Vector-like Fermions in a Minimal Composite Higgs Model
aa r X i v : . [ h e p - ph ] M a y Vector-like Fermions in a MinimalComposite Higgs Model
Haiying Cai
Department of Physics, Peking University, Beijing 100871, China
Abstract
We consider the scenario where the composite Higgs arising as a pNGB in atwo-site model with a non-local term included. Constraints from pion scatteringand electroweak precision test are considered. We discuss the effects of compositeresonances, in particular the one from composite vector-like fermions, on theoblique parameters. It is noticed that the gluon fusion production of Higgs bosonis suppressed with respect to the Standard Model for about 6% after imposingthe unitarity and electroweak bounds.
Introduction
With the discovery of the 125 GeV Higgs boson at the LHC [1], one of the mostcrucial task is to unveil the nature of this scalar particle. Current measurement ofHiggs couplings in various channels reports certain deviation from the SM expectation,implicating that new physics may exist beyond the TeV energy scale, although there isno direct signature that confirms the existence of new particles. One of the theoreticallyplausible frameworks for the BSM new physics is the traditional SUSY, which aims tosolve the hierarchy problem and propose mechanisms for the electroweak symmetrybreaking. In this paper, we are interested to explore the composite Higgs scenario asan alternative option, since it will provide a little Higgs candidate from an underlyingstrong dynamic sector. There are many varieties of composite Higgs models, withthe original one realized in an extra dimensional scenario following the AdS/CFTcorrespondence [2]. The Higgs potential is calculable in a 5D description of compositeHiggs Model, but the particle spectrum in this framework is much more complicatedand not easy to make contact with the LHC measurement. Therefore recent effortis more focused on the 4D construction of Composite Higgs theory (CHM) [3, 4, 5].Since only the lowest lying states are accessible in the future LHC, it is adequate toformulate a predictive description without resorting to an UV completing picture.One effective approach to qualitatively describe a strong dynamic theory is employ-ing the Callan-Coleman-Wess-Zumino (CCWZ) formalism [6], where the full globalsymmetry is nonlinearly realized and the Lagrangian is constructed by covariant ob-jects transforming under a local symmetry group. The CCWZ prescription capturesthe common features in a generic CHM e.g. modified Higgs couplings, while on theother hand it hides the dynamic origin of the partial compositeness and leaves themasses of vector bosons to be less correlated. Here we are going to use the deconstruc-tion method proposed by [4, 5] to parametrize the scenario where one composite Higgsis realized as a pNGB from a spontaneous breaking global symmetry. We are interestedin exploring the simplest case, a two-site model with an enlarged global symmetry of SO (5) × SO (5) . In such a description, only the first level of composite resonances1s available. In contrast to the CCWZ formalism, the symmetry is realized in a linearway via the deconstruction method, with the partial compositeness being manifestedas the result of symmetry breaking in the composite sector. In particular non-localterms are possible to be introduced into this two-site model according to the symme-try principle. The existence of non-local terms has crucial impact on the unitarity of W L W L scattering and could change the sign of S parameter under certain condition.Composite vector-like fermions in SO (5) representations are necessarily to be incorpo-rated so that the SM fermions will gain the masses. One important motivation for thispaper is to explore the influence of composite fermions on the electroweak precisiontest and estimate their contribution to the Higgs production.The paper is organized in the following way. We starts from a review of the modelset up in the two-site description in Section 2. For the gauge bosons, the spectrum iscalculated in the unitary gauge and we investigate the influence of non-local term onthe ππ → ππ scattering. While for the fermion sector, composite fermions in a basic SO (5) representation are included and we are going to explore their mixing