TIFR/TH/12-11

Large mass splittings for fourth generation fermions allowed by LHC Higgs bosonexclusion

Amol Dighe,1, ∗ Diptimoy Ghosh,1, † Rohini M Godbole,2, ‡ and Arun Prasath2, §

1Tata Institute of Fundamental Research, 1, Homi Bhabha Road, Mumbai 400 005, India2Centre for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India

(Dated: July 4, 2012)

In the context of the standard model with a fourth generation, we explore the allowed massspectra in the fourth-generation quark and lepton sectors as functions of the Higgs mass. Using theconstraints from unitarity and oblique parameters, we show that a heavy Higgs allows large masssplittings in these sectors, opening up new decay channels involving W emission. Assuming thatthe hints for a light Higgs do not yet constitute an evidence, we work in a scenario where a heavyHiggs is viable. A Higgs heavier than ∼ 800 GeV would in fact necessitate either a heavy quarkdecay channel t′ → b′W/b′ → t′W or a heavy lepton decay channel τ ′ → ν′W as long as the mixingbetween the third and fourth generations is small. This mixing tends to suppress the mass splittingsand hence the W -emission channels. The possibility of the W -emission channel could substantiallychange the search strategies of fourth-generation fermions at the LHC and impact the currentlyreported mass limits.

PACS numbers: 14.65.Jk,11.80.Et,13.66.Jn,14.60.Hi

I. INTRODUCTION

The number of fermion generations in the standardmodel (SM) is three, though there is no fundamentalprinciple restricting it to this number. The data on Zdecay only puts a lower bound of mZ/2 on their masses[1]. Direct searches at the Tevatron [1–6] and the LHC[7–14] further put lower bounds on the masses of fourth-generation charged fermions, by virtue of them not hav-ing been observed at these colliders. These limits aresubject to certain assumptions about the decay channelsfor these quarks. The most conservative, almost model-independent limits for fourth-generation quarks are givenin [15]. Indirect limits on the masses and mixing of thesefermions are also obtained through the measurementsof the oblique parameters S, T, U [16]. Theoretical con-straints like the perturbativity of the Yukawa couplingsand the perturbative unitarity of heavy fermion scatter-ing amplitudes [17] bound the masses from above. Inspite of the rather strong bounds from all the above di-rections, there is still parameter space available for thefourth generation that is consistent with all the data [18–29]. The fourth-generation scenario (SM4) is thus stillviable even after the recent Tevatron and LHC results[30, 31].

The discovery of a fourth-generation of fermionswill have profound phenomenological consequences [32].Some of the experimental observations that deviate some-what from the SM expectations – like the CP-violatingphases in the neutral B mixing [33–36] – could be in-

∗Electronic address: amol@theory.tifr.res.in†Electronic address: diptimoyghosh@theory.tifr.res.in‡Electronic address: rohini@cts.iisc.ernet.in§Electronic address: arunprasath@cts.iisc.ernet.in

terpreted as radiative effects by the fourth-generationfermions. The implications of a fourth family for ob-servables in charmed decays [37] and lepton-flavor violat-ing decays [38] as well as flavor constraints on the quarksector[39] have also been discussed.

The Cabibbo-Kobayashi-Maskawa (CKM) structure of4 generations, with 3 observable phases and large Yukawacouplings, may also provide enough source of CP viola-tion for the matter-antimatter asymmetry of the Uni-verse, although the issue of the order of the electroweakphase transition still has to be resolved [40]. The exis-tence of a fourth-generation is also intimately connectedwith the Higgs physics. The Higgs in the SM4 with amass of about 800 GeV is consistent with precision elec-troweak (EW)data when mt′ ,mb′ ∼ 500 GeV, with themixing between third and fourth-generation quarks of theorder of 0.1 [25]. This raises an interesting possibilityof Higgs being a composite scalar of fourth-generationquarks [41–43] with interesting phenomenological im-plications including an enhancement of flavor-changingas well as flavor-diagonal Higgs decays into third andfourth-generation fermions. Implications of a stronglyinteracting fourth-generation quark sector on LHC Higgssearches has been discussed in [44]. Phenomenology ofthe lepton sector of the SM4 has been studied in [45–48].

The Higgs production cross section at hadron collid-ers is affected strongly by the fourth-generation throughthe gg → h channel due to the heavy masses of the newfermions [49–52]. The branching ratios of Higgs into dif-ferent channels are affected too [49, 52, 53]. As a con-sequence, the direct search limits on the Higgs mass arestronger in the presence of the fourth-generation. Higgsproduction and decay cross sections in the context of fourgenerations, with next-to-leading order EW and QCDcorrections, have been calculated in [50–52]. The produc-tion cross section is enhanced for a light Higgs (mh < 200GeV) by an order of magnitude. However the enhance-

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ment may be somewhat reduced for a heavier Higgs [54].The ATLAS experiment at the LHC has excluded theHiggs boson of SM4 with a mass between 119 GeV and593 GeV [55], while the CMS exclusion limits are from120 GeV to 600 GeV [56]. These experiments include theone loop EW corrections to Higgs production from thefourth-generation fermions1.

