W plus two jets from a quasi-inert Higgs doublet

We show that, the result recently reported by the CDF collaboration showing an excess in the invariant mass distribution of jet pairs produced in association with a W-boson can be explained by a simple extension of the Standard Model (SM) with an additional quasi-inert Higgs doublet. The two additional neutral Higgs states H^0 and A^0 have a mass of about 150 GeV and decay into a pair of jets. W^\pm H^0/A^0 pairs are produced from the decay of the heavier charged Higgs boson H^\pm. Depending on the precise masses of the neutral and charged Higgs bosons, the model is shown to be in agreement with constraints from electroweak precision tests and from flavor physics for a broad range of the Standard Model-like Higgs mass from 100 GeV to several hundreds of GeV. Other possible signals of this model at the Tevatron and the LHC are discussed.

1 Introduction except λ 5 and µ 2 12 . We note also that vacuum stability imposes the requirements These conditions are easily fulfilled for the range of parameters relevant to this work. We assume that Φ 1 and Φ 2 interact with the SM quarks via the Yukawa interaction where Q = (u L , d L ) c denotes the SM quark doublet while u and d are the SM quark singlets.
To avoid experimental constraints from the lepton sector, at this point we choose not to couple Φ 2 to leptons.
We demand that Φ 2 develops a vanishing (or negligible) vacuum expectation value (VEV). In other words Φ 2 is not involved in the electroweak symmetry breaking. The SM SU (2) × U (1) Y gauge symmetry is spontaneously broken by φ 0 = v = 174 GeV. In order to forbid a VEV for Φ 2 at tree-level it suffices to set the µ 12 term in the scalar potential to zero. At one loop however, the µ 12 term will be generated by radiative corrections: where Λ is a cutoff scale that is naturally expected to be around a few TeV. Assuming that the Φ 2 couplings are flavor independent, and of order 0.1, we get a mixing contribution of the order of 1000 GeV 2 . This mixing term would induce a vacuum expectation value Φ 2 of order 10 GeV, leading to too large contributions of Φ 2 to the light quark masses. In order to avoid such a large contribution, a mixing mass µ 2 12 smaller than about 10 GeV 2 would be required, what may be obtained in the case of flavor independent couplings by a cancellation between the tree-level and loop-induced contributions, requiring a large fine tuning. It is however possible that Φ 2 only couples to the first generation quarks. In such case, provided that µ 2 12 is set to zero at the tree-level by some symmetry, no large fine tuning would be required.
To explicitly forbid the mixing between the two Higgs doublets, we can impose a Z 2 parity symmetry under which Φ 2 is odd [20,21], leading also to a justification of the suppression of the quartic couplings with odd powers of the fields Φ 1,2 in the scalar potential (Eq. (1)). This is similar to what is obtained in R-Parity violating models, with Φ 2 replaced by a slepton [11]. However, R-Parity violating models of this type are strongly constrained by the possible generation of neutrino masses (for a recent analysis, see Ref. [22]), a constraint that is not present in the two Higgs doublet model case.
To allow a coupling of Φ 2 to quarks, we further require that the right-handed up quark is odd under this new parity symmetry, while all the other SM fields are even. A consequence of this is that Φ 1 would not couple to the right-handed up quark and therefore the up quark would remain massless at tree-level. Were this parity symmetry preserved, it would lead to a potential solution of the strong CP-problem, which seems to be disfavored by results from chiral perturbation and lattice gauge theory [23,24,25] (see, however, Ref. [26]).
In order to generate a value of the up quark mass of about a few MeV, a soft breakdown of the Z 2 -symmetry, via a non-vanishing µ 2 12 term of about 10 GeV 2 , would be necessary.
Assuming this approximate Z 2 -symmetry in the Yukawa Lagrangian of Eq. (3), we can write down the Yukawa couplings that we use for our analysis as Note that the index α can only take the values α ∈ {2, 3} since a coupling of the righthanded up quark to Φ 1 is forbidden by the parity symmetry. We conclude this section with the Higgs spectrum of the theory: The lone Higgs boson of the Standard Model is of course h, whereas H ± , H 0 , and A 0 are the components of the new scalar doublet. Using these expressions for the masses of the scalar fields, we can rewrite the condition on the vacuum stability (Eq. (2)) as a condition on the µ 2 parameter Finally, the Z-pole precision measurement sets the limit There are also bounds coming from the direct searches for the neutral Higgs particles at LEP2, which may increase the above limit in Eq. (11) from M Z to values close to 200 GeV.
