K pi vector form factor, dispersive constraints and tau -> nu_tau K pi decays
aa r X i v : . [ h e p - ph ] D ec UAB-FT-653
K π vector form factor, dispersive constraintsand τ → ν τ K π decays
Diogo R. Boito , Rafel Escribano , and Matthias Jamin , Grup de F´ısica Te`orica and IFAE, Universitat Aut`onoma de Barcelona,E-08193 Bellaterra (Barcelona), Spain. Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA).
Abstract:
Recent experimental data for the differential decay distribution of the de-cay τ − → ν τ K S π − by the Belle collaboration are described by a theoretical model whichis composed of the contributing vector and scalar form factors F Kπ + ( s ) and F Kπ ( s ). Bothform factors are constructed such that they fulfil constraints posed by analyticity and unitar-ity. A good description of the experimental measurement is achieved by incorporating twovector resonances and working with a three-times subtracted dispersion relation in order tosuppress higher-energy contributions. The resonance parameters of the charged K ∗ (892) me-son, defined as the pole of F Kπ + ( s ) in the complex s -plane, can be extracted, with the result M K ∗ = 892 . ± . K ∗ = 46 . ± . λ ′ + = (24 . ± . · − and λ ′′ + = (12 . ± . · − of the vector form factor F Kπ + ( s ) directly from the data.PACS: 13.35.Dx, 11.30.Rd, 11.55.FvKeywords: Decays of taus, chiral symmetries, dispersion relations Introduction
Hadronic decays of the τ lepton provide a fruitful environment to study low-energy QCDunder rather clean conditions [1–5]. Many fundamental QCD parameters can be determinedfrom investigations of the τ hadronic width as well as invariant mass distributions. A primeexample in this respect is the QCD coupling α s [6–8]. In addition, fundamental parame-ters of the strange sector in the Standard Model can also be obtained from the τ strangespectral function. The experimental separation of the Cabibbo-allowed decays and Cabibbo-suppressed modes into strange particles [9–11] paved the way for a new determination of thequark-mixing matrix element | V us | [12–15] as well as the mass of the strange quark [16–23].Cabibbo-suppressed τ -decays into strange final states are dominated by τ → ν τ Kπ . Inthe past, ALEPH [9] and OPAL [10] have measured the corresponding distribution functionbut lately B -factories have become a new source of high-statistics data for this reaction.Recently, the Belle experiment published results for the τ → ν τ Kπ spectrum [24] and a newdetermination of the total branching fraction became available from BaBar [25–27]. In thefuture, there are good prospects for results on the full spectrum both from BaBar and BESIII.Theoretically, the general expression for the differential decay distribution of the decay τ → ν τ Kπ can be written as [28]dΓ Kπ d √ s = G F | V us | M τ π s S EW (cid:18) − sM τ (cid:19) "(cid:18) sM τ (cid:19) q Kπ | F Kπ + ( s ) | + 3∆ Kπ s q Kπ | F Kπ ( s ) | , (1)where isospin invariance is assumed and we have summed over the two possible decay channels τ − → ν τ K π − and τ − → ν τ K − π , with the individual decays contributing in the ratio 2 : 1respectively. Furthermore, S EW = 1 . Kπ ≡ M K − M π , and q Kπ is the kaon momentum in the rest frame of the hadronic system, q Kπ ( s ) = 12 √ s r(cid:16) s − ( M K + M π ) (cid:17)(cid:16) s − ( M K − M π ) (cid:17) · θ (cid:16) s − ( M K + M π ) (cid:17) . (2)Finally, we denote by F Kπ + ( s ) and F Kπ ( s ) the vector and scalar Kπ form factors respectively,which we will discuss in detail below.In eq. (1), the prevailing contribution is due to the Kπ vector form factor F Kπ + ( s ), and inthe energy region of interest, this form factor is by far dominated by the K ∗ (892) meson. Adescription of F Kπ + ( s ) based on the chiral theory with resonances (R χ T) [30,31] was providedin ref. [32], analogous to a similar description of the pion form factor presented in refs. [33–35].Then in ref. [36] this description was employed in fitting Belle data for the spectrum of thedecay τ → ν τ K S π − [24]. The additionally required scalar Kπ form factor F Kπ ( s ) had beencalculated in the same R χ T plus dispersive constraint framework in a series of articles [37–39],and