Electroweak precision observables have provided a precise test of the standard model (SM) at the loop level [1, 2], which is consistent with the observations of a 125 GeV SM-like Higgs [3, 4]. However, the SM could not provide satisfactory solutions to dark matter, neutrino mass, baryogenesis, etc. [5−8]. Furthermore, the naturalness problem in the SM points to new physics beyond the SM [9].

One of the simplest extensions of the SM Higgs sector is the two-Higgs doublet model (2HDM) [10], which has been studied extensively. The 2HDM can be further extended by an additional singlet field, which is the N2HDM with a real singlet [11−13], and the 2HDM+S with a complex singlet [14, 15]. The 2HDM+S matches the next-to minimal supersymmetric standard model (NMSSM) [16] at a low energy scale and can provide a dark matter candidate [17−19], as well as accommodate the possible 95 GeV excess at the LEP and LHC [15]. The phenomenological properties of the 2HDM+S have only been explored in some specific scenarios, whereas the more general cases of the 2HDM+S have not yet been studied in detail. In this study, we explore the implications of the electroweak precision measurements on the 2HDM+S parameter space. In particular, we focus on the oblique parameters S, T, and U, which are sensitive to the new physics contributions to the W and Z self-energies [20, 21].

The scalar sector of the 2HDM+S includes two doublets and a complex singlet. The singlet field does not couple to the SM gauge bosons and fermions. After the neutral components achieve vacuum expectation values (vev), assuming no CP-violation, the mass spectrum of the Higgs sectors includes 3 CP-even scalars, two CP-odd scalars, and a pair of charged Higgses. In particular, the CP-even and CP-odd singlet components mix with the corresponding ones in the doublets, which leads to the couplings of the singlet-like scalars to the SM gauge bosons, as well as modifications of the couplings of the doublet-like scalars to the SM sector. The most general 2HDM+S Higgs potential has 27 free parameters, and 11 of these can be chosen to be the masses of the Higgs bosons, as well as the mixing angles between Higgses. The remaining parameters in the Higgs potential are the Higgs self-couplings, which do not directly contribute to the oblique parameters. Therefore, in our study, we only focus on the STU constraint and the relevant parameters, including these 11 mass and mixing parameters. We parameterize such mixing parameters by , the mixing of the CP-even singlet with the 125 GeV SM-like Higgs h, , the mixing of the CP-even singlet with the 2HDM CP-even Higgs H, and , the mixing of the CP-odd singlet with the 2HDM CP-odd Higgs A.

While the general formalism for the contributions of various Higgses to the oblique parameter exists in literature [22], the analyses of electroweak precision constraints in the 2HDM+S could be complex given the enlarged parameter space. In our analyses, we performed a systematic study of the impacts of each mixing angle on the oblique parameters. Including the usual 2HDM mixing angle of the CP-even Higgses α, we introduce five basic benchmark scenarios, Case-0 for the 2HDM alignment limit and Cases-I−IV in which only one mixing angle is set to be nonzero. We analyze the contributions to the oblique parameters in each case and study the 95% C.L. allowed region in the relevant parameter spaces under the oblique parameters. After the discussion of these five benchmark scenarios, we discuss the cases with a non-zero singlet mixing angle away from the alignment limit.

The implications of electroweak precision measurements in the 2HDM and singlet extended SM have been studied in the literature [22−27]. Our study offers a comprehensive electroweak precision analysis of the 2HDM+S and identifies the impact of each singlet mixing angle. As only the couplings between the Higgses and the SM gauge bosons enter the oblique parameters, our results are universal to the 2HDM+S models, with or without further symmetry assumption of the Higgs potential. In addition, we explore the complementarity of the electroweak precision analyses with the Higgs precision measurements. Note that, if we start from the parameters in the Higgs potential for a specific 2HDM+S model, and impose the theoretical considerations of successful electroweak symmetry breaking, vacuum stability, perturbativity, and unitarity, the resulting values of the mixing angles and mass differences might be restricted to a certain range. These ranges would depend on the particular symmetry assumption of the Higgs potential, and could also be relaxed with the variation of other model parameters. In our analyses, we consider a model independent approach and use the various mixing angles and physics Higgs masses as our relevant model parameters for the STU study. We let the mixing angles vary over the whole range and the mass difference up to approximately 1 TeV, which allows a straightforward mapping of a particular Higgs potential scenario to the general results of the electroweak precision constraints that we studied herein.

