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Polarized observables to probe Z' at the e+e

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IC/98/206

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

arXiv:hep-ph/9811328v1 14 Nov 1998

A. A. Babich 1
International Centre for Theoretical Physics, Trieste, Italy

A. A. Pankov 2
International Centre for Theoretical Physics, Trieste, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy

N. Paver 3
Dipartimento di Fisica Teorica, Universit` di Trieste, Trieste, Italy a Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy

Abstract We study the sensitivity to the Z ′ couplings of the processes e+ e? → l+ l? , ? and bb √ cc at the linear collider with s = 500 GeV with initial beam polarization, for typ? ical 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 ′ chiral couplings to fermions. We discuss the derivation of separate, model-independent limits on the couplings in the case of no observed indirect Z ′ 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 ′ masses where the individual models can be distinguished from each other as the source of the e?ect.

Permanent address: Department of Mathematics, Technical University, Gomel, 246746 Belarus. Email: BABICH@GPI.GOMEL.BY 2 Permanent address: Department of Physics, Technical University, Gomel, 246746 Belarus. E-mail: PANKOV@GPI.GOMEL.BY 3 Also supported by the Italian Ministry of University, Scienti?c Research and Technology (MURST).

1

1

Introduction

The existence of extra neutral heavy gauge bosons Z ′ is the natural consequence of the extensions of the Standard Model (SM) based on larger gauge symmetry groups [1, 2]. Indeed, the search for the Z ′ is included in the physics programme of all the present and future high energy collider facilities. In particular, the strategies for the experimental determination of the Z ′ 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 [1]-[7]. Taking into account the limit MZ ′ > 600 ? 700 GeV from ‘direct’ searches at the Tevatron [8], only ‘indirect’ (or virtual) manifestations of the Z ′ can be expected at LEP2 √ [9] and at the planned e+ e? linear collider (LC) with CM energy s = 500 GeV [10, 11]. Such e?ects would be represented by deviations from the calculated SM predictions of the measured observables relevant to the di?erent processes. In this regard, of particular interest for the LC is the annihilation into fermion pairs ? e+ + e? → f + f , (1)

that gives information on the Z ′ f f interaction. In the case of no observed signal within the experimental accuracy, limits on the Z ′ parameters to a conventionally de?ned con?dence level can be derived, either from a general analysis taking into account the full set of possible Z ′ couplings to fermions, or in the framework of speci?c models where characteristic relations among the couplings strongly reduce the number of independent free parameters. Clearly, completely model-independent limits can result only in the optimal situation where the di?erent couplings can be disentangled, by means of suitable observables, and analysed independently so as to avoid potential cancellations. The essential role of the initial electron beam polarization has been repeatedly emphasized in this regard, and the potential of the linear collider along these lines has been extensively reviewed, e.g., in Refs. [6, 7]. The same need of a procedure to disentangle the di?erent Z ′ couplings arises in the case where deviations from the SM were experimentally observed. Indeed, in this situation, the numerical values of the individual couplings must be extracted from the measured deviations in order to identify the source of these e?ects and to make tests of the various theoretical models. In what follows, we discuss the role of two particular, polarized, variables σ+ and σ? in the analysis of the Z ′ f f interaction from both points of view, namely, the derivation of model-independent limits in the case of no observed deviation and the sensitivity to individual couplings and model identi?cation in the hypothesis of observed deviations. These observables could directly distinguish the helicity cross sections of process (1) and, therefore, depend on a minimal number of independent free parameters (basically, the product of the Z ′ chiral couplings to electrons and to the fermionic ?nal state). They have been previously introduced to study Z ′ e?ects at LEP2 (no polarization there) [12, 13] and manifestations of four-fermion contact interactions at the LC [14]. Here, we extend the analysis of [12, 13] to the case of the LC with polarized beams. For illustration, we will explicitly consider a speci?c class of E6 -motivated models and of Left-Right symmetric models. 1

