G2HDM: Gauged Two Higgs Doublet Model

Journal of High Energy Physics, Apr 2016

A novel model embedding the two Higgs doublets in the popular two Higgs doublet models into a doublet of a non-abelian gauge group SU(2) H is presented. The Standard Model SU(2) L right-handed fermion singlets are paired up with new heavy fermions to form SU(2) H doublets, while SU(2) L left-handed fermion doublets are singlets under SU(2) H . Distinctive features of this anomaly-free model are: (1) Electroweak symmetry breaking is induced from spontaneous symmetry breaking of SU(2) H via its triplet vacuum expectation value; (2) One of the Higgs doublet can be inert, with its neutral component being a dark matter candidate as protected by the SU(2) H gauge symmetry instead of a discrete Z 2 symmetry in the usual case; (3) Unlike Left-Right Symmetric Models, the complex gauge fields (W 1 ′  ∓ W 2 ′ ) (along with other complex scalar fields) associated with the SU(2) H do not carry electric charges, while the third component W 3 ′ can mix with the hypercharge U(1) Y gauge field and the third component of SU(2) L ; (4) Absence of tree level flavour changing neutral current is guaranteed by gauge symmetry; and etc. In this work, we concentrate on the mass spectra of scalar and gauge bosons in the model. Constraints from previous Z′ data at LEP and the Large Hadron Collider measurements of the Standard Model Higgs mass, its partial widths of γγ and Zγ modes are discussed.

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G2HDM: Gauged Two Higgs Doublet Model

Revised: March G2HDM: Gauged Two Higgs Doublet Model Wei-Chih Huang 0 1 3 5 7 Yue-Lin Sming Tsai 0 1 3 6 Tzu-Chiang Yuan 0 1 2 3 4 London 0 1 3 Hsinchu 0 1 3 Taiwan 0 1 3 0 Nangang , Taipei 11529 , Taiwan 1 Kashiwa , Chiba 277-8583 , Japan 2 Institute of Physics , Academia Sinica 3 44221 Dortmund , Germany 4 Physics Division, National Center for Theoretical Sciences 5 Fakultat fur Physik, Technische Universitat Dortmund 6 Kavli IPMU (WPI), University of Tokyo 7 Department of Physics and Astronomy, University College London A novel model embedding the two Higgs doublets in the popular two Higgs doublet models into a doublet of a non-abelian gauge group SU(2)H is presented. The Standard Model SU(2)L right-handed fermion singlets are paired up with new heavy fermions to form SU(2)H doublets, while SU(2)L left-handed fermion doublets are singlets under SU(2)H . Distinctive features of this anomaly-free model are: (1) Electroweak symmetry breaking is induced from spontaneous symmetry breaking of SU(2)H via its triplet vacuum expectation value; (2) One of the Higgs doublet can be inert, with its neutral component being a dark matter candidate as protected by the SU(2)H gauge symmetry instead of a discrete Z2 symmetry in the usual case; (3) Unlike Left-Right Symmetric Models, the complex gauge elds (W10 iW20) (along with other complex scalar elds) associated with the SU(2)H do not carry electric charges, while the third component W30 can mix with the hypercharge U(1)Y gauge eld and the third component of SU(2)L; (4) Absence of tree level avour changing neutral current is guaranteed by gauge symmetry; and etc. In this work, we concentrate on the mass spectra of scalar and gauge bosons in the model. Constraints from previous Z0 data at LEP and the Large Hadron Collider measurements of the Standard Model Higgs mass, its partial widths of Beyond Standard Model; Higgs Physics - and Z modes are discussed. 2.1 2.2 2.3 3.1 3.2 3.3 4.1 4.2 4.3 4.4 4.5 4.6 3 Spontaneous symmetry breaking and mass spectra Spontaneous symmetry breaking Scalar mass spectrum SU(2)H U(1)X gauge boson mass spectrum 4 Phenomenology Numerical solutions for scalar and gauge boson masses Z0 constraints Constraints from the 125 GeV SM-like Higgs Dark matter stability Dark matter relic density EWPT | S, T and U 5 Conclusions and outlook A Analytical expression for v, v and v B Decay width of SM Higgs to and Z 1 Introduction 2 G2HDM set up persymmetry (SUSY) chiral structure. In the inert two Higgs doublet model (IHDM) [6], since the second Higgs doublet is odd under a Z2 symmetry its neutral component can be { 1 { for in the SM since the CKM phase in the quark sector is too small to generate su cient baryon asymmetry [11{13], and the Higgs potential cannot achieve the strong rst-order electroweak phase transition unless the SM Higgs boson is lighter than 70 GeV [14, 15]. A general two Higgs doublet model (2HDM) contains additional CP -violation source [16{20] in the scalar sector and hence it may circumvent the above shortcomings of the SM for the baryon asymmetry. The complication associated with a general 2HDM [21] stems from the fact that there exist many terms in the Higgs potential allowed by the SM gauge symmetry, including various mixing terms between two Higgs doublets H1 and H2. In the case of both doublets develop vacuum expectation values (vevs), the observed 125 Higgs boson in general would be a linear combination of three neutral scalars (or of two CP -even neutral scalars if the Higgs potential preserves CP symmetry), resulting in avour-changing neutral current (FCNC) at tree-level which is tightly constrained by experiments. This Higgs mixing e ects lead to changes on the Higgs decay branching ratios into SM fermions which must be confronted by the Large Hadron Collider (LHC) data. One can reduce complexity in the 2HDM Higgs potential by imposing certain symmetry. The popular choice is a discrete symmetry, such as Z2 on the second Higgs doublet in IHDM or 2HDM type-I [22, 23], type-II [23, 24], type-X and type-Y [25{27] where some of SM fermions are also odd under Z2 unlike IHDM. The other choice is a continuous symmetry, such as a local U(1) symmetry discussed in [28{31]. The FCNC constraints can be avoided by satisfying the alignment condition of the Yukawa couplings [32] to eliminating dangerous tree-level contributions although it is not radiatively stable [33]. Alternatively, the aforementioned Z2 symmetry can be used to evade FCNC at tree-level since SM fermions of the same quantum number only couple to one of two Higgs doublets [34, 35]. Moreover, the Higgs decay branching ratios remain intact in the IHDM since H1 is the only doublet which obtains the vacuum expectation value (vev), or one can simply make the second Higgs doublet H2 much heavier than the SM one such that H2 essentially decouples from the theory. All in all, the IHDM has many merits, including accommodating DM, avoiding stringent collider constraints and having a simpler scalar potential. The required Z2 symmetry, however, is just imposed by hand without justi cation. The lack of explanation prompts us to come up with a novel 2HDM, which has the same merits of IHDM with an simpler two-doublet Higgs potential but naturally achieve that H2 does not obtain a vev, which is reinforced in IHDM by the arti cial Z2 symmetry. In this work, we propose a 2HDM with additional SU(2)H U(1)X gauge symmetry, where H1 (identi ed as the SM Higgs doublet) and H2 form an SU(2)H doublet such that the two-doublet potential itself is as simple as the SM Higgs potential with just a quadratic mass term plus a quartic term. The