Flavorful Two Higgs Doublet Models with a Twist

We explore Two Higgs Doublet Models with non-standard flavor structures. In analogy to the four, well studied, models with natural flavor conservation (type 1, type 2, lepton-specific, flipped), we identify four models that preserve an approximate $U(2)^5$ flavor symmetry acting on the first two generations. In all four models, the couplings of the 125 GeV Higgs are modified in characteristic flavor non-universal ways. The heavy neutral and charged Higgs bosons show an interesting non-standard phenomenology. We discuss their production and decay modes and identify the most sensitive search channels at the LHC. We study the effects on low energy flavor violating processes finding relevant constraints from $B_d$ and $B_s$ meson oscillations and from the rare decay $B_s \to \mu^+ \mu^-$. We also find that lepton flavor violating $B$ meson decays like $B_s \to \tau \mu$ and $B \to K^{(*)} \tau \mu$ might have branching ratios at an observable level.

original twin Higgs model includes light twin fermions and a massless twin photon. These light degrees of freedom lead to the mirror twin Higgs model having tension with early universe cosmology [1,6].
Twin Higgs models can be reconciled with cosmological bounds for example in nonstandard cosmologies [6][7][8][9], or by relaxing the mirror symmetry so that there are no light degrees of freedom in the twin sector. One realization of the second approach is the fraternal twin Higgs (FTH) model [10]. In this model the twin sector is constructed with the minimal amount of new physics needed in order to solve the little hierarchy problem in a consistent way.
The minimal twin sector required to stabilize the Higgs up to a scale of O(10) TeV contains a twin Higgs doublet, the twin third generation of fermions, and a twin SU (3) c × SU (2) L gauge symmetry.
In the fraternal twin Higgs model the third generation and the first and second generations are inherently treated differently. We wish to motivate the distinction between these generations. We propose that the visible sector is actually realized as a 2 Higgs doublet model (2HDM) with a flavorful Yukawa structure [29,30]. One Higgs doublet is responsible for the mass of the third generation fermions and the other doublet is responsible for the mass of the first and second generations. In such a flavorful 2HDM (F2HDM), the mass of the first and second generation of fermions is set by the vacuum expectation value (vev) of the second Higgs that can be considerably smaller than the vev of the first Higgs. Combining the flavorful 2HDM with the twin Higgs mechanisms thus offers the possibility to partially address the hierarchical structure of the quark and charged lepton masses and, at the same time, to stabilize the electroweak scale up to O(10) TeV.
We consider two setups of this "twinned" flavorful two Higgs doublet model. In the fist setup, the twin sector is realized in a similar fashion to the mirror twin Higgs model, with a fully mirrored 2HDM structure. In the second setup, we consider a minimal twin sector similar to that of a fraternal twin Higgs model. We show under which conditions these two setups can be mapped onto each other.
The paper is organized as follows: we briefly summarize twin Higgs models in sec. II; in sec. III we describe the details of the setup of our twin F2HDM and discuss the resulting physical Higgs mass eigenstates and their couplings to both the SM and twin sector particles; in sec. IV we discuss the bounds on the model from Higgs signal strength measurements and the most important flavor constraint, the B s → µµ decay; finally, in sec. V we look at the phenomenology of this model, particularly focusing on displaced decays occurring in regions of parameter space that are unique to this setup; we conclude in sec. VI.

II. TWIN HIGGS MODELS
The twin Higgs mechanism stabilizes the Higgs mass up to some moderate scale, Λ, usually considered to be around 10 TeV. Above this scale some additional new physics is invoked to protect the Higgs mass up to the Planck scale. The largest contributions to the Higgs mass are the 1-loop top quark correction, the 1-loop SU (2) L correction, and the 2-loop QCD correction. In the twin Higgs model a twin sector exists with new degrees of freedom which cancel these contributions. Here we briefly review two versions of the twin sector: the mirror model and the fraternal model. More detailed discussions of these models and the underlying protection mechanism can be found in [1] and [10], respectively.
