FR0005562
LAL 99-78 December 1999
Hadronic r decays and QCD
M. Davier
Laboratoire de l'Accelerateur Lineaire IN2P3-CNRS et Universite de Paris-Sud, BP 34, F-91898 Orsay Cedex
Invited talk given at the &h International Symposium on Heavy Flavour Physics University of Southampton, England, 25-29 July 1999
U.E.R Institut National de de Physique Nucleaire PUniversite Paris-Sud et de Physique des Particules
32/ 05 B.P. 34 ~ Bailment 200 - 91898 ORSAY CEDEX Gestion IH\S Doc. Enreg. I© .iU4/. N* TRN &&.O.O.OSS LAL 99-78 December 1999
Hadronic r decays and QCD
Michel Davier Laboratoire de VAccel6rateur Lineaire IN2P3/CNRS et University de Paris-Sud 91898 Orsay, France E-mail: davierQlal.in2p3.fr
ABSTRACT: Hadronic decays of the r lepton provide a clean source 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 T decays are borne out of distance effects and thus expected to be essen- the charged weak current, i.e. out of the QCD tially final-state independent. vacuum. This property garantees that hadronic Hadronic r 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 r mass. Many hadronic modes have systems have 1 = 1 and spin-parity Jp = 0+, 1~ p + been measured and studied, while some earlier (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 x 105 measured decays respective branching ratios and corrected for the are available in each LEP experiment and CLEO. r decay kinematics. For a given spin-1 vector Conditions for low systematic uncertainties are decay, one has particularly well met at LEP: measured samples have small non-r backgrounds (< 1%) and large mt B(T~ -> V- uT) vi(s) = 2 selection efficiency (92%), for example in ALEPH. 61Vud\ SEW B{T Recent results in the field are discussed in . dN v (1.1) this report. ' Nyds 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 Isospin symmetry (CVC) connects the r and + The decay r -» vTir~Tr° 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 Aira? are now available [3] with the mass spectrum T/=i (S) = vi,x-(s) (1.2) given in figure 1 dominated by the p(770) res- Michel Davier onance. Good agreement is observed between (24.65±0.35)% [9]. This 1.8<7 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, p(1400) through interference with the dominant such as radiative corrections in e+e~ data and amplitude. Fits also include a p(1700) contribu- 7T° reconstruction in t data. CVC violations are tion, taken from e+e~ data as the value of the r of course expected at some point: hadronic vio- 2 mass does not allow r data alone to tie down the lation should be very small (~ {mu - md) /m*), 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 p lineshape, with a prefer- the difference between the charged and neutral ence given to the Gounaris-Sakurai parametriza- p widths should only be at the level of (0.3 ± tion [4] over that of Kiihn-Santamaria [5]. 0.4)% [10].
' • ' ' 1 ' ' • ' i > >i ' ' [••i i • v-f • . • i j T ,"\-r-rx- II .
- Mj I C V -» 7t* It' Sr T I A ALEI'll t • CMD-2 1 ? I « NA7 - I o OLVA ii " l01'' i It 4 CMD ft, n DM1 - JI
r* we Invariant Mass
Figure 1: Mass distribution in CLEO r ->• VTK TT° s (CJcV) events. The solid curve overlaid is the result of the Kiihn-Santamaria fit, while the dashed curve has the Figure 2: Cross section for e+e~ - p(1400) contribution turned off. pared to the ALEPH r data using CVC with p — w interference built in (shaded band). + The quality of data on e e~ -> TT+TT" has also recently improved with the release of the The 4TT 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 r data is given in figure 2 (for this exercise ments which disagree well beyond their quoted the p — w interference has to be artificially intro- systematic uncertainties. A new CLEO analy- duced in the r data). Although the agreement sis studies the resonant structure in the STTTT0 looks impressive, it is possible to quantify it by channel which is shown to be dominated by unr computing a single number, integrating over the and aiTr contributions. The ton spectral func- complete spectrum. It is convenient for this to tion shown in figure 3 is in good agreement with use the branching ratio B(r -* vT%~ir°) as di- CMD-2 results and it is interpreted by a sum of rectly measured in r decays and computed from p-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 tion of CVC. Using B(T -> Vyh'-rr0) = (25.79 ± with the value (1406± 14)MeV from the fit of the 0.15)% [7] and subtracting out B(T -* i/TK~ir°) = 2TT spectral function. This point has to be clar- (0.45 ± 0.02)% [8], one gets B{T -» uTic-ir°) = ified. Following a limit of 8.6% obtained earlier (25.34 ± 0.15)%, somewhat larger than the CVC by ALEPH [12], CLEO sets anew 95% CL limit value using all available e+e~ data, of 6.4% for the relative contribution of second- Michel Davier
class currents in the decay r -> vr-K LJ from the (a) : hadronic angular decay distribution. - T 1°3 8 / i
a 3 V(q) i.A(1523),,o-(17OO) / 10 0.050 :
0.8 1.2 16
q(om) [GeV/c2]
Figure 3U).: Fits of the CLEO spectral function for
2.2 Axial-vector states
