QCD Description of Hadronic Tau Decays

0.19 MB·English

by  Antonio Pich

1 QCD Description of Hadronic Tau Decays
1
0 A. Picha
2
aDepartament de F´ısica Teo`rica, IFIC, Univ. Val`encia–CSIC, Apt. 22085, E-46071Val`encia, Spain
n
a
J The QCD analysis of hadronic τ decays is reviewed and a summary of the present phenomenological status is
1 presented. The following topics are discussed: the determination of αs(m2τ)=0.338±0.012 from the inclusive τ
1 hadronicwidth,themeasurement of|Vus|throughtheCabibbo-suppressed decaysoftheτ,andtheextractionof
chiral-perturbation-theory couplings from thespectral tau data.

]
h
p
-
p
1. THEORETICAL FRAMEWORK invariant mass s of the final-state hadrons:
e
h The hadronic τ decays turn out to be a beau- m2τ ds s 2
[ R = 12π 1−
tiful laboratory for studying strong interaction τ m2 m2
1 Z0 τ (cid:18) τ(cid:19)
effects at low energies [1–3]. The τ is the s
v × 1+2 ImΠ(1)(s)+ImΠ(0)(s) . (3)
7 only known lepton massive enough to decay into m2
0 hadrons. Its semileptonic decays are then ideally (cid:20)(cid:18) τ(cid:19) (cid:21)
The appropriate combinations of correlators are
1 suited to investigate the hadronic weak currents.
2 The inclusive character of the total τ hadronic Π(J)(s) ≡ |V |2 Π(J) (s)+Π(J) (s)
1. width renders possible an accurate calculation of ud ud,V ud,A
0 the ratio [4–8] + |V |2 (cid:16)Π(J) (s)+Π(J) (s)(cid:17). (4)
1 us us,V us,A
1 Γ[τ− →ντhadrons] Thecontributions(cid:16)comingfromthefirstt(cid:17)woterms
v: Rτ ≡ Γ[τ− →ντe−ν¯e] = Rτ,V+Rτ,A+Rτ,S. correspond to Rτ,V and Rτ,A respectively, while
i R contains the remaining Cabibbo-suppressed
X τ,S
The theoretical analysis involves the two-point contributions.
ar correlationfunctionsforthevector Viµj =ψ¯jγµψi The integrand in Eq. (3) cannot be calculated
and axial-vector Aµ = ψ¯ γµγ ψ colour-singlet atpresentfromQCD.Neverthelesstheintegralit-
ij j 5 i
quark currents (i,j =u,d,s): selfcanbecalculatedsystematicallybyexploiting
the analytic propertiesofthe correlatorsΠ(J)(s).
Πµν (q)≡i d4xeiqxh0|T(Jµ(x)Jν(0)†)|0i,(1) They are analytic functions of s except along the
ij,J ij ij
positive real s-axis, where their imaginary parts
Z
havediscontinuities. R canthen be writtenas a
which have the Lorentz decompositions τ
contour integral in the complex s-plane running
Πµijν,J(q) = −gµνq2+qµqν Π(ij1,)J(q2) counter-clockwise around the circle |s|=m2τ [6]:
(cid:0)+qµqνΠ(0) (q2)(cid:1), (2) ds s 2
ij,J R = 6πi 1−
τ m2 m2
where the superscript (J = 0,1) denotes the an- I|s|=m2τ τ (cid:18) τ(cid:19)
s s
gular momentum in the hadronic rest frame. × 1+2 Π(0+1)(s)−2 Π(0)(s) .(5)
The imaginary parts of Π(J) (q2) are pro- (cid:20)(cid:18) m2τ(cid:19) m2τ (cid:21)
ij,J
portional to the spectral functions for hadrons This expression requires the correlators only
with the corresponding quantum numbers. The for complex s of order m2, which is signifi-
τ
hadronic decay rate of the τ can be written as cantly larger than the scale associated with non-
an integral of these spectral functions over the perturbativeeffects. UsingtheOperatorProduct
1
2
Expansion (OPE), Π(J)(s) = C(J)/(−s)D/2, firmed by ALEPH [17], CLEO [18] and OPAL
D D
to evaluate the contour integral, R can be ex- [19]. The most recent analysis gives [20]
τ
pressed as an expansion in powPers of 1/m2. The
τ δ = −0.0059±0.0014. (8)
uncertainties associatedwith the use of the OPE NP
near the time-like axis are heavily suppressed by The QCD prediction for R is then com-
τ,V+A
the presence in (5) of a double zero at s=m2. pletelydominatedbyδ ;non-perturbativeeffects
τ P
In the chiral limit (m = 0), the vector being smaller than the perturbative uncertain-
u,d,s
and axial-vector currents are conserved. This ties from uncalculated higher-order corrections.
