Excited heavy meson decays to light vector mesons: implications for spectroscopy

We analyze strong decays of excited charmed and beauty mesons into a light vector meson, exploiting the effective field theory based on heavy quark (HQ) symmetries for heavy mesons, and on the hidden gauge symmetry to incorporate light vector mesons. HQ symmetries allow to classify the heavy mesons in spin doublets, and to relate decays of excited states. We build effective Lagrangian terms governing the ${\cal H}_i \to P^{(*)} V$ modes, with ${\cal H}_i$ an excited $s$, $p$, $d$ and $f$-wave heavy-light quark meson, $P,\,P^*$ the lowest-lying $J^P=(0^-,\,1^-)$ heavy-light mesons, and V a light vector meson. Predictions are provided for ratios of decay widths that are independent of the strong couplings in the effective Lagrangian terms. A classification of the newly observed heavy-light mesons is proposed.


I. INTRODUCTION
Great progress has been achieved in heavy hadron spectroscopy, thanks to the efforts of several experimental groups at different facilities which have provided new pieces of information [1]. In the open-charm meson spectrum the two lowest-lying (1S) and the four 1P orbital excitations are identified, both for non-strange and strange mesons [2]. Information is available for larger mass states which could be identified either with higher orbital or radial excitations. Experimental observations are less abundant in the case of beauty mesons: the established states are the two lowest-lying (1S) states and two among the four 1P orbital excitations, both with and without strangeness [2]. There is progress also in baryon spectroscopy, with the observation of five new narrow Ω c resonances [3] and of the doubly-charmed Ξ cc [4]: however, in this paper we are only concerned with mesons.
Prompt production and production in B decays, the main production mechanisms of excited charmed mesons, provide us with different and complementary information. Prompt production allows to establish if a state has natural (J P = 0 + , 1 − , 2 + , . . . ) or unnatural (J P = 0 − , 1 + , 2 − , . . . ) parity, while spin-parity can be determined by Dalitz plot analyses in B decay production. On the other hand, it is possible to measure ratios of branching fractions of strong decay modes, an information that can be used to classify the decaying meson, as we are going to discuss.
Several observed open charm mesons are awaiting for a proper identification. In Table I we include the resonances observed by BaBar Collaboration (in 2010) in the inclusive production of D + π − , D 0 π + and D * + π − [5]. LHCb Collaboration has performed a similar analysis (in 2013), with the findings in Table II [6]. It has also carried out (in 2016) a Dalitz plot analysis of B − → D + π − π − , reporting evidence of the resonances in Table III [7]. Many of the states found in the different analyses are likely to be the same, namely the BaBar D 0 (2550) and D * 0 (2600) states in Table I coincide with the LHCb ones D 0 J (2580) and D * 0 J (2650) in Table II. D * 0 1 (2680) in Table III is probably different from D * 0 (2600), although presumably they both have J P = 1 − . The identification of D 0 (2750) in Table I with D 0 J (2740) in Table II is also plausible. Two different resonances are present in the mass range around 2760 MeV: Table I. Mass, width and spin-parity of charmed resonances observed by BaBar Collaboration [5].
While spin-parity of charmed mesons can be established by the amplitude analyses in production in B decays, for beauty mesons the quantum number assignment is more difficult. In addition to the above mentioned established states, recent observations are due to CDF and LHCb Collaborations. CDF found a state named B(5970) [13], likely the same as B 0,+ J (5960) observed by LHCb together with B 0,+ J (5840) decaying to B + π − , B 0 π + [14]. Spin-parity is not established, and mass and width are affected by large uncertainties: the values from PDG fits [2] are in Table V. The identification as 2S excitations has been proposed [14]. New results on B s1 (5830) and B * s2 (5840) have also been obtained by CMS [15].
In [16] a comprehensive analysis of the open charm and open beauty mesons was performed based on the classification scheme in the heavy quark limit, attempting to fit the observed states in this scheme. Information on the strong decay modes to D (s) M or D * (s) M , with M a light pseudoscalar meson, was exploited, and the states in Table I and most of  those in Table IV were considered. In the same approach, studies for the states in Tables II and III observed after the analysis in [16] have been carried out in [17,18].
