Where are the Next Higgs Bosons?

TL;DR

The paper predicts new Higgs doublets with masses up to 5 TeV, coupled to third-generation fermions, and discusses their potential detectability at current and future colliders.

Contribution

It introduces a symmetry-based prediction of new Higgs doublets with universal Yukawa couplings, extending the Standard Model to include additional Higgs states.

Findings

Prediction of a new $H_b$ doublet accessible at LHC energies.

Extension to leptons predicts $H_ au$ and possibly $H_{ u_ au}$.

Masses of new Higgs doublets are estimated to be up to 5 TeV.

Abstract

Simple symmetry arguments applied to the third generation lead to a prediction: there exist new sequential Higgs doublets with masses of order $\lesssim 5 $ TeV, with approximately universal Higgs-Yukawa coupling constants, $g\sim 1$. This is calibrated by the known Higgs boson mass, the top quark Higgs-Yukawa coupling, and the $b$-quark mass. A new massive weak-isodoublet, $H_b$, coupled to the $b$-quark with $g\sim 1$ is predicted, and may be accessible to the LHC at $13$ TeV, and definitively at the energy upgraded LHC of $26$ TeV. The extension to leptons generates a new $H_\tau$ and a possible $H_{\nu_\tau}$ doublet. The accessibility of the latter depends upon whether the mass of the $\tau$-neutrino is Dirac or Majorana.

Equations

$$G=\mathrm{SU}(2{)}_{\mathrm{L}}\times \mathrm{SU}(2{)}_{\mathrm{R}}\times \mathrm{U}(1{)}_{B-L}\times \mathrm{U}(1{)}_A.$$

$$\displaystyle V_{HY}=g\overline{\Psi}_{\mathrm{L}}\Sigma\Psi_{\mathrm{R}}\;+\;\mathrm{h.c.}\;\quad\text{where}\quad\Psi=\left(\begin{array}[]{c}t\\ b\end{array}\right).$$

$$\Sigma =(H_0,H_b^c),$$

$$V_{HY}=g_t{\left(\overline{t},\overline{b}\right)}_{\mathrm{L}}{H}_0{t}_{\mathrm{R}}+g_b{\left(\overline{t},\overline{b}\right)}_{\mathrm{L}}{H}_b^c{b}_{\mathrm{R}}+h.c.,$$

$$H_b\to ϵH_0,ϵ=\frac{m_b}{m_t}=\mathrm{0.0234.}$$

$$V_{\text{Higgs}}=M_0^2{H}_0^{†}{H}_0+\frac{\lambda }{2}(H_0^{†}{H}_0{)}^2,$$

$$V$$

$$H_b=-\frac{{\mu }^2}{M_b^2}{H}_0+O(\lambda ,{\lambda }^{\prime }).$$

$$V$$

$$M_0^2=M_H^2-\frac{{\mu }^4}{M_b^2}.$$

$$\displaystyle H_{0}=\left(\begin{array}[]{c}v+\frac{1}{\sqrt{2}}h\\ 0\end{array}\right),\qquad v=$174\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$,$$

$$v^2=-M_0^2/\lambda ,m_h=\sqrt{2}|M_0|=125\text{makebox}GeV,$$

$$H_{b}\rightarrow H_{b}-\frac{\mu^{2}}{M_{b}^{2}}\left(\begin{array}[]{c}v+\frac{1}{\sqrt{2}}h\\ 0\end{array}\right),$$

$$\left(\begin{array}{cc}{M}_H^2& {\mu }^2\\ {\mu }^2& M_b^2\end{array}\right).$$

$$m_b=g_b(m_b)v\frac{{\mu }^2}{M_b^2}=m_t\frac{g_b(m_b){\mu }^2}{g_t(m_t)M_b^2},$$

$$R_b=\frac{g_b(m_b)}{g_t(m_t)}\simeq \mathrm{1.5.}$$

$$m_b=g_b(m_b)v\frac{{\mu }_b^2}{M_b^2}=m_t{R}_b\frac{{\mu }_b^2}{M_b^2}.$$

$$M_{b}=\frac{m_{t}}{m_{b}}R_{b}|M_{0}|\simeq$5.5\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$,$$

$$V_{HY\ell }$$

$${\Sigma }_{\ell }=(H_{\nu },H_{\tau }^c),\text{where}\Psi =\left(\genfrac{}{}{0ex}{}{\nu }{\tau }\right),$$

$$g=g_t(\Lambda )=g_b(\Lambda )=g_{\tau }(\Lambda )=g_{\nu }(\Lambda ),$$

$$R_{\tau }=\frac{g_{\tau }(m_{\tau })}{g_t(m_t)}{R}_{\nu }=\frac{g_{\tau }(m_{\nu })}{g_t(m_t)}$$

$$V_0^{\prime }$$

$$V_1^{\prime }$$

$${\Sigma }_{\ell }^c=i{\sigma }_2{\Sigma }_{\ell }^{\ast }(-i{\sigma }_2),$$

$${\Sigma }_{\ell }^c=(H_{\tau },H_{\nu }^c).$$

$$V_2^{\prime }$$

$$H_{\nu }=-\frac{{\mu }_1^2}{M_{\nu }^2}{H}_0,\delta M_0^2=-\frac{{\mu }_1^4}{M_{\nu }^2}\simeq 0.$$

$$V_{HY\ell }$$

Full text

††preprint: \minibox[]FERMILAB-PUB-19-143-T

CP3-Origins-2019-15 DNRF90

I Introduction

Understanding flavor physics will likely involve the discovery of new particles, associated with the mystery of the origin of the small parameters of the Standard Model. For example, the observed Higgs-Yukawa coupling of the $b$-quark is small, $y_{b}\simeq 0.024$. This is “technically natural” ’t Hooft (1980) because as $y_{b}\rightarrow 0$, a chiral symmetry emerges, $b_{\mathrm{R}}\rightarrow e^{i\theta}b_{\mathrm{R}}$, and a nonzero $y_{b}$ is never perturbatively regenerated. It is therefore not possible to induce these small parameters as perturbative radiative corrections, starting from zero, and it becomes challenging to understand how they are generated. However, they may have a perturbative origin, arising from virtual effects involving new particles with larger couplings, i.e., a small parameter is induced by a large parameter that is power-law suppressed via mixing. Such new physics may be accessible to the LHC, or its energy doubled upgrade.

