Threshold effects in P-wave bottom-strange mesons
Pablo G. Ortega, Jorge Segovia, David R. Entem, Francisco Fernández
aa r X i v : . [ h e p - ph ] D ec Threshold effects in P-wave bottom-strange mesons
Pablo G. Ortega, ∗ Jorge Segovia, † David R. Entem, ‡ and Francisco Fern´andez § Instituto de F´ısica Corpuscular (IFIC),Centro Mixto CSIC-Universidad de Valencia,ES-46071 Valencia, Spain Physik-Department, Technische Universit¨at M¨unchen,James-Franck-Str. 1, 85748 Garching, Germany Grupo de F´ısica Nuclear and Instituto Universitario de F´ısica Fundamental y Matem´aticas (IUFFyM),Universidad de Salamanca, E-37008 Salamanca, Spain (Dated: November 10, 2018)Using a nonrelativistic constituent quark model in which the degrees of freedom are quark-antiquark and meson-meson components, we have recently shown that the D ( ∗ ) K thresholds play animportant role in lowering the mass of the physical D ∗ s (2317) and D s (2460) states. This observationis also supported by other theoretical approaches such as lattice-regularised QCD or chiral unitarytheory in coupled channels. Herein, we extend our computation to the lowest P -wave B s mesons,taking into account the corresponding J P = 0 + , 1 + and 2 + bottom-strange states predicted bythe naive quark model and the BK and B ∗ K thresholds. We assume that mixing with B ( ∗ ) s η andisospin-violating decays to B ( ∗ ) s π are negligible. This computation is important because there is noexperimental data in the b ¯ s sector for the equivalent j Pq = 1 / + ( D ∗ s (2317), D s (2460)) heavy-quarkmultiplet and, as it has been seen in the c ¯ s sector, the naive theoretical result can be wrong by morethan 100 MeV. Our calculation allows to introduce the coupling with the D -wave B ∗ K channel andto compute the probabilities associated with the different Fock components of the physical state. PACS numbers: 12.39.Jh, 14.40.Nd, 14.40.RtKeywords: Nonrelativistic quark model, Bottom mesons, Exotic mesons
I. INTRODUCTION
Despite many of the B s states should be accessible bythe B -factories (CLEO, BaBar, Belle) and also by theproton–anti-proton colliders (CDF and D0), much of the b ¯ s excitation spectrum remains to be observed. Onlythe ground S -wave spin-singlet and spin-triplet states( B s and B ∗ s ) and the orbitally excited B s (5830) and B ∗ s (5840) mesons are presently well established [1]. Itis expected that the situation will change in the nearfuture thanks to the LHCb experiment and to the futurehigh-luminosity flavour and p − ¯ p facilities.In the heavy quark limit ( m Q → ∞ ), flavour and spinsymmetries hold and the open-flavoured heavy mesonscan be classified in doublets. This is because in the heavyquark limit the spin s Q of the heavy quark and the totalangular momentum of the light quark j q decouple and areseparately conserved in strong interaction processes [2].Therefore, Q ¯ q mesons can be classified according to thevalue of j q , and can be collected in doublets; the twostates of each doublet are spin partners with total spin J = j q ± and parity P = ( − ℓ +1 , with ℓ the orbitalangular momentum of the light degrees of freedom suchas ~j q = ~ℓ + ~s q ( s q is the light antiquark spin).The well established B s (5830) and B ∗ s (5840) states ∗ pgortega@ific.uv.es † [email protected] ‡ [email protected] § [email protected] belong to the j Pq = 3 / + doublet and therefore theunambiguous experimental determination of the b ¯ s j Pq =1 / + doublet is of particular interest because, followingthe predictions of heavy quark symmetry(HQS), theyshould be almost degenerated and broad. However, weknow that the experimental values of the masses andwidths of the D ∗ s (2317) [3] and D s (2460) [4] mesons,which belong to the same multiplet but in the c ¯ s sector,do not accommodate into the theoretical estimates.We have recently