Precision Determination of α S (m Z ) from Thrust Data

Transcription

1 Precision Determination of α (m Z ) from Thrust Data Z Michael Fickinger University of Arizona Phys.Rev. D83 (2011) Taskforce: A. Hoang - U. Vienna I. tewart,v. Mateu& R. Abbate -MIT M. Fickinger - UA Frontiers in CD, INT, 6 th October 2011

2 The trong Coupling Constant α α is a key parameter for the analysis of all collider experiments CD: α (M Z )=0.1184(07). Bethke, Eur. Phys. J. C64 (2009) JADE GeV Bethke et al. ALEPH GeV Dissertori et al.

3 Event shape variable:assigns a number to the shape of an event

4 Event shape variable:assigns a number to the shape of an event r p j j tˆ Thrust: T max t ˆ r use = 1 -T p j j

5 Event shape variable:assigns a number to the shape of an event r p j j tˆ Thrust: T max t ˆ r use = 1 -T p j j Measures 2-jet likeness ideal 2-jet event spherical event

6 Measures 2-jet likeness Event shape variable:assigns a number to the shape of an event j j j j t p t p T r r ˆ max ˆ Thrust: use = 1 -T ideal 2-jet event spherical event ( ) K + > + + = ) 0 ( ) ( ) ( δ α δ σ σ R c d d Observable is very sensitive to α!

7 Outline Results Theory Numerical Analysis Moment Fits ummary

8 Results

9 α (m Z ) = ± (0.0002) expt ± (0.0005) hadr ± (0.0009) pert with 2 χ dof = 0.91 = 91.2 GeV

10 α (m Z ) = ± (0.0002) expt ± (0.0005) hadr ± (0.0009) pert with 2 χ dof = 0.91 = 91.2 GeV

11 Fit to Distribution: α (m Z ) = ± (0.0002) expt ± (0.0005) hadr ± (0.0009) pert with 2 χ dof = data points Ω ( µ, R ) = R = 2GeV 1 ± µ = preliminary Fit to First Moment: α (m Z ) = ± (0.0015) expt ± (0.0005) hadr ± (0.0007) pert with 2 χ dof = data points Ω ( µ, R 1 ) = ± 0.061

12 α (m Z ) = ± (0.0002) expt ± (0.0005) hadr ± (0.0009) pert with 2 χ dof = 0.91

13 Theory

14 hard scale(cm energy) jet scale(invariant mass of all energetic particles in one hemisphere) soft scale(uniform soft radiation)

15 hard scale(cm energy) jet scale(invariant mass of all energetic particles in one hemisphere) soft scale(uniform soft radiation) >> Λ / CD >> Λ CD >> >> >> Λ CD >> Λ CD

16 oft Collinear Effective Theory (CET) Bauer, Fleming, Luke, Pirjol, Rothstein, tewart Describes light-like particles (collinear) interacting with a low energetic background (soft) Expansion in Power counting: soft: collinear: λ Λ CD ( ) ( p, p, p, λ, λ ) + ( ) ( 2 p, p, p λ,1 λ) pµ = λ p µ = +,

17 Perturbativeeries 1 σ 0 dσ 2 3 = δ ( ) + α R1 ( ) + α R2 ( ) + α R3 ( ) +K d

18 Perturbativeeries 1 σ 0 dσ 2 3 = δ ( ) + α R1 ( ) + α R2 ( ) + α R3 ( ) +K d Problems if: α t1 coefficients not of O(1)

19 Perturbativeeries Problems if: α t1 coefficients not of O(1) +K = ) ( ) ( ) ( ) ( α α α δ σ σ R R R d d ) ( ln ) ( n i R

20 Perturbativeeries Problems if: α t1 coefficients not of O(1) +K = ) ( ) ( ) ( ) ( α α α δ σ σ R R R d d CET gives factorization: ) ( ln ) ( n i R ( ) µ ( ) J µ J ( ) H µ H ( ) ( ) ( ) ( ) CD J H s J H d d Λ mod 0 ˆ 1 µ µ µ σ σ

