Weak Mixing Angle in the Thomson Limit

We present a calculation of the weak mixing angle in the $\overline{\rm\small MS}$ renormalization scheme which is relevant for experiments performed at very low energies or momentum transfers. We include higher orders in the perturbative QCD expansion, as well as updated phenomenological and theoretical input, and obtain the result $\sin^{2}{\theta}_W(0) = 0.23868(5)(2)$ for the reference values $\alpha_s(M_Z) = 0.1182$ and $m_c(m_c) = 1.272$ GeV. The first quoted error is from the current Standard Model evaluation of the mixing angle at the $Z$ boson mass scale. The second error represents the theoretical and parametric uncertainties induced by the evolution to the Thomson limit and is discussed in detail.

. RGE contributions of different particle types, where the minus sign is indicative for the asymptotic freedom in non-Abelian gauge theories.
the calculation of the singlet contribution to the weak mixing angle, with some details given in Appendix B. In Section 5 the flavor separation (contributions of light and strange quarks) is addressed and threshold masses are calculated. In Section 6 theoretical uncertainties are discussed in detail, and Section 7 offers our final results and conclusions.

Renormalization group evolution
In an approximation in which all fermions are either massless and active or infinitely heavy and decoupled, the RGE for the electromagnetic coupling in the MS scheme [24],α, can be written in the form [2], where the sum is over all active particles in the relevant energy range. The Q i are the electric charges, while the γ i are constants depending on the field type and shown in Table 1. The K i and σ contain higher-order corrections and are given by [25],   with n q the number of active quarks and N c i = 3 the color factor for quarks. For leptons one substitutes N c i = 1 andα s = 0, while K i = 1 for bosons. We can relate the RGE ofα to that of sin 2θ W since both, the γZ mixing tensor Π γZ and the photon vacuum polarization functionΠ γγ are pure vector-current correlators. Including higher order corrections, the RGE for the Z boson vector coupling to fermion f , v f = T f − 2Q f sin 2θ W , where T f is the third component of weak isospin of fermion f , is then (2.4) Eqs. (2.1) and (2.4) can be used [2] to obtain where the λ i are known [2] constants given in Table 2 and the explicit K i dependence has disappeared. Theσ terms, represent the singlet contributions to the RGE evolution of the weak mixing angle at four and five loop order. These terms arise from quark-antiquark annihilation (disconnected) diagrams (see Figure 1) and are suppressed in perturbative QCD (PQCD). In the nonperturbative domain these give rise to so-called OZI-rule [20][21][22] violations. Eq. (2.5) together with the solution of the four-loop QCD β-function [26,27] represents a complete solution, as long as all matching scales µ at which an active particle decouples are known, because there the λ i change their values. The matching scales of all bosons [28], charged leptons, and heavy (t, b, and c) quarks [29][30][31] can be calculated as what we call threshold massesm q , where the QCD corrections to the matching relations vanish by definition.

Matching conditions
At each particle threshold the RGE coefficients need to be modified to reflect the particle content of the associated effective field theory (EFT), and in the MS scheme it is also convenient to change the definitions ofα andŝ to correspond to this same EFT. This is analogous to the usual treatment ofα s and leads to very small matching discontinuities in the RGE running of the couplings.
Denoting the electromagnetic coupling with and without the fermion near the threshold byα(m f ) + andα(m f ) − , respectively 2 , the matching condition forα reads [29][30][31], The first three lines derive from heavy quark vector-current correlators. The last line involves a sum over all quarks with m m q , and arises from the decoupling of the heavy quark q propagating in inner loops of multi-bubble type diagrams in which the outer loop (the one coupled to the currents) is occupied by a light quark . The corresponding contribution at orderα 3 s is parametrized by the coefficients K i and is unknown at present. The knownα 2 s term for the charm and bottom quarks, and theα 3 s terms from the charm and bottom quark vector-current correlators amount to about 9 × 10 −6 and −9 × 10 −6 , respectively. Taking these as conservative bounds on the unknown higher-order terms and combining them in quadrature results in an estimated truncation error of ±1.3 × 10 −5 inα. The matching conditions ofŝ 2 andα can also be related [2], Applying the numerical analysis of the previous paragraph to Eq. (2.9), we find 2.4 × 10 −6 and −1.4 × 10 −6 , respectively, and we estimate a truncation error related to the matching of about ±3 × 10 −6 inŝ 2 . For completeness we recall that integrating out the W ± bosons induces the one-loop matching condition [2,28], (2.10)

