Bound-state effects on kinematical distributions of top quarks at hadron colliders

First we present a theoretical framework to compute the fully differential cross sections for the top-quark productions and their subsequent decays at hadron colliders, incorporating the bound-state effects which are important in the t\bar{t} threshold region. We include the bound-state effects such that the cross sections are correct in the LO approximation both in the threshold and high-energy regions. Then, based on this framework we compute various kinematical distributions of top quarks as well as of their decay products at the LHC, by means of Monte-Carlo event-generation. These are compared with the corresponding predictions based on conventional perturbative QCD. In particular, we find a characteristic bound-state effect on the (bW^+)-(\bar{b}W^-) double-invariant-mass distribution, which is deformed to the lower invariant-mass side in a correlated manner.


Introduction
At the CERN Large Hadron Collider (LHC), the top quark will be produced copiously. The cross section for the top-quark pair-production amounts to several hundred pb [1,2,3], and order 10 6 top-quark events will be observed each year if the LHC runs with 14 TeV collision energy and achieves the designed luminosity. Collecting these top-quark events, detailed analysis on the properties of the top quark will be possible, such as precise determinations of its mass and width, structure of electroweak and strong interactions, and its spin properties [4]. The current world average of the top-quark mass measurements from the combined analysis of CDF and D0 collaborations at the Fermilab Tevatron reads m t = 173.1 ± 1.3 GeV [5] (see also [6]). Furthermore, the top-quark production process is considered as a standard candle process. Namely, it serves understanding detector performances, e.g. jet energy calibrations, from comparisons of experimental measurements with theoretically reliable or well-controllable predictions, for observables including jet topologies, backgrounds and underlying events.
Recently, tt invariant-mass (m tt ) distribution near threshold has been investigated incorporating the bound-state effects [20,21]. The effects are found to be significant at the LHC, since (in contrast to the Tevatron) the gluon-fusion channel dominates the cross section and there are significant contributions from (the remnant of) the color-singlet tt resonances.
In this paper we compute the fully differential cross sections for the top-quark pair productions and their subsequent decays at the LHC. In particular, we incorporate the bound-state effects, which are important in the tt threshold region, into the cross sections. We extend the studies of [20,21] and present a theoretical frame to incorporate the bound-state effects to the differential cross sections. Using this result, we compute various kinematical distributions of the top quarks and their decay products at hadron colliders, by developing a Monte-Carlo (MC) event-generator incorporating the bound-state effects. (There exist similar MC event-generators for computing the top-quark cross-sections in the tt threshold region at future e + e − colliders [22,23,24].) Through the analysis, we elucidate the nature of the bound-state effects at various stages: at the partonic matrix-element level, both with and without including the decay of the top quark, and in the kinematical distributions after incorporating the initial-state radiation (ISR) effects. Theoretically the fully differential cross sections contain more information on the bound-state effects than just the tt invariant-mass distribution; for instance, it is known that the top momentum distribution is sensitive to the resonance wave functions in momentum space [25,26]. From a practical point of view, the differential cross sections are useful for studying effects of various kinematical cuts, detector acceptance corrections, detector calibrations, etc.
The method for incorporating tt bound-state effects has been developed mainly in the studies of tt productions in e + e − collisions [27,28,29]. Formally, in the limit where we neglect the top-quark width, Γ t → 0, bound-state effects can be incorporated by resummation of the Coulomb singularities (α s /β) n , where β is the velocity of the top quark in the tt c.m. frame. In contrast to the e + e − collision, at hadron colliders, tt pairs are produced in both color-singlet and octet states, and the (partonic) collision energy is not fixed. Due to the latter reason, we have to set up a theoretical framework which is valid both in the threshold region (m tt ≃ 2m t ) and in the high-energy region (m tt ≫ 2m t ). The former region is where the bound-state effects (Coulomb corrections) become significant and where the non-relativistic approximation is valid. On the other hand, in the latter region, the bound-state effects are not significant and the top quarks are relativistic. We present a framework which takes into account all the leading-order (LO) corrections in both regions. Namely, we incorporate all the (α s /β) n terms in the threshold region, while we include all the β n terms in the relativistic region. (Some of the important subleading corrections are also incorporated.) Furthermore, we interpolate the two regions smoothly in a natural way.
Another important aspect in computing the differential cross sections for the top-quark productions and decays is to construct full amplitudes corresponding to the bW +b W − final distributions in tt production, using the MC simulation which implements the ingredients explained in the previous section. In Sec. 4, we summarize our results. To avoid complexity in the main body of the paper, several detailed discussions are presented in the Appendices. In App. A, we identify the tt Green function in a Feynman amplitude. In App. B, we derive the off-shell suppression factor. In App. C, the color decomposition of the amplitude is explained. In App. D, we examine the leptonic decays of W 's from top quarks with and without spin correlations.

Inclusion of Bound-state Effects
In this section we present a theoretical investigation of how to include the tt bound-state effects in the matrix elements for gg → bW +b W − and qq → bW +b W − . In particular we include the effects such that the amplitude is correct in the leading-order approximation both in the tt threshold region and in the high energy region. Inclusion of several different effects is explained in steps: In Sec. 2.1 we explain how to incorporate the bound-state effects; in Sec. 2.2 important higher-order effects of the large top-quark decay-width are incorporated; in these subsections, we consider only the partonic S-matrix elements. In Sec. 2.3 we incorporate the ISR effects and the K-factors in the corresponding partonic differential cross sections. For later convenience, we divide each amplitude into two parts, the tt (double-resonant) part and the non-resonant part, as where I = gg or qq represents initial-state partons, and c represents the color (c = 1 and 8 for the singlet and octet, respectively) of I, or equivalently, of bb in the final-state. The first term on the right-hand side represents the sum of the diagrams which contain both t andt as an intermediate state. This part of the amplitude consists of I → tt processes followed by subsequent decays of t andt. The second term represents the sum of the rest of the diagrams, which consists of single(-top)-resonant diagrams and non-resonant diagrams. Fig. 1 shows the tree-level Feynman diagrams for the processes gg → tt and qq → tt. Some examples of the tree-level diagrams included in each part for I = gg are shown in Fig. 2.
In general each part is gauge-dependent. In this paper, we work in Feynman gauge for SU (3) c and in unitary gauge for the broken electroweak symmetry. In computing the tree-level Feynman diagrams which contain the top-quark propagators, we include the (on-shell) top-quark decay-width Γ t in the propagator denominator as (2.2) nr , respectively. The term "resonant" is used to refer to the t ort quark propagator (shown with double line) which can become close to on-shell. The blob in the left diagram represents the first three diagrams in Fig. 1.

