COMPLETE DESCRIPTION OF POLARIZATION EFFECTS IN TOP QUARK DECAYS INCLUDING HIGHER ORDER QCD CORRECTIONS a

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1 COMPLETE DESCRIPTION OF POLARIZATION EFFECTS IN TOP QUARK DECAYS INCLUDING HIGHER ORDER QCD CORRECTIONS a BODO LAMPE Department o Physics, University o Munich, Theresienstrasse 37, D Munich bol@mppmu.mpg.de The complete set o matrix elements or all polarization conigurations in top quark decays is presented including higher order QCD corrections. The analysis is done in the ramework o the helicity ormalism. The results can be used in a variety o circumstances, e.g. in the experimental analysis o top quark production and decay at Tevatron, LHC and NLC. Relations to LEP1 and LEP physics are pointed out. 1 Introduction Since its discovery in 1995 the top quark has been an object o increasing interest. The production process or top quarks has been analyzed in various theoretical studies both or proton and e + e collisions. Early reerences on the lowest order cross section are 1 or pp collsions and or e + e annihilation. Higher order corrections to the cross section (total cross section, p T distribution etc.) have been calculated by several groups, 3,4 or pp and 5,6 or e + e. These total cross sections do not involve inormation on the top quark polarization. Such spin eects only come in, i one studies distributions o top quark decay products. In some cases, spin eects have been studied, i.e. the distribution o the top quark spin vector, in re. 7 or pp and in re. 8 or e + e collisions. These latter studies have not been extended to higher orders yet. However, an interesting step in this direction has been taken in re. 6, where a Monte Carlo progam including inal state spin terms has been written. Unortunately, that paper is in act concerned with higher order corrections to τ + τ production in e + e annihilation at lower energies and does not take into account axial vector couplings. More general results, including axial vector couplings, can be ound in 9, but not in the orm o a Monte Carlo program. Nothing is known about higher order spin corrections to the processes which induce top quark production in proton collisions (light quark annihilation q q t t and gluon gluon usion q q t t). In contrast to e + e annihilation, there are many higher order diagrams involved in these processes, so that the calculations are very diicult. a invited talk presented at the International Workshop on QCD and New Physics, Hiroshima,

2 O course, in studying distributions o top quark decay products one has to take into account the higher order top quark decay matrix elements, too. The spin averaged matrix elements have been calculated including higher order corrections. For example, oneloop QCD corrections are known to decrease the total width o the top quark by about 8%. I one studies distributions o top quark decay products with reerence to the production process, one needs in addition the spin correlations in the top quark decay matrix elements. In lowest order, these spin correlations are known, but in higher order only rudimentary results exist 10,11. In those calculations, higher order corrections to special decay product distributions were calculated. This corresponds to certain linear combinations o the polarized matrix elements. The present article describes how all the polarized matrix elements can be obtained in a complete and systematic way in the ramework o the so called helicity ormalism. Within the Standard Model, all couplings o the top quark to other particles are completely ixed by its mass and by a ew quantum numbers. For example, the coupling o the top quark to gluons is a pure vector coupling with strength g s, the coupling to the W boson is a V A coupling etc. The coupling to the W boson is particularly interesting or this article, because it induces the decay t W + b. b quark mass terms O(m b /m t ) can very probably be neglected in higher order corrections to the decay process, because they are known to be small ( 1%) in leading order. In this approximation it can be shown that the b quark is always let handed in the top quark decay process, both in lowest order and in higher order QCD. As will be seen, this strongly reduces the number o independent helicity amplitudes and makes the results quite intuitive. Helicity vs. Spin Vector Description Helicity amplitudes have been considered in many applications o phenomenological importance in high energy physics, like jet production 1, nonstandard eects in top quark processes 13, and many others. The idea is, irst to separate a given process into simpler subprocesses and then to explicitly evaluate all the possible spin amplitudes or the subprocesses in special Lorentz and Dirac rames. The results can aterwards be put together with the help o a master ormula (to be given below). For example, consider the lowest order helicity amplitudes or top quark decay in the Standard Model. They are given by e s W A(h t, h W ) = g ū 1/ (p b )γ µ 1 (1 γ 5)u ht (p t )ɛ µ h W (1) where g = and h t and h W label the spins or the top quark and the W boson, h t = ± 1 and h W = 0, ±1. The helicity o the (massless) bottom

