Partial Compositeness in Higgs Models

• Partial compositeness is a mechanism that generates fermion masses by mixing elementary fermions with composite resonances, leading to dynamical Yukawa hierarchies.
• Composite Higgs realizations leverage this mechanism, especially in the top sector, where effective Yukawas are controlled by the product of left- and right-handed mixing angles.
• UV completions, such as Fundamental Partial Compositeness and partial unification, address flavor hierarchies and precision observables while predicting distinctive collider signatures.

Partial compositeness is a mechanism for fermion-mass generation in which a Standard Model fermion does not acquire its mass from a direct elementary Yukawa coupling to the Higgs, but from a linear mixing with a fermionic operator or resonance of a new strong sector. In composite Higgs constructions, the same strong sector also contains the Higgs as a composite state, often a pseudo-Nambu–Goldstone boson, so the physical light fermions are admixtures of elementary and composite states, with a “degree of compositeness” set by the mixing angle $s_f$. In its standard effective form one writes $$\mathcal L_{\rm mix}\sim \lambda_f\,\bar f_{\rm SM}\,\mathcal O_{\rm comp}+{\rm h.c.}$$, while effective Yukawas are controlled by the product of left- and right-handed compositeness mixings (Stangl, 2019, Goertz et al., 2023).

1. Core mechanism and effective description

In composite Higgs models the low-energy theory is organized as an elementary sector, a composite sector, and a mixing sector,

$$\mathcal{L}_{\rm CHM}=\mathcal{L}_{\rm elementary}+\mathcal{L}_{\rm composite}+\mathcal{L}_{\rm mixing}\,.$$

This is the operational definition of partial compositeness in the effective description: the light fermions are not purely elementary fields but mixtures of elementary chiral fermions and composite fermions of a strong sector (Stangl, 2019). The Higgs couples directly to the composite fermions, while flavor structure is encoded in the elementary–composite mixings. A standard consequence is that effective SM Yukawas depend on the product of a left-handed and a right-handed degree of compositeness (Stangl, 2019).

When the strong sector has a walking or almost-conformal regime, the mixings run with anomalous dimensions. In the functional-RG formulation of fundamental partial compositeness, the dimensionless coupling satisfies

$$\overline \lambda_f (\Lambda_{c}) = \overline \lambda_f(\Lambda_{\rm UV}) \left(\frac{\Lambda_c}{\Lambda_{\rm UV}}\right)^{\gamma_{\lambda_f}},$$

so small differences in scaling exponents can be amplified into large Yukawa hierarchies (Goertz et al., 2023). This feature is the dynamical core of the mechanism: hierarchies need not be inserted directly into the IR Yukawa matrices if they can be generated by RG flow.

A central distinction within the literature concerns the nature of the composite fermionic operator. For three-fermion baryon-like operators the canonical dimension is $9/2$, which makes linear mixing strongly irrelevant unless anomalous dimensions are large. In fundamental partial compositeness, where the composite is a fermion–scalar bound state ${\cal B}\sim{\cal S}{\cal F}$, the canonical dimension is $5/2$, so the linear mixing is much less suppressed (Goertz et al., 2023, Goertz et al., 2023). This difference is one of the main reasons FPC is treated as a distinct ultraviolet strategy rather than merely a variant notation.

2. Composite Higgs realizations and the top sector

Partial compositeness is most closely tied to composite Higgs models in which the Higgs is a pNGB of a spontaneously broken global symmetry. The supplied literature includes realizations based on $SO(5)/SO(4)$, $SU(5)/SO(5)$, and $SU(4)/Sp(4)$, all of which retain custodial structure while accommodating fermionic resonances that can mix with SM quarks (Caracciolo et al., 2012, Ferretti, 2014). In these models electroweak symmetry breaking is described by vacuum misalignment,

$$\mathcal L_{\rm mix}\sim \lambda_f\,\bar f_{\rm SM}\,\mathcal O_{\rm comp}+{\rm h.c.}$$0

with $$\mathcal L_{\rm mix}\sim \lambda_f\,\bar f_{\rm SM}\,\mathcal O_{\rm comp}+{\rm h.c.}$$1 the decay constant of the strong sector (Franzosi et al., 2019).

