QCD condensates and $\alpha_{s}$ from $\tau$-decay

We shall be concerned with the two-point correlator :

built from the T-product of the bilinear axial-vector current of $u,d$ quark fields:

The upper indices (0) and (1) correspond to the spin of the associated hadrons. The two-point function obeys the dispersion relation:

where $\cdots$ are subtraction constants polynomial in $q^{2}$ and $t>$ is the hadronic threshold.

Within the SVZ-expansion [3], the two-point function can be expressed in terms of the sum of higher and higher quark and gluon condensates:

where $H\equiv V(A)$, $\mu$ is the subtraction scale which separates the long (condensates) and short (Wilson coefficients) distance dynamics and $m_{q}$ is the quark mass.

$\diamond~{}$In the chiral limit [4, 5]:

where $\rho$ measures the deviation from the factorization of the 4-quark condensates and the last equality for $d_{8}$ is based on vacuum saturation estimate of some classes of computed dimension-8 diagrams.

$\diamond~{}$The perturbative expression of the spectral function is known to order $\alpha_{s}^{4}$ [6, 7]. It reads for 3 flavours:

where :

where $-\beta_{1}=(1/2)(11-2n_{f}/3)$ is the first coefficient of the $\beta$-function and $n_{f}$ is the number of quark flavours; $\cdots$ stands for higher order terms which can e.g. be found in [8]. We shall use the value $\Lambda=(342\pm 8)$ MeV for $n_{f}=3$   from the PDG average [9].

We shall use the recent ALEPH data [10] in Fig. 1 for the spectral function $a_{1}(s)$.

Like in the case of the vector spectral function from $e^{+}e^{-}\to$ Hadrons [2], we subdivide the region from $3\pi$ threshold to $M_{\tau}^{2}=3.16$ GeV² into different subregions in $s$ (units in [GeV²]):

and fit the data with 3rd order polynomials using the optimized Mathematica program FindFit except for $[1.29,1.42]$ where a 2nd order polynomial is used. We show the different fits in the Fig. 2.

To order $\alpha_{s}^{4}$, the perturbative (PT) expression of the lowest moment reads [1]:

Then, taking its derivative in $\tau$, one gets ${\cal L}_{1}(\tau)$ and then their ratio ${\cal R}_{10}(\tau)$.

From Eq. 4, one can deduce the non-perturbative contribution to the lowest LSR moment :

from which one can deduce ${\cal L}^{NPT}_{1}$ and ${\cal R}^{A}_{10}$.

We show in Fig. 3a) the $\tau$ behaviour of the phenomenological side of ${\cal R}^{A}_{10}$ (experiment $\oplus$ QCD continuum beyond the threshold $t_{c}$) for values of $t_{c}$ around the physical $\tau$-lepton mass squared where the effect of $t_{c}$ is negligible.

For determining $d_{6,A}$ and $d_{8,A}$, we use as input the more precise value of $\langle\alpha_{s}G^{2}\rangle$ from the heavy quark mass-splittings and some other sum rules [14, 15]:

and perform a two-parameter fit $(d_{6,A},d_{8,A})$ by confronting the phenomenological and QCD side of $R^{A}_{10}$ for different values of $\tau$. The QCD coupling $\alpha_{s}$ is evaluated at the LSR sum rule scale $\tau$ where we use $\Lambda=(342\pm 8)$ MeV for $n_{f}=3$ flavours deduced from the PDG world average [9]. The results of the analysis are shown in Fig. 3. There is an inflexion point around $\tau\simeq 2.5$ GeV^-2 at which we extract the optimal values of the condensates. One should note that at this scale the OPE converges quite well in the vector channel [1]. Then, we may expect that for the axial-vector the same feature occurs. We obtain the optimal result:

where the 1st error comes from the fitting procedure and the 2nd one from the localization of $\tau\simeq(2.5\pm 0.1)$ GeV^-2.

We use the previous values of $d_{6,A}$ and $d_{8,A}$ into ${\cal R}^{A}_{10}$ and we re-extract $\langle\alpha_{s}G^{2}\rangle$ using a one-parameter fit. The analysis is shown in Fig. 4. We obtain:

in good agreement with the one in Eq. 12 used previously as input. This result also indicates the sef-consistency of the set of condensates entering in the analysis.

Using previous fits of the data, we show in Fig. 5 the behaviour of the lowest moment ${\cal R}_{0/A}$ versus an hypothetical $\tau$-lepton mass squared $s_{0}\equiv M^{2}_{0}$. At the observed mass value: $M_{\tau}=1.777$ GeV, one obtains:

It reproduces with high accuracy the ALEPH value [10] :

which is an important self-consistency test of our fitting procedure.

