Renormalization group improved $m_s$ and $\vert V_{us}\vert$ determination from hadronic $\tau$ decays

We determine the strange quark mass ($m_s$) and quark mixing element $\vert V_{us}\vert $, and their joint determination from the Cabibbo suppressed hadronic $\tau$ decays in various perturbative schemes. Compared to the previous analysis based on the optimal renormalization or the renormalization group summed perturbation theory (RGSPT) scheme, we have improved this analysis by replacing the theoretical longitudinal contributions with phenomenological parametrization, and the RGSPT coefficients are used for the dimension-4 Adler functions. The improved analysis results in the extraction of $m_s(2\,\mathrm{GeV})=98\pm19\,\mathrm{MeV}$ and $\vert V_{us}\vert=0.2191\pm0.0043$ from the RGSPT scheme.


I. INTRODUCTION
The hadronic decays of the τ leptons have been of constant interest for determining various parameters of the Standard Model (SM) of particle physics. The availability of experimental data on the strange and non-strange decay modes for the hadronic τ decays has opened the window for the determination of various parameters relevant for the quantum chromodynamics (QCD), namely the strong coupling constant α s , the strange quark mass m s , the vacuum condensates, the low-energy chiral couplings, and the quark mixing element |V us | of the Cabibbo-Kobayashi-Maskawa (CKM) matrix (see ref. [1,2] for details).
On the theoretical side, the QCD contributions to the hadronic τ decays are studied by evaluating the current correlator using the Operator Product Expansion (OPE) [3]. The OPE factorizes the long-and shortdistance contributions. The long-distance information is encoded into the vacuum condensates. The short distance part is written as the perturbative series in the strong coupling constant and quark masses. The vacuum condensates can also be evaluated using chiral perturbation theory (ChPT) [4], lattice QCD [5], and Renormalization Group (RG) optimized perturbation theory [6,7]. The short distance contributions require the evaluation of the Feynman diagrams. It is also known that some of the contributions to the hadronic vacuum polarization function are not captured by the OPE, and it is a quarkhadron duality violation. These duality violating terms are parameterized in a model-dependent way and fitted to experimental data and should also be added to the OPE contributions [8].
The longitudinal component of the QCD Adler function, corresponding to the zero angular momentum state, has been calculated to O α 4 s [9][10][11][12][13][14]. It has poor convergence behavior and raises the question of the method's applicability in the extraction of strange quark mass. * anant@iisc.ac.in † diganta.das@iiit.ac.in ‡ alam.khan1909@gmail.com This problem can be cured by replacing these contributions with their phenomenological input and has been used in ref. [15][16][17][18][19] for m s and |V us | extractions from the experimental moment data. These improvements have resulted in much better control over the theoretical uncertainties in the m s and |V us | determinations. The hadronic τ decays have been extensively studied using various perturbative schemes. These schemes differ in how strong coupling constant and quark masses are evaluated along the contour in the complex plane using their renormalization-group (RG) properties. The most commonly used schemes in the extraction of strange quark mass and CKM matrix element from the Cabibbo suppressed hadronic τ decay are fixed-order perturbation theory (FOPT) and contour-improved perturbation theory (CIPT). For the hadronic τ decays, the FOPT suffers from the problem of large logarithms along the contour in the complex energy plane, and the higher-order spectral moments are very sensitive to scale variations. In the CIPT scheme, direct numerical evaluation of coupling constant and masses along complex contour using their RGE does not suffer from the problem of the large logarithms. However, scale dependence is still the major source of theoretical uncertainties for higher moments.
Recently, the optimal renormalization or RG-summed perturbation theory (RGSPT) has been used by two of us in ref. [20] in the strange quark mass determination. The behavior of polarization and Alder functions in the complex contour was also studied for RGSPT, CIPT, FOPT, and the method of effective charges (MEC) in great detail. However, the numerical impact of the theoretical uncertainties from perturbation series truncation, and scale dependence were excluded. We improve the previous analysis by: determinations for the first time using RGSPT.
• The effects of the variation of the m s and |V us | with the variation of the moments calculated at different energies (s 0 < M 2 τ ) is also included and found to be constituting an important source of uncertainty.
• Using the five-loop QCD β−function and anomalous dimensions for the running of the strong coupling constant and quark masses.
The article is organized as follows: Section II provides a brief overview of the various quantities that are needed for the extraction of m s and |V us |. A short introduction to RGSPT is given in section III. Section IV explains the OPE contributions to the Adler function. The behavior of leading-order mass corrections to the Adler functions in different schemes used in this article is studied in the section V. The higher-order term of the perturbation series becomes very important for the higher moments, and two prescriptions for the truncation of the perturbation series are also defined in this section. In section VI, the phenomenological parametrization of longitudinal contributions is briefly discussed. Then, we move to section VII where strange quark mass is extracted using only the perturbative QCD (pQCD) contributions calculated from OPE. The weighted average results for the m s (M 2 τ ) extraction using this method in CIPT, FOPT and RGSPT schemes are presented in table IV. The details of uncertainties can be found in the appendix E 1. In section VIII the m s (M 2 τ ) determination using the phenomenological parametrization for the longitudinal component is performed and results are presented in table V. Details of the strange quark mass determinations from moments are presented in the appendix E 2. The determination of |V us | using external input for m s is performed in section IX. The weighted average results using the OPAL and ALEPH data are presented in table VIII and table VI, respectively. The details of determinations from the moments as well as the uncertainties coming from various sources are presented in appendix E 3. In section X the joint extraction of m s and |V us | is performed. We provide a summary and conclusion in section XI. We also provide supplementary inputs needed for this article in the appendix A,B,D,C. Details of m s and |V us | determinations from moments can be found in appendix E.