the recent update of F Kπ ( s ) [40] was incorporated as well.A slight drawback of the description for the vector form factors of refs. [32, 33] is thatthe form factors only satisfy the analyticity constraints in a perturbative sense, that is upto higher orders in the chiral expansion. Though the violation of analyticity is expected toonly be a small correction (of order p in the chiral expansion in the case at hand) it is1ertainly worthwhile to corroborate this assumption. A coupled channel analysis of the Kπ vector form factor, which would allow for such a test, was performed in ref. [41]. However,in ref. [41] the theoretical description was not really fitted to the experimental data, so thatit is difficult to decide if differences of the coupled channel analysis [41] as compared to thedescription [32, 36] already show up in the current experimental data. Even more so as thefits performed in ref. [36] provided a satisfactory description of the Belle spectrum.Below, we shall investigate the related questions in a more modest approach. In the regionof the K ∗ (892) meson, elastic unitarity is still expected to hold. Since this meson dominatesthe Kπ vector form factor, an ansatz implementing elastic unitarity should result in a goodapproximation. For the pion vector form factor such an approach was pursued in ref. [34]and in the present work we perform an analogous investigation for F Kπ + ( s ). Even thoughpossible coupled-channel contributions are not explicitly included in our parametrisation ofthe Kπ vector form factor, their influence can be studied through the sensitivity of our ansatzwhen changing the number of subtractions in the dispersion relation that the form factorsatisfies, because a larger number of subtractions entails a stronger suppression of higher-energy contributions. As a benefit of our approach, we are able to extract the resonanceparameters of the K ∗ (892) from the pole of F Kπ + ( s ) in the complex s -plane, which should beregarded as more model-independent than the Breit-Wigner type parameters extracted in theprevious analyses [24, 36], as well as the first three slopes in the Taylor expansion of F Kπ + ( s )around s = 0. Kπ vector form factor The Kπ vector form factor F Kπ + ( s ) is an analytic function in the complex s -plane, exceptfor a cut along the positive real axis, starting at the Kπ threshold s Kπ ≡ ( M K + M π ) , whereits imaginary part develops a discontinuity. The analyticity and unitarity properties of theform factor result in the fact that it satisfies an n -subtracted dispersion relation, explicatedin more detail for example in refs. [33, 34]. In the elastic region below roughly 1 . F Kπ + ( s ) = P n ( s ) exp ( s n π ∞ Z s Kπ ds ′ δ Kπ ( s ′ )( s ′ ) n ( s ′ − s − i ) , (3)which corresponds to performing the n subtractions at s = 0, and where P n ( s ) = exp ( n − X k =0 s k k ! d k ds k ln F Kπ + ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 ) (4)is the subtraction polynomial. More general formulae with subtractions at an arbitrary point s = s can for example be found in ref. [43]. Furthermore, δ Kπ ( s ) is the P-wave I = 1 / Kπ phase shift. As on general grounds the phase shift is expected to go to an integermultiple of π , at least one subtraction ( n = 1) is required in eq. (3) to make the integralconvergent. Let us first dwell on this case in more detail, before turning to the case with alarger number of subtractions. 2hile in the approach of refs. [32, 33] to the vector form factor the real part of the one-loop integral function e H ( s ) (for its definition see [32]) is resummed into an exponential, strictunitarity is only maintained if this piece, together with the imaginary part which providesthe width of the resonance, is resummed in the denominator of the form factor [37, 44]. Theresulting expression of the vector form factor corresponding to one single resonance then reads F Kπ + ( s ) = m K ∗ m K ∗ − s − κ e H Kπ ( s ) . (5)Of course, up to order p in the chiral expansion, resumming the real part in the denominatoror an exponential is fully equivalent. Differences of both approaches first start to appear at O ( p ). In eq. (5) the parameter m K ∗ is to be distinguished from the true mass of the K ∗ meson M K ∗ , which later will be identified with the real part of the pole position of F Kπ + ( s )in the complex s -plane. Identifying the imaginary part in the denominator of eq. (5) with − m K ∗ γ K ∗ ( s ), wherethe s -dependent width of the K ∗ meson takes the generic form of a vector resonance, γ K ∗ ( s ) = γ K ∗ sm K ∗ σ Kπ ( s ) σ Kπ ( m K ∗ ) , (6)with γ K ∗ ≡ γ K ∗ ( m K ∗ ), the dimensionful constant κ has to take the value: κ = 192 πF K F π σ Kπ ( m K ∗ ) γ K ∗ m K ∗ . (7)In eqs. (6) and (7), the phase space function σ Kπ ( s ) is given by σ Kπ ( s ) = 2 q Kπ ( s ) / √ s . Theform factor F Kπ + ( s ) can therefore be written in the equivalent form F Kπ + ( s ) = m K ∗ m K ∗ − s − κ Re e H Kπ ( s ) − i m K ∗ γ K ∗ ( s ) . (8)From eq. (8) the normalisation F Kπ + (0) at s = 0 which is needed in order to calculate thereduced form factor e F Kπ + ( s ) ≡ F Kπ + ( s ) /F Kπ + (0), is given by F Kπ + (0) = m K ∗ m K ∗ − κ e H Kπ (0) . (9)The phase δ Kπ ( s ) of the form factor F Kπ + ( s ) is found to be:tan δ Kπ ( s ) ≡ Im F Kπ + ( s )Re F Kπ + ( s ) = m K ∗ γ K ∗ ( s ) m K ∗ − s − κ Re ˜ H Kπ ( s ) . (10)It is a straightforward matter to verify that with the phase δ Kπ ( s ) of eq. (10), the form factorof eq. (8) satisfies the Omn`es relation (3) with n = 1 and P ( s ) = F Kπ + (0). Therefore, it is The renormalisation scale µ appearing in e H Kπ ( s ) will be set to the physical resonance mass µ = M K ∗ .
3n accord with the analyticity and unitarity requirements. Finally, the reduced form factor iswritten as e F Kπ + ( s ) = m K ∗ − κ e H Kπ (0) m K ∗ − s − κ Re e H Kπ ( s ) − i m K ∗ γ K ∗ ( s ) , (11)which resembles the non-strange form factor of Gounaris and Sakurai [45]. (For comparison,see also ref. [46].) However, while in the Breit-Wigner resonance shape of ref. [45] the real partof the inverse propagator is obtained through a twice subtracted dispersion relation combinedwith proper subtractions to fix mass and width of the resonance, the form factor in eq. (11)is found from a once subtracted dispersion relation satisfying analyticity and unitarity.As has already been discussed in ref. [36], in eq. (1) only the reduced form factor e F Kπ + ( s )has to be modelled, as the normalisation of F Kπ + ( s ) only appears in the product | V us | F Kπ + (0).This combination is determined most precisely from the analysis of semi-leptonic kaon decays.A recent average was presented by the FLAVIAnet kaon working group, and reads [47] | V us | F K π − + (0) = 0 . ± . . (12)In what follows, we work with form factors normalised to one at the origin and assume thevalue (12) for the overall normalisation. We remark that the normalisation for the scalar andthe vector form factors is the same and that (12) already corresponds to the K π − channel,which was analysed by the Belle collaboration [24]. Consequently, possible isospin-breakingcorrections to F K π − + (0) are properly taken into account. Our fits to the Belle τ − → ν τ K S π − spectrum [24] will be performed in complete analogy tothe recent analysis of ref. [36]. Let us briefly review the main strategy for these fits. Thecentral fit function is taken to have the form12 · · . / bin] N T · τ ¯ B Kπ dΓ Kπ d √ s . (13)The factors 1 / / K S π − channel has been analysed. Then, 11 . N T = 53110 is the total number ofobserved signal events. Finally, Γ τ denotes the total decay width of the τ lepton and ¯ B Kπ aremaining normalisation factor that will be deduced from the fits. The normalisation of ouransatz (13) is taken such that for a perfect agreement between data and fit function, ¯ B Kπ would correspond to the total branching fraction B Kπ ≡ B [ τ − → ν τ K S π − ] which is obtainedby integrating the decay spectrum. All further numerical input parameters have been chosenas in ref. [36].Numerically, our first fit of the Belle data [24] with the vector form factor F Kπ + ( s ) accordingto eq. (8), and including data up to √ s = 1 . Kπ threshold. For the scalar formfactor F Kπ ( s ) we have used the recent update [40], employing the central parameters given4q. (8) for F Kπ + ( s )¯ B Kπ ( B Kπ ) 0 . ± . . m K ∗ . ± .
