$\alpha_s$ and $\rm V_{cs}$ determination, and CKM unitarity test, from W decays at NNLO

The hadronic ($\Gamma^{\rm W}_{\rm had}$) and total ($\Gamma^{\rm W}_{\rm tot}$) widths of the W boson, computed at least at next-to-next-to-leading-order (NNLO) accuracy, are combined to derive a new precise prediction for the hadronic W branching ratio ${\cal B}^{\rm W}_{\rm had} \equiv \Gamma^{\rm W}_{\rm had}/\Gamma^{\rm W}_{\rm tot}$ = $0.682 \pm 0.011_{\rm par}$, using the experimental Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, or ${\cal B}^{\rm W}_{\rm had} = 0.6742 \pm 0.0002_{\rm th} \pm 0.0001_{\rm par}$ assuming CKM unitarity, with uncertainties dominated by the input parameters of the calculations. Comparing the theoretical predictions and experimental measurements for various W decay observables, the NNLO strong coupling constant at the Z pole, $\alpha_s(m_Z) = 0.117 \pm 0.042_{\rm exp} \pm 0.004_{\rm th} \pm 0.001_{\rm par}$, as well as the charm-strange CKM element, $\rm V_{cs}$ = 0.973 $\pm$ 0.004$_{\rm exp}$ $\pm$ 0.002$_{\rm par}$, can be extracted under different assumptions. We also show that W decays provide today the most precise test of CKM unitarity for the 5 quarks lighter than $m_{\rm W}$, $\sum_{ u,c,d,s,b} |V_{ ij}|^2 = 1.999 \pm 0.008_{\rm exp} \pm 0.001_{\rm th}$. Perspectives for $\alpha_s$ and $\rm V_{cs}$ extractions from W decays measurements at the LHC and future $e^+e^-$ colliders are presented.

The hadronic (Γ W had ) and total (Γ W tot ) widths of the W boson, computed at least at next-to-next-to-leadingorder (NNLO) accuracy, are combined to derive a new precise prediction for the hadronic W branching ratio B W had ≡ Γ W had /Γ W tot = 0.682 ± 0.011 par , using the experimental Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, or B W had = 0.6742 ± 0.0002 th ± 0.0001 par assuming CKM unitarity, with uncertainties dominated by the input parameters of the calculations. Comparing the theoretical predictions and experimental measurements for various W decay observables, the NNLO strong coupling constant at the Z pole, α s (m 2 Z ) = 0.117 ± 0.042 exp ± 0.004 th ± 0.001 par , as well as the charm-strange CKM element, |V cs | = 0.973 ± 0.004 exp ± 0.002 par , can be extracted under different assumptions. We also show that W decays provide today the most precise test of CKM unitarity for the 5 quarks lighter than m W , u,c,d,s,b |V ij | 2 = 1.999 ± 0.008 exp ± 0.001 th . Perspectives for α s and |V cs | extractions from W decays measurements at the LHC and future e + e − colliders are presented.