More data are still needed for classification, which is a non trivial task for the newly observed mesons. If the resonance mass is large enough, several decay channels are open, in particular those with a light final vector meson which provide important new piece of information. This paper is devoted to such a phenomenology.
In the next Section we restate the theoretical framework based on heavy quark (HQ) symmetries to describe spectrum and decay processes. For transitions into light pseudoscalars, effective Lagrangians are written exploiting the HQ symmetries and the (spontaneoulsy broken) chiral symmetry holding in QCD for massless u, d, s quarks, with the light pseudoscalar mesons being the pseudo-Goldstone bosons. The approach can be extended to incorporate the light vector mesons, treated as gauge fields of a hidden local symmetry. In Section III we construct the effective Lagrangians describing strong heavy-light meson decays with the emission of a light vector meson, generalizing the analysis in [19,20]. We give the expressions for H i → P ( * ) V decay rates, with V a light vector meson, P ( * ) the lowest-lying J P = (0 − , 1 − ) heavy-light mesons, and H i either a orbital or a radial a heavy-light excitation. Sections IV and V contain numerical analyses for charmed and beauty mesons, considering states requiring identification and making predictions for heavier excitations. Relations among decay rates, independent of the hadronic couplings, are constructed: they are suitable for experimental measurements and for classifications. The conclusions are presented in the last Section. The physics of hadrons containing a single heavy quark can be systematically analyzed considering the m Q → ∞ heavy quark (HQ) mass limit, formalized in the Heavy Quark Effective Theory (HQET) [21]. In such a limit, two symmetries emerging in QCD can be exploited: the heavy quark spin symmetry, allowing to relate the properties of hadrons which only differ for the heavy quark spin orientation, and the heavy quark flavour symmetry, relating the properties of hadrons which only differ for the heavy quark flavour. The classification of heavy-light Qq mesons (q a light antiquark) in the HQ limit is based on the decoupling of the heavy quark spin s Q from the total angular momentum s of the light degrees of freedom (light quarks and gluons). Since such angular momenta are separately conserved in strong interaction processes, the heavy mesons can be classified in doublets of different s . Each doublet comprises two states, spin partners, with total spin J = s ± 1 2 and parity P = (−1) +1 , being the orbital angular momentum of the light degrees of freedom and s = + s q (s q the light antiquark spin). In the HQ limit the spin partners in each doublet are degenerate and, due to flavour symmetry, the properties of the states in a doublet are related to those of the corresponding states differing for the heavy flavour.
Effective Lagrangians describing the strong interactions of such mesons can be constructed introducing effective fields for each doublet, following e.g. the procedure based on the covariant representation of the states [22]. We denote by H a (a = u, d, s a light flavour index) the s P = 1 2 − doublet, S a and T a the s P = 1 2 + and s P = 3 2 + doublets, X a the s P = 3 2 − , X a the s P = 5 2 − and F a the s P = 5 2 + doublets. The corresponding effective fields read: v is the meson four-velocity, conserved in strong interactions. The operators in Eq.
(1) contain a factor √ m Q , have dimension 3/2 and annihilate mesons with four-velocity v.
The octet of light pseudoscalar mesons is introduced defining ξ = e iM fπ and Σ = ξ 2 , with the matrix M comprising π, K and η fields (f π = 132 MeV): and under the chiral group SU (3) L × SU (3) R the transformation properties are: With the definition at the leading order in the heavy quark mass expansion and in the light meson momentum, the effective Lagrangian terms invariant under heavy quark spin-flavour and light quark chiral transformations can be constructed [23][24][25][26][27]: withH = γ 0 H † γ 0 and Λ χ a scale parameter. The coupling constants g, h, h , k , k 1,2 ,p 1,2 can be inferred from experiment, indeed bounds have been found [16]. Theoretical determinations using nonperturbative approaches are available, namely for g and h [28][29][30][31][32][33][34]. The expressions for the H i → P ( * ) M decay widths, with P ( * ) in the H doublet and M a light pseudoscalar meson, can be found in [16], with the exception of decaying mesons in the F doublet, obtained from Eq. (14) 1 : where η ( ) is the polarization tensor (vector),p the combination of the couplingsp =p 1 +p 2 , p M the three-momentum of M , and the factor C M is different for the various mesons, C π + = C K + = 1, C π 0 = C K S = 1 2 and C η = 2 3 .

