Study of the lowest tensor and scalar resonances in the τ→πππ ν τ decay
SStudy of the lowest tensor and scalar resonancesin the τ → πππν τ decay Olga Shekhovtsova , ,(cid:63),(cid:63)(cid:63) , Juan José Sanz-Cillero , and T. Przedzi ´nski Institute of Nuclear Physics PAN, Cracow, Poland NSC KIPT Institute for Theoretical Physics, Kharkov, Ukraine Departamento de Física Teórica and Instituto de Física Teórica, IFT-UAM/CSIC, Universidad Autónoma deMadrid, Cantoblanco, 28049 Madrid, Spain The Faculty of Physics, Astronomy and Applied Computer Science, Jagellonian University, Cracow, Poland
Abstract.
In this note we present a new parametrization of the hadronic current for thedecay τ → πππν τ derived from the chiral lagrangian with explicit inclusion of resonances.We have included both scalar, vector and axial-vector resonances. For the first time, thelowest tensor resonance ( f (1270)) is included as well. Both single and double-resonancecontributions to the hadronic form factors are taken into account. To satisfy the correcthigh energy behaviour of the hadronic form factors, constraints on numerical values ofthe vertex constants are obtained. Hadronic decay modes of τ -lepton gives information about the hadronization mechanism and reso-nance dynamics in the energy region, where the pQCD methods are not applicable. In the last yearssubstantial progress for the simulation of the process τ → πν τ was achieved.The progress [1] wasrelated to a new parametrization of the hadronic current based on the Resonance Chiral Lagrangian(RChL) and to the recent availability of the unfolded distributions from preliminary BaBar analysis [2]for all invariant hadronic masses for the three-prong mode. The lowest-energy scalar resonance wasadded phenomenologically and, as a result, the corresponding hadronic current does not reproducethe correct chiral low-energy behaviour and the π π π − and π − π − π + amplitudes do not reproduce theisospin relation [3]. Comparison with the data has demonstrated also a hint on the missing tensorresonance ( f (1270)).The goal of this note is to outline a consistent model to describe the tau-lepton decys into threepions based on RChL with scalar ( J PC = ++ ) and tensor resonances ( J PC = ++ ) and that fulfill thehigh-energy QCD and low-energy chiral limits for the hadronic form-factors. The detail descriptionof the model and calculation of the hadronic form-factor will be presented [4]. (cid:63) Speaker, e-mail: [email protected] (cid:63)(cid:63)
IFJPAN-IV-2016-22 a r X i v : . [ h e p - ph ] O c t Three pion hadronic current. Axial-vector form-factors related withscalar and tensor resonances.
The most general Lorenz invariant current for τ − → π − (0) ( p ) π − (0) ( p ) π − ( + ) ( p ) ν τ H πµ ( p , p , p ) = iP T ( q ) αµ (cid:16) ( p µ − p ) µ F ( s , s , q ) + ( p µ − p ) µ F ( s , s , q ) (cid:17) + iq α F P ( s , s , q )with s i = ( p j − p k ) , q = ( p + p + p ) and, due to the Boson symmetry, the hadronic form-factorsare related: F ( s , s , q ) = F ( s , s , q ). The longitudinal form-factor F P is suppressed by m π / q compared to F , and in this note we will neglect it. In Fig. 1, we show the three relevant diagrams Figure 1.
