Determination of α s from the QCD static energy

Transcription

1 Determination of α s from the QCD static energy Antonio Vairo Technische Universität München

2 Bibliography (1) A. Bazavov, N. Brambilla, X. Garcia i Tormo, P. Petreczky, J. Soto and A. Vairo Determination of α s from the QCD static energy: an update Phys. Rev. D90 (2014) 7, arxiv: (2) X. Garcia i Tormo Review on the determination of α s from the QCD static energy Mod. Phys. Lett. A28 (2013) arxiv: (3) A. Bazavov, N. Brambilla, X. Garcia i Tormo, P. Petreczky, J. Soto and A. Vairo Determination of α s from the QCD static energy Phys. Rev. D86 (2012) arxiv: (4) N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairo Precision determination of r 0 Λ MS from the QCD static energy Phys. Rev. Lett. 105 (2010) arxiv: (5) N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairo The QCD static energy at NNNLL Phys. Rev. D80 (2009) arxiv: (6) N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairo The logarithmic contribution to the QCD static energy at N 4 LO Phys. Lett. B647 (2007) 185 arxiv:hep-ph/

3 α s in 2015

4 Status of α s Moch et al arxiv:

5 Status of α s C-parameter global fit HKMS'14 Lattice HPQCD'14 Thrust global fit FOPT CIPT AFHMS'10 t decays BGMOP'14 global PDF DIS ABM'12 global PDF NNPDF'11 MSTW'09 Static Energy BBGPS'14 Thrust moments Electroweak Gfitter' as Hm Z L AFHMS'12 Hoang Kolodrubetz Mateu Stewart PRD 91 (2015)

6 PDG average Uncertainties of α s are not fully reflected in the PDG average of Α s M Z year... but they will in 2016: for the first time in over 20 years there will be an increase in the error of α s! Bethke ISMD2015

7 Theory

8 Static energy E 0 (r) = lim T i T ln ; = exp j I ig dz µ A µ ff Perturbation theory describes E 0 (r) in the short range (rλ 1, α s (1/r) < 1): E 0 (r) = Λ s C F α s r E 0 (r) is known at three loops. (1+#α s +#α 2 s +#α 3 s +#α 3 s ln α s +#α 4 s ln 2 α s +#α 4 s ln α s +... ) ln α s signals the cancellation of contributions coming from different energy scales: ln α s = ln µ 1/r + ln α s/r µ Brambilla Pineda Soto Vairo PRD 60 (1999)

9 Energy scales In the short range the static Wilson loop is characterized by a hierarchy of energy scales: 1/r V o V s Λ; V s C F α s r, V o 1 2N α s r r 0 scale (V o V s ) α s (V o V s ) 1/r 1 Λ MS = 0.602/r r r 0 (r fm)

10 Effective Field Theories EFTs allow the factorization of contributions from different energy scales. = QCD pnrqcd E 0 (r) = Λ s + V s (r, µ) i g2 N V 2 A Z 0 dt e it(v o V s ) Trr E(t)r E(0) (µ) +... res. mass potential ultrasoft contribution Brambilla Pineda Soto Vairo NPB 566 (2000) 275 The µ dependence cancels between V s ln rµ, ln 2 rµ,... ultrasoft contribution ln(v o V s )/µ, ln 2 (V o V s )/µ,... ln rµ, ln 2 rµ,...

11 V A The first contributing diagrams are of the type: Therefore V A (r, µ) = 1 + O(α 2 s)

12 Chromoelectric field correlator: E(t)E(0) Is known at two loops. (a) (b) (c) (d) LO (e) (f) Eidemüller Jamin PLB 416 (1998) 415 (g) NLO (h)

13 Static octet potential lim T i T ln φ adj ab = 1 2N α s r (1 + #α s + #α 2 s + #α 3 s + #α 3 s ln µr +... ) Is known at three loops. Anzai Prausa A.Smirnov V.Smirnov Steinhauser PRD 88 (2013)

14 Static singlet potential at N 4 LO ( α s (1/r) V s (r, µ) = C F 1 + α «s(1/r) αs (1/r) 2 a 1 + a 2 r 4π 4π «αs (1/r) 3» 16 π 2 + CA 3 4π 3 ln rµ + a 3 «αs (1/r) 4» + a L2 4 ln 2 rµ + a L «4π 9 π2 CA 3 β 0( ln2) ln rµ +... ) + Anzai Kiyo Sumino PRL 104 (2010) A.Smirnov V.Smirnov Steinhauser PRL 104 (2010)

