Precise predictions of charmed-bottom hadrons from lattice QCD

We report the ground state masses of hadrons containing at least one charm and one bottom quark using lattice quantum chromodynamics. These include mesons with spin (J)-parity (P) quantum numbers J(P): 0(-), 1(-), 1(+) and 0(+) and the spin-1/2 and 3/2 baryons. Among these hadrons only the ground state of 0(-) is known experimentally and therefore our predictions provide important information for the experimental discovery of all other hadrons with these quark contents.

We report the ground state masses of hadrons containing at least one charm and one bottom quark using lattice quantum chromodynamics. These include mesons with spin (J)-parity (P ) quantum numbers (J P ): 0 − , 1 − , 1 + and 0 + and the spin-1/2 and 3/2 baryons. Among these hadrons only the ground state of 0 − is known experimentally and therefore our predictions provide important information for the experimental discovery of all other hadrons with these quark contents.
Investigations of such hadrons are highly appealing, as they provide a unique laboratory to explore the heavy quark dynamics at multiple scales: 1/m b , 1/m c (m Q : mass of heavy quark), 1/Λ QCD , and hence may reveal more information on strong interactions which may be concealed in other hadrons. Decay constants and form factors of bc mesons are still unknown but are quite important because of their relevance to investigate the physics beyond the standard model, particularly in view of the recent measurement of R(J/ψ) [34]. The information on spin splittings and decay constants can shed light on the structure of these states and help us to understand the nature of strong interactions at multiple scales. Moreover, bc baryon decays can aid in studying b → c transition and |V cb | element of the CKM matrix.
However, to date the discovery of these hadrons is limited to only two observations: B c (0 − ) with mass 6275(1) MeV [35] and B c (2S)(0 − ) at 6842(6) MeV [36] while the latter has not yet been confirmed [37]. On the other hand, LHC being an efficient factory for producing bc hadrons [38,39], one would envisage for their discovery and study their decays in near future. Precise theoretical predictions related to the energy spectra and decay of these hadrons are thus utmost essential to guide the discovery of these states.
In fact multitude of model calculations exist in literatures on bc mesons [40][41][42][43][44][45][46] and baryons [47][48][49][50][51][52]. However, those predictions vary widely, e.g. prediction for 1Shyperfine splitting in B c (bc) mesons spread over a range of 40-90 MeV [40][41][42][43][44][45][46]. The prediction for bc baryons and excited states are even more scattered. Naturally first principle calculations using lattice QCD with controlled systematics, which have already proved to be successful in predicting the masses of low lying [53][54][55][56][57][58] and excited hadrons [59][60][61], are quite essential to study these hadrons. However, unlike quarkonia, lattice study of bc hadrons are confined only to a few calculations [57,[62][63][64][65]. In this work we carry out a detailed lattice calculation of the ground state energy spectra of all low lying bc hadrons (showed in Table I) with very good control over systematics and predict their masses most precisely to this date. Lattice QCD investigations are subject to various lattice artefacts. Of these the most relevant one in a study of heavy hadrons is the discretization error due to large masses of heavy quarks. It is thus essential to take a controlled continuum extrapolation of the results attained at finite lattice spacings. To that goal we obtain results at three sets of lattices with spacings a ∼ 0.12, 0.09 and 0.06 fm, and then are able to perform such extrapola-arXiv:1806.04151v1 [hep-lat] 11 Jun 2018 tions. Below we elaborate our numerical procedure. Numerical details: A. Lattice ensembles: We use three dynamical 2+1+1 flavours (u/d, s, c) lattice ensembles generated by the MILC collaboration [66] with HISQ fermion action [67]. The lattices are with sizes 24 3 × 64, 32 3 × 96 and 48 3 × 96 at gauge couplings 10/g 2 = 6.00, 6.30 and 6.72, respectively. The details of these gauge configurations, which are currently being extensively used by MILC-Fermilab-HPQCD collaborations, are summarized in Ref. [66]. The measured lattice spacings, obtained from r 1 parameter, for the set of ensembles being used here are 0.1207(11) 0.0888(8) and 0.0582(5) fm, respectively [66].

B. Quark actions:
For valence quark propagators, from light to charm quarks, we use the overlap action which has exact chiral symmetry at finite lattice spacings [68][69][70] and no O(ma) error. Using multimass algorithm of overlap action [71,72], we are able simulate light to charm quarks in a single lattice formalism with little computational overhead. A wall source is utilized as smearing function for calculating the quark propagators on Coulomb gauge-fixed lattices.
