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Z′, new fermions and flavor changing processes.

arXiv:hep-ph/9209223v1 9 Sep 1992UM-TH 92–19Z′, new fermions and flavor changing processes.Constraints on E6 models from µ −→eee.Enrico NardiRandall Laboratory of Physics, University of Michigan, Ann Arbor, MI 48109–1120AbstractWe study a new class of flavor changing interactions, which can arise in models basedon extended gauge groups (rank >4) when new charged fermions are present togetherwith a new neutral gauge boson. We discuss the cases in which the flavor changingcouplings in the new neutral current coupled to the Z′ are theoretically expected tobe large, implying that the observed suppression of neutral flavor changing transitionsmust be provided by heavy Z′ masses together with small Z-Z′ mixing angles.

Con-centrating on E6 models, we show how the tight experimental limit on µ →eee impliesserious constraints on the Z′ mass and mixing angle. We conclude that if the valueof the flavor changing parameters is assumed to lie in a theoretically natural range, inmost cases the presence of a Z′ much lighter than 1 TeV is unlikely.PACS number(s): 13.10.+q,12.10.Dm,12.15.Cc,14.60.Jj——————————————–E-mail: nardi@umiphys.bitnetUM-TH 92–19August 1992

I. IntroductionThe large set of accurate measurements performed during the last few years hasestablished that the standard electroweak theory provides an excellent descriptionof the particle physics phenomena up to about 100 GeV. It should also be stressedthat the present theory accommodates in a satisfactory way the whole spectrumof known particles, and only two states, the top–quark and the Higgs scalar thatare necessary for the consistency of the model, have not been discovered yet.Another feature of the standard model (SM) that explains in a satisfactory waya large set of experimental limits on rare processes, is the Glashow-Iliopoulos-Maiani suppression of flavor changing neutral currents (FCNC) in the quark sector,together with the absence of lepton flavor violating (LFV) currents.All thesefeatures are quite peculiar of the SM, and in general most of its possible extensionspredict a larger spectrum of states as well as larger rates for FCNC processes.Clearly the direct detection of any new unconventional particle would be a majorbreakthrough towards the identification of a new and more fundamental theory,but at the same time it is not unlikely that some hints on the existence of newphysics will come from the observation of rare processes at rates larger than whatexpected in the standard theory.The aim of this paper is to analyze a new class of FCNC interactions thatare generally present in most of the extensions of the SM which predict one (ormore) additional neutral gauge boson Z1 together with new charged fermions, andthat are induced by Z1 interactions.

In general the Z1 is expected to be mixedwith the standard Z0, and as a consequence both the resulting mass eigenstates,that we will denote as Z and Z′, will have flavor changing couplings to the knowncharged fermions.1

In particular we will concentrate on LFV interactions, which are strictlyforbidden in the SM, and we will show that they could provide a clear signaturefor this kind of new physics. In turn, the existing limits on LFV processes implyserious constraints for several interesting models.

To stress the power of theseconstraints we will apply them to a class of E6 grand unified theories (GUTs),which are a well known example of theories where additional fermions and newneutral gauge bosons are simultaneously present, and provide a good frame forillustrating the kind of effects that we want to explore. In an attempt of quantifyingthe constraints, we will assume for the relevant E6 LFV couplings a range of valuesthat we believe is theoretically natural, and we will then turn the existing limitson LFV decays into limits on the Z′ parameters.

Though model dependent, ourbounds turn out to be much tighter than the present limits obtained from directsearches at colliders [1] or as derived from the analysis of other Z′ indirect effects[2–4].The interesting possibility of violating the conservation of lepton familynumber as a consequence of Z1 interactions was already briefly analyzed in [5] fora general class of extended electroweak models. It was shown that LFV decaysof the standard neutral gauge boson like Z →ℓiℓj, (i ̸= j, hereafter understood)can be induced by the presence of a Z1i) if a sizeable mixing with the Z1 ispresent, and ii) if the new neutral gauge boson do not couple universally to thefermion generations.

While the first condition is a natural feature of extendedgauge models, it could seem that the second one is somewhat more difficult torealize. However, we will show that both these conditions are naturally satisfied inextended gauge models that, like E6, predict also new charged fermions.

