Axial meson exchange and the Z_c(3900) and Z_{cs}(3985) resonances as heavy hadron molecules
Mao-Jun Yan, Fang-Zheng Peng, Mario Sánchez Sánchez, Manuel Pavon Valderrama
aa r X i v : . [ h e p - ph ] F e b Axial meson exchange and the Z c (3900) and Z cs (3985) resonancesas heavy hadron molecules Mao-Jun Yan, Fang-Zheng Peng, Mario S´anchez S´anchez, and Manuel Pavon Valderrama ∗ School of Physics, Beihang University, Beijing 100191, China Centre d’ ´Etudes Nucl´eaires, CNRS / IN2P3, Universit´e de Bordeaux, 33175 Gradignan, France (Dated: February 26, 2021)Early speculations about the existence of heavy hadron molecules were grounded on the idea that light-meson exchanges forces could lead to binding. In analogy to the deuteron, the light-mesons usually consideredinclude the pion, sigma, rho and omega, but not the axial meson a (1260). Though it has been argued in thepast that the coupling of the axial meson to the nucleons is indeed strong, its mass is considerably heavier thanthat of the vector mesons and thus its exchange ends up being suppressed. Yet, this is not necessarily the case inheavy hadrons molecules: we find that even though the contribution to binding from the axial meson is modest,it cannot be neglected in the isovector sector where vector meson exchange cancels out. This might providea natural binding mechanism for molecular candidates such as the Z c (3900), Z c (4020) or the more recentlyobserved Z cs (3985). However the Z cs (3985) is much more dependent on the nature of scalar meson exchange(in particular whether the sigma has a sizable coupling to the strange quark) than on the contribution of the axialmesons. This later observation might also be relevant for the strange hidden-charm pentaquark P cs (4459). I. INTRODUCTION
Heavy hadron molecules were originally theorized as ananalogy to the deuteron [1, 2]. The argument is that the sametype of forces binding two nucleons together might bind otherhadrons as well. Since then a continuous inflow of ideasfrom nuclear physics has enriched our understanding of heavymolecular states, ranging from phenomenological approachessuch as light-meson exchanges [3–6] to modern e ff ective fieldtheory (EFT) formulations [7–11]. This is not at all surpris-ing: in both cases we are dealing with hadrons, where nucle-ons happen to be the most well-studied of all hadrons.Yet the origin and derivation of nuclear forces has itself atortuous and winding history, in which many competing ideashave been proposed but few have succeeded [12]. The rea-sons behind the failures are important though, as they mightbe specific to nucleons. If we focus on light-meson exchangeforces, the idea is that the nuclear potential can be derivedfrom the exchange of a few light mesons, which usually in-clude the pion, the sigma, the rho and the omega, i.e. the oneboson exchange model (OBE) [13, 14]. Mesons heavier thanthe nucleon are generally not expected to have a sizable con-tribution to the nuclear force: their Compton wavelength isshorter than the size of the nucleon and the forces generatedby their exchange are heavily suppressed.A prominent example is the axial meson a (1260), whichis expected to have a considerably strong coupling to the nu-cleons [15–17]. It is also heavier than the vector mesons: theratio of the masses of the axial and rho mesons, m a and m ρ ,is m a / m ρ ∼ .
6. In fact it is even heavier than the nucleonand its influence on the description of the nuclear force hasturned out to be rather limited [18]. However this is not nec-essarily the case for heavy hadrons: on the one hand, they areheavier than axial mesons and nucleons, and on the other vec-tor meson exchange cancels out in a few specific molecular ∗ [email protected] configurations, which increases the relative importance of theaxial meson.The axial meson has a particularly interesting feature: itsquantum numbers I G ( J PC ) = − (1 ++ ) indicate that it can mixwith the axial current of the pions. That is, we can modify theaxial pion current by including a term proportional to the axialmeson ∂ µ π → ∂ µ π + λ m a a µ , (1)where π a and a µ are the pion and axial meson fields and λ a proportionality constant, which we expect to be in the λ ∼ (1 . − .
1) range. From this the coupling of the axial mesonwith the charmed hadrons D and D ∗ will be proportional totheir axial coupling to the pions, g : g Y(a ) = √ λ m a f π g ∼ (8 − g , (2)where the coupling is defined by matching the axial mesonexchange potential to a Yukawa (check Eqs. (15) and (22)).Depending on the configuration of the two-hadron system un-der consideration, this exchange potential might be remark-ably attractive and even explain binding if other light-mesonexchanges are suppressed.The reason why we are interested in the axial meson is aspecific di ffi culty when explaining the isovector hidden-charm Z c (3900) [19], Z ∗ c (4020) [20] and Z cs (3985) [21] resonances ashadronic molecules: while their closeness to the D ∗ ¯ D , D ∗ ¯ D ∗ and D ∗ ¯ D s - D ¯ D ∗ s thresholds suggest the molecular nature of the Z c ’s [22–26] and the Z cs [27–32], the reasons why their in-teraction is strong remains elusive. It happens that rho- andomega-exchange cancel out for the Z c and Z ∗ c , which in turnrequires a binding mechanism not involving the vector mesons(where this cancellation does not happen with axial mesons,as we will see). Possible explanations include one-pion andsigma exchange [33–35], two-pion exchanges (the correlatedpart of which is sometimes interpreted as sigma exchange)and charmonium exchanges [36–39]. Here we will investi-gate how axial meson exchange works as a binding mecha-nism within the OBE model.The manuscript is structured as follows: in Sect. II we willderive the axial meson exchange potential for the charmedmeson-antimeson system. In Sect. III we will review scalarand vector meson exchange and the potentials their gener-ate, which are still an important part of the OBE potential.In Sect. IV we will investigate how the inclusion of the axialmeson makes the simultaneous description of the X (3872) andthe Z c (3900) more compatible with each other. In Sect. V wewill consider how the previous ideas apply to the Z cs (3985)and how the OBE potential can be made compatible with theexpectations from SU(3)-flavor symmetry. Finally in Sect. VIwe will explain our conclusions. II. AXIAL MESON EXCHANGE
First we will derive the potential generated by axial mesonexchange. We will begin with the interaction Lagrangian be-tween heavy mesons and pions and from it we will derive theLagrangian and potential for axial mesons.The quark content of the heavy mesons is Q ¯ q , with Q = c , b a heavy-quark and q = u , d , s a light-quark. The properties ofheavy mesons and their interactions are expected to be inde-pendent of the heavy-quark spin, which is usually referred toas heavy-quark spin symmetry (HQSS) [40, 41]. The con-sequences of HQSS for heavy-hadron molecules are impor-tant and have been extensively explored in the literature [9–11, 23, 42–46]. For S-wave heavy mesons (e.g. the D and D ∗ charmed mesons) the standard way to take into account HQSSis to define a superfield H Q as H Q = √ h P + ~σ · ~ P ∗ i , (3)with P and ~ P ∗ the J P = − and 1 − heavy mesons, the 2x 2 identity matrix and ~σ the Pauli matrices, where our def-inition of H Q corresponds to the non-relativistic limit of thesuperfield defined in Ref. [47]. This field has good propertieswith respect to heavy-quark spin rotation, i.e. the heavy-quarktransformation | Q i → e − i ~ S H · θ | Q i induces the superfield trans-formation: H Q → e − i ~ S H · θ H Q , (4)from which it is clear that H † Q H Q O field combinations, with O some operator in the form of a 2 x 2 matrix, will be inde-pendent of heavy-quark spin rotations. With this formalism,the interaction of S-wave heavy mesons with the pion can bewritten as L = g √ f π Tr h H † Q H Q ~σ · ~ a i , (5)where g = . g = . ± . ± .
07 as extracted from the D ∗ → D π decay [48, 49]), f π ≃
132 MeV the pion weak decayconstant and ~ a is the (reduced) axial current, which tradition-ally only includes the pion ~ a = ~ ∇ π , (6) where we implicitly include the SU(2)-isospin indices in thepion field, i.e. π = τ c π c with c an isospin index.Alternatively, instead of grouping the P and ~ P ∗ into a singlesuperfield with good heavy-quark rotation properties, we no-tice that the heavy-quark spin degrees of freedom do not comeinto play in the description of heavy-light hadron interactions.This allows to write interactions in terms of a fictitious light-quark subfield , a heavy field with the quantum numbers of thelight-quark within the heavy meson [50]. If we call this e ff ec-tive field q L , the corresponding Lagrangian will read L = g √ f π q † L ~σ L · ~ a q L , (7)with ~σ L the spin operators (Pauli matrices) as applied to thelight-quark spin. When this operator acts on the light-quarkdegrees of freedom it can be translated into the correspondingspin operator acting on the heavy meson field with the rules h P | ~σ L | P i = , (8) h P | ~σ L | P ∗ i = ~ǫ , (9) h P ∗ | ~σ L | P ∗ i = ~ S , (10)with ~ǫ the polarization vector of the P ∗ heavy meson and ~ S thespin-1 matrices. From now on we will work in this notation.The previous light-quark subfield Lagrangian leads to thenon-relativistic potential V π ( ~ q ) = − ζ g f π ~τ · ~τ ~σ L · ~ q ~σ L · ~ q ~ q + m π , = − ζ g f π ~τ · ~τ ~σ L · ~σ L ~ q ~ q + m π − ζ g f π ~τ · ~τ (cid:16) ~σ L · ~ q ~σ L · ~ q − ~σ L · ~σ L q (cid:17) ~ q + m π , (11)with ~ q the exchanged momentum, m π ≃
138 MeV the pionmass and ζ = ± + −
1) for the meson-meson(meson-antimeson) potential (which comes from the G-parityof the pion). In the second and third lines we separate thepotential into its S-wave and S-to-D-wave components (i.e.spin-spin and tensor pieces): owing to the exploratory natureof the present manuscript, we will be only concerned with theS-wave components of hadronic molecules and will ignore theD-waves.Now, to include the axial meson we simply modify the axialcurrent ~ a as follows ~ a = ~ ∇ π + λ m a ~ a , (12)with the isospin indices again implicit, i.e. a = τ c a c , and λ a parameter describing how the axial meson mixes with thepion axial current (which value we will discuss later). Thisreadily leads to the potential V a ( ~ q ) = − λ g m a f π ~τ · ~τ h ~σ L · ~σ L ~ q + m a + m a ~σ L · ~ q ~σ L · ~ q ~ q + m a i = − λ g m a f π ~τ · ~τ ~σ L · ~σ L ~ q + m a (1 + ~ q m a ) + . . . (13)where in the last line we isolate the S-wave component.Finally we are interested in the r-space expressions of thepion and axial exchange potentials. For this we Fourier-transform into r-space, which in the pion case yields V π ( ~ r ) = ζ g m π f π ~τ · ~τ ~σ L · ~σ L e − m π r π r + . . . , (14)where the dots represent tensor (i.e. S-to-D-wave) andcontact-range (i.e. Dirac-delta) terms (which we also ignoreowing to their short-range nature). For the axial meson ex-change we have instead V a ( ~ r ) = − ζ λ g m a f π ~τ · ~τ ~σ L · ~σ L e − m a r π r + . . . , (15)where the dots indicate again contact and tensor terms.The coupling of the axial meson to the hadrons depends on λ , which could be deduced from the matrix elements of theaxial current A µ : h | A µ | π i = f π q µ , h | A µ | a i = f a m a ǫ µ , (16)with q µ the momentum of the pion, f π and f a the weak decayconstants of the pion and axial meson, m a the mass of theaxial meson and ǫ µ its polarization vector. From Eq. (1) wearrive at the identification λ = f a f π , (17)but f a is not particularly well-known. Di ff erent estimationsexists, of which a few worth noticing are:(i) The Weinberg sum rules [51] or the Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin (KSFR) relations [52,53], both of which lead to m a = √ m ρ = .
