Polarized observables to probe Z′ at the e+e− linear collider

22 April 1999

Physics Letters B 452 Ž1999. 355–363

Polarized observables to probe Z at the eqey linear collider X

A.A. Babich

a,1

, A.A. Pankov

a,b,2

, N. Paver

c,b,3

a

b

International Centre for Theoretical Physics, Trieste, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy c Dipartimento di Fisica Teorica, UniÕersita` di Trieste, Trieste, Italy

Received 24 November 1998; received in revised form 8 February 1999 Editor: R. Gatto

Abstract We study the sensitivity to the Z X couplings of the processes eqey™ lqly,bb and cc at the linear collider with 's s 500 GeV with initial beam polarization, for typical extended model examples. To this aim, we use suitable integrated, polarized, observables directly related to the helicity cross sections that carry information on the individual Z X chiral couplings to fermions. We discuss the derivation of separate, model-independent limits on the couplings in the case of no observed indirect Z X signal within the expected experimental accuracy. In the hypothesis that such signals were, indeed, observed we assess the expected accuracy on the numerical determination of such couplings and the consequent range of Z X masses where the individual models can be distinguished from each other as the source of the effect. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction The existence of extra neutral heavy gauge bosons Z X is the natural consequence of the extensions of the Standard Model ŽSM. based on larger gauge symmetry groups w1,2x. Strategies for the experimental determination of the Z X couplings to the ordinary SM degrees of freedom, and the relevant discovery limits, have been discussed in the large, and still growing, literature on this subject w1–7x. 1

Permanent address: Department of Mathematics, Technical University, Gomel, 246746 Belarus; E-mail: [email protected]. 2 Permanent address: Department of Physics, Technical University, Gomel, 246746 Belarus. E-mail: [email protected]. 3 Also supported by the Italian Ministry of University, Scientific Research and Technology ŽMURST..

The current limit MZ X ) 600–700 GeV from ‘direct’ searches at the Tevatron w8x allows only ‘indirect’ Žor virtual. manifestations of the Z X at LEP2 w9x and at the planned eq ey linear collider ŽLC. with CM energy 's s 500 GeV w10,11x. Such signals would be represented by deviations from the calculated SM predictions of the measured observables relevant to the different processes. In this regard, of particular interest for the LC is the annihilation into fermion pairs eqq ey™ f q f ,

Ž 1. X

that gives information on the Z ff interaction. In the case of no observed signal within the experimental accuracy, limits on the Z X couplings to a conventionally defined confidence level can be derived. Clearly, completely model-independent limits can result only in the optimal situation where the

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 2 8 4 - 1

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356

different couplings can be disentangled, by means of suitable observables, and analysed independently so as to avoid potential cancellations. The essential role, to this aim, of the initial electron beam polarization has been repeatedly emphasized, e.g., in Refs. w6,7x. The same need of a procedure to disentangle the different Z X couplings occurs in the alternative case where deviations from the SM were experimentally observed. In this situation, one would like to determine the numerical values of the individual couplings from the measured deviations and eventually make tests of the various Z X models. In what follows, we discuss the role of two particular, integrated, polarized cross sections sq and sy in the analysis of the Z X ff interaction under both kinds of circumstance. These observables are directly connected to the helicity cross sections of process Ž1. and, therefore, depend on a minimal set of independent free parameters. They have been previously introduced to study Z X signals at LEP2 Žno polarization there. w12,13x and potential manifestations of four-fermion contact interactions at the LC w14x. Here, we extend the analysis of w12,13x to the case of the LC with polarized beams. For definiteness, we explicitly consider a specific class of E6motivated and of Left-Right Z X models, and give particular attention to the quantitative assessment of the expected sensitivity of these observables to the individual Z X couplings and to the Z X model identification in the case of observed deviations. 2. Polarized observables for the Z X

and sab s NC 4p a e .m . < Aa b < 2 with Aa b the helicity 3s amplitudes Ž a , b s L, R .. Moreover, u is the angle between the initial electron and the outgoing fermion in the CM frame; NC f 3Ž1 q a srp . for quarks and NC s 1 for leptons; Pe and Pe are the degrees of electron and positron longitudinal polarization. Clearly, the cross sections for the different combinations of helicities, that carry the information on the individual Z X ff couplings, can be disentangled Õia the measurement of s˜q and s˜y with different choices of the initial beams polarization. Instead, the total cross section s s s F q s B and the forward-backward asymmetry s A FB s s F y s B depend on linear combinations of all helicity cross sections, even for longitudinally polarized initial beams. One can notice the relation 2

s˜ "s 0.5 s Ž 1 " 43 A FB . s 76 s F ,B y 16 s B ,F .

