Revisiting Vector-like Quark Model with Enhanced Top Yukawa Coupling

TL;DR

This paper explores a vector-like quark model with an enhanced top Yukawa coupling, showing that such scenarios are compatible with LHC and precision data and revealing a relation between loop contributions in Higgs pair production.

Contribution

It demonstrates the viability of models with large Yukawa couplings and negative VLQ Yukawa, and uncovers a specific relation between triangle and box diagram contributions in gg → hh.

Findings

Parameter space consistent with LHC bounds is wide.

Enhanced top Yukawa can be realized in strongly interacting theories.

Relation between triangle and box diagram contributions in Higgs pair production.

Abstract

We revisit a scenario with an enhanced top yukawa coupling in vector-like quark (VLQ) models, where the top yukawa coupling is larger than the standard model value and the lightest VLQ has a negative yukawa coupling. We find that the parameter space satisfying the LHC bounds of the Higgs signal strengths consistently with the precision measurements is rather wide. Because the Lagrangian parameters of the yukawa couplings are large, such scenario can be realized in some strongly interacting theories. It also turns out that there is a noticeable relation between the contributions of the triangle and box diagrams in the $gg \to hh$ process by using the lowest order of the $1/M$ expansion where $M$ is the heavy mass running in the loops.

Tables

Table 1. Table 1: Charge assignment of the VLQ model.

Equations

$${\mathcal{L}}_Y$$

$${\mathcal{L}}_{\mathrm{VM}}$$

$${\cal L}_{M}=-(\bar{t}_{L}\;\bar{T}_{L}\;\bar{U}_{L})\;{\cal M}\;\left(\begin{array}[]{c}t_{R}\\ T_{R}\\ U_{R}\end{array}\right)-m_{22}\bar{X}_{L}X_{R}$$

$${\cal M}\equiv\frac{v}{\sqrt{2}}{\mathbf{Y}}\oplus{\mathbf{M}}=\frac{v}{\sqrt{2}}\left(\begin{array}[]{ccc}0&0&y_{13}\\ y_{21}&0&y_{23}\\ 0&y_{32}&0\end{array}\right)+\left(\begin{array}[]{ccc}0&0&0\\ 0&m_{22}&0\\ m_{31}&0&m_{33}\end{array}\right),$$

$${\mathcal{M}}_{\mathrm{diag}}=V_L^{†}\mathcal{M}{V}_R,{\mathcal{M}}_{\mathrm{diag}}=\mathrm{diag}(m_1,m_2,m_3),$$

$$\left(\begin{array}[]{@{}c@{}}t_{L,R}\\ T_{L,R}\\ U_{L,R}\end{array}\right)=V_{L,R}\left(\begin{array}[]{@{}c@{}}t^{\prime}_{L,R}\\ T^{\prime}_{L,R}\\ U^{\prime}_{L,R}\end{array}\right),$$

$${\mathcal{L}}_Y$$

$${\mathbf{G}}^h=\frac{1}{\sqrt{2}}{V}_L^{†}\mathbf{Y}{V}_R.$$

$${\cal M}=M_{X}\left(\begin{array}[]{ccc}0&0&\epsilon\\ 0&1&\xi\\ \epsilon&\xi&a\end{array}\right),$$

$$V_L=({\vec{v}}_t{\vec{v}}_T{\vec{v}}_U),V_R=(-{\vec{v}}_t{\vec{v}}_T{\vec{v}}_U)=V_L\mathrm{diag}(-1,1,1),$$

$$v_{t1}\equiv {\vec{v}}_t^{(1)}=\mathcal{O}(1),v_{T2}\equiv {\vec{v}}_T^{(2)}=\mathcal{O}(1),v_{U3}\equiv {\vec{v}}_U^{(3)}=\mathcal{O}(1),$$

$$\frac{m_t}{M_X}\equiv {\lambda }_1=\mathcal{O}\left(\frac{{ϵ}^2}{a}\right),\frac{M_T}{M_X}\equiv {\lambda }_2<1,\frac{M_U}{M_X}\equiv {\lambda }_3>a.$$