51 MeV γ K ∗ . ± .
79 MeV χ / n . d . f . F Kπ + ( s ) according to eq. (8), as well as the scalar form factor F Kπ ( s ) [40].there. One observes that due to the not very satisfactory quality of the fit, the fit parameter¯ B Kπ and the integrated branching fraction B Kπ display a marked deviation, which is howeverwithin the uncertainties. Furthermore, because of the real part of the loop-integral in thedenominator, m K ∗ and γ K ∗ turn out rather different from their physical values M K ∗ andΓ K ∗ . For the final results of the parameters in our description of the vector form factor,in our conclusion we shall also present values for the physical parameters M K ∗ and Γ K ∗ , asobtained from the pole position in the complex s -plane.In order to make the fit less sensitive to deficiencies of our description in the higher-energyregion, a larger number of subtractions can be applied to the dispersion relation. Besides,employing an n -subtracted dispersion relation has the advantage that the slope parameterswhich appear as subtraction constants are determined more directly from the data. It shouldbe pointed out, however, that the form factor with a larger number of subtractions n ≥ F Kπ + ( s ) should vanish as 1 /s . Since we only employ the vector form factor up to about √ s ≈ . F Kπ + ( s ) is a decreasing function of s . On the other hand, fits with only onesubtraction were generally found to only provide a poor description of the experimental data.Let us present our results for the case n = 3 in detail, but below, we shall also brieflycomment on the cases n = 2 and n = 4. The importance of the high-energy region can bestudied by introducing a cutoff s cut as the upper limit of the integration in the Omn`es integral(3). Incorporating three subtractions, the reduced form factor e F Kπ + ( s ) then takes the form: e F Kπ + ( s ) = exp ( α sM π − + 12 α s M π − + s π s cut Z s Kπ ds ′ δ Kπ ( s ′ )( s ′ ) ( s ′ − s − i ) , (14)where the phase δ Kπ ( s ) corresponds to the expression of eq. (10). The parameters α and α can easily be related to the slope parameters λ ( n )+ , which appear in the Taylor expansion of e F Kπ + ( s ) around s = 0 : e F Kπ + ( s ) = 1 + λ ′ + sM π − + 12 λ ′′ + s M π − + 16 λ ′′′ + s M π − + . . . . (15)5xplicitly, the relations for the linear and quadratic slope parameters λ ′ + and λ ′′ + then takethe form: λ ′ + = α , λ ′′ + = α + α . (16)Below, we shall also compute the cubic slope parameter λ ′′′ + from the dispersive integral. s cut = 3 .
24 GeV s cut = 4 GeV s cut = 9 GeV s cut → ∞ ¯ B Kπ . ± .
045 % 0 . ± .
046 % 0 . ± .
046 % 0 . ± .
046 %( B Kπ ) (0 .
389 %) (0 .
391 %) (0 .
393 %) (0 .
393 %) m K ∗ [MeV] 943 . ± .
57 943 . ± .
58 943 . ± .
58 943 . ± . γ K ∗ [MeV] 66 . ± .
87 66 . ± .
89 66 . ± .
89 66 . ± . λ ′ + × . ± . . ± . . ± . . ± . λ ′′ + × . ± . . ± . . ± . . ± . χ / n . d . f . F Kπ + ( s ) according to eq. (8), as well as the scalar form factor F Kπ ( s ) [40].The results of our fits with the three-subtracted dispersion relation, employing four valuesof √ s cut , namely 1 . s cut → ∞ , are given in table 2. For these fits,we have included experimental data up to the data point 50 at √ s = 1 . s cut , implying that the higher-energy region is well suppressed. Compared with the fitof table 1, which only employed a single subtraction, with χ / n . d . f . ≈ . B Kπ and B Kπ turn out muchcloser. However, the slope parameter λ ′ + is not well determined from our fit. This originatesin the fact that the slope parameters are almost 100 % correlated with the total branchingfraction ¯ B Kπ , and this parameter has relatively large uncertainties. Therefore, in our bestestimate of the model parameters below, we shall impose the experimental measurement ofthe total branching fraction B Kπ .Though we only present explicit results for the three-subtracted dispersion relation, wehave also investigated the cases n = 2 and n = 4. In the case n = 2, a still somewhat strongerdependence on the cutoff s cut is observed, which is why we do not discuss the correspondingresults in detail. The four-subtracted dispersion relation, on the other hand, yields almostunchanged central values for the fit parameters without any improvement in the χ / n . d . f . , butwith larger parameter errors due to the additional degree of freedom. From this we concludethat the case n = 3 discussed above is an optimal choice as far as the number of subtractionsis concerned. Next, we shall also include a second vector resonance, the K ∗ (1410) into ourdescription of the Kπ vector form factor. 