The twin Higgs mechanism is based on an approximate SU (4) symmetry that is respected by the scalar sector. An SU (4) fundamental scalar Φ contains two doublets φ andφ, parameterized as with the potential Besides the SU(4) symmetric mass term µ 2 and the quartic coupling λ, the potential includes a soft SU (4) breaking term σ, which allows a misalignment of the SM and twin vevs, v and v, and the parameters κ andκ are hard breaking terms, which help to reduce fine tuning [12]. We identify φ as the SU (2) L Higgs doublet in the SM sector andφ is the corresponding doublet in the twin sector.
After symmetry breaking and rotating to the physical mass eigenstates results in two physical scalar bosons that we identify as a SM-like Higgs (h) and a twin Higgs (ĥ) which are mixed states of S andŜ. The mixing angle is of order O(v/v).
The particle content of the twin sector is where the mirror and fraternal realizations of the twin Higgs mechanism differ. We first consider the mirror twin Higgs model where the twin sector is an exact copy of the SM sector containing the same forces, particles, and couplings that the SM does.
The Higgs mass receives loop contributions from both fermions and twin fermions, as shown in fig. 1. The twin top contribution comes with a relative minus sign as compared to the top contribution causing these two diagrams to cancel. In a similar fashion to the top quarks the twin contributions from the weak gauge bosons and two loop gluon contributions to the Higgs mass are exactly the same as the SM contribution with a relative minus sign. This is the fundamental mechanism that stabilizes the Higgs mass in twin Higgs models. The total correction to the Higgs mass from these loops is [10] whereĝ 2 is the strength of the twin SU (2) L ,ĝ 3 is the strength of the twin SU (3), andŷf are the twin Yukawa couplings. The color factor N c = 3 for quarks and N c = 1 for leptons. In the mirror twin Higgs model the couplings in the twin sector and the visible sector are set to be equal, thus leading to δm 2 h above being zero. However, the many light degrees of freedom in the mirror twin sector (in particular the light twin fermions and the massless twin photon) lead to tensions with cosmology. This inspired a minimal version of the twin Higgs model known as the fraternal twin Higgs model.
The fraternal twin Higgs model adds the minimum new physics in the twin sector necessary to stabilize the Higgs. The particle content in the twin sector consists of a twin top, a twin SU (2) L and a twin SU (3). In this setup the couplings of these particles are free parameters.
From eq. (3) we see that to ensure that the Higgs mass is not significantly tuned up to Λ ∼ 10 TeV one requires In order for the twin SU To ensure the twin fermions other than the top do not reintroduce large corrections to the Higgs mass one has to demand that where in the last step we neglected the small SM Yukawas y f . The above criterion translates intoŷf being no larger than ∼ 0.05, with the precise value depending on the maximum acceptable choice for δm h .

III. TWIN TWO HIGGS DOUBLET MODELS
Both the mirror and fraternal twin Higgs models successfully stabilize the Higgs mass up to order Λ. However, the mirror twin Higgs needs additional physics which can reconcile the model with cosmology, while the fraternal twin Higgs model leaves us with no explanation for the lack of the first two generations in the twin sector. Here we describe how the addition of a new source of mass generation in the form of a second Higgs doublet might provide a resolution to these issues. We also show how the mirror and fraternal version of a 2HDM setup can be mapped onto one another.

A. Mirror Setup
A well studied setup that provides additional sources of mass generation and distinguishes between the first two generations and the third generation is the flavorful 2HDM [29,30].
This model contains a SM-like doublet which primarily provides mass to the third generation fermions, and an additional doublet that primarily provides mass to the first and second generations. We propose a mirror twin Higgs inspired model where both the visible sector and the twin sector are realized as flavorful 2HDMs.
In this realization we have four doublets φ 1 ,φ 1 , φ 2 and,φ 2 , where φ 1 and φ 2 live in the visible sector andφ 1 andφ 2 live in the twin sector. The fields are arranged into SU (4) We will consider a scenario in which φ 1 andφ 1 couple to the third generation particles in the visible and twin sector, respectively, and φ 2 andφ 2 couple to the first two generations in the visible and twin sector, respectively. The most generic potential for Φ 1 and Φ 2 looks like where the soft breaking terms σ andσ and the hard breaking terms κ 1 , κ 2 ,κ 1 , andκ 2 are introduced in analogy to the usual twin Higgs setup. The terms containing m 2 12 andm 2 12 are mass parameters that mix the doublets φ 1 andφ 1 with φ 2 andφ 2 , respectively. m 2 12 =m 2 12 is another source of soft symmetry breaking.