3 quark is ixed to be 1/ by the V A nature o the interaction. Thus, in the Standard Model there are six amplitudes to be considered, and this number remains the same in higher order QCD (ater integrating over the gluon degrees o reedom). The six amplitudes can be used to deine the 36 elements o the density matrix ρ(h t, h W, h t, h W) := A(h t, h W )A(h t, h W). () Note that the amplitudes are only determined up to an overall phase, and that this arbitrariness goes away when orming the density matrix elements. Unortunately, higher order QCD corrections cannot be ully calculated on the level o amplitudes, but only on the level o the density matrix which contains in principle 36 degrees o reedom. As will be discussed below, hermiticity and CP invariance reduce this number, but still leave an appreciable set o matrix elements. The ull density matrix is in act needed in the master ormula, i one considers some combined production and decay process or the top quark. Namely, assume that top quarks are produced in some process ab t t and then decay according to t W + b and t W b, where the W s urther decay to light (massless) ermions, W + 1 and W 3 4. The cross section is then given by the master ormula σ = EXT INT A(h a, h b, h t, h t)a(h t, h W +)A(h t, h W )A(h W +)A(h W ) (3) where EXT = h a, h b denotes the spins o the external particles and INT = h t, h t, h W +, h W the spins o the internal particles o the process. A(h a, h b, h t, h t) are the helicity amplitudes or the production process, A(h t, h W ) or the decay o the antitop quark, and A(h W +) and A(h W ) are the amplitudes or the decay o the W + and W, respectively. Note that beore Eq. (3) the W + spin has been denoted by h W instead o h W +, and or simplicity it will again be denoted by h W below. The act that the ormula makes no explicit reerence to the spins o the massless ermions i, i=1,,3,4, has the same reason as the non appearance o the b quark spins. Namely, the h i are ixed by the V A nature o the W decays. Furthermore, i one is not interested in the decay o the antitop or o the W s, the corresponding amplitudes and helicities will not appear in the above ormulas. In that case, the t and/or the W s will be one o the external (EXT) particles, whose spins have to be summed over ater taking the square in Eq. (3). Besides neglecting the b quark mass, it is also a good approximation in Eq. (3) to take the internal particles on shell, because o-shell contributions 3

4 are suppressed by powers o the width Γ t and Γ W. More precisely, one has 1 (P M ) + M Γ = π MΓ δ(p M ) + O( Γ M ) (4) or a particle o mass M and 4 momentum P. For a top quark o mass 175 GeV, the width is about 1.5 GeV, so that terms o order Γ M can be neglected, in Γ particular in higher orders where O(α s M ) is o the order o a permille. Similar considerations apply to the W boson. In higher order the narrow width approximation Eq. (4) is o particular use in reducing the number o diagrams to be calculated. The reason is that all diagrams where a gluon runs rom the production part o the process to the decay part, and also the corresponding intererence diagrams, give contributions which are suppressed by powers o Γ M. Thereore, in this approximation the process can be really decomposed into a number o building blocks, the production block, the t decay blocks and the W decay blocks. In the narrow width approximation, these blocks are interrelated by spin indices, but not by gluon exchange. When one carries out the modulus squared in Eq. (3), it becomes apparent that in general the ull density matrix A(h t, h W )A(h t, h W ) is needed to calculate the cross section o the decay products. I have calculated the six amplitudes A(h t, h W ) using the chiral representation o γ matrices, in which γ 5 =diag( 1, 1, 1, 1) etc. This representation makes calculations with massless ermions (the b quark in the case at hand) quite transparent. Furthermore, I shall present results in two dierent Lorentz rames, in the rest rame o the top quark and the rest rame o the W boson. Both rames have special virtues, so it is worthwhile to study them both. The top quark rest rame is o course the natural rame to study top decays, and to look at distributions in the energies o the decay products etc. The W rest rame has the particular virtue that the amount o longitudinal W s can be read o most easily in this rame. I am quite sure there is a (complicated) transormation between the amplitudes in both rames. However, I was not able to derive it and, urthermore, ound it reasonably convenient to do the calculations in both rames separately. There is a popular alternative to the helicity ormalism, where use is made by the act that the spin o a ermion with 4 momentum P can be described by a spin vector S, a pseudo 4 vector which ulills S = 1 and S P = 0. In this ormalism one does not calculate amplitudes but (squared) matrixelements. The matrix element or the production o a t t pair with spin vectors s t and s t has the generic orm M P tr(p/ t + m t)(1 + γ 5 s/ t )...(p/ t + m t)(1 + γ 5 s/ t )... 4

