EFI-15-2 The Twin Higgs mechanism and Composite Higgs Matthew Lowa, Andrea Tesib, Lian-Tao Wanga aDepartment of Physics, Enrico Fermi Institute, and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637 5 1 bDepartment of Physics, Enrico Fermi Institute, University of Chicago, Chicago, IL 60637 0 2 y a Abstract M WecombinetheTwinHiggsmechanismwiththeparadigmofCompositeHiggsmodels. In 2 thisclassofmodelstheHiggsisapseudo-Nambu-Goldstonebosonfromastronglycoupled 2 sector near the TeV scale, and it is additionally protected by a discrete symmetry due ] h to the twin mechanism. We discuss the model building issues associated with this setup p and quantify the tuning needed to achieve the correct electroweak vacuum and the Higgs - p mass. In contrast to standard Composite Higgs models, the lightest resonance associated e h with the top sector is the uncolored mirror top, while the colored top partners can be [ made parameterically heavier without extra tuning. In some cases, the vector resonances 2 are predicted to lie in the multi-TeV range. We present models where the resonances – v both fermions and vectors – being heavier alleviates the pressure on naturalness coming 0 9 from direct searches demonstrating that theories with low tuning may survive constraints 8 from the Large Hadron Collider. 7 0 . 1 0 5 1 : v i X r a E-mail: [email protected], [email protected], [email protected] Contents 1 Introduction 1 2 The SO(8)/SO(7) model 3 2.1 Representations of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The non-linear σ-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Ingredients for minimal tuning 7 4 The breaking of Z and electroweak symmetry 9 2 4.1 Z -breaking in the top sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 4.2 Z -breaking in the lighter quarks . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 4.3 Z -breaking in the gauge sector . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 5 Concrete models 14 5.1 Computing the Higgs mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 Model A: Z -breaking in the gauge sector . . . . . . . . . . . . . . . . . . . . . . 16 2 5.3 Model B: Z -breaking in the lighter quarks . . . . . . . . . . . . . . . . . . . . . 18 2 6 Phenomenology 19 7 Conclusions 21 A Technicalities of SO(8)/SO(7) 23 1 Introduction There are several possibilities to naturally stabilize the electroweak scale and the Higgs mass against large UV corrections. However, after the discovery of a light Standard Model (SM)-like Higgs boson and increasing limits on new particles from the first run of the LHC, it is hard to find models less tuned than 5 10%. The challenges faced by fully natural models are ∼ − becoming several, one of them in particular is the lack of any sign of new colored particles around the weak scale. This is particularly true for the case of Natural SUSY [1,2], where light stops are generally required. On the other hand, in strongly coupled scenarios such as Composite Higgs (CH), where the Higgsisapseudo-Nambu-Goldstoneboson(pNGB)ofagivencosetG/H [3],newcoloredvector- likequarksareexpectedtoliewithinafewhundredsofGeVoftheHiggs. Morespecificallythey are expected to be close to f, where f is the Goldstone scale. This is a direct consequence of the partial compositeness mechanism implemented in these models to solve the flavor problem [4, 5]. In this framework, the SM quarks are an admixture of elementary quarks and composite fermions, which we will write generically as Ψ, with the same SM gauge quantum numbers. In 1 this case, the Higgs mass is [6–8] N y2v2m2 m2 c t Ψ. (1) h (cid:39) 2π2 f2 From these estimates, as well as from several other checks [9–12], the prediction of natural CH models is the presence of light fermionic resonances, i.e. with a mass m f. This prediction