-
The first interesting application is about the polarization amplitudes to
$ \tau $ leptonic decays[4]. In this work we extend the formalism to a case, with the operator$ \gamma^\mu -\alpha\gamma^\mu \gamma_5 $ , that can account for different models BSM. We study the$ \tau^- \!\to\! \nu_{\tau} K^{*0} K^{-} $ reaction for the different polarizations of the$ K^{*0} $ [4].The final transition amplitudes for the
$J \!=\! 1,\; J'\! =\! 0$ (vector- pseudoscalar) are obtained by1)
$M \!= 0 \!$ 2) for
$\!M = 1 \!$ 3) for
$ M = -1 $ where
$ p $ is the momentum of the$ \tau $ in the$ M_1 M_2 $ (with$ M1, M2 $ pseudoscalar or vector mesons) rest frame given bywith
$ \lambda(x,y,z) = x^2+y^2+z^2-2xy-2xz-2yz $ .Finally the differential width is given
where
$ p_\nu $ is the neutrino momentum in the$ \tau $ rest frame and$ {\widetilde p_1} $ the momentum of$ K^{*0} $ in the$ K^{*0}K^- $ rest frame.The total differential width is
We find that the polarization amplitudes vary very much with different values of
$ \alpha $ . In particular we find a magnitude$\frac{1}{R} \big[\frac{{\rm{d}} \varGamma}{{\rm{d}} M_{\rm{inv}}^{(K^{*0} K^-)}}\big|_{M = +1}- \frac{{\rm{d}} \varGamma}{{\rm{d}} M_{\rm{inv}}^{(K^{*0} K^-)}}\big|_{M = -1} \big]$ , which is extremely sensitive to$ \alpha $ . The results are shown in Fig. 2, it is shown that this magnitude is very sensitive to the$ \alpha $ parameter and even changes sign for some value of$ \alpha $ . It is obviously to be most suited to investigate possible physics BSM. -
Now firstly we open up a new direction to test the nature of scalar resonances
$ f_0(980) $ and$ a_0(980) $ in the$ \tau $ decays. In the present work the$ \tau^- \to \nu_\tau \pi^- f_0(980) $ and$ \tau^- \to \nu_\tau \pi^- a_0(980) $ reactions are investigated[7].In Fig. 3 we show the triangle mechanism to produce the
$ f_0(980) $ state via$ \tau \to \nu_\tau K^{*0} K^- $ ($ K^0 K^{*-} $ ) followed by$ \bar{K}^{*} \to \pi^- K $ and the posterior fusion of$ K\bar{K} $ . Similarly, the process producing the$ a_0(980) $ state is depicted in Fig. 4.Figure 3. Diagram for the decay of
$\tau^- \to \nu_\tau \pi^- \pi^+ \pi^- $ . The double line, labeled R, indicating the$K\bar K\to\pi^+\pi^-$ scattering amplitudes. The brackets in figure (a) indicate the momenta of the particles.We obtain that the
$ M_0 $ terms cancel for the production of$ f_0(980) $ and add for the production of$ a_0(980) $ . This is, the$ f_0(980) $ production proceeds via the$ N_i $ term and the$ a_0(980) $ production via the$ M_0 $ term. Since$ \pi^- f_0(980) $ has negative$ G $ -parity and$ \pi^- a_0(980) $ positive$ G $ -parity, we confirm that the$ M_0 $ term in the loop corresponds to positive$ G $ -parity and the$ N_i $ term to negative$ G $ -parity[3].For the production of
$ \pi^- f_0(980) $ Similarly, for the production of
$ \pi^- a_0(980) $ For
$ \tau^- \to \nu_\tau \pi^- \pi^+ \pi^- $ decay, the double differential mass distribution for$ M_{\rm{inv}}(\pi^+ \pi^-) $ and$ M_{\rm{inv}}(\pi^- f_0) $ is given bywith
Similarly, for the
$ \tau^- \to \nu_\tau \pi^- \pi^0 \eta $ decay, we can get the double differential mass distribution for$ M_{\rm{inv}}(\pi^0 \eta) $ and$ M_{\rm{inv}}(\pi^- a_0) $ .In the calculation, the factor
$ \frac{{\cal{C}}^2}{\varGamma_{\tau}} = (2.10\pm0.40)\times 10^{-5}\,{\rm{MeV}}^{-3} $ is obtained by the experimental branching ratio of$ {\cal{B}} (\tau \to \nu_\tau K^{*0} K^-) $ decay and thus we can provide absolute values for the mass distributions.The loop function in Fig. 3 (a) is given by
with
$ P^0 = M_{\rm{inv}}(\pi^- f_0) $ ,$ \omega_{K^-} = \sqrt{{{q}}^2+m_K^2} $ ,$ \omega_{K^+} = \sqrt{({{{q}}}+{{k}})^2+m_{K}^2} $ , and$ \omega_{K^*} = \sqrt{{{q}}^2+m_{K^\ast}^2} $ ,Similarly, we can get the triangle amplitude for the
$ \pi^- a_0 $ case. Note that an$ i \epsilon $ in the propagators involving$ \omega_{K^{*}} $ is replaced by$ i\frac{\varGamma_{K^{*}}}{2} $ .In Fig. 5 we show the contribution of the triangle loop, the real, imaginary and the absolute value parts of
