The recent ALEPH measurements of the inclusive Cabibbo--suppressed decay
width of the $\tau$ and several moments of its invariant mass distribution are
used to determine the value of the strange quark mass. We obtain, in the
$\bar{\rm MS}$ scheme, $m_s(M_\tau^2) = (119\pm 24)$ MeV, which corresponds to
$ m_s(1 {\rm GeV}^2) = (164 \pm 33) {\rm MeV}, m_s(4 {\rm GeV}^2) = (114\pm 23)
{\rm MeV} $.Comment: 5pages. Invited talk at QCD'99 (Montpellier, July 1999

Pich, A.

Prades, J.

English

arXiv

6 páginas, 3 tablas.-- Comunicación presentada al QCD 99 Euroconference celebrado en Julio de 1999 en Montpellier (Francia).-- arXiv:hep-ph/9909559v1The recent ALEPH measurements of the inclusive Cabibbo--suppressed decay width of the $\tau$ and several moments of its invariant mass distribution are used to determine the value of the strange quark mass. We obtain, in the $\bar{\rm MS}$ scheme, $m_s(M_\tau^2) = (119\pm 24)$ MeV, which corresponds to $ m_s(1 {\rm GeV}^2) = (164 \pm 33) {\rm MeV}, m_s(4 {\rm GeV}^2) = (114\pm 23) {\rm MeV} $.This work has been supported
in part by the European Union TMR Network
EURODAPHNE (Contract No. ERBFMXCT98-
0169), by CICYT, Spain, under grants No.
AEN-96/1672 and PB97-1261, and by Junta de
Andalucía, Grant No. FQM-101.Peer reviewe

