QCD perturbative series in tau decays: scheme variations of the coupling

Transcription

1 A new TauLepton determination Beijing, of s from China decays QCD perturbative series in tau decays: scheme variations of coupling São Carlos Institute of Physics University of São Paulo -- In coll. with Matthias Jamin and Ramon Miravitllas (UAB/IFAE, Spain) [arxiv: ] (to appear in PRL)

2 Introduction

3 Strong coupling α s (Q 2 ) τ decays (N 3 LO) DIS jets (NLO) Heavy Quarkonia (NLO) e + e jets & shapes (res. NNLO) e.w. precision fits (NNLO) ( ) pp > jets (NLO) pp > tt (NNLO) October QCD α s (M z ) = 0.8 ± Q [GeV] 000 Not a physical observable: scheme an scale dependent

4 Key idea of talk 2 Perturbative expansions in QFTs are (at best) asymptotic Different schemes lead to different asymptotic series freedom to exploit and look for optimal series Perturbative order

5 Key idea of talk 2 Perturbative expansions in QFTs are (at best) asymptotic Different schemes lead to different asymptotic series freedom to exploit and look for optimal series Perturbative order

6 A few definitions 3 The scale evolution is given by QCD beta function a Q = s Q da Q dq (a Q)= a 2 Q + 2 a 3 Q + 3 a 4 Q + 4 a 5 Q + 5 a 6 Q... A scale invariant (but scheme dependent) QCD scale " Za Q # 2 Q e a 2 da Q [aq ] exp, (a) where we introduced combination 0 Baikov, Chetyrkin, Kühn 6 (mind different conventions) can be defined as (a) (a) a a

7 A few definitions 3 The scale evolution is given by QCD beta function a Q = s Q da Q dq (a Q)= a 2 Q + 2 a 3 Q + 3 a 4 Q + 4 a 5 Q + 5 a 6 Q... A scale invariant (but scheme dependent) QCD scale " Za Q # 2 Q e a 2 da Q [aq ] exp, (a) where we introduced combination 0 Baikov, Chetyrkin, Kühn 6 (mind different conventions) can be defined as See next talk by J. Khün! (a) (a) a a

8 A few definitions 3 The scale evolution is given by QCD beta function a Q = s Q da Q dq (a Q)= a 2 Q + 2 a 3 Q + 3 a 4 Q + 4 a 5 Q + 5 a 6 Q... scheme independent A scale invariant (but scheme dependent) QCD scale " Za Q # 2 Q e a 2 da Q [aq ] exp, (a) where we introduced combination 0 Baikov, Chetyrkin, Kühn 6 (mind different conventions) can be defined as See next talk by J. Khün! (a) (a) a a

9 A few definitions 3 The scale evolution is given by QCD beta function a Q = s Q da Q dq (a Q)= a 2 Q + 2 a 3 Q + 3 a 4 Q + 4 a 5 Q + 5 a 6 Q... scheme independent A scale invariant (but scheme dependent) QCD scale " Za Q # 2 Q e a 2 da Q [aq ] exp, (a) where we introduced combination 0 Baikov, Chetyrkin, Kühn 6 (mind different conventions) can be defined as See next talk by J. Khün! (a) (a) a a In anor scheme one would have ã = a + c a 2 + c 2 a 3 + c 4 a 4 + = e c / Celmaster & Gonsalves '79

10 Why varying scheme? 4 Higher-loop calculations allow us to gain a better understanding of series at intermediate orders. Scheme variations give us an extra handle on perturbative series. Optimisation of series in spirit of asymptotic expansions. Applications (not limited to) tau decays (FOPT vs CIPT), Higgs decays,.

11 Scheme variations

12 Scheme variations 5 In large- 0 limit its convenient to redefine coupling as Beneke '99 Q ln Q + C = + â 2 a Q 2 C + Q az We propose following generalisation to QCD coupling (C-scheme) + 2 â Q ln â Q ln Q + C 2 az = a Q + 2 C + 2 ln a Q Za Q 0 da (a) â Q â Q (C)

13 Scheme variations 5 In large- 0 limit its convenient to redefine coupling as Beneke '99 Q ln Q + C = + â 2 a Q 2 C + Q az We propose following generalisation to QCD coupling (C-scheme) + 2 â Q ln â Q ln Q + C 2 az = a Q + 2 C + 2 ln a Q Za Q 0 da (a) free parameter â Q â Q (C) In this new scheme beta function takes simple form: Q dâ Q dq ˆ(â Q )= â 2 Q. 2 âq only scheme independent coefficients appear

14 The coupling ba(m 2 6 ) The new coupling as a function of C for s (m 2 )=0.360 PDG '6 â Q â Q (C) [a(m 2 )=0.006(32)] (From numerical solution, not a perturbative result.)

