The $K\pi$ vector form factor, $F_+^{K\pi}$, used to reproduce the Belle
spectrum of $\tauKpi$ decays is described by means of a three-times subtracted
dispersion relation also incorporating constraints from $K_{l3}$ decays. The
slope and curvature of $F_+^{K\pi}$ are fitted to the data yielding
$\lambda_+'=(25.49 \pm 0.31) \times 10^{-3}$ and $\lambda_+"= (12.22 \pm 0.14)
\times 10^{-4}$. The pole parameters of the $K^*(892)^\pm$ are found to be
$m_{K^*(892)^\pm}= 892.0\pm 0.5$ MeV and $\Gamma_{K^*(892)^\pm}= 46.5 \pm1.1$
MeV. The phase-space integrals relevant for $K_{l3}$ analyses and the $P$-wave
isospin-1/2 $K\pi$ phase-shift threshold parameters are also calculated.Comment: 3 pages, uses aipproc style. Talk presented by R. Escribano at the IX
  International Conference on Quark Confinement and Hadron Spectrum (QCHS09),
  Madrid, Spain, 30.8-3.9.201

Boito, Diogo R.

Escribano, Rafel

Jamin, Matthias

English

arXiv

Altres ajuts: this work has been supported in part under the EU Contract No. MRTN-CT-2006-035482 "FLAVIAnet"Talk presented by R. Escribano at the IX International Conference on Quark Confinement and Hadron Spectrum (QCHS09), Madrid, Spain, 30.8-3.9.2010The Kπ vector form factor, FKπ+, used to reproduce the Belle spectrum of τ→Kπντ decays is described by means of a three-times subtracted dispersion relation also incorporating constraints from Kl3 decays. The slope and curvature of FKπ+ are fitted to the data yielding λ'+ = (25.49±0.31)×10-3 and λ''+ = (12.22±0.14)×10-4. The pole parameters of the K*(892)± are found to be mK* (892)± = 892.0±0.5 MeV and ΓK*(892)± = 46.5±1.1 MeV. The phase-space integrals relevant for Kl3 analyses and the P-wave isospin-1/2 Kπ phase-shift threshold parameters are also calculate