21 Perturbativeeries 1 σ 0 dσ 2 3 = δ ( ) + α R1 ( ) + α R2 ( ) + α R3 ( ) +K d Problems if: Ri ( ) n ln ( ) α t1 CET gives factorization: H ( µ H ) coefficients not of O(1) U i from RGE J ( µ J ) 1 σ 0 d ˆ σ s d H ( µ ) J ( µ ) ( µ ) ( Λ ) H J mod CD ( µ ) CET reorganizes the power series in terms of (α m ln n )

22 Factorization Theorem for Thrust dσ d = dk d ˆ σ s d + d ˆ σ ns d + d ˆ σ b d ( ) ( ) ( ) k mod α Λ k 2 + O σ 0 CD Methode to simultaneously describe all three regions and multiple : profile functions ingular partonic: NNLO matching, including full 3-loop hard function N 3 LL resummationof large logs Nonsingular partonic: fixed order thrust distribution (subleading orders in CET) Nonperturbative soft function: nonperturabtive effects treated within field theory Operator Product Expansion in tail Ω 1 Interface between power and perturbative corrections: renormalon subtraction b-mass effects (~2% effect) ED effects (~2% effect) axial anomaly at O(α 2 ) (~1% effect)

23 Factorization Formula for all Thrust peak: sum large logs, nonperturbative soft fct. Profile Functions: >> Λ / CD >> Λ CD tail: sum large logs, series of nonperturbative power corrections >> >> >> Λ CD far tail: fixed order perturbation theory, power corrections >> Λ CD cales must equal in the far tail: turns of resummation

24 Far Tail

25 Factorization Theorem for Thrust dσ d = dk d ˆ σ s d + d ˆ σ ns d + d ˆ σ b d ( ) ( ) ( ) k mod α Λ k 2 + O σ 0 CD Methode to simultaneously describe all three regions and multiple : profile functions Moch, Vermaseren, Vogt ingular partonic: Baikov et al. Becher, Neubert NNLO matching, including full 3-loop hard function chwartz; Fleming et al. N 3 LLresummationof large logs Becher, chwartz; Hoang, Kluth Dasgupta, alam Nonsingular partonic: Becher, chwartz fixed order thrust distribution (subleading orders in CET) Nonperturbative soft function: nonperturabtive effects treated within field theory Operator Product Expansion in tail Ω 1 Interface between power and perturbative corrections: renormalon subtraction b-mass effects (~2% effect) ED effects (~2% effect) axial anomaly at O(α 2 ) (~1% effect)

26 umming large Logarithms H ( µ H ) J ( µ J ) ( µ ) H J ln n µ H ( µ ) ln ( ) H ( ) ( ) n µ µ ln J n µ ( µ ) ln ( ) ( ) ( ) ( ) n µ n µ n µ U ln,ln,ln ( ) i H J J etting μ H,μ J,μ to their natural scales much better convergence!

27 Factorization Theorem for Thrust dσ d = dk d ˆ σ s d + d ˆ σ ns d + d ˆ σ b d ( ) ( ) ( ) k mod α Λ k 2 + O σ 0 CD Methode to simultaneously describe all three regions and multiple : profile functions ingular partonic: NNLO matching, including full 3-loop hard function N 3 LL resummationof large logs Nonsingular partonic: Ellis et al., Catani et al., Gehrmann et al., Weinzierl fixed order thrust distribution (subleading orders in CET) Nonperturbative soft function: nonperturabtive effects treated within field theory Operator Product Expansion in tail Ω 1 Interface between power and perturbative corrections: renormalon subtraction b-mass effects (~2% effect) ED effects (~2% effect) axial anomaly at O(α 2 ) (~1% effect) d ˆ σ ns d = dσ d fixed order d ˆ σ s d no resum.