One can directly determine s˜q and s˜y as differences of integrated observables. To this aim, we define z ) ) 0 such that

ž

yz )

1

Hyz

y )

Hy1

/Ž

2

1 y cos u . dcos u s 0.

žH žH

s˜qs 14 Ž 1 q Pe . Ž 1 y Pe . sR R q Ž 1 y Pe . Ž 1 q Pe . sL L ,

s˜ys

Ž 3.

1 4

Ž 1 q Pe . Ž 1 y Pe . sR L q Ž 1 y Pe . Ž 1 q Pe . sL R ,

Ž 4.

/

H

z)

y1

where, in terms of helicity cross sections:

Ž 6.

Numerically, z ) s 2 2r3 y 1 s 0.59, corresponding to u ) s 548, and, for a reduced angular range
sy' s 1yy s 2ys

Process Ž1. with f / e, t is described in Born approximation by the s-channel g , Z and Z X exchanges. Neglecting terms of order m fr 's : ds 2 2 s 38 Ž 1 q cos u . s˜qq Ž 1 y cos u . s˜y , dcos u Ž 2.

Ž 5.

y

1

Hz

)

/

ds dcos u

dcos u

Ž 8.

are such that

s˜ "s

1 3Ž 2

2r3

y 2 1r3 .

s "s 1.02 s " .

Ž 9.

Therefore, for practical purposes one can identify s "( s˜ " to a very good approximation. Although the two definitions are practically equivalent from the mathematical point of view, in the next Sections we prefer to use s ", that are found to be more convenient for the discussion of the expected uncertainties and the corresponding sensitivities to the Z X couplings. Also, it turns out numerically that z ) s 0.59 in Ž7. and Ž8. maximizes the statistical significance of the results.

A.A. Babich et al.r Physics Letters B 452 (1999) 355–363

3. Model independent Z X search and discovery limits

The helicity amplitudes can be written as Aab s Ž Q e . a Ž Q f . b q g ae g bf x Z q g aX e g bX f x Z X ,

Ž 10 . in the notation where the general neutral-current interaction is written as yL NC s eJgmAm q g Z JZm Zm q g ZmX JZmX ZmX .

Ž 11 .

(

Here, e s 4pa e .m . ; g Z s ersW cW and g Z X are the Z and Z X gauge couplings; in Eq. Ž10., x i s srŽ s y Mi2 q iMi Gi . with i s Z,Z X . The fermion currents that couple to the neutral gauge bosons are expressed as Jim s Ý f c f g m Ž L if PL q R if PR . c f , with PL, R s Ž1 . g 5 .r2 the projectors onto the left- and righthanded fermion helicity states. With these definitions, the SM couplings are Rgf s Q f

;

Lgf s Q f

According to Eqs. Ž3., Ž4. and Ž9., by the measurements of sq and sy for the different initial electron beam polarizations, one determines the cross sections related to definite helicity amplitudes Aab . From Eq. Ž10., one can observe that the Z X manifests itself in these amplitudes by the combination of the product of couplings g aX e g bX f with the propagator x Z X . Indeed, if 's is much smaller than MZ X , the deviation from the SM of each helicity cross section is well-approximated by the ‘linear’ interference term

Dsa b ' sa b y saSbM s NC

4pa e2.m . 3s

Ž Q e Q f q gae gbf xZ . P Ž g ae g bf x Z) . . =2 Re

;

X

R Zf s yQ f sW2 ;

LZf s I3f L y Q f sW2 ,

357

Ž 12 .

