The CDF W-mass, muon g-2, and dark matter in a $U(1)_{L_\mu-L_\tau}$ model with vector-like leptons

We study the CDF $W$-mass, muon $g-2$, and dark matter observables in a local $U(1)_{L_\mu-L_\tau}$ model in which the new particles include three vector-like leptons ($E_1,~ E_2,~ N$), a new gauge boson $Z'$, a scalar $S$ (breaking $U(1)_{L_\mu-L_\tau}$), a scalar dark matter $X_I$ and its partner $X_R$. We find that the CDF $W$-mass disfavors $m_{E_1}= m_{E_2}={m_N}$ or $s_L=s_R=0$ where $s_{L(R)}$ is mixing parameter of left (right)-handed fields of vector-like leptons. A large mass splitting between $E_1$ and $E_2$ is favored when the differences between $s_L$ and $s_R$ becomes small. The muon $g-2$ anomaly can be simultaneously explained for appropriate difference between $s_L$ $(m_{E_1})$ and $s_R$ $(m_{E_2})$, and some regions are excluded by the diphoton signal data of the 125 GeV Higgs. Combined with the CDF $W$-mass, muon $g-2$ anomaly and other relevant constraints, the correct dark matter relic density is mainly obtained in two different scenarios: (i) $X_IX_I\to Z'Z',~ SS$ for $m_{Z'}(m_S)<m_{X_I}$ and (ii) the co-annihilation processes for $min(m_{E_1},m_{E_2},m_N,m_{X_R})$ closed to $m_{X_I}$. Finally, we use the direct searches for $2\ell+E_T^{miss}$ event at the LHC to constrain the model, and show the allowed mass ranges of the vector-like leptons and dark matter.

The two anomalies both call for new physics beyond SM. There have been many works explaining the CDF W -mass .
In this paper, we study the CDF W -mass, the muon g − 2, and the DM observables in a local U (1) Lµ−Lτ model in which a singlet vector-like lepton, a doublet vector-like lepton and a complex singlet X field are introduced in addition to the U (1) Lµ−Lτ gauge boson Z [72] and a complex singlet S breaking U (1) Lµ−Lτ symmetry. As the lightest component of X, X I is a candidate of dark matter (DM) and its heavy partner is X R . The gauge boson self-energy diagrams exchanging the vector-like leptons in the loop can give additional contributions to the oblique parameters (S, T, U ), and explain the CDF W -mass. The interactions between the vector-like leptons and muon mediated by the X I (X R ) can enhance the muon g − 2 [73][74][75][76][77][78][79][80][81][82][83]. These new particles can affect the DM relic density via the DM pair-annihilation and various co-annihilations processes.
Our work is organized as follows. In Sec. II we introduce the model. In Sec. III and Sec. IV we study the W-boson mass, muon g-2 anomaly, and the DM observables imposing relevant theoretical and experimental constraints. Finally, we give our conclusion in Sec. V.

