Mirror fermion model with electroweak scale non-sterile right-handed neutrinos is an interest-ing extension of the SM. Aside from its aesthetically appealinterest-ing to restorinterest-ing parity symmetry at higher energy scale, it can have immediate impacts for experiments in both complementary frontiers of high energy and high intensity searching for new physics of CLFV.
In this chapter, we have explored the low-energy experimental constraints on the pa-rameter space of the EW-nRmodel. We find out that the current experimental limits on the electron and the neutron EDMs impose the most stringent constraints on the parameter space of the model, as compared with CLFV process observables likeµ ! eg and µ e conversion in nuclei. However, projected sensitivities ofµ ! eg from MEG and of µ e conversion experiments at Mu2e, Mu2e II, COMET and PRISM, can provide comparable if not more stringent constraints on the parameters of the mirror fermion model.
The region of parameter space that can “explain" the muon anomalous magnetic dipole moment in the mirror fermion model is not favored by the current limits of these CLFV processes and the electron EDM from various experiments, which suggest much smaller couplings of order 10 4to 10 5.
On the other hand, the parameter space that can be probed by current and near future experiments for these CLFV processes and the electron EDM is close to the region where the mirror leptons, when produced at the LHC, have the decay length of about 1 mm. Besides missing energies, the search strategies for these mirror leptons at the LHC [23] may have to include displaced vertices located at distances from 1 mm to 1 cm away from the beam axises. It is interesting to note that SM background is expected to be small in this region and signatures for mirror leptons could be distinctive. It is also interesting to see how, in the mirror model, low energy experiments (rare processes, electron EDM) guide the direct searches at high energy experiments (like the LHC) for new particles such as the mirror leptons. Similar comments for the mirror quarks can be obtained.
Chapter 5
Double Higgs Boson Production in G2HDM at the LHC
5.1 The G2HDM Model
5.1.1 Motivation
It is certainly not the end of particle physics after the 125 GeV Higgs boson, the long sought last particle in SM, was discovered at the LHC in 2012. We still need to determine whether the observed Higgs boson is indeed the one predicted in SM. Moreover, many questions in SM, such as whether Higgs boson is a CP-even or CP-odd particle, what is the shape of the Higgs potential, are there any other scalars in Nature, the naturalness and hierarchy problems, matter-antimatter asymmetry, neutrino masses, dark matter, dark energy, etc. remain unanswered. To address these issues many models beyond the Standard Model (BSM) had been proposed by just extending the scalar sector of the SM. Among the many BSM models in the literature, the general two Higgs doublet model (2HDM) is a simple extension [116] by just adding one more Higgs doublet to the SM. One particular type of 2HDM, the inert Higgs doublet model (IHDM) [117–120], proposed the neutral component of the second Higgs doublet to be a DM candidate. The stability of this DM candidate is ensured by imposing a discrete Z2symmetry on the scalar potential of the model. However, it is commonly believed that, regardless of being discrete or continuous, global symmetry is considered to be strongly violated by gravitational effects [121, 122]. To remedy these unappealing features, a recent study in [10] has proposed a model called Gauged Two Higgs Doublet model (G2HDM) in which the two Higgs doublets H1and H2are gauged to form a doublet H = (H1,H2)Tof a new non-abelian SU(2)H gauge group. The neutral component of H2is stable under protection of SU(2)H gauge symmetry and hence can be DM candidate.
94 Double Higgs Boson Production in G2HDM at the LHC
However, we need to introduce more particles in G2HDM including a SU(2)H doublet, a SU(2)H triplet, and heavy SU(2)L singlet Dirac fermions. Moreover, the SM right-handed fermions are grouped with new heavy right-handed fermions to form SU(2)Hdoublets. After SSB, the VEV of SU(2)H doublet gives masses to these heavy fermions. In addition, in order to simplify the Higgs potential (which will be discussed more in 5.1.3), an abelian gauge group U(1)X was also introduced. Let us list the crucial features of G2HDM here:
• It is free of gauge and gravitational anomalies;
• It is renormalizable;
• Without resorting to the previous ad-hoc Z2symmetry, an inert Higgs doublet H2can be naturally realized, providing a DM candidate;
• No flavour changing neutral currents at tree level for the SM sector;
• the VEV of the triplet can trigger SU(2)L symmetry breaking while that ofFH can provide masses to the new fermions through SU(2)H-invariant Yukawa couplings;
• etc.