5 = a + bs t + cs t + ds t s t (5) where, or example, d is a tensor with two Lorentz indices and b and d are 4 vectors. Similarly, the matrix elements or the decay o the top quarks have the orm M D = e + s t M D = ē + s t (6) and the ull matrix element or production and decay is then given by M = aeē bē ce + d. (7) According to this ormula, the cross section will not be just a product o a production and o a decay piece, but is given by a sum o such products. In act, Eq. (7) can be shown to be equivalent to Eq. (3) The Method The results to be presented were obtained in the helicity ormalism. Furthermore, they concern the oneloop QCD corrections to the lowest order matrix elements. The lowest order expressions are usually simple, whereas the oneloop expressions, in particular or the case o hard gluons are quite lenghty or arbitrary spin orientations. Furthermore, the hard gluon contributions cannot be treated on the level o amplitudes, because they have to be integrated over the gluon s energy and angles. One really has to go to the level o the (spin) density matrix, Eq. (), to do the phase space integrations. There is, however, one circumstance which simpliies the task. This is related to the act, that the oneloop QCD corrections to the total (spin averaged) width o the top quark are known 15. This allows to get rid o the inrared and collinear singularities present in the matrix elements, by orming suitable singularity ree combinations o the spin dependent and the spin averaged expressions. The point is that the inrared and collinear singularities are universal, i.e. independent o the spin direction, so that they will drop out in suitable dierences. We shall discuss our procedure in more detail in the next section, where QCD corrections to W q q are considered as a rather simple warming up exercise. 4 QCD Corrections to W q q This process is simpler because both outgoing quarks can be considered to be massless particles, so that their helicities are ixed by the V A nature o the decay. Thereore, the decay amplitudes depend only on the W spin and in lowest order are given by A(0) = sinθ A(±1) = 1 ± cosθ e ±iφ (8) 5

6 where θ and φ are the (polar and azimuthal) angles between the z direction (deined as the direction o the outgoing quark q) and the direction o the W as given in some LAB rame (it may also be considered as the direction to which the W spin points). These ormulae hold in the rest rame o the em W boson. An overall actor W has been let out in the amplitudes. Note s W that the integrated W width Γ W = e m W can be obtained rom the trace o 48πs W the density matrix h W A(h W )A(h W ) by multiplying with the square o the em actor W and by dividing by the well known 16πm 16 s W. W Using Eq. (8), the corresponding density matrix D hw,h A(h W)A(h W W ) can easily be calculated in lowest order to yield D lo (0, 0) = sin θ D lo (±1, ±1) = 1 (1 + cos θ)± cosθ D lo (+1, 1) = 1 sin θe iφ D lo (±1, 0) = (± cosθ sin θ sinθ) e±iφ. (9) The underlined terms reer to parity violating eects and cannot be measured in hadronic W decays, because the outgoing quark is detected in the orm o a jet and its lavor and charge cannot be identiied. The unctions D(h W, h W ) are sometimes called the decay unctions o the W. An upper index lo has been introduced in Eq. (9) in order to make clear that these are the Born level contributions to the density matrix. The aim is then to calculate higher order corrections to the decay unctions/density matrix, in the orm D(h W, h W ) = Dlo (h W, h W ) + α s π Dho (h W, h W ) (10) To accomplish this calculation, use was made o the well known higher order result or the spin averaged decay unction, the trace o the spin density matrix, which in our normalization is given by The point to notice is that one has D total = Dtotal lo + α s π Dho total = (1 + α s ). (11) π D(h W, h W ) = Dlo (h W, h W ) D total Dtotal lo + α s π D ho (h W, h W )Dlo total Dlo (h W, h W )Dho total (D lo total ) (1) 6

7 where the dierence D ho (h W, h W )Dlo total Dlo (h W, h W )Dho total is completely ree o singularities separately or hard and virtual gluons. In act, the diagrams with virtual gluon exchange contribute nothing to the ratio D(hW,h W ) D total. (In top decay, where we shall proceed similarly, the virtual diagrams will contribute, but only a inite amount.) The real gluon processes W q q g can be explicitly seen to give a inite contribution to the above dierence, i.e. the result is inite or E g 0 and θ g 0, where E g and θ g are the gluon energy and angle with respect to the quark direction 17. The integration over E g and θ g is thereore straightorward and one obtains the corrections to the W decay unctions in a very compact orm 17 D(h W, h W) = (1 + α s π )( α s π )[Dlo (h W, h W) α s π δ h W,h W ]. (13) This representation can only be obtained ater neglecting the irrelevant parity violating terms. Furthermore, in higher orders the angles θ and φ have been deined as to reer to the thrust instead o the quark momentum direction. These results have been applied to W pair production at LEP 17 and NLC 18. According to a master ormula similar to (3), the cross section or W pair production and decay in e + e annihilation was calculated including the higher order corrections Eq. (13) and nonstandard contributions. Our main motivation in studying this cross section was twoold: irst o all we wanted to know how QCD corrections to angular correlations o W decay products dier rom the naive expection o a constant K actor 1 + αs π. We ound that depending on the kinematic point, the deviations rom a constant K actor can be appreciable (o the order o a ew percent). Unortunately, at LEP with its ew thousand W events these eects are just at the edge to become visible. The situation is dierent at NLC with its larger statistics, where the QCD corrections really become relevant. secondly we have proven that QCD correction can mimic the presence o nonstandard physics. As has been shown by Monte Carlo studies 19, NLC is sensitive to nonstandard couplings as small as This is well below the magnitude o the QCD eects induced by Eq. (13) 18. In addition to the QCD corrections presented above, there are o-shell W corrections (with and without QCD 0 ) which are particularly important at LEP, i.e. near threshold. Unortunately, there is no space here to discuss them in detail. 7