The top sector is the canonical arena for partial compositeness because generating $$\mathcal L_{\rm mix}\sim \lambda_f\,\bar f_{\rm SM}\,\mathcal O_{\rm comp}+{\rm h.c.}$$2 typically requires the largest elementary–composite mixing. In the $$\mathcal L_{\rm mix}\sim \lambda_f\,\bar f_{\rm SM}\,\mathcal O_{\rm comp}+{\rm h.c.}$$3 hypercolor completion centered on the $$\mathcal L_{\rm mix}\sim \lambda_f\,\bar f_{\rm SM}\,\mathcal O_{\rm comp}+{\rm h.c.}$$4 coset, the relevant composite fermions transform as an $$\mathcal L_{\rm mix}\sim \lambda_f\,\bar f_{\rm SM}\,\mathcal O_{\rm comp}+{\rm h.c.}$$5 fundamental and decompose into one state of electric charge $$\mathcal L_{\rm mix}\sim \lambda_f\,\bar f_{\rm SM}\,\mathcal O_{\rm comp}+{\rm h.c.}$$6, three of charge $$\mathcal L_{\rm mix}\sim \lambda_f\,\bar f_{\rm SM}\,\mathcal O_{\rm comp}+{\rm h.c.}$$7, and one of charge $$\mathcal L_{\rm mix}\sim \lambda_f\,\bar f_{\rm SM}\,\mathcal O_{\rm comp}+{\rm h.c.}$$8 (Ferretti, 2014). These states furnish the top-partner multiplet that mixes linearly with $$\mathcal L_{\rm mix}\sim \lambda_f\,\bar f_{\rm SM}\,\mathcal O_{\rm comp}+{\rm h.c.}$$9 and $$\mathcal{L}_{\rm CHM}=\mathcal{L}_{\rm elementary}+\mathcal{L}_{\rm composite}+\mathcal{L}_{\rm mixing}\,.$$0. In that construction the top mass is generated by partial compositeness, while the remaining SM fermions are instead assigned standard quadratic Higgs couplings (Ferretti, 2014).

This restriction of explicit partial compositeness to the top sector is common in concrete UV completions. A plausible implication is that “partial compositeness” is often a sector-specific mechanism rather than an all-fermion statement: many fully specified confining models can realize the top-partner structure and pNGB Higgs simultaneously, but extending the same microscopic construction to all SM fermions is substantially harder (Ferretti, 2014, Caracciolo et al., 2012).

3. Ultraviolet completions and fundamental formulations

Several distinct UV strategies appear in the literature. In the supersymmetric Seiberg-dual constructions based on $$\mathcal{L}_{\rm CHM}=\mathcal{L}_{\rm elementary}+\mathcal{L}_{\rm composite}+\mathcal{L}_{\rm mixing}\,.$$1, the low-energy composite sector is identified with an IR-free magnetic dual. The essential point for partial compositeness is that UV trilinear couplings of the form $$\mathcal{L}_{\rm CHM}=\mathcal{L}_{\rm elementary}+\mathcal{L}_{\rm composite}+\mathcal{L}_{\rm mixing}\,.$$2 flow in the IR to linear elementary–composite mass mixings $$\mathcal{L}_{\rm CHM}=\mathcal{L}_{\rm elementary}+\mathcal{L}_{\rm composite}+\mathcal{L}_{\rm mixing}\,.$$3, reproducing the standard partial-compositeness structure dynamically rather than postulating it directly in a two-site EFT (Caracciolo et al., 2012). These models also make explicit that fermion and vector resonance masses can be governed by different couplings, $$\mathcal{L}_{\rm CHM}=\mathcal{L}_{\rm elementary}+\mathcal{L}_{\rm composite}+\mathcal{L}_{\rm mixing}\,.$$4 and $$\mathcal{L}_{\rm CHM}=\mathcal{L}_{\rm elementary}+\mathcal{L}_{\rm composite}+\mathcal{L}_{\rm mixing}\,.$$5, allowing relatively light fermion partners alongside heavier vectors (Caracciolo et al., 2012).