The decay r -> I/T3TT is the cleanest place to Figure 4: (a) 3TT mass spectrum from CLEO in the study axial-vector resonance structure. The spec- TT~ 2TT° channel with the lineshape from the resonance + trum is dominated by the 1 a\ state, known to fit (zoom on the K*K threshold). In (b), the -y/s- decay essentially through fm. A comprehensive dependent a\ width is plotted with the different con- analysis of the ir~2ir° channel has been presented tributions considered. by CLEO. First, a model-independent determi- nation 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-7r0 decays and the proper plitude analysis was performed [14]: while the treatment of final states with a KK pair. The dominant (m 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 {a, /0(1470), /2(1270)) behaviour, dominated by the lowest p and ai was found in the 2TT system. states, with a tendancy to converge at large mass The a\ —> 7r~27r° 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 a\ width is clearly seen. The while the axial-vector one lies below. Thus, the derived branching ratio, B(a\ -> K*K) = (3.3 ± two spectral functions are clearly not 'asymp- 0.5)% is in good agreement with ALEPH results totic' at the r mass scale. on the KKir 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 (ai) dominated with B(a\ ->• 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 higher mass state {a\) 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 r nonstrange spectral functions have been ture will be quantitatively discussed in the next measured by ALEPH [2,15] and OPAL [16]. The section. Michel Davier
0, + a, : • ALEPH: 2.5
• f* . T-->(V,A,(=1)VT ; parton model prediction 2 perturbative QCD (massless)
1.5 • -
-^"***^« ulillili- i U 1 : . ,•: X^tii {.[::
0.5 2 2.5 3 • s (GeV2) 0 0.5 1 1.5 2 IS 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 ex- pectation from the naive parton model, while the solid one is from massless perturbative QCD using aa(M|)= 0.120.
This favourable situation could be spoiled by the fact that the Q2 scale is rather small, so that questions about the validity of a perturbative ap- proach can be raised. At least two levels are to be considered: the convergence of the perturbative expansion and the control of the nonperturbative contributions. Happy circumstances make these contributions indeed very small[17, 18]. Figure 6: Inclusive nonstrange axial-vector spectral 4.2 Theoretical prediction for Rr function from ALEPH and OPAL. The dashed line is the expectation from the naive parton model. The imaginary parts of the vector and axial-vector two-point correlation functions Tiidy/^{s), with the spin J of the hadronic system, are propor- 4. QCD analysis of nonstrange r de- tional to the r hadronic spectral functions with cays corresponding quantum numbers. The non-strange ratio RT can be written as an integral of these 4.1 Motivation spectral functions over the invariant mass-squared The total hadronic r width, properly normalized s of the final state hadrons [19]: to the known leptonic width,
#r(s0) = 12TT5EW / — 1 (4.2) P(r ->• hadrons uT) RT = (4.1) 1 should be well predicted by QCD as it is an inclu- x / l + 2— J Irnn^ ) (s + ie) + imE^ (s + ie) sive observable. Compared to the similar quan- tity defined in e+e~ annihilation, it is even twice By Cauchy's theorem the imaginary part inclusive: not only all produced hadronic states is proportional to the discontinuity across the at a given mass are summed over, but an inte- positive real axis. gration is performed over all the possible masses The energy scale so for so = vr?r is large from m-n to mT. enough that contributions from nonperturbative Michel Davier
effects be small. It is therefore assumed that one 2 can use the Operator Product Expansion (OPE) ALEPH aa(m r)
ments (Oud). duce large correlations, especially between the The perturbative expansion (FOPT) is known fitted nonperturbative operators. to third order [22]. A resummation of all known One notices a remarkable agreement within 2 higher order logarithmic integrals improves the statistical errors between the as^m .) values us- convergence of the perturbative series (contour- ing vector and axial-vector data. The total non- improved method FOPTci} [23]. As some ambi- perturbative power contribution to i?r>v+^i is com- guity persists, the results are given as an average patible with zero within an uncertainty of 0.4%, of the two methods with the difference taken as that is much smaller than the error arising from systematic uncertainty. the perturbative term. This cancellation of the nonperturbative terms increases the confidence 2 4.3 Measurements on the a^m .) determination from the inclusive (V + A) observables. The ratio RT is obtained from measurements of The final result is : the leptonic branching ratios: 2 as{m T) = 0.334 ± 0.007exp ± 0.021th (4.6) RT = 3.647 ±0.014 (4.5) where the first error accounts for the experimen- using a value B(T -» e~ uevT) = (17.794 ± tal uncertainty and the second gives the uncer- 0.045)% which includes the improvement in ac- tainty of the theoretical prediction of RT and the curacy provided by the universality assumption spectral moments as well as the ambiguity of the of leptonic currents together with the measure- theoretical approaches employed. ments of B(T~ -> e~ ueuT), B(T~ -» JJT v^Vr) 4.5 Test of the running of aa(s) at low en- and the r lifetime. The nonstrange part of RT is obtained subtracting out the measured strange ergies contribution (see last section). Using the spectral functions, one can simulate Michel Davier
