Leading hadronic contributions to the running of the electroweak coupling constants from lattice QCD

The quark-connected leading-order hadronic contributions to the running of the electromagnetic fine structure constant, $\alpha_{\rm QED}$, and the weak mixing angle, $\theta_W$, are determined by a four-flavour lattice QCD computation with twisted mass fermions. Full agreement of the results with a phenomenological analysis is observed with an even comparable statistical uncertainty. We show that the uncertainty of the lattice calculation is dominated by systematic effects which then leads to significantly larger errors than obtained by the phenomenological analysis.


Introduction
Finding hints for new physics beyond the standard model (SM) has been a major objective of particle physics over the past decades. A very promising strategy to detect such effects are high precision experimental measurements which are matched by equally precise theoretical predictions. An important ingredient for the precision attainable in a theoretical calculation is the knowledge of the coupling constants since they enter the quantum loop corrections.
In this article, we investigate the leading-order hadronic contributions for two of these couplings, the electromagnetic fine structure constant, α QED , and the SU (2) L coupling constant, α 2 , both related by the weak mixing angle, θ W . An accurate knowledge of these hadronic contributions is mandatory to accomplish sufficiently precise predictions for future high-energy colliders [1] or low energy experiments [2]. However, the hadronic contributions to the running of α QED turn out to be only poorly known at the scale of the Z-boson mass. Compared to α QED at zero momentum transfer, there is a five orders of magnitude loss of precision when α QED is taken at the Z-scale turning α QED (M 2 Z ) into one of the least determined input parameters of the standard model [3]. Phenomenologically, the leading hadronic contribution to the running of α QED originating from hadronic vacuum polarisation effects, ∆α hvp QED , is determined from a dispersion relation and experimental e + e − scattering data for the hadronic cross-sections [1,3,4].
Although new data has recently become available, the present analysis does not lead to a sufficient improvement of the error which would be needed for the requirements of future collider experiments [3]. In principle, lattice QCD calculations would be an ideal tool to determine the hadronic contributions to electroweak observables such as α QED or α 2 considered here. However, presently the precision that can be obtained from such lattice QCD computations is usually still lower than from the phenomenological analyses. Nevertheless, the steady progress which is taking place in lattice QCD calculations promises to make it an expedient alternative to the phenomenological results in the future. In fact, as we will demonstrate here, even with our present simulations the statistical uncertainty already matches the phenomenological error of ∆α hvp QED and ∆ hvp sin 2 θ W . ∆α hvp QED has first been investigated on the lattice for two dynamical twisted mass fermions [5]. Preliminary results incorporating also dynamical strange and charm quarks for one selected momentum value have been reported in [6]. Another determination of ∆α hvp QED , following the approach suggested in [4] has been performed in [7]. Here, we present our results obtained on the N f = 2 + 1 + 1 ensembles of the European twisted mass collaboration [8,9]. We will include an estimate of the systematic uncertainties originating from the continuum limit and from the extrapolation to the physical point for energies ranging from 0 to 10 GeV 2 . In contrast to ∆α hvp QED , the hadronic contributions to the running of the weak mixing angle, θ W , have not been studied on the lattice so far. Such a calculation is important since the phenomenological determination at low energies cannot only be based on data but also needs some assumptions such as a partial flavour separation of the cross-section data [10]. Lattice calculations can, in contrast, provide a first-principle evaluation of the weak mixing angle in the low-momentum region, where several measurements exist [11][12][13][14]. In addition, due to the great potential of such low energy experiments for unveiling the nature of physics beyond the SM, there are also newly planned experimental facilities [15][16][17], see also [18] for a discussion on such experiments. Here, we present the first lattice QCD calculation of the leading hadronic contribution to the weak mixing angle, ∆ hvp sin 2 θ W .
2 The fine structure constant α QED Radiative corrections lead to charge renormalisation and thus to the running of the fine structure constant obtained by summing the one-particle irreducible bubble insertions in the photon propagator [10] Here, α 0 is the value at vanishing momentum transfer Q 2 = 0, α −1 0 = 137.035999173(35) [19]. The leading-order hadronic contribution is given by [1] and is thus proportional to the subtracted vacuum polarisation function As mentioned in the introduction, this is usually [1,10,19] determined by a phenomenological approach relying on the once-subtracted dispersion relation [20] which for Euclidean momenta Q 2 reads and experimental cross-section data for . (2.5) Lattice QCD represents an ab-initio alternative for the calculation of Π R (Q 2 ), since the hadronic vacuum polarisation tensor can be obtained directly in Euclidean space-time from the correlator of two electromagnetic vector currents.