II. THE MODEL
In addition to the U (1) Lµ−Lτ gauge boson Z , we introduce a complex singlet S breaking U (1) Lµ−Lτ , a complex singlet X, and the following vector-like lepton fields, Their quantum numbers under the gauge group SU Table I.
Where D µ is the covariant derivative and g Z is the gauge coupling constant of the U (1) Lµ−Lτ group. The field strength tensor Z µν = ∂ µ Z ν − ∂ ν Z µ , and the kinetic mixing term of gauge bosons of U (1) Lµ−Lτ and U (1) Y is ignored. V and L Y indicate the scalar potential and Yukawa interactions.
The scalar potential V is written as where the SM Higgs doublet H, the complex singlet fields S and X are Here H and S respectively acquire vacuum expectation values (VEVs), v h = 246 GeV and v S , and the VEV of X field is zero. The parameters µ 2 h and µ 2 S are determined by the minimization conditions for Higgs potential, The complex scalar X is split into two real scalar fields X R and X I by the µ term after the S field acquires VEV v S . Their masses are Because the X field has no VEV, there is a remnant discrete Z 2 symmetry which makes the lightest component X I to be stable and as a candidate of DM.
The λ HS term leads to a mixing of h 1 and h 2 , and their mass eigenstates h and S are obtained from following relation, with α being the mixing angle. From the λ HS term and λ HX term, we can obtain the 125 GeV Higgs h coupling to a pair of DM. In order to escape the strong bounds of the DM indirect detection and direct detection experiments, we simply assume the hX I X I coupling to be absent, namely choosing λ HS = 0 and λ HX = 0. Thus we obtain The gauge boson Z acquires a mass after S breaks the U (1) Lµ−Lτ symmetry, The Yukawa interactions with the U (1) Lµ−Lτ symmetry are given as where L µ = (ν µL , µ L ).
Since the X field has no VeV, there is no mixing between the vector-like leptons and the muon lepton. However, there is a mixing between the vector-like leptons E" and E after the H acquires the VeV, v h = 246 GeV, and their mass matrix is given as We take two unitary matrices to diagnolize the mass matrix, where c 2 L,R +s 2 L,R = 1. The E 1 and E 2 are the mass eigenstates of charged vector-like leptons, and the mass of neutral vector-like lepton N is From the Eq. (12), we can obtain the interactions between the charged vector-like leptons and muon mediated by X R and X I , (16) and the 125 GeV Higgs interactions to the charged vector-like leptons E 1 and E 2 , III. THE S, T, U PARAMETERS, W -MASS, AND MUON g − 2 In addition to m h = 125 GeV, v h = 246 GeV, λ HS = 0, λ HX = 0, there are many new parameters in the model. We take κ 1 , and κ 2 as the input parameters, which can be used to determine other parameters.
In order to maintain the perturbativity, we conservatively take The mixing parameters s L and s R are taken as We scan over the input mass parameters in the following ranges: The mass of neutral vector-like lepton N is determined by m E 1 , m E 2 , s L and s R , we require m N > m X I . We choose 0 < g Z /m Z ≤ (550 GeV) −1 to satisfy the bound of the neutrino trident process [84]. We take -2 < q x ≤ 2, and require | g Z (1 − q x ) |≤ 1 and g Z ≤ 1 to respect the perturbativity of the Z couplings.
The tree-level stability of the potential in Eq. (5) impose the following bounds, The H → γγ decay can be corrected by the loops of the charged vector-like leptons E 1 and E 2 . We impose the bound of the diphoton signal strength of the 125 GeV Higgs [2], The model contains the interactions of gauge bosons and vector-like leptons, where L ij and R ij are with Where s W ≡ sin θ W and c W = 1 − s 2 W , and θ W is the weak mixing angle. The gauge boson self-energy diagrams exchanging the vector-like leptons in the loop can give additional contributions to the oblique parameters (S, T, U ) [85,86], which are calculated as [85][86][87] where the Π NP function is given in Appendix A.
Analyzing precision electroweak data including the new CDF W -mass, Ref. [8] gave the values of S, T and U , S = 0.06 ± 0.10, T = 0.11 ± 0.12, U = 0.14 ± 0.09 (29) with correlation coefficients The W -boson mass is given as [86], We perform a global fit to the values of S, T, U , and require χ 2 < χ 2 min + 6.18 with χ 2 min denoting the minimum of χ 2 . These surviving samples mean to be within the 2σ range in any two-dimension plane of the model parameters fitting to the S, T , and U parameters.
In Fig. 1, we show the samples explaining the CDF W -boson mass within 2σ range while satisfying the constraints of the oblique parameters and theoretical constraints. Fig. 1 shows that the explanation of the CDF W -mass requires appropriate mass splittings among E 1 , E 2 and N , which do not simultaneously equal to zero. For example, when m E 2 = m N , the mass splitting between m E 1 and m E 2 (m N ) is required to be larger than 100 GeV. From the right panel of Fig. 1, we see that the measurement of CDF W -mass disfavors s L and s R to approach to zero simultaneously, and favors E 1 and E 2 to have a large mass splitting when the difference between s L and s R becomes small. The model can give additional corrections to the muon g − 2 via the one-loop diagrams containing the interactions between muon and E 1 (E 2 ) mediated by X R and X I , and the main corrections are calculated as [73,75,88] where the function Eq. (32) shows that the correction of the model to the muon g − 2 is absent We respectively take s L = s R and m E 1 = m E 2 , and show the samples explaining the muon g − 2 anomaly within 2σ range while satisfying the constraints "pre-muon g − 2" (denoting the theory, the oblique parameters, and the CDF W -mass) in Fig. 2. From Fig. 2, we see that the explanation of the muon g − 2 anomaly favors | s L | to decrease with increasing After imposing the constraints of the diphoton signal data of the 125 GeV Higgs and "pre-muon g − 2", we scan over the parameter space, and project the samples explaining the muon g − 2 anomaly in Fig. 3. We find that the diphoton signal data of the 125 GeV Higgs exclude some samples explaining the muon g − 2 anomaly, and favors s L and s R to have same sign, especially for large | s L | and | s R |. When s L and s R have same sign, the terms of hĒ 1 E 1 (hĒ 2 E 2 ) coupling in Eq. (17) are canceled to some extent, which suppresses the corrections of E 1 and E 2 to the h → γγ decay.

