Supersymmetric Twin Higgs Models

• Supersymmetric Twin Higgs models combine supersymmetry with the twin Higgs mechanism to regulate Higgs mass sensitivity and reduce fine-tuning.
• They utilize non-decoupling D-term contributions from new gauge symmetries to generate a large SU(4)-preserving quartic essential for realistic electroweak breaking.
• These frameworks enhance the tree-level Higgs mass while predicting distinctive signatures and spectra, including modified Higgs couplings and light stop scenarios.

Supersymmetric Twin Higgs denotes a class of ultraviolet completions of the Twin Higgs mechanism in which supersymmetry regulates the quadratic sensitivity of the Higgs sector to very high scales, while the observed Higgs is simultaneously realized as a pseudo-Nambu–Goldstone boson of an approximate global symmetry relating the visible and twin sectors. In the formulation reviewed in “Natural supersymmetric Twin Higgs,” the central model-building problem is to generate a large enough $SU(4)$-preserving quartic for the Higgs sector while retaining a realistic Higgs mass, acceptable electroweak symmetry breaking, and perturbativity to very high scales; the distinctive solution is a non-decoupling $D$-term of a new gauge symmetry, with a non-abelian and ultimately asymptotically free completion permitting perturbativity up to the Planck scale (Badziak et al., 2019).

1. Conceptual basis and low-energy structure

Supersymmetric Twin Higgs models are motivated by the post-LHC form of the little hierarchy problem. Supersymmetry addresses the big hierarchy problem by regulating quadratic sensitivity through superpartners, but ordinary SUSY models such as the MSSM require very heavy stops and/or large stop mixing to obtain $m_h \simeq 125$ GeV, with corresponding fine-tuning in electroweak symmetry breaking. The Twin Higgs mechanism addresses the little hierarchy problem by making the observed Higgs a pNGB of an approximate global symmetry, so that sensitivity of the weak scale to heavy colored states is reduced (Badziak et al., 2019).

A standard low-energy Twin Higgs potential is

$$V = \lambda (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + \Delta \lambda(|H'|^4 + |H|^4) + \Delta m^2 |H|^2 \,.$$

Here $H$ is the SM Higgs doublet and $H'$ is the twin Higgs doublet. The first two terms are $\mathbb{Z}_2$-symmetric and approximately $SU(4)$-symmetric, $\Delta\lambda$ breaks $SU(4)$ while preserving $D$0, and $D$1 breaks $D$2 explicitly. The vacuum expectation values satisfy

$D$3

and spontaneous breaking of the approximate global symmetry yields the pNGB Higgs (Badziak et al., 2019).

The alignment $D$4 required by Higgs coupling measurements is not automatic. It is obtained only after introducing explicit $D$5 breaking, with the associated irreducible Twin Higgs tuning

$D$6

Current Higgs data require roughly $D$7, so the misalignment tuning is moderate rather than negligible (Badziak et al., 2019). Earlier work on the mirror-MSSM realization described the combined mechanism as “double protection”: supersymmetry removes quadratic divergences above the soft scale, while the twin symmetry enforces an accidental approximate $D$8 in the Higgs sector, allowing the measured Higgs mass, couplings, and percent-level naturalness to coexist with stops at $D$9 TeV and higgsinos at $m_h \simeq 125$0 TeV (Craig et al., 2013).

2. The quartic problem and early supersymmetric realizations

A successful supersymmetric Twin Higgs model requires a large $m_h \simeq 125$1-invariant quartic $m_h \simeq 125$2, because the tuning relative to a non-twinned model is relaxed by roughly

$m_h \simeq 125$3

The central issue is therefore how to generate a large $m_h \simeq 125$4-preserving quartic while also obtaining $m_h \simeq 125$5 GeV (Badziak et al., 2017).

