The strong coupling from ALEPH tau decays

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Nuclear and Particle Physics Proceedings 282–284 (2017) 144–148 www.elsevier.com/locate/nppp

The strong coupling from ALEPH tau decays Antonio Rodr´ıguez S´anchez Departament de F´ısica Te`orica, IFIC, Universitat de Val`encia – CSIC, Apt. Correus 22085, E-46071 Val`encia, Spain,

Abstract The strong coupling from ALEPH tau decays. We use the publically available non-strange spectral function from ALEPH tau decays to critically analyze the different determinations of α s (m2τ ) that can be found in the literature and the numerical impact of their possible weaknesses. We also introduce some novel approaches. We find that perturbative uncertainties dominate. Our results with different approaches are very stable. Our final value is α s (m2τ ) = 0.328±0.013 Keywords: QCD, Strong Coupling, Tau Decays

1. Introduction One of the most powerful tests of asymptotic freedom of QCD comes from the determination of the strong coupling from inclusive τ decays [1, 2]. In this work we summarize our recently made determination of Ref. [3]. The hadronic decay width can obtained from [4] −

Γ[τ → ντ hadrons] Γ[τ− → ντ e− νe ]  2  m2τ s ds 1 − = 12π S EW m2τ m2τ 0    s 1 + 2 2 Im Π(1) (s) + Im Π(0) (s) , mτ

Rτ ≡

(1)

where S EW = 1.0201 ± 0.0003 contains the renormalization-group-improved electroweak correction [5–7] and   (J) |Vuq |2 Π(J) (s) + Π (s) , (2) Π(J) (s) ≡ uq,V uq,A q=d,s

are the two-point correlation function of quark currents. Since one can extract the invariant-mass and spin of the final hadronic system of the τ decays, one has experimental access to the different spectral http://dx.doi.org/10.1016/j.nuclphysbps.2016.12.027 2405-6014/© 2016 Published by Elsevier B.V.

J (s). We make use of the most prefunctions Im Πuq,J cise non-strange spectral functions available ρ(s) = (1+0) 1 1 π Im ΠJ (s) ≡ π Im Πud,J (s), coming from the last update of the ALEPH collaboration [8].

2. Theoretical Framework 2.1. Operator Product Expansion (OPE) of the correlators For large-euclidean momentum, the correlators can be expanded into series of local operators weighted by their Wilson coefficients, which can be calculated using perturbative QCD [9]. At Q2 ∼ m2τ , the numerical contribution to the different observables is dominated by the purely perturbative part. In order to compare the theoretical OPE prediction with the experimental data, we can make use of the analytic extension of the correlator, which is well defined in all the complex the plane but in the hadronic cut in the positive real axis. Using this, it is straightforward to obtain the exact relation [4, 10, 11]  s0 ds ω ω(s) Im ΠV/A (s) AV/A (s0 ) ≡ sth s0

i ds = ω(s) ΠV/A (s) , (3) 2 |s|=s0 s0

A. Rodríguez Sánchez / Nuclear and Particle Physics Proceedings 282–284 (2017) 144–148

where ω(s) is any weight function analytic in all complex plane except in the positive real axis. We will be able to extract the experimental AωV/A (s0 ) from the second term of Eq. (3) and the theoretical one from the third term of the same equation using the OPE of the correlator. 2.2. Perturbative contribution The dominant contribution to AωV/A (s0 ∼ m2τ ) is purely perturbative. It is precisely this fact, as well as the high sensitivity of this contribution to α s (s ∼ m2τ ), which allows such a precise determination. In order to calculate the purely perturbative contribution, one can make use of the Adler function [12], known up to 4 loops [13–18]: D(s) ≡ −s

d ΠP (s) 1  ˜ = Kn (ξ) ans (−ξ2 s) , (4) ds 4π2 n=0

where K˜ n are known up to n = 4 and a s (m2τ ) ≡ satisfies the renormalization group equation 2

s da s  = βn ans (s) . a s ds n=1

α s (m2τ ) π

(5)