Implementation of experimental input
The perturbative treatment of the previous section cannot be applied at hadronic energy scales and experimental input is required. This is usually taken from R(s), i.e., the cross section σ(e + e − → hadrons) normalized to σ(e + e − → µ + µ − ). Additional information on R(s) is encoded in hadronic τ decay spectral functions [32]. The traditional method to implement the R(s) measurements is through a subtracted dispersion integral, which gives the hadronic contribution (with the top quark removed) to the Z scale value of the electromagnetic coupling in the on-shell scheme. One supplements the input data with the theoretical (perturbative) prediction for R(s) at s ≥ s 0 , with s 0 large enough to be able to trust QCD perturbation theory. A variant [33] of this approach evaluates Eq. (3.1) in the space-like region, ∆α (5) had. (−M 2 Z ), and obtains ∆α (5) had. (M 2 Z ) in a second step. More details about how different groups get the running of alpha are given in Appendix A.
In the MS scheme it is more natural to use an unsubstracted dispersion relation [24], where the superscript indicates that we focus here on the currents produced by the three light quarks (bosons, leptons, charm and bottom quarks are included following Sec. 2). The upper integration limit can in principle be chosen as an arbitrary perturbative scale µ 0 , but in practice we take µ 2 0 to coincide with the cut-off value s 0 used in the traditional method, since this allows us to recycle results obtained there. Indeed [24], for µ 0 2 GeV. Using the results of Ref. [16] including inputs from τ decays which we correct for γ-ρ mixing [17], we obtain, We compute the second term in Eq. (3.2) at the scale µ = 2 GeV perturbatively [34], extending the O(α 2 s ) result of Ref. [24] to O(α 3 s ), where the F i (m c ,m b ) are correction terms from the charm and bottom quarks. The explicit analytical expression for F 2 (m c ,m b ) −0.2348 is given in Ref. [24], while that for F 3 (m c ,m b ) −0.390 will appear in ref. [34]. The numerical evaluation in the last line of Eq.
The uncertainty is the size of the O(α 3 s ) term, and we have defined to display the dependence onα s . Thus, from Eqs. (3.2)-(3.5) we obtain,

Singlet contribution
We recall that Eq. (2.6) exhibits an explicit dependence on α s , which in the non-perturbative domain gives rise to the QCD induced OZI-rule [20][21][22] violations. These have to be known independently, since they affectα andŝ 2 differently. Thus, in addition to a quark flavor separation, one also needs a singlet piece separation, even though the singlet piece is expected to be small. To do so, we first relate ∆ discα , the disconnected part in ∆α (3) (2 GeV), to the one entering the low energy weak mixing angle, ∆ discŝ 2 . Non-singlet and singlet contributions are separately gauge-invariant, and to gain information on ∆ discα , we will adopt a lattice QCD calculation [23] of the disconnected quark line contributions to the anomalous magnetic moment of the muon, a µ . By construction, theσ terms in Eq. (2.5) are related to the σ terms in Eq. (2.1), On the other hand, isolating the ∆ discα term in Eq. (2.1) we obtain (working here in lowest order in α), where the last step applies for µ <m c (we are assuming approximate isospin symmetry which eliminates the intervalm u < µ <m d ). Then, These relations are general, but there is a subtle point. In general, the singlet pieces effectively decouple at renormalization scalesm disc q that may differ from the scalesm q at which the non-singlet pieces decouple. This would generate various energy intervals with generally different values for λ 1 . Implementing strong isospin symmetry in the form m u =m d andm disc u =m disc d , as well as accepting the physical mass orderingsm s ≥m u and m disc s ≥m disc u , there remain a total of six different orderings.
As an example, consider the case, For scales µ >m disc s there are three active quarks with Q u + Q d + Q s = 0 and the singlet contributions vanish. For scales in the rangem disc s > µ >m s we obtain the value λ 1 = 1/2. Similarly, form s > µ >m u and form u >m disc u we find λ 1 = 9/20 and 1/4, respectively. Belowm disc u all singlet contributions vanish by definition. Inserting these results into Eq. (4.4) and summing the contributions from all intervals, we find the constraint, where we have anticipated that ∆ discα < 0 (see below). The other five cases are dealt with in the same way, and one can check that the inequality (4.6) is never violated. For the three mass orderings satisfyingm disc u ≥m u , or generally if we can neglect the presumably small rangem u > µ >m disc u , we find the much stronger constraint, Since we do not expect them disc q to be numerically very different from them q we choose our central value to correspond tom disc q =m q , and we include twice the range in Eq. (4.7) as the uncertainty due to possiblem disc q =m q effects. Thus, which can be inserted into Eq. (2.5). Notice, however, that Eq. (2.5) also contains an implicit singlet contribution from each of the two terms in the first line. Taken together, the λ 1 term cancels exactly the central value in Eq. (4.8) and we finally arrive at In Appendix B we compute ∆ disc α in the on-shell scheme by exploiting the lattice gauge theory calculation [23] of the corresponding contribution to a µ with the result, (4.10) Note that because the sum of the charges of the three light quarks vanishes, and we enter the perturbative domain where the singlet piece is known to be tiny, we expect an asymptotically stable value at higher energies for ∆ disc α(q). This is supported by Figure 2, showing that ∆ disc α(q) is nearly q-independent for q 1.2 GeV. We also remark that the dominance of low scales notwithstanding, the sign in Eq. (4.10) coincides with that of the singlet piece in the perturbative regime. Also shown in Figure 2 is the step function approximation of ∆ disc α(q), with the step defined as the value of q where it reaches half of its asymptotic value in Eq. (4.10). We interpret this as the value where the strange quark decouples from singlet diagrams, so thatm disc s ∼ 350 MeV. Our central value ofm s to be derived in the next section,m s = 342 MeV, is numerically very close to this providing evidence form disc s ≈m s . Eq. (4.9) and Eq. (4.10) refer to quantities in the MS and on-shell schemes, respectively, and in general these may differ. However, since we are working here in the three quark theory and the sum of the charges of three light quarks vanishes, the change of schemes is trivial. We can therefore use Eq. (4.10) in Eq. (4.9) and obtain, where the uncertainty combines the errors from Eq. (4.9) and the one induced by the lattice calculation [23].