This wasalready noted in a recent paper [2], where the consequences of the simultaneouspresence of new neutral gauge bosons and new fermions were analyzed, and ageneral formalism for taking into account their combined effects was outlined.In models like E6, where several new neutral leptons are present and aquite general form for the corresponding mass matrices and mixing patterns ispossible, LFV processes like Z →ℓiℓj arise naturally due to the contributions of2

loop diagrams [6]. A Z0ℓiℓj vertex for the gauge eigenstate Z0 boson can howeverappear already at the tree level, due to the presence of new exotic charged leptonsthat could be mixed with the standard ones.

A general discussion of this kind offlavor changing vertices is given e.g. in [7], while the particular case of E6 models isanalyzed in [8].

The interest of considering flavor changing Z1 interactions stemsfrom the fact that while in general the Z0 flavor changing vertices are stronglysuppressed as the ratio between the masses of the known and of the knew fermions,in some class of models no suppression factors are expected for the Z1fifj vertices.Then, in the context of these models, the new interactions due to the additionalgauge boson could well represent the main source of flavor changing transitions.In Sec. II we will first review the essential formalism for dealing with gaugeboson and fermion mixing effects [2] and we will also discuss the theoretical expec-tations for the various flavor changing parameters.

In Sec. III we will concentrateon E6 models.

We will confront the theoretical expression for the decay µ →eeewith the extremely stringent experimental limit Br(µ+ →e+e+e−) < 1 · 10−12obtained by the SINDRUM collaboration [9], and we will derive constraints on themass of the new gauge boson and on the Z0–Z1 mixing angle as a function of theLFV parameters. Finally, in Sec.

IV we will draw our conclusions.II. Z′ and new fermion effects.

FormalismWe will now review the formalism for describing the combined effects dueto the presence of a new neutral gauge boson, and of new fermions that could bemixed with the known ones. We will follow closely the presentation given in [2]concentrating mainly on the charged fermions sector and on flavor changing effects,and we refer to [2] for a more general discussion.

We will assume a low energy3

gauge group of the form GSM × U1(1), where GSM = SU(2)L × U(1)Y × SU(3)Cis the usual SM gauge group. Then in the gauge basis, the corresponding neutralcurrent Lagrangian reads−LNC = eJµemAµ + g0Jµ0 Z0µ + g1Jµ1 Z1µ.

(2.1)In (2.1) Z0 is the SM neutral gauge boson, which couples with strengthg0 = (4√2GF M 2Z0)1/2(2.2)to the usual combination of the neutral isospin and electromagnetic currentsJµ0 = Jµ3 −s2W Jµem(2.3)where s2W = sin2 θW is the weak mixing angle. The new Z1 corresponds to theadditional U1(1) factor, and couples to the new J1 current with strength g1.

Inparticular, if we assume that the U1(1) originates at low energy from a GUTbased on a simple group, then the q1(f) charges of the fermions are fixed by thegauge group. Normalizing the new generator Q1 to the hypercharge axis Y/2 andassuming a similar renormalization group evolution for the two abelian couplings g1and gY down to the electroweak scale, the coupling strength of the new interactioncan be written asg1 ≃g0sW .

(2.4)In general, after spontaneous symmetry breaking the Z0–Z1 mass matrix turns outto be non diagonal. The Z and Z′ gauge bosons mass eigenstates then correspondto two orthogonal combinations of Z0 and Z1 that we will parametrize in term ofan angle φ.

As a result, the currents that couple with strength g0 to the physicalZ and Z′ are:JµZJµZ′=cφsφ−sφcφ Jµ0sW Jµ1. (2.5)We see that the main effects of the presence of the new gauge boson are theadditional contribution to NC amplitudes, described by the third term in the4

Lagrangian (2.1), and the mixing between the J0 and J1 currents in (2.5). We willnot discuss additional indirect effects, as e.g.

the shifts induced by Z0–Z1 mixingin the value of the weak mixing angle sW and in the overall coupling strength g0when expressed as a function of the Z mass [2-4], since they are irrelevant for thepresent analysis.We now discuss the form of the two currents J0 and J1, and in particularthe effects that could originate from the presence of additional charged fermions.We will henceforth denote as new, possible degrees of freedom in addition to thestandard 15 known fermions per generation. Since no new fermions have beendirectly observed yet, the new states must be rather heavy, and mnew >∼MZ/2 canbe taken as the present model independent limit on the new masses.

Accordingly,we will denote the corresponding mass eigenstates as heavy, while the known masseigenstates will be labeled as light. In the presence of additional fermions, thelight mass eigenstates will correspond to superpositions of the known and newstates.

