Asymmetric di-Higgs signals of the next-to-minimal 2HDM with a $U(1)$ symmetry

The two-Higgs-doublet model with a $U(1)_H$ gauge symmetry (N2HDM-$U(1)$) has several advantages compared to the ``standard'' $Z_2$ version (N2HDM-$Z_2$): It is purely based on gauge symmetries, involves only spontaneous symmetry breaking, and is more predictive because it contains one parameter less in the Higgs potential, which further ensures $CP$ conservation, i.e., avoiding the stringent bounds from electric dipole moments. After pointing out that a second, so far unknown version of the N2HDM-$U(1)$ exists, we examine the phenomenological consequences for the Large Hadron Collider (LHC) of the differences in the scalar potentials. In particular, we find that while the N2HDM-$Z_2$ predicts suppressed branching ratios for decays into different Higgs bosons for the case of the small scalar mixing (as suggested by Higgs coupling measurements), both versions of the N2HDM-$U(1)$ allow for sizable rates. This is particularly relevant in light of the CMS excess in resonant Higgs-pair production at around $650\,$GeV of a Standard Model Higgs boson subsequently decaying to photons and a new scalar with a mass of $\approx90\,$GeV subsequently decaying to bottom quarks (i.e., compatible with the CMS and ATLAS $\gamma\gamma$ excesses at $95\,$GeV and $\approx 670\,$GeV). As we will show, this excess can be addressed within the N2HDM-$U(1)$ in case of a nonminimal Yukawa sector, predicting an interesting and unavoidable $Z+ b\bar b$ signal and motivating further asymmetric di-Higgs searches at the LHC.


I. INTRODUCTION
The discovery of the Brout-Englert-Higgs boson [1][2][3][4][5][6] by ATLAS [7] and CMS [8] established, for the first time, the existence of a fundamental scalar particle within the Standard Model (SM).This observation motivates the existence of more scalar particles and, in turn, the experimental search for them.While the 125 GeV Higgs (h) boson has approximately SM-like properties [9][10][11][12][13][14], this does not exclude the existence of additional scalar bosons, as long as their role in the breaking of the electroweak (EW) gauge symmetry and the mixing with the SM-like Higgs boson are sufficiently small.
In this context, strong constraints on new physics (NP) models are provided by the ρ parameter that relates the EW gauge couplings to the W and Z masses and is defined to be unity in the SM at tree level.This singles out models with SU (2) L -singlet or SU (2) L -doublet extensions of the SM Higgs sector whose vacuum expectation values (VEVs) conserve the custodial symmetry, such that the additional scalars only give loop-level effects in the ρ parameter. 1 The most studied extensions of the SM scalar sector are the two-Higgs-doublet models (2HDMs) [16][17][18][19].
Here, usually a discrete Z 2 symmetry is imposed to both solve the problem of flavor changing neutral currents [20,21] (resulting in four different versions with natural flavor conservation [22,23]) and to provide (accidental) CP conservation in the Higgs potential.However, for phenomenological reasons, i.e., to give VEVindependent masses to the additional scalars, the Z 2 symmetry must be broken.In order to avoid domain wall problems caused by a spontaneous discrete symmetry breaking [24], the Z 2 symmetry is usually softly broken by a dimension-two term [25].This operator (in case of a nonvanishing and non-aligned λ 5 term) in general gives rise to CP violation within the Higgs sector, with potentially dangerously significant effects in electric dipole moments [26].
Reference [27] proposed to solve these problems by replacing the discrete Z 2 symmetry with a U (1) H gauge symmetry, which can mimic the effect of the Z 2 symmetry but forbids the explicit soft-breaking term.However, if the Z ′ boson originating from the U (1) H gauge is required to be heavier than the EW scale, one has to supplement the model with an additional scalar charged un-der U (1) H ; minimally a complex scalar ϕ that is a singlet under the SM gauge group.Because its CP -odd component becomes (in the vanishing scalar mixing limit) the longitudinal component of the Z ′ , the scalar potential effectively resembles the one of the Next-to-Minimal 2HDM (N2HDM) with a real scalar (see, e.g., Refs.[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]).In particular, the VEV of ϕ gives rise to the m2 12 term that softly breaks the Z 2 symmetry.
