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Tracing CP violation in the production of top quark pairs by multiple TeV proton-proton col


SLAC–PUB–6403 PITHA 93/43 December 1993 T/E hep-ph/9312210

arXiv:hep-ph/9312210v1 2 Dec 1993

Tracing CP violation in the production of top quark pairs by multiple TeV proton–proton collisions
Werner Bernreuther Institut f. Theoretische Physik Physikzentrum, RWTH Aachen 52056 Aachen, Germany Arnd Brandenburg
2

1

Stanford Linear Accelerator Center, P.O. Box 4349 Stanford, California 94309, USA

Abstract We investigate the possibilities of searching for non-standard CP violation in pp → ?X at multiple TeV collision energies. A general kinematic analysis of the underlying tt ? ? partonic production processes gg → tt and q q → tt in terms of their density matrices ? is given. We evaluate the CP-violating parts of these matrices in two-Higgs doublet extensions of the standard model (SM) and give results for CP asymmetries at the parton level. We show that these asymmetries can be traced by measuring suitable observables constructed from energies and momenta of the decay products of t and ? t. We ?nd CP-violating e?ects to be of the order of 10?3 and show that possible contaminations induced by SM interactions are savely below the expected signals.

1 2

Work supported by the Department of Energy, contract DE-AC03-76SF00515 Supported by Max Kade Foundation

1

Introduction

A high energy and high luminosity proton-proton collider, such as the planned Large Hadron Collider (LHC) at CERN, would be capable of producing millions of top and antitop quarks. This would o?er the unique possiblity to explore in detail the physics of these quarks – which have not been discovered yet, but for whose existence there is indirect evidence [1]. Speci?cally, since the top is known to be heavy, mt > 113 GeV ? [2], precision studies based on large samples of tt events may serve as a probe, through sizeable top-Yukawa couplings, to the electroweak symmetry breaking sector (Higgs sector for short). This sector may have a richer structure than the one conceived in the standard model (SM) — as is the case in many of its extensions. As a consequence a number of new phenomena may exist. A particularly intriguing one is a new “source” of CP violation provided by the Higgs sector1 which is unrelated to the KobayashiMaskawa phase [3]. This is possible already in the two-Higgs doublet extensions of the SM [5]–[7]: here neutral Higgs boson exchange leads to CP-violating e?ects in fermionic amplitudes, and these e?ects would show up most pronouncedly in reactions involving top quarks [8]–[11]. The subject of this paper is to investigate in detail2 the ? manifestation and the magnitude of neutral Higgs particle CP violation in pp → ttX. (For other studies on CP violation in top quark production and decay see [12]–[25].) The outline of our paper is as follows: In section 2 we give a general kinematic ? ? analysis of the reactions gg → tt and q q → tt which are the leading partonic processes ? ?X. We study these reactions in terms of their production density matrices in pp → tt and describe the properties of these matrices under various symmetry transformations including CP transformations. In section 3 we evaluate the CP-violating parts of the density matrices in a speci?c model, namely the two-Higgs doublet extensions of the SM with CP-nonconserving neutral Higgs boson exchange. Correlations which are sensitive to CP violation at the parton level are identi?ed and results for their expectation values are presented. In section 4 we show that these CP asymmetries ? can be traced in pp → ttX by looking at simple observables which involve energies ? and/or momenta of the decay products of t and t. Moreover, possible contaminations by CP-conserving interactions are discussed and shown to be much smaller than the expected signals. In an appendix we list the analytic results of our calculations.

This source was shown to be of interest in attempts to explain the cosmological baryon asymmetry [4]. 2 A short account of our work was given in [11].

1

1

2

? Production density matrices for gg → tt and q q → ? ? tt

Because a heavy top has an extremely short lifetime (τt < 10?23 s if mt > 100 GeV), ? the polarization of and spin–spin correlations between t and t are not severely diluted by hadronization [34]. These are “good” observables in the sense that e?ects involving ? the spins of t and t can be treated perturbatively. Therefore we will discuss the reac? tion pp → ttX, respectively the underlying partonic processes in terms of production ? density matrices. We will consider only unpolarized pp collisions. At LHC energies tt pairs are produced mainly by gluon gluon fusion. This reaction dominates over quark ? antiquark annihilation into tt. We will ?rst discuss the (unnormalized) production ? density matrix for the reaction g(p1 ) + g(p2 ) → t(k1 ) + t(k2 ) in the gluon–gluon center of mass system. It is de?ned by
g Rα1 α2 , β1 β2 (p, k) ?1 = Ng colors,gluon spin

? t(k1 , α1 ), t(k2 , β1 )|T |g(p1 ), g(p2) ? t(k1 , α2 ), t(k2 , β2 )|T |g(p1), g(p2 )

?

(2.1)

where α, β are spin indices, p = p1 , k = k1 and Ng = 256. We sum here over the gluon spins and colors since we are only interested in analysing the polarizations of ? the t and t and their spin–spin correlations. The matrix Rg can be decomposed in ? the spin spaces of t and t as follows:
g Rg = Ag 1l ? 1l + Big+ σ i ? 1l + Big? 1l ? σ i + Cij σ i ? σ j .

(2.2)

The ?rst (second) factor in the tensor products of the 2 × 2 unit matrix 1l and of the ? Pauli matrices σ i refers to the t (t) spin space. Because of rotational invariance, the functions Big± and Cij can be further decomposed: Big± = ? b± pi + b± ki + b± ni g1 ? g2 g3 ? ? Cij = cg0 δij + ?ij? (cg1 p? + cg2 k? + cg3 n? ) ? ? ?? +cg4 pi pj + cg5 ki kj + cg6 (?i kj + pj ki ) ?? p? ?? ?? ? ? +cg7 (?i nj + pj ni ) + cg8 (ki nj + kj ni ). p? ?? (2.3)

Here the hat denotes a unit vector and n = p×k. The structure functions Ag , b± and gi 2 ? ? cgi depend only on s = (p1 + p2 ) and on the cosine of the scattering angle, z = p · k. ? g Next we discuss the properties of R under various symmetry transformations. Since the initial gg state is Bose symmetric, Rg must satisfy 2

Rg (?p, k) = Rg (p, k).

(2.4)

The initial gg state, when averaged over colors and spins, is a CP eigenstate in its center of mass system. It is therefore possible to classify the individual terms in Rg according to their CP transformation properties. If the interactions were CPinvariant, the matrix Rg would have to satisfy
g Rβ1 β2 , α1 α2 (p, k) g = Rα1 α2 , β1 β2 (p, k).

