Parity Violation and Rare Higgs Decays from a Dark Force

TL;DR

This paper explores a hypothetical dark Z boson that could influence parity violation measurements and rare Higgs decays, potentially explaining existing experimental tensions and providing new avenues for discovery at colliders.

Contribution

It introduces a generalized dark Z model with mass mixing, analyzing its effects on low Q^2 parity violation and rare Higgs decay processes, and discusses experimental prospects.

Findings

Dark Z could explain the 1.8σ tension in weak mixing angle measurements.

The model predicts observable effects in low Q^2 parity violation experiments.

Potential signals in rare Higgs decays at the LHC are identified.

Abstract

We outline the phenomenology of the "dark" $Z$, denoted by $Z_d$, which is a generalization of the "dark" photon hypothesis. Whereas the dark photon interacts with the Standard Model through kinetic mixing, $Z_d$ is assumed also to have mass-mixing with the $Z$ boson. In particular, we highlight the possibility of $Z_d$ contributions to low $Q^2$ parity violation measurements and rare Higgs decays $H\to Z Z_d \to 4 \ell$, where $\ell$ is a charged lepton. The parity violation effects of a $Z_d$ with an intermediate mass $\sim 10-35$ GeV can in principle relieve the mild $\sim 1.8\sigma$ tension among various measurements of the weak mixing angle $\theta_W$. We briefly comment on the prospects for future parity violation experiments at low $Q^2$ to probe this scenario, which could have correlated signals in rare Higgs decays at the LHC.

Figures

Equations

$$\frac{1}{2}\frac{\epsilon }{\cos {\theta }_W}{B}_{\mu \nu }{Z}_d^{\mu \nu },$$

$$-e\epsilon J_{em}^{\mu }{Z_d}_{\mu },$$

$${\epsilon }_Z=(m_{Z_d}/m_Z)\delta ,$$

$${\mathcal{L}}_{\text{int}}=(-e\epsilon J_{\mu }^{em}-\frac{g}{2\cos {\theta }_W}{\epsilon }_Z{J}_{\mu }^{NC}+\dots )Z_d^{\mu },$$

$$\Delta {\sin }^2{\theta }_W(Q^2)=-\epsilon \delta \left(\frac{m_Z}{m_{Z_d}}\right)\frac{\cos {\theta }_W\sin {\theta }_W}{1+Q^2/m_{Z_d}^2}.$$

$${\sin }^2{\theta }_W(m_Z{)}_{\overline{\mathrm{MS}}}=0.2283(20)\text{(APV)}$$

$${\sin }^2{\theta }_W(m_Z{)}_{\overline{\mathrm{MS}}}=0.2329(13)\text{(Moller E158)}$$

$${\sin }^2{\theta }_W(m_Z{)}_{\overline{\mathrm{MS}}}=0.2356(16)\text{(NuTeV)}$$

$${\sin }^2{\theta }_W(m_Z{)}_{\overline{\mathrm{MS}}}=0.23124(12)\text{(SM)}$$

$$\Delta {\sin }^2{\theta }_W\simeq 0.0016(9).$$

$${\delta }^{\prime }\simeq \delta +\frac{m_{Z_d}}{m_Z}\epsilon \tan {\theta }_W.$$

$$-0.0010<\epsilon {\delta }^{\prime }<-0.0003,$$

$$|{\delta }^{\prime }|\lesssim 0.02\text{(ATLAS}),$$

$$\text{Br}(Z_d\to l^{+}{l}^{-})\approx 0.15,$$

$$|\epsilon {\delta }^{\prime }|\lesssim \mathrm{0.0008.}$$

Full text

Only about $5\%$ of the energy budget of the Universe is in the form of the ordinary matter made of atoms. Nearly $27\%$ of that energy budget is made up of some unknown substance referred to as dark matter (DM) Ade et al. (2016) and the rest is a type of energy density that is responsible for the accelerated expansion of the Universe, often called dark energy. While dark energy could be the cosmological constant, its nature remains as yet uncertain.

All the evidence that we so far have for DM comes from its gravitational effects and is gleaned from various astronomical and cosmological observations. What we know about DM, broadly speaking, is quite basic: it is a from of matter that is stable on cosmological time scales and does not interact with ordinary matter with significant strength if at all. This leaves a wide range of possible scenarios for what DM could be and how it was produced in the early Universe. The strong evidence for DM is one of the main motivations to expect new physics, since none of the known particles within the Standard Model (SM) has the requisite properties to be a viable candidate.

