Hadronic Τ Decays and QCD

Heavy Flavours 8, Southampton, UK, 1999

PROCEEDINGS

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]

Abstract: 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 on exclusive channels are reviewed. Inclusive spectral functions are the basis for QCD analyses, delivering an accurate determination of the strong coupling constant and quantitative information on nonperturbative contributions. Strange decays yield a determination of the strange quark mass.

1. Introduction Radiative corrections violate CVC, as contained in the SEW factor which is dominated by short- Hadrons produced in τ decays are born out of distance effects and thus expected to be essen- i.e. the charged weak current, out of the QCD tially final-state independent. vacuum. This property guarantees that hadronic Hadronic τ decays are then a clean probe of physics factorizes in these processes which are hadron dynamics in an interesting energy region then completely characterized for each decay chan- dominated by resonances. However, perturbative nel by spectral functions as far as the total rate is QCD can be seriously considered due to the rel- concerned . Furthermore, the produced hadronic atively large τ mass. Many hadronic modes have P + − systems have I = 1 and spin-parity J =0 ,1 been measured and studied, while some earlier P − + (V) and J =0 ,1 (A). The spectral functions discrepancies (before 1990) have been resolved are directly related to the invariant mass spectra with high-statistics and low-systematics exper- of the hadronic final states, normalized to their iments. Samples of ∼ 4 × 105 measured decays respective branching ratios and corrected for the are available in each LEP experiment and CLEO. τ decay kinematics. For a given spin-1 vector Conditions for low systematic uncertainties are decay, one has particularly well met at LEP: measured samples 2 − → − have small non-τ backgrounds (< 1%) and large ≡ mτ B(τ V ντ ) v1(s) 2 − − 6 |Vud| SEW B(τ → e ν¯eντ ) selection efficiency (92%), for example in ALEPH. " # 2 −1 Recent results in the field are discussed in × dNV − s 2s this report. 1 2 1+ 2 (1.1) NV ds mτ Mτ where Vud denotes the CKM weak mixing matrix 2. Specific final states element and SEW =1.0194 0.0040 accounts for electroweak radiative corrections [1]. 2.1 Vector states − 0 Isospin symmetry (CVC) connects the τ and The decay τ → ντ π π is now studied with large + − e e annihilation spectral functions, the latter statistics of ∼ 105 events. Data from ALEPH being proportional to the R ratio. For example, have been published [2]. New results from CLEO 2 are now available [3] with the mass spectrum I=1 4πα σ + − 0 (s)= v1,X−(s) (1.2) e e →X s given in figure 1 dominated by the ρ(770) res- Heavy Flavours 8, Southampton, UK, 1999 Michel Davier onance. Good agreement is observed between (24.650.35)% [9]. This 1.8σ discrepancy should the ALEPH and CLEO data and the lineshape be further investigated with a detailed examina- fits show strong evidence for the contribution of tion of the respective possible systematic effects, ρ(1400) through interference with the dominant such as radiative corrections in e+e− data and amplitude. Fits also include a ρ(1700) contribu- π0 reconstruction in τ data. CVC violations are tion, taken from e+e− data as the value of the τ of course expected at some point: hadronic vio- 2 2 mass does not allow τ data alone to tie down the lation should be very small (∼ (mu − md) /mτ ), corresponding resonance parameters. Thanks to while significant effects could arise from long- the high precision of the data, fits are sensitive to distance radiative processes. Estimates show that the exact form of the ρ lineshape, with a prefer- the difference between the charged and neutral ence given to the Gounaris-Sakurai parametriza- ρ widths should only be at the level of (0.3 tion [4] over that of Kühn-Santamaria [5]. 0.4)% [10].

1400 e+e- → π+ π– 105 – 0 τ → π π ντ

1200 ALEPH

Cross Section (nb) CMD-2 NA7 104 1000 OLYA TOF CMD DM1 800 DM2 103

600 Events per 0.025 GeV 102 400

0.3 0.5 0.7 0.9 1.1 1.3 1.5 200 ππ0 Invariant Mass [GeV]