There are, of course, many theoretical ways to achieve this, but we are also motivated by the hypothesis that a single lone Higgs boson is unlikely to exist—there may be a rich spectrum of Higgs bosons, presenting a new spectroscopy in nature. Thus, the “new particles” we will consider are exclusively new massive Higgs iso-doublets.

Hence, a plausible origin of the small $y_{b}$ is via a new heavy Higgs iso-doublet, $H_{b}$, coupled as $g_{b}\overline{T}_{\mathrm{L}}H_{b}b_{\mathrm{R}}$ where $T_{\mathrm{L}}=(t,b)_{\mathrm{L}}$. Most importantly, the new coupling $g_{b}$ is large, owing to a symmetry such as we propose below, where it is of order the top quark Higgs-Yukawa coupling, $g_{b}=O(1)$. The observed $y_{b}$ is then“effective,” that is, $y_{b}\sim g_{b}\,(\mu_{b}^{2}/M_{b}^{2})$, where the power law suppression arises from the large mass of the new $H_{b}$, $M_{b}^{2}H^{\dagger}_{b}H_{b}$, and its mixing with the Standard Model Higgs doublet $H_{0}$, $\mu_{b}^{2}H_{0}^{\dagger}H_{b}+\mathrm{h.c.}$, and $\mu_{b}^{2}/M_{b}^{2}\ll 1$. The breaking of the $b_{\mathrm{R}}$ chiral symmetry is then governed, not by $y_{b}$, but rather by a large $g_{b}\sim 1$, yet the small observed $y_{b}\sim 10^{-2}$ arises naturally. A UV completion with symmetries and nontrivial renormalization group evolution has a better chance of predicting the large $g_{b}$ than the small $y_{b}$. One would hope to directly observe the heavy $H_{b}$ at its mass, $M_{b}$. As a bonus, a larger $g_{b}$ enhances the production and detection possibilities for $H_{b}$ at the LHC, or any other collider.

Presently we examine this scenario in detail, which we believe describes the first sequential new Higgs doublets that could emerge at the LHC. We will focus, for simplicity, upon the third generation, and will ignore masses and mixings involving the lighter two generations. We note that, in a previous study, Hill et al. (2019), the flavor constraints on this system were found to be consistent.

We assume the top-bottom subsystem is approximately invariant under a simple extension of the Standard Model symmetry group structure

[TABLE]

By “approximately” we mean that, if we turn off the $U(1)_{Y}$ gauging, $g_{1}\rightarrow 0$, the symmetry $G$ is exact in the $d=4$ operators (kinetic terms, Higgs-Yukawa couplings, and potential terms). The Standard Model (SM) gauging is the usual, $\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{Y}$, and is a subgroup of $G$, where the $\mathrm{U}(1)_{Y}$ generator is now $I_{3R}+(B-L)/2$. This electroweak gauging weakly breaks the symmetry $\mathrm{SU}(2)_{\mathrm{R}}\times\mathrm{U}(1)_{B-L}\rightarrow\mathrm{U}(1)_{Y}\times\mathrm{U}(1)_{B-L}$, however, the $\mathrm{SU}(2)_{\mathrm{R}}$ remains as an approximate global symmetry of the $d=4$ operators. In addition, a global $\mathrm{U}(1)_{A}$ arises as well.

To implement $G$ in the $(t,b)$ sector we require the Standard Model Higgs doublet, $H_{0}$, couples to $t_{\mathrm{R}}$ with coupling $y_{t}\equiv g_{t}$ in the usual way, and a second Higgs doublet, $H_{b}$, couples to $b_{\mathrm{R}}$ with coupling $g_{b}$. The symmetry $G$ then dictates that there is only a single Higgs-Yukawa coupling $g=g_{t}=g_{b}$ in the quark sector. This coupling is thus determined by the known top quark Higgs-Yukawa coupling, $y_{t}=g\simeq 1$. A schematic proposal for a UV completion theory that leads to $G$, based upon far UV compositeness, was proposed in Hill et al. (2019). In that scheme $y_{t}=g\simeq g_{b}\simeq 1$ is actually predicted by the infrared fixed point Pendleton and Ross (1981); Hill (1981). Our current discussion is a simplified subsector of that larger theory. We do not presently require the ingredients of the larger theory, and we simply calibrate $g\simeq 1$ from experiment.

Since $m_{b}/m_{t}\ll 1$, the $\mathrm{SU}(2)_{\mathrm{R}}$ must be broken. Here we follow the old rules of chiral dynamics, deploying “soft” symmetry breaking through bosonic mass terms. The symmetry breaking in the $d=2$ Higgs mass terms preserves the universality of quark Higgs-Yukawa couplings $g$. We remark that these explicit $d=2$ symmetry-breaking mass terms could arise from $d=4$ scale invariant interactions involving additional new fields and hence $G$ could then be broken spontaneously. However, we are interested presently in the simplest phenomenological scheme and will be content to insert the $d=2$ symmetry breaking mass terms by hand.