II. FORMALISM
An important quantity for the study of hadronic τ decay width [21][22][23] is the two-point current correlator:  (1) where |Ω⟩ denotes the physical vacuum, J V /A µ,ij (y) = (q j γ µ / (γ µ γ 5 ) q i ) (x) is the hadronic vector/axial current, and the indices i and j denote the flavors of light quarks.
The current correlator can be calculated perturbatively using OPE [3] as a power expansion in 1/p, and the corresponding coefficients of are the operator of that dimension. Purely perturbative corrections appear up to dimension-2 in the OPE expansion, and the longdistance corrections corresponding to the vacuum condensates start from dimension-4.
Using Lorentz decomposition, the current correlator in eq. (1) can be decomposed into the longitudinal and transverse components with angular momentum J = 0 and J = 1 as: (2) The L/T correlators are related to experimentally measurable semi-hadronic τ decay rate (R τ ), defined by: and it is related to the imaginary part of the current correlators in eq. (2) by: It should be noted that these current correlators also carry the information about mixing among the quark flavors and can be written as: and |V ij | are the elements of the CKM matrix.
To study the invariant mass distribution of final-state hadrons, we need moments from the hadronic τ decay rate, defined by [24]: Using integration by parts, we can convert eq. (6) into the following form: where x c = s/s 0 and D L+T (s) are the Adler functions. Usually, the experimental value of the moments defined above are provided for s 0 = M 2 τ in the literature. However, their values at different energy can be calculated using the experimental data on the spectral functions provided in the refs. [25][26][27][28]. The Adler function satisfies the homogeneous renormalization group equation (RGE) and is related to the current correlators by the relation: The resulting quantity in eq. (7) is an expansion in the strong coupling constant, quark masses, and condensates of higher dimension operators. It explicitly depends on the CKM matrix element and the electroweak corrections. These terms are not shown in eq. (7) but factored out in eq. (12) and eq. (13). The kinematic kernels F kl L+T (x c ) and F kl L (x c ) appearing in the eq. (7) are given by: and their explicit form used in this article is presented in the table I. Performing the contour integral defined in eq. (7), we can write R kl τ [21] as: where ud,us [21] carry the information of contour integrals evaluated in eq. (7), δ ′ EW = 0.0010 and S EW = 1.0201 ± 0.0003 are one-loop RG-improved electroweak corrections [29,30].
The most important quantity of interest in the determination of the strange quark mass is the SU (3) breaking terms δR kl τ (s 0 ) [31] defined as which is free from instanton and renormalon contributions and vanishes in the chiral limit. It is an experimentally measurable quantity that is used as input from table II along with the theoretical quantities appearing in eq. (13) in the strange quark mass determination. The value of strong coupling constant α s (M 2 Z ) = 0.1179 ± .0010 has been taken from ref. [32] and evolved to τ lepton mass scale using five-loop β−function using package REvolver [33]. Its value at τ lepton mass is α s (M 2 τ ) = 0.3187 ± 0.0083 and has been used in this article.