Boito, Diogo

Diposit Digital de Documents de la UAB

This is the accepted version of the journal article:Boito, Diogo R.; Escribano, Rafel; Jamin, Matthias. «Improving the Kpi vectorform factor through Kl3 constraints». AIP Conference Proceegings, Vol. 1343,issue 1 (May 2011), p. 271–273. DOI 10.1063/1.3574999This version is available at https://ddd.uab.cat/record/275018under the terms of the licenseImproving the Kp vector form factor through Kl3 constraints1Diogo R. Boito⇤, Rafel Escribano⇤ and Matthias Jamin†⇤Grup de Física Teòrica and IFAE, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain†Institució Catalana de Recerca i Estudis Avançats (ICREA), IFAE and Grup de Física Teòrica,Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), SpainAbstract. The Kp vector form factor, FKp+ , used to reproduce the Belle spectrum of t ! Kpnt decays is described by meansof a three-times subtracted dispersion relation also incorporating constraints from Kl3 decays. The slope and curvature ofFKp+ are fitted to the data yielding l 0+ = (25.49±0.31)⇥10 3 and l 00+ = (12.22±0.14)⇥10 4. The pole parameters of theK⇤(892)± are found to be mK⇤(892)± = 892.0±0.5 MeV and GK⇤(892)± = 46.5±1.1 MeV. The phase-space integrals relevantfor Kl3 analyses and the P-wave isospin-1/2 Kp phase-shift threshold parameters are also calculated.Keywords: t decays, K decays, dispersion relationsPACS: 13.35.Dx, 13.25.Es, 11.55.FvINTRODUCTIONThe non-perturbative physics of K ! plnl (Kl3) andt ! Kpnt decays is governed by two Lorentz-invariantKp form factors, namely the vector, denoted FKp+ (q2),and the scalar, FKp0 (q2). A good knowledge of theseform factors paves the way for the determination of manyparameters of the Standard Model, such as the quark-mixing matrix element |Vus| obtained from Kl3 decays[1], or the strange-quark mass ms determined from thescalar QCD strange spectral function [2].Until recently, the main source of experimental infor-mation on Kp form factors have been Kl3 decays. Lately,five experiments have collected data on semileptonic andleptonic K decays: BNL-E865, KLOE, KTeV, ISTRA+,and NA48. Additional knowledge on the Kp form factorscan be gained from the dominant Cabibbo-suppressed tdecay: the channel t ! Kpnt . A detailed spectrum fort ! KSp nt produced and analyzed by Belle was pub-lished in 2007 [3]. Also, preliminary BaBar spectra withsimilar statistics have appeared recently in conferenceproceedings [4] and, finally, BESIII should produce re-sults for this decay in the future [5]. The new data setsprovide the substrate for up-to-date theoretical analysesof the Kp form factors. In Ref. [6] we have performeda reanalysis of the t ! Kpnt spectrum of [3]. More re-cently, we carried out an analysis with restrictions fromKl3 experiments [7].On the theory side, the knowledge of these form fac-tors consists of two tasks. The first of them is to deter-mine their value at the origin, F+,0(0), crucial in orderto disentangle the product |Vus|F+,0(0). Historically, chi-1 Speaker: R. Escribano.ral perturbation theory has been the main tool to studyF+,0(0), but recently lattice QCD collaborations haveproduced more accurate results for this quantity [8]. Sec-ond, one must know the energy dependence of the formfactors, which is required when calculating phase-spaceintegrals for Kl3 decays or when analyzing the detailedshape of the t ! Kpnt spectrum. In our work we con-centrate on the latter aspect of the problem and thereforeit is convenient to introduce form factors normalized toone at the origin2F̃+,0(q2) = F+,0(q2)/F+,0(0) . (1)A salient feature of the form factors in the kinematicalregion relevant for Kl3 decays, i.e. m2l < q2 < (mK  mp)2, is that they are real. Within the allowed phase-space they admit a Taylor expansion and the energydependence is customarily translated into constants l (n)+,0defined asF̃+,0(q2) = 1+l 0+,0q2m2p +12l 00+,0 q2m2p !2+ · · · . (2)In t !Kpnt decays, however, since (mK +mp)2 < q2 <m2t , one deals with a different kinematical regime inwhich the form factors develop imaginary parts, render-ing the expansion of Eq. (2) inadmissible. One must thenresort to more sophisticated treatments. Moreover, in or-der to fully benefit from the available experimental data,it is desirable to employ representations of the form fac-tors that are valid for both Kl3 and t ! Kpnt decays.Dispersive representations of the form factors provide apowerful tool to achieve this goal.2 From now on we refrain from writing the superscript Kp on the formfactors.From general