This simple $(t,b)$ quark scenario with $G$ is then predictive, because of the universal coupling. As we shall observe below, the natural mass scale for $H_{b}$ is found to be $\sim$5\text{,}\mathrm{T}\mathrm{e}\mathrm{V}$$. This prediction involves the assumption that the input Higgs mass, $M_{H}^{2}$, is very small, or a “no–fin–tuning” argument similar to no–fine–tuning of the Higgs mass in the MSSM.

Here we have three a priori unknown renormalized parameters, $M_{H}^{2}$ (the input Standard Model Higgs potential mass), $\mu_{b}^{2}$ (the mixing of $H_{b}$ with the Standard Model Higgs) and $M_{b}^{2}$ (the heavy $H_{b}$ mass). We do not specify a UV completion and thus do not solve the naturalness problems of these quantities nor explain their origin. However, once these are specified, we obtain the $b$-quark mass $m_{b}\sim m_{t}(\mu_{b}^{2}/M_{b}^{2})$, and the Higgs potential mass, $M_{0}^{2}=-($88.4\text{,}\mathrm{G}\mathrm{e}\mathrm{V}$)^{2}$ $=M_{H}^{2}-\mu_{b}^{4}/M_{b}^{2}$ (note: $\left|M_{0}\right|=$125\text{,}\mathrm{G}\mathrm{e}\mathrm{V}$/\sqrt{2}$). The tachyonic (negative) $M_{0}^{2}$ can thus be generated from an unknown input value, $M_{H}^{2}$, but we should not fine tune the difference, $M_{H}^{2}-\mu^{4}/M^{2}$. This requirement establishes the scale of $M_{b}\sim$5\text{,}\mathrm{T}\mathrm{e}\mathrm{V}. Or, if we postulate that the input Higgs mass is small, $M_{H}^{2}\ll M_{b}^{2}$, we immediately obtain $M_{b}\simeq$5.5\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}.

However, this result is modified when we extend the theory to include the third generation leptons $(\nu_{\tau},\tau)$. This requires two additional Higgs fields, $H_{\tau}$ and $H_{\nu_{\tau}}$ with masses $M_{\tau}$ and $M_{\nu}$. The presence of the third generation leptonic Higgs fields with an extension of the symmetry, $G$, has the general effect of reducing all the heavy Higgs masses, e.g., of $H_{b}$ to $\sim$3.6\text{,}\mathrm{T}\mathrm{e}\mathrm{V}$$ with $H_{\tau}$, and possibly, $H_{\nu_{\tau}}$, nearby.

Remarkably, the new Higgs spectrum is then systematically dependent upon whether neutrinos receive only Dirac masses, via a “Dirac-Higgs seesaw mechanism,” or Majorana masses in addition to Dirac masses through the Type-I seesaw mechanism. If neutrino masses are pure Dirac, the new $H_{\nu_{\tau}}$ must be ultra-heavy to produce a seesaw in $\mu^{2}/M_{\nu}^{2}$, and only the $H_{\tau}$ is detectable; if on the other hand the neutrino mass is Majorana, then we expect $M_{\nu}\sim M_{\tau}$. Hence, the physics of neutrinos and the new Higgs spectrum are intimately interwoven here.

II The top-bottom sub-system

The assumption of the symmetry, $G$, of Eq. (1) for the top-bottom system leads to a Higgs-Yukawa (HY) structure that is reminiscent of the chiral Lagrangian of the proton and neutron (or the chiral constituent model of up and down quarks Manohar and Georgi (1984))

[TABLE]

In Eq. (4), $\Sigma$ is a $2\times 2$ complex matrix and $V_{HY}$ is invariant under $\Sigma\rightarrow U_{\mathrm{L}}\Sigma U^{\dagger}_{\mathrm{R}}$ with ${\Psi}_{\mathrm{L}}\rightarrow U_{\mathrm{L}}{\Psi}_{\mathrm{L}}$ and ${\Psi}_{\mathrm{R}}\rightarrow U_{\mathrm{R}}{\Psi}_{\mathrm{R}}$. Note that the $\mathrm{U}(1)_{Y}$ generator $Y$ of the SM now becomes $Y=I_{3R}+(B-L)/2$ and furthermore $\Sigma$ is neutral under $B-L$ which implies its weak hypercharge $[Y,\Sigma]=\Sigma\,I_{3R}$.

An additional $\mathrm{U}(1)_{A}$ axial symmetry arises as an overall phase transformation of $\Sigma\rightarrow e^{i\theta}\Sigma$, accompanied by ${\Psi}_{\mathrm{L}}\rightarrow e^{i\theta/2}{\Psi}_{\mathrm{L}}$ and ${\Psi}_{\mathrm{R}}\rightarrow e^{-i\theta/2}{\Psi}_{\mathrm{R}}$. Keeping or breaking this symmetry is an arbitrary option for us (though in the $(u,d)$ subsystem this is the Peccei-Quinn symmetry and is required if one incorporates the axion).

$\Sigma$ can be written in terms of two column doublets,

[TABLE]

where $H^{c}=i\sigma_{2}H^{*}$. Under $\mathrm{SU}(2)_{\mathrm{L}}$ we have $H\rightarrow U_{\mathrm{L}}H$, and $H^{c}\rightarrow U_{\mathrm{L}}H^{c}$ where we note that the weak hypercharge eigenvalues of $H$ and $H^{c}$ have opposite signs. The HY couplings of Eq. (4) become

[TABLE]

where the $\mathrm{SU}(2)_{\mathrm{R}}$ symmetry has forced $g=g_{t}=g_{b}$.

