Composite Higgs Models with a Hidden Sector

We discuss the phenomenology of composite Higgs models that naturally produce a Standard Model-like Higgs boson with a mass of $126$ GeV. The effective theory below the compositeness scale is weakly coupled in these models, and the goldstone sector acts as a portal between the third generation of quarks and a hidden gauge sector. The addition of hidden-sector fermions gives rise to a calculable effective scalar potential with a naturally light scalar resonance. The generic prediction of these theories is the existence of additional pseudo-Nambu Goldstone bosons with electroweak-scale cross sections and masses. In this paper we analyze the collider signatures for some simple concrete realizations of this framework. We find that despite the existence of additional weakly and strongly coupled particles that are kinematically within reach of current experiments, the generic signatures are difficult to resolve at the LHC, and could remain well-hidden in the absence of an $e^+ e^-$ Higgs factory such as the CEPC or a surface detector such as MATHUSLA.


I. INTRODUCTION
Despite its success in discovering a Standard Model-like Higgs boson, the Large Hadron Collider (LHC) has yet to provide a satisfying explanation for the mechanism of electroweak symmetry breaking (EWSB). To date there is no discovery leading the way to new physics, and many of the popular explanatory frameworks are becoming constrained into finely tuned regions of their parameter spaces. Theoretical development over the last several decades has largely been motivated by criteria of naturalness and parsimony [1]. While there is a strong logical and historical motivation for this notion of naturalness, there is also an arguably comparable motivation for cautious skepticism in our conceptions about parsimony. In this paper we study a class of effective theories that generically give rise to a composite Higgs boson with Standard Model (SM)-like properties, and analyze their collider signatures. A light Higgs with a large quartic is achieved via the introduction of a hidden-sector that couples to the SM through the Higgs portal, thus preserving naturalness at the expense of parsimony. This work adds to a growing abundance of effective models for electroweak symmetry breaking with highly subtle hadron-collider signatures, and motivates several concrete analyses for further scrutiny.
The composite Higgs hypothesis [2] offers an attractive and radically conservative [3] possibility for the origin of a naturally macroscopic electroweak scale. In this framework the compositeness scale is naturally higher than the electroweak symmetry breaking scale, as the Higgs arises as a pseudo-Nambu-Goldstone boson (pNGB) [4][5][6][7][8][9][10][11][12][13][14][15][16] from the spontaneous breaking of an approximate global symmetry by the compositeness dynamics. These models retain a weakly coupled description of the EWSB sector above the chiral symmetry breaking scale f , with a full UV completion appearing at a higher scale associated with the new strong dynamics Λ [17,18]. The most parsimonious constructions of composite Higgs theories have historically found tension with measurements of precision electro-weak (PEW) observables due to their generic assumption of additional gauge structure above the scale f . The Intermediate Higgs (IH) or "natural composite" scenario [19][20][21][22] relaxes this assumption, which results in a quadratic sensitivity to the compositeness scale Λ from loops of gauge bosons. However this contribution is numerically small enough to avoid serious fine tuning for a new physics scale as high as Λ ∼ 10 TeV. The large radiative corrections to the Higgs mass from the top sector are canceled by new vector-like quarks with TeV-scale masses, utilizing the mechanism of collective symmetry breaking, and resulting in partially composite third generation [23].
The recently measured mass of the SM Higgs boson m h = 126 GeV [24,25] provides a significant piece of additional data for the composite Higgs hypothesis. In the IH scenario, the scalar potential is highly constrained by the explicit symmetry breaking pattern of the approximate global symmetries. Again the most parsimonious constructions have historically found difficulty generating a sufficiently light Higgs mass or an otherwise viable Higgs potential [26][27][28][29][30][31][32]. In this paper we consider the effects of extending the composite sector to include additional hidden-sector interactions via Yukawa couplings between the pNGB matrix and a new multiplet of hidden-sector fermions. We find that the radiative corrections to the Higgs potential from these states at f generically produce a pattern of EWSB that is consistent with current experimental data. The invariant prediction of these theories is the existence of new vector-like quarks with masses of O(f ) and new weakly interacting scalars with masses of O(gf ). All of the pNGB scalars remain uneaten in these models and manifest as physical resonances, however their weak scale cross sections are typically below threshold for discovery via direct production at the LHC. Furthermore the extended Yukawa sector can generically lead to nonstandard decays of the new quarks that significantly weakens the limits from LHC searches for simplified top-partner models [33,34]. The variety of possible coset spaces thus provides a wide range of theoretically motivated collider signatures that should now be considered more seriously.
In this paper we select two contrasting examples for detailed analysis. Section II discusses a model based on the SU (5)/SO(5) coset space [5], which generates a rich variety of new scalar resonances. Here we focus on the low-energy phenomenology as well as the conditions for a realistic pattern of EWSB. Large couplings between the pNGBs and the new vector-like quarks in this model lead to an enhanced production of high-multiplicity tops and bottoms, resulting in cascade decays with many leptons and b-quarks in the final state. The hiddensector interactions result in a fairly generic hidden-valley phenomenology which we review in the context of these models. Section III discusses the SU (4)/Sp(4) coset space [5], which generates only one new pNGB resonance beyond the Higgs multiplet. We briefly discuss the tension between the fermionic sector of this model and electroweak precision constraints on the bottom quark interactions and how to alleviate this. We then focus on the difficulties in resolving the goldstone sector of this theory, which could remain well hidden until future Higgs factories come online. In Section IV we conclude with a summary of these results.

II. SU (5)/SO(5) INTERMEDIATE MODEL
The symmetry breaking pattern SU (5)/SO(5) produces 24 − 10 = 14 pseudo Nambu-Goldstone Bosons (pNGBs). The sigma field transforms under SU (5) as V ΣV T where V is an SU (5) matrix. It is convenient to specify an SO(5) symmetric background field Σ 0 , under which the unbroken SU (5) generators T and the broken generators X satisfy The SO(5) symmetry contains the global custodial SO(4) c = SU (2) L ⊗ SU (2) R subgroup, where SU (2) R is approximate and contains U (1) Y . The generators of this global symmetry L a and R a can be expressed as To describe the quantum numbers, we embed the SU (2) L ⊗ U (1) Y gauge symmetry of the Standard Model into the global SU (2) L ⊗ SU (2) R with Y = R 3 . These generators remain unbroken in the reference vacuum Σ = Σ 0 and the pNGB's are fluctuations about this background in the direction of the broken generators, Π ≡ π a X a . Under under the gauged SU (2) L ⊗ U (1) Y they transform as 1 0 ⊕ 2 ±1/2 ⊕ 3 0 ⊕ 3 ±1 and may be parameterized as All of the goldstone modes in this theory remain uneaten and appear as physical fluctuations. These include the SM Higgs doublet H = (H + H 0 ), a parity-odd electroweak singlet η, and a set of Georgi-Machacek scalars Φ and Φ [35,36] which may be written as Under the SU (2) L ⊗ SU (2) R global symmetry the pNGBs transform as (2, 2) ⊕ (3, 3) ⊕ (1, 1). The SM Higgs field transforms as a bi-doublet, while the real and complex triplets Φ and Φ transform together as a bi-triplet. The fermionic sector of this theory gives a large negative contribution to the mass of the Higgs, while the remaining goldstone masses are dominated by a positive contribution from the one-loop gauge interactions. The resulting vacuum breaks the gauge symmetry down to U (1) EM and can be parameterized by a single angle θ.
The SU (5) global symmetry is explicitly broken by the SU (2) L × U (1) Y gauge couplings. The covariant derivative can be expanded to second order in the vacuum giving tree level masses to the weak gauge bosons in terms of the SU (2) L ×U (1) Y couplings g, g . The vacuum expectation value of the Higgs bi-doublet thus gives mass to the W and Z bosons which preserves the remnant custodial SU (2) c , thus guaranteeing tree level relation m W /m Z = cos θ w .

