A Flavorful Composite Higgs Model :
Connecting the B anomalies with the hierarchy problem

The Standard Model (SM) of particle physics successfully describes all known elementary particles and their interactions. However, there are still a few puzzles that have yet to be understood. One of them is the well-known hierarchy problem. With the discovery of light Higgs bosons in 2012 [1, 2], the last missing piece of the SM seemed to be filled. 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 require an extremely sensitive cancellation to get a $125$ GeV Higgs boson. 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.

One 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 [3, 4]. Through the analogy to 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 $H$ at 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.

Among all types of CHMs with different cosets, the CHMs with fundamental gauge dynamics featuring only fermionic matter fields are of interest in many studies [5, 6, 7, 8], which is known as the fundamental composite Higgs model (FCHM). In this type of CHMs, hyperfermions $\psi$ are introduced as the representation of hypercolor (HC) group $G_{HC}$. Once the HC group becomes strongly coupled, hyperfermions form a condensate, which breaks the global symmetry. However, they always introduce more than four pNGBs, which means more light states are expected to be found. The minimal FCHM, which is based on the $SU(4)/Sp(4)$ coset [9, 10, 11], contains five pNGBs. The four of them formes the SM Higgs doublet, and the fifth one, as a SM singlet, could be a light scalar boson (if the symmetry is global) or a TeV-scale $Z^{\prime}$ boson (if the symmetry is local). No matter which, it should lead to some deviations in low energy phenomenology.

Although the direct searches by ATLAS and CMS haven’t got any evidence of new particles, LHCb, which does the precise measurement of B meson properties, shows interesting hints of new physics. There are discrepancies in several measurements of semileptonic B meson decays, especially the tests of lepton flavor universality (LFU), which are so-called the neutral current B anomalies [12, 13, 14, 15, 16, 17, 18]. Each anomaly is not statistically significant enough to reach the discovery level, but the combined analysis shows a consistent deviation from the SM prediction [19, 20, 21, 22, 23, 24]. These anomalies might be the deviation we are looking for.

One of the popular explanations is through a new $Z^{\prime}$ vector boson which has flavor-dependent interactions with SM fermions. Many different types of $Z^{\prime}$ models with diverse origins of $U(1)^{\prime}$ gauge symmetry have been proposed [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. Depending on its couplings with fermions, the mass of the $Z^{\prime}$ can range from sub-TeV to multi-TeV. For a $Z^{\prime}$ boson at the TeV scale, it is natural to try to connect it with the hierarchy problem ¹¹1For our interest, we would like to mention some researches aiming at explaining the B anomalies within composite Higgs models. Different studies using different features of composite theory to address the problem, such as additional composite leptoquarks [54, 55, 57, 58] or composite vector resonances [59, 60, 61, 62, 63, 64, 65]. However, they are all different from this study, where we introduce a new $Z^{\prime}$ boson..

In this paper, we realize this idea using a $SU(4)/Sp(4)$ FCHM, where an $U(1)^{\prime}$ subgroup within $SU(4)$ is gauged. The corresponding $Z^{\prime}$ boson only couples to the third generation SM fermions $F_{3}$ and the hyperfermions $\psi$ through the terms

where $g_{Z^{\prime}}$ was normalized such that SM fermions $F_{3}$ carry a unit charge and hyperfermions carry charge $Q_{HC}$. When the hypercolor group becomes strongly coupled, the global symmetry $SU(4)$ and its gauged $U(1)^{\prime}$ subgroup are broken. The $5^{th}$ pNGB is eaten by the $U(1)^{\prime}$ gauge boson, which results in a TeV-scale $Z^{\prime}$ boson. We will test the potential for this $Z^{\prime}$ boson to explain the neutral current B anomalies. The parameter space allowed by different experimental constraints, mainly from neutral meson mixings and lepton flavor violation decays, will be discussed. The bounds on $M_{Z^{\prime}}$ from the LHC direct searches are also shown.

This paper is organized as follows. In section II, we introduce the $SU(4)/Sp(4)$ FCHM. The calculations of the gauge sector, including SM gauge group and $U(1)^{\prime}$ gauge symmetry, are presented. To study its phenomenology, we specify the transformation between flavor basis and mass basis in section III. The resulting low energy phenomenology is discussed in section IV, including the B anomalies and other experimental constraints. Section V focuses on the direct searches, which play an important role in constraining a TeV-scale $Z^{\prime}$ boson. Section VI and Section VII contains our discussions and conclusions.

