Coupled-channel approach to Tcc+ including three-body effects

Coupled-channel approach to T+ cc including three-body effects Meng-Lin Du ,1,* Vadim Baru ,2,3,†Xiang-Kun Dong ,4,5,‡Arseniy Filin ,2Feng-Kun Guo ,4,5,§ Christoph Hanhart ,6,∥Alexey Nefediev ,7,8,¶ Juan Nieves ,1,** and Qian Wang 9,10,11,†† 1Instituto de Física Corpuscular (centro mixto CSIC-UV), Institutos de Investigación de Paterna, Apartado 22085, 46071 Valencia, Spain 2Institut für Theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum, Germany 3Institute for Theoretical and Experimental Physics NRC “Kurchatov Institute”, Moscow 117218, Russia 4CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China 5School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 6Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany 7P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 119991, Leninskiy Prospect 53, Moscow, Russia 8Moscow Institute of Physics and Technology, 141700, Institutsky lane 9, Dolgoprudny, Moscow Region, Russia 9Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China 10Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 11Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 510006, China (Received 5 November 2021; accepted 3 January 2022; published 24 January 2022) A coupled-channel approach is applied to the charged tetraquark state Tþ cc recently discovered by the LHCb Collaboration. The parameters of the interaction are fixed by a fit to the observed line shape in the three-body D0D0πþchannel. Special attention is paid to the three-body dynamics in the Tþ cc due to the finite life time of the D. An approach to the Tþ cc is argued to be self-consistent only if both manifestations of the three-body dynamics, the pion exchange between the Dand Dmesons and the finite Dwidth, are taken into account simultaneously to ensure that three-body unitarity is preserved. This is especially important to precisely extract the pole position in the complex energy plane whose imaginary part is very sensitive to the details of the coupled-channel scheme employed. The D0D0and D0Dþinvariant mass distributions, predicted based on this analysis, are in good agreement with the LHCb data. The low-energy expansion of the DDscattering amplitude is performed and the low-energy constants (the scattering length and effective range) are extracted. The compositeness parameter of the Tþ cc is found to be close to unity, which implies that the Tþ cc is a hadronic molecule generated by the interactions in the DþD0and D0Dþ channels. Employing heavy-quark spin symmetry, an isoscalar DDmolecular partner of the Tþ cc with JP¼1þis predicted under the assumption that the DD-DDcoupled-channel effects can be neglected. DOI: 10.1103/PhysRevD.105.014024 I. INTRODUCTION The quest of exotic hadrons with configurations beyond the naive quark-model picture of a pair of quark-antiquark for a meson and three quarks for a baryon has been a central issue in the study of nonperturbative quantum chromodynamics (QCD) for decades. A breakthrough along this path was the discovery of the Xð3872Þ(also known as χc1ð3872Þ according to the contemporary classification scheme by the Particle Data Group [1]) by the Belle Collaboration in 2003 [2]. It resides extremely close to the threshold of a pair of neutral charmed mesons D0¯ D0. This exotic state is generally considered to be an excellent candidate for a *[email protected].es †v[email protected] ‡[email protected] §[email protected] ∥[email protected] ¶[email protected] **[email protected] ††[email protected].edu.cn 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, 014024 (2022) 2470-0010=2022=105(1)=014024(19) 014024-1 Published by the American Physical Society hadronic molecule, which is a composite object formed by at least a pair of hadrons via the strong interaction in analogy to atomic nuclei. However, since the quantum numbers of the Xð3872Þ,JPC ¼1þþ, are also accessible for a generic ¯ cc charmonium or a compact tetraquark, debates regarding its internal structure and production mechanisms last since its discovery; see, for example, Refs. [3–11] and references therein for the discussion. Quite recently, the LHCb Collaboration announced the discovery of a double-charm exotic candidate, Tþ cc, which reveals itself as a high-significance peaking structure in the D0D0πþinvariant mass distribution just below the nominal DþD0threshold [12]. Further studies of the Tþ cc performed by LHCb [13] demonstrate quite intriguing properties of this state and allow for several conclusions concerning its nature: (i) Narrow near-threshold structures are observed in the D0D0and DþD0mass spectra, which supports