2

Polarized observables for the Z ′

The polarized di?erential cross section for process (1) with f = e, t is given in Born approximation by the s-channel γ, Z and Z ′ exchanges. Neglecting mf with respect to the √ CM energy s, it has the form 3 dσ (1 + cos θ)2 σ+ + (1 ? cos θ)2 σ? , ? ? = d cos θ 8 where, in terms of helicity cross sections σ+ = ? σ? = ? with (α, β = L, R) σαβ = NC σpt |Aαβ |2 . (5) In these equations, θ is the angle between the initial electron and the outgoing fermion in the CM frame; NC the QCD factor NC ≈ 3(1 + αs ) for quarks and NC = 1 for leptons, π respectively; Pe and Pe are the degrees of longitudinal electron and positron polarization; ? 2 σpt ≡ σ(e+ e? → γ ? → l+ l? ) = (4παe.m. )/(3s); Aαβ are the helicity amplitudes. According to Eqs. (3) and (4), the cross sections for the di?erent combinations of helicities, that carry the information on the individual Z ′ f f couplings, can be disentangled via the measurement of σ+ and σ? with di?erent choices of the initial beams polarization. ? ? Instead, the total cross section and the forward-backward asymmetry, de?ned as: σ = σF + σB; AFB = (σ F ? σ B )/σ, (6) 1 [(1 + Pe )(1 ? Pe ) σRR + (1 ? Pe )(1 + Pe ) σLL ] , ? ? 4 1 [(1 + Pe )(1 ? Pe ) σRL + (1 ? Pe )(1 + Pe ) σLR ] , ? ? 4 (3) (4) (2)

0 with σ F = 01 (dσ/d cos θ)d cos θ and σ B = ?1 (dσ/d cos θ)d cos θ, depend on linear combinations of all helicity cross sections even for longitudinally polarized initial beams. One can notice the relation

7 1 4 σ± = 0.5 σ 1 ± AFB = σF,B ? σB,F . ? 3 6 6

(7)

Alternatively, one can directly project out σ+ and σ? from Eq. (2), as di?erences of ? ? ? integrated observables. To this aim, we de?ne z > 0 such that
1 ?z ? ?z ?

?

?1

(1 ? cos θ)2 d cos θ = 0.

(8)

Numerically, z ? = 22/3 ? 1 = 0.59, corresponding to θ? = 54? ,4 and for this value of z ? :
1 ?z ?
4

?z ?

?

?1

(1 + cos θ)2 d cos θ = 8 22/3 ? 21/3 .

(9)

In the case of a reduced angular range | cos θ| < c, one has z ? = (1 + 3c2 )1/3 ? 1.

2

From Eq. (2) one can easily see that the observables
1 ?z ?

σ+ ≡ σ1+ ? σ2+ = σ? ≡ σ1? ? σ2? = are such that σ± = ?

?z ? z? ?1

? ?

?1 1 z?

dσ d cos θ, d cos θ dσ d cos θ d cos θ

(10) (11)

1 σ± = 1.02σ± . (12) 3 (22/3 ? 21/3 ) Therefore, for practical purposes one can identify σ± ? σ± to a very good approximation. =? Although the two de?nitions are practically equivalent from the mathematical point of view, in the next Section we prefer to use σ± , that are found more convenient to discuss the expected uncertainties and the corresponding sensitivities to the Z ′ couplings. Also, it turns out numerically that z ? = 0.59 in (10) and (11) maximizes the statistical signi?cance of the results. The helicity amplitudes Aαβ in Eq. (5) can be written as
e f Aαβ = (Qe )α (Qf )β + gα gβ χZ + g ′α g ′ β χZ ′ , e f

(13)

in the notation where the general neutral-current interaction is written as
? ? ? ? ′ ? LN C = eJγ A? + gZ JZ Z? + gZ ′ JZ ′ Z? .

