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Asymptotic structure of perturbative series for tau lepton observables

We analyze tau lepton decay observables, namely moments of the hadronic spectral density in the finite energy interval (0,M_\tau), within finite order perturbation theory including \alpha_s^4 corrections. The start of asymptotic growth of perturbation theory series is found at this order in a scheme invariant manner. We establish the ultimate accuracy of finite order perturbation theory predictions and discuss the construction of optimal observables.


Introduction
The study of τ lepton decays provides a wealth of information on low energy hadronic physics where the accuracy of experimental data is permanently improving [1,2]. The central quantity of interest is the hadronic spectral density. The spectral density has been calculated with a very high degree of accuracy within perturbation theory (e.g. [3,4,5]). The structure of observables -related to the two-point correlator of hadronic currents with well established and simple analytic properties -makes the comparison of experimental data with theoretical calculations very clean. All these features make τ lepton physics an important area of particle phenomenology where theory (QCD) can be confronted with experiment to a very high precision [6,7,8,9,10].
In the present note we show that within the finite order perturbation theory analysis the ultimate theoretical precision has been reached already now. The limit of precision exists due to the asymptotic nature of the perturbation theory series.
The actual magnitude of this limiting precision depends on the numerical value of the coupling constant which is the expansion parameter. We perform our analysis and reach our conclusions in a renormalization scheme invariant way.
The normalized τ lepton decay rate into hadrons h is given by with R exp τ = 3.649 ± 0.014 and δ exp = 0.216 ± 0.005 .
The first term in eq. (1) is the parton model result while the second term δ represents the effects of QCD interaction. In this paper we neglect electroweak corrections altogether. The theoretical expression for the rate is given by The spectral density ρ(s) is related to Adler's D-function through the dispersion relation D(Q 2 ) = Q 2 ρ(s)ds (s + Q 2 ) 2 .
where the running coupling is normalized at the scale µ = Q. The light quarks u, d and s are taken to be massless. Eqs. (3)(4)(5) constitute the full theoretical information necessary for the perturbation theory analysis of the τ system. The fourth order MS-scheme coefficient k 3 is not known at present.
In the present note we do not systematically discuss non-perturbative effects stemming from standard power corrections [11]. Also, the infinite resummation of the perturbation theory series different from the standard renormalization group improvement is used only as a toy example [12]. The standard power corrections due to nonvanishing vacuum expectation values of local operators within operator product expansion are relatively small and can be simply accounted for if desired. The coefficient functions of local operators are known in low orders of the perturbation theory expansions and there is no necessity to thoroughly analyze their convergence properties. It is the high precision achieved in the experimental analysis of τ decays and the rather advanced stage of theoretical description that calls for a detailed analysis of the physics of the τ system.

τ system observables
The central quantity of interest in the τ system is the hadronic spectral density which can be measured in the finite energy interval (0, M τ = 1.777 GeV). Being a distribution (in theory) or a rapidly varying function in the vicinity of resonances (in experiment) the hadronic spectral density cannot be analyzed pointwise within perturbation theory. The appropriate quantities to be analyzed are the moments (or generalized Fourier components over a chosen complete set of test functions). We define moments of the spectral density by (with M τ chosen to be the unity of mass) Due to the completeness of the basis {s n : n = 0, . . . , ∞} the moments m n contain the entire information about ρ(s). The invariant content of the investigation of the spectrum, i.e. independent of any definition of the charge, is the simultaneous analysis of all the moments. Note that within finite order perturbation theory the moments eq. (6) coincide with the results of contour integration [13,14,15,16] because of analytic properties of the functions ln p s.
In order to get rid of artificial scheme-dependent constants in the perturbation theory expressions for the moments we define an effective coupling a(s) directly on the physical cut through the relation ρ(s) = 1 + a(s) .
All the constants that may appear due to a particular choice of the renormalization scheme are absorbed into the definition of the effective charge e.g. [17,18,19,20].
