- Email: [email protected]
Spectral functions from hadronic τ decays
PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proc. Suppl.)98 (2001) 305-313
Spectral
Functions
from Hadronic
www.elsevier.nVlocate/npe
r Decays
Michel Davier Laboratoire de l’Acdl&-ateur Lin6aire IN2PS/CNRS et Universitd de Paris-Sud 91898 Orsay, France E-mail: [email protected] Hadronic decays of the T lepton provide a clean environment dominated exclusive
by resonances, channels
determination
are reviewed.
of the strong
T vector
spectral
vacuum
polarization
the running
with the interesting
functions
Inclusive
coupling
information
spectral
constant
functions
integrals
occuring
of the electromagnetic
in the calculations
in the spectral
functions.
are the basis for QCD analyses,
and quantitative
for the 27~ and 47r final states
in an energy regime
to study hadron dynamics
captured
information
Recent
delivering
on nonperturbative
are used together of the anomalous
with e+emagnetic
data
results
contributions. in order
moment
on
an accurate The
to compute
of the muon
and
coupling.
1. Introduction
Hadrons produced in T decays are born out of the charged weak current, i.e. out of the QCD vacuum. This property guarantees that hadronic physics factorizes in these processes which are then completely characterized for each decay channel by spectral functions as far as the total rate is concerned . Furthermore, the produced hadronic systems have I = 1 and spin-parity Jp = Of, l- (V) and Jp = O-, l+ (A). The spectral functions are directly related to the invariant mass spectra of the hadronic final states, normalized to their respective branching ratios and corrected for the r decay kinematics. For a given spin-l vector decay, one has
However, perturbative inated by resonances. QCD can be seriously considered due to the relatively large T mass. Many hadronic modes have been measured and studied, while some earlier discrepancies (before 1990) have been resolved with high-statistics and low-systematics experiments. Samples of N 4 x lo5 measured decays are available in each LEP experiment and CLEO. Conditions for low systematic uncertainties are particularly well met at LEP: measured samples have small non-r backgrounds (< 1%) and large selection efficiency (92%), for example in ALEPH. Recent results in the field are discussed in this report. 2. Specific Final States 2.1. 2.1.1.
(1) where Vud denotes the CKM weak mixing matrix element and SEW accounts for electroweak radiative corrections. Isospin symmetry (CVC) connects the 7 and e+e- annihilation spectral functions, the latter being proportional to the R ratio. Hadronic T decays are a clean probe of hadron dynamics in an interesting energy region dom-
The
2n Vector
State
The Data
The decay r + v,K-~’ is now studied with Data from large statistics of lo5 events. ALEPH have been published [l]. Results from CLEO are now available [2] with the mass spectrum given in Fig. 1 dominated by the ~(770) resonance. Good agreement is observed between the ALEPH and CLEO data and the line shape fits show strong evidence for the contribution of p( 1400) through interference with the dominant amplitude.
0920-S632/01/$ - see front matter 0 2001 Elsevier Science B.V All rights reserved PII SO920-S632(01)01239-7
306
0.3
0.5 m"
0.7
0.9
1.1
Invariant Mass
1.3
1.5
[GeV]
Figure 1. Mass distribution in CLEO ; t v, 7r- 7r” sample. The solid curve overlaid is the result of the Kiihn-Santamariafit, while the dashed curve has the ~(1400) contribution turned off.
Figure 2. Cross section for e+e- + 7riT+7r-compared to the ALEPH T data using CVC with p-w interference built in (shaded band).
bution 6~lV,~125’mv B, --1 dI-C(s)v_ (s) The quality of the e+e- _$ ~+n- data has also F ds (’ * r-ro”) = m? B,,o recently improved with the release of CMD-2 reu-(s) = !$,F;(s),2 sults [3]. A comparison of the mass spectrum as (3) measured in e+e- and r data is given in Fig. 2. For this exercise the p - w interference has to be C(s) = (I-$) (I+$) artificially introduced in the r data. SU(2) symmetry (CVC) implies 2.1.2. Testing CVC v_(s) = vo(s) (4) It is useful to carefully write down all the factors involved in the comparison of e+e- and r Three levels of CVC breaking can be identified: spectral functions in order to understand possiradiative corrections to r decays produce ble sources of CVC breaking. On the e+e- side the SEW factor [4,5] which is dominated we have by short-distance effects. As such it is ex4ncY2 pected to be weakly dependent on the spefl(e+e- + 7r+n-) = -00(s) cific hadronic final state, as verified by deS tailed computations in the r ---_) (r,K)v, $lF;(s)12 Vo(S) = channels [6]. These considerations provide a conservative estimate of the corresponding where /3:(s) is the threshold kinematic factor and uncertainty F:(s) the pion form factor. On the r side, the SEW = 1.0104 & 0.0040 physics is contained in the hadronic mass distri(5)
M. Llnuier/Nuclear
Physics 6: (Ptwc.
Suppl.)
98 (2001)
307
305-313
u
OPAL 98
I*
ALEPH 96
m
CLE094
I
WA .I ..._..._...._.___......_._._.. DELPHI 92
-
... ... .. .. .. .. .._............_..... , I
L3 95
-
ARGUS 92 -
r I
0.25
0.75
0.5 s
1
l
r
22
pion
mass splitting, which is almost completely from electromagnetic origin, directly breaks isospin symmetry in the spectral functions [7,8] since /?- (s) # ,&(s).
is also broken in the pion form [7]. The p width is affected by K and p mass splittings and by explicit electromagnetic decays such as ~7, ny, 1+1and nry. Isospin violation in the strong amplitude is expected to be negligible because of the small absolute mass difference between u and d quarks.
symmetry factor
Although the agreement between 27r spectral functions looks impressive, a closer look can be taken by removing the strongly varying p line shape. This is conveniently achieved by comparing the available e+e- data sets to a fit performed to the ALEPH spectral function (discussed in the next section). Fig. 3 shows that e+e- experiments agree with 7 data within their systematic uncertainties, except DMl [9] and NA7 [lo]. The latter data set is problematic [ll].
