Updated determination of chiral couplings and vacuum condensates from hadronic tau decay data
IIFIC/16-08FTUV/16-0219
Updated determination of chiral couplings andvacuum condensates from hadronic τ decay data Mart´ın Gonzalez-Alonso , a Antonio Pich b and Antonio Rodr´ıguez-S´anchez ba IPN de Lyon, CNRS and Universit´e Lyon 1, F-69622 Villeurbanne, France b Departament de F´ısica Te`orica, IFIC, Universitat de Val`encia – CSIC,Apt. Correus 22085, E-46071 Val`encia, Spain,
Abstract
We analyze the lowest spectral moments of the left-right two-point correlation func-tion, using all known short-distance constraints and the recently updated ALEPH V − A spectral function from τ decays. This information is used to determine the low-energycouplings L and C of chiral perturbation theory and the lowest-dimensional con-tributions to the Operator Product Expansion of the left-right correlator. A detailedstatistical analysis is implemented to assess the theoretical uncertainties, includingviolations of quark-hadron duality. a r X i v : . [ h e p - ph ] F e b Introduction
The hadronic decays of the τ lepton provide very valuable information on low-energy prop-erties of the strong interaction, allowing us to analyze important perturbative and non-perturbative aspects of QCD [1]. A very precise determination of the strong coupling canbe extracted from the inclusive hadronic τ decay width [2–7], while the SU(3)-breaking cor-rections to the ∆ S = 1 decay width [8, 9] are very sensitive to the Cabibbo quark mixing | V us | [10, 11]. In this paper we are interested in the difference between the vector ( V ) andaxial-vector ( A ) τ spectral functions, which gives a direct access to non-perturbative pa-rameters related with the spontaneous chiral symmetry breaking of QCD [12–27]. A verydetailed phenomenological study of the non-strange V − A spectral function, using the 2005release of the ALEPH τ data [28], was already done in Refs. [22–24]. The recent update ofthe ALEPH non-strange τ spectral functions [29] motivates an updated numerical analysis,based on the strategies developed in those references, which we present here. A comparisonwith other works available in the literature that employ different theoretical approaches willalso be performed.Compared to the 2005 ALEPH data set, the new public version of the ALEPH τ dataincorporates an improved unfolding of the measured mass spectra from detector effects andcorrects some problems [30] in the correlations between unfolded mass bins. The improvedunfolding brings an increased statistical uncertainty near the edges of phase space. It hasalso reduced the number of bins in the spectral distribution, as a larger bin size has beenadopted.The starting point of our analysis is the two-point correlation function of the left-handedand right-handed quark currents:Π µνud,LR ( q ) ≡ i (cid:90) d x e iqx (cid:104) | T (cid:16) L µud ( x ) R ν † ud (0) (cid:17) | (cid:105) = ( − g µν q + q µ q ν ) Π (1) ud,LR ( q ) + q µ q ν Π (0) ud,LR ( q ) , (1)where L µud ( x ) ≡ ¯ u ( x ) γ µ (1 − γ ) d ( x ) and R µud ( x ) ≡ ¯ u ( x ) γ µ (1 + γ ) d ( x ). Owing to the chiralinvariance of the massless QCD Lagrangian, this correlator vanishes identically to all ordersin perturbation theory when m u,d = 0. The non-zero value ofΠ( s ) ≡ Π (0+1) ud,LR ( s ) ≡ Π (0) ud,LR ( s ) + Π (1) ud,LR ( s ) = 2 f π s − m π + Π( s ) (2)originates in the spontaneous breaking of chiral symmetry by the QCD vacuum, which resultsin different vector and axial-vector two-point functions. Thus, Π( s ) is a perfect theoreticallaboratory to test non-perturbative effects of the strong interaction, without perturbativecontaminations. The perturbative corrections induced by the non-zero quark masses are tinyand can be easily taken into account. In Eq. (2) we have made explicit the contribution ofthe pion pole to the longitudinal axial-vector two-point function. We will work in the isospinlimit m u = m d ≡ m q where the longitudinal part of the vector correlator vanishes.The correlator Π( s ) is analytic in the entire complex s plane, except for a cut on thepositive real axis that starts at the threshold s th = 4 m π . Applying Cauchy’s theorem in the2 s th Re I q M Im I q M Figure 1: Analytic structure of Π( s ).circuit in Fig. 1 to the function ω ( s ) Π( s ), one gets the exact expression [24]: (cid:90) s s th ds ω ( s ) 1 π Im Π( s ) + 12 πi (cid:73) | s | = s ds ω ( s ) Π( s ) = 2 f π ω ( m π ) + Res[ ω ( s )Π( s ) , s = 0] , (3)which relates the correlator in the Euclidean region, where it can be approximated by itsshort distance Operator Product Expansion (OPE) [31, 32],Π OPE ( s ) = (cid:88) k O k ( − s ) k , (4)with its imaginary part at Minkowskian momenta, accessible experimentally at low energies.For s th ≤ s ≤ m τ , the spectral