I. INTRODUCTION
The strong coupling α s is one of the fundamental parameters of the Standard Model (SM), setting the scale of the strength of the strong interaction theoretically described by Quantum Chromodynamics (QCD). At the reference Z pole mass scale, its value amounts to α s (m 2 Z ) = 0.1181 ± 0.0013 [1] as determined from different experimental observables confronted to perturbative QCD (pQCD) calculations at (at least) next-to-next-to-leading-order (NNLO) accuracy [2]. Given its current δα s /α s ≈ 1% uncertainty-orders of magnitude larger than that of the gravitational (δG/G ≈ 10 −5 ), Fermi (δG F /G F ≈ 10 −7 ), and QED (δα/α ≈ 10 −10 ) couplings-the strong coupling is the least precisely known of all interaction strengths in nature. Improving our knowledge of α s is a prerequisite to reduce the theoretical uncertainties in the calculations of all high-precision pQCD processes whose cross sections or decay rates depend on higher-order powers of α s , as is the case for virtually all those measured at the LHC. In the Higgs sector, in particular, the α s uncertainty is currently the second major contributor (after the bottom mass) to the parametric uncertainties of the calculations of its prevalent H→ bb decay, the leading one for the H→ cc, gg modes [3], and it also introduces a 3.7% uncertainty on theoretical NNLO cross sections for the (dominant) Higgs production channel via gluon-gluon fusion [2].
The hadronic decay widths of the electroweak bosons, Γ W,Z had , are high-precision theoretical and experimental observables from which an accurate determination of α s can be obtained. On the one hand, the hadronic Z width -measured with 0.1% experimental uncertainty in e + e − collisions, and theoretically known up to next-to-NNLO (N 3 LO), i.e. O α 4 s QCD corrections-provides, combined with other Z-pole observables, a powerful constraint on the current α s world average [4]. On the other hand, the hadronic W width has not been used so far in any α s extraction. The reasons for that are twofold. First, the Γ W had experimental uncertainties -of order 2%, or 0.4% in the case of the more precisely known B W had ≡ Γ W had /Γ W tot branching fraction-are much larger than the corresponding ones for Γ Z had , whereas the α s sensitivity of the W and Z hadronic decays comes only through small higher-order loop corrections. Secondly, a complete expression of Γ W had including all computed higher-order terms was lacking until recently. This situation changed with the work of [5] [11] QCD corrections. Despite the progress, the work of [5] still contains a range of approximations (such as e.g. one-loop α s running between m W and m Z , and massless quarks), plus no real estimation of the associated uncertainties, which hinder its use to extract α s from a comparison to the data.
The purpose of this letter is twofold. First, by improving upon the N 3 LO theoretical derivation of the W hadronic width, removing various of the approximations applied in previous works, and by combining it with the total W decay width known at NNLO accuracy [12,13], we obtain a theoretical expression of the hadronic W branching ratio with a sound determination of all associated uncertainties. We then compare the theoretical predictions with the experimental data, and thereby determine α s . Secondly, since the hadronic decay width is directly proportional to the sum over the first two rows of the CKM matrix, u,c,d,s,b |V ij | 2 (the top quark is kinematically forbidden in W decays), we can also extract -by fixing now α s to its current world average-a precise independent value of the charm-strange quark mixing CKM element |V cs |, which currently has an experimental uncertainty of 1.6% (|V cs,exp | = 0.986 ± 0.016) [1]. We demonstrate, at the same time, that the measurements of W decays provide today the most stringent test of CKM arXiv:1603.06501v2 [hep-ph] 10 Oct 2016 matrix unitarity for all quarks lighter than the top quark. The developments presented here should motivate highquality measurements of W decays using the large datasets available at the LHC, as well as improve the α s extraction benchmarks expected from W measurements at future e + e − colliders such as ILC [14], FCC-ee [15], and CEPC [16].