price to pay is to introduce additional scalars: one SU(2)H triplet and one SU(2)H doublet (which are all singlets under the SM gauge groups) with their vevs providing masses to the new gauge bosons. At the same time, the vev of the triplet induces the SM Higgs vev, breaking SU(2)L U(1)Y down to U(1)Q, while H2 do not develop any vev and the neutral component of H2 could be a DM candidate, whose stability is protected by the SU(2)H gauge symmetry and Lorentz invariance. In order { 2 { to write down SU(2)H U(1)X invariant Yukawa couplings, we introduce heavy SU(2)L singlet Dirac fermions, the right-handed component of which is paired up with the SM right-handed fermions to comprise SU(2)H doublets. The masses of the heavy fermions come from the vev of the SU(2)H doublet. In this setup, the model is anomaly-free with respect to all gauge groups. In what follows, we will abbreviate our model as G2HDM, the acronym of gauged 2 Higgs doublet model. We here concentrate on the scalar and additional gauge mass spectra of G2HDM and various collider constraints. DM phenomenology will be addressed in a separate publication. As stated before, the neutral component of H2 or any neutral heavy fermion inside an SU(2)H doublet can potentially play a role of stable DM due to the SU(2)H gauge symmetry without resorting to an ad-hoc Z2 symmetry. It is worthwhile to point out that the way of embedding H1 and H2 into the SU(2)H doublet is similar to the Higgs bi-doublet in the Left-Right Symmetric Model (LRSM) [36{40] based on the gauge group U(1)B L, where SU(2)L gauge bosons connect elds within H1 or H2 , whereas the heavy SU(2)R gauge bosons transform H1 into H2 . The main di erences between G2HDM and LRSM are: The charge assignment on the SU(2)H Higgs doublet is charged under U(1)Y U(1)X while the bi-doublet in LRSM is neutral under U(1)B L. Thus the corresponding Higgs potential are much simpler in G2HDM; All SU(2)H gauge bosons are electrically neutral whereas the WR of SU(2)R carry electric charge of one unit; The SM right-handed fermions such as uR and dR do not form a doublet under SU(2)H unlike in LRSM where they form a SU(2)R doublet, leading to very di erent phenomenology. On the other hand, this model is also distinctive from the twin Higgs model [41, 42] where H1 and H2 are charged under two di erent gauge groups SU(2)A (identi ed as the SM gauge group SU(2)L) and SU(2)B respectively, and the mirror symmetry on SU(2)A and SU(2)B, i.e., gA = gB, can be used to cancel quadratic divergence of radiative corrections to the SM Higgs mass. Solving the little hierarchy problem is the main purpose of twin Higgs model while G2HDM focuses on getting an inert Higgs doublet as the DM candidate without imposing an ad-hoc Z2 symmetry. Finally, embedding two Higgs doublets into a doublet of a non-abelian gauge symmetry with electrically neutral bosons has been proposed in refs. [43{45], where the non-abelian gauge symmetry is called SU(2)N instead of SU(2)H . Due to the E6 origin, those models have, nonetheless, quite di erent particle contents for both fermions and scalars as well as varying embedding of SM fermions into SU(2)N , resulting in distinct phenomenology from our model. This paper is organized as follows. In section 2, we specify the model. We discuss the complete Higgs potential, Yukawa couplings for SM fermions as well as new heavy fermions and anomaly cancellation. In section 3, we study spontaneous symmetry breaking conditions (section 3.1), and analyze the scalar boson mass spectrum (section 3.2) and the extra gauge boson mass spectrum (section 3.3). We discuss some phenomenology { 3 { 1/6 2/3 1=3 1=2 0 1 1=3 0 1 2/3 1/2 0 0 1 1 0 1 0 1 0 0 0 0 1 0 1 HJEP04(216)9 QL = (uL dL) UR = uR uH T R DR = dRH dR LL = ( L eL) T T T NR = ER = eRH eR H T R T R u d e H = (H1 H2) 0 3=2 p T p p= 2 3=2 1 A which contains just two terms (1 mass term and 1 quartic term) as compared to 8 terms (3 mass terms and 5 quartic terms) in general 2HDM [21]; V ( H ) = V ( H ) = 2 y H H + and nally the mixed term Vmix (H; H ; H ) = + MH Hy H H 2 2) + before performing the minimization of it to achieve spontaneous symmetry breaking (see next section). 2 H 2 T H U(1)X is introduced to simplify the Higgs potential V (H; H ; H ) in eq. (2.1). For example, a term H H obeying the SU(2)H symmetry would be allowed in the absence of U(1)X . Note that as far as the scalar potential is concerned, treating U(1)X as a global symmetry is su cient to kill this and other unwanted terms. In eq. (2.4), if 2 < 0, SU(2)H is spontaneously broken by the vev h 3i = v 6= 0 with h p;mi = 0 by applying an SU(2)H rotation. The quadratic terms for H1 and H2 have the following coe cients 1 2 MH v + 1 doublet H2 does not obtain a vev, its lowest mass component can be potentially a DM candidate whose stability is protected by the gauge group SU(2)H . 2 , H1 can still develop a vev (0 v=p2)T H Similarly, the quadratic terms for two elds 1 and 2 have the coe cients 1 2 M v + 1 2 v 2 + 1 2 may acquire nontrivial vev and h 1i = 0 with the help of (2.7) (2.8) respectively. The eld a large second term. 2.2 Yukawa couplings We start from the quark sector. Setting the quark SU(2)L doublet, QL, to be an SU(2)H singlet and including additional SU(2)L singlets uRH and dRH which together with the SM righthanded quarks uR and dR, respectively, to form SU(2)H doublets, i.e., URT = (uR uRH )2=3 and DRT = (dRH dR) 1=3, where the subscript represents hypercharge, we have2 LYuk ydQL (DR H) + yuQL UR H + H:c:; t = ydQL dRH H2 dRH1 yuQL uRH~1 + uRH H~2 + H:c:; (2.9) where H t (H~2 u and d obtain their masses but uRH and dRH remain massless since H2 does not get a vev. To give a mass to the additional species, we employ the SU(2)H scalar doublet ( 1 2)T , which is singlet under SU(2)L, and left-handed SU(2)L;H singlets u and d as H~1)T with H~1;2 = i 2H1;2. After the EW symmetry breaking hH1i 6= 0, LYuk = yd0 d (DR H yd0 d dR 2 H ) + yu0 u UR ~ H + H:c:; dR 1 yu0 u uR 1 + uRH 2 + H:c:; (2.10) 2A B is de ned as ijAiBj where A and B are two 2-dimensional spinor representations of SU(2)H . { 6 { where has Y = 0, Y ( u) = Y (UR) = 2=3 and Y ( d) = Y (DR) = 1 )T . With h 2i = v =p2, uRH ( u) and dRH ( d) obtain masses yu0v =p2 and yd0v =p2, 1=3 with ~ H = ( 2 respectively. Note that both v and v contribute the SU(2)H gauge boson masses. The lepton sector is similar to the quark sector as paired up with L having Dirac mass MD = y v=p2, while where ERT = (eRH eR) 1, NRT = ( R RH )0 in which R and RH are the right-handed neutrino and its SU(2)H partner respectively, while e and are SU(2)L;H singlets with Y ( e) = 1 and Y ( ) = 0 respectively. Notice that neutrinos are purely Dirac in this setup, i.e., R RH paired up with having H = y0 v =p2. As a result, the lepton number is conserved, implying vanishing neutrinoless double beta decay. In order to generate the observed neutrino masses of order sub-eV, the Yukawa couplings for L and R are extremely small ( compared to the electron Yukawa coupling. The smallness can arise from, for example, the 10 11) even small overlap among wavefunctions along a warped extra dimension [47, 48]. Alternatively, it may be desirable for the