B. Incorporating light vector mesons
There are several ways to incorporate the light vector mesons in the effective Lagrangian describing heavy meson decays. Here we reconsider the hidden gauge symmetry approach [35][36][37][38] applied in [19,39,40].
The hidden local symmetry method, which dates back to applications to supergravity theories [41,42], exploits the equivalence of the non-linear sigma model based on a group G spontaneously broken to a subgroup H, to another model having G as global symmetry group and H as a local symmetry. This allows to introduce the gauge bosons of the local symmetry, which are identified with the light vector mesons in applications to chiral theory. In this formulation the vector fields transform inhomogeneously under nonlinear realization of the chiral symmetry, while in alternative approaches to incorporate the vector mesons (Weinber [43] and Callan, Coleman, Wess and Zumino [44]), the vector fields transform homogeneously. The different methods are shown to be equivalent [45].
In the hidden gauge symmetry framework one writes The The action of the group SU (3) H is hidden when one considers the field Σ. One now defines Fixing the gauge in such a way that ξ L = ξ R = ξ, these fields can be identified with the ones in Eqs. (4) and (3). Their transformation properties under SU (3) H are given by Eqs. (6)- (7), identifying U with U H . The octet of light vector mesons plays the role of gauge fields of the a hidden symmetry, and is introduced writing whereρ µ is a Hermitian matrix defined in analogy to the matrix M of pseudoscalar fields (2): The constant g V is set to g V 5.8 to satisfy the KSRF relations [46,47]. The observed vector mesons ω and φ correspond to a mixing between the octet component φ (8) in (22) and the singlet component φ (0) : The angle θ V ArcTan 1 √ 2 realizes the ideal mixing allowing to identify ω and φ with the flavour eigenstates φ q =ū u +dd √ 2 and φ s =ss. In terms of these, in (22) one can replace 1 a replacement becoming exact in the large N c limit [48]. The antisymmetric field tensor is defined: ρ µ transforms as V µ : while the difference V µ − ρ µ , as well as F µν transform homogenously as A µ : The covariant derivative D α can be defined, such that satisfies the previous relation. W α = V α , or W α = ρ α , or a linear combination of them can be chosen, but for our purpose it is irrelevant to fix W , since at the leading order in the effective theory and for processes describing heavy-light meson decays to another heavy one and a single light vector meson, only the partial derivative in (29) contributes to the amplitude.

