Top Seesaw with a Custodial Symmetry, and the 126 GeV Higgs

The composite Higgs models based on the top seesaw mechanism commonly possess an enhanced approximate chiral symmetry, which is spontaneously broken to produce the Higgs field as the pseudo-Nambu-Goldstone bosons. The minimal model with only one extra vector-like singlet quark that mixes with the top quark can naturally give rise to a 126 GeV Higgs boson. However, without having a custodial symmetry it suffers from the weak-isospin violation constraint, which pushes the chiral symmetry breaking scale above a few TeV, causing a substantial fine-tuning for the weak scale. We consider an extension to the minimal model to incorporate the custodial symmetry by adding a vector-like electroweak doublet of quarks with hypercharge +7/6. Such a setup also protects the $Zb\bar{b}$ coupling which is another challenge for many composite Higgs models. With this addition, the chiral symmetry breaking scale can be lowered to around 1 TeV, making the theory much less fine-tuned. The Higgs is a pseudo-Nambu-Goldstone boson of the broken O(5) symmetry. For the Higgs mass to be 126 GeV, the hypercharge +7/6 quarks should be around or below the chiral symmetry breaking scale, and are likely to be the lightest new states. The 14 TeV LHC will significantly extend the search reach of these quarks. To probe the rest of the spectrum, on the other hand, would require a higher-energy future collider.


Introduction
The nature and properties of the Higgs boson have become the focus of particle physics research since its discovery in 2012. The relatively light Higgs boson of 126 GeV suggests that it is either an elementary particle or a pseudo-Nambu-Goldstone boson (pNGB) of some spontaneously broken symmetry if it is a composite degree of freedom of some strong dynamics [1][2][3][4][5][6]. Other than the Higgs boson, the Large Hadron Collider (LHC) so far has not discovered any new physics yet. The couplings of the Higgs boson are consistent with their standard model (SM) values, though some significant deviations are still possible. If there exists new physics responsible for the origin of the electroweak symmetry breaking (EWSB), the current experimental results indicate that it is probably close to the decoupling limit. On the other hand, the naturalness argument strongly prefers the new physics to be near the weak scale to avoid excessive fine-tuning. The tension between these two requirements has becomes a severe challenge for any model that attempts to explain the electroweak (EW) scale.
In a previous paper [7], it was found that in a top seesaw model of dynamical EWSB [8][9][10][11], the Higgs boson arises naturally as a pNGB of the spontaneously broken U (3) L symmetry, which relates the left-handed top-bottom doublet and a new quark χ L . The top seesaw model fixes the problem of the top quark being too heavy in the top condensation model [12][13][14][15][16] by mixing the top quark with a new vector-like quark χ. It was shown that, in the presence of the approximate U (3) L symmetry, the Higgs boson mass is highly correlated with, and generically smaller than the top quark mass. The experimental value of 126 GeV can be obtained with natural values of the parameters of this model. A drawback of this model is that the U (3) L does not contain a custodial symmetry. As a result, the constraint on the weak-isospin violation requires the chiral symmetry breaking scale f to be above 3.5 TeV. Some significant fine-tuning is needed to obtain the weak scale at v ≈ 246 GeV. Such a high chiral symmetry breaking scale also implies that none of the new states are predicted to be reachable at the LHC. A collider of much higher center of mass energy (∼ 100 TeV) would be needed to have any chance to see some of the new states.
It is therefore desirable to consider extensions of the minimal top seesaw model to include a custodial symmetry. A straightforward extension to the top seesaw model in Ref. [7] is to introduce "bottom seesaw" by adding a vector-like singlet bottom partner ω.
The spontaneously broken U (4) L symmetry can produce 2 light Higgs doublets. Without 1/2 -1/2 1/2 -1/2 T 3 R 1/2 1/2 -1/2 -1/2 Table 1: The quantum numbers of X L , T L , t L , b L under SU (2) L × SU (2) R . additional contributions, the mass of the Higgs boson made of the bottom and ω quarks is related to the bottom quark mass and hence is too light. To avoid this situation, one could introduce scalar mass terms (which come from 4-fermion interactions in the UV theory) to explicitly break the U (4) L chiral symmetry of (t L , b L , χ L , ω L ) down to Sp (4).
While the Sp(4) contains the SU (2) C custodial symmetry which can be used to protect the weak-isospin, such a model suffers from the constraint on Z → bb branching ratio. The most recent results suggest that the SM prediction for Z → bb branching ratio (R b ) is 2.4σ smaller than the measured value [17]. When the bottom quark mixes with a heavy singlet, as required for the bottom seesaw mechanism, the Zb LbL coupling is reduced (becomes less negative) while the Zb RbR coupling is not modified. As a result, the Z → bb branching ratio is further reduced. This puts a constraint on the mixing angle (θ b L ) between b L and ω L , which pushes the mass of ω to be very large [11,18]. In order not to have a large weak-isospin violation, the masses of χ and ω should be close, again implying a large chiral symmetry breaking scale. By playing with the model parameters, the chiral symmetry breaking scale may only be slightly reduced compared to the original top seesaw model, which means that such an extension still require stiff fine-tunings.