CBALL 91 I
24
CELLO 90
26
B(hz’)
(GeV2)
Figure 3. Relative deviation between the square of the pion form factor measured in e+e- annihilation and the fit to the ALEPH data using the Gounaris-Sakurai parametrization.
l
20
1.25
ALEPH 92
I
28
30
(%)
Figure 4. Measurements of the branching ratio for the decay B(r -+ v,h-no) [la]. The average (WA) of the four most precise values is given.
It is possible to quantify the comparison by computing a single number, integrating over the complete spectrum. It is convenient for this to use the branching ratio B(r + Y,K rr”) as directly measured in T decays and the corresponding quantity computed from the e+e- spectral function under the assumption of CVC. Fig. 4 displays the current situation [12] on B(’ + y7h-n0), yielding an average value of (25.76 f 0.15)%. Subtracting out B(’ + u,K-no) = = (0.45 * 0.03)% [13], one gets B(r + v7r-rro) (25.31*0.15)%. Using all the available published e+e- data, except DMl and NA7, we obtain B&vc = (24.67 Jo 0.39 f 0.12 * O.lO)% where the quoted errors are respectively from (i) e+e- experiments, (ii) SEW, Vud and B,, and (iii) isospin violation in the p widthr. Corrections for isospin breaking amount to only 0.01% (+0.20% from ,B3, -0.21% from F,). One finally gets B;,,
- BFTvC = (0.64 ;t 0.45)%
showing ‘A more CMD-2
(6)
fair agreement. precise
value using still unpublished
is given in Ref.
[14]
results
from
M. Dmier/Nucleur
308
2.1.3.
Fitting
the 2~ Spectral
Physics B (Proc. Suppl.) 98 (2001) 305-313
Function
The 27r spectral function is dominated by the This provides us with an opporp resonance. tunity to study the relevant description of a wide hadronic resonance. Two parametrizations are used in practice: Breit-Wigner with energytraditionally called Kiihndependent width, Santamaria
and Gounaris-Sakurai
[15] (KS),
[16]
(GS) which incorporates analyticity through finite width corrections. The latter is expected to be a more accurate description. The line shape has to be modified
to account
p - w interference. The pion form factor
for the effect
is fitted
of
with interfering
1.2
data. The GS parametrization is clearly preferred and provides a good description from threshold to 3-4 GeV2. The KM line shape is acceptable in the CLEO fit but yields a rather unphysical normalization more precise threshold
at s = 0, in contradiction data from ALEPH obtained
to test r,o =
as predicted (Im,2.2.
in the
in these fits is not good
the predicted isospin violation (1.1 f 0.6) MeV), but the re-
sults are quite consistent served p- and p” masses by Chiral
with it. Also the obare equal within errors
Perturbation
Theory
State
The 47r final states have also been studied Tests of CVC are severely hampered 181. large
[17]
- mpoI < 0.5 MeV). The 4n Vector
deviations
between
different
e+e-
Figure
5. Fits of the CLEO
spectral
function
for
7 + u,?r-w.
with the
and e+e-
region.
The precision enough (r,-
1.6
q(w$ [GeV/c’l
pieces from p, p’ and p” with relative amplitudes 1, p and y. Table 1 presents the results of fits and T (ALEPH and CLEO) to e+e(CMD-2)
[l, by
experi-
CLEO sets a new 95% CL limit of 6.4% for the relative contribution of second-class currents in the decay r +
v,r-
w from the hadronic
angular
decay distribution. 2.3. The
Axial-vector decay
States
T -+ v737r is the cleanest
study axial-vector resonance structure. trum is dominated by the lt al state,
place
to
The specknown to
decay essentially through pn. A comprehensive analysis of the 7r-27r” channel has been presented by CLEO. First, a model-independent determination
of the hadronic
structure
no evidence for non-axial-vector
functions
gave
contributions
(<
ments which disagree well beyond their quoted systematic uncertainties. A new CLEO analysis
17% at 90% CL) [20]. S econd, a partial-wave amplitude analysis was performed [21]: while the
studies the resonant structure in the 37r7r” channel which is shown to be dominated by wn and aln
came as a surprize
contributions.
The wz spectral
function
shown in
Fig. 5 is in good agreement with CMD-2 results and it is interpreted by a sum of plike amplitudes. The mass of the second state is however found at (1523 & lO)MeV, in contrast with the value (1406 f 14)MeV from the fit of the 27~ spectral function. This point has to be clarified. Following a limit of 8.6% obtained earlier by ALEPH [19],
dominant
pi
mode
was of course that
confirmed,
an important
tion (- 20%) from scalars (u, f0(1470), was found in the 21r system.
it
contribufi(l270))
The al + n-27r0 precisely determined line shape shows the opening of the K’K decay mode in the total al width. The derived branching ratio, B(al agreement modes
+
K*K)
with
=
ALEPH
(3.3 f 0.5)% is in good results on the I
which were indeed
shown
(with
the help
M. Dnuier/Nuclenr
1
CMD-2 GS
)
Physics B (Pmc.
ALEPH GS
SuppI.)
9X (2001)
1 ALEPH KS 1
305-313
300
CLEO GS
1
CLEO KS
1
22.9123 23.2123 34144 51157 134157 1 [ X2bf I Table 1 Results of fits to the pion form factor squared from e + e - CMD-2 data and r data (ALEPH and CLEO). The parametrizations of the p line shape (KS or GS) are described in the text. The errors from the ALEPH KS fit are not given in view of the very bad x2.
of e+e- data and CVC) to be axial-vector (ur) dominated with B(ar + I(*h’) = (2.6 f 0.3)% [13]. No conclusive evidence for a higher mass state (ui) was found in this analysis.