This aspect is now more relevant in light of the ongoing and intensified Large Hadron Collider (LHC) searches for new scalar bosons (see, e.g., Ref. [72] for a recent review).While no unequivocal evidence for a new particle has been observed, interesting hints for new scalars with masses around 95 GeV [46,[73][74][75][76][77][78][79][80][81][82][83][84][85][86][87], 151 GeV [88][89][90][91][92] and 670 GeV [76,[93][94][95] have been reported. 4In particular, the CMS excess for a ≈ 650 GeV scalar decaying into a ≈ 90 GeV one (i.e., compatible with the 95 GeV hints mentioned above due to the limited detector resolution for bottom jets) and a SM Higgs [76] needs multiple new Higgs bosons with a mass hierarchy (which is only possible if they are not within the same SU (2) L multiplet while respecting perturbativity bounds) 56 .Furthermore, as it is an asymmetric di-Higgs decay, it requires interactions among the three different scalars.Here, asymmetric decay means the decay of a heavy scalar into two different lighter scalars.While the rates of such asymmetric di-Higgs decays are in general small in the MSSM [100,101], 2HDMs [102] and also in the N2HDM-Z 2 [103], we will show that for the N2HDM-U (1) they are naturally siz-able.

II. THE MODEL
As outlined in the Introduction, a Z 2 symmetry is commonly used to construct the four versions of the 2HDMs with natural flavor conservation and, at the same time, constrain the scalar potential.In the N2HDM, even two Z 2 symmetries are usually employed to prevent tree level flavor changing neutral currents and eliminate most sources of CP violation.We want to use instead a single U (1) H gauge symmetry under which at least two of the scalar fields are charged.
We start with the scalar potential for the two SU (2) L doublets H 1 and H 2 with hypercharge 1/2 (where according to the usual 2HDM conventions H 2 contains most of the SM Higgs for the case of small mixing angles).If the U (1) H charges of H 1 and H 2 are different, operators with an odd number of these fields are forbidden, leading to This potential is CP conserving as it does not contain the soft-breaking term Next, we add a complex scalar SM singlet ϕ that is charged under the U (1) H gauge symmetry.Its selfinteractions, as well as the ones with two identical doublets are allowed independently of the U (1) H charges.In addition, there are two options for charge assignments under the U (1) H symmetry (for or ϕ replaced by ϕ † , depending on the sign on the is gauge invariant.Case (a) was already proposed in Ref. [27], while case (b) is novel, to the best of our knowledge.
Note that we have normalized the prefactors of these potentials in such a way, that once we decompose η S (mostly) becomes the longitudinal mode of the Z ′ and the terms involving Ŝ match the N2HDM-Z 2 .Here, v S is the VEV of ϕ, and one can choose it to be real and positive without loss of generality.Therefore, disregarding the Z ′ boson, which could be heavy or weakly coupled, the N2HDM-U (1) resembles the N2HDM-Z 2 with the important differences that the m 2 12 and λ 5 terms are only effectively generated by v S and Z ′ exchange, respectively, similar to the µ term in the next-to-minimal supersymmetric Standard Model (NMSSM) [104][105][106][107][108][109].Moreover, this guarantees the absence of CP violation in the scalar potential (even when the Z ′ is integrated out), while this, in general, is inevitable in both the N2HDM-Z 2 and NMSSM.
We know from Higgs signal strength measurements that the mixing among the SM-like Higgs boson and the other two CP -even scalars should be rather small.Therefore, we will label the CP -even mass eigenstates, contained in the absence of mixing within H 2 , H 1 , and ϕ as h, H, and S, respectively. 7Importantly, the mixing among H, h and S in the N2HDM-U ( 1) is related to the masses m H , m H ± , and m A (where H ± and A denote the charged and CP -odd Higgs boson, respectively) because they all involve the effective m 2  12 term originating from µ or λ ϕ12 .This means that the effective m 2  12 term automatically leads to H-S mixing as can be inferred from the CP -even mass matrix (in the large tan β limit, i.e., assuming v 1 to be small) where tan β = v 2 /v 1 and Note that the effects of λ 1 , λ 3 , λ 4 and λ ϕ1 on the CP -even Higgs masses become negligible in the large tan β limit.The full expressions for the minimization, the mass matrices, etc., can be found in Appendix A.
