Heavy Baryon Spectrum on the Lattice with NRQCD Bottom and HISQ Lighter Quarks

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 ⃗ ⃗ ⃗ ⃗ 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 ˜ ˜ ˜ ˜ 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 ⃗ ⃗ ⃗ ⃗ 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 Δ 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 Δ ¼ ðΔþ þ Δ−Þ Hamiltonian, where 2X 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˜ ¼ Δ − ΔþΔΔ− 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. However, γ matrices of Δ− − † − ρˆ − ρˆ a ρ FμνðxÞ¼FμνðxÞ Uρðx a ÞFμνðxÞUρðx a Þ these two representations are related by the unitary trans- a4 formation of the form, ˜ − ΔþΔ− ΔþΔ− gFμνðxÞ¼gFμνðxÞ 6 ½ μ μ þ ν ν gFμνðxÞ: ð7Þ 1 σ σ γMILC † γNR ffiffiffi y y ˜ ˜ S μ S ¼ μ where ;S¼ p : ð13Þ The chromoelectric E and chromomagnetic B fields in 2 −σy σy δHð3Þ and δHð4Þ of Eq. (3) are thus Oða4Þ improved.

B. HISQ charm and lighter quarks III. TWO-POINT FUNCTIONS For the lighter quarks—charm, strange, and up/down— In this section, we discuss the construction of the bottom relativistic HISQ action [14] is used. Apart from anything baryons by combining spin and color indices of the else, from a practical point of coding the bottom-light appropriate quark fields to form necessary baryon operators operators ðlbb; llb; l1l2bÞ, using the same relativistic action and two-point functions. The b quark field is universally for all lighter quarks offers a great degree of simplification. represented with Q throughout the paper. It is defined later The HISQ action is given by in the Eq. (17). X ¯ μ HISQ S l x γ Dμ m l x ; 8 ¼ ð Þð þ Þ ð Þ ð Þ A. Bottom-bottom meson two-point function x After the b quark propagators are generated according to where Eq. (4), we calculate the masses of bottomonium states 2 from the exponential falloff of two-point functions, i.e., HISQ Δ − a 1 ϵ Δ3 Dμ ¼ μðWÞ 6 ð þ Þ μðXÞð9Þ correlators of the state with quantum numbers of interest. The meson creation operators are constructed from two HISQ † μ U ψ with WμðxÞ¼F UμðxÞ and XμðxÞ¼ FμUμðxÞ. The component quark and antiquark creation operators h and HISQ χ† 3¯ Fμ has the form, h [19,20]. As antiquarks transform as under color χ ≡ χ X 2 2 rotation, we rename the antiquark spinor as h h a δρ HISQ − ð Þ U [13]. The meson creation operator is, thus, Fμ ¼ Fμ 2 Fμ: ð10Þ ρ≠μ O ψ † Γχ hhðxÞ¼ hðxÞ hðxÞ: ð14Þ Here, the U is the unitarizing operator, it unitarizes what- Heavy-heavy, i.e., bottom-bottom meson two-point func- ever it acts on, and the smearing operator Fμ is given by tion is then given by [13,21] Y 2 ð2Þ X a δρ † 1 C ðp;⃗ tÞ¼ hO ðxÞO ð0Þi Fμ ¼ þ 4 : ð11Þ hh hh hh ρ≠μ Xx⃗ ip⃗·x⃗ † 0 Γ† 0 Γ ð2Þ ¼ e Tr½G ðx; Þ sinkGðx; Þ src; ð15Þ The δρ and δρ in the Eqs. (10) and (11) are covariant first x⃗ and second order derivatives. Because HISQ action reduces 2 Γ Γ σ Oðαsa Þ discretization errors found in Asqtad action, it is sink ¼ src ¼ I and i for the pseudoscalar and vector well suited for s and u=d quarks. The parameter ϵ in the mesons, respectively. The heavy-heavy propagator Gðx; 0Þ coefficient of the Naik term can be appropriately tuned to is a 2 × 2 matrix in spin space. If we think of Gðx; 0Þ as a use the action for c quarks. For s and u=d quarks, the ϵ ¼ 0. 4 × 4 matrix with vanishing lower components then we can Later, in the Table IV, we listed the parameters rewrite the above Eq. (15) as [9] used for HISQ quarks. We have taken the values of ϵ X ⃗ ip⃗·x⃗ γ † 0 γ Γ† 0 Γ from [16] and used MILC subroutines for generating HISQ Chhðp; tÞ¼ e Tr½ 5G ðx; Þ 5 sinkGðx; Þ src; ð16Þ propagators. x⃗

094503-3 PROTICK MOHANTA and SUBHASISH BASAK PHYS. REV. D 101, 094503 (2020)

Γ Γ Γ γ γ γ γ α where matrices now changed to sink ¼ src ¼ 5 and i where C ¼ 4 2. Here, a, b, c are the color indices, is the for pseudoscalar and vector mesons, respectively. In spinor index, and k is the Lorentz index, which runs from 1 Eq. (16), we have used the nonrelativistic Dirac represen- to 3. The zero momentum two-point function reads [9] tation of γ matrices. In the Eqs. (15), (16), and (19), the X trace is taken over both the spin and color indices. hhh Ohhh Ohhh 0 † Cjk;αδðtÞ¼ h½ j ðxÞα½ k ð Þδi Xx⃗ B. Heavy-light meson two-point function ch ¼ ϵabcϵfghGαδðx; 0Þ As discussed above, b quark field ψ h has only two spin x⃗ components. We convert it to a four-component spinor bg afT ×Tr½Cγ G ðx; 0Þγ γ2G ðx; 0Þ: ð21Þ having vanishing lower components, j k ψ h In the above Eq. (21) and the subsequent ones, the Q ¼ : ð17Þ 0 transpose and traces are taken over spin indices. Baryon operators having Cγk in the diquark component have 3 1 This helps us to combine the b and light quark fields in the overlap with both spin-2 and 2 states. For example, the usual way, correlator defined in Eq. (21) can be written explicitly as an 3 1 ¯ overlap with both spin-2 and 2 states [23], OhlðxÞ¼QðxÞΓlðxÞ; ð18Þ † 2 − 3=2 2 − 1=2 ¯ γ hhh E3=2tΠ E1=2tΠ where lðxÞ stands for the light quark fields, Q ¼ Q 4, and Cij ðtÞ¼Z3=2e Pij þ Z1=2e Pij ; ð22Þ depending on pseudoscalar and vector mesons Γ ¼ γ5 and γ i, respectively. Note that in the Dirac, i.e., the NR where Π ¼ð1 þ γ4Þ=2 and the spin projection operators representation of γ matrices γ4Q ¼ Q. The zero momentum 3=2 1=2 P ¼ δ − γ γ =3 and P ¼ γ γ =3. The individual bottom-light two-point function becomes [10,22] ij ij i j ij i j X contribution to the respective spin states can be obtained O† O 0 by taking appropriate projections, ChlðtÞ¼ h hlðxÞ hlð Þi x⃗ X 3=2 2 − 3=2 † † hhh Π E3=2t γ 0 γ Γ 0 Γ Pij Cjk ¼ Z3 2 e P ¼ Tr½ 5M ðx; Þ 5 sinkGðx; Þ src; ð19Þ = ik ⃗ 1=2 2 − 1=2 x hhh Π E1=2t Pij Cjk ¼ Z1=2 e Pik : ð23Þ where Mðx; 0Þ is the light quark propagator. It has the usual full 4 × 4 spin structure. As before, Gðx; 0Þ is the b quark In this paper, we use these projections to separate the propagator having vanishing lower components. However, different spin states. We would like to point out that the G x; 0 1 before implementing Eq. (19), ð Þ has to be rotated to spin-2 state of triply bottom baryon is not a physical state as the MILC basis. it violates Pauli exclusion principle even though we can take the projection in practice. C. Bottom baryon two-point functions The bottom quark field Q has vanishing lower compo- 2. Bottom-bottom-light baryon nents and hence, can be projected to positive parity states Interpolating operator for baryons, having two b quarks only. Besides, the use of Γ ¼ Cγ5 in a diquark operator T and a light quark, can be constructed in two ways based on made from same flavor, i.e., l Cγ5l is not allowed by the Pauli exclusion principle. In other words, the insertion of how the diquark component is formed [24], Cγ5 between two quark fields of the same flavor creates a Ohhl ϵ aT γ b c combination which is antisymmetric in spin indices, while ð k Þα ¼ abcðQ C kQ Þlα ð24Þ the presence of ϵabc makes the combination antisymmetric in color indices. This makes the overall operator become Ohlh ϵ aT γ b c ð k Þα ¼ abcðQ C kl ÞQα: ð25Þ symmetric under the interchange of the same flavored quark fields. Keeping this in mind, the constructions The corresponding baryon correlators are of various bottom baryon two-point functions are described X below. hhl Ohhl Ohhl 0 † Cjk;αδðtÞ¼ h½ j ðxÞα½ k ð Þδi Xx⃗ 1. Triply bottom baryon ch ¼ ϵabcϵfgh½M ðx; 0Þγ4αδ A triply bottom baryon operator is defined by x⃗ γ γ γ bg 0 γ γ afT 0 Ohhh ϵ aT γ b c ×Tr½ 4 2 jG ðx; Þ k 2G ðx; Þ ð26Þ ð k Þα ¼ abcðQ C kQ ÞQα; ð20Þ

