Hyperfine Splitting in the Bottomonium System on the Lattice and in the Continuum
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1 Hyperfine Splitting in the Bottomonium System on the Lattice and in the Continuum Nikolai Zerf in collaboration with M. Baker, A. Penin, D. Seidel Department of Physics University of Alberta Radcor-Loopfest, LA 2015 Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
2 Overview 1 Motivation & Basics 2 Experimental Results 3 Theory Predictions 4 Determination of E hfs 5 Summary Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
3 Motivation M(Υ 1S ) M(η b ) = E hfs Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
4 Motivation M(Υ 1S ) M(η b ) = E hfs Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
5 Motivation M(Υ 1S ) M(η b ) = E hfs Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
6 Motivation M(Υ 1S ) M(η b ) = E hfs Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
7 Motivation M(Υ 1S ) M(η b ) = E hfs Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
8 Motivation M(Υ 1S ) M(η b ) = E hfs Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
9 Motivation M(Υ 1S ) M(η b ) = E hfs Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
10 Motivation M(Υ 1S ) M(η b ) = E hfs Is perturbative QCD breaking down? Is new physics needed? [Domingo et al. 2009] Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
11 Basics Quantum configuration: q s SU(3) Color SU(2) Spin n L I G (J PC ) η b bb (0 + ) Υ 1S bb (1 ) h b bb (1 + ) χ b bb (1 ++ ) Υ 1S production: s = M(Υ1S ) = ± 0.26 MeV [E288 collaboration 1977] [PDG 2014] Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
12 Basics Quantum configuration: q s SU(3) Color SU(2) Spin n L I G (J PC ) η b bb (0 + ) Υ 1S bb (1 ) h b bb (1 + ) χ b bb (1 ++ ) Υ 1S production: s = M(Υ1S ) = ± 0.26 MeV [PDG 2014] Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
13 Basics Quantum configuration: q s SU(3) Color SU(2) Spin n L I G (J PC ) η b bb (0 + ) Υ 1S bb (1 ) h b bb (1 + ) χ b bb (1 ++ ) Υ 1S production: s = M(Υ1S ) = ± 0.26 MeV η b production: [PDG 2014] [BaBar 2008] Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
14 Experimental Results BaBar 2008: Inklusive photon spectrum 3 10 Entries / ( GeV ) (a) E γ (GeV) Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
15 Experimental Results BaBar 2008: (non-peaked background subtracted) Entries/ (0.005 GeV) (b) (GeV) E γ Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
16 Experimental Results BaBar 2008: (non-peaked background subtracted) Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
17 Experimental Results BaBar 2008: (non-peaked background subtracted) Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
18 Experimental Results BaBar 2008: (non-peaked background subtracted) Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
19 Experimental Results BaBar 2008: (η b signal) Entries/ (0.020 GeV) (c) E γ (GeV) Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
20 Experimental Results BaBar 2008: E hfs = ± 2.7 MeV. Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
21 Experimental Results Belle 2012: [Belle 2012] Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
22 Experimental Results Belle 2012: h b (1P) yield, 10 3 / 10 MeV/c 2 h b (2P) yield, 10 3 / 10 MeV/c (a) (b) h b (1P) γη b (1S) h b (2P) γη b (1S) h b (2P) yield, 10 3 / 10 MeV/c (c) h b (2P) γη b (2S) M (n) miss(π + π - γ), GeV/c 2 Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
