W -boson Mass Anomaly from High-Dimensional Scalar Multiplets

,


I. INTRODUCTION
Recently, the CDF-II Collaboration has reported the most precise measurement of the Wboson mass, showing that the observed W -boson mass of m CDF−II W = 80433.5± 9.4 MeV [1] deviates the latest Standard Model (SM) prediction of m SM W = 80357 ± 6 MeV [2].The significance of this anomaly is more than 7σ, which indicates new physics (NP) beyond the SM (BSM).Therefore, it is a natural and pressing concern to introduce NP models to account for this anomalous W -boson mass.
Among a multitude of BSM scenarios, extending the SM Higgs sector by incorporating extra SU (2) L multiplets  is a promising avenue, since the modification of the scalar sector is intricately connected to the underlying mechanism for the electroweak (EW) gauge symmetry breaking and the related hierarchy problem, which can be studied through the measurement of EW oblique parameters [52][53][54][55][56][57].Furthermore, the added scalar multiplet has the potential to resolve many puzzles in the SM, such as the nature of dark matter (DM) [58][59][60][61][62][63], the origin of the matter-antimatter asymmetry [64][65][66][67], and the characteristics of the EW phase transition as well as its related stochastic gravitational wave signals [68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85].Therefore, comprehending the structure of the scalar sector could lead to a more profound understanding of the big picture of the SM and the physics beyond it.Extensive studies have been carried out in the literature to explain the CDF-II W -boson mass anomaly with low-dimensional scalar multiplets, which include a scalar singlet [3][4][5][6], a second Higgs doublet , and a scalar triplet [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][46][47][48][49][50][51].More recently, we have explored scalar multiplet scenario up to a maximum of a septuplet [45].In this letter, we aim to explain the W -boson mass anomaly with higher dimensional multiplets, at both tree and one-loop levels.In particular, for the one-loop solution, we shall focus on the case of a scalar octuplet with Y = 7/2 and zero vacuum expectation value (VEV), which is the highest dimension for a complex scalar allowed by the perturbative unitarity constraint [86].This paper is organized as follows: In Sec.II, we investigate the possibility of the treelevel explanation of the CDF-II W -boson mass excess with a high-dimensional multiplet.In Sec.III, we present a detailed phenomenological study of a scalar octuplet with Y = 7/2, which is of physical interest to explain the CDF-II W -boson mass anomaly at the one-loop level.Finally, we conclude in Sec.IV.

II. TREE-LEVEL EXPLANATION OF W -BOSON MASS ANOMALY
One may easily explain the W -boson mass anomaly by introducing a SU (2) L scalar multiplet with its VEV of the neutral component inducing an additional mass correction to the W -boson mass.However, such an idea is hampered by the fact that, since the W and Z-bosons have the common origin from the EW gauge symmetry breaking, the associated Z-boson mass should also be corrected, which was strongly constrained by the current experiments [2].In this section, we show that if the added scalar multiplet is in an odd-dimensional presentation of SU (2) L with zero hypercharge, the above problem can be resolved automatically.
Let us begin by considering a real SU (2) L multiplet ξ of dimension n = 2k + 1 with k as a positive integer denoting the weak isospin SU (2) L representation.The hypercharge of ξ is fixed to Y = 0.When ξ and the SM Higgs doublet obtain their VEVs, v ξ , and v H , from the spontaneous breaking of the EW gauge symmetry, the W and Z-boson mass terms can be written as follows where D µ denotes the covariant derivatives of the scalar fields H and ξ with g and g ′ the SM SU (2) L and U (1) Y gauge couplings, respectively.Therefore, the SM gauge boson masses are given by indicating that the Z-boson mass is not corrected in the presence of the extra multiplet scalar VEV of v ξ .It can be understood by the fact that, for Y = 0, the couplings of various components in the multiplet with the Z-boson are proportional to their electric charges, so that the neutral component and the associated VEV cannot interact with the Z boson.As a consequence, this scalar-multiplet-extended model can avoid the strong constraint from the Z-boson mass measurement [2].Further, if we take the CDF-II value of the W -boson mass as the one in our model, then the VEV of the multiplet can be estimated as follows where the SM Higgs doublet VEV is taken to be v H = 246.22GeV [2].Moreover, note that the model can be further extended by introducing a series of SU (2) L scalar multiplets with vanishing hypercharges.In this case, we can still keep the salient feature that the Z-boson mass is not modified so that the related constraint is weak.
Another critical constraint on the added scalar multiplet is provided by the constraint of the ρ parameter, which can be related to the oblique parameter T as follows [2,87] ∆ρ = αT.
In the light of the current constraint on T given by the updated electroweak global fit in Ref. [43] which includes the CDF-II W -boson mass and fixes S = U = 0, we have ∆ρ ∈ [0.00101, 0.00133] , at 2σ C.L.
Theoretically, our model predicts this important quantity as follows [2] According to Eq. ( 5), we can obtain the following 2σ region for (∆v Obviously, the multiplet VEV range in Eq. ( 7) allowed by the ρ parameter constraints is not in conflict with the CDF-II W -boson mass signal region in Eq. (3).Consequently, the CDF-II m W anomaly can be solved in this multiplet scalar extension of the SM at the tree-level, while the data of ρ strongly constrains the allowed parameter space.

