α s and the τ hadronic width

1 α s and the τ hadronic width Based on the recent article: Martin Beneke, MJ JHEP 09 (2008) 044, arxiv:0806.3156

Introduction 2 Investigations of hadronic τ decays already contributed tremendously for fundamental QCD parameters like α s, the strange mass and non-perturbative condensates. In particular: (Davier et al. 2008) α s (M τ ) = 0.344 ± 0.005 exp ± 0.007 th, leading to α s (M Z ) = 0.1212 ± 0.0011. This should be compared to the recent average: (Bethke 2007) α s (M Z ) = 0.1185 ± 0.0010, being significantly lower.

Hadronic τ decay rate 3 Consider the physical quantity R τ : (Braaten, Narison, Pich 1992) R τ Γ(τ hadronsν τ (γ)) Γ(τ e ν e ν τ (γ)) = 3.640±0.010. R τ is related to the QCD correlators Π T,L (z): (z s/m 2 τ) 1 R τ = 12π dz(1 z) 2[ (1+2z)ImΠ T (z)+imπ L ] (z) 0 with the appropriate combinations Π J (z) = V ud 2[ Π V,J ud +Π A,J ] ud + V us 2[ Π V,J us +Πus A,J ].,

Hadronic τ decay moments 4 Additional information can be inferred from the moments R kl τ 1 0 Theoretically, R kl τ R kl τ = N c S EW dz (1 z) k z l dr τ dz = Rkl τ,v +R kl τ,a +R kl τ,s. + D 2 can be expressed as: [ { ( V ud 2 + V us 2 ) V ud 2 δ kl(d) ud [1+δ kl(0)] + V us 2 δ kl(d) ] } us. δ kl(d) ud and δ kl(d) us are corrections in the Operator Product Expansion, the most important ones being m 2 s and m s qq.

Hadronic τ decay rate 5 Im q 2 Light quarks s 0 Hadrons տ Zero Re q 2 Complex q 2 -Plane

Adler function 6 The perturbative part δ (0) is related to the Adler function D(s): D(s) s d ds Π V (s) = N c 12π 2 n=0 a n µ n+1 k=1 k c n,k ln k 1( ) s µ 2 where a µ α s (µ)/π. Resumming the Log s with the scale choice µ 2 = s Q 2 : D(Q 2 ) = N c 12π 2 n=0 c n,1 a n (Q 2 ) As a consequence, only the coefficients c n,1 are independent: c 0,1 = c 11 = 1, c 2,1 = 1.640, c 3,1 = 6.371, c 4,1 = 49.076!! (Baikov, Chetyrkin, Kühn 2008)

RG-improvement 7 Fixed order perturbation theory amounts to choose µ 2 =M 2 τ : δ (0) FO = n=0 a n (M 2 τ) n+1 k=1 k c n,k J k 1 = n=0 [c n,1 +g n ]a n (M 2 τ) A given perturbative order n depends on all coefficients c m,1 with m n, and on the coefficients of the QCD β-function. Contour improved perturbation theory employs µ 2 = M 2 τx: (Pivovarov; Le Diberder, Pich 1992) δ (0) CI = c n,1 J a n(m 2 τ) with n=0 J a n(m 2 τ) = 1 2πi x =1 dx x (1 x)3 (1+x)a n ( M 2 τx)

Contour integrals 8 2 1.5 J n a (Mτ 2 )/a(mτ 2 ) n 1 0.5 0-0.5-1 -1.5 0 5 10 15 20 25 30 Perturbative order n α s (M τ ) = 0.34.

Numerical analysis 9 Employing α s (M τ ) = 0.34, the numerical analysis results in: a 1 a 2 a 3 a 4 a 5 δ (0) FO = 0.108+0.061+0.033+0.017(+0.009) = 0.220 (0.229) δ (0) CI = 0.148+0.030+0.012+0.009(+0.004) = 0.198 (0.202) Contour improved PT appears to be better convergent. The difference between both approaches amounts to 0.022! From the uniform convergence of δ (0) FO, and the assumption that the series is not yet asymptotic, one may also infer c 5,1 283, leading to a difference of δ (0) FO δ (0) CI = 0.027.

Toy models 10 0.24 0.22 δ (0) 0.2 0.18 0.16 FO perturbation theory CI perturbation theory 2 4 6 8 10 12 14 16 18 Perturbative order n c n,1 = 0 for n 6

Borel transform 11 To further investigate the difference between CI and FOPT, we propose to model the Borel-transformed Adler function. 4π 2 D(s) 1+ D(s) 1+ n=0 r n α s (s) n+1, where r n =c n+1,1 /π n+1. The Borel-transform reads: D(α s ) = 0 dt e t/α s B[ D](t) ; B[ D](t) = n=0 r n t n n!. Generally, the Borel-transform B[ D] developes poles and cuts at integer values p of the variable u β 0 t. (Except at u=1.) The poles at negative p are called UV renormalon poles and the ones at positive p IR renormalons.

Large-β 0 12 0.3 0.25 δ (0) 0.2 0.15 Borel sum FO perturbation theory CI perturbation theory Smallest term 0.1 0 2 4 6 8 10 12 14 Perturbative order n δ(0) = 0.263 ±i0.009, δ (0) FO = 0.259, δ (0) CI = 0.227.

