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Home , Quarkonium, Strange B meson

EPJ Web of Conferences 175, 05031 (2018) https://doi.org/10.1051/epjconf/201817505031 Lattice 2017

Heavy spectra on lattice using NRQCD

Protick Mohanta1, and Subhasish Basak1 1National Institute of Science Education and Research Bhubaneswar, P.O. Jatni, Khurda 752050, Odisha, India

Abstract. Comparison of radial excitation energies to masses show that the velocity of b is very non-relativistic in bottomonium states. In a mixed system like charmed B , the b quark has less velocity than it has in bottomonium states and in strange it is even slower. So one can use NRQCD for the b quark in those systems. Using overlap and hisq action for the s and c and NRQCD for b quark we simulated spectra of charmed and strange B and also few other .

1 Introduction Lattice methods are powerful techniques in analyzing the spectrum of hadrons. However for hadrons containing heavy quarks particularly are difficult to analyze. For spectrum calculation it is necessary that aM << 1. For light quarks it is true but for aMc > 0.7 and for bottom quark aMb > 2 with lattice spacing a = 0.12fm. However in hadrons containing heavy quarks the velocities of heavy quarks are non-relativistic. Comparison of masses of heavy to the masses of heavy quarks also indicates the same. For example MΥ = 9390 MeV whereas 2 M = 8360 MeV (MS Scheme) and M = 3096 MeV whereas 2 M = 2580 × b J/ψ × c MeV. This is in contrast with mesons and baryons that are made up of light quarks for example mass of M 140 MeV whereas masses of up or is few MeV. So one can use π ∼ effective theories like non relativistic QCD(NRQCD) to study hadrons containing heavy quark/quarks.

In this report we present some of preliminary results on charmed and and baryons. For the bottom quark we used NRQCD action corrected up to order O(v6), for strange and charm quarks we used overlap and hisq actions. We simulated the ground state of Bc and Bs meson, hyperfine 3 + 3 + 1 + splitting of Υ and ηb. We also simulated ground states of Ωbbb( 2 ), Ωbbs∗ ( 2 ) and Λbss( 2 ) baryons.

2 Simulation Details 2.1 NRQCD Action We used NRQCD for the bottom quark. The NRQCD Lagrangian [1], [2]corrected up to O[(v/c)6] is given by = + δ 4 + δ 6 where L L0 Lv Lv D 2 = ψ(x)† iD + ψ(x) (1) L0 0 2m   Speaker, e-mail: [email protected]    

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 175, 05031 (2018) https://doi.org/10.1051/epjconf/201817505031 Lattice 2017

1 4 g δ 4 = c ψ†D ψ + c ψ†(D.E E.D)ψ Lv 1 8m3 2 8m2 − ie g +c ψ†σ. (D E E D)ψ + c ψ†σ. Bψ (2) 3 8m2 × − × 4 2m

2 g 2 ig δ 6 = c ψ† D , σ. B ψ c ψ†(σ. E E)ψ Lv 5 8m3 { } − 6 8m3 × 3 ig 2 +c ψ† D , σ. (D E E D) ψ (3) 7 64 m4 { × − × }

Tree level value of all these coefficients c1, c2, c3, c4, c5, c6 and c7 is 1. Here D denotes the continuum covariant derivative. The lattice version of NRQCD is obtained by replacing the continuum derivatives by lattice derivatives:

+ a∆ ψ(x) = U (x)ψ(x + aµˆ) ψ(x);a∆−ψ(x) = ψ(x) U†(x aµˆ)ψ(x aµˆ) µ µ − µ − µ − − Symmetric derivative and Laplacian is related to forward and backward derivative as

1 + 2 + + ∆± = (∆ +∆−);∆ ∆ ∆− = ∆−∆ 2 ≡ i i i i i i Now for coarser lattices in order to extract physical quantities we need to improve the discretization error. Simply by Taylor expansion of symmetric derivative and the Laplacian operator we can find their forms corrected up to O(a4)[1]

2 2 a + 2 2 a + 2 ∆˜ ± f =∆± f ∆ ∆±∆− f ; ∆˜ =∆ [∆ ∆−] i i − 6 i i i − 12 i i i Similarly using cloverleaf instead of plaquette one can improve the gauge fields up to O(a4)

