A More Natural Composite Higgs Model

The Standard Model (SM) of particle physics successfully describes all known elementary particles and their interactions. At the center of SM is the mechanism of electroweak symmetry breaking (EWSB), which is responsible for the masses of gauge bosons and fermions. The discovery of Higgs bosons in 2012 Chatrchyan:2012xdj ; Aad:2012tfa filled in the last missing piece of the SM. However, the Higgs boson itself brings new questions and puzzles that need to be answered. As a minimal model to realize EWSB, the Higgs field is characterized by the potential

$$V(H)=-\mu^{2}|H|^{2}+\lambda|H|^{4}$$

(1)

with just two parameters. The two parameters are now fixed by the observed Higgs vacuum expectation value (VEV) $v\simeq 246$ GeV and Higgs boson mass $M_{h}\simeq 125$ GeV as

$$\mu^{2}\simeq\left(88\text{ GeV}\right)^{2},\qquad\lambda\simeq 0.13~{}.$$

(2)

However, SM does not address the UV-sensitive nature of scalar bosons. The Higgs mass-squared receives quadratically divergent radiative corrections from the interactions with SM fields, which leads to the well-known hierarchy problem. To avoid the large quadratic corrections, the most natural way is to invoke some new symmetry such that the quadratic contributions cancel in the symmetric limit. This requires the presence of new particles related to SM particles by the new symmetry, such as top partners, in order to cut off the divergent loop contributions.

One such appealing solution to the hierarchy problem is the composite Higgs model (CHM), where the Higgs doublet is the pseudo-Nambu-Goldstone boson (pNGB) of a spontaneously broken global symmetry of the underlying strong dynamics Kaplan:1983fs ; Kaplan:1983sm . Through the analogy of the chiral symmetry breaking in quantum chromodynamics (QCD), which naturally introduces light scalar fields, i.e., pions, we can construct models with light Higgs bosons in a similar way. In a CHM, an approximate global symmetry $G$ is spontaneously broken by some strong dynamics down to a subgroup $H$ with a symmetry breaking scale $f$. The heavy resonances of the strong dynamics are expected to be around the compositeness scale $\sim 4\pi f$ generically. The pNGBs of the symmetry breaking, on the other hand, can naturally be light with masses $<f$ as they are protected by the shift symmetry. The potential of the Higgs field arises from the explicit symmetry breaking effects, such as the interactions with other SM fields. The largest coupling of the Higgs field in SM is to the top quark. As a result, for naturalness, the top partners which regulate the top loop contribution to the Higgs potential should not be too heavy. The top loop contribution to the Higgs mass term can be estimated as

$$\Delta\mu^{2}\sim\frac{N_{c}}{8\pi^{2}}y_{t}^{2}M_{T}^{2}\sim\left(220\text{ GeV}\right)^{2}\left(\frac{M_{T}}{1.2\text{ TeV}}\right)^{2},$$

(3)

where $M_{T}$ is the top partner mass. On the other hand, the bounds on the SM colored top partners have reached beyond 1 TeV from the collider searches Sirunyan:2018omb ; Aaboud:2018pii . Compared with Eq.(2), we see that the models with colored top partners (including both the minimal supersymmetric standard model (MSSM) and the CHM) already require some unavoidable $\mathcal{O}(10\%)$ tuning, albeit not unimaginable.

In most CHMs, however, the tuning is much worse than that is shown in Eq. (3). Depending on the coset $G/H$ and the representations of composite operators that couple to the top quarks, the strongly interacting resonances of the top sector in the UV often give a bigger contribution to the Higgs potential than Eq. (3), which requires more tuning to cancel. Another problem is that, unlike the pions, the Higgs field needs to develop a nonzero VEV $v$. The current experimental constraints require $v<f/3$. On the other hand, for a generic pNGB potential, the natural VEV for the pNGB is either 0 or $f$. To obtain a VEV much less than $f$, a significant quartic Higgs potential compared to the quadratic term is needed. In little Higgs models ArkaniHamed:2001nc ; ArkaniHamed:2002qx ; ArkaniHamed:2002qy , a Higgs quartic term can be generated without inducing a large quadratic term from the collective symmetry breaking. Such a mechanism is not present in most CHMs, which is another cause of the fine-tuning issue.

