cS2HDM: Complex Singlet-Extended 2HDM

• cS2HDM is a scalar-sector extension combining two Higgs doublets with a complex singlet, accommodating both CP conservation and violation.
• The model supports multiple realizations via distinct symmetries, Yukawa alignments, and vacuum configurations, leading to varied phenomenological benchmarks.
• Predictions include distinctive Higgs cascades, alignment limits, and potential strong first-order electroweak transitions that connect EDM constraints with collider signals.

The complex singlet-extended 2-Higgs-doublet model (cS2HDM), also denoted 2HDM+S or 2HDMS in much of the literature, is the class of scalar-sector extensions of the Standard Model built from two electroweak doublets and one complex gauge singlet. In its broadest sense, the field content is $\Phi_1\sim({\bf 2},1/2)$, $\Phi_2\sim({\bf 2},1/2)$, and $S\sim({\bf 1},0)$, but the literature contains several inequivalent realizations distinguished by their discrete or global symmetries, Yukawa assumptions, and dark-sector assignments. Consequently, the cS2HDM is not a single canonical model: some versions are CP-conserving, some admit explicit or spontaneous CP violation, some contain a stable singlet-derived dark matter state, and some are best regarded as restricted subclasses rather than the fully general complex singlet extension of the 2HDM (Baum et al., 2018, Biekötter et al., 1 Sep 2025, Lahiri et al., 23 Mar 2026).

1. Model class, nomenclature, and scope

The unifying feature of the cS2HDM is the coexistence of a 2HDM doublet sector with a complex singlet sector. Beyond that common field content, the defining assumptions vary substantially. A fully general CP-conserving 2HDM+S scalar potential was analyzed without imposing an extra scalar symmetry, while recent dark-matter-oriented benchmarks instead organize the singlet sector around a softly broken global $U(1)$ so that the singlet imaginary component becomes a pseudo-Nambu-Goldstone dark matter state. Other constructions use a singlet-sector $Z'_2$, or retain CP conservation but arrange the singlet as a separate dark sector. The literature therefore uses “cS2HDM” both for the general field-content class and for narrower benchmark realizations (Baum et al., 2018, Biekötter et al., 1 Sep 2025, Ziegler et al., 2023).

Realization	Defining restriction	Immediate consequence
General CP-conserving 2HDM+S (Baum et al., 2018)	No extra scalar symmetry imposed	$3$ CP-even, $2$ CP-odd, $H^\pm$
pNG-DM cS2HDM benchmark (Biekötter et al., 1 Sep 2025)	Softly broken singlet $U(1)$, flavour alignment	stable $\chi$, explicit CPV allowed
Exact-alignment CPV complex 2HDMS (Lahiri et al., 23 Mar 2026)	Yukawa alignment, hard $\Phi_2\sim({\bf 2},1/2)$0-breaking couplings kept	CPV can survive exact alignment
Degenerate-scalar 2HDMS (Cho et al., 2024)	CP-even masses nearly degenerate	direct-detection cancellation
MPP-constrained 2HDMS (Cho et al., 6 Jan 2026)	tree-level electroweak/singlet vacuum degeneracy	tension between MPP and DM blind spot
Type-II $\Phi_2\sim({\bf 2},1/2)$1-structured 2HDMS (Ziegler et al., 2023)	stable singlet pseudoscalar $\Phi_2\sim({\bf 2},1/2)$2	three CP-even Higgs bosons plus DM
$\Phi_2\sim({\bf 2},1/2)$3 dark-sector subclass (Dutta et al., 2022)	singlet does not acquire a vev	no singlet–doublet mixing
Inert-doublet plus complex singlet cousin (Krawczyk et al., 2015)	one doublet inert	not the usual active-active cS2HDM

A recurrent terminological ambiguity is that “complex singlet” refers to the field content, not automatically to CP violation. Several analyses explicitly take all scalar-potential coefficients to be real and work in a CP-conserving limit, with the complex nature of the singlet reflected only in the presence of distinct real and imaginary singlet components (Cho et al., 6 Jan 2026, Cho et al., 2024).

2. Symmetries, Yukawa structures, and scalar potentials

The most general CP-conserving 2HDM+S scalar potential discussed in the literature combines the usual renormalizable 2HDM potential with singlet operators of the forms $\Phi_2\sim({\bf 2},1/2)$4, $\Phi_2\sim({\bf 2},1/2)$5, $\Phi_2\sim({\bf 2},1/2)$6, $\Phi_2\sim({\bf 2},1/2)$7, $\Phi_2\sim({\bf 2},1/2)$8, $\Phi_2\sim({\bf 2},1/2)$9, $S\sim({\bf 1},0)$0, $S\sim({\bf 1},0)$1, and $S\sim({\bf 1},0)$2, with all parameters chosen real in the CP-conserving case (Baum et al., 2018). This is the broadest scalar-sector definition represented here.

