A short overview on low mass scalars at future lepton colliders

In this manuscript, I give a short summary on scenarios with new physics scalars that could be investigated at future $e^+e^-$ colliders. I concentrate on cases where at least one of the additional scalar has a mass below 125 GeV, and discuss both models where this could be realized, as well as studies which focus on such scenarios. This work is based on several overview talks I recently gave at the CEPC workshop, FCC week and ECFA future collider workshop, as well as a Snowmass White Paper.


I. INTRODUCTION
The discovery of a scalar which so far largely agrees with predictions for the Higgs boson of the Standard Model (SM) has by now been established by the LHC experiments, with analyses of Run II LHC data further confirming this.In the European Strategy Report [1,2], a large focus was put on future e + e − colliders, especially so-called Higgs factories with center-of-mass (com) energies around 240 − 250 GeV.While these will on the on hand further help to determine properties of the scalar discovered at the LHC, and especially will help to determine in detail the parameters and shape of the scalar potential, it is also interesting to investigate theis potential to search for additional scalar states.Many new physics models still allow for extra scalar states, including those which have masses 125 GeV.
In this whitepaper, I give a short summary of presentations I gave at various recent meetings and workshops.I give an overview on some models that allow for such light states, and point to phenomenological studies investigating such models.This should be viewed as an encouragement for further detailed studies in this direction.

II. MODELS A. Singlet extensions
In singlet extensions, the SM scalar potential is enhanced by additional scalar states that are singlets under the SM gauge group.In such scenarios, the coupling of the novel scalar to SM particles is typically inherited via mixing, i.e. mass-eigenstates are related to gauge eigenstates via a unitary mixing matrix.The corresponding couplings and interactions are mediated via a simple mixing angle.
In [3], the authors present the status of current searches for the process which for such models can be read as a bound in We display these results in figure 1.
We show an example of the allowed parameter space in a model with two additional singlets, the two real scalar extension studied in [4].In this model, three CP-even neutral scalars exist that relate the gauge and mass eigenstates h 1,2,3 via mixing.One of these states has to have couplings and mass complying with current measurements of the SM-like scalar, the other two can have higher or lower masses.In figure 2, we show two cases where either one (high-low) or two (low-low) scalar masses are smaller than 125 GeV.On the y-axis, the respective mixing angle is shown.Complete decoupling would be designated by sin α = 0 in the notation used in this figure.
FIG. 3. Allowed regions in the 2HDM, from a scan presented in [14].

B. Two Higgs Doublet Models
Two Higgs doublet models (2HDMs) constitute another example of new physics models allowing for low mass scalar states.A general discussion of such models is e.g.given in [13] and will not be repeated here.In general, such models contain, besides the SM candidate, two additional neutral scalars which differ in CP properties as well as a charged scalar, so the particle content is given by h, H, A, H ± , where one of the two CP-even neutral scalars h, H needs to be identified with the 125 GeV resonance discovered at the LHC.Couplings to the fermions in the Yukawa sector distinguish different types of 2HDMs.
In [14], the authors perform a scan including bounds from theory, experimental searches and constraints, as e.g.electroweak observables, as well as B-physics.Examples for these scan results are shown in figure 3, taken from that reference.We see that for all regions solutions for either one or several low mass scenarios exist and are viable for the constraints discussed in that reference.Unfortunately, the information about the cos (β − α) regions in these scenarios is not available in that reference.Depending on the Yukawa couplings considered, the limits on the absolute value of this rescaling angle vary between 0.05 and 0.25 [15].

C. Other extensions
The scalar sector of the SM can be extended by an arbitrary number of additional scalar fields, such as singlets, doublets, etc.One option which is also often consider is the extension of this sector by both singlets and doublets.

N2HDM
In [16], the authors consider a model where the SM scalar sector is extended by an additional doublet as well as a real singlet.This model has 3 CP even neutral scalar particles, out of which one needs to have the properties in compliance with LHC measurements of the 125 GeV scalar.The authors perform an extensive scan and find regions in parameter space where either one or both of the additional scalars have masses below 125 GeV.We show an example of the allowed parameter space in figure 4 2. Lepton-specific IDM In [17], the authors consider a model where the SM scalar sector is augmented by an additional doublet, where they impose an exact Z 2 symmetry.This symmetry is then broken by a specific coupling to the fermionic sector.The authors identify regions in the models parameter space that agree with current searches as well as anomalous magnetic momenta of electron and muon.They identify regions in the models parameter space where the second CP-even scalar can have a mass 30 GeV.We display these regions in figure 5. Allowed regions in the parameter space of the model discussed in [17], taken from that reference, where squares denote allowed and bullets excluded regions in the models parameter space.CP-even neutral scalars with low masses are viable within this model.

Scalar triplet model
Finally, we want to discuss a model containing scalar triplets, leading to a rich particle content as well as the possibility of CP violating terms.The model has been presented in [18].This model contains 5 neutral, 3 charged, and 2 doubly charged mass eigenstates.The authors present regions in parameter space where masses for some of these can be 125 GeV.We display these results in figure 6.

