Approximate Alignment in Two Higgs Doublet Model with Extra Yukawa Couplings

TL;DR

This paper explores how a two Higgs doublet model with extra Yukawa couplings can naturally explain the observed Higgs boson properties, potentially enable electroweak baryogenesis, and suggests new particles could be detectable at the LHC.

Contribution

It introduces a scenario where Higgs quartic couplings are of order one, linking alignment with new physics and proposing testable predictions for collider experiments.

Findings

Alignment can arise with Higgs quartic couplings of order one.

Extra top Yukawa couplings can facilitate electroweak baryogenesis.

Exotic Higgs bosons may be sub-TeV and yet undetected at current colliders.

Abstract

With discovery of the 125 GeV boson $h^0$, the existence of a second doublet is very plausible. We show that the "alignment" phenomenon, that $h^0$ is found to resemble closely the Standard Model Higgs boson, may correspond to Higgs quartic couplings $\eta_i$ that are ${\cal O}(1)$ in strength. If the exotic bosons of the second doublet possess extra top Yukawa couplings, which are the least constrained by data, such a two Higgs doublet model could drive electroweak baryogenesis, as well as further "protect" the apparent alignment. The exotic Higgs bosons can be sub-TeV in mass while remaining well hidden so far, with broad parameter space for search at the Large Hadron Collider.

Equations

$$V(\Phi ,{\Phi }^{\prime })={\mu }_{11}^2|\Phi {|}^2+{\mu }_{22}^2|{\Phi }^{\prime }{|}^2-\left({\mu }_{12}^2{\Phi }^{†}{\Phi }^{\prime }+h.c.\right)$$

$$+\frac{{\eta }_1}{2}|\Phi {|}^4+\frac{{\eta }_2}{2}|{\Phi }^{\prime }{|}^4+{\eta }_3|\Phi {|}^2|{\Phi }^{\prime }{|}^2+{\eta }_4|{\Phi }^{†}{\Phi }^{\prime }{|}^2$$

$$+\left\{\frac{{\eta }_5}{2}({\Phi }^{†}{\Phi }^{\prime }{)}^2+\left[{\eta }_6|\Phi {|}^2+{\eta }_7|{\Phi }^{\prime }{|}^2\right]{\Phi }^{†}{\Phi }^{\prime }+h.c.\right\}.$$

$${\mu }_{11}^2=-\frac{1}{2}{\eta }_1{v}^2,{\mu }_{12}^2=\frac{1}{2}{\eta }_6{v}^2.$$

$$m_{H^{\pm }}^2$$

$$m_A^2$$

$$\displaystyle M_{\textrm{even}}^{2}=\left[\begin{array}[]{cc}\eta_{1}v^{2}&\eta_{6}v^{2}\\ \eta_{6}v^{2}&\mu_{22}^{2}+\frac{1}{2}(\eta_{3}+\eta_{4}+\eta_{5})v^{2}\\ \end{array}\right],$$

$$\displaystyle R^{T}_{\gamma}M_{\textrm{even}}^{2}R_{\gamma}=\left[\begin{array}[]{cc}m_{H}^{2}&0\\ 0&m_{h}^{2}\\ \end{array}\right],\quad R_{\gamma}=\left[\begin{array}[]{cc}c_{\gamma}&-s_{\gamma}\\ s_{\gamma}&c_{\gamma}\\ \end{array}\right],$$

$$c_{\gamma }^2=\frac{{\eta }_1{v}^2-m_h^2}{m_H^2-m_h^2},\sin 2\gamma =\frac{2{\eta }_6{v}^2}{m_H^2-m_h^2}.$$

$$c_{\gamma }$$

$$|{\eta }_6|\sim \mathcal{O}(1),m_H^2-m_h^2\gtrsim \mathrm{several}{v}^2.$$

$$\Delta T^{2HDM}=\Delta T_{SS}+\Delta T_{SV},$$

$${\eta }_4={\eta }_5\equiv {\eta }^{\prime },(\mathrm{custodial}\mathrm{SU}(2))$$

$${\mu }_{22}^2/v^2+{\eta }_3/2\simeq 2,3.7,6,$$

Full text

Introduction.—

The 125 GeV scalar boson $h^{0}$ discovered h125_discovery in 2012 resembles closely Khachatryan:2016vau the Higgs boson of the Standard Model (SM), but no sign of physics beyond SM has emerged so far at the Large Hadron Collider (LHC). This includes supersymmetry (SUSY), where its scale could be Craig ; Athron:2017yua at several TeV, with some light state(s) perhaps still elusive.