23 Experimental Results BaBar 2008: Belle 2012: E hfs = ± 2.7 MeV E hfs = 57.9 ± 2.3 MeV Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
24 Theory prediction for E hfs Phenomenologic potential models fitted potentials = QCD potentials? no systematic treatment of radiative corrections Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
25 Theory prediction for E hfs Phenomenologic potential models fitted potentials = QCD potentials? no systematic treatment of radiative corrections Perturbative (pnr)qcd calculation systematic ordering of radiative corrections in α s, v available up to NNLL non-perturbative effects missing Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
26 Theory prediction for E hfs Phenomenologic potential models fitted potentials = QCD potentials? no systematic treatment of radiative corrections Perturbative (pnr)qcd calculation systematic ordering of radiative corrections in α s, v available up to NNLL non-perturbative effects missing Lattice QCD simulation non-perturbative effects demanding for hardware (L 1/Λ QCD & a 1/m b ) Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
27 Theory prediction for E hfs Phenomenologic potential models fitted potentials = QCD potentials? no systematic treatment of radiative corrections Perturbative (pnr)qcd calculation systematic ordering of radiative corrections in α s, v available up to NNLL non-perturbative effects missing Lattice QCD simulation non-perturbative effects demanding for hardware (L 1/Λ QCD & a 1/m b ) Lattice NRQCD simulation non-perturbative effects less demanding for hardware (scales separation) systematic ordering of radiative corrections in α s, v Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
28 Perturbative pnrqcd calculation EFT-Tower: QCD NRQCD Ψ(i /D m)ψ integrating out m ψ (id 0 + D2 2m g s c F 2mσ B)ψ +... integrating out vm (v = velocity) pnrqcd Ĥ φ = E φ, Ĥ = 2 m C Fα s r ( ) + V S +... Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
29 Perturbative pnrqcd calculation EFT-Tower: QCD NRQCD Ψ(i /D m)ψ integrating out m ψ (id 0 + D2 2m g s c F 2mσ B)ψ +... integrating out vm (v = velocity) pnrqcd Ĥ φ = E φ, Ĥ = 2 m C Fα s r ( ) + V S +... Spin dependence: c NRQCD L σ = g F s 2m ψ C σ Bψ + (ψ χ c ) + d F α s σ ψ σψχ cσχ c pnrqcd V S (q 2 ) = (c 2 F 3 2 d σ) 4πC Fα s(q 2 )S 2 3m m 2 Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
30 Perturbative pnrqcd calculation EFT-Tower: QCD NRQCD Ψ(i /D m)ψ integrating out m ψ (id 0 + D2 2m g s c F 2mσ B)ψ +... integrating out vm (v = velocity) pnrqcd Ĥ φ = E φ, Ĥ = 2 m C Fα s r ( ) + V S +... Spin dependence: c NRQCD L σ = g F s 2m ψ C σ Bψ + (ψ χ c ) + d F α s σ ψ σψχ cσχ c pnrqcd V S (q 2 ) = (c 2 F 3 2 d σ) 4πC Fα s(q 2 )S 2 3m E hfs calculated within pnrqcd using TIPT m 2 Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
31 Perturbative pnrqcd calculation E hfs result: [Kniehl, Penin, Pineda, Soto, Steinhauser, Smirnov 2004] E hfs (MeV) ν (GeV) NLL NLO LO LL E hfs = 41 ± 11(th) +9 8 (δα s) Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
32 Lattice NRQCD Simulation Basic idea take vm < 1/a < m as NRQCD factorization scale use NRQCD on the lattice (lnrqcd) to simulate E hfs at finite a extrapolate to a 0 Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