where Q f are fermion electric charges, and the couplings in Eq. Ž10. are normalized as gZ f gZ f g Lf s LZ , g Rf s R , e e Z g ZX f g ZX f g LX f s LZ X , g RX f s R X. Ž 13 . e e Z We will consider, as representative examples, models inspired by GUT scenarios, superstring-motivated ones, and those with Left-Right symmetric origin w3x. Specifically, the x model originating from the breaking SO Ž10. ™ SUŽ5. = UŽ1.x , the c model originating from E6 ™ SO Ž10. = UŽ1.c , and the h model which is encountered in superstring-inspired models in which E6 breaks directly to a rank-5 group. As an example of Left-Right models, we consider the ones with k s g Rrg L s 1. For all such grand-unified E6 and Left-Right models, the Z X gauge coupling in Ž11. is g Z X s g Z sW w3x. Due to the current mass limits, new vector boson effects at the LC are expected to be small and, therefore, should be disentangled from the radiative corrections to the SM Born predictions for the cross section. To this aim, in our numerical analysis we follow the strategy of Refs. w15,16x, in particular we use the improved Born approximation that accounts for the electroweak one-loop corrections.

X

X

Ž 14 .

Therefore, for each individually observable deviation Dsa b , the above mentioned combination g aX e g bX f x Z)X can be considered as a single ‘effective’ Z X coupling. Actually, since in an analysis of experimental data for sab based on a x 2 procedure a one-parameter fit is involved, in principle one might get a slightly better sensitivity to the Z X parameters, compared to other kinds of observables where more couplings appear simultaneously. Our x 2 procedure defines a x 2 function for any observable O : 2

x s

DO

ž /

2

, Ž 15 . dO where D O ' O Ž Z X . y O Ž SM . and d O is the expected uncertainty on the considered observable combining both statistical and systematic uncertainties. The domain allowed to the Z X parameters by the non-observation of the deviations D O within the accuracy d O will be assessed by imposing x 2 2 2 xcrit , where the actual value of xcrit specifies the desired ‘confidence’ level. In the ‘linear’ approximation Ž14., this condition easily translates into an upper limit on the Z X ‘effective’ coupling mentioned above, in terms of xcrit and on the expected uncertainty on the SM helicity cross section dsab . The estimate of the dsab must take into account that, in the realistic case, the beams longitudinal

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polarization cannot exactly be "1. Thus, ultimately, the separation of the pure helicity cross sections will be obtained by inverting the linear system of Eqs. Ž3. and Ž4. corresponding to the measured sq and sy with different values of the initial electron and positron polarizations. In our discussion, we consider the situation Pe s "P Žwith P - 1. and Pe s 0, so that Eq. Ž3. gives:

sR R s

1qP P 1qP

sL L s

P

sq Ž P . y

1yP

sq Ž yP . y

P

sq Ž yP . ,

(ž

1q P

GRf s R Zf X

1yP P

P

2

/

Ž dsq Ž P . . 2q

GLf s L Zf X

Ž 16 .

sq Ž P . .

Ž 17 .

From these relations, adding the uncertainties dsq Ž"P . on sq Ž"P . in quadrature: dsR R s

TER w18x, with input values m top s 175 GeV and m H s 300 GeV. The obtained, model-independent, upper bounds on the ‘effective’ couplings of the Z X refer to the general parameterization of the Z X-exchange interaction used, e.g., in Refs. w7,12x:

ž

1y P P

2

/ Ž ds

q Žy P . .

2

,

Ž 18 . and dsL L can be expressed quite similarly, by replacing dsq Ž P . l dsq ŽyP .. Similar expressions hold for the separation of sR L and sL R using the data on sy. Numerically, we assume the following values for the expected identification efficiencies Ž e . and systematic uncertainties Ž d sys . for the various fermionic final states w17x: e s 100% and d sys s 0.5% for leptons; e s 60% and d sys s 1% for b quarks; e s 35% and d sys s 1.5% for c quarks. We assume that s 1q Ž"P . and s 2q Ž"P . have the same systematic 2 2 error d sys , dsq Ž"P . sys s d sys Ž s 1q Ž "P . q s 2q 1r2 Ž"P .. , and combine also statistical and systematic uncertainties in quadrature. In our analysis, we 2 take xcrit s 3.84 as typical for 95% C.L. with a one-parameter fit. We consider 's s 0.5 TeV and a one-year run with L int s 50 fby1 . For polarized beams, we assume 1r2 of the total integrated luminosity quoted above for each value of the electron polarization, Pe s "P. Concerning polarization, in the numerical analysis presented below we take three different values, P s 1, 0.8 and 0.5, in order to test the dependence of the bounds on this variable. Our numerical analysis has been performed by means of the program ZEFIT, adapted to the present discussion, which has to be used along with ZFIT-