(14)

√ Here, e = 4παe.m. ; gZ = e/sW cW (s2 = 1 ? c2 ≡ sin2 θW ) and gZ ′ are the Z and Z ′ W W gauge couplings, respectively. Moreover, in (13), χi = s/(s ? Mi2 + iMi Γi ) are the gauge boson propagators with i = Z and Z ′ , and the g’s are the left- and right-handed fermion couplings. The fermion currents that couple to the neutral gauge boson i are expressed as f ? Ji? = f ψf γ ? (Lf PL + Ri PR )ψf , with PL,R = (1 ? γ5 )/2 the projectors onto the left- and i right-handed fermion helicity states. With these de?nitions, the SM couplings are
f Rγ = Qf ;

Lf = Qf ; γ

f RZ = ?Qf s2 ; W

f Lf = I3L ? Qf s2 , Z W

(15)

where Qf are fermion electric charges, and the couplings in Eq. (13) are normalized as
f gL =

gZ f L , e Z

f gR =

gZ f R , e Z

g ′L =

f

gZ ′ f L ′, e Z

g′R =

f

gZ ′ f R ′. e Z

(16)

In what follows, we will limit ourselves to a few representative models predicting new gauge heavy bosons. Speci?cally, models inspired by GUT inspired scenarios, superstringmotivated ones, and those with Left-Right symmetric origin [3]. These are the χ model occurring in the breaking SO(10) → SU(5) × U(1)χ , the ψ model originating in E6 → SO(10) × U(1)ψ , and the η model which is encountered in superstring-inspired models in which E6 breaks directly to a rank-5 group. As an example of Left-Right model, we consider the particular value κ = gR /gL = 1, corresponding to the most commonly considered case of Left-Right Symmetric Model (LR). For all such grand-uni?ed E6 and Left-Right models the Z ′ gauge coupling in (14) is gZ ′ = gZ sW [3]. 3

As they are constrained from present low-energy data and from recent data from the Tevatron [8], new vector boson e?ects at the LC are expected to be quite 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. [15][16], in particular we use the improved Born approximation accounting for the electroweak one-loop corrections.

3

Model independent Z ′ search and discovery limits

According to Eqs. (3), (4) and (12), by the measurements of σ+ and σ? for the di?erent initial electron beam polarizations one determines the cross sections related to de?nite helicity amplitudes Aαβ . From Eq. (13), one can observe that the Z ′ manifests itself in ′e ′f these amplitudes by the combination of the product of couplings gα gβ with the propagator √ χZ ′ . In the situation s ? MZ ′ we shall consider here, only the interference of the SM term with the Z ′ exchange is important and the deviation of each helicity cross section from the SM prediction is given by
SM ?σαβ ≡ σαβ ? σαβ = NC σpt 2 Re e f Qe Qf + gα gβ χZ · g ′α g ′ β χ? ′ Z e f

.

(17)

As one can see, ?σαβ depend on the same kind of combination of Z ′ parameters and, correspondingly, each such combination can be considered as a single ‘e?ective’ nonstandard parameter. Therefore, in an analysis of experimental data for σαβ based on a χ2 procedure, a one-parameter ?t is involved and we may hope to get a slightly improved sensitivity to the Z ′ with respect to other kinds of observables. As anticipated, in the case of no observed deviation one can evaluate in a modelindependent way the sensitivity of process (1) to the Z ′ parameters, given the expected experimental accuracy on σ+ and σ? . It is convenient to introduce the general parameterization of the Z ′ -exchange interaction used, e.g., in Refs. [7, 12]: Gf = Lf ′ L Z
2 2 gZ ′ MZ , 2 4π MZ ′ ? s f Gf = RZ ′ R 2 2 gZ ′ MZ . 2 4π MZ ′ ? s

(18)

An advantage of introducing the ‘e?ective’ left- and right-handed couplings of Eq. (18) is that the bounds can be represented on a two-dimensional ‘scatter plot’, with no need to specify particular values of MZ ′ or s. Our χ2 procedure de?nes a χ2 function for any observable O: ?O χ = δO
2 2

,

(19)

where ?O ≡ O(Z ′ ) ? O(SM) and δO is the expected uncertainty on the considered observable combining both statistical and systematic uncertainties. The domain allowed to the Z ′ parameters by the non-observation of the deviations ?O within the accuracy δO will be assessed by imposing χ2 < χ2 , where the actual value of χ2 speci?es the desired crit crit ‘con?dence’ level. The numerical analysis has been performed by means of the program 4