the conjecture that the Tþ cc decays through a formation of the Dmeson at the intermediate stage of the reaction with its subsequent decays to the Dπand Dγfinal states, Tþ cc →D0Dþ →D0D0πþ=D0Dþπ0; Tþ cc →DþD0→DþD0π0=DþD0γ: To produce a visible near-threshold signal in the line shape, the DDpair in the Tþ cc has to be in S-wave. This hints at the quantum numbers of the Tþ cc to be JP¼1þ. (ii) No signal is found in the DþD0πþinvariant mass distribution, nor is any structure seen in the DþDþ mass spectrum. This precludes the existence of the Tþþ cc isospin jI¼1;I3¼1istate and hints at the Tþ cc being an isoscalar. (iii) The parameters of the resonance extracted from a generic constant-width Breit-Wigner fit built by LHCb in Ref. [12] are δmBW ¼−273 61 5þ11 −14 keV; ΓBW ¼410 165 43þ18 −38 keV; where δmBW defines the mass shift, derived from the Breit-Wigner parametrization, with respect to the DþD0threshold. However, since approximately 90% of the D0D0πþevents contain a genuine Dþ meson [13], it is natural to expect (see the discussions in Refs. [14–16]) that the width of the Tþ cc should be smaller than that of the Dþ, which is only (83.41.8) keV [1]. Thus the value of ΓBW quoted above is way too large, suggesting that a more rigorous data analysis is required. (iv) A more profound data analysis reported by LHCb in Ref. [13], based on a unitarized Breit-Wigner parametrization with a momentum-dependent width, allowed to extract the pole position of the amplitude on the second Riemann sheet, ffiffiffis ppole ¼½−360 40þ4 −0−ið24 1þ0 −7Þ keV;ð1Þ where, as before, the real part is given relative to the DþD0threshold. The imaginary part of the pole in Eq. (1) appears to be in a good qualitative agreement with the natural expectation discussed above. (v) Only a lower bound was established for a crucial parameter of the model, g, which defines the coupling strength of the Tþ cc to the DDchannel, jgj>5.1ð4.3ÞGeV at 90ð95Þ%CL:ð2Þ The problem is rooted in the intrinsic properties of the resonance, which is quite narrow and located very close to the threshold. In such circumstances, the amplitude demonstrates a scaling behavior [17]. In addition, once the employed parametrization is convolved with the energy resolution, the resulting shape appears to be hardly distinguishable from the Breit-Wigner distribution with a constant width. As a consequence, the effective range parameter r, which crucially depends on the value of g,was extracted to be 0≤−r<11.9ð16.9Þfm at 90ð95Þ%CL:ð3Þ Similarly, using the formula proposed in Ref. [18], the Weinberg Z-factor of the Tþ cc, that is, the probability to find a compact component in the Tþ cc wave function (a component other than DþD0), was computed using the scattering length and effective range, Z<0.52ð0.58Þat 90ð95Þ%CL:ð4Þ This implies that the properties of the Tþ cc known so far indicate that this state is generally consistent with a molecular nature. Thus from here on we assume that the Tþ cc is an isoscalar state and investigate if all of its properties can be described within a model that treats it as a hadronic molecule. In the particle basis, the isoscalar character is manifested in the equality of the magnitudes of the couplings of the Tþ cc to the channels DþD0and DþD0, while having opposite signs. However, the Tþ cc wave function is dominated by the DþD0component due to the incredible proximity of the mass of this exotic state to the threshold of this channel. The discovery of the Tþ cc quickly spurred a lot of phenomenological studies [14–16,19–36]. However, contrary to the case of the Xð3872Þwith JPC ¼1þþ, where the static one pion exchange (OPE) interaction is attractive, MENG-LIN DU et al. PHYS. REV. D 105, 014024 (2022) 014024-2 which allowed Törnqvist to correctly predict its mass long before its experimental discovery [37], the static OPE provides repulsive and weakly attractive potentials in the isoscalar and isovector DDchannels with JP¼1þ, respectively [37,38], It should be noted, however, that the static approximation for the OPE in charmonium/ double-charm systems of interest here is not justified, since the three-body intermediate state involving the exchanged pion can go on shell. This leads to a potential that has different signs at different values of the momenta. An isoscalar bound state in the DDsystem was predicted in quark model [39,40] and in hadronic-level (with the short-distance potential modeled by meson exchanges) calculations [25,41–46]. In addition, compact double-charm tetraquarks were also predicted in Refs. [41,47–89]. In particular, it was suggested in Ref. [90] to search for the Tþ cc in the channels D0D0πþ and D0Dþγas the