09 GeV, f a = f π and λ = τ → π ν τ decay involves the axial meson as anintermediate state, and have been used in the past todetermined f a :(ii.a) Three decades ago Ref. [54] obtained m a f a = (0 . ± .
02) GeV , (18)for m a = .
22 GeV, which translates into λ = . ± .
12. Later Ref. [55] made the observationthat m a f a shows a simple dependence on the τ → π ν τ branching ratio, from which it updated theprevious value to m a f a = (0 . ± .
20) GeV ,yielding λ = . ± .
12. (ii.b) Ref. [17] noticed a result from Ref. [56], whichcontains a phenomenological relation between m a f a and the relative branching ratios for the τ → πν τ and τ → πν τ decay. This led the au-thors of Ref. [17] to the estimation m a f a = .
33 GeV , (19)which is equivalent to λ = . τ → πν τ de-cay [57, 58], usually yield λ ∼ . − . m a ∼ . m a f a = (0 . ± .
02) GeV , (20)and m a = . ± .
08 GeV, from which we extract λ = . ± . ± .
12 where the first and second error referto the uncertainties in m a f a and m a , respectively.(iv) Ref. [59] uses QCD sum rules to obtain f a = (238 ±
10) MeV , (21)that is, λ = . ± . λ arelarge. But we can reduce its spread if we concentrate on thedeterminations of λ for which m a is close to its value in theReview of Particle Physics (RPP) [60], i.e. m a = .
23 GeV(which is also the value we will adopt for the mass of the axialmeson). In this case we end up with the λ ∼ (1 . − . λ = .
8. This is thevalue we will use from now on.At this point it is interesting to compare the strengths of theresulting Yukawa-like piece of the previous potentials V Y ( ~ r ) = ± g Y π O I O S e − mr r , (22)where g Y is an e ff ective Yukawa-like coupling, O I = ~τ · τ and O S = ~σ L · σ L the usual isospin and spin operators,while m is the mass of the exchanged meson. For the pion andaxial meson exchange potentials we have that the strength ofthe e ff ective Yukawas are g π ) π ≃ . · − and g ) π ≃ . − . , (23)which gives an idea of the relative strength of axial mesonexchange with respect to the pion. Provided it is attractive,the condition for this e ff ective Yukawa-like potential to bindis 2 µ m g Y π |hO I ihO S i| ≥ . , (24) We notice that Ref. [17] uses the f π ∼
93 MeV normalization for the decayconstants, i.e. a √ with µ the reduced mass of the two hadron system. If weconsider the I G ( J PC ) = + (1 + − ) D ∗ ¯ D system, which is theusual molecular interpretation of the Z c (3900), the potentialis indeed attractive and the previous condition is fulfilled for λ ≥ . m a = .
23 GeV).
III. SCALAR AND VECTOR MESON EXCHANGE
Besides the pion and the axial mesons, usually the otherimportant exchanged light-mesons in the OBE model are thescalar σ and the vector mesons ρ and ω . In the following lineswe will discuss the potentials they generate. A. Scalar meson
For the scalar meson we write a Lagrangian of the type L S = g σ Tr h H † Q H Q i σ (25) = g σ q † L σ q L , (26)depending on the notation (superfield / subfield in first / secondline), with g σ the coupling of the scalar meson to the charmedhadrons. From this Lagrangian we derive the potential V σ ( ~ q ) = − g σ ~ q + m σ , (27)which is attractive and where m σ is the scalar meson mass.Finally, if we Fourier-transform into coordinate space we willarrive at V σ ( ~ r ) = − g σ e − m σ r π r . (28)The parameters in this potential are the coupling g σ andthe mass m σ . For the coupling we will rely on the linearsigma model (L σ M) [61], which we briefly review here asit will prove useful for the discussion on the Z cs (3985) later.The L σ M is a phenomenological model in which originallywe have a massless nucleon field that couples to a combina-tion of four boson fields, i.e. this model contains a nucleoninteraction term of the type L N σ Lint = g ¯ ψ N ( φ + i γ ~τ · ~φ ) ψ N , (29)where ψ N is the relativistic nucleon field and g a couplingconstant. By means of spontaneous symmetry breaking weend up with three massless bosons ~φ , which might be inter-preted as pions, while the isospin scalar φ acquires a vac-uum expectation value ( h φ i = f π / √
2) which also providesthe nucleons with mass. The σ field is defined as a pertur-bation of the φ field around its vacuum expectation value( φ = f π / √ + σ ). This model provides a relation between f π , the nucleon mass M N ≃
940 MeV and the couplingsof the scalar mesons and the pions to the nucleon, where g = g σ NN = g π NN = √ M N / f π = . g π NN = g A √ M N / f π with g A = .
26, which means that the linearsigma model is o ff by about a 26% (or a 30% once we takeinto account the Goldberger-Treiman discrepancy). Thus thisis the expected uncertainty that we should have for g σ NN . Forcomparison purposes, the L σ M gives g σ NN / π = . g σ NN / π = . − . σ M). For the charmed mesons, which containonly one light-quark, we will assume the quark model relation g σ = g σ NN / ≃ . g σ qq = √ m con q / f π ≃ . m con q the constituent q = u , d quark mass).For the mass of the sigma the OBE model of nuclear forcesuses m σ =
550 MeV, but it is also common to find m σ =
600 MeV in a few recent implementations of the OBE modelfor hadronic molecules [33–35, 63, 64]. Nowadays the RPPdesignation of the σ is f (500) and the mass is in the 400 −
550 MeV range [60]. However this does not necessarily implythat the mass of the f (500) pole should be used for the scalarmeson exchange, owing in part to its large width and in partto its relation with correlated two-pion exchange, as has beenextensively discussed [65–68]. Direct fits of g σ NN and m σ canalso lead to more than one solution, though they are usuallycompatible with the RPP mass range of the sigma and withthe expected 30% uncertainty for the coupling in the L σ M.For instance, a renormalized OBE fit to NN data [69] leadsto two solutions, one with m σ =
