$\tau^{-}\to\ell_{i}^{-}\ell_{i}^{+}\ell_{j}^{-}\bar{\nu}_{j}\nu_{\tau}$ decays with a magnetic dipole term

We found that the SM five body decays $\tau^{-}\to\ell_{i}^{-}\ell_{i}^{+}\ell_{j}^{-}\bar{\nu}_{j}\nu_{\tau}$ whose branching ratios, at the tree level, have been calculated by Dicus and Vega Dicus , Volobouev of the CLEO Collaboration CLEO and more recently by López Castro et. al. Roig . However Dicus and Roig differ from the predictions of CLEO whose various branching ratios are $\sim 7\%$ higher. The CLEO II experiment has searched for the $\tau^{-}\to e^{-}e^{+}e^{-}\bar{{\nu}_{e}}\nu_{\tau}$ decay, whose measured value is PDG :

The statistical, systematic and background uncertainties are $\sim 70\%$ but forthcoming BELLE measurements are expected to improve to $\sim 2.7\%$ and $\sim 6.5\%$ on the statistical and systematic uncertainties junya , respectively. The importance of our independent calculation is that it is useful to overcome this discrepancy. Furthermore, we analyze the possibility of limiting the $a_{\tau}$ value by using current and future experimental measurements by BELLE or other future collaborations.

Leptons offer some of the cleanest signals that can be obtained in collider physics experiments and might also provide new insight into physics beyond the standard model (SM). In particular, with the yet unsolved origin of the discrepancy between the experimental and theoretical SM prediction of the muon anomalous magnetic dipole moment $a_{\mu}$. Although much work has already been devoted to $a_{\mu}$, the $a_{\tau}$ has recently become the source of theoretical and experimental interest, which is why in the present work we suggest a tau decay as a means to obtain further insight into $a_{\tau}$. The current lower and upper bounds on $a_{\tau}$, $-0.052\leq a_{\tau}\leq 0.013$ with 95$\%$ C.L. DELPHI , were obtained via the process $e^{+}e^{-}\to e^{+}e^{-}\tau^{+}\tau^{-}$ by the DELPHI collaboration. These bounds differ from the theoretical value predicted by the SM by one order of magnitude: $a_{\tau}^{\rm Theor.}=1177.21(5)\times 10^{-6}$ SMMomMagTAU . Measurements of the electron and muon Anomalous Magnetic Dipole Moments (AMDM) were obtained by means of spin precession experiments, however, in the case of the tau lepton this class of measurements are troublesome due to its short lifetime, $(290.3\pm 0.5)\times 10^{-15}s$. Even though there is a plethora of tau decays, only a few of them are viable candidates to constrain $a_{\tau}$. Whereas two- and three-body decays do not involve the $\gamma\tau^{-}\tau^{+}$ coupling, the $\tau^{\pm}\to\ell^{\pm}\bar{\nu_{\ell}}\nu_{\tau}\gamma$ decay, as suggested in Ref. tauTOfourbodies , is constrained by the tau lifetime and is only sensitive to large values of $a_{\tau}$. The use of tau decays to constrain $a_{\tau}$ is particularly relevant because the Belle-II experiment is expected to produce about $10^{9}$ tau leptons annually, which greatly exceeds the previous CLEO-II experiment, in which there were $3\times 10^{6}$ produced tau leptons.

The dominant Feynman diagrams for this decay within the SM are shown in Fig. 1, where the dot represents the QED contribution along with an extra contribution from the tau AMDM. All other tree-level diagrams give a negligible contribution.

In this work we will consider this branching ratio to obtain a bound on $a_{\tau}$. To this aim we will consider the following effective vertex of the photon to a charged lepton pair respecting Lorentz invariance:

where $q=p_{1}-p_{2}$ is the photon transferred four-momentum. Here ${F_{V}^{\gamma}}$ is the tau electric charge form factor (${F_{V}^{\gamma}}=1$ at the tree level), whereas the five-dimensional $CP$-conserving and $CP$-violating terms correspond to the static anomalous magnetic dipole moment $a_{\ell}$ and the electric dipole moment $d_{\ell}$, which are given by:

We will assume $CP$ invariance and take $F_{E}^{\gamma}$=0.