A. Fermion Sector
If the SU (5) symmetry arises accidentally from the dynamics of a strongly coupled theory then the Yukawa sector could contain interactions between the Σ field and composite fermions in SU (5) multiplets. Yukawa couplings that softly break the global symmetry can result in potentially large contributions to the scalar potential. However these interactions can naturally be constructed with a collective symmetry breaking property that guarantees the absence of quadratic divergences to the Higgs mass from fermion loops. Despite the absence of such quadratic divergences, constructing soft-symmetry breaking interactions that reproduce the measured properties of the Higgs boson remains non-trivial. We find that this goal can be achieved in a relatively simple way by extending the fermion sector of this theory to include two vector-like multiplets of fermions in the fundamental representation of SU (5). One of the multiplets (ψ,ψ) is color-charged and mixes with the SM quarks resulting in a partially composite third generation. The other multiplet (χ,χ) is assumed to be charged under a hidden-sector gauge group G that confines at some scale Λ < f .

Color-charged Fermions
The most serious quadratic divergence to the Higgs potential from the top quark can be eliminated by extending the fermion sector of the Standard Model to include new vector-like quarks (ψ,ψ) in the fundamental representation of SU (5). The states in this multiplet mix with the massless chiral third generation q and (t,b) to produce partially composite top and bottom mass eigenstates. The gauge quantum numbers of the vector-like quarks are fixed by the requirement of partial compositeness and the proposed embedding of SU (2) L ×U (1) Y ⊂ SU (5). The corresponding embedding of the gauge eigenstate components in (ψ,ψ) is given in Equation 9, and their transformation properties are given in Table I  The most general gauge invariant fermion interactions include terms that softly break the SU (5) global symmetry. For simplicity we assume that the SM bottom quark mass term arises from the Yukawa interactions of an incomplete SU (5) multiplet. This introduces an explicit source of SU (5) breaking and a quadratic contribution to the Higgs potential from loops of bottom quarks. However due to the relatively small value of the bottom quark Yukawa coupling, these effects are numerically negligible compared to the leading logarithmic contribution from the top sector. We thus express the Yukawa interactions for the color-charged fermions as The top and bottom quark mass matrices M T , M B can be expressed in terms of the Higgs vacuum expectation value θ defined in Equation (6), and the UV-insensitivity of the Yukawa interactions is guaranteed by the following identity Diagonalizing M T and M B gives seven mass eigenstates. The up-type sector contains four charge ±2/3 states which we label in order of descending mass as t , t , t , t, and the lightest of these states corresponds to the SM top quark. The down-type sector contains three mass eigenstates. Two of these b , b have charge ∓1/3 and include the SM bottom quark, and there is additionally a "peculiar" charge ±5/3 quark p , which can lead to interesting phenomenological signatures. The heavy mass eigenstates are independent of the Higgs VEV at leading order, and the numerically diagonalized mass spectrum is shown in Figure (1). Their analytic expressions have been computed in Appendix B to leading order in sin 2 2θ and are approximated by the following scalings.
The top-sector mass matrix M T also has the property that det M † T M T ∝ sin 2 2θ, thus producing one massless state when θ = nπ/2 for n ∈ Z. The states corresponding to the SM top and bottom quarks t and b remain massless at zeroth order in θ, and couple to the Higgs via the following Yukawa couplings y t = y 1 y 2 y 3

Hidden-Sector Fermions
A realistic pattern of EWSB via radiative effects can be achieved throughout this parameter space by introducing an additional multiplet of vector-like fermions (χ,χ). In this analysis we will assume that (χ,χ) are charged under a hidden-sector gauge group G, transforming in the ( , 5) ⊕ ( , 5) representation of G × SU (5). If the hidden-sector gauge group G confines at some scale Λ < f , then the low energy spectrum will be comprised of hidden-sector mesons, baryons, and glueballs. Due to a conserved U (1) B in the hidden-sector, the baryons can be stable on cosmological time scales and may thus be an attractive candidate for dark matter if the lightest baryon is electrically neutral. The direct and indirect detection signatures for these scenarios have been well studied in the context of asymmetric dark matter [37][38][39][40][41][42][43] and will not be discussed further. Here we choose a hypercharge assignment for the hidden-sector fermions that can naturally accomodate a number of low-lying neutral mass eigenstates. The hidden-sector baryons in this framework thus become a plausible candidate for dark matter, although in general this will depend on the discrete Z N subgroup at the center of G and the mass splitting interactions of the hidden baryon spectrum. The embedding of the gauge eigenstate components in (χ,χ) is given in Equation 14 and the gauge quantum numbers of these new states are given in Table III.
A realistic Higgs potential from one-loop fermion corrections can be obtained by introducing symmetry breaking interactions between the hidden-sector SU (5) multiplet (χ,χ) and an additional massless vector-like pair of hidden-fermions (n,n). The Yukawa interactions thus include an SU (5) symmetric coupling to the Σ-field in addition to terms that softly break the global symmetry L χ =ỹ 1 f χ Σχ +ỹ 2 f n N +ỹ 3 f N n + h.c. In this scenario the mass matrix for the neutral hidden-sector states M N has the property det M † N M N ∝ cos 2 2θ, resulting in one massless particle when θ = (n + 1/2)π/2 for n ∈ Z. The absence of a quadratic senstivity to the Higgs mass from loops of hidden-sector fermions is guaranteed by the identity The hidden-sector mass matrix for M N produces four electrically neutral mass eigenstates which we label in in order of descending mass as n , n , n , n. The lightest of these states is generically comparable in mass to the heaviest goldstone modes, and thus stable against pair production via resonant scalar decay. The down-type hidden-sector mass eigenstates include a charge ±1 vector-like pair (x + ,x + ) and a charge ∓1 vector-like pair (y − ,ȳ − ). Since these hidden-sector states are color-neutral, their relatively low production cross sections make all but the lightest state n inconsequential for collider phenomenology. For the remainder of this section we thus consider a simplified parameter space for this sector in whichỹ 3 =ỹ 2 =ỹ 1 . Analytic expressions for general couplings have been derived in Appendix B to leading order in sin 2 2θ. With this simplified parameter space the hidden-sector fermion masses scale approximately as