477 MeV, g σ NN = .
76 andanother with m σ =
556 MeV, g σ NN = .
04. What we will dothen is to investigate binding as a function of the σ mass. B. Vector mesons
The interaction of the vector mesons with hadrons is anal-ogous to that of the photons and it can be expanded in a mul-tipole expansion. For the S-wave charmed mesons the spin ofthe light-quark degree of freedom is S L = , which admits anelectric charge ( E
0) and magnetic dipole ( M
1) moment, fromwhich the Lagrangian reads L V = L E + L M = g V Tr h H † Q H Q i V + f V M ǫ i jk Tr h H † Q H Q σ i i ∂ j V k (30) = q † L " g V V + f V M ǫ i jk σ Li ∂ j V k q L , (31)depending on the notation (superfield or subfield), where g V and f V are the electric- and magnetic-type couplings with theS-wave charmed mesons, M is a mass scale (it will prove con-venient to choose this mass scale equal to the nucleon mass,i.e. M = M N ) and V µ = ( V , ~ V ) the vector meson field.For notational convenience we have momentarily ignored theisospin factors. From this the vector-meson exchange poten-tials are also expressible as a sum of multipole components V V ( ~ q ) = V E ( ~ q ) + V M ( ~ q ) , (32)which read V E ( ~ q ) = + g V ~ q + m V , (33) V M ( ~ q ) = − f V M ( ~σ L × ~ q ) · ( ~σ L × ~ q ) ~ q + m V = − f V M ~σ L · ~σ L ~ q ~ q + m V + . . . , (34)with m V the vector meson mass and where the second line ofthe M1 contribution to the potential isolates its S-wave com-ponent. After Fourier-transforming into coordinate space weend up with V V ( ~ r ) = g V + f V m V M ~σ L · ~σ L e − m V r π r + . . . , (35)where the dots indicate contact-range and tensor terms, whichwe are ignoring. If we particularize for the ρ meson, we willhave to include isospin factors V ρ ( ~ r ) = ~τ · ~τ g ρ + f ρ m ρ M ~σ L · ~σ L e − m ρ r π r + . . . . (36)For the ω no isospin factor is required, but there is a sign com-ing from the negative G-parity of this meson V ω ( ~ r ) = ζ " g ω + f ω m ω M ~σ L · ~σ L e − m ω r π r + . . . , (37)where, as usual, ζ = + −
1) for the meson-meson (meson-antimeson) potential.The determination of the couplings with the vector mesonsfollows the same pattern we have used for the axial mesons.The neutral vector mesons, the ω and the ρ (where refers tothe isospin index, i.e. the neutral ρ ), have the same quantumnumbers as the photon and thus can mix with the electromag-netic current. It is convenient to write down the mixing in theform ρ µ → ρ µ + λ ρ eg A µ , (38) ω µ → ω µ + λ ω eg A µ , (39)with e the electric charge of the proton and g = m V / f π ≃ . ff ectively encapsulate Sakurai’s universalityand vector meson dominance [52, 53, 70].The proportionality constants can be determined frommatching with the electromagnetic Lagrangian of the light-quark components of the hadrons. To illustrate this idea,we can apply the substitution rules to the E0 piece of theLagrangian describing the interaction of the neutral vectormesons with the charmed antimesons, i.e. L E = Tr h H † ¯ c (cid:16) g ρ τ ρ + g ω ω (cid:17) H ¯ c i , (40) where we have chosen the antimesons because they containlight-quarks. After applying Eqs. (38) and (39), we end upwith L e . m . (L) E = e Tr " H † ¯ c g ρ g λ ρ τ + g ω g λ ω ! H ¯ c A = e Tr h H † ¯ c (cid:16) λ ρ τ + λ ω (cid:17) H ¯ c i A . (41)where in the second line we have used that g ρ = g ω = g .This is to be matched with the contribution of the light-quarksto the E0 electromagnetic Lagrangian L e . m . E = e Tr (cid:20) H † ¯ Q ( Q H + Q L ) H ¯ Q (cid:21) A . (42)where Q H and Q L are the electric charges of the heavy-antiquark and light-quarks in the isospin basis of the super-field H ¯ Q , of which only Q L is relevant for matching purposes Q L = − ! . (43)which implies that λ ρ = / λ ω = /
6. Alternatively wecould have determined λ ρ and λ ω from the nucleon couplingsto the vector mesons ( g ρ = g , g ω = g ) and their electriccharges, leading to the same result.Given λ ρ and λ ω and repeating the same steps but nowfor the M1 part of the Lagrangian, we can readily infer themagnetic-type coupling f V of the charmed antimesons withthe vector mesons, which turn out to be f V = g κ V with κ V = Me ! µ L ( D ∗ + ) , (44)where µ L ( D ∗ + ) refers to the light-quark contribution to themagnetic moment of the D ∗ + charmed antimeson, which inthe heavy-quark limit will coincide with the total magneticmoment of the heavy meson. From the quark model we ex-pect this magnetic moment to be given by the u-quark, i.e. Me ! µ L ( D ∗ + ) = M N e ! µ u ≃ . , (45)where we have taken M = M N ≃
940 MeV (i.e. the nucleonmass) as to express the magnetic moments in units of nuclearmagnetons.The outcome is g V = . κ V = .
8, which are the val-ues we will use here. Besides this determination, the vectormeson dominance model of Ref. [71] leads to g V = . κ V = . g V = . ± . ± . g V = . IV. DESCRIPTION OF THE X (3872) AND Z c (3900 / Now we will consider the X (3872), Z c (3900) and Z c (4020)from the OBE model perspective. The problem we want toaddress is: can they be described together with the same setof parameters? We will find that(i) the axial meson indeed favors the compatible descrip-tion the X and Z c resonances,(ii) yet the e ff ect of axial mesons depends on the choice ofa mass for the scalar meson in the OBE model.In general, lighter scalar meson masses will diminish the im-pact of axial meson exchange and eventually even vector me-son exchange, leading to the binding of both the X and Z c for m σ →
400 MeV. This is not necessarily a desired fea-ture, as the Z c in the molecular picture is not necessarilya bound state but more probably a virtual state or a reso-nance [26, 27]. That is, we expect the strength of the charmedmeson-antimeson potential to be short of binding for the Z c and Z ∗ c . However, as the mass of the scalar meson increasesand reaches the standard values traditionally used in the OBEmodel, m σ ∼ −
600 MeV, the importance of the axialmeson becomes clearer, where the a meson might be the dif-ference between a virtual state close to threshold or not. A. General structure of the potential
Before considering the light-meson exchange potential indetail, we will review the general structure of the S-wave po-tential. For the D ( ∗ ) ¯ D ( ∗ ) system there are two relevant symme-tries — SU(2)-isospin and HQSS — from which we decom-pose the potential into V = ( V a + τ W a ) + ( V b + τ W b ) ~σ L · ~σ L , (46)with τ = ~τ · ~τ . In this notation, the X and Z potentials read V X = ( V a − W a ) + ( V b − W b ) , (47) V Z = ( V a + W a ) − ( V b + W b ) . (48)However, it will be more useful to define the isoscalar andisovector contributions to the potential as follows V (0) a = V a − W a , (49) V (0) b = V b − W b , (50) V (1) b = V b + W b , (51) V (1) a = V a + W a , (52)from which the general structure of the potential is the oneshown in Table I. In the following lines we will explain whatare the contributions of each light-meson to the potential, yetwe can advance that(i) V ( I ) a receives contributions from σ and E0 ρ / ω and itsattractive for D ( ∗ ) ¯ D ( ∗ ) molecules.(ii) V ( I ) b receives contributions from the pion, the axial me-son and M1 ρ / ω exchange:(ii.a) V (0) b is dominated by M1 vector meson exchangeand its sign is negative, making the X (3872) themost attractive isoscalar molecular configuration. (ii.b) V (1) b is dominated by axial meson exchange andits sign is positive, implying that the Z c (3900) and Z c (4020) are among the most attractive isovectormolecular configurations.(ii.c) The most attractive isovector configuration shouldbe the I G ( J PC ) = − (0 ++ ) D ∗ ¯ D ∗ molecule, thoughno molecular state has been found yet with thesequantum numbers.Finally, the potential for the Z c (3900) and Z c (4020) are iden-tical, which explains the evident observation that they comein pairs [23]. For this reason from now on we will ignorethe Z c (4020) and concentrate in the Z c (3900), as results in thelater automatically apply to the former. B. The OBE potential
In the OBE model the H c H ′ c and H c ¯ H ′ c potentials (where H c , H ′ c represent the S-wave charmed mesons) can be writtenas the sum of each light-meson contribution V OBE = V ( ζ ) π + V σ + V ρ + V ( ζ ) ω + V ( ζ ) a , (53)where the individual contributions have been already dis-cussed in this manuscript (for the approximation in which thehadrons are point-like):(i) V ( ζ ) π and V ( ζ ) a in Eqs. (14) and (15),(ii) V σ in Eq. (28),(iii) V ρ and V ( ζ ) ω in Eqs. (36) and (37).We have included the superscript ( ζ ) as a reminder that the con-tributions stemming from exchange of negative G-parity lightmesons ( π , ω , a ) change sign depending on whether we areconsidering the meson-meson ( ζ = +
1) or meson-antimeson( ζ = −
1) systems. These signs have been already included inthe definition of the potential contributions, i.e. in Eqs. (14),(15) and (37). For convenience we review our choice of cou-plings in Table II.Here we consider only the S-wave component of the light-meson exchange potential, i.e. we ignore the tensor (S-to-D-wave) components. This choice allows a simpler analysis ofthe factors involved in binding.
C. Form-factors and Regulators
As mentioned, the previous form of the potential assumespoint-like hadrons. The finite size of the hadrons involvedcan be taken into account with di ff erent methods, e.g. formfactors. The inclusion of form factors amounts to multiplyeach vertex involving a heavy hadron and light meson by afunction of the exchanged momentum, i.e. A R ( H → HM ( q )) = f M ( q ) A ( H → HM ( q )) , (54)where A and A R are the point-like and regularized ampli-tudes, respectively, and f ( q ) the form factor. In terms of the System I G ( J P ( C ) ) Potential Candidate System I G ( J P ( C ) ) Potential Candidate D ¯ D + (0 ++ ) V (0) a - D ¯ D − (0 ++ ) V (1) a - D ∗ ¯ D − (1 + − ) V (0) a − V (0) b - D ∗ ¯ D + (1 + − ) V (1) a − V (1) b Z c (3900) D ∗ ¯ D + (1 ++ ) V (0) a + V (0) b X (3872) D ∗ ¯ D − (1 ++ ) V (1) a + V (1) b - D ∗ ¯ D ∗ + (0 ++ ) V (0) a − V (0) b - D ∗ ¯ D ∗ − (0 ++ ) V (1) a − V (1) b - D ∗ ¯ D ∗ − (1 + − ) V (0) a − V (0) b - D ∗ ¯ D ∗ + (1 + − ) V (1) a − V (1) b Z c (4020) D ∗ ¯ D ∗ + (2 ++ ) V (0) a + V (0) b - D ∗ ¯ D ∗ − (2 ++ ) V (1) a + V (1) b -TABLE I. SU(2)-isospin and HQSS structure of the S-wave potential in the heavy meson-antimeson molecules. “System” indicates the specificcharmed meson-antimeson molecule, I ( J PC ) its quantum numbers, “Potential” the potential and “Candidate” refers to known experimentalresonances that might be explained by the specific configuration considered. V ( I ) a and V ( I ) b are the central and spin-spin pieces of the potential,with I = , V (0) a < V (0) b < ++ and 2 ++ the most promising configurations for binding (in the absences of other binding factors, e.g. coupled channels, nearby charmonia,etc.). For the isovector sector we expect V (1) a < V (1) b >
