Slac-Pub-14810 Hiding a Heavy Higgs Boson at the 7 TeV LHC
A heavy Standard Model Higgs boson is not only disfavored by electroweak precision observables but is also excluded by direct searches at the 7 TeV LHC for a wide range of masses. Here, we examine scenarios where a heavy Higgs boson can be made consistent with both the indirect constraints and the direct null searches by adding only one new particle beyond the Standard Model. This new particle should be a weak multiplet in order to have additional contributions to the oblique parameters. If it is a color singlet, we find that a heavy Higgs with an intermediate mass of 200 - 300 GeV can decay into the new states, suppressing the branching ratios for the standard model modes, and thus hiding a heavy Higgs at the LHC. If the new particle is also charged under QCD, the Higgs production cross section from gluon fusion can be reduced significantly due to the new colored particle one-loop contribution. Current collider constraints on the new particles allow for viable parameter space to exist in order to hide a heavy Higgs boson. We categorize the general signatures of these new particles, identify favored regions of their parameter space and point out that discovering or excluding them at the LHC can provide important indirect information for a heavy Higgs. Finally, for a very heavy Higgs boson, beyond the search limit at the 7 TeV LHC, we discuss three additional scenarios where models would be consistent with electroweak precision tests: including an additional vector-like fermion mixing with the top quark, adding another gauge boson and modifying triple-gauge boson couplings.
The exciting LHC era will soon answer one of the most important questions in particle physics: the existence or nonexistence of a light Standard Model (SM) Higgs boson. This will be the most valuable result in particle physics in the last thirty years. The discovery of a SM Higgs boson will complete the SM and the argument for the existence of new physics will be solely from a naturalness viewpoint. On the other hand, nonexistence of a SM Higgs boson will be more interesting in a sense that it gives us hints of new particles or new dynamics at the TeV scale. Discovering those additional particles and dynamics in the absence of a SM Higgs boson would be a subsequent focus of the LHC program.
From the viewpoint of simplicity, the Higgs mechanism is an economical way to provide the and gauge boson masses as well as fermion masses in the SM. The Higgs couplings to gauge bosons and fermions are hence dictated by electroweak symmetry breaking (EWSB) and should not be modified too much from physics at a higher scale. The null result for the SM Higgs from the LHC searches does not immediately lead to the conclusion that no fundamental Higgs field is responsible for EWSB. Actually, there are two generic possibilities to explain the null 7 TeV LHC Higgs searches: the Higgs boson has a new non-standard decay channel that suppresses the branching ratios of the SM decay channel, or the production cross section of the Higgs boson from gluon fusion is suppressed because of other QCD charged particles contributing to the effective operator between the Higgs boson and two gluons. For sure, another plausible possibility to explain the non-existence of a SM Higgs boson at the LHC would be no Higgs boson at all and use new strong dynamics like the Technicolor models  or Higgsless models  to explain EWSB.
Mechanisms to hide the SM Higgs boson is not new at all in the literature. There are numerous activities that concentrate on a light Higgs boson with a mass below 200 GeV (see  for a recent review). However, less attention has been paid to the case of a heavy Higgs boson, which will be the main focus of this paper. One motivation to consider a heavy Higgs boson is that the fine-tuning problem becomes less stringent as for a lighter one . Another motivation actually comes from the electroweak precision test (EWPT). As is well known, a heavy Higgs boson is not preferred by the electroweak precision data. For example, the oblique parameters , and [5, 6] prefer a lighter Higgs boson, assuming there are no new contributions. Therefore, a heavy Higgs boson should always be accompanied by new particles beyond the SM to be consistent with the EWPT. It is not hard to imagine that these new particles could change the Higgs properties as well. Taking simplicity as a guidance, in this paper we consider adding only one new particle at a time charged under the SM gauge group for both to obtain consistency with the electroweak precision observables and to hide a heavy Higgs boson from direct searches.
Considerable efforts have been spent on relaxing the electroweak constraints on the Higgs boson mass, which were summarized into three scenarios in Ref.  ten years ago. The first scenario is to add particles whose vacuum polarization integral shifts in the negative direction. The main example of this is given by scalar fields in several specific multiplets of , where the first is the weak interaction gauge group and the second one is the custodial symmetry group [8, 9]. The second method is to add heavy vector bosons to shift all three oblique corrections [10, 11, 12, 13, 14, 15]. Finally, one could add new particles that produce a nonzero, positive with or without changing . This have been implemented in quite a few new physics models, for instance, the ‘topcolor seesaw’ where EWSB arises from a heavy singlet fermion . Here, we will loosely follow  and introduce new scalars or fermions which are charged under the electroweak gauge group and modify the and parameters at the same time.