− 0 0 Figure 1: Mass distribution in CLEO τ → ντ π π 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s (GeV2) events. The solid curve overlaid is the result of the Kühn-Santamaria fit, while the dashed curve has the Figure 2: Cross section for e+e− → π+π− com- ρ(1400) contribution turned off. pared to the ALEPH τ data using CVC with ρ − ω interference built in (shaded band). The quality of data on e+e− → π+π− has also recently improved with the release of the The 4π final states have also been studied [2, CMD-2 results from Novosibirsk [6]. A compar- 11]. Tests of CVC are severely hampered by ison of the mass spectrum as measured in e+e− large deviations between different e+e− experi- and τ data is given in figure 2 (for this exercise ments which disagree well beyond their quoted the ρ − ω interference has to be artificially intro- systematic uncertainties. A new CLEO analy- duced in the τ data). Although the agreement sis studies the resonant structure in the 3ππ0 looks impressive, it is possible to quantify it by channel which is shown to be dominated by ωπ computing a single number, integrating over the and a1π contributions. The ωπ spectral func- complete spectrum. It is convenient for this to tion shown in figure 3 is in good agreement with − 0 use the branching ratio B(τ → ντ π π ) as di- CMD-2 results and it is interpreted by a sum of rectly measured in τ decays and computed from ρ-like amplitudes. The mass of the second state the e+e− spectral function under the assump- is however found at (1523 10)MeV, in contrast − 0 tion of CVC. Using B(τ → ντ h π )=(25.79 with the value (140614)MeV from the fit of the − 0 0.15)% [7] and subtracting out B(τ → ντ K π )= 2π spectral function. This point has to be clar- − 0 (0.45 0.02)% [8], one gets B(τ → ντ π π )= ified. Following a limit of 8.6% obtained earlier (25.34 0.15)%, somewhat larger than the CVC by ALEPH [12], CLEO sets a new 95% CL limit + − CV C value using all available e e data, B2π = of 6.4% for the relative contribution of second-

2 Heavy Flavours 8, Southampton, UK, 1999 Michel Davier

− 3100199-009 class currents in the decay τ → ντ π ω from the ( a ) 3

hadronic angular decay distribution. 10 I 1

2 10

(0.025 GeV) 3 10 3 1 10 N / m

1 1.31.4 1.5

0.6 1.0 1.4 1.8 m (GeV) 3

( b ) 3

tot

2 neutral 3 (s) (GeV) charge Figure 3: Fits of the CLEO spectral function for 1 3 − τ → ντ π ω. K*K

0 0.8 1.6 2.4 3.2 2 2.2 Axial-vector states s (GeV ) → The decay τ ντ 3π is the cleanest place to Figure 4: (a) 3π mass spectrum from CLEO in the study axial-vector resonance structure. The spec- π−2π0 channel with the lineshape from the resonance + ∗ √ trum is dominated by the 1 a1 state, known to fit (zoom on the K K threshold). In (b), the s- decay essentially through ρπ. A comprehensive dependent a1 width is plotted with the different con- analysis of the π−2π0 channel has been presented tributions considered. by CLEO. First, a model-independent determination of the hadronic structure functions gave procedure requires a careful separation of vector no evidence for non-axial-vector contributions (< (V) and axial-vector (A) states involving the re- 17% at 90% CL) [13]. Second, a partial-wave am- construction of multi-π0 decays and the proper plitude analysis was performed [14]: while the treatment of final states with a KK¯ pair. The dominant ρπ mode was of course confirmed, it V and A spectral functions are given in figures 5 came as a surprize that an important contribu- and 6, respectively. They show a strong resonant ∼ tion ( 20%) from scalars (σ, f0(1470), f2(1270)) behaviour, dominated by the lowest ρ and a1 was found in the 2π system. states, with a tendancy to converge at large mass − 0 The a1 → π 2π lineshape is displayed in toward a value near the parton model expecta- ∗ figure 4 where the opening of the K K decay tion. Yet, the vector part stays clearly above mode in the total a1 width is clearly seen. The while the axial-vector one lies below. Thus, the ∗ derived branching ratio, B(a1 → K K)=(3.3 two spectral functions are clearly not ’asymp- 0.5)% is in good agreement with ALEPH results totic’ at the τ mass scale. on the KKπ¯ modes which were indeed shown The V + A spectral function, shown in fig- + − (with the help of e e data and CVC) to be ure 7 has a clear converging pattern toward a ∗ axial-vector (a1) dominated with B(a1 → K K)= value above the parton level as expected in QCD. (2.6 0.3)% [8]. No conclusive evidence for a In fact, it displays a textbook example of global 0 higher mass state (a1) is found in this analysis. duality, since the resonance-dominated low-mass region shows an oscillatory behaviour around the 3. Inclusive spectral functions asymptotic QCD expectation, assumed to be valid in a local sense only for large masses. This fea- The τ nonstrange spectral functions have been ture will be quantitatively discussed in the next measured by ALEPH [2, 15] and OPAL [16]. The section.