4 a + + gF˜µν(x) = gFµν(x) [∆ ∆− +∆ ∆−]gFµν(x) − 6 µ µ ν ν NRQCD propagators of the quarks are obtained by the time evolution of the Green’s function n n aH0 aδH aδH aH0 G(x, t + 1;0, 0) = 1 1 U (x, t)† 1 1 G(x, t;0, 0) (4) − 2n − 2 t − 2 − 2n          with G(x, t; 0, 0) = 0 for t < 0 and G(x, t; 0, 0) = δx,0 for t = 0. From the above equation it is evident that n > 3 . In NRQCD quarks and anti-quarks are decoupled. Here G(x, t;0, 0) is a 2 2 matrix in 2m × spinor space. The propagators of heavy anti-quarks can be obtained by taking the dagger of the quark propagators [2].

2.2 Correlators When using NRQCD, all energies obtained from fits to hadronic two-point functions are shifted by some common constant, as the rest mass is not included in the Lagrangian eq(1),eq(2) and eq(3). Thus, to tune the b quark mass we have to use the ’kinetic mass’ (ap)2 (a∆E)2 aM = − (5) kin 2a∆E

2 EPJ Web of Conferences 175, 05031 (2018) https://doi.org/10.1051/epjconf/201817505031 Lattice 2017

In order to tune kinetic mass given in eq(5) we need to calculate the energy of the for a given momentum. Correlator for Υ or ηb is [3, 4] 1 4 g δ 4 = c ψ†D ψ + c ψ†(D.E E.D)ψ Lv 1 m3 2 m2 − ip.x 8 8 C(p, t) = e Tr[G†(x, 0)Γsk(x)G(x, 0)Γsc† (0)] (6) ie g x + ψ†σ.     ψ + ψ†σ. ψ c3 2 (D E E D) c4 B (2) 8m × − × 2m Here Γ(x) =Ωφ(x). φ is the smearing operator and Ω is a 2 2 matrix in space. Ω=I for × pseudoscalar and Ω=σi for vector particles [5]. g ig2 2 Correlator for heavy-light mesons for example Bs or Bc meson is given as [6] δ 6 = c ψ† D , σ. B ψ c ψ†(σ. E E)ψ Lv 5 3 { } − 6 3 × 8m 8m ip.x C(p, t) = e Tr[γ5M(x, 0)†γ5ΓG(x, 0)Γ] (7) 3 ig 2 +c7 ψ† D , σ. (D E E D) ψ (3) − 64 m4 { × − × } x Γ=γ for pseudoscalar and Γ=γ for vector particles. M(x, 0) is the propagator of the relativis- Tree level value of all these coefficients c , c , c , c , c , c and c is 1. Here D denotes the continuum 5 i 1 2 3 4 5 6 7 tic charm or . G(x, 0) is now a 4 4 matrix in spinor space having vanishing lower covariant derivative. The lattice version of NRQCD is obtained by replacing the continuum derivatives × components but it is in Dirac representation of gamma matrices. We can convert it to MILC gamma by lattice derivatives: representation by a unitary transformation + a∆µ ψ(x) = Uµ(x)ψ(x + aµˆ) ψ(x);a∆µ−ψ(x) = ψ(x) Uµ†(x aµˆ)ψ(x aµˆ) 1 σ σ − − − − S = y y √ σy σy Symmetric derivative and Laplacian is related to forward and backward derivative as 2 −  + Ω 3 =  aT γ b c = γ γ 1 + 2 + + For bbb( 2 ) the interpolator is ( k)α abc(Q C kQ )Qα with C 4 2, and the correlator is ∆± = (∆ +∆−);∆ ∆ ∆− = ∆−∆ O 2 ≡ i i i i given by [7] i i C (t) = 0 [ (x, t)] [ †(0, 0)] 0 Now for coarser lattices in order to extract physical quantities we need to improve the discretization ij,αβ  | Oi α O j β|  x error. Simply by Taylor expansion of symmetric derivative and the Laplacian operator we can find T 4 ch bg af their forms corrected up to O(a )[1] = abc f ghGαβ(x, 0)Tr[CγiG (x, 0)CγjG (x, 0)] (8) x 2 2 a + 2 2 a + 2 ∆˜ ± f =∆± f ∆ ∆±∆− f ; ∆˜ =∆ [∆ ∆−] Here i, j are the indices of the gamma matrices(γi,γj) and α, β are the spinor indices. In the last line of i i − 6 i i i − 12 i i i the above equation we haven taken transpose only over spin indices. The correlator has overlap with E / t 3/2 E / t 1/2 1 both spin 3/2 and spin 1/2 states: C (t) = Z / e− 3 2 ΠP + Z / e− 1 2 ΠP , where Π= (1 + γ ), 4 ij 3 2 ij 1 2 ij 2 4 Similarly using cloverleaf instead of plaquette one can improve the gauge fields up to O(a ) 3/2 1 1/2 1 3/2 1/2 Pij = δij 3 γiγ j, Pij = 3 γiγ j and Pij .P jk = 0. We can decouple their contributions by using: 4 − ˜ a + + gFµν(x) = gFµν(x) [∆µ ∆µ− +∆ν ∆ν−]gFµν(x) 3/2 3/2 3/2 2 E3/2t − 6 P .Cxx + P .Cyx + P .Czx = Z3/2Πe− (9) xx xy xz 3 NRQCD propagators of the quarks are obtained by the time evolution of the Green’s function 3 + aT b c For Ωbbs∗ ( 2 ) the interpolator is ( k)α = abc(Q CγkQ )sα and the correlator is given by [8] n n O  aH0 aδH aδH aH0  G(x, t + 1; 0, 0) = 1 1 Ut(x, t)† 1 1 G(x, t; 0, 0) (4) C (t) = 0 [ (x, t)] [ †(0, 0)] 0 − 2n − 2 − 2 − 2n ij,αβ  | Oi α O j β|         x with G(x, t;0, 0) = 0 for t < 0 and G(x, t;0, 0) = δ for t = 0. From the above equation it is evident ch bg afT x,0 = abc f gh[M (x, 0).γ4]αβTr[CγiG (x, 0)CγjG (x, 0)] (10) that n > 3 . In NRQCD quarks and anti-quarks are decoupled. Here G(x, t;0, 0) is a 2 2 matrix in 2m × x spinor space. The propagators of heavy anti-quarks can be obtained by taking the dagger of the quark This correlator also has overlap with both spin 3/2 and spin 1/2 states and can be decoupled in the propagators [2]. similar way. + Finally we write the interpolator of Λ ( 1 ) , which is =  (saT Cγ sb)Qc and the bss 2 Oα abc 5 α 2.2 Correlators correlator is given by [8]  When using NRQCD, all energies obtained from fits to hadronic two-point functions are shifted Cαβ(t) = 0 [ (x, t)]α[ †(0, 0)]β 0  | O O |  by some common constant, as the rest mass is not included in the Lagrangian eq(1),eq(2) and eq(3). x ch bg afT Thus, to tune the b quark mass we have to use the ’kinetic mass’ = abc f gh[G (x, 0)]αβTr[γ4γ2γ5M (x, 0)γ5γ2γ4M (x, 0)] (11) (ap)2 (a∆E)2 x aMkin = − (5) / 2a∆E This correlator does not have any overlap with spin 3 2 state.

3 EPJ Web of Conferences 175, 05031 (2018) https://doi.org/10.1051/epjconf/201817505031 Lattice 2017

3 Results 3 We used a ensemble of 40, 24 64 HISQ lattices with N f = 2 + 1 + 1 having ml/ms = 1/5, ×2 amc = 0.635 and gauge coupling 10/g = 6.00 generated by MILC collaboration. The lattice spacing is a = 0.12fm. Details of these lattices can be found here [9]. We set all the coefficients of the NRQCD action eq(2),eq(3) c1, c2, c3, c4, c5, c6 and c7 to be 1 and nr-loop parameter of eq(4) n = 3. The mass parameter m of the Lagrangian eq(1),eq(2), eq(3) is tuned to produce the kinetic mass of ηb = 9.42 GeV. We find that the mass differences of Υ and ηb to be 131 MeV.

101

100 (t)|} b η

10-1 (t)|/|C Υ ln{|C

10-2

10-3 0 10 20 30 40 50 60 70 time slice(t)

Figure 1. Hyperfine splitting of bottomonium 1S states.

