Is χc1 (3872) generated from string breaking?

Is χc1ð3872Þgenerated from string breaking? R. Bruschini 1,* and P. González 1,2,† 1Unidad Teórica, Instituto de Física Corpuscular, Universidad de Valencia–CSIC, E-46980 Paterna, Valencia, Spain 2Departamento de Física Teórica, Universidad de Valencia, E-46100 Burjassot, Valencia, Spain (Received 2 December 2021; accepted 4 March 2022; published 28 March 2022) We show, from a diabatic analysis of lattice results for string breaking, that mixing of Q¯ Qwith openflavor meson-meson configurations may be expressed through a mixing potential which is order 1=mQ. A relation between the minimum string breaking energy gap and the string tension comes out naturally. Using this relation, and matching the energy gap for b¯ bwith lattice QCD data, we study the mixing in the c¯ ccase without any additional parameter. A 1þþ bound state very close below the D0¯ D0threshold, in perfect correspondence with χc1ð3872Þ, is predicted. DOI: 10.1103/PhysRevD.105.054028 A new era in heavy-quark meson spectroscopy begun in 2003 with the discovery by the Belle collaboration of the Xð3872Þ[1], now labeled χc1ð3872Þby the Particle Data Group (PDG) [2],aJPC ¼1þþ meson containing a c¯ c component, but with properties at odds with those expected for a conventional charmonium state. Since then, many other unconventional quarkoniumlike mesons have been discovered, the relevant role played by open-flavor mesonmeson components has been recognized, and an enormous theoretical effort has been dedicated to their description, see, for instance, [3–14] for a comprehensive review. Notwithstanding this, almost 20 years after the discovery of χc1ð3872Þ, a deep dynamical understanding of unconventional heavy-quark mesons is still lacking. In this paper, we aim at contributing to this understanding through the analysis of the dynamical role played by string breaking. We show from lattice results that string breaking may be the essential dynamical ingredient for a QCD-based explanation of the new states, in particular χc1ð3872Þon which we focus. For this purpose, we use the recently developed diabatic approach in QCD, which allows for a unified and consistent description of conventional and unconventional heavyquark mesons from lattice inputs [15,16]. To be more precise, let us consider a heavy-quark meson characterized by quantum numbers JPC, made of a heavy quark-antiquark Q¯ Q(with Qa heavy quark, bor c) component and two open-flavor meson-meson components, MðiÞ¯ MðiÞwith i¼1, 2, where ¯ MðiÞdoes not indicate necessarily the antiparticle of MðiÞ, interacting with the light (gluon and sea quark) fields. From this case, the generalization to a different number of meson-meson components is straightforward. Notice that the mass mMof any open-flavor meson can be written as mM¼mQþd, where mQis the heavy quark mass and dincludes the light quark mass and the binding energy. Hence, for d=mQ≪1, as it will be our case, the center of mass of MðiÞ¯ MðiÞpractically coincides with that of Q¯ Q. On the other hand, given the large ratio of the heavyquark mass, mQ, to the QCD energy scale associated to the light fields, ΛQCD, the instantaneous configuration of these fields can be approximately determined by considering static color sources Qand ¯ Q. Then, in the Q¯ Qcenter of mass frame one can write the multichannel Schrödinger equation ðKþVÞjψi¼Ejψi;ð1Þ where jψiis the heavy-quark meson state jψi¼0 B @ jψð0Þi jψð1Þi jψð2Þi 1 C A with the superscript 0 and ireferring, respectively, to its Q¯ Qand MðiÞ¯ MðiÞcomponent, K the relative kinetic energy, V the (diabatic) potential, and Ethe energy. In order to do a systematic study of (1), we proceed to an analysis of the Hamiltonian in terms of powers of the inverse of the heavy quark mass, 1=mQ. In this manner, the *[email protected].es †[email protected] 