3 Heavy Flavours 8, Southampton, UK, 1999 Michel Davier

v(s) υ + a 1 1 ALEPH 2.5 OPAL 98 2.5 τ– → ν ALEPH 98 (V,A, I=1) τ 2 parton model prediction 2 perturbative QCD (massless) 1.5 1.5

1 0.5

0.5 0 0 0.5 1 1.5 2 2.5 3 s (GeV2) 0 0 0.5 1 1.5 2 2.5 3 3.5 Figure 5: Inclusive nonstrange vector spectral func- Mass2 (GeV/c2)2 tion from ALEPH and OPAL. The dashed line is the expectation from the naive parton model Figure 7: Inclusive V + A nonstrange spectral function from ALEPH. The dashed line is the expectation from the naive parton model, while the solid one is from massless perturbative QCD using

a(s) 1.4 2 αs(MZ)=0.120. 1.2 OPAL 98

ALEPH 98 1 is performed over all the possible masses from

0.8 mπ to mτ . This favourable situation could be spoiled by 0.6 the fact that the Q2 scale is rather small, so that 0.4 questions about the validity of a perturbative ap- 0.2 proach can be raised. At least two levels are to be

0 considered: the convergence of the perturbative 0 0.5 1 1.5 2 2.5 3 s (GeV2) expansion and the control of the nonperturbative contributions. Happy circumstances make these Figure 6: Inclusive nonstrange axial-vector spectral contributions indeed very small [17, 18]. function from ALEPH and OPAL. The dashed line is the expectation from the naive parton model. 4.2 Theoretical prediction for Rτ The imaginary parts of the vector and axial-vector (J) two-point correlation functions Πud,V/A¯ (s), with 4. QCD analysis of nonstrange τ de- the spin J of the hadronic system, are proportional to the τ hadronic spectral functions with cays corresponding quantum numbers. The non-strange ratio Rτ can be written as an integral of these 4.1 Motivation spectral functions over the invariant mass-squared The total hadronic τ width, properly normalized s of the final state hadrons [19]: to the known leptonic width, s Z 0 2 ds s − → − Rτ (s0)=12πSEW 1 − (4.2) Γ(τ hadrons ντ ) s0 s0 Rτ = − − (4.1) 0 Γ(τ → e ν¯eντ ) s (1) (0) should be well predicted by QCD as it is an inclu- × 1+2 ImΠ (s + i) + ImΠ (s + i) s0 sive observable. Compared to the similar quan- tity deﬁned in e+e− annihilation, it is even twice By Cauchy’s theorem the imaginary part of Π(J) inclusive: not only are all produced hadronic states is proportional to the discontinuity across the at a given mass summed over, but an integration positive real axis.

4 Heavy Flavours 8, Southampton, UK, 1999 Michel Davier

2 The energy scale s0 for s0 = mτ is large 2 enough that contributions from nonperturbative ALEPH αs(mτ ) δNP effects be small. It is therefore assumed that one V 0.330 0.014 0.018 0.020 0.004 . . . − . . can use the Operator Product Expansion (OPE) A 0 339 0 013 0 018 0 027 0 0004 . . . − . . to organize perturbative and nonperturbative con- V+A 0 334 0 007 0 021 0 003 0 0004 tributions [20] to Rτ (s0). 2 Table 1: Fit results of αs(mτ ) and the OPE nonper- The theoretical prediction of the vector and turbative contributions from vector, axial-vector and axial-vector ratio Rτ,V/A canthusbewrittenas: (V + A) combined fits using the corresponding ratios Rτ and the spectral moments as input parameters. 3 2 Rτ,V/A = |Vud| SEW (4.3) The second error is given for theoretical uncertainty.  2  X ×  (0) 0 (2−mass) (D)  1+δ + δEW + δud,V/A + δud,V/A obtained by subtracting out the measured strange D=4,6,8 contribution (see last section). with the residual non-logarithmic electroweak cor- Two complete analyses of the V and A parts 0 have been performed by ALEPH [15] and OPAL [16]. rection δEW =0.0010 [21], neglected in the following, and the dimension D = 2 contribution Both use the world-average leptonic branching (2−mass) δ from quark masses which is lower than ratios, but their own measured spectral functions. ud,V/A 2 (0) The results on αs(m ) are therefore strongly cor- 0.1% for u, d quarks. The term δ is the purely τ (D) related and indeed agree when the same theoreti- perturbative contribution, while the δ are the −D/2 cal prescriptions are used. Here only the ALEPH OPE terms in powers of s0 of the following results are given. form X hO i (D) ∼ ud V/A δud,V/A D/2 (4.4) − 0 dimO=D( s ) 4.4 Results of the fits where the long-distance nonperturbative effects The results of the fits are given in table 1. The are absorbed into the vacuum expectation ele- limited number of observables and the strong cor- ments hOudi. relations between the spectral moments intro- The perturbative expansion (FOPT) is known duce large correlations, especially between the to third order [22]. A resummation of all known fitted nonperturbative operators. higher order logarithmic integrals improves the One notices a remarkable agreement within 2 convergence of the perturbative series (contour- statistical errors between the αs(mτ ) values us- improved method FOPTCI) [23]. As some ambi- ing vector and axial-vector data. The total non- guity persists, the results are given as an average perturbative power contribution to Rτ,V +A is com- of the two methods with the difference taken as patible with zero within an uncertainty of 0.4%, a systematic uncertainty. that is much smaller than the error arising from the perturbative term. This cancellation of the 4.3 Measurements nonperturbative terms increases the confidence 2 on the αs(mτ ) determination from the inclusive The ratio Rτ is obtained from measurements of the leptonic branching ratios: (V + A) observables. The final result is : Rτ =3.647 0.014 (4.5) 2 αs(mτ )=0.334 0.007exp 0.021th (4.6) − − using a value B(τ → e ν¯eντ )=(17.794 0.045)% which includes the improvement in ac- where the first error accounts for the experimen- curacy provided by the universality assumption tal uncertainty and the second gives the uncer- of leptonic currents together with the measure- tainty of the theoretical prediction of Rτ and the − − − − ments of B(τ → e ν¯eντ ), B(τ → µ ν¯µντ ) spectral moments as well as the ambiguity of the and the τ lifetime. The nonstrange part of Rτ is theoretical approaches employed.

