Vacuum alignment in a composite 2HDM
VVacuum alignment in a composite 2HDM
Chengfeng
Cai and Hong-Hao
Zhang ∗ School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
Giacomo
Cacciapaglia † Universit´e de Lyon, F-69622 Lyon, France: Universit´e Lyon 1,Villeurbanne CNRS/IN2P3, UMR5822,Institut de Physique Nucl´eaire de Lyon.
We study in detail the vacuum structure of a composite two Higgs doublet modelbased on a minimal underlying theory with 3 Dirac fermions in pseudo-real represen-tations of the condensing gauge interactions, leading to the SU(6) / Sp(6) symmetrybreaking pattern. We find that, independently on the source of top mass, the mostgeneral CP-conserving vacuum is characterised by three non-vanishing angles. Aspecial case occurs if the Yukawas are aligned, leading to a single angle. In the lattercase, a Dark Matter candidate arises, protected by a global U(1) symmetry.
Preprint: LYCEN 2017-05 ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] J a n I. INTRODUCTION
Since the discovery of a Higgs boson at the LHC in 2012, the main priority in particle physicshas been to establish if the new state really is the last missing particle [1] predicted by theStandard Model (SM) of particle physics. Measuring the properties of the new boson can be seenas a new tool to test theories Beyond the SM. One very attractive and time-honoured possibilityis that the Higgs boson arises as a composite state of a more fundamental confining theory.This is the only experimentally well tested mechanism of spontaneous symmetry breaking, asit appears at low energies in quantum chromodynamics (QCD). In fact, as soon as the SMwas proposed, models of dynamical symmetry breaking were born [2, 3]. While the initialproposals were essentially Higgs-less, i.e. featuring heavy scalar states like in QCD, theoriesfeaturing light composite scalars are rather common. A Higgs-like boson can arise either as anaccidentally light scalar resonance (being light, for instance, because of a conformal dynamicsat high energies [4, 5] and/or radiative corrections [7]) or as a pseudo-Nambu-Goldstone boson(pNGB) [8, 9].In this work we are mainly interested in the latter possibility, which received renewed atten-tion in the last decade following the conjectured correspondence between anti-de-Sitter extradimensions and conformal field theories [10]. Boldly using the conjecture without supersym-metry, the dynamical properties of composite near-conformal theories have been associatedwith extra dimensional constructions where the Higgs boson arises as an additional polarisa-tion of a gauge boson [11]. The phenomenology of composite pNGB Higgses has been widelyexplored in the recent literature (see for instance the reviews [12–14]). The properties of thecomposite sector mainly depend on the symmetry breaking pattern associated to the strongdynamics. A useful guiding principle is thus provided by the definition of a simple underlyinggauge-fermion theory that confines in the infra-red, like QCD. The existence of a FundamentalComposite Dynamics (FCD) [15] restricts the cosets to 3 basic possibilities: SU(2 N ) / Sp(2 N )for fermions in a pseudo-real representation of the confining gauge group, SU( N ) / SO( N ) forreal and SU( N ) / SU( N ) for complex ones. It follows that the minimal model is based onSU(4) / Sp(4) , which enjoys a very simple FCD based on G TC = Sp(2) with 2 Dirac fermionsin the fundamental representation [22, 23] (in the following, for historical reasons, we will de-note the underlying dynamics as Technicolor [3] – TC). This theory has also been studied onthe lattice, where it has been proven to condense [24] and its spectrum has been obtained [25–27]. An analysis based on a Nambu-Jona-Lasinio modelling can be found in Ref. [28]. Also,minimality can now be defined not in terms of the number of pNGBs the coset contains (aka“coset-ology”), but in terms of the degrees of freedom of the underlying theory.Going non-minimal has its own advantages: having more light pNGBs allows to study morecomplex Higgs sectors. For instance, a composite two Higgs doublet model (2HDM) can beobtained from a QCD-like G TC = SU( N ) theory with 4 fundamental Dirac fermions [29] (forother composite 2HDMs, see Refs [30–32]). The additional pNGBs can also play the role ofDark Matter [21] if the coset allows for a protecting symmetry [33–36]. In this work we are See also the recent results in Ref. [6]. A more minimal coset would be SO(5) / SO(4) [16], which however does not have a simple FCD [17, 18]. TheSU(4) / Sp(4) ∼ SO(6) / SO(5) coset has also been studied in Refs [19–21]. interested in an extension of the minimal FCD model based on the coset SU(6) / Sp(6), whichcan be minimally obtained in an Sp(2) model with 3 Dirac fermions and leads naturally toa composite 2HDM. The main difference with the FCD in Ref. [29], based on SU(4) / SU(4),is about the action of the custodial symmetry on the two doublets. Including a custodialsymmetry in the global symmetry is crucial in order to obtain realistic models [37]. In modelswith two doublets, the custodial symmetry can be defined in two inequivalent ways [38, 39]: inthe model of Ref. [29] the two doublets transform as a complex bi-doublet under the gaugedSU(2) L and the global SU(2) R . We will see that in the SU(6) / Sp(6) model under considerationone can define two global SU(2) R ’s acting, individually, on the two doublets.We finally remark that an underlying theory similar to the one we are interested in has beenused in Refs [40, 41] to construct a Little Higgs model, where the light Higgs also arises as apNGB from the SU(6) / Sp(6) coset. The main difference between the two approaches is thatin Little Higgs models the gauged subgroup is more extended than the electroweak (EW) one,and the additional gauge generators are assumed to be broken at the condensation scale.
A. Preliminaries:
SU(6) → Sp(6)
The unbroken group Sp(6), of rank 3, contains 3 commuting SU(2) subgroups: it is thusnatural to try to identify them with the EW sector of the SM, i.e. the gauged SU(2) L andthe partly-gauged SU(2) R . We insist on being able to identify an SU(2) R in order to obtain amodel that respects a custodial symmetry. Furthermore, the 14 Goldstone bosons in the cosettransform, under the 3 SU(2)’s, as: Sp(6) → (2 , , ⊕ (2 , , ⊕ (1 , , ⊕ (1 , , ⊕ (1 , , . (1)A composite Higgs boson can, thus, be identified with any one of the 3 bi-doublets. To classifythe various possibilities, it is useful to list the quantum number assignments of a fundamental of SU(6), ψ , that will correspond to the techni-fermions in the underlying gauge-fermion model.Each sextet ψ can be decomposed into 3 doublets ψ i , i = 1 , ,
3, each one transforming under adifferent global SU(2) i . Note that ψ is a left-handed Weyl spinor, and each doublet ψ i forms aDirac spinor (which can be assigned an SU(2)-invariant mass). There are, therefore, 2 possibleassignments that lead to a custodial invariant composite Higgs, whose properties are listed inTable I.In Case A, the pNGBs in the multiplets (1 , ,
2) and (2 , ,
2) contain scalars with electro-magnetic charges Q = ± ± ξ , so that only choosing integer odd values for ξ avoids states withnon-integer charges. In particular, for ξ = 1, the model contains two Higgs doublets.In Case B, there are always 2 Higgs doublets, however the (2 , ,
1) pNGBs contain anisotriplet: if the vacuum misalignment projects a vacuum expectation value on the triplet,the custodial symmetry would be broken, thus incurring in strong constraints. Furthermore,the contribution to precision tests, in particular to the S parameter, is sensitive to the numberof EW doublets [42], thus one would expect milder constraints in Case A where the underlyingtheories contain only one doublet. SU(2) L U(1) Y SU(2) L Y HiggsCase A ψ T + ξT (2 , , , ,
2) if ξ = 1] ψ ± / ψ ± ξ/ ψ + SU(2) T (2 , ,
2) + (1 , , ψ ψ ± / T i are the diagonalgenerators of each SU(2) i . For the above reasons, in the following we will focus on the family of models in Case A, dis-cussing in particular the case ξ = 1 that provides a composite 2HDM. The paper is organisedas follows: in Section II we introduce the model and study the simplest vacuum alignment.We comment on various mechanisms that generate the top mass, and find that the only con-sistent configuration entails misalignment along one of the two Higgs doublets and a DarkMatter candidate protected by a global U(1) symmetry. Then, in Section III, we study a moregeneral vacuum alignment, naturally twisted along a singlet direction. We present some nu-merical results, mainly focussed on the mass of the Higgs boson candidate, before presentingour conclusions and perspectives in Section IV. II. THE MODEL
The underlying model consists of 3 Dirac fermions ψ i in a pseudo-real representation ofthe strongly interacting group G TC , that we expect to condense and confine in the infrared.The minimal model, in terms of degrees of freedom, respecting these properties is G TC =SU(2) with fermions in the fundamental. The global symmetry of the strong sector is SU(6)(times an anomalous U(1)): the EW gauge symmetry is embedded as in Case A with ξ = 1,i.e. SU(2) L is identified with SU(2) acting on ψ , while the hypercharge is identified withthe diagonal generator of the two remaining SU(2)’s. The pions (pNGBs) generated by thesymmetry breaking SU(6) → Sp(6) include two Higgs doublet candidates transforming underSU(2) R ≡ SU(2) and SU(2) R ≡ SU(2) respectively. One can thus identify two custodialsymmetries, depending on which bi-doublet breaks the EW symmetry:SU(2) L × SU(2) R (cid:124) (cid:123)(cid:122) (cid:125) SO(4) × SU(2) R or SU(2) L × SU(2) R (cid:124) (cid:123)(cid:122) (cid:125) SO(4) × SU(2) R . (2)In general, the vacuum can be misaligned along both doublets, thus care is needed in order topreserve custodial symmetry.The underlying Lagrangian for the techni-fermions thus reads: L = i ¯ ψσ µ D µ ψ − ψ M ψ ψ − ¯ ψ M † ψ ¯ ψ , (3)where ψ is a 6-component Weyl spinor, and M ψ is a mass term that can be written, withoutloss of generality, as: M ψ = m L ( iσ ) 0 00 m R ( − iσ ) 00 0 m R ( − iσ ) . (4)The signs of the mass terms have been chosen arbitrarily, following the choice for the vacuumalignment explained below, and we also chose to work in the basis where the masses of the twoSU(2) L singlets are diagonal. This mass term breaks SU(6) → Sp(2) , however the symmetryis enhanced to Sp(2) × Sp(4) if two masses are equal and to Sp(6) is all masses are equal (recallthat Sp(2) ∼ SU(2)).The covariant derivative contains both the FCD gauge interactions and the EW ones. Thelatter ones are embedded in the following way : T iL = 12 σ i (cid:124) (cid:123)(cid:122) (cid:125) SU(2) L generators , T iR = 12 − σ Ti
00 0 0 (cid:124) (cid:123)(cid:122) (cid:125)
SU(2) R generators and T iR = 12 − σ Ti (cid:124) (cid:123)(cid:122) (cid:125) SU(2) R generators , i = 1 , , , (5)where Y = T R + T R is identified with the hypercharge generator. The above choice (underwhich ψ and ψ are actually anti-doublets of the corresponding SU(2)’s) is consistent with thefollowing choice for the condensate alignment that preserves the EW symmetry: (cid:104) ψψ (cid:105) ∼ Σ = iσ − iσ
00 0 − iσ . (6)A complete list of the 14 broken generators X i and of the 21 unbroken ones S j can be foundin Appendix A 1. Under the EW symmetry SU(2) L × U(1) Y , the pNGBs transform as Sp(6) = ± / + ± / + ± + + + + , (7)where the charged and 2 neutral singlets belong to the bi-doublet under the SU(2) R symmetries.The model contains three bi-doublets but only two are doublets of SU(2) L and may play therole of a Brout-Englert-Higgs doublet. The pion matrix is embedded in U ( φ ) = e i Π( φ ) /f , with Π( φ ) = (cid:88) i =1 φ i X i . (8)One can now perturb the EW preserving vacuum Σ along the broken directions to make thepNGBs appear Σ ( φ ) = U ( φ ) · Σ . (9)The object we defined above, which transforms linearly under SU(6), can be used to define thechiral Lagrangian L ( p ) = f Tr (cid:2) ( D µ Σ( φ )) † · D µ Σ( φ ) (cid:3) − Tr (cid:2) χ · Σ † ( φ ) + χ † · Σ( φ ) (cid:3) , (10)where the spurion χ contains the techni-fermion mass term: χ = 2 BM † ψ , (11) B being a low energy constant that can be computed on the lattice. A. Vacuum misalignment