the physics of a hypothetical r lepton with a mass Good agreement is observed with the four-loop ,/s^ smaller than mT through equation (4.2) and RGE evolution using three quark flavours. hence further investigate QCD phenomena at low The experimental fact that the nonpertur- energies. Assuming quark-hadron duality, the bative contributions cancel over the whole range 2 2 evolution of RT(so) provides a direct test of the 1.2 GeV < s0 < m . leads to confidence that running of as (so), governed by the RGE /^-function. the as determination from the inclusive (V + A) On the other hand, it is a test of the validity of data is robust. the OPE approach in r decays.
4.6 Discussion on the determination of as(rri^
The evolution of the as{ml) measurement from the inclusive (V + .4) observables based on the Runge-Kutta integration of the differential equa- tion of the renormalization group to N3LO [24, 26] yields
as(Mj) = 0.1202±0.0008exp±0.0024th±0.0010evoi (4.7) where the last error stands for possible ambigu- ities in the evolution due to uncertainties in the Figure 8: The ratio RT:V+A versus the square "r matching scales of the quark thresholds [26]. mass" so- The curves are plotted as error bands to emphasize their strong point-to-point correlations in
so. Also shown is the theoretical prediction using the '•HALEPH) M, • /.(LEP + SLD) results of the fit at so = m?. |o,
o.i -
— 0.6 I Dafa (cxp. snrors only) i 0.25 a" •
\ ^Q*?Js£-. I ""•I*" KGB Evolution (ni=2) i 0.2
0.15
1 1.25 1.5 1.75 2.25 2.5 2.75 3 O.I 2 s0 (GeV )
Encrsy (GeV)
Figure 9: The running of as(so) obtained from the fit of the theoretical prediction to RT,V+A(SO)- The Figure 10: Evolution of the strong coupling mea- shaded band shows the data including experimental sured at ra?) to M% predicted by QCD compared errors. The curves give the four-loop RGE evolution to the direct measurement. The evolution is carried for two and three flavours. out at 4 loops, while the flavour matching is accom- plished at 3 loops at 2 mc and 2 mi thresholds. The functional dependence of RTy+A(so) is plotted in figure 8 together with the theoretical The result (4.7) can be compared to the de- prediction using the results of table 1. Below termination from the global electroweak fit. The 2 1 GeV the error of the theoretical prediction variable Rz has similar advantages as RT, but of RT,V+A(SO) starts to blow up due to the in- it differs concerning the convergence of the per- creasing uncertainty from the unknown fourth- turbative expansion because of the much larger order perturbative term. Figure 9 shows the plot scale. It turns out that this determination is corresponding to Fig. 8, translated into the run- dominated by experimental errors with very small ning of as(s0), i.e., the experimental value for theoretical uncertainties, i.e. the reverse of the as(so) has been individually determined at ev- situation encountered in r decays. The most re- ery so from the comparison of data and theory. cent value[27] yields a,(M^) = 0.119 ± 0.003, Michel Davier in excellent agreement with (4.7). Fig. 10 illus- These two facts have direct applications to trates well the agreement between the evolution calculations of hadroiiic vacuum polarisation which of as(ml) predicted by QCD and as(M%). involve the knowledge of the vector spectral func- tion: the muon magnetic anomaly and the run- ning of a. In both cases, the standard method in- 5. Applications to hadronic vacuum volves a dispersion integral over the vector spec- polarization tral function taken from the e+e~ -> hadrons data. Eventually at large energies, QCD is used 5.1 Improvements to the standard calcula- to replace experimental data. Hence the preci- tions sion of the calculation is given by the accuracy From the studies presented above we have learned of the data, which is poor above 1.5 GeV. Even that: at low energies, the precision can be significantly improved at low masses by using r data [10]. • the 7=1 vector spectral function from r + The next breakthrough comes about using decays agrees with that from e e~ annihi- the prediction of perturbative QCD far above lation, 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 p dure involves a proper treatment of the quark decays and SU(2)-breaking in the -K and p masses in the QCD prediction [25]. masses. Finally, it is still possible to improve the con- tributions from data by using analyticity and • the description of RT by perturbative QCD 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 Ref. [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 still improve the calculations [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 1.8 GeV shaded bands depict the regions where data are for integrals over the vector spectral func- used instead of theory to evaluate the respec- tive integrals. Good agreement between data and tion such as RT,v- QCD is found above 8 GeV, while at lower ener- gies systematic deviations are observed. The R
I 11 11 11 I' '' I ' ' '' I' '' ' I' '' ' I ' ' ' ' measurements in this region are essentially pro- ->(V,I=l)v, (ALEPH) vided by the 772 [31] and MARK I [32] collabo- :eV-*(V, 1=1) rations. MARK I data above 5 GeV lie system- atically above the measurements of the Crystal Ball [33] and MD1 [34] Collaborations as well as the QCD prediction.