with theSM fermions. In Section 3, the contribution to S and T parameter from the vectorand fermion resonances are illustrated in detail, which is further compared with theexperimental data by a numerical scanning of the parameter space in this two-sitemodel. Finally in Section 4, we estimate the reduced Higgs production rate from thegluon fusion process by imposing the EW constraints on the composite scale f . Let us first review the basic model set up. The two-site model is the simplest scenario todescribe a composite Higgs boson, with one site imitating the UV brane and the othersite imitating the bulk in a warped extra dimension. Since the symmetry breaking isgeneralized to be SO (5) × SO (5) /SO (4) as depicted in Fig. 1, there are in principletwo sets of non-linear sigma fields in order to describe the coset space. One linkfield Ω mediating the interaction between the site-1 and the site-2 will break the SO (5) × SO (5) into a diagonal one SO (5) D , while another scalar field Φ = Ω φ G G H Figure 1: moose diagram for a two-site model, where the global symmetry is G × G = SO (5) × SO (5) . The two sites are connected by a link field Ω and the SO (5) symmetryis spontaneously broken by a local field Φ = Ω φ into a subgroup H = SO (4). located at the site-2 is responsible to break the SO (5) D into SO (4). Thus we obtain atotal of 10 + 4 NGBs. In order to get rid of the redundant NGBs, we are going to gaugea subgroup SU (2) L × U (1) Y at the site-1 and the full SO (5) symmetry at the site-2.Therefore after the spontaneous symmetry breaking, this two-site model delivers onlyone copy of Higgs boson.The two-site model interprets the holographic nature of a composite Higgs arisingfrom a strong dynamics. We would like to illustrate this point by connecting thedeconstruction method with the CCWZ formalism. Since elementary fields in the SMwill be put at the site-1, they play the roles of the source fields for operators constructedby composite fields residing at the site-2. Let us simply set Ω = 1 and rescale thekinetic terms to be canonically normalized, thus we obtain an effective Lagrangianequivalent to the CCWZ description. However for the phenomenology relevance inthe following discussion, we prefer to investigate the particle spectrum in the unitarygauge, where only the physical degrees of freedom appear in the Lagrangian. First of all, we need to figure out the gauge interactions with those NGBs. In theunitary gauge, the pion fields in the coset of SO (5) /SO (4) are parametrized in thesigma fields Ω and Φ . As pointed in the Ref. [7], there should be another scalar fieldΩ X , whose existence is necessary to lift one combination of extra U (1) gauge fields.Under the nearest gauge interaction principle, the Lagrangian for the gauge bosons3nd nonlinear sigma fields is: L − site = f T r | D µ Ω | + f D µ Φ ) T D µ Φ + f X | D µ Ω X | − T rw µν w µν − b µν b µν − T rρ µν ρ µν − X µν X µν . (1)with the covariant derivative terms defined as: D µ Ω = ∂ µ Ω − ig w aµ T aL Ω − ig ′ b µ T R Ω + ig ρ Ω ρ Aµ T A D µ Ω X = ∂ µ Ω X − ig ′ b µ Ω X + ig X X µ Ω X D µ Φ = ∂ µ Φ − ig ρ ρ Aµ T A Φ (2)where T A is the generator for SO (5) group and T aL , T R are the generators in thesubgroup SU (2) L and SU (2) R . The broken generator in the coset of SO (5) /SO (4) isdenoted as T ˆ a , with ˆ a = 1 , , ,
4. In the unitary gauge, we will simply set Ω X = 1,thus the remaining sigma fields are parametrized as:Ω = exp (cid:20) i f f f + f π ˆ a T ˆ a (cid:21) , Φ = exp (cid:20) i f f f + f π ˆ a T ˆ a (cid:21) φ , (3)with φ t = (0 , , , ,
1) indicating a spontaneous symmetry breaking. Defining the fieldΦ = Ω Φ , we obtain the physical sigma field for this model set-up:Φ T = 1 π sin( π/f π ) ( π , π , π , π , π cot( π/f π )) (4)Furthermore the following two-derivative kinetic term is possible to add into the La-grangian [5], which is a non-local term, but allowed by the symmetries. L nl = f D µ Φ) T D µ Φ D µ Φ = ∂ µ Φ − ig w aµ T aL Φ − ig ′ b µ T R Φ (5)Since the non-local term contributes to the pion kinetic term, it will modify the piondecay constant. Combining the results from L − site + L nl and demanding it normalizedaccording to (cid:0) ∂ µ π ˆ a (cid:1) , we obtain the following expression for f : f = f + f f f + f . (6)4he particle