ALEPH Tau Spectral Functions and QCD
a r X i v :h e p -p h /0701170v 1 20 J a n 2007
ALEPH Tau Spectral Functions and QCD
M.Davier a ,A.H¨o cker b ,and Z.Zhang a
a Laboratoire de l’Acc′e l′e rateur Lin′e aire,Universit′e Paris-Sud 11,91898Orsay,France b
Physics Division,CERN,1211Geneva 23,Switzerland
Hadronic τdecays provide a clean laboratory for the precise study of quantum chromodynamics (QCD).Observables based on the spectral functions of hadronic τdecays can be related to QCD quark-level calculations to determine fundamental quantities like the strong coupling constant,quark and gluon https://www.sodocs.net/doc/b515208069.html,ing the ALEPH spectral functions and branching ratios,complemented by some other available measurements,and a revisited analysis of the theoretical framework,the value αS (m 2τ)=0.345±0.004exp ±0.009th is obtained.Taken
together with the determination of αS (M 2
Z )from the global electroweak ?t,this result leads to the most accurate
test of asymptotic freedom:the value of the logarithmic slope of α?1
s (s )is found to agree with QCD at a precision
of 4%.The value of αS (M 2Z )obtained from τdecays is αS (M 2
Z )=0.1215±0.0004exp ±0.0010th ±0.0005evol =0.1215±0.0012.
1.Introduction
The τis the only lepton of the three-generation Standard Model (SM)that is heavy enough to de-cay into hadrons.It is therefore an ideal labora-tory for studying the charged weak hadronic cur-rents and QCD.Observables based on the spec-tral functions of hadronic τdecays can be re-lated to QCD quark-level calculations to deter-mine fundamental quantities like the strong cou-pling constant,quark and gluon condensates.We report here the results of a QCD analysis of the ?nal ALEPH spectral functions [1]using a revis-ited theoretical framework [2].
2.Tau hadronic spectral functions 2.1.De?nitions
The spectral function v 1(a 1,a 0),where the subscript refers to the spin J of the hadronic sys-tem,is de?ned for a nonstrange (|?S |=0)or strange (|?S |=1)vector (axial-vector)hadronic decay τ?→V ?ντ(A ?ντ).The spectral function is obtained from the normalized invariant mass-squared distribution (1/N V/A )(dN V/A /ds )for a given hadronic mass
√6|V ud |2S EW B (τ?→V ?/A ?ντ)
νe ντ)
×dN V/A m 2τ
2 1+
2s
6|V ud |2S EW B (τ?→π?(K ?)ντ)
νe ντ)
×
dN A
m 2τ
?2,(2)
where S EW accounts for electroweak radiative corrections [3].Since CVC is a very good ap-proximation for the nonstrange sector,the J =0contribution to the nonstrange vector spectral function is put to zero,while the main contribu-tions to a 0are from the pion or kaon poles,with (1/N A )dN A /ds =δ(s ?m 2π,K ).They are con-nected through partial conservation of the axial-vector current (PCAC)to the corresponding de-cay constants,f π,K .The spectral functions are normalized by the ratio of the vector/axial-vector branching fraction B (τ?→V ?/A ?ντ)to the branching fraction of the massless leptonic,i.e.,electron,channel.The direct value for B e and the two derived values from B μand ττusing lep-ton universality are in good agreement with each
1
2M.Davier,A.H¨o cker,and Z.Zhang other,providing a consistent and precise com-
bined ALEPH result for the electronic branching
fraction,
B uni e=(17.818±0.032)%.(3)
Using unitarity and analyticity,the spectral
functions are connected to the imaginary part of
the two-point hadronic vacuum polarization func-
tions
Πμν
ij,U
= ?gμνq2+qμqν Π(1)ij,U(q2)
+qμqνΠ(0)ij,U(q2)(4)
of vector(Uμij=Vμij=
q jγμγ5q i)color-singlet quark cur-
rents,and for time-like momenta-squared q2>0.
Lorentz decomposition is used to separate the cor-
relation function into its J=1and J=0parts.