ZEFIT, adapted to the present discussion, which has to be used along with ZFITTER [17], with input values mtop = 175 GeV and mH = 300 GeV. In the real case, the longitudinal polarization of the beams will not exactly be ±1 and, consequently, instead of the pure helicity cross section, the experimentally measured σ± will determine the linear combinations on the right hand side of Eqs. (3) and (4) with |Pe | (and |Pe |) less than unity. Thus, ultimately, the separation of σRR from σLL will be ? obtained by solving the linear system of two equations corresponding to the data on σ+ for, e.g., both signs of the electron longitudinal polarization. The same is true for the separation of σRL and σLR using the data on σ? . √ In the ‘linear’ approximation of Eq. (17), and with MZ ′ ? s, the constraints from the condition χ2 < χ2 can be directly expressed in terms of the e?ective couplings (18) crit as: SM δσαβ M2 αe.m. e f 2 |ASM | Z . (20) |Gα Gβ | < χcrit αβ SM 2 σαβ s We need to evaluate the expected uncertainties δσαβ . To this aim, starting from the discussion of σ+ , we consider the solutions of the system of two equations corresponding to Pe = ±P and Pe = 0 in Eq. (3): ? σRR = σLL = 1+P 1?P σ+ (P ) ? σ+ (?P ), P P (21)

1?P 1+P σ+ (?P ) ? σ+ (P ). (22) P P From these relations, adding the uncertainties δσ+ (±P ) on σ+ (±P ) in quadrature, δσRR has the form δσRR = 1+P P
2

1?P (δσ+ (P )) + P
2

2

(δσ+ (?P ))2 ,

(23)

and δσLL can be expressed quite similarly. Also, we combine statistical and systematic uncertainties in quadrature. In this case, if σ+ (±P ) are directly measured via the difference (10) of the integrated cross sections σ1+ (±P ) and σ2+ (±P ), one can see that
stat δσ+ has the simple property: δσ+ (±P )stat = σ SM (±P )/?Lint , where Lint is the timeintegrated luminosity, ? is the e?ciency for detecting the ?nal state under consideration and σ SM (±P ) is the polarized total cross section. For the systematic uncertainty, we use 2 2 δσ+ (±P )sys = δ sys σ1+ (±P ) + σ2+ (±P ) , assuming that σ1+ (±P ) and σ2+ (±P ) have sys the same systematic error δ . One can easily see that δσLL can be obtained by changing δσ+ (P ) ? δσ+ (?P ) in (23) and that the expression for δσRL and δσLR also follow from this equation by δσ+ → δσ? . Numerically, to exploit Eq. (17) with δσαβ expressed as above, we assume the following values for the expected identi?cation e?ciencies and systematic uncertainties on the various fermionic ?nal states [18]: ? = 100% and δ sys = 0.5% for leptons; ? = 60% and δ sys = 1% for b quarks; ? = 35% and δ sys = 1.5% √ c quarks. Also, χ2 = 3.84 as typical for 95% C.L. for crit with a one-parameter ?t. We take s = 0.5 TeV and a one-year run with Lint = 50 f b?1 . For for polarized beams, we assume 1/2 of the total integrated luminosity quoted above 1/2 1/2

5

Table 1: 95% C.L. model-independent upper limits at LC with Ec.m. = 0.5 TeV. For polarized beams, we take Lint = 25 f b?1 for each possibility of the electron polarization, Pe = ±P . couplings observables process P e+ e? → l+ l? 1.0 e+ e? → l+ l? 0.8 e+ e? → l+ l? 0.5 e+ e? → bb 1.0 e+ e? → bb 0.8 e+ e? → bb 0.5 e+ e? → cc 1.0 e+ e? → cc 0.8 e+ e? → cc 0.5 |Ge Gf |1/2 R R (10?3 ) σRR 2.1 2.3 2.7 1.9 2.2 3.0 2.3 2.5 3.2 |Ge Gf |1/2 L L (10?3 ) σLL 2.1 2.3 2.7 2.0 2.1 2.3 2.6 2.7 3.0 |Ge Gf |1/2 R L (10?3) σRL 3.0 3.3 3.9 2.5 2.8 3.7 4.1 4.5 5.5 |Ge Gf |1/2 L R (10?3 ) σLR 3.2 3.4 4.0 4.6 4.8 5.7 3.9 4.1 4.6

