Spectroscopy of Charmed and Bottom Hadrons using Lattice QCD
SSpectroscopy of Charmed and Bottom Hadrons using LatticeQCD
Sourav
Mondal , M. Padmanath , and Nilmani
Mathur ,(cid:63) Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai. Institüt fur Theoretische Physik, Universität Regensburg,Universitässtrase 31, 93053 Regensburg, Germany.
Abstract.
We present preliminary results on the light, charmed and bottom baryon spectrausing overlap valence quarks on the background of 2 + + ff erentgauge ensembles at three lattice spacings ( a ∼ Energy spectra of heavy hadrons as well as the spin splittings between them play an important rolein understanding the fundamental strong interactions. With the recent experimental discovery of thedoubly charmed baryon, Ξ ++ cc [1], and various Ω c resonances [2] by the LHCb collaboration, there hasbeen resurgence of scientific interest in the study of heavy baryons. Anticipating discovery of manymore heavy hadrons at ongoing experiments at LHCb, BESIII and future experiments at Belle, firstprinciple calculations such as using Lattice QCD are essential to investigate the energy spectra and theproperties of these heavy hadrons in a model independent way. Not only that these calculations arecrucial in understanding the structure and interactions of these hadronic excitations, but also can makeprecise predictions that can guide future discoveries. However, lattice QCD study of heavy hadronsis severely a ff ected by the large discretization errors due to relatively larger heavy quark masses.Therefore, it is important to systematically approach the continuum limit to reduce and quantify thesediscretization uncertainties. In this talk, we present updated results from our ongoing study of thehadron spectra. Particular emphasis is given on the calculations of the charmed and bottom baryons.Results are extracted at three lattice spacings and then are extrapolated to the continuum limit. We use three N f = + + ×
64, 32 ×
96 and 48 ×
144 and at gauge couplings β = . , .
30 and 6 .
72 respectively. The (cid:63)
Speaker, e-mail: [email protected] a r X i v : . [ h e p - l a t ] D ec etails of these gauge configurations are summarized in Ref. [3]. We use Ω sss baryon mass to calculatelattice spacings [4, 5] and those are found to be consistent with 0 . , . . r parameter.For the light and charm quarks we use a unified approach adopting overlap fermions, which doesnot have O ( ma ) errors and has exact chiral symmetry at finite lattice spacing. Details of the action, itsnumerical implementation, mass tuning are given in Refs. [4, 5]. The tuned bare charm quark masses( am c ) are found to be 0.528, 0.425 and 0.29 on coarser to fine lattices respectively [5].While we intend to treat bottom quarks with the same formalism in future, for the current workwe use a non-relativistic formulation [6]. This NRQCD Hamiltonian is improved by including spin-independent terms through O ( v ). For the coarser two ensembles, we use the values of the improve-ment coe ffi cients, c to c , as estimated non-perturbatively by the HPQCD collaboration [7] on thesame ensembles. For the fine lattice, we use tree level coe ffi cients. The details of the NRQCD actionand tuning is given in Ref. [8, 9]. The spin averaged 1 S bottomonium kinetic mass is utilized to tunethe bottom quark mass [8]. The observed hyperfine splitting (64 ± S bottomonium isfound to be in good agreement with its experimental value (62 . ± We have calculated hadron spectra over a wide range of pseudoscalar meson masses from the physicalpion mass to pseudoscalar meson mass at about 5.5 GeV. On fine lattices pseudoscalar meson massranges are from about 300 MeV to around 6 GeV. In future we intend to extend pseudoscalar mesonmass range towards η b on a hyperfine lattice. While that calculation is underway, using non-relativisticbottom quarks we have extended this study to calculate the energy spectra of bottom hadrons. Whilea comprehensive analysis with existing data set is also underway, here we present our preliminaryresults on the ground state energy spectra for light as well as heavy hadrons with particular emphasison charmed and charmed-bottom baryons. In Figure 1, we present our results on the light hadron spectra. Figure 1(a) shows pseudoscalar mesonmasses as a function of quark masses on the coarser lattice ( a ∼ . ff ect). In Figure 1(b) we show the hyperfine splittings between vector (1 − ) and pseudoscalar(0 − ) mesons on this wide range of pseudoscalar meson masses. In Figure 1(c) we show the similarhyperfine splittings (in MeV) between ∆ -baryon ( J P = + ) and nucleon ( J P = + ) at three latticespacings and also over a wide range of pseudoscalar meson masses. Figure 1(d) shows our preliminaryresults on the ground state spectra of the low lying octet baryons on 32 ×
96 lattice at the lattice spacing ∼ .
088 fm. As expected energies of these baryons coincide at the SU(3) flavour symmetric point anddeviate from each other on its both sides. An analysis with the chiral and continuum extrapolation ofthese baryons is ongoing.