In this study, our goal is to find a more natural CHM by removing the additional tuning beyond Eq. (3). We first identify the cosets and the composite operator representations that couple to the top quarks, which can preserve a larger symmetry for the resonances to suppress the UV contribution to the Higgs potential. Next, we implement the collective symmetry breaking to generate a Higgs quartic potential while keeping the quadratic term at the level of Eq. (3). In this way we can naturally separate the scales of $v$ and $f$, resulting in a more natural CHM.

This paper is organized as follows. In section 2, we review the tuning problems in CHMs and identify the sources of the extra tuning, using the $SO(5)/SO(4)$ CHMs as an example. In section 3, we introduce the $SU(6)/Sp(6)$ CHM, including the interactions that produce the SM Yukawa couplings, and show how the large UV contribution to the Higgs potential is avoided. We then move on to the next step to generate an independent Higgs quartic term from collective symmetry breaking in section 4. The resulting Higgs potential of the 2HDM is discussed in section 5. The complete potential and spectrum of all the pNGBs in our model are summarized in section 6 with numerical estimation. Section 7 and Section 8 are devoted to the phenomenology of this model. Section 7 focuses on the collider searches and constraints. The analyses of the indirect constraints from the precision experimental measurements are presented in Section 8. Section 9 contains our summaries and conclusions. In Appendix A we briefly discuss the possibility of constructing a similar model based on the $SU(5)/SO(5)$ coset. We point out the differences and some drawbacks of such a model. Appendix B contains the details of the interactions between elementary fermions and composite operators for a realistic implementation of the $SU(6)/Sp(6)$ CHM model.

We first give a brief review of the tuning problem of the Higgs potential in general CHMs, which was comprehensively discussed in Ref. Panico:2012uw ; Panico:2015jxa . This will help to motivate possible solutions. As an illustration, we consider the Minimal Composite Higgs Models (MCHMs) Agashe:2004rs with the symmetry breaking $SO(5)\to SO(4)$. The four pNGBs are identified as the SM Higgs doublet. The SM gauge group $SU(2)_{W}\times U(1)_{Y}$ is embedded in $SO(5)\times U(1)_{X}$, with the extra $U(1)_{X}$ accounting for the hypercharges of SM fermions. The explicit breaking of the global symmetry introduces a pNGB potential such that at the minimum the $SO(5)$ breaking VEV $f$ is slightly rotated away from the direction that preserves the $SU(2)_{W}\times U(1)_{Y}$ gauge group. The misalignment leads to the EWSB at a scale $v\ll f$.

The explicit global symmetry breaking comes from SM gauge interactions and Yukawa interactions. The SM Yukawa couplings arise from the partial compositeness mechanism Kaplan:1991dc : elementary fermions mix with composite operators of the same SM quantum numbers from the strong dynamics,

$$\mathcal{L}=\lambda_{L}\bar{q}_{L}{O}_{R}+\lambda_{R}\bar{q}_{R}{O}_{L},$$

(4)

where $q_{L}$, $q_{R}$ are elementary fermions and ${O}_{L}$, ${O}_{R}$ are composite operators of some representations of $G$ ($=SO(5)$ in MCHMs). The values of couplings $\lambda_{L}$, $\lambda_{R}$ depend on the UV theory of these interactions and are treated as free parameters to produce viable models. With these interactions, the observed SM fermions will be mixtures of elementary fermions and composite resonances. The SM fermions can then couple to the Higgs field through the portion of the strong sector with couplings given by

$$y\simeq\frac{\lambda_{L}\lambda_{R}}{g_{\psi}}\simeq\epsilon_{L}\cdot g_{\psi}\cdot\epsilon_{R}~{},$$

(5)

where $g_{\psi}$ is a coupling of the strong resonances and is expected to be $\gg 1$, $\epsilon_{L,R}$ are ratios $\lambda_{L,R}/g_{\psi}$, which are expected to be small. The resonances created by ${O}_{L,R}$ have masses $\sim g_{\psi}f$, and play the roles of SM fermion partners. They cut off the divergent contributions to the Higgs potential and make it finite. Notice that the operators belong to representations of the global symmetry $G$, but the resonances are divided into representations of $H$ after the symmetry breaking. Because the elementary fermions in general do not fill the whole representations of $G$, the partial compositeness couplings $\lambda_{L}$, $\lambda_{R}$ explicitly break the global symmetry $G$ and generate a nontrivial Higgs potential.