A more microscopic route is Fundamental Partial Compositeness. In the minimal FPC construction, the strong sector contains technifermions $$\mathcal{L}_{\rm CHM}=\mathcal{L}_{\rm elementary}+\mathcal{L}_{\rm composite}+\mathcal{L}_{\rm mixing}\,.$$6 and techniscalars $$\mathcal{L}_{\rm CHM}=\mathcal{L}_{\rm elementary}+\mathcal{L}_{\rm composite}+\mathcal{L}_{\rm mixing}\,.$$7, repackaged as $$\mathcal{L}_{\rm CHM}=\mathcal{L}_{\rm elementary}+\mathcal{L}_{\rm composite}+\mathcal{L}_{\rm mixing}\,.$$8, with renormalizable Yukawa couplings

$$\mathcal{L}_{\rm CHM}=\mathcal{L}_{\rm elementary}+\mathcal{L}_{\rm composite}+\mathcal{L}_{\rm mixing}\,.$$9

After confinement, these couplings induce linear mixing between SM fermions and fermionic bound states of schematic form $$\overline \lambda_f (\Lambda_{c}) = \overline \lambda_f(\Lambda_{\rm UV}) \left(\frac{\Lambda_c}{\Lambda_{\rm UV}}\right)^{\gamma_{\lambda_f}},$$0, so the analogue of the usual $$\overline \lambda_f (\Lambda_{c}) = \overline \lambda_f(\Lambda_{\rm UV}) \left(\frac{\Lambda_c}{\Lambda_{\rm UV}}\right)^{\gamma_{\lambda_f}},$$1 interaction is generated from a renormalizable UV Lagrangian (Cacciapaglia et al., 2017). In this framework the effective fermion masses are bilinear in the fundamental Yukawas, while four-fermion operators are quartic in them (Cacciapaglia et al., 2017, Stangl, 2019). The MFPC flavor analyses exploit exactly this structure.

A third UV route is partial unification. In the Techni-Pati-Salam proposal based on

$$\overline \lambda_f (\Lambda_{c}) = \overline \lambda_f(\Lambda_{\rm UV}) \left(\frac{\Lambda_c}{\Lambda_{\rm UV}}\right)^{\gamma_{\lambda_f}},$$2

the four-fermion operators needed for partial compositeness are generated by heavy vectors and scalars associated with staged gauge breaking, and the low-energy hypercolor theory is predicted to be an $$\overline \lambda_f (\Lambda_{c}) = \overline \lambda_f(\Lambda_{\rm UV}) \left(\frac{\Lambda_c}{\Lambda_{\rm UV}}\right)^{\gamma_{\lambda_f}},$$3 model (Cacciapaglia et al., 2019). This framework is explicitly intended as a renormalizable completion extending toward the Planck scale, while still using composite baryons of the hypercolor theory as the fermionic operators entering partial compositeness (Cacciapaglia et al., 2019).

These UV constructions also clarify recurrent obstacles. In the supersymmetric $$\overline \lambda_f (\Lambda_{c}) = \overline \lambda_f(\Lambda_{\rm UV}) \left(\frac{\Lambda_c}{\Lambda_{\rm UV}}\right)^{\gamma_{\lambda_f}},$$4 models, extending partial compositeness to all SM fermions is obstructed by low Landau poles in the SM gauge couplings (Caracciolo et al., 2012). In the partial-unification scenario, the viability of the mechanism depends on the walking dynamics and anomalous dimensions of the predicted $$\overline \lambda_f (\Lambda_{c}) = \overline \lambda_f(\Lambda_{\rm UV}) \left(\frac{\Lambda_c}{\Lambda_{\rm UV}}\right)^{\gamma_{\lambda_f}},$$5 hypercolor theory, which are explicitly identified as nonperturbative questions (Cacciapaglia et al., 2019).

4. Flavor hierarchies, LFU violation, and the shift from anarchy to symmetry

Partial compositeness is tightly connected to flavor because non-universal degrees of compositeness automatically induce non-universal couplings to composite vectors. In the composite-Higgs flavor-anomaly analyses, if muons are more composite than electrons, a neutral composite vector $$\overline \lambda_f (\Lambda_{c}) = \overline \lambda_f(\Lambda_{\rm UV}) \left(\frac{\Lambda_c}{\Lambda_{\rm UV}}\right)^{\gamma_{\lambda_f}},$$6 couples more strongly to muons than to electrons, so LFU violation becomes a generic consequence rather than an extra assumption (Stangl, 2019). In the rare-$$\overline \lambda_f (\Lambda_{c}) = \overline \lambda_f(\Lambda_{\rm UV}) \left(\frac{\Lambda_c}{\Lambda_{\rm UV}}\right)^{\gamma_{\lambda_f}},$$7 context this naturally favors the semileptonic pattern