The EW symmetry breaking is triggered, in this model, by the pNGBs acquiring a vacuumexpectation value (VEV). In general, all the scalars can develop it, however only two correspondto the real part of the neutral components of the two Higgs doublets: (cid:104) φ (cid:105) = v , (cid:104) φ (cid:105) = v . (12)The potential for the pNGBs is generated by all explicit SU(6) breaking terms. Our strategywill be to apply the ansatz that only the two above scalars are involved, and then check underwhich conditions this is realised by the potential.A pion acquiring a VEV generates a misalignment of the vacuum, i.e. it is possible to definean SU(6) transformation that rotates the vacuum Σ to the true minimum of the theory. Inour case, this is defined by a rotation Ω( θ, β ), such thatΣ θ,β = Ω( θ, β ) · Σ · Ω T ( θ, β ) , (13)where tan β = v /v and θ = √ v + v √ f . Note that the pion matrix on the new vacuum is definedas Π( φ ) θ,β = Ω · Π( φ ) · Ω † , U ( φ ) θ,β = Ω · U ( φ ) · Ω † , (14)and Σ( φ ) = U θ,β ( φ ) · Σ θ,β = Ω · U ( φ ) · Σ · Ω T . (15)The rotation Ω that applies to our ansatz can be decomposed as:Ω( θ, β ) = R β · Ω θ · R † β , (16)where Ω θ is a rotation generated by X :Ω θ = e i √ θ X = cos θ I sin θ iσ θ iσ cos θ I
00 0 I . (17)The rotation R β is generated by S : R β = e i √ β S = I β I − sin β I β I cos β I . (18)The fact that the rotation R β is generated by an unbroken generator of the vacuum Σ , allowsto write the pNGB-dependent vacuum matrix asΣ( φ ) = R β · Ω θ · U ( φ (cid:48) ) · Σ · Ω Tθ · R Tβ , (19)where U ( φ (cid:48) ) = R † β · U ( φ ) · R β . The rotation R β reshuffles the broken generators, so U ( φ (cid:48) ) issimply a change of basis, furthermore it drops out from the first term of the chiral Lagrangianin Eq.(10) as the gauged generators are left invariant. It is therefore convenient to study thetheory in a vacuum that only depends on θ , with the pNGB basis φ (cid:48) , so that in the followingΣ( φ ) = Ω θ · U ( φ (cid:48) ) · Σ · Ω Tθ . (20)All the spurions that explicitly break SU(6) thus need to be rotated by R † β : if they are invariant,the β -dependence will be completely removed, indicating that the vacuum of the theory can becharacterised in terms of a single angle; if not, the dependence on β will be carried-off by thespurions themselves.The pNGB matrix can be conveniently expressed as follows: i Π( φ (cid:48) ) · Σ = 12 − (cid:16) √ η + √ η (cid:17) σ H H − H T − (cid:16) √ η − √ η (cid:17) σ G − H T − G T − (cid:113) η σ (21)with η , being the singlet pseudo-scalars (1 , , H , H , G being, respectively, theSU(2) L × SU(2) R , SU(2) L × SU(2) R and SU(2) R × SU(2) R bi-doublets. In components,they can be written as: H = √ (cid:32) h + iπ iπ + π iπ − π h − iπ (cid:33) , H = √ (cid:32) h + iA √ H + −√ H − h − iA (cid:33) ,G = √ (cid:32) −√ η − ϕ − iη ϕ + iη √ η + (cid:33) . (22)In this basis, H is the doublet that develops a VEV and breaks the EW symmetry, so that h will play the role of the pNGB Higgs and π i are the exact Goldstones eaten by the massive W and Z , while H is the second doublet. G , on the other hand, contain two neutral singlets ( η being a pseudo-scalar) and a charged one. More details on the CP properties of the pNGBs areprovided in Appendix A 2, while the lowest order pNGB couplings, in the new basis, are givenin Appendix A 3 and explicitly show no dependence on β . The LO chiral Lagrangian generatesthe following masses for the W and Z : m W = 2 g f sin θ , m Z = m W cos θ W , (23)so that v SM = 2 √ f sin θ = 246 GeV (24)fixes the relation between the decay constant f and the EW scale via the alignment angle θ .The field h is the only one that has linear couplings to the vector bosons and thus can playthe role of the Higgs boson. Its couplings are given by: g h W W = g h ZZ cos ( θ W ) = √ g f sin θ cos θ = g SM hW W cos θ . (25)The results above match the ones from more minimal models [15, 16].The second term in Eq.(10), which contains the techni-fermion mass, can also be written interms of the θ -vacuum following Eq.(20) by defining a rotated mass: M (cid:48) ψ = R † β · M ψ · R β = m L ( iσ ) 0 00 ( m R + δm R c β )( − iσ ) δm R s β ( iσ )0 δm R s β ( iσ ) ( m R − δm R c β )( − iσ ) , (26)where we employ the compact notation s x = sin x and cos x = c x , and we defined m R = m R + m R , δm R = m R − m R . (27)We see that the β -dependence drops if δm R = 0, i.e. when the two SU(2) R doublets aredegenerate.The misalignment of the vacuum, and the masses of the remaining pNGBs, are determinedby the effective potential, that is dominated by the effect of loops of tops, i.e. by the explicitbreaking of the global symmetry introduced by the interactions giving rise to the mass of thetop. In the following, we will first discuss the effect of the effective top (and bottom) Yukawas,before presenting general results on the potential. B. Finding a minimum: the top (and bottom) Yukawas
As a warm up, we will first consider the top mass generated by 4-fermion interactions,bilinear in the SM fields Q L and t cR (defined here as left-handed Weyl spinors): y (cid:48) t Λ t ( Q L t cR ) † α ( ψ T P α ψ ) + y (cid:48) t Λ t ( Q L t cR ) † α ( ψ T P α ψ ) , (28)where α is an SU(2) L index and P α , P α are projectors (defined in Appendix B 1) that selectthe components of (cid:104) ψ T ψ (cid:105) transforming as the Higgs doublets. This is a generalisation of thefour-fermion interactions introduced in Refs [15, 23]. The couplings y (cid:48) t and y (cid:48) t are the pre-Yukawas, while Λ t is a new dynamical scale where the 4-fermion interactions are generated.This scenario is inspired by the old Extended Technicolor [43] idea, where such interactions weregenerated by an extended gauge sector. A generic problem with this approach, that renders itsomewhat unlikely, is the fact that other four-fermion interactions, including flavour violatingones in the SM fields, are generated at the same scale. This is a problem as Λ t cannot betoo large without suppressing the top (and other quarks’) mass, unless the ψ T ψ operators hasa large anomalous dimension . One easy way out would be to consider models where thetop mass is generated at a different scale from the other quarks and leptons: this would beenough to suppress dangerous flavour violation [47, 48], however complete models of this kindare still lacking (for a bookkeeping example, see [49]). In a following subsection we will discusshow other mechanisms generating fermion masses would affect the conclusions we reach in thissection. For now, we take this simple mechanism for the top, without worrying about the fullflavour structure, and analyse the impact on the vacuum alignment.First, we note that the effect of the two pre-Yukawas can be embedded into a single spurionthat is a linear combination of the two projectors, i.e. y t P α + y t P α , where we removed theprime to distinguish the couplings of the effective chiral Lagrangian from the couplings of thefour-fermion interactions, to which they are related via form factors. To define the theory inthe θ -vacuum of Eq.(20), we need to rotate the spurion as follows: R β · ( y t P α + y t P α ) · R † β = Y t P α + Y t P α , (29)where Y t = c β y t + s β y t , Y t = − s β y t + c β y t , (30)carry the dependence on β (we remark that ∂∂β Y t = Y t and ∂∂β Y t = − Y t ). The operator inthe chiral Lagrangian generating the top mass is thus: L ( p ) ⊃ f ( Q L t cR ) † α ( Y t Tr[ P α · Σ( φ (cid:48) )] + Y t Tr[ P α · Σ( φ (cid:48) )]) ∼ − f sin θ Y t ( t L t cR ) † − (cid:20) Y t √ (cid:18) c θ h + i √ s θ η (cid:19) + Y t √ (cid:16) c θ ( h + iA ) + s θ ( ϕ + iη ) (cid:17)(cid:21) ( t L t cR ) † + (cid:20) Y t c θ H − − Y t s θ η − (cid:21) ( b L t cR ) † + . . . (31)where the dots include couplings to the pNGBs. The top mass is thus proportional to Y t ( m t = Y t f sin θ ), while Y t generates couplings of other pNGBs. The mass for the bottomquark can be generated by a very similar operator, defined in terms of two other projectors P b and P b (also defined in Appendix B 1): L ( p ) ⊃ f ( Q L b cR ) † α ( Y b Tr[ P α b · Σ( φ (cid:48) )] + Y b Tr[ P α b · Σ( φ (cid:48) )]) ∼ − f sin θ Y b ( b L b cR ) † − (cid:20) Y b √ (cid:18) c θ h + i √ s θ η (cid:19) + Y b √ (cid:16) c θ ( h − iA ) − s θ ( ϕ − iη ) (cid:17)(cid:21) ( b L b cR ) † − (cid:20) Y b c θ H + + Y b s θ η + (cid:21) ( t L b cR ) † + . . . (32)where the β -dependence is encoded in Y b = c β y b + s β y b , Y b = − s β y b + c β y b . (33) The tension is in the fact that the bilinear operator ψ T ψ , that enters in the top mass, needs a dimensionclose to 1 (i.e. that of an elementary scalar field), while the dimension of ( ψ T ψ ) , that induces a mass for thepNGB Higgs, needs to be close to 4. Bootstrap techniques have been used to prove the existence of an upperlimit on the anomalous dimensions of scalar operators that seem to disprove the above situation [44, 45].However, the bound applies to the anomalous dimension of the operator with smallest dimension, which maynot be the one associated to the Higgs mass (see Ref. [46] for a counter-example). The feasibility of thisscenario is, therefore, still an open question. V Yuk = − C t f (cid:88) α (cid:0) | Y t Tr[ P α · Σ] + Y t Tr[ P α · Σ] | + | Y b Tr[ P αb · Σ] + Y b Tr[ P αb · Σ] | (cid:1) . (34)Expanding the above operator up to linear terms in the pNGBs: V Yuk = − C t f (cid:26)(cid:0) | Y t | + | Y b | (cid:1) s θ + h √ f (cid:0) | Y t | + | Y b | (cid:1) s θ + h √ f ( (cid:60) Y t Y ∗ t + (cid:60) Y b Y ∗ b ) c θ s θ + ϕ √ f ( (cid:60) Y t Y ∗ t − (cid:60) Y b Y ∗ b ) s θ s θ + A √ f ( (cid:61) Y t Y ∗ t − (cid:61) Y b Y ∗ b ) c θ s θ + η √ f ( (cid:61) Y t Y ∗ t + (cid:61) Y b Y ∗ b ) s θ s θ (cid:27) , (35)where (cid:60) and (cid:61) indicate, respectively, the real and imaginary parts. The above form of thepotential can be generalised to including all SM fermions by summing over the respectiveYukawa couplings. The potential generates tadpoles for many pNGBs, as a sign that thevacuum alignment is not well defined. Some of them are, however, easily accounted for: thetadpole for the Higgs-like state h vanishes once the potential is minimised in θ , while theone for h vanishes after minimisation w.r.t. β , as it is proportional to ∂∂β V (0)Yuk . Hence, thecancellation of the h tadpole determines β in terms of the pre-Yukawa couplings. On theother hand, the tadpoles for A , η and ϕ do not automatically vanish: either the Yukawas arealigned in such a way that the coefficient vanish at the minimum for θ and β , or the vacuumneeds to be misaligned along the generators associated to the 3 pNGBs.Both tadpoles for the pseudo-scalars A and η are proportional to residual phases in thepre-Yukawas, thus it would be enough to assume that the two couplings have the same phasesfor them to vanish. Note that misaligning the vacuum along A is particularly dangerous, as itleads to a breaking of the custodial symmetry (and a contribution to the ρ parameter, like instandard 2HDMs [38, 39]). The vanishing of the tadpole for the scalar ϕ , on the other hand,would require a non-trivial relation between the real parts of the Yukawa couplings: (cid:60) Y t Y ∗ t = (cid:60) Y b Y ∗ b = 0 , ⇒ tan β = y t y t = y b y b , (36)where the latter relation is valid if the pre-Yukawas have the same phase (i.e., if the tadpolesfor A and η also vanish). In other words, a sufficient condition for the vanishing of theadditional tadpoles is that the pre-Yukawas are aligned, i.e. each SM fermion couples to thesame combination of the two SU(2) R fermions. Then, the rotation R β would tell us that thedoublet that develops the EW-breaking VEV is the same one that couples to all SM fermions. It is enough to replace Y ti Y ∗ tj → (cid:80) u Y ui Y ∗ uj , where the sum included the 3 up-type quarks, and Y bi Y ∗ bj → (cid:80) d Y di Y ∗ dj + (cid:80) l Y li Y ∗ lj , with sums over the down-type quarks and leptons (the factor of 3 compensating thecolour multiplicity). V m = 2 B Tr[ M (cid:48) ψ · Σ] + h.c. = − B (cid:26) M (1 − ∆ + 2 c θ ) − δm R c β (1 − c θ ) − h √ f (2 M + δm R c β ) s θ + h √ f δm R s β s θ + . . . (cid:27) (37)where we have expanded up to linear terms in the pNGBs, and defined M = m L + m R , ∆ = m L − m R m L + m R . (38)Besides the tadpole for the Higgs related to the minimisation of the potential for θ , a tadpolefor h emerges proportional to the mass difference between the two Dirac SU(2) L singlets. Inpresence of δm R (cid:54) = 0, therefore, the value of β at the minimum is modified by the presence ofthe potential above and it is not possible any more to cancel the tadpole for ϕ with a simplealignment of the top and bottom pre-Yukawas, like in Eq. (36).Finally, one needs to consider the contribution of the gauge loops: V g = − C g f (cid:8) g Tr (cid:2) T iL Σ (cid:0) T iL Σ (cid:1) ∗ (cid:3) + g (cid:48) Tr [ Y Σ ( Y Σ) ∗ ] (cid:9) == − C g f (cid:26) g (cid:48) g + g (cid:48) c θ − h √ g + g (cid:48) s θ + . . . (cid:27) . (39)This term will only affect the minimisation for θ . Furthermore, it does not depend on β nor itwill change our conclusions on the tadpole cancellations for other pNGBs. Summary of findings
