Spectral Functions from Hadronic Τ Decays And

7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

Spectral Functions from Hadronic τ Decays and QCD Michel Davier Laboratoire de l’Accélérateur Linéaire IN2P3/CNRS et Université de Paris-Sud 91898 Orsay, France E-mail: [email protected]

Hadronic decays of the τ lepton provide a clean environment to study hadron dynamics in an energy regime dominated by resonances, with the interesting information captured in the spectral functions. Recent results from ALEPH on exclusive channels are presented, with some emphasis on the ππ0 ﬁnal state which plays a crucial role for the determination of the hadronic contribution to the muon anomalous magnetic moment. Inclusive spectral functions are the basis for QCD analyses, delivering an accurate determination of the strong coupling constant, quantitative information on nonperturbative contributions and a measurement of the mass of the strange quark.

1. Introduction inated by resonances. However, perturbative QCD can be seriously considered due to the rel- Hadrons produced in τ decays are born out of atively large τ mass. Many hadronic modes have the charged weak current, i.e. out of the QCD been measured and studied, while some earlier vacuum. This property guarantees that hadronic discrepancies (before 1990) have been resolved physics factorizes in these processes which are with high-statistics and low-systematics experi- then completely characterized for each decay ments. Samples of 4 105 measured decays channel by spectral functions as far as the total are available in each ∼LEP×experiment and CLEO. rate is concerned . Furthermore, the produced Conditions for low systematic uncertainties are hadronic systems have I = 1 and spin-parity P + − P − + particularly well met at LEP: measured samples J = 0 , 1 (V) and J = 0 , 1 (A). The spec- have small non-τ backgrounds ( 1%) and large tral functions are directly related to the invariant selection efficiency (92%), for example∼ in ALEPH. mass spectra of the hadronic final states, normal- Recent results in the field are discussed in this ized to their respective branching ratios and cor- report. rected for the τ decay kinematics. For a given spin-1 vector decay, one has 2. New ALEPH Spectral Functions m2 B(τ − V − ν ) v s τ τ ( ) 2 − → − Preliminary spectral functions based on the full ≡ 6 Vud SEW B(τ e ν¯eντ ) | | → −1 LEP1 statistics are available from ALEPH. The dN s 2 2s corresponding results for the branching fractions V 1 1 + (1) 2 2 are given separately [2]. The analysis uses an ×NV ds " − mτ Mτ # improved treatment of photons as compared to where V = 0.9748 0.0010 denotes the CKM the published analyses based on a reduced sam- ud weak mixing matrix element [1] and SEW ac- ple [3,4]. Spectral functions are unfolded from counts for electroweak radiative corrections (see the measured mass spectra after background sub- below). Isospin symmetry (CVC) connects the traction using a mass-migration matrix obtained τ and e+e− annihilation spectral functions, the from the simulation in order to account for detec- latter being proportional to the R ratio. tor and reconstruction biases. Backgrounds from Hadronic τ decays are a clean probe of hadron non-ττ events are small (< 1% and subtractions dynamics in an interesting energy region dom- are dominated by τ decay feedthroughs. This is

WE03 135 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

o τ→ππ 2 0.0225 2 4 2 10 10 4 MC+τ-bg (all) τ-bg (π) Measured 0.02 3 τ 3 10 -bg (all) 10 τ -bg (e) Unfolded

