Masses of a Fourth Generation with Two Higgs Doublets
Abstract
We use sampling techniques to find robust constraints on the masses of a possible fourth sequential fermion generation from electroweak oblique variables. We find that in the case of a light () Higgs from a single electroweak symmetry breaking doublet, inverted mass hierarchies are possible for both quarks and leptons, but a mass splitting more than in the quark sector is unlikely. We also find constraints in the case of a heavy () Higgs in a single doublet model. As recent data from the Large Hadron Collider hints at the existence of a resonance at and a single Higgs doublet at that mass is inconsistent with a fourth fermion generation, we examine a type II two Higgs doublet model. In this model, there are ranges of parameter space where the Higgs sector can potentially counteract the effects of the fourth generation. Even so, we find that such scenarios produce qualitatively similar fermion mass distributions.
pacs:
14.65.JkFERMILABPUB12251PPD
I Introduction
Adding a sequential fourth generation of fermions (4G) is one of the simplest possible extensions to the Standard Model. Indeed, although the width of the limits the number of active light neutrinos to three, there can in principle be a fourth neutrino generation which is much heavier. A recent and extensive literature examines the impact of how 4G would reduce tensions in recent measurements in the sector and create distinctive phenomena in kaon decays bib:SoniEtTu (); bib:BurasEtal (), as well as providing a potential scenario for the baryon asymmetry of the universe bib:WS_Hou (). As a more top down motivation, simple string constructions often lead to toy models with an even number of generations. Of course, achieving three chiral generations is also possible in a wide class of examples, and some stringy models of flavor physics predict that more than three generations would be inconsistent with the measured three generation quark mixing matrix bib:Heckman:2008qa (). More generally, one can view the 4G scenario as a simple template for scenarios of physics beyond the Standard Model in which states of some extra sector receive a mass proportional to the Higgs vev.
In light of these considerations, it is clearly of interest to study the viability of the 4G scenario. In addition to the possibility of direct detection of such states, the contributions of these additional states enter as loop corrections to various Standard Model processes. For example, a fourth generation tends to produce a positive contribution to both and , the oblique electroweak parameters. By contrast, in a single Higgs doublet model, increasing the mass of the Higgs generates a positive contribution to and a negative contribution to . Thus, while cancellation for the parameter is possible, the contributions to the parameter typically move in the same (positive) direction, though mass hierarchies in the fourth generation can reduce the size of this contribution. The extra generation also affects the phenomenology of the Higgs, leading to an increase in , and a decrease in .
Though less wellstudied, even simple extensions of the Higgs sector can counteract (or exacerbate) some of the effects of a chiral fourth generation with an appropriate tuning of parameters. For example, in two Higgs doublet models (2HDM), the contributions to and can have either sign (see e.g. bib:Heckman:2011bb (); bib:Funk:2011ad (); bib:HaberONeil ()). General values of the Higgs mixing angles also allow for changes in and , independently, relative to the 4G scenario.
In this paper we study the available parameter space for the 4G scenario and its extension to 2HDM models. The full parameter space of 4G is too large for easy visualization, but much of what we need to know to understand existing experimental results and to inform future searches can be expressed with two pairs of numbers: the two quark masses and the two lepton masses. We therefore seek, by sampling this fourparameter space and comparing the samples with constraints on electroweak oblique parameters to determine the most likely mass spectrum for 4G, should it exist. Similar earlier analyses of this type may be found in bib:HePoSu (); bib:Tim (); bib:GFitter (), but new experimental data has appeared since these publications. Other similar studies have appeared recently bib:GoodInj (); bib:CloseCall (); bib:A_VV ().