Our main focus, however, is to explore how the new physics required by the EWPT modifies the properties of a heavy Higgs,in particular how the Higgs can be hidden at the 7 TeV LHC. If the new particles are also charged under QCD, the production cross section of the heavy Higgs boson from gluon fusion can be modified and even reduced dramatically compared to the SM rate. One such example we will discuss in detail is a colored scalar with a negative quartic coupling to the Higgs. After taking into account the current collider constraints of these new colored scalar particles, we find that a viable model exists to reduce the gluon-fusion production cross section of the Higgs boson by as much as 90%. Hence, a heavy Higgs boson consistent with EWPT could still be allowed by the 7 TeV LHC searches. Since these colored particles have large production cross sections at the LHC, performing a specific search for these states at the LHC can indirectly provide constraints on a heavy Higgs boson.
For new QCD-singlet particles, the production cross section can not be modified dramatically, but new decay channels of a heavy Higgs boson can open up. However, this way of hiding a Higgs can only work for a Higgs boson with an intermediate mass below 400 GeV, above which the Higgs boson SM decay width becomes so large that the partial width of the new decay channels could not dominate in any perturbative model. Below, we will check the current collider constraints on these new QCD-singlet particles and discuss various viable non-standard decays of a heavy Higgs boson.
Our paper is organized as following. In Section 2, we first review the current status about electroweak precision measurements with an emphasis on the oblique parameters. Then, we discuss how to hide a heavy Higgs boson by including a new color-singlet particle in Section 3, where we will first check the electroweak precision constrains on the masses of different isospin states in Section 3.1 and then study the collider signatures as well as constraints in Section 3.2. For QCD-charged particles in Section 4, we first consider the QCD charged scalar and consider their constraints from the EWPT as well as from colliders in Section 4.1. The modifications on the Higgs production cross section from gluon fusion will be discussed in Section 4.1.1. We then consider the fermion case in Section 4.2 by mixing a new fermion with the top quark. After that, we also consider collider constraints on an additional gauge boson mixing with the boson and hence modifying the electroweak precision observables in Section 5. For the last case of a non-linearly realized EWSB, we discusssed a scenario to transfer the constraints from oblique parameters to triple-gauge boson couplings in Section 6. Finally, we conclude in Section 7.
2 Oblique parameter analysis
The usual wisdom to prefer a lighter Higgs boson is because a light Higgs boson is more consistent with the EWPT. Using the recent results from the Gfitter group , the Higgs mass is constrained to be GeV by the standard fit and GeV by a complete fit including the LEP data, the Tevatron and 2010 LHC null results of direct Higgs searches. The upper mass constraint for a SM Higgs boson is 169 GeV (200 GeV) at 95% (99%) C.L. from the standard fit and 143 GeV (149 GeV) from the complete fit. The shortly-coming LHC direct serches with a luminosity of 5-10 fb should cover all the mass range of a light Higgs boson.
In many new physics models, additional particles can easily modify the electroweak precision observables. So, a more proper attitude towards a heavy Higgs boson around or above 200 GeV is to include additional new heavier particles in the EWPT. In this paper, we are going to take this attitude and consider minimal models by including only one new particle at a time. The common approach to constrain physics beyond the SM with the precision electroweak data is through the formalism of oblique parameters: , and [5, 6]. The parameter measures new physics contributions to the derivate differences of gauge current vacuum polarizations at zero momenta. The parameter indicates the difference between the new physics contributions of neutral and charged vacuum polarization at low energies, i.e., it is sensitive to weak isospin violation. Generally as the new physics predicts a negligible contribution to with a few exceptions such as models with anomalous interactions , one could fix and only consider the constraints from the and parameters.