spectrum for the gauge bosons is easy to be identified before theEWSB, which are mildly corrected after setting h h i 6 = 0. For simplicity, we assumethat f X = f and g X = g ρ , the mass eigenstates for those partial composite massivestates are: ˜ W ± , µ = 1 q g + g ρ (cid:0) g w ± , µ − g ρ ρ ± , Lµ (cid:1) , B µ = 1 √ (cid:0) ρ Rµ − X µ (cid:1) B µ = 1 q g ′ + 2 g ρ (cid:0) g ρ ρ Rµ + g ρ X µ − g ′ b µ (cid:1) (7)with their masses squared calculated to be: m ρL = ( g ρ + g ) f , m B = g ρ f and m B = ( g ρ + 2 g ′ ) f . There are another six massive gauge bosons, which do not mixwith other fields at the leading order approximation. The mass squared for two chargedones ρ ± Rµ is m ρR = g ρ f , whereas the mass squared for four axial ones a ˆ iµ , ˆ i = 1 , , , m a = g ρ ( f + f ). It should be noticed that with the existence of the non-localterm, we can set f <
0, and demanding f + f > f < f as indicated by Eq. [6].For the gauge sector, it is worthwhile to investigate whether the unitarity for thepion scattering is partially restored in a two-site framework after the adding of vectorresonances. Some works in this direction have already been well done in [8, 9, 10]. Weare interested to derive the ρ L,R - π - π and π vertices which are relevant to the ππ → ππ scattering. From the first term in L − site , we can extract the interaction: L (1) ρπ + π = ( f − f ) f f g ρ (cid:2) ε ijk π i ∂ µ π j ρ kLµ + (cid:0) π k ∂ µ π − π ∂ µ π k (cid:1) ρ kLµ (cid:3) + ( f − f ) f f g ρ (cid:2) ε ijk π i ∂ µ π j ρ kRµ − (cid:0) π k ∂ µ π − π ∂ µ π k (cid:1) ρ kRµ (cid:3) + ( f − f ) f f h ( π a ∂ µ π a ) − (cid:0) π a ∂ µ π b (cid:1) i (8)and from the second term in L − site , we obtain a similar result: L (2) ρπ + π = ( f − f ) f f g ρ (cid:2) ε ijk π i ∂ µ π j ρ kLµ + (cid:0) π k ∂ µ π − π ∂ µ π k (cid:1) ρ kLµ (cid:3) + ( f − f ) f f g ρ (cid:2) ε ijk π i ∂ µ π j ρ kRµ − (cid:0) π k ∂ µ π − π ∂ µ π k (cid:1) ρ kRµ (cid:3) + ( f − f ) f f h ( π a ∂ µ π a ) − (cid:0) π a ∂ µ π b (cid:1) i (9)5ith the index i = 1 , , a, b = 1 , , ,
4. While in the non-local term,there exits additional π self interaction term: L nlπ = f f h ( π a ∂ µ π a ) − (cid:0) π a ∂ µ π b (cid:1) i (10)Following the standard procedure described in [10], we get the partial wave expansionfor the pion elastic scattering: a ( s ) ( ππ ) = 116 π ( f − f ) f f + ( f − f ) f f + f f ! s + g s π ( f − f ) f f + ( f − f ) f f ! (cid:20) (cid:18) m ρ s + 2 (cid:19) log (cid:18) sm ρ + 1 (cid:19) − (cid:21) , (11)with the approximation m ρ = m ρR ≃ m ρL in the limit g ≪ g ρ . Here we ignore thewidth effect from the vector resonance. Depending on the sign choice for the f , twodistinct scenarios will occur for the unitarity bound. It is noticed from the aboveequation that when we choose f >
0, the linear and logarithmic divergent terms areboth positive, which leads to the result that unless f and f are large enough, theunitarity bound | Re a ( s ) ( ππ ) | ≤ will be saturated very quickly before the effectivecut off scale is approached. However in the other scenario f <
0, one should set thelinear divergence to be almost vanishing, so that the high-energy behavior for the pionscattering is mainly determined by the mild logarithmic growing term. In Fig. 2, weplot the unitarity bound in the parameter space ( f , f ) for the two opposite situationsby fixing the cut off scale to be a few TeV. It turns out that in the case with f > f /f (prefered by the S bound) without too much raising the composite scale f . On theother hand, in the case with f <
0, the partial cancelation in the linear s term wouldhelp restore the perturbative unitarity. But as we should observe from the figure, itlargely reduces the unitarity conserving region as compared to the previous case. In this sector, we discuss the embedding of fermions in the framework of a two-sitemodel. Respecting the full global symmetry, the fermion is supposed to be put into an6
TeV7 TeV6 TeV
600 800 1000 1200 1400 1600 1800 2000600800100012001400160018002000 f GeV f G e V H a L Π Π ® Π Π H f < f L
600 700 800 900 1000200022002400260028003000 - i f GeV f G e V H b L Π Π ® Π Π H f > f L Figure 2:
Parameter-space region where the unitarity bound | Re a ( s ) ( ππ ) | ≤ is violated atenergies s ≤ Λ , for Λ = 6 . , . , . g ρ = 2 .
