1 Introduction
###### Abstract

We discuss in detail top quark polarization in above-threshold production at a polarized linear collider. We pay particular attention to the minimization and maximization of the polarization of the top quark by tuning the longitudinal polarization of the and beams. The polarization of the top quark is calculated in full next-to-leading order QCD. We also discuss the beam polarization dependence of the longitudinal spin–spin correlations of the top and antitop quark spins.

MZ-TH/10-47

1012.4600 [hep-ph]

December 2010

A survey of top quark polarization

at a polarized linear collider

S. Groote Loodus- ja Tehnoloogiateaduskond, Füüsika Instituut,

Tartu Ülikool, Riia 142, EE–51014 Tartu, Estonia

J.G. Körner Institut für Physik der Johannes-Gutenberg-Universität,

Staudinger Weg 7, D–55099 Mainz, Germany

B. Melić Rudjer Bošković Institute, Theoretical Physics Division,

Bijenička c. 54, HR–10000 Zagreb, Croatia

S. Prelovsek Physics Department at University of Ljubljana

and Jozef Stefan Institute, SI–1000 Ljubljana, Slovenia

## 1 Introduction

A future linear collider offers the cleanest conditions for studying top quark properties, such as the top quark mass, its vector and axial couplings, and possible magnetic and electric dipole moments. Apart from these static properties, also the polarization of the top quark can be studied with great precision. The top decays sufficiently fast so that hadronization effects do not spoil the polarization which it has at its birth. The large number of top quark pairs expected to be produced at the ILC, e.g., at (based on a luminosity of  [1, 2]), will enable one to precisely determine the top quark polarization from an angular analysis of its decay products in the dominant decay . The expected statistical errors in the angular analysis are below the level. Therefore very precise measurements of the angular distributions and correlations of the decay products of and will shed light on the polarization of the top quarks and on the spin–spin correlations of the top and antitop quark pairs which are imprinted on the top and antitop quarks by the -production mechanism. In addition, the measurement of the top polarization will make it possible to precisely determine the electroweak Standard Model (SM) parameters or to study a variety of new phenomena beyond the SM.

It is well known that the top quarks from annihilations are polarized even for unpolarized beams due to the presence of parity-violating interactions in the Standard Model (SM). One also knows from the work of Ref. [3, 4] that the polarization of the top quark in polarized annihilations can become quite large when the beam polarization is adequately tuned. This is illustrated in Fig. 1 where we display the energy dependence of the mean longitudinal and transverse polarization of the top quark in the helicity system for different values of the effective polarization defined by

 Peff=h−−h+1−h−h+. (1)

In Eq. (1), and are the longitudinal polarization of the electron and positron beams, respectively. Note that, for unpolarized positron beams , one has . For a given value of , even small values of positron polarization of opposite sign will enhance the effective beam polarization. We shall return to this point in Sec. 2. As compared to the work of Ref. [3], Fig. 1 now includes the radiative corrections. Large single-spin polarization effects due to beam polarization effects are also implicit in the work of Parke and Shadmi [5]. Although Ref. [5] is designed for the analysis of top–antitop quark spin correlations, it is easily adapted to single-spin polarization effects as also discussed in Ref. [6].

We shall see that the polarization of the top quark is governed by three parameters: the velocity , the effective polarization , and the cosine of the scattering angle . At the respective boundaries of the three parameters the description of the polarization phenomena becomes reasonably simple, in particular, at the Born term level. The limits (threshold) and (high-energy limit) are discussed in Sec. 4. In Sec. 5 we discuss the limiting cases .

The two respective limiting cases and contain in a nutshell much of the information that we want to discuss in the remaining part of the paper for intermediate values of these parameters. Many of the qualitative features of our results can be understood from extrapolations away from the two respective limits.

We shall also address the question of how to maximize and minimize the polarization of the top quark by tuning the beam polarization. Whereas a maximum polarization is optimal for the experimental determination of polarization effects, it is often desirable to gauge the quality of a polarization measurement against the corresponding unpolarized decay analysis. For some measurements it may even be advantageous to eliminate polarization effects altogether.