neutrinos to have a Majorana mass term which can be easily incorporated by introducing a SU(2)H scalar triplet N with X( N ) = Then a renormalizable term gN N Rc N NR with a large h N i 6= 0 will break lepton number and provide large Majorana masses MN = gN h N i to Rs (and also for the Ls can be realized via the type-I seesaw mechanism which allows one large mass RH s). Sub-eV masses of order MN and one small mass of order (MD)2=MN . For MD sub-eV neutrino masses can be achieved provided that y 1:28 y v and v 10 7pMN =GeV. 246 GeV, We note that only one SU(2)L doublet H1 couples to two SM fermion elds in the above Yukawa couplings. The other doublet H2 couples to one SM fermion and one nonSM fermion, while the SU(2)H doublet H couples to at least one non-SM fermion. As a consequence, there is no avour changing decays from the SM Higgs in this model. This is in contrast with the 2HDM where a discrete Z2 symmetry needed to be imposed to forbid avour changing Higgs decays at tree level. Thus, as long as H2 does not develop a vev in the parameter space, it is practically an inert Higgs, protected by a local gauge symmetry instead of a discrete one! 2.3 Anomaly cancellation We here demonstrate the aforementioned setup is anomaly-free with respect to both the SM and additional gauge groups. The anomaly cancellation for the SM gauge groups SU(3)C U(1)Y is guaranteed since addition heavy particles of the same hypercharge form Dirac pairs. Therefore, contributions of the left-handed currents from cancel those of right-handed ones from uH , dRH , RH and eRH respectively. R u, d and e Regarding the new gauge group SU(2)H , the only nontrivial anomaly needed to be checked is [SU(2)H ]2U(1)Y from the doublets UR, DR, NR and ER with the following { 7 { result 2Tr[T afT b; Y g] =2 ab X Yl l X Yr r ! = 2 ab X Yr r = 2 ab (3 2 Y (UR) + 3 2 Y (DR) + 2 Y (NR) + 2 Y (ER)) (2.12) where 3 comes from the SU(3)C color factor and 2 from 2 components in an SU(2)H doublet. With the quantum number assignment for the various elds listed in table 1, one can check that this anomaly coe cient vanishes for each generation. In terms of U(1)X , one has to check [SU(3)C ]2U(1)X , [SU(2)H ]2U(1)X , [U(1)X ]3, rst three terms are zero due to cancellation between UR and DR and between ER and NR with opposite U(1)X charges. For [U(1)Y ]2U(1)X and [U(1)X ]2U(1)Y , one has respectively 2 2 3 3 Y (UR)2X(UR) + Y (DR)2X(DR) + Y (ER)2X(ER) ; X(UR)2Y (UR) + X(DR)2Y (DR) + X(ER)2Y (ER) ; both of which vanish. One can also check the perturbative gravitational anomaly [49] associated with the hypercharge and U(1)X charge current couples to two gravitons is proportional to the following sum of the hypercharge 3 (2 Y (QL) + Y ( u) + Y ( d) 2 Y (UR) 2 Y (DR)) + 2 Y (LL) + Y ( ) + Y ( e) 2 Y (NR) 2 Y (ER); and U(1)X charge X(UR) + X(DR) + X(ER) + X(NR); which also vanish for each generation. Since there are 4 chiral doublets for SU(2)L and also 8 chiral doublets for SU(2)H for each generation, the model is also free of the global SU(2) anomaly [50] which requires the total number of chiral doublets for any local SU(2) must be even. We end this section by pointing out that one can also introduce QLH = (uLH dLH )T to pair up with QL and LLH = ( LH eLH )T to pair up with LL to form SU(2)H doublets. Such possibility is also interesting and will be discussed elsewhere. 3 Spontaneous symmetry breaking and mass spectra After specifying the model content and fermion mass generation, we now switch to the scalar and gauge boson sector. We begin by studying the minimization conditions for spontaneous symmetry breaking, followed by investigating scalar and gauge boson mass spectra. Special attention is paid to mixing e ects on both the scalars and gauge bosons. 3[SU(2)L]2U(1)X anomaly does not exist since fermions charged under U(1)X are singlets under SU(2)L. (2.13) (2.14) (2.15) { 8 { To facilitate spontaneous symmetry breaking, let us shift the elds as follows G+ v+2h + iG0 p ! H1 = ; H = p GH are the physical elds. and H2 = (H2+ H20)T . Here v, v of the potential; G and v are vevs to be determined by minimization fG+; G3; GpH ; G0H g are Goldstone bosons, to be absorbed by the fh; H2; 1; 2; 3; pg Minimization of the potential in eq. (3.2) leads to the following three equations for the vevs 4 v 3 Note that one can solve for the non-trivial solutions for v2 and v2 in terms of v and other parameters using eqs. (3.3) and (3.4). Substitute these solutions of v2 and v2 into eq. (3.5) leads to a cubic equation for v which can be solved analytically (See appendix A). 3.2 Scalar mass spectrum The scalar boson mass spectrum can be obtained from taking the second derivatives of the potential with respect to the various elds and evaluate it at the minimum of the potential. The mass matrix thus obtained contains three diagonal blocks. The rst block is 3 the basis of S = fh; 3; 2g it is given by 0 H vv v2 (MH v2 (M 2 H v ) 2 v ) M20 = BB v2 (MH 1 2 H v ) 4v 8 v 3 + MH v2 + M v 2 v2 (M This matrix can be diagonalized by a similar transformation with orthogonal matrix O, Oij jmij with i and j referring to the avour and mass eigenwhich de ned as jf ii states respectively, OT M20 O = Diag(m2h1 ; m2h2 ; m2h3 ) ; where the three eigenvalues are in ascending order. The lightest eigenvalue mh1 will be identi ed as the 125 GeV Higgs h1 observed at the LHC and the other two mh2 and mh3 { 9 { H vv 2 v 2 2 v )CC : 1 A (3.6) (3.7) 0 M 0 v p M002 = BB 12 M v 1 4v are for the heavier Higgses h2 and h3. The physical Higgs hi is a linear combination of the three components of S: hi = OjiSj . Thus the 125 GeV scalar boson could be a mixture of the neutral components of H1 and the SU(2)H doublet H , as well as the real component 3 of the SU(2)H triplet H . The SM Higgs h1 tree-level couplings to f f , W +W , ZZ and H2+H2 pairs, each will be modi ed by an overall factor of O11, resulting a reduction by jO11j2 on the h1 decay branching ratios into these channels. On the other hand, as we shall see later, h1 ! and Z involve extra contributions from the 3 and 2 components, which could lead to either enhancement or suppression with respect to the SM prediction. The second block is also 3 p 3. In the basis of G = fGH ; p; H20 g it is given by D can be a DM candidate in G2HDM. Note that in the parameter space where the quantity inside the square root of eq. (3.9) is very small, e would be degenerate with D. In this case, we need to include coannihilation processes for relic density calculation. Moreover, it is possible in our model to have RH or decays to SM lepton and Higgs) to be DM candidate as well. ( R either is too light or is not stable since it The nal block is 4 4 diagonal, giving It is easy to show that eq. (3.8) has a zero eigenvalue, associated with the physical Goldstone boson, which is a mixture of GH , of two physical elds e and D. They are given by p and H20 . The other two eigenvalues are the masses M e ;D = 1 n 8v h MH MH v2 + 4 (MH + M ) v2 + M v 2 v2 +4v2 +M v2 +4v2 2 16MH M v 2 v2 + 4v2 + v2 i 2 for the physical charged Higgs H2 , and m2 H2 = MH v ; m2G = m2G0 = m2G0H = 0 ; for the three Goldstone boson elds G , G0 and G0H . Note that we have used the minimization