It was pointed out in Ref. [19] that the custodial symmetry which protects the weak isospin can also protect the Zb LbL coupling under certain conditions. Namely, the new physics needs to be invariant under an O(4) global symmetry, which is the familiar SU (2) L × SU (2) R of the SM Higgs sector together with a parity defined as the interchange L ↔ R (P LR ); also, b L needs to be charged under both SU (2) L and SU (2) R with T L = T R = 1/2, T 3 L = T 3 R = −1/2 . This implies that the SM (t L , b L ) doublet needs to be embedded into a (2, 2) representation of SU (2) L × SU (2) R , together with a new doublet quark (X L , T L ) of hypercharge +7/6, with the quantum numbers given in Table 1.
To adopt this setup we introduce an SU (2) W -doublet vector-like quarks, Q ≡ (X, T ), with hypercharge +7/6, in addition to the vector-like SU (2) W -singlet quark χ which is responsible for the top seesaw mechanism.
The underlying strong dynamics is assumed to approximately respect the U (5) L × U (4) R symmetry among the five left-handed quarks (t L , b L , X L , T L , χ L ) and the four right-handed quarks (X R , T R , t R , χ R ). To avoid too many light pNGBs after the chiral symmetry breaking, gauge invariant scalar mass terms (arising from 4-fermion interactions in the UV) can be introduced to explicitly break U (4) R symmetry and also U (5) L down to O(5).
In this way, only one light Higgs doublet arises from the chiral symmetry breaking of O(5) → O(4). An important difference between our model and the setup in Ref. [19] is that in our model, the custodial symmetry that protects both the weak isospin and the Zb LbL coupling is only approximately preserved by the new physics, which violates the conditions in Ref. [19]. Nevertheless, we found that within some regions of the parameter space, both corrections are within experimental constraints, while the chiral symmetry breaking can be as low as ∼ 1 TeV, significantly ameliorating the fine-tuning of the weak scale.
The rest of this paper is organized as follows. In Section 2, we write down the effective theory with composite scalars below the compositeness scale with U (5) L × U (4) R symmetric dynamics of the extended quark sector. In Section 3, we focus on the theory at the TeV scale and show that the Higgs boson arises as a pNGB of the chiral symmetry breaking. We derive an approximate analytic formula for the mass of the Higgs boson (M h ) and discuss various possible corrections. It can naturally be around 126 GeV for model parameters within reasonable ranges. In Section 4, we further verify the results in Section 3 with numerical studies. We show that in this model the chiral symmetry breaking scale can be lowered to ∼ 1 TeV without large weak-isospin violation, and a 126 GeV Higgs boson mass can easily be obtained. We also comment on the search of the new states at the LHC and future colliders. The conclusions are drawn in Section 5. The two appendices collect the formula of T parameter from fermion loops and the estimates of some model parameters.

Composite Scalars with a Custodial Symmetry
As in the usual composite Higgs models, we assume that at a scale Λ 1 TeV there are no fundamental scalars. To implement the custodial symmetry in the top seesaw dynamics, we introduce an SU (2) W -singlet vector-like quark, χ, of electric charge +2/3 and SU (2) W -doublet vector-like quarks, Q ≡ (X, T ), with hypercharge +7/6, in addition to the SM gauge group and fermions. For the doublet quarks, T has electric charge +2/3, same as the SM top quark t, while X has electric charge +5/3. We assume that these new quarks, the left-handed (t L , b L ) doublet and the right-handed t R in the SM (but not b R ) have some new non-confining strong interactions, which can be represented by 4-fermion interactions with strength proportional to 1/Λ 2 . The strong dynamics is further assumed to approximately preserve the U (5) L × U (4) R chiral symmetry of the five left-handed fermions Ψ L ≡ (t L , b L , X L , T L , χ L ) and the four right-handed fermions 1 The strong dynamics among the fermions at scale Λ is given by We assume the 4-fermion interactions in Eq.
which also preserves the approximate U (5) L × U (4) R symmetry. The scalar field Φ is a For each of the 20 complex scalars, the superscript denotes the electric charge and the subscript indicates the fermion constituents of the scalar. For example, σ − tX ∼ (t L X R ), and has electric charge −1. The fields that contain χ R are labelled differently (φ instead of σ) because they contain the light scalars which will be the focus of our study. It is useful to classify the scalar fields in Eq. (2.4) into the following categories: are EW doublets; • σ 0 χt and φ 0 χχ are EW singlets; contains one EW triplet and one singlet, which can be parameterized The vector-like fermions can possess gauge invariant masses, which may be generated by the physics at some higher scale than Λ: These fermion mass terms explicitly break the U (5) L ×U (4) R symmetry. They are assumed to be small compared to Λ so that they do not affect the strong dynamics. Below the compositeness scale, these mass terms are matched to the tadpole terms of the composite scalars.