LO cross section valid from threshold to high energies
In this subsection we include the bound-state effects in the tt amplitude M (c) tt . We consider the narrow-width limit of top-quark width in this subsection. Namely, we take into account only the leading contributions as Γ t → 0. Important subleading effects by the finite topquark width will be investigated separately in the next subsection.
We start by reviewing the conventional method for including the bound-state effects in the matrix element (or in the fully differential cross section) of e + e − → tt → bW +b W − close to the threshold of tt pair productions. At the leading-order, this is achieved by multiplying the tree-level amplitude corresponding to the diagrams e + e − → tt → bW +b W − by an enhancement factor as [25,26] Here, the non-relativistic Green functions are defined by § is the c.m. energy measured from the threshold; m t is the pole mass of the top quark; r denotes the relative coordinate of t andt, while p denotes the three-momentum of t (or minus the three-momentum oft), both defined in the c.m. frame; V (c) QCD (r) is the QCD potential between the tt pair in the color-singlet (c = 1) or color-octet (c = 8) channel. In e + e − collisions, tt pairs are produced in the color-singlet channel, hence c = 1 in Eq. (2.3). The free non-relativistic Green function G 0 (E + iΓ t , p) is obtained from G (c) (E + iΓ t , p) by ‡ There are two tree-level diagrams for e + e − → tt → bW +b W − with γ and Z boson intermediate states.
M(e + e − → tt → bW +b W − )tree denotes the sum of them.
§ In this study, G (c) (E + iΓt, p) is computed numerically by solving the Schrödinger equation in coordinate space and taking Fourier transform [25]. Alternatively, one may solve the Schrödinger equation in momentum space directly [26].
setting V (c) QCD (r) → 0. Formally the above Green function can be expressed as using an operator notation in quantum mechanics. By definition, the above Green function contains only the S-wave contributions. There are two methods to compute the total cross section for e + e − → tt → bW +b W − incorporating the bound-state effects. One way is to integrate the absolute square of the matrix element given in Eq. (2.3) over the phase-space of the final bW +b W − state. The other method is to use the optical theorem (unitarity relation) and take the imaginary part of a current-current correlator (two-point function). At the leading-order, the latter method leads to the formula [27,28]: where the Green function in coordinate space is defined in Eq. (2.4). σ tot (e + e − → tt) tree denotes the Born cross-section for the production of on-shell top quarks. Note that in the denominator we set Γ t to zero inG 0 , whereas Γ t is retained in the denominator in Eq. (2.3 incorporate all the leading-order corrections ∼ (α s /β) n in the threshold region E ≪ m t . On the other hand, the formulas are not valid at higher c.m. energies E > ∼ m t , since relativistic corrections ∼ β n , which grow with energy, are neglected in these formulas. Now we turn to the partonic cross sections for the top-quark productions in hadron collisions, I → tt → bW +b W − with I = gg and qq. Unlike e + e − collisions, the collision energy of the initial state cannot be fixed. Thus, we need to consider both threshold and high-energy regions. The formulas which we propose, valid in both regions within the leading-order approximation, can be summarized as follows: (1) The tt amplitude for I → tt → bW +b W − is given by with The Feynman diagrams which contribute to the tree-level amplitude M (c) tt (I → tt → bW +b W − ) tree are those shown in Fig. 1, after attaching the decay vertices t → bW + andt →bW − to each diagram. There are both color-singlet (c = 1) and color-octet (c = 8) channels in the case I = gg, while there is only the color-octet channel in the case I = qq. Here, E is defined from the tt invariant-mass m tt as E = m tt − 2m t . The only essential difference from the corresponding formula for the e + e − collision, Eq. (2.3), is the use of the modified energy E ′ [Eq. (2.9)] instead of E.
(2) We may compute the tt invariant-mass distribution by integrating the absolute square of the above amplitude |M tt (I → tt → bW +b W − )| 2 over the bW +b W − phase-space for each fixed m tt . Alternatively we may obtain the formula for the tt invariantmass distribution using the optical theorem, similarly to the e + e − → tt case. In the leading-order approximation, this readŝ .

(2.10)
As in the e + e − → tt case,σ (c) tot (I → tt) tree is the Born cross-section for the on-shell top quarks; accordingly Γ t is set to zero inG 0 (E ′ , r = 0 . We note that the tt invariant-mass distribution obtained from Eq. (2.10) does not exactly coincide with that obtained by integrating |M (c) tt (I → tt → bW +b W − )| 2 over the bW +b W − phasespace; the difference is O(Γ t /m t ) and will be discussed in the next subsection.
In the following we sketch our theoretical consideration which led to the above formulas (2.8) and (2.10), where some details are relegated to App. A. As explained in that appendix, part of the Feynman amplitude for I → tt → bW +b W − can be identified with a Green function that dictates the time evolution of the tt system. (Such an identification is possible in all kinematical regions.) In the c.m. frame of tt, for the initial-state | i and final-state | f of the tt system, this Green function can be written formally as where m tt is the c.m. energy of tt (tt invariant-mass). The full QCD Hamiltonian is denoted by H. Because of this property, the amplitude for I → tt → bW +b W − , which incorporates the tt bound-state effects, can be obtained from the tree-level amplitude by multiplying an enhancement factor: (2.12) ¶ As already stated, treatment of the top-quark width is correct only in the leading order.
H 0 denotes the Hamiltonian H after setting α s → 0, i.e. the free Hamiltonian. As before, r denotes the relative coordinate of t andt, while p denotes the three-momentum of t, both defined in the c.m. frame of the tt system. Corrections to Eq. (2.12), which come from the non-resonant part of the amplitude (i.e. that vanish as Γ t → 0), are neglected; see Appendix A.
In the above equation, we have taken advantage of the fact that we work in the leadingorder approximation and set the initial-state of the Green functions as | r = 0 . This follows from the following consideration. Naively, the tt system cannot be regarded as being created by contact interaction (i.e. at the same point r = 0) in the t-and u-channel diagrams of gg → tt, in which the top-quark is exchanged between the initial gg (see Fig. 1). In the nonrelativistic region, however, we may set p → 0 in the t-or u-channel top-quark propagator in the leading-order approximation. The denominator of the propagator effectively reduces to a constant close to the threshold, so that tt can be regarded as being created at the same point. On the other hand, in the relativistic region the enhancement factor in Eq. (2.12) reduces to 1 + O(α s ). Hence, it is justified to evaluate the Green functions with the initialstate | r = 0 within our present approximation.