15 Relations between different schemes 7 The perturbative relation between two schemes up to fifth order is â(a) =a + c a 2 + c 2 a 3 + c 3 a 4 + c 4 a 5 + The first four coefficients read c = 9 4 C c 2 = C 8 6 C2 c 3 = C 45 2 C C c 4 = C C C3 C C

16 Application to tau decays

17 Applications to hadronic tau decays 8 R = [! hadrons ] [! e e ] Theoretical computation: optical orem + Cauchy integration µ (q) =i Z d 4 xe iqx 0 T {J µ (x)j } R (th) = 2 i I dz w(z) (z) m 2 z =s 0 Braaten, Narison, Pich '92

18 The Adler function in C-scheme 9 Perturbative QCD contribution to Adler function Gorishnii, Kataev, Larin 9 Surguladze&Samuel '9 ( 2 ) ( s 3 ) ( s s 4 ) Baikov, Chetyrkin, Kühn 08 MS result bd(a Q )= X n= c n, a n Q = a Q a 2 Q a 3 Q a 4 Q +... In C-scheme we have bd(â Q )=â Q +( C) â 2 Q +( C C 2 ) â 3 Q + ( C C C 3 ) â 4 Q +

19 The Adler function in C-scheme 9 Perturbative QCD contribution to Adler function Gorishnii, Kataev, Larin 9 Surguladze&Samuel '9 ( 2 ) ( s 3 ) ( s s 4 ) Baikov, Chetyrkin, Kühn 08 MS result bd(a Q )= X n= c n, a n Q = a Q a 2 Q a 3 Q a 4 Q +... In C-scheme we have bd(â Q )=â Q +( C) â 2 Q +( C C 2 ) â 3 Q + ( C C C 3 ) â 4 Q +

20 The Adler function in C-scheme 0 Using C dependence we are able to kill fifth order contribution c 5, = 283 ± 283 Beneke & Jamin 08 [a(m 2 )=0.006(32)] extreme case bd(â M,C = 0.783) = ± ± O(â 4 ) s

21 Dependence on fifth order coefficient 0.20 Adler Function Sum c_5 (Shown only uncertainty due to truncation)

22 Dependence on fifth order coefficient 0.20 Adler Function Sum C= C= c_5 (Shown only uncertainty due to truncation)

23 Tau decays into hadrons 2 Perturbative contribution to observable R R = [! hadrons ] [! e e ] = N c S EW ( V ud 2 + V us 2 )(+ + ) Prescriptions for RG improvement Fixed Order PT (FOPT) µ = s FOPT CIPT MS FO,w i = X n= a(s 0 ) n n X k= kc n,k J FO,w i k δ Contour Improved PT (CIPT) µ = s 0 x CI,w i = X n= c n, J CI,w i n (s 0,a s ) Le Diberder & Pich '92 R = [! hadrons ] [! e e ] Perturbative order n

24 Tau decays into hadrons 2 Perturbative contribution to observable R R = [! hadrons ] [! e e ] = N c S EW ( V ud 2 + V us 2 )(+ + ) Prescriptions for RG improvement Fixed Order PT (FOPT) µ = s FOPT CIPT MS FO,w i = X n= a(s 0 ) n n X k= kc n,k J FO,w i k δ Contour Improved PT (CIPT) µ = s 0 x CI,w i = X n= c n, J CI,w i n (s 0,a s ) Le Diberder & Pich '92 R = [! hadrons ] [! e e ] Perturbative order n

25 Tau decays into hadrons 2 Perturbative contribution to observable R R = [! hadrons ] [! e e ] = N c S EW ( V ud 2 + V us 2 )(+ + ) Prescriptions for RG improvement Fixed Order PT (FOPT) µ = s FOPT CIPT MS FO,w i = X n= a(s 0 ) n n X k= kc n,k J FO,w i k δ Contour Improved PT (CIPT) µ = s 0 x CI,w i = X n= c n, J CI,w i n (s 0,a s ) Le Diberder & Pich '92 R = [! hadrons ] [! e e ] Perturbative order n

26 Tau decays into hadrons: FOPT 3 Perturbative contribution to observable R = [! hadrons ] [! e e ] = N c S EW ( V ud 2 + V us 2 )(+ + ) MS FOPT result: FO (a Q)=a Q a 2 Q a 3 Q a 4 Q +... C-scheme FOPT result: FO (â Q)=â Q +( C) â 2 Q + ( C C 2 ) â 3 Q + ( C + 0.5C C 3 ) â 4 Q +...