4.
3. Inclusive
R
Spectral
Functions
The T nonstrange spectral functions have been measured by ALEPH [1,22] and OPAL [23]. The procedure requires a careful separation of vector (V) and axial-vector (A) states involving the reconstruction of multi-no decays and the proper treatment of final states with a h’l;’ pair. The V and A spectral functions are given in Fig. 6. They show a strong resonant behaviour, dominated by the lowest p and al states, with a tendancy to converge at large mass toward a value near the parton model expectation. Yet, the vector part stays clearly above while the axial-vector one lies below. Thus, the two spectral functions are clearly not ‘asymptotic’ at the ‘T mass scale. The V+A spectral function, shown in Fig. 7 has a clear pattern converging toward a value above the parton level as expected in &CD. In fact, it displays a textbook example of global duality, since the resonance-dominated low-mass region shows an oscillatory behaviour around the asymptotic QCD expectation, assumed to be valid in a local sense only for large masses. This feature will be quantitatively discussed in the next section.
QCD
4.1.
Analysis
of Nonstrange
7 Decays
Motivation
The total hadronic T width, properly ized to the known leptonic width, = T(r7
F(r-
+ hadrons-
normal-
vs)
+ e- V,z+)
(7)
should be well predicted by QCD as it is an inclusive observable. Compared to the similar quantity defined in e+e- annihilation, it is even twice inclusive: not only are all produced hadronic states at a given mass summed over, but an integration is performed over all the possible masses from m, to mT. 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 approach 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 latter components indeed very small [24,25]. 4.2. Theoretical Prediction for R, The imaginary parts of the vector and axial-vector two-point correlation functions HS!“,, (S)? with the spin J of the hadronic system, are proportional to the 7 hadronic spectral functions with corresponding quantum numbers. The non-strange ratio R, can be written
310
i”“l”“l”“TT?“l”“i””
Ul+al
ALEPH
*
2.5 + .
*
1-
z- -+ @‘,A,
I=l) v,
..-...
parton model prediction
-
perturbative QCD (massless)
-
.
: d ” ”
0 0
0.5
”
‘1” 1
L
‘1” 1.5
s”
IL”’ 2
1 j 1.5
‘1”
”
3
3.5
Mass* (GeV/c2)2
1.5
2
2.5
3
s (GeV’)
c x1.4
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 o, (M,2) = 0.120.
0.8
as an integral of these spectral functions the invariant mass-squared s of the final hadrons [26]:
0.6
over state
0.2
R,(Q) O0
0.5
1
1.5
2
2.5
=
127T&w /$(l-%,’
(8)
3
s (GA’*)
x [(1+2$)ImII~Pl(x+ir) Figure 6. Inclusive nonstrange vector (top) and axial-vector (bottom) spectral functions from ALEPH and OPAL. The dashed line is the expectation from the naive parton model.
+ Imnl”)(s+ir)]
By Cauchy’s theorem the imaginary part of IItJl is proportional to the discontinuity across the positive real axis. The energy scale so = rn: is large enough so that contributions from nonperturbative effects are small. It is therefore assumed that one can use the Operator Product E.xpansion (OPE) to organize perturbative and nonperturbative contributions [27] to R,(so). The theoretical prediction of the vector and
M. Daoier/Nuclrar
ratio Rr,v/A can thus be written
axial-vector
R r,V/A
Ph,vsic.sB (PKx.
as:
~lvudl%v
=
.711
SuppI.) 98 (2001) 305-313
(9)
with the residual non-logarithmic electroweak correction Sk, = 0.0010 [5], neglected in the following, and the dimension D = 2 contribution 6(2- mass1 from quark masses which is lower than ud,VIA
ALEPH V A V-l-A
JNP os(m2,) 0.020 f 0.004 0.330 f 0.014 f 0.018 110.339 & 0.013 f 0.018 1 -0.027 f 0.004 I] 0.334 f 0.007 f 0.021 1 -0.003 f 0.004
Table 2 Fit results of o,(mp) and the OPE nonperturbative contributions from vector, axial-vector and (V + A) combined fits using the corresponding ratios R, and the spectral moments as input parameters. The second error is given for theoretical uncertainty.
0.1% for U, d quarks. The term J(O) is the purely perturbative contribution, while the S(O) are the OPE form
terms
@ ud,V/A
-
in powers
c
of saD’2
of the following
(%d)V/A
~i~O=D(-so)D’2
(10)
where the long-distance nonperturbative effects are absorbed into the vacuum expectation elements (Oud). The perturbative expansion (FOPT) is known to third order [28]. A resummation of all known higher order logarithmic integrals improves the convergence of the perturbative series (contourimproved method FOPTcr) [29]. As some ambiguity persists, the results are given as an average of the two methods with the difference taken as a systematic uncertainty. 4.3. Measurements The ratio R, is obtained from measurements the leptonic branching ratios: R,
=
3.047 f 0.014
of
(II)
using a value B(T+ e- VeVT) = (17.794 It 0.045)% which includes the improvement in accuracy provided by the universality assumption of leptonic currents together with the measurements of a(~+ e- Y,y,), B(rt 1-1~VpvT) and the T lifetime. The nonstrange part of R, is obtained by subtracting out the measured strange contribution (see last section). Two complete analyses of the V and A parts have been performed by ALEPH [22] and OPAL [23]. Both use the world-average leptonic
branching ratios, but their own measured spectral functions. The results on o,(mf) are therefore strongly correlated and indeed agree when the same theoretical prescriptions are used. 4.4. Results of the Fits The results of the fits are given in Table 2 for the ALEPH analysis. Similar results are obtained by OPAL. It is worth emphasizing that the nonperturbative contributions are found to be very small, as expected. The limited number of observables and the strong correlations between the spectral moments introduce large correlations, especially between the fitted nonperturbative operators. One notices a remarkable agreement within statistical errors between the cr,(m;) values using vector and axial-vector data. The total nonperturbative power contribution to Rr,v+~ is compatible with zero within an uncertainty of 0.4%, that is much smaller than the error arising from the perturbative term. This cancellation of the nonperturbative terms increases the confidence on the o,(m:) determination from the inclusive (V + A) observables. The final result from ALEPH is : o&n”,)
= 0.334 f 0.007,,,
Zh 0.02I,h
(12)
where the first error accounts for the experimental uncertainty and the second gives the uncertainty of the theoretical prediction of R, and the spectral moments as well as the ambiguity of the theoretical approaches employed.