II. HADRONIC W DECAY WIDTH AT N 3 LO ACCURACY
The hadronic decay width of the W boson can be decomposed into the following contributions: where Γ (0) denotes the Born decay width, O(α i s ) the higher-order QCD corrections, Γ ewk the electroweak corrections of order O(α), and Γ mixed the mixed electroweak+QCD corrections of order O(αα s ). In the massless quark limit, the zeroth-order decay width reads where N c = 3 is the number of colours, G F is the Fermi constant, m W is the W boson mass, and |V ij | the CKM matrix element ij summed over quark pairs (ij = ud, us, ub, cd, cs, cb). The first QCD correction to the tree-level width is The calculation of Γ W had can be factorized as a product of the Born width, Eq.
(2), times the remaining terms: where the c (i) QCD coefficients can be obtained from the perturbative expansion in α s of the well-known e + e − cross-section ratio R = σ(e + e − → hadrons) σ(e + e − →µ + µ − ) , calculated up to O(α 4 s ) in [11,17], with coefficients (for N f = 5 flavours): Numerically, the relative weights of the different partial widths in Eq. (1) are: (Table I). In Ref. [5], the first-order QCD corrections of Eq. (4) were obtained assuming zero quark masses, i.e. directly from the coefficients of Eq. (5), and the higher-order corrections and renormalization constants in the QCD, electroweak and mixed terms were obtained setting the CKM matrix to unity. Since Γ (0) + Γ (1) QCD numerically amount to ∼100% of Γ W had , a first improvement over [5] consists in computing the exact results for the Born width and the first QCD correction using finite quark masses, rather than through the first two coefficients of R. In our calculations, we thus replace Eq. (2) with the exact expression for the decay width with full quark masses m q,i [18], namely where κ(x, y, z) is the Källén function. Such an improved evaluation of the Born width also directly impacts the most important QCD correction obtained through Eq. (3). We have cross checked that our implementation of Eq. (6) matches numerically the result of Eq. (2) in the limit m q,i , m q ,j → 0, as well as the exact leading order calculation of [6]. For the remaining higher-order QCD corrections, starting from O(α 2 s ), we use the coefficients given by Eq. (5), while the electroweak and mixed corrections are those computed in [5]. Since the main motivation of the analysis is to obtain a precise value of α s , a second direct improvement with respect to the LO α s expression used in [5] is achieved by evaluating α s at the relevant scales here (m W and m Z ) including up to three loops (i.e. NNLO) in the renormalization group β function [19]. Also, for our numerical evaluations we use the latest values of the SM parameters with their associated uncertainties [1]: Here, m u , m d and m s correspond to current-quark masses, and m c , m b and m t to pole masses [1]. The Higgs boson mass corresponds to the most recent LHC average value [20]. When not left free, the QCD coupling is taken at its current world average, α s (m 2 Z ) = 0.1181 ± 0.0013 [1]. The experimental values of the CKM matrix elements used are which approximately satisfy the unitarity condition i V ij V * ik = δ jk and j V ij V * kj = δ ik . From the values (8), we have u,c,d,s,b |V ij | 2 = 2.024 ± 0.032 (i.e. with a 1.6% uncertainty, dominated by the |V cs | value), although in various cases below we will assume exact CKM unitarity, i.e. we will take u,c,d,s,b |V ij | 2 ≡ 2. Table I lists the partial and total hadronic widths obtained with and without assuming CKM unitarity. The results are compared (bottom rows) to the values of Ref. [5] obtained for zero quark masses, using the 2013 PDG SM input parameters, and without full determination of the associated uncertainties. Our result, without imposing CKM unitarity, is lower by about 30 MeV compared to that in [5], mostly due to the updated PDG parameters (the most important are the changes in |V cs | and |V cd | which result in width variations of −28 and −1.6 MeV respectively), whereas the inclusion of finite quark masses results in less than a ∼1 MeV decrease of the width.
Partial widths (MeV)  Our computed W hadronic width, listed in the last column of Table I, includes two type of uncertainties. The first "parametric" one, clearly dominant, is associated with the uncertainties of the various input parameters used in the calculations (mostly |V cs |, m W , and α s ). The second "theoretical" one is due to uncertainties mostly from missing higher-order corrections. The parametric uncertainties have been determined as follows. For each parameter p = |V ij |, m W , α s , ... we have calculated the decay width for p, p + ∆p and p − ∆p, while all other parameters are kept fixed at their central values. The error on the width is then determined by The total parametric errors have been obtained by adding in quadrature the parametric errors from the N parameter variations. The dominant parametric uncertainty is due to the |V cs | quark coupling strength, whose relative uncertainty of 1.6% [1] propagates into ±22 MeV in Γ W had . If one assumes CKM unitarity (or, equivalently, negligible |V ij | uncertainties) the second most important source of parametric uncertainty is that from m W which propagates into ±0.7 MeV in Γ W had . The theoretical uncertainties of our calculations are clearly much smaller than the parametric ones. They are obtained from the quadratic sum of missing higher-order QCD corrections, considered to be of the same size, ±0.019 MeV, as the O(α 5 s ) corrections assessed for the hadronic Z boson width [11]; plus missing higher-order electroweak and electroweak+QCD terms estimated to be ±0.012 MeV and ±0.029 MeV based on [5]. Non-perturbative effects -suppressed by O Λ 4 QCD /m 4 W power corrections-, zero quark mass approximations beyond LO [21] -estimated to be O(m 2 q /m 2 W ) and amounting to ±0.001 MeV at O(α 2 s ) and ±0.002 MeV at O(α)-, as well as residual effects due to the dependence on the CKM matrix renormalization scheme -evaluated in [22]-, are much smaller and neglected here. In Fig. 1 (left) (Table II).

III. HADRONIC W BRANCHING RATIO AT NNLO ACCURACY
The W hadronic branching fraction, given by the ratio of hadronic to all W decays, is a very simple and robust experimental observable. It is as inclusive as the total W cross section measureable in p-p or e + e − collisions but much free from experimental (e.g. normalization) uncertainties. We obtain its theoretical numerical value from the ratio B W had = Γ W had /Γ W tot , where Γ W had is the value computed in the previous Section, and the total decay width is that obtained from the NNLO calculation of [12] as parametrized in [13]. Using the input parameters (7)-(8) and the same procedure to compute parametric and theoretical uncertainties as for Γ W had , we obtain Γ W tot = 2093.4 ± 1.2 par ± 0.8 th MeV, which agrees well with the experimental value, Γ W tot,exp = 2085 ± 42 MeV [1], as well as with the indirect determination from the full electroweak fit Γ W tot,fit = 2091 ± 1 MeV [4]. The theoretical NNLO hadronic branching ratio amounts thus to B W had = 0.682 ± 0.011 par (using the experimental CKM matrix), with negligible theoretical compared to parametric uncertainties, and B W had = 0.6742 ± 0.0002 th ± 0.0001 par (assuming CKM matrix unitarity). Note also that the m W parametric uncertainty cancels out in the B W had ratio of hadronic to total W widths. Both results are in very good accord with the experimental value of B W had,exp = 0.6741 ± 0.0027, as shown in the right plot of Fig. 1   , and hadronic-to-leptonic ratio R W with their associated theoretical and parametric uncertainties (using the full calculation with experimentally-measured V ij elements where needed, or assuming CKM unitarity); compared to the current experimental world averages (last column).