28 Factorization Theorem for Thrust dσ d = dk d ˆ σ s d + d ˆ σ ns d + d ˆ σ b d ( ) ( ) ( ) k mod α Λ k 2 + O σ 0 CD Methode to simultaneously describe all three regions and multiple : profile functions ingular partonic: NNLO matching, including full 3-loop hard function N 3 LL resummationof large logs Nonsingular partonic: fixed order thrust distribution (subleading orders in CET) Nonperturbative soft function: nonperturabtive effects treated within field theory Operator Product Expansion in tail Ω 1 Interface between power and perturbative corrections: renormalon subtraction b-mass effects (~2% effect) ED effects (~2% effect) axial anomaly at O(α 2 ) (~1% effect) Hoang, tewart Ligeti, tewart, Tackmann Lee, terman Korchemsky, terman

29 NonperturbativeCorrections oft function from CET factorization: ( k, µ ) 0 TrY Y δ ( k i ) Y Y = N n n n n c Factorization in perturbative and non-perturbative part: 0 i θ ( in in ) in + θ ( in in ) in = part ( k, µ ) dk' ( k k', µ ) mod ( k') complete basis of functions Hoang, tewart Ligeti, tewart, Tackmann OPE: part part d ( k, µ ) = ( k, µ ) 2Ω1 ( k, µ ) dk +K

30 Are non-perturbativeeffects negligible? Leading effect: shift in the thrust distribution 2 Λ/ Λ ( ) 1 dσ = h 2 σ d Manohar, Wise; Webber; Dokshitzer, Weber; Akhoury, Zakharov; Nason, eymour; Korchemsky, terman; Movilla Fernandez, Bethke, Biebel, Kluth

31 Are non-perturbativeeffects negligible? Leading effect: shift in the thrust distribution 2 Λ/ 1 dσ = h σ d Λ Λ ( 2 ) h( ) 2 h' ( ) Λ/ `1

32 Are non-perturbativeeffects negligible? Leading effect: shift in the thrust distribution 2 Λ/ 1 dσ = h σ d hproportional to α => Λ Λ ( 2 ) h( ) 2 h' ( ) δα α h Λ/ `1 Λ ( ) h( ) h h' ( ) 2 ( ) h ( ) 2 Λ

33 Are non-perturbativeeffects negligible? Leading effect: shift in the thrust distribution 2 Λ/ 1 dσ = h σ d hproportional to α => Λ Λ ( 2 ) h( ) 2 h' ( ) δα α h Λ/ `1 Λ ( ) h( ) h h' ( ) 2 ( ) h ( ) 2 Λ

34 Are non-perturbativeeffects negligible? Leading effect: shift in the thrust distribution 2 Λ/ 1 dσ = h σ d hproportional to α => Λ Λ ( 2 ) h( ) 2 h' ( ) δα α h Λ/ `1 Λ ( ) h( ) h h' ( ) 2 ( ) h ( ) 2 Λ Λ 0.3 GeV and h /h -14±4 in tail region(see figure) δα /α -(9±3)%

35 Are non-perturbativeeffects negligible? Leading effect: shift in the thrust distribution 2 Λ/ 1 dσ = h σ d hproportional to α => Λ Λ ( 2 ) h( ) 2 h' ( ) δα α h Λ/ `1 Λ ( ) h( ) h h' ( ) 2 ( ) h ( ) 2 Λ Λ 0.3 GeV and h /h -14±4 in tail region(see figure) δα /α -(9±3)% NOT negligible for 2% analysis

36 Non-perturbativeeffects Use tail fit results to predict peak

37 Non-perturbativeeffects Use tail fit results to predict peak much better prediction for peak

38 Non-perturbativeeffects Use tail fit results to predict peak much better prediction for peak α from full analysisis approximately 9% smaller than α without model, as predicted.

39 Factorization Theorem for Thrust dσ d = dk d ˆ σ s d + d ˆ σ ns d + d ˆ σ b d ( ) ( ) ( ) k mod α Λ k 2 + O σ 0 CD Methode to simultaneously describe all three regions and multiple : profile functions ingular partonic: NNLO matching, including full 3-loop hard function N 3 LL resummationof large logs Nonsingular partonic: fixed order thrust distribution (subleading orders in CET) Nonperturbative soft function: nonperturabtive effects treated within field theory Operator Product Expansion in tail Ω 1 Interface between power and perturbative corrections: renormalon subtraction b-mass effects (~2% effect) ED effects (~2% effect) axial anomaly at O(α 2 ) (~1% effect) Hoang, tewart Hoang, Jain, cimemi, tewart

