Selected Topics on Tau Physics ††thanks: Invited talk at the XXXI International Conference of Theoretical Physics, “Matter to the Deepest”, Ustroń, Poland, 5-11 September 2007
The B Factories have generated a large amount of new results on the lepton. The present status of some selected topics on physics is presented: charged-current universality tests, bounds on lepton-flavour violation, the determination of from the inclusive hadronic width, and the measurement of through the Cabibbo-suppressed decays of the lepton.
PACS numbers: 14.60.Fg, 13.35.-r, 12.15.-y, 12.38.-t
The known leptons provide clean probes to perform very precise tests of the Standard Model and search for signals of new dynamics. The electroweak gauge structure has been successfully tested at the 0.1% to 1% level, confirming the Standard Model framework . Moreover, the hadronic decays turn out to be a beautiful laboratory for studying strong interaction effects at low energies [2, 3, 4, 5]. The is the only known lepton massive enough to decay into hadrons. Its semileptonic decays are then ideally suited for studying the hadronic weak currents in very clean conditions. Accurate determinations of the QCD coupling, and the strange quark mass have been obtained with decay data.
The huge statistics accumulated at the B Factories allow to explore lepton-flavour-violating decay modes with increased sensitivities beyond , which could be further pushed down to few at future facilities. Moreover, BESIII will soon start taking data at the new Beijing Tau-Charm Factory. With the excellent experimental conditions of the threshold region, complementary information on the should be obtained, such as an improved mass measurement. Thus, physics is entering a new era, full of interesting possibilities and with a high potential for new discoveries.
2 Tests on charged-current universality
In the Standard Model all lepton doublets have identical couplings to the boson. Comparing the measured decay widths of leptonic or semileptonic decays which only differ in the lepton flavour, one can test experimentally that the interaction is indeed the same, i.e. that . As shown in Table 1, the present data verify the universality of the leptonic charged-current couplings to the 0.2% level.
The leptonic branching fractions and the lifetime are known with a precision of . A slightly improved lifetime measurement could be expected from BaBar and Belle . For comparison, the lifetime is already known with an accuracy of , which should be further improved to by the MuLan experiment at PSI . The universality tests require also a good determination of , which is only known to the level . Two new measurements of the mass have been published recently:
Belle  has made a pseudomass analysis of decays, while KEDR  measures the threshold production, taking advantage of a precise energy calibration through the resonance depolarization method. In both cases the achieved precision is getting close to the previous BES-dominated value, . KEDR aims to obtain a final accuracy of 0.15 MeV. A precision better than 0.1 MeV should be easily achieved at BESIII , through a detailed analysis of at threshold [12, 13, 14].
Table 1 shows also the contraints obtained from pion  and kaon decays , applying the recently calculated radiative corrections at NLO in chiral perturbation theory [16, 17]. The accuracy achieved with data is already comparable to the one obtained from or decays.
Owing to the limited statistics available, the decays only test universality at the 1% level. At present, is larger than . The stringent limits on from -mediated decays make unlikely that this is a real effect.
3 Lepton-flavour violating decays
We have now clear experimental evidence that neutrinos are massive particles and there is mixing in the lepton sector. The smallness of neutrino masses implies a strong suppression of neutrinoless lepton-flavour-violating processes, which can be avoided in models with other sources of lepton flavour violation, not related to . The scale of the flavour-violating new-physics interactions can be constrained imposing the requirement of a viable leptogenesis. Recent studies within different new-physics scenarios find interesting correlations between and lepton-flavour-violating decays, with often expected to be close to the present exclusion limit [21, 22, 23, 24, 25, 26, 27].
The B Factories are pushing the experimental limits on neutrinoless lepton-flavour-violating decays beyond the level [19, 20], increasing in a drastic way the sensitivity to new physics scales. Future experiments could push further some limits to the level , allowing to explore interesting and totally unknown phenomena. Complementary information will be provided by the MEG experiment, which will search for events with a sensitivity of . There are also ongoing projects at J-PARC aiming to study conversions in muonic atoms, at the  or even  level.
4 The inclusive hadronic width of the tau lepton
The hadronic decays of the lepton provide a very clean laboratory to perform precise tests of the Standard Model . The inclusive character of the total hadronic width renders possible an accurate calculation of the ratio [32, 33, 34, 35, 36]
using analyticity constraints and the Operator Product Expansion. One can separately compute the contributions associated with specific quark currents: and correspond to the Cabibbo-allowed decays through the vector and axial-vector currents, while contains the remaining Cabibbo-suppressed contributions.