312
M. Dnuirr/Nuclenr
Physics
13 (Proc. Suppl.)
9X (2001) 305-313
1
1.25
1.5
1.75
2
2.25
2.5
2.75 sg
Figure 8. The ratio Rr,v+~ versus the square “r mass” se. 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 results of the fit at so = m:.
In the OPAL analysis the corresponding results are quoted within three prescriptions for the perturbative expansion, respectively FOPTcr, FOPT and with renormalon chains, i.e. =
o&n:)
4.5.
Test
0.348 f 0.009,,,
f 0.019th
(13)
=
0.324 zt 0.006,,,
f 0.013th
(14)
=
0.306 f 0.005,,,
f O.OlIth
(15)
of the Running
of (Y,(S)
at Low
3 (GM
Figure 9. The running of cr,(so) obtained from the fit of the theoretical prediction to R,,v+A(so). The shaded band shows the data including experimental errors. The curves give the four-loop RGE evolution for two and three flavours.
perturbative term. Fig. 9 has the same physical content as Fig. 8, but translated into the running of o,(so), i.e., the experimental value for o’s (so) has been individually determined at every sn from the comparison of data and theory. Good agreement is observed with the four-loop RGE evolution using three quark flavours. The experimental fact that the nonperturbative contributions cancel over the whole range 1.2 GeV2 5 SO < rn: leads to confidence that the o, determination from the inclusive (V + A) data is robust. 4.6.
Discussion
on
the
Determination
of
Energies
Using the spectral functions, one can simulate the physics of a hypothetical r lepton with a mass fi smaller than m, through equation (8) and hence further investigate QCD phenomena at low energies. Assuming quark-hadron duality, the evolution of R,(so) provides a direct test of the running of o,(so), governed by the RGE pfunction. On the other hand, it is a test of the validity of the OPE approach in r decays. The functional dependence of R,,v+~(sg) is plotted in Fig. 8 together with the theoretical prediction using the results of Table 2. Below 1 GeV’ the error of the theoretical prediction of RT,v+~(so) starts to blow up due to the increasing uncertainty from the unknown fourth-order
The evolution of the crJ (m:) measurement from the inclusive (V + A) observables based on the Runge-Kutta integration of the differential equation of the renormalization group to N3L0 [30,32] yields for the ALEPH analysis o,(M;)
= 0.1202~0.0008,,,~0.0024thf0.0010,,,~(16)
where the last error stands for possible ambiguities in the evolution due to uncertainties in the matching scales of the quark thresholds [32]. The result (16) can be compared to the determination from the global electroweak fit. The variable Rz has similar advantages to R,, but it differs concerning the convergence of the perturbative expansion because of the much larger scale.
M. Dauier/Nucleur
Pllysics B (Proc. Suppl.)
313
98 (2001) 305-313
Mass’
Figure 10. Evolution of the strong coupling (measured at rn:) to Mi predicted by QCD compared to the direct measurement. The evolution is carried out at 4 loops, while the flavour matching is accomplished at 3 loops at 2 m, and 2 mb thresholds.
It turns out that this determination is dominated by experimental errors with very small theoretical uncertainties, i.e. the reverse of the situation encountered in r decays. The most recent value [33] yields crJ(Mi) = 0.1183 f 0.0027, in excellent agreement with (16). Fig. 10 illustrates well the agreement between the evolution of os(m~) predicted by QCD and crJ (A&?). 5. Applications
to Hadronic
Vacuum
Po-
larization 5.1.
Improvements lat ions
From the learned that: l
studies
to the Standard
presented
above
Calcu-
we have
the I = 1 vector spectral function from T decays agrees with that from e+e- annihilation, while it is more precise for masses less than 1.6 GeV as can be seen on Fig. 11. Small CVC violations are expected at a few 10m3 level as discussed above, essentially from SU(2)-breaking in the r masses.
(G~V/C~)~
Figure 11. Global test of CVC using r and e+evector spectral functions.
l
the description of R, by perturbative QCD works down to a scale of 1 GeV. Nonperturbative contributions at 1.8 GeV are well below 1 % in this case. They are larger (- 2 %) for the vector part alone, but reasonably well described by OPE. The complete (perturbative + nonperturbative) description is accurate at the 1 % level at 1.8 GeV for integrals over the vector spectral function such as R,,".