Ψ ∼ also applies to electroweak composite vector resonances, but their masses can be larger than m due to the smallness of the weak gauge coupling, g, compared to y . While this simple Ψ t scenario points to straightforward and testable LHC signals, it is useful to assess the robustness of such a connection. There are, however, composite Higgs scenarios where one can deviate from this conclusion, i.e. models where m = 125 GeV without the need for light top partners, h but those models are severely fine tuned [13]. An interesting possibility to disentangle, without additional tuning, the strong connection between a light Higgs and light colored top partners, while keeping the scale f as small as possible,1 can be offered by the Twin Higgs (TH) mechanism [16], see also [17]. As far as the SMaloneisconsidered, theTHmechanismconsistsofsimplymirroringtheSMlagrangian, viaa Z -symmetry, resulting in the SM and its copy, SM(cid:48) [18]. The scalar potential has an accidental 2 U(4)/U(3) symmetry, and radiative corrections do not break it at the level of the quadratic action thanks to the Z symmetry [16]. The embedding of this mechanism in calculable models, 2 like CH models (see [19–21] for the supersymmetric case), serves as a test as to how well the twin mechanism works to realize the weak scale with low tuning.2 As we will show in this paper, CH models augmented with the TH mechanism provide a correct Higgs mass with minimal tuning and without any light colored top partner (see also [24]).3 In this scenario the lightest top partner is an uncolored mirror top from the mirror sector, with mass y f, which replaces Ψ in the prediction of eq. (1). At a practical level, the t ∼ simplest example of a Composite Twin Higgs (CTH) – just a larger class of CH models – relies on the global symmetry breaking SO(8)/SO(7). The global symmetry is explicitly broken by couplings to SM SM(cid:48). As in standard CH, the explicit breaking induces radiative electroweak × symmetry breaking (EWSB). The novelty here is that, roughly speaking, the overall scale of the potential is suppressed thanks to the protection of the additional Z symmetry. This crucially 2 depends on the coupling to a mirror elementary sector (both to gauge fields and fermions), as shown in figure 1. However, in the limit of an exact Z symmetry, we expect v = f. Due to the 2 strong constraints on f, having v f is necessary. Hence, Z needs to be broken to provide 2 (cid:28) realistic realizations of CTH models. Depending on the actual breaking of Z the Higgs potential can be relatively insensitive 2 to the mass of the composite fermions, while being set by the scale of the uncolored mirror top. If this is the case we can achieve the correct Higgs mass without colored top partners and trigger EWSB with minimal tuning. A key result of this paper is to point out a class 1An orthogonal possibility is offered by the Little Higgs mechanism, where the model building allows for a larger separation between f and v without large tuning [14] (see references therein), but we will not discuss them here. See also [15] for a comparison between Little Higgs and Composite Higgs. 2See also orbifold models [22,23]. 3Or conversely, TH models embedded inside the CH framework offer a more UV friendly setup. 