D0D0π+ mass distribution in the production of the Tcc exotic state

D0D0π+mass distribution in the production of the Tcc exotic state A. Feijoo * Departamento de Fsica Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigación de Paterna, Aptdo. 22085, 46071 Valencia, Spain and Nuclear Physics Institute, 25068 Rez, Czech Republic W. H. Liang† Department of Physics, Guangxi Normal University, Guilin 541004, China and Guangxi Key Laboratory of Nuclear Physics and Technology, Guangxi Normal University, Guilin 541004, China Eulogio Oset‡ Departamento de Fsíca Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigación de Paterna, Aptdo. 22085, 46071 Valencia, Spain and Department of Physics, Guangxi Normal University, Guilin 541004, China (Received 12 August 2021; revised 7 October 2021; accepted 2 November 2021; published 10 December 2021) We perform a unitary coupled channel study of the interaction of the DþD0;D 0Dþchannels and find a state barely bound, very close to isospin I¼0. We take the experimental mass as input and obtain the width of the state and the D0D0πþmass distribution. When the mass of the Tcc state quoted in the experimental paper from raw data is used, the width obtained is of the order of the 80 keV, small compared to the value given in that work. Yet, when the mass obtained in an analysis of the data considering the experimental resolution is taken, the width obtained is about 43 keV and both the width and the D0D0πþmass distribution are in remarkable agreement with the results obtained in that latter analysis. DOI: 10.1103/PhysRevD.104.114015 I. INTRODUCTION The recent discovery of the Tcc state by the LHCb Collaboration [1–4] has added a new exotic hadron state to an already long list of states discovered in the latest years that challenge the q¯ qnature of the standard mesons or qqq of the standard baryons. The novelty with respect to many states containing hidden charm is that now there are two charm quarks open. This finding follows the discovery of the X0ð2866Þand X1ð2904Þwhich have an open charm quark and a strange quark with manifestly tetraquark structure [5]. On the theory side there have been quite a few works devoted to the study of tetraquarks with two heavy quarks [6–22], with quite a wide range of predictions going from about 250 MeV below the energy reported for the Tcc to 250 MeV above in the case of two open charmed quarks. The mass of the Tcc state is remarkably close to the DþD0and D0Dþthresholds, its value is given by [1–4] mexp ¼3875.09 MeV þδmexp;ð1Þ where 3875.09 MeV is the threshold of the DþD0state and δmexp ¼−273 61 5þ11 −14 keV:ð2Þ The width reported for the Tcc state is [1–4] Γ¼410 165 43þ18 −38 keV:ð3Þ As we can see, the mass is very close to the DþD0 threshold and the width is very small. The results of Ref. [4] were accompanied by a theoretical analysis of the data by the LHCb Collaboration [23], using a unitary Breit-Wigner amplitude and taking into account the experimental resolution. The results obtained differ somewhat from those of Eqs. (2) and (3) and the new values reported from the pole position of the state are *[email protected].es †[email protected] ‡[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 104, 114015 (2021) 2470-0010=2021=104(11)=114015(7) 114015-1 Published by the American Physical Society δmexp ¼−360 40þ4 −0keV;ð4Þ Γ¼48 2þ0 −14 keV:ð5Þ The closeness to the DDthreshold makes one think immediately about the possibility that this state could be a molecular state of DD, and in fact such structure was anticipated in Refs. [16,21,24]. Independent of the structure of the Tcc state, the proximity of the DDthreshold makes unavoidable the explicit consideration of the DDchannels in its study, as shown in the detailed study of threshold structures in Ref. [25]. The possible DDbound state would have an analogous structure to the DDmolecular state already studied in Ref. [26], where such open charm molecular structures were reported for the first time. It is interesting to mention that also in Ref. [26] predictions were made for another exotic state of D¯ Knature that matches correctly the X0ð2866Þstate reported in Ref. [5] (see update in Ref. [27]). The reaction of the theory community to the experimental finding has been fast. In Ref. [28] a reminder was given that in Ref. [16] a prediction for a molecular DD state