Explanations of the Tentative New Physics Anomalies and Dark Matter in the Simple Extension of the Standard Model (SESM)

We revisit the Simple Extension of the Standard Model (SESM) which can account for various tentative new physics anomalies and dark matter (DM). We consider a complete scalar potential which is needed to address the $W$ boson anomaly. Interestingly, the SESM can simultaneously explain the B physics anomaly, muon anomalous magnetic moment, $W$ mass anomaly, and dark matter, etc. Also, we study the unitarity constraint in this model. We perform the systematic study, and find the viable parameter spaces which can explain these anomalies and evade all the current experimental constraints. To be complete, we briefly comment on the neutrino masses and mixings, baryon asymmetry, and inflation.

After the discovery of Higgs boson at the LHC in 2012 [1,2], the Standard Model (SM) has been confirmed to be a correct effective theory at the low energy scale.However, we do have a few evidences of new physics Beyond the SM (BSM), for instance, dark energy, dark matter (DM), neutrino masses and mixings, baryon asymmetry, and inflation, etc [3].Also, there exist some fine-tuning problems in the SM, for example, gauge hierarch problem, and strong CP problem, etc.Thus, the SM is not complete, and we need to explore the new physics.
In recent years, the LHCb Collaboration has declared the results in rare decays of B mesons.The persistent discrepancies between the Standard Model and the experimental measurements imply that there might be new physics.Such B physics anomalies can be observed in the angular distribution of B → Kµ + µ − and Lepton Flavor Universality (LFU) ratios R K = BR(B → µµ)/BR(B → ee) [4][5][6][7][8].Also, there exists a 4.2 σ discrepancy for the muon anomalous magnetic moment (muon g − 2) a µ = (g µ − 2)/2 between the experimental results and theoretical predictions [9].Although the hadronic contribution might induce the discrepancy, muon g − 2 is still a promising hint for new physics beyond the SM [12][13][14], and has been studied extensively [15][16][17][18][19]. Recently, the CDF Collaboration announced a state-of-the-art measurement of the W boson mass, which shows 7 σ deviation from the prediction of the SM [20].For recent studies, see Refs. .Moreover, the DM is a crucial problem in both particle physics and astronomy, and the observations from astrophysics and cosmology provide the overwhelming evidence.The Weakly Interacting Massive Particle (WIMP) provide an excellent DM candidate to account for the relic density observed by the experiment of Cosmic Microwave Background (CMB).This is the so called "WIMP miracle".
To address the above anomalies and dark matter, we revisit the Simple Extension of the Standard Model (SESM) [61].We consider a complete scalar potential which is needed to account for the W boson anomaly.Interestingly, the SESM can simultaneously explain the B physics anomaly, muon anomalous magnetic moment, W boson mass anomaly, and dark matter, etc.Also, we study the unitarity constraint in this model.We perform the systematic study, and find the viable parameter spaces which can explain these anomalies and evade all the current experimental constraints.To be complete, similar to the New Minimal SM (NMSM) [3], we can introduce two right-handed neutrinos to explain the neutrino masses and mixing as well as baryon asymmetry, and introduce a real scalar to address inflation.Because of the strong constraint on the tensor-to-scalar ratio, we need to choose the proper inflaton potential, for example, the inflaton potentials in Section 2 of Ref. [62].
The XENON1T excess [63] will be studied elsewhere as well.
This paper is organized as follows.In Section II, we briefly review the SESM and present the complete scalar potential.In Section III, the constraints of unitarity on the parameters in the Yukawa sector are studied.In Section IV, we explain the above anomalies and dark matter.In Section V, the W boson mass anomaly is investigated.We conclude in Section VI.

