ALEPH Tau Spectral Functions and QCD

0.32 MB·English

by  Michel Davier

ALEPH Tau Spectral Functions and QCD
7
0 M. Daviera, A. Ho¨ckerb, and Z. Zhanga
0
2 aLaboratoire de l’Acc´el´erateurLin´eaire, Universit´e Paris-Sud 11, 91898 Orsay, France

n
bPhysics Division, CERN, 1211 Geneva 23, Switzerland
a
J

0 Hadronic τ decays provide a clean laboratory for the precise study of quantum chromodynamics (QCD).
2 Observablesbased on thespectral functionsof hadronic τ decayscan berelated toQCDquark-levelcalculations
to determine fundamental quantities like the strong coupling constant, quark and gluon condensates. Using the

1 ALEPH spectral functions and branching ratios, complemented by some other available measurements, and a
v revisitedanalysisofthetheoreticalframework, thevalueαS(m2τ)=0.345±0.004exp±0.009th isobtained. Taken
0 togetherwiththedeterminationofαS(MZ2)fromtheglobalelectroweak fit,thisresultleadstothemostaccurate
7 testofasymptoticfreedom: thevalueofthelogarithmicslopeofα−s1(s)isfoundtoagreewithQCDataprecision
1
1 of 4%. The value of αS(MZ2) obtained from τ decays is αS(MZ2) =0.1215±0.0004exp ±0.0010th ±0.0005evol =
0.1215±0.0012 .
0
7
0
/
h
1. Introduction priate kinematic factor
p
- m2 (τ− V−/A−ν )
p Theτ istheonlyleptonofthethree-generation v (s)/a (s) = τ B → τ
e StandardModel(SM)thatisheavyenoughtode- 1 1 6 Vud 2SEW (τ− e−νeντ)
| | B →
:h cay into hadrons. It is therefore an ideal labora- dNV/A s 2 2s −1
v toryfor studying the chargedweak hadroniccur- 1 1+ , (1)
Xi rents and QCD. Observables based on the spec- ×NV/Ads "(cid:18) − m2τ(cid:19) (cid:18) m2τ(cid:19)#
tral functions of hadronic τ decays can be re-
r m2 (τ− π−(K−)ν )
a lated to QCD quark-level calculations to deter- a (s) = τ B → τ
0 6 V 2S (τ− e−ν ν )
mine fundamental quantities like the strong cou- | ud| EW B → e τ
−2
plingconstant,quarkandgluoncondensates. We dN s
A
1 , (2)
report here the results of a QCD analysis of the ×N ds − m2
A (cid:18) τ(cid:19)
final ALEPH spectral functions [1] using a revis-
where S accounts for electroweak radiative
EW
ited theoretical framework [2].
corrections [3]. Since CVC is a very good ap-
proximation for the nonstrange sector, the J =0
contribution to the nonstrange vector spectral
function is put to zero, while the main contribu-
tions to a are from the pion or kaon poles, with
2. Tau hadronic spectral functions 0
(1/N )dN /ds = δ(s m2 ). They are con-
A A − π,K
2.1. Definitions nected through partial conservation of the axial-
The spectral function v (a , a ), where the vector current (PCAC) to the corresponding de-
1 1 0
subscriptrefersto the spinJ ofthe hadronicsys- cay constants, f . The spectral functions are
π,K
tem, is defined for a nonstrange (∆S = 0) or normalizedbytheratioofthevector/axial-vector
| |
strange(∆S =1)vector(axial-vector)hadronic branching fraction (τ− V−/A−ν ) to the
τ
| | B →
decayτ− V−ν (A−ν ). Thespectralfunction branching fraction of the massless leptonic, i.e.,
τ τ
→
is obtained from the normalized invariant mass- electron, channel. The direct value for and
e
B
squared distribution (1/N )(dN /ds) for a the two derived values from and τ using lep-
V/A V/A µ τ
B
givenhadronic mass √s multiplied by the appro- ton universalityare in goodagreementwith each
1
2 M. Davier, A. Ho¨cker, and Z. Zhang
other, providing a consistent and precise com- of Fig. 1, respectively. The left hand plots give
bined ALEPH result for the electronic branching the ALEPH results [5–7] together with its most
fraction, important exclusive contributions, and the right
handplotscompareALEPHwithOPAL[4]. The
uni =(17.818 0.032)% . (3)
Be ± agreement between the experiments is satisfying.