The $h^{0}$ boson belongs to the weak doublet $\Phi$ that has a vacuum expectation value (VEV) $v\cong 246$ GeV, inducing electroweak symmetry breaking (EWSB). Since all weak fermion doublets come in three copies, it is reasonable that a second scalar doublet $\Phi^{\prime}$ exists, and the physical $H^{0}$, $A^{0}$ and $H^{\pm}$ bosons should be pursued at the LHC. But where are they? In fact, SUSY implies two Higgs doublets, but a recent minimal SUSY (MSSM) fit places the exotic scalars at 5 TeV Athron:2017yua , which is in the decoupling limit Gunion:2002zf , and would be bad news for the LHC search. It echoes also the apparent “alignment”, the observed proximity Khachatryan:2016vau of $h^{0}$ to the SM Higgs boson. The MSSM fit, however, is actually quite “flat” MWhite in $m_{\Phi^{\prime}}$, and in any case does not stop phenomenological studies Craig:2013hca ; Carena:2013ooa ; Carena:2014nza ; Haber:2015pua ; Bernon:2015qea ; Bechtle:2016kui ; Han:2017pfo from probing below the TeV scale.

Most phenomenological and experimental studies for the LHC are in the SUSY-inspired two Higgs doublet model (2HDM), called 2HDM-II, where up- and down-type quarks receive mass from separate scalar doublets. In this way, mass and Yukawa matrices are diagonalized simultaneously, and flavor changing neutral Higgs (FCNH) couplings are absent. This is the Natural Flavor Conservation (NFC) criteria Glashow:1976nt of Glashow and Weinberg, where a $Z_{2}$ symmetry is invoked for the Higgs and fermion fields. As such, 2HDM-II (and also the so-called 2HDM-I) has no new Yukawa couplings.

It is, however, the unique $CP$ violating (CPV) phase in quark mixing, rooted in Yukawa couplings, that accounts for all laboratory-observed CPV phenomena. Given the deep shortfall from explaining the baryon asymmetry of the Universe (BAU), it is natural to ask whether extra Yukawa couplings could be considered without the usual $Z_{2}$. It was noted long ago Cheng:1987rs that, given the trickle-down pattern or mass suppression of off-diagonal quark mixings, NFC (hence the $Z_{2}$ symmetry) may not be necessary, while $t\to ch$ decay induced by the least constrained $tch$ coupling could serve as hallmark Hou:1991un . After the $h^{0}$ discovery, it was further advocated Chen:2013qta that, much like flavor couplings within SM, one should just let Nature decide on the FCNH couplings.

Starting from the general 2HDM without the $Z_{2}$ symmetry, we show that approximate alignment may reflect ${\cal O}(1)$ couplings in the Higgs potential. Such couplings could bring about first order electroweak phase transition, which in fact could work also for 2HDM-II. But the bonus Fuyuto:2017ewj for the general 2HDM is that it could attain electroweak baryogenesis (EWBG), i.e. generate BAU, where new ${\cal O}(1)$ top quark Yukawa couplings provide new sources of CPV. We argue that NFC protection against FCNH can be replaced by approximate alignment, together with a flavor organizing principle reflected in SM itself. Interestingly, the ${\cal O}(1)$ extra top Yukawa coupling may help protect alignment Hou:2017vvp .