The polarization functionsΠμν
ij,U
(s)have a branch
cut along the real axis in the complex s=q2
plane.Their imaginary parts give the spectral
functions de?ned in(1),for nonstrange quark cur-
rents
ImΠ(1)
ud,V/A (s)=
1
2π
a0(s).(5)
The analytic vacuum polarization function
Π(J) ij,U (q2)obeys,up to subtractions,the disper-
sion relation
Π(J) ij,U (q2)=
1
s?q2?iε,(6)
where the unknown but in general irrelevant sub-traction constants can be removed by taking the derivative ofΠij,U(q2).The dispersion relation allows one to connect the experimentally accessi-ble spectral functions to the correlation functions Π(J)
ij,U
(q2),which can be derived from QCD.
2.2.Inclusive nonstrange spectral func-
tions
2.2.1.Vector and axial-vector spectral
functions
The inclusiveτvector and axial-vector spectral functions are shown in the upper and lower plots of Fig.1,respectively.The left hand plots give the ALEPH results[5–7]together with its most important exclusive contributions,and the right hand plots compare ALEPH with OPAL[4].The agreement between the experiments is satisfying. The curves in the left hand plots of Fig.1rep-resent the parton model prediction(dotted)and the massless perturbative QCD prediction(solid), assuming the relevant physics to be governed by short distances.The di?erence between the two curves is due to higher order terms in the strong coupling(αS(s)/π)n with n=1,2,3.At high en-ergies the spectral functions are assumed to be dominated by continuum production,which lo-cally agrees with perturbative QCD.This asymp-totic region is not yet reached at s=m2τfor the vector and axial-vector spectral functions.
2.2.2.Inclusive V±A spectral functions For the total v1+a1hadronic spectral function it is not necessary to experimentally distinguish whether a given event belongs to one or the other current.The one,two and three-pion?nal states dominate and their exclusive measurements are added with proper accounting for anticorrelations due to the feedthrough.The remaining contribut-ing topologies are treated inclusively,i.e.,with-out separation of the vector and axial-vector de-cay modes.This reduces the statistical uncer-tainty.The e?ect of the feed-through betweenτ?nal states on the invariant mass spectrum is de-scribed by the Monte Carlo simulation and res-olution e?ects are corrected by data unfolding. In this procedure the simulated mass distribu-tions are iteratively corrected using the exclusive vector/axial-vector unfolded mass spectra.Also, one does not have to separate the vector/axial-vector currents of the K Kππmodes. The v1+a1spectral functions for ALEPH and OPAL are plotted in the left hand plot of Fig.2. The improvement in precision when comparing to a sum of the two parts in Fig.1is signi?cant at higher mass-squared values.
One nicely identi?es the oscillating behavior of the spectral function and it is interesting to ob-serve that,unlike the vector/axial-vector spec-tral functions,it does approximately reach the asymptotic limit predicted by perturbative QCD
ALEPH Tau Spectral Functions and QCD 3
0.511.52
2.5300.51 1.52
2.53
3.5
s (GeV 2
)v 1(s )
s (GeV 2
)
v 1(s
)
00.20.40.60.81
1.21.4s (GeV 2
)a 1(s )
s (GeV 2
)
a 1(s )
Figure 1.Left hand plots:
comparison of the inclusive vector (upper)and axial-vector (lower)
spectral functions obtained by ALEPH and OPAL [4].
4
M.Davier,A.H¨o cker,and Z.Zhang
0.511.522.53
s (GeV 2
)(v 1 + a 1)(s )
-1
-0.500.511.522.53
00.51 1.5
2
2.53
3.5
s (GeV 2
)
(v 1 – a 1)(s )
Figure 2.Inclusive vector plus axial-vector (left)and vector minus axial-vector spectral function (right)as measured in [5](dots with errors bars)and [4](shaded one standard deviation errors).The lines show the
predictions from the parton model (dotted)and from massless perturbative QCD using αS (M 2
Z )=0.120(solid).They cancel to all orders in the di?erence.at s →m 2τ.
Also,the V +A spectral func-tion,including the pion pole,exhibits the features expected from global quark-hadron duality:de-spite the huge oscillations due to the prominent π,ρ(770),a 1and ρ(1450)resonances,the spectral function qualitatively averages out to the quark contribution from perturbative QCD.
