Probing the top Yukawa coupling at the LHC via associated production of single top and Higgs

We study Higgs boson production associated with single top or anti-top via $t$-channel weak boson exchange at the LHC. The process is an ideal probe of the top quark Yukawa coupling, because we can measure the relative phase of $htt$ and $hWW$ couplings, thanks to the significant interference between the two amplitudes. By choosing the emitted $W$ momentum along the polar axis in the $th\,(\bar{t}h)$ rest frame, we obtain the helicity amplitudes for all the contributing subprocesses analytically, with possible CP phase of the Yukawa coupling. We study the azimuthal asymmetry between the $W$ emission and the $Wb\,(\bar{b})\to t(\bar{t})\,h$ scattering planes, as well as several $t$ and $\bar{t}$ polarization asymmetries as a signal of CP violating phase in the $htt$ coupling. Both the azimuthal asymmetry and the polarization perpendicular to the scattering plane are found to have the opposite sign between the top and anti-top events. We identify the origin of the sign of asymmetries, and propose the possibility of direct CP violation test in $pp$ collisions by comparing the top and anti-top polarization at the LHC.


I. INTRODUCTION
The top quark Yukawa coupling of the 125 GeV Higgs boson (h) is the largest of the Standard Model (SM) couplings, and the precise measurement of its magnitude and properties is the important target of the LHC experiments. Measurements of the loop-induced hgg and hγγ transitions constrain the top Yukawa, or htt coupling indirectly, if only the SM particles contribute to the vertices with the SM couplings. The observation of the associated production of the Higgs boson and the top quark pair at the LHC [1,2] determines the htt coupling directly, constraining its magnitude to be within about 10% of the SM prediction.
In this paper, we study the possibility of measuring a possible CP violating phase of the htt coupling in the Higgs boson production associated with single top or anti-top at the LHC. The cross section is dominated by the so-called t-channel W exchange process, where the W boson emitted from a quark or anti-quark in a proton scatters with a b orb quark in the other proton to produce a pair of h and t, ort. The process is particularly sensitive to the phase of the htt coupling, because we can measure the real and imaginary part of the htt coupling through the interference between the amplitudes with the htt and hW W couplings which have the same order of magnitude with opposite sign [3,4] in the SM limit. We can therefore measure the phase of the htt coupling with respect to that of the hW W coupling, whose magnitude and phase have already been constrained rather well [5][6][7] and will be determined precisely in the HL-LHC era.
We adopt the following minimal non-SM modification to the top Yukawa coupling, where we introduce the positive κ factor as g htt = (m t /v)κ htt > 0 (2) for the normalization of the coupling. The Lagrangian expressed in terms of the chiral two-spinors t L and t R show that ξ htt is the CP phase of the Yukawa interactions. Its defined range is −π < ξ htt π (4) with respect to the hW W coupling term L hW W = g hW W hW − µ W +µ (5) for which we take the real positive value g hW W = (2m 2 W /v)κ hW W > 0. (6) CP violation in the htt coupling, ξ htt = 0, with κ htt = 1 can arise by radiative effects in the htt vertex due to new interactions which violate CP, or in models with two or more Higgs doublets when the Higgs interactions violate CP. Once the underlying new physics model is fixed, we often obtain correlations among the non-SM effective couplings, such as κ hW W , κ htt , ξ htt , and also for the other hf f couplings as well as the loop induced hgg, hγγ and hZγ vertices. In this report, we set κ htt = κ hW W = 1 (7) in all the numerical results, in order to focus on the observable CP violating effects for relatively small phase |ξ htt | 0.1π.
In Fig. 1, we show the total cross section of the Higgs boson production with single t ort via t-channel W exchange in pp collisions at √ s = 13 TeV for the effective htt coupling of Eq. (1), with κ htt = 1 and |ξ htt | between 0 (SM) and π. Also shown is the total cross section for h and a tt pair in the same model. They are obtained by MadGraph [8] with the effective Lagrangian of Eq. (1) in Feynrules [9]. Here, and in all the following numerical calculations, we set m h = 125 GeV, m t = 173 GeV, m W = 80.4 GeV, v = 246 GeV, 4π/e 2 = 128 and sin 2 θ W = 0.233 for the electroweak couplings. Factorization scale is set at µ = (m t + m h )/4 for the ht and ht production via t-channel W exchange processes in 5-flavor QCD, following Ref. [10]. As for the QCD production of htt processes, we set the factorization √ s = 13 TeV) for the sum of pp → th and pp →th production via t-channel W exchange as a function of the CP phase |ξ htt | for κ htt = 1. Also shown is the pp → tth production cross section in the same model. and renormalization scales both at µ = (2m t + m h )/2, following Ref. [11]. The QCD coupling at µ = m Z is set at α s (m Z ) = 0.118 [12].
As is well known, the cross sections for the Higgs production with single t ort are sensitive to the relative sign of the htt and the hW W couplings, which becomes 13 times larger than the SM value at |ξ htt | = π where the sign of the htt coupling is reversed [3]. Because of this huge enhancement factor, LHC experiments [13][14][15][16][17] have ruled out the region around |ξ htt | ∼ π for κ htt = 1. It is worth noting, however, that we focus our attention in this paper on a relatively small magnitude of the CP phase |ξ htt | 0.1π, where the total cross sections do not deviate much from the SM values, σ(th +th) = 60.85 fb and σ(tth) = 406.26 fb in the LO, as shown in Fig. 1.
The paper is organized as follows. In section II, we give helicity amplitudes for all the four LO subprocesses analytically. In section III, we study event distributions of ht and ht production with a tagged forward jet, and show the exchanged W helicity decomposition in Q (the virtual W mass) and W (the invariant mass of the th orth system) distributions. In section IV, we study the azimuthal angle asymmetry between the W emission plane and the W + b → th or W −b →th production plane about the W momentum direction. In section V, we study t andt polarizations in the t (t) rest frames, as a function of Q, W and the W + b → th (W −b →th) scattering angle θ * in the th (th) rest frame. In section VI, we study consequences of T and CP transformations, and show the possibility that CP violation signal can be distinguished from T-odd asymmetry arising from the final state scattering phase in pp collisions, by measuring the t andt polarizations perpendicular to the scattering plane. The last section VII gives a summary of our findings and remarks on possible measurements at HL-LHC. Appendix A gives the relation between the helicity amplitudes and t andt spin polarizations, and Appendix B gives polarized t andt decay distributions.

II. HELICITY AMPLITUDES
In the SM, four subprocess contribute to single top plus Higgs production in the leading order and also to single anti-top plus Higgs production; db → uth (sb → cth) (10a) ub →dth (cb →sth) (10b) FIG. 2: Feynman diagrams of ub → dth subprocess. The four momenta q µ and q ′µ along the W + and P µ th along the top propagators are shown with arrows.