principles, the form factors must sat-isfy a dispersion relation. Supplementing this constraintwith unitarity, the dispersion relation has a well-knownclosed-form solution within the elastic approximation re-ferred to as the Omnès representation [9]. Although sim-ple, this solution requires the detailed knowledge of thephase of F+(s) up to infinity, which is unrealistic. An ad-vantageous strategy to circumvent this problem is the useof additional subtractions, as done, for instance, for thepion form factor in Ref. [10]. Subtractions in the disper-sion relation entail a suppression of the integrand in thedispersion integral for higher energies. The outcome ofthese tests is that for our purposes an optimal descrip-tion of F+(s) is reached with three subtractions and tworesonances. Here we quote the resulting expressionF̃+(s) = exp"a1sm2p +12a2s2m4p +s3pscutZsKpds0d (s0)(s0)3(s0   s  i0)#. (3)In the last equation, sKp = (mK0 + mp )2 and the twosubtraction constants a1 and a2 are related to the Taylorexpansion of Eq. (2) as l 0+ = a1 and l 00+ = a2 + a21 . Itis opportune to treat them as free parameters that captureour ignorance of the higher energy part of the integral.The constants l 0+ and l 00+ can then be determined throughthe fit. The main advantage of this procedure, advocatedfor example in Refs. [6, 10, 11], is that the subtractionconstants turn out to be less model dependent as theyare determined by the best fit to the data. It is importantto stress that Eq. (3) remains valid beyond the elasticapproximation provided d (s) is the phase of the formfactor, instead of the corresponding scattering phase. But,of course, in order to employ it in practice we must have amodel for the phase. As described in detail in Ref. [6], wetake a form inspired by the RChT treatment of Refs. [12]with two vector resonances. For the detailed expressionswe refer to the original works. With Eq. (3), the transitionfrom the kinematical region of t ! Kpnt to that of Kl3decays is straightforward and the dominant low-energybehavior of F+(s) is encoded in l 0+ and l 00+. The cut-offscut in the dispersion integral is introduced to quantify thesuppression of the higher energy part of the integrand.The stability of the results is checked varying this cut-offin a wide range from 1.8GeV < scut < •.In t ! Kpnt decays, the scalar form factor is sup-pressed kinematically. Albeit marginal, the contributionfrom F0 cannot be neglected in the lower energy part ofthe spectrum. Here, we keep this contribution fixed us-ing the results for F0 from the coupled-channel dispersiveanalysis of Refs. [2, 13].FITS TO t !Kpnt WITHCONSTRAINTS FROM Kl3The analysis of the spectrum for t ! Kpnt producesa wealth of physical results, many of them with greataccuracy, e.g., the mass and width of the K⇤(892). Wehave advocated by means of Monte Carlo simulationsthat a joined analysis of t ! Kpnt and Kl3 spectrafurther constrains the low-energy part of the vector form-factor yielding results with a better precision [6]. Thisidea was pursued in our recent work [7].In order to include the experimental information avail-able from Kl3 decays —and for the want of true unfoldeddata sets from these experiments— we adopt the follow-ing strategy3. In our fits, the c2 that is to be minimizedcontains a standard part from the t ! Kpnt spectrumand a piece which constrains the parameters l (0,00)+ us-ing information from Kl3 experiments. For the latter ex-perimental values we employ the results of the compila-tion of KL analyses performed by Antonelli et al. for theFlaviaNet Working Group on Kaon Decays in Ref. [14]:l 0exp+ = (24.9±1.1)⇥10 3, l00exp+ = (16±5)⇥10 4 andrl 0+,l 00+ = 0.95.ResultsFrom the minimization of the c2 a collection of phys-ical results can be derived. Some of them are obtaineddirectly from the fit, such as l 0+ and l 00+ and the mass andwidth of the K⇤(892). With the form factor under control,one can then obtain other results such as the phase-spaceintegrals for Kl3 decays. Here, we present the main re-sults of Ref. [7]. A careful comparison with other resultsfound in the literature can be found in that reference.We start by quoting our final results for the mass andthe width of the K⇤(892)±mK⇤(892)± = 892.03± (0.19)stat ± (0.44)sys MeV ,GK⇤(892)± = 46.53± (0.38)stat ± (1.0)sys MeV . (4)These results are obtained from the complex pole po-sition on the second Riemann sheet, sK⇤ , following thedefinitionpsK⇤ = mK⇤   (i/2)GK⇤ [15]. It is importantto stress that the mass and width thus obtained are ratherdifferent from