Hints of Dynamical Symmetry Breaking

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UTPT-96-12hep-ph/9606338Hints of Dynamical Symmetry Breaking?B.Holdom 1Department of Physics University of Toronto Toronto,Ontario Canada M5S 1A7There is current interest in a possible new massive gauge boson X which mixes slightly with the Z boson and accounts for certain anomalies in the LEP data.We show why constraints on models in which the

X boson does not couple to the ?rst two families suggest dynamical electroweak symmetry breaking.The associated TeV mass fermions make up a fourth family.Constraints on the e?ects of the fourth left-handed neutrino also suggest a dynamical origin for its Majorana mass.We ?nally comment on related implications for the origin of quark masses.

1Z–X Mixing

We have learned by experience that discrepancies between experiment and the stan-dard model tend to go away over time,and so it is natural to take a cautious attitude toward the present set of anomalies in the data.But we may still ask,from a more theoretical point of view,which of the present anomalies are most likely to survive? To this question we are motivated to consider seriously the following two anomalies in R b andαs.[1]

R b=0.2202±0.0016when R c=R SM

c

=0.2156±0.003(1) R SM

b

αs(M Z)=0.126±0.005±0.002using LEP R?only

αs(M Z)=0.113±0.005from deep inelastic scattering(2) The main reason why these particular anomalies are intriguing is that the same piece of new physics would account for both anomalies;namely new physics which slightly enhances the Zb

b vertex.If the main e?ect of new physics is in the Zb

mixed slightly with the Z and thus shifted the Zb

g X≡?θg X.(3)

M2X

To be speci?c we de?ne the shift in the Z couplings to be?δg Z(bγμγ5b)and the X coupling to be?Xμ(bγμγ5b).(The reason for axial X couplings to quarks will become clear below.)SinceδM2is the o?-diagonal element of the2×2 mass-squared matrix,θis the Z–X mixing angle which we may assume to be small.

The Z–X mixing also induces a shift in the Z mass,which translates into a contribution toδρ,

M2X

δρ≈θ2

A possible Z–X mixing in the kinetic terms would contribute a term of the opposite sign,[7,8]but we will assume that this may be neglected.By insertingθfrom(3)into (4)and requiring a large enoughδg Z to account forδR b,we have the upper bound

M X

sc H t 2

g2X

,D

Q L Q R and

L

Q

Q L Q Q

We have noted that M X/g X≈1TeV,which implies that there is some physics at a TeV which breaks the U(1)X gauge symmetry.We now note that if t′and b′have axial X couplings then their masses do not respect U(1)X.In this case the existence of these masses is naturally linked to the breakdown of the U(1)X at a TeV, implying that the t′and b′masses are of order a TeV.Given that these new fourth family quarks have conventional weak charges their masses,if fairly degenerate,would imply appropriate masses for the W and Z.We are being led to consider dynamical electroweak symmetry breaking.

Given this prompting,let us remove all elementary scalar?elds.There is then no tree level contribution to the Z–X mass-mixing.There is also little contribution from t′and b′loops due to the required degeneracy of the t′and b′masses.That is, the X couplings to t′and b′are the same whereas the axial Z couplings to t′and b′are equal and opposite,implying that the two contributions in the mass-mixing loop will cancel.The mixing must then come from the t-loop.It is interesting that the t-loop contribution would vanish if the X had purely vector couplings to the t,and so this allows us to reject the vector coupling possibility.

The shift in the Z coupling from the t-loop is the same as in(6),but with H t replaced by a quantity f t determined by the t-loop.f t is normalized such that(f t/v)2, with v≈240GeV,gives the fractional contribution of the t-loop to M2Z.The point is that the t-loop involves a momentum dependent t mass function which we may assume is fairly constant up to the scale of new physics at a TeV,at which point it falls.We thus calculate the loop with a1TeV cuto?and?nd[9]

f2t≈3

m2t ≈(60GeV)2.(8)

This value is consistent with the constraints we found before on H t .The di?erence is that f t is calculated here,whereas H t was a free parameter.We conclude that the t-loop produces Z–X mixing of the correct magnitude and sign to produce the desired shift in the Zb

to leptons to cancel anomalies,but it is nevertheless easy to motivate such couplings in the context of quark-lepton uni?cation.It is simplest to expect that the U(1)X gauge symmetry commutes with the quark-lepton gauge symmetry present at some higher scale,in which case the X boson should couple similarly to quarks and leptons (at least in some basis).This in turn will shift the Z couplings to the third(and fourth)family leptons,and the question is whether such shifts are still allowed by the data.In this connection we note that the Z couplings to charged leptons is mostly axial.Thus if the shifts occurred mostly in the vector couplings then the strongly constrained leptonic partial decay widths of the Z would be little a?ected.

