Determination of Λ MS from the static QQ potential in momentum space

Transcription

1 Determination of Λ MS from the static QQ potential in momentum space Antje Peters Goethe Universität Frankfurt January 16, 2014

2 Overview 1 Introductory notes and motivation 2 My investigations 3 Preliminary results 4 Difficulties Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

3 Introductory notes and motivation QCD problems are difficult... G n (x 1,.., x n ) = Z 0 Z δ n δj(x 1 )..δj(x n ) e(s I[ δ δj ]) exp( 1 2 d 4 x d 4 yj(x) F (x, y)j(y)) J=0 Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

4 Introductory notes and motivation QCD problems are difficult... G n (x 1,.., x n ) = Z 0 Z δ n δj(x 1 )..δj(x n ) e(s I[ δ δj ]) exp( 1 2 d 4 x d 4 yj(x) F (x, y)j(y)) J=0 Since S I is not a quadratic term no chance for exact analytical calculation. Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

5 Introductory notes and motivation QCD problems are difficult... G n (x 1,.., x n ) = Z 0 Z δ n δj(x 1 )..δj(x n ) e(s I[ δ δj ]) exp( 1 2 d 4 x d 4 yj(x) F (x, y)j(y)) J=0 Since S I is not a quadratic term no chance for exact analytical calculation. perturbation theory F = F 0 + ɛf 1 + ɛ 2 F 2 + ɛ 3 F Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

6 Introductory notes and motivation QCD problems are difficult... G n (x 1,.., x n ) = Z 0 Z δ n δj(x 1 )..δj(x n ) e(s I[ δ δj ]) exp( 1 2 d 4 x d 4 yj(x) F (x, y)j(y)) J=0 Since S I is not a quadratic term no chance for exact analytical calculation. perturbation theory F = F 0 + ɛf 1 + ɛ 2 F 2 + ɛ 3 F lattice gauge theory d 4 x n a4 Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

7 Motivation Introductory notes and motivation The parameter Λ defines the scale for dimensionful perturbative results (cf. running coupling α s (Q) 1 ) log Q2 Λ 2 Analogous to the lattice spacing a for lattice results Results not reasonable without a scale! In the MS renormalization scheme: Λ MS Determine a result by a lattice computation and a perturbative calculation solve for Λ MS suitable observable : QQ potential simple computation on the lattice (no quark propagators) precise analysis in perturbation theory Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

8 Overview Introductory notes and motivation overall aim: calculate Λ MS method: compare lattice and perturbation theory results in the small range of QQ separation where both approaches are reliable approach: lattice computation in momentum space (more precise than position space 1?) 1 K. Jansen et al. [ETM Collaboration], JHEP 1201 (2012) 025 [arxiv: [hep-ph]]. Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

9 Introductory notes and motivation The story so far... Determination of Λ MS in position space (i.e. comparison of lattice and perturbation theory results in position space) Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

10 Introductory notes and motivation The story so far... Determination of Λ MS in position space (i.e. comparison of lattice and perturbation theory results in position space) Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

11 Introductory notes and motivation The story so far... Determination of Λ MS in position space (i.e. comparison of lattice and perturbation theory results in position space) Result: Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

12 Introductory notes and motivation The story so far... Determination of Λ MS in position space (i.e. comparison of lattice and perturbation theory results in position space) Result: Λ MS = 315(30)MeV Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

13 Introductory notes and motivation Why momentum space? Lattice theory lives in position space, but perturbation theory lives in momentum space. Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

14 Introductory notes and motivation Why momentum space? Lattice theory lives in position space, but perturbation theory lives in momentum space. First idea: Fourier transform perturbation theory before comparison to lattice results! Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

15 Introductory notes and motivation Why momentum space? Lattice theory lives in position space, but perturbation theory lives in momentum space. First idea: Fourier transform perturbation theory before comparison to lattice results! Problem in position space: V pert (r) = d 3 pe ipr Ṽ pert (p) is wrong for p < Λ ( 300 MeV) Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

16 Introductory notes and motivation Why momentum space? Lattice theory lives in position space, but perturbation theory lives in momentum space. First idea: Fourier transform perturbation theory before comparison to lattice results! Problem in position space: V pert (r) = d 3 pe ipr Ṽ pert (p) is wrong for p < Λ Unavoidable mistakes! ( 300 MeV) Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

17 Introductory notes and motivation Why momentum space? Lattice theory lives in position space, but perturbation theory lives in momentum space. First idea: Fourier transform perturbation theory before comparison to lattice results! Problem in position space: V pert (r) = d 3 pe ipr Ṽ pert (p) is wrong for p < Λ Unavoidable mistakes! ( 300 MeV) New idea: get V (p) directly and consider it only for large momenta p Λ Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

18 Introductory notes and motivation The static QQ-potential Ṽ (p) = C F 4π p 2 α V [α s (µ), L(µ, p)] with L(µ, p) = ln µ2 p 2 L C F = 4 3 Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

