The Bosonic Sector of the Electroweak Interactions, Status and Tests at Present and Future

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The status of the Standard Model (SM) is reviewed. We emphazize the fact that in spite of the success of the SM for the descrition of the fermionic sector, the status of the bosonic sector (gauge and scalar) suffers from many theoretical deficiencies and f

a r X i v :h e p -p h /9501305v 1 16 J a n 1995PM/95-01

January 1995

The Bosonic Sector of the Electroweak Interactions,Status and Tests at Present and Future Colliders F.M.Renard Physique Math´e matique et Th´e orique,CNRS-URA 768Universit´e Montpellier II,F-34095Montpellier Cedex 5.Lectures given at Regensburg University,January 1995Extended version of a talk given at the Festkolloqium Dieter Schildknecht,Bielefeld,Oct.14th 1994Abstract The status of the Standard Model (SM)is reviewed.We emphazize the fact that in spite

of the success of the SM for the descrition of the fermionic sector,the status of the bosonic sector (gauge and scalar)su?ers from many theoretical de?ciencies and from the lack of empirical support.This situation,which leaves room for several types of extensions or alternatives to SM,strongly motivates the pursue of intense e?orts for ?nding hints of New Physics (NP)e?ects.We present a phenomenological description valid for energies lying below the NP scale.We discuss the indirect constraints established from high precision tests at LEP1,as well as the direct tests that could be performed at future machines.

1Introduction,the status of the Standard Model

It is a common leitmotiv to say that the Standard Model(SM)is largely successful.On the one hand it is already remarkable that this model is able to make de?nite and unambiguous preditions for all processes involving usual particles.This property is the consequence of the gauge principle which allows to predict the dynamics once a classi?cation group has been chosen.The simpler QED case with the U(1)EM has been extended to the non-abelian cases of QCD with SU(3)colour and to the electroweak interactions with SU(2)×U(1).However the speci?c feature of electroweak interactions is the fact that W,Z bosons are massive.The gauge principle has to be completed with a mass generation mechanism. In the Standard Model it is chosen as the Higgs mechanism of spontaneous symmetry breaking(SSB).It is this last property that makes the SM a renormalizable theory which allows to compute high order e?ects and to make the accurate predictions mentioned above.These predictions practically agree with all available experimental results.In spite of this success many questions arise.

Let us?rst quickly review the status of the SM by clearly separating the caracteristics of its three sectors:

a)The fermionic sector

consists in theγ,W±,Z and the8gluons,as generated by the SU(3)×SU(2)×U(1)gauge group.

At this stage both fermions and bosons are massless states.They are coupled through gauge interactions.Self-boson interactions appear through the non-abelian Yang-Mills kinetic terms of the W±,3gauge bosons.

c)The scalar sector

the SM(Technicolour mechanism(TC),Grand Uni?ed Theories(GUT),substructures). New concepts like those introduced with superstrings may also be necessary.In any case it is not obvious at what energy scale(between the TeV range and the Planck mass)these features may originate.

The status of the bosonic sector is not yet empirically established because it is still not possible to perform signi?cant direct tests.The agreement of the SM predictions with experiments in fermionic processes(LEP1,low energy experiments)is often taken as a sign of general validity of the SM including the bosonic sector,because high order terms indirectly involve gauge boson and also higgs boson self-interactions.However as we will see in Sect.4,these indirect tests?rst su?er from a lack of accuracy,but also from many ambiguities which prevent to give well-de?ned model-independent statements. Many extensions of,or alternatives to,SM are also consistent with the fermionic results.

On another hand the bosonic sectors su?er from much more serious questions and de?ciencies.They concern the origin of three a priori independent gauge couplings(that one would like to unify),the origin of SSB,i.e.the origin of the scalar potential(not generated by the gauge principle but put by hand),and of the Fermi scale v(the basic mass scale of the electroweak interactions),the restricted choice of Higgs doublets,as well as the unpredicted value of the Higgs mass(seen either as an unpredicted coupling constant for theφ4term or as a new mass scale).