At scales µ < Λ, the Yukawa couplings give rise to the quartic couplings and corrections to the masses of the scalars. We assume that there are additional explicit U (4) R breaking effects which distinguish t R , χ R and Q R . Since mass terms are quadratically sensitive to the UV physics, such effects could induce a large relative splitting of the masses for Σ X,T , Σ t and Φ χ . Combining the quartic couplings, mass terms and tadpole terms, the scalar potential below scale Λ is given by Because Q R ≡ (X R , T R ) is an EW doublet, Σ X , Σ T have the same mass-squared M 2 Σ X,T , and σ 0 XX , σ 0 T T have the same tadpole coefficient C Q . (This guarantees that the VEV of triplet scalars are suppressed.) Matching at the scale Λ, the size of the tadpole terms are related to the fermion mass terms by When the scalars are integrated out at the cutoff scale, the fermion mass terms are recovered. This means that at scale µ < Λ we do not need to include the explicit fermion mass terms in Eq. (2.1). They will appear from the scalar VEVs in the low energy effective theory. The quartic coupling λ 1 is generated by fermion loops and becomes nonperturbative near Λ. λ 2 , on the other hand, is not induced by fermion loops at the leading order and vanishes at Λ in the large N c limit. At scales µ < Λ, scalar loops generate a non-zero value for λ 2 and give corrections to λ 1 . Nevertheless, we expect λ 1 |λ 2 |. The spontaneous breaking of the chiral symmetry requires at least one of the scalars to have a negative mass-squared. To obtain the correct SM limit, we require M 2 Φχ < 0, while M 2 Σt and M 2 Σ X,T are assumed to be positive for simplicity. The theory below the compositeness scale Λ is given by Eq. (2.2) and Eq. (2.7).
Overall, the scalar sector contains 2 complex triplets, 5 complex doublets and 4 complex singlets. The full theory is rather complicated. However, the main focus of this paper is the low energy (µ Λ) phenomenology, in particular, the mass of the Higgs boson and the constraint from the weak-isospin violation T parameter. To produce the correct top seesaw mechanism, the SM Higgs doublet is required to be mostly the linear combination necessarily ruled out by current experimental constraints, from a naturalness point of view it is more reasonable to assume that their masses are not much smaller than Λ, so that all the degrees of freedom in them are heavy and can be integrated out for µ Λ to obtain a low energy effective theory with Φ χ only. We will focus on this low energy theory for the rest of this paper.

Higgs Boson as a PNGB of the O(5) Symmetry
We now study the effective theory at scale µ Λ obtained by integrating out the heavy modes in Σ X , Σ T and Σ t . For simplicity we will sometimes label them collectively as Σ X,T,t and their masses as M Σ X,T,t . In the effective theory, the lowest order contribution of Σ X,T,t simply comes from the VEVs of σ 0 XX , σ 0 T T and σ 0 χt , induced by the tadpole terms in Eq. (2.7). The subleading corrections, including the VEVs of other neutral fields in Σ X,T,t , are suppressed by 1/M 2 Σ X,T,t . We will first consider the contributions from VEVs of σ 0 XX , σ 0 T T and σ 0 χt only and study the O(1/M 2 Σ X,T,t ) corrections later in Section 3.4.
The scalar potential at µ Λ can be written as w and u t are defined as At the lowest order, σ XX and σ T T have the same VEVs since they have the same tadpole terms. This guarantees that the triplet scalar does not develop a VEV at the lowest order, which may otherwise cause a large weak isospin violation.
Eq. (3.9) has an U (5) L chiral symmetry which is explicitly broken by the heavy field VEVs w and u t and the tadpole term C χχ . Without the explicit breaking terms, U (5) L is spontaneously broken to U (4) L due to a negative mass-squared M 2 Φχ , and Φ χ contains 9 NGBs which includes two massless Higgs doublets. If the explicit breaking is small, the theory will have two light Higgs doublet. Although the possibility of additional light scalars is not ruled out, such a theory will not have an EWSB minimum that approximately preserves the custodial symmetry. As we will see in Section 3.1, the VEV w is constrained by the search of the charge +5/3 quark to be at least several hundred GeV.
