Single- and double-heavy Hadronic Molecules

TL;DR

This paper reviews the concept of hadronic molecules, arguing that certain heavy mesons like $T_{cc}(3875)^+$ and $X(3872)$ are best described as molecular states rather than quark-antiquark pairs, supported by phenomenology and lattice data.

Contribution

It provides evidence that some enigmatic heavy mesons are hadronic molecules and discusses the potential existence of an isovector partner state of $X(3872)$ as further support.

Findings

$T_{cc}(3875)^+$ and $X(3872)$ are likely hadronic molecules.

Diquark--anti-diquark models conflict with phenomenology and lattice data.

An isovector partner of $X(3872)$ could confirm molecular structure.

Abstract

In this presentation the notion of hadronic molecules is reviewed and it is argued that some of the enigmatic single and double heavy mesons, namely the lowest lying positive parity open charm states, the $T_{cc}(3875)^+$ and the $\chi_{c1}(3872)$ aka $X(3872)$, that do not fit into the conventional quark--anti-quark scheme in fact qualify as hadronic molecules. For the single heavy states we show that an alternative explanation as diquark--anti-diquark structure is at odds with either phenomenology or lattice data. For the $X(3872)$ we discuss also the claimed isovector partner state, whose properties would provide additional strong support for a molecular structure of the $J^{PC}=1^{++}$ states near the $D\bar D^{*}$ thresholds. Its existence could be confirmed by, e.g., a high statistics measurement of the $J/\psi\pi^+\pi^-$ lineshape from $B^0\to K^0 J/\psi\pi^+\pi^-$.

Equations

$$T_{\mathrm{NR}}=\frac{g_0^2}{E+E_B+g_0^2\mu /(2\pi )(ip+\gamma )},$$

$$g_0^2=\frac{2\pi \gamma }{{\mu }^2}\left(\frac{1}{{\lambda }^2}-1+\mathcal{O}\left(\frac{\gamma }{\beta }\right)\right),$$

$$T_{\mathrm{NR}}(E)=-\frac{2\pi }{\mu }\frac{1}{1/a+(r/2)p^2-ip},$$

$$\frac{1}{a}$$

$$r$$

$${\mathcal{A}}_a=C_a{E}_{\varphi a}/F_0^2(F_0=\pi \text{mbox}decayconstant),$$

$$[\bar{3}]\otimes [8]=[\bar{3}]\oplus [6]\oplus [\overline{15}].$$

Full text

1 Introduction

Since the beginning of this century hadron spectroscopy regained focus of the physics community triggered by a series of spectacular observations in the sector of single and double heavy hadrons—for reviews, putting emphasis on different aspects, see Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Many of those states are located close to $S$-wave thresholds of pairs of long lived or, with respect to the strong interaction, stable particles and thus appear to be good candidates for hadronic molecules or, more generally, two-hadron (or even few-hadron) states. A state qualifies as hadronic molecule, if all its properties are consistent with its structure being composed of two (or more) hadrons—a well known example of such a state is the deuteron. Since the hadron pair can go on the mass shell above the threshold, generating an imaginary part in the scattering amplitude that is absent below threshold, there is a branch point at the threshold (for a detailed discussion of the analytic properties of scattering amplitudes see Refs. [11, 13]), which leaves an imprint in properly chosen observables. In contrast to this, diquarks carry a color charge and therefore they are confined within the hadron. This implies that there is no prominent cut in the amplitude, if mesons were dominated by diquark–anti-diquark structures and the above mentioned observables are predicted to be different.

In this writeup I present the overwhelming evidence, why the lowest lying positive parity open charm states as well as the $\chi_{c1}(3872)$ aka $X(3872)$ should be classified as hadronic molecules. The analyses performed employ chiral effective field theory as well as, for the former case, lattice QCD. Moreover, the Weinberg criterion and extensions thereof are used to identify the nature of the states. Accordingly, the write-up is structured as follows: In the next section the Weinberg criterion is introduced. Then we present a short description of chiral perturbation theory (ChPT) and the unitarisation thereof (lattice QCD was presented in depth in other presentations at the conference), before we turn to the discussion of the examples. The presentation closes with a short summary.

