PoS(EPS-HEP 2013)413. Strong coupling from the τ-lepton hadronic width

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1 Strog couplig from the τ-lepto hadroic width Istitute of Mathematical Scieces, Cheai, Idia B. Aathaaraya Cetre for High Eergy Physics, Idia Istitute of Sciece, Bagalore , Idia Iriel Caprii Horia Hulubei Natioal Istitute for Physics ad Nuclear Egieerig, P.O.B. MG-6, Magurele, Romaia Ja Fischer Istitute of Physics, Academy of Scieces of the Czech Republic, CZ Prague 8, Czech Republic The strog couplig is determied usig a ovel reormalizatio-group-summed perturbatio theory ad a o power reormalizatio-group-summed perturbatio theory with a tamed large-order behaviour. I this talk, the mai features of these two schemes are preseted ad are applied to the extractio of α s M 2 τ) from the hadroic decay width of the τ lepto. While the reormalizatiogroup-summed perturbatio theory leads to a value of strog couplig close to that from cotourimproved perturbatio theory, the o power reormalizatio-group-summed perturbatio theory yields a value closer to the stadard fixed-order perturbatio theory. The Europea Physical Society Coferece o High Eergy Physics July, 2013 Stockholm, Swede Speaker. c Copyright owed by the authors) uder the terms of the Creative Commos Attributio-NoCommercial-ShareAlike Licece.

2 1. Itroductio The strog couplig is oe of the most importat parameters of the Stadard Model. Its precise determiatio at various scales is crucial for testig the asymptotic freedom of QCD, the moder theory of strog iteractios. The iclusive hadroic decay width of the τ lepto provides a clea way to determie α s at low eergies [1, 2, 3, 4]. The R ratio for the τ decays is defied as: R τ Γ[τ hadrosν τ ] Γ[τ e. 1.1) ν e ν τ ] Here we are iterested i the τ decay rate ito the lightest u ad d) quarks, which proceeds either through the vector V ) or axial-vector curret A). This quatity ca be expressed theoretically i the form R τ,v/a = N [ ] c 2 S EW V ud 2 1+δ 0) + δ EW+ δ D) ud, 1.2) D 2 where S EW = ± ad δ EW = ± are electroweak correctios, ad δd) ud is the cotributio from the higher D-dimesioal operators which arise i the operator product expasio. Our mai iterest is i the perturbative correctio δ 0) which ca be writte as [5] δ 0) = 1 2πi s =M 2 τ ds 1 s ) 3 s M 2 1+ s ) ˆD τ M 2 pert a,l), 1.3) τ where a aµ 2 ) α s µ 2 )/π, L l s µ 2 ad ˆD pert a,l), is the perturbatio expasio of the Adler fuctio. I the fixed-order perturbatio theory FOPT) this expasio has the form [6] ˆD FOPT a,l) = a =1 k=1 k c,k L k 1, 1.4) while i reormalizatio-group-improved or cotour-improved perturbatio theory CIPT) the series is writte as [7, 8] ) αs s) ˆD CIPT α s s)/π,0)= c,1. 1.5) π I the above expasios the coefficiets c,1 are kow for 4 from perturbative calculatios i the MS-reormalizatio scheme see [9] ad refereces therei). The umerical values for f = 3 flavours are: =1 c 1,1 = 1, c 2,1 = 1.640, c 3,1 = 6.371, c 4,1 = For the ext coefficiet c 5,1 estimates are give i [6, 10, 11, 12]. The coefficiets c, j for j > 1 are determied i terms of c,1 ad the coefficiets β j of the β-fuctio of the reormalizatio group equatio RGE). At preset this fuctio is calculated to four loops i the MS-reormalizatio scheme see [13, 14] ad refereces therei), where the first coefficiets are: β 0 = 9/4, β 1 = 4, β 2 = , β 3 =