094503-4 HEAVY BARYON SPECTRUM ON THE LATTICE WITH NRQCD … PHYS. REV. D 101, 094503 (2020) X hlh Ohlh Ohlh 0 † Ohl1l2 ϵ aT γ b c Cjk;αδðtÞ¼ h½ j ðxÞα½ k ð Þδi ð k Þα ¼ abcðl1 C kl2ÞQα; ð29Þ Xx⃗ and the corresponding two-point function is ϵ ϵ Gch x; 0 ¼ abc fgh αδð Þ X x⃗ hl1l2 hl1l2 hl1l2 † C ;αδðtÞ¼ h½O ðxÞα½O ð0Þδi bg afT jk j k ×Tr½γ4γ2γ M ðx; 0Þγ γ2G ðx; 0Þ: ð27Þ ⃗ j k Xx ¼ ϵ ϵ Gchðx; 0Þ The propagator Gðx; 0Þ is required to be converted to the abc fgh αδ ⃗ MILC basis using the unitary matrix S defined in Eq. (13). x 1 hlh γ γ γ bg 0 γ γ γ afT 0 An additional spin-2 operator can be defined for the O ×Tr½ 4 2 jM2 ðx; Þ k 2 4M1 ðx; Þ: ð30Þ type operator as In HISQ formalism for light quarks, the corresponding hlh aT b c ðO5 Þα ¼ ϵabcðQ Cγ5l ÞQα: ð28Þ propagators M1ðx; 0Þ and M2ðx; 0Þ have the same Kawamoto-Smit multiplicative factor ΩðxÞ. Hence, they The two-point function for this operator is obtained by have the same spin structure irrespective of color indices. γ γ γ γ replacing j and k by 5 in Eq. (27). We cannot have a C 5 As a result, the trace over spin indices in Eq. (30) vanishes Ohhl 3 between two Q in diquark and hence, no 5 . if γj ≠ γk. Therefore, we can not separate the two spin-2 and In Table I, we tabulated the full list of triple and double 1 2 states. If we want to use the same diquark structure as in bottom baryon operators that are used in this work. We Eq. (29), for baryons having different light quark flavors, have broadly followed the nomenclature adopted in [11] we can define the spin-1 operator by but with certain modifications as needed for this work. The 2 P baryons having the same quark content and J are obtained hl1l2 aT b c ðO Þα ¼ ϵ ðl Cγ5l ÞQα; ð31Þ in two different ways, as mentioned above. The operators 5 abc 1 2 “ ” Ω˜ 1 2þ with tilde , for instance, bbð = Þ, are obtained by and the corresponding two-point function is γ 1=2 projecting the relevant ðQC klÞQ operator with Pij . The X hl1l2 hl1l2 hl1l2 † “ ” Ω0 1 2þ C ðtÞ¼ h½O ðxÞα½O ð0Þ i operators with prime , such as bbð = Þ, are obtained 55;αδ 5 5 δ ⃗ from ðQCγ5lÞQ diquark construction. The “prime” states Xx “ ” ch so constructed on lattice correspond to the prime con- ¼ ϵabcϵfghGαδðx; 0Þ Ξ0 Ω0 tinuum states, such as b or till unobserved cb etc. It is x⃗ obvious that baryon states calculated by projecting out bg afT ×Tr½γ4γ2γ5M ðx; 0Þγ5γ2γ4M ðx; 0Þ: ð32Þ definite spin states from a two-point function share the 2 1 same interpolating operator. The starred baryons are for Besides, instead of Eq. (29), we can choose our (hl1l2) P þ J ¼ 3=2 states. operator as

Ohl2l1 ϵ aT γ b c 3. Bottom-light-light baryon ð k Þα ¼ abcðQ C kl2Þl1α: ð33Þ The natural choice for an interpolating operator, as The two-point function is now motivated by the Heavy Quark Effective Theory (HQET) X [25],inllh-baryon kind is hl2l1 ϵ ϵ ch 0 γ Cjk;αδðtÞ¼ abc fgh½M1 ðx; Þ 4αδ x⃗ bg afT TABLE I. Operators for triple and double bottom baryons. Q is ×Tr½γ4γ2γjM2 ðx; 0Þγkγ2G ðx; 0Þ: ð34Þ used for b field and l for any of the c, s, u=d lighter quarks. Because the light quark propagators M1ðx; 0Þ and M2ðx; 0Þ Baryon Quark content JP Operator are proportional to each other, the relative positions Ω 3þ 1þ aT b c bbb bbb 2 ; 2 ϵabcðQ CγkQ ÞQ of the quark fields l1 and l2 in Eq. (33) is irrelevant. Ω⋆ Ω 3þ 1þ ϵ aT γ b c cbb; cbb cbb 2 ; 2 abcðQ C kQ Þl The interpolating operator defined in Eq. (33) has overlap Ω˜ ⋆ Ω˜ 3þ 1þ ϵ aT γ b c 3 1 cbb; cbb cbb 2 ; 2 abcðQ C kl ÞQ with both spin-2 and 2 states and can be projected out by Ω0 1þ ϵ aT γ b c 1=2;3=2 cbb cbb 2 abcðQ C 5l ÞQ appropriate projection operators Pij . Ω⋆ Ω 3þ 1þ ϵ aT γ b c bb; bb sbb 2 ; 2 abcðQ C kQ Þl 1 As before, we can also define an additional spin-2 Ω˜ ⋆ Ω˜ sbb 3þ 1þ ϵ aT γ b c bb; bb 2 ; 2 abcðQ C kl ÞQ operator here too, Ω0 1þ ϵ aT γ b c bb sbb 2 abcðQ C 5l ÞQ Ξ⋆ Ξ 3þ 1þ ϵ aT γ b c hl2l1 aT b c bb; bb ubb 2 ; 2 abcðQ C kQ Þl ðO5 Þα ¼ ϵabcðQ Cγ5l2Þl1α: ð35Þ Ξ˜ ⋆ Ξ˜ 3þ 1þ ϵ aT γ b c bb; bb ubb 2 ; 2 abcðQ C kl ÞQ Ξ0 1þ ϵ aT γ b c The two-point function for this operator has the same form bb ubb 2 abcðQ C 5l ÞQ as in Eq. (34) with γj and γk replaced by γ5.