had been done matching perfectly the mass found in the experiment. In Ref. [29] the width of the Tcc state is studied with the DþD0and D0Dþcoupled channels and found small compared with the experimental one.1The same conclusion is obtained in Ref. [31] where a single channel DDmolecule is assumed. The QCD sum rules method also brings its contribution to the subject showing that such a state appears at a central value of 3868 MeV for the mass, with the typical large uncertainties of the sum rules method, about 124 MeV in this case [32]. In the present work we report on how a molecular state of DþD0;D 0Dþnature naturally emerges from the interaction of these two coupled channels, and we make a study of the invariant mass distribution of D0D0πþin the production of this state, which is the mode where it has been observed in Refs. [1–4]. We use as a source of interaction the exchange of vector mesons provided by the local hidden gauge approach [33–36]. In the case of VP (vector-pseudoscalar) interaction one can also have the exchange of pseudoscalar mesons, but comparatively to the vector exchange their contribution is very small [37–39] (see detailed calculations in Appendix A of Ref. [37]). In any case the coupled channels unitary approach requires the use of the Gfunctions, the loop functions of the intermediate DDstates, which have to be regularized, and missing pieces of the interaction can be accommodated by means of an appropriate choice of the cutoff or the subtraction constant, in the cutoff or dimensional regularization methods, which are fine tuned to the precise value of the mass of the state. II. FORMALISM AND RESULTS We use a unitary method with the coupled channels DþD0and D0Dþ, paying attention to the exact masses and widths. The interaction is obtained from the extended local hidden gauge Lagrangians [33–36] and they correspond to the exchange of vector mesons in the diagrams of Fig. 1. The Lagrangians used are LVPP ¼−igh½P; ∂μPVμi; LVVV ¼ighðVν∂μVν−∂μVνVνÞVμi; g¼MV 2f;ðMV¼800 MeV;f¼93 MeVÞ;ð6Þ with hi meaning the trace of the matrices in the SU(4) space, where Pand Vstand for the pseudoscalars and vectors respectively, and they correspond to the qi¯ qj matrices written in terms of the corresponding mesons, which can be found in Ref. [40]. Since we are close to the DDthreshold we neglect the ϵ0components of the vectors and work with the vector polarizations ϵ;ϵ0. Calling DþD0;D 0Dþthe 1,2 channels, the interaction that we obtain is Vij ¼Cijg2ðp1þp3Þ·ðp2þp4Þϵ·ϵ0 →Cijg21 23s−ðM2þm2þM02þm02Þ − 1 sðM2−m2ÞðM02−m02Þϵ·ϵ0;ð7Þ where M,mare the initial vector, pseudoscalar masses and M0;m 0the corresponding final ones. The second expression (a) (b) (c) FIG. 1. Diagrams considered for the interaction VP. 1The couplings of Tcc to the DþD0and D0Dþchannels obtained in Ref. [29] (version v1 of the ArXiv) are under revision [30]. A. FEIJOO, W. H. LIANG, and EULOGIO OSET PHYS. REV. D 104, 114015 (2021) 114015-2 in Eq. (7) follows after projection in s-wave, which is what we study. The matrix Cij is given by Cij ¼0 B @ 1 M2 J=ψ 1 m2 ρ 1 m2 ρ 1 M2 J=ψ 1 C A:ð8Þ In diagrams Fig. 1(a) and (b) one can exchange ρ0and ω, but one can see that the product of the couplings for ρor ω exchange is the same, yet with opposite sign, and assuming equal masses for the ρand ω, there is an exact cancellation. One finds that individually the DþD0;D 0Dþstates have a weak and repulsive interaction due to J=ψexchange and hence they do not bind by themselves, but the coupled channels have the virtue of making a bound state possible. Indeed, if we take the isospin combinations [our isospin doublets are ðDþ;−D0Þand ðDþ;−D0Þ] jDD; I ¼0i¼− 1 ffiffiffi 2 pðDþD0−D0DþÞ; jDD; I ¼1;I3¼0i¼− 1 ffiffiffi 2 pðDþD0þD0DþÞ;ð9Þ we find (the indices indicating the isospin) C00 ¼1 M2 J=ψ − 1 m2 ρ ;C11 ¼1 M2 J=ψþ1 m2 ρ ;C01 ¼0; ð10Þ which means that we find an attraction, and not weak, in I¼0and repulsion in I¼1. An approximate solution can be obtained using the single channel with I¼0, which implies using average masses for D’s and D’s, but we wish to be accurate and will use the coupled channels method with the exact masses. The term with 1 M2 J=ψ in Eq. (8) is kept since it comes from the extended local hidden gauge formalism, but being just a 6% of the 1 m2 ρterm it can be safely neglected with no appreciable change in the