Anomalies will cancel within the lepton sector if the two families of leptons (νL,τL,τR;ν′L,ν′L,τ′R)have X charges(+,+,+;?,?,?)or(+,+,?;?,?,+).The di?erence again is related to choice of mass eigenstates.As for the quarks we may de?ne the?elds(E L,E R)and(E R)to have equal and opposite vector X charge. Unlike for quarks these?elds must correspond to the mass eigenstates,so that we have vector X couplings to theτ(E)andτ′(E

t(Lμ?Rμ)t+τ(Lμ+Rμ)τ+

the discrepancy between the results may suggest that it is too early to completely rule out new physics in the Z coupling toτ.

We note that predicted shifts in other quantities due to Z–X mixing,+0.2%,?1.5%,and?0.5%inΓτ,Γντ,and A b respectively,[2]are all compatible with current measurements.Additional small corrections toΓτfrom vertex corrections and toδρfrom two-loop graphs are discussed in[10]and[11]respectively.In the event that the X couples only to quarks,then only the shifts inΓb and A b remain.We also note [2,4]that?avor changing neutral currents induced by nonuniversal Z couplings are acceptable as long as most mass mixing occurs in the up-quark sector.

2Neutrino Mass and Quark Mass

We have seen that the Z–X mixing can be induced by a Higgs as long as that Higgs has a large Yukawa coupling to the t quark.But then we saw that the t-loop would su?ce by itself,so that we could remove the Higgs.We now note that a similar situation occurs when we consider the massive fourth-family neutrino.We treat the case where all right-handed neutrinos are absent from the e?ective theory at a TeV,since we note that the large Majorana masses for right-handed neutrinos are allowed by the U(1)X gauge symmetry.We must then consider the electroweak corrections induced by the fourth-family left-handed neutrino.Other analyses[12] either have Dirac neutrino masses or right-handed neutrinos involved in a see-saw mechanism.But in the dynamical symmetry breaking context it appears that right-handed neutrinos much more massive than a TeV will more naturally completely decouple.

The fourth-family left-handed neutrinoνL must have a Majorana mass greater than M Z/2.If this mass were to come from a vacuum expectation value of a SU(2)

-triplet scalar?eld H M ,then we have a tree level contribution toαT≈L

?( H M /125GeV)2.This puts a severe upper bound on H M ,which in turn im-plies a very largeνL Yukawa coupling.We therefore are in the same situation as before,leading us again to remove scalar?elds and consider the dynamical genera-tion of mass.Of interest now is the neutrino-loop contribution to T.

6

We have considered[8]the contributions to S,T,and U from the fourth family leptons(νL,τ′)=(N,E)(we omit the underlines)for a range of masses m N and m E.The result for T depends on the e?ective cuto?in the neutrino loop(similar to(8));this cuto?is supplied by the momentum dependence of the dynamical mass, and we have usedΛ=1.5m N andΛ=2m N.From Figs.(1)and(2)we see ranges of masses for which the new contributions to T are negative with reasonable size, while the new contributions to S and U are simultaneously small.We therefore have what appears to be a natural source of negative T within the dynamical symmetry breaking context.This may be useful,given the fact that a dynamically generated t-quark mass is typically accompanied by positive contributions to T.

Let us consider quark masses further.t′and b′have received TeV masses via dyanamics associated with the breakdown of U(1)X.In the absence of scalars there must be four-fermion operators which will feed mass down to the other quarks,and the largest such operator will be the one which provides the t mass.We recall our previous notation for the two quark doublets having equal and opposite vector X charge:Q≡(U,D)and Q≡(U).The fourth family quarks have the form

L Q R,which breaks U(1)X,and we need an operator to feed this down to

R

.

