Higgs boson as a top-mode pseudo-Nambu-Goldstone boson

Hidenori S. Fukano,1,* Masafumi Kurachi,1,† Shinya Matsuzaki,2,3,‡ and Koichi Yamawaki1,§1Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya University,

Nagoya 464-8602, Japan2Institute for Advanced Research, Nagoya University, Nagoya 464-8602, Japan

3Department of Physics, Nagoya University, Nagoya 464-8602, Japan(Received 26 November 2013; published 9 September 2014)

In the spirit of the top-quark condensation, we propose a model which has a naturally light compositeHiggs boson, “tHiggs” (h0t ), to be identified with the 126 GeV Higgs discovered at the LHC. The tHiggs, abound state of the top quark and its flavor (vectorlike) partner, emerges as a pseudo-Nambu–Goldstoneboson (NGB), “top-mode pseudo-Nambu-Goldstone boson,” together with the exact NGBs to be absorbedinto theW and Z bosons as well as another (heavier) top-mode pseudo-Nambu-Goldstone bosons (CP-oddcomposite scalar, A0

t ). Those five composite (exact/pseudo-) NGBs are dynamically produced simulta-neously by a single supercritical four-fermion interaction having Uð3Þ × Uð1Þ symmetry which includesthe electroweak symmetry, where the vacuum is aligned by a small explicit breaking term so as to break thesymmetry down to a subgroup, Uð2Þ × Uð1Þ0, in a way not to retain the electroweak symmetry, in sharpcontrast to the little Higgs models. The explicit breaking term for the vacuum alignment gives rise to a massof the tHiggs, which is protected by the symmetry and hence naturally controlled against radiativecorrections. Realistic top-quark mass is easily realized similarly to the top-seesaw mechanism byintroducing an extra (subcritical) four-fermion coupling which explicitly breaks the residual Uð2Þ0 ×Uð1Þ0 symmetry with Uð2Þ0 being an extra symmetry besides the above Uð3ÞL × Uð1Þ. We present aphenomenological Lagrangian of the top-mode pseudo-Nambu-Goldstone bosons along with the StandardModel particles, which will be useful for the study of the collider phenomenology. The coupling property ofthe tHiggs is shown to be consistent with the currently available data reported from the LHC. Severalphenomenological consequences and constraints from experiments are also addressed.

DOI: 10.1103/PhysRevD.90.055009 PACS numbers: 12.60.Rc, 14.80.Ec

I. INTRODUCTION

The ATLAS [1] and CMS collaborations [2] havediscovered a new scalar particle at around 126 GeV havingthe properties compatible with the Higgs boson in theStandard Model (SM). However, the origin of mass is still amystery, since we do not yet understand detailed features ofthe 126 GeV Higgs, in particular the dynamical origin ofthe mass of the 126 GeV Higgs itself which is just a freeparameter 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 approxi-mate scale invariance and large anomalous dimensionγm ≃ 1, which predicts a technidilaton as a compositepseudo-Nambu–Goldstone boson (NGB) of the approxi-mate scale invariance [3,4]. It in fact was shown to beconsistent with the current LHC data for the 126 GeVHiggs [5,6], and a recent lattice study [7] showed indicationof such a very light flavor-singlet scalar meson as acandidate for the technidilaton in the walking technicolor.

To accommodate a large top-quark mass in the walkingtechnicolor, however, we would need even larger anoma-lous dimension γm > 1 for the technicondensate relevant tothe top mass, possibly realized by the additional strongfour-fermion interaction (strong extended technicolor)[8–10]. An alternative composite Higgs model based onsuch a strong four-fermion coupling is the top-quarkcondensate model [11–16] with γm ≃ 2. Here we proposea variant of the top-quark condensate model based on thestrong four-fermion couplings, which yields a naturallylight Higgs boson to be identified with the 126 GeV Higgsboson at the LHC.Actually, among masses of the SM fermions, the top-

quark mass (mt ≃ 173 GeV) is the only one roughly of theorder of the electroweak symmetry breaking (EWSB) scale(vEW ≃ 246 GeV). Furthermore, the mass of the LHCHiggs boson (mh ≃ 126 GeV) is also roughly of the orderof the EWSB scale. This coincidence may imply that thetop quark plays a crucial role for both the EWSB and thegeneration 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 closerelation among the top-quark mass, the EWSB scale andthe Higgs mass, based on the phase structure of the gauged

*fukano@kmi.nagoya‑u.ac.jp†kurachi@kmi.nagoya‑u.ac.jp‡synya@hken.phys.nagoya‑u.ac.jp§yamawaki@kmi.nagoya‑u.ac.jp

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Nambu–Jona–Lasinio (NJL) model [17,18]. The four-fermion interactions in the TMSM are written in theSM-gauge-invariant form [11,12]:

L4fTMSM ¼ GtðqiLtRÞðtRqLiÞ þ GbðqiLbRÞðbRqLiÞ

þGtbðqiLqkRÞðiτ2Þijðiτ2ÞklðqjLqlRÞ þ H:c:; ð1Þ

where qLðRÞ ¼ ðt; bÞTLðRÞ and τ2 is the second component ofthe Pauli matrices. It is straightforward to extend this toinclude all the three generations of the SM fermions[11,12]. The four-fermion interactions are arrangedto trigger the top-quark condensate, htti ≠ 0, withoutother condensates such as the bottom condensate,hbbi; htbi;… ¼ 0, in such a way that

Gt > Gcrit > Gb; � � � ; ð2Þ

where the critical coupling Gcrit is given asGcrit ¼ 4π2=ðNcΛ2Þ, with Nc being the number of QCDcolor and Λ the cutoff scale of the theory, up to smallcorrections from the SM gauge interactions as implied bythe phase structure of the gauged NJL model [17,18]. Thesolution of the gap equation indicates that the top-quarkmass can be much smaller than the cutoff scale Λ, mt ≪ Λ,by tuning the four-fermion coupling close to the criticalcoupling, 0 < Gt=Gcrit − 1 ≪ 1. The TMSM producesthree NGBs which are absorbed into the W and Z bosonswhen the electroweak gauge interactions are switched onand predicts the top-quark mass to be on the order of theEWSB scale, vEW ≃ 246 GeV, through the Pagels–Stokarformula [19] for the decay constant Fπð¼ vEWÞ of theNGBs, which are evaluated with the solution of the gapequation 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 mt ¼ 220–250 GeV[11,12,16], which is somewhat larger than the experimentalvalue mexp

t ¼ 173 GeV [20]. If we assume Λ to be a fewTeV to avoid excessive fine-tuning to reproduce the EWSBscale, we would face a disastrous situation where the top-quark mass is too large: mt ∼ 600 GeV (top mass problem).ii) The TMSM predicts a Higgs boson as a tt bound state(the “top-Higgs boson,” Ht) with mass in a range ofmt < mHt

< 2mt. Such a top-Higgs boson cannot be iden-tified as the Higgs boson with the mass ≃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 (vectorlike)SUð2ÞL-singlet quark (seesaw partner of the top quark)is introduced to mix with the tR, which pulls the top-quarkmass down to the desired value ≃173 GeV, so that thetop-mass problem is resolved. However, it turns out thatthe top-Higgs boson is still heavy, mt < mHt

< 2mt, andtherefore the Higgs mass problem still remains in thetop-seesaw model.

The Higgs mass problem was recently resolved by thetop-seesaw assisted technicolor model [24,25], which isa hybrid version combining the top-seesaw model withtechnicolor. In this model a light top-Higgs boson withmHt

< mt was realized by sharing the top-quark masswith the technicolor sector. It was shown, however, that thecoupling properties of the top-Higgs boson are quitedifferent from those of the Higgs boson in the SM, andthe model has currently been disfavored by the LHC data,most notably by the results on the diphoton decay channel[1,2] and production cross section through the vector bosonfusion process [26,27].In this paper, in the spirit of the top-quark condensation,

we propose a model which solves the Higgs mass problemin a natural way, while keeping the solution of the topmass problem by the top-seesaw mechanism. The lightcomposite Higgs, what we call tHiggs, emerges as one ofthe composite pseudo-NGBs, dubbed top-mode pseudo-Nambu-Goldstone bosons, associated with the spontaneousbreaking of an approximate global symmetry, which istriggered by strong (supercritical) four-fermion interactions.The model is constructed from the third-generation

quarks in the SM q ¼ ðt; bÞ and a (vectorlike) χ quarkwhich is a flavor partner of the top quark and has the sameSM charges as those of the right-handed top quark. Thefour-fermion interaction term takes the form

L4f ¼ Gðψ iLχRÞðχRψ i

LÞ; ð3Þ

where ψ iL ≡ ðtL; bL; χLÞTi; ði ¼ 1; 2; 3Þ. The four-fermion

interaction in Eq. (3) possesses a global symmetryUð3ÞL ×Uð1ÞχR . For the supercritical setting, G > Gcrit,the symmetry is spontaneously broken down to Uð2ÞL ×Uð1ÞV by the quark condensates hχRtLi ≠ 0 andhχRχLi ≠ 0, which are realized in the vacuum aligned withthe additional explicit breaking terms mentioned below,while hχRbLi is gauged away to hχRbLi ¼ 0 by theelectroweak gauge symmetry when it is switched on.Note that the electroweak gauge symmetry is spontaneouslybroken when Uð3ÞL ×Uð1ÞχR → Uð2ÞL ×Uð1ÞV , in sharpcontrast to the little Higgs models. It should also be notedthat the Lagrangian has Uð2ÞqR symmetry [see belowEq. (47)] of ðtR; bRÞ not broken by the condensate, whichwill not be explicitly mentioned unless it becomes 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 modeof the usual NJL model is also formed. Three of theseNGBs will be eaten by the electroweak gauge bosons whenthe subgroup of the symmetry is gauged by the electroweaksymmetry, while two of them remain as physical states.Those two NGBs, top-mode pseudo-Nambu-Goldstonebosons, acquire their masses due to additional terms whichexplicitly break the Uð3ÞL × Uð1ÞχR symmetry in such away that the vacuum aligns to break the electroweak

