HHiggs as a Top-Mode Pseudo
Hidenori S. Fukano, ∗ Masafumi Kurachi, † Shinya Matsuzaki,
2, 3, ‡ and Koichi Yamawaki § Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI)Nagoya University, Nagoya 464-8602, Japan. Institute for Advanced Research, Nagoya University, Nagoya 464-8602, Japan. Department of Physics, Nagoya University, Nagoya 464-8602, Japan.
In the spirit of the top quark condensation, we propose a model which has a naturally lightcomposite Higgs boson, “tHiggs” ( h t ), to be identified with the 126 GeV Higgs discovered at theLHC. The tHiggs, a bound state of the top quark and its flavor (vector-like) partner, emerges as apseudo Nambu-Goldstone boson (NGB), “Top-Mode Pseudo”, together with the exact NGBs to beabsorbed into the W and Z bosons as well as another (heavier) Top-Mode Pseudo (CP-odd compositescalar, A t ). Those five composite (exact/pseudo) NGBs are dynamically produced simultaneouslyby a single supercritical four-fermion interaction having U (3) × U (1) symmetry which includes theelectroweak symmetry, where the vacuum is aligned by small explicit breaking term so as to breakthe symmetry down to a subgroup, U (2) × U (1) (cid:48) , in a way not to retain the electroweak symmetry,in sharp contrast to the little Higgs models. The explicit breaking term for the vacuum alignmentgives rise to a mass of the tHiggs, which is protected by the symmetry and hence naturally controlledagainst radiative corrections. Realistic top quark mass is easily realized similarly to the top-seesawmechanism by introducing an extra (subcritical) four-fermion coupling which explicitly breaks theresidual U (2) (cid:48) × U (1) (cid:48) symmetry with U (2) (cid:48) being an extra symmetry beside the above U (3) L × U (1).We present a phenomenological Lagrangian of the Top-Mode Pseudos along with the standard modelparticles, which will be useful for the study of the collider phenomenology. The coupling propertyof the tHiggs is shown to be consistent with the currently available data reported from the LHC.Several phenomenological consequences and constraints from experiments are also addressed. I. INTRODUCTION
The ATLAS [1] and CMS collaborations [2] have discovered a new scalar particle at around 126 GeV having theproperties compatible with the Higgs boson in the Standard Model (SM). However, the origin of mass is still amysterious, since we do not yet understand detailed features of the 126 GeV Higgs, in particular the dynamical originof the mass of the 126 GeV Higgs itself which is just a free parameter in the SM.A straightforward way to understand the dynamical origin of the Higgs boson in the explicit underlying theorybeyond the SM is the walking technicolor having approximate scale invariance and large anomalous dimension γ m (cid:39) γ m > γ m (cid:39)
2. Here we propose a variant of the top quark condensate model based on the strong four-fermion couplings,which yields a naturally light Higgs boson to be identified with the 126 GeV Higgs boson at LHC.Actually, among masses of the SM fermions, the top quark mass ( m t (cid:39)
173 GeV) is the only one roughly of theorder of the electroweak symmetry breaking (EWSB) scale ( v EW (cid:39)
246 GeV). Furthermore, the mass of the LHCHiggs boson ( m h (cid:39)
126 GeV) is also roughly of the order of the EWSB scale. This coincidence may imply that thetop quark plays a crucial role for both the EWSB and the generation of the mass of the Higgs boson. In fact, beforethe top quark was discovered with the mass being this large, the top quark condensation (Top-Mode Standard Model;TMSM) was proposed [11, 12] to predict such a close relation among the top quark mass, the EWSB scale and theHiggs mass, based on the phase structure of the gauged Nambu-Jona-Lasinio (NJL) model [17, 18]. The four-fermion ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - ph ] A ug interactions in the TMSM are written in the SM-gauge-invariant form [11, 12]: L f TMSM = G t (¯ q iL t R )(¯ t R q Li ) + G b (¯ q iL b R )(¯ b R q Li ) + G tb (¯ q iL q kR )( iτ ) ij ( iτ ) kl (¯ q jL q lR ) + h.c. , (1)where q L ( R ) = ( t, b ) TL ( R ) and τ is the second component of the Pauli matrices. It is straightforward to extend thisto include all the three generations of the SM fermions [11, 12]. The four-fermion interactions are arranged to triggerthe top quark condensate, (cid:104) ¯ tt (cid:105) (cid:54) = 0, without other condensates such as the bottom condensate, (cid:10) ¯ bb (cid:11) , (cid:104) ¯ tb (cid:105) , · · · = 0, insuch a way that G t > G crit > G b , · · · , (2)where the critical coupling G crit is given as G crit = 4 π / ( N c Λ ), with N c being the number of QCD color and Λ thecutoff scale of the theory, up to small corrections from the SM gauge interactions as implied by the phase structureof the gauged NJL model [17, 18]. The solution of the gap equation indicates that the top quark mass can bemuch smaller than the cutoff scale Λ, m t (cid:28) Λ, by tuning the four-fermion coupling close to the critical coupling,0 < G t /G crit − (cid:28)
1. The TMSM produces three NGBs which are absorbed into the W and Z bosons when theelectroweak gauge interactions are switched on, and predicts the top quark mass to be on the order of the EWSBscale, v EW (cid:39)
246 GeV, through the Pagels-Stokar formula [19] for the decay constant F π (= v EW ) of the NGBs, whichare evaluated with the solution of the gap equation of the gauged NJL model.However, the original TMSM has a few problems: i) Even if we assume the cutoff scale, Λ, is the Planck scale, thetop quark mass is predicted to be m t = 220 −
250 GeV [11, 12, 16], which is somewhat larger than the experimentalvalue m exp t = 173 GeV [20]. If we assume Λ to be a few TeV to avoid excessive fine-tuning to reproduce the EWSBscale, we would face a disastrous situation where the top quark mass is too large: m t ∼
600 GeV (top mass problem);ii) the TMSM predicts a Higgs boson as a t ¯ t bound state (the “top-Higgs boson”, H t ) with mass in a range of m t < m Ht < m t . Such a top-Higgs boson cannot be identified as the Higgs boson with the mass (cid:39)
126 GeV whichwas discovered at the LHC [1, 2] (Higgs mass problem).The top-seesaw model [21–23] can solve the top mass problem. In the top-seesaw model, a new (vector-like) SU (2) L -singlet quark (seesaw partner of the top quark) is introduced to mix with the t R , which pulls the top quark mass downto the desired value (cid:39)
173 GeV, so that the top mass problem is resolved. However, it turns out that the top-Higgsboson is still heavy, m t < m Ht < m t , and therefore the Higgs mass problem still remains in the top-seesaw model.The Higgs mass problem was recently resolved by the top-seesaw assisted technicolor model [24, 25], which is ahybrid version combining the top-seesaw model with technicolor. In this model a light top-Higgs boson with m Ht < m t was realized by sharing the top quark mass with the technicolor sector. It was shown, however, that the couplingproperties of the top-Higgs boson are quite different from those of the Higgs boson in the SM, and the model hascurrently been disfavored by the LHC data, most notably by the results on the diphoton decay channel [1, 2] andproduction cross section through the vector boson fusion process [26, 27].In this paper, in the spirit of the top quark condensation, we propose a model which solves the Higgs massproblem in a natural way, while keeping the solution of the top mass problem by the top-seesaw mechanism. Thelight composite Higgs, what we call “tHiggs”, emerges as one of the composite pseudo NGBs, dubbed “Top-ModePseudos”, associated with the spontaneous breaking of an approximate global symmetry, which is triggered by strong(supercritical) four-fermion interactions.The model is constructed from the third generation quarks in the SM q = ( t, b ) and a (vector-like) χ -quark whichis a flavor partner of the top quark, and has the same SM charges as those of the right-handed top quark. Thefour-fermion interaction term takes the form, L f = G ( ¯ ψ iL χ R )( ¯ χ R ψ iL ) , (3)where ψ iL ≡ ( t L , b L , χ L ) T i , ( i = 1 , , U (3) L × U (1) χ R . For the supercritical setting, G > G crit , the symmetry is spontaneously broken down to U (2) L × U (1) V by thequark condensates (cid:104) ¯ χ R t L (cid:105) (cid:54) = 0 and (cid:104) ¯ χ R χ L (cid:105) (cid:54) = 0, which are realized in the vacuum aligned with the additional explicitbreaking terms mentioned below, while (cid:104) ¯ χ R b L (cid:105) is gauged away to (cid:104) ¯ χ R b L (cid:105) = 0 by the electroweak gauge symmetrywhen it is switched on. Note that the electroweak gauge symmetry is spontaneously broken when U (3) L × U (1) χ R → U (2) L × U (1) V , in sharp contrast to the little Higgs models. It should also be noted that the Lagrangian has U (2) q R symmetry (see below Eq.