Multilepton Higgs Decays through the Dark Portal

The U(1)_D gauge sector containing one dark Higgs boson h_D and one dark photon \gamma_D may be explored through the decays of the 126 GeV particle discovered at the Large Hadron Collider (LHC), assumed here as the heavier mass eigenstate h_1 in the mixing of the standard model h with h_D. The various decays of h_1 to \gamma_D \gamma_D, h_2 h_2, h_2 \gamma_D \gamma_D and h_2 h_2 h_2 would yield multilepton final states through the mixing of \gamma_D with the photon and the decay h_2 \to \gamma_D \gamma_D, where h_2 is the lighter dark Higgs. Future searches for signals of multilepton jets at the LHC may reveal the existence of this possible dark sector governed simply by the original Abelian Higgs model.


I. INTRODUCTION
The original Higgs model [1] of spontaneous symmetry breaking involves just one complex scalar field χ and one vector gauge field C. As χ acquires a nonzero vacuum expectation value (VEV), the physical spectrum of this model consists of a massive vector boson γ D and a massive real scalar boson h D , and the only interactions between them are of the form h D γ D γ D and h 2 D γ D γ D . The analog of h D in the electroweak SU (2) × U (1) extension [2] of this original model, commonly called the Higgs boson h, is presumably the 126 GeV particle observed at the Large Hadron Collider (LHC) [3,4]. Is this the whole story? Perhaps not, because the original Higgs model may still be realized physically, but in a sector which connects with the standard model (SM) of particle interactions only through h D − h mass mixing and γ D − γ kinetic mixing [5]. If so, the 126 GeV particle may be identified with the heavier mass eigenstate h 1 and decays such as h 1 to γ D γ D , h 2 h 2 , h 2 γ D γ D and h 2 h 2 h 2 would result in multilepton final states via γ D →ll or h 2 → γ D γ D and then followed by γ D →ll, where h 2 is the lighter dark Higgs and l is the SM lepton.
In Sec. 2 we set up our model. Phenomenology based on similar model has been studied before, see for example Refs. [6][7][8][9][10][11] and references therein. In Sec. 3 we consider mixing effects in the scalar sector as well as the gauge boson sector. We show the h D − h mixing in detail and present all the relevant trilinear and quadrilinear couplings of the physical h 1 and h 2 bosons. We also briefly discuss the mixings between the three neutral gauge bosons in the model as studied previously in Ref. [11]. In Sec. 4 we discuss the possible decay modes of the SM Higgs outside those of the SM and their several kinematic regions. In Sec. 5 we present numerical results for various branching ratios of the non-standard decay modes of the SM Higgs, identified here as h 1 . In Sec. 6 we study the signals of multilepton jets of the model at the LHC-14. We conclude in Sec. 7.
We extend the electroweak SM by including the original Abelian Higgs model for a dark U (1) D [6][7][8][9][10][11]. The bosonic part of the Lagrangian density is with L scalar = |D µ Φ| 2 + |D µ χ| 2 − V scalar (Φ, χ) , and where W µ , B µ and C µ are the gauge potentials of the SU (2) L , U (1) Y and U (1) D with gauge couplings g, g and g D respectively, and is the kinetic mixing parameter between the two U (1)s [5]. The scalar potential in (3) is given by We pick the unitary gauge and expand the scalar fields around the vacuum with the VEVs v and v D fixed by minimisation of the potential to be In terms of the shifted fields h and h D , the scalar potential V scalar can then be decomposed as Here V 0 is a cosmological constant and will be discarded from now on; the tadpole term V 1 vanishes with v and v D given by Eq. (8); V 2 is quadratic in the fields h and h D , and we have to diagonalize the mass matrix M 2 S in Eq. (11) to get the physical Higgs fields h 1 and h 2 (see next section); and V 3 and V 4 are the trilinear and quadrilinear self couplings among the two Higgs fields. Since χ is a SM singlet, the W and Z bosons acquire their masses through the SM Higgs doublet VEV v entirely which implies v = 246 GeV.