Theory Space
The fermion interactions introduce a number of new parameters to the Standard Model. The new color-charged interactions introduce four Yukawa couplings y i , two of which are fixed by the Standard Model top and bottom quark masses as implied by Equation 13. This leaves two free parameters y 1 and y 2 which are bound from below by the top-quark Yukawa coupling y t < y 1 y 2 / y 2 1 + y 2 2 . The large observed top quark mass in this framework thus descends from the O(1) values of the composite sector Yukawa couplings. The new hidden-sector interactions introduce three new Yukawa couplings which we have reduced to a single parameter by assuming y 1 =ỹ 2 =ỹ 3 . The fermion content of this theory has been chosen to allow for a pattern of electroweak symmetry breaking that can be tuned to agree with the measured properties of the Higgs boson. In this simplified parameterization, for each point in the two-dimensional parameter space of (y 1 , y 2 ) there exists a unique value forỹ 1 which gives a Higgs mass and quartic that is consistent with Standard Model predictions. The conditions of EWSB thus generically imply a mass scale for the new hidden-fermions that is O(few) times lower than the masses of the new color-charged states. The masses of the new fermions as a function of the chiral symmetry breaking scale f are illustrated for two benchmark points in Figure 1.
These Yukawa interactions also introduce additional sources of SU (2) c violation which can generate large contributions to the T -parameter at loop-level. These contributions will vanish as the chiral symmetry breaking scale f increases due to the decoupling of these vector-like states, but can become unacceptably large at low values of f . The T -parameter may be computed from the fermionic contributions to the W and Z-boson self-energies Π W W and Π ZZ as defined in Equation 18. We find that these SU (2) c violating interactions provide unacceptable contributions to the T -parameter when f becomes on the order of 600 GeV, as illustrated in Figure 2.

B. Electroweak Symmetry Breaking
The scalar effective potential in this theory is sculpted by the radiative corrections from SU (5) breaking interactions at f , which may be extracted from the quadratically and logarithmically sensitive terms in the Coleman Weinberg potential. The absence of additional gauge structure above the chiral symmetry breaking scale implies that the contribution from W and Z loops is quadratically sensitive to the compositeness scale f , which we take to be Λ ∼ 4πf . The form of this gauge contribution δV G can be extracted from the covariant derivative and expressed in terms of the gauge boson mass matrix Here c is a UV sensitive constant which parameterizes the leading gauge contribution to the effective potential for the pNGBs. Dimensional analysis gives c of order 1, while a technicolorlike UV completion of the model requires c to be positive, that is, the gauge contribution to vacuum alignment prefers vanishing gauge boson masses. The analogous term in the effective low energy description of QCD gives the (positive) mass squared splitting between the π + and π 0 [16]. The absence of quadratically divergent one-loop contributions from the Yukawa interactions follows from the properties of the color-charged and hidden-sector fermion mass matrices in Equation 11 and Equation 16. The most significant contribution to the scalar potential from fermions δV F in this effective theory thus comes from the logarithmically sensitive term. The above relations again guarantee that the Λ dependent terms cancel order by order in θ. The result is thus sensitive only to scales of O(f ), and can be written in terms of ratios of mass eigenstates.
The collective symmetry breaking property of these mass matrices guarantees that these contributions to the effective potential are extremely well approximated by their leading order terms in det M † F M F . As argued in Appendix B, higher order terms in det M † F M F are negligible due to corresponding suppression by powers of (tr M † F M F ) 4 . The most significant logarithmic contributions from the color-charged top-like sector δV T and the neutral hidden-sector δV N thus enter with opposite sign, and the θ dependence is numerically well approximated by δV F ∝ ± cos 4θ. The dominant contribution to the Higgs potential from fermions may thus be parameterized by two scales f T and f N , which are functions of the color-charged and hidden-sector Yukawa couplings respectively.
In this simplified parameter space, the hidden-sector fermion mass spectrum is fixed by the masses of the color-charged fermions and the conditions of electroweak symmetry breaking. Theỹ i dependence of this potential for arbitrary hidden-sector Yukawa couplings are derived in Appendix B.
Since all of the Yukawa and gauge interactions are invariant under the axial U (1) a ⊂ SU (5), the goldstone excitation η in this direction is left massless by the contributions considered thus far. In order to give a mass to this gauge-singlet large enough to avoid constraints from hadronic physics, we introduce an explicit source of U (1) a violation in the form of a spurion term that proportional to the reference vacuum M 0 ≡ m 0 Σ 0 . Such a contribution would descend naturally from fermion mass terms in the UV completion.
In the absence of additional symmetry breaking terms, the scalar potential is fully parameterized by the Yukawa coupings y i andỹ i , the 1-loop gauge coefficient c, and spurion coefficient m 0 . This potential has a generic EWSB vacuum when f 2

and can be expressed as
The order parameter θ 2 is assumed to develop a vacuum expectation value at v 2 = 4f 2 θ 2 = (246 GeV) 2 . The observed Higgs mass m h = 126 GeV thus fixes the relation between the colorcharged and hidden sector Yukawa couplings, as well as between the gauge renormalization coefficient c and the axial-breaking spurion coefficient m 0 . The mass spectrum of the non-SM scalars can also be extracted from these higher dimensional operators. The fermionic contributions to the potential of the scalars Φ and Φ are suppressed by O(θ 2 ). Their masses are thus roughly independent of the Yukawa couplings y i and dominated by the positive contribution from gauge interactions.
The mass of the gauge-singlet η is simply proportional to the coefficient of the spurion operator m 0 , which is a function of the Higgs mass m h and the gauge renormalization coefficient c. The mass splitting between the Georgi-Machacek scalars Φ/Φ and the gauge-singlet η thus grows as a function of c, with η developing a vacuum expectation value at c ∼ 0.35. These features of the scalar mass spectrum are shown in Figure (3).