3 2. Reormalizatio-Group-Summed Perturbatio Theory We use a method based o the explicit summatio of all reormalizatio group accessible logarithms [15, 16, 17], which was recetly applied to the hadroic width of the τ lepto [18, 19, 20, 21]. I this method, the Adler fuctio is writte as where D al) ˆD RGSPT a,l)= k= =1 a D al), 2.1) k +1)c k,k +1 al) k. 2.2) We ow exploit the fact that the Adler fuctio defied by the expasio 1.4) is scale idepedet µ 2 d dµ 2[ ˆD FOPT a,l)]=0 βa) ˆD FOPT a The relevat reormalizatio group equatio RGE) is writte as: 0= =1 k=2 kk 1)c,k a L k 2 β 0 a 2 + β 1 a 3 + β 2 a β l a l ) =1 ˆD FOPT L k=1 = ) kc,k a 1 L k ) By extractig the aggregate coefficiet of a L p we obtai the recursio formula p) p 2 0= p+2)c, p+2 + l 1)β l c l 1, p ) l=0 Multiplyig both sides of 2.5) by p+1)al) p ad summig from = p to, we obtai a system of first-order liear differetial equatio for the fuctios defied i 2.2), writte as 1 ) dd dal) + d β l al) dal) + l D l = 0, 2.6) l=0 for 1, with the iitial coditios D 0) = c,1 which follow from 2.2). The solutio of the above equatios ca be foud iteratively i a aalytical closed form. The first two solutios are D 1 al)= c 1,1 y, D 2aL)= c 2,1 y 2 β 1c 1,1 ly β 0 y 2, y=1+β 0 al. 2.7) By isertig i the itegral 1.3) the RGSPT expasio 2.1) of the Adler fuctio we obtai the expasio of δ 0) δ 0) RGSPT = amτ) 2 d, where d = 1 =1 2πi s =M 2 τ ds 1 s ) 3 s M 2 1+ s ) τ M 2 D al). 2.8) τ I Table 1 we show the behaviour of the FOPT, CIPT ad RGSPT as fuctios of the trucatio order N of the series. For N = 4, the differece betwee the results of the RGSPT ad the stadard FOPT is , ad the differece from the RGSPT ad CIPT is , which cofirms that the RGSPT gives results close to those of the CIPT. 3

4 δ 0) FOPT δ 0) CIPT δ 0) RGSPT N = N = N = N = N = Table 1: Predictios of δ 0) i the stadard FOPT, CIPT ad the RGSPT, for various trucatio orders N usig α s M 2 τ) = Determiatio of α s from RGSPT expasio We adopt as iput the recet pheomeological value [6, 12] δ 0) phe = ± exp± PC. 3.1) Usig the expasio 2.8) trucated at N = 5, with the kow coefficiets c,1 from 1.6) ad the coservative choice c 5,1 = 283±283 as i [6], we obtai [18] α s M 2 τ)=0.3378± exp ± PC c 5,1) scale) β 4), 3.2) where the errors from various sources are idicated separately i the last term we used the estimate β 4 = ±β 2 3 /β 2 for ext coefficiet i the expasio of the β fuctio). Combiig the errors i quadrature α s M 2 τ)=0.338± ) 4. RGS No-Power Perturbatio Theory Oe of the ambiguites i the extractio of α s from the hadroic τ decays arises from the large-order behaviour of the QCD perturbative series. The large-order behaviour of the RGSPT expasio was ivestigated i [19] i a model of the Adler fuctio proposed i [6]. I this model, the RGS expasio of the QCD Adler fuctio has a behaviour which is similar to that of CIPT ad evetually exhibits large oscillatios, thereby showig the diverget character of the QCD perturbative series. We improve this behaviour of the RGSPT expasio by the aalytical cotiuatio i the Borel plae. The method was applied to FOPT ad CIPT by Caprii ad Fischer [22, 23, 24]. The large-order behaviour of the perturbatio theory is ecoded i the sigularities of the Borel trasform Bu), defied startig from the expasio 1.5) as Bu)= u c +1,1 =0 β0!. 4.1) The fuctio Bu) has sigularities placed o the real axis alog the lies u 1 ad u 2. Therefore, the Taylor expasio 4.1) coverges oly i the disk u < 1, limited by the earest sigularity at u = 1 of the expaded fuctio. The regio of covergece ca be elarged if 4