8 5 Complete Lowest Order Analysis o the Spin Density Matrix or t bw The decay o the top quark has some extremely interesting physics eatures, which are related to the spin decomposition o the matrix element. In particular, it is well known that, due to the large top quark mass value, t decay is dominated by longitudinal W s. In act, the total width o the W is given by Γ t = Γ L + Γ T = G F m 3 t 8 π (1 m W m ) (1 + m W t m ) (14) t and the ratio o the number o longitudinal over transverse W s is given by Γ L /Γ T = m t. Note that the QCD corrections to Γ t, Γ L and Γ T are known m W 15,11 to be about -9% resp. 5%, and that all other correction eects like electroweak, b mass and inite width eects contribute roughly 1%. Note urther that Γ L is the most interesting source o loop corrections to the amous R b value [the partial width Γ(Z b b) measured at LEP1]. The point is that the exchange o longitudinal W s between the b-quarks gives rise to the celebrated corrections o order O(G F m t) m t. Unortunately, at LEP1 this m W is only a small loop eect which is o the order o 1%, because it is suppressed 1 by a actor 16π. In contrast, in t decays the longitudinal W s enter as the most dominant leading order eect, so that they can be studied much clearer. Top quark decay may be looked at in dierent rames; particularly interesting are the rest rames o the t quark or that o the W. In the W rest rame, or example, there is a very simple way to experimentally determine the ratio Γ L /Γ T and other spin dependent observables 11. O course, the two systems are related just by a simple boost. However, the transormation ormula between the spin amplitudes in the two systems is quite complicated, so that I preer to give results in the two systems separately. Let s start with the t rest system. In order to calculate the amplitudes (1), I have choosen the ollowing parametrization o momenta, polarization vectors and spinors. First, the momenta: p t = (m t, 0) p W = m t ( +, 0, 0, ) p b = p t p W (15) where ± = 1 ± and = m W m and the W boson has been chosen to deine t the z direction. The most general top quark spinor is given by u +1/ (p t ) = m t (cos θ, sin θ eiφ, cos θ, sin θ eiφ ) u 1/ (p t ) = m t ( sin θ e iφ, cos θ, sin θ e iφ, cos θ ) (16) 8

9 where θ and φ reer to some direction, e.g. to the direction o the top quark in some lab system. One may put φ = 0 without restriction, because this corresponds to deining the y direction. The possible b and W polarizations are ixed to be ū 1/ (p b ) = m t (0, 0,, 0) (17) and ɛ 1 = 1 (0, ±1, i, 0) ɛ 0 = 1 (, 0, 0, + ). (18) This leads to the ollowing amplitudes or t decays in the top quark rest rame: A lo t ( 1, 0) = 1 sin θ e iφ A lo t (+1, 0) = 1 cos θ A lo t ( 1, +1) = 0 A lo t ( 1, 1) = cos θ Alo t (+1, +1) = 0 A lo t (+1, 1) = sin θ eiφ (19) where the upper index reers to lowest order and the lower index to top quark decay in the t rest rame. A universal spin independent coeicient c 0 = emt s W has been let out in all the amplitudes. Note that the amplitudes are only determined up to an overall phase, and that this arbitrariness goes away when orming the density matrix elements. Since the amplitudes are explicitly given, it is straightorward to obtain the density matrix in lowest order. Its trace is easily obtained rom the above expressions to be A lo t (h t, h W )A lo t (h t, h W ) = c h t,h W (0) and one can reproduce rom this the total width o the top quark, Eq. (14) by dividing by the phase space actor 16 16πm t. There are lots o other combinations o density matrix elements to describe interesting physics. For example, the above mentioned ratio ΓL Γ T is obtained as Γ L Γ T = h t=± h t=± 1,hW=±1 Alo 1,hW=0 Alo t (h t, h W ) t (h t, h W ) = 1. (1) Let us now repeat the same analysis in the W rest rame. This time I chose to deine the z direction by the direction o the top quark, i.e. p W = (m W, 0) p t = m t ( +, 0, 0, ) p b = p t p W. () 9