II. THE SIMPLE EXTENSION OF THE STANDARD MODEL
Following Ref. [61] we introduce a complex singlet scalar Φ S , a doublet scalar Φ D , and two vectorlike pairs of Weyl fermions (that combine into two Dirac fermions) with the same quantum numbers as the SM quark and lepton doublets Q and L .Under a discrete Z 2 symmetry, these extra fields are all odd while the SM fields are even.The quantum numbers of extotic fields under SM gauge group are The new vectorlike fermions and scalars can be written as The Lagrangian involving the new fields is given by where we have systematically considered the scalar potential relative to the phenomenology that we are interested in.We denote the left-handed exotic quarks, right-handed exotic quarks, left-handed exotic leptons, right-handed exotic leptons, left-handed quark doublets, right-handed up-type quarks, right-handed down-type quarks, left-handed lepton doublets, and right-handed down-type leptons as , and E i (i=1,2,3), respectively.
After the breaking of the electroweak symmetry, the terms involved λ SH and λ DH will contribute to masses of Φ S and Φ D , respectively, while the term involved a H will induce mixing of S 0 s and S 0 d .For simplicity, we choose a H , λ 1 and λ 2 to be zero.Therefore, the modified mass matrix of the BSM neutral scalars can be written as where v 246 GeV.The corresponding mass eigenstates S 1 and S 2 have physical masses of with The mixing of S 1 and S 2 is given by The couplings between up and down type quarks have a relative misalignment, and we choose where V is the CKM matrix.
In our model, we impose that the lightest BSM particle is S 1 , which is the DM candidate.
But the mass hierarchies between M Q , M L , and M S 2 are not mandatory.

III. THE UNITARITY CONSTRAINT
The partial wave amplitude of any 2→2 scattering process in the high-energy massless limit is [64], where θ is the scattering angle in the center of mass frame, and standing for the helicities of initial and final particles, respectively.
The unitarity of S matrix respects SS † = 1.By considering elastic scattering and imposing that the intermediate states are two-particle states, the condition required by the unitarity of S matrix can be obtained.For general unitarity bound, one has At tree level, the above unitarity bound becomes To compute the unitarity bounds in the Yukawa sector, the helicity amplitude approach demonstrated in [64] is employed, and the general form is given in TABLE I. T is the nontrivial part of S matrix.The absence of T in C ( ) , E ( ) , and F ( ) is due to the angular momentum conversation.In the massless limit, the entities in A ( ) and B ( ) vanish exactly since T is proportional to the mass.Besides, the entity in D is present when considering the scattering of two scalars.
One can decompose the T matrix into the following structure where is the group factor.To obtain the partial wave amplitude, we should consider a concrete elastic scattering process which can occur in different channels and representations.Our strategy is 1) Considering which partial wave we are interested in; 2) Calculating the group factors of the corresponding elastic scattering process; 3) Combining Eqs. 9, 11, and 12 to obtain the constraint of unitarity on this elastic scattering process.According to the classification in Ref. [64], the term 3 corresponds to the first model of dirac type theory with respect to SU (3) C and SU (2) L .Since Q i denotes a left-handed field, Q should be a right-handed field.The term for the second generation is In the massless limit, (anti-)fermions described by the left-handed and righthanded spinors have helicities of − 1 2 ( 1 2 ) and 1 2 (-1 2 ), respectively.
1. J = 0 partial wave TABLE.I shows that only T ++++ and its conjugation contribute to the J = 0 partial wave.Since Q R and Q 2 are in the fundamental and anti-fundamental representations of The elastic scattering process is mediated by the singlet complex scalar Φ S .Thus, only the channel of the 1 representation of SU (2) L × SU (3) C is present.As for the group factor of this process, we have where N comes from that both fermions involved are in the (anti-)fundamental representation of the SU (N ) group (there are more details in Appendix A) which means that N 2 = 2 and N 3 = 3 in our case.In the basis of ( , based on Eqs. 9 and 12, we get where the last step means that we find the largest eigenvalue after the diagonalization of the matrix.Besides, we have used d 0 00 (θ) = 1 and in [64].Then, the perturbation unitarity condition of Eq. 11 leads to the bound

B. A brief summary of the unitarity constraints in Yukawa sector
The elaborate constraints on the parameters of Yukawa sector in our model are given in TABLE II, and the details can be found in Appendix A. The unitarity constraints on Yukawa couplings eventually lead the following bounds Of course, we need to consider the perturbative bounds on the Yukawa couplings, i.e., all the Yukawa couplings are smaller than √ 4π.

