# The Contribution of Charged Bosons with Right-Handed Neutrinos to the Muon g − 2 Anomaly in the Twin Higgs Models

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## Abstract

**:**

## 1. Introduction

## 2. The TH Models and the Relevant Couplings

#### 2.1. The Charged Higgses and the Yukawa Couplings to the Third Generation in Twin Higgs Models

**8**of $SO\left(8\right)$

#### 2.2. Flavor-Changing Couplings of Leptons in TH Models

## 3. The Rough Ranges of the Relevant Parameters in TH Models

- (1)
- The first constraint comes from the signal data of the 125 GeV Higgs, which is important, since the couplings of the SM-like Higgs with the fermions and the bosons in TH models can deviate from the SM largely and the production and decay modes of the SM-like Higgs may be modified severely. In the paper, we will perform the calculation of ${\chi}_{h}^{2}$ for the signal strengths of the 125 GeV Higgs, which will be shown in Section 5 and Section 6.
- (2)
- The constraints on parameter f come from the joint effects of the Z-pole precision measurements, the low-energy neutral current process, and the high-energy precision measurements off the Z-pole indirectly, and according to all these data, f should be larger than 500–600 GeV [74]. On the other hand, to control in a mild fine tuning, f should not be too large, since the fine tuning is more severe for large f. The constraints for f can also come from the flavor-changing decay $\mu \to e\gamma $: With the experimental constraints, [75,76], $\mathrm{BR}(\mu \to e\gamma )<4.2\times {10}^{-13}$, the flavor-changing decay $\mu \to e\gamma $ will give $f\sim [0.6\u20132]$ TeV.So after we take the above constraints from the electroweak precision measurements and the LHC data into account, we can assume that $500\le f\le 2000$ GeV. In our numerical evaluations, however, we have not taken f as a free parameter. Instead, we assume the characteristic mass and coupling of the composite resonances is set by m and g, respectively, which are related by the symmetry-breaking order parameter, f, as $m=gf$.
- (3)
- In the present experiments, ${m}_{{W}_{H}^{\pm}}$ has been constrained stringently [77,78,79]. The ATLAS experiment has presented the first search for dilepton resonances based on the full Run 2 data-set [77,79] and set limits on the ${W}^{\prime}$ production cross-section times branching fraction in the process$$\sigma (pp\to {W}^{\prime}X)\times BR({W}^{\prime}\to \nu \ell )$$This analysis, however, is based on the simplest models [80] such as the sequential standard model proposed by Altarelli et al. [81], which is usually taken as a convenient benchmark in the experiments. In the simplest models, the gauge particles are considered to be the copies of the SM gauge bosons, and their couplings to fermions are in the same mode as those of the SM gauge bosons, but they miss trilinear couplings such as ${W}^{\prime}WZ$ and ${Z}^{\prime}WW$, etc. Therefore, the situation that the sequential standard model [81] has acted as a reference for experimental extended gauge boson searches may be changed, and the results may be re-interpreted in the context of other new physics models [82]. In the following computation, we will check the sensitivity of the charged heavy-gauge boson with the mass range $1\le {m}_{{W}_{H}^{\pm}}\le 20$ TeV.
- (4)
- Regarding the top Yukawa ${y}_{t}$, at the EW scale, it should be the same as that in the SM, but at the higher scale, each will be different. Since in general, we assume that the top quark is connected to electroweak symmetry breaking and sensitive to new physics models, we scan the top Yukawa ${y}_{t}$ from zero to $1.5$ times of the SM top Yukawa ${y}_{t}^{SM}$. The heavy-gauge boson couplings to the lepton ${V}_{\mu}$ is also from zero to $1.5$ times of the SM couplings ${V}_{\mu}^{SM}$.From the relationship of the masses of the ordinary neutrino and the right-handed neutrino, ${m}_{\nu}$∼$\frac{{Y}_{\mu}^{2}{v}^{2}}{{m}_{{\nu}_{R}}}$ [20], we can estimate the Yukawa coupling ${Y}_{\mu}$ is roughly ${10}^{-5}$–${10}^{-3}$ when the right-handed neutrino masses are taken in the order of TeV and the ordinary neutrino masses are assumed as ${10}^{-3}$–${10}^{-1}$ eV. Or, if the right-handed mass ${m}_{{\nu}_{R}}$ is free, this relationship may serve as a constraint on ${Y}_{\mu}$.