As shown in [31] this setup is a self consistent extension of the twin Higgs model and provides the same cancellations as in the traditional twin Higgs setup. However, we now have extra sources of mass generation in the SM sector from φ 2 and the twin sector fromφ 2 .
Constraints from cosmology (in particular N ef f ) can be avoided by making the twin degrees of freedom sufficiently heavy, i.e. heavier than O(1 GeV).
The first two generations in the visible and twin sector have masses Due to the mirror symmetry the only way to make the first two generations of twin fermions heavy is to make the vacuum expectation valuev 2 much larger than v 2 (see [32] for a different mechanism to raise the masses of the fermions in the twin sector.) Characteristic values for v 1 /v 2 ≈ 10 (motivated to explain the hierarchy between the third and the second generation of SM fermions) and requiring that the lightest mirror particles to be at least O(1 GeV), leads tov 2 ≈ 10 TeV. We thus envision the following set of vevs We can approximate the amount of fine tuning needed to put the vevs in this hierarchical structure as [10,33] v This means the tuning of v 1 vs.v 1 is order percent level, but the tuning of v 2 vs.v 2 is substantial, of order 10 −6 .
In addition to the fermions, also the twin photon needs to be sufficiently heavy to avoid cosmological bounds. Two options to do this are: breaking electromagnetism in the mirror sector, or simply removing the U (1) hypercharge in the twin sector. In both cases the mirror symmetry of the model is weakened. In the following we will follow the scenario where there is no twin U (1) hypercharge.
The setup we have described so far leads to a large number of O(1 GeV) particles in the twin sector resulting in a complicated, yet rich set of dynamics. We leave a detailed discussion of this scenario to future work. Instead, we focus on a simplified setup which takes the twin Yukawas as free parameters in order to make the first and second generation twin fermions sufficiently heavy to be irrelevant for the Higgs phenomenology that we will discuss below.
By takingŷ f to be free parameters (up to the bound imposed by eq. (5)) we can push the masses of the twin first and second generation particles to O(1 TeV), forv 2 ∼ 10 TeV. In such a setup, the low energy phenomenology will be determined by the twin third generation, the twin SU (3) andφ 1 , while all the other twin states are effectively decoupled.

B. Fraternal Setup
Another approach to a twin flavorful 2HDM is to construct a model inspired by the fraternal twin Higgs. Starting from a flavorful 2HDM with doublets φ 1 and φ 2 , we add a third doubletφ 1 , with φ 1 andφ 1 being part of an approximate SU (4)  The twin sector consists of a Higgs doublet, a twin SU (2), a twin SU (3), and the third generation of twin fermions.

C. Twin F2HDM
The two approaches mentioned above both result in the same particle content and forces at low scales. Regardless of the high scale setup we will refer to the low energy simplified model as the twin F2HDM. 1 The potential for the twin F2HDM can be derived from eq. (7) withφ 2 integrated out.
This leaves an effective three Higgs doublet potential for the fields 1 One difference between the two discussed setups (mirror and fraternal twin sector) is that the mass of the twin weak gauge bosons will be set by a combination ofv 1 andv 2 in the mirror setup, but only set byv 1 in the fraternal setup. Generically, becausev 2 v 1 then twin weak gauge bosons in the mirror setup will be much heavier than in the fraternal setup. However, in both cases the twin weak gauge bosons will be O(1 TeV) or heavier, leaving no noticeable difference in the low energy phenomenology we will discuss in the remainder of this paper.
After electroweak symmetry breaking we are left with 6 massive modes: three scalar Higgs bosons, two charged Higgs bosons, and one pseudoscalar Higgs boson. The three scalars S 1 , S 1 , S 2 are related to the mass basis counterparts h 1 ,ĥ 1 , and h 2 (identified as the SM-like, twin, and heavy Higgs) by where the three mixing angles (s α i = sin(α i ), c α i = cos(α i )) are approximately given by .