Lattice calculation
The strategy for computing and analysing the hadronic vacuum polarisation function is the same as in [22,23]. In particular, we employ the same set of ensembles [8,9], which is presented in table 1. Additionally, we have checked our chiral extrapolations of the light quark contribution by comparing the results with those obtained on a N f = 2 ensemble featuring the physical pion mass [24][25][26]. The parameters of this ensemble are given in table 2. As in [22,23], we use the conserved point-split vector current at source and sink and we restrict our considerations to the quark-connected contributions. In this case, the total vacuum polarisation function is obtained by summing the single-flavour contributions which we define without the charge factors.
For each ensemble and each flavour f, we first fit the temporal vector current correlator to determine the vector meson masses, m i , and their couplings, g i . Then we fit the hadronic vacuum polarisation function obtained from the current correlator as detailed in [22] to the following functional form where Θ(x) is the Heaviside step function. The low-momentum fit function for 0 ≤ Q 2 ≤ Q 2 match is given by Parameters of the N f = 2 + 1 + 1 flavour gauge field configurations that have been analysed in this work. β denotes the gauge coupling, a the lattice spacing, L a 3 × T a the spacetime volume and m P S is the value of the light pseudoscalar meson mass. The values for m P S have been determined in [8]. L is the spatial extent of the lattices. The lattice spacings are taken from [21]. The last column gives the number of statistically independent gauge configurations used to estimate the light/strange/charm contribution to the polarisation function.
The number of terms and thus the fit function is characterised by M, N, B, and C. The ansatz in Eq. (2.8) consists of three parts: a series of poles at energies m i and with residual g 2 i m 2 i , i = 1, . . . , M , an additive constant a 0 and further polynomial terms a i (Q 2 ) i for i ≥ 1. The poles characterised by (m i , g i ) are identified with the exponential contributions to the time-dependent vector-current 2-point correlation function at zero spatial momentum. This is the reason why the parameters (m i , g i ) are obtained from a fit of the latter and inserted into the fit of the vacuum polarisation function under preservation of all error correlations. The ansatz in Eq. (2.8) is valid for any four-momentum Q 2 , in particular for Q = Q 0 , Q = 0 with Q 2 = Q 2 0 . Given such a momentum configuration, the above identification follows from the Fourier transform of the polarisation tensor. With the limited statistical precision of the 2-point vector correlator, the number of exponentials we can resolve in practice is limited to 2. Contributions from states with even larger energies are effectively accounted for by the polynomial terms. The ansatz in Eq. (2.9) is chosen to provide an adequate parametrisation of the polarisation function. This is the only requirement in the high-momentum region Q 2 match < Q 2 ≤ Q 2 max . While in the low-momentum region the extrapolation beyond the lowest non-zero lattice momentum to zero momentum is of physical significance as it predicts the curvature of the polarisation function in this interval, in the high-momentum region we only need the ansatz in Eq. (2.9) to interpolate the available lattice data. In the following, we use Q 2 max = 100 GeV 2 and choose Q 2 match = 2 GeV 2 . Varying Q 2 match by 1 GeV 2 to the left and to the right gives compatible results. We perform extrapolations of the subtracted polarisation function in the light quark mass and the lattice spacing only for momenta in the interval 0 < Q 2 ≤ 10 GeV 2 since for larger Q 2 perturbative calculations are expected to yield more precise determinations. Since our four-flavour ensembles feature unphysically large pion masses, an extrapolation to the physical point has to be performed. The pion mass dependence of the single-flavour contributions can be assessed by looking at the leading vector meson contribution obtained in chiral perturbation theory [27,28] (2.10) The spectral properties of the heavy vector mesons hardly depend on the pion mass and also the coupling constant g V of the ρ-meson has been found to be well-described by a linear fit in the squared pion mass, m 2 PS , cf. [5]. However, the ρ-meson mass, m V , strongly depends on the value of the light quark masses, taken to be degenerate in our calculation, and thus the squared pion mass [22]. This is illustrated in Fig. 1 left by a model extrapolation which is constrained by requiring the ρ-meson mass to attain its experimental value [29] at the physical point, similarly to the one used in the two-flavour case in Ref. [30]. Eq. (??) (2.10) implies that a similar non-linear behaviour can be expected for the lightquark hadronic vacuum polarisation function as we indeed observe for the lower set of data points in Fig. 1 right. From Eq. (2.10) we also see that we can eliminate this non-linear dependence on the squared pion mass to a large extent by employing the lattice redefinition presented in Ref. [5] for the light-quark contribution to the vacuum polarisation function, The lattice data obtained with the definition ∆α hvp QED in Eq. (2.12) can be sufficiently well described already by a linear dependence on the squared pion mass. Since we use only this definition throughout this work and no confusion is possible, we henceforth omit the bar in ∆α hvp QED .

