Preliminaries

An additional U(1)_D gauge symmetry added to the SM is theoretically well-motivated and occurs in many top-down and bottom-up extensions of the SM. The U(1)_D vector boson (the "dark photon" or the "dark-Z") is usually referred to as A′, Z′, γ_D, or Z_D in the literature and various possibilities exist to connect the additional U(1)_D to the SM (see e.g. [1,2,3,4] for reviews). The U(1)_D can couple to the SM sector via a small gauge kinetic mixing term [1/2] ϵF′_μν B^μν [5,6,7] between the dark photon and the hypercharge gauge boson. This renormalizable interaction can be generated at a high scale in a grand unified theory or in the context of string theory with a wide range of ϵ ∼ 10⁻¹⁷ − 10⁻². This term effectively gives SM matter a dark milli-charge and allows for dark photon decay to SM particles and possible experimental detection. To avoid the tight constraints on new long-range forces, a `dark Higgs' S with a non-zero vacuum expectation value can give a non-zero mass to the A′. An A′ with a sub-GeV mass can be probed at beam dumps and colliders, and with measurements of the muon anomalous magnetic moment, supernova cooling, and rare meson decays, see Fig. 1 and e.g. [4] for a recent review. A broken U(1)_D can also lead to exotic Higgs decays, especially if there is mixing between the two Higgs sectors. In this context we refer to the corresponding vector field as Z_D. The possibility of h → Z_D Z_D through Higgs-to-dark-Higgs mixing or h → Z Z_D through Z-Z_D mass mixing (which is also induced by the above-mentioned kinetic mixing) was discussed in [8] and [9,10], respectively, with both occurring, for example, in hidden valley models [11,12]. To examine the range of possible exotic Higgs phenomena due to a U(1)_D sector we examine the model of [8], but with m_h set to 125 GeV and allowing for the full range of dark Higgs and dark-Z masses relevant to exotic Higgs decay phenomenology.¹ This includes Higgs-to-dark-Higgs mixing and kinetic mixing between the B boson and the dark vector Z_D, but no explicit mass mixing between the Z and Z_D.² We will assume prompt Z_D decays, which requires m_ZD >~10  MeV given the current constraints shown in Fig. 1.    

Model Details

The model is defined by a U(1)_D gauge sector and a SM singlet S that has unit charge under the U(1)_D. The kinetic terms of the hypercharge and U(1)_D gauge bosons (adopting mostly the notation of [9]) are

(1)

with ∧B_μν = ∂_μ ∧B_ν − ∂_ν ∧B_μ, ∧Z_Dμν = ∂_μ ∧Z_Dν − ∂_ν ∧Z_Dμ, and cosθ_W = g/√{g² + g′²} is the usual Weinberg mixing angle. The hatted quantities are fields before diagonalizing the kinetic term. The Higgs potential is

(2)

The dark Higgs S acquires a vacuum expectation value and gives Z_D, which `eats' the pseudoscalar component of S, some mass m_ZD. There are two connections between the dark and the SM sectors: the gauge kinetic mixing ϵ and the Higgs mixing ζ. The phenomenology depends on which one dominates. The SM gauge boson Z_μ = − sinθ_W B_μ + cosθ_W W³_μ is no longer a mass eigenstate and to leading order in ϵ the mass eigenstates with masses m_Z, m_ZD + O(ϵ²) are

 (3)

(Henceforth, we omit the tildes and will refer to the mass eigenstates unless otherwise noted.) Therefore, there are interaction terms of the form 2 ϵ_Z [(m_ZD²)/v] h Z_μ Z_D^μ and ϵ_Z² [(m_ZD⁴)/( m_Z² v)] h Z_Dμ Z_D^μ which lead to h → Z_D Z and h → Z_D Z_D decays (though the latter is strongly suppressed), see Figure 3. If Z_D is the lightest state in the dark sector it will decay to SM particles. For m_ZD >~ GeV the Z_D, branching ratios are easily computed to lowest order and without QCD corrections, and are shown in Figure 2 (a). For m_ZD <~ GeV, non-perturbative QCD effects are important. They can be computed from the QCD contribution to the imaginary part of the electromagnetic two-point function, which in turn is determined from cross-section measurements of e⁺ e⁻ →  hadrons [14]. The resulting branching ratios are shown in Figure 2 (b). The most important qualitative difference to the scalar decays considered in the SM+S and 2HDM+S scenarios is that branching ratios are ordered by gauge coupling instead of Yukawa coupling, meaning decays to e⁺ e⁻ and μ⁺ μ⁻ remain large above the τ thresholds. Prompt Z_D decay requires ϵ >~10⁻⁵ − 10⁻³, as indicated in Figure 1, which summarizes the constraints on Z_D kinetic mixing for our regime of interest.