implies sΠ(0)(s) = 0. Therefore, only the cor- Assuming lepton universality, the measured val-
relator Π(0+1)(s) contributes to Eq. (5). Since ues of the τ lifetime and leptonic branching ra-
(1 − x)2(1+ 2x) = 1− 3x2 +2x3 [x ≡ s/m2], tios imply R = 3.6291±0.0086 [21]. Subtract-
τ τ
Cauchy’s theorem guarantees that, up to tiny ing the Cabibbo-suppressed contribution R =
τ,S
logarithmic running corrections, the only non- 0.1613 ± 0.0028 [21], one obtains R =
τ,V+A
perturbative contributions to the circle integra- 3.4678±0.0090. Using |V |=0.97425±0.00022
ud
tionin(5)originatefromoperatorsofdimensions [22] and (8), the pure perturbative contribution
D = 6 and 8. The usually leading D = 4 opera- to R is determined to be:
τ
torscanonlycontributetoR withanadditional
τ δ =0.1997±0.0035. (9)
suppression factor of O(α2), which makes their P
s
effect negligible [6]. The predicted value of δ turns out to be very
P
sensitive to α (m2), allowing for an accurate de-
s τ
α
2. DETERMINATION OF s termination of the fundamental QCD coupling
[5,6]. The calculation of the O(α4) contribu-
s
TheCabibbo-allowedcombinationRτ,V+A can tion [12] has triggered a renewed theoretical in-
be written as [6] terest on the α (m2) determination, since it al-
s τ
lows to push the accuracy to the four-loop level.
R = N |V |2S {1+δ +δ }, (6)
τ,V+A C ud EW P NP However,asshowninTable1,therecenttheoret-
ical analyses slightly disagree on the final result.
whereN =3isthenumberofquarkcoloursand
C
ThedifferencesarelargerthantheclaimedO(α4)
SEW = 1.0201±0.0003 contains the electroweak s
accuracy and originate in the different inputs or
radiative corrections [9–11]. The dominant cor-
theoretical procedures which have been adopted.
rection (∼ 20%) is the perturbative QCD con-
tribution δ , which is already known to O(α4)
P s 2.1. Perturbative contribution to R
τ
[6,12]. Quark mass effects [6,13,14] are tiny
In the chiral limit, the result is more con-
for the Cabibbo-allowed current and amount to
veniently expressed in terms of the logarithmic
a negligible correction smaller than 10−4 [6,15].
derivative of the two-point correlation function
Non-perturbative contributions are suppressed of the vector (axial) current, Π(s)= 1Π(0+1)(s),
by six powers of the τ mass [6] and, therefore, 2
which satisfies an homogeneous renormalization–
are very small. Their numerical size has been
group equation:
determined from the invariant-mass distribution
ofthefinalhadronsinτ decay,throughthestudy d 1 αs(−s) n
D(s)≡−s Π(s)= K .(10)
of weighted integrals [16], ds 4π2 n π
n=0 (cid:18) (cid:19)
X
Rτkl ≡ m2τ ds 1− ms2 k ms2 l ddRsτ , (7) Wallitlhogtahreitchhmoiiccecoofrrreecntoiormnsa,lipzraotpioonrtsiocanlaelµt2o=po−ws-
Z0 (cid:18) τ(cid:19) (cid:18) τ(cid:19) ers of log(−s/µ2), have been summed into the
whichcanbe calculatedtheoreticallyinthe same running coupling. For three flavours, the known
way as R , but are more sensitive to OPE cor- coefficients take the values: K = K = 1; K =
τ 0 1 2
rections. The predicted suppression [6] of the 1.63982; K (MS) = 6.37101 and K (MS) =
3 4
non-perturbativecorrectionsto R hasbeencon- 49.07570 [12].