0. The left panel is for the case with f > f = 800 GeV, wherethe region in the right and upper direction is allowed. The right panel is for the case with f < f = 1200 GeV, where the narrow region between the same color lines is allowed. irreducible SO (5) representation. Here we are going to focus on the top quark sectorand the simplest choice would be the basic representation. In order to let the fermionsacquire the right hypercharge assignments, an extra U (1) X symmetry is necessary suchthat quantum numbers are determined by: Y = T R + X , and Q = T L + T R + X .Since SO (5) is spontaneously broken into SO (4) ∼ SU (2) L × SU (2) R , the followingdecomposition will apply: 5 = (2 , ⊕ (1 , ψ ±± with T L , T R charges ( ± , ± ) and one singlet ψ . For the elementary fermions in the site-1,i.e. q L = ( t L , b L ), t R and b R , they are embedded into incomplete SO (5) representations7ith the non-dynamic spurion fields being turned off : ξ uL = 1 √ b L − ib L t L it L / , ξ uR = it R / (12)Let us simply introduce one set of composite fermions in the site-2, which should beaccommodated in a complete SO (5) representation: ψ = 1 √ X + b iX − ib − T + t iT + it √ iT / (13)In the above construction we get one doublet ( t , b ), one non-standard doublet ( X , T ),where the exotic quark X carries an electric charge of 5 /
3, and one singlet top T .Due to the composite nature of our Higgs field, we can not directly couple two SMfermions with one Higgs field. However a bilinear mix interaction is permitted, that iswe can use the link field Ω to connect the SM field in site-1 with the composite fieldin site-2. It is also possible for us to write down the SO (5) invariant terms constructedwith only the composite fermion fields and pNGBs, so that the SM fermions willgain mass via the partial compositeness. As far as the top quark is concerned, theLagrangian for the fermion sector is: L top = ¯ q L i Dq L + t R i Dt R + ¯ ψi D ρ ψ + c tR ¯ ξ uR Ω ψ L + c qL ¯ ξ uL Ω ψ R − y T ¯ ψ L Φ Φ T ψ R − m Y ¯ ψ L ψ R + h.c. (14)where after the EWSB in the unitary gauge, the Ω takes the following simple form:Ω = I × cos ff h sin ff h − sin ff h cos ff h , (15)8nd the explicit expression for the other scalar field is:Φ t = (cid:18) , , , sin (cid:20) f f + f hf (cid:21) , cos (cid:20) f f + f hf (cid:21)(cid:19) . (16)There are four top quarks with electric charge 2 / t, t , T , T ),with the relevant mass term: L m = t L t L T L T L M top t R t R T R T R + h.c. (17)Since v < f , , f , let us expand the M top to the order of O ( v/f ), M top = c qL c qL f v √ f ( f + f ) − c tR f v √ f ( f + f ) − m Y − y T f v √ f ( f + f ) − c tR f v √ f ( f + f ) − m Y − y T f v √ f ( f + f ) c tR − y T f v √ f ( f + f ) − y T f v √ f ( f + f ) − m Y − y T . (18)The top quark mass matrix is easy to be analytically diagonalized if we set the HiggsVEV to be zero. From the Lagrangian L top , the mass matrix for the two bottom quarks( b, b ) can also be extracted. But unlike the top quark case, the bottom quark massmatrix has no dependence on the Higgs field. Since the mixing pattern for the lefthanded top and bottom quarks coincides with each other at h h i = 0, the followingrotation simultaneoulsy tranforms them into the mass eigenstates:˜ t L = m Y q c qL + m Y t L + c qL q c qL + m Y t L , ˜ t L = − c qL q c qL + m Y t L + m Y q c qL + m Y t L , (19)˜ b L = m Y q c qL + m Y b L + c qL q c qL + m Y b L , ˜ b L = − c qL q c qL + m Y b L + m Y q c qL + m Y b L . (20)9hile for the right handed top quarks, it is the two singlets ( t R , T R ) that would mixwith each other and the corresponding rotation is:˜ t R = m Y + y T p c tR + ( m Y + y T ) t R + c tR p c tR + ( m Y + y T ) t R , ˜ t R = − c tR p c tR + ( m Y + y T ) t R + m Y + y T p c tR + ( m Y + y T ) t R . (21)Defining the mixing angles sin θ L = c qL √ c qL + m Y and sin θ R = c tR √ c tR +( m Y + y T ) , thus to theleading order expansion, the top quark mass can be approximated as, m t ≃ | y T | sin θ L sin θ R √ vf . (22)where sin θ L and sin θ R indicate the degree of compositeness for t L and t R respectively.Notice that though the term proportional to y T in Eq. [14] only gives mass to a singlettop T , it is necessary to be present for the SM top t to gain the observed mass.Furthermore, suppose that | y T | is of a few TeV energy scale, we need the left handedor the right handed top to mostly origin from the composite sector. The masses forthree heavy top quarks are determined by: m t = c qL + m Y , m T = m Y , m T = c tR + ( m Y + y T ) (23)For the composite sector, we generally will set the parameter y T to be positive. Insuch a case, the SU (2) L partner for T , i.e. an exotic quark X , would be the lightestfermionic resonance, since it gets no further correction of O ( v/f ) after the EWSB. Onthe other hand, when we choose a negative y T , the lightest fermion could either be thesinglet top T or the exotic quark X . S and T Oblique parameters associated with the electroweak precision test puts a severe boundon the parameter space for “universal” models beyond the Standard Model. Let usfirst recall the definitions of oblique parameters, which are extracted from the two-pointfunctions of weak currents for the gauge bosons. S , T and U correspond to the residue10 % % % - - - - - S T % % % - - - - - S T Figure 3:
S-T plane for the composite resonance parameter scanning. The ellipses are atthe 68% (1 σ ), 95% (2 σ ) and 99% (3 σ ) confidence levels. The strong coupling is fixed to be g ρ = 2 .
