# SM + Vector

## 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′, γD, or ZD 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 Bμν [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 ZD.
The possibility of h → ZD ZD through Higgs-to-dark-Higgs mixing or h → Z ZD through Z-ZD 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 mh 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 ZD, but no explicit mass mixing between the Z and ZD.2 We will assume prompt ZD decays, which requires mZD >~10  MeV given the current constraints shown in Fig. 1.

Figure 1: Constraints on ϵ, mZD for pure kinetic mixing (no additional source of Z-ZD mass mixing) for mZD  ∼ MeV-10 GeV. The black dashed line separates prompt (cτ < 1 μm) from non-prompt decays. The three blue lines are contours of Br(h → Z ZD) of 10−4, 10−5, 10−6 respectively. Shaded regions are existing experimental constraints, see e.g. [4] for a recent review.

## 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 Bμν = ∂μ Bν − ∂ν Bμ, ZDμν = ∂μ Z − ∂ν Z, and cosθW = 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 ZD, which eats' the pseudoscalar component of S, some mass mZD. 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 Zμ = − sinθW Bμ + cosθW W3μ is no longer a mass eigenstate and to leading order in ϵ the mass eigenstates with masses mZ, mZD + O2) 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 ϵZ [(mZD2)/v] h Zμ ZDμ and ϵZ2 [(mZD4)/( mZ2 v)] h Z ZDμ which lead to h → ZD Z and h → ZD ZD decays (though the latter is strongly suppressed), see Figure 3.
If ZD is the lightest state in the dark sector it will decay to SM particles. For mZD >~ GeV the ZD, branching ratios are easily computed to lowest order and without QCD corrections, and are shown in Figure 2 (a). For mZD <~ 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 ZD decay requires ϵ >~10−5 − 10−3, as indicated in Figure 1, which summarizes the constraints on ZD kinetic mixing for our regime of interest.

Figure 2:

(a) Branching ratios for ZD decay, to lowest order and without QCD corrections, assuming decays to the dark sector are kinematically forbidden. l stands for either e or μ. Hadronization effects likely invalidate our simple calculation in the shaded region. (b) Branching ratios for ZD decay for mZD <~3  GeV, including non-perturbative QCD effects.

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(mh3 + 2 mh ms2)/(√{16 λ} (mh2 − ms2)), and 2 ϵh [(mZD2)/w] h ZD μ ZDμ, which lead to exotic Higgs decays h → s s and h → ZD ZD, see Figure 3. The vertex h s ZD is present but is suppressed by both mixings.
Figure 3: The dominant exotic Higgs decays in the SM+V model. The h → Z ZD matrix element is proportional to the gauge kinetic mixing ϵ, while h → ZD ZD and h → s s are controlled by the Higgs mixing parameter ζ. The vertex h s ZD is present but 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 ZD. To leading order in mZD2/mZ2 the partial width is

(7)

This agrees with the full analytical expression to  ∼ 10% for mh − mZ − mZD > 1  GeV. Figure 1 shows contours of Br(h → Z ZD) = 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 ZD 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 → ZD ZD, s s are both possible, depending on the spectrum of the dark sector. (We still assume that ϵ is large enough for ZD to decay promptly.) The partial decay widths to leading order in ζ are

(8)

Different regions of of the (mZD, ms) mass plane are shown in Figure 4, along with the size of the Higgs mixing ζ ∼ 10−3 − 10−2 required for Br(h → ZDZD , s s) = 10% and the relative rates of h → ss vs h → ZD ZD decays when both are allowed.
In Region A (ms > mh/2, mZD < mh/2) the only relevant exotic Higgs decay is h → ZD ZD. This allows for spectacular h → 2l2l′ decays (l, l′ = e or μ) with a reconstructed ZD resonance above the τ- or b-thresholds.
Region B allows exotic Higgs decays both to ZD ZD and ss. The presence of two resonances below half the Higgs mass gives a rich exotic decay phenomenology. h → s s → 4 ZD occurs with roughly equal probability as h → ZD ZD and can result in spectacular final states with as many as 8 leptons. Note that, in this simplified model, there is no corresponding  ZD → 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→ ZD ZD) <~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 mZD, allowing Br(h→ Z ZD)  ∼ 10−4. The branching ratios for h→ ZD ZD, s s will be similar if ζ ∼ 10−4.

Figure 4: Left: Regions of the (mZD, ms) mass plane in the SM+V model with different exotic Higgs decays for ζ >> ϵ (i.e. when the mixing between the Higgs and dark-Higgs dominates over the kinetic mixing). The black contours are the values of ζ×103 required for Br(h → ZD ZD, s s) = 10%. Region A is the case examined by [8] (the dotted red line indicates mh = ms). Region C has no exotic Higgs decays. Region D reproduces the SM+S model. Region B has both h → s s and h → ZD ZD decays, with the h → ss fraction of exotic decays shown on the right. In the upper left shaded region, s → ZD ZD is the dominant decay mode of the dark scalar. This allows the Higgs to decay to up to 8 SM fermions.

## Summary

In summary, the SM+V setup allows for many different kinds of exotic Higgs decays, including h → Z ZD, h→ ZD ZD, and h→ ss, with ZD → ff , and s → ff or s → ZD ZD → (ff)(ff). This leads to final states of Z + (ff), (ff)(ff), and ((ff)(ff))((ff)(ff)), where parentheses around a set of particles denotes a resonance (all final-state particles combined will form the Higgs resonance). Since the ZD (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.

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.

## 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.)

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### Footnotes:

1Ref. [13] appeared while this work was being completed, performing a similar analysis with a different focus on constraining the couplings of the extended Higgs potential for relatively low mZD < 5  GeV.
2The constraints shown in Fig. 1 are altered in the presence of such pure mass mixing, which requires additional Higgs doublets that also carry dark charge. The resulting ZDSM decays would be more Z-like and lead to additional constraints from rare meson decays as well as new parity-violating interactions [9]. However, we stress that the exotic Higgs phenomenology would not be qualitatively different.
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