5 Heavy Flavours 8, Southampton, UK, 1999 Michel Davier

4.5 Test of the running of αs(s)atlowen- responding to ﬁgure 8, translated into the run- ergies ning of αs(s0), i.e., the experimental value for Using the spectral functions, one can simulate αs(s0) has been individually determined at ev- ery s0 from the comparison of data and theory. the physics of a hypothetical τ lepton with a mass √ Good agreement is observed with the four-loop s0 smaller than mτ through equation (4.2) and hence further investigate QCD phenomena at low RGE evolution using three quark flavours. energies. Assuming quark-hadron duality, the The experimental fact that the nonperturbative contributions cancel over the whole range evolution of Rτ (s0) provides a direct test of the 2 2 1.2GeV ≤s0 ≤mτ leads to confidence that running of αs(s0), governed by the RGE β-function. On the other hand, it is a test of the validity of the αs determination from the inclusive (V + A) the OPE approach in τ decays. data is robust. 2 4.6 Discussion on the determination of αs(mτ )

) 4.2 0

(s 2 Data (exp. err.) ALEPH The evolution of the αs(mτ ) measurement from ,V+A

τ 4 Theory (theo. err.) R the inclusive (V + A) observables based on the 3.8 Runge-Kutta integration of the differential equa- 3 3.6 tion of the renormalization group to N LO [24, 26] yields 3.4 2 s exp th evol 3.2 α (MZ)=0.1202 0.0008 0.0024 0.0010

0.5 1 1.5 2 2.5 3 (4.7) 2 s0 (GeV ) where the last error stands for possible ambigu- ities in the evolution due to uncertainties in the Figure 8: The ratio Rτ,V +A versus the square “τ matching scales of the quark thresholds [26]. mass” s0. The curves are plotted as error bands to emphasize their strong point-to-point correlations in 0.4 τ (ALEPH) s0 Mτ . Also shown is the theoretical prediction using the Z (LEP + SLD) (Energy) 2 s 0.35 results of the ﬁt at s0 = mτ . α

0.3

0.25

) 0.6 0 (s

s Data (exp. errors only) α 0.5 RGE Evolution (nf=3) 0.2

RGE Evolution (nf=2)

0.4 0.15 MZ ALEPH 0.3 0.1 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 2 10 10 2 s0 (GeV ) Energy (GeV)

Figure 9: The running of αs(s0) obtained from the Figure 10: Evolution of the strong coupling (mea- 2 2 fit of the theoretical prediction to Rτ,V +A(s0). The sured at mτ )toMZ predicted by QCD compared shaded band shows the data including experimental to the direct measurement. The evolution is carried errors. The curves give the four-loop RGE evolution out at 4 loops, while the flavour matching is accom- for two and three flavours. plished at 3 loops at 2 mc and 2 mb thresholds.