5.2 Muon magnetic anomaly By virtue of the analyticity of the vacuum polar- ization correlator, the contribution of the hadronic
vacuum polarization to aM can be calculated via 0 0.5 I 1.5 2 2.5 3 3.5 the dispersion integral [35] Mass2 (GeV/c2)2
00 + If Figure 11: Global test of CVC using r and e e" ™ = 4TT3 / dsa^d{s)K(s) (5.1) vector spectral functions. 4m.2 Michel Davier
Aa{s)=-4iraRe [U'y{s) - 11^(0)], one has
a(0) a{s) = (5-2) 1 - Aa(s) where 47ra(0) is the square of the electron charge in the long-wavelength Thomson limit. The leading order leptonic contribution is equal to 314.2 x 10~4. Using analyticity and uni- tarity, the dispersion integral for the contribution from the light quark hadronic vacuum polariza-
tion Aaimti (M|) reads [36]
da- T2 a - it Ami (5-3) 2 2 Figure 12: Inclusive hadronic cross section ratio where a{s) = 16ir a {s)/s • Imll^s) from the in e+e~ annihilation versus the cm. energy y/s. optical theorem. In contrast to a£ad, the inte- Additionally shown is the QCD prediction of the gration kernel favours cross sections at higher continuum contribution from Ref. [28] as explained masses. Hence, the improvement when includ- in the text. The shaded areas depict regions were ing r data is expected to be small. experimental data are used for the evaluation of The top quark contribution can be calculated Aahad(Mz) and a£ad in addition to the measured using the next-to-next-to-leading order a% pre- narrow resonance parameters. The exclusive e+e~ diction of the total inclusive cross section ratio cross section measurements at low cm. energies are taken from DM1,DM2,M2N,M3N,OLYA,CMD,ND R from perturbative QCD [22, 37]. The evalua- and T data from ALEPH (see Ref. [10] for detailed tion of the integral (5.3) with mtop = 175 GeV 4 information). yields Aatop(M|) = -0.6 x 10~ . 5.4 Results + Here a\md(s) is the total e e -fhadrons cross The combination of the theoretical and experi- section as a function of the cm. energy-squared mental evaluations of the integrals (5.3) and (5.1) s, and K(s) denotes a well-known QED kernel. yields the results The function K(s) decreases monotonically 4 with increasing s. It gives a strong weight to the Aahad(M|) = (276.3 ± l.lexp ± l.lth) x 10~ x low energy part of the integral (5.1). About 91% a~ {Ml) = 128.933 ± 0.015exP ± 0.015ti, ad of the total contribution to a£ is accumulated ad 10 ojl = (692.4 ± 5.6exp ± 2.6th) x lO" at cm. energies y/s below 2.1 GeV while 72% of af* = (11659159.6 ahad js covered by the two-pion final state which 10 is dominated by the p(770) resonance. The new ±5.6exp ± 3.7th) x 10- (5.4) information provided by the ALEPH 2- and 4- and 4iad = (187.5±1.7 ±0.7th) x 10~w for the pion spectral functions can significantly improve oxp ad leading order hadronic contribution to a - The the a£ determination. e total a^M value includes additional contributions 5.3 Running of the electromagnetic cou- from non-leading order hadronic vacuum polar- ization (summarized in Refs.[38, 10]) and light- pling by-light scattering [39, 40] contributions. Fig- In the same spirit we evaluate the hadronic con- ures 13 and 14 show a compilation of published tribution Aa(s) to the renormalized vacuum po- results for the hadronic contributions to ct{M%) larization function n^s) which governs the run- and aM. Some authors give the hadronic contri- ning of the electromagnetic coupling a(s). With bution for the five light quarks only and add the Michel Davicr
top quark part separately. This has been cor- with the 'standard' a(M|) [37], and rected for in Fig. 13. log MH = l-97+°|2 (5.6)