2 G/H SM SM 0 Z 2 Figure1. PictorialrepresentationofthedynamicsofaCompositeHiggsmodelprotectedbytheTwin Higgs mechanism. In the model under consideration, G/H = SO(8)/SO(7). This global symmetry is explicitly broken by the interactions with the two external copies of the SM (exchangeable under a Z 2 symmetry). of models with this property. A schematic drawing of the resulting spectrum, as well as a comparison with usual composite Higgs models, for minimal tuning f2/v2, is shown in figure 2. We provide estimates for the Higgs mass and tuning, as well several examples where we will be able to explicitly compute the Higgs potential. In order to do this we rely on purely four dimensional models (see [25] for a holographic realization). To emphasize what is truly related to the TH mechanism, as opposed to standard CH models, we will consider several fermionic representations for the elementary quarks in order to show their effect on the Higgs potential. Therestofthepaperisorganizedasfollows. Insection2wedefinetheframework, discussing thecosetstructure, thegaugingofSM SM(cid:48), fermionrepresentations, andtheformoftheHiggs × potential. A general parameterization of the Higgs potential as well as possible mechanisms of Z -breaking will be shown in section 4, after a brief introduction of the basic points in section 2 3. Two concrete examples are provided in section 5. We discuss the main phenomenology in section 6 and we conclude in section 7. We refer to appendix A for technical details. 2 The SO(8)/SO(7) model The global symmetry breaking pattern of the simplest CTH model is SO(8)/SO(7).4 The subgroup SO(7) allows for an unbroken SO(4) custodial group. The 7 pNGBs are encoded in the field U obtained by the exponentiation of the fluctuations associated with the broken 4The minimal coset used in linear realizations is U(4)/U(3), which delivers the same number of pNGBs, but does not contain a residual custodial symmetry. Moreover, with U(4)/U(3) in the non-linear case the twin mechanism is not realized in the gauge sector (as first observed in [26]). Groups larger than SO(8) could be fine as well, but there one expects the presence of extra physical pNGBs in addition to the Higgs. An earlier work [27] recognized the usefulness of SO(8)/SO(7) to prevent a large custodial breaking in composite models implementingthetwinmechanism. Thismodeldiffersfromoursinthatitisbasedonleft-rightsymmetrywhere the top partners are still colored under SM color. 3 E 4πf Ψ Ψ mirror top f h h v Composite Higgs Composite Twin Higgs Figure 2. Comparison between the spectra of minimally tuned Composite Higgs and Composite Twin Higgs models. generators, Π U = expi . (2) f Given the basis of generators chosen in appendix A, the pNGB matrix Π containing the 7 goldstones can be written as Π = √2πaˆTaˆ, aˆ = 1,...,7, (3) where Taˆ are the broken generators, defined in appendix A, and πaˆ are the goldstone fields in the 7 of SO(7). The transformation under SO(8) U g U h(g,Π)T, → · · lets us write the two indices of U as U¯j (we follow the notation of [28]). The index i is linear i under G, while ¯j = J,8 is non-linear under G but split into a 7 of SO(7) (index J) and a { } singlet. Later we will make use of UJ and Σ U8, i i ≡ i sin π ΣT = f (πˆ1,πˆ2,πˆ3,πˆ4,πˆ5,πˆ6,πˆ7,πcot π), π √πaˆπaˆ. (4) π f ≡ However, in CTH these are not the only global symmetries of the composite sector. Indeed, to realize the TH mechanism we have to include at the level of the composite sector a mirror copy of QCD, which amounts to having an unbroken SU(3) SU(3)(cid:48) Z 5. Formally this means c × c × 2 5Note that this is in contrast to orbifold-based models in which QCD and mirror QCD descend from an SU(6) group or larger [23,25]. 