094503-5 PROTICK MOHANTA and SUBHASISH BASAK PHYS. REV. D 101, 094503 (2020)

TABLE II. Operators for single bottom baryons. Q is used for b IV. SIMULATION DETAILS field as before. Interchange in the position of two lighter quarks keeps the operator unchanged. We calculated the bottom baryon spectra using the publicly available Nf ¼ 2 þ 1 Asqtad gauge configura- Baryon Quark content JP Operator tions generated by the MILC Collaboration. Details about Ω˜ ⋆ Ω˜ 3þ 1þ ϵ aT γ b c these lattices can be found in [26]. It uses Symanzik- ccb; ccb ccb 2 ; 2 abcðQ C kc Þc Ω0 1þ ϵ aT γ b c improved Lüscher-Weisz action for the gluons and Asqtad ccb ccb 2 abcðQ C 5c Þc Ω scb 1þ ϵ aT γ b c action [27,28] for the sea quarks. The lattices we choose cb 2 abcðs C 5c ÞQ 1 5 Ω˜ ⋆ Ω˜ 3þ 1þ ϵ aT γ b c have a fixed ratio of aml=ams ¼ = and lattice spacings cb; cb scb 2 ; 2 abcðQ C kc Þs 0 1þ aT b c ranging from 0.15 to 0.09 fm corresponding to the same Ω scb ϵ Q Cγ5c s cb 2 abcð Þ physical volume. We have not determined the lattice Ξ ucb 1þ ϵ aT γ b c cb 2 abcðu C 5c ÞQ spacings independently but use those given in [26].In Ξ˜ ⋆ Ξ˜ ucb 3þ; 1þ ϵ QaTCγ cb uc cb; cb 2 2 abcð k Þ Table III, we listed the ensembles used in this work. Ξ0 ucb 1þ ϵ aT γ b c cb 2 abcðQ C 5c Þu In NRQCD, the rest mass term does not appear in Ω˜ ⋆ Ω˜ ssb 3þ 1þ ϵ aT γ b c b; b 2 ; 2 abcðQ C ks Þs Eq. (3), and therefore, we cannot determine hadron masses Ω0 1þ ϵ aT γ b c b ssb 2 abcðQ C 5s Þs from their energies at zero momentum directly from the Ξ 1þ ϵ aT γ b c b usb 2 abcðu C 5s ÞQ exponential falloff of the correlation functions. Instead, we Ξ˜ ⋆ Ξ˜ usb 3þ 1þ ϵ aT γ b c b; b 2 ; 2 abcðQ C ks Þu calculate the kinetic mass Mk of heavy-heavy mesons from Ξ0 1þ ϵ aT γ b c 2 b usb 2 abcðQ C 5s Þu its energy-momentum relation, which to Oðp Þ is [15], Σ˜ ⋆ Σ˜ 3þ 1þ ϵ aT γ b c b; b uub 2 ; 2 abcðQ C ku Þu qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1þ aT b c 2 2 Σ uub ϵ Q Cγ5u u 0 − b 2 abcð Þ EðpÞ¼Eð Þþ p þ Mk Mk 1þ aT b c Λb udb ϵ ðu Cγ5d ÞQ 2 abc Eð0Þ ⇒ EðpÞ2 ¼ Eð0Þ2 þ p2: ð39Þ Mk In Table II, we tabulate our full list of single bottom We calculate the EðpÞ at different values of lattice momenta 2 π 0 0 0 1 0 0 1 1 0 1 1 1 baryon operators that we made use of in this work. The p ¼ n =L, where, n ¼ð ; ; Þ;ð ; ; Þ;ð ; ; Þ;ð ; ; Þ; 2 0 0 2 1 0 “tilde” and “prime” states that appear in the table have been ð ; ; Þ;ð ; ; Þ, and (2,1,1). explained before in the context of multi bottom baryon operators. A. mb tuning The b quark mass is tuned from the spin average ϒ and η 4. Light baryon b masses, We occasionally need charmed baryon states like qcc 3 Mϒ þ Mηb or qqc, where q is any of s or u=d quarks or both; hence, Mbb¯ ¼ 4 ; ð40Þ we include a discussion on charmed baryon operators. ϒ η The c quark in the present case is relativistic. For the using kinetic mass for both and b. The experimental reason discussed above, with HISQ action for lighter value to which Mbb¯ is tuned to is not 9443 MeV, as ϒ η quarks we can define only the spin-1 operators. Consider obtained from spin averaging (9460 MeV) and b 2 (9391 MeV) experimental masses, but to an appropriately a ðl1l2l3Þ-baryon where at least two quarks are differently 1 adjusted value of 9450 MeV [16], which we denote as flavored, say l1 ≠ l2. The spin- operator and the corre- 2 Mmod later in the Eq. (41). The reasons being, firstly, sponding two-point function in such case is phys electromagnetic interaction among the quarks are not considered here. Secondly, the disconnected diagrams while Ol1l2l3 ϵ aT γ b c ð 5 Þα ¼ abcðl1 C 5l2Þl3α ð36Þ computing a two-point function are also not considered ¯ X thus, not allowing b, b quarks to annihilate to gluons. And l1l2l3 ϵ ϵ ch 0 γ C55;αδðtÞ¼ abc fgh½M3 ðx; Þ 4αδ x⃗ TABLE III. MILC configurations used in this work. The gauge β bg afT coupling is , lattice spacing a, u=d and s sea quark masses are ml ×Tr½γ4γ2γ5M2 ðx; 0Þγ5γ2γ4M 1 ðx; 0Þ: ð37Þ 3 and ms, respectively, and lattice size is L × T. The Ncfg is the number of configurations used in this work. The two light baryon states ðJP ¼ 1=2þÞ that we are 2 3 interested in this work are β ¼ 10=g a ðfmÞ aml ams L × T Ncfg 6.572 0.15 0.0097 0.0484 163 × 48 400 Σ ∶ ϵ aT γ b c 3 cðuucÞ abcðc C 5u Þu 6.76 0.12 0.01 0.05 20 × 64 400 3 aT b c 7.09 0.09 0.0062 0.031 28 × 96 300 ΞccðuccÞ∶ ϵabcðc Cγ5u Þc : ð38Þ

094503-6 HEAVY BARYON SPECTRUM ON THE LATTICE WITH NRQCD … PHYS. REV. D 101, 094503 (2020)