The pNGB Higgs field parametrizes the coset $G/H$ so the potential is periodic in the Higgs field. The Higgs potential can be expanded in $\sin(H/f)$ and up to the quartic term it takes the form

$$V(H)=-\hat{\alpha}f^{2}\text{sin}^{2}\frac{H}{f}+\hat{\beta}f^{2}\text{sin}^{4}\frac{H}{f}~{},$$

(6)

where $\hat{\alpha}$ and $\hat{\beta}$ have mass dimension two and $\hat{\alpha}$ corresponds to the mass-squared parameter of the Higgs field while $\hat{\beta}/f^{2}$ will contribute to the quartic term. By expanding $\sin(H/f)$, higher powers of $H$ can be generated from each term, but for convenience, we will simply call the first term quadratic term and the second term quartic term. The parameters $\hat{\alpha}$ and $\hat{\beta}$ are model dependent and are generated by explicit breaking parameters, like $\lambda_{L}$ and $\lambda_{R}$. Given the potential, we can get the VEV and Higgs mass parameterized as

$$v=\sqrt{\frac{\hat{\alpha}}{2\hat{\beta}}}~{}f,\qquad M_{h}^{2}=8\hat{\beta}\frac{v^{2}}{f^{2}}(1-\frac{v^{2}}{f^{2}})~{}.$$

(7)

The misalignment of the minimum from the SM gauge symmetry preserving direction is parametrized by

$$\xi\equiv\frac{v^{2}}{f^{2}}=\text{sin}^{2}\langle\theta\rangle=\frac{\hat{\alpha}}{2\hat{\beta}}\ll 1~{},$$

(8)

where angle $\langle\theta\rangle\equiv\langle h\rangle/f$. Therefore, for a realistic model, we need $\hat{\alpha}\ll\hat{\beta}$ and at the same time, the correct size of $\hat{\beta}$ to get the observed Higgs boson mass $M_{h}\simeq 125$ GeV.

From the most explicit symmetry breaking effects of the composite Higgs models, one typically gets $\hat{\alpha}>\hat{\beta}$, which is the source of the tuning problem. For example, in MCHM₅ Agashe:2004rs ; Panico:2015jxa , the SM fermions mix with composite operators $O_{L}$, $O_{R}\in\mathbf{5}$ of $SO(5)$. After the symmetry breaking, the composite resonances split into $\mathbf{4}$ and $\mathbf{1}$ representations of $SO(4)$. The mass difference between $\mathbf{4}$ and $\mathbf{1}$ resonances generates a Higgs potential at the compositeness (UV) scale with

	$\displaystyle\hat{\alpha}\sim\frac{N_{c}}{16\pi^{2}}\lambda_{L,R}^{2}M_{\psi}^{2}\sim\epsilon_{L,R}^{2}\frac{N_{c}g_{\psi}^{4}}{16\pi^{2}}f^{2},$		(9a)
	$\displaystyle\hat{\beta}\sim\frac{N_{c}}{16\pi^{2}}\lambda_{L,R}^{4}f^{2}\sim\epsilon_{L,R}^{4}\frac{N_{c}g_{\psi}^{4}}{16\pi^{2}}f^{2}.$		(9b)

The quartic term coefficient $\hat{\beta}$ arises at a higher order in $\epsilon$ than $\hat{\alpha}$, so generically $\hat{\beta}\ll\hat{\alpha}$ is expected instead. It is then required more fine-tuning to achieve the correct EWSB. In some models, it is possible to have $\hat{\alpha}\sim\hat{\beta}$. For example, MCHM₁₄ Panico:2012uw with $O_{L}$, $O_{R}\in\mathbf{14}$ of $SO(5)$ can lead to the potential with

