Higgs Quadruplet Impact on W Mass Shift, Dark Matter, and LHC Signatures

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The Standard Model (SM) of quarks and leptons has one Higgs doublet Φ = (ϕ + , ϕ 0 ).When the gauge symmetry SU (2) L ×U (1) Y undergoes spontaneous symmetry breaking due to ⟨ϕ 0 ⟩ = v 0 , a well-established tree-level condition emerges: ρ = M 2 W /M 2 Z cos 2 θ W = 1.The recent measurement of the W boson's mass by the CDF experiment is given by m CDF W = 80.4335 ± 0.094 GeV [1], which shows 7σ deviation from the SM prediction: m SM W = 80.357 ± 0.006 GeV.This has prompted theorists to speculate about potential new physics contributing to this unexpected increase in the W boson's mass.
If an additional Higgs multiplet with an isospin I is introduced to the SM, the vacuum expectation value v 1 of its neutral component, having third component of isospin I 3 , contributes 2(g 2 / cos 2 θ W )I 2  3 v 2 1 to M 2 Z , and g 2 [I(I + 1) − I 2  3 ]v 2 1 to M 2 W .As a result, ρ deviates from unity except for certain specific values of I and I 3 .Notably, the ρ parameter derived from electroweak global fits is ρ = 1.0002 ± 0.0009 [2].Therefore, this leads to stringent constraints on the value of v 1 .
In this letter, we study the effect of adding a Higgs quadruplet ζ = (ζ ++ , ζ + , ζ 0 , ζ − ) to the SM.The motivation for extending the SM with a Higgs quadruplet primarily stems from the quest to resolve the neutrino mass problem.The contribution of ⟨ζ This could then explain the W mass shift, in agreement with the recent precision measurement [1].Whereas ζ is necessary for the quintuplet neutrino seesaw mechanism [3,4], it may also be the connector in Type III seesaw [5].In our model, another use of ζ is proposed, as the connector to the dark sector consisting of a neutral Majorana fermion singlet N and a Dirac fermion quadruplet Consider the Higgs quadruplet ζ.It interacts with 2 , where (1) For the fermion quadruplet Σ, the corresponding inter- We assume that Σ is odd under a dark Z 2 symmetry together with a neutral singlet fermion N .Hence ζ † ΣN is allowed and ζ becomes the connection between the SM and the dark sector.The Higgs potential consisting of Φ and ζ is given by [4] where the sum over i = 1, 2, 3 represents the components of the tensor product of the scalar fields.There are in principle 4 quartic terms of the type corresponding to the pairings 1 × 1, 3 × 3, 5 × 5, and 7 × 7, where 1 × 1 indicates the multiplication between singlet components of each product, whereas 3 × 3 rep-resents the multiplication between triplet components, and so on.However, the 5 × 5 term is proportional to 1 × 1.The sum of 3 × 3 and 7 × 7 is also proportional to 1×1.Hence there are only two independent terms.As for the terms mixing ζ with Φ, ( Let This is the analog of the scalar seesaw studied previously [6,7].
The linear combination proportional to v 0 Im(ϕ 0 ) + v ζ Im(ζ 0 ) becomes the Goldstone boson for Z whereas its orthogonal combination has mass-squared = (λ The linear combination proportional to neglecting terms of order v 0 v ζ and smaller.The doubly charged scalar boson The other relevant renormalizable interactions can be written as Let us now discuss the impact of our model in light of the CDF collaboration measurement of the W boson mass M W = 80433.5± 9.4 MeV [1] collected at the CDF-II detector of Fermilab Tevatron collider.The recently measured value of W −mass has a 7σ departure from the SM expectation (M W = 80357 ± 6 MeV).This has led to various different proposals on the feasible implications and interpretations related to electroweak precision parameters [8][9][10][11], BSM physics like DM [12][13][14][15][16][17], additional scalar fields [18][19][20][21][22][23][24][25][26][27][28][29], effective field theory [30,31], supersymmetry [32][33][34][35][36] and several others [37][38][39][40][41][42][43][44][45][46][47][48][49][50].We study the fit to the new measurement [1] of the W boson mass, using newly added quadruplet ζ.The new physics contributions to W −boson mass anomaly can be parametrised in terms of the oblique parameters S, T, U [51,52].Considering the U parameter to be vanquished, any BSM physics contribution to W − boson mass can be parametrised in terms of S and T parameters.Taking the fine-structure constant α, the Fermi constant G F , and Z boson mass M Z as input parameters, the fitting of S, T parameters in view of the recent W -mass anomaly has been discussed in [31].It is very important to note that, a change in the oblique parameters due to BSM physics will also change the precisely measured weak mixing angle θ W .The sin 2 θ W (m Z ) M S and the mass of W boson m W can be expressed in terms of S and T parameters as [53], From the above two equations, it can be inferred that the compatibility of the new measurement of W −boson mass with the θ W requires both S and T parameters to be non-zero, also seen from the fits shown in [31].However, in the case of a scalar quadruplet ζ with hypercharge 1/2, we will get a correction to the T parameter only as also shown in [9].To take into account the enhanced W − boson mass, the required limit on the T parameter is T = 0.17 ± 0.020889.