Pich, Antonio

Prades, Joaquim

Digital.CSIC

arXiv:hep-ph/9909559v1  29 Sep 1999Tau–Decay Determination of the Strange Quark MassAntonio Picha and Joaquim PradesbaDepartament de F´ısica Teo`rica, IFIC, Universitat de Vale`ncia – CSIC,Dr. Moliner 50, E-46100 Burjassot (Vale`ncia), Spain∗bDepartamento de F´ısica Teo´rica y del Cosmos, Universidad de Granada,Campus de Fuente Nueva, E-18002 Granada, SpainThe recent ALEPH measurements of the inclusive Cabibbo–suppressed decay width of the τ and several momentsof its invariant mass distribution are used to determine the value of the strange quark mass. We obtain, in theMS scheme, ms(M2τ) = (119 ± 24) MeV, which corresponds to ms(1GeV2) = (164 ± 33) MeV, ms(4 GeV2) =(114± 23) MeV.FTUV/99-63IFIC/99-661. INTRODUCTIONThe precise numerical value of the strangequark mass is a controversial issue, with impor-tant implications for low–energy phenomenology.The Particle Data Group [1] quotes a rather widerange of ms values, reflecting the large uncertain-ties in the present determinations of this param-eter from QCD Sum Rules and Lattice calcula-tions.The high precision data on tau decays [2] col-lected at LEP and CESR provide a very powerfultool to analyse strange quark mass effects in acleaner environment. The QCD analysis of theinclusive tau decay width,Rτ ≡Γ[τ− → ντ + hadrons (γ)]Γ[τ− → e− νe ντ (γ)] , (1)has already made possible [3] an accurate mea-surement of the strong coupling constant at theτ mass scale, αs(M2τ ), which complements andcompetes in accuracy with the high precisionmeasurements of αs(M2Z) performed at LEP.More recently, detailed experimental studies ofthe Cabibbo–suppressed width of the τ havestarted to become available [4,5], allowing to per-form a systematic investigation of the correctionsinduced by the strange quark mass in the τ decaywidth [6–8].What makes a ms determination from τ datavery interesting is that the hadronic input does∗Invited talk at QCD’99, Montpellier, July 1999not depend on any extra hypothesis; it is a purelyexperimental issue, which accuracy can be sys-tematically improved. The major part of the un-certainty will eventually come from the theoret-ical side. However, owing to its inclusive char-acter, the total Cabibbo–suppressed tau decaywidth can be rigorously analyzed within QCD,using the Operator Product Expansion (OPE).Therefore, the theoretical input is in principle un-der control and the associated uncertainties canbe quantified.2. THEORETICAL FRAMEWORKThe theoretical analysis of the inclusivehadronic tau decay width [9–12] involves the two–point correlation functionsΠµνij,J (q) ≡ i∫d4x eiqx 〈0|T(J µij(x)Jνij(0)†)|0〉for the vector, J µij = Vµij (x) ≡ qjγµqi, and axial–vector, J µij = Aµij(x) ≡ qjγµγ5qi, colour–singletquark currents (i, j = u, d, s). These correlatorshave the Lorentz decompositionsΠµνij,V/A(q) =(−gµν q2 + qµqν)ΠTij,V/A(q2)+ qµqν ΠLij,V/A(q2) , (2)where the superscript in the transverse and lon-gitudinal components denotes the correspondingangular momentum J = 1 (T) and J = 0 (L) inthe hadronic rest frame.(k, l) FklL+T (x) FklL (x)(0,0) (1− x)3 (1 + x) (1− x)3(1,0) 110 (1− x)4 (7 + 8x) 34 (1− x)4(2,0) 215 (1− x)5 (4 + 5x) 35 (1− x)5(1,1) 16 (1− x)4 (1 + 2x)2 320 (1 − x)4 (1 + 4x)(1,2) 1210 (1− x)4 (13 + 52x+ 130x2 + 120x3) 120 (1− x)4 (1 + 4x+ 10x2)Table 1Explicit values of the relevant kinematical kernels.The semi-hadronic decay rate of the τ lepton,can be expressed as an integral of the spectralfunctions ImΠT (s) and ImΠL(s) over the invari-ant mass s of the final–state hadrons as follows:Rτ = 12π∫ M2τ0dsM2τ(1−sM2τ)2×[(1 + 2sM2τ)ImΠT (s) + ImΠL(s)]. (3)Moreover, according to the quantum numberscontent of the two–point function correlatorsΠJ (s) ≡ |Vud|2[ΠJV,ud(s) + ΠJA,ud(s)]+ |Vus|2[ΠJV,us(s) + ΠJA,us(s)], (4)we can decompose Rτ intoRτ ≡ Rτ,V +Rτ,A +Rτ,S , (5)where Rτ,V and Rτ,A correspond to the first twoterms in Eq. (4), while Rτ,S contains the remain-ing Cabibbo–suppressed contributions.The measurement of the invariant mass distri-bution of the final hadrons provides additionalinformation on the QCD dynamics, through themoments [12]Rklτ ≡∫ M2τ0ds(1−sM2τ)k (sM2τ)ldRτds, (6)which include Rτ ≡ R00τ as a particular case.Exploiting the analytic properties of ΠJ (s), wecan express these moments as contour integralsin the complex s-plane running counter-clockwisearound the circle |s| = M2τ :Rklτ = −πi∮|x|=1dxx{3FklL+T (x)DL+T (M2τ x)+ 4FklL (x)DL(M2τ x)}. (7)We have used integration by parts to rewrite Rklτin terms of the logarithmic derivativesDL+T (s) ≡ −sdds[ΠL+T (s)], (8)DL(s) ≡sM2τdds[sΠL(s)], (9)which satisfy homogeneous renormalization groupequations. All kinematical factors have been ab-sorbed into the kernels FklL+T (x) and FklL (x). Ta-ble 1 shows the explicit form of these kernels forthe moments which we are going to analyze in thefollowing sections.For large enough −s, the contributions toDJ (s) can be organized with the OPE in a se-ries of local gauge–invariant scalar operators ofincreasing dimension D = 2n, times the appro-priate inverse powers of −s. This expansion isexpected to be well behaved along the complexcontour |s| = M2τ , except in the crossing pointwith the positive real axis [13]. As shown in Ta-ble 1, the region near the physical cut is stronglysuppressed by a zero of order 3 + k at s = M2τ .Therefore, the uncertainties associated with theuse of the OPE near the time–like axis are verysmall. Inserting this series in (7) and evaluatingthe contour integral, one can rewrite Rklτ as anexpansion in inverse powers of M2τ [9],Rklτ ≡ 3[|Vud|2 + |Vus|2]SEW{1 + δ′EW + δkl (0)+∑D=2,4,···(cos2 θC δkl (D)ud + sin2 θC δkl (D)us)},where sin2 θC ≡ |Vus|2/[|Vud|2 + |Vus|2] andwe have pulled out the electroweak correctionsSEW = 1.0194 [14] and δ′EW ≃ 0.0010 [15].The dimension–zero contribution δkl (0) is thepurely perturbative correction, neglecting quarkmasses, which, owing to chiral symmetry, is iden-tical for the vector and axial–vector parts. Thesymbols δkl (D)ij ≡ [δkl (D)ij,V + δkl (D)ij,A ]/2 stand forthe average of the vector and axial–vector con-tributions from dimension D ≥ 2 operators; theycontain an implicit suppression factor 1/MDτ .3. SU(3) BREAKINGThe separate measurement of the Cabibbo–allowed and Cabibbo–suppressed decay widths ofthe τ [4] allows one to pin down the SU(3) break-ing effect induced by the strange quark mass,through the differencesδRklτ ≡Rklτ,V+A|Vud|2−Rklτ,S|Vus|2= 3SEW∑D≥2[δkl (D)ud − δkl (D)us]. (10)The leading contributions to δRklτ are quark–mass corrections of dimension two [6,7]; they arethe dominant SU(3) breaking effect, generatingthe wanted sensitivity to the strange quark mass.The corrections of O(m4) are very tiny [7]. Themain D = 4 contribution comes from the SU(3)–breaking quark condensateδO4 ≡ 〈0|ms s¯s−md d¯d|0〉 . (11)Neglecting the small O(m4) terms and D ≥ 6contributions, δRklτ can be written as [7]:δRklτ ≈ 24SEW{m2s(M2τ )M2τ(1− ǫ2d)∆(2)kl (aτ )−2π2δO4M4τQkl(aτ )}, (12)where ǫd ≡ md/ms = 0.053± 0.002 [16] and aτ ≡αs(M2τ )/π.The perturbative QCD expansions ∆(2)kl (aτ )and Qkl(aτ ) are known to O(a2τ ). Moreover, theO(a3τ ) contributions to ∆(2)kl (aτ ) coming from thelongitudinal correlatorDL(s) have been also com-puted. Using the value of the (MS) strong cou-pling determined by the total hadronic τ decaywidth [3], αs(M2τ ) = 0.35 ± 0.02, one gets thenumerical results shown in Table 2 [7].