Disentangling Higgs-Top Couplings in Associated Production

In the presence of CP violation, the Higgs-top coupling may have both scalar and pseudoscalar components, $\kappa_t$ and ${\tilde \kappa_t}$, which are bounded indirectly but only weakly by the present experimental constraints on the Higgs-gluon-gluon and Higgs-$\gamma$-$\gamma$ couplings, whereas upper limits on electric dipole moments provide strong additional indirect constraints on ${\tilde \kappa_t}$, if the Higgs-electron coupling is similar to that in the Standard Model and there are no cancellations with other contributions. We discuss methods to measure directly the scalar and pseudoscalar Higgs-top couplings by measurements of Higgs production in association with ${\bar t} t$, single $t$ and single ${\bar t}$ at the LHC. Measurements of the total cross sections are very sensitive to variations in the Higgs-top couplings that are consistent with the present indirect constraints, as are invariant mass distributions in ${\bar t} t H$, $tH$ and ${\bar t} H$ final states. We also investigate the additional information on $\kappa_t$ and ${\tilde \kappa_t}$ that could be obtained from measurements of the longitudinal and transverse $t$ polarization in the different associated production channels, and the ${\bar t} t$ spin correlation in ${\bar t} t H$ events.

the the Standard Model values. The dependences on these couplings of the cross section and Higgs branching ratios as well as the search strategy have been studied in [36,37] assuming CP-conserving interactions.
The structure of this paper is as follows. In Section 2 we introduce the scalar and pseudoscalar H-top couplings κ t andκ t and discuss the current indirect experimental constraints, paying particular attention to those provided by LHC constraints on the H-gluon-gluon and H-γ-γ couplings, taking their correlations into account [40]. Section 3 presents calculations of the total cross sections for H production in association withtt, single t and singlet. We show that, within the region of the (κ t ,κ t ) plane allowed at the 68% CL, the total cross section forttH production may be considerably smaller than in the Standard Model, whereas the cross sections for tH andtH may be considerably larger. As we show in Section 4, thē ttH, tH andtH invariant mass distributions may also be very different from those expected in the Standard Model. We proceed in Section 5 to discuss the possibilities for t polarization measurements at the LHC.
Our results indicate that the LHC operating at 13/14 TeV may soon be able to provide interesting direct constraints on κ t andκ t , including direct constraints on CP violation in the top sector.

