Detecting Fourth Generation Quarks at Hadron Colliders

Detecting Fourth Generation Quarks at Hadron Colliders

David Atwood Dept. of Physics and Astronomy, Iowa State University, Ames, IA 50011    Sudhir Kumar Gupta Dept. of Physics and Astronomy, Iowa State University, Ames, IA 50011    Amarjit Soni Theory Group, Brookhaven National Laboratory, Upton, NY 11973
Abstract

Although there is no compelling evidence, at present, against the Standard Model (SM), in the past few years, a number of 2-3 sigma tensions have appeared which could be alleviated simply by adding another generation of fermions. Furthermore, a fourth generation could help resolve the issue of baryogenesis and the understanding of the hierarchy problem.

In this paper, we consider the phenomenology of the fourth generation heavy quarks which would be pair produced at the LHC. We show that if such a quark with a mass in the phenomenologically interesting range of 400 GeV–600 GeV decays to a light quark and a W-boson, it will produce a signal in a number of channels which can be seen above the background from the three generation Standard Model processes. In particular, such quarks could be seen in channels where multiple jets are present with large missing momentum and either a single hard lepton, an opposite sign hard lepton pair or a same sign lepton pair.

In the same sign dilepton channel there is little background and so an excess of such pairs at large invariant mass will indicate the presence of heavy down type quarks. More generally, in our study, the main tool we use to determine the mass of the heavy quark in each of the channels we consider is to use the kinematics of the decay of such quarks to resolve the momenta of the unobserved neutrinos. We show how this can be carried out, even in cases where the kinematics is under-determined by use of the approximation, which holds quite well, that the two heavy quarks are nearly at rest in the center of mass frame.

Since it is very likely that at least the lightest heavy quark decays in the mode we consider, this means that it should be observed at the LHC. Indeed, it is expected that the mass splitting between the quarks is less than so that if the Cabbibo-Kobayshi-Maskawa (CKM) matrix element between the fourth and lower generations are not too small, both members of the fourth generation quark doublet will decay in this way. If this is so, the combined signal of these two quarks will make the signal for the fourth generation somewhat more prominent.

pacs:
11.30.Er, 12.60.Cn, 13.25.Hw, 13.40.Hq

I Introduction

The Standard Model with three generations (SM3) has been very successful in explaining all experimental results to date, in particular CP violation in the K- and B-meson systems is well understood, to an accuracy of about 15-20%, in terms of the CKM matrix of that theory cref (); kmref (). Recently, however, some “possible evidence” for deviations from SM3 in B decays Lunghi:2007ak (); Lunghi:2008aa (); Lunghi:2009sm (); Lenz:2006hd (); Bona:2008jn (); Bona:2009cj () is claimed. Although these effects can be explained by several physics beyond the Standard Model APS1 (); APS2 (); AJB081 (); AJB082 (); MN08 (); lang (); paridi () scenarios, the simplest viable explanation seems to be an extension of the Standard Model to 4 generations (SM4)  Hung:2007ak (); Frampton:1999xi (); Soni:2010xh (); AS08 (); AJB10 (); GH10 (); Eberhardt:2010bm () where the mass of the new heavy quarks is in the range 400-600 GeV.

If further studies in the B system continue to show deviations from the SM3 predictions, it may be difficult to resolve which extension of SM3 is responsible. The most direct way to determine the nature of the new physics which may be involved is to produce it “on shell”. Indeed, hadron colliders, particularly the LHC, are ideally suited for this task. Since the LHC generates a significant rate of parton interactions up to energies of 1 TeV it may be able to produce direct evidence of the new physics although this is not guaranteed in all cases. For instance, if the explanation lies in warped space ideas APS1 (); AJB082 (); MN08 () then it appears that the relevant particles have to be at least approximately 3 TeV AMS03 () rendering their detection at LHC rather difficult ADPS07 (); shri1 (); shri2 ().

