Interpreting the 3 TeV $WH$ Resonance as a $W'$ boson

Motivated by a local $3.2-3.4$ sigma resonance in $WH$ and $ZH$ in the ATLAS Run 2 data, we attempt to interpret the excess in terms of a $W'$ boson in a $SU(2)_1 \times SU(2)_2 \times U(1)_X$ model. We stretch the deviation from the alignment limit of the Equivalence Theorem, so as to maximize $WH$ production while keeping the $WZ$ production rate below the experimental limit. We found a viable though small region of parameter space that satisfies all existing constraints on $W' \to jj, t \bar b, WZ$, as well as the precision Higgs data. The cross section of $W' \to WH$ that we obtain is about $5-6$ fb.


I. INTRODUCTION
Recently, the ATLAS Collaboration [1] reported an experimental anomaly in W H or ZH production in the qqbb final state at √ s = 13 TeV with an apparent excess at around 3 TeV resonance mass region. Note that CMS also searched for the same channels [2]. Though they did not claim observing anything peculiar, we can see that there is a visible peak of more than 2σ at around 2.7 TeV. Currently, the CMS observation does not support the 3 TeV excess of ATLAS base on the narrow width resonance analysis. The broad width analysis has not been fully studied, and so it is hard to make conclusion for broad width resonance case. We shall focus on interpreting the ATLAS result while we emphasize that the CMS result does not falsify the ATLAS result. The excessive cross section is roughly [1] (which is estimated from the 95% CL upper limits on the cross section curves) A similar excess was seen in ZH production. The local excesses are at about 3.2 − 3.4σ for both W H and ZH channels at around 3 TeV, while the global significance is about 2.2σ.
Nevertheless, the boosted hadronic decays of W and Z have substantial overlap at about 60% level, which means that it is difficult to differentiate between the W and Z bosons. In the following, we focus on the excess interpreted as a 3 TeV W H resonance.
We attempt to interpret that there is a 3 TeV spin-1 resonance W that decays into W H.
The W can arise from a number of extended symmetric models, e.g., SU (2) 1 × SU (2) 2 × U (1) X [3,4]. With an additional SU (2) symmetry, which is broken at the multi-TeV scale, there will be extra W and Z bosons, whose masses may be similar or differ depending on the symmetry-breaking pattern. Then the decay W → W H can explain the excess with a resonance structure. Similarly, the Z → ZH can explain the excess in ZH production.
Here we focus on the W H channel.
The W boson couples to the right-handed fermions with a strength g R , independent of the  [5]. In the model that we are considering, it is indeed true in the alignment limit β → π/2 + α. Here we attempt to explore how much we can deviate from the alignment limit so that the W H channel can be enhanced while suppressing the W Z, thus satisfying the constraint from W Z [6-10], dijet [11,12], and precision Higgs data [13]. 1 The organization of this note is as follows. In the next section, we describe the SU (2) 1 × SU (2) 2 × U (1) X model that we consider in this work. In Sec. III, we demonstrate the deviation from the alignment limit. In Sec. IV, we discuss all the relevant constraints. We present the results in Sec. V, and conclude and comment in Sec. VI.
metry. In addition to the SM fermions and gauge bosons, this model also contains new gauge bosons W , Z , the right-handed neutrinos N R , and also some extra scalars from the extended Higgs sector: a complex SU (2) R triplet T and a complex SU (2) L × SU (2) R bidoublet Σ. We summarize the particle contents and gauge charges in Table I of this We focus on the extended Higgs sector to study the mass and mixing of new gauge bosons W , Z . There are two steps of symmetry breaking from two sets of complex scalar fields, separately. First, the SU (2) R triplet scalar T = (T ++ , T + , T 0 ) breaks SU (2) R × U (1) B−L to U (1) Y by acquiring a large vacuum-expectation value (VEV) at the multi-TeV scale.
The heavy masses of W and Z are set by u T . Second, the SU (2) L × SU (2) R bidoublet scalar, develops a VEV at the electroweak scale v = (v 2 The phase α Σ is CP-violating, and we do not include its effects in this work. The ratio tan β = v 2 /v 1 of two VEV's follows the same notation as two-Higgs-doublet models (2HDM). This symmetry breaking induces a small mixing between the charged gauge bosons.
Explicitly, the field content of Σ is given by with the H being the observed 125 GeV Higgs boson, H the heavy Higgs boson, H ± the charged Higgs boson, A the pseudoscalar Higgs boson, and G ± , G 0 the Nambu-Goldstone bosons.
We are interested in the energy scale u T much larger than the electroweak scale v. Therefore, the scalar fields from the triplet T are decoupled from the electroweak scale. At the energy scale lower than u T , the scalar sector only consists of the bidoublet Σ, which is the same as the 2HDM with the doublet fields H T . The electrically-charged states, W ± L and W ± R , of the SU (2) L and SU (2) R symmetries will mix to form physical gauge bosons, W ± and W ± , The W ± L − W ± R mixing angle φ w satisfies and the W and W masses are given by where g L and g R are the SU (2) L × SU (2) R gauge couplings. We assume that the mass of the right-handed neutrino is heavier than the W , such that the decay W → l R N R is kinematically forbidden.
There are other possible decay modes for the W into other Higgs bosons [3] if they are kinematically allowed: e.g., Such decay widths depend on the mass parameters and are highly model dependent, and so we treat the sum of these decay widths as a restricted variable parameter denoted by Γ other W .