(2.5)

In table 1 we give a complete list of the transformation properties of the structure functions under P, CP, and exchange of the initial gluons (“Bose”). It is also instructive to collect the properties of these functions under time reversal (T) and CPT transformations neglecting, just for this purpose, non-hermitean parts of the scattering matrix. To give an example, table 1 is then to be read as follows: for a T-invariant interaction one has b± (z) = ?b± (z), i.e., b± = 0 only at the Born level, whereas at g3 g3 g3 higher orders absorptive parts render this function non-zero. Because of Bose symmetry, the structure functions Ag , b± , cg0 , cg2 , cg4 , cg5 and cg7 g2 are even functions of z, the other functions are odd in z. The contributions to Rg can be decomposed into a CP–even and a CP–odd part:
g g Rg = Reven + RCP .

(2.6)

g As can be read o? from table 1, the CP–even term Reven in general has the following structure:

g ? Reven = Ag 1l ? 1l + (beven pi + beven ki + beven ni )(σ i ? 1l + 1l ? σ i ) g2 g3 ? g1 ? ?? +(cg0 δij + cg4 pi pj + cg5 kikj + cg6 (?i kj + pj ki ) ?? p? ??

?? ? ? +cg7 (?i nj + pj ni ) + cg8 (ki nj + kj ni ))σ i ? σ j . p? ??

(2.7)

Nonzero beven , beven , cg7 , cg8 can be induced only by parity-violating interactions, g1 g2 cg7 , cg8 need in addition absorptive parts in the scattering amplitude when the interactions are CPT-invariant. The structure function beven can only get contributions g3 from absorptive parts induced by parity-invariant interactions. g The CP–odd term RCP reads
g ? RCP = (bCP pi + bCP ki + bCP ni )(σ i ? 1l ? 1l ? σ i ) g1 ? g2 g3 ? ? ? +?ijk (cg1 pi + cg2 ki + cg3 ni )σ j ? σ k . ?

(2.8)

3

CP-violating interactions which are also parity-violating can give contributions to bCP , bCP , cg1 , cg2 . Nonzero bCP , bCP require in addition absorptive parts. C- and CPg1 g2 g1 g2 violating interactions can induce non–vanishing structure functions bCP , cg3 , where g3 cg3 = 0 requires in addition absorptive parts. The above discussion of the transformation properties of the structure functions holds to all orders of pertubation theory. ? The production density matrix Rq for q q → tt is de?ned in complete analogy to ? (2.1) as
q Rα1 α2 , β1 β2 (p, k) ?1 = Nq colors,q q spins ?

? t(k1 , α1 ), t(k2 , β1 )|T |q(p1 ), q(p2 )

?

? t(k1 , α2 ), t(k2 , β2 )|T |q(p1 ), q(p2 ) ,

(2.9)

? where Nq = 36. The decomposition of Rq in the spin spaces of t and t is exactly the g same as for R (equ. (2.2) , (2.3)) as is the splitting into CP–even and odd terms (equ. (2.6)– (2.8)). The transformation properties of the structure functions Aq , . . . , cq8 of Rq are the same as the respective ones for Ag , . . . , cg8 of Rg given in table 1. Thus all conclusions derived from these transformation properties — except those from Bose symmetry, of course — are also valid for the structure functions of Rq .

3

CP violation and density matrices in two-Higgs doublet models

Up to now our discussion has been independent of any speci?c model. Su?ce it to say that the Kobayashi-Maskawa mechanism of CP violation [3] induces only tiny e?ects in the ?avor-diagonal reactions of sect. 2. In the following we will concentrate on CP-violating e?ects generated by two–Higgs doublet extensions of the SM with CP violation in the scalar potential [6]. We brie?y recall the features of these models relevant for us. CP violation in the scalar potential induces mixing of CP–even and –odd scalars, thus leading to three physical mass eigenstates |?j (j = 1, 2, 3) with no de?nite CP parity. That means, these bosons couple both to scalar and pseudoscalar fermionic currents. For the top quark these couplings are (in the notation of [8]): √ LY = ?( 2GF )1/2
3

? ? ? (ajt mt tt + ajt mt tiγ5 t)?j ,

(3.1)

j=1

where GF is Fermi’s constant, mt is the top mass, ajt = d2j / sin β, ajt = ?d3j cot β, ? (3.2)

tan β = v2 /v1 is the ratio of vacuum expectation values of the two doublets, and d2j , d3j are the matrix elements of a 3 × 3 orthogonal matrix which describes the 4

mixing of the neutral states [8]. In the following we assume that the couplings and masses of ?2,3 are such that their e?ect on all quantities discussed below is negligible. Then the measure of CP violation generated by ? ≡ ?1 exchange in ?avor–diagonal ? ? ? reactions like gg → tt, q q → tt is γCP ≡ ?a? = d21 d31 cot β/ sin β, a (3.3)

where we have put a = a1t , a = a1t . So far, data from low energy phenomenology, ? ? in particular the experimental upper bounds on the electric dipole moments of the neutron [26] and of the electron [27] do not severely constrain this parameter: γCP may be of order one. We note here that the couplings of the ?j to quarks and leptons induce CP violation already at the Born level. The especially interesting case of a ? Higgs boson ? decaying into tt was shown in [11] to lead to CP-violating spin–spin correlations which may be as large as 0.5. (For other discussions of the CP properties of neutral Higgs bosons see [28]–[33].) g q We will now discuss the structure of the matrices RCP and RCP in these models. ? ? The Higgs boson contributions to the processes gg → tt and q q → tt discussed in ? section 2 are shown, together with the leading SM diagrams, in ?gs. 1, 2. Since the CP-nonconserving neutral Higgs exchange is, in particular, parity-violating, the relations g(q) g(q) RCP (?p, ?k) = ?RCP (p, k) (3.4) hold as long as RCP results from interference of these Higgs exchange amplitudes with amplitudes from parity-invariant interactions. This forces bCP , cg3 and bCP , cq3 g3 q3 to be zero in these models. Furthermore, the virtual intermediate gluon produced by annihilation of unpolarized q and q cannot have a vector polarization. Thus the ? contributions of ?g. 2 to Rq are invariant with respect to the substitution p = p1 → g q ?p. Hence the structures of RCP and RCP are the same; the functions bCP , cg1 , bCP , q1 g1 cq1 of eqn. (2.8) are odd under z → ?z, whereas bCP , cg2 , bCP , cq2 are even functions g2 q2 of z. The explicit results for the matrices Rg and Rq evaluated from the diagrams of ?g. 1, 2, respectively, are given in the appendix. The width of ? must be taken into account in the calculation of Rg if ? > 2mt , since in that case the contribution from ?g. 1h can become resonant. Because for ? > 2mt the width of ? is not very small as compared to its mass it is important to note that the narrow width approximation cannot be applied in this case. In view of the above discussion it is now very easy to identify the correlations at the parton level which trace the various CP–odd parts of the production density matrices. The expectation value of an observable O for the respective parton reactions is de?ned as O i=
1 ?1

dztr(Ri O) 1 4 ?1 dzAi 5

(i = g, q)