$$-{\lambda }_1+{\lambda }_2+{\lambda }_3=1+a,{\lambda }_1{\lambda }_2{\lambda }_3={ϵ}^2,-G_{tt}^h+G_{TT}^h+G_{UU}^h=0,$$

$$G_{tt}^h$$

$$G_{TT}^h$$

$$\xi^{2}>\frac{1+\lambda_{0}}{2}\Big{(}\epsilon^{2}-(a-1)\lambda_{0}\Big{)},$$

$${\lambda }_0\equiv \frac{4{ϵ}^2}{2a-{ϵ}^2+\sqrt{(2a-{ϵ}^2{)}^2+8{ϵ}^2(1+a)}}.$$

$${\lambda }_1=\frac{{ϵ}^2}{a},{\lambda }_2=1-{\delta }^2,{\lambda }_3=a+\frac{{ϵ}^2}{a}+{\delta }^2,$$

$${\delta }^2\equiv \frac{2{ϵ}^2}{a(a-1)+{ϵ}^2+\sqrt{\{a(a-1)+{ϵ}^2{\}}^2+4a{ϵ}^2}},$$

$$\vec{v}_{t}=v_{t1}\left(\begin{array}[]{c}1\\ \frac{\epsilon^{2}}{a\sqrt{a+\epsilon^{2}}}\\ -\frac{\epsilon}{a}\end{array}\right),\quad\vec{v}_{T}=v_{T2}\left(\begin{array}[]{c}-\frac{\lambda_{3}\delta^{2}}{\sqrt{a+\epsilon^{2}}}\\ 1\\ -\frac{\epsilon}{(\lambda_{3}-1)\sqrt{a+\epsilon^{2}}}\end{array}\right),\quad\vec{v}_{U}=v_{U3}\left(\begin{array}[]{c}\frac{\epsilon(1-\delta^{2})}{a}\\ \frac{\epsilon\sqrt{a+\epsilon^{2}}}{a(\lambda_{3}-1)}\\ 1\end{array}\right),$$

$$v_{t1}^{-1}$$

$$v_{T2}^{-1}$$

$$v_{U3}^{-1}$$

$$(1-{\lambda }_2)({\lambda }_3-1)={\delta }^2({\lambda }_3-1)=\frac{{ϵ}^2}{a},{\lambda }_2{\lambda }_3={\lambda }_3(1-{\delta }^2)=a.$$

$$G_{tt}^h$$

$$G_{TT}^h$$

$$G_{UU}^h$$

$${\mu }_F^{VV}\equiv {\mu }_{\mathrm{ggF}+\mathrm{ttH}}^{VV}$$

$${\mu }_F^{\gamma \gamma }\equiv {\mu }_{\mathrm{ggF}+\mathrm{ttH}}^{\gamma \gamma }$$

$${\kappa }_g$$

$${\kappa }_{\gamma }$$

$${\kappa }_t$$

$${\kappa }_T$$

$$A_1(x)$$

$$A_{\frac{1}{2}}(x)$$

$$f(x)\equiv\left\{\begin{array}[]{lcc}\arcsin^{2}\sqrt{x}&\mbox{for}&x\leq 1,\\[5.69054pt] {\displaystyle-\frac{1}{4}\bigg{[}\,\ln\frac{1+\sqrt{1-x^{-1}}}{1-\sqrt{1-x^{-1}}}-i\pi\,\bigg{]}^{2}}&\mbox{for}&x>1\,.\\ \end{array}\right.$$