argued in Ref. [5] that the disagree-ment between theory and experiment for the D ∗ s (2317)and D s (2460) mesons can be explained as a delicatecompromise between the bare masses and the couplingwith their respective DK and D ∗ K thresholds. Thestudy of Ref. [5] taught us that the traditional quarkmodel do not reproduce the masses and widths of the D ∗ s (2317) and D s (2460) mesons. When, following theproposal of Ref. [6], we include one-loop corrections tothe one-gluon exchange potential as derived by Gupta et al. [7], these α s corrections affect basically the 0 + D ∗ s (2317), bringing this state closer to the DK thresh-old. However, the same correction leaves the D s (2460)almost degenerated with the D s (2536). It is the cou-pling with the DK and D ∗ K channels which lowers thebare masses of the D ∗ s (2317) and D s (2460) mesons tothe observed experimental values. Since their masses arebelow their respective DK and D ∗ K thresholds, theiropen-flavoured strong decays are forbidden and thus thestates are narrower than expected theoretically. Similarconclusions were drawn by dynamical coupled-channelapproaches [8, 9] and by lattice-regularised QCD com- B ∗ s mesonRelativistic quark model [13] 5804Relativistic quark model [14] 5833Relativistic quark model [15] 5830Relativized quark model [16] 5805Bardeen, Eichten, Hill [17] 5718(35)Q.-F. L¨u et al. [18] 5756LQCD: q ¯ q + BK [12] 5713(11)(19)LQCD: q ¯ q [19] 5752(16)(5)(25)Covariant (U)ChPT [20] 5726(28)NLO UHMChPT [21] 5696(20)(30)LO UChPT [22, 23] 5725(39)LO χ -SU(3) [24] 5643HQET + ChPT [25] 5706 . . B ∗ s meson predicted bydifferent theoretical approaches. B ′ s mesonRelativistic quark model [13] 5842Relativistic quark model [14] 5865Relativistic quark model [15] 5858Relativized quark model [16] 5822Bardeen, Eichten, Hill [17] 5765(35)Q.-F. L¨u et al. [18] 5801LQCD: q ¯ q + B ∗ K ( S ) [12] 5750(17)(19)LQCD: q ¯ q [19] 5806(15)(5)(25)Covariant (U)ChPT [20] 5778(26)NLO UHMChPT [21] 5742(20)(30)LO UChPT [22, 23] 5778(7)LO χ -SU(3) [24] 5690HQET + ChPT [25] 5765 . . TABLE II. Mass, in MeV, of the B ′ s meson predicted bydifferent theoretical approaches. putations [10, 11].In this work we closely follow Ref. [5] and addressthe mass-shifts, due to the B ( ∗ ) K thresholds, of thelowest lying P -wave b ¯ s states with total spin and parityquantum numbers J P = 0 + , 1 + , 2 + .A first lattice-regularised QCD study of the lowest-lying P -wave bottom-strange mesons in which B ( ∗ ) K operators are explicitly incorporated in the interpolatorbasis has been recently released [12]. We shall mainlycompare our results with the ones obtained by them, buta number of phenomenological model and effective fieldtheory (EFT) mass determinations of the same states canbe found in Tables I and II.The first conclusion one can obtain from such tablesis that the naive quark models, including the recentcalculation of Godfrey et al. , predict masses above the BK and B ∗ K thresholds, respectively. The authors of Ref. [18] use a nonrelativistic quark model which include α s -corrections. Taken a set of parameters fitted to thecharm sector, they succeed to bring the 0 + b ¯ s state belowthe BK threshold. To break the degeneration betweenthe B ′ s and B s mesons, they adjust a mixing angle thatalso place the B ′ s below the B ∗ K threshold. The resultof Bardeen et al. is calculated using a set of parametersfitted to the D ∗ s (2317).The second conclusion one can make is that the resultsof lattice computations, which take explicitly B ( ∗ ) K operators in the interpolator basis, and the estimates ofEFTs, that study dynamically generated bound statesfrom the B ( ∗ ) K scattering, are 80 MeV lower than thequark model predictions.Despite the significant progress made by lattice cal-culations incorporating open-flavoured thresholds, twomain drawbacks