We work in 5-flavor QCD with massless b-quark distribution in the proton, where the matching with the 4-flavor QCD with massive b-quark has been shown for the single t plus h processes in the NLO level [11,19]. The subprocesses in the parenthesis with second generation quarks have exactly the same matrix elements when we ignore quark mass and CKM mixing effects.
The Feynman diagrams of the subprocess ub → dth in Eq.(9a) are shown in Fig. 2. The left diagram (a) has the hW W coupling, while the right diagram (b) has the htt coupling. The u → dW + emission part is common to both diagrams. The amplitudes for all the other subprocesses in Eq. (9) are obtained by replacing the u → dW + emission current by c → sW + ,d →ūW + ands →cW + current, respectively. The Feynman diagrams for anti-top plus Higgs production in Eq. (10) are obtained simply by replacing the W + emission currents by the W − emission currents, and by reversing the fermion-number flow along the b to t transitions to make themb tot transitions.
In pp collisions, valence quark initiated subprocesses ub → dth (3a) and db → uth (4a) dominate the single top and anti-top production cross sections, respectively. The amplitudes for the subprocess ub → dth in Fig. 2 are simply with for the effective top Yukawa coupling of Eq.(1) and the SM hW W coupling of Eq. (6). The propagator factors and D νρ W (q ′ ) are the W -propagators, with D W (q) = (q 2 − m 2 W ) −1 , and D t (P th ) = (P 2 th − m 2 t ) −1 is for the top quark. The four momenta are depicted in Fig. 2 as In the limit of neglecting all the quark masses except the top quark mass, m t , the amplitudes depend only on the top quark helicity σ/2 for σ = ±1, since only the left-handed quarks and right-handed anti-quarks contribute to the SM charged currents in the massless limit. Because the W + emission current of massless u and d quarks is conserved, only the spin 1 components of off-shell W + propagates in the common D µν W (q) term in Eq. (13): where λ denotes the helicity of virtual W + , and the (−1) λ+1 factor appears for q 2 < 0. By replacing the covariant propagation factor in the common W + propagator with Eq. (15), we can express the amplitudes Eq. (11) as a sum over the contributions of the three W + helicity states: 3: Scattering anglesθ, φ and θ * . The polar angleθ is defined in the Breit frame, whereas θ * is defined in the W + b rest frame, for the common polar axis along the W momentum direction. The azimuthal angle φ is the angle between the emission plane and the scattering plane. with and We calculate the helicity amplitudes T λσ for W + b → th process in the th or W + b rest frame. Therefore all the polarization asymmetries presented below refer to the top quark helicity in the th rest frame, see Fig. 3. On the other hand, because massless quark helicities are Lorentz invariant, and the W + helicity is boost invariant along the W + momentum direction, which we take as the polar axis in Fig. 3, we can evaluate the u → dW + emission amplitudes in the Breit frame [33], where the W + four momentum has only the helicity axis component with Q > 0 and Q 2 = −q 2 . The u and d quark four momenta are where their common energyω and the reflecting momentum along the polar axis are, respectively, withŝ = (p u + p b ) 2 and W = P 2 th = (p t + p h ) 2 . In Eq. (20) and in Fig. 3, the u → dW + emission plane is rotated by −φ about the z-axis, so that the top quark azimuthal angle measured from the u → dW + emission plane is φ. The u → dW + emission amplitudes have very compact and intuitive expressions in the Breit frame: Here we adopt for the three polarization vectors, which differs by the sign of the λ = +1 vector from the standard Jacob-Wick convention. The convention dependence cancels in the product, and our choice makes CP transformation properties of the sub-amplitudes, J λ and T λσ , simple because It is interesting to note [33] that the u → dW + emission amplitudes can be expressed in terms of Wigner's d-functions.
In terms of the invariants, they are expressed as is the energy fraction of the d-quark in the ub collision rest frame. It should be noted that for typical events with x 0.1, the ordering holds among the magnitudes of the d-functions. In particular, J − for the helicity λ = −1 W + dominates over J + , because left-handed u-quark tends to emit a left-handed W -boson in the forward direction. The helicity amplitudes T λσ for W + λ b → t σ h process are calculated in the th rest frame. We first express T λσ (18) in terms of chiral two-spinors [34] where we denote P = P th , ξ = ξ htt , and σ µ ± = (1, ± σ) are the chiral four-vectors of σ matrices. We note that the chirality flip term for the right-handed top with e −iξ phase factor grows with P , while the chirality non-flip term for the left-handed top with e iξ is proportional to m t , because of the chirality flip by the Yukawa interactions. As for the W -exchange amplitudes, the chirality flip right-handed top proportional to m t is non-negligible because of the scalar component of the exchanged W boson, which has the 1/m 2 W factor. In the th rest frame, where the W + momentum is along the positive z-axis, the four momenta are given by momentum p * of t and h in the th rest frame in units of W/2. The amplitudes T λσ can be calculated straightforwardly, giving where we denote m = m t and E * = E * t . Note that the term √ E * + p * appears when the top helicity matches its chirality, while √ E * − p * when they mismatch. The amplitude for λ = +1 does not have the top Yukawa coupling contribution because the angular momentum along the z-axis is J z = + 3 2 for the left-handed b-quark, which cannot couple to J = 1/2 top quark. For λ = −1 and λ = 0 W + , both W and t exchange amplitudes contribute. Most importantly, the λ = 0 amplitudes are enhanced by the factor of W/Q, which is a consequence of the boost factor of the longitudinally polarized λ = 0 W + wave function. The polarization vectors in Eq. (23) in the Breit frame are invariant for λ = ±1, but the longitudinal vector becomes in the th rest frame, where both q * and q 0 * are the order of W/2 as in Eq. (29a). Summing over the three W polarization contributions, we find the amplitudes [31] where the factors are chosen such that they are positive definite. The ǫ, δ, and δ ′ factors are where ǫ ≃ 0.21, δ and δ ′ are all small at large W, and in particular, δ ≃ δ ′ holds rather accurately except in the vicinity of th production threshold, W ≃ m t + m h . At W 400 GeV, the amplitudes are well approximated as where we have dropped λ = +1 contributions which are suppressed at high W/Q. The above approximations show that the leading λ = 0 contributions with the W/Q enhancement factor are proportional Because both A and B terms are positive definite, their magnitudes are smallest at ξ = 0 (SM), where the W exchange term A and the t-exchange term B interfere destructively, whereas they become largest at |ξ| = π where the two terms interfere constructively, explaining the order of magnitude difference in the total cross section between ξ = 0 and |ξ| = π shown in Fig. 1. This strong interference between the two amplitudes gives the opportunity to accurately measure the htt Yukawa coupling with respect to the hW W coupling. Another important observation from the above approximation is that the CP-violation (CPV) effects proportional to sin ξ are significant only in the amplitude of the right-handed helicity top quark, M + , because M − is proportional to e −iξ + e +iξ = 2 cos ξ at large W where δ = δ ′ . We note here that M + is the leading helicity amplitude at large W, where the chirality flip Yukawa interactions give right-handed top quark from the left-handed b-quark. The negative helicity amplitudes M − is suppressed by an additional chirality flip of the top quark, indicated by the factor δ = m/W in Eq. (35b).