In order to solve the integral of Eq. (3) one can either expand Aω,P (s0 ) in a fixed order in α s (ξ2 s0 ) (FOPT) or use the exact solution to the differential equation (5) in the βn>5 = 0 approximation, which resums large logarithms (CIPT) [10, 19]. For a given perturbative approach, FOPT or CIPT, we cut Eq. (4) in n = 5 taking K5 = 275 ± 400 [20] and varying the scale dependence that arises from the fact we are cutting the series in the interval ξ2 = {0.5, 2} as a conservative estimates of perturbative uncertainties. The difference between FOPT and CIPT is for some moments Aω (s0 ) larger than these perturbative uncertainties precisely because of the large logarithms that CIPT resums. Taking this into account, and in the absence of a better understanding of higher perturbative corrections, we average the FOPT and CIPT and add quadratically half of the difference between both values to give a conservative final value. 2.3. Non-Perturbative contribution The non-perturbative contributions due to the D ≥ 4 operators to a given moment Aω,P can be safely aproximated as functions of effective dimensional condensates OD  OD, V/A Aω,NP a−1, D D/2 , (6) V/A (s0 ) = π s0 D with

ω(−s0 x) =



145

an, D xn+D/2 .

(7)

n

The moment associated to Rτ , whose weigth function is ω(s0 x) = (1 − x2 )(1 + 2x), is only sensitive the D = 6 and the D = 8 condensates, supressed by m6τ and m8τ . Together with the cancellation of O6 in the V + A channel, the D ≥ 4 contribution to Rτ happens to be very supressed [4, 21]. In addition to the non-perturbative OPE contributions, one has to take into account the differences between the physical correlators and their OPE aproximant. These differences are known as quark-hadron duality violations (DVs) [22–29]. Using Eq. (3), the contribution of duality violations to the physical observables studied are given by

 i ds (s ) ≡ ω(s) ΠV/A (s) − ΠOPE ΔAω,DV 0 V/A (s) V/A 2 |s|=s0 s0  ∞ ds ω(s)ΔρDV (8) = −π V/A (s) . s0 s0 These DVs are reduced using pinched weight functions [4, 11], which are functions that avoid the contribution to the integral of the region near the cut in the positive real axis, where the OPE is badly defined. Additionally, they decrease very fast with the opening of the higher multiplicity hadronic thresholds [30]. Therefore, they become very small at s0 ∼ m2τ , specially in the more inclusive channel V + A. We are able to make reliable and conservative estimates of DV uncertainties looking at the stability of the strong coupling determination both by taking moments that depend on these DVs in different ways and changing s0 . 3. Results 3.1. ALEPH-like fits First we reproduce the determination of the ALEPH collaboration [8]. They take s0 = m2τ and the moments associated to the weight functions 

s ωkl (s) = 1 − 2 mτ

2+k 

s m2τ

l 

 2s 1+ 2 , mτ

(9)

with (k, l) = {(0, 0), (1, 0), (1, 1), (1, 2), (1, 3)}. For these moments, duality violations are supressed by, at least, double pinching. They depend on α s and O4,6,8...16 . The fit becomes possible when one neglects the contribution of the higher energy condensates, whose contribution is supressed by powers of m2τ (Eq. 6).

A. Rodríguez Sánchez / Nuclear and Particle Physics Proceedings 282–284 (2017) 144–148

146

α s (m2τ )

Method

3.2. Other fits, same results

CIPT

FOPT

Average

ALEPH moments

0.339 +− 0.019 0.017

0.319 +− 0.017 0.015

0.329 +− 0.020 0.018

Modified ALEPH moments

0.338 +− 0.014 0.012

0.319 +− 0.013 0.010

0.329 +− 0.016 0.014

0.336 +− 0.018 0.016

0.317 +− 0.015 0.013

0.326 +− 0.018 0.016

A

(2,m)

moments

s0 dependence Borel transform

0.335 ± 0.014 0.323 ± 0.012 0.329 ± 0.013 0.328 +− 0.014 0.013

0.318 +− 0.015 0.012

0.323 +− 0.015 0.013

Table 1: Summary of the most reliable determinations of α s (m2τ ), performed in the V + A channel.