The three Higgs boson masses are approximately The SM-like Higgs mass can be set by fixing κ 1 , κ 2 , and λ 1 . The heavy Higgs mass is primarily set by the parameter m 2 12 , and the twin Higgs mass is primarily set byv 1 , both of which can be taken as free parameters.
The most generic Yukawa Lagrangian can be written as The Yukawa matrices in the SM sector λ f i are determined by the flavor structure imposed on φ 1 and φ 2 in the flavorful setup [29,30]. The couplings λ u 1 , λ d 1 , and λ e 1 are rank one matrices, providing mass only to the third generation, while λ u 2 , λ d 2 , and λ e 2 have full rank and provide mass for the remaining fermions as well as CKM mixing in the quark sector. We find that the couplings of the Higgs bosons to the up-type quarks in the fermion mass eigenstate basis are given by where v = v 2 1 + v 2 2 = 246 GeV and s β = sin β, c β = cos β with tan β = v 1 /v 2 . The mass parameters m u i u j are given by the Yukawa couplings λ u 2 in the fermion mass eigenstate basis. For the flavor indices i or j equal to 1, the mass parameters m u i u j are of the order of the up quark mass and of the order of the charm quark mass otherwise (see [30] for their explicit expressions).
The above expressions for the couplings hold analogously for the down-type quarks and leptons. The couplings of the SM fermions to the charged Higgs bosons are the same as in the standard versions of the F2HDMs [29]. In our setup discussed here, the scalar Higgs bosons (and charged Higgs bosons) couple in addition also to the twin sector fermions as Finally, the couplings of the Higgs bosons to the vector bosons (hW W and hZZ) are given by the following expressions where Y SM V are the corresponding couplings of the Higgs boson in the Standard Model.

IV. CONSTRAINTS
The introduction of two additional Higgs doublets alters the couplings of the SM-like Higgs boson h 1 as shown in eqs. (17)- (19). The ATLAS and CMS experiments at the LHC have taken measurements of the production and decays of the Higgs boson and we must ensure that our model is consistent with the existing experimental results. Additionally, we will also consider the impact of projected sensitivities from the high luminosity (HL) LHC.
To determine these constraints we construct a χ 2 function are the experimental measurements, the Standard Model predictions, and our BSM predictions for the production cross sections times branching ratio of the various measured channels.
We use the SM predictions from [34]. As in [30],  [44,45], and top associated production [46,47]. The projected sensitivities are taken from [48], and correspond to 3000 fb −1 of data collected at 14 TeV. the twin Higgs, as seen in eq. (14). In addition, there is also weak dependence of the Higgs couplings on the mass parameters m f i f j (see eq. (17) and text below). We let those mass parameters vary up to a factor of 3 around their expected values, as was also done in [30].
Generally, the sensitivities that are expected at the HL-LHC can potentially constrain the twin vevv 1 much stronger than the current bound. Previous studies found the constraint v 1 3v, while future experiments favorv 1 to be closer to an order of magnitude larger than v, at least for moderate values of tan β.
As shown in [29], the flavorful structure we impose on the φ 1 and φ 2 couplings leads to flavor violating Higgs couplings for the SM-like Higgs and the heavy Higgs. The SU (2) 5 flavor symmetry, that is preserved by the rank 1 Yukawa couplings of the doublet φ 1 , protects flavor changing transitions between the first and second generation of quarks and leptons that typically plague 2HDMs without flavor conservation [49]. However, we still find strong and robust constraints from the rare decay B s → µµ. 2 In the limitv 1 v, the expression for the B s → µµ branching ratio in our model can be easily generalized from the expression in [30] with α → α 2 .
The SM prediction and the current experimental measurements are [50] BR(B s → µµ) SM = (3.67 ± 0.15) × 10 −9 , For the future experimental sensitivities to B s → µµ, we assume that the central value for the branching ratio stays consistent with the current experimental value, while we take an uncertainty of ±0.16 × 10 −9 [51]. It is important to note that there is some tension (at the 2σ level) between the SM prediction and current experimental value, and assuming that the experimental central value holds there will be very significant discrepancy from future experiments.