40 Renormalonubstraction M perturbativeseries includes fluctuations with arbitrarily small momenta large unphysical corrections Both part and Ω 1 suffer from renormalon Introduce gap parameter Δ: mod ( k) mod ( k 2 ) Hoang, tewart and = ( R, µ ) +δ ( R, µ ) renormalon free soft function: ( k, ) = dk' k µ ( ) 2δ part mod e ( k k', ) ( k' 2 ) µ

41 renormalon substracted renormalon NOT substracted M perturbative series includes fluctuations with arbitrarily small momenta large unphysical corrections

42 Factorization Theorem for Thrust dσ d = dk d ˆ σ s d + d ˆ σ ns d + d ˆ σ b d ( ) ( ) ( ) k mod α Λ k 2 + O σ 0 CD Methode to simultaneously describe all three regions and multiple : profile functions ingular partonic: NNLO matching, including full 3-loop hard function N 3 LL resummationof large logs Nonsingular partonic: fixed order thrust distribution (subleading orders in CET) Nonperturbative soft function: nonperturabtive effects treated within field theory Operator Product Expansion in tail Ω 1 Interface between power and perturbative corrections: renormalon subtraction b-mass effects (~2% effect) Fleming, Hoang, Mantry, tewart ED effects (~2% effect) axial anomaly at O(α 2 ) (~1% effect)

43 Factorization Theorem for Thrust dσ d = dk d ˆ σ s d + d ˆ σ ns d + d ˆ σ b d ( ) ( ) ( ) k mod α Λ k 2 + O σ 0 CD Methode to simultaneously describe all three regions and multiple : profile functions ingular partonic: NNLO matching, including full 3-loop hard function N 3 LL resummationof large logs Nonsingular partonic: fixed order thrust distribution (subleading orders in CET) Nonperturbative soft function: nonperturabtive effects treated within field theory Operator Product Expansion in tail Ω 1 Interface between power and perturbative corrections: renormalon subtraction b-mass effects (~2% effect) ED effects(~2% effect) axial anomaly at O(α 2 ) (~1% effect)

44 Factorization Theorem for Thrust dσ d = dk d ˆ σ s d + d ˆ σ ns d + d ˆ σ b d ( ) ( ) ( ) k mod α Λ k 2 + O σ 0 CD Methode to simultaneously describe all three regions and multiple : profile functions ingular partonic: NNLO matching, including full 3-loop hard function N 3 LL resummationof large logs Nonsingular partonic: fixed order thrust distribution (subleading orders in CET) Nonperturbative soft function: nonperturabtive effects treated within field theory Operator Product Expansion in tail Ω 1 Interface between power and perturbative corrections: renormalon subtraction b-mass effects (~2% effect) ED effects (~2% effect) axial anomaly at O(α 2 ) (~1% effect) Kniehl, Kuhn Hagiwara, Kuruma, Yamada

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46 Comparison with similar analysis sum logs power corrections data α (M Z ) Dissertori et al. no Monte Carlo (MC) ALEPH (34) Dissertori et al. NLL Monte Carlo ALEPH (39) Becher, chwartz N3LL uncertainty from MC ALEPH, OPAL (21) Davison, Webber NLL effective coupling model Most of data (28) Bethke et al. NLL Monte Carlo JADE (51) Becher, chwartz: no nonperturbative soft function different profile function different way of calculating binned cross section

47 Numerical Analysis

48 Two parameter fit in tail region(factorization formula valid for all tau) Ω 1 is the 1 st nonperturbativepower correction LEP LAC DEY KEK Experiment ALEPH DELPHI OPAL L3 LD TAO JADE AMY Values of {91.2, 133.0, 161.0, 172.0, 183.0, 189.0, 200.0, 206.0} {45.0, 66.0, 76.0, 89.5, 91.2, 93.0, 133.0, 161.0, 172.0, 183.0, 189.0, 192.0, 196.0, 200.0, 202.0, 205.0, 207.0} {91.0, 133.0, 161.0, 172.0, 177.0, 183.0, 189.0, 197.0} {41.4, 55.3, 65.4, 75.7, 82.3, 85.1, 91.2, 130.1, 136.1, 161.3, 172.3, 182.8, 188.6, 194.4, 200.0, 206.2} {91.2} {(14.0), (22.0), 35.0, 44.0} {35.0, 44.0} {55.2}