remain: i) the thresholds are added onlyas S -wave channels and ii) no statement can be madeabout the probabilities of the different Fock componentsin the physical state. Our approach solves these two is-sues and allows e.g. to introduce the coupling with the D -wave B ∗ K channel in the 1 + b ¯ s sector and to computethe amount of B ( ∗ ) K component in the meson.Our theoretical framework is a nonrelativistic con-stituent quark model in which quark-antiquark andmeson-meson degrees of freedom are incorporated.The constituent quark model (CQM) was proposed inRef. [26] (see references [27] and [28] for reviews).In order to keep the predictive power of the formalismwe do not change any model parameter. Moreover, itis worth to emphasize here that the quark model hasbeen applied to a wide range of hadronic observables,e.g. Refs. [29–36], and thus the model parameters arecompletely constrained.The manuscript is arranged as follows. In Sec. II wedescribe briefly the main properties of our theoreticalformalism. Section III is devoted to present our resultsfor the lowest lying P -wave B s states with total spin andparity quantum numbers J P = 0 + , 1 + , 2 + . We finishsummarizing and giving some conclusions in Sec. IV. II. CONSTITUENT QUARK MODEL
Constituent light quark masses and Goldstone-bosonexchanges, which are consequences of dynamical chi-ral symmetry breaking in Quantum Chromodynamics(QCD), together with the perturbative one-gluon ex-change and a nonperturbative confining interaction arethe main pieces of our constituent quark model [26, 28].A simple Lagrangian invariant under chiral transfor-mations can be written in the following form [37] L = ¯ ψ ( i /∂ − M ( q ) U γ ) ψ , (1)where M ( q ) is the dynamical (constituent) quark massand U γ = e iλ a φ a γ /f π is the matrix of Goldstone-bosonfields that can be expanded as U γ = 1 + if π γ λ a π a − f π π a π a + . . . (2)The first term of the expansion generates the constituentquark mass while the second gives rise to a one-boson exchange interaction between quarks. The maincontribution of the third term comes from the two-pionexchange which has been simulated by means of a scalar-meson exchange potential.In the heavy quark sector chiral symmetry is explicitlybroken and Goldstone-boson exchanges do not appear.However, it constrains the model parameters through thelight-meson phenomenology [38] and provides a naturalway to incorporate the pion exchange interaction in themolecular dynamics.The one-gluon exchange (OGE) potential is generatedfrom the vertex Lagrangian L qqg = i √ πα s ¯ ψ γ µ G µc λ c ψ , (3)where λ c are the SU (3) colour matrices and G µc is thegluon field. The resulting potential contains central,tensor and spin-orbit contributions.To improve the description of the open-flavour mesons,we follow the proposal of Ref. [6] and include one-loopcorrections to the OGE potential as derived by Gupta et al. [7]. These corrections show a spin-dependent termwhich affects only mesons with different flavour quarks.It is well known that multi-gluon exchanges producean attractive linearly rising potential proportional tothe distance between infinite-heavy quarks. However,sea quarks are also important ingredients of the stronginteraction dynamics that contribute to the screening ofthe rising potential at low momenta and eventually to thebreaking of the quark-antiquark binding string [39]. Ourmodel tries to mimic this behaviour using the followingexpression: V CON ( ~r ) = (cid:2) − a c (1 − e − µ c r ) + ∆ (cid:3) ( ~λ cq · ~λ c ¯ q ) , (4)where a c and µ c are model parameters.Explicit expressions for all the potentials and the valueof the model parameters can be found in Ref. [26],updated in Ref. [40]. Meson eigenenergies and eigenstatesare obtained by solving the Schr¨odinger equation usingthe Gaussian Expansion