$H\rightarrow\tau^{+}\tau^{-}\gamma$ as a Probe of the $\tau$ Magnetic Dipole Moment

The discovery of the Higgs boson by the LHC Aad:2012tfa ; Chatrchyan:2012xdj in the $\sqrt{s}=7$ and $8~\rm{TeV}$ datasets represents a key milestone in the ongoing study of the Standard Model (SM). The fact that the observed boson has features which, at least in the broad brush, match with the SM expectations confirms that it is the principle agent of electroweak symmetry breaking. A major focus of the LHC run II is to establish its properties to high precision to either confirm the SM vision for electroweak breaking or to find the influence of new physics Morrissey:2009tf ; Chang:2008cw ; Curtin:2013fra ; Belusca-Maito:2016axk .

As the agent of the electroweak breaking, the observed Higgs boson is connected to any process for which the chirality of a SM fermion must flip. To maintain $SU(2)\times U(1)$ gauge invariance, such processes must contain an insertion of an electro-weak breaking vacuum expectation value (VEV), and the Higgs boson, as the fluctuations around the electroweak vacuum, has interactions connected with all such terms. A particularly interesting class of such observables are the electroweak dipole moments of the SM fermions, dimension five operators which can be sensitive probes of physics beyond the Standard Model. In fact, a long-standing mystery surrounds the magnetic dipole moment of the muon, whose experimental determination defies the best available theoretical predictions of the SM at more than two sigma Bennett:2006fi .

The anomalous magnetic dipole moment of a fermion $\psi$ is usually written in terms of the mass $m_{\psi}$ and the electromagnetic coupling $e$ as:

$$a_{\psi}~\frac{e}{2m_{\psi}}~\bar{\psi}\sigma^{\mu\nu}\psi~F_{\mu\nu}$$

(1)

where $F_{\mu\nu}$ is the electromagnetic field strength tensor and the real quantity $a_{\psi}$ parameterizes the size of the anomalous magnetic moment. The chiral structure of $\sigma^{\mu\nu}$ demands that one of $\psi$ or $\bar{\psi}$ be left-chiral (and thus part of an $SU(2)$ doublet), and the other right-chiral (and thus an $SU(2)$ singlet). In the UV theory, it descends from a pair of dimension six operators combining $\bar{\psi}_{L}\sigma^{\mu\nu}\psi_{R}$ with an additional Higgs doublet (and field strengths for the $U(1)$ and $SU(2)$ gauge bosons). This obscured dependence on electroweak symmetry breaking is the origin of the well known fact that, in a theory where the SM Yukawa interactions account for all of the chiral symmetry breaking, the anomalous magnetic moment is proportional to $m_{\psi}$ itself. It also implies that $a_{\psi}$ maps uniquely (in a theory with a single source of electroweak breaking) to an interaction between the Higgs boson, $\psi\bar{\psi}$, and a photon.

In this article, we focus on the magnetic dipole moment of the $\tau$ lepton. As the heaviest of the SM leptons¹¹1Quark dipole moments are more subtle, manifesting as properties of the hadrons into which they are bound., the $\tau$ is a natural place to search for new physics, and the fact that neutrino masses imply some kind of physics beyond the Standard Model may be a further indication that the lepton sector is a good place to look for its influence. Indeed, if the anomalous magnetic moment of the muon is indeed a manifestation of such physics, one could hope that even larger relative deviations could be present in the $\tau$ sector. At the same time, measurements of the $\tau$ dipole moment are currently relatively mildly constrained, leaving room for large deviations.