A. Higgs Mass Eigenstates and Their Self Interactions
The mass matrix M 2 S in Eq. (11) for the scalar bosons is Its eigenvalues are The physical Higgs (h 1 , h 2 ) are related to the original (h, h D ) as with the mixing angle We will identify the heavier Higgs h 1 with mass m 1 = 126 GeV as the new boson observed at the LHC [3,4], while the lighter one h 2 has been escaped detection thus far. The SM Higgs couplings with the SM fermions and gauge bosons are thus modified by a factor of cos α.
In terms of the physical Higgs fields h 1 and h 2 , the cubic term V 3 is given by with the trilinear couplings and the quartic term V 4 is given by with the quadrilinear couplings

B. Kinetic and Mass Mixing of the neutral gauge bosons
In additional to the mass mixing of the three neutral gauge bosons arise from the spontaneously electroweak symmetry breaking given by with the following mass matrix we also have the kinetic mixing between the two U (1)s from the last term in Eq. (2). Both the kinetic and mass mixings can be diagonalized simultaneously by the following mixed transformation [11]  where A µ , Z µ and A µ are the physical dark photon, Z boson and the photon respectively.
Here K is a general linear transformation that diagonalizes the kinetic mixing where β = 1/(1− 2 ) 1/2 ( ≤ 0), and O is a 3×3 orthogonal matrix which can be parametrized as with the mixing angles given by [11] tan θ = g g , tan φ = − β, tan ψ = ± tan φ cos θ tan θ After the K transformation, the gauge boson mass matrix is The O matrix diagonalizes thisM 2 matrix For small kinetic parameter mixing , the Z and γ D masses can be approximated by For couplings of these physical neutral gauge bosons with the SM fermions, we refer the readers to Ref. [11].

IV. NON-STANDARD DECAYS OF h 1
The global fits [12][13][14][15] for the signal strengths of the various SM Higgs decay channels from the LHC data imply the total width of the SM Higgs is about 4.03 MeV and the nonstandard width for the SM Higgs can be at most 1.2 MeV; in other words the non-standard branching ratio for the SM Higgs must be less than 22%. One can use this result to constrain the parameter space of the model.
We will compute the following non-standard processes Each of the h 2 in the final state of these processes will decay into two dark photons and each dark photon will give rise to two leptons through its mixing with the photon 2 . These non-standard processes will provide multiple leptons in the final state of the standard model Higgs decay [9]. The contribution to the heavier Higgs width from these non-standard processes is 3 Thus the total width of the heavier Higgs h 1 is modified as 1 For the case of M γ D > M Z , we will have M 2 γ D ,Z = (q ± p)/2 as was studied by the authors in [11]. 2 We note that h 2 can decay to SM particles as well through its mixing with h 1 and hence they are suppressed.
We take the branching ratio of h 2 → γ D γ D to be 100%. See discussion after Eq. (50). 3 The four lepton modes from the first term h 1 → γ D γ D followed by γ D → ll (l = e, µ) were studied in details in [6].
whereΓ h is the width of the SM Higgs h, which has a theoretical value of 4.03 MeV. The branching ratio for the non-standard modes of the heavier Higgs decay is which should be constrained to be less than 22% or so. The partial decay width for the two body decays are given bŷ and For the three body decay with the following differential decay rate where the matrix element is given by with µ = m 2 2 /m 2 1 and x 1 + x 2 + x 3 = 2. The range of integration for x 1 and x 2 is confined by The matrix element for the three body process h 1 → h 2 γ D γ D is rather long, we will not present the expression here but it is included entirely in our numerical work. Now the dark Higgs h 2 decays into γ D γ D and SM particles with coefficients cos 2 α and sin 2 α respectively, so its branching fraction into γ D γ D is given by whereΓ(h 2 → γ D γ D ) can be obtained from Eq. (43) with the following substitution m 1 → m 2 , andΓ SM h 2 is the partial decay width of h 2 into SM particles. SinceΓ SM h 2 are suppressed by a factor of sin 2 α, the above branching fraction is close to unity.