2 The Weinberg criterion

Before discussing the implications of a molecular or more general few-hadron structure of certain families of states in some depth, we first need a criterion to identify those states from observable quantities. Thus a criterion was provided by Weinberg already in the 1965 [14] (inelastic channels were later included in Ref. [15]). We here give a very short summary of the key result. For more details see Refs. [3, 10].

For a single channel system the scattering amplitude for two-hadrons in the presence of a near threshold bound-state pole at $E=-E_{B}$, where $E$ denotes the kinetic energy of the hadron pair, can be expressed as [3]

[TABLE]

in the region very close to threshold. In Eq. (1) $\gamma=\sqrt{2\mu E_{B}}$ denotes the binding momentum with $\mu=m_{1}m_{2}/(m_{1}+m_{2})$ for the reduced mass. The term proportional to $\gamma$ needs to be introduced to cancel the analytic continuation of the $ip$ term a $E=-E_{B}$111A bound state is a pole on the physical sheet, where for negative energies $p\to i\sqrt{2\mu|E|}$; for the unphysical sheet we have for the analytic continuation $p\to-i\sqrt{2\mu|E|}$.—for virtual states this term needs to be changed to $-\gamma$ [3]. With this, the finding of Ref. [14] can be condensed into

[TABLE]

where $\beta$ denotes the intrinsic momentum scale of the vertex functions, which may be identified with the mass of the lightest exchange particle allowed. The quantity $\lambda^{2}$ is the probability to find the compact component inside the hadronic state—accordingly we have $g_{0}^{2}\to 0$ for a purely compact state and $g_{0}^{2}\to\infty$ for a purely molecular state. Thus Eq. (2) provides a relation between the coupling of a bound state to the continuum and the structure of the state. To relate this to observables we may employ the effective range expansion (ERE), defined via

[TABLE]

with $a$ and $r$ for the scattering length and the effective range, respectively222We use the particle physics sign convention for the scattering length., that can be determined from fits to data. Matching Eq. (3) to Eq. (1) gives

[TABLE]

using Eq. (2). Thus, for a pure molecule ($\lambda^{2}=0$) we have $a=-1/\gamma+{\cal O}\left({1}/{\beta}\right)$ and $r={\cal O}\left({1}/{\beta}\right)$ and for a purely compact structure ($\lambda^{2}=1$) $a={\cal O}\left({1}/{\beta}\right)$ and $r\to-\infty$. In single channel scattering the corrections to $r$ are larger than zero [16, 17] such that a positive effective range is an unambiguous signature of a purely molecular state [14, 17, 18, 19, 20]. In general, however, not even a sizeable negative effective effective range provides a unique signature of a compact structure, since coupled channel effects also induce a negative contribution to the effective range that needs to be corrected for [18] before applying the Weinberg criterion. Note that one should not use $a$ and $r$ as input to extract $\lambda^{2}$ [21], but more use the relations of Eqs. (4) and (5) to check if, within errors, the properties of a given system are consistent with e.g. a pure molecule ($\lambda^{2}=0$). For generalizations of Weinberg’s criterion also to virtual states and resonances see Refs. [15, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].

3 Chiral effective field theory

Chiral perturbation theory (ChPT) is a well established effective field theory for the Standard Model at low energies—for a recent review see [34]. It not only allows for the evaluation of hadronic amplitudes with controllable accuracy, but also provides a basis for a study of hadronic resonances—for the alternative approach via the recently developed BOEFT see Ref. [35]. ChPT exploits systematically the implications of the Goldstone theorem related to the spontaneous symmetry breaking of the chiral symmetry group of massless QCD, $SU(3)_{L}~{\times}~SU(3)_{R}~{\to}~SU(3)_{V}$, namely that the masses and the interactions of the pseudoscalar ground states, below generically called $\phi$, that can be identified with the Goldstone bosons, vanish for vanishing up-, down- and strange-quark masses and hadron momenta. Then ChPT allows one to include systematically corrections for the finite quark masses (as an expansion in $(m_{q}/\Lambda_{\rm QCD})$ that in hadronic terms translates to $(M_{\rm ps}/\Lambda_{\chi})$, where $M_{\rm ps}$ denotes the masses of the pseudo-scalar mesons that are now pseudo-Goldstone bosons and $\Lambda_{\chi}\sim 1$ GeV the typical hadronic mass scale) as well as the finite hadron momenta (as an expansion in $(p/\Lambda_{\chi}$)). While the theory evolving in this way is formally non-renormalisable and characterised by infinitely many terms, it gets predictive once a proper power counting is introduced that at the same time allows for a renormalisation order-by-order. The two-pion system is one of the best studied system in this context. For a review see Ref. [36].