49 ln Order Counting dσ k k k k ( ) ~ ( α ln) ln+ ( α ln) + α ( α ln) + α 2 ( α ln) + K dy LL NLL NNLL N 3 LL cusp non-cusp matching alphas LL 1 - tree 1 NLL 2 1 tree 2 NNLL N 3 LL 4 pade LL 1 - tree 1 NLL NNLL N 3 LL 4 pade Primed counting is better if fixed order results are important

50 Experimental and Hadronic Uncertainty σ +σ Ω 2 2 exp 1

51 PerturbativeUncertainty 12 theory parameters: 6 parameters for the variation of the renormalization scales 3 parameters related to the statistical uncertainties of numerical fixed order calculations 3 parameters for Padéapproximants of unknown constants 1. Flat random scan over theory parameters 2. Fitting for each parameter set 3. Range of best fits perturbative uncertainty

52 = 91.2 GeV α (m Z ) = ± (0.0002) expt ± (0.0005) hadr ± (0.0009) pert with 2 χ dof = 0.91

53 same analysis for other event shape variables: Heavy Jet Mass AHM+chwartz fits to moment data AFHM

54 Moment Fits

55 n th moment: purely perturbative ( ) p n n CD O p d d dp d M Λ + = α σ ) ( ) ( ˆ mod 0 0 max ( ) k n k n k n k n CD O M k n M Λ = + Ω = α 2 ˆ 0 where is defined as for the distribution Higher sensitivity for higher moments with primed moments: ) ( 2 mod p p dp i i = Ω ( ) n n n n CD O M M Λ + Ω + = α 2 ˆ Ω = Ω Ω 1 Ω 1 = Ω Ω + Ω Ω = Ω Ω appear to also be 1/ n!!!!

56 Global fits to M 1 data N 3 LL N 3 LL NNLL NLL

57 Global fits to M 1 data theory errors only experimental errors are dominant N 3 LL NNLL NLL

58 Fits for (2/) i Ω i at α (m Z )=0.114, M scheme: cancelation for primed moments no new information from higher moment data M 1 ˆM 1 M ˆM 3 3 Ω 3 =4.5±3.7 GeV Ω 1 =0.37±0.02 GeV M ˆM 4 4 [GeV] Ω 4 =38±42 GeV M ˆM 2 2 [GeV] Ω 2 =0.54±0.28 GeV M ˆM 5 5 [GeV] Ω 5 =200±550 GeV [GeV] [GeV]

59 ( ) k n k n k Gehrmann n P M k n M = = 2 ˆ 0 no Ω i>1! Comparison to Gehrmann with dispersive model for power corrections: ( ) ( ) { } K R R C I I R F O M P µ β µ µ π β π α µ α µ α α = ) ( 1 ln ) ( ) ( k n k n k n k n M k n M = Ω = 2 ˆ 0 fits to the higher moments are all sensitive to the same 1 ˆM

60 ummary α = ±(0.0002) expt ±(0.0005) hadr ±(0.0009) pert CET can improve convergence and therefore precision CET allows to include non-perturbativeeffectsin a systematic manner Profile functions allow to combine several kinematic regions

61 Backup lides

62 Why do we need a global fit? in the tail α (m Z ) and Ω 1 are degenerated for a single degeneracy is lifted by simultaneously fitting multiple

63 ED and b-mass effects b-mass: horizontal shift of thrust distribution towards larger lower Ω 1 ED: effective increase of coupling strength lower α

64 Ω 2 effects increase min Cut on Dataset decrease of Δ increase of statistical uncertainty max missing α Λ CD / effects become important

65 Theory Parameters profile function (variation of renormalization scales) Padé approximats non-singular stat. uncertainty

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