: Lynn. Pcaw. Veragnassi. "87 with the QCD-improved value [30]. i Eidetnun. Icneilctmer '95 Buifctordi. Ketrtyfc '95 : The interest in reducing the uncertainty in ! . Manin, Zeppenfelrf 95 d ! : : Swanz '96 the hadronic contribution to a}™ is directly linked Alenuny. Davier. JlOcker '97 ! " • Davier. Hacker "97 to the possibility of measuring the weak contri- Kflhn, Sicinluuser '9S Croote 41 si. '•>$ bution: Erler "98 Davier. HiWUer '98 aSM = 270 275 230 285 290 295 (5.7) where a%BD = (11658470.6 ± 0.2) x lO"10 is Figure 13: Comparison of Aai,ad(A^z) evaluations. the pure electromagnetic contribution (see [51] The values are taken from Refs. [41, 37, 42, 43, 44, and references therein), a^a<1 is the contribution 10, 28, 45, 46, 30]. from hadronic vacuum polarization, and a™eak = (15.1 ± 0.4) x 10~10 [51, 52, 53] accounts for corrections due to the exchange of the weak in-
Barkov el il. SS teracting bosons up to two loops. The present | M 1 Kinoshiu. Niac.Okanwto 'S + | I ; | -4-: Cwas. Lopez. Yndurfin '85 value from the combined fi and [i~ measure- MM Eidebnui. Jeiterlehner '9S ill! Brown, Worslell "96 ments [54], M ! !_ Alenuny. D»vier. Keeker '97 •~~i— i I Ehvier. Mocker '97 -10 ! : i-h Davier. Hecker '98 ati = (11659 230 ±85) x 10 (5.8) i : i—+*- i i ; i i i i 720 (x!0-1 should be improved to a precision of at least 4 x 10~10 by a forthcoming Brookhaven exper- aH Figure 14: Comparison of a|} evaluations. The iment (BNL-E821) [55], well below the expected values are taken from Refs. [47, 48, 49, 37, 50, 10, weak contribution. Such a programme makes 28, 30]. 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 direction. These results have direct implications for phe- nomenology and on-going experimental programs. Most of the sensitivity to the Higgs boson 6. Strange r 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 RTts 2 2 pairs and in fine from (sin Ow)e/f = s . Un- As previously demonstrated in Ref. [56], the in- fortunately, this approach is limited by the fact clusive r decay ratio into strange hadronic final that the intrinsic uncertainty on a{M^) in the states, standard evaluation is at the same level as the experimental accuracy on s2. The situation has T(r -T- hadronss__j vT) Rr,S = (6.1) completely changed with the new determination T(r~ -?• e~ vevr) of a(Mz) which does not limit anymore the ad- can be used due to its precise theoretical pre- justment of the Higgs mass from accurate ex- 2 diction [19, 57] to determine ms(s) at the scale perimental determinations of sin #w- The im- 2 s = M . Since then it was shown [58] that provement in precision can be directly appreci- the perturbative expansion used for the massive ated on the relevant variable log MH with term in Ref. [57] was incorrect. After correction in GeV/c2 [27]: the series shows a problematic convergence be- haviour [58, 59, 60]. log MH = 1.88t°o3f9 (5.5) Michel Davier
6.1.1 New ALEPH results on strange de- cays ALEPH has recently published a comprehensive study of r decay modes including kaons (charged, K% and K°L) [61, 62, 63, 8] up to four hadrons in
BU0-3) the final state. A comparison with the published results is given in figure 15. •i -» K°K"vt 0 CLEO94 0 CLEOM(KS) The total branching ratio for r into strange 0 ALEPHOT (K?) -•- ALBPHM final states, Bs, is —•— ALEPH99(K&
2.S 3 BdO"3) Bs = (2.87 ±0.12)% (6.2) T"-»i?itVv, —O— CLEOM(KS) O— L3K(K°) corresponding to —O— CLEO9«(Kjl —0 ALEP!»8(Ks) —0— ALBPH98 (Ks) -*- AU3PH»
