Extended Color Models with a Heavy Top Quark

arXiv:hep-ph/9203205v1 9 Mar 1992February 1992McGill/92-06hep-ph@xxx/9203205Extended Color Models with a Heavy Top QuarkOscar F. Hern´andez†Department of Physics, ERP Building3600 University St.McGill UniversityMontr´eal, Qu´ebec, Canada H3A 2T8AbstractWe present a class of models in which the top quark, by mixing with new physics at ahigher energy scale, is naturally heavier than the other standard model particles. Wetake this new physics to be extended color.

Our models contain new particles withmasses between 100 GeV and 1 TeV, some of which may be just within the reachof the next generation of experiments. In particular one of our models implies theexistence of two right handed top quarks.

These models demonstrate the existenceof a standard model-like theory consistent with experiment, and leading to newphysics below the TeV scale, in which the third generation is treated differentlythan the first two.PACS numbers: 12.15.-y, 12.10.Dm, 12.38.-t† internet: oscarh@physics.mcgill.ca ,decnet: 19191::oscarh

1. IntroductionUpon examining the fundamental particle spectrum one is immediately impressedby how much heavier the top quark is than all the other quarks.

In the standardmodel this is achieved in an unnatural way. One tunes most of the Yukawa couplingsto be very small and allows one of them to be of order one.

Here we propose a classof models in which the much higher mass of the top quark (with respect to the otherfundamental fermions) is a result of the dynamics and symmetry breaking in thetheory. The key feature of the models we propose is that while the top quark getsits mass from electroweak symmetry breaking, this quark also mixes with physicsat a higher energy scale.

We propose the physics of this higher energy scale to bean extended color group. In particular, we have in mind models like SU(4)c [1] andSU(5)c color [2].

The SU(5)c model was particularly successful in generating newphenomena at the multi-GeV energy scale while agreeing with experiment to thesame extent as the standard model [3]-[8].In the extended color model scenario, the constraint of reproducing the stan-dard model at energies below 100 GeV led to SU(5)c as the only possiblility [2],[9].This constraint meant that none of the standard model particles received a contribu-tion to their mass from the breaking of the extended color group. We are no longerrestricted to the group SU(5)c; however that is the group we will use to describeour scenario.

In section 2 we present the basic ideas behind two different modelsthat allow for a heavy top quark and discuss their common features. Sections 3 and4 concentrate on the specific features of each of the two respective models.

Finallyin section 5 we present our criticisms and conclusions.2. The modelsWe would like to reformulate the SU(5)c color model [2] in such a way that itleads to a heavy top quark and yet retains all its previous successes.

Non-standardmodel alternatives to the color sector are especially important avenues to explore.For example, hadronic physics has nothing comparable to the stunning theoreticalagreement with precision LEP measurements, whereas revising the leptonic sectorwould almost most certainly contradict LEP results (or force us to place the newphysics at a much higher energy scale).Our models therefore always leave theleptonic sector unchanged.One loophole which we exploit is that experiments have very little to say aboutthe right handed top quark, tR, since it is an SU(2)L singlet. It is actually tR which1

we will couple to the extended color physics coming from the higher energy scale.This comes about because tR is in a higher dimensional representation of SU(5)c,whereas the other quarks are in the fundamental. This is one of the major differencesbetween these models and the original SU(5)c of reference [2].

We present twopossiblities for the SU(5)c representation of tR, the 10 and the 10.Consider a theory with gauge groupSU(5)color ⊗SU(2)L ⊗U(1)′y. (2.1)As stated above the leptonic sector is unchanged, and under (2.1) they have thequantum numbersfL ∼(1, 2, −1),eR ∼(1, 1, −2),(2.2)while the first two quark generations have the formQL ∼(5, 2, yQ),uR ∼(5, 1, yQ + 1),dR ∼(5, 1, yQ −1),(2.3)just as in the original SU(5)c color model [2].

However the third quark generation,and in particular the SU(2)L singlet top quark, have different quantum numbers.We will be studying two types of models. The “ten bar” model has tR in the 10 ofSU(5)c,TL ∼(5, 2, yQ),tR ∼(10, 1, ytR),bR ∼(5, 1, yQ −1),(2.4)and the “ten” model has tR in the 10 of SU(5)c,TL ∼(5, 2, yQ),tR ∼(10, 1, ytR),bR ∼(5, 1, yQ −1).

(2.5)Just as in [2] we break SU(5)c with the Higgsχ ∼(10, 1, 2yQ). (2.6)However we use a colored Higgs, H, to break the electroweak symmetry, and givemasses to the W and Z bosons and the top quark.

Its quantum numbers areH ∼(10, 2, yQ −ytR),(2.7)for tR ∼10 andH ∼(5, 2, yQ −ytR),(2.8)for tR ∼10. The Yukawa Lagrangian isLYuk = λT LHtR + λ′T Lχ(TL)c + λ1QLχ(QL)c + λ2uRχ(dR)c + h.c.

