Sébastien Descotes-Genon

Transcription

1 α s (CIPT) Sébastien Descotes-Genon Laboratoire de Physique Théorique CNRS & Université Paris-Sud 11, 9145 Orsay, France α s workshop - February Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 1

2 Introduction Motivation Improved perturbative computation of Adler function [P. Baikov et al. 28] ALEPH τ spectral functions with some improved branching ratios and V/A separation [BABAR and Belle 28] M. Davier et al, Eur. Phys. J. C 56 (28) 35 [hep-ph/83.979] Reanalysis of τ spectral moments Extraction of α s (m 2 τ ) using Contour Improved Perturbation Theory Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 2

3 Tau spectral functions Unitarity: R τ,v /A = B V /A ν τ /B e can be expressed as s ) 2 ) ] R τ,v /A (s ) = 12πS EW V ud 2 ds (1 [(1 ss + 2 ss ImΠ (1) (s) + ImΠ () (s) s S EW = ±.6 short-dist. electroweak in terms of 2-pt correlator of axial/vector U ij = q i γ µ (γ 5 )q j i d 4 x e iqx T [U µ ij (x)uν ij () ] = [ g µν q 2 + q µ q ν ]Π (1) ij,u (q2 ) + q µ q ν Π () ij,u (q2 ) with cut along real positive axis : spectral functions of given spin ImΠ (1,) ud(s),v /A (s) = 1 2π v 1/a 1, (s), experimentally accessible [ ( mτ 2 B V v 1 (s)/a 1 (s) = /A ν τ dn V /A 6 V ud 2 1 s ) 2 ( S EW B e N V /A ds mτ s ) ] 1 mτ 2 a (s) id without (1 + 2s/m 2 τ ) Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 3

4 V /A separation V /A separation simple for π channels due to G-parity Main limitation used to be from strange channel K K π ALEPH 1999 e + e DM1 and DM2 f A,CVC = CLEO 2 a 1 in τ ν τ π 2π + unitar. f A,a1 = 1.3 ±.24 1 CLEO 24 partial wave K K + π f A,K K π =.56 ±.1 ALEPH db/ds (%) Babar (CVC) DM1 (CVC) DM2 (CVC) s (GeV 2 ) = Previous analyses: f A (K K π) =.75 ±.25 Analysis of Babar e + e + CVC f A,CVC (K K π) =.833 ±.24 Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 4

5 Update of inclusive hadronic τ widths New measurements of τ strange decays (K π, K S π, and K π + π ) [Babar, Belle] R τ,s =.1615 ±.4 = R τ (R τ,v + R τ,a ) New axial fraction for K K π R τ,v = ±.11 ±.2 R τ,a = ±.11 ±.2 R τ,v +A = ±.11 R τ,v A =.87 ±.18 ±.3 1st error : experimental 2nd error : V /A separation, dominated by K K ππ Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 5

6 Update of the ALEPH spectral functions v 1 (s) ALEPH [mod-8] τ V ν τ π π π 3π, 2π π + π, (6π) ωπ, ηπ π, (KK-bar(π)) QCD prediction parton model a 1 (s) QCD prediction Parton model ALEPH [mod-8] τ A ν τ π 2π, 2π π + (5π) (KK-barπ) s (Gev 2 ) s (Gev 2 ) (v 1 + a 1 )(s) τ (V, A) ν τ QCD prediction parton model ALEPH (v 1 a 1 )(s) τ (V, A) ALEPH ν τ [mod-8] QCD pred. = parton model Sébastien Descotes-Genon (LPT-Orsay) 2 α s(cipt) 2 9/2/11 6

7 Contour integral and OPE Convert integral along cut < s iɛ < s into a contour s = s OPE assuming quark-hadron duality holds even close to real cut R τ,v /A = 3 ( 2 S EW V ud δ () + δ EW + δ(2,mq) ud,v /A + δ (D) ud,v /A D=4,6,... with δ (D) ud,v /A = dimo=d C V /A (s, µ) O D(µ) V /A and EW corrections s D/2 R τv /A dominated by perturbative contribution, related to α s (mτ 2 ) Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 7 )

8 RGE for α s 1+δ () = 2πi s =s ] ds [1 2 ss + 2 s3 s s 3 s4 s 4 D(s) D(s) = 1 4π 2 K n (ξ)as( ξs n ) K...4 known for Adler function D(s) = sdπ/ds [P. Baikov, 28] a s (s) = α s (s)/π on the contour obtained by RGE, expansion in a s = a s (s ) and η = log(s/s ) a s (s) a s = a 2 sβ η +a 3 s[ β 1 η +β 2 η2 ] +a 4 s[ β 2 η β β 1 η 2 β 3 η3 ] +a 5 s[ β 3 η +( 3 2 β β β 2 )η β2 β 1η 3 +β 4 η4 ] +a 6 s[ β 4 η +( 7 2 β 1β β β 3 )η 2 ( 35 6 β β β2 β 2)η β3 β 1η 4 β 5 η5 ] +... unknown β n 5 (set to zero) involved at higher orders central value for β 4 assuming geometrical growth Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 8 n=