function ρ ( s ) ≡ π Im Π( s ) is determined by the differ-ence between the vector and axial-vector hadronic spectral functions measured in τ decays.Choosing different weight functions ω ( s ) one can change the sensitivity to different kine-matical domains. We have only assumed that ω ( s ) is an arbitrary analytic function in thewhole complex plane except maybe at the origin where it can have poles, generating thecorresponding residue Res[ ω ( s )Π( s ) , s = 0]. The pion pole contribution is given by the term2 f π ω ( m π ).The OPE expresses the correlator as an expansion in inverse powers of momenta, whichapproximates very well Π( s ) in the complex plane, away from the real axis, at large valuesof | s | . Therefore, it provides a very reliable short-distance tool to compute the integral alongthe circle | s | = s , for sufficiently large values of s . The main source of uncertainty is theintegration region near the real axis, but it can be suppressed with adequately chosen weightfunctions [7]. In order to account for the small difference between the physical (exact)correlator and its OPE representation along the circle integration [16, 23, 33–35], one canintroduce the correction [23, 35, 36]: δ DV [ ω ( s ) , s ] ≡ πi (cid:73) | s | = s ds ω ( s ) (cid:2) Π( s ) − Π OPE ( s ) (cid:3) = (cid:90) ∞ s ds ω ( s ) ρ ( s ) , (5)3hich becomes zero at s → ∞ . A non-zero value of δ DV [ ω ( s ) , s ] signals a violation ofquark-hadron duality in the spectral integration between s th and s . We will discuss laterthe best strategy to control and minimize this kind of theoretical uncertainty.Taking ω ( s ) = s n with non-negative values of the integer power n , the pion pole is theonly singularity within the contour. Therefore, the integral over the spectral function from s th to s is equal to the pion pole term 2 f π m nπ , plus the OPE contribution ( − n O n +1) generated by the integration along the circle, up to duality violations (DV). However, in thechiral limit ( m q = 0) and owing to the short-distance properties of QCD, Π OPE ( s ) containsonly power-suppressed terms from dimension d = 2 k operators, starting at d = 6 [37], whichimplies a vanishing OPE contribution for n = 0 , (cid:90) s s th ds π Im Π( s ) = 2 f π − δ DV [1 , s ] , (6) (cid:90) s s th ds s π Im Π( s ) = 2 f π m π − δ DV [ s, s ] . (7)The superconvergence properties of Π( s ) guarantee that the DV corrections to both sumrules approach zero very fast for increasing values of s . When s → ∞ , there is no dualityviolation and one gets the well-known first and second Weinberg Sum Rules (WSRs) satisfiedby the physical spectral functions [38]. With non-zero quark masses taken into account, thefirst relation is still exact, while the second gets a negligible correction of O ( m q ).For higher values of the power n , Eq. (3) gives relations involving the different OPEcoefficients: (cid:90) s s th ds s n π Im Π( s ) = ( − n O n +1) + 2 f π m nπ − δ DV [ s n , s ] ( n ≥ . (8)For negative values of n = − m <
0, the OPE does not give any contribution to theintegration along the circle s = s , but there is a non-zero residue at the origen proportionalto the ( m − s ) at s = 0. At low values of s the correlator can berigorously calculated within chiral perturbation theory ( χ PT) [39–43]. At present Π( s ) isknown to O ( p ) [44], in terms of the so-called chiral low-energy couplings (LECs) that wecan determine through the relations: (cid:90) s s th ds s − π Im Π( s ) = 2 f π m π + Π(0) − δ DV [ s − , s ] ≡ − L eff10 − δ DV [ s − , s ] , (9) (cid:90) s s th ds s − π Im Π( s ) = 2 f π m π + Π (cid:48) (0) − δ DV [ s − , s ] ≡ C eff87 − δ DV [ s − , s ] . (10)The explicit expression of the correlator Π( s ) at O ( p ) in χ PT is given in appendix A.The relation between the effective parameters L eff10 and C eff87 and their χ PT counterparts, theLECs L and C , will be discussed in section 5.4igure 2: L eff10 and C eff87 from Eqs. (9) and (10), neglecting DVs, for different values of s . Using the updated ALEPH spectral function [29], we can determine L eff10 and C eff87 withEqs. (9) and (10). As a first estimate, we neglect the DV terms and show in Fig. 2 theresulting effective couplings, for different values of s . As expected and as it was alreadyobserved in Ref. [22], the results exhibit a strong dependence on s at low energies, wherethe duality-violation corrections are not negligible. At larger momentum transfers the curvesstart to stabilise, indicating that the violations of duality become smaller. However, espe-cially for L eff10 , the curves are not yet horizontal lines at s near m τ , which implies thatduality-violation effects are still present.Instead of weights of the form s n , we can try to reduce DV effects using pinched weightfunctions [7, 16, 45], which vanish