τ
3
Table 1
O(α4) determinations of α (m2). The assumed value of δ is also given (if quoted by the authors).
s s τ P
Reference Method δ α (m2) α (M2)
P s τ s Z
Baikov et al. [12] CIPT, FOPT 0.1998±0.0043 0.332±0.016 0.1202±0.0019
Davier et al. [23] CIPT 0.2066±0.0070 0.344±0.009 0.1212±0.0011
Beneke-Jamin [15] BSR + FOPT 0.2042±0.0050 0.316±0.006 0.1180±0.0008
Maltman-Yavin [24] PWM + CIPT — 0.321±0.013 0.1187±0.0016
Menke [25] CIPT, FOPT 0.2042±0.0050 0.342+0.011 0.1213±0.0012
−0.010
Caprini-Fischer [26] BSR + CIPT 0.2042±0.0050 0.320+0.011 —
−0.009
Cvetiˇc et al. [27] β + CIPT 0.2040±0.0040 0.341±0.008 0.1211±0.0010
exp
Pich [1] CIPT 0.2038±0.0040 0.342±0.012 0.1213±0.0014
Table 2
Exact results for A(n)(α ) (n ≤ 4) at different β-function approximations, and corresponding values of
s
δ = 4 K A(n)(α ), for a ≡α (m2)/π =0.11. The last row shows the FOPT estimates at O(a4).
P n=1 n s τ s τ τ
A(1)(α ) A(2)(α ) A(3)(α ) A(4)(α ) δ
P s s s s P
β =0 0.14828 0.01925 0.00225 0.00024 0.20578
n>1
β =0 0.15103 0.01905 0.00209 0.00020 0.20537
n>2
β =0 0.15093 0.01882 0.00202 0.00019 0.20389
n>3
β =0 0.15058 0.01865 0.00198 0.00018 0.20273
n>4
O(a4) 0.16115 0.02431 0.00290 0.00015 0.22665
τ
The perturbative component of R is given by 4 K A(n)(α ), taking a = 0.11. The per-
τ n=1 n s τ
turbativeconvergenceisverygoodandtheresults
δP = KnA(n)(αs), (11) Pare stable under changes of the renormalization
n=1 scale. The errorinduced by the truncationof the
X
where the functions [7] β function at fourth order can be conservatively
estimated through the variation of the results at
A(n)(α ) = 1 ds αs(−s) n five loops, assuming β5 =±β42/β3 =∓443, i.e. a
s 2πi s π geometric growth of the β function.
I|s|=m2τ (cid:18) (cid:19)
Higher-order contributions to the Adler func-
s s3 s4
× 1−2 +2 − (12) tion D(s) will be taken into account adding the
m2 m6 m8
(cid:18) τ τ τ(cid:19) fifth-ordertermK A(5)(α )withK =275±400.
5 s 5
arecontourintegralsinthe complexplane,which Moreover, we will include the 5-loop variation
only depend on a ≡ α (m2)/π. Using the with changes of the renormalization scale in the
τ s τ
exact solution (up to unknown β contribu- range µ2/(−s) ∈ [0.5,1.5]. Adopting this very
n>4
tions) for α (−s) given by the renormalization- conservativeprocedure,theexperimentalvalueof
s
group β-function equation, they can be numer- δP given in Eq. (9) implies
ically computed with a very high accuracy [7].
α (m2)=0.338±0.012. (13)
Table 2 gives the numerical values for A(n)(α ) s τ
s
(n ≤ 4) obtained at the one-, two-, three- and The result is slightly lower than the one given in
four-loop approximations (i.e. β =0, β = Ref. [1], due to the smaller value of δ .
n>1 n>2 P
0, β = 0 and β = 0, respectively), to- The strong coupling measured at the τ mass
n>3 n>4
gether with the corresponding results for δ = scale is significantly larger than the values ob-
P
4
tained at higher energies. From the hadronic de- which is only convergent for a < 0.14. At
τ
caysoftheZ,onegetsα (M2)=0.1190±0.0027 the four-loop level the radius of convergence is
s Z
[28], which differs from α (m2) by 18σ. After slightly smaller than the physical value of a .
s τ τ
evolutionuptothescaleM [29],thestrongcou- Thus, FOPT gives rise to a pathological non-
Z
pling constant in (13) decreases to convergent series. The long running along the
circle makes compulsory to resum the large loga-
αs(MZ2) = 0.1209±0.0014, (14) rithms, logn(−s/m2), using the renormalization
τ
group. This is precisely what CIPT does.
in excellent agreement with the direct measure-
ments at the Z peak and with a better accuracy.