Indirect Constraints on Top-Higgs Couplings
We write the top-H couplings in the form where v = 246 GeV is the conventional Higgs vacuum expectation value (v.e.v.) and κ t = 1 andκ t = 0 in the Standard Model. As observed in [29], theκ t coupling makes an important contribution to the electric dipole moment of the electron d e via a two-loop diagram of the Barr-Zee type. Assuming that the H coupling to the electron is the same as in the Standard Model, and that there are no other significant contributions to d e , the recent upper bound |d e /e| < 8.7 × 10 −29 cm [41] can be used to set the indirect constraint |κ t | < 0.01. However, we note that there is no experimental information on the electron-H coupling, that no direct information on this couplings is likely to become available in the foreseeable future, and that there could in principle be other contributions to d e that might cancel the two-loop top contribution, e.g., in supersymmetric models. We therefore seek bounds on κ t andκ t that are less model-dependent.
As already commented in the Introduction, the data from ATLAS and CMS on H production at the LHC with E CM = 7 and 8 TeV provide indirect bounds on the coupling parameters κ t andκ t via the constraints they impose on the H-gluon-gluon and H-γ-γ couplings, which have also been explored in [29,38]. The interpretation of the H-gluon-gluon and H-γ-γ constraints is also somewhat model-dependent, since they are obtained from data on H production and decay into γγ final states, and must rely on assumptions about the H couplings to other particles. In considering these constraints, we assume here that the couplings to other fermions and bosons are the same as in the Standard Model, i.e., κ f = 1 andκ f = 0 for f = t, and κ W = κ Z = 1. This assumption is purely phenomenological but motivated by the following reasons. There are several processes which can constrain κ W and κ Z independently from κ t andκ t [39] at the time when the luminosity required in this study is accumulated. The effect of κ f andκ f (f = t) is almost negligible unless κ f 1 because of the suppression proportional to the Yukawa couplings of the light fermions. Moreover such a possibility will be ruled out for the bottom and tau by the relatively precise H →bb and H → τ τ measurements available at the time.
Under these assumptions, the available ATLAS and CMS data on H production and decay were analyzed in [40] and constraints on the H-gluon-gluon and H-γ-γ couplings were derived, taking into account the correlations imposed by the measurements: see the left panel of Fig. 4 of [40]. The ratios are represented there by c 2 g and c 2 γ , respectively. Including the contribution to the Hgg loop amplitude of the b quark and the contribution to the Hγγ loop amplitude of the b quark, τ lepton and W bosons, following [29] one has in the notation of [40] The left panel of Fig. 4 of [40] displays regions in the (c γ , c g ) plane that are allowed by the LHC data at the 68, 95 and 99% CL. There we see explicitly the anticorrelation between c g and c γ due to the fact that one may, to some extent, compensate for a possible enhancement in the LHC H → γγ signal 1 by suppressing σ(gg → H), though this possibility is restricted by the LHC measurements of the strengths of the other H signatures if one assumes that κ f = 1 andκ f = 0 for f = t, and κ W = κ Z = 1 as done here. 1 We recall that this possibility is suggested by the ATLAS data, but not by the CMS data, so that the Standard Model value of the Hγγ coupling is allowed at the 68% CL [40].  Figure 1: The regions of the (κ t ,κ t ) plane allowed by the analysis of [40] at the 68 and 95% CL (solid and dotted red contours, respectively). Also shown for comparison is the region discussed in [29] (solid black contour). Black dots represent the simulated model points.
We display in Fig. 1 the regions of the (κ t ,κ t ) plane that are allowed at the 68, and 95% CL according to the analysis of [40]. At the 68% CL, the allowed region is a crescent with apex close to the Standard Model point (κ t ,κ t ) = (1, 0), bounded by the solid red contour, whereas at the 95% CL a complete annulus is allowed, bounded by the dotted red contour.
For convenience we define the CP violation phase in thettH coupling by For comparison, we also display the (smaller) crescent discussed in [29,38], bounded by the solid black contour. As already mentioned, if one assumes the Standard Model value of the electron-H coupling and there are no other important contributions to the EDM of the electron d e , the experimental upper limit on its value imposes |κ t | < 0.01. Here we consider the capability of future LHC measurements to constrain κ t andκ t directly, considering for illustration the full crescent allowed by the analysis of [40] at the 68% CL.
We have simulated the production ofttH, tH andtH final states at the LHC in leading order 2 using MadGraph [42] All the results presented below are for a centre-of-mass energy of 14 TeV. We consider first the effects on the total cross sections for H production in association withtt, single t and singlet, taking into account the present LHC constraints discussed in the previous Section, and discuss other possible LHC measurements in subsequent Sections.

Cross Sections forttH Production
The leading tree-level diagrams forttH production at the LHC are displayed in the upper panel of Fig. 2, and the left panel of Fig. 3 displays the corresponding values of σ(ttH) at the LHC at 14 TeV, using colour-coding to represent the ratio to the Standard Model cross section. The contributions of thettH andtγ 5 tH couplings to σ(ttH) do not interfere, so the iso-σ contours are ellipses in the (κ t ,κ t ) plane. For equal values of κ t andκ t , the latter yields a smaller cross section, with the result that the major axes of these ellipses are aligned with theκ t axis, as seen in the left panel of Fig. 3. Also shown there is the crescent-shaped region allowed by the present LHC data at the 68% CL. It is clear that in this region σ(ttH) is in general smaller than in the Standard Model, as we discuss in more detail later.
The left panel of Fig

Cross Sections for tH andtH Production
We now discuss the total cross sections for the associated production of H with a single t ort and a light-quark jet via the tree-level diagrams are shown in the lower panel of Fig. 2. We neglect s-channelqq → tHb andtHb production, since their cross sections are an order of magnitude smaller than the processes with light quarks [36]. In Fig. 2 Figure 2: Leading diagrams contributing tottH production at the LHC (upper panel) and to single t ort production (lower panel). The red and blue dots correspond to thettH and W W H couplings, respectively.
relative magnitude and sign of the scalarttH and W W H couplings, with the result that σ(tH) and σ(tH) are minimized around the Standard Model value κ t = 1 [36]. 3 As in the case of σ(ttH), iso-σ contours for tH andtH production are also ellipses whose major axes are aligned with theκ t axis, as we see in the right panel of Fig. 3, where colour-coding is used to represent the ratio to the Standard Model cross section. As a consequence, σ(tH) and σ(tH) increase along the 68% CL crescent as κ t decreases andκ t increases in magnitude. This effect is also seen clearly in the left panel of Fig. 4, where we see that σ(tH) and σ(tH) reach more than 3 times the Standard Model values when ζ t > 60 o . A measurement at the Standard Model level with a factor of two uncertainty would determine ζ t ∼ 0 ± 45 o . As seen in the right panel of Fig. 4, the combination of the decrease in σ(ttH) and the increases in σ(tH) and σ(tH) along the crescent imply that the ratio σ(tH +tH)/σ(ttH) increases by a factor of more than 20 along the crescent, compared to its value in the Standard Model, ∼ 0.06.