These two facts have direct applications to calculations of hadronic vacuum polarisation which involve the knowledge of the vector spectral function: the muon magnetic anomaly and the running of cr. In both cases, the standard method involves a dispersion integral over the vector spectral function taken from the e+e-+ hadrons data. Eventually at large energies, QCD is used to replace experimental data. Hence the precision of the calculation is given by the accuracy of the data, which is poor above 1.5 GeV. Even at low energies, the precision can be significantly improved by using r data [7]. The next breakthrough comes about using the prediction of perturbative QCD far above quark thresholds, but at low enough energies (compati-
Spectral
Functions
from Hadronic
www.elsevier.nVlocate/npe
r Decays
Michel Davier Laboratoire de l’Acdl&-ateur Lin6aire IN2PS/CNRS et Universitd de Paris-Sud 91898 Orsay, France E-mail: [email protected] Hadronic decays of the T lepton provide a clean environment dominated exclusive
by resonances, channels
determination
are reviewed.
of the strong
T vector
spectral
vacuum
polarization
the running
with the interesting
functions
Inclusive
coupling
information
spectral
constant
functions
integrals
occuring
of the electromagnetic
in the calculations
in the spectral
functions.
are the basis for QCD analyses,
and quantitative
for the 27~ and 47r final states
in an energy regime
to study hadron dynamics
captured
information
Recent
delivering
on nonperturbative
are used together of the anomalous
with e+emagnetic
data
results
contributions. in order
moment
on
an accurate The
to compute
of the muon
and
coupling.
1. Introduction
Hadrons produced in T decays are born out of the charged weak current, i.e. out of the QCD vacuum. This property guarantees that hadronic physics factorizes in these processes which are then completely characterized for each decay channel by spectral functions as far as the total rate is concerned . Furthermore, the produced hadronic systems have I = 1 and spin-parity Jp = Of, l- (V) and Jp = O-, l+ (A). The spectral functions are directly related to the invariant mass spectra of the hadronic final states, normalized to their respective branching ratios and corrected for the r decay kinematics. For a given spin-l vector decay, one has
However, perturbative inated by resonances. QCD can be seriously considered due to the relatively large T mass. Many hadronic modes have been measured and studied, while some earlier discrepancies (before 1990) have been resolved with high-statistics and low-systematics experiments. Samples of N 4 x lo5 measured decays are available in each LEP experiment and CLEO. Conditions for low systematic uncertainties are particularly well met at LEP: measured samples have small non-r backgrounds (< 1%) and large selection efficiency (92%), for example in ALEPH. Recent results in the field are discussed in this report. 2. Specific Final States 2.1. 2.1.1.
(1) where Vud denotes the CKM weak mixing matrix element and SEW accounts for electroweak radiative corrections. Isospin symmetry (CVC) connects the 7 and e+e- annihilation spectral functions, the latter being proportional to the R ratio. Hadronic T decays are a clean probe of hadron dynamics in an interesting energy region dom-
The
2n Vector
State
The Data
The decay r + v,K-~’ is now studied with Data from large statistics of lo5 events. ALEPH have been published [l]. Results from CLEO are now available [2] with the mass spectrum given in Fig. 1 dominated by the ~(770) resonance. Good agreement is observed between the ALEPH and CLEO data and the line shape fits show strong evidence for the contribution of p( 1400) through interference with the dominant amplitude.
0920-S632/01/$ - see front matter 0 2001 Elsevier Science B.V All rights reserved PII SO920-S632(01)01239-7
306
0.3
0.5 m"
0.7
0.9
1.1
Invariant Mass
1.3
1.5
[GeV]
Figure 1. Mass distribution in CLEO ; t v, 7r- 7r” sample. The solid curve overlaid is the result of the Kiihn-Santamariafit, while the dashed curve has the ~(1400) contribution turned off.
Figure 2. Cross section for e+e- + 7riT+7r-compared to the ALEPH T data using CVC with p-w interference built in (shaded band).
bution 6~lV,~125’mv B, --1 dI-C(s)v_ (s) The quality of the e+e- _$ ~+n- data has also F ds (’ * r-ro”) = m? B,,o recently improved with the release of CMD-2 reu-(s) = !$,F;(s),2 sults [3]. A comparison of the mass spectrum as (3) measured in e+e- and r data is given in Fig. 2. For this exercise the p - w interference has to be C(s) = (I-$) (I+$) artificially introduced in the r data. SU(2) symmetry (CVC) implies 2.1.2. Testing CVC v_(s) = vo(s) (4) It is useful to carefully write down all the factors involved in the comparison of e+e- and r Three levels of CVC breaking can be identified: spectral functions in order to understand possiradiative corrections to r decays produce ble sources of CVC breaking. On the e+e- side the SEW factor [4,5] which is dominated we have by short-distance effects. As such it is ex4ncY2 pected to be weakly dependent on the spefl(e+e- + 7r+n-) = -00(s) cific hadronic final state, as verified by deS tailed computations in the r ---_) (r,K)v, $lF;(s)12 Vo(S) = channels [6]. These considerations provide a conservative estimate of the corresponding where /3:(s) is the threshold kinematic factor and uncertainty F:(s) the pion form factor. On the r side, the SEW = 1.0104 & 0.0040 physics is contained in the hadronic mass distri(5)
M. Llnuier/Nuclear
Physics 6: (Ptwc.
Suppl.)
98 (2001)
307
305-313
u
OPAL 98
I*
ALEPH 96
m
CLE094
I
WA .I ..._..._...._.___......_._._.. DELPHI 92
-
... ... .. .. .. .. .._............_..... , I
L3 95
-
ARGUS 92 -
r I
0.25
0.75
0.5 s
1
l
r
22
pion
mass splitting, which is almost completely from electromagnetic origin, directly breaks isospin symmetry in the spectral functions [7,8] since /?- (s) # ,&(s).
is also broken in the pion form [7]. The p width is affected by K and p mass splittings and by explicit electromagnetic decays such as ~7, ny, 1+1and nry. Isospin violation in the strong amplitude is expected to be negligible because of the small absolute mass difference between u and d quarks.
symmetry factor
Although the agreement between 27r spectral functions looks impressive, a closer look can be taken by removing the strongly varying p line shape. This is conveniently achieved by comparing the available e+e- data sets to a fit performed to the ALEPH spectral function (discussed in the next section). Fig. 3 shows that e+e- experiments agree with 7 data within their systematic uncertainties, except DMl [9] and NA7 [lo]. The latter data set is problematic [ll].