B (HP3) Since the QCD expectation for a massless quarks is 0.1809 ±0.0036, the result (6.3) is evidence for Figure 15: Some of the new ALEPH results on a massive s quark. decay modes with kaons compared to the published The strange spectral function is given in fig- data. ure 16, dominated at low mass by the K* reso- nance. Similarly to the nonstrange case, the QCD prediction is given by equations (4.2) and (4.3) where the attention is now turned to the • the '1+0'method singles out the well-behaved 6.1.2 The ALEPH analysis for m8 7=1 + 0 part by subtracting the exper- imentally determined J = 0 longitudinal As proposed in Ref. [64] and successfully applied component from data. The measurement in as(ml) analyses, the spectral function is used is then less inclusive and the sensitivity to to construct moments ms is significantly reduced; however, the 82 perturbative expansion is under control and the corresponding theoretical uncer- tainty is reduced. (6.4) 10 Michel Davier (6) In order to reduce the theoretical uncertain- <5 = 0.039 ± 0.016exp ± 0.014th (6.8) w ties one considers the difference between non- S = -0.021 ± 0.014exp ± 0.008th(6.9) strange and strange spectral moments, properly normalized with their respective CKM matrix el- with additional (smaller) uncertainties from the ements: fitting procedure and the determination of the J = 0 part. The quoted theoretical errors are 1 1 mostly from the Vu, uncertainty. The D = 6 \Vud and D = 8 strange contributions are found to be for which the massless perturbative contribution surprizingly larger than their nonstrange coun- vanishes so that the theoretical prediction now terparts. If the fully inclusive method is used reads (setting m = rrid = 0) instead (with the problematic convergence) the u 2 result ma{ml) = (149 ^t^) MeV/c is ob- tained. Akl oc I rfci(2-mass) AT = 3OEW I — 0s The result (6.7) can be evolved to the scale of 73=4,6,... 1 GeV using the four-loop RGE 7-function [66], (6.6) yielding For the CKM matrix elements the values \Vud\ = 2 2 0.9751 ± 0.0004 and \VUS\ = 0.2218 ± 0.0016 [65] m.(l GeV ) = (234t?J) MeV/c (6.10) are used, while the errors are included in the the- oretical uncertainties. This value of m« is somewhat larger than previous determinations [67], but consistent with Figure 17 shows the weighted integrand of 0 them within errors. the lowest moment A? 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 r -» vT + hadrons constitute a clean and powerful way to study hadronic physics up to y/s ~ 1.8 GeV. Beautiful resonance analyses have already been done, providing new insight ALEPH into hadron dynamics. Probably the major sur- prize has been the fact that inclusive hadron pro- duction is well described by perturbative QCD with very small nonperturbative components at the r mass. Despite the fact that this low-energy massless pQCD region is dominated by resonance physics, meth- ods based on global quark-hadron duality work indeed very well. The measurement of the vector and axial- s (GeV/c2)2 vector spectral functions has provided the way for quantitative analyses. Precise determinations Figure 17: Integrand of Eq. (6.5) for (k=0, 1=0), of aB agree for both spectral functions and they i.e., difference of the Cabibbo corrected non-strange also agree with all the other determinations from and strange invariant mass spectra. the Z width, the rate of Z to jets and deep in- elastic lepton scattering. Indeed from r decays Subtracting out the experimental J = 0 con- a (M%) = 0.1202 ±0.0027 tribution (essentially given by the single K chan- s T (7.1) nel), a fit to 5 moments is performed with the in excellent agreement with the average from all safer J = 1 + 0 method, yielding other determinations [68] ms(ml) = (6.7) a.