results. We solve then the Bethe-Salpeter equation in coupled channels and have in matrix form T¼½1−VG−1V; ð11Þ with G¼diag½G1;G 2, where Giare the DDloop functions, which we regularize using dimensional regularization (DR) as in Ref. [41], GDR l¼1 16π2αHþlog M2 1 μ2þM2 2−M2 1þs 2slog M2 2 M2 1 þp ffiffiffi s pðlogðs−M2 2þM2 1þ2pffiffiffi s pÞ −logð−sþM2 2−M2 1þ2pffiffiffi s pÞ þlogðsþM2 2−M2 1þ2pffiffiffi s pÞ −logð−s−M2 2þM2 1þ2pffiffiffi s pÞÞð12Þ where M1;2are the masses of the two particles and pthe on shell three momentum of the two mesons, with the value of the renormalization energy scale μ¼1500 MeV and the subtraction constant αHhaving a value close to αH¼−1.15. The parameter μis a typical hadronic energy scale and only αHis a free parameter, since GDR goes as αH−log μ2. The particular value of αHwas taken in Ref. [41] to obtain the open charm molecular spectrum. Alternatively, one can also employ the cutoff regularization scheme Gcut l¼Zqmax 0 q2dq ð2πÞ2 ω1þω2 ω1ω2½ðP0Þ2−ðω1þω2Þ2þiϵ;ð13Þ where qmax stands for the cutoff in the three momentum, ωi¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  q2þM2 i pand P02 ¼s. The value of αHis fine tuned to get the experimental binding of the Tcc state. Yet, to get a finite width for the state below the DDthresholds we need to consider the width of the Dstates. This is accomplished performing a convolution of the Gfunctions with the spectral function (mass distribution) of the Dstates, as done in Ref. [42] (see Eqs. (4),(5) of that reference), with the width of the D states showing the energy dependence: ΓDþðMinvÞ¼ΓðDþÞmDþ Minv 2 · 2 3pπ pπ;on3 þ1 3p0 π p0 π;on3;ð14Þ where pπis the πþmomentum in Dþ →D0πþdecay with Dþ mass Minv, and pπ;on the same one with the physical mass of Dþ taken from the PDG [43]. Analogously, p0 π;p 0 π;on are the same magnitudes for Dþ →Dþπ0. The width ΓðDþÞis taken from the PDG, ΓðDþÞ¼83.4keV. For the D0, we take ΓD0ðMinvÞ¼ΓðD0ÞmD0 Minv2 · 0.647pπ pπ;on3 þ0.353;ð15Þ where the second term corresponds to the D0→D0γ decay, which does not change appreciably with the small changes in Minv of our problem, and we have taken the branching fractions from the PDG and the value of ΓðD0Þ¼55.3keV from Ref. [44]. The values of pπ;p π;on correspond now to the D0→D0π0decay. The results are practically indistinguishable if we use ΓðD0Þ¼ 55.9keV from Ref. [45] or 77.7 keV from Ref. [46], indicating that the DþD0channel is the one playing a major role in the state given its proximity to the DþD0 D0D0πþMASS DISTRIBUTION IN THE …PHYS. REV. D 104, 114015 (2021) 114015-3 threshold. Indeed, if we change ΓðD0Þfrom 55.9 keV to 77.7 keV, the results change in the fourth decimal. In Fig. 2we show the results for jTDþD0;DþD0j2as a function of ffiffiffi s pfor the case of two channels using the mass of Eq. (2) as input.2 We have taken two subtraction constants αH¼−0.863 for DþD0and αH¼−1.03 for D0Dþ, and we find a neat peak around the experimental mass. We see that the explicit consideration of the Dwidths provides a width at the peak. The width of the peak for the two channels case is Γ≃80 keV:ð16Þ This value is small compared with the experimental width of Eq. (3), even considering the large errors, and is about 60% larger than those obtained in Refs. [29] and [31] of the order of 50 keV, using the couplings of Tcc to the DD components and the input of the Tcc mass given by Eq. (2). The explicit consideration of the coupled channels with the convolutions done using the energy dependent widths is responsible for this increased width. However, it is about double than the value obtained in Ref. [23], Eq. (5), after the consideration of the experimental resolution. Nonetheless, we shall see later that when we use the mass of Eq. (4) as input, the width is considerably reduced and is in agreement with Eq. (5). It is interesting to see which are the couplings of the resonance in the case of two channels for the state corresponding to the peak of Fig. 2. They are obtained from T11 and T12 as g2 1¼lims→sRðs−sRÞT11, g2¼g1T21=T11. In the easy case of neglecting the width of the Dstates where the state appears as bound and there is no problem in defining the Riemann sheet we get gTcc;DþD0¼3658.30 MeV; gTcc;D0Dþ¼−3921.04 MeV;ð17Þ and, as we can see, they are basically opposite to each other indicating that we have indeed a quite good I¼0state, in