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symmetry by hχRtLi ≠ 0. One of them is a CP-even scalar(tHiggs, h0t ), which is identified as the 126 GeV Higgsboson discovered at the LHC, while the other is a heavyCP-odd scalar (A0

t ), which is similar to the CP-odd Higgsin the two Higgs doublet models (there is an essentialdifference from the two-doublet Higgs model, though). Wefind a notable relation between masses of those top-modepseudo-Nambu-Goldstone bosons:

mh0t¼ mA0

tsin θ;

�tan θ ¼ hχRtLi

hχRχLi�; ð4Þ

where the angle θ is related to the presence of thecondensate, hχRqLi ≠ 0, which causes the electroweaksymmetry breaking.It will be shown that the tHiggs couplings to the SM

particles coincide with those of the SM Higgs boson in thelimit sin θ → 0 (where vEW ≃ 246 GeV is kept fixed). Evenif the tHiggs coupling coincides with that of the SM Higgs,the virtue of our model is that the tHiggs h0t is a bound stateof the top quark and χ quark and is natural in the sense thatits mass is protected by the symmetry, in sharp contrast tothe SM Higgs. One notable feature of our model is theprediction of the heavy CP-odd Higgs (without additionalcharged heavy Higgs in contrast to the two-doublet Higgsmodels), 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 Uð3ÞL × Uð1ÞχR to showthat five NGBs emerge due to the spontaneous breaking ofthe global symmetry by the quark condensate generated bythe supercritical four-fermion dynamics. Additional explicitbreaking terms are then introduced to give mass to the two ofthe five NGBs (top-mode pseudo-Nambu-Goldstone bosons;h0t and A0

t ) and also to the top quark. We estimate the mass ofthe top-mode pseudo-Nambu-Goldstone bosons based onthe current algebra to find the mass formula Eq. (4). Theinteraction property of the tHiggs (h0t ) and the stability of themass against the radiative corrections are addressed incomparison with the SM Higgs boson case. The extensionof the model to incorporate masses of light fermions is alsodiscussed in Sec. II. Several phenomenological constraintsare given in Sec. III. Section IV is devoted to the summary ofthis paper including some discussions. In Appendix A, weprovide a straightforward derivation of the top-mode pseudo-Nambu-Goldstone boson masses by directly solving thebound state problem in the four-fermion dynamics based onthe auxiliary field method. Appendix B is devoted to somedetails of computations for one-loop corrections to the top-mode pseudo-Nambu-Goldstone bosons arising from the topand its flavor partner, t0-quark loops.

II. MODEL

In this section we propose a model based on four-fermion dynamics constructed from the top and bottom

quarks q ¼ ðt; bÞ with the flavor partner of top quark ðχÞ.The model possesses an approximate global symmetrywhich is spontaneously broken by the quark condensateshχRqLi ≠ 0 and hχRχLi ≠ 0 generated by the four-fermiondynamics. Five NGBs emerge as bound states of the quarksassociated with the spontaneous breaking of the global sym-metry, in addition to a composite heavy Higgs boson whichcorresponds to the σ mode in the usual NJL model. Two ofthe NGBs [what we call top-mode pseudo-Nambu-Goldstonebosons (h0t and A0

t )] obtain their masses due to the introduc-tion of additional four-fermion interactions which explicitlybreak the global symmetry, while other three remain masslessto be eaten by the W and Z bosons once the electroweakcharges are turned on. Themass of h0t turns out to be protectedby the global symmetry, and the coupling property is shownto be consistent with the currently reported Higgs bosonwith mass around ≃126 GeV. We will call h0t tHiggs.The present model, which is based on the strong four-

fermion interactions with a vectorlike 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 modelmakes it clear that the 126 GeV Higgs exists as a pseudo-NGB associated with the global symmetry breaking causedby four-fermion interactions. For the purpose of makingthis point clearer, in Sec. II A, we introduce a simplifiedmodel in which all the explicit breaking terms are turnedoff. In that simplified model, five NGBs which exist in themodel are all massless. Then, in Sec. II B, we introduceexplicit breaking terms into the Lagrangian to give massesto two of the NGBs, top-mode pseudo-Nambu-Goldstonebosons (h0t , A0

t ), which are identified as a 126 GeV Higgsboson and its CP-odd partner. Sections II C and II D aredevoted to explaining fermion masses, Yukawa interactionsas well as the nature of the tHiggs.

A. Structure of symmetry breaking

Let us consider an NJL-like model constructed fromthe third-generation quarks in the SM, q ¼ ðt; bÞ, and anSUð2ÞL singlet quark ðχÞ. The left-handed quarks qL and χLform a flavor triplet ψ i

L ≡ ðtL; bL; χLÞTiði ¼ 1; 2; 3Þ underthe flavor Uð3ÞψL

group, while the right-handed top andbottom quarks qiR ≡ ðtR; bRÞTiði ¼ 1; 2Þ and χR are adoublet and singlet under the Uð2ÞqR group, respectively.Turning off the SM gauge interactions momentarily, we thuswrite the Lagrangian having the global Uð3ÞψL

×Uð2ÞqR ×Uð1ÞχR symmetry:

Lkin þ L4f ¼ ψLiγμ∂μψL þ qRiγμ∂μqR þ χRiγμ∂μχR

þ Gðψ iLχRÞðχRψ i

LÞ; ð5Þ

where G denotes the four-fermion coupling strength.We can derive the gap equations for fermion dynamical

masses mtχ and mχχ through the mean field relations

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mtχ ¼ −GhχRtLi and mχχ ¼ −GhχRχLi in the large Nclimit1:

mtχ ¼ mtχNcG8π2

�Λ2 − ðm2

tχ þm2χχÞ ln

Λ2

m2tχ þm2

χχ

�; ð6Þ

mχχ ¼ mχχNcG8π2

�Λ2 − ðm2

tχ þm2χχÞ ln

Λ2

m2tχ þm2

χχ

�; ð7Þ

where Λ stands for the cutoff of the model which is to beof Λ ≫ Oð1Þ TeV. There exist nontrivial solutionsmtχ ≠ 0and mχχ ≠ 0 when the following criticality condition issatisfied:

G > Gcrit ¼8π2

NcΛ2; ð8Þ

under which we have the nonzero dynamical masses as wellas the nonzero condensates,

hχRqLi ≠ 0; hχRχLi ≠ 0: ð9Þ

Note, however, that the two gap equations, Eqs. (6) and (7),cannot determine the ratio of two condensates; those twogap equations with the criticality condition in Eq. (8) justlead to the nontrivial solution for the squared-sum of twomasses, ðm2

tχ þm2χχÞ ≠ 0, so that the vacuum with mtχ ≠ 0

is degenerate with that with mtχ ¼ 0. To lift the degeneracyfor breaking the electroweak symmetry by mtχ ≠ 0, weshall later introduce explicit breaking to align the vacuum,which also gives rise to the mass of two NGBs (top-modepseudo-Nambu-Goldstone bosons) out of five NGBs, withthe rest three being exact NGBs to be absorbed into the Wand Z bosons. Also note that the condensatembχ ≠ 0 can begauged away when the model is gauged by the electroweaksymmetry.To make the structure of the symmetry breaking clearer,

we may change the flavor basis of fermions by introducingan orthogonal rotation matrix R:

~ψL ¼

0B@

~tL~bL~χL

1CA≡ R · ψL; R ¼

0B@

cos θ 0 − sin θ

0 1 0

sin θ 0 cos θ

1CA;

tan θ≡ mtχ

mχχ¼ hχRtLi

hχRχLi: ð10Þ

The two gap equations, Eqs. (6) and (7), are then reduced toa single gap equation,

1 ¼ NcG8π2

�Λ2 −m2

~χχ lnΛ2

m2~χχ

�; ð11Þ

with

m2~χχ ≡m2

tχ þm2χχ ≠ 0: ð12Þ

Accordingly, the associated two condensates in Eq. (9) arereduced to a single nonzero condensate on the basis of ~ψL:

hχR ~χLi ≠ 0: ð13ÞWe thus see that, with the criticality condition in Eq. (8)satisfied, the four-fermion dynamics triggers the followingglobal symmetry breaking pattern:

Uð3Þ ~ψL×Uð1ÞχR → Uð2Þ ~qL ×Uð1ÞV¼~χLþχR

: ð14ÞThe broken currents associated with this symmetry break-ing are found to be

J4;μ3L ¼ ~ψLγμλ4 ~ψL

¼ ~tLγμ ~χLþ ~χLγμ~tL

¼ðtLγμtLþ χLγμχLÞsin2θþðtLγμχLþ χLγ

μtLÞcos2θ;ð15Þ

J5;μ3L ¼ ~ψLγμλ5 ~ψL

¼ i½−~tLγμ ~χL þ ~χLγμ~tL�

¼ −iðtLγμχL − χLγμtLÞ; ð16Þ

J6;μ3L ¼ ~ψLγμλ6 ~ψL

¼ ~bLγμ ~χLþ ~χLγμ ~bL

¼ ðbLγμtLþ tLγμbLÞ sinθþðbLγμχLþ χLγμbLÞcosθ;

ð17Þ

J7;μ3L ¼ ~ψLγμλ7 ~ψL

¼ i½− ~bLγμ ~χL þ ~χLγμ ~bL�

¼ −iðbLγμtL − tLγμbLÞ sin θ− iðbLγμχL − χLγ

μbLÞ cos θ; ð18Þ

and

JμA ≡ 1

4

�Jμ1R −

1ffiffiffi6

p J0;μ3L þ 1ffiffiffi3

p J8;μ3L

�

¼ 1

4ðχRγμχR − ~χLγ

μ ~χLÞ

¼ 1

4½χRγμχR − tLγμtLsin2θ − χLγ

μχLcos2θ

− ðtLγμχL þ χLγμtLÞ sin θ cos θ�; ð19Þ

1We have put mbχ ¼ 0, by gauging it away by the electroweakgauge symmetry. Otherwise ðm2

tχ þm2χχÞ in Eqs. (6) and (7)

should read ðm2tχ þm2

bχ þm2χχÞ because of Uð3ÞψL

symmetry,with mbχ also satisfying the same type of gap equation.

