Towards a new paradigm for heavy-light meson spectroscopy

Since 2003 many new hadrons, including the lowest-lying positive-parity charm-strange mesons ${D_{s0}^*(2317)}$ and ${D_{s1}(2460)}$, were observed that do not conform with quark model expectations. It was recently demonstrated that various puzzles in the charm meson spectrum find a natural resolution, if the SU(3) multiplets for the lightest scalar and axial-vector states, amongst them the ${D_{s0}^*(2317)}$ and the ${D_{s1}(2460)}$, owe their existence to the nonperturbative dynamics of Goldstone-Boson scattering off $D_{(s)}$ and $D^*_{(s)}$ mesons. Most importantly the ordering of the lightest strange and nonstrange scalars becomes natural. In this work we demonstrate for the first time that this mechanism is strongly supported by the recent high quality data on the ${B^-\to D^+\pi^-\pi^- }$ provided by the LHCb experiment. This implies that the lowest quark-model positive-parity charm mesons, together with their bottom counterparts, 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.

Since 2003 many new hadrons, including the lowest-lying positive-parity charm-strange mesons D * s0 (2317) and Ds1(2460), were observed that do not conform with quark model expectations. It was recently demonstrated that various puzzles in the charm meson spectrum find a natural resolution, if the SU(3) multiplets for the lightest scalar and axial-vector states, amongst them the D * s0 (2317) and the Ds1(2460), owe their existence to the nonperturbative dynamics of Goldstone-Boson scattering off D (s) and D * (s) mesons. Most importantly the ordering of the lightest strange and nonstrange scalars becomes natural. In this work we demonstrate for the first time that this mechanism is strongly supported by the recent high quality data on the B − → D + π − π − provided by the LHCb experiment. This implies that the lowest quark-model positive-parity charm mesons, together with their bottom counterparts, 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. First principle lattice QCD (LQCD) calculations are indispensable in this regard. In many cases, one can efficiently bridge the computationally intensive LQCD framework with complicated experimental observables using chiral perturbation theory (ChPT)-the effective field theory for QCD at low energies-and its unitarization to fulfill probability conservation. In this work we demonstrate how such a combination leads to the resolution of a number of longstanding puzzles in charmmeson spectroscopy. It also paves the way towards a new paradigm in the spectroscopy for heavy-light mesons.
Until the beginning of the millennium heavy-hadron spectroscopy was assumed to be well understood by means of the quark model [1,2], which describes the positive-parity ground state charm mesons as bound systems of a heavy quark and a light antiquark in a Pwave. This belief was put into question in 2003, when the charm-strange scalar (J P = 0 + ) and axial-vector (1 + ) mesons D * s0 (2317) [3] and D s1 (2460) [4] were discov- * fkguo@itp.ac.cn ered (for recent reviews on new hadrons, see Refs. [5][6][7][8][9][10][11]), since the states showed properties at odds with the quark model. Moreover, attempts to adjust the quark model raised more questions [12]. Various alternative proposals were put forward about the nature of these new states including D ( * ) K hadronic molecules (loosely bound states of two colorless hadrons) [13,14], tetraquarks (compact states made of two quarks and two antiquarks) [15] and chiral partners (doublets due to the chiral symmetry breaking of QCD in heavy-light systems) [16,17]. The situation became more obscure in 2004, when two new charm-nonstrange mesons, the D * 0 (2400) [18] and D 1 (2430) [19], were observed. In brief, the experimental discoveries brought up three puzzles: (1) Why are the D * s0 (2317) and D s1 (2460) masses much lower than the quark model expectations for the lowest 0 + and 1 + cs mesons?
(2) Why is the mass difference between the D s1 (2460) and the D * s0 (2317) equal to that between the ground state vector meson D * + and pseudoscalar meson D + within 2 MeV?
(3) Why are the D * 0 (2400) and D 1 (2430) masses almost equal to or even higher than their strange siblings, a relationship exploited in many works [20][21][22][23], although states with an additional strange quark are typically at least 100 MeV heavier since m s /m d 20, see, e.g., Ref. [ 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.