A large w can raise the masses of one of the Higgs doublet by explicit breaking the U (5) L chrial symmetry down to an approximate U (3) L symmetry of (φ 0 t , φ − b , φ 0 χ ). However, the U (3) L symmetry does not contain the SU (2) custodial symmetry and we just recover the minimal model of Ref. [7] in this limit, which makes the extension of the hypercharge +7/6 quarks (X and T ) and the corresponding composite scalars totally pointless! To solve this problem, we introduce the following mass terms (parameterized by the masssquared parameter K 2 ) that also explicitly break U (5) L , and A χ is the CP-odd field in φ 0 χ shown later in Eq. (3.16). They can come from gauge invariant 4-fermion operators in the UV theory. We require K 2 to be positive. Eq. (3.12) lifts up the masses of A χ and one linear combination of the two Higgs doublets, hence breaks U (5) L down to O(5). The custodial symmetry will approximately hold as long as the value of K 2 is large enough (K 2 λ 1 w 2 ). (More explicitly discussion will be done in Section 3.2.) In this case, the theory has only 4 pNGBs that forms the light SM-like Higgs doublet from spontaneous breaking of O(5) to O(4). At the same time an approximate custodial symmetry is also retained.
Combining Eq. (3.9) and Eq. (3.12), the scalar potential is VEVs from tadpoles, heavy field VEVs and the negative mass squared M 2 Φχ . We parameterize them as The electroweak VEV, v = v 2 t + v 2 T , is required to be about 246 GeV. Due to the explicit breaking from the VEV w, v t > v T is required for the potential to be at a minimum. For u χ is a singlet VEV which is expected to be significantly larger than the electroweak VEV.
which is conventionally called the chiral symmetry breaking scale.

Extended top seesaw
Once the scalar fields develop VEVs as in Eq. (3.11) and (3.16), the Yukawa couplings in Eq. (2.2) generate the following mass terms of the fermions: The X quark has electric charge +5/3 and does not mix with any other fermions. Its mass is given by The most recent CMS search has excluded the charge +5/3 quark with a mass below 800 GeV at 95% confidence level (CL), assuming that they decay exclusively to tW [20]. 4 This constrains the value of w to be at least a few hundred GeV. The T quark, on the other hand, mixes with t and χ so that the 2 × 2 mass matrix in the usual top seesaw model is extended to a 3 × 3 mass matrix. We denote the three mass eigenstates as t 1 , t 2 and t 3 , ordered by m t 1 ≤ m t 2 ≤ m t 3 . Given that w can not be too small (w 300 GeV for ξ ∼ 3.6), the top quark is always the the lightest mass eigenstate t 1 , and its mass (m top ≡ m t 1 ) is approximately given by As we will see later f w is required to obtain the correct Higgs mass. The lighter toppartner t 2 is mainly T , its mass m t 2 is almost degenerate with m X due to the small mixing.
There is also a bound on m t 2 from the searches of the heavy top-like quarks [21,22], similar to but slightly weaker than the bound of m X . The heavier top partner t 3 is mostly the EW singlet χ, with a mass given by m t 3 ∼ ξf / √ 2. Finally, to obtain the correct top mass in Eq. (3.21), we have the following constraint where y t is the SM top Yukawa coupling, define as m 2 top ≡ With the addition of the X and T quarks, (t L , b L ) and (X L , T L ) form a (2, 2) representation under SU (2) L × SU (2) R , which contains the SU (2) C custodial symmetry after EWSB. In the limit that the vector-like mass µ Q vanishes (or equivalently w = 0), there is no explicit violation of the custodial symmetry in the (t L , b L , X L , T L ) sector, which implies a negative T parameter relative to the SM value because it removes the SM contribution On the other hand, if µ Q → ∞, then (X, T ) decouples and we recover the fermion sector of the minimal model, which gives a large positive contribution to T if the chiral symmetry breaking scale is low. We expect that in a suitable range of the X, T masses, the T parameter can be small and consistent with the EW measurements. In Appendix A, we provide the full expression for the T -parameter calculated from fermion loops, which we use in the numerical calculations in Section 4. Other contributions, such as the contribution from triplet scalar VEVs, are very small as long as the masses of heavy scalars M Σ X,T,t are sufficiently large. In principle there could be additional modeldependent contributions from unknown UV physics. Here we assume that the custodial symmetry is a good symmetry in the UV and all major explicit breaking effects have been parameterized in our low energy effective theory, so that they are negligible.
Since we only add vector-like quarks, the calculable contributions to the S parameter is negligible. However, there could be UV contributions from heavy vector states [23][24][25][26][27].