(47)) of ( t R , b R ) not broken by the condensate, which will not be explicitly mentioned unlessbecomes relevant.Associated with this symmetry breaking, five NGBs emerge as bound states of the quarks. Besides those, acomposite heavy Higgs boson corresponding to the σ mode of the usual NJL model is also formed. Three of theseNGBs will be eaten by the electroweak gauge bosons when the subgroup of the symmetry is gauged by the electroweaksymmetry, while two of them remain as physical states. Those two NGBs, Top-Mode Pseudos, acquire their massesdue to additional terms which explicitly break the U (3) L × U (1) χ R symmetry in such a way that the vacuum alignsto break the electroweak symmetry by (cid:104) ¯ χ R t L (cid:105) (cid:54) = 0. One of them is a CP-even scalar (tHiggs, h t ), which is identifiedas the 126 GeV Higgs boson discovered at the LHC, while the other is a heavy CP-odd scalar ( A t ), which is similarto the CP-odd Higgs in the two Higgs doublet models (there is an essential difference from the two-doublet Higgsmodel, though). We find a notable relation between masses of those Top-Mode Pseudos: m h t = m A t sin θ , (cid:18) tan θ = (cid:104) ¯ χ R t L (cid:105)(cid:104) ¯ χ R χ L (cid:105) (cid:19) , (4)where the angle θ is related to the presence of the condensate, (cid:104) ¯ χ R q L (cid:105) (cid:54) = 0, which causes the electroweak symmetrybreaking.It will be shown that the tHiggs couplings to the SM particles coincide with those of the SM Higgs boson in thelimit sin θ → v EW (cid:39)
246 GeV is kept fixed). Even if the tHiggs coupling coincides with that of the SM Higgs,the virtue of our model is that the tHiggs h t is a bound state of the top quark and χ -quark, and is natural in the sensethat its mass is protected by the symmetry, in sharp contrast to the SM Higgs. One notable feature of our modelis the prediction of the heavy CP-odd Higgs (without additional charged heavy Higgs in contrast to the two-doubletHiggs models), which will be tested in future collider experiments.This paper is organized as follows: In Sec. II, we start with a simplified model based on four-fermion dynamics whichhas the (exact) global symmetry U (3) L × U (1) χ R to show that five NGBs emerge due to the spontaneous breakingof the global symmetry by the quark condensate generated by the supercritical four-fermion dynamics. Additionalexplicit breaking terms are then introduced to give masse to the two of the five NGBs (Top-Mode Pseudos; h t and A t )and also to the top quark. We estimate the mass of the Top-Mode Pseudos based on the current algebra to find themass formula Eq.(4). The interaction property of the tHiggs ( h t ) and the stability of the mass against the radiativecorrections are addressed in comparison with the SM Higgs boson case. The extension of the model to incorporatemasses of light fermions are also discussed in Sec. II. Several phenomenological constraints are given in Sec. III. Sec. IVis devoted to the summary of this paper including some discussions. In appendix. A, we provide a straightforwardderivation of the Top-Mode Pseudo masses by directly solving the bound state problem in the four-fermion dynamicsbased on the auxiliary field method. Appendix. B is devoted to some details of computations for one-loop correctionsto the Top-Mode Pseudos arising from the top and its flavor partner, t (cid:48) -quark loops. II. MODEL
In this section we propose a model based on four-fermion dynamics constructed from the top and bottom quarks q = ( t, b ) with the flavor partner of top quark ( χ ). The model possesses an approximate global symmetry which isspontaneously broken by the quark condensates (cid:104) ¯ χ R q L (cid:105) (cid:54) = 0 and (cid:104) ¯ χ R χ L (cid:105) (cid:54) = 0 generated by the four-fermion dynamics.Five NGBs emerge as bound states of the quarks associated with the spontaneous breaking of the global symmetry,in addition to a composite heavy Higgs boson which corresponds to the σ mode in the usual NJL model. Two ofthe NGBs (what we call Top-Mode Pseudos ( h t and A t )) obtain their masses due to the introduction of additionalfour-fermion interactions which explicitly break the global symmetry, while other three remain massless to be eatenby the W and Z bosons once the electroweak charges are turned on. The mass of h t turns out to be protected by theglobal symmetry and the coupling property is shown to be consistent with the currently reported Higgs boson withmass around (cid:39)
126 GeV. We will call h t tHiggs.The present model, which is based on the strong four-fermion interactions with vector-like χ -quark, can actuallybe viewed as a version of so-called top-seesaw model [21, 22]. The crucial difference between existing top-seesawmodels and the present model is that the present model makes it clear that the 126 GeV Higgs exists as a pseudoNGB associated with the global symmetry breaking caused by four-fermion interactions. For the purpose of makingthis point clearer, in subsection II A, we introduce a simplified model in which all the explicit breaking terms areturned off. In that simplified model, five NGBs which exist in the model are all massless. Then, in subsection II B,we introduce explicit breaking terms into the Lagrangian to give masses to two of NGBs, Top Mode Pseudos ( h t , A t ), which are identified as 126 GeV Higgs boson and its CP-odd partner. Subsections II C and II D are devoted toexplaining fermion masses, Yukawa interactions as well as the nature of the tHiggs. A. Structure of symmetry breaking
Let us consider an NJL-like model constructed from the third generation quarks in the SM, q = ( t, b ), and an SU (2) L singlet quark ( χ ). The left-handed quarks q L and χ L form a flavor triplet ψ iL ≡ ( t L , b L , χ L ) T i ( i = 1 , , U (3) ψ L group, while the right-handed top and bottom quarks q iR ≡ ( t R , b R ) T i ( i = 1 ,
2) and χ R area doublet and singlet under the U (2) q R group, respectively. Turning off the SM gauge interactions momentarily, wethus write the Lagrangian having the global U (3) ψ L × U (2) q R × U (1) χ R symmetry: L kin . + L f = ¯ ψ L iγ µ ∂ µ ψ L + ¯ q R iγ µ ∂ µ q R + ¯ χ R iγ µ ∂ µ χ R + G ( ¯ ψ iL χ R )( ¯ χ R ψ iL ) , (5)where G denotes the four-fermion coupling strength.We can derive the gap equations for fermion dynamical masses m tχ and m χχ through the mean field relations m tχ = − G (cid:104) ¯ χ R t L (cid:105) and m χχ = − G (cid:104) ¯ χ R χ L (cid:105) in the large N c limit: m tχ = m tχ N c G π (cid:20) Λ − (cid:0) m tχ + m χχ (cid:1) ln Λ m tχ + m χχ (cid:21) , (6) m χχ = m χχ N c G π (cid:20) Λ − (cid:0) m tχ + m χχ (cid:1) ln Λ m tχ + m χχ (cid:21) , (7)where Λ stands for the cutoff of the model which is to be of Λ (cid:29) O (1) TeV. There exist nontrivial solutions m tχ (cid:54) = 0and m χχ (cid:54) = 0 when the following criticality condition is satisfied: G > G crit = 8 π N c Λ , (8)under which we have the nonzero dynamical masses as well as the nonzero condensates, (cid:104) ¯ χ R q L (cid:105) (cid:54) = 0 , (cid:104) ¯ χ R χ L (cid:105) (cid:54) = 0 . (9)Note, however, that the two gap equations, Eqs.(6) and (7), cannot determine the ratio of two condensates: Thosetwo gap equations with the criticality condition in Eq.(8) just lead to the nontrivial solution for the squared-sum oftwo masses, ( m tχ + m χχ ) (cid:54) = 0, so that the vacuum with m tχ (cid:54) = 0 is degenerate with that with m tχ = 0. In order tolift the degeneracy for breaking the electroweak symmetry by m tχ (cid:54) = 0, we shall later introduce explicit breaking toalign the vacuum, which also gives rise to the mass of two NGBs (Top-Mode Pseudos) out of five NGBs, with therest three being exact NGBs to be absorbed into the W and Z bosons. Also note that the condensate m bχ (cid:54) = 0 canbe gauged away when the model is gauged by the electroweak symmetry.In order to make the structure of the symmetry breaking clearer, we may change the flavor basis of fermions byintroducing an orthogonal rotation matrix R :˜ ψ L = ˜ t L ˜ b L ˜ χ L ≡ R · ψ L , R = cos θ − sin θ θ θ , tan θ ≡ m tχ m χχ = (cid:104) ¯ χ R t L (cid:105)(cid:104) ¯ χ R χ L (cid:105) . (10)The two gap equations, Eqs.(6) and (7), are then reduced to a single gap equation,1 = N c G π (cid:34) Λ − m χχ ln Λ m χχ (cid:35) , (11)with m χχ ≡ m tχ + m χχ (cid:54) = 0 . (12)Accordingly, the associated two condensates in Eq.(9) are reduced to a single nonzero condensate on the basis of ˜ ψ L : (cid:104) ¯ χ R ˜ χ L (cid:105) (cid:54) = 0 . (13)We thus see that, with the criticality condition in Eq.(8) satisfied, the four-fermion dynamics triggers the followingglobal symmetry breaking pattern: U (3) ˜ ψ L × U (1) χ R → U (2) ˜ q L × U (1) V =˜ χ L + χ R . (14) We have put m bχ = 0, by gauging it away by the electroweak gauge symmetry. Otherwise ( m tχ + m χχ ) in Eqs.(6) and (7) should read( m tχ + m bχ + m χχ ) because of U (3) ψ L symmetry, with m bχ also satisfying the same type of gap equation. The broken currents associated with this symmetry breaking are found to be J ,µ L = ¯˜ ψ L γ µ λ ˜ ψ L = ¯˜ t L γ µ ˜ χ L + ¯˜ χ L γ µ ˜ t L = (¯ t L γ µ t L + ¯ χ L γ µ χ L ) sin 2 θ + (¯ t L γ µ χ L + ¯ χ L γ µ t L ) cos 2 θ , (15) J ,µ L = ¯˜ ψ L γ µ λ ˜ ψ L = i (cid:104) − ¯˜ t L γ µ ˜ χ L + ¯˜ χ L γ µ ˜ t L (cid:105) = − i (¯ t L γ µ χ L − ¯ χ L γ µ t L ) , (16) J ,µ L = ¯˜ ψ L γ µ λ ˜ ψ L = ¯˜ b L γ µ ˜ χ L + ¯˜ χ L γ µ ˜ b L = (cid:0) ¯ b L γ µ t L + ¯ t L γ µ b L (cid:1) sin θ + (cid:0) ¯ b L γ µ χ L + ¯ χ L γ µ b L (cid:1) cos θ , (17) J ,µ L = ¯˜ ψ L γ µ λ ˜ ψ L = i (cid:104) − ¯˜ b L γ µ ˜ χ L + ¯˜ χ L γ µ ˜ b L (cid:105) = − i (cid:0) ¯ b L γ µ t L − ¯ t L γ µ b L (cid:1) sin θ − i (cid:0) ¯ b L γ µ χ L − ¯ χ L γ µ b L (cid:1) cos θ , (18)and J µA ≡ (cid:18) J µ R − √ J ,µ L + 1 √ J ,µ L (cid:19) = 14 ( ¯ χ R γ µ χ R − ¯˜ χ L γ µ ˜ χ L )= 14 (cid:2) ¯ χ R γ µ χ R − ¯ t L γ µ t L sin θ − ¯ χ L γ µ χ L cos θ − (¯ t L γ µ χ L + ¯ χ L γ µ t L ) sin θ cos θ (cid:3) , (19)where the Gell-Mann matrices λ a ( a = 1 , · · · ,