In summary, we have found that it is not enough to describe the misalignment in terms of theEWSB angle θ and of β . While choosing real pre-Yukawas (more precisely, it suffices that y f y ∗ f is real for all SM fermions so that the SM phase of the CKM matrix can be accommodated)allows us to avoid tadpoles for the pseudo-scalars A and η , it is not possible, in general, toeliminate the tadpole for the scalar ϕ . In the next section, we will therefore define a moreappropriate vacuum that includes a misalignment along the EW singlet.The only simple, and physically relevant case, where the tadpole for ϕ is absent, is whenall pre-Yukawas are aligned with one of the two SU(2) R doublets, i.e. y f = 0 (or y f = 0)for which the minimum solution β = 0 (or β = π/
2) assures Y f = 0 and the vanishing of alltadpoles. Note that for δm R = 0 and aligned pre-Yukawas, i.e. when y f y f is a fermion-universalquantity, one can have the same situation. In fact, the rotation in Eq. (18) can be used toeliminate β from all physical quantities and align the pre-Yukawas with only one of the twodoublets. The vacuum is, therefore, simply parametrised by a single angle θ . The misalignmentequation can then be easily solvedcos θ = 16 BM/f + 8 Bδm R /f C t ( | y t | + | y b | ) − C g (3 g + g (cid:48) ) . (40)2This is the same result from the minimal case SU(4) / Sp(4) [15]. Furthermore, the seconddoublet decouples from the fermions and does not develop a VEV, thus it will act as an inertdoublet.
C. Variations on the top mass and the vacuum misalignment
In the previous subsection we have followed the simplifying assumption that the top andbottom masses come from bilinear couplings to the underlying techni-fermions. In practice,the key assumption is that the spurion generating the top and bottom masses is embeddedin the global symmetries of the model in the same way as the Higgs bosons themselves. Thisscenario may be generated by an Extended Technicolor sector [43]. Another possibility ifoffered by Bosonic Technicolor [50, 51], where the couplings between SM and techni-fermionsare mediated by a scalar field, neutral under the FCD interactions. As the mediator couples tothe techni-fermions via Yukawa-like interactions, in the simplest incarnation the spurion willhave the same symmetry properties as the case we studied. Note that this case is also realisedin the so-called models of half-composite Higgs [52, 53], where the mediator (which has thequantum numbers of the SM Higgs) is assumed to be as light as the composite states. In therest of this section we will explore how other mechanisms for the SM fermion mass generationmay affect our conclusions on the vacuum misalignment.A more recent scenario, which gained popularity in the past decade, is the mechanismof partial compositeness [54], where the elementary SM fermions mix directly with a spin-1/2 composite state. The masses of the SM fermions, thus, are generated after integratingout the heavy resonances. The symmetry properties of the spurions, now, depend on therepresentation of the global symmetry the composite operators belong to. From an effectivefield theory (EFT) point of view, there are infinite possible choices of increasing complexity andonly phenomenological considerations may help preferring one over the other (see, for instance,the discussion in Ref. [20]). Instead, here we will rely on the indications spawning from concreteunderlying models. This will not only fix the representation of the composite fermions, butalso provide a controlled power counting for the construction of effective operators.As a first example, we will follow the proposal from Ref. [55], where techni-scalars are addedto the theory. The fermionic bound states are thus made of one fermion and one scalar, andthe linear mixing of the SM fermions is added in terms of renormalisable Yukawa interactions.The main advantage of this class of models is that the interactions giving rise to fermionmasses are always relevant without the need for large anomalous dimensions of the compositefermions. Moreover, hierarchies in the SM fermion masses can be easily obtained by choosinghierarchical Yukawa couplings or introducing large masses for the scalars (leading to multi-scaleflavour generation [47, 48]). The price to pay is that the underlying theory is not natural, as themasses of the techni-scalars remain unprotected: one possible solution might be that the theorybecomes asymptotically safe at high energies [56]. While we limit ourselves to fields that givemass to the top and bottom, additional scalars can be introduced to give mass to leptons andthe light quarks without destabilising the theory [55, 57, 58]. The most minimal model thatextends our SU(6) → Sp(6) model would thus contain a complex techni-scalar S t transforming3 G TC SU(3) c SU(2) L U(1) Y global sym.Model 1: Sp(2) TC with techni-scalars, ψ = S t ¯3 1 − / ψ × Sp(6) S Model 2: Sp(2 N ) TC with techni-fermions, ψ = χ / ψ × SU(6) χ × U(1)˜ χ ¯3 1 − / TC or SO(13) TC with techni-fermions, ψ = Spin χ / ψ × SU(6) χ × U(1)˜ χ ¯3 1 − / ψ ’s and an additional fermion χ in a different FCDrepresentation. In all models, the component of ψ transform under the SM interactions as in Table I. in the same representation as the techni-fermion under the FCD interactions, and that carriesSM charges as indicated in Table II (Model 1). A prediction of this class of models is that thecomposite fermions, containing only one techni-fermion, always transform as the fundamentalof the global symmetry SU(6). The Yukawa couplings in the underlying theory read: − L FPCYuk = ˜ y L Q L S t ψ − ˜ y R t cR S ∗ t ψ ( d )2 − ˜ y R t cR S ∗ t ψ ( d )3 + ˜ y bR b cR S ∗ t ψ ( u )2 + ˜ y bR b cR S ∗ t ψ ( u )3 , (41)where the indices ( u ) and ( d ) indicate the up and down components of the SU(2) R anti-doublets ψ , , and the appropriate gauge contractions are left understood. A general operator analysisof the low energy EFT for this class of theories has been recently presented in Ref. [57]. It hasbeen shown that, at leading order in the chiral expansion, operators corresponding to Eqs (31)and (32) are generated, with y t / = 14 π C Yuk ˜ y L ˜ y R / , y b / = 14 π C Yuk ˜ y L ˜ y bR / , (42)where C Yuk is a form factor. An operator contributing to the vacuum misalignment can alsobe constructed at order ˜ y L ˜ y R , and it matches with the form of Eq. (35). This matching forboth effective Yukawa and potential can be easily explained by the observation that the anti-symmetric spurions we used in the previous section (given in Appendix B 1) can be constructedstarting from the spurions in the fundamental that we use to describe partial compositenessin this model [59]. This model, however, contains new contributions to the potential in theform of operators at order ˜ y L and ˜ y R [57, 59]: we checked that while the one proportional to ˜ y R vanishes, the one of order ˜ y L generates a potential term proportional to c θ but it only contains atadpole for the would-be Higgs h . This means that, while the spectrum will receive additionalcontributions from this operator, the discussion on the vacuum alignment remains unaffected.Another possibility to construct underlying theories of partial compositeness is to assumethat the fermionic bound states are made of 3 techni-fermions. In Ref. [60] it has been In principle, techni-fermion/techni-gluon bound states as well as multi-fermion ones may form. For the former χ , in a different representation of the FCD gaugesymmetry. The only two cases allowed by asymptotic freedom are listed in Table II. Themodel based on Sp(2 N ) TC has been first proposed in Ref. [61]. Interestingly, the two modelswould be indistinguishable in an EFT approach, except for the properties of the singlet mesonsassociated to the U(1) global symmetry [62, 63] and, of course, by the mass spectra [28, 64]. Abig limitation of this approach is that the multiplicity of the top partners is strongly limitedby the loss of asymptotic freedom , so that it is unlikely to be able to give mass to all SMfermions via partial compositeness. In this class of models, the representation of the compositefermions depends on their fermion content. Being always made of two techni-fermions ψ andone χ , they can only belong to the following representations of SU(6):= SU(6) , = SU(6) , = SU(6) , (43)i.e. all possible 2-index representations. In the following we will only consider the symmetric ,as it’s the only case where operators with two pre-Yukawa insertions are forbidden. In an EFTanalysis, the pre-Yukawas are spurions that carry power counting in the derivative expansion,thus operators with larger number of spurion insertions are higher order compared to onescontaining a smaller number of spurions. Their coefficient is thus expected to be naturallysuppressed. Operators with two pre-Yukawa insertions appear at the lowest possible order andthey are expected to give too large contributions to the pNGB masses unless they are fine-tuned(see, for instance, Refs [59, 68]). One possible way to fine-tune them is to assume that they aredominantly generated by loops of the composite fermions [69, 70], which are assumed to be lightand weakly coupled to the pNGBs [71]. We will follow, here, the complementary approach ofselecting a case where the operators are naturally appearing at higher order, leading to naturalchoices for the form factors. This is the case of the symmetric representation of Sp(6), forwhich operators with two spurion insertions are forbidden [59], and the operators only ariseat the order of four spurion insertions. The symmetric representation decomposes, under thesymmetry SU(2) L × SU(2) R × SU(2) R as: SU(6) = (2 , , ⊕ (2 , , ⊕ (1 , , ⊕ (3 , , ⊕ (1 , , ⊕ (1 , , . (44)The left-handed quarks can couple linearly with either one of the SU(2) L doublets, thus allowingfor two pre-Yukawa couplings y L / . On the other hand, for the right-handed top, there arefour possible choices: either one of the two SU(2) R triplets, or the two neutral components ofthe (1 , ,