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where the Gell-Mann matrices λaða ¼ 1;…; 8Þ are nor-malized as tr½λaλb� ¼ 2δab, and λ0 ¼ ffiffiffiffiffiffiffiffi

2=3p

13×3. Theassociated NGBs emerge with the decay constant f as

h0jJaμðxÞjπbt ðpÞi ¼ −ifδabpμe−ip·x; a; b ¼ 4; 5; 6; 7; A:

ð20Þ

The decay constant f is calculated through the Pagels–Stokar formula [19]:

f2 ¼ Nc

8π2m2

~χχ lnΛ2

m2~χχ

: ð21Þ

The five NGBs ðπat Þ can be expressed as composite fields(interpolating fields) made of the fermion bilinears on thebasis of ( ~ψL, χR) or (ψL; χR):

π4t ∼ χR~tL − ~tLχR

¼ ðχRtL − tLχRÞ cos θ − ðχRχL − χLχRÞ sin θ;π5t ∼ −iðχR~tL þ ~tLχRÞ

¼ −iðχRtL þ tLχRÞ cos θ þ iðχRχL þ χLχRÞ sin θ;π6t þ iπ7t ∼ ðχR ~bL − ~bLχRÞ þ ðχR ~bL þ ~bLχRÞ

¼ 2χRbL;

π6t − iπ7t ∼ ðχR ~bL − ~bLχRÞ − ðχR ~bL þ ~bLχRÞ¼ −2bLχR;

πAt ∼ χR ~χL − ~χLχR

¼ ðχRtL − tLχRÞ sin θ þ ðχRχL − χLχRÞ cos θ:

Besides these composite NGBs, there exists a compositescalar (H0

t ) corresponding to the σ mode in the usual NJLmodel,

H0t ∼ χR ~χL þ ~χLχR

¼ ðχRtL þ tLχRÞ sin θ þ ðχRχL þ χLχRÞ cos θ;

with the mass

m2H0

t¼ 4m2

~χχ ¼ 4ðm2tχ þm2

χχÞ: ð22Þ

The H0t will be regarded as a heavy Higgs boson with the

mass of Oð1Þ TeV, not the light Higgs boson at around126 GeV.With the electroweak gauge interactions turned on, the

W and Z bosons turn out to couple to the broken currentsðJ6;μ3L∓iJμ;73L Þ and ðJμ;43L cos θ þ Jμ;A3L sin θÞ, respectively. Thecorresponding would-be NGBs (w�

t ; z0t ) eaten by theelectroweak gauge bosons are then found to be

z0t ≡ π4t cos θ þ πAt sin θ

∼ χRtL − tLχR;

w−t ≡ 1ffiffiffi

2p ðπ6t þ iπ7t Þ

∼ffiffiffi2

pχRbL;

wþt ≡ 1ffiffiffi

2p ðπ6t − iπ7t Þ

∼ −ffiffiffi2

pbLχR:

On the other hand, the following two NGBs remain asphysical states:

h0t ≡ π5t

∼ −iðχR~tL þ ~tLχRÞ¼ −iðχRtL þ tLχRÞ cos θ þ iðχRχL þ χLχRÞ sin θ;

A0t ≡ −π4t sin θ þ πAt cos θ

∼ χRχL − χLχR:

The correspondence between the broken currents andNGBs along with the CP transformation property issummarized in Table I. The two massless NGBs ðh0t ; A0

t Þwill become pseudo-NGBs, called top-mode pseudo-Nambu-Goldstone bosons, obtaining their masses onceexplicit breaking effects are introduced (see Sec. II B).We will identify the CP-even top-mode pseudo-Nambu-Goldstone boson, h0t , as the 126 GeV Higgs, called tHiggs.We may integrate out the heavy Higgs boson H0

t [withmass of Oð1Þ TeV] to construct the low-energy effectivetheory governed by the five NGBs ðπat Þ described by anonlinear sigma model based on the coset space,

GH

¼ Uð3Þ ~ψL×Uð1ÞχR

Uð2Þ ~ψL×Uð1ÞV¼χLþχR

: ð23Þ

For this purpose we introduce representatives (ξL;R) of theG=H which are parameterized by NGB fields as

TABLE I. The list of the broken currents and NGBs associatedwith the spontaneous symmetry breaking in Eq. (14). w�

t and z0tare eaten by the W� and Z bosons once the electroweak gaugesare turned on, while the remaining two A0

t and h0t become pseudo-NGBs (top-mode pseudo-Nambu-Goldstone bosons) by explicitbreaking effects [see Eqs. (40) and (41)].

Broken current Corresponding NGB CP property

J4;μ3L π4t ¼ z0t cos θ − A0t sin θ Odd

J5;μ3L π5t ¼ h0t Even

J6;μ3L � iJ7;μ3L π6t � iπ7t ¼ffiffiffi2

pw∓t –

JμA πAt ¼ z0t sin θ þ A0t cos θ Odd

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ξL ¼ exp

�−if

� Xa¼4;5;6;7

πat λa þ πAt

2ffiffiffi2

p λA��

;

ξR ¼ exp�if

πAt2

ffiffiffi2

p λA�;

where

λA ¼

0B@

0 0 0

0 0 0

0 0ffiffiffi2

p

1CA:

We further introduce the “chiral” field U,

U ¼ ξ†L · Σ · ξR with Σ ¼ 1ffiffiffi2

p λA: ð24Þ

The transformation properties of ξL;R and U under G aregiven by

ξL → hðπt; ~gÞ · ξL · g†~3L;

ξR → hðπt; ~gÞ · ξR · g†1R;

U → g ~3L · U · g†1R; ð25Þwhere ~g ¼ fg ~3L; g1Rg; g ~3L ∈ Uð3Þ ~ψL

; g1R ∈ Uð1ÞχR andhðπt; ~gÞ ∈ H. Thus, we find the G-invariant Lagrangianwritten in terms of the NGBs to the lowest order ofderivatives of Oðp2Þ:

LNLσM ¼ f2

2tr½∂μU†∂μU�: ð26Þ

When the electroweak symmetry turned on, the covariantderivative acting on U is given by

DμU ≡ R

264∂μ − ig

X3a¼1

Waμ

0B@

0

τa=2 0

0 0 0

1CA

þ ig0Bμ

0B@

1=2 0 0

0 1=2 0

0 0 0

1CA375RT · U; ð27Þ

where Wμ and Bμ are the SUð2ÞL and Uð1ÞY gauge bosonfields with the gauge couplings g and g0. Then theLagrangian Eq. (26) is changed to the covariant form:

LNLσM ¼ f2

2tr½DμU†DμU�: ð28Þ

From this one can read off the W an Z boson masses as

m2W ¼ 1

4g2f2sin2θ; m2

Z ¼ 1

4ðg2 þ g02Þf2sin2θ;

which lead to

v2EW ¼ f2sin2θ

¼ Nc

8π2m2

tχ lnΛ2

m2tχ þm2

χχ

≃ ð246 GeVÞ2; ð29Þ

where use has been made of Eq. (21). Thus, imposing theEWSB scale vEW gives a nontrivial relation between mtχand mχχ .Note that switching on the electroweak gauge interaction

explicitly breaks the Uð3Þ ~ψL×Uð1ÞχR symmetry. Such

explicit breaking effects would generate masses to theNGBs at the loop level, as a part of the 1=Nc subleadingeffect, which are, however, negligibly small since the sizeof effects is suppressed by the small electroweak gaugecoupling, as will be discussed later [see the discussionbelow Eq. (80)].As we mentioned earlier the criticality G > Gcrit implies

hχRtLi2 þ hχRχLi2 ≠ 0, but not necessarily hχRtLi ≠ 0which is responsible for the electroweak symmetry break-ing. The electroweak gauge interaction itself can contributeto lifting the degeneracy between the vacuum withhχRtLi ¼ 0 and that with hχRtLi ≠ 0 in principle by someextreme fine-tuning of the critical coupling of the gaugedNJL model [17,18].A more natural way will be to introduce extra effective

four-fermion interactions to explicitly break the Uð3Þ ~ψL×

Uð1ÞχR symmetry, which can align the vacuum to havehχRtLi ≠ 0 and simultaneously give the mass of the tHiggson the right amount. Such an explicit breaking may beinduced from a strong Uð1Þ gauge interaction distinguishinghχRtLi from hχRχLi. This we perform in the next subsection.

B. top-mode pseudo-Nambu-Goldstone bosons

Here we incorporate explicit breaking terms into theLagrangian Eq. (5) to give masses to the top-mode pseudo-Nambu-Goldstone bosons ðh0t ; A0

t Þ:

Lkin þ L4f þ Lh; ð30Þ

where

Lh ¼ −½Δχχ χRχL þ H:c:� −G0ðχLχRÞðχRχLÞ: ð31Þ

Similarly to Eqs. (6) and (7), we derive the gap equationsfor fermion dynamical masses mtχ and mχχ :

mtχ ¼ mtχNcG8π2

�Λ2 −m2

~χχ lnΛ2

m2~χχ

�; ð32Þ

mχχ ¼ Δχχ þmχχNcðG −G0Þ

8π2

�Λ2 −m2

~χχ lnΛ2

m2~χχ

�; ð33Þ

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where m2~χχ ¼ m2

tχ þm2χχ . Note that the nonzero Δχχ and G0

allow us to determine the ratio of two dynamical massesmtχ and mχχ , i.e. tan θ ¼ mtχ=mχχ , in contrast to theprevious gap equations, Eqs. (6) and (7), which onlydetermine the squared-sum of two, m2

tχ þm2χχ .