8) are normalized as tr[ λ a λ b ] = 2 δ ab , and λ = (cid:112) / × . Theassociated NGBs emerge with the decay constant f as (cid:10) (cid:12)(cid:12) J aµ ( x ) (cid:12)(cid:12) π bt ( p ) (cid:11) = − if δ ab p µ e − ip · x , a, b = 4 , , , , A . (20)The decay constant f is calculated through the Pagels-Stokar formula [19]: f = N c π m χχ ln Λ m χχ . (21)The five NGBs ( π at ) can be expressed as composite fields (interpolating fields) made of the fermion bilinears on thebasis of ( ˜ ψ L , χ R ) or ( ψ L , χ R ): π t ∼ ¯ χ R ˜ t L − ¯˜ t L χ R = ( ¯ χ R t L − ¯ t L χ R ) cos θ − ( ¯ χ R χ L − ¯ χ L χ R ) sin θ ,π t ∼ − i (cid:16) ¯ χ R ˜ t L + ¯˜ t L χ R (cid:17) = − i ( ¯ χ R t L + ¯ t L χ R ) cos θ + i ( ¯ χ R χ L + ¯ χ L χ R ) sin θ ,π t + iπ t ∼ (cid:16) ¯ χ R ˜ b L − ¯˜ b L χ R (cid:17) + (cid:16) ¯ χ R ˜ b L + ¯˜ b L χ R (cid:17) = 2 ¯ χ R b L ,π t − iπ t ∼ (cid:16) ¯ χ R ˜ b L − ¯˜ b L χ R (cid:17) − (cid:16) ¯ χ R ˜ b L + ¯˜ b L χ R (cid:17) = − b L χ R ,π At ∼ ¯ χ R ˜ χ L − ¯˜ χ L χ R = ( ¯ χ R t L − ¯ t L χ R ) sin θ + ( ¯ χ R χ L − ¯ χ L χ R ) cos θ . Besides these composite NGBs, there exists a composite scalar ( H t ) corresponding to the σ mode in the usual NJLmodel, H t ∼ ¯ χ R ˜ χ L + ¯˜ χ L χ R = ( ¯ χ R t L + ¯ t L χ R ) sin θ + ( ¯ χ R χ L + ¯ χ L χ R ) cos θ , with the mass m H t = 4 m χχ = 4( m tχ + m χχ ) . (22)The H t will be regarded as a heavy Higgs boson with the mass of O (1) TeV, not the light Higgs boson at around 126GeV.With the electroweak gauge interactions turned on, the W and Z bosons turn out to couple to the broken currents( J ,µ L ∓ iJ µ , L ) and ( J µ , L cos θ + J µ ,A L sin θ ), respectively. The corresponding would-be NGBs ( w ± t , z t ) eaten by theelectroweak gauge bosons are then found to be z t ≡ π t cos θ + π At sin θ ∼ ¯ χ R t L − ¯ t L χ R ,w − t ≡ √ π t + iπ t ) ∼ √ χ R b L ,w + t ≡ √ π t − iπ t ) ∼ −√ b L χ R . On the other hand, the following two NGBs remain as physical states: h t ≡ π t ∼ − i (cid:16) ¯ χ R ˜ t L + ¯˜ t L χ R (cid:17) = − i ( ¯ χ R t L + ¯ t L χ R ) cos θ + i ( ¯ χ R χ L + ¯ χ L χ R ) sin θ ,A t ≡ − π t sin θ + π At cos θ ∼ ¯ χ R χ L − ¯ χ L χ R . The correspondence between the broken currents and NGBs along with the CP transformation property is summarizedin Table.I. The two massless NGBs ( h t , A t ) will become pseudo NGBs, called “Top-Mode Pseudos”, obtaining theirmasses once explicit breaking effects are introduced (see Sec. II B). We will identify the CP-even Top-Mode Pseudo, h t , as the 126 GeV Higgs, called tHiggs. Broken current corresponding NGB CP-property J ,µ L π t = z t cos θ − A t sin θ odd J ,µ L π t = h t even J ,µ L ± iJ ,µ L π t ± iπ t = √ w ∓ t – J µA π At = z t sin θ + A t cos θ oddTABLE I: The list of the broken currents and NGBs associated with the spontaneous symmetry breaking in Eq.(14). w ± t and z t are eaten by the W ± and Z bosons once the electroweak gauges are turned on, while the remaining two A t and h t becomepseudo NGBs (Top-Mode Pseudos) by explicit breaking effects (see Eqs.(40) and (41)). We may integrate out the heavy Higgs boson H t (with mass of O (1) TeV) to construct the low-energy effectivetheory governed by the five NGBs ( π at ) described by a nonlinear sigma model based on the coset space, GH = U (3) ˜ ψ L × U (1) χ R U (2) ˜ ψ L × U (1) V = χ L + χ R . (23)For this purpose we introduce representatives ( ξ L,R ) of the G / H which are parameterized by NGB fields as ξ L = exp (cid:34) − if (cid:32) (cid:88) a =4 , , , π at λ a + π At √ λ A (cid:33)(cid:35) , ξ R = exp (cid:20) if π At √ λ A (cid:21) , where λ A = √ . We further introduce the “chiral” field U , U = ξ † L · Σ · ξ R with Σ = 1 √ λ A . (24)The transformation properties of ξ L,R and U under G are given by ξ L → h ( π t , ˜ g ) · ξ L · g † ˜3 L , ξ R → h ( π t , ˜ g ) · ξ R · g † R , U → g ˜3 L · U · g † R , (25)where ˜ g = { g ˜3 L , g R } , g ˜3 L ∈ U (3) ˜ ψ L , g R ∈ U (1) χ R and h ( π t , ˜ g ) ∈ H . Thus we find the G -invariant Lagrangianwritten in terms of the NGBs to the lowest order of derivatives of O ( p ): L NL σ M = f (cid:2) ∂ µ U † ∂ µ U (cid:3) . (26)When the electroweak symmetry turned on, the covariant derivative acting on U is given by D µ U ≡ R ∂ µ − ig (cid:88) a =1 W aµ τ a /
00 0 0 + ig (cid:48) B µ / / R T · U , (27)where W µ and B µ are the SU (2) L and U (1) Y gauge boson fields with the gauge couplings g and g (cid:48) . Then theLagrangian Eq.(26) is changed to the covariant form: L NL σ M = f (cid:2) D µ U † D µ U (cid:3) . (28)From this one can read off the W an Z boson masses as m W = 14 g f sin θ , m Z = 14 ( g + g (cid:48) ) f sin θ , which lead to v EW = f sin θ = N c π m tχ ln Λ m tχ + m χχ (cid:39) (246 GeV) , (29)where use has been made of Eq.(21). Thus imposing the EWSB scale v EW gives a nontrivial relation between m tχ and m χχ .Note that switching on the electroweak gauge interaction explicitly breaks the U (3) ˜ ψ L × U (1) χ R symmetry. Suchexplicit breaking effects would generate masses to the NGBs at the loop level, as a part of the 1 /N c sub-leading effect,which are, however, negligibly small since the size of effects is suppressed by the small electroweak gauge coupling, aswill be discussed later (see the discussion below Eq.(80)).As we mentioned earlier the criticality G > G crit implies (cid:104) ¯ χ R t L (cid:105) + (cid:104) ¯ χ R χ L (cid:105) (cid:54) = 0, but not necessarily (cid:104) ¯ χ R t L (cid:105) (cid:54) = 0which is responsible for the electroweak symmetry breaking. The electroweak gauge interaction itself can contributeto lifting the degeneracy between the vacuum with (cid:104) ¯ χ R t L (cid:105) = 0 and that with (cid:104) ¯ χ R t L (cid:105) (cid:54) = 0 in principle by some extremefine tuning of the critical coupling of the gauged NJL model [17, 18].More natural way will be to introduce extra effective four-fermion interactions to explicitly break the U (3) ˜ ψ L × U (1) χ R symmetry, which can align the vacuum to have (cid:104) ¯ χ R t L (cid:105) (cid:54) = 0 and simultaneously give the mass of the tHiggson the right amount. Such an explicit breaking may be induced from a strong U (1) gauge interaction distinguishing (cid:104) ¯ χ R t L (cid:105) from (cid:104) ¯ χ R χ L (cid:105) . This we perform in the next subsection. B. Top-Mode Pseudos