Mass Distributions
We now examine the information that can be obtained from measurements of the invariant masses of combinations of the final-state t,t and H particles. In the case of thettH final state, there are three distinct combinations that can be measured: the total invariant mass Mt tH , the tH (ortH) invariant mass M tH (or Mt H ), and thett invariant mass Mt t . In the case of single t ort production, there is also a forward jet j corresponding to the quark from which the virtual W was emitted, as seen in the lower panel of Fig. 2. Hence there are again three final-state particles t (ort), H and j, and therefore four measurable invariant masses

Total Invariant Mass Distributions
The left panel of Fig. 5 displays the normalized Mt tH distributions for ζ t = arc tan(κ t /κ t ) = 0 (in black), ±π/4 (in dotted red) and ±π/2 (in solid red). We see that the Mt tH distribution that is most peaked towards small masses is that for the Standard Model case ζ t = 0. That for ζ t = ±π/4 is less peaked, and that for ζ t = ±π/2 is substantially broader.
The central panel of Fig. 5 displays the M tHj distributions for ζ t = 0, ±π/4 and ±π/2 using the same colour-coding. In this case, we see that the invariant mass distribution is least peaked for the Standard Model case ζ t = 0, more peaked for ζ t = ±π/4 and particularly for ζ t = ±π/2. of ζ t . We note that Mt tH is maximized for |ζ t | = π/2: the value for |ζ t | = π would be the same as in the Standard Model.

Two-Particle Invariant Mass Distributions
More information may be obtained from two-particle invariant mass distributions, and we start by showing the two-body mass distributions inttH production events. The left and central panels of Fig. 6 show the invariant mass distributions oftt and tH, respectively, with the same colour-coding as in Fig. 5. The peaks of the distributions are lowest for the SM and highest for ζ t = ±π/2 in both thett and tH cases. The right panel of Fig. 6 shows the variation with ζ t of Mt t (solid black) and M tH (solid red) along a contour passing trough the middle of the crescent-shape allowed region in Fig. 1. The means of the two-particle invariant mass distributions take their lowest values in the Standard Model case and their maximum values for ζ t = ±π/2 in both thett and tH cases, as observed in the total invariant mass distribution. The difference between Mt t and M tH is more than 100 GeV, despite the difference between m t and m H being less than 50 GeV, and is almost independent of ζ t .
We do not show thetH invariant mass distribution as it is identical to that for tH. We now turn to the two-particle invariant mass distributions in the tHj andtHj production events. The left, central and right panels of Fig. 7 show thett, tH and tj invariant mass distributions, respectively. In thett case, the two-body invariant mass distribution is  case, whilst M Hj is minimized in this case. Although the threshold of the tj invariant mass is smaller than that for the tH invariant mass, M tj is larger than M tH and is indeed the largest among the three two-particle invariant masses in the 68% CL. allowed region. In the Standard Model case, M Hj is smaller than M tH . However this relation becomes reversed near the two tips of the crescent-shape allowed region because of the increase and decrease in M Hj and M tH , respectively, as ζ t deviates from the Standard Model value.
We also show the variation with ζ t of M Hj (dotted black), Mt H (solid blue) and Mt j (dotted blue) intHj production events in the right panel of Fig. 8. As can be seen, the features we have discussed above for the tHj events are also found fortHj production events. However, we also note a tendency for the two-particle invariant masses intHj production events to be somewhat smaller than the corresponding invariant masses in tHj production events. This can be traced back to the different initial-state parton distributions involved in tHj andtHj production.

Top Polarization Measurements
We now consider the additional information on the top-H couplings that could be obtained from measurements of the top-(anti)quark polarization(s). In principle, there are two classes of measurements: single-spin