Dark-matter-oriented cS2HDM formulations usually impose extra structure. In the pNG benchmark, the singlet sector is organized by a global $S\sim({\bf 1},0)$3 acting on $S\sim({\bf 1},0)$4, exact in all singlet-dependent terms except for a single soft dimension-two breaking term. The scalar potential contains the ordinary 2HDM quartics $S\sim({\bf 1},0)$5, singlet quartic $S\sim({\bf 1},0)$6, portal quartics $S\sim({\bf 1},0)$7, doublet mixing $S\sim({\bf 1},0)$8, and a soft singlet-breaking term proportional to $S\sim({\bf 1},0)$9. In that benchmark, $U(1)$0, $U(1)$1, $U(1)$2, $U(1)$3, $U(1)$4, and $U(1)$5 can be complex, while the remaining parameters are real (Biekötter et al., 1 Sep 2025).

A different explicitly CP-violating complex 2HDMS retains the general 2HDM part with $U(1)$6 potentially complex and extends it by singlet couplings $U(1)$7, also potentially complex. In that construction, the portal operators $U(1)$8, $U(1)$9, and $Z'_2$0 are central for heavy-sector CP mixing (Lahiri et al., 23 Mar 2026).

The doublet Yukawa sector is likewise non-unique. Standard natural-flavour-conservation realizations employ the usual 2HDM $Z'_2$1 symmetry, softly broken by the doublet mixing term, with Type-I, II, X, or Y Yukawa assignments (Cho et al., 6 Jan 2026, Cho et al., 2024). By contrast, the 2025 cS2HDM benchmark deliberately uses flavour alignment,

$Z'_2$2

with $Z'_2$3 in general complex. The explicit purpose is to avoid tree-level FCNCs without imposing the discrete $Z'_2$4 that would otherwise remove CP-violating scalar effects in the alignment limit (Biekötter et al., 1 Sep 2025).

3. Vacuum structure, mass eigenstates, and alignment

A common CP-conserving vacuum choice is

$Z'_2$5

with

$Z'_2$6

In this setting, the CP-even fields $Z'_2$7 mix into three physical CP-even scalars $Z'_2$8, the doublet CP-odd fields $Z'_2$9 rotate into $3$0 and $3$1, the charged fields into $3$2 and $3$3, and the singlet-imaginary component $3$4 remains unmixed if CP is preserved (Cho et al., 2024, Cho et al., 6 Jan 2026).

The resulting mass formulas in the CP-conserving pNG-DM construction include

$3$5

and

$3$6

The $3$7 CP-even mass matrix is diagonalized by an orthogonal matrix $3$8, parameterized by angles $3$9, and the portal couplings $2$0 can be reconstructed from masses and mixings through

$2$1

These relations are especially important in the degenerate-scalar scenario, where orthogonality suppresses $2$2 and $2$3 (Cho et al., 2024).

In the general CP-conserving 2HDM+S, the neutral spectrum instead separates into three CP-even eigenstates $2$4 and two CP-odd eigenstates $2$5, together with the charged Higgs pair. The paper introducing this parameterization emphasizes an extended Higgs basis in which only $2$6 carries the electroweak vev, while $2$7, $2$8, $2$9, and $H^\pm$0 describe orthogonal doublet and singlet directions (Baum et al., 2018).

Once explicit CP violation is admitted while keeping $H^\pm$1 real to preserve dark-matter stability, the neutral sector enlarges. In the pNG benchmark the physical neutral basis is

$H^\pm$2

and an orthogonal $H^\pm$3 matrix $H^\pm$4 diagonalizes the neutral mass matrix into four Higgs mass eigenstates $H^\pm$5, while the singlet-imaginary field $H^\pm$6 remains a stable dark matter particle (Biekötter et al., 1 Sep 2025).