The remainder of this paper is organized as follows. In Section II, we introduce the theoretical framework of the 2HDM+S, as well as five benchmark cases. In Section III, we introduce the electroweak oblique parameters and the contributions from the Higgs sector in the 2HDM+S. In Section IV, we present 95% C.L. STU allowed regions in the 2HDM+S parameter spaces of the five benchmark cases. In Section V, we study the cases beyond the alignment limit. In Section VI, we show the complementarity of electroweak precision analyses with Higgs precision measurements. We conclude this paper in Section VII.

The 2HDM+S is the singlet extension of the 2HDM, which has the following scalar contents:

1

where and are the SU(2) _L doublets with hypercharge , and S is the gauge singlet. The general Higgs potential of the 2HDM+S has been introduced in [14], whereas the simplified version of the 2HDM+S potential can be found in [15] when certain symmetries are imposed. After electroweak symmetry breaking, the neutral components of , , and S develop non-zero vacuum expectation values, , , and , with GeV. We also introduce with . Assuming no CP-violation, the mass spectrum of the 2HDM+S includes three neutral CP-even scalars, two neutral CP-odd scalars, and one pair of charged Higgs bosons.

The neutral CP-even states, mix together to form three mass eigenstates: the non-SM-like H, the SM-like Higgs h, and the singlet-like , with the rotation matrix R

2

The R matrix is parameterized using three mixing angles α, , and , which characterize the mixing angle between the two neutral components of the Higgs doublets , and the mixing angles between the singlet with the 2HDM CP-even Higgses:

3

where we use the shorthand notations and . For the CP-odd states, we have

4

where is the neutral Goldstone boson, and the angle is the mixing between the 2HDM pseudoscalar and the singlet pseudoscalar . In addition, the charged sector of the 2HDM+S is the same as that of the 2HDM, containing one pair of charged Higgses and the Goldstone bosons . Each of the mixing angles varies in the range of

5

When , the mixing between the two Higgs bosons reaches maximum, and the properties of the two corresponding scalars flip when . Note that the effects of different signs of the mixing angles appear only when all four mixing angles are nonzero. When at least one mixing angle is nonzero, the properties of the Higgs bosons are independent of the sign of the mixing angles. When the theoretical considerations of successful electroweak symmetry breaking, vacuum stability, perturbativity, and unitarity are imposed on the Higgs potential, the resulting values of the mixing angles might be restricted to a smaller range. These ranges would depend on the particular symmetry assumption of the Higgs potential. We consider the whole range of these mixing angles, which allows a straightforward mapping of a particular Higgs potential scenario to the general results of the electroweak precision constraints that we investigate in this study.

After the diagonalization of the Higgs mass matrices, there are 11 free parameters for the mass eigenstates: six Higgs boson masses, , and four mixing angles. As only the couplings between the Higgses and the SM gauge bosons enter the oblique parameters, we focus on the following nine free parameters for our study of the oblique parameters:

6

Using the mixing matrices, one can obtain the couplings of physical Higgses to the gauge bosons, which are denoted by the following effective couplings:

7

where represents all possible neutral CP-even states, including h, H, and , and . The normalized couplings are shown in Table 2.

In addition, the gauge boson can couple to two different Higgs bosons: the Z boson couples to two Higgs bosons with different CP properties, and the W bosons couple to neutral and charged Higgs bosons. These interactions can be parameterized as

8

9

10

where and correspond to different types of Higgs bosons: φ includes neutral states, and ϕ includes charged Higgs ^① . Furthermore, the Higgs bosons can couple to gauge bosons via the quartic interactions, which are

11

Given the complexity of the 2HDM+S scalar sectors and the appearance of multiple mixing angles, we consider five benchmark cases to disentangle the impact of each mixing angle. For Case-0, we have all the mixing angles set to be 0, which is the 2HDM alignment limit case. For other cases, only one mixing angle is nonzero, whereas the others are fixed to 0, as shown in Table 1.