Recently, experiments at the Tevatron and the LHChave reported an excess of events around mh ≈ 125 GeVwith a local significance of about 2σ−3σ in the search forthe standard model Higgs boson [63–65]. Reference[66]interprets these “hints” for a light Higgs in the frameworkof SM4. We on the otherhand, assume that these hintsdo not yet constitute an evidence and may well be astatistical fluctuation. Under this assumption, a heavyHiggs with mass & 600 GeV is a viable scenario.

Recent explorations of possible effects of a fourth-generation on the Higgs mass, precision observables,quark mixing matrix, and flavor-changing neutral currentphenomena have yielded interesting results. It has beenpointed out [67] that the existence of a fourth generationallows for a heavier Higgs to be consistent with the pre-cision measurements. The constraints on the mixing be-tween the third and fourth generation have been obtainedfrom the precision EW data [25], and a fit to the flavor-physics data [68]. The latter shows that, while the mixingof the fourth-generation quarks to the three SM genera-tions is consistent with zero and restricted to be small,observable effects on K and B decays are still possible.Large masses of the fourth-generation fermions lead tononperturbativity of Yukawa couplings at a low scaleΛ�MGUT as well as instability of the vacuum. This hasbeen investigated in the context of models without super-symmetry (SUSY) [41, 67, 69–72] and with SUSY [73],after taking into account various bounds from precisionEW data as well as collider and direct search experimentson sequential heavy fermions.

In this article, we revisit the electroweak precisionconstraints from the oblique parameters S, T, U on thefourth-generation, taking into account the mixing withthe third generation. We perform a χ2 analysis, varyingthe fourth-generation quark as well as lepton masses intheir experimentally allowed ranges, and obtain a quan-titative measure for the fourth-generation fermion massspectrum preferred by the measurements of these param-eters. In the light of the heavy Higgs preferred by theLHC data, we focus on the implications of a heavy Higgsfor the mass spectrum. We also study the effect of themixing between third and fourth-generation on this massspectrum, and try to understand these effects analyti-cally. As we will see later, the correlation between themass splittings in quark and lepton sectors is strongly

1 These bounds can be circumvented either by a suitable extensionof the scalar sector [57] or by having Higgs decay to stable invis-ible particles which could be candidates for dark matter[58–62].However this is not the minimal SM4 we focus on in this paper.

influenced by this mixing angle.The paper is organized as follows. In Sec. II, we dis-

cuss the bounds on the fourth-generation fermion massesfrom direct searches and the theoretical requirement ofperturbative unitarity. We also analyze the structure ofconstraints from the measurements of oblique parame-ters. In Sec. III, we perform a χ2-fit to the oblique pa-rameters and obtain constraints on the mass splittings inthe quark and lepton sectors, focusing on a heavy Higgs.Sec. IV discusses the collider implications, while Sec. Vsummarizes our results.

II. CONSTRAINTS ON THEFOURTH-GENERATION FERMION MASSES

A. Lower bounds on masses from direct searches

The direct search constraints presented by CDF [2, 3]on the masses of t′ and b′ quarks have been gener-alized to more general cases of quark mixing by [15],and a lower bound mt′ ,mb′ > 290 GeV has been ob-tained. The currently quoted exclusion bounds by CDF,DØ, CMS and ATLAS collaborations [6, 8–10, 12, 14]are 400-500 GeV, however as stated earlier, they arebased on specific assumptions on branching ratios of thefourth-generation quarks and mass differences betweenthe fourth-generation fermions.

The limits on the masses of heavy charged fermions areobtained from the nonobservation of their expected de-cay modes. The choice of analyzed decay modes affectsthe bounds to a large extent. Since we would like ourresults to be independent of assumptions about the mix-ing angles, mass differences, and hence branching ratios,in our analysis we shall use the bounds from [15] for thequark masses. For the fourth-generation leptons τ ′ andν′, we take the bounds mτ ′ > 101.0 GeV and mν4 > 45.0GeV [1]. The mixing of the fourth-generation leptons isrestricted to be very small, so it would not affect ouranalysis.

B. Upper bounds on masses from unitarity

The direct search constraints imply that the fourth-generation quarks are necessarily heavy (mF �MW ,MZ). For such heavy fermions F , the tree-level amplitudes of certain processes like FF →FF ,WW,ZZ,ZH,HH, in a spontaneously brokenSU(2)L × U(1)Y gauge theory, tend to a constant valueGFm

2F at center-of-mass energies s � m2

F . For largevalue of mF , the term GFm

2F can be O(1). In that case,

the tree-level unitarity of the S-matrix is saturated andin order to regain a unitary S-matrix, higher order ampli-tudes need to contribute significantly. This necessitatesa strong coupling of these fermions to the gauge bosons,which makes the perturbation theory unreliable. Thiswas first studied in the context of the SM with ultraheavy

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fermions in [17, 74].The corresponding analysis in thecontext of the minimal supersymmetric standard modelwith a sequential fourth generation was performed in [75],in the limit of vanishing mixing between the fourth gen-eration quarks and the first three generations.