The current constraint (left) and future sensitivity (right) from B s → µµ is shown in the plane of the heavy Higgs mass m h 2 vs. tan β in fig. 3, with the Higgs fit constraints overlayed in red. Based on current constraints masses as low as 300 GeV are consistent with both B s → µµ and Higgs signal strengths measurements for a moderate tan β 5. However, future projections push this lower bound on the mass up to around 450 GeV. For somewhat larger tan β 10, the expected bound on the heavy Higgs mass is around 700 GeV. This is significant as the production of the heavy Higgs becomes quickly suppressed as its mass m h 2 increases.
Our model can moderate the tension between the theoretical prediction and experimental value of the B s → µµ branching ratio by the additional contributions of the heavy and pseudoscalar Higgs. In particular the pseudoscalar Higgs contribution interferes destructively with the SM amplitude and thus can lower the B s → µµ rate, reconciling the theoretical prediction with the experimental central value. This is evident in the shape of the allowed region of the plots in fig. 3, where the band represents the region of parameter space that removes unwanted tension. In the scenario that tan β becomes too large the rate of B s → µµ also becomes too large, violating the 2σ bound. While if m h 2 becomes too large, our theoretical prediction matches back onto the SM prediction and is disfavored.
In addition to Higgs fit and the B s → µµ constraints, there exists constraints from heavy Higgs searches performed by ATLAS and CMS. The most relevant constraints come currently from H → µµ searches [52,53], but are weak compared to the B s → µµ and Higgs signal strength measurements as shown in previous work [30]. For a detailed description of the twin bottomonium and glueball spectrum see [10]. We will assume that the the twin taus and neutrinos are sufficiently heavy such that the bottomonia and glueballs do not decay into them. Some of the bottomonia and glueballs (in particular the lightest glueball) can mix with the Higgs bosons in the visible and therefore decay into SM particles. The lifetime of the glueballs can be sizeable and one often finds displaced decays in the discussed scenario.
Displaced events occur as a result of the production of twin sector states (bottomonia and/or glueballs) through one of the three scalar Higgs bosons or the pseudoscalar Higgs boson. We assume that the twin spectrum is such that there are glueball states with mass below half the mass of the twin bottomonia mĜ < m [bb] /2 and assume that all decays in the twin sector result in at least one lightest glueballĜ 0 . The lightest glueball has the same quantum numbers as the SM-like Higgs allowing it to mix back into the visible sector and decay, in particular to bb. In the viable region of our parameter space, the corresponding lifetime of the glueball can be approximated as [10] cτ ≈ 18m × 10GeV mĜ which depends very sensitively on the glueball mass.
We can break down the phenomenology of this scenario into three distinct regions: SM-like Higgs dominated, twin Higgs dominated, and heavy Higgs dominated. The SM-like Higgs dominates the phenomenology when the twin vevv 1 and twin bottom Yukawaŷb take values such that the twin bottomonia and the twin glueballs are lighter than half the Higgs mass, m [bb] < m h 1 /2. As the SM-like Higgs is produced at the LHC at a much higher rate than the heavy Higgs or twin Higgs, displaced decays from the SM-like Higgs dominate the phenomenology 3 . In this case the phenomenology of our model is similar to that of the original FTH model. For this reason we forgo an analysis of this scenario here and instead point the reader to [10,11].
The twin Higgs dominates the displaced phenomenology whenv 1 andŷb take on values such that m [bb] > m h 1 /2, while the twin Higgsĥ 1 is still moderately light. In this case the twin Higgs is produced at a high enough rate that its production of twin sector hadrons is much larger than that of the heavy Higgs, so again the phenomenology follows a similar path of the original FTH model, and the addition of the heavy Higgs has little impact on the phenomenology. For this reason we forgo an analysis of this scenario here as well, and instead point the reader to [10,12,13,54].