While such contributions are model-dependent, they can be estimated to be [28] S whereŜ = g 2 /(16π) S and m ρ is the mass scale of the heavy vector state. We expect such states to exist, as mentioned later in Secion 3.3, which sets the scale where gaugeloop contributions are cut off. For m ρ = 3 TeV, a typical value for f ∼ 1 TeV, we have S ∼ 0.08, within the 68% CL of the experimental constraint [17]. A larger value of S (up to ∼ 0.27) may still be allowed if we arrange a larger value for T as well, which can be easily achieved in this model.
The Zbb coupling has been a long-standing issue in beyond SM model building, particularly for composite Higgs models. The measured value of the Z → bb branching ratio (R b ) [29] was known to be larger than the SM prediction. A recent calculation of R b including two-loops corrections [30] suggests that the SM prediction for Z → bb branching ratio (R b ) is 2.4σ smaller than the measured value [17]. On the other hand, the forwardbackward asymmetry of the bottom quark A b F B measured at the Z-pole exhibits a 2.5σ discrepancy with the SM prediction [17]. The two notable discrepancies together prefer a larger Zb RbR coupling compared with the SM value and a Zb LbL coupling very close to the SM value [31].
Our model, by construction, does not introduce any modification to the Zbb coupling at tree level. However, there are corrections to Zb LbL at loop levels, since the custodial symmetry that protects the Zb LbL coupling is only approximately preserved by the new T R , t R and χ R respectively and will induce corrections to the Zb LbL coupling at one-loop level. These corrections are suppressed, either by the large masses of the scalars or due to the vectorlike nature of X, T and χ. 6 We found these corrections to be much smaller than the allowed deviation on the Zb LbL coupling. Another contribution to the Zb LbL coupling comes from the mixing of the top with its vector-like partners. The mixing between t and T is negligible in our model. The correction due to the mixing between t and χ, though suppressed by v 2 /f 2 , could become non-negligible for small f . Nevertheless, to fulfill the experimental constraints on the Zbb coupling, one needs to introduce additional new physics which enhances the Zb RbR coupling. If b R also couples strongly to the new physics, it is possible to arrange it in some representation under the custodial symmetry that gives an significant enhancement to the Zb RbR coupling [19,[33][34][35]. We will not discuss this possibility in this paper.

Mass of the Higgs boson(s)
Using the extremization conditions (requiring the linear terms of h t , h T , h χ to vanish), one can write the dimensionful parameters M 2 Φχ , K 2 and C χχ in the scalar potential in Eq. (3.15) in terms of the VEVs and quartic couplings, The second equation in Eq. (3.24) can be written as which explicitly shows that tan β > 1 as K 2 λ 1 w 2 is positive, and that the custodial symmetric limit tan β → 1 corresponds to K 2 λ 1 w 2 . The constraint on the weak-isospin violation T parameter puts an upper bound on tan β. In Section 4 it will be shown that tan β can not be much larger than 1 if the chiral symmetry breaking scale is close to the weak scale (f ∼ 1 TeV).
Substituting Eq. (3.24) back to the potential in Eq. (3.15), we can write the Higgs mass in terms of the VEVs and quartic couplings, which is the smallest eigenvalue of the 3 × 3 mass-squared matrix of the CP-even neutral scalars (h t , h T , h χ ). It is useful to switch to the basis (h 1 , h 2 , h χ ) with the following rotation where the electroweak VEV is purely associated with h 1 . In this basis, the mass-squared .
One can see that in this basis h 2 is already a mass eigenstate. The 126 GeV Higgs boson, on the other hand, should correspond to the lighter eigenstate of (h 1 , h χ ). At the leading order of v 2 /f 2 , the Higgs mass (M h ) is given by Since λ 2 is not generated by the fermion loops, we expect that |λ 2 /λ 1 | 0.] To obtain the correct top quark mass through the top seesaw mechanism, we need u 2 Eq. (3.29) also shows that the Higgs mass in independent of λ 2 at the leading order.
Combining it with Eq. (3.20) and Eq. (3.22), we obtain where tan β ≡ v t /v T , y t is the SM top Yukawa coupling and m X is the mass of the heavy quark with charge +5/3. As mentioned earlier, for the case of small f which we are interested in, tan β is restricted to be slightly larger than 1. The correct Higgs mass (126 GeV) corresponds to λ h = M 2 h /v 2 ≈ 0.26 at the weak scale. It is typically obtained for For the other CP-even neutral scalars, h 2 is already a mass eigenstate with mass- . Due to the O(5) symmetry, the masses of the heavy doublet CP-even neutral (h 2 ), CP-odd neutral, and charged scalars all have the same mass at the lowest order, which we denote collectively as .
A large K 2 , required if f is small, would imply that these scalars are significantly heavier than the hypercharge +7/6 quarks, beyond any current experimental bounds. The heavier eigenstate of (h 1 , h χ ) is mostly the EW singlet. Its mass-squared is approximately (λ 1 + λ 2 )f 2 which is also much larger than the current bounds.