(3.5)

Contributions of the functions bCP , bCP are picked up by taking expectation values g1,g2 q1,q2 of ? k · (s+ ? s? )fe (z), or ? or linear combinations thereof, where s+ , s? are the spin operators of t and t, respectively, fe (z) is an even function of z and fo (z) is odd in z. One has for example ? k · (s+ ? s? )
g

(3.6) (3.7)

? p · (s+ ? s? )fo (z),

=

4

1 ?1

dz(zbCP + bCP ) g1 g2 , 1 g 4 ?1 dzA

(3.8)

? and likewise for k · (s+ ? s? ) q . ? The result for the basic longitudinal polarization asymmetry k · (s+ ? s? ) g is plotted in ?g. 3 as a function of the parton CM energy for mt = 150 GeV and two values of the Higgs boson mass: The dashed curve corresponds to m? = 100 GeV and γCP = 1. For m? of the order of 2mt or larger, the shape of the resulting graph depends, for ?xed m? , on the strength of the Higgs couplings a, a and on ? + ? the couplings to W W , ZZ determining the width of the ?. (See eqn. (A.10) for details.) For the solid line we have chosen m? = 350 GeV, |a| = |?| = γCP = 1 a ? and Γ? = 47 GeV. The asymmetry k · (s? ? s+ ) corresponds to the asymmetry ?L ) ? N(tR tR )]/(all tt) studied in [10]. We reproduce the numerical ? ? ?NLR = [N(tL t √ results of [10] for ?NLR if we neglect Γ? and use γCP = 1/ 2 which corresponds to √ the parameter Im(A2 ) = 2 used in [10]. The functions cg1,g2 , cq1,q2 generate nonzero expectation values of the triple product correlations ? k · (s+ × s? )he (z), (3.9) ? p · (s+ × s? )ho (z), (3.10)

and of their linear combinations, where he and ho are even and odd functions of z, respectively. For example, ? k · (s+ × s? )
g

=

2

1 ?1

dz(zcg1 + cg2 ) . 1 4 ?1 dzAg

(3.11)

In ?g. 4 we plot this basic CP–odd and T–odd spin–spin correlation with the same choice of parameters as in ?g. 3. It reaches values of up to about two percent. ? For completeness, we show in ?gs. 5 and 6 the expectation values k · (s+ ? s? ) q ? and k · (s+ × s? ) q , respectively, again for the same choice of parameters as in ?g. 3. Here the CP asymmetries get smaller with growing Higgs masses.

6

4

? The CP-violating spin-momentum correlations for t and t of the previous section must ? decay. In this section we discuss a be traced in the ?nal states into which t and t few observables which allow to do this. The charged lepton from t → W b → ?+ ν? b is an e?cient analyzer of the top spin [35]. We will therefore conider only decay chains where at least one of the top quarks decays semileptonically. We shall use the SM decay density matrices as given in [16, 18]. ? Observables in pp → ttX cannot be classi?ed as being even or odd with respect to CP, because the initial state is not a CP eigenstate. However, they can be classi?ed as being T–even or T–odd (i.e. even or odd under re?ection of momenta and spins). Their expectation values will in general be contaminated by contributions from CPconserving interactions. ? The asymmetries in the t and t polarizations in the production plane, as given in eqns. (3.6) and (3.7), translate into T–even observables formed by energies and/or momenta of the ?nal states. As an example, we have investigated the expectation values of the following two observables (another one was given in [10]): A1 = E+ ? E? A2 = kt · ?+ ? kt · ?? . ? (4.1)

? CP observables for pp → ttX

(4.2) ? Here E± , ?± are the energies and momenta of the leptons in t → ?+ ν? b and t → ?? ν?? ?b in the laboratory frame and kt(t) is the top (antitop) momentum in this system. ? To measure A2 , one has to select events where the t decays leptonically and the ? ? t hadronically, which in principle allow to reconstruct the t momentum [36], and vice versa. We will give explicit expressions for the expectation values of A1 and A2 below when we discuss contaminations by CP-conserving interactions. In calculating these expectation values we have used the narrow width approximation for the top: because the top width is much smaller than its mass (in view of the experimental upper bound on mt which is of the order of 200 GeV), the approximation of the on– ? shell production of t and t followed by their weak decays yields a good description of the reactions considered here. We also neglected CP violation in the decays of ? t and t (for comments on this, see [11]). For the parton distributions entering the calculation of expectation values in pp collisions we have used the parametrization of [37]. In order to assess the statistical sensitivity of the observables A1 and A2 , we have computed the signal–to–noise ratios Ai /?Ai (i = 1,√ for Higgs masses 2) √ 100 GeV ≤ m? ≤ 450 GeV both for s = 15 TeV (LHC) and s = 40 TeV. Here A2 ? Ai 2 denotes the width of the distribution of Ai . We present our ?Ai = i results in ?gs. 7 and 8 for the same parameter set as used in calculating the partonic asymmetries, that is, a = ?? = 1, gV V = 1 (for the de?nition of gV V see appendix, a ? ? eqn. (A.9)). We integrate here over the whole phase space. Both observables have signal–to–noise ratios of order 10?3. LHC o?ers larger e?ects due to the fact that √ ?A1,2 is larger for s= 40 TeV. 7

? If both t and t decay leptonically, one can look at the T–odd observable ? T2 = (b ? b) · (?+ × ?? ) (4.3)