spite of using different masses for the components and being close to thresholds. This finding is in agreement with the conclusions in Ref. [23]. Next we make a study of the D0D0πþmass distribution in the decay of the resonance, the channel observed in the experiment. This corresponds to a diagram like the one depicted in Fig. 3. The decay of whichever object producing the Tcc and decaying to D0D0πþcan be obtained with the standard formula dΓ dM2 12dM2 23 ¼1 2 1 ð2πÞ3 1 s3=2jtj2;ð18Þ where tis obtained from the diagram of Fig. 3, symmetrizing over the two D0momenta and the factor 1 2is added in the formula. The amplitude tfor this process is given by t¼CTDþD0;D þD0ðffiffiffi s pÞ ϵ·ð p1−  p2Þ M2 12 −m2 Dþ þiM12ΓDþðM12Þ þ  ϵ·ð p3−  p2Þ M2 23 −m2 Dþ þiM23ΓDþðM23Þ; ð19Þ where Cis an arbitrary constant and ϵstands for the polarization vector of the Tccð1þÞ. Upon summing over the polarizations ϵin jtj2for Tcc at rest, we have terms X pol:ðϵ· q1Þðϵ· q2Þ¼ϵμqμ 1ϵνqν 2¼−gμν þPμPν M2 Tcc qμ 1qν 2; ð20Þ where,  q1and  q2can be ð p1−  p2Þand ð p3−  p2Þ.We convert the terms  q1· q2into invariants which can be 3873 3874 3875 3876 3877 3878 3879 3880 s1/2 [MeV] 2×109 4×109 6×109 8×109 1×1010 |T(D *+D0 -->D *+D0)|2 D *+D0 D *0D+ FIG. 2. jTDþD0;DþD0j2as a function of ffiffiffi s p. Dashed vertical line, DþD0threshold. Continuous vertical line, D0Dþ threshold. FIG. 3. Mechanism for D0πþD0decay of the Tcc state. The diagram with D0Dþdecay does not lead to final D0D0πþ. 2The first version of the paper was done before Refs. [4,23] were made public. We show these results first and later we show the new results if we take the value of Eq. (4) for δmexp. A. FEIJOO, W. H. LIANG, and EULOGIO OSET PHYS. REV. D 104, 114015 (2021) 114015-4 written in terms of M12 and M23 using that M2 12 þM2 12 þ M2 23 ¼M2 Tcc þm2 D0þm2 D0þm2 πþ. Integrating Eq. (18) over M12 and M23, using the limits of the PDG for the Dalitz boundary, we obtain the mass distribution Γðffiffiffi s pÞ shown in Fig. 4. Next we repeat the calculations using the mass of Ref. [23], Eq. (4), as input. The new mass is obtained taking αH¼−0.870 for DþD0and αH¼−1.03 for D0Dþ. Note that the relevant channel is the DþD0which is closer to the pole. Hence, the important parameter αHis the one for this channel. In Fig. 5and 6, we show now jTDþD0;DþD0j2and the D0D0πþspectrum respectively. We see that now the width at half the strength of the peak is Γ≃43 keV;ð21Þ which is in good agreement with the findings of Ref. [23] in Eq. (5). The shape of the spectrum in Fig. 6is practically identical to the one obtained from analysis of the LHCb data in Ref. [23] taking into account the experimental resolution, and also has a small contribution between the two thresholds, which is tied to the D0D0πþspectrum and does not show up in jTj2. The couplings that we get now are gTcc;DþD0¼3884.68 MeV; gTcc;D0Dþ¼−4144.26 MeV:ð22Þ One should stress that these couplings are also consistent with those determined by the experimental analysis [23] jgj>5.1 ffiffiffi 2 p¼3.6GeV4.3 ffiffiffi 2 p¼3.0GeVat 90ð95%ÞCL and close by to them. It should be noticed that what we call gTcc;DDin Eq. (22) is called gTcc;DD=ffiffiffi 2 pin Eq. (M2a) and Eq. (M2b) in [23]. It is also interesting to see what do we get if we assume just one channel with exact I¼0as assumed in the analysis of Ref. [23], Eq. (9), and averaged masses for the Dmesons and Dmesons. The calculations are done with a single channel with Vgiven by Eq. (7) with C00 of Eq. (10). The convolved Gfunction is taken as the average of GDþD0and GD0Dþusing average masses and the values of the widths, Eqs. (14),(15) for each Gfunction and a single αHfor the two channels. The results do not change qualitatively from those of Figs. 5and 6, only the width of the state is now Γ≃55 keV and the strength of the D0D0πþspectrum between the thresholds, while still very small is about three times bigger than before, as a consequence of the change of the thresholds when employing average masses. It is interesting to remark the coincidence in practice of the approach of Ref. [23] making fits to the data with a unitary Breit-Wigner amplitude and our approach based upon the use of the DDscattering matrix, with the G 3873 3874 3875 3876 3877 3878 3879 3880 s1/2 [MeV] 0.05 0.1 0.15 0.2 0.25 Γ(T cc-->D0D0π +) [arbitrary units] D *+D0 threshold D *0D+ threshold FIG. 4. Γðffiffiffi s pÞfor the decay of the Tcc into D0D0πþ. 3874 3875 3876 3877 3878 3879 3880 s1/2 [MeV] 0 1×1010 2×1010 3×1010 4×1010 |T(D *+D0 -->D *+D0)|2 D *+D0 D *0D+ FIG. 5. jTDþD0;DþD0j2as a function of ffiffiffi s p. Dashed vertical line, DþD0threshold. Continuous vertical line, D0Dþ threshold. 