The anomalous four-point interaction in the radiative leptonic $\tau$ decay

TL;DR

This paper explores potential anomalous four-point interactions involving tau leptons, W bosons, neutrinos, and photons, extending the Standard Model and constraining their coupling constants through tau decay measurements.

Contribution

It introduces gauge-invariant dimension-five operators for scalar and tensor interactions and constrains their coupling constants using experimental decay data.

Findings

Scalar coupling: -4.9 to 9.4 at 95% CL

Tensor coupling: -1.4 to 2.8 at 95% CL

Provides bounds on anomalous interactions in tau decays

Abstract

As one of the extensions of the Standard Model, we investigate the anomalous four-point $\tau-W-\nu_\tau-\gamma$ scalar- and tensor-type interactions, which originate from the gauge invariant dimension-five operators. The coupling constants are constrained by the measured branching ratio of the $\tau^- \rightarrow \mu^- \nu_{\tau} \bar{\nu_\mu} \gamma$ decay: $-4.9<\kappa_\tau^S<9.4$ and $-1.4<\kappa_\tau^T<2.8$ at the 95% confidence level for the scalar and tensor interactions, respectively.

Equations

$${\left(\frac{c_{\mu }}{c_e}\right)}^2=\frac{\mathcal{B}({\tau }^{-}\to {\mu }^{-}{\nu }_{\tau }{\bar{\nu }}_{\mu })}{\mathcal{B}({\tau }^{-}\to e^{-}{\nu }_{\tau }{\bar{\nu }}_e)}\frac{f(m_e^2/m_{\tau }^2)}{f(m_{\mu }^2/m_{\tau }^2)},$$

$$\mathcal{L}\supset \frac{g}{\sqrt{2}}$$

$$|{\mathcal{M}}_{\mathrm{tot}}{|}^2$$

$$=|{\mathcal{M}}_{\mathrm{SM}}{|}^2+2\Re \{{\mathcal{M}}_{\mathrm{SM}}{\mathcal{M}}_N^{\ast }\}+|{\mathcal{M}}_N{|}^2.$$

$$\frac{\mathrm{d}{\Gamma }_N({\tau }^{-}\to {\ell }^{-}{\nu }_{\tau }{\bar{\nu }}_{\ell }\gamma )}{\mathrm{d}x\mathrm{d}y\mathrm{d}{\Omega }_{\ell }^{\ast }\mathrm{d}{\Omega }_{\gamma }^{\ast }}=\frac{4m_{\tau }^5{G}_F^2\alpha }{3(4\pi {)}^6}\frac{x{\beta }_{\ell }^{\ast }}{z}\left[{\kappa }_{\tau }^N{F}_1^N(x,y,z)+({\kappa }_{\tau }^N{)}^2{F}_2^N(x,y,z)\right],$$

$$\Gamma ({\tau }^{-}\to {\ell }^{-}{\nu }_{\tau }{\bar{\nu }}_{\ell }\gamma {)}_{E_{\gamma }^{\ast }>10\mathrm{MeV}}={\Gamma }_{E_{\gamma }^{\ast }>10\mathrm{MeV}}^{\mathrm{SM}}\left[1+c_{\ell }^N{\kappa }_{\tau }^N+d_{\ell }^N({\kappa }_{\tau }^N{)}^2\right],$$

$$-4.9<{\kappa }_{\tau }^S$$

$$-1.4<{\kappa }_{\tau }^T$$

$$F_1^S$$

$$\displaystyle\hskip 85.35826pt+(y-z)(x-z-2\lambda^{2})\big{\}}$$

$$+3y(z-2{\lambda }^2)(1+{\lambda }^2-x-y+z),$$

$$F_2^S$$

$$F_1^T$$

$$+6z^3{\lambda }^2\{-18y+18xy+18y^2+z(8-3x-37y)$$

$$+9z^2\}-18{\lambda }^{4}y,$$

$$F_2^T$$

Full text

\subjectindex

B40, B50

1 Introduction

The decays of $\tau$ lepton provide unique opportunities to search for the effects beyond the Standard Model (BSM) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The large mass of $\tau$ ($m_{\tau}=(1776.86\pm 0.12)$ MeV/$c^{2}$ [12]), in comparison with that of the electron or muon, allows one to expect an essential enhancement in the sensitivity to the effects of New Physics (NP) [13].

Of all tau decays, the leptonic ones are precisely calculated within electroweak sector of the SM, hence they offer a clean laboratory to search for the effects of NP.

Through the measurement of Michel parameters, $\rho$, $\eta$, $\xi$ and $\xi\delta$, in ordinary leptonic decays $\tau^{-}\to\ell^{-}\nu_{\tau}\bar{\nu}_{\ell}$ ($\ell=e,~{}\mu$), the experimental verification of the Lorentz structure of the charged weak interaction was carried out [12]. The most precision measurements were done by ALEPH [14] and CLEO [15] collaborations.