We reevaluate the bounds given in [74], considering theJ = 0 partial-wave channel of the tree-level amplitudesof the color-neutral and charge-neutral processes FiFi →FjFj . In [74] only the amplitudes involving two heavyfermions of a SU(2)L doublet were analyzed. The secondSU(2)L doublet of heavy fermions only provided a sourcefor mixing included in the analysis. However, we includeall the relevant channels involving all the heavy quarks-t′, b′ and t, and take into account mixing between thethird and fourth-generations. The lowest critical valueof the fermion mass is obtained by equating the largesteigenvalue of this submatrix to unity. Expressions for thepartial-wave matrices are given in the Appendix. Theresults are shown in Fig. 1 for sin θ34 = 0.0, 0.3. One caneasily observe from the figure an improvement of about6% in the bounds compared to those in [74]. The boundsare affected due to the inclusion of the top quark in theanalysis, which introduces more scattering channels. Itmay be seen from Fig. 1 that the bounds are not verysensitive to the actual value of the mixing.

In the lepton sector, only fourth-generation leptons(τ ′ and ντ ′) are relevant for the perturbative unitar-ity constraints as all the first three generation fermionsare light compared to MW . The mixing between thefourth-generation leptons and the first three generationsis constrained by experimental bounds on lepton- flavorviolating processes [38, 76, 77]. The 2σ lower boundon the (4,4) element of the Pontecarvo-Maki-Nakagawa-Sakata matrix in SM4, Uτ ′ν′ , is very close to unity:|Uτ ′ν′ | > 0.9934 [76]. Moreover, we do not have anyheavy fermion in the first three generations in contrastto the quark sector which has the top quark. Therefore,we do not have any new channel in addition to thosethat are considered in [74].Hence we do not expect anyimprovement over the bounds given in [74] in the case ofleptons.

C. Constraints from oblique parameters

The oblique parameters S, T, U [78] are sensitiveprobes of the fourth-generation masses and mixing pat-tern. The definitions of S, T, U parameters are given by

S = 16π[Π′33(0)−Π′3Q(0)] , (1)

T =4π

s2W (1− s2W )m2Z

[Π11(0)−Π33(0)] , (2)

U = 16π[Π′11(0)−Π′33(0)] , (3)

where s2W = sin2 θW , and Π(q2) are the vacuum polariza-tion Π-functions. The suffixes 1, 2, 3 refer to the gener-ators of SU(2)L, and the suffix Q to that of the electro-magnetic current. The contribution to these parameters

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Our result, sinθ34 =0.0

Our result, sinθ34 =0.3

Chanowitz et al, sinθ34 =0.0

Chanowitz et al, sinθ34 =0.3

FIG. 1: The perturbative unitarity bounds on(mt′ ,mb′) are shown above as dashed-lines for

sin θ34 = 0.0 (green/light) and 0.3 (blue/dark). Thesolid lines are obtained from the analytical expression

for the unitarity bound given in [74] by substituting therespective values of sin θ34. The corresponding bounds

on the fourth-generation lepton masses are mτ ′, ν′ < 1.2TeV [74].

from a fermion doublet (u, d) is [78]

S(xu, xd) =NC6π

[1− Y log

(xuxd

)], (4)

T (xu, xd) =NC

16πs2W (1− s2W )×[

xu + xd −2xuxdxu − xd

log

(xuxd

)], (5)

U(xu, xd) =NC6π

[−5x2u − 22xuxd + 5x2d

3(xu − xd)2+

x3u − 3x2uxd − 3xux2d + x3d

(xu − xd)3log

(xuxd

)], (6)

where xf ≡ m2f/m

2Z . Here NC is the number of colors

of the fermions (NC = 3 for quarks and NC = 1 forleptons), and Y is the hypercharge of the fermion doublet.Note that when the quarks in the doublet are almostdegenerate, i.e. ∆ ≡ |mu −md| � mu,md,

T (xu, xd) =1

12πs2W (1− s2W )

(∆2

m2Z

). (7)

These expressions can be readily generalized to the caseof additional sequential generations.

Following Gfitter [16], we fix the masses of the topquark and the Higgs boson at their reference values ofmt = 173.1 GeV and mh = 120 GeV. With these masses,in the SM we have S = 0, T = 0 and U = 0. The devi-ation from these values are denoted by ∆S, ∆T , ∆U re-spectively. The effect of the Higgs mass appears through

4

the dependence [78]

∆SH =1

6πlog

(mh

mh

), (8)

∆TH = − 3

8π(1− s2W )log

(mh

mh

), (9)

∆UH ≈ 0 . (10)

The contribution from the fourth-generation offermions to these parameters, after taking into ac-count the mixing between third and fourth-generationof quarks, can be expressed as

∆S4 = S(xt′ , xb′) + S(xν′ , xτ ′) , (11)

∆T4 = −s234T (xt, xb) + s234T (xt′ , xb) + s234T (xt, xb′) +

c234T (xt′ , xb′) + T (xν′ , xτ ′) , (12)

∆U4 = −s234U(xt, xb) + s234U(xt′ , xb) + s234U(xt, xb′) +

c234U(xt′ , xb′) + U(xν′ , xτ ′) , (13)

where s234 = sin2 θ34, c234 = cos2 θ34 and θ34 is the mix-ing angle between the third and the fourth-generationquarks. Note that here we neglect the mixing of thefourth-generation quarks with the first two generations,since the bounds on this mixing are rather strong [68].We also neglect any mixing of the fourth leptonic gener-ation. We work in an approximation in which we neglectthe nonoblique corrections to precision EW observables.This allows us to use the S, T, U values provided by fitsto the precision EW observables, for example the onesprovided by the Gfitter group [16].