The scheme can be extended to include heavy sources like nucleons [37, 38] or hadrons that contain heavy quarks [39]. It then turns out that, e.g., the amplitude for scattering of the pseudoscalar ground states off heavy sources for a given set of flavor quantum numbers $a$ at leading order in $S$-wave in the cms can be expressed as

[TABLE]

where the parameters $C_{a}$ can be determined from Group Theory and $F_{0}$ from the weak decay of charged pions 333Formally $F_{0}$ is the pion decay constant in the chiral limit, but differences to the physical value or of higher order in the expansion. Here $E_{\phi}$ denotes the energy of the pseudo-scalar mesons. Thus, there is no free parameter at leading order, which is known as the Weinberg-Tomozawa theorem [40, 41]. Especially, if the chiral expansion converges, the sign of the leading interaction, which is fixed from the start, decides, if the interaction is attractive or repulsive. This observation is crucial for the discussion of heavy-light systems in the next section. For the scattering of pseudo-scalars off heavy sources, at next-to-leading order additional tree level contributions enter with a few parameters to be determined from fits to experiment or to lattice data. Loops enter one order higher.

As a direct consequence of the Goldstone theorem, Eq. (6) contains two features crucial for the following discussion: The leading interaction grows quickly as the energy is increased and the interaction of kaons and eta mesons is stronger than that of pions. The former has the consequence that the amplitude hits the unitarity limit rather quickly. Then the range of applicability of the theory needs to be extended by a proper unitarisation [42, 43] that can be understood in the spirit of a dispersive representation of the inverse scattering amplitude with the subtraction terms fixed by ChPT [44, 45]. The respective studies of heavy-light systems of interest below were initiated in Refs. [46, 47]. As a welcome side effect this at the same time enables the equations to dynamically generate poles in the scattering amplitude making the formalism ideal for the study of hadronic molecules. The latter provides the reason, why in the multiplets of two-hadron states those with kaons and eta mesons show significantly more attraction that those with pions—for a comprehensive discussion of the implications for systems with kaons and heavy mesons see Ref. [48].

The situation is somewhat different for the scattering of two heavy sources: Here the leading-order interaction, that now contains a free parameter (the strength of a contact term with 4 external legs) and, in the formulation we shall employ below, the one-pion exchange, is energy independent. However, still a resummation is necessary, since the intermediate propagator of the two heavy fields that comes in in this way gets enhanced by a factor $\pi\mu/p$—it is this enhancement that can be understood as the origin of the existence of nuclei [49] (for a comprehensive review see Ref. [50]). Especially for the study of doubly-heavy systems in the charm sector it is important to note that the one-pion exchange contribution introduces a three-body cut at the nominal $D\bar{D}^{*}$ threshold. Already for pion masses only slightly above the physical value it also generates a left-hand cut that then starts very close to the physical axis and thus needs to be considered for the chiral extrapolation of the pertinent systems [51, 52, 53, 54]. At next-to-leading order pion mass and $p^{2}$ dependent contact terms enter as well as pion loops—note that it was shown by an explicit calculation that the latter can be absorbed into the counter terms [55].

4 Singly heavy states: The family of positive parity open charm states

With the discovery of $D_{s0}^{*}(2317)$ in 2003 [57] and of its non-strange partner the $D_{0}^{*}(2300)$ [58] shortly afterwards began a quest for the nature of these states. The driving questions, mostly illustrated in Fig. 2, can be summarised as

1. $(i)$

Why is $M(D_{s1}){-}M(D_{s0}^{*})\simeq M(D^{*}){-}M(D)$? 2. $(ii)$

Why are $M(D_{s1})\&M(D_{s0}^{*})$ so much lighter than expected from the quark models (see, e.g., Ref. [56]) that assume a $c\bar{q}$ structure? 3. $(iii)$

Why are $M(D_{0}^{*})\simeq M(D_{s0}^{*})$ and $M(D_{1})\simeq M(D_{s1})$ although usually the presence of strange quarks enhance hadron masses by at least 100 MeV? 4. $(iv)$

Why do unitarised ChPT and lattice analyses provide masses for the lightest $D_{0}^{*}$ state so much smaller than the mass extracted from experiment?