● Case-0 with is the 2HDM alignment limit, where the singlet components are decoupled, and the 125 GeV Higgs h is the same as the SM Higgs. In this case, all the couplings of the singlet Higgs bosons , to SM particles are zero, and the beyond the SM (BSM) Higgs coupling HVV is zero. However, the BSM Higgs bosons can still couple to gauge bosons via AHZ, , , HHVV, AAVV, and couplings.

● Case-I with is the 2HDM limit, when the singlet components are completely decoupled. The mixing between H and h is parameterized by α, as in the usual 2HDM.

● Case-II with represents the case when the 125 GeV h mixes with the singlet Higgs ; thus, the SM-like Higgs properties are similar to those of the singlet extended SM (SSM). However, the BSM doublet components are the same as the alignment limit of the 2HDM.

● Case-III with represents the case when the non-SM H mixes with the singlet Higgs , whereas the 125 GeV Higgs h is completely SM-like.

● Case-IV with represents the case when A mixes with the singlet pseudoscalar , where the CP-even sector is the same as the alignment limit of the 2HDM, plus a decoupled singlet scalar S.

In Table 2, we list the couplings between the Higgses and the SM gauge bosons, which are relevant for the calculation of the oblique parameters. The general expressions are given in the second column, as well as the couplings in the individual Case-0 − Case-IV. As the STU parameters only depend on couplings between the Higgses and gauge bosons, the fermionic couplings of the Higgs bosons are irrelevant in this study. Therefore, the contributions to the STU parameters are independent of the specific structure of the Yukawa couplings. In particular, when the singlet CP-odd Higgs is decoupled by , the 2HDM+S is similar to the N2HDM (the real singlet extension of the 2HDM [11]). Note that the and couplings are always zero for these benchmark cases, as multiple non-zero mixing angles are needed to couple the CP-odd singlet Higgs to the CP-even Higgs h and . In addition, the quartic coupling is zero for the benchmark cases and is non-zero only when and are both non-zero.

As the oblique parameters STU are constructed with the W and Z self-energies [21], as shown in Eqs. (A2)− (A3), they receive contributions from the Feynman diagrams in Fig. 1. The three-point vertices (including , , , and ), as well as the four-point vertices (including , , and ), contribute to the self-energies of the gauge bosons.

The contributions to the STU parameters from various Higgses can be found in Ref. [22]. Using those expressions, the STU parameters in the 2HDM+S are given by

12

13

14

where GeV is the reference mass of the SM Higgs. The functions F, G, and can be found in Eqs. (A4), (A5), and (A6) in Appendix A.

For the T parameter, the contributions from the quartic couplings are canceled out, as are the same as and the T observable is defined by the self-energy difference between the W boson and Z boson (see Eq. (A1)). Thus, the T observable only receives the contribution from the , , and couplings. Furthermore, the S parameter mainly represents the Z boson self-energy, and receives contributions from the interaction via and the interaction via . In addition, the quartic couplings , , and enter into the S parameter via the logarithmic functions. For the U parameter, the contributions of the quartic interactions are canceled out again. Furthermore, the U parameter is related to the dim-8 operator, which is usually suppressed. Therefore, in our discussion below, we mostly focus on the S and T parameters, which are more sensitive to the BSM effects.

The experimental measurements for the electroweak precision observables yield the following best-fit values of STU [28] for GeV:

15

where , , and are the correlation coefficients between S, T, and U. The contributions to the oblique parameters STU in the 2HDM+S, i.e., Eqs. (13), (12), and (14), can be used to obtain the value [26, 29],

16

where

17

The two-dimensional fit to the STU parameters at 95% C.L. corresponds to .

In the 2HDM, the STU parameters play an important role in constraining the mass splittings between the BSM neutral Higgses and the charged Higgses . In the 2HDM+S, the singlet field enters via the mixing, which further changes the dependence of the STU parameters on the model parameters. In this section, we explore the impacts of electroweak constraints on the mixing angles, , , , and , as well as various mass splittings. For convenience, we define the following mass splittings, which are relevant for the STU constraints:

18

With a general scan of the model parameters in the Higgs potential with theoretical considerations taken into account, we find that a relatively large range of mass differences is allowed, particularly with the variation of the soft breaking mass parameter in the Higgs potential.