2.3. Renormalon hypothesis
Thecomparisonofthesetwodeterminationsofα
s The perturbative expansion of the Adler func-
intwoverydifferentenergyregimes,m andM ,
τ Z tion is expected to be an asymptotic series. If its
providesabeautifultestofthe predictedrunning Borel transform, B(t) ≡ K tn/n!, were
of the QCD coupling; i.e., a very significant ex- n=0 n+1
well-behaved, one could define D(s) through the
perimental verification of asymptotic freedom. P
Borel integral
2.2. Fixed-order perturbation theory 1 ∞
The integrals A(n)(α ) can be expanded in D(s)= 1+ dte−πt/αs(s)B(t) . (17)
s 4π2
powers of a , A(n)(α ) = an + O(an+1). One (cid:26) Z0 (cid:27)
τ s τ τ
recovers in this way the naive perturbative ex- However, B(t) has pole singularities at positive
pansion [7] (infrared renormalons) and negative (ultravio-
let renormalons) integer values of the variable
δP = (Kn+gn)anτ ≡ rnanτ . (15) u ≡ −β1t/2, with the exception of u = 1 [31].
nX=1 nX=1 The infrared renormalons at u = +n are related
This approximation is known as fixed-order per- to OPE corrections of dimension D = 2n. The
turbation theory (FOPT), while the improvedex- renormalon poles closer to the origin dominate
pression(11),keepingthenon-expandedvaluesof the large-order behaviour of D(s).
A(n)(α ), is usually called contour-improved per- Ithasbeenarguedthat,onceintheasymptotic
s
turbation theory (CIPT) [7,30]. regime (large n), the renormalonic behaviour
As shown in the last row of Table 2, even at of the Kn coefficients could induce cancelations
O(a4τ), FOPT gives a rather bad approximation with the running gn corrections, which would be
to the integrals A(n)(α ), overestimating δ by missed by CIPT. In that case, FOPT could ap-
s P
12% at a = 0.11. The long running of α (−s) proach faster the ‘true’ result provided by the
τ s
along the circle |s| = m2 generates very large g Borel summation of the full renormalon series
τ n
coefficients, which depend on K and β (BSR) [15]. This happens actually in the large–
m<n m<n
[7]: g1 = 0, g2 = 3.56, g3 = 19.99, g4 = 78.00, β1 limit[32,33],whichhoweverdoesnotapproxi-
g5 = 307.78. These corrections are much larger mate well the known Kn coefficients. A model of
than the original K contributions and lead to higher-order corrections with this behaviour has
n
valuesofα (m2)smallerthan(13). FOPTsuffers been recently advocated [15]. The model mixes
s τ
fromalargerenormalization-scaledependence[7] three different types of renormalons (n = −1, 2
and its actual uncertainties are much largerthan and3)plusalinearpolynomial. Itcontains5free
usually estimated [25]. parameters which are determined by the known
The origin of this bad behaviour can be un- values of K1,2,3,4 and the assumption K5 = 283.
derstood analytically at one loop [7]. In FOPT One gets in this way a larger δP, implying a
one makes within the contour integral the series smaller value for αs(m2τ). The result looks how-
expansion (log(−s/m2)=iφ, φ∈[−π,π]) ever model dependent [34].
τ
The implications of a renormalonic behaviour
n
αs(−s) aτ i have been put on more solid grounds, using an
≈ ≈a β a φ ,(16)
τ 1 τ
π 1−iβ a φ/2 2 optimal conformal mapping in the Borel plane,
1 τ n (cid:18) (cid:19)
X
5
whichachievesthebestasymptoticrateofconver- a different estimate of the non-perturbative con-
gence,andproperlyimplementing the CIPTpro- tributions. Unfortunately, Ref. [24] does not
cedurewithintheBoreltransform[26]. Assuming quote any explicit values for δ and δ . From
NP P
thattheknownfourth-orderseriesisalreadygov- the information given in that reference, I deduce
erned by the u=−1 and u=2 renormalons,the δ = 0.012±0.018. Although compatible with
NP
conformal mapping generates a full series expan- (8), the central value is larger and has the op-
sion (K = 256, K = 2929 ...) which results, posite sign. This shift implies a smaller δ and,
5 6 P
afterBorelsummation,inalargervalueofδ ;i.e. therefore, a slightly smaller strong coupling.