Higgs decays thus furnish an opportunity to study the $\tau$ magnetic dipole, through the rare decay,

$$h\to\tau^{+}\tau^{-}\gamma$$

(2)

which we find is potentially observable during high luminosity running of the LHC. As explained above, this process is related through gauge invariance to the more traditional probes of the $\tau$’s electroweak interactions via precision measurements of its production and/or decay Silverman:1982ft ; Almeida:1991hq ; delAguila:1991rm ; Samuel:1992fm ; Aeppli:1992tp ; Escribano:1993pq ; Escribano:1996wp ; GonzalezSprinberg:2000mk ; Bernabeu:2007rr ; Bernabeu:2008ii ; Atag:2010ja ; Peressutti:2012zz ; Hayreter:2013vna ; Fael:2013ij ; Hayreter:2015cia ; Eidelman:2016aih , and we will see that Higgs decays provide both an opportunity to discover physics beyond the Standard model, or to provide some of the best constraints on an anomalous contribution to the $\tau$ magnetic moment.

This work is structured as follows. In Section II we discuss the operators which parameterize new physics contributions to the $\tau$ magnetic moment, and review the existing constraints. In Section III, we outline the search strategy and discuss backgrounds, and in Section IV we discuss the potential for discovery or limits in the case no excess of signal events is found. We conclude in Section V.

Anomalous contributions to the electroweak dipole moments of the tau (at low-energy scales) originate from dimension six operators:

$$c_{1}~\bar{\tau}_{R}\sigma^{\mu\nu}B_{\mu\nu}~H^{\dagger}L_{3}+c_{2}~\bar{\tau}_{R}\sigma^{\mu\nu}~H^{\dagger}W_{\mu\nu}L_{3}+h.c.$$

(3)

where $L_{3}$ is the third family left-handed $SU(2)$ doublet, $H$ is the Higgs doublet, $B_{\mu\nu}$ and $W_{\mu\nu}$ are the field strengths for the hypercharge and $SU(2)$ gauge bosons, and $c_{1}$ and $c_{2}$ are complex numbers with units of $(\rm{energy})^{-2}$. After electroweak symmetry-breaking, linear combinations of these operators lead to anomalous magnetic and electric dipole interactions, and analogous terms involving the $Z$ boson.

Currently, the most stringent constraints on the $\tau$ dipole operators are from LEP2, where the DELPHI collaboration searched for $e^{+}e^{-}\to e^{+}e^{-}\tau^{+}\tau^{-}$ events, at various collision energies between $183~\rm{GeV}$ and $208~\rm{GeV}$ with a dataset corresponding to an integrated luminosity of $650~\rm{pb}^{-1}$ Abdallah:2003xd . The results were found to be consistent with the SM expectations, leading to the constraint

$$-0.052<a^{\gamma}_{\tau}<0.013,\quad 95\%~{\rm CL}.$$

(4)

Similarly, the ALEPH collaboration searched for $e^{+}e^{-}\to\tau^{+}\tau^{-}$ events on the $Z$-pole with a dataset corresponding to an integrated luminosity of $155~\rm{pb}^{-1}$ Heister:2002ik , and obtained the limit

$$a^{Z}_{\tau}<1.14\times 10^{-3},\quad 95\%~{\rm CL}.$$

(5)

Since the $Z$ magnetic dipole interaction is much more severely constrained, we choose to focus on $a_{\tau}^{\gamma}$ from here on.

The combination related to the magnetic moment is described by,

$${1\over\Lambda^{2}}\bigg{\{}v~\bar{\tau}\sigma^{\mu\nu}~\tau F_{\mu\nu}+h~\bar{\tau}\sigma^{\mu\nu}~\tau F_{\mu\nu}\bigg{\}}$$

(6)

where $h$ is the Higgs boson and $\Lambda^{2}$ is a real parameter with dimensions of $(\rm{energy})^{2}$. The anomalous magnetic moment $a_{\tau}^{\gamma}$ is related to $\Lambda$ and the Higgs VEV $v$ via:

$\displaystyle a^{\gamma}_{\tau}$

$\displaystyle=$

$\displaystyle-\frac{4m_{\tau}}{e}~{v~\over\Lambda^{2}}.$

(7)

In terms of $\Lambda$, the LEP bound (4) implies

$$|\Lambda|>\begin{array}[]{l}333~\rm{GeV}\\ 666~\rm{GeV}\end{array}~~~~~~~~~{\rm for:}~~~\begin{array}[]{r}~\Lambda^{2}>0\\ \Lambda^{2}<0\end{array}.$$

(8)

The second term of Eq. (6) leads to the rare Higgs decay $h\to\tau^{+}\tau^{-}\gamma$. Measurements of this decay therefore translate into measurements of the dipole moment. In fact, the SM contribution to this decay mode has the same chirality structure as the dipole operator, allowing for constructive interference, and resulting in a relative enhancement of the new physics contribution compared to the direct search by LEP.