TABLE IV. Tuned b, c, and s quark bare masses for lattices TABLE V. D and B meson masses in MeV with the tuned amb, used in this work. For the s-quark mass, we mentioned the amc, and ams. particle states to which it is tuned to. The values of ϵ parameter used for the c quark are given in the last column. Ds 3 L × TBc ηs Bs a (fm) amb amc ams (ηs) ams (Bs) ϵ [16] 163 × 48 6260(8) 1994(3) 2197(2) 0.15 2.76 0.850 0.065 0.215 −0.34 203 × 64 6263(12) 1977(4) 2172(2) 0.12 2.08 0.632 0.049 0.155 −0.21 283 96 6255(10) 1971(3) 2167(2) 0.09 1.20 0.452 0.0385 0.114 −0.115 × PDG [8] 6275 1968 finally, we do not have sea c quarks in our simulation. For a more detailed discussion on this, see [16]. above and found to agree with the experimental Ds (1968 MeV). The b quark mass mb and the coefficient c4 in Eq. (3) are then tuned to obtain a modified spin average mass and the Next, we explore, tuning ms when a s quark is in a bound state with a heavy b quark. Here, we are assuming that the hyperfine splitting of ϒ and ηb, which is ∼60 − 65 MeV [29]. In order to achieve the desired hyperfine splittings, we potential experienced by the s quark in the field of a b quark in a B meson remains the same in other strange bottom tuned only c4 since at Oð1=mbÞ, this is the only term that s contains Pauli spin matrices. Therefore, it allows the baryons, and there is no spin-spin interactions taking place between the quarks. In the infinite mass limit, the HQET mixing of spin components of ψ h. This term contributes maximally to the hyperfine splitting compared to the others Lagrangian becomes invariant under arbitrary spin rota- that contain Pauli spin matrices. The one-loop radiative tions [25]. Thereby, we can argue that for sbb and scb correction to c4 [30] has been used to tune hyperfine systems the spin-spin interactions do not contribute sig- splitting in [31], where it was found to change, but only nificantly in spectrum calculation. However, this argument is perhaps not valid in systems like bss or bsd but still with mildly, over lattice spacings ∼0.15 − 0.09 fm for Nf ¼ 2 þ 1 þ 1 HISQ gauge configurations. In our present s quark thus tuned; we possibly can obtain their masses close to their physical masses without resorting to any mixed action study, the changes in the tuned c4 on various lattice ensembles are small enough. Taking an average of extrapolation. In this paper, we will present our results obtained at these those values, we choose c4 ¼ 1.9 for which the hyperfine m splittings obtained on three different lattices 0.15, 0.12, and two different values of s. In Table IV, we listed these s B 0.09 fm are 60.6, 61.1, and 61.8 MeV, respectively. values of -quark masses. In the Table V, we calculate c D b c s All other coefficients c in Eq. (3) are set to 1.0. We set and s mesons using tuned , , and masses. As is seen, i when m is tuned with η the D mass obtained is fairly stability factor n ¼ 4 throughout our simulation. The s s s close to the PDG value, whereas when tuned to B , we see Table IV lists the values of m used in this work. s b an upward shift by an average 200 MeV. We have observed similar differences when s quark appears together with c in B. mc tuning (scb)-baryon masses. The c-quark mass is tuned pretty much in the same way as mb, except that Mcc¯ is tuned to the spin average of J=ψ D. u=d quarks η and c experimental masses. In this case, however, the For the valence u=d quark mass, we used a range of bare adjustment to the spin averaged value due to the absence of quark masses varying from the lightest sea quark masses all electromagnetic interaction, c quarks in sea, and discon- the way to a little above, where the s mass is tuned to Bs. nected diagrams are very small and hence, neglected. The Whenever the mass of a bottom baryon containing u=d bare c-quark masses used in this work are given in Table IV. quark(s) is quoted, it will correspond to the u=d quark mass tuned at the B mass. Since we are not including either C. ms tuning

The s-quark mass is tuned to two different values. In the TABLE VI. Values of amu=d used in this work. first case, we tune to the mass of fictitious ss¯ pseudoscalar 3 L × T amu=d meson ηs while in the second case, to the mass of Bs. The ηs is a fictitious meson that is not allowed to decay through ss¯ 163 × 48 0.065, 0.10, 0.13, 0.14, 0.155, annihilation. Hence, no disconnected diagrams arise in the (0.15 fm) 0.165, 0.185, 0.215, 0.225 3 two-point function calculation. From chiralpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi perturbation 20 × 64 0.05, 0.075, 0.090, 0.10, 2 2 (0.12 fm) 0.115, 0.13, 0.155, 0.165 theory, its mass is estimated to be mη ¼ 2m − mπ ¼ s K 283 96 0.04, 0.07, 0.085, 0.09, 689 MeV [32,33]. The s-quark mass thus tuned is checked × (0.09 fm) 0.10, 0.114, 0.12, 0.13 against Ds meson, making use of the c-quark mass obtained

094503-7 PROTICK MOHANTA and SUBHASISH BASAK PHYS. REV. D 101, 094503 (2020)

Σ Ω˜ Ω˜ 283 96 FIG. 1. Tuning of s and u=d quark masses in various lattices. FIG. 2. b, b and ccb correlators in × lattice. 0 0 The experimental values of B and Bs are shown by bands whose thickness are to enhance visibility and have nothing to do with experimental errors. electromagnetic or isospin breaking in our calculation, we do not distinguish between u and d quarks, and it is always amu ¼ amd. This tuning of u=d mass to B works well in capturing the b-baryon states containing a single u=d, such as (usb)or(ucb) baryons, when compared to either PDG or other works. In Table VI, we listed u=d quark masses (amq) against the lattice spacing. We show, in the Fig. 1, our strategy used to tune ms and mu=d. The tuned amu=d for different lattices to use with b quarks are

163 × 48∶ 0.165 203 × 64∶0.115 283 × 96∶0.085: FIG. 3. Effective mass plots for the states in Fig. 2. The bands The s and u=d masses so tuned, ms=mu=d turns out to be are placed over what we consider plateau. ∼1.3 compared to ∼6 that we get when ms is obtained from ηs and mu=d is the bare sea quark mass. For states containing two u=d quarks, such as ΣbðuubÞ and ΛbðudbÞ, we have mixed success with the above γ Σ approach. When b and u form diquarks ðQC fk;5guÞ for b state, the masses obtained are consistent with other lattice studies. However, this tuning scheme involving B fails for Λb where the diquark part is ðuCγ5dÞ (see the Table II). Hence, for Λb containing both u and d quarks, we have to resort to different tuning to account for 190 MeV of mass difference with Σb. The mass of Λb is obtained from this 0 Λ specially tuned mu=d (to be used only for b). Thus, tuned Ξ − Λ Σ⋆ − Λ differently, the mass differences b b and b b are well within 2σ of PDG values while Λb − B is about 60 MeV higher; see the Table XIV.

V. RESULTS AND DISCUSSION FIG. 4. Variation of single b baryon masses in MeV against the same light quark masses as in Fig. 1. mq ¼ 0.085 and 0.114 are In order to extract the masses of the bottom baryons, we the tuned u=d and s quark masses, respectively, indicated by perform a two-exponential uncorrelated fit to the two-point dashed vertical lines. The bands correspond to the PDG values, functions. We then cross-checked it with fitting the except for Ξcb, which is taken from [11].

094503-8 HEAVY BARYON SPECTRUM ON THE LATTICE WITH NRQCD … PHYS. REV. D 101, 094503 (2020)

ϒ η effective masses over the same range of time slices. and b and is equal to 9450 MeV, and Mlatt is the lattice However, this zero momentum energy does not directly bottom baryon mass in MeV. give us the mass of the bottom baryons because of In the calculation of mass splittings, this shift in energies unphysical shift in zero of energy. To account for it, the is canceled by subtraction among energies of hadrons mass is obtained considering energy splittings, having equal number of b quarks (nb) in them. For this calculation, we use a jackknifed ratio of the correlation n η functions for fitting [34], M E b Mmod − E b ; latt ¼ latt þ 2 ð phys lattÞ ð41Þ Y − C ðtÞ − − Y X ∼ ðMY MXÞt C ðtÞ¼ X e : ð42Þ where Elatt is the lattice zero momentum energy in MeVand C ðtÞ nb is the number of b quarks in the bottom baryon. For ¯ bottom mesons ðblÞ, nb is obviously always 1. As dis- Below in the Fig. 2, we show a few correlators for single b mod cussed before, Mphys is the modified spin average mass of baryons containing exclusively either two c or s or u=d.