for each value of the electron polarization, Pe = ±P . Concerning polarization, in the numerical analysis presented below we take three di?erent values, P =1, 0.8 and 0.5, in order to test the dependence of the bounds on this variable. As already noticed, in the general case where process (1) depends on all four independent Z ′ f f couplings, only the products Ge Gf and Ge Gf can be constrained by the R R L L σ+ measurement via Eq. (17), while the products Ge Gf and Ge Gf can be analogously R L L R bounded by σ? . The exception is lepton pair production (f = l) with (e ? l) universality of Z ′ couplings, in which case σ+ can individually constrain either Ge or Ge . Also, it is interL R esting to note that such lepton universality implies σRL = σLR and, accordingly, for Pe = 0 ? electron polarization drops from Eq. (4) which becomes equivalent to the unpolarized one, with a priori no bene?t from polarization. Nevertheless, the uncertainty in Eq. (23) still depends on the longitudinal polarization P . The 95% C.L. upper bounds on the products of lepton couplings (without assuming lepton universality) are reported in the ?rst three rows of Table 1. For quark-pair production (f = c , b), where in general σRL = σLR due to the appearance of di?erent fermion couplings, the analysis takes into account the reconstruction e?ciencies and the systematic uncertainties previously introduced, and in Table 1 we report the 95% C.L. upper bounds on the relevant products of couplings. Also, for illustrative purposes, in Fig. 1 we show the 95% C.L. bounds in the plane e (GR , Gb ), represented by the area limited by the four hyperbolas. The shaded region is R obtained by combining these limits with the ones derived from the pure leptonic process with lepton universality. Thus, in general we are not able to constrain the individual couplings to a ?nite region. On the other hand, there would be the possibility of using Fig. 1 to constrain the quark couplings to the Z ′ to a ?nite range in the case where some ?nite e?ect were observed in the lepton-pair channel. The situation with the other 6

Figure 1: 95% C.L. upper bounds on the model independent Z ′ couplings in the e b plane (GR , GR ) determined by σRR . The areas enclosed by vertical straight lines are obtained from the process e+ e? → l+√ , while those enclosed between hyperbolas are from l? ?1 e+ e? → ? at Lint = 50 fb and s = 500 GeV. The dot-dash, solid and dotted conbb tours are obtained at P = 1, 0.8, 0.5, respectively. The shaded region is derived from the combination of e+ e? → l+ l? and e+ e? → ? at P = 0.8. bb couplings, and/or the c quark, is similar to the one depicted in Fig. 1. Table 1 shows that the integrated observables σ+ and σ? are quite sensitive to the indirect Z ′ e?ects, with upper limits on the relevant products |Ge · Gf | ranging from α β 2.2 · 10?3 to 4.8 · 10?3 at the maximal planned value P = 0.8 of the electron longitudinal polarization. In most cases, the best sensitivity occurs for the ? ?nal state, while the bb worst one is for cc. Decreasing the electron polarization from P = 1 to P = 0.5 results in ? worsening the sensitivity by as much as 50%, depending on the ?nal fermion channel. Regarding the role of the assumed uncertainties on the observables under consideration, in the cases of e+ e? → l+ l? and e+ e? → ? the expected statistics are such that the bb 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 e+ e? → cc both ? statistical and systematic uncertainties are important. Moreover, as Eqs. (3) and (4) show, a further improvement on the sensitivity to the various Z ′ couplings in Table 1 would obtain if both initial e? and e+ longitudinal polarizations were available [11].