It is worth mentioning that the imaginary parts of $c_{1}$ and $c_{2}$ would also lead to a CP-violating electric dipole moment (and its $Z$ analogue) for the $\tau$. While interesting in their own right Zhang:2002zw ; Bower:2002zx ; Desch:2003mw ; Harnik:2013aja ; Berge:2008wi ; Berge:2008dr ; Berge:2011ij ; Berge:2013jra ; Hagiwara:2016zqz , CP symmetry prevents these new physics amplitudes from interfering with the SM contribution to $h\to\tau^{+}\tau^{-}\gamma$, thus leading to greatly decreased LHC sensitivity (see also Chen:2014ona ). For this reason, we choose here to focus on the magnetic dipole moment for which Higgs decays are a more sensitive probe.

In this section we estimate the LHC sensitivity reach to the $\tau$ magnetic dipole moment through the decay $h\to\tau^{+}\tau^{-}\gamma$. This is a challenging signal to reconstruct at a hadron collider for a number of reasons. First, $\tau$ leptons decay promptly, producing missing momentum along with either a charged lepton or a handful of hadrons. Events containing more than one of them can at best be incompletely reconstructed. Decays involving neutral pions also contain energetic photons, which can potentially fake the additional $\gamma$ which distinguishes our rare decay mode from background associated with $h\to\tau^{+}\tau^{-}$. Photons themselves receive non-perturbative contributions to their production from QCD, which are not very well understood. Minimizing these uncontrolled contributions to photon production typically requires tight isolation cuts, which can be inefficient for events with dense angular population of energy in the detector. In addition, jets can fake both taus and photons at a rate which is intimately tied to the detector response, which is beyond our ability to reliably estimate. To work around these difficulties, we base our $h\to\tau^{+}\tau^{-}\gamma$ on the existing CMS search for $h\to\tau^{+}\tau^{-}$ Chatrchyan:2014nva , allowing us to draw from its detailed background study. We augment this search by requiring an additional energetic final state photon that along with the $\tau^{+}\tau^{-}$ system reconstructs the Higgs.

We analyze the reach of the LHC running at $\sqrt{s}=13~\rm{TeV}$, with the reconstruction strategy outlined above. In Fig. 1 we show the expected $M_{\tau\tau}$ and $M_{\tau\tau\gamma}$ distributions for the signal (with $\Lambda=343~\rm{GeV}$, $\Lambda^{2}>0$) and $Z+\gamma$ and $Z+j$ background processes, in the $\tau_{h}\tau_{e}$ and $\tau_{h}\tau_{\mu}$ topologies, for an integrated luminosity of ${\cal L}_{\rm int}=300~\rm{fb}^{-1}$. All analysis cuts with the exception of the $M_{\tau\tau}<60~\rm{GeV}$ cut are applied. Evident from the upper plots is the motivation for the $M_{\tau\tau}<60~\rm{GeV}$ cut to separate the signal from the backgrounds. The $M_{\tau\tau\gamma}$ distributions after this cut are presented in Fig. 2, and demonstrate its dramatic efficacy, with only a handful of background events left in the $120~\rm{GeV}\leq M_{\tau\tau\gamma}<140~\rm{GeV}$ signal region.