TABLE VII. Masses, in lattice unit, of baryons involving single b quark and no s quark. The bare u=d-quark masses are 0.165 for 163 × 48, 0.115 for 203 × 64, and 0.085 for 283 × 96.

Baryons 163 × 48 (0.15 fm) 203 × 64 (0.12 fm) 283 × 96 (0.09 fm) Average (MeV) Ω˜ ⋆ ccb 2.954(5) 2.497(4) 2.088(3) 7807(11) ˜ Ωccb 2.933(5) 2.482(3) 2.078(3) 7780(9) Ω0 ccb 2.952(4) 2.497(3) 2.078(3) 7797(11) Ξ˜ ⋆ cb 2.222(6) 1.899(4) 1.648(6) 6835(20) Ξcb 2.177(11) 1.881(4) 1.623(5) 6787(12) ˜ Ξcb 2.199(8) 1.886(4) 1.631(4) 6805(16) Ξ0 cb 2.227(6) 1.904(4) 1.653(6) 6843(19) Σ˜ ⋆ b 1.468(8) 1.292(5) 1.189(5) 5836(22) ˜ Σb 1.460(7) 1.290(3) 1.174(6) 5820(21) Σ0 b 1.470(7) 1.305(3) 1.194(9) 5848(18) Λb 1.322(7) 1.208(6) 1.109(9) 5667(14)

TABLE VIII. Masses, in lattice unit, of baryons containing single b quark and s quark(s).

Baryons Tuning 163 × 48 (0.15 fm) 203 × 64 (0.12 fm) 283 × 96 (0.09 fm) Average (MeV) η Ω˜ ⋆ s 2.035(5) 1.782(5) 1.542(3) 6611(9) cb Bs 2.292(7) 1.957(6) 1.693(4) 6930(19) ηs 2.010(8) 1.754(5) 1.532(3) 6578(9) Ωcb Bs 2.248(11) 1.937(7) 1.684(2) 6893(16) η ˜ s 2.012(7) 1.765(5) 1.536(3) 6587(10) Ωcb Bs 2.267(8) 1.943(7) 1.686(2) 6906(17) η Ω0 s 2.052(5) 1.785(5) 1.548(3) 6625(8) cb Bs 2.297(6) 1.966(6) 1.705(2) 6946(17) η Ξ˜ ⋆ s 0.987(4) 0.945(2) 0.918(3) 5237(8) b Bs 1.541(8) 1.352(6) 1.235(6) 5935(22) ηs 0.986(5) 0.947(2) 0.909(4) 5231(11) Ξb Bs 1.520(9) 1.345(3) 1.207(6) 5901(20) η ˜ s 0.978(5) 0.944(2) 0.904(5) 5222(13) Ξb Bs 1.532(11) 1.350(4) 1.224(4) 5921(19) η Ξ0 s 0.987(4) 0.948(3) 0.913(5) 5235(11) b Bs 1.544(10) 1.366(4) 1.238(6) 5946(16) η Ω˜ ⋆ s 1.129(5) 1.058(3) 1.012(4) 5430(11) b Bs 1.611(8) 1.412(6) 1.264(3) 6019(20) η ˜ s 1.118(7) 1.050(3) 0.997(4) 5410(10) Ωb Bs 1.600(11) 1.411(7) 1.264(3) 6014(17) η Ω0 s 1.131(9) 1.057(3) 1.007(2) 5427(9) b Bs 1.615(8) 1.425(7) 1.295(2) 6051(15)

094503-9 PROTICK MOHANTA and SUBHASISH BASAK PHYS. REV. D 101, 094503 (2020)

The fitting range is typically chosen (i) looking at positions of what we consider plateau in the effective mass plots and (ii) exponential fits of the correlators. Both fittings return same masses over suitably chosen range. In the effective mass plot Fig. 3, the zero momentum energies and the errors of the same three states as in the Fig. 2 are represented as bands. In these figures, we choose to present the data from 283 × 96 lattices but the data from 163 × 48 and 203 × 64 are similar. Just to remind, in order to obtain the masses in MeV from these, we need the Eq. (41).

A. Single bottom baryons A couple of baryon states containing one b quark have FIG. 5. The ratio of correlators for the calculation of the two been listed in the PDG [8], such as ΛbðudbÞ, ΩbðssbÞ, splittings shown in Table IX. The bands overlaid on the data Ξ0 bðusbÞ etc., and they provide a good matching oppor- points represent single exponential fits. tunity. In the Fig. 4, we show the agreement of some of m m these baryon masses with PDG at the tuned u=d and s. Next, we determine mass differences in single bottom In the following Tables VII, VIII, X, and XI, we present sector, including the hyperfine splittings. our results of the single and multi b baryon states The mass splittings are calculated using a ratio of corresponding to the operators given in the Tables I and correlators as given in the Eq. (42). As an example, in II. In the columns corresponding to various lattice ensem- Fig. 5, we provide the plots for the ratio of correlators, bles, we show the masses in lattice unit, aElatt of the Ω0 − Λ Ξ − Λ b b and b b, for comfortable viewing because of Eq. (41). In the last column of each table, we provide the their relatively large mass differences; i.e., slopes are average Mlatt and the statistical errors, calculated assuming prominent and well separated. In the case of smaller the lattice configurations of different lattice spacings are ⋆ ⋆ differences, for instance, Ω˜ − Ω˜ or Ξ˜ − Ξ˜ , the slopes statistically uncorrelated. Additionally, we include b b cb cb of the ratio of correlators are rather small and not quite Tables XV and XVI for M for the bottom baryons for latt visible. In the Table IX above, we collect the results of each lattice spacing in the Appendix. single b baryons mass splittings. We collect our results for single bottom baryons, not A heavy quark basically acts as a static color source, and containing s quark(s), in the Table VII and those with s therefore, we expect that the hyperfine splittings between quark(s) in Table VIII. For the m , we state the results u=d states containing single or multiple s and u=d quark(s) to when the valence ml gives physical B meson mass. depend only weakly on the tuning of m and m .For Since s quark has been tuned in two different ways, we s u=d m ≤ 0.085 and two values of m , we show this pattern quote both the b-baryon masses at η , B points. u=d s s s for a few hyperfine splittings in Fig. 6. As is evident from our results, the numbers coming from the s quark tuned to η are about 300 MeV smaller from s B. Double bottom baryons those tuned to Bs (600 MeV in baryons with two s). If we take Ωb (ssb) and compare with the PDG value 6046 MeV, For the baryons containing more than one b quark, the it becomes obvious. data are relatively less noisy than those containing a single

TABLE IX. Single bottom baryons mass splittings in MeV.