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Figure 2. Spectral functions in the ππ0 mode from ALEPH before (histogram) and after Figure 1. Feedthrough in the ππ0 channel from (points) the unfolding procedure. other τ decay modes in the ALEPH analysis, as estimated from the Monte Carlo simulation. The light-shaded histogram indicates the total feedthrough, while the dark-shaded histograms in 3. Specific Final States each plot show the dominant individual sources. 3.1. The 2π Vector State 3.1.1. The Data − 0 The spectral function from τ ντ π π in the 0 illustrated in Fig. 1 for the ππ channel, where full-LEP1 ALEPH analysis ( 10→5 events) is pre- the main feedthrough contributions are from the sented in Fig 3. It is dominated∼ by the ρ(770) π mode at large masses (extra photon from radia- resonance, with some broad structure around tion and fake photons from hadronic interactions 1.25 GeV. in the electromagnetic calorimeter) and mostly Results from CLEO [6] are in good agreement 0 0 the a1 π2π mode (one π lost). with the ALEPH data. The statistics is compara- → − 0 The raw and unfolded spectra for the π π ble in the two cases, however whereas the accep- mode are given in Fig. 2. Apart from the normal- tance is very flat across the mass spectrum in the ization effects studied in the branching ratio anal- case of ALEPH, a strong increase is observed for ysis [2], systematic biases affecting the mass spec- the CLEO case. As a consequence, ALEPH data tra, are studied separately. The corresponding are more precise below the ρ peak, while CLEO is uncertainties and correlations in mass are incor- more precise above. Thus the two pieces of data porated in covariance matrices for every consid- are complementary. The comparison is given in ered channel. According to the number of pions Fig. 4. in the final state, the spectral functions are sep- arated into vector and axial-vector components. 3.1.2. ππ Spectral Functions and π Form As for final states involving KK pairs, specific in- Factors put is required to achieve the V A separation [5]. It is useful to carefully write down all the fac- Strange final states are treated−separately. tors involved in the comparison of e+e− and τ

WE03 136 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

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Figure 4. The comparison between ALEPH and CLEO spectral functions in the ππ0 mode. − 2 Figure 3. The pion form factor squared Fπ (s) measured by ALEPH. Because of the unfolding| | s 2s procedure, the errors in nearby bins are strongly C(s) = 1 1 + correlated (only diagonal errors are shown). − m2 m2 τ τ SU(2) symmetry (CVC) implies

spectral functions in order to make explicit the v−(s) = v0(s) (4) possible sources of CVC breaking. On the e+e− It is important to note that experiments on side we have τ decays measure the rate inclusive of radiative 2 0 + − + − 4πα photons, i.e. for τ ντ ππ (γ). The measured σ(e e π π ) = v0(s) (2) → ∗ −→ s spectral function is thus v−(s) = v−(s) G(s), 3 where G(s) is a radiative correction. Also, as β0 (s) 0 2 v0(s) = F (s) 0 2 12π | π | Fπ (s) should only involve the pion hadronic structure,| | it must not contain vacuum polariza- 3 where β0 (s) is the threshold kinematic factor and tion in the photon propagator. 0 Fπ (s) the pion form factor. On the τ side, the Three levels of SU(2) breaking can be identi- physics is contained in the hadronic mass dis- fied: tribution and one can write correspondingly to Eq. (3) electroweak radiative corrections to τ decays • are contained in the SEW factor [7,8] which 1 dΓ − 0 is dominated by short-distance effects. As (τ π π ντ ) = Γ ds −→ such it is expected to be weakly depen- 2 6π Vud SEW Be dent on the specific hadronic final state, C s v− s | |2 ( ) ( ) (3) mτ Bππ0 as verified by detailed computations in the β3 s τ (π, K)ντ channels [9]. Recently, de- −( ) − 2 −→ v−(s) = Fπ (s) tailed calculations have been performed for 12π | |