P
theK termsgiveapositivecontributiontoδ The so-called duality violation effects, i.e. the
n>4 P
implying a smaller α (m2) [26]. uncertainties associated with the use of the OPE
s τ
Renormalons provide an interesting guide to to approximate the exact correlator, have been
possible higher-order corrections, making appar- also investigated [23,35]. Owing to the presence
ent that the associated uncertainties have to be in (5) of a double zero at s = m2, these effects
τ
carefully estimated. However,one shouldkeep in arequitesuppressedinR . Theyaresmallerthan
τ
mindthe adoptedassumptions. Infact,thereare the errorsinducedbyδ ,whichareinturnsub-
NP
no visible signs of renormalonic behaviour in the dominantwithrespecttotheleadingperturbative
presently known series: the n = −1 ultraviolet uncertainties.
renormalon is expected to dominate the asymp-
toticregime,implyinganalternatingseries,while 3. |Vus| DETERMINATION
allknownK coefficientshavethesamesign. One
n
The separate measurement of the |∆S| = 0
could either assume that renormalons only be-
and |∆S| = 1 tau decay widths provides a very
come relevant at higher orders, for instance at
cleandeterminationofV [36,37]. To a firstap-
n = 7, and apply the conformal mapping with us
proximation the Cabibbo mixing can be directly
arbitrary input values for K and K . Differ-
5 6
obtainedfromexperimentalmeasurements,with-
entassumptionsaboutthese twounknowncoeffi-
out any theoretical input. Neglecting the small
cients would result in different central values for
α (m2). SU(3)-breaking corrections from the ms − md
s τ
quark-mass difference, one gets:
A different reshuffling of the perturbative se-
ries,notrelatedtorenormalons,hasbeenrecently
1/2
R
proposed [27]. Instead of the usual expansion in |V |SU(3) = |V | τ,S = 0.210±0.002.
us ud
powers of the strong coupling, one expands in (cid:18)Rτ,V+A(cid:19)
termsoftheβ functionanditsderivatives(β ),
exp The new branching ratios measured by BaBar
which effectively results in a different estimate of
and Belle are all smaller than the previous world
higher-order corrections. One gets in this way a
averages, which translates into a smaller value
weaker dependence on the renormalization scale
of R and |V |. For comparison, the previ-
and a value of α (m2) similar to the standard τ,S us
s τ ous value R = 0.1686± 0.0047 [20] resulted
CIPT result. τ,S
in |V |SU(3) =0.215±0.003.
us
This rather remarkable determination is only
2.4. Non-perturbative corrections
slightly shifted by the small SU(3)-breaking con-
Atthepresentlyachievedprecision,oneshould
tributions induced by the strange quark mass.
worry about the small non-perturbative correc-
These effects can be estimated through a QCD
tions. In fact, a proper definition of the infrared
analysis of the differences [13,14,36–43]
renormalon contributions is linked to the corre-
sponding OPE corrections with D = 2n. A re- Rkl Rkl
cent re-analysis of the ALEPH data [24], with δRkl ≡ τ,V+A − τ,S . (18)
τ |V |2 |V |2
ud us
pinched-weight moments of the hadronic distri-
bution (PWM) and CIPT, obtains α (m2) = The onlynon-zerocontributionsareproportional
s τ
0.321± 0.013. This smaller value originates in to the mass-squared difference m2 − m2 or to
s d
6
vacuumexpectationvaluesofSU(3)-breakingop- malized difference between the two sets of mea-
erators such as δO ≡ h0|m s¯s − m d¯d|0i ≈ surements is −1.36σ. Thus,the result(20)could
4 s d
(−1.4 ± 0.4)· 10−3 GeV4 [13,36]. The dimen- easily fluctuate in the near future. In fact, com-
sions of these operators are compensated by cor- bining the measured Cabibbo-suppressed τ dis-
responding powers of m2, which implies a strong tribution with electroproduction data, a slightly
τ
suppression of δRkl [13]: larger value of |V | is obtained [46].