## 4. The One-Loop Muon Anomalous Magnetic Moment $\mathit{g}-\mathbf{2}$

## 5. The Analytic Expressions for the Two-Loop Muon $\mathit{g}-\mathbf{2}$ Anomaly and the 125 GeV Higgs Global Fit in TH Models

#### 5.1. The Analytic Expressions for Two-Loop Barr–Zee Muon $g-2$

#### 5.2. Global Fit of the 125 GeV Higgs

## 6. The Calculation of the Barr–Zee Diagrams and the Final Total Constraints from the One- and Two-Loop Contribution

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The triangle (

**a**,

**b**) and the penguin-type (

**c**,

**d**) diagrams for the muon anomalous magnetic moment at the one-loop level. The solid lines, wavy lines and dash lines denote the fermions, the gauge bosons and the charged Higgs, respectively, which are the same as those in Figure 3.

**Figure 2.**The one-loop muon anomalous magnetic moment for ${V}_{\mu}=0.6$, ${Y}_{\mu}=0.4$, ${y}_{\mu}=0.4$, ${y}_{t}=0.5$. The green shadow area is the discrepancy between the SM and the measurement for the anomalous magnetic moment $\Delta {a}_{\mu}$.

**Figure 3.**The potential two-loop Barr-Zee muon $g-2$ contributions from the charged Higgs ${H}^{\pm}$ and the gauge boson with fermions loop ($ff{f}^{\prime}$) in TH models, with $f=t,\phantom{\rule{3.33333pt}{0ex}}b,\phantom{\rule{3.33333pt}{0ex}}\ell $ and ${f}^{\prime}=b,\phantom{\rule{3.33333pt}{0ex}}t,\phantom{\rule{3.33333pt}{0ex}}{\nu}_{R}$, respectively, where (

**a**) is for one charged Higgs and one gauge boson, and (

**b**) is for either two charged Higgses or two gauge bosons, connecting with the triangle loops.

**Figure 4.**The comparison among the two-loop $\Delta {a}_{\mu}$ contributions of the inner lines of the charged gauge bosons and the TH heavy charged Higgs boson. The contribution from the right-handed neutrino loop with the SM charged leptons are also considered. The green shadow area is the discrepancy between the SM and the measurement for the anomalous magnetic moment $\Delta {a}_{\mu}$.

**Figure 5.**$\Delta {a}_{\mu}$ varies with ${y}_{\mu}$, ${V}_{\mu}$ and ${y}_{t}$ for ${m}_{{H}^{\pm}}=300$ GeV.

**Figure 6.**The surviving samples within 3$\sigma $ ranges of ${\chi}_{h}^{2}$ on the planes of ${V}_{\mu}\sim {y}_{\mu}$, ${y}_{\mu}\sim {y}_{t}$ and ${V}_{\mu}\sim {y}_{t}$.

**Figure 7.**The samples satisfying the constraints of Higgs global fit ${\chi}_{h}^{2}$ within 3$\sigma $ range and of the $\Delta {a}_{\mu}$ from the discrepancy between the experiments and theoretical calculation, on the planes of ${y}_{t}\sim {m}_{{H}^{\pm}}$, ${y}_{t}\sim {V}_{\mu}$, ${V}_{\mu}\sim {y}_{\mu}$ and ${y}_{t}\sim {y}_{\mu}$.

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**MDPI and ACS Style**

Liu, G.-L.; Zhou, P.
The Contribution of Charged Bosons with Right-Handed Neutrinos to the Muon *g* − 2 Anomaly in the Twin Higgs Models. *Universe* **2022**, *8*, 654.
https://doi.org/10.3390/universe8120654

**AMA Style**

Liu G-L, Zhou P.
The Contribution of Charged Bosons with Right-Handed Neutrinos to the Muon *g* − 2 Anomaly in the Twin Higgs Models. *Universe*. 2022; 8(12):654.
https://doi.org/10.3390/universe8120654

**Chicago/Turabian Style**

Liu, Guo-Li, and Ping Zhou.
2022. "The Contribution of Charged Bosons with Right-Handed Neutrinos to the Muon *g* − 2 Anomaly in the Twin Higgs Models" *Universe* 8, no. 12: 654.
https://doi.org/10.3390/universe8120654