Concerning the fermion Yukawa sector, the most natural choice is probably to assume that the SM fermions are uncharged under U (1) H , or to assign equal charges to left-handed and right-handed fields (such as B − L or L µ − L τ ) in order to avoid gauge anomalies.In this setup, the doublet H 2 would be U (1) H neutral, while H 1 carries some U (1) H charge Q H .This then leads to a type-I Yukawa sector, which also has the advantage of being quite unconstrained in the large tan β and small α (Higgs mixing angle) limit.However, also the other three types of 2HDMs with natural flavor conservation, as well as the generic type-III model,8 can be obtained even in an anomaly free fashion if the fermion sector is extended [27].The discovery of this process, for which the CMS measurement constitutes a first hint, would be a smoking gun for the N2HMD-U (1).Here, the black circle denotes the loop induced effective coupling to gluons.However, note that the heavy top limit cannot be used because mt ≪ mH and we use the expression for a dynamical top quark in our numerical analysis.

III. PHENOMENOLOGY
The N2HDM-U ( 1) is in general more predictive than the N2HDM-Z 2 as it contains one parameter less and has no sources of CP violation in the Higgs potential.However, what is the most striking difference regarding LHC observables between the different N2HDMs, even when the Z ′ predicted by the N2HDM-U (1) is disregarded, as it might be heavy or weakly coupled?
To answer this, let us consider the limit of vanishing mixing among the neutral CP -even scalars, in which h is purely SM-like, H only couples to W ± H ∓ and ZA, and S is sterile.Now, µ (λ ϕ12 ) in Eq. (3) (Eq.( 4)) has to be nonvanishing to give masses to H, A, and H ± that are above the EW scale, i.e., assuming λ 4 to be small.This then, at the same time, suppresses H-h and H-S mixings, and from Eq. ( 6) we see that h-S mixing can be avoided, for large tan β with λ ϕ2 = 0.This means that for m H ≫ v the only unsuppressed decay of H, in the absence of Yukawa coupling of H 1 , is H → Sh for case (a), and in addition H → SS for case of (b), if m H ≫ m S and m H ± ≈ m H . Therefore, in the large tan β limit, the N2HDM-U (1) predicts sizable branching ratios for H → Sh (and also H → SS in case (b)).As this decay in the N2HDM-Z 2 is suppressed by small mixing angles, the discovery of an asymmetric di-Higgs signal would be a smoking gun for the N2HDM-U (1). 9 Let us now illustrate this observation more quantitatively in the context of the hint for the ≈ 650 GeV boson decaying into a ≈ 90 GeV scalar and the SM Higgs boson with a global (local) significance of 2.8 σ (3.8 σ) [76].
Here, the ≈ 90 GeV resonance decays into b b and the SM Higgs boson into γγ.Because the detector resolution for bottom jets is not so good, this ≈ 90 GeV excess could be compatible with the γγ [74], τ τ [75] and the LEP ZH measurement [73] pointing towards a mass of ≈ 95 GeV, as well as with the γγ [94] and ZZ excesses [93] at around ≈ 670 GeV.This makes this asymmetric di-Higgs signal particularly interesting and effectively eliminates the look-elsewhere effect of the CMS di-Higgs analysis.