As outlined below, in recent works it was demonstrated that analyses combining effective field theory methods with LQCD allows one to resolve all those puzzles. These analyses suggest that all low-lying positive-parity heavy open-flavor mesons qualify as hadronic molecules. In this paper we add two crucial pieces to the existing line of reasoning, namely we propose a lattice QCD study at unphysical quark masses that will allow one to see the twomeson character of the mentioned states more explicitly and we demonstrate that recent data on B − → D + π − π − show a nontrivial structure fully in line with the proposed dynamical picture.
One reason why the analyses that led to the D * 0 (2400) and D 1 (2430) resonance parameters in the Review of Particle Physics (RPP) [24] 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 energydependent interactions even for S-wave couplings. The standard Breit-Wigner (BW) resonance shapes used in the experimental analyses correspond, however, to constant couplings. Moreover, the energy range of these states overlaps with various S-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 [25][26][27][28][29][30][31][32][33][34]. In this approach, ChPT 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. [35]. Although there is no unique method for unitarization, different methods do not differ much as long as there are no prominent left-hand cuts [36], as is the case here. It should be mentioned that any algebraic unitarization generates logarithms at higher order with wrong coefficients -this is discussed for the case of ππ scattering in Ref. [37] (for a more recent discussion see Ref. [38]). As long as the unitarization is set up as for the amplitudes employed here, those appear only at orders higher than the order of the potential. However, there is no a priori way to estimate their significance and the reliability of the amplitudes must be tested, e.g., by a comparison with lattice data or experiment. We will employ here the next-toleading order (NLO) version whose free parameters have been fixed to the Goldstone-Boson-charm-meson scattering lengths determined in fully dynamical LQCD in channels without disconnected diagrams [32]. Later it was demonstrated [39] that these coupled-channel amplitudes properly predict the energy levels generated in LQCD (with a pion mass M π 391 MeV) for the isospin-1/2 channel even beyond the threshold [40]. This means that now the scattering amplitudes for the coupled Goldstone-Boson-charm-meson system are available that are 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 [41] is employed. The predicted masses for the lowest charm-strange positiveparity mesons are fully in line with the well-established measurements, and those for the bottom-strange mesons are consistent with LQCD results with an almost physical pion mass [42], see Table I where the uncertainties quoted stem from the one-sigma uncertainties of the parameters in the NLO UChPT determined in Ref. [32].
It should be noted that in principle in addition to the mentioned uncertainty there is potentially an additional uncertainty stemming from the truncation of the chiral expansion at NLO -in other words terms of order (M K /Λ χ ) 3 ∼ 10% were neglected. It is difficult to translate this uncertainty into an uncertainty of the pole locations since the parameters of the NLO potential were fixed to LQCD data such that higher order effects are effectively absorbed into the parameters. However, we did check that a variation of the strength of the potential by 10% simply moved the poles by 10-20 MeV -but neither the number of poles nor their sheets did change as a result of this variation.
The first two of the puzzles listed above are solved in this picture: the D ( * ) K hadronic molecules do not correspond to the quark model cs states; spin symmetry predicts that the binding energies are independent of the heavy meson spin up to an uncertainty of about 10%, as the leading spin symmetry breaking interaction is also of NLO in the chiral expansion. Moreover, there are two  Table II. 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 D * 0 (2400) 0 and D 1 (2430) 0 given in RPP are also listed.
This pattern of two I = 1/2 states emerges naturally in the underlying formalism since already leading order ChPT interactions are attractive in two flavor multiplets, to which the two states belong: the anti-triplet and the sextet [25,39]. These two scalar isospin I = 1/2 states were predicted in the earlier works of Refs. [25,26], where, however, less refined amplitudes were employed.
Given the above discussion, it is important to test the scenario outlined above as much as possible. In this work we discuss two possible paths: On the one hand we propose a numerical experiment on the lattice, on the other hand we demonstrate that recent experimental data provide additional support of the nontrivial dynamics that leads to the existence of the light positive-parity opencharm states.