Of course, in the tuning process one has to bear in mind to keep the production rate at an acceptable level. This problem is not unrelated to the one of the original motivations of including beam polarization in linear colliders, namely, the gain in rate through beam polarization effects. We shall also address this question.

Our paper is structured as follows. In Sec. 2 we present the spin formalism of polarized beam production of top–antitop quark pairs including the polar angle dependence of the various spin components and longitudinal beam polarization effects. We present Born term and loop formulas for the relevant structure functions and collect general expressions necessary for the numerators and the denominator of the polarization observables. Section 3 contains numerical next-to-leading (NLO) results on the angle-integrated rate and on polar angle distributions of the rate including beam polarization effects. We also provide numerical results on the left–right polarization asymmetry . In Sec. 4 we discuss the limiting cases and at the Born term level. In Sec. 5 we describe the simplifications that occur for maximal effective beam polarizations which correspond to the beam configurations. In Sec. 6 we discuss beam polarizations effects on the three components of the top quark polarization vector. Section 7 contains a discussion of the magnitude and the orientation of the polarization vector of the top quark. In Sec. 8 we present numerical NLO results on beam polarization effects on longitudinal spin–spin correlations of the top and antitop quark. Finally, Sec. 9 contains a summary of our results and our conclusions. In an Appendix we list the electroweak coupling coefficients used in this paper and relate them to the chiral electroweak coupling coefficients used e.g. in Ref. [5].

Many of the quantitative arguments presented in this paper are based on Born term level results for which we give explicit alpha-numerical expressions for . We emphasize, though, that all numerical results presented in the plots include the full radiative corrections where we have integrated over the full gluon phase space. By comparing the graphical NLO results with the numerical LO results, one can assess the size of the radiative corrections, at least for the representative energy of . In general, the corrections to polarization observables are small (up to several percent) but can become much larger in some areas of phase space. A case in point is the longitudinal polarization of the bottom quark produced on the at the backward point which obtains a correction when  [4]. As we shall see later on, the corrections to -production can amount up to (see Sec. 7). In addition, there are polarization observables that are zero at the Born term level and become populated only at . Among these are the normal component of the polarization (see Sec. 6) and the longitudinal polarization produced from a longitudinal intermediate vector boson (see Sec. 2).

## 2 Spin formalism of polarized beam production

The production of top quark pairs at a linear -collider proceeds via - and -exchange,

 e−e+\lx@stackrelγ,Z→t¯t. (2)

At the center of mass energies which are being envisaged at the ILC (), , top quark pairs will be produced with nonrelativistic velocities in the threshold region () up to relativistic velocities of at the highest energy ).111In the first stage of the ILC, one will reach energies up to with an optional second stage upgrade to  [1, 2]. For the multi-TeV collider CLIC one foresees energies up to  [7]. This enables the study of the complete production phenomena with different polarization and correlation effects that reach from the nonrelativistic to the relativistic domain. For unpolarized beams the total rate is dominated by the diagonal () and the () rates which contribute at the same order of magnitude. The () interference contribution to the total rate is suppressed due to the smallness of the vector coupling (). The () interference contribution can, however, become quite sizable for polarized beams, for the polar angle dependent rates and for top quark polarization effects.

We mention that, at threshold, there will be the opportunity for very precise measurements of the top quark mass and width, as well as of the strong coupling . In this region, perturbative QCD is no longer applicable. One has to solve the Schrödinger equation for the relevant Green functions in a nonrelativistic approximation for a Coulombic potential, i.e. the nonrelativistic QCD (NRQCD) method, described first in Ref. [8] and later applied to the calculation of various different quantities at threshold (see for example the discussion in Ref. [9, 10] and references therein). In this paper we shall discuss top–antitop production well above threshold where perturbation theory can be safely applied. For our purposes we take the perturbative regime to start approximately above threshold. Throughout this paper we shall take the top quark mass to have a nominal value of . Therefore, we shall consider c.m. beam energies starting from .