In our second determination, we apply the same procedure than in section 3.1 but removing the factor (1 + 2 ms2 ), so that the new fit does not depend on O16 τ and there is a re-weighting in the dependence on the neglected and non-neglected condensates and in the possible DVs. The strong coupling value obtained (line two of Table 1) is in perfect agreement with the previous one. If one wants to be free of the dependence on the less supressed condensate O4 but keeping the double pinch, one can use the set of weights: ω

(2,m)

(s0 x) = (1 − x)

2

m 

(k + 1) xk

k=0

= 1 − (m + 2) xm+1 + (m + 1) xm+2 . (10) with m ≥ 1. Again, even with a completely different dependence of the moment fitted on the neglected and non-neglected condensates and on DVs, the determination of the strong coupling (line 3 of Table 1) is in total agreement with the previous one, which again reinforces the idea that they are conservatively accounted in the quoted errors. Figure 1: Dependence on s0 of the experimental moment associated to the pinched weight functions ω(s), together with their purely perturbative predictions with α s (m2τ ) = 0.329 +− 0.020 0.018 . Data points are shown for the V (red), A (green) and 12 (V + A) (blue) channels.

The small tension observed for the results of the semiinclusive channels V and A motivates new checks in order to see the role of the neglected nonperturbative uncertainties. In order to estimate them, we repeat the same fit but including one more condensate in the fit. The small variation on the fitted value of the strong coupling has been included as an additional uncertainty. The results are shown in the first line of Table 1. In Ref. [31] this kind of determinations is criticized because of the lack of stability under s0 . In Figure 1 we perform a simple test. We take the only moment independent on the dimensional condensates, associated to the weight function ω(s) = 1. Even being an unprotected moment against DVs, so they are expected to be larger than for the moments we used in the fit, we can see how the purely perturbative predictions perfectly agree with the data at s0 ∼ m2τ . Particularly, the perturbative predicition is valid up to very low energies for the most inclusive channel V + A, the one we use for the strong coupling determination, showing a stability under s0 even better than expected.

3.3. Playing with the s0 -dependence One can try to find a way of fitting the strong coupling without having to neglect any higher-dimensional condensate and without removing the pinching protection. In order to do that, one can fix a moment, for example  2 ω(2,0) (s) = 1 − ss0 , and take all the Aω(2,0) (s0 ) with s0 above a threshold sˆ0 to fit α s , a sGG and O6 . We observe a good agreement among channels for the largests values of sˆ0 . However, when we go to lower energies, the fit quality becomes really bad and, due to duality violations, we are not able to extract a reliable determination of α s in the separate V and A channels if we require stability under s0 . However, the most inclusive channel, V + A, presents a very good stability, which allows to extract α s (m2τ ) adding as an additional uncertainty the small fluctuations observed when changing the number of fitted bins. One could think that the plateau might be temporary, but in Figure 2 we show how the stability extend to very low energies. A determination based on these ideas but taking additional moments (see Ref. [3] for more details), leads to the values of fourth line of Table 1, again in a very good agreement with the previous determinations. One can easily explain why DVs affect so much to the quality fit. The problem is that fitting m n-pinched

A. Rodríguez Sánchez / Nuclear and Particle Physics Proceedings 282–284 (2017) 144–148

147

Figure 3: FOPT determination of α s (m2τ ) from the s0 dependence of A(0,0) V (s0 ), fitting all s0 bins with s0 > sˆ0 , as function of sˆ0 , using the approach of Ref. [31].

Figure 2: Determination of the strong coupling using all the points with s0 > sˆ0 for the moment Aω(2,0) (s0 ) in the V + A channel ignoring DVs in CIPT.

consecutive points A(n,0) (s0 ) is equivalent to a fit to:

(n,0) A (s0 ) , A(n−1,0) (s0 ) , · · · , A(0,0) (s0 ) ,  ρ(s0 ) , ρ(s0 + Δs0 ) , ..., ρ(s0 + (m − n − 2)Δs0 ) (11) Therefore, we are removing pinch and then fitting the spectral function. Since the OPE is not expected to reproduce the s0 -dependence of the spectral function, the bad quality of the fit is well understood. 3.4. Modeling Duality Violations For a given moment Aω (s0 ), DV contributions can be written as (Eq. 8)  ∞ ds (s ) = −π ω(s)ΔρDV ΔAω,DV 0 V/A (s) . V/A s0 s0 Under some assumptions, ΔρDV V/A (s) is expected to be a combination of exponential times oscillatory functions at very high energies. In Ref. [31] the ad-hoc functional form, DV = e−δV/A −γs sin(αV/A + βV/A s) ΔρV/A

s > sˆ0 ,

Figure 4: V + A determinations of α s (m2τ ) from different moments, as function of s0 , fitted ignoring all perturbative and non-perturbative corrections, extracted from A(1,n) (s0 ) with {n = 0, ..., 6}. FOPT fits are on the left and CIPT on the right.