) )

g Z2X

MZ2

4p MZ2X y s g Z2X

MZ2

4p MZ2X y s

,

.

Ž 19 .

An advantage of introducing Eq. Ž19. is that the bounds on the left- and right-handed couplings can be represented on a two-dimensional ‘scatter plot’ with no need to specify particular values of MZ X or s. As already noticed, in the general case where process Ž1. depends on all four independent Z X ff couplings, only the products GRe GRf and GLe GLf can be constrained by the sq measurement Õia Eq. Ž14., while the products GRe GLf and GLe GRf can be analogously bounded by sy. The exception is lepton pair production Ž f s l . with Ž e y l . universality of Z X couplings, in which case sq can individually constrain either GLe or GRe . Also, it is interesting to note that such lepton universality implies sR L s sL R and, accordingly, for Pe s 0 the electron polarization drops from Eq. Ž4. which becomes equivalent to the unpolarized one, with a priori no benefit from polarization. Nevertheless, the uncertainty in Eq. Ž18. still depends on the longitudinal polarization P. The 95% C.L. upper bounds on the products of lepton couplings Žwithout assuming lepton universality. are reported in the first three rows of Table 1. For quark-pair production Ž f s c ,b ., where in general sR L / sL R due to the appearance of different fermion couplings, the analysis takes into account the reconstruction efficiencies and the systematic uncertainties previously introduced. In Table 1 we report the 95% C.L. upper bounds on the relevant products of couplings. Table 1 shows that the integrated observables sq and sy are quite sensitive to the indirect Z X effects, with upper limits on the relevant products < Gae P Gbf < ranging from 2.2 P 10y3 to 4.8 P 10y3 at the maximal planned value P s 0.8 of the electron longitudinal polarization. In most cases, the best sensitivity oc-

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359

Table 1 95% C.L. model-independent upper limits at LC with Ec.m .s 0.5 TeV. For polarized beams, we take Lint s 25 fby1 for each possibility of the electron polarization, Pe s "P Process

P

Couplings < GRe GRf < 1r 2 Ž10y3 . sR R

< GLe GLf < 1r2 Ž10y3 . sL L

< GRe GLf < 1r2 Ž10y3 . sR L

< GLe GRf < 1r2 Ž10y3 . sL R

eq ey™ lq ly eq ey™ lq ly eq ey™ lq ly

1.0 0.8 0.5

2.1 2.3 2.7

2.1 2.3 2.7

3.0 3.3 3.9

3.2 3.4 4.0

eq ey™ bb eq ey™ bb eq ey™ bb

1.0 0.8 0.5

1.9 2.2 3.0

2.0 2.1 2.3

2.5 2.8 3.7

4.6 4.8 5.7

eq ey™ cc eq ey™ cc eq ey™ cc

1.0 0.8 0.5

2.3 2.5 3.2

2.6 2.7 3.0

4.1 4.5 5.5

3.9 4.1 4.6

curs for the bb final state, while the worst one is for cc. Decreasing the electron polarization from P s 1 to P s 0.5 results in worsening the sensitivity by as much as 50%, depending on the final fermion channel. Regarding the role of the assumed uncertainties on the observables under consideration, in the cases of eq ey™ lq ly and eq ey™ bb the expected statistics are such that the uncertainty turns out to be dominated by the statistical one, and the results are almost insensitive to the value of the systematical uncertainty. Conversely, for eq ey™ cc both statistical and systematic uncertainties are important. Moreover, as Eqs. Ž3. and Ž4. show, a further improvement on the sensitivity to the various Z X couplings in Table 1 would obtain if both ey and eq longitudinal polarizations were available w11x.