Alignment is a unifying phenomenological requirement. In the pNG benchmark, a convenient alignment limit is

$H^\pm$7

so that $H^\pm$8 is SM-like (Biekötter et al., 1 Sep 2025). In the exact-alignment complex 2HDMS, the conditions

$H^\pm$9

ensure that the $U(1)$0 GeV Higgs is exactly SM-like and can be identified as $U(1)$1 with $U(1)$2 (Lahiri et al., 23 Mar 2026). In the CP-conserving 2HDM+S, approximate alignment without decoupling is instead formulated through the suppression of the off-diagonal mass-matrix elements $U(1)$3 and $U(1)$4 (Baum et al., 2018).

4. CP violation and its realization in the cS2HDM

CP violation in the cS2HDM can be spontaneous or explicit, depending on the symmetry structure. A useful related example is the inert-doublet plus complex-singlet construction, in which all scalar-potential parameters are real and CP violation arises spontaneously through a complex singlet vev,

$U(1)$5

In the simplified one-doublet-plus-complex-singlet subsystem used there, the minimization conditions imply

$U(1)$6

with $U(1)$7 and $U(1)$8. The conclusion stated there is that viable CP violation requires at least two among $U(1)$9, $\chi$0, and $\chi$1 to be nonzero, and the abstract summarizes the requirement as one non-zero cubic term in the singlet potential being needed for CP violation (Krawczyk et al., 2015).

The explicitly CP-violating cS2HDM benchmark takes a different route. After imposing the vacuum conditions, the singlet soft-breaking parameter $\chi$2 is real in the chosen vacuum, but four independent CP-violating phases remain in the scalar potential, and in the parameter basis used later $\chi$3 is retained as the independent explicit CP-violating scalar-potential parameter (Biekötter et al., 1 Sep 2025). A central structural point is that CP violation can survive the alignment limit: unlike in a $\chi$4-symmetric CP-violating 2HDM, heavy-sector and Yukawa CP violation need not disappear when the 125 GeV Higgs becomes SM-like (Biekötter et al., 1 Sep 2025).

The exact-alignment complex-singlet extension of the aligned 2HDM sharpens this observation. After imposing exact alignment and the dark-matter stability conditions, the neutral basis $\chi$5 has a block structure in which the $\chi$6 GeV Higgs $\chi$7 and the dark state $\chi$8 decouple, while the nontrivial $\chi$9 mixing occurs in $\Phi_2\sim({\bf 2},1/2)$00. The CP-even/CP-odd mixing entry

$\Phi_2\sim({\bf 2},1/2)$01

is the additional source of CP violation absent in the ordinary aligned 2HDM. The corresponding phase is conveniently written as

$\Phi_2\sim({\bf 2},1/2)$02

That analysis identifies three CPV sources in exact alignment: $\Phi_2\sim({\bf 2},1/2)$03, the phase of $\Phi_2\sim({\bf 2},1/2)$04 generating CP-violating scalar self-interactions, and the phases of the Yukawa alignment parameters $\Phi_2\sim({\bf 2},1/2)$05 (Lahiri et al., 23 Mar 2026).

This enlarged CP structure directly affects EDM phenomenology. The 2025 cS2HDM benchmark computes the electron EDM from generalized two-loop Barr-Zee diagrams and emphasizes cancellations between scalar-sector and Yukawa-sector CP phases as one of the model’s defining advantages (Biekötter et al., 1 Sep 2025). The 2026 exact-alignment study reaches a parallel conclusion: relative to the aligned 2HDM, the extra scalar CP phase substantially enlarges the EDM-allowed region, with the electron EDM remaining the dominant CP-sensitive constraint (Lahiri et al., 23 Mar 2026).

5. Dark matter realizations

Dark matter is optional in the cS2HDM, but several important constructions use the singlet imaginary component as the dark state. In the pNG benchmark, the real part of $\Phi_2\sim({\bf 2},1/2)$06 acquires a vev $\Phi_2\sim({\bf 2},1/2)$07, the imaginary part $\Phi_2\sim({\bf 2},1/2)$08 becomes a pseudo-Nambu-Goldstone boson, and a remnant $\Phi_2\sim({\bf 2},1/2)$09 symmetry stabilizes it. The model predicts a pseudo-Nambu-Goldstone DM candidate whose interactions with nuclei are naturally suppressed, while the annihilation phenomenology is controlled by the trilinear couplings

$\Phi_2\sim({\bf 2},1/2)$10

A notable result is that CP violation opens annihilation through a dominantly CP-odd heavy scalar funnel, which can lower the relic density near $\Phi_2\sim({\bf 2},1/2)$11 while keeping the 125 GeV Higgs almost exactly SM-like (Biekötter et al., 1 Sep 2025).