0, from which the 0 ++ and 1 + − configurations are the most promising. However V (1) b is really weak, making this conclusion contingent on other factors (e.g. isospin breaking in vector meson exchange).Coupling Value Relevant to meson(s) g π , a g σ σ g V ρ , ωκ V ρ , ωλ a TABLE II. Couplings of the light-mesons we are considering in thiswork ( π , σ , ρ , ω and a ) to the charmed mesons. For the massesof the light-mesons we will use m π =
138 MeV, m σ =
550 MeV, m ρ =
770 MeV, m ω =
780 MeV and m a = M =
938 MeV. For thecharmed mesons we will consider their isospin-averaged masses, m D = m D ∗ = potential, the inclusion of a form factor is equivalent to thesubstitution rule V M ( ~ q ) → f M ( ~ q ) V M ( ~ q ) . (55)Here we will use multipolar form-factors, i.e. f M ( q ) = Λ − m Λ − q ! n P , (56)with Λ the form-factor cuto ff , q = − q + ~ q the exchanged 4-momentum of the meson M , m the mass of said meson and n P the multipole momentum. In general this procedure requiresthat Λ > m . For the inclusion of the axial meson this conditionentails n P ≥ V M ( ~ r ) = g Y π O I X i c i O iS e − mr r = g Y O I X i c i O iS m W Y ( mr ) , (57)with g Y the e ff ective Yukawa coupling, O I and O S isospin andspin operators, m the mass of the exchanged light-meson andwhere the exact potential could involve a sum of di ff erent spinoperators with c i their coe ffi cients. In the second line we haveincluded the dimensionless function W Y ( x ) = e − x π x , (58)which is the only thing that changes when a multipolar formfactor is included W Y ( x ) → W Y ( x , λ ; k P ) , (59)where λ = Λ / m , k P = n P and with W Y ( x , λ, k P ) = Z d z (2 π ) λ − λ + z ! k P e i ~ z · ~ x + z . (60)The general form of W Y for integer k P ≥ W Y as W Y ( x , λ ; k P ) = ( λ − k P I Y ( x , λ ; k P ) , (61)then I Y follows the recursive relation I Y ( x , λ ; 1) = λ − W Y ( x ) − λ W Y ( λ x )] , (62) I Y ( x , λ ; k P > = λ k P dd λ [ I Y ( x , λ ; k P − , (63)from which we can find the form of the potential for arbitrarymultipolar form factors.For the choice of the polarity n P , we have to choose at leasta dipolar form factor ( n P ≥ ff will be lighter than the axial meson, rendering it im-possible the inclusion of said meson with a multipolar formfactor. D. Mass gaps and e ff ective meson masses When the light-meson exchange potential entails a transi-tion between two hadrons of di ff erent masses, i.e. there is avertex of the type H → H ′ M , (64)with H , H ′ the initial and final hadrons and M the light-meson,we will have to modify the e ff ective mass of the in-flight lightmeson. In this case the light-meson exchange potential is notdiagonal and entails a transition between the HH ′ and H ′ H configurations V M ( ~ q ) = V M ( ~ q , HH ′ → H ′ H ) , (65)for which the light-meson propagator in the exchange poten-tial will change to 1 m + ~ q → µ + ~ q , (66)where µ is the e ff ective mass of the light-meson, i.e. µ = m − ∆ , (67)with ∆ = m ( H ′ ) − m ( H ).If we are dealing with a charmed meson-antimeson system,this correction will only have to be taken into account for the D ∗ ¯ D and D ¯ D ∗ systems, i.e. for the X (3872) and Z c (3900).Only the spin-spin part of the potential will be a ff ected, as thecentral part cannot generate a transition between the D and D ∗ charmed mesons. For the pion and axial meson exchangepotentials, the correction is trivial V π ( ~ r ) = ζ g µ π f π ~τ · ~τ ~σ L · ~σ L e − µ π r π r , (68) V a ( ~ r ) = − ζ λ g m a f π ~τ · ~τ ~σ L · ~σ L e − µ a r π r , (69)with µ π and µ a the e ff ective pion and axial meson masses,where we notice that for axial meson exchange the m a → µ a substitution is limited to the long-range decay exponent (butnot the m a factor involved in the strength of the potential).For the vector meson exchange potential the correction onlya ff ects its spin-spin piece V V ( ~ r ) = g V e − m V r π r + f V µ V M ~σ L · ~σ L e − µ V r π r . (70)Finally for the sigma meson no modification is required.If we combine this modification with a multipolar form fac-tor, from direct inspection of Eq. (56) we find that besidesmodifying the e ff ective mass of the light-meson, we also haveto modify its cuto ff m → µ and Λ → √ Λ − ∆ . (71)Taking into account that m ( D ∗ ) − m ( D ) ∼
140 MeV, the onlylight-meson that is substantially a ff ected by this change is thepion, for which its spin-spin contribution essentially vanishesfor the X (3872) and Z c (3900) molecules. E. The X (3872) and Z c (3900) cuto ff s Now, with the OBE model written down here, for a dipolarform factor ( n P =
2) we can reproduce the mass of the X with Λ ( X ) = .
37 (1 . − .
41) GeV , (72)for which we have used the isospin averaged D and D ∗ masses(in this limit the X is bound by 4 MeV). The central valuecorresponds to m σ =
550 MeV, which is the σ mass usedin the original OBE model for nuclear forces [13, 14], whilethe spread represents the mass range m σ = −
600 MeV,which covers most of the plausible choices for its mass. For m σ =
450 MeV the dipolar cuto ff is merely a bit above theaxial meson mass, m a = .
23 GeV. In contrast the cuto ff forwhich the Z c binds at threshold is Λ ( Z c ) = .
82 (1 . − .
99) GeV , (73)for which we get the ratio Λ ( Z c ) Λ ( X ) = .
33 (1 . − . . (74)Now if we assume that the cuto ff s for the X and the Z c arethe same, modulo HQSS violations, this ratio should be onewithin a 15% uncertainty Λ Z Λ X = ± . = (0 . − . , (75)which means that the existence of the X is compatible withinone standard deviation with a Z c binding at threshold for thelower σ meson mass range. This lower σ mass range basi-cally gives cuto ff s that are barely larger than the axial mesonmass, which indicates that we should consider larger dipolarmomenta for a better comparison. This is done in Table III,where we extend the comparison to the n P = , Z c is expected to be a virtual stateor a resonance near threshold, which means that the amountattraction in the 1 + (1 + − ) D ∗ ¯ D system is not enough to bind thecharmed meson and antimeson at threshold. That is, the ratioshould be larger than one (but still of O (1)), though it is di ffi -cult to estimate how much larger. Thus the previous ratio andthe ones in Table III are probably compatible with a molecular Z c .However the most interesting comparison is against theOBE model without the axial meson, which will reveal theconditions under which the axial meson might be relevant. Ifwe remove the axial meson, the cuto ff s we get are Λ / a X = .
37 (1 . − .
41) GeV , (76) Λ / a Z = .
99 (1 . − .
38) GeV , (77)yielding the ratio Λ / a Z Λ / a X = .
45 (1 . − . , (78)which results in larger relative cuto ff s as the σ gets heavier, in-creasing the discrepancy with HQSS to the three standard de-viation level (if we assume a molecular Z c at threshold, whichis probably too restrictive). The ratios also grow larger forhigher polarity n P , see Table III. Again lighter σ masses re-sult in cuto ff s that are not completely satisfactory if we wantto take the axial mesons into account. Finally in Fig. (1) weshow the dependence of the cuto ff ratio with the mass of the σ for a dipolar form factor, where we can see again that the im-pact of the axial meson increases with the mass of the scalarmeson.At first sight the comparison between the axial-full andaxial-less OBE models indicates a modest contribution fromthe axial mesons. But the observation that the cuto ff ratios in-crease with larger scalar meson masses, Fig. (1), and with itthe compatibility of the molecular description of the X and the Z c decreases, indicates that the previous conclusion dependson the strength of scalar meson exchange and the parametersused to describe the later. We actually do not know the cou-pling of the σ to the charmed baryons very precisely, but withconsiderable errors: the L σ M and the quark model suggest g σ = . ± .
0, where this uncertainty turns out to be impor-tant. If the attraction provided by the σ falls short of binding,the axial meson will be the di ff erence between the charmedmeson-antimeson interaction being weak or strong. Indeed, ifthere is no axial meson, the condition for the isovector D ∗ ¯ D system to bind is g σ ≥ .
45 (2 . − . , (79)which is within the expected uncertainties for the scalar cou-pling. That is, σ exchange is by itself no guarantee that the Z c (3900) can be explained in terms of the charmed meson-antimeson interaction alone. In Fig. (2) we visualize the de-pendence of the Λ ( Z c ) / Λ ( X ) ratio as a function of g σ , whichfurther supports the previous interpretation of axial meson ex-change as the factor guaranteeing the required molecular in-teraction necessary for the Z c . Finally for a σ -less theory withaxial mesons the Z c will still bind for large cuto ff s, with con-crete calculations yielding Λ /σ ( X ) = .
55 GeV and Λ /σ ( Z c ) = .
62 GeV , (80)for which the ratio is 2 .
33. In this later scenario the uncer-tainty of the factor involved in the mixing of the pion and ax-ial meson current ( λ = . − .
05) might be relevant, as thiserror induces the ratio to move within the 2 . − .
99 range.
V. DESCRIPTION OF THE NEW Z cs (3985) Finally we turn our attention to the Z cs (3985), recently dis-covered by BESIII [21]. The existence of this resonancecan be readily deduced from the Z c (3900) and Z c (4020) andSU(3)-flavor symmetry, as the latter dictates that the D ∗ ¯ D in-teraction in the I = D ∗ s ¯ D system [27, 73]. However the realization ofSU(3)-flavor symmetry in the OBE model is not automaticand depends on two conditions R ( Z c / X ) m σ w/ a wo/ a FIG. 1. Cuto ff ratio R ( Z c / X ) as a function of the mass of the scalarmeson for the OBE model with (solid line) and without (dashed-dotted line) axial meson exchange. R ( Z c / X ) is defined as the ratio ofthe cuto ff for which a molecular Z c will be a bound state at thresholdover the cuto ff for which the mass of the X (3872) is reproduced asa I G ( J PC ) = + (1 ++ ) D ∗ ¯ D bound state. If the Z c were to be a boundstate at threshold, the ratio would be expected to be R ( Z c / X ) = . ± .
15, which we show in the figure as a dashed line and a series ofbands representing one, two and three standard deviations (shown inincreasingly light colors). The scalar coupling is taken to be g σ = .