The functional dependence of Rτ,V +A(s0)is The result (4.7) can be compared to the de- plotted in figure 8 together with the theoretical termination from the global electroweak fit. The prediction using the results of table 1. Below variable RZ has similar advantages to Rτ , but 1GeV2 the error of the theoretical prediction of it differs concerning the convergence of the per- Rτ,V +A(s0) starts to blow up due to the increas- turbative expansion because of the much larger ing uncertainty from the unknown fourth-order scale. It turns out that this determination is perturbative term. Figure 9 shows the plot cor- dominated by experimental errors with very small

6 Heavy Flavours 8, Southampton, UK, 1999 Michel Davier

theoretical uncertainties, i.e. the reverse of the 1 υ τ– → ν situation encountered in τ decays. The most re- (V, I=1) τ (ALEPH) 2 2.5 e+e– → (V, I=1) cent value [27] yields αs(MZ)=0.119 0.003, in excellent agreement with (4.7). Fig. 10 illus- 2 trates well the agreement between the evolution 2 2 of αs(mτ ) predicted by QCD and αs(MZ). 1.5

5. Applications to hadronic vacuum 1 polarization 0.5 5.1 Improvements to the standard calcula- 0 tions 0 0.5 1 1.5 2 2.5 3 3.5 Mass2 (GeV/c2)2 From the studies presented above we have learned that: Figure 11: Global test of CVC using τ and e+e− • the I = 1 vector spectral function from τ vector spectral functions. decays agrees with that from e+e− annihilation, while it is more precise for masses quark thresholds, but at low enough energies (com- less than 1.6 GeV as can be seen on fig- patible with the remarks above) in place of non- ure 11. Small CVC violations are expected −3 competitive experimental data [28]. This proce- at a few 10 level [10] from radiative ρ dure involves a proper treatment of the quark decays and SU(2)-breaking in the π and ρ masses in the QCD prediction [25]. masses. Finally, it is still possible to improve the con- • the description of Rτ by perturbative QCD tributions from data by using analyticity and works down to a scale of 1 GeV. Nonper- QCD sum rules, basically without any additional turbative contributions at 1.8 GeV are well assumption. This idea, advocated in reference [29], ∼ below 1 % in this case. They are larger ( 3 has been used within the procedure described %) for the vector part alone, but reason- above to improve the calculations further [30]. ably well described by OPE. The complete The experimental results of R(s) and the the- (perturbative + nonperturbative) descrip- oretical prediction are shown in figure 12. The tion is accurate at the 1 % level at 1.8GeV shaded bands depict the regions where data are for integrals over the vector spectral func- used instead of theory to evaluate the respec- tion such as Rτ,V . tive integrals. Good agreement between data and These two facts have direct applications to QCD is found above 8 GeV, while at lower ener- calculations of hadronic vacuum polarisation which gies systematic deviations are observed. The R involve the knowledge of the vector spectral func- measurements in this region are essentially pro- tion: the muon magnetic anomaly and the run- vided by the γγ2 [31] and MARK I [32] collabo- ning of α. In both cases, the standard method in- rations. MARK I data above 5 GeV lie system- volves a dispersion integral over the vector spec- atically above the measurements of the Crystal tral function taken from the e+e− → hadrons Ball [33] and MD1 [34] Collaborations as well as data. Eventually at large energies, QCD is used the QCD prediction. to replace experimental data. Hence the precision of the calculation is given by the accuracy of the data, which is poor above 1.5GeV.Even 5.2 Muon magnetic anomaly at low energies, the precision can be significantly improved at low masses by using τ data [10]. By virtue of the analyticity of the vacuum polar- The next breakthrough comes about using ization correlator, the contribution of the hadronic the prediction of perturbative QCD far above vacuum polarization to aµ can be calculated via