2HDM-II: Alignment without Decoupling.—

In want of additional scalar bosons at sub-TeV scale, the issue of “alignment without decoupling” has certainly been a topic of discussion Gunion:2002zf ; Craig:2013hca ; Carena:2013ooa ; Carena:2014nza ; Haber:2015pua ; Bernon:2015qea ; Bechtle:2016kui ; Han:2017pfo . Let us define alignment more precisely: in notation of 2HDM-II, near alignment means the mixing angle $\cos(\beta-\alpha)$ between $h^{0}$ and the heavy $CP$-even scalar $H^{0}$ is small, where $\tan\beta=v_{2}/v_{1}$ is the ratio of VEVs of the two doublets, distinguished by the $Z_{2}$ symmetry that enforces NFC, while $\alpha$ is a mixing parameter arising from the Higgs potential.

It is useful to clarify some confusion brought about by this notation. Given that $\beta$ and $\alpha$ have rather different origins, one may ask: Why should $\beta\simeq\alpha\pm\pi/2$? This is exacerbated by claimed “solutions” for alignment without decoupling. For example, whether in the form of Eq. (4.8) or Eq. (4.14) of Ref. Carena:2013ooa , the proposed solutions equate $\tan\beta$ with some combination of quartic Higgs couplings, be it from Higgs potential, or from loop effects. One has alignment without decoupling if the equality holds approximately, but the question then is why would $\tan\beta$ match the right-hand side. Efforts have been made to elucidate Carena:2013ooa ; Carena:2014nza ; Bechtle:2016kui the situation in MSSM, where one needs fine-tuned cancellations between tree level and up to two-loop corrections Bechtle:2016kui from top (squark) sector. As we will discuss shortly, $\beta-\alpha$ should be viewed as a single angle between the so-called Higgs basis Botella:1994cs and neutral Higgs mass basis HHaber .

The sense of “fine-tuning” might reflect some underlying symmetry. Ref. Dev:2014yca found three maximal symmetries (with SO(5) the simplest) of the Higgs potential for two Higgs doublets, that “naturally realize the alignment limit” without need of large $\tan\beta$, hence may be an improvement. Renormalization group effects would break this symmetry, hence deviate from alignment. In this Letter, we take a simpler approach: rather than more symmetry that may link to high scales, we focus on the Higgs potential and remove all symmetry, which is opposite the direction of exploration from Ref. Dev:2014yca . In the following, we explore the Higgs sector, and return to flavor and phenomenology issues subsequently.

General Higgs Potential and Alignment.—

The Higgs potential of the 2HDM without $Z_{2}$ symmetry is,

[TABLE]

The usual notation for Higgs potential with $Z_{2}$ symmetry, where $\Phi\to\Phi$ and $\Phi^{\prime}\to-\Phi^{\prime}$ under $Z_{2}$ is in form of $m^{2}_{ij}$ and $\lambda_{i}$, with $\lambda_{6}=\lambda_{7}=0$. We take $V$ as $CP$ conserving, hence all parameters are real for simplicity. Without $Z_{2}$, $\Phi$ and $\Phi^{\prime}$ cannot be distinguished (and $\tan\beta$ is unphysical), we can choose the (Higgs) basis where $\Phi$ generates $v$ for EWSB, with minimization conditions Davidson:2005cw ; Haber:2006ue ; Haber:2010bw

[TABLE]

The second condition reduces the parameter count by one. To be explicit, Eq. (1) has the parameters $v$, $\mu_{22}^{2}$ (positive definite) and $\eta_{i}$ with $i=1$–7; with $Z_{2}$, the parameters are $v$, $\tan\beta$, $m_{12}^{2}$ (soft breaking) and $\lambda_{i}$ with $i=1$–5. As the Higgs basis always exist, a relation (Eq. (54) of Ref. Haber:2015pua ) between $\eta_{i}$’s must be satisfied to guarantee one could rotate to the “$Z_{2}$ basis” Haber:2015pua where $\lambda_{6}=\lambda_{7}=0$, through which the $\tan\beta$ value can be found HHaber . Thus, our following discussion applies also to 2HDM-II.