4. Resolving power and model identification If a Z X is indeed discovered, perhaps at a hadron machine, it becomes interesting to determine as accurately as possible its parameters from process Ž1. at the LC, and make tests of the various extended gauge models. To assess the accuracy, the same procedure as in the previous section can be applied to the determination of Z X parameters, by simply replacing the SM cross sections in Eqs. Ž15. and Ž18. by the ones expected for the ‘true’ values of the

parameters Žnamely, the extended model ones., and evaluating the x 2 variation around them in terms of the expected uncertainty on the cross section. 4.1. Z X couplings to leptons We now examine the bounds on the Z X couplings for MZ X fixed at some value. Starting from the leptonic process eqey™ lqly, let us assume that a Z X signal is detected by means of the observables sq and sy. Using Eqs. Ž16. and Ž17., the measurement of sq for the two values Pe s "P will allow to extract sR R and sL L which, in turn, determine independent and separate values for the right- and left-handed Z X couplings R Ze X and LeZ X Žwe assume lepton universality.. The x 2 procedure determines the accuracy, or the ‘resolving power’ of such determinations in terms of the expected experimental uncertainty Žstatistical plus systematic.. In Table 2 we give the resolution on the Z X leptonic couplings for the typical model examples introduced in Section 2, with MZ X s 1 TeV. In this regard, one should recall that the two-fold ambiguity intrinsic in process Ž1. does not allow to distinguish the pair of values of Ž g aX e, g bX f . from the one Žyg aX e, yg bX f ., see Eq. Ž14.. Thus, the actual signs of the couplings R Ze X and LeZ X cannot be determined from the data Žin Table 2 we have chosen the signs dictated by the relevant models.. In principle, the sign ambiguity of fermionic couplings might be re-

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Table 2 The values of the Z X leptonic and quark chiral couplings for typical models with MZ X s1 TeV and expected 1-s error bars from combined statistical and systematic uncertainties, as determined at the LC with Ec.m .s 0.5 TeV and P s 0.8 R Ze X LeZ X R Zb X LbZ X R Zc X LcZ X

x

c

h

LR

0.204q0.042 y0.069 q0.020 0.612y0.020 q0.110 y0.612y0.111 q0.040 y0.204y0.042 q0.092 0.204y0.090 q0.059 y0.204y0.064

y0.264q0.052 y0.043 q0.042 0.264y0.052 q0.111 y0.264y0.172 q0.158 0.264y0.103 q0.138 y0.264y0.207 q0.222 0.264y0.149

y0.333q0.038 y0.035 q0.102 y0.166y0.061 q0.096 0.166y0.075 q0.230 0.333y0.168 q0.114 y0.333y0.145 q0.577 0.333y0.326

y0.438q0.029 y0.028 q0.036 0.326y0.039 q0.116 y0.874y0.138 q0.080 y0.110y0.085 q0.122 0.656y0.104 q0.106 y0.110y0.134

solved by considering other processes such as, e.g., eqey™ WqWy. Another interesting question is the potential of the leptonic process Ž1. to identify the Z X model underlying the measured signal, through the measurement of the helicity cross sections sR R and sL L . Such cross sections only depend on the relevant leptonic chiral coupling and on MZ X , so that the resolving power clearly depends on the actual value of the Z X mass. In Figs. 1a and 1b we show this dependence for the E6 and the LR models of interest here. In these figures, the horizontal lines represent the values of the couplings predicted by the various models, and the lines joining the upper and the lower ends of the vertical bars represent the expected experimental uncertainty at the 95% CL. The intersection of the lower such lines with the MZ X axis determines the discovery reach for the corresponding model: larger values of MZ X would determine a Z X signal smaller than the experimental uncertainty and, consequently, statistically invisible. Also, Figs. 1a and 1b show the complementary roles of sL L and sR R to set discovery limits: while sL L is mostly sensitive to the ZxX and has the smallest sensitivity to the ZhX , sR R provides the best limit for the ZLX R and the worst one for the ZxX . As illustrated by Figs. 1a and 1b, the different models can be distinguished by means of sq and sy as long as the uncertainty of the coupling of one model does not overlap with the value predicted by the other ones. Thus, the identification power of the leptonic process Ž1. is determined by the minimum MZ X value at which such ‘confusion region’ starts. For example, Fig. 1a shows that the x model cannot be distinguished from the LR, c and h models at Z X