$$\overline \lambda_f (\Lambda_{c}) = \overline \lambda_f(\Lambda_{\rm UV}) \left(\frac{\Lambda_c}{\Lambda_{\rm UV}}\right)^{\gamma_{\lambda_f}},$$8

because it only requires sizable left-handed muon compositeness and avoids the strong tension between a large muon vector current and the smallness of $$\overline \lambda_f (\Lambda_{c}) = \overline \lambda_f(\Lambda_{\rm UV}) \left(\frac{\Lambda_c}{\Lambda_{\rm UV}}\right)^{\gamma_{\lambda_f}},$$9 (Stangl, 2019).

In the simplified partial-compositeness setup for rare-$9/2$0 anomalies, the preferred region is approximately

$9/2$1

after balancing the semileptonic effect against $9/2$2-mixing and electroweak constraints (Stangl, 2019). In the MFPC effective theory, the strongest flavor constraint is $9/2$3, but viable parameter points still survive all bounds and can explain the observed LFU violation in $9/2$4 at the $9/2$5 level (Stangl, 2019, Sannino et al., 2017). By contrast, the same framework cannot reproduce the large central values of $9/2$6 while satisfying LEP constraints on $9/2$7 couplings, because the needed degree of $9/2$8 compositeness is too strongly constrained (Stangl, 2019, Sannino et al., 2017).

Recent work reframes the flavor problem by combining UV anarchy with emergent IR symmetry. In this picture the strong sector is flavor-anarchic in the far UV, where RG running generates hierarchies through operator dimensions, but it flows near the compositeness scale to an IR regime with accidental flavor and CP symmetries (Agashe et al., 7 Jul 2025). The phenomenological payoff is substantial: anarchic partial compositeness is summarized as requiring the compositeness scale to be at least $9/2$9, while flavor-symmetric strong dynamics can permit ${\cal B}\sim{\cal S}{\cal F}$0, and about ${\cal B}\sim{\cal S}{\cal F}$1 in less symmetric variants (Agashe et al., 7 Jul 2025). This suggests that the long-standing opposition between “anarchic PC” and “flavor-protected PC” can be interpreted as an RG history rather than a dichotomy of unrelated models.

5. Precision observables, collider probes, and the limits of local EFT

Leptonic observables provide especially sharp probes of partial compositeness. In the composite-Higgs explanation of the muon ${\cal B}\sim{\cal S}{\cal F}$2 anomaly, the muon mass is induced by

${\cal B}\sim{\cal S}{\cal F}$3

and the same structure produces a chirality-enhanced one-loop contribution to ${\cal B}\sim{\cal S}{\cal F}$4 from composite muon partners. In the near-degenerate doublet–singlet limit the dominant contribution is the Higgs-exchange diagram, and the paper states that for ${\cal B}\sim{\cal S}{\cal F}$5 and ${\cal B}\sim{\cal S}{\cal F}$6 TeV the anomaly can be readily explained (Xu et al., 2022). After indirect constraints, the viable numerical region is roughly

${\cal B}\sim{\cal S}{\cal F}$7

and the 14 TeV HL-LHC with ${\cal B}\sim{\cal S}{\cal F}$8 can discover ${\cal B}\sim{\cal S}{\cal F}$9 GeV at $5/2$0 and exclude $5/2$1 GeV at about $5/2$2 through Drell–Yan production (Xu et al., 2022). The same analysis stresses that this mechanism is conceptually different from simply adding vectorlike leptons: the large coupling is the strong-sector Yukawa $5/2$3, the Higgs is composite, and the dipole operator effectively appears as a dimension-five operator within the partial-compositeness structure (Xu et al., 2022).

Top partial compositeness also leads beyond the usual EFT truncation. In the form-factor framework for a partially composite top, the gluonic current is written in terms of Dirac and Pauli chromo-form factors $5/2$4 and $5/2$5, and the paper emphasizes an intermediate regime

$5/2$6

in which the top is already non-pointlike while strong-sector resonances are not yet produced (Franzosi et al., 2019). In that regime a low-order local EFT misses shape information. For the benchmark $5/2$7, $5/2$8, $5/2$9 TeV, $SO(5)/SO(4)$0, and $SO(5)/SO(4)$1, the full form factor modifies the $SO(5)/SO(4)$2 distribution by about a $SO(5)/SO(4)$3 enhancement in the total $SO(5)/SO(4)$4-initiated cross section at 13 TeV and about $SO(5)/SO(4)$5 if $SO(5)/SO(4)$6 TeV (Franzosi et al., 2019).