0. In the left contour parameters are in the range (GeV): 3000 < c tR , y T < < m Y , f < < f <
830 and f = 0 .
0; In the right contour parametersare in the range (GeV): 3400 < c tR < < y T < < m Y , f < < − if <
830 and f = 1200. The remaining parameter c qL is calculated with an inputtop quark mass and the points which are located in the ellipses pass the EWPT. coefficients for expansion up to the order of p after fixing gauge couplings, Higgs VEVand imposing the U (1) em gauge invariance [11, 12]. Roughly speaking, heavy fieldsfrom the EW symmetry breaking sector additively contribute to the S , while the effectof isospin breaking is counted by the T and U . In the two-site composite Higgs model,there is a tree level mixing between the elementary gauge fields and composite gaugefields. Thus after integrating out heavy spin-1 resonances (both the vector and axialbosons), we find the dominating contribution to S parameter is:∆ S = − π · Π ′ W B (0)= 8 πg ρ (cid:20) − f ( f + f ) (cid:21) v f (24)As we should notice that although this effect is proportional to v /f , the tree leveldeviation still imposes a stringent constraint due to a factor of 8 π unless the strong11oupling 1 < g ρ < π is large enough. It is argued in ref. [13] that the S parameter isgenerally positive in most extra dimension scenarios. However, in the two-site modelwith the existence of non-local term, it is possible to enforce the condition 0 < f + f 12 2 m W v ( v + a h ) W + µ W − µ + 12 m Z v ( v + a h ) Z µ Z µ + O ( h ) . (26)with the masses of gauge bosons and the parameter | a | < m W = g f sin ( v/f )4 , m Z = ( g + g ′ ) f sin ( v/f )4 , a = cos( v/f ) (27)In the SM, there is an exact cancelation of logarithmic divergence for the S and T ,which is spoiled by the reduced Higgs couplings with gauge bosons. Therefore the IRcontribution in fact describes a running effect from the EW scale till the compositescale in the effective theory, where by the NDA estimation Λ NDA = 4 πf .∆ S IR = 16 π (cid:20) sin ( v/f ) log (cid:18) Λ m h (cid:19) + log (cid:18) m h m h,ref (cid:19)(cid:21) , (28)∆ T IR = − πc w (cid:20) sin ( v/f ) log (cid:18) Λ m h (cid:19) + log (cid:18) m h m h,ref (cid:19)(cid:21) (29)12nder the condition that f ∼ . f , the IR contribution generally constitutes asizable portion to both S and T .Finally let us discuss more about the fermion loop correction to EW precision test.The evaluation for the S and T from various types of vector-like quarks is given inthe reference [14]. Some detail studies of T parameter constraint on vector-like singlet,doublet and triplet quarks could also be found in [15, 16]. I will give the analyticexpressions of S parameter from vector-like quark loops, especially for the nonstandarddoublet scenario in the appendix. It is worth to point out that since the exotic quark X carries an electric charge of , the function ψ + defined in the original work [14] willbe modified. For the S parameter, the virtual fermion effect is generally subleading,but not totally irrelevant, which leads to the consequence that the S is less modeldependent on the mixing parameters. By contrast the fermion contribution to the T ismore important and needs to take into consideration. We expect that the T obtains asubstantial positive shift though quark mixings, so that it will partially compensate thenegative IR correction. The magnitude of T ferm in a generic composite Higgs modelcan be estimated in the limit | f | ≪ | f | as:∆ T ferm ∼ N c πs W g y T sin θ L,R M T f sin v f , (30)where M T collectively stands for the mass of a composite vector-like quark. In orderto get a quantitative understanding, we could assume that sin θ L,R ∼ . M T ∼ | y T | ∼ . f ∼ . T ferm ∼ . 