4

Resolving power and model identi?cation

If a Z ′ is indeed discovered, perhaps at a hadron machine, it becomes interesting to measure as accurately as possible its couplings and mass at the LC, and make tests of the various 7

Table 2: The values of the Z ′ leptonic and quark chiral couplings for typical models with MZ ′ = 1 TeV and expected 1-σ error bars from combined statistical and systematic uncertainties, as determined at the LC with Ec.m. = 0.5 TeV and P = 0.8. χ
e RZ ′

ψ ?0.264+0.052 ?0.043 0.264+0.042 ?0.052

η ?0.333+0.038 ?0.035 ?0.166+0.102 ?0.061 0.166+0.096 ?0.075

LR ?0.438+0.029 ?0.028 0.326+0.036 ?0.039

0.204+0.042 ?0.069

Le ′ Z

0.612+0.020 ?0.020

b RZ ′

?0.612+0.110 ?0.111 ?0.204+0.040 ?0.042 0.204+0.092 ?0.090

?0.264+0.111 ?0.172 0.264+0.158 ?0.103

?0.874+0.116 ?0.138 ?0.110+0.080 ?0.085 0.656+0.122 ?0.104

Lb ′ Z

0.333+0.230 ?0.168

c RZ ′

?0.264+0.138 ?0.207 0.264+0.222 ?0.149

?0.333+0.114 ?0.145 0.333+0.577 ?0.326

Lc ′ Z

?0.204+0.059 ?0.064

?0.110+0.106 ?0.134

extended gauge models. To assess the accuracy, the same procedure as in the previous section can be applied to the determination of Z ′ parameters by simply replacing the SM cross sections in Eqs. (19) and (23) by the ones expected for the ‘true’ values of the parameters (namely, the extended model ones), and evaluating the χ2 variation around them in terms of the expected uncertainty on the cross section.

4.1

Z ′ couplings to leptons

We now examine bounds on the Z ′ couplings for MZ ′ ?xed at some value. Starting from the leptonic process e+ e? → l+ l? , let us assume that a Z ′ signal is detected by means of the observables σ+ and σ? . Using Eqs. (21) and (22), the measurement of σ+ for the two values Pe = ±P will allow to extract σRR and σLL which, in turn, determine independent and e separate values for the right- and left-handed Z ′ couplings RZ ′ and Le ′ (we assume lepton Z 2 universality). The χ procedure determines the accuracy, or the ‘resolving power’ of such determinations given the expected experimental uncertainty (statistical plus systematic). In Table 2 we give the resolution on the Z ′ leptonic couplings for the typical model examples introduced in Section 2, with MZ ′ = 1 TeV. In this regard, one should recall that the two-fold ambiguity intrinsic in process (1) does not allow to distinguish the pair ′f ′e ′f ′e of values of (gα , gβ ) from the one (?gα , ?gβ ), see Eq. (17). Thus, the actual sign of the e couplings RZ ′ and Le ′ cannot be determined from the data (in Table 2 we have chosen Z 8

Figure 2: Resolution power at 95% C.L. for the absolute value of the leptonic Z ′ couplings, e |Le ′ | (a) and |RZ ′ | (b), as a function of MZ ′ , obtained from σLL and σRR , respectively, in Z process e+ e? → l+ l? . The error bars combine statistical and systematic uncertainties. Horizontal lines correspond to the values predicted by typical models. the signs dictated by the relevant models). In principle, the sign ambiguity of fermionic couplings might be resolved by considering other processes such as, e.g., e+ e? → W + W ? . Another interesting question is the potential of the leptonic process (1) to identify the Z ′ model underlying the measured signal, through the measurement of the helicity cross sections σRR and σLL . Such cross sections only depend on the relevant leptonic chiral coupling and on MZ ′ , so that such resolving power clearly depends on the actual value of the Z ′ mass. In Figs. 2a and 2b we show this dependence for the E6 and the LR models of interest here. In these ?gures, 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 ′ axis determines the discovery reach for the corresponding model: larger values of MZ ′ would determine a Z ′ signal smaller than the experimental uncertainty and, consequently, statistically invisible. Also, Figs. 2a and 2b show the complementary roles of σLL and σRR to set discovery limits: while σLL is ′ ′ mostly sensitive to the Zχ and has the smallest sensitivity to the Zη , σRR provides the best ′ ′ limit for the ZLR and the worst one for the Zχ . As Figs. 2a and 2b show, the di?erent models can be distinguished by means of σ± as long as the uncertainty of the coupling of one model does not overlap with the value predicted by the other model. Thus, the identi?cation power of the leptonic process (1) is determined by the minimum MZ ′ value at which such ‘confusion region’ starts. For example, Fig. 2a shows that the χ model cannot be distinguished from the LR, ψ and η 9