Method [41].The quark-antiquark bound state can be stronglyinfluenced by nearby multiquark channels. In this work,we follow Refs. [35, 36] to study this effect in thespectrum of the bottom-strange mesons and thus we needto assume that the hadronic state is given by | Ψ i = X α c α | ψ α i + X β χ β ( P ) | φ A φ B β i , (5)where | ψ α i are b ¯ s eigenstates of the two-body Hamil-tonian, φ M are wave functions associated with the A and B mesons, | φ A φ B β i is the two meson state with β quantum numbers coupled to total J P quantum num-bers and χ β ( P ) is the relative wave function between thetwo mesons in the molecule. To derive the meson-mesoninteraction from the quark-antiquark interaction we usethe Resonating Group Method (RGM) [42].The coupling between the quark-antiquark and meson-meson sectors requires the creation of a light quark pair.The operator associated with this process should describealso the open-flavour meson strong decays and is givenby [43] T = − √ X µ,ν Z d p µ d p ν δ (3) ( ~p µ + ~p ν ) g s m µ √ π ×× (cid:20) Y (cid:18) ~p µ − ~p ν (cid:19) ⊗ (cid:18)
12 12 (cid:19) (cid:21) a † µ ( ~p µ ) b † ν ( ~p ν ) , (6)where µ ( ν ) are the spin, flavour and colour quantumnumbers of the created quark (antiquark). The spin ofthe quark and antiquark is coupled to one. The Y lm ( ~p ) = p l Y lm (ˆ p ) is the solid harmonic defined in function of thespherical harmonic. We fix the relation of g s with thedimensionless constant giving the strength of the quark-antiquark pair creation from the vacuum as γ = g s / m ,being m the mass of the created quark (antiquark).From the operator in Eq. (6), we define the transitionpotential h βα ( P ) within the P model as [36] h φ A φ B β | T | ψ α i = P h βα ( P ) δ (3) ( ~P cm ) , (7)where P is the two-meson relative momentum.The usual version of the P model gives vertices thatare too hard specially when we work at high momenta.Following the suggestion of Ref. [44], we use a momentumdependent form factor to truncate the vertex as h βα ( P ) → h βα ( P ) × e − P , (8)where Λ = 0 .
84 GeV is the value used herein.Adding the coupling with bottom-strange states weend-up with the coupled-channels equations c α M α + X β Z h αβ ( P ) χ β ( P ) P dP = Ec α , X β Z H β ′ β ( P ′ , P ) χ β ( P ) P dP ++ X α h β ′ α ( P ′ ) c α = Eχ β ′ ( P ′ ) , (9)where M α are the masses of the bare b ¯ s mesons and H β ′ β is the RGM Hamiltonian for the two-meson statesobtained from the q ¯ q interaction. Solving the couplingwith the b ¯ s states, we arrive to a Schr¨odinger-typeequation X β Z (cid:0) H β ′ β ( P ′ , P )+ V eff β ′ β ( P ′ , P ) (cid:1) ×× χ β ( P ) P dP = Eχ β ′ ( P ′ ) , (10) State J P The. ( α s ) The. ( α s ) Exp. B s − . ± . B ∗ s − . ± . B ∗ s + B ′ s + B s (5830) 1 + . ± . B ∗ s (5840) 2 + . ± . P -wave bottom-strange mesons predicted by the constituent quark model ( α s )and those including one-loop corrections to the one-gluonexchange potential ( α s ). For completeness, our predictionsfor the 0 − and 1 − states that belong to the j Pq = 1 / − doubletare included. Experimental data are taken from Ref. [1].State Mass Width P [ q ¯ q ( P )] P [ BK ( S − wave )](MeV) (MeV) (%) (%) B ∗ s . . . B Rs . . . . J P = 0 + b ¯ s sector. where V eff β ′ β ( P ′ , P ; E ) = X α h β ′ α ( P ′ ) h αβ ( P ) E − M α . (11) III. RESULTS