If the new physics is an additional sequential fourth generation, there would be two new heavy quarks, a heavy charge +2/3 quark () and charge -1/3 quark (). These quarks will be produced at the LHC predominantly by gluon-gluon fusions and should be produced at the LHC with appreciable rates MGM08 (). For example at 10 TeV center of mass energy cross-section for pair producing 500 GeV quarks is large, 1 pb, rising to around 4 pb at 14 TeV and LHC experiments may well be able to study up to about 1 TeV MGM08 (); Whiteson10 (), which is well above the perturbative bound chanowitz_furman_hinchliffe (); cfl (). We also note that experiments have already been searching for the heavier quarks and provided (95% CL) bounds:  GeV;  GeV Murat_ICHEP10 (); Whiteson10 (); Conway_BF10 (); CDF09 (); Aaltonen:2011vr (); Abazov:2011vy (); Chatrchyan:2011em (). These bounds are a little higher than the one quoted in Ref. Nakamura:2010zzi () of GeV; GeV at CL.

For the analysis to follow an important characteristic of the fourth generation quark doublet is that the mass splitting between the - and -quarks is constrained by electroweek precision tests to be small, likely less than splitting ().

In addition to resolving the phenomenological hints of physics beyond SM3 that are alluded to in the above discussion, if a fourth generation is present, it may be helpful in explaining the long standing hierarchy problem within the SM. With very massive quarks in the new generation it has been proposed that electroweak symmetry breaking may well become a dynamical feature of the model  Holdom:2009rf (); Holdom:1986rn (); King:1990he (); Hill:1990ge (); Carpenter:1989ij (); Hung:2009hy (); Hung:2010xh (); Burdman:2007sx ().

Another issue which is naturally addressed by the presence of a fourth generation is the origin of baryogenesis in the early universe. The tiny amount of CP violation allowed in the context of the CKM matrix of SM3 is much too small to supply the CP violation necessary for baryogenesis in the early universe. However, if a fourth generation is present, then there are two more additional phases in the CKM matrix. The effects of these new phases is significantly enhanced Hou:2008xd () by the larger masses of the new generation and so the natural CP violation of SM4 is perhaps large enough Ham:2004xh (); Fok:2008yg (); Kikukawa:2009mu () to satisfy that Sakharov Sakharov:1967dj () condition for baryogenesis. It has been pointed out Fok:2008yg (), however, that in the Standard Model with a fourth generation there might not be a first order phase transition hence it may still be the case that additional physics is required to satisfy that condition for baryogenesis. The presence of the two additional phases in the SM4 mixing matrix also can have many interesting phenomenological implications especially in observables that in SM3 are predicted to yield null results  Soni:2010xh (); AS08 (); AJB10 (); GH10 (); Gershon:2006mt (); Eilam:2009hz ().

Motivated by these considerations, in this paper we will consider the strategies for detecting fourth generation quarks at the hadron colliders. In Section II we present the expressions for the decay distributions of the decay of a fourth generation quarks considering, in particular the energy spectrum of the lepton that is produced by the decay of such a quark.

In Section III we discuss the various event samples which are most likely to be useful in obtaining signals of heavy quarks. In particular, we will consider signals consisting of multiple hard jets with missing momentum in combination with either one hard lepton, an opposite sign dilepton pair or a same sign dilepton pair. We also set out a set of basic cuts which are helpful in enhancing the signal with respect to the SM3 background. For each of the three kinds of event samples, we then discuss how the kinematics can be used to determine the mass of the heavy quark. In general a significant signal versus the background will be seen in a histogram of the reconstructed mass. In general we highlight two important challenges in reconstruction of the mass. First of all, in the presence of a large number of jets, there will be a potentially large combinatorial background; secondly in some cases there are not enough kinematic constraints to reconstruct the neutrino momenta which make up the missing momentum. In the case of the same sign dilepton pairs the SM3 background is much smaller than the signal which arises in -pair production thus an excess of high invariant mass same sign pairs is a clear signal for new physics in general and could be produced by -quarks.

In Section IV we discuss in detail the Standard Model (i.e. SM3) backgrounds which contribute to the event samples while in Section V we present our conclusions.

Ii Decay Rates of Heavy Quarks

Let us now consider the main decay modes for the - and - quarks. A similar discussion is found in Das:2010fh (); Arhrib:2006pm () and in Holdom:2007ap (); Chao:2011th () the threshold effects at the interface between two body and three body decays are discussed.