Early supersymmetric constructions generated the quartic from a singlet $m_h \simeq 125$6-term. In that class of models,

$m_h \simeq 125$7

This quartic is maximized at $m_h \simeq 125$8 and falls at large $m_h \simeq 125$9, whereas the ordinary SUSY Higgs mass benefits from large $$V = \lambda (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + \Delta \lambda(|H'|^4 + |H|^4) + \Delta m^2 |H|^2 \,.$$0. The same analyses emphasized a further difficulty: singlet-Higgs mixing proportional to $$V = \lambda (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + \Delta \lambda(|H'|^4 + |H|^4) + \Delta m^2 |H|^2 \,.$$1 gives a negative correction to the Higgs mass, so a heavy singlet or very light higgsino is often required, worsening naturalness (Badziak et al., 2017).

The major structural alternative was the supersymmetric $$V = \lambda (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + \Delta \lambda(|H'|^4 + |H|^4) + \Delta m^2 |H|^2 \,.$$2-term Twin Higgs with a new $$V = \lambda (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + \Delta \lambda(|H'|^4 + |H|^4) + \Delta m^2 |H|^2 \,.$$3. In that framework the $$V = \lambda (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + \Delta \lambda(|H'|^4 + |H|^4) + \Delta m^2 |H|^2 \,.$$4-invariant quartic is

$$V = \lambda (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + \Delta \lambda(|H'|^4 + |H|^4) + \Delta m^2 |H|^2 \,.$$5

so it is maximized at large $$V = \lambda (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + \Delta \lambda(|H'|^4 + |H|^4) + \Delta m^2 |H|^2 \,.$$6, aligning the requirements of a large Twin-Higgs quartic, a sufficiently heavy Higgs, and moderate stop masses. In the $$V = \lambda (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + \Delta \lambda(|H'|^4 + |H|^4) + \Delta m^2 |H|^2 \,.$$7 model, the 125 GeV Higgs mass can be obtained for stop masses below $$V = \lambda (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + \Delta \lambda(|H'|^4 + |H|^4) + \Delta m^2 |H|^2 \,.$$8 TeV, and the tuning required to obtain the correct electroweak scale can be as low as $$V = \lambda (|H'|^2 + |H|^2)^2 -m^2 (|H'|^2 + |H|^2) + \Delta \lambda(|H'|^4 + |H|^4) + \Delta m^2 |H|^2 \,.$$9; a stop mass of about $H$0 TeV is also possible with tuning of order $H$1 (Badziak et al., 2017).

3. $H$2-term realizations and the move to non-abelian completions

The class of models reviewed in 2019 organizes supersymmetric Twin Higgs realizations into three closely related constructions: Abelian $H$3 $H$4-term Twin Higgs, non-abelian $H$5 $H$6-term Twin Higgs, and asymptotically free $H$7 Twin Higgs (Badziak et al., 2019).

Realization	$H$8-invariant quartic	UV behavior
$H$9	$H'$0	Low Landau pole
$H'$1	$H'$2	Slower running
$H'$3	$H'$4	Asymptotically free

In the $H'$5 model, the extra gauge symmetry is broken by

$H'$6

leading, after integrating out the heavy fields, to

$H'$7

The phenomenological problem is perturbativity: the Abelian coupling runs rapidly to a Landau pole. In the benchmark charge assignment,

$H'$8

for the mirror (fraternal) model, while electroweak precision and LEP di-muon constraints require

$H'$9

This is the immediate reason later papers turned to non-abelian completions (Badziak et al., 2019).

The minimal non-abelian model replaces $\mathbb{Z}_2$0 with $\mathbb{Z}_2$1, with the visible and twin up-type Higgses embedded into bifundamentals $\mathbb{Z}_2$2 and $\mathbb{Z}_2$3. The same symmetry-breaking structure generates

$\mathbb{Z}_2$4

and only a small set of particles is charged under $\mathbb{Z}_2$5, allowing the model to be perturbative around the Planck scale. A distinctive further feature is that the new gauge interaction drives the top Yukawa coupling small at higher energy scales, which also reduces the tuning (Badziak et al., 2017).

The asymptotically free completion duplicates the new gauge factor,

$\mathbb{Z}_2$6

with a bifundamental $\mathbb{Z}_2$7 breaking the product to a diagonal subgroup at a scale of order a few tens of TeV. This reduces the matter charged under each factor and makes the gauge interactions asymptotically free. The resulting construction was presented as the first SUSY Twin Higgs model in which the TH mechanism is introduced by a new asymptotically free gauge interaction, with natural electroweak symmetry breaking for squarks and gluino heavier than $\mathbb{Z}_2$8 TeV even if supersymmetry breaking is mediated around the Planck scale (Badziak et al., 2017).

