the binding energies were bigger. The small binding found for the Tcc FIG. 10. Squared amplitude jTD sD→D sDj2with Λ¼ 1200 MeV. The vertical line indicates the D sDthreshold at 4122.46 MeV. FIG. 11. The same as Fig. 8but with a smaller range of qmax as in [8]. FIG. 12. The same as Fig. 10 but with a smaller range of qmax as in [8]. FIG. 9. Squared amplitude jTD sD→D sDj2with qmax ¼ 750 MeV. PREDICTION OF NEW TCC STATES OF DD…PHYS. REV. D 105, 016029 (2022) 016029-9