We are going to discuss the most general case of the polarization of the top quark with arbitrary longitudinal polarizations of the - and -beams. The rate depends on the set of four parameters or, equivalently, on the set , where we shall call the gain factor. We have indicated the range of the parameter values in square brackets. In contrast to the rate, the polarization of the top quark depends only on the set of the three parameters . When discussing our predictions we shall attempt to explore the whole four- and three-dimensional parameter space for the rate and polarization, respectively. We mention that the beam polarizations envisaged at the ILC are for electrons and for positrons [11].

We will see that beam polarizations significantly influence the polarization phenomena of a top quark. In addition, adequately tuned beam polarization can enhance the top–antitop quark signal and suppress other background processes such as -pair production (see discussion in Ref. [12]).

In what follows, we concentrate on the polarization of the top quark, i.e. we sum over the polarization of the antitop quark. The polarization of the antitop quark can be obtained from the corresponding polarization components of the top quark using CP invariance as will be discussed in the summary section. Even more structure is revealed when one considers joint top-antitop polarization. In order to reveal this structure, one must perform a joint analysis of the decay products of the top and antitop quark. spin–spin correlations will be briefly discussed in Sec. 8 at the end of the paper.

The general expression of the cross section for production in collisions is given by222The spin kinematics of collisions has been formulated in a number of papers. These include the unpublished DESY report [13] of which the portions relevant to this paper have been summarized in Ref. [14]. Other papers on the subject are Refs. [4, 12, 15, 16, 17, 18].

 dσ(m)=2πe4s24∑i,j=1gijLiμνHj(m)μνdPS. (3)

is the lepton tensor, is the hadron tensor encoding the hadronic production dynamics, is the phase space factor, and the are the elements of the electroweak coupling matrix which are defined in the Appendix. The sum runs over the four independent configurations of products of the vector and axial vector currents, i.e.  for ,   for , and for for the product of lepton and quark currents, and denotes one of the possible polarization configurations of the top quark: longitudinal (), transverse () in the beam scattering plane and normal () to the beam scattering plane. Our choice of the three orthonormal spin directions are given by

 →e(tr)=(→pe−×→pt)×→pt|(→pe−×→pt)×→pt|,→e(n)=→pe−×→pt|→pe−×→pt|,→e(ℓ)=→pt|→pt|. (4)

In Fig. 2 we have drawn the directions of and for a generic top quark direction; the vector shows out of the plane. For the unpolarized top quark case the superscript is dropped in Eq. (3). The explicit definitions for all the above quantities together with explicit analytical expressions for the radiative corrections can be found in Refs. [14, 19, 20, 21] (see also Ref. [4]).

We proceed with the discussion in the helicity basis, i.e. we take the direction of the top quark to define the direction of the hadronic system. For unpolarized beams the angular decomposition of the differential polarized cross section can be written as

 dσ(m)dcosθ = 38(1+cos2θ)σ(ℓ)U+34sin2θσ(ℓ)L+34cosθσ(ℓ)F (5) −3√2sinθcosθσ(tr,n)I−3√2sinθσ(tr,n)A,

where, at NLO of QCD,

 σ(m)a=πα2v3s24∑j=1g1j(Hj(m)a(Born\/)+Hj(m)a(αs))for a=U,L,I (6)

and

 σ(m)a=πα2v3s24∑j=1g4j(H4(m)a(Born\/)+H4(m)a(αs))for a=F,A. (7)

In Eq. (5) we have rewritten the covariant representation (3) in terms of helicity structure functions . The angle is the polar angle between the momentum of the top quark and the electron momentum (see Fig. 2). For example, in the purely electromagnetic case one obtains the LO formula

 dσcosθ=2πNcQ2fvα24s(1+cos2θ+(1−v2)sin2θ) (8)

using the LO born term expressions listed later in Eq. (2). The distribution (8) agrees with Eq. (41.2) in the PDG booklet. We mention that our corrections agree with those in Ref. [4] after correcting a sign mistake in the normal polarization (see Erratum in Ref. [21]).