7 Heavy Flavours 8, Southampton, UK, 1999 Michel Davier

6 R ω Φ + - J/ψ ψ 5.3 Running of the electromagnetic cou- e e → hadrons 1S 2S ψ 5 QCD 3770 pling 4

3 In the same spirit we evaluate the hadronic con-

2 tribution ∆α(s) to the renormalized vacuum po- Crystal B. 0 MARK I 1 larization function Πγ (s) which governs the run- exclusive data γγ2 PLUTO 0 ning of the electromagnetic coupling α(s). With 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0 √s (GeV) ∆α(s)=−4πα Re Πγ(s) − Πγ(0) , one has 6 R ϒϒ 1S 2S 3S 4S 5 α(0) ϒ 4 10860 α(s)= (5.2) ϒ − 11020 1 ∆α(s) 3 MARK I MD1 2 PLUTO JADE where 4πα(0) is the square of the electron charge e+e- → hadrons LENA MARK J 1 QCD Crystal B. CESR in the long-wavelength Thomson limit. 0 567891011121314 The leading order leptonic contribution is √s (GeV) equal to 314.2×10−4. Using analyticity and uni- tarity, the dispersion integral for the contribution Figure 12: Inclusive hadronic cross section ratio from the light quark hadronic vacuum polariza- + − √ 2 in e e annihilation versus the c.m. energy s. tion ∆αhad(MZ) reads [36] Additionally shown is the QCD prediction of the Z∞ continuum contribution from reference [28] as ex- 2 (5) 2 − MZ σhad(s) plained in the text. The shaded areas depict regions ∆αhad(MZ)= 2 Re ds 2 4π α s − MZ − i 2 where experimental data are used for the evaluation 4mπ 2 had of ∆αhad(MZ)andaµ in addition to the measured (5.3) + − 2 2 0 narrow resonance parameters. The exclusive e e where σ(s)=16πα(s)/s · ImΠγ (s) from the had cross section measurements at low c.m. energies are optical theorem. In contrast to aµ , the inte- taken from DM1,DM2,M2N,M3N,OLYA,CMD,ND gration kernel favours cross sections at higher τ and data from ALEPH (see reference [10] for de- masses. Hence, the improvement when includ- tailed information). ing τ data is expected to be small. The top quark contribution can be calculated 3 using the next-to-next-to-leading order αs pre- the dispersion integral [35] diction of the total inclusive cross section ratio R from perturbative QCD [22, 37]. The evalua- Z∞ tion of the integral (5.3) with mtop = 175 GeV had 1 a = ds σhad(s) K(s) (5.1) 2 − × −4 µ 4π3 yields ∆αtop(MZ)= 0.6 10 . 2 4mπ 5.4 Results

+ − Here σhad(s) is the total e e → hadrons cross The combination of the theoretical and experi- section as a function of the c.m. energy-squared mental evaluations of the integrals (5.3) and (5.1) s,andK(s) denotes a well-known QED kernel. yields the results The function K(s) decreases monotonically 2 −4 ∆αhad(MZ) = (276.3 1.1exp 1.1th) × 10 with increasing s. It gives a strong weight to the −1 2 α (M ) = 128.933 0.015exp 0.015th low energy part of the integral (5.1). About 91% Z had −10 had exp th × of the total contribution to aµ is accumulated aµ = (692.4 5.6 2.6 ) 10 √ SM at c.m. energies s below 2.1 GeV while 72% of aµ = (11 659 159.6 had −10 aµ is covered by the two-pion final state which 5.6exp 3.7th) × 10 (5.4) is dominated by the ρ(770) resonance. The new had −14 information provided by the ALEPH 2- and 4- and ae = (187.51.7exp0.7th)×10 for the pion spectral functions can significantly improve leading order hadronic contribution to ae.The had SM the aµ determination. total aµ value includes additional contributions

8 Heavy Flavours 8, Southampton, UK, 1999 Michel Davier

2 from non-leading order hadronic vacuum polar- of α(MZ) which does not limit anymore the ad- ization (summarized in references [38, 10]) and justment of the Higgs mass from accurate exper- 2 light-by-light scattering [39, 40] contributions. Fig- imental determinations of sin θW. The improve- ures 13 and 14 show a compilation of published ment in precision can be directly appreciated by 2 results for the hadronic contributions to α(MZ) considering the relevant variable log MH with 2 and aµ. Some authors give the hadronic contri- MH in GeV/c [27]: bution for the five light quarks only and add the +0.31 top quark part separately. This has been cor- log MH =1.88−0.39 (5.5) rected for in ﬁgure 13. 2 with the ’standard’ α(MZ ) [37], and

+0.22 Lynn, Penso, Verzegnassi, ´87 log MH =1.97−0.25 (5.6) Eidelman, Jegerlehner ´95 Burkhardt, Pietrzyk ´95 Martin, Zeppenfeld ´95 with the QCD-improved value [30]. Swartz ´96 Alemany, Davier, Höcker ´97 The interest in reducing the uncertainty in Davier, Höcker ´97 had Kühn, Steinhauser ´98 the hadronic contribution to aµ is directly linked Groote et al. ´98 to the possibility of measuring the weak contri- Erler ´98 Davier, Höcker ´98 bution: 270 275 280 285 290 295 ∆α (M2) (× 10– 4 ) SM QED had weak had Z aµ = aµ + aµ + aµ (5.7)