. (2.9)2

Consider the SU(5) tensor products5 × 10 = 10 + 405 × 10 = 5 + 45,(2.10)and the following SU(5)c branching rules under SU(2)′ × SU(3)c × ˜U(1).10 =(1, 1)(6) + (1, 3)(−4) + (2, 3)(1)40 =(2, 1)(9) + (1, 3)(−4) + (2, 3)(1) + (3, 3)(−4) + · · ·5 =(2, 1)(3) + (1, 3)(−2)45 =(2, 1)(3) + (1, 3)(−2) + (3, 3)(−2) + (1, 3)(8) + · · · . (2.11)In order for the Higgs H to give a mass to the top, the ˜U(1) charges must satisfythe constraint ˜yH −˜yTL + ˜ytR = 0.

Eqs. (2.9), (2.10), (2.11) show that we have nochoice but to pick H in the 10 for tR in the 10.

If we take tR in the 10, we can pickH to be either in the 5 or 45 representation; however we will only consider H inthe 5.The symmetry breaking pattern for the ten bar model is:SU(5)c⊗SU(2)L ⊗U(1)′y↓⟨χ⟩∼wSU(2)′ ⊗SU(3)c ⊗SU(2)L ⊗U(1)y↓⟨H⟩∼uSU(2)′⊗SU(3)c ⊗U(1)em(2.12)That for the ten model is:SU(5)c⊗SU(2)L ⊗U(1)′y↓⟨χ⟩∼wSU(2)′ ⊗SU(3)c ⊗SU(2)L ⊗U(1)y↓⟨H⟩∼uSU(3)c ⊗U(1)em(2.13)The important point is that unlike the standard model, electroweak symmetrybreaking takes place with a Higgs that carries color. Within the context of thismodel the problem of fermion masses becomes the question of why the other fivequarks and leptons have small nonzero masses (in relation to the electroweak sym-metry breaking scale).

We do not seek a fundamental solution to this problem in3

this paper. Instead we will add the usual standard model Higgs φ and imagine thatits VEV is fifty times smaller than the VEV of H. This offers a partial explana-tion of light fermion masses, and maybe renormalization explains why ⟨H⟩is muchgreater than ⟨φ⟩since the parameters for H run faster.

We write the standardmodel Yukawa couplings to generate these masses,L0 = λ3fLφeR + λ4QLφdR + λ5QLφcuR + λ′T LφbR + h.c. .

(2.14)At this point we mention that all the hypercharge assignments have been normalizedin such a way that the standard model Higgs hypercharge is one, i.e. φ ∼(1, 2, 1)under eq.

(2.1).In order to give both components of the top the correct electric charge, wemust have ytR = 2 −2yQ for tR in the 10 and ytR = (3 −yQ)/2 for tR in the 10. Theelectromagnetic charge operator isQem = I3 + 12(Y ′ + (1 −3yQ)T)(2.15)One can show that Y = Y ′ + (1 −3yQ)T gives the familiar values for the hyper-charges of the color triplet quarks.T is an SU(5)c generator in ˜U(1) with thefollowing representation when acting on the five colors of ordinary plus exotic quarksT = Diag13, 13, 13, −12 , −12.

(2.16)Further characteristics of the 10 and the 10 model differ substantially and wewill discuss each one separately in the following two sections.3.The 10 modelIn this section we study in more detail the tR ∼(10, 1, 2 −2yQ) model. When theSU(2)′ × SU(3)c singlet piece of χ and H gets a VEV we are left with SU(2)′ ×SU(3)c × U(1)em as the unbroken gauge group.

SU(2)′ is the same exotic forcesector discussed in [2] and in much more detail in [8]. The SU(3)c singlet quarksthat come from the 5 of SU(5)c have charges (5yQ −1 ± 2)/4.

These quarks areconfined by the SU(2)′ force and will lead to charge 0, ±1, 10yQ ± 12, 10yQ −32 exoticmesons.The field tR in the 10 will contain a charge 2/3 color triplet which is just theusual right handed top quark. The exotics that come from the tR have charges(3 −5yQ)/2 for the singlets and (13 −15yQ)/12 for the SU(2)′ doublet color anti-triplets, which will be bound into a charge-0 meson.4

The [SU(2)L]2U(1)′ anomaly equation implies yQ = 1/5. Thus all the exoticmesons will have integer charge.

However since tR is in the 10 of SU(5)c, cancel-lation of the color anomaly is no longer automatic. This forces us to add exotics.Recall that the anomaly of the 5 and 10 of SU(5) are equal.

Our philosophy is toassociate the higher dimensional representations of SU(5)c with the higher energyphysics. Thus we cancel the anomaly by adding two electroweak singlets in the 10.We also need to give large mass to the exotic components of tR and the other exoticfields.

One possible exotic particle content isaR ∼(10, 1, ya)cR ∼(10, 1, yc)ER ∼(24, 1, −2)PR ∼(75, 1, −ya + 2/5)QR ∼(75, 1, −yc + 2/5),(3.1)with the following Yukawa termsλtχEcRtR + λaχcP cRaR + λcχQcRcR+λEρEEcRER + λPρPP cRPR + λQρQQcRQR + h.c. .