at s = s (or in the vicinity) where the OPE breaksdown. Following Ref. [22], we employ the WSRs in Eqs. (6) and (7) and take ω − , ( s ) = s − (1 − s/s ) and ω − ( s ) = s − (1 − s/s ) for estimating L eff10 , and ω − , ( s ) = s − (1 − s /s )and ω − ( s ) = s − (1 − s/s ) (1 + 2 s/s ) for estimating C eff87 . Again, neglecting the DV terms,we plot the values of the effective couplings for different s in Fig. 3. We observe that usingthese pinched weights the results converge and become stable below s = m τ . This suggeststhat DV effects are negligible at s ∼ m τ , when these pinched weight functions are used.Assuming that, we obtain: L eff10 = − (6 . ± . · − , (11) C eff87 = (8 . ± . · − GeV − . (12) The stability under changes of s of the L eff10 and C eff87 determinations is a necessary conditionfor vanishing duality violations. However the plateau could be accidental and disappearat slightly higher values of s where experimental data are not available. Although this5igure 3: L eff10 and C eff87 at different values of s , using pinched weight functions and neglectingDVs.possibility looks rather unlikely, we want to gain confidence on our numerical results andperform a reliable estimation of the uncertainties associated with violations of duality, usingEq. (5). The problem is that the spectral function is experimentally unknown above s = m τ .Fortunately, there are strong theoretical constraints on ρ ( s ) that originate in the specialchiral-symmetry-breaking properties of Π( s ), implying its very fast fall-off at large momenta.In addition to the two WSRs, the spectral function should satisfy the so-called Pion SumRule ( π SR), which determines the electromagnetic pion mass splitting in the chiral limit [46]: (cid:90) ∞ s th ds s log (cid:16) s Λ (cid:17) π Im Π( s ) (cid:12)(cid:12)(cid:12)(cid:12) m q =0 = (cid:0) m π − m π + (cid:1) em π α f π (cid:12)(cid:12) m q =0 . (13)Owing to the second WSR, the π SR does not depend on the arbitrary scale Λ. The r.h.s ofthis equation is well-known in χ PT and, within the needed accuracy, we can identify in thel.h.s the spectral function in the chiral limit with the physical ρ ( s ) because m q correctionsare tiny. All the theoretical and phenomenological knowledge we have about Π( s ) can be used to getan estimate of the DV uncertainties. In order to do that, let us adopt the following ansatzfor the spectral function at large values of s [23, 24, 47]: ρ ( s > s z ) = κ e − γs sin { β ( s − s z ) } , (14)with four free parameters κ , γ , β and s z . This parametrization incorporates the expectedstrong fall-off when s → ∞ and the oscillating behaviour predicted in resonance-basedmodels [33,48,49]. We will split the spectral integrations in two parts, using the experimentaldata in the lower energy range and the ansatz (14) at higher energies. From the ALEPHdata we know that the V − A spectral function has a zero around s z ∼ , which is6epresented in Eq. (14) through the s z parameter. We will take this zero as the separationpoint between the use of the data and the use of the model.Our parametrization is compatible with the ALEPH spectral function above s z . Fittingthe parameters given in (14) to the ALEPH data in the interval s ∈ (1 . , m τ ), weobtain a very good fit with χ / d . o . f . = 8 . /
9. In fact, the fit with the updated ALEPHdata looks more reliable compared to the previous one, where a value of χ / d . o . f . (cid:28) s -dependence of the spectral function in the high-energy region cannot be derived from first principles. The ansatz (14) is just a convenientparametrization, consistent with present knowledge, that we are going to use to estimatetheoretical uncertainties associated with violations of quark-hadron duality. Imposing that ρ ( s ) should satisfy all known theoretical and experimental constraints, the free parametersin the ansatz will allow us to measure how much freedom remains for the spectral functionshape and, therefore, to obtain a reliable estimate of the associated uncertainty.There is an inherent systematic error in any work that estimates DV effects, namelythe dependence on the chosen parametrization. The comparison with other works thatparametrize the data in a different way, such as Refs. [25, 27], represents an important stepin this regard. Following the procedure described in [23], we generate 3 · tuples of the parameters( κ, γ, β, s z ), randomly distributed in a rectangular region large enough to contain all thepossible acceptable tuples. Among all generated tuples, we select those satisfying the fol-lowing four physical conditions:– The tuples must be consistent with the ALEPH data above s = 1 . , i.e. , theymust be contained within the 90% C.L. region in the fit to the experimental ALEPHspectral function described before: χ < χ + 7 .