4, the value predicted in the L σ M. The m σ =
550 MeV ratios arehighlighted as a round dot. (i) the pseudo Nambu-Goldstone meson current mixingwith the octet part of the axial mesons,(ii) the scalar meson coupling with similar strengths to the q = u , d , s light-quarks.The first condition is required in order for the axial mesonexchange to be non-trivial in the isovector molecules: if theaxial mesons form a clear nonet with almost ideal decouplingof strange and non-strange components, as happens with thevector mesons, then axial meson exchange will cancel out inboth the Z c (3900) and Z cs (3985). We warn though that thestatus and nature of the axial mesons — a (1260), f (1285), f (1420), K (1270) — is not clear: they might be compos-ite [74], the f (1420) might not exist [75, 76], etc. Here wewill not discuss these issues, but simply point out the con-ditions under which they will help explain the Z c (3900) and Z cs (3985).The second condition is required for SU(3)-flavor symme-try to be respected between the Z c (3900) and Z cs (3985): if thescalar σ meson does not coupled with the strange quarks, thena sizable part of the attraction in the Z c (3900) system will sim-ply not be present in the Z cs (3985). As happened with the ax-ial mesons, the nature of the scalar mesons is not clear either:they might be q ¯ q or tetraquark or a superposition of both, themixing angle between the singlet and octet components is notknown or it might violate the OZI (Okubu-Zweig-Iizuka) rule.We will note that the binding of the Z cs (3985) as a hadronic0 Polarity ( n P ) Λ ( X ) Λ / a ( X ) Λ ( Z c ) Λ / a ( Z c ) R ( Z c / X ) R / a ( Z c / X )2 1.37 (1.27-1.41) 1.37 (1.27-1.41) 1.82 (1.37-1.99) 1.99 (1.37-2.38) 1.33 (1.08-1.41) 1.45 (1.08-1.69)3 1.65 (1.53-1.69) 1.65 (1.53-1.70) 2.19 (1.67-2.40) 2.44 (1.68-2.92) 1.33 (1.09-1.42) 1.48 (1.10-1.72)4 1.95 (1.82-2.00) 1.97 (1.83-2.03) 2.72 (2.10-2.96) 3.26 (2.16-4.02) 1.39 (1.15-1.48) 1.65 (1.18-1.98)TABLE III. Cuto ff s required to reproduce the X (3872) and to bind a molecular Z c at threshold in a OBE model with and without axial mesonsfor di ff erent masses of the scalar meson. We use a multipolar form factor with polarity n P = , , Λ ( X / Z c ) showsthe X (3872) and Z c cuto ff s for the OBE model including the axial meson, while for the axial-less case we add the superscript / a . Finally weshow the ratio between the X (3872) and Z c cuto ff s. The central value represents m σ =
550 MeV and the intervals (in parentheses) correspondto m σ = −
600 MeV. R ( Z c / X ) g σ w/ a wo/ a FIG. 2. Cuto ff ratio R ( Z c / X )as a function of the coupling tothe scalar meson for the OBE model with (solid line) and without(dashed-dotted line) axial meson exchange, where we refer to Fig. (2)for further details. The scalar mass is taken to be m σ =
550 MeV, itstraditional value in the OBE model for nuclear forces. The g σ = . g σ < .
4, i.e.70% of the coupling predicted by the L σ M and within the uncertain-ties expected within this model (namely 30% or g σ = . ± . Z c does not bind without axial meson exchange. molecule requires a σ that couples with similar strength to thenon-strange and strange light-quarks, which is not implausi-ble. A. Flavor structure of the potential
The D ( ∗ ) and D ( ∗ ) s charmed mesons belong to the ¯3 SU(3)-flavor representation. From this the flavor structure of the D ( ∗ ) a ¯ D ( ∗ ) a potential, where a = , , D , D + and D + s , is expected to be 3 ⊗ ¯3 = ⊕
8, i.e. the sum of a singletand octet contributions: V ( D ( ∗ ) a ¯ D ( ∗ ) a ) = λ ( S ) V ( S ) + λ ( O ) V ( O ) , (81)with the superscript ( S ) and ( O ) referring to the singlet and octetand where the specific decomposition is shown in Table IV. Ofcourse the flavor structure compounds with the HQSS struc-ture , that is, the singlet and octet potential can be further de- System
I S VD ( ∗ ) ¯ D ( ∗ ) V ( S ) + V ( O ) D ( ∗ ) s ¯ D ( ∗ ) s V ( S ) + V ( O ) D ( ∗ ) ¯ D ( ∗ ) V ( O ) D ( ∗ ) ¯ D ( ∗ ) s -1 V ( O ) TABLE IV. SU(3)-flavor structure of the charmed meson-antimesonpotential. The charmed mesons (antimesons) belong to the ¯3 (3) rep-resentation of SU(3)-flavor symmetry, from which the potential ac-cepts a 3 ⊗ ¯3 = ⊕ composed into a central and a spin-spin part V ( S ) = V ( S ) a + V ( S ) b ~σ L · ~σ L , (82) V ( O ) = V ( O ) a + V ( O ) b ~σ L · ~σ L . (83)Finally, the relation between the singlet and octet compo-nents and the isospin components we previously defined forthe X (3872) and the Z c (3900) is V ( S ) = V ( I = − V ( I = , (84) V ( O ) = V ( I = . (85)From the flavor decomposition of the potential (Table IV)it is apparent that the potential for a molecular Z c (3900) and Z cs (3985) are identical (provided their tentative identificationswith the I G ( J PC ) = + (1 + − ) D ∗ ¯ D and D ∗ ¯ D s - D ¯ D ∗ s systems arecorrect). This in turn is compatible with the their experimen-tal masses, as can be deduced from the qualitative argumentthat their interaction is the same. This conclusion has indeedbe checked by concrete EFT calculations [27], which do notmake hypotheses about the binding mechanism but simply as-sume that the Z c and Z cs are bound states. The question wewill explore now is what are the conditions under which weexpect the OBE model to respect this SU(3)-flavor structure. B. Flavor structure of the light-meson exchanges
When extending the present formalism from SU(2)-isospinto SU(3)-flavor, a problem appears regarding the coupling ofthe charmed and light mesons: the isoscalar ( I =
0) light-mesons, being q ¯ q states, can be either in a flavor singlet or1octet configuration. The singlet and isoscalar octet states, hav-ing the same quantum numbers, can mix and this mixing mostoften works out as to separate the ( u ¯ u + d ¯ d ) / √ s ¯ s com-ponents of these two type of mesons almost perfectly. Thisis what happens for instance with the vector mesons ω and φ .However, the other light mesons we are considering here arefurther away from decoupling. The easiest case will be thepseudoscalar mesons, for which the singlet and octet almostdo not mix. The axial meson case will be the most complexone, as it entails non-trivial mixing angles that have to be com-bined with the fact that the axial meson mixes with the pioncurrent. In the following lines we will consider each case indetail.
1. Pseudoscalar meson octet
We will begin with the pseudoscalar mesons for which thesinglet and octet ( η and η ) members can be identified withthe η ′ and η mesons, as the mixing angle is small. Thus inpractice we can consider the pseudoscalar mesons as forminga standard octet M = π √ + η √ π + K + π − − π √ + η √ K K − ¯ K − η √ . (86)From this, the interaction term of the pseudoscalars with thecharmed mesons can be written as L flavor = g f π Tr h H a † ¯ c ~σ · ~∂ M ab H b ¯ c i , (87)where a , b are flavor indices, which are ordered as ¯ D a = ( ¯ D , D − , D − s ), and H ¯ c the heavy-superfield for the charmed an-timesons (as with this choice we have light-quarks). Alterna-tively, if we use the light-subfield notation we will have L flavor = g f π q a † L ~σ L · ~∂ M ab q bL , (88)with q aL = ( u L , d L , s L ) for a = , ,
2. Vector meson nonet
Next we consider the vector mesons, for which the non-strange and strange components of the singlet ( ω ) and octet( ω ) decouple almost perfectly to form the ω and φ mesons.While the light-quark content of the singlet and isoscalar octetmesons is expected to be | ω i = √ h | u ¯ u i + | d ¯ d i + | s ¯ s i i , (89) | ω i = √ h | u ¯ u i + | d ¯ d i − | s ¯ s i i , (90)for the physical ω and the φ meson we have | ω i ≃ √ h | u ¯ u i + | d ¯ d i i , (91) | φ i ≃ | s ¯ s i , (92) which means that the relation between the physical and SU(3)eigenstates is φω ≃ q − q q q ω ω . (93)This matrix is actually a rotation, as can be seen by directinspection.In principle there should be two independent couplings forthe singlet and the octet vector meson components, i.e. g (1) V for ω and g (8) V for ω and ρ (or f (1) V and f (8) V for the magnetic-type couplings). These two couplings are reduced to one oncewe consider the OZI rule: the coupling of hadrons that do notcontain strange quarks to the φ meson should be suppressed.This in turn generates a relation between g (1) V and g (8) V (there isonly one independent coupling owing to the OZI rule), fromwhich we can deduce the relation g ρ = g ω .Alternatively, if we consider that the mixing is indeed ideal,we can write down a vector meson nonet matrix V = ρ √ + ω √ ρ + K ∗ + ρ − − ρ √ + ω √ K ∗ ¯ K ∗− ¯ K ∗ φ , (94)and notice that the structure of the interaction Lagrangian inflavor space will be L flavor ∝ ¯ D † a V ab ¯ D b , (95)with a , b flavor indices and ¯ D a the anticharmed meson fieldin flavor space, i.e. ¯ D a = ( ¯ D , D , D − s ) for a = , ,