IV. THE TENTATIVE NEW PHYSICS ANOAMLIES AND DARK MATTER
In this Section, we will address the R K and B s mixing, muon g − 2, and dark matter relic density simultaneously.Random scan of the model parameters is employed, and the benchmark points satisfying all the current constraints are selected.
A. R K and B s mixing According to [61], the contribution to R K can be described by the following effective Lagrangian, where the normalization is given by Contributions to C bsµµ 9,10 from the BSM particles are with loop function According to the up-to-date fitting [4] The contribution to B s − Bs oscillation can be induced from the following operators The Q − Φ S box diagram gives Wilson coefficients where the loop function is The bound given in [10,11] is

B. Muon g-2
The contribution of BSM particles in this model to a µ is where the loop function is The latest (1σ) discrepancy is given by [9]

C. Dark matter phenomenology
We consider the lightest neutral scalar S 1 as the DM candidate, and the typical freeze-out mechanism controls the DM production.In addition, the annihilation and co-annihilation processes can deplete the DM relic density.In our random scan, the ranges of parameters have to satisfy the bounds of unitarity given in TABLE II.In addition, we set the couplings to the right-handed quarks (λ U i and λ D i ) and the left-handed quarks of the first generation (λ Q 1 ) to be zero, and only consider the second family couplings in the lepton sector (λ L 2 and λ E 2 ).The elaborate scan ranges of parameters are given in TABLE III.  2) Co-annihilations of S 1 and other particles (S 2 , L , and Q ) will occur when their mass difference are small, which are shown in the second column of FIG. 2.
Ten benchmark points are given in  In addition, we investigate the large mixing and relative small coupling case.We can see from Eqs. 20 and 21, ∆C 9 and ∆C 10 are quadratically depended in λ L 2 .To reduce the λ L 2 , one needs small M Q and M L , which are already near the lower edge of experimental search.After a somehow simple analysis, one cannot have a small λ L 2 to account for all the aforementioned anomalies.In the left panel of FIG. 3, we can see that only Higgs resonance can explain the R K , B s mixing, muon g − 2, and the saturated DM relic density.The right panel of FIG. 3 shows the S 1 and L co-annihilation process with large mixing (a H = 500 GeV).In this case, the R K , B s mixing, and muon g − 2 can be explained simultaneously and the DM relic density is undersaturated.In FIG. 4 we display viable parameter space with different a H .In top left panel, the flavor observables and DM relic density can be explained as well.As a H increases, the DM relic density is undersaturated while the constraints of flavour observables are satisfied.
We intend to give some comments on the parameter space on the edge of unitarity.
In FIG. 5 we employ the conservative parameters which are consistent with the unitarity bounds in Eq. 17 and avoid large quantum corrections.Roughly speaking, we find the upper bounds on the masess of exotic particles, M Q = 17.5 TeV and M L = 2.4 TeV which can account for the R K , B s mixing, muon g − 2, and the saturated DM relic density.

V. W BOSON MASS
The oblique corrections contributing to W boson mass are given by [65,66] where α is the fine structure constant, c=cosθ W , s=sinθ W .The expressions of ∆S and ∆T are ∆S = 1 2π with the the loop function For convenience, we consider the small mixing between scalar singlet and doublet.To interpret the W boson mass anomoly, the scalar potential is essential to give an appropriate mass splitting between neutral and charge part of the doublet.We find that W boson mass cannot be explained if we consider λ 2 only.However, if the λ DH , λ 2 , λ 1 are involved simultaneously, the W mass amomoly can be account for in a straightforward way.We can fit the up-to-date measurement of W boson mass (1 σ) by using Eqs.31, 32 and 33.FIG.6 shows the corresponding oblique parameters between ∆S and ∆T , and the mass splitting between neutral and charge part of the scalar doublet.

VI. CONCLUSION
We have revisited the SESM which can address various tentative new physics anomalies and DM.The mass splitting between the charged and neutral parts of scalar doublet via scalar potential can account for the W boson mass anomaly.Moreover, we considered the unitarity constraints in the Yukawa sector.Also, we employed the random scan approach, and obtained the viable parameter spaces which can explain the B physics anomaly, muon anomalous magnetic moment, W mass anomaly, and dark matter relic density simultaneously.To be concrete, we select some benchmark points.The various DM (co)-annihilation and resonance processes are investigated, and the benchmark points whose DM relic density are around or smaller than the observed DM relic density are demonstrated.In particular, the Higgs pole, S 1 and S 1 annihilation, S 1 and S 2 , L , Q co-annihilation are demonstrated as well, and all these points can evade the current flavor and XENON1T direct detection constraints.
where ψ ψ|ψ ψ 1 denotes the elastic scattering process in singlet channel, and ψ 1 stands for one component of ψ in the fundamental representation.