The cross section for the pp → b bγγ excess is ≈ 0.35 +0.17 −0.13 fb according to the CMS analysis [76].However, the CMS analysis of pp → b bτ τ [115] finds an upper limit on the corresponding cross section of ≈ 4 fb for ≈ 650 GeV boson search with m bb ≈ 90 GeV.With BR(h → τ τ )/BR(h → γγ) ≈ 20, one can obtain ≈ 0.2 fb as the upper bound on pp → b bγγ, and this translates into the limit σ(pp → (650) → (95) h)×BR((95) → b b) ⪅ 90 fb, taking into account that BR(h → γγ) ≈ 0.23%.Therefore, the b bγγ excess cannot be fully explained, but it is still possible to account for it within 2 σ and we will aim at There are two options within the N2HDM-U (1); one can A) identify the ≈ 95 GeV state with H and the ≈ 650 GeV one with S (pp → S → Hh) or B) vice versa (pp → H → Sh).However, in case A), the µ term is naturally small because H is light, such that also the branching ratio is suppressed, unless one chooses very small mixing angles among the CP -even scalars.Let us, therefore, consider option B), i.e., pp → H → Sh (shown in Fig. 1) in the following, again in the limit of small mixing and large tan β.To obtain a sufficient production cross section of H we will consider the case of a nonminimal flavor structure and assume that H has a (effective) coupling to top quarks originating from the Lagrangian term − Ỹ t QL H1 t R . 10This coupling then also leads to unsuppressed decays of H → t t (and A → t t).
For the numerical analysis we use that a SM Higgs boson with a mass of ≈ 650 GeV would have a gluon fusion production cross section of ≈ 1.35 pb at NNLO [118][119][120][121][122][123].This means that a coupling to top quarks is needed, that is around one quarter of the one of the SM Higgs boson, i.e., Ỹt ≈ Y t /4/( BR(H → Sh) BR(S → b b)).Therefore, assuming that S decays SM-like 11 (BR(S → b b) ≈ 10 This coupling can be induced at tree level by vector like quarks mixing with SM ones via coupling to S. Alternatively, an effective coupling to gluons could be loop induced by colored new heavy fermions or scalars.In fact, CMS observed an excess in di-di-jet searches [116] that point towards new colored particles at the TeV scale [117]. 11This procedure is justified since the other couplings of S are given  8) is explained.The gray region is excluded by the requirement of perturbative couplings, while the red region is excluded by the t tt t search [132], assuming mA ≈ mH .Note that the b bγγ excess cannot be explained in the top-right region of the green dashed line.0.8) results with Eq. ( 8) in σ(pp → H) ≈ 84 fb/BR(H → Sh).Based on the Goldstone boson equivalence theorem [124,125], we also expect BR(A → SZ) ≈ BR(H → Sh) leading to pp → A → ZS → Zb b (and also pp → A → Zh → Zb b) with cross sections around 1.5 × 70 fb, 12searched for by ATLAS [127][128][129] and CMS [130,131].Note that in fact, Ref.
[131] finds a mild excess within the relevant region.
Furthermore, we can predict the cross section of H → t t and A → t t, as well as pp → t tH → t tt t and pp → t tA → t tt t as a function of tan β and v S (assuming λ ϕ2 = 0 as well as m H ≈ m A ) and compare this to the limits on the resonant t t production of CMS [133] and ATLAS [134] as well as to t tt t production measured by CMS [135] and ATLAS [132].This is illustrated in Fig. 2 where we show the predicted cross section for pp → A → t t in units of pb as a function of tan β and v S .The red region is excluded by the pp → t tt t search of AT-LAS and the gray region by the requirement of positive eigenvalues of the mass matrix as well as perturbative couplings.Since σ(pp → H → Sh) ≈ 84 fb is required, when BR(S → γγ) ≈ 0.15% (again assuming that S has by the mixing with the other scalars and hence aligned to the SM-like coupling structure. SM-like branching ratios) we obtain for the inclusive cross section σ(pp → S + X → γγ + X) ≈ 0.1 fb which is compatible with the current limits but insufficient to explain the γγ excess at 95 GeV of ≈ 50 fb [74].Therefore, direct production of S would be required in addition to explain the γγ excess, e.g., via gluon fusion from the mixing of S with H or h [82].We would like to emphasize again that although the current excesses are not fully explained in the minimal model, the asymmetric di-Higgs decay is a key prediction and further experimental analyses are encouraged.

IV. CONCLUSIONS
While Higgs physics in the N2HDM with two discrete Z 2 symmetries (N2HDM-Z 2 ) has been studied in detail in the literature, this phenomenological aspect of the N2HDM with a U (1) H gauge symmetry has received little to no attention so far.While both versions have desirable features such as natural flavor conservation, there are even several advantages of the N2HDM-U (1) H over the N2HDM-Z 2 : • Only one U (1) H gauge symmetry is needed instead of two Z 2 symmetries.