27 Tau decays into hadrons: FOPT 4 FOPT results c 5, = 283 ± 283 [a(m 2 )=0.006(32)] FO (â M,C = 0.882) = ± ± 0.033

28 Tau decays into hadrons 5 Perturbative contribution to observable R R = [! hadrons ] [! e e ] = N c S EW ( V ud 2 + V us 2 )(+ + ) Prescriptions for RG improvement FOPT µ = s FOPT CIPT MS FO,w i = X n= a(s 0 ) n n X k= kc n,k J FO,w i k δ CIPT µ = s 0 x CI,w i = X n= c n, J CI,w i n (s 0,a s ) R = [! hadrons ] [! e e ] Perturbative order n

29 Tau decays into hadrons: CIPT 6 CIPT results c 5, = 283 ± 283 [a(m 2 )=0.006(32)] CI (â M,C =.246) = ± ±

30 Conclusions

31 Conclusions 7 The C-scheme gives an extra handle on pt. series: optimisation Can be applied to many different processes Jamin, Miravitllas 6 Having result for fifth order coefficient would be excellent. Unfortunately, CIPT vs FOPT problem persists CI (â M,C =.246) = ± ± FO (â M,C = 0.882) = ± ± Optimised results corroborate renormalon models of Adler function bd(â M,C = 0.783) = ± ± bd(a M )=0.354 ± ± [this work] [Renormalon based description] Beneke & Jamin 08, DB, Beneke, Jamin '3 Uncertainties tend to be smaller with more terms in pt. series.

32

33 Extra

34 D(a M, C = 0.783) = ± ± , (3) our central predictio red and blue dots, O(a 5 ) vanishes, and O(a 4 ) is taken as and conservative. + ( C + 0.5C Comparison where second error originates fromexplanation, uncertainty in text. uncertainty. For furr see prisingly, studie result We also (a Q ) may as a be function of C. yellow band arises 2 CI (3) s (M ).FIG. The 4. result compared to The direct centr [a(m 0.006(32)] tion (2) ) = of hadronic width, In Fig. 3, we display (a Q ) e Comparison with from eir removing or doubling fifth-order term. In MS FO MS prediction (), which reads 55 4 hence providing som suming c = 283, yellow b our central predic red and blue dots, O(a ) vanishes, and O(a ) is taken as 5, and C =.629, O(a ) correction vanishes, and The disparity be 4 uncertainty. For furr explanation, see text. prisingly, resu to removing or doubling O(a b MS O(a ) term is taken as uncertainty. The point to D(a ) = 0.36 ± ± (4) M for is not resolv a nice plateau is found for C tion (2) of ce right has a substantially smaller error, and yields 5 bfirst Eq. (22), CIPT [this work], C =is obtained 0.783) = ± ± Here, D(a by removing or doubling and n doubling O(a ) res M error hence providing 5 Csecond =M,.629, O(a )0.2047±0.0034±0.033 correction vanishes, and not (asshow is case for t (a C = 0.882) =. (9) c5,, andand error again corresponds to that does stability. He s FO The disparity 4 b M [Renormalon based description] D(a ) = ± ± O(a ) term is taken as uncertainty. The point to tored investigat uncertainty. is disfavored.return In dots,reso w for is not Oncehas more, second error covers uncertainty of right a substantially smaller error, and yields Beneke & Jamin 08, DB, Beneke, coupling Jamin '3 a, in order Eq. (22), CIP s (M ). In this case, direct MS prediction of Eq. (7) such models. This c is case fo (a M to, Cbe = 0.882) = ±0.0034± (9) of(as isfo found s from hadronic return to investig (am second ) = ± (20)of Once more, error± covers uncertainty FO Helpful discussio coupling a, in ord s (M ). In this case, direct MS prediction of Eq. (7) fully suchacknowledged models. Thi This value is somewhat lower, but within of is found to be been in of ssupported from hadro higher-order uncertainty. Comparing, on or hand, model (BM) result of [7], which is given by CICYT-FEDER-FP to Borel (a ) = 0.99 ± ± (20) MS M FO Helpfulprogram discus excellence fully acknowledg description] ± , [Renormalon This valuebmis(asomewhat lower, but ±within of(2) based M ) = and Secretaria d U beend Economia supported i higher-order uncertainty. Comparing, on or hand, ment C = that 0.882)(9) = ± ± similar. it Mis,found and (2) are surprisingly FO (a CICYT-FEDERto Borel model (BM) result of [7], which is given by Catalunya under G In both cases, parametric s uncertainty is substan=.246) = ± ± excellence progra ported by Sa o P CI (a M, C tially larger(a than) higher-order one especially given (2) grant and 5/ , Secretaria ad BM M = ± ± 0.030, recent increase in s uncertainty provided by ment d Economia itpdg is found that (9) and (2) are surprisingly similar. [6] which underlines good potential of s Catalunya under Inextractions both cases, hadronic parametric s uncertainty is substanfrom decays. Boito porteddiogo by Sa o

35 Tau decays into hadrons: FOPT FOPT Sum c_5 ( ) δ