Full text

I Introduction

The LHC experiments have discovered a Higgs boson and revealed that its properties are similar to that of the Standard Model (SM) Khachatryan:2016vau . It is thus essential to explore any signatures of physics beyond the SM (BSM). One of the hint is that the observed signal strength of $pp\to t\bar{t}h$ channel is deviated from the SM value about twice, \mu_{tth}=2.3\raisebox{-2.15277pt}{\stackrel{{\scriptstyle+0.7}}{{\mbox{\scriptsize$-0.6$}}}} for the Run 1 combined data Khachatryan:2016vau , \mu_{tth}=1.8\raisebox{-2.15277pt}{\stackrel{{\scriptstyle+0.7}}{{\mbox{\scriptsize$-0.7$}}}} for the ATLAS Run 2 ATLAS:2016axz , and \mu_{tth}=1.5\raisebox{-2.15277pt}{\stackrel{{\scriptstyle+0.5}}{{\mbox{\scriptsize$-0.5$}}}} for the CMS Run 2 in the multilepton final states111 A combined result of the CMS Run 2 has not yet been reported. For other decay channels, \mu_{tth}=1.91\raisebox{-2.15277pt}{\stackrel{{\scriptstyle\tiny+1.5}}{{\mbox{$-1.2$}}}} in the $h\to\gamma\gamma$ decay channel CMS:2016ixj , \mu_{tth}=-0.19\raisebox{-2.15277pt}{\stackrel{{\scriptstyle\tiny+0.80}}{{\mbox{$-0.81$}}}} in the $h\to b\bar{b}$ decay channel CMS:2016zbb , and \mu_{tth}=0.00\raisebox{-2.15277pt}{\stackrel{{\scriptstyle\tiny+1.19}}{{\mbox{$-0.00$}}}} in the $h\to ZZ\to 4\ell$ channel CMS:2017jkd .  CMS:2017vru , although the uncertainties are still large.

These experiments provide a reason for considering models based on strongly interacting theories. In this direction222 Although the top condensate model Miransky:1988xi and the chiral fourth generation 4family directly predict large yukawa couplings, they had been severely constrained. , widely studied are vector-like quark (VLQ) models Lavoura:1992np ; Anastasiou:2009rv ; Aguilar-Saavedra:2013qpa ; Cacciapaglia:2011fx ; Cacciapaglia:2010vn ; Alok:2015iha ; Alok:2014yua ; Cacciapaglia:2015ixa ; Angelescu:2015kga ; Biekotter:2016kgi ; Chen:2017hak , the minimal composite Higgs models (MCHMs) Agashe:2004rs ; Contino:2006qr , and the Little Higgs models ArkaniHamed:2002qy ; Schmaltz:2005ky . We easily find, however, the top yukawa coupling is always suppressed in the VLQ model having only one up-type quark Aguilar-Saavedra:2013qpa . For example, introducing the VLQ $U_{L,R}$ having $+2/3$ electric charge as in the top-seesaw model TSS , the top yukawa coupling is modified as $c_{L}^{2}g_{\bar{t}th}^{\rm SM}$, where $c_{L}\equiv\cos\theta_{L}$ represents the cosine of the mixing angle between $t_{L}$ and $U_{L}$, and we defined the SM top yukawa coupling by $g_{\bar{t}th}^{\rm SM}=m_{t}/v$ with $m_{t}$ and $v$ being the top mass and the vacuum expectation value (VEV) of the Higgs field, respectively. It is the case333 Quite recently, it is shown that the MCHMs with the fermions of the ${\bf 5}+{\bf 10}$ or ${\bf 14}$ representations can have the enhanced or suppressed $tth$ coupling Liu:2017dsz .

for the MCHMs such as MCHM4, MCHM5, and MCHM10 where the fermions are embedded in the spinorial, ${\bf 5}$, and ${\bf 10}$ representations of $SO(5)$, respectively Espinosa:2010vn ; Carena:2014ria . Nevertheless one should not jump to a conclusion: A simple model is effective for a benchmark, but it might be misguided if simplified too much.

In this paper, we reconsider a scenario that the top yukawa coupling is larger than the SM value by ${\cal O}(10\%)$, and the lightest VLQ with the mass around 1 TeV has a negative yukawa coupling of the order of $-m_{t}/v$, introducing more than one up-type VLQ Cheng:2014dwa ; Angelescu:2015kga ; Cacciapaglia:2015ixa . In our scenario, owing to the cancellation among the yukawa couplings, the Higgs signal strengths can be consistent with the experiments. A similar analysis444 In the framework of the two Higgs doublet model, the cancellation mechanism via the light stop was considered in Ref. Badziak:2016exn ; Badziak:2016tzl . See also Ref. Das:2017scg .

was performed in Ref. Angelescu:2015kga . Although the allowed region looked narrow in Ref. Angelescu:2015kga , we find that our scenario is possible in a rather wide parameter space.