(A6)
This corresponds to the bound 2. The Unitarity Constraint on λ Q

3
The λ Q 3 in Eq. 3 is the first model of dirac type theory in both SU (3) C and SU (2) L as well, and thus the constraint on λ Q 3 is exact the same as λ Q 2 .We have the strict bound |λ Q 3 | ≤ 2.05 from J = 0 partial wave.
As we have claimed in the quark sector before, the Lagrangian becomes λ This kind of interaction can be classified by the first type of dirac theory in SU (2) L .
a. J = 0 Partial Wave One can consider the elastic scatting processes The Lagrangian is given by λ Since E is the SU (2) L singlet and Φ D is SU (2) L doublet, this kind of interaction can be classified into the second type of dirac theory.
a. J = 0 Partial Wave Two particle state is defined (E is SU (2) L singlet fermion) as The group factor is We have + + + + or − − − − F s, L L EL L E = F s, L L EL L E = 1.
In (L L E, L L E) basis, we have

2 1. 1 . 2 Wave 2 . 3 3. 2 a
Simple Extension of the Standard Model III.The Unitarity Constraint A. The unitarity constraint of λ Q J = 0 partial wave B. A brief summary of the unitarity constraints in Yukawa sector IV.The Tentative New Physics Anoamlies and Dark Matter A. R K and B s mixing B. Muon g-2 C. Dark matter phenomenology V. W Boson Mass VI.Conclusion Acknowledgments A. The Unitarity Constraint on λ Q 2 The Unitarity Constraint on λ Q 2 a. J = 0 Partial Wave b.J = 1 Partial The Unitarity Constraint on λ Q The Unitarity Constraint on λ L

FIG. 1 :
FIG. 1: DM annihilation mediated by Higgs resonance.The black points satisfy the constraints of R K (2σ) and B s − Bs , the blue points satisfy the constraints of R K (2σ), B s − Bs and muon g-2, the red points satisfy the constraints of R K (2σ), B s − Bs , muon g-2 and DM relic density, and the green points satisfy the constraints of R K (2σ), B s − Bs , muon g-2, DM relic density and XENON1T direct detection.

FIG. 1 1 )
FIG.1shows the DM relic density versus the mass of S 1 .The conditions from R K (2σ) and B s − Bs (black dot), the muon g-2 (1σ) constraint (blue dot), the constraint of (rescaled) DM relic density (red dot), and the constraint from XENON1T experiment (green dot) are applied in order.It can be seen from FIG.1that the Higgs mediated DM annihilation

2 | 2 | 1 L= F s, 1 L
< 5.01 .(A15) Group factor is +0 + 0 or −0 − 0 F s, E Φ D E Φ D = F s, E Φ D E Φ D = 1.In (E Φ D , E Φ D ) base,we have (We should claim that Φ D is in the fundamental representation.)< 7.09 .(A17)However,an alternative basis (L L Φ D , L L Φ D ) gives stronger constraint since the group factoris +0 + 0 or −0 − 0 F s,L Φ D L L Φ D L Φ D L L Φ D = N .In this case, we get a factor enhancement, the unitarity bound becomes |λ E

TABLE I :
Scattering matrix T .
A. The unitarity constraint of λ Q 2

TABLE II :
Unitarity constraint in Yukawa sector.

TABLE III :
The ecan ranges of parameters employed in the random scan.New parameters of this model which are not shown in this table are set to be zero.
(10)S 1 .The Higgs mediated DM annihilation is demonstrated by Points 3 and 4. Point 5 (6), Point 7 (8), and Point 9(10)are characterized by the co-annihilation between S 1 and S 2 , L , and Q , respectively.In addition, the dark matter candidate S 1 can explain some of the observed relic density and evade XENON1T experiment as well.
TABLE IV, and all these points can evade the experimental constraints from R K and B s mixing, and muon g-2.Points 1 and 2 correspond to annihilation of

TABLE IV :
Benchmark points in DM annihilation and co-annihilation.