Note that this becomes identical to the Standard Model (SM) if we make the identification

[TABLE]

The HY coupling of the $b$-quark to $H_{0}$ is then $y_{b}=\epsilon g_{b}$, but with $\epsilon\ll 1$ the $\mathrm{SU}(2)_{\mathrm{R}}$ symmetry is then lost. The SM Higgs boson (SMH), $H_{0}$, in the absence of $H_{b}$, has the usual SM potential:

[TABLE]

where $M_{0}^{2}\simeq-(88.4)$ GeV2 (note the sign), and $\lambda\simeq 0.25$. Minimizing this, we find that the Higgs field $H_{0}$ acquires its usual VEV, $v=|M_{0}|/\sqrt{\lambda}=174$ GeV, and the observed physical Higgs boson, $h$, acquires mass $m_{h}=\sqrt{2}|M_{0}|\simeq 125$ GeV. In our present scheme Eqs. (7) and (8) arise at low energies dynamically.

Given Eqs. (4)and (5), we postulate a new potential of the form

[TABLE]

We have written the potential in the $\Sigma$ notation in order to display the symmetries more clearly. In the above, $\sigma_{3}$ acts on the $\mathrm{SU}(2)_{\mathrm{R}}$ side of $\Sigma$, hence in the limit $M_{2}^{2}=0$ the potential $V$ is invariant under $SU(2)_{R}$, while the $\mu^{2}$ term breaks the additional $\mathrm{U}(1)_{A}$. The associated CP-phase can be removed by field redefinition in the present model. If the exact $\mathrm{U}(1)_{A}$ symmetry is imposed on the $d=4$ terms in the potential then operators such as, $e^{i\alpha}(\det\Sigma)\mathop{\rm Tr}\left(\Sigma^{\dagger}\Sigma\right)$, $e^{i\alpha^{\prime}}(\det\Sigma)^{2}$, etc., are forbidden. Noting the identity $(\mathop{\rm Tr}(\Sigma^{\dagger}\Sigma))^{2}=\mathop{\rm Tr}(\Sigma^{\dagger}\Sigma)^{2}+2\det\Sigma^{\dagger}\Sigma$, we obtain only the two indicated $d=4$ terms as the maximal form of the invariant potential.

Using Eq. (5), $V$ can be written in terms of $H_{0}$ and $H_{b}$:

[TABLE]

where $\lambda_{2}=\lambda^{\prime}+\lambda$, $M_{H}^{2}=M_{1}^{2}-M_{2}^{2}$, and $M_{b}^{2}=M_{1}^{2}+M_{2}^{2}$. In the limit $M_{b}^{2}\rightarrow\infty$ the field $H_{b}$ decouples, and Eq. (10) reduces to the SM if $M_{H}^{2}\rightarrow M_{0}^{2}$, and $\lambda\rightarrow 0.25$ thereby recovering Eq. (8).

We assume the quartic couplings, following the SM value $\lambda\sim 0.25$, contribute negligibly small effects, and set $\theta=0$. Then, varying the potential with respect to $H_{b}$, the low momentum components of $H_{b}$ are locked to $H_{0}$:

[TABLE]

Substituting back into $V$ we recover the SMH potential

[TABLE]

with

[TABLE]

Note that, even with $M_{H}^{2}$ positive, $M_{0}^{2}=-(88.4)^{2}$ GeV2 can be driven to its negative value by the mixing with $H_{b}$ (level repulsion). We minimize the SMH potential and define

[TABLE]

in the unitary gauge. The minimum of Eq. (16) yields the usual SM result

[TABLE]

where $m_{h}$ is the propagating Higgs boson mass. We can then write

[TABLE]

for the full $H_{b}$ field. This is a linearized (small angle) approximation to the mixing, and si reasonably insensitive to the small $\lambda,\lambda^{\prime}\ll 1$.

Note the effect of “level repulsion” of the Higgs mass, $M_{0}^{2}$, downward due to the mixing with heavier $H_{b}$. The level repulsion in the presence of $\mu^{2}$ and $M_{b}^{2}$ occurs due to an approximate “seesaw” Higgs mass matrix

[TABLE]

The input value of the mass term $M_{H}^{2}$ is unknown and in principle arbitrary, and can have either sign. We can presumably bound $M_{H}^{2}\gtrsim$1\text{,}\mathrm{G}\mathrm{e}\mathrm{V}^{2} from below, since QCD effects will mix glueballs and the QCD-$\sigma$-meson with $H_{0}$ in this limit. As $M_{H}^{2}$ is otherwise arbitrary, we might then expect that the most probable value is $M_{H}^{2}\sim$1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}^{2}, and $|M_{H}^{2}|\ll(\mu^{2},M_{b}^{2})$.

Let us consider the case $M_{H}^{2}=0$. Then Eq. (19) has eigenvalues $M_{0}^{2}=-\mu^{4}/M_{b}^{2}$, and $M_{b}^{2}$. Thus, in the limit of small, nonzero $|M_{H}^{2}|$, we see that a negative $M_{0}^{2}$ arises naturally, and to a good approximation the physical Higgs mass is generated entirely by the negative mixing term.

The mass mixing causes the neutral component of $H_{b}$ to acquire a small VEV (“tadpole”) of $-v(\mu^{2}/M_{b}^{2})$. This implies that the SM HY-coupling of the $b$-quark is induced with the small value $y_{b}=g_{b}(\mu^{2}/M_{b}^{2})$ (note $g_{b}$ is positive with a phase redefinition of $b_{\mathrm{R}}$). The $b$-quark then receives its mass from $H_{b}$,

[TABLE]

where we have indicated the renormalization group (RG) scales at which these couplings should be evaluated.