WE03 137 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

the ππ0 channel [10] which also confirm the considered, since the relative phase of the ρ and relative smallness of the long-distance con- ρ0 amplitude is a priori unknown. In practice we 0 + − tributions. One can write the total correc- fit the Fπ (s) form factor from e e data and the − tion as Fπ (s) form factor from the τ spectral function duly corrected for SU(2) breaking, however only Shad Shad in the spectral function phase space and in the S = EW EM (5) EW lep S factor. In this way, the two form factors can SEM EW be readily compared, i.e. the masses and widths had of the dominating ρ resonance in the two isospin where SEW is the leading-log short-distance electroweak factor (which vanishes for lep- states can be unambiguously determined. On the + − had,lep e e side, all available data have been used (com- tons) and SEM are the nonleading electromagnetic corrections. The latter correc- plete references can be found in Ref. [13], except tions are calculated in Ref. [8] at the quark those of NA7 [15] which are questionable [16]. ρ level and in Ref. [10] at the hadron level for However, in the mass region the new precise the ππ0 decay mode, and in Ref. [7,8] for results from CMD-2 [17] are dominating. On the τ leptons. The total correction amounts [13] side, the accurate data from ALEPH and CLEO inclu are used. to SEW = 1.0198 0.0006 for the inclusive ππ0 The systematic uncertainties are included in hadron decay rate and SEW = (1.0232 0 the fits through appropriate covariance matrices. ππ 0 0.0006) SEM (s) for the ππ decay mode, The ρ mass systematic uncertainty in τ data is ππ0 where SEM (s) is an s-dependent radiative mostly from calibration (0.7 MeV for ALEPH correction [10]. and 0.9 MeV for CLEO). The corresponding uncertainties on the ρ width are 0.8 and 0.7 MeV. pion mass splitting, which is almost com- Both mass and width determinations are system- • pletely from electromagnetic origin, directly atically limited in τ data, however they are uncor- breaks isospin symmetry in the spectral related between ALEPH and CLEO. Due to the functions [11,12] since β−(s) = β (s). 6 0 large event statistics, the fits are quite sensitive to symmetry is also broken in the pion form the precise line shape and to the interference be- • factor ρ π tween the different amplitudes. Of course, ρ ω [11,10]. The width is affected by + − − and ρ mass splittings and by explicit elec- interference is included for e e data only and πγ ηγ l+l− the corresponding amplitude (αρω) fitted. Fits tromagnetic decays such as , , + − and ππγ. Isospin violation in the strong are performed separately for the τ and e e form factors. The upper range of the fit is taken at 3.6 amplitude is expected to be negligible be- 2 + − 2 cause of the small absolute mass difference GeV for e e , but at 2.4 GeV for τ to avoid between u and d quarks. the extreme edge of the spectral function where large corrections are applied. 3.1.3. Fitting the 2π Spectral Function It turns out that the fitted ρ masses and widths The 2π spectral function is dominated by the (essentially the latter) are quite sensitive to the 0 00 ρ resonance. This provides us with an opportu- strength of the ρ and ρ amplitudes, β and γ. So, nity to study the relevant description of a wide depending on the type of fit, the derived values − hadronic resonance. The Gounaris-Sakurai [14] exhibit some systematic shifts. The e+e and 0 (GS) parametrization satisfies analyticity and τ fits yield significantly different ρ amplitudes unitarity through finite width corrections. The and phases: the phase in the τ fit is consistent line shape has to be modified to account for the with 1800, while it comes out 300 smaller in the − effect of ρ ω interference. e+e fit. This is merely a reflection of the dis- The pion−form factor is fitted with interfering crepancy between the measurements of the pion amplitudes from ρ, ρ0 and ρ00 vector mesons with form factor, especially between 0.8 and 1.0 GeV. relative strengths 1, β and γ. A phase φβ is also A fit to both data sets is also performed, keep-

WE03 138 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

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0 2 − 2 Figure 5. The pion form factor squared Fπ (s) Figure 6. The pion form factor squared Fπ (s) − | | from e+e data and the combined ﬁt from Ta- from τ data and the combined ﬁt from T|able 1.| ble 1.

3.2. The 4π Vector State The 4π final states have also been studied [3, 20]. Tests of CVC are severely hampered by large deviations between different e+e− experi- ing common values for the ρ0 and ρ00 parameters ments which disagree beyond their quoted sys- (second-order with respect to the dominant in- tematic uncertainties. A CLEO analysis studies vestigated ρ parameters). Table 1 presents the the resonant structure in the 3ππ0 channel which results of the common fits, the quality of which is shown to be dominated by ωπ and a1π contri- can be inspected in Figs. 5 and 6. butions. The ωπ spectral function is interpreted The differences found between the masses and by a sum of ρ-like amplitudes. The mass of the widths of the charged and neutral ρ’s are some- second state is however found at (1523 10)MeV, what larger than predicted by Chiral Perturba- in contrast with the value (1406 14)MeV from tion Theory (χPT) [18]( m − m 0 < 0.7 MeV) the fit of the 2π spectral function. This point | ρ − ρ | and isospin breaking (Γρ− Γρ0 = (0.7 0.3) has to be clarified. Following a limit of 8.6% ob- MeV). However, if the mass−difference (2.6MeV) tained earlier by ALEPH [21], CLEO sets a new is taken as an experimental fact, then the larger 95% CL limit of 6.4% for the relative contribution − observed width difference (3.1 MeV) could be ex- of second-class currents in the decay τ ντ π ω plained by χPT (0.7 MeV from isospin breaking from the hadronic angular decay distribution.→ + 1.5 MeV from the mass difference). Whether The new ALEPH spectral function in the 3ππ0 or not these discrepancies are of experimental or mode is shown in Fig. 7: it compares well with theoretical nature is still open [13,19] and needs the CLEO published results [20], except maybe to be further investigated. in the threshold region near 1 GeV.