τ us
The final error of the V determination from
m2(m2) us
δRkl ≈ 24S s τ 1−ǫ2 ∆ (α ) τ decay is dominated by the experimental uncer-
τ EW (cid:26) m2τ d kl s tainties. IfRτ,S ismeasuredwitha 1%precision,
δO (cid:0) (cid:1) the resulting V uncertainty will get reduced to
−2π2 4 Q (α ) , (19) us
m4 kl s around 0.6%, i.e. ±0.0013, making τ decay the
τ (cid:27)
best source of information about V .
us
where ǫ ≡ m /m = 0.053± 0.002 [44]. The
d d s An accurate measurement of the invariant-
perturbativecorrections∆ (α )andQ (α )are
kl s kl s massdistributionofthefinalhadronscouldmake
known to O(α3) and O(α2), respectively [13,14].
s s possible a simultaneous determination of V
us
The J = 0 contribution to ∆ (α ) shows a
00 s and the strange quark mass, through a corre-
rather pathologicalbehaviour,with clear signs of latedanalysisofseveralweighteddifferencesδRkl.
τ
being a non-convergent perturbative series. For-
However, the extraction of m suffers from theo-
s
tunately, the corresponding longitudinal contri-
reticaluncertaintiesrelatedtothe convergenceof
bution to δR ≡ δR00 can be estimated phe-
τ τ the perturbative series ∆ (α ). A better under-
kl s
nomenologically with a much better accuracy,
standing of these corrections is needed.
δR |L = 0.1544±0.0037 [36,45], because it is
τ
dominated by far by the well-known τ → ν π
τ 4. CHIRAL SUM RULES
and τ → ν K contributions. To estimate the re-
τ
maining transverse component, one needs an in- When m = 0, the QCD Lagrangian has
u,d,s
put value forthe strangequarkmass. Takingthe an independent SU(3) flavour invariance for the
range ms(mτ) = (100±10)MeV [ms(2GeV) = left and right quark quiralities. The two quirali-
(96±10)MeV], which includes the most recent tieshaveexactlythesamestronginteraction,but
determinations of ms from QCD sum rules and they are completely decoupled. This chiral in-
latticeQCD[45],onegetsfinallyδRτ,th =0.216± variance guarantees that the two-point correla-
0.016 [37], which implies tionfunction ofa left-handedandaright-handed
quark currents, Π (s) = Π(0+1)(s)−Π(0+1)(s),
1/2 LR ud,V ud,A
Rτ,S vanishes identically to all orders in perturbation
|V | =
us R|Vτ,uVd+|2A −δRτ,th! tcheievoeryid(etnhteicvaelctpoerratunrdbaaxtiivael-vceocnttorribcuotriroenlas)t.orsTrhee-
= 0.2166±0.0019 ±0.0005 . (20)
exp th non-zero value of Π (s) originates in the spon-
LR
A largercentralvalue, |V |=0.2217±0.0032,is taneousbreakingofchiralsymmetrybytheQCD
us
obtained with the old world average for R . vacuum. At large momenta, the corresponding
τ,S
Sizeable changes on the experimental determi- OPE only receives contributions from operators
nation of R could be expected from the full with dimension d≥6,
τ,S
analysis of the huge BaBar and Belle data sam-
O O
ples. In particular, the high-multiplicity decay ΠOPE(s)=− 6 + 8 +··· (21)
LR s3 s4
modes are not well known at present. The re-
centdecreaseofseveralexperimentaltaubranch- The non-zero up and down quark masses induce
ing ratios is also worrisome. As pointed out by tiny corrections with dimensions two and four,
the PDG [22], 15 of the 16 branching fractions which are negligible at high energies.
measured at the B factories are smaller than the At very low momenta, Chiral Perturbation
previous non-B-factory values. The average nor- Theory (χPT) dictates the low-energy expansion
7
ofΠ (s)intermsofthepiondecayconstantand sum rule of Das et al. [53] that determines the
LR
the χPT couplings L [O(p4)] and C [O(p6)]. pion electromagnetic mass difference.
10 87
Analyticityrelatestheshort-andlong-distance Ref. [52] performs a statistical analysis, scan-
regimes through the dispersion relation ning the parameter space (κ,γ,β,s ) and select-
z
ing those ‘acceptable’ spectral functions which
1 s0
dsw(s)Π (s) = − dsw(s)ρ(s) satisfy the experimental and theoretical con-
LR
2πi
I|s|=s0 Zsth straints. From a generated initial sample of
+2f2w(m2)+Res[w(s)Π (s),s=0] ,(22) 160,000tuples, one finds 1,789acceptable distri-
π π LR
butions compatiblewithQCDandthe data. The
where ρ(s) ≡ π1 ImΠLR(s) and w(s) is an arbi- differencesamongthemdeterminehowmuchfree-
traryweightfunctionthatisanalyticinthewhole domisleftforthe behaviourofthe spectralfunc-
complex plane except in the origin (where it can tion beyond the kinematical end of the τ data.