(k, l) ∆(2)kl (aτ ) Qkl(aτ )(0,0) 2.0± 0.5 1.08± 0.03(1,0) 2.4± 0.7 1.52± 0.03(2,0) 2.7± 1.0 1.93± 0.02(1,1) −0.39± 0.26 −0.41± 0.02(1,2) 0.07± 0.06 −0.02± 0.01Table 2Numerical values [7] of the relevant perturbativeexpansions for αs(M2τ ) = 0.35± 0.02.The rather large theoretical uncertainties of∆(2)kl (aτ ) ≡14{3∆L+Tkl (aτ ) + ∆Lkl(aτ )}, (13)have their origin in the bad perturbative be-haviour of the longitudinal contribution. Themost important higher–order corrections can beresummed [6], using the renormalization group,but the resulting “improved” series is still ratherbadly behaved. For instance,∆L00(0.1) = 1.5891+1.1733+1.1214+1.2489+ · · ·which has O(a2) and O(a3) contributions of thesame size. On the contrary, the J = L+ T seriesconverges very well:∆L+T00 (0.1) = 0.7824 + 0.2239 + 0.0831 + · · ·Fortunately, the longitudinal contribution to∆(2)kl (aτ ) is parametrically suppressed by a factor1/3. Thus, the combined final expansion looksstill acceptable for the first few terms:∆(2)00 (0.1) = 0.9840 + 0.4613 + 0.3427+(0.3122− 0.000045 cL+T3)+ · · · (14)Nevertheless, after the third term the series ap-pears to be dominated by the longitudinal contri-bution, and the bad perturbative behaviour be-comes again manifest. Taking the unknownO(a3)coefficient of the DL+T (s) perturbative series ascL+T3 ∼ cL+T2(cL+T2 /cL+T1)≈ 323, the fourthterm becomes 0.298; i.e. a 5% reduction only.Since the longitudinal series seems to reach anasymptotic behaviour at O(a3), the central val-ues of ∆(2)kl (aτ ) have been evaluated adding tothe fully known O(a2) result one half of the lon-gitudinal O(a3) contribution. To estimate the as-sociated theoretical uncertainties, we have takenone half of the size of the last known perturba-tive contribution plus the variation induced by achange of the renormalization scale in the rangeξ ∈ [0.75, 2] (added in quadrature).The SU(3)–breaking condensate δO4 could beextracted from the τ decay data, together withms, through a combined fit of different δRklτ mo-ments. However, this is not possible with theactual experimental accuracy. We can estimatethe value of δO4 using the constraints providedby chiral symmetry. To lowest order in ChiralPerturbation Theory, δO4 is fully predicted interms of the pion decay constant and the pionand kaon masses: δO4 ≃ −f2pi(m2K −m2pi)≃−1.9× 10−3GeV4. Taking into account the lead-ing O(p4) corrections through the ratio of quarkvacuum condensates [17,18]vs ≡〈0|ss|0〉〈0|dd|0〉= 0.8± 0.2 , (15)one gets the improved estimate,δO4 ≃ −ms2mˆ(vs − ǫd) f2pim2pi≃ −(1.5± 0.4)× 10−3 GeV4 , (16)where we have used the known quark mass ratio[16] ms/mˆ = 24.4± 1.5.Strictly speaking, δO4 and vs are scale depen-dent. This dependence cancels with the O(m4)contributions [7] and is then of O(p8) in the chiralexpansion. The numerical effect is smaller thanthe accuracy of (16) and has been neglected to-gether with the tiny O(m4) corrections.4. NUMERICAL ANALYSISThe ALEPH collaboration has measured [4] theweighted differences δRklτ for five different valuesof (k, l). The experimental results are shown inTable 3, together with the correspondingms(M2τ )values. Since the QCD counterparts to the mo-ments (k, l) = (1,1) and (1,2) have theoretical un-certainties larger than 100%, we only use the mo-ments (k, l) = (0,0), (1,0), and (2,0).The experimental errors quoted in Table 3 donot include the present uncertainty in |Vus|. To(k, l) δRklτ ms(M2τ ) (MeV)(0,0) 0.394± 0.137 143± 31± 18(1,0) 0.383± 0.078 121± 17± 18(2,0) 0.373± 0.054 106± 12± 21(1,1) 0.010± 0.029 –(1,2) 0.006± 0.015 –Table 3Measured [4] moments δRklτ and correspondingms(M2τ ) values [7]. The first error is experimentaland the second theoretical.estimate the corresponding error in ms, we takethe following numbers from ALEPH: R00τ,V+A =3.486 ± 0.015, R00τ,S = 0.1610 ± 0.0066, |Vud| =0.9751±0.0004 and |Vus| = 0.2218±0.0016. Thisgives δR00τ = 0.394 ± 0.135 ± 0.047, where thesecond error comes from the uncertainty in |Vus|and translates into an additional uncertainty of10 MeV in the strange quark mass. We will putthe same |Vus| uncertainty