WE03 139 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

τ e+e− combined

m 0 - 773.3 0.6 772.7 0.5 ρ Γ 0 - 145.2 1.3 146.4 0.9 ρ m − 775.0 0.6 - 775.3 0.6 ρ Γρ− 149.5 1.1 - 149.5 0.8 − − α - (2.02 0.10) 10 3 (1.98 0.10) 10 3 ρω β 0.195 0.028 0.123 0.011 0.172 0.006 φ 173.0 7.0 139.4 6.5 178.2 4.5 β m 0 1440 34 1337 35 1415 15 ρ Γ 0 597 102 569 81 528 42 ρ γ 0.095 0.029 0.048 0.008 0.072 0.006 φγ 0. 0. 0. m 00 1713 1713 15 1741 20 ρ Γρ00 235 235 235 Table 1 Results of ﬁts to the pion form factor squared to τ and e+e− data (ALEPH and CLEO) separately, then combined. The parametrization of the ρ line shape (Gounaris-Sakurai) is described in the text. All mass and width values are in MeV and the phase φβ in degrees.

The τ spectral function in the four-pion modes plitude analysis was performed [23]: while the can be used to predict the corresponding isospin- dominant ρπ mode was of course confirmed, it rotated e+e− cross sections using the SU(2) sym- came as a surprize that an important contribu- metric relations tion ( 20%) from scalars (σ, f (1470), f (1270)) ∼ 0 2 2 was found in the 2π system. I=1 4πα − σ + −→ + − + − = 2 vπ− 3π0 , a π π0 e e π π π π · s The 1 2 precisely determined line shape shows→the opening of the K∗K decay mode 2 I=1 4πα in the total a1 width. The derived branching ra- σ + −→ + − 0 0 = [v2π−π+π0 vπ− 3π0 ] ∗ e e π π π π s − tio, B(a1 K K) = (3.3 0.5)% is in good → ¯ Comparisons can be performed after applying agreement with ALEPH results on the KKπ modes which were indeed shown (with the help known isospin-breaking corrections [12,13]. As + − of e e data and CVC) to be axial-vector (a1) shown in Figs. 8 and 9, agreement is observed in ∗ + − dominated with B(a K K) = (2.6 0.3)% the 2π 2π mode, whereas the situation is con- 1 → + − 0 [5]. No conclusive evidence for a higher mass state fused in the π π 2π mode where a large spread 0 exists between the different e+e− results (refer- (a1) was found in this analysis. ences can be found in Ref. [13]). 4. Inclusive Nonstrange Spectral Func- 3.3. Axial-vector States tions The decay τ ντ 3π is the cleanest place to study axial-vector→resonance structure. The spec- The τ nonstrange spectral functions have been + trum is dominated by the 1 a1 state, known to measured by ALEPH [3,4] and OPAL [24]. The decay essentially through ρπ. A comprehensive procedure requires a careful separation of vector analysis of the π−2π0 channel has been presented (V) and axial-vector (A) states involving the re- by CLEO. First, a model-independent determi- construction of multi-π0 decays and the proper nation of the hadronic structure functions gave treatment of final states with a KK¯ pair. The no evidence for non-axial-vector contributions (< V and A spectral functions are given in Fig. 10. 17% at 90% CL) [22]. Second, a partial-wave am- They show a strong resonant behaviour, domi-

WE03 140 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

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e+e− π+ π− 0 Figure 8. The cross section for 2 2 Figure 7. The spectral function for the 3ππ from e+e− experiments and derived→from the mode as determined by ALEPH and CLEO. π3π0 spectral function measured by ALEPH.