3. Fromthis we end up with a unique g V and f V , and thus g ρ = g ω and f ρ = f ω automatically.
3. Axial meson octet vs nonet
The SU(3)-extension of the present formalism to the axialmesons will encounter three problems. The first is which arethe flavor partners of the a (1260) meson, which we will sim-ply assume to be the f (1285), f (1420) and K (1270) (wewill further discuss this point later).The second problem is the singlet and octet mixing: if weconsider that the isoscalar partners of the a are the f (1285)and f (1420), they will be a non-trivial mixture of a singletand octet axial meson f (1285) f (1420) = cos θ sin θ − sin θ cos θ f f , (96)where f and f are the singlet and octet components of thetwo f ’s. As a matter of fact, the f (1285) and f (1420) are rel-atively far away from the mixing angle which e ff ectively sep-arates them into non-strange and strange components. If wedefine the decoupling mixing angle as θ dec = atan(1 / √ ∼ . ◦ , the θ angle can be expressed as θ = θ dec + α , (97)2where there is a recent determination of this angle by theLHCb, α = ± (24 . + . − . + . − . ) ◦ [77] (which is the value we willadopt), and previously in the lattice α = ± (31 ± ◦ [78].Third, the axial neutral mesons are J PC = ++ states buttheir strange partners do not have well-defined C-parity. De-pending on their C-parity there are two possible types ofaxial mesons: J PC = ++ and 1 + − mesons originate from P and P quark-antiquark configurations, where we haveused the spectroscopic notation S + L J , with S , L and J beingthe spin, orbital and total angular momentum of the quark-antiquark pair. For a quark-antiquark system the C-parity is C = ( − L + S , which translates into C = + −
1) for P ( P ).The J PC = ++ and 1 + − neutral axial mesons correspond withthe a , f and b , h , respectively, of which only the a , f can mix with the pseudo Nambu-Goldstone boson axial cur-rent. The strange partners of the a and b axial mesons arereferred to as the K A and K B , but for the strange axial mesonsC-parity is not a well-defined number and the physical statesare a mixture of the P and P configurations K (1270) K (1400) = cos θ K sin θ K − sin θ K cos θ K K B K A , (98)with most determinations of θ K usually close to either 30 ◦ or60 ◦ [79–81].The mixing of the axial pseudo Nambu-Goldstone mesoncurrent (i.e. the SU(3) extension of the axial pion current) hasto happen with the axial meson octet (instead of the physicalaxial mesons) ∂ µ M ab → ∂ µ M ab + λ m A ab , (99)where M ab and A ab would be the pseudoscalar and axial me-son octets, the first of which is given by Eq. (86) and the sec-ond by A = a √ + f √ a + K + ∗ A a − − a √ + f √ K ∗ A K −∗ A ¯ K ∗ A − f √ . (100)Thus the specific relations for the contribution of the f and K A to the axial meson currents, i.e. ∂ µ η → ∂ µ η + λ m f , (101) ∂ µ K → ∂ µ K + λ m K A , (102)have to be translated into the physical basis by undoing therotations. For the f we will get ∂ µ η → ∂ µ η + λ m (cid:0) sin θ f + cos θ f ∗ (cid:1) , (103)which determines the coupling of the f and f ∗ with the D and D s mesons, while for the K A we will get instead ∂ µ K → ∂ µ K + λ m (cid:0) sin θ K K + cos θ K K ∗ (cid:1) . (104)Owing to the form-factors, the exchange of the heavier vari-ants of the axial mesons ( f ∗ , K ∗ ) are expected to be suppressedwith respect to the lighter ones ( f , K ). From this observationand the previous relations, the most important contribution for axial-meson exchange will come from the couplings of the f and K mesons, which are proportional to sin θ and sin θ K ,respectively.The K meson deserves a bit more of discussion as it canhelp us to get a sense of the accuracy of the previous relationfrom a comparison with the K axial meson decay constant,which can be extracted from experimental information. Thecurrent mixing relation implies that the decay constant will be h | A µ | K i = f K m K with f K = λ cos θ K f K , (105)with from θ K = − ◦ or 55 − ◦ and f K =
160 MeV yields f K = −
150 MeV or f K = −
230 MeV, respectively,which is to be compared with f K = ±
19 MeV [82] (whichin turn is extracted from the experimental data of Ref. [83]).The two possible mixing angles are in principle compatiblewith the previous determination of f K : though it is possibleto argue that the higher angle might be a slightly better choice,this is based on the assumption that f K A , f K B = P axial mesons b and h have to be zero owing to their negativeC-parity, i.e. f b = f h =
0, this is not true for the K B for which f K B = f K B , ff ect, though small, is enough as to make the comparison ofmixing angles more ambiguous [82, 84].Finally, it is worth stressing that the structure of the ax-ial mesons is not particularly well-known and there exist in-teresting conjectures about their nature in the literature. Afew hypotheses worth noticing are: (i) the axial mesons mightbe dynamically generated (i.e. molecular) [74, 85], (ii) the K (1270) resonance might actually have a double pole struc-ture [86, 87] and (iii) the f (1420) might simply be a K ¯ K de-cay mode of the f (1285) [76]. All of them might potentiallyinfluence the theoretical treatment of the axial mesons: (i) ac-tually was considered in Ref. [88] four decades ago for the a (1260), where it was determined that it would not stronglyinfluence the form of the potential. It is worth noticing that(iii) would imply that there are not enough axial mesons toform a nonet, but only an octet. This would be interesting, asin this scenario it might be plausible to identify the f (1285)with f in Eq. (96), leading to θ = ◦ . However, thoughinteresting, we will not consider the multiple ramifications ofthe previous possibilities in this work.
4. Scalar meson singlet vs nonet
The lightest scalar meson nonet is formed by the σ (or f (500)), a (980), f (980) and K ∗ (700). If we are consider-ing light-meson exchange the most important of the scalarswill be the lightest one, i.e. the σ (see Ref. [89] for an exten-sive review about the status of this meson). While the a and K ∗ are pure octets, the σ and f (980) are a mixture of singletand octet, i.e. f (500) f (980) = cos θ sin θ − sin θ cos θ S S , (106)3where S and S represent the pure singlet and octet states.The meaning of S and S depends however on the inter-nal structure of the scalar mesons: if the σ were to be a q ¯ q state, the light-quark content of S and S would be analo-gous to that of the vector mesons, i.e. to | ω i and | ω i inEqs. (89) and (90). But if the σ were to be a qq ¯ q ¯ q state, thelight-quark content of S and S would be di ff erent (yet easilyobtainable from the substitutions u → ¯ d ¯ s , d → ¯ u ¯ s , s → ¯ u ¯ d ,which assumes the diquark-antidiquark structure proposed byJa ff e [90], where the antidiquark and diquark are in a tripletand antitriplet configuration, respectively).If g and g are the coupling of the charmed mesons to thesinglet and octet scalar, respectively, we will have that the cou-pling of the σ to the non-strange and strange charmed mesonswill be g σ = g σ DD = cos θ g + √ θ g , (107) g ′ σ = g σ D s D s = cos θ g − √ θ g . (108)Independently of whether the σ is a q ¯ q or qq ¯ q ¯ q scalar, if weassume a mixing angle that decouples the non-strange andstrange components, we will end up with g σ D s D s = σ meson exchange will badly broke SU(3)symmetry between the Z c (3900) and Z cs (3985). However thisconclusion depends on the previous assumptions, which arenot necessarily correct. In the following lines we will discusshow the observed SU(3) symmetry can still be preserved withscalar meson exchange.The most obvious solution would be a flavor-singlet σ , asthis would provide roughly the same attraction for a molecular Z c (3900) and Z cs (3985). In this regard it is relevant to noticeRef. [91], which analyzed the σ pole in unitarized chiral per-turbation theory and obtained a mixing angle θ = ± ◦ . Thiswould translate into a σ that is mostly a flavor-singlet.The interpretation of the σ as a singlet would also be com-patible with the following naive extension of the L σ M fromSU(2)-isospin to SU(3)-flavor, in which an originally masslessbaryon octet interacts with a total of nine bosons by means of L N σ L ′ int = g Tr h ¯ B ( φ + i γ λ a φ a ) B i , (109)with B the baryon octet ( N , Λ , Σ , Ξ ), φ and φ a the bosonicfields, λ a with a = , . . . , g acoupling constant. In the standard L σ M the nucleon field ac-quires mass owing to the the spontaneous symmetry breakingand the subsequent vacuum expectation value of the φ field.Here it is completely analogous, with h φ i = f P / √
2, the re-definition of φ = f P / √ + σ and the reinterpretation of the φ a fields as the pseudoscalar octet ( π , K , η ). This procedurewill give g σ B B = g φ B B = √ M / f P , with M the averagedmass of the octet baryons and f P representing either f π , f K or f η , which are all identical in the SU(3) symmetric limit. The g σ B B thus obtained is basically compatible with the previousSU(2) value for g σ NN ( ≃ . F / ( D + F )-ratiowould be α = / F = D ), tobe compared for instance with the SU(6) quark model value α = / σ to strange and non-strange hadrons alike,resulting in the same attraction strength for both the Z c (3900)and Z cs (3985). In fact if we assume the relation g σ qq = √ m q / f P at the quark level, take m q = q = u , d , s constituent quark masses and choose f P = f π for q = u , d and f P = f K for q = s , we would get g σ uu ≃ g σ dd ≃ . g σ ss ≃ .
3, leading to the counter-intuitive conclusion that the coupling of the σ to the strangequark is larger than to the u , d quarks. However if we subtractthe mass of the quarks g σ qq = √ m con q − m q ) / f P (with m con q and m q the constituent and standard quark masses), we willobtain g σ ss ≃ . g σ uu and g σ dd ).But the SU(3) extension of the L σ M we have presentedhere is not the only possible one. In fact it could just be con-sidered as a simplified version of the chiral quark model [92]in which the scalar octet is removed. It happens that the in-clusion of the scalar octet in the chiral quark model makes itperfectly possible to have a non-strange σ and still explain themass of the light baryons.However the problem might not necessarily be whether the σ contains a sizable strange component or not, but whetherit couples to the strange degrees of freedom. In this regardit has been suggested that the OZI rule does not apply in thescalar 0 ++ sector [93, 94]. This in turn might be the mostrobust argument in favor of a sizable coupling of the σ mesonwith the strange-charmed mesons, as it does not depend onthe flavor structure or the strange content of the σ . This inprinciple implies that the g σ and g ′ σ couplings to the D ( ∗ ) and D ( ∗ ) s charmed mesons would be independent parameters.Without the OZI rule there is no reason for the g and g couplings to have comparable sizes: while the applicationof the OZI rule implies that g = √ / θ g (which forthe q ¯ q and qq ¯ q ¯ q ideal decoupling angles would translate into g = √ / g ≃ . g and g = − √ g ≃ − . g , respec-tively), without OZI g and g are independent parameters.Now, if it happens that g ≫ g the result will be indistin-guishable from a flavor-singlet σ : the g σ and g ′ σ couplingscan be approximated by g σ ≃ g ′ σ ≃ g cos θ , resulting in ap-proximately the same couplings to the D ( ∗ ) and D ( ∗ ) s charmedmesons. As to whether the g ≫ g condition is met or not, ithappens that g cos θ can be identified with g / σ M of Eq. (109), giving it a relatively largevalue, while there is no reason why g should be as large. Be-sides, g ≫ g would also justify not including the scalar octetin the OBE model.Yet, we might get a better sense of the sizes of g and g from a comparison with previous determinations of the σ coupling in the light-baryon sector. While the Nijmegenbaryon-baryon OBE models originally considered a flavor-singlet σ [95], latter this idea was put aside in favor of a morestandard singlet-octet interpretation for the σ [96]. Their de-scription depended on a singlet and octet couplings, g B B and g B B , the mixing angle θ and the F / ( F + D )-ratio (whichis necessary in the light-baryon octet but not for the antitripletcharmed mesons). They obtained g B B = . g B B ,4while for the later Nijmegen soft-core baryon-baryon OBEmodel [97] this ratio is g B B = . g B B . Thus it wouldnot be a surprise if the relation g ≫ g also happens for thecharmed mesons.The comparison of the coupling constants to the lightbaryons might provide further insights too. If we considerthe J¨ulich hyperon-nucleon OBE model [65], their results are g σ ΛΛ ≃ .