Plethora of experimental discoveries have been made over the last two decades in the heavy hadronsector, part of which are understood theoretically, while the nature of the rest continues to be puz-zling [10]. Investigations using first principle calculations, such as lattice QCD, are crucial to under-stand the structure and interactions of these excitations as well as to guide future discoveries of moresubatomic particles. As mentioned earlier, being heavy, the energy spectra of heavy hadrons on the .0 0.2 0.4 0.6 0.8 m q a ( m π a ) (a) m q a ∆ M a ( − − − ) (b) m π (GeV ) ∆ M ( ∆ ( / + ) − N ( / + )) ( M e V ) : a ~ 0.1207 fm: a ~ 0.088 fm: a ~ 0.06 fm (c) m π a M a : N : Λ : Σ : Ξ (d) Figure 1. (a) Square of pseudoscalar meson masses as a function of quark masses from physical pion massto about 5.5 GeV. (b) Hyperfine splittings between vector and pseudoscalar mesons over the same range of pionmasses. (c) Hyperfine splittings between ∆ -baryon ( J P = + ) and the nucleon ( J P = + ) at three lattice spacingsand over a wide range of pseudoscalar meson masses. (d) Ground state energy spectra of the low lying octetbaryons on 32 ×
96 ensemble. lattice are subject to strong discretization errors and thus lattice calculations at more than one latticespacings followed by a systematic continuum extrapolation is quite essential.In Figure 2 we plot our preliminary results on continuum extrapolations for charm-strange mesons(Figure 2(a)) and charmonia (Figure 2(b)). To reduce discretization errors, we calculate splitting of ameson from the respective 1 S spin average masses. It is interesting to note that di ff erent mesons havedi ff erent slopes towards the continuum limit. The continuum extrapolation is carried out using termsup to O (( m q a ) ) with Bayesian priors (there is no O ( m q a ) term for overlap action). We observe thatexcept for the hyperfine splittings, the coe ffi cients for O (( m q a ) ) terms are very small.Next we present our preliminary results for the ground state of the positive parity charmed baryons.In order to reduce discretization errors, we perform chiral extrapolation on the ratios of baryon masses( M b a ) to the 1 S spin-average mass ( M S a ), i.e., M rb = M b a / nM S a , where n = / M rb = A + B . ( m π a ) , as well as with a chiral extrapolation form (equations .000 0.005 0.010 0.015 0.020 a ∆ M c ¯ s ( M e V ) : Expt : − − − : − − cc/ : + − cs : + − cs (a) a ∆ M c ¯ s ( M e V ) : Expt : − − − : − − cc/ : + − cs : + − cs (b) Figure 2.
Energy splittings in (a) the charm-strange and (b) the charmonium ground state spectra plotted againstthe square of the lattice spacings at three lattice spacings. Bands represent one-sigma errorbars in the continuumextrapolations. below) using heavy baryon chiral perturbation theory (HBChPT), as described in Ref. [11]. m Λ c m spina v. = m Λ c m spina v. + σ Λ c (4 π f π ) m spina v. m π − g (4 π f π ) ( m spina v. ) (cid:32) F ( m π , ∆ Λ c Σ c ,µ ) + F ( m π , ∆ Λ c Σ ∗ c ,µ ) (cid:33) (1) m Ξ c m spina v. = m Ξ c m spina v. + σ Ξ c (4 π f π ) m spina v. m π − g (4 π f π ) ( m spina v. ) (cid:32) F ( m π , ∆ Ξ c Ξ (cid:48) c ,µ ) + F ( m π , ∆ Ξ c Ξ ∗ c ,µ ) (cid:33) (2)for Λ c and Ξ c respectively. The chiral function F in eqn. 2 is defined as, F ( m , ∆ , µ ) = ( ∆ − m + i (cid:15) ) / ln ∆ + √ ∆ − m + i (cid:15) ∆ − √ ∆ − m + i (cid:15) − ∆ m ln (cid:32) m µ (cid:33) − ∆ ln (cid:32) ∆ m (cid:33) , (3)with F ( m , , µ ) = π m π . Splittings ∆ used in the extrapolation formula are obtained by extrapolatingthe splittings between two baryons to the physical pion masses using trivial extrapolation form ∆ i j = ∆ i j + A ( m π a ) , (4)where i and j are the baryons under consideration. For Λ c and Ξ c we could use HBChPT with χ / do f ∼
1. We then perform continuum extrapolation of the chirally extrapolated ratios with aform up to O ( a ) terms. Finally to obtain the physical values we multiply the extrapolated values by nM ph y (1 S : cc ).In Figure 3 we show our preliminary chiral and continuum extrapolated results for the groundstate singly charm baryons. We compare our results with experimental values of these baryons [10]and also with other lattice results [11–15].Here we would like to point out that the LHCb Collaboration has recently reported observationof five new resonances based on the invariant mass distribution of Ξ + c K − in the energy range between M c ( M e V ) Λ c Σ c Ξ c Σ ∗ c Ω c Ω ∗ c : Expt (PDG): This work: Alexandrou et. al [12]: Brown et. al [13] : Perez-Rubio et. al [14]: Briceno et. al [11]: Namekawa et. al [15] Figure 3.