2) bi-doublet. In the following, we will choose to embed it in both the SU(2) R triplets:the explicit form of the spurions S L and S R , which are a generalisation of the ones discussed the problem is that the techni-fermions need to be in the adjoint representation, thus their multiplicity isstrongly limited by asymptotic freedom. In the latter case, the linear mixing with the SM fermions wouldarise at dimension larger than 9, implying a stronger suppression. By increasing the number of fermions, before losing asymptotic freedom, the model will enter the conformalwindow [65, 66], thus losing the formation of a condensate and mass gap. This further limits the Model 2 to N c = 2 [67] for the minimal fermion content in Table II. The pre-Yukawas are couplings generating the linear mixing of the SM fermions with the composite fermions.They are 4-fermion interactions in the underlying theory, like in the case we considered in the previous section,but only linear in the SM fields. / Sp(4) model, are shown in Appendix B 1. The top mass isgenerated by an operator in the form [59, 72]: − C t f π Tr[ S L · Σ † · S R · Σ † ] = − Y t f s θ + . . . (45)where y t / = 14 π C t y L / y R . (46)Thus, the top mass has a different θ –dependence from the case of bi-linear operators. We furtherobserve that the linear couplings (which are included in the dots) have the same structure asthe ones in Eq.(31), except for a different θ –dependence and the appearance of a couplings of η proportional to the top mass. An operator that generates a potential for the misalignmentangle can be constructed following Ref. [59]. We have checked that tadpoles for the non-HiggspNGBs are generated by the operators proportional to y L and y L y R , however they have asimilar structure as Eq.(35), except for a different dependence on θ . Thus, for this casetoo, our considerations relative to the vacuum misalignment stay unchanged. The spectrum ofthe pNGBs would, however, be different. We should remark that a different choice for the t R spurion would change the dependence on β of the potential.We thus conclude that our conclusions about the vacuum misalignment, summarised at theend of Section II B, are solid and not specific to the mechanism generating the top and bottommasses we chose in the previous section. We, in fact, provide examples of similar misalignmentpotential generated for models with partial compositeness for the top, and ones bases on half-composite Higgs. D. A composite inert 2HDM
Before concluding the section, we would like to further investigate the aligned case that leadsto a composite inert 2HDM. We recall that this situation is achieved when all fermions coupleto the same SU(2) R doublet (thus, we can choose y f = 0 without loss of generality), or whenthe top and bottom Yukawas are aligned and m R = m R (in which case, the Yukawas can berotated on the first doublet for free). The minimum is thus characterised by β = 0 , ⇒ Y f = 0 , Y f = y f . (47)As already mentioned, the minimisation of the vacuum imposes a relation between the angle θ and the average mass M very similar to the one in the SU(4) / Sp(4) case; similarly, we findthat the would-be Higgs boson h does not mix to the other CP-even scalars h and ϕ and itsmass and couplings follow the same formulas as in the SU(4) / Sp(4) case [15]: m h = C t m t − C g
16 (2 m W + m Z ) , g hXX g SM hXX = c θ . (48) The operator of order y L , which does not contain y R , has an overall normalisation factor Y t /y R . C t ∼ R acting on the second doublet is not broken either bythe techni-fermion mass or by the Yukawa couplings, however it’s broken to U (1) by the partialgauging of hypercharge. Thus, within the minimal set of spurions, we can identify an unbro-ken U (1) under which some of the PNGBs are charged and stable (being the elementary SMfields uncharged). The presence of this unbroken symmetry is guaranteed as long as the onlyspurions in the models are the ones described above, i.e. gauging of the electroweak symmetry,techni-fermion masses and Yukawa couplings (involving only the first Higgs doublet). Higherorder operators containing such spurions will preserve the symmetry at all orders. If additionalspurions are added, the symmetry is preserved as long as they are invariant under SU(2) R .Thus, the presence of the U (1) is a rather solid.If we defined the dark matter (DM) charge Q DM = 2 T R , analysing the pNGB structure inEq.(21), we see that the following fields have charge Q DM = 1: H = h − iA √ , η = ϕ − iη √ , H + , η + . (49)The two additional singlets, η and η , are not charged under U(1) DM .Both the DM-charged and neutral pairs of states feature a mixing: to calculate the spectrum,we solve the minimum condition in terms of the average mass M , and leave the value of theminimum angle θ as a free parameter. The resulting mass in the DM-charged neutral sector,in the basis ( H , η ), reads M . = C t f (cid:32) Y t (1 + c θ ) − K δ Y t s θ Y t s θ Y t (1 + c θ − c θ ) − − ∆) K δ − C g C t (3 g + g (cid:48) )(1 − ∆) c θ (cid:33) , (50)where, for convenience of notation, we have defined K δ = 2 Bδm R C t f . (51)Note that, as expected, the mixing depends linearly on the EW symmetry breaking parameter s θ , so it is suppressed at small misalignment angles. The system of the two charged states( H ± , η ± ) have an additional contribution from the gauging of hypercharge, which is thus re-sponsible for generating a small mass splitting between them: M . = M . + C g g (cid:48) f (cid:32) (1 − c θ ) 00 (1 + c θ ) (cid:33) . (52)As C g >
0, the charged states will always be slightly heavier than the neutral ones. Finally, forthe neutral singlets in the basis ( η , η ), the mass matrix reads M η = m h s θ (cid:32) √ ∆ c θ √ ∆ c θ (2(1 − ∆) + c θ ) c θ (cid:33) − C t f K δ (cid:32) √ √ − ∆3 (cid:33) , (53)7 - - - - -
20 0 - - - K δ Δ FIG. 1. Allowed parameter space as a function of ∆ and K δ for θ = 0 . where we have used the relation between C t and the Higgs mass to simplify the expression.We remark that, once C t is fixed to reproduce the measured Higgs mass, the model has 3 freeparameters: ∆, K δ and C g . The misalignment angle θ is also not fixed, but provides a directrelation between the EW scale and the condensation scale. However, once we fix a small θ , notall values of the other parameters are physical. This can be seen from the fact that the massmatrices, in particular the one for the singlets η / that does not depend on C g , develops negativeeigenstates. This signals that the vacuum has not been properly chosen and, as the fields aresinglets of the EW symmetry, it indicates that the EW preserving vacuum should be modified.In Fig. 1 we show the allowed parameter space for θ = 0 . √ f = 1 . −
1. The shape andsize of the allowed parameter space depend only mildly on the misalignment angle. From thedefinition of ∆, in Eq.(38), we see that its value is bound between 1 and − − m R (cid:29) m L . Thus, values ∆ < − m R < − m L .On the other hand, large negative values for K δ are possible for large values of the masses m R and m R : the throat, therefore, can be consistently achieved for m R very large (for ∆ > − m R large and negative (for ∆ < − θ = 0 . C g = 1 / C t . We remark that below a critical value of ∆, that depends on K δ , the mass ofthe lightest state becomes independent on ∆: this behaviour can be traced back in the formof the mass matrix in Eq.(50), as the entry corresponding to the SU(2) L doublet does notdepend on ∆. Thus, the region with vertical lines corresponds to the lightest mass eigenstatedominantly made of the doublet. Note that this region is very likely to be excluded by directdetection, if this state saturates the relic abundance [74]. This behaviour is matched by the massdifference between the lightest charged and neutral states, which is shown in the right panel ofFig.2. We see that a sub-GeV mass splitting is observed in the region where the lightest neutralstate belongs to the doublet, in agreement with the mass matrix in Eq.(52). We can thereforeconclude that the interesting parameter space for Dark Matter will correspond to values of ∆larger than −
1, which is also consistent with our initial assumption of positive masses. To The factor of 1 / - - - - - - - - K δ Δ - - - - - - - - K δ Δ FIG. 2. Contours, as a function of ∆ and K δ , of the mass of the Dark Matter candidate (left) and themass splitting with the nearby charged scalar (right). All values are in GeV, and we fixed C g = 1 / C t and θ = 0 . √ f = 1 . - - - Δ M [ G e V ] K δ = - - - Δ δ M [ G e V ] K δ = - - - Δ M [ G e V ] K δ =- - - - Δ δ M [ G e V ] K δ =- FIG. 3. (Left) Masses of the even (dashed red) and odd (blue) scalars as a function of ∆. (Right)Mass splitting between the charged and neutral scalars in the two odd tears, with the light one insolid lines. The two rows correspond to K δ = 0 . and − C g = 1 / C t and θ = 0 . √ f = 1 . further clarify this behaviour, in Fig. 3 we slice the parameter space for fixed values of K δ = 0(upper row) and K δ = − m h /s θ ∼
600 GeV,which becomes tachyonic close to the unaccessible region.As already mentioned, the SU(2) L doublet cannot be a thermal DM candidate because of thecoupling to the Z boson. However, in this model the singlet η also features a coupling to the Z boson due to non-linear effects of the misalignment, as shown in Appendix A 3. The couplingof the singlet is proportional to (1 − c θ ), thus for small θ it arises at order s θ , as expected.Integrating out the Z boson, this coupling will generate a coupling of the DM candidate with avector quark current, which contributes to the spin-independent scattering cross section of theDM off nucleons. Following Ref. [75], we estimate the cross sections off a nucleon N = p, n tobe σ V,η N = (1 − c θ ) g m N πc W m Z × (cid:40) (cid:0) − s W (cid:1) , for protons , N = p ; (cid:0) (cid:1) , for neutrons , N = n . (54)Numerically, this leads to σ V,η p ∼ . · − (1 − c θ ) cm , σ V,η n ∼ . · − (1 − c θ ) cm , (55)where the smallness of the proton cross section is due to a cancellation in the Weinberg angledependent factor. By comparing with the strongest upper bound from direct detection fromXENON 1T [76] (see also LUX [77] and PandaX-II [78] results), which ranges between 4 · − cm (for m DM = 30 GeV) and 9 · − cm (for m DM = 1 TeV), we see that the scatteringto neutrons would require s θ ∼ − in order to evade the bound, thus rendering the finetuning very severe. One possible way out would be to consider very low masses in the few GeVrange, where the direct detection experiments loose sensitivity, however this scenario wouldbe excluded by indirect detection, like for instance Fermi-LAT observation of photons fromglobular galaxies [79]. Note finally that the mixing between the two states only makes thesituation worse, as the couplings of the lightest eigenstate would be proportional to (1 − c α c θ ), α being the mixing angle between η and H . The model also contains a coupling of η to theHiggs and a direct coupling to fermions, however we checked that they give spin-independentcross sections below the exclusion limits.One possible way out from this issue is to consider the case where the thermal relic abundanceis very small, but an asymmetry in the DM number, generated during the phase transition atthe condensation of the strong dynamics, saturates the relic abundance. In this case, no indirectdetection bounds apply and low mass DM may be able to avoid the strong constraints fromthe Z -mediated scattering. Also, if the thermal relic density of the stable pNGB is smallerthan the measured one, direct detection bounds can be avoided at the price of the scalar beingonly a minor component of Dark Matter. An extension of the model that contains other DarkMatter candidates would be necessary in such case. We leave a more detailed study of thephenomenology of the Dark Matter candidate and of the other pNGBs for future investigation.In the next section, we will analyse a more general vacuum structure of the model.0 III. MOST GENERAL (REAL) VACUUM ALIGNMENT
As we have seen in the previous section, unless the Yukawa couplings of the top and bottomquarks are aligned, a misalignment along a direction corresponding to a singlet is forced bythe quark loops. In this section, we will therefore study this vacuum and its effects on thecomposite Higgs physics. For simplicity, we will neglect the effect from the bottom quarks,which we assume to be numerically small, and focus on the top, gauge and techni-fermion masscontributions.