$\displaystyle\hat{\alpha}\sim\hat{\beta}\sim\frac{N_{c}}{16\pi^{2}}\lambda_{L,R}^{2}M_{\psi}^{2}\sim\epsilon_{L,R}^{2}\frac{N_{c}g_{\psi}^{4}}{16\pi^{2}}f^{2},$

(10)

where $\hat{\beta}$ arises at the same order as $\hat{\alpha}$. It requires less tuning to achieve $\xi\ll 1$. This has been called “minimal tuning.” But even so, the UV contribution of Eq. (10) to $\hat{\alpha}$ is larger than the IR contribution from the top quark loop

$\displaystyle\Delta m^{2}_{\text{IR}}\sim\frac{N_{c}}{16\pi^{2}}y_{t}^{2}M_{T}^{2}\sim\epsilon_{L,R}^{4}\frac{N_{c}g_{\psi}^{4}}{16\pi^{2}}f^{2},$

(11)

which already requires some levels of fine-tuning as shown in Eq. (3). This additional UV contribution actually worsens the condition and requires more tuning. A less-tuned scenario is to have a composite right-handed top quark (which is a singlet of $G$). In this case, $\epsilon_{R}\sim 1$ but does not contribute to the Higgs potential. The Higgs potential is controlled by $\lambda_{L}\sim y_{t}$, which can be smaller.

From the above discussion, one can see that to obtain a more natural Higgs potential in CHM, it would be desirable to suppress the contribution from the composite top-partner resonances to the quadratic term. For example, a maximal symmetry was proposed in Ref. Csaki:2017cep to keep the degeneracy of the whole $G$ representation of the top-partner resonances. However, the maximal symmetry is somewhat ad hoc within a simple model and its natural realization requires more complicated model constructions by doubling the global symmetry groups or invoking a holographic extra dimension Csaki:2018zzf ; Blasi:2020ktl . We will look for cosets $G/H$ such that the representation of the top-partner resonances do not split even after the symmetry breaking of $G\to H$ so that it preserves a global symmetry $G$ in any single partial compositeness coupling to prevent unwanted large contributions to the Higgs potential. Besides, we need some additional contribution to the quartic term without inducing the corresponding quadratic term simultaneously to make $\hat{\beta}>\hat{\alpha}$ naturally. This may be achieved by the collective symmetry breaking of the little Higgs mechanism ArkaniHamed:2001nc ; ArkaniHamed:2002qx ; ArkaniHamed:2002qy . Previous attempts include adding exotic elementary fermions to an $SU(5)/SO(5)$ CHM model Vecchi:2013bja and a holographic model with double copies of the global symmetry Csaki:2017eio . Another way of generating the quartic term without the quadratic term using the Higgs dependent kinetic mixing requires both new elementary fermions and an enlarged global symmetry or an extra dimension Csaki:2019coc . We will take a more economical approach by implementing the little Higgs mechanism without adding exotic elementary fermions or invoking multiple copies of the global symmetry, but simply using the couplings that mix SM fermions with composite resonances.

Among the possible cosets, the cosets $SU(5)/SO(5)$ and $SU(6)/Sp(6)$ are potential candidates to realize the ideas discussed at the end of the previous section. If the composite operator $O_{L,R}\in\mathbf{5}(\mathbf{6})$ of $SU(5)(SU(6))$, the corresponding resonances do not split under the unbroken subgroup $SO(5)(Sp(6))$.¹¹1Naïvely they can split into two real representations, but if they carry charges under the extra $U(1)_{X}$ gauge group which is required to obtain the correct hypercharge, they need to remain complex. Since they are still complete multiplets of $G$, there is an enhanced symmetry for each mixing coupling $\lambda_{L,R}$, which protects the pNGB potential. The cosets were also some earliest ones employed in little Higgs models ArkaniHamed:2002qy ; Low:2002ws where the collective symmetry breaking for the quartic coupling was realized. In CHMs, it requires different explicit implementations if no extension of the SM gauge group or extra elementary fermions are introduced. The $SU(5)/SO(5)$ model has a general problem that an $SU(2)$ triplet scalar VEV violates the custodial $SU(2)$ symmetry, leading to strong experimental constraints. We will focus on the $SU(6)/Sp(6)$ model²²2A CHM with the $SU(6)/Sp(6)$ coset were considered in Ref. Cai:2018tet , but for a different prospect. here and leave a brief discussion of the $SU(5)/SO(5)$ model in Appendix A.