6 Fits with two vector resonances
For our final fits, we aim at a description of the τ → ν τ Kπ spectrum in the full energyrange up to the τ mass. To this end, we also introduce a second vector resonance, the K ∗ ′ = K ∗ (1410), into our model for the vector form factor F Kπ + ( s ). As it is unclear howto directly incorporate a second vector resonance in the phase δ Kπ ( s ), we have followed asomewhat indirect approach, which should be sufficient for our purposes, as the contributionof the K ∗ ′ resonance is suppressed by phase space.Like for the case of the single resonance, as a starting point we assume the form of F Kπ + ( s )given in eq. (5) of ref. [36], however resumming the real part of e H Kπ ( s ) in the denominatorof the form factor. This leads to the following expression: e F Kπ + ( s ) = m K ∗ − κ K ∗ e H Kπ (0) + γsD ( m K ∗ , γ K ∗ ) − γsD ( m K ∗′ , γ K ∗′ ) , (17)where D ( m n , γ n ) ≡ m n − s − κ n Re e H Kπ ( s ) − i m n γ n ( s ) . (18)For both resonances, γ n ( s ) is given equivalently to the form of eq. (6), and the corresponding κ n can be deduced in analogy to eq. (7). Like in eq. (10), the phase δ Kπ ( s ) can be calculatedfrom the relation tan δ Kπ ( s ) = Im F Kπ + ( s )Re F Kπ + ( s ) . (19)This is the phase that we then employ in the Omn`es integral representation (3) for the formfactor to perform our fits. s cut = 3 .
24 GeV s cut = 4 GeV s cut = 9 GeV s cut → ∞ ¯ B Kπ . ± . . ± . . ± . . ± . B Kπ ) (0 . . . . m K ∗ [MeV] 943 . ± .
58 943 . ± .
57 943 . ± .
57 943 . ± . γ K ∗ [MeV] 66 . ± .
90 66 . ± .
87 66 . ± .
86 66 . ± . m K ∗′ [MeV] 1392 ±
41 1369 ±
30 1361 ±
28 1361 ± γ K ∗′ [MeV] 273 ±
137 224 ±
101 212 ±
93 212 ± γ × − . ± . − . ± . − . ± . − . ± . λ ′ + × . ± . . ± . . ± . . ± . λ ′′ + × . ± . . ± . . ± . . ± . χ / n . d . f . F Kπ + ( s ) according to eqs. (17) to (19), as well as the scalar form factor F Kπ ( s )of ref. [40].Our first fit with two vector resonances proceeds in complete analogy to the second fitin the single-resonance case. Again, we use the three-subtracted dispersion relation, and7nvestigate four values of s cut , in order to study the dependence of our fits on the higher-energy contributions. Now, the Belle data [24] have been included up to √ s = 1 . The corresponding fit results are presented in table 3.The general picture of our fits with two resonances is quite satisfying. The theoreticalmodel provides a good description of the experimental data in the full energy range. The χ / n . d . f . for all values of s cut turns out smaller than one. The dependence of the resultingfit parameters on s cut is small and within the uncertainties, though clearly visible. The fitsprovide a precise determination of the parameters of the lowest lying K ∗ vector resonance,and a still reasonable accuracy for the second K ∗ ′ resonance. However, the fit uncertaintiesfor the branching fraction ¯ B Kπ are relatively large. Accordingly, also the error on the slopeparameter λ ′ + is found large, because these two parameters are almost 100 % correlated. s cut = 3 .
24 GeV s cut = 4 GeV s cut = 9 GeV s cut → ∞ m K ∗ [MeV] 943 . ± .
59 943 . ± .
58 943 . ± .
57 943 . ± . γ K ∗ [MeV] 66 . ± .
88 66 . ± .
86 66 . ± .
85 66 . ± . m K ∗′ [MeV] 1407 ±
44 1374 ±
30 1362 ±
26 1362 ± γ K ∗′ [MeV] 325 ±
149 240 ±
100 216 ±
86 215 ± γ × − . ± . − . ± . − . ± . − . ± . λ ′ + × . ± .
74 24 . ± .
69 24 . ± .
68 24 . ± . λ ′′ + × . ± .
20 11 . ± .