conditions eqs. (3.3), (3.4) and (3.5) to simplify various matrix elements of the above mass matrices. Altogether we have 6 Goldstone particles in the scalar mass spectrum, we thus expect to have two massless gauge particles left over after spontaneous symmetry breaking. One is naturally identi ed as the photon while the other one could be interpreted as dark photon D. (3.8) (3.9) 1 : (3.10) (3.11) After investigating the spontaneous symmetry breaking conditions, we now study the mass spectrum of additional gauge bosons. The gauge kinetic terms for the H , and H are L Tr h D0 H y D0 H i + D0 y D0 + D0 H y D0 H ; with and D0 H igH W 0 ; H ; D0 = W 0pT p + W 0mT m igH W 03T 3 igX X ; HJEP04(216)9 D0 H = D 1 W 0pT p + W 0mT m igH W 03T 3 igX X H ; 1 2 i pgH 2 i pgH 2 3 a=1 1 2 where D is the SU(2)L covariant derivative, acting individually on H1 and H2, gH (gX ) is the SU(2)H (U(1)X ) gauge coupling constant, and W 0 = X W 0aT a = p W 0pT p + W 0mT m + W 03T 3; in which T a = (W 01 iW 0 2)=p2, and a=2 ( a are the Pauli matrices acting on the SU(2)H space), W 0 (p;m) = T p = 1 + i 2 = ; T m = 1 i 2 = 1 2 obtained its mass entirely from v, so it is given by same as the SM. The SU(2)H gauge bosons W 0a and the U(1)X gauge boson X receive masses from h 3i, hH1i and h 2i. The terms contributed from the doublets are similar with that from the standard model. Since H transforms as a triplet under SU(2)H , i.e., in the adjoint representation, the contribution to the W 0a masses arise from the term L gH2 Tr W 0 ; H y W 0 ; H : All in all, the W 0(p;m) receives a mass from h 3i, h 2i and hH1i m2W 0(p;m) = 14 gH2 v2 + v2 + 4v2 ; while gauge bosons X and W 03, together with the SM W 3 and U(1)Y gauge boson B, acquire their masses from h 2i and hH1i only but not from h H i: 1 8 v 2 2gX X + gH W 03 gW 3 + g0B 2 + v2 2gX X + gH W 03 2 ; (3.21) where g0 is the SM U(1)Y gauge coupling. Note that the gauge boson W 0(p;m) corresponding to the SU(2)H generators T do not carry the SM electric charge and therefore will not mix with the SM W B and X will become massive, by absorbing the imaginary part of H10 and and X do mix with the SM W 3 and B bosons via hH1i. In fact, only two of W 3, W 03, 2. To avoid undesired additional massless gauge bosons, one can introduce extra scalar elds charged under only SU(2)H U(1)X but not under the SM gauge group to give a mass to W 03 and X, without perturbing the SM gauge boson mass spectrum. Another possibility is to involve the Stueckelberg mechanism to give a mass to the U(1)X gauge boson as done in refs. [51{54]. Alternatively, one can set gX = 0 to decouple X from the theory or simply treat U(1)X as a global symmetry, after all as mentioned before U(1)X is introduced to bosons while W 03 simplify the Higgs potential by forbidding terms like SU(2)H but not U(1)X . in the basis V 0 = B; W 3; W 03; X : T H H H , which is allowed under From eq. (3.21), one can obtain the following mass matrix for the neutral gauge bosons g0g v2 4 g2v2 4 As anticipated, this mass matrix has two zero eigenvalues corresponding to m = 0 and m D = 0 for the photon and dark photon respectively. The other two nonvanishing eigenvalues are where 1 8 M 2 = v2 + v 2 q v2 + v A strictly massless dark photon might not be phenomenologically desirable. One could have a Stueckelberg extension of the above model by including the Stueckelberg mass term [51, 52] = g2 + g02 + gH2 + 4gX2 ; = gH2 + 4gX2 ; 2 = gH 4gX2 : 1 2 + MY B )2 ; (3.22) (3.24) (3.25) where MX and MY are the Stueckelberg masses for the gauge elds X and B of U(1)X and U(1)Y respectively, and a is the axion eld. Thus the neutral gauge boson mass matrix is modi ed as M21 = BBB 0 B B g02v2 + MY2 4 It is easy to show that this mass matrix has only one zero mode corresponding to the photon, and three massive modes Z; Z0; Z00. This mass matrix can be diagonalized by an orthogonal matrix. The cubic equation for the three eigenvalues can be written down analytically similar to solving the cubic equation for the vev v given in the appendix A. However their expressions are not illuminating and will not presented here. As shown in ref. [52], MY will induce the mixing between U(1)Y and U(1)X and the resulting massless eigenstate, the photon, will contain a U(1)X component, rendering the neutron charge, Qn = Qu + 2Qd, nonzero unless u's and d's U(1)X charges are zero or proportional to their electric charges. In this model, however, none of the two solutions can be satis ed. Besides, left-handed SM elds are singlets under U(1)X while right-handed ones are charged. It implies the left-handed and right-handed species may have di erent electric charges if the U(1)X charge plays a role on the electric charge de nition. Here we will set MY to be zero to maintain the relations Q = I3 + Y and 1=e2 = 1=g02 + 1=g2 same as the SM in order to avoid undesired features. As a result, after making a rotation in the 1 2 plane by the Weinberg angle w, the mass matrix M12 can transform into a block diagonal matrix with the vanishing rst column and rst row. The nonzero 3-by-3 block matrix can be further diagonalized by an orthogonal matrix O, characterized by three rotation angles ( 12; 23; 13), 0 Z 1 Z00 0 Z 1 Z00 0ZSM 1 X (3.27) 0ZSM 1 X where ZSM is the SM Z boson without the presence of the W30 and X bosons. In this model, the charged current mediated by the W boson and the electric current by the photon are exactly the same as in the SM: f g 2 L( ) = X Qf ef f A ; L(W ) = p ( L eL + uL dL) W + + H:c: ; where Qf is the corresponding fermion electric charge in units of e. On the other hand, neutral current interactions, including ones induced by W 0, take the following form (for illustration, only the lepton sector is shown but it is straightforward to include the quark sector) LNC = L(Z) + L(Z0) + L(Z00) + L(W 0) ; (3.28) (3.29) and gH 2 X f=e; JZSM = 1 cos w JW 03 = J X = X X f=NR;ER f=NR;ER fR (I3H )fR ; QfX fR fR ; with I3 (I3H ) being the SU(2)L (SU(2)H ) isospin and QfX the U(1)X charge. Detailed analysis of the implications of these extra gauge bosons is important and will be presented elsewhere. 4 Phenomenology In this section, we discuss some phenomenology implications of the model by examining the mass spectra of scalars and gauge bosons, Z0 constraints from various experiments, and Higgs properties of this model against the LHC measurements on the partial decay widths of the SM Higgs boson, which is h1 in the model. 