III. DEVIATIONS FROM THE ALIGNMENT LIMIT
In this section, we would derive the W W Z and W W H couplings in this SU (2) L × SU (2) R × U (1) B−L model, using the 2HDM convention, by rewriting the bidoublet Σ in . The deviation from the ET, Γ(W → W Z) = Γ(W → W H), can be realized, if the mixing angles α and β in 2HDM stays away from the alignment limit. Or vice versus, the ET is restored when β → α + π/2. The mass mixing term between W and W comes from the bidoublet and is given by, with the VEV's of the decomposed doublets denoted by v 1 = v cos β and v 2 = v sin β, Note that the factor of 2 in front comes from two ways of matchings. So the induced mixing is described by Similarly, there is mixing between Z and Z . The mixing angle φ w induces the coupling The two contributions sum up to However, the leading vertex W † W H is given not explicitly from the mixing, but derived by the following steps, Therefore, Similarly, the Goldstone boson G 0 , associated with Z, also accompanies with H.
In summary, g W W G 0 = ig R m W sin 2β, and g W W H = g R m W cos(α + β). Thus, we obtained the decay widths for W → W Z and W → W H in the limit m W m W,Z,H .
In the alignment limit, α → β − π 2 , the two widths above become equal. As ET identifies G 0 with the longitudinal Z, we expect the relations, We are going to illustrate the operation of the ET. The longitudinal W + is identified with G + = cos βΦ + 1 + sin βΦ + 2 in Eq. (4). The action of W moves entries within the same row in the 2 × 2 matrix form of the bidoublet. Therefore the amplitude The factor (p + −p 0 ) corresponds to the Feynman amplitude for the convective current, which is contracted with the polarization vector of W . The above amplitude should give the On the other hand, we can start from the tri-gauge coupling of the anti-symmetric Lorentz form, Now using the ET, we concentrate at the longitudinally polarized W of + ≈ p + /m W and Z of 0 ≈ p 0 /m Z . Up to an over factor 1 m W m Z , we obtain Therefore, the longitudinal amplitude from Eq.(10) agrees with the other calculation based Integrating out the angular parameter θ, the decay width is which is in agreement with Eq. (15).
Following the similar method, we can verify the coupling of W W H in this model by using Then the coupling of W W H is In the alignment limit, β → α + π/2, the W W H coupling goes back to the SM Higgs-gauge boson coupling.
Gauge-boson and fermonic couplings of the 125 GeV Higgs boson are now well measured by ATLAS and CMS, especially, the couplings to the massive gauge bosons. The deviations from the SM values shall be less than about 10%, i.e | sin(β − α)| > ∼ 0.9. That implies the allowed range of | cos(β − α)| < ∼ 0.44. Weaker limits for the couplings to up-and down-type quarks from Higgs precession data also dictate the α and β's parameter region. Therefore, in this model framework, the Higgs precision data would set the boundary on the deviation from the alignment limit, and thus restrict the ratio of Γ(W → W Z) and Γ(W → W H).
The robust and detailed allowed region of α and β from Higgs precision data depends on different types of 2HDM's. For the allowed parameter region, we refer to Ref. [13], where Type-I, -II, Lepton-specific, and Flipped 2HDMs have been studied. The universal feature from their results, in the small tan β 0.1 region, the allowed cos(β − α) is close to the alignment limit, i.e | cos(β − α)| < ∼ 0.05. This is because the universal up-type quark Yukawa coupling among the 2HDMs is enhanced by factor 1/ sin β. For tan β > ∼ 2 region, only the Type-I case allows more dramatic deviation from the alignment limit. For instance, taking tan β = 2.5, the allowed range from Higgs precision data is −0.37 < cos(β − α) < 0.42.
Because only in Type-I case, all the up-, down-quark and leptonic Yukawa couplings deviate from SM values by the same factor (cos α/ sin β), such that larger tan β would not enhance any of these couplings, and they are therefore less constrained by Higgs precision data. We shall use the results of Type-I 2HDM obtained in Ref. [13] to restrict the parameter of our model.