19 11 . ± .
19 11 . ± . χ / n . d . f . F Kπ + ( s ) according to eqs. (17) to (19), as well as the scalar form factor F Kπ ( s )[40]. The total branching fraction B Kπ has been fixed to the experimental value (20), andthe corresponding uncertainty is included in quadrature.In order to provide a better determination of the slope parameters, a possible way toproceed is to fix the total branching fraction B Kπ to the experimental measurement. A veryrecent update of the world average has been presented in ref. [24, 26, 27], with the finding: B [ τ − → ν τ K S π − ] = 0 . ± .
011 % . (20)Thus, for our final fits, we have fixed B Kπ to take this value. The resulting fit parameters forthe remaining quantities are displayed in table 4. Here, the quoted uncertainties include thestatistical fit errors as well as a variation of B Kπ in the range given by (20). From table 4 weinfer that due to the reduction in the uncertainty of B Kπ , correspondingly also the uncertaintyin the slope parameters is much reduced, while the errors of the remaining parameters to agood approximation stay as before. Also the χ / n . d . f . is practically unchanged, still remainingbelow one for all values of s cut . Although the full Belle data set consists of 100 data points, we follow a suggestion of the experimentaliststo only fit the data up to point 90 [48]. .6 0.8 1 1.2 1.4 1.6 1.8 √ s [GeV] N E v e n t s Figure 1: Main fit result to the Belle data [24] for the differential decay distribution of thedecay τ − → ν τ K S π − . Our theoretical description corresponds to the fit of table 4 with s cut = 4 GeV . The full fit including vector form factor F Kπ + ( s ) and scalar form factor F Kπ ( s ) is displayed as the solid line. The separate vector and scalar contributions are shownas the dashed and dotted lines respectively.A graphical account of our central result is displayed in figure 1. The solid line correspondsto the fit of table 4 including vector form factor F Kπ + ( s ) as well as the scalar form factor F Kπ ( s ) at s cut = 4 GeV . The separate contributions of F Kπ + ( s ) and F Kπ ( s ) are shown asthe dashed and dotted lines respectively. As is apparent, apart from the data points 5, 6 and7 in the low-energy region, our model provides a perfect description of the experimental databy the Belle collaboration. Also, from an inspection of the region below the K ∗ resonance it isevident that a contribution from the scalar form factor F Kπ ( s ) is required, though, like in theanalysis of ref. [36], the sensitivity to F Kπ ( s ) is not strong enough to allow for a determinationof the corresponding model parameters. Hadronic τ decays provide a means to obtain information on low-energy QCD as well ashadron phenomenology. In this work, we have studied recent data on the decay channel τ − → ν τ K S π − by the Belle collaboration [24]. The measured decay spectrum allows totest models for the vector and scalar Kπ form factors F Kπ + ( s ) and F Kπ ( s ), and to deduce9he corresponding model parameters for the vector form factor. For F Kπ + ( s ), we have useda model which incorporates the constraints on the form factor from analyticity and elasticunitarity. Furthermore, we investigated n -subtracted dispersive integrals with a cutoff s cut ,where n ranges from 1 to 4 and √ s cut was varied between 1 . F Kπ ( s ), we have employedthe description of ref. [38], which is based on solving dispersion relations for a two-bodycoupled-channel problem, and was recently updated in [40].Let us begin with summarising our final results for the parameters of the K ∗ and K ∗ ′ vector resonances. As our central results, we quote the values of table 4 at s cut = 4 GeV .To the uncertainty given in table 4, we add an error for the variation of our results whenchanging s cut . The resonance mass and width parameters are then found to be: m K ∗ = 943 . ± .
59 MeV , γ K ∗ = 66 . ± .
87 MeV , (21) m K ∗′ = 1374 ±
45 MeV , γ K ∗′ = 240 ±
131 MeV , (22)while the mixing parameter for the second resonance reads γ = − . ± . K ∗ and K ∗ ′ resonances, we have to compute thepositions of the poles of the vector form factor in the complex s -plane. From the pole position s p we can then read off the physical mass and width of the respective resonance according tothe relation √ s p = M R − i R . (23)Calculating the pole positions along the lines of the approach outlined in ref. [49] yields: M K ∗ = 892 . ± .