We numerically show that our scenario is realized, roughly speaking, when the Lagrangian parameters $y_{ij}$ of the yukawa interactions are large, say, $|y_{ij}|\gtrsim 2$. This may suggest the existence of the underlying strongly interacting models where the dynamically generated yukawa couplings are typically around $3\sim 5$ TSS . As for the di-Higgs production process $gg\to hh$ Kniehl:1995tn ; Falkowski:2007hz ; Low:2009di ; Grober:2010yv ; Gillioz:2012se ; Dawson:2012mk ; Chen:2014xwa ; Grober:2016wmf , we find a noticeable relation between the contributions of the triangle and the box diagrams in the lowest order of the $1/M$ expansion, where $M$ is the relevant heavy mass running in the loops. The di-Higgs production process may give information on the off-diagonal yukawa couplings.

The paper is organized as follows: In Sec. II, we introduce the VLQ model. In Sec. III, we first describe the existence proof of our scenario in an analytical approach, and next show a numerical calculation. Sec. IV is devoted to summary. In Appendix A, the oblique parameters Peskin:1990zt in our model are presented. Analytical expressions of the triangle and the box contributions to the $gg\to hh$ process in the lowest order of the $1/M$ expansion are given in Appendix B.

II Vector-like quark model

Let us introduce two types of the VLQ’s, $U_{L,R}$ and $Q_{L,R}=(X,T)_{L,R}$, having the hypercharges $\frac{2}{3}$ and $\frac{7}{6}$, respectively. (See also Table 1.) Because of no mixing between the bottom quark and VLQ’s, the flavor constraints such as $Z\to b\bar{b}$, etc. can be suppressed in this model. Assuming one Higgs doublet model, the mass terms and the yukawa interactions are

[TABLE]

with $q_{L}=(t,b)_{L}$, and $\tilde{H}\equiv i\tau_{2}H^{*}$. The SM term of $y_{11}\bar{q}_{L}\tilde{H}t_{R}$ was rotated away via the $t_{R}$–$U_{R}$ mixing like in the top seesaw model TSS , while $m_{31}$ is removed in literature Angelescu:2015kga ; Cacciapaglia:2015ixa . We here abbreviated the SM part such as the light quark sector, gauge kinetic terms, etc..

After the electroweak symmetry breaking (EWSB), the mass matrix is then

[TABLE]

with $\langle H\rangle=(0,\;\frac{v}{\sqrt{2}})^{T}$, $v=246$ GeV, and

[TABLE]

where the mass of the $X$ quark with $+5/3$ electric charge is not affected by the EWSB, i.e., $M_{X}=m_{22}$. We diagonalize ${\cal M}$ by

[TABLE]

with $0\leq m_{1}\leq m_{2}\leq m_{3}$, and

[TABLE]

where $(t,T,U)_{L,R}$ and $(t^{\prime},T^{\prime},U^{\prime})_{L,R}$ represent the gauge and mass eigenstates, respectively. Each up-type quark mass is identified by $m_{t}=m_{1}$, $M_{T}=m_{2}$, and $M_{U}=m_{3}$. The yukawa coupling matrix ${\mathbf{G}}^{h}$ in the mass eigenstates is given by

[TABLE]

with

[TABLE]

III Analytical and numerical studies

III.1 Analytical study with crude approximation

We schematically show our scenario having ${\mathbf{G}}^{h}_{11}>m_{t}/v$ and ${\mathbf{G}}^{h}_{22}<0$ is possible in an analytical approach. For this purpose, we employ a crude approximation in this subsection. A numerical study without such approximation will be shown in the next subsection.