Above the top quark threshold, one is sufficiently far away from the -boson pole to neglect the imaginary part of the boson pole propagator. This can be appreciated from the Breit-Wigner line shape of the propagator, viz.

 χZ=1s−M2Z+iMZΓZ=1s−M2Z(1−iMZΓZs−M2Z)/(1+M2ZΓ2Z(s−M2Z)2). (9)

The factor determines the ratio of the imaginary and real parts of the propagator . It is already quite small at threshold () and falls off with . Dropping the imaginary part contribution of the propagator implies that we neglect contributions proportional to in Eq. (6) and in Eq. (7). We shall also neglect the width dependence in the real part of the propagator because it is negligibly small.

The nonvanishing unpolarized Born term contributions read

 H1U(Born\/)=2Ncs(1+v2), H1L(Born\/)=Ncs(1−v2) = H2L(Born\/), H2U(Born\/)=2Ncs(1−v2), H4F(Born\/)=4Ncsv. (10)

One has showing that the longitudinal rate falls off with a power behaviour relative to the transverse rates . The longitudinally polarized contributions read

 H4(ℓ)U(Born\/)=4Ncsv, H1(ℓ)F(Born\/)=2Ncs(1+v2), H4(ℓ)L(Born\/)=0, H2(ℓ)F(Born\/)=2Ncs(1−v2). (11)

Note that one has the Born term relations

 H4(ℓ)U(Born\/) = H4F(Born\/), H1,2(ℓ)F(Born\/) = H1,2U(Born\/), (12)

which are due to angular momentum conversation in the back-to-back configuration at the Born term level. It is quite clear that Eqs. (2) no longer hold true in general at since quark and antiquark are no longer back-to-back in general due to additional gluon emission. The relations (2) will be useful in our subsequent discussion of the longitudinal polarization at the forward and backward points. Notable also is the relation in Eq. (2) which is again related to the LO back-to-back configuration. The radiative corrections to the corresponding polarization component have been studied in Ref. [20] and have been found to be small of when averaged over gluon phase space. For small top quark energies can become as large as at .

For the transverse polarization components, one has [21]

 H4(tr)I(Born\/)=2Ncsvmt√2s,H1(tr)A(Born\/)=2Ncsmt√2s=H2(tr)A(Born\/). (13)

The only nonnegligible contribution to the normal polarization component comes from the imaginary part of the one-loop contribution ()

 H1(n)I(loop\/) = 2NcsαsCF4ππvmt√2s=H2(n)I(loop), (14) H4(n)A(loop\/) = 2NcsαsCF4ππ(2−v2)mt√2s. (15)

As already mentioned in the Introduction, the transverse and normal polarization components can be seen to fall off with a power behaviour of relative to the longitudinal polarization components.

The corrections to the polarized structure functions and the unpolarized structure function ( is the polarization vector of the top quark) are too lengthy to be listed here. They can be found in Refs. [14, 19, 20, 21], or, in a very compact two-page analytical representation, in Sec. 5 of Ref. [22].

The longitudinal polarization of the electron and positron beams enter the above formulas as [14]333Transverse beam polarization effects will not be discussed in this paper because present plans call for longitudinal beam polarization at the ILC. Transverse beam polarization effects can be included as described e.g. in Ref. [14].

 g1j → g4j → (16)

where is defined in Eq. (1). In Eq. (2) denotes the electron’s and denotes the positron’s longitudinal polarization which can take values between . An electron with will be referred to as the totally polarized left-handed (right-handed) electron (). Similarly, a right-handed positron () has and a left-handed positron () has . From the definition of (see Eq. (1)) it is clear that large values of can be reached even for nonmaximal values of and , as Fig. 3a shows. For example, the large value of can be achieved with ; , and correspondingly, can be reached with ; . These two examples have been marked off in Fig. 3. Both sets correspond to a gain factor of .