Phenomenology of G2HDM at the LHC had been explored previously in [10, 123] for Higgs physics and in [124] for the new gauge bosons. Recently, a detailed study on theoretical and Higgs phenomenological constraints in G2HDM has been presented in [125].
In the following sections we will briefly review the matter content, the mass spectra and various theoretical and Higgs phenomenological constraints in G2HDM.
5.1.2 The Matter Content
The gauge groups in the G2HDM consist of SU(3)C⇥ SU(2)L⇥U(1)Y⇥ SU(2)H⇥U(1)X. The scalar sector includes not only the two SU(2)L Higgs doublets H1, H2 which form a doublet H = (H1,H2)T in the SU(2)H gauge group, but also a tripletDH and a doubletFH
of this new gauge group. Note that,DHandFH are both singlets under the SM gauge group.
Furthermore, H andFH are assigned to carry U(1)X charge. For the fermion sector, the SM left-handed SU(2)L doublets are singlets under SU(2)H, while the SM right-handed SU(2)L singlets are now paired up with new right-handed singlets to form doublets under SU(2)H. Furthermore new heavy left-handed fermions are needed which are singlets under both SU(2)Land SU(2)H gauge groups. In Table 5.1, we summarize the matter content and their quantum number assignments in G2HDM.
5.1 The G2HDM Model 95 Matter Fields SU(3)C SU(2)L SU(2)H U(1)Y U(1)X
QL= (uL dL)T 3 2 1 1/6 0
UR= uR uHR T 3 1 2 2/3 1
DR= dRH dR T 3 1 2 1/3 1
uHL 3 1 1 2/3 0
dLH 3 1 1 1/3 0
LL= (nL eL)T 1 2 1 1/2 0
NR= nR nRH T 1 1 2 0 1
ER= eHR eR T 1 1 2 1 1
nLH 1 1 1 0 0
eHL 1 1 1 1 0
H = (H1 H2)T 1 2 2 1/2 1
DH=✓ D3/2 Dp/p 2 Dm/p
2 D3/2
◆
1 1 3 0 0
FH= (F1 F2)T 1 1 2 0 1
Table 5.1 Matter content and their quantum number assignments in G2HDM.
5.1.3 Higgs Potential
The most general Higgs potential which invariant under both SU(2)L⇥U(1)Y and SU(2)H⇥ U(1)X can be written down as follows
VT =V (H) +V (FH) +V (DH) +Vmix(H,DH,FH) , (5.1) where
V (H) =µH2 HaiHai +lH HaiHai 2+1
2lH0 eabegd HaiHgi ⇣
Hb jHd j⌘ ,
=µH2⇣
H1†H1+H2†H2⌘ +lH
⇣H1†H1+H2†H2⌘2
+lH0 ⇣
H1†H1H2†H2+H1†H2H2†H1⌘ , (5.2) with (a, b, g, d) and (i, j) refer to the SU(2)H and SU(2)L indices respectively, all of which run from one to two, and Hai=Hai⇤ ;
V (FH) =µF2F†HFH+lF⇣
F†HFH⌘2
,
=µF2(F⇤1F1+F⇤2F2) +lF(F⇤1F1+F⇤2F2)2 , (5.3)
96 Double Higgs Boson Production in G2HDM at the LHC
whereFH = (F1F2)T;
V (DH) = µD2Tr D2H +lD Tr D2H 2 ,
= µD2✓1
2D23+DpDm
◆
+lD✓1
2D23+DpDm
◆2
, (5.4)
where
DH= D3/2 Dp/p 2 Dm/p
2 D3/2
!