III. REVIEW OF OPTIMAL RENORMALIZATION
The optimal renormalization technique is used to resum the running logarithms present in the perturbation series using RGE [34][35][36]. The resulting summed series shows reduced scale dependence and hence a reduction in the theoretical uncertainty in the extraction of a quantity of interest. In the case of hadronic τ decays, where weighted integrals along the complex contour are involved, these running logarithms become very important. Their summation is necessary to perform the perturbative analysis properly. RGSPT resum these logarithms, and the resulting fixed order truncated series has less sensitivity to scale variations even for higher moments than FOPT and CIPT.
The perturbative series describing a QCD process is given by: where x ≡ x(µ 2 ), L ≡ log(µ 2 /q 2 ) and m = m(µ 2 ). We can rewrite the series as follows: where the S i [xL]-coefficients are given by: The RGE for eq. (15) is given by: where γ a (x) = i=0 x i+1 γ (i) a is the anomalous dimension associated with W (x, m). We can collect the terms corresponding to summed coefficients defined in eq. (17). This process results in a set of coupled differential equations for S i [w], which can be summarized as: Here we have substituted w = 1 − β 0 xL, which simplifies the solutions of differential equations. For further details on RG summation, we refer to ref. [20,[34][35][36][37][38][39].
The solution to the first three summed coefficients, relevant for dimension-0 and dimension-2 Adler functions, appearing in the eq. (20) can be found in the appendix B. It should be noted that the RGE for dimension-4 operators mix perturbative coefficients with condensates; hence they do not obey eq. (16). The RG-summed perturbative coefficients are relevant for eq. (33) and eq. (34) in the appendix C.

A. Leading order contribution
Dimension-zero is the leading perturbative contribution to the current correlator in the massless limit and has been calculated to O α 4 s [40][41][42][43][44][45][46][47][48][49]. It receives a contribution only from the transverse piece of the current correlator, which is identical for both vector and axialvector channels and thus cancels in the eq. (13). The Adler functions obtained using the OPE can be organized as follows: where ij are the flavor indices and the Adler functions in the RHS of the above equations are expansion in α s , m q , and the quark and gluon condensate terms. Their definition gets clearer if we take a contour integration along s = M 2 τ e iϕ and the coefficient of (M 2 τ ) −n are called operators of dimension 2n. The massless Adler functions are given by where x(−s) = x(q 2 ) = α s (q 2 )/π andK L+T i are the coefficient of Adler function at i th −loop which can be found in appendix D. The RG running of dimension-zero "L + T "−component of the Adler function is given by: where L = log( µ 2 −s ), and the QCD beta function (β(x)) is defined as: The coefficients of the beta function β i 's are known up to the five-loops and are presented in the appendix A.

B. The Dimension-2 contributions to the Adler Function
The leading order mass corrections to hadronic τ decay rate come from the dimension-2 Adler function. The D L+T (s) Adler function is known to be O α 3 s . [50][51][52][53][54][55] while D L is known to O α 4 s [9][10][11][12][13][14] and their analytic expression can be found in the appendix D 2. The RG running of dimension-2 operators is given by: where the QCD beta function and the quark mass anomalous dimension (γ m ) are known to the five-loop and can be found in the appendix A. The SU (3) breaking contributions from the Adler function in the determination of quark masses is the difference: where J = (L+T )/L and the analytic expressions can be found in the appendix D 2. These contributions are used in eq. (13) to evaluate the leading order mass correction term δ The renormalization group running of different coefficients for CIPT and FOPT coefficients can be found in the ref. [22,56]. The RG-summed coefficients can be obtained from appendix B by setting {n 1 , n 2 } = {0, 2}.