The charged and $CP$-odd Higgs masses are,

[TABLE]

For the $CP$-even Higgs mass matrix, one has

[TABLE]

which is diagonalized by

[TABLE]

with convention similar to 2HDM-II: $c_{\gamma}\equiv\cos\gamma$ replaces $\cos(\beta-\alpha)$, and $s_{\gamma}\equiv\sin\gamma$. It is now clear that $\gamma\equiv\beta-\alpha$ is the relative angle between the Higgs basis and the neutral Higgs mass basis, and is basis-independent.

Taking $m_{h}\cong 125$ GeV the observed value, we do not give the detailed formula for $m_{H}$, but note the mixing angle $c_{\gamma}$ satisfies two relations,

[TABLE]

In alignment limit of $c_{\gamma}\to 0$, $s_{\gamma}\to-1$, one has $\eta_{1}\to m_{h}^{2}/v^{2}\simeq 0.26$. For $c_{\gamma}$ small but nonvanishing, $\eta_{1}v^{2}-m_{h}^{2}$ can be weighed down by $m_{H}^{2}-m_{h}^{2}$. Since $s_{\gamma}\to-1$ always holds better than $c_{\gamma}\to 0$, the second relation gives

[TABLE]

which appears e.g. in Refs. Bernon:2015qea ; Bechtle:2016kui .

The discussion in Ref. Bernon:2015qea is more general (see below), but both Refs. Bernon:2015qea ; Bechtle:2016kui focused on $|\eta_{6}|\ll 1$ in their discussions. We stress, however, that small $\eta_{6}$ is in general not needed in the 2HDM: $c_{\gamma}$ can be small for

[TABLE]

Note that a low $m_{h}^{2}/v^{2}\simeq 0.26$ is not required, i.e. $c_{\gamma}$ can be small even for $m_{h}\sim 300$ GeV, so long that $m_{H}^{2}-m_{h}^{2}\gtrsim{\rm several}\;v^{2}$. But $m_{H}^{2}-m_{h}^{2}\gtrsim{\rm several}\;v^{2}$ implies that at least one of $\eta_{3}$, $\eta_{4}$, $\eta_{5}$ is ${\cal O}(1)$, or (inclusive) $\mu_{22}^{2}>v^{2}$. Furthermore, for $\eta_{6}\sim{\cal O}(1)$, level repulsion could drive $m_{H}^{2}-m_{h}^{2}$ apart and push $m_{h}^{2}/v^{2}$ down from $\eta_{1}\sim{\cal O}(1)$. On the other hand, given the observed $m_{h}^{2}/v^{2}\sim 0.26$, if one allows the tuning of $\eta_{6}\lesssim 1/4$, one could get extreme alignment, i.e. $c_{\gamma}\to 0$, resulting in $\eta_{1}\to 0.26$.

But in general, $\eta_{6}$ need not be small! This can in fact be seen from Fig. 1 of Ref. Bernon:2015qea , the numerical study of 2HDM-II and 2HDM-I: $|Z_{6}|\sim 1$ can still give $|c_{\beta-\alpha}|\sim 10\%$ (or more) for $m_{H}\lesssim$ TeV, where $Z_{6}$ is equivalent to our $\eta_{6}$. However, as mentioned, this study focused on very small $|Z_{6}|$ values, and $|Z_{6}|\sim 1$ was barely discussed.

Numerical Illustration with Custodial SU(2).—

Our presentation so far has been general, and as we have just commented, does not really depend on whether a $Z_{2}$ symmetry is imposed or not. But with $\eta_{i}\sim{\cal O}(1)$, one should check precision electroweak constraints. There are two corrections to the $T$ parameter,

[TABLE]

where the $SS$ term involves only scalars, while the $SV$ term involves both scalars and vectors. We note that

[TABLE]

gives $m_{A}=m_{H^{+}}$ hence restores the custodial SU(2) symmetry Pomarol:1993mu ; custodial , which can be verified through $V(\Phi,\,\Phi^{\prime})$. Assuming Eq. (17), the $\Delta T_{SS}$ term vanishes. However, $\Delta T_{SV}$ does not vanish because $M_{W}\neq M_{Z}$, and provides some constraint as it is proportional to $c_{\gamma}^{2}$.