Conservation of the electric and color charges forbids a mixing betweengauge eigenstates with different U(1)em and SU(3)c quantum numbers, and, inturn, this implies that the electromagnetic and color currents of the light masseigenstates are not modified in the presence of the new states.However, theneutral isospin generator T3 and the new generator Q1 are spontaneously broken,then a mixing between gauge eigenstates with different t3 and q1 eigenvalues isallowed, and as a result the couplings of the light mass eigenstates to the Z0 andZ1 will be affected.Since in the gauge currents chirality is conserved too, it is convenient togroup the fermions with the same electric charge and chirality α = L, R in acolumn vector of the known (K) and new (N ) gauge eigenstates Ψoα = (ΨoK, ΨoN)Tα.The relation between the gauge eigenstates Ψoα and the corresponding light andheavy mass eigenstates Ψα = (Ψl, Ψh)Tα is then given by a unitary transformationΨoKΨoNα= UαΨlΨhαwhereUα =AGFHα,α = L, R.(2.6)The submatrices A and F describe the overlap of the light eigenstates with the5

known and the new states respectively, and from the unitarity of U we haveA†A + F †F = AA† + GG† = I. (2.7)Note that we have not introduced an extra index to label the electric charge;nevertheless we will treat Ψoα and Ψα as vectors corresponding to a definite valueof qem.In terms of the fermion mass eigenstates the neutral current correspondingto a (broken) generator Q readsJµQ =Xα=L,R¯ΨαγµU †αQαUαΨα,(2.8)where Qα represents a generic diagonal matrix of the charges for the chiralfermions.

Here we have to consider only the mixing effects in J3 appearing in(2.3) and in J1, since the term proportional to Jem in J0 is not modified byfermion mixing. Hence in (2.8) Q = T3, Q1 and the elements of the correspondingmatrices are given by the eigenvalues t3 and q1.From (2.8) we see that if in one subspace of states with equal electric chargeand chirality the matrix Qα is proportional to the identity, then U †αQαUα = Qαand for these fermions the corresponding current is not modified in going to thebase of the mass eigenstates.

In the SM for example, for a given electric charge andchirality the t3 eigenvalues of the fermions are indeed the same, and this implies inparticular the absence (at the tree level) of FCNC. In models with new fermionsin contrast, the diagonal matrices Qα have the general form Qα = diag(QKα, QNα)and do not commute with U.To put in evidence the indirect effects of fermion mixings in the couplingsof the light mass eigenstates, we now project (2.8) on the light components Ψl,obtainingJµlQ =Xα=L,R¯Ψlαγµ A†αQKαAα + F †αQNαFαΨlα.

(2.9)This equation is quite general, and describes the effects of fermion mixings in theneutral–currents of light–states for a wide class of models. If the gauge group is6

generation independent, all the known states appearing in one vector Ψoα have thesame eigenvalues with respect to the generators of the gauge symmetry, and hencewe have QKα = qKαI with qKα = t3(f Kα ), q1(f Kα ). The same is not true in general forthe new states; however, if we consider the particular case when the mixing is withonly one type of new fermions with the same qNα charges, then we have QNα = qNαIas well.

Under these conditions, and by means of the unitarity relation (2.7), (2.9)reduces to the simple formJµlQ =Xα=L,R¯Ψlαγµ qKαI + (qNα −qKα)F †αFαΨlα. (2.10)We will restrict ourself to this equation, which is general enough for describingthe mixing with the additional charged fermions in E6, and we refer to [2] for adiscussion of more general cases.A few consideration are now in order.

In (2.10) the first term qKαI insidethe square brackets gives the couplings of a particular light fermion in the absenceof mixing effects. The second term represents the modifications due to fermionmixings.

The matrix F †F appearing in this term is in general not diagonal, andwhile the magnitude of the diagonal elements will affect the strength of the flavordiagonal couplings of the mass eigenstates, the offdiagonal terms will induceFCNC. Clearly whenever the coefficient (qNα −qKα) vanishes, the mixing effectsare absent.When t3(f Nα ) ̸= t3(f Kα ) the J0 current is modified, and then theexisting low–energy and on–resonance NC data as well as the current limits onrare processes, can be used to constrain directly the corresponding elements ofF †F.

Model independent limits on the diagonal elements (F †F)ii affecting Z0interactions were first given in [7] and subsequently updated in [10], and the mostrecent limits in the frame of E6 models can be found in [2]. Bounds on the offdiagonal terms (F †F)i̸=j have been given in [7] as well.All these limits turnout to be very stringent, usually at the level of 1% or better.In contrast, ift3(f Nα ) = t3(f Kα ) the J0 current is not modified, but since in general we still haveq1(f Nα ) ̸= q1(f Kα ), sizable effects could indeed be present in J1.

