_{ L }scalar multiplet

Supported in part by the National Key Research and Development Program of China (2021YFC2203003, 2020YFC2201501) and the National Natural Science Foundation of China (NSFC) (12005254, 12147103)

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

We explain the

Article funded by SCOAP^{3}

The CDF-II Collaboration recently reported their new measurement of the

However, the most recent prediction of

Thus, it is clear that the discrepancy between the latest CDF-II value and SM calculation of

In light of the potential importance of the scalar multiplet extension of the SM, we focus on its solution for the ^{
①
}, the dominant contributions to the

The paper is organized as follows: In Sec. II, we begin by setting the notation of a general

Generally, we label the scalar multiplet as

where

The EW covariant derivative after EW symmetry breaking can be written as

where

where

The gauge interaction terms stemming from the kinetic term

In this type of model, the potential constructed by the Higgs doublet

Here, we only list the most general interaction terms for any scalar multiplet

However, as shown in Appendix B, they are not independent because they can be represented as linear combinations of terms proportional to

NP effects in the EW sector are usually imprinted by three oblique parameters,

where ^{
②
}

where the functions

where

In the present model, the scalar multiplet would induce additional contributions to the

One-loop Feynman diagrams for the

One-loop Feynman diagrams for the

where the function

In this study, we rederive the above scalar contributions to

According to Eqs. (14) and (15), mass differences among components in the additional scalar multiplet

Consequently, we focus on this term in our discussions on specific models. Furthermore, note that

With the general expression of the one-loop correction to the

First, we consider the scalar multiplet to be in a real representation under the weak isospin

where the Latin indices with the values of 0 or 1 denote those under the

We can also write the multiplet in terms of its components as

Thus, the transformation in Eq. (17) can be expressed by

After EW symmetry breaking, the SM Higgs doublet obtains its VEV and can be written in the unitary gauge as

Then, we have

where the term on the right-hand side of the last relation gives rise to mass splitting among the scalar components from

which leads to the following relations:

According to Eq. (24), for any integer

Note that the case with a real scalar multiplet without its VEV is usually regarded as a natural DM candidate [

For a complex multiplet, Eq. (22) shows that each component in the scalar multiplet can obtain the following mass correction from

Because

so that

In light of Eq. (15), the scalar multiplet

(color online) Parameter spaces in the

(color online) Parameter spaces in the

In this subsection, we pay attention to the models in which a complex scalar multiplet with

Type A: If

Type B: If

(color online) Parameter spaces in the

(color online) Legend is the same as in

In light of Secs. III B and III C, we find that the value of

(color online) Legend is the same as in

(color online) Legend is the same as in

From

In contrast, as shown in

where the subscript

Finally, we conclude this section by mentioning that the EW global fit constraints we apply here from Refs. [

In light of the recent measurement of the

It was argued in Ref. [

Finally, we comment on the possible collider signatures for such scalar multiplet extensions of the SM. In particular, note that for a high-dimensional representation, the multiplet contains highly electrically charged states in the spectrum, which would give us spectacular collider signals at the LHC. We take an example with an extra scalar multiplet of

Dominant production channels for new scalar particles in the multiplet in the Type-A models with

Decay chains of the scalar particles when the scalar multiplet has a VEV in the Type-A model with

Decay chains of the scalar particles when the high-dimensional operators are introduced in the Type-A model with

In this appendix, we explicitly write the EW gauge couplings of a general scalar multiplet in the Lagrangian, which are useful in our calculation of the EW oblique parameters

and the weak charged current part is given by

where the coefficient

We can derive the Feynman rules for these EW gauge couplings from the Lagrangian, in which the three-point vertices are shown in

Three-point EW gauge couplings for components in a multiplet.

Four-point EW gauge couplings for components in a multiplet.

In this appendix, we show that the possible potential terms given in Eq. (10) can be written as linear combinations of other terms already existing in Eq. (9). We begin our discussion by expanding the operator

where

in which the lowercase Latin indices are those for the fundamental

When computing the oblique parameters

in which

where

where

Moreover, to obtain the corrections to the oblique parameter

where

with

and

In this appendix, we rederive the analytic expressions for the one-loop contributions to the oblique parameters

where

so we can extract

The vacuum polarization of the

so we have

Therefore, the contribution of the scalar multiplet

In contrast, the expression for

where

where

based on Eq. (C7). Then, we calculate the

Finally, we calculate the

Therefore, the contribution of a scalar multiplet

Because the sum over the third isospin components

the UV divergences in Eq. (D12) are canceled. Thus, the contribution of a scalar multiplet

Note that the expressions for

The models by introducing a scalar multiplet with its VEV have already been studied in

Throughout this paper, we use the definition of the oblique parameters