4.1 Numerical solutions for scalar and gauge boson masses We rst study the SM Higgs mass (mh1 ) dependence on the parameters in the mass matrix in eq. (3.6). As we shall see later the vev v has to be bigger than 10 TeV ( v = 246 GeV). In light of LEP measurements on the e+e ! e+e cross-section [55], the mass matrix will exhibit block-diagonal structure with the bottom-right 2-by-2 block much bigger than the rest and h basically decouple from 2 and 3 . To demonstrate this behavior, we simplify the model by setting to zero and then choose that one can investigate how the Higgs mass varies as a function of H , H , equal is very small compared to v , the Higgs mass is simply the (1,1) element of the mass matrix, 2 H v2, and h1 is just h, i.e., O121 ' 1. Nonetheless, when becomes comparable to v , and the (1,2) element of the mass matrix gives rise to a sizable but negative contribution to the Higgs mass, requiring a larger value of H than (3.30) (3.31) 1 0 3 e M ) V mass is basically determined by two parameters MH and H only. Other parameters are set as follows: SM-like: O121 ' 1. Therefore, one has to measure the quartic coupling double Higgs production to be able to di erentiate this model from the SM. the SM one so as to have a correct Higgs mass. In this regime, jO11j is, however, still very H through the For the analysis above, we neglect the fact all vevs, v , v and v are actually functions of the parameters s, M s and s in eq. (2.1), the total scalar potential. The analytical solutions of the vevs are collected in appendix A. As a consequence, we now numerically diagonalized the matrices (3.6) and (3.8) as functions of , M and , i.e., replacing all vevs by the input parameters. It is worthwhile to mention that v has three solutions as eq. (3.5) is a cubic equation for v . Only one of the solutions corresponds to the global minimum of the full scalar potential. We, however, include both the global and local minimum neglecting the stability issue for the latter since it will demand detailed study on the nontrivial potential shape which is beyond the scope of this work. In order to explore the possibility of a non-SM like Higgs having a 125 GeV mass, we further allow for nonzero mixing couplings. We perform a grid scan with 35 steps of each dimension in the range 10 2 10 2 10 2 10 1 1:0 1:05 H H H MH = GeV v =v 5 ; 5 ; 5 ; 5 ; 2 5:0 : 104 ; In order not to overcomplicate the analysis, from now on we make = 0:8MH , unless otherwise stated. In table 2, we show 6 representative benchmark points (3 global and 3 local minima) from our grid scan with the dark matter mass mD and the charged Higgs mH of order few hundred GeVs, testable m Z ′ θ Z ′ − g H × ′ . Z0 mixing (blue dotted line) and the LEP constraints on the electron-positron scattering cross-section (black dashed line) in the mZ0 gH plane. v for M understood as in the near future. It is clear that Global scenario can have the SM Higgs composition 2 signi cantly di erent from 1, as O11 0:8 in benchmark point A, but h1 is often just h in Local case. On the other hand, the other two heavy Higgses are as heavy as TeV because their mass are basically determined by v and v . For the other mass matrix eq. (3.8), we focus on the mass splitting between the dark matter and the charged Higgs and it turns out the mass splitting mostly depends on v and v . In gure 2, we present the mass di erence normalized to mD as a function of and v = 10 TeV used in table 2. The behavior can be easily mH2 mD mD ' = v 2 1 v2 + jv 2 8 v2 + 2 v2 v2 =v2 v2 j 4 + v2 =v2 (4.2) where we have neglected terms involving v since we are interested in the limit of v , v Note that this result is true as long as M = 0:8MH and v > v v regardless of the other parameters. 4.2 Z0 constraints Since performing a full analysis of the constraints on extra neutral gauge bosons including all possible mixing e ects from ; Z; Z0; Z00 is necessarily complicated, we will be contented in this work by a simple scenario discussed below. The neutral gauge boson mass matrix in eq. (3.22) can be simpli ed a lot by making U(1)X a global symmetry. In the limit of gX = MX = 0, the U(1)X gauge boson X decouples from the theory, and the 3-by-3 mass matrix in the basis of B, W 3 and W 03 can be rotated into the mass basis of the SM , Z and the SU(2)H Z0 by: 0 0 4 4 where Rij refers to a rotation matrix in the i j block; e.g., R12 ( w) is a rotation along the z-axis (W 03) direction with cos w as the (1; 1) and (2; 2) element and sin w as the (1; 2) element (sin w for (2; 1)). The mixing angles can be easily obtained as sin w = sin ZZ0 = g0 pg2 + g02 ; p2pg2 + g02 gH v 2 1=4 (g2 + g02)v2 + gH2 v2 + v2 + 1=2 1=2 ; where = have the approximate result g2 + g02 + gH2 v2 + gH2 v2 2 4gH2 g2 + g02 v2v2 . In the limit of v sin ZZ0 pg2 + g02 v2 gH v 2 ; mZ pg2 + g02 ; mZ0 v 2 v gH 2 : and A couple of comments are in order here. First, the SM Weinberg angle characterized by w is unchanged in the presence of SU(2)H . Second, the vev ratio v2=v2 controls the mixing between the SM Z and SU(2)H Z0. However, the Z Z0 mixing for TeV Z0 is constrained to be roughly less than 0:1%, results from Z resonance line shape measurements [56], electroweak precision test (EWPT) data [57] and collider searches via the W +W nal states [58, 59], depending on underlying models. Direct Z0 searches based on dilepton channels at the LHC [60{62] yield stringent constraints on the mass of Z0 of this model since right-handed SM fermions which are part of SU(2)H doublets couple to the Z0 boson and thus Z0 can be produced and decayed into dilepton at the LHC. To implement LHC Z0 bounds, we take the Z0 constraint [62] on the Sequential Standard Model (SSM) with SM-like couplings [63], rescaling by a factor of gH2 (cos2 w=g2): It is because rst SU(2)H does not have the Weinberg angle in the limit of gX = 0 and second we assume Z0 decays into the SM fermions only and branching ratios into the heavy fermions are kinematically suppressed, i.e., Z0 decay branching ratios are similar to those of the SM Z boson. The direct search bound becomes weaker once (4.3) (4.4) (4.5) v, we (4.6) (4.7) gH2 (cos2 the couplings. e+e ! `+` heavy fermion nal states are open. Note also that Z0 couples only to the right-handed SM elds unlike Z0 in the SSM couples to both left-handed and right-handed elds as in the SM. The SSM Z0 left-handed couplings are, however, dominant since the right-handed ones are suppressed by the Weinberg angle. Hence we simply rescale the SSM result by w=g2) without taking into account the minor di erence on the chiral structure of In addition, Z0 also interacts with the right-handed electron and will contribute to processes. LEP measurements on the cross-section of e+e ! `+` can be translated into the constraints on the new physics scale in the context of the e ective four-fermion interactions [55] = 0 (1) for f 6= e (f = e) and ij = 1 ( 1) corresponds to constructive (destructive) interference between the SM and the new physics processes. On the other hand, in our model for mZ0 TeV the contact interactions read Le = (1 + ) g 2 mH2Z0 eR eRfR fR : + + L R ! e e L R with (4.8) (4.9) (4.10) It turns out the strongest constraint arises from e e = 8:9 TeV and = 1 [55], which implies gH mZ0 . 0:2 TeV and v & 10 TeV : v 4.3 In gure 3, in the plane of gH and mZ0 , we show the three constraints: the Z Z0 mixing by the blue dotted line, the heavy narrow dilepton resonance searches from CMS [62] in red and the LEP bounds on the electron-positron scattering cross-section of e+e in black, where the region above each of the lines is excluded. The direct searches are dominant for most of the parameter space of interest, while the LEP constraint becomes most stringent toward the high-mass region. From this gure, we infer that for 0:1 . gH . 