The curves in the left hand plots of Fig. 1 rep-
Using unitarity and analyticity, the spectral
resent the parton model prediction (dotted) and
functions are connected to the imaginary part of
themasslessperturbativeQCDprediction(solid),
thetwo-pointhadronicvacuumpolarizationfunc-
assuming the relevant physics to be governed by
tions
short distances. The difference between the two
Πµν = gµνq2+qµqν Π(1) (q2) curves is due to higher order terms in the strong
ij,U − ij,U
coupling (α (s)/π)n with n=1,2,3. At highen-
(cid:0) +qµqνΠ(0) (q2(cid:1)) (4) S
ij,U ergies the spectral functions are assumed to be
of vector (Uµ = Vµ = q γµq ) or axial-vector dominated by continuum production, which lo-
ij ij j i
(Uµ = Aµ = q γµγ q ) color-singlet quark cur- callyagreeswithperturbativeQCD.Thisasymp-
renijts, andijfor timj e-li5keimomenta-squaredq2 >0. totic region is not yet reached at s = m2τ for the
vector and axial-vector spectral functions.
Lorentzdecompositionisusedtoseparatethecor-
relation function into its J = 1 and J = 0 parts.
ThepolarizationfunctionsΠµν (s)haveabranch 2.2.2. Inclusive V A spectral functions
ij,U ±
Forthetotalv +a hadronicspectralfunction
cut along the real axis in the complex s = q2 1 1
it is not necessary to experimentally distinguish
plane. Their imaginary parts give the spectral
whetheragiveneventbelongstooneortheother
functionsdefinedin(1),fornonstrangequarkcur-
current. The one, two and three-pion final states
rents
dominate and their exclusive measurements are
1
ImΠ(1) (s) = v /a (s) , addedwithproperaccountingforanticorrelations
ud,V/A 2π 1 1 duetothefeedthrough. Theremainingcontribut-
1
ImΠ(0) (s) = a (s) . (5) ing topologies are treated inclusively, i.e., with-
ud,A 2π 0 out separation of the vector and axial-vector de-
The analytic vacuum polarization function cay modes. This reduces the statistical uncer-
Π(J) (q2) obeys, up to subtractions, the disper- tainty. The effect of the feed-through between τ
ij,U
sion relation final states onthe invariantmass spectrum is de-
scribed by the Monte Carlo simulation and res-
1 ∞ ImΠ(J) (s) olution effects are corrected by data unfolding.
Π(J) (q2)= ds ij,U , (6)
ij,U π s q2 iε In this procedure the simulated mass distribu-
Z0 − − tions are iteratively correctedusing the exclusive
wherethe unknownbutingeneralirrelevantsub- vector/axial-vector unfolded mass spectra. Also,
traction constants can be removed by taking the one does not have to separate the vector/axial-
derivative of Π (q2). The dispersion relation vector currents of the KKπ and KKππ modes.
ij,U
allows one to connect the experimentally accessi- The v1 +a1 spectral functions for ALEPH and
ble spectralfunctionstothecorrelationfunctions OPAL are plotted in the left hand plot of Fig. 2.
Π(J) (q2), which can be derived from QCD. Theimprovementinprecisionwhencomparingto
ij,U a sum of the two parts in Fig. 1 is significant at
2.2. Inclusive nonstrange spectral func- higher mass-squared values.
tions One nicely identifies the oscillatingbehaviorof
2.2.1. Vector and axial-vector spectral the spectral function and it is interesting to ob-
functions serve that, unlike the vector/axial-vector spec-
Theinclusiveτ vectorandaxial-vectorspectral tral functions, it does approximately reach the
functions are shown in the upper and lower plots asymptotic limit predicted by perturbative QCD
ALEPH Tau Spectral Functions and QCD 3
3 3
ALEPH
t – → V–n t – → V–n (ALEPH)
2.5 p –p 0 t 2.5 t – → V–n t (OPAL)
t
p –3p 0, 2p –p +p 0, (6p )–
2 wp –, hp –p 0, (KK–(p ))– 2
v(s)1 1.5 QpaCrtDo np rmedoidcetilon v(s)1 1.5
1 1
0.5 0.5
0 0
0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5
s (GeV2) s (GeV2)
1.4 1.4
QCD prediction ALEPH
1.2 Parton model t – → A–n t 1.2 tt –– →→ AA––nn tt ((AOLPEAPLH))
1 p –2p 0, 2p –p + 1
(5p )–
s) 0.8 (KK–p )– s) 0.8
(1 (1
a 0.6 a 0.6
0.4 0.4
0.2 0.2
0 0
0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5
s (GeV2) s (GeV2)
Figure 1. Left hand plots: the inclusive vector (upper) and axial-vector (lower) spectral functions as
measuredin[5]. The shadedareasindicatethecontributingexclusiveτ decaychannels. Thecurvesshow
the predictions from the parton model (dotted) and from massless perturbative QCD using α (M2) =
S Z
0.120 (solid). Right hand plots: comparison of the inclusive vector (upper) and axial-vector (lower)
spectral functions obtained by ALEPH and OPAL [4].