In the previous section, we show that in the $SU(6)/Sp(6)$ CHM the UV contribution from the strong dynamics to the Higgs potential is suppressed, minimizing the tuning of the quadratic term. However, we need some additional quartic Higgs potential to further reduce the tuning and to obtain a 125 GeV Higgs boson, as the IR contribution from the top quark loop to the Higgs quartic term is not enough. Generating a Higgs quartic coupling without inducing the corresponding quadratic term is the hallmark of the little Higgs mechanism. For example, in the original $SU(6)/Sp(6)$ little Higgs model Low:2002ws , a Higgs quartic term from the collective symmetry breaking can be generated by gauging two copies of $SU(2)$, with generators given by

$$Q_{1}^{a}=\frac{1}{2}\begin{pmatrix}\sigma^{a}&0&0&0\\ 0&0&0&0\\ 0&0&0_{2\times 2}&0\\ 0&0&0&0\\ \end{pmatrix}\quad\text{and}\quad Q_{2}^{a}=-\frac{1}{2}\begin{pmatrix}0_{2\times 2}&0&0&0\\ 0&0&0&0\\ 0&0&\sigma^{a*}&0\\ 0&0&0&0\\ \end{pmatrix}$$

(41)

and gauge couplings $g_{1}$ and $g_{2}$. The two $SU(2)$’s are broken down to the diagonal $SU(2)_{W}$ by the $\Sigma$ VEV. The potential for the pNGBs generated by the two gauge couplings takes the form

$$g_{1}^{2}f^{2}\left|s+\frac{i}{2f}\tilde{H_{2}}^{\dagger}H_{1}\right|^{2}+g_{2}^{2}f^{2}\left|s-\frac{i}{2f}\tilde{H_{2}}^{\dagger}H_{1}\right|^{2}.$$

(42)

The $g_{1}^{2}$ term preserves the $SU(4)$ symmetry of the $3,4,5,6$ entries which contains the shift symmetry of $H_{1}$ and $H_{2}$. If only the first term of the potential exists, the $\tilde{H_{2}}^{\dagger}H_{1}$ dependence can be absorbed into $s$ by a field redefinition and the term just corresponds to a mass term for $s$. Similarly, the $g_{2}^{2}$ term preserves the $SU(4)$ symmetry of the $1,2,3,6$ entries under which $H_{1}$ and $H_{2}$ remain as Nambu-Goldstone bosons, but with a different shift symmetry. The combination of both terms breaks either of the shift symmetries, and a quartic Higgs potential is generated after integrating out the $s$ field,

$$\lambda\left|\tilde{H_{2}}^{\dagger}H_{1}\right|^{2}\quad\text{with}\quad\lambda=\frac{g_{1}^{2}g_{2}^{2}}{g_{1}^{2}+g_{2}^{2}}~{}.$$

(43)

The possibility of gauging two copies of $SU(2)$ gauge group is subject to the strong experimental constraints on $W^{\prime}$ and $Z^{\prime}$. We would like to generate the quartic Higgs potential without introducing additional elementary fields to the $SU(6)/Sp(6)$ CHM, so we will consider the collective symmetry breaking from the interactions between the elementary fermions and the resonances of the strong dynamics.

From the discussion of the previous section, we see that the elementary quark doublets may couple to composite operators of $SU(6)$ representations $\mathbf{6}$ and/or $\mathbf{\bar{6}}$, and each contains two doublets of the same SM quantum numbers:

	$$\mathbf{6}_{1/6}=\mathbf{2}_{1/6}\oplus\mathbf{1}_{2/3}\oplus\mathbf{\bar{2}}_{1/6}\oplus\mathbf{1}_{-1/3},$$		(44a)
	$$\mathbf{\bar{6}}_{1/6}=\mathbf{\bar{2}}_{1/6}\oplus\mathbf{1}_{-1/3}\oplus\mathbf{2}_{1/6}\oplus\mathbf{1}_{2/3}~{}.$$		(44b)

Both operators can create the same resonances which belong to $\mathbf{6}$ of the $Sp(6)$ group.

