A new paradigm for heavy-light meson spectroscopy
Understanding the hadron spectrum is one of the premier challenges in particle physics. For a long time, the quark model has served as an ordering scheme and brought systematics into the hadron zoo. However, many new hadrons that were observed since 2003, including the lowest-lying positive-parity charm-strange mesons and , do not conform with quark model expectations. Various modifications to the quark model and alternative approaches have been proposed ever since to explain their low masses and decay properties. Here, we demonstrate that if the lightest scalar and axial-vector open heavy-flavor states, amongst them the and the , are assumed to owe their existence to the nonperturbative dynamics of Goldstone-Boson scattering off and mesons, various puzzles in the spectrum of the charm mesons find a natural resolution. Most importantly the ordering of the lightest strange and nonstrange scalars becomes natural. Furthermore it is demonstrated that the well constrained amplitudes for Goldstone-Boson scattering off charm mesons are fully consistent with recent high quality data on the final states provided by the LHCb experiment. This implies that the lowest quark-model positive-parity charm mesons, together with their bottom cousins, if realized in nature, do not form the ground-state multiplet. This is similar to the pattern that has been established for the scalar mesons made from light up, down and strange quarks, where the lowest multiplet is considered to be made of states not described by the quark model. In a broader view, the hadron spectrum must be viewed as more than a collection of quark model states.
One of the currently most challenging problems in fundamental physics is to understand the nonperturbative regime of the Quantum Chromodynamics (QCD), the fundamental theory for the interaction of quarks and gluons. However, since the quark and gluon fields are confined inside color-neutral hadrons, what needs to be achieved is a quantitative understanding of the hadron spectrum. In this work we demonstrate how chiral perturbation theory, which is the effective field theory for QCD at low energies, combined with a unitarization procedure to fulfill probability conservation and lattice QCD resolves a number of long-standing puzzles in charm-meson spectroscopy and paves the way for a new paradigm in spectroscopy for heavy-light mesons, where heavy refers to the charm and bottom quarks whereas light denotes the up, down and strange quarks.
Until the beginning of the millennium heavy-hadron spectroscopy was assumed to be well understood by means of the quark model GellMann:1964nj ; Godfrey:1985xj , which describes the positive-parity ground state charm mesons as bound systems of a heavy quark and a light antiquark in a -wave. This belief was put into question in 2003, when the charm-strange scalar and axial-vector mesons Aubert:2003fg and Besson:2003cp were discovered (for recent reviews on new hadrons, see Refs. Chen:2016qju ; Chen:2016spr ; Lebed:2016hpi ; Esposito:2016noz ; Guo:2017jvc ; Ali:2017jda ; Olsen:2017bmm ). Not only are both states significantly lighter than the quark model predictions, but also their splitting is different Godfrey:1985xj . Since attempts to adjust the quark model raised more questions Cahn:2003cw , various alternative proposals were put forward about the nature of these new states including hadronic molecules (loosely bound states of two colorless hadrons) Barnes:2003dj ; vanBeveren:2003kd , tetraquarks (compact states made of two quarks and two antiquarks) Chen:2004dy and chiral partners (doublets due to the chiral symmetry breaking of QCD in heavy-light systems) Bardeen:2003kt ; Nowak:2003ra . The situation became more obscure in 2004, when two new charm-nonstrange mesons, the Link:2003bd and Abe:2003zm , were observed. Their quantum numbers suggest that they should be the SU(3) partners of the and , respectively, which was in fact the basis of many theoretical studies, see, e.g., Ref. Mehen:2005hc ; Colangelo:2012xi ; Cheng:2017oqh . However, this assignment implies a near degeneracy of strange and nonstrange states at odds with the common pattern reflecting that the strange quark is significantly heavier than its light siblings, , see e.g. Ref. Patrignani:2016xqp , typically leading to mass differences of the order of 100 MeV for the corresponding hadrons. In brief, the experimental discoveries brought up three puzzles in charm mesons:
Why are the masses of the and much lower than the quark model expectations for the lowest scalar and axial-vector charm-strange mesons?
Why is the mass difference between the and the equal to that between the ground state vector meson and pseudoscalar meson within 2 MeV?
Why are the masses of the nonstrange mesons and almost equal to or even higher than their strange siblings?
Although their bottom cousins are still being searched for in high-energy experiments, it is natural to ask whether such puzzles will be duplicated there and in other sectors. In fact, we will predict the masses of the corresponding bottom mesons in the following, and those for the strange ones are in a remarkable agreement with recent lattice QCD results.