1, the Z0 mass has to be of order O(TeV). Note that mZ0 gH v =2 and it implies ! e+e [55] 30 TeV for mZ0 1:5 TeV but v can be 10 TeV for mZ0 & 2:75 TeV. Constraints from the 125 GeV SM-like Higgs We begin with the SM Higgs boson h1 partial decay widths into SM fermions. Based on the Yukawa couplings in eqs. (2.9), (2.10) and (2.11), only the avor eigenstate H1 couples to two SM fermions. Thus, the coupling of the mass eigenstate h1 to the SM fermions is rescaled by O11, making the decay widths universally reduced by O121 compared to the SM, which is di erent from generic 2HDMs. For the tree-level (o -shell) h1 ! W +W , since the SM W bosons do not mix with additional SU(2)H (and U(1)X ) gauge bosons and 2 and 3 are singlets under the SU(2)L gauge group, the partial decay width is also suppressed by O121. h1 ! ZZ receives additional contributions from the Z On the other hand, Z0 mixing and thus the 3 and and h ! Z in this model. Due to the fact the jO11j 1 and H is always positive, R is often less than the SM prediction while RZ ranges from 0:9 to 1, given the ATLAS and CMS measurements on R : : 1:17 0:27 (ATLAS [64]) and 1:13 0:24 (CMS [65]). Only ATLAS result is shown which completely covers the CMS 1 con dence region. The left (right) panel is for Global (Local) scenario. 2 components charged under SU(2)H will also contribute. The mixing, however, are constrained to be small (. 10 3) by the EWPT data and LEP measurement on the electronpositron scattering cross-section as discussed in previous subsection. We will not consider the mixing e ect here and the decay width is simply reduced by O121 identical to other tree-level decay channels. Now, we are in a position to explore the Higgs radiative decay rates into two photons and one Z boson and one photon, normalized to the SM predictions. For convenience, we de ne RXX to be the production cross section of an SM Higgs boson decaying to XX divided by the SM expectation as follows: (4.11) (4.12) Note that rst additional heavy colored fermions do not modify the Higgs boson production cross section, especially via gg ! h1, because the SU(2)H symmetry forbids the coupling of h1f f , where f is the heavy fermion. Second, in order to avoid the observed Higgs invisible decay constraints in the vector boson fusion production mode: Br(h1 ! invisible) < 0:29 [66, 67], we constrain ourself to the region of 2mD > mh so that the Higgs decay channels are the same as in the SM. As a consequence, eq. (4.11) becomes, R = (h1 ! SM(h1 ! ) ) ; RZ = (h1 ! SM(h1 ! Z) ) ; similar to the situation of IHDM. As mentioned before due to the existence of SU(2)H , there are no terms involving one SM Higgs boson and two heavy fermions, which are not SU(2)H invariant. Therefore, the RXX (pp ! h1) Br(h1 ! XX) SM(pp ! h1) BrSM(h1 ! XX) : only new contributions to h1 ! and h1 ! Z arise from the heavy charged Higgs boson, H2 . In addition, h1 consists of 3 and 2 apart from H10. With the quartic interactions 2 H H2yH2H1yH1 + H H2yH2 32=2 + H H2yH2 2 2, there are in total three contributions to the H2 -loop diagram. The Higgs decay widths into and Z including new scalars can be found in refs. [68{71] and they are collected in appendix B for convenience. The results of h ! (red circle) and h ! Z (blue square) are presented in gure 4 for both the Global minimum case (left) and additional points which having the correct Higgs mass only at Local minimum (right). All the scatter points were selected from our grid scan described in subsection 4.1. It clearly shows that the mass of the heavy charged Higgs has to be larger than 100 GeV in order to satisfy the LHC measurements 3jH1j2 jH2j2 can be either positive or negative, in this model we H as a quartic coupling has to be positive to ensure the potential is bounded from below. It implies that for jO11j2 being very close to 1 like the benchmark point C in table 2, h1 is mostly h, the neutral component in H1, and R is determined by H . In this case, R , corresponding to the red curve, is always smaller than 1 in the region of interest where mH2 values below the SM prediction as well. charged Higgs in the context of IHDM [72]), while RZ denoted by the blue curve also has > 80 GeV (due to the LEP constraints on the In contrast, there are some points such as the benchmark point A where the contributions from 2 and 3 are signi cant, they manifest in gure 4 as scatter points away from the lines but also feature R , RZ < 1. 4.4 Dark matter stability In G2HDM, although there is no residual (discrete) symmetry left from SU(2)H symmetry breaking, the lightest particle among the heavy SU(2)H fermions (uH , dH , eH , H ), the SU(2)H gauge boson W 0 (but not Z0 because of Z Z0 mixing) and the second heaviest eigenstate in the mass matrix of eq. (3.8)4 is stable due to the gauge symmetry and the p Lorentz invariance for the given particle content. In this work, we focus on the case of the second heaviest eigenstate in the mass matrix of eq. (3.8), a linear combination of H20 , p and GH , being DM by assuming the SU(2)H fermions and W 0 are heavier than it. At tree level, it is clear that all renormalizable interactions always have even powers of the potential DM candidates in the set D fuH ; dH ; eH ; H ; W 0; H20 ; p; GpH g. It implies they always appear in pairs and will not decay solely into SM particles. In other words, the decay of a particle in D must be accompanied by the production of another particle in D, rendering the lightest one among D stable. Beyond the renormalizable level, one may worry that radiative corrections involving DM will create non-renormalizable interactions, which can be portrayed by high order e ective operators, and lead to the DM decay. Assume the DM particle can decay into solely SM particles via certain higher dimensional operators, which conserve the gauge symmetry but spontaneously broken by the vevs of H1, H , H . The SM particles refer 4The lightest one being the Goldstone boson absorbed by W 0. to SM fermions, Higgs boson h and 2 (which mixes with h).5 The operators, involving one DM particle and the decay products, are required to conserve the gauge symmetry. We here focus on 4 quantum numbers: SU(2)L and SU(2)H isospin, together with U(1)Y and U(1)X charge. DM is a linear combination of three avour states: H20 , First, we study operators with H20 decaying leptonically into n1 Ls, n2 eLs, n3 Rs, n4 eRs, n5 hs and n6 2s.6 One has, in terms of the quantum numbers (I3H ; I3; Y; X), p and GH . p This implies the number of net fermions (fermions minus anti-fermions) is n1+n2+n3+n4 = 1, where positive and negative ni correspond to fermion and anti-fermion respectively. In other words, if the number of fermions is odd, the number of anti-fermions must be even; and vice versa. Clearly, this implies Lorentz violation since the total fermion number (fermions plus anti-fermions) is also odd. Therefore, the avor state H20 can not decay solely into SM particles. It is straightforward to show the conclusion applies to GpH as well. As a result, the DM candidate, a superposition of H20 , p and GpH , is stable p and as long as it is the lightest one among the potential dark matter candidates D. Before moving to the DM relic density computation, we would like to comment on e ective operators with an odd number of D, like three Ds or more. It is not possible that this type of operators, invariant under the gauge symmetry, will induce operators involving only one DM particle, by connecting an even number of Ds into loop. It is because those operators linear in D violate either the gauge symmetry or Lorentz invariance and hence can never be (radiatively) generated from operators obeying the symmetries. After all, the procedure of reduction on the power of D, i.e., closing loops with proper vev insertions (which amounts to adding Yukawa terms), also complies with the gauge and Lorentz symmetry. 