3874 3875 3876 3877 3878 3879 3880 s1/2 [MeV] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Γ(T cc-->D0D0π +) [arbitrary units] D *+D0 thresh. D *0D+ thresh. FIG. 6. Γðffiffiffi s pÞfor the decay of the Tcc into D0D0πþ.The inset in the figure is a zoom to illustrate the mass distribution between DþD0and D0Dþthresholds. D0D0πþMASS DISTRIBUTION IN THE …PHYS. REV. D 104, 114015 (2021) 114015-5 function convolved to account for the Dwidths, and the explicit mechanism of Fig. 3to account for the D0D0πþ production. The formalisms look rather different but the physics contained in them coincide. We would expect the same results for other mass distributions discussed in Ref. [23]. In particular, the most relevant, the lack of any signal in channels related to a possible I¼1state, is guaranteed in our approach since we already showed that we do not get any state for I¼1, since the interaction is repulsive there. Another interesting information can be extracted using the cutoff regularization of Eq. (13) and we obtain qmax ¼ 415 MeV when taking the mass of Eq. (2) and 418.6 MeV when using the mass of Eq. (4). This value is somewhat small compared with the 600 MeV that one needs in the study of the low lying scalar mesons (σ,f0,a0)ofqmax ¼ 600 MeV [48]. Since, in one channel one has T¼ ½V−1−G−1, a decrease of jVjby means of a repulsive interaction induces an increase in jGjto get the pole at the same place, implying a larger cutoff. We perform the test of increasing the repulsion of J=ψexchange by a factor 3 in Eq. (8), which still makes this contribution only about 18% of the ρexchange and we need qmax ¼476 MeV, closer to the 600 MeV used in Ref. [48]. Note that our approach generates an accurate interaction from the exchange of light vectors, which is consistent with heavy quark symmetry, but subleading terms, like the exchange of heavy vectors which do not follow that rule, are less accurate. III. SUMMARY In summary, we get a molecular state of DþD0;D 0Dþ, with a mixing that corresponds very approximately to an I¼0state. No signal is seen for the orthogonal, approximately I¼1state, as one can see in Fig. 2. This is in contrast to the suggestion made in Ref. [47] that two states could come from isospin mixing. Actually, the small bump in the D0D0πþspectrum suggested in Ref. [47] as a possible new state, also shows up in our spectrum of D0D0πþin Fig. 4, but since this small bump does not appear in jTj2in Fig. 2, it has to be associated to the decay channel D0D0πþand its phase space and not to a physical state. This also means that measuring other decay channels would provide new and valuable information, as it has been shown in Ref. [23]. The analysis of the LHCb data done in Ref. [23], considering the experimental resolution and using a unitary Breit Wigner amplitude, has been most useful since it allows to compare with theoretical calculations as the one we have done. We have shown that when using the mass obtained in Ref. [23] as input, the width obtained for the Tcc state and the D0D0πþmass distribution are in remarkable agreement with the results reported in Ref. [23]. ACKNOWLEDGMENTS This work is partly supported by the National Natural Science Foundation of China under Grants No. 11975083 and No. 12047567. This work is also partly supported by the Spanish Ministerio de Economia y Competitividad (MINECO) and European FEDER funds under Contracts No. FIS2017-84038-C2-1-P B, No. PID2020–112777 GBI00, and by Generalitat Valenciana under Contract No. PROMETEO/2020/023. This project has received funding from the European Union Horizon 2020 research and innovation programme under the program No. H2020INFRAIA-2018-1, Grant agreement No. 824093 of the “STRONG-2020”project. The present work has been also partially supported by the Czech Science Foundation, GACR Grant No. 19-19640S. [1] F. Muheim, the European Physical Society conference on high energy physics 2021, https://indico.desy.de/event/ 28202/contributions/102717/. [2] I. 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