4. Higgs mass, tuning, and perturbativity

A central quantitative result across these constructions is the enhanced tree-level Higgs mass,

$\mathbb{Z}_2$9

Compared to the MSSM relation $SU(4)$0, this gives an approximate enhancement by a factor of $SU(4)$1 in $SU(4)$2 when $SU(4)$3, thereby reducing the stop mass needed to reach $SU(4)$4 GeV (Badziak et al., 2019).

Naturalness is conventionally quantified by

$SU(4)$5

with tuning in percent given by $SU(4)$6 (Badziak et al., 2019). The extra gauge sector contributes an irreducible threshold,

$SU(4)$7

in the Abelian case, and three times larger in the non-abelian case, so increasing $SU(4)$8 improves the Twin Higgs quartic but also worsens radiative corrections to Higgs soft masses (Badziak et al., 2019).

The main quantitative conclusions are model-dependent but structurally consistent. In the $SU(4)$9 model, tuning at the level of $\Delta\lambda$0 is possible for stop masses above $\Delta\lambda$1 TeV, and the Higgs mass can point to stops around $\Delta\lambda$2 GeV with zero stop mixing for $\Delta\lambda$3, while stops of $\Delta\lambda$4 TeV remain possible with moderate $\Delta\lambda$5 (Badziak et al., 2019). In the non-abelian $\Delta\lambda$6 model, the Higgs mass is in agreement with the measured value in most of parameter space for a representative point with $\Delta\lambda$7 TeV, $\Delta\lambda$8, and $\Delta\lambda$9 TeV; tuning remains $SU(4)$0 for low mediation scales and only a few percent for high scales (Badziak et al., 2019).

The strongest result belongs to the asymptotically free $SU(4)$1 completion. For

$SU(4)$2

the tuning is better than $SU(4)$3 even if the mediation scale is as large as the Planck scale, and the tuning is relaxed by two to three orders of magnitude relative to the MSSM (Badziak et al., 2019). This gain is attributed simultaneously to the Twin Higgs mechanism, the enhanced tree-level Higgs mass, and suppression of $SU(4)$4 by the large new gauge interaction (Badziak et al., 2019).

5. Spectrum, signatures, and experimental constraints

Supersymmetric Twin Higgs models generically predict a new gauge boson or gauge bosons associated with $SU(4)$5, $SU(4)$6, or $SU(4)$7; scalar fields $SU(4)$8 and, in the asymptotically free model, $SU(4)$9; additional Higgs-sector fields such as $D$00, $D$01, $D$02, $D$03; vectorlike or exotic matter for anomaly cancellation; and twin-sector copies of most MSSM matter (Badziak et al., 2019). Typical benchmark superpartner scales discussed in the literature are $D$04–$D$05 TeV, $D$06 TeV, and $D$07 GeV (Badziak et al., 2019).

The most model-independent constraints are Higgs coupling measurements, which require $D$08, and electroweak precision constraints, which in the Abelian model imply $D$09 TeV (Badziak et al., 2019). Earlier mirror-MSSM analyses highlighted a corresponding Higgs-sector phenomenology rather than conventional natural-SUSY signatures: modifications of Higgs couplings, a modest invisible Higgs width, resonant Higgs pair production, and an invisibly-decaying heavy Higgs were identified as primary signs of naturalness (Craig et al., 2013).

The non-abelian high-scale models also predict distinctive flavor structure. In the asymptotically free $D$10 construction, the right-handed up quark is embedded together with the right-handed top in $D$11, leading to an effective flavor-violating coupling for

$D$12

The paper estimates

$D$13

quotes the current bound $D$14, and notes a projected High-Luminosity LHC sensitivity of $D$15 (Badziak et al., 2017). The same flavor structure implies unusual heavy-Higgs production and the near-degeneracy of the right-handed stop and right-handed up squark (Badziak et al., 2017).