Furthermore, it turns out that these Δχχ and G0 terms donot affect the criticality of the four-fermion dynamics at all;assuming mtχ ≠ 0 and mχχ ≠ 0, we find that the followingrelation is required so as to keep the self-consistency in thegap equations:

Δχχ ¼G0

Gmχχ ¼ −G0hχRχLi: ð34Þ

By taking this into account, the two gap equations,Eqs. (32) and (33), are reduced to a single one,

1 ¼ NcG8π2

�Λ2 −m2

~χχ lnΛ2

m2~χχ

�: ð35Þ

Thus, the presence of the nontrivial solution is controlledsolely by G > Gcrit, which is the same criticality conditionas in Eq. (8). The Lagrangian Eq. (31) is also rewritten as

Lh ¼ −G0ðχLχR − hχLχRiÞðχRχL − hχRχLiÞ: ð36Þ

This implies that the explicit breaking effects can also beexpressed only by the four-fermion interaction. We can thusapproximately keep the global Uð3Þ ~ψL

×Uð1ÞχR invarianceby taking G0 perturbatively to be

0 <G0

G≪ 1 and Gcrit < G: ð37Þ

We have explicitly checked that in the presence of thisexplicit breaking the vacuum with mtχ ≠ 0 is preferred tothat with mtχ ¼ 0 in the phenomenologically interestingparameter space to be discussed later.As in Sec. II A, the global Uð3Þ ~ψL

×Uð1ÞχR symmetry isspontaneously broken down to Uð2Þ ~ψL

×Uð1ÞV¼~χLþχRdue

to the four-fermion interaction in L4f, resulting in thepresence of five NGBs πaða ¼ 4; 5; 6; 7; AÞ. Then theexplicit breaking terms in Lh force the vacuum to choosea specific direction, hχR ~χLi ≠ 0, and give masses to someof the NGBs. Note that the Lh term is invariant under thechiral transformation associated with the broken currentsðJ6;μ3L � iJ7;μ3L Þ and ðJ4;μ3L cos θ þ JA;μ3L sin θÞ, but not for J5μ3Land ð−J4;μ3L sin θ þ JA;μ3L cos θÞ. Hence, the Lh term givesmasses only to the NGBs associated with latter two, i.e. thetop-mode pseudo-Nambu-Goldstone bosons A0

t and h0t .Estimation of the masses of the pseudo-NGBs can be

done by the traditional approach based on the currentalgebra [28]:

m2ab ¼

1

f2h0j½iQa; ½iQb;−Lh��j0i; ð38Þ

where Qa is the Noether charge associated with the brokencurrents Eqs. (15), (16), (17), (18) and (19), and f is givenby Eq. (21). From Eq. (38), together with the gap equations,Eqs. (32), (33) and (34), we obtain

m2z0t¼ m2

w�t¼ 0; ð39Þ

m2A0t¼ 2hχR ~χLihχRχLi

f2 cos θ

≃ G0

G2×2ðm2

tχ þm2χχÞ

f2; ð40Þ

m2h0t¼ 2hχR ~χLihχRχLi

f2 cos θ× sin2θ

¼ m2A0tsin2θ; ð41Þ

where the second equation of Eq. (40) is obtained byexpanding in terms of G0=G ≪ 1 and taking the leadingnontrivial order. The mass of h0t is proportional to mtχassociated with the EWSB scale vEW as in Eq. (29), just likethe case of the SM Higgs boson, while the mass of A0

t isnot. We may set the mass of h0t to ≃126 GeV;

mh0t¼ mA0

tsin θ≃ 126 GeV: ð42Þ

Note also that G0 > 0 assumed in Eq. (37) ensuresthe positiveness of squared masses for the top-modepseudo-Nambu-Goldstone bosons, m2

h0t> 0 and m2

A0t> 0.

In Appendix A, we present an alternative derivation of thepseudo-NGBs masses and the heavy Higgs massmH0

tbased

on the approach used in Ref. [16].As was done in Eq. (28), we may construct a nonlinear

Lagrangian valid for scales belowmH0tdescribed by the five

NGBs based on the coset space in Eq. (23) including theexplicit breaking effect from the G0- and Δχχ-terms inEq. (31). To this end, we introduce the spurion fields χ1 andχ2 to write the Oðp2Þ potential terms corresponding toEq. (31):

ΔLNLσM ¼ f2tr½c1ðRTUÞ†χ1ðRTUÞþ c2ðχ†2ðRTUÞ þ ðRTUÞ†χ2Þ�; ð43Þ

where χ1 and χ2 transform under the G-symmetry as

χ1 → g ~3L · χ1 · g†~3L; χ2 → g ~3L · χ2 · g

†1R: ð44Þ

The G-symmetry is explicitly broken when the spurionfields acquire the vacuum expectation values,

hχ1i ¼ hχ2i ¼ Σ; ð45Þ

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so that the c1 and c2 terms break the G down to Uð2ÞqL ×Uð1ÞχL ×Uð1ÞχR and Uð2ÞqL ×Uð1ÞV¼χRþχL

, respectively,in the same way as the G0 and Δχχ terms in Eq. (31) do.Matching to the tHiggs mass formula in Eq. (41), we thenfind the coefficients c1 and c2 to be

c1 ¼ −1

2m2

A0t; c2 ¼

1

2m2

A0tcos θ:

C. Fermion masses and Yukawa couplings

1. Top and t0 quark

Let us consider the top-quark mass based on theLagrangian Eq. (30). After the spontaneous symmetrybreaking by the nontrivial solutions of the gap equations,Eqs. (32), (33) and (34), the mass terms of the top-quark tand its flavor partner χ look like

LkinþL4f þLhjmass ¼−ð tL χL Þ�0 mtχ

0 mχχ

��tRχR

�þH:c:

ð46Þ

From this we find the fermion mass eigenvalues as

m2t0 ¼ m2

χχ þm2tχ ; m2

t ¼ 0; ð47Þ

where the mass eigenstates t0 and t are given by

t0L ¼ ~χL ¼ tL sin θ þ χL cos θ;

tL ¼ ~tL ¼ tL cos θ − χL sin θ:

Thus, the top quark does not “feel” the EWSB by sin θ ≠0ðmtχ ≠ 0Þ and is still massless. This is essentially due tothe residual symmetry, Uð2ÞqR ×Uð1ÞV¼~χLþχR

(Uð2ÞqR∶tR↔bR), which forbids the couplings between tR and χL inthe Lagrangian Eq. (30); hence, there is no mass term forχLtR in Eq. (46).To make the model more realistic, we introduce a

four-fermion interaction term which breaks the residualsymmetry so as to allow tR to couple to χL:

Lkin þ L4f þ Lh þ Lt; ð48Þ

where

Lt ¼ G00ðχLχRÞðtRχLÞ þ H:c: ð49Þ

As was done in Eq. (37), we also treat theG00 coupling to beperturbative,

0 <G00

G< 1; ð50Þ

so that the symmetry breaking pattern G=H ¼ ½Uð3Þ ~ψL×

Uð1ÞχR �=½Uð2Þ ~ψL×Uð1ÞV � is not destroyed (the vacuum

aligned to this manifold): htRχLiG00¼0 ¼ htRtLiG00¼0 ¼ 0.The Dashen formula Eq. (38) with the G00 term in Eq. (49)leads to

m2h0tjG00 ¼

1

f2h0j½iQ5; ½iQ5;−Lt��j0i

¼ G00

f2½ðcos2θ − sin2θÞh ~χLtRihtR ~χLi þ 2 sin θ cos θh ~χLχRihtR~tLi�

≃G00½hχRχLiG00¼0htRχLiG00¼0 − hχRtLiG00¼0htRtLiG00¼0� þOððG00Þ2Þ¼ 0þOððG00Þ2Þ; ð51Þ

similarly for the mass of the CP-odd top-mode pseudo-Nambu-Goldstone boson, A0

t . Here we have granted thevacuum saturation valid at the leading order of 1=Nc. Thus,the top-mode pseudo-Nambu-Goldstone boson’s massesare stable against the leading order correction of theexplicit-breaking G00 term as dictated by the Dashenformula Eq. (51). The next-to-leading order in the G00-perturbation, i.e., ðG00=GÞ2 corrections (which are also ofthe leading order of 1=Nc), will affect the masses, as will bediscussed later [see around Eq. (79)].After the spontaneous symmetry breaking by

hχR ~χLi ≠ 0, i.e. m2~χχ ¼ m2

tχ þm2χχ ≠ 0, we thus have the

fermion mass matrix,

Lkin þ L4f þ Lh þ Ltjmass

¼ −ð tL χL Þ�

0 mtχ

μχt mχχ

��tRχR

�þ H:c:; ð52Þ

where μχt ¼ −G00hχRχLi is a dynamical mass coming fromLt in Eq. (48). Note that the fermion mass matrix is

identical to that discussed in top-seesaw models [21,22].