(3.2)It is straightforward to check that aR, cR and the exotic components of tR get massesof order the SU(5)c scale without affecting the mass of the ordinary color tripletquarks. The ρ Higgses are SU(5)c,SU(2)L singlets whose U(1)′ charges are chosenso as to allow the Yukawa coupling above.

We imagine these ρ’s getting a VEV atthe same time as H thus giving ER, PR, QR masses of order the top quark mass.We still have to cancel the [SU(5)c]2U(1)′ and [U(1)′]3 anomalies. This willdetermine the hypercharges ya and yc.

The equations that correspond to cancellingthe above two anomalies are−1185+ 47(ya + yc) = 0375225+ 36(ya + yc) −90(y2a + y2c) + 65(y3a + y3c) = 0. (3.3)These equations can be combined to give a quadratic equation whose solutions areirrational and are given approximately by ya = 1.64 and yc = −1.14.4.The 10 modelWe now consider tR in the 10 of SU(5)c. One key difference between the ten andten bar model is that H also breaks the SU(2)′ subgroup of SU(5)c. Thus theunbroken gauge group is the same as in the standard model.

Since the couplingin SU(2)′ is given by the strong coupling constant, the three heavy gauge bosons5

from that sector will be slightly heavier that the Z. Also these SU(2)′ gauge bosonswill not couple to ordinary matter at tree level so that their production in hadroncolliders will be suppressed.This model has two right handed top quarks with charge 2/3 that coupleto the broken SU(2)′ generators.

The singlet and anti-triplet exotics of tR havecharges (5yQ + 1)/4 and (13 −15yQ)/12, respectively. As in the ten bar model, the[SU(2)L]2U(1)′ anomaly equation implies yQ = 1/5.

While exotics are no longerneeded to cancel the color anomaly, we need them to cancel the U(1)′ anomaliesand to give the exotic components of tR a mass. Thus we add the following particlecontentER ∼(75, 1, −1)PR ∼(24, 1, yP )QR ∼(75, 1, yQ)(4.1)with the following Yukawa termsλtχcEcRtR + λEρEEcRER + λPρPP cRPR + λQρQQcRQR + h.c.

. (4.2)The ρ Higgses behave in the same way as they did in the ten bar model.The values for yP and yQ are determined from anomaly cancellation, whichcan be summarized in the following equations:47 −10yP −50yQ = 0106725−24y3P −75y3Q = 0.

(4.3)This has one real irrational solution given approximately by yP = −1.67 and yQ =1.27.5. ConclusionsWe have presented two models which lead to a heavy top quark within the frame-work of extended color models [2].In our approach, the first two generations are exact copies of each other.

How-ever the third-generation right-handed top quark is in a nonfundamental represen-tation of the extended color group. It is this property which we use to characterizethe physics at the higher energy scale.Both models lead to new particles at the color symmetry breaking scale.

Evenwithout a detailed calculation, one expects that this scale will satisfy the sameexperimental bounds of the original SU(5)c model [2]. These bounds can be aslow as 300 GeV [3]–[7].

Certain theoretical prejudices may put it at the 1-10 TeV6

scale [8], yet the exotic SU(2)′ force sector has been sufficiently changed in our“ten” model, that the constraints from [8] do not apply. In any case we find itmore interesting to consider the possibility of a multi-GeV color breaking scale,which would then naturally lead to our exotics having masses below a TeV.

The tenmodel in particular will have a top-like right handed quark with the same electriccharge, which gets its mass by coupling to the exotic sector only.However, independant of how high one chooses the color symmetry breakingscale, both models predict that the ER, PR, QR exotics will have masses of order thetop quark mass. There are obvious ways around this prediction, such as choosing adifferent mass generating mechanism than the one suggested here, or considering adifferent exotic particle content.Another feature of our models is the irrational electric charges of some of theexotic particles.This was forced upon us by anomaly cancellation, and by thedesire to keep the number of new fields to a minimum.

Irrational charges certainlyrun against current folklore, though it leads to no contradiction with acceleratorexperiments. We do not know if irrational charges are a generic feature of suchmodels or if perhaps some cleverer model builder will be able to construct an exoticsector with rational charges only.Unfortunately our model offers no insights into the origin of mass for the lightergenerations, nor does it explain why higher dimensional representations should in-volve higher energy physics.

Instead we present this model as an existence prooffor standard model-like theories in which the third generation is treated differentlythan the first two. The models we presented are consistent with experiment andlead to new physics at the 100 GeV to 1 TeV scale.

We hope these models can serveas a possible direction of study into the origin of fermion masses.AcknowledgementsI would especially like to thank Jim Cline, for many useful discussions andfor helping me trace down my algebra mistakes. Special thanks are also due CliffBurgess and Robert Foot.

I would like to acknowledge fruitful conversations withJean-Ren´e Cudell, Keith Dienes, C. S. Lam, Bernie Margolis, and the rest of theMcGill High Energy Theory group.Finally I would also like to thank GordonKane whose talk at the CINVESTAV Workshop on High Energy Phenomenologyin Mexico City, 1991, started me thinking about this problem.7

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