19 Introductory notes and motivation The static QQ-potential with More than a series expansion: Ṽ (p) = C F 4π p 2 α V [α s (µ), L(µ, p)] L(µ, p) = ln µ2 p 2 L C F = 4 3 { α V [α s (µ), L(µ, p)] =α s (µ) 1 + α ( ) 2 s(µ) 4π P αs (µ) 1(L) + P 2 (L) 4π ( ) 3 αs (µ) + [P 3 (L) + a 3ln ln α s (µ)] +...} 4π Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

20 Introductory notes and motivation The static QQ-potential with More than a series expansion: Ṽ (p) = C F 4π p 2 α V [α s (µ), L(µ, p)] L(µ, p) = ln µ2 p 2 L C F = 4 3 { α V [α s (µ), L(µ, p)] =α s (µ) 1 + α ( ) 2 s(µ) 4π P αs (µ) 1(L) + P 2 (L) 4π ( ) 3 αs (µ) + [P 3 (L) + a 3ln ln α s (µ)] +...} 4π Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

21 Introductory notes and motivation A renormalization group invariant α V [α s (µ), L(µ, p)] is a renormalization group invariant, so it fulfils Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

22 Introductory notes and motivation A renormalization group invariant α V [α s (µ), L(µ, p)] is a renormalization group invariant, so it fulfils ( L + α ) s(µ) β[α s (µ)] α V [α s (µ), L(µ, p)] = 0 2 α s (µ) Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

23 Introductory notes and motivation A renormalization group invariant α V [α s (µ), L(µ, p)] is a renormalization group invariant, so it fulfils ( L + α ) s(µ) β[α s (µ)] α V [α s (µ), L(µ, p)] = 0 2 α s (µ) The QCD β function fulfils Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

24 Introductory notes and motivation A renormalization group invariant α V [α s (µ), L(µ, p)] is a renormalization group invariant, so it fulfils ( L + α ) s(µ) β[α s (µ)] α V [α s (µ), L(µ, p)] = 0 2 α s (µ) The QCD β function fulfils β[α s (µ)] = µ dα s (µ) α s (µ) dµ Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

25 Introductory notes and motivation A renormalization group invariant α V [α s (µ), L(µ, p)] is a renormalization group invariant, so it fulfils ( L + α ) s(µ) β[α s (µ)] α V [α s (µ), L(µ, p)] = 0 2 α s (µ) The QCD β function fulfils β[α s (µ)] = µ dα s (µ) α s (µ) dµ Integration yields to leading order ( β[α s(µ)] = β0 2π Λ µ 1 µ dµ = Λ = µe 2π β 0 αs (µ) α s (µ) αs (µ) β0): 2π 1 α s(µ ) 2 dαs(µ ) Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

26 Introductory notes and motivation A renormalization group invariant α V [α s (µ), L(µ, p)] is a renormalization group invariant, so it fulfils ( L + α ) s(µ) β[α s (µ)] α V [α s (µ), L(µ, p)] = 0 2 α s (µ) The QCD β function fulfils β[α s (µ)] = µ dα s (µ) α s (µ) dµ Integration yields to leading order : Λ = µe 2π β 0 αs (µ) µ = Λ = α s (µ) = (for perturbation theory: µ Λ) Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

27 Introductory notes and motivation The quantity Λ MS Λ MS was introduced as a parameter of integration Determination in modified Minimal Subtraction Scheme Λ MS Λ MS tells us in which energy range perturbation theory is valid. Λ MS is necessary to interpret a QCD result properly. Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

28 Starting point My investigations use n f = 2 gauge link configurations generated by the ETMC a = fm β = 4.35 (L/a) 3 T /a = Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

29 My investigations Extrapolation of lattice data extrapolation in position space V(r)a lattice data 500 extrapolated data data for fit r/a Extrapolation of lattice data Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

30 My investigations Transformation to momentum space Fourier transform Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

31 My investigations Transformation to momentum space Fourier transform Ṽ ( k) = 1 n 3 N 1 n x,n y,n z =0 V (n)e 2πi k n N Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

32 My investigations Transformation to momentum space Fourier transform Ṽ ( k) = 1 n 3 N 1 n x,n y,n z =0 V (n)e 2πi k n N cylinder cut to avoid lattice spacing errors (non-physical!) 0 cylinder cut -2e-08 V(p) in MeV -2-4e-08-6e-08-8e-08-1e p in MeV Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

33 My investigations Symanzic improvement Symanzik s effective theory: handle lattice artefacts calculate potential at tree level (Symanzic action) compute the quark propagator with a recursive algorithm compare with the perturbation theory result in momentum space at leading order 1 p V ( p) p perturbative (leading order) propagator p in MeV Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

34 My investigations Symanzic improvement Symanzik s effective theory: handle lattice artefacts calculate potential at tree level (Symanzic action) compute the quark propagator with a recursive algorithm compare with the perturbation theory result in momentum space at leading order 1 p 2 match tree level data 0-5e-06 improved original V(p) in MeV -2-1e e-05-2e e-05-3e p in MeV Antje Peters (Goethe improved Universität potential Frankfurt) in momentum Determinationspace of Λ January 16, / 22