9 α s along the contour CIPT = step-wise integration along contour with previous exp FOPT = one-step determination with RGE cut at η 5, a 6 s FOPT ++ = id., but including all known orders up to η 5 [including higher-order contributions a 7 s a 1 s ] Re(α s (m 2 τ e iφ )) FOPT FOPT (= FOPT + ) CIPT φ (rad) [Ref value: α s (m 2 τ ) =.34] Re[α s (m 2 τ e iφ )] FOPT/FOPT ++ : η k 5 dominate when included, large deviations near cut CIPT CIPT ++ j th step of CIPT with N steps up to and including a n s leads to error O(j/N n+1 ) = CIPT error 1/N n -suppr. compared to FOPT Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 9

10 Fixed order truncation Re(4π 2 D(m 2 τ e iφ )-1) FOPT ++ FOPT CIPT FOPT Re(Integrand(m 2 τ e iφ )) FOPT ++ FOPT CIPT ( FOPT + ) φ (rad) φ (rad) Re[4π 2 D(m 2 τ e iφ ) 1] Real part integrand for 1 + δ () FOPT+ : truncation for a s, but not for the integral yielding δ () FOPT+/CIPT similar : truncation after integration makes difference for FOPT/CIPT Crucial role of the pinching weight to suppress cut region Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 1

11 Dependence on the renormalisation scale Scale for perturbative computation of Adler function O(mτ 2 ) D(s) = 1 K 4π 2 n (ξ)as( ξm n τ 2 ) n= = signficant uncertainty for δ () and thus α s δ () δ () th order 5th order 4th order 3rd order th order 5th order 4th order 3rd order ξ ξ FOPT CIPT Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 11

12 Numerical tests Computation of δ () for α S (m 2 τ ) =.34 Unknown K 5,6 and β 4 by geometric growth (higher set to ) Variation of unknown coefficients δβ 4, K 5, K 6 ± 1% Variation of scale δξ ±.63 (i.e. scale m 2 τ ±.63) n = 1 n = 2 n = 3 n = 4 (n = 5) (n = 6) P 4n=1 P 5n=1 P 6n=1 FOPT δ(β 4 ) ±.6 ±.6 δ(k 5 ) ±.56 ±.18 ±.56 ±.164 δ(k 6 ) ±.47 ±.47 δ(ξ) CIPT δ(β 4 ) δ(k 5 ) ±.49 ±.49 ±.49 δ(k 6 ) ±.2 ±.2 δ(ξ) Convergence near the cut, scale dependence = CIPT preferred [FOPT/CIPT : M. Beneke ; S. Menke ; I Caprini and J. Fischer 21] Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 12

13 Models of quark-duality violation (1) in spite of pinched weight, contribution near cut where OPE fails soft contributions exp( λq k )/Q l for Q 2 < missed by OPE translated into oscillatory behaviour for Q 2 > Models of duality violation, providing oscillatory behaviour for v, a [M. Shifman 2] Instanton background field of size ρ (sin(ρ s) oscillation) Π (I) = C I Q 2 K 1(Qρ)K 1 (Qρ) Comb of resonances with growing width (sin(s/σ) oscillation) [ ] ( Π (II) ψ(z) + 1/z Q 2 (Q) = C II 4π 2 (1 B/3π) Π(OPE) (Q) z = σ 2 ) 1 B (3π) Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 13

14 Models of quark-duality violation (2) Parameters determined to vanish around s =1.6 GeV 2 and to match smoothly v + a from s 2 GeV 2 Instanton : 2.4 ρ 4.4 GeV 1 = δ () Resonances : 1.65 σ 2 2 GeV 2,.3 < B <.6 = δ () < (v 1 + a 1 )(s) τ (V, A) ν τ QCD prediction parton model δ () = 2πi ds s s =s s (Gev 2 ) ALEPH [ 1 2 s s + 2 s3 s ] s4 D(s) s 4 Resonance Instanton 3. s [M. Golterman s talk] Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 14