However, the precise bound on the sum of the neutral Higgs boson masses depends strongly on the decay properties of these particles. In this article, we will be interested in masses for these particles that strongly exceed the precision measurement and LEP2 bounds.

Electroweak precision constraints and flavor
We perform the analysis of the ElectroWeak Precision Tests (EWPTs) in the S-T plane, with the experimental contours taken from [27]. Since, as we will show in the next section, we have to require a rather large mass splitting between the scalars H 0 , A 0 and the charged Higgs H ± to fit the W + 2jets CDF excess, we expect a large positive New Physics (NP) contribution to the T parameter. To calculate the NP effects to S and T, we use the general formula valid for 2HDMs [28,29], under the assumptions of no mixing between the two Higgs doublets and of negligible VEV of the second doublet Φ 2 . In order to generate a W + 2j signature consistent with observation, we shall fix the mass of one of the scalars to be close to 150 GeV. In Fig would be required to be in agreement with EWPTs at the 2σ level (see the red dotted curve in Fig. 1). An alternative way of reducing the required SM-like Higgs boson mass is to assume a splitting between the A 0 and H 0 masses through a non-vanishing value of λ 5 1 . In Fig. 1, we show our results once we fix M H 0 = 150 GeV, M A 0 = 210 GeV and the charged mass to M H ± = 300 GeV (solid blue curve) or M H ± = 250 GeV (solid red curve). As we can observe from the figure, a more moderate value of the Higgs mass M h is required in order to be in accordance with EWPTs: the best fit is obtained for M h = 450 GeV (with 2σ boundary given by (290 − 660) GeV) in the case of a 300 GeV charged Higgs boson, and for M h = 150 GeV (with 2σ boundary given by (90 − 250) GeV) in the case of M H ± = 250 GeV. Observe that a heavy SM-like Higgs boson with mass up to 400 GeV will be tested at the LHC through its decay into massive gauge bosons within the next two years [30].
Furthermore flavor adds additional constraints on this model. In the mass eigenbasis, the quark-Φ 2 Higgs interaction Lagrangian involving the right-handed up quark is Clearly the GIM mechanism is violated strongly and an assumption about the flavor structure of the Yukawa matrix is needed to avoid large NP effects in flavor violating observables. We choose to make the simplifying assumption, y i1 u,2 = y u,2 δ i1 . We shall analyze two possible limits for the rotation matrix, U ij u,L = δ ij and U ij u,L = (V ij CKM ) † . Under these assumptions the flavor violating Higgs mediated couplings are For the case U ij u,L = δ ij , the charged Higgs φ − 2 interactions lead to a new source of flavor violation, beyond the SM one. These flavor violating interactions are governed by the couplings y u,2 V 1k CKM , and depending on the y u,2 value, tree-level flavor violating processes like K + → π + π 0 can put strong constraints on this scenario. The contribution to the K + → π + π 0 mode due to the charged Higgs, as compared to the Standard Model, is naively suppressed by the factor (y u,2 /g 2 ) 4 (m K /m π ) 2 (M W /M H ± ) 4 . Additionally, NP contributions to Kaon mixing appear only at the loop-level. The constraints from Kaon physics are satisfied for values of the Yukawa couplings smaller than about 0.1 and charged Higgs masses larger than 250 GeV, as required to describe the W + 2jets data within this model.
For the case of U ij u,L = V * ji CKM , the flavor violating coupling of the c-and u-quarks to φ 0 2 leads to a generically large tree-level contribution to the D −D mixing when the scalar and pseudoscalar masses are non-degenerate. In all generality, we can define y cu the coupling φ 0 2ū L,2 u 1 . In Fig. 2, we show the constraint on y cu as a function of the mass splitting The fit is performed using the experimental values given by HFAG [31]. As Fig. 2 suggests, even a few GeV splitting between M H 0 and M A 0 requires that y cu 10 −3 . This bound can be read as a bound on the coupling y u,2 for the scenario with U ij u,L = V * ji CKM (y cu = y u,2 V * 21 CKM ). We can conclude that for this scenario to be viable we need H 0 and A 0 to be almost degenerate in mass.