masses larger than 2165 GeV, 2270 GeV and 2420 GeV, respectively. The identification power for the typical models are indicated in Figs. 1a and 1b by the symbols circle, diamond, square and triangle. The corresponding MZ X values at 95% C.L. for the typical E6 and LR models are reported in Table 3, where the Z X models listed in first columns should be distinguished from the ones listed in the first row assumed to be the origin of the observed Z X signal. For this reason, Table 3 is not symmetric. Analogous considerations hold also for sL R and sR L . These cross sections are found to give qualita-

Fig. 1. Ža. Resolution power at 95% C.L. for the absolute value of the left-handed leptonic Z X coupling LeZ X as a function of MZ X , as obtained from sL L in eq ey ™ lq ly. The error bars combine statistical and systematic uncertainties. Horizontal lines correspond to the values predicted by the Z X models considered here. Žb. Same as in Fig. 1a, but for < R Ze X < extracted from sR R .

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Table 3 Identification power of process eq ey™ ff at 95% C.L. expressed in terms of MZ X Žin GeV. for typical E6 and LR models at Ec.m .s 0.5 TeV and Lint s 25 fby1 for each value of the electron polarization, Pe s "0.8

sR R q y

q y

sL L

e e ™l l

c

h

c h x LR

– 950 830 1160

960 – 1165 1220

x

LR

c

1470 1210 1615 –

– 960 1170 915

eq ey™ bb

c

h

c h x LR

– 700 1075 1210

725 – 1100 1100

x

LR

c

1080 1210 – 1540

2345 2410 2130 –

– 750 1130 940

eq ey™ cc

c

h

x

c h x LR

– 880 760 1050

865 – 1050 1280

LR

c

830 970 – 970

800 880 – 880

tively similar results for the product LeZ X R Ze X , but with weaker constraints because of smaller sensitivity.

1740 1580 1840 –

– 645 935 780

h

x

LR

2270 2420 – 2165

920 1220 1400 –

h

x

LR

710 – 1140 760

1120 1250 – 1370

h

x

840 – 840 840

620 – 940 685

935 1035 – 1135

940 750 950 – LR 800 665 810 –

contours in the upper-left and the lower-right corners of Fig. 2. Then, to get finite regions for the quark couplings, one must combine the hyperbolic regions

4.2. Z X couplings to quarks In the case of process Ž1. with qq pair production Žwith q s c, b ., the analysis is complicated by the fact that the relevant helicity amplitudes depend on three parameters Ž g aX e, g bX q and MZ X . instead of two. Nevertheless, there is still some possibility to derive general information on the Z X chiral couplings to quarks. Firstly, by the numerical procedure introduced above one can determine from the measured cross section the products of electrons and final state quark couplings of the Z X , from which one derives allowed regions to such couplings in the independent, two-dimensional, planes Ž LeZ X , LqZ X . and Ž LeZ X , R Zq X .. The former regions are determined through sL L , and the latter ones through sL R . As an illustrative example, in Fig. 2 we depict the bounds from the process eq ey™ bb in the Ž LeZ X , LbZ X . and the Ž LeZ X , R Zb X . planes for the Z X of the x model, with MZ X s 1 TeV. Taking into account the above mentioned two-fold ambiguity, the allowed regions are the ones included within the two sets of hyperbolic

Fig. 2. Allowed bounds at 95% C.L. on Z X couplings with MZ X s1 TeV Ž x model. in the two-dimension planes Ž LeZ X , LbZ X . and Ž LeZ X , R Zb X ., as obtained from the helicity cross sections sL L Žsolid lines. and sL R Ždashed lines., respectively. The shaded and hatched regions are derived from the combination of eq ey ™ lq ly and eq ey ™ bb processes. The two allowed regions for each helicity cross section correspond to the two-fold ambiguity discussed in text.