Relevant diagrams for the τ -lepton decay into S π and T π . Single straight lines are for pions, the waveline is for an incoming W − . that must be taken into account: a) the direct production; b) the intermediate π − production and c)the double resonance production through the intermediate a axial-vector resonance. To calculatethe corresponding diagrams we use the RChL approach [5] for the vector and axial-vector ( A ) reso-nances combined wothw the Lagrangian including interaction of a tensor ( T ) multiplet and pions [6].Moreover we add the operators with two resonances: • AS π interaction ∆ L AS π = λ AS (cid:104){∇ µ S , A µν } u ν (cid:105)• AT π interaction ∆ L AT π = λ AT π (cid:104){ A αβ , ∇ α T µµ } u β (cid:105) + λ AT π (cid:104){ A αβ , ∇ µ T µα } u β (cid:105) ,where for the axial-vector field A αβ we apply the antisymmetric tensor representation [5], S is thescalar field, a tensor multiplet is T µν = f µν / √ ∗ dia g (1 , ,
0) (we will assume the ideal mixing in thetensor nonet and that the f (1270) resonance is pure u ¯ u + d ¯ d ).The π π π − and π − π − π + amplitudes obey the isospin relation [3] that leads to F −− + ( s , s , q ) = F − ( s , s , q ) − F − ( s , s , q ) − F − ( s , s , q ) . (1)For the three-pion form-factor caused by the intermediate σ -resonance we have: F ( s , s , q ) − = F aS π ( q ; s ) G S ππ ( s ) , (2)where the AS π form-factor and propagation of the σ -resonance and its decay into ππ are F aS π ( q ; k ) = c d F π + √ F A λ AS F π q M A − q , G S ππ ( s ) = √ c d F π ( s − m π ) M S − s and qp j = ( m π + q − s j ) /
2. Requiring F aS π ( q ; k ) → q → ∞ we got F A λ AS = √ c d .To include a σ – f (980) splitting and non-zero width of the resonances we follow [7]1 M S − s −→ cos φ S M σ − s − f σ ( s ) − iM σ Γ σ ( s ) + sin φ S M f − s − iM f Γ f , (3)where φ S is the scalar mixing angle. For the f parameters we will use the numerical values M f =
980 MeV, φ S = − ◦ [7]. As a first approach we also consider the Breit-Wigner function for the σ -propagator in our numerical study.chematically the form-factor related with the intermediate tensor resonance state is written as F ( s , s , q ) − = H ( q , s , s )( M A − q )( M f − s ) + H ( q , s , s )( M A − q ) + H ( q , s , s )( M f − s ) + H ( q , s , s ) , (4)where H i ( q , s , s ) are non-singular functions. We would like to stress that for q = M A and s = M f our expression (4) reproduces the corresponding contribution of Eq. (A.3) of [8] and that in [9]. How-ever, for an arbitrary o ff -shell momentum of the intermediate tensor resonance we have a more generalmomentum structure of the hadronic current, which also ensures the right low energy behaviour andthe transversality of the matrix element in the chiral limit. As a result it brings three additional func-tions H , , ( q , s , s ) in (4) (see for discussion [4]).To obtain the π − π − π + form-factors we apply the relation (1) for (2) and (4). Exact formulae arepresented in [4].The hadronic form-factors (2) and (4) have been implemented in the Monte Carlo Tauola [1]. Toget the model parameters the one-dimentional spectra d Γ / ds , d Γ / ds and d Γ / dq with the hadronicform-factors (2) and (4) in addition to [10] have been fitted to the preliminary π − π − π + BaBar data [2].The results are presented in Fig. 2 (as an example we present the result for the Breit-Wigner σ -mesonpropagator). For the first approach we have fixed the tensor resonance parameters to their PDG values.The di ff erence between the data and the theoretical distributions is less than 5 − Figure 2.
The τ − → π − π − π + ν τ decay invariant mass distribution. The preliminary BaBar data [2] are presentedby points and the line corresponds to the model.The work of J.J.S.C is partially supported by grant FPA2013-44773-P and the Centro de Excelencia SeveroOchoa Programme (Spanish Ministry MINECO) SEV-2012-0249, the research of O.Sh. was supported in partby funds of the Foundation of Polish Science grant POMOST / / References [1] I. M. Nugent, T. Przedzinski, P. Roig, O. Shekhovtsova and Z. Was, Phys. Rev. D88 (2013)093012.[2] I. M. Nugent [BaBar Collaboration], Nucl. Phys. Proc. Suppl. (2014) 38.[3] L. Girlanda and J. Stern, Nucl. Phys. B 575 (2000) 285.[4] J.J. Sanz-Cillero, O. Shekhovtsova, in preparation.[5] G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B321 (1989) 311.[6] G. Ecker and C. Zauner, Eur.Phys.J. C 52 (2007) 315.[7] R. Escribano, P. Masjuan and J. J. Sanz-Cillero, JHEP (2011) 094.[8] D. M. Asner et al. [CLEO Collaboration], Phys. Rev. D 61 (2000) 012002.[9] G. L. Castro and J. H. Munoz, Phys. Rev. D83