have poles). The last term in (22) accounts for For each acceptable spectral function one calcu-
the possible residue at the origin. lates the parameters L , C , O and O , ob-
10 87 6 8
For s0 ≤ m2τ, the integral along the real axis tained through the dispersion relation (22) with
can be evaluated with the measured tau spectral the appropriate weight functions. The resulting
functions. Taking w(s)=sn with n≥0, there is statisticaldistributionsdeterminetheirfinallyes-
noresidueattheoriginand,withs0 largeenough timated values; the dispersion of the numerical
sothattheOPEcanbeappliedintheentirecircle results provides a good quantitative assessment
|s|=s0, the OPE coefficients are directly related of the actual uncertainties.
to the spectral function integration. With n = 0 The study has been also performed with
and 1, there is no OPE contributionin the chiral pinched weight functions of the form w(s) =
limitandonegetsthe celebratedfirstandsecond sn(s − s )m (m > 0) that vanish at s = s .
z z
Weinberg sum rules [47]. For negative values of As expected, these weights are found to mini-
the integer n, the OPE does not contribute ei- mize the uncertainties from duality-violation ef-
ther while the residues atzero are determined by fects,allowingforamoreprecisedeterminationof
theχPTlow-energycouplings,whichcanbethen thehadronicparameters. Onefinallyobtains[52],
experimentally determined [48].
Moreover, the absence of perturbative contri- Lr10(Mρ) = −(4.06±0.39)·10−3,
butions makes (22) and ideal tool to investigate Cr (M ) = (4.89±0.19)·10−3 GeV−2,
87 ρ
possible quark-hadron duality effects, formally O = (−4.3+0.9)·10−3 GeV6,
6 −0.7
defined through [49–51]
O = (−7.2+4.2)·10−3 GeV8. (25)
8 −5.3
1
DVw ≡ 2πi dsw(s) ΠLR(s)−ΠOLRPE(s) ThedeterminationofthetwoχPTcouplingsis
I|s|=s0 in good agreement with (but more precise than)
∞ (cid:0) (cid:1)
recent theoretical calculations, using Resonance
= ds w(s) ρ(s). (23)
Chiral Effective Theory [54] and large–N tech-
Zs0 C
niques at the next-to-leading order [55], which
ThishasbeenthoroughlystudiedinRef.[52],us-
predict [56]: Lr (M ) = −(4.4±0.9)·10−3 and
ingforthespectralfunctionbeyonds ∼2.1GeV 10 ρ
z Cr (M )=(3.6±1.3)·10−3 GeV−2. Italsoagrees
the parametrization [35,49–51] 87 ρ
withthepresentlatticeestimatesofLr (M )[57].
10 ρ
ρ(s≥s ) = κ e−γssin(β(s−s )), (24) Duality-violationeffectshaveverylittleimpact
z z
on the determinationof L and C because the
10 87
and finding the region in the 4-dimensional corresponding sum rules are dominated by the
(κ,γ,β,s ) parameter space that is compatible low-energy region where the data sits. Thus, one
z
with the most recent experimental data [17] and obtains basically the same results with pinched
the following theoretical constraints at s → ∞: and non-pinched weight functions. This is no-
0
first and second Weinberg sum sules [47] and the longer true for O and O , which are sensitive to
6 8
8
Ref.@52DH95%L Prades, who sadly passed away recently. Ximo
Ref.@52DH68%L
has made many relevant contributions to the
Almasyetal.,2008
Masjuan&Peris,2007 physics of the tau lepton, some of which have
Almasyetal.,2007
been mentioned here. This work has been sup-
Bordesetal.,2006
ALEPHColl.,2005 ported by MICINN, Spain (grants FPA2007-
Narison,2005 60323 and Consolider-Ingenio 2010 CSD2007-
Friotetal.,2004
Zyablyuk,2004 00042, CPAN) by the EU Contract MRTN-CT-
Latorre&Rojo,2004 2006-035482 (FLAVIAnet) and by Generalitat
Ciriglianoetal.,2003
Bijnensetal.,2001 Valenciana (PROMETEO/2008/069).
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