? where b, b denote the momenta of the b and ? jets in the laboratory frame. (This b observable was also discussed in [11].) The expectation value of T2 traces the spin– spin correlations of (3.9) and (3.10). In ?g. 9 we show the signal–to–noise ratio as a function of the Higgs mass for this observable, again with the same choice of parameters as for A1,2 . The e?ect is also of the order of 10?3 . We will now discuss in some detail possible contaminations of the observables A1,2 and T2 due to CP-conserving interactions. Such contaminations arise in particular because the pp initial state is not a CP eigenstate. One can give general arguments why these contaminations should be small. Most importantly, the dominant subprocess is gluon fusion which does not induce any CP-conserving contributions to our observables (cf. [11] and below). Furthermore, T–odd observables like T2 do not receive contributions from CP-invariant interactions at the Born level but only from 3 absorptive parts. The main background in this case comes from order αs and order 2 ? αs αweak absorptive parts in q q → tt which generate nonzero functions beven in Rq . ? 3 However, numerical simulations show that these contributions are smaller than 10?6 , i.e. about three orders of magnitude smaller than the signal shown in ?g. 9. Potentially more dangerous are CP–even contributions to A1 and A2 , because, as will be shown below, they can already be generated by weak interactions at the Born level. Integrating over the whole phase space we can actually give explicit analytic formulae for the expectation values of A1 and A2 in terms of the structure functions. This is very illuminating for identifying possible contaminations. We have carried out our calculations within the naive parton model (which neglects intrinsic transverse momenta of the incoming partons) and restricted ourselves again to gg and q q initial ? states. Furthermore, we have used the narrow width approximation desribed above ? and have taken into account only the SM decays of t and t. Then we ?nd: 1 1 g(m2 /m2 ) W t σ 2s 4π p p 1 Ng (x1 ) 1 Ng (x2 ) 1 x1 + x2 E1 CP dx1 dx2 dzβ √ (zbg1 + bCP ) g2 x1 x2 2 x1 x2 3 0 0 ?1 p p 1 Nq (x1 ) 1 Nq (x2 ) 1 x1 + x2 E1 CP ? dx2 dzβ √ (zbq1 + bCP ) +2 dx1 q2 x1 x2 2 x1 x2 3 0 ?1 0 1 x1 ? x2 E1 βzAq + ([(1 ? z 2 )mt + z 2 E1 ]beven + E1 zbeven ) , (4.4) + √ q1 q2 2 x1 x2 3

A1

=

8

A2

=

1 1 g(m2 /m2 ) W t σ 2s 4π p p 1 Ng (x1 ) 1 Ng (x2 ) 1 E 2β dx1 dx2 dzβ ? 1 (zbCP + bCP ) g1 g2 x1 x2 3 0 0 ?1 (x1 ? x2 )2 E1 β + (1 ? z 2 )((E1 ? mt )zbCP + E1 bCP ) g1 g2 4x1 x2 3 p p 1 Nq (x1 ) 1 Nq (x2 ) 1 E2β ? +2 dx1 dx2 dzβ ? 1 (zbCP + bCP ) q1 q2 x1 x2 3 0 0 ?1 (x1 ? x2 )2 E1 β (1 ? z 2 )[(E1 ? mt )zbCP + E1 bCP ] + q1 q2 4x1 x2 3 x2 ? x2 mt 2 + 1 ([(1 ? z 2 )E1 + z 2 mt ]beven + mt zbeven ) . (4.5) q1 q2 4x1 x2 3

? Here σ is the total cross section for pp → ttX, s is the pp collision energy squared, p 2 p g(y) = (1 + 2y + 3y )/(2 + 4y), Ng , Nq(?) denote the gluon and quark (antiquark) q distribution functions of the proton, E1 is the energy of the top quark in the partonic 2 CM and β = (1 ? m2 /E1 )1/2 . t Equations (4.4) and (4.5) exhibit several interesting features: - One can see explicitly that gluon fusion generates no CP-even contributions to the observables. - Quark–antiquark annihilation produces several contaminations: In A1 a term ? zAq (z) appears which, after integrating over z, is nonzero only if Aq (z) has a part ? which is odd in z; that is, if q q → tt has a forward–backward asymmetry. Such an ? 3 asymmetry is induced in order αs . (In [10] this potential source of contaminations was discussed.) Possibly more important are the terms beven and beven which appear q1 q2 in both expectation values above because these terms can be generated at the Born ? level via q q → Z → tt. We calculated their contributions and found that for both ? observables they are suppressed by more than two orders of magnitude in comparison to the signals shown in ?gs. 7 and 8. A future multiple TeV and high luminosity collider like the LHC has the potential ? of producing more than 107 tt pairs. If it were for statistics alone detection of e?ects of a few permil which we found might be feasible. More detailed (Monte Carlo) studies including judicious choices of phase space cuts are required in order to explore the possiblity of enhancing the signals by some factor. A crucial issue will eventually be whether detector e?ects can be kept at the level of 10?3 .

9

5

Conclusions

In this article we have studied the possibility of detecting CP violation in top quark pair production at future hadron colliders. We have given a general kinematic analysis of the underlying dominant partonic subprocesses and identi?ed the relevant CP asymmetries at the parton level. We have further computed these asymmetries in two-Higgs doublet extensions of the SM where CP violation is generated through neutral Higgs boson exchange. Whereas at the parton level these models can induce asymmetries of the order of a few percent, realistic observables built up from energies and/or momenta of the ?nal states into which the top quarks decay give signal–to– noise ratios of up to a few ×10?3 . Contaminations by CP-conserving interactions were shown to be much smaller than the signals. Since the issue of CP violation is of fundamental interest detailed investigations of the experimental feasibility of an observation of these e?ects would certainly be worthwhile.

Acknowledgments A. B. would like to thank the SLAC theory group for the hospitality extended to him.