IV. EXTRACTION OF α s
The theoretical dependencies on α s of the hadronic W decay width and branching fraction are shown in Fig. 2 imposing CKM unitarity (solid curves) or using the measured values of the CKM elements (dashed curves). The vertical lines indicate the current experimental values for both quantities while the grey bands indicate their associated uncertainties. Fixing all SM parameters except α s to their PDG values, and equating the theoretical expressions for Γ W had (α s ) and B W had (α s ) to their corresponding experimental measurements, the strong coupling can be extracted. The corresponding results are listed in the top rows of Table III, where the obtained α s (m 2 W ) values (second column) are evolved to the Z scale (last column) with the NNLO running coupling expression. As expected, the much larger uncertainty of Γ W had (±2%) compared to B W had (±0.4%) results in a more precise α s extraction from the latter. Yet, the current experimental and parametric uncertainties on Γ W had and B W had propagate into very large α s uncertainties in both cases. Clearly, those results call first for higher precision measurements of Γ W tot and B W had . Indicatively, for each MeV of reduced uncertainty on Γ W had,exp the precision of the extracted α s value would improve by approximately 2%. Secondly, a competitive extraction of α s requires also a reduction of the parametric uncertainties of the calculations. The impact of measuring |V cs | with better precision can be seen by comparing the α s values extracted with and without assuming CKM unitarity. Having |V cs | measured with a precision comparable to that of |V ud | today, namely 5·10 −4 , would make of m W the leading source of parametric uncertainty on the α s value extracted from W hadronic decays.    Fig. 3, as obtained imposing CKM unitarity (solid curve) or using experimental CKM elements (dashed curve). The theoretical predictions for the hadronic-toleptonic W branching ratio are R W = 2.069 ± 0.002 th ± 0.001 par (assuming CKM unitarity) and R W = 2.15 ± 0.11 par (experimental CKM), in very good agreement with the empirical result: R W,exp = 2.068 ± 0.025. The corresponding derived values of α s are listed in the bottom rows of Table III. The final most precise extraction of the QCD coupling from W decays is α s (m 2 Z ) = 0.117 ± 0.042 exp ± 0.004 th ± 0.001 par , with a relative uncertainty of 35%, obtained from R W imposing CKM unitarity.

V. EXTRACTION OF |V cs |, AND CKM MATRIX UNITARITY TEST
The hadronic W width, Eq. (6), involves a sum over the first two rows of the CKM matrix, i.e. the six CKM elements involving quarks lighter than m W listed in (8). Among these, the |V ud | and |V cs | terms are the most important in W hadronic decays and, as shown previously, the least precisely known (|V cs |) contributes to the largest uncertainty in the calculation of Γ W had and B W had . From the theoretical expressions and the experimental values of the hadronic width and branching ratio, fixing all SM parameters to their world-averages except |V cs |, we can extract the charm-strange mixing parameter. The corresponding results are listed in the middle column of Table IV. The associated experimental, parametric, and theoretical |V cs | uncertainties are propagated as explained before for the α s determination. The Γ W had,exp and B W had,exp uncertainties propagate into ±2% and ±0.4% respectively, the parametric uncertainties are of order ±0.2%, and the theoretical ones are negligible (±0.0004 th ) and not quoted. Our most precise extraction, combining hadronic and leptonic branching fractions through the R W ratio, yields |V cs | = 0.973±0.004 exp ±0.002 par , with a 0.5% uncertainty, improving by a factor of four the precision of the current world-average experimental value, |V cs,exp | = 0.986±0.016 [1]. As a matter of fact, the W decays provide the most stringent test of CKM unitarity today. Indeed, leaving free the sum u,c,d,s,b |V ij | 2 in the theoretical expression for B W had , the hadronic-to-leptonic ratio measurement of R W,exp = 2.069 ± 0.018 implies u,c,d,s,b |V ij | 2 = 1.999 ± 0.008 exp ± 0.001 th (with negligible ±0.0002 par parametric uncertainty).