362

A.A. Babich et al.r Physics Letters B 452 (1999) 355–363

so obtained with the determinations of the leptonic Z X couplings from the leptonic process Ž1., represented by the two vertical strips. The corresponding shaded areas represent the determinations of LbZ X , while the hatched areas are the determinations of R Zb X . Notice that, in general, there is the alternative possibility of deriving constraints on quark couplings also in the case of right-handed electrons, namely,

from the determinations of the pairs of couplings Ž R Ze X , LbZ X . and Ž R Ze X , R Zb X .. However, as observed with regard to the previous analysis of the leptonic process, the sensitivity to the right-handed electron coupling turns out to be smaller than for LeZ X , so that the corresponding constraints are weaker. The determinations of the Z X couplings with the c and b quarks for the typical E6 and LR models with MZ X s 1 TeV, are given in Table 2 where the combined statistical and systematic uncertainties are taken into account. Furthermore, similar to the analysis presented in Section 4.1 and the corresponding Figs. 1a and 1b, we depict in Figs. 3a and 3b the different models identification power as a function of MZ X , for the reaction eq ey™ bb as a representative example. The model identification power of the bb and cc pair production processes are reported in details in Table 3.

5. Concluding remarks

Fig. 3. Ža. Resolution power at 95% C.L. for < LeZ X LbZ X < 1r 2 as a function of MZ X obtained from sL L in eq ey ™ bb. The error bars combine statistical and systematic errors. Horizontal lines correspond to the values predicted by the Z X models considered here. Žb. Same as in Fig. 3a, but for < R Ze X R Zb X < 1r 2 extracted from sR R .

We briefly summarize our findings concerning the Z X discovery limits and the model identification power of process Ž1. Õia the separate measurement of the helicity cross sections sab at the LC, with 's s 0.5 TeV and Lint s 25 fby1 for each value Pe s "P the electron longitudinal polarization. Given the present experimental lower limits on MZ X , only indirect effects of the Z X can be studied at the LC. In general, the helicity cross sections allow to extract separate, and model-independent, information on the individual ‘effective’ Z X couplings Ž Gae P Gbf .. As depending on the minimal number of free parameters, these cross sections may be expected to show some convenience with respect to other observables in an analysis of the experimental data based on a x 2 procedure. In the case of no observed signal, i.e., no deviation of sab from the SM prediction within the experimental accuracy, one can directly obtain model-independent bounds on the leptonic chiral couplings of the Z X from eqey™ lqly and on the products of couplings Gae P Gbq from eq ey™ qq Žwith l s m ,t and q s c,b .. From the numerical point of view, as far as the actual sensitivity is concerned,

A.A. Babich et al.r Physics Letters B 452 (1999) 355–363

sab are found to just have a complementary role with respect to other observables, like s and A FB . In the case where Z X manifestations are observed as deviations from the SM, with MZ X of the order of 1 TeV, the role of sab is more interesting, specially regarding the problem of identifying the various models as potential sources of such non-standard effects. Indeed, in principle, these observables provide a unique possibility to disentangle and extract numerical values for the chiral couplings of the Z X in a general way Žmodulo the aforementioned sign ambiguity., avoiding the danger of cancellations, so that Z X model predictions can be tested. Data analyses with other observables may involve combinations of different coupling constants and need some assumption to reduce the number of independent parameters in the x 2 procedure. In particular, by the analysis combining sab Ž lq ly . and sa b Ž qq . one can obtain information of the Z X couplings with quarks without making assumptions on the values of the leptonic couplings. Numerically, as displayed in the previous Sections, for the class of E6 and Left-Right models considered here, the couplings would be determined to about 3 y 60% for MZ X s 1 TeV. Of course, the considerations above hold only in the case where the Z X signal is seen in all observables. Finally, one can notice that for 's < MZ X the energy-dependence of the deviations Dsab is determined by the SM and that, in particular, the definite sign Dsaa Ž lq ly . - 0 Ž a s L, R . is typical of the Z X . This property might be helpful in order to identify the Z X as the source of observed deviations from the SM in process Ž1..

Acknowledgements A.A.B. and A.A.P. gratefully acknowledge the support of the University of Trieste.

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