If the is more massive than the and the splitting is greater than then the main decay mode of the will be . If the splitting is less than then it can decay through the three body modes where is a virtual which either decays leptonically or hadronically. The heavy top can also decay through the two body mode to lighter quarks, etc.. As we shall show below, unless the two body mode will generally dominates over the three body mode. In this scenario the lighter should undergo a two body decay to u- or c-quarks with the relative branching ratios depending on the values of .

If the is lighter than the ; the will decay via a 2 body mode to generation 1-3 quarks; in analogy with the case above, the relative branching ratios depend on the values of . If the splitting is greater than , the will decay via . If the splitting is less than the dominant decay mode might either be the three body mode or the two body mode to generation 1-3 quarks, depending on the value of . Again, the two body mode will dominate unless .

Six scenarios for the complete decay chains of a fourth generation quark are thus possible depending on whether the relative mass of these quarks and the CKM matrix coupling to the lighter generations:

1. in which case and where (with branching ratio depending on the CKM elements).

2. If and is relatively large, the dominant decay of is with a small branching ratio to the three body decay . Again where .

3. If but is small enough, the dominant decay of is . Again where .

4. If in which case and where (with branching ratio depending on the CKM elements).

5. If and is relatively large, the dominant decay of is with a small branching ratio to the three body decay . Again where .

6. If but is small enough, the dominant decay of is . Again where .

In the scenarios where the dominant decay is , there is an important distinction between the case where and . In the case the quark will just manifest as a single jet while in the case, the top subsequently decays to and the may in turn decay leptonically or hadronically. The collider signature will thus depend on the nature of the decay.

If the CKM matrix is not unitary (i.e. the fourth generation is not a genuine sequential generation of the SM) or if there is a further fifth generation (thus rendering the CKM submatrix non-unitary) then there are some possible modifications to the above scenarios. If is small, then in scenarios 1 and 4 it might not be the case that or are the dominant decay modes and these cases will resemble scenarios 2 and 5 respectively.

In the case where is small enough to suppress the two body decay mode as in scenario 3, it could happen that for is sufficiently large that the mode becomes important. If the CKM matrix is unitary then:

 ∑i=1,3|Vt′i|2=∑i=1,3|Vib′|2 (1)

therefore a large will imply a large for some or and so the two body mode must dominate. The analogous argument also applies for the decay in scenario 6.

ii.1 Decay Rates

Let us now consider the two body decay of a heavy quark where is a fourth generation quark and is either the other fourth generation quark or a 1-3 generation quark.

The total decay rate at tree level is:

 Γ(q1→q2W) = |V12|2Γ2(m1)Δ(1,x21,xW1)(Δ(1,x21,xW1)2+(3+2x21−3xW1)xW1) (2)

where

 x21 = (m2m1)2     xW1=(mWm1)2 Δ(a,b,c) = √∣∣a2+b2+c2−2ab−2bc−2ca∣∣ Γ2(m1) = GF8π√2m31 (3)

Note that in the limit that which would be the case if , we can approximate Eqn. (2) by:

 Γ(q1→q2W) ≈ |V12|2Γ2(m1)(1−x1W)2(1+2x1W)+O(x21) (4)

Conversely if is not small (i.e. for decays between the two heavy quarks or to ) one can expand in this expression in :

 Γ(q1→q2W) ≈ |V12|2Γ2(m1)((1−x21)3+x21xW1(1−x21))+O(x2W1) (5)

Let us now consider the three body decay where, in this paper, we will generally consider and to be the fourth generation quarks and are light fermion pairs which arise from the virtual so , , , , , or .