Here we incorporate explicit breaking terms into the Lagrangian Eq.(5) to give masses to the Top-Mode Pseudos( h t , A t ): L kin . + L f + L h , (30)where L h = − [∆ χχ ¯ χ R χ L + h.c.] − G (cid:48) ( ¯ χ L χ R ) ( ¯ χ R χ L ) . (31)Similarly to Eqs.(6) and (7), we derive the gap equations for fermion dynamical masses m tχ and m χχ : m tχ = m tχ N c G π (cid:34) Λ − m χχ ln Λ m χχ (cid:35) , (32) m χχ = ∆ χχ + m χχ N c ( G − G (cid:48) )8 π (cid:34) Λ − m χχ ln Λ m χχ (cid:35) , (33)where m χχ = m tχ + m χχ . Note that the nonzero ∆ χχ and G (cid:48) allow to determine the ratio of two dynamical masses m tχ and m χχ , i.e. tan θ = m tχ /m χχ , in contrast to the previous gap equations, Eqs.(6) and (7), which only determinethe squared-sum of two, m tχ + m χχ .Furthermore, it turns out that these ∆ χχ and G (cid:48) terms do not affect the criticality of the four-fermion dynamics atall: Assuming m tχ (cid:54) = 0 and m χχ (cid:54) = 0, we find that the following relation is required so as to keep the self consistencyin the gap equations: ∆ χχ = G (cid:48) G m χχ = − G (cid:48) (cid:104) ¯ χ R χ L (cid:105) . (34)By taking this into account, the two gap equations, Eqs.(32) and (33), are reduced to a single one,1 = N c G π (cid:34) Λ − m χχ ln Λ m χχ (cid:35) . (35)Thus the presence of the nontrivial solution is controlled solely by G > G crit , which is the same criticality conditionas in Eq.(8). The Lagrangian Eq.(31) is also rewritten as L h = − G (cid:48) ( ¯ χ L χ R − (cid:104) ¯ χ L χ R (cid:105) ) ( ¯ χ R χ L − (cid:104) ¯ χ R χ L (cid:105) ) . (36)This implies that the explicit breaking effects can also be expressed only by the four-fermion interaction. We can thusapproximately keep the global U (3) ˜ ψ L × U (1) χ R invariance by taking G (cid:48) perturbatively to be0 < G (cid:48) G (cid:28) G crit < G . (37)We have explicitly checked that in the presence of this explicit breaking the vacuum with m tχ (cid:54) = 0 is preferred to thatwith m tχ = 0 in the phenomenologically interesting parameter space to be discussed later.As in Sec. II A, the global U (3) ˜ ψ L × U (1) χ R symmetry is spontaneously broken down to U (2) ˜ ψ L × U (1) V =˜ χ L + χ R due to the four-fermion interaction in L f , resulting in the presence of five NGBs π a ( a = 4 , , , , A ). Then theexplicit breaking terms in L h force the vacuum to choose a specific direction, (cid:104) ¯ χ R ˜ χ L (cid:105) (cid:54) = 0, and give masses to someof the NGBs. Note that the L h term is invariant under the chiral transformation associated with the broken currents( J ,µ L ± iJ ,µ L ) and ( J ,µ L cos θ + J A ,µ L sin θ ), but not for J µ L and ( − J ,µ L sin θ + J A ,µ L cos θ ). Hence L h term givesmasses only to the NGBs associate with latter two, i.e., the Top-Mode Pseudos A t and h t .Estimation of the masses of the pseudo NGBs can be done by the traditional approach based on the currentalgebra [28]: m ab = 1 f (cid:10) (cid:12)(cid:12)(cid:2) iQ a , (cid:2) iQ b , −L h (cid:3)(cid:3)(cid:12)(cid:12) (cid:11) , (38)where Q a is the Noether’s charge associated with the broken currents Eqs.(15), (16), (17), (18) and (19), and f isgiven by Eq.(21). From Eq.(38), together with the gap equations, Eqs.(32), (33) and (34), we obtain m z t = m w ± t = 0 , (39) m A t = 2 (cid:104) ¯ χ R ˜ χ L (cid:105) (cid:104) ¯ χ R χ L (cid:105) f cos θ (cid:39) G (cid:48) G × m tχ + m χχ ) f , (40) m h t = 2 (cid:104) ¯ χ R ˜ χ L (cid:105) (cid:104) ¯ χ R χ L (cid:105) f cos θ × sin θ = m A t sin θ , (41)where the second equation of Eq.(40) is obtained by expanding in terms of G (cid:48) /G (cid:28) h t is proportional to m tχ associated with the EWSB scale v EW as in Eq.(29), just like the case ofthe SM Higgs boson, while the mass of A t is not. We may set the mass of h t to (cid:39)
126 GeV; m h t = m A t sin θ (cid:39)
126 GeV . (42)Note also that G (cid:48) > m h t > m A t >
0. In Appendix. A, we present an alternative derivation of the pseudo NGBs masses and theheavy Higgs mass m H t based on the approach used in [16].As was done in Eq.(28), we may construct a nonlinear Lagrangian valid for scales below m H t described by thefive NGBs based on the coset space in Eq.(23) including the explicit breaking effect from the G (cid:48) - and ∆ χχ -terms inEq.(31). To this end, we introduce the spurion fields χ and χ to write the O ( p ) potential terms corresponding toEq.(31): ∆ L NL σ M = f tr (cid:104) c ( R T U ) † χ ( R T U ) + c (cid:16) χ † ( R T U ) + ( R T U ) † χ (cid:17)(cid:105) , (43)where χ and χ transform under the G -symmetry as χ → g ˜3 L · χ · g † ˜3 L , χ → g ˜3 L · χ · g † R . (44)The G -symmetry is explicitly broken when the spurion fields acquire the vacuum expectation values, (cid:104) χ (cid:105) = (cid:104) χ (cid:105) = Σ , (45)so that the c and c terms break the G down to U (2) q L × U (1) χ L × U (1) χ R and U (2) q L × U (1) V = χ R + χ L , respectively,in the same way as the G (cid:48) and ∆ χχ terms in Eq.(31) do. Matching to the tHiggs mass formula in Eq.(41), we thenfind the coefficients c and c to be c = − m A t , c = 12 m A t cos θ . C. Fermion masses and Yukawa couplings
1. top and t (cid:48) -quark Let us consider the top quark mass based on the Lagrangian Eq.(30). After the spontaneous symmetry breakingby the nontrivial solutions of the gap equations, Eqs.(32), (33) and (34), the mass terms of the top quark t and itsflavor partner χ look like L kin . + L f + L h (cid:12)(cid:12) mass = − (cid:0) ¯ t L ¯ χ L (cid:1) (cid:18) m tχ m χχ (cid:19) (cid:18) t R χ R (cid:19) + h.c. . (46)From this we find the fermion mass eigenvalues as m t (cid:48) = m χχ + m tχ , m t = 0 , (47)0where the mass eigenstates t (cid:48) and t are given by t (cid:48) L = ˜ χ L = t L sin θ + χ L cos θ , t L = ˜ t L = t L cos θ − χ L sin θ . Thus the top quark does not “feel” the EWSB by sin θ (cid:54) = 0 ( m tχ (cid:54) = 0) and is still massless. This is essentially due tothe residual symmetry, U (2) q R × U (1) V =˜ χ L + χ R ( U (2) q R : t R ↔ b R ), which forbids the couplings between t R and χ L in the Lagrangian Eq.(30), hence no mass term for ¯ χ L t R in Eq.(46).To make the model more realistic, we introduce a four-fermion interaction term which breaks the residual symmetryso as to allow t R to couple to χ L : L kin . + L f + L h + L t , (48)where L t = G (cid:48)(cid:48) ( ¯ χ L χ R ) (¯ t R χ L ) + h.c. . (49)As was done in Eq.(37), we also treat the G (cid:48)(cid:48) coupling to be perturbative,0 < G (cid:48)(cid:48) G < , (50)so that the symmetry breaking pattern G / H = [ U (3) ˜ ψ L × U (1) χ R ] / [ U (2) ˜ ψ L × U (1) V ] is not destroyed (the vacuumaligned to this manifold): (cid:104) ¯ t R χ L (cid:105) G (cid:48)(cid:48) =0 = (cid:104) ¯ t R t L (cid:105) G (cid:48)(cid:48) =0 = 0: The Dashen formula Eq.(38) with the G (cid:48)(cid:48) term in Eq.(49)leads to m h t (cid:12)(cid:12)(cid:12) G (cid:48)(cid:48) = 1 f (cid:10) (cid:12)(cid:12)(cid:2) iQ , (cid:2) iQ , −L t (cid:3)(cid:3)(cid:12)(cid:12) (cid:11) = G (cid:48)(cid:48) f (cid:2) (cos θ − sin θ ) (cid:104) ¯˜ χ L t R (cid:105) (cid:104) ¯ t R ˜ χ L (cid:105) + 2 sin θ cos θ (cid:104) ¯˜ χ L χ R (cid:105) (cid:10) ¯ t R ˜ t L (cid:11)(cid:3) (cid:39) G (cid:48)(cid:48) [ (cid:104) ¯ χ R χ L (cid:105) G (cid:48)(cid:48) =0 (cid:104) ¯ t R χ L (cid:105) G (cid:48)(cid:48) =0 − (cid:104) ¯ χ R t L (cid:105) G (cid:48)(cid:48) =0 (cid:104) ¯ t R t L (cid:105) G (cid:48)(cid:48) =0 ] + O (( G (cid:48)(cid:48) ) )= 0 + O (( G (cid:48)(cid:48) ) ) , (51)similarly for the mass of the CP-odd Top-Mode Pseudo, A t . Here we have granted the vacuum saturation validat the leading order of 1 /N c . Thus the Top-Mode Pseudo’s masses are stable against the leading order correctionof the explicit-breaking G (cid:48)(cid:48) term as dictated by the Dashen formula Eq.(51). The next-to leading order in the G (cid:48)(cid:48) -perturbation, i.e., ( G (cid:48)(cid:48) /G ) corrections (which are also of the leading order of 1 /N c ), will affect the masses, as will bediscussed later (see around Eq.(79)).After the spontaneous symmetry breaking by (cid:104) ¯ χ R ˜ χ L (cid:105) (cid:54) = 0, i.e. m χχ = m tχ + m χχ (cid:54) = 0, we thus have the fermionmass matrix, L kin . + L f + L h + L t (cid:12)(cid:12) mass = − (cid:0) ¯ t L ¯ χ L (cid:1) (cid:18) m tχ µ χt m χχ (cid:19) (cid:18) t R χ R (cid:19) + h.c. , (52)where µ χt = − G (cid:48)(cid:48) (cid:104) ¯ χ R χ L (cid:105) is a dynamical mass coming from L t in Eq.(48). Note that the fermion mass matrixis identical to that discussed in top-seesaw models [21, 22]. The top quark and t (cid:48) -quark masses are given as theeigenvalues of the mass matrix in Eq.(52), m t (cid:48) = m tχ + m χχ + µ χt (cid:118)(cid:117)(cid:117)(cid:116) − m tχ µ χt (cid:0) m tχ + m χχ + µ χt (cid:1) , (53) m t = m tχ + m χχ + µ χt − (cid:118)(cid:117)(cid:117)(cid:116) − m tχ µ χt (cid:0) m tχ + m χχ + µ χt (cid:1) . (54)Now the top quark mass becomes nonzero and is proportional to m tχ which breaks the electroweak symmetry, similarlyto the mass of h t in Eq.(41), while the t (cid:48) -quark mass is not. The corresponding mass eigenstates ( t, t (cid:48) ) Tm are relatedto the gauge (current) eigenstates ( t, χ ) Tg by the orthogonal rotation keeping m t , m t (cid:48) ≥ (cid:32) t L t (cid:48) L (cid:33) m = (cid:32) c tL − s tL s tL c tL (cid:33) (cid:32) t L χ L (cid:33) g = O L · (cid:32) t L χ L (cid:33) g , (cid:32) t R t (cid:48) R (cid:33) m = (cid:32) − c tR s tR s tR c tR (cid:33) (cid:32) t R χ R (cid:33) g = O R · (cid:32) t R χ R (cid:33) g , (55)1where c tL ( R ) ≡ cos θ tL ( R ) and s tL ( R ) ≡ sin θ tL ( R ) which are given up to O (( G (cid:48)(cid:48) /G ) ) as c tL = 1 √ (cid:34) m χχ − m tχ + µ χt m t (cid:48) − m t (cid:35) / (cid:39) cos θ (cid:34) (cid:18) G (cid:48)(cid:48) G (cid:19) cos θ sin θ (cid:35) , (56) s tL = 1 √ (cid:34) − m χχ − m tχ + µ χt m t (cid:48) − m t (cid:35) / (cid:39) sin θ (cid:34) − (cid:18) G (cid:48)(cid:48) G (cid:19) cos θ (cid:35) , (57) c tR = 1 √ (cid:34) m χχ + m tχ − µ χt m t (cid:48) − m t (cid:35) / (cid:39) − (cid:18) G (cid:48)(cid:48) G (cid:19) cos θ , (58) s tR = 1 √ (cid:34) − m χχ + m tχ − µ χt m t (cid:48) − m t (cid:35) / (cid:39) G (cid:48)(cid:48) G cos θ . (59)Including the effect of the explicit breaking G (cid:48)(cid:48) term, we may thus add the fermion sector to the nonlinear Lagrangianconstructed from Eqs.