CBALL 91 I
24
CELLO 90
26
B(hz’)
(GeV2)
Figure 3. Relative deviation between the square of the pion form factor measured in e+e- annihilation and the fit to the ALEPH data using the Gounaris-Sakurai parametrization.
l
20
1.25
ALEPH 92
I
28
30
(%)
Figure 4. Measurements of the branching ratio for the decay B(r -+ v,h-no) [la]. The average (WA) of the four most precise values is given.
It is possible to quantify the comparison by computing a single number, integrating over the complete spectrum. It is convenient for this to use the branching ratio B(r + Y,K rr”) as directly measured in T decays and the corresponding quantity computed from the e+e- spectral function under the assumption of CVC. Fig. 4 displays the current situation [12] on B(’ + y7h-n0), yielding an average value of (25.76 f 0.15)%. Subtracting out B(’ + u,K-no) = = (0.45 * 0.03)% [13], one gets B(r + v7r-rro) (25.31*0.15)%. Using all the available published e+e- data, except DMl and NA7, we obtain B&vc = (24.67 Jo 0.39 f 0.12 * O.lO)% where the quoted errors are respectively from (i) e+e- experiments, (ii) SEW, Vud and B,, and (iii) isospin violation in the p widthr. Corrections for isospin breaking amount to only 0.01% (+0.20% from ,B3, -0.21% from F,). One finally gets B;,,
- BFTvC = (0.64 ;t 0.45)%
showing ‘A more CMD-2
(6)
fair agreement. precise
value using still unpublished
is given in Ref.
[14]
results
from
M. Dmier/Nucleur
308
2.1.3.
Fitting
the 2~ Spectral
Physics B (Proc. Suppl.) 98 (2001) 305-313
Function
The 27r spectral function is dominated by the This provides us with an opporp resonance. tunity to study the relevant description of a wide hadronic resonance. Two parametrizations are used in practice: Breit-Wigner with energytraditionally called Kiihndependent width, Santamaria
and Gounaris-Sakurai
[15] (KS),
[16]
(GS) which incorporates analyticity through finite width corrections. The latter is expected to be a more accurate description. The line shape has to be modified
to account
p - w interference. The pion form factor
for the effect
is fitted
of
with interfering
1.2
data. The GS parametrization is clearly preferred and provides a good description from threshold to 3-4 GeV2. The KM line shape is acceptable in the CLEO fit but yields a rather unphysical normalization more precise threshold
at s = 0, in contradiction data from ALEPH obtained
to test r,o =
as predicted (Im,2.2.
in the
in these fits is not good
the predicted isospin violation (1.1 f 0.6) MeV), but the re-
sults are quite consistent served p- and p” masses by Chiral
with it. Also the obare equal within errors
Perturbation
Theory
State
The 47r final states have also been studied Tests of CVC are severely hampered 181. large
[17]
- mpoI < 0.5 MeV). The 4n Vector
deviations
between
different
e+e-
Figure
5. Fits of the CLEO
spectral
function
for
7 + u,?r-w.
with the
and e+e-
region.
The precision enough (r,-
1.6
q(w$ [GeV/c’l
pieces from p, p’ and p” with relative amplitudes 1, p and y. Table 1 presents the results of fits and T (ALEPH and CLEO) to e+e(CMD-2)
[l, by
experi-
CLEO sets a new 95% CL limit of 6.4% for the relative contribution of second-class currents in the decay r +
v,r-
w from the hadronic
angular
decay distribution. 2.3. The
Axial-vector decay
States
T -+ v737r is the cleanest
study axial-vector resonance structure. trum is dominated by the lt al state,
place
to
The specknown to
decay essentially through pn. A comprehensive analysis of the 7r-27r” channel has been presented by CLEO. First, a model-independent determination
of the hadronic
structure
no evidence for non-axial-vector
functions
gave
contributions
(<
ments which disagree well beyond their quoted systematic uncertainties. A new CLEO analysis
17% at 90% CL) [20]. S econd, a partial-wave amplitude analysis was performed [21]: while the
studies the resonant structure in the 37r7r” channel which is shown to be dominated by wn and aln
came as a surprize
contributions.
The wz spectral
function
shown in
Fig. 5 is in good agreement with CMD-2 results and it is interpreted by a sum of plike amplitudes. The mass of the second state is however found at (1523 & lO)MeV, in contrast with the value (1406 f 14)MeV from the fit of the 27~ spectral function. This point has to be clarified. Following a limit of 8.6% obtained earlier by ALEPH [19],
dominant
pi
mode
was of course that
confirmed,
an important
tion (- 20%) from scalars (u, f0(1470), was found in the 21r system.
it
contribufi(l270))
The al + n-27r0 precisely determined line shape shows the opening of the K’K decay mode in the total al width. The derived branching ratio, B(al agreement modes
+
K*K)
with
=
ALEPH
(3.3 f 0.5)% is in good results on the I
which were indeed
shown
(with
the help
M. Dnuier/Nuclenr
1
CMD-2 GS
)
Physics B (Pmc.
ALEPH GS
SuppI.)
9X (2001)
1 ALEPH KS 1
305-313
300
CLEO GS
1
CLEO KS
1
22.9123 23.2123 34144 51157 134157 1 [ X2bf I Table 1 Results of fits to the pion form factor squared from e + e - CMD-2 data and r data (ALEPH and CLEO). The parametrizations of the p line shape (KS or GS) are described in the text. The errors from the ALEPH KS fit are not given in view of the very bad x2.
of e+e- data and CVC) to be axial-vector (ur) dominated with B(ar + I(*h’) = (2.6 f 0.3)% [13]. No conclusive evidence for a higher mass state (ui) was found in this analysis.