Baryon splittings 163 × 48 (MeV) 203 × 64 (MeV) 283 × 96 (MeV) Average (MeV) Ω˜ ⋆ − Ω˜ ccb ccb 28(3) 23(2) 26(3) Ω˜ ⋆ − Ω˜ cb cb 59(8) 62(13) 61(22) 61(15) Ξ˜ ⋆ − Ξ˜ cb cb 37(6) 44(5) 44(9) 42(7) Ω˜ ⋆ − Ω˜ b b 29(5) 28(11) 29(4) 29(7) Ω0 − Λ b b 396(4) 391(9) 406(10) 398(9) Ξ˜ ⋆ − Ξ b b 138(20) 122(38) 138(46) 133(36) ˜ Ξb − Λb 170(9) 166(11) 163(6) 166(9) Λb − B 391(20) 431(20) 397(22) 406(21) Σ˜ ⋆ − Σ˜ b b 30(9) 30(8) 29(8) 30(8) Σ˜ ⋆ − Λ b b 224(13) 203(12) 175(13) 201(13)

094503-10 HEAVY BARYON SPECTRUM ON THE LATTICE WITH NRQCD … PHYS. REV. D 101, 094503 (2020)

Ω⋆ The plot for bbb appears counterintuitive since being the heaviest, it is showing a lower mass compared to the other two. However, it receives a large correction because of shift in the rest mass of three b quarks. We tabulate our results for double bottom nonstrange baryons in the Table X, while those containing a s quark in Table XI. It is to note that Ωbbb is a spin-3=2 state having no spin- 1=2 counterpart. But in practice, we can take a spin-1=2 projection to get such a fictitious state. Therefore, we label 3 2 Ω⋆ the physical ðbbbÞ spin- = state with bbb to keep consistency with our remaining notation. In this case, none of the states have PDG entries. We would like to point out that the variation of the FIG. 6. Hyperfine splittings at various ms and mu=d for a ΞbbðubbÞ masses with mu=d is almost absent as the major 3 selected few bottom baryons on 28 × 96 lattice. Horizontal contribution to these baryons are coming from the two b bands are the average values of the splittings and are used to quarks. Similarly, from the Table XI, we see that the guide the eye. different tuning of the s quark has significantly less influence on the double bottom baryon masses, a situation unsurprisingly similar to double bottom baryons with a u=d quark. The splittings in double bottom sector are tabulated in the Table XII. In the double bottom sector, the splittings between the spin-3=2 and 1=2 states are particularly interesting because HQET relates this mass differences with hyperfine splittings of heavy-light mesons, which in the heavy-quark limit [35],

ΔMbaryon 3 bb → : 43 Δ meson 4 ð Þ mb

Ω⋆ Ω˜ ⋆ Ω˜ This behavior is consistent with our results within errors as FIG. 7. , cbb and ccb effective masses. bbb can be seen in Table XIII. b. The effective mass plots in Fig. 7, shown for only 163 × A few GMO mass relations involving b quarks are 48 lattices but similar for two other lattices, are an evidence provided in Ref. [36], which we try to verify in this work, for this. The fitting ranges are chosen the same way as is MΩ⋆ − MΩ ≈ MΩ⋆ − MΩ ð44Þ done for Fig. 3. ccb ccb cbb cbb

TABLE X. Triple and double bottom nonstrange baryon masses.

Baryon 163 × 48 (0.15 fm) 203 × 64 (0.12 fm) 283 × 96 (0.09 fm) Average (MeV) Ω⋆ bbb 1.983(4) 2.031(3) 2.154(4) 14403(7) Ωbbb 1.974(4) 2.023(5) 2.148(4) 14390(8) Ω⋆ cbb 2.429(16) 2.259(4) 2.117(3) 11081(21) Ωcbb 2.409(16) 2.246(5) 2.110(3) 11060(23) Ω˜ ⋆ cbb 2.431(8) 2.255(4) 2.113(3) 11077(14) ˜ Ωcbb 2.432(10) 2.251(4) 2.113(3) 11075(13) Ω0 cbb 2.434(8) 2.250(4) 2.114(4) 11076(12) Ξ⋆ bb 1.721(12) 1.643(10) 1.666(5) 10103(24) Ξbb 1.700(12) 1.640(7) 1.664(5) 10091(17) Ξ˜ ⋆ bb 1.720(12) 1.635(8) 1.668(3) 10100(27) ˜ Ξbb 1.703(16) 1.634(8) 1.661(4) 10087(22) Ξ0 bb 1.704(16) 1.635(10) 1.672(3) 10096(24)

094503-11 PROTICK MOHANTA and SUBHASISH BASAK PHYS. REV. D 101, 094503 (2020)

TABLE XI. Double bottom strange baryon spectra.

Baryon Tuning 163 × 48 (0.15 fm) 203 × 64 (0.12 fm) 283 × 96 (0.09 fm) Average (MeV) η Ω⋆ s 1.545(11) 1.536(6) 1.576(4) 9902(12) bb Bs 1.791(12) 1.703(11) 1.716(3) 10203(22) ηs 1.553(9) 1.527(7) 1.570(4) 9896(13) Ωbb Bs 1.768(12) 1.699(8) 1.715(3) 10190(17) η Ω˜ ⋆ s 1.542(9) 1.529(7) 1.575(4) 9896(12) bb Bs 1.791(12) 1.693(10) 1.717(3) 10199(28) η ˜ s 1.541(12) 1.527(7) 1.571(4) 9891(13) Ωbb Bs 1.789(9) 1.695(7) 1.714(3) 10197(24) η Ω0 s 1.552(9) 1.539(7) 1.578(3) 9908(11) bb Bs 1.782(9) 1.699(7) 1.720(3) 10200(20)

TABLE XII. Double bottom baryon mass splittings in MeV. None of the splittings have PDG entries.

Baryon splittings 163 × 48 (MeV) 203 × 64 (MeV) 283 × 96 (MeV) Average (MeV) Ω˜ ⋆ − Ω˜ cbb cbb 25(5) 35(2) 30(5) Ω⋆ − Ω bb bb 34(5) 25(8) 37(9) 32(7) Ξ⋆ − Ξ bb bb 25(4) 39(7) 32(5)

TABLE XIII. Ratio of hyperfine splittings of doubly heavy baryons to heavy mesons in the heavy quark limit in 283 × 96 lattice.

Baryon splittings Our results (MeV) Meson splittings Our results (MeV) Ratio Ω˜ ⋆ − Ω˜ ⋆ − bbc bbc 35(2) Bc Bc 46(4) 0.76(4) Ω⋆ − Ω ⋆ − bb bb 37(9) Bs Bs 45(9) 0.82(9) Ξ⋆ − Ξ ⋆ − bb bb 39(7) B B 47(7) 0.83(8)

We have discussed the construction of one and two MΣ⋆ − MΣ ≈ MΞ⋆ − MΞ : ð45Þ b b bb bb bottom baryon operators in details and pointed out the difficulty for constructing operators motivated by HQET For the GMO relation (44), both sides are expected to be for single bottom baryons. Consequently, we modified the approximately 31 MeV. In our case, for 203 × 64 lattice, for operators accordingly. For some of the baryons, we have which we have data for both the sides, they are approx- multiple operators for the same state, i.e., baryons having imately equal but are around 24 MeV as against 31 MeV the same quantum numbers. It would be natural in such given in [36]. Our lattice data are also consistent with the cases to construct correlation matrices and obtain the lowest approximate GMO relation (45), where each side is about lying, i.e., ground states by solving the generalized eigen- 30 MeV against 20 MeV calculated in [36]. value method. Single bottom baryons can have isodoublets with the VI. SUMMARY same overall quantum numbers JP. For instance, there exist three isodoublets of Ξ , which are not radially or orbitally In this paper, we presented the lattice QCD determi- b excited states [37]. These states have been categorized by nation of masses of the baryons containing one or more b the spin of the us or ds diquark denoted by j and the spin- quark(s) using NRQCD action for the b quark and HISQ parity of the baryon. These baryons are referred to as action for the c, s, and u=d quarks. This combination of Ξ j 0;JP 1þ Ξ0 j 1;JP 1þ Ξ⋆ j 1; NRQCD and HISQ has previously been employed in [16] bð ¼ ¼ 2 Þ, bð ¼ ¼ 2 Þ, and bð ¼ P 3þ Ξ for bottom mesons; however, the exact implementation was J ¼ 2 Þ. The same pattern is observed in c states [8]. Ξ0 Ξ rather different. In this work, we converted the one The mass difference between b and b is about 150 Mev. component HISQ propagators to 4 × 4 matrices by the So depending upon the choice of the wave function having Kawamoto-Smit transformation and the two component the same overall quantum numbers, we can have different T NRQCD propagators to 4 × 4 matrices using the prescrip- baryon states. If we choose ðs Cγ5dÞQ as our j ¼ 0 baryon tion suggested in [11]. operator, then we will be simulating Ξb state, and if we