Large-β 0 13 Quite generally, δ (0) FO can be written as: δ (0) FO = n=1 [ c n,1 +g n ] a(m 2 τ) n. The large-order behaviour of the two components reads: c n+1,1 = g n+1 = ( ) β1 n [ n! 2 ( ) β1 n [ n! 2 4 9 e 5/3 ( 1) n ( n+ 7 ) + e10/3 ] 2 2 +..., n 4 9 e 5/3 ( 1) n( n+ 16 2 ) e10/3 2 n +... ]. Thus, large cancellations of the leading UV (u = 1) and IR (u = 2) renormalon contributions are seen to take place.

Borel model 14 B[ D](u) = B[ D UV 1 ](u) +B[ D IR 2 ](u) +B[ D IR 1 ](u) where B[ D p ](u) = +d PO 0 +d PO 1 u, d p (p±u) 1+γ [ 1 +b 1(p±u) +b 2 (p±u) 2 ]. Our model incoporates the leading UV pole (u = 1), as well as the two leading IR renormalons (u = 2,3). It should reproduce the exactly known c n,1, n 4. For both UV and IR, the residues d p are free while γ, b 1,2 depend on anomalous dimensions and β-coefficients.

Adler function 15 0.3 ^D(αs ) 0.25 0.2 0.15 Borel sum Perturbative series Smallest term 0.1 0.05 0 2 4 6 8 10 12 14 Perturbative order n c 5,1 = 283, α s (M τ ) = 0.34.

Adler function 16 0.15 Re[ ^D FO (ϕ)] 0.1 0.05 Borel sum FOPT up to 4th order FOPT up to 5th order FOPT up to 6th order FOPT up to 7th order 0 0 1 2 3 4 5 6 ϕ [rad] c 5,1 = 283, α s (M τ ) = 0.3156.

Adler function 17 0.15 Re[ ^D CI (ϕ)] 0.1 0.05 Borel sum CIPT up to 4th order CIPT up to 5th order CIPT up to 6th order CIPT up to 7th order 0 0 1 2 3 4 5 6 ϕ [rad] c 5,1 = 283, α s (M τ ) = 0.3156.

Borel model 18 0.26 0.24 0.22 δ (0) 0.2 0.18 0.16 0.14 Borel sum FO perturbation theory CI perturbation theory Smallest term 2 4 6 8 10 12 14 16 Perturbative order n c 5,1 = 283, α s (M τ ) = 0.34.

α s analysis 19 Employing the hadronic decay rate into light quarks R τ,v+a = N c V ud 2 S EW [ 1 +δ (0) +δ NP V+A with δ NP V+A = ( 7.1 ± 3.1) 10 3, one finds ] δ (0) = R τ,v+a 3 V ud 2 S EW 1 δ NP V+A = 0.2042(38)(33) The first uncertainty is due to R τ,v+a, while the remaining error is dominated by δ NP V+A. Scanning over plausible models and adjusting α s such as to reproduce δ (0), one finally obtains α s (M τ ) = 0.3156(30)(51) α s (M Z ) = 0.1180(8)

Conclusions 20 FOPT povides the closer approach to the true (Borel resummed) value for the perturbative correction δ (0). CIPT fails to yield the correct result, as it misses the large cancellations between the coefficients c n,1 and the g n. This behaviour is particular for R τ, due to the strong suppression of the leading IR renormalon at u = 2. Our central result is α s (M τ ) = 0.3156(59), leading to α s (M Z ) = 0.1180(8) after evolution to M Z, significantly lower than previous α s values from R τ. Work on m 2 and scalar contributions in process with F. Schwab.

Conclusions 20 FOPT povides the closer approach to the true (Borel resummed) value for the perturbative correction δ (0). CIPT fails to yield the correct result, as it misses the large cancellations between the coefficients c n,1 and the g n. This behaviour is particular for R τ, due to the strong suppression of the leading IR renormalon at u = 2. Our central result is α s (M τ ) = 0.3156(59), leading to α s (M Z ) = 0.1180(8) after evolution to M Z, significantly lower than previous α s values from R τ. Work on m 2 and scalar contributions in process with F. Schwab. Thank You for Your attention!

Maltman moments 21 0.26 0.24 0.22 0.2 Maltman weight w 2 Borel sum FO perturbation theory CI perturbation theory Smallest term δ (0) 0.18 0.16 0.14 0.12 0.1 2 4 6 8 10 12 14 Perturbative order n c 5,1 = 283, α s (M τ ) = 0.3156.

Maltman Moments 22 0.26 0.24 0.22 0.2 Borel sum FO perturbation theory CI perturbation theory Smallest term Maltman weight w 4 δ (0) 0.18 0.16 0.14 0.12 0.1 2 4 6 8 10 12 14 16 Perturbative order n c 5,1 = 283, α s (M τ ) = 0.3156.

Maltman Moments 23 0.26 0.24 0.22 0.2 Borel sum FO perturbation theory CI perturbation theory Smallest term Maltman weight w 6 δ (0) 0.18 0.16 0.14 0.12 0.1 2 4 6 8 10 12 14 16 Perturbative order n c 5,1 = 283, α s (M τ ) = 0.3156.