Flavor separation
In this section we perform a flavor separation of the contributions of up-type from down-type quarks, or -given that up and down quarks are linked by the approximate strong isospin symmetry -a separation of s from u and d quarks. Our strategy consists of first using exclusively the experimental electro-production data as tabulated in Ref. [16] to constrain the contribution ∆ s α of the strange quark to ∆α. We then exploit the lattice gauge theory results in Refs. [18,19] to confirm and refine the purely data driven analysis. Then we introduce the threshold massm q of a quark q as the value of the 't Hooft scale where the QCD contribution to the corresponding decoupling relation becomes trivial.m c andm b are treated in perturbation theory, while for u, d, and s quarks we derive bounds using phenomenological and theoretical constraints.  Table 3. Channels associated with the strange quark external current (top) and possible further channels originating from it (bottom).

Experimental data
To obtain ∆ s α we use Ref. [16] where the contribution of each hadronic channel to a µ and ∆α for energies up to 1.8 GeV is given. The main idea is to determine for each channel whether it was produced by anss or a first generation quark current. For reasons that will become clear later, we consider both, ∆ s α, and the strange quark contribution to the anomalous magnetic moment, a s µ . We begin by listing in the upper part of Table 3 the experimental channels [16] which we associate with an ss current. Up to OZI-rule violating φ-ω and φ-ρ mixing effects, the φ meson can be identified with strange quarks. We calculate its contribution using a Breit-Wigner shape with s-dependent total and partial widths, adopting the PDG values [35] for the φ meson branching ratios and applying a small correction for φ-ω mixing. As for the φ(1680), the main decay channel is KK * with K * mesons decaying almost entirely into Kπ. As can be seen from data [16], the KKπ channel is indeed virtually saturated by φ(1680) decays. The η-φ channel also arises dominantly from the strange quark current since the contribution to this channel from light quarks is Zweig rule suppressed. Conversely, we expect channels involving an η meson accompanied by non-strange states to be mainly due to light quark currents. For a s µ we need to add the contribution from energies above 1.8 GeV. It can be computed within PQCD and taken as one sixth of the corresponding light quark contribution [36] of 43.8×10 −10 . The lower part of Table 3 shows further channels involving strange quarks to which first generation quark currents could conceivably contribute, and we conservatively assign (50 ± 50)% of these to the ss current. The table also shows the corresponding contributions to a µ . Adding the totals in this way we find, The first errors are experimental [16] where we accounted for correlations. The second errors allow for differences in parametrizations when decay parameters are extracted from experimental data by different groups. The last errors are half of the totals in Table 3, but we expect the ss current to virtually saturate the kaon channels in Table 3 because the larger strange quark mass should suppress the probability amplitude to produce an ss sea quark pair relative to first generation quark pairs. The uncertainty in Eq. (5.2) is already about three times smaller than in the past [2]. We can reduce it further by quantifying our expectation that the strange quark current actually saturates the kaon channels listed in the bottom part of Table 3. For this, we re-write Eqs.  Table 3 arise from the strange (first generation) quark current. In order to confirm that indeed κ ≈ 0 and to compute an uncertainty for possible κ = 0 effects, we can use results on a s µ from lattice gauge theory, as we show next.