A. Defining the γ -vacuum The new vacuum is defined in terms of three parameters: besides the two angles identifiedwith the Higgs directions X and X , a misalignment along the singlet associated to the gen-erator X is activated. The new vacuum is thus defined in terms of an SU(6) rotation thatdepends on three angles: Σ θ,β,γ = Ω( θ, β, γ ) · Σ · Ω † ( θ, β, γ ) . (56)The rotation can be written asΩ( θ, β, γ ) = R β · R γ · Ω θ · R † γ · R † β , (57)where R β and Ω θ are defined in the previous Section in Eqs (17) and (18), and R γ = e − i √ γ S = cos γ I γ σ I − sin γ σ γ I . (58)This property is due to the fact that the two sets of three generators { X , X , S } and { X , X , S } form two partly-broken SU(2) subgroups of SU(6) (more details on the deriva-tion of the above expression can be found in Appendix C 1). As both R β and R γ are definedin term of unbroken generators around the vacuum Σ , the pNGBs can be described similarlyto Eq.(19), as: Σ( φ ) = R β · R γ · Ω θ · U (cid:48) ( φ ) · Σ · Ω Tθ · R Tγ · R Tβ . (59)The role of β in this vacuum is very similar to the one in the previous case (with γ = 0), asit can be projected on the spurions. The same can be done, in principle, for γ : however, thegenerator S belongs to a doublet of SU(2) L , thus its presence will affect the breaking of theEW symmetry, and it will rotate in a non-trivial way the EW gauge generators. From theabove vacuum, the relation between the EW scale and the compositeness scale reads v SM √ f = 2 cos γ sin θ (cid:114) − cos γ sin θ ≡ sin( τ ) , (60)where we define sin τ ≡ cos γ sin θ . (61)1We thus see that the EW scale does depend on γ , and the angle τ replaces the function playedby θ in describing the misalignment of the vacuum. We will thus systematically replace θ with τ . Once we redefine h (cid:48) = τ (cid:0) cos γ cos θ h − sin γ ϕ (cid:1) ,ϕ (cid:48) = τ (cid:0) sin γ h + cos γ cos θ ϕ (cid:1) , (62)we find that only h (cid:48) couples to the W and Z bosons with couplings proportional to the SMones g h (cid:48) W W g SM hW W = g h (cid:48) ZZ g SM hZZ = cos τ , (63)thus reproducing Eq. (25). In this basis, therefore, it is h (cid:48) that plays the role of the would-beHiggs boson. Similarly, the Goldstones that give mass to W and Z are now a superposition ofthe π i and η and η ± (the explicit expressions can be found in Appendix C 2). Thus, in thefollowing the physical scalars will be denoted as η (cid:48) and η (cid:48)± .In the SM fermion sector, using the new vacuum in the interaction in Eq. (31), we canextract the mass of the top quark m t = 2 cos γ θ (cid:18) Y t sin γ Y t cos γ θ (cid:19) = f sin( τ ) Y top , (64)which leads to defining two new combinations of the couplings in Eq. (30): Y top = 1cos τ (cid:18) Y t sin γ sin θ Y t cos θ (cid:19) , Y = 1cos τ (cid:18) Y t cos θ − Y t sin γ sin θ (cid:19) . (65)Note that the coupling Y top is fixed by requiring the observed value of the top mass. TheYukawa couplings will also include couplings of the tops to pNGBs, that include − ( t L t cR ) (cid:26) Y top √ c τ h (cid:48) + Y √ s τ ϕ (cid:48) + c τ h ) + . . . (cid:27) + . . . (66)where we omitted couplings to other pNGBs and higher order ones coming from non-linearities.We see that the coupling of h (cid:48) differs from the ones of a SM Higgs by a factor cos τ , however,contrary to the case of gauge bosons, the other two scalars also couple to t ¯ t . This is relevantbecause, as we will see in the next section, the three scalars h (cid:48) , h and ϕ (cid:48) mix via mass terms. B. Strategy for the minimisation of the potential
The potential that fixes the vacuum alignment is the same as in the previous section, withthe difference of inserting the new γ -dependent vacuum: it thus includes the top loops, Eq. (35),the techni-fermion mass contribution, Eq. (37), and the gauge loops, Eq. (39). As discussed inthe previous section, we will assume real Yukawas so that the tadpoles for the pseudo-scalarsare absent. The potential, expressed in terms of τ and up to constant terms, reads V ( τ, β, γ ) = − C t f Y t s τ − C g f g + g (cid:48) c τ + 16 B (cid:18) M − ∆ s γ c γ + δm R c β (cid:19) s τ . (67) Note that cos τ ≡ (cid:113) cos θ + sin γ sin θ = (cid:113) sin γ + cos γ cos θ . γ only for ∆ (cid:54) = 1 while β always comes with δm R , while an implicit dependence is hidden in the definition of Y t . From the same expressions,we can compute the tadpoles for the three scalars h (cid:48) , h and ϕ (cid:48) , and setting them to zero willdetermine the value of the three angles at the minimum. In fact, we note that the tadpoles arealways proportional to derivatives of the potential with respect to the angles. Explicitly, wefind: Tadpole( h ) : ⇒ √ f s τ ∂V∂β , Tadpole( h (cid:48) ) : ⇒ √ f (cid:32) c γ c θ ∂V∂θ − s γ c γ s τ ∂V∂γ (cid:33) , (68)Tadpole( ϕ (cid:48) ) : ⇒ √ f (cid:32) s γ ∂V∂θ + c γ c θ c γ s τ ∂V∂γ (cid:33) . Instead of determining the angles at the minimum for every choice of the input parameters,in order to simplify the analysis our strategy is to determine three of the initial parameters inthe potential as a function of the angles. Then, we check the allowed range of values of thethree angles for which the solution is indeed a minimum. This last requirement is equivalentto checking that no tachyons are present in the pNGB spectrum. The most convenient choiceof parameters is Y , ∆ and M , for which we find the following relations: Y = − K δ s β Y t c τ , (69)∆ = 1 + C t f BM K δ s β s τ c γ (cid:112) c γ + c τ ) s γ , (70)8 BMC t f = (cid:18) − C g C t g + g (cid:48) (cid:19) c τ − K δ (cid:32) c β − s β s γ s τ (cid:112) c γ + c τ ) c τ (cid:33) . (71)Besides the three angles, the only free parameters are K δ and C g , as Y t is fixed by the top massand C t by fitting the Higgs mass. We remark that for δm R = 0 (i.e., K δ = 0), the dependence on β disappear (besides the implicit dependence in Y t ): as we have seen in the previous section, inthis case β can be removed and the vacuum described in terms of a single angle τ ≡ θ . Anotherspecial point is for ∆ = 1, which corresponds to m R = 0 (i.e., m R = − m R ). From Eq.(70) itfollows that either K δ = 0 or β = 0 , π/ Y = 0 determines the value of γ ). However, wechecked that there always is a tachyon in the pseudo-scalar spectrum, unless K δ = 0, signallingthat the wrong vacuum has been selected. For K δ = 0, the vacuum can be aligned to the onealready described in the previous section. This simple analysis confirms that both β and γ arerequired to be non-zero.One of the most interesting features of this vacuum is that a mass mixing between the threescalars h (cid:48) , h and ϕ (cid:48) is always generated by the potential. Due to the mixing with heavierstates, the mass of the would-be Higgs h (cid:48) is parametrically reduced compared to the value inthe θ -vacuum. As a consequence, the value of the low energy constant C t that determines itsvalue can be larger than what we found in the previous section, in Eq.(48), and it is not fixedto a single value any more. This kind of mechanism was proposed to reduce the fine tuning in3the Higgs mass in the minimal model SU(4) / Sp(4) in Refs [80, 81], where the authors addedan additional parameter in the partial compositeness mixing of the right-handed top quark.In the SU(6) / Sp(6) case we consider, however, the mixing arises naturally from the potential.Furthermore, the mixing does not involve CP-odd scalars [59]. The complete mass matrixfor the three scalars is reported in Appendix C 2, and it contains mixing proportional to K δ between the would-be Higgs h (cid:48) and the other two: remarkably, the one with the EW singlet ϕ (cid:48) arises at order s τ in an expansion for small τ . It is instructive to expand the mass spectrumfor small s τ and small K δ . Besides one state (approximately associated to h ) that receives amass of order f , i.e. unsuppressed by s τ , the spectrum contains two lighter states with masses: m / = C t f s τ (cid:20)(cid:18) Y t − C g C t g + g (cid:48) (cid:19) Q ± ( γ ) − K δ c β Q ∓ ( γ ) + O ( K δ ) (cid:21) ∓ C t f s τ K δ s β (cid:112) cot γ + 16 Q ∓ ( γ ) + O ( s τ ) , (72)where the functions Q ± ( γ ) are defined as Q ± ( γ ) = 1 ± cot γ (cid:112) cot γ + 16 . (73)For γ → + , we note that Q + → Q − → O ( γ ), while the roles are exchanged for γ → − . Thus, the would be Higgs can be identified with m for positive γ , and m for negative γ . For γ ∼
0, therefore, the mass of the would-be Higgs recovers the result we obtained in theprevious section. For non-zero γ , however, an additional correction of order s τ is induced bythe term in the second line. This can be either positive or negative, depending on the sign of β and δm R , thus it can allow a larger or smaller value of C t compared to C t ∼
2. We recallhere that C t is determined by the strong dynamics and by the interactions in the UV thatgenerate the top mass operator, so that it helps to have some flexibility regarding its value.The most interesting point is that the additional correction is enhanced at small τ , thus it ismore important for small alignment angles.In the following section, we will study this case numerically to determine the allowed rangefor the coefficient C t that fits the Higgs mass. In fact, constraints arise both from the range ofvalidity of the parameters, and from the modifications of the would-be Higgs couplings due tothe mixing between h (cid:48) and ϕ (cid:48) . C. Numerical results
For concreteness, we focus on a benchmark model with τ = 0 .
1, corresponding to a compos-iteness scale of 2 √ f = 2 . C g = 1 / C t . We then study the parameter space asa function of γ , β and K δ : the allowed one is given by the region where no tachyons exist inthe spectrum. The result is shown in Fig. 4, where the three colours delimitate the safe regionsfor the neutral scalars (blue), pseudo-scalars (green) and charged ones (red). We see that theallowed region, indicated by an orange arrowhead, is mainly determined by the pseudo-scalarsand charged scalars developing tachyons. Furthermore, the region with γ → γ K δ tan β = / γ tan β = γ tan β = FIG. 4. The allowed parameter space for τ = 0 . C g = 1 / C t is indicated by the orangearrowhead. The blue, green and red contours indicate where the neutral scalars, pseudo-scalars andcharged scalars respectively develop a tachyonic spectrum. Below the red line, the theory will fall toa charge-violating vacuum. γ m [ G e V ] tan β = / γ tan β = γ tan β = FIG. 5. Spectra for τ = 0 . C t = 2 and C g = 1 / C t . The blue, green and red lines correspond toneutral scalars, pseudo-scalars and charged scalars respectively. We fix K δ = 0 . β = 1 /
3, 1 and 3. due to pseudo-scalars, while K δ → γ = 0and K δ = 0, the vacuum collapses to the one described in the previous section, where β can berotated away. We remark that the red line delimits the region where charged tachyons appear:this implies that for parameter values below the red line the vacuum will be misaligned alonga charge-violating direction.To study the spectrum, we find useful to slice the parameter space further and show charac-teristic plots as a function of a single variable. In Fig. 5 we fix K δ = 0 . γ for tan β = 1 /
3, 1 and 3. We fix C t = 2 so that for small γ the Higgs mass is recovered for thelightest neutral scalar. We recognise that very small values of γ are ruled out by a pseudo-scalarbecoming tachyonic, while the maximal allowed value of γ depends on a pseudo-scalar becoming5 γ m [ G e V ] K δ = γ K δ = γ K δ = FIG. 6. Same as Fig. 5 for fixed tan β = 1 /
3, showing spectra for K δ = 0 .