At tree level,

 Γ(q1→q2ff′)=|V12|2∣∣Vff′∣∣2Nc(ff′)Γ3(m1)I(x21,xW1) (6)

for quark pairs and 1 for leptons; is the appropriate CKM element for quark pairs and 1 for lepton pairs and

 Γ3(m1) = G2F192π3m51. (7)

The factor is given by

 I(x21,xW1) = 12x1W(13(1−x21)(2V2−6xW1U+xW1W)+x2W1Ulog1x21 (8) +x2W12U2−x21V[arctan1−UV+arctanU−x21V])

where

 U = 12(1+x21−xW1) V = 12Δ(1,x21,xW1) W = 12(1+x21+xW1) (9)

In the scenarios we are most interested in where the splitting between the two heavy quarks is not very large, the expression in Eqn.(8) is well approximated by:

 I(x21,xW1) = (1−x12)5(25+15(1−x12)+4xW1+335xW1(1−x12)2+O((1−x12)3)) (10)

ii.2 Decay Kinematics and Lepton Energy Spectrum for Two Body Decays

The two body kinematics together with the V-A structure of the W couplings determine the energy spectrum of the lepton arising from the two body and .

In the case of the kinematic limits of the lepton energy in the rest frame of the are:

 Eℓmax = 14mb′[m2b′+m2W−m2t+Δ(m2b′,m2t,m2W)] Eℓmin = 14mb′[m2b′+m2W−m2t+Δ(m2b′,m2t,m2W)] (11)

Within this kinematic region, the distribution is given by:

 dΓdEℓ ∝ Eℓ(E0−Eℓ) (12)

where .

From this we can calculate the average lepton energy:

 ¯¯¯¯Eℓ = mb′41−3xt+xW+3x2t+x2W−3x3W−x3t−xWx2t+5xtx2W1+xW−2xt+x2t+xtxW−2x2W (13) = mb′4(1−xt)+O(x2t,x2W)

where and .

When a pair is produced, the matrix element and the structure functions tend to drive it to lower . The transverse momentum distribution of the lepton should thus be well approximated by the transverse energy distribution of the at rest. The average transverse momentum is thus given by:

 ¯¯¯¯PℓT = π4¯¯¯¯Eℓ≈π16mb′ (14)

The distribution in the limit where the is at rest is:

 dΓdPT = ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩PT(E0cosh−1EmaxPT−√E2max−P2T)if Emin

In -quark decay a lepton of the opposite sign may also be produced through the cascade through the top quark, . Again if the is at rest, we can develop an analytic expression for the energy spectrum of this lepton.

It is useful to divide the spectrum into three segments by the energies:

 E0 = 12mWe−(θt+θW) E1 = 12mWe−|θt−θW| E2 = 12mWe+|θt−θW| E3 = 12mWe+(θt+θW)

where

 θW = arccoshxt+xW2√xW θt = arccosh1+xt−xW2√xt. (17)

The energy spectrum is thus

 dΓdEℓ ∝ ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩F(y,zmax)−F(y,xW+4y24y)if E0

where

 zmax = √xWcoshθt+θW zmin = √xWcoshθt−θW (19)

and

 F(y,z) = y4η((2yxt−(2y+6yxt+xt)z+8xt(1+y)z2+4xtz3) (20) +xW((1+6y+11xt+1−yxt)−(3−6yxt+5xt)z−4(1−xt)z2) +x2W((4−6yxt−10y−7xt)+(5+2yxt+4y+3xt)z+4z2) +x3W((2y−6−xt)−(1+2y)z)+x4W) −y2arccosh(y)(xt(1+4y)+2xW(1+xt+y−yxt)−x2W(3+xt+2y))

and

Iii Event Samples

In order to find evidence for heavy quark pair production at hadronic colliders, we use the decay scenarios discussed above to suggest which signals should be searched for. In this discussion we would like to highlight three issues which lead to acceptance cuts that apply all the signal classes we consider. First of all, we will introduce a set of basic cuts which will preserve the signal but reduce the SM background to a manageable level (about signal). Next, we will consider extracting a reconstructed from the kinematics of the events that pass the basic cuts. We will see that a histogram in the reconstructed generally separates the signal from background and allows the determination of the heavy quark mass. In order to most effectively use this method, however, it is useful to reduce the combinatorial background, particularly if the signal consists of a large multiplicity of jets. As we will show, this combinatorial background can be greatly reduced due to the fact that most of the jets are in pairs resulting from W-boson decay.