C. Phenomenology
In this SU (5)/SO(5) model the spectrum of new states contains strongly interacting quarks with masses that scale as M T ∼ y i f , hidden-sector fermions with masses that scale as M N ∼ỹ i f , and weakly interacting scalars with masses that scale as M S ∼ gf . At √ s = 13 TeV the neutral spin-0 states are singly produced via gluon fusion with O(pb) cross sections, and there is also zoo of new particle states that are pair produced with O(fb) cross sections. These include Drell-Yan production of the charged goldstone scalars, strong production of the lightest quark partners t t /p p , as well as weak production of the lightest hidden-fermions nn. The lightest hidden-sector states will be a spectrum of glueballs which could be long-lived, and produced via nn annihilation or through Higgs decay. The direct production cross section for these states at two benchmark points are shown in Figure 4. Despite the abundance of new particle states in this theory, their production cross sections and dominant decay paths would make them extremely difficult to resolve in current LHC data sets. Drell-Yan production of the doubly charged scalarφ ++ is appreciable in large parts of the parameter space due to the Q 2 ++ enhancement to its cross section relative to the other charged states. However, existing limits on this process typically assume an O(1) branching fraction to pairs of same sign leptons [44,45] rendering them inapplicable to the goldstone scalars in this model, which are leptophobic by assumption. Pairs of the lightest top quark partner t t and peculiar quark p p can also be strongly produced with cross sections as high as ∼ 5 fb in the viable regions of parameter space. The strongest limits on vector-like quarks are set by the CMS search at √ s =13 TeV in the single lepton channel [33,34]. These limits are driven by kinematically optimized searches for tt + X and bb + X where X = W, Z, h is highly boosted, and are interpreted in the context of simplified models that assume BR(t Z t) + BR(t h t) + BR(t W b) = 1. In this model the lowest lying vector-like partners are composed mostly of the hypercharge-7/6 doublet (P T , P B ) and their branching fractions are dominated by decays to top and bottom quarks in association with the scalar states in Φ. These generic experimental signatures are thus qualitatively different from those of simplified models of top and bottom partners, rendering existing limits on vector-like quarks largely inapplicable.

Resonant tt Final States
Gluon fusion of neutral pseudo-scalars φ 0 , η and the complex scalarφ 0 generically leads to a resonant tt final state. The CP properties of the pNGB matrix forbids the two-body decays of although the gauge singlet η can also have a significant branching fraction to gg and bb when its mass is below the top threshold as computed in Section III C. Currently the strongest limit on the process gg →φ 0 /φ 0 → tt comes from the ATLAS search at √ s = 8 TeV in the single lepton channel [46]. The gluon fusion process is mediated by loops of heavy vector-like quarks. Their effects may thus be parameterized by a set of dimension-5 couplings c S gg , which can be expressed as a sum over the well known gluon vertex function for the fermion triangle graph Searches for scalar resonances decaying to tt pairs is complicated by destructive interference between the spin-0 and spin-1 mediated s-channel diagrams in Figure 5. This results in a "peak-dip" structure for the invariant mass distribution of the tt pairs rather than the usual Breit-Wigner resonance. The analysis performed by ATLAS searches for this kinematic feature in resolved tt pairs at invariant masses in the range of m tt = 500 − 800 GeV 1 . The results are interpreted in the context of a Two-Higgs-Doublet model (2HDM) and the cuts are kinematically optimized for the case of a scalar, a pseudo-scalar, and the case of a degenerate scalar-pseudoscalar pair. The limits are reported as a function of the 2HDM parameter tan β in the alignment limit where the W/Z couplings are equal to their SM values. In this limit, tan β can be interpreted as the ratio of the top quark Yukawa coupling to the SM Higgs and heavy Higgs bosons, tan β → y H tt /y H tt . Here H and H refer to the SU (2) L doublets containing the SM and heavy Higgses respectively.
The strength of this kinematic feature is a function of ratios of the amplitudes for the two processes, which include factors involving the vertex function V H gg . For each scalar mass eigenstate S we can thus define an analagous angle β S eff by replacing the vertex function in the numerator with its analogous dimension-5 coupling as shown in Equation 41. If the vector-like quark masses are heavy and decoupled, the contribution to the gluon fusion vertex function is dominated by the top quark loop. In this regime the O(1) top-quark Yukawa coupling implies the approximate relation cot β S eff ∼ y S tt /y H tt .
The ATLAS limits are strongest in the mass range of O(500 GeV), but are insensitive to new scalars with Yukawa couplings to the top quark that are below O(1). These limits have been reparameterized as a function of cot β S eff in Figure 6, along with the translated 2σ ATLAS limits at √ s = 8 TeV. The masses of the Georgi-Machacek scalars φ 0 andφ 0 and the gauge-singlet η can be varied as a function of the spurion coefficient m 0 throughout the mass range of interest, and are roughly independent of the magnitude of their Yukawa couplings. Throughout the parameter space of this model we find that cot β φ 0 eff < cot βφ 0 eff 1. The ability to resolve these neutral scalars are thus well beyond the reach of near-term LHC limits in this channel.

Multi-top Final States
Pair production of doubly-charged scalars and heavy vector-like quarks democratically provide an O(fb) contribution to σ(pp tttt) via final states that include (W b) n with n ≥ 4. Despite the kinematic suppression of these production modes, the abundance of new particle states can lead to a measurable multi-top signature when combined over all processes. An anomalous production of final states with a high multiplicity top and bottom quarks is thus a general expectation for this class of theories. The most sensitive searches for these final states are driven by multi-lepton analyses that look for anomalously high b-jet multiplicities within samples containing three or more leptons, or two leptons of the same sign [47]. The dominant contributions to the multi-top final state come from diagrams such as those in Figure 7.
In this model Drell-Yan production of the charged scalars are mediated by current interactions that can be extracted from the covariant derivative as shown in Equation 46. The form of these couplings agree with those computed in the low energy effective theory for Georgi-Machacek scalars [36] in the limit where only the Higgs gets a vacuum expectation value.
These current interactions also mediate transitions between the different scalar mass eigenstates via "weak-strahling" processes. However these branching fractions are sub-leading relative to their decays to pairs of fermions via O(1) Yukawa couplings. The mass, charge, and CP properties of theφ ++ scalar in this model forbid all two-body decays and its branching fraction is thus dominated by the 3-body decay BR(φ ++ W + tb) ∼ 1. Pair production of this doubly charged scalar will thus always result in the final stateφ ++φ−− → (W b) 4 , effectively providing an O(fb) contribution to the four top-quark cross section from the process shown in Figure 7. The other major contribution to an anomalous multi-top cross section comes from strong production of pairs of heavy vector-like quarks, which are the lightest top quark partner t t and peculiar quark p p . The lightest top partner t decays mostly via the neutral current t →φ 0 t but there is also a sub-leading branching fraction to the top quark via emission of Z and h bosons. These two sub-leading contributions are approximately equal due to the goldstone equivalence theorem, as shown in Figure 8. Flavor-changing charged-current decays of t are suppressed due to the fact that it mostly an electroweak doublet implying that BR(t W + b) ∼ 0. The lightest bottom-type quark partner p has a peculiar charge of 5/3 and therefore decays exclusively via emission of charged bosons. Pair production of these heavy color-charged fermions thus generically results in final states involving pairs of Φ scalars and additional third generation quarks. At large values of the SU (5) symmetric Yukawa coupling y 1 , the mass splitting between the new fermion and scalar states becomes large enough to attenuate the kinematic advantage of twobody final states. In this regime we find an increasingly significant contribution from 3-body decays, which can result in exotic signatures containing a very high multiplicity of leptons and b-jets. In this model the contribution to the multi-top cross section from any one production mode is generally below threshold for detection by current experimental searches. However we find that combining the effective contribution to this final state across multiple channels makes these searches sensitive to large parts of the parameter space. The various combinations of decay paths that contribute to the effective tttt cross section are enumerated in Appendix C. The aggregate contribution from all of these channels allows for a direct translation of the limit on anomalous 4-top production to the parameter space of this theory, as illustrated in Figure  9.