We stress that7

the present experimental data cannot be used to set limits on these mixings, sincethey only affect a new hypothetical interaction, and we can only rely on theoreticalspeculations to estimate their magnitude.According to these considerations it is clear that from a phenomenologi-cal point of view it is convenient to classify possible new fermions in terms oftheir transformation properties under SU(2)L. Since we are only interested infermions with conventional electric charges, the new states must be singlets ordoublets of weak–isospin. A rather heterodox exception is that of a gauge tripletof fermions [11], but we will not consider this possibility here.

According to thenomenclature in use [2,7,10], we denote the particles with unconventional isospinassignments (left–handed singlets or right–handed doublets) as exotic fermions.All the standard fermions, as well as all the new states that have conventionalSU(2)L assignments, are referred to as ordinary. For example mirror fermions,having opposite SU(2)L assignments from those of the known fermions, are ex-otic.

Sequential fermions are simply repetition of the new fermions. They could bepresent in a complete new family or as components of large fermion representationsand are clearly classified as ordinary.

In this paper we will mainly concentrate onvector multiplets of new fermions for which the L and R components have thesame SU(2)L transformation properties, and hence always contain both ordinaryand exotic states. From the previous discussion, we see that while ordinary–exoticfermion mixings are tightly constrained due to the effects induced in the J0 cur-rent, no limits can be given for the ordinary–ordinary mixings, since they affectonly J1.This classification is also very convenient for discussing the possible formof the fermion mass matrices and the expected magnitude of the mixings betweenthe known and the new fermions.

To give an example, let us introduce for eachfermion family a vector gauge singlet of new fermions (XoEL, XoOR)i (E = exotic,O = ordinary, i = 1, 2, 3) with the same electric and color charges than the known8

fermions (f oOL, f oOR)i. Then in the gauge eigenstate basis the mass term readsLmass = ( ¯f oO, ¯XoE)L M f oOXoOR+ h.c.(2.11)where e.g.

f o = (f o1, f o2, f o3)T etc.. The non diagonal mass matrix M takes theformM =DD′S′S,(2.12)where D and D′ are 3×3 matrices generated by vacuum expectation values (vevs)of doublets multiplied by Yukawa couplings, while S and S′ are generated by vevsof singlets.As a general rule, while the mass terms which couple ordinary L-fermions to ordinary R-fermions (or exotic L-fermions to exotic R-fermions) arisefrom vevs of Higgs doublets, the entries which couple ordinary fermions to theexotic ones are generated by vevs of singlets.

Then in general Higgs singlets areresponsible for the large masses of new heavy fermions in vector multiplets and,in most cases, also contribute to the mass of the new heavy gauge boson; hence itis natural to assume S, S′ ∼Λ ≫D, D′.The diagonal mass matrix M is obtained via a biunitary transformationacting on the L and R sectors:M2 = UO−EL(MM†)UO−EL†= UO−OR(M†M)UO−OR†. (2.13)Since D/Λ, D′/Λ ∼ε ≪1 , the order of magnitude of the different entries inMM† and M†M isMM† ∼Λ2ε2εε1(2.14)M†M ∼Λ2Λ2Λ2Λ2.

(2.15)Given the form of MM† in (2.14) and keeping in mind the expression (2.6)for the matrices U, we see that for the matrix describing the ordinary–exotic9

mixings in (2.13) it is natural to expect that the submatrices F and G wouldacquire an overall suppression factor ε, of the order of the ratio of the light toheavy mass scale.In contrast, since all the entries in (2.15) are of the sameorder of magnitude, such a suppression is not present for the ordinary-ordinary Fand G mixing terms. Now, since it is precisely F †F in (2.10) which affects theflavor diagonal couplings and also induces FCNC, the suppression of the ordinary–exotic mixings explains in a natural way the non–observations of these effectsin the Z0 interactions.On the other hand for the ordinary–ordinary mixingsthere is no reason to expect the elements of F †F to be particularly small, andaccordingly flavor changing processes can be expected to occur at a sizeable ratein Z1 interactions.Written explicitly, the flavor diagonal chiral couplings of a light f fermionto the Z0 and Z1 gauge bosons areεf0α = t3(fα) −s2W qem(f) + [t3(fNα ) −t3(fKα )] (F †αFα)ffεf1α = q1(fα) + [q1(fNα ) −q1(fKα )] (F †αFα)ff,α = L, R.(2.16)while the fi fj flavor changing couplings readκij0α = [t3(fNα ) −t3(fKα )] (F †αFα)ijκij1α = [q1(fNα ) −q1(fKα )] (F †αFα)ij,α = L, R.(2.17)The corresponding couplings to the physical Z and Z′ bosons, that we will denoteas εfα, ε′ fα and κijα , κ′ ijα , can be readily obtained via the transformation (2.5).