Even more serious are the following two problems concerning the Higgs sector.One is called”triviality”and expresses the fact that the renormalized coupling constant of theφ4 term of the Higgs potential tends to zero when one wants to get rid of the cut-o?introduced for regularization(as one usually does in renormalizable?eld theories).This would imply that SSB disappears in this limit.The second one is called”the naturalness problem”and corresponds to the fact that,at1-loop,the Higgs mass depends quadratically of the cut-o?and is no longer controlled by the tree level value M0H.Its value then only depends on an outside scale.This is opposite to what happens in the”natural”fermion mass case where the mass shift is proportional to the tree level mass term and only weakly (logarithmically)depends on the cut-o?.This seems to indicate that the description of the scalar sector of the SM is not in a fundamental stage but must be considered as an e?ective

2Questions to be asked about the bosonic sector The W±and Z bosons have been discovered in a range of mass which precisely agrees with the one expected from the properties of the weak interactions found in low energy experiments.The high precision tests which followed their discovery have con?rmed that their couplings to leptons and quarks agree up to a few permille with the SM predictions [1].Can one from that conclude that W and Z have exactly the gauge nature that the SM assumes for them?Does it also mean that the Higgs mechanism is necessarily responsible for mass generation?

Certainly not!In fact many options for non-standard(NP)models are still allowed by the presently limited empirical knowledge and one can ask the following questions, classi?ed into three types.

a)The nature of the W±,Z bosons.

Are they true gauge bosons?In that case what is the precise gauge group?SU(2)×U(1)or a larger one like SU(2)×SU(2)×U(1)or SU(2)×U(1)×U(1)?Such extensions like Left-Right symmetry,E6symmetry[4]are obtained on the way of a Grand Uni?ed Theory (GUT)[3]or in certain alternative mass generation mechanisms based on a strongly interacting sector[5],[6].

More drastically departing from the SM picture,W±and Z may be kind of massive vector states(hadron like)whose interactions respect some global symmetry.This is what happens in compositeness schemes where the global symmetry originates from the subconstituent structure.This ensures that the couplings to leptons and quarks are similar to the SM ones[9].Mass may here simply originate from con?nement e?ects.

b)The precise spectrum of weak bosons.

Vector bosons

Does the Higgs boson exist at all?This question arises because there exist alternative models without Higgs(Technicolour-like[8]or compositeness inspired[9]).If the Higgs exists,is it an elementary or a composite state[8]?.If it exists as an elementary state,mainly because of the naturalness problem[2],the question arises whether it is light(close to M Z)or heavy(close to the unitarity limit in the TeV range).If it is light,is it accompanied by other neutral H0states and charged H±states(as claimed by Supersymmetry in order to cancel the quadratic divergences)[2]?

One can also raise the question weather there exist higher spin(J≥2)bosonic states?

c)The precise structure of the bosonic interactions?

This question is motivated by the fact that any extension or modi?cation of the SM should lead to”anomalous”interactions among usual bosons.In the vector boson sub-sector,the basic W,Z,γself-interactions can be di?erent from the Yang-Mills ones.In

particular new forms and new multi-boson interactions could appear.Couplings involving longitudinal W L states may have special features related to the fact that they are created by the mass generation mechanism(MGM).This feature is a genuine one as compared to the QED or QCD cases where SSB does not occur.Within the SM structure it is already known that a very heavy Higgs is a source of strong W L W L interactions[7].New Physics structures may also introduce further di?erences between W T and W L interactions[10].

Obviously the Higgs sector should be directly a?ected by the existence of a di?erent MGM,especially Higgs self-interactions because they re?ect the structure of the potential. Scalar boson-Vector boson couplings would also be modi?ed if the origin of the scalar boson is non standard,for example like in TC[8]or in any other compositeness schemes[9].

In order to answer these questions,precision tests of the bosonic sectors(gauge and scalar)have to be performed.Because of the rich variety of possible NP schemes,the analyses of present and future experiments must be done in the most possible unbiased and model independent way.This is the aim of the phenomenological description presented in the next Section.

3Phenomenological description of the bosonic sector Searches for NP e?ects can be divided into2classes:

(A)search for new particles which cannot?t into the SM classi?cation(not a new family of leptons and quarks,not a Higgs scalar),and

(B)search for anomalous interactions among usual particles due to residual e?ects of

NP.

In both cases we can look for direct as well as for indirect e?ects of these new particles

or interactions.The characteristic scale of NP is generally expected to lie in the TeV

range(following arguments based on unitarity,on the TC mechanism or simply on present experimental limits).If this is true,then new particles should more probably have masses

in this TeV range so that their direct production requires high energy colliders.It is however not excluded that some states have lower masses and can be found earlier.If

this is not the case one can nevertheless indirectly try,from their virtual e?ects in certain

processes(mixing e?ects with usual particles,e?ects through loop diagrams),to?nd hints of their existence.Similarly the existence of new interactions can be directly observed

in processes involving gauge bosons and Higgs bosons.But they could also be detected through indirect e?ects in fermionic processes(like loops involving self-boson couplings),

measured with a very high accuracy as it is the case at Z peak.