C. The Dimension-4 contributions to the Adler Function
The OPE expansion at dimension-4 involves contributions from perturbative and quark, and gluon condensates [22,57]. However, these contributions are suppressed by a factor of ( 1 M 2 τ ) 2 , and they have the following form: where the upper/lower sign corresponds to the V/A component. TheΩ L+T /L coefficients are given by: where ⟨m iqj q j ⟩ ≡ ⟨0|m iqj q j |0⟩(−ξ 2 s), m i = m j (−ξ 2 s), ⟨G 2 ⟩ ≡ ⟨0|G 2 |0⟩(−ξ 2 s) and ξ is the scale parameter to keep track of the dependence of the renormalization scale. The RG-evolution of the perturbative coefficients and the condensates can be found in the ref. [22]. The relevant OPE corrections to strange quark mass determination are as follows: where δO 4 = ⟨0|m s ss − m d dd|⟩(−ξ 2 s) with ϵ u = m u /m s and ϵ u = m u /m s . Using the numerical values v s = 0.738 ± 0.029 [58] , f π = 92.1 ± 0.8 MeV , m π = 139.6 MeV [32] ϵ d = 0.053 ± 0.002 , ϵ u = 0.029 ± 0.003 [59] , δO 4 can be estimated similar to ref. [22] as: The FOPT and CIPT are the two versions of perturbative theory for the QCD analysis of the τ decay frequently used in the literature, and it has been further extended by including an RGSPT version of perturbation theory [20,37]. It has been shown in the ref. [37] that the δ (0) contributions from the RGSPT scheme approach CIPT at higher orders of the perturbation theory, and the corresponding numerical value of the strong coupling constant lies closer to the CIPT value. Similar behavior is also observed in this article for the higher dimensional operators, but with the advantage that the scale dependence for higher moments is under control in the case of RGSPT compared to the FOPT and CIPT. Before moving on to the strange quark mass determination, the convergence behavior of the leading order mass corrections must be analyzed carefully for different schemes. This exercise is performed in the rest of the section.
The leading order mass corrections to moment δR kl,2 τ in eq. (13) are given by: where, and the ∆ J kl (x, ξ) are the contributions from the Adler functions δD J involving eq. (D5),(D8),(D11)) evaluated along a contour in the complex plane with the kernels presented in table I. These functions are calculated differently in various schemes explained in the later subsections.
It should be noted that the leading-order mass corrections are presented to remember where perturbative series is truncated in prescription I.

A. CIPT scheme
In CIPT, the masses and the strong coupling evolved along the contour in the complex plane by solving the RGE numerically. By construction, it does not suffer from the problem of large logarithm along the contour. Following the ref. [22,56,60], dimension-2 contribution to "L + T "−component moments can be organized in terms of contour integrals: and for the longitudinal component: The dimension-2 contributions to ∆ L+T kl for x(M 2 τ ) = 0.3187/π contributions of different orders are given by: It is clear from the eq. (43) that the pathological longitudinal contributions are a restricting factor in getting any reliable determination from CIPT unless we truncate the perturbative series to the minimum term.

B. FOPT scheme
In FOPT, the perturbative series for the Adler function is truncated to a given order in α s (µ), and running logarithms are integrated analytically along the contour in the complex energy plane [56,61]. The ∆ J kl for FOPT is evaluated by inserting eq. (D6),(D9) in eq. (37) and can be written as: where H kl,J i (ξ) are evaluated analytically: Evaluating the above integrals, the ∆ L+T kl contribution, using FOPT, at different orders of perturbative series is given by: We can see that the longitudinal piece has a bad convergence in the FOPT scheme. The total contribution ∆ kl is: We can see that the convergence behavior of dimension-2 contribution in eq. (49) is not very different from the CIPT scheme in eq. (43).