(28) and (43), L t,t (cid:48) yuk . = − f √ (cid:2) y ¯ ψ L ( R T U ) ψ R + y χt ¯ ψ L ( χ R T U χ ) ψ R + h.c. (cid:3) , (60)where ψ R = ( t R , b R , χ R ) T . The spurion fields χ and χ have been introduced in Eq.(60), which transform as χ → g L · χ · g † L , χ → g R · χ · g † R , (61)so that the Lagrangian Eq.(60) is invariant under the G -symmetry, U (3) ˜ ψ L × U (1) χ R , and U (2) q R symmetry. Thesesymmetries are explicitly broken by the vacuum expectation values of the spurion fields, (cid:104) χ (cid:105) = Σ , (cid:104) χ (cid:105) = λ , (62)in which the (cid:104) χ (cid:105) breaks the U (3) ψ L symmetry down to U (2) ψ L × U (1) χ L and the (cid:104) χ (cid:105) does the U (2) q R × U (1) χ R down to U (1) χ R = t R . We thus see that the first term in Eq.(60) corresponds to the G -four fermion term in Eq.(5),which is invariant under the G -symmetry, U (3) ˜ ψ L × U (1) χ R and U (2) q R symmetry, while the second term does tothe G (cid:48)(cid:48) -four fermion term, which explicitly breaks the G and U (2) q R down to U (2) q L × U (1) χ L and U (1) t R = χ R . TheYukawa couplings y and y χt can be fixed by matching to the fermion mass matrix in Eq.(52) as y = 2( m tχ + m χχ ) f , y χt = y (cid:18) G (cid:48)(cid:48) G (cid:19) = 2 µ χt f cos θ . (63)
2. Fermions other than top and t (cid:48) -quark In order to give masses to SM fermions other than the top quark, inspired by [11, 12], we may add the followingfour-fermion interactions to the Lagrangian Eq.(48): L others = (cid:88) α =1 , G αtu (¯ q iL χ R )(¯ u αR q α,iL ) + (cid:88) α =1 , , G αtd (¯ q iL χ R )( iτ ) ij (¯ q α,jL d αR ) + (cid:88) α =1 , , G αte (¯ q iL χ R )( iτ ) ij (¯ l α,jL e αR ) + h.c , (64)which are SM gauge invariant, where α = 1 , , i, j = 1 , (cid:104) ¯ χ R t L (cid:105) (cid:54) = 0 ( m tχ (cid:54) = 0) breaking the electroweak gauge symmetry, theseSM fermions acquire their masses as m u α = − G αtu (cid:104) ¯ χ R t L (cid:105) , m d α = − G αtd (cid:104) ¯ χ R t L (cid:105) , m e α = − G αte (cid:104) ¯ χ R t L (cid:105) . Since the NGBs arise from ¯ χ R ψ L as ¯ χ R q iL ∼ (cid:104) ¯ χ R ˜ χ L (cid:105) [ R T U ] i , (65)2one can read off the fermion couplings to h t : L othersyuk . (cid:12)(cid:12) h t = − cos θ (cid:34) (cid:88) α =1 , m u α v EW h t ¯ u α u α + (cid:88) α =1 , , m d α v EW h t ¯ d α d α + (cid:88) α =1 , , m e α v EW h t ¯ e α e α (cid:35) . (66)A full set of the particle content in the present model with the SM charge assignment free from the gauge anomaly islisted in Table. II. field SU (3) c SU (2) L U (1) Y q L = (cid:18) t L b L (cid:19) t R b R l L = (cid:18) ν τL τ L (cid:19) τ R χ L χ R D. tHiggs
We here discuss the coupling property of the tHiggs h t and the stability of the mass against radiative corrections.From Eqs.(28), (43), (60) and (66), we find the h t couplings to the SM particles, L NL σ M + ∆ L NL σ M + L t,t (cid:48) yuk . + L othersyuk . (cid:12)(cid:12)(cid:12) h t = g hV V v EW (cid:18) g h t W + µ W − µ + g + g (cid:48) h t Z µ Z µ (cid:19) − g hhh m h t v EW (cid:0) h t (cid:1) − g hhhh m h t v EW (cid:0) h t (cid:1) − g htt m t v EW h t ¯ tt − g hbb m b v EW h t ¯ bb − g hττ m τ v EW h t ¯ τ τ + · · · , (67)where g hV V = g hhh = g hbb = g hττ = cos θ , (68) g hhhh = 1 −
73 sin θ , (69) g htt = v EW m t y √ (cid:20) ( c tL cos θ + s tL sin θ ) s tR − s tL c tR sin θ (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) = 2 cos θ − θ + O (cid:32)(cid:18) G (cid:48)(cid:48) G (cid:19) (cid:33) . (70)From these, we see that the h t couplings to the W, Z bosons and to the SM fermions become the same as the SMHiggs ones when we take the limit cos θ →
1, i.e., g hV V = g hhh = g hhhh = g htt = g hbb = g hττ = g SM (= 1) , Those Yukawa interactions would give quadratically divergent corrections to the h t mass, which are, however, small enough due to thesmall Yukawa coupling for the lighter fermions, to be negligible compared to the terms in Eq.(79) arising from the Top-Mode Pseudos, t and t (cid:48) -loops. θ = m tχ (cid:113) m tχ + m χχ = v EW f → , by f → ∞ with v EW = 246 GeV fixed . (71)Actually, this limit turns out to be favored by the current experiments as will be discussed in Sec. III A.From Eqs.(28), (43), (60) and (66), we can also evaluate the quadratic divergent corrections to the h t mass at theone-loop level. The one-loop corrections of the leading order of the explicit breaking parameters G (cid:48) /G , G (cid:48)(cid:48) /G and α em = e / (4 π ) with e being the electromagnetic coupling, can be evaluated at the one-loop level of the nonlinearsigma model constructed from terms in Eqs.(28), (43), (60) and (66). The correction of O ( G (cid:48)(cid:48) /G ) from t and t (cid:48) loopsexactly cancel and do not contribute to the tHiggs mass, which is consistent with the Dashen formula in Eq. (51).Hence the leading order corrections in the perturbation with respect to the explicit breaking couplings only comefrom terms of O ( G (cid:48) /G ) = O ( m h t ) and O ( α em ), in which the quadratic divergent contributions dominate. Of theseleading corrections, the electroweak gauge terms are actually highly suppressed by α em compared to the corrections of O (( G (cid:48) /G )Λ χ / (4 π ) ) which arises from the Top-Mode Pseudo’s ( h t and A t ) self-interaction sector, where Λ χ ∼ m H t is the cutoff of the nonlinear sigma model. Thus we evaluate the leading order corrections in the perturbation of G (cid:48) /G , G (cid:48)(cid:48) /G and α em to the tHiggs mass: m h t (cid:12)(cid:12)(cid:12) O ( G (cid:48) /G,G (cid:48)(cid:48) /G,α em )1 − loop = m h t (cid:34) χ (4 π ) v EW (cid:35) + O (cid:32) α em Λ χ π (cid:33) . (72)The size of G (cid:48) /G correction (the second term in the square bracket) could be O (1), when we took Λ χ ∼ m H t ∼ πv EW ,and hence is potentially a large correction to the tHiggs mass, which might need some fine tuning.Actually, the top and t (cid:48) -loop corrections arising as the next to leading order of O (( G (cid:48)(cid:48) /G ) ) will be more significantto give the sizable contribution to the tHiggs mass at the one-loop order since G (cid:48)(cid:48) /G is numerically not very small inorder to realize the reality, though those terms are potentially suppressed in terms of the ( G (cid:48)(cid:48) /G )-perturbation. Theconcrete estimate of the size of corrections to the tHiggs mass from those terms will be addressed in Sec. IV.One might naively suspect from Eqs.(72) that the presence of quadratic divergent corrections to the h t mass causesthe fine-tuning problem just like the SM Higgs boson case. However, it is not the case because the h t is natural inaccordance with the original argument in [29]: If one takes the massless Higgs boson limit ( m h t →
0) correspondingto G (cid:48) → , G (cid:48)(cid:48) → G αtu → , G αtd → , G αte →
0) and electroweak gauge interactions are turned off in Eqs.(28),(43) and (60) (and Eq.(66)), then the global G -symmetry is restored, meaning that the symmetry is enhanced. Inthis limit the quadratic divergences disappear as well. Thus the h t mass is protected by the G -symmetry just likethe QCD pseudo NGBs ( π, K · · · ). In contrast, as is well known, the SM Higgs sector is unnatural since even if onetakes the massless Higgs boson limit ( m h →
0) the symmetry of the SM is not enhanced. In this sense, the h t is anatural Higgs.Before closing this section, it is also worth mentioning the difference between the present model and little Higgsmodels [30–33]. They are similar in the sense that one of NGBs associated with the spontaneous global symmetrybreaking ( G / H ) is identified as the Higgs boson, and the Higgs mass is generated through the explicit breaking effect.Therefore, the Higgs mass is under control in both models as explained in the previous paragraph. The crucialdifference, though, is that, in the case of little Higgs models, the electroweak symmetry is embedded as a subgroupof H , while in the case of the present model, it is outside of H , namely, the electroweak symmetry is broken by thedynamics which breaks G down to H . III. PHENOMENOLOGIES
In this section, we discuss several constraints from existing experimental results and phenomenological implicationsof the model. A. Phenomenological constraints on Top-Mode Pseudos
Examining Eqs.(67) and (68), we see that the couplings of the tHiggs h t to the W and Z bosons deviate from theSM Higgs ones by κ V ≡ g hV V g SM hV V = cos θ , (73)4where V = W and Z . The current LHC data give the constraint on κ V to be κ V > .