In the case of the v 1?a 1spectral function,un-certainties on the V/A separation are reinforced due to their relative anticorrelation.Similarly,the presence of anticorrelations in the branch-ing fractions between τ?nal states with adja-cent numbers of pions increase the errors.The v 1?a 1spectral functions for ALEPH and OPAL are shown in the right hand plot of Fig.2.The oscillating behavior of the respective v 1and a 1spectral functions is emphasized and the asymp-totic regime is not reached at s =m 2τ.How-ever again,the strong oscillation generated by the hadron resonances to a large part averages out to zero,as predicted by perturbative QCD.
3.HADRONIC TAU
DECAYS
AND
QCD
3.1.Generalities
Proposed tests of QCD at the τmass scale [8–11]and the precise measurement of the strong coupling constant αS ,carried out for the ?rst time by the ALEPH [12]and CLEO [13]collabo-rations have triggered many theoretical develop-ments.They concern primarily the perturbative expansion for which innovative optimization pro-cedures have been suggested.Among these are contour-improved (resummed)?xed-order per-turbation theory [14,15],e?ective charge and min-imal sensitivity schemes [16,17],the large-β0ex-pansion [18,19],and combinations of these ap-proaches.They mainly distinguish themselves in how they deal with the fact that the perturbative series is truncated at an order where the missing part is not expected to be small.
One could wonder how τdecays may at all allow us to learn something about perturbative QCD.The hadronic decay of the τis dominated by resonant single particle ?nal states.The corre-sponding QCD interactions that bind the quarks and gluons into these hadrons necessarily involve long distance scales,which are outside the domain
ALEPH Tau Spectral Functions and QCD5
of perturbation theory.Indeed,it is the inclu-sive character of the sum of all hadronicτdecays that allows us to probe fundamental short dis-tance physics.Inclusive observables like the total hadronicτdecay rate Rτcan be accurately pre-dicted as function ofαS(m2τ)using perturbative QCD,and including small nonperturbative con-tributions within the framework of the Operator Product Expansion(OPE)[20].In e?ect,Rτis a doubly inclusive observable since it is the result of a summation over all hadronic?nal states at a given invariant mass and further over all masses between mπand mτ.The scale mτlies in a com-promise region whereαS(m2τ)is large enough so that Rτis sensitive to its value,yet still small enough so that the perturbative expansion con-verges safely and nonperturbative power correc-tions are small.
If strong and electroweak radiative corrections are neglected,the theoretical parton level predic-tion for SU C(N C),N C=3reads
Rτ=N C |V ud|2+|V us|2 =3,(7) and we can estimate a perturbative correction to this value of approximately21%.One realizes the increase in sensitivity toαS compared to the Z hadronic width,where because of the three times smallerαS(M2Z)the perturbative QCD correction reaches only about4%.
The nonstrange inclusive observable Rτcan be theoretically separated into contributions from speci?c quark currents,namely vector(V)and axial-vector(A)us quark currents.It is therefore appropriate to decompose
Rτ=Rτ,V+Rτ,A+Rτ,S,(8) where for the strange hadronic width Rτ,S vec-tor and axial-vector contributions are so far not separated because of the lack of the correspond-ing experimental information for the Cabibbo-suppressed modes.Parton-level and perturba-tive terms do not distinguish vector and axial-vector currents(for massless partons).Thus the corresponding predictions become Rτ,V/A= (N C/2)|V ud|2and Rτ,S=N C|V us|2,which add up to Eq.(7).
A crucial issue of the QCD analysis at theτmass scale is the reliability of the theoretical de-
scription,i.e.,the use of the OPE to organize the perturbative and nonperturbative expansions, and the control of unknown higher-order terms in these series.A reasonable stability test is to continuously vary mτto lower values√
s0represents the new mass of theτ.
3.2.Theoretical prediction of Rτ
According to Eq.(5)the absorptive parts of the vector and axial-vector two-point correla-tion functionsΠ(J)
ud,V/A
(s),with the spin J of the hadronic system,are proportional to the τhadronic spectral functions with correspond-ing quantum numbers.The nonstrange ratio Rτ,V+A can therefore be written as an integral of these spectral functions over the invariant mass-squared s of the?nal state hadrons[10]
Rτ,V+A(s0)=12πS EW
s0
ds
s0 2×(9) 1+2s
2i |s|=s0ds w(s)Π(s),(10)
where w(s)is an arbitrary analytic function, and the contour integral runs counter-clockwise around the circle from s=s0+i?to s=s0?i?as indicated in Fig.3.
6M.Davier,A.H¨o cker,and Z.Zhang
Figure3.Integration contour for the r.h.s.in
Eq.(10).