4.
3. Inclusive
R
Spectral
Functions
The T nonstrange spectral functions have been measured by ALEPH [1,22] and OPAL [23]. The procedure requires a careful separation of vector (V) and axial-vector (A) states involving the reconstruction of multi-no decays and the proper treatment of final states with a h’l;’ pair. The V and A spectral functions are given in Fig. 6. They show a strong resonant behaviour, dominated by the lowest p and al states, with a tendancy to converge at large mass toward a value near the parton model expectation. Yet, the vector part stays clearly above while the axial-vector one lies below. Thus, the two spectral functions are clearly not ‘asymptotic’ at the ‘T mass scale. The V+A spectral function, shown in Fig. 7 has a clear pattern converging toward a value above the parton level as expected in &CD. In fact, it displays a textbook example of global duality, since the resonance-dominated low-mass region shows an oscillatory behaviour around the asymptotic QCD expectation, assumed to be valid in a local sense only for large masses. This feature will be quantitatively discussed in the next section.
QCD
4.1.
Analysis
of Nonstrange
7 Decays
Motivation
The total hadronic T width, properly ized to the known leptonic width, = T(r7
F(r-
+ hadrons-
normal-
vs)
+ e- V,z+)
(7)
should be well predicted by QCD as it is an inclusive observable. Compared to the similar quantity defined in e+e- annihilation, it is even twice inclusive: not only are all produced hadronic states at a given mass summed over, but an integration is performed over all the possible masses from m, to mT. 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 approach 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 latter components indeed very small [24,25]. 4.2. Theoretical Prediction for R, The imaginary parts of the vector and axial-vector two-point correlation functions HS!“,, (S)? with the spin J of the hadronic system, are proportional to the 7 hadronic spectral functions with corresponding quantum numbers. The non-strange ratio R, can be written
310
i”“l”“l”“TT?“l”“i””
Ul+al
ALEPH
*
2.5 + .
*
1-
z- -+ @‘,A,
I=l) v,
..-...
parton model prediction
-
perturbative QCD (massless)
-
.
: d ” ”
0 0
0.5
”
‘1” 1
L
‘1” 1.5
s”
IL”’ 2
1 j 1.5
‘1”
”
3
3.5
Mass* (GeV/c2)2
1.5
2
2.5
3
s (GeV’)
c x1.4
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 o, (M,2) = 0.120.
0.8
as an integral of these spectral functions the invariant mass-squared s of the final hadrons [26]:
0.6
over state
0.2
R,(Q) O0
0.5
1
1.5
2
2.5
=
127T&w /$(l-%,’
(8)
3
s (GA’*)
x [(1+2$)ImII~Pl(x+ir) Figure 6. Inclusive nonstrange vector (top) and axial-vector (bottom) spectral functions from ALEPH and OPAL. The dashed line is the expectation from the naive parton model.
+ Imnl”)(s+ir)]
By Cauchy’s theorem the imaginary part of IItJl is proportional to the discontinuity across the positive real axis. The energy scale so = rn: is large enough so that contributions from nonperturbative effects are small. It is therefore assumed that one can use the Operator Product E.xpansion (OPE) to organize perturbative and nonperturbative contributions [27] to R,(so). The theoretical prediction of the vector and
M. Daoier/Nuclrar
ratio Rr,v/A can thus be written
axial-vector
R r,V/A
Ph,vsic.sB (PKx.
as:
~lvudl%v
=
.711
SuppI.) 98 (2001) 305-313
(9)
with the residual non-logarithmic electroweak correction Sk, = 0.0010 [5], neglected in the following, and the dimension D = 2 contribution 6(2- mass1 from quark masses which is lower than ud,VIA
ALEPH V A V-l-A
JNP os(m2,) 0.020 f 0.004 0.330 f 0.014 f 0.018 110.339 & 0.013 f 0.018 1 -0.027 f 0.004 I] 0.334 f 0.007 f 0.021 1 -0.003 f 0.004
Table 2 Fit results of o,(mp) and the OPE nonperturbative contributions from vector, axial-vector and (V + A) combined fits using the corresponding ratios R, and the spectral moments as input parameters. The second error is given for theoretical uncertainty.
0.1% for U, d quarks. The term J(O) is the purely perturbative contribution, while the S(O) are the OPE form
terms
@ ud,V/A
-
in powers
c
of saD’2
of the following
(%d)V/A
~i~O=D(-so)D’2
(10)
where the long-distance nonperturbative effects are absorbed into the vacuum expectation elements (Oud). The perturbative expansion (FOPT) is known to third order [28]. A resummation of all known higher order logarithmic integrals improves the convergence of the perturbative series (contourimproved method FOPTcr) [29]. As some ambiguity persists, the results are given as an average of the two methods with the difference taken as a systematic uncertainty. 4.3. Measurements The ratio R, is obtained from measurements the leptonic branching ratios: R,
=
3.047 f 0.014
of
(II)
using a value B(T+ e- VeVT) = (17.794 It 0.045)% which includes the improvement in accuracy provided by the universality assumption of leptonic currents together with the measurements of a(~+ e- Y,y,), B(rt 1-1~VpvT) and the T lifetime. The nonstrange part of R, is obtained by subtracting out the measured strange contribution (see last section). Two complete analyses of the V and A parts have been performed by ALEPH [22] and OPAL [23]. Both use the world-average leptonic
branching ratios, but their own measured spectral functions. The results on o,(mf) are therefore strongly correlated and indeed agree when the same theoretical prescriptions are used. 4.4. Results of the Fits The results of the fits are given in Table 2 for the ALEPH analysis. Similar results are obtained by OPAL. It is worth emphasizing that the nonperturbative contributions are found to be very small, as expected. The limited number of observables and the strong correlations between the spectral moments introduce large correlations, especially between the fitted nonperturbative operators. One notices a remarkable agreement within statistical errors between the cr,(m;) values using vector and axial-vector data. The total nonperturbative power contribution to Rr,v+~ is compatible with zero within an uncertainty of 0.4%, that is much smaller than the error arising from the perturbative term. This cancellation of the nonperturbative terms increases the confidence on the o,(m:) determination from the inclusive (V + A) observables. The final result from ALEPH is : o&n”,)
= 0.334 f 0.007,,,
Zh 0.02I,h
(12)
where the first error accounts for the experimental uncertainty and the second gives the uncertainty of the theoretical prediction of R, and the spectral moments as well as the ambiguity of the theoretical approaches employed.