Fixing , the most updated global electroweak fit at the reference point GeV and GeV is 
with a correlation coefficient of 111The fit using only low-energy experiment data such as atomic parity violation and lepton scattering prefers a larger value for the parameter . In this paper, we consider the result of fit including also high energy experiment data.. Shift in the reference point has to be compensated by shifts in the and parameters. For a Higgs boson heavier than 120 GeV, the central value of from new physics contribution is required to be reduced by while the central value of from new physics is increased by . For instance, at GeV and GeV, the new physics should have the following contributions to the oblique parameters
Thus for theories with a heavy Higgs to be compatible with the precision data, there should be new particles shifting in the positive direction and/or pushing to be negative. The allowed regions in the and plane with GeV and GeV are presented in Fig. 1. One can easily see from Fig. 1 that a heavy Higgs boson without other new physics is inconsistent with the electroweak observables at more than 3 level.
In general, one can introduce additional weak charged particles and adjust the mass differences of their isospin components to fix the parameter while keeping the -parameter almost untouched. Yet the constraints from fitting the parameter can not set a constraint on the absolute mass scales. Interestingly, by requiring those particles to modify the Higgs decays or production cross sections, one can also fix the masses of those particles and hence have a pretty concrete prediction for the LHC. For sure, this kind of prediction is only possible due to our simplicity assumption that only one new particle is relevant for both EWPT and Higgs phenomenologies.
3 Hiding a Heavy Higgs Using a New Color-singlet Particle
For color-singlet and charged particles, the production cross section of the Higgs boson can not be modified significantly. So, in this section we will concentrate on the parameter space where the heavy Higgs boson has a new decay channel dominant over the SM channels. In principle, the new particles could be scalars or fermions. However, to fix the electroweak precision observables we found that a large mass splitting is required. Thus the fermion case is not preferred as no renormalizable operators can be written down to achieve that and a large modification of the parameter is not anticipated. On the contrary, renormalizable operators coupling the SM Higgs field to the new weak multiplet exists to generate a sizable splitting inside the scalar multiplet. Therefore, in this section we only study the scalars and consider two models with a weak doublet or a weak triplet.
3.1 Electroweak Precision Test
3.1.1 Scalar Doublet
We first consider the weak doublet model, which has been studied before with an emphasis on the dark matter phenomenology under the name of inert doublet models [4, 20]. Here, we will not use the dark matter relic abundance as a constraint on the parameter space but rather consider more generic collider consequences of those new particles. It is shown that these new scalars could produce a negative as long as the lightest state in the multiplet also has the smallest spin [8, 9]. In [8, 9], the custodial symmetry is always preserved by the interactions of the new electroweak multiplets and hence the parameter is not modified. Below we will consider a more general model with custodial breaking operators, in which both and will be modified.
The model contains an addition scalar doublet transforming as under . For simplicity, we impose an approximate or exact parity on the new doublet and first consider only conserving operators. For , this is exactly the inert doublet model considered in . Further studies on this model could be found in [20, 21, 22]. The scalar potential of Higgs and is
Notice that the last operator is only present when and breaks the continuous Peccei-Quinn symmetry enjoyed by the other operators down to the . The operator with the coefficient splits the masses of components with different isospins while the last operator with the coefficient further breaks the degeneracy between the real and axial neutral scalars. Throughout this paper, we will always assume is small such that the real and axial neutral scalars have approximately equal masses. The potential is bounded from below if and only if
Under this condition, the minimum with is stable and the global one provided all the masses of the scalar fields are positive. All the parameters in Eq. (3) would be renormalized and the potential stays perturbative up to a reasonably high scale 2 TeV provided the quartic couplings are not too big. Among the quartic couplings, only affects the self-interactions of and will always taken to be smaller than 1. As we will show, is fixed by the EWPT and is also small . ’s beta function is . We will require so that the radiative correction to will not exceed 30% of its tree level value given the cutoff of the model is 2 TeV. Parametrizing the field as and after EWSB, we have
where we neglected the ’s contribution to those masses in the limit . Here, denotes the mass of . is the Higgs vacuum expectation value (VEV) and GeV.
Choosing , we show the constraints on the masses of the two different components of in Fig. 2. One can see that there are two prefered horizontal bands with the mass splitting around 100 - 140 GeV, which is almost independent of the scalar mass and the heavy Higgs mass. We also checked that the contributions to the parameter is small for the range of parameter in Fig. 2 and a fit including the parameter does not modify the conclusion above.