When the mixing of the fourth and third generationquarks is nonzero, the decay width for Z → bb receivescontribution from fourth-generation quarks through ver-tex corrections, in addition to the oblique corrections.To study the effect of fourth generation fermions onthe precision EW observables in general mixing scenar-ios, one should, in principle include both the vertex andthe oblique corrections to precision EW observables [79].However, as mentioned above , in order to use the Gfit-ter results on S, T, U which were obtained in the limit ofvanishing mixing ,we use only the values of the mixingangles that are consistent with the Z → bb constraints[68].

In our analysis, we evaluate the S, T, U parameters nu-merically using FeynCalc [80] and LoopTools [81]. Thenwe take the experimentally measured values of these pa-rameters [16] and perform a χ2-fit to six parameters: thefour combinations of masses

mq ≡ (mt′ +mb′)/2 , ∆q ≡ mt′ −mb′ ,

m` ≡ (mν′ +mτ ′)/2 , ∆` ≡ mν′ −mτ ′ ,

the Higgs mass mh, and the mixing sin θ34. We take theranges of mq and m` to be those allowed by the uni-tarity constraints and the direct search bounds statedabove. For the other parameters, we take |∆q| < 200GeV, |∆`| < 200 GeV, 100 GeV < mh < 800 GeV,| sin θ34| < 0.3, unless explicitly specified otherwise.

We present our results in terms of the goodness-of-fitcontours for the joint estimates of two parameters at atime, where the other four parameters are chosen to getthe minimum of χ2. For the purposes of this investiga-tion, we show contours of p = 0.0455, which correspondto χ2 = 6.18, or a confidence level(C.L) of 95%.

III. CONSTRAINTS ON THE MASSSPLITTINGS ∆q AND ∆`

We now focus on the constraints on the mass splittings∆q and ∆`. We scan over the allowed values of the otherparameters, and take only those parameter sets that areconsistent with all the data currently available.

A. Constraints on ∆q

In the top left panel of Fig. 2, we show the 95% C.L.contours in the mh – ∆q plane marginalizing over othernew-physics (NP) parameters (mq,ml,∆`, θ34). Fromthis panel, it is observed that at large mh values, thevalue of |∆q| can exceed MW .

We now explore the effect of lepton mass splitting (∆`)and the fourth-generation quark mixing (θ34) in more de-tail. The bottom left panel shows the effect of restricting|∆`| to MW , while the top right panel shows the effect ofvanishing θ34. It is observed that there is no significantchange in the allowed parameter space.

However, when both the conditions of vanishing θ34and |∆`| < MW are imposed, the character of the con-straints changes dramatically, as can be seen from thebottom right panel. In this case, not only is |∆q| > MW

allowed at large mh, one has to have |∆q| ≥ MW formh & 800 GeV. At such large mh values, then, either|∆`| > mW or |∆q| > mW . At least one of the decaysvia W emission (t′ → b′W , b′ → t′W , τ ′ → ν′W andν′ → τ ′W ) then must take place.

In most analyses of a fourth-generation scenario witha single higgs doublet, |∆q| had been assumed to be . 75GeV due to the need to satisfy precision EW constraints.As seen above, the LHC exclusion of Higgs masses uptomh ∼ 600 GeV in fact allows larger mass differences inthe fourth-generation quark doublet. Since |∆q| > MW

is allowed, the channel t′ → b′W , or b′ → t′W becomesallowed. This condition will have strong implications forthe direct searches of the fourth generation scenario.

The correlation between ∆q and ∆` observed abovemay be understood analytically as follows. First, notethat the functions T (xu, xd) in Eq. (5) are positivesemidefinite [78]. The only negative contribution to ∆T4of fermions is then from the first term of Eq. (12), how-ever it is compensated by the next term which is larger inmagnitude and positive. [In particular, when xt′ � xb,the first two terms in ∆T4 add up to a positive quan-tity: 3s234(xt′ − xt)/(16πs2W c

2W ).] Thus, the contribu-

tion from fermions to ∆T4 is positive for all values of

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FIG. 2: The 95% C.L. allowed regions in the mh – ∆q plane. In the top two panels, ∆` is varied over |∆`| ≤ 200GeV, while in the bottom two panels ∆` is restricted to be less than MW . In the left two panels, θ34 is varied oversinθ34 ≤ 0.3, while in the right two panels, θ34 has been fixed to zero (no mixing). The parameters mq and ml are

varied over their complete allowed range. The grey shaded region is excluded at 95% by the LHC data. The dashedblue lines correspond to |∆q| = mW .

fermion masses. On the other hand, Eq. (9) showsthat the contribution from Higgs to ∆T is negative formh > mh = 120 GeV. In order to be consistent withprecision electroweak data, the negative contribution to∆TH from Higgs should be compensated adequately bythe positive contribution ∆T4 from fermions. (Althoughwe also have S,U parameters, the effect on T dominatesthe behavior of our results.) This contribution comesfrom a combination of T (xu, xd) in the quark and leptonsectors, leading to a strong correlation between the quarkand lepton mass splittings.

