Impact of the CDF $W$-mass anomaly on two Higgs doublet model

We consider the implication of the recent CDF $W$-mass anomaly in the general framework of two Higgs doublet model. We find that the large deviation of the $S$ and $T$ parameters from their SM values of zero leads to the upper limit of about $1$ TeV on the heavy charged and neutral Higgs bosons when it is combined with the theoretical constraints from the perturbative unitarity and for the Higgs potential to be bounded from below.

In this Letter, taking the framework of two Higgs doublet model (2HDM), we report that the large deviation of the T parameter, especially, from its SM value of zero results in the upper limit of about 1 TeV on the masses of the heavy charged and neutral Higgs bosons when it is combined with the theoretical constraints from the perturbative unitarity and for the Higgs potential to be bounded from below.
which contains 3 dimensionful quadratic and 7 dimensionless quartic parameters of which four parameters are complex. In this work we consider the CP-conserving case assuming m(Y 3 ) = m(Z 5,6,7 ) = 0 but without imposing the so-called Z 2 symmetry. 1 The complex SU(2) L doublets of H 1 and H 2 can be parameterized as where

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GeV and G ±,0 and H ± stand for the Goldstone and charged Higgs bosons, respectively. For the neutral Higgs bosons, A denotes a CP-odd mass eigenstate and the two states ϕ 1 and ϕ 2 result in two CP-even mass eigenstates through mixing and one of them should play the role of the SM Higgs boson. The tadpole conditions relate the quadratic parameters Y 1,3 to Z 1,6 as follows: The 2HDM Higgs potential includes the mass terms which can be cast into the form consisting of three in terms of the charged Higgs bosons H ± , the neutral CP-odd Higgs boson A, and the two neutral CP-even scalars ϕ 1,2 . The charged and CP-odd Higgs boson masses are given by while the 2 × 2 mass-squared matrix of the neutral Higgs bosons M 2 0 takes the form Note that the quartic couplings Z 2 and Z 7 have nothing to do with the masses of Higgs bosons and the mixing of the neutral ones. We further note that ϕ 1 does not mix with ϕ 2 in the Z 6 = 0 limit, and its mass squared is simply given by 2Z 1 v 2 which gives Z 1 0.13. 1 The Z 2 symmetry in the Higgs basis might be realized by requiring the invariance of the Higgs potential under the transformations H 1 → −H 1 and H 2 → +H 2 . Therefore, by imposing it strictly, the Y 3 , Z 6 and Z 7 terms have to be removed from the Higgs potential. Without imposing the Z 2 symmetry, there appear the Higgs-mediated flavorchanging neutral currents at tree level which could be avoided, for example, by considering the models in which the Yukawa matrices describing the couplings of the two Higgs doublets to the SM fermions are aligned in the flavor space [32][33][34].
With the 2 × 2 real and symmetric mass-squared M 2 0 is given, the mixing is described by introducing the mixing angle γ between the two CP-even states ϕ 1 and ϕ 2 . Then the quartic couplings in terms of the four masses M h,H,A,H ± and the mixing angle γ. We observe that, in the alignment limit of sin γ = 0, Z 1 = M 2 h /2v 2 and Z 6 = 0 and Z 4 and Z 5 are determined by the mass differences of [35][36][37][38][39]. For the study of the CP-conserving case, one may choose one of the following two equivalent sets: each of which contains 9 real degrees of freedom, and the convention of |γ| ≤ π/2 can be taken without loss of generality resulting in c γ ≥ 0 and sign(s γ ) = sign(Z 6 ) assuming M H > M h = 125.5 GeV. The heavy Higgs masses squared are scanned up to (1.5 TeV) 2 and the quartic couplings Z 2 , |Z 3 |, and |Z 7 | up to 3, 10, 5, respectively.