5. QCD Analysis of Nonstrange τ Decays 5.1. Motivation The total hadronic τ width, properly normal- nated by the lowest ρ and a1 states, with a ten- ized to the known leptonic width, dancy to converge at large mass toward a value − near the parton model expectation. Yet, the vec- Γ(τ − hadrons ν ) R = → τ (6) tor part stays clearly above while the axial-vector τ Γ(τ − e− ν¯ ν ) one lies below. Thus, the two spectral functions → e τ are clearly not ’asymptotic’ at the τ mass scale. should be well predicted by QCD as it is an inclusive observable. Compared to the similar quan- The new inclusive ALEPH V and A spectral tity deﬁned in e+e− annihilation, it is even twice functions are given in Figs. 11 and 12 with a inclusive: not only are all produced hadronic breakdown of the respective contributions. states at a given mass summed over, but an inte- The V + A spectral function, shown in Fig. 13 gration is performed over all the possible masses has a clear pattern converging toward a value from mπ to mτ . above the parton level as expected in QCD. In This favourable situation could be spoiled by fact, it displays a textbook example of global du- the fact that the Q2 scale is rather small, so that ality, since the resonance-dominated low-mass re- questions about the validity of a perturbative ap- gion shows an oscillatory behaviour around the proach can be raised. At least two levels are to be asymptotic QCD expectation, assumed to be considered: the convergence of the perturbative valid in a local sense only for large masses. This expansion and the control of the nonperturbative feature will be quantitatively discussed in the contributions. Happy circumstances make these next section. latter components indeed very small [25,26].

WE03 141 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

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π3π and 3ππ spectral functions measured by a(s) 1.4 ALEPH. 1.2 OPAL 98

ALEPH 98 1 5.2. Theoretical Prediction for Rτ The imaginary parts of the vector and 0.8 axial-vector two-point correlation functions (J) 0.6 Πu¯d,V/A(s), with the spin J of the hadronic system, are proportional to the τ hadronic spec- 0.4 tral functions with corresponding quantum num- bers. The non-strange ratio Rτ can be written 0.2 as an integral of these spectral functions over 0 the invariant mass-squared s of the final state 0 0.5 1 1.5 2 2.5 3 2 hadrons [27]: s (GeV )

s0 ds s 2 Rτ (s0) = 12πSEW 1 (7) s0 − s0 Figure 10. Inclusive nonstrange vector (top) Z0 and axial-vector (bottom) spectral functions from s 1 + 2 ImΠ(1)(s + i) + ImΠ(0)(s + i) ALEPH and OPAL. The dashed line is the expec- × s 0 tation from the naive parton model. By Cauchy’s theorem the imaginary part of Π(J) is proportional to the discontinuity across the positive real axis. 2 The energy scale s0 = mτ is large enough so that contributions from nonperturbative effects are small. It is therefore assumed that one can

WE03 142 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

1 1 a v 0 3 ALEPH 91-95 Preliminary ALEPH 91-95 π2π ,3π 1.2 Preliminary

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Figure 11. Preliminary ALEPH inclusive vector Figure 12. Preliminary ALEPH inclusive axial- spectral function with its diﬀerent contributions. vector spectral function with its diﬀerent contri- The dashed line is the expectation from the naive butions. The dashed line is the expectation from parton model. the naive parton model.

use the Operator Product Expansion (OPE) to are absorbed into the vacuum expectation ele- organize perturbative and nonperturbative con- ments ud . tributions [28] to Rτ (s0). The hOperturbativi e expansion (FOPT) is known The theoretical prediction of the vector and to third order [29]. A resummation of all known axial-vector ratio Rτ,V/A can thus be written as: higher order logarithmic integrals improves the