p-value is too low, does not allow us to give a competitive value based on this approach. Even if, ignoring instabilities, we take as reference sˆ0 = 1.55 GeV2 we obtain better p-values and different values for the strong coupling slightly changing the DV ansatz, which teach us that this extraction is model dependent. In Ref. [31] another fit with the same ansatz is made. However, once one removes linearly dependent points of it, the fit turns out to be equivalent to the previous one (2,1) but with two additional points A(1,1) V (s0 ) and AV (s0 ). Since one has two extra parameters to be fitted, there is no additional information in this fit about the strong coupling. Actually, the extra two parameters O6 V and O8 V of this fit are forced to compensate the distorsion due to the DV model, so that the absolute value of the fitted value is tipically larger than the physical one.

(12)

is to be exactly true and a fit to A1V (s0 ) =  s0assumed ds s0 Im ΠV (s), s0 > sˆ0 is made. From Equation (11), this fit is equivalent to a fit to the s0 -dependence of the spectral function plus a point A1V ( sˆ0 ), where the OPE is expected to work worse. We display the results we have reproduced in Fig. 3 for FOPT. Even being an ad-hoc model with 5 parameters, the s0 stability displayed by changing the starting point of the fit sˆ0 is much worse than the QCD one in the more inclusive channel V + A. This fact, together with the fact that the

3.5. Borel Transform If one takes the set of weight functions ω1,n (s) =  n+1 and extract the strong coupling ignoring all 1 − ss0 perturbative and non-perturbative uncertainties one obtain the results of Figure 4.The lack of splitting tells us that power corrections are small at s0 ∼ m2τ . We can make use of it to try to make reliable estimates in the V and A channels, where duality violations are larger. In order to reduce their effects we add a exponential term to the weight function,

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148





n+1 ω(1,n) e−ax ; a (x) = 1 − x

References x=

s . s0

(13)

Now all the condensates contribute to every moment. However, for a ∼ 1 the factorial supression should be enough not to enhance its contribution, so that it should still be small. We have observed how increasing a one obtains more stable results in the separate channels before power corrections become important. Accepting for each moment all values of α s (m2τ ) in the Borel-stable region, including the information from all moments, and adding as additional theoretical uncertainties the differences among moments and the variations in the region s0 ∈ [2, 2.8] GeV2 , one gets values for the strong coupling for the separate channels in very good agreement with Table 1. If we do the same for the V + A channel, we obtain the results from last row of Table 1, again in agreement with the previous ones. 3.6. Summary Different approaches and tests, which use moments that depend on perturbative and non-perturbative contribution in very different ways, have been used to extract the strong coupling from the nonstrange ALEPH spectral functions. The more competitive values obtained are summarized in Table 1. They are in total agreement, which reinforces their reliability. Averaging the determinations but keeping the lowest uncertainties, we get α s (m2τ ) = 0.328 ± 0.013 ,

(14)

which after evolution becomes (n f =5)

αs

(MZ2 ) = 0.1197 ± 0.0015 ,

(15)

in excelent agreement with the direct measurement at the Z peak from the Z hadronic width. Acknowledgments We want to thank the organizers for their effort to make this conference such a successful event. We also thank Michel Davier, Andreas Hoecker, Bogdan Malaescu, Changzheng Yuan and Zhiqing Zhang for making publicly available the updated ALEPH spectral functions, with all the necessary details about error correlations. This work has been supported in part by the Spanish Government and ERDF funds from the EU Commission [Grants No. FPA2014-53631-C2-1-P and FPU14/02990], by the Spanish Centro de Excelencia Severo Ochoa Programme [Grant SEV-2014-0398] and by the Generalitat Valenciana [PrometeoII/2013/007].

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