33 Lattice NRQCD Simulation Basic idea take vm < 1/a < m as NRQCD factorization scale use NRQCD on the lattice (lnrqcd) to simulate E hfs at finite a extrapolate to a 0 Matching QCD to lnrqcd QCD d 4 x Ψ(i /D m)ψ integrating out m NRQCD d 4 x ψ (id 0 + D2 2m g s c F 2mσ B)ψ +... descretizing/introducing lattice spacing a lnrqcd a 3 ψ c (x)(1 U 0 + ag F s 2mσ B)ψ(.) +... x Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
34 Lattice NRQCD Simulation Basic idea take vm < 1/a < m as NRQCD factorization scale use NRQCD on the lattice (lnrqcd) to simulate E hfs at finite a extrapolate to a 0 Matching QCD to lnrqcd QCD d 4 x Ψ(i /D m)ψ integrating out m NRQCD d 4 x ψ (id 0 + D2 2m g s c F 2mσ B)ψ +... descretizing/introducing lattice spacing a lnrqcd a 3 ψ c (x)(1 U 0 + ag F s 2mσ B)ψ(.) +... x Determination of MCs Lattice Perturbation Theory Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
35 Spin Dependent Matching Status c F C F α s L σ = g s 2m ψ σ Bψ + (ψ χ c ) + d σ m 2 ψ σψχ cσχ c Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
36 Spin Dependent Matching Status c F C F α s L σ = g s 2m ψ σ Bψ + (ψ χ c ) + d σ m 2 ψ σψχ cσχ c c F (chromomagnetic moment) known including O(α s ) [Hammant, Hart, Hippel, Horgan, Monahan 2011] can be cross checked against lattice extraction Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
37 Spin Dependent Matching Status c F C F α s L σ = g s 2m ψ σ Bψ + (ψ χ c ) + d σ m 2 ψ σψχ cσχ c c F (chromomagnetic moment) known including O(α s ) [Hammant, Hart, Hippel, Horgan, Monahan 2011] can be cross checked against lattice extraction d σ (local four-quark interaction) result available at O(αs ) currently sign of log depends on specific publication (typo) we do not agree with constant of QCD amplitude finite quark mass are kept includes uncontrolled HO terms in 1/m Coulomb singularity numerically subtracted Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
38 Naive Lattice Action Wilson s glue action S g = β 2 x µ,ν ( 1 2N F tr [ ] ) µν + µν 1, β = 2N F /g 2 s, µν = U +µ U +ν U µ U ν, U ±µ (x) = exp{iag s T a A a µ(x ± 1 2 aê µ)}. Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
39 Naive Lattice Action Wilson s glue action S g = β 2 x µ,ν ( 1 2N F tr [ ] ) µν + µν 1, β = 2N F /g 2 s, µν = U +µ U +ν U µ U ν, U ±µ (x) = exp{iag s T a A a µ(x ± 1 2 aê µ)}. Static quark action S ψ = x a 4 ψ D 0 ψ. D + µ = 1 a (U +µ 1), D µ = 1 a (1 U µ). Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
40 Naive Lattice Action Wilson s glue action S g = β 2 x µ,ν ( 1 2N F tr [ ] ) µν + µν 1, β = 2N F /g 2 s, µν = U +µ U +ν U µ U ν, U ±µ (x) = exp{iag s T a A a µ(x ± 1 2 aê µ)}. Static quark action S ψ = x a 4 ψ D 0 ψ. Spin flip action S ψσ = x a 4 i 2m ɛijk ψ 1 σ k 2 (D i D + j + D + i D j ) ψ }{{} G ij D + µ = 1 a (U +µ 1), D µ = 1 a (1 U µ). Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
41 Lattice Action g s 0 Feynman-rules Expansion of a single gauge link for small g s x ± = x ± 1 2 aê µ, U ±µ (x) = 1 + iag s T a A a µ(x ± ) a2 g 2 s 2 Ta T b A a µ(x ± )A b µ(x ± ) +... Switching to momentum space φ(x) = 1 e ip x φ(p), φ {A, ψ, χc }, V 1.BZ : π a < p i π a. Feynman-rules propagators: Inverse of 2-field Fourier coefficient n-vertex: n-field Fourier coefficient (symmetrized) x Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
42 Analytic Integral Treatment Evaluation in the limit aλ 0 [Becher, Melnikov 2002] Full integral I = π a π a d 4 k 1 (2π) 4 [ ( 2 µ a sin(ak µ/2) ) 2 + λ 2 ]. 2 Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
43 Analytic Integral Treatment Evaluation in the limit aλ 0 [Becher, Melnikov 2002] Full integral I = π a π a d 4 k 1 (2π) 4 [ ( 2 µ a sin(ak µ/2) ) 2 + λ 2 ]. 2 α Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