Let us take the mass matrix as a symmetric one,

[TABLE]

where we scaled the mass matrix by $M_{X}=m_{22}$. For the perturbation theory, the parameters arising from the yukawa couplings should not be so large, i.e., $\epsilon^{2},\xi^{2}\ll 1$. We also assume $a\geq 1$. We diagonalize ${\cal M}$ by the matrices of

[TABLE]

where the first, the second and the third components ($v_{t1}$, $v_{T2}$ and $v_{U3}$) of $\vec{v}_{t}$, $\vec{v}_{T}$ and $\vec{v}_{U}$ are order of unity,

[TABLE]

respectively, and then obtain the mass eigenvalues,

[TABLE]

In general, by taking the trace and the determinant, we find

[TABLE]

where we defined the diagonal components of the yukawa couplings in the mass basis as ${\mathbf{G}}^{h}_{11,22,33}\equiv G_{tt,TT,UU}^{h}$. More explicitly, the yukawa couplings of $G_{tt}^{h}$ and $G_{TT}^{h}$ are given by

[TABLE]

and the situation of $G_{tt}^{h}>m_{t}/v$ is realized when

[TABLE]

with

[TABLE]

An analytic solution is frequently useful. Let us take $\xi=\frac{\epsilon}{a}\sqrt{a+\epsilon^{2}}$, for example. In this case, we find that the eigenvalues are

[TABLE]

with

[TABLE]

and the corresponding eigenvectors are

[TABLE]

with

[TABLE]

The following relations might be useful:

[TABLE]

Substituting the above results for Eqs. (16)–(18), we explicitly obtain

[TABLE]

In this way, our scenario can be realized in the framework of the VLQ model. The parameter space should be constrained by the $S$ and $T$-parameters, however.

III.2 Numerical study without approximation

Without assuming the symmetric mass matrix (12), we now calculate numerically the signal strengths in our model:

[TABLE]

where $VV=WW$ and $ZZ$, and the scaling factors $\kappa_{g,\gamma,f}$ are defined by

[TABLE]

with $x_{i}\equiv m_{h}^{2}/(4m_{i}^{2})$. The loop functions for spin 1 and $1/2$ are represented by $A_{1}(x)$ and $A_{1/2}(x)$, respectively Gunion:1989we ; Djouadi:2005gi ,

[TABLE]

with

[TABLE]

In our model, the scaling factor $\kappa_{V}$ of the $hWW$ and $hZZ$ couplings is SM-like, $\kappa_{V}=1$. Since we do not change the down quark and lepton sectors, the scaling factors of the bottom and tau are also $\kappa_{b}=\kappa_{\tau}=1$.

By using the results of the LHC Run 1 via the six-parameter fit shown in Ref. Khachatryan:2016vau ,

[TABLE]

we read the $2\sigma$ constraints as

[TABLE]

because of $\mu_{F}^{WW}=\mu_{F}^{ZZ}\equiv\mu_{F}^{VV}$ in our model. On the other hand, the best fit values of $(\sigma\cdot B)^{ZZ}_{ggF}$, $\sigma_{VBF}/\sigma_{ggF}$ and $B^{\gamma\gamma}/B^{ZZ}$ yield \mu_{ggF}^{ZZ}=1.42\raisebox{-2.15277pt}{\stackrel{{\scriptstyle+0.35}}{{\mbox{\scriptsize$-0.31$}}}} and \mu_{ggF}^{\gamma\gamma}=0.67\raisebox{-2.15277pt}{\stackrel{{\scriptstyle+0.25}}{{\mbox{\scriptsize$-0.21$}}}} in the ATLAS Run 2 ATLAS:2016hru . The signal strengths in the CMS Run 2 are \mu_{ggF}^{ZZ}=1.20\raisebox{-2.15277pt}{\stackrel{{\scriptstyle+0.22}}{{\mbox{\scriptsize$-0.21$}}}} CMS:2017jkd and \mu_{ggF}^{\gamma\gamma}=0.77\raisebox{-2.15277pt}{\stackrel{{\scriptstyle+0.25}}{{\mbox{\scriptsize$-0.23$}}}} CMS:2016ixj . One should keep in mind that both of the Run 2 results for $\mu_{ggF}^{\gamma\gamma}$ are much smaller than that of the LHC Run 1.