The energy scale s0=m2τis large enough that contributions from nonperturbative e?ects are ex-pected to be subdominant and the use of the OPE is appropriate.The kinematic factor(1?s/s0)2 suppresses the contribution from the region near the positive real axis whereΠ(J)(s)has a branch cut and the OPE validity is restricted due to large possible quark-hadron duality violations.
The theoretical prediction of the vector and axial-vector ratio Rτ,V/A can hence be written as Rτ,V/A=
3
(?√
ds
.(13)
The function D(s),calculated in perturbative QCD within the
d ln s
=β(a s)=?a2s nβn a n s,(14) with a s=αS/π.Expressed in the
ALEPH Tau Spectral Functions and QCD7
μ+μ?(γ))[24,25]
1
D(s)=
2πi |s|=s0ds
+2 s s0 4 a n s(?ξs).
s0
3.2.2.Fixed-order perturbation theory
(FOPT)
Inserting the RGE solution for a s(s)into
Eq.(16)to evaluate the contour integral,and col-
lecting the terms with equal powers in a s leads to
the familiar expression[14]
δ(0)= n ?K n(ξ)+g n(ξ) αS(ξs0)
8M.Davier,A.H¨o cker,and Z.Zhang
tion scaleξ.Both methods yield K4～27,but it has been shown[2]that the precision of this estimate is seriously limited(～100%)by the lack of knowledge of the unknown higher order parameters in the perturbative series(K5,...). Signi?cant e?orts are underway with the goal to calculate the K4coe?cient.Although the large number of?ve-loop diagrams that are needed to calculate the two-point current corre-lator at this order may appear discouraging,the results on two gauge invariant subsets are already available.The subset of order O(α4s n3f)was eval-uated long ago through the summation of renor-malon chains[29],while the much harder subset O(α4s n2f)was recently calculated[30].Following these investigations,the value K4=25±25is used in our analysis.
https://www.sodocs.net/doc/b515208069.html,parison of the perturbative
methods
To study the convergence of the perturbative series,we give in Table1the contributions of the di?erent orders in PT toδ(0)for the various approaches usingαS(m2τ)=0.35.A geometric growth,K n～K2n?1/K n?2,is assumed for all unknown PT and RGE coe?cients.In the case of CIPT the results are given for the various tech-niques used to evolveαS(s).
Faster convergence is observed for CIPT com-pared to FOPT yielding a signi?cantly smaller er-ror associated with the renormalization scale am-biguity.Our coarse extrapolation of the higher order coe?cients could indicate that minimal sen-sitivity is reached at n～5for FOPT,while the series further converges for CIPT.Although the Taylor expansion in the CIPT integral exhibits signi?cant deviations from the exact solution on the integration circle,the actual numerical e?ect from this onδ(0)is small(cf.second and third col-umn in Table1).The convergence of the ECPT series is much worse than for FOPT and CIPT. Consequently,the di?erence between truncation at n=4and n=6may be signi?cant.A similar instability may occur for the large-β0expansion. The CIPT series is found to be better behaved than FOPT(as well as ECPT)and is therefore to be preferred for the numerical analysis of the τhadronic width.As a matter of fact,the di?er-ence in the result observed when using a Taylor expansion and when truncating the perturbative series after integrating along the contour(FOPT) with the exact result at given order(CIPT)ex-hibits the incompleteness of the perturbative se-ries.However,it is even worse than that since large known coe?cients are neglected in FOPT so that the di?erence between CIPT and FOPT may actually overstate the perturbative truncation un-certainty(certainly it is not a good measure of the latter uncertainty).This can be veri?ed by study-ing the behavior of this di?erence for the various orders in perturbation theory given in Table1. The CIPT-vs.-FOPT discrepancy increases with the addition of each order,up to order four where a maximum is reached.Adding the?fth order does not reduce the e?ect,and only beyond?fth order the two evaluations may become asymptotic to each other.As a consequence varying the un-known higher order coe?cients and using the dif-ference between FOPT and CIPT as indicator of the theoretical uncertainties overemphasizes the truncation e?ect.