15 Static energy at N 4 LO ( E 0 (r) = Λ s C F α s (1/r) 1 + α s(1/r) [a 1 + 2γ E β 0 ] r 4π «αs (1/r) 2» π 2 «+ a 2 + 4π 3 + 4γ2 E β0 2 + γ E (4a 1 β 0 + 2β 1 ) «αs (1/r) 3» 16π 2 + 4π 3 C3 A ln C Aα s (1/r) + ã 3 2 «αs (1/r) 4» + a L2 4 ln 2 C Aα s (1/r) + a L 4 ln C Aα s (1/r) π 2 2 ) +

16 Renormalization group equations 8 µ d dµ V s = 2 3 C F α s π r2 V 2 A [V o V s ] α s π c >< µ d dµ V o = 1 N µ d dµ V A = 0 α s π r2 V 2 A [V o V s ] α s π c >: µ d dµ α s = α s β(α s ); c = 5n f + C A (6π ) 108

17 Static singlet potential and energy at N 3 LL V s (r, µ) = V s (r, 1/r) C F C 3 A 6β 0 α 3 s(1/r) r j «α s (1/r) a 1 4 π β1 αs (µ) 6c 4β 0 π ln α s(1/r) α s (µ) α s(1/r) π ff Summed to the ultrasoft contribution at two loops, it provides the static energy at N 3 LL.

18 Mass renormalon The perturbative expansion of V s is affected by a renormalon ambiguity of order Λ. This ambiguity does not affect the slope of the potential (and the extraction of α s ). It may be eliminated from the perturbative series either by subtracting a (constant) series in α s to V s and reabsorb it in a redefinition of the residual mass Λ s, or by considering the force: F(r, α s (ν)) = d dr E 0(r, α s (ν)) The force F(r, α s (1/r)) could be directly compared with lattice, or integrated and compared with the static energy E 0 (r) = Z r r dr F(r, α s (1/r )) up to an irrelevant constant fixed by the overall normalization of the lattice data. Note that there are no ln νr (ν = renormalization scale).

19 Analysis

20 Lattice We use 2+1-flavor lattice QCD obtained from tree-level improved gauge action and Highly-Improved Staggered Quark (HISQ) action by the HotQCD collaboration. m s was fixed to its physical value, while m l = m s /20. This corresponds to a pion mass of about 160 MeV in the continuum limit. β r 1 /a 5.172(34) 6.336(56) 7.690(58) Volume The largest gauge coupling, β = 7.825, corresponds to lattice spacings of a = fm. Bazavov et al PRD 90 (2014) The lattice spacing was fixed using the r 1 scale defined as r 2 de 0(r) = 1.0; dr r=r1 r 1 = ± fm from the pion decay constant f π. Bazavov et al PoS LATTICE 2010 (2010) 074

21 Procedure We use data for each value of the lattice spacing separately, and at the end perform an average of the different obtained values of α s with the following procedure. Perform fits to the lattice data for the static energy E 0 (r) at different orders of perturbative accuracy. The parameter of the fits is Λ MS. Repeat the above fits for each of the following distance ranges: r < 0.75r 1, r < 0.7r 1, r < 0.65r 1, r < 0.6r 1, r < 0.55r 1, r < 0.5r 1, and r < 0.45r 1. Use ranges where the reduced χ 2 either decreases or does not increase by more than one unit when increasing the perturbative order, or is smaller than 1. To estimate the perturbative uncertainty of the result, repeat the fits by varying the scale in the perturbative expansion, from ν = 1/r to ν = 2/r and ν = 1/( 2r), by adding/subtracting a term ±(C F /r 2 )α n+2 s to the expression at n loops. Take the largest uncertainty.

22 Data ranges r1e0const Β Β Β rr 1

23 χ 2 /d.o.f. for β = Χ 2 d.o.f r0.75r 1 r0.7r 1 r0.65r 1 r0.6r 1 r0.55r 1 r0.5r 1 r0.45r 1 Β N ref 7 tree level 1 loop 2 loop 3 loop Χ 2 d.o.f r0.75r 1 r0.7r 1 r0.65r 1 r0.6r 1 r0.55r 1 r0.5r 1 r0.45r 1 Β N ref 8 tree level 1 loop 2 loop 3 loop Β N ref 9 Χ 2 d.o.f r0.75r 1 r0.7r 1 r0.65r 1 r0.6r 1 r0.55r 1 r0.5r 1 r0.45r 1 tree level 1 loop 2 loop 3 loop Fits for r < 0.6r 1 are acceptable. In the final result we will use only fits for r < 0.5r 1. The fitting curve has been normalized on the 7th, 8th and 9th lattice point respectively.