Before starting discussions about signals at the LHC, let us complete all the helicity amplitudes of the contributing subprocess for both th andth productions. First, the amplitudes for the subprocesses cb → sth are the same as those for ub → dth in our approximation of neglecting quark masses and CKM mixing: as summarized in Eqs. (32). There are two additional contributions to th production from the anti-quark distributions of proton where thed →ūW + emission amplitudes are In the Breit frame, they are expressed as We note the relation between J λ and J λ . The matrix elements for the W + emission from anti-quarks differ from those from quarks by simply replacing 1 ±c by 1 ∓c, thereby changing the preferred helicity of W + from λ = −1 (for u → dW + and c → sW + ) to λ = +1 (ford →ūW + ands →cW + ). The λ = 0 amplitude remains the same. Note that our special phase convention for the vector boson polarization vectors in Eq. (23) allows the identities in Eq. (41) to hold without the λ-dependent sign factor, (−1) λ , that appears in the standard Jacob-Wick convention. Now theth production amplitudes are where J λ and J λ are the same as in Eqs. (22) and (40), respectively, and the W −b →th amplitudes T λσ are obtained from T λσ by CP transformation Note that the first identity above tells the invariance of the amplitudes when all the initial and final states are CP transformed, along with the reversal of the sign of the CP phase. The latter equality is valid in our tree-level expressions Eqs. (30) where absorptive parts of the amplitudes (including the top quark width) are set to be zero. i It is instructive to compare the amplitudes of the two subprocesses which are related by CP transformation, such as between the amplitudes (32) or (37) for ub → dth and those of Eq. (42b) forūb →dth, By using the identities (41) and (43) among the sub-amplitudes, we find the relation between M σ and M σ , whenσ = −σ. It is worth noting that if we ignore the absorptive phase of the amplitudes, such as the top quark width in the propagator, the above identity gives because both φ and ξ appear in the amplitudes only as the phase factor, e ±iφ and e ±iξ . This tells that all the distributions of the CP transformed processes are identical even in the presence of CP-violating phase, ξ = 0, if we ignore the absorptive amplitudes from the final state interactions. We will discuss the origin of this somewhat unexpected property among the amplitudes in section V.
In pp collisions, the dominant subprocess for single production of Higgs and anti-top quark comes from the collision of valence down-quark scattering withb quark, whose amplitudes are given by Eq. (42a). Since the properties of thē th production processes are governed by these amplitudes, we give their explicit form by using the same A and B factors of Eq. (33): where the d → uW − emission amplitudes J λ are the same as the u → dW + emission amplitudes in the ub → dth subprocess amplitudes, Eq. (32), while the W −b →th amplitudes T λσ are obtained from the W + b → th amplitudes T λσ by CP transformation in Eq. (43). The chirality favored helicity oft from right-handedb is now −1/2, and the corresponding amplitude M − has the leading e iξ factor from the t † L t R term in the Lagrangian Eq. (1), while the contribution of the e −iξ t † R t L term is doubly chirality suppressed, by the δδ ′ factor. In the helicity +1/2 amplitude i Note that the sign factor, −σ, in the identities (43) is a consequence of the phase convention of Ref. [34,35] where the v-spinors for anti-fermions are expressed as See also Appendix. B of Ref. [36].
M + , single chirality flip (in addition to the flip due to the Yukawa interaction) is necessary, either in the spinor wave function (giving δ), or in the top quark propagator (giving δ ′ ). Summing up, we find M − to have significant imaginary part proportional to sin ξ, whereas M + is almost proportional to cos ξ, which are opposite of what we find for the single t and h production amplitudes.

III. DIFFERENTIAL CROSS SECTIONS
The differential cross section in pp collisions from the subprocess ub → dth is given at leading order by where D u and D b are the PDF of the u and b quark, respectively, in the protons. The colliding parton momenta in LHC laboratory frame are in the first term of Eq. (48), whereas the u-and b-quark four momenta are exchanged in the second term. Therefore, the b-quark momentum is negative along the z-axis for half of the events and positive for the other half. In order to perform the azimuthal angle or polarization asymmetry measurements proposed in [31], we should identify the momentum of the virtual W + emitted from the u (or c,d,s) quark. This is possible only when we can identify the sign of the b-quark momentum.

A. Selecting the b andb momentum direction
Because valence quark distributions are harder than the sea quark distributions, we expect that the subprocess with negative b-quark momentum should have positive rapidity of the hard scattering system (p The black curves are the sum of the rapidities from the four subprocesses. The quark and anti-quark jets from t-channel W emission are tagged with cuts p j T > 30 GeV, and |ηj | < 4.5. p j T > 30 GeV, and |η j | < 4.5. Events with negative momentum b-quark are shown by solid curves, whereas those with positive b-quark momentum are given by dashed curves. The solid black curve shows the total sum of all thj events. The blue curves give the sum of ub → dth and cb → sth subprocess contributions (that have exactly the same matrix elements), and the red curves are for the sum ofdb →ūth andsb →cth subprocess contributions. As expected, events with Y (thj) > 1 are mostly from the negative momentum b-quark (blue and red solid curves). Although the purity (the probability) of negative b-momentum is 95%, only 41% of the total events satisfy the Y (thj) > 1 cut, leaving (59%) of the events with mixed kinematics which results in significant reduction of observable asymmetries and polarizations. The situation is much worse forthj production processes, as shown in Fig. 4(b). With the same Y (thj) > 1 cut, the purity is only 89% and only 31% of the events are kept. It is mainly because the down quark is not as hard and populous as the up quark in the proton. All results in our study are calculated with the CTEQ14 PDF in the LO [37] with the factorization scale µ = mt+m h 4 , following Ref. [10]. In Fig. 5(a) and 5(b), we show the tagged jet pseudo-rapidity distributions. Now the separation of events with negative b momentum (shown by blue and red thick curves) and those with positive b momentum (shown by blue and red thin curves) is clearer for both thj (a) andthj (b). In Tables I and II, we show the purity and the survival rate of several η j selection cuts for choosing events with negative b orb momentum events, respectively, for thj andthj processes. Even for η j > 0, when all events are used in the analysis, the purity is higher than 96% for both thj and thj events. In this report, we adopt the selection cut      for the jet tag. Since the purity is higher than 99% for both thj (Table I) andthj (Table II), we can safely neglect contribution from events with the wrong b-quark momentum direction, whose analysis requires additional kinematical considerations. Needless to say, events with η j < −1 have exactly the same signal with those with η j > 1 because there is no distinction between the two colliding proton beam. From Tables I and II, we find that the selection cut |η j | > 1 allow us to study 90% of thj and 88% ofthj events with full kinemetical reconstruction. In the following analysis, we assume that a significant fraction of h and single t ort production via t-channel W exchange can be kinematically reconstructed, and define observables whose properties are directly determined by the helicity amplitudes of Section II.