The top-quark and t0-quark masses are given as the

eigenvalues of the mass matrix in Eq. (52),

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m2t0 ¼

m2tχ þm2

χχ þ μ2χt2

2641þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

4m2tχμ

2χt

ðm2tχ þm2

χχ þ μ2χtÞ2s 3

75;ð53Þ

m2t ¼

m2tχ þm2

χχ þ μ2χt2

2641 −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

4m2tχμ

2χt

ðm2tχ þm2

χχ þ μ2χtÞ2s 3

75:ð54Þ

Now the top-quark mass becomes nonzero and is propor-tional to mtχ which breaks the electroweak symmetry,similarly to the mass of h0t in Eq. (41), while the t0-quarkmass is not. The corresponding mass eigenstates ðt; t0ÞTm arerelated to the gauge (current) eigenstates ðt; χÞTg by theorthogonal rotation keeping mt;mt0 ≥ 0 [23]:�tLt0L

�m

¼�ctL −stLstL ctL

��tLχL

�g

¼ OL ·

�tLχL

�g

;

�tRt0R

�m

¼�−ctR stR

stR ctR

��tRχR

�g

¼ OR ·

�tRχR

�g

; ð55Þ

where ctLðRÞ ≡ cos θtLðRÞ and stLðRÞ ≡ sin θtLðRÞ which aregiven up to OððG00=GÞ2Þ as

ctL ¼ 1ffiffiffi2

p�1þm2

χχ −m2tχ þ μ2χt

m2t0 −m2

t

�1=2

≃ cos θ

�1þ

�G00

G

�2

cos2θsin2θ

�; ð56Þ

stL ¼ 1ffiffiffi2

p�1 −

m2χχ −m2

tχ þ μ2χtm2

t0 −m2t

�1=2

≃ sin θ

�1 −

�G00

G

�2

cos4θ

�; ð57Þ

ctR ¼ 1ffiffiffi2

p�1þm2

χχ þm2tχ − μ2χt

m2t0 −m2

t

�1=2

≃ 1 −1

2

�G00

G

�2

cos4θ;

ð58Þ

stR ¼ 1ffiffiffi2

p�1 −

m2χχ þm2

tχ − μ2χtm2

t0 −m2t

�1=2

≃ G00

Gcos2θ: ð59Þ

Including the effect of the explicit breaking G00 term, wemay thus add the fermion sector to the nonlinearLagrangian constructed from Eqs. (28) and (43),

Lt;t0yuk ¼−

fffiffiffi2

p ½yψLðRTUÞψRþyχtψLðχ1RTUχ3ÞψRþH:c:�;

ð60Þ

where ψR ¼ ðtR; bR; χRÞT . The spurion fields χ1 and χ3have been introduced in Eq. (60), which transform as

χ1 → g3L · χ1 · g†3L;

χ3 → g1R · χ3 · g†1R; ð61Þ

so that the Lagrangian Eq. (60) is invariant under theG-symmetry, Uð3Þ ~ψL

×Uð1ÞχR and Uð2ÞqR symmetry.These symmetries are explicitly broken by the vacuumexpectation values of the spurion fields,

hχ1i ¼ Σ; hχ3i ¼ λ4; ð62Þ

in which the hχ1i breaks the Uð3ÞψLsymmetry down to

Uð2ÞψL×Uð1ÞχL and the hχ3i does the Uð2ÞqR ×Uð1ÞχR

down to Uð1ÞχR¼tR. We thus see that the first term in

Eq. (60) corresponds to the G four-fermion termin Eq. (5), which is invariant under the G-symmetry,Uð3Þ ~ψL

×Uð1ÞχR and Uð2ÞqR symmetry, while thesecond term does to the G00 four-fermion term, whichexplicitly breaks the G and Uð2ÞqR down to Uð2ÞqL ×Uð1ÞχL and Uð1ÞtR¼χR

. The Yukawa couplings y and yχtcan be fixed by matching to the fermion mass matrix inEq. (52) as

y2 ¼ 2ðm2tχ þm2

χχÞf2

;

y2χt ¼ y2�G00

G

�2

¼ 2μ2χtf2cos2θ

: ð63Þ

2. Fermions other than top and t0 quark

To give masses to SM fermions other than the top quark,inspired by Refs. [11,12], we may add the following four-fermion interactions to the Lagrangian Eq. (48):

Lothers ¼Xα¼1;2

GαtuðqiLχRÞðuαRqα;iL Þ

þX

α¼1;2;3

GαtdðqiLχRÞðiτ2Þijðqα;jL dαRÞ

þX

α¼1;2;3

GαteðqiLχRÞðiτ2Þijðlα;jL eαRÞ þ H:c:; ð64Þ

which are SM gauge invariant, where α ¼ 1; 2; 3 denotesthe index for the fermion generation and i; j ¼ 1; 2 for theweak isospin. With the nonzero condensate hχRtLi ≠ 0(mtχ ≠ 0) breaking the electroweak gauge symmetry, theseSM fermions acquire their masses as

muα ¼ −GαtuhχRtLi;

mdα ¼ −GαtdhχRtLi;

meα ¼ −GαtehχRtLi:

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Since the NGBs arise from χRψL as

χRqiL ∼ hχR ~χLi½RTU�3i; ð65Þ

one can read off the fermion couplings to h0t2:

Lothersyuk j

h0t¼ − cos θ

�Xα¼1;2

muα

vEWh0t uαuα þ

Xα¼1;2;3

mdα

vEWh0t dαdα

þX

α¼1;2;3

meα

vEWh0t eαeα

�: ð66Þ

A full set of the particle content in the present model withthe SM charge assignment free from the gauge anomaly islisted in Table. II.

D. tHiggs

We here discuss the coupling property of the tHiggs h0tand the stability of the mass against radiative corrections.From Eqs. (28), (43), (60) and (66), we find the h0tcouplings to the SM particles,

LNLσM þ ΔLNLσM þ Lt;t0yuk þ Lothers

yuk jh0t¼ ghVV

vEW2

�g2h0t Wþ

μ W−μ þ g2 þ g02

2h0t ZμZμ

�

− ghhh3m2

h0t

vEW

ðh0t Þ33!

− ghhhh3m2

h0t

v2EW

ðh0t Þ44!

− ghttmt

vEWh0t tt − ghbb

mb

vEWh0t bb − ghττ

mτ

vEWh0t ττ

þ � � � ; ð67Þ

where

ghVV ¼ ghhh ¼ ghbb ¼ ghττ ¼ cos θ; ð68Þ

ghhhh ¼ 1 −7

3sin2θ; ð69Þ

ghtt ¼vEWmt

yffiffiffi2

p�ðctL cos θ þ stL sin θÞstR − stLc

tR sin θ

�G00

G

��

¼ 2cos2θ − 1

cos θþO

��G00

G

�2�: ð70Þ

From these, we see that the h0t couplings to theW;Z bosonsand to the SM fermions become the same as the SM Higgsones when we take the limit cos θ → 1, i.e.,

ghVV ¼ ghhh ¼ ghhhh ¼ ghtt ¼ ghbb ¼ ghττ ¼ gSMð¼ 1Þ;

when

sin θ ¼ mtχffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

tχ þm2χχ

q ¼ vEWf

→ 0; by f → ∞ with

vEW ¼ 246 GeV fixed: ð71Þ

Actually, this limit turns out to be favored by the currentexperiments 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 h0t mass at theone-loop level. The one-loop corrections of the leadingorder of the explicit breaking parameters G0=G;G00=G andαem ¼ e2=ð4πÞ with e being the electromagnetic couplingcan be evaluated at the one-loop level of the nonlinearsigma model constructed from terms in Eqs. (28), (43), (60)and (66). The correction of OðG00=GÞ from t and t0 loopsexactly cancels and does not contribute to the tHiggs mass,which is consistent with the Dashen formula in Eq. (51).Hence, the leading order corrections in the perturbationwith respect to the explicit breaking couplings only comefrom terms of OðG0=GÞ ¼ Oðm2

h0tÞ and OðαemÞ, in which

the quadratic divergent contributions dominate. Of theseleading corrections, the electroweak gauge terms are

2Those Yukawa interactions would give quadratically diver-gent corrections to the h0t mass, which are, however, smallenough due to the small Yukawa coupling for the lighterfermions, to be negligible compared to the terms in Eq. (79)arising from the top-mode pseudo-Nambu-Goldstone bosons, tand t0 loops.

TABLE II. Full particle contents and the charge assignmentsunder the SM gauge group. All the fermions are represented interms of the electroweak gauge eigenbasis.

Field SUð3Þc SUð2ÞL Uð1ÞYqL ¼

� tLbL

�3 2 1/6

tR 3 1 2/3

bR 3 1 −1/3lL ¼

� ντLτL

�1 2 −1/2

τR 1 1 −1χL 3 1 2/3χR 3 1 2/3

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actually highly suppressed by αem compared to thecorrections of OððG0=GÞΛ2

χ=ð4πÞ2Þ which arises fromthe top-mode pseudo-Nambu-Goldstone boson’s (h0t andA0t ) self-interaction sector, where Λχ ∼mH0

tis the cutoff of

the nonlinear sigma model. Thus, we evaluate the leadingorder corrections in the perturbation of G0=G;G00=G andαem to the tHiggs mass:

m2h0tjOðG0=G;G00=G;αemÞ1-loop

¼ m2h0t

�1þ Λ2

χ

ð4πÞ2v2EW23

16

�

þO�αemΛ2

χ

4π

�: ð72Þ

The size of G0=G correction (the second term in the squarebracket) could be Oð1Þ, when we took Λχ ∼mH0

t∼ 4πvEW,

and hence is potentially a large correction to the tHiggsmass, which might need some fine-tuning.Actually, the top and t0-loop corrections arising as the

next-to-leading order of OððG00=GÞ2Þ will be more signifi-cant to give the sizable contribution to the tHiggs mass atthe one-loop order since G00=G is numerically not verysmall in order to realize the reality, though those terms arepotentially suppressed in terms of the ðG00=GÞ perturbation.The concrete estimate of the size of corrections to thetHiggs 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 h0t masscauses the fine-tuning problem just like the SM Higgsboson case. However, it is not the case because the h0tis natural in accordance with the original argument inRef. [29]: If one takes the massless Higgs boson limit(mh0t

→ 0) corresponding to G0 → 0; G00 → 0 (andGα

tu → 0; Gαtd → 0; Gα

te → 0) and electroweak gauge inter-actions are turned off in Eqs. (28), (43) and (60) [andEq. (66)], then the global G-symmetry is restored, meaningthat the symmetry is enhanced. In this limit the quadraticdivergences disappear as well. Thus, the h0t mass isprotected by the G-symmetry just like the QCD pseudo-NGBs (π; K…). In contrast, as is well known, the SMHiggssector is unnatural since even if one takes the masslessHiggs boson limit (mh0SM

→ 0) the symmetry of the SM isnot enhanced. In this sense, the h0t is a natural 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 ofNGBs associated with the spontaneous global symmetrybreaking (G=H) is identified as the Higgs boson, and theHiggs mass is generated through the explicit breakingeffect. Therefore, the Higgs mass is under control in bothmodels as explained in the previous paragraph. The crucialdifference, though, is that, in the case of little Higgsmodels, the electroweak symmetry is embedded as asubgroup of H, while in the case of the present model,it is outside of H; namely, the electroweak symmetry isbroken by the dynamics which breaks G down to H.