Let us first consider the possible heavy quark decay modes and then turn our attention to the analysis of the cases which are likely to be of greatest experimental interest.

In general, if we assume that the heavy quark decays dominantly through a two body decay mode, the net decay of the heavy quark will be to a light quark plus 1-3 of W-bosons

If is the heavy quark (either a - or - quark) and one of the five lightest quarks , the decay chain will assume one of the following forms:

 (1)Q→qW(2)Q→qWW(3)Q→qWWW (21)

In particular, decay channel (1) occurs when a decays to a , or quark and a W-boson or when a decays to a or and a W-boson. Case (1) would be the dominant decay mode in the mixing scenario where was much smaller than and/or .

Conversely, the quark will decay dominantly through decay channel (2) if the first decay in its cascade is where the top quark then decays to . The could also decay through a channel like this in the scenario where and the decayed through channel (1), thus or .

Decay channel (3) would apply if the cascaded down to the which in turn decayed via channel (2). Thus .

If the mixing between the fourth and third generations is sufficiently small and the quark mass splitting is less than then two other decay modes involving three body decay channels may be important:

 (4)Q→Q′W∗→qWW∗(5)t′→b′W∗→tWW∗→bWWW∗ (22)

(=virtual W-boson) where in channel (4) is the heavier fourth generation quark and is the lighter fourth generation quark.

Since the fourth generation quark is pair produced, depending on which of the channels above controls the decay, there are potentially up to 6 W-bosons plus 2 light quark jets in the final state. If all those W-bosons decayed hadronically, that would result in a final state with up to 14 jets. In any case the QCD background to a purely hadronic final state is likely overwhelming so we must consider cases where at least one of the W-bosons decays leptonically.

In particular, we will consider the prospect of signals where one or two of the W-bosons decay leptonically. In such a case, the signal will be a final state with one or two hard lepton(s) and significant missing momentum.

Depending on which of the channels 1-5 is dominant for each quark species, it is not unreasonable to suppose that signals of this type will received contributions from both species of quarks. This is a natural situation if the two species are roughly degenerate and so both and quarks will be produced at roughly comparable rates, especially if the masses of the - and -quarks are around 400-600 GeV and the collisions are at LHC energies,  TeV. Such a state of affairs can be helpful in building a signal indicating a fourth generation even before the individual contributions from and are separately identified.

The case where one or two W-bosons decay leptonically therefore leads to three signal channels for the fourth generation of quarks. We will consider the signals in each of these channels in order to determine the signal to background ratio and how the signal may be used to reconstruct the mass of the quark. These issues are related in that the difference in kinematics between SM backgrounds and the heavy quark signals means that if the mass of the quark can be reconstructed, this will provide a good mechanism for separating signals from background.

The three event samples which we consider are as follows:

1. Single lepton sample: The signature of this sample is where is a lepton, either or , means jets and means missing transverse momentum.

2. Like sign di-lepton sample: The signature of this sample is

3. Opposite sign di-lepton sample: The signature of this sample is . Note that for some of the potential SM backgrounds, it might be helpful to consider . Only heavy quarks that cascade to at least two W-bosons will contribute to this sample.

In our analysis we will consider mainly the scenario where the splitting between the two heavy quark masses is less than and the CKM element between the fourth generation quarks and lighter quarks is large enough that the dominant decay mode is the two body decay to lighter quarks. Thus the decay modes for the and quarks we are mainly considering are:

 t′→bW b′→tW→bWW (23)

which are the modes that apply if the dominant mixing of the fourth generation is with the third. The analysis we carry out however easily generalizes to the case where this assumption is weakened. Thus if the decay in fact proceeds through

 t′→d or sW b′→u or cW

the kinematics in these cases will be identical to the decay to and so the analysis we discuss below will apply to all of these cases. In addition, if b-tagging can be carried out then we can distinguish the final state from .

For our event analysis we first generated the signal and background events with the aid of MadGraph Stelzer:1994ta () and later interfaced these to PYTHIA6 pythia () for further analysis including decays of the top and W’s.