Note that if there was no running (as in the conformal limit of QCD with vanishing β-function or at the infrared fixed point) then the whole physics of the τ system in the massless approximation (without strange particles, for instance, and including only perturbative corrections without possible power corrections) would reduce to the determination of a single number a(M τ ) ≡ a and consequently there would not be any problems with the convergence of the perturbation theory series. Because of the running of a(s), however, different observables, i.e. different moments of the spectral density, generate different perturbation theory series from the original object ρ(s) in eq. (7). The whole set of moments needs to be analyzed in a scheme invariant way e.g. [21,22,23]. Note that the introduction of a natural internal coupling parameter such as the effective charge a(s), allows one to extend the perturbation theory series needed for the description of relations between observables by one more term as compared to the analysis in e.g. the MS scheme (e.g. [21,24]). When defining the effective charge directly through ρ(s) itself we get theoretical perturbative corrections to the moments only because of running. Without running one would have and the perturbation theory analysis would be over (we neglect power corrections for the moment!). In any given order of perturbation theory the running of the coupling a(s) defined in eq. (7) contains only logarithms of s with coefficients given by an where a = a(M 2 τ ), L = ln(M 2 τ /s). The contributions of powers of logarithms to the normalized moments are (n + 1) 1 0 s n ln p (1/s)ds = p! (n + 1) p .
Therefore, at fixed order of perturbation theory the effects of running die out for large n improving the convergence of perturbation theory series. With the definition of the charge according to eq. (7) all high order corrections vanish at n → ∞ at any fixed order of perturbation theory. With running one has instead of eq. (8) For large n the moments behave better because the infra-red region of integration is suppressed. Note that the coefficients of the series in eq. (11) are saturated with the lowest power of logarithm for large n for a given order of perturbation theory, i.e.
they are saturated with the highest coefficient of the effective β-function.
Higher moments are not welcome from the experimental point of view. They are dominated by the contributions coming from the high energy end of the τ decay spectrum (therefore they converge better perturbatively) but experimental accuracy for the moments basically deteriorates with increasing n because poorly known contributions close to the right end of the interval are enhanced. To suppress experimental errors from the high energy end of the spectrum the modified system of mixed mo- can be used e.g. [25]. The weight function (1−s) k s l has its maximum ats = l/(l +k).
The integral in eq. (12) is dominated by contributions from around this value. A disadvantage of choosing such moments is that the (1 − s) k factor enhances the infrared region strongly and ruins the perturbation theory convergence. As an example one hasm The values of the coefficients in the series eq. (13) can be found in a concise form for arbitrary k at any giving finite order of perturbation theory. For instance, the contribution of the log-term is given by In contrast to eq. (10) it increases as ln(k) for large k making the coefficients of the perturbation theory series large. The contribution of log 2 -term reads and can be seen to grow as ln 2 (k) for large k.
In practical applications our formal criterion of the accuracy which the series provides is given by the numerical magnitude of the last term of the series. However, this criterion should be applied with great caution. Because of the freedom of the redefinition of the expansion parameter the last term of the series can always be made arbitrary small for any given observable. One can give an invariant meaning to the quality of the perturbation theory expansion only for a set of observables.
Before proceeding we would like to comment on the contribution of power corrections to the systems of moments eqs. (6,12) and the interplay between the magnitude of this contribution and the structure of the perturbation theory series. For the system of moments in eq. (6) the contribution of power corrections reduces to a single term (neglecting the weak log(Q) dependence of coefficient functions of the operator product expansion which is a common practice) of the form (Λ 2 /M 2 τ ) n which decreases very fast with n (Λ is a typical scale of power corrections related to the non-perturbative scale of QCD and Λ < M τ ). This makes the perturbation theory contribution dominant in the total result. This perturbative term for the large n moments is saturated with high energy contributions and therefore converges perturbatively. The convergence becomes even better with increasing n. The moments are perturbatively dominated and, therefore, precise. On the contrary, for the system of mixed moments in eq. (12) with l ∼ 0 the large k moments are saturated with low energy contributions, i.e. basically with the contribution of the ground state resonance, and therefore are completely nonperturbative which is reflected in the fast deterioration of perturbative convergence. The contribution of power corrections to the moments eq. (12) picks out many terms in all orders from n = 2 to n = k + 1.
Nothing definite can be said about such a sum of power corrections in any realistic case. This also indicates the importance of power corrections for mixed moments. The perturbation theory series is the same both for the vector and axial vector channels while the lowest resonance contributions are completely different (the pion instead of the ρ-meson). Therefore no method of summation of perturbation theory series can bring it to agree with experiment. In this case perturbation theory is in trouble and the power corrections provide the correct result for large k. Large k mixed moments, therefore, are not usable within the perturbation theory framework even if they are preferable from the experimental point of view.