IV. CONSTRAINTS FROM EXISTING DATA
Recently, both ATLAS and CMS collaborations have published their W searches with different decay channels, including fermionic final states l ± ν [15], dijet [11,12], tb [16], and also bosonic final states W ± Z [6-10] at 13 TeV. Here we list all the constraints from these searches in Table II. Here j includes all light flavors, l includes (e, µ) and ν includes (ν e , ν µ , ν τ ). Finally, J means large-R jets (W jet or Z jet).
As we can see from Table II that the strongest constraint comes from W ± → l ± ν searches, but here we choose the leptophobic version of the model such that this constraint will not cause serious effects on our results. On the other hand, the dijet constraints from both the ATLAS and CMS analyses rely on the acceptance (A) and the width-to-mass ratio (Γ/M ) effects. Note that the dijet limits quoted in Table II analyses and the width-to-mass ratio effects are from Table 2 in [11] for ATLAS analysis and Table 4 in for CMS analysis [17] to rescale in our case. 2 Another set of constraints come from the precision Higgs boson data, including the gauge-Higgs couplings, Yukawa couplings, and the Hγγ and Hgg factors. In 2HDMs, such constraints can be recast in terms of tan β and cos(β −α). The excluded region in the parameter space of Type-I 2HDM is shown explicitly in the upper-left panel in Fig. 1 [13].

V. RESULTS
In Fig. 1, we show the aforementioned experimental constraints on the parameter space of SU (2) L × SU (2) R × U (1) B−L model, and include the non-standard W decay width Γ other W . The red-dotted points satisfy the requirement on the signal cross section  Table II, and the dijet upper limit adapted for the broad-width resonance. The cyan (green) hatched region was excluded by the Higgs precision data of Type-I 2HDM [13,14]. evaluated in the narrow-width approximation, and the upper limits listed in Table II, except for the dijet upper limit. The dijet limits are adapted to the broad-width-resonance case, following the instructions in Ref. [11,17]. The excess bump in the m JJ distribution of the 2-tag W H channel from ATLAS [1] is not necessarily a narrow resonance, likewise, we do not restrict the width of W to be narrow. The cyan (green) hatched region was excluded by the combined 7 and 8 TeV ATLAS and CMS signal strength data (the ATLAS data only) [13,14]. The shifting of the hatched region is mainly due to the change in the diphoton signal strength from µ γγ (ggF ) = 1.32 ± 0.38 to 1.10 +0.23 −0.22 . Most of the red-dotted points are ruled out by this constraint, yet there exists a small region that satisfies all the existing constraints and Higgs precision data.
Nevertheless, as shown in Fig 1 there exists a small region of parameter space that is not excluded by the aforementioned constraints, including all those listed in Table II ( can be rewritten as Also, from the lower-right panel we can see that without the non-standard decay of W , the A few comments are offered as follows.

Below the symmetry breaking scale of SU
the Higgs field can be recast into two doublet Higgs fields, in a manner similar to the conventional 2HDM. Therefore, the model is also subject to the constraints from the precision Higgs data. The ATLAS publication [13] has presented the excluded region in various 2HDM's. We adopted the least restricted one -Type I -in this work, and showed the excluded region in the upper-left panel Fig. 1 couplings are proportional to cos α/ sin β. Therefore, based on the constraint from Higgs precision data, sin(β − α) ≈ 1, and so that α β − π/2. It implies that cos α/ sin β 1. Hence, there is no tan β enhancement in contrast to the Type II model. Therefore, as long as tan β > ∼ 1, the constraint on the charged Higgs mass is rather weak. Another important constraint is the ρ parameter (or ∆T ) being very close to 1 -the custodial limit. It can be fulfilled by taking the mass splitting among A, H , H + to be small. We therefore set m A ≈ m H ≈ m H + .
3. We have adopted the leptophobic condition for the W boson, or by assuming the right-handed neutrino is heavier than the mass of W .
4. Note that the boson jets for W and Z bosons are overlapping at 60%. We do not work out for the Z → ZH boson in this work, but it can be done similarly. However, leptophobic version is a must for the Z to avoid the very strong leptonic limit. 5. The dijet limit of pp → W → jj presented the most stringent constraint to the model.
We have to adopt other decay modes in order to dilute the branching ratio into dijets.
Possible decay modes are W + → H + A, ZH + , W + H , W + A, H + H, H + H . Searches for these modes serve as further checks on the model. 6. The ATLAS data (and also the CMS data) did not indicate a narrow resonance at 3 TeV. Therefore, we assume one more parameter (somewhat restricted) Γ other W to alleviate the constraint from dijet. As shown in Fig. 2, the resonance width is rather wide. Currently, we obtained the total width Γ W = 0.3 × m W .
7. Although there are some direct searches on A and H from the LHC [18], the constraints for Type-I 2HDM are not strong enough. Conservatively, we can focus on heavy Higgs bosons around 500 − 1000 GeV, and the interesting signatures for this mass range can