24 aλ MS at different orders of perturbative accuracy for β = Β Nref r0.5r a MS tree level 1 loop 2 loop 3 loop 0.04

25 r 1 Λ MS at three-loop accuracy loop N ref 7 Β Β Β loop N ref 8 Β Β Β r1 MS r1 MS r0.6r 1 r0.55r 1 r0.5r 1 r0.45r 1 r0.6r 1 r0.55r 1 r0.5r 1 r0.45r 1 3 loop N ref Β Β Β r1 MS r0.6r 1 r0.55r 1 r0.5r 1 r0.45r The band shows the determination of 2012.

26 Statistical error vs perturbative error Β Nref 7 3 loop Β Nref 7 3 loop a MS a MS r0.6r 1 r0.55r 1 r0.5r 1 r0.45r r0.6r 1 r0.55r 1 r0.5r 1 r0.45r 1 Β Nref 7 3 loop a MS r0.6r 1 r0.55r 1 r0.5r 1 r0.45r 1 The statistical error is estimated by taking values of Λ MS at one χ 2 unit above minimum.

27 Analysis with the force Χ 2 d.o.f r0.75r 1 r0.7r 1 r0.65r 1 r0.6r 1 r0.55r 1 r0.5r 1 r0.45r 1 Β 7.825; 7.596; Force tree level 1 loop 2 loop 3 loop The band shows the determination of 2012.

28 The counting of the ultrasoft contributions 1.5 finite 3 loops r 2 Fr,1r finite 2 loops 0.5 Pert. terms 0.0 α s LL LL rr 1 Leading-ultrasoft resummation included along with the three-loop terms is consistent with the observed size of the terms. This goes in our final result. We chose µ = 1.26r GeV, for the ultrasoft factorization scale. Variations of µ only produce small effects on the results.

29 Short-distance points vs long-distance points 3 loop N ref Β Β Β r1 MS r0.75r 1 r0.7r 1 r0.65r 1 r0.6r 1 r0.55r 1 r0.5r 1 r0.45r The band shows the determination of 2012.

30 Looking for condensates Β Nref 7 3 loop r 3 term a MS = r 3 monomial r0.75r 1 r0.7r 1 r0.65r 1 r0.6r 1 r0.55r 1 r0.5r 1 r0.45r By repeating the fits adding a monomial term proportional to r 3 and r 2, which could be associated with gluon and quark local condensates, and also a term proportional to r, we do not find evidence for a significant non-perturbative term at short distances and the value of Λ MS remains unchanged.

31 Results

32 Λ MS Results at three-loop plus leading-ultrasoft resummation for the r < 0.5r 1 fit range. The final result is the weighted average of different βs with linearly added errors. r 1 Λ MS = which converts to Λ MS = MeV

33 Static energy vs lattice data 0.0 Β Nref 7 3 looplead. us Β Nref 7 3 looplead. us r1e0const datatheory r 1 Λ MS = rr rr 1 Perturbation theory agrees with lattice data up to about 0.2 fm.

34 Static energy at different perturbative orders vs lattice data r1e0rconst tree level 1 loop 2 loop 3 loop 3 loopus. res. r 1 Λ MS = datatheory tree level loop 2 loop 3 loop loopus. res r r 1 rr 1 Lattice data with β from to are displayed. The red error bars correspond to the errors of the lattice data (include normalization).

35 Force vs lattice data r 2 F Β Β Β tree level loop r 1 Λ MS = loop 3 loop 3 loopus. res r r 1

36 α s α s (1.5 GeV, n f = 3) = which corresponds to α s (M Z, n f = 5) = from four-loop running, m c = 1.6 GeV and m b = 4.7 GeV.

37 Comparison with other determinations C-parameter global fit HKMS'14 Lattice HPQCD'14 Thrust global fit FOPT CIPT AFHMS'10 t decays BGMOP'14 global PDF DIS ABM'12 global PDF NNPDF'11 MSTW'09 Static Energy BBGPS'14 Thrust moments Electroweak Gfitter' as Hm Z L AFHMS'12

38 Outlook Not all of the presently available perturbative information has been used. More precise lattice data on finer lattices and with more data points at short distances could take advantage of it. It would be important, in order to reduce possible systematic effects, to perform the same study on Wilson loops computed on different lattices with different actions. A possible systematic effect is due to the finite lattice spacing. A continuum extrapolation would reduce this effect and allow for a precise determination of the force between static charges along the same lines developed by Necco and Sommer (2001) for the quenched case. Compute the force directly from the lattice: F(r) = lim T D Tr Pˆr ge(t,r)exp nig H oe r T dzµ A µ D Tr Pexp nig H oe r T dzµ A µ