B. Q and W distributions
The differential cross section for the subprocess ub → dth is is the probability to find left-handed u and b quarks inside their PDFs, the color factor is unity for t-channel color-singlet exchange between the colliding quarks, and the three-body Lorentz invariant phase space can be parametrized as as a convolution of the two-body phase space integrated over the invariant mass W of the t + h system The j + th phase space is parametrized in the ub or thj rest frame, where the four momenta are parametrized as The forward peak in the η j distribution in Fig. 5 is due to the square of the common t-channel W propagator in the amplitude, which grows towards cosθ ∼ 1, subject to the jet p T cut The t + h phase space in the th rest frame is where the participating four momenta are parametrized as When evaluating the amplitudes M σ , we rotate the frame about the virtual W momentum axis so that the top three momentum is in the x-z plane, as in Eq. (58) and the azimuthal angle is given to the u → dW emission plane as in Eq. (20) in the Breit frame.  We show in Fig. 6 the distributions with respect to the momentum transfer Q, Eq. (58). Contributions from λ = 0 and λ = ±1 W ′ s separately and their sum are shown. Because the momentum transfer Q does not depend on the azimuthal angle, integration over φ about the W -momentum axis (the common z-axis in Fig. 3) projects out the W helicity states and the interference among different λ contributions vanish. It is clearly seen that the longitudinal W (λ = 0) contribution in red solid curves dominates at small Q (Q 100 GeV) both for thj andthj. This is a consequence of the W/Q enhancement of the λ = 0 amplitudes as shown explicitly in Eqs. (32) for thj, and in Eq. (47) forthj. Among the transverse W contributions, λ = −1 (solid green) dominates over λ = +1 (dashed green) for thj, but they are comparable forthj.
This somewhat different behaviour of the transverse W contribution between thj andthj processes needs clarification, and we show in Fig. 7 the distribution with respect to W, the invariant mass of the th system. The upper plots (a) and (b) are for Q < 100 GeV, and the lower plots (c) and (d) are for Q > 100 GeV. The left figures (a) and (c) are for thj, while the right ones (b) and (d) are forthj. Again contributions from the three helicity states of the exchanged W are shown separately. It is clearly seen that at small Q (Q < 100 GeV) and large W, W 500 GeV, the longitudinal W (λ = 0) contribution dominates the cross sections of both thj (a) andthj (b) production. The transverse W contributions are significant at large Q (Q > 100 GeV), where the left handed (λ = −1) W dominates over the longitudinal W (λ = 0) at W 400 GeV for thj.
On the other hand, the right-handed W − dominatesthj production at small W, especially at large Q (Q > 100) GeV, see Fig. 7(d). This is because the λ = +1 W − collides with the right-handedb-quark, giving J z = + 1 2 initial state with noβ suppression, as can be seen from the first terms in Eqs. the d → uW − splitting amplitudes J λ , Eq. (22).
Summing up, the λ = +1 W − contribution is significant near the threshold (W 400 GeV) forthj production, while the λ = −1 W − contribution takes over at larger W because of dominant d-quark contribution. In contrast, for the thj production, the λ = +1 contribution (green dashed lines) is deeply suppressed, as the disfavored helicity emitted from left-handed u quark at large W and by the p-wave threshold suppression at small W, making them very small both at small (a) and large Q (b).

IV. AZIMUTHAL ANGLE ASYMMETY
In Fig. 8(a), we show distributions of the azimuthal angle between the emission (u → dW + ,d →ūW + , etc) plane and the W + b → th production plane about the common W + momentum direction in the W + b rest frame; see Fig. 3. Shown in Fig. 8(b) are the same distributions for pp →thj process, where the azimuthal angle is between the W − emission plane and the W −b →th production plane about the common W − momentum direction. The results are shown at W = 400 and 600 GeV for large Q (Q > 100 GeV). The black, red and green curves are for the SM (ξ = 0), ξ = ±0.05π, and ±0.1π, respectively. Solid curves are for ξ ≥ 0 while dashed curves are for ξ < 0. The φ distributions are proportional to where the top polarization is summed over.Likewise, they are proportional to |M + | 2 + |M − | 2 forthj events. Analytic expression for the amplitudes, M ± and M ± are given in Eqs. (32) and (47), respectively, where we can tell that azimuthal angle dependences are in the λ = ±1 W ± exchange amplitudes. The asymmetry is large at small W and large Q because the transverse W ± amplitudes are significant there, see Fig. 7. The asymmetry remains significant at W = 400 GeV, however, even for events with Q < 100 GeV [31]. We show in Fig. 9(a) the azimuthal angle distribution of right-handed and left-handed top quark separately, in green and red curves respectively, at W = 400 GeV for events with Q > 100 GeV and ξ = 0.1π. Their sum, given by the black curve agree with the corresponding curve in Fig. 8(a). As expected from the analytic expressions Eqs. (32) and (35), |M − | 2 is almost symmetric about φ = 0, and the asymmetry is mainly from |M + | 2 . Likewise, for thethj events, shown in Fig. 9(b), the asymmetry is mainly from left handedt quark, depicted by the red |M − | 2 curve.