III. STUDIES AT 90 GEV
For this center-of-mass energy, several searches exist which have already been performed at LEP and are summarized in [19,20], concentrating on Z h, h 1 h 2 , and h 1 h 1 h 1 final states, where h i signifies novel scalars.Possible new studies could build upon these searches.We want to note that the luminosity at FCC-ee and CEPC at this center of mass energy is exceeding LEP luminosity by several orders of magnitude [21,22].
We also want to present one specific study which investigates several composite models at FIG. 6. Allowed regions in the parameter space of the model discussed in [18], taken from that reference.For neutral and charged new scalars, masses 125 GeV are achievable.
a com energy of 91 GeV [23].The authors consider the process e + e − → + − τ + τ − , where the tau-pair stems from an additional pseudoscalar a radiated off one of the fermion lines in the pair-production.They apply a cut-based study as well as an improved analysis using machine learning techniques; for the latter, the authors are able to achieve a 3 σ exclusion for benchmarks with masses M a ∼ 20 GeV.We display event rates for the various benchmark scenarios in figure 7. Cross section for light SM-like scalar at e+e-, sqrts=250 GeV Zh ν ν h ν ν h via Zh FIG. 8. Leading order production cross sections for Z h and h ν ν production at an e + e − collider with a com energy of 240 GeV(left) and 250 GeV (right) using Madgraph5 for an SM-like scalar h.Shown is also the contribution of Z h to ν ν h using a factorized approach for the Z decay.

IV. STUDIES AT 240-250 GEV
Throughout this work, we show for reference leading order predictions for Zh production at e + e − colliders for low mass scalars which are SM-like.These results were obtained using Madgraph5 [24] and are given for approximate reference only.We also display the VBF-type production of e + e − → h ν ν .Note that the latter signature also contains contributions from Z h production, where Z → ν ν .
Figure 8 shows the production cross sections for these processes for a center-of-mass energy of ∼ 240 − 250 GeV.Using these predictions, and taking into account standard rescaling ∼ 0.1, around 10 5 − 10 6 events could be produced with ILC, FCC-ee, and CEPC design luminosities [21,22,25].
A. Dedicated studies

Light scalars in Zh production
Not many dedicated studies exist that investigate low-mass scalars at Higgs factories.We here point to a study [26] that investigates the sensitivity of the ILC for low-mass scalars in Z h production, either using pure Z recoil ("recoil method") or taking the light scalar decay into b b into account.The y-axis shows the 95 % CL limit for agreement with a background only hypothesis, which can directly be translated into an upper bound on rescaling.The authors validate their method by reproducing LEP results [20,27] for these channels prior to applying their method to the ILC.Their predictions are shown in figure 9.A more detailed study along similar lines using the recoil method only and comparing different detector options has been presented recently in [28].We display their results in figure 10.The authors perform their analysis in a model where the coupling of the new resonance is rescaled by a mixing angle sin θ; therefore, their results can directly be compared with the ones presented in [26] and figure 9.
95 % confidence bounds on branching ratios for Higgs decay into a pair of lighter particles, for a com energy of 240 GeVand L = 5 ab −1 .Taken from [29].

Higgsstrahlung and decay into two light scalars
In [29], the authors consider Higgs-strahlung at a 240 GeVe + e − collider, where the Higgs subsequently decays into two light scalar states.The give 95 % confidence level bounds for the branching ratios into the decay productions of the two light scalars as a function of the light scalar masses for an integrated luminosity of L = 5 ab −1 following a detailed study.Their results are subsequently used by many authors as standard reference.We show their results for various channels in figure 11.
A more recent study [30] investigates the same process, but for 4 τ final states, for the same center-of-mass energy and integrated luminosity.The results, for varying values of tracking efficiency, are shown in figure 12.Note that curent constraints on the invisible branching ratio of the Higgs, the signal strength, as well as SM-like decays of the light scalars currently render a bound 10 −3 .
Several works make use of the bounds derived in [29].In [31], the authors investigate the allowed parameter space in the scNMSSM, an NMSSM extension that relaxes unification requirements at the GUT scale [32][33][34][35], also known as NUHM, which contains in total 5 scalar particles; if CP is conserved, 3 are CP-even and 2 CP-odd.The authors investigate various bounds on the models parameter space, and show the allowed scan points and predictions for the above channels for various scalar combinations.We show their results in figure 13.Taken from [30].Finally, in simple singlet extensions it is possible to test regions in the models parameter space which can lead to a strong first-order electroweak phase transition.Several authors have worked on this; we here show results from [36], where in addition several collider sensitivity projections are shown, including the bounds derived in [29].From figure 14, it FIG.14. Expected bounds on Higgs production via Higgs strahlung and subsequent decay into two light scalars, in the singlet extension scenario discussed in [36].The blue band denotes the region where a strong first-order electroweak phase transition is possible.We see that e + e − Higgs factories are required on order to confirm or exclude such scenarios.Figure taken from [36].
becomes obvious that e + e − Higgs factories would be an ideal environment to confirm or rule out such scenarios.
Related work, with a spontaneous breaking of the imposed Z 2 symmetry, has been presented in [37].