A1)Direct production of new particles

The rate for new particle production in a collider is essentially controlled by the productσ×B of the production cross section times the branching ratio of the new particle

decay mode into the channel that is detected.When no candidate event is observed a

mass limit for the new particle is given.This is signi?cant only if the coupling of the new particle to the initial and to the?nal states consisting of usual particles is su?ciently

strong so thatσ×B reaches the observability limit of the experiment.This is a very model dependent question and it explains why mass limits given in the literature are so strongly

process dependent and why the results are so largely spred out.As one essentially uses

fermionic processes,limits appear to be especially low for those states that are weakly coupled to usual leptons or quarks,i.e.M H≥60GeV from LEP1[11],M V≥250GeV for the V bosons generated by the strongly interacting sector[5],[6].On the opposite,in other cases they approach the TeV range[13].The low values quoted above illustrate the

fact that indeed,at present,the bosonic sector is still very weakly constrained.

A2)Indirect e?ects of new particles

As an example of indirect e?ect of heavier particles we shall treat the Z?Z′mixing case which has been extensively studied at LEP1[12].We shall?rst present a rather general model-independent description and then look at speci?c models.

If the Z0mixes with a higher Z′0vector boson with a mixing angleθM

Z=Z0cosθM+Z′0sinθM(1) its vector g V f and axial g Af couplings get modi?ed as follows

δg V f=G′θM c f+d f

(2)

depending on the Z′0f¯f couplings de?ned as

?i eG′2c f+1+γ5

4A2l.

For the4parameters of the quark sector one can take the following two partial widths Γ4=Γu+Γd+Γc+Γs,Γb and the two asymmetries A c,A b.

OnlyΓ4is presently available with a high accuracy.It is in fact more convenient[12]

to use the combination[14]

D=Γ4

s2)A l(4)

The forward-backward asymmetries A F Bq=3

4A q for q=c,b

in order to get a meaningful result[12],[15],[16].

Application to speci?c models

M′

Z .In Sect.4we will see how LEP1results allow to give upper limits for

θM and hence to give lower mass limits for the Z′.

B)Residual bosonic interactions below New Physics threshold.

We now present the description of residual interactions among usual particles.We anticipate the discussion of results from Z peak physics which strongly constrain(at the permille level)all non SM e?ects involving light fermions.We restrict to couplings involving W±,Z,γand Higgses,avoiding those which involve lepton and quark?elds. The case of couplings involving a heavy top quark is still an opened question which is under study[17].Let us start by recalling the basic SM bosonic couplings.

SM self-couplings at tree level

<WµνWµν>(5)

Higgs boson couplings with gauge bosons are given by the covariant derivative of the scalar kinetic terms.

LΦ=(DµΦ+)(DµΦ)=

2v2(Φ+Φ?

2µ2=?

M2H

(8)

Our notations are the following ones:

W aµν=?µW aν??νW aµ?g?abc W bµW cµ(9)

Wµ=?→Wµ·?→τ

,(10)

Φ= φ+

(v+H+iφ0) ,(11)

Dµ=(?µ+i g1Y Bµ+i g2Wµ),

U=v2U=( Φ,Φ),(12)

where Φ=iτ2Φ?and A ≡T rA.

Standard radiative corrections

General Lorentz and U(1)invariant forms for ZW +W ?and γW +W ?couplings

M 2

W V νλ W ?λµ W

+µ

ν(15)4)ez V

M 2W ?Z νλ W

+λµ

W ?µ

ν(18)7)eg SM V K V (?µZ ν+?νZ µ)W +µW ?

ν(19)where the abelian W a µν=?µW a ν??νW a µis used as well as the dual

?Z µν=1

As summarized in Table1the three?rst terms are C-and P-conserving(charge, magnetic moment and quadrupole moment),the fourth one is C-and P-violating but

CP-conserving(anapole term)and the last three ones are CP-violating.The speci?c helicity properties[20],[21]of the W+W?state for each type of coupling are also given in the last three lines of Table1.The identi?cation of these properties is particularly useful

for experimental analysis as it gives a way to disentangle the various forms.

Table1:Space-time properties of the seven3-boson coupling forms

P P P

C C

CP CP

LT LT LT

NP can contribute to such new couplings and form factors.This may happen in

various ways.The basic W,Z structure may di?er from the SM one if one uses an alternative description,for ex.if W,Z are massive vector bosons not directly generated by the gauge principle(composite states like hadronicρ,ω,...vector mesons)[27],[29]. In these cases tree level modi?cations of the self-boson couplings(?niteδκ,λ,...)may exist.In less drastic pictures in which the SU(2)×U(1)system is kept but extended or coupled to a new additional sector,tree level modi?cations may still appear through mixing of W,Z with higher vector bosons(especially if these ones pertain to a strongly interacting sector like SU(2)V)[5],[6].In any case at1-loop,NP e?ects will always appear through contributions of virtual states.They can even be enhanced by non-perturbative e?ects(hypercolour factors,resonant e?ects,...).The peculiarities of the terms generated in this way[37](for example the speci?c sectors that they a?ect,charged versus neutral states,transverse versus longitudinal ones,Higgs versus no-Higgs?nal states,...)and the symmetries that they respect should re?ect their origin and help to identify the nature of NP through detailed analyses of the processes.