In a larger framework with a UV completion, such as Ref. Hill et al. (2019), at a mass scale, $m>M_{b}$, both $g_{t}(m)$ and $g_{b}(m)$ will have a common renormalization group equation modulo $\mathrm{U}(1)_{Y}$ effects. We implicitly assume that, at some very high scale $\Lambda\gg v$, the $\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{SU}(2)_{\mathrm{R}}$ is a good symmetry. This implies g(\Lambda)=g_{t}(\Lambda)=g_{b}(\Lambda)\mathrel{\vbox{\hbox{>}\nointerlineskip\hbox{\sim}}}1. Then, we will predict $g(M_{b})\simeq g_{t}(M_{b})\simeq g_{b}(M_{b})$, e.g., where the values at the mass scale $M_{b}$ are determined by the RG fixed point Pendleton and Ross (1981); Hill (1981) with small splittings due to $\mathrm{U}(1)_{Y}$.

Furthermore, we find that $g_{t}(m)$, and moreso $g_{b}(m)$, increase somewhat as we evolve downward from $M_{b}$ to $m_{t}$ or $m_{b}$; the top quark mass is then $m_{t}=g_{t}(m_{t})v$ where $v$ is the SM Higgs VEV. From these effects we obtain the ratio

[TABLE]

The $b$-quark then receives its mass from the tadpole VEV of $H_{b}$.

[TABLE]

In the case that the Higgs mass, $M_{0}^{2}$, is due entirely to the level repulsion by $H_{b}$, i.e. $M_{H}^{2}=0$, and using Eqs. (13), (21) and (22), we obtain a predicted mass of the $H_{b}$,

[TABLE]

with $m_{b}=$4.18\text{,}\mathrm{G}\mathrm{e}\mathrm{V}, $m_{t}=$173\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}, and $|M_{0}|=$88.4\text{,}\mathrm{G}\mathrm{e}\mathrm{V}$$. We remind the reader we have ignored the effects of the quartic couplings $\lambda$, which we expect are small. Moreover, the quartic couplings do not enter the mixing, because terms such as, $H^{\dagger}_{0}H_{0}H^{\dagger}_{0}H_{b}$ are forbidden by our symmetry. The remaining terms only act as slight shifts in the masses, never larger than $\sim\lambda v^{2}$ and can be safely ignored.

This is a key prediction of the model (see also Porto and Zee (2008); BenTov and Zee (2013)). In fact, we can argue that with $M_{H}^{2}$ nonzero, but with small fine tuning (see below), the result M_{b}\mathrel{\vbox{\hbox{<}\nointerlineskip\hbox{\sim}}}5.5\text{,}\mathrm{T}\mathrm{e}\mathrm{V}$$ is obtained. This mass scale is accessible to the LHC with luminosity and energy upgrades, and we feel represents an important target for discovery of the first sequential Higgs Boson.

However, we will see in the next section that this result is reduced when the third generation leptons are included.

III Inclusion of the $(\nu_{\tau},\tau)$ leptons

The simple $(t,b)$ system described above can be extended to the third generation leptons $(\nu_{\tau},\tau)$. Presently we will abbreviate $\nu_{\tau}$ to $\nu$, and ignore neutrino mixing in this third generation scheme (see also Porto and Zee (2009)). We consider two distinct mechanisms to generate the small neutrino mass scale.

Remarkably the predictions for the mass spectrum are sensitive to the mechanism of neutrino mass generation. The next sequential massive Higgs iso-doublet, in addition to $H_{b}$, is likely to include the $H_{\tau}$, and possibly also $H_{\nu}$, which is dependent upon whether neutrino masses are Majorana or Dirac in nature.

We introduce the Higgs-Yukawa couplings for the leptons in a the $G$ invariant form

[TABLE]

where we’ve introduced a second “leptonic” chiral field

[TABLE]

and $\Sigma_{\ell}$ transforms as $\Sigma$ under $G$.

The couplings $g_{\nu}$ and $g_{\tau}$ are assumed to have the common value at the high scale $\Lambda$,

[TABLE]

Since the RG equations for the leptons do not involve QCD, we typically find $g_{\tau}(m_{\tau})=g_{\nu}(m_{\tau})\simeq 0.7$ for $\Lambda\sim M_{\mathrm{Planck}}$ Hill et al. (2019). More generally we define the parameters

[TABLE]

We extend the potential of Eq. (II), $V\rightarrow V+V^{\prime}$ to incorporate the $\Sigma_{\ell}$ mass terms:

[TABLE]

where the ellipsis refers to quartic terms which we will ignore altogether. Moreover, we assume the new iso-doublets are dormant, $M_{\nu}^{2},M_{\tau}^{2}>0$.

We can also introduce a mixed term that involves both $\Sigma_{\ell}$ and $\Sigma$ of Eq. (5):

[TABLE]

Note that this term mixes the Higgs fields as $H_{0}\leftrightarrow H_{\nu}$ and $H_{b}\leftrightarrow H_{\tau}$. Such mixing would lead to the $\tau$ acquiring its mass sequentially via mixing with $H_{b}$, which directly mixes with $H_{0}$ as in Eq. (10). Although such a scenario has potentially interesting physics, we do not pursue it presently

However, we can introduce a second term consistent with $G$ that leads to direct mixing of $H_{0}^{\dagger}H_{\tau}$. This can be constructed using a charge conjugated $\Sigma^{c}$, where

[TABLE]

and note that $\Sigma^{c}\rightarrow U_{\mathrm{L}}\Sigma^{c}U_{\mathrm{R}}$, transforms identically to $\Sigma\rightarrow U_{\mathrm{L}}\Sigma U_{\mathrm{R}}$ under the $\mathrm{SU}(2)$ groups. The effect of the conjugation is to flip the column and charge conjugation assigments of Higgs iso-doublets in $\Sigma_{\ell}$,

[TABLE]

Therefore, a term that permits direct mixing $H_{0}\leftrightarrow H_{\tau}$ and $H_{b}\leftrightarrow H_{\nu}$ is

[TABLE]

It should be noted that this term violates the $\mathrm{U}(1)_{A}$ symmetry, but trivially not $\mathrm{U}(1)_{B-L}$ since both $\Sigma^{\prime}s$ are sterile under $B-L$. For simplicity we will simply set the CP phases to zero, $\theta=\theta^{\prime}=\phi^{\prime}=0$.