4 that the global symmetry is actually6 G SU(3) SU(3)(cid:48) Z SO(8) = c × c × 2 × . (5) H SU(3) SU(3)(cid:48) Z SO(7) c × c × 2 × This guarantees that we can partially gauge the global symmetry G by two identical copies of the SM, SM SM(cid:48)(including QCD and its mirror copy). The gauging of SM SM(cid:48)proceeds in × × the following way (see appendix A for details): we identify the two SO(4)’s inside SO(8) and within each SO(4) SU(2) SU(2) we gauge SU(2) U(1) . Hereafter primed objects L R L Y ∼ × × refer to quantities and fields of the mirror sector. Given this gauging, the Σ field, in terms of the only physical fluctuation πˆ4 = h, reduces to ΣT = (0,0,0,s ,0,0,0,c ), (6) h h where s sin(h/f) and c cos(h/f). While the Z is evident in the exchange of the two h h 2 ≡ ≡ SO(4)’s inside SO(8), it acts non-linearly on the pNGBs. Indeed it can act on Σ as (cid:32) (cid:33) 0 1 Z : Σ R Σ, R = 4 4 . (7) 2 → · 1 0 4 4 The physical mode h shifts by discrete values under this symmetry π Z : h h+f s c . (8) 2 h h → − 2 ↔ → 2.1 Representations of fields Given the global symmetries in the model, we can expect the presence of resonances charged under all of them. In this work, however, we will only be interested in those that couple directly to SM fields. The couplings comes from the partial compositeness mechanism which couples “elementary” quarks to “composite” resonances via a linear mixing term.7 This fixes the quantum numbers of the composite operators y f q¯ Ψ+y f u¯ Ψ+(mirror). mix L L R R L ∼ · · The elementary fields are in the usual representations of the (elementary) SM and likewise for the mirror fields. Table 1 summarizes the possible irreducible representations of SO(8) for SM fields, mirror fields, and the composite resonances, as well as their decompositions under the relevant subgroups. From Table 1, it is clear that q (and q(cid:48) ) can only be embedded in the L L 6There must be an unbroken U(1) to obtain the correct hypercharge for each SM fermion. It is defined by X Y =T3+X separately for both the sectors, where T3 comes from each SU(2) of each SO(4). The charges are R R R 2/3 and 1/3 for up-type and down-type quarks, respectively. We similarly omit any discussion of the leptons − as their effect on the Higgs potential is typically negligible. 7Hereafter, the word “elementary” will refer to q and u fields that appear in . In this basis they are L R mix L neither mass eigenstates nor eigenstates of the SM gauge groups, but rather the “elementary” SM gauge group. In the mass basis both gauge fields and fermions are partially composite, in general. 5 8, while the right-handed quarks and their mirror partners have several options, namely the 1, the 28, or the 35. The gauge fields are in the adjoint representation and couple to composite vectors in the 28. The SM gauge fields acquire the typical masses proportional to the scale of electroweak symmetry breaking, v. The mirror gauge fields, on the other hand, are not inside of SO(7) and acquire masses proportional to the goldstone scale, f, instead. We expect a spectrum of the form m gv, m gf, m (1 4π)f, W W(cid:48) ρ ∼ ∼ ∼ O − (9) m y v, m y f, m (1 4π)f. t t t(cid:48) t Ψ ∼ ∼ ∼ O − SM SO(8) SO(7) SO(4) SO(4)(cid:48) SU(3) SU(3)(cid:48) Z × c × c × 2 q - - (4,1) (3,1) L u - - (1,1) (3,1) R W - - (6,1) (1,1) Mirror SO(8) SO(7) SO(4) SO(4)(cid:48) SU(3) SU(3)(cid:48) Z × c × c × 2 q(cid:48) - - (1,4) (1,3) L u(cid:48) - - (1,1) (1,3) R W(cid:48) - - (1,6) (1,1) Resonances SO(8) SO(7) SO(4) SO(4)(cid:48) SU(3) SU(3)(cid:48) Z × c × c × 2 Ψ 8 7 1 (4,1) (1,4) (3,1) (1,3) L ⊕ ⊕ ⊕ Ψ 1 1 (1,1) (3,1) (1,3) R ⊕ Ψ 35 27 7 1 (9,1) (1,9) (4,4) (1,1) (3,1) (1,3) R ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ Ψ 28 21 7 (6,1) (1,6) (4,4) (3,1) (1,3) R ⊕ ⊕ ⊕ ⊕ ρ 28 21 7 (6,1) (1,6) (4,4) (1,1) ⊕ ⊕ ⊕ Table 1. Possible representations of the resonances. Note that the composite fermions are charged under SU(3) SU(3)(cid:48) in a Z -invariant way. Also note that the SM and mirror fields are embedded in 2 × incomplete representations. The visible sector resonances are singlets of SO(4)(cid:48), e.g. the (9,1), and likewise the mirror resonances are singlets of SO(4). 