3.3.Results
It was shown in[11]that one can exploit the shape of the spectral functions to ob-tain additional constraints onαS(s0)and—more importantly—on the nonperturbative e?ective operators.Theτspectral moments at s0=m2τare de?ned by
R k?τ,V/A=
m2τ
ds 1?s m2τ ?dRτ,V/A
qq 2and O D for dimension D=4,6and8,respectively. Due to the large correlations between the di?er-ently weighted spectral integrals,only?ve mo-ments are used as input to the?t.
ALEPH Tau Spectral Functions and QCD9 Table1
Massless perturbative contribution to Rτ(m2τ)for the various methods considered,and at orders n≥1 withαS(m2τ)=0.35.The value of K4is set to25,while all unknown higher order K n>4andβn>3 coe?cients are assumed to follow a geometric growth.Details are given in Ref.[2].
FOPT(ξ=1)0.11140.06460.03650.01590.0010?0.00860.22830.2208 CIPT(Taylor RGE,ξ=1)0.15730.03170.01260.00420.00110.00010.20580.2070 CIPT(full RGE,ξ=1)0.15240.03110.01290.00460.00130.00020.20090.2025 CIPT(full RGE,ξ=0.4)0.2166?0.01330.0006?0.00070.0010?0.00070.20320.2048 ECPT0.14420.2187?0.1195?0.0344?0.0160?0.01200.20900.1810 Large-β0expansion0.11140.06350.03980.02410.01550.00930.23880.2636
K pair,fully anticorrelated beetwen Rτ,V and Rτ,A.To re-duce the model dependence of the analysis,one ?ts simultaneously the nonperturbative opera-tors,which is possible since the correlations be-tween these andαS turn out to be small enough. The main theoretical uncertainties are due to K4 (25±25)and to the renormalization scale,which is varied around mτfrom1.1to2.5GeV(the vari-ation over half of the range taken as systematic uncertainty).
The?t results are given in Table2.There is a remarkable agreement within statistical errors be-tween theαS(m2τ)determinations using the vec-tor and axial-vector data.This provides an im-portant consistency check of the results,since the two corresponding spectral functions are experi-mentally independent and manifest a quite dif-ferent resonant behavior.However it must be mentioned that theαS(m2τ)determination using either the V and A spectral functions is more dependent on the validity of the OPE approach since their nonperturbative contributions are sig-ni?cantlty larger than for V+A.Indeed the lead-ing nonperturbative contributions of dimension D=6and D=8approximately cancel in the inclusive sum.This cancellation of the nonper-turbative terms increases the con?dence in the αS(m2τ)determination from the inclusive V+A observables.Averaging CIPT and FOPT,the re-sult quoted by ALEPH is
αS(m2τ)=0.340±0.005exp±0.014th,(25) The gluon condensate is determined by the?rst k=1,?=0,1moments,which receive lowest order contributions.The values obtained in the V and A?ts are not very consistent,which could indicate problems in the validity of the OPE ap-proach used once the nonperturbative terms be-come signi?cant.Taking the value obtained in the V+A?t,where nonperturbative e?ects are small, and adding as systematic uncertainties half of the
10M.Davier,A.H¨o cker,and Z.Zhang Table2
Results[5]forαS(m2τ)and the nonperturbative contributions for vector,axial-vector and V+A combined ?ts using the corresponding experimental spectral moments as input parameters.Where two errors are given the?rst is experimental and the second theoretical.Theδ(2)term is theoretical only with quark masses varying within their allowed ranges(see Ref.[2]).The quark condensates in theδ(4)term are obtained from PCAC,while the gluon condensate is determined by the?t.The total nonperturbative contribution is the sumδNP=δ(4)+δ(6)+δ(8).Full results are listed only for the CIPT perturbative prescription,except forαS(m2τ)where the results using both CIPT and FOPT are given(See Ref.[1] for the complete results).
αS(m2τ)(CIPT)0.355±0.008±0.0090.333±0.009±0.0090.350±0.005±0.009αS(m2τ)(FOPT)0.331±0.006±0.0120.327±0.007±0.0120.331±0.004±0.012 a s GG (GeV4)(CIPT)(0.4±0.3)×10?2(?1.3±0.4)×10?2(?0.5±0.3)×10?2δ(6)(CIPT)(2.85±0.22)×10?2(?3.23±0.26)×10?2(?2.1±2.2)×10?3 TotalδNP(CIPT)(1.99±0.27)×10?2(?2.91±0.20)×10?2(?4.8±1.7)×10?3
di?erence between the vector and axial-vector?ts
as well as between the CIPT and FOPT results,
ALEPH measures the gluon condensate to be
a s GG =(0.001±0.012)GeV4.(26)
This result does not provide evidence for a
nonzero gluon condensate,but it is consistent
with and has comparable accuracy to the inde-
pendent value obtained using charmonium sum
rules and e+e?data in the charm region,(0.011±
0.009)GeV4in a combined determination with
the c quark mass[31].