LHCb had already collected a sufficient number of events for the discovery of Tþ cc. For a brief review of the literature, see Ref. [25]. In this paper, we present a theoretical analysis of the LHCb data which improves the experimental analysis by LHCb, and the existing theoretical ones in several aspects. (i) We proceed beyond the simplest approach based solely on the short-range contact interactions (see, for example, the most recent work of Ref. [35]) and nonperturbatively include long-range interactions provided by the OPE mechanism. We study its effect on the properties of the Tþ cc under various assumptions about its form. (ii) We study three-body effects in the Tþ cc state which are expected to have a strong impact on its properties. This is because the D’s are unstable and the corresponding three-body DDπthresholds lie very close to and below the two-body DDones, and the Tþ cc resides between the mentioned twoand threebody thresholds. The interplay of those thresholds in the case of the Xð3872Þwas studied in detail in Ref. [91]. (iii) We reliably extract the parameters of the effective range expansion from the low-energy scattering amplitude and discuss the consequences for the compositeness of the Tþ cc. Thus, in this work, we investigate the properties of the Tþ cc in the framework of a nonrelativistic effective field theory constrained with the requirements of isospin and heavy-quark spin (HQSS) symmetries (the leading isospin symmetry breaking is taken into account by using the physical masses of the involved mesons). The paper is organized as follows. In Sec. II, we define our coupled-channel framework including the details of the three-body dynamics. In Sec. III, we introduce different fitting schemes and analyse the LHCb data in the D0D0πþ channel using these schemes. Then we make predictions for the Tþ cc spin partners in the complementary channels. In Sec. IV, a low-energy expansion of the scattering amplitude is performed and the low-energy constants (scattering length and effective range) are extracted. An evidence that the Tþ cc is a composite object is presented in Sec. V. We discuss the results obtained and conclude in Sec. VI. Generalization to the light flavor SU(3) group is outlined in Appendix Aand the effect of a finite width on the effective range is discussed in Appendix B. II. FRAMEWORK A. Interactions 1. Contact potentials The leading-order (LO) DðÞDðÞ interaction in the chiral effective field theory follows from the effective Lagrangian which contains only Oðp0Þcontact potentials, with p denoting a small momentum scale [92], LHH ¼−D00 8TrðH† aHbH† bHaÞ−D01 8TrðσiH† aHbσiH† bHaÞ −D10 8TrðτA aa0H† a0HbτA bb0H† b0HaÞ −D11 8TrðτA aa0σiH† a0HbτA bb0σiH† b0HaÞ;ð5Þ where the subscripts að0Þ,bð0Þ denote flavor indices, τA¼1;2;3 are the isospin Pauli matrices, and the D00;10;01;11 are four low-energy constants (LECs) describing the contact interactions between the heavy-light mesons grouped into the superfield, Ha¼PaþVa·σ;ð6Þ with Paand Vaannihilating the ground-state pseudoscalar and vector charmed mesons, respectively, which in the flavor space are written explicitly as Pa¼D0 Dþa ;Va¼D0 Dþ a :ð7Þ The proximity of the Tþ cc to the DDthresholds suggests that the dominating component of its wave function consists of a Dand Dmeson pair in a relative S-wave. The quantum numbers of such a system, JP¼1þ, perfectly match the findings of the LHCb Collaboration [12,13]. Then we build the DDisoscalar (I¼0) and isovector (I¼1) combinations as jDD; I ¼0i¼−1 ffiffiffi 2 pðDþD0−D0DþÞ; jDD; I ¼1i¼−1 ffiffiffi 2 pðDþD0þD0DþÞ;ð8Þ COUPLED-CHANNEL APPROACH TO Tþ CC INCLUDING …PHYS. REV. D 105, 014024 (2022) 014024-3 and employ the Lagrangian of Eq. (5) to find the corresponding S-wave contact potentials, VI¼0 CT ðDD→DD;1 þÞ¼−2ðD01 −3D11Þ≡v0;ð9Þ VI¼1 CT ðDD→DD;1 þÞ¼D00 þD01 þD10 þD11 ≡v1; ð10Þ where 1þstands for the spin and parity JP. Then, in the particle basis fDþD0;D 0Dþg, the contact potential reads VCTðDD→DD;1 þÞ¼cd dc ;ð11Þ where the diagonal and off-diagonal matrix elements are c¼1 2ðv1þv0Þ;d¼1 2ðv1−v0Þ:ð12Þ According to the claim by LHCb [13], the Tþ cc is an isoscalar state, so in what follows we stick to the potential of Eq. (9) and set to zero the contact isovector interaction, that is, v1¼0in Eq. (10), or equivalently d¼−c,to reduce the number of free parameters. The contact potentials in the complementary spin-parity DðÞDðÞ channels, as well as their generalization to the light quark flavor SU(3) group, can be found in Appendix A. In this paper, we work in the strict isospin limit for the contact potentials and take the isospin breaking effects into account through the mass difference of the charged and neutral DðÞ mesons as well as that of the pions. 