4 M. Davier, A. Ho¨cker, and Z. Zhang
3 3
t – → (V, A)–n (ALEPH) 2.5 t – → (V, A)–n (ALEPH)
2.5 t – → (V, A)–n tt (OPAL) 2 t – → (V, A)–n tt (OPAL)
QCD prediction QCD pred. = parton model
2 1.5
s) parton model s)
)(1 )(1 1
a 1.5 a
+ 1 – 1 0.5
v v
( 1 ( 0
-0.5
0.5
-1
0
0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5
s (GeV2) s (GeV2)
Figure2. Inclusivevectorplusaxial-vector(left)andvectorminusaxial-vectorspectralfunction(right)as
measuredin[5](dotswitherrorsbars)and[4](shadedonestandarddeviationerrors). Thelinesshowthe
predictions from the parton model (dotted) and from masslessperturbative QCD using α (M2)=0.120
S Z
(solid). They cancel to all orders in the difference.
at s m2. Also, the V + A spectral func- 3. HADRONIC TAU DECAYS AND
→ τ
tion,includingthepionpole,exhibitsthefeatures QCD
expected from global quark-hadron duality: de-
3.1. Generalities
spite the huge oscillations due to the prominent
Proposed tests of QCD at the τ mass scale [8–
π,ρ(770),a andρ(1450)resonances,thespectral
1
11] and the precise measurement of the strong
function qualitatively averages out to the quark
coupling constant α , carried out for the first
contribution from perturbative QCD. S
time by the ALEPH [12]and CLEO [13]collabo-
Inthe caseofthe v a spectralfunction,un-
1 1
− rations have triggered many theoretical develop-
certainties on the V/A separation are reinforced
ments. They concern primarily the perturbative
due to their relative anticorrelation. Similarly,
expansionfor which innovative optimization pro-
the presence of anticorrelations in the branch-
cedures have been suggested. Among these are
ing fractions between τ final states with adja-
contour-improved (resummed) fixed-order per-
cent numbers of pions increase the errors. The
turbationtheory[14,15],effectivechargeandmin-
v a spectralfunctions for ALEPHand OPAL
1 1
− imal sensitivity schemes [16,17], the large-β ex-
are shown in the right hand plot of Fig. 2. The 0
pansion [18,19], and combinations of these ap-
oscillating behavior of the respective v and a
1 1
proaches. They mainly distinguish themselves in
spectral functions is emphasized and the asymp-
totic regime is not reached at s = m2. How- howtheydealwiththefactthattheperturbative
τ series is truncated at an order where the missing
everagain,thestrongoscillationgeneratedbythe
part is not expected to be small.
hadronresonancestoalargepartaveragesoutto
One could wonder how τ decays may at all
zero, as predicted by perturbative QCD.
allow us to learn something about perturbative
QCD. The hadronic decay of the τ is dominated
byresonantsingleparticlefinalstates. Thecorre-
sponding QCD interactions that bind the quarks
and gluons into these hadrons necessarily involve
longdistancescales,whichareoutsidethedomain
ALEPH Tau Spectral Functions and QCD 5
of perturbation theory. Indeed, it is the inclu- scription, i.e., the use of the OPE to organize
sivecharacterofthe sumofallhadronicτ decays theperturbativeandnonperturbativeexpansions,
that allows us to probe fundamental short dis- and the control of unknown higher-order terms
tancephysics. Inclusiveobservableslikethe total in these series. A reasonable stability test is to
hadronic τ decay rate R can be accurately pre- continuously vary m to lower values √s m
τ τ 0 τ
≤
dicted as function of α (m2) using perturbative forboththeoreticalpredictionandmeasurement,
S τ
QCD, and including small nonperturbative con- which is possible since the shapes of the τ spec-
tributions within the framework of the Operator tralfunctions areavailable. The kinematicfactor
ProductExpansion(OPE)[20]. In effect, R is a that takes into account the τ phase space sup-
τ
doubly inclusive observable since it is the result pression at masses near to m is correspondingly
τ
ofa summationoverallhadronicfinalstates at a modified so that √s represents the new mass of
0
given invariant mass and further over all masses the τ.
