We set up a beyond SM model which include a kinetic mixing between U (1)Y and U (1)D, and the kinetic mixing mechanism will produce non-SM process in multi-leptons final state, that is a topic in this section. In Sec.V A we set up our model. Phenomenology based on similar model has been studied before, see for example Refs. [7–9, 33, 34] and references therein. In Sec.V B we consider mixing effects in the scalar sector as well as the gauge boson sector. We show the hDh mixing in detail and present all the relevant trilinear and quadrilinear couplings of the physical h1 and h2 bosons. We also briefly discuss the mixings between the three neutral gauge bosons in the model as studied previously in Ref. [9]. In Sec.V C we discuss the possible decay modes of the SM Higgs outside those of the SM and their several kinematic regions. In Sec.V D we present numerical results for various branching ratios of the non-standard decay modes of the SM Higgs, identified here as h1 . In Sec.V E we study the signals of multilepton jets of the model at the LHC-14. We conclude in Sec.V F.
A. SU (2)L× U (1)Y × U (1)D Model
We extend the electroweak SM by including the original Abelian Higgs model for a dark U (1)D [7, 8, 33, 34]. The bosonic part of the Lagrangian density is
LB= Lgauge+ Lscalar (152)
with
Lgauge = −1 4
W~µν · ~Wµν − 1
4BµνBµν −1
4CµνCµν−
2BµνCµν , (153) Lscalar = |DµΦ|2+ |Dµχ|2− Vscalar(Φ, χ) , (154) and
DµΦ =
∂µ+ ig1
2σaWaµ+ i1 2g0Bµ
Φ , (155)
Dµχ = (∂µ+ igDCµ) χ , (156)
where ~Wµ, Bµand Cµare the gauge potentials of the SU (2)L, U (1)Y and U (1)D with gauge couplings g, g0 and gD respectively, and is the kinetic mixing parameter between the two
U (1)s [5]. The scalar potential in (154) is given by Vscalar = −µ2ΦΦ†Φ + λΦ Φ†Φ2
− µ2χχ∗χ + λχ(χ∗χ)2+ λΦχ Φ†Φ (χ∗χ) . (157) We pick the unitary gauge and expand the scalar fields around the vacuum
Φ(x) = 1 with the VEVs v and vD fixed by minimisation of the potential to be
Vscalar(v, vD) = 1
In terms of the shifted fields h and hD, the scalar potential Vscalar can then be decomposed as
Here V0 is a cosmological constant and will be discarded from now on; the tadpole term V1 vanishes with v and vD given by Eq.(160); V2 is quadratic in the fields h and hD, and we have to diagonalize the mass matrix MS2 in Eq.(163) to get the physical Higgs fields h1 and h2 (see next section); and V3 and V4 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.
B. Mixing Effects
1. Higgs Mass Eigenstates and Their Self Interactions
The mass matrix MS2 in Eq.(163) for the scalar bosons is MS2 =
We will identify the heavier Higgs h1 with mass m1 = 126 GeV as the new boson observed at the LHC [35, 36], while the lighter one h2 has been escaped detection thus far.
In terms of the physical Higgs fields h1 and h2, the cubic term V3 is given by
and the quartic term V4 is given by
2. Kinetic and Mass Mixing of the neutral gauge bosons
In additional to the mass mixing of the three neutral gauge bosons arise from the spon-taneously electroweak symmetry breaking given by
Lm = 1
with the following mass mixing matrix
M2 =
we also have the kinetic mixing between the two U(1)s from the last term in Eq.153. Both the kinetic and mass mixings can be diagonalized simultaneously by the following mixed transformation [76]
where AD, , Z and A are the physical dark photon, Z boson and the photon respectively.
Here K is a general linear transformation that diagonalizes the kinetic mixing
K = with the mixing angles defined as
tan θ = gY After the K transformation, the gauge bosons mass matrix is
M˜2 = KT · M2· K =
The O matrix diagonalize this ˜M2 matrix
MDiag2 = OT · ˜M2· O =
with the following eigenvalues (assuming mγD ≤ mZ5) m2γ = 0, m2z,γ For small kinetic parameter mixing , the Z and γD masses can be approximated by mz ≈p(g2+ gY2)v/2 and mγD ≈ gDvD.