In Fig. 1, the various regions of kinematics in the (m γ D , m 2 ) plane that exhibits the very rich Higgs phenomenology in this model are schematically shown. The different regions can be described briefly as follows: • Clearly, the two lines 2m γ D = m 1 (left of which h 1 → γ D γ D is open) and 2m 2 = m 1 (below of which h 1 → h 2 h 2 is open) defines our region of interest (un-shaded).
• Other lines correspond to 2, 4, or 6 dark photons coming from the decays of to the left of the 5 lines 2m γ D +m 2 = m 1 , 2m γ D +2m 2 = m 1 , 4m γ D + m 2 = m 1 , 4m γ D = m 1 and 6m γ D = m 1 correspond to the openings of the 5 • Lastly, the special line m 2 = 2m γ D emanated from the coordinate origin separates the γ D γ D pair coming from either a on-shell h 2 or off-shell h * 2 for these multi-γ D processes.
Since the dark photon γ D will mix with the photon, through either kinetic mixing [5] or a gauge invariant Stueckelberg mass term [16], it will communicate with the SM fermions eventually. If the dark photon mass is larger than twice the electron or muon mass, theses processes will lead to multileptons in the final states of the h 1 decay. These lepton jets can be distinguished from the QCD jets by imposing cuts on the electromagnetic ratio and charge ratio, as proposed in [9]. Supersymmetric models with or without R-parity can also give rise to multilepton events as experimentalists had searched for such signals and placed exclusion limits on the masses of supersymmetric particles [17]. LHC search for multilepton Higgs decay modes in the dark portal model will be discussed later in Sec. 6.

V. BRANCHING RATIOS
In our numerical work, we will restrict our interest where both the dark photon and dark Higgs have masses smaller than 126 GeV. In particular, we will pay special attention to the small mass region where their masses are in the range of 0.5 to a few GeV. In this range, final states of τ pair and light quarks pairs (pion and kaon pairs) from the dark photon decay are also possible, but they are harder to detect at the LHC.
Limit for invisibly decay of a Higgs boson with mass as low as 1 GeV had been reported by OPAL [18] 4 . For a 1 GeV Higgs boson mass, an upper limit for the mixing angle of |α| ≤ 3 × 10 −2 can be extracted from the Fig. 5 in Ref. [18]. However the exclusion curve on the Higgs mass versus mixing angle plot given in [18] was obtained under the assumption that invisible branching ratio of the Higgs boson decay is 100%. Relaxing this assumption would lead to larger mixing angle for a given Higgs mass. In the present case, the branching ratio in Eq. (42) must be less than 20% or so.
In Fig. 2, we plot the fundamental couplings λ Φ , λ χ and λ Φχ that entered in the Lagrangian density and their combination (4λ Φ λ χ −λ 2 Φχ ) as function of (m γ D , m 2 ) in the small mass region up to 5 GeV with fixed values of g D = 0.01 and α = 0.03. As one can easily see that λ Φ is not sensitive to these input parameters and very close to its SM value of m 2 1 /2v 2 = 0.13. We note the following hierarchy λ χ λ Φχ λ Φ in this small mass region from the first three plots of this figure. Moreover, the positiveness of the combination (4λ Φ λ χ − λ 2 Φχ ) in the last plot of this figure implies the scalar potential is bounded from below at tree level.
In Fig. 3, we plot the trilinear couplings λ 4 that are relevant to the three body processes h 1 → h 2 γ D γ D and h 1 → h 2 h 2 h 2 as function of (m γ D , m 2 ) in the small mass region up to 5 GeV with fixed values of g D = 0.01 and α = 0.03. We note that the matrix element (Eq. (47)) for the three body process h 1 → h 2 h 2 h 2 involves one term proportional to the quartic coupling λ respectively, while the matrix element for the three body process 3 sin α or g 4 D sin α cos α. The dark gauge coupling g D alone in general is not too severely constrained by experiments 5 . On the 3 /v and λ other hand, since the 126 GeV new boson observed at the LHC behaves very much SM-like, the mixing angle α is constrained to be quite small. Thus the three body decay h 1 → h 2 γ D γ D is expected to be more relevant than h 1 → h 2 h 2 h 2 . In our analysis, we include both of these three body modes and find that the mode h 1 → h 2 h 2 h 2 is indeed negligible.