3 2 convergence of the perturbative series (contour- Rτ,V/A = Vud SEW (8) 2| | improved method FOPTCI) [30]. As some ambiguity persists, the results are given as an average (0) 0 (2−mass) (D) of the two methods with the diﬀerence taken as 1 + δ + δEW + δud,V/A + δud,V/A ×   a systematic uncertainty. D=4X,6,8 with the residual non-logarithmic electroweak 5.3. Measurements 0 correction δEW = 0.0010 [8], neglected in the fol- The QCD analysis of the τ hadronic width has lowing, and the dimension D = 2 contribution not yet been completed with the ﬁnal ALEPH (2−mass) δud,V/A from quark masses which is lower than spectral functions. Results given below corre- 0.1% for u, d quarks. The term δ(0) is the purely spond to the published analyses with a smaller perturbative contribution, while the δ(D) are the data set. −D/2 The ratio R is obtained from measurements of OPE terms in powers of s of the following τ 0 the leptonic branching ratios: form

(D) ud V/A R = 3.647 0.014 (10) δ hO i (9) τ ud,V/A ∼ ( s )D/2 dimO=D 0 − − X − using a value B(τ e ν¯eντ ) = (17.794 where the long-distance nonperturbative effects 0.045)% which includes→the improvement in ac-

WE03 143 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

1 +a 1 v 3 ALEPH 91-95 Preliminary 2 ALEPH αs(mτ ) δNP

Naive parton model prediction V 0.330 0.014 0.018 0.020 0.004 2.5 A 0.339 0.013 0.018 −0.027 0.004 V+A 0.334 0.007 0.021 −0.003 0.004

2 Table 2 2 Fit results of αs(mτ ) and the OPE nonperturba-

1.5 tive contributions from vector, axial-vector and (V + A) combined ﬁts using the corresponding ratios Rτ and the spectral moments as input pa- 1 rameters. The second error is given for theoretical uncertainty. 0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 ators. Mass2 (GeV2) One notices a remarkable agreement within sta- 2 tistical errors between the αs(mτ ) values using vector and axial-vector data. The total nonper- Figure 13. The preliminary inclusive V + A turbative power contribution to Rτ,V +A is com- nonstrange spectral function from ALEPH. The patible with zero within an uncertainty of 0.4%, dashed line is the expectation from the naive par- that is much smaller than the error arising from ton model. the perturbative term. This cancellation of the nonperturbative terms increases the conﬁdence 2 on the αs(mτ ) determination from the inclusive (V + A) observables. curacy provided by the universality assumption The final result from ALEPH is : of leptonic currents together with the measure- − − − − 2 ments of B(τ e ν¯ ν ), B(τ µ ν¯ ν ) αs(mτ ) = 0.334 0.007exp 0.021th (11) → e τ → µ τ and the τ lifetime. The nonstrange part of Rτ is where the first error accounts for the experimen- obtained by subtracting out the measured strange tal uncertainty and the second gives the uncer- contribution (see last section). tainty of the theoretical prediction of Rτ and the Two complete analyses of the V and A spectral moments as well as the ambiguity of the parts have been performed by ALEPH [4] and theoretical approaches employed. OPAL [24]. Both use the world-average leptonic In the OPAL analysis the corresponding re- branching ratios, but their own measured spec- sults are quoted within three prescriptions for the tral functions. The results on α (m2 ) are there- s τ perturbative expansion, respectively FOPTCI, fore strongly correlated and indeed agree when FOPT and with renormalon chains, i.e. the same theoretical prescriptions are used. α (m2 ) = 0.348 0.009 0.019 (12) s τ exp th 5.4. Results of the Fits = 0.324 0.006exp 0.013th (13) The results of the ﬁts are given in Table 2 for = 0.306 0.005 0.011 (14) the ALEPH analysis. Similar results are obtained exp th by OPAL. It is worth emphasizing that the non- 5.5. Test of the Running of αs(s) at Low perturbative contributions are found to be very Energies small, as expected. The limited number of ob- Using the spectral functions, one can simulate servables and the strong correlations between the the physics of a hypothetical τ lepton with a spectral moments introduce large correlations, es- mass √s0 smaller than mτ through equation (7) pecially between the fitted nonperturbative oper- and hence further investigate QCD phenomena