1, 0 .
25 and 0 . K δ m [ G e V ] tan β = / K δ tan β = K δ tan β = FIG. 7. Same as Figs 5 and 6 for fixed γ = 0 .
02, showing spectra for tan β = 1 /
3, 1 and 3. tachyonic for tan β <
1, and a charged scalar for tan β >
1. Note also that the heaviest neutralscalar is nearly degenerate with the heavier charged one and with one of the pseudo-scalars.We also remark significant reduction of the would-be Higgs mass for non-zero values of γ : thisneeds to raise the value of C t ∝ /m in order to keep the fit with the experimentally measuredmass. This plot already shows a proof of principle that the mixing can allow for coefficientslarger than 2. We also note that tan β ∼ K δ we made, the line wherethe pseudo-scalar spectrum becomes tachyonic is close to the line where the lightest scalar massapproaches zero, as it can be seen in the central plot of Fig. 4.For completeness, in Figs 6 and 7, we show the spectra as a function of γ for tan β = 1 / K δ = 0 .
1, 0 .
25 and 0 .
5, and as a function of K δ for γ = 0 .
02 and tan β = 1 /
3, 1 and 3. Acommon feature of all these plots is the presence of light scalars, whose masses are below 1 TeV,even if the condensation scale of the benchmark point we chose is 2 . IV. FINAL REMARKS AND CONCLUSIONS
In this paper, we have analysed in detail the vacuum alignment for a model of compositeHiggs based on the coset SU(6) / Sp(6). In terms of the underlying theory, this is a simplegeneralisation of the minimal case SU(4) / Sp(4) by the addition of one further Dirac fermion.However, the global symmetry is significantly enlarged, allowing for a second Higgs doublet inthe pNGB spectrum. We focus in particular on a model with a single SU(2) L doublet in orderto avoid the presence of EW triplets in the spectrum. The underlying fermions thus consist ofone SU (2) L doublet and two SU(2) R doublets.We studied the vacuum alignment problem in presence of bilinear operators at the origin ofthe SM fermion masses, but we checked that our conclusions also apply to models with partialcompositeness. In absence of CP violating phases, we find that the vacuum can be misalignedonly in two exclusive ways, depending on the alignment of the Yukawa couplings and underlyingfermion masses:a) The misalignment is characterised by a single angle θ , which corresponds to a rotationalong one of the two Higgs doublets. This situation is achieved if the Yukawas are alignedin the SU(6) space and the two SU(2) R doublets are degenerate, or if the Yukawas onlyinvolve one of the two SU(2) R doublets.b) The misalignment depends on 3 angles, θ , β and γ . While the first two correspond to anordinary two-Higgs-doublet model, γ is generated by a singlet. All three angles must benon-vanishing in this scenario.In the first case a), a global U(1) symmetry remains unbroken, thus a subset of pNGBs isprevented from decaying into a pair of SM states. The spectrum therefore contains two setsof nearly degenerate neutral and electromagnetically charged states that carry the global U(1)charge, and the lightest one can be associated with a Dark Matter candidate. Besides thewould-be Higgs, which has properties similar to the minimal SU(4) / Sp(4) model, the spectrumalso contains two pseudo-scalars that decay into SM gauge bosons via topological anomalies.We provide details of the spectrum of the theory, where a pseudo-scalar lighter than the DMcandidate is a common occurrence, and a complete list of the relevant couplings among pNGBs.The detailed study of the properties of the Dark Matter candidate are left for future investiga-tions, with all the necessary tools already provided in this paper.In the second case, there is no stable pNGB, and all the neutral scalars, pseudo-scalarsand charged scalars mix with each other. In particular, a mixing between the would-be Higgsboson and a scalar singlet is always generated, and its presence tends to parametrically reducethe value of the physical Higgs mass at the price of facing constraints from the consequentreduction of the Higgs couplings to SM states. Note that the current determination of theHiggs couplings at the LHC still allows for a 10 ÷
20% deviation. This mechanism, alreadydiscussed in the literature, has the potential of reducing the tuning in the Higgs mass byallowing for larger form factors coming from the strong dynamics. The novelty of this result inthe SU(6) / Sp(6) model is that this mixing is not introduced ad-hoc, but it is a necessary feature7of the most general vacuum. Furthermore, the mixing only involves CP-even scalars, contraryto the case in the minimal SU(4) / Sp(4) model that has been studied in the literature. Oncelattice results determining the value of the form factors are available, our results will allow tofurther constrain the parameter space of the model and thus predict the phenomenology of thespectrum of scalars. One common feature we already identify is the presence of light states,with masses below the TeV, even when the compositeness scale is pushed in the multi-TeVrange.
ACKNOWLEDGEMENTS
We thank M.Lespinasse for his contribution in the initial stages of this project. GC acknowl-edges partial support from the Labex-LIO (Lyon Institute of Origins) under grant ANR-10-LABX-66 (Agence Nationale de la Recherche) and FRAMA (FR3127, F´ed´eration de Recherche“Andr´e Marie Amp`ere”). This work has also been supported by the LIA FCPPL (France-ChinaParticle Physics Laboratory). HHZ and CC are supported by the National Natural ScienceFoundation of China (NSFC) under Grant Nos. 11875327 and 11375277, the China Postdoc-toral Science Foundation under Grant No. 2018M643282, the Natural Science Foundation ofGuangdong Province under Grant no. 2016A030313313, and the Sun Yat-Sen University Sci-ence Foundation. GC thanks the Sun Yat-Sen University for hospitality during the completionof this work.
Appendix A: Properties of the model1. Broken and un-broken generators in the vacuum Σ The 21 unbroken generators of Sp(6) are: S = 12 σ , S = 12 σ , S = 12 σ , (A1) S = 12 − σ T
00 0 0 , S = 12 − σ T
00 0 0 , S = 12 − σ T
00 0 0 , (A2) S = 12 − σ T , S = 12 − σ T , S = 12 − σ T , (A3) S = 12 √ iσ − iσ , S = 12 √ iσ − iσ S = 12 √ iσ − iσ , S = 12 √ I I , (A4) S = 12 √ iσ − iσ , S = 12 √ iσ − iσ , S = 12 √ iσ − iσ , S = 12 √ I I , (A5) S = 12 √ σ σ , S = 12 √ σ σ , S = 12 √ σ σ , S = 12 √ i I − i I , (A6) The first 3 correspond to the gauged SU(2) L , while the following 6 to the two partly-gaugedSU(2) R ’s. The 3 sets of generators S ,... , S ,... and S ,... transform as bidoublets.8The 14 broken generators are: X = 12 √ I − I
00 0 0 , X = 12 √ I I
00 0 − I , (A7) X = 12 √ σ σ , X = 12 √ σ σ , X = 12 √ σ σ , X = 12 √ i I − i I , (A8) X = 12 √ σ σ , X = 12 √ σ σ , X = 12 √ σ σ , X = 12 √ i I − i I , (A9) X = 12 √ iσ − iσ , X = 12 √ iσ − iσ , X = 12 √ iσ − iσ , X = 12 √ I I . (A10) The first two correspond to the singlets, while the following 3 groups of 4 (in each row)correspond to the 3 bi-doublets.
2. CP properties and Wess-Zumino-Witten topological term
The Wess-Zumino-Witten topological term [82, 83] reads: L W ZW = d FCD g V V √ π f (cid:18) c θ η + 1 √ c θ η (cid:19) (cid:15) µνρσ V µν V ρσ , (A11)where d FCD is the dimension of the FCD representation of the underlying fermions ( d FCD = 2in the minimal SU(2) TC model), and g W W = g , g ZZ = ( g − g (cid:48) ) , g Zγ = gg (cid:48) . (A12)This result shows that the model has the same anomaly structure as the minimal SU(4) / Sp(4)model [73], as a linear combination of the two singlets have the same WZW couplings. Inparticular, note the absence of couplings to two photons. The couplings above also show that η and η are pseudo-scalars under CP.