III. EFFECTIVE LAGRANGIAN TERMS AND STRONG DECAY WIDTHS
We now construct the effective Lagrangian terms governing the decays H i → P ( * ) V , where H i is a heavy-light meson, V a light vector meson and P, P * the lowest-lying heavy-light J P = (0 − , 1 − ) mesons. For the doublets corresponding to = 0 and = 1 such Lagrangians have been derived in [19,20]. We denote by H µ1µ2...µ k the spin doublet which the decaying heavy meson H i belongs to. The effective Lagrangian describing the transition H i → P ( * ) V can have two structures: with the minus sign in (30) included for later convenience. The two structures Γ µ1...µ k α and Γ µ1...µ k αβ are chosen in such a way that the Lagrangians are invariant under heavy quark symmetry and hidden gauge symmetry transformations, parity, charge conjugation and time reversal. Invariance under heavy quark velocity reparametrization must also be preserved [49]. The heavy quark symmetry imposes further constraints, since in the decays of the two members in a spin doublet to the ones of the lowest-lying doublet, the light meson must be emitted in the same orbital state. This reduces the number of terms in the effective Lagrangian. Beyond the leading order in the HQ expansion, additional Lagrangian terms must be included [50,51]. Considering the doublets in (1), in the effective Lagrangian terms (30) and (31) we are concerned with indices having k = 0, 1, 2, that we discuss in turn. For the decay mode Before discussing in details the transitions of states in the various doublets, we remark that the effective Lagrangian approach is in principle applicable when the emitted light particle is soft. This is guaranteed when the mass difference between the decaying meson and the final heavy-light meson is not too large. When decays of heavier excitations are considered, it is possible that corrections from higher order terms in the effective Lagrangians could become sizeable. Nevertheless, we push our predictions also for large values of the mass of the decaying particle, considering the symmetries as the main guidelines in the description of the heavy-light meson phenomenology.
A. TransitionsH → HV , withH = (P ,P * ) When the decaying meson belongs to the H doublet we have k = 0 in Eqs. (30) and (31). Decays to P ( * ) V are not kinematically allowed for the n = 1 H doublet, hence we consider the radially excitedH doublet (n = 1 is relevant for processes with intermediate virtual mesons [19]). The decays occur in p-wave, and the terms (30) and (31) fulfilling the constraints are: with the parameter Λ introduced to render the couplings dimensionless. We set Λ = 1 GeV. In the previous expressions, the replacement of a single γ matrix with the four-velocity v produces terms that either give the same result or vanish, a remark holding for all cases considered below. The Lagrangians (32) and (33) coincide with those obtained in [19], and from them the decay widths are worked out: with V and ,˜ light and heavy meson polarization vectors,m V = m V Λ , and C V = 1 for V = ρ ± , K * ± , K * 0 ,K * 0 , ϕ, Relations among the decay widths, not involving the coupling constants, can be constructed: Other relations independent of the couplings can be worked out considering modes with different final light vector mesons, as discussed in Sect. IV.
B.S → HV , withS = (P * 0 ,P 1 ) When the decaying meson belongs to the S doublet one has k = 0 in Eqs. (30) and (31). The P ( * ) V phase space is closed for n = 1, therefore we consider radial excitations inS doublet. The transitions occur in s-wave, and the effective Lagrangian terms (30) and (31) read: as also obtained in [20]. The decay widths read: with η polarization vector. The transition For the decays of the states in the T doublet one has k = 1 in (30) and (31). The transitions to P ( * ) V is not kinematically allowed for the n = 1 T doublet, hence consider n = 2T . The transitions proceed in d-wave, and the effective Lagrangian reads: with the covariant derivative acting on the light vector meson field tensor. The resulting decay widths are: The relations are fulfilled: Ratios of decay rates for modes with different final light vector mesons, independent of the coupling constant, will be constructed below.
For X doublet one has k = 1 in (30) and (31). No candidates belonging to such a doublet have been observed, and we do not know whether the P ( * ) V channels are open for the n = 1 states. The transitions occur in p-wave and are governed by the Lagrangian with the covariant derivative acting on the light vector meson field tensor. The decay widths are given by: Coupling-independent ratios of decay widths are: while ratios of decay widths for processes with different final light vector meson are discussed in Sect. IV.
E. X → HV , with X = (P 2 , P * 3 ) For the decays of the members of the X doublet one has to consider k = 2 in (30) and (31). The processes occur in f -wave, with Lagrangian The decay widths read: For this doublet the relations are fulfilled: The case of the F doublet requires k = 2 in (30) and (31). The transitions occur in d-wave with effective Lagrangian and decay widths A relation independent of the couplings connects various modes: Moreover, for V A and V B two light vector mesons one simply has:

IV. NUMERICAL ANALYSIS: CHARM
The expressions in the previous Section allow to construct quantities useful for the classification of high mass charmed and beauty states. In Table VI we and For decaying strange mesons we define For different final heavy mesons we consider The P V and P * V thresholds of neutral non-strange and of strange charmed mesons are shown in Fig. 1. For charged non-strange charmed mesons the thresholds are almost coincident with the neutral ones.

A. States inH doublets
There are candidates of radial excitations of (D (s) , D * (s) ) in the H doublet. In particular, D * s1 (2700) observed by Belle [52] and BaBar [53], with mass and width in Table IV, can be identified with the n = 2 excitation of D * s (2112). Indeed, the measurement [11] with D ( * ) K = D ( * )0 K + + D ( * ) + K 0 S , agrees with the prediction for the first radial excitation of D * s (2112) [54]. The situation is unclear for the states without strangeness. Two resonances can be identified with the members of the n = 2H doublet, (D(2550), D * (2600)) in Table I, most likely coinciding with (D 0 J (2580), D * 0 J (2650)) in Table II. However, this classification needs to be further corroborated [16]. [���]

(89)
This relation is shown for V = ρ in Fig. 4, varying the mass mD of J P = 0 − radial excitation in the range [2900, 3200] MeV, and setting the spin splitting mD * − mD = 40 ± 20 MeV.