2. OPE potential The most important portion of the experimental signal is localized within just 1 MeV below the DþD0threshold, while the splitting between the DþD0and D0Dþthresholds is around 1.41 MeV, which implies that the isospin breaking effects can be significant. We stick to the notations mðÞ 0and mðÞ cfor the masses of the neutral and charged DðÞ mesons, respectively, and take their values to be [1] m0¼1864.84 MeV;m c¼1869.66 MeV; m 0¼2006.85 MeV;m  c¼2010.26 MeV:ð13Þ The LO Lagrangian for the DDπinteraction reads [38,93–95] L¼1 4gTrðσ·uabHbH† aÞ;ð14Þ where u¼−∇Φ=fπwith Φ¼π0ffiffiffi 2 pπþ ffiffiffi 2 pπ−−π0:ð15Þ Here fπ¼92.1MeV is the pion decay constant and the coupling g¼0.57 is determined from the experimentally measured Dþ →D0πþdecay width. The OPE potential can be naturally decomposed into two contributions which correspond to the two different orderingsintheframeworkofthetime-orderedperturbationtheory (TOPT)—see Fig. 1. It should also be noticed that, given the very limited energy and momentum ranges covered by the theory, it is sufficient to employ a nonrelativistic approach for all particles involved, including the pion. Thus, for the propagator of the pion of mass mπ, we use DπðM; p; p0;zÞ¼−1 2mπ½Dπ 1ðM; p; p0;zÞþDπ 2ðM; p; p0;zÞ; Dπ 1ðM; p; p0;zÞ¼miþmjþmπþp2 2miþp02 2mjþp2þp02−2pp0z 2mπ −M−iϵ−1 ; Dπ 2ðM; p; p0;zÞ¼m iþm jþmπþp2 2m iþp02 2m jþp2þp02−2pp0z 2mπ −M−iϵ−1 ;ð16Þ where Mis the total energy, pand p0stand for the incoming and outgoing three-momenta, respectively, with pð0Þ their magnitudes, z¼ðp·p0Þ=ðpp0Þand mðÞ idenotes the mass of the DðÞ in theith channel. To guarantee a proper treatment of the three-body effects, all recoil terms need to be kept in Eq. (16), as discussed in detail inRefs. [96–98] in the context of the πNN and KNN intermediate states. This is especially relevant in the double-charm system at hand given that, near the DþD0threshold, 2mD0þmπþ−M≈mD0þmπþ− mDþ ≈−6MeV, and hence the effective parameter which FIG. 1. The two TOPT contributions (V1and V2, respectively) to the OPE potential between D(single solid line) and D(double solid line) mesons. The dashed line stands for the pion and the vertical thin line shows the relevant intermediate state. MENG-LIN DU et al. PHYS. REV. D 105, 014024 (2022) 014024-4 governs the pion exchange [see the first ordering Dπ 1in Eq. (16)], μπ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mπðmD0þmπþ−mDþÞ p≈i41 MeV;ð17Þ is not only quite small, but also imaginary. The matrix of the OPE potential VOPEðM; p; p0Þin the particle basis, fDþD0ðSÞ;D 0DþðSÞ;D þD0ðDÞ;D 0DþðDÞg;ð18Þ where Sand Din the parentheses indicate the corresponding partial waves, reads VOPE ¼g2 12f2 π 0 B B B B B B @ Vπþ SS −1 2Vπ0 SS −ffiffiffi 2 pVπþ SD 1 ffiffi2 pVπ0 SD −1 2Vπ0 SS Vπþ SS 1 ffiffi2 pVπ0 SD −ffiffiffi 2 pVπþ SD −ffiffiffi 2 pVπþ DS 1 ffiffi2 pVπ0 DS 1 2Vπþ DD −1 4Vπ0 DD 1 ffiffi2 pVπ0 DS −ffiffiffi 2 pVπþ DS −1 4Vπ0 DD 1 2Vπþ DD 1 C C C C C C A ;ð19Þ with the individual partial-wave-projected components given by Vπ SSðM; p; p0Þ¼Z1 −1 dzDπðM; p; p0;zÞðp2þp02−2pp0zÞ; Vπ SDðM; p; p0Þ¼Z1 −1 dzDπðM; p; p0;zÞ1 2p2ð3z2−1Þþp02−2pp0z; Vπ DSðM; p; p0Þ¼Z1 −1 dzDπðM; p; p0;zÞ1 2p02ð3z2−1Þþp2−2pp0z; Vπ DDðM; p; p0Þ¼Z1 −1 dzDπðM; p; p0;zÞ½2ðp2þp02Þð3z2−1Þ−pp0zð9z2−1Þ:ð20Þ It is important to notice that, since the D→Dπvertex is P-wave, the OPE interaction in the DDsystem at hand contains a short-range contribution and, therefore, is well defined only in the presence of the contact potential introduced above [99]. B. Lippmann–Schwinger equation The dynamics of the system under study can be described in terms of a coupled-channel LippmannSchwinger equation (LSE) for the DD→DDT-matrix (amplitude) satisfying three-body unitarity, TαγðM;p; p0Þ¼VαγðM;p;p0Þ−X βZd3q ð2πÞ3VαβðM;p; qÞ ×GβðM;qÞTβγðM;q; p0Þ;ð21Þ where the Greek indices run from 1 to 4 accounting for the channels listed in Eq. (18) and the potential is treated as a sum of the OPE and contact terms [91,100], VðM; p; p0Þ¼VCT þVOPEðM; p; p0Þ:ð22Þ The contact potential in the extended basis (18) takes a matrix form, VCT ¼v0 20 B B B @ 1−100 −1100 0000 0000 1 C C C A ;ð23Þ where, as explained above, we have set to zero the contact isovector interaction and thus taken c¼−d¼v0=2—see Eqs. (11) and (12). The pion exchange potential VOPE is quoted in Eq. (19). The full DDpropagators incorporating both the effect of the self-energy from the Dπloop functions and the contributions from Dγdecay channels can be expressed as G1ðM;pÞ¼G3ðM; pÞ ¼m cþm0þp2 2μc0 −M−i 2ΓcðM;pÞ−1 ; G2ðM;pÞ¼G4ðM; pÞ ¼m 0þmcþp2 2μ0c −M−i 2Γ0ðM;pÞ−1 ; ð24Þ where the reduced masses are μc0¼m cm0=ðm cþm0Þand μ0c¼m 0mc=ðm 0þmcÞ, and the energy-dependent widths read [91] COUPLED-CHANNEL APPROACH TO Tþ CC INCLUDING …PHYS. REV. D 105, 014024 (2022) 014024-5 ΓcðM;pÞ¼ΓðDþ →DþγÞþ g2m0 12πf2 πm c ΣD0πþD0ðM;p;μc0Þ þg2mc 24πf2 πm c ΣDþπ0D0ðM;p;μc0Þ;ð25Þ Γ0ðM;pÞ¼ΓðD0→D0γÞþ g2m0 24πf2 πm 0 ΣD0π0DþðM; p; μ0cÞ þg2mc 12πf2 πm 0½ΣDþπ−DþðM;p; μ0cÞ −ΣDþπ−Dþðmcþm 0;0;μ0cÞ;ð26Þ where ΣijkðM;p;μÞ¼2μijM−mi−mj−mk−p2 2μ3=2 ;ð27Þ with μij ¼mimj=ðmiþmjÞ. Notice that D0has a mass below the Dþπ−threshold and the two contributions in the last term in Eq. (26) cancel against each other at the point p¼0and M¼mcþm 0to ensure that m 0represents the physical mass of the D0. The three-body formalism used here incorporates the full three-body dynamics. It is, however, not Lorentz covariant. Meanwhile, since the missing diagrams appear only at higher order in the power counting, their omission is justified as shown in Ref. [101]. For alternative treatments of the three-body dynamics, see, for example, Ref. [102] and references therein. Since the momentum integrals in the LSE, Eq. (21), diverge, we regularize them with a sharp cutoff Λ.The numerical results presented below correspond to Λ¼ 0.5GeV. However, we have verified that the physical observables are almost Λ-independent in a reasonably wide range of Λfrom 0.3 to 1.2 GeV, consistent with treating OPE explicitly while effectively integrating out all higher degrees of freedom into contact terms, as given in Eq. (22). A very weak Λ-dependence of the results obtained should not come as a surprise given a very large separation of scales involved. Indeed, the signal in the data is localized within Δ≈1MeV from the DDthreshold, so that a typical soft scale for the problem at hand can be estimated as Q≃ffiffiffiffiffiffiffiffi mΔ p≃ 40…50 MeV ≪Λ, where mis a DðÞ-meson mass given in Eq. (13). Therefore, in the entire interval of the cutoffs used, the EFT expansion parameter Q=Λappears to be extremely small, which makes it possible to absorb the leading-order cutoff dependence in a single momentumindependent contact term. Note however that subleading corrections, which scale with the inverse power of the cutoff, still contribute to the problem and give rise to some model dependence of the effective range in this leading-order calculation, as discussed at the end of Sec. IV. C. Line shape in the D0D0π+channel To describe the D0D0πþmass distribution, we first proceed from the scattering amplitudes to the production ones, so that the production amplitude in the αth channel, UαðM;pÞ, takes the form UαðM;pÞ¼Pα−X βZd3q ð2πÞ3TαβðM; p; qÞGβðM; qÞPβ; ð28Þ wherePαis a pointlike production source for theαth channel. In therelativelynarrow energy region ofinterest, weconsider only an S-wave production. In addition, in the isoscalar channel isospin symmetry requires that P2¼−P1. Finally, since the parameter P1can always be absorbed by the overall normalization factor, without loss of generality we set P1¼1, so that the vector of the sources reads Pα¼ð1;−1;0;0Þ. Then, the production rate for the three-body D0D0πþ channel (see the corresponding diagrams shown in Fig. 2) is calculated as [91] dBr½D0D0πþ dM¼NZpmax 0 pdpZ¯ pmax ¯ pmin ¯ pd¯ pjqπU1ðM;pÞ ×G1ðM;pÞþ¯ qπU1ðM; ¯ pÞG1ðM; ¯ pÞj2; ð29Þ where Nis a normalization constant, qπ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μD0πþM−2m0−mπþ−p2 2μp s; ¯ qπ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μD0πþM−2m0−mπþ−¯ p2 2μp s; and pmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μpðM−2m0−mπþÞ q; ¯ pmin;max ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μD0πþM−2m0−mπþ−p2 2μp s∓m0p m0þmπþ ; with μp¼jm0ðm0þmπþÞ=ð2m0þmπþÞj. FIG. 2. Graphical representation for the production amplitude in the D0D0πþchannel with the DDfinal state interaction. The symbol ⊗stands for the pointlike production source Pα, the filled circle is for the full production amplitude Uα[see Eq. (28)] while the filled squares stand for the DDinteractions described by the solution of the LSE quoted in Eq. (21). MENG-LIN DU et al. PHYS. REV. D 105, 014024 (2022) 014024-6 D. Invariant mass distributions in the D0D0 and D0D+channels Invariant mass distributions for the selected D0D0(see Fig. 2) and D0Dþ(see Fig. 3) candidates can be obtained as dBr½D0D0 dm00 ¼N0ZMmax m00þmπþ dMZmmax 23 mmin 23 dm23jqπU1ðM;pÞ ×G1ðM;pÞþ¯ qπU1ðM; ¯ pÞG1ðM; ¯ pÞj2; dBr½D0Dþ dm0c¼N00ZMmax m0cþmπ0 dMZmmax 23 mmin 23 dm23jqπU1ðM;pÞ ×G1ðM;pÞ−¯ qπU2ðM; ¯ pÞG2ðM; ¯ pÞj2;ð30Þ where N0and N00 are normalization constants, m00 and m0c are the invariant masses of D0D0and D0Dþ, respectively, and the particles in the final D0D0πþand D0Dþπ0states are