III. PHENOMENOLOGIES

In this section, we discuss several constraints fromexisting experimental results and phenomenological impli-cations of the model.

A. Phenomenological constraints on top-mode pseudo-Nambu-Goldstone bosons

Examining Eqs. (67) and (68), we see that the couplingsof the tHiggs h0t to theW and Z bosons deviate from the SMHiggs ones by

κV ≡ ghVVgSMhVV

¼ cos θ; ð73Þ

where V ¼ W and Z. The current LHC data give theconstraint on κV to be κV > 0.94 at 95% C.L. forthe 126 GeV Higgs boson [34]. Therefore, we obtain thefollowing constraint on the angle θ:

cos θ > 0.94; sin θ < 0.34: ð74Þ

As noted in Eq. (71), the tHiggs couplings to the SMparticles coincide with the SM Higgs ones when θ ≪ 1; thedeviation in the ratio of the couplings to fermions to thoseof the W and Z bosons, ghff=ghVV , can be expanded inpowers of θ as

ghffghVV

¼ 1; for f ¼ τ; b;

ghttghVV

≃ 1 −3

2θ2:

This and the current bound on θ in Eq. (74) imply that thehighly precise measurement (by about 5% accuracy) wouldbe required to distinguish the coupling properties between theSM Higgs and the tHiggs. It might be possible to make it bythe high luminosity LHC or international linear collider [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 modelsince the top-seesaw model is often described by using atwo-Higgs doublet model as the low-energy effectivetheory [23,24]. One difference is in the low-energy massspectrum; As seen in Sec. II B, the low-energy massspectrum in the present model has no charged Higgsbosons which the two-Higgs doublet model posseses.The other would be the tree-level mass relation amongthe neutral Higgs bosons (A0

t and h0t ) as in Eq. (42), whichis absent in the usual two-Higgs doublet model andtherefore may distinguish two models.

B. S, T parameters and the constraint on t0-quark mass

Looking at the current bound on θ in Eq. (74), we seethat the coupling property of the t0 quark arises mainly fromthe SUð2ÞL singlet χ quark which carries exactly the same

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charge as that of the right-handed top quark. The mass ofthe t0 quark can therefore be constrained from the Peskin–Takeuchi S; T parameters [36,37].3

By taking mt0 ≫ mt ≫ mb, the one-loop contributionsfrom the t0 quark to the S; T parameters are evaluated as[22]

S ¼ 3

2πðstLÞ2

�−1

9lnxt0

xt− ðctLÞ2Fðxt; xt0 Þ

�; ð75Þ

T ¼ 3

16πs2Wc2WðstLÞ2

�ðstLÞ2xt0 − ð1þ ðctLÞ2Þxt

þ ðctLÞ22xt0xtxt0 − xt

lnxt0

xt

�; ð76Þ

where sW ≡ sin θWðc2W ≡ 1 − s2WÞ is the weak mixing angleand xa ≡m2

a=m2Z; ða ¼ t; t0Þ. The function Fðx; yÞ is given

by [38]

Fðx; yÞ ¼ 5ðx2 þ y2Þ − 22xy9ðx − yÞ2 þ 3xyðxþ yÞ − x3 − y3

3ðx − yÞ3 lnxy:

The S; T parameters in Eqs. (75) and (76) are calculated as afunction of the two parameters, stLðctLÞ and mt0 once we fixmZ, sWðcWÞ and mt to be the experimental values [20].Expanding Eqs. (53) and (54) in powers of G00=G ¼μχt=mχχ < 1 we express the ratio mt=mt0 to the next-to-leading order of G00=G:

mt

mt0≃ sin θ cos θ

�G00

G

��1 − cos4θ

�G00

G

�2�

¼ μχtmχχ

sin θ cos θ

�1 − cos4θ

�μχtmχχ

�2�: ð77Þ

Using this and Eq. (57), we can rewrite stLðctLÞ in terms of θandG00=G to evaluate S and T as a function of θ andG00=G.In Fig. 1 (left panel), we thus plot the S; T parameters vscos θ for several values of G00=G, together with a 95% C.L.allowed region (inside of the ellipsis) on the ðS; TÞ 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 allowedt0-quark mass,

mt0 ≥

8>><>>:

8.11 TeV; cos θ ≥ 0.997 for G00G ¼ 0.3;

3.23 TeV; cos θ ≥ 0.991 for G00G ¼ 0.5;

1.19 TeV; cos θ ≥ 0.952 for G00G ¼ 0.7;

ð78Þ

where the lower mass limits corresponds to the 95% C.L.upper limit of the ST ellipsis in the left panel of Fig. 1. Notethat the limit cos θ → 1ðsin θ → 0Þ corresponds to thedecoupling limit of the t0 quark, mt0 → ∞ where stL → 0.The stringent phenomenological constraints on the t0 quarkthus come from the contribution to the S; T parameters,which limit the mass to be ≳O ðTeVÞ. Here vEW ¼246 GeV is realized when the cutoff scale of NJL dynamicsis set as Λ≃ 66; 223; 480 TeV for G00=G ¼ 0.3; 0.5; 0.7.

IV. SUMMARY AND DISCUSSION

In the spirit of the top-quark condensation, we proposeda model which has a naturally light composite Higgs bosonto be identified with the 126 GeV Higgs discovered at theLHC. The tHiggs, a bound state of the top quark and its

0 0.005 0.010.00

0.05

0.10

0.15

0.20

0.25

0 0.2 0.4 0.6 0.8

1

2

5

10

20

FIG. 1 (color online). Left panel: The S; T constraints from Eqs. (75) and (76) on the t0-quark mass in the ðS; TÞ plane for G00=G ¼ 0.3(dotted), 0.5 (dashed), 0.7 (solid). The 95% C.L. allowed region corresponds to an area lower than the solid curve (inside the S-Tellipsis). The regions of S < 0 and T < 0 are not displayed due to ctL ≤ 1. Right panel: The t0-quark mass vs G00=G allowed by the S; Tconstraints of the left panel. The 95% C.L. allowed region corresponds to an area upper than the blue curve.

3Another possible constraint would be t0-quark contribution tothe ZbLbL coupling, which, however, turns out to be much milderthan that from the S; T parameters. This is due to the fact that thepresent model does not include b0-like particle usually arising in aclass of top-seesaw models with so-called bottom-seesaw mecha-nism [23,24].

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flavor (vectorlike) partner, emerges as a pseudo-Nambu–Goldstone boson, top-mode pseudo-Nambu-Goldstoneboson, together with the exact NGBs to be absorbed intothe W and Z bosons as well as another (heavier) top-modepseudo-Nambu-Goldstone boson (CP-odd composite sca-lar, A0

t ). Those five composite (exact/pseudo-) NGBs aredynamically produced simultaneously by a single supercriti-cal four-fermion interaction having the Uð3Þ ~ψL

×Uð1ÞχRsymmetry which includes the electroweak symmetry,where the vacuum is aligned by small explicit breakingterm so as to break the symmetry down to a subgroup,Uð2Þ ~ψL

×Uð1ÞV¼χLþχR, in a way not to retain the electro-

weak symmetry, in sharp contrast to the little Higgs models.The h0t couplings to the SM particles coincide with those

of the SM Higgs boson in the limit sin θ ¼ vEW=f → 0with vEW being finite [Eqs. (67) and (71)]. Even if thetHiggs coupling coincides with that of the SM Higgs, thevirtue of our model is that the tHiggs h0t is a bound state ofthe top quark and χ quark and is natural in the sense that itsmass is protected by the symmetry, in sharp contrast to theSM Higgs.One notable feature of our model is the prediction of the

heavy CP-odd Higgs (without additional charged heavyHiggs in contrast to the two-doublet Higgs models). Themass of the A0

t is related to the tHiggs mass [Eq. (42)] at thetree level of perturbations with respect to the explicitbreaking effects, involving the size of deviation of couplingsðsin θÞ to the electroweak gauge bosons from the SM Higgsones. The CP-odd top-mode pseudo-Nambu-Goldstoneboson A0

t does not couple to the W and Z bosons due tothe CP-symmetry, and the couplings to other SM particlesare generically suppressed by sin θð< 0.34Þ. Hence, the A0

t isdistinguishable from that of the SM-like Higgs boson in thehigh-mass SM Higgs boson search at the LHC.As noted around Eq. (72) the tHiggs would get the

significant corrections of higher order in G00=G to the massfrom the top and t0 quark as well as the top-mode pseudo-Nambu-Goldstone bosons’ loops. In particular, the mostsizable corrections would come from the top and t0 loopsonly at subleading order OððG00=GÞ2Þ; those one-loopcorrections are dominated by the quadratic divergent termsas follows (for details of the computations, see Appendix B):

m2h0tjt;t01-loop

¼ m2h0tþ Nc

8π2y2�G00

G

�2

ð2cos2θ − 1ÞΛ2χ

¼ m2h0tþ Nc

8π2

� ffiffiffi2

pmt

vEW

�2�2cos2θ − 1

cos2θ

�

× Λ2χ

�1þO

��G00

G

�2��

; ð79Þ

where the first line is an exact result without further higher-order corrections in G00=G and in the second line weusedm2

t ¼ ðy2f2=2Þsin2θcos2θðG00=GÞ2½1þOððG00=GÞ2Þ�which can be derived from Eq. (54) with Eq. (63) and

vEW ¼ f sin θ, and the chiral symmetry breaking scaleΛχ ∼mH0

tis the cutoff of the nonlinear sigma model

constructed from Eqs. (28), (43), (60) and (66). FromEq. (79), we see that the perturbative G00=G correctionscontribute to the tHiggs mass at the order of OððG00=GÞ2Þ,in accord with the Dashen formula Eq. (51). Note theamazing cancellation of the quadratic divergent termsamong the top- and t0-quark loops when the mixing angleθ reaches an ideal amount, cos θ ¼ 1=

ffiffiffi2

p. However, to be

consistent with the current Higgs coupling measurement atthe LHC, the θ is actually strongly constrained to becos θ > 0.94 [see Eq. (74)], which is somewhat far fromcos θ ¼ 1=

ffiffiffi2

p ≃ 0.71. Thus, the ideal mixing cannotreproduce the reality.One possibility to make the present model realistic

would be to pull the t0-quark mass down to a low scalein such a way that the t0 quark can be integrated out. In thatcase, we may take the t0-quark mass to be the cutoff of thenonlinear sigma model,Λχ , sayΛχ ¼ mt0 ≃ 1.2 TeV whichis consistent with the S; T-parameter constraints in Eq. (78)for cos θ ¼ 0.952. Then the one-loop quadratic divergentcorrections only come from the tHiggs and A0

t loops as inEq. (72) and top loops involving some effective h0t -h0t -t-tvertices induced from integrating out the t0 quark (seeAppendix B). Thus, we find the mass shift (with Λχ

replaced by mt0 ),

m2h0tjt;h0t ;A0

t

1-loop¼ m2

h0t

�1þ m2

t0

ð4πÞ2v2EW23

16

�−

3

8π2

� ffiffiffi2

pmt

vEW

�2

×1 − 6cos2θ þ 6cos4θ

cos2θm2

t0

�1þO

��G00

G

�2��

:

ð80ÞNote the negative correction from the top quark forcos θ ¼ 0.952, in the absence of the t0-quark loop con-tributions. Note also that the result differs from that comingonly from the top loop, since the t0-quark effects are nottotally decoupled via equation of motion as manifested inthe induced vertex which never exists in the theory havingno t0 quark from the onset. We thus achieve the desiredtHiggs mass around≃126 GeV at the one-loop level, whenthe cutoff Λχ ¼ mt0 is set to ≃1.2TeV in which case wehave the tree-level mass mh0t

jtree ≃ 200 GeV.As seen from the explicit one-loop computation for the

A0t mass given in Appendix B, the mixing angle cos θ ¼

0.952 is large enough to highly suppress the one-loop

corrections to the A0t mass, so that we may take mA0

tjtree ≃

mA0tjt;h0t ;A0

t1-loop from Eq. (80). Using the tree-level mass relation

among h0t and A0t together with the tree-level tHiggs mass

≃200 GeV, we then find

mA0tjtree

≃mA0tjt;h0t ;A0

t1-loop ≃ 700 GeV: ð81Þ

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055009-13

The scenario in the above would be phenomenologicallyinteresting, where the top-mode pseudo-Nambu-Goldstonebosons (h0t ; A0

t ) (and heavy top Higgs Ht0 with the mass

≃2mt0 ≃ 2.4 TeV) as well as the t0 quark have massesaccessible at the LHC. Actually, the direct searches for thet0 quark at the LHC [40–43] have placed the limitmt0 ≥ 1 TeV, which is available also to the t0 quark inthe present model. However, in addition to usual t0-quarksearches as reported in Refs. [40–43], a decay channel t0 →tA0

t would be a characteristic signature of the t0 quark in thepresent model. More on the detailed phenomenologicalstudy is to be pursued in the future.To summarize, the present model predicts the following

five masses:

m2H0

t½Eq: ð22Þ�; m2

A0t½Eq: ð40Þ�; m2

h0t½Eqs: ð41Þ and ð80Þ�;

m2t0 ½Eq: ð53Þ�; m2

t ½Eq: ð54Þ�: ð82Þ

These masses are controlled by the five model parameters:

G; G0; Δχχ ; G00; Λ: ð83Þ

By tuning these model parameters, we can thus realize themass hierarchies:

Λ2 > m2H0

t∼m2

t0 ≫ m2A0t> m2

h0t∼m2

t : ð84Þ

The hierarchy m2H0

t≫ m2

A0t

is realized by tuningð0 <ÞG0=G ≪ 1 as in Eq. (40). The CP-even top-modepseudo-Nambu-Goldstone boson (tHiggs) mass mh0t

issmaller than the CP-odd top-mode pseudo-Nambu-Goldstone boson mass mA0

tdue to the mass relation in

Eq. (41). The fermion mass hierarchy mt0 > mt is realizedby taking ð0 <ÞG00=G < 1 [see Eqs. (53) and (54)].When we set Λ ¼ 480 TeV and take G=Gcrit − 1≃ 10−4

to get mt0 ≃ 1.2 TeV; mH0t≃ 2mt0 ≃ 2.4 TeV. Taking

vEW ¼ 246 GeV, which determines mtχ throughEq. (29), and setting G0=G≃ 10−5 and G00=G≃ 0.7, wehave mt ≃ 173 GeV and mh0t

jt;h0t ;A0t

1-loop ≃ 126 GeV wheremh0t

jtree ≃ 200 GeV. Thus, the phenomenologicallyfavored situation as above can be realized from the originalfour-fermion dynamics.

ACKNOWLEDGMENTS

We would like to thank Hiroshi Ohki for useful dis-cussions. We thank H. C. Cheng for pointing out the erroromitting, ctL ≤ 1, obvious condition in Fig. 1. This workwas supported by the JSPS Grant-in-Aid for ScientificResearch Grants No. (S) #22224003 and No. (C)#23540300 (K.Y.).

APPENDIX A: ALTERNATIVE DERIVATION OFTOP-MODE PSEUDO-NAMBU-GOLDSTONE

BOSON MASS FORMULAS

In this Appendix we derive the mass formulas forthe top-mode pseudo-Nambu-Goldstone bosons inEqs. (40) and (41) based on the Bardeen–Hill–Lindnerapproach [16].Introducing the auxiliary fields ϕtχ ∼ χRtL, ϕbχ ∼ χRbL

and ϕχχ ∼ χRχL, we rewrite the Lagrangian Eq. (30) into alinear sigma model-like form including the 1=Nc-leadingcorrections renormalized at the scale μð< ΛÞ,

LBHL ¼ Lkin −1ffiffiffiffiZ

p�ð tL bL Þ

�ϕtχ

ϕbχ

�χR

þ χLϕχχχR þ H:c:

�þ LLσM; ðA1Þ

where

LLσM ¼Dμ

�ϕtχ

ϕbχ

�2 þ j∂μϕχχ j2 − VðϕÞ;

Dμ

�ϕtχ

ϕbχ

�¼

�∂μ − igWa

μσa

2þ ig0

1

2Bμ

��ϕtχ

ϕbχ

�; ðA2Þ

and

VðϕÞ ¼ M2½ϕ†tχϕtχ þ ϕ†

bχϕbχ þ ϕ†χχϕχχ �

þ λ½ϕ†tχϕtχ þ ϕ†

bχϕbχ þ ϕ†χχϕχχ �2

þ ΔM2ϕ†χχϕχχ − Cχχ ½ϕ†

χχ þ ϕχχ �;

Z ¼ 1

λ¼ Nc

16π2lnΛ2

μ2; M2 ¼ 1

Z

�1

G−

Nc

8π2Λ2

�;

ΔM2 ¼ 1

Z

�1

G −G0 −1

G

�; Cχχ ¼

1ffiffiffiffiZ

p Δχχ

G −G0 : ðA3Þ

We define the vacuum expectation values corresponding toEq. (9),

hϕtχi ¼f sin θffiffiffi

2p ≡ vtχffiffiffi

2p ; hϕbχi ¼ 0;

hϕχχi ¼f cos θffiffiffi

2p ≡ vχχffiffiffi

2p ; ðA4Þ

and the dynamical masses,

mAB ¼ 1ffiffiffiffiZ

p hϕABi for A; B ¼ t; b; χ: ðA5Þ

The stationary conditions, corresponding to the gap equa-tions Eqs. (32) and (33), are obtained from the potentialVðϕÞ to be

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055009-14

∂V∂vtχ ¼ 0 ⇔ mtχ ¼ mtχ

NcG8π2

�Λ2 − ðm2

tχ þm2χχÞ ln

Λ2

m2tχ þm2

χχ

�; ðA6Þ

∂V∂vχχ ¼ 0 ⇔ mχχ ¼ Δχχ þmχχ

NcðG −G0Þ8π2

�Λ2 − ðm2

tχ þm2χχÞ ln

Λ2

m2tχ þm2

χχ

�: ðA7Þ

We next parametrize the neutral scalar fields ϕtχ and ϕχχ as

ϕtχ ¼vtχ þ Reϕtχ þ iImϕtχffiffiffi

2p ; ϕχχ ¼

vχχ þ Reϕχχ þ iImϕχχffiffiffi2

p : ðA8Þ

Taking into account the stationary conditions Eqs. (A6) and (A7), we find the mass terms of ðReϕtχ ;Reϕχχ ; ImϕχχÞ in theeffective potential Eq. (A3),

−1

2m2

A0tðImϕχχÞ2 −

1

2ðReϕtχ Reϕχχ Þ

� 4m2tχ 4mtχmχχ

4mtχmχχ 4m2χχ þm2

A0t

��Reϕtχ

Reϕχχ

�; ðA9Þ

where Imϕχχ ≡ A0t and the A0

t mass mA0tis given by expanding terms in powers of G0=G ≪ 1 as

m2A0t¼ ΔM2

¼�

1

G −G0 −1

G

�×

16π2

Nc lnðΛ2=ðm2tχ þm2

χχÞÞ

≃ 16π2

GNc lnðΛ2=ðm2tχ þm2

χχÞÞ�G0

G

��1þO

��G0

G

�2��

≃ 2ðm2tχ þm2

χχÞGf2

�G0

G

��1þO

��G0

G

�2��

; ðA10Þ

where the renormalization scale μ has been set to ðm2tχ þm2

χχÞ1=2 and use has been made of the Pagels–Stokar formula forthe decay constant f given in Eq. (21). For G0=G ≪ 1 we may takemA0

t≪ m~χχ ¼ ðm2

tχ þm2χχÞ1=2 so that the scalar mass in

the last term of Eq. (A9) can be diagonalized up to terms of Oðm4A0t=m2

~χχÞ as� 4m2

tχ 4mtχmχχ

4mtχmχχ 4m2χχ þm2

A0t

�≃

�cos θ sin θ

− sin θ cos θ

��m2A0tsin2θ 0

0 4ðm2tχ þm2

χχÞ þm2A0tcos2θ

��cos θ − sin θ

sin θ cos θ

�;

where tan θ≡mtχ=mχχ and the corresponding mass eigen-states h0t and H0

t with mH0t> mh0t

are given as

�h0tH0

t

�¼

�cos θ − sin θ

sin θ cos θ

��Reϕtχ

Reϕχχ

�:

Thus, we find the masses of the two CP-even neutral scalarmesons up to terms of Oðm4

A0t=m2

~χχÞ:

m2h0t≃m2

A0tsin2θ; m2

H0t≃ 4ðm2

tχ þm2χχÞ: ðA11Þ

Equations (A10) and (A11) exactly reproduce the top-modepseudo-Nambu-Goldstone boson mass formulas inEqs. (40) and (41) obtained from the nonlinearLagrangian with the heavy top Higgs integrated out.