III.1 $H_{\tau}$, $H_{b}$, and a Dirac Neutrino Seesaw

One interesting possibility is that the $\mathrm{SU}(2)_{\mathrm{R}}$ symmetry is badly broken in the lepton sector with $M_{\nu}^{2}\gg M_{\tau}^{2}$ of Eq. (28). In this limit the $H_{\nu}$ has become ultra-massive and non-detectable at collider energies.

Through the $V^{\prime}_{1}$ term we observe that $H_{\nu}$ acquires a tiny tadpole VEV, but now has negligible feedback on the Higgs mass:

[TABLE]

where $\mu_{1}^{2}/M^{2}_{\nu}\ll 1$. Hence the neutrino acquires a tiny Dirac mass through the induced coupling to the Higgs:

[TABLE]

wheer $R_{\nu}\simeq O(1)$. In such a scenario a Majorana mass term is not necessary, as we have a (Dirac) seesaw mechanism to naturally generate a tiny neutrino mass.

The $H_{\tau}$ mixes directly to the SMH through the $V^{\prime}_{2}$ term. $H_{\tau}$ then acquires a VEV, and feeds back upon the Higgs mass as

[TABLE]

Hence, the $\tau$ mass is given by

[TABLE]

and we expect $R_{\tau}\sim 0.7$.

$H_{\tau}$ and $H_{b}$ now simultaneously contribute to the SMH mass

[TABLE]

(we remind the reader that $M_{0}^{2}=-(88.4)^{2}$ GeV2 is negative as defined in Eq.(8)). Using Eqs. (20) and (39) this yields an elliptical constraint on the heavy Higgs masses $M_{b}$ and $M_{\tau}$,

[TABLE]

for fixed $M_{H}^{2}$. Bear in mind that the Higgs potential input mass, $M_{H}^{2}$, is a priori unkown, while $M_{0}^{2}=-($88.4\text{,}\mathrm{G}\mathrm{e}\mathrm{V}$)^{2}$ is known from the Higgs boson mass, $m_{h}=\sqrt{2}|M_{0}|=125$ GeV.

If we make the assumption $M_{H}^{2}=0$ the ellipse is shown as the red-dashed line in Fig.(1). If we further assume both $H_{b}$ and $H_{\tau}$ contribute equally to the SMH mass, then we obtain from Eq. (41)

[TABLE]

Of course, we can raise (lower) these masses by introducing the bare positive (negative) $M_{H}^{2}$. However, we do not want to excessively fine tune the difference $M_{H}^{2}-\sum_{i}\mu_{i}^{4}/M_{i}^{2}$.

As an alternative way of estimating the Higgs masses we follow the same procedure for estimating the fine tuning as is sometimes used in the MSSM or composite Higgs models. To account for the contribution to the fine tuning from all parameters $x_{i}$ on an observable $O$ we use the measure (see e.g. Barnard and White (2015))

[TABLE]

That is to say, the fine tuning is the length of the gradient of the logarithmic observable in the space of logarithmic parameters (other fine tuning definitions have been explored and lead to quantitatively similar results).

Considering the case of two dormant Higgses, $H_{b}$ and $H_{\tau}$, we measure the log-derivative sensitivities of the known value of $M_{0}^{2}$, to the five input parameters $M_{H}^{2}$, $\mu^{2}_{b}$, $\mu_{\tau}^{2}$, $M_{b}^{2}$ and $M_{\tau}^{2}$. The quadrature combined sensitivities yield the fine tuning parameter, $\Delta$. The fine tuning is summarized in Fig. (1) where the red-dashed curve indicates $M^{2}_{H}=0$. Every point on this figure is a bonafide fit to the known values of $M_{0}^{2}$, $m_{b}$ and $m_{\tau}$.

Examining just the theory with $H_{b}$ alone, we can combine the log-derivative sensitivities of $M_{0}^{2}$, $m_{b}$ to three input parameters $M_{H}^{2}$, $\mu^{2}_{b}$, and $M_{b}^{2}$. Then we find the “no fine tuning limit” $\Delta\leq 1$ translates into the more restrictive constraint

[TABLE]

The lightest values of $M_{b}$ (and $M_{\tau}$) correspond to $-|M_{0}^{2}|=M_{H}^{2}$ with no contribution from $M_{b}^{2}$ ($M_{\tau}^{2}$). Indeed, with more heavy Higgses we have generically smaller masses and the various become more easily accessible to the LHC.

III.2 $H_{\nu}$, $H_{\tau}$, $H_{b}$, and Majorana Neutrinos

We can also consider the more conventional possibility that the neutrino has a Majorana mass term, which is an extension of the physics beyond the minimal model. The lepton sector Higgs masses may then be comparable, $M^{2}_{\nu}\sim M^{2}_{\tau}$, and we have direct coupling to the SMH by both $H_{\nu}$ and $H_{\tau}$, through the $\mu_{1}^{2}$ and $\mu_{2}^{2}$ terms. In the absence of a Majorana mass term, this would imply $G$ is a good symmetry, with the neutrino having a large Dirac mass $m_{\nu}=m_{D\nu}$ comparable to $m_{\tau}$.