Since our aim is to include the leading-order contributions both in the relativistic and non-relativistic regions, the full form of the Hamiltonian is not necessary. In the region where t andt are relativistic, m tt − 2m t > ∼ m t , the leading-order contribution in the Hamiltonian reads as shown in Appendix A. It is nothing but the sum of the energies of free on-shell t andt. The above equation also indicates how the next-to-leading order effects enter the Hamiltonian. On the other hand, in the non-relativistic region, E = m tt − 2m t ≪ m t , the leading-order contributions in the Hamiltonian can be written explicitly as 14) It is indicated that the next-to-leading order corrections enter as O(β) relativistic corrections or O(α s ) corrections. A natural choice of the Hamiltonian, which incorporates the leading-order contributions in both regions and smoothly interpolates these regions, is given by In fact, it is well known that, when computing higher-order corrections in Coulombic bound-state problems, part of them (relativistic corrections) contribute exactly in the above form [37]. Thus, in principle, one may determine the enhancement factor in Eq. (2.12) using the above Hamiltonian. Due to technical reasons, however, we use an alternative form of the enhancement factor, which is equivalent within the present approximation. By substituting E = m tt − 2m t to the on-shell relation Therefore, if we define with E ′ defined by Eq. (2.9), the position of the pole of G (c) (E ′ + iΓ t , p) is the same as that of p|(m tt − H LO + iΓ t ) −1 | r = 0 in the limit α s → 0. Furthermore, in the non-relativistic region, evidently G (c) (E ′ , p) is the same as p|(m tt − H LO + iΓ t ) −1 | r = 0 in the leadingorder approximation. Hence, we may compute the matrix element by the formula Eq. (2.8).
One may be worried that, although the replacement E → E ′ correctly accounts for the pole position, the residue of the pole may be altered significantly from that of Eq. (2.12) in the relativistic region. This is not the case, since the change of the residue will be canceled in the ratios of the Green functions in Eqs. (2.8) and (2.10). The advantage of using G (c) (E ′ + iΓ t , p) is that one can obtain it from the conventional non-relativistic Green function with a minimal modification E → E ′ = E + E 2 /(4m t ). In particular, properties of G (c) are fairly well known.
Let us comment on the dependence of the Green function on the top-quark width Γ t in Eq. (2.18). In Eq. (2.2) the shift of the pole position of the top-quark propagator due to the finite top-quark width can be incorporated simply by a replacement m 2 t → m 2 t − im t Γ t . If we apply it to Eq. (2.16), one finds that iΓ t will be added to the left-hand side of Eq. (2.17). Hence, inclusion of Γ t as in Eq. (2.18) is correct in the leading-order approximation.
Throughout our analysis, we include an important subleading correction to the boundstate effects, in order to make our analysis more realistic. This is the NLO (1-loop) correction to the static potentials V    The difference between the tt invariant-mass distributions using Eqs. (2.8) and (2.10) (solid and dotted green lines) is due to O(Γ t /m t ) corrections. The replacement E ′ → E in Eq. (2.10) changes the tt invariant-mass distribution slightly above the tt threshold; compare the green dotted and red dotted lines. The difference between the two cross sections is about 2.5% in the large m tt region.
On the other hand, the effect of the replacement E ′ → E in Eq. (2.8) is much more pronounced above the tt threshold. There exist a large enhancement which amounts to In Secs. 2.1 and 2.2, the partonic tt invariant-mass distribution (before including the effects of ISR and parton distribution function) is proportional to the partonic total cross-section and delta function, dσ/dm tt ∝σδ(ŝ − m 2 tt ); c.f. eq. (17) of [20]. Hence, we plotσ(ŝ = m 2 tt ) instead of dσ/dm tt in these subsections. tot,tree in Eq. (2.10) does not contain the t ort propagator, so that this mismatch problem does not occur when we replace E ′ by E in Eq. (2.10); compare the green and red dotted lines.] In fact, the mechanism of this abnormally large deviation is closely tied to a characteristic bound-state effect on the invariant-mass distributions of the bW + andbW − systems. We will investigate this issue in detail in Sec. 3, in which we examine closely the differential distributions. Nevertheless, even without going into these details, the present comparison clearly shows the necessity of a proper treatment of the relativistic kinematics, when we include the bound-state effects to the fully differential cross section of the process I → tt → bW +b W − .
Since our formulas are correct in the narrow width limit Γ t → 0, the unitarity relation should be restored in this limit. In order to check this, in Fig. 4 we plot the ratios of dσ/dm tt computed using Eqs. (2.8) and (2.10) as we vary the value of Γ t at a fixed m tt of 360 GeV. * * We confirm that the ratio approaches to unity as Γ t is reduced, in the case that we use our relativistic formulas (green solid line) or in the case that we use the tree-level cross sections (black dotted line). In sharp contrast, the ratio does not approach to unity in the case that we replace E ′ by E (red dashed line) due to the mismatch problem. It shows invalidity of the non-relativistic approximation far above the threshold, especially for the fully differential cross section. * * As we vary the value of Γt in the t propagator and the Green functions, the value of the weak gauge coupling constant gW in the tbW vertex is varied consistently, such that the tree-level top-quark width takes the correct value. As is well known, the leading-order bound-state effects in the tt threshold region are contained in the S-wave part of the amplitude. In the case of gg → tt, the S-wave contributions reside in the J = 0 amplitude both for the color-singlet and color-octet channels. Hence, it may be more appropriate to include the bound-state effects only in the J = 0 amplitude, rather than multiplying the whole tt amplitude by the enhancement factor as in Eq. (2.8). Theoretically, the difference between the two prescriptions is subleading. We examine this feature by comparing the tt invariant-mass distributions computed in both ways. In Fig. 5, we plot the tt invariant-mass distributions for gg → tt process. Each solid line represents the cross section computed using Eq. (2.10), namely, the whole Born crosssection (the sum of the Born cross-sections for all J's) is multiplied by the enhancement factor. Each dashed line represents the sum of the cross sections for all J's, where only the J = 0 cross section is multiplied by the enhancement factor. The red (solid and dashed) lines represent the cross sections for gg → tt in the color-singlet channel, while the blue (solid and dashed) line represent those in the color-octet channel.
The cross section in the singlet channel is more enhanced if we use the overall prescription, Eq. (2.10), since the force between the color-singlet tt pair is attractive and hence the enhancement factor is larger than one. The difference of the two prescriptions is sizable only above the tt threshold and becomes maximal around m tt ≃ 400 GeV, where the difference is about 7%. On the other hand, the cross section in the octet channel is more reduced if we use the overall prescription, since the force between the color-octet tt pair is repulsive and the enhancement factor is (slightly) less than one. The difference of the two prescriptions is at most 2%. The black lines (solid and dashed) represent the sum of the cross sections for gg → tt in the above two channels. The difference of the two prescriptions in this case is at most about 1%, since the differences have opposite signs in the two channels and are largely canceled. Thus, the difference of the two prescriptions is rather small and much smaller than other subleading corrections which we neglect in our analysis. Furthermore, we have checked that the above tendencies are not changed significantly by the ISR effects. Therefore, for simplicity of our analysis, we will adopt the overall prescription in the following analysis, namely, we will not decompose the amplitude into different J's.