Exclusion limits on the mass of a Standard Modellike Higgs impose additional constraints on the 4G scenario. In 4G, the gluon fusion production cross section for the Higgs is markedly increased over the three generation scenario. Both LHC collaborations bib:LHC_Higgs4 () independently exclude, using a combination of channels, the range when there is a fourth generation. The LEP II lower limit of is independent of the number of fermion generations bib:LEP_Higgs (). Because a fourth generation of fermions contributes to roughly quadratically with and , and because a large corresponds to a large , values of as high as are allowed by electroweak constraints bib:TwoFits (); bib:GFitter () in 4G. However, studies of the stability and triviality bounds on in 4G bib:MichioAkin () prohibit unless there is also some other new phenomena on a scale below .
Most recently, there are “hints” of a Higgs with mass bib:JensWeighs () of from the LHC bib:LittleBump1 () and supporting evidence from the Tevatron bib:LittleBump2 (). The hint is strongest in the channel , where the ATLAS experiment reports an excess above background of 2.8 standard deviations. The statistical significance of these results is not enough to declare discovery or even strong evidence for a Higgs, but is strong enough to provoke discussion. This mass is within the bounds ruled out by the LHC when supposing a fourth fermion generation. To leading , it is possible to retain 4G if one supposes only the channel’s hint remains significant with the addition of more data, but including exact nexttoleading order electroweak corrections makes this difficult bib:gamgamOK1 (); bib:gamgamOK2 (). Consequently, the 4G hypothesis is valid only if (a) the hints turn out to be statistical fluctuations or (b) the hints are due to something beyond a single Higgs doublet, such as a two Higgs doublet model.
The rest of this paper is organized as follows. In Section II we treat the 4G single Higgs doublet case. We give results for GeV (the “baseline” scenario) and for GeV (the “highmass” scenario). The baseline scenario is appropriate for the case considered in bib:gamgamOK1 (); bib:gamgamOK2 (); the difference in our results between GeV and GeV is small. In Section III we extend the analysis to consider a Type II model with its parameters adjusted to match the hint. Section IV provides a summary.
Ii Single Higgs Doublet Scenarios
ii.1 Method
We have updated constraints on the oblique electroweak bib:PandT () parameters , and , as found by the Global Analysis of Particle Properties (GAPP) bib:GAPP () using data available in October 2011. In our sampling procedure, each sample is assigned a weight corresponding to the probability density function for these three parameters. We employ the oneloop contributions to the oblique parameters, assuming small mixing with the extra family, as in bib:HePoSu (). See bib:Eberhardt:2012sb () for some recent discussion of the more general case of potentially large mixing effects.
The sampling distribution in this type of analysis plays the role of a Bayesian prior; we are taking the probability of a specific value for , and given an assumed set of four fermion masses, and weighting it in our result as the probability density created by our sampling of the fermion spectrum. We interpret the result as a probability density function for the fermion mass spectrum, but that interpretation is only valid in the context of that assumed sampling distribution. The peril in this process  the validity of the assumed prior  thus has the advantage of requiring explicit description.
We draw 50 million uniformly distributed samples in the fermion mass spectrum with lower bounds set by direct experimental constraints described below. The upper bound is limited by unitarity arguments bib:unitarity () to , but this is a rough bound and we raise it to for clarity in the resulting figures.
The lower bound on the sampled mass range is, in our baseline scenario, = from LEP II bib:L3 (). This limit is the weakest of the limits obtained under the assumption of decay to each of the three known charged leptons; if , then we would obtain a stronger limit. The lower bound on the sampled range is ; again, this is the weakest limit obtained in all the possible decay scenarios. These results are therefore robust against all assumptions about the lepton mass hierarchies. On the other hand, lepton mixing parameters are important considerations in searches for the leptons of 4G at the LHC which have been discussed bib:4thG_L () but have not yet been carried out.
Obtaining robust lower bounds on 4G quark masses and mixing angles is a little more complicated. Dramatic results ^{1}^{1}1All limits are at the 95% confidence level from the LHC are indeed available bib:PIConf (), and new ones are appearing constantly. The CMS collaboration has searched for:

with samecharge leptons and trileptons in a sample bib:CMS_p (), obtaining a limit of .

both and using a simplified model with a range of final states, all containing 2 quarks, in of data bib:CMS_2 (). All of the diagrams considered have or . Lower limits of were obtained.

pair produced in the “lepton with jets” channel, wherein a decay to having the same signature as a event but with a different primary quark mass is sought. The analysis reconstructed in each event. A lower limit was found using only of data bib:CMS_3 ().

pair produced in the “dilepton” channel, wherein also a decay to having a topquark signature but different mass is sought. A weaker constraint than that which was obtained in the “lepton with jets” analysis, , was found using of data bib:CMS_4 ().
The ATLAS collaboration has searched for:

pair produced in of data bib:ATLAS_1 (), as part of an inclusive search for exotic production of the samecharge dilepton signature.

pair produced or decaying to , where or , appearing with oppositecharge dileptons and missing transverse momentum in of data bib:ATLAS_2 (). An approximate event reconstruction is done. The resulting limit is = .

with one lepton, at least six jets, and large missing momentum transverse to the beamline on a sample, obtaining bib:ATLAS_3 () a limit of .

pair produced or appearing with samecharge dileptons, large missing transverse momentum, and at least two jets in of data bib:ATLAS_4 (). A limit of was obtained.
See bib:wierdLHC () for recent searches of more exotic fermions.
These search results, while impressive, are all built upon specific decay, i.e. CKM mixing angle, assumptions. With the exception of bib:ATLAS_2 (), mixing of the fourth generation into anything other than the third generation is not considered. Furthermore, (or in an inverted hierarchy, ) will be an additional contribution to (or ) production which will not necessarily appear in any specific signature as a result of the products; the contribution from this channel can be significant if the mass splitting is small.
Additionally, there are constraints on the possible mixing parameters. For example, the mixing parameters for the quark sector may be constrained bib:BurasEtal (); bib:Mixing1 () with data from neutral mesons, the transition, existing constraints on the threegeneration quark mixing matrix and limits on Br( ). Reference bib:Mixing1 () concludes that large mixings of the fourth generation with the three known generations are not ruled out, but bib:Mixing2 (); bib:Mixing3 (), which considers constraints from corrections to the vertex from a fourth generation conclude that these mixings could be comparable to Cabibbo mixing. The quark mixing matrix can also be constrained with precision electroweak data and mixing bib:Mixing4 (). In any case however, there is the possibility that 4G fermions could decay to either third or lower generation fermions with varying branching ratios.
A method for producing experimental limits that are mixingangle independent ^{2}^{2}2P.Q. Hung and M. Sher, Phys. Rev. D 77, 037302 (2008) point out that for very small mixing angles, 4G quarks are charged massive particles, with signatures very different from those typically used in 4G searches. and that allows for the contributions of both 4G quarks to any particular signature was applied to the results of CDF searches bib:Irvine (), resulting in lower limits of for and for . We use these lower but mixing independent values here while strongly advocating the application of these techniques to the more recent LHC results. Such an analysis could soon sharply constrain or even rule out the 4G hypothesis.
ii.2 Results
Figures 1 and 2 show the lepton and quark mass spectra in our baseline and high scenarios. In these and similar Figures, the color for each bin represents a probability density integrated over the bin, and normalized so as to give unit probability when summed over the entire plot. For the baseline (high mass) case, in over 99% (90%) of our samples; transitions between 4G quarks will produce offshell bosons. The lepton mass splitting is less than with probability 69% (24%). Normal mass hierarchies are more likely than not, but by no means certain; in the lepton sector the probability of a normal mass hierarchy is 70% (93%), and in the quark sector, it is 59% (69%).
Masses just over the existing limit for the leptons are heavily favored, and this tendency is greater in the high scenario. Being able to predict this parameter relatively precisely makes it a valuable target for future searches.
In Figure 3 we show the lepton and quark mass splittings. We see that, with perhaps a twofold ambiguity, the mass splittings in the two sectors are tightly related.
Carpenter and Rajaraman bib:RHnu4 () revisited the LEP II results in a scenario with both left and righthanded neutrinos. They conclude that as low as is possible. Some recent studies bib:GoodInj (); bib:CloseCall (); bib:gamgamOK2 () also consider low values of . We find that lowering the bound on to does not produce much change relative to our baseline scenario. Figure 4 shows distributions that have the same probabilities of mass splittings less than and the same probabilities of normal mass hierarchies as our baseline scenario to within about 2%.
Iii Two Higgs Doublet Scenario
iii.1 Method
Should the hints of a Higgs boson with GeV solidify with more data, the 4G hypothesis is only tenable if an extended electroweak symmetry breaking sector exists. As an example of such an extension we consider a second Higgs doublet bib:Branco () in conjunction with a fourth sequential fermion generation (2HD4G). Two identical complex scalar doublet fields and , both of hypercharge are postulated. To forbid flavor changing neutral currents, we select the Type II Yukawa coupling pattern, in which quarks couple to one doublet and quarks and charged leptons to the other. This restriction permits a symmetry to distinguish from . We restrict consideration to the gauge invariant, renormalizable and conserving potential
(1) 
where all the parameters and are real. This system and its vacua preserve an additional symmetry. There are two even neutral bosons, and () a odd neutral boson, , and the charged bosons in this model.
This model is different from the similarlynamed “4G2HDM” model of bib:differs (); however, bib:RGE_2HD4G () analyzed a similar model prior to the appearance of the GeV hint. The presence of large fourth generation Yukawas can lead to large radiative corrections which can potentially destabilize the form of the Higgs potential. Here we assume that the 2HDM effective potential is stabilized by some effect near the TeV scale, so that we can focus on the resulting effective theory below the TeV scale.