5 the series i powers of u is replaced by a series i powers of a optimal" variable wu) that coformally maps the holomorphy domai of Bu), i.e. the u-plae with cut alog u 2 ad u 1, oto the uit disk w <1 of the complex plae w wu). This also accelerates the covergece rate at all poits i the holomorphy domai [24, 25]. The Borel trasform of the RGSPT expasio 2.1) is writte as [19] B RGSPT u,y)=bu)+ =0 u β 0! c j,1 d +1, j y), 4.2) j=1 where y=1+β 0 al. We cosider a geeral class of coformal mappigs 1+u/l 1 u/m w lm u)=, l 1,m 2 4.3) 1+u/l+ 1 u/m where l,m are positive itegers satisfyig l 1 ad m 2. The fuctio w lm u) maps the u-plae cut alog u l ad u m oto the disk w lm <1 i the plae w lm w lm u). We defie further the class of compesatig factors of the simple form where the expoets S lm u)= 1 w ) l) lmu) γ 1 1 w lmu) w lm 1) w lm 2) γ l) 1 = γ 1 1+δ l1 ), γ m) 2 = γ 2 1+δ m2 ), ) γ m) 2, 4.4) γ 1 = 1.21, γ 2 = 2.58, 4.5) are chose such that S lm u) cacels the domiat sigularities of B RGSPT u,y), situated at u= 1 ad u=2. We further expad the product S lm u)b RGSPT u,y) i powers of the variable w lm u), as S lm u)b RGSPT u,y)= c lm),rgspt y) w lmu)). 4.6) 0 We are thus led to the class of RGSNPPT expasios where D RGSNPPT s)= c lm) lm),rgspt y)w,rgspt s), 4.7) 0 W lm),rgspt s)= 1 PV β 0 0 exp ) u wlm u)) du. 4.8) β 0 ã s s) S lm u) The coefficiets c lm),rgspt y) are defied by the expasio 4.6) ad the couplig ã s s) appearig i the Laplace-Borel itegral is the oe-loop solutio of the RGE. 5. High-order behaviour of RGSNPPT expasios I Table 2 we record the remarkable supressio of the diverget behaviour of the RGSPT expasio through aalytic cotiuatio i the Borel plae. 5

6 N CIPT FOPT RGSPT RGSNPPT w 12 RGSNPPT w 13 RGSNPPT w 1 RGSNPPT w Table 2: The differece δ 0) -δ 0) exact for the model proposed i [6] for α s M 2 τ)=0.34 with the stadard CIPT, FOPT ad RGSPT expasios, ad the ew RGSNPPT expasios for various coformal mappigs w lm, trucated at order N. The exact value δ 0) exact = Determiatio of α s from RGSNPPT expasios Usig the pheomeological iput 3.1) ad the RGSNPPT expasios 4.7) we obtai [19] α s M 2 τ)=0.3189± exp ± PC c 5,1) ± β4, 6.1) ad after combiig the errors i quadrature By evolvig to the scale of M Z our predictio reads 7. Coclusio α s M 2 τ)= ) α s M 2 Z)= ) This work is motivated by the well-kow discrepacy betwee the predictios of α s M 2 τ) from the stadard FOPT ad CIPT expasios. We have show that the summatio of all the logarithms accesible by reormalizatio group ivariace provides a systematic expasio of the Adler fuctio with a good behaviour i the complex eergy plae [18]. The results of the ew RGSPT expasio are similar to those obtaied with CIPT. We further tamed the diverget character of the perturbative series by the method of coformal mappigs of the Borel plae, defiig the RGS o-power expasios [19], similar to the FONPPT ad CINPPT defied i [22, 24]. The RGSNPPT expasios lead to a predictio for α s M 2 τ) similar to that obtaied with stadard FOPT ad with CINPPT. As show recetly [26], the good large-order properties of the reormalizatiogroup improved o-power expasios are valid also for a large class of momets of the spectral fuctios the oe associated with the hadroic width beig a special oe). Therefore, CINPPT ad RGSNPPT provide a solid theoretical framework i momet aalyses for the simultaeous determiatio of the strog couplig ad other parameters of QCD from hadroic τ decays. 6