betweenm andm . The scalem lies inacom-
promise regπion wheτre α (m2) is laτrge enough so 3.2. Theoretical prediction of Rτ
S τ According to Eq. (5) the absorptive parts of
that R is sensitive to its value, yet still small
τ
the vector and axial-vector two-point correla-
enough so that the perturbative expansion con-
verges safely and nonperturbative power correc- tion functions Πu(Jd),V/A(s), with the spin J of
tions are small. the hadronic system, are proportional to the
If strong and electroweak radiative corrections τ hadronic spectral functions with correspond-
areneglected,the theoreticalpartonlevelpredic- ing quantum numbers. The nonstrange ratio
tion for SUC(NC), NC =3 reads Rτ,V+A canthereforebe writtenasanintegralof
these spectral functions over the invariant mass-
R =N V 2+ V 2 =3 , (7)
τ C ud us squared s of the final state hadrons [10]
| | | |
and we ca(cid:0)n estimate a pe(cid:1)rturbative correctionto s0 2
thisvalueofapproximately21%. Onerealizesthe ds s
R (s ) = 12πS 1 (9)
τ,V+A 0 EW
increase in sensitivity to αS compared to the Z Z s0 (cid:18) − s0(cid:19) ×
hadronicwidth, wherebecause ofthe three times 0
smallerαS(MZ2)theperturbativeQCDcorrection 1+2 s ImΠ(1)(s+iǫ) + ImΠ(0)(s+iǫ) ,
reaches only about 4%. (cid:20)(cid:18) s0(cid:19) (cid:21)
ThenonstrangeinclusiveobservableR canbe
τ where Π(J) can be decomposed as Π(J) =
theoretically separated into contributions from
V 2 Π(J) +Π(J) . The lower integration
specific quark currents, namely vector (V) and | ud| ud,V ud,A
axial-vector (A) ud and us quark currents. It is limitis(cid:16)zerobecauseth(cid:17)e pionpoleisatzeromass
therefore appropriate to decompose in the chiral limit.
The correlationfunctionΠ(J) isanalyticinthe
R =R +R +R , (8)
τ τ,V τ,A τ,S complex s plane everywhere except on the posi-
where for the strange hadronic width R vec- tive real axis where singularities exist. Hence by
τ,S
tor and axial-vector contributions are so far not Cauchy’s theorem, the imaginary part of Π(J) is
separated because of the lack of the correspond- proportionalto the discontinuity across the posi-
ing experimental information for the Cabibbo- tive real axis
suppressed modes. Parton-level and perturba- s0
1
tive terms do not distinguish vector and axial- dsw(s)ImΠ(s)= dsw(s)Π(s) , (10)
−2i
vector currents (for massless partons). Thus
Z I
the corresponding predictions become R = 0 |s|=s0
τ,V/A
(N /2)V 2 and R = N V 2, which add where w(s) is an arbitrary analytic function,
C ud τ,S C us
| | | |
up to Eq. (7). and the contour integral runs counter-clockwise
A crucial issue of the QCD analysis at the τ around the circle from s = s +iǫ to s = s iǫ
0 0
−
mass scale is the reliability of the theoretical de- as indicated in Fig. 3.
6 M. Davier, A. Ho¨cker, and Z. Zhang
the vacuum expectation elements (µ) , from
ud
5 Im(s) hO i
the short-distance effects that are included in
the Wilson coefficients C (s,µ). Note that
ud,V/A
δ(D) =(δ(D) +δ(D) )/2.
ud,V+A ud,V ud,A
2.5
3.2.1. The Perturbative Prediction
Theperturbativepredictionusedbytheexperi-
s
0 mentsfollowstheworkof[14]. Effectsfromquark
0
Re(s) masseshavebeencalculatedin[22]andarefound
to be well below 1% for the light quarks. As a
consequence, the contributions from vector and
-2.5 axial-vector currents coincide to any given order
of perturbation theory and the results are flavor
independent.
For the evaluation of the perturbative series,
-5
-5 -2.5 0 2.5 5 it is convenient to introduce the analytic Adler
function
Figure 3. Integration contour for the r.h.s. in dΠ(s)
D(s) s . (13)
Eq. (10). ≡ − ds
The energy scale s =m2 is large enough that
0 τ The function D(s), calculated in perturbative
contributionsfromnonperturbativeeffectsareex-
QCD within the MS renormalizationscheme, de-
pectedtobesubdominantandtheuseoftheOPE
pends on a non-physical parameter µ occurring
is appropriate. The kinematic factor (1 s/s )2
suppresses the contribution from the reg−ion ne0ar as ln(µ2/s). Furthermore it is a function of αS.