In the case of qq → tt, there is only the J = 1 color-octet channel at tree level. Hence, the enhancement factor multiplies the whole amplitude also in this case.

Effects of large Γ t
In this subsection we describe how we incorporate part of the subleading corrections that are induced by the large top-quark width. As an inevitable consequence of numerically integrating the fully differential cross sections for I → tt → bW +b W − , there is a significant phase-space-suppression effect. We partly compensate this effect, which is related to a gauge cancellation inherent in the inclusive cross section.
First, we briefly review existing theoretical studies on the treatment of the top-quark width, in the cases with and without bound-state effects. In the latter case, many schemes have been proposed for incorporating the top-quark width. The use of the top-quark propagator in Eq. (2.2) is called the fixed-width scheme (FWS). It is widely used in simple analysis of the cross sections which include the top quark as an unstable intermediate particle. It is known, however, that subleading electroweak effects are not properly treated in this scheme. At present, the complex-mass scheme (CMS) [39] seems to be most advanced from a practical point of view, due to the simplicity of its implementation. In fact, for the process e + e − → W + W − , which is kinematically similar to tt productions, the fully differential cross section has been computed incorporating the effects of W -boson width with NLO accuracy in this scheme, basically in all kinematical regions [40]. For tt productions in hadron collisions, the fully differential cross sections is computed incorporating top-quark width with LO accuracy in CMS, and various differential cross sections in different schemes were compared [30]. In particular, the study has shown an agreement within errors between all the calculated cross sections in CMS and in FWS. (FWS is simpler but less sophisticated than CMS.) Regarding tt productions in the threshold region including the bound-state effects, studies on the finite-width effects are most advanced in the total cross section for e + e − → tt. The finite-width effects have been incorporated with NNLO accuracy * [41,42], using the velocity-Non-Relativistic QCD (vNRQCD) effective field theory framework [43,44].
Recently an NLO correction to the total cross section arising from the single-top resonance region has been pointed out and computed in [31], using unstable-particle effective field theory [45,46]. On the other hand, in the corresponding fully differential cross section the width effects are incorporated only up to NLO accuracy [47] (apart from the contributions from the single-top resonance region). MC generators, developed specifically for simulation * In the tt threshold region, it is customary to count Γt/mt studies in the threshold region of the e + e − → tt process, have incorporated both boundstate effects and finite-width effects in the LO approximation [22,24]. For tt productions in hadron collisions, only the tt invariant-mass distributions have been computed with NLO accuracy, incorporating the bound-state effects, finite-width effects and ISR effects in the threshold region [20,21].
One effect is known to be particularly important in computing the fully differential cross sections in the threshold region of tt productions. It is the phase-space-suppression effect [25,26,48], which is formally an NNLO effect of the top-quark width, but it seriously modifies the shape of the sharply rising S-wave cross section as a function of m tt , after integrating the differential cross section over the final-state phase-space [25]. Let us briefly explain this effect. The tt cross section starts to rise below the tt threshold m tt = 2m t as a result of formation of tt resonances. This means that the dominant kinematical configuration is such that one of t andt is on-shell and the other is off-shell. Therefore, the phase-space of bW which decayed from the off-shell t ort is reduced as compared to the on-shell case. This suppresses the production cross section, and this effect is automatically incorporated if we integrate the LO differential cross section numerically over the phasespace of the final bW +b W − . A remarkable feature is that there is another effect at NNLO which exactly cancels the phase-space-suppression effect, for the integrated cross section at each tt invariant-mass [49,50]. This is the Coulomb-enhancement effect due to gluon exchanges between t andb (decayed fromt) and betweent and b. The cancellation is guaranteed by gauge invariance (Ward identity). It protects the tt resonance widths from being determined by gauge-dependent off-shell width of the top quark. Consequently the only surviving NNLO effect to the tt resonance widths turns out to be the time-dilatation effect due to the relative motion of t andt inside the resonances, which is gauge independent. † Thus, we face a problem when we compute differential cross sections in the tt threshold region by a MC generator: The phase-space-suppression effect is automatically incorporated, while the Coulomb-enhancement effect due to gluon exchanges between t andb (ort and b) is difficult to incorporate in a MC generator. (This is not yet achieved even in theoretical computations of the e + e − → tt differential cross sections.) Our prescription in this study is only effective. Since we know that the phase-space-suppression effect is canceled in the inclusive tt cross section, we multiply the tt amplitude M   At the differential level, the above treatment of the cancellation of the phase-spacesuppression effect is only effective, since the Coulomb-enhancement effect does not cancel the phase-space-suppression effect at each kinematical point. Nevertheless, we consider that a higher priority should be given to the gauge cancellation mechanism that is inherent in the inclusive cross section. We also note that the replacement E → E ′ in the Green function, Eqs. (2.8) and (2.9), automatically incorporates the time-dilatation effects to the resonance widths. ‡ In Fig. 6, we compare the tt invariant-mass distributions for gg → tt and qq → tt in all the channels, which are computed by integrating the following four cross sections over the bW +b W − phase-space, where the tt invariant-mass m tt is defined as the invariant-mass of the final bW +b W − system: (i) |M nr | 2 (red dot-dashed), and (iv) the absolute square of our formula Eq. (2.22) (green solid). Comparing the distributions for (i) and (ii), or, (iii) and (iv), we see that the phase-space-suppression effects are sizable especially close to the threshold of gg → tt in the color-singlet channel. This is consistent with the explanation given above. In particular, in each figure the difference between (iii) and (iv) is hardly visible at large m tt . We may also compare the distributions for (i) and (iii), or, (ii) and (iv), to see the contributions of the non-resonant amplitude. The contributions become comparatively larger at high energies for gluon-fusion channels, since the contribution of the s-channel diagram in M In Fig. 7(a) we plot the tt invariant-mass distributions for gg → bW +b W − (sum of the color-singlet and octet channels), using Eq. (2.22) after integrating over the bW +b W − phase-space (red solid line). The tt invariant-mass distribution at the Born level for gg → ‡ The relation E ′ + iΓt = p 2 /mt (corresponding to m 2 t → m 2 t − imtΓt) is relativistically correct, so that the time-dilatation effect enters the lifetime of the tt system. It can also be seen by the fact  The tt invariant-mass distributions for qq → bW +b W − are also plotted in Fig. 7(b). The enhancement factor is smaller than unity because of the repulsive force, whose values are about 0.96 and 0.98 at m tt = 400 GeV and 500 GeV, respectively.