The origin of the azimuthal angle asymmetry comes from the interference between transverse W amplitudes with the e ±iφ phase factor for λ = ±1 W and the longitudinal W (λ = 0) amplitudes as shown in Eq. (32) for ub → thj and Eq. (47) for db → uth. We show in Fig. 10 the azimuthal angle distribution of |M + | 2 for the subprocess ub → dth, separately. The three squared terms, |M(λ)| 2 , for λ = +1, −1 and 0, give no φ dependence, while the interference terms between M + (0) and M + (−1) amplitudes give terms proportional to sin φ sin ξ with positive coefficients, leading to positive sin φ for sin ξ > 0. It is clearly seen from Fig. 10 that |M + (λ = −1)| 2 ≃ |M + (λ = 0)| 2 ≫ |M − (λ = +1)| 2 at W = 400 GeV for Q > 100 GeV for the subprocess ub → dth, consistent with the trend expected from the SM amplitudes at ξ = 0, shown in Fig. 7(c). It is therefore the interference between the M + (λ = −1) and M + (λ = 0) amplitudes, shown by the orange curve in Fig. 10, which determines the asymmetry sin φ . The interference between the λ = ±1 W exchange amplitudes give terms of the form sin 2φ sin ξ, which gives rise to another asymmetry sin 2φ . Because |M − (λ = +1)| is generally small at all Q and W regions, as shown in Figs. 7(a) and (c), the asymmetry sin 2φ turns out to be small in our analysis. We therefore do not show results on sin 2φ in the following, but note that its  measurement should improve the ξ sensitivity at a quantitive level, and that it should be sensitive to other type of new physics that affects mainly the transversally polarized W amplitudes. It may be worth noting that asymmetry sin 2φ is larger inthj process, because both λ = ±1 transversally polarized W contributions are significant, as can be seen from Fig. 7(d), especially at large Q and small W.
In Fig. 11, we show the azimuthal asymmetry integrated over φ, as a function of the invariant mass W of the th orth system for ξ = 0 (SM), ±0.05π (red curve) and ±0.1π (green curve). The asymmetry for large Q (Q > 100) GeV events is shown by solid curves, while those for small Q (Q < 100 GeV) is shown by dashed curves. The positive aysmmetry is found for thj events, while negative asymmetry is found forthj, in accordance with the observation from the φ distribution in Fig. 8. Generally speaking, the asymmetry is large for large Q events at around W ∼ 400 GeV where the magnitudes of the transverse and longitudinal W exchange amplitudes are comparable in Fig. 7(c) and (d). For small Q, (Q < 100 GeV), the asymmetry is significant only near the threshold, W ∼ m t + m h , where the transverse W amplitudes are non-negligible in Fig. 7(a) and (b). Because the asymmetry due to the term linear in sin ξ are nearly absent in |M − | 2 for thj and in |M + | 2 forthj, as can be seen from Eqs. (32b) and (47a) for δ ≃ δ ′ approximation, we can expect enhancement of the asymmetry by selecting right-handed top and the left-handed anti-top. This can easily be achieved when t andt decay semileptonically, where the charged-lepton decay angular distribution in the t ort rest frame takes the form [38] dΓ(t → blν) d cos θl ∼ (1 + σ cos θl) 2 , (67a) about the helicity axis, where σ andσ are twice the helicities of t andt, respectively, in the th orth rest frame. For instance, if we select those events with cos θl, cos θ ℓ > 0, then dσ/dW/dφ is proportional to and the asymmetry is significantly larger, as shown in Fig. 12 for ξ = 0.1π when Q > 100 GeV. The asymmetries shown by the green curves are when no cuts are applied, and they agree with the corresponding curves in Fig. 11. The asymmetry grows to A φ (W) ∼ 0.22 for thj events and A φ (W) ∼ −0.23 forthj events, both at around W ∼ 450 GeV with the selection cut of the t andt decay charged lepton angles in Eq. (68).

V. POLARIZATION ASYMMETRIES
We are now ready to discuss the polarization of the top quark in the single top+h production processes. We first note that the helicity amplitudes M + and M − in Eq. (32) for the subprocess ub → dth, and those in Eq. (47) for db → uth are purely complex numbers when production kinematics ( √ŝ , Q, W, cosθ, cos θ * , φ) are fixed. This is a peculiar feature of the SM where only the left-handed u, d, and b quarks, and their anti-particles with right-handed helicities contribute to the single t andt production process via W exchange. It implies that the produced top quark polarization state is expressed as the superposition in the top quark rest frame, where the quantization axis is along the top momentum direction in the th rest frame, where the top quark helicity is defined. The top quark is hence in the pure quantum state with 100% polarization, with its orientation fixed by the complex number M − /M + . Its magnitude |M − /M + | determines the polar angle and the phase arg(M − /M + ) determines the azimuthal angle of the top spin direction ii . Therefore, the kinematics dependence of the polarization direction can be exploited to measure the CP phase ξ, e.g. by combining matrix element methods with the polarized top decay density matrix iii . Exactly the same applies for thet spin polarization, whose quantum state can be expressed as in Eq. (70) where the helicity amplitudes M ± are replaced by M ± . In this report, we investigate the prospects of studying CP violation in the htt coupling through the top and anti-top quark polarization asymmetries in the single th andth processes respectively, with partial integration over the final state phase space. For this purpose, we introduce a complex matrix distribution ii See Appendix A for a pedagogical review of quantum mechanics. iii The top quark decay polarization density matrices for its semi-leptonic and hadronic decays are given in Appendix B.
Note that the matrix (71) is normalized such that its trace gives the differential cross section of Eq. (48).
Here we denote λ/2 and λ ′ /2 for the top helicity, and the 1/4 factor accounts for the colliding parton spin average, just as in Eq. (53) for the subprocess cross section. All the other subprocesses which contribute to the same thj final state, cb → sth,db →ūth,sb →cth, whose matrix elements are given in Eqs. (37) and (38) should be summed over in the matrix (71). The integration over phase space and the summation over different subprocess contribution make the top quark in the mixed state and its polarization density matrix is given by for an arbitrary distribution. The coefficients of the three σ matrices makes a three-vector, P = (P 1 , P 2 , P 3 ), whose magnitude P = | P | gives the degree of polarization (P = 1 for 100% polarization, P = 0 for no polarization), while its spatial orientation gives the direction of the top quark spin in the top rest frame. The polarization vector P in (73) can be obtained directly from the matrix distribution (71) as follows where the integral over the phase space can be chosen appropriately in order to avoid possible cancellation of polarization asymmetries. For the helicity amplitudes (32) calculated in the th rest frame, the z-axis is along the top momentum in the th rest frame, and the y-axis is along the q × p t direction, perpendicular to the W + b → th scattering plane. In Appendix A, we obtain the orientation of the top quark spin in terms of the helicity amplitudes for a pure state and for general mixed states. The polarization oft quark is obtained also from the matrix distribution (71) with thethj amplitudes MλM * λ ′ , simply by replacing λλ ′ byλλ ′ in the density matrix (73). The orientation of the polarization vector is measured in the same frame, where the z-axis is now along thet quark momentum direction in theth rest frame and the y-axis is along the q × pt direction.
We show in Fig. 13 the three components (P 1 , P 2 , P 3 ) of the polarization vector P as a function of the top (anti-top) scattering angle cos θ * in the th (th) rest frame, at W = 400 GeV (upper four plots) and 600 GeV (lower four plots), when all the other kinematical variables are integrated over subject to the constraint Q < 100 GeV (a), (e), (c), (g) and Q > 100 GeV (b), (f), (d), (h). The left-hand side of Fig. 13 gives the top polarization in thj processes, while the right-hand side plots give thet polarization inthj processes.