=D†H with Dm= (Dp)⇤ and (D3)⇤=D3; (5.5)
and the last term
Vmix(H,DH,FH) = +MHD⇣
H†DHH⌘
MFD⇣
F†HDHFH⌘ +lHF⇣
H†H⌘⇣
F†HFH
⌘+lHF0 ⇣ H†FH
⌘⇣F†HH⌘
+lHD⇣ H†H⌘
Tr D2H +lFD⇣ F†HFH
⌘Tr D2H . (5.6)
Eq. (5.6) can be expanded further in terms of the component fields of H,DH and FH as follows
Vmix(H,DH,FH) = +MHD✓ 1
p2H1†H2Dp+1
2H1†H1D3+ 1
p2H2†H1Dm 1
2H2†H2D3
◆
MFD✓ 1
p2F⇤1F2Dp+1
2F⇤1F1D3+ 1
p2F⇤2F1Dm 1
2F⇤2F2D3
◆
+lHF⇣
H1†H1+H2†H2⌘
(F⇤1F1+F⇤2F2) +lHF0 ⇣
H1†H1F⇤1F1+H2†H2F⇤2F2+H1†H2F⇤2F1+H2†H1F⇤1F2⌘ +lHD⇣
H1†H1+H2†H2⌘✓1
2D23+DpDm
◆
+lFD(F⇤1F1+F⇤2F2)✓1
2D23+DpDm
◆
. (5.7)
Note that without U(1)X imposed in this model, many unwanted terms, for example FTHDHFH, would be allowed in the above Higgs potential. We note that the coefficient of µD2in V (DH)has an opposite sign compared with the coefficients of µH2 andµF2 in V (H) and
5.1 The G2HDM Model 97 V (FH), respectively. We have the coefficients of the quadratic terms for H1and H2as
µH2 1
2MHD· vD+1
2lHD· v2D+1
2lHF· v2F, (5.8)
µH2 +1
2MHD· vD+1
2lHD· v2D+1
2(lHF+lHF0 )· v2F, (5.9) respectively. It is easy to see that these coefficients can be either positive or negative values, even with a positive µH2. So if we choose Eqs. (5.8) and (5.9) to be negative and positive respectively, one can achieve hH1i 6= 0 and hH2i = 0 to break SU(2)L.
Similarly, the coefficients of the quadratic terms forF1andF2are µF2+1
2MFD· vD+1
2lFD· v2D+1
2(lHF+lHF0 )· v2, (5.10) µF2 1
2MFD· vD+1
2lFD· v2D+1
2lHF· v2, (5.11)
respectively. As in the above cases of H1and H2, even with a positiveµF2, one can achieve hF1i = 0 and hF2i 6= 0 by making judicious choices of the parameters.
In (5.4), ifµD2>0, SU(2)H is spontaneously broken by the VEV hD3i = vD6= 0 with hDp,mi = 0 by using an SU(2)H rotation. In fact, this also triggers the symmetry breaking of the other gauge symmetries.
Note that the scalar potential in G2HDM is CP-conserving due to the fact that all terms in V (H), V (FH), V (DH)and Vmix(H,DH,FH)are Hermitian, implying all the coefficients are necessarily real.
5.1.4 Spontaneous Symmetry Breaking and Scalar Mass Spectrum
Spontaneous Symmetry Breaking
First, let us parameterize the fields as follows
H1= G+
v+hp 2 +ipG0
2
!
, H2= H+ H20
!
, FH= GHp
vFp+f2 2 +iGp0H
2
!
, DH =
vD+d3
2 p1
2Dp
p1
2Dm vD2d3
! . (5.12) where v, vF and vD are VEVs to be determined by minimization of the potential. The set YG⌘ {G0,G+,G0H,GHp} are Goldstone bosons.
98 Double Higgs Boson Production in G2HDM at the LHC
Then, inserting the VEVs, v, vF, vD into the potential VT in Eq. (5.1) leads to VT(v,vD,vF) = 1
4⇥lHv4+lFv4F+lDv4D+2 µH2v2+µF2v2F µD2v2D
MHDv2+MFDv2F vD+lHFv2v2F+lHDv2v2D+lFDv2Fv2D⇤ . (5.13) We will obtain the following equations by minimizing the potential in Eq. 5.13:
2lHv2+2µH2 MHDvD+lHFv2F+lHDv2D = 0 , (5.14) 2lFv2F+2µF2 MFDvD+lHFv2+lFDv2D = 0 , (5.15) 4lDv3D 4µD2vD MHDv2 MFDv2F+2vD lHDv2+lFDv2F = 0 . (5.16) By solving this set of coupled equations, one can get solutions for v, vF and vDin terms of other parameters in the potential.