The parameter space of the mass matrix (4) is constrained by the precision measurements Peskin:1990zt . Especially, owing to the mixing among $t$, $T$ and $U$, the $T$-parameter is potentially large. We explicitly show the expression of the $S$ and $T$-parameters in our model in Appendix A Lavoura:1992np ; Anastasiou:2009rv . Fixing $U=0$, we impose the constrains Olive:2016xmw ,

[TABLE]

where $m_{t}=173.3$ GeV, $m_{b}=4.2$ GeV, and $m_{h}=125.1$ GeV Olive:2016xmw .

We now describe the numerical results. In the following analysis, we take the Higgs mass, the pole mass of the top, the $\overline{\rm MS}$ mass of the bottom, and the CKM matrix element for $t$ and $b$ as $m_{h}=125.1$ GeV, $m_{t}^{\rm pole}=173.2$ GeV, $m_{b}^{\overline{\rm MS}}=4.2$ GeV, and $|V_{tb}|=0.95$, respectively. The relation $m_{1}=m_{t}$ must hold. Although strong couplings are acceptable in our scenario, we may impose $|y_{ij}|<5$ for the Lagrangian parameters in Eq. (4). Even in this case, there is still wide parameter space, as we will see below. Considering the lower mass bound for the $T$ quark ATLAS:2016btu , we fix $M_{T}=1.2$ TeV and take the mass range for the heavier VLQ to $1.5\leq M_{U}\leq 3.5$ TeV.

The signal strengths of $\mu_{F}^{VV}$ and $\mu_{F}^{\gamma\gamma}$ are depicted in Figs. 1 and 2. For the red points, the $S,T$ constraints and the $2\sigma$ bounds (46) of the Higgs signal strengths are satisfied, while the green points are outside of the $2\sigma$ bounds (46). For the blue points in Figs. 1 and 2, $G_{tt}^{h}>g_{tth}^{\rm SM}$ and $G_{TT}^{h}>0$. We did not plot the data with $G_{tt}^{h}<g_{tth}^{\rm SM}$ in our model, although they exist. We also show the results for MCHM4 and MCHM5, where the scaling parameters are $\kappa_{V}=\sqrt{1-\xi}$ for both and $\kappa_{f}=\sqrt{1-\xi}$ and $\kappa_{f}=(1-2\xi)/\sqrt{1-\xi}$ for MCHM4 and MCHM5, respectively, with $\xi=v^{2}/f^{2}$ and $f$ being the typical scale of the MCHMs Espinosa:2010vn ; Carena:2014ria ; Kanemura:2016tan ; Liu:2017dsz ; Sanz:2017tco . The $2\sigma$ constraint of the top yukawa coupling from the Run 1 combined data Khachatryan:2016vau , which reads $1.05<\kappa_{t}<1.92$, is also shown in Figs. 1 and 2 with the proviso that the Run 2 data do not restrict the top yukawa so much yet, within the $2\sigma$ bounds, $0.63<\kappa_{t}<1.79$ (ATLAS Run2 ATLAS:2016axz ) and $0.71<\kappa_{t}<1.58$ (CMS Run2 CMS:2017vru ). In passing, we comment that there is a parameter space inside of the $2\sigma$ bounds (46), even if we take $M_{T}=2.0$ TeV. Although the window is closed at $M_{T}=2.4$ TeV under the condition $|y_{ij}|<5$, the parameter space still exists even for $M_{T}=3.0$ TeV, if we allow $|y_{ij}|>5$.