VI. MULTILEPTON JETS AT THE LHC
We will study some collider signatures for the model in this section. In particular, we will focus on the 4 lepton-jets and 2 lepton-jets modes in our analysis. We consider the following four processes which may lead to signals of multilepton jets at the LHC: FIG. 6. Some topologies of 4 (left) and 2 (right) lepton-jets for process III. The 4 lepton-jets can also be coming from the SM of process I with h 1 replaced by the SM h. The immediate state of h 2 h 2 for the 2 lepton-jets is not shown since the branching ratio for h 2 → l + l − is very tiny.
We compute the matrix elements of these processes using FeynRules 6 [20,21] and Mad-Graph [22]. We pass these matrix elements to the event generator MadEvent [23] to obtain our event samples. The set of parton distribution functions used is CTEQ6L1 [24]. For illustration, we will choose several benchmark points in the dark portal as shown in Table I. If the kinetic mixing parameter is smaller than 10 −5 , the dark photon will have a very long lifetime and it may decay outside the detector. We will choose it to be 10 −4 as used by previous analyses by theorists [9] as well as experimentalists [25]. The mass of dark photon is chosen to be less than 2 GeV in these benchmark points. With such relatively low mass the opening angle of the lepton pair from the decay of the dark photon will be small which may lead to multilepton jets. Such low mass dark photon may also be desirable for indirect dark matter searches, since the allowed decay γ D → e + e − may be used to explain the positron excess [26][27][28], while γ D → pp is kinematically disallowed in accord with observation that the cosmic anti-proton flux is consistent with the background [26,29]. We also choose sin 2 α = 10 −3 in consistent with the analysis of the invisible branching ratio of h 1 in previous section (see also [30] and [31,32]). At these benchmark points, we see from the last three columns of Table I that (1) the invisible decay branching ratio of the SM Higgs is consistent with global fit results, (2) the decay of the dark Higgs is almost 100% into pair of dark photons, and (3) the branching ratio of the dark photon into light lepton pairs can be as large as 50%. Due to the smallness of the two mixing parameters α and , the production cross section of h 1 at the LHC remains to be very close to its SM value.
For the kinematic cuts for the 2 and 4 lepton-jets, we follow Refs. [6,9] and [25]. For the basic cuts that we will impose in all processes, we have Basic cuts: (4 leptons case) p T l ≥ 20, 10, 10, 10 GeV, |η l | < 2.3; (8 leptons case) p T l ≥ 20, 10, 10, 10, 0, 0, 0, 0 GeV, |η l | < 2.3, where p T l and η l are the transverse momenta and pseudo-rapidity of the lepton respectively. On top of the basic cuts, we employ the following lepton-jets cuts 4 lepton-jets cuts: Here ∆R d jj denotes the cone radius between two different lepton-jets and ∆R s ll denotes the cone radius between two different leptons in the same lepton jet, as depicted in Fig. 8. M Invariant denotes the invariant mass of all final state particles due to the decay chain of the SM Higgs boson resonance, give or take 10 GeV from the central value of 126 GeV.
The number of events versus the total invariant mass M Invariant for the four processes I, II, III and IV at the LHC-14 without any cuts are shown in Fig. 9 for the benchmark point B. We can see that before imposing any cuts the number of events around the Higgs boson resonance for the two processes III (red) and IV (yellow, 8 leptons) from the dark portal can stand above the SM processes of I (blue) and II (black). However away from the resonance region, the 4 leptons SM background from process I (black) is 2 to 3 order of magnitudes above the signals from process IV (green).