Fermionic UV completions frequently predict additional pseudo-scalars that are diagnostically tied to the hyperfermion structure needed for top partial compositeness. In the “standard candle” analysis, the generic states are two neutral singlets and a colored octet, all with anomaly-induced diboson couplings (Belyaev et al., 2016). Their searches are complementary to EWPT and Higgs-coupling deviations; for the octet, pair production excludes $SO(5)/SO(4)$7–$SO(5)/SO(4)$8 GeV from 4-jet searches when the top coupling vanishes, and $SO(5)/SO(4)$9 GeV when $SU(5)/SO(5)$0 dominates and 4-top searches apply (Belyaev et al., 2016). A plausible implication is that, in realistic confining completions, the first experimentally accessible states need not be the top partners themselves.

6. Nonperturbative tests, emergent composites, and extensions to neutrinos

Partial compositeness depends on strong-sector matrix elements that cannot be fixed by symmetry alone. The lattice study of the $SU(5)/SO(5)$1 hypercolor theory with fundamental and sextet fermions makes this explicit by computing the vacuum-to-baryon overlap factors entering

$SU(5)/SO(5)$2

In that model the continuum and sextet-chiral extrapolation gives

$SU(5)/SO(5)$3

which is too small to generate a realistic top Yukawa while preserving the hierarchy $SU(5)/SO(5)$4 (Ayyar et al., 2018). The conclusion is model-specific rather than a no-go theorem, but it shows that the existence of the right baryon quantum numbers is not sufficient; the overlap factors must also be large enough.

Functional-RG studies pursue the same question from a different angle by following emergent fermion–scalar composites through the flow. In the FPC analysis, the key composite anomalous dimension satisfies

$SU(5)/SO(5)$5

and the indicative estimates for the minimal setup extended by extra Dirac fermions are

$SU(5)/SO(5)$6

for 1, 2, and 3 additional Dirac fermions, respectively (Goertz et al., 2023). The case with two additional Dirac fermions, $SU(5)/SO(5)$7, is highlighted as an interesting benchmark for lighter SM generations (Goertz et al., 2023, Goertz et al., 2023). This suggests that moderate anomalous scaling of the emergent composite field, rather than extremely large anomalous dimensions of purely fermionic baryons, may be enough for realistic flavor hierarchies in FPC.

The neutrino sector shows that partial compositeness is not restricted to the composite-Higgs top problem. In low-scale neutrino partial compositeness, the portal

$SU(5)/SO(5)$8

matches in the IR to $SU(5)/SO(5)$9, with

$SU(4)/Sp(4)$0

so light neutrino masses emerge through an inverse seesaw with composite singlet states $SU(4)/Sp(4)$1 (Chacko et al., 2020). This framework predicts long-lived composite singlet neutrinos, multiple displaced vertices at colliders and beam dumps, observable lepton-flavor violation, and, if $SU(4)/Sp(4)$2 lies below about $SU(4)/Sp(4)$3 MeV, a suppression of neutrinoless double beta decay by form factors that may reduce the rate by an order of magnitude or more (Chacko et al., 2020). A recent extension adds electromagnetic transition dipoles with

$SU(4)/Sp(4)$4

leading to $SU(4)/Sp(4)$5 and, for lighter compositeness scales or higher beam energies, multi-photon final states as a qualitative signature of composite dynamics (Assi et al., 18 Jun 2026).

Taken together, these results indicate that partial compositeness is less a single model than a family of mechanisms linking fermion masses to strong dynamics through linear elementary–composite mixing. Its most developed realizations tie the Higgs sector, top partners, flavor hierarchies, and precision observables into one structure; its main open questions concern the actual strong dynamics required for realistic anomalous dimensions and overlap factors, the size of residual flavor and CP violation, and the extent to which UV-complete constructions can remain both calculable and phenomenologically viable (Ayyar et al., 2018, Agashe et al., 7 Jul 2025).