labeled as 1, 2 and 3, in order, while the relative negative sign in the D0Dþchannel is due to the isospin relation gDþ→D0πþ¼ffiffiffi 2 pgD0→D0π0¼−ffiffiffi 2 pgDþ→Dþπ0.1 Further, in Eq. (30), qπ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μ1ðM−m23 −m1Þ p;¯ qπ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μ2ðM−m2−m31Þ p; m31 ¼Mm123 −m12ðm1þm2Þ−m23ðm2þm3Þþm2 1þm2 2þm2 3 m1þm3 ;ð31Þ with μi¼miðm123 −miÞ=m123,m123 ¼m1þm2þm3, and mij the invariant mass of particles iand j. The limits of integration mmin 23 and mmax 23 are determined as mmin 23 ¼m2þm3þμ23½2μ12m123 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ3ðM−m12 −m3Þ p−m1m3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μ12ðm12 −m1−m2Þ p2 2μ2 12m2 3ðm1þm2Þ2; mmax 23 ¼m2þm3þμ23½2μ12m123 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ3ðM−m12 −m3Þ pþm1m3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μ12ðm12 −m1−m2Þ p2 2μ2 12m2 3ðm1þm2Þ2; with μij ¼mimj=ðmiþmjÞ. III. DATA ANALYSIS A. Strategy and fitting schemes From the consideration of the previous section one can easily see that the DDπvertex, described by the Lagrangian of Eq. (14), gives rise to two complementary effects: pion exchange in the DDsystem and a momentum-dependent self-energy of the D, encoded in the momentum-dependent widths given in Eqs. (25) and (26). Therefore, a self-consistent treatment of the three-body dynamics requires that both above effects be simultaneously included in order not to violate threebody unitarity [103]. In particular, keeping a nontrivial momentum dependence of the Dself-energy, thus accounting for the virtual dressing process D→ Dπ→D, while neglecting the pion exchange between the Dand Das the Tþ cc constituents breaks three-body unitarity, which may cause troubles when precisely extracting and interpreting the parameters of the resonance and the low-energy constants in the effective range expansion of the amplitude. Thus, in order to assess the role of the three-body effects, we consider the following three different fit schemes: (i) Scheme I (no three-body effects): only the LO contact potentials in the DDchannels are employed with the constant Dwidths, Γ0ðM; pÞ¼53.7keV and ΓcðM; pÞ¼82.5keV. This scheme is similar in spirit to the one used in Ref. [35]. (ii) Scheme II (partial three-body effects): the dynamical widths of the Dmesons, as given in Eqs. (25) and FIG. 3. Graphical representation for the two contributions to the production amplitude in the D0Dþπ0channel (filled circle) defined in a similar way to that depicted in Fig. 2. 1In Eq. (30), dBr½D0D0=dm00 and dBr½D0Dþ=dm0cshould be understood as the number-of-events distributions measured experimentally. Thus, the overall normalization constants depend on the detection efficiency and vary for different final states. The symmetry factor 1=2in the phase space integration for the D0D0πþchannel due to the presence of identical D0mesons is absorbed by N0. Similarly, the 1=2factor due to the isospin relation for the axial coupling constants, which appears in the amplitude squared of the D0Dþπ0channel relatively to that of the D0D0πþchannel, is absorbed by N00. COUPLED-CHANNEL APPROACH TO Tþ CC INCLUDING …PHYS. REV. D 105, 014024 (2022) 014024-7 (26), are implemented while the OPE potential is not included. (iii) Scheme III (full three-body effects): the complete potential of Eq. (22), which incorporates both the contact and OPE interactions, is used to ensure that the full three-body dynamics is self-consistently taken into account, and in this way three-body unitarity is preserved. We would like to mention that a direct comparison of our results with those from the LHCb analysis of Ref. [13] is not possible, since that analysis includes some OPE effects, but not all [104]. Once the parameters are fixed from the best fit to the data, we search for poles of the amplitude in the complex energy plane. Then the physical Tþ cc state is associated with the corresponding pole, and its effective coupling gα to the channel αis obtained from the residue of the scattering amplitude, gαgβ¼lim M→MpoleðM2−M2 poleÞTαβðMÞ:ð32Þ Here the on-shell scattering amplitude TαβðMÞis evaluated from Eq. (21) by imposing the condition that pð0Þ is the pole momentum of the two-body propagator GαðβÞðM; pÞ, which reduces to Eq. (39) given below for a constant Dwidth. B. Fits to the D0D0π+spectrum and predictions for the D0D0and D0D+line shapes The expression of Eq. (29) for the line shape in the channel D0D0πþcontains only two free parameters: the strength of the contact potential v0from Eq. (9) and the overall normalization factor N,withtheshape depending only on the former one. The signal