94 at 95% C . L . for the 126 GeVHiggs boson [34]. Therefore, we obtain the following constraint on the angle θ :cos θ > . , sin θ < . . (74)As noted in Eq.(71), the tHiggs couplings to the SM particles coincide with the SM Higgs ones when θ (cid:28)
1: Thedeviation in the ratio of the couplings to fermions to those of the W and Z bosons, g hff /g hV V , can be expanded inpowers of θ as g hff g hV V = 1 , for f = τ, b,g htt g hV V (cid:39) − θ . This and the current bound on θ in Eq.(74) imply that the highly precise measurement (by about 5% accuracy) wouldbe required to distinguish the coupling properties between the SM Higgs and the tHiggs. It might be possible to makeit by the high luminosity LHC or ILC [35].Here, it is also worth giving some comments on the difference between the present low-energy effective theory(Eqs.(28),(43) and (60)) and the two-Higgs doublet model since the top-seesaw model is often described by usinga two Higgs doublet model as the low-energy effective theory [23, 24]. One difference is in the low-energy massspectrum: As seen in Sec.II B, the low-energy mass spectrum in the present model has no charged Higgs bosons whichthe two-Higgs doublet model posses. The other would be the tree-level mass relation among the neutral Higgs bosons( A t and h t ) as in Eq.(42), which is absent in the usual two-Higgs doublet model and therefore may distinguish twomodels. B. S , T parameters and the constraint on t (cid:48) -quark mass Looking at the current bound on θ in Eq.(74), we see that the coupling property of the t (cid:48) -quark arises mainly fromthe SU (2) L singlet χ -quark which carries exactly the same charge as that of the right-handed top quark. The massof t (cid:48) -quark can therefore be constrained from the Peskin-Takeuchi S, T -parameters [36, 37] .By taking m t (cid:48) (cid:29) m t (cid:29) m b , the one-loop contributions from the t (cid:48) -quark to the S, T -parameters are evaluatedas [22] S = 32 π ( s tL ) (cid:20) −
19 ln x t (cid:48) x t − ( c tL ) F ( x t , x t (cid:48) ) (cid:21) , (75) T = 316 πs W c W ( s tL ) (cid:20) ( s tL ) x t (cid:48) − (cid:0) c tL ) (cid:1) x t + ( c tL ) x t (cid:48) x t x t (cid:48) − x t ln x t (cid:48) x t (cid:21) , (76)where s W ≡ sin θ W ( c W ≡ − s W ) is the weak mixing angle and x a ≡ m a /m Z , ( a = t, t (cid:48) ). The function F ( x, y ) isgiven by [38] F ( x, y ) = 5( x + y ) − xy x − y ) + 3 xy ( x + y ) − x − y x − y ) ln xy . The
S, T -parameters in Eqs. (75) and (76) are calculated as a function of the two parameters, s tL ( c tL ) and m t (cid:48) once we fix m Z , s W ( c W ) and m t to be the experimental values [20]. Expanding Eqs.(53) and (54) in powers of G (cid:48)(cid:48) /G = µ χt /m χχ < m t /m t (cid:48) to the next to leading order of G (cid:48)(cid:48) /G : m t m t (cid:48) (cid:39) sin θ cos θ (cid:18) G (cid:48)(cid:48) G (cid:19) (cid:34) − cos θ (cid:18) G (cid:48)(cid:48) G (cid:19) (cid:35) = µ χt m χχ sin θ cos θ (cid:34) − cos θ (cid:18) µ χt m χχ (cid:19) (cid:35) . (77)Using this and Eq.(57), we can rewrite s tL ( c tL ) in terms of θ and G (cid:48)(cid:48) /G to evaluate the S and T as a function of θ and G (cid:48)(cid:48) /G . In Fig. 1 (left panel), we thus plot the S, T -parameters versus cos θ for several values of G (cid:48)(cid:48) /G , together with Another possible constraint would be t (cid:48) -quark contribution to the Zb L ¯ b L coupling, which, however, turns out to be much milder thanthat from the S, T parameters. This is due to the fact that the present model does not include b (cid:48) -like particle usually arising in a classof top-seesaw models with so-called bottom-seesaw mechanism [23, 24]. . L . allowed region (inside of the ellipsis) on the ( S, T )-plane [39]. We have taken into account the constraints oncos θ in Eq.(74) from the current LHC Higgs search. From the right panel of Fig. 1, we read off the allowed t (cid:48) -quarkmass, m t (cid:48) ≥ .
11 TeV , cos θ ≥ .
997 for G (cid:48)(cid:48) G = 0 . , .
23 TeV , cos θ ≥ .
991 for G (cid:48)(cid:48) G = 0 . , .
19 TeV , cos θ ≥ .
952 for G (cid:48)(cid:48) G = 0 . , (78)where the lower mass limits corresponds to the 95% C . L . upper limit of the ST -ellipsis in the left panel of Fig. 1.Note that the limit cos θ → θ →
0) corresponds to the decoupling limit of t (cid:48) -quark, m t (cid:48) → ∞ where s tL → t (cid:48) -quark thus come from the contribution to the S, T parameters,which limit the mass to be (cid:38) O (TeV). Here v EW = 246 GeV is realized when the cutoff scale of NJL dynamics is setas Λ (cid:39) , ,
480 TeV for G (cid:48)(cid:48) /G = 0 . , . , . ST S - T ellispsis G /G = 0 . . . m t and cos ✓ G /G m t ( T e V ) FIG. 1: Left panel: The
S, T constraints from Eqs.(75) and (76) on the t (cid:48) -quark mass in the ( S, T )-plane for G (cid:48)(cid:48) /G = 0 . . . . L . allowed region corresponds to an area lower than the solid curve (inside the S - T ellipsis). The region of S <
T < c tL ≤ a . Right panel: The t (cid:48) -quark mass versus G (cid:48)(cid:48) /G allowed by the S, T constraints of the left panel. The 95% C . L . allowed region corresponds to an area upper than the bluecurve. IV. SUMMARY AND DISCUSSION
In the spirit of the top quark condensation, we proposed a model which has a naturally light composite Higgs bosonto be identified with the 126 GeV Higgs discovered at the LHC. The tHiggs, a bound state of the top quark and itsflavor (vector-like) partner, emerges as a pseudo Nambu-Goldstone boson (NGB), Top-Mode Pseudo, together withthe exact NGBs to be absorbed into the W and Z bosons as well as another (heavier) Top-Mode Pseudo (CP-oddcomposite scalar, A t ). Those five composite (exact/pseudo) NGBs are dynamically produced simultaneously by asingle supercritical four-fermion interaction having the U (3) ˜ ψ L × U (1) χ R symmetry which includes the electroweaksymmetry, where the vacuum is aligned by small explicit breaking term so as to break the symmetry down to asubgroup, U (2) ˜ ψ L × U (1) V = χ L + χ R , in a way not to retain the electroweak symmetry, in sharp contrast to the littleHiggs models.The h t couplings to the SM particles coincide with those of the SM Higgs boson in the limit sin θ = v EW /f → v EW being finite (Eqs.(67) and (71)). Even if the tHiggs coupling coincides with that of the SM Higgs, the virtueof our model is that the tHiggs h t is a bound state of the top quark and χ -quark, and is natural in the sense that itsmass is protected by the symmetry, in sharp contrast to the SM Higgs. a We thank H. C. Cheng for pointing out the error omitting this obvious condition. A t is related to the tHiggs mass (Eq.(42)) atthe tree-level of perturbations with respect to the explicit breaking effects, involving the size of deviation of couplings(sin θ ) to the electroweak gauge bosons from the SM Higgs ones. The CP-odd Top-Mode Pseudo A t does not coupleto the W and Z bosons due to the CP-symmetry and the couplings to other SM particles are generically suppressedby sin θ ( < . A t is distinguishable from that of the SM-like Higgs boson in the high-mass SM Higgsboson search at the LHC.As noted around Eq.(72) the tHiggs would get the significant corrections of higher order in G (cid:48)(cid:48) /G to the mass fromthe top and t (cid:48) -quark as well as the Top-Mode pseudos’ loops. In particular, the most sizable corrections would comefrom the top and t (cid:48) -loops only at sub-leading order O (( G (cid:48)(cid:48) /G ) ): Those one-loop corrections are dominated by thequadratic divergent terms as follows (for details of the computations, see Appendix. B): m h t (cid:12)(cid:12)(cid:12) t,t (cid:48) − loop = m h t + N c (4 π ) y (cid:18) G (cid:48)(cid:48) G (cid:19) (2 cos θ − χ = m h t + N c π (cid:32) √ m t v EW (cid:33) (cid:18) θ − θ (cid:19) Λ χ (cid:34) O (cid:32)(cid:18) G (cid:48)(cid:48) G (cid:19) (cid:33)(cid:35) , (79)where the first line is an exact result without further higher order corrections in G (cid:48)(cid:48) /G and in the second linewe used m t = ( y f /
2) sin θ cos θ ( G (cid:48)(cid:48) /G ) [1 + O (( G (cid:48)(cid:48) /G ) )] which can be derived from Eq.(54) with Eq.(63) and v EW = f sin θ , and the chiral symmetry breaking scale Λ χ ∼ m H t is the cutoff of the nonlinear sigma model constructedfrom Eqs.(28), (43), (60) and (66). From Eq.(79), we see that the perturbative G (cid:48)(cid:48) /G corrections contribute to thetHiggs mass at the order of O (( G (cid:48)(cid:48) /G ) ), in accord with the Dashen formula Eq.(51). Note the amazing cancellationof the quadratic divergent terms among the top and t (cid:48) -quark loops when the mixing angle θ reaches an ideal amount,cos θ = 1 / √
2. However, to be consistent with the current Higgs coupling measurement at the LHC, the θ is actuallystrongly constrained to be cos θ > .