Thus, one faces the usual clash between experimental and theoretical accuracy which is reflected in our case in the range of (k, l) values for the mixed moments that are chosen as optimal observables. Having explicit perturbation theory formulas at hand one can establish the ultimate theoretical accuracy implied by the asymptotic character of perturbation theory series for a given experimental observable with any stated precision. This allows one to conclude which error -experimental or theoretical -dominates the uncertainty of an observable related to τ decay physics.
As mentioned before the numerical value of the coefficient k 3 is unknown at present.
In some of the following evaluations we want to fix its value to have a feeling of the importance of the last term of the perturbation theory series. One popular value is Formally, the convergence still persists even in eq. (17) if one only requires subsequent terms of the series to decrease but the convergence is very slow. Also, the total contribution of the four higher order terms is more than 60% of the leading one. For the first moment the convergence is considerably better The convergence for the moments m 1 -m 4 (and for n > 4) is fine. The total contribution of higher order corrections is small. The worst series is the one for the zeroth order moment. Eq. (20) shows that no choice of k 3 yields accuracy for both m 0 and m 1 which is essentially better than the fourth order term. In fact, there is a narrow window 40 < k 3 < 60 where the formal criterion of convergence is satisfied for both m 0 and m 1 but we do not find it natural to rely on such a fine tuning and even then the accuracy of the zero moment is only about 10%. This is an indication that the ultimate accuracy of the perturbation theory expansion for the zeroth moment has been reached. If the moment m 0 is excluded the choice k 3 ∼ 100 allows one to make the convergence fast even to fifth order and no conclusion about an asymptotic growth is possible.
The perturbation theory expansions for the system of moments with (1 − s) k weight shows worse behavior. With the above criterion of accuracy, the precision which is given by the series from eq. (13) is of order 10% -20% for the numerical value of a. This is not enough for a comparison with experiment at the present level of precision. For instance, an expansion of the higher moments in eq. (13) in terms of the first one (which is the most perturbative one for this system) goes as follows These series possess a formal accuracy of from 6% to 25% and the contribution of higher order terms can be as large as the leading term. Because of the slow convergence there is no sign of improvement with higher order of the perturbation theory: the series expansions do not allow any reliable estimate of accuracy for large mixed moments. Also while for the moments eq. (6) the total contribution of corrections is small, the situation is different here. The total change of the leading order result due to higher order corrections is considerable and strongly differs for various moments. This is another indication that the set of mixed moments is not commensurate perturbatively.

τ decay rate
The τ decay width is given by a specific linear combination of moments. Because of the factor (1 − s) 2 present in eq. (3) the convergence property of the total decay rate observable is not optimal. The (1 − s) 2 factor enhances the infra-red region of integration, i.e. the relative magnitude of the contributions of logarithms ln(M 2 τ /s) at small energy. The concrete shape of the weight function with the weight factor (1 − s) 2 is the main source of slow convergence. One has r τ = a + 3.563a 2 + 24.97a 3 + 174.8a 4 + (1041. + 7.125k 3 )a 5 .
Using a = 0.111 and k 3 = 25 r τ = 0.111 + 0.044 + 0.034 + 0.027 + "0.021 ′′ Formally the consecutive terms decrease but the decrease is very slow. One can see that the pattern of convergence mainly follows that of the momentm 20 from eq. (13) because of the factor (1 − s) 2 in eq. (3).
Eqs. (2) and (23) show the essence of the problem we are addressing. In the finite order perturbation theory analysis one has to compare δ exp with δ th 0.216 ± 0.005 = δ exp vs δ th = 0.111 + 0.044 + 0.034 + 0.027 + "0.021 ′′ (24) and the uncertainty of the theoretical expression "0.021" (or even 0.027) is much larger than the experimental error 0.005. Thus, the theoretical uncertainty due to the truncation of the perturbation theory series is much larger than the experimental error of the corresponding observable. The common practical expectation for theoretical perturbation theory expansions to be useful is the smallness of the total higher order corrections if nothing is known about the convergence of the expansion. For the rate observable the corrections increase the leading order result by a factor of 2.
Note that one can improve the explicit convergence of the rate observable by a special redefinition of the expansion parameter due to renormalization scheme freedom.
However, then the first moment of the differential decay rate will behave wildly. It is this feature that prompts us to reach definite conclusions about the asymptotic growth of perturbation theory expansion independent of any scheme. Two different sets of observables where one set includes the moment m 0 and the other set does not include it are not perturbatively connected with an accuracy required by experiment.
