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Recent progress in hadronic τ decays 1 1 Matthias Jamina 0 2 a Institucio´ Catalana de Recerca i Estudis Avan¸cats (ICREA), n Institut de F´ısica d’Altes Energies (IFAE), Campus UAB, E-08193 Bellaterra, Barcelona,Spain a J 4 The determination of αs from hadronic τ decays is impeded by the fact that two choices for the renormalisation group resummation, namely fixed-order (FOPT) and contour-improved perturbation theory (CIPT), yield ] systematically differing results. On the basis of a model for higher-order terms in the perturbative series, which h incorporates well-known structurefrom renormalons, it is found that FOPT smoothly approachestheBorel sum p fortheτ hadronicwidth,whileCIPTisunabletoaccountfortheresummedseries. Anexampleforthebehaviour - of QCD spectral function moments, displaying a similar behaviour, is presented as well. p e h [ 1 1. INTRODUCTION The dispersion in these results dominantly origi- v nates from different treatments of the renormali- 1 Hadronicdecays ofthe τ lepton providean ex- sation group (RG) resummation of the perturba- 8 cellent playground for the study of QCD at low tive series, namely fixed-order perturbation the- 6 energies. Its mass of M ≈ 1.8GeV is in an en- τ ory (FOPT), or contour-improved perturbation 0 ergy regionwhere perturbationtheory is still ap- . theory (CIPT) [9,10], being systematically larger 1 plicable, but also non-perturbative effects come than the last included term in the expansion, 0 into play and need to be included. These may whichoften inasymptoticseriesprovidesanesti- 1 arisefromvacuum-condensatetermsintheframe- 1 mateoftheuncertaintyduetohigher-orderterms work of the operator product expansion (OPE) : not included in the partial sum. This points to v andfromso-calledduality-violating contributions i the necessity of investigating the influence of dif- X close to the physical, Minkowskianenergy axis. ferent RG resummations in more detail, which Intheseminalarticle[1],thestrategyforapre- r can for example be performed within exactly re- a cise determination of the QCD coupling α from s summable models for the perturbative series. the total τ hadronic width Most suitable for the α determination is the s Γ[τ− →hadronsντ(γ)] τ decay rate into light u and d quarks Rτ,V/A R ≡ = 3.640(10), (1) τ Γ[τ− →e−ν ν (γ)] via a vector or axialvector current, since in this e τ case power corrections are especially suppressed. wasdeveloped,whileinthesubsequentyearsmo- Theoretically, R takes the form [1] τ,V/A ments of spectral τ-decay distributions were in- corporatedinto the analyses as well [2,3,4]. R = Nc S |V |2 1+δ(0) Theanalyticalcomputationoftheperturbative τ,V/A 2 EW ud h order α4s correction [5] has recently revived the +δ′ + δ(D) , (3) interest in αs analyses from hadronic τ decays EW DX≥2 ud,V/Ai which after evolution to the Z boson mass scale resulted in the following determinations: where S = 1.0198(6) [11] and δ′ = EW EW 0.0010(10) [12] are electroweak corrections, δ(0) 0.1202(6) (18) [5] , exp th comprises the perturbative QCD correction, and αs(MZ2)=00..11211820((54))eexxpp((97))tthh [[67]] ,, (2) dthimeeδnu(sDdi,o)Vn/aAl dopeneoratetorqucaorrkrecmtiaosnssawnhdichhigahreisreDin- 0.1187(6) (15) [8] . the framework of the OPE.  exp th 1 2 2. PERTURBATIVE CORRECTION δ(0) Thus,allcontributionsproportionaltothe coeffi- cientc whichinFOPTappearatallperturba- n,1 Below, only the purely perturbative correction tive ordersequalorgreaterthannareresummed δ(0) shallbeconsidered,whichgivesthedominant into a single term. contribution to R . In FOPT it takes the τ,V/A Numerically, the two approaches lead to sig- general form nificant differences. Employing the recent aver- ∞ n age αs(MZ) = 0.1184 [13], leading to αs(Mτ) = δ(0) = a(M2)n kc J , (4) 0.3186, in eqs. (4) and (8), one finds FO τ n,k k−1 nX=1 Xk=1 δ(0) = 0.1959(0.2022), (10) where a(µ2) ≡ a ≡ α (µ)/π, and c are the FO µ s n,k coefficients which appear in the perturbative ex- δ(0) = 0.1814(0.1847), (11) CI pansion of the vector correlation function, where the firstnumber inboth casesemploysthe ∞ n+1 known coefficients up to O(α4) [5] and the num- N −s s Π (s) = − c an c lnk . (5) bers inbracketsinclude anestimate ofthe O(α5) V 12π2 µ n,k (cid:18)µ2(cid:19) s nX=0 Xk=0 term with c5,1 ≈ 283 [7]. Inspecting the