4.5 Dark matter relic density We now show this model can reproduce the correct DM relic density. As mentioned above, the DM particle will be a linear combination of GH , p p and H20 . Thus (co)-annihilation channels are relevant if their masses are nearly degenerated and the analysis can be quite involved. In this work, we constrain ourselves in the simple limit where the G2HDM becomes IHDM, in which case the computation of DM relic density has been implemented in the software package micrOMEGAs [73, 74]. For a recent detailed analysis of IHDM, see ref. [10] and references therein. In other words, we are working in the scenario that the 5The SM gauge bosons are not included since they decay into SM fermions, while inclusion of 3 will not change the conclusion because it carries zero charge in term of the quantum numbers in consideration. 6The conclusion remains the same if quarks are also included. DM being mostly H20 and the SM Higgs boson h has a minute mixing with 3 and 2. The IHDM Higgs potential reads, VIHDM = 21jH1j2 + 22jH2j2 + 1jH1j4 + 2jH2j4 + 3jH1j2jH2j2 + 4jH1yH2j2 + 2 5 n(H1yH2)2 + h:c:o ; (4.15) H2 H20 ! W the fact H20 (DM particle) also receives a tiny contribution from where 1 = 2 = 2 3 = H with 4 = 5 = 0 when compared to our model. It implies the degenerate mass spectrum for H2 and H20. The mass splitting, nonetheless, arises from p and GpH as well as loop contributions. In the limit of IHDM the mass splitting is very small, making paramount (co)-annihilations, such as H2+H2 ! W +W , H20H20 ! W +W , (H2+H2 ; H20H20) ! ZZ, , and thus the DM relic density is mostly determined by these channels. Besides, from eq. (4.15), H is seemingly xed by the Higgs mass, 1 = m2h=2v2 0:13, a distinctive feature of IHDM. In G2HDM, however, the value of H can deviate from 0.13 due to the mixing among h, 3 and 2 as shown in gure 1, where the red curve corresponds to the Higgs boson mass of 125 GeV with H varying from 1.2 to 1.9 as a function of MH , which controls the scalar mixing. To simulate G2HDM but still stay close to the IHDM limit, we will treat H as a free parameter in the analysis and independent of the SM Higgs mass. We note that micrOMEGAs requires ve input parameters for IHDM: SM Higgs mass ( xed to be 125 GeV), H2 mass, H20 mass (including both CP-even and odd components which are degenerate in our model), 2 (= H ) and ( 3 + 4 5)=2 ( H ). It implies that only two of them are independent in G2HDM, which can be chosen to be mDM and H . Strictly speaking, mDM is a function of parameters such as M , MH , v , v , etc., since it is one of the eigenvalues in the mass matrix of eq. (3.8). In this analysis, we stick to the scan range of the parameters displayed in eq. (4.1) with slight modi cation as follows 0:12 0:8 M H =MH 0:2 ; 1:0 ; (4.16) see the IHDM limit (OD2 and also demand the mixing (denoted by OD) between H20 and the other scalars to be less than 1%. In the exact IHDM limit, OD should be 1. Besides, the decay time of H2 into H20 plus an electron and an electron neutrino is required to be much shorter than one second in order not to spoil the Big Bang nucleosynthesis. Our result is presented in gure 5. For a given H , there exists an upper bound on the DM mass, above which the DM density surpasses the observed one, as shown in the left panel of gure 5. The brown band in the plot corresponds to the DM relic density of 0:1 < h2 < 0:12 while the yellow band refers to h2 < 0:1. In the right panel of gure 5, we show how well the IHDM limit can be reached in the parameter space of M versus v =v , where the red band corresponds to 0:8 < OD2 < 0:9, the green band refers to 0:9 < OD2 < 0:99 (green) and while the light blue area represents 0:99 < OD2. One can easily =MH 1) can be attained with M . MH . For M > MH , we .08 .072 Te ) V ( M D .012 .014 .018 .02 0:99 < OD2, respectively. See the text for detail of the analysis. with the relic density between 0.1 and 0.12 but the lower yellow region is the relic density less than 0.1. Right: accepted DM mass region projected on (v =v , M =MH ) plane. The red, green and light blue regions present the DM inert Higgs fraction 0:8 < OD2 < 0:9, 0:9 < OD2 < 0:99 and have mH2 < mH20 , implying H20 can not be DM anymore. Finally, we would like to point out that the allowed parameter space may increase signi cantly once the small H20 mixing constraint is removed and other channels comprising GpH and It is worthwhile to mention that not only H20 but also p are considered. H (right-handed neutrino's SU(2)H partner) can be the DM candidate in G2HDM, whose stability can be proved similarly based on the gauge and Lorentz invariance. If this is the case, then there exists an intriguing connection between DM phenomenology and neutrino physics. We, however, will leave this possibility for future work. 4.6 In light of the existence of new scalars and fermions,7 one should be concerned about the EWPT, characterized by the oblique parameters S, T and U [75]. For scalar contri butions, similar to the situation of DM relic density computation, a complete calculation in G2HDM can be quite convoluted. The situation, however, becomes much simpler when one goes to the IHDM limit as before, i.e., H20 is virtually the mass eigenstate and the inert doublet H2 is the only contribution to S, T and U , since H and H are singlets under the SM gauge group. Analytic formulas for the oblique parameters can be found in ref. [8]. All of them will vanish if the mass of H2 is equal to that of H20 as guaranteed by the SU(2)L invariance. 7For additional gauge bosons in the limit of U(1)X being a global symmetry, the constrain considered above, sin ZZ0 < 10 3, are actually derived from the electroweak precision data combined with collider bounds. As a consequence, we will not discuss their contributions again. On the other hand, the mass splitting stems from the H20 mixing with GpH and p therefore, in the limit of IHDM, the mass splitting is very small, implying a very small deviation from the SM predictions of the oblique parameters. We have numerically checked that all points with the correct DM relic abundance studied in the previous section have negligible H2 contributions, of order less than 10 3, to T and Finally for the heavy fermions, due to the fact they are singlets under SU(2)L, the corresponding contributions will vanish according to the de nition of oblique parameters [75]. Our model survives the challenge from the EWPT as long as one is able to suppress the extra scalar contributions, for instance, resorting to the IHDM limit or having cancellation among di erent contributions. 