3.1.2 Scalar Triplet
where with as the Pauli matrices; are the generators for spin-1 representation with . For , there is an additional renormalizable operator , where and . It could be forbidden by a symmetry acting on . The physical fields appear in the parameterization of the triplets as follows: and each of them has electric charge . Only the last operator in the potential splits the masses of different components inside a complex triplet with a non-zero . For , this operator vanishes identically as the triplet is real. Thus the real triplet does not contribute to the and parameters. From now on, we will only consider a complex triplet with a non-zero whose components have masses
with . The condition for the potential to be bounded from below and the existence of a global minimum at is
Similar to the doublet model case, we require to preserve perturbativity up to 2 TeV.
The new contribution to the parameter from the mass splitting of the triplet components is
In the small mass splitting limit, we have with . The contribution to the parameter from the mass splittings is
In the limit , one has . Choosing , we have the allowed regions for and shown in Fig. 3. We can see from Fig. 3 that a triplet with a mass splitting around 50 GeV can be consistent with the EWPT for a heavy Higgs boson.
3.2 Collider Phenomenologies
The additional scalars charged under lead to interesting collider signals. They will be produced either indirectly from Higgs decays if kinematically allowed, or they could be paired-produced via weak gauge boson exchanges. The collider signatures highly depend on whether the symmetry is broken or not. Below we will first discuss several possibilities of these scalars’ decays by coupling them to the SM particles in different ways. Then we will show that they could modify the heavy Higgs decays significantly and thus impact the Higgs searches. We will point out some interesting signatures from the cascade decays of the Higgs boson. Finally, we consider the direct productions of those new scalars and work out the current collider constraints on different decay channels. In this section, we will focus on two benchmark models where scalars have specific hypercharges: the doublet model and the triplet model .
3.3 Decays of Scalars
If the lightest state inside the scalar is stable due to the unbroken symmetry as in the inert models, it would contribute to the dark matter (DM) density. Thus we have to take the lightest state neutral to avoid the stringent constraints on charged relics. However, unlike the discussions of the inert models, we will not restrict ourselves to the parameter region with the right DM relic abundance. Instead, we will focus on a larger parameter space where the Higgs decay is modified. If the is broken by couplings of a single field to SM fermions and/or gauge bosons, we could have in principle the lightest state to be either electrically charged or neutral. Without loss of generality, we will assume the lightest state to be the electrically neutral one inside the multiplet.
In the doublet model with , consists of one charged and two neutral particles and can be parametrized as . As shown in the previous section, 100 GeV from the EWPT. Therefore, the charged state decays to the neutral ones plus an on-shell gauge boson: . There could be three possibilities of decays :
or is stable. A splitting between and must be generated by a non-zero . Otherwise, and have an unsuppressed vector-like interaction with the boson, which lead to a large spin-independent elastic cross section scattering off nucleus, many orders of magnitude above the current direct detection limit . Notice that this is true even in mass regions where the relic density of and is small. On the other hand, a non-zero splitting above 1 MeV, the kinetic energy of DM in our galactic halo, is not sufficient to fulfill the inelastic scattering. At colliders, this means the heavier neutral scalar, e.g., the axial one , would decay to the lighter one plus an off-shell , . For a small splitting ( 10 GeV), the decay products from are soft and could not be triggered on unless a hard jet from initial state radiation is present to boost the decay products . The decay length is estimated to be
where is the gauge coupling. If the mass splitting is smaller than a few hundred MeV, which means , both neutral scalars are collider stable.
. At the renormalizable level, one could write down
where the flavor indexes are not shown. Those operators induce to decay into two jets or two leptons depending on the strengthes of Yukawa couplings. To avoid any potential flavor problem, we assume the Yukawa couplings follow the pattern of Minimal Flavor Violation (MFV) to match the SM Higgs Yukawa coupling pattern to fermions. For , is the dominate decay channel. Notice that these operators break parity and induce mass mixing terms such as in the scalar potential at the one-loop level. Without considering any accidental cancellation between the tree level and the loop-level contributions, the magnitude of is estimated from naive dimensional analysis as
This radiative contribution would mix and and modify the spectrum. To avoid a large mixing between and , we require the Yukawa couplings to be small, . Thus the heavier state decaying to two SM fermions are suppressed and has a smaller width compared to the decay into the neutral states plus the gauge boson. If , the decay length is of order meters and the lightest state is collider stable.