In this letter we demonstrate that a modern theoretical analysis that combines effective field theory methods with lattice QCD allows one to resolve all those puzzles: Not only emerge the and with the right properties as and molecular states, respectively, from nonperturbative scattering between charm and light mesons ( here), but also additional poles appear in the positive-parity nonstrange sector. In particular, one finds two scalar isospin states with the lighter one located more than 100 MeV below its corresponding strange partner. This pattern emerges naturally in the underlying formalism as is demonstrated in the earlier works of Refs. Kolomeitsev:2003ac ; Guo:2006fu , where, however, less refined amplitudes were employed. This coherent picture clearly calls for a change of paradigm for the positive-parity open-flavor heavy mesons: The scalar and axial-vector ground states need to be considered as dynamically generated two-hadron states as opposed to a simple quark-antiquark structure. This necessarily implies that the lowest quark-model positive-parity charm mesons, together with their bottom cousins, if realized in nature, do not form the ground-state multiplet.
One reason why the analyses that led to the resonance parameters of the and the in the Review of Particle Physics (RPP) Patrignani:2016xqp should be questioned is that the amplitudes used were inconsistent with constraints from the chiral symmetry of QCD. As its chiral symmetry is spontaneously broken, the pions, kaons and eta arise as Goldstone Bosons with derivative and thus energy-dependent interactions even for -wave couplings. The standard Breit–Wigner (BW) resonance shapes used in the experimental analyses correspond, however, to constant couplings. Moreover, the range of energy in which these states are found overlaps with various -wave thresholds that necessarily need to be considered in a sound analysis, as these thresholds can leave a remarkable imprint on observables as will be shown below. A theoretical framework satisfying such requirements is provided by the unitarized chiral perturbation theory (UCHPT) for heavy mesons Kolomeitsev:2003ac ; Hofmann:2003je ; Guo:2006fu ; Guo:2006rp ; Gamermann:2006nm ; Flynn:2007ki ; Guo:2009ct ; Liu:2012zya ; Altenbuchinger:2013vwa ; Albaladejo:2015kea ; Du:2017ttu . In this approach, chiral perturbation theory at a given order is used to calculate the interaction potential which is then resummed in a scattering equation to fulfill exact two-body unitarity and allows for the generation of resonances as pioneered in Ref. Truong:1988zp . We will employ here the next-to-leading order (NLO) version whose free parameters have been fixed to the Goldstone-Boson–charm-meson scattering lengths determined in fully dynamical lattice QCD in channels without disconnected diagrams Liu:2012zya . Later it was demonstrated Albaladejo:2016lbb that these coupled-channel amplitudes properly predict the energy levels generated in lattice QCD for the isospin-1/2 channel even beyond the threshold Moir:2016srx . This means that now the scattering amplitudes for the coupled Goldstone-Boson–charm-meson system are available that are thoroughly based on QCD. Moreover, those amplitudes allow us to identify the poles in the complex energy plane reflecting the lowest positive-parity meson resonances of QCD in the charm sector as well as in the bottom sector, once heavy quark flavor symmetry Neubert:1993mb is employed. The predicted masses for the lowest charm-strange positive-parity mesons are fully in line with the well-established measurements, and those for the bottom-strange mesons are consistent with lattice QCD results with an almost physical pion mass Lang:2015hza , see Table 1 where the uncertainties quoted stem from the one-sigma uncertainties of the parameters in the NLO UCHPT determined in Ref. Liu:2012zya .
The first two of the puzzles listed above are solved in this picture, since binding energies of the order of 40 MeV are natural and spin symmetry demands that these are independent of the heavy meson spin up to an uncertainty of about , as the leading spin symmetry breaking interaction is also of NLO in the chiral expansion. Moreover, there are two poles, corresponding to two resonances, in the , channel, with the strangeness quantum number. The predicted poles, located at the complex energies , for both scalar and axial-vector charm and bottom mesons are listed in Table 2. The masses for the lower nonstrange resonances are smaller than those for the strange ones, leading to the solution to the third puzzle. For comparison the currently quoted masses and widths of the and given in the RPP are listed in the last column.
|lower pole||higher pole||RPP|
In the following, we demonstrate that our resolution to these puzzles is backed by the data collected in the LHCb experiment by showing that the amplitudes with the two states are fully consistent with the LHCb data for the reaction Aaij:2016fma , which are at present the best data providing access to the system and thus to the nonstrange scalar charm mesons. Therefore, all the available theoretical, experimental and lattice QCD knowledge is consistent with the existence of two states in the mass region where there was believed to be only one .