Forthe flavor changing couplings we have for exampleκijα = cφ κij0α + sφsw κij1ακ′ ijα= −sφ κij0α + cφsw κij1αα = L, R,(2.18)and analogous expressions hold for εfα and ε′ fα too. In terms of the flavor non–diagonal couplings (2.18), the FCNC Lagrangian for the light f i and f j fermionsin the mass eigenstate basis finally reads−LijFC = g0Xα=L,R ¯f iαγµ κijα f jα Zµ +¯f iαγµ κ′ijα f jαZ′µ.

(2.19)10

From the first equation in (2.18) we see that even in the case when onlyordinary–ordinary mixing effects are present and hence κij0α = 0, the Z boson canstill mediate flavor changing transitions, suppressed now by a factor proportionalto the Z0–Z1 mixing [5]. However, for several models the existing limits on φ arerather stringent: |φ| <∼0.02 [2-4] and then we can expect that if the Z′ is not tooheavy, FCNC processes would be mainly induced by Z′ exchange.III.

Constraints on E6 models from µ →eee.In the previous section we have shown that in the presence of new charged fermionsthe new neutral current J1 could induce sizeable flavor changing transitions eithervia Z′ interactions, or as a consequence of a non–vanishing Z0–Z1 mixing. Thiskind of new physics would manifest itself in affecting the rates for several processeswhich are forbidden or highly suppressed in the SM.

In the quark sector it couldenhance the branchings for the leptonic decays of mesons like K0, D0, B0 →ℓ+ℓ−,and it would also affect the neutral meson mixings and mass differences. In thelepton sector it would induce several LFV neutrinoless τ decay modes like τ →eee,µµµ, µee, eµµ, µπ, µρ which are all constrained at the level of <∼few×10−5 [12],but in particular it would also give rise to the decay µ →eee for which the existinglimit [9] is much more stringent:Br(µ+ →e+e+e−) < 1.0 · 10−12(at90% c.l.

). (3.1)Given the LFV Lagrangian (2.19), the expression for this decay rate relative tothe charged current decay µ →νe¯ν is11

Br(µ →eee)Br(µ →νe¯ν) = 2h3(ε2R + ε2L)(κ2R + κ2L) + (ε2R −ε2L)(κ2R −κ2L)i+2( M 2ZM 2Z′ )2h3(ε′2R + ε′2L)(κ′2R + κ′2L) + (ε′2R −ε′2L)(κ′2R −κ′2L)i+(3.2)4 M 2ZM 2Z′3(εRε′R + εLε′L)(κRκ′R + κLκ′L) + (εRε′R −εLε′L)(κRκ′R −κLκ′L)where for κR,L and κ′R,L defined in (2.18) we have dropped the indices i = e andj = µ, and εR,L and ε′R,L refer to the electron couplings.As a first result, by confronting (3.1) and (3.2) we can derive the limits onthe Zµe LFV vertices. Assuming that the Z′ is completely decoupled from thelow energy physics (MZ′ →∞and φ →0) and taking s2W = 0.23, we obtain:|κeµL | < 1.1 · 10−6|κeµR | < 1.2 · 10−6.

(3.3)As we have discussed, if these couplings do originate from some kind of mixing ofthe electron and muon with heavy exotic leptons, we expect them to be stronglysuppressed, e.g. by a factor of the order m2µ/M 2Z ∼10−6 or smaller, and thenwe see that the existence of FCNC induced by ordinary–exotic mixings does notconflict with the stringent limits in (3.3).Now, in order to study the effects of the Z′ flavor changing vertices, we firstneed to fix the q1 charges of the fermions, which in turn determine the coefficient ofthe flavor changing mixings.

This can be done by choosing a specific GUT, and wewill carry out our analysis in the frame of E6. Since E6 has rank 6, while the SMgauge group GSM has rank 4, the breaking of E6 to the SM will lead to extra Z′s.We will consider the possibility that either E6 breaks directly to rank 5, or that oneof the two extra Z′s is heavy enough so that its effects on the low energy physics arenegligible, and in these cases the formalism developed in the previous section canbe straightforwardly applied.