The measurement of the ratio of the branching fractions $\mathcal{Q}\equiv\mathcal{B}(\tau^{-}\rightarrow\mu^{-}\nu_{\tau}\bar{\nu}_{\mu})/\mathcal{B}(\tau^{-}\rightarrow e^{-}\nu_{\tau}\bar{\nu}_{e})$ is used to test the lepton universality:

[TABLE]

where $c_{\mu}$ and $c_{e}$ are the couplings of $\tau$ with $\mu$ and $e$, respectively, $m_{\ell}$ ($\ell=e,~{}\mu$) is outgoing lepton mass, and $f(m_{\ell}^{2}/m_{\tau}^{2})$ is a known function [16]. The most precise measurement at $BABAR$, $\mathcal{Q}=0.9796\pm 0.0016\pm 0.0036$ [17], is consistent with lepton universality.

Radiative leptonic decay $\tau^{-}\to\ell^{-}\nu_{\tau}\bar{\nu}_{\ell}\gamma$ (unless specified otherwise, charge-conjugated decays are implied throughout the paper) provides an additional promising tool to search for NP. Feynman diagrams of this decay in the SM are presented in Fig 1. The presence of the radiation exposes the internal structure of $\tau$ decays differently from the ordinary leptonic decays. For instance, the measurement of the spectra of outgoing lepton and photon allows one to access three more Michel parameters, $\bar{\eta}$, $\eta^{\prime\prime}$ and $\xi\kappa$ [19, 20, 21]. In this note we consider the anomalous four-point $\tau-W-\nu_{\tau}-\gamma$ scalar and tensor couplings.

2 Anomalous four-point scalar and tensor interactions

We suggest to consider anomalous four-point $\tau-W-\nu_{\tau}-\gamma$ scalar and tensor interactions (see Fig. 2), the modified Lagrangian of the charged weak interaction of $\tau$ is written as:

[TABLE]

where the first term is the SM Lagrangian, $\kappa_{\tau}^{S}$ and $\kappa_{\tau}^{T}$ characterize the magnitudes of the scalar and tensor interactions, respectively, $A_{\mu}$ is the electromagnetic field, $\sigma_{\mu\nu}=i[\gamma_{\mu},\gamma_{\nu}]/2$, $e$ is electron charge. The introduced terms appear from the gauge invariant dimension-five operators $\bar{\psi}(\nu_{\tau})D_{\mu}D^{\mu}\psi(\tau)$ and $\bar{\psi}(\nu_{\tau})\sigma_{\mu\nu}i[D^{\mu},D^{\nu}]\psi(\tau)$, where $D_{\mu}$ is the $\mathrm{U}(1)\otimes\mathrm{SU}(2)$ gauge covariant derivative, $[D^{\mu},D^{\nu}]\equiv D^{\mu}D^{\nu}-D^{\nu}D^{\mu}$.

The total matrix element squared of the $\tau^{-}\rightarrow\ell^{-}\nu_{\tau}\bar{\nu}_{\ell}\gamma$ decay in the presence of the anomalous amplitude $\mathcal{M}_{N}~{}({N}=S,~{}T)$ is written as:

[TABLE]

The contribution to the differential decay width from the $2\Re\{\mathcal{M}_{\mathrm{SM}}\mathcal{M}_{N}^{*}\}+|\mathcal{M}_{N}|^{2}$ is given by:

[TABLE]

where $G_{F}=1.1663787(6)\times 10^{-5}$ GeV*-2*[12] is the Fermi constant and $\alpha=1/137.035999139(31)$[12] is the fine-structure constant, $p_{\ell}=(E_{\ell},\vec{p}_{\ell})$ and $p_{\gamma}=(E_{\gamma},\vec{p}_{\gamma})$ are four-momenta of the outgoing charged lepton and photon, respectively; $\beta_{\ell}=|\vec{p}_{\ell}|/E_{\ell}$, $\Omega_{\ell}$ and $\Omega_{\gamma}$ are the solid angles of the final charged lepton and photon, respectively; $x=2E_{\ell}^{*}/m_{\tau}$, $y=2E_{\gamma}^{*}/m_{\tau}$, $z=2(p_{\gamma}\cdot p_{\ell})/m_{\tau}^{2}=xy(1-\beta_{\ell}^{*}\cos\theta_{\ell\gamma}^{*})/2$, $\cos\theta_{\ell\gamma}=(\vec{p}_{\ell}\cdot\vec{p}_{\gamma})/(|\vec{p}_{\ell}|\cdot|\vec{p}_{\gamma}|)$ (asterisks indicate parameters measured in the $\tau$ rest frame) and $\lambda=m_{\ell}/m_{\tau}$; $F_{1}^{N}$ and $F_{2}^{N}$ are the form factors (see Appendix for the explicit formulae). Integrating the differential decay width numerically, we obtain:

[TABLE]

where the coefficients $c_{\ell}^{N}$ and $d_{\ell}^{N}$ $(N=S$ or $T)$ are: $c_{e}^{S}=(-2.06\pm 0.01)\times 10^{-3}$, $c_{\mu}^{S}=(-8.46\pm 0.03)\times 10^{-3}$, $c_{e}^{T}=(-6.18\pm 0.02)\times 10^{-3}$, $c_{\mu}^{T}=(-3.19\pm 0.006)\times 10^{-2}$, $d_{e}^{S}=(3.77\pm 0.02)\times 10^{-4}$, $d_{\mu}^{S}=(1.85\pm 0.006)\times 10^{-3}$, $d_{e}^{T}=(4.51\pm 0.03)\times 10^{-3}$ and $d_{\mu}^{T}=(2.21\pm 0.006)\times 10^{-2}$, where the error is statistical uncertainty of numerical integration. In this calculation, we use the photon energy threshold of $10~{}\mathrm{MeV}$ in the $\tau$ rest frame. Taking into account the SM prediction $\mathcal{B}^{\rm Exc}(\tau^{-}\rightarrow\mu^{-}\nu_{\tau}\bar{\nu}_{\mu}\gamma)=(3.572\pm 0.007)\times 10^{-3}$ [22] and the PDG average $\mathcal{B}(\tau^{-}\rightarrow\mu^{-}\nu\bar{\nu}\gamma)=(3.68\pm 0.10)\times 10^{-3}$ [12], we get the constraint at 95% CL: $-0.025<(c^{N}_{\mu}\kappa^{N}_{\tau}+d^{N}_{\mu}(\kappa^{N}_{\tau})^{2})<0.085$, or:

[TABLE]

Moreover, the four-point $\tau-W-\nu_{\tau}-\gamma$ scalar and tensor interactions can be searched for at the electron-positron colliders, in the process of the $e^{+}e^{-}$ annihilation $e^{+}e^{-}\to\gamma^{*}\to\tau^{-}\bar{\nu}_{\tau}(W^{+}\to\ell^{+}\nu_{\ell},{\rm hadron(s)}^{+})$. This anomalous coupling is responsible for the production of the single $\tau$ lepton below $\tau^{+}\tau^{-}$ production threshold. Such a mechanism can be searched for at the low energy $e^{+}e^{-}$ colliders like Beijing Electron-Positron Collider (BEPC), Cornell Electron Storage Ring (CESR) and VEPP-2000 [23, 24], as well as at the B-factories, Belle [25]/KEKB [26] and $BABAR$ [28]/PEP-II, in the processes of $e^{+}e^{-}$ annihilation with the initial state radiation [27]. For example, at VEPP-2000 in the center-of-mass energy range from about $1.8$ GeV up to $2.0$ GeV (near the production threshold of single tau), $\tau^{-}$ lepton is produced almost at rest, accompanied with $e^{+}\nu_{e}\bar{\nu}_{\tau}$, $\mu^{+}\nu_{\mu}\bar{\nu}_{\tau}$ or $\pi^{+}\bar{\nu}_{\tau}$. As a result, besides $e-\mu$ events, a clear signature of the production of single $\tau^{-}$ will be the monochromatic $\pi^{-}$ or $\rho^{-}$ (from $\tau^{-}\to\pi^{-}\nu_{\tau}$ or $\tau^{-}\to\rho^{-}\nu_{\tau}$ decays).

3 Summary

The precision measurement of the properties of leptonic decays of $\tau$ lepton offers unique opportunity to search for the physics beyond the Standard Model. The radiative leptonic decay $\tau^{-}\rightarrow\ell^{-}\nu_{\tau}\bar{\nu}_{\ell}\gamma$ provides an additional tool to probe the internal structure of the weak interaction. The anomalous four-point $\tau-W-\nu_{\tau}-\gamma$ scalar and tensor interactions are simple extensions of the Standard Model, which affect the spectra of the daughter particles in the radiative leptonic decays of tau. We calculated the corresponding differential and the total decay widths. The world average value of the branching ratio of the $\tau^{-}\rightarrow\mu^{-}\nu_{\tau}\bar{\nu}_{\mu}\gamma$ decay and its SM prediction constrain the magnitudes of the scalar and tensor couplings to be $-4.9<\kappa_{\tau}^{S}<9.4$ and $-1.4<\kappa_{\tau}^{T}<2.8$ (at 95% CL), respectively. In this note, we extract the constraints on the $\kappa_{\tau}^{S}$ and $\kappa_{\tau}^{T}$ coupling constants from the total widths, but the more sensitive method is to fit the full differential decay width of the radiative leptonic decay of tau.

Acknowledgments

We are grateful to Andrey Pomeransky (Budker Institute of Nuclear Physics) for the fruitful discussions.

Appendix A Form factors

[TABLE]

[TABLE]