The starting point for the analysis of these new LHCb data is the effective Lagrangian for the weak decays to with the emission of two light pseudoscalar mesons, induced by the Cabibbo-allowed transition . In the phase space region near the threshold, chiral symmetry puts constraints on one of the two pions while the other one moves fast and can be treated as a matter field. Moreover, its interaction with the other particles in the final state can be safely neglected. Then the relevant leading order chiral effective Lagrangian reads,
Here, and are the fields for bottom and charm mesons, denotes the trace in the SU(3) light-flavor space, and is the axial current derived from chiral symmetry. The Goldstone Bosons are represented non-linearily via , with
in terms of the pions (), the kaons () and the , and denotes the Goldstone-Boson decay constant in the chiral limit. In addition, is a spurion field with Savage:1989ub
for Cabibbo-allowed decays. The matter field , having the same form as , describes the fast moving light meson. The ) are low-energy constants (LECs). This effective Lagrangian considers both chiral, for the regime with soft Goldstone Bosons, and SU(3) constraints, the latter of which has been considered in Ref. Savage:1989ub .
The Feynman diagrams for the decay amplitude for are shown in Fig. 1, where the filled square denotes the final-state interaction (FSI). All the channels (, , and ) coupled to need to be considered in the intermediate state. The decay amplitude in the mass region up to 2.6 GeV, which is sufficient to study the low-lying scalar states, can be decomposed into -, - and -waves,
where correspond to the amplitudes with in the -, - and -waves, respectively, and the are the Legendre polynomials. For the - and -wave amplitudes we use the same BW form as in the LHCb analysis Aaij:2016fma . However, for the -wave we employ
where and , c.f. Eq. (1), and are the energies of the light mesons. Here, the are the -wave scattering amplitudes for the coupled-channel system with total isospin , where are channel indices with and 3 referring to , and , respectively. These scattering amplitudes can be found in Ref. Liu:2012zya where also all the parameters were fixed. The unitarity relation
with the two-body phase space factor in channel-, is satisfied as long as , which allows us to represent via a once-subtracted dispersion relation Oller:1998zr . The corresponding subtraction constant is taken to be the same for all channels as in Ref. Liu:2012zya . The amplitude of Eq. (5) embodies chiral symmetry constraints and coupled-channel unitarity, and thus has a sound theoretical foundation.
The so-called angular moments contain important information about the partial-wave phase variations. The unnormalized angular moments are defined as the event distribution weighted by the Legendre polynomials,
and the first few moments are given in Eq. (8). The angular moments are unnormalized, but we use the same relative normalization as in the LHCb analyses Aaij:2014baa ; Aaij:2016fma . Neglecting partial waves with , which is perfectly fine in the energy region of interest as shown by the LHCb data, the first few moments are given by
where are the phase differences of - and -waves relative to the -wave, respectively. Notice that instead of and we analyze the linear combination, since it only depends on the - interference up to and is particularly sensitive to the -wave phase motion.
We fit to the data of the moments defined in Eq. (8) up to GeV for the decay measured by the LHCb Collaboration Aaij:2016fma . Except for the -wave given in Eq. (5), we include the resonances and in the -wave and in the -wave. Their masses and widths are fixed as the central values in the LHCb analysis Aaij:2016fma , and their phase parameters are denoted by , and , respectively. The best fit has a and the parameter values are , , , , and . We do not show the four normalization parameters (three for these resonances and one for the -wave). A comparison of the best fit with the LHCb data is shown in Fig. 2 together with the best fit provided by the LHCb collaboration using cubic splines (dashed). In Ref. Aaij:2016fma the amplitude and phase of the -wave were fit freely within five predefined bins. We remark that in total there are 12 bins, but here only the number of bins relevant for the mass range considered is used. The bands in Fig. 2 reflect the one-sigma errors of the parameters in the Goldstone-Boson–charm-meson scattering amplitudes determined in Ref. Liu:2012zya . It is worthwhile to notice that in , where the does not play any role, the data show a significant variation between 2.4 and 2.5 GeV. Theoretically this feature can now be understood as a signal for the opening of the and thresholds at 2.413 and 2.462 GeV, respectively, which leads to two cusps in the amplitude. This clearly highlights the importance of a coupled-channel treatment for this reaction. An updated analysis of the LHC Run-2 data is called for to confirm the prominence of the two cusps. Notice that the shape of the -wave is determined only two real parameters (, ), while its phase motion is largely determined from unitarity, Eq. (6).