• Like the SM, the N2HDM-U (1) H is built on local gauge symmetries and spontaneous symmetry breaking (i.e., unlike the N2HDM-Z 2 no softbreaking is needed).
• The N2HDM-U (1) H symmetry is more predictive than the N2HDM-Z 2 because it contains one parameter less.
If the Z ′ boson is decoupled from phenomenology, either because it is heavy or weakly interacting, the scalar sector of the N2HDM-U (1) H is close to the one of the N2HDM-Z 2 , however, there are important differences: • In the N2HDM-U (1) no λ 5 term is allowed, leading to CP conservation.This feature is even conserved when the Z ′ is integrated out because of an automatic phase alignment avoiding potentially dangerous effects in electric dipole moments..
• The m 2 12 term is absent before spontaneous symmetry breaking and induced by the VEV of ϕ, either from the term µH † 1 H 2 ϕ or λ ϕ12 (H † 1 H 2 )ϕ 2 , depending on the charge assignment.Please note that the latter option was, to the best of our knowledge, not proposed before in the literature.
• While in N2HDM-Z 2 , if H is heavy, only symmetric decays into Higgs pairs; i.e., H → SS and H → hh are possible in the limit of zero mixing, in the N2HDM-U (1) one expects naturally large branching ratios for H → Sh.Note that while in case (a), only asymmetric decays are unsuppressed, in case (b) also decays to identical scalars (e.g., H → SS) can be sizable.
The last difference has important implications for the asymmetric ≈ 650 GeV excess in b bγγ.While even if H is equipped with a sufficiently high production cross section (e.g., from direct top-quark Yukawa couplings of H 1 ), the N2HDM-Z 2 could not account for the preferred central value of the measurement as BR(H → Sh) could not be sizable enough, taking into account the limits on the scalar mixing from Higgs coupling strength measurements at the LHC.However, the N2HDM-U (1) can address this measurement, predicting signatures in pp → H(A) → t t, pp → t tH(A) → 4t and pp → A → SZ, not far away from the current experimental limits.
Finally, let us point out that Z-Z ′ mixing, in general present in this model, can naturally account for the higher than expected value of the W mass [136], as suggested by the measurement of the CDF-II Collaboration [137].Together with the previous arguments this strongly motivates detailed studies of the N2HDM-U (1) (including its Higgs sector) which, in our opinion, should be considered to be (at least) at the same level of interest as the standard N2HDM-Z 2 and therefore be examined with the same scrutiny theoretical and experimental in the future, motivating more asymmetric di-Higgs searches at the LHC.
N ote Added-Recently, CMS presented updated results for low mass searches for new scalars decaying into γγ [138], confirming the previous excess.The minimization conditions are where λ 345 = λ 3 + λ 4 + λ eff 5 , and λ eff 5 ̸ = 0 is only generated if the Z ′ is integrated out.The scalar squared-mass matrices are which are defined via the bilinear potential terms The eigenvalues of the CP -odd and charged-Higgs masses are then given by The minimization conditions in this case are and the squared-mass matrices are given by (A12) The eigenvalues of the CP -odd and charged-Higgs masses are then given by )

FIG. 1 .
FIG. 1. Feynman diagram showing resonant asymmetricHiggs pair production.The discovery of this process, for which the CMS measurement constitutes a first hint, would be a smoking gun for the N2HMD-U (1).Here, the black circle denotes the loop induced effective coupling to gluons.However, note that the heavy top limit cannot be used because mt ≪ mH and we use the expression for a dynamical top quark in our numerical analysis.

FIG. 2 .
FIG.2.Predictions for σ(pp → A → t t)[pb] as a function of tan β and vS in the N2HMD-U (1) for case (a), assuming that the CMS excess b bγγ in Eq. (8) is explained.The gray region is excluded by the requirement of perturbative couplings, while the red region is excluded by the t tt t search [132], assuming mA ≈ mH .Note that the b bγγ excess cannot be explained in the top-right region of the green dashed line.