7 Refereces [1] J. Beriger et al. Particle Data Group), Phys. Rev. D86, ). [2] A. Pich, PoS CofiemetX 2012) 022, arxiv: [hep-ph]. [3] M. Jami, PoS CofiemetX 2012) 098, arxiv: [hep-ph]. [4] G. Altarelli, PoS Corfu ) 002, arxiv: [hep-ph]. [5] E. Braate, S. Nariso ad A. Pich, Nucl. Phys. B 373, ). [6] M. Beeke ad M. Jami, JHEP 09, ), arxiv: [hep-ph]. [7] A.A.Pivovarov, Z. Phys. C 53, ), [Sov. J. Nucl. Phys. 54, )], hep-ph/ [8] F. Le Diberder ad A. Pich, Phys. Lett. B 289, ). [9] P.A. Baikov, K.G. Chetyrki ad J.H. Küh, Phys. Rev. Lett. 101, ), arxiv: [hep-ph]. [10] M. Davier, S. Descotes-Geo, A. H ocker, B. Malaescu ad Z. Zhag, Eur. Phys. J. C56, ), arxiv: [hep-ph]. [11] A. Pich, Tau decay determiatio of the QCD couplig, i Workshop o Precisio Measuremets of α s, ed. S. Bethke et al, page 21, arxiv: [hep-ph]. [12] M. Beeke ad M. Jami, Fixed-order aalysis of the hadroic τ decay width, i Workshop o Precisio Measuremets of α s, ed. S. Bethke et al, page 25, arxiv: [hep-ph]. [13] S.A. Lari, T. va Ritberge ad J.A.M. Vermasere, Phys. Lett. B400, ), hep-ph/ ; S.A. Lari, T. va Ritberge ad J.A.M. Vermasere, Phys. Lett. B404, ), hep-ph/ [14] M. Czako, Nucl. Phys. B710, ), hep-ph/ [15] C.J. Maxwell ad A. Mirjalili, Nucl. Phys. B 577, ), hep-ph/ ; Nucl. Phys. B 611, ), hep-ph/ [16] M.R. Ahmady, F. A. Chishtie, V. Elias, A. H. Fariboz, N. Fattahi, D. G. C. McKoe, T. N. Sherry, ad T.G. Steele, Phys. Rev. D 66, ), hep-ph/ [17] M.R. Ahmady,, F. A. Chishtie, V. Elias, A. H. Fariboz, D. G. C. McKoe, T. N. Sherry, A. Sqires, ad T.G. Steele, Phys. Rev. D 67, ), hep-ph/ [18] G. Abbas, B. Aathaaraya ad I. Caprii, Phys. Rev. D 85, ), arxiv: [hep-ph]. [19] G. Abbas, B. Aathaaraya, I. Caprii ad J. Fischer, Phys. Rev. D 87, ), arxiv: [hep-ph]. [20] I. Caprii, Mod. Phys. Lett. A 28, ), arxiv: [hep-ph]. [21] G. Abbas, B. Aathaaraya ad I. Caprii, Mod. Phys. Lett. A 28, ), arxiv: [hep-ph]. [22] I. Caprii ad J. Fischer, Eur. Phys. J. C64, ), arxiv: [hep-ph]. [23] I. Caprii ad J. Fischer, Nucl. Phys. B Proc. Suppl., 218, ), arxiv: [hep-ph]. [24] I. Caprii ad J. Fischer, Phys. Rev. D 84, ), arxiv: [hep-ph]. [25] S. Ciulli ad J. Fischer, Nucl. Phys. 24, ). [26] G. Abbas, B. Aathaaraya, I. Caprii ad J. Fischer, Phys. Rev. D 88, ), arxiv: [hep-ph]. 7