decays to two gauge bosons through dimension-six operators
where are field strengths of SM gauge groups. More operators can be written down with covariant derivatives, which may lead to the similar final sates. From those operators, one could have
However, this is not the whole story. At the one loop order, all these operators would generate , which by naive dimension analysis is of order with the parameter including various powers of SM gauge couplings as well as the coupling of to new particles which generate these dimension six operators. To avoid the case that develops a very large VEV, we assume a very tiny here. The induced mixing between and would then cause light decaying to pair with a partial width estimated to be
with as the SM Higgs coupling to the bottom quark. The ratio between and the width of decaying to two gauge bosons, e.g., , scales as
where the dependences on the coefficient cancel out. Yet, one should bear in mind that there could be large uncertainties in this evaluation by ignoring the inputs of UV physics. If the effective cutoff is lowered 1 TeV, the estimate above leads to comparable branching ratios. Thus, we still keep decaying to two gauge bosons as one possibility.
For the triplet case with , consists of a doubly charged state, a single charged state and a complex neutral state . The single charged state has a mass 50 GeV from Fig. 3 and decays as . The doubly charged state is even heavier, , which gives a mass difference smaller than 50 GeV. Thus the doubly charged state also decays to an off-shell with the single charged state, . Analogous to the neutral state in the doublet model, there are three possibilities for decays:
one component of is stable. To avoid the constraints from DM direct detection experiments, we need to include a dimension six operator to split the real and axial components of the neutral scalar. This mass splitting could be naturally small , which is about 1 GeV for 100 GeV and 2 TeV. Again the axial component can decay into the real one, which could be a stable particle, plus an off-shell gauge boson.
mediated by an operator , assuming MFV and . There are two other similar operators . To avoid large radiative generated term, , we require the Yukawa couplings to be small.
decays to two gauge bosons through dimension six operators such as
3.3.1 Higgs decays
The existence of additional weak-scale scalars opens up new Higgs decay channels. The partial decay widths of Higgs to additional scalars depend on the quartic couplings between two Higgs and two ’s in the potential.
First we consider the doublet model with . The partial width of Higgs decaying to the scalars is given by
For the triplet case, we have
where and .
The branching fractions of are presented in Fig. 4. The coefficients that give rise to the mass splitting are fixed by EWPT and we plot the branching faction as a function of the remaining coefficient ( in both cases) that preserves the custodial symmetry. From Fig. 4, one can see that for Higgs in the mass range 200 - 300 GeV, the Higgs decaying to the new scalars could easily dominate over the Higgs decaying to ’s or ’s, e.g., . For an even heavier Higgs boson, the width/mass ratio of Higgs becomes order of unit if one add new decay channels to suppress the SM branching ratios. Therefore, we only concentrate on the intermediate mass ranges in this section. In the mass range GeV, the current Higgs searches with 1 - 2.3 data exclude . It is projected that 5 data could push the limit down to . As we can see from Fig. 4, a lot of parameter space associated with the new scalar particles exist to reduce and hide a heavy Higgs boson at the 7 TeV LHC.
Although the Higgs boson can be hided in the existing searches by adding a new weak-charged scalar, new signatures from Higgs decays are predicted at the 7 TeV LHC. Taking into account of different decays, we could have several interesting possibilities
The first one is the Higgs invisible decay, which could be searched for in the monojet channel, the plus missing energy channel and two forward jets plug missing energy channel from -boson-fusion productions. The second and the third possibilities, to hide Higgs in four jets or “bare” them in four photons have already been discussed in the context of hiding light Higgs [30, 31, 3]. Notice that in the context of hiding light Higgs, the intermediate particles are always very light pesudo-scalars and the final jets or photons could be boosted and collimated while in our scenario, as is not very light, the final state particles are not necessarily close to each other. In the triplet model, there could be a small region of parameter space where .