III. ONE-LOOP LEVEL EXPLANATION OF THE W -BOSON MASS ANOMALY
In this section, we discuss the explanation of the W -boson mass anomaly with only the one-loop corrections from the addition of high-dimensional scalar multiplet fields.After a general review of the relation between EW oblique parameters and the one-loop W -boson mass correction, we shall focus as a concrete example on the case of a scalar octuplet with Y = 7/2 and a vanishing VEV.

A. Oblique Parameters and the W -Boson Mass
The NP effects in the EW sector are usually encoded by the three oblique parameters, namely S, T , and U [52,53], which are defined at the one-loop level as follows: where the functions ) refer to the vacuum polarizations for EW gauge bosons V (′) = γ, W, Z.Moreover, as shown in Refs.[7,53], the general one-loop corrections of non-SM scalars to the W -boson mass squared can be expressed in terms of these oblique parameters S, T , and U as follows where s W (c W ) are (co)sine of the Weinberg angle.
However, as demonstrated in Ref. [88], the parameters T and S are usually generated by dimension-6 operators, while U can only be induced by a dimension-8 operator, resulting in its significant suppression.Therefore, in a NP model augmented by a scalar multiplet, it is expected that the dominant one-loop contribution to the W -boson mass arises from T and S, and the impact of U can be neglected.Besides, Ref. [45] has explicitly confirmed this expectation by demonstrating that when the masses of extra scalars exceed 300 GeV and their multiplet dimensions are restricted to be smaller than 10, the correction of U to the W mass is at least one order of magnitude smaller than the leading ones from T and S.
According to Refs.[45,56], the contribution of a scalar multiplet ξ to T can be expressed as: where referred to the electric charge, and the function F (A, B) is given by On the other hand, the correction of ξ to S can be expressed as follows: where ξ Q I represents components of the scalar multiplet ξ with its electric charge Q.

B. Scalar octuplet Explanation of the W -boson Mass Anomaly
According to Ref. [86], the highest dimension of a complex scalar multiplet allowed by the perturbative unitarity constraint is eight.Moreover, Ref. [45] has shown that a real scalar multiplet is unable to generate the mass splitting to produce nonzero values of T and S, which is required to explain the W -mass anomaly.Consequently, in the following, we will focus on the model by adding a complex octuplet scalar with Y = 7/2, which has not been discussed so far in the literature.
Note that the potential in this model can be constructed as follows with the Higgs doublet H and the scalar octuplet ξ Here, we present the most generic interaction terms for a scalar octuplet ξ.It is worth noting that the potential given in Eq.( 13) exhibits an Z 2 symmetry that arises naturally without the need for any additional assumptions.Note that the mass splitting can be only generated in Eq. ( 13) by the following term: Therefore, we will concentrate on this term in our subsequent phenomenological studies.
For the model with a scalar octuplet ξ of Y =7/2, the phenomenology can be classified into the following two distinct types based on the sign of λ 4 : • Type A: when λ 4 > 0, the lightest particle in the octuplet is the most electrically charged one with its mass denoted as M C , so that M L = M C .
• Type B: when λ 4 < 0, the lightest particle in the octuplet is the electrically neutral one with its mass denoted as M 0 , indicating M L = M 0 .
In light of the mass splittings among scalars from O 4 , this Y = 7/2 scalar octuplet has the potential to explain the W -mass anomaly with its following nonzero corrections to the parameters T and S: Figs. 1 and 2 display the parameter spaces in the M L -√ ∆m 2 plane for the Type-A and Type-B scalar octuplet models, respectively.The horizontal axis denotes the mass of the lightest particle in the octuplet (M L ), ranging from 1700 GeV to 5000 GeV, while the vertical axis denotes the mass splitting between adjacent components.The pink region in the figure indicates the parameter space allowed by EW global fits for T and S at the 2σ CL when U = 0, as reported in Ref. [88].The cyan area corresponds to the parameter space that can explain the measured W -boson mass by CDF-II within the 2σ CL.The red solid line represents the scalar mass difference corresponding to the perturbative limit |λ 4 | = 10.The results show that the Type-A model has a substantial amount of parameter space that can solve the CDF-II m W anomaly while satisfying the EW global fits and perturbative limits.
On the other hand, for the Type-B model, it is seen from Fig. 2 that the CDF-II preferred region that explains the W -boson mass excess is entirely ruled out by the global fits of EW precision observables.