C. RGSPT Scheme
In Optimal renormalization, masses and coupling are fixed at some renormalization scale, but the RG-summed running logarithms are evolved around the contour. Interestingly, contour integration can be done analytically, similar to FOPT. However, due to the summation of the running logarithms, the resulting perturbative contributions are very much closer to the CIPT numbers and can be seen later in this subsection.
The perturbative series in RGSPT scheme for dimension-2 Adler function has form: which is obtained by inserting eq. (D11) into eq. (37) and the corresponding contour integrals K kl,J n,m (x, ξ) have the following form: The ∆ L+T i,0 contributions for different moments are given by:   [31,62] and OPAL [27]. OPAL moments are calculated using current value of |V us | = 0.2243 ± 0.0008 quoted in the PDG [32]. and the ∆(x) behave as: We can see from the numerical values provided in eq. (42), eq. (48) and eq. (54) that the longitudinal contributions have a convergence issue, and it is difficult to get the reliable determinations using them as input. However, the important ingredient in the mass determination is ∆ i,j , defined in eq. (38). We can see from the numerical values presented in eq. (43), eq. (49) and eq. (55) that these inputs can be taken in the mass determination if we truncate the perturbation series to the term which gives minimum contribution to it. This minimum term of the perturbative series is taken as the truncation uncertainty. This prescription has already been advocated in ref. [62], and we have termed this procedure of truncation as prescription I. Another choice is to use all available terms of the perturbation series coefficients of the Adler function, including the estimate for the unknown O α 4 s term of the "L + T "−component of the dimension-2 Adler function and termed as prescription II. These prescriptions have some advantages and disadvantages, which will be discussed later.

VI. PHENOMENOLOGICAL CONTRIBUTION TO THE LONGITUDINAL SECTOR
We can see from the section V that although the contributions from the "L + T " part of the dimension-2 has a better convergence for CIPT and RGSPT relative to the FOPT, the longitudinal contributions are forcing us to truncate the higher-order terms. These pathological contributions get enhanced for higher moments and restrict one to use only the leading-order term of the perturbation series. This problem is cured by replacing the longitudinal perturbative series contributions with the phenomenological contributions from the chiral perturbation theory [15][16][17][18][19]. These contributions carry significantly less theoretical uncertainty and agree well with the corresponding pQCD results, as shown in ref. [15]. With these advantages at hand, the strange quark mass determination using the pQCD contribution from the "L+T "−component of the Adler function combined with phenomenological contributions for longitudinal contributions in section VIII can be performed.
The relevant quantities of interest for phenomenological contributions to R kl,L ij,V /A , the longitudinal component of eq. (7), are vector/axial-vector spectral functions ρ V /A ij (s). They are related by: The pseudoscalar spectral function receives contributions from pion and kaon mass poles and higher resonances in the strange and non-strange channels. We are using the Maltman and Kambor [16] parametrization of the pseudoscalar spectral function for the us and ud channels in our analysis, which is given by: Here f i and M i are the decay constants and masses of resonances, and B i (s) is the Briet-Wigner resonance function taking the form: where Γ i is the decay width of the resonances. The spectral function for the ud channel is obtained by replacing the kaon terms with the pion in eq. (57). For the resonance contributions to pseudoscalar ud and us channels appearing eq. (57), we have used the following data:   [32] and decay constants from ref. [17].
The vector component of spectral function receives dominant contributions from the scalar channels Kπ,Kη and Kη ′ and the spectral function has the following form [19]: The phase space factor σ Ki (s) are given by: The strangeness changing scalar form factors F Ki (s) are defined by: and can be found in ref. [18]. Detailed discussion on the application of these form factors in the extraction of strange quark mass can be found in the refs. [15,19].