IV. THE DM OBSERVABLES
In the model, in addition to X I X I → µ + µ − , and the DM pair-annihilation processes X I X I → Z Z , SS will be open for m Z (m S ) < m X I . When the masses of E 1 , E 2 , N and X R are closed to m X I , their various co-annihilation processes will play important roles in the DM relic density. We use FeynRules [89] to generate model file, and employ micrOMEGAs DM in the universe, Ω c h 2 = 0.1198 ± 0.0015 [91].
After imposing the constraints of "pre-muon g − 2", the diphoton signal data of the 125 GeV Higgs, and the muon g − 2 anomaly, we project the samples achieving the DM relic density within 2σ range in Fig. 4. Due to the constraints of muon g − 2 on the interactions between the vector-like leptons and muon mediated by X I , it is not easy to obtain the correct DM relic density only via the X I X I → µ + µ − annihilation process, and other processes are needed to accelerate the DM annihilation. As shown in Fig. 4, for min(m Z , m S ) < m X I , the X I X I → Z Z or SS will be open and play a main role in achieving the correct relic density. Then the masses of X R , E 1 , E 2 and N are allowed to have sizable deviation from m X I . When min(m Z , m S ) is larger than m X I and the X I X I → Z Z , SS processes are kinematically forbidden, min(m E 1 , m E 2 , m N , m X R ) is required to be closed to m X I so that the correct DM relic density is obtained via their co-annihilation processes.
The X I has no interactions to the SM quark, and its couplings to the muon lepton and vector-like leptons are constrained by the muon g − 2 anomaly. Therefore, the model can easily satisfy the bound from the direct detection of DM. At the LHC, the vector-like leptons are mainly produced via electroweak processes, then the decay modes include If kinematically allowed, the following decay modes will be open, The 2µ+E miss T event searches at the LHC can impose strong constraints on the vector-like leptons and DM. The production processes of 2µ+E miss T in our model are very similar to the electroweak production of charginos and sleptons decaying into final states with 2 + E miss T analyzed by ATLAS with 139 fb −1 integrated luminosity data [92]. Therefore, we will use this analysis to constrain our model, which is implemented in the MadAnalysis5 [93][94][95].
We perform simulations for the samples using MG5 aMC-3.
If the DM relic density is achieved via the co-annihilation processes of vector-like lepton, the mass of vector-like lepton is required to be closed to m X I . As a result, the µ from the vector-like lepton decay is too soft to be distinguished at detector, and the scenario can easily satisfy the constraints of the direct searches at LHC. Here, we employ the ATLAS analysis of 2 + E miss T in Ref. [92] to constrain another scenario in which 1 < m X R /m X I < 1.15 and m Z (m S ) > m X R is taken, and the co-annihilation processes of X R can play a main role in achieving the correct relic density. Thus, the masses of the vector-like leptons are allowed to be much larger than m X I .
We impose the constraints of "pre-muon g − 2", the diphoton signal data of the 125 GeV Higgs, the muon g −2 anomaly, the DM relic density, and the direct searches for 2 +E miss min(m E 1 , m E 2 ) is favored to be larger than 500 GeV for min(m E 1 , m E 2 ) − m X I > 300 GeV.
The DM mass is allowed to be as low as 100 GeV if min(m E 1 , m E 2 ) − m X I < 60 GeV or min(m E 1 , m E 2 ) − m X I > 400 GeV.

V. CONCLUSION
In this paper we discussed the CDF W -mass, the muon g − 2, and the DM observables Here the coupling constants g f 1 f 2 LX and g f 1 f 2 RX are from (A5)