10 The top and bottom quark spinors are then ixed as u +1/ (p t ) = m t ( 1/4, 0, 1/4, 0) u 1/ (p t ) = m t (0, 1/4, 0, 1/4 ) (3) and ū 1/ (p b ) = m t (0, 0, 0, ) (4) whereas the W polarization direction is arbitrary: ɛ 1 = eiφ (0, cosφcosθ i sinφ, sin φcos θ + i cosφ, sinθ) ɛ +1 = ɛ 1 ɛ 0 = (0, sinθ cosφ, sin θ sinφ, cos θ). (5) The angles θ and φ reer to some arbitrary direction, e.g. to the direction o the W boson in some lab system. Note that although I am using the same symbols θ and φ as in Eqs. (19) and (8), the meaning o these angles is completely dierent in the three cases. As beore, one may in principle put φ = 0 without restriction, because this corresponds to deining the y direction. One is lead to the ollowing amplitudes or t decays in the W rest rame: A lo W ( 1, 0) = 1 cosθ A lo W (+1, 0) = sin θeiφ A lo W ( 1, +1) = 1 sin θe iφ A lo W (+1, +1) = 1 (1 + cosθ) A lo W ( 1, 1) = 1 sinθe iφ A lo W (+1, 1) = 1 (1 cosθ)e iφ (6) Again, the universal coeicient c 0 = emt s W has been let out in all the amplitudes. From the trace o the corresponding density matrix one can reconstruct the total width o the top quark, just as in (0). However, the ratio (0) can only be obtained or θ = 0, because otherwise the notion o longitudinal does not reer to the heavy quark direction. 6 QCD Corrections to the Spin Density Matrix o t bw rom the Exchange o Virtual Gluons The oneloop QCD Corrections to t decay are somewhat more complicated than to W decay because even neglecting the b mass there is one more mass parameter involved. This is despite the act that the Feynman diagrams needed 10

11 are exactly the same as or W decay (with the directions o the W and one o the quarks interchanged). Namely, there are the virtual gluon vertex and sel energy diagrams and two real diagrams with gluon emission rom one o the quark legs. As discussed in the introduction, there are 36 density matrix elements () which will be considered in the normalized orm ρ norm (h t, h W, h t, h W ) = ρ(h t, h W, h t, h W ) ρ total (7) because this helps to cancel universal contributions, in analogy to the case o W decay, Eq. (1). ρ total h t,h W ρ(h t, h W, h t, h W ) is deined to be the trace o the density matrix and related to the total width Γ t o the top quark as discussed above. Let us start the discussion with the virtual contributions, because they can easily be obtained rom the corrections to the V A vertex calculated a long time ago 1 Γ µ (t bw) = ie { 1 Hγµ sw (1 γ iσ µν p ν W 1 5) + α d H + m t (1 + γ 5) } (8) where the known unction H = 1 + O(α s ) renormalizes the V A structure, and contains all the ultraviolet, inrared and collinear singularities, and H + = ln (9) is a regular unction o = m W /m t. Note that the appropriate expansion α parameter in the case o top quark decay is α d = C s F π. It should urther be noted that the contribution o H to the normalized density matrix ρ norm vanishes because it cancels between the numerator and denominator in Eq. (7). The argument works in the same way as was discussed in the case o the W boson. The only dierence is that now a inite contribution H + rom the σ µν term survives due the nonvanishing (top )quark mass. In turn, one may conclude that the contribution rom real gluon emission to the normalized density matrix is inite, too, because o the Lee Nauenberg theorem, which says that any such singularity cancels between real and virtual corrections. One may write down the contribution H + to the amplitudes in the orm A t (h t, h W ) = A lo t (h t, h W ) + α d A + t (h t, h W )H + (30) where the lowest order amplitudes A lo t were given in (19) and the higher order amplitudes A + t originating rom the σ µν term are given by A + t ( 1, 0) = sin θ e iφ A + t (+1, 0) = cos θ 11

12 A + t ( 1, +1) = 0 A+ t (+ 1, +1) = 0 A + t ( 1, 1) = 1 cos θ A + t (+ 1, 1) = 1 sin θ eiφ (31) Again, the universal coeicient c 0 = emt s W has been let out in all the amplitudes. These results imply a higher order contribution due to the σ µν term on the level o the density matrix ρ(h t, h W, h t, h W ). Including the lowest order piece it will be o the orm A lo t (h t, h W )A lo t (h t, h W ) + α d H + [ A lo t (h t, h W )A + t (h t, h W ) +A + t (h t, h W )A lo t (h t, h W ) ] (3) Note that to calculate ρ norm, one has to divide aterwards by ρ total, whose contribution rom the σ µν term (including lo) can be calculated to be A lo t (h t, h W ) + α s π H [ + A lo t (h t, h W )A + t (h t, h W ) h t,h W h t,h W +A + t (h t, h W )A lo t (h t, h W ) ] = c { α d H + } (33) This result was obtained by summing over all spin conigurations and using the explicit representation o the amplitudes given above. It shows explicitly that the contribution rom virtual gluon exchange to ρ norm is completely under control. The amplitudes corresponding to the σ µν term in the vertex may also be evaluated in the W rest rame A + W ( 1, 0) = cosθ A+ W (+1, 0) = 1 sinθeiφ (34) A + W ( 1, +1) = sin θe iφ A + W (+1, +1) = 1 (1 + cosθ) A + W ( 1, 1) = sin θeiφ A + W (+1, 1) = 1 (1 cosθ)eiφ and analgous relations as (3) and (33) apply. As discussed beore, the angles θ and φ have completely dierent meanings in the t and W rest rames. 7 QCD Corrections to the Spin Density Matrix o t bw rom Real Gluon Emission The amplitudes and density matrix or real gluon emission are the most diicult part o the higher order calculation and are also the most diicult to document. 1