VII. STRANGE QUARK MASS DETERMINATION FROM PQCD
The strange quark mass determination in this section is based on the method used in refs. [20,22,62]. In addition, we have employed different schemes to perform the comparative study. The strange quark mass determination from hadronic τ decays using RGSPT has been performed ref. [20]. However, the uncertainties coming from the truncation of perturbative series and the scale dependence of strange quark masses were neglected. We have improved the previous determination using pQCD inputs by including these uncertainties and the determinations made in the two prescriptions mentioned in section V.
It should be noted that the higher dimensional OPE contributions (d > 4) to the Adler functions, which are numerically small [22] and not considered in this analysis. The strange quark mass is determined by supplying experimental and theoretical inputs to eq. (13). The RHS of the equation is provided with theory inputs from dimension-2 contributions eqs. (D5), (D8) and dimension-4 with eqs. (33), (34). These quantities are evaluated along the complex contour in different schemes, as explained in section V. We present our weighted averaged determinations for m s (M 2 τ ) from different moments in the table IV for different schemes. The details of various sources of uncertainty in the two prescriptions are presented in appendix E 1.
We can see from table IV that the strange quark mass determination from different schemes agrees with each other within uncertainty. It is also evident from the tables presented in appendix E 1 that the uncertainties in the final strange quark mass are higher in prescription I than in prescription II mainly due to the truncation of the perturbative series. We also emphasize that the systematic comparison of the behavior of perturbative series in different schemes can only be made in prescription II, where the same order information is used. The RGSPT provides better control over the theoretical uncertainty by minimizing the renormalization scale dependence. The scale dependence of the strange quark mass for various moments is shown in the figure 1, plotted using prescription II. These plots indicate that the strange mass from the RGSPT scheme is stable for a wider range of scale variations for the moments under consideration. It should be noted that the uncertainties associated with renormalization scale dependence are included only in the range ξ ∈ [0.75, 2.0] in the strange mass determination in table IV.
It should be noted that the poor convergence of the longitudinal contributions restricts this method to be applicable in the lower energies s 0 < M 2 τ . Additional uncertainties in the determinations of m s arise due to the variations of the upper limit of the moment in the integral s 0 defined in eq. (4). These are estimated using the phenomenological determination, discussed in the next section, and are also included in the table IV. Further details on the numerical uncertainties using pQCD inputs can be found in the appendix E 1.   The determination of the strange quark mass in this section is similar to the one used in section VII, but now the longitudinal Adler function is replaced with the phenomenologically parameterized contributions, as discussed in section VI. It should be noted that the "L + T " component of the Adler function at dimension-2 is known to O α 3 s and we do not use its estimate for the O α 4 s coefficient in the determination of m s (M 2 τ ) using phenomenological inputs. Contributions from the last known term of the perturbation series of the Adler function are taken as the total truncation uncertainty, similar to that of the previous section.
Following the discussions of section VI, we now have all the necessary ingredients for the strange quark mass determination. Using the transverse contributions used in section VII and combining them with the input from section VI, we determine the strange quark in different schemes. We present our result for the weighted average in the table V and the further details of the determinations from moments in the appendix E 2. The scale dependence in the m s (M 2 τ ) is presented in the figure 2 using prescription II. As observed in the previous section, the strange quark mass determinations from the RGSPT scheme are stable over the wider range of scale variation for moments under consideration. The determination of the m s from the traditional spectral moments is sensitive to the variation of the s 0 . A typical 5% variation of the s 0 from M 2 τ in the range s 0 ∈ 3, M 2 τ induces variations of ∼ 6−13% in the m s determinations from moments using the OPAL data. Unfortunately, such variations can not be calculated for the ALEPH moments as the strange spectral function is not publicly available. These uncertainties are estimated from the determinations using the OPAL data.