? Table 3: Identi?cation power of process e+ e? → ff at 95% C.L. expressed in terms of MZ ′ (in GeV) for typical E6 and LR models at Ec.m. = 0.5 TeV and Lint = 25 f b?1 for each value of the electron polarization, Pe = ±0.8. σRR σLL e e →l l ψ η χ LR + ? e e →? bb ψ η χ LR + ? e e → cc ? ψ η χ LR
+ ? + ?

ψ — 950 830 1160 ψ — 700 1175 1210 ψ — 880 760 1050

η χ LR 960 830 1470 — 970 1210 1165 — 1615 1220 970 — η χ LR 725 1180 2345 — 1210 2410 1100 — 2130 1100 1540 — η χ LR 865 800 1740 — 880 1580 1050 — 1840 1280 880 —

ψ η χ LR — 840 2270 920 960 — 2420 1220 1170 840 — 1400 915 840 2165 — ψ η χ LR — 710 1120 940 750 — 1250 750 1130 1140 — 950 940 760 1370 — ψ η χ LR — 620 935 800 645 — 1035 665 935 940 — 810 780 685 1135 —

models at Z ′ masses larger than 2165 GeV, 2270 GeV and 2420 GeV, respectively. The identi?cation power for the typical models are indicated in Figs. 2a and 2b by the symbols circle, diamond, square and triangle. The corresponding MZ ′ values at 95% C.L. for the typical E6 and LR models are listed in Table 3, where the Z ′ models listed in ?rst columns should be distinguished from the ones listed in the ?rst row assumed to be the origin of the observed Z ′ signal. For this reason Table 3 is not symmetric. Analogous considerations hold also for σLR and σRL . These cross sections give quale itatively similar results for the product Le ′ RZ ′ , but with weaker constraints because of Z smaller sensitivity.

4.2

Z ′ couplings to quarks

In the case of process (1) with q q pair production (with q = c, b), the analysis is compli? ′e cated by the fact that the relevant helicity amplitudes depend on three parameters (gα , ′q gβ and MZ ′ ) instead of two. Nevertheless, there is still some possibility to derive general information on the Z ′ chiral couplings to quarks. Firstly, by the numerical procedure introduced above one can determine from the measured cross section the products of electrons and ?nal state quark couplings of the Z ′ , from which one derives allowed regions to such q couplings in the independent, two-dimensional, planes (Le ′ ,Lq ′ ) and (Le ′ ,RZ ′ ). The former Z Z Z regions are determined through σLL , and the latter ones through σLR . As an illustrative example, in Fig. 3 we depict the bounds from the process e+ e? → ? in the (Le ′ ,Lb ′ ) and bb Z Z ′ b e ′ = 1 TeV. Taking into account the (LZ ′ ,RZ ′ ) planes for the Z of the χ model, with MZ above mentioned two-fold ambiguity, the allowed regions are the ones included within the 10

