PHYSICAL REVIEW D 101, 094503 (2020) Heavy baryon spectrum on the lattice with NRQCD bottom and HISQ lighter quarks Protick Mohanta and Subhasish Basak School of Physical Sciences, National Institute of Science Education and Research, Homi Bhabha National Institute, Odisha 752050, India (Received 9 November 2019; accepted 21 April 2020; published 12 May 2020) We determine the mass spectra of heavy baryons containing one or more bottom quarks along with their hyperfine splittings and various mass differences on MILC 2 þ 1 Asqtad lattices at three different lattice spacings. NRQCD action is used for bottom quarks whereas relativistic HISQ action for the lighter up/ down, strange, and charm quarks. We consider all possible combinations of bottom and lighter quarks to construct the bottom baryon operators for the states JP ¼ 1=2þ and 3=2þ. DOI: 10.1103/PhysRevD.101.094503 I. INTRODUCTION the above studies employed different heavy quark actions for charm quark, HISQ action [14] is becoming an Lattice QCD has been extensively employed to study B increasingly popular choice for the charm quark. This physics phenomenology, especially the decay constants and approach of simulating bottom quark with NRQCD and the mixing parameters needed for CKM matrix elements and rest of the quarks i.e., charm, strange, and up/down with the mass differences in the meson sector [1]. The B mesons HISQ for calculation of bottom baryon spectra has been spectroscopy and mass splittings have undergone thorough adopted in this work. investigations on lattice, see [1] and references therein, with In this paper, we present our lattice QCD results of heavy an increasing impact on heavy flavor phenomenology; see, baryons involving one and two bottom quarks. We consider for instance, [2]. However, studying heavy baryons with all possible combinations of bottom quark(s) with charm, bottom quark(s) on lattice is relatively a recent pursuit. strange, and up/down lighter quarks of the form lbb , llb , Some of the early studies of heavy baryons on lattice can be ð Þ ð Þ and ðl1l2bÞ, where b is the bottom quark and l are the lighter found in [3–7]. Of late, a slew of low lying JP ¼ 1=2þ charm, strange, and up/down quarks. We are addressing the bottom baryons, such as Λ , Σ , Ξ0 , and Ω , have made b b b b charm quark as “light” quark in the sense that we have used entries in Particle Data Group (PDG) [8]. The possibilities relativistic action for it. The action for the lighter quarks is of the discoveries of JP 3=2þ are rather high, whereas ¼ HISQ [14] and NRQCD [12,13] for the bottom. We discuss doubly and triply bottom baryons are right now beyond the these actions in Sec. II. The propagators generated using reaches of present experiments. In this state, lattice QCD nonrelativistic and relativistic actions are required to be can provide an insight into the masses, mass splittings, and combined to construct baryon states of appropriate quantum other properties of such bottom baryons from the first numbers. A discussion to achieve this combination is spread principle. To this end, quite a few lattice investigations of over both Secs. II and III. The bottom baryon operators are heavy baryons containing one, two, or three bottom quarks described in details in Sec. III. In the following Sec. IV,we have been undertaken using a range of light quark actions present the simulation details including the lattice ensembles [9–11]. For an extensive list on contemporary lattice used, various parameters, and tuning of different quark literature on heavy baryons, see [11]. masses. The lattice calculations are carried out at three These studies on heavy hadrons with bottom quark(s) are different lattice spacings with fixed m =m value and largely made possible by the use of nonrelativistic QCD u=d s action, proposed and formulated in [12,13], because of the several quark masses. We assemble our bottom baryon well-known fact that the current lattice spacings, even for as spectrum results along with hyperfine and various other low as 0.045 fm, render am ≳ 1. Although almost all of mass splittings in the Sec. V. Finally, we conclude and b summarize in Sec. VI, which also includes a comparison of our results to the existing ones. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. II. QUARK ACTIONS Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, As of now, the bottom (b) quark masses are not small, 3 and DOI. Funded by SCOAP . i.e., amb≮1 in units of the lattice spacings available. 