Scalar Mass Spectrum
After SSB, we obtained three diagonal blocks in the mass matrix. The first 3 ⇥ 3 block with the basis of S = {h,f2,d3} given as
MH2 = 0 B@
2lHv2 lHFvvF 2v(MHD 2lHDvD) lHFvvF 2lFv2F v2F(MFD 2lFDvD)
v2(MHD 2lHDvD) v2F(MFD 2lFDvD) 4v1D 8lDv3D+MHDv2+MFDv2F 1 CA .
(5.17) This matrix can be diagonalized by an orthogonal matrix OH,
(OH)T · MH2· OH=Diag(m2h1,m2h2,m2h3) . (5.18) The lightest eigenvalue mh1 is the mass of h1 which is identified as the 125 GeV Higgs boson observed at the LHC, while mh2 and mh3 are the masses of heavier Higgses h2and h3respectively. The physical Higgs hi(i = 1,2,3) is a mixture of the three components of S: hi=OHjiSj. Thus the SM-like Higgs boson in this model is a linear combination of the neutral components of the two SU(2) doublets H1 andFH and the real component of the SU(2)H tripletDH.
5.1 The G2HDM Model 99
The second block is also 3 ⇥ 3. In the basis of D = {GHp,H20⇤,Dp}, it is given by
MD2= 0 B@
MFDvD+12lHF0 v2 12lHF0 vvF 12MFDvF
12lHF0 vvF MHDvD+12lHF0 v2F 12MHDv
12MFDvF 12MHDv 4v1D MHDv2+MFDv2F 1
CA . (5.19)
This matrix can also be diagonalized by an orthogonal matrix OD
(OD)T· MD2· OD=Diag(m2G˜p,m2D,m2D) . (5.20) One eigenvalue of Eq. (5.19) is zero (i.e. mG˜p =0) and identified as the unphysical Goldstone boson ˜Gp. The mDand m˜D(mD<m˜D) are masses of two physical fields D and ˜D respectively.
The D could be a DM candidate in G2HDM. We note that the neutral fieldsnLH,nRHor W0(p,m) can be a DM candidate as well, depending on which one is the lightest. In this work, we will assume D is the lightest among them.
The final one is a 4 ⇥ 4 diagonal block with m2H± =MHDvD 1
2lH0v2+1
2lHF0 v2F, (5.21) m2G± =m2G0 =m2G0
H=0 , (5.22)
where mH± is mass of the physical charged Higgs H±, and mG±,mG0,mG0
H are masses of the four Goldstone boson fields G±, G0and G0H, respectively. Note that we have used the minimization conditions Eqs. (5.14), (5.15) and (5.16) to simplify various matrix elements of the above mass matrices. The six Goldstone particles G±, G0, G0H and eGp,mwill be absorbed by the longitudinal components of the massive gauge bosons W±, Z, Z0 and W0(p,m) after SSB.
5.1.5 Theoretical and Higgs Phenomenological Constraints [125]
In this subsection, we summarize the theoretical and Higgs phenomenological constraints on the parameters space of the model which have been studied in [125].
From the study in [125], the allowed parameter space in the model have been identified under the theoretical constraints from vacuum stability (VS), perturbative unitarity (PU) and the 125 GeV Higgs physics (HP) constraints including the Higgs boson mass and signal strengths of Higgs boson decays into diphoton andt+t from the LHC. It turns out that among the eightl parameters only two of them lH andlHF are significantly constrained by (VS+PU+HP). Some of the couplings likelH0,lHF0 andlFD are loosely constrained. We
100 Double Higgs Boson Production in G2HDM at the LHC
note that some of the parameters such as MHD, MFD and the VEVs are constrained only by HP but not by (VS+PU). In the numerical results of [125], the two parameters MHD, MFD were set to be varied in the range of [ 1,1] TeV, vD2 [0.5,20] TeV, while v and vF were fixed to be 246 GeV and 10 TeV, respectively.
We show a summary of allowed regions of parameter space in Fig. 5.1. The diagonal panels indicate the allowed ranges of the eight couplingslH,F,D,lH0, andlHF,HD,FD,lHF0 under the combined constraints of (VS+PU+HP). The upper red triangular block corre-sponds to (VS+PU) constraints, while the lower magenta triangular block correcorre-sponds to the (VS+PU+HP) constraints.