On the other hand, since D(s) is connected to a
the positive real axis where Π(J)(s) has a branch
physical quantity, the spectral function ImΠ(s),
cutandtheOPEvalidityisrestrictedduetolarge
it cannot depend on the subjective choice of µ.
possible quark-hadronduality violations.
This can be achievedif α becomes a function of
The theoretical prediction of the vector and S
µ providing independence of D(s) of the choice
axial-vectorratioR canhence be written as
τ,V/A of µ. Nevertheless,in the realistic case of a trun-
3 catedseries,someµdependenceremainsandrep-
R = V 2S 1+δ(0)+
τ,V/A 2| ud| EW resents an irreducible systematic uncertainty.
(cid:18) The s dependence of the QCD coupling con-
δ′ +δ(2,mq) + δ(D) , (11) stant is obtained from the renormalizationgroup
EW ud,V/A ud,V/A
D=4,6,... (cid:19) equation (RGE)
X
with the massless perturbative contribution δ(0), da
the residual non-logarithmic electroweak correc- dlnss = β(as)=−a2s βnans , (14)
tion δ′ = 0.0010 [21], and the dimension D = n
EW X
2 perturbative contribution δ(2,mq) from quark
ud,V/A with a = α /π. Expressed in the MS renormal-
s S
masses. The term δ(D) denotes the OPE con- ization scheme and for three active quark flavors
tributions of mass dimension D attheτ massscale,the β coefficientsareknown
n
(µ) to four loops (n=3) [23].
δ(D) = C (s,µ)hOud iV/A , (12)
ud,V/A ud,V/A ( √s )D The perturbative expansion of the Adler func-
0
dimXO=D − tion can be inferred from the 3-loop calcula-
where the scale parameter µ separates the long- tion of the e+e− inclusive cross section ratio
distance nonperturbative effects, absorbed into Re+e−(s) = σ(e+e− → hadrons(γ))/σ(e+e− →
ALEPH Tau Spectral Functions and QCD 7
µ+µ−(γ)) [24,25] to infinite) order logarithmic power terms of the
expansion that are functions of the known β
∞ n≤3
1
D(s)= 4π2 K˜n(ξ)ans(−ξs) , (15) panerdtuKrbna≤t3ivoenelxy.paInnsieoffneoctf,thbeeyAodnldertfhuencutsieonof(1t5h)e,
n=0
X
twoapproximationshavebeenusedtoobtainthe
wheretheK˜ (ξ)canbeexpressed[14]asfunction
n FOPT series (19): (i) the RGE (14) has been
ofK andβ parameters(bothknownuptoorder
n n Taylor-expandedandtermshigherthanthegiven
n = 3), and ξ, an independent scale parameter.
FOPT order have been truncated, and (ii) this
This leads to the perturbative expansion
Taylorexpansionisusedtopredicta ( s)onthe
s
−
5 entire s =s0 contour.
δ(0) = K˜ (ξ)A(n)(a ) , (16) | |
n s
n=1 3.2.3. Contour-improved fixed-order per-
X
with the functions turbation theory (CIPT)
A more promising approach to the solution of
1 ds
A(n)(a ) = (17) the contour integrals Eq.(17) is to perform a di-
s
2πi s ×
I rectnumericalevaluationbymeansofsingle-step
|s|=s0
integration and using the solution of the RGE
"1−2ss0 +2(cid:18)ss0(cid:19)3−(cid:18)ss0(cid:19)4#ans(−ξs) . atot feoaucrhloinotpesgraastioinnpusttefpor[1t5h,e14r]u.nnIitngimaps(li−ciξtsly)
provides a partial resummation of the (known)
3.2.2. Fixed-order perturbation theory
higher order logarithmic integrals and improves
(FOPT)
the convergenceofthe perturbativeseries. While
Inserting the RGE solution for a (s) into
s for instance the third order term in the expan-
Eq.(16)toevaluatethecontourintegral,andcol-
sion(19)contributeswith17%tothetotal(trun-
lectingthetermswithequalpowersina leadsto
s cated) perturbative prediction, the correspond-
the familiar expression [14]
ing term of the numerical solution amounts to
δ(0) = K˜ (ξ)+g (ξ) αS(ξs0) n , (18) only 6% (assuming αS(m2τ) = 0.35). This nu-
n n merical solution of Eq. (16) is referred to as
π
Xn h i(cid:18) (cid:19) contour-improvedfixed-orderperturbationtheory
wheretheg arefunctionsofK˜ andβ , (CIPT) in the following. Single-step integration
n m<n m<n−1
and of elementary integrals with logarithms of alsoavoidsthe Taylorapproximationofthe RGE
power m < n in the integrand. Setting ξ = 1 on the entire contour, since as is iteratively com-
andreplacingallknownβ andK coefficients by putedfromthepreviousstepusingthefullknown
i i
their numerical values, Eq. (18) simplifies to RGE.