10

Appendix
? In this appendix we list our analytic results for the structure functions of the tt spin g density matrix R de?ned in equ. (2.1)–(2.3) and decomposed into CP–even and CP–odd parts in equ. (2.6)–(2.8) and also the corresponding functions in Rq de?ned in (2.9). All calculations are carried out in the two-Higgs doublet extensions of the SM described in section 3. The relevant Feynman diagrams are shown in ?gures 1a–h ? ? for the process gg → tt and in ?gures 2a,b for q q → tt. ? q R is obtained very easily: The CP–even part is determined to good approximation q by the Born diagram ?g. 2a, whereas RCP results from the interference of ?g. 2b (with couplings a? = ?γCP ) with ?g. 2a. The nonzero CP–even structure functions a of Rq read: Aq = cq0 = cq4 = cq5 = cq6 =
4 gs (β 2 (z 2 ? 1) + 2) 18 4 gs β 2 (z 2 ? 1) 18 4 gs 9 4 2 gs β 2 β 2 z 2 E1 +1 9 (E1 + mt )2 4 ?gs β 2 zE1 . 9(E1 + mt )

(A.1)

Here and in the following gs denotes the strong coupling constant, E1 is the energy 2 of the top in the CM system of the incoming partons and β = 1 ? m2 /E1 . Recall t ? ? that z = p · k. The CP–odd contributions are: √ 4 ?m3 2GF γCP 4gs E1 βz t = ImG(?), s 2 9 √ 8π 4 m3 2GF γCP 4gs E1 β (z 2 ? 1)β 2E1 t + 1 ImG(?), s bCP = q2 8π 2 9 E1 + mt √ 4 m3 2GF γCP 4gs E1 βz t ReG(?), s cq1 = 8π 2 9 √ 4 ?m3 2GF γCP 4gs E1 β (z 2 ? 1)β 2 E1 t cq2 = + 1 ReG(?). s 8π 2 9 E1 + mt Here GF is Fermi’s constant and bCP q1 G(?) = s ?(m2 C0 (?, m2 , m2 , m2 ) + B0 (?, m2 , m2 ) ? B0 (m2 , m2 , m2 )) s ? t t s t t ? t ? t , sβ 2 ? 11 (A.2) (A.3) (A.4) (A.5)

(A.6)

where C0 (?, m2 , m2 , m2 ) = s ? t t 1 1 1 d4 l . (A.7) 2 l2 ? m2 + i? (l + k )2 ? m2 + i? (l + k ? p ? p )2 ? m2 + i? iπ 1 1 1 2 t t ?

is a standard three-point scalar integral which can be reduced to dilogarithms [38]. We note here that for the models of section 3 all structure functions are ultraviolet ?nite. In particular, the scalar two-point functions B0 show up in all our results only as di?erences of the form
2 2 B0 (q1 , m2 , m2 ) ? B0 (q2 , m2 , m2 ) = 1 t 2 t 2 2 1 x2 q1 + x(m2 ? m2 ? q1 ) + m2 ? i? t 1 1 dxlog 2 2 ? . 2 x q2 + x(m2 ? m2 ? q2 ) + m2 ? i? 0 t 2 2

(A.8)

This completes our results for the matrix Rq . The computation of Rg is more involved since the contribution of ?g. 1h becomes resonant if m? > 2mt . (Fig. 1h actually represents four amplitudes: two CP– ? conserving ones with couplings a2 and a2 , respectively, and two CP–violating ones with couplings a?.) The width of ? must therefore be taken into account in the ? a ? propagator. We compute Γ? by summing the partial widths for ? → W + W ? , ZZ, tt in the two–Higgs doublet model which contains (3.1). At the Born level only the CP = +1 component of ? couples to W + W ? and ZZ. The couplings are given by the respective SM couplings times the factor gV V = (d11 cos β + d21 sin β). ? Explicitly, Γ? = ΓW + ΓZ + Γt (A.9)

√ gV V m3 2GF βW 2 ?2 m4 ? ΓW = Θ(m? ? 2mW ) βW + 12 W 16π m4 ? √ 2 3 gV V m? 2GF βZ 2 ? m4 βZ + 12 Z ΓZ = Θ(m? ? 2mZ ) 8π m4 ? √ 2 3m? mt 2GF βt 2 2 Γt = Θ(m? ? 2mt ) (βt a + a2 ). ? (A.10) 8π Here we have used the notation βW,Z,t = (1 ? 4m2 /m2 )1/2 . In order to incorporate W,Z,t ? g the resonance region we have determined Reven from the squared Born amplitudes ?gs. 1a, 1b, the interference of ?g. 1a with the CP–even amplitudes of ?g. 1h, and the squared amplitudes of ?g. 1h. We denote the Born contributions by a lower index “Born” and the other two contributions by a lower index “resonance” in the g following. The results for the nonzero structure functions of Reven are: 12

Ag = Ag + Ag Born resonance 4 gs (7 + 9β 2 z 2 ) 4 2 2 4 (E1 + 2E1 m2 ? 2m4 ? 2β 2 E1 m2 z 2 ? β 4 E1 z 4 ) Ag t t t Born = 4 2 z 2 )2 192E1 (?1 + β √ 4 gs 1 m3 2GF mt t g Aresonance = ? 2 )2 + Γ2 m2 2 (? ? m? s 16 8π ?1 + β 2 z 2 ? ? +ImC0 (?, m2 , m2 , m2 )Γ? m? ] ? 4a2 β 2 (? ? m2 ) s t t t s ? √ 2 3 3 mt 2GF s3 (a2 a2 β 2 + a4 )|C0 (?, m2 , m2 , m2 )|2 ? ? ? s t t t + 2 32 8π +?(a4 β 2 + a2 a2 )|2 ? sβ 2 C0 (?, m2 , m2 , m2 )|2 , s ? ? s t t t (A.11) 2?(a2 β 4 + a2 )[ReC0 (?, m2 , m2 , m2 )(? ? m2 ) s ? s t t t s ?

cg0 = cg0,Born + cg0,resonance 4 ?gs (7 + 9β 2z 2 ) 2 cg0,Born = (E 4 ? 2E1 m2 + 2m4 t t 4 192E1 (?1 + β 2 z 2 )2 1 4 2 4 ?2β 2 E1 z 2 + 2β 2 E1 m2 z 2 + β 4 E1 z 4 ) t √ 4 gs 1 m3 2GF mt t cg0,resonance = ? 2 )2 + Γ2 m2 2 (? ? m? s 16 8π ?1 + β 2 z 2 ? ? s +ImC0 (?, m2 , m2 , m2 )Γ? m? ] ? 4a2 β 2 (? ? m2 ) s t t t ? √ 2 3 m3 2GF t + s3 (a2 a2 β 2 ? a4 )|C0 (?, m2 , m2 , m2 )|2 ? ? ? s t t t 32 8π 2 +?(a4 β 2 ? a2 a2 )|2 ? sβ 2 C0 (?, m2 , m2 , m2 )|2 , s ? ? s t t t
4 gs β 2 (7 + 9β 2 z 2 ) (1 ? z 2 ) , 32(?1 + β 2 z 2 )2

2?(a2 β 4 ? a2 )[ReC0 (?, m2 , m2 , m2 )(? ? m2 ) s ? s t t t s ?