O(5) breaking from electroweak interactions
where we have assumed that the quartic terms are U (4) R symmetric for simplicity and parameterized the scalar fields as in Eq. (3.10). Assuming that the SU (2) W × U (1) Y gauge interactions are the only O(5) breaking contribution besides the tadpole terms, the parameters ∆M 2 and κ 1(2) , κ 1(2) in Eq. (3.34) are estimated to be and It is straightforward to repeat the analysis in Section 3.2 by including Eq. (3.34).
Additional O(5) breaking effects may exist besides the SU (2) W × U (1) Y gauge interactions. In principle, these effects could break the U (4) L symmetry, but in order to avoid large violation of custodial symmetry, they should at least approximately preserve O(4).
If the U (4) L breaking effects are mainly in the mass term, it effectively causes a shift of the K 2 terms in Eq. (3.15) except for A 2 χ , and results in a splitting between the mass of A 2 χ and the mass of the heavy Higgs doublet. Figure 1: The tree-level diagram which corresponds to the dimensional-six operators of the form λ 2 The thin lines represent Φ χ , the thick line represents the heavy field Σ, and the thick dash lines are the heavy field VEVs Σ (i.e. σ 0 XX , σ 0 T T or σ 0 χt ).

Corrections from heavy scalars masses
In Sec. 3.2 we have only included the lowest order contributions from heavy scalar fields Σ X,T,t , which are the VEVs of σ 0 XX , σ 0 T T and σ 0 χt . We now study the corrections that are proportional to 1/M 2 Σ X,T,t . As long as M 2 Σ X,T,t are large, Σ X,T,t can be integrated out and the dominate contributions come from the dimension-six operators of the form They are generated by the tree-level diagram in Fig. 1, where we use Σ and λ to denote a general heavy field and a general quartic coupling. Replacing the heavy fields with their VEVs, these operators generate quartic couplings of the Φ χ fields that explicitly break O(5) and hence modify the Higgs mass.
With the quartic couplings in Eq. (3.15), we can write down the terms generated by Fig. 1. For simplicity, we assume all the fields in Σ X and Σ T have mass M Σ X,T and all the fields in Σ t has mass M Σt , which is an good approximation for large M Σ X,T,t where the corrections from the tadpoles of σ 0 XX , σ 0 T T and σ 0 χt are negligible. For simplicity, we also ignore the contributions from EW interactions discussed in Section 3.3. (At the lowest order, different contributions add up linearly.) Thus, the leading correction from heavy scalars masses to the scalar potential is (3.38) In the limit λ 2 → 0, the above expression is simplified to Again, it is straightforward to calculate the effects of Eq. (3.38) on the Higgs mass by repeating the analysis in Section 3.2. For simplicity we set λ 2 = 0. Keeping the lowest orders in terms of 1/M 2 Σ X,T and 1/M 2 Σt , we have . The other contribution comes from the VEVs of the other neutral components of Σ X,T,t , which are σ 0 tT , σ 0 χT , σ 0 tt and σ 0 χt in Eq. (2.4). These fields do not have tadpole terms generated by gauge invariant fermion masses. However, once other fields develop VEVs, the quartic couplings will induce VEVs for these fields that are suppressed by 1/M 2 Σ X,T,t . Compared to the leading order corrections in Eq. (3.40) that are proportional to λ 1 f 2 2M 2 , the effects coming from these quartic-coupling-induced VEVs are further suppressed by at least a factor of w 2 /f 2 or v 2 /f 2 . The contribution to S and T parameters from the triplet scalar VEVs are also negligible as long as M 2 Σ X,T is significantly large. We will ignore these effects for simplicity.

Numerical Studies and Phenomenology
In this section, we perform numerical studies of this model to obtain predictions and preferred ranges of the parameters, given the experimental constraints. They serve to verify the approximate analytic results obtained in the previous sections. We also discuss possible phenomenologies at the LHC or future colliders.
We start with an enumeration of the parameters of this model. At energy scale µ Λ, the theory is described by the scalar potential Eq.  [36]. w is related to the mass of the charge +5/3 quark m X by m X = ξw/ √ 2. Hence, the spectrum is fully determined by the following parameters, We choose the ratios of couplings λ 1 /(2ξ 2 ) and λ 2 /λ 1 as the independent parameters because they are more convenient and better constrained. To calculate M h , we match the theory to the SM at the scale of the heavier top partner m t 3 , compute the quartic Higgs coupling λ h , and then evolve λ h down to the weak scale.