094503-12 HEAVY BARYON SPECTRUM ON THE LATTICE WITH NRQCD … PHYS. REV. D 101, 094503 (2020)

FIG. 8. Comparison of our single bottom baryon spectra with FIG. 9. Comparison of our triple and double bottom baryon Brown et al. [11], Burch [10], Mathur et al. [6], and PDG [8], spectra with Brown et al. [11] and Burch [10]. where available. the standard action of choice for the b quark in these three T cited studies, but the actions used for c are all different— project out the spin-1=2 state of j ¼ 1 operator ðs CγkdÞQ, Ξ0 NRQCD, Clover-Wilson, and relativistic heavy quark then we will get the b. For the reason discussed before, we can not define j ¼ 1 light-light diquark state. In our action [38]. Whatever differences we see in the results Ξ0 T γ for single b baryons with c quark(s), particularly in the case, the wave function corresponds to b is ðQ C 5sÞd. Constructing an operator in this way allows the s and d quarks cases with two c, possibly have stemmed from the to have parallel spin configurations. By simple physical differences in actions. However, in this work, we do not reasoning, we can argue that the explicit construction of j ¼ 0 address the systematics involved, which could be significant, because of these differences. (The study of such diquark for Ξb is more likely to have a significant overlap Ξ Ξ0 1 systematics needed to arrive at phenomenologically rel- with physical b compared to b (j ¼ ) upon gauge T evant numbers will be reported elsewhere.) But otherwise, averaging. However, the operator ðQ Cγ5sÞd is expected the results of bottom baryon spectra in the present study to have a good overlap with the Ξ0 state, and this is also b with NRQCD b quark and HISQ c; s; u=d quarks appear to supported by our result. For an antiparallel s and d spin agree with each other. We would like to emphasize again configuration, Ξ0 can also have an overlap with the Ξ state. b b that the errors we reported here are only statistical. On lattice, operators for states having the same quantum The comparison of the hyperfine splittings is shown numbers can mix, and therefore, a detailed generalized in Fig. 10. eigenvalue problem analysis can only resolve the issue of Apart from the hyperfine splittings, a few other mass Ξ Ξ0 mutual overlap of b and b states, which we did not include splittings calculated in this work are assembled in in this work. This is perhaps the reason we see discrepancies Table XIV. The bottom baryon spectra and various mass in their values with PDG and others in Fig. 8. splittings reported in this paper and those appearing in The b mass has been tuned to modified ϒ − ηb spin average mass, while c quark to J=ψ − ηc spin average mass. The s quark is required to be tuned to both the fictitious ηs and Bs mass since we expect the bottom- strange bound state to be more appropriate than the s − s¯ bound state in bottom baryons. For the light u=d quarks, we have considered a wide range of bare masses and tune it using B meson. However, this scheme of tuning u=d quarks has not worked for Λb. There, u=d are tuned to capture the 190 MeV mass difference Σb − Λb. This specially tuned 0 Λ mu=d, which is used only for b, gives it a mass of 5667 MeV. The PDG value for Λb mass is 5620 MeV. We demonstrated the variation of bottom baryons as well as hyperfine splittings against varying ms and mu=d.We showed that the hyperfine splittings are almost independent of s and u=d quark masses. FIG. 10. Hyperfine splittings of bottom baryons calculated in We compare our bottom baryon results with other works, this work and compared with Brown et al. [11], Burch [10], mostly with [6,10,11], in Figs. 8 and 9. NRQCD has been Mathur et al. [6], and PDG [8], where available.

094503-13 PROTICK MOHANTA and SUBHASISH BASAK PHYS. REV. D 101, 094503 (2020)

TABLE XIV. Bottom baryon mass differences in MeV. PDG Department of Atomic Energy (DAE), Government of values without error are simply the differences of the two states. India. A significant part of this work has been carried out in the Proton cluster funded by DST-SERB Project No. Mass splittings This work Brown et al. [11] PDG [8] SR/S2/HEP-0025/2010. The authors acknowledge useful Ω0 − Λ b b 398(9) 426.4(2.2) discussions with Dipankar Chakrabarti (IIT-Kanpur, India) Ξ⋆ − Ξ b b 133(36) 189(29) 155.5 and Stefan Meinel (University of Arizona, U.S.) on bottom Ξ − Λ b b 166(9) 172.5(0.4) baryon operator construction. One of the authors (P. M.) Λb − B 406(21) 339.2(1.4) Σ⋆ − Λ thanks DAE for financial support. b b 201(13) 251(46) 213.5

[8,11] are well comparable given the wide choice of actions APPENDIX: BOTTOM BARYON and tuning employed in achieving them. MASSES IN MEV For the reason of completeness, we provide the bottom ACKNOWLEDGEMENTS baryon masses in MeV for different lattices used in this The numerical part of this work has been performed at work. In these tables, we provide only Bs tuned strange the HPC facility in NISER (Kalinga cluster) funded by bottom baryon masses and not ηs tuned values.

TABLE XV. Single bottom baryon masses in MeV, corresponding to the Tables VII and VIII.

163 × 48 203 × 64 283 × 96 163 × 48 203 × 64 283 × 96 Baryons (0.15 fm) (0.12 fm) (0.09 fm) Average Baryons (0.15 fm) (0.12 fm) (0.09 fm) Average Ω˜ ⋆ Ω˜ ⋆ ccb 7816(6) 7794(6) 7805(6) 7807(11) cb 6945(9) 6906(10) 6938(9) 6930(19) ˜ Ω Ωccb 7788(6) 7769(5) 7782(6) 7780(9) cb 6887(14) 6873(11) 6919(4) 6893(16) Ω0 ˜ ccb 7813(5) 7794(5) 7782(6) 7797(11) Ωcb 6912(10) 6883(11) 6923(4) 6906(17) Ξ˜ ⋆ Ω0 cb 6853(8) 6810(6) 6840(13) 6835(20) cb 6951(8) 6921(10) 6965(4) 6946(17) Ξ Ξ˜ ⋆ cb 6794(14) 6781(6) 6785(11) 6787(12) b 5957(10) 5911(10) 5934(13) 5935(22) ˜ Ξ Ξcb 6822(10) 6789(6) 6802(9) 6805(16) b 5929(12) 5900(5) 5873(13) 5901(20) Ξ0 ˜ cb 6859(8) 6819(6) 6851(13) 6843(19) Ξb 5945(14) 5908(7) 5910(9) 5921(19) Σ˜ ⋆ Ξ0 b 5861(10) 5812(8) 5833(11) 5836(22) b 5961(13) 5934(7) 5941(13) 5946(16) Σ˜ Ω˜ ⋆ b 5850(9) 5809(5) 5800(13) 5820(21) b 6049(10) 6010(10) 5998(7) 6019(20) Σ0 ˜ b 5864(9) 5834(5) 5844(20) 5848(18) Ωb 6035(14) 6008(11) 5998(7) 6014(17) Λ Ω0 b 5669(9) 5674(10) 5658(20) 5667(14) b 6054(10) 6031(11) 6066(4) 6051(15)

TABLE XVI. Triple and double bottom baryon masses in MeV, corresponding to the Tables X and XI.