78 = 16 . . (15)Although we will only use the ansatz above s z ∼ , we impose the compatibilitywith the data from 1 . to ensure the continuity of the spectral function in thematching region between the data and the model.– The tuples must satisfy within the experimental uncertainties up to s z the first andsecond WSRs with: (cid:90) s z ds ρ ( s ) ALEPH + (cid:90) ∞ s z ds ρ ( s ; κ, γ, β, s z ) = 17 . · − GeV , (16) (cid:90) s z ds s ρ ( s ) ALEPH + (cid:90) ∞ s z ds s ρ ( s ; κ, γ, β, s z ) = 0 . · − GeV , (17)where the right-hand-side errors are omitted as they are negligible compared to theleft-hand-side ones. 7 The tuples must satisfy within the experimental uncertainties the π SR: (cid:90) s z ds s log (cid:16) s (cid:17) ρ ( s ) ALEPH + (cid:90) ∞ s z ds s log (cid:16) s (cid:17) ρ ( s ; κ, γ, β, s z )= − (10 . ± . · − GeV . (18)The quoted error in the π SR takes into account that quark masses do not vanishin nature and we are using real data instead of chiral-limit one. We estimate thisuncertainty taking for the pion decay constant the range f = (87 ±
5) MeV [23],which includes the physical value and its estimated value in the chiral limit [50]. Wealso include a small uncertainty coming from the residual scale dependence of thelogarithm, which is proportional to the second WSR.We accept only those tuples that fulfil the four conditions. This requirement constrainsthe regions in the parameter space of the ansatz (14) that are compatible with both QCDand the data. From the initial set of 3 · randomly generated tuples we obtain 3716satisfying our set of minimal conditions. They represent the possible shapes of the spectralfunction beyond s z , as shown in Fig. 4. In Fig. 5, we plot the statistical distribution of theparameters ( κ, γ, β, s z ) for the accepted tuples. ( GeV ) ρ ( s ) Figure 4: Updated ALEPH V − A spectral function [29] (blue points) and all the “acceptable”spectral functions (red band above 1.7 GeV ) that follow our parametrization and satisfythe physical conditions described in the main text. For every selected tuple we have an acceptable spectral function ∗ that can be used to esti-mate the different physical parameters through the corresponding spectral integrals. Using ∗ Given by the ALEPH data below s z and by the parametrization (14) above that value. κ , γ , β , s z ) that satisfy the physical constraints.GeV units are used for dimensionful quantities.Eqs. (9), (10) and (8) (for n = 2 ,
3) with s = s z , we determine L eff10 , C eff87 , O and O foreach of the 3716 accepted tuples. The statistical distributions of the calculated parametersare shown in Fig. 6 (light gray).We can reduce both the experimental and the DV uncertainties using the followingpinched weight functions [24]: (cid:90) s s th ds ρ ( s ) s (cid:18) − ss (cid:19) (cid:18) ss (cid:19) = 16 C eff87 − f π s + 4 f π m π s − δ DV [ ω − , s ] , (19) (cid:90) s s th ds ρ ( s ) s (cid:18) − ss (cid:19) = − L eff10 − f π s + 2 f π m π s − δ DV [ ω − , s ] , (20) (cid:90) s s th ds ρ ( s ) ( s − s ) = 2 f π s − f π m π s + 2 f π m π + O − δ DV [ ω , s ] , (21) (cid:90) s s th ds ρ ( s ) ( s − s ) ( s + 2 s ) = − f π m π s + 4 f π s + 2 f π m π − O − δ DV [ ω , s ] . (22)Following the same method with these relations, we obtain new distributions of acceptable9igure 6: Statistical distribution of L eff10 , C eff87 , O y O for the tuples accepted, using s n weights (light gray) and pinched weight (dark gray) functions.physical parameters, which are also shown in Fig. 6 (dark gray). From these new distributionswe get: L eff10 = ( − . + 0 . − . ± . · − = ( − . ± . · − , (23) C eff87 = (8 . + 0 . − . ± . · − GeV − = (8 . ± . · − GeV − , (24) O = ( − . + 0 . − . ± . · − GeV = ( − . + 0 . − . ) · − GeV , (25) O = ( − . ± . ± . · − GeV = ( − . ± . · − GeV , (26)where the first errors correspond to DV uncertainties, computed from the dispersion of thehistograms, and the second errors are the experimental ones.We observe that pinched weight functions reduce indeed the DV effects, and that they arenegligible for L eff10 and C eff87 at s ∼ m τ , compared with the experimental uncertainties. Theresults obtained for these two LECs are in perfect agreement with our first determinationsin Eqs. (11) and (12) that did not include any estimate of DV. The corresponding spectralintegrals contain weight functions with negative powers of s that suppress the contributionfrom the upper end of the integration range, making DV irrelevant. This is no-longer true forthe vacuum condensates O and O , which are determined with weight functions growingwith positive powers of s . The use of pinched weights is then essential to suppress the10igure 7: Values of the condensates O and O , at different values of s , obtained fromEqs. (21) and (22) ignoring duality violations.contributions