In our scenario, we require $G_{tt}^{h}>g_{tth}^{\rm SM}$ and $G_{TT}^{h}<0$ in order for the Higgs signal strengths to be consistent with the experiments. For this cancellation mechanism among the diagonal yukawa couplings in the gluon fusion process, we show the normalized yukawa couplings of $\kappa_{T,U}=G_{TT,UU}^{h}/g_{tth}^{\rm SM}$ vs $\kappa_{t}=G_{tt}^{h}/g_{tth}^{\rm SM}$ in Fig. 3. At the red points, the conditions of $G_{tt}^{h}>g_{tth}^{\rm SM}$ and $G_{TT}^{h}<0$, the $2\sigma$ constraints (46), and the $S,T$-constraints (47) are satisfied. On the other hand, the green points are outside of the $2\sigma$ constraints (46). We find that the cancellation mechanism works up to $\kappa_{t}\lesssim 1.4$.

The Lagrangian parameters are important for the model-building. We depict them in Fig. 4. The entry of $y_{21}$ can be either positive or negative. The vanishing $y_{21}$ is also possible. This is consistent with the analytical approach in the previous subsection. Although $y_{23}$ barely takes negative or small positive values, $y_{ij}$ except for $y_{21}$ are positive and large, say, $y_{ij}\gtrsim 2$, in the wide parameter space. Thus the VLQ model discussed here might be provided from some underlying theories based on strongly interacting systems.

In the end of this subsection, we comment on the di-Higgs production via the gluon fusion. The off-diagonal yukawa couplings can be extracted from the decay channels such as $T\to th$. Also, they contribute to the box diagram of the di-Higgs production, so that the $gg\to hh$ process may give us further information on the model parameters. In the lowest order of the $1/M$ expansion (LET), the triangle and box contributions normalized by the SM values are Kniehl:1995tn ; Falkowski:2007hz ; Low:2009di ; Gillioz:2012se ; Dawson:2012mk ; Chen:2014xwa

[TABLE]

respectively. Note that $R_{gg\to h}^{\rm tri}\approx\kappa_{g}$, because of $A_{\frac{1}{2}}(x_{t})\simeq A_{\frac{1}{2}}(x_{T})\simeq A_{\frac{1}{2}}(x_{U})\approx A_{\frac{1}{2}}(0)=4/3$. The numerical result is depicted in Fig. 5. Also, we can analytically obtain the expressions of $R_{gg\to h}^{\rm tri}$ and $R_{gg\to hh}^{\rm box}$ in the general case (4) as

[TABLE]

where $b_{22}$ denotes the $(2,2)$ element of the dimensionless mass matrix inverse $M_{X}{\cal M}^{-1}$. See also Appendix B. This analytic result is shown as the blue curve in Fig. 5. Under the symmetric mass matrix assumption (12) in the previous subsection, we find $R_{gg\to h}^{\rm tri}=R_{gg\to hh}^{\rm box}=1$ owing to $b_{22}=1$. It then turns out that the box contribution $R_{gg\to hh}^{\rm box}$ is decreasing with respect to the increasing triangle contribution $R_{gg\to h}^{\rm tri}$ under the $2\sigma$ constraints (46). Since the box is destructive in $gg\to hh$, it means that the di-Higgs production is either much enhanced or suppressed. In Fig. 6, we show the ratio $\sigma_{\rm LET}/\sigma_{\rm LET}^{\rm SM}$ of the total cross section of the Higgs pair production through $gg$ in $pp$ collisions normalized by the SM one under the LET approximation Kniehl:1995tn ; Falkowski:2007hz ; Low:2009di ; Gillioz:2012se ; Dawson:2012mk ; Chen:2014xwa . We took the LHC center of mass energy as $\sqrt{s}=14$ TeV and used the CT14 LO PDF set Dulat:2015mca . The renormalization scale ($\mu$) and the factorization scale ($Q$) are chosen equal to the invariant mass ($M_{hh}=\sqrt{\hat{s}}$) of the Higgs pair, $\mu=Q=M_{hh}$. We find that the ratio can be increasing/decreasing about 40%, depending on the values of $R_{gg\to h}^{\rm tri}\approx\kappa_{g}$. This might be striking, although one should take notice of inaccuracy of the LET approximation. A detailed analysis will be performed elsewhere.