17, whichis numerically competitive with the IR correction. The sign for the T parameter isdetermined by the isospin of the heavy top quark. As we have shown in the massmatrix for top-like quarks, the mixing is entangled with each other. For simplicity,we consider that the top quark separately mixes with one type of vector-like quarkeach time. Using the equations presented in [16], it is possible to exactly evaluate the T parameter for various scenarios. For the SM top t mixing with one singlet T , ormixing with one doublet ( t , b ), ∆ T ferm is always positive. The situation becomesopposite for the nonstandard doublet ( X , T ), as its modification to the W µ W µ form13actor is bigger than the other two cases. In the small mixing limit, the last typemixing gives a negative contribution to the T parameter. In this two-site model, wecan find that because the t mixing with T is larger than with t , under the condition M T < M t , the negative contribution from nonstandard doublet will overcome thepositive one from the doublet. However provided that additional positive contributionobtained from the mixing with T is large enough, the combining result ∆ T ferm wouldstill be positive, as we can prove this point using the numerical scanning.In Fig. 3, we show the numerical results for S and T by scanning the parameterset ( c qL , c tR , m Y , y T , f , f , f ) in typical ranges. It is observed that the S is mainlysensitive to f and f , , whereas the T is more dependent on the mixing parametersfor the fermions. We are intending to find out viable parameter regions which arecompatible with experimental values, i.e. S = 0 . ± . 10 and T = 0 . ± . 12 with acorrelation coefficient of ρ col = 0 . 89. In the left contour, we illustrates one case withno non-local term, i.e. f = 0, therefore the S parameter is bound to be positive dueto the condition f < f + f . Combining all the mixing effects from the SM chiraltops with composite vector-like quarks, we find out that there is enough possibility forthe T to be shifted into the positive region. While in the right contour, another casewith a non-local term is present. Since we enforce an opposite condition f > f + f ,the sign of S parameter is tuned to be negative. In the latter case, the points with anegative T are more consistent with the electroweak data. Since the Yukawa couplings for the composite fermions are crucial for the gluon fusionprocess, we are going to briefly comment their contribution to the Higgs production.For all the fermion fields in the mass eigenstates, let us assume that they are interactingwith the Higgs field in the following way: L h = X M i ¯ ψ i ψ i + X Y ii h ¯ ψ i ψ i (31)where M i and Y ii are the mass and Yukawa coupling for each fermion. Notice that14 .090.110.13 0.130.13 800 1000 1200 1400 1600 1800 2000 22002200240026002800300032003400 f GeV m Ρ G e V S Parameter - - - - - f GeV m Ρ G e V S Parameter Figure 4: S parameter contour and unitarity bound with Λ = 8 TeV on the plane of ( m ρ , f ).In the left contour, we set f = 800 GeV and demand that f + f > f . The orange linecorresponds to the unitarity bound, where the region in the right direction is allowed. The S = +0 . 13 bound (blue dashed line) and the unitarity bound (red dashed line) at f = 0 . f = 1600 GeV and insteadenforce f + f < f . The orange line represents the unitarity bound, where the region inthe lower direction is allowed. The S = − . 07 bound (blue dashed line) and the unitaritybound (red dashed line) at f = 1550 GeV is shown for comparision. in our simplified model, only the top-like quarks have interaction with the Higgs field.The production rate for a Higgs boson from fermion loops is proportional to: σ (cid:0) gg → h (cid:1) ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i Y ii M i A / ( τ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , τ i = m h M i (32)For the case of a light Higgs boson, we generally have M Q,t ≫ m h ≫ m b , with theindex Q stands for the heavy vector-like quarks. In order to make contact with thelow energy phenomenology, one just needs to take advantage of the approximation for A / ( τ i ) in certain limits, i.e. A / ( τ ) → / 3, for τ → 0, and A / ( τ ) → 0, for τ → ∞ .15urthermore relating the sum of P y ii /M i to the determinant of the mass matrix: X i Y ii M i = ∂ log (det M ) ∂v , (33)we can effectively evaluate the