Results
In order to show that the above redefinition in Eq. (2.11) indeed provides the expected benefit for the chiral extrapolation of the light quark contribution to the running of the fine structure constant, we show the data for both Eqs. (2.2) and (2.11) with H = m V in Fig. 2 left for a single momentum value Q 2 = 1 GeV 2 . The upper set of data points obtained with the redefinition Eq. (2.11) evidently is much easier to extrapolate to the physical value of the pion mass than the lower points procured from the standard definition Eq. (2.2). Since we do neither observe lattice spacing artefacts nor finite size effects in these data at Q 2 = 1 GeV 2 , we can actually compare our results computed on the four-flavour ensembles linearly extrapolated in the squared pion mass, m 2 PS , with those obtained from the N f = 2 ensemble featuring the physical pion mass. Additionally to the standard analysis, we have performed a correlated [1,1] Padé fit [31], possessing the same number of parameters, up to Q 2 max = 1.5 GeV 2 such that Q 2 = 1 GeV 2 is safely covered. As expected, the values for the pole parameters determined from the temporal correlator in our standard approach and from the Padé fit are compatible   (50) 0.003097(88) 0.003062(77) Table 3. Comparison of the chirally extrapolated result for ∆α hvp,ud QED (1 GeV 2 ) obtained on the N f = 2 + 1 + 1 ensembles with those obtained on the N f = 2 ensemble at the physical point. For the latter, we have performed our standard analysis but without the redefinition and also tested a [1,1] Padé fit.
Including also the heavy quark contributions by using Eq. (2.12), a dependence on the lattice spacing is clearly visible, especially in the high-Q 2 region shown in the right panel of Fig. 2. This is accounted for by combining the chiral extrapolation with taking the continuum limit and employing the following fit function to the four-flavour results obtained on individual ensembles with fit parameters A, B, C for each momentum value Q 2 ∈ {0, 0.02, 0.04, . . . , 10} GeV 2 .
In [32] we have shown that automatic O(a) improvement is at work for our definition of the hadronic vacuum polarisation function. Thus, performing the continuum extrapolation without a term linear in the lattice spacing a in Eq. (2.14) is justified. The ansatz in Eq. (2.14) neglects any dependence on the finite lattice extent which has been found to be smaller than our current statistical uncertainties and will be discussed when assessing the systematic effects of our calculations.
∆α from Jegerlehner's alphaQED package lattice data linearly extrapolated to m π in CL The results are depicted in Fig. 3 together with the results obtained by a phenomenological analysis [1]. Here, for both the lattice calculation and the phenomenological analysis only the statistical errors are shown. Over the whole momentum range, perfect agreement with comparable statistical uncertainties is found. We will discuss the systematic uncertainties of our lattice QCD determination below. An updated phenomenological analysis including all data published till the end of 2014 will soon be available [34]. The lattice data also agree with those results featuring even smaller uncertainties.

Systematic uncertainty from the choice of vector meson fit ranges
As mentioned before, the first step in our analysis is the determination of the masses and the coupling constants of the vector mesons from the vector two-point functions at zero momentum. The values of the spectral parameters differ when varying the fit range. We have repeated the complete analysis for various vector meson fit ranges for the light, strange and charm quark currents propagating the uncertainty to the final results. In the light quark sector depicted in Fig. 4, we observe systematic uncertainties depending on whether we start fitting the vector meson correlator at 0.6 fm or at 0.7 fm whereas changing the upper border of the fit interval by 0.1 fm does not lead to observable effects. The dependence on the lower starting point of the fit can be attributed to excited state contamination of the ρ-meson correlator. When stating the final results for selected momentum values below, we take for these systematic uncertainties half the difference between the central values that are furthest apart from each other. For the heavy flavours changing the fit interval by 0.1 fm to the left and to the right of both the lower and the upper time slice of the fit ranges does not lead to observable differences. This is shown in Fig. 5