5 For the case of mγD > mz, we will have m2z,γ
D = (q ∓ p)/2 the case which has been studied previously [76], which diagonal method is different with [33, 34]
C. Non-standard Decays of h1
The global fits [9, 11, 12, 60] 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 non-standard 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 h1 → γDγD, h1 → h2h2, h1 → h2h∗2 → h2γDγD and h1 → h2h2h2. Each of the h2 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 6. These non-standard processes will provide multiple leptons in the final state of the standard model Higgs decay [7]. The contribution to the heavier Higgs width from these non-standard processes is 7
ΓN Sh1 = sin2αˆΓ(h1 → γDγD)+Γ(h1 → h2h2)+Γ(h1 → h2γDγD)+Γ(h1 → h2h2h2)+· · · (192) Thus the total width of the heavier Higgs h1 is modified as
Γh1 = cos2αˆΓh+ ΓN Sh
1 , (193)
where ˆΓh 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
BhN S1 = ΓN Sh
1
Γh1 , (194)
which should be constrained to be less than 22% or so. The partial decay width for the two body decays are given by
Γ(hˆ 1 → γDγD) = gD2m2γ
6 We note that h2can decay to SM particles as well through its mixing with h1and hence they are suppressed.
We take the branching ratio of h2→ γDγD to be 100%. See discussion after Eq.(202).
7 The four lepton modes from the first term h1 → γDγD followed by γD → l¯l (l = e, µ) were studied in details in [33].
For the three body decay h1 → h2h2h2, we obtain
with the following differential decay rate dΓ (h1 → h2h2h2)
dx1dx2 = m1
1536π3|M|2 (198)
where the matrix element is given by M = λ(3)4 + 1 present the expression here but it is included entirely in our numerical work.
Now the dark Higgs h2 decays into γDγD and SM particles with coefficients cos2α and sin2α respectively, so its branching fraction into γDγD is given by
B(h2 → γDγD) = cos2αˆΓ(h2 → γDγD) cos2αˆΓ(h2 → γDγD) + sin2αˆΓSMh
2
, (202)
where ˆΓ(h2 → γDγD) can be obtained from Eq.(195) with the following substitution m1 → m2, and ˆΓSMh2 is the partial decay width of h2 into SM particles. Since ˆΓSMh2 are suppressed by a factor of sin2α, the above branching fraction is close to unity.
• Clearly, the two lines 2mγD = m1 (left of which h1 → γDγD is open) and 2m2 = m1
(below of which h1 → h2h2 is open) defines our region of interest (un-shaded).
• Below the line 3m2 = m1 , the 3-body process h1 → h2h2h2 is open too.
2m2=m1
FIG. 17: The kinematical regions in the (mγD, m2) plane for the non-standard decays of the heavier Higgs h1, identified as the 126 GeV boson observed at LHC. See the last paragraph of
section 4 for illustrations.
• Other lines correspond to 2, 4, or 6 dark photons coming from the decays of h2 in h1 → h2h2orh1h2h2h2 : i.e. to the left of the 5 lines 2mγD + m2 = m1 , 2mγD + 2m2 = m1
, 4mγD + m2 = m1 , 4mγD = m1 and 6mγD = m1 correspond to the openings of the 5 processes h1 → h2γDγD , h1 → h2h2γDγD , h1 → h2γDγDγDγD , h1 → γDγDγDγD and h1 → γDγDγDγDγDγDγDγD respectively.
• Lastly, the special line m2 = 2mγD emanated from the coordinate origin separates the γDγD pair coming from either a on-shellh2 or off-shell h2 for these multi-γD processes.
Since the dark photon D will mix with the photon, through either kinetic mixing [5] or a gauge invariant Stueckelberg mass term[13], 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 [7]. 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 [14]. LHC search for multilepton Higgs decay modes in the dark portal model will be discussed later in Sec.V E
D. 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 [38] 8. 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. [38]. However the exclusion curve on the Higgs mass versus mixing angle plot given in [38] 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.(194) must be less than 20% or so.
In Fig.18, we plot the fundamental couplings λΦ, λχ and λΦχ that entered in the La-grangian density and their combination (4λΦλχ− λ2Φχ) as function of (mγD, m2) in the small mass region up to 5 GeV with fixed values of gD = 0.01 and α = 0.03. As one can easily see that λΦ is not sensitive to these input parameters and very close to its SM value of m21/2v2 = 0.13. We note the following hierarchy λχ λΦχ λΦ in this small mass re-gion from the first three plots of this figure. Moreover, the positiveness of the combination (4λΦλχ − λ2Φχ) in the last plot of this figure implies the scalar potential is bounded from below at tree level.