production rate without rotating into the mass eigen-state. Since the X / and b in this model do not couple to the Higgs field, by neglectingthe bottom quark contribution we obtain a concise result : σ ( gg → h ) CHM σ ( gg → h ) SM ≃ vf cot (cid:18) vf (cid:19) = 1 − v f (34)Therefore in the case with only one multiplet of composite quarks, the gluon fusionproduction of Higgs boson is generally reduced with respect to the SM scenario, aspointed out by the pioneer work [17]. However it is noticed in ref. [18] that throughintroducing the h dependent bottom quark mixing with composite quarks, an enhanced h gg coupling is possible to be realized in the composite Higgs scenario.Through constraining the decay constant using the unitarity and EWPT, we willbe able to estimate the reduced percentage in this model. The S parameter, especiallyits dominate part ∆ S fit ≃ ∆ S tree + ∆ S IR , imposes a stringent bound on m ρ and v /f by requiring that − . < S < . 13. On the other hand, we should also require theunitarity bound to be conserved till a relative large effective cut off scale, e.g. 8 TeV.The Fig. 4 interprets the S parameter and unitarity bound in the ( m ρ , f ) plane. Forthe case f + f > f , it turns out that a larger f leads to a growing f , but themass scale m ρ will be lowered, resulting in more parameter region compatible withthe experimental data. However, for the opposite case f + f < f , the unitaritybound intends to push f to be smaller, while the S parameter requires a larger f .Thus one has to increase the f in order to gain more compatible region. Undereither situation, we will find out that after imposing necessary constraints, an upperbound v /f < . 042 is generally permitted, which in turn translates into a roughestimation σ CHM /σ SM > . 94 for the reduced Higgs production rate. Notice that alower production rate is applicable if one reduces the Λ eff for the unitarity bound.16 Conclusion In this paper, we study the minimal SO (5) /SO (4) Composite Higgs Model in a two-sitescenario, where only the first level of composite resonances is present in comparisionwith the KK modes in an extra dimension theory. In addition to the nearest neighborinteraction, we especially investigate the effects of non-local term on the perturbativeunitarity and the EW precision test. In the scenario f > 0, the existence of non-localterm will lead to a lower bound for the vector rosonance mass m ρ , which amelioratesthe compatibility of this two-site model with experimental data. On the other hand,the non-local term is necessary to be added in the scenario f < 0. Under such asituation, there is a tension between the unitarity requirement and the negative S parameter bound. Thus we should increase the f in order to relieve this tension andachieve a relative large effective cut off scale Λ eff .For simplicity, vector-like composite fermions are embedded in the SO (5) basicrepresentations, which will mix the SM quarks via bilinear interactions. Under thesituation of a positive S parameter, we prefer a positive T for a better fit with theEW precision test. Through the parameter scanning, we have shown that there isenough possibility that the positive correction to the T from vector-like fermions coulddominate the negative IR contribution from the reduced Higgs couplings. Furthermore,the effects of vector-like fermions on the gluon fusion Higgs production is discussed.It turns out the production rate will at most be reduced around 6% provided that wedemand a strict unitarity and EWPT to be satisfied. Acknowledgments H.Cai is supported by the postdoctoral foundation under the Grant No. 2012M510001,and in part supported by National Nature Science Foundations of China (NSFC) underthe Contract No. 10925522. 