The approximate cancellation of the nonper-
turbative contributions in the V+A case was
predicted[10]for D=6assuming vacuum satu-
ration for the matrix elements of four-quark oper-
ators,which yieldsδ(6)
V /δ(6)
A
=?7/11=?0.64,in
fair agreement with the result?0.90±0.18.The
estimate[10]forδ(6)
V =(2.5±1.3)×10?2agrees
with the experimental result.
The total nonperturbative V+A correction,δNP,V+A=(?4.3±1.9)×10?3,is an order of magnitude smaller than the corresponding values
in the V and A components,δNP,V=(2.0±0.3)×10?2andδNP,A=(?2.8±0.3)×10?2.
3.3.2.Running ofαS(s)below m2
τ
Using the spectral functions,one can simulate
the physics of a hypotheticalτlepton with a mass √
ALEPH Tau Spectral Functions and QCD
11
3.3
3.4
3.53.63.7
3.83.9
s 0 (GeV 2
)
R τ,V +A (s 0)
0.3
0.350.40.450.5
0.550.6
s 0 (GeV 2
)
αs (s 0)
Figure 4.Left:
The running of
αS (s 0)obtained from the ?t of the theoretical prediction to R τ,V +A (s 0)using CIPT.The shaded band shows the data including only experimental errors.The curve gives the expected four-loop RGE evolution for three ?avors.
tions are reinforced by the original experimental and theoretical correlations.Below 1GeV 2the error of the theoretical prediction of R τ,V +A (s 0)starts to blow up because of the increasing sensi-tivity to the unknown K 4perturbative term;er-rors of the nonperturbative contributions are not contained in the theoretical error band.Figure 4(right)shows the plot corresponding to Fig.4(left),translated into the running of αS (s 0).Only experimental errors are shown.Also plotted is the four-loop RGE evolution using three quark ?avors.
It is remarkable that the theoretical predic-tion using the parameters determined at the τmass and R τ,V +A (s 0)extracted from the mea-sured V +A spectral function agree down to s 0～0.8GeV 2.The agreement is good to about 2%at 1GeV 2.This result,even more directly il-lustrated by the right hand plot of Fig.4,demon-strates the validity of the perturbative approach down to masses around 1GeV,well below the τmass scale.The agreement with the expected scale evolution between 1and 1.8GeV is an inter-esting result,considering the relatively low mass range,where αS is seen to decrease by a factor of 1.6and reaches rather large values ～0.55at the lowest masses.This behavior provides con?-dence that the αS (m 2τ)measurement is on solid phenomenological ground.
3.3.3.Final assessment on the αS (m 2τ)de-termination
Although this evaluation of αS (m 2τ)represents the state-of-the art,several remarks can be made:
?The analysis is based on the ALEPH spec-tral functions and branching fractions,en-suring a good consistency between all the observables,but not exploiting the full ex-perimental information currently available from other experiments.Because the re-sult on αS (m 2τ)is limited by theoretical un-certainties,one should expect only a small improvement of the ?nal error in this way,however it can in?uence the central value.?One example for this is the evaluation of the strange component.Some discrepancy
12M.Davier,A.H¨o cker,and Z.Zhang
is observed between the ALEPH measure-
ment of the(Kππ)?νmode and the CLEO
and OPAL results.Although this could
still be the result of a statistical?uctuation,
their average provides a signi?cant shift in
the central value compared to using the
ALEPH number alone.Another improve-
ment is the substitution of the measured
branching fraction for the K?νmode by the
more precise value predicted fromτ–μuni-
versality.Both operations have the e?ect
to increase Rτ,S,the ratio of theτdecay
width into strange hadronic?nal states to
the electronic width,from0.1603±0.0064,
as obtained by ALEPH,to0.1686±0.0047
for the world average.