Hidden-Sector Final States
The lightest states in the hidden-sector are a spectrum of glueballs which may be labeled by their J P C quantum numbers. The masses and widths of these glueballs take a large contribution from the non-perturbative dynamics of the hidden-sector gauge group G, thus preventing a detailed quantitative analysis of their collider phenomenology. However naive dimensional anal- ysis implies that the glueball masses should be proportional to the hidden-sector confinement scale Λ, and their widths should be proportional to some high power of Λ. If the hidden-sector gauge group is G = SU (3), computations on the lattice indicate the existence of at least a dozen stable glueball mass eigenstates [48][49][50]. The lightest state has J P C = 0 ++ and a mass m 0 ++ ∼ 7 Λ, while the higher excited states all lie within O(few) × m 0 ++ . Despite the relative compression of this mass spectrum, if m 0 ++ Λ then the production of higher-mass glueballs will be sub-dominant due to a Boltzmann suppression in the thermal partition function. We will thus assume going forward that an O(1) fraction of all glueballs produced are of the type 0 ++ . The lowest-lying glueball 0 ++ can decay back to Standard Model particles through the Higgs portal and its branching ratios are thus approximately equal to those of a Higgs boson of the same mass. The widths of low-mass Higgs bosons are well known [51] and are illustrated in Figure 10. The parametric form of the glueball width has been worked in out in [52], and is given by the expression in Equation 47.
Here κ is an O(1) constant that parameterizes the non-perturbative contribution to the width, and lattice computations for G = SU (3) give κ ∼ 3 [49,53]. For this analysis we remain agnostic about the details of the hidden-sector gauge group by treating κ and m 0 ++ as the input quantities that parameterize the relevant glueball properties. If m 0 ++ > m h /2 then the primary production mode of the 0 ++ glueball will be through decays of a broad nn-onium resonance, which has an O(fb) production cross section in the bulk of this parameter space, often exceeding the pair production rate of the new color-charged states t t and p p . Weak production of nn pairs is enhanced by an abundance of new production modes 1/2 (right). Here we define "prompt" decays as those corresponding to cτ < 10 µm and "invisible" as those corresponding to cτ > 10 m. via s-channel diagrams involving the new goldstone states, and also by the conditions of EWSB which generically result in a hierarchy between the color-charged and hidden-sector Yukawa couplings m n < m t . However in this regime, the hidden-sector states would be extremely difficult to resolve in any near-term LHC search due to the overwhelming QCD backgrounds to its generic final states. For example if the hidden-sector gauge group is QCD-like, then a relatively light glueball mass m 0 ++ m n ∼ 0.5 TeV may result in a nn-onium state that decays to a high multiplicity of glueballs, the lightest of which would decay promptly with an O(1) branching fraction to pairs of bottom quarks. This O(fb) contribution to high-multiplicity bottom-quark final states would be extremely challenging to probe. However given the impressive recent advances on object identification using deep-learning algorithms, it is conceivable that future analysis techniques may become sensitive to the resonance structure of such glueball decays between correlated bb pairs. As the lightest glueball mass begins to saturate the kinematic limit m 0 ++ → m n , the nn-onium state will decay promptly to pairs of the 0 ++ glueball, which has a prompt O(1) branching fraction to W + W − in this regime. The hidden sector will thus provide an O(fb) contribution to the cross section σ(pp nn W + W − W + W − ) that would be similarly challenging to resolve. The non-resonant same-sign dilepton and multi-lepton final states are suppressed by O(10 −2 ) branching fractions and the rates are thus orders of magnitude too low to be constrained by current non-resonant multi-lepton searches [54,55]. In the regime m 0 ++ > 2m W where all W -bosons are on-shell, then the fully hadronic decay modes of the W + W − W + W − final state would contain interesting hierarchical resonance structures that could also conceivably be exploited by future advances in analysis techniques.
As a consequence of Equation 47, the lifetime of the 0 ++ glueball is a strong function of the hidden-sector confinement scale Λ and its lifetime can either be prompt or extremely longlived. The decay lengths at a generic value of the Yukawa couplings are illustrated in Figure  11, with the middle plot corresponding to the QCD-like scenario with κ = 3. Here we define "prompt" decays as those corresponding to cτ < 10 µm and "invisible" as those corresponding to cτ > 10 m. In the regime where m 0 ++ < m h /2 then the 0 ++ glueball can be very longlived [56][57][58][59], and Big-Bang Nucleosynthesis (BBN) places a limit on long-lived particles for which cτ > 10 7 m. For a QCD-like hidden-sector this would place a lower limit on the lightest glueball mass m 0 ++ of about O(few) GeV. A simplified parameterization of the decay length relative to the BBN limit is given by Equation 48.
In this parameter space cτ (0 ++ ) ∼ 10 m corresponds approximately to m 0 ++ ∼ 10 GeV, thus placing the glueball decay vertex outside of the LHC detectors at lower masses. For simplicity we thus restrict our analysis to glueball masses m 0 ++ > 2m b where we have approximately BR(0 ++ bb) ∼ O(1), although similar limits hold at lower masses when BR(0 ++ τ + τ − ) ∼ O(1). The primary production mode of the 0 ++ glueball in this regime occurs via decays of the Higgs mediated by loops of hidden-sector fermions. The branching ratio of the Higgs boson to hidden-sector glueballs in this framework can be estimated by the Higgs partial width to pairs of hidden-sector gluons, and is parametrically given by the relation in Equation 49.
This branching ratio BR(h → 0 ++ 0 ++ ) depends on the running of the hidden-sector coupling constantα from the hidden confinement scale Λ to the Higgs mass m h . In this model we have assumed that the lightest states charged under the hidden-sector gauge group G have masses on the order of f > Λ and the β-function forα thus depends purely on the quadratic Casimir of G. Weak-scale perturbativity of the coupling constantα(m h ) < 0.5 generically implies a small branching fraction for h → 0 ++ 0 ++ that is below sensitivity projections for the HL-LHC [60][61][62][63][64]. However future Higgs factories such as the proposed Circular Electron-Positron Collider (CEPC) or the International Linear Collider (ILC) [65][66][67] will be able to exclude a wide range of values forα(m h ). The expected sensitivities to a variety of rare Higgs decays have been projected for a long-term run at the CEPC [68], which will be sensitive to the decay topology BR(h 0 ++ 0 ++ bbbb) ∼ 6 × 10 −4 , as illustrated in Figure 12. Here we have restricted these limits to regions in which the 0 ++ glueballs decay within the tracker radius of the proposed CEPC detectors ∼ 1810 mm, and assume that object identification would be inefficient from calorimeter data alone. The large region of parameter space with macroscopic glueball decay lengths also provides additional motivation for a more careful exploration of the lifetime frontier via surface detectors such as the proposed MATHUSLA experiment [69]. Preliminary studies indicate that such a surface detector could exclude rare branching fractions of the Higgs boson as low as BR(h XX) ∼ 10 −5 for detector-optimized values of cτ (0 ++ ). The long-term projections for these rare Higgs decays assuming 3 ab −1 of MATHUSLA data are also illustrated in Figure 12.