92 MeV , Γ K ∗ = 46 . ± .
38 MeV , (24) M K ∗′ = 1276 + 72 − MeV , Γ K ∗′ = 198 + 61 − MeV . (25)The uncertainties are calculated by assuming a Gaussian error propagation while simultane-ously varying both m R and γ R . The mass of the charged K ∗ meson turns out rather close tothe value advocated by the PDG, but more than 3 MeV lower than the Breit-Wigner reso-nance parameters obtained in refs. [24, 36]. On the other hand, the K ∗ width Γ K ∗ of eq. (24)nicely agrees with the result of [24], but it is more than 4 MeV lower than the PDG average.To shed further light on the comparison with previous works let us calculate the poleposition of the K ∗ (892) for the best fit of ref. [36]. The model employed in this referenceamounts, as far as the poles are concerned, to removing the term proportional to Re e H Kπ ( s )from eq. (18) while keeping the energy dependent width. Denoting the respective fit param-eters with a tilde we have e m K ∗ = 895 . ± .
20 MeV and e γ K ∗ = 47 . ± .
41 MeV [36]. Thecorresponding pole position is given as the second line in table 5, being perfectly consistentwith our results provided in eq. (24) and the first line of table 5. The same exercise can berepeated for the original analysis performed by the Belle collaboration [24]. Here, we haveemployed the fit parameters corresponding to the second fit given in table 3, which are closeto their final result for the K ∗ parameters. In this case the corresponding pole position is10isplayed as the last line in table 5. Again it turns out rather close to the previous results.To summarise, the pole position is found to be rather stable since different models yield com-patible values for the physical parameters M K ∗ and Γ K ∗ as defined in eq. (23). Concerningthe parameters of the second resonance, they are in general agreement with the findings ofref. [36], especially after the pole position is computed for the latter results. Due to the largeuncertainties, however, we cannot make any more definite statement.Model Parameters Pole Positions( m K ∗ , γ K ∗ ) [MeV] ( M K ∗ , Γ K ∗ ) [MeV]This work (943 . ± . , . ± .
87) (892 . ± . , . ± . . ± . , . ± .
41) (892 . ± . , . ± . . ± . , . ± .
57) (892 . ± . , . ± . K ∗ (892) meson. For definiteness, from ref. [24] we have employed the parameters ofthe second fit of their table 3 and consider only statistical uncertainties.Our fits to the τ − → ν τ K S π − spectrum also allow to determine the slope parameters of thevector form factor F Kπ + ( s ). The advantage of using a three-subtracted dispersion relation isthat the parameters λ ′ + and λ ′′ + are directly determined from the data, making the extractionmore model independent. The disadvantage being that therefore the uncertainties for λ ′ + turn out larger than for example in ref. [36], where these parameters are a direct consequenceof the form factor model. Higher slope parameters can of course also be calculated throughdispersive integrals. For example in the case of λ ′′′ + one has the relation: λ ′′′ + = α + 3 α α + M π − π s cut Z s Kπ ds ′ δ Kπ ( s ′ )( s ′ ) . (26)Together with the explicit fit results, this leads to λ ′ + = (24 . ± . · − , λ ′′ + = (11 . ± . · − , λ ′′′ + = (8 . ± . · − , (27)where again the uncertainty due to the variation of s cut has been included in quadrature.Within the given errors, the value (27) for λ ′ + is in good agreement to the result of ref. [36],as well as the determination from an average of current experimental data for K l decays [47].On the other hand, both, the quadratic slope λ ′′ + , and the cubic slope λ ′′′ + , are found somewhatlower than the corresponding results of ref. [36].To conclude, differential decay spectra of hadronic τ decays provide important informationfor testing form factor models and extracting the corresponding model parameters, therebyaccessing QCD in the realm of low energies. It will be very interesting to see if our findings arecorroborated by additional experimental data in the future. Furthermore, when comparingparameters of hadronic resonances, even when employing Breit-Wigner type parametrisations11ith an energy-dependent width, pole positions in the complex s -plane should be provided inorder to arrive at more model independent results. Acknowledgements
MJ is most grateful to the Belle collaboration, in particular S. Eidelman, D. Epifanov andB. Shwartz, for providing their data and for useful discussions. He should also like to thank thereferee of ref. [36] for a question which initiated the present study. This work was supportedin part by the Ramon y Cajal program (RE), the Ministerio de Educaci´on y Ciencia un-der grant FPA2005-02211, the EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”, theSpanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), and the Generalitatde Catalunya under grant 2005-SGR-00994.
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