POLARIZATION EFFECTS IN RADIATIVE DECAY OF A POLARIZED r LEPTON
G. I. Gakha'h, M. I. Konchatnija* A. Yu. Korchina<b, N. P. Merenkova'h**
a NSC "Kharkov Institute of Physics and Technology" 61108, Kharkov. Ukraine
b Karazin Kharkiv National University 61022, Kharkov, Ukraine
Received Jule 30, 2014
The polarization effects in the one-meson radiative decay of a polarized r lepton, —► are investigated.
The inner bremsstrahlung and structural amplitudes are taken into account. The asymmetry of the differential decay width caused by the r-lepton polarization and the Stokes parameters of the emitted photon itself are calculated depending on the polarization of the decaying r lepton. These physical quantities are estimated numerically for an arbitrary direction of the r lepton polarization 3-vector in the rest frame. The vector and axial-vector form factors describing the structure-dependent part of the decay amplitude are determined using the chiral effective theory with resonances (R\T).
DOI: 10.7868/S0044451015020078
1. INTRODUCTION
As is known, the r lepton is the only existing lepton that, duo to its large mass, can decay into final states containing hadrons. The energy region of these decays corresponds to the hadron dynamics described by the nonperturbative QCD. Since the complete theory of nonperturbative QCD is absent at present, the phenomena in this energy region are described using various phenomenological approaches. To test different theoretical models, it is important to experimentally investigate the hadronization processes of weak currents. The semileptonic r-lepton decays are very suitable for such investigations because the leptonic weak interaction is well understood in the Standard Model (SM). A review of the present status of r physics can be found in Ref. [1].
In the last decade, experimental investigations of the r lepton decays have been strongly extended due to the construction of the B-factories (BaBar, Belle) with a very high luminosity L « 1034 cm"2 [2]. At present, experiments at the B-factories led to the accumulation of the data sets of more than 10B r-lepton pairs [3]. Interesting results obtained at the B-factories
* E-mail: konchatnij'fflkipt.kharkov.ua
E-mail: merenkovfflkipt.kharkov.ua
revived the plans for constructing new facilities such as SuperKEKB (Japan) and Super c r (Russia) [2, 4, 5]. These projects will use a new technique to collide the electron positron beams, which permits increasing the existing luminosity by one or two orders of magnitude. The designed luminosity is L k. (1^2)-1035 cm"2 • s-1 for the Super c r and L « 1036 cm"2 • s-1 for the SuperKEKB [2]. Besides, the Super c r and SuperKEKB factories can have a longitudinally polarized electron beam with the polarization degree of more than 80%, which guarantees production of polarized r leptons.
This very high luminosity of the planned r factories will allow performing precise measurements of various decays of the r lepton and hence searching for the manifestations of the new physics beyond the SM, such as the lepton flavor violation, CP/T violation in the lepton sector, and so on.
The simplest semileptonic r-lepton decay is t~ —¥ —¥ 7t_(A'_)i/t, but in this case, the hadronization of the weak currents is described by form factors at a fixed value of the momentum transfer squared t (the difference of the t~ and vT 4-nionienta squared). The dependence of the form factors on this variable can be determined, in principle, in the transition W —¥ tt(K)j, where t is the squared invariant mass of the tt(A') 7 system. This transition can be investigated in the r-lepton radiative decay t~
The ono-psoudoscalar-moson radiative r-lopton do-cays have boon investigated in a number of papers [4 9]. The authors of Ref. fC] obtained an expression for the double-differential decay rate for the —¥ vttt~7 decay in terms of the vector v(t) and axial-vector u(t) form factors. The numerical estimates were done for real parameterizations of these form factors using the vector-meson dominance approach. But the assumption that the form factors are real functions is not generally true because these form factors are complex functions in the time-like region of the momentum transfer squared, which is the case for the considered decay.
The author of Ref. [7] has studied the radiative decays t- —¥ i/T7r_7 and t- —¥ vTp~7, obtained analytic formulas for the differential decay rates, and evaluated them assuming that the form factors are constant. The authors of Ref. [8] have studied the decays t- —¥ vTn~(K~)~/. They obtained the photon energy spectrum, the meson photon invariant mass distribution, and the integrated rates. The inner bremsstrahlung contribution to the decay rate contains infrared divergences and that is why the integrated decay rates must depend on the photon energy cut-off (or the meson photon invariant mass cut-off). For the photon energy cut-off 100 MeV, the integrated decay-rates R = T(t i/T7T7)/r(r vttx) = 1.4 • lO"2 [6] and R = 1.0 • 10-2 [8] were obtained. We note that the leptonic radiative decay of the r lepton —¥ was measured with the branching ratio 3.6 • 10-3 [10]. Hence, we can expect that the one-pseudoscalar-meson radiative r-lopton decay r- —¥ vttt~7 can also be measured experimentally, because theory predicts the value for its branching ratio of the same order as for the r- —¥ irP^p-rj decay.