III. SU (4)/Sp(4) INTERMEDIATE MODEL
The symmetry breaking pattern SU (4)/Sp(4) produces 15 − 10 = 5 pseudo Nambu-Goldstone Bosons (pNGBs). The nonlinear sigma model describing the low energy effective theory may be expressed in terms of an antisymmetric unitary matrix Σ, which transforms under SU (4) as V ΣV T where V is an SU (4) matrix. It is convenient to specify a background field Σ 0 , which is invariant under the Sp(4) subgroup containing SU (2) L ⊗ SU (2) R . The generators for the SU (2) L ⊗ U (1) Y gauge symmetry L a and Y are embedded into an approximate global custodial symmetry via Y = R 3 , which remains unbroken in the reference vacuum Σ = Σ 0 .
The pNGB's thus transform under the electroweak gauge group as 1 0 ⊕ 2 ±1/2 giving one real electroweak singlet and a SM Higgs doublet. The Nambu-Goldstone bosons are fluctuations about this background in the direction of the broken generators, Π ≡ π a X a , and may be parameterized as Here H = (H + H 0 ) and H c = iσ 2 H * . Vacuum misalignment with respect to the weak gauge interactions can be parameterized by an angle θ. This vacuum, which is invariant under the custodial SU (2) c generated by R a + L a , can be written as The gauge couplings explicitly break the SU (4) global symmetry. The covariant derivative can be expanded to second order in the vacuum giving tree level masses to the weak gauge bosons in terms of the SU (2) L × U (1) Y couplings g, g . The tree level relation m W /m Z = cos θ w is guaranteed by the preservation of the remnant custodial symmetry SU (2) c ⊂ Sp(4)

A. Fermion Sector
In this model the Yukawa sector of the Standard Model is extended to include interactions between the Σ field and composite fermions in complete SU (4) multiplets. Additional gauge invariant interactions softly break the global symmetry, generating a significant negative contribution to the Higgs potential, thus driving electroweak symmetry breaking. The mass and quartic couplings of the Higgs can be tuned to their experimentally measured values by extending the fermion sector of this theory to include two vector-like multiplets of fermions in the fundamental representation of SU (4). The first multiplet (ψ,ψ) is charged under SU (3) c and mixes with the third generation of Standard Model quarks resulting in a partially composite top and bottom. The second multiplet (χ,χ) is charged under a hidden-sector gauge group G that confines at some scale Λ < f .

Color-charged Fermions
The new color-charged vector-like states (ψ,ψ) mix with the chiral third-generation of SM quarks, which are singlets under the global SU (4) symmetry. The gauge eigenstate components of these multiplets are fixed by the embedding of SU (2) L × U (1) Y ⊂ SU (4), and are shown in Equation 54. The gauge quantum numbers of the new color-charged fermions are given in Table V. The most general gauge invariant Yukawa couplings for the color-charged fermions are given by Equation 55, and collective symmetry breaking properties of these interactions guarantee the absence of quadratic divergences to the scalar potential from fermion-loops. The resulting top-like mass matrix M T has the property det M † T M T ∝ sin 2 θ, thus producing one massless state when θ = nπ for n ∈ Z.
Note that a non-zero top quark mass requires y 1 , y 2 , y 3 > 0 simultaneously, revealing the collective symmetry breaking properties of the top sector mass matrix. The masses of the colorcharged fermions are computed in Appendix A to second order in sin 2 θ, and the numerically diagonalized mass spectrum for the heavy fermions are shown in Figure (13). To leading order in sin 2 θ these masses scale approximately as

Hidden-Sector Fermions
The new hidden-sector vector-like states (χ,χ) transform in the ( , 4)⊕( , 4) representation of G × SU (4). The hidden-sector gauge group G confines at some scale Λ < f resulting in a low energy spectrum comprised of unstable mesons, stable baryons, and potentially long-lived glueballs. The lightest baryons in this sector are a natural candidate for dark matter if they are electrically neutral, however a detailed analysis of hidden-baryon spectrum is beyond the scope of this analysis. The gauge quantum numbers of the hidden-fermions are given in Table  VII. doublet X = (X + X 0 ), one charged singlet C, and two neutral singlets N and n.
A Standard Model-like Higgs boson can be obtained from fermionic loop-corrections by introducing symmetry breaking interactions between the multiplet (χ,χ) and an additional massless vector-like pair of hidden-sector fermions (n,n). The Yukawa interactions thus include an SU (4) symmetric coupling to the Σ-field in addition to terms that softly break the global symmetry.
The hidden-sector interactions are constructed in such a way that the neutral mass matrix M N obeys the condition det M † N M N ∝ cos 2 θ, thus producing one massless state when θ = (n + 1/2)π. Diagonalizing M N produces three electrically neutral mass eigenstates which we label in in order of descending mass as n , n , n. Diagonalizing the charged hidden-sector mass matrix M C results in a charge ±1 vector-like pair (x + ,x + ) and a charge ∓1 vector-like pair (c − ,c − ). The precision electroweak constraints described in the following section push the fermion masses of this theory to very high scales, and the details of their mass spectrum are thus irrelevant for hadron-collider phenemenology. We thus consider a simplified parameter space in whichỹ 3 =ỹ 2 =ỹ 1 , though the masses are computed for general Yukawa couplings in Appendix A to second order in sin 2 θ. In this simplified parameter space the eigenvalues of the mass matrix scale approximately as