from the region around s in the contour integration. This is clearly reflectedin the strong reduction of uncertainties observed in the two lower panels of Fig. 6.Actually, ignoring completely DV effects, from the double-pinched weight functions inEqs. (21) and (22) one obtains values for O and O that are perfectly compatible with ourdeterminations in Eqs. (25) and (26), although with much larger experimental uncertainties.This is illustrated in Fig. 7, which shows how the extracted condensates stabilize at large s ,around the right values but with very large error bars. The implementation of short-distanceconstraints (WSRs and π SR), through the procedure described in the previous section, hasmade possible to better pin down the spectral function in that region and obtain the moreprecise values in Eqs. (25) and (26).Our results are in good agreement with those obtained previously in Ref. [24] with the2005 ALEPH data set. Thus, the improvements incorporated in the 2014 release of theALEPH data do not introduce sizeable modifications of the physical outputs. Similar resultshave been obtained recently in Ref. [27], using also the 2014 ALEPH data set.Ref. [27] emphasises the existence of a slight tension with the results obtained in Ref. [25]with the 1999 OPAL data set [51]. In view of this, we have repeated our numerical analyseswith the OPAL spectral function [51]. As happened with the 2005 ALEPH data set, the fit ofthe ansatz (14) to the OPAL data in the interval s ∈ (1 . , m τ ) has a χ min / d . o . f . (cid:28) L eff10 = ( − . ± . · − , (27) C eff87 = (8 . ± . · − GeV − , (28) O = ( − . +1 . − . ) · − GeV , (29) O = (0 . +0 . − . ) · − GeV . (30)Owing to the larger uncertainties of the OPAL data, specially at higher values of s , theextracted parameters are less precise than those obtained with the ALEPH data. Neverthe-11ess, comparing Eqs. (27)-(30) with (23)-(26), we observe a good agreement between bothsets of results, the differences being only 0 . σ , 0 . σ , 1 . σ and 1 . σ for L eff10 , C eff87 , O and O , respectively. We conclude that the much larger fluctuations obtained in Refs. [25, 27]between the results extracted from the two data sets are a consequence of the particularapproach adopted in their DV analyses, which does not look optimal to us. † Finally, we can use double-pinched weight functions in order to estimate higher-dimensionalcondensates: (cid:90) s s th ds ρ ( s ) ( s − s ) ( s + 2 s s + 3 s )= − f π m π s + 6 f π s + 2 f π m π + O − δ DV [ ω , s ] , (31) (cid:90) s s th ds ρ ( s ) ( s − s ) ( s + 2 s s + 3 s s + 4 s )= − f π m π s + 8 f π s + 2 f π m π − O − δ DV [ ω , s ] , (32) (cid:90) s s th ds ρ ( s ) ( s − s ) ( s + 2 s s + 3 s s + 4 s s + 5 s )= − f π m π s + 10 f π s + 2 f π m π + O − δ DV [ ω , s ] , (33) (cid:90) s s th ds ρ ( s ) ( s − s ) ( s + 2 s s + 3 s s + 4 s s + 5 s s + 6 s )= − f π m π s + 12 f π s + 2 f π m π − O − δ DV [ ω , s ] . (34)Using these equations with the same method, we obtain from the ALEPH data: O = (5 . ± . ± . · − GeV = (5 . ± . · − GeV , (35) O = ( − . + 0 . − . ± .
02) GeV = ( − . + 0 . − . ) GeV , (36) O = (0 . + 0 . − . ± .
06) GeV = (0 . + 0 . − . ) GeV , (37) O = ( − . + 0 . − . ± .
13) GeV = ( − . + 0 . − . ) GeV . (38) Our final results for L eff10 , C eff87 , O and O are compared in Table 1 with recent (post-2005) phenomenological determinations of these parameters, obtained with different data † In Refs. [25] and [27] the exact s -dependence of the resonance-based model (14) is assumed to be truefor the V and A channels separately and a complex analysis involving 9 parameters, including a model-dependent determination of the strong coupling, is performed. In this way, uncertainties related to an α s determination from the V and A spectral distributions are introduced in the analysis of the correlator Π( s ),which does not contain any perturbative contribution. Moreover the LECs and vacuum condensates aredirectly extracted from the fitted V and A spectral functions without imposing any further requirement(WSRs and π SR are only checked to be satisfied within errors a posteriori). Since DV is not very relevantfor the extraction of the LECs, similar values are obtained for L eff10 and C eff87 with the two data sets. However,sizeable differences show up in their determinations of O and O where DV is more important. · L eff10 · C eff87 · O · O Reference Comments (GeV − ) (GeV ) (GeV ) − . ± .
06 – − . ± . − . ± . − . + 2 . − . . + 2 . − . ASS’08 [21] ALEPH’05 + DV= 0 − . ± .
06 8 . ± .
14 – – GPP’08 [22] ALEPH’05 + DV= 0 − . ± .
05 8 . ± . − . ± . − . ± . V − A − . ± .
09 8 . ± . − . ± . . ± . V/A − . ± .
10 – − . ± . − . ± . − . ± .
05 8 . ± . − . ± . − . ± . V/A − . ± .
10 8 . ± . − . + 1 . − . . + 0 . − . this work OPAL’99 + DV V − A − . ± .