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, 054028 (2022) 2470-0010=2022=105(5)=054028(6) 054028-1 Published by the American Physical Society relative relevance of the different interactions in the Hamiltonian can be established. This is of interest to identify the dominant dynamical mechanisms not only in this case, but also if additional components and interactions were considered. So, for the relative kinetic energy, we have K¼0 B B B @ p2 mQ 00 0p2 2μð1Þ0 00p2 2μð2Þ 1 C C C A with pstanding for the relative momentum operator and μ for the reduced mass. Notice that, up to order 1=mQ, one has 1=ð2μðiÞÞ≃1=mQand K ≃1p2=mQ, with 1the identity matrix. The potential V results from integrating out the light degrees of freedom. This Hermitian matrix is the representation of Hstatic, the light-field Hamiltonian containing also the interaction with the Q¯ Qand MðiÞ¯ MðiÞcomponents, in the basis fjζQ¯ Qig ∪fjζMðiÞ¯ MðiÞigi¼1;2, formed by the light field eigenstates in the absence of any mixing interaction [15]. It reads V¼0 B B @ V00 V01 V02 V† 01 V11 0 V† 02 0V22 1 C C A with V00 ¼hζQ¯ QjHstaticjζQ¯ Qi; Vii ¼hζMðiÞ¯ MðiÞjHstaticjζMðiÞ¯ MðiÞi; V0i¼hζQ¯ QjHstaticjζMðiÞ¯ MðiÞi: Let us note that we have put V12 ¼0, meaning that no direct mixing between different meson-meson components through Hstatic is considered, and consequently no direct coupling of jψð1Þiwith jψð2Þiis present. A key point of this diabatic formalism is that the functional form of elements of the diabatic potential matrix V can be obtained from currently available lattice results. Specifically, the diagonal element V00 has been calculated ab initio in quenched (no sea quarks) lattice QCD [17],as the energy of the ground state of the gluon field in the presence of Qand ¯ Qcolor sources placed at a relative distance r. Up to order 1=mQ(for the specific form of the 1=m2 Qspin dependent terms, see [18]), it corresponds to a central potential whose form mimics the phenomenological Cornell potential VCðrÞ¼σr−χ rþ2mQ−β: Notice that using quenched lattice calculations for V00 is justified because, as we shall see explicitly later on, the diabatic potential coincides with the quenched one in the limit mQ→∞. It is also worth remarking that in phenomenological applications of this potential, as the calculation of the spectrum of conventional quarkonium, the string tension σ, color Coulomb strength χ, and heavy quark mass mQare effective parameters whose values may be incorporating some effect from the terms of order 1=m2 Qand higher. The values of σand χare usually assumed to be flavor independent, and this can also be the case for the constant βthrough a convenient choice of the values of the quark masses. For the other diagonal elements Vii we shall follow, for the sake of simplicity, a free meson-meson approximation, implying Vii ¼TðiÞ; where TðiÞ¼mMðiÞþm¯ MðiÞ¼2mQþdðiÞþ¯ dðiÞis the meson-meson threshold. To first order in 1=mQ, it can be expressed as TðiÞ≃2mQþaþb=mQwhere adoes not depend on mQand bhas at most some logarithmic dependence [19]. As for the off-diagonal elements V0i, their radial dependence can be derived from the results of unquenched (with sea quarks) lattice calculations. More concretely, the energy levels of the light fields in the presence of Qand ¯ Q color sources placed at a relative distance r, when jζQ¯ Qi and jζMðiÞ¯ MðiÞimix, have been calculated [20,21].To connect these lattice results with V0i, let us particularize, for reasons that shall be made clear later on, to the case Q≡b,Mð1Þ¯ Mð1Þ≡BþB−, and Mð2Þ¯ Mð2Þ≡B0¯ B0. Let us also realize that the BþB−and B0¯ B0thresholds are approximately degenerate, TBþB−≈TB0¯ B0¼TB¯ Bwith TB¯ Ba shorthand notation for the degenerate threshold mass. Then, we can build light-field states with definite isospin, (I¼0;Iz¼0) and (I¼1;Iz¼0): jζðB¯ BÞI¼0i¼ 1 ffiffiffi 2 pðjζBþB−iþjζB0¯ B0−iÞ; jζðB¯ BÞI¼1i¼ 