We shall discuss these questions in a precise manner through the e?ective lagrangian method.If the characteristic scaleΛof NP is su?ciently larger than M W,e?ective lagrangians among usual particles are obtained by integrating out all heavy degrees of freedom.They can be written in the form

L=Σi ¯f i

dimensionless.A priori such a series can be in?nite and one needs restrictions in order to have in practice a useful description.These restrictions must be done on a physical basis because often an apparently”harmless”mathematical property can have very important physical consequences.As already said and motivated by LEP1results we restrict O i to not involve lepton and quark?elds.The next restriction comes from the dimension. IfΛ>>M W it is natural to expect observable e?ects only from the lowest dimensions d=4,6,perhaps8.

Global symmetries

xγ(26)

No quadrupole coupling is generated at this level.This set of free parameters can be further reduced if one considers the high energy behaviour of boson-boson scattering amplitudes.Because of these non-standard terms,they grow like s2.Demanding that these terms cancel,one obtains certain relations among the four free parameters which ?nally reduce to only one[28].

x Z=?

m2t

by the gauge coupling of the B?eld.This is why it is often discussed only in the limit g′→0.

Local symmetries

O DW=4 ([Dµ,Wµρ])([Dν,Wνρ]) ,(29)

O DB=(?µBνρ)(?µBνρ),(30)

O BW=Φ?BµνWµνΦ,(31)

OΦ1=(DµΦ)?ΦΦ?(DµΦ).(32) O W=13 WνλWλµWµν ,(33)

O UW=1

OΦ2=(?µ U U? )(?µ U U? ),(38)

OΦ3= U U? 3.(39) As presented in Table2,one can regroup the operators into sets which have basically di?erent physical consequences and behaviours under certain symmetries.The?rst four of them are called non-blind[34]because they involve2-point gauge boson functions.They would then directly a?ect the observables measured at LEP1.Consequently their coupling constants must have a strongly reduced strength in order to avoid direct observation.The next?ve ones are the”blind”ones in the sense that LEP1is blind to them at tree level. They can only a?ect the LEP1observables through1-loop.The resulting constraints are very mild and allow for large values of the coupling constants.The last two ones only involve Higgs?elds and has been dubbed”super-blind”[37]because they are almost unconstrainable by present and future machines.

Table2:Properties of the eleven bosonic operators

O DB

OΦ1

O UW x x

O WΦx

OΦ2x

Demanding a strict application of the custodial symmetry for the NP e?ects,strongly restricts the list of operators,see Table2.From the11above ones only?ve of them

are SU(2)c invariant namely,¯O DW,O W, O UW and the two superblind Oφ2and OΦ3. SU(2)c symmetry restricts the5blind ones to only two.Remember that one of them, O W was already obtained from the SU(2)L global symmetry which is for the pure W sector,a remnant of the full SU(2)c.The other one is O UW which also involves Higgs ?elds.The justi?cation for this strict use of custodial symmetry is that NP is supposed to be intimately related to the origin of the scalar sector and should therefore respects the same symmetries.

Chiral descriptions

v(40) E?ective lagrangians invariant under SU(2)×U(1)resulting from integrating out the e?ects of this sector can be constructed as combinations of gauge boson?elds,U matrices and their covariant derivatives.At present energies it is meaningful to make an expansion with respect to the number p of derivatives or of gauge?elds(U being dimensionless).At lowest(p2)order one?nds the SM part eq.(6).New couplings appear at order p4,p6,...etc. In this way one can again generate all possible bosonic operators.In the physical gauge, they produce the set of anomalous3-boson couplings listed above as well as higher multi-boson couplings.However the di?erence with the linear representation presented before is the absence of a physical Higgs?eld and a di?erent ordering in magnitude of the anomalous self-boson couplings.For exampleδZ,κγandκZ appear at order p4,through the operators called L9L and L9R and satisfy eq.(26)

L=?igL9L WµνDµUDνU? ?ig′L9R BµνDµUDνU? (41)

δZ=

c xγ=?