For bottom propagators we utilize a non-relativistic QCD (NRQCD) formulation [73] in which we incorporate all terms up to 1/M 2 0 and the leading term of the order of 1/M 3 0 , where M 0 = am b is the bare mass for the bottom quark [74,75]. This Hamiltonian is improved by including spin-independent terms through O(α s v 4 ) with non-perturbatively tuned improvement coefficients, as estimated in Ref. [76] for these ensembles. Such an action has been utilized by HPQCD collaboration over many years and found to be very effective for bottom quark physics [57,62,[76][77][78]. For the coarser two ensembles, we study the spectrum using "improved" coefficients as well as tree level coefficients ("unimproved"). For the finer lattice, we study only with tree level coefficients. C. Quark mass tuning: Following the Fermilab prescription for heavy quarks [79] we tune the heavy quark masses by equating the spin-averaged kinetic mass of the 1S quarkonia states (M kin (1S) = 3 4 M kin (1 − ) + 1 4 M kin (0 − )) to their respective physical values. A momentum induced wall-source, which is found to be very efficient compared to point or smeared sources [80], is utilized to obtain energy values from the correlators with finite momenta and helps to obtain kinetic masses precisely with significantly little statistics. Details of the charm and bottom quark mass tuning are described in Ref. [81] and Ref. [75], respectively. The tuned bare charm quark masses are found to be 0.528, 0.427 and 0.290 on coarse to fine lattices respectively, which also satisfy m c a << 1, a necessary condition for using heavy relativistic quark to reduce discretization effects. We tune strange quark mass, following Ref. [82], by equating the unphysicalss pseudoscalar mass to 688.5 MeV [80,81]. D. Hadron interpolators: For mesons, we utilize the local meson (bΓc) interpolators, where Γ, corresponding to different spin(J) and parity (P ) quantum numbers, J P , are : γ 5 (0 − ), γ i (1 − ), I(0 + ) and γ 5 γ i (1 + ). We work with the assumption that the extracted ground state with γ 5 γ i is 1 + and is unaffected by a possible nearby 2 + level [57]. For baryons, we utilize the conventional interpolators given by P + [(q T 1 CΓq 2 )q 3 ] (discussed in detail in Refs. [64,65,83]). The ordering of quark flavours (q i ) in these interpolators are the same as mentioned by the quark content inside the bracket corresponding to each baryon in Table I.
With this set up we calculate bc hadron masses of Table  I and below we elaborate our results. Results: To cancel out bare quark mass term which enters additively into the NRQCD Hamiltonian we calculate the mass differences between energy levels, rather than masses directly. To obtain the mass of a hadron (M c H ) we first calculate subtracted masses on the lattice as where 1S b and 1S c are the lattice calculated spin average 1S bottomonia and charmonia masses respectively, whereas n b and n c are the number of b and c valence quarks in the hadron. At each lattice spacing we calculate this subtracted mass and then perform the continuum extrapolation to get its continuum value ∆M c H . Finally the physical result is obtained by adding the physical values of spin average masses to ∆M c H as Since the B c (0 − ) mass is known experimentally we also utilize a dimensionless ratio, which is then extrapolated to the continuum limit (R c H ) and the final hadron mass is obtained from These procedures of utilizing dimensionless ratios as well as mass differences for the continuum extrapolations substantially reduce the systematic errors arising from mass tuning as well as for the terms which enter masses additively. We use both equations (2) and (4) and found consistent results and added the difference in systematics. Below we discuss results for bc mesons and baryons.
Mesons: In Figure 1 we plot the subtracted mass (∆M H ), as defined in Eq. (1), for B c (0 − ) as a function of the lattice spacings (a). Blue circles represent unimproved and red squares represent improved results. We extrapolate unimproved results using fit forms as well as C f = A + a 3 B. Two bands corresponds to one sigma error for these fittings (purple : Q f , green : C f ). The continuum extrapolated result and the experimental value are shown by red and blue stars respectively. As expected the improved results are closer to the continuum limit result than that of unimproved ones. With only two improved points no continuum extrapolation is performed of their own but we fit these together with the unimproved point of the fine lattice and that aids in determining the central value and the extent of discretization error. We also draw a horizontal cyan band, to show the proximity of the improved results from the continuum result. To see the consistency in fits we also use a constrained fit with both forms together by loosely constraining A values from previous fits and any difference in fitting is included in discretization error. As in Figure 1, throughout this paper we follow the same conventions for symbols and color coding. In Figure 2 (top), we plot the hyperfine splitting of 1S B c mesons. After the continuum extrapolation we obtain (3) MeV which is consistent with previous lattice calculations [57,63]. In the bottom figure we show the subtracted ratios, as defined in Eq.