Furthermore, the data for the angular moments for Aaij:2014baa can be easily reproduced in the same framework with the interaction fixed from Ref. Liu:2012zya again, which has the as a dynamically generated state. We focus on the angular moments as functions of the invariant mass which were measured in the LHCb experiment Aaij:2014baa . The decay mechanism is similar to the one in Fig. 1, and the final state can be generated from , , and intermediate states. Considering isospin symmetry, the -wave part of the decay amplitude for this process can be written as
where in terms of the LECs in Eq. (1), the channel labels 1 and 2 refer to the and channels, respectively, and the superscript of the -matrix refers to the isospin. Note that the Lagrangian in Eq. (1) does not yield a term contributing to . Taking the central values of and the subtraction constant as determined from fitting to the data, there is only one free parameter in the -wave amplitude, which is (we choose to serve as the normalization constant for the -wave contribution). For the - and -waves, we again take the same BW resonances as the LHCb analysis Aaij:2014baa , i.e., and for the -wave and for the -wave with their masses and widths fixed to the central values in Ref. Aaij:2014baa . The best fits to the angular moments , and for the LHCb data Aaij:2014baa up to 2.7 GeV leads to , and the only free parameter in the -wave amplitude is determined to be . A comparison of the fit to the data is shown in Fig. 3.
To summarize, we have demonstrated that amplitudes fixed from QCD inputs for the Goldstone-Boson scattering off charm mesons not only resolve some longstanding puzzles in charm-meson spectroscopy but also are at the same time fully consistent with recent LHCb data on decays which provide by far the most precise experimental information for the and systems. The amplitudes have a pole corresponding to the in the isoscalar strangeness channel, and two poles in the nonstrange channel Albaladejo:2016lbb . The latter pair of poles should replace the lowest charm nonstrange meson, , listed in the RPP Patrignani:2016xqp .
It should be stressed that the observation that certain scattering amplitudes employ poles does not necessarily imply that the corresponding states need to be interpreted as molecular states. However, a time-honored analysis by Weinberg Weinberg:1965zz showed that for a near-threshold state the molecular admixture of a given state can be quantified from the scattering length directly. Applying this argument to the scattering length in the channel, predicted in Ref. Liu:2012zya and determined using lattice QCD Mohler:2013rwa , reveals that the molecular component of the wave function is larger than 70%, a conclusion confirmed later in Ref. Torres:2014vna ; Bali:2017pdv for both states, the and the . Therefore one is bound to conclude that all poles generated from the scattering equations discussed above are to be envisioned as generated from coupled-channel two-hadron dynamics. It should be stressed that an unambiguous experimental test of this claim would be a measurement of the hadronic width of , predicted to be above 100 keV for molecular states Lutz:2007sk ; Liu:2012zya while below 10 keV for non-molecular states. For a recent review on hadronic molecules, see Ref. Guo:2017jvc .
What has been discussed above for should also apply to the case of as well: The broad listed in RPP should also be replaced by two states. It is thus crucial to test the amplitude with the two states in Table 2 with data in the channel with statistics much higher than the old measurement by Belle Abe:2003zm . Treating other narrow heavy mesons, such as the and , as matter fields leads to additional molecular states such as the Guo:2011dd and its nonstrange partners. In addition, in view that the interaction between Goldstone Bosons with heavy mesons, which is even suppressed at low energies due to chiral symmetry, can still generate hadronic molecular states, one would naturally expect the -wave attractive interaction of other hadrons with heavy mesons can produce poles as well, unless suppressed by, e.g. the Okubo–Zweig–Iizuka rule. These poles probably cannot be the exclusive origin of higher resonances, but they are important contributors to the hadron zoo and, unavoidably must be included. Given more and more -wave thresholds at higher energies, quark models describing mesons as quark-antiquark and baryons as three-quark systems are expected to become less and less useful.
We therefore conclude that the long accepted paradigm underlying open-flavor heavy meson spectroscopy that identifies all ground states with or quark model states, is no longer tenable. In a broader view, the hadron spectrum must be viewed as more than a collection of quark model states, but rather as a manifestation of a more complex dynamics that leads to an intricate pattern of various types of states that can only be understood by an interplay of theory and experiment.
Acknowledgements We acknowledge Tim Gershon, Jonas Rademacker, Mark Peter Whitehead and Yan-Xi Zhang for discussions on the LHCb data. This work is partially supported by the National Natural Science Foundation of China (NSFC) and Deutsche Forschungsgemeinschaft (DFG) through funds provided to the Sino–German Collaborative Research Center “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No. 11621131001, DFG Grant No. TRR110), by the NSFC (Grant No. 11647601), by the Thousand Talents Plan for Young Professionals, by the CAS Key Research Program of Frontier Sciences (Grant No. QYZDB-SSW-SYS013), by the CAS President’s International Fellowship Initiative (PIFI) (Grant No. 2017VMA0025), by Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund under contracts FIS2014-51948-C2-1-P, FIS2017-84038-C2-1-P and SEV- 2014-0398, by Generalitat Valenciana under contract PROMETEOII/2014/0068, and by the “Ayudas para contratos predoctorales para la formación de doctores” program (BES-2015-072049) from the Spanish MINECO and ESF.
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