In fundamental composite Higgs models, additional hyperfermions $\psi$ are added to generate composite Higgs. The hyperfermions are representations of hypercolor group $G_{HC}$, whose coupling becomes strong around the TeV scale. The hyperfermions then form a condensate, which breaks the global symmetry and results in the pNGBs as the Higgs doublet. In this paper, we study the minimal fundamental composite Higgs model based on the global symmetry breaking $SU(4)\to Sp(4)$. The fermionic UV completion of a $SU(4)/Sp(4)$ FCHM only require four Weyl fermions in the fundamental representation of the $SU(2)=Sp(2)$ hypercolor group [7, 8]. The four Weyl fermions transform under $G_{SM}=SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ as

Next, we rewrite the two right-handed hyperfermions as $\tilde{U}_{L}=-i\sigma^{2}C\bar{U}^{T}_{R}$ and $\tilde{D}_{L}=-i\sigma^{2}C\bar{D}^{T}_{R}$. Since all the four Weyl fermions are according to the same representation of the hypercolor group, we can recast them together as

which has a $SU(4)$ global symmetry (partially gauged). The hypercolor group becomes strongly coupled at the TeV scale, which forms a non-perturbative vacuum and breaks the $SU(4)$ down to $Sp(4)$. In CHMs, the condensate $\langle\psi\psi\rangle\propto\Sigma_{0}$ is chosen such that electroweak symmetry is preserved. It will be broken after the Higgs interactions and loop-induced potentials are taken into account. However, we will only focus on some key ingredients here and leave the complete analysis to the future.

To discuss the phenomenology, we need to first rewrite the $Z^{\prime}$ interaction terms in eq. (1) to cover all generations and separate different chirality as

where $F=(F_{1},F_{2},F_{3})$ includes SM fermions of all the three generations with superscript $f$ for flavor basis. The $3\times 3$ charge matrices in the flavor basis look like

However, to study phenomenology, we need to transform them to the mass basis $F_{L/R}^{m}$ through the mixing matrices as $F^{f}_{L/R}=U_{F_{L/R}}F^{m}_{L/R}$. After the transformation, we get

where the charge matrices becomes

Therefore, we need to know all the $U_{F_{L/R}}$ to determine the magnitude of each interaction. However, The only information about these unitary transformation matrices is the CKM matrix for quarks and PMNS matrix for leptons. The two relations that need to be satisfied are

which only tells us about the left-handed part with no information about the right-handed part. Even with these two constraints, they only give the difference between two unitary transformations, but not the individual one. Therefore, we need to make some assumptions about the matrices so there won’t be too many parameters.

To simplify the analysis, we assume all the $U_{F_{R}}$ are identity matrices. Therefore, for right-handed fermions, only the third generation joins in the interaction with no flavor changing at all. The couplings are the same for all the right-handed fermions it couples to with coupling strength $g_{Z^{\prime}}$.

For the left-handed side, due to the observation of $V_{CKM}$ and $V_{PMNS}$, there is a guarantee minimal transformation for $U_{F_{L}}$. Because we only care about the transition between the second and third generation down-type quarks and charged leptons, we will only specify the rotation $\theta_{23}$ between the second and third generation of $U_{d_{L}}$ and $U_{e_{L}}$ as

where $F=d,\,e$. Keeping only the angle $\theta_{23}$ is a strong assumption but a good example case for phenomenological study because it avoids some of the most stringent flavor constraints from light fermions and leaves a simple parameter space for analysis. Following this assumption, the rest of the matrices are fixed as $U_{u_{L}}=V_{CKM}^{\dagger}U_{d_{L}}$ and $U_{\nu_{L}}=V_{PMNS}^{\dagger}U_{e_{L}}$. Notice that, although they looks similar, the magnitude we expect for the two angles are quite different. For $\theta_{d}$, we expect it to be CKM-like, i.e. sin $\theta_{d}\sim\mathcal{O}(0.01)$. However, for $\theta_{e}$, it could be as large as sin $\theta_{e}\sim 1$.

We can then calculate the charge matrices as

where $F=d,\,e$, and write down all the coupling for left-handed fermions. To study the B anomalies, two of them, $g_{sb}$ and $g_{\mu\mu}$, are especially important, so we further define

We will see later that constraints will be put on the three key parameters: the scale $f^{\prime}$, the mixings $\epsilon_{sb}$, and $\epsilon_{\mu\mu}$.

With the specified mixing matrices, we can then discuss the parameter space allowed to explain the B anomalies. Also, the constraints from other low energy experiments are presented in this section.