To estimate the eventual sensitivity of the high luminosity LHC to new physics in the tau magnetic dipole, we write the amplitude for the signal process as

$${\cal M}_{\rm sig}={\cal M}_{\rm{SM}}+\frac{1}{\Lambda^{2}}{\cal M}_{\rm{NP}}$$

(9)

with the $\Lambda$ dependence explicitly factored out. The yield of signal events (for ${\cal L}_{\rm int}=300~\rm{fb}^{-1}$) after cuts is:

$\displaystyle N_{\rm sig}$

$\displaystyle=$

$\displaystyle N_{\rm{SM}}+\frac{2}{\Lambda^{2}}\hat{N}_{\rm INT}+\frac{1}{\Lambda^{4}}\hat{N}_{\rm NP}.$

(10)

The sizes of these three coefficients, after all analysis cuts, are shown in Table 2. We present $N_{\rm sig}$ as a function of $\Lambda$ for both signs of $\Lambda^{2}$ in Fig. 3, including the $\tau_{h}\tau_{e}$ and $\tau_{h}\tau_{\mu}$ channels, and also the combined number of events in both signal categories, $\tau_{h}\tau_{\ell}$.

Under the Assumption that no signal is observed, and the mean number of background events, 6.7, is obtained, one may place a lower limit on the new physics scale $\Lambda$. The $95\%$ CL bound on $\Lambda$ in that case would be given by:

$$|\Lambda|>\begin{array}[]{l}634~\rm{GeV}\\ 739~\rm{GeV}\end{array}~~~~~~~~~{\rm for:}~~~\begin{array}[]{r}~\Lambda^{2}>0\\ \Lambda^{2}<0\end{array}.$$

(11)

which translates into a projected limit on the anomalous moment of

$$-0.0144<a^{\gamma}_{\tau}<0.0106.\quad(95\%~{\rm CL}),$$

(12)

approximately a factor of two improvement on $\Lambda$ for $\Lambda^{2}>0$ or $10\%$ for $\Lambda^{2}<0$.

As measurements of the Higgs boson become more sophisticated, we move into a regime where it becomes a tool in its own right to search for new physics. In particular, low energy observables involving a chiral flip of the SM fermions necessarily invoke electroweak symmetry-breaking, and thus imply a modification of the properties of the Higgs. In this work, we have examined the possibility that one can place bounds on the electromagnetic moments of the $\tau$ by searching for the rare decay of the Higgs into $\tau^{+}\tau^{-}\gamma$. Given the longstanding discrepancy between measurements of the muon’s magnetic moment and SM predictions, one could hope that $a^{\gamma}_{\tau}$ might also be a likely target for which to search for manifestations of new physics.

We find the promising result that the LHC with a large data set should be sensitive to modifications of the $\tau$ magnetic dipole moment beyond the current bounds extracted from LEP – by about a factor of two on the new physics scale if $\Lambda^{2}>0$. Given these promising results, it would be worthwhile to follow up with a study based on more realistic detector simulations and including effects beyond our ability to reliably simulate, such as pile-up. We hope that our study will motivate the experimental collaborations to carry out this effort.

Another interesting direction for the future would be to study the prospects at a future $e^{+}e^{-}$ collider such as the ILC or a future circular collider Ozguven:2016rst . While the rate for $hZ$ production at such colliders is considerably smaller than the inclusive Higgs production at the LHC, new physics saturating the LEP bound nonetheless allows for a handful of events, and the prospects depend sensitively on the ability to efficiently reconstruct the signal events and reject backgrounds.

The discovery of the Higgs is a triumph of run I of the LHC. We look forward to run II and beyond to follow up with precision measurements that reveal the deep secrets that reflect its character.

The authors are supported in part by National Science Foundation grants PHY-1316792 and PHY-1620638. We thank Enrique Kajomovitz, Olivier Mattelaer, Andrew Nelson, Yoram Rozen, Yael Shadmi, Yotam Soreq, Philip Tanedo and Scott Thomas for useful discussions; Christian Veelken for his assistance with setting up the tau reconstruction platform and for useful discussions; and David Cohen for assistance in grid-based computing. IG and TMPT are grateful to the Mainz Institute for Theoretical Physics for its hospitality and its partial support during the completion of this work.