Many models of DM are based on a single particle that resides in some extension of the SM. However, it is also possible that DM is part of a “dark sector” that has its own matter content and forces, in close analogy with the SM (which comprises the sub-dominant sector in the cosmic energy budget). Such a dark sector could then interact with the SM indirectly, for example through suppressed mixing.

In what follows, we will assume that the dark sector contains a vector boson $Z_{d}$ that is the mediator of a dark $U(1)_{d}$ force. Such a vector boson has been invoked in connection with a potential resolution of the $\sim 3.5\sigma$ deviation in the measured value of the muon anomalous magnetic moment $g_{\mu}-2$ Fayet (2007); Pospelov (2009) and also in interpreting certain astrophysical data as signals of DM Arkani-Hamed et al. (2009).

A popular framework for the interaction of the dark vector boson $Z_{d}$ with the SM is kinetic mixing Holdom (1986) between $U(1)_{d}$ and hypercharge $U(1)_{Y}$ through the dim-4 operator

[TABLE]

where $\varepsilon$ is the mixing parameter (assuemd to be small), $B_{\mu\nu}\equiv\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$, with $B_{\mu}$ the hypercharge gauge field, $Z_{d\mu\nu}\equiv\partial_{\mu}{Z_{d}}_{\nu}-\partial_{\nu}{Z_{d}}_{\mu}$, and $\theta_{W}$ is the weak mixing angle. With the above coupling, after proper diagonalization of the Lagrangian, one finds that $Z_{d}$ couples to the electromagnetic current $J_{em}^{\mu}$ via

[TABLE]

that is like a photon, but suppressed by $\varepsilon$. In this case, the $Z_{d}$ is often referred to as the “dark” photon.

The above fairly simple setup can in principle lead to an explanation of the $g_{\mu}-2$ deviation, if the mass of $Z_{d}$ is in the range $m_{Z_{d}}\lesssim 1$ GeV and $\varepsilon\lesssim 10^{-2}$. This possibility has provided a well-motivated search target for various experiments to look for the potential signals of $Z_{d}$ in low energy experiments. If the lightest decay final states for $Z_{d}$ are in the SM sector, with significant branching fractions into $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$, the dark photon explanation of $g_{\mu}-2$ is basically ruled out by various null search results; see for example Ref. Alekhin et al. (2016).

If the dark sector includes states that are lighter than $m_{Z_{d}}/2$ and have non zero $U(1)_{d}$ charge, then it is typically expected that $Z_{d}$ would decay dominantly into those states and therefore end up being largely “invisible.” In that case, other constraints will be applicable, which will also remove much of the allowed parameter space for explaining the $g_{\mu}-2$ anomaly; see for example Refs. Essig et al. (2013); Davoudiasl et al. (2014). (Since the presentation of the original talk, the “invisible” dark photon explanation of the $g_{\mu}-2$ anomaly has been essentially ruled out, for $m_{Z_{d}}\leq 8$ GeV, by the BaBar collaboration Lees et al. (2017).)

Regardless of its utility in addressing the $g_{\mu}-2$ anomaly, an interesting implication of the above invisibly decaying $Z_{d}$ scenario is the possibility to produce and detect light dark sector states in fixed target (or beam dump) experiments. Here, the $Z_{d}$ bosons produced in the collisions of a proton Batell et al. (2009) or electron Izaguirre et al. (2013) beam with the target would decay in flight into dark sector states (perhaps DM). These particles would form a “dark beam,” due to the typically large boost of the initial $Z_{d}$ bosons, and go through a shield (such as Earth) unimpeded. A suitable detector down stream of the beam can then detect the dark states, via the $Z_{d}$ couplings to the SM (e.g. kinetic mixing in the case of a dark photon $Z_{d}$).

One could extend the above picture of $Z_{d}$ as a “dark” photon by including $Z_{d}$-$Z$ mass mixing effects parameterized by

[TABLE]

as may be realized in a 2-Higgs doublet model with a doublet charged under $U(1)_{d}$ Davoudiasl et al. (2012). With this addition, the couplings of $Z_{d}$ to the SM are given by Davoudiasl et al. (2012)

[TABLE]

where $J_{\mu}^{NC}$ is the SM neutral current. Given the new $Z$-like coupling to the neutral current, we refer to $Z_{d}$ as “dark” $Z$. In Eq. (4), $\ldots$ denotes other $Z_{d}$ couplings, such as $HZZ_{d}$ Davoudiasl et al. (2012).