For the -pair event generation, we wrote the MadGraph model files to incorporate the and and their interactions. We use CTEQ6L Pumplin:2002vw () to evaluate parton densities. The renormalization scale, , and the factorization scale, are fixed at

 μR=√^s=μF (25)

Jet formation has been done using default PYTHIA scheme implemented through PYCELL. We also incorporate effects of initial state radiation (ISR) and final state radiation (FSR) using the same simulation package.

The basic cuts that apply in these tables on leptons, and, jets, j (including b’s) which consists of

• Lepton should have  GeV and , to ensure that they lie within the coverage of the detector.

• jets should have  GeV and

• Spatial resolution between lepton - lepton, lepton - jet, and, jet - jet should be , , respectively, (where , , ), such that the leptons and jets are well separated in space.

• A missing transverse energy cut, to enhance the likelihood that leptons are due to decay.

In addition to the cuts mentioned above, we apply the following cuts to reduce the background further:

• A minimum cut on the scalar sum of transverse momenta () of the final state lepton, jets, and the missing transverse energy of 350 GeV. is defined to be:

 HT=pTvisible+⧸ET=∑i=l,jpTi+⧸ET. (26)

Effects of the aforementioned cuts are shown in Table 1 where results for both quark species are considered and results are given in the cases of , and  TeV. In all cases we consider the signals for heavy quark masses and  GeV. In Table 1 we did not put any isolation cuts on the jets. In Table 2 we further demand that all the jets are separated with . We see that this latter requirement does not alter the numbers very much.

In our approach, to further enhance the signal to background ratio and characterize the fourth generation quarks, we will first consider the reconstruction of the heavy quark mass from the kinematics of the event. In this endeavor we must deal with the combinatorial background that results from the high jet multiplicity; in particular, the kinematic role of each of the jets in the event is not a priori known. This problem is most acute in in the high jet multiplicities which result from -pair events. We then discuss the various methods which result in the reduction of this combinatorial background. We now detail our analysis in each of the signal types.

iii.1 Analysis of Single Lepton Sample

Under our assumptions, both and decays can contribute to signals of this type.

For each event of this type, the key to determining the kinematics and therefore the heavy quark mass is the partitioning of the jets between the two ’s in the initial state. In particular, in this case one of the heavy quarks decays only to hadrons while the others decay includes the one lepton observed. We will denote these two heavy quarks by and respectively. Thus if there are jets, of which should originate from the then an initial combinations must be tried where only one of the possible partitions will be ”correct”.

In the correct partition, it is possible to determine the momentum of the unobserved neutrino. If we denote by the total 4-momentum of all the jets assigned to the side of the event and all the jets assigned to the side of the event then the following five constraints apply to the four undetermined neutrino 4-momentum:

 (1)(ν+ℓ)2=m2W(2)(ν+ℓ+jℓ)2=j2h   (=m2Q)(3)(ν)2=0(4)νx=\hbox{p}\hbox to 0.0pt{/ }x(5)νy=\hbox{p}\hbox to 0.0pt{/ }y (27)

Note that if we combine equation (1) and (2) with (3) we can rephrase these two conditions as:

 (1′)ν⋅ℓ=12m2W(2′)2ν⋅(ℓ+jℓ)+(ℓ+jℓ)2=j2h   (=m2Q) (28)

which are linear in

Generally, we can use two of conditions 1-3 to solve for the - and -components of the neutrino momentum while 4 and 5 give the - and -components. The remaining condition acts as a check to ensure that we have a consistent partitioning of the jets. For example if for a given partition of the jets in each event we solve then (because the equations are linear) we will have a unique determination of and therefore and . Only for the correct partitions of events will the reconstruction give . Thus if we plot all possible reconstructions on a scatter plot of versus and accept only those events in a strip near we should find an accumulation of events near the real value of . Those events outside of the strip are presumably background or wrongly partitioned events. If there were data from two (or more) quarks mixed together in the sample, then there would be multiple peaks corresponding to the masses of each of the quarks present. Since all of the decay channels 1-5 can contribute to this sample, this method will ultimately determine all of the fourth generation masses regardless of which mixing scenario applies.

In Figure 3 we show Leggo plots of the reconstructed versus reconstructed for a number of different scenarios. For the correct partitioning of the jets the events would indeed be at the physical values of and but not so for the incorrect partitions.