VI. FUTURE PROSPECTS
A precise determination of the strong coupling, as well as stringent SM tests such as CKM unitarity, require measurements of W decays of higher precision than those available today. The total W width has been directly measured via maximum-likelihood fits of (i) the Breit-Wigner W mass distribution in e + e − → W + W − , yielding Γ W tot,exp = 2195 ± 83 MeV [23], as well as of (ii) the tail of the W transverse mass m T ( ν) spectrum in leptonic W→ ν decays in p-p, p-p → W + X collisions, yielding Γ W tot,exp = 2046 ± 49 MeV [24] (their combination yielding the experimental world average quoted in Table II). The branching fraction Γ W had can only be measured with small uncertainties in e + e − → W + W − [23], although a competitive Γ W had = 1 − Γ W lep value can be obtained from precise measurements of the total W width and the leptonic branching ratio exploiting the large W data samples at p-p, p-p colliders [24,25]. Measurements at the LHC and future e + e − colliders will provide Γ W had , B W had and R W with higher accuracy and precision. In the hadron collider determinations of Γ W tot and B W lep , the leading source of systematic uncertainties comes from the proton parton distributions functions (PDF), amounting to 70% and 60% respectively [24,25]. At the LHC, a maximum factor of four reduction of the current uncertainties on the derived value of B W had,exp can be assumed thanks to our improved knowledge of PDFs, and the much higher statistics available in measurements of the large-m T ( ν) spectra (Fig. 4, left). Combining all upcoming W decays measurements at the LHC with the currently available results, can thereby reduce the propagated α s experimental uncertainty to the 10% level, but going below this can only be achieved through high-precision e + e − measurements. In e + e − → W + W − at the FCC-ee, the total W width Γ W tot can be accurately measured through a threshold scan around √ s = 2m W , and also the W hadronic branching ratio B W had would profit from the huge sample of 5 × 10 8 W bosons (a thousand times more than the 5 × 10 5 W's collected at LEP) [15] which would reduce the statistical uncertainty of B W had to around 0.005%. Thus, neglecting parametric uncertainties, a B W had measurement at the FCC-ee would significantly improve the extraction of α s with propagated experimental uncertainties of order 0.4%. The α s uncertainty could be further lowered down to ∼0.2% through the measurement of the R W ratio in three e + e − → W + W − final states, such as ν ν, ν qq, qq qq. Indeed, the ratio of cross sections σ(WW → qq qq)/σ(WW → ν ν) is proportional to (R W ) 2 , thereby gaining a factor two in statistical sensitivity, and being totally independent of potential modifications of the weak coupling running and free from cross section normalization uncertainties [15]. Figure 4 shows the estimated α s extractions from the expected improved measurements of Γ W had alone at the LHC (left), and R W at FCC-ee (right). Also the obtained ratios of hadronic-to-leptonic branching fractions, R W = 2.069 ± 0.002 th ± 0.001 par (assuming CKM unitarity) and R W = 2.15 ± 0.11 par (experimental CKM elements), are in very good agreement with the measured value R W,exp = 2.068 ± 0.025. By comparing the experimental results to the theoretical expectations, we have extracted the strong coupling α s , and the charmstrange CKM element |V cs | under different assumptions. The current experimental and parametric uncertainties on Γ W had , B W had and R W are too large today to allow for a precise determination of α s (the best result obtained is α s (m 2 Z ) = 0.117 ± 0.042 exp ± 0.004 th ± 0.001 par , assuming CKM matrix unitarity) although upcoming high-statistics W measurements at the LHC could reduce the α s extraction uncertainties to the ∼10% level. Our study shows that a future high-luminosity e + e − collider such as FCC-ee running at √ s ≈ 2m W will allow for an α s determination with uncertainties as low as 0.2%.

VII. SUMMARY
We have also quantified the constraints that the hadronic W decays impose on the quark mixing parameters as encoded in the CKM matrix of the Standard Model. By fixing all SM parameters, including α s , to their default values and leaving free |V cs | in the theoretical expressions for B W had , we can determine the charm-strange coupling with a 0.5% uncertainty, |V cs | = 0.973 ± 0.004 exp ± 0.002 par , which is four times better than the current world-average experimental value, |V cs,exp | = 0.986 ± 0.016. Similarly, the experimental values of the hadronic and leptonic W branching fractions imply u,c,d,s,b |V ij | 2 = 1.999 ± 0.008 exp ± 0.001 th , providing today the most stringent test of CKM unitarity for the five lightest quarks.