2 Figure 13: Comparison of ∆αhad(MZ) evaluations. QED −10 where aµ = (11 658 470.6 0.2) × 10 is The values are taken from references [41, 37, 42, 43, the pure electromagnetic contribution (see [51] 44, 10, 28, 45, 46, 30]. had and references therein), aµ is the contribution weak from hadronic vacuum polarization, and aµ = (15.1 0.4) × 10−10 [51, 52, 53] accounts for Barkov et al. ´85 corrections due to the exchange of the weak in- Kinoshita, Nizic, Okamoto ´85 Casas, Lopez, Ynduráin ´85 teracting bosons up to two loops. The present Eidelman, Jegerlehner ´95 + − Brown, Worstell ´96 value from the combined µ and µ measure- Alemany, Davier, Höcker ´97 ments [54], Davier, Höcker ´97 Davier, Höcker ´98 −10 680 700 720 aµ = (11 659 230 85) × 10 (5.8) had × – 10 a µ ( 10 ) should be improved to a precision of at least ahad Figure 14: Comparison of µ evaluations. The 4 × 10−10 by a forthcoming Brookhaven exper- values are taken from references [47, 48, 49, 37, 50, iment (BNL-E821) [55], well below the expected 10, 28, 30]. weak contribution. Such a programme makes sense only if the uncertainty on the hadronic term is made sufficiently small. The improvements de- 5.5 Outlook scribed above represent a significant step in this These results have direct implications for phe- direction. nomenology and on-going experimental programs. Most of the sensitivity to the Higgs boson 6. Strange τ decays and ms mass originates from the measurements of asym- + − metries in the process e e → Z → fermion 6.1 The strange hadronic decay ratio Rτ,S 2 2 pairs and in fine from (sin θW )eff =¯s.Un- As previously demonstrated in reference [56], the fortunately, this approach is limited by the fact inclusive τ decay ratio into strange hadronic final that the intrinsic uncertainty on α(M 2)inthe Z states, standard evaluation is at the same level as the 2 − − experimental accuracy ons ¯ . The situation has Γ(τ → hadronsS=−1 ντ ) Rτ,S = − − (6.1) completely changed with the new determination Γ(τ → e ν¯eντ )

9 Heavy Flavours 8, Southampton, UK, 1999 Michel Davier can be used due to its precise theoretical pre- component from data. The measurement diction [19, 57] to determine ms(s)atthescale is then less inclusive and the sensitivity to 2 s=Mτ. Since then it was shown [58] that the ms is signiﬁcantly reduced; however, the perturbative expansion used for the massive term δ2 perturbative expansion is under control in reference [57] was incorrect. After correction and the corresponding theoretical uncer- the series shows a problematic convergence be- tainty is reduced. haviour [58, 59, 60]. 6.2 New ALEPH results on strange decays ALEPH has recently published a comprehensive

 study of τ decay modes including kaons (charged, τ– → K–ν τ– → 0π–ν τ K τ 0 0 DELCO84 L3 95 (K0) KS and KL) [61, 62, 63, 8] with up to four hadrons 0 CLEO94 CLEO96 (KS) 0 in the final state. A comparison with the pub- DELPHI94 ALEPH98 (KS) 0 ALEPH99 ALEPH99 (KL) lished results is given in figure 15. 4 6 8 10 12 7.5 10 12.5 15 17.5 The total branching ratio for τ into strange B (10–3) B (10–3) S τ– → –π0ν – 0 – final states, B ,is K τ τ → K K ντ CLEO94 0 CLEO96 (KS) 0 ALEPH98 (KS) BS =(2.87 0.12)% (6.2) ALEPH99 0 ALEPH99 (KL)

4681 1.5 2 2.5 3 corresponding to B (10–3) B (10–3)  τ– → 0 –π0ν τ– → 0π–π0ν K K τ K τ Rτ,S =0.1610 0.0066 (6.3) 0 0 L3 95 (K ) CLEO96 (KS) CLEO96 (K0) 0 S ALEPH98 (KS) 0 ALEPH98 (KS) 0 Since the QCD expectation for a massless quarks ALEPH99 (KL) 0 ALEPH99 (KL) is 0.1809 0.0036, the result (6.3) is evidence for 12345246810 B (10–3) B (10–3) a massive s quark. The strange spectral function is given in ﬁg- ure 16, dominated at low mass by the K∗ reso- Figure 15: Some of the new ALEPH results on decay modes with kaons compared to the published nance. data.