Theory Space
The new color-charged fermion interactions introduce four Yukawa couplings y i , two of which are fixed by the Standard Model top and bottom quark masses as implied by Equation 55. This leaves two free parameters y 1 and y 2 which are bound from below by the top-quark Yukawa coupling y t < y 1 y 2 / y 2 1 + y 2 2 . The new hidden-sector interactions introduce three new Yukawa couplings, which we reduce to a single parameter by assumingỹ 1 =ỹ 2 =ỹ 3 . The conditions of EWSB described in the following section thus fixes a unique value forỹ 1 at each point in the two-dimensional parameter space of (y 1 , y 2 ). In this framework, the experimentally measured Higgs mass implies a mass scale for the new hidden-sector fermions that is O(few) times lower than their color-charged counterparts. The masses of the new fermionic states as a function of the chiral symmetry breaking scale f are illustrated for two benchmark points in Figure 13 The most constraining precision electroweak observable for the SU (4)/Sp(4) model comes from the branching fraction of Z → bb. The doublet-singlet mixing between the SM bottom quark and its vector-like partners will generically induce large corrections to the bottom quark neutral currents if y 2 is significantly larger than y 1 . The extreme precision of experimental measurements on the parameter R b = Γ(Z → bb)/Γ(Z → had) = 0.21629 ± 0.00066 thus excludes all regions except those in which y 1 y 2 , as shown in Figure 14. The precision electroweak constraints from R b thus push the masses of the vector-like fermions to very high scales that are likely beyond the reach of a 14 TeV machine via direct production. These constraints may be relaxed by introducing additional states with bottom-like gauge quantum numbers that reduce the doublet-singlet mixing. However, for the remainder of this paper we will assume the minimal matter content for this model in which the fermions are heavy and decoupled. The contributions to the T -parameter from fermion loops are less constraining in this model and are also shown in Figure 14.

B. Electroweak Symmetry Breaking
The one-loop corrections to the scalar effective potential can be extracted from the quadratically and logarithmically sensitive terms in the Coleman Weinberg potential. The Higgs mass receives a large positive contribution from gauge loops that is quadratically sensitive to the compositeness scale Λ, and proportional to the gauge boson mass matrix squared. Taking Λ ∼ 4πf we find Collective symmetry breaking ensures the absence of quadratically divergent contributions from the Yukawa sector, and a logarithmic contribution that is well approximated by the leading order terms in det M † T M T and det M † N M N . As argued in Appendix A the higher order contributions are suppressed by corresponding powers of (tr M † T M T ) 3 , and are thus proportional to a function of the Yukawa couplings that has a numerically negligible global maximum. The leading contribution to the Higgs effective potential from the color-charged and hidden-sector fermions are thus numerically well approximated by a simple δV F ∝ ± cos 2θ dependence, and contribute with opposite sign. Analytical expressions for the coefficients of this potential are computed for general values of the Yukawa couplings y i andỹ i in Appendix A, as well as higher order corrections. In the simplified parameter space, the fermionic contribution to the scalar effective potential may be expressed to leading order as A realistic pattern of EWSB in this model requires the introduction of a U (1) a violating spurion analogous to the operator of Equation 30. This spurion descends naturally from fermion mass terms in the UV complete theory, and gives a contribution to the Higgs potential that is periodic in θ mod 2π = −f 2 f 2 0 cos θ In the absence of additional matter content at the scale f , the leading contributions to the scalar effective potential are fixed by the Yukawa coupings y i andỹ i , the 1-loop gauge coefficient c, and spurion coefficient m 0 . This potential has a generic EWSB vacuum when f 2 T −f 2 N −f 2 G f 2 0 > 0 and the vacuum expectation value of the order parameter θ can be expressed as a function of these scales.
Fixing v 2 = 2f 2 θ 2 = (246 GeV) 2 and expanding about this symmetry breaking vacuum, we find that the coefficients of the one-loop corrections are fixed by the measured value of the physical Higgs boson m h = 126 GeV.
In this framework the mass of the gauge-singlet pseudoscalar η takes a large contribution from the spurion coefficient m 0 via Equation 72. At leading order in θ, the size of this coefficient is fixed by the Higgs mass to be m 0 ∼ f m 2 h /v 2 resulting in a heavy pseudoscalar m η ∼ O(TeV). The pseudoscalar mass can be made into a free parameter by introducing an additional symmetry breaking spurion that is proportional to an alternative Sp(4) preserving background field M 0 ≡m 0 Σ 0 = 8f 2m2 0 η 2 sin 2 α α 2 (74) The spurion given by Equation 74 would descend naturally from a four-fermion interaction in the UV completion at the compositeness scale Λ. This term provides a contribution to the potential for η only. In the absence of additional sources of symmetry breaking terms, the pseudoscalar mass is given as a function of the spurion coefficients m 0 andm 0 In the SU (4)/Sp(4) model, precision electroweak constraints push the masses of the BSM fermions to very high values, putting them firmly beyond the reach of near-term LHC limits. These constraints can be relaxed by introducing additional vector-like partners in a way that suppresses mixing between the SM bottom quark and the heavy singlet. However the branching fractions of the lightest BSM states to the singlet η are generically small. The lightest colorcharged states in such models would thus be virtually indistinguishable from the top and bottom partners of simplified models [33,34] and their phenomenology will not be discussed further here. The phenomenological signatures of of the hidden-sector in this model are similar to those described in Section II C 3 and will also not be reiterated here. In the absence of kinematically accessible BSM fermions, the production of new particles is limited to weak production of the goldstone mode η, which can be singly produced via gluon fusion, or pair produced via Higgs decay h → ηη if it is sufficiently light. The large tree-level Yukawa couplings in this model result in BR(η tt) ∼ 1 when m η > 2m t and BR(η bb) ∼ 1 when m η < 2m t .
In the high-mass regime the discovery of such a state would be extremely challenging due to the low production cross section and the interference effects of its two-body decays, as described in Section II C. The loop-induced interactions of the η boson are roughly independent of the Yukawa-couplings due to the dominant invariant contribution from the top-quark loop. The cross sections and branching fractions of the η boson are shown in Figure 15 for a generic value of the Yukawa couplings. The pseudo-scalar η only has tree level couplings to the third generation L ⊃ 1 4 η c η ww W µνW µν +c η zz Z µνZ µν +c η zγ Z µνF µν +c η γγ F µνF µν +c η gg G µνG µν (77)

Rare Higgs Decays
If the pseudo-scalar mass is comparable to that of the Higgs boson then its dominant branching fraction to bottom quarks would make it vulnerable to sophisticated searches for boosted bb pairs such as those recently performed by CMS [71,72]. However these techniques are unable to resolve boosted objects with low invariant masses m η < 50 GeV due to overwhelming multijet backgrounds [73]. In this regime, the most viable kinematic pathway for resolving the pseudoscalar η would be via its interactions with the Higgs boson, which can decay to pairs of η bosons via loop induced couplings as shown in Figure 16. The branching ratio BR(h → ηη) is thus loop-suppressed and such decays would likely be impossible to resolve at a future hadron collider such as the HL-LHC, which will constrain the decay modes of the Higgs at a relative precision not exceeding O(10%) [60][61][62][63][64]. The presence of such a low-mass goldstone mode provides additional motivation for precision measurements at a future Higgs factory. One existing proposal for such a machine is the Circular Electron-Positron Collider (CEPC), which is expected to measure various couplings of the Higgs boson at a relative precision of O(0.1 − 1%) [65][66][67]. The partial width for the rare Higgs decay h → ηη is an increasing function of the Yukawa couplings y i and is saturated at the kinematic limit, as shown in Figure 17. The branching fraction for this process lies in the range of BR(h ηη) ∼ 10 −5 − 10 −3 throughout the parameter space of this model. The dominant branching fraction of the pseudoscalar to bb pairs would thus lead to a rare h → (bb)(bb) signal that could be effectively probed at an e + e − collider. An estimate of the expected reach for this process at the CEPC has been computed at 5 ab −1 [68], and we find that translating these limits puts a large part of this parameter space within reach of a long-term run at the CEPC, as shown in Figure 17.