The orientation-dependent longitudinal, transverse, and normal polarization components which we are interested in are defined by

 P(m)(cosθ)=dσ(m)/dcosθdσ/dcosθm=ℓ,tr,n, (17)

where is the unpolarized differential cross section. Of course, there is an additional dependence of the above quantities on the c.m. beam energy , and on the beam polarizations and to be discussed later on. The unpolarized cross section is given by the first three terms in Eq. (5) dropping, of course, the label .

Dropping the common factor in the ratio (17), we shall represent the polarization components by the ratios

 P(m)(cosθ)=N(m)(cosθ)D(cosθ),m=ℓ,tr,n. (18)

In particular, the gain factor has canceled out in the ratio (18) implying that the polarization only depends on .

The numerator factors in Eq. (18) are given by

 N(ℓ)(cosθ) = 38(1+cos2θ)(g14+g44Peff)H4(ℓ)U+34sin2θ(g14+g44Peff)H4(ℓ)L (19) +34cosθ((g41+g11Peff)H1(ℓ)F+(g42+g12Peff)H2(ℓ)F),
 N(tr)(cosθ) = −3√2sinθcosθ(g14+g44Peff)H4(tr)I (20) −3√2sinθ((g41+g11Peff)H1(tr)A+(g42+g12Peff)H2(tr)A),

and by

 N(n)(cosθ) = −3√2sinθcosθ((g11+g41Peff)H1(n)I(loop\/)+(g12+g42Peff)H2(n)I(loop\/)) (21) −3√2sinθ(g44+g14Peff)H4(n)A(loop\/).

For the denominator, one has

 D(cosθ) = 38(1+cos2θ)((g11+g41Peff)H1U+(g12+g42Peff)H2U) (22) +34sin2θ((g11+g41Peff)H1L+(g12+g42Peff)H2L) +34cosθ(g44+g14Peff)H4F.

At the forward (FP) and backward (BP) point the transverse and normal polarization components vanish. Referring to the relations (2), at Born term level the longitudinal polarization component takes a very simple form at the forward (FP) and backward (BP) point for the maximal values of the effective polarization . One has

 FP:P(ℓ)(cosθ=+1) = ±1, BP:P(ℓ)(cosθ=−1) = ∓1, (23)

in agreement with angular momentum conservation. It is clear that these relations no longer hold true in general at NLO due to hard gluon emission.

It is useful to define the left–right polarization asymmetry through the relation

 dσ(Peff)−dσ(−Peff)dσ(Peff)+dσ(−Peff)=−ALRPeff, (24)

where

 ALR=−38(1+cos2θ)(g41H1U+g42H2U)+34sin2θ(g41H1L+g42H2L)+34cosθg14H4F38(1+cos2θ)(g11H1U+g12H2U)+34sin2θ(g11H1L+g12H2L)+34cosθg44H4F (25)

Of interest is the angle enclosed by the momentum and the polarization of the top quark projected onto the scattering plane (see Fig. 2).444For the present purposes we neglect the normal component of the polarization vector which is quite small. Note that, in general, one needs two angles to describe the orientation of the polarization vector instead of the one angle defined in Eq. (26). The angle is determined by

 tanα(cosθ)=N(tr)(cosθ)N(ℓ)(cosθ). (26)

Equation (26) assumes a simple form at threshold and in the high-energy limit as discussed in Sec. 4, and for as will be discussed in Sec. 5. The correlations between and implied by Eq. (26) will be discussed in Secs. 4, 5 and 7.