If the mentioned states wereqc states, they would all be members of the flavor anti-triplet -the presence of the sextet is a nontrivial prediction emerging from the meson-meson dynamics that the picture presented above is based on. We notice that while we predicted two I = 1/2 states (see Table II), the Hadron Spectrum Collaboration reported only one, located just below the Dπ threshold, in their lattice calculation at M π 391 MeV [40]. This is in line with the expectation that the lower pole would become a bound state at M π 350 MeV [32]. The authors of Ref. [40] report that they "do not find any further poles in the region where ... [their lattice analysis] constrained the amplitudes". This does on the other hand not exclude the presence of the second pole advocated in Ref. [39] as well as above: The quote simply reflects the fact that while various of the amplitudes employed in the analysis of Ref. [40] contained a second pole, its location was strongly parametrization-dependent [43]. With the quark masses used in Ref. [40], the predicted sextet pole is located deep in the complex plane [39], and thus it is not captured easily. The advantage of our amplitudes compared to those employed in the analysis of Ref. [40] is that they are constrained by both the pattern of chiral symmetry breaking of QCD as well as lattice data in additional channels. To further test our explanation for the light positive parity open charm states, we propose to search for them in lattice studies at a SU(3) symmetric point, with a relatively large quark mass m u = m d = m s , such that the lightest pseudoscalar-meson masses will be near or above m φ 475 MeV. We predict that the sextet pole will become a virtual state below threshold for such large quark masses, and that it would even become a bound state for higher quark masses. This behaviour is illustrated in Fig. 1, where one can see that now the pole is close to threshold, and it should be easy to detect in a lattice calculation. Note that the trajectory of the pole displayed in Fig. 1, in particular that in a certain parameter range resonance poles exist in the complex energy plane below threshold, is common for two-meson states in a relative S-wave. This feature is discussed in quite general terms in Ref. [44] (see also Refs. [45,46] for the f 0 (500) case) and was first presented for the open flavor states in the focus here in Ref. [31].
In the following, we show that our resolution to these puzzles is backed by precise experimental data by showing that the amplitudes with the two D * 0 states are fully consistent with the LHCb measurements of the reaction B − → D + π − π − [48], which are at present the best data providing access to the Dπ system and thus to the nonstrange scalar charm mesons. Therefore, all the available theoretical, experimental and LQCD knowledge is consistent with the existence of two D * 0 states in the mass region where there was believed to be only one D * 0 (2400). The Feynman diagrams for the decay amplitude for B − → D + π − π − are shown in Fig. 2. All the channels (D + π − , D 0 π 0 , D 0 η and D + s K − ) coupled to D + π − need to be considered in the intermediate states. The decay amplitude in the energy region up to 2.6 GeV, which is sufficient to study the low-lying scalar states, can be decomposed into S-, P -and D-waves, where A 0,1,2 (s) correspond to the amplitudes with D + π − in the S , P and D waves, respectively, and P L (z) are the Legendre polynomials. For the P -and D-wave amplitudes we use the same BW form as in the LHCb analysis [48]. However, for the S-wave we employ where A and B are two independent couplings following from SU (3) flavor symmetry [49], and E π,η,K are the energies of the light mesons. The effective Lagrangian for the production vertex leading to the above amplitude can be found in Appendix A. Here the T I ij (s) are the S-wave scattering amplitudes for the coupled-channel system with total isospin I, where i, j are channel indices with 1, 2 and 3 referring to Dπ, Dη and D sK , respectively. These scattering amplitudes can be found in Ref. [32] where also all the parameters were fixed. The unitarity relation with ρ j (s) the two-body phase space factor in channel-j, is satisfied as long as Im G i (s) = −ρ i (s), which allows us to represent G i (s) via a once-subtracted dispersion relation [50]. The same subtraction constant a A is taken for all channels. The amplitude of Eq. (2) embodies chiral symmetry constraints and coupled-channel unitarity, and thus has a sound theoretical foundation. Here the final state interaction between the two π − mesons is neglected because the two pions are in an isospin-tensor state, and they have a large relative momentum so that they quickly fly away from each other. The so-called angular moments, see, e.g. [48,51], contain important information about the partial-wave phase variations. Neglecting partial waves with L ≥ 3, which is perfectly fine in the energy region of interest as indicated by the LHCb data, the first few moments are given by where ∆δ 1,2 are the phase differences of P -and D-waves relative to the S-wave, respectively. Instead of P 1 and P 3 we propose to analyze the linear combination P 13 , since it only depends on the S-P interference up to L = 2 and