Now consider two elementary quark doublets couple to the first two components of the composite operators of $\mathbf{6}$ and $\mathbf{\bar{6}}$ respectively, while both representations contain the same resonances:

$$\lambda_{L}\bar{q}_{La}{\Lambda^{a}}_{i}O_{R}^{i}=\lambda_{L}\bar{q}_{La}{\Lambda^{a}}_{i}\left({\xi^{i}}_{\alpha}Q_{R}^{\alpha}\right),$$

(45)

where

$${(\Lambda)^{a}}_{i}=\Lambda=\begin{pmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ \end{pmatrix}~{},$$

(46)

and

$$\lambda^{\prime}_{L}\bar{q}^{\prime}_{La}\epsilon^{ab}{\Omega_{b}}^{i}O^{\prime}_{Ri}=\lambda^{\prime}_{L}\bar{q}^{\prime}_{La}\epsilon^{ab}{\Omega_{b}}^{i}\left({\xi^{*}_{i}}^{\beta}\Sigma_{0\beta\alpha}Q_{R}^{\alpha}\right),$$

(47)

where

$${(\Omega)_{a}}^{i}=\Omega=\begin{pmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ \end{pmatrix}~{}.$$

(48)

The combination of the two interactions breaks the $SU(6)$ global symmetry explicitly but preserves an $SU(4)$ symmetry of the $3,4,5,6$ entries. It leads to a potential for the pNGBs at $\mathcal{O}(\lambda_{L}^{2}\lambda_{L}^{\prime 2})$ of the form

$$[{(\Lambda)^{a}}_{i}{(\Omega^{*})^{b}}_{j}\Sigma^{ij}][{{(\Omega)_{b}}^{m}}{(\Lambda^{*})_{a}}^{n}\Sigma^{*}_{mn}]~{},$$

(49)

which can easily be checked by drawing a one-loop diagram, with $q_{L}$, $q^{\prime}_{L}$, $Q_{R}$ running in the loop.

After expanding it we obtain

$$\sim\frac{N_{c}}{8\pi^{2}}\lambda_{L}^{2}\lambda_{L}^{\prime 2}f^{4}\left|{(\Lambda)^{a}}_{i}{(\Omega^{*})^{b}}_{j}\Sigma^{ij}\right|^{2}\quad\to\quad\frac{N_{c}}{4\pi^{2}}\lambda_{L}^{2}\lambda_{L}^{\prime 2}f^{2}\left|s+\frac{i}{2f}\tilde{H_{2}}^{\dagger}H_{1}\right|^{2}~{}.$$

(50)

(The factor of 2 comes from the trace which reflects the degrees of freedom running in the loop, as both elementary fermions are doublets.) This is one of the terms needed for the collective symmetry breaking. The coefficient is estimated from the dimensional analysis. Notice that we have chosen different (generations of) elementary quark doublets, $q_{L}$ and $q_{L}^{\prime}$ in the two couplings. If $q_{L}$ and $q_{L}^{\prime}$ were the same, the loop can be closed at $\mathcal{O}(\lambda_{L}\lambda_{L}^{\prime})$ and a large $s$ tadpole term and Higgs quadratic term will be generated,

$$\sim\frac{N_{c}}{8\pi^{2}}\lambda_{L}\lambda_{L}^{\prime}f^{4}\left(\epsilon_{ab}{(\Lambda)^{a}}_{i}{(\Omega^{*})^{b}}_{j}\Sigma^{ij}\right)\quad\to\quad\frac{N_{c}}{4\pi^{2}}\lambda_{L}\lambda_{L}^{\prime}g_{\psi}^{2}f^{3}\left(s+\frac{i}{2f}\tilde{H_{2}}^{\dagger}H_{1}\right)~{}.$$

(51)

Such a term is actually needed for a realistic EWSB, but it would be too large if it were generated together with Eq. (50) that will produce the Higgs quartic term. It can be generated of an appropriate size in a similar way involving some other different fermions and composite operators with smaller couplings.