As we will demonstrate, not only provides the hadronic molecule picture—or, for resonances like the $D_{0}^{*}$ one should better say two-hadron structures—a natural solution to all those questions, but also are recent lattice data inconsistent with both a $c\bar{q}$ as well as a diquark–anti-diquark structure.

If indeed the lightest positive parity open-charm states owe their existence to non-pertrubative hadron-hadron dynamics, it is natural that their masses are not dictated by the quark model (item $(ii)$), but get correlated with the pertinent thresholds. Moreover, in this picture the binding energy of kaons and $D$-mesons should equal that of kaons and $D^{*}$-mesons up to spin symmetry violating effects that scale as $(\Lambda_{\rm QCD}/m_{c})\approx 10$%, solving item $(i)$. To address items $(iii)$ and $(iv)$ one needs to have a closer look at the driving interaction given in Eq. (6). First of all one understands immediately, why the $KD^{(*)}$ systems develop bound states, while the $\pi D^{(*)}$ systems only resonances—for a detailed discussion of the pole trajectories of systems containing pseudo-Goldstone bosons as the quark masses get changed, see Refs. [59, 60, 61]. Moreover, from the Group Theory point of view the scattering of the pseudo-scalar mesons off $D$-mesons calls for the decomposition

[TABLE]

The emerging strength parameters $C_{a}$, see Eq. (6), turn out such that the $[\mathbf{\bar{3}}]$ feels the strongest attraction, a somewhat weaker one is found for $[\mathbf{6}]$ and the $[{\mathbf{\overline{15}}}]$ is repulsive [46]. Employing a chiral-unitary amplitude [62], whose parameters were fixed by fits to lattice data for the pion mass dependence of certain heavy light scattering lengths, the authors of [63] demonstrated that not only the recent lattice data for $\pi D$ scattering above threshold of Ref. [64] at 391 MeV pion mass are fully consistent with the amplitude determined earlier at threshold, but also contains two poles, one at 2264 GeV (belonging predominantly to the $[\mathbf{\bar{3}}]$) and one at 2468 MeV (belonging predominantly to the $[\mathbf{6}]$)444The lattice group claimed only a single pole in their analysis—in Ref. [65] it is explained, why the higher one was overlooked. instead of a single one (for more refined analyses of the finite momentum lattice data see Refs. [66, 67, 68, 69]). Moreover, when chirally extrapolated to the physical pion mass the two poles were located at 2105 and 2451 MeV, respectively—in line with the earlier findings of Ref. [47]. The reason can be read off, e.g., Fig. 3 showing the results of Ref. [70]: A naive Breit-Wigner fit of the data will naturally return a mass at 2.3 GeV. However, in the molecular picture this structure appears as the interplay of the mentioned two poles, one above and one below 2.3 GeV. Moreover, chiral symmetry constraints pull the pole location to lower values [71] already in the absence of a second pole. This is backed by an analysis of the phase of the $\pi D$ amplitude also extracted in Ref. [72] which is found inconsistent with a Breit-Wigner pole at $2.3$ GeV, but fully consistent with the amplitude of Ref. [62] as demonstrated in Refs. [73]. Thus, the SU(3)-flavor violation within the multiplets of the molecular states is characterised by the quark mass dependence of the pseudo-Goldstone bosons, which is build into the energy dependence of Eq. (6), while group theory decides, which multiplets exist and their hirarchy—for a more general discussion of two-pole structures see Ref. [74].