δ(0) = as(s0)+(1.6398+3.5625)a2s(s0) 3.2.4. Other schemes and the value of K4
+(6.371+19.995)a3s(s0) (19) Other approaches for evaluating the perturba-
+(K +78.003)a4(s ) tive prediction have been presented, such as the
4 s 0
effective charge perturbation theory (ECPT, see
+(K +14.250K 391.54)a5(s ) ,
5 4− s 0 forinstanceRefs.[16,26])andthelarge-β expan-
0
where for the purpose of systematic studies we sion [18,19]. Whereas these methods are of con-
havekepttermsuptofifthorder. Whenonlytwo siderable theoretical interest, they are not suited
numbers are given in the parentheses, the first for precision analyses [2].
numbercorrespondstoK ,andthesecondtog . ECPT has been largely used to estimate the
n n
TheFOPTseriesistruncatedatgivenorderde- value of the first uncalculated term K [27].
4
spite the fact that parts of the higher coefficients AnotherapproachtoestimatingK [28]is,within
4
g (ξ) are known to all orders and could be re- CIPT, to enhance the sensitivity to higher order
n>4
summed. These known parts are the higher (up perturbative terms by reducing the renormaliza-
8 M. Davier, A. Ho¨cker, and Z. Zhang
tion scale ξ. Both methods yield K 27, but ence in the result observed when using a Taylor
4
∼
it has been shown [2] that the precision of this expansion and when truncating the perturbative
estimate is seriously limited ( 100 %) by the seriesafterintegratingalongthecontour(FOPT)
∼
lack of knowledge of the unknown higher order with the exact result at given order (CIPT) ex-
parameters in the perturbative series (K ,...). hibits the incompleteness of the perturbative se-
5
Significant efforts are underway with the goal ries. However, it is even worse than that since
to calculate the K coefficient. Although the largeknowncoefficientsareneglectedinFOPTso
4
large number of five-loop diagrams that are thatthedifferencebetweenCIPTandFOPTmay
needed to calculate the two-point current corre- actuallyoverstatetheperturbativetruncationun-
lator at this order may appear discouraging, the certainty(certainlyitisnotagoodmeasureofthe
resultsontwogaugeinvariantsubsetsarealready latteruncertainty). Thiscanbeverifiedbystudy-
available. The subset of order (α4n3) was eval- ing the behavior of this difference for the various
O s f
uated long ago through the summation of renor- orders in perturbation theory given in Table 1.
malon chains [29], while the much harder subset The CIPT-vs.-FOPT discrepancy increases with
(α4n2) was recently calculated [30]. Following theadditionofeachorder,uptoorderfourwhere
O s f
these investigations, the value K = 25 25 is a maximum is reached. Adding the fifth order
4
±
used in our analysis. does not reduce the effect, and only beyond fifth
orderthetwoevaluationsmaybecomeasymptotic
3.2.5. Comparison of the perturbative to each other. As a consequence varying the un-
methods knownhigherordercoefficientsandusingthe dif-
To study the convergence of the perturbative ference between FOPT and CIPT as indicator of
series, we give in Table 1 the contributions of the theoretical uncertainties overemphasizes the
the different orders in PT to δ(0) for the various truncation effect.
approaches using α (m2) = 0.35. A geometric
S τ
growth, K K2 /K , is assumed for all 3.3. Results
unknown PnT∼andnR−G1E cno−e2fficients. In the case It was shown in [11] that one can exploit
ofCIPTtheresultsaregivenforthevarioustech- the shape of the spectral functions to ob-
niques used to evolve αS(s). tain additional constraints on αS(s0) and—more
Faster convergence is observed for CIPT com- importantly—on the nonperturbative effective
paredtoFOPTyieldingasignificantlysmallerer- operators. The τ spectral moments at s0 = m2τ
rorassociatedwiththerenormalizationscaleam- are defined by
biguity. Our coarse extrapolation of the higher
ordercoefficientscouldindicatethatminimalsen- m2τ s k s ℓ dR
sitivity is reached at n 5 for FOPT, while the Rkℓ = ds 1 τ,V/A (20)
∼ τ,V/A − m2 m2 ds
series further converges for CIPT. Although the Z (cid:18) τ(cid:19)(cid:18) τ(cid:19)
0
Taylor expansion in the CIPT integral exhibits
significant deviations from the exact solution on where R00 =R . The factor (1 s/m2)k
τ,V/A τ,V/A − τ
the integration circle, the actual numerical effect suppresses the integrand at the crossing of the
fromthisonδ(0)issmall(cf.secondandthirdcol- positive real axis where the validity of the OPE
umn in Table 1). The convergence of the ECPT less certain and the experimental accuracy is
series is much worse than for FOPT and CIPT. statistically limited. Its counterpart (s/m2)ℓ
τ
Consequently, the difference between truncation projects upon higher energies. The spectral in-
at n=4 and n=6 may be significant. A similar formation is used to fit simultaneously α (m2)
S τ
instability may occur for the large-β expansion. andtheeffectiveoperators a GG ,ρα qq 2 and
0 h s i Sh i
The CIPT series is found to be better behaved for dimension D =4, 6 and 8, respectively.