the parameters that enter our descriptionof the phase of F+(s). When comparing results from dif-ferent works one must always be sure that the same def-inition is used in all cases. In Ref. [7], we showed thatour results are compatible with others provided the poleposition prescription is employed for all the analyses.3 For a detailed discussion of the fit procedure we refer to [7].The final results for the parameters l 0+ and l 00+ readl 0+⇥103 = 25.49± (0.30)stat ± (0.06)scut ,l 00+⇥104 = 12.22± (0.10)stat ± (0.10)scut . (5)In this case, the uncertainty from the variation of scutcontributes as indicated. From the expansion of Eq (3)we can calculate the third coefficient of a Taylor seriesof the type of Eq. (2). We findl 000+ ⇥105 = 8.87± (0.08)stat ± (0.05)scut . (6)These results are in good agreement with other analy-ses but have smaller uncertainties since our fits are con-strained by t ! Kpnt and Kl3 experiments.In the extraction of |Vus| from the Kl3 decay widths,one must perform phase-space integrals where the form-factors play the central role. The integrals are definedin Ref. [1, 14]. From our form-factors we obtain thefollowing resultsIK0e3 = 0.15466(17) , IK0µ3 = 0.10276(10) ,IK+e3 = 0.15903(17) , IK+µ3 = 0.10575(11) . (7)The uncertainties were calculated with a MC sampleof parameters obeying the results of our fits with thecorrelations properly included. The final uncertaintiesare competitive if compared with the averages of [14]and the central values agree.Another interesting result that can be extracted fromthe t ! Kpnt spectrum is the Kp isospin-1/2 P-wavescattering phase. The decay in question is indeed a veryclean source of information about Kp interactions, sincethe hadrons are isolated in the final state. Below inelasticthresholds, the phase of the form-factor is the scatteringphase, as dictated by Watson’s theorem. From the expan-sion of the corresponding partial-wave T -matrix elementin the vicinity of the Kp threshold one can determine theKp P-wave threshold parameters. With our results, thefirst three readm3p  a1/21 ⇥10 = 0.166(4) ,m5p  b1/21 ⇥102 = 0.258(9) ,m7p  c1/21 ⇥103 = 0.90(3) . (8)ACKNOWLEDGMENTSR. E. would like to express his gratitude to the QCHS9Organizing Committee for the opportunity of present-ing this contribution, and for the pleasant and interest-ing conference we have enjoyed. We are grateful to theBelle collaboration in particular to S. Eidelman, D. Epi-fanov and B. Shwartz, for providing their data. This workwas supported in part by the Ministerio de Ciencia e In-novación under grant CICYT-FEDER-FPA2008-01430,the EU Contract No. MRTN-CT-2006-035482, “FLA-VIAnet”, the Spanish Consolider-Ingenio 2010 Pro-gramme CPAN (CSD2007-00042), and the Generalitatde Catalunya under grant SGR2009-00894.REFERENCES1. H. Leutwyler and M. Roos, Z. Phys. C 25 (1984) 91.2. M. Jamin, J. A. Oller and A. Pich, Phys. Rev. D 74, (2006)074009 [arXiv:hep-ph/0605095].3. D. Epifanov et al. [Belle Collaboration], Phys. Lett. B654 (2007) 65 [arXiv:0706.2231 [hep-ex]].4. S. Paramesvaran [BaBar Collaboration], proceedingsof Meeting of the Division of Particles and Fields ofthe American Physical Society (DPF 2009), Detroit,Michigan, 26-31 Jul 2009, arXiv:0910.2884 [hep-ex].5. D. M. Asner et al., “Physics at BES-III,” arXiv:0809.1869[hep-ex]. Edited by K. T. Chao and Y. F. Wang, Int. J. ofMod. Phys. A 24 Supplement 1, (2009).6. D. R. Boito, R. Escribano and M. Jamin, Eur. Phys. J. C59, (2009) 821 [arXiv:0807.4883 [hep-ph]]; PoS EFT09(2009) 064 [arXiv:0904.0425 [hep-ph]]7. D. R. Boito, R. Escribano and M. Jamin, JHEP 1009(2010) 031 [arXiv:1007.1858 [hep-ph]]; arXiv:1010.0100[hep-ph].8. G. Colangelo et al., arXiv:1011.4408 [hep-lat].9. R. Omnès, Nuovo Cim. 8 (1958) 316.10. A. Pich and J. Portolés, Phys. Rev. D 63 (2001) 093005[arXiv:hep-ph/0101194].11. V. Bernard, M. Oertel, E. Passemar and J. Stern, Phys.Lett. B 638 (2006) 480 [arXiv:hep-ph/0603202]; Phys.Rev. D 80 (2009) 034034 [arXiv:0903.1654 [hep-ph]].12. M. Jamin, A. Pich and J. Portolés, Phys. Lett. B 640(2006) 176 [arXiv:hep-ph/0605096]; Phys. Lett. B 664(2008) 78 [arXiv:0803.1786 [hep-ph]].13. M. Jamin, J. A. Oller and A. Pich, Nucl. Phys. B 622(2002) 279 [arXiv:hep-ph/0110193].14. M. Antonelli et al., Eur. Phys. J. C 69 (2010) 399[arXiv:1005.2323 [hep-ph]].15. R. Escribano, A. Gallegos, J. L. Lucio M, G. Moreno andJ. Pestieau, Eur. Phys. J. C 28 (2003) 107 [arXiv:hep-ph/0204338].

Improving the Kpi vector form factor through Kl3 constraints

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