Finally we would like to comment why existing experimental and lattice data are in striking conflict with both a quark–anti-quark structure of the states as well as with a diquark–anti-diquark structure for the states discussed above. To exclude the former it is sufficient to fully establish the existence of the states in the flavor $[\mathbf{6}]$—this was done e.g. in Refs. [75, 76] via lattice simulations with pion masses high enough that even this state gets nearly bound as predicted in Ref. [70]. The striking difference between two-hadron structures and the diquark–anti-diquark picture is that in the former there are no states in the $[{\mathbf{\overline{15}}}]$ representation, while in the latter they should be there, if both axial-vector and scalar diquarks exist as relevant degrees of freedom [77]—in Ref. [78] a phenomenology consistent with that of the two-hadron states is found, however, at the cost of abandoning the axial-vector diquarks that on the other hand are needed for a consistent diquark phenomenology in other systems [79]. The key effect of states in the $[{\mathbf{\overline{15}}}]$ representation that would emerge in the diquark–anti-diquark picture is that low the lying $D$-meson spectra in the $0^{+}$ and $1^{+}$ sector should be very different, while they would be very similar in the molecular scenario. The reason for the pattern for the compact tetrquarks is that the mass difference of about 200 MeV between the scalar and the axial-vector diquark is roughly the same for heavy-light and light-light diquarks. Accordingly are the states in the $[{\mathbf{\overline{15}}}]$ representation in the scalar sector about 400 MeV heavier than the state in the $[\mathbf{\bar{3}}]$ and in the axial-vector sector about equally light, simply since scalar (axial-vector) quantum numbers are reached by either combining two scalar or two axial-vector diquarks (a scalar and an axial-vector diquark). Since the lattice simulations of Ref. [80] in a flavor symmetric setting show consistent energy levels for both quantum numbers, the diquark picture is excluded for the lightest positive parity open charm states.

5 Doubly heavy states: $X(3872)$ and its isovector partner $W_{c1}$

The first works studying the $X(3872)$ and the $T_{cc}(3875)$ with non-perturbative pions are Ref. [81] and Ref. [82], which also form the basis of the results presented here—alternatively Ref. [83] propose to work without pions and Ref. [84] to include them perturbatively, but the energy range of applicability of those approaches is more limited and the effects of isospin violation cannot be captured completely, as is illustrated by Fig. 4.

The $T_{cc}(3875)$, discovered by LHCb [85], is seen as a prominent structure in the $D^{0}D^{0}\pi^{+}$ invariant mass distribution resulting from prompt $pp$ collisions—it thus contains two charm quarks and has no strong decay channel besides two $\pi DD$ channels (see upper plot of Fig. 4). On the one hand this makes it an ideal playground for theoretical studies both within EFT as well as lattice QCD, on the other hand it calls for a careful treatment of three-body effects, which need both an energy dependent $D^{*}$ width as well as a full treatment of the pion propagator including its energy dependence and non-locality induced by the $D$-meson recoil terms (for a detailed discussion of this effect see Sec. 3 in Ref. [86]). This is especially necessary to ensure that the imaginary parts that the two contributions induce show the onset of a three-body phase space and that the intermediate two-$D$ state that is present while the pion is in flight has the proper Bose-symmetry. As a result of this the imaginary part of the pole location is very sensitive to the formalism: the pole location that emerges from the full calculation of Ref. [82] is $(-356\pm 39-i(28\pm 1))$ keV, while an omission of the cut in the potential changes the imaginary part to $-(18\pm 1)$ keV [82]. The effective range extracted from the full calculation, corrected for isospin violation following Ref. [18], is $r_{0}=(1.38\pm 0.85)$ fm—thus the $T_{cc}$–phenomenology is fully consistent with its being a $DD^{*}$ hadronic molecule. Moreover, already for pion masses only slightly above the physical value the $D^{*}$ gets stable, such that the pion exchange induces a very close by left-hand branch point [51] and, through this, the chiral extrapolation gets very non-trivial, employing the chiral effective field theory described above [87]. A confirmation of this behavior from careful lattice QCD studies for very near physical pion masses should provide further strong evidence for a molecular nature of $T_{cc}(3875)$.