IX. DETERMINATION OF |Vus|
The data on strange and non-strange spectral moments for the hadronic τ decay provided by ALEPH [26,63,64], HFLAV [65], and OPAL [27] collaborations can be used to determine the CKM matrix element |V us |. These experimental moments along with the theoretical moments calculated with strange quark mass as input from other sources can be used to determine |V us | using the following relation: where R kl τ,S and R kl τ,V +A are experimental inputs and δR kl τ,th is the theory input, in which m s (2 GeV) is taken as an external input. This method has already been used previously in refs. [15,[66][67][68][69][70], and it has been observed that the uncertainties are dominated by the experimental data available for the strange component. An additional source of uncertainties is pointed out in ref. [66] due to s 0 variations which can be cured using a different analysis based on the non-spectral weight functions. However, we have restricted this analysis only to the traditional weight functions.
The latest branching fraction of hadronic τ decays into non-strange and slightly more precise strange component from HFLAV [65] can also be used to get more precise determination of |V us | from this method. The results for different schemes are presented in table VII.
The uncertainties shown in these tables are dominated by those coming from the variation of s 0 ∈ 2.5, M 2 τ and    [65] are based on (0, 0)−moment. A detailed analysis for higher moments can be performed using the OPAL [27] data, where (k, 0) moments for k = 0, · · · , 4 are also available. These moments are correlated and their correlation should also be considered in the full analysis. Given the large uncertainties in their strange components and unknown precise higher dimensional OPE corrections, we have neglected these correlations among various moments in our determinations from this data.
Using the strange and non-strange moments from OPAL and the theoretical inputs for δR kl τ,th from Section VII, we present the weighted average of the determination |V us | from prescription I and prescription II in the table VIII. Details of the |V us | from moments along with the sources of uncertainties can be found in table XX and table XXI, respectively. We can observe from these tables that the RGSPT is slightly more sensitive to the strange quark mass taken as input, but the overall theory uncertainty coming in this scheme is lesser than CIPT and FOPT in prescription II. We can also see that the divergent nature of the longitudinal component is still an issue causing a large theoretical uncertainty dominating in the higher moments in prescription II in the table XXI.
These shortcomings are slightly improved in the phenomenological determination and can be seen in table XXII and table XXIII. Again, these determinations suffer from large s 0 and theoretical uncertainties, especially those coming from the strange quark mass in the higher moments. It is worth emphasizing that prescription I reduces the dependence on the spectral moments in the |V us | determination. The weighted average of these results for |V us | are presented in the table VIII. The RGSPT scheme is slightly more sensitive to the variation of the s 0 , which, compared to CIPT, dominates in the final average presented in the

X. JOINT ms AND |Vus| DETERMINATION
The experimental moments provided by the OPAL collaboration in ref. [27] can be used for the joint extraction of m s and |V us |. It should be noted that the moments provided in the ref. [27] are correlated, and a proper analysis will require their correlations to be taken into account. Given the uncertainties present in the data, we are disregarding these correlations and restrict ourselves to simplified analysis .
Using the phenomenological parametrization for the longitudinal contributions and perturbative "L + T "−component from section VI and section IV, we fit the m s (2 GeV) and |V us | to eq. (62) for moments (k, 0) with k = 0, 1, · · · , 4. The central values of the joint fit are presented in table IX. These joint fits give smaller values for m s (2 GeV) and |V us | compared to the PDG average [32], but very close to the findings of Gamiz et al. [67] for CIPT and RGSPT.

XI. SUMMARY AND CONCLUSION
The hadronic τ decays are important ingredients for extracting various QCD parameters. We have used perturbative schemes CIPT, FOPT, and RGSPT in the extraction of m s , |V us |, and their joint determinations from the experimental inputs available from ALEPH [26,63,64], HFLAV [65], and OPAL [27]   hadronic τ decays. To reach the goal, we first calculate the RGSPT coefficients for the dimension-4 operator and use them for determinations of m s and |V us |. Dimension-6 OPE corrections are known to NLO and their RG-improvement is discussed in ref. [71][72][73]. Four quark condensates present in these contributions are estimated using the vacuum saturation approximation [74] and found to be numerically very small and are not considered in this article. Higher dimensional OPE corrections are not fully known and are neglected in this article. The moments calculated using the perturbation theory suffer from convergence issues, so we have employed two prescriptions. The central values of the strange quark mass determinations from prescription I are less spread for different moments than in prescription II. The theoretical uncertainties arising from truncation and scale dependence dominate for higher moments in the prescrip-tion I for CIPT and FOPT. However, RGSPT has better control over the scale dependence for a wider range of scale variation, even for higher moments as shown in figure 1 and figure 2. This improvement results in a more precise determination in RGSPT compared to FOPT and CIPT schemes.
The important results of this article for the m s (M 2 τ ) determination are presented in table (IV),(V) and for |V us | determinations in tables (VIII), (VI) and (VII). The joint m s and |V us | determination results are presented in table IX. It should be noted that the ALEPH moments used in the m s determinations in this article are based on the old |V us | calculated in ref. [62]. The strange quark mass determinations from the moments are very sensitive to the value of |V us |, and hence we do not consider them in the final average. However, the experimental data for the strange and non-strange moments provided by the OPAL collaboration in ref. [27] with the current value of |V us | = 0.2245 ± 0.0008 [32] as an input can be used to provide the most updated determination of the strange quark mass.
The values obtained for |V us | using OPAL data agrees with the PDG average within uncertainties. However, the |V us | determination from ALEPH [26,63,64] and HFLAV [65] are more than 1.2σ and 1.9σ away from the PDG [32] average and CKM unitarity fit value. It should be noted that the PDG average is already in tension with 2.2 σ with the CKM unitarity.
The dependence of our determinations on the choice of the moments and their correlation is not considered in this article, and we expect that these can be further improved using non-spectral weights used by Maltman et al. in ref. [75][76][77].