(A.12)

cg4 =

(A.13)

cg5 = cg5,Born + cg5,resonance 4 gs β 2 (7 + 9β 2 z 2 ) cg5,Born = (E1 4 + 2E1 3 mt ? E1 2 mt 2 2 2 2 z 2 )2 96E1 (E1 + mt ) (?1 + β ?4E1 mt 3 ? 2mt 4 ? 2β 2 E1 3 mt z 2 ? 2β 2 E1 2 mt 2 z 2 ? β 4 E1 4 z 4 ) 13

cg5,resonance

√ 4 gs 1 m3 2GF mt t = (? ? m2 )2 + Γ2 m2 16 8π 2 ?1 + β 2 z 2 s ? ? ? +ImC0 (?, m2 , m2 , m2 )Γ? m? ] ? 8a2 β 2 (? ? m2 ) s t t t s ? √ 2 3 m3 2GF t 2?3 a2 a2 β 2 |C0 (?, m2 , m2 , m2 )|2 s ? s t t t ? 2 32 8π +2?a4 β 2 |2 ? sβ 2 C0 (?, m2 , m2 , m2 )|2 , s ? s t t t cg6 =
4 gs β 4 z (7 + 9β 2 z 2 ) (z 2 ? 1) . 96 (E1 + mt ) (?1 + β 2 z 2 )2

4?a2 β 4 [ReC0 (?, m2 , m2 , m2 )(? ? m2 ) s s t t t s ?

(A.14)

(A.15)

The scalar three point function C0 appearing in equations (A.11), (A.12) and (A.14) is given by C0 (?, m2 , m2 , m2 ) = s t t t 1 1 1 d4 l . 2 l2 ? m2 + i? (l ? p )2 ? m2 + i? (l ? p ? p )2 ? m2 + i? iπ 1 1 2 t t t

(A.16)

g Numerically, Reven is dominated by the Born contributions. This completes our disg cussion of Reven . g The CP–violating part RCP results from the interference of the Born diagrams with the amplitudes of ?gs. 1c–1h (with couplings a? = ?γCP ) and the interference a of the CP–even and –odd amplitudes of ?g. 1h. We found that if m? is of the order g of 2mt or larger, RCP is dominated in the resonance region by the contributions from ?g. 1h. Since the complete expressions are rather lengthy, we have split them with (h) respect to the contributions from the individual diagrams. For example, bg2 means (d,c) the contribution from ?g. 1h to the function bCP and cg1 denotes the part of cg1 g1 that is generated by the diagrams of ?gs. 1d and 1e. The function bCP gets nonzero contributions only from the box diagrams of ?g. g1 1c:

bCP = g1

(c) bg1

2 +(7 ? 9βz) ? ImDs (?)(1 ? β 2 ) ? 2ImD11 (?)βzE1 (1 ? β 2 ) u u 2 +2(ImD11 (?) + ImD21 (?))β 3 zE1 (z 2 ? 1) . u u

2 ? ? +2(ImD11 (t) + ImD21 (t))β 3 zE1 (z 2 ? 1)

√ 4 m3 2GF γCP ?gs E1 = t 8π 2 96(?1 + β 2 z 2 ) 2 ? ? (7 + 9βz) ImDs (t)(1 ? β 2 ) ? 2ImD11 (t)βzE1 (1 ? β 2 )

(A.17)

14

In this expression, ? ? t ? Ds (t) = D0 (t)(m2 ? t) + C0 (?, m2 , m2 , m2 ) s ? t t

Ds (?) = D0 (?)(m2 ? u) + C0 (?, m2 , m2 , m2 ), u u ? s ? t t t

(A.18)

? (where C0 (?, m? , mt , mt ) is de?ned in (A.7)), t = (p1 ? k1 )2 , u = (p2 ? k1 )2 and s ? ? ? ? D0 (t); D? (t); D?ν (t) = d4 l 1; l? ; l? lν 1 2 l2 ? m2 + i? (l + k )2 ? m2 + i? iπ 1 t ? 1 1 2 ? m2 + i? (l + k ? p ? p )2 ? m2 + i? (l + k1 ? p1 ) 1 1 2 t t

? ? ? ? D? (t) = D11 (t)k1? ? D12 (t)p1? ? D13 (t)p2? ? ? ? ? D?ν (t) = D21 (t)k1? k1ν + D22 (t)p1? p1ν + D23 (t)p2? p2ν ? ? ? ? ?D24 (t)k1? p1ν ? D25 (t)k1? p2ν + D26 (t)p1? p2ν + D27 (t)g?ν .

(A.19)

D0 (?); D? (?); D?ν (?) are obtained from (A.19) by interchanging p1 and p2 . The u u u functions D11 , . . . , D27 can be reduced to expressions which contain only the scalar two–, three– and four–point functions B0 , C0 , D0 (see e.g.[39] ). The function bCP reads: g2 bCP = bg2 + bg2 + bg2 g2 √ 4 gs β m3 2GF γCP (c) t bg2 = 8π 2 96(?1 + β 2 z 2 ) ? mt (E1 + mt + βzE1 ) (7 + 9βz) ? ImDs (t) E1 + mt ? ? +2ImD11 (t)mt E1 (?E1 + E1 z 2 ? mt z 2 ) + 4ImD27 (t)mt 2 ? ? +2(Im(D11 t) + ImD21 (t))β 2 E1 (z 2 ? 1)(2mt + E1 z 2 ? mt z 2 ) mt (E1 + mt ? βzE1 ) +(7 ? 9βz) ? ImDs (?) u E1 + mt +2ImD11 (?)mt E1 (?E1 + E1 z 2 ? mt z 2 ) + 4ImD27 (?)mt u u bg2
(g) (c) (g) (h)

2 +2(ImD11 (?) + ImD21 (?))β 2 E1 (z 2 ? 1)(2mt + E1 z 2 ? mt z 2 ) u u √ 4 s m3 2GF γCP 3gs mt β 3 z 2 ImG(?) t = 2 2z2) 8π 8(?1 + β