Finally we report briefly the most recent and most precise study of the $X(3872)$ reported in Ref. [88]. Also in this work the full three-body dynamics as well as the leading isospin violating effects are taken care off, with the latter being mass differences within isospin multiplets of the participating particles (pions and $D$-mesons) as well as $\rho{-}\omega$ mixing. A fit to the data for $B^{+}\to K^{+}(J/\psi\pi^{+}\pi^{-})$ by LHCb [89, 90] and $e^{+}e^{-}\to\gamma(D^{0}\bar{D}^{0}\pi^{0}/J/\psi\pi^{+}\pi^{-})$ by BESIII [91] confirmed the claim of Ref. [92] that in addition to the mostly isocalar $X(3872)$ there needs to be a predominantly isovector state, dubbed $W_{c1}$, present. The pole of the $X(3872)$ is found to be a quasi-bound state with a significance of $2.8\,\sigma$ at $(-160^{+57}_{-74}-125^{+23}_{-38}\,i)\rm keV$, relative to the nominal $D^{0}\bar{D}^{*0}$ threshold and the $W_{c1}^{0}$ pole is found at $(3.1\pm 0.7+1.3^{+1.9}_{-0.6}\,i)\ \rm MeV$ relative to the $D^{+}D^{*-}$ threshold on a so called hidden Riemann sheet—see below for a more detailed discussion. In the existing high statistics data the latter state appears merely as a slight modulation of the the line shape provided by the $X(3872)$ alone, however, as a very non-trivial prediction it is demonstrated in Ref. [88] that this signal should be significantly more prominent in $B^{0}\to K^{0}\left(\left.D^{0}\bar{D}^{0}\pi^{0}\right/J/\psi\pi^{+}\pi^{-}\right)$, which can be studied at both Belle-II and LHCb. The origin of the difference is a combination of the expectation that in $B^{0}\to K^{0}(X/W_{c1}^{0})$ decays, contrary to $B^{+}\to K^{+}(X/W_{c1}^{0})$, the charged intermediate states, $D^{\pm}D^{*\,\mp}$, are more frequently produced than the neutral ones and that the $X(3872)$ and the $W_{c1}^{0}$ sit rather close to the neutral and charged threshold, respectively. Fig. 5 shows the fit to the existing LHCb data for $B^{+}\to K^{+}J/\psi\pi^{+}\pi^{-}$ (left panel), where the fit returned for the ratio of couplings of the source to the charged channel over that to the neutral one 0.5, as well as the prediction for $B^{0}\to K^{0}J/\psi\pi^{+}\pi^{-}$ (right panel), where the same ratio is estimated to be 2. In the figure the red solid line shows the full result, while the green dashed the one of the $X(3872)$ only, which is given by a Flatte parametrisation with parameters fixed to reproduce the pole location as well as the pole residues as suggested in Ref. [93]. For the black dot-dashed curve the $X(3872)$ contribution was subtracted from the full result on the amplitude level—it thus shows the $W_{c1}$ signal.