15 Spectral moments Joint fit of (k, l) = (1, ), (1, 1), (1, 2), (1, 3) for D kl τ,v /A = Rkl τ,v /A /R τ,v /A m2 τ Rτ,V kl /A = ds ( 1 s ) k ( ) s l dr τ,v /A mτ 2 mτ 2 ds R τ,v /A = R τ,v /A expressed through OPE in terms of D = : α s D = 2 : quark masses (PDG) D = 4 : quark condensate qq (F 2 π M 2 π via Gell-Mann-Oakes-Renner) and gluon condensate α s GG D = 6 : four quark operators down to ρ qq 2 by vacuum saturation D = 8 : single (effective) condensate D 1 : contributions neglected, even for higher moments Rτ,V /A : contributions from D/2 =, 1, 3, 4 : contributions from D/2 =, l + 1, l + 2, l + 3, l + 4, l + 5 R 1l τ,v /A Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 15

16 Correlations V Dτ,V 1 Dτ,V 11 Dτ,V 12 Dτ,V 13 R τ,v Dτ,V Dτ,V Dτ,V A Dτ,A 1 Dτ,A 11 Dτ,A 12 Dτ,A 13 R τ,a Dτ,A Dτ,A Dτ,A V+A Dτ,V 11 +A Dτ,V 12 +A Dτ,V 13 +A Dτ,V 1 +A Dτ,V 11 +A Dτ,V 12 +A (1, ) dominated by low-energy peaks (ρ or π, a 1 ) (1, l > ) more and more sensitive to higher energies = increasing negative correl. with (1, ) [positive among them] Negligible correlation between R τ,v +A and D ij τ,v +A Point-by-point correlelations? Impact on R, D matrix? [D. Boito et al., 21] Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 16

17 Outcome of the fit (1) Param. Vector (V ) Axial-Vector (A) V + A α S(m 2 τ ).3474 ± ± ± δ ().293 ± ± ±.7 δ (2) ( 3.2 ± 3.) 1 4 ( 5.1 ± 3.) 1 4 ( 4.3 ± 2.) 1 4 a sgg (.8 ±.4) 1 2 ( 2.2 ±.4) 1 2 ( 1.5 ±.3) 1 2 δ (4) (.1 ± 1.5) 1 4 ( 5.9 ±.1) 1 3 ( 3. ±.1) 1 3 δ (6) (2.68 ±.2) 1 2 ( 3.46 ±.21) 1 2 ( 3.7 ± 1.7) 1 3 δ (8) ( 8. ±.5) 1 3 (9.5 ±.5) 1 3 (8.1 ± 3.6) 1 4 Total δ NP (1.89 ±.25) 1 2 ( 3.11 ±.16) 1 2 ( 5.9 ± 1.4) 1 3 χ 2 /DF Small negative gluon condensate Opposite-sign contributions δ (6) and δ (8) for V and A Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 17

18 Outcome of the fit (2) Experimental spectral moments and theoretical counterparts (after fit) D 1l l = l = 1 l = 2 l = 3 τ,v l (theo) Dτ,V exp Dτ,V 1l D 1l τ,a l (theo) Dτ,A exp Dτ,A 1l D 1l τ,v +A l (theo) Dτ,V +A exp Dτ,V 1l +A = Difficulties for axial fit for (k, l) = (1, ), (1, 1) (low s) Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 18

19 α s (CIPT ) α s,cipt (m 2 τ ) =.344 ±.5 exp ±.7 th with th. errors : K 5 [.62], ξ[ ], S EW [.7], V ud [.5]! s (µ)! s (M Z ) DIS (Bj-SR) DIS (GLS-SR) " decays $ decays QQ # states (lattice) DIS (e/µ; F 2 ) [ ] e + e # (% had ) e + e # (jet & event-shape) pp (#) &bb # X, 'X e + e # (% had ) e + e # (jet & event-shape) e + e # (Z width) µ scale (GeV) N 3 LO N 2 LO NLO e + e # (jet & event-shape) α s,cipt (M Z ) =.1212 ±.5 exp ±.8 th ±.5 evol where evolution adds in quadrature uncertainties from b, c-quark masses matching scales truncation in matching truncation in RGE Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 19

20 Back-up Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 2

21 CIPT/FOPT and higher orders Model for higher-order PT for D (1 UV and 2 IR renormalons) ajusted to match known orders [M. Beneke and M. Jamin 29 ; M. Beneke] FOPT converges more slowly than CIPT, but it hits exact value given by Borel sum = FOPT favoured w.r.t CIPT Statement to consider w.r.t model and integration weight [I. Caprini and J. Fischer 21 ; SDG and B. Malaescu 21].26.2! () CIPT FOPT FOPT(BJ) BOREL moment k=1 l= CIPT FOPT FOPT(BJ) BOREL Perturbative order n Perturbative order n Different low-order structure w (k=1,l=1) Sébastien Descotes-Genon (LPT-Orsay) α s(cipt) 9/2/11 21