Lattice data
Two groups [18,19]  which is consistent with, but more precise than Eq. (5.2). We assigned the uncertainty from κ 1σ symmetrically around κ = 0, which is both the physically favored and most probable value (the peak of the distribution). This rather conservative treatment effectively doubles the error from κ 1σ , and is meant to account for the fact that the kernels of ∆α and a µ differ.
The experimental values of a µ and ∆α are correlated, possibly impacting Eq. (5.7). However, we found that even assuming them to be fully correlated changes the central value only very slightly and reduces the uncertainty modestly. Thus, we keep Eq. (5.7) as our final result on ∆ s α(1.8 GeV). As an additional cross-check we used the vacuum polarization function of another lattice calculation [19] of a s µ (expressed as a Padé approximant which is the source of the largest uncertainty [19]) to first reproduce their results, and then we computed ∆ s α which yields, , (5.8) in excellent agreement with Eq. (5.7).

Heavy quarks
We can computem c andm b in perturbation theory by reincorporating the RGE summable logarithms of the form lnm q /m q into Eq. (2.8), and then solving form q by setting the contribution from quark q equal to zero. Sincem q →m q forα s → 0, these logarithms are at most of orderα s and can be ignored in theα 3 s coefficient. Thus, we can use a previous analysis [24] where the logarithms up to orderα 2 s are given. We find, It will be useful for later to define quantities ξ q [2] as ratios between the threshold mass of quark q and the 1Sqq bound state mass, This definition implies that ξ q → 1 form q → ∞ and ξ q → 0 form q → 0. We expect ξ q to be a monotonically increasing in the sense that ξ 1 > ξ 2 ifm 1 >m 2 . Using the PDG values for the bound state masses [35] we find ξ c = 0.766 and ξ b = 0.844, and thus ξ b > ξ c as expected.

Light quarks
Next we constrain the individual contributions of the light quarks to ∆α, evaluated atm c . Using the RGE and the starting value given in Eq. given by one sixth of the continuum contribution [16] of (3.31 ± 0.26) × 10 −4 between the two scales. The uncertainty is the difference to using PQCD instead of data and accounts for quark-hadron duality violations. Changing to the MS scheme and employing again the RGE gives, ∆ sα (m c ) = (8.71 ± 0.32) × 10 −4 .

(5.15)
Since the threshold mass is the value of the 't Hooft scale corresponding to trivial matching conditions regarding the QCD contribution, we can write, where we defined a scale dependent factor K q QCD (µ) as the average QCD correction to the β function betweenm q and the scale µ. Eq. (5.16) has two unknowns, K s QCD (m c ) andm s , and it shows that increasing K s QCD (m c ) forces the logarithm to decrease and in turnm s to increase. Thus, smaller (larger) values of K s QCD (m c ) correspond to a smaller (larger) values ofm s . On the other hand, if we have two quarks with massesm 1 >m 2 , we expect the average QCD contribution betweenm 2 and µ to be larger than that betweenm 1 and µ, since α s is larger at lower scales. Thus, and we must have, where we used α = α(m s ) ≈ 1/135. We can also obtain an upper bound onm s ,  Table 4. Theoretical uncertainties in the low energy mixing angle.
implying K s QCD (m c ) < 1.50. We can summarize these results by writing, m u andm d can be obtained in a similar way. We have, where the quark connected contribution to ∆α (3) (m c ) is given by, Following the same steps as form s we find, Finally, accounting for the squares of the electric charges we obtain the contributions from the first generation quarks at the scalem s , where we only quote the uncertainty from the flavor separation in Eq. (5.15).

Theoretical uncertainties
In addition to parametric uncertainties, there are five sources of theoretical uncertainties for the weak mixing angle at low energies affecting our calculation. They are summarized in Table 4 and discussed in the following.
The three light quarks enter with different electroweak weights intoŝ 2 (0) and ∆α (3) (m c ). The flavor separation uncertainty is due to the imperfect knowledge of how much s quarks relative to u and d quarks contribute to ∆α (3) (m c ). It is given by [2], where we used δ∆α (2) (m s ) = ±1.9 × 10 −4 from Eq. (5.25). The flavor separation assumed isospin symmetry in the formm u =m d . To estimate the uncertainty associated with isospin breaking, we first consider the idealized case in which SU (2) isospin violation was as large as SU (3) breaking. This would occur form d =m s , so that from Eq. (5.25) the u quark current could at most contribute To propagate this uncertainty toŝ 2 (0) we can use [2], A measure of the breaking of SU (2) relative to SU (3) is given by the ratio, This error is asymmetric because we assumem d ≥m u , but it is convenient and conservative to treat it symmetrically in Table 4. The uncertainty arising from the singlet contribution is given in Eq. (4.11). The last entry in Table 4 combines the truncation error from the perturbative matching conditions with the scheme conversion error shown as the second uncertainty in Eq. (3.7).