To better understand the phenomenology of the $X$ and the $W_{c1}$—and in particular why the $W_{c1}$ was not seen prominently in data yet—and to see why it strongly points at a molecular nature of both states, we need to discuss in more detail the pole locations of $W_{c1}^{\pm}$, $W_{c1}^{0}$ and $X$, respectively, largely following Ref. [92]. The natural explanation for the charged $W_{c1}$ states not producing a prominent signal in data is, that they are virtual states—those are below threshold poles on the unphysical Riemann sheet, labeled $R_{-}$, where the subscript denotes the sign of the imaginary part of the two-hadron momentum. Virtual states, which are necessarily of molecular nature [22], can leave a significant imprint in observables only, if they are located very close to the threshold (see, e.g., the neutron-neutron system, which has a virtual pole only 66 keV below threshold, leading to a scattering length of 18.5 fm). The $W_{c1}^{\pm}$ poles are found $8^{+8}_{-5}$ MeV below the $D^{0}D^{*\,\pm}$ threshold555For simplicity in this discussion we neglect the effects from the $D^{*}$ decay and other inelastic channels., such that their impact on the respective lineshapes is small. In particular, in comparison with $X(3872)$, which shows its full peak below threshold as a quasi-bound state, the peak height prior to convolution with the energy resolution is orders of magnitude smaller [92]. If there were no isospin violating effects (for charged and neutral $D^{(*)}$-mesons being mass degenerate and an isospin-symmetric interaction), the $W_{c1}^{0}$ would be at the same location. If we assume in addition that the isovector and isoscalar interactions are of equal strength (note that this assumption allows for a successful phenomenology for light and medium heavy nuclei [94]), also the pole of the $X$ appears as a virtual state. Thus the system hosts two poles, one close to the neutral $D\bar{D}^{*}$ threshold, one close to the charged one—the transitions between the two channels, which scale with the difference in isoscalar and isovector interaction strength, are at this point assumed negligible; an assumption to be lifted below. Introducing isospin violation (most importantly by the mass differences of the neutral and the charged $D$-mesons) affects the charged $W_{c1}$ states only mildly, however, in the neutral sector the thresholds move apart and the two poles start to feel each other—the system becomes a coupled channel problem with, already for infinitesimal channel coupling, the two poles sitting on sheets $R_{-+}$ and $R_{+-}$, respectively. This channel-coupling has two important effects: On the one hand the two poles repel each other—the now lower lying pole feels more attraction, eventually moving from the $R_{-+}$ sheet to the $R_{++}$ sheet, explaining naturally why the physical $X(3872)$ appears as bound state in data. Note that this trajectory is also possible only, when the $X$ is a molecular state, since it starts as a virtual state and needs a strong channel coupling. This claim is consistent with a direct evaluation of the compositness of the $X$ using the method of Ref. [19] (a generalisation of the Weinberg criterion to include range corrections), which gives a probability of $97\pm 2$ % to find the two-hadron component in the $X$ wave function. At the same time the higher lying pole, identified with the $W_{c1}^{0}$, gets pushed up and acquires a width, since it can now decay into the lower lying neutral $D\bar{D}^{*}$ channel. In other words: the $W_{c1}^{0}$ pole moves into the complex plane. However, since it was sitting on the $R_{+-}$ sheet, it stays there with no direct connection to the physical axis besides exactly at the charged $D\bar{D}^{*}$ threshold: it sits on a hidden sheet. The scenario outlined here agrees to that of scenario V4 of Ref. [95]. That the lower lying $X$ state is predominantly isoscalar and the higher lying $W_{c1}$ is predominantly isovector follows most probably from the observation that the isocalar interaction is in reality more attractive than its isovector counter part [88].

6 Summary

These are clearly exciting times for the spectroscopy of single- and double-heavy states, since high quality data meet sophisticated theoretical tools, namely lattice QCD and effective field theories. Here we focussed on two sectors, namely the lowest lying positive parity open charm states and the states sitting close to the $D\bar{D}^{*}$ and $DD^{*}$ thresholds, respectively. The effective field theory ideal to study hadronic molecules is unitarised chiral perturbation theory, which was employed here in all cases. In the first part of the presentation we demonstrated that the phenomenology and the lattice QCD data of the open charm states in the $0^{+}$ and $1^{+}$ sectors not only are fully consistent with what is expected from a two-hadron nature of the states, but also that they are inconsistent with a $c\bar{q}$ as well as diquark–anti-diquark structure. In the second part, where the doubly heavy states with $J^{PC}=1^{++}$ 666Clearly, for the isovector states the $C$ assignment can be applied to the neutral state only. were discussed, evidence was presented that there exits an isovector partner of the $X(3872)$ whose existence can be confirmed by studies of $B^{0}\to K^{0}(X/W_{c1}^{0})$. Moreover, also for these states compelling evidence is presented that they can be considered molecular with the same level of rigour as the deuteron.

Acknowledgments—The author thanks the organisers of Hadron2025 conference for the excellent organisation of a very educating conference, Vadim Baru, Xiang-Kun Dong, Menglin Du, Evgeny Epelbaum, Feng-Kun Guo, Teng Ji, Ulf-G. Meißner, Alexey Nefediev and many others for the pleasant and productive collaboration on the work presented, Feng-Kun Guo and Ulf-G. Meißner for valuable comments on the manuscript, and the CAS President’s International Fellowship Initiative (PIFI) under Grant No. 2025PD0087 for partial support. Part of the work was supported by the MKW NRW under funding code NW21-024-A.