D
hO i
than FOPT (as well as ECPT) and is therefore Due to the large correlations between the differ-
to be preferred for the numerical analysis of the ently weighted spectral integrals, only five mo-
τ hadronic width. As a matter of fact, the differ- ments are used as input to the fit.
ALEPH Tau Spectral Functions and QCD 9
Table 1
Massless perturbative contribution to R (m2) for the various methods considered, and at orders n 1
τ τ ≥
with α (m2) = 0.35. The value of K is set to 25, while all unknown higher order K and β
S τ 4 n>4 n>3
coefficients are assumed to follow a geometric growth. Details are given in Ref. [2].
δ0
Pert. Method n=1 n=2 n=3 (n=4) (n=5) (n=6) 4 6
n=1 n=1
FOPT (ξ =1) 0.1114 0.0646 0.0365 0.0159 0.0010 0.0086 0P.2283 0P.2208
−
CIPT (Taylor RGE, ξ =1) 0.1573 0.0317 0.0126 0.0042 0.0011 0.0001 0.2058 0.2070
CIPT (full RGE, ξ =1) 0.1524 0.0311 0.0129 0.0046 0.0013 0.0002 0.2009 0.2025
CIPT (full RGE, ξ =0.4) 0.2166 0.0133 0.0006 0.0007 0.0010 0.0007 0.2032 0.2048
− − −
ECPT 0.1442 0.2187 0.1195 0.0344 0.0160 0.0120 0.2090 0.1810
− − − −
Large-β expansion 0.1114 0.0635 0.0398 0.0241 0.0155 0.0093 0.2388 0.2636
0
In analogy to R , the contributions to the uncertainty).
τ
moments originating from perturbative and non- The fit results are given in Table 2. There is a
perturbative QCD are decomposed through the remarkableagreementwithinstatisticalerrorsbe-
OPE. tween the α (m2) determinations using the vec-
S τ
tor and axial-vector data. This provides an im-
3.3.1. The ALEPH determination of
portantconsistencycheckoftheresults,sincethe
α (m2) and nonperturbative con-
S τ two corresponding spectral functions are experi-
tributions
mentally independent and manifest a quite dif-
Combined fits to experimental spectral mo-
ferent resonant behavior. However it must be
mentsandtheextractionofαS(m2τ)togetherwith mentioned that the α (m2) determination using
the leading nonperturbative operators have been S τ
either the V and A spectral functions is more
performed by ALEPH, CLEO and OPAL [4,5,7,
dependent on the validity of the OPE approach
12,13] using similar strategies and inputs. This
since their nonperturbativecontributions aresig-
analysis uses the final and complete data on
nificantltylargerthanforV +A. Indeedthelead-
branching fractions and spectral functions from
ing nonperturbative contributions of dimension
ALEPH [5], yielding
D = 6 and D = 8 approximately cancel in the
R = 3.482 0.014 , (21) inclusive sum. This cancellation of the nonper-
τ,V+A
±
turbative terms increases the confidence in the
R = 1.787 0.011 0.007 , (22)
τ,V ± ± α (m2) determination from the inclusive V +A
R = 1.695 0.011 0.007 , (23) S τ
τ,A ± ± observables. AveragingCIPT and FOPT,the re-
Rτ,V−A = 0.092 0.018 0.014 , (24) sult quoted by ALEPH is
± ±
where the second error originates from the V/A α (m2)=0.340 0.005 0.014 , (25)
separation in final states with a KK pair, fully S τ ± exp± th
anticorrelated beetwen R and R . To re- The gluon condensate is determined by the first
τ,V τ,A
duce the model dependence of the analysis, one k = 1, ℓ = 0,1 moments, which receive lowest
fits simultaneously the nonperturbative opera- order contributions. The values obtained in the
tors, which is possible since the correlations be- V andA fits arenotveryconsistent,whichcould
tween these and α turn out to be small enough. indicate problems in the validity of the OPE ap-
S
The main theoreticaluncertaintiesare due to K proach used once the nonperturbative terms be-
4
(25 25) and to the renormalization scale, which comesignificant. Takingthevalueobtainedinthe
±
isvariedaroundm from1.1to2.5GeV(thevari- V+Afit,wherenonperturbativeeffectsaresmall,
τ
ation over half of the range taken as systematic andaddingassystematicuncertaintieshalfofthe
10 M. Davier, A. Ho¨cker, and Z. Zhang
Table 2
Results[5]forα (m2)andthenonperturbativecontributionsforvector,axial-vectorandV+Acombined
S τ
fits using the corresponding experimental spectral moments as input parameters. Where two errors are
given the first is experimental and the second theoretical. The δ(2) term is theoretical only with quark
masses varying within their allowed ranges (see Ref. [2]). The quark condensates in the δ(4) term are
obtained from PCAC, while the gluon condensate is determined by the fit. The total nonperturbative
contribution is the sum δ = δ(4) +δ(6) +δ(8). Full results are listed only for the CIPT perturbative
NP
prescription, except for α (m2) where the results using both CIPT and FOPT are given (See Ref. [1]
S τ
for the complete results).