III. ANALYSIS
First, we consider the perturbative unitarity (UNIT) conditions and those for the Higgs potential to be bounded from below (BFB) to obtain the primary theoretical constraints on the potential parameters or, equivalently, the constraints on the Higgs-boson masses including correlations among them and the mixing angle γ. For the unitarity conditions, we closely follow Ref. [40] taking into account three scattering matrices which are expressed in terms of the quartic couplings Z 1−7 . Using the set I in Eq. (12) for the input parameters, all the seven quartic couplings are fixed exploiting the relations given by Eq. (11). For the details of the implementation of the UNIT conditions, we refer to Ref. [41].
For the BFB constraints, we require the following 5 necessary conditions for the Higgs potential to be bounded-from-below [42]: Second, we consider the electroweak (ELW) oblique corrections to the so-called S, T and U parameters [43,44] which provide significant constraints on the quartic couplings of the 2HDM. Fixing U = 0 which is suppressed by an additional factor M 2 Z /M 2 BSM 2 relative to S and T , the S and T parameters are constrained as follows [2,45] ( with R 2 = 2.3, 4, 61, 5.99, 9.21, 11.83 at 68.3%, 90%, 95%, 99%, and 99.7% confidence levels (CLs), respectively. For our numerical analysis, we take the 95% CL limit. For the central values S 0 and T 0 and the standard deviations σ S,T , we adopt those given in Ref. [3], see Eq. (2).
Using the set I for the input parameters, the S and T parameters take the following forms [42,[46][47][48]: The one-loop functions are given by 3 In the left panel of Fig. 1, we show the S and T parameters imposing the UNIT, BFB, and ELW constraints abbreviated by the combined UNIT⊕BFB⊕ELW 95% ones. Note that the 95% CL ELW limits are adopted and the heavy Higgs masses squared are scanned up to (1.5 TeV) 2 . We find that S takes values in the range between −0.03 and 0.05 whose absolute values are smaller than σ S = 0.08, see Eq. (2). Note that the narrow region around 0 with radius about 0.004 is not allowed since the misalignment between M H ± and M A is required to achieve the sizable central value of the T parameter, see the first line of Eq. (15). Note that S is negative (positive) when M H ± > (<)M A . The T parameter takes its value between 0.12 and 0.24. Note that T is positive definite and sizable and, accordingly, M H ± = M A is not allowed. Actually, we find that the region −40 < ∼ (M H ± − M A )/GeV < ∼ 20 is ruled out at 95% CL. In the right panel of Fig. 1 panel of Fig. 2. Requiring the ELW constraint in addition to the UNIT⊕BFB ones, we find that Z 1 and γ take values near to 0 less likely while Z 5 positive ones more likely, see the right panel of Fig. 2. On the other hand, the Z 2 and Z 7 distributions remain almost the same since they are irrelevant to the masses of Higgs bosons and the mixing angle γ. The Z 3 and Z 6 distributions undergo some changes but the Z 4 distribution changes most drastically excluding the region |Z 4 | < ∼ 1. This could be understood by looking into the expression for Z 4 given in Eq. (11). Taking γ = 0 and M H = M A for the convenience of discussion, one may have with ∆ M ≡ M A − M H ± and M ≡ (M A + M H ± )/2. Therefore Z 4 can not vanish in the presence of the misalignment between M H ± and M A , which is required to achieve the sizable central value of the T parameter. We recall that the mass difference |∆ M | approaches to 100 GeV as M H ± grows, see the right panel of Fig. 1. On the other hand, the above relation could be rewritten for M as which implies that there exists the absolute upper limit on the masses of the heavy charged and neutral Higgs bosons with |Z 4 | max and |∆ M | min from the UNIT⊕BFB and ELW 95% constraints, respectively. We observe that |Z 4 | tends to increase as M H ± grows until it reaches ∼ 6 where M H ± takes its maximum value of about 1 TeV. When M H ± approaches to 1 TeV, |∆ M | converges to 100 GeV while taking its minimum value of about 30 GeV around M H ± = 450 GeV, see the upper plots in the right panel of Fig. 1 and the plot of M H ± versus M A in the left panel of Fig. 2. 4 Taking into account the full correlations among the masses of heavy Higgs bosons and the mixing angle, we find that (Z 4 /∆ M ) max ∼ 6/(100 GeV) leading to the upper limit of about 1 TeV as clearly shown in the left panel of Fig. 2.
Finally, in order to assess the reliability of our main result, we consider the variation of the upper limit on the heavy Higgs-boson masses by shifting the central value of the T parameter by the amount of ±σ T . We find that M H ± < ∼ 1, 000 −100 +400 GeV taking T 0 = 0.27 ± 0.06. We observe that the upper limit is quite stable when T 0 is larger than the nominal value of 0.27 while it grows faster for the smaller values of T 0 .

IV. CONCLUSIONS
We consider the implication of the recent CDF W -mass anomaly in the framework of 2HDM. We find that the large deviation of the S and T parameters from their SM values of zero leads to the upper limit of about 1 TeV on the masses of the heavy charged and neutral Higgs bosons when it is combined with the theoretical constraints from the perturbative unitarity and for the Higgs potential to be bounded from below.
Note added: After the completion of our work, we have received Ref. [50] in which the CDF W -mass anomaly studied in the framework of 2HDMs. They also find the upper bounds of about 1 TeV on the masses of the heavy Higgs bosons by including phenomenological constraints from flavor observables, Higgs precision data, and direct collider search limits in addition to theoretical ones. We observe that the upper limit of about 1 TeV on the masses of heavy Higgs bosons are largely unaffected by the phenomenological constraints.  4 Note that the precise value of |∆ M | and its parametric dependence depend on the order to which the S and T parameters are computed. In Ref. [3] from which we are adopting the central values and the standard deviations of the S and T parameters, the electroweak oblique parameters are computed at the one-loop order as in this work. In Ref. [26], a two-loop calculation of the W -boson mass has been performed and it is found that |∆ M | > ∼ 50 GeV.