6. Dark matter, thermal history, and related developments

Supersymmetric Twin Higgs model building has extended beyond electroweak naturalness into dark matter and finite-temperature cosmology. “Natural Twin Neutralino Dark Matter” showed that a twin bino-like neutralino can be the LSP and can obtain the observed relic abundance through standard thermal freeze-out without the tuning usually required for bino dark matter in the MSSM. In that framework naturalness requires $D$16 to keep fine-tuning around the $D$17 level, and the thermal relic contour $D$18 is obtained over broad parameter space once roughly

$D$19

Most of the viable parameter space can be probed by future direct-detection experiments such as LZ and by LHC searches for staus and higgsinos, potentially with displaced vertices (Badziak et al., 2019).

A more unconventional possibility is charged dark matter in the twin sector. “Charged Dark Matter in Supersymmetric Twin Higgs models” showed that the twin stau is a viable candidate even if the twin electromagnetic gauge symmetry is unbroken, with thermal relic abundance naturally matching the observed dark matter abundance. The twin stau has a mass in the range of $D$20–$D$21 GeV, a wide parameter space satisfies the constraints on dark matter self-interactions, and in the minimal scenario the visible-sector stau can have a decay length long enough to be observed as a disappearing track or a long-lived particle at the LHC (Badziak et al., 2022).

Finite-temperature studies initially concluded that electroweak baryogenesis is hardly realized in typical Twin Higgs models, while in the supersymmetric case there remain some parameter spaces in which the higher-scale global-symmetry-breaking transition is first order, although the resulting stochastic gravitational-wave background is impossible to be detected by DECIGO or BBO in the linear realization and decoupling limit (Fujikura et al., 2018). Later work revisited this conclusion. “First-order phase transitions in Twin Higgs models” found strong FOPTs in models with hard $D$22 breaking in the scalar potential and in models with enhanced twin lepton Yukawa couplings; in the supersymmetric UV completion of the second scenario, light twin sleptons strengthen the transition and can bring the gravitational-wave signal close to the reach of AEDGE and Einstein Telescope (Badziak et al., 2022). The 2025 study of electroweak symmetry non-restoration in supersymmetric Twin Higgs models pushed the thermal program further: with hard $D$23-breaking in twin Yukawas and light twin sfermions, the visible electroweak symmetry can remain broken up to temperatures of order the twin electroweak scale, dark radiation can be reduced to the level consistent with CMB data, and tuning can improve to better than $D$24 in the RHN and sneutrino scenario (Badziak et al., 21 Aug 2025).

Several neighboring constructions clarify the boundaries of the subject. “Spontaneous Twin Symmetry Breaking” showed that replacing explicit by spontaneous $D$25 breaking does not remove the familiar tuning of order $D$26; in the minimal exact-$D$27 model one finds $D$28, while twin vector-like leptons can lower the scale to $D$29 TeV (Jung, 2019). “Twin Turtles” raised the Higgs-sector cutoff by making the radial mode of twin symmetry breaking itself a pNGB, with robustness demonstrated in two supersymmetric completions and with multiple Higgs-like scalars identified as the characteristic signature (Asadi et al., 2018). “The Hyperbolic Higgs” is not a supersymmetric Twin Higgs in the usual sense, but it showed how SUSY tools can realize neutral naturalness with Standard-Model-neutral scalar top partners and a non-compact accidental $D$30 symmetry, thereby clarifying which ingredients of SUSY are transferable to Twin-like protection (Cohen et al., 2018). At the level of SUSY breaking and mediation, a gravity-mediated mirror Twin Higgs model with a $D$31-odd Polonyi field showed that tree-level gaugino masses are compatible with a Polonyi field odd under the exchange symmetry and that the same structure may serve as an origin of the $D$32-breaking of the Higgs potential, while avoiding the Polonyi problem (Randle et al., 2023).

Taken together, these developments define supersymmetric Twin Higgs as a research program centered on a technically specific claim: neutral naturalness becomes substantially more effective when the twin pNGB mechanism is embedded into a supersymmetric UV completion with a large $D$33-preserving quartic, and the most successful known route to that quartic is a non-decoupling non-abelian $D$34-term, especially in asymptotically free realizations that remain perturbative up to the Planck scale (Badziak et al., 2019).