When θ34 is zero or extremely small, the only contribu-tions to ∆T4 are from T (xt′ , xb′) and T (xν′ , xτ ′). Thesequantities then have to be sufficiently large to compen-sate for the large ∆TH appearing at large mh, necessi-tating a large mass splitting either in quark or in leptonsector. As the Higgs mass increases, the compensatingcontribution ∆T4, and hence the required mass splittings,also increase, exceeding MW for mh & 800 GeV. On the

other hand, when θ34 is near its maximum allowed valueof sin θ34 = 0.3, the first three terms in ∆T4 also con-tribute, as a result of which the mass splitting in fourth-generation quarks is restricted and cannot exceed MW .

Based on the above discussion, the features of Fig. 2can be easily understood. The insensitivity of the allowedvalues of |∆q| to | sin θ34| in the top panels is mainly dueto the fact that the lepton mass-splitting |∆`| is variedover a sufficiently large range. This ensures that the con-tribution of Higgs to ∆T is compensated by the contribu-tion from fermions even in the absence of an enhancementof quark contribution by a nonzero value of sin θ34. Nowin the bottom-right panel, |∆`| is restricted to be lessthan MW along with sin θ34 = 0. Here the absence of anenhanced quark contribution to ∆T4 due to sin θ34 = 0,as well as insufficient contribution from leptons to ∆T4due to the restriction on |∆`|, lead to the exclusion of|∆q| . MW at large mh. In the bottom-left panel theexcluded regions |∆q| .MW become allowed as the con-

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tribution from the quarks to ∆T4 is enhanced by sin θ34.Note that earlier works [25, 28, 67] that had predicted

the mass splitting in the quark sector to be less thanMW had focused on a light Higgs. That a lighter Higgswould only allow a small splitting should be clear fromour figures and analytical arguments.

B. Constraints on ∆`

In the top-left panel of Fig. 3, we show the 95% C.L.contours in the mh – ∆` plane, marginalizing over otherNP parameters(mq,ml,∆q, θ34). It is observed that whileat low mh values ∆` can take any sign, at large mh valuesit is necessarily negative, i.e. mτ ′ > mν′ . Moreover, |∆`|can take values as large as 180 GeV.

The effect of setting sin θ34 = 0 on the allowed valuesof |∆`| is not significant, as can be seen in the top-rightpanel. If |∆q| is restricted to be less than MW , whileallowing any mixing sinθ34 ≤ 0.3, the parameter space isagain not affected much as the bottom left panel shows.However, if |∆q| is restricted to be less than MW , alongwith setting sin θ34 = 0 as shown in the bottom-rightpanel, |∆`| is required to be large in magnitude. Formh & 800 GeV, |∆`| > MW and the decay τ ′ → ν′W isbound to occur. The features of the four panels regardingthe role of sin θ34 and |∆q| can be understood by thearguments given in the previous section.

The negative sign of ∆` may be understood as follows.The Higgs contribution ∆SH to the S parameter is al-ways positive, as can be seen from Eq. (8). It increases asmh increases. Since leptons have a negative hypercharge,the contribution to S parameter from leptons can be re-duced if mτ ′ > mν′ for appropriate quark masses.

The interplay of the contributions to ∆S from thefourth-generation leptons and from the Higgs is also re-sponsible for the asymmetry in the allowed region about|∆`| = 0 in the ∆` – mh plane. The T parameter isapproximately symmetric with respect to the masses ofup and down-type fermions when the mass difference offermions is small compared to their masses, as can beseen from Eq. (7). But for large mh, minimizing the lep-tonic contributions to S parameter becomes importantfor consistency. This causes the allowed regions to prefer∆` < 0 compared to ∆` > 0, even though the T param-eter tends to produce symmetric allowed regions.

In contrast to leptons, for quarks, the T parameter be-comes important in constraining |∆q|, as the hyperchageof quarks is positive. This makes allowed regions sym-metric about ∆q = 0.

C. Constraints on (∆q,∆`) and the effect of θ34

The left panel of Fig. 4 shows the allowed parameterspace in the ∆q – ∆` plane for θ34 = 0, for differentmh values. It can be easily seen that with increasingmh, the allowed difference mτ ′ −mν′ increases. This is

consistent with the arguments in Sec. III B that used thecontribution to ∆S from fermions and Higgs. Also, whenthe lepton splitting |∆`| is small, the quark splitting |∆q|has to be large to compensate for the Higgs contributionto ∆T , as argued earlier in Sec. III A. Indeed at largeenough mh values, the allowed region is outside the cen-tral square and hence always corresponds to |∆q| > MW

or |∆`| > mW , implying that the W -emission channel isnecessarily active. Therefore, in case further direct con-straints increase the lower bound onmh to be & 900 GeV,the W -emission signal is not observed in either quark orlepton channel, and θ34 is restricted by independent ex-periments to be very small (say sin θ34 < 0.05), then themodel with four generations can be ruled out at 95%confidence level.