13 The point is not only that the amplitudes are quite complicated expressions, but also that one has to deal with the phase space integration over the real gluon s degrees o reedom on the level o the spin density matrix. Furthermore, the cancellations o the singularities in ρ norm in Eq. (7) require a subtle understanding o the interplay between lowest order and irst order QCD. All this enorces the use o an algebraic computer program like FORM, REDUCE or MATHEMATICA to handle the long and complicated expressions. Ater calculating the spin density matrix, the real gluon s degrees o reedom have to be integrated over. This integration can be done without regularization, because according to the last section the correction to ρ norm rom real gluons is inite and o the generic orm, c. Eq. (1), ρ norm (h t, h W, h t, h W ) = ρlo norm + α ρ ho ρ lo total ρlo ρ ho total d (ρ lo (35) total ) α where α d = C s F π. The dependence on the gluon spin is not considered, because it cannot be determined experimentally. Accordingly, the gluon polarizations have been summed in the standard ashion. To be more explicit, let us write the 4 momenta relevant or the process t Wbg in the rest system o the top quark as where p t = (m t, 0) p W = m t +x W (1, 0, 0, β W ) p b = p t p W p g p g = m t x g (1, sin θ g cosφ g, sin θ g sin φ g, cosθ g ) (36) βw = 1 4 +x W and where θ g is given in terms o the other variables according to (37) x g + (1 x W ) = 1 +x W x g (1 β W cosθ g ) (38) Note that the meaning o,, + etc is as in lowest order. The integrations over the gluon s degrees o reedom then have to be done with the phase space dps 3 (t Wbg) = m t π xg(1 xg) + (1 xg) π dφ g dx g d(1 x W ) 0 0 π (39) The top quark spinors may be taken as in Eq. (16), with φ = 0 because the y z plane has not yet been speciied. However, the W polarization vectors 13

14 are dierent rom the lowest order expressions, Eq. (18), because the W momentum has changed. More precisely, the transverse polarization vectors are let unchanged, but the longitudinal polarization vector now reads ɛ 0 = 1 ( +x W β W, 0, 0, + x W ) (40) The h b = 1/ spinor or the b quark changes, too, but or the density matrix one needs only the combination u 1/ ū 1/ = 1 (1 γ 5)b/. I have calculated all the 6 6 = 36 spin density matrix elements using these parametrizations (1 + )ρ norm ( 1, 0, 1, 0) = c ( α d ) c + α d (41) (1 + )ρ norm (+ 1, 0, +1, 0) = c + ( α d ) c α d (4) (1 + )ρ norm ( 1, 0, +1, 0) = s 0 ( α d ) (43) (1 + )ρ norm (+ 1, 0, 1, 0) = s 0 ( α d ) (44) (1 + )ρ norm ( 1, 1, 1, 1) = c + ( α d ) c α d (45) (1 + )ρ norm (+ 1, 1, +1, 1) = c ( α d ) c + α d (46) (1 + )ρ norm ( 1, 1, +1, 1) = s 0 ( α d ) (47) (1 + )ρ norm (+ 1, 1, 1, 1) = s 0 ( α d ) (48) (1 + )ρ norm ( 1, +1, 1, +1) = c + α d c α d (49) (1 + )ρ norm (+ 1, +1, +1, +1) = c α d c + α d (50) (1 + )ρ norm ( 1, +1, +1, +1) = s 0 α d (51) (1 + )ρ norm (+ 1, +1, 1, +1) = s 0 α d (5) (1 + )ρ norm ( 1, 0, 1, +1) = s 0 α d (53) (1 + )ρ norm ( 1, +1, 1, 0) = s 0 α d (54) (1 + )ρ norm (+ 1, 0, +1, +1) = s 0 α d (55) 14

15 (1 + )ρ norm (+ 1, +1, +1, 0) = s 0 α d (56) (1 + )ρ norm ( 1, 0, +1, +1) = c + α d (57) (1 + )ρ norm (+ 1, +1, 1, 0) = c + α d (58) (1 + )ρ norm (+ 1, 0, 1, +1) = c α d (59) (1 + )ρ norm ( 1, +1, +1, 0) = c α d (60) (1 + )ρ norm ( 1, 0, 1, 1) = s 0 ( αd ) (61) (1 + )ρ norm ( 1, 1, 1, 0) = s 0 ( αd ) (6) (1 + )ρ norm (+ 1, 0, +1, 1) = +s 0 ( αd ) (63) (1 + )ρ norm (+ 1, 1, +1, 0) = +s 0 ( αd ) (64) (1 + )ρ norm ( 1, 0, +1, 1) = c ( αd ) (65) (1 + )ρ norm (+ 1, 1, 1, 0) = c ( αd ) (66) (1 + )ρ norm (+ 1, 0, 1, 1) = +c + ( αd ) (67) (1 + )ρ norm ( 1, 1, +1, 0) = +c + ( αd ) (68) (1 + )ρ norm ( 1, +1, 1, 1) 0 (69) (1 + )ρ norm ( 1, 1, 1, +1) 0 (70) (1 + )ρ norm (+ 1, +1, +1, 1) 0 (71) (1 + )ρ norm (+ 1, 1, +1, +1) 0 (7) (1 + )ρ norm (+ 1, +1, 1, 1) 0 (73) (1 + )ρ norm ( 1, 1, +1, +1) 0 (74) (1 + )ρ norm ( 1, +1, +1, 1) 0 (75) 15