44 Analytic Integral Treatment Evaluation in the limit aλ 0 [Becher, Melnikov 2002] Full integral I = π a π a d 4 k 1 (2π) 4 [ ( 2 µ a sin(ak µ/2) ) 2 + λ 2 ]. 2 α Soft region k λ expand in ak I soft = d 4 k 1 (2π) 4 [k 2 + λ 2 ] 2 α + O(a2 λ 2 ). Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
45 Analytic Integral Treatment Evaluation in the limit aλ 0 [Becher, Melnikov 2002] Full integral I = π a π a d 4 k 1 (2π) 4 [ ( 2 µ a sin(ak µ/2) ) 2 + λ 2 ]. 2 α Soft region k λ expand in ak I soft = d 4 k 1 (2π) 4 [k 2 + λ 2 ] 2 α + O(a2 λ 2 ). Hard region k 1/a expand in aλ I hard = π a π a d 4 k 1 (2π) 4 [ ( 2 µ a sin(ak µ/2) ) 2 + ] 2 α O(a2 λ 2 ). Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
46 Analytic Integral Treatment Evaluation in the limit aλ 0 [Becher, Melnikov 2002] Full integral I = π a π a d 4 k 1 (2π) 4 [ ( 2 µ a sin(ak µ/2) ) 2 + λ 2 ]. 2 α Soft region k λ expand in ak I soft = 1 8π 4 [ 1 2α log λ +... Hard region k 1/a expand in aλ I hard = 1 8π 4 [ 1 2α log a +... ] + O(a 2 λ 2 ). ] + O(a 2 λ 2 ). Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
47 Analytic Integral Treatment Evaluation in the limit aλ 0 [Becher, Melnikov 2002] Full integral I = I soft + I hard = 1 8π 4 [log(aλ) +... ] + O(a2 λ 2 ). Soft region k λ expand in ak I soft = 1 8π 4 [ 1 2α log λ +... Hard region k 1/a expand in aλ I hard = 1 8π 4 [ 1 2α log a +... ] + O(a 2 λ 2 ). ] + O(a 2 λ 2 ). Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
48 4-Fermion Operator Matching QCD lnrqcd = q 2 = M QCD 1PI = C Fαs 2 [ CA ( m 2 q 2 log mq ) ( + (ln 2 1) T F + 1 2πm ) ] q C F ψ σψχ λ 3λ cσχ c, M lnrqcd 1PI = C Fαs 2 [ m 2 q? + 0 T F 2πm ] q 3λ C F ψ σψχ cσχ c + O ( a 2), Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
49 4-Fermion Operator Matching QCD = q 2 = 0 lnrqcd +... M QCD 1PI = C Fαs 2 [ CA ( m 2 q 2 log mq ) ( + (ln 2 1) T F + 1 2πm ) ] q C F ψ σψχ λ 3λ cσχ c, M lnrqcd 1PI = C Fαs 2 [ m 2 (δ + 12 ) ln (λa) C A 2πm ] q q 3λ C F ψ σψχ cσχ c + O ( a 2), d σ = α s [( δ L ) C A + (ln 2 1) T F + C F ]. Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
50 4-Fermion Operator Matching Naive action δ naive = π2 b 2 256π 2 b 3 = calculated analytically naive action simulation has large discretization error Improved lattice action are in use for improved a 0 limit no analytic calculation numeric calculation Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
51 Action Improvements Glue action Wilson Lüscher & Weisz Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
52 Action Improvements Glue action Wilson Lüscher & Weisz Naive G ij cloverleaf G ij improved cloverleaf G ij Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
53 Action Improvements Glue action Wilson Lüscher & Weisz Naive G ij cloverleaf G ij improved cloverleaf G ij Additional smoothing time slice in quark-/anti-quark action x a 4 i 2m ɛijk ψ σ k G ij ψ x a 4 i 2m ɛijk ψ σ k (U t G ij + G ij U t )ψ Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
54 Numeric Setup fastcol.m [NZ] Full automatic lattice amplitude generator (Mathematica) QGRAF [Nogueira 91] diagram generator (FORTRAN) COLOR [Ritbergen,Schellekens,Vermaseren 99] precalculating color amplitudes (FORM) HIPPY [Hart,Hippel,Horgan,Muller 09] numeric expansion coefficients for gauge link configurations numeric Feynman-rules (Python) HPSRC [Hart,Hippel,Horgan,Muller 09] numeric vertex & propagator functions (FORTRAN) CUBA [Hahn 05] integrator package with multicore support (FORTRAN) Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