However, following Eq.( 11), any change in the T parameter with S = 0 will put the weak mixing angle sin 2 θ W in tension with the LEP data.The abovementioned range of T would imply that sin 2 θ W should lie in between 0.230746 − 0.230854.An additional contribution to S, T parameters would be required to reduce this tension.The T parameter will get a new contribution at tree-level from the vev of ζ (v ζ ) and can be written as T = 6v decide the mass of the doubly charged scalar ζ ++ and we have discussed the impact of ζ ++ in collider in the later part of this letter.As mentioned above, the dark fermions N and Σ = (Σ ++ , Σ + , Σ 0 , Σ − ) are connected to the SM through ζ with the Yukawa coupling y N ζ † Σ L N L + H.c. and the presence of an unbroken Z 2 symmetry ensures the stability of either N or the neutral component of Σ to be a viable dark matter candidate of the model.As both ζ and Σ are charged under SM gauge symmetry, they can be thermalized with the SM bath through their gauge interactions.
Assuming N is lighter than Σ, N also can be thermally produced in the early universe for a sizable y N and eventually freezes out when its interaction rate drops below the expansion rate of the universe.Assuming, ζ is lighter than the dark matter N , it can dominantly annihilate into the pair of ζ particles via the Yukawa interaction shown in equation (10).There can also be significant contributions from the coannihilation with different components of Σ.The dominant annihilation and coannihilation channels of N for different final states are shown in figure 2. As a result, the important parameters which can affect the relic abundance are the dark matter mass (M N ), the Yukawa coupling (y N ), and the mass splitting between Σ and N which is defined as ∆M = M Σi − M N .
In figure 3, we have shown the allowed parameter space in M N − ∆M plane as the variation of y N is shown through the colour code.It is important to note that the relic density can only be satisfied for larger mass splitting ∆M .For smaller mass splitting there can be huge coannihilation which suppresses the relic abundance which also sets a correlation with the Yukawa coupling y N .The smaller the mass splitting stronger the coannihilation and one needs to reduce the Yukawa coupling to reduce the cross-section.
Another interesting dark matter phenomenology can arise in the case of tiny Yukawa coupling (y N ).This can lead to the non-thermal production of dark matter.For tiny y N , dark matter can be directly produced from the direct decay of SM Higgs.The 3 × 3 mass matrix spanning Assuming that m N is of order GeV and m Σ is very much heavier, the N − Σ mixing is y N v ζ /m Σ .Consider now the decay of the SM Higgs h to N N .It does so first through h − H mixing which is roughly v ζ /3v 0 , then through N − Σ mixing as just noted.The effective coupling is The decay rate of where x = m N /m h .The correct dark matter relic abundance is obtained [57] if  tively suppress the background while preserving our signal.Table I presents the comprehensive list of all cuts employed to eliminate the backgrounds.The concluding distributions of these events are depicted in Fig 5, showcasing the invariant mass distribution for two same-sign leptons both pre and post the application of cuts.A notable observation is that following the implementation of cuts, the signal exhibits a higher count of events compared to the background.
We also explore another scenario to probe ζ ±± via the process pp → ζ ++ ζ −− → 2W + 2W − → 4l (l = e, µ)+ MET.However, in this scenario, the cross sections for masses of 400 and 500 GeV are notably small, amounting to 0.016 (0.0056) fb for m ζ ±± = 400 (500) GeV.These extremely low cross sections (in fractions of fb) result in an exceedingly small number of events compared to the relevant background.

FIG. 3 :
FIG.3:The allowed parameter space in ∆M −MN plane from the relic density constraints where the colour code represents the variation of the Yukawa coupling yN .
provided that the reheat temperature of the universe is above m h but well below m H and m Σ .For m N ∼ 1 GeV, this implies m Σ /y 2 N ∼ (v ζ /GeV) 2 (1.2 × 10 8 GeV).We now turn to probe ζ ±± at the LHC.The sole method of producing the heavy Higgs Quadruplet ζ ±±