We will choose the embedding of GSM into E6 troughthe maximal subalgebra chain E6 →U(1)ψ × SO(10) →U(1)χ × SU(5) →GSM,12

then an effective extra U1(1) could arise at low energy as a combination of theU(1)ψ and U(1)χ factors. We will parametrize this combination in terms of anangle β, and this will define an entire class of Z′ models in which each fermion fis coupled to the new boson through the effective chargeq1(f) = qψ(f) sinβ + qχ(f) cosβ.

(3.4)Particular cases that are commonly studied in the literature [2–4,13] correspondto sin β = −p5/8, 0, 1 and are respectively denoted Zη, Zχ and Zψ models. Zψoccurs in E6 →SO(10), while Zη occurs in superstring models when E6 directlybreaks down to rank 5.

As we will see this model plays a peculiar role in thepresent analysis, since it evades completely the kind of constraints that we areinvestigating. Finally, a Zχ boson occurs in SO(10)→SU(5) and couples to theconventional fermions in the same way than the Z′ present in SO(10) GUTs,however since SO(10) does not contain additional charged fermions, the kind offlavor changing effects that we are studying here is absent.

In contrast, new chargedquarks and leptons are present in E6. In the GUTs based on this gauge group thefermions are assigned to the fundamental 27 representation that contains, beyondthe standard 15 degrees of freedom, 12 additional states for each generation, amongwhich we have a vector doublet of new leptons (N E−)TL, (E+ N c)TL on which wewill now concentrate.The chiral couplings of the leptons to the Z1 and the coefficient of the LFVterm, are determined by the qψ and qχ charges of the new and known states, whichareqψ(EL) = −qψ(ER) = −13r52qχ(EL) = qχ(ER) = −13r32qψ(eL) = −qψ(eR) = 16r52qχ(eL) = 3qχ(eR) = 12r32.

(3.5)With respect to the SU(2)L transformation properties, the E+L heavy leptons areexotic, and than the mixing of their CP conjugate states E−R with eR, µR and τRare constrained by Z0 interactions. From (3.3) we have for example(F †RFR)eµ < 2.4 · 10−6,(3.6)13

while the 90% c.l. limits on the flavor diagonal mixings given in [2] are respectively(F †RFR)ee < 1.3 · 10−2(3.7)(F †RFR)µµ < 1.1 · 10−2.

(3.8)Due to the tight bound (3.6) it is reasonable to neglect the LFV couplings in theR-sector and (conservatively) set κeµR = κ′eµR = 0 in (3.2). According to (3.7), itis also justified to neglect the effects of the fermion mixings in the flavor diagonalcouplings of the R-electrons.In contrast, the E−L leptons are ordinary, and no bounds exist on theirmixing with the light leptons.

We will still neglect the diagonal term (F †LFL)eesince it is reasonable to expect that if this term were so large as to spoil theapproximation εf1α ≃q1(fα) the value of the off-diagonal term (F †LFL)eµ wouldalso be large, possibly leading to even stronger limits than the ones derived here.Due to the approximations made, for each value of the parameter β in (3.4)the branching ratio (3.2) depends on the values of M ′Z, φ and Feµ ≡(F †LFL)eµ.However, it is easy to see that since the gauge boson mixing effects in the diagonalelectron couplings are in any case very small, being |φ| <∼0.02 [2-4], the relevantvariables are actually only two, namely Feµ·(M 2Z/M 2Z′) and Feµ·φ. Moreover oncethe Higgs sector of the model is specified, MZ′ and φ are no more independentquantities.

For example an approximate relation that holds for small mixings andwhen MZ′ (≫MZ) originates from a large Higgs singlet vev [3] readsφ ≃−M 2ZM 2Z′ sWPi ti3qi1|⟨φi⟩|2Pi ti32|⟨φi⟩|2 ,(3.9)and in this case the branching ratio (3.2) is in practice only a function ofFeµ · (M 2Z/M 2Z′).As in the SM for the Cabibbo-Kobayashi-Maskawa (CKM) matrix, also inE6 we don’t have a clue for predicting the values of the fermion mixing parame-ters. Without attempting to push too far an analogy between the mixings we are14

interested in and the CKM matrix, we will merely note that both these cases in-volve mixings among ordinary fermions, and that we do not expect in the presentcase any additional suppression factor. We also note that all the CKM matrixelements are > 10−3 and that in particular the mixing between the first and thesecond generation is rather large.