Taking the $T$ parameter constraint into account, we illustrate approximate alignment for the simplified case of custodial SU(2), i.e. $m_{A}^{2}=m_{H^{+}}^{2}$. We plot $m_{H}$ vs $c_{\gamma}$ in Fig. 1[left] for fixed $m_{A}=m_{H^{+}}=350$, 475, 600 GeV, and for $\eta^{\prime}$ (or $\eta_{4}=\eta_{5}$) ranging from 0.5 to 2. We see that the 95% C.L. limit from $S$-$T$ parameter data Baak:2012kk cuts the solution space off around $-c_{\gamma}\sim 0.3$, illustrated by solid lines that turn into dotted lines. This corresponds to $-s_{\gamma}\simeq 0.95$, which is still close to alignment. To be conservative, one can take $-c_{\gamma}<0.2$, or $-s_{\gamma}>0.98$, which is rather close to the alignment limit.

In Fig. 1[right] we plot $\eta_{1}$ vs $\eta_{6}$, where $-c_{\gamma}=0.1,\ 0.2,\ 0.3$ ($-s_{\gamma}\cong 0.995,\ 0.980,\ 0.954$) are marked by filled circles. We see that, for $m_{A}=m_{H^{+}}=350$, 475, 600 GeV, one has $-c_{\gamma}\lesssim 0.2$ for $\eta_{6}\lesssim 0.5$, 1, 1.5, respectively, i.e. close to alignment, while for $\eta_{6}\lesssim m_{h}^{2}/v^{2}\cong 0.26$, one is very close to the alignment limit. We note further that, while $\eta_{1}<\eta_{6}$ in general, there is large parameter space where both are ${\cal O}(1)$, and $m_{h}^{2}$ is pushed down from larger $\eta_{1}v^{2}$ values to what is observed by level repulsion. We stress that Fig. 1 is an illustration, not yet a scan.

Although $m_{A}=m_{H^{+}}=350$, 475, 600 GeV do not distinguish between $\mu_{22}^{2}$ and $\eta_{3}v^{2}/2$, they imply

[TABLE]

respectively. If one assumes equal share between the two terms, one has $\eta_{3}\sim 2$ for 350 GeV, while for 600 GeV, $\eta_{3}\sim 6$ is becoming sizable. If we keep $\eta_{3}\lesssim 3$ for perturbativity, then $\mu_{22}^{2}\gtrsim(367\ {\rm GeV})^{2},\ (519\ {\rm GeV})^{2}$, respectively, for $m_{A}=m_{H^{+}}=475$, 600 GeV. Thus, although our alignment discussion is not affected, the inertial $\mu_{22}^{2}$ would take up more role for $m_{A}$, $m_{H^{+}}\gtrsim 500$ GeV. In this context we remark that, just having one $|\eta_{i}|\sim 1$ implies the Landau pole is not far above 10 TeV. Thus, ${\cal O}(1)$ Higgs quartic couplings imply a nearby strong coupling scale, which would be a boon to a 100 TeV $pp$ collider. Some Higgs quartic couplings could be negative, however, hence the Landau pole scale is rather uncertain.