94 (see Eq.(74)), which is somewhat far from cos θ = 1 / √ (cid:39) .
71. Thus theideal mixing cannot reproduce the reality.One possibility to make the present model realistic would be to pull the t (cid:48) -quark mass down to a low scale insuch a way that the t (cid:48) -quark can be integrated out. In that case, we may take the t (cid:48) -quark mass to be the cutoff ofthe nonlinear sigma model, Λ χ , say Λ χ = m t (cid:48) (cid:39) . S, T -parameter constraints inEq.(78) for cos θ = 0 . A t loopsas in Eq.(72) and top loops involving some effective h t - h t - t - t vertices induced from integrating out the t (cid:48) -quark (seeAppendix. B). Thus we find the mass shift (with Λ χ replaced by m t (cid:48) ), m h t (cid:12)(cid:12)(cid:12) t,h t ,A t − loop = m h t (cid:20) m t (cid:48) (4 π ) v EW (cid:21) − π (cid:32) √ m t v EW (cid:33) − θ + 6 cos θ cos θ m t (cid:48) (cid:34) O (cid:32)(cid:18) G (cid:48)(cid:48) G (cid:19) (cid:33)(cid:35) . (80)Note the negative correction from the top quark for cos θ = 0 . t (cid:48) -quark loop contributions.Note also that the result differs from that coming only from the top loop, since the t (cid:48) -quark effects are not totallydecoupled via equation of motion as manifested in the induced vertex which never exists in the theory having no t (cid:48) -quark from the onset. We thus achieve the desired tHiggs mass around (cid:39)
126 GeV at the one-loop level, when thecutoff Λ χ = m t (cid:48) is set to (cid:39) . m h t | tree (cid:39)
200 GeV.As seen from the explicit one-loop computation for the A t mass given in Appendix. B, the mixing angle cos θ = 0 . A t mass, so that we may take m A t | tree (cid:39) m A t | t,h t ,A t − loop from Eq.(80). Using the tree-level mass relation among h t and A t together with the tree-level tHiggs mass (cid:39)
200 GeV,we then find m A t (cid:12)(cid:12)(cid:12) tree (cid:39) m A t (cid:12)(cid:12)(cid:12) t,h t ,A t − loop (cid:39)
700 GeV . (81)The scenario in the above would be phenomenologically interesting, where the Top-Mode Pseudos ( h t , A t ) (andheavy top Higgs H t with the mass (cid:39) m t (cid:48) (cid:39) . t (cid:48) -quark have masses accessible at the LHC.Actually, the direct searches for t (cid:48) -quark at the LHC [40–43], have placed the limit, m t (cid:48) ≥ t (cid:48) -quark in the present model. However, in addition to usual t (cid:48) -quark searches as reported in [40–43], adecay channel t (cid:48) → tA t would be a characteristic signature of the t (cid:48) -quark in the present model. More on the detailedphenomenological study is to be pursued in the future.To summarize, the present model predicts the following five masses: m H t [Eq.(22)] , m A t [Eq.(40)] , m h t [Eqs.(41) and (80)] , m t (cid:48) [Eq.(53)] , m t [Eq.(54)] . (82)7These masses are controlled by the five model parameters: G , G (cid:48) , ∆ χχ , G (cid:48)(cid:48) , Λ . (83)By tuning these model parameters, we can thus realize the mass hierarchies:Λ > m H t ∼ m t (cid:48) (cid:29) m A t > m h t ∼ m t . (84)The hierarchy m H t (cid:29) m A t is realized by tuning (0 < ) G (cid:48) /G (cid:28) m h t is smaller than the CP-odd Top-Mode Pseudo mass m A t due to the mass relation in Eq.(41). Thefermion mass hierarchy m t (cid:48) > m t is realized by taking (0 < ) G (cid:48)(cid:48) /G < G/G crit − (cid:39) − to get m t (cid:48) (cid:39) . , m H t (cid:39) m t (cid:48) (cid:39) . v EW = 246 GeV, which determines m tχ through Eq.(29), and setting G (cid:48) /G (cid:39) − and G (cid:48)(cid:48) /G (cid:39) .
7, we have m t (cid:39)
173 GeV and m h t | t,h t ,A t − loop (cid:39)
126 GeV where m h t | tree (cid:39)
200 GeV. Thus the phenomenologically favored situationas above can be realized from the original four-fermion dynamics.
Acknowledments
We would like to thank Hiroshi Ohki for useful discussions. This work was supported by the JSPS Grant-in-Aidfor Scientific Research (S)
Appendix A: Alternative derivation of Top-Mode Pseudo mass formulas
In this appendix we derive the mass formulas for the Top-Mode Pseudos in Eqs.(40) and (41) based on the Bardeen-Hill-Lindner approach [16].Introducing the auxiliary fields φ tχ ∼ ¯ χ R t L , φ bχ ∼ ¯ χ R b L and φ χχ ∼ ¯ χ R χ L , we rewrite the Lagrangian Eq.(30) intoa linear sigma model-like form including the 1 /N c -leading corrections renormalized at the scale µ ( < Λ), L BHL = L kin . − √ Z (cid:20)(cid:0) ¯ t L ¯ b L (cid:1) (cid:18) φ tχ φ bχ (cid:19) χ R + ¯ χ L φ χχ χ R + h.c. (cid:21) + L L σ M , (A1)where L L σ M = (cid:12)(cid:12)(cid:12)(cid:12) D µ (cid:18) φ tχ φ bχ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + | ∂ µ φ χχ | − V ( φ ) , (A2) D µ (cid:18) φ tχ φ bχ (cid:19) = (cid:18) ∂ µ − igW aµ σ a ig (cid:48) B µ (cid:19) (cid:18) φ tχ φ bχ (cid:19) , and V ( φ ) = M (cid:104) φ † tχ φ tχ + φ † bχ φ bχ + φ † χχ φ χχ (cid:105) + λ (cid:104) φ † tχ φ tχ + φ † bχ φ bχ + φ † χχ φ χχ (cid:105) +∆ M φ † χχ φ χχ − C χχ (cid:2) φ † χχ + φ χχ (cid:3) , (A3) Z = 1 λ = N c π ln Λ µ , M = 1 Z (cid:18) G − N c π Λ (cid:19) , ∆ M = 1 Z (cid:18) G − G (cid:48) − G (cid:19) , C χχ = 1 √ Z ∆ χχ G − G (cid:48) . We define the vacuum expectation values corresponding to Eq.(9), (cid:104) φ tχ (cid:105) = f sin θ √ ≡ v tχ √ , (cid:104) φ bχ (cid:105) = 0 , (cid:104) φ χχ (cid:105) = f cos θ √ ≡ v χχ √ , (A4)8and the dynamical masses, m AB = 1 √ Z (cid:104) φ AB (cid:105) for A, B = t, b, χ . (A5)The stationary conditions, corresponding to the gap equations Eqs.(32) and (33), are obtained from the potential V ( φ ) to be ∂V∂v tχ = 0 ⇔ m tχ = m tχ N c G π (cid:20) Λ − (cid:0) m tχ + m χχ (cid:1) ln Λ m tχ + m χχ (cid:21) , (A6) ∂V∂v χχ = 0 ⇔ m χχ = ∆ χχ + m χχ N c ( G − G (cid:48) )8 π (cid:20) Λ − (cid:0) m tχ + m χχ (cid:1) ln Λ m tχ + m χχ (cid:21) . (A7)We next parametrize the neutral scalar fields φ tχ and φ χχ as φ tχ = v tχ + Re φ tχ + i Im φ tχ √ , φ χχ = v χχ + Re φ χχ + i Im φ χχ √ . (A8)Taking into account the stationary conditions Eqs.(A6) and (A7), we find the mass terms of (Re φ tχ , Re φ χχ , Im φ χχ )in the effective potential Eq.(A3), − m A t (Im φ χχ ) − (cid:0) Re φ tχ Re φ χχ (cid:1) (cid:18) m tχ m tχ m χχ m tχ m χχ m χχ + m A t (cid:19) (cid:18) Re φ tχ Re φ χχ (cid:19) , (A9)where Im φ χχ ≡ A t and the A t mass m A t is given by expanding terms in powers of G (cid:48) /G (cid:28) m A t = ∆ M = (cid:18) G − G (cid:48) − G (cid:19) × π N c ln(Λ / ( m tχ + m χχ )) (cid:39) π GN c ln(Λ / ( m tχ + m χχ )) (cid:18) G (cid:48) G (cid:19) (cid:34) O (cid:32)(cid:18) G (cid:48) G (cid:19) (cid:33)(cid:35) (cid:39) m tχ + m χχ ) Gf (cid:18) G (cid:48) G (cid:19) (cid:34) O (cid:32)(cid:18) G (cid:48) G (cid:19) (cid:33)(cid:35) . (A10)where the renormalization scale µ has been set to ( m tχ + m χχ ) / and use has been made of the Pagels-Stokar formulafor the decay constant f given in Eq.(21). For G (cid:48) /G (cid:28) m A t (cid:28) m ˜ χχ = ( m tχ + m χχ ) / so that thescalar mass in the last term of Eq.(A9) can be diagonalized up to terms of O ( m A t /m χχ ) as (cid:32) m tχ m tχ m χχ m tχ m χχ m χχ + m A t (cid:33) (cid:39) (cid:32) cos θ sin θ − sin θ cos θ (cid:33) (cid:32) m A t sin θ