The rest of this presentation is organized as follows. In Section II we will present the unpolarized square amplitude for the $\tau^{-}\to e^{-}e^{+}e^{-}\bar{\nu}_{e}\nu_{\tau}$ decay via the massive helicity formalism dittmaier , which considerably simplifies the calculation. The numerical integration of the $\tau^{-}\to e^{-}e^{+}e^{-}\bar{\nu}_{e}\nu_{\tau}$ decay width and the resulting bound on the $a_{\tau}$ are presented in Sec. III, whereas Sec. IV is devoted to the conclusions and outlook. Details of the calculation are presented in Appendix A.

We will calculate the average square amplitude of the $\tau^{-}\to\ell_{i}^{-}\ell_{i}^{+}\ell_{j}^{-}\bar{\nu}_{j}\nu_{\tau}$ decay using the massive helicity formalism for which usual treatments, such as dixon or elvang , only deal with the massless case. Here we need to take into account the mass of the tau lepton, so must go further. In particular, in dittmaier the two main ways to deal with massive helicity amplitudes are presented, although with a somewhat old-fashioned notation. We use here the approach that consists in performing a light-cone decomposition to write the four-momentum of a massive particle as a linear combination of two light-like momenta. For a detail account we refer the interested reader to Ref. proceedings , where this formalism is presented in a self-contained manner (with opposite metric signature convention). We will present a brief outline here for convenience.

Let $p$ be the four-momentum of a massive particle of mass $m>0$, which can always be written in terms of two light-like ones $q^{\mu}$ and $r^{\mu}$ as

where $q$ satisfies $p\cdot q\neq 0$ but is otherwise arbitrary, and $r$ is defined through (5). The positive frequency momentum space Dirac equation has two linearly independent solutions, which we label with the subindices $+$ and $-$. It is easily checked that they are given (maybe up to a phase) by

while for the negative frequency we have

Here $|a]$ and $|b\rangle$ are 2-component Weyl spinors linked to the light-like momenta $a_{\mu}$ and $b_{\mu}$ proceedings . We will use these solutions to calculate the helicity amplitudes $\mathcal{M}_{a}^{h_{1}h_{2}h_{3}h_{4}h_{5}h_{6}}$ for the $\tau^{-}(p_{1})\to\ell_{i}^{-}(p_{2})\ell_{i}^{+}(p_{3})\ell_{j}^{-}(p_{4})\bar{\nu}_{j}(p_{5})\nu_{\tau}(p_{6})$ decay, where $h_{k}$ is the helicity of the particle $k$ with four-momentum $p_{k}$, and the subindex $a$ labels each Feynman diagram of Fig. 1: it runs from 1 to 4 in the case that the charged leptons $\ell_{i}^{-}$ and $\ell_{j}^{-}$ are identical, such as in the $\tau^{-}\to e^{-}e^{+}e^{-}\bar{\nu}_{e}\nu_{\tau}$ decay, but it only runs from from 1 to 2 if $\ell_{i}^{-}$ and $\ell_{j}^{-}$ are distinguishable.

To determine the individual non-zero helicity amplitudes we will neglect the mass of the outgoing charged leptons, which is a good approximation as their mass is negligible as compared to the tau mass. In this limit, each amplitude will vanish unless the helicities of particles 4, 5 and 6 have certain fixed signs because of its spinor index structure. After doing that, we present the sum of the squared modulus of the helicity amplitudes and then all of the interferences.

In the unitary gauge, the amplitude for the first Feynman diagram of Fig. 1, dubbed (1), is

where we use the short-hand notation $(i)\equiv(p_{i})$ and the four-momenta $k_{i}$ are the ones of the virtual particles of this diagram. Note that we are using the effective interaction (2) for the $\gamma\bar{\tau}\tau$ vertex but in our calculation below we will use the tree-level value $F_{V}^{\gamma}=1$. In addition

with $p_{ij}=p_{i}\cdot p_{j}$. The helicity structure of the amplitude fixes $h_{4}=-$, $h_{5}=+$, and $h_{6}=-$. We then write

We now choose $q_{1}\equiv p_{3}$ and define

thus the individual helicity amplitudes are given by

and

To obtain the helicity amplitudes of the the third Feynman diagram, obtained from the first one after the exchange of identical particles, we simply exchange the momenta and helicities of particles 2 and 4, both in the helicity amplitudes and in the definitions of $D_{1}$ and $G_{1}$, leading to new coefficients that we denote as $D_{3}$ and $G_{3}$, respectively.

By an analogous procedure, we obtain for the Feynman diagram (2) of Fig. 1

where

We straightforwardly obtain the corresponding amplitudes for the fourth diagram after the exchange of the momenta and helicities of particles 2 and 4.