The QCD expression of ${\cal R}_{0,A}$ including the dimension six $(d_{6,A})$ and eight $(d_{8,A})$ condensates within the SVZ expansion [3] can be deduced from [4]. To simplify for the reader, we give its expression, in the chiral limit ²²2We use the complete expression including quark masses and $\alpha_{s}$ corrections in the numerical analysis.:

where the electroweak factors and corrections are :

The QCD corrections copied from BNP are:

$\diamond~{}$Perturbative corrections to order $\alpha_{s}^{4}$:

for fixed order (FO) [4] and contour improved (CI)  [16].

Observing that the PT series grows geometrically [17] from the calculated coefficients in different channels, we estimate the $a_{s}^{5}$ coefficient to be [1] :

which one can consider either to be the error due to the unknown higher order terms of the series or (more optimistically) to be the estimate of the uncalculated $\alpha_{s}^{5}$ coefficient.

$\diamond~{}$Power corrections up to $d_{8,A}$:

They read:

where $s_{0}\equiv M_{0}^{2}$ while $d_{6,A}$ and $d_{8,A}$ have been defined in Eqs. 5 . We have assumed $\langle\bar{\psi}_{u}\psi_{u}\rangle=\langle\bar{\psi}_{d}\psi_{d}\rangle$. We shall not include the $D=2$ contribution due to an eventual tachyonic gluon mass within the standard OPE. Using duality [17], this term can be included in the estimate of the non-calculated higher order terms of the PT series discussed previously. Instanton contributions are expected to have higher dimensions and their contributions can be safely neglected [1].

We confront the experimental and QCD sides of ${\cal R}_{0,A}$ for different values of $s_{0}$. Like in the case of the ratio of Laplace sum rules, we use a two-parameter fit ($d_{6,A},d_{8,A}$) to extract the values of these condensates using as input the values of the $d_{4}$ condensates and light quark masses. We consider the PT series up to order $\alpha_{s}^{4}$ and evaluate $\alpha_{s}$ at the hypothetical $\tau$-mass squared $s_{0}=M_{0}^{2}$. We use $\Lambda=(342\pm 8)$ MeV for $n_{f}=3$ flavours from the PDG world average [9]. The results versus $s_{0}$ are shown in Fig. 6a).

We consider as a reliable value the one from $s_{0}=1.93$ GeV² beyond the peak of the $A_{1}$ meson. We notice a stability (minimum) around 3 GeV² just below the physical $\tau$ mass which we consider as our optimal value:

One can notice from the analysis that the absolute values of the condensates are slightly higher for FO than for CI. To be conservative, we take the arithmetic average of the FO and CI values and add as a systematic the largest distance between the mean and the individual value :

We use the previous values of $d_{6,A}$ and $d_{8,A}$ together with the one of $\langle\alpha_{s}G^{2}\rangle$ in Eq.12 as inputs in the lowest BNP moment ${\cal R}_{0,A}$ in order to determine $\alpha_{s}(M_{\tau})$. We show in Fig. 7, the behaviour of $\alpha_{s}(M_{\tau})$ versus an hypothetical $\tau$ mass squared $M_{0}^{2}\equiv s_{0}$. One can notice an inflexion point in the region $2.5^{+0.10}_{-0.15}$ GeV² at which we extract the optimal result. The conservative result from $s_{0}=2.1$ GeV² to $M_{\tau}^{2}$ (see Fig. 7) is :

The 1st error in $\alpha_{s}(M_{\tau})|_{A}$ comes from the fitting procedure. The 2nd one comes from an estimate of the $\alpha_{s}^{5}$ contribution from Ref. [1]. At the scale $s_{0}=$2.5 GeV² the sum of non-perturbative contributions to the moment normalized to the parton model is:

One can notice that extracting $\alpha_{s}(M_{\tau})|_{A}$ at the observed $\tau$-mass tends to overestimate the result:

These values extracted at $M_{\tau}$ agree with the ones from Ref. [18] obtained at the same scale. The same feature has been observed in the case of $e^{+}e^{-}\to$ Hadrons.

We compare our results with the ones from $e^{+}e^{-}$ and $\tau$-decay Vector channel [1] and with the results obtained by different authors in the Axial-Vector channel:

$\diamond~{}$Our values of $d_{6,A}$ and $d_{8,A}$ are in good agreement within the errors with the ones of Ref. [18] but about two times larger than the ones of Ref.[10].

$\diamond~{}$The value of $d_{8,A}$ suggests that the assumption in Eq. 5:

is not satisfied by the fitted values given in Table 1.