WE03 144 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

) 4.2 ) 0.6 0 0 (s (s Data (exp. err.) ALEPH s Data (exp. errors only) α

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3.2

0.5 1 1.5 2 2.5 3 2 Figure 15. The running of αs(s0) obtained from s0 (GeV ) the ﬁt of the theoretical prediction to Rτ,V +A(s0). The shaded band shows the data including experimental errors. The curves give the four-loop

Figure 14. The ratio Rτ,V +A versus the square “τ RGE evolution for two and three ﬂavours. mass” s0. The curves are plotted as error bands to emphasize their strong point-to-point correlations in s0. Also shown is the theoretical predic- 2 tion using the results of the ﬁt at s0 = m . τ 5.6. Discussion on the Determination of 2 αs(mτ ) 2 The evolution of the αs(mτ ) measurement from the inclusive (V + A) observables based on the at low energies. Assuming quark-hadron duality, Runge-Kutta integration of the differential equation of the renormalization group to N3LO [31,33] the evolution of Rτ (s0) provides a direct test of yields for the ALEPH analysis the running of αs(s0), governed by the RGE β- function. On the other hand, it is a test of the validity of the OPE approach in τ decays. α (M 2) = 0.1202 0.0008 0.0024 0.0010 (15) s Z exp th evol

The functional dependence of Rτ,V +A(s0) is where the last error stands for possible ambigu- plotted in Fig. 14 together with the theoretical ities in the evolution due to uncertainties in the prediction using the results of Table 2. Below matching scales of the quark thresholds [33]. 1 GeV2 the error of the theoretical prediction of Rτ,V +A(s0) starts to blow up due to the increas- The result (15) can be compared to the pre- ing uncertainty from the unknown fourth-order cise determination from the measurement of the perturbative term. Fig. 15 has the same phys- Z width, as obtained in the global electroweak ical content as Fig. 14, but translated into the ﬁt. The variable RZ has similar advantages to running of αs(s0), i.e., the experimental value for Rτ , but it diﬀers concerning the convergence of αs(s0) has been individually determined at ev- the perturbative expansion because of the much ery s0 from the comparison of data and theory. larger scale. It turns out that this determination Good agreement is observed with the four-loop is dominated by experimental errors with very RGE evolution using three quark flavours. small theoretical uncertainties, i.e. the reverse of The experimental fact that the nonperturba- the situation encountered in τ decays. The most 2 tive contributions cancel over the whole range recent value [34] yields αs(MZ) = 0.1183 0.0027, 1.2 GeV2 s m2 leads to confidence that in excellent agreement with (15). Fig.16 illus- ≤ 0 ≤ τ the αs determination from the inclusive (V + A) trates well the agreement between the evolution 2 2 data is robust. of αs(mτ ) predicted by QCD and αs(MZ).

WE03 145 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

0.4

Mτ τ (ALEPH)

(Energie) 0.35 s α Z (LEP + SLD) 6 ALEPH v+a ± 0.3 Kπ 5 ± Kππ 0.25 ± 4 K3π+K-η (MC) ± 3 K4π (MC) 0.2 ± K5π (MC) 2 0.15 M Z 1

0.1 2 0 10 10 0.5 1 1.5 2 2.5 3 2 2 2 Energie (GeV) Mass (GeV/c )

Figure 16. Evolution of the strong coupling (mea- 2 2 Figure 17. The ALEPH strange spectral function sured at mτ ) to MZ predicted by QCD compared to the direct measurement. The evolution is car- with various contributions indicated. Higher- ried out at 4 loops, while the ﬂavour matching is mass contributions from ﬁnal states with larger multiplicities lead to very small branching frac- accomplished at 3 loops at 2 mc and 2 mb thresholds. tions and are estimated.