The fifth order term is larger than the fourth order term for k 3 = 25. No choice of k 3 can simultaneously make all these three observables convergent at fifth order. If one chooses k 3 ∼ 100 in order to guarantee for a better convergence of the higher moments (which is physically motivated) one almost destroys the perturbation theory series for the decay rate eq. (22).
The (1 − s) n moments of the differential decay rate suppress poorly known high energy experimental data. Taking n = 1 as an example one has which gives only about 30% accuracy and more than a factor 2 change of the leading order term. We conclude that the theoretical precision cannot compete with the experimental precision.
There are two distinct problems in analyzing τ decays: one is to describe the set of observables of the system using its internal coupling parameter defined to get the highest precision and establish whether the set is perturbatively commensurate, while another is to extract the standard MS parameters. It can happen that the set of observables is perturbatively connected with some given accuracy but the MS coupling α s is not the best parameter for the expansion. This is the case here. In internal terms the τ system is described with higher accuracy in terms of the number of perturbation theory terms than in the MS scheme. However, at this level of expansion one sees the asymptotic growth of the perturbation series for the numerical value of the expansion parameter fixed by experiment.
The expression for the decay rate in the MS scheme possesses only O(α 3 s ) accuracy where the only free parameter is k 3 because the β 3 coefficient in the MS scheme is known [26].
The best estimate is formally given by the sum of the first ten terms m f ix,best according to the formal prescription for the the evaluation of precision. The exact result m f 1 = 0.130954, however, does not fall into the tiny interval given by the error bars in eq. (37). Therefore the formal criterion of the accuracy is violated in this case: the discrepancy m f ix,best 1 − m f 1 = 0.00184 is not controlled by the smallest term of the asymtpotic expansion eq. (36). Still this discrepancy is small and the actual accuracy for the first moment given by the asymptotic expansion eq. (36) is 1.3%. This suffices for practical purposes. One can easily see the difference between these two observables which reflects the different numerical magnitude of the infra-red contributions. Note also that the numerical magnitude of the smallest term of the expansion eq. (36) is very sensitive to the value of the third coefficient of the effective β-function. For the decay rate we find r f τ = 0.1946 and r f τ (1) = 0.1527 to be compared with eq. (22,23,24) and eqs. (25,26). ¿From eq. (34) one sees that an infra-red fixed point exists also in fourth order of the effective β-function for any k 3 < 95.9. For these numerical values the effect of k 3 on the exact moments within the model is rather weak. The pattern of "convergence" for the decay rate is mainly determined by the contribution of the zeroth order moment (or even by the mixedm 20 moment) and is very close to the expressions in eqs. .
In this model there are ways of accelerating the convergence with nice results but they definitely cannot be justified for use in the general case. Still our conclusion about the achievable precision within finite order perturbation theory in fifth order remains valid.

Conclusions
Using the standard estimate of the accuracy of an asymptotic series we have found that the theoretical precision in the perturbative description of τ -decay is already limited by the asymptotic growth of the coefficients in fifth order of perturbation theory. This is a scheme invariant statement. The accuracy of perturbative expansions for a reasonably general set of observables cannot be better than 5% -10%.
Taking a stricter attitude we claim that the zeroth order moment is not computable within perturbation theory. Any consistent description of τ -decay data at fifth order of perturbation theory requires exclusion of the zeroth order moment from the list of observables (or it should not constitute a dominant contribution). At fifth order of perturbation theory and with the present numerical value of the coupling, the first two moments of the spectral density are too different to be simultaneously treated by perturbation theory with an accuracy better than 5% -10%. Therefore one has to go beyond finite order perturbation theory to compare these two observables if one requires a theoretical accuracy that exceeds present experimental accuracy. This implies the use of some procedure of resummation. The resummation procedure is not defined uniquely and the result depends on the prescription chosen [16,24,27].
Moreover, if one resums the infinite number of perturbation theory terms the condensates have no invariant meaning anymore and their numerical values may change [24]. Therefore, improving the theoretical accuracy for this system seems to require the creation of a new paradigm.
The extraction of α s from the τ -decay rate and its comparison with α s -values determined from other experiments does not appear to be the best test of perturbation theory for the τ system. The crucial test of the applicability of perturbation theory for the τ system would be the simultaneous calculation of two observables (moments) with an appropriate accuracy. If the set of moments includes the zeroth moment then the ultimate accuracy of finite order perturbative expansions has been already reached.