For Case-I, we consider the non-alignment limit with all the single mixing angles set to be zero. For Cases-II − IV, we focus on the scenario with only one mixing angle set to be non-zero under the alignment limit. In this section, we consider the cases with a non-zero singlet mixing angle beyond the alignment limit.

We first explore the interplay between with the singlet− mixing . In the left plot of Fig. 8, we show the 95% C.L. STU allowed region in the vs. plane for various . For (region enclosed by the solid blue curve), is constrained to be less than 0.275, independent of . However, the singlet admixture can enlarge the allowed region in , as shown by the two elliptical rings for and . The interaction can compensate for the contribution of in Eq. (24) for larger .

In the right plot of Fig. 8, we present the 95% C.L. STU allowed region in the vs. plane for various . For increasing , the allowed region shifts to the left. For GeV, the allowed range of reduces, whereas for GeV, larger values of are allowed. For slightly above 0.275, is no longer allowed, and two branches appear.

We explore the interplay between and singlet-double CP-even Higgs H mixing in Fig. 9. The left panel shows the 95% C.L. STU allowed region in the vs. plane for (blue), (orange), and (green). The blue line in the left panel of Fig. 9 is consistent with the blue curve in the left panel of Fig. 4 (Case-I). For larger , the allowed range of shrinks for but expands for . In this case, , , and . As increases, the HVV contribution to the T observable decreases, whereas the contribution increases. Therefore, the allowed region shrinks in the dominate region ( ) and enlarges in the HVV dominate region ( ). When 800 GeV, the term plays the same role as HVV. The constraints of this point are independent of and all curves cross at . For , H becomes the pure singlet Higgs and does not contribute to the STU parameters. Therefore, the STU limit of is independent of . As the roles of H and switch when , the parameter space of vs. is the same as that of the left panel of Fig. 9 with .

The right panel of Fig. 9 presents the vs. plane for =0 (blue), 0.25 (orange), and 0.375 (green). For , almost the entire region of the parameter space is allowed, except for a small open region with relatively small and large . The allowed region reduces when increases. For , only a small region with GeV and is allowed. This is due to the increased contribution from the HVV term at larger . Only when is lighter and close to 125 GeV would the HVV contribution be small enough to be allowed. Similar to the left panel, the parameter space of vs. is the same as that of the right panel of Fig. 9 with .

We explore the interplay between and singlet-double CP-odd Higgs A mixing in Fig. 10. The left panel presents the 95% C.L. allowed region in vs. for =0 (blue), (orange), and (green). The blue region in the left panel of Fig. 10 for is consistent with the blue region in the right panel of Fig. 4. For larger , the allowed regions shift to the left, whereas the bounds at both and become larger, owing to the suppression of both the and terms by . However, the term is enhanced by , which compensates for the suppression of and . Therefore, the STU limit is relaxed faster at where is less dominant in this region. For , only the contribution is left, and the contribution from A is decoupled. The 95% C.L. allowed region for the limit is a constant and independent of . As the roles of A and switch when , the parameter space of vs. is the same as that in the left panel of Fig. 10 with .

The right panel of Fig. 10 presents the 95% C.L. allowed region in vs. for =0 (blue), 0.25 (orange), and 0.375 (green). The blue line indicates the same behavior of the STU dependence on as for . However, these two cases differ when singlet admixture enters for . The CP-odd Higgs A enters via and s, where these contributions are suppressed when is close to . In the case where , the allowed regions that only appear in the region of large are non-zero , as in this area is already dominated by the doublet properties. Similar to the left panel, the parameter space of vs. is the same as that in the right panel of Fig. 10 with .