APPENDIX B: t; t0-LOOP CORRECTIONS TO THETOP-MODE PSEUDO-NAMBU-GOLDSTONE

BOSON MASSES

In this Appendix, we shall compute the quadratic diver-gent corrections to the top-mode pseudo-Nambu-Goldstonebosons ðh0t ; A0

t Þ arising from the top and t0 loops as the1=Nc-leading contribution in Eqs. (79) and (80).We start with the Yukawa sector in Eq. (60),

Lt;t0yuk ¼ −

fffiffiffi2

p ½yψLðRTUÞψR þ yχtψLðχ1RTUχ3ÞψR þH:c:�:

ðB1ÞFrom this Lagrangian, we find the couplings relevant to theone-loop corrections in the basis of the mass eigenstatesðt; t0Þm:

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055009-15

Lt;t0yukjh0t ¼ −yhtth0t tt − yht0t0h0t t0t0 − ghtLt0Rh

0t ðtLt0R þ H:c:Þ − ghtRt0Lh

0t ðtRt0L þ H:c:Þ

− ghhtth0t h0t tt − ghht0t0h0t h0t t0t0 þ � � � ; ðB2Þ

and

Lt;t0yukjA0

t¼ −yAttA0

t tγ5t − yAt0t0A0t t0γ5t0 − gAtLt0RA

0t tγ5t0 − gAtRt0LA

0t tγ5t0

− gAAttA0t A0

t tγ5t − gAAt0t0A0t A0

t t0γ5t0 þ � � � ; ðB3Þ

where

yhtt ¼yffiffiffi2

p�ðctL cos θ þ stL sin θÞstR − stLc

tR sin θ

�G00

G

��;

ðB4Þ

yht0t0 ¼yffiffiffi2

p�ðstL cos θ − ctL sin θÞctR − ctLs

tR sin θ

�G00

G

��;

ðB5Þ

ghtLt0R ¼ yffiffiffi2

p�ðctL cos θ þ stL sin θÞctR þ stLs

tR sin θ

�G00

G

��;

ðB6Þ

ght0LtR ¼ yffiffiffi2

p�ðstL cos θ − ctL sin θÞstR þ ctLc

tR sin θ

�G00

G

��;

ðB7Þ

ghhtt ¼y

2ffiffiffi2

pf

�ðctL sin θ − stL cos θÞstR þ stLc

tR cos θ

�G00

G

��;

ðB8Þ

ghht0t0 ¼y

2ffiffiffi2

pf

�ðstL sin θ þ ctL cos θÞctR þ ctLs

tR cos θ

�G00

G

��;

ðB9Þ

and

yAtt ¼iyffiffiffi2

p�−stLstR þ stLc

tR

�G00

G

��; ðB10Þ

yAt0t0 ¼iyffiffiffi2

p�ctLc

tR þ ctLs

tR

�G00

G

��; ðB11Þ

gAtLt0R ¼ iyffiffiffi2

p�−stLctR − stLs

tR

�G00

G

��; ðB12Þ

gAt0LtR ¼ iyffiffiffi2

p�ctLs

tR − ctLc

tR

�G00

G

��; ðB13Þ

gAAtt ¼ ghhtt þ3y

4ffiffiffi2

pfsin θ cos θ

×�−ðctL cos θ þ stL sin θÞstR þ stLc

tR sin θ

�G00

G

��;

ðB14Þ

gAAt0t0 ¼ ghht0t0 þ3y

4ffiffiffi2

pfsin θ cos θ

×

�−ðstL cos θ − ctL sin θÞctR þ ctLs

tR sin θ

�G00

G

��:

ðB15Þ

The one-loop corrections arise from Feynman graphsinvolving the top and t0 loops as depicted in Fig. 2. Thequadratic divergent corrections to the top-mode pseudo-Nambu-Goldstone boson masses are thus calculated to be

δm2h0t ;A

0t¼ Nc

4π2Λ2χ · Ch;A; ðB16Þ

where the chiral symmetry breaking scaleΛχ is the cutoff ofthe nonlinear sigma model constructed from Eqs. (28),(43), (60) and (66) and

Ch ¼ −y2htt − y2ht0t0 − g2htLt0R− g2ht0LtR

þ g2hhtt þ g2hht0t0

¼ y2

2

�G00

G

�2

ð2cos2θ − 1Þ; ðB17Þ

FIG. 2. The one-loop diagrams contributing to h0t ; A0t masses as

the quadratic divergent corrections up to OððG00=GÞ2Þ.

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055009-16

CA ¼−y2Att− y2At0t0 − g2AtLt0R− g2At0LtR

þ ghhttgAAttþ ghht0t0gAAt0t0

¼ y2

2

�G00

G

�2 1

2ð1− cos2 θÞð3cos2 θ− 2Þ: ðB18Þ

Here we used the orthogonality relations among the mixingangles stL;R (ctL;R) which follows from the diagonalizationof the fermion mass matrix in Eq. (52) with the rotationmatrices in Eq. (55):

stLstR cos θ

�G00

G

�¼ ðctL sin θ − stL cos θÞctR;

ctLctR cos θ

�G00

G

�¼ ðstL sin θ þ ctL cos θÞstR: ðB19Þ

Thus, we have Eq. (79) and the associated formula for A0t :

δm2h0tjt;t0 ¼ Nc

4π2y2

2

�G00

G

�2

ð2cos2θ − 1ÞΛ2χ

¼ Nc

8π2

� ffiffiffi2

pmt

vEW

�2�2cos2θ − 1

cos2θ

�

× Λ2χ

�1þO

��G00

G

�2��

¼ Last term in Eq: ð79Þ; ðB20Þ

δm2A0tjt;t0 ¼ Nc

4π2y2

2

�G00

G

�2 1

2ð1 − cos2θÞð3cos2θ − 2ÞΛ2

χ

¼ Nc

8π2

� ffiffiffi2

pmt

vEW

�2 ð1 − cos2θÞð3cos2θ − 2Þ2cos2θ

× Λ2χ

�1þO

��G00

G

�2��

; ðB21Þ

where the first lines in Eqs. (B20) and (B21) are exactresults without further higher-order corrections in G00=Gand in the second lines in Eqs. (B20) and (B21) weused m2

t ¼ðy2f2=2Þsin2θcos2θðG00=GÞ2½1þOððG00=GÞ2Þ�which can be derived from Eq. (54) with Eq. (63) andvEW ¼ f sin θ. As noted around Eq. (79), from Eq. (B20),we see that the perturbative G00=G corrections contribute tothe tHiggs mass at the order of OððG00=GÞ2Þ, no correctionof OðG00=GÞ, in accord with the Dashen formula Eq. (51),as well as the A0

t mass in Eq. (B21).In the limit where mt ≪ mt0 ∼ Λχ , the t0 quark may be

integrated out to induce the effective h0t -h0t -t-t andA0t -A0

t -t-t-couplings,

g0hhtt ¼y

2ffiffiffi2

pf

ghtLt0Rght0LtRghht0t0

; g0AAtt ¼−y

2ffiffiffi2

pf

gAtLt0RgAt0LtRghht0t0

:

ðB22Þ

In that case, Ch;A in Eq. (B16) become, up to order of ðG00=GÞ2,

Ch ¼ −y2htt þ g2hhtt þ ghhttg0hhtt

¼ y2

2

�G00

G

�2

ð−1þ 6 cos2θ − 6 cos4θÞ�1þO

��G00

G

�2��

; ðB23Þ

CA ¼ −y2Att þ g2AAtt þ ghhttg0AAtt

¼ y2

2

�G00

G

�2 1

2ð1 − cos2θÞð−2þ 7 cos2θ − 6 cos4θÞ

�1þO

��G00

G

�2��

: ðB24Þ

Thus, we find Eq. (80) and the associated result for A0t :

δm2h0tjt ¼ Nc

4π2y2

2

�G00

G

�2

ð−1þ 6 cos2θ − 6 cos4θÞm2t0

�1þO

��G00

G

�2��

¼ −3

8π2

� ffiffiffi2

pmt

vEW

�2 1 − 6 cos2θ þ 6 cos4θcos2θ

m2t0

�1þO

��G00

G

�2��

¼ Last term in Eq:ð80Þ; ðB25Þ

δm2A0tjt ¼ Nc

4π2y2

2

�G00

G

�2 1

2ð1 − cos2θÞð−2þ 7 cos2θ − 6 cos4θÞm2

t0

�1þO

��G00

G

�2��

¼ −3

8π2

� ffiffiffi2

pmt

vEW

�2 ð1 − cos2θÞð2 − 7 cos2θ þ 6 cos4θÞ2 cos2θ

m2t0

�1þO

��G00

G

�2��

: ðB26Þ

HIGGS BOSON AS A TOP-MODE … PHYSICAL REVIEW D 90, 055009 (2014)

055009-17

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