individual contributions from each order, up to O(α5) At each perturbative order, the coefficients c s n,1 the CIPT series appears to be better convergent. can be considered independent, while all other However, around the seventh order, the contour c with k ≥2 are calculable fromthe RGequa- n,k integralsJa(M2)changesignandthusatthisor- tion. Further details can for example be found n τ derthecontributionsareboundtobecomesmall. in ref. [7]. Finally, the J are contour integrals in l Therefore, the faster approach to the minimal the complex s-plane, which are defined by term does not necessarily imply that CIPT gives 1 dx the closer approach to the true result for the re- J ≡ (1−x)3(1+x)lnl(−x). (6) l 2πi I x summed series. |x|=1 The first three, being required up to O(α3), take 3. A PHYSICAL MODEL s the numerical values To investigate whether FOPT or CIPT results J = 1, J = −19, J = 265−1π2. (7) in a better approximation to δ(0), one requires a 0 1 12 2 72 3 physically motivated model for its series. Such a At order αns FOPT contains unsummed loga- model was constructed in ref. [7] and is based on rithms of order lnl(−x) ∼ πl with l < n related the Boreltransformofthe Adler function D (s): V tothecontourintegralsJ . CIPTsumstheselog- l arithms, which yields d Nc D (s) ≡ −s Π (s) ≡ 1+D(s) . (12) V ds V 12π2 ∞ (cid:2) (cid:3) δC(0I) = cn,1Jna(Mτ2) (8) Inthefollowingdiscussionitisslightlybmorecon- nX=1 venient to utilise the related function D(s). Its intermsofthecontourintegralsJna(Mτ2)overthe BoreltransformB[D](t)isdefinedbythebrelation running coupling, defined as: ∞ b D(α) ≡ dte−t/αB[D](t). (13) 1 dx Ja(M2) ≡ (1−x)3(1+x)an(−M2x). Z n τ 2πi I x τ b 0 b |x|=1 TheintegralD(α),ifitexists,givestheBorelsum (9) of the originaldivergentseries. Itwasfound that b In contrast to FOPT, for CIPT each order n theBorel-transformedAdlerfunctionB[D](t)ob- justdependsonthecorrespondingcoefficientc . tains infrared (IR) and ultraviolet (UV) renor- n,1 b 3 malon poles at positive and negative integer val- c c c c c 2,1 3,1 4,1 5,1 6,1 ues of the variable u ≡ 9t/(4π), respectively IR −77.8 82.4 100.4 135.9 97.5 2 [14,15,16]. (With the exception of u=1.) IR 152.0 28.7 −10.0 −20.2 −13.3 Apartfromveryloworders,whereadominance 3 UV 22.5 −11.2 9.7 −15.6 15.8 ofrenormalonpolesclosetou=0hasnotyetset 1 in, intermediate orders should be dominated by Table 1 the leading IR renormalon poles, while the lead- Relative contributions (in %) of the different IR ing UV renormalon, being closest to u = 0, dic- and UV renormalon poles to the Adler-function tates the large-order behaviour of the perturba- coefficients c to c for the Borel model (14). tive expansion. Assuming that only the first two 2,1 6,1 orders are not yet dominated by the lowest IR pole to the coefficients c to c is tabulated. renormalons,one is led to the ansatz 2,1 6,1 The sum of the contributions to c is close to 2,1 B[D](u) = B[DUV](u)+B[DIR](u)+ 100%,implyingagainthatthepolynomialtermis 1 2 b B[DbIR](u)+dPOb+dPOu, (14) alreadysmall. Then, fromc3,1 to c6,1 the leading 3 0 1 IR pole at u = 2 is dominating the coefficients, whichincludesoneUbVrenormalonatu=−1,the before the leading UV pole at u=−1 takes over two leading IR renormalons at u = 2 and u = 3, atevenhigherorders. (Forthecentralmodel(14) aswellaspolynomialtermsforthetwolowestper- this happens around the 10th order [7].) turbative orders. Explicit expressionsfor the UV and IR renormalon pole terms B[DUV](u) and p 0.26 B[DIR](u) can be found in section 5 of ref. [7]. p b 0.24 Apart from the residues dUV and dIR, the full b p p structure of the renormalon pole terms is dic- 0.22 tated by the OPE and the RG. Therefore, the 0.2 model(14)depends onfive parameters,the three (0)δ0.18 residuadUV,dIR anddIR,aswellasthetwopoly- Borel sum 1 2 3 0.16 FO perturbation theory nomial parameters dPO and dPO. These parame- CI perturbation theory 0 1 0.14 Smallest term terscanbefixedbymatchingtotheperturbative expansionofD(s)uptoO(α5). Thereby,alsothe 0.12 2 4 6 8 10 12 14 16 s Perturbative order n estimate for c is used. The parameters of the b5,1 model (14) are then found to be: Figure 1. Results for δ(0) (full circles) and δ(0) FO CI (grey circles) at α (M )=0.3186,employing the dUV = −1.56·10−2, dIR = 3.16, dIR = −13.5, s τ 1 2 3 model (14), as a function of the order n up to which the