05 8 . ± . − . + 0 . − . − . ± . V − A Table 1: Compilation of recent determinations of the LECs and vacuum condensates.sets [28, 29, 51] and various DV parametrizations. ‡ There is an excellent agreement among the different values quoted for the effective LECs L eff10 and C eff87 , showing that these determinations are very solid and do not get affected byDV effects. In fact, as shown in Table 1, the precision has not changed in the last ten years.Nonetheless, the robustness of these determinations has increased significantly thanks to thethorough studies of DV effects with different approaches. The values obtained from differentdata sets are also in good agreement, although one can notice a 1 σ shift of the C eff87 centralvalue when changing from the old (2005) to the updated (2014) ALEPH data.The different results for O and O are also in reasonable agreement, within the quoteduncertainties. A good control of DV effects is more important for these vacuum condensates.The use of pinched weights allows to sizeably reduce their impact and obtain more reliabledeterminations. With the ALEPH’14 data one reaches a 20% accuracy for O , but theerror remains still large (40%) for O . As commented before, we do not see any significantdiscrepancy between the results obtained from the OPAL and ALEPH data samples. ‡ A complete list including theoretical estimates [52–54] and previous phenomenological determinationsof these quantities (and of higher-dimensional condensates) [13–18, 20, 28, 34, 51, 55–59] can be found inRefs. [24, 60]. χ PT couplings
The effective couplings L eff10 and C eff87 can be rewritten in terms of O ( p ) and O ( p ) couplingsof the χ PT Lagrangian [22, 44]: L eff10 ≡ −
18 Π(0)= L r ( µ ) + 1128 π (cid:20) − log (cid:18) µ m π (cid:19) + 13 log (cid:18) m K m π (cid:19)(cid:21) −
18 ( C r + C r ) ( µ ) − µ π + µ K ) ( L r + 2 L r )( µ ) + G L ( µ, s = 0) + O ( p ) , (39) C eff87 ≡
116 Π (cid:48) (0)= C r ( µ ) − π f π (cid:20) − log (cid:18) µ m π (cid:19) + 13 log (cid:18) m K m π (cid:19)(cid:21) L r ( µ )+ 17680 π (cid:18) m K + 2 m π (cid:19) − G (cid:48) L ( µ, s = 0) + O ( p ) , (40)where the factors µ i = m i log( m i /µ ) / (16 π f π ) originate from one-loop corrections and G L ( µ, s = 0) and G (cid:48) L ( µ, s = 0) are two-loop functions, whose numerical values are givenin the appendix. We have also defined C r = 32 m π ( C − C + C ) , (41) C r = 32 ( m π + 2 m K ) ( C − C + C ) . (42)To first approximation the effective parameters correspond to the chiral couplings L and C , which appear at O ( p ) and O ( p ), respectively, in the χ PT expansion. The scaledependence of L r ( µ ) is cancelled by the one-loop logarithmic terms in the second line ofEq. (39), which are suppressed by one power of 1 /N C with respect to L r ( µ ), where N C isthe number of QCD colours. The remaining contributions in Eq. (39) contain the O ( p )corrections, which unfortunately introduce other O ( p ) and O ( p ) chiral couplings (thirdline). The corrections to C r ( µ ) in Eq. (40) only involve one additional LEC, L r ( µ ), througha one-loop correction with the O ( p ) chiral Lagrangian.It is convenient to give the following compact numerical form of these equations to easetheir future use: L eff10 = L r − . O ( p ) , (43) L eff10 = 1 . L r + 0 . L r − . −
18 ( C r + C r ) + O ( p ) , (44) C eff87 = C r + 0 . L r + 0 . O ( p ) , (45)where we have used µ = M ρ as the reference value for the χ PT renormalization scale. Theuncertainties in these numbers are much smaller than those affecting the different LECs andcan therefore be neglected. 14orking with O ( p ) precision, the determination of L r ( µ ) is straightforward and wefind: L r ( M ρ ) = − (5 . ± . · − [ O ( p ) analysis] . (46)As mentioned before, an O ( p ) determination of L r requires to know some next-to-next-to-leading-order (NNLO) LECs, § namely those in C r , . This has motivated some interest inthese quantities in the last few years. Here we briefly review the different approaches.In the first O ( p ) determination of L r [22], C r was extracted from a combination ofphenomenological ( C r , ) [62–65] and theoretical ( C r , R χ T) [44, 66] inputs, namely ¶ C r ( M ρ ) = (1 . ± . · − GeV − [62, 64, 65] , (47) C r ( M ρ ) = (0 . ± . · − GeV − [63] , (48) C r ( M ρ ) = (2 . ± . · − GeV − [44, 66] , (49)whereas C r , which was completely unknown at the time, was estimated using | C r − C r − C r | ≤ | C r − C r − C r | , (50) i.e. , a simple educated guess based on the fact that those LECs are suppressed by a factor1 /N C . Using these numbers and Eq. (44), we obtain the results shown in Table 2 (5th row)and Fig. 8 (magenta point), which supersede those found in Ref. [22].An alternative sum rule involving L r and C r was recently derived in Ref. [65] from ananalysis of the flavour-breaking left-right correlator Π (0+1) ud − us,LR (0), namely (cid:107) (cid:104) Π (0+1) ud,LR (0) − Π (0+1) us,LR (0) (cid:105) LEC = − . L r + 1 . L r + 2 . L r − m K − m π m π C r = 0 . , (51)again at µ = M ρ . Combining this constraint with the sum rule ∗∗ in Eq. (44) and the naiveinequality in Eq. (50), we obtain the results shown in Table 2 (6th row) and Fig. 8 (darkblue region). We see that L r is in excellent agreement with the value obtained using Eqs.(47-49) and has a smaller error. Concerning the NNLO LECs, almost the same value isobtained for C r , whereas a 1.8 σ tension is present in the C r case. § It also requires L r , which we take from Ref. [61]: L r ( M ρ ) = 5 .