A comparison of these two flavor scenarios suggests that the flavor constraints on the U ij u,L = δ ij scenario are weaker and a substantial splitting in the Higgs masses is allowed. With sizable splittings between the Higgs masses M H 0 and M A 0 , the electroweak precision tests are consistent with a lighter Standard Model-like Higgs, as is shown in Fig. 2. For the U ij u,L = V * ji CKM scenario, instead, the flavor constraints typically require a small splitting between the H 0 and A 0 Higgs masses and hence a heavier Standard Model-like Higgs boson is preferred in this case.

W + H 0 /W + A 0 production and W+jets Excess
The signal process of interest to us is where the charged lepton comes from the W -boson decay. This is the dominant source of W ± H 0 /A 0 production since the charged Higgs is produced resonantly. In addition there   are t-channel processes that contribute to the total signal rate. Inverting the hierarchy between the neutral and charged Higgs boson, one could also consider the process This will have essentially a very similar collider signature and comparable effects in flavor and electroweak measurements. In addition to the desired signal, our model also contributes to the Z + jj and γ + jj channels, through the processes Since these processes are not mediated by on-shell resonances, they are suppressed compared to the W ± jj channel. The tree-level cross sections for these channels at the Tevatron are listed in Tab. 1. The mass of the scalar H 0 is fixed to 150 GeV in order to reproduce the excess in the dijet invariant mass spectrum. We calculate the cross sections for W ± H 0 /A 0 , ZH 0 /A 0 and γH 0 /A 0 at the Tevatron for two values of the charged Higgs mass, M H ± = 250 GeV and M H ± = 300 GeV. Larger splittings between the neutral and charged Higgs states are disfavored by EWPTs. The remaining free parameter is the up quark Yukawa coupling y 11 u,2 , which is chosen such that the required amount of signal events is obtained. As discussed in the previous section, we shall assume that both y 21 u,2 and y 31 u,2 are smaller than y 11 u,2 and hence they have no impact in the particle production at the Tevatron. We shall briefly discuss on the possible effects of a different coupling arrangement in the conclusions of this article.
The signal is generated using CalcHEP/CompHEP [32,33] (and checked independently with MadGraph [34]), and then processed through Pythia [35] for parton showering and hadronization and through PGS for a detector simulation, for which the CDF parameter set is used.
We implement the cuts of the CDF analysis: for the leptons, we require p T > 20 GeV and |η| < 1. In addition, a missing energy greater than 30 GeV is required, and the transverse mass of the lepton+E / T system is required to be larger than 30 GeV. Jets are reconstructed with a cone size R = 0.4 and required to have p T > 30 GeV, |η| < 2.4 and ∆η jj < 2.5. Further the transverse momentum of the dijet system is required to have |p T j 1 + p T j 2 | > 40 GeV. Events with more than two jets are rejected, as well as events with an additional lepton with p T > 20 GeV. We also reject events where the lepton is within a ∆R ≤ 0.52 cone around either of the two jets.
To correct the jet energy scale, we also simulate the W + 2jets signal from W + W − and W ± Z production. Then we multiply the jet energy by 1.1 such that the mass of the hadronically decaying W -boson is reconstructed at 80 GeV. The same scaling is applied to the signal.
The result of the simulation for di-boson background only and for signal plus di-boson background is shown in Fig. 3. To match the height of the W-boson peak to data we multiply the tree-level cross sections for W W and W Z production by a K-factor of 1.4. For M H ± = 300 GeV, 17% of the events with W → ν , ( = e, µ) pass the cuts, while for M H ± = 250 GeV this number drops to 8.7%. A good fit to the dijet invariant mass spectrum is found using 300 signal events, corresponding to a coupling y 11 u,2 = 0.06 for both M H ± = 300 GeV and M H ± = 250 GeV.
In Ref. [10] a different set of cuts was suggested for the CDF analysis that may improve the signal to background ratio by a factor of two. In particular, to be more sensitive to the W +resonance structure of the event, they suggest a cut on the sum of lepton and missing transverse momentum, |p T + p T ν | > 60 GeV, and require the angular separation between the jets to be ∆φ(j 1 , j 2 ) > 1.75. For our model with M H ± = 300 GeV, these cuts reduce the signal by 32%. Comparing with the results of Ref. [10] we expect a similar increase of the signal to background ratio in our case. Let us finally comment on the effects of splitting the masses of the neutral Higgs bosons H 0 and A 0 . Small mass differences have only a minor influence on the signal rate. For large mass differences, δm H 0 A 0 ≥ 50 GeV, as required for a significant improvement of the fit to precision electroweak data for moderate values of the SM Higgs boson mass, the dominant signal rate is associated with the lightest of these Higgs bosons, while the other one leads to a small or no signal contribution. This is, for instance, the case for the values M H 0 150 GeV, M A 0 210 GeV, and M H ± 300 GeV, used in Fig. 1. Although the charged Higgs may decay to on-shell W and A 0 bosons, the phase space suppression is strong enough to lead to only a very minor increase of the signal in the M jj 210 GeV region. This region can however be populated if also the mass of the charged Higgs is increased (see Appendix).