As cos θ * deviates from −1, P 3 deviates from −1 according to the growth of |M + | 2 /|M − | 2 , but |P 1 | (and also |P 2 | when ξ = 0) grows quickly as they are linear in M + . The polarization P 2 normal to the scattering plane can become   as large as 0.6 even for ξ = 0.05π, when Q < 100 GeV at W = 600 GeV; see Fig. 13(c). This is because at small Q and large W, the longitudinal (λ = 0) W contribution dominates over the transverse (λ = ±1) W contributions, and hence the integration over the azimuthal angle φ does not diminish much the degree of top polarization.
Likewise, thet polarization is shown in the right hand side of Fig. 13, for the same configuration of W = 400 GeV (e), (f ) and 600 GeV (g), (h), for Q < 100 GeV (e), (g) and Q > 100 GeV (f ), (h). P 3 is now almost unity at cos θ * = −1, because |M − (θ * )| = |M + (θ * )| ≈ 0 at θ * = π. As cos θ * deviates from −1, P 3 decreases rapidly and the polarization perpendicular to the helicity axis, P 1 inside the scattering plane and P 2 normal to the scattering plane when ξ = 0 grows, just as in the case of top polarizations shown in the left-hand plots of the figure. Most notably, the magnitude of all three polarization components P 1 , P 2 , P 3 behave very similar as functions of cos θ * between the top and the anti-top polarizations for the same CP phase, whereas their signs are all opposite. As for P 2 , the magnitude becomes the largest for Q < 100 GeV events at W = 600 GeV, as shown in Fig. 13(c) for top and (g) for anti-top. As we will explain carefully in the next section, this is a consequence of CP violation in CPT invariant theory in the absence of rescattering phase in the amplitudes.
Before we move on studying t andt polarization after integration over cos θ * , we note in Fig. 13(c) and (g) for Q < 100 GeV at W = 600 GeV, the magnitudes of P 2 are predicted to be larger for ξ = 0.05π (dashed red curve) than those for ξ = 0.1π (dash-dotted curve) in the cos θ * > 0 region. This non-linear behavior was not expected for relatively small phase of |ξ| ≤ 0.1π, and we study the elements of matrix dσ λλ ′ carefully for ten values of ξ in the range 0 < ξ < 0.1π. Shown in the left plot of Fig. 16 is the thj production differential cross section, σ ++ + σ −− , with respect to cos θ * at W = 600 GeV for Q < 100 GeV events. The cross section is smallest at ξ = 0, and grows with ξ almost linearly in the region cos θ * −0.5. The cross section near cos θ * = −1 is dominated by the W exchange amplitudes (with the A factor), and hence does not depend on the htt coupling. In the middle plot, Fig. 14(b), we show Im(σ(+, −)) v.s. cos θ * . Its magnitude grows with ξ, but it changes sign at around cos θ * = 0 and the growth of the magnitude is very slow at cos θ * > 0. The average polarization P 2 is obtained as their ratio −2Imσ(+, −)/(σ(+, +) + σ(−, −)) in Eq. (74b), which is shown in Fig. 14(c). In the cos θ * > 0 region, the magnitude grows from ξ = 0 up to ξ ≃ 0.05π, but decreases to the orange curve at ξ = 0.1π. This study shows that the polarization P 2 has strong sensitivity to the CP phase ξ, whose magnitude can reach 20% even for ξ = ±0.01π.
As can be seen from Fig. 14(a), the differential cross section decreases sharply as cos θ * deviates from cos θ * = −1, and hence the polarization asymmetry integrated over cos θ * is determined by the sign and magnitude in the cos θ * < 0 region. Shown in Fig. 15 are the polarization asymmetry P 2 for top (above zero) and antitop (below zero), for the events with cos θ * < 0, plotted against the th (th) invariant mass W. The results for Q > 100 GeV are shown by solid curves, while those for Q < 100 GeV are shown by dashed curves. The red curves are for ξ = 0.05π, while green curves are for ξ = 0.1π. Although the ad-hoc selection cut cos θ * < 0 is not optimal, we can observe the general trend that the magnitude of the polarization asymmetry P 2 grows with the CP phase ξ, and the sign of P 2 is positive for t, but it is negative fort, when ξ > 0.
We may tempt to conclude that the same physics governs the sign of A φ in Fig. 11 and that of P 2 in Fig. 15, since both asymmetries change sign between thj andthj events. We will study the cause of this similar behaviour in the next section. Before discussing consequences of CPT invariance in the next section, let us introduce a slightly more complicated top quark polarization asymmetries whose signs also measure the sign of ξ. We recall that the t polarization perpendicular to the W + b → th scattering plane P 2 can be expressed as a triple three-vector product (75), which is naive T-odd (T-odd), since it changes the sign when we changes the signs of both the three momentum and spin. In the absence of final state re-scattering phase,T-odd observables measure T-violation, or CP-violation in quantum field theories (QFT). Therefore, we examine pentuple products which are clearlyT-odd polarization asymmetries, whose expectation values should vanish at ξ = 0 in the tree level. We note that the three-vector ( q × p j ) × ( q × p h ) points toward the direction of q, while its sign changes when the azimuthal angle between the W + emission plane and the W + b → th scattering plane changes sign, between −π < φ < 0 and 0 < φ < π. Likewise, ( p b × p j ) × ( p b × p h ) points either along or opposite of p b direction, depending on the same azimuthal angle between the two planes, because the W + momentum q and the b momentum p b are back to back in the frames which define the emission and the scattering planes, see Fig. 3. In the top quark rest frame, the two three-vectors, q and p b span the scattering plane, which is chosen as the x-z plane in our analysis. Therefore, if we define the azimuthal asymmetry of the top quark polarization vector as where P k (φ > 0) and P k (φ < 0) denotes, respectively, the top quark polarization of events with φ > 0 and φ < 0, P A 1 and P A 3 areT-odd. This is because the x-and z-axis vectors are linear combination of q and p b in the t-rest frame. We show in Fig. 16 all three polarization asymmetries, P A k for k = 1, 2, 3, for pp → thj events in the left two panels (a), (b), and for pp →thj in the right panels. The upper plots in Fig. 16 (a), (c) are for W = 400 GeV, while the bottom plots (b), (d) are for W = 600 GeV, both for Q > 100 GeV. As expected, P A 1 = P A 3 = 0 for the SM (ξ = 0). We find that P A 3 > 0 for ξ = 0.05π (dashed curves) and 0.1π (dash-dotted curves) in all the regions of cos θ * , W, and Q that we study, including the four cases shown in Fig. 16. This follows our observation that P 3 is large and opposite in sign between t andt, see Fig. 13, and that azimuthal angle asymmetry is also opposite in sign, see Fig. 11. The magnitude of P A 1 is small near cos θ * = −1 where the cross section is large.