15

(h) bg2

√ 4 1 m3 2GF γCP gs mt β t = (? ? m2 )2 + Γ2 m2 s 8π 2 4(?1 + β 2 z 2 ) ? ? ? 2m2 ReC0 (?, m2 , m2 , m2 )Γ? m? s t t t t ?ImC0 (?, m2 , m2 , m2 )(? ? m2 ) + Γ? m? . s t t t s ? (A.20)

As can be seen seen explicitly from these formulae, all contributions to the functions bCP and bCP result either from absorptive parts of the one loop amplitudes or from g1 g2 (h) terms of the form: width Γ? times dispersive terms (which is present only in bg2 ). This is in agreement with the general statements made in section 2. One can also check the relations following from Bose symmetry as given in table 1. The functions cg1 and cg2 arise from dispersive parts in the one loop amplitude or width terms times (h) absorptive parts (which is present only in cg2 below). They read: cg1 = cg1 + cg1 + cg1 √ 4 m3 2GF γCP ?gs E1 (c) t cg1 = 8π 2 96(?1 + β 2 z 2 ) 2 ? ? (7 + 9βz) ReDs (t)(1 ? β 2 ) ? 2ReD11 (t)βzE1 (1 ? β 2 )
2 ? ? ?2(ReD11 (t) + ReD21 (t))β 3 zE1 (z 2 ? 1) 2 +(7 ? 9βz) ? ReDs (?)(1 ? β 2 ) ? 2ReD11 (?)βzE1 (1 ? β 2 ) u u (c) (d,e) (f)

cg1

(d,e)

2 ?2(ReD11 (?) + ReD21 (?))β 3 zE1 (z 2 ? 1) u u √ 4 ?gs E1 m3 2GF γCP t = 8π 2 96(?1 + β 2 z 2 ) ? ? t t (9βz + 7) 2C0 (t, m2 , m2 , m2 )(β 2 ? 1)

+

cg1

(f)

βz Cs (?)(β 2 z 2 ? 2β 2 + 1) u βz + 1 √ 4 ?gs m3 2GF γCP t = 8π 2 192E1 (?1 + β 2 z 2 ) +

+(9βz ? 7) 2C0 (?, m2 , m2 , m2 )(β 2 ? 1) u ? t t

βz ? Cs (t)(β 2 z 2 ? 2β 2 + 1) βz ? 1

? ? t β ?1 (7 + 9βz)(B0 (m2 , m2 , m2 ) ? B0 (t, m2 , m2 )) t ? t βz ? 1 β2 ? 1 , +(7 ? 9βz)(B0 (m2 , m2 , m2 ) ? B0 (?, m2 , m2 )) u ? t t ? t βz + 1 where we used the notation 16

2

(A.21)

? ? t B0 (m2 , m2 , m2 ) ? B0 (t, m2 , m2 ) t ? t ? m2 ? t t B0 (m2 , m2 , m2 ) ? B0 (?, m2 , m2 ) u ? t t ? t Cs (?) = C0 (?, m2 , m2 , m2 ) + u u ? t t , 2 mt ? u ? ? ? ? t t Cs (t) = C0 (t, m2 , m2 , m2 ) + and ? ? t t C0 (t, m2 , m2 , m2 ) = 1 1 d4 l 1 . 2 l2 ? m2 + i? (l + k )2 ? m2 + i? (l + k ? p )2 ? m2 + i? iπ 1 1 1 t t ? C0 (?, m2 , m2 , m2 ) is obtained from (A.23) by the replacement p1 → p2 . u ? t t Finally, cg2 = cg2 + cg2 + cg2 + cg2 + cg2 , √ 4 m3 2GF γCP ?gs β (c) t cg2 = 8π 2 96(?1 + β 2 z 2 ) ? mt (E1 + mt ? βzE1 ) (7 + 9βz) ? ReDs (t) E1 + mt ? ? +2ReD11 (t)mt E1 (E1 ? E1 z 2 + mt z 2 ) + 4ReD27 (t)mt 2 ? ? +2(ReD11 (t) + ReD21 (t))β 2E1 (z 2 ? 1)(2mt + E1 z 2 ? mt z 2 ) mt +(7 ? 9βz) ? ReDs (?) u (E1 + mt + βzE1 ) E1 + mt +2ReD11 (?)mt E1 (E1 ? E1 z 2 + mt z 2 ) + 4ReD27 (?)mt u u cg2
(d,e) (c) (d,e) (f) (g) (h)

(A.22)

(A.23)

2 +2(ReD11 (?) + ReD21 (?))β 2 E1 (z 2 ? 1)(2mt + E1 z 2 ? mt z 2 ) u u √ 4 ?gs β m3 2GF γCP t = 8π 2 96(?1 + β 2 z 2 ) mt ? ? t t (7 + 9βz) 2C0 (t, m2 , m2 , m2 ) (E1 + mt ? βzE1 ) E1 + mt 2 2 2 2 ? ?mt ? E1 z + mt z (E1 ? 2m2 ? β 2 E1 z 2 ) +Cs (t) t 2 E1 (1 ? βz) mt +(7 ? 9βz) 2C0 (?, m2 , m2 , m2 ) u ? t t (E1 + mt + βzE1 ) E1 + mt ?mt ? E1 z 2 + mt z 2 2 2 (E1 ? 2m2 ? β 2 E1 z 2 ) +Cs (?) u t 2 E1 (1 + βz)

17

(f) cg2

√ 4 m3 2GF γCP gs t = 8π 2 192(?1 + β 2 z 2 ) mt E1 + mt ? βzE1 2 E1 + mt E1 (1 ? βz) +(7 ? 9βz)(B0 (m2 , m2 , m2 ) ? B0 (?, m2 , m2 )) u ? t t ? t mt E1 + mt + βzE1 2 E1 + mt E1 (1 + βz) √ 4 s m3 2GF γCP ?3gs mt β 3 z 2 ReG(?) t = 2 2z2) 8π 8(?1 + β √ 4 1 m3 2GF mt β gs γCP t ? = 2 )2 + Γ2 m2 2 (? ? m? s 16 8π ?1 + β 2 z 2 ? ? +ImC0 (?, m2 , m2 , m2 )Γ? m? ] ? 4(? ? m2 ) s t t t s ? √ 2 3 m3 2GF t + [2?3 β?2 |C0 (?, m2 , m2 , m2 )|2 s a s t t t 2 32 8π +2?βa2 |2 ? sβ 2 C0 (?, m2 , m2 , m2 )|2 ] . s ? s t t t (A.24) 2?(1 + β 2 )[ReC0 (?, m2 , m2 , m2 )(? ? m2 ) s s t t t s ? ? ? t (7 + 9βz)(B0 (m2 , m2 , m2 ) ? B0 (t, m2 , m2 )) t ? t

cg2 cg2

(g)