One-loop Protection.—

So far our discussion does not depend on imposing a $Z_{2}$ symmetry or not. We wish, however, to further point out a mechanism Hou:2017vvp , within 2HDM without $Z_{2}$, where alignment could be “protected” at loop level. With all $\eta_{i}\sim{\cal O}(1)$, the one loop corrections to, e.g. $\Gamma(h\to ZZ^{*})$ by the extra scalars are significant, resulting in reduction. But while Yukawa couplings of $\Phi$ (Eq. (1)), the mass-giving doublet, are diagonalized with fermion masses, those of $\Phi^{\prime}$ are not, and the point Chen:2013qta of general 2HDM is that Nature has some flavor organizing principle that suppresses FCNH involving low mass fermions. Given $\lambda_{t}\cong 1$, however, the top-related extra Yukawa couplings $\rho_{tt}$ and $\rho_{ct}$ could be ${\cal O}(1)$ Chen:2013qta . We have shown Hou:2017vvp that, for $\rho_{tt}c_{\gamma}<0$, the top loop effect enhances $\Gamma(h\to ZZ^{*})$, and could compensate a negative deviation due to bosonic loops. Thus, approximate alignment may be only apparent, although $\rho_{tt}c_{\gamma}$ of wrong sign would be in the wrong direction.

In Fig. 2 we plot $\kappa_{Z}$, the one-loop corrected $hZZ$ coupling Hou:2017vvp in the general 2HDM normalized to SM, vs $\rho_{tt}$ for $m_{A}=m_{H^{+}}$ as before, and for $\eta^{\prime}=0.5$, 2 and several $\eta_{6}\leq 1$ values. As $c_{\gamma}<0$, we see that $\rho_{tt}>0$ could bring $\kappa_{Z}$ back to 1, but for $\rho_{tt}<0$, the top and bosonic loops together can push $\kappa_{Z}$ sufficiently below SM expectation, which is not observed Khachatryan:2016vau . Note that for heavier $H^{0}$, the $\rho_{tt}<0$ case is still allowed at present. We have checked that the trend for $h\to\gamma\gamma$ is similar, but because the experimental errors Khachatryan:2016vau are more forgiving, they do not constrain the plots in Fig. 2.

Thus, 2HDM without $Z_{2}$ can not only give rise to small $|c_{\gamma}|$, but provide a loop protection mechanism.

Phenomenology.—

For the phenomenology of 2HDM without $Z_{2}$ symmetry, one first needs to address the recent bound Misiak:2017bgg of $m_{H^{+}}\gtrsim 580$ GeV. This holds actually for 2HDM-II, as the $H^{+}$ effect is always bsga2HDM constructive with SM in the $b\to s\gamma$ decay amplitude, with leading piece independent of $\tan\beta$. Thus, with the Belle update Belle:2016ufb of $B\to X_{s}\gamma$ being slightly below SM expectation, a more stringent limit follows. In the 2HDM without $Z_{2}$, however, there is no $\tan\beta$, while the new Yukawa couplings $\rho_{ct}$, $\rho_{tt}$ (even $\rho_{cc}$) and $\rho_{bb}$ enter the loop Chen:2013qta , with $\rho_{bb}$ constrained to be small ($|\rho_{bb}|\lesssim\lambda_{b}$, which is reasonable) because of chiral enhancement. These couplings are further complex, hence $b\to s\gamma$ carves out a solution space in the aforementioned parameters, but the 580 GeV bound on $m_{H^{+}}$ valid for 2HDM-II does not apply.

With $\tan\beta$ removed as a parameter, the extra Higgs boson search strategy should be revised. For example, $H^{0}/A^{0}\to\tau^{+}\tau^{-}$ and $b\bar{b}$ search are in the wrong direction, as the extra Yukawa couplings should be rather small. With $\rho_{tt}\sim{\cal O}(1)$, however, the neutral states $A^{0}$ and $H^{0}$ are produced via gluon fusion and decay strongly to $t\bar{t}$, which is hard to disentangle Frederix:2007gi ; Carena:2016npr from the large $gg\to t\bar{t}$ background, due to distortion from interference and detector smearing. The ATLAS study ATLAS:2016pyq with 8 TeV data takes interference effects into account, but leverage is mainly for $\tan\beta\lesssim 0.7$ in 2HDM-II, i.e. with enhanced top couplings, so our $\rho_{tt}\sim{\cal O}(1)$ parameter space is not yet constrained. Likewise, the ATLAS study of $pp\to tbH^{\pm}$ ($H^{+}\to t\bar{b}$) ATLAS:2016qiq at 13 TeV, or $t\bar{t}$ final states with additional heavy flavor jets ATLAS:2016btu , are again sensitive to low $\tan\beta$ values and do not yet constrain our scenario. But with fast accumulation of Run 2 data, further discernment by ATLAS and CMS are encouraged.