6. Strange Spectral Function and Strange Quark Mass where the flavour-independent perturbative part The spectral function for strange final states and gluon condensate cancel. Fig. 18 shows the 00 has been determined by ALEPH [5]. Fig. 17 interesting behaviour of ∆τ expressed differen- shows that it is dominated by the vector K∗(890) tially as a function of s. and higher mass (mostly axial-vector) resonances, kl The leading QCD contribution to ∆τ is a term leading into a poorly defined continuum region, proportional to the square of the strange quark however in agreement with the quark model. mass at the τ energy scale. Special attention must The total rate for strange final states, using be devoted to the perturbative expansion of this the complete ALEPH analyses supplemented by term which has the early behaviour of an asymp- results from other experiments [35] is determined totic series. We quote here the recent result from −3 to be B(τ ντ hadronsS=−1) = (29.3 1.0) 10 , the analysis of Ref. [36], yielding leading to→

2 Rτ,S = 0.163 0.006. (16) m (m ) = (120 11 8 19 ) MeV(18) s τ exp Vus th Spectral moments are again useful tools to un- ravel the different components of the inclusive where the dominant uncertainty is from the- rate. Since we are mostly interested in the spe- ory, mostly because of the poor convergence be- cific contributions from the us strange final state, haviour. Stability checks have been performed it is useful to form the difference and their effect has been conservatively taken into

kl 1 kl 1 kl account in the estimate of the theoretical uncer- ∆ R R − (17) τ ≡ V 2 τ,S=0 − V 2 τ,S= 1 tainty. | ud| | us|

WE03 146 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

the computed eﬀect of isospin symmetry break- π ALEPH ρ ing and requires further experimentation. The 5 massless pQCD determination of the widths of the charged and a 1 the neutral ρ is quite sensitive to the ﬁt condi- /ds

00 0 tions, even though the data cover a very large R

δ K 1 mass range. The measurement of the vector and axial- -5 vector spectral functions has provided the way * K K for quantitative QCD analyses. These spectral 0 1 2 3 functions are very well described in a global way s (GeV2) 3 by O(αs) perturbative QCD with small nonperturbative components. Precise determinations of αs agree for both spectral functions and they also agree with all the other determinations from the 00 Figure 18. Differential rate for ∆τ , differ- Z width, the rate of Z to jets and deep inelastic ence between properly normalized nonstrange lepton scattering. Indeed from τ decays and strange spectral functions (see text for de- 2 αs(M )τ = 0.1202 0.0027 (19) tails). The contribution from massless pertur- Z bative QCD vanishes. To guide the eye, the in excellent agreement with the average from all solid line interpolates between bins of constant other determinations [37] 2 2 0.1 GeV width. α (M ) − = 0.1187 0.0020 (20) s Z non τ The strange spectral function yields a very competitive value for the strange quark mass 7. Conclusions which can be evolved to the usual comparison scale The decays τ ντ + hadrons constitute a → +20 clean and powerful way to study hadronic physics ms(2 GeV) = (116−25) MeV (21) up to √s 1.8 GeV. Beautiful resonance analy- ∼ The use of the vector τ spectral function and ses have already been done, providing new insight the QCD-based approach as tested in τ decays into hadron dynamics. Probably the major sur- improve the calculations of hadronic vacuum po- prise has been the fact that inclusive hadron pro- larization considerably. Significant results have duction is well described by perturbative QCD been obtained for the running of α to the Z with very small nonperturbative components at mass and the muon anomalous magnetic moment. τ the mass. Despite the fact that this low-energy Both of these quantities must be known with high region is dominated by resonance physics, meth- precision as they have the potential to give access ods based on global quark-hadron duality work to new physics. indeed very well. The new ALEPH preliminary results using the Acknowledgments full LEP1 sample have been presented. Sat- isfactory agreement with CLEO is observed in I would like to thank my colleagues Shaomin the ππ0 and 3ππ0 decay modes. The τ spec- Chen, Andreas Höcker, Changzheng Yuan tral functions have now reached a precision level and Zhiqing Zhang for their many contribu- where detailed investigations are possible, partic- tions to this work. Fruitful discussions with ularly in the most interesting ππ0 channel. The V. Cirigliano, G. Ecker, S.I. Eidelman, W. Mar- breaking of SU(2) symmetry can be directly de- ciano, H. Neufeld, A. Pich and A. Weinstein are termined through the comparison between e+e− acknowledged. Congratulations to Abe Seiden and τ spectral functions. The difference between and his staff for organizing an exciting Tau2002 the two spectral functions does not agree with Workshop.

WE03 147 7th International Workshop on Tau Lepton Physics (Tau02), Santa Cruz, CA, 10-13 September, 2002

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