The theoretical prediction for can be expressed as 
where is the number of quark colours and contains the electroweak radiative corrections [37, 38, 39]. The dominant correction () is the perturbative QCD contribution , which is fully known to  and includes a resummation of the most important higher-order effects [35, 40].
Non-perturbative contributions are suppressed by six powers of the mass  and, therefore, are very small. Their numerical size has been determined from the invariant-mass distribution of the final hadrons in decay, through the study of weighted integrals ,
which can be calculated theoretically in the same way as . The predicted suppression  of the non-perturbative corrections has been confirmed by ALEPH , CLEO  and OPAL . The most recent analysis  gives
The QCD prediction for is then completely dominated by the perturbative contribution; non-perturbative effects being smaller than the perturbative uncertainties from uncalculated higher-order corrections. The result turns out to be very sensitive to the value of , allowing for an accurate determination of the fundamental QCD coupling [33, 34]. The experimental measurement implies 
The strong coupling measured at the mass scale is significantly larger than the values obtained at higher energies. From the hadronic decays of the , one gets , which differs from by more than twenty standard deviations. After evolution up to the scale , the strong coupling constant in ( ‣ 4) decreases to 
in excellent agreement with the direct measurements at the peak and with a similar accuracy. The comparison of these two determinations of in two very different energy regimes, and , provides a beautiful test of the predicted running of the QCD coupling; i.e., a very significant experimental verification of asymptotic freedom.
5 determination from Cabibbo-suppressed tau decays
The separate measurement of the and tau decay widths provides a very clean determination of [47, 48]. To a first approximation the Cabibbo mixing can be directly obtained from experimental measurements, without any theoretical input. Neglecting the small SU(3)-breaking corrections from the quark-mass difference, one gets:
We have used , and the value , which results from the most recent BaBar  and Belle  measurements of Cabibbo-suppressed tau decays . The new branching ratios measured by BaBar and Belle are all smaller than the previous world averages, which translates into a smaller value of and . For comparison, we give in brackets the result obtained with the previous value .
This rather remarkable determination is only slightly shifted by the small SU(3)-breaking contributions induced by the strange quark mass. These corrections can be theoretically estimated through a QCD analysis of the differences [47, 48, 52, 53, 54, 55, 56, 57, 58, 59, 60]
Since the strong interactions are flavour blind, these quantities vanish in the SU(3) limit. The only non-zero contributions are proportional to powers of the quark mass-squared difference or to vacuum expectation values of SU(3)-breaking operators such as [53, 47]. The dimensions of these operators are compensated by corresponding powers of , which implies a strong suppression of :
The theoretical analysis of involves the two-point correlation functions of vector and axial-vector quark currents, which can be separated into their transverse () and longitudinal () components. The longitudinal contribution to shows a rather pathological behaviour, with clear signs of being a non-convergent perturbative series. Fortunately, the corresponding longitudinal contribution to can be estimated phenomenologically with a much better accuracy, [47, 62], because it is dominated by far by the well-known and contributions. To estimate the remaining transverse component, one needs an input value for the strange quark mass. Taking the range , which includes the most recent determinations of from QCD sum rules and lattice QCD , one gets finally , which implies 
Again, the first number is the updated value, including the recent BaBar and Belle data, while the number inside brackets gives the previous result.
Sizeable changes on the experimental determination of are to be expected from the full analysis of the huge BaBar and Belle data samples. In particular, the high-multiplicity decay modes are not well known at present and their effect has been just roughly estimated or simply ignored. Thus, the result ( ‣ 5) could easily fluctuate in the near future. However, it is important to realize that the final error of the determination from decay is completely dominated by the experimental uncertainties. If is measured with a 1% precision, the resulting uncertainty will get reduced to around 0.6%, i.e. , making decay the best source of information about .
An accurate experimental measurement of the invariant-mass distribution of the final hadrons in Cabibbo-suppressed decays could make possible a simultaneous determination of and the strange quark mass, through a correlated analysis of several weighted differences . The extraction of suffers from theoretical uncertainties related to the convergence of the perturbative series , which makes necessary a better understanding of these corrections. Further work in this direction is in progress .
I want to thank Michal Czakon, Henryk Czyż and Janusz Gluza for organizing an enjoyable conference. This work has been supported by the Spanish Ministry of Education (Grant FPA2004-00996), by Generalitat Valenciana (GVACOMP2007-156), by EU funds for regional development and by the EU MRTN-CT-2006-035482 (FLAVIAnet).
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