We used overlap action to simulated Bc and Bs mesons and we used hisq action for Bs meson only. Simulations of Bc using hisq action is under way.

3.1 Simulations with overlap

In overlap formalism charm quark mass has been tuned to produce kinetic mass of the ηc to be 3.05 GeV. Strange quark has been directly tuned to produce ss¯ pseudoscalar pole mass to be 686 MeV. Here we used mass formula predicted by chiral perturbation theory to calculate the mass of pseudoscalar s-quarkonium state M = 2M2 M2. This part of the tuning of strange and charm ss¯ K − π quark mass using overlap action has been done by the TIFR, Mumbai group [10]. As we used kinetic mass in tuning the bottom and charm masses the following formula has been used to calculate the mass of Bc. 1 1 M = E + (Mη Eη ) + (Mη Eη ) (12) Bc Bc 2 b − b 2 c − c

Here EBc , Eηb , Eηc are the simulated masses and Mηb , Mηc are their pdg values. We find M = 6256.5 Mev with error = 20 MeV. Bc ± For Bs we used the following formula 1 M = E + (Mη Eη ) (13) Bs Bs 2 b − b and we found M = 5096 MeV 30 MeV. Bs ±

4 EPJ Web of Conferences 175, 05031 (2018) https://doi.org/10.1051/epjconf/201817505031 Lattice 2017

3 Results 10-9

3 -10 We used a ensemble of 40, 24 64 HISQ lattices with N f = 2 + 1 + 1 having ml/ms = 1/5, 10 ×2 amc = 0.635 and gauge coupling 10/g = 6.00 generated by MILC collaboration. The lattice spacing 10-11 is a = 0.12fm. Details of these lattices can be found here [9]. We set all the coefficients of the -12 NRQCD action eq(2),eq(3) c1, c2, c3, c4, c5, c6 and c7 to be 1 and nr-loop parameter of eq(4) n = 3. 10 η The mass parameter m of the Lagrangian eq(1),eq(2), eq(3) is tuned to produce the kinetic mass of b -13 = ff Υ η 10 9.42 GeV. We find that the mass di erences of and b to be 131 MeV. ln|c(t)|

10-14 101 10-15

10-16 100 10-17 16 18 20 22 24 26 (t)|} b time slice(t) η

10-1

(t)|/|C Figure 2. Bc meson correlators obtained at zero momentum using eq(7). Υ ln{|C

-2 10 3.2 Simulations with hisq fermions

Using hisq action we find the mass of Bs meson to be MBs = 5040 MeV. For baryons we used only

-3 the hisq action for strange quark/quarks. 10 + 0 10 20 30 40 50 60 70 For Ω ( 3 ) we used the following formula to calculate its mass time slice(t) bbb 2 3 MΩ = EΩ + (Mη Eη ) Figure 1. Hyperfine splitting of bottomonium 1S states. bbb bbb 2 b − b We find M =14.38 GeV with error = 20 MeV. Ωbbb ± We used overlap action to simulated Bc and Bs mesons and we used hisq action for Bs meson only. 100 Simulations of Bc using hisq action is under way.

10-10 3.1 Simulations with overlap fermions

10-20 In overlap formalism charm quark mass has been tuned to produce kinetic mass of the ηc to be

3.05 GeV. Strange quark has been directly tuned to produce ss¯ pseudoscalar pole mass to be 686 -30

(t)| 10

MeV. Here we used mass formula predicted by chiral perturbation theory to calculate the mass of bbb Ω 2 2 10-40 pseudoscalar s-quarkonium state M = 2M M . This part of the tuning of strange and charm ln|C ss¯ K − π quark mass using overlap action has been done by the TIFR, Mumbai group [10]. As we used kinetic 10-50 mass in tuning the bottom and charm masses the following formula has been used to calculate the mass of Bc. 10-60 1 1 MBc = EBc + (Mηb Eηb ) + (Mηc Eηc ) (12) 2 − 2 − 10-70 0 10 20 30 40 50 60 70 Here EBc , Eηb , Eηc are the simulated masses and Mηb , Mηc are their pdg values. time slice(t) We find M = 6256.5 Mev with error = 20 MeV. Bc ± For Bs we used the following formula Figure 3. Ωbbb baryon correlators obtained by using eq(9). Spin indices has been summed over: CΩbbb (t) = [P3/2.C ] . Here we assumed the summation convention for repeated indices. 1 αβ xk kx αβ M = E + (Mη Eη ) (13) Bs Bs 2 b − b + and we found M = 5096 MeV 30 MeV. For Ω ( 3 ) we find M = E + (M E )=9.81 GeV with error= 40 MeV. Bs ± bbs∗ 2 Ωbbs Ωbbs ηb − ηb ±