1 ffiffiffi 2 pðjζBþB−i−jζB0¯ B0−iÞ; so that hζb¯ bjHstaticjζðB¯ BÞI¼0i¼ðV01 þV02Þ=ffiffiffi 2 pand hζb¯ bjHstaticjζðB¯ BÞI¼1i¼ðV01 −V02Þ=ffiffiffi 2 p. As b¯ bhas I¼0and Hstatic does not contain any isospin breaking term (once the thresholds of BþB−and B0¯ B0are taken to be equal), jζðB¯ BÞI¼1icannot couple to b¯ b. This implies V01 ¼V02, so that, in the isospin basis, V can be written as R. BRUSCHINI and P. GONZÁLEZ PHYS. REV. D 105, 054028 (2022) 054028-2 0 B B B @ VCffiffiffi 2 pV01 0 ffiffiffi 2 pV† 01 TB¯ B0 00TB¯ B 1 C C C A : This makes clear that the effect of the degenerate BþB−and B0¯ B0thresholds can be taken into account through only one isospin-zero threshold, which we shall call henceforth B¯ B, whose interaction potential with b¯ bcontains an additional factor ffiffiffi 2 pas compared to that of a nondegenerate threshold like Bs¯ Bs. Actually, in [20] the radial dependence of the eigenstates and eigenvalues of VCffiffiffi 2 pV01 ffiffiffi 2 pV† 01 VB¯ B; where VB¯ Bdiffers from TB¯ Bin the incorporation of the meson-meson interaction, has been calculated. For the eigenstates, it can be expressed as jζ−i¼cosθðrÞjζb¯ biþsin θðrÞjζB¯ Bið2aÞ jζþi¼−sinθðrÞjζb¯ biþcos θðrÞjζB¯ Bi;ð2bÞ where θðrÞis the mixing angle. Following [20], we shall consider that the effect of the meson-meson interaction, VB¯ B−TB¯ B,inθðrÞis the appearance of a bump for low values of r(in [21] no bump appears). A schematic representation of θðrÞwithout the bump is presented in Fig. 1. A look at this figure makes clear that the maximum or ideal (θ¼π=4) light field configuration mixing occurs at approximately the crossing radius rB¯ B, defined by VCðrB¯ BÞ¼TB¯ B, and that the mixing is only significant around rB¯ B. The corresponding eigenvalues, V−and Vþ, are schematically represented in Fig. 2. The minimal energy gap between them, ffiffiffi 2 pΔb, occurring at rB¯ Bis also drawn. From hζjHstaticjζi¼V,hζjHstaticjζ∓i¼0, and Eq. (2), the radial dependence of the mixing potential V01, which we shall call Vmix B¯ BðrÞ, can be straightforwardly extracted (for the complete spatial dependence of V01, see [16]). It reads Vmix B¯ BðrÞ¼−sin 2θðrÞ 2ffiffiffi 2 pðVþðrÞ−V−ðrÞÞ; so that its absolute value is significant only around rB¯ B, where it reaches a maximum jVmix B¯ BðrB¯ BÞj ¼ Δb=2with Δb¼ðVþðrB¯ BÞ−V−ðrB¯ BÞÞ=ffiffiffi 2 p. The term ffiffiffi 2 pVmix B¯ BðrÞdetermines the mixing between the b¯ band B¯ Bcomponents. The physical mechanism underlying this mixing is string breaking. As the static energy of b¯ b,VCðrÞ, approaches that of B¯ B,TB¯ B, an interaction between the two components, from the creation of a light quark pair q¯ qand its recombination with b¯ b, becomes more and more probable. This makes, see Fig. 2, the static energies, which in the absence of string breaking would cross each other at rB¯ B, experience an avoided crossing characterized by a nonvanishing energy gap ffiffiffi 2 pΔb¼ VþðrB¯ BÞ−V−ðrB¯ BÞ≠0. In general, string breaking is expected to occur as a consequence of the Q¯ Q-meson-meson threshold interaction. According to our discussion above, we expect Vmix MðiÞ¯ MðiÞðrÞto be a function of VCðrÞ−TðiÞ. Besides, lattice results for θðrÞand VþðrÞ−V−ðrÞare approximately symmetric with respect to the crossing radius (see Figs. 16 and 17 in [20]). Then, the simplest parametrization may be Vmix MðiÞ¯ MðiÞðrÞ≃− ΔQ 2fðVCðrÞ−TðiÞÞ; where fis a positive even function with an absolute maximum, fð0Þ¼1, which vanishes for VCðrÞ≪TðiÞ FIG. 1. Mixing angle θ, in radians, as a function of the distance r. The crossing radius is highlighted by the dashed vertical line. FIG. 2. Schematic representation of the static energies near the avoided crossing. Solid lines: unquenched ground and excited state light-field energies. Dashed line: quenched ground state static light-field energy. Dotted line: meson-meson threshold. IS