## 3 Beam polarization dependence of the rate

We begin our numerical discussion with the rate proportional to the denominator expression in Eq. (18). The effect of longitudinally polarized beams on the polar averaged rate (called total rate) can be obtained from the form

 σ=σ(Peff=0)(1−h−h+)⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1+Peffg41g111+g42g41H2U+LH1U+L1+g12g11H2U+LH1U+L⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (27)

which, at the Born level and at , gives

 (28)

From Eq. (28) it is evident that the total rate becomes maximal on two counts: (i) large values of the gain factor , requiring ; and (ii) large negative values of , which can be achieved with large negative and positive values of and , respectively. The maximal enhancement of the rate will be obtained for and such that and . At , this leads to a maximal enhancement factor of over the unpolarized case. It is interesting to note that for production at the effective enhancement through beam polarization effects is slightly larger than in the case. For production the last factor in Eq. (27) is replaced by the simpler expression since, at , the ratio is practically zero for bottom quark production. Using the results of the Appendix applied to the case, one finds leading to an overall enhancement factor of for the optimal choice of parameters and () at .

In Fig. 3b we show some contour lines for fixed values of the gain factor in the -plane. Clearly, quadrants 2 and 4 are favoured if one wants to obtain a gain factor exceeding one, i.e. . As concerns the rate dependence on (rightmost factor in Eq. (28)), a further rate enhancement is achieved for negative values of , i.e. one would have to choose points lying to the left of the line in Fig. 3a. The optimal choice as concerns the rate would thus be quadrant 2 in the -plane. One notes that large negative values of can readily be achieved for nonmaximal values of the beam polarization, as illustrated in Fig. 3a, where we have plotted some contour lines in the plane corresponding to fixed values of . One notes that the regions of large and large negative have a large overlap. We mention that one may have to give up the optimal choice in the plane if one wants to achieve other goals such as minimizing the polarization.

The QCD one-loop corrections to the total cross section are well-known (see e.g. Ref. [18]) and add about at to the Born total cross section, where the percentage increase has very little dependence on the beam polarization. We mention that the electroweak corrections to the total rate are smaller, and amount to about of the QCD corrections [23]. For the strong coupling we use two-loop running adjusted to the value and fitted at .555For we take the value . If one uses a running , for example , the cross sections in Fig. 4 would increase by . Close to threshold the corrections become larger and amount to about of the total cross section at e.g. . The c.m. energy dependence of the total cross section is shown in Figs. 4a and 4b. In Fig. 4a we take and show the energy dependence of the total cross section varying over its whole range . One notes a strong dependence on apart from the standard falloff of the total cross section with beam energy. Since for the gain factor is equal to and since , the rate depends linearly on as displayed in Eqs. (27) and (28). The rate is largest for and then linearly drops to its lowest value at . In Fig. 4b we show the energy dependence of the rate for the three pairs of beam polarizations . If one translates this into the representation, one has . The hierarchy of rates in Fig. 4b can be seen to be mostly determined by the gain factor in Eq. (27).

Next we turn to the differential rate distribution with respect to . In order to illustrate the forward dominance of the differential -distribution we plot against . Note that the dependence on the gain factor drops out in the ratio. In Fig. 5a we plot the differential rate distribution for a fixed value of and for . One sees a pronounced forward dominance of the differential distribution which does not depend much on the value of . In Fig. 5b we keep the effective beam polarization fixed at and vary through several values. At threshold one has a flat distribution . When the energy is increased, the forward rate clearly dominates over the backward rate. The forward dominance becomes even stronger for increasing energies.

Of related interest is the rate into the forward (F) and backward (B) hemispheres. Again, the gain factor drops out in the ratio. At , one numerically obtains

 ⟨σ⟩F⟨σ⟩B=⟨σ⟩F⟨σ⟩B∣∣∣Peff=01−0.34Peff1−0.43Peff=⎧⎪⎨⎪⎩+2.73Peff=+1+2.36Peff=+0+2.21Peff=−1⎫⎪⎬⎪⎭. (29)

The mean forward rate clearly dominates over the mean backward rate . The dependence of the rate ratio on is not very pronounced.

In Fig. 6 we plot the polar angle dependence of the NLO left-right polarization asymmetry for different energies. At GeV the dependence already starts to deviate from the flat Born term behaviour at threshold given by . The left-right polarization asymmetry peaks toward the backward region and reaches at the backward point for the highest energy in Fig. 6.