We now factor all the dependence on the form factors ${F_{V}^{\gamma}}$ and ${F_{M}^{\gamma}}$ by defining the following coefficients:

from which we obtain the sum of the squared helicity amplitudes of the first diagram:

where $p_{i_{1}i_{2}\cdots i_{N}}$ is defined in Appendix A. For the second diagram we get

The corresponding expression for the fourth diagram is attained by exchanging $p_{2}$ and $p_{4}$.

On the other hand, the non-zero interferences are given by

and

where $\mathcal{I}_{ij}$ stands for the interference between the amplitudes of diagrams $i$ and $j$. Explicit expressions for the $\left(\mathcal{P},\mathcal{Q},\mathcal{R}\right)^{(p)}_{ij}$ functions are given in Appendix A.

The full unpolarized squared amplitude for the $\tau^{-}\to e^{-}e^{+}e^{-}\bar{\nu}_{e}\nu_{\tau}$ decay is given by

It is worth noting that this method is straightforward and yields compact results easy to handle in the numerical integration.

We now turn to compute the branching ratio of the tau five-body decay $\tau^{-}\to e^{-}e^{+}e^{-}\bar{\nu}_{e}\nu_{\tau}$. The decay width is given by the usual formula

After dividing by the tau total width $\Gamma_{\tau}=1/\tau_{\tau}$ we obtain the corresponding branching ratio

According to Ref.EspFase the four-momenta of the involved particles in a five-body decay can be related to eight independent Lorentz invariant parameters through the relations:

The phase-space integral (32) was numerically computed over these kinematic variables via Monte-Carlo integration by using the VEGAS routines Lepage . A cross-check of our results was done by implementing the electromagnetic vertex in the CalcHEP package Belyaev:2012qa , which performs all the numerical calculation.

We first consider that the impact of the magnetic form factor is negligible, i.e. we use $F_{M}^{\gamma}=0$ and make a comparison of our numerical results for the widths of the allowed $\tau^{-}\to\ell_{i}^{-}\ell_{j}^{+}\ell_{i}^{-}\bar{{\nu}_{\ell}}_{j}\nu_{\tau}$ decays with those obtained in previous studies. The results are shown in Table 1. The uncertainties arise from the numerical integration. Our results are in good agreement with these predictions, though are closer to those of Ref. Roig , which could be attributed to the fact that we used the same values of the tau mass and mean lifetime. Additionally, for completeness, we computed the $\mu^{-}\to e^{-}e^{+}e^{-}\bar{\nu_{e}}\nu_{\mu}$ for which we obtain $BR(\mu^{-}\to e^{-}e^{+}e^{-}\bar{\nu_{e}}\nu_{\mu})=(3.599\pm 0.002)\times 10^{-5}$.

In this work we present the results of a numerical calculation for the predictions of the branching ratio of the allowed $\tau^{-}\to\ell_{i}^{-}\ell_{i}^{+}\ell_{j}^{-}\bar{\nu}_{j}\nu_{\tau}$ decays, which are consistent with previous results reported by Dicus and Vega and López Castro et. al. In addition, we use the current experimental measurement on the branching ratio of the $\tau^{-}\to e^{-}e^{+}e^{-}\bar{{\nu}_{e}}\nu_{\tau}$ decay to obtain an upper limit on the tau anomalous magnetic dipole moment by using an effective electromagnetic vertex including a dipole term. We find that the effect of the magnetic dipole form factor $F_{M}^{\gamma}$ on the reported value by CLEO II colaboration for the branching ratio of the $\tau^{-}\to e^{-}e^{+}e^{-}\bar{{\nu}_{e}}\nu_{\tau}$ decay allows to extract an upper bound of $a_{\tau}\leq 0.0056$, which is below the current upper bound by the DELPHI collaboration. With the next measurements by BELLE II results are expected with $\sim 2.7\%$ and $\sim 6.5\%$ of statistical and systematic uncertainties. Under this scenario, we find that the effect of $F_{M}^{\gamma}$ on the $BR(\tau^{-}\to e^{+}e^{+}e^{-}\bar{\nu}_{e}\nu_{\tau})$ is such that the lower bound is given by $-0.02\leq a_{\tau}$ but it is not possible to extract a good upper bound. Finally, assuming an accurate measurement of the $2\%$ of statistical and systematic uncertainties, it is found that our best bounds are $-0.0032\leq a_{\tau}\leq 0.0061$.