In Fig.19, we plot the trilinear couplings λ(2)3 /v, λ(3)3 /v and λ(4)3 /v normalized to the VEV v, and the quadrilinear coupling λ(3)4 that are relevant to the three body processes h1 → h2γDγD and h1 → h2h2h2 as function of (mγD, m2) in the small mass region up to
8 We would like to thank W. Y. Keung bringing us the attention of this experimental paper.
ΛF (Eq.(199)) for the three body process h1 → h2h2h2 involves one term proportional to the quartic coupling λ(3)4 and two other terms proportional to the product of cubic couplings λ(2)3 · λ(3)3 and λ(3)3 · λ(4)3 respectively, while the matrix element for the three body process h1 → h2γDγD involves diagrams proportional to gD2 sin α cos α, gD2λ(3)3 cos α, gD2λ(4)3 sin α or gD4 sin α cos α. The dark gauge coupling gD alone in general is not too severely constrained by experiments 9. On the 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 h1 → h2γDγD is expected to be more relevant than h1 → h2h2h2. In our analysis, we include both of these three body modes and find that the mode h1 → h2h2h2 is indeed negligible.
9 At the low mass region of the dark photon and dark Higgs that we are interested in, the BABARexperiment [39] had only obtained the limit for the product αD· 2, where αD = gD2/4π and is the kinetic mixing parameter in Eq.(153), as a function of the dark Higgs mass or dark photon mass.
Λ3H2Lv
In Fig.4, we plot the contour of the non-standard branching ratio BhN S
1 (Eq.(194)) = 0.1 (left) and 0.2 (right) of the heavier Higgs h1 in the (mγD, m2) plane up to 126 GeV in both directions with the following parameter input: sin2α = 0.0009 and gD = 0.05, 0.1, 0.2, 0.4 and 0.8. Comparing with the kinematics regions shown in Fig.(17), we can find out what processes are contributing in a given region.
In Fig.21, we plot the contour of the non-standard branching ratio BhN S1 (Eq.(194)) = 0.1 (left) and 0.2 (right) of the heavier Higgs h1 in the (mγD, m2) plane for the small mass region of 0.5 to 5 GeV in both directions for sin2α = 0.0009 and gD = 0.005, 0.009, 0.013 and 0.017.
FIG. 20: Contour plot of the non-standard branching ratio BN Sh
1 (Eq.(194)) = 0.1 (left) and 0.2 (right) of the heavier Higgs h1 in the (mγD, m2) plane up to 126 GeV in both directions for
sin2α = 0.0009 and gD = 0.05, 0.1, 0.2, 0.4 and 0.8.
FIG. 21: Contour plot of the non-standard branching ratio BN Sh
1 (Eq.(194)) = 0.1 (left) and 0.2 (right) of the heavier Higgs h1 in the small mass region of 0.5 to 5 GeV in the (mγD, m2) plane for
sin2α = 0.0009 and gD = 0.005, 0.009, 0.013 and 0.017.
E. MultiLepton-Jets At 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:
(I) pp → h1 → ZZ → l+l−l+l−
(II) pp → V V → l+l−l+l− (V V = ZZ, γγ, Zγ)
(III) pp → h1 → XX → l+l−l+l− (XX = ZZ, γDγD , h2h2 ) (IV) pp → h1 → h2h2 → γDγDγDγD → l+l−l+l−l+l−l+l−
where l = eorµ. Processes (I) and (II) are coming entirely from the SM, process (III) can be arise from either SM (with modified Higgs-ZZ coupling) or the dark portal (see Fig.23),and process (IV) is purely from the dark portal (see Fig.23).
FIG. 22: 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 h1 replaced by the SM h. The immediate state
of h2h2 for the 2 lepton-jets is not shown since the branching ratio for h2 → l+l− is very tiny.
We compute the matrix elements of these processes using FeynRules10 [17, 18] and Mad-Graph [16]. We pass these matrix elements to the event generator MadEvent [19] to obtain our event samples. The set of parton distribution functions used is CTEQ6L1 [20].
FIG. 23: Some topologies of 4 (left) and 2 (right) lepton-jets for process IV.