However, we can then suppress the physical neutrino mass, $m_{\nu}$, by allowing a the usual Type I seesaw. We thus postulate a large Majorana mass term for the ungauged $\nu_{\mathrm{R}}$:

[TABLE]

Integrating out $\nu_{\mathrm{R}}$ we then have an induced $d=5$ operator that generates a Majorana mass term for the left-handed neutrinos Weinberg (1979) through the VEV of $H_{\nu}$,

[TABLE]

In this scenario there is no restriction that requires the $H_{\nu}$ mass to be heavy, and the neutrino physical mass is now small, given by $m_{\nu}\sim m_{D\nu}^{2}/M\sim m_{\tau}^{2}/M$ where $m_{D\nu}\sim m_{\tau}$ is the neutrino Dirac mass.

Through the $V^{\prime}_{1}$ term we find that $H_{\nu}$ acquires a VEV, and has comparable feedback on the Higgs mass as $H_{\tau}$,

[TABLE]

We now have an ellipsoid for the masses $M_{b}$, $M_{\tau}$ and $M_{\nu}$

[TABLE]

With additional light Higgs fields, the elliptical constraint forces all of the Higgs masses to smaller values. We emphasize that the mass bounds of Fig. (1) should be viewed as upper limits on the Higgs mass spectrum that could be explored at the LHC.

IV Phenomenology of the Sequential Higgs Bosons

Presently we touch upon the collider phenomenology of this model and refer the reader to Hill et al. (2019) for further discussion. Flavor physics bounds have also been considered in Hill et al. (2019).

$H_{b}$ is an iso-doublet with neutral $h^{0}_{b}$ and charged $h^{\pm}_{b}$ complex field components coupling to $(t,b)$ as $g_{b}h^{0}_{b}\overline{b}_{\mathrm{L}}b_{\mathrm{R}}+\mathrm{h.c.}$ and $g_{b}h_{b}^{+}\overline{t}_{\mathrm{L}}b_{\mathrm{R}}\;+\;\mathrm{h.c.}\;$ with $g_{b}\simeq 1$. At the LHC the $h_{b}^{0}$ is singly produced in $pp$ by perturbative (intrinsic) $b$ and $\overline{b}$ components of the proton,

[TABLE]

Cross-sections for $13$ TeV, $26$ TeV and $100$ TeV are given for various $M_{b}$ in Hill et al. (2019) and in Fig. (2).

The cross-section for production at the LHC of $h_{b}^{0}$ is $\sigma(h^{0}_{b}\rightarrow b\overline{b})\sim${10}^{-4}\text{,}\mathrm{p}\mathrm{b} at $13$ TeV (or $\sigma(h^{0}_{b}\rightarrow b\overline{b})\sim${10}^{-2}\text{\,}\mathrm{p}\mathrm{b} at $26$ TeV) for a mass of $M_{b}={3.5}$ TeV. The decay width of $h_{b}^{0}$ is large, $\Gamma=3M/16\pi\simeq$210\text{,}\mathrm{G}\mathrm{e}\mathrm{V}$$, for $M_{b}=3.5$ TeV. To reduce the background we impose a $100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ $p_{T}$ cut on the each $b$ jet in the quoted cross-sections. Note that the charged $h_{b}^{+}$ would be produced in association with $\overline{t}b$, has a significantly smaller cross-section and we have not analyzed it.

We estimate that a $5\sigma$ excess in $S/\sqrt{B}$, in bins spanning twice the full-width of the Breit-Wigner, requires an integrated luminosity or $M_{b}=3.5$ TeV of $\sim$20\text{,}\mathrm{a}\mathrm{b}^{-1} at $13$ TeV, or $\sim$100\text{\,}\mathrm{f}\mathrm{b}^{-1} at $26$ TeV.111 1\text{\,}\mathrm{a}\mathrm{b}$=$1\text{\,}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{n}$=${10}^{-3}\text{\,}\mathrm{f}\mathrm{b} (femtobarn). Similarly, at a mass of $5$ TeV one requires $\sim$3\text{,}\mathrm{a}\mathrm{b}^{-1}$$ at $26$ TeV for a $5\sigma$ discovery. This assumes double $b$-tagging efficiencies of order $50\%$. Here the background is assumed to be mainly $gg\rightarrow\overline{b}b$, but the large fake rate from $gg\rightarrow\overline{q}q$ must be reduced to \mathrel{\vbox{\hbox{<}\nointerlineskip\hbox{\sim}}}3\% for this to apply.

Note that we have not attempted any significant optimization of the $S/\sqrt{B}$ in this study. Hence according to our estimates, while meaningful bounds are acheiveable at the current $13$ TeV LHC, particularly M_{b}\mathrel{\vbox{\hbox{<}\nointerlineskip\hbox{\sim}}}3.5 TeV, the energy doubler is certainly favored for this physics and could cover the full mass range up to $\sim 7$ TeV.

Remarkably, the $h^{0}_{\tau}$ (and $h^{0}_{\nu}$), neutral components of the associated iso-doublets, $H_{\tau}$ and $H_{\nu}$, may also be singly produced because they can mix with $H_{b}$ through the $\mu_{1}^{2}$ and $\mu_{2}^{2}$ terms of Eq. (29) and (32) respectively. This implies a total cross section

[TABLE]

for the mixing angle, $\theta$, between either the states $h^{0}_{\tau}$ or $h^{0}_{\nu}$ and $h^{0}_{b}$. Although $\theta$ is unknown it could easily be large, $\theta\sim 0.3$. The $h^{0}_{\tau}\rightarrow\overline{\tau}\tau$ is visible with $\tau$-tagging, and the background is also slightly suppressed since the peak is narrower by a factor of $(g_{\tau}^{2}/3g_{b}^{2})\sim 0.16$. Hence at the $26$ TeV LHC discovery is in principle possible for a $3.5$ TeV state with integrated luminosity of order $2$ ab*-1*.

These states are also pair produced by electroweak processes at the LHC or through $\gamma+Z^{0}$ at a high energy lepton collider:

[TABLE]

where $x$ denotes either $\tau$ or $\nu$.