function is supplemented with the combinatorial background taken directly from the LHCb analysis of Refs. [12,13]. In addition, the experimental energy resolution is taken into account using the resolution function given in Ref. [13].2Thefitresultsforthethree schemes introduced in Sec. III A are shown in Fig. 4, and the best fit parameters are collected in Table I.The quality of each fit can be assessed through the corresponding value of χ2=d:o:f:quoted in the same table. The pole position responsible for the Tþ cc in each scheme is given in Table II. The real and imaginary parts of the pole position are treated as the binding energy and half of the width, respectively. In Scheme I, where a constant width of the Dis employed and hence the three-body cut does not show up, the Riemann surface contains four sheets. Then the Tþ cc pole is located on the first (physical) Riemann sheet (RS-I), just below the DþD0threshold and thus it describes a shallow bound state. This Riemann sheet (RS) corresponds to positive values of the imaginary part of all involved momenta. It has to be noticed, however, that by including a constant Dwidth (or, equivalently, a complex Dmass), one distorts the two-body cut, which does not any longer spread along the real axis, so that the bound state pole below the threshold on RS-I naturally acquires an imaginary part. FIG. 4. Fitted line shapes before (left plot) and after (right plot) convolution with the energy resolution function—see footnote for its explicit form. The background is taken from the LHCb analysis [12,13]. The experimental binning with the bin size of 200 keV is included in the fits. 2The resolution function for the Tþ cc mass distribution, RLHCb, is parameterized by a sum of two Gaussian functions, RLHCbðM; M0Þ¼αGðM;M0;σ1Þþð1−αÞGðM;M0;σ2Þ; Gðx; μ;σÞ¼ 1 ffiffiffiffiffi 2π pσexp −ðx−μÞ2 2σ2;ð33Þ with the parameters σ1¼1.05 ×263 keV, σ2¼2.413 ×σ1, and α¼0.778 taken from Ref. [13]. MENG-LIN DU et al. PHYS. REV. D 105, 014024 (2022) 014024-8 In Schemes II and III, when the three-body channels (DDπ) are included explicitly, the three-body cuts appear with their branch points at the three-body thresholds. Thus, in these two schemes, the Tþ cc pole is located in the lower half plane of the second Riemann sheet (RS-II)3and its position can be accessed through the analytic continuation of the self-energy [105], ΣijkðM; p; μÞ→(−ΣijkðM;p;μÞ;ImM<0&ReðM−mi−mj−mk−p2 2μÞ>0; ΣijkðM;p;μÞ;ImM<0&ReðM−mi−mj−mk−p2 2μÞ<0;ð34Þ with Σijk defined in Eq. (27). Then, in the energy range near the Tþ cc pole, the D0D0πþand DþD0π0channels are on their unphysical RSs while the DþDþπ−is on its physical RS. A comment on the role played by the three-body dynamics in the Tþ cc is in order here. As mentioned above, the imaginary part of the pole can be treated as half of the Tþ cc width. It is, therefore, instructive to notice that neglecting the three-body dynamics due to the finite life time of the Done overestimates the Tþ cc width by up to a factor of 2 (compare the imaginary parts of the pole positions for Schemes I and II quoted in Table II). This shift is partially overcome once also the three-body cut is included in the scattering potential. If after neglecting the three-body effects, one employs the static approximation for the OPE, together with a constant width of the D, the half width of the Tþ cc would turn out to be around 70 keV thus overestimating the full result in Scheme III by a factor of 2.5. These results agree with the conclusions obtained in Ref. [91] about the role of the three-body D¯ Dπdynamics in the Xð3872Þ. In Table III, we compile the values of the effective coupling constants to the different channels extracted as detailed in Eq. (32). For convenience, we also define these couplings in the isospin basis. In Schemes I and II, where only isospin conserving contact interactions are retained, the coupling of the Tþ cc to the isovector channel vanishes exactly, while in Scheme III, because of the isospin symmetry breaking effects in the OPE driven by the mass difference between the charged and neutral pions, there is a non-vanishing, though very small, admixture of the isovector component. The D-wave couplings in Scheme III (given only in the isospin basis) are strongly suppressed compared with the S-wave ones. This should not come as a surprise given the very limited near-threshold energy range spanned by the signal. With the contact interaction parameter v0[see Eq. (9)] determined from the fit, it is straightforward to predict the invariant mass distributions in the D0D0and D0Dþ channels as given in Eq. (30), and the results are presented in Figs. 5and 6. In