XII. ACKNOWLEDGMENTS
We thank Prof. Diogo Boito for providing help with the experimental data on the spectral functions. BA is partly supported by the MSIL Chair of the Division of Physical and Mathematical Sciences, Indian Institute of Science. DD would like to thank the DST, Govt. of India for the INSPIRE Faculty Award (grant no IFA16-PH170). DD also thanks the Institute for Theoretical Physics III, University of Stuttgart for kind hospitality during various stages of the work. AK thanks Shiuli Chatterjee, Rhitaja Sengupta, and Prasad Hegde for their valuable discussions. AK is supported by a fellowship from the Ministry of Human Resources Development, Government of India.

The Dimension-2 Corrections
The dimension-2 correction to the Adler function for the quark flavor i and j is known to O α 3 s [50][51][52][53][54][55] and the analytic expression reads: where upper and lower signs correspond to vector and axial-vector components, respectively, and this convention is used for the Alder functions in this section. It should be noted that the O(α 4 s ) correction to m i m j term has been obtained from eq(15) of ref. [50]. The longitudinal component of the dimension-2 operator is known to O α 4 s [9-14] and has the form: From the above dimension-2 Alder functions, the important piece relevant for Cabibbo suppressed strange quark mass determination [22] using eq. (13) are: and the corresponding contribution from the longitudinal component is: The RGSPT coefficients for the dimension-2 operators can be written in the following form: .
Appendix E: Details of ms and |Vus| determinations in different schemes and the details of sources of uncertainty

Strange quark mass determinations using pQCD inputs
In this section, the strange quark mass is calculated using the longitudinal component calculated using OPE as described in sec. V. As mentioned before, these contributions are poorly convergent, and the strange quark mass determinations will suffer from the large truncation uncertainties in addition to significant dependence on the moments used. The dependence on the moment can be slightly reduced by using the prescription I at the cost of enhanced truncation uncertainty. This behavior in the different perturbative schemes and the details of various sources of uncertainties are discussed in the later subsections. τ ) determination and can be seen in the tables[X,XI]. However, the main advantage of using prescription-I is that the masses from various moments, using different experimental inputs, agree within the uncertainty, which is not the case in using prescription-2. For this reason, we are presenting m s (M 2 τ ) from both prescriptions for different schemes in other schemes too.

The strange quark mass determinations using phenomenological inputs
In this section, the strange quark mass is calculated using the phenomenological parametrization along with the perturbative "L + T "−contributions described in section VI and section IV. The quark mass is calculated in the two prescriptions for various moments, and the details of sources of uncertainties are presented in the tables.

a.
Strange quark mass determination using CIPT scheme The CIPT determination of m s (M 2 τ ) in this section makes use of the full dimension-2 results of O α 3 s as the series presented in eq. (41) is convergent for all moments, and prescription I and prescription II yield the same determinations. The results are shown in the table XVI.

Details of the |Vus| determinations from OPAL data
The CKM matrix element |V us | is calculated using eq. (62) from the available data on moments for strange and non-strange components. Details of extraction from these moments and associated uncertainties from the inputs parameters are presented in this section. Purely pQCD inputs for the longitudinal component is used to extract |V us | in the table XX and table XXI. Large theoretical uncertainties in prescription II come from the truncation of perturbative series. Determination of |V us | from the phenomenological inputs for longitudinal contribution is presented in the table XXII and table XXIII for prescriptions I and II,