Preliminary results for the positive parity singly charmed baryons. Results from this work are comparedwith experimental values [10] and other lattice results [11–15]. − Ω c baryon. Be-fore the discovery of these resonances we studied the excited state spectra of Ω c baryons in detail [16–18]. It is quite satisfying to see that our prediction matches very well with the experimental resultsand strongly indicates that the observed states Ω c (3000) and Ω c (3050) have spin-parity J P = / − ,the states Ω c (3066) and Ω c (3090) have J P = / − , whereas Ω c (3119) is possibly a 5 / − state [19].This identification is crucial to decipher the nature of these resonances. M cc ( M e V ) Ξ cc Ξ ∗ cc Ω cc Ω ∗ cc : Expt [1]: This work: Padmanath et.al [20]: Mathur et.al [21, 22]: Alexandrou et.al [12] : Brown et.al [13]: Perez-Rubio et.al [14]: Briceno et.al [11]: Namekawa et. al [15] Figure 4.
Preliminary results for the positive parity doubly charmed baryons. Results from this work are com-pared with experimental values (where available) and other lattice results [11–15, 20–22] n Figure 4, we show our preliminary results for doubly charmed baryons. It is noteworthy topoint out that all lattice results including the current work are predictions before the experimentaldiscovery of Ξ ++ cc by the LHCb collaboration [1]. So far Ξ ++ cc is the only doubly charmed baryondiscovered experimentally. In that context lattice predictions for other doubly charmed baryons arevery interesting for their future discovery. Except the pseudoscalar B c and B c (2 S ) mesons, no other hadron has been discovered yet with b and c quark content together. Anticipating discoveries of such hadrons in the near future, we present ourresults on charmed-bottom hadrons in this section. For the calculation of bc hadrons we use NRQCDpropagators for bottom quarks which are contracted with overlap propagators to obtain correlatorsfor various bc hadrons. We have already presented our predictions for the hyperfine splitting of B c meson to be 55 ± B ∗ c meson. Here we present thepreliminary results on the ground state spectrum of the positive parity charmed bottom Ω baryons,namely, Ω ccb ( + ), Ω ∗ ccb ( + ), Ω cbb ( + ) and Ω ∗ cbb ( + ). Works on the spectra of Ξ cbu and Ξ cbs baryons aswell on negative parity baryons are ongoing.To reduce the relative discretization errors due to heavy charm and bottom quarks we presentthe mass of the bc baryons as : M sub a = Ma − n c Ma ( cc ) − n b Ma ( bb ), where Ma ( cc ) and Ma ( bb )are spin-average masses of the 1S charmonia and bottomonia respectively, whereas n c and n b are thenumber of charm and bottom quarks in bc hadrons. One would expect that these subtractions wille ff ectively remove the heavy quark content and will reduce the discretization errors. In Figure 5 weshow subtracted energy levels for the charmed-bottom Ω baryons at two lattice spacings. Our resultsare compared with those obtained in Ref. [13]. Horizontal bars are possible errors one may expectafter continuum extrapolation of hadron masses with non-relativistic bottom quarks. In order to obtainphysical values we need to add back physical value of n c M ( cc ) + n b M ( bb ) to above subtracted masses.In future we will add numbers from another lattice spacing and then will do continuum extrapolationto obtain final numbers for these baryons. We report preliminary results on the ground state energy spectra for various hadrons with light to bot-tom quark content. We incorporate a unified approach to treat light to charm quarks uniformly usingoverlap fermions. A relativistic overlap action is used on the background of 2 + + η b using a hyperfine lattice to treat light to bottom quarks with the same action. In this calculation forbottom quark we use a non-relativistic action with non-perturbatively tuned coe ffi cients with terms upto O ( v ). For the coarser two ensembles, we use the values of the improvement coe ffi cients, c to c ,as estimated non-perturbatively by the HPQCD collaboration [7], while for the fine ensemble, we usetree level coe ffi cients. The charm and bottom masses are tuned by equating the spin-averaged kineticmasses of the 1 S charmonia and bottomonia states, respectively, to their physical values.We present the hyperfine splittings between vector (1 − ) to pseudoscalar (0 − ) mesons as well asbetween ∆ -baryon ( + ) to nucleon ( + ) in a large range of quark masses. In future we will fit thesesplittings with appropriate formulae to find out their variations at di ff erent quark mass ranges. As iswell known that these hyperfine splittings provide very useful information about the spin-spin inter-actions within the strongly interacting theory and are invaluable ingredients for any potential model a) Ω ccb (b) Ω ∗ ccb (c) Ω cbb (d) Ω ∗ cbb Figure 5.