## 4 Born term simplifications at threshold and in the high-energy limit

Before turning to the numerical analysis of the polarization of the top quark, in this section we shall first discuss Born term simplifications of the polarization of the top quark at threshold and in the high-energy limit. In Sec. 5 we discuss Born term simplifications that occur for .

At threshold and in the high-energy limit , the polarization expressions become quite simple. At threshold, the polarization of the top quark is parallel to the beam axis, regardless of the polar orientation of the top quark (see e.g. Ref. [24]). In fact, a large part of the beam polarization gets transferred to the polarization of the top quark at threshold. For the Born term contributions the top quark polarization at threshold can be calculated from Eqs. (19), (20) and (22) (see also Ref. [18, 25]). It is nominally given by666As discussed in Sec. 2, QCD binding effects significantly modify the naive threshold results in the threshold region.

 →P=Peff−ALR1−PeffALR^ne−, (30)

where is the left-right beam polarization asymmetry at threshold (see Eq. (25)) and is a unit vector pointing into the direction of the electron momentum. In terms of the electroweak coupling parameters (see the Appendix), the nominal polarization asymmetry at threshold is given by . The simplification at threshold arises from the fact that, from the four amplitudes describing the production of a spin-1/2 pair, only the -wave amplitude survives at threshold. The suffices and denote vector current (V) and axial vector current (A) production. Correspondingly, the combinations and contain only the vector current coupling on the quark side.

The magnitude of the threshold polarization is given by

 |→P|=∣∣∣Peff−ALR1−PeffALR∣∣∣. (31)

The threshold polarization is independent of , i.e. . The polarization vanishes for independent of .777Threshold simplifications for production have also been discussed in Ref. [26]. Similar simplifications for polarization observables occur for the threshold production of gauge boson pairs [27]. For and one has and , respectively, such that and . In particular, one has a threshold polarization of the top quark for with .

Extrapolations away from are more stable for than for as the slope of Eq. (31) at shows. One has

 d|→P|dPeff=±1±ALR1∓ALR. (32)

For one has a slope of while one has a much larger positive slope of for . This substantiates the statement made above and in Sec. 1 about the stability of extrapolations away from . For example, keeping only the linear term in the Taylor expansion of Eq. (31), one has for , while drops to for .

For energies above threshold the slope Eq. (32) becomes energy and angle dependent. We do not show plots of the slope at higher energies. We have, however, checked numerically that the above statement about the stability of the result at against variations of remains true at higher energies in the whole angular range, where the slope in the backward region has a tendency to be smaller than in the forward region.

As mentioned above, minimal polarization occurs for for all values of . This again shows that an extrapolation away from is more stable than an extrapolation from since one is much closer to the polarization zero in the latter case. This observation will carry over to the -dependence at higher energies.

In Fig. 7a we show the threshold correlation of the angles and for different values of . Starting at the two angles are related by up to the longitudinal polarization zero at after which the correlation becomes .

As the beam energy increases, the polarization vector of the top quark slowly turns into the direction of its momentum (or opposite to it). Finally, in the high-energy limit , when , the polarization of the top becomes purely longitudinal in the helicity system such that since its transverse and normal components involve a spin flip amplitude and thus vanish as . Note that, although is asymptotically suppressed, it is still sizable at as Fig. 1 shows.

In fact, in the high-energy limit, one has with

 P(ℓ)(cosθ) = (g14+g41+Peff(g11+g44))(1+cosθ)2+(g14−g41−Peff(g11−g44))(1−cosθ)2(g11+g44+Peff(g14+g41))(1+cosθ)2+(g11−g44−Peff(g14−g41))(1−cosθ)2

for the surviving longitudinal polarization. In the same limit, the electroweak coupling coefficients take the numerical values , , , , , and . When it is more convenient to switch to the chiral electroweak coefficients defined in the Appendix. One has (; )

 P(ℓ)(cosθ)=−1−bLR1+bLRwithb