IV Summary and discussions

We revisited the scenario with the enhanced top yukawa coupling in the framework of the VLQ model. We found that the scenario can be realized in the rather wide parameter space. Since the Lagrangian parameters of the yukawa couplings except for $y_{21}$ are positive and large, such VLQ model can be obtained from some underlying strong dynamics. We also calculated the ratios of the triangle and box diagrams to the SM values in the $gg\to hh$ process and found the noticeable relation. The detailed studies will be done in future.

Appendix A $S,T$-parameters in the VLQ model

The parameter space of the VLQ models is severely restricted by the oblique corrections Peskin:1990zt . In particular, the $T$-parameter is essential. In our model, it reads Lavoura:1992np ; Anastasiou:2009rv ,

[TABLE]

where $N_{c}=3$, and we defined $s_{W}\equiv\sin\theta_{W}$ and $c_{W}\equiv\cos\theta_{W}$ with $\theta_{W}$ being the weak mixing angle, and also

[TABLE]

[TABLE]

and

[TABLE]

with $\Delta$ being the divergent term in the dimensional regularization. The mass eigenvalues of the up-type quarks are $m_{1}=m_{t}$, $m_{2}=M_{T}$ and $m_{3}=M_{U}$. The rotation matrices are defined by

[TABLE]

where $(t,T,U)_{L,R}$ and $(t^{\prime},t^{\prime},U^{\prime})_{L,R}$ are the gauge and mass eigenstates, respectively.

We have left the divergent term $\Delta$ for checking of the calculations Anastasiou:2009rv . By using the unitarity and the mass relations

[TABLE]

and also

[TABLE]

we can confirm that the divergent term $\Delta$ is exactly canceled out, as it must be.

The deviation from the SM is given by

[TABLE]

with

[TABLE]

Throughout the paper, we take the $1\sigma$ constraint, $\Delta T=0.10\pm 0.07$ Olive:2016xmw .

The $S$-parameter constraint is not so severe, compared with the $T$-parameter. The expression for the $S$-parameter in our model is as follows Lavoura:1992np :

[TABLE]

where we defined

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Note that $\chi_{+}(y_{i},y_{i})=\chi_{-}(y_{i},y_{i})=0$.

The deviation from the SM is given by

[TABLE]

with

[TABLE]

Throughout the paper, we take the $1\sigma$ constraint, $\Delta S=0.07\pm 0.08$ Olive:2016xmw .

Appendix B Analytical expression of $R_{gg\to h}^{\rm tri}$ and

$R_{gg\to hh}^{\rm box}$

For the general mass matrix (4) in our model, we define the dimensionless matrices as follows:

[TABLE]

where we scaled by $M_{X}=m_{22}$. We assume $a_{13}\neq 0$ for getting $m_{t}\neq 0$. We then read Eqs. (48) and (49) as

[TABLE]

Let us determine the dimensionless mass matrix inverse $\widetilde{{\cal M}}^{-1}$. The definition of the inverse matrix, $\widetilde{{\cal M}}\widetilde{{\cal M}}^{-1}=\widetilde{{\cal M}}^{-1}\widetilde{{\cal M}}=\mathrm{diag}(1,1,1)$, yields the expression for $\widetilde{{\cal M}}^{-1}$ as

[TABLE]

and the matrix elements $b_{ij}$ satisfy

[TABLE]

Therefore we analytically obtain $b_{ij}$,

[TABLE]

and

[TABLE]

By using $a_{21}b_{12}=a_{32}b_{23}=1-b_{22}$, we find

[TABLE]

and

[TABLE]

Eliminating $b_{22}$ from the above equations, we obtain Eq. (50),

[TABLE]

When $a_{21}=0$ as in Sec. III.1, we immediately find $b_{22}=1$ and thereby obtain $R_{gg\to h}^{\rm tri}=R_{gg\to hh}^{\rm box}=1$.

Acknowledgements.

The author thank to A. Deandrea and G. Cacciapaglia for useful comments. Numerical computation in this work was carried out at the Yukawa Institute Computer Facility. This work is supported by JSPS Grant-in-Aid for Scientific Research No. 17K05423 and partially by the France-Japan Particle Physics Lab (TYL/FJPPL).