$ t_L $ as a function of$ M_{\rm{inv}}(\pi^- R) $ , with$ M_{\rm{inv}}(R) $ fixed at 985 MeV ($ R $ standing for$ f_0(980) $ or$ a_0(980) $ ). It can be observed that$ {\rm{Re}}(t_T) $ has a peak around 1 393 MeV, and$ {\rm{Im}}(t_T) $ has a peak around 1 454 MeV, and there is a peak for$ |t_T| $ around 1 425 MeV. The peak of the real part is related to the$ K^* K $ threshold and the one of the imaginary part, that dominates for the larger$ \pi^- R $ invariant masses, to the triangle singularity. This triangle mechanism has the same origin as the explanations for the COMPASS peak in$ \pi f_0(980) $ that was initially presented as the new resonance “$ a_1(1420) $ ”[10].Figure 5. (color online)Triangle amplitude
${\rm{Re}} (t^{}_L)$ ,${\rm{Im}}(t^{}_L)$ and$|t^{}_L|$ , taking$M_{\rm{inv}}(R)$ = 985 MeV.In order to give a branching ratio for what an experimentalist would brand as
$ \pi^- f_0(980) $ or$ \pi^- a_0(980) $ decay, we integrate the strength of the double differential width and obtain$\frac{1}{\varGamma_{\tau}}\frac{{\rm d} \varGamma}{{\rm d} M_{\rm{inv}}(\pi^- R)}$ which is shown in Fig. 6. We see a clear peak of the distribution around$ 1\,423 $ MeV for$ \pi^- f_0 (980) $ production and$ 1\,412 $ MeV for$ \pi^- a_0 (980) $ production.Integrating
$ \frac{{\rm{d}} \varGamma}{{\rm{d}} M_{\rm{inv}}(\pi^- R)} $ over$ M_{\rm{inv}}(\pi^- R) $ , we obtain the branching fractionsThese numbers are within measurable range, since branching ratios of
$ 10^{-5} $ are quoted in the PDG. -
Now in the
$ \tau $ decays we firstly test the nature of axial-vector resonances. The decays of$ \tau \to \nu_\tau \pi A $ reactions, with$ A $ an axial-vector resonance, are investigated[9].The process proceeds through a triangle mechanism where a vector meson pair is first produced from the weak current and then one of the vectors produces two pseudoscalars, one of which reinteracts with the other vector to produce the axial resonance. The
$ G $ -parity is important to explicitly filter different states, and finally we obtain analytic formulas as follow:a)
$ G $ -parity positive axial states:b)
$ G $ -parity negative axial states:The differential mass distribution for
$ M_{\rm{inv}}(\pi^- A) $ is given bywith
and
$ p $ is the momentum of the$ \tau $ in the$ \pi^- A $ rest frame.We make predictions for invariant mass distributions. In Fig. 7 we show the mass distribution for
$ \tau^- \to \nu_\tau \pi^- b_1(1235) $ decay, with$ G $ -parity positive$ b_1(1235) $ state.In Fig. 8 we also show the mass distribution for
$ \tau^- \to \nu_\tau \pi^- a_1(1260) $ decay, with$ G $ -parity negative$ a_1(1260) $ state.We also give the decay of
$ \tau \to \nu_\tau K K_1(1270) $ . The axial-vector resonance$ K_1(1270) $ corresponds two states, one of them at$ 1195 $ MeV coupling mostly to$ K^* \pi $ , and another one at$ 1284 $ MeV coupling mostly to$ \rho K $ . Proceeding analogously to the previous cases, we obtain:a)
$ K_{1} (1) $ state:b)
$ K_{1} (2) $ state:Then in Figs. 9 and 10 and we get the differential mass distributions for
$ M_{\rm{inv}}(K^- K_1(1270)) $ and$ M_{\rm{inv}}(K^- K_2(1270)) $ cases, respectively,From the experimental branching ratio
$ {\cal{B}} (\tau \to \nu_\tau K^{*0} K^{*-}) = (2.1 \pm 0.5)\times 10^{-3} $ , we obtain$ \frac{{\cal{C}}^2}{\varGamma_\tau} = 5.0 \times 10^{-4} \; {\rm{MeV}}^{-1} $ .Then we integrate the differential mass distributions for different
$ \tau\to \nu_\tau P A $ decay channels to obtain the branching ratios, which are listed in Table 1.The results show that these numbers are within measurable range. It would be very interesting to measure these branching ratios in the possible future STCF large research facility.
${\cal{B}}$ $h_1(1170)$ $ 3.1 \times 10^{-3}$ $a_1(1260)$ $ 1.3 \times 10^{-3}$ $b_1(1235)$ $ 2.4 \times 10^{-4}$ $f_1(1285)$ $ 2.4 \times 10^{-4}$ $h_1(1380)$ $ 3.8 \times 10^{-5}$ $K_1 (1)$ $ 2.1 \times 10^{-5}$ $K_1 (2)$ $ 4.1 \times 10^{-6}$ Table 1. The branching ratios for
$\tau \to P A$ decays
τ Lepton Decays and Applications
doi: 10.11804/NuclPhysRev.37.2019CNPC24
- Received Date: 2019-12-29
- Rev Recd Date: 2020-04-02
- Available Online: 2020-09-30
- Publish Date: 2020-09-20
-
Key words:
- τ decay /
- angular momentum algebra /
- polarization amplitude /
- G-parity /
- resonance state
Abstract: A novel algebraic approach recently proposed is presented in this paper for investigating the
Citation: | Lianrong DAI. τ Lepton Decays and Applications[J]. Nuclear Physics Review, 2020, 37(3): 698-704. doi: 10.11804/NuclPhysRev.37.2019CNPC24 |