χc1ð3872ÞGENERATED FROM STRING …PHYS. REV. D 105, 054028 (2022) 054028-3 and VCðrÞ≫TðiÞ, and we have assumed ΔQto be the same for any of the corresponding thresholds. Notice that in the limit mQ→∞, which implies mMðiÞ→∞, a single-channel (i.e., no mixing) approximation must be recovered since, as shown in [15], the nonadiabatic coupling terms breaking it are weighted by a factor 1=mQ, see Eq. (17) in [15]. Hence, limmQ→∞ΔQ¼0 and we can expand ΔQin powers of 1=mQas ΔQ¼ α=mQþOð1=m2 QÞ, where αis a constant with dimensions of energy squared. Then, at order 1=mQ, we predict Δcmc≃Δbmb:ð3Þ Let us also realize that αhas the same dimensions as σ, and that if there were no confining interaction, no avoided crossing could ever take place. Therefore, it is natural to express αas σtimes a dimensionless constant γ, so that, up to order 1=mQ, we have ΔQ≃γσ mQð4Þ and Vmix MðiÞ¯ MðiÞðrÞ≃− γσ 2mQ fðVCðrÞ−TMðiÞ¯ MðiÞÞ:ð5Þ Equations (3)–(5), where γ≈1as we show next, represent a main outcome of this paper. They tell us that in a systematic expansion of the Hamiltonian of a heavyquark meson in powers of 1=mQ, the dominant order in the string breaking mixing potential is 1=mQwith a strength proportional to the string tension. In practical applications, as previously mentioned, σ,χ,β and mQare effective parameters which are fine-tuned phenomenologically. As for γ, it may be fixed by matching ΔQwith the energy gap ΔLat calculated in lattice QCD, at some finite heavy quark mass that we call ˜ mQ. As a matter of fact, using standard phenomenological values [22] σ¼ ð427.4MeVÞ2and χ¼0.52, a slightly modified quark mass mb¼5215 MeV to get the same β¼855 MeV for charmonium and bottomonium, and ffiffiffi 2 pΔb¼51 MeV ≈ ΔLat from [20], a nice (diabatic) description of the spectrum and properties of bottomoniumlike mesons comes out [23] (we have checked that the slight difference in the value of Δbused in [23] with respect to ffiffiffi 2 pΔb¼51 MeV gives rise to spectral mass differences of 2 MeV at most). Hence, we may identify ˜ mQ≈mb, so that using (4) we get γ≈1.03. Then, taking into account that for the same standard values of σ,χand β, the charm quark mass is quite constrained from electromagnetic transitions in charmonium [24] to be mc≈1840 MeV, we predict from (3) Δc≈102.2MeV. In order to check whether this value of Δcmay give rise to χc1ð3872Þ, we proceed to solve (1), expanding the Hamiltonian up to order 1=mQ, to get the JPC ¼1þþ bound states in exactly the same manner as done in [15],to which we refer for details. There are, however, two crucial differences. First, the value of Δcis now a theoretical input, instead of a parameter fitted to get the χc1ð3872Þ. Second, the different masses of the D0¯ D0and DþD−thresholds are implemented, instead of an effective isospin-zero D¯ D component, where D0¯ D0,DþD−, and D¯ D, is the common shorthand notation for the C-parity eigenstates. For the sake of completeness, we also list thevalues of the meson masses: mD0¼1864.8MeV, m¯ D0¼2006.9MeV, mDþ¼1869.5MeV, mD−¼2010.3MeV. It is worth commenting that the use of these open-charm meson experimental masses has implicit the assumption that terms of order 1=m2 cand higher are negligible for them. As for the radial mixing potential, we assume the Gaussian form, Vmix MðiÞ¯ MðiÞðrÞ¼− Δc 2exp−ðVCðrÞ−TðiÞÞ2 2Λ2; where Λ¼σρ with ρ≈0.3fm the radial scale for the mixing, which we assume to be the same for Q≡cand Q≡b. Notice that from this Gaussian form the corrections of order 1=m2 cand higher in ðVCðrÞ−TMðiÞ¯ MðiÞÞ2, which cannot be easily separated in the calculation, hardly play any quantitative role. We find a JPC ¼1þþ bound state at 3871.6 MeV, which can be assigned to χc1ð3872Þ. The calculated probabilities for the different components, Pc¯ c¼4%,PD0¯ D0¼93%, and PDþD−¼3%, indicate that χc1ð3872Þis very dominantly a D0¯ D0state. However, it