Inclusion of ISR effects and K-factors
In this subsection we explain how we incorporate the ISR effects in the cross sections in our framework. In addition, we determine the K-factors to match our predictions for the tt invariant-mass distributions to the available NLO predictions.
In hadron collisions, it is important to include the ISR effects. In our framework, they are incorporated by connecting the differential cross sections computed from the matrix elements Eq. (2.8) to a parton-shower simulator such as PYTHIA [51] or HERWIG [52]. In addition, we include "K-factors" as the normalization constants of the cross sections for I → bW +b W − in the individual channels. § The K-factors are determined such that the tt invariant-mass distribution for each channel in the threshold region matches the corresponding NLO prediction in the threshold region. We also extrapolate these Kfactors to the large m tt region. The main reason to do so is a lack of the NLO predictions in the large m tt region for the individual channels: The present theoretical prediction for the NLO tt invariant-mass distribution in the large m tt region is provided numerically only for the color-summed cross section for on-shell top-quark productions [12]. Theoretically, by naive extrapolation of the K-factors, we reproduce the double-logarithmic terms of the cross sections correctly in the large m tt region, due to the universal structure of softgluon emissions; on the other hand, we do not reproduce the single-logarithmic and nonlogarithmic terms.
The NLO corrections to the tt invariant-mass distributions in the threshold region are known for the individual channels [20,21]. The corrections are given in terms of the hard-correction factors and the gluon radiation functions. The major difference of the predictions of [20] and [21] is that in the latter predictions contributions from high √ŝ § Note that the parton shower simulators incorporate ISR effects by way of stochastic processes, and that the tt invariant-mass distributions handed to the simulators at the parton level are not affected by the ISR effects.
(the c.m. collision energy of the initial partons) are included more accurately. ¶ Hence, we use the latter predictions to compute the K-factors . We can determine the K-factors by taking the ratios of these NLO partonic cross sections and our (LO) partonic cross sections given in Sec. 2.2. Since the NLO cross sections [20,21] do not include contributions from non-resonant diagrams M In general, the K-factors depend on m tt . We first examine m tt -dependences of the K-factors as we choose different renormalization and factorization scales, µ R and µ F . The renormalization scale µ R enters the NLO formula as the scale of the strong coupling constant and also through the logarithmic term in the hard-vertex function; see Eq. (3.2) of [21]. On the other hand, the factorization scale µ F enters the NLO formula as the scale of the parton distribution functions (PDFs) and through the terms with ln m 2 tt /µ 2 F in the gluon radiation functions; see Eqs. (3.4-3.7) of [21]. We find that, the m tt -dependences of the K-factors can be relatively flat in the threshold region, with appropriate choices of µ R and µ F . In this case, extrapolation of the K-factors from the threshold region to the high m tt region can be performed trivially. Indeed, for simplicity of our analysis, we take the K-factors to be independent of m tt . In Fig. 8, we plot the K-factors of the tt invariant-mass distributions in the individual channels at the LHC with √ s = 14 TeV, in the cases that we choose the scales as µ R = µ F = κm t with κ = 0.5, 1, 2. As can be seen, the m tt -dependence of the K-factors are mild. We have also examined the K-factors corresponding to the LHC with √ s = 7 TeV and Tevatron with √ s = 1.96 TeV; we find that the K-factors are only mildly dependent on m tt also in these cases. In Table 1, we list the numerical values of the K-factors for all the channels corresponding to the LHC √ s = 14 TeV, 7 TeV and Tevatron √ s = 1.96 TeV, obtained at m tt = 2m t for µ R = µ F = κm t with κ = 0.5, 1, 2. The values of the K-factors for κ = 1 will be used in the following.
We check consistency of our K-factor normalization in the large m tt region, by comparing our prediction for the color-summed tt invariant-mass distribution with the NLO prediction. In Fig. 9 we plot the ratio of these two cross sections. In the former distribution, we include the K-factors, while we do not include the non-resonant diagrams M [c.f. (ii) of Fig. 6]. The latter distribution is computed for the on-shell tt productions, by MC@NLO [34,35] with CTEQ6M PDFs with a scale choice µ F = µ R = m 2 t + p 2 T,t . As can be seen, both cross sections are mutually consistent within 2% accuracy up to m tt = 800 GeV.
Including non-resonant diagrams in a way that the gauge cancellation holds effectively, ¶ In the gluon radiation functions, the terms enhanced by plus-distributions or delta-functions as z → 1 are common in [20] and [21], while non-enhanced terms differ.
In [21], effects of resummation of threshold logs are also examined and found to enhance the normalization at 10% level. See also [53,54,55] Table 1: K-factor normalization constant for each channel (gg color-singlet, octet and qq) for the LHC √ s = 14 TeV, 7 TeV and the Tevatron, with setting the factorization and renormalization scales to µ R = µ F = κm t with κ = 0.5, 1, 2.

Event Generation and Top-Quark Distributions
In this section, we present numerical evaluations of various kinematical distributions of the top-quark computed from the pp → bW +b W − cross section, using the theoretical framework explained in the previous section. In particular we study the bound-state effects on these distributions. Our numerical calculations are carried out based on the MadGraph output [58] which makes use of the HELAS subroutines [59] for helicity-amplitude calculations. The original MadGraph output code has been modified to implement the color-decomposition and to include the bound-state effects via the Green functions. For the convenience of the readers, we collect the formulas necessary for decomposing amplitudes into the color-singlet and octet components of the tt (or bb) system in Appendix C. In particular, we discuss how to implement the color decomposition into the MadGraph notation. The bound-state correction factor c.f. Eq. (2.8), is pre-tabulated to save time for computing the momentum-space Green functions * . We perform phase-space integrations using BASES/SPRING [60], or alternatively by adapting our code to MadEvent [32,33] utilities (ver. 4.4.42), where both tools are able to generate unweighted events at the partonic final-state level. For each event, we assign the specific color-flow according to an ordinary manner, except the color-singlet channel, as explained in Appendix C. The generated events can be subsequently provided e.g. to PYTHIA for simulations of parton-showering and hadronizations.
In the main body of this paper, we do not consider the decay of W 's but consider only the observables constructed from the bW +b W − final state. The W -boson decays can be incorporated at the PYTHIA stage, where however the polarization of W -bosons cannot be taken into account. Alternatively, one can calculate the helicity amplitudes including the decay of W -bosons by specifying a decay mode for each W -boson. In Appendix D, as a sample case, we examine the distributions of dileptons in the dilepton mode, where both W 's decay leptonically, and study the effects of W -boson polarization and bound-state corrections.
