Strong coupling from. τ lepton decays. Matthias Jamin. Strong coupling from τ decays Matthias Jamin, ICREA & IFAE, UA Barcelona

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1 1 Strong coupling from τ lepton decays Matthias Jamin

2 α s measurements 2 PDG 2012 For0.6 % precision atm Z need only 2 % atm τ.

3 Hadronic τ decay rate 3 Consider the physical quantityr τ : (Braaten, Narison, Pich 1992) R τ Γ(τ hadronsν τ (γ)) Γ(τ e ν e ν τ (γ)) = (94). (HFAG 2012) R τ is related to the QCD correlatorsπ (1,0) (x): (x s/m 2 τ) 1 R τ = 12π dx(1 x) 2[ (1+2x)ImΠ (1) (x)+imπ (0) ] (x) 0 with appropriate combinations of mesonic 2-point correlators Π (J) (x) = V ud 2[ Π V,J ud +Π A,J ] ud + V us 2[ Πus V,J +Π A,J ] us.,

4 τ decay spectra 4 Experimental information can be inferred from the moments s 0 R w τ(s 0 ) dsw(s) d R τ ds = Rw τ,v +R w τ,a+r w τ,s. 0 w τ (s) = (1 s ) 2 (1+2 s ). s 0 s 0 v 1 (s) ALEPH [mod-08] τ V ν τ π π 0 π 3π 0, 2π π + π 0, (6π) ωπ, ηπ π 0, (KK-bar(π)) QCD prediction parton model v(s) OPAL π π 0 3π π 0, π 3π 0 MC corr. perturbative QCD (massless) naïve parton model s (Gev 2 ) s (GeV 2 )

5 Hadronic τ decay moments 5 Theoretically,R τ is calculated via the contour-integral: R w V/A(s 0 ) 6πi ds s =s 0 s 0 w(s) [ Π (1+0) V/A (s)+ 2s (s 0 +2s) Π(0) V/A (s) ]. Generally,R w τ takes the structure: R w τ = N c S EW {( V ud 2 + V us 2 ) + D 2 [ [1+δ w(0)] V ud 2 δ w(d) ud + V us 2 δ w(d) us ] }. δ w(d) ud andδ w(d) us are corrections in the Operator Product Expansion, the most important ones being m 2 s andm s qq.

6 Contour integral 6 Im q 2 Light quarks s 0 Hadrons տ Zero Re q 2 Complex q 2 -Plane

7 Adler function 7 The perturbative partδ (0) is related to the Adler functiond(s): D(s) s d ds Π V(s) = N c 12π 2 n=0 a n µ n+1 k=1 kc n,k ln k 1( ) s µ 2 wherea µ α s (µ)/π. Resumming the Log s with the scale choiceµ 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 coefficientsc n,1 are independent: c 0,1 = c 11 = 1, c 2,1 = 1.640, c 3,1 = 6.371, c 4,1 = !! (Baikov, Chetyrkin, Kühn 2008)

8 RG-improvement 8 Fixed-order perturbation theory amounts to chooseµ 2 =M 2 τ : δ (0) FO = n=1 a n (M 2 τ) n+1 k=1 kc n,k J k 1 = n=1 [c n,1 +g n ]a n (M 2 τ) A given perturbative orderndepends on all coefficientsc m,1 withm n, and on the coefficients of the QCDβ-function. Contour-improved perturbation theory employsµ 2 = M 2 τx: (Pivovarov; Le Diberder, Pich 1992) δ (0) CI = n=1 J a n(m 2 τ) = 1 2πi c n,1 J a n(m 2 τ) x =1 with dx x (1 x)3 (1+x)a n ( M 2 τx)

9 FOPT vs CIPT problem 9 The purely perturbative contributionδ (0) is plagued by differences for different RG-resummations. (FOPT vs CIPT.) Usingα s (M τ )=0.3186, the numerical analysis results in: a 1 a 2 a 3 a 4 a 5 δ (0) FO = (+0.006) = 0.196(0.202) δ (0) CI = (+0.003) = 0.181(0.185) Contour-improved PT appears to be better convergent. The difference between both approaches is 0.015(0.017)! This problematic entails a 6% difference forα s (M τ ).

10 Borel transform 10 To further investigate the difference between CI and FOPT, let us consider the Borel-transformed Adler function. 4π 2 D(s) 1+ D(s) 1+ n=0 r n α s (s) n+1, wherer n =c n+1,1 /π n+1. The Borel-transform reads: D(α s ) = 0 dte t/α s B[ D](t); B[ D](t) = n=0 r n t n n!. Generally, the Borel-transformB[ D] developes poles and cuts at integer valuespofu β 1 t/(2π). (Except atu=1.) The poles at negativepare called UV renormalon poles and the ones at positive p IR renormalons.