3.3.2 Direct collider searches
The neutral states could not be lighter than around 45 GeV; otherwise, the boson could decay to them and the total boson width will be modified, which is highly constrained. For a lighter with a mass below 100 GeV but above 45 GeV, they could be paired-produced directly at LEP with a cross section 
where is the weak coupling constant and is the weak angle. For center of mass energy GeV, 60 GeV, pb. If is stable, it would lead to the mono-photon + MET signal . However, the cross section is small with , e.g., pb for 50 GeV, which is beyond the sensitivity of LEP experiments . However, if decays to SM particles, LEP results put stringent constraints on the parameter space that is kinematically accessible. If decays to 100% of times, the 4 jet final state search with both LEP1 and LEP2 data rules out up to 90 GeV . More concretely, the 4 jet search conducted by the DELPHI collaboration rules out a rate as large as that of paired-production of CP odd state and Higgs in a two Higgs doublet model in the Higgs mass range from 40 to 90 GeV (see Fig. 11 in ), assuming ; and 100% branching into 4. In our case, the cross section of the 4 final state is the same as in the doublet model and three times larger in the triplet model. Similarly, if decays to two photons, , below 90 GeV is ruled out by multi-photon searches at LEP .
At the hadron colliders, all states of could be produced through electroweak interactions. For the scalar doublet,
while for the scalar triplet,
|Relevant final states||Possible signals|
|stable||MET, + MET, +MET||mono-jet+MET, jets+MET, 1 +MET|
|’s||4 , ,||4 , +’s, OS+MET, 1 + jets + MET|
|’s + ’s + ’s||SS, multi-leptons, multi-photons, multi-jets|
|Relevant final states||Possible signals|
|stable||MET, ’s + MET,||mono-jet+MET, OS+MET, SS+MET, multi-leptons, multi-jets|
|’s||4 , ’s+ 4,||OS+MET, SS+MET, multi-leptons, multi-jets|
|’s + ’s + ’s||multi-photons, OS+MET, SS+MET, multi-leptons, multi-jets|
Given the small electroweak production rates, we found that most of the current searches are not sensitive to these new scalars. For instance, one would worry about constraints on the production of and from the same-sign (SS) lepton searches at both Tevatron and the LHC. For instance, the Tevatron SS search adopts a set of very loose cuts 
|at least two SS leptons|
which already imposes a strong constraint on the doubly-charged scalar inside an electroweak triplet to have a mass above 245 GeV. However, this limit is set by assuming 100% branching ratio of to or . In our case, however, the leptons are from the (off-shell) decays in the long cascade decay chain . Thus the cross section of the SS final state is reduced by a factor of , where 0.2 is the leptonically decay branching fraction and the factor 2 takes into account that SS leptons could come from either decay chain in the pair production. Besides, the invariant mass of the two SS leptons does not reconstruct a bump at . Thus we conclude that the region GeV, or equivalently, GeV is not constrained. The CMS SS searches require a much stronger set of cuts . The preselection cuts are
Among the final four signal regions, the search region with low but high cuts, 80 GeV and GeV is most sensitive to the case where is stable and contributes to the missing energy. The high low search region is most sensitive to an unstable (decaying to two jets). We used the FeynRules package  to generate our new physics models and then feed them into MadGraph 5  which calculated the matrix elements and simulated events. The events are then showered using Pythia 6.4 . For a stable , we found that the acceptance of the signal is 7% for (50, 132) GeV and there are 2 SS events after cuts for 1 luminosity. For , we found that the acceptance of the signal could be as large as 50% for (100, 187) GeV, which yields 3 events at 1 luminosity. They are below the observed upper limits on event yields from new physics , which is 7.5 for GeV and 6 for GeV.
In summary, if is stable at the collider scales, there is no constraints for the mass regions we are interested in GeV. If decays promptly to the SM final states, the allowed mass region shrinks to GeV due to the LEP constraints. Although the current LHC searches with 1 have not provided a constraint on the models we are considering here, more data would allow us to probe the parameter space in interest and to close this way of hiding Higgs. Especially, the SS lepton searches could set interesting limits on the triplet model very soon.
4 Hiding a Heavy Higgs using a New QCD-Charged Particle
4.1 QCD Charged Scalars
Restricting ourselves to the fundamental and adjoint representations of and , we have four choices of scalars: , , and . Depending on the hypercharges of those scalars, we have different consequences for the electroweak precision observables and couplings to SM particles. For the color-octet scalars, we consider and as two examples. The former was considered in Ref.  as the only choice other than to realize the MFV in the quark sector at the renormalizable level. For the color-triplets, we choose , which can couple to both up-type and down-type quarks, as the representative. For the electroweak triplet, we consider the representation .