16 (1 + )ρ norm (+ 1, 1, 1, +1) 0 (76) where c ± = 1 (1 ± cosθ) and s 0 = 1 sin θ and where the numerical coeicients α in order α d = C s F π have been obtained with m t = 175 GeV, or which = 0.1. With the help o my intergation programs I have shown that the m t dependence o these coeicients is in all cases moderate. For example, the coeicient in Eq. (41) depends on mt m W = 1 in the way depicted in Figure 1. The m t dependence o the other independent QCD coeicients (denoted by 0.046, 0.01, , , , , , 0.16, 0.36 and , i.e. by their values at m t = 175 GeV) are given in the igures that ollow. Note that only lowest order and real gluons are incorporated in Eqs. (41) (76). The virtual corrections Eqs. (3) and (31) rom the σ µν term have to be added. The actors 1 + appearing on the let hand side are a relic o the act that I am presenting the density matrix normalized to the total width/trace. There are several possibilities to make checks on this list. For example, I have checked that h t,h W ρ norm (h t, h W, h t, h W ) 1 (77) is true including the oneloop QCD corrections. Furthermore, I have also checked that the ratio h t ρ norm (h t, 0, h t, 0) h t [ρ norm (h t, +1, h t, +1) + ρ norm (h t, 1, h t, 1)] = Γ L = 1 Γ T (1 + α s ) (78) reproduces the ratio o longitudinal over transverse W s as calculated in 11 including higher order QCD corrections. Finally, there is the check as to the hermiticity o the density matrix, ρ norm (h t, h W, h t, h W ) = ρ norm(h t, h W, h t, h W ). Note that the density matrix is real in the present case, because in the considered rame there is not azimuthal dependence. I have carried through a second analogous calculation in the rest system o the W and obtained the real gluon QCD corrections to the density matrix ρ norm (h t, h W, h t, h W ) also in that system. The momenta are now parametrized as ollows p W = (m t, 0) p t = m t +x t (1, 0, 0, β t ) p g = p t p W p b p b = m t x b (1, sinθ b cosφ b, sin θ b sin φ b, cosθ b ) (79) 16

17 m(t)/m(w) Figure 1: m(t)/m(w) Figure : 17

18 m(t)/m(w) Figure 3: m(t)/m(w) Figure 4: 18

19 m(t)/m(w) Figure 5: m(t)/m(w) Figure 6: 19

20 m(t)/m(w) Figure 7: m(t)/m(w) Figure 8: 0

21 m(t)/m(w) Figure 9: m(t)/m(w) Figure 10: 1

22 m(t)/m(w) Figure 11: where β t = x t and where θ b is given in terms o the other variables according to (80) x b + + (1 x t ) = 1 +x t x b (1 β t cosθ b ). (81) The integrations over the gluon s degrees o reedom are encoded as integrations over x t, x b and φ b and have to be done with the phase space dps 3 (t Wbg) = m t π x b (1 x b ) + (1 (1 xg)) π dφ b d(1 x b ) d(1 x t ) 0 0 π. (8) About the eects o gluon emission on polarization: Since the W momentum is unchanged as compared to lowest order, the orm o the W polarization vectors remains as in (5). However, the top and bottom spinors are modiied. They now read u +1/ (p t ) = m t (a, 0, a +, 0) u 1/ (p t ) = m t (0, a, 0, a + ) (83)