Subtracted masses for the charmed bottom Ω baryons. The red points are the preliminary results fromthis work and the blue points are the continuum limit result of Ref. [13] .calculations. We also show our preliminary results on octet baryons. In future we will perform sys-tematic chiral and continuum extrapolations to get physical results for these observables.We also report our preliminary results on charmed hadrons. Controlling the discretization error isa major challenge for heavy quarks and using results at three lattice spacings we are able to perform asystematic continuum extrapolation. Extracted value (115 ± S charmonia agrees very well to its physical value (113 . ± . Ξ ++ cc baryon. These preciseand successful predictions of Ξ ++ cc baryon demonstrate the capability of lattice investigations to rightlyguide the future experimental discoveries. We also report our preliminary results on hadrons contain-ing both charm and bottom quarks. The hyperfine splitting of B c meson is found to be 55 ± bc baryons we present results for Ω ccb , Ω ∗ ccb , Ω cbb , and Ω ∗ cbb . In future we will address other bc baryons including negative parity baryons. Computations are carried out using computing resources of the Indian Lattice Gauge Theory Initiativeand the Department of Theoretical Physics, TIFR. We thank A. Salve, K. Ghadiali and P. Kulkarnifor technical supports. S. M. and N. M. would like to acknowledge support from the Departmentof Theoretical Physics, TIFR. M. P. acknowledges support from Deutsche ForschungsgemeinschaftGrant No. SFB / TRR 55 and EU under grant no. MSCA-IF-EF-ST-744659 (XQCDBaryons). We aregrateful to the MILC collaboration and in particular to S. Gottlieb for providing us with the HISQlattices.
References [1] R. Aaij et al. (LHCb), Phys. Rev. Lett. , 112001 (2017), [2] R. Aaij et al. (LHCb), Phys. Rev. Lett. , 182001 (2017), [3] A. Bazavov et al. (MILC), Phys. Rev.
D87 , 054505 (2013), [4] S. Basak, S. Datta, M. Padmanath, P. Majumdar, N. Mathur, PoS
LATTICE2012 , 141 (2012), [5] S. Basak, S. Datta, A.T. Lytle, M. Padmanath, P. Majumdar, N. Mathur, PoS
LATTICE2013 ,243 (2014), [6] G.P. Lepage et al. , Phys.Rev.
D46 , 4052 (1992),[7] R.J. Dowdall et al. (HPQCD), Phys. Rev.
D85 , 054509 (2012), [8] N. Mathur, M. Padmanath, R. Lewis, PoS
LATTICE2016 , 100 (2016), [9] R. Lewis, R.M. Woloshyn, Phys. Rev.
D79 , 014502 (2009), [10] C. Patrignani et al. (Particle Data Group), Chin. Phys.
C40 , 100001 (2016)[11] R.A. Briceno, H.W. Lin, D.R. Bolton, Phys. Rev.
D86 , 094504 (2012), [12] C. Alexandrou, C. Kallidonis, Phys. Rev.
D96 , 034511 (2017), [13] Z.S. Brown, W. Detmold, S. Meinel, K. Orginos, Phys. Rev.
D90 , 094507 (2014), [14] P. Perez-Rubio, S. Collins, G.S. Bali, Phys. Rev.
D92 , 034504 (2015), [15] Y. Namekawa et al. (PACS-CS), Phys. Rev.
D87 , 094512 (2013), [16] M. Padmanath, R.G. Edwards, N. Mathur, M. Peardon, Proceedings
Charm (2013), [17] P. Madanagopalan, R.G. Edwards, N. Mathur, M.J. Peardon, PoS
LATTICE2014 , 084 (2015), [18] M. Padmanath, N. Mathur, Proceedings
Charm (2015), [19] M. Padmanath, N. Mathur, Phys. Rev. Lett. , 042001 (2017), [20] M. Padmanath, R.G. Edwards, N. Mathur, M. Peardon, Phys. Rev.
D91 , 094502 (2015), [21] N. Mathur, R. Lewis, R.M. Woloshyn, Phys. Rev.
D66 , 014502 (2002), hep-ph/0203253 [22] R. Lewis, N. Mathur, R.M. Woloshyn, Phys. Rev.
D64 , 094509 (2001),, 094509 (2001),