95 (0 . g σ NN and g σ ΣΣ ≃ .
13 (1 . g σ NN in whatis referred to as model A(B) in Ref. [65], where these cou-plings are supposed to represent correlated and uncorrelated(correlated) processes in the scalar channel. It is worth notic-ing that the J¨ulich model [65] predated the rediscovery of the σ as a pole in the pion-pion scattering amplitude [98, 99],and consequently treated the σ as a fictitious degree of free-dom. From a modern point of view in which the σ is not afictitious meson, their results would be compatible (within theexpected 30% error of the L σ M) both with a σ that only cou-ples with the non-strange q = u , d quarks ( g NS σ ΛΛ = . g NS σ NN and g NS σ ΣΣ = . g NS σ NN , where NS indicate that it couples onlyto the non-strange quarks) and with a σ that couples withequal strength to the q = u , d , s quarks ( g FS σ ΛΛ = g FS σ NN and g FS σ ΣΣ = g FS σ NN , where FS indicates that the coupling is flavor-symmetric).In short, there are theoretical arguments in favor of a sizablecoupling of the σ meson to the D s and D ∗ s charmed mesons, g ′ σ . In what follows we will consider the problem form a phe-nomenological point of view, i.e. we will simply consider the g ′ σ coupling to be a free parameter and discuss which are thevalues which allow for a simultaneous description of the Z c and Z cs , without regard as to which is the theoretical reasonbehind this. C. Light-meson exchange for the Z c and Z cs Now that we have reviewed the flavor structure of the pseu-doscalar, scalar, vector and axial mesons, we can write downthe resulting light-meson exchange potential for the Z c and Z cs . We will begin with the pseudoscalar mesons, for whichthe singlet and octet can be considered as e ff ectively decou-pled. For the Z c there will be π - and η -exchange, while forthe Z cs only η -exchange will be possible. The pseudoscalar-exchange potential can be written as V P ( D ∗ ¯ D ) = ζ ~τ · ~τ W π ( r ) + W η ( r ) , (110) V P ( D ∗ s ¯ D ) = − W η ( r ) , (111)where W π and W η are the π - and η -exchange potentials oncewe have removed the flavor and G-parity factors, i.e. W π ( r ) = g f π ~σ L · ~σ L µ π W Y ( µ π ) , (112) W η ( r ) = g f η ~σ L · ~σ L µ η W Y ( µ η ) . (113)In the flavor-symmetric limit we will have m π = m η and f π = f η , leading to identical π - and η -exchange potentials, W π = W η . However in the real world, m η > m π and f η > f π ,implying a suppression of the η -exchange potential relative tothe pion-exchange one.For the vector mesons there is instead an almost ideal mix-ing between the singlet and octet, from which the ω and φ areclose to being purely non-strange and strange, respectively.The structure of the potential will be V V ( D ∗ ¯ D ) = ζ ~τ · ~τ W ρ ( r ) + W ω ( r ) , (114) V V ( D ∗ s ¯ D ) = , (115)with W ρ and W ω the ρ - and ω -exchange potential once theflavor and G-parity factors have been removed W ρ = g ρ m ρ W Y ( m ρ r ) + f ρ m ρ M ~σ L · ~σ L µ ρ W Y ( µ ρ r ) , (116) W ρ = g ω m ω W Y ( m ω r ) + f ω m ω M ~σ L · ~σ L µ ω W Y ( µ ω r ) . (117)For the scalar meson we will consider it to generate two in-dependent couplings for the non-strange and strange charmedmesons: V σ ( D ∗ ¯ D ) = − g σ m σ W Y ( m σ r ) , (118) V σ ( D ∗ s ¯ D ) = − g ′ σ g σ m σ W Y ( m σ r ) , (119)as this choice allows to explore the conditions for which weexpect the Z cs to bind, provided that the Z c binds. We suspectthat g σ and g ′ σ are of the same order of magnitude, yet pro-vided | g ′ σ | is not much smaller than | g σ | , the Z c and Z cs willbe related to each other.For the axial mesons, the f (1285) and f (1420) are prob-ably a non-standard mixture between a singlet and and octet,where the mixing angles will have to be taken into accountexplicitly V A ( D ∗ ¯ D ) = ζ ~τ · ~τ W a ( r ) + h sin θ W f ( r ) + cos θ W f ∗ ( r ) i , (120) V A ( D ∗ s ¯ D ) = − h sin θ W f ( r ) + cos θ W f ∗ ( r ) i . (121)The reduced W a , W f and W ∗ f potentials are given by W a = − λ g m a f π ~σ · ~σ µ a W Y ( µ a r ) , (122) W f = − λ g m f f η ~σ · ~σ µ f W Y ( µ f r ) , (123) W ∗ f = − λ g m ∗ f f η ~σ · ~σ µ ∗ f W Y ( µ ∗ f r ) , (124)with m a , m f and m ∗ f the masses of the a , f and f ∗ axialmesons (while µ a , µ f and µ ∗ f are their e ff ective masses whenthere is a mass gap). In general the exchange of the f and f ∗ mesons will be moderately suppressed owing to f η > f π .5 D. The Z c (3900) and Z cs (3985) cuto ff s With the light-flavor structure of the OBE potential athand, we simply have to choose the parameters (couplings,masses and mixing angles), compare the cuto ff s for which the Z c (3900) and Z cs (3985) bind and check whether they are com-patible with each other. This is analogous to what we havealready done with the X (3872) and the Z c (3900), though nowthe focus is the preservation of SU(3)-flavor symmetry, fromwhich we expect Λ ( Z cs ) Λ ( Z c ) ≃ . . (125)Of course this relation is approximate: HQSS and SU(3)-flavor violations will generate deviations from this cuto ff ra-tio. We expect HQSS and SU(3)-flavor breaking e ff ects to beof the order of 15% and 20% (i.e. Λ QCD / m c and the di ff erencebetween f π and f K ), respectively, which add up to 25% if wesum them in quadrature, i.e. Λ ( Z cs ) Λ ( Z c ) = ± . = (0 . − . . (126)For the SU(3)-flavor breaking, an extra factor to be taken intoaccount is the relative sizes of the D and D s mesons, which arenot necessarily the same. If we use the electromagnetic radiias a proxy of the matter radii, although they have not been ex-perimentally measured, there are theoretical calculations: inRef. [100] they are estimated to be p r e . m ∼ .
43 and 0 .
35 fmfor the D and D s , respectively. This indicates that the strangecharmed meson D s is 0 .
82 the size of its non-strange partner,from which it would be expectable for the form-factor cuto ff of a D s ¯ D s molecule to be a 22% larger than a D ¯ D molecule.This figure is in fact compatible with the f K and f π ratio wementioned before, but indicates a bias in the flavor uncertain-ties: the naive expectation will be a larger cuto ff for the Z cs than for the Z c . The D s ¯ D molecules would be in between,with deviations at the 10% level expected for the cuto ff (bi-ased towards larger cuto ff s), from which we might revise therange of acceptable cuto ff ratios to Λ ( Z cs ) Λ ( Z c ) ≃ . ± . = (0 . − . , (127)i.e. we have moved the expected central value from 1 to 1 . σ that does not couple with the strangecharmed meson D ( ∗ ) s , for n P = Z c and Z cs eventuallybind for large enough cuto ff s, though the ratio is too large Λ σ (NS) ( Z cs ) Λ σ (NS) ( Z c ) (cid:12)(cid:12)(cid:12)(cid:12) θ + = .
65 (4 . − . , (128) Λ σ (NS) ( Z cs ) Λ σ (NS) ( Z c ) (cid:12)(cid:12)(cid:12)(cid:12) θ − = .
31 (3 . − . , (129)where θ ± = θ dec ± α , from which it can be appreciated thatthe dependence on the θ mixing angle is weak. If we insteadconsider a σ that couples with the same strength to the non-strange and strange quarks, i.e. a σ with a flavor-symmetric R ( Z c / X ) g σ ’ / g σ w/ a wo/ a FIG. 3. Cuto ff ratio between the Z cs (3985) and the Z c (3900) as afunction of the ratio g ′ σ / g σ , where g σ and g ′ σ are the couplings of thescalar meson to the non-strange and strange charmed baryons D and D s . We assume g σ = .
4, its expected central value from the L σ Mand the quark model. We show the ratios in the OBE model with andwithout axial mesons, where we notice that for the axial-less casethe Z ′ cs does not bind for g ′ σ < .
7. We compare the cuto ff ratio R ( Z cs / Z c ) to the expected ratio derived from SU(3)-flavor, HQSS andcorrections from the strange charmed meson size, R ( Z cs / Z c ) ≃ . ± . coupling, we will get instead the ratios Λ σ (FS) ( Z cs ) Λ σ (FS) ( Z c ) (cid:12)(cid:12)(cid:12)(cid:12) θ + = .
04 (1 . − . , (130) Λ σ (FS) ( Z cs ) Λ σ (FS) ( Z c ) (cid:12)(cid:12)(cid:12)(cid:12) θ − = .