(h)

18

References
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19

[17] D. Atwood, A. Soni: Phys. Rev. D 45 (1992) 2405; D. Atwood, A. Aeppli, A. Soni: Phys. Rev. Lett. 69 (1992) 2754; D. Atwood, G. Eilam, A. Soni: Phys. Rev. Lett. 71 (1993) 492. [18] J.P. Ma, A. Brandenburg: Z. Phys. C 56 (1992) 97; A. Brandenburg, J.P. Ma: Phys. Lett. B 298 (1993) 211. [19] C.R. Schmidt: Phys. Lett. B 293 (1992) 111. [20] B. Grzadkowski: Phys. Lett. B 305 (1993) 384; B. Grzadkowski, W.-Y. Keung: CERN preprint CERN-TH.6917/93 (1993) [21] W. Bernreuther, P. Overmann: Heidelberg preprint HD-THEP-93-11 (1993). [22] J. Liu: Phys. Rev. D 47 (1993) R1741 [23] A. Pilaftsis, M. Nowakoski: Mainz preprint MZ-TH-92-56 (1992). [24] D. Chang, W.-Y. Keung: Phys. Lett. B 305 (1993) 261. [25] D. Chang, W.-Y. Keung, I. Phillips: CERN preprint CERN-TH-6658.92-REV (1993). [26] K.F. Smith et al.: Phys. Lett. B 234 (1990) 191. [27] S.A. Murthy et al.: Phys. Rev. Lett. 63 (1989) 965; K. Abdullah et al.: Phys. Rev. Lett. 65 (1990) 2347. [28] X.G. He, J.P. Ma, B. McKellar: Melbourne preprint UM-P-93-11 (1993); Phys. Lett. B 304 (1993) 285. [29] A. Mendez, A. Pomarol: Phys. Lett. B 272 (1991) 313. [30] J.F. Gunion, T.C. Yuan, B. Grzadkowski: Phys. Rev. Lett. 71 (1993) 488. [31] A. Ilakovac, B.A. Kniehl, A. Pilaftsis: preprint MAD-PH-787 (1993). [32] A. Djouadi, B.A. Kniehl: preprint UDEM-LPN-TH-93-171 (1992). [33] C. Kao: Florida preprint FSU-HEP-930924 (1993). [34] J. K¨ hn: Acta Phys. Austriaca Suppl. XXIV (1982) 203; I. Bigi, Y. Dokshitzer, u V. Khoze, J. K¨ hn, P. Zerwas: Phys. Lett. B 181 (1986) 157. u [35] A. Czarnecki, M.Jezabek, J.H. K¨ hn: Nucl. Phys. B 351 (1991) 70. u [36] G.A. Ladinsky: Phys. Rev. D 46 (1992) 3789; D 47 (1993) 3086 (E). [37] J.F. Owens: Phys. Lett. B 266 (1991) 126. 20

[38] G. ’t Hooft, M. Veltman: Nucl. Phys. B 153 (1979) 365. [39] G. Pasarino, M. Veltman: Nucl. Phys. B 160 (1979) 151.

21

Table Caption
Table 1: Transformation properties of the structure functions de?ned in (2.2)–(2.3).

22

Table 1

CP

P

T CPT (ImT = 0) (ImT = 0) Ag (z) b± (z) g1 b± (z) g2 ± ?bg3 (z) cg0 (z) ?cg1 (z) ?cg2 (z) cg3 (z) cg4 (z) cg5 (z) cg6 (z) ?cg7 (z) ?cg8 (z) Ag (z) b? (z) g1 b? (z) g2 ? ?bg3 (z) cg0 (z) cg1 (z) cg2 (z) ?cg3 (z) cg4 (z) cg5 (z) cg6 (z) ?cg7 (z) ?cg8 (z)

“Bose” Ag (?z) ?b± (?z) g1 ± bg2 (?z) ?b± (?z) g3 cg0 (?z) ?cg1 (?z) cg2 (?z) ?cg3 (?z) cg4 (?z) cg5 (?z) ?cg6 (?z) cg7 (?z) ?cg8 (?z)

Ag (z) Ag (z) Ag (z) b± (z) b? (z) ?b± (z) g1 g1 g1 ± ? bg2 (z) bg2 (z) ?b± (z) g2 ± ? ± bg3 (z) bg3 (z) bg3 (z) cg0 (z) cg0 (z) cg0 (z) cg1 (z) ?cg1 (z) ?cg1 (z) cg2 (z) ?cg2 (z) ?cg2 (z) cg3 (z) ?cg3 (z) cg3 (z) cg4 (z) cg4 (z) cg4 (z) cg5 (z) cg5 (z) cg5 (z) cg6 (z) cg6 (z) cg6 (z) cg7 (z) cg7 (z) ?cg7 (z) cg8 (z) cg8 (z) ?cg8 (z)

23

Figure Captions
Fig. 1: Born level QCD and ? exchange Feynman diagrams which contribute to the ? production density matrix for gg → tt. Diagrams with crossed gluons are not shown. ? Fig. 3: Expectation value k · (s+ ? s? ) g as a function of the parton CM energy for m? = 100 GeV (dashed curve) and m? = 350 GeV (solid curve). ? Fig. 4: Expectation value k · (s+ × s? ) Fig. 3.
g

? Fig. 2: Born level QCD and relevant ? exchange Feynman diagrams for q q → tt. ?

for the same choice of Higgs masses as in

? Fig. 5: Same as Fig. 3, but for k · (s+ ? s? ) q . ? Fig. 6: Same as Fig. 4, but for k · (s+ × s? ) q . Fig. 7: Signal–to–noise ratio for the observable A1 (de?ned in (4.1)) as a function of √ the Higgs boson mass m? for proton–proton CM energies s = 15 TeV (solid curve) √ and s = 40 TeV (dashed curve). Here mt = 150 GeV, a = ?? = 1, gV V = 1. a ? Fig. 8: Same as ?g. 7, but for the observable A2 (de?ned in (4.2)). Fig. 9: Same as ?g. 7, but for the observable T2 (de?ned in (4.3)).

24

This figure "fig1-1.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9312210v1

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