The measurements from flavor physics in the last section can only put the constraints on the mixings and the scale $f^{\prime}=M_{Z^{\prime}}/g_{Z^{\prime}}$. The direct searches, on the other hand, can give the lower bound on the mass of $M_{Z^{\prime}}$ directly. A general $Z^{\prime}$ collider search has been discussed in [75]. In this section, we will focus on the scenario determined by our model.

In this study, we are interested in the value of $f^{\prime}$, which is related to the breaking scale $f$, and the bound on $M_{Z^{\prime}}$, which is important for the collider searches. In the last section, we found that a certain straight line (such as the blue line) in Fig. 1 corresponding to a predicted cross section $\sigma_{\mu\mu}(f^{\prime}_{0})$, which is given by

where $f^{\prime}_{0}$ represents the slope of the line, e.g. for the blue line in Fig. 1, $f^{\prime}_{0}=7.7$ TeV. Using this relation, we can calculate the cross section $\sigma_{\mu\mu}$ for each point in the parameter space in Fig. 1 with a certain value of $M_{Z^{\prime}}$. It allows us to combine “the constraints in the parameter space in $f^{\prime}$ v.s. $\epsilon_{\mu\mu}$ plot” (as shown in Fig. 1) with “the direct $\mu\mu$ channel search results from the ATLAS [78]” into “the viable parameter space in $f^{\prime}$ v.s. $M_{Z^{\prime}}$ plot” as shown in Fig. 3.

The blue region is excluded by the $B_{s}-\bar{B}_{s}$ meson mixing, which gives the lower bound $M_{Z^{\prime}}\gtrsim 1.2$ TeV. The bright blue line corresponds to the same parameter space as in Fig. 1 with $M_{Z^{\prime}}\sim 1.2$ TeV. The yellow region, also excluded by the $B_{s}-\bar{B}_{s}$ meson mixing, sets the maximum value for $f^{\prime}$ as shown in eq. (32), which can also be found directly in Fig. 1. Once the stronger constraint from $B_{s}-\bar{B}_{s}$ meson mixing is placed, the yellow line will move downward and the blue line will move rightward. The red region, which is excluded by $\tau\to 3\mu$, restricts the parameter space from below. It places the lower bound on $f^{\prime}$, which will be pushed upward if the constraint becomes stronger. We can also see the data fluctuations in dimuon search become the fluctuations on the red curve. The strength of the coupling $g_{Z^{\prime}}$ with three different values is also labeled as the black straight line in the plot.

There are two regions worth noticed in the plot: (1) The region with the light $Z^{\prime}$ that corresponds to a small $g_{Z^{\prime}}$ but a large $f^{\prime}$ region, i.e. $(g_{Z^{\prime}},f^{\prime})\sim{(0.2,7\text{ TeV})}$. (2) For a natural CHM without a large fine-tuning, a smaller $f$ (and thus $f^{\prime}=4f$) is preferred, which corresponds to a larger $g_{Z^{\prime}}$ region, such as $(g_{Z^{\prime}},f^{\prime})\sim{(0.5,4\text{ TeV})}$ with a heavier $Z^{\prime}$. Both regions are around the boundary. The direct searches will extend both blue and red exclusion regions rightward, so both points we mentioned will be probed soon. The lower bound on $M_{Z^{\prime}}$ will be pushed to 2 TeV and most of the interesting parameter space will be explored during the HL-LHC era [79, 80].

In this paper, we presented a new $Z^{\prime}$ solution to the B anomalies, whose scale is related to the symmetry breaking scale of the underlying strong dynamics. We found that the anomaly-free $U(1)^{\prime}$ symmetry can arise from $SM_{3}-HF$, the difference between the third generation SM fermion number and the hyperfermion number. This type of $U(1)^{\prime}$ is naturally broken at the TeV scale in many fundamental composite Higgs models, which allow us to connect it with the hierarchy problem. We constructed a concrete model based on $SU(4)/Sp(4)$ minimal FCHM. The relation $f^{\prime}=2\,|Q_{HC}|f=4f$ connects the flavor anomalies scale $f^{\prime}$ with the symmetry breaking scale $f$ in the FCHM.

The potential for the $Z^{\prime}$ boson to explain the B anomalies is discussed in detail. Other flavor physics measurements, like neutral meson mixings and lepton flavor violation decays, put constraints on the allowed parameter space as shown in Fig. 1. The direct searches also give the bound on the mass of $Z^{\prime}$ as $M_{Z^{\prime}}\gtrsim 1.2$ TeV. The combined constraints on the scale $f^{\prime}$ v.s. mass $M_{Z^{\prime}}$ are shown in Fig. 3, which gives a clear picture about how the parameter space will be probed in the future. Some attractive regions are still viable and will be tested during the HL-LHC era.