Systematic uncertainty from the choice of vacuum polarisation fit function
Performing the whole analysis with different numbers of terms in our vacuum polarisation fit functions also leads to observable differences in the light quark contribution as shown in Fig. 6. These are larger than the effects from the fit ranges of the vector meson fits discussed in the preceding subsection and thus present the dominant systematic uncertainty in our calculation. It might be possible to improve the situation by e.g. the method of analytic continuation [35,36] or by taking momentum derivatives of the vacuum polarisation function [37]. The situation for the heavy quarks is shown in Fig. 7. Here, almost no systematic deviations are visible. Furthermore the contributions from the heavy quarks are about an order of magnitude smaller than the light-quark one. Hence, we do not take systematic effects from the variation of the second-generation quark fit functions into account in our final error estimate.

Finite size effects
In lattice QCD, typically m PS L 4 is required to minimise systematic effects due to the finite lattice volumes, where L denotes the spatial extent of the lattice. The

Systematic uncertainty from including heavy pion masses
In order to extrapolate to the physical point, m π ≈ 140 MeV, often not too high pion masses should be included in the fit. The ensembles entering the standard analysis comprise pion masses up to m PS ≈ 491 MeV. Using only the ensembles with m PS < 400 MeV yields fully compatible results for ∆α hvp QED as can be seen in the right panel of Fig. 8. Therefore, we do not account for a systematic uncertainty related to the usage of pion masses above 400 MeV. Table 2.2.5 contains our final results compared to those of a phenomenological analysis [1] utilising the once-subtracted dispersion relation Eq. (2.4). The first error denotes the statistical and the second error the systematic uncertainty of our results. The latter constitutes the dominant source of uncertainty of which the biggest part originates from the choice of the vacuum polarisation fit function. This might change when lowering the statistical uncertainty, because then the vacuum polarisation fit gets more constrained. Alternatively, avoiding to fit the vacuum polarisation might be considered.

Final results for selected momentum values
3 The weak mixing angle sin 2 θ W The weak mixing or Weinberg angle, θ W , is one of the fundamental parameters of the electroweak standard model defined by where g is the SU (2) L coupling constant and g the U (1) Y coupling constant. The second equality is the electroweak unification condition e 2 = g 2 sin 2 θ W for the positron charge e.  Thus, the running of the weak mixing angle can be obtained from the running of the fine structure constant and the SU (2) L coupling α 2 . In the leading logarithmic approximation this is given by [38] The value of sin 2 (θ 0 ) has essentially been measured by the Boulder group studying atomic parity violation in Cesium [11], the latest value is sin 2 (θ 0 ) = 0.2356(20) [39]. The standard model prediction in the MS scheme is sin 2 (θ 0 ) = 0.23871(9) [18,40] which is the value employed in the analysis below in order to gain fully theoretical results without experimental input. In the computation of this value the Higgs boson mass determined by the LHC experiments [41,42] has been used. A phenomenological value of the leading hadronic contribution to the running of sin 2 θ W between 0 and the Z-scale has been computed for the first time in [43] relying on results of [44]. The method has been described and used with an older dispersive analysis [45] before [46]. In [40] the error has been reduced with respect to the original rather conservative estimate of the uncertainty by about an order of magnitude. The leading hadronic contribution to the running of the SU (2) L coupling constant originates from Z-γ mixing Z had γ From the expressions for the hadronic currents of up-type (u) and down-type (d) quarks where 3 refers to the third component of the weak isospin current and γ to the electromagnetic current discussed already above, we see that to leading order and thus the leading hadronic contribution to the running of α 2 is given by [1,10] ∆α hvp As for the purely electromagnetic current correlator, Π 3γ denotes the transverse part of the vacuum polarisation function.
Beyond the leading log approximation, ∆α hvp QED and ∆α hvp 2 become renormalisation scheme dependent. Additional hadronic contributions to these corrections at the scale of the Wmass and the Z-mass originate from chiral symmetry breaking. They have been shown to be calculable in perturbation theory and to be at least two orders of magnitude smaller and thus negligible compared to the leading contributions [10]. Thus, having computed ∆α QED before, all that is left to do to leading order is to compute ∆α 2 as given in Eq. (3.7).