We first consider the fit to the eletroweak precision observables. For electroweak doublets with the mass splitting , the modifications on , and are approximately
with for color octets and for color triplets. Similarly for electroweak triplets, we have
Without performing a numerical study, we can already know the constraints from and on the mass splitting and the lightest state mass. Since only depends on the mass splitting , required values to fit the observed value of predict a constant value of that is independent of the overall mass. Once the parameter is satisfied, the constraints from can only impose a lower bound on the overall mass scale. The modification on the parameter has one more power of the heavy weak multiplet mass in the denominator than the parameter. This could be understood as when writing all new physics contributions as high dimensional operators in terms of the SM fields, the parameter starts to get contribution from dimension six operators while the parameter is already modified by dimension five operators. The modifications of the parameter are numerically small and will be neglected in this section. Assuming , we show the numerically fitted results in Fig. 6 from just fitting the and parameters.
One can see from those two plots that the allowed mass splittings are always below the gauge boson mass. The heavier state may decay into the light state plus an off-shell , which will be discussed later for the collider phenomenologies.
4.1.1 Modifications on the Higgs Boson Production
In the SM, the Higgs boson is mainly produced from gluon fusion through the Higgs-gluon effective operator after integrating out the top quark
If other heavy colored particles exist, they will also contribute to . According to the low energy Higgs theorem, each new colored particle with mass would contribute 
where is the particle’s contribution to the gauge coupling function coefficient which equals 2/3 or 1/6 respectively for a Dirac fermion and a complex scalar and is the Dynkin index. The sign of thus only depends on the dependence of the new colored state’s mass. If decreases with the Higgs VEV, would be negative and the interference between SM and new particle would be destructive.
In the presence of new colored scalars or , one could write down in the Lagrangian the following quartic operators coupling , to the Higgs,
After EWSB, the mass is then , where the constant is from the quadratic mass term (). According to the argument at the beginning of this section, when the quartic coupling , one may have destructive interference and reduce the SM Higgs boson production cross section in the gluon fusion channel. Notice that should always be bigger than to forbid a negative mass which will trigger the spontaneous breaking of . Requiring the radiative corrections to to be below 30% of its tree-level value, the coupling is constrained to be .
So far the existing literature mainly focus on the enhancement of the Higgs production in the gluon fusion channel due to the colored scalars . However, we are interested in the destructive interference region where the coupling is reduced by the colored scalar loop (see Section A for its formula). Note that when , the SM contribution to contains both a real part and an imaginary part. To reduce the absolute value of by a certain amount, both parameters and the mass have to be fixed. Only for a lighter Higgs boson with mass below both and , is real and the reduction of Higgs productions only constrain the ratio of for a small splitting inside the scalar multiplet. In the limit that is much less than the masses of particles in the loop and neglecting , the Higgs production cross section vanishes if the following relation between and is satisfied
where GeV; is the number of colored states and is 4 for a weak doublet and 6 for a complex weak triplet; for fundamental representations and for adjoint representations. In Fig. 7 and Fig. 8, we show the ratios of the Higgs production cross section from gluon fusion in the SM plus a new colored state over that in the SM.
For a heavy Higgs boson with 500 GeV mass, the colored states are predicted to be between around 200 GeV to 250 GeV, if the SM Higgs boson production cross section is observed to be one tenth of the SM production cross section. For a lighter Higgs boson with a 250 GeV mass, the new heavy colored state can be as heavy as 500 GeV.
Since we have only considered here the reduction of Higgs production cross sections at the leading order in , one may worry about how stable this reduction is at the next-to-leading order (NLO) . Taking the heavy top quark and heavy colored-particle limit, using Eq. (68) at NLO we present the modifications on the relation of and for different (defined as the term in the Lagrangian) in Fig. 9 for GeV, and a color-octet weak-double scalar.
Comparing the blue and orange regions in Fig. 9, we have found the relation between and is fairly stable to reduce the Higgs production cross section. For fixed , the relative modification on the new scalar mass is around 10%.
4.1.2 The properties of the colored particles
For the color octet and weak doublet particle , one can write down the Yukawa couplings to SM quarks at the renormalizable level. As discussed in Ref. , the MFV assumptions can be realized for this particle and one can have the following interactions in the Lagrangian
with as the generators and are up-type (down-type) quark masses. Since the third-generations have the largest Yukawa couplings, the color octets prefer to decay into or quarks. The decays widths are