When operators with d>4are considered,they generally lead to boson-boson scat-tering amplitudes which grow fastly with the center of mass energy.For example d=6 terms lead to partial wave amplitudes growing like s or s2.This means that for a given value of the coupling constant the amplitudes reach the unitarity limit at a certain energy scale.At this point unitarity saturation e?ects(resonances or new particle creation,...)

must occur.So the unitarity relations which are obtained for each of the operators have two meanings.

1.For a given coupling constant one obtains a value for the scale at which unitarity saturation occurs(this can be considered as a practical de?nition of the NP scale),

2.For a given NP scale one can set upper limits for the coupling constants in order to satisfy unitarity in the whole s≤Λ2domain.

For the5blind operators the unitarity constraints read[39],[40]

|f B|≤98M2W s,|λW|<～19M2W

(45)

|d|<～17.6M2W√

s +1070

M3W

s?1123

M3W

4Status after the high precision tests at LEP1

It is interesting to discuss how far the high precision tests done at Z peak with fermionic

processes can be used to test the bosonic sector.In order to achieve this goal it is essential to use a description of the Z exchange processes which is su?ciently general in order to

account for possible NP e?ects but also to cover in an accurate way the SM radiative

correction e?ects(W,Z self-energies,vertex and box corrections).For this reason the usual description[45],[50]of the e?ective Z exchange amplitude in e+e?→f¯f has been somewhat generalized[24].

A)Formalism

We write it in the form

A Z=

√

q2?M2Z?iM ZΓZ(q2)[1+δs.e.][¯v eγµ([g V e+?g V e]?γ5[g Ae+?g Ae])u e]××[¯u fγµ([g V f+?g V f]?γ5[g Af+?g Af])v f](50)

The SM part at1-loop is fully taken into account through the three inputsα(0),Gµ, M Z,and through the shiftsδs.e.,?g V f and?g Af.From the inputs one derives

s21c21=πα(0)

2GµM2Z

(51)

and the basic g V f and g Af couplings

g V f=I3f?2s21Q f,g Af=I3f.(52) The shifts contain the SM radiative correction e?ects(in particular the large m t and M H dependent terms)and the NP contributions.We have already seen in Sect.3how Z?Z′mixing e?ects modify the Z couplings,i.e.addδg V f andδg Af for f=ν,l,u,d(assuming

universality).Non universal e?ects(i.e.b quark terms di?erent from s quark terms) already appear within SM because of large m2t e?ects in Zb¯b couplings[26].NP can add further non universal terms which can be described by eq(50).This leads us to separately discuss the various subsectors.

a)charged leptonic processes

2s21

(?g V l?(1?4s21)?g Al)(55)

and can be experimentally measured through two ”good”observables

Γl =G µM 3Z 2[1+?l 1][1+(1?4¯s 2l )2](56)

and

A l =2(1?4¯s 2l )

4A 2l .

b)light quark processes

2s 2[δg V l ?vδg Al +32?4s 2)δg Au ](62)

δ′d =?14

A l A q but can only be measured with a su?cient accuracy through polarized e ±beams

with A pol (q )F B =3The only process available at Z peak is e +e ?→Z →b ¯b .It contains two additional parameters [25]that we identify through the departures from universality with the two ?rst families:δg V b =δg V d +δg Heavy V b

(64)δg Ab =δg Ad +δg Heavy Ab (65)

that can be determined through the two new observables

Γb=Γd[1+δbV](66)

A b=A d[1+ηb](67) where the coe?cients correspond to

δbV=?

v d(1+v2d)[δg Heavy

V b?v dδg Heavy

](69)

with v d=1?4

Γhad

(70) and

A pol(b) F

B = 3

(72)

d)W mass

c2M2Z

=1+δξ(73)

δξ=?

s2?1+2?2+

c2?s2

Ref.[45]Ref.[46]Ref.[47]Vac.pol.

αT A33(0)?A11(0)?2

4s2

F11(M2W)?F33(M2Z)

This table contains a dictionary for the various notations which had been introduced in the past for the leading vacuum polarization(universal or”oblique”)contributions written as

Πij(q2)=Aij(0)+q2F ij(q2)(76) Let us also recall the de?nition

?α=Fγγ(0)?Fγγ(M2Z)(77) and notice one recent notation[30]

?x=?1??2?y=??2?=??3(78) We emphazize that the inclusion of non universal SM or NP terms requires the use of the more general parametrization de?ned above with at least7+1free parameters.

Brief summary of LEP1constraints

[3δg V u+9δg Au?6δg V d+4δg V e+23δg Ae]≤0.008(83)

and in the heavy sector[53]

δbV=0.0414±0.0110(84)

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