The heavy-quark doublet j Pq = 3 / + is well establishedin the PDG, with the B s (5830) and B ∗ s (5840) mesonsbelonging to this doublet. Table III shows the predictedmasses within the naive quark model; one can seeour results taking into account the one-gluon exchangepotential ( α s ) and including its one-loop corrections ( α s ).In both cases our values are slightly higher than theexperimental figures but compatible.The mass of the B ∗ s state obtained using the naivequark model and without the one-loop spin correctionsto the OGE potential is 5851 MeV, which is compatiblewith the quark model predictions shown in Table Ibut, nevertheless, is much higher than the averagevalue predicted by lattice and EFT approaches. Themass associated to the B ∗ s state is sensitive to the α s -corrections of the OGE potential. This effect bringsdown its mass about 50 MeV but still is incompatiblewith the lattice and EFT results. The mass-shift dueto the α s -corrections allows the 0 + state to be closerto the BK threshold. This makes the BK coupling arelevant dynamical mechanism in the formation of the B ∗ s bound state. When we couple the 0 + b ¯ s ground state with the BK threshold the mass goes down towards5741 MeV (Table IV), in good agreement with lattice andEFT estimations.We turn now to discuss the probabilities of the differentFock components in the physical state. The lattice-regularised QCD study of Ref. [12] is only able toremark that both quark-antiquark and meson-mesonlattice interpolating fields have non-vanishing overlapswith the physical state. Our wave function probabilitiesare given in Table IV which reflects that the B ∗ s meson ismostly of quark-antiquark nature. This is in agreementwith the fact that lattice-regularised QCD computationsobserve this state even with only q ¯ q interpolators [19].In our model the probability of the BK state dependsbasically on three quantities: the bare meson mass, the P coupling constant and the residual BK interaction.Obviously, as neither of the three are observables, theycan take different values depending on the dynamics,making the results, and hence the BK probability, modeldependent. No model parameters have been changedfrom our study of the D ∗ s (2317) meson in Ref. [5].Moreover, the above mentioned quantities have beenconstrained in previous works by reproducing otherphysical observables like strong decays [43] (the P coupling constant), bottomonium spectrum (the baremass) [45] and N N and p ¯ p interactions (the BK residualinteraction) [46, 47].The scattering length is sensitive to the BK compos-iteness in the B ∗ s wave function. Therefore, to completethe analysis of the J P = 0 + b ¯ s sector, we have calculatedthe value of the scattering length from the value of the T -matrix at threshold, obtaining a BK = − .
18 fm. Thisvalue is compatible with the one reported by Ref. [12],that is, a BK = ( − . ± .
10) fm, where the differencemay originate from the binding energies predicted in eachstudy.In addition to the B ∗ s state below the BK threshold,we find a resonance with mass 5 .
88 GeV and width300 MeV which is denoted as B Rs in Table IV. Theresonance is a ∼
84 % b ¯ s state which comes from theresidual bare pole, that does not disappear but it isdressed with the BK interaction and quickly moves intothe complex plane. This resonance can be potentiallyobserved in the BK channel, but its large width is ahandicap for the experiments.The quark model including α s -corrections to the OGEpotential predicts that the states corresponding to the B ′ s and B s (5830) mesons are almost degenerated, withmasses close to the experimentally observed mass ofthe B s (5830). This B ′ s result goes in the sameline than the ones predicted by other phenomenologicalmodels (see Tables II and III). However, the average ofphenomenological model predictions, including our naiveone, is around 80 MeV higher than lattice studies andEFT estimations.We couple the two 1 + b ¯ s states associated with the B ′ s and B s (5830) mesons with the B ∗ K threshold.Following lattice criteria, we couple first the B ∗ K State Mass Width P [ q ¯ q ( P )] P [ q ¯ q ( P )] P [ B ∗ K ( S − wave )] P [ B ∗ K ( D − wave )](MeV) (MeV) (%) (%) (%) (%) B ′ s . .
000 13 .
5% 42 .
3% 44 .
2% - B s (5830) 5850 . .
024 37 .
2% 12 .
8% 50 .
0% - B Rs . . .
4% 58 .
8% 19 .