00 4( m tχ + m χχ ) + m A t cos θ (cid:33) (cid:32) cos θ − sin θ sin θ cos θ (cid:33) , where tan θ ≡ m tχ /m χχ and the corresponding mass eigenstates h t and H t with m H t > m h t are given as (cid:32) h t H t (cid:33) = (cid:32) cos θ − sin θ sin θ cos θ (cid:33) (cid:32) Re φ tχ Re φ χχ (cid:33) . Thus we find the masses of the two CP-even neutral scalar mesons up to terms of O ( m A t /m χχ ): m h t (cid:39) m A t sin θ , m H t (cid:39) m tχ + m χχ ) . (A11)Eqs.(A10) and (A11) exactly reproduce the Top-Mode Pseudo mass formulas in Eqs.(40) and (41) obtained from thenonlinear Lagrangian with the heavy top-Higgs integrated out.9 Appendix B: t, t (cid:48) -loop corrections to the Top-Mode Pseudo masses
In this appendix, we shall compute the quadratic divergent corrections to the Top Mode Pseudos ( h t , A t ) arisingfrom the top and t (cid:48) -loops as the 1 /N c -leading contribution in Eqs.(79) and (80).We start with the Yukawa sector in Eq.(60), L t,t (cid:48) yuk . = − f √ (cid:2) y ¯ ψ L ( R T U ) ψ R + y χt ¯ ψ L ( χ R T U χ ) ψ R + h.c. (cid:3) . (B1)From this Lagrangian, we find the couplings relevant to the one-loop corrections in the basis of the mass-eigenstates( t, t (cid:48) ) m : L t,t (cid:48) yuk . (cid:12)(cid:12)(cid:12) h t = − y htt h t ¯ tt − y ht (cid:48) t (cid:48) h t ¯ t (cid:48) t (cid:48) − g ht L t (cid:48) R h t (¯ t L t (cid:48) R + h.c.) − g ht R t (cid:48) L h t (¯ t R t (cid:48) L + h.c.) − g hhtt h t h t ¯ tt − g hht (cid:48) t (cid:48) h t h t ¯ t (cid:48) t (cid:48) + · · · , (B2)and L t,t (cid:48) yuk . (cid:12)(cid:12)(cid:12) A t = − y Att A t ¯ tγ t − y At (cid:48) t (cid:48) A t ¯ t (cid:48) γ t (cid:48) − g At L t (cid:48) R A t ¯ tγ t (cid:48) − g At R t (cid:48) L A t ¯ tγ t (cid:48) − g AAtt A t A t ¯ tγ t − g AAt (cid:48) t (cid:48) A t A t ¯ t (cid:48) γ t (cid:48) + · · · , (B3)where y htt = y √ (cid:20) ( c tL cos θ + s tL sin θ ) s tR − s tL c tR sin θ (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) , (B4) y ht (cid:48) t (cid:48) = y √ (cid:20) ( s tL cos θ − c tL sin θ ) c tR − c tL s tR sin θ (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) , (B5) g ht L t (cid:48) R = y √ (cid:20) ( c tL cos θ + s tL sin θ ) c tR + s tL s tR sin θ (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) , (B6) g ht (cid:48) L t R = y √ (cid:20) ( s tL cos θ − c tL sin θ ) s tR + c tL c tR sin θ (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) , (B7) g hhtt = y √ f (cid:20) ( c tL sin θ − s tL cos θ ) s tR + s tL c tR cos θ (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) , (B8) g hht (cid:48) t (cid:48) = y √ f (cid:20) ( s tL sin θ + c tL cos θ ) c tR + c tL s tR cos θ (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) , (B9)and y Att = iy √ (cid:20) − s tL s tR + s tL c tR (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) , (B10) y At (cid:48) t (cid:48) = iy √ (cid:20) c tL c tR + c tL s tR (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) , (B11) g At L t (cid:48) R = iy √ (cid:20) − s tL c tR − s tL s tR (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) , (B12) g At (cid:48) L t R = iy √ (cid:20) c tL s tR − c tL c tR (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) , (B13) g AAtt = g hhtt + 3 y √ f sin θ cos θ (cid:20) − ( c tL cos θ + s tL sin θ ) s tR + s tL c tR sin θ (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) , (B14) g AAt (cid:48) t (cid:48) = g hht (cid:48) t (cid:48) + 3 y √ f sin θ cos θ (cid:20) − ( s tL cos θ − c tL sin θ ) c tR + c tL s tR sin θ (cid:18) G (cid:48)(cid:48) G (cid:19)(cid:21) . (B15)The one-loop corrections arise from Feynman graphs involving the top and t (cid:48) -loops as depicted in Fig. 2. The quadraticdivergent corrections to the Top-Mode Pseudo masses are thus calculated to be δm h t ,A t = N c π Λ χ · C h,A , (B16)0 h t , A t h t , A t t, t h t , A t h t , A t t, t FIG. 2: The one-loop diagrams contributing to h t , A t masses as the quadratic divergent corrections up to O (( G (cid:48)(cid:48) /G ) ). where the chiral symmetry breaking scale Λ χ is the cutoff of the nonlinear sigma model constructed from Eqs.(28),(43), (60) and (66) and C h = − y htt − y ht (cid:48) t (cid:48) − g ht L t (cid:48) R − g ht (cid:48) L t R + g hhtt + g hht (cid:48) t (cid:48) = y (cid:18) G (cid:48)(cid:48) G (cid:19) (2 cos θ − , (B17) C A = − y Att − y At (cid:48) t (cid:48) − g At L t (cid:48) R − g At (cid:48) L t R + g hhtt g AAtt + g hht (cid:48) t (cid:48) g AAt (cid:48) t (cid:48) = y (cid:18) G (cid:48)(cid:48) G (cid:19)
12 (1 − cos θ )(3 cos θ − . (B18)Here we used the orthogonality relations among the mixing angles s tL,R ( c tL,R ) which follows from the diagonalizationof the fermion mass matrix in Eq.(52) with the rotation matrices in Eq.(55): s tL s tR cos θ (cid:18) G (cid:48)(cid:48) G (cid:19) = (cid:0) c tL sin θ − s tL cos θ (cid:1) c tR , c tL c tR cos θ (cid:18) G (cid:48)(cid:48) G (cid:19) = (cid:0) s tL sin θ + c tL cos θ (cid:1) s tR . (B19)Thus we have Eq.(79) and the associated formula for A t : δm h t (cid:12)(cid:12)(cid:12) t,t (cid:48) = N c π y (cid:18) G (cid:48)(cid:48) G (cid:19) (2 cos θ − χ = N c π (cid:32) √ m t v EW (cid:33) (cid:18) θ − θ (cid:19) Λ χ (cid:34) O (cid:32)(cid:18) G (cid:48)(cid:48) G (cid:19) (cid:33)(cid:35) = Last term in Eq.(79) , (B20) δm A t (cid:12)(cid:12)(cid:12) t,t (cid:48) = N c π y (cid:18) G (cid:48)(cid:48) G (cid:19)
12 (1 − cos θ )(3 cos θ − χ = N c π (cid:32) √ m t v EW (cid:33) (1 − cos θ )(3 cos θ − θ Λ χ (cid:34) O (cid:32)(cid:18) G (cid:48)(cid:48) G (cid:19) (cid:33)(cid:35) , (B21)where the first lines in Eqs.(B20) and (B21) are exact results without further higher order corrections in G (cid:48)(cid:48) /G andin the second lines in Eqs.(B20) and (B21) we used m t = ( y f /
2) sin θ cos θ ( G (cid:48)(cid:48) /G ) [1 + O (( G (cid:48)(cid:48) /G ) )] which canbe derived from Eq.(54) with Eq.(63) and v EW = f sin θ . As noted around Eq.(79), from Eq.(B20), we see that theperturbative G (cid:48)(cid:48) /G corrections contribute to the tHiggs mass at the order of O (( G (cid:48)(cid:48) /G ) ), no correction of O ( G (cid:48)(cid:48) /G ),in accord with the Dashen formula Eq.(51), as well as the A t mass in Eq.(B21).In the limit where m t (cid:28) m t (cid:48) ∼ Λ χ , the t (cid:48) -quark may be integrated out to induce the effective h t - h t - t - t and A t - A t - t - t -couplings, g (cid:48) hhtt = y √ f g ht L t (cid:48) R g ht (cid:48) L t R g hht (cid:48) t (cid:48) , g (cid:48) AAtt = − y √ f g At L t (cid:48) R g At (cid:48) L t R g hht (cid:48) t (cid:48) . (B22)1In that case, C h,A in Eq.(B16) become, up to order of ( G (cid:48)(cid:48) /G ) , C h = − y htt + g hhtt + g hhtt g (cid:48) hhtt = y (cid:18) G (cid:48)(cid:48) G (cid:19) ( − θ − θ ) (cid:34) O (cid:32)(cid:18) G (cid:48)(cid:48) G (cid:19) (cid:33)(cid:35) , (B23) C A = − y Att + g AAtt + g hhtt g (cid:48) AAtt = y (cid:18) G (cid:48)(cid:48) G (cid:19)
12 (1 − cos θ )( − θ − θ ) (cid:34) O (cid:32)(cid:18) G (cid:48)(cid:48) G (cid:19) (cid:33)(cid:35) . (B24)Thus we find Eq.(80) and the associated result for A t : δm h t (cid:12)(cid:12)(cid:12) t = N c π y (cid:18) G (cid:48)(cid:48) G (cid:19) ( − θ − θ ) m t (cid:48) (cid:34) O (cid:32)(cid:18) G (cid:48)(cid:48) G (cid:19) (cid:33)(cid:35) = − π (cid:32) √ m t v EW (cid:33) − θ + 6 cos θ cos θ m t (cid:48) (cid:34) O (cid:32)(cid:18) G (cid:48)(cid:48) G (cid:19) (cid:33)(cid:35) = Last term in Eq.(80) , (B25) δm A t (cid:12)(cid:12)(cid:12) t = N c π y (cid:18) G (cid:48)(cid:48) G (cid:19)
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