Though it is beyond the scope of this study, the combination of two Higgs doublets with a fourth sequential fermion generation creates a rich phenomenology for which constraints from the kaon and sector could be derived. For example, the coupling constants vertex will obtain corrections which depend on , , and (depending on chirality) either or its inverse; these contributions can be constrained experimentally.
Two important parameters of this model are , the ratio of the vacuum expectation values of the two doublets and , the angle which diagonalizes the masssquared matrix of the even bosons. Values of less than 1 are disfavored experimentally assuming three fermion generations; more generally, results from the requirement that the top quark Yukawa coupling not exceed the perturbative limit bib:Branco (). Requiring perturbativity of the fourth generation Yukawa interactions can impose additional constraints. For the sake of generality, however, we do not impose this additional restriction in our scans. We sample in a scaleindependent way, i.e., the distribution of is uniform. The angle is scanned uniformly but samples are weighted according to the value of as described below; the masses of the 4G fermions and the bosons , and are selected with an initially uniform distribution. The mass of the lightest even boson is set to . For further discussion on the phenomenology of two Higgs doublets with a fourth fermion generation, including the case where is stable and contributes invisible decays to either or , see bib:ChenHe ().
The hint is strongest in the channel , where the ATLAS experiment reports an excess above background of 2.8 standard deviations. The second most significant hints are in the channels , where the ATLAS results have a significance of 2.1 and 1.4 standard deviations for respectively. The combination of ATLAS and CMS data correspond to a production rate about times the prediction of the Standard Model bib:Carmi2012 (); for , it is about 0.8times the Standard model rate.
For each scanned value of , we calculate
and
for the 2HD4G scenario and form a of these values against these experimental values. We weight each sample according to that . We do not consider constraints from decays of the , and which are very parameter dependent in the 2HD4G scenario.
The dominant production mechanism at both the LHC and the Tevatron is gluon fusion through loop diagrams involving the colored states. In a 2HDM, this includes the contributions from both the , as well as and . The Standard Model normalized cross section of the gluon fusion production cross sections is:
(2) 
where and denote and respectively, and is the threshold correction of a spin particle to the vertex for a GeV Higgs, with notation as in bib:tHHG (). A similar expression holds for the Standard Model normalized decay rate . In a 2HDM, this will include terms from loops containing , , and and charged fourth generation fermions, as well as a contribution from , which all depend on the mixing angles. The total width of the Higgs in 2HD4G including the mixing angle dependence is fixed by similar considerations. Much as in bib:Heckman:2012nt (), the overall normalization can be extracted from the recently updated values for the Standard Model GeV Higgs partial widths bib:Barger:2012hv () by including the mixing angle dependence and contribution from extra states in the various 2HD4G partial widths.
Constraints on two doublet models are readily available bib:THDMC () through the package 2HDMC. We observe the constraints of treelevel unitarity bib:unitMC (), perturbativity (i.e., the magnitudes of all the quartic Higgs couplings must be less than ), and the absence of runaway directions, as implemented in 2HDMC. Contributions to the oblique electroweak parameters bib:obieMC () are also provided as part of 2HDMC.
iii.2 Results
Figures 5 and 6 show the lepton and quark mass spectra in our two Higgs doublet scenario. The quark (lepton) mass splittings are less than in 99% (65%) of our samples; normal mass hierarchies in the quark (lepton) sector occur with a probability of 59% (72%).
Low values of are likely in 2HD4G; in Figure 6, in 46% of the final probability density function. Figure 7 shows the distribution of Higgs boson masses. There is a strong correlation between the masses and largely but not entirely created by requiring as well as production to be in agreement with experiment. It is amusing to note that the most likely values for the mass of the second even boson are just over , and masses corresponding to a small excess in the channel at are not improbable.
While the extended Higgs sector does alter the results from the single Higgs double scenarios, the broad features of the mass splitting structures and preference for low masses, particularly for , remain. These features are largely a result of the structure of the contributions to the electroweak oblique parameters from the fourth generation of sequential fermions. Similar results might be expected in almost any extension to the Higgs sector that is broadly consistent with a Standard Modellike Higgs.
Iv Summary
While stringent limits on and have been found in specific decay modes by the LHC, completely ruling out the fourth generation hypothesis requires an analysis bib:Irvine () that combines the results from a number of modes to obtain a result that is independent of quark mixing in the fourth generation.
We have used sampling methods to determine the probability densities of the masses of a possible fourth sequential generation of fermions in scenarios with one or two Higgs doublets. With a single Higgs doublet and a low ( or ) Higgs mass, fourth generation mass splitting in the quark sector is less than (see also bib:GoodInj ()). Quark sector mass splittings less than are favored but less certain if the Higgs mass is . A fourth generation is on the verge of being ruled out in the case of a single Higgs doublet bib:LHC_Higgs4 (), but a Type II two Higgs doublet model can be designed to reproduce the hints at from the LHC and the Tevatron. In that case, quark mass splittings less than are still favored. In all of our scenarios, the most favored values for are just above the experimental limit of GeV, making searches for a fourth generation charged lepton an interesting possibility.
V Acknowledgements
We gratefully thank the authors of 2HDMC in particular for developing a new version of their code. We also thank P. Kumar and P. Langacker for helpful discussions. J.J.H thanks the Simons Center for Geometry and Physics for hospitality during the completion of this work. The work of L.B. and E.R.H. is supported by DOE contract DEAC0207CH11359; the work of J.E. is supported by CONACyT (México) contracts 82291–F and 151234; the work of J.J.H. is supported by NSF grant PHY0969448 and by the William Loughlin membership at the Institute for Advanced Study.
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