We note that the LHC now has the capability of ruling out an $H_{b}$ with $g_{b}\sim 1$ of mass $\sim 1$ TeV, with current integrated luminosities, $\sim$200\text{,}\mathrm{f}\mathrm{b}^{-1}$$.

V Conclusions

We believe it is likely that an extended spectrum of Higgs bosons exists in nature. The expected masses of iso-doublet Higgs bosons have been discussed here with the prominent new field, $H_{b}$, couped to $b_{R}$ in the range \mathrel{\vbox{\hbox{<}\nointerlineskip\hbox{\sim}}}5 TeV. We emphasize that this is an upper limit, and experiments should search and quote limits in the full mass range for $H_{b}$. Additional new states, such as leptonic Higgs iso-doublets, or an extended spectroscopy as in Hill et al. (2019), may emerge.

We strongly emphasize that the LHC already has the capability of ruling out an $H_{b}$ with $g_{b}\sim 1$ of mass $\sim 1$ TeV, with current integrated luminosities, $\sim$200\text{,}\mathrm{f}\mathrm{b}^{-1}$$. The main reason is that the very plausible $g_{b}\simeq 1$ significantly enhances access to these states. We are unaware of any limits for these in the literature to date. We think it is important for the collaborations to develop an analysis strategy for limiting or discovery of these states.

This multi-Higgs scenario connects with the traditional way in which physics has evolved from atoms to nuclei to hadrons, thus presenting yet another spectroscopy Weisskopf (1970). Observation of the $H_{b}$, and measurement of $g_{b}\simeq 1$, would constitute compelling evidence of such an extended Higgs spectrum and validity of the logic behind these schemes.

Here we have shown that a subsector of a more general scalar democracy Hill et al. (2019) can be described simply in the context of the restricted, yet most observable, third generation by applying a simple extension of the Standard Model symmetry group. We think the overall simplicity of this idea is compelling, and we emphasize that the keystone is the universal Higgs-Yukawa coupling of order unity, controlled by a symmetry.

Above all, this theory is testable and should provide motivation to go further and deeper into the energy frontier with LHC upgrades, possibly with a future machine at the $\sim 100$ TeV scale and/or high energy lepton colliders. This scenario also suggests remarkable synergies with the ultra-weak scale of neutrinos. For example, if the $H_{\nu}$ were inferred to exist with a mass in the \mathrel{\vbox{\hbox{<}\nointerlineskip\hbox{\sim}}}10 TeV range, then neutrinos must have Majorana masses.

The idea that small effective Higgs-Yukawa couplings are dynamical, as $y\sim g(\mu^{2}/M^{2})$, is an old idea inherited from, e.g., extended technicolor models Eichten and Lane (1980); Dimopoulos and Susskind (1979). The primacy of the top quark in this scheme, with the calibrating large Higgs-Yukawa coupling, $g\simeq 1$, was anticipated long ago by the top infrared quasi-fixed point Pendleton and Ross (1981); Hill (1981). The fixed point corresponds to a Landau pole in $y_{t}$ Hill (1981) and the Higgs then naturally arises as a $\overline{t}t$ composite state Miransky et al. (1989); Bardeen et al. (1990); Hill and Simmons (2003) (i.e., the fixed point Hill (1981) is the solution to top condensation models) The model of Hill et al. (2019), which contains the present model as a subgroup, has the SMH as a composite $H^{0}\sim\overline{T}t$, and $H_{b}\sim\overline{T}b$, $H_{\tau}\sim\overline{L}\tau$, etc. The fixed point prediction becomes concordant with the top quark mass in our extended scheme, and even probes the spectrum of Higgs bosons.

Many of the structural features of our potentials are similar to a chiral constituent quark model Manohar and Georgi (1984), though presently $\Sigma$ is subcritical, with only the SM Higgs condensing. Indeed, the $\mu^{2}$ mixing term above that generates the $b$-quark mass is equivalent to a ‘t Hooft determinant in chiral quark models, or in topcolor models Hill (1995).

Extended Higgs models are numerous, but those that most closely presage our present discussion are Porto and Zee (2008, 2009); BenTov and Zee (2013); Hill (2014); Rodejohann and Saldaña-Salazar (2019). In Hill et al. (2019) we suggested a universal composite system of scalars, which we dubbed “scalar democracy,” however, in the more minimal third generation scenario presented here, which is most important for immediate experimental observation, these features are determined only by the symmetry group $G$. This mainly makes all of the HY couplings of quarks and leptons universal, modulo RG running, and “hyperfine splitting” by the small SM gauge interactions. We also find it compelling that a negative $M_{0}^{2}$ mass can be generated by this mixing effect starting from a positive or very small input $M_{H}^{2}$.

The universal value of all HY couplings, $g\simeq 1$, is of central importance to these models, and also enhances their predictivity. The observed small parameters of the SM, such as $y_{b}$, $y_{\tau}$, etc., are given essentially by one large universal parameter $g$ multiplied by a suppressing power law, $\sim\mu^{2}/M^{2}$, together with smaller perturbative renormalization effects $R_{x}$. In this way, the technically natural small couplings, and in principle the full CKM structure Hill et al. (2019), can be understood to emerge from the UV and the inverted spectrum of Higgs bosons, and may possibly be accessible to experiment.

A large extended spectrum of Higgs bosons with symmetry properties that unify all HY couplings to large common values is a phenomenologically rich idea, worthy of further theoretical study and the development of search strategies at the LHC and other future high energy colliders.

Acknowledgements

This work was done at Fermilab, operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy. AET would like to thank Fermilab, and gratefully acknowledges financial support from the Danish Ministry of Higher Education and Science through an EliteForsk Travel Grant. The CP3-Origins centre is partially funded by the Danish National Research Foundation, grant number DNRF90.