to the other two mo-ments, for which the ALEPH collaboration doesnot quote the separate values ofRklτ,V+A andRklτ,S .Taking the information from the three mo-ments into account, we get our final result [7]:ms(M2τ ) = (119± 12± 18± 10)MeV= (119± 24)MeV . (17)The first error is experimental, the second reflectsthe QCD uncertainty and the third one is fromthe present uncertainty in |Vus|. Since the threemoments are highly correlated, we have taken thesmaller individual errors as errors of the final av-erage. Our determination (17) corresponds toms(1GeV2) = (164± 33) MeV (18)andms(4GeV2) = (114± 23) MeV . (19)5. COMPARISON WITH ALEPHThe ALEPH collaboration has performed aphenomenological analysis of the δRklτ momentsin Table 3, which results in larger ms values [4]:ms(M2τ ) ={149+24exp−30exp+21th−25th± 6fit MeV,176+37exp−48exp+24th−28th± 8fit ± 11J=0 MeV.To derive these numbers, ALEPH has used ourpublished results in refs. [6], [9] and [12]. Sincewe have analyzed the same data with improvedtheoretical input [7], it is worthwhile to under-stand the origin of the numerical difference.ALEPH makes a global fit to the five measuredmoments, including the last two which are unre-liable (100% theoretical errors). In view of theasymptotic behaviour of ∆Lkl(aτ ), they truncatethis perturbative series at O(aτ ), neglecting theknown and positive O(a2τ ) and O(a3τ ) contribu-tions. Thus, they use a smaller value of ∆(2)kl (aτ )and, therefore, get a larger result for ms (the firstvalue above) because the sensitivity to this pa-rameter is through the product m2s(M2τ )∆(2)kl (aτ ).Since they put rather conservative errors, theirresult is nevertheless consistent with ours.ALEPH has made a second analysis subtract-ing the J = L contribution. Unfortunately,only the pion and kaon contributions are known.Using the positivity of the longitudinal spectralfunctions, this pole contributions provide lowerbounds on ImΠLud(s) and ImΠLus(s), which trans-late into lower limits on the corresponding J = Lcontribution to δRklτ . Subtracting this contribu-tion, one gets upper bounds on δRklτ,L+T [7] whichimply ms(M2τ ) < 202 MeV [7].However, besides subtracting the pion and kaonpoles, ALEPH makes a tiny ad-hoc correction toaccount for the remaining unknown J = L con-tribution, and quotes the resulting number as ams(M2τ ) determination [the second value above].Since they add a generous uncertainty, their num-ber does not disagree with ours. However, it isactually an upper bound on ms(M2τ ) and not adetermination of this parameter.ACKNOWLEDGEMENTSWe have benefit from useful discussions withShaomin Chen, Michel Davier, Andreas Ho¨ckerand Edwige Tournefier. This work has been sup-ported in part by the European Union TMR Net-work EURODAPHNE (Contract No. ERBFMX-CT98-0169), by CICYT, Spain, under grants No.AEN-96/1672 and PB97-1261, and by Junta deAndaluc´ıa, Grant No. FQM-101.REFERENCES1. C. Caso et al., Eur. Phys. J. C 3 (1998) 1.2. Proceedings of the Fifth Workshop on TauLepton Physics –TAU 98– (Santander, 14–17September 1998), eds. A. Pich and A. Ruiz,Nucl. Phys. B (Proc. Suppl.) 76 (1999).3. A. Pich, Tau Physics, in “Heavy Flavours II”,eds. A.J. Buras and M. Lindner, AdvancedSeries on Directions in High Energy Physics– Vol. 15 (World Scientific, Singapore, 1998),p. 453 [hep-ph/9704453].4. R. Barate et al. (ALEPH), hep-ex/9903015;hep-ex/9903014;E. Tournefier, these proceedings.5. A. Ho¨cker, Nucl. Phys. B 55C (Proc. Suppl.)(1997) 379; M. Davier, Nucl. Phys. B 55C(Proc. Suppl.) (1997) 395; S. Chen, Nucl.Phys. B 64 (Proc. Suppl.) (1998) 256; S.Chen, M. Davier, and A. Ho¨cker, Nucl. Phys.B 76 (Proc. Suppl.) (1999) 369.6. A. Pich and J. 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Tau Decay Determination of the Strange Quark Mass

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