11 Borel model 11 To proceed, realistic modelb[ D](u): (Beneke, MJ 2008) B[ D](u) = B[ D UV 1 ](u)+b[ D IR 2 ](u)+b[ D IR 3 ](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 ]. Main model incorporates the leading UV pole (u= 1), as well as the two leading IR renormalons (u=2,3). It should reproduce the exactly knownc n,1, n 4. For both UV and IR, the residuesd p are free whileγ,b 1,2 depend on anomalous dimensions and β-coefficients.

12 Central Borel model 12 δ (0) Borel sum FO perturbation theory CI perturbation theory Smallest term Perturbative order n α s (M τ ) = , c 5,1 =283. (Beneke, MJ 2008)

13 Other moments 13 w(x) = 1: α s (M τ ) = , c 5,1 =283. (Beneke, Boito, MJ 2012)

14 Other moments 14 w(x) = 1 x: α s (M τ ) = , c 5,1 =283. (Beneke, Boito, MJ 2012)

15 Other moments 15 w(x) = (1 x) 3 x 2 (1+2x): α s (M τ ) = , c 5,1 =283. (Beneke, Boito, MJ 2012)

16 Model dependence 16 The behaviour of the Borel model crucially depends on the residue of the gluon-condensate renormalon pole. Assuming some sensitivity to the u=2 pole at intermediate orders (3-5), a fit to the knownc n,1 yieldsd IR For smalld IR 2, models can be constructed for which Contour-improved PT is the preferred resummation. Hence, to make progress the value ofd IR 2 corroborated. Two possible routes: should be i) As the renormalon ambiguity is universal, employ PT series of other correlators to obtain additional information. ii) Determined IR 2 from the lattice. Not possible directly for the Adler function, but for the plaquette.

17 Duality violations 17 In the OPE, close to the Minkowskian axis (s > 0), so-called Duality Violations (DV s) can appear. They can be studied on the basis of a toy-model: (Shifman et al ) (Catà, Golterman, Peris 2005/2008) where ( M 2 Π V (s)= ψ V +u(s) u(s)=λ 2 ( s Λ 2 ) ζ Λ 2 and ) + const.. ζ = 1 a πn c. The model is based on large-n c QCD and Regge-theory. M V =770MeV, Λ=1.2GeV, a=0.4.

18 Duality violations 18 The OPE corresponds to the asymptotic expansion of the ψ-function for large s (large u). ψ(z) lnz 1 2z n=1 B 2n, Rez>0. 2nz2n In the Minkowskian region, an additional term arises: π[cot(πz)±i], Rez<0, Imz > < 0. Formally, this term is exponentially suppressed, but it is enhanced by the poles of the ψ-function.

19 Asymptotic expansion 19 Re[ψ(z)], Im[ψ(z)] arg(z) arg(z) z=1.5 exp(iϕ)

20 ψ-function moments R w (s) s 0 [GeV 2 ] ψ-function moment for w(z) = 1.

21 ψ-function moments R w (s) s 0 [GeV 2 ] ψ-function moment forw(z)=(1 z) 2.

22 Fits to OPAL data 22 In fits to experimental data, a model for DV s should be included. The ψ-function model suggests an oscillating, decaying exponential, which can be chosen of the form: ρ DV V/A(s)=κ V/A e γ V/As sin(α V/A +β V/A s). The fit quantities are the w-moments of the exp spectra. R w τ,v/a(s 0 ) s 0 0 dsw(s)ρ V/A (s). The cleanest moment turns out to bew(s)=1. Fitting combinations of several moments is complicated by very strong correlations.

23 OPAL V-moment 23 w(s)=1 (Boito, Golterman, MJ, Mahdavi, Maltman, Osborne, Peris 2012)

24 OPAL V-spectrum 24 w(s)=1 (Boito, Golterman, MJ, Mahdavi, Maltman, Osborne, Peris 2012)

25 Summary 25 Presently, the most reliable value ofα s fromτ s including DV s comes from the trivial momentw(s)=1. α s (M τ )=0.325±0.016±0.007 (FOPT) α s (M τ )=0.347±0.024±0.005 (CIPT) This moment only has very small contaminations from QCD condensate contributions. These values should be compared to the World Average (PDG 2012): α s (M τ )=0.3186±

26 Main messages 26 The ambiguity in the perturbative resummation should be overcome. Possible routes: dedicated moments and/or better understanding of Borel models. Use of several moments to fitα s and condensates for consistency should include duality violations. Due to strong correlations in moment fits, better τ spectral data, e.g. from B-factories would be extremely helpful to resolve the theoretical issues.

27 Main messages 26 The ambiguity in the perturbative resummation should be overcome. Possible routes: dedicated moments and/or better understanding of Borel models. Use of several moments to fitα s and condensates for consistency should include duality violations. Due to strong correlations in moment fits, better τ spectral data, e.g. from B-factories would be extremely helpful to resolve the theoretical issues. Thank You for Your attention!