3. Lowest order pNGB couplings
The chiral Lagrangian at LO contains couplings to one and two gauge bosons, via the covari-ant derivatives. In the following, we use the short notation c W = cos θ W , c W = cos 2 θ W , t W =tan θ W and drop the exact Goldstones eaten by the W and Z (in the θ – β vacuum). Also, forshort notation, we define (Cf. Section II D): H = h − iA √ , η = ϕ − iη √ . (A13)The V ∂φφ interactions read L V ∂φφ = g √ W + µ [(1 + c θ ) H − i ↔ ∂ µ H − (1 − c θ ) η − i ↔ ∂ µ η ] + h.c.+ g c W Z µ [(1 + c θ − s W ) H − i ↔ ∂ µ H + + (1 − c θ − s W ) η − i ↔ ∂ µ η + +(1 + c θ ) H i ↔ ∂ µ ( H ) ∗ + (1 − c θ ) η i ↔ ∂ µ ( η ) ∗ ]+ eA µ [ H − i ↔ ∂ µ H + + η − i ↔ ∂ µ η + ] ; (A14)9where φ ↔ ∂ µ φ = φ ( ∂ µ φ ) − ( ∂ µ φ ) φ .The V V φφ interactions read L V V φφ = e A µ A µ [ H + H − + η + η − ]+ g W + µ W − ,µ [ c θ h − s θ η + (1 + c θ ) c θ ( | H | + H + H − ) − (1 − c θ ) c θ ( | η | + η + η − )]+ g c W Z µ Z µ [(2 c W − (1 − c θ )( c θ + 2 c W )) H + H − + (2 c W + (1 + c θ )( c θ − c W )) η + η − + (1 + c θ ) c θ | H | − (1 − c θ ) c θ | η | + c θ h − s θ η ]+ eg c W A µ Z µ [(2 c W − c θ ) H + H − + (2 c W − − c θ ) η + η − ]+ eg √ A µ W + µ [(1 + c θ ) H − H − (1 − c θ ) η − η ] + h.c. − g √ c W Z µ W + µ [(1 + c θ ) H − H + (1 − c θ ) η − η ] + h.c. (A15)In addition, there are couplings of two (massive) vectors with the Higgs boson L V V h = √ g f s θ c θ W + µ W − ,µ h + ( g (cid:48) + g ) f √ s θ c θ Z µ Z µ h . (A16)The chiral Lagrangian also contains self-interactions of the pNGBs, starting with quarticterms. Here we will only report the couplings containing the Higgs boson, as they are themost relevant ones for the production at collider of the additional scalars, and for studying theproperties of the Dark Matter candidate. The relevant interactions consist of couplings bilinearin the Higgs boson h , that read L h φφ∂ = 196 f (cid:2) h ∂ µ h ( φ ∗ ∂ µ φ + φ∂ µ φ ∗ ) − h ∂ µ φ ∗ ∂ µ φ − ∂ µ h ∂ µ h φ ∗ φ (cid:3) ; (A17)where φ = H + , η + , H , η , √ η . The couplings linear in the Higgs read L h φ ∂ = − i f ( h ↔ ∂ µ η ) [ η − ↔ ∂ µ H + + η ↔ ∂ µ ( H ) ∗ ] + h.c. − i √ f ( h ∂ µ η + η ∂ µ h ) [ H − ∂ µ η + + η + ∂ µ H − + H ∂ µ ( η ) ∗ + ( η ) ∗ ∂ µ H ] + h.c.+ i f h η [ ∂ µ η + ∂ µ H − + ∂ µ ( η ) ∗ ∂ µ H ] + h.c. (A18)0 Appendix B: Spurions and the θ –vacuum1. Projectors for the top mass The projectors used in Eq. 31 to define the effective operator generating the top mass are: P = 12 − P = 12 − (B1) P = 12 − P = 12 − (B2) Similarly, for the bottom mass in Eq. 32, we have: P b = 12 − P b = 12 − (B3) P b = 12 − P b = 12 − (B4) For the case of partial compositeness with the SU(6) , the spurions containing the pre-Yukawas are give by the following matrices: S L = y L y L
00 0 0 0 0 0 y L y L S L = y L y L y L y L S R = y R . (B5)
2. Couplings of the pNGBs to tops
Besides the linear couplings of the pNGBs to top and bottom, reported in Eqs (31) and(32), the effective interactions contain higher order terms that become relevant for productionand annihilation of the pNGBs. Here we report the couplings involving two pNGBs, which arerelevant for Dark Matter and collider phenomenology: L φφff ⊃ Y t f ( t L t cR ) † (cid:20) − i √ c θ h η + (1 − c θ )( H − η + + ( η ) ∗ H ) + (1 + c θ )( H + η − + η ( H ) ∗ )+ s θ (cid:18) η + η − + H + H − + | H | + | η | + h + η + 13 η (cid:19)(cid:21) + Y b f ( b L b cR ) † (cid:20) − i √ c θ h η − (1 + c θ )( H − η + + ( η ) ∗ H )) − (1 − c θ )( H + η − + η ( H ) ∗ )+ s θ (cid:18) η + η − + H + H − + | H | + | η | + h + η + 13 η (cid:19)(cid:21) + Y t f ( b L t cR ) † (cid:2) η − H − H − η (cid:3) + Y b f ( t L b cR ) † (cid:2) η + ( H ) ∗ − H + ( η ) ∗ (cid:3) + h.c. (B6)
3. Self-couplings of the pNGBs
The potential that determines the misalignment of the vacuum also generates self interactionsamong the pNGBs, which do not involve derivatives and are proportional to the spurionsexplicitly breaking the global symmetry in the strong sector. Here we will report the results inthe vacuum of Section II D, after imposing the minimum condition (and solving for the averagetechni-fermion mass M as a function of the minimum misalignment angle θ ). Furthermore, wedefine Y ± ≡ | Y t | ± | Y b | (B7)and work in the Unitary gauge.First, the trilinear and quartic Higgs couplings read: L h + h = f s θ √ (cid:2) C t Y + 32 C g g (3 + t W ) (cid:3) h − (cid:2) C t Y ( − c θ ) + C g g (3 + t W )(1 + 9 c θ ) (cid:3) h . (B8)We also report all the couplings that may be relevant for the calculation of production andscattering rates of the pNGBs (useful for collider studies and Dark Matter ones). The potentialgenerates couplings of a single Higgs field with two other pNGB, in the form L hφφ = f √ g hφ φ h φ φ , (B9)2with g h H + H − = (cid:2) (14 C t Y + 3 C g g (3 + t W )) c θ + 3 C g g (3 − t W ) (cid:3) s θ ,g h ( H ) ∗ H = (cid:2) (14 C t Y + 3 C g g (3 + t W )) c θ + 3 C g g (3 + t W ) (cid:3) s θ ,g h η + η − = (cid:2) (14 C t Y + 3 C g g (3 + t W )) c θ − C g g (3 − t W ) (cid:3) s θ ,g h ( η ) ∗ η = (cid:2) (14 C t Y + 3 C g g (3 + t W )) c θ − C g g (3 + t W ) (cid:3) s θ ,g h η − H + = g h η + H − = g h ( η ) ∗ H = g h η ( H ) ∗ = 6 C t Y − c θ ,g h η = (cid:2) C t Y + 3 C g g (3 + t W ) (cid:3) c θ s θ ,g h η = (cid:2) C t Y − C g g (3 + t W ) (cid:3) c θ s θ . (B10)For completeness, the following terms exhaust the list of trilinear couplings: L φ = if s θ √ (cid:110) ( η + H − − η − H + ) (cid:2) C g g (1 − t W ) η + √ C t Y + C g g (3 + t W )) c θ η (cid:3) +(( η ) ∗ H − η ( H ) ∗ ) (cid:2) C g g (3 + t W ) η + √ C t Y + C g g (3 + t W )) c θ η (cid:3)(cid:111) . (B11)Quartic couplings that are bilinear in the Higgs field have the form L h φ φ = − g h φ φ h φ φ , (B12)with g h H + H − = (cid:2) − C t Y (2 + c θ − c θ ) + 8 C t K δ + C g g (3 + t W )(1 + 4 c θ + 5 c θ ) + 2 C g g t W (1 − c θ ) (cid:3) ,g h ( H ) ∗ H = (cid:2) − C t Y (2 + c θ − c θ ) + 8 C t K δ + C g g (3 + t W )(1 + 4 c θ + 5 c θ ) (cid:3) ,g h η + η − = (cid:2) − C t Y (2 + (1 − c θ − c θ ) + 4 C t K δ (1 − ∆) + C g g (3 + t W )(1 − (3 + ∆) c θ + 5 c θ )+2 C g g t W (1 + 7 c θ ) (cid:3) ,g h ( η ) ∗ η = (cid:2) − C t Y (2 + (1 − c θ − c θ ) + 4 C t K δ (1 − ∆) + C g g (3 + t W )(1 − (3 + ∆) c θ + 5 c θ ) (cid:3) ,g h η − H + = g h η + H − = g h ( η ) ∗ H = g h η ( H ) ∗ = − C t Y − s θ ,g h η = 12 (cid:2) C t Y ( − c θ ) + C g g (3 + t W )( − c θ ) (cid:3) ,g h η = (cid:2) − C t Y + C g g (3 + t W ) (cid:3) c θ ,g h η η = 2 √ (cid:2) − (2 C t Y − C g g (3 + t W ))∆ c θ + 4 C t K δ (1 + ∆) (cid:3) . (B13)Quartic couplings linear in the Higgs are also generated in the form L h φ = i h (cid:2) ( H ) ∗ η − H ( η ) ∗ (cid:3) (cid:2) g h η η + g h η η (cid:3) + i h (cid:2) H + η − − H − η + (cid:3) (cid:2) g h η + − η + g h η + − η (cid:3) , (B14)3with g h η = (cid:2) (2 C t Y ∆ + C g g (3 + t W )(3 − ∆)) c θ − C t K δ (1 + ∆) (cid:3) ,g h η = 1 √ (cid:2) C t Y (2 − (1 − ∆) c θ + 5 c θ ) − C t K δ ( − − C g g (3 + t W )(2 − (1 − ∆) c θ − c θ ) (cid:3) ,g h η + − = (cid:2) (2 C t Y ∆ + C g g (3 + t W )(3 − ∆)) c θ − C t K δ (1 + ∆) + 12 C g g t W c θ (cid:3) ,g h η + − = 1 √ (cid:2) C t Y (2 − (1 − ∆) c θ + 5 c θ ) − C t K δ ( − − C g g (3 + t W )(2 − (1 − ∆) c θ − c θ ) + 12 C g g t W (cid:3) . (B15) Appendix C: General vacuum1. Parameterisation of the vacuum
The form of the most general vacuum misalignment can be simplified by using the followingproperty of the SU(6) generators.
Consider three generators { A, B, C } of SU( N ) (i.e. N × N hermitian matrices) that forman SU(2) algebra (cid:2)
A, B (cid:3) = iκ C , (cid:2) B, C (cid:3) = iκ A , (cid:2) C, A (cid:3) = iκ B . (C1) Then, the following relation holds: e i ( aA + bB ) = e − iκϕ C · e iρ A · e iκϕ C , (C2) where a = ρ cos ϕ and b = ρ sin ϕ . The most general CP-conserving vacuum, misaligned along the three directions identified inSection II, is defined by the rotationΩ = e i √ θ X + θ X + θ X ) . (C3)We remark that the following triplets of generators form (different) SU(2) subgroups of SU(6) { X , X , S } , { X , X , S } , (C4)with κ = 2 √
2. Thus, applying the relation in Eq.(C2), the rotation can be written asΩ = e − i √ γ ( c β S − s β S ) · e i √ θ ( c β X + s β X ) · e i √ γ ( c β S − s β S ) , (C5)where θ = θc γ s β , θ = θc γ s β and θ = θs γ . The relation in Eq.(C2) can be applied again tothe two exponentials, as the following triplets also form SU(2) subgroups of SU(6) (with thesame value of κ ): { X , X , S } , { S , S , S } . (C6)4Thus: e i √ θ ( c β X + s β X ) = e i √ βS · e i √ θX · e − i √ βS , (C7) e − i √ γ ( c β S − s β S ) = e i √ βS · e − i √ γS · e − i √ βS . (C8)(The former relation explains Eq.(16).) Putting these relations together allows us to writeΩ = e i √ βS · e − i √ γS · e i √ θX · e i √ γS · e − i √ βS , (C9)which coincides with the one used in Eq.(57) of Section III.
2. Masses of the pNGBs
The Goldstones eaten by the massive W and Z are defined as π (cid:48) = 1cos τ (cid:18) cos γ cos θ π + sin γ η (cid:19) , (C10) π (cid:48)± = 1cos τ (cid:18) cos γ cos θ π ± iπ √ γ η ± (cid:19) , (C11)while the orthogonal combinations remain as physical states: η (cid:48) = 1cos τ (cid:18) cos γ cos θ η − sin γ π (cid:19) , (C12) η (cid:48)± = 1cos τ (cid:18) cos γ cos θ η ± + sin γ π ± iπ √ (cid:19) . (C13)The mass matrix for the 3 scalars h (cid:48) , h and ϕ (cid:48) at the minimum of the vacuum as definedin Section III B can be written, in matrix form, as12 C t f h (cid:48) , h , ϕ (cid:48) ) · N · h (cid:48) h ϕ (cid:48) , (C14)with the symmetric matrix N having entries: N = (cid:18) Y t − C g C t g + g (cid:48) (cid:19) s τ , (C15) N = N = 4 K δ s β s τ c τ , (C16) N = N = − K δ s β c τ s τ c τ , (C17) N = Y t c τ − K δ c β − K δ s β Y t c τ , (C18) N = N = Y t s τ − K δ s β Y t c τ c τ , (C19) N = Y t s τ + K δ s β c τ − c γ (3 − c τ ) (cid:112) c γ + c τ ) s τ c τ − K δ s β Y t s τ c τ . (C20)5Note that in absence of mixing, i.e. K δ (cid:28)
1, the mass of h (cid:48) , m h (cid:48) = C t f N = C t m t − C g m W + m Z , (C21)matches the result of the minimal SU(4) / Sp(4) model. [1] P. W. Higgs, Phys. Rev. Lett. , 508 (1964).[2] S. Weinberg, Phys. Rev. D13 , 974 (1976).[3] S. Dimopoulos and L. Susskind,
Proceedings: International Conference on High Energy Physics,Geneva, Switzerland, Jun 27 - Jul 4, 1979: In 2 volumes , Nucl. Phys.
B155 , 237 (1979),[2,930(1979)].[4] F. Sannino and K. Tuominen, Phys. Rev.
D71 , 051901 (2005), arXiv:hep-ph/0405209 [hep-ph].[5] S. Catterall and F. Sannino, Phys. Rev.
D76 , 034504 (2007), arXiv:0705.1664 [hep-lat].[6] J. Rantaharju, C. Pica, and F. Sannino, Phys. Rev.