terms in the perturbative series have dPO = 0.781, dPO = 7.66·10−3. (15) been summed. The straight line represents the 0 1 result for the Borel sum of the series. The fact that the parameter dPO turns out to be 1 small implies that the coefficient c is already 2,1 The implications of the model (14) for δ(0) in reasonablywelldescribedbytherenormalonpole FOPTandCIPTisgraphicallyrepresentedinfig- contribution, although it was not used to fix the residua. Therefore, one could set dPO = 0 and ure 1. The full circles denote the result for δF(0O) actually work with a model which o1nly has four and the grey circles the one for δ(0), as a func- CI parameters. The predicted value c = 280 then tion of the order n up to which the perturbative 5,1 turnsoutveryclosetotheestimate,whichcanbe series has been summed. The straight line cor- viewed as one test of the stability of the model. responds to the principal value Borel sum of the This is also corroborated in table 1, where series, δ(0) = 0.2080, and the shaded band pro- BS therelativecontributionsofacertainrenormalon vides an error estimate based on its imaginary 4 part divided by π. The order at which the series hand for low coefficients there are huge cancella- havetheirsmallesttermsis indicatedby thegrey tions between the IR renormalon poles and also diamonds. As is obvious from figure 1, FOPT c is not well described at all. This entails that 2,1 displays the behaviour expected from an asymp- a large, additional polynomial term is required. totic series: the terms decrease up to a certain Besides, it appears unnatural that the residue of order around which the closest approach to the thefirstIRrenormalonpoleatu=2issmall,and resummedresultisfound,andforevenhigheror- stillthereshouldbeanaturalsizecontributionof ders, the divergent large-order behaviour of the the gluon condensate to the Adler function. series sets in. For CIPT, on the other hand, the A graphicalaccountof the model with dIR =0 2 asymptotic behaviour sets in earlier, and the se- is presented in figure 2. As anticipated, now riesisneverabletocomeclosetotheBorelsum.1 CIPT provides a good description of the Borel sum,whileFOPTisabletocomereasonablyclose to it around its minimal term, but generally is c c c c c rather badly behaved.2 Nonetheless, again, the 2,1 3,1 4,1 5,1 6,1 behaviour observed in table 2 and figure 2 ap- IR −743.3 −140.5 49.1 98.9 99.1 3 pearsunnaturalfromtheperspectiveofthestruc- IR 662.8 244.2 47.7 6.3 −7.2 4 ture of the Boreltransform of the Adler function UV1 7.5 −3.7 3.2 −5.2 8.1 and should be considered less likely than the behaviour of the model (14) with the residues (15). Table 2 Relative contributions (in %) of the different IR and UV renormalon poles to the Adler-function 0.22 coefficients c to c for the Borel model with 3,1 6,1 dI2R =0. 0.2 AsthebehaviourofCIPTversusFOPThinges (0)δ0.18 Borel sum on the contributionof the leading IR renormalon 0.16 FO perturbation theory at u = 2, in principal also models can be con- CI perturbation theory structed for which CIPT provides a better ac- 0.14 Smallest term count of the Borel sum. These would generally 0.12 2 4 6 8 10 12 14 16 be models where dIR is much smaller than the Perturbative order n 2 value quoted in eq. (15). While such models can Figure 2. Results for δ(0) (full circles) and δ(0) at present not be excluded, the pattern of the FO CI (grey circles) at α (M )=0.3186,employing the individual contributions appears more unnatural s τ model (14) with dIR = 0 and an additional IR than in the main model (14): the knowncn,1 can 2 pole at u = 4, as a function of the order n up to onlybereproducedwhenoneallowsforlargecan- which the terms in the perturbative series have cellations between the individual terms. Thus, been summed. The straight line represents the the behaviour generally expected from the pres- result for the Borel sum of the series. ence of renormalon poles, namely dominance of leadingIRpolesatintermediateorders,wouldbe lost. In the standard α determinations from s Thisisapparentfromtable2,whichistheana- hadronic τ decays [2,3], besides the total decay logoftable1,butforamodelwheredIR isforced rate also moments of the spectral decay distri- 2 to be zero and an additional IR pole at u = 4 butions are employed. Historically, the so-called is added, in order to be able to reproduce the known Adler function coefficients. On the one 2Asimilarbehaviourwasfoundinref.