The degenerate-scalar scenario provides a different suppression mechanism. Here the three CP-even mediators are taken nearly degenerate,

$\Phi_2\sim({\bf 2},1/2)$12

so that the tree-level $\Phi_2\sim({\bf 2},1/2)$13 amplitude mediated by CP-even Higgs exchange cancels by orthogonality of the mixing matrix. Representative benchmarks use

$\Phi_2\sim({\bf 2},1/2)$14

and exhibit strong suppression of direct detection while remaining compatible with the observed relic abundance (Cho et al., 2024). When the tree-level Multiple Point Principle is imposed on the same 2HDMS class, the electroweak and singlet vacua are required to be degenerate, $\Phi_2\sim({\bf 2},1/2)$15, which pushes the model toward large $\Phi_2\sim({\bf 2},1/2)$16 and small $\Phi_2\sim({\bf 2},1/2)$17. That competes with the degenerate-scalar direct-detection blind spot, yet viable regions remain in the Higgs-resonance regime $\Phi_2\sim({\bf 2},1/2)$18 and in a heavy-DM regime $\Phi_2\sim({\bf 2},1/2)$19 (Cho et al., 6 Jan 2026).

A distinct Type-II realization uses a $\Phi_2\sim({\bf 2},1/2)$20-structured singlet sector in which

$\Phi_2\sim({\bf 2},1/2)$21

the CP-even singlet scalar mixes with the CP-even doublet fields, and the singlet pseudoscalar $\Phi_2\sim({\bf 2},1/2)$22 remains unmixed and stable. In that model the main annihilation channels highlighted are

$\Phi_2\sim({\bf 2},1/2)$23

with resonances around $\Phi_2\sim({\bf 2},1/2)$24 and $\Phi_2\sim({\bf 2},1/2)$25 (Ziegler et al., 2023).

At the opposite extreme, a DM-oriented subclass sets $\Phi_2\sim({\bf 2},1/2)$26 so that the singlet does not mix with the doublets at all. The visible scalar sector then remains exactly the CP-conserving 2HDM spectrum, while the real and imaginary singlet components $\Phi_2\sim({\bf 2},1/2)$27 and $\Phi_2\sim({\bf 2},1/2)$28 form the dark sector. In that framework the light-Higgs funnel near $\Phi_2\sim({\bf 2},1/2)$29 and another viable region around $\Phi_2\sim({\bf 2},1/2)$30 are emphasized, and direct detection is found to be especially constraining for low-mass dark matter unless the portal couplings are small (Dutta et al., 2022).

6. Constraints, collider signatures, and electroweak phase transition

Across its variants, the cS2HDM is constrained by the standard combination of vacuum stability or bounded-from-below requirements, perturbative unitarity, electroweak precision data, flavour observables, Higgs signal strengths, direct searches for additional Higgs bosons, dark-matter relic abundance and direct detection, and—once CP violation is present—EDMs. In the 2025 benchmark, bounded-from-below is imposed numerically through bilinear minimization because analytic BFB conditions are unavailable in the full cS2HDM, perturbative unitarity is enforced through scalar $\Phi_2\sim({\bf 2},1/2)$31 scattering eigenvalues $\Phi_2\sim({\bf 2},1/2)$32, and collider constraints are handled through HiggsSignals/HiggsTools and HiggsBounds/HiggsTools, together with a modified HDECAY library and micrOMEGAs interface collected in the public package cs2hdmTools (Biekötter et al., 1 Sep 2025).

The collider phenomenology of the CP-conserving 2HDM+S is dominated by Higgs cascades. Near alignment, the decays

$\Phi_2\sim({\bf 2},1/2)$33

are not alignment-suppressed in the same way as $\Phi_2\sim({\bf 2},1/2)$34 and $\Phi_2\sim({\bf 2},1/2)$35, and large branching ratios into two lighter Higgs bosons or a light Higgs plus a $\Phi_2\sim({\bf 2},1/2)$36 are described as ubiquitous in the 2HDM+S. The collider study concludes that combining different final states arising from Higgs cascades would allow most of the interesting region of parameter space with Higgs masses up to $\Phi_2\sim({\bf 2},1/2)$37 to be probed at the LHC with $\Phi_2\sim({\bf 2},1/2)$38 (Baum et al., 2018).