D96 , 014512 (2017), arXiv:1704.03977[hep-lat].[7] R. Foadi, M. T. Frandsen, and F. Sannino, Phys. Rev.
D87 , 095001 (2013), arXiv:1211.1083[hep-ph].[8] D. B. Kaplan and H. Georgi, Phys. Lett.
B136 , 183 (1984).[9] D. B. Kaplan, H. Georgi, and S. Dimopoulos, Phys. Lett.
B136 , 187 (1984).[10] J. M. Maldacena, Int. J. Theor. Phys. , 1113 (1999), [Adv. Theor. Math. Phys.2,231(1998)],arXiv:hep-th/9711200 [hep-th].[11] R. Contino, Y. Nomura, and A. Pomarol, Nucl. Phys. B671 , 148 (2003), arXiv:hep-ph/0306259[hep-ph].[12] R. Contino, in
Physics of the large and the small, TASI 09, proceedings of the Theoretical AdvancedStudy Institute in Elementary Particle Physics, Boulder, Colorado, USA, 1-26 June 2009 (2011)pp. 235–306, arXiv:1005.4269 [hep-ph].[13] B. Bellazzini, C. Cs´aki, and J. Serra, Eur. Phys. J.
C74 , 2766 (2014), arXiv:1401.2457 [hep-ph].[14] G. Panico and A. Wulzer, Lect. Notes Phys. , pp.1 (2016), arXiv:1506.01961 [hep-ph].[15] G. Cacciapaglia and F. Sannino, JHEP , 111 (2014), arXiv:1402.0233 [hep-ph].[16] K. Agashe, R. Contino, and A. Pomarol, Nucl. Phys. B719 , 165 (2005), arXiv:hep-ph/0412089[hep-ph].[17] F. Caracciolo, A. Parolini, and M. Serone, JHEP , 066 (2013), arXiv:1211.7290 [hep-ph].[18] G. von Gersdorff, E. Pont´on, and R. Rosenfeld, JHEP , 119 (2015), arXiv:1502.07340 [hep-ph].[19] E. Katz, A. E. Nelson, and D. G. E. Walker, JHEP , 074 (2005), arXiv:hep-ph/0504252[hep-ph].[20] B. Gripaios, A. Pomarol, F. Riva, and J. Serra, JHEP , 070 (2009), arXiv:0902.1483 [hep-ph].[21] M. Frigerio, A. Pomarol, F. Riva, and A. Urbano, JHEP , 015 (2012), arXiv:1204.2808 [hep-ph].[22] T. A. Ryttov and F. Sannino, Phys. Rev. D78 , 115010 (2008), arXiv:0809.0713 [hep-ph]. [23] J. Galloway, J. A. Evans, M. A. Luty, and R. A. Tacchi, JHEP , 086 (2010), arXiv:1001.1361[hep-ph].[24] A. Hietanen, R. Lewis, C. Pica, and F. Sannino, JHEP , 116 (2014), arXiv:1404.2794 [hep-lat].[25] V. Drach, A. Hietanen, C. Pica, J. Rantaharju, and F. Sannino, Proceedings, 33rd Interna-tional Symposium on Lattice Field Theory (Lattice 2015): Kobe, Japan, July 14-18, 2015 , PoS
LATTICE2015 , 234 (2016), arXiv:1511.04370 [hep-lat].[26] R. Arthur, V. Drach, M. Hansen, A. Hietanen, C. Pica, and F. Sannino, Phys. Rev.
D94 , 094507(2016), arXiv:1602.06559 [hep-lat].[27] R. Arthur, V. Drach, A. Hietanen, C. Pica, and F. Sannino, (2016), arXiv:1607.06654 [hep-lat].[28] N. Bizot, M. Frigerio, M. Knecht, and J.-L. Kneur, Phys. Rev.
D95 , 075006 (2017),arXiv:1610.09293 [hep-ph].[29] T. Ma and G. Cacciapaglia, JHEP , 211 (2016), arXiv:1508.07014 [hep-ph].[30] J. Mrazek, A. Pomarol, R. Rattazzi, M. Redi, J. Serra, and A. Wulzer, Nucl. Phys. B853 , 1(2011), arXiv:1105.5403 [hep-ph].[31] E. Bertuzzo, T. S. Ray, H. de Sandes, and C. A. Savoy, JHEP , 153 (2013), arXiv:1206.2623[hep-ph].[32] S. De Curtis, S. Moretti, K. Yagyu, and E. Yildirim, Proceedings, 6th International Workshopon Prospects for Charged Higgs Discovery at Colliders (CHARGED 2016): Uppsala, Sweden,October 3-6, 2016 , PoS
CHARGED2016 , 018 (2016), arXiv:1612.05125 [hep-ph].[33] M. Chala, JHEP , 122 (2013), arXiv:1210.6208 [hep-ph].[34] Y. Wu, T. Ma, B. Zhang, and G. Cacciapaglia, JHEP , 058 (2017), arXiv:1703.06903 [hep-ph].[35] G. Ballesteros, A. Carmona, and M. Chala, Eur. Phys. J. C77 , 468 (2017), arXiv:1704.07388[hep-ph].[36] R. Balkin, M. Ruhdorfer, E. Salvioni, and A. Weiler, JHEP , 094 (2017), arXiv:1707.07685[hep-ph].[37] H. Georgi and D. B. Kaplan, Phys. Lett. B145 , 216 (1984).[38] A. Pomarol and R. Vega, Nucl. Phys.
B413 , 3 (1994), arXiv:hep-ph/9305272 [hep-ph].[39] B. Grzadkowski, M. Maniatis, and J. Wudka, JHEP , 030 (2011), arXiv:1011.5228 [hep-ph].[40] I. Low, W. Skiba, and D. Tucker-Smith, Phys. Rev. D66 , 072001 (2002), arXiv:hep-ph/0207243[hep-ph].[41] T. Brown, C. Frugiuele, and T. Gregoire, JHEP , 108 (2011), arXiv:1012.2060 [hep-ph].[42] M. E. Peskin and T. Takeuchi, Phys. Rev. D46 , 381 (1992).[43] E. Eichten and K. D. Lane, Phys. Lett. , 125 (1980).[44] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, JHEP , 031 (2008), arXiv:0807.0004[hep-th].[45] V. S. Rychkov and A. Vichi, Phys. Rev. D80 , 045006 (2009), arXiv:0905.2211 [hep-th].[46] O. Antipin, E. Mølgaard, and F. Sannino, JHEP , 030 (2015), arXiv:1406.6166 [hep-th].[47] G. Cacciapaglia, H. Cai, T. Flacke, S. J. Lee, A. Parolini, and H. Serˆodio, JHEP , 085 (2015),arXiv:1501.03818 [hep-ph].[48] G. Panico and A. Pomarol, JHEP , 097 (2016), arXiv:1603.06609 [hep-ph]. [49] G. Cacciapaglia and F. Sannino, Phys. Lett. B755 , 328 (2016), arXiv:1508.00016 [hep-ph].[50] E. H. Simmons, Nucl. Phys.
B312 , 253 (1989).[51] S. Samuel, Nucl. Phys.
B347 , 625 (1990).[52] O. Antipin and M. Redi, JHEP , 031 (2015), arXiv:1508.01112 [hep-ph].[53] A. Agugliaro, O. Antipin, D. Becciolini, S. De Curtis, and M. Redi, Phys. Rev. D95 , 035019(2017), arXiv:1609.07122 [hep-ph].[54] D. B. Kaplan, Nucl. Phys.
B365 , 259 (1991).[55] F. Sannino, A. Strumia, A. Tesi, and E. Vigiani, JHEP , 029 (2016), arXiv:1607.01659 [hep-ph].[56] G. M. Pelaggi, F. Sannino, A. Strumia, and E. Vigiani, Front.in Phys. , 49 (2017),arXiv:1701.01453 [hep-ph].[57] G. Cacciapaglia, H. Gertov, F. Sannino, and A. E. Thomsen, (2017), arXiv:1704.07845 [hep-ph].[58] F. Sannino, P. Stangl, D. M. Straub, and A. E. Thomsen, (2017), arXiv:1712.07646 [hep-ph].[59] T. Alanne, N. Bizot, G. Cacciapaglia, and F. Sannino, Phys. Rev. D97 , 075028 (2018),arXiv:1801.05444 [hep-ph].[60] G. Ferretti and D. Karateev, JHEP , 077 (2014), arXiv:1312.5330 [hep-ph].[61] J. Barnard, T. Gherghetta, and T. S. Ray, JHEP , 002 (2014), arXiv:1311.6562 [hep-ph].[62] A. Belyaev, G. Cacciapaglia, H. Cai, T. Flacke, A. Parolini, and H. Serˆodio, Phys. Rev. D94 ,015004 (2016), arXiv:1512.07242 [hep-ph].[63] A. Belyaev, G. Cacciapaglia, H. Cai, G. Ferretti, T. Flacke, A. Parolini, and H. Serodio, JHEP , 094 (2017), arXiv:1610.06591 [hep-ph].[64] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai, and D. Vadacchino, JHEP , 185 (2018), arXiv:1712.04220 [hep-lat].[65] F. Sannino, Phys. Rev. D79 , 096007 (2009), arXiv:0902.3494 [hep-ph].[66] T. A. Ryttov and F. Sannino, Int. J. Mod. Phys.
A25 , 4603 (2010), arXiv:0906.0307 [hep-ph].[67] G. Ferretti, JHEP , 107 (2016), arXiv:1604.06467 [hep-ph].[68] C. Csaki, T. Ma, and J. Shu, Phys. Rev. Lett. , 131803 (2017), arXiv:1702.00405 [hep-ph].[69] O. Matsedonskyi, G. Panico, and A. Wulzer, JHEP , 164 (2013), arXiv:1204.6333 [hep-ph].[70] D. Marzocca, M. Serone, and J. Shu, JHEP , 013 (2012), arXiv:1205.0770 [hep-ph].[71] R. Contino, D. Marzocca, D. Pappadopulo, and R. Rattazzi, JHEP , 081 (2011),arXiv:1109.1570 [hep-ph].[72] M. Golterman and Y. Shamir, Phys. Rev. D97 , 095005 (2018), arXiv:1707.06033 [hep-ph].[73] A. Arbey, G. Cacciapaglia, H. Cai, A. Deandrea, S. Le Corre, and F. Sannino, Phys. Rev.
D95 ,015028 (2017), arXiv:1502.04718 [hep-ph].[74] T. Hambye, F. S. Ling, L. Lopez Honorez, and J. Rocher, JHEP , 090 (2009), [Erratum:JHEP05,066(2010)], arXiv:0903.4010 [hep-ph].[75] Z.-H. Yu, J.-M. Zheng, X.-J. Bi, Z. Li, D.-X. Yao, and H.-H. Zhang, Nucl. Phys. B860 , 115(2012), arXiv:1112.6052 [hep-ph].[76] E. Aprile et al. (XENON), (2018), arXiv:1805.12562 [astro-ph.CO].[77] D. S. Akerib et al. (LUX), Phys. Rev. Lett. , 021303 (2017), arXiv:1608.07648 [astro-ph.CO].[78] X. Cui et al. (PandaX-II), Phys. Rev. Lett. , 181302 (2017), arXiv:1708.06917 [astro-ph.CO]. [79] M. L. Ahnen et al. (Fermi-LAT, MAGIC), JCAP , 039 (2016), arXiv:1601.06590 [astro-ph.HE].[80] J. Serra, JHEP , 176 (2015), arXiv:1506.05110 [hep-ph].[81] A. Banerjee, G. Bhattacharyya, and T. S. Ray, Phys. Rev. D96 , 035040 (2017), arXiv:1703.08011[hep-ph].[82] J. Wess and B. Zumino, Phys. Lett. , 95 (1971).[83] E. Witten, Nucl. Phys.