Some polarization observables in the decay r- —¥ —¥ i/T7T_7 were considered in Ref. [9]. The general expressions for the Stokes parameters of the produced photon have been calculated. The influence of the possible anomalous magnetic moment of the r lepton and the existence of excited neutrinos on the matrix element of this decay are briefly discussed. The authors showed that a measurement of the dependence of the differential decay rate on the photon energy (at a fixed sum of the photon and pion energies) allows determining the moduli and phases of the form factors as functions of the variable t.
The r-lopton radiative decays r- —¥ vTn~(K~)~/ were also studied in Refs. [11,12] in the case of unpo-larized particles. The light front quark model was used to evaluate the form factors v(t) and u(t) describing the structure-dependent contribution to these decays [11]. In the SM, the decay width was found to be
T = 1.62 • 10-2(3.86 • 10_3)F(r pit)
for the photon energy cut-off 50 (400) MeV. The same decays were studied in Ref. [12]. The photon energy spectrum, the pion photon invariant mass distribution, and the integrated decay rate were calculated without free parameters and the authors obtained the decay-width 1.46 • 10-2(2.7 • 10_3)r(r —¥ pit) for the same cut-off conditions.
The r-lopton decay in the case of a virtual photon that converted into a lepton antilepton pair was investigated in Refs. [13,14]. This decay was not measured up to now, but the cross channel (namely, tt+ —¥
e+i/ce+e~) has already been measured [15,16]. This decay and the decay r —¥ ttUvt probe the transition M"* —¥ 7T7*, where both bosons (W" and the photon) are virtual. These decays complement the decay we consider in this paper, which can be a source of information about the transition W"* —¥ n"f. The vector and axial-vector form factors are functions of two variables (instead of one as in our case) due to the virtuality of the photon, and a third form factor appears in this case. The authors of [13,14] calculated the branching ratios and di-lepton invariant mass spectra. They predicted that the process with / = should be measured soon at B-factorics.
Because the SuperKEKB and Super c r factories plan to have a longitudinally polarized electron beam with the polarization degree about 80%, it is worthwhile to investigate the effects caused by the polarization of the r lepton. In this paper, we investigate the polarization effects in the one-meson radiative decay of the r lepton, r~ The decay polarization
asymmetry and the Stokes parameters of the emitted photon itself are calculated for a polarized r lepton. These observables are estimated numerically for an arbitrary polarization of the r lepton.
The vector and axial-vector form factors (which are of theoretical and experimental interest), describing the structure-dependent part of the decay amplitude, are determined in the framework of the chiral effective theory with resonances (R;\T) [17, 18]. The R\T is an extension of the chiral perturbation theory to the region of energies around 1 GeV, which explicitly includes the meson resonances. The corresponding Lagrangian contains a few free parameters, or coupling constants, and at the same time has a good predictive power. This approach has further theoretical developments, e. g., in Refs. [19, 20], and applications to various aspects of the meson phenomenology (see, e.g., review [21]). Here, we mention earlier studies of the e+e~ annihilation to a pair of pseudoscalar mesons with final-state radia-
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tion [22], radiative decays with light scalar mesons [23], and two-photon form factors of the tt°, j/, and //' mesons and throo-pion of r- lopton [24].
This paper is organized as follows. In Sec. 2, the matrix element of the decay r~ yr7T~7 is considered, the phase-space factor of the final particles is introduced for an unpolarizod and polarized r lopton, and Stokes parameters and spin-correlation parameters of the photon are defined. This is done with the help of the current tensor T'11' and two unit space-like orthogonal 4-vectors that describe polarization states of the photon and which we express via particle 4-momenta. In Sec. 3, the current tensor is calculated in terms of the particle 4-momenta and the r lopton polarization 4-vector. In Sec. 4, we describe the chosen model for the vector and axial-vector form factors that enter the structural part of the decay amplitude. In Sec. 5, results of some analytic and numerical calculations are presented and illustrated. Section 6 contains a discussion and conclusion. In A
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