We now discuss the impact of imposing the multilepton jets cuts on the cross sections. The topologies of imposing the 4 and 2 lepton-jets cuts for processes III and IV are shown in Figs. 6 and 7 respectively. In Table II LHC-14 with the basic, 4 and 2 lepton-jets cuts for the 4 benchmark points listed in Table I. The following statements can be drawn from the results shown in Table II: • The 4 and 2 lepton jets cuts have strong and different impact for the SM processes I and II. For process I, since the intermediate state is the Z boson with a relatively high mass, its decay products can be produced at a relatively large angle with respect to the original Z boson direction. Thus it favors 4 lepton-jets in the final state (see left diagram in Fig. 6) and 2 lepton-jets is vanishing small for process I. On the other hand, SM process II has a cross section of about 700 times larger than process I with just the basic cuts imposed. Imposing the 4 and 2 lepton-jets cuts reduce the cross section of process II by a factor of 4.7 × 10 −3 and 1.1 × 10 −3 respectively. We note that the ZZ intermediate state in process II arises from the tree level parton processes of quark-quark annihilation while in process I it is connected with the loop-induced gluon fusion mechanism of Higgs production.
• For process III since the dark photon mass is small (1.5 GeV for benchmark points A and C, and 1.8 GeV for benchmark points B and D) the contribution from intermediate state of γ D γ D will give rise mainly to 2 lepton jets (see right diagram in Fig. 6). Thus imposing the 4 lepton jets cuts for process III will suppress this intermediate state and only the contribution from ZZ intermediate state will survive (see left diagram in Fig. 6). Since this ZZ contribution is very similar to the SM process I, they should have very similar cross sections after imposing 4 lepton-jets cuts as clearly seen in Table II. On the other hand, imposing 2 lepton-jets cuts will suppress the ZZ intermediate state but keep the γ D γ D . However, the contribution of ZZ intermediate state for process III is negligible. The 2 lepton-jets cross sections of process III are several orders of magnitudes larger than the corresponding cross sections of SM process II.
• For process IV, with just basic cuts its cross section is about a factor 4 (benchmark points A and B) to 5 (benchmark points C and D) smaller than that of process III. However, due to the small mass of the dark photon (compared with Z boson mass), one can has either 4 or 2 lepton-jets in the final state. Imposing the 4 and 2 lepton-jets cuts in addition to the basic cuts for process IV have more nontrivial effects on the cross section depending on the benchmark points. For 4 lepton-jets the cross sections can reach about 3 and 1 femtobarn for benchmark points C and D respectively. For 2 lepton-jets, the cross section can reach 2 femtobarn for benchmark point A only. At these benchmark points, these cross sections are an order of magnitude larger than the corresponding cross sections of the SM process II. Other benchmark points have negligible cross sections for 4 and 2 lepton-jets as can be clearly seen in the last column of Table II.

VII. CONCLUSIONS
We have studied a simple extension of the SM by adding a dark sector described by the original Abelian Higgs model. The communication between the visible sector and the dark sector is due to the mixing between the SM and dark Higgses and/or mixing between the SM and dark photons. We study various non-standard decay modes of the heavier Higgs h 1 in this model, identified as the 126 GeV new boson observed recently at the LHC. Multilepton modes in the final states of this heavier Higgs decay are possible. For the case of h 1 → h 2 h 2 followed by h 2 → γ D γ D and γ D →ll, there could be eight leptons in the final states. The three body process h 1 → h 2 γ D γ D is found to be significant and could lead to eight leptons final state as well. On the other hand, the other three body process h 1 → h 2 h 2 h 2 has an insignificant branching ratio; otherwise, it would lead up to a even more spectacular twelve leptons final state. The signals of 4 and 2 lepton-jets in this model are already quite unique and spectacular. We show that there are parameter space in this simple dark portal model satisfying the current constraint of the non-standard decay width of the 126 GeV Higgs and may give rise to interesting signals of multilepton jets at the LHC-14. Experiments at the LHC should therefore search for multilepton modes in the Higgs decay in order to probe for the possible existence of a U (1) D dark sector governed by the original Abelian Higgs model.