CP-violating realizations add qualitatively new heavy-sector observables. In the exact-alignment complex 2HDMS, the simultaneous occurrence of the channels

$\Phi_2\sim({\bf 2},1/2)$39

is proposed as a direct test of heavy-sector CP violation, because each process is forbidden in the CP-conserving limit for the corresponding benchmark pairing. The computed cross sections are small but conceptually clean, and the paper emphasizes their dependence on different CP phases, with $\Phi_2\sim({\bf 2},1/2)$40 განსაკუთრებით sensitive to $\Phi_2\sim({\bf 2},1/2)$41 and $\Phi_2\sim({\bf 2},1/2)$42 to $\Phi_2\sim({\bf 2},1/2)$43 (Lahiri et al., 23 Mar 2026).

Electroweak phase-transition studies show that the enlarged scalar sector can support a strong first-order transition in dark-matter-compatible regions. In the degenerate-scalar 2HDMS, the conventional criterion

$\Phi_2\sim({\bf 2},1/2)$44

is satisfied with benchmark values around $\Phi_2\sim({\bf 2},1/2)$45, even though the singlet vev changes very little across the transition; the conclusion is that the first-order transition is loop-driven by the enlarged Higgs sector rather than tree-level-singlet-driven (Cho et al., 2024). The MPP analysis sharpens this: tree-level MPP forbids a tree-level-driven first-order transition, but thermal loop effects computed with CosmoTransitions and Parwani resummation still yield $\Phi_2\sim({\bf 2},1/2)$46, preserving compatibility with electroweak-baryogenesis-motivated SFOEWPT criteria (Cho et al., 6 Jan 2026).

Restricted dark-sector subclasses exhibit different collider profiles. In the $\Phi_2\sim({\bf 2},1/2)$47 2HDMS, representative invisible-heavy-Higgs signatures are found to be weak at the HL-LHC, while a $\Phi_2\sim({\bf 2},1/2)$48 $\Phi_2\sim({\bf 2},1/2)$49 collider with $\Phi_2\sim({\bf 2},1/2)$50 yields $\Phi_2\sim({\bf 2},1/2)$51 for one benchmark in the $\Phi_2\sim({\bf 2},1/2)$52 channel (Dutta et al., 2022). In the Type-II $\Phi_2\sim({\bf 2},1/2)$53-structured realization, the HL-LHC sensitivity to $\Phi_2\sim({\bf 2},1/2)$54 is poor for the quoted benchmark, whereas future lepton colliders are described as more promising (Ziegler et al., 2023).

7. Restricted subclasses, adjacent frameworks, and common misconceptions

Several recurrent misconceptions arise from conflating all complex-singlet extensions of the 2HDM. First, the cS2HDM is not synonymous with CP violation. The 2HDMS studies based on real scalar-potential coefficients explicitly stress that “complex singlet” means the field $\Phi_2\sim({\bf 2},1/2)$55 is complex, not that the scalar potential necessarily violates CP (Cho et al., 6 Jan 2026).

Second, not every model with two doublets and one complex singlet is the usual active-active cS2HDM. The inert-doublet plus complex-singlet construction has the same total field content, but one doublet is $\Phi_2\sim({\bf 2},1/2)$56-odd and remains inert,

$\Phi_2\sim({\bf 2},1/2)$57

Because the inert doublet neither acquires a vev nor mixes with the Higgs-singlet sector, that framework is structurally nearer to an inert 2HDM plus complex singlet than to a generic singlet-extended 2HDM with two active doublets (Krawczyk et al., 2015).

Third, some dark-matter-oriented 2HDMS realizations lie at the boundary of the broader cS2HDM notion. The $\Phi_2\sim({\bf 2},1/2)$58 subclass studied for Higgs-portal dark matter is a specific CP-conserving, Type-II, $\Phi_2\sim({\bf 2},1/2)$59-symmetric limit with no singlet–doublet mixing, so it is better described as a dark-sector subclass than as the generic cS2HDM (Dutta et al., 2022). Likewise, the $\Phi_2\sim({\bf 2},1/2)$60-structured Type-II realization with stable $\Phi_2\sim({\bf 2},1/2)$61 is accurately characterized as a restricted cS2HDM realization rather than the fully general model (Ziegler et al., 2023).

A plausible implication of this model diversity is that the cS2HDM is best understood as a framework family rather than a unique Lagrangian. Within that family, the main theoretical themes are stable singlet-derived dark matter, alignment-compatible heavy-sector CP violation, Higgs-cascade-dominated collider phenomenology, and electroweak-phase-transition dynamics strengthened by the enlarged scalar sector. The detailed spectrum, couplings, and admissible observables are determined not by the field content alone, but by which symmetry assumptions are imposed on the doublet and singlet sectors.