should be pointed out that at short distances, r≲1fm, the c¯ cand D0¯ D0probability densities are comparable, as shown in Fig. 3, which may be instrumental to explain the electromagnetic decays of χc1ð3872Þ. In contrast, at distances r≳3fm, only the D0¯ D0component survives. The large value of the root mean square radius, ffiffiffiffiffiffiffiffi hr2i p≈14 fm, supports the physical image of χc1ð3872Þas a loosely bound D0¯ D0molecule. Moreover, the equal content of isospin 0 and isospin 1 in D0¯ D0may provide us with a natural explanation for the approximately equal measured decay rates of χc1ð3872Þto ωJ=ψand πþπ−J=ψ. Let us emphasize that this state is a prediction, not a fit, as the value of Δcis completely fixed from the matching of Δbwith ΔLat. It is nevertheless important to study the robustness of this prediction against small variations of Δcaround its nominal value. A numerical analysis shows that for Δc≳101 MeV the bound state prediction is stable, its binding energy increasing with Δc. Moreover, for Δc∈½101;103MeV, the predicted bound state mass is perfectly compatible with the PDG average experimental mass of χc1ð3872Þ, 3871.65 0.06 MeV. R. BRUSCHINI and P. GONZÁLEZ PHYS. REV. D 105, 054028 (2022) 054028-4 The resulting physical image of χc1ð3872Þis in line with the one from molecular models, where hadronic molecules are the result of nonperturbative meson-meson interactions (see, for example, [6,7,10] and references therein). A more quantitative connection with these models can be done by realizing that, as explicitly shown in a recent study of meson-meson scattering in the diabatic framework [16], the diabatic potential gives rise to a nonperturbative mesonmeson interaction mediated by Q¯ Q[see Fig. 1(b) in Ref. [16]]. Concretely, we have repeated the calculation carried out in [16], but considering D0¯ D0and DþD−as separate channels with nondegenerate thresholds, instead of an effective D¯ Dchannel with isospin zero. Although this analysis is completely out of the scope of this paper, and will be included in a future publication, let us just advance that for Δc≳101 MeV the calculated S matrix consistently reflects the presence of the bound state close below threshold, while for Δc≲101 MeV we get instead a virtual state. It is also important to emphasize that, although we restrict here our analysis to the χc1ð3872Þ, a consistent description of the whole charmoniumlike spectrum, with no significant difference with respect to the one obtained in [15], comes out. Furthermore, systematic corrections could be incorporated through higher order terms in the Hamiltonian expansion. Therefore, we conclude that χc1ð3872Þ(as well as other unconventional heavy-quark mesons) may be generated from the mixing of Q¯ Qwith open-flavor meson-meson components induced by string breaking. ACKNOWLEDGMENTS This work has been supported by Ministerio de Ciencia e Innovación and Agencia Estatal de Investigación of Spain MCIN/AEI/10.13039/501100011033 and European Regional Development Fund Grant No. PID2019–105439 GB-C21, by EU Horizon 2020 Grant No. 824093 (STRONG-2020), and by Conselleria de Innovación, Universidades, Ciencia y Sociedad Digital, Generalitat Valenciana GVA PROMETEO/2021/083. R. B. acknowledges a Formación de Personal Investigador fellowship from Ministerio de Ciencia, Innovación y Universidades of Spain, under Grant No. BES-2017-079860. [1] S.-K. Choi et al. (Belle Collaboration), Observation of a Narrow Charmoniumlike State in Exclusive B→ Kπþπ−J=ψDecays, Phys. Rev. Lett. 91, 262001 (2003). [2] P. A. Zyla et al. (Particle Data Group), Review of particle physics, Prog. Theor. Exp. Phys. 2020, 083C01 (2020). [3] N. Brambilla et al., Heavy quarkonium: Progress, puzzles, and opportunities, Eur. Phys. J. C 71, 1534 (2011). [4] G. T. Bodwin, E. Braaten, E. 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