93 (43) · − . Let us notice that this isthe value used also in all other O ( p ) extractions of L r from tau data. ¶ This value of C r comes from a flavour-breaking finite-energy sum rule involving the correlatorΠ (0+1) ud − us,V V (0). The original result [62] has been updated recently [65], finding32 ( m K − m π ) C + 1 . L r = 0 . . Since L r appears in this relation only at one loop, i.e. at O ( p ), we can use here an O ( p ) determinationof L r to extract C r . We can indeed see that the L r contribution to the C r error is subdominant. We usethe conservative value L r = − . C r . (cid:107) We use the value obtained in Ref. [65] using 1999 OPAL data for the non-strange part, 0 . . L eff10 . ∗∗ We use L r ( M ρ ) = (1 . ± . · − [68] and, once again, L r ( M ρ ) = 5 .
93 (43) · − [61]. r ( M ρ ) C r ( M ρ ) C r ( M ρ ) Reference Input × × × -4.06 (39) +0.54 (42) 0 (5) GPP’08 [22] Π(0) + C pheno/R χ T0 + 1 /N c -3.10 (80) -0.81 (82) 14 (10) Boito’12 [25] Π(0) + Π( s ) latt -3.46 (32) -0.34 (13) 8.1 (3.5) Boyle’14, GMP’14 [65, 67] Π(0) + Π( s ) latt + ∆Π(0)-3.50 (17) -0.35 (10) 7.5 (1.5) Boito’15 [27] Π(0) + Π( s ) latt + ∆Π(0)-4.08 (44) +0.21 (34) 0 (5) this work Π(0) + C pheno/R χ T0 + 1 /N c -4.17 (35) -0.43 (12) -1 (6) this work Π(0) + ∆Π(0) + 1 /N c Table 2: Compilation of recent determinations of the LECs. The determinations of L eff10 , i.e. Π(0), are obtained as explained in Table 1. 1 /N c refers to Eq. (50), whereas ∆Π(0) refers tothe sum rule given in Eq. (51). Additional details are given in the text.Another interesting development was performed in Ref. [67], where additional constraintson L r , C r and C r were obtained from lattice simulations of the correlator Π( s ) at unphysicalmeson masses. As shown in Table 2, the lattice data allow for a more accurate determinationof the LECs, making unnecessary the use of the naive guess in Eq. (50). However, toderive the lattice constraints one needs to assume that the O ( p ) χ PT expansion reproduceswell the correlator at s ∼ − .
25 GeV , the energy region with smaller lattice uncertainties,which dominates these constraints. Unfortunately, it was shown in Ref. [25] that O ( p ) χ PT does not approximate well enough Π( s ) at these energies, taking into account the lowuncertainties we are dealing with, and one needs to incorporate the so-far unknown O ( p )chiral corrections.In order to take advantage of the most precise lattice constraint, Ref. [27] makes the strongassumption that the missing O ( p ) chiral contributions are dominated by mass-independentterms, i.e. , Π( s ) ≈ Π χ PT O ( p ) + D s , so that they cancel in the lattice-continuum differenceΠ χ PTlattice − Π χ PTphysical . It is worth noting that this is not a good approximation at the previouschiral order, O ( p ), since more than 25% of the O ( p ) correction proportional to s comesfrom known mass-dependent chiral terms. Therefore, the uncertainties associated with theselattice constraints seem at present underestimated.Additionally, correlations between the continuum and the lattice sum rules ( e.g. dueto L r ) are not publicly available. It is worth mentioning nonetheless that if we implementthese lattice constraints †† (instead of the inequality in Eq. (50)), neglecting such correlations,we reproduce the results of Ref. [27] except for the uncertainties associated to L r and L r ,for which the neglected correlations are likely to be relevant. Such an agreement is notsurprising, as our determinations of the effective coupling L eff10 were very close.From Table 2 and Fig. 8 we see that the determinations obtained with the lattice con-straints are (in most cases) significantly more precise than those using instead the inequalityof Eq. (50). The agreement is reasonable (in the 0 . − . σ range depending on the quantity),taking into account that Eq. (50) is nothing but a naive educated guess, while the lattice †† We find that the constraint associated to the third lattice ensemble used in [27] fully dominates the fits. C r , , at µ = M ρ .We follow the same notation as in Table 2. The region allowed by the inequality of Eq. (50),inspired by large- N c arguments, is indicated in light blue, whereas the light gray area aroundit (dashed) simply represents a naive estimate of its error, namely 33%.improvement suffers from additional uncertainties not yet included in the quoted errors.The determination of C r from C eff87 at O ( p ) does not involve any unknown LEC. Therelation (40) contains a one-loop correction of size − (3 . ± . · − , which only dependson L r ( M ρ ) and the pion and kaon masses, and small non-analytic two-loop contributionscollected in the term G (cid:48) L ( M ρ , s = 0) = − . · − GeV − . In spite of its 1 /N C suppression,the one-loop correction is very sizeable, decreasing the final value of the O ( p ) LEC: C r ( M ρ ) = (5 . ± . · − GeV − . (52) Our phenomenological determinations of L r ( M ρ ) and C r ( M