The scenario when the mixing angle θ34 is significant isshown in the right panel of Fig. 4, where the value of θ34corresponds to the current upper bound on it. It can beseen that in such a case, |∆q| > MW is forbidden, while|∆`| > mW is allowed.

Our results are consistent with those obtained earlierin [82]. In addition, we have shown the pattern of al-lowed mass differences of fermions as a function of mh

and quark mixing.

IV. COLLIDER IMPLICATIONS

According to our results, mt′ > mb′ + MW may beallowed for large mh values. In that case, the branch-ing ratio BR(t′ → b′W ) will depend on sin θ34. Such ascenario was considered in [15] to generalize the directsearch experiment limits on mt′ and mb′ to include theeffect of mixing of fourth-generation quarks to the threeexisting generations. In our case, we have zero mixingbetween fourth-generation quarks and the first two gen-erations in contrast to the assumption of [15]. But itdoes not affect the BR(t′ → b′W ) [although the decayst′ → qW (q = d, s) will be forbidden] as long as theb-quark mass can be neglected. Therefore our use ofthe results of [15] for the model-independent bounds onthe quark masses is still justified. The consideration ofmt′ > mb′ + MW scenario by [15] was motivated by theresult of [82] which stated that in a two-Higgs doubletmodel with a fourth generation, mt′−mb′ can be greaterthan MW and also be consistent with precision EW data.Reference [82] also shows thatmt′−mb′ > MW is possiblewith one Higgs doublet of SM4 when mτ ′ −mν′ ≤MW .However, it assumes no 3-4 mixing in the quark sec-tor. Similar conclusions hold also for the mass differencemb′ − mt′ , which may be greater than MW , leading tothe possibility of the decay channel b′ → t′W .

We have shown that |mt′−mb′ | > MW is allowed evenafter marginalizing over the lepton masses and θ34. Ourresult, in addition to the result of [82], justifies consider-ing |mt′ −mb′ | > MW for interpreting direct search dataon fourth generation quarks, as was done in [15]. Our re-sult also means that the conditions |mt′−mb′ | > MW can

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200

150

100

50

0

50

100

150

200

∆` (G

eV)

∆`=MW

∆`=−MW

|∆q | 200 GeV

sinθ34 =0

100 200 300 400 500 600 700 800mh (GeV)

200

150

100

50

0

50

100

150

200

∆` (G

eV)

∆`=MW

∆`=−MW

|∆q | MW

100 200 300 400 500 600 700 800mh (GeV)

200

150

100

50

0

50

100

150

200

∆` (G

eV)

∆`=MW

∆`=−MW

|∆q | MW

sinθ34 =0

FIG. 3: The 95% C.L. allowed regions in the mh – ∆` plane. In the top two panels, ∆q is varied over |∆q| ≤ 200GeV, while in the bottom two panels ∆q is restricted to be less than MW . In the left two panels, θ34 is varied oversinθ34 ≤ 0.3, while in the right two panels, θ34 has been fixed to zero (no mixing). The parameters mq and ml are

varied over their complete allowed range. The grey shaded region is excluded at 95% by the LHC data. The dashedblue lines correspond to |∆`| = mW .

200 150 100 50 0 50 100 150 200∆q (GeV)

200

150

100

50

0

50

100

150

200

∆` (

GeV

)

sinθ34 =0.0

200 150 100 50 0 50 100 150 200∆q (GeV)

200

150

100

50

0

50

100

150

200

∆` (

GeV

)

sinθ34 =0.3

FIG. 4: The 95 % C.L contours in the ∆q – ∆` plane, for mh = 400 GeV (black dotted lines), mh = 600 GeV (reddashed lines) and mh = 800 GeV (blue/solid), respectively. The left (right) panel shows the results when

sin θ34 = 0.0(0.3). All the other parameters are varied over their 2σ allowed ranges. The vertical blue dashed linescorrespond to |∆q| = MW , while the horizontal green (dotted) lines correspond to |∆`| = MW .

8

0.00 0.05 0.10 0.15 0.20 0.25sinθ34

0.0

0.2

0.4

0.6

0.8

1.0

B.R

(t→bW

)

mh =600 GeV

0.00 0.05 0.10 0.15 0.20 0.25sinθ34

0.0

0.2

0.4

0.6

0.8

1.0

B.R

(b→tW

)

mh =600 GeV

FIG. 5: Shaded region in the left (right) panel indicates the values of the branching ratios of t′ → b′W (b′ → t′W )allowed at 95% C.L. by all the constraints considered in this paper.

0.00 0.05 0.10 0.15 0.20 0.25sinθ34

200

150

100

50

0

50

100

150

200

mt-m

b (G

eV)

mt =mb +MW

mb =mt +MW

mh =600 GeV

FIG. 6: Allowed values of mt′ −mb′ (shaded region) formh = 600 GeV as a function of sin θ34 aftermarginalizing over the other NP parameters

be met even in the case of one Higgs doublet. In Fig.5,we plot the allowed values of branching ratios of the de-cays t′ → b′W and b′ → t′W as functions of sin θ34. Onecan easily see that the branching ratio of the decay ofa fourth-generation quark into another fourth-generationquark can be close 100%. This emphasizes the need toconsider these decay modes in direct search experimentswhich search for fourth-generation quarks.