Parameter Vector (V) Axial-Vector(A) V + A
αS(m2τ) (CIPT) 0.355±0.008±0.009 0.333±0.009±0.009 0.350±0.005±0.009
αS(m2τ) (FOPT) 0.331±0.006±0.012 0.327±0.007±0.012 0.331±0.004±0.012
δ(2) (CIPT) (−3.3±3.0)×10−4 (−5.1±3.0)×10−4 (−4.4±2.0)×10−4
hasGGi (GeV4) (CIPT) (0.4±0.3)×10−2 (−1.3±0.4)×10−2 (−0.5±0.3)×10−2
δ(4) (CIPT) (4.1±1.2)×10−4 (−5.7±0.1)×10−3 (−2.7±0.1)×10−3
δ(6) (CIPT) (2.85±0.22)×10−2 (−3.23±0.26)×10−2 (−2.1±2.2)×10−3
δ(8) (CIPT) (−9.0±0.5)×10−3 (8.9±0.6)×10−3 (−0.3±4.8)×10−4
Total δ (CIPT) (1.99±0.27)×10−2 (−2.91±0.20)×10−2 (−4.8±1.7)×10−3
NP
χ2/DF (CIPT) 0.52 4.97 3.66
differencebetweenthevectorandaxial-vectorfits magnitude smallerthanthe correspondingvalues
as well as between the CIPT and FOPT results, intheV andAcomponents,δ =(2.0 0.3)
NP,V
± ×
ALEPH measures the gluon condensate to be 10−2 and δ =( 2.8 0.3) 10−2.
NP,A
− ± ×
a GG =(0.001 0.012)GeV4. (26)
h s i ± 3.3.2. Running of α (s) below m2
S τ
This result does not provide evidence for a Using the spectral functions, one can simulate
nonzero gluon condensate, but it is consistent thephysicsofahypotheticalτ leptonwithamass
with and has comparable accuracy to the inde- √s0 smaller than mτ through Eq. (9). Assum-
pendent value obtained using charmonium sum ingquark-hadronduality,theevolutionofR (s )
τ 0
rulesande+e− datainthecharmregion,(0.011 provides a direct test of the running of α (s ),
0.009)GeV4 in a combined determination wit±h governed by the RGE β-function. On theSoth0er
the c quark mass [31]. hand, it is also a test of the stability of the OPE
The approximate cancellation of the nonper- approach at small scales. The studies performed
turbative contributions in the V + A case was in this section employ only CIPT.
predicted [10] for D = 6 assuming vacuum satu- The measured function R (s ) is plotted
τ,V+A 0
rationforthematrixelementsoffour-quarkoper- in the left hand plot of Fig. 4 together with the
ators,whichyieldsδ(6)/δ(6) = 7/11= 0.64,in theoreticalpredictionusingtheresultsofTable2.
V A − −
fair agreement with the result 0.90 0.18. The The correlations between two adjacent points in
− ±
estimate [10] for δ(6) = (2.5 1.3) 10−2 agrees s are large as the only new information is pro-
V ± × 0
with the experimental result. vided by the small mass difference between the
The total nonperturbative V + A correction, two points andthe slightly modified weightfunc-
δ = ( 4.3 1.9) 10−3, is an order of tions under the integral. Moreover the correla-
NP,V+A
− ± ×