A suitable operator which will feed mass from the b′to the t is(Q R). (The antisymmetric?has SU(2)L indices.)Note that this operator is composed of two Lorentz scalars which are also singlets under U(1)X.This operator is thus of a form which could be expected to be enhanced by strong U(1)X interactions.If these interactions are of the walking-coupling type,then there can be signi?cant anomalous scaling enhancement of this operator relative to other operators.

This operator has a partner,(Q R).This operator will feed mass from the t′to the b,and it must thus be suppressed relative to the previous operator.The attractive feature is that the isospin breaking implied by the di?erent sizes of these operators does not feed into the TeV quark masses or T in a direct way.In fact four insertions of these operators are needed to produce a contribution to T.Other isospin breaking operators which could feed more directly into T are not so enhanced by the anomalous scaling e?ects.We thus have an example[3]of electroweak breaking physics being protected to some extent from the isospin breaking physics,which is feeding down from higher scales in four-fermion operators.

7

We may bring in the two light families by embedding U(1)X into a larger gauge symmetry at a higher scaleΛ(say100–1000TeV),which connects the light families to heavy families.We may then write down various four-fermion operators which can arise at the scaleΛ.It may be shown that such operators are su?cient to produce nontrivial mass mixing and a potentially realistic mass matrix.[11]We?nd that the existence of these operators may also be associated with the breakdown of the U(1)X gauge symmetry.

Hierarchies develop in the quark mass matrices for three reasons.1)The various operators have di?erent numbers of?elds to which the U(1)X couples,and thus are enhanced by varying amounts due to anomalous scaling induced by the U(1)X.2) Some entries in the mass matrices receive mass fed down from a t orτ′rather than the heavier t′or b′.3)Some entries only receive contributions from loops involving more than one4-fermion operator.We note from[11]that contributions feeding down from theτ′are important for obtaining realistic quark mass matrices,and that these contributions only arise for the choice of mass eigenstates forτandτ′which we have already motivated.

Acknowledgments

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada.

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References

[1]The LEP Electroweak Working Group,LEPEWWG/96-01.

[2]B.Holdom,Phys.Lett.B339(1994)114.

[3]B.Holdom,Phys.Lett.B336(1994)85.

[4]P.Bamert,C.P.Burgess,J.M.Cline,D.London,E.Nardi,McGill-96/04,hep-

ph/9602438.

[5]P.Chiappetta et al,PM/96-05,hep-ph/9601306;G.Altarelli et al,CERN TH/96-

20,hep-ph/9601324;K.S.Babu,C.Kolda and J.March-Russell,IASSNS-HEP-96/20,hep-ph/9603212;K.Agashe et al,LBL-38569,hep-ph/9604266;Alon E.

Faraggi and Manuel Masip,UFIFT-HEP-96-11,hep-ph/9604302;T.Gehrmann and W.Stirling,DTP/96/24,hep-ph/9603380;V.Barger,K.Cheung,and P.

Langacker,MADPH-96-936,hep-ph/9604298;J.L.Lopez and D.V.Nanopoulos, DOE/ER/40717-27,hep-ph/9605359.

[6]P.H.Frampton,M.B.Wise,B.D.Wright,IFP-722-UNC,hep-ph/9604260.

[7]B.Holdom,Phys.Lett.B259(1991)329.

[8]B.Holdom,UTPT-96-01,hep-ph/9602248,to appear in Rapid Communications,

Phys.Rev.D.

[9]R.S.Chivukula,B.A.Dobrescu,J.Terning,Phys.Lett.B353(1995)289.

[10]B.Holdom Phys.Lett.B351(1995)279.

[11]B.Holdom,UTPT-95-28,hep-ph/9512298,to appear in Phys.Rev.D.

[12]S.Bertolini and A.Sirlin,Phys.Lett.B257(1991)179;E.Gates and J.Terning,

Phys.Rev.Lett.67(1991)1840.

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m E m N

Figure 1:Lines of constant T as a function of the N and E masses in TeV.Thick and thin lines are for Λ=1.5m N and Λ=2m N respectively.In each case,from top to bottom,T =?2,?1,0.

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m E m N

Figure 2:Thick and thin lines are lines of constant S and U respectively as a function of the N and E masses in TeV.From top to bottom in each case S =1/6π,0,?1/6πand U =?1/12π,0,1/6π.

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