Figure 5 shows that for sin θ34 & 0.15 there existsno point in the parameter space which passes all theconstraints (direct search, S, T, U) for which the decay(b′ → t′W or t′ → b′W ) is possible. This can be under-stood from Fig. 6, where we show the allowed values of ∆q

as a function of θ34. For large θ34 values, sin θ34 & 0.15,the mass splitting goes below MW and the W-emissionchannel is forbidden.

V. SUMMARY AND OUTLOOK

We have explored the allowed mass spectra of fourth-generation fermions, calculating the constraints from di-rect searches at the colliders, the theoretical requirementof perturbative unitarity, and electroweak precision mea-surements. We take into account the masses of fourth-generation quarks as well as leptons, and possible mixingof the fourth-generation quarks with the third generationones. The other mixings of the fourth-generation quarksand leptons are more tightly constrained, and hence ne-glected.

Our perturbative unitarity calculation with the inclu-sion of all the J = 0 channels for 2→ 2 fermion scatteringtightens the earlier upper bounds on fourth-generationquark masses by about 6%, while keeping the constraintson the fourth-generation lepton masses unaffected. Thesebounds are relatively insensitive to the precision elec-troweak observables S, T, U . The mixing between thethird and fourth-generation quarks is constrained primar-ily by the flavor-physics data, to sin θ34 < 0.3. The per-turbative unitarity bounds depend only weakly on thismixing.

Performing a χ2-fit to the measured values of theprecision electroweak parameters S, T, U , we find thatlarge values of the Higgs mass, mh & 600 GeV as in-dicated by the current LHC data, allow the mass split-ting between the fourth-generation quarks or leptons tobe greater than MW . In the case of the quark splitting∆q = mt′ −mb′ , the possibility |∆q| > MW starts beingallowed at 95% C.L. for mh & 200 GeV. For the leptonsplitting ∆` = mν′ − mτ ′ , the possibility ∆` > MW isallowed at 95% for mh . 450 GeV, while ∆` < −MW

is allowed for all values of mh. Moreover, if θ34 is small,either |∆q| > MW or |∆`| > MW is necessary for valuesof mh as large as 800 GeV. We present correlations be-tween the values of ∆q and ∆`, as well as the constraintson them as functions of mh and θ34.

9

Most of the above observations may be explained qual-itatively through the analytic expressions for the con-tribution to the S and T parameters by the Higgs andthe fourth-generation fermions, and their interference. Inparticular, the requirement of |∆q| > MW or |∆`| > MW

at large mh for small θ34, and the relaxation of this forlarge θ34, can be easily motivated. These expressionsalso allow an understanding of the asymmetric boundson ±∆`, and why mτ ′ > mν′ is necessary at large mh.No such hierarchy of masses can be predicted in the quarksector.

The unique feature of our analysis are the simultane-ous consideration of the lepton masses, the quark mixing,and the recent indication of the heavy Higgs. The ma-jor consequence of our result is the opening up of theW -emission channels t′ → b′W , b′ → t′W , or τ ′ → ν′Wfor large values of mh. This will have major implica-tions for the direct collider searches for fourth-generationfermions which currently are performed assuming for ex-ample, t′ → bW/b′ → tW as the dominant decay modes.Indeed, since the branching ratios of these decay modes,when kinematically allowed, are large, they can have im-pact on the currently stated exclusion bounds, whichhave been arrived at by assuming that these decays arenot allowed. In order to get model-independent boundson the masses of the fourth-generation fermions, it is nec-essary to analyze the data keeping open the possibilityof large branching ratios in the W -emission channels.

We also find in our analysis, that in case further directconstraints increase the lower bound on mh to be & 900GeV, the W -emission signal is not observed in eitherquark or lepton channel, and θ34 is restricted by indepen-dent experiments to be very small (say sin θ34 < 0.05),then the model with four generations can be ruled out at95% confidence level.

In conclusion, the fourth generation is currently aliveand well. However if the corresponding standard modelHiggs is heavy, it presents the possibility of an early di-rect detection of the fourth-generation fermions and alsoaffects the search strategies as well as possible exclusionsof the fourth-generation scenario strongly. Either waywill lead to an important step ahead in our understand-ing of the fundamental particles.

Acknowledgments

We would like to thank A.Lenz and A.Djouadi for use-ful comments. RMG wishes to acknowledge the Depart-ment of Science and Technology of India, for financialsupport under the J.C. Bose Fellowship scheme underGrant No. SR/S2/JCB-64/2007.

Appendix A: Partial-wave amplitudes

In this appendix, we give the expressions for the J = 0partial-wave amplitudes of 2 → 2 scattering processes

of the heavy quarks of the SM4. Unitarity of the S-matrix and the validity of perturbative expansion of theS-matrix at high center-of-mass energies constrain thebehavior of scattering amplitudes at high center-of-massenergies. For example, in the case of 2 → 2 scatteringof scalars, the tree-level amplitude is restricted to lessthan unity, |M0| < 1. If the scattering particles haveother quantum numbers, the tree-level amplitudes forma matrix in the space of the quantum numbers. Theanalogous criterion for the perturbative unitarity of theS-matrix will be that the absolute value of the maximumeigenvalue of the matrix-valued amplitude should be lessthan unity.