35 Fitting procedure My investigations fit perturbation theory fomula to lattice results leading order (LO) up to next to-next to-next to-leading order (NNNLO) in coupling α s (µ) dependency of scale parameter µ? set µ = p estimate error Jackknife Method find proper fit range lattice assumptions and perturbation theory both reasonable? Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

36 A first result My investigations In leading order, lattice data confirms former result Λ MS = 315 MeV 4e-07 2e-07 N=256 LO, Λ=315 MeV V(p) in MeV e-07-4e-07-6e-07-8e-07-1e p in MeV good agreement at first glance Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

37 Results Preliminary results different fitting procedures yield different results estimate average value for Λ MS Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

38 Results Preliminary results different fitting procedures yield different results estimate average value for Λ MS Λ - MS in MeV fit range width 700 MeV, 3 parameters 250 A 200 B 150 C 100 D MeV center of fit range in momentum space in MeV Λ - MS in MeV fit range width 700 MeV, 4 parameters A 150 B C 100 D MeV center of fit range in momentum space in MeV Λ - MS in MeV fit range width 1500 MeV, 3 parameters 250 A 200 B 150 C 100 D MeV center of fit range in momentum space in MeV Λ - MS in MeV fit range width 1500 MeV, 4 parameters A 150 B C 100 D MeV center of fit range in momentum space in MeV Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

39 Conclusions Preliminary results Λ MS in right order of magnitude: Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

40 Conclusions Preliminary results Λ MS in right order of magnitude: fit range width 700 MeV, 4 parameters Λ - MS in MeV A B C D 315 MeV center of fit range in momentum space in MeV Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

41 Conclusions Preliminary results Λ MS in right order of magnitude: fit range width 700 MeV, 4 parameters Λ - MS in MeV A B C D 315 MeV center of fit range in momentum space in MeV 250 MeV MeV 2 (preliminary!) = (335 ± 50) MeV Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

42 Conclusions Preliminary results Λ MS in right order of magnitude: fit range width 700 MeV, 4 parameters Λ - MS in MeV A B C D 315 MeV center of fit range in momentum space in MeV 250 MeV MeV 2 (preliminary!) = (335 ± 50) MeV Wide spreading of values No better precision (?) Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

43 Conclusions Preliminary results Λ MS in right order of magnitude: fit range width 700 MeV, 4 parameters Λ - MS in MeV A B C D 315 MeV center of fit range in momentum space in MeV 250 MeV MeV 2 (preliminary!) = (335 ± 50) MeV Wide spreading of values No better precision (?) But the former result was definitely confirmed. Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

44 Conclusions Preliminary results Λ MS in right order of magnitude: fit range width 700 MeV, 4 parameters Λ - MS in MeV A B C D 315 MeV center of fit range in momentum space in MeV 250 MeV MeV 2 (preliminary!) = (335 ± 50) MeV Wide spreading of values No better precision (?) But the former result was definitely confirmed. But there are still almost 3 months left!!! Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

45 Difficulties The wiggly line-quest alpha lattice alpha perturbative 0.6 α(p) p in 10 3 MeV wiggly behaviour of α Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

46 Difficulties 1 analysis of toy models to reproduce wiggly line potential potential in position space gap 0 gap 0.01 gap gap r/a 2 gaussian blur 3 adapted fitting procedures for extrapolation in position space Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

47 Difficulties 1 analysis of toy models to reproduce wiggly line V(p) 1e-07 5e-08 0 potential in momentum space gap 0 gap 0.01 gap gap e-08-1e p in MeV 2 gaussian blur 3 adapted fitting procedures for extrapolation in position space Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

48 Difficulties 1 analysis of toy models to reproduce wiggly line 2 gaussian blur V(p) 3e e-05 2e e-05 1e-05 5e e-06 potential in momentum space 0 it 1 it, σ Gauss =sqrt(0.5) 5 it, σ Gauss =sqrt(0.5) 1 it, σ Gauss =1.0 5 it, σ Gauss =1.0 1 it, σ Gauss =sqrt(2.0) 5 it, σ Gauss =sqrt(2.0) -1e p in MeV 3 adapted fitting procedures for extrapolation in position space Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

49 Difficulties 1 analysis of toy models to reproduce wiggly line 2 gaussian blur 3 adapted fitting procedures for extrapolation in position space 2 parameters: V (r) = π 1 + σr + 12 r r 2 3 parameters: V (r) = π 1 + σr r r 2 r 3 4 parameters: V (r) = π 1 + σr r r 2 r 3 r 4 Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

50 Difficulties 1 Limiting effects like in position space. 2 Momentum space brings its own problems. Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

51 Difficulties 1 Limiting effects like in position space. 2 Momentum space brings its own problems. 3 But there are still almost 3 months left!!! Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22

52 Difficulties Thanks for your attention. Antje Peters (Goethe Universität Frankfurt) Determination of Λ January 16, / 22