Returning to Eqn. 27 we can also use two slightly different approaches to reconstructing the quark mass which may offer some advantages. If we start with constrains then on each event we extract the apparent masses of each side of the event: and . We then check condition 2 by constructing an versus scatter plot. The correct partitions will be near the diagonal , implementing condition 2, and the quark mass or masses will be revealed as accumulations in . In this method, for each event the neutrino momentum is determined independently of the partitioning of the jets. However, since the equations are quadratic, there is a two fold ambiguity in the solution so on the scatter plot two points must be plotted for each partitioning of each event increasing the combinatorial background.

In Figure 4 we show a histogram of versus for the and cases. Again the peaks at the correct value of correspond to correct partitioning of the jets.

In Figure 5 we illustrate another approach which is to solve the equations and then reconstruct the mass as . As with the above method the equations here are quadratic giving an additional 2 fold ambiguity. In this approach the resulting scatter plot will be in and where the correct partition will be in a strip near . The heavy quark mass or masses can be extracted from accumulations in . This approach has the advantage that if there were another (beyond the SM) particle playing the role of the -boson in the decays such as a charged Higgs, an additional W boson or a Kaluza-Klein excitation of the W-boson, this would be evident in additional accumulations of events in the variable.

Conversely, using this method, if decay channels (4) or (5) are significant, the case where virtual W-boson decays to will lead to additional points on the scatter plot with the correct value of but with .

In enumerating the various jet partitions for a given event, it is useful to combine equations (1’) and (2’) which leads to the inequality:

 j2h≥m2W+2jℓ⋅ℓ+j2ℓ (29)

where this inequality will eliminate at least half of the possible partitions.

Let us now see how to use such kinematics to enhance the signal to background. For this, we will start with the method where we solve 1,3,4,5 to determine and . For a given partition of the jets, we define . For a given event, we will select the reconstructed value of , to be the value of corresponding to the partition with the minimum value of .

Another cut which may be helpful in limiting the combinatorial background is to first pair up the jets in pairs with roughly the W mass. For example in the case there must exist one pair of jets which results from the decay of a W-boson and therefore should have an invariant mass of . Let us denote the deviation of such a jet pair from by, , so using this cut, we would only consider partitions of jets where was smaller than some threshold. In the case of this cut is more constraining since it would apply to three different jet pairs in a given jet assignment.

In Figure 6 we show a histogram of the reconstructed mass using this method. In the cut on the upper left we just use the basic cuts. In the two plots on the right we use the cut  GeV while in the lower two plots we impose the cut. Thus the graph on the lower right has both cuts imposed. The plots are shown for the SM background and for , and  GeV. Clearly the signal peaks well above background and the cut appears helpful in enhancing this further.

In Figure 7 we apply the same method to the case of and again the mass peak is well above the background and the signal is further enhanced by the cut. In Figure 8 we consider the same method in the case where we consider the total signal where both species contribute. As an illustration here we are assuming that  GeV and  GeV. The close mass of the two quarks gives a signal which is larger than we would get if just one quark contributed (see also Holdom:2011uv ()).

iii.2 Dilepton signals

If two of the W-bosons in the decay chains we are considering decay leptonically there will consequentially be two leptons in the final state. If those two W-bosons are of opposite sign then the lepton pair will therefore be of opposite sign while if the two W-bosons are of the same sign then the pair will likewise be of the same sign. Due to the simplifying assumptions we are making, this latter case only occurs in the decay chain of -quark pairs. Below we will consider the problem of extracting the heavy quark mass from the kinematics in a dilepton signal while here we will consider the characteristics of the signal itself.

Hard leptons are often part of the signal of new physics. In this case, the dileptons would be produced in association with jets and missing momentum. To study this signal (for opposite sign and like sign), we selected Montecarlo events passing the basic cuts described above and also imposed the cut that  GeV.

In Figure 9 we plot the invariant mass spectrum for opposite sign dilepton pairs in the case of - and -quarks of mass 450 and 600 GeV as well as the case with both species present where  GeV and  GeV.