The precision measurements of the couplings of the 125 GeV Higgs at the LHC also place strong constraints on the parameter space of the 2HDM+S, in particular, on , the singlet- mixing , and . We perform the fit for 125 GeV Higgs properties with [31−35]. In Fig. 11, we present both the 95% C.L. STU allowed region in the vs. plane for various (regions enclosed by solid curves) and the 95% C.L. allowed region by 125 GeV Higgs precision measurements for various (region enclosed by dashed curves) for Type-I (left panel) and Type-II (right panel). As the STU constraints only depend on the couplings of the Higgses with the gauge bosons, which is the same for different types of 2HDM, the solid curves are the same at both panels. For GeV, the allowed range in is independent of . This is because and contribute the same for , and is not constrained as shown in the left plot of Fig. 5. For GeV, a larger region of can be accommodated for , as a larger can be compensated for by with lighter . In contrast, for GeV, the allowed region in shrinks, where and have the same sign in this mass region, which leads to tighter constraints.

For the Higgs precision on the Type-I 2HDM+S in the left panel, the allowed range of becomes weaker for larger . For , the electroweak precision measurements provide a stronger constraint on in the negative region, whereas the Higgs precision measurements constrain the value of better for GeV. For , the STU constraint on positive can be stronger than the Higgs precision measurement at . Thus, in the Type-I model would be constrained to be less than 0.3 by the coupling measurements, where the STU can provide a stronger limit for GeV.

For the Higgs precision on the Type-II 2HDM+S in the right panel, the allowed region in is constrained to be much tighter, to only a thin region around . The constraints from the coupling measurements are the weakest at and become stronger as increases or decreases. is constrained to be approximately 0.4, which is less dependent on the values of . The electroweak precision measurements provide a tight bound on the range of for GeV. A combination of the electroweak precision measurements and the Higgs precision measurements could help us narrow down the parameter space of the 2HDM+S.

We studied the implications of the oblique parameters, in particular, the T parameter, on the parameter space of the 2HDM+S model. Nine model parameters enter, including five masses , , , , and and four mixing angles , , , and . To systematically study the impact of each mixing angle, we identified five benchmark scenarios, Case-0 with and all the singlet mixing angles being 0 (the 2HDM alignment limit), and Cases-I to IV with only one of the mixing angles being non-zero. We studied the 95% C.L. STU allowed region in the relevant parameter spaces. We observed that

● Case-0

Other than the well known conclusion that the electroweak precision constraints are satisfied for or , there is an upper limit on the mass splitting of 900 GeV for GeV and , coming from the S parameter constraint. This upper limit also varies with .

● Case-I with

The constraint on is weak for GeV, , or and GeV. The parameter space in or is significantly reduced for or away from 0.

● Case-II with

is unconstrained for GeV and 0. The allowed region shifts to larger and for .

● Case-III with

The STU constraint can be satisfied for or .

● Case-IV with

The STU constraint can be satisfied for or .

We further explored Cases-II−IV with non-zero and observed that the singlet extension could compensate for the contribution and extend the allowed parameter space. However, a larger typically leads to more constrained mass vs. mixing angle parameter space.

We also studied the complementarity between the electroweak precision analyses and Higgs coupling measurements. We observed that, for the Type-I scenario, the electroweak precision measurements provide stronger constraints on for GeV, whereas the Higgs coupling measurements constrain better for . For the Type-II scenario, the electroweak precision measurements provide a tight bound on the range of for GeV, whereas the Higgs coupling measurements constrain better for all values of .

In summary, the singlet extension of the 2HDM opens up the allowed parameter space when constraints from the electroweak precision measurements are considered. It also provides a complementary reach when combined with Higgs precision measurements. While our study examined benchmark scenarios with only one singlet mixing angle being nonzero, it identified the main features of each mixing case and provided a more comprehensive understanding in the most general mixing cases. Note that we adopted the set of the model parameters including physical Higgs masses and mixing angles. When studying a particular 2HDM+S scenario with a specific symmetry assumption of the Higgs potential, theoretical considerations might restrict the range of the mixing angles and mass differences. Our analyses were performed in a model independent way so that it is straightforward to map our results to a particular 2HDM+S model with a restricted range of mixing angles and mass differences.

The STU observables are defined by

A1

A2

A3

where the F, G, and functions are defined as [22]

A4

A5

A6

with

A7