One can show Davoudiasl et al. (2012) that the above new interactions lead to effects in parity violating processes, as well as rare decays of mesons and the Higgs, such as $H\to ZZ_{d}$, where $Z_{d}$ could be light and hence on-shell. For a light dark $Z$, its longitudinal mode will dominate the rare decay processes, as the corresponding amplitudes are enhanced by $\sim E/m_{Z_{d}}$, where $E$ is the typical energy of $Z_{d}$ in the process. Flavor physics data, from $K$ and $B$ decays imply $|\delta|\lesssim 10^{-3}$, assuming on-shell $Z_{d}$ final states, depending on its“visible” branching fractions.

Assuming both kinetic and mass mixing induced couplings, it can be shown that the contribution of $Z_{d}$ to neutral current amplitudes induces a variation of the weak mixing angle $\theta_{W}$ with momentum transfer $Q^{2}$, given by

[TABLE]

Th above shift could yield measurable deviations in the value of the mixing angle for $Q^{2}\lesssim m_{Z_{d}}^{2}$.

There is currently a slight tension, at $\sim 1.8\sigma$, between the SM prediction and the measured weak mixing angle at various $Q^{2}\ll m_{Z}^{2}$. These measurements, extrapolated to the $\overline{\rm MS}$ scale $\mu=m_{Z}$, are:

[TABLE]

[TABLE]

[TABLE]

at $\langle{Q}\rangle=2.4~{}\text{MeV}$, $\langle{Q}\rangle=160~{}\text{MeV}$, and $\langle{Q}\rangle\approx 5~{}\text{GeV}$, respectively. The deviation from the SM prediction

[TABLE]

from the weighted average of the above values is given by (see Ref. Davoudiasl et al. (2015) for details):

[TABLE]

It is interesting to consider how the contribution of an intermediate mass $Z_{d}$, with $m_{Z_{d}}\sim 10-35$ GeV, may help bring these measurements into better agreement with the SM prediction. For these values of $m_{Z_{d}}$, the mass mixing parameter $\delta$ also receives a non-negligible contribution from kinetic mixing, and we have $\delta\to\delta^{\prime}$ with

[TABLE]

As an example for illustrative purposes, using the above mass-mixing parameter and setting $m_{Z_{d}}=15$ GeV, a 1-$\sigma$ fit to Eq. (10) would require Davoudiasl et al. (2015)

[TABLE]

as presented in Fig. 1. This range of parameters is largely allowed by the existing constraints on $\varepsilon$ and $\delta^{\prime}$, as discussed below.

Constraints from the ATLAS collaboration search for $H\to ZZ_{d}\to l_{1}^{+}l_{1}^{-}l_{2}^{+}l_{2}^{-}$, with $l_{1,2}=e,\mu$ Aad et al. (2015), place a 2-$\sigma$ bound on mass mixing

[TABLE]

roughly constant over the $m_{Z_{d}}$ range considered here, assuming that the total branching fraction

[TABLE]

with $l=e,\mu$. The bound $|\varepsilon|\lesssim 0.04$ is obtained from precision and production constraints Davoudiasl et al. (2015) (allowing some cancellation with $\delta^{\prime}$-dependent contribution). These constraints suggest the bound

[TABLE]

Hence, a narrow band (light blue in Fig. 1) in the lower end of the 1-$\sigma$ range (12) considered in the above example has some tension with the above bound.

In the near future, the full results of the Qweak experiment at JLAB are expected to yield a low $Q^{2}$ determination of $\sin^{2}\theta_{W}(Q^{2})$ with an uncertainty at the $\pm 0.0007$ level. This would lower the uncertainty of the aforementioned weighted average to $\pm 0.00055$, which assuming the central value based on the published results, could lead to a $\sim 3\sigma$ deviation. At the same time, future LHC search results for the rare Higgs decay $H\to ZZ_{d}$ could probe the relevant parameter space suggested by such a deviation in $\sin^{2}\theta_{W}(Q^{2})$. Further reductions in the uncertainty of the weighted average of $\sin^{2}\theta_{W}(m_{Z})_{\rm\overline{MS}}$ at low $Q^{2}$ are expected to be achieved by future measurements of polarized $ee$ Moller scattering at JLAB, as well as polarized $ep$ scattering (P2) at MESA, in Mainz.

In conclusion, we see that the dark $Z$ framework presented here could in principle lead to an interesting interplay between various low $Q^{2}$ effects and high energy LHC signals, originating from the contributions of an intermediate mass $Z_{d}$.