Other channels
In [38], the authors investigate a slightly different channel, i.e. tau pair-production where a light pseudoscalar is radiated off one on the outgoing fermion lines and decays again into τ + τ − , leading to a 4 τ final state.They are investigating this within a type X 2HDM, which in addition allows them to explain the current discrepacy between theoretical prediction and experiment for the anomalous magnetic momentum of the muon.They perform a detailed study including background and determine 2 and 5 σ countours in the m A , tan β plane, where tan β denotes the ratio of the vevs of the two doublets.Their results are shown in figure 15.
It is also interesting to investigate models with give the possibility of light charged scalars.A corresponding study has been performed in [39], where the authors consider charged scalar

B. Cross section predictions
Inspired by possible low-mass excesses in at LEP [20] and CMS [40], in [41] several models are fitted to these excesses that contain singlet and doublet extensions of the SM scalar sector; in particular, they consider models with an additional doublet as well as a (complex) singlet, labelled N2HDM and 2HDMs, respectively.For both models, as well as varying tan β ranges (where tan β denotes the ratio of the vevs in the 2HDM part of the models), the authors consider the possibility to explain the observed accesses and give rate predictions for a 250 GeV collider with a total luminosity of L = 2 ab −1 .We display their results in figure 17.We see that also other final states for the h decay, as e.g.τ + τ − , gg, or W + W − can render sizeable rates.

V. OTHER CENTER OF MASS ENERGIES
The FCC-ee and CEPC colliders are supposed to also run with a center-of-mass energy of ∼ 160 GeV, already tested at LEP.In analogy to figure 8, in figure 18 we show again cross section predictions for the process Z h in dependence of the mass of h, assuming a SM-like scalar.Note we here assume onshell production of Z h, which leads to a hard cutoff for M h ∼ 70 GeV.Detailed studies should in turn assume contributions from offshell Zs and hs as well.
We see that for this lower com energy, there is basically no contribution to the ν ν h final state that does not originate from Z h.Using FCC-ee target luminosity for this energy, and again assuming a general suppression factor ∼ 0.1 stemming from signal strength, we expect up to 10 6 events depending on the mass of the additional scalar.
For this center-of-mass energy, several searches exist which have already been performed at LEP and are summarized in [19,20], concentrating on Z h, h 1 h 2 , and h 1 h 1 h 1 final states, which could be further pursued in future collider studies.We want to note that the luminosity at FCC-ee at this center of mass energy is exceeding LEP luminosity by several orders of magnitude.
Finally, we present a study that investigates various types of 2HDMs containing several neutral scalars [42], for a collider energy of 350 GeV.The authors perform a scan of the allowed parameter space and render predictions for the Higgs-strahlung process as well as ν ν h final states with the scalar decaying into b b pairs.We show their results in figure 19.

VI. CONCLUSION
In this short note, I have presented several models and searches that investigate the sensitivity of future e + e − machines for scalars with masses 125 GeV.This corresponds to a summary of several talks I have recently given and is thereby not meant to be inclusive.
I have pointed to models that allow for such light scalars, as well as several references that either provide rates or pursue dedicated studies.I have also pointed to the connection of low-scalar searches at such colliders and the electroweak phase transition within certain models.My impression is that further detailed studies are called for, with a possible focus on so-called Higgs factories with center-of-mass energies around 240-250 GeV.

FIG. 1 . 3 FIG. 2 .
FIG.1.Limits on the process in eqn (1, taken from[3].This displays current constraints which can especially be easily reinterpreted in extended scalar sector models, in particular models where couplings are inherited via a simple mixing angle.

FIG. 4 .
FIG.4.Scan results in the N2HDM, taken from[16].There are regions in the models parameter space where either one or two of the additional scalars have masses 125 GeV.
FIG.5.Allowed regions in the parameter space of the model discussed in[17], taken from that

FIG. 13 .
FIG. 13.Allowed rates for various Higgsstrahlung processes with successive decays into two light scalars.Top: 2 b 2 τ final state.Bottom: 4τ final state.Also shown are expected upper bounds for various collider machines, with projections from [29]. Figure taken from [31].
FIG. 15.Exclusion and discovery regions in the 2HDM type X model, in the m A , tan β plane.The color region additionally explains the current g µ − 2 discrepancy.Taken from[38].

FIG. 17 .
FIG. 17. Left: Points in the 2HDMs that agree with both CMS and LEP excess and which can be probed at the ILC.Right: predicted rates in the 2HDMS and N2HDM at 250 GeVusing full target luminosity.
FIG.18.As figure18, for a com of 160 GeV.We assume onshell final states.