{M%)non-r = 0.1187 ± 0.0020 (7.2) 11 Michel Davier The use of the vector r spectral function [14] D.M. Asncr et al. (CLEO Collaboration), and the QCD-based approach as tested in r de- CLNS 99/1635, CLEO 99-13. cays improve the calculations of hadronic vac- [15] ALEPH Collaboration, Eur.Phys.J. C4 (1998) uum polarization considerably. Significant re- 409. sults have been obtained for the running of a [16] OPAL Collaboration, CERN-EP/98-102, June to the Z mass and the muon anomalous mag- 1998. netic moment. Both of these quantities must be [17] E. Braaten, Phys. Rev. Lett./ 60 (1988) 1606. known with high precision as they give access to [18] S. Narison and A. Pidi, Phys. Lett./ B211 new physics. (1988) 183. Finally the strange spectral function has been [19] E. Braaten, S. Narison and A. Pich, Nucl. measured, providing a determination of the strange Phys. B373 (1992) 581. quark mass. [20] M.A. Shifman, A.L. Vainshtein and V.I. Za- kharov, Nucl. Phys. B147 (1979) 385, 448, Acknowledgments 519. [21] E. Braaten and C.S. Li, Phys. Rev. D42 (1990) The author would like to thank my ALEPH col- 3888. leagues S.M. Chen, A. Hocker and C.Z. Yuan [22] L.R. Surguladze and M.A. Samuel, Phys. for their precious collaboration, A. Weinstein for Rev. Lett. 66 (1991) 560; S.G. Gorishny, fruitful discussions on the CLEO data and the A.L. Kataev and S.A. Larin, Phys. Lett. B259 organizers of Heavy Flavours 8 for a nice confer- (1991) 144. ence. [23] F. Le Diberder and A. Pich, Phys. Lett. B286 (1992) 147. References [24] S.A. Larin, T. van Ritbergen and J.A.M. Ver- maseren, Phys. Lett. B400 (1997) 379; [1] W. Marciano and A. Sirlin, Phys. Rev. Lett. K.G. Chetyrkin, B.A. Kniehl and M. Stein- 61(1988)1815. hauser, Nucl. Phys. B51O (1998) 61; W. Bern- [2] R. Barate et al. (ALEPH Collaboration), Z. reuther, W. Wetzel, Nucl. Phys. B197 (1982) Phys. C76 (1997) 15. 228; W. Wetzel, Nucl. Phys. B196 (1982) 259; [3] S. Anderson et al. (CLEO Collaboration), hep- W. Bernreuther, PITHA-94-31 (1994). ex/9910046. [25] K.G. Chetyrkin, B.A. Kniehl and M. Stein- [4] G.J. Gounaris and J.J. Sakurai, Phys. Rev. hauser, Phys. Rev. Lett. 79 (1997) 2184; Lett. 21 (1968) 244. [26] G. Rodrigo, A. Pich, A. Santamaria, FTUV- [5] J.H. Kiihn and A. Santamaria, Z. Phys. C48 97-80 (1997). (1990) 445. [27] A. Blondel, in Rencontres de Blois Frontiers of [6] R.R. Akhmetshin et al., hep-ex/9904027. Matter, Blois (June 28-July 3 1999). [7] B.K. Heltsley, Nucl. Phys. B (Proc. Suppl.) 64 [28] M. Davier and A. Hocker, Phys. Lett. B419 (1998) 391. (1998) 419. [29] S. Groote, J.G. Korner, N.F. Nasrallah and [8] ALEPH Collaboration, hep-ex/9903015. K. Schilcher, Report MZ-TH-98-02 (1998). [9] S.I. Eidelman, Workshop on Lepton Moments, [30] M. Davier and A. Hdcker, Phys. Lett. B435 Heidelberg (June 10-12 1999). (1998) 427. [10] R. Alemany, M. Davier and A. Hocker, Europ. [31] C. Bacci et al. (772 Collaboration), Phys. Lett. Phys. J. C2 (1998) 123. B86 (1979) 234. [11] K.W. Edwards et al. (CLEO Collaboration), [32] J.L. Siegrist et al. (MARK I Collaboration), hep-ex/9908024. Phys. Rev. D26 (1982) 969. [12] D. Buskulic et al. (ALEPH Collaboration), Z. [33] Z. Jakubowski et al. (Crystal Ball Collabora- Phys. C74 (1997) 263. tion), Z. Phys. C40 (1988) 49; C. Edwards et [13] T.E. Browder et al. (CLEO Collaboration), al. (Crystal Ball Collaboration), SLAC-PUB- hep-ex/9908030. 