We will then assume that the LFV term Feµlies in the range 10−2–10−4. Under this assumption the presence of a too lightZ′ as well as a too large amount of Z0–Z1 mixing will clearly conflict with thelimit (3.1).

The bounds that can be derived in this way are indeed very strong,but obviously they cannot replace the direct [1] or indirect [2-4] limits on the Z′parameters, since for very small values of the LFV Z′ couplings (Feµ <∼10−6) theywould in fact be weaker. We will nevertheless present our constraints in the formof numerical limits on MZ′ and φ, since, in doing so, the strength of the argumentsthat have been discussed here is put in clear evidence.Our results are collected in Figs.1 and 2.Figure 1 shows the boundson M ′Z, the thick solid line depicts the bounds obtained for Feµ = 10−2 and bysetting the gauge boson mixing angle φ to zero.

The decay µ →eee is due onlyto Z′ exchange in this case. The limits for different values of Feµ can be red offthis line as well, by assuming for the vertical axis units of of GeV·[100Feµ]12 .

Thethick dashed line, drawn here for convenience, shows the bounds corresponding toφ = 0 and Feµ = 10−3, vertical units are again in GeV in this case. We see thatfor Feµ > 10−3 a Z′ below 1 TeV would be excluded for most of the values ofβ.

Also, it is clear that Feµ ≃10−4 still leads to significative bounds, being MZ′constrained to values >∼400GeV for large part of the sin β axis.To study the possible effects on these results of a non vanishing mixingangle φ, i.e. when both the Z′ and Z bosons contribute to the decay, we haveused (3.9) assuming two doublets of Higgs fields hNc and hN with vevs ¯v andv.

Since ¯v and v give mass respectively to the t and b quarks, σ ≡¯v2/v2 > 1is theoretically preferred. The bounds on MZ′ obtained by allowing for a Z0–Z1mixing consistent with this minimal Higgs sector are shown Fig.

1 by the dottedand dot-dashed lines, which correspond to σ = 1 and ∞respectively. It is apparent15

that by allowing for a non vanishing value of φ, the limits on the Z′ mass are onlyslightly affected.Figure 2 depicts the constraints on the Z0–Z1 mixing angle φ. The solidline shows the bounds obtained by assuming Feµ = 10−2 and taking the limitMZ′ →∞.

In this case the decay µ →eee is mediated only by the Z boson,and is due to the mixing between the Z0 and the Z1. The dotted line shows thebounds for Feµ = 10−3 in the same limit.

The limits for different choices of Feµare easily obtained from the solid [dashed] lines by rescaling the vertical units by(102Feµ)−1 [(103Feµ)−1].The dotted (σ = 1) and dot–dashed lines (σ = ∞) enclose the regions of thelimits obtained assuming a minimal Higgs sector. In this case the value of MZ′ isfinite and consistent, according to (3.9), with the values of φ at the bound.

Wesee that with this additional constraint significant limits are found for Feµ = 10−4as well.From Fig. 2 it is apparent that for a minimal Higgs sector the limits on φare significantly tighter than in the MZ′ →∞limit, showing that (3.1) in firstplace gives direct constraints on the Z′ mass, while the bounds on the Z0–Z1mixing obtained independently of MZ′ are weaker.

We note that this behaviouris opposite to what is encountered in deriving limits on the Z′ parameters fromprecise electroweak data [2-4], where in fact the best bounds on the Z′ mass areobtained from the tight limits on φ implied by the LEP measurements.From Figs. 1 and 2 it is apparent that for the η model, corresponding tosin β = −p5/8, both the Z′ mass and the Z0–Z1 mixing angle are not constrainedby the present analysis.

This is due to the fact that in this model, for the ordinary–ordinary fermion mixings besides tK3 = tN3 we also have qKη = qNη , implying thatboth the coefficients of the F †LFL term in the J3 and in the J1 currents vanish. Thishappens also in the quark sector and in the neutral sector, hence the unsuppressedflavor changing vertices are completely absent for the Zη boson, and this insuresthat besides the decay µ →eee no other processes can be found for implementingthis kind of constraints for the η model.

The reason for this can be understood by16

considering the decomposition E6 →SU(6) × SU(2)I, where SU(6) contains theSM group, while the SU(2)I is “inert” in the sense that I3I does not contributeto the Qem generator [14,15]. I3I corresponds to β = arctanp3/5 in (3.4) andis orthogonal to Qη [β = arctan(−p5/3)] which is then contained in the SU(6)factor as well.