We use a two-parameter fit to extract $(d_{10,A},d_{12,A})$ from ${\cal R}_{2,A}$, $(d_{12,A},d_{14,A})$ from ${\cal R}_{3,A}$. The $s_{0}$ behaviour of the results is given in Fig. 8.

We deduce from ${\cal R}_{2,A}$:

inside the stability region 1.93 GeV² to $M_{\tau}^{2}$. We take as a final value of $d_{10,A}$ the mean from ${\cal R}_{1,A}$ in Eq. 25 and the one from ${\cal R}_{2,A}$ in Eq. 34:

We do the same procedure as previously for higher dimension condensates $n\geq 6$. The $s_{0}$ behaviours of the condensates are shown in Figs. 9,  10. The results are summarized in Table 2.

$\diamond~{}$One can notice that the condensates in the A channel has alternate sign.

$\diamond~{}$Their size is almost constant and more accurate than previous determinations in the literaure. There is not also any sign of an exponential growth. This feature in the Euclidian region does not favour a sizeable duality violation in the time-like region [19].

Its corresponds to the V–A quark current:

In the chiral limit, the QCD expression of the corresponding two-point function is similar to the one in Eq. 5 where:

The spectral function has been measured by ALEPH [10] which we show in Fig.,̇11.

In order to fit the data, we proceed like in the case of the axial-vector channel by subdividing the region into 5 subregions :

in units of GeV². We use 3rd order polynomials. The results of the fits are given in Fig. 12.

In the chiral limit their expression is similar to the previous ones for the axial-vector current modulo an overall factor 2 and the change into the condensates contributions $d_{D,V-A}$ of dimension $D$. We show the $s_{0}$ behaviour of the moment in Fig.13.

For $s_{0}=M_{\tau}^{2}$, we obtain :

to be compared with the ALEPH data [10]:

where our error is larger which may due to the fact that we have separately fitted the upper and lower values of the data.

We shall use the values of the corresponding condensates from the $e^{+}e^{-}\to$ Hadrons in Ref. [1, 2] and the ones from the axial-vector channel obtained in the previous section. They read:

where the error is the largest relative % error from each channel. We note (as can be found in Ref. [4]) that for ${\cal R}_{0,V-A}(s_{0})$, the contribution of $\langle\alpha_{s}G^{2}\rangle$ is $\alpha_{s}$ suppressed while the one of $d_{6,V-A}$ is smaller than in the individual V and A channels. Then, we expect that V–A is a golden channel for extracting $\alpha_{s}$.

Using the value of $\langle\alpha_{s}G^{2}\rangle$ in Eq. 12 and the previous values of $d_{6,V-A}$ and $d_{8,V-A}$ in Eq, 41, we extract the value of $\alpha_{s}(M_{\tau})$ as a function of $s_{0}$ (see Fig. 14) from ${\cal R}_{0,V-A}(s_{0})$. We notice a stable result in the region $s_{0}\simeq(2.5\sim 2.6)$ GeV² though not quite convincing. The, we consider as a conservative value the one obtained from a least-square fit of the values inside the region $[2,5,M_{\tau}^{2}]$. The optimal result corresponds $s_{0}=2.8$ GeV² (see Fig. 14):

The 1st error in $\alpha_{s}(M_{\tau})|_{V-A}$ comes from the fitting procedure. The 2nd one from an estimate of the $\alpha_{s}^{5}$ contribution from Ref. [1]. At this scale the sum of non-perturbative contributions to the moment normalized to the parton model is:

which is completely negligible. One can notice from Fig. 14 that extracting $\alpha_{s}(M_{\tau})|_{V-A}$ at the observed $M_{\tau}$-mass tends to overestimate its value :

In Table 3, we compare our results with some other determinations [10, 18, 20] ³³3Some estimates including renormalon within a large $\beta$ approximation [ resp. duality violation] effects can be e.g. found in Refs. [21] [resp. [22]]. obtained at the scale $s_{0}=M_{\tau}^{2}$.

$\diamond~{}$There is a quite good agreement for $d_{6,V-A}$ but not for $d_{8,V-A}$ from different determinations.

$\diamond~{}$The central values of the condensates given in Table 7 of Ref. [18] using a truncation of the OPE up to $D=10$ are systematically smaller than the ones in their Table 8 using the OPE up to $D=12$ though they agree within the errors.

$\diamond~{}$Extracting $\alpha_{s}$ at $M_{\tau}$, there is a good agreement among different determinations where the values are slightly higher than the ones from the optimal region given in the first row  (see Fig. 14).