23 with and x t + a ± = 1 ± βt (84) ū 1/ (p b ) = m t (0, 0, b 1, b ) (85) where b 1 and b are given indirectly by u 1/(p b )ū 1/ (p b ) = 1 (1 γ 5)b/. The 36 elements o the normalized density matrix obtained in this rame are given by (1 + )ρ norm ( 1, 0, 1, 0) = cos θ + ( 0.48 z z + )α d (86) (1 + )ρ norm (+ 1, 0, +1, 0) = sin θ + ( z z + )α d (87) (1 + )ρ norm ( 1, 0, +1, 0) = sin θ cosθ ( α d ) (88) (1 + )ρ norm (+ 1, 0, 1, 0) = sin θ cosθ ( α d ) (89) (1 + )ρ norm ( 1, 1, 1, 1) = 1 sin θ (90) +(0.188 z z cosθ)α d (91) (1 + )ρ norm (+ 1, 1, +1, 1) = (1 cosθ) (9) +( z 0.05 z cosθ)α d (93) (1 + )ρ norm (+ 1, 1, 1, 1) = (1 + )ρ norm( 1, 1, +1, 1) (94) = [ sinθ ( α d) + sinθ cosθ ( α d )] (95) (1 + )ρ norm ( 1, +1, 1, +1) = 1 sin θ (96) +(0.188 z z cosθ)α d (97) (1 + )ρ norm (+ 1, +1, +1, +1) = (1 + cosθ) (98) +( z 0.05 z cosθ)α d (99) (1 + )ρ norm (+ 1, +1, 1, +1) = (1 + )ρ norm( 1, +1, +1, +1) (100) = [ sin θ ( α d) + sin θ cosθ ( α d )] (101) (1 + )ρ norm ( 1, 0, 1, +1) = (1 + )ρ norm( 1, +1, 1, 0) (10) 3

24 = 1 sin θ cosθ( α d ) α d sin θ (103) (1 + )ρ norm (+ 1, 0, +1, +1) = (1 + )ρ norm(+ 1, +1, +1, 0) (104) = [ sinθ (1 0. α d ) + sin θ cosθ ( α d )] (105) (1 + )ρ norm ( 1, 0, +1, +1) = (1 + )ρ norm(+ 1, +1, 1, 0) (106) = cosθ (1 + cosθ) + ( z z cosθ)α d (107) (1 + )ρ norm (+ 1, 0, 1, +1) = (1 + )ρ norm( 1, +1, +1, 0) (108) = sin θ + ( z z cosθ)α d (109) (1 + )ρ norm ( 1, 0, 1, 1) = (1 + )ρ norm( 1, 1, 1, 0) (110) = 1 sinθ cosθ ( α d ) α d sin θ (111) (1 + )ρ norm (+ 1, 0, +1, 1) = (1 + )ρ norm(+ 1, 1, +1, 0) (11) = [ sinθ (1 0. α d ) sin θ cosθ ( α d )] (113) (1 + )ρ norm ( 1, 0, +1, 1) = (1 + )ρ norm(+ 1, 1, 1, 0) (114) = cosθ (1 cosθ) + ( z z cosθ)α(115) d (1 + )ρ norm (+ 1, 0, 1, 1) = (1 + )ρ norm( 1, 1, +1, 0) (116) = sin θ + ( z z cosθ)α d (117) (1 + )ρ norm ( 1, +1, 1, 1) = (1 + )ρ norm( 1, 1, 1, +1) (118) = z ( α d ) (119) (1 + )ρ norm (+ 1, +1, +1, 1) = (1 + )ρ norm(+ 1, 1, +1, +1) (10) = z ( α d ) (11) (1 + )ρ norm ( 1, +1, +1, 1) = (1 + )ρ norm(+ 1, 1, 1, +1) (1) 4

25 = sin θ (1 cosθ )( α d) (13) (1 + )ρ norm (+ 1, +1, 1, 1) = (1 + )ρ norm( 1, 1, +1, +1) (14) = sinθ (1 + cosθ )( α d) (15) where z ± = 1 (1 ± cos θ). The QCD coeicients have again been obtained by numerical integration with m t = 175 GeV. The m t dependence o these coeicients is again moderate. In act, it can be shown that the coeicients in the rest rame o the top quark, Eqs. (41) (76), and o the W boson, Eqs. (86) (15), are related. I was not able to derive a general ormula, but I have ound, or example, that [Eq.(49)]= 0.096[Eq.(91)] 0.0 [Eq.(91)] and [Eq.(49)]= 0.05[Eq.(93)] [Eq.(93)]. Corresponding equalities are true or all other values o m t, i.e. they hold or the coeicients in general. There are some other relations which I do not want to quote here. 8 Summary In this report I have summarized a recent new calculation o a complete spin analysis o the Standard Model top quark decay including higher order QCD corrections. The QCD corrections to the normalized density matrix are in general quite small, o the order o 1%, in particular or the real gluon contribution in the rest rame o the top quark, c. Eqs. (41) (76) and o the W, c. Eqs. (86) (15). The contribution rom virtual gluons is somewhat larger, c. Eq. (9). Note that the relative magnitude o the QCD corrections can be very large in all cases, where the lowest order contribution vanishes, like Eqs. (49) (60). In other cases, the symmetry requirement o CP gives vanishing matrix elements beyond the leading order, c. Eqs. (69) (76). Complete results including azimuthal dependence, numerical analysis and physical applications have not been included here. However, I plan to write a long article with J. Körner and his group, in which not only this, but also analytical ormulae or all the QCD coeicients will be given. Acknowledgments Discussions with Joseph Abraham, Jürgen Körner and Bohdan Grzadskowski are grateully acknowledged. 5

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