For illustration, we will choose several benchmark points in the dark portal as shown in TableV. If the kinetic mixing parameter is smaller than 105, the dark photon will have a very long lifetime and it may decay outside the detector. We will choose it to be 104 as used by previous analyses by theorists [7] as well as experimentalists [35]. 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
10 We include both gluon and photon fusion ggh1 and γγh1 computed at next-to-leading-order.
TABLE V: Several benchmark points of the dark portal used to calculate the signals of multilepton jets. ( = 10−4andsin2α = 10−3)
Point gD sin2α MγD M2 Brh1→Dark Brh2→γDγD BrγD→l+l−
A 0.005 10−3 1.5 4 ∼ 16% 99% 50%
B 0.009 10−3 1.8 10 ∼ 20% 100% 50%
C 0.005 10−3 1.5 40 ∼ 15% 99% 50%
D 0.005 10−3 1.8 40 ∼ 11% 99% 50%
lead to multilepton jets. Such low mass dark photon may also be desirable for indirect dark matter searches, since the allowed decay γD → e+e− may be used to explain the positron excess [40, 41], while γD → p¯p is kinematically disallowed in accord with observation that the cosmic anti-proton flux is consistent with the background [41]. We also choose sin2α = 103 in consistent with the analysis of the invisible branching ratio of h1 in previous section. At these benchmark points, we see from the last three columns of TableV 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 and , the production cross section of h1 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.[7, 33] and [35]. For the basic cuts that we will impose in all processes, we have
Basic cuts:
(4 leptons case) pTl ≥ 20, 10, 10, 10 GeV; |ηl| < 2.3,
(8 leptons case) pTl ≥ 20, 10, 10, 10, 0, 0, 0, 0 GeV; |ηl| < 2.3,
where pTl and ηl are the transverse momenta and pseudo-rapidity of the lepton respec-tively. On top of the basic cuts, we employ the following lepton-jets cuts
4 Lepton-Jets cuts: ∆Rdj
ijj > 0.7, ∆Rsl
ilj < 0.2, Minvariant= Mh1 ± 10 GeV, 2 Lepton-Jets cuts: ∆Rdj1j2 > 0.7, ∆Rsl
ilj < 0.2, Minvariant= Mh1 ± 10 GeV.
Here ∆Rjjddenotes the cone radius between two different lepton-jets and ∆Rsj
1j2 denotes the cone radius between two different leptons in the same lepton jet, as depicted in Fig.24.
MInvariant 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.25 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.22 and 23 respectively. In TableVI, we show the cross sections of the 4 processes at the
FIG. 24: Graphical illustrations for the kinematic cuts on the cone radius ∆R of final state leptons. The 2 and 4 lepton-jets cases are shown in the left and right figures respectively.
LHC-14 with the basic, 4 and 2 lepton-jets cuts for the 4 benchmark points listed in TableV. The following statements can be drawn from the results shown in TableVI
• 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.22) 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
FIG. 25: Number of events versus MInvariant [GeV] with the basic cuts for benchmark point B at LHC-14 with a fixed luminosity of 10f b−1 . Histogram of blue strip is for process I, black dash is for process II, red solid is for process III, yellow strip is for process IV of 8 leptons, and green
dash is for process IV of 4 leptons.
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 γDγD will give rise mainly to 2 lepton jets (see right diagram in Fig.22). 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.22).
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 TableVI. On the other hand, imposing 2 lepton-jets cuts will suppress the ZZ intermediate state but keep the γDγD . 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 TableVI.
TABLE VI: Cross sections (in unit of fb) at the LHC-14 for the background processes (I and II) and dark sector processes (III and IV) with the basic, 4 and 2 lepton-jets cuts at
the 4 benchmark points.
Cuts Decision I II III IV
A 0.118 70.7 95.3 23.2
Basic cuts only
B 0.118 70.7 204 45.8
C 0.118 70.7 96.7 19.2
D 0.118 70.7 68.3 13.1
A 9.63×10−3 0.337 9.86×10−3 ≤ 10−10 4 Lepton-Jets
B 9.63×10−3 0.337 9.80×10−3 ≤ 10−10 C 9.63×10−3 0.337 9.93×10−3 3.05 D 9.63×10−3 0.337 9.84×10−3 0.92
A ≤ 10−10 0.08 95.3 1.75
2 Lepton-Jets
B ≤ 10−10 0.08 201 ≤ 10−10
C ≤ 10−10 0.08 95.8 ≤ 10−10
D ≤ 10−10 0.08 68.2 ≤ 10−10
F. Conclusions for Multilepton Higgs Decays
F. Conclusions for Multilepton Higgs Decays