both figures, the left and right panels show the corresponding distributions before and after convolution with the experimental energy resolution function, respectively. For the latter, we use the same form as that for the experimental D0D0πþmass distribution with the two Gaussian functions given in Eq. (33)—see footnote 2. Both distributions present a narrow peak just above the corresponding DD threshold, which is a reflection of the Tþ cc, as a very shallow bound state of the Dand D, decaying into DDπthrough the intermediate off-shell D with a very small energy released in the transition D→Dπ. Thus the form of the DD mass distributions supports the interpretation of the Tþ cc as a weakly bound TABLE I. Values of χ2=d:o:f:for the three fit schemes, together with the fitted values of v0[see Eq. (9)]. The cutoff in the LSE is set to Λ¼0.5GeV. Scheme I II III χ2=d:o:f:0.79 0.74 0.71 v0[GeV−2−23.34 0.08 −22.88þ0.08 −0.06 −5.04þ0.10 −0.08 TABLE II. The pole position of the Tþ cc relative to the DþD0threshold and the Riemann sheet (RS) where the pole is located in each scheme (see the text for details). The errors are statistical propagated from fitting to the LHCb data while the uncertainties from the cutoff variation are well within the errors quoted here. Scheme I II III Pole [keV] −368þ43 −42 −ið37 0Þ(RS-I) −333þ41 −36 −ið18 1Þ(RS-II) −356þ39 −38 −ið28 1Þ(RS-II) 3Note that here RS-II is not the second Riemann sheet in the usual sense in two-body scattering. Instead, we use it to refer to the unphysical RS, specified by Eq. (34), with respect to the branch points at the three-body thresholds. COUPLED-CHANNEL APPROACH TO Tþ CC INCLUDING …PHYS. REV. D 105, 014024 (2022) 014024-9 labeled as X0,anab pair in their relative S-wave, and the a½cdstate, where the particles cand dare the products of the decay b→cþdwhich proceeds in a partial wave defined by the angular momentum l(the discussion around Eq. (38) corresponds to a P-wave decay with l¼1). Thus the wave function of the resonance Xcan be written as jXi¼0 B @ CjX0i χðpÞjabi φðp;qÞja½cdi 1 C A;ðB1Þ where pand qare the center-of-mass momenta in the ab and cd subsystems, respectively. The twoand three-body thresholds, Mð2Þ thr and Mð3Þ thr , respectively, are split by ER¼Mð2Þ thr −Mð3Þ thr >0;ðB2Þ and the width of the unstable constituent bis ΓR¼glElþ1=2 R;ðB3Þ where glis a coupling constant which governs the decay b→cþd. It also proves convenient to define a dimensionless ratio λ¼ΓR 2ER ;ðB4Þ which in what follows will be assumed to be small, λ≪1, corresponding to the case of a narrow constituent. The system of coupled-channel Lippmann–Schwinger equations for the state (B1) was solved in Ref. [113] for the interaction between channels exhausted by the transitions X0↔ab and ab ↔acd described by momentumdependent form factors whose explicit form is not important. As a result, the line shapes in the three-body final state acd were obtained and compared with the predictions of simple prescriptions found in the literature. The scattering amplitude in the S-wave two-body channel was found to be Tðaþb→aþbÞ¼π μ g E−EXþGXðEÞ;ðB5Þ where the self-energy GXðEÞof the resonance Xcan be written as ReðGXðEÞÞ ¼ 1 2gκeffðEÞ;ImðGXðEÞÞ ¼ 1 2gkeffðEÞ; ðB6Þ with gbeing an effective coupling constant. In the limit ΓR→0the quantities keff and κeff turn to the usual two-body momentum k¼ffiffiffiffiffiffiffiffiffi 2μE pabove the two-body threshold and its analytical continuation below the threshold, respectively. Here we choose the energy Eto be counted from the two-body threshold Mð2Þ thr (note also that it is counted from the lower-lying three-body threshold in Ref. [113], so an energy shift by ERis required to arrive from the formulas derived in Ref. [113] to those quoted below). Since here we are interested in the contribution to the effective range r0coming from the width of the unstable constituent b, it is sufficient to consider only the function κeffðEÞintroduced in Eq. (B6). Then, according to the findings of Ref. [113], κeffðEÞ¼κ1ðEÞþκ2ðEÞ−κ1ðEXÞ−κ2ðEXÞ;ðB7Þ where κ1ðEÞ¼−1 πμZ∞ 0 p2dpE−p2 2μ ðE−p2 2μÞ2þg2 l 4ðEþER−p2 2μÞ2lþ1 ðB8Þ and κ2ðEÞ¼−gl 2πμZ∞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μðERþEÞ pp2dp ×ðp2 2μ−ER−EÞð2lþ1Þ=2 ðE−p2 2μÞ2þg2 l 4ðEþER−p2 2μÞ2lþ1:ðB9Þ Notice that κ2ðEÞis suppressed as compared to κ1ðEÞby a factor gl∝λ, which is small by assumption, thus in what follows we use that κeffðEÞ≈κ1ðEÞ−κ1ðEXÞ;ðB10Þ where the last, constant term does not contribute to the effective range evaluated as a derivative in the energy from κeffðEÞ. Then, using the explicit expression for κ1ðEÞfrom Eq. 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