Before starting the numerical calculations, we first examine the expected ranges of input parameters listed in Eq. (4.42). The Yukawa coupling ξ is expected to be ∼ 3 − 4 in a strongly coupled theory. We will use ξ ≈ 3.6 as the standard reference value [7].
The ranges of λ 1 /(2ξ 2 ) , λ 2 /λ 1 are discussed in Appendix B and are expected to be 0.35 λ 1 /(2ξ 2 ) 1 and −0.15 λ 2 /λ 1 0. Since the focus of this paper is to reduce the chiral symmetry breaking scale f without violation of experimental constraints, we will consider lower values of f ( 5 TeV). We often take f = 1 TeV as a benchmark point. As we will see later, to obtain a correct Higgs mass f can not be much smaller than 1 TeV. In Section 3, we already saw that tan β > 1 is required for the potential to be at a minimum. 7 For small f , we expect tan β to be not much larger than 1 from the T parameter constraint. For the effective theory below the composite scale Λ to be a valid description, the states in the theory should have masses below Λ ∼ 4πf . Furthermore, for the effective theory at µ Λ described in Section 3 to be a valid description, the heavy scalar masses M Σ X,T,t need to be much larger than f . Thus, we require M ρ 4πf and f M Σ X,T,t 4πf . Finally, the current bound from LHC requires m X > 0.8 TeV.
In this model, we incorporate the custodial symmetry by introducing a vector-like EW doublet (X, T ), in order to reduce the chiral symmetry scale f without introducing large weak isospin violation. We first would like to verify whether this can indeed be achieved. In Fig. 2, we show the Higgs boson mass M h as a function of m X and tan β, by fixing f = 1 TeV and other parameters to some typical values, ξ = 3.6, λ 1 /(2ξ 2 ) = 0.7, For simplicity, we set the heavy scalar masses to be M Σ X,T,t = 10f , a value close to the compositeness scale. We also show the contours of the T parameter calculated using the expressions in Appendix A. The regions −0.06 < T < 0.1 and −0.11 < T < 0.15 roughly correspond to the 68% and 95% CL (fixing S = 0) [17], which are shown on the plots with different colors. We see that, indeed, there is a region for which the T parameter is within the constraint, while a 126 GeV Higgs boson mass can also be obtained. This demonstrates that the chiral symmetry breaking scale can be lowered from multi-TeV in the minimal model [7] to ∼ 1 TeV, which greatly reduces the tuning. In Section 3.2 we argued that with small f , tan β can not be much larger than 1, as otherwise the custodial symmetry is badly broken. This is verified in Fig. 2, as one can see the 68% CL bound of the T parameter requires tan β < 1.4. On the other hand, a small custodial breaking is needed to account for the (t L , b L ) contribution in the SM, which translates into a lower bound on tan β when m X is small.
The Higgs boson mass is sensitive to λ 1 /(2ξ 2 ) and M ρ /f . To study the effects of these two parameters, we choose a point in Fig. 2, m X = 0.9 TeV and tan β = 1.25, then vary λ 1 /(2ξ 2 ) and M ρ /f and plot the Higgs boson mass as a function of these two parameters.
The result is shown in the left panel of Fig. 3. Due to the running effects, the Higgs boson mass-squared does not vary linearly with λ 1 /(2ξ 2 ) as naïvely indicated from Eq. (3.30).
The dependence is somewhat less sensitive. The Higgs mass decreases as one increases M ρ as expected from Eq. (3.37). If M ρ is not too large (M ρ 7f ), its effect can be compensated by different choices of other parameters to obtain the correct Higgs mass.
The Higgs boson mass also receives corrections from the masses of heavy scalars Σ X,T,t .
We repeat the same exercise (choosing the point in Fig. 2  unless m X /f is very small. This suggests that the mass of the heavier top partner, Higgs boson mass and other parameters are fixed. This is different from the predictions of many other composite Higgs models that contain more than one top partners, such as MCHM 5 and MCHM 10 in Ref. [33,37]. In practice, the required ratio m X /f depends on other parameters that affect the Higgs boson mass, such as λ 1 /(2ξ 2 ) and M ρ , which are not known a priori. Nevertheless, for any reasonable set of other parameters, we could find the corresponding value of m X /f to give M h = 126 GeV. In the right panel of Fig. 4, symmetry. For f ∼ 1 TeV, the constraint on the T parameter requires tan β 1.5 (from Fig. 2), which gives K 2 1.2λ 1 w 2 1.7m 2 X so that M H 1.3m X . The CP-even (mostly) singlet scalar has a mass ∼ √ λ 1 f , which is related to the mass of the heavier top partner m t 3 ∼ ξf / √ 2 by the standard NJL relation. We have also assumed that the scalars in Σ X,T,t have masses much larger than f . Therefore, the hypercharge +7/6 quarks (X, T ), being the lightest states in the model and carrying color, will be the first particles to be discovered if this model is realized in nature. Such hypercharge +7/6 quarks (X, T ) are a generic prediction of a composite Higgs model with a low chiral symmetry breaking scale and a custodial symmetry to avoid the T parameter and Zbb coupling constraints. To unravel the underlying theory we would still need to find the other states and study their properties. On the other hand, if the hypercharge +7/6 quarks (X, T ) are excluded up to a few TeV, in our model the chiral symmetry breaking scale would need to be at least as large, making the model as fine-tuned as the minimal model [7], then such an extension will be less motivated. The estimated reach and exclusion regions on these quarks for the 14 and 33 TeV LHC can be found in the Snowmass 2013 report [38], which is ∼ 1. To produce the Higgs boson mass at 126 GeV, we found that f needs to be somewhat larger than the X quark mass. The current LHC bound on X quark mass of 800 GeV renders a lower bound on f of the order of 1 TeV. The tuning, measured by v 2 /f 2 , can be improved to ∼ 5%, compared to 0.5% in the minimal model.