As explained in the previous sections, the asymmetries A φ , P 2 , P A 1 and P A 3 , whose signs measure the sign of the CP violating phase ξ are all so-called T-odd asymmetries. We found in section IV that the asymmetry A φ has opposite sign between the pp → thj events and pp →thj events, and we found in section V the polarization asymmetry P 2 has the opposite sign between the thj andthj events. In this section, we study consequences of the invariance under the discrete unitary transformationsT and CP, and CPT.
We adopt the symbolT for the unitary transformation under which all the three momenta p and the spin vectors s reverse their sign, in order to distinguish it from the time reversal transformation T, which reverses the sign of the time direction, and hence is anti-unitary. In the absence of the final state interaction phases of the amplitudes,T-odd asymmetries are proportional to T violation, or equivalently CP violation in QFT. Fig. 17 illustrates theT and CP transformations of the subprocess ub → dth, whose three momenta are the same as those in Fig. 3, or Eqs. (20) and (29). We add the helicities of external massless quarks (u, d, b) and also along the W + momentum direction, where the λ = −1 state is chosen for illustration. The top polarization, or its decay charged lepton momentum, is normal to the scattering plane along the positive y-axis. UnderT transformation, all the three momenta and spin polarizations change sign, as shown in (b), which can be viewed as (d) by making the 180 degree rotation about the y-axis. Comparing (a) and (d), we find that the initial state remains the same while in the final state φ → −φ and P 2 → −P 2 (78) underT transformation. Therefore the observation ofT-odd asymmetries such as implies either T-violation or the presence of an absorptive phase of the scattering amplitudes or both [39,40]. Likewise, the configuration (c) or (e) after the R y (π) rotation, is obtained by CP transformation from the configuration (a). All the particles are transformed to anti-particles and their helicities and three momenta are reversed. If we define the asymmetries A φ and P 2 for the processpp →thj, then CP-invariance between (a) and (e) implies A φ = −A φ and P 2 = P 2 .
(80) Violation of the above identities hence gives CP-violation. Finally, the configuration (f ) in Fig. 17 is obtained from (d) by applying CP, or from (e) by applyingT, together with the rotation R y (π). In short, (f ) is obtained from our original configuration (a) by CPT transformation [41]. By comparing (a) and (f ), CPT invariance, or the absence of the absorptive phase in QFT amplitudes should give As an illustration of how absorptive phases of the amplitudes in T or CP invariant theory contribute toT-odd asymmetries, we examine the impacts of the top-quark width in the s-channel propagator D t (P th ) in Eq. (12), or in the B factor of Eq. (33b). The width of Breit-Wigner propagator gives absorptive parts to our amplitudes, and since the top quark width appears only in the amplitudes with htt coupling, it can give rise toT-odd asymmetries, A φ and P 2 . We show in Fig. 18 the asymmetries A φ (a) and P 2 (b) in the CP-invariant SM (ξ = 0) for Γ t = 0 (blue), Γ t = 1.35 GeV (red), the SM value, and for 10 times the SM width Γ t = 13.5 GeV (green). We find that the asymmetries are both zero when Γ t = 0 as expected. Furthermore, we confirm the relations (80) between the asymmetries of pp → thj events, A φ and P 2 , and those ofpp →thj events, A φ and P 2 , respectively. This is a consequence of CP invariance, as can be viewed from the illustration by comparing the configurations (a) and (e). If CP is conserved, the amplitudes for the configuration (e) should have the same magnitude with those of the original configuration (a). The azimuthal angle between the W emission plane and the scattering plane is reversed, whereas thet spin polarization should be the same as the t spin polarization.
It is worth noting here that instead of top and anti-top spin polarization vector, P 2 and P 2 , if we use the decay charged lepton momentum normal to the scattering plane in the t ort rest frame, we find as a consequence of P 2 = P 2 (80) in CP invariant theory. Here, we assume that the t andt decay angular distributions follow the SM, where the charged leptons are emitted preferably along the t spin polarization direction, whereas they are emitted in the opposite of thet spin polarization direction. This is simply because only the right-handed l + and the left-handed l − are emitted from t andt decays, respectively, in the SM. The above spin-momentum correlation is CP invariant, and hence the identity (82) is also a consequence of CP invariance. In Fig. 19, we show comparisons of the asymmetries between pp → thj andpp →thj events for CP violating theory (ξ = 0) in the approximation of no absorptive parts in the amplitudes, i.e., we set Γ t = 0. We confirm the relations (81) for the same value of ξ, as a consequence of CPT invariance. The relations between the asymmetries in pp → thj andpp →thj are opposite between Fig. 18 and Fig. 19, as expected from Eqs. (80) and (81).
All the above relations between pp andpp may seem to be just formal rules since we will not have app collider with the LHC energy and luminosity. However, we find the above rules useful in testing our amplitudes, especially in fixing the relative sign between the two helicity amplitudes which determines the top and anti-top spin polarization directions away from their helicity axis. Furthermore, we find that it is possible to disentangleT-odd effects coming from the SM re-scattering effects (that give rise to the absorptive amplitudes) from CP violating new physics effects in pp collisions at the LHC by measuring the polarization asymmetry P 2 of t andt precisely.
Let us examine Fig. 15 again, where we show P 2 for thj andthj events at the LHC as a function of W, the th or th invariant mass. The polarization asymmetry P 2 have opposite sign between t andt. More quantitatively, we note that the magnitudes of the asymmetry is almost the same for small Q events (Q < 100 GeV) at large W (W 600 GeV). This is a consequence of CPT invariance of our tree-level amplitudes with Γ t = 0, because at small Q and large W, the events are dominated by the contributions of the longitudinally polarized W bosons; see Fig. 7 (a) and (b). Therefore, in this region of the phase space, we can regard the single top or anti-top plus Higgs production processes as which are CP conjugates of each other. Their amplitudes are given in Eqs. (30c) and (43), and we can obtain the polatization asymmetries directly from these amplitudes, which are independent of parton distribution functions in pp collisions. Because the absorptive amplitudes contribute to the polarization asymmetry P 2 with the same sign as shown in Fig. 18, we can further tell that the difference, measures CP violation, whereas the sum P 2 (thj events) + P 2 (thj events) measures the CPT-odd effects from the absorptive amplitudes in the region of small Q and large W. We find in the SM the leading contributions for the absorptive amplitudes appear at one-loop level in QCD and in the electroweak theory [42]. The top quark width that we adopted in this section for illustration is a part of the electroweak corrections. The sign of the polarization asymmetry P 2 remains the same and the magnitudes are larger at smaller W and large Q. This can be understood qualitatively also from Fig. 7, where the sub-dominant contributions are at small W (W 500 GeV) especially at large Q (Q > 100 GeV). The above subprocesses are again CP-conjugate to each other, and hence follow the rule (81) from CPT invariance.