(3), and their continuum extrapolation for the ground states of 1 − , 1 + and 0 + B c mesons. Taking the experimental values for B c (0 − ) and 1S quarkonia [35] masses, we obtain the ground state masses for these mesons and tabulated those in the top part of Table II. Baryons: Here we first discuss the Ξ cb baryons. Presence of a valence light quark in Ξ cb demands a chiral extrapolation first. Use of multimass algorithm allows to simulate a range of pion masses and for 24 3 × 64 lattice we even calculate quark propagators close to the physical one. In Figure 3 (top), for this lattice we plot Ξ cb masses at various pion masses which clearly show a quadratic variation starting from the physical pion mass to ∼ 600 MeV. We thus use a chiral extrapolation of the form A + m 2 π B. Within the limit of acceptable χ 2 /dof, variations in chiral extrapolation forms, as in Ref. [64], do not change the final value. The same procedure is repeated for Ξ cb and Ξ * cb at three lattices. These chiral extrapolated values are then used to calculate the subtracted masses and are plotted in the bottom part of Figure 3. These subtracted masses are then extrapolated to the continuum limit to get final values for the ground state masses of these baryons and are tabulated in Table  II Table II. Error estimation: Below we address the estimation of various errors related to this work. Statistical : The use of wall source reduces the statistical errors substantially and facilitates wide and stable fit ranges even for baryons. We find that the statistical error is always below percent level and is maximum for the Ξ cb baryons which is about 0.4%. Discretization: Adaptation of overlap fermions ensures no O(ma) error for light to charm quarks. The value of ma for charm quarks (0.528, 0.427 and 0.290 on three lattices) are rather small compared to unity and hence implies smaller error from higher orders in ma. The utilization of energy splittings and ratios, for the con- ? Ξ cb (cbu)(1/2 + ) 6966(23) (14) ? Ξ * cb (cbu)(3/2 + ) 6989 (24)(14) ? Ω cb (cbs)(1/2 + ) 6994 (15) (13) ? Ω cb (cbs)(1/2 + ) 7045(16) (13) ? Ω * cb (cbs)(3/2 + ) 7056 (17)(13) ? Ω ccb (1/2 + ) 8005 (6)(11) ?
known to be quite susceptible to this error and an excellent agreement between our and experimental values assures good control over discretization and hence a reliable estimation of masses of other heavy hadrons. Different fitting methods, quadratic, cubic in lattice spacing as well as both together in constrained fits, help to find possible discretization effects in continuum extrapolations. The largest discretization error is found to be for Ξ cb baryons which is about 6-7 MeV. Scale setting: We independently calculate lattice spacings from Ω sss baryon mass and obtained those as 0.1192 (14), 0.0877(10) and 0.0582(5) fm, which are consistent with the values measured by MILC collaboration [66]. With these estimates of lattice spacings, the largest error in mass splittings due this scale uncertainty are within 6 MeV. Finite size: The lattice volume in this study is about 3f m and we believe it is large enough for all the hadrons studied here. These hadrons are quite heavy and stable to strong decays, and hence the finite volume effects are expected to be within a few MeV as estimated in Ref. [64] on similar lattice volume. Chiral extrapolation: In this study only Ξ cb baryons are subjected to this error. Due to the use of multimass algorithm we could calculate these baryons at a large number of pion masses, as shown in Figure 3, which help to per-form extrapolations to the physical limit in a controlled and reliable way. Our results are found to be quite robust with respect to different chiral extrapolation forms. NRQCD errors: Since we have included terms up to α s v 4 , higher order terms, such as spin dependent as well as spin independent terms (α 2 s v 4 and α s v 6 ) will contribute to the systematics. For bc mesons, these errors are 4 MeV as estimated in Ref. [57] on similar lattices. As in Ref. [64], we also estimate these errors to be 5, 5 and 6 MeV for bcq, bcc and bbc baryons, respectively. Other errors : Errors due to quark mass tuning are expected to be negligible in these results, considering the precision and rigor that enter into heavy quark mass tuning procedure. In a previous study we also found that the mixed action effects, which would vanish at the continuum limit, to be small [84] within this lattice set up. As discussed in Ref. [57,82,85] for similar lattices, the effect due to unphysical sea quark masses could be less than a percent level. Other errors due to electromagnetism, isospin breaking and the absence of dynamical bottom quarks are expected to be within 2-4 MeV [57].
Finally we summarize our results in Table II. These are also plotted in Figure 5.

11.
12. Conclusions: In this Letter, we present precise predictions of the ground state masses of bc hadrons using lattice QCD simulations with very good control over systematics. These hadrons have not been discovered yet and considering the recent interests on them, particularly for their relevance to the physics beyond the Standard Model, these predictions provide important information for their future discovery. We have used three sets of gauge configurations at three different lattice spacings, finest one being 0.0582 fermi, which help us to obtain precise results at the continuum limit. The overlap fermions, which have exact chiral symmetry at finite lattice spacings and no O(ma) errors, are used for the light, strange as well as for the charm quarks. For the bottom quark, we use a non-relativistic formulation with non-perturbatively tuned coefficients with terms up to O(α s v 4 ). Utilization of a wall source helps to keep the statistical error below percent level. Use of mass differences as well as ratios, in which the extent of discretization effects are significantly lesser for the continuum extrapolation, enables us to predict the masses of these states more precisely. We have also addressed other possible systematic errors in detail, which when added in quadrature are found to be smaller than the statistical error in most cases. Our final results for the ground state masses of all bc hadrons are tabulated in Table II and also showed in Figure 5.