To account for the absence of signal for sub-TeV scalar bosons, we have therefore chosen our lowest mass, $m_{A}=m_{H^{+}}=350$ GeV, to be above the $t\bar{t}$ threshold, which should be revisited by experiment. Associated production, such as $t\bar{t}H^{0}/A^{0}$ and $\bar{t}bH^{+}(t\bar{b}H^{-})$ followed by $H^{0}/A^{0}\to t\bar{t}$ and $H^{+}\to t\bar{b}(H^{-}\to\bar{t}b)$ decays would be the search strategy ATLAS:2016qiq ; ATLAS:2016btu ; Craig:2015jba at high luminosity. An intriguing alternative, as studied in Ref. Altunkaynak:2015twa , may be $gg\to H^{0}/A^{0}\to t\bar{c}(\bar{t}c)$ with decay mediated by $\rho_{ct}$ coupling. Despite being suppressed by $c_{\gamma}^{2}$, search for $t\to ch^{0}$ Chen:2013qta should certainly continue. Note that $A^{0}\to h^{0}Z^{0}$, $H^{\pm}\to h^{0}W^{\pm}$ and $H^{0}\to h^{0}h^{0}$ decays are suppressed by $c_{\gamma}^{2}$, and can be ignored above $t\bar{t}$ threshold.

Close to alignment and with custodial SU(2), $m_{H}^{2}>m_{A}^{2}=m_{H^{+}}^{2}$ by $\eta^{\prime}v^{2}$ (see Eq. (11)). So for $\eta^{\prime}\lesssim(2m_{Z}m_{A}+m_{Z}^{2})/v^{2}$, one has $m_{H}-m_{A}\lesssim m_{Z}$. Thus, for $m_{A}=350$, 475 GeV, $m_{H}-m_{A,\,H^{+}}\gtrsim m_{Z}$ for $\eta^{\prime}\gtrsim 1.19$, 1.57, while $m_{H}-m_{A}<m_{Z}$ for $m_{A}=600$ GeV, unless $\eta^{\prime}\gtrsim 2$. Thus, for lower $m_{A}=m_{H^{+}}$ that enjoy larger cross sections, there are interesting $H^{0}\to A^{0}Z$, $H^{\pm}W^{\mp}$ decays. The former could lead to $t\bar{t}Z$ signature, with potential mass peaks in both $m_{t\bar{t}}$ and $m_{t\bar{t}Z}$. The $H^{\pm}W^{\mp}\to t\bar{b}W^{-}(\bar{t}bW^{+})$ final state, however, may be swamped by $t\bar{t}$ background. In any case, these processes would likely be subdominant, unless $\rho_{tt}$ turns out to be far less than 1.

After completion of the original work of this Letter, the process $cg\to tS^{0}\to tt\bar{c}$, $tt\bar{t}$ ($S^{0}=H^{0},\,A^{0}$) was explored Kohda:2017fkn , which proceed via the extra $\rho_{ct}$ and $\rho_{tt}$ couplings. It was found that the same-sign top Hou:1997pm or triple top signatures can be probed already with full LHC Run 2 data, in contrast with the need of very high luminosity for the four-top signature Dev:2014yca ; ATLAS:2016qiq ; ATLAS:2016btu ; Craig:2015jba .

The general 2HDM with its FCNH couplings has much impact on flavor physics and CPV (FPCP), and our parameter range in fact overlaps well with the possibility Fuyuto:2017ewj of EWBG. We refer to Ref. Fuyuto:2017ewj for discussion of related FPCP phenomenology, such as electron (and neutron) electric dipole moment, $h\to\mu\tau$ and $\tau\to\mu\gamma$, as well as corrections to $h\to\gamma\gamma$ and $\lambda_{hhh}$.