is particularly sensitive to the S-wave phase motion. We fit to the data of the moments defined in Eq. (4) up to M D + π − = 2.54 GeV for the decay B − → D + π − π − measured by the LHCb Collaboration [48]. Except for the S-wave Dπ given in Eq. (2), we include the resonances D * and D * (2680) in the P -wave and D 2 (2460) in the D-wave. Their masses and widths are fixed to the central values in the LHCb analysis [48], and their phase parameters are denoted by δ D * , δ D * and δ D2 , respectively. The best fit has χ 2 /d.o.f. = 1.7 and the parameter values are B/A = −3.6 ± 0.1, a A = 1.0 ± 0.1, δ D * = −0.42±0.04, δ D * = 1.1±0.2, and δ D2 = −0.83±0.07. We do not show the four normalization parameters (three for these resonances and one for the S-wave). A comparison of the best fit with the LHCb data is shown in Fig. 3 together with the best fit provided by the LHCb collaboration [48] (dashed) where cubic splines were used to interpolate between certain anchor point -below we detail this method further. The bands in Fig. 3 reflect the one-sigma errors of the parameters in the scattering amplitudes determined in Ref. [32]. It is worthwhile to notice that in P 13 , where the D 2 (2460) 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 D 0 η and D + s K − thresholds at 2.413 and 2.462 GeV, respectively, which leads to two cusps in the amplitude. This effect is amplified by the higher pole which is relatively close to the D sK threshold on the unphysical sheet.
One might wonder if the discrepancy between our amplitude and the data for P 13 at low energies points at a deficit of the former. Fortunately the LHCb Collaboration provides more detailed information on their S-wave amplitude in Ref. [48]: In the analysis of the data a series of anchor points were defined where the strength and the phase of the S-wave amplitude were extracted from the data. Then cubic splines were used to interpolate between these anchor points. In Fig. 4 the S-wave amplitude fixed as described above is compared to the LHCb anchor points. Not only shows this figure very clearly that the strength of the S-wave amplitude largely determined by the fits to lattice data is fully consistent with the one extracted from the data for B − → D + π − π − , the amplitude fixed in experiment also shows clear structures at both the Dη and D sK thresholds. From our point of view the most natural explanation of those structures is that they are the mentioned cusps enhanced in impact by the pole located nearby. Thus the comparison of the S-wave amplitude extracted by the LHCb Collaboration with our result shows the role of the higher pole in the I = 1/2 and S = 0 channel even more clearly than the angular moments discussed above. 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 S-wave is determined by only   sK spectrum in the latter [37].
• Measuring the hadronic width of the D * s0 (2317), predicted to be of about 100 keV in the molecular scenario [32,55], while much smaller otherwise. This will be measured by the PANDA experiment.
• Checking the existence of the sextet pole in LQCD with a relatively large SU(3) symmetric quark mass.

ACKNOWLEDGMENTS
We acknowledge Tim Gershon, Jonas Rademacker, Mark Peter Whitehead and Yan-Xi Zhang for discussions on the LHCb data, and thank Sinéad Ryan and David Wilson for communications regarding the LQCD calculation carried out in Ref. [38]. This work is partially supported by the National Natural Science two real parameters (B/A, a A ), while its phase motion is largely determined from unitarity, Eq. (3). Furthermore, the data for the angular moments for B 0 s →D 0 K − π + [51] can be easily reproduced in the same framework with theDK interaction fixed from Ref. [32] again, which has theD * s0 (2317) as a dynamically generated state. We focus on the angular moments as functions of theD 0 K − invariant mass which were measured in the LHCb experiment [51]. The decay mechanism is similar to the one in Fig. 2, and the final stateD 0 K − can be generated fromD 0 K − , D −K 0 ,D s η andD s π 0 intermediate states. Considering isospin symmetry, the S-wave part of the decay amplitude for this process can be written as where C = √ 2(c 2 + c 4 )/F in terms of the LECs in Eq. (A.1), the channel labels 1 and 2 refer to the DK and D s η channels, respectively, and the superscript of the T -matrix refers to the isospin. Note that the Lagrangian in Eq. (A.1) does not yield a term contributing to B 0 s →D s π 0 π + . Taking the central values of B/A and the subtraction constant a A as determined from fitting to the B − → D + π − π − data, there is only one free parameter in the S-wave amplitude, which is C/A (we choose A to serve as the normalization constant for the S-wave contribution). For the P -and D-waves, we again take the same BW resonances as the LHCb analysis [51], i.e., D * s and D * s (2700) for the P -wave and D s2 (2573) for the D-wave with their masses and widths fixed to the central values in Ref. [51]. The best fits to the angular moments P 0 , P 2 and P 1 − 14 P 3 /9 for the LHCb B 0 s →D 0 K − π + data [51] up to 2.7 GeV leads to χ 2 /d.o.f. = 1.6, and the only free parameter in the S-wave amplitude is determined to be C/A = 4.8 +3.4 −1.7 . A comparison of the fit to the data is shown in Fig. 5.