B. States in X doublets
Ratios of decay rates independent of the coupling constant can be written for (D * 1 , D 2 ) belonging to the X doublet. They are plotted in Fig. 5 varying the mass of the decaying particle. There are two candidates for the lowest-lying X doublet: D * + J (2760) observed in the decay to D 0 π + [6], that is likely to have J P = 1 − , and D * s1 (2860) [10]. Their parameters are in Tables III and IV. Since the P V and not P * V modes are kinematically allowed, we display in Fig. 5 only the ratio R   At the chosen order in the effective Lagrangian approach the strong decay widths of the members of the X doublet with a light final pseudoscalar meson depend on the constant k in Eq. (12). Neglecting phase-space suppressed channels (e.g. decays to excited doublets), the widths of the members of the X doublet are determined by the couplings k and h X . Saturating the widths of D * + (2760) and D * + s1 (2860) by the modes the couplings k and h X can be constrained in the region in Fig. 7, with the bound |k | < 0.16.    Figure 6. Ratios (84)-(86) for decaying particle in the X doublet.

C. States in X doublet
In 2006 BaBar observed the D sJ (2860) meson decaying to DK [53], which was proposed as the J P = 3 − state in the cs X doublet [55]. A subsequent LHCb analysis supported this classification and showed that another state, D * s1 (2860) with J P = 1 − , is present in same mass region, likely the member of the X doublet [10]. The parameters of the J P = 3 − resonance are in Table IV. LHCb observed another candidate for the X doublet, D * 0 3 (2760) with parameters in Table III, that can be identified with the non-strange partner of D * − s3 (2860) [7]. Finally, BaBar and LHCb found a resonance that might be the J P = 2 − state in the X doublet: this is D 0 (2750) decaying to D * + π − [5], with parameters in Table I. The LHCb D 0 (2740) state, observed in D * + π − [6] (see Table II), likely coincides with it.
For the two J P = 3 − states, allowed decays to light vector mesons are D * 0 3 (2760) → D + ρ − , D 0 ρ 0 , D 0 ω and D * + s3 →D 0 K * + , D − K * 0 . We plot in Fig. 8 ratios of widths independent of the coupling constant, varying the mass of the decaying particle. In correspondence to the measured D * 0 3 (2760) mass we predict: and Analogous ratios for the J P = 2 − member of the X doublet, with and without strangeness, are shown in Fig. 8. In the non-strange case, the candidate is D 0 (2740). The D + ρ − , D 0 ρ 0 , D 0 ω channels are open, and we predict:     Figure 8. Ratios in Eqs. (78)-(81) (left panels) and (82),(83) (right panels), evaluated when the decaying particle is the decaying particle D * (s)3 (top row) and D (s)2 (bottom row) in the X doublet. The gray regions (enlarged in the inset) correspond to the measured mass of D * 0 3 (2760) candidate as D * 3 , and of D 0 (2740) candidate for the D 2 in the X doublet.
In the same figure R is plotted versus the mass of D 2 , with the gray region corresponding to the D 0 (2740) measured mass. Ratios involving the same final light vector meson are displayed in Fig. 9.
In the effective Lagrangian approach, the strong decay widths of the members of the X doublet to a light pseudoscalar meson are controlled by k = k 1 + k 2 , with k 1 and k 2 in Eq. (13). Neglecting phase-space suppressed modes, the widths of the members of the X doublet are determined by the couplings k and k X . If (D 0 (2750), D * 3 (2760))and D * s3 (2860) belong to X doublet, their widths impose constraints on the two constants, as shown in the left panel of Fig. 10 obtained assuming the full widths saturated by For |k X | < 1 the coupling region is also shown in Fig. 10 Figure 9. Ratios (84)-(86) for a decaying particle belonging to X .  Figure 10. Left: Constraints on the couplings k, k X from the measured widths of D 0 (2740) [6], D * 3 (2760) [7] and D * s3 (2860) [10], candidates for the X doublet. In the dark blue region all constraints are fulfilled. Right: coupling region for |k X | < 1.
slightly above the value obtained in [16] using the BaBar data [5,53].