ρ ) from τ decay data are ingood agreement with the large- N C estimates based on lowest-meson dominance [44, 69–73]: L = − F V M V + F A M A ≈ − f π M V ≈ − . · − ,C = F V M V − F A M A ≈ f π M V ≈ . · − GeV − . (53)They also agree with the C determinations based on Pade approximants [53, 74], which arehowever unable to fix the renormalization-scale dependence that is of higher-order in 1 /N C .The resonance chiral theory (R χ T) Lagrangian [70, 71, 75, 76] was used to analyze theleft-right correlator at NLO in the 1 /N C expansion in Ref. [54]. Matching the effectivefield theory description with the short-distance QCD behavior, both LECs are determined,17eeping full control of their µ dependence. The predicted values [54] L r ( M ρ ) = − (4 . ± . · − ,C r ( M ρ ) = (3 . ± . · − GeV − , (54)are in good agreement with our determinations, although they are less precise.Lattice determinations of the χ PT LECs have improved considerably in recent times,although they are still limited to O ( p ) accuracy. The most recent simulations find: L r ( M ρ ) = (cid:26) − (5 . ± . ± . · − [77] , − (5 . ± . + 0 . − . ) · − [78] . (55)These lattice results are in good agreement with our determinations, but their accuracy isstill far from the phenomenological precision. We have determined the LECs L eff10 and C eff87 , using the recently updated ALEPH spectralfunctions [29], with the methods developed in Refs. [22–24]. Our final values, obtained usingpinched weight functions with a statistical analysis that includes possible DV uncertainties,are: L eff10 = ( − . ± . · − , (56) C eff87 = (8 . ± . · − GeV − . (57)These results are in excellent agreement with the values extracted with non-pinched weightsand with those determined neglecting DV in Eqs. (11) and (12). Thus, DV does not play anysignificant role in the determination of LECs, where the weight functions strongly suppressthe high energy region of the spectral integrations. Our results are in good agreement withthe ones obtained previously with the 2005 release of the ALEPH τ data [24]: L eff10 = ( − . ± . · − , (58) C eff87 = (8 . ± . · − GeV − . (59)The improvements introduced in the 2014 ALEPH data set did not bring major changesin these parameters. The values in Eqs. (56) and (57) also agree with the results obtainedrecently with the same experimental data but with a different approach in Ref. [27].The statistical approach adopted in our analysis allows for a precise determination of thedimension-6 and 8 terms in the OPE of the left-right correlator Π( s ). We obtain: O = ( − . + 0 . − . ) · − GeV , (60) O = ( − . ± . · − GeV , (61)also compatible with the determinations performed in Refs. [24] (with non-updated ALEPHdata) and [27] (with a different approach for estimating DV effects). Using the same method,some higher-dimensional terms in the OPE have also being estimated in Eqs. (35)-(38).18he numerical determination of the effective couplings L eff10 and C eff87 has allowed us toderive the corresponding LECs of the χ PT Lagrangian. At O ( p ), we find L r ( M ρ ) = − (4 . ± . · − , (62) C r ( M ρ ) = (5 . ± . · − GeV − . (63)The final value quoted for L r ( M ρ ) takes into account our two different estimates in Table 2,keeping conservatively the individual errors in view of the present uncertainties induced bythe NLO LECs. Acknowledgments
This work has been supported in part by the Spanish Government and ERDF funds fromthe EU Commission [Grants No. FPA2014-53631-C2-1-P and FPU14/02990], by the SpanishCentro de Excelencia Severo Ochoa Programme [Grant SEV-2014-0398] and by the Gener-alitat Valenciana [PrometeoII/2013/007]. M.G.-A. is grateful to the LABEX Lyon Instituteof Origins (ANR-10-LABX-0066) of the Universit´e de Lyon for its financial support withinthe program ANR-11-IDEX- 0007 of the French government.
A Low-energy expansion of the left-right correlationfunction
At low energies, the correlator Π( s ) can be expanded in powers of momenta over the chiralsymmetry-breaking scale. The series expansion has been calculated to O ( p ) in χ PT [40, 41,44]: Π( s ) = 2 f π s − m π − L r − B ππV ( s ) − B KKV ( s )+ 16 C r s − m π ( C r − C r − C r ) −
32 ( m π + 2 m K ) ( C r − C r − C r )+ 16 (cid:18) (2 µ π + µ K )( L r + 2 L r ) − (cid:2) B ππV ( s ) + B KKV ( s ) (cid:3) L r sf π (cid:19) − G L ( s ) , (64)where B iiV ( s ) ≡ − π (cid:18) σ i (cid:20) σ i log (cid:18) σ i − σ i + 1 (cid:19) + 2 (cid:21) − log (cid:18) m i µ (cid:19) − (cid:19) , (65) σ i = (cid:114) − m i s , (66) µ i ≡ m i log( m i /µ ) / (16 π f π ) , (67)19nd G L ( s ) is the two-loop contribution. The analytic expression of G L ( s ) is too large tobe given here, even in the s → µ = M ρ , thenumerical values for its contribution and its derivative at s = 0 are: G L (0) = − . · − , (68) G (cid:48) L (0) = − . · − GeV − . (69) References [1] Antonio Pich. Precision Tau Physics.
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