Below we show the results at the partonic bW +b W − final-state level. We do not discuss the parton-showering and hadronization effects, in order to concentrate on the examination of bound-state effects. For the parton distribution functions, we use the CTEQ6L1 parameterization with the LO evolution (1-loop running) of the QCD coupling constant. We set the renormalization and factorization scales to µ R = µ F = m t and incorporate the K-factors obtained in Sec. 2.3 to the cross sections in the individual channels. [The final formula for the matrix element is given by Eq. (2.25).] We set the top-quark pole-mass, the (tree-level) on-shell top-quark width and the strong coupling constant as m t = 173 GeV, Γ t = 1.49 GeV and α s (M z ) = 0.1298, respectively.
In Fig. 10(a), we plot the tt invariant-mass distribution in pp → bW +b W − production at √ s = 14 TeV. The tt invariant-mass m tt is defined as the invariant-mass of the final bW +b W − system. The green solid line represents the full result which includes the boundstate effects as well as the K-factors, and the blue dashed line represents the Born-level result (the LO prediction in the conventional perturbative QCD approach). Fig. 10(b) shows a magnification of the same cross sections in the threshold region. As shown in [20,21], theoretically the bound-state effects can be seen most clearly in the shape of the m tt distribution in the threshold region. One can see that the cross section is enhanced over the Born cross-section significantly by the bound-state effects, and there appears a broad peak below the threshold corresponding to the 1 S 0 resonance state in the color-singlet tt channel. Far above the threshold, the bound-state effects disappear and the cross section approaches the Born-level distributions, up to the K-factor normalization. * The S-wave Green function depends only on | p| but not on the direction of the three-momenta.  In the same figures, we also compare our prediction with the NLO m tt distribution computed by MC@NLO [34,35] with CTEQ6M PDFs and the scale choice of µ F = µ R = m 2 t + p 2 T,t . The latter prediction includes the full NLO QCD corrections (but not the Coulomb resummation) for the on-shell tt productions; we switched on an option of MC@NLO to incorporate off-shellness of the top-quarks effectively by re-weighting the cross section by skewed Breit-Wigner functions [61], so that the cross section is non-zero below the threshold. (However, non-resonant diagrams are not incorporated.) Below and near the threshold, our prediction is much larger than the MC@NLO prediction, due to the bound-state formation. The two cross sections become approximately equal from around m tt ∼ 370-380 GeV up to larger m tt . Note that, in Fig. 9, the contributions from non-resonant diagrams are not included in our full prediction, whereas in Figs. 10 they are included.
Integrating the distributions over m tt , the total cross section by our full (Born-level) calculation is estimated as σ bW +b W − = 855 pb (633 pb), while we obtain σ tt = 816 pb as the MC@NLO prediction.
The shape of the m tt distribution at the LHC 7 TeV, shown in Fig. 11, is similar to that for the LHC 14 TeV. The total cross sections are estimated to be 158 pb, 106 pb and 146 pb by our full, Born-level calculations and MC@NLO, respectively.
Let us examine other distributions of the top quark. From Figs. 10 and 11, it is obvious that the phase-space region, where the bound-state effects are important, corresponds to a rather limited portion of the full top-quark events produced at the LHC. Thus, in various distributions formed by the full events, the bound-state effects may well be negligible in practice. In order to examine the bound-state effects closely, in the following we consider the events restricted by m tt ≤ 370 GeV (except where otherwise stated), instead of considering the full events. They amount to about 9% (8%) of the full events according to our calculation with (without) the bound-state corrections at the LHC 14 TeV. Due to the large tt cross sections at the LHC, still a large number of such near-threshold events would be accumulated. In this paper, we do not discuss the important subject of how to measure m tt in real experiments, which requires detailed studies of errors and fake solutions; one may find them in earlier studies [15,17].
One observes a characteristic bound-state effect in the (bW + )-(bW − ) double-invariantmass distribution. In Figs. 12, we show the density plots of the invariant-masses of the bW + andbW − systems, given by (a) the Born-level prediction and (b) our full prediction. In each figure, the number of events is normalized to 100,000 in total, and the number of events per bin (0.2 GeV×0.2 GeV) is plotted with graded colors. The Born-level prediction (a) is essentially determined by the product of the Breit-Wigner functions, hence the distribution is almost reflection symmetric with respect to the on-shell lines (p b + p W + ) 2 = m 2 t and (pb + p W − ) 2 = m 2 t . By contrast, the distribution by our full prediction (b) is not symmetric and biased towards the configuration, where one of t ort is on-shell and the other has an  Table 2: A fraction of events which satisfy |m bW − m t | ≤ kΓ t for both or either of the bW invariant-masses. The events with m tt ≤ 370 GeV as well as the full events at the LHC 14 TeV are considered. In the bracket is shown the result in Born-level.
invariant-mass smaller than m t . In fact, such a configuration is known to be the dominant configuration just below the threshold in e + e − → tt [25,24], although in that case deviation from the double Breit-Wigner distribution [ Fig. 12(a)] is more prominent. (Note that, below the threshold, t andt cannot become simultaneously on-shell.) In order to quantify the correlated deformation of the double-invariant-mass distribution, we count the fraction of the events for which both or either of the bW + andbW − invariant-masses satisfy where k = 1, 2, . . . , 5. These fractions are tabulated in Table 2 for our full prediction and for the Born-level prediction. The bound-state effect reduces the fraction for which both invariant-masses are close to on-shell more than the fraction for which either of the invariant-masses is close to on-shell. In the former case, the change of the fraction by the bound-state effect amounts up to about 4%. For comparison, we also tabulate the same fractions for the full events; in this case, the variation of the fractions are small and at most 1%. In any case, a proper understanding of this effect would be important, since it potentially biases the mass cut and may affect, for instance, the top-quark mass measurement with high accuracy. Let us explain the mechanism how the bound-state effects alter the double-invariantmass distribution. As shown in Appendix A, the leading part of the tt amplitude M  the case that tt is in the singlet channel (c = 1), the potential energy between t andt is negative, V QCD (r) < 0. Therefore, the denominator of the Green function become close to zero (hence, the Green function is most enhanced) ifp is somewhat larger than the on-shell momentum p OS ≡ m 2 tt /4 − m 2 t , i.e.,p > p OS . On the other hand, the second factor (p 0 t − p 2 + m 2 t ) −1 + (p 0 t − p 2 + m 2 t ) −1 is most enhanced whenp = p OS , since p 0 t + p 0 t = m tt . Thus, there is a competition between the two factors on the right-hand side of Eq. (3.3). As a consequence, the dominant configuration is the one in which neither of the two factors are maximal. In fact, in the dominant configuration one of t andt is on-shell and the other is off-shell:p 2 t = m 2 t ,p 2 t < m 2 t , or,p 2 t < m 2 t ,p 2 t = m 2 t . The effect is opposite in the case that tt is in the octet channel (c = 8). Since the magnitude of the octet potential is much smaller than the singlet potential, V , for the color-singlet and octet channels, respectively. To see essential features, only the tt diagrams are taken into account and the K-factors are not included. In each figure, the black solid (blue dotted) line shows our full prediction (Born-level prediction). The peak momentum for each distribution is shown with a vertical line. The peak momenta of the Born-level distributions are (to a good approximation) the on-shell momentum,p peak ≈ p OS = 65.5 GeV. We see that the bound-state effects shift the peak momentum by about 0.7 GeV to a larger value for the color-singlet distribution, while the peak momentum of the color-octet channel is shifted only by 50 MeV to a smaller value. In the color-summed cross section, the peak momentum is shifted to a larger value. Consequently, one of the invariant-masses of the bW + andbW − systems is reduced below m t . The integral of this effect over the region m tt ≤ 370 GeV can be seen in Fig. 12(b).