The fermions in the 27 of E6 with the same SM quantum numbers(qem, t3, color) form multiplets (singlets and doublets) of SU(2)I and clearly thesemultiplets also carry definite values of the Qη charge. All the ordinary fermionswith the same color and electric charges, being members of the same SU(2)I mul-tiplet, have also the same qη, and this implies the absence of both the diagonal andthe flavor changing ordinary–ordinary mixing effects.

In contrast ordinary–exoticfermion mixing could still give rise to FCNC also in the η model, but as alreadydiscussed the corresponding transitions are expected to be largely suppressed, anddo not imply any useful constraint.According to this discussion, if an additional Z′ with a mass of a few hun-dreds GeV is found together with new fermions that could fit in the 27 of E6,the observed absence of unsuppressed FCNC would suggest that it could mostprobably be a Zη.IV. ConclusionsWe have carried out an analysis of models that predict a new neutral gauge bosonand new charged fermions from the point of view of FCNC processes.

We haveargued that in most of these models unsuppressed flavor changing couplings of thelight fermions to the new Z′ can be present as a consequence of a mixing betweenthe known and the new charged fermions. By assuming that these flavor changingvertices should not be unnaturally small, we have inferred that the observed sup-pression of FCNC processes can still be explained in a natural way if the new gaugeboson is sufficiently heavy and almost unmixed with the standard Z.

Also we have17

attempted a semi-quantitative analysis of this kind of new physics in the frame ofE6 models, by confronting the theoretical expectations for the LFV effects with theextremely stringent limits on the µ →eee decay mode. Our conclusions are thatthe existence of Z′ bosons from E6 much lighter than 1 TeV is unlikely, with thenoticeable exception of the superstring inspired η model which evades completelyour constraints.

At the same time, our analysis suggests that the observation ofFCNC processes at rates larger than the SM predictions could be interpreted asa hint for the simultaneous presence of additional gauge bosons and new chargedfermions. Indeed these new states could manifest themselves indirectly via thiskind of flavor changing effects well before they are directly produced.18

References[1] CDF Collaboration, F. Abe et.al., Phys. Rev.

Lett. 67, 2609 (1991); ibid.68, 1463 (1992).

[2] E. Nardi, E. Roulet and D. Tommasini, Report No. FERMILAB PUB-92/127-A 1992, to appear in Phys.

Rev. D46, (1 october 1992).

[3] P. Langacker and M. Luo, Phys. Rev.

D45, 278 (1992). [4] J. Layssac, F.M.

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Gonzalez Garc´ıa and J.W.F. Valle; Phys.

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[7] P. Langacker and D. London, Phys. Rev.

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[15] For a review on the group E6, see R. Slansky; Phys. Rep. 79, (1981) 1.20

Figure captionsFig. 1: Limits on MZ′ for a general E6 neutral gauge boson, as a function ofsin β and for different values of the lepton flavor violating term Feµ ≡(F †LFL)eµ.The thick solid line is obtained by setting the Z0–Z1 mixing angle φ to zero, andassuming Feµ = 10−2.

The limits for different values of Feµ can be read offthisline by assuming for the vertical axis units of GeV·(100Feµ)12 . The thick dashedline depicts the limits corresponding to Feµ = 10−3 (vertical units in GeV).

Thebounds obtained by allowing for a non-vanishing Z0–Z1 mixing, consistent withthe values of M ′Z when a minimal Higgs sector is assumed, are also shown. Thedotted lines correspond to equal vevs of the two Higgs doublets present in themodel, i.e.

σ ≡¯v/v = 1 while the dot–dashed lines correspond to σ = ∞.Fig. 2: Limits on the Z0–Z1 mixing angle φ for a general Z1 from E6, as a functionof sin β and for different values of the lepton flavor violating term Feµ ≡(F †LFL)eµ.The thick solid and dashed lines are obtained in the limit MZ′ →∞assumingFeµ = 10−2 and Feµ = 10−3 respectively.

Limits for different values of Feµ canbe obtained from these lines by rescaling the vertical units by (102Feµ)−1 and(103Feµ)−1 respectively. The dotted (σ = 1) and dot-dashed (σ = ∞) lines showthe limits obtained for a finite Z′ mass and assuming a minimal Higgs sector.

Inthis case the bounds are tighter and are essentially determined by the correspond-ing limits on MZ′ through eq. (3.9).

Also the limits corresponding to Feµ = 10−4are shown in this case.21

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