Similarly to the nonstrange case, the QCD prediction is given by equations (4.2) and (4.3) 6 ALEPH where the attention is now turned to the δ(2−mass) v+a – Kπ 5 – term, important for the relatively heavy strange Kππ – quark. The corresponding perturbative expan- 4 K3π+K-η (MC) – sion is known to second order for the J =1+0 3 K4π (MC) – part and to the third order for J = 0 [57, 58]. K5π (MC) While the J = 1+0 series behaves well, the J =0 2 expansion in fact diverges after the second term. 1

Following these observations, two methods 0 2 0.5 1 1.5 2 2.5 3 can be considered in order to determine ms(mτ ): Mass2 (GeV/c2)2 • in the inclusive method, the inclusive strange hadronic rate is considered and both J = Figure 16: The strange spectral function measured 1+0andJ = 0 are included with their re- by ALEPH. spective convergence behaviour taken into account in the theoretical uncertainties. 6.3 The ALEPH analysis for ms • the ‘1+0’ method singles out the well-behaved J = 1 + 0 part by subtracting the exper- As proposed in reference [64] and successfully ap- 2 imentally determined J = 0 longitudinal plied in αs(mτ ) analyses, the spectral function is

10 Heavy Flavours 8, Southampton, UK, 1999 Michel Davier used to construct moments nel), a fit to 5 moments is performed with the M2 safer J = 1 + 0 method, yielding Z τ k l kl s s dRτ,S R ≡ ds 1 − 2 +37 +24 2 τ,S 2 2 ms(m ) = (176 )MeV/c (6.7) Mτ Mτ ds τ −48exp−28th 0 ˜(6) (6.4) δ =0.039 0.016exp 0.014th (6.8) (8) In order to reduce the theoretical uncertain- δ˜ = −0.021 0.014exp 0.008th(6.9) ties one considers the difference between nonstrange and strange spectral moments, properly with additional (smaller) uncertainties from the normalized with their respective CKM matrix el- fitting procedure and the determination of the ements: J = 0 part. The quoted theoretical errors are mostly from the Vus uncertainty. The D =6 kl ≡ 1 kl − 1 kl ∆τ 2 Rτ,S=0 2 Rτ,S=−1 (6.5) and D = 8 strange contributions are found to be |Vud| |Vus| surprisingly larger than their nonstrange coun- for which the massless perturbative contribution terparts. If the fully inclusive method is used vanishes so that the theoretical prediction now instead (with the problematic convergence) the reads (setting mu = md =0) 2 +24 +21 2 result ms(m ) = (149 )MeV/c is ob-   τ −30exp−25th X tained. kl  kl(2−mass) kl(D) ∆τ =3SEW − δS + δ˜ The result (6.7) can be evolved to the scale of D=4,6,... 1 GeV using the four-loop RGE γ-function [66], (6.6) yielding For the CKM matrix elements the values |Vud| = 2 +61 2 0.9751 0.0004 and |Vus| =0.2218 0.0016 [65] ms(1 GeV ) = (234 −76)MeV/c (6.10) are used, while the errors are included in the the- This value of ms is somewhat larger than oretical uncertainties. previous determinations [67], but consistent with Figure 17 shows the weighted integrand of 00 them within errors. the lowest moment ∆τ from the ALEPH data, as a function of the invariant mass-squared, and for which the expectation from perturbative QCD 7. Conclusions vanishes. The decays τ → ντ + hadrons constitute a clean and√ powerful way to study hadronic physics up to s ∼ 1.8 GeV. Beautiful resonance analyses

π have already been done, providing new insight /ds

τ ALEPH 00 into hadron dynamics. Probably the major sur- ∆

d ρ 5 prise has been the fact that inclusive hadron pro-

a1 duction is well described by perturbative QCD

0 with very small nonperturbative components at K 1 the τ mass. Despite the fact that this low-energy massless pQCD region is dominated by resonance physics, meth- -5 ods based on global quark-hadron duality work K K* indeed very well. 0123 s (GeV/c2)2 The measurement of the vector and axial- vector spectral functions has provided the way for quantitative analyses. Precise determinations Figure 17: Integrand of equation (6.5) for (k=0, of αs agree for both spectral functions and they l=0), i.e., diﬀerence of the Cabibbo-corrected non- also agree with all the other determinations from strange and strange invariant mass spectra. the Z width, the rate of Z to jets and deep in- elastic lepton scattering. Indeed from τ decays Subtracting out the experimental J =0con- 2 tribution (essentially given by the single K chan- αs(MZ)τ =0.1202 0.0027 (7.1)

11 Heavy Flavours 8, Southampton, UK, 1999 Michel Davier

in excellent agreement with the average from all [12] D. Buskulic et al. (ALEPH Collaboration), Z. other determinations [68] Physik C74(1997) 263 et al. 2 [13] T.E. Browder (CLEO Collaboration), αs(MZ)non−τ =0.1187 0.0020 (7.2) hep-ex/9908030

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