One may suspect that the above shifts of the invariant-masses (or the shift of the peak momentum) may be an artifact of our specific method to interpolate the tt cross sections in the threshold region and in the higher m tt region. To check this, let us estimate the size of the shift of the peak momentum at m tt = 370 GeV and compare it with the above prediction. The distance a top-quark propagates before it decays is estimated as γcτ = m tt /(2m t Γ t ) = 0.72 GeV −1 = 1/(1.4 GeV). This distance is considered to be within the range where the potential V Hence, the effect seen in Fig. 13(a) seems to be physical.
We note that the effect elucidated here is a kind of effect that can never be seen in perturbative QCD computations for the on-shell tt productions, such as those given in [34,35,36]. This is because, the effect originates from the exchange of Coulomb gluons between off-shell t and on-shellt (or vice versa). Our full prediction correctly incorporates the (gauge-independent) LO off-shellness of the top quark as dictated by the exchange of Coulomb gluons, which is crucial for predicting the deformations of the top-quark momentum distribution and the double-invariant-mass distribution of the bW + andbW − systems.
Now we are in a position to understand the origin of the abnormally large enhancement of the cross section, which we observed in Sec. 2.1, in the case that we use the nonrelativistic formula for the differential cross section at large m tt ; see the red dot-dashed line in Fig. 3. The non-relativistic formula corresponds to replacing the Hamiltonian H = 2 p 2 + m 2 t + V . Thus, the non-relativistic formula overestimates the kinetic energy of the tt system in the large m tt region, 2m t +p 2 /m t > 2 p 2 + m 2 t . For this reason, the two factors on the righthand side of Eq. (3.3) can be brought close to maximal simultaneously with a nearly on-shell momentum,p ≃ p OS , since all the denominators in this expression nearly vanish. Since the individual factors are made of pole-type functions, applying an inaccurate kinematical relation only in one of the denominators can lead to a substantial overestimate of the cross section. In Fig. 13(a) we also plot the top-quark-momentum distribution computed with the non-relativistic formula, Eq. (2.8) after the replacement E ′ → E for the singlet channel (red dashed line). As can be seen, the peak momentum approaches the on-shell momentum and the distribution is more enhanced around the peak, compared to our full prediction.
Other top-quark distributions are less affected by the bound-state effects. In Fig. 14(a), we show the normalized distribution of the top-quark momentum p = | p t | = | p b + p W + | in the laboratory frame. In Fig. 14(b), we show the normalized distribution of the invariantmass of bW (bW + orbW − ). The Born-level and full predictions are shown by the green solid and blue dashed lines, respectively. These lines in Fig. 14(b) correspond to the projections of Figs. 12(a,b) to the m bW + (or mb W − ) axis. All the histograms in Figs. 14(a,b) are normalized, such that their integrals take the same value.
In e + e − collisions, the top-quark momentum distribution in the threshold region is known to be proportional to the absolute square of the momentum-space Green function [25,26], whose shape is strongly influenced by the bound-state effects. At hadron colliders, the top-quark momentum is boosted along the beam direction, and also the partonic collision energy is not fixed. As a result, even if we limit the events to those with m tt ≤ 370 GeV, the distribution of the top-quark momentum (p, defined in the lab. frame) is not much affected by the bound-state effects at hadron colliders. The m bW distribution is important for the determination of the top-quark mass, hence it should be understood well. The bound-state effects deform the Born-level m bW distribution towards the lower side. The mean values of m bW over the range |m bW − m t | < 5 GeV are estimated to be 172.7 GeV and 172.9 GeV, for the full and Born-level predictions, respectively. The change of the mean value is about −200 MeV, for the restricted events with m tt ≤ 370 GeV. At the LHC 7 TeV, we obtain almost the same result as in the 14 TeV case. At the Tevatron 1.96 TeV, where the qq color-octet channel dominates, the mean values of m bW are estimated as 172.96 GeV and 172.98 GeV, respectively. Thus, the variation of the mean value is rather small. Note that MC@NLO predicts a m bW distribution similar to the Born-level distribution, since it simply re-weights the on-shell tt cross-section by skewed Breit-Wigner functions.

Summary
In the first part of this paper (Sec. 2), we explain our theoretical framework for including the bound-state effects in the fully differential cross sections for the top-quark production and their subsequent decay processes at hadron colliders.
We formulate a theoretical basis to compute the fully differential top-quark crosssections, which are valid at leading-order both in the threshold and high-energy regions, and which smoothly interpolate between the two regions. The tree-level tt double-resonant amplitude for each process is multiplied by a correction factor, which is written in terms of the (well-known) non-relativistic Green function, but using a modified energy. This prescription preserves the required unitarity relation between the total and differential cross sections, which would be seriously violated had we used the naive non-relativistic formula for the differential cross sections at higher energies.
We also include into the cross sections two important subleading corrections induced by the large top-quark width. (i) In addition to the tt double-resonant diagrams, which receives the above bound-state corrections, we include the contributions of non-resonant diagrams, whose effects are comparatively larger at higher energies. (This is more or less trivial.) (ii) As long as we perform numerical integrations of the differential cross sections, a sizable phase-space-suppression effect in the threshold region is inevitable, due to the sizable off-shellness of top quarks. In order to effectively account for the gauge cancellation by the Coulomb enhancement, we compensate the phase-space-suppression effect (by hand).