In the case of SM4, the J = 0 partial-wave ampli-tude of 2 → 2 fermion scattering receives contributionfrom processes of the type FF → F ′F ′ where F andF ′ are two heavy fermions of the SM4. The scatteringamplitudes depend on the helicity configurations of theinitial and final state fermions. Since the couplings offermions to bosons in the SM are of scalar, vector oraxial-vector type, the number of helicity configurationsat high center-of-mass energies which have nonvanishingJ = 0 amplitudes reduces to four: {++→ ++}, {++→−−}, {−− → ++}, {−− → −−}. The 2 → 2 scatteringamplitude of quarks is a matrix with both helicity andcolor indices. We consider only the amplitudes wherethe initial and final states are color-neutral and charge-neutral.

Let t, t′, b′ be denoted by the indices i = 1, 2, 3, respec-tively, and let mi be their masses. In the limit wherethe center-of-mass energy of the scattering s � mimj ,the tree-level amplitudes for the processes FF → F ′F ′

may be written in the form of a 30× 30 matrix M . Thismatrix may be conveniently represented in the basis

(tR+ tR+, t

R− t

R−, t

′R+ t

′R+ , t

′R− t

′R− , b

′R+ b

′R+ , b

′R− b

′R− ,

tR+ t′R+ , tR− t

′R− , t

′R+ tR+, t

′R− tR−,

tG+ tG+, tG− t

G−, t

′G+ t

′G+ , t

′G− t

′G− , b

′G+ b

′G+ , b

′G− b

′G− ,

tG+ t′G+ , tG− t

′G− , t

′G+ tG+, t

′G− tG−,

tB+ tB+, tB− tB−, t

′B+ t

′B+ , t

′B− t

′B− , b

′B+ b

′B+ , b

′B− b

′B− ,

tB+ t′B+ , tB− t

′B− , t

′B+ tB+, t

′B− tB−

),

where R,G,B represent the three colors. In this basis,

M =

A B BB A BB B A

(A1)

Where A,B are 10×10 matrices which describe the scat-tering amplitudes. Taking the mixing between the thirdand fourth-generations into account, the matrices A andB may be written as

10

A = −√

2GF8π

m11 0 m12 0 0 −m13x 0 0 0 00 m11 0 m12 −m13x 0 0 0 0 0m21 0 m22 0 0 −m23y 0 0 0 0

0 m21 0 m22 −m23y 0 0 0 0 00 −m13x 0 −m23y m33 0 0 zm13 0 z∗m23

−m13x 0 −m23y 0 0 m33 zm32 0 z∗m31 00 0 0 0 0 z∗m32 0 0 0 00 0 0 0 z∗m31 0 0 0 0 00 0 0 0 0 zm31 0 0 0 00 0 0 0 zm32 0 0 0 0 0

,

(A2)

B = −√

2GF8π

m11 0 m12 0 0 −m13 0 0 0 00 m11 0 m12 −m13 0 0 0 0 0m21 0 m22 0 0 −m23 0 0 0 0

0 m21 0 m22 −m23 0 0 0 0 00 −m13 0 −m23 m33 0 0 0 0 0

−m13 0 −m23 0 0 m33 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

,

(A3)

where mij = mimj , x = 1− |Vtb′ |2, y = 1− |Vt′b′ |2, z = Vtb′V∗t′b′ , and z∗ = V ∗tb′Vt′b′ . Block-diagonalising M, we get

M ′ =

A+ 2B 0 00 A−B 00 0 A−B

(A4)

The maximum eigenvalue is obtained from A+ 2B [74], which is given by

A+ 2B = −3√

2GF8π

m11 0 m12 0 0 −m13δx 0 0 0 00 m11 0 m12 −m13δx 0 0 0 0 0m21 0 m22 0 0 −m23δy 0 0 0 0

0 m21 0 m22 −m23δy 0 0 0 0 00 −m13δx 0 −m23δy m33 0 0 δzm13 0 δ∗zm23

−m13δx 0 −m23δy 0 0 m33 δzm32 0 δ∗zm31 00 0 0 0 0 δ∗zm32 0 0 0 00 0 0 0 δ∗zm31 0 0 0 0 00 0 0 0 0 δzm31 0 0 0 00 0 0 0 δzm32 0 0 0 0 0

,

(A5)

where δx = 1− (1/3)|Vtb′ |2, δy = 1− (1/3)|Vt′b′ |2, δz = (1/3)Vtb′V∗t′b′ and δ∗z = (1/3)V ∗tb′Vt′b′ .

Note that the presence of a nonzero mixing between thethird and fourth-generation quarks is responsible for theappearance of the channels tt′ → b′b′ and t′t→ b′b′.These

channels are directly proportional to the mixing matrixelements in contrast to other channels where the effect ofmixing matrix elements is not significant.

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