5160 (1990). 12 Michel Davier [34] A.E. Blinov et al. (MD-1 Collaboration), Z. [56] M. Davier, Nucl. Phys. C (Proc. Suppl.) 55 Phys. C49 (1991) 239; A.E. Blinov et al. (MD- (1997) 395; S.M. Chen, Nucl. Phys. B (Proc. 1 Collaboration), Z. Phys. C70 (1996) 31. Suppl.) 64 (1998) 265; S.M. Chen, M. Davier [35] M. Gourdin and E. de Rafael, Nucl. Phys. BIO and A. Hocker, Nucl. Phys. B (Proc. Suppl.) (1969) 667; S.J. Brodsky and E. de Rafael, 76 (1999) 369. Phys. Rev. 168 (1968) 1620. [57] K.G. Chetyrkin and A. Kwiatkowski, Z. Phys. [36] N. Cabibbo and R. Gatto, Phys. Rev. Lett. 4 C59 (1993) 525. (1960) 313; Phys. Rev. 124 (1961) 1577. [58] K. Maltman, hep-ph/9804298. [37] S. Eidelman and F. Jegerlehner, Z. Phys. C67 [59] A. Pich and J. Prades, hep-ph/9804462. (1995) 585. [60] K.G. Chetyrkin, J.H. Kiihn and A.A. Pivo- [38] B. Krause, Phys. Lett. B390 (1997) 392. varov, hep-ph/9805335. [39] M. Hayakawa, T. Kinoshita, Phys. Rev. D57 [61] ALEPH Collaboration, Eur. Phys. J. Cl (1998) 465. (1998) 65. [40] J. Bijnens, E. Pallante and J. Prades, Nucl. [62] ALEPH Collaboration, Eur. Phys. J. C4 Phys. B474 (1996) 379. (1998) 29. [41] B.W. Lynn, G. Penso and C. Verzegnassi, [63] ALEPH Collaboration, hep-ex/9903014. Phys. Rev. D35 (1987) 42. [64] F. Le Diberder and A. Pich, Phys. Lett. B289 [42] H. Burkhardt and B. Pietrzyk, Phys. Lett. (1992) 165. B356 (1995) 398. [65] R.M. Barnett et al., Particle Data Group, Eur. [43] A.D. Martin and D. Zeppenfeld, Phys. Lett. Phys. J. C3 (1998) 1. B345 (1995) 558. [66] S.A. Larin, T. van Ritbergen and J.A.M. Ver- [44] M.L. Swartz, Phys. Rev. D53 (1996) 5268. maseren, Phys. Lett. B405 (1997) 327. [45] J.H. Kiihn and M. Steinhauser, MPI-PHT-98- [67] J. Gasser and H. Leutwyler, Phys. Rep. 87 12 (1998). (1982) 77; M. Jamin and M. Miinz, Z. Phys. [46] J. Erler, UPR-796-T, 1998. C66 (1995) 633; K.G. Chetyrkin et al., Phys. [47] L.M. Barkov et al. (OLYA, CMD Collabora- Lett. B404 (1997) 337; C. Becchi et al., Z. tion), Nucl Phys. B256 (1985) 365. Phys. C8 (1981) 335; C.A. Dominguez and E. de Rafael, Ann. Phys. 174 (1987) 372; [48] T. Kinoshita, B. Nizic and Y. Okamoto, Phys. C.A. Dominguez et al., Phys. Lett. B253 Rev. D31 (1985) 2108. (1991) 241; M. Jamin, Nucl. Phys. B (Proc. [49] J.A. Casas, C. L6pez and F.J. Yndurain, Phys. Suppl.) 64 (1998) 250; A.L. Kataev et al., Rev. D32 (1985) 736. Nuovo. Cim. A76 (1983) 723; P. Colangelo et [50] D.H. Brown and W.A. Worstell, Phys. Rev. al, Phys. Lett. B408 (1997) 340; S. Narison, D54 (1996) 3237. Phys. Lett. B216 (1989) 191; K.G. Chetyrkin et al., Phys. Rev. D51 (1995) 5090; S. Nar- [51] A. Czarnecki, B. Krause and W.J. Marciano, ison, Phys. Lett. B358 (1995) 113; K. Malt- Phys. Rev. Lett. 76 (1995) 3267; Phys. Rev. man, Phys. Lett. B428 (1998) 179; C.R. Allton D52 (1995) 2619. et al., Nucl. Phys. B431 (1994) 667; R. Gupta [52] T.V. Kukhto, E.A. Kuraev, A. Schiller and and T. Bhattacharya, Phys. Rev. D55 (1997) Z.K. Silagadze, Nucl. Phys. B371 (1992) 567. 7203; N. Eicker et al., SESAM-Collaboration, [53] R. Jackiw and S. Weinberg, Phys. Rev. D5 hep-lat/9704019; B.J. Gough et al., Phys. Rev. (1972) 2473. Lett. 79 (1994) 1662. [54] J. Bailey et al., Phys. Lett B68 (1977) 191; [68] M. Davier, '98 Rene. Moriond on Elec- F.J.M. Farley and E. Picasso, Advanced Series troweak Interactions, Ed. J. Tran Thanh Van, on Directions in High Energy Physics - Vol. 7 Frontieres, Paris (1998). Quantum Electrodynamics, ed. T. Kinoshita, World Scientific 1990. [55] B. Lee Roberts, Z. Phys. C56 (Proc. Suppl.) (1992) 101. 13