[18]inmodelswhere thehigher-orderAdlerfunctioncoefficients wereassumed 1Thesameconclusionshadalreadybeendrawninref.[17] to be small, in contrast to the expected asymptotic be- onthebasisofthelarge-β0 approximationforΠV(s). haviourofthisseriesinQCD. 5 (k,l)-momentswereused,forwhichapolynomial prescriptions in the computation of the total τ (1−x)kxl is multiplied tothe kinematicalweight hadronic width, as well as related moments of function from phase-space. In the analyses [2,3], τ decay spectral distributions. The most promi- moments with k =1 and l =0,1,2,3 were taken nentmethodsarefixed-orderperturbationtheory into account, such that also condensate contri- (FOPT) and the so-calledcontour-improvedper- butions up to dimension-8 could be extracted in turbation theory (CIPT), which performs a par- additiontoα . Therefore,itisofinteresttoanal- tial resummation of running effects of the QCD s yse the behaviour of these moments in models of coupling α in the integration along the complex s higher orders of perturbation theory as well. contour in the s-plane. A physically motivated model for the higher- order behaviour of the Adler function was pre- 0.014 sented in ref. [7], and is given in eq. (14). The 0.012 model was based on the general structure of the 0.01 Borel transform of the Adler function and the 0.008 renormalisation group equation. Furthermore, (0)δ0.006 knowledge on the first four analytically available coefficients c to c , as well as an estimate for 0.004 1,1 4,1 the fifth coefficient c , were incorporated. 0.002 5,1 Resultsforδ(0)inthemainmodel(14)weredis- 0 2 4 6 8 10 12 14 16 played in figure 1, and it is observed that while Perturbative order n FOPT provides a good account of the full Borel Figure 3. Results for (1,2) moments δ(0,12) (full summation, CIPT is never able to come close to (0,12) FO the resummed value.3 This general behaviour circles) and δ (grey circles) at α (M ) = CI s τ hinges on the size of the residue of the first IR 0.3186, employing the model (14), as a function renormalon pole at u = 2. In a model in which of the order n up to which the terms in the per- this residue is set to zero by hand, on the con- turbative series havebeen summed. The straight traryCIPTwelldescribestheBorelsum,whereas linerepresentsthe resultforthe Borelsumofthe FOPT, though approaching the resummed value series. arounditsminimalterm,generallyisratherbadly behaved. Models in which dIR ≈ 0, however, are Anexampleofsuchananalysisisshowninfig- 2 only able to reproduce the known Adler function ure 3 for the moment (1,2). It is observed that, coefficients through large cancellations between like for δ(0), CIPT is unable to provide a rea- different IR renormalon contributions, which ap- sonable account of the full Borel sum. In con- pears unnatural. trast, FOPT comes close to the resummed result In the standardexperimentalextractions ofα forperturbativeordersaroundtheminimalterm, s from hadronic τ decays, also moments of the de- though it is obvious that this particular moment cay spectra are employed [2,3]. An example of displays a very bad behaviour of the asymptotic the behaviour of such a moment was shown in series, with only unsatisfactory convergence up figure 3. The repeatedly unacceptable behaviour to its minimal term. The example of the (1,2)- of CIPT in this case and the also unsatisfactory moment should motivate an exhaustive investi- convergenceofFOPT,suggestthatsuchmoments gation of the moments employed in previous α s shouldbeinvestigatedsystematically,beforetheir analyses from hadronic τ decays, which will be presented in the near future. 3The general behaviour of the Borel model is also supported by an independent approach where the perturba- 4. CONCLUSIONS tive series is conformally transformed into a series which displays better convergence properties than the original Models of higher orders of perturbation the- seriesinpowersofαs [19],though inthiscaseamodified ory allow for the study of different resummation CIPTbetter converges towardsthefullresult. 6 usefulness in α determinations from hadronic τ 3. OPAL Collaboration, K. Ackerstaff et al., s decays is corroborated. Eur. Phys. J. C7 (1999) 571. 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