Figure 2:

  The Higgs potential is minimized by vacuum expectation values of H⁰ and S

(4)

The mass eigenstates

 (5)

have masses

(6)

(Again we drop the tildes from now on and always refer to the mass eigenstates.) The effective Lagrangian contains terms of the form κh s s where κ = ζ(m_h³ + 2 m_h m_s²)/(√{16 λ} (m_h² − m_s²)), and 2 ϵ_h [(m_ZD²)/w] h Z_{D μ} Z_D^μ, which lead to exotic Higgs decays h → s s and h → Z_D Z_D, see Figure 3. The vertex h s Z_D is present but is suppressed by both mixings.   We can now discuss the relevant limits of this theory for exotic Higgs phenomenology:

• Gauge mixing dominates:For ϵ >> ζ the dominant exotic Higgs decay is h → Z Z_D. To leading order in m_ZD²/m_Z² the partial width is

 (7)

This agrees with the full analytical expression to  ∼ 10% for m_h − m_Z − m_ZD > 1  GeV. Figure 1 shows contours of Br(h → Z Z_D) = 10⁻⁴, 10⁻⁵, 10⁻⁶. The largest Br allowed by indirect electroweak precision constraints is  ∼ 10⁻⁴. In this limit, the SM+V theory leads to ff + Z exotic Higgs signatures , and exotic branching ratio sensitivities better than  ∼ 10⁻³ are possible at LHC8. Bounds from LHC14 could become competitive with indirect limits. For very light Z_D above the electron threshold this would also lead to lepton-jets + Z signatures, see here and here [15].  

• Higgs mixing dominates:When ζ >> ϵ and Higgs mixing dominates then h → Z_D Z_D, s s are both possible, depending on the spectrum of the dark sector. (We still assume that ϵ is large enough for Z_D to decay promptly.) The partial decay widths to leading order in ζ are

(8)

Different regions of of the (m_ZD, m_s) mass plane are shown in Figure 4, along with the size of the Higgs mixing ζ ∼ 10⁻³ − 10⁻² required for Br(h → Z_DZ_D , s s) = 10% and the relative rates of h → ss vs h → Z_D Z_D decays when both are allowed. In Region A (m_s > m_h/2, m_ZD < m_h/2) the only relevant exotic Higgs decay is h → Z_D Z_D. This allows for spectacular h → 2l2l′ decays (l, l′ = e or μ) with a reconstructed Z_D resonance above the τ- or b-thresholds. Region B allows exotic Higgs decays both to Z_D Z_D and ss. The presence of two resonances below half the Higgs mass gives a rich exotic decay phenomenology. h → s s → 4 Z_D occurs with roughly equal probability as h → Z_D Z_D and can result in spectacular final states with as many as 8 leptons. Note that, in this simplified model, there is no corresponding  Z_D → ss  decay in the lower right hand corner of that mass plane. However, a (pseudo)scalar pair could be produced from dark vector decay in e.g. a 2HDM+V framework, resulting in final states with as many as 8 b-quarks. Already with current data, limits of Br(h→ Z_D Z_D) <~10⁻⁴ can be achieved. Each of the above cases may, for suitable masses, also lead to interesting `lepton jet' signatures, see here.  

• Intermediate Regime:Here the decays induced by kinetic and Higgs mixing are comparable. For example, Figure 1 shows that ϵ ∼ 10⁻² is not excluded for some values of m_ZD, allowing Br(h→ Z Z_D)  ∼ 10⁻⁴. The branching ratios for h→ Z_D Z_D, s s will be similar if ζ ∼ 10⁻⁴.

   

Summary

In summary, the SM+V setup allows for many different kinds of exotic Higgs decays, including h → Z Z_D, h→ Z_D Z_D, and h→ ss, with Z_D → f―f , and s → f―f or s → Z_D Z_D → (f―f)(f―f). This leads to final states of Z + (f―f), (f―f)(f―f), and ((f―f)(f―f))((f―f)(f―f)), where parentheses around a set of particles denotes a resonance (all final-state particles combined will form the Higgs resonance). Since the Z_D (although not the s) couples to the fermions' gauge charges, final states with several light leptons have sizable branching fractions over the entire kinematically permitted mass range. Certain spectra can produce interesting lepton-jet signatures.

MadGraph Model

MadGraph Model

We have constructed a validated MadGraph 5 model for the SM + vector + dark higgs scenario outlined here. This can be used for signal generation in experimental and phenomenological studies.

Link to MadGraph5 model for SM + S + V

Branching Ratio Tables

In "Illuminating Dark Photons with High-Energy Colliders" (Curtin, Essig, Gori, Shelton, arXiv:1412.0018) we compute the dark photon total width and branching ratio to leptons Br(ZD->ll) including QCD corrections, using PDG experimental data for mZD < 12 GeV, and by repurposing the calculations of hep-ph/0005139 for mZD > 12 GeV. We also compute the exotic higgs decay branching ratios Br(h->ZdZ*->4l) and Br(h->ZdZd->4l). All of these computed widths and branching ratios can be downloaded as a text table here. (See 1412.0018 for more details.)      

References