17 ppendix In this appendix, we are going to collect the contribution to S parameter from vector-like fermions in singlet, doublet and nonstandard doublet scenarios. Consider the topquark mixing with each type of vector-like fermion in the following way: L top ⊃ − m t ¯ t L t R − x T ¯ t L T R − x t ¯ t R t L − x T ¯ t R T L − M T ¯ T T − M t ¯ t t − M T ¯ T T + h.c. (35)For a singlet vector-like fermion, the mixing angles are determined by diagonalizingthe mass matrix,sin θ Lu = M T x T q ( M T − m t ) + M T x T , sin θ Ru = x T M T sin θ Lu . (36)While for a doublet or nonstandard doublet, the L.H. and R.H. mixing angles areexchanged with respect to the singlet scenario,sin θ Ru = M t ( T ) x t ( T ) q ( M t ( T ) − m t ) + M t ( T ) x t ( T ) , sin θ Lu = x t ( T ) M t ( T ) sin θ Ru . (37)In terms of mixing angles, the S parameter for each kind of scenario can be expressedby the following equations:∆ S t − T = 32 π (cid:2) sin θ Lu ψ + ( y T , y b ) − sin θ Lu ψ + ( y t , y b ) − cos θ Lu sin θ Lu χ + ( y T , y t ) (cid:3) ∆ S t − t = 32 π (cid:2) sin θ Lu ψ + ( y t , y b ) − sin θ Lu ψ + ( y t , y b )+ (cos θ Lu + cos θ Ru ) ψ + ( y t , y b ) + (sin θ Lu + sin θ Ru ) ψ + ( y t , y b )+ 2 cos θ Lu cos θ Ru ψ − ( y t , y b ) + 2 sin θ Lu sin θ Ru ψ − ( y t , y b ) − cos θ Ru sin θ Ru χ + ( y t , y t ) (cid:3) ∆ S t − T = 32 π (cid:2) sin θ Lu ψ + ( y T , y b ) − sin θ Lu ψ + ( y t , y b )+ (sin θ Lu + sin θ Ru ) ψ + ( y X , y t ) + (cos θ Lu + cos θ Ru ) ψ + ( y X , y T )+ 2 sin θ Lu sin θ Ru ψ − ( y X , y t ) + 2 cos θ Lu cos θ Ru ψ − ( y X , y T ) − (4 cos θ Lu sin θ Lu + cos θ Ru sin θ Ru ) χ + ( y t , y T ) − θ Lu sin θ Lu cos θ Ru sin θ Ru χ − ( y t , y T ) (cid:3) (38)18here the rescaled mass squared is defined as y i = M i /m Z , with M i representing themass of vector-like quark and the functions of χ + , χ − , ψ + and ψ − are: χ + ( y , y ) = 5 ( y + y ) − y y y − y ) + 3 y y ( y + y ) − y − y y − y ) ln y y ,χ − ( y , y ) = −√ y y (cid:18) y + y y y − y + y ( y − y ) + 2 y y ( y − y ) ln y y (cid:19) ,ψ + ( y α , y i ) = 13 − 13 ( Q α + Q i ) ln y α y i , ψ − ( y α , y i ) = − y α + y i √ y α y i . (39)where our function ψ + ( y α , y i ), with the index α for an up-type quark and the index i for a down-type quark, is in fact dependent on the sum of electric charges Q α + Q i ,which generalizes the previous result reported in the Ref. [14]. Notice that for themixing of t with T , the argument y X should be put in front of the argument y T or y t , due to the opposite isospin assignment in a non-standard doublet. References [1] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716 (2012) 1 arXiv:1207.7214[hep-ex]; S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716 (2012) 30arXiv:1207.7235 [hep-ex].[2] R. Contino, Y. Nomura and A. Pomarol, Nucl. Phys. B , 148 (2003)[hep-ph/0306259]; K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B ,165 (2005) [hep-ph/0412089].[3] C. Anastasiou, E. Furlan and J. Santiago, Phys. Rev. D , 075003 (2009)[arXiv:0901.2117 [hep-ph]].[4] G. Panico and A. Wulzer, JHEP , 135 (2011) arXiv:1106.2719 [hep-ph];[5] S. De Curtis, M. Redi and A. Tesi, JHEP , 042 (2012) [arXiv:1110.1613[hep-ph]].[6] S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. 177, 2239 (1969); C. G. CallanJr., S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. 177, 2247 (1969).197] M. Carena, L. Da Rold and E. Ponton, arXiv:1402.2987 [hep-ph].[8] R. Contino, D. Marzocca, D. Pappadopulo and R. Rattazzi, JHEP , 081(2011) [arXiv:1109.1570 [hep-ph]].[9] B. Bellazzini, C. Csaki, J. Hubisz, J. Serra and J. Terning, JHEP , 003 (2012)[arXiv:1205.4032 [hep-ph]].[10] H. Cai, Phys. Rev. D , no. 3, 035018 (2013) [arXiv:1303.3833 [hep-ph]].[11] M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. , 964 (1990); Phys. Rev. D ,381 (1992);[12] R. Barbieri, A. Pomarol, R. Rattazzi and A. Strumia, Nucl. Phys. B , 127(2004) [hep-ph/0405040].[13] K. Agashe, C. Csaki, C. Grojean and M. Reece, JHEP , 003 (2007)[arXiv:0704.1821 [hep-ph]].[14] L. Lavoura and J. P. Silva, Phys. Rev. D 47 (1993) 2046;[15] G. Cacciapaglia, A. Deandrea, D. Harada and Y. Okada, JHEP , 159 (2010)[arXiv:1007.2933 [hep-ph]].[16] H. Cai, JHEP , 104 (2013) arXiv:1210.5200 [hep-ph].[17] A. Falkowski, Phys. Rev. D , 055018 (2008) [arXiv:0711.0828 [hep-ph]].[18] A. Azatov and J. Galloway, Phys. Rev. D85