5 EPJ Web of Conferences 175, 05031 (2018) https://doi.org/10.1051/epjconf/201817505031 Lattice 2017

+ For Λ ( 1 ) we find M = E + 1 (M E )= 5.45 GeV with error = 30 MeV. bss 2 Λbss Λbss 2 ηb − ηb ±

4 Summary We presented our preliminary results of charmed and strange heavy mesons and baryons. For bottom quark we used NRQCD action corrected up to O(v6) in velocity and up to O(a4) in lattice spacing. For strange and charm quarks we used relativistic overlap and hisq actions. We simulated on N f = 2 + 1 + 1 HISQ gauge configurations generated by MILC collaboration. We found that mass we obtained for Bc meson is in agreement with its experimental value. Also the mass of Ωbbb baryon matches with the value reported by other groups. The strange B meson mass deviates from its experimental value. However it is interesting to see splitting of Bs and B∗s.

5 Acknowledgement Simulations have been carried out on the cluster of School of Physical Sciences, NISER. We thank TIFR, Mumbai group for their overlap propagators. We also thank MILC collaboration for providing HISQ lattices.

6 EPJ Web of Conferences 175, 05031 (2018) https://doi.org/10.1051/epjconf/201817505031 Lattice 2017

1 + 1 For Λbss( ) we find MΛ = EΛ + (Mη Eη )= 5.45 GeV with error = 30 MeV. References 2 bss bss 2 b − b ± [1] G.P. Lepage, L. Magnea, C. Nakhleh, U. Magnea, K. Hornbostel, Phys. Rev. D46, 4052 (1992), 4 Summary hep-lat/9205007 [2] B.A. Thacker, G.P. Lepage, Phys. Rev. D43, 196 (1991) We presented our preliminary results of charmed and strange heavy mesons and baryons. For [3] H.D. Trottier, Phys. Rev. , 6844 (1997), hep-lat/9611026 bottom quark we used NRQCD action corrected up to O(v6) in velocity and up to O(a4) in lattice D55 spacing. For strange and charm quarks we used relativistic overlap and hisq actions. We simulated [4] C.T.H. Davies, K. Hornbostel, G.P. Lepage, A.J. Lidsey, J. Shigemitsu, J.H. Sloan, Phys. Rev. D52, 6519 (1995), hep-lat/9506026 on N f = 2 + 1 + 1 HISQ gauge configurations generated by MILC collaboration. We found that [5] C.T.H. Davies, K. Hornbostel, A. Langnau, G.P. Lepage, A. Lidsey, J. Shigemitsu, J.H. Sloan, mass we obtained for Bc meson is in agreement with its experimental value. Also the mass of Ωbbb baryon matches with the value reported by other groups. The strange B meson mass deviates from its Phys. Rev. D50, 6963 (1994), hep-lat/9406017 [6] T. Burch (2015), 1502.00675 experimental value. However it is interesting to see splitting of Bs and B∗s. [7] S. Meinel, Phys. Rev. D82, 114514 (2010), 1008.3154 [8] S. Meinel, W. Detmold, C.J.D. Lin, M. Wingate, PoS LAT2009, 105 (2009), 0909.3837 5 Acknowledgement [9] A. Bazavov et al. (MILC), Phys. Rev. D87, 054505 (2013), 1212.4768 Simulations have been carried out on the Proton cluster of School of Physical Sciences, NISER. [10] S. Basak, S. Datta, A.T. Lytle, M. Padmanath, P. Majumdar, N. Mathur, PoS LATTICE2013, We thank TIFR, Mumbai group for their overlap propagators. We also thank MILC collaboration for 243 (2014), 1312.3050 providing HISQ lattices.

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