Prediction of positive parity B s mesons and search for the X(5568)
PPrediction of positive parity B s mesons and searchfor the X ( ) Daniel Mohler ∗ Helmholtz-Institut Mainz, 55099 Mainz, GermanyJohannes Gutenberg Universität Mainz, 55099 Mainz, GermanyE-mail: [email protected]
C. B. Lang
Institute of Physics, University of Graz, A–8010 Graz, AustriaE-mail: [email protected]
Sasa Prelovsek
Department of Physics, University of Ljubljana, 1000 Ljubljana, SloveniaJozef Stefan Institute, 1000 Ljubljana, SloveniaE-mail: [email protected]
We use a combination of quark-antiquark and B ( ∗ ) K interpolating fields to predict the mass oftwo QCD bound states below the B ∗ K threshold in the quantum channels J P = + and 1 + . Themesons correspond to the b-quark cousins of the D ∗ s ( ) and D s ( ) and have not yet beenobserved in experiment, even though they are expected to be found by LHCb. In addition to thesepredictions, we obtain excellent agreement of the remaining p-wave energy levels with the known B s ( ) and B ∗ s ( ) mesons. The results from our first principles calculation are comparedto previous model-based estimates. More recently the D0 collaboration claimed the existence ofan exotic resonance X ( ) with exotic flavor content ¯ bs ¯ du . If such a state with J P = + exists,only the decay into B s π is open which makes a lattice search for this state much cleaner andsimpler than for other exotic candidates involving heavy quarks. We conclude, however, that wedo not find such a candidate in agreement with a recent LHCb result. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] N ov ositive parity B s mesons and the X ( ) Daniel Mohler
In these proceedings we summarize two recently published lattice QCD studies [1, 2] of statesclose to multi-particle thresholds.
1. Prediction of positive parity B s mesons The discovery of the D ∗ s ( ) by BaBar [3] and the subsequent discovery of the D s ( ) more than 10 years ago revealed an unexpected peculiarity: unlike expected by potential models,these states turned out to be narrow states below the DK and D ∗ K thresholds. Moreover their massis roughly equal to the mass of their non-strange cousins, which immediately sparked speculationsabout their structure in terms of quark content, with popular options including both tetraquark andmolecular structures.The corresponding J P = + and 1 + states in the spectrum of B s hadrons have not been estab-lished in experiment. Given the success of recent lattice QCD calculations of the D ∗ s ( ) and D s ( ) [4, 5], it is therefore interesting to see if a prediction of these positive parity B s statesfrom lattice QCD is feasible. N L × N T N f a [fm] L [fm] m π [MeV] m K [MeV]32 ×
64 2+1 0.0907(13) 2.90 196 156(7)(2) 504(1)(7)
Table 1:
Gauge configurations used for the simulations in these proceedings.
For this study we use the 2+1 flavor gauge configurations with Wilson-Clover quarks generatedby the PACS-CS collaboration [6]. Table 1 shows details of the ensemble used in our simulation.Our quark sources are smeared with a Gaussian-like envelope as produced by use of the stochasticdistillation technique [7]. For the heavy b-quarks in the Fermilab interpretation [8], we tune theheavy-quark hopping parameter κ b for the spin averaged kinetic mass M B s = ( M B s + M B ∗ s ) / φ meson and η s to setthe strange quark mass, obtaining κ s = . D s spectrum [4, 5], where this approach allowed us to obtain reliable energy levels forthe D ∗ s ( ) and D s ( ) . For elastic s-wave scattering the Lüscher relation [9] relating thefinite volume spectrum to the phase shift δ of the infinite volume scattering amplitude is given by p cot δ ( p ) = √ π L Z ( q ) ≈ a + r p . (1.1)1 ositive parity B s mesons and the X ( ) Daniel Mohler
Lattice [MeV] Exp. [MeV] m B ∗ − m B m B s ∗ − m B s . + . − . m B s − m B m Y − m η b m B − m ¯ bb m B s − m ¯ bb m B c − m η b − m η c Table 2:
Selected mass splittings (in MeV) of mesons involving bottom quarks compared to the values fromthe PDG [11]. A bar denotes spin average. Errors are statistical and scale-setting only.