Prospects at Tevatron and LHC
At hadron colliders, the 150 GeV dijet resonance can be produced also in association with a Z boson or with a photon. In our model, the main production of the W + jets signal is through resonant charged Higgs production. Hence, the γ + jets and Z + jets signals at the Tevatron, which are only produced in the t-channel, are suppressed by more than an order of magnitude with respect to the W + 2jets one (c.f. Tab. 1). Assuming a similar signal acceptance, and taking into account the small branching fraction of the Z boson to lepton pairs, we do not expect the Tevatron to see any relevant excess in the (Z → )jj channel [36].
In the photon plus dijet channel at the Tevatron, we expect up to 40 events for a photon with p T > 20 GeV. This channel is dominated by QCD dijet production with an additional photon. A search in this channel has been performed with very mild cuts on the jet energies [37], and our model is consistent with these observations. A more refined analysis with optimized cuts could provide important constraints on this and other models for the W +jets CDF excess.
At the LHC, it is possible to search for all of the above signals. In addition, the larger center of mass energy allows for sizable production cross sections also for ZH ± , W ± H ∓ and for pair production of scalars. The cross sections at the LHC with √ s = 7 TeV are given in Tab. 2.
One promising signal comes from Drell-Yan production of H + H − pairs, that gives rise to W + W − +4jets signals. The main background for this process is tt production with GeV. Shown are the production cross sections (in pb) at LHC with √ s = 7 TeV for several promising channels. For the photon p T > 20 GeV is required. additional jets. This background can be reduced by explicitly asking for four hard jets, and demanding that the invariant mass of pairs of jets is close to the mass of the resonance. In addition, a veto on b tagged jets may be useful to discriminate between the signal and background.

Conclusions
We have presented a simple renormalizable model that can explain the CDF excess in W +2jets that was recently observed by the CDF collaboration. Our model is in agreement with all constraints coming from electroweak precision measurements, and depending on the precise values of the new charged and neutral Higgs boson masses, allows a broad range of the Standard Model Higgs mass from 100 GeV to several hundreds of GeV.
The model predicts no significant signal in the Z+2jets and in the W bb channel, in agreement with current experimental constraints on these channels. At the LHC, the model can be searched for in dijets produced in association with an electroweak gauge boson. In addition, we also expect a sizable production cross section for W W +jets and ZW +jets at the LHC. These signals might be easier to separate from the notoriously large QCD background.
Given the steadily growing number of models that attempt to explain the CDF excess, it might be worth considering possibilities to discriminate between different models. In principle angular observables should be suitable for this task, since angular distributions typically depend on the spin of intermediate resonances and on the production mode.
More concretely, we expect t-channel Z models to give a different p T spectrum (softer and more forward) than in the s-channel production considered in this work. Moreover, no resonant feature on the total invariant mass of the W + jj system is expected in those models. Considering there is only one leptonic W , we can fully reconstruct the event and study the spin of various resonances. In our particular case, the spin of H + may be studied using the W H 0 angular distribution in the rest frame of H + . As stressed before, the relative ratio of W + jj rate to the Z + jj (or γ + jj) is a very good diagnostic that may distinguish different models.
We finally want to mention that, had we assumed y 31 u,2 to be the most significant coupling, a relevant signal could still be obtained for large values of this coupling, y 31 u,2 1.5, via the ub → H + → H 0 /A 0 +jj and gu → tH 0 /A 0 channels. We studied this possibility and found that the rate in the former channel is a factor four larger than the latter and therefore the signal characteristics are very similar to the one studied in this article. Similarly to what happens in the model of Ref. [13], such a coupling arrangement leads also to an increase of the top-quark forward-backward asymmetry, but is subject to constraints coming from flavor physics and top-quark decays [38]. Further discussion on these possibilities will be studied elsewhere.