D. States inT doublet
We analyze theT doublet before F since there is a state that can fit in both of them, and this sequence in the discussion is convenient. For each one of the two states in theT spin doublet we construct ratios of decay rates independent of strong couplings. A J P = 2 + meson has been observed [7], D * 2 (3000), that could fit in theT or in the F doublet. Hence, we compute the various ratios varying the mass of the decaying particle, then we specialize to the mass of the candidate, as shown in Fig. 11. For D * 2 (3000) belonging to this doublet we predict  The LHCb assignment for this particle is J P = 2 + [7], and mass and width are compatible with D * 0 J (3000) [6] (Tables II and III). It could be identified with the lowest lying J P = 2 + n = 2 stateD * 2 inT doublet, or with D * 2 belonging to the n = 1 F doublet. Predictions for the masses of the two states have been worked out in quark models. For example, using the chiral quark model developed in [56,57], the values mD * 2 = 3035 GeV and m D * 2 = 3101 GeV have been predicted [58]. As for the identification of D * 2 (3000), no consensus is reached adopting variants of the quark model. Using a model with instantaneous Bethe-Salpeter potential, identification withD * 2 is supported [59], while D * 2 (3000) is preferably interpreted as D * 2 on the basis of the 3 P 0 model for strong decays [60].      Figure 12. Ratios (84)-(86) when the decaying particle belongs to theT doublet.  Figure 13. Bounds on the couplingsh in (11) and h T in (45) from the measured width of D * 2 (3000), assuming that the state belongs toT .
Decay modes of the the strange partner of D * 2 (3000) are D * s2 → D ( * ) s η, D ( * )+ K S , D ( * )0 K + , with ratios : The results are different if one identifies D * 2 (3000) withD * 2 in theT or with D * 2 in the F doublet. We fix the D * 0 2 mass to the value in Table III with Figure 14. Ratio (73) for different final states when the decaying particle is D (s)3 in the F doublet. The two assignments lead to predictions for the spin partner of D * 2 (3000). For D * 2 (3000) identified withD * 2 , the spin partner is the J P = 1 + stateD 1 , while the spin partner of D * 2 is D 3 with J P = 3 + . In the two cases we construct the ratios of decay widths Varying conservatively the mass ofD 1 in the range [m D * 2 (3000) − 100 MeV, m D * 2 (3000) ] and the mass of D 3 in [m D * 2 (3000) , m D * 2 (3000) + 100 MeV] we obtain:

V. NUMERICAL ANALYSIS: BEAUTY
The flavour symmetry allows to extend the analysis to the beauty sector. The H i → P ( * ) V thresholds, for H i a neutral beauty or a beauty-strange meson, are displayed in Fig. 15. No one of the observed excited beauty mesons are above the P ( * ) V thresholds, therefore our predictions hold for higher excitations. We define ratios of decay widths   for charged and for neutral non-strange decaying beauty meson: and Ratios of decay widths can also be constructed for beauty mesons with strangeness: Ratios of decay widths with the same final V meson are also independent of strong couplings: H doublet The ratios (110)-(113) and (114), (115) evaluated when the decaying particle inH, are in Fig. 16 T doublet Presenting the results in Fig. 16 we do not distinguish the decayingB 1 orB * 2 , which have the same expressions for the ratios. The observables in (116), (118) are displayed in Fig. 17.

X doublet
Ratios of decay rates for beauty mesons in the X doublet are in Fig. 16. When the decaying particle is B * 1 , the two ratios R

X doublet
The considered ratios have the same expressions for the two members in the X doublet. Those defined in Eqs. (110)-(113) and (114), (115) are displayed in Fig. 16, those with the same final vector meson in Fig. 17.

VI. CONCLUSIONS
The construction of a QCD-based framework to classify the excited resonances with open charm and beauty and to describe their decays is needed in view of the ongoing and forthcoming esperimental investigations. Since orbital and radial excitations can be above the thresholds for decays to light vector mesons, we have worked out effective Lagrangian terms governing the strong transition of a heavy meson to a light vector meson and a member of the lowest-lying heavy-light spin doublet, in the HQ limit. We have defined observables independent of the couplings in the Lagrangian, and made predictions varying the mass of the decaying particle. The HQ limit is considered as the guideline for the description in the actual cases. Our methods can be exploited for a few observed states with uncertain identification, namely D * 2 (3000) for which we have compared predictions corresponding to two different classifications. Among the various tasks left to new analyses there are the computation of the various strong couplings and the classification of the subleading Lagrangian terms, which is particularly interesting in case of charm.