IV. CONCLUSION AND DISCUSSION
In order to explain the excess of the W -boson mass recently observed by the CDF-II Collaboration, we have studied two promising candidate scenarios both involving highdimensional SU (2) L scalar multiplets.The first scenario is to introduce an odd-dimensional scalar multiplet with Y = 0 so that the W mass correction from its nonzero scalar VEV can explain the CDF-II excess at the tree level.One salient feature of this mechanism is that the scalar VEV does not contribute to the mass of Z boson, so that it can easily evade the associated strong constraint from m Z .By estimating the experimental bound on the ρ parameter through its relation to the T parameter, we find that the regions permitted by the ρ parameter is enlarged by about one order of magnitude, and thus the tree-level scalar multiplet scenario is available to explain the CDF-II W mass anomaly even though the allowed parameter regions is strongly reduced.Then, we turn to the second scenario in which the CDF-II anomaly could be possibly solved by one-loop effects generated by the additional scalar multiplet.We take a scalar octuplet as a concrete example to investigate this possibility.By making use of the octuplet scalars' contributions to the oblique parameters S and T , it is found that the CDF-II measured W -boson mass can be easily obtained.We also take into account the constraints from the EW global fits and the perturbativity limit in this model.For the Y = 7/2 case, the current data from the EW precision measurements has excluded the Type-B model in which the lightest particle is the electrically neutral one.
In contrast, for the Type-A model in which the lightest scalar is the most charged one, it is found that there is a substantial parameter region explaining the CDF-II m W anomaly while still allowed by the experimental and theoretical constraints.In the latter case, the lightest scalar mass is found to lie in the range from 1800 to 5000 GeV.
In this letter, we have exclusively investigated two representative cases, namely the highmultiplet scenarios with tree-level corrections and one-loop effects.In more generic cases, both mechanisms should work to solve the W -boson mass anomaly like a model added by a scalar triplet with Y = 0 [31,43,46].Finally, we would like to mention that, for the available Type-A octuplet model with Y = 7/2 and a vanishing scalar VEV, due to the accidental Z 2 symmetry, the lightest but also most charged particle ξ ±7 should be nearly stable.This particle together with its less charged but heavier partners in the octuplet can be pair produced via the Drell-Yan processes.By introducing an additional high-dimensional effective operators such as ones in Ref. [45], the Z 2 symmetry is explicitly broken and the lightest particle ξ ±7 can decay into multiple-W or multiple-lepton final states.However, owe to the suppression from the high dimension of operators as well as the multiplicity of produced particles in the final states, a simple estimate shows that ξ ±7 should be long-lived with its lifetime of O(s).This implies that ξ ±7 at the LHC should be stable and only leaves charged tracks in the detectors [90,91].Currently, the most recent search at the LHC for such a highly charged particle has been performed in Ref. [92], which gives the lowest bound on its mass to be around 1700 GeV.For other charged counterparts such as ξ ±i with i ≤ 6, they would decay in a cascade into ξ ±7 by emitting several W -bosons, giving a signature of multiple W 's plus charged tracks, which potentially serves as a key signal.The detailed investigation of these LHC signatures is well beyond the scope of our work, so that we leave them for future studies.

Y = 7 / 2 ,FIG. 1 .
FIG. 1.The parameter region in the M L − |∆M 2 | plane for the Type-A scalar octuplet model, in which the lightest scalar is the most charged one, i.e., M L = M C .The horizontal axis denotes the mass of the lightest particle in the octuplet (M L ), ranging from 1700 GeV to 5000 GeV, while the vertical axis denotes the mass splitting between adjacent components.The pink region in the figure indicates the parameter space allowed by electroweak global fits for T and S at the 2σ CL when U = 0, as reported in Ref. [88].The solid cyan area corresponds to the parameter space that can explain the measured W -boson mass by CDF-II within the 2σ CL.The red solid line represents the mass difference for |λ 4 | = 10, and the region below this line satisfies the perturbativity condition.