VII. SUMMARY AND DISCUSSIONS
We studied associated production of single top (or anti-top) and the Higgs boson via t-channel W exchange at the LHC. We obtained analytically the helicity amplitudes for all the tree-level subprocesses with massless b (orb) quark PDF in the proton, and studied consequences of possible CP violation in the Higgs Yukawa coupling to the top quark. By choosing the momentum direction of the W ± exchanged in the t-channel, the helicity amplitudes are factorized into the W ± emission amplitudes from light quarks or anti-quarks, and the W + b → th or W −b →th production amplitudes. We find that the amplitudes for the right-handed top quark and those of the left-handed anti-top quark are sensitive to the sign of the CP violating phase ξ in the effective Yukawa interaction Lagrangian of Eq. (1). This is because the right-handed top quark is produced by the t † R t L operator with the e −iξ phase without chirality suppression, whereas the contribution of the t † L t R operator with the e iξ phase is doubly suppressed. For the anti-top production, the role of the two operators are reversed. On the other hand, the other amplitudes for the left-handed top and the right-handed anti-top productions are almost proportional to e iξ + e −iξ = 2 cos ξ because both terms in the Lagrangian contribute with one chirality suppression, either in the top quark propagator or from the helicity-chirality mismatch in the wave function, δ ′ and δ in Eq. (34), respectively.
We studied mainly the azimuthal angle asymmetry A φ between the W ± emission plane and the W + b → th or W −b →th production plane, and the t ort spin polarization normal to the scattering plane, P 2 , as observables which are sensitive to the sign of the CP phase ξ. The asymmetry A φ arises from the interference between the amplitudes with longitudinal and transversely polarized W ± contributions, and hence is significant when the exchanged momentum transfer Q is relatively large and the th orth invariant mass W is not too large, where both of the interfering amplitudes are significant. The magnitude of the asymmetry can be enhanced by selecting the chirality favored top or anti-top quark helicity, e.g. by selecting those events with charged lepton momentum along the top or anti-top momentum direction in the th orth rest frame; see Fig. 11.
On the other hand, the polarization asymmetry P 2 is obtained as the interference between the two helicity amplitudes of t ort. We find that the amplitudes are dominated by the collision of longitudinally polarized W ± and b orb when the momentum transfer Q is small and the invariant mass W of the th orth system is large. Therefore in such kinematical configuration, the asymmetry P 2 of the top and the anti-top can be regarded as the direct test of CP violation between the CP-conjugate processes, W + (λ = 0) + b → t + h and W − (λ = 0) +b →t + h. Because of the dominance of the longitudinally polarized W ± exchange amplitudes, all the differences in the quark and anti-quark PDF's of the colliding protons drop out in the polarization asymmetry.
All the analytic and numerical results presented in this report are done strictly in the tree-level, in order to clarify the symmetry properties of observable asymmetries that are sensitive to the sign of the CP violating phase ξ htt . In order to show their observability at the HL-LHC with its 3 ab −1 of integrated luminosity, we should perform the following studies.
Most importantly, we should identify the top and the Higgs decay modes which can be used to measure the asymmetries, since we may have different radiative corrections and background contributions for each set of the decay modes. We expect that semi-leptonic decays of t andt when the Higgs decays into modes without missing energy are favorable because the lepton charge identify t vs.t, and the charged lepton decay anglular distribution measures the t andt polarization with maximum sensitivity. Hadronically decaying t andt events can have sensitivity to the asymmetries, because their decay density matrix polarimeter introduced in Ref. [36] retains strong sensitivity to the t andt polarizations, and also because the CP asymmetry of the polarizations, P 2 (thj) ≈ −P 2 (thj) in Fig. 15 tells that the observable asymmetries in the decay distributions are the same between t andt events even if we cannot distinguish between them. Although the direct test of CP violation cannot be made in the hadronic decay modes, the sensitivity to the sign and the magnitude of the CP violating phase ξ can be improved by assuming the SM radiative contribution to the asymmetries [42].
We believe that the associated production of the Higgs boson and single t ort via t-channel W ± exchange at the LHC can be an ideal testing ground of the top quark Yukawa coupling, because the amplitudes with the htt Yukawa coupling and those of the hW W coupling interfere strongly. We studied the sensitivity of the process to possible CP violation in the Yukawa coupling. We anticipate that our studies based on the analytic form of the helicity amplitudes will be useful in the test of various scenarios of physics beyond the SM. For the mixed states, it is useful to introduce the density matrix where the summation is over all the processes and kinematical configurations that contribute to the top quark which we observe. Because the matrix is Hermitian and has trace 1, we can parametrize it as ρ = 1 2 1 + P · σ (A7) by using the σ matrices. We find , which for the pure state (A1) gives (A2).
In general, we can parametrize the density matrix (A7) as which is a sum of unpolarized top quark with the probability 1 − | P |, and the fully polarized top quark with its spin polarization orientation along P = | P |(sin θ cos φ, cos θ sin φ, cos θ) (A10) with the probability | P |. We find it convenient to show the general polarization vector P (A10) by using an arrow of length | P | in the polar coordinate defined as −π < θ ≤ π, −π 2 < φ ≤ π 2 . (A11) When the imaginary part of M − /M + is small, we tend to have small |φ|, and with the above definition we can show φ > 0 and φ < 0 as pointing up and down in the z-x plane [31] .
andρ ′d is obtained from (B9) by replacing thed and u four momentum (B5) in the W + rest frame. This simple density matrix distribution reduces to the charged lepton distribution (B1a) in the Pd u = 1 limit. The decay density distribution fort →bℓν is obtained similarly as dρ(t →bℓν) = 6B(t →bdū) where the density matrixρ is obtained from the d-quark momentum (B7b) in the t-rest frame, andρ ′ d is obtained by exchanging the d andū four momenta (B6) in the same event.
The decay angular distribution of arbitrary polarized t andt are then obtained simply by taking the ′ trace ′ Note that the decay distributions for t → bsc andt →bsc are the same as (B12a) and (B12b), respectively, where instead ofd and d momenta we haves and s momenta, while the identification probability Ps c = P sc may be significantly larger than 0.5, the most pessimistic value which was assumed in Ref. [36]. Finally, we find it encouraging that the t andt decay angular asymmetries have the same sign when as suggested from approximate CPT invariance in section VI and from Figs. 13 and 15 in section V. This tells that the polarization asymmetry can be measured even if we cannot distinguish t fromt, which may often be the case for hadronic decays.