Discussion and Conclusion.—

The parameter space for alignment in 2HDM is larger than we discussed. Custodial SU(2), or $m_{A}=m_{H^{+}}$, used to control $\Delta T$, need not be required: the $\Delta T$ constraint would just carve out some “slices” in $m_{H}$, $m_{A}$ and $m_{H^{+}}$ space. The derived condition that some $\eta_{i}\sim{\cal O}(1)$ echoes the need for first order EWPT, while ${\rm Im}\,\rho_{tt}$ drives the CPV source for EWBG Fuyuto:2017ewj . But ${\cal O}(1)$ couplings are also not strictly necessary. A full account of the phenomenology associated with alignment is beyond the scope of this work.

We have argued from Eq. (18) that, to keep $\eta_{3}$ perturbative, $\mu_{22}^{2}/v^{2}$ has to grow in value for $m_{H}\gtrsim 500$ GeV, which would eventually lead to decoupling. Thus, for sake of EWBG, Fig. 1 reflects the mass range of interest for exotic Higgs search at the LHC.

As an example of different parameter values, suppose we let $\eta^{\prime}<0$ in Eq. (17). For $m_{A}=m_{H^{+}}=550$ GeV and $\eta^{\prime}\in(-1,\,-1/4)$, $\eta_{6}\simeq 1/4$, we have $m_{H}\in(493,\,537)$ GeV and $c_{\gamma}\simeq-0.067$ to $-0.056$, still close to alignment. More negative $\eta^{\prime}$ values would move $m_{H}$ lower. Another example is “twisted custodial” symmetry Gerard:2007kn , $m_{H^{0}}^{2}=m_{H^{+}}^{2}$, where $\eta_{4}+\eta_{5}=0$ instead of Eq. (17). The situation is not so different from Fig. 1: $m_{A^{0}}>m_{H^{0}}=m_{H^{+}}$ for $\eta_{4}>0$, and $m_{A^{0}}<m_{H^{0}}=m_{H^{+}}$ for $\eta_{4}<0$, which is more constrained by $\Delta T$ but less aligned.

Small Cabibbo mixing, $\sin\theta_{C}\lesssim 1/4$, as well as the full quark mass-mixing pattern, are not understood. In contrast, approximate alignment may reflect ${\cal O}(1)$ Higgs quartic couplings. In retrospect, NFC Glashow:1976nt was an overkill against FCNH. Even the Cheng-Sher ansatz Cheng:1987rs was only an intermediate step. The exotic $H^{0}/A^{0}$ can have FCNH couplings, but approximate alignment suppresses such couplings for $h^{0}$. The observed mass-mixing pattern, rooted in the Yukawa couplings of the SM-like doublet, further modulates the strength of Yukawa couplings (including FCNH) of the exotic doublet. There is much to look forward to if such Higgs bosons are discovered.

In conclusion, approximate alignment arises in the general two Higgs doublet model if some of the dimensionless parameters in the Higgs potential, including $\mu_{22}^{2}/v^{2}$, are ${\cal O}(1)$. There is no need to impose small parameters, and it does not depend on whether a $Z_{2}$ discrete symmetry is imposed or not. This revitalizes the outlook for sub-TeV $H^{0}$, $A^{0}$ and $H^{\pm}$ bosons. Discovery of such bosons with ${\cal O}(1)$ Higgs couplings would point to a new scale below 10–20 TeV. We have advocated the two Higgs doublet model without $Z_{2}$ symmetry, as it offers hope for electroweak baryogenesis, precisely for Higgs quartic and extra top Yukawa couplings that are ${\cal O}(1)$ in strength.

Acknowledgments  We thank J.-M. Gérard, M. Kohda, E. Senaha, M. White and T. Yanagida, and especially H.E. Haber, for discussions. This research is supported by grants MOST 104-2112-M-002-017-MY2, 105-2112-M-002-018, 106-2811-M-002-010, and NTU 106R8811, 106R104022. WSH is grateful to E. Gardi and F. Muheim for a fruitful stay at University of Edinburgh.