Finally we incorporate ISR and the K-factors. ISR effects are incorporated by connecting our differential cross sections to a parton-shower simulator such as PYTHIA or HERWIG. We determine the K-factors for the cross sections in the individual channels by matching them to the corresponding NLO tt invariant-mass distributions in the threshold region. With an appropriate choice of scales and using m tt -independent K-factors, we have checked that the color-summed tt invariant-mass distribution agrees with the conventional NLO prediction by MC@NLO reasonably well at high energies.
In the latter part of the paper (Sec. 3), using the above fully differential cross sections, we compute numerically various kinematical distributions of the top quark, constructed from the momenta of the bW +b W − final state (at the parton level). Our computations are carried out by MC event-generation using MadGraph, after implementing the color decomposition and the bound-state corrections to the output codes.
We confirmed that our prediction reproduces the known NLO predictions for the tt invariant-mass distribution in the threshold region (which include bound-state effects) at the LHC 7 TeV or 14 TeV; in particular it exhibits the 1S resonance peak below threshold. Furthermore, our prediction approaches smoothly to the conventional NLO prediction (without bound-state effects) at higher invariant-masses, from around 30 GeV above the threshold.
We restrict the events to those with m tt ≤ 370 GeV (in the case 2m t = 346 GeV), corresponding to about 10% of the full events, and examine kinematical distributions other than the m tt distribution. In particular, a characteristic bound-state effect on the (bW + )-(bW − ) double-invariant-mass distribution is observed. The distribution is deformed from the double Breit-Wigner shape, towards the configuration in which one of the tt pair is close to on-shell and the other has a smaller invariant-mass than m t .
The effect can be understood as a consequence of a competition between the contributions from the (dominant) color-singlet Green function and from the t andt propagators. If the top-quark width were tiny, the Breit-Wigner distribution would tend to a deltafunction, and the top quarks would be forced to on-shell. Due to the large decay width, however, the binding effect (towards off-shell mass) and the Breit-Wigner constraint (towards on-shell mass) remain to be competitive up to a few tens GeV above the threshold. This effect lowers the mean value of each bW invariant-mass by a few hundred MeV for the above restricted set of events. The correlated deformation of the double-invariant-mass distribution may affect the mass cut and eventually the top quark mass measurement at the LHC. This requires further careful investigations. It would be worth emphasizing that the bound-state effect elucidated here can never be seen in the conventional perturbative QCD corrections to the on-shell tt productions, since the off-shellness of the top quark by the LO Coulomb binding effects plays a crucial role, and therefore it signifies a unique aspect of the present study.
We examine other distributions, namely the (single) bW invariant-mass distribution and top-quark-momentum distribution. The bound-state effects on these distributions as a whole are not very significant, although there are certain systematic tendencies in the small deformations of the distributions, such as the aforementioned shift of the mean value of the invariant-mass. Furthermore, the dilepton distributions are examined including the leptonic decays of W 's at the matrix-element level (Appendix D). We have confirmed the previous observations that the effects of W -boson polarization are quite significant, whereas we find that the bound-state effects are much smaller. == i p 0 t − ω( p) + iǫ 0 | ψ(0) | t; p + (terms without a single particle pole).
We may also express θ(x 0 3 − x 0 4 ) t ; − p | e −iH(x 0 g g g gt t g g t t g gt t Figure 15: Color-flow diagrams in the gg → tt amplitudes. Left diagram representing the colorfactor δ ab δ ij is for the color-singlet case, middle and right diagrams representing the color-factor (T a T b ) ij and (T b T a ) ij , respectively, are for the color-octet case.
Alternatively, we may express the amplitude in the following basis; M ab ij (p k , λ k ) = T a T b kinematical distributions of the decay daughters of W 's. As an example, we show some distributions in the dilepton decay mode pp → bW +b W − → bℓ + ν ℓb ℓ −ν ℓ , where ℓ = e, µ.
We calculate the amplitudes from the set of Feynman diagrams obtained by just adding the W → ℓν ℓ vertex to the Feynman diagrams for the bW +b W − final state. We generate the dilepton events at the LHC √ s = 14 TeV with standard kinematical cuts for the lepton momenta, |η ℓ | ≤ 2.5 and p T,ℓ ≥ 10 GeV. To avoid a singularity due to the vanishing running top-quark width, see Eq. (B.5) in Appendix B, we restrict the phase-space integral region to m bW > m b + m W . Note that the matrix-elements for m bW < m b + m W are very suppressed. We set the W -boson decay-width to Γ W = 2.05 GeV.
In Figs. 16, we plot three different distributions of kinematical variables constructed from the four-momenta of dileptons: (a1, a2) the invariant-mass of the two leptons, (b1, b2) the distance in the η-φ plane, ∆R ℓℓ = ∆η 2 ℓℓ + ∆φ 2 ℓℓ , and (c1, c2) the difference of the azimuthal angles, ∆φ ℓℓ . The first three graphs (a1, b1, c1) correspond to the events from all the m tt region, while the last three (a2, b2, c2) to the events with m tt ≤ 370 GeV. The solid lines represent our full prediction, and the dashed lines represent the Bornlevel predictions. To see the effects of the non-zero W -boson polarization, we also plot the distributions computed from the bW +b W − events followed by the leptonic decays of on-shell unpolarized W -bosons.
We find that, due to the bound-state corrections, all three distributions are shifted to the lower side, although the variations are fairly small even for the events with m tt ≤ 370 GeV. By contrast, the effects of the non-zero W -boson polarization is pronounced, especially for the events with m tt ≤ 370 GeV. The most evident difference can be seen in the ∆φ ℓℓ distribution, where the full calculation predicts that the number of events at ∆φ ℓℓ = 0 is more than twice than the number of events at ∆φ ℓℓ = π, while almost flat distribution by assuming the unpolarized W -bosons.
This finding is consistent with the similar study in Ref. [17] where the importance of the top-quark spin correlation in the ∆φ ℓℓ distribution is examined for the events with m tt ≤ 400 GeV. They compare the ∆φ ℓℓ distribution fully taking into account the top-quark spin correlation with that assuming the spherical top-quark decay into bW 's followed by the correlated W -boson decay. Our calculation assuming unpolarized W -boson decays includes the correct angular distributions in t → bW decays, but forcing spherical distributions in W → ℓν ℓ decays. Thus, to predict the dilepton observables, all the spin correlations in decays of top-quarks and also W -bosons are required. We confirm their finding that the difference in ∆R ℓℓ distribution comes from mainly the difference in ∆φ ℓℓ distribution.