Lattice calculation
Since our ensembles feature mass-degenerate up and down quarks, m u = m d , light-quark disconnected contributions cannot occur in Π 3γ due to the isospin symmetry of the vacuum. Without those interference terms, single-flavour contributions to Eq. (3.6) in the continuum limit have the general structure Π 3γ,f =< (V − A)V >, where V and A denote vector and axial vector currents, respectively. Since QCD conserves parity, no mixing between vector and axial vector currents occurs such that without quark-disconnected contributions we obtain for up-type quarks twice the contribution of down-type quarks Combining this with ∆α QED , the leading-order hadronic contribution to the running of the weak mixing angle from the two light flavours reads Neglecting disconnected contributions also for the heavy flavours, we have for the strange and the charm quark contributions respectively. Hence, the single flavour contributions are all proportional to the hadronic vacuum polarisation function but with different prefactors than for ∆α hvp QED . In order to treat both contributions to ∆ hvp sin θ W consistently, we use in the light sector the same redefinition of the vacuum polarisation function for ∆α hvp 2 as for ∆α hvp QED . Thus, in our lattice calculation we consider ∆ hvp sin 2 θ W (Q 2 ) = ∆ hvp,ud (Q 2 · H 2 /H 2 phys ) + ∆ hvp,s (Q 2 ) + ∆ hvp,c (Q 2 ) . (3.12)

∆α hvp 2
As stated above, the leading hadronic contribution to the running of the weak mixing angle in the leading logarithmic approximation is obtained from the difference of the corresponding contributions of the electromagnetic and the SU (2) L coupling constants, α QED and α 2 . In contrast to ∆α hvp QED , it is not straightforward to extract ∆α hvp 2 from experimental e + e − → hadrons data, since the data comprising the three lightest quarks would have to be separated either in up-type (u) and down-type (d and s) quarks or assuming isospin symmetry in light and strange quark contributions. This problem has no unique solution, e.g. final states involving kaons could either originate directly from a strange quark current or from a gluon that could be radiated off light quarks. Another possibility is to assume SU (3) f symmetry and thus only split the data into information attributed to the three lightest quarks and the rest. The contributions from charm and heavier quarks can be computed in perturbation theory. Fig. 9 shows our results after combined extrapolation to the physical point and to vanishing lattice spacing compared to the results of [1]. There, two ways of flavour separation have been implemented, one is assuming approximate SU (3) f and the other one  SU (2) f symmetry neglecting OZI violating terms. Our results clearly prefer the SU (3) f flavour separation and thus indicate that the latter assumption is not tenable as has also been observed in [47] in a different context. In Fig. 9 we have multiplied the data from [33] with sin 2 θ W (M Z )/ sin 2 (θ 0 ) to account for the different reference values employed. As mentioned before sin 2 (θ 0 ) = 0.23871 (9) and the value used by Jegerlehner is sin 2 θ W (M Z ) = 0.23153 (16) which has been measured at LEP [48]. The flavour separation performed for the data set including very recent e + e − measurements is based on isospin symmetry relations [34] and the results are much closer to the ones based on SU (3) f flavour separation in Fig. 9 than to the old SU (2) f curve. Thus, our lattice results are also compatible with the newest phenomenological analysis based on an isospin SU (2) f flavour separation, however, not assuming flavour non-diagonal elements to be small.

∆ hvp sin 2 θ W
Having determined the four-flavour contributions to ∆α hvp QED and ∆α hvp 2 , it is straightforward to obtain the leading-order hadronic vacuum polarisation contribution to the running of the weak mixing angle ∆ hvp sin 2 θ W (Q 2 ) = ∆α hvp QED (Q 2 ) − ∆α hvp 2 (Q 2 ) . The results are shown in Fig. 11. Complying with the indication from the previous subsection, we have employed the results for ∆α hvp 2 obtained from SU (3) f flavour separation in Fig. 11 together with the factor needed to take the different reference values into account.
Since we do not have information on the correlation of the data in [33], we have simply added the uncertainties of ∆α hvp QED and ∆α hvp 2 in quadrature and may thus overestimate the errors of the phenomenological determination.