IV. CONCLUSIONS AND OUTLOOK
A pseudo Nambu-Goldstone Higgs with a top coupling arising from mixing with top partners offers the theoretically compelling possibility of a natural and calculable model for electroweak symmetry breaking. We have considered two of the most economical scenarios, expanded to include a Higgs coupling to a hidden fermion sector. The addition of a hidden sector coupled to the pseudo Nambu-Goldstone bosons allows for a relatively simple way to obtain a Higgs potential that is consistent with the experimentally measured masses of Higgs and weak gauge bosons, as well as a possible dark matter candidate. We have performed analyses on the most relevant collider signatures of the new fermions and new spinless particles, and find that for most of the viable parameter space a new particle discovery will be challenging at the LHC. Given low energy precision electroweak constraints, the lack of an LHC discovery of any nonstandard model particles to date would thus be a general expectation for both of these composite Higgs models, though for different reasons. Nonetheless our analysis suggests several options for probing these nonstandard phenomena at the LHC and future colliders.
In the SU (5)/SO(5) model these challenges arise due to the serendipitous subtlety of the final state signatures which arise from electroweak interactions that are preferential to the third generation of quarks. The backgrounds from QCD processes generically overwhelm or interfere with the most obvious processes that could be used to resolve new particle states beyond the Standard Model. However the high multiplicity of new BSM states will generally aggregate to produce a measurable deviation in the production of final states with a high multiplicity of third generation quarks in the high-luminosity limit. The presence of a confining hiddensector gauge group could also lead to a potentially measurable branching fraction of the Higgs boson to displaced pairs of bottom quarks, thus providing additional motivation for experiments focused on the long-lived particle frontier, such as the proposed MATHUSLA experiment. In the SU (4)/Sp(4) model, the experimental challenges arise primarily due to the stark absence of new particle states involved with the EWSB dynamics. The goldstone sector of this theory consists a single spin-0 gauge-singlet state, and precision electroweak constraints imply a mass range for new strongly interacting particles that is likely well beyond the reach of a 14 TeV hadron collider. However the Higgs boson generically has a decay rate to the gauge-singlet goldstone mode that could be resolved at a future Higgs factory such as the CEPC.

ACKNOWLEDGMENTS
The work of A.E.N. is partially supported by the DOE under grant DE-SC0011637 and by the Kenneth K. Young Memorial Endowed Chair. The work of D.G.E.W. was partially supported by Burke faculty fellowship. We would like to thank Calibourne D. Smith and Scott Thomas for helpful conversations. A.E.N. acknowledges the hospitality of the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. We thank the Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial support during the completion of this work. This work was partially supported by a grant from the Simons Foundation (341344, LA).

Appendix A: Fermion Contribution to SU (4)/Sp(4) Higgs Potential
In the SU (4)/Sp(4) model, the top-sector mass matrix for the color-charged quarks M T takes the form given in Table IX. The mass matrices can be expressed in terms of a single angle θ ≡ v/ √ 2f , representing the magnitude of the Higgs VEV relative to the chiral symmery breaking scale. The UV insensitivity of the Yukawa sector provides powerful constraints on the form of the scalar effective potential, and is guaranteed by the following properties of the mass matrix The fermion masses are periodic with respect to θ mod 2π. The color-charged mass matrix M T produces one massless state when θ = nπ for n ∈ Z. For notational simplicity we define the following combinations of Yukawa couplings y 2 ± = (y 2 1 + y 2 2 ) ± (y 2 1 + y 2 3 ) y 2 123 ≡ y 2 1 y 2 2 y 2 3 (A2) In terms of these Yukawa parameters, the traces and determinants of the fermion mass matrices take a simple form Computing to fourth order in sin θ, we find that the color-charged fermion mass eigenstates take the following form The UV insensitivity of the Yukawa sector additionally guarantees that their contributions to the scalar effective potential can be expressed in terms of ratios of mass eigenstates. Computing to fourth order in sin θ, we find that the top-sector contribution to the Higgs effective potential takes the following form q (y i , θ)| 2 log |m 2 q (y i , θ)| (A12) The coefficients of the θ dependent terms may be expressed as The collective symmetry breaking properties of M T guarantees that the potential is proportional to det M † T M T . Higher order contributions are thus suppressed by powers of y 2 123 /y 6 + , which is a function of the Yukawa couplings that has a global maximum at 1/54 For the color-charged contribution we thus find that the potential is numerically well approximated by the second order approximation in sin θ, which has been plotted against the numerical result in Figure 18. The fermionic contribution to the Higgs potential is thus extremely well approximated by a cos 2θ dependence.
δV T (y i , θ) = −2f 2 f 2 T sin 2 θ (A18) ∼ f 2 f 2 T cos 2θ (A19) The contribution to the Higgs effective potential from hidden-sector fermions follows a similar story to the color-charged case. The neutral hidden-sector fermion mass matrix M N is given in Table X, and the insensitivity of the Higgs potential to loops of hidden-sector fermions is guaranteed by Equation A21 The mass matrix for the neutral states M N produces one massless state when θ = (n + 1/2)π, for n ∈ Z which can be seen from the form of the mass determinant. Similarly for the hidden-sector fermions, we find that the contribution is well approximated by a cos 2θ dependence, but with an opposite sign that descends from the form of the determinants in Equation A23 δV N (ỹ i , θ) = −2f 2 f 2 N cos 2 θ (A27) ∼ −f 2 f 2 N cos 2θ (A28) in terms of ratios of mass eigenstates, and are extremely well approximated by their leading order terms in sin 2θ, as shown in Figure 19. This contribution to the Higgs potential is thus approximately given by the cos 4θ dependence in Equation B4. The fermion contribution to the Higgs potential is thus a function of two independent scales f 2 T and f 2 N for the color-charged and hidden Yukawa sectors respectively Production Mode Decay Final State gg → t t t t → (φ 0 t)(φ 0t ) → (ttt)(ttt) (W b) 6