163 × 48 203 × 64 283 × 96 163 × 48 203 × 64 283 × 96 Baryons (0.15 fm) (0.12 fm) (0.09 fm) Average Baryons (0.15 fm) (0.12 fm) (0.09 fm) Average Ω⋆ Ω⋆ bbb 14399(5) 14405(5) 14403(9) 14403(7) bb 10217(16) 10177(18) 10216(7) 10203(22) Ωbbb 14388(5) 14392(8) 14390(9) 14390(8) Ωbb 10187(16) 10171(13) 10214(7) 10190(17) Ω⋆ Ω˜ ⋆ cbb 11056(21) 11091(6) 11095(7) 11081(21) bb 10217(16) 10161(16) 10218(7) 10199(28) Ω ˜ cbb 11029(21) 11070(8) 11080(7) 11060(23) Ωbb 10214(12) 10164(11) 10212(7) 10197(24) Ω˜ ⋆ Ω0 cbb 11058(10) 11085(7) 11086(7) 11077(14) bb 10205(12) 10171(11) 10225(7) 10200(20) ˜ Ωcbb 11060(13) 11078(7) 11086(7) 11075(13) Ω0 cbb 11062(10) 11076(7) 11088(9) 11076(12) Ξ⋆ bb 10124(16) 10078(16) 10106(11) 10103(24) Ξbb 10097(16) 10073(11) 10102(11) 10091(17) Ξ˜ ⋆ bb 10123(16) 10065(13) 10111(7) 10100(27) ˜ Ξbb 10101(21) 10063(13) 10095(9) 10087(22) Ξ0 bb 10102(21) 10065(16) 10119(7) 10096(24)

094503-14 HEAVY BARYON SPECTRUM ON THE LATTICE WITH NRQCD … PHYS. REV. D 101, 094503 (2020)

[1] C. T. H. Davies, Proc. Sci., LATTICE2011 (2011) 019. [21] H. D. Trottier, Phys. Rev. D 55, 6844 (1997). [2] E. Gamiz, C. T. H. Davies, G. P. Lepage, J. Shigemitsu, and [22] M. Wingate, J. Shigemitsu, C. T. H. Davies, G. P. M. Wingate, Phys. Rev. D 80, 014503 (2009). Lepage, and H. D. Trottier, Phys. Rev. D 67, 054505 [3] A. A. Khan, T. Bhattacharya, S. Collins, C. T. H. Davies, R. (2003). Gupta, C. Morningstar, J. Shigemitsu, and J. Sloan, Phys. [23] K. C. Bowler, R. D. Kenway, O. Oliveira, D. G. Richards, Rev. D 62, 054505 (2000). P. Ueberholz, L. Lellouch, J. Nieves, C. T. Sachrajda, N. [4] R. Woloshyn, Phys. Lett. B 476, 309 (2000). Stella, and H. Wittig (UKQCD Collaboration), Phys. Rev. D [5] R. Lewis, N. Mathur, and R. M. Woloshyn, Phys. Rev. D 64, 54, 3619 (1996). 094509 (2001). [24] J. M. Flynn, F. Mescia, and A. S. B. Tariq, J. High Energy [6] N. Mathur, R. Lewis, and R. M. Woloshyn, Phys. Rev. D 66, Phys. 07 (2003) 066. 014502 (2002). [25] J. G. Korner, M. Kramer, and D. Pirjol, Prog. Part. Nucl. [7] H. Na and S. A. Gottlieb, Proc. Sci., LATTICE2006 (2006) Phys. 33, 787 (1994). 191; LATTICE2007 (2007) 124; LATTICE2008 (2008) 119. [26] A. Bazavov et al., Rev. Mod. Phys. 82, 1349 (2010). [8] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, [27] K. Orginos and D. Toussaint, Phys. Rev. D 59, 014501 030001 (2018) and 2019 update. (1998). [9] S. Meinel, Phys. Rev. D 82, 114514 (2010). [28] K. Orginos, D. Toussaint, and R. L. Sugar, Phys. Rev. D 60, [10] T. Burch, arXiv:1502.00675. 054503 (1999). [11] Z. S. Brown, W. Detmold, S. Meinel, and K. Orginos, Phys. [29] N. Zerf, Proc. Sci., RADCOR2015 (2015) 089. Rev. D 90, 094507 (2014). [30] T. C. Hammant, A. G. Hart, G. M. von Hippel, R. R. [12] G. P. Lepage, L. Magnea, C. Nakhleh, U. Magnea, and K. Horgan, and C. J. Monahan, Phys. Rev. Lett. 107, 112002 Hornbostel, Phys. Rev. D 46, 4052 (1992). (2011). [13] B. A. Thacker and G. P.Lepage, Phys. Rev. D 43, 196 (1991). [31] R. J. Dowdall, C. T. H. Davies, T. C. Hammant, and R. R. [14] E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. P. Horgan, Phys. Rev. D 86, 094510 (2012). Lepage, J. Shigemitsu, H. Trottier, and K. Wong, Phys. Rev. [32] C. Bernard, T. Burch, K. Orginos, D. Toussaint, T. A. D 75, 054502 (2007). DeGrand, C. DeTar, S. Datta, S. Gottlieb, U. M. Heller, [15] A. X. El-Khadra, A. S. Kronfeld, and P. B. Mackenzie, and R. Sugar, Phys. Rev. D 64, 054506 (2001). Phys. Rev. D 55, 3933 (1997). [33] C. T. H. Davies, E. Follana, I. D. Kendall, G. P. Lepage, and [16] E. B. Gregory, C. T. H. Davies, I. D. Kendall, J. Koponen, K. C. McNeile, Phys. Rev. D 81, 034506 (2010). Wong, E. Follana, E. Gámiz, G. P. Lepage, E. H. Müller, H. [34] S. R. Beane, K. Orginos, and M. J. Savage, Phys. Lett. B Na, and J. Shigemitsu (HPQCD Collaboration), Phys. Rev. 654, 20 (2007). D 83, 014506 (2011). [35] N. Brambilla, A. Vairo, and T. Rösch, Phys. Rev. D 72, [17] www.physics.utah.edu/ detar/milc/. 034021 (2005). [18] N. Kawamoto and J. Smit, Nucl. Phys. B192, 100 (1981). [36] X.-Z. Weng, X.-L. Chen, and W.-Z. Deng, Phys. Rev. D 97, [19] C. T. H. Davies, K. Hornbostel, A. Langnau, G. P. Lepage, 054008 (2018). A. Lidsey, J. Shigemitsu, and J. Sloan, Phys. Rev. D 50, [37] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 114, 6963 (1994). 062004 (2015). [20] C. T. H. Davies, K. Hornbostel, G. P. Lepage, A. J. Lidsey, J. [38] S. Aoki, Y. Kuramashi, and S. Tominaga, Prog. Theor. Phys. Shigemitsu, and J. Sloan, Phys. Rev. D 52, 6519 (1995). 109, 383 (2003).

094503-15