2.2 The non-linear σ-model The 7 of SO(7) of pNGBs contains a 4 under the visible SO(4), while the other 3 pNGB’s form a broken multiplet of the SM(cid:48) and are eaten by the mirror W(cid:48)±,Z(cid:48).8 Using the basis of appendix A, the low energy non-linear σ-model is given by f2 = (D Σ)TDµΣ, (10) µ L 2 8Note that under Z the mirror photon remains massless. One possibility to remove the mirror photon is to 2 break the Z in hypercharge by not gauging the mirror hyperchange [26,29]. 2 6 where D = ∂ ig(AaTa +A(cid:48)aT(cid:48)a) with T and T(cid:48) as the generators of SU(2) and SU(2)(cid:48) . µ µ − µ L µ L L L L L As previously noted, there is a relation between the masses of gauge bosons and their mirror partners, g2f2 g2f2 m2 (h) = s2, m2 (h) = c2. (11) W 4 h W(cid:48) 4 h This expression for m fixes the value of h , W (cid:104) (cid:105) (cid:18) (cid:19) h v = f sin (cid:104) (cid:105) . (12) f We move onto the fermion sector which, given the size of the top Yukawa, gives the leading contribution to the Higgs potential. The lagrangian of the top sector is = q¯ iD/q +u¯ iD/u +y f(q¯8)iΣ u1 +h.c.+(mirror). (13) L L L R R t L i R In order to write down the Yukawa-like term in the above lagrangian we have assigned the SM SM(cid:48)quarks to representations of SO(8). The notation in eq. (13) means that q 8 and L × ∈ u 1. The field u is shown in the 1 but as shown in Table 1 other representations can be R R ∈ used. The embeddings are 1 (q8)i = (ib ,b ,it , t ,0,0,0,0)i, u1 = u . (14) L √2 L L L − L R R From eq. (13), the top and its mirror partner have masses y fs y fc t h t h m (h) = , m (h) = . (15) t t(cid:48) √2 √2 The ratio of these, m /m = c /s , is typical of the mirror sector. t(cid:48) t h h From eqs. (11) and (15), it is possible extract the Higgs couplings to SM vectors V, SM fermions f, mirror vectors V(cid:48), and mirror fermions f(cid:48). Normalizing the couplings to the values they have in the ordinary (unmirrored) SM, they read (cid:112) (cid:112) c = 1 v2/f2, c = 1 v2/f2, hVV hff − − (16) (cid:112) c = 1 v2/f2(g(cid:48)2/g2), c = (v/f)(y(cid:48)/y), hV(cid:48)V(cid:48) hf(cid:48)f(cid:48) − − − where g(cid:48) and y(cid:48) denote the mirror gauge couplings and mirror Yukawa couplings, respectively, and show the effect of Z -breaking in the couplings. The Higgs couplings to SM particles are 2 of the usual form as in standard CH models and they are induced by the non-linearities of the σ-model. Notice that there is a universal rescaling for the couplings of both the SM vectors and the SM fermions to the Higgs [24]. 3 Ingredients for minimal tuning When radiatively generating the Higgs potential there are two sources of tuning, obtaining the correct vacuum v and obtaining the correct Higgs mass. In realistic composite models, getting 7 the correct vacuum requires a separation of scales, v f; this tuning is always present. Tuning (cid:28) in the Higgs mass is model dependent and is often worse and/or requires light top partners [13]. A natural model should then aim to tune the Higgs mass no more than the vacuum expectation value (VEV). This scenario is called minimal tuning f2 ∆ = . (17) |minimal v2 The Higgs potential of CTH can satisfy minimal tuning, without the need for light colored top partners, provided some important ingredients are included. Section 4 presents a systematic discussion, but here we highlight the two most important aspects. For illustration we use the simple, though incomplete, non-linear σ-model of the previous section. From eq. (13), the Coleman-Weinberg potential is [30] N y4f4 (cid:20) (cid:18) 2Λ2 (cid:19) (cid:18) 2Λ2 (cid:19)(cid:21) V(h) = c t c4 log +s4 log . (18) nlσm 64π2 h y2f2c2 h y2f2s2 t h t h Due to theZ invariance, the minimum is at h /f = π/4, which is unviable for phenomenology. 