The final region of parameter space is characterized by a phenomenology that is dominated by the heavy Higgs. This happens whenv 1 andŷb take on values such that m [bb] > m h 1 /2 and at the same time the twin Higgsĥ 1 is very heavy (motivated byv 1 being large). In this regime both the heavy Higgs and the twin Higgs participate in producing twin sector particles, but the production rate of the twin Higgs becomes very small. Asv 1 rises, the decay of the heavy Higgs to the twin sector is also suppressed. However, the production rate of the twin Higgs drops more quickly than the branching ratio of heavy Higgs decays into  GeV. The production cross sections and decay rates are rather robust against order one changes to the parameters of this benchmark point other than λ 3 . We can see from eq. (14) that λ 3 controls the mixing of the heavy Higgs with the twin sector and therefore has a substantial impact on the branching ratio into the twin sector BR(H →bb). We choose a large value of λ 3 = 5 as a representative example. For the Yukawa coupling of the twin bottom we chooseŷb = 3y SM b , within the range allowed by naturalness arguments, see eq. (5). The branching ratios in this model are similar to that of the type 1B F2HDM [30], with the addition of a small, but important, branching ratio to the twin bottom. In the left plot we show the production cross sections of the heavy Higgs. Also the cross sections are similarly to the type 1B F2HDM and we see that over most of parameter space the main production modes are charm-charm fusion, gluon-gluon fusion, and vector boson fusion. Generally, gluon-gluon fusion is dominant at small tan β and charm-charm fusion is dominant at high tan β. For moderate values of tan β, the production cross section for a 500 GeV heavy Higgs can be around 100 fb.  absent. In that case, whenv 1 is larger than about 2000 GeV, we see that the lifetime of the glueballs is of the order of at least millimeters which, given a typical boost factor of a few, falls into the decay lengths of interest for displaced signatures at the LHC.
We see that for the lower mass choices of the heavy Higgs 500 GeV (at the LHC) and 700 GeV (at the HL-LHC) O(10)s of events could occur with O(few mm) displaced decays. As we push the scale ofv 1 to larger values, we see that this number drops down to a handful of decays. The HL-LHC will generically produce more displaced decays at a given heavy Higgs mass, but the stronger expected constraints on the parameter space roughly balance out the increase. So, for masses that are not indirectly probed by flavor constraints or Higgs coupling strength measurements, we see a similar amount of expected displaced decays. Similarly such an observation also holds for the higher mass scenarios that we considered. We see that for heavier Higgs mass m h 2 the estimated number of events is reduced to several and below as the production of the heavy Higgs is suppressed at these higher masses.
Searches for long lived particles have been explored to some degree at the LHC [55,56].
The existing searches do currently not put strong constraints on the displaced decays we have considered in this model. For sizable displacements of the order O(1 cm − 1 m) the expected sensitivities from the LHC could cover sizable regions of parameter space (see [54] for a detailed study of the fraternal twin Higgs model). However, in the scenarios discussed in this paper, the displacement is typically of the order O(few mm) 4 , making it much more challenging to search for the displaced signatures, for example due to triggering difficulties (see e.g. [10,56]). Future improvements in searching for displaced decays with O(few mm) displacement would be necessary to further explore the models described in this work.

VI. CONCLUSION
The little hierarchy problem and the SM flavor puzzle are two longstanding problems in particle physics. We have discussed a setup which attempts to address both of them (at least partially). We considered a 2HDM with a flavorful Yukawa structure, where one Higgs doublet is responsible for the mass of the third generation fermions and the other doublet is responsible for the mass of the first and second generations. A hierarchy in vevs can explain the mass hierarchy between the third and first two generations. We combined this setup with the twin Higgs mechanism which stabilizes the Higgs mass up to O(10 TeV), considering both a mirror twin and fraternal twin setup.
In the visible sector, the flavorful Yukawa structure of this model leads to modifications of the B s → µµ branching ratio. Large values of tan β are already strongly constrained.
We showed that the current mild tension that exists between the SM prediction and the experimental results can be solved in our setup for moderate values of tan β. This is of particular interest in view of the expected future sensitivities to B s → µµ from LHCb, which could turn the current tension into a very significant discrepancy. The prediction of this scenario are slightly displaced decays at length scales of few millimeters which are challenging to detect experimentally. We find that for a twin vevv 1 of at least 2000 GeV that the heavy Higgs can naturally dominate the displaced phenomenology with as many as O(30) displaced decays predicted to have taken place at the LHC already.
Anticipating improved indirect constraints on the model parameter space from future experimental results on Higgs signal strengths measurements and the B s → µµ decay, we find that there is still viable region of parameter space which can produce O(30) displaced decays at the HL-LHC.