Figure 3: Allowed bounds at 95% C.L. on Z ′ couplings with MZ ′ = 1 TeV (χ model) b in the two-dimension planes (Le ′ ,Lb ′ ) and (Le ′ ,RZ ′ ) obtained from helicity cross sections Z Z Z σLL (solid lines) and σLR (dashed lines), respectively. The shaded and hatched regions are derived from the combination of e+ e? → l+ l? and e+ e? → ? processes. Two allowed bb regions for each helicity cross section correspond to the two-fold ambiguity discussed in text. two sets of hyperbolic contours in the upper-left and in the lower-right corners of Fig. 3. Then, to get ?nite regions for the quark couplings, one must combine the hyperbolic regions so obtained with the determinations of the leptonic Z ′ couplings from the leptonic process (1), represented by the two vertical strips. The corresponding shaded areas represent the b determinations of Lb ′ , while the hatched areas are the determinations of RZ ′ . Notice that, Z 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 b e e of couplings (RZ ′ ,Lb ′ ) and (RZ ′ ,RZ ′ ). However, as observed with regard to the previous Z analysis of the leptonic process, the sensitivity to the right-handed electron coupling turns out to be smaller than for Le ′ , so that the corresponding constraints are weaker. Z The determinations of the Z ′ couplings with the c and b quarks for the typical E6 and LR models with MZ ′ = 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. 2a and 2b, we depict in Figs. 4a and 4b the di?erent models identi?cation power as a function of MZ ′ , for the reaction e+ e? → ? as a representative example. The model identi?cation power of the ? and cc bb bb ? pair production processes are reported in Table 3.

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b e Figure 4: Resolution power at 95% C.L. for |Le ′ Lb ′ |1/2 (a) and |RZ ′ RZ ′ |1/2 (b) as a Z Z + ? bb. function of MZ ′ obtained from σLL and σRR , respectively, in process e e → ? The error bars combine statistical and systematic errors. Horizontal lines correspond to the values predicted by typical models.

5

Concluding remarks

We brie?y summarize our ?ndings concerning the Z ′ discovery limits and the models identi?cation power of process (1) via the separate measurement of the helicity cross sections √ σαβ at the LC, with s = 0.5 TeV and Lint = 25 f b?1 for each value Pe = ±P the electron longitudinal polarization. Given the present experimental lower limits on MZ ′ , only indirect e?ects of the Z ′ can be studied at the LC. In general, the helicity cross sections allow to extract separate, and model-indpendent, information on the individual ‘e?ective’ Z ′ couplings (Ge · Gf ). As depending on the minimal number of free parameters, they may α β be expected to show some convenience with respect to other observables in an analysis of the experimental data based on a χ2 procedure. In the case of no observed signal, i.e., no deviation of σαβ from the SM prediction within the experimental accuracy, one can directly obtain model-independent bounds on the leptonic chiral couplings of the Z ′ from e+ e? → l+ l? and on the products of couplings Ge · Gq from e+ e? → q q (with l = ?, τ and q = c, b). From the numerical point of view, ? α β σαβ are found to just have a complementary role with respect to other observables like σ and AFB . In the case Z ′ manifestations are observed as deviations from the SM, with MZ ′ of the order of 1 TeV, the role of σαβ is more interesting, specially as regards the problem of identifying the various models as potential sources of such non-standard e?ects. Indeed, in principle, they provide a unique possibility to disentangle and extract numerical values for 12

the chiral couplings of the Z ′ in a general way (modulo the aforementioned sign ambiguity), avoiding the danger of cancellations, so that Z ′ model predictions can be tested. Data analyses with other observables may involve combinations of di?erent coupling constants and need some assumption to reduce the number of independent parameters in the χ2 procedure. In particular, by the analysis combining σαβ (l+ l? ) and σαβ (?q) one can obtain q ′ information of the Z 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 the 3 ? 60% for MZ ′ = 1 TeV. Of course, the considerations above hold only in√ case where ′ the Z signal is seen in all observables. Finally, one can notice that for s ? MZ ′ the energy-dependence of the deviations ?σαβ is determined by the SM and that, in particular, the de?nite sign ?σαα (l+ l? ) < 0 (α = L, R) is typical of the Z ′ . This property might be helpful in order to identify the Z ′ as the source of observed deviations from the SM in process (1).

Acknowledgements
AAB and AAP gratefully acknowledge the support of the University of Trieste.

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