8% - B ′ s . .
000 13 .
2% 42 .
6% 44 .
2% 0 . B s (5830) 5832 . .
058 35 .
4% 12 .
1% 15 .
9% 36 . B Rs . . .
4% 58 .
8% 19 .
8% 0 . J P = 1 + b ¯ s sector.Results with and without coupling of the D -wave B ∗ K channel are listed. channel in an S -wave. One can see in Table V that thestate associated with the B ′ s meson goes down in thespectrum and it is located below B ∗ K threshold. Ourvalue, 5793 MeV, is compatible within errors with latticedata and with most of the EFT estimates. The stateassociated with the B s (5830) meson is almost insensitiveto the B ∗ K S -wave coupling. The mass predicted in thiscase is m B s (5830) = 5850 MeV, which is the same thanthe one obtained using quark model without coupling.This is because the B s (5830) state is the J P = 1 + member of the j Pq = 3 / + doublet predicted by HQS andthus it couples mostly in a D -wave to the B ∗ K threshold.It is important to highlight that the lattice computationof Ref. [12] does not take into account the coupling ofthe B s (5830) meson to the B ∗ K threshold either in S -or D -wave. This is because the coupling of the B ∗ K in D -wave is not easy to implement and, moreover, theyassume that the coupling to B ∗ K in S -wave is small.The coupling in D -wave of the B ∗ K threshold istrivially implemented in our approach. Table V showsthat the state associated with the B ′ s meson experiencea negligible modification. This is understandable becauseit is almost the | / , + i eigenstate of HQS. The stateassociated with B s (5830) meson suffers a moderatemass-shift with a final mass of 5833 MeV. This valuecompares nicely with the one reported in Ref. [12]:(5831 ±
10) MeV, but the coupling in D -wave of the B ∗ K threshold in the lattice result needs to be incorporatedin order to make a stronger statement. In any case,our result for the B s (5830) is compatible with othertheoretical approaches and with the experimental valuereported in PDG.Table V shows the probabilities of the different Fockcomponents in the physical B ′ s and B s (5830) states.When the B ∗ K threshold is coupled in an S -wave, themeson-meson component is 44% for the B ′ s and 50% forthe B s (5830). It is also interesting to point out thatthe relation between the quark-antiquark partial wavecomponents is close to the predictions of HQS, being the B ′ s meson a dominant j q = 1 / B s (5830) a j q = 3 / B ∗ K threshold in D -wave has very little effect for the formation of the B ′ s meson, keeping the probabilities pretty much the same.However, the D -wave coupling of the B ∗ K channel isrelatively important in the formation of the B s (5830)with a contribution to its physical wave function ofabout 37%. Moreover, in the case of the B s (5830), thedistribution of the meson-meson component in S - and D -wave channels does not affect very much the probabilitiesof the quark-antiquark components, being still dominantthe j q = 3 / J P = 0 + b ¯ s sector, we can predict the valueof the scattering length for the S -wave B ∗ K channel, a B ∗ K = − .
35 fm, and find that is slightly higher butcompatible with the a B ∗ K = ( − . ± .
16) fm reportedby Ref. [12]. Again, the different mass for the B ′ s predicted in each study can account for the differencein the predicted scattering lengths.In the J P = 1 + b ¯ s sector we also find a broad resonanceabove the B ∗ K threshold. It is originated from the j Pq = 1 / + component of the bare b ¯ s pole, which islargely dressed with the B ∗ K S -wave interaction. Thisresonance, labelled B Rs in Table V, is located at the massregion of 5 .