05 (1 . − . , (131)which are basically independent of θ and compatible withEq. (127).The conclusion is that some coupling of the σ meson to thestrange charmed meson is required for a coherent moleculardescription of the Z c and Z cs . Thus we might consider thequestion of what is the g ′ σ / g σ ratio which is compatible withthe upper bound for the cuto ff ratio, i.e. with Eq. (127). Thishappens to be g ′ σ g σ ≥ . − . , (132)which is weakly dependent of m σ (that is why we do not in-clude a braket showing the m σ spread) and somewhat depen-dent on θ , with θ + ( θ − ) yielding 0 .
64 (0 . ff erent interpretations of the σ wehave discussed: provided the σ has a sizable coupling to thestrange components, it should be a plausible outcome. Forobtaining the cuto ff ratio suggested by the strange and non-strange charmed meson size comparison ( ≃ .
1) we will needinstead g ′ σ g σ ≃ . − . , (133)6which again is nearly independent of m σ and weakly depen-dent on θ ( θ + ( θ − ) gives 0 . . ff ratio R ( Z cs / Z c ) with the g ′ σ / g σ ratio isshown in Fig. 3, from which can see again that axial mesonexchange becomes important if scalar meson exchange hap-pens to be weak. Owing to the weak dependence of this ratioon θ , Fig. 3 only shows the θ + resultsFinally, if we remove the sigma completely we will still geta ratio compatible with Eq. (127), Λ /σ ( Z cs ) Λ /σ ( Z c ) (cid:12)(cid:12)(cid:12)(cid:12) θ + = .
47 (1 . − . , (134) Λ /σ ( Z cs ) Λ /σ ( Z c ) (cid:12)(cid:12)(cid:12)(cid:12) θ − = .
50 (1 . − . , (135)where the intervals now reflect the uncertainty in λ = . − .
05. Sigma-less molecular descriptions include mostworks which use vector-meson exchange (usually within thehidden-gauge approach) to predict molecular states (e.g. the X (3872) [101], the hidden-charm pentaquarks [102, 103] or,recently, more general descriptions of the molecular spec-trum [39]). However these descriptions traditionally requirea di ff erent binding mechanism for the Z c (3900) resonance,which might include two-pion exchange or charmonium ex-change [36–39]. Here we note that axial meson exchangecould be a useful complementary addition to these models,but if we want these models to simultaneously reproduce the X (3872) with the same set of parameters the σ is probably arequired addition. E. The scalar meson and the P cs (4459) pentaquark The couplings of the scalar meson are not only impor-tant for a unified molecular description of the Z c (3900) and Z cs (3985), but also for the new strange hidden-charm pen-taquark P cs (4459) [104] when considered in comparison tothe other three molecular pentaquark candidates the P c (4312), P c (4440) and P c (4457) [105].The P cs (4459) is 19 . D ∗ Ξ c threshold— 4478 . P cs (4459) has been conjectured to be a ¯ D ∗ Ξ c molecule [106–110]. The charmed baryon Ξ c is a flavor an-titriplet with quark content csu ( Ξ + c ) and csd ( Ξ c ), where thelight-quark pair within the Ξ c is in a S L = S L the total light-quark spin. As a consequence pionexchange, axial meson exchange and the M1 part of vectormeson exchange do not contribute to the ¯ D ∗ Ξ c interaction.This observation also applies to the ¯ D Σ c system [46, 111–119], which is the most common molecular explanation forthe P c (4312): the P c (4312) pentaquark is merely 8 . D Σ c threshold.The question is whether this is compatible with the ex-pected binding of the P cs (4459) as a ¯ D ∗ Ξ c molecule. Owing tothe lack of explicit light-spin dependence, the only di ff erencebetween the OBE descriptions of the P c (4312) and P cs (4459)is scalar meson exchange (the strength of vector meson be-ing identical in both cases). While the Σ c baryon contains two non-strange light-quarks the Ξ c contains only one, and ifwe assume a σ that does not couple to the non-strange light-quarks we will have g NS σ Σ c Σ c ≃ g NS σ Ξ c Ξ c , (136)which will translate into considerably less attraction (andbinding) for the P cs (4459). In contrast if the σ couples withapproximately the same strength to the strange quark withinthe Ξ c , we will have g FS σ Σ c Σ c ≃ g FS σ Ξ c Ξ c , (137)where the superscript FS indicates that the coupling is nowflavor-symmetric (in the sense of identical coupling strengthswith the q = u , d , s quarks). That is, if the σ couples equally toall the light-quarks, the binding of the P c (4312) and P cs (4459)molecules will be approximately the same.However the experimental determination of the mass of the P cs (4459) indicates that it is more bound than the P c (4312) byabout 11 . B ( P c ) = . B ( P cs ) = . , (138)for the P c (4312) and P cs (4459) pentaquarks, respectively.This could be interpreted as g σ Ξ c Ξ c > g σ Σ c Σ c , which wouldbe somewhat surprising but still within the realm of possi-bility. Yet the comparison we have done does not take intoaccount that there is a factor that generates spin-dependencein the P cs (4459) pentaquark: the coupled channel dynamicswith the nearby ¯ D Ξ ′ c and ¯ D Ξ ∗ c thresholds, where the J = / J = /
2) ¯ D ∗ Ξ c molecule will mix with the ¯ D Ξ ′ c ( ¯ D Ξ ∗ c ) chan-nel. Owing to the relative location of the thresholds, this gen-erates repulsion and attraction for the J = / / ff ect with acontact-range theory, yielding B CC ( ¯ D ∗ Ξ c , J =
12 ) = (2 . − .
9) MeV , (139) B CC ( ¯ D ∗ Ξ c , J =
32 ) = (13 . − .
3) MeV , (140)depending on the cuto ff used, where the superscript CC standsfor “coupled channels”. That is, we expect a hyperfine split-ting of ∆ B CC = (7 . − .
4) MeV, i.e. of the same orderof magnitude of the aforementioned 11 . ff erence inbinding between the P c and P cs binding. For comparison, arecent phenomenological calculation provides a similar hyper-fine splitting [110] of ∆ B CC = (2 . − .
0) MeV.But this contact-range theory provides us with a very inter-esting advantage: we can explicitly turn o ff the coupled chan-nel e ff ects within the theory, which following the formalismof Ref. [107] would lead to B SC ( ¯ D ∗ Ξ c , J = ,
32 ) = (9 . − .
1) MeV , (141)with the superscript SC indicating “single channel” and wherenow the two spin states are degenerate and have binding ener-gies compatible with that of the P c (4312) as a ¯ D Σ c molecule.7The conclusion seems to be that, if the P cs (4459) pen-taquark is a ¯ D ∗ Ξ c molecule (and assume that the previousprocedures e ff ectively isolate the single and coupled channelcontributions), it is probably more compatible with a σ thatcouples to all the light quarks with equal strength than with a σ that does not couple with the strange quark. But this con-clusion depends on the size of the coupled channel e ff ects: ifthey were to be larger than estimated in Refs. [107, 110], thenthe P cs (4459) pentaquark might be more compatible with anon-strange sigma instead. VI. CONCLUSIONS
In this manuscript we consider the problem of describ-ing the Z c (3900), Z c (4020) and Z cs (3985) as heavy hadronmolecules from a phenomenological perspective. Of course,we do not know for sure whether they are molecular or not.We are instead interested in what their binding mechanism is(provided they are molecular). Regarding the problem of theirnature, the closeness of these resonances to the D ∗ ¯ D , D ∗ ¯ D ∗ and D ∗ s ¯ D thresholds suggest a molecular nature. The suc-cess of EFT formulations in describing the Z c ’s [23, 26, 73]and Z cs [27] further points towards the plausibility of themolecular nature. Yet, tetraquark explanations are also plau-sible [120–123]. What is not trivial to explain though inthe molecular picture is the binding mechanism: while the Z c ’s should not be there in vector meson exchange models,OBE models usually require relatively large cuto ff s for thesetwo-body systems to bind [33–35] (or might simply not binddepending on the choice of couplings), prompting other ex-planations such as two-pion exchange or charmonium ex-change [36–39].Here we consider a new factor in the molecular descrip-tion of the Z c ’s and Z cs : axial meson exchange. The exchangeof axial mesons is strongly suppressed in the two-nucleon sys-tem, partly owing to the fact that the axial meson mass is largerthan the nucleon’s and partly owing to vector meson exchangebeing a more dominant factor than the axial mesons. But thisis not necessarily the case for charmed mesons, promptinga reevaluation of the role of axial mesons. We find that theinclusion of the axial mesons makes the molecular descrip-tion more plausible for the Z c ’s, as they indeed provide addi-tional attraction. But their importance depends on the strength of scalar meson exchange: if the coupling of the charmedhadrons to the scalar meson is smaller than suggested by phe-nomenological models, axial mesons will become the bind-ing mechanism. Conversely, if the scalar coupling is largeenough, axial mesons will become irrelevant. For molecu-lar candidates in which vector meson exchange is strong, forinstance the X (3872), the axial meson exchange contributionis negligible. Thus we expect the relevance of axial mesonexchange to be limited to molecules where rho- and omega-exchange cancel out, as is the case with the Z c ’s.Yet, besides the axial meson, the nature of sigma exchangeis probably the most important factor for a coherent moleculardescription of the Z c (3900) and Z cs (3985). If the sigma mesoncouplings breaks SU(3)-flavor symmetry to a large degree, theshort-range of the axial meson, combined with its non-trivialSU(3)-flavor structure, might be insu ffi cient to explain the Z cs as the molecular SU(3)-partner of the Z c in the molecular pic-ture. Thus a molecular Z cs requires a non-trivial coupling ofthe strange charmed mesons D s and D ∗ s with the sigma me-son. This is not improbable though, as there are theoreticalreasons (in particular the suspicion that the OZI rule mightnot apply to the scalar mesons [93, 94]) why the sigma mesoncould have a sizable coupling to the strange degrees of free-dom The bottom-line though is that a molecular Z cs requires anon-negligible coupling of the sigma to the strange hadrons inthe OBE model, independently of which is the origin of thiscoupling. This might not only be a requirement for the Z cs tobe molecular but also for the recently discovered P cs (4459),which interpretation as a ¯ D ∗ Ξ c bound state might also requirea coupling of the Ξ c strange charmed baryon to the sigma sim-ilar to that of the non-strange Σ c . ACKNOWLEDGMENTS
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