2470-0010=2020=101(9)=094503(15) 094503-1 Published by the American Physical Society PROTICK MOHANTA and SUBHASISH BASAK PHYS. REV. D 101, 094503 (2020) Δ2 2 The use of improved NRQCD is the action of choice for ð1Þ ð Þ 6 δH −c1 the b quarks. We have used Oðv Þ NRQCD action in ¼ 8 3 mb this paper. The charm (c) quark is also similarly heavy ð2Þ ig ⃗Æ ⃗ ⃗ ⃗Æ enough for existing lattices, but the Fermilab proposal δH c2 Δ E − E Δ ¼ 8 2 ð · · Þ [15] made it possible to work with relativistic actions mb g ˜ Æ ˜ ˜ ˜ Æ for the c quark, provided we trade the pole mass with δ ð3Þ − σ⃗ Δ⃗ ⃗− ⃗ Δ⃗ H ¼ c3 8 2 · ð × E E × Þ the kinetic mass. Subsequently, HISQ action [14] mb became available for relativistic c quark. In this paper, g ˜ δ ð4Þ − σ⃗ ⃗ we choose HISQ action for the c quark along with s and H ¼ c4 · B 2mb u=d quarks. In this work, as because we use the same g δ ð5Þ − Δ2 σ⃗ ⃗ relativistic HISQ action for all quarks except the bottom, H ¼ c5 3 f ; · Bg 8m we use the word light quarks to refer to c, s,andu=d b 3 ð6Þ g 2 ⃗Æ ⃗ ⃗ ⃗Æ quarks. This is similar to what has been done in [16] for δH −c6 Δ ; σ⃗ Δ E − E Δ ¼ 64 4 f · ð × × Þg the B meson states calculation. Besides, one of the big mb advantages of this choice of action is the ability to use 2 ð7Þ ig ⃗ ⃗ δH −c7 σ⃗ E E: the MILC code [17] for this bit. ¼ 8 3 · × ð3Þ mb A. NRQCD action and b quark The b quark propagator is generated by the time evolution In order to perform a lattice QCD computation of of this Hamiltonian, hadrons containing bottom quarks in publicly available relatively coarse lattices, NRQCD [12,13] is perhaps the n aH0 aδH † most suitable and widely used quark action for the bottom. Gðx;⃗ t þ 1; 0; 0Þ¼ 1 − 1 − U4ðx;⃗ tÞ 2n 2 As is understood, the typical velocity of a b quark inside a hadron is nonrelativistic. Comparison of masses of botto- aδH aH0 n × 1 − 1 − Gðx;⃗ t;0; 0Þ monium states to the mass of b quark supports the fact that 2 2n the velocity of b quark inside hadron (v2 ∼ 0.1) is much smaller than the bottom mass. For example Mϒ ¼ ð4Þ 9460 MeV whereas 2 × mb ¼ 8360 MeV in the MS scheme. For bottom hadrons containing lighter valence with quarks, the velocity of the bottom quark is even smaller. This allows us to study the b quark with a nonrelativistic 0 for t<0 effective field theory. NRQCD will remain the action of Gðx;⃗ t;0; 0Þ¼ : δx;⃗0 for t ¼ 0 choice for the b quark until finer lattices with amb < 1 become widely available. In NRQCD, the upper and lower components of the The tree level value of all the coefficients c1, c2, c3, c4, c5, Dirac spinor decouple, and the b quark is described by a c6, and c7 is 1. Here, n is the factor introduced to ensure 3 2 two component spinor field, denoted by ψ h. The NRQCD numerical stability at small amb [13], where n> = mb. Lagrangian has the following form: The symmetric derivative ΔÆ and Laplacian Δ2 in terms of forward and backward covariant derivatives are L ψ † ⃗ ψ ⃗ 1 − ψ ⃗ ψ ⃗ ¼ hðx; tÞ½U4ðxÞ hðx; t þ Þ hðx; tÞþaH hðx; tÞ; þ ð1Þ aΔμ ψðxÞ¼UμðxÞψðx þ aμˆÞ − ψðxÞ − † aΔμ ψðxÞ¼ψðxÞ − Uμðx − aμˆÞψðx − aμˆÞ where a is the lattice spacing and U4ðxÞ is the temporal δ 1 gauge link operator. H ¼ H0 þ H is the NRQCD ΔÆ ¼ ðΔþ þ Δ−Þ Hamiltonian, where X2 X Δ2 ¼ ΔþΔ− ¼ Δ−Δþ: ð5Þ Δ˜ 2 Δ2 2 X i i i i a ð Þ ðiÞ i i H0 − − δH δH : ¼ 2 4 4 2 and ¼ ð2Þ mb n mb i By Taylor expanding the symmetric derivative and the 2 4 6 The H0 is the leading Oðv Þ term; the Oðv Þ and Oðv Þ Laplacian operator, we can find their forms corrected up to 4 terms are in δH with coefficients c1 through c7, Oða Þ [12] that are used in the above Eq. (3), 094503-2 HEAVY BARYON SPECTRUM ON THE LATTICE WITH NRQCD … PHYS. REV. D 101, 094503 (2020) a2 Δ˜ Æ ¼ ΔÆ − ΔþΔÆΔ− Since HISQ action is diagonal in spin space, propagators i i 6 i i i obtained do not have any spin structure. The full 4 × 4 spin a2 X structure can be regained by multiplying the propagators by Δ˜ 2 Δ2 − ΔþΔ− 2 ¼ 12 ½ i i : ð6Þ the Kawamoto-Smit multiplicative phase factor [18], i 4 4 Y In the same way, the gauge fields are improved to Oða Þ xμ x1 x2 x3 x4 ΩðxÞ¼ ðγμÞ ¼ γ1 γ2 γ3 γ4 : ð12Þ using cloverleaf plaquette, μ¼1 þ † γ aΔρ FμνðxÞ¼UρðxÞFμνðx þ aρˆÞUρðxÞ − FμνðxÞ MILC library uses a different representation of matrices than the ones used in NRQCD.