55 δ for Improved Action Numeric evaluation of the difference δ = δ improved δ naive Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
56 δ for Improved Action 0.04 Λ Λ 2 in a (0) = 6 π 2 δ (λ 2 ) = (0) + c 1 λ 2 + c 2 λ 2 log(λ 2 ) Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
57 Determination of E hfs Lattice simulations with d σ = 0 O(v 4 ) action [HPQCD 12] O(v 6 ) action [HPQCD 14] Four quark operator contribution 4C F α s E hfs = d σ m 2 ψ(0) 2 q mb, α s, ψ(0) 2 from the lattice Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
58 Determination of E hfs Ehfs MeV v 4 action v 6 action New value: a 2 fm 2 Ehfs th = 52.9 ± 5.5 MeV Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
59 Summary of E hfs results Experiment Υ(3S) decays (Babar, 2008) (stat) ± 2.7(syst) Υ(2S) decays (Babar, 2009) (stat) ± 2.0(syst) Υ(3S) decays (CLEO, 2010) 68.5 ± 6.6(stat) ± 2.0(syst) h b (1P) decays (Belle, 2012) 57.9 ± 1.5(stat) ± 1.8(syst) PDG average 62.3 ± 3.2 Theory NRQCD, NLL (2004) 41 ± 11(th) +9 8 (δα s) Lattice QCD (MILC, 2009) 54.0 ± Lattice NRQCD O(v 4 ) (HPQCD, 2012) 70 ± 9 Lattice NRQCD O(v 6 ) (HPQCD, 2013) 62.8 ± 6.7 Lattice NRQCD, new result 52.9 ± 5.5 Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
60 Summary of E hfs results Babar 2008 Babar 2009 CLEO 2010 Belle 2012 PDG average NRQCD NLL, KPPSS 2004 lqcd MILC 2009 lnrqcd HPQCD 2012 lnrqcd HPQCD 2013 lnrqcd, new result E hfs MeV Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
61 Summary 1 employed perturbative lnrqcd to establish a radiative UV improvement (4FO) on top of lnrqcd simulations 2 Ehfs th = 52.9 ± 5.5 MeV 3 QCD works / no evidence for new physics 4 lattice amplitude generator Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L / 21
62 M. Buffer Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L
63 Checks Naive action analytic & numeric setup yield corrections to the static potential analytic result for dσ is gauge independent numeric calculation λ 0 = analytic calculation analytic Feynman rules checked independently (MATHEMATICA) integrands of amplitudes checked against MATHEMATICA implementation Improved action numeric log coefficient = analytic log coefficient Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L
64 Numeric Setup Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L
65 Feynman-rules for naive action a = 1 δi j δs 2 s1 1 e ip 0 g s T a δ µ0 δ s 2 s 1 e i(p k 0) δ abδ µν ˆk 2 +λ 2 gs 2m Ta δ µi ɛ ilm (σ m ) s 2 s 1 k l ρ kil g2 s 2m δ µ 1 iδ µ2 j(σ m ) s [ 2 s 1 f a 1 a 2 c CA c ( ) ɛ ijm k 1+2 ji ζ ij ( + δ a1 a 2 C S ɛ ijm k 1+2 ji ζ ij δ ij ɛ ilm )] k 1+2 ll ζ jl Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L
66 Shorthands k µ = 2 sin ( 1 2 k µ), ρ kil = cos{ 1 2 (k i k l + 2p i 2p l )}, ζ ij = sin{( 1 2 (k 1i + k 2i k 1j k 2j + 2p i 2p j )}, ζ ij = cos{( 1 2 (k 1i + k 2i k 1j k 2j + 2p i 2p j )}, k 1+2 ij = 2 sin{( 1 2 (k 1i + k 2j )}, k 1+2 ij = 2 cos{( 1 2 (k 1i + k 2j )}, C c A = i 2 Tc, C S = I 2,F N F I NF N F δ 1 ab d abct c. Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L
67 Numeric Data L&W 4ICL action Feynman gauge:ξ 0 fbox Λ numeric log c fixed log fit free log fit : numeric data Λ 2 in a Nikolai Zerf (Dep. of Phys. UofA) E hfs = M(Υ 1S ) M(η b a, a 0 R-L
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