94 GeV and has a width of 271 MeV, whichcomplicates but not forbids its possible experimentalobservation.Only quark-antiquark operators were used in thelattice study [12] of the B ∗ s (5840) meson. They obtaineda value of (5853 ±
13) MeV for the mass. This is inqualitative agreement with experiment and with ournaive quark model prediction, confirming that this statecan be described well within the b ¯ s picture.Finally, let us clarify that we have omitted the possiblecoupling of the states to the B ( ∗ ) s π channels since theyviolate isospin and thus the effect should be subleading.We also neglect effects coming from the B ( ∗ ) s η coupling,partially motivated by the threshold lying O (140 MeV)above the B ( ∗ ) K threshold. Similar criteria has beenfollowed by the lattice analysis of Ref. [12]. IV. EPILOGUE
Within the formalism of a nonrelativistic constituentquark model in which quark-antiquark and meson-mesoncomponents are incorporated, we have performed acoupled-channel computation taking into account the J P = 0 + , 1 + and 2 + bottom-strange states predicted bythe naive quark model and the BK and B ∗ K thresholds.Our method allows to introduce the coupling with the D -wave B ∗ K channel and to compute the probabilitiesassociated with the different Fock components of thephysical state, features which cannot be addressednowadays by other theoretical approaches.Our study has been motivated by the fact that thereare no experimental evidences of the b ¯ s mesons whichbelong to the doublet j Pq = 1 / + and, as it has beenseen in the c ¯ s sector [5], the naive theoretical result canbe wrong by more than 100 MeV. In order to keep thepredictive power of the formalism we do not change anyparameter of the calculation in Ref. [5]. Moreover, it isworth to emphasize again that the quark model has beenapplied to a wide range of hadronic observables and thusthe model parameters are completely constrained.The level assigned to the B ∗ s meson within the naivequark model is much higher than the ones predicted bylattice and EFT approaches. However, it has been shownthat the value is compatible with other phenomenologicalmodel predictions. The one-loop corrections to the OGEpotential brings down this level and locates it slightlyabove the BK threshold. This makes the coupling withthe nearby threshold to acquire an important dynamicalrole. When coupling, the level is down-shifted againtowards the average mass obtained by lattice and EFTformalisms. We predict a probability of around 40% forthe BK component of the B ∗ s wave function. LatticeQCD can only state that both quark-antiquark andmeson-meson operators have important overlaps with thephysical state.The B ′ s and B s (5830) mesons appear almost degen-erated using the naive quark model that includes theone-loop corrections to the OGE potential. We havecoupled the two 1 + b ¯ s states associated with the B ′ s and B s (5830) mesons with the B ∗ K threshold. When coupling the B ∗ K channel in a S -wave, the B ′ s stategoes down in the spectrum and it is located below B ∗ K threshold with a mass compatible with lattice and EFTpredictions. The B s (5830) meson is almost insensitiveto this coupling because it is the | / , + i state predictedby HQS and thus couples mostly in a D -wave to the B ∗ K .When such coupling is included the state associated with B s (5830) meson suffers a moderate mass-shift and it isin very good agreement with other theoretical approachesand with the value reported in PDG. We observe thatthe meson-meson component is around 50% for both B ′ s and B s (5830) mesons, taking the quark-antiquark par-tial waves the other 50%.It is worth mentioning that, while finishing thepreparation of this work, a new study was published [48]supporting the idea that the B ∗ s and the B ′ s should bein the mass region of 5720 and 5770 MeV, respectively.Such values are compatible with other EFT predictionsand with the results presented in this work.Finally, the mass of the B ∗ s (5840) meson is pre-dicted reasonably well within our quark model approachtaking into account only quark-antiquark degrees-of-freedom. The same conclusion has been drawn by lattice-regularised QCD computations. ACKNOWLEDGMENTS
This work has been partially funded by Ministeriode Ciencia y Tecnolog´ıa under Contract no. FPA2013-47443-C2-2-P, by the Spanish Excellence Network onHadronic Physics FIS2014-57026-REDT, and by theJunta de Castilla y Le´on under Contract no. SA041U16.P.G.O. acknowledges the financial support from theSpanish Ministerio de Econom´ıa y Competitividad andEuropean FEDER funds under the contract no. FIS2014-51948-C2-1-P. J.S. acknowledges the financial supportfrom Alexander von Humboldt Foundation. [1] K. A. Olive et al. (Particle Data Group),Chin. Phys.
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