5 Conclusions and outlook In this work, we propose a novel framework to embed two Higgs doublets, H1 and H2 into a doublet under a non-abelian gauge symmetry SU(2)H and the SU(2)H doublet is charged under an additional abelian group U(1)X . To give masses to additional gauge bosons, we introduce an SU(2)H scalar triplet and doublet (singlets under the SM gauge group). The potential of the two Higgs doublets is as simple as the SM Higgs potential at the cost of additional terms involving the SU(2)H triplet and doublet. The vev of the triplet triggers the spontaneous symmetry breaking of SU(2)L by generating the vev for the rst SU(2)L Higgs doublet, identi ed as the SM Higgs doublet, while the second Higgs doublet does not obtain a vev and the neutral component could be the DM candidate, whose stability is guaranteed by the SU(2)H symmetry and Lorentz invariance. Instead of investigating DM phenomenology, we have focused here on Higgs physics and mass spectra of new particles. To ensure the model is anomaly-free and SM Yukawa couplings preserve the additional U(1)X symmetry, we choose to place the SM right-handed fermions and new heavy right-handed fermions into SU(2)H doublets while SM SU(2)L fermion doublets are singlets under SU(2)H . Moreover, the vev of the SU(2)H scalar doublet can provide a mass to the new heavy fermions via new Yukawa couplings. Di erent from the left-right symmetric model with a bi-doublet Higgs bosons, in which WR carries the electric charge, the corresponding W 0 bosons in this model that connects H1 and H2 are electrically neutral since H1 and H2 have the same SM gauge group charges. On the other hand, the corresponding Z0 actually mixes with the SM Z boson and the mixing are constrained by the EWPT data as well as collider searches on the W nal state. Z0 itself is also confronted by the direct resonance searches based on dilepton or dijet channels, limiting the vev of the scalar doublet to be of order O(TeV). By virtue of the mixing between other neutral scalars and the neutral component of H1, the SM Higgs boson is a linear combination of three scalars. So the SM Higgs boson treelevel couplings to the SM fermions and gauge bosons are universally rescaled by the mixing angle with respect to the SM results. We also check the h ! decay rate normalized to the SM value and it depends not only on the mass of H2 but also other mixing parameters. As a result, H2 has to be heavier than 100 GeV while the h ! Z decay width is very close to the SM prediction. We also con rm that our model can reproduce the correct DM relic abundance and stays unscathed from the EWPT data in the limit of IHDM where DM is purely the second neutral Higgs H20. Detailed and systematic study will be pursued elsewhere. As an outlook, we brie y comment on collider signatures of this model, for which detailed analysis goes beyond the scope of this work and will be pursued in the future. Due to the SU(2)H symmetry, searches for heavy particles are similar to those of SUSY partners of the SM particles with R-parity. In the case of H20 being the DM candidate, one can have, for instance, uRuR ! W 0pW 0m via t-channel exchange of uH , followed R by W 0p ! uRuRH ! uRH20uL and its complex conjugate, leading to 4 jets plus missing transverse energy. Therefore, searches on charginos or gauginos in the context of SUSY may also apply to this model. Furthermore, this model can also yield mono-jet or monophoton signatures: uRuR ! H20H20 plus or g from the initial state radiation. Finally, the recent diboson excess observed by the ATLAS Collaboration [76] may be partially explained by 2 TeV Z0 decays into W +W via the Z0 Z mixing. Phenomenology of G2HDM is quite rich. In this work we have only touched upon its surface. Many topics like constraints from vacuum stability as well as DM and neutrinos physics, collider implications, etc are worthwhile to be pursued further. We would like to return to some of these issues in the future. A Analytical expression for v, v and v From eqs. (3.3) and (3.4), besides the trivial solutions of v2 = v2 = 0 one can deduce the following non-trivial expressions for v2 and v2 respectively, v2 = v 2 = H (2 H H H 2 H 2 H 2 )v2 + ( H M MH )v + 2(2 2 H H )v2 + ( H MH Substituting the above expressions for v2 and v2 into eq. (3.5) leads to the following cubic equation for v : v 3 + a2v 2 + a1v + a0 = 0 ; where a2 = C2=C3, a1 = C1=C3 and a0 = C0=C3 with C0 = 2 ( H M C1 = 2 2 (2 H +2 4 H C2 = 3 [( H C3 = 4 H H 2 2 H 2 H MH ) 2H + 2 ( H MH ) 2H + 2 (2 H 2 + H M 2 ) M + ( H H H H MH + 2 H H M H ) 2 ) MH ] ; 2 H M H2 : 2 ) ; 2 ) H : (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) HJEP04(216)9 The three roots of cubic equation like eq. (A.3) are well-known since the middle of 16th century where with v 1 = v 2 = v 3 = 3 a2 + (S + T ) ; 1 1 p D ; D ; Q3 + R2 ; 1 2 1 2 (S + T ) + 1 ip3 (S (S + T ) 2 2 1 ip3 (S T ) ; T ) ; Q R 3a1 9 9a1a2 a 2 2 ; 27a0 54 (h1 ! ) = with B Decay width of SM Higgs to and Z Below we summarize the results for the decay width of SM Higgs to and Z [68{71], including the mixing among h, 3 and 2 characterized by the orthogonal matrix O, i.e., (h; 3; 2) T = O (h1; h2; h3)T . In general one should include the mixing e ects among Z and Z0 (and perhaps Z00) as well. As shown in section IV, these mixings are constrained to be quite small and we will ignore them here. Taking into account H2 contributions, the partial width of h1 ! is GF 1282pm23h1 O3121 Ch m2 H v2 H2 A0 ( H2 )+A1 ( W )+X NcQf2 A1=2( f ) ; The form factors for spins 0, 12 and 1 particles are given by Ch = 1 O21 2 H v + MH O11 4 H v + O31 H v O11 2 H v : A0 ( ) = [ A1=2( ) = 2[ + ( ; A1 ( ) = [2 2 + 3 + 3(2 ; (A.8) (A.9) (A.10) (A.11) (A.12) (A.13) (A.14) (A.15) 2 (B.1) (B.2) (B.3) HJEP04(216)9 with the function f ( ) de ned by f ( ) = < 8> arcsin2 p > > > : > > 1 4 log 1 1 1 i #2 ; for 1 ; ; for > 1 : The parameters i = m2h1=4mi2 with i = H2 ; f; W masses of the heavy particles in the loops. Including the H2 contribution, we have (h1 ! Z ) = G2F m2W m3h1O121 64 4 Ch m2 H v2 H2 1 m2Z !3 m2h1 v H2 A0Z ( H2 ; H2 ) + A1Z ( W ; W ) + X Nc cW f with the function f ( ) de ned in eq. (B.4) and the function g( ) can be expressed as g( ) = < p 8 p > > > > > > : 1 1 2 1 arcsin p 1 " log 1 + p 1 p 1 1 1 1 ; for 1 ; # i ; for < 1 : The corresponding SM rates SM(h1 ! ) and SM(h1 ! Z ) can be obtained by omitting the H2 contribution and setting O11 = 1 in eqs. (B.1) and (B.5). with v H2 The loop functions are A0Z ( H ; H ) = I1( H ; H ) ; where I1 and I2 are de ned as I1( ; ) = I2( ; ) = 2( 2( 2( + 2 2 )2 f ( 1 ) f ( 1 ) + )2 g( 1 ) g( f ( 1 ) f ( 1) ; I2( ; )+ 1+ 5 + 2 I1( ; ) ; 2 s Acknowledgments The authors would like to thank A. Arhrib, F. Deppisch, M. Fukugita, J. Harz and T. Yanagida for useful discussions. WCH is grateful for the hospitality of IOP Academia Sinica and NCTS in Taiwan and HEP group at Northwestern University where part of this work was carried out. TCY is grateful for the hospitality of IPMU where this project was completed. 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Wei-Chih Huang, Yue-Lin Sming Tsai, Tzu-Chiang Yuan. G2HDM: Gauged Two Higgs Doublet Model, Journal of High Energy Physics, 2016, 19, DOI: 10.1007/JHEP04(2016)019