2 (cid:104) (cid:105) On the other hand, the Z symmetry ensures that the terms quadratically divergent in the cut- 2 off Λ are absent because they are proportional to y2Λ2(s2 + c2) which is accidentally SO(8) t h h invariant. Additionally, the overall scale of the potential is suppressed because it is generated at (y4) O t the order at which the Z no longer results in accidental SO(8) invariance. In order to attain 2 s 1 the Z symmetry needs to be broken. Assuming that the Z breaking terms have the h 2 2 (cid:104) (cid:105) (cid:28) same parametric dependence on y and f, the potential becomes t N y4f4 V(h) = V(h) + c t bs2. (19) nlσm 32π2 h Ifbisamodel-dependent (1)coefficient, electroweaksymmetryisbrokenwithminimaltuning. O The Higgs mass is N m2m2 (cid:20) (cid:18)Λ2(cid:19) (cid:18) Λ2 (cid:19) (cid:21) m2 c t t(cid:48) log +log + (1) , (20) h (cid:39) 2π2 f2 m2 m2 O t t(cid:48) where m and m are the top and mirror top masses, respectively, and Λ is the scale where t t(cid:48) resonances will appear. In this case, minimal tuning is achieved, independent of Λ. This example shows that the basic ingredients for a minimally tuned CTH model, without light colored top partners, are an overall scale of the potential proportional to y4f4. • t Z -breaking terms of the same numerical size of the Z -preserving ones. 2 2 • In the next section we systematically study CTH for several representations of the composite fermions and different patterns for the breaking of Z . 2 8 4 The breaking of Z and electroweak symmetry 2 As we will soon see, generating the correct Higgs potential relies on breaking Z at the right 2 order in partial compositeness couplings y and y . We start by considering these couplings, L R especially those of the top sector, which usually give the largest contribution to the Higgs potential, = y fq¯ UΨ+y fu¯ UΨ+h.c.+ (Ψ,U,m )+mirror. (21) L L R R comp Ψ L L Thereareseveralimportantconsequencesthatalreadyfollowfrompartialcompositeness. First, the parameters y and y break the global symmetries. This implies that in the limit y 0 L R L,R → the Higgs potential vanishes and that the contributions start at order y2 (hereafter we power count in y y y ). With the addition of the mirror sector, contributions that are Z L R 2 ∼ ∼ symmetric will start at order y4. There are several possible functional forms for the Higgs potential, which are determined by the elementary quark embeddings; a discussion of the different expressions is presented in appendix A. In this section we closely follow [13,31]. In the cases of interest, the 1-loop Higgs potential generated by the top sector is (cid:20) (cid:21) N V(h) c (yf)2nm2(2−n) aF (h/f)+bF (h/f) , n = 1,2 (22) TH (cid:39) 16π2 Ψ Z2 Z/2 where for a given function, F, subleading terms in y have been dropped. The functions F Z2 and F specify the Z -preserving and Z -breaking parts of the potential, respectively. For Z/ 2 2 2 illustration one can consider F s2c2 and F s2. Z2 ∼ h h Z/2 ∼ h As eq. (21) suggests, the Yukawa couplings for the quarks are fk−1 y yk k = 1,2. (23) SM (cid:39) mk−1 Ψ Different quark representations provide different values of n and k which are summarized in Table 2. The case of k = 1 versus k = 2 simply reflects whether u is fully composite or not, R which can be realized for u in the total singlet, given that this embedding does not break the R global symmetries and hence the natural size of the elementary-composite mixing is y f m . R Ψ ∼ That n = 1 for the 35 is a peculiarity of the 35 (see Table 1 and the discussion in [32]). n k V(h) y TH SM q8 u1 2 1 y4f4 y L R ∼ q8 u28 2 2 y4f4 y2(f/m ) L R ∼ Ψ q8 u35 1 2 y2f2m2 y2(f/m ) L R ∼ Ψ Ψ Table 2. Valuesofnandk forseveralrepresentationsofright-handedquarksasdescribedineqs.(22) and (23). The most favorable case is when n = 2 and k = 1 and the values of a and b both (1). ∼ O With this choice and with a and b of the same size, the Higgs mass is not sensitive to m upon Ψ 9