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Heavy Hadrons Spectra on Lattice Using NRQCD

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Heavy Hadrons Spectra on Lattice Using NRQCD EPJ Web of Conferences 175, 05031 (2018) https://doi.org/10.1051/epjconf/201817505031 Lattice 2017 Heavy hadrons 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 quark is very non-relativistic in bottomonium states. In a mixed system like charmed B meson, the b quark has less velocity than it has in bottomonium states and in strange B meson 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 quarks and NRQCD for b quark we simulated spectra of charmed and strange B mesons and also few other baryons. 1 Introduction Lattice methods are powerful techniques in analyzing the spectrum of hadrons. However for hadrons containing heavy quarks particularly bottom quark are difficult to analyze. For spectrum calculation it is necessary that aM << 1. For light quarks it is true but for charm quark 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 quarkonium 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 pions M 140 MeV whereas masses of up or down quark 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 strange B meson 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 particle 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 spin space. Ω=I for × pseudoscalar particles 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 strange quark. 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.
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