In summary, 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 B decays, which provide by far the most precise experimental information for the Dπ system. The amplitudes have a pole corresponding to the D * s0 (2317) in the isoscalar FIG. 5. Fit to the angular moments as a function of theD 0 K − invariant mass for the process B 0 s →D 0 K − π + provided by LHCb [51]. The solid lines represent the present work with the bands corresponding to the one-sigma uncertainty propagated from the input scattering amplitudes and the dashed lines show the LHCb fit [51]. strangeness S = 1 channel, and two poles in the I = 1/2 nonstrange channel [39]. The latter pair of poles should replace the lowest J P = 0 + charm nonstrange meson, D * 0 (2400), listed in the RPP [24]. Similarly, the broad D 1 (2430) listed in RPP should also be replaced by two J P = 1 + states. 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, the S-wave molecular admixture of a near-threshold state can be quantified from the scattering length directly [52]. Applying this argument to the DK scattering length in the D * s0 (2317) channel, predicted in Ref. [32] and determined using LQCD [53], reveals that the molecular component of the D * s0 (2317) is larger than 70%, a conclusion confirmed later in Ref. [47,54,55] for both the D * s0 (2317) and the D s1 (2460). All the poles listed in Tables I and II are spin-flavor partners, due to approximate QCD symmetries. Therefore, they should be envisioned as to have the same origin, i.e., hadronic molecules generated from coupled-channel two-hadron chiral dynamics.
Treating other narrow heavy mesons, such as the D 1 (2420) and the D 2 (2460), as matter fields leads to additional molecular states such as the J P = 1 − D * s1 (2860) [56] and its partners. In fact, the interactions of Goldstone bosons with matter fields are relatively weak at low energies because of the chiral symmetry of QCD. Even though, hadronic molecular states can be still generated. One would expect that the S-wave attractive interaction of other hadrons with heavy mesons, not suppressed by chiral symmetry, may produce hadron-hadron states as well, analogous to nuclei. These states are not the exclusive origin of higher resonances, but they are important contributors to the hadron zoo. Given more and more S-wave thresholds at higher energies, quark models are expected to become less and less reliable.
We therefore conclude that the long accepted paradigm underlying open-flavor heavy meson spectroscopy that identifies all ground states with cq or bq 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 a joint effort from experiment, LQCD and phenomenology. We close the paper by summarising a few suggestions that will provide further, non-trivial tests of the scenario proposed here: • Measuring the angular moments, in particular P 1 − 14 P 3 /9, with unprecedented accuracy for the B → D ( * ) ππ and B → D ( * ) sK π reactions. This can be done at LHCb and Belle-II. We expect to see nontrivial cusp structures at the D ( * ) η and D ( * ) sK thresholds in the former, and near-threshold enhancement in the D ( * ) sK spectrum in the latter [39].
• Measuring the hadronic width of the D * s0 (2317), predicted to be of about 100 keV in the molecular scenario [32,57], while much smaller otherwise. This will be measured by the PANDA experiment.
• Checking the existence of the sextet pole in LQCD with a relatively large SU(3) symmetric quark mass.
• Searching for the predicted analogous bottom positive-parity mesons both experimentally and in LQCD.