Systematic uncertainties
Since the systematic uncertainties stem from the same sources as for ∆α hvp QED discussed before, the relative errors are the same and only the absolute numbers differ due to the different prefactors of the renormalised vacuum polarisation function. Naturally, also the plots all look very similar. Therefore, we refrain from discussing the systematic effects separately and only summarise the general findings. As before, due to the light quark contribution being an order of magnitude bigger than the contributions from the heavy quarks, we only need to take systematic uncertainties of this part into account. The dominant source of systematic errors is again the choice of the vacuum polarisation fit function as depicted in Fig. 12. The only other relevant effect comes from the excited state contamination of the ρ-meson correlator and is shown in Fig. 13. Finite volume effects and the choice of rather heavy pion masses in the chiral extrapolation seem to be negligible in our calculation as outlined be-∆ sin 2 θ from Jegerlehner's alphaQED package lattice data linearly extrapolated to m π in CL  Figure 11. N f = 2 + 1 + 1 contribution to the leading-order hadronic contribution ∆ hvp sin 2 θ W compared to the difference of the data collected in [33].The lattice data are extrapolated to the physical point and to the continuum limit (CL). fore. The only unknown systematic effect is the heavy-flavour disconnected contributions which we have neglected here.

Final results for selected momentum values
In table 6 we collect our results for ∆ hvp sin 2 θ W with statistical as well as systematic uncertainties for selected momentum values. Experiments which have measured or will measure the weak mixing angle in the respective momentum region are also indicated. The outcome of the E158 experiment at the SLAC linear accelerator was the first successful measurement of parity violation in electron-electron (Møller) scattering [12]. The momentum transfer was Q 2 = 0.026 GeV 2 . The Qweak experiment conducted at JLAB in 2012 measured parity violation in electron-proton scattering at almost exactly the same    Table 6. We tabulate ∆ hvp sin 2 θ W for selected values of Q 2 . The first error of the lattice results is statistical, the second systematic. Several low energy experiments sensitive to the respective momentum regions are indicated in the last column. momentum transfer [14]. The data is still being analysed. The predicted final uncertainty is about 5% or 0.7 · 10 −3 taking the central SM value. Another JLAB experiment performed by the PVDIS collaboration determined the weak mixing angle from parityviolating deep inelastic scattering [13,49] which effectively means electron-quark scattering at Q 2 = 1.085 GeV 2 and Q 2 = 1.901 GeV 2 . The envisioned successor of the PVDIS experiment which also measures parity violation in electron-quark scattering is the SoLID spectrometer proposed at JLAB [16]. It can study about 20 kinematic points with Q 2 ranging from about 2 GeV 2 to about 10 GeV 2 . Its target accuracy is 6 · 10 −4 . Our results in table 6 indicate that it will be essential to take the leading QCD corrections into account in order to deploy the whole potential of the experiment in the search for new physics beyond the SM.

Summary and Outlook
Hadronic contributions to the running of electroweak parameters nowadays constitute the major uncertainties of their values even at high energies thus also limiting the precision achievable in predictions for future high-energy colliders. Here we have considered the running of α QED and of the weak mixing angle which represents one of the most important parameters of the SM and provides a sensitive probe of new physics over a large energy range.
Lattice QCD provides a most valuable tool to compute these hadronic contributions from first principles alone. As we have demonstrated in this article, lattice QCD can be used to compute to a good precision the leading-order hadronic contribution to the running of α QED . In particular, we have carried out the first dynamical four-flavour calculation of the leading-order hadronic contribution to the running of the fine structure constant and the first lattice QCD calculation of the leading hadronic contribution to the shift of the weak mixing angle at energies between 0 and 10 GeV 2 . In both cases the chiral as well as continuum extrapolations have been performed. A main effort has been undertaken to assess systematic uncertainties on a quantitative level. For both quantities, agreement of our results with a phenomenological determination is observed with an even comparable statistical uncertainty. However, we have found that the systematic effects of the calculation still exceed the statistical errors. The dominant systematic uncertainty has been found to be the choice of fit function. Thus, methods which try to avoid fitting the vacuum polarisation are promising to reduce the overall uncertainty. A further improvement can be achieved by increasing the statistical precision which would more strongly constrain the vacuum polarisation fit. Such improvements might be accomplished by the use of the all-mode-averaging [50] or the exact deflation [51,52] techniques. In this article, we have provided further successful examples for the programme to determine hadronic contributions to electroweak observables from lattice QCD. The steady progress in lattice QCD with ever increasing statistical accuracy and better understanding and control of systematic uncertainties makes the lattice approach to compute these hadronic contributions very promising and gives hope that lattice results can directly be used for future low energy and collider experiments.