Results and conclusions
Eq. (2.5) together with the Z pole value of the weak mixing angle from a global fit to the SM [35], sin 2θ W (M Z ) = 0.23129 (5), can now be used to compute the weak mixing angle at zero momentum transfer, sin 2θ W (0) = 0.23868 ± 0.00005 ± 0.00002, (7.1) where the second error is the total theoretical uncertainty from Table 4.
When our result for the weak mixing angle in the Thomson limit or some other low momentum scale is used for the calculation of physical observables, there will generally be further process-dependent radiative corrections which need to be addressed. We expect this to be possible with theoretical uncertainties well below those in sin 2θ W (0) summarized in Table 4. Thus, we reduced the total theoretical uncertainty in the weak mixing angle at low energies from 7 × 10 −5 [2] to less than 2 × 10 −5 which can safely be neglected for any current or planned experiment.
In summary, we developed a new way of calculating the flavor separation which involved both e + e − → hadrons data and results from lattice gauge theory. We also better control now the uncertainty in the contribution of disconnected diagrams where we exploited results of Ref. [23] on the anomalous magnetic moment. Furthermore, we extended various formulas to the next order in perturbation theory, reducing the perturbative uncertainty. There has also been significant progress in the evaluations of ∆α [16,38] and ∆m c (m c ) [37]. The theoretical uncertainty in sin 2θ W (0) is now at a negligible level.

A Calculations of α(M 2 Z )
Three independent groups presented recent evaluations of the hadronic contribution to the scale dependence of α. In this appendix we briefly compare their approaches and results.
In the Adler function approach [33,38], one uses the relations, where Q 2 = −q 2 , and where the dispersion integral in the latter expression can be used to implement experimental data up to some cut-off M 0 . One can then write, where the last two terms are computed using the operator product expansion (OPE) of R(s), i.e., including the leading non-perturbative condensate corrections. Demanding consistency with the OPE of the Adler function itself suggests that a value of M 0 as low as 2 GeV appears to be a safe choice. Using this approach implies [38] for the on-shell definition, The approach of Ref. [39] is mostly data driven. Experimental data were used up to 11.09 GeV (except for the interval between 2.6 GeV and 3.73 GeV) and PQCD beyond that. The dispersion relation (3.1) then implied, Here, we rely on the data handling of this work as it includes much more recent data than Ref. [39]. Moreover, the breakdown of individual channels and energy ranges is more explicit compared to Ref. [38]. Finally, changing our own result, withα(M 2 Z ) −1 = 127.959 ± 0.010, based on the direct application of the renormalization group and matching equations and including τ decay data, from the MS scheme to the on-shell scheme including the top quark contribution, we find, α(M 2 Z ) −1 = 128.949 ± 0.010 . (A.6) The numerical difference of our result to Ref. [16] arises mostly from the different 3 value of α s and our treatment of the charm quark contribution [37]. Thus, in view of the rather different approaches and differences in data sets, all numerical results are in good agreement with each other.

B Calculation of ∆ discα
In the on-shell scheme one has [40], where, C(t) has been computed [23] in units set by the lattice cut-off scale a −1 = 1.73 GeV. To obtain Eq. (4.10), we plotted ∆ disc α(q) as a function of T and observe a plateau near T = 20, which closely mirrors the result for the case of a µ . The value of the plateau is interpreted as the physical value [23]. As an independent check we compute the ratio ρ(a µ ) of the disconnected contribution to the anomalous magnetic moment [23], a disc µ = −9.6 × 10 −10 , to the total hadronic contribution [16] for energies up to 1.8 GeV, obtaining ρ(a µ ) = −0.015. The integration kernel of a µ enhances contributions from low q 2 momenta, and recalling that Q u +Q d +Q s = 0, the disconnected piece also predominantly arises from such momenta. On the other hand, the integration kernel for ∆α has greater support at higher scales compared to a µ , so that ρ(a µ ) should imply an upper bound on the disconnected contribution to ∆α. Numerically, where ∆ had α(1.8 GeV) = 55.26 × 10 −4 . This confirms the finding in Eq. (4.10) that ∆ disc α is very small.