Naturalness does not come without a price. To reduce fine-tuning and to avoid the experimental constraints, we are forced to introduce the X and T quarks and the corresponding composite scalars, making the structure of the theory much more complicated. As a matter of fact, the minimal model in Ref. [7] and the extended model studied in this paper are another example of the so-called "crossroads" situation, and one has to choose between fine-tuning and complexity. Ultimately, both models need to be tested by experiments. The search for the X and T quarks at the 14 TeV (and possibly the 33 TeV) LHC can provide important clues in discriminating the two scenarios. However, to fully probe either model, one needs to go beyond the LHC. There has been discussion of a future 100 TeV Hadron Collider that could be built either at CERN [40] or in China [41].
Such a collider, if realized, will further probe the origin of the EWSB and tell us which road our Mother Nature takes.

Acknowledgments
We would like to thank Bogdan Dobrescu and Ennio Salvioni for discussion. This work is supported by the Department of Energy (DOE) under contract no. DE-FG02-91ER40674.

A T parameter from fermion loops
In Section 3.1 we argued that the leading contribution to the T parameter is captured by the fermion loops. In this appendix, we provide an expression for the T parameter calculated from fermion loops. In terms of SU (2) W eigenstates, the contribution comes from the fermions (t L , b L ), (X L , T L ) and (X R , T R ) [since (X R , T R ) is also a SU (2) W doublet]. The charge +2/3 fermions, t, T and χ form a 3 × 3 mass matrix, as shown in Eq. (3.19).
We denote the three mass eigenstates as t 1 , t 2 and t 3 , ordered by m t 1 ≤ m t 2 ≤ m t 3 , and denote the left-handed and right-handed rotation matrices as The contribution to the T parameter from fermion loops is and quartic couplings λ 1 , λ 2 in Eq. (3.15). It was shown in the previous paper [7] that the ratios of couplings, λ 1 /(2ξ 2 ) and λ 2 /λ 1 , are better estimated than their individual values. At the same time the predictions of the model, such as the mass of the Higgs boson, also have stronger dependences on the ratio of the couplings. This is also true for the model studied in this paper. With the addition of the (X, T ) quarks, the estimated coupling ratios are slightly modified from the minimal model [7], while the derivations remain the same. Here we provide a short summary of the results and refer the reader to the appendix of [7] for more details of this study.
In the fermion bubble approximation, the ratio λ 1 /(2ξ 2 ) is predicted to be 1, while λ 2 is zero since it is not generated by the fermion loops. These results are modified once the gauge loop corrections and the back reaction of the scalar self-interactions are included, for example, by using the full one loop RG equations [16]. If the chiral symmetry breaking scale f is not much smaller than the compositeness scale Λ, as in the case that we are interested, one cannot trust the RG analysis because the couplings are strong and the logarithms are only O(1). Nevertheless, it may provide us some ideas of the possible range of the coupling ratios λ 1 /(2ξ 2 ) and λ 2 /λ 1 .
N f is the number of quark flavors. We solve these equations numerically for our model which has N L = 5, N R = 4, N c = 3 and N f = 9. We set the initial conditions λ 1 = 2ξ 2 , λ 2 = 0 and choose several different initial values for ξ.
The results are shown in Fig. 5. The ratios of couplings are quickly driven to some approximate fixed point values, though we should not trust the exact evolution near Λ due to potentially large higher loop contributions.
If the chiral symmetry breaking scale is not far below the compositeness scale, we can not trust the 1-loop RG results. However, if we assume a smooth evolution, the ratios of couplings are expected to lie in between their initial values and the quasi-infrared fixed point values: We adopt these ranges in Section 3 and 4.