312
M. Dnuirr/Nuclenr
Physics
13 (Proc. Suppl.)
9X (2001) 305-313
1
1.25
1.5
1.75
2
2.25
2.5
2.75 sg
Figure 8. The ratio Rr,v+~ versus the square “r mass” se. 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 results of the fit at so = m:.
In the OPAL analysis the corresponding results are quoted within three prescriptions for the perturbative expansion, respectively FOPTcr, FOPT and with renormalon chains, i.e. =
o&n:)
4.5.
Test
0.348 f 0.009,,,
f 0.019th
(13)
=
0.324 zt 0.006,,,
f 0.013th
(14)
=
0.306 f 0.005,,,
f O.OlIth
(15)
of the Running
of (Y,(S)
at Low
3 (GM
Figure 9. The running of cr,(so) obtained from the fit of the theoretical prediction to R,,v+A(so). The shaded band shows the data including experimental errors. The curves give the four-loop RGE evolution for two and three flavours.
perturbative term. Fig. 9 has the same physical content as Fig. 8, but translated into the running of o,(so), i.e., the experimental value for o’s (so) has been individually determined at every sn from the comparison of data and theory. Good agreement is observed with the four-loop RGE evolution using three quark flavours. The experimental fact that the nonperturbative contributions cancel over the whole range 1.2 GeV2 5 SO < rn: leads to confidence that the o, determination from the inclusive (V + A) data is robust. 4.6.
Discussion
on
the
Determination
of
Energies
Using the spectral functions, one can simulate the physics of a hypothetical r lepton with a mass fi smaller than m, through equation (8) and hence further investigate QCD phenomena at low energies. Assuming quark-hadron duality, the evolution of R,(so) provides a direct test of the running of o,(so), governed by the RGE pfunction. On the other hand, it is a test of the validity of the OPE approach in r decays. The functional dependence of R,,v+~(sg) is plotted in Fig. 8 together with the theoretical prediction using the results of Table 2. Below 1 GeV’ the error of the theoretical prediction of RT,v+~(so) starts to blow up due to the increasing uncertainty from the unknown fourth-order
The evolution of the crJ (m:) measurement from the inclusive (V + A) observables based on the Runge-Kutta integration of the differential equation of the renormalization group to N3L0 [30,32] yields for the ALEPH analysis o,(M;)
= 0.1202~0.0008,,,~0.0024thf0.0010,,,~(16)
where the last error stands for possible ambiguities in the evolution due to uncertainties in the matching scales of the quark thresholds [32]. The result (16) can be compared to the determination from the global electroweak fit. The variable Rz has similar advantages to R,, but it differs concerning the convergence of the perturbative expansion because of the much larger scale.
M. Dauier/Nucleur
Pllysics B (Proc. Suppl.)
313
98 (2001) 305-313
Mass’
Figure 10. Evolution of the strong coupling (measured at rn:) to Mi predicted by QCD compared to the direct measurement. The evolution is carried out at 4 loops, while the flavour matching is accomplished at 3 loops at 2 m, and 2 mb thresholds.
It turns out that this determination is dominated by experimental errors with very small theoretical uncertainties, i.e. the reverse of the situation encountered in r decays. The most recent value [33] yields crJ(Mi) = 0.1183 f 0.0027, in excellent agreement with (16). Fig. 10 illustrates well the agreement between the evolution of os(m~) predicted by QCD and crJ (A&?). 5. Applications
to Hadronic
Vacuum
Po-
larization 5.1.
Improvements lat ions
From the learned that: l
studies
to the Standard
presented
above
Calcu-
we have
the I = 1 vector spectral function from T decays agrees with that from e+e- annihilation, while it is more precise for masses less than 1.6 GeV as can be seen on Fig. 11. Small CVC violations are expected at a few 10m3 level as discussed above, essentially from SU(2)-breaking in the r masses.
(G~V/C~)~
Figure 11. Global test of CVC using r and e+evector spectral functions.
l
the description of R, by perturbative QCD works down to a scale of 1 GeV. Nonperturbative contributions at 1.8 GeV are well below 1 % in this case. They are larger (- 2 %) for the vector part alone, but reasonably well described by OPE. The complete (perturbative + nonperturbative) description is accurate at the 1 % level at 1.8 GeV for integrals over the vector spectral function such as R,,".
These two facts have direct applications to calculations of hadronic vacuum polarisation which involve the knowledge of the vector spectral function: the muon magnetic anomaly and the running of cr. In both cases, the standard method involves a dispersion integral over the vector spectral function taken from the e+e-+ hadrons data. Eventually at large energies, QCD is used to replace experimental data. Hence the precision of the calculation is given by the accuracy of the data, which is poor above 1.5 GeV. Even at low energies, the precision can be significantly improved by using r data [7]. The next breakthrough comes about using the prediction of perturbative QCD far above quark thresholds, but at low enough energies (compati-