Figure 1:
Plots of ap cot δ ( p ) vs. ( ap ) for B ( ∗ ) K scattering in s -wave. Circles are values from our simu-lation; red lines indicate the error band following the Lüscher curves (broken lines). The full line gives thelinear fit to the points. Below threshold | p | is added and the zero of the combination indicates the boundstate position in infinite volume. Displayed uncertainties are statistical only. We perform an effective range approximation with the s-wave scattering length a and effec-tive range r . The resulting parameters and the mass of the resulting binding momentum (fromcot ( δ ( p )) = i ) are shown in Figure 1. We obtain a BK = − . ( ) fm a B ∗ K = − . ( ) fm (1.2) r BK = . ( ) fm r B ∗ K = . ( ) fm M B ∗ s = . ( ) GeV M B s = . ( ) GeVwhere the uncertainty on the bound state mass is statistical only. A full uncertainty estimate isgiven in Table 3 and explained in more detail in [1]. B s mesons Figure 2 shows our final results for the spectrum of s-wave and p-wave B s states. For values ofmasses in MeV we quote M = ∆ M lat + M exp B s where we substitute the experimental B s spin averagein accordance with our tuning. The states with blue symbols result from a naive determination ofthe finite volume energy levels (statistical uncertainty only). Notice that the j = states agree wellwith the experimental B s ( ) and B ∗ s ( ) as determined by CDF/D0 and LHCb [11]. The B s states with magenta symbols indicate the bound state positions extracted using Lüscher’s method2 ositive parity B s mesons and the X ( ) Daniel Mohler source of uncertainty expected size [MeV]heavy-quark discretization 12finite volume effects 8unphysical Kaon, isospin & EM 11b-quark tuning 3dispersion relation 2spin-average (experiment) 2scale uncertainty 13 pt vs. 2 pt linear fit 2total (added in quadrature) 19
Table 3:
Systematic uncertainties in the mass determination of the below-threshold states with quantumnumbers J P = + , + . The heavy-quark discretization effects are quantified by calculating the Fermilab-method mass mismatches and employing HQET power counting [10] with Λ =
700 MeV. The finite volumeuncertainties are estimated conservatively by the difference of the lowest energy level and the complex poleposition. The last line gives the effect of using only the two points near threshold for the effective range fit.The total uncertainty has been obtained by adding the single contributions in quadrature. m [ G e V ] Lat: energy levelLat: bound state from phase shift m π = 156 MeV B * KB K B s B s* B s0* B s1 B s1 ’ B s2 J P : 0 - - + + + + Figure 2:
Spectrum of s-wave and p-wave B s states from our simulation. The blue states are naive energylevels, while the bound state energy of the states in magenta results from an effective range approximationof the phase shift data close to threshold. The black lines are the energy levels from the PDG [11]. The errorbars on the blue states are statistical only, while the errors on the magenta states show the full (statisticalplus systematic) uncertainties. ositive parity B s mesons and the X ( ) Daniel Mohler N e v en t s / M e V / c D0 Run II, 10.4 fb
DATAFit with background shape fixedBackgroundSignal a) ] ) [GeV/c ± π S0 (B m R e s i dua l s ( D a t a F i t ) C and i da t e s / ( M e V ) Claimed X(5568) stateCombinatorial ) > 10 GeV s0 B ( T p LHCb (MeV) ) – p s m(B Figure 3:
Left pane: B s π ± invariant mass distribution from D0 [12] (after applying a cone cut). Rightpane: B s π ± invariant mass distribution by LHCb [13] shown in black symbols with a signal componentcorresponding to ρ x = .
6% as observed by D0 shown in red. and taking into account the sources of uncertainty detailed in Table 3. Notice that our Lattice QCDcalculations yields bound states well below the B ( ∗ ) K thresholds. B s π + scattering and search for the X(5568) Recently, the D0 collaboration has reported evidence for a peak in the B s π + invariant mass notfar above threshold [12]. This peak is attributed to a resonance dubbed X(5568)with the resonancemass m X and width Γ x , m X = . ± . + . − . MeV , (2.1) Γ X = . ± . + . − . MeV . L [fm] E [ G e V ] B s (n) π (-n)B(n)K(-n) m Bs +m π m B +m K m X +/- Γ X /2 Figure 4:
Analytic predictions for energies E ( L ) ofeigenstates as a function of lattice size L . Decay of this resonance into B s π + im-plies an exotic flavor structure with the min-imal quark content ¯ bs ¯ du . Most model stud-ies which accommodate a X(5568) proposespin-parity quantum numbers J P = + . Shortafter D0 reported their results, the LHCb col-laboration investigated the cross-section as afunction of the B s π + invariant mass with in-creased statistics and did not find any peakin the same region [13]. Figure 3 shows boththe plot from D0 (left pane) and the data fromLHCb (right pane), where the red shaded re-gion illustrates the signal expectation giventhe ratio of yields ρ x determined by D0. 4 ositive parity B s mesons and the X ( ) Daniel Mohler B s π + The presence of an elastic resonance with the parameters of the X ( ) would lead to acharacteristic pattern of finite volume energy levels corresponding to QCD eigenstates with givenquantum numbers for finite spatial size L . E [ G e V ] a ll A a ll A a ll - A a ll - A a ll - B a ll - B choice-0.200.2 a [f m ] Figure 5:
The eigenenergies of the ¯ bs ¯ du systemwith J P = + from a lattice simulation for thechoices detailed in the text. Figure 4 shows analytic predictions for ener-gies of eigenstates for an elastic resonance in B s π (with J P = + ) as a function of the lattice size L as determined from Lüscher’s formalism [9]. Redsolid lines are B s π eigenstates in the scenario withresonance X ( ) ; orange dashed lines are B s π eigenstates when B s and π do not interact; bluedot-dashed lines are B + ¯ K eigenstates when B + and ¯ K do not interact; the grey band indicatesthe position of X ( ) from the D0 experiment[12]. The lattice size L = . E (cid:39) m X (red solid), while there is no sucheigenstate for L = − B s and π + (or-ange dashed). In the unlikely scenario of a deeplybound BK state, the simulation would result in aneigenstate with E ≈ m X up to exponentially smallcorrections in L . In our simulation we use the PACS-CS en-semble [6] from Table 1. The interpolator basis O B s ( ) π ( ) , = (cid:2) ¯ b Γ , s (cid:3) ( p = ) (cid:2) ¯ d Γ , u (cid:3) ( p = ) , O B s ( ) π ( − ) , = ∑ p = ± e x , y , z π / L (cid:2) ¯ b Γ , s (cid:3) ( p ) (cid:2) ¯ d Γ , u (cid:3) ( − p ) , O B ( ) K ( ) , = (cid:2) ¯ b Γ , u (cid:3) ( p = ) (cid:2) ¯ d Γ , s (cid:3) ( p = ) , consisting of both B s π and BK interpolators, is employed.Figure 5 shows the eigenstates determined from our simulation for various choices. The setswith full symbols are from correlated fits while open symbols result from uncorrelated fits. No-tation “all” refers to the full set of gauge configurations while “all-4” refers to the set with four(close to exceptional) gauge configurations removed. Set A is from interpolator basis O B s ( ) π ( ) , O B s ( ) π ( − ) , O B ( ) K ( ) while set B results from a larger basis O B s ( ) π ( ) , O B s ( ) π ( − ) , , O B ( ) K ( ) , . Allchoices consistently result in a small scattering length a consistent with 0 within error.5 ositive parity B s mesons and the X ( ) Daniel Mohler
Figure 6 shows the eigenenergies of the ¯ bs ¯ du system with J P = + calculated on the lattice(left pane) compared to the analytic prediction based on the X ( ) as observed by D0 (rightpane). Unlike expected for the case of a resonance with the parameters of the X(5568), our latticesimulation at close-to-physical quark masses does not yield a second low-lying energy level. Ourresults therefore do not support the existence of X ( ) with J P = + . Instead, the results appearcloser to the limit where B s and π do not interact significantly, leading to a B s π scattering lengthcompatible with 0 within errors. E [ G e V ] m Bs +m π m B +m K m X +/- Γ X /2 (a) (b) Figure 6: (a) The eigenenergies of the ¯ bs ¯ du sys-tem with J P = + from our lattice simulation and(b) an analytic prediction based on X ( ) , both atlattice size L = . B s ( ) π + ( ) , B + ( ) ¯ K ( ) and B s ( ) π + ( − ) in absence of interactions; momenta inunits of 2 π / L are given in parenthesis. The pane (a)shows the energies E = E latn − E latB s + E expB s with thespin-averaged B s ground state set to its experimentvalue. The pane (b) is based on the experimental massof the X ( ) [12], given by the grey band, and ex-perimental masses of other particles. References [1] C. B. Lang, D. Mohler, S. Prelovsekand R. M. Woloshyn, Phys. Lett. B ,17 (2015) doi:10.1016/j.physletb.2015.08.038.[2] C. B. Lang, D. Mohlerand S. Prelovsek, Phys. Rev. D , 074509(2016) doi:10.1103/PhysRevD.94.074509.[3] B. Aubert et al. [BaBar Collaboration],Phys. Rev. Lett. , 242001 (2003).[4] D. Mohler, C. B. Lang,L. Leskovec, S. Prelovsek and R. M. Woloshyn,Phys. Rev. Lett. , no. 22, 222001 (2013).[5] C. B. Lang, L. Leskovec,D. Mohler, S. Prelovsek and R. M. Woloshyn,Phys. Rev. D , no. 3, 034510 (2014).[6] S. Aoki et al. , Phys. Rev. D , 034503 (2009).[7] C. Morningstar et al. , Phys. Rev. D , 114505 (2011).[8] A. X. El-Khadra, A. S. Kronfeld andP. B. Mackenzie, Phys. Rev. D , 3933 (1997).[9] M. Lüscher, Commun.Math. Phys. , 153 (1986); Nucl. Phys. B , 531 (1991); Nucl. Phys. B , 237 (1991).[10] M. B. Oktay andA. S. Kronfeld, Phys. Rev. D , 014504 (2008).[11] K. A. Olive et al. [Particle Data Group Collaboration],Chin. Phys. C , 090001 (2014).[12] V. M. Abazov et al. [D0 Collaboration],Phys. Rev. Lett. , no. 2, 022003 (2016).[13] R. Aaij et al. [LHCb Collaboration],Phys. Rev. Lett. , no. 15, 152003 (2016)., no. 15, 152003 (2016).