?One can likewise substitute the world av-
erage value for the universality-improved
value of the electronic branching fraction,
B uni e=(17.818±0.032)%,to the ALEPH
result,B e=(17.810±0.039)%,with little
change in the central value,but some im-
provement in the precision.
From this analysis,one?nds the new value for the nonstrange ratio,
Rτ,V+A=Rτ?Rτ,S
=(3.640±0.010)?(0.1686±0.0047)
=3.471±0.011.(27) The result(27)translates into the following de-termination ofαS(m2τ)from the inclusive V+A component using the CIPT approach
αS(m2τ)=0.345±0.004exp±0.009th,(28) with improved experimental and theoretical pre-cision over the ALEPH result.Most of the the-oretical uncertainty originates from the limited knowledge of the perturbative expansion,only predicted to third order.Following the dicussion above we take the result from the CIPT expan-sion,not introducing any additional uncertainty spanning the di?erence between FOPT and CIPT results.The dominant theoretical errors are from the uncertainty on K4and from the renormaliza-tion scale dependence,both covering the e?ect of truncating the series after the estimated fourth order.3.3.4.Evolution to M2
Z
It is customary to compareαS values,obtained at di?erent renormalization scales,at the scale of the Z-boson mass.
The evolution of theαS(m2τ)measurement from the inclusive V+A observables given in Eq.(28),based on Runge-Kutta integration of the RGE(14)to N3LO,and three-loop quark-?avor matching,gives
αS(M2Z)=0.1215(4exp)(10)th(5)evol,
=0.1215±0.0012.(29) The?rst two errors originate from theαS(m2τ)de-termination given in Eq.(28).The last error re-ceives contributions from the uncertainties in the c-quark mass(0.00020,m c varied by±0.1GeV) and the b-quark mass(0.00005,m b varied by ±0.1GeV),the matching scale(0.00023,μvar-ied between0.7m q and3.0m q),the three-loop truncation in the matching expansion(0.00026) and the four-loop truncation in the RGE equation (0.00031),where we used for the last two errors the size of the highest known perturbative term as systematic uncertainty.These errors have been added in quadrature.The result(29)is a deter-mination of the strong coupling at the Z mass scale with a precision of1%.
The evolution path ofαS(m2τ)is shown in the upper plot of Fig.5.The two discontinuities are due to the quark-?avor matching atμ=2m q. One could prefer to have an(almost)smooth matching by choosingμ=m q.However,in this case,one must?rst evolve from mτdown to
m b.The e?ect onαS(M2Z)from this ambiguity is within the assigned systematic uncertainty for the evolution.
The comparison with the other determinations ofαS(M2Z)is given in Fig.5using compiled re-sults from Ref.[32].
3.3.5.A measure of asymptotic freedom
between m2
τ
and M2
Z
Theτ-decay and Z-width determinations have comparable accuracies,which are however very di?erent in nature.Theτvalue is dominated by theoretical uncertainties,whereas the determina-tion at the Z resonance,bene?ting from the much
ALEPH Tau Spectral Functions and QCD
13
αs (μ)
1
10
10
μ scale (GeV)
αs (M Z )
Figure 5.Top
:The corresponding
extrapolated αS values at M Z .The shaded band displays the τdecay result within https://www.sodocs.net/doc/b515208069.html,rger energy scale and the correspondingly small uncertainties from the truncated perturbative ex-pansion,is limited by the experimental precision on the electroweak observables,essentially the ra-tio of leptonic to hadronic peak cross sections.The consistency between the two results provides the most powerful present test of the evolution of the strong interaction coupling,as it is predicted by the nonabelian nature of the QCD gauge the-ory.This test extends over a range of s spanning more than three orders of magnitude.The di?er-ence between the extrapolated τ-decay value and the measurement at the Z is:
ατS (M 2Z )?αZ S
(M 2
Z )=0.0029(10)τ(27)Z (30)
which agrees with zero with a relative precision of 2.4%.
In fact,the comparison of these two values is valuable since they are among the most precise single measurements and they are widely spaced in energy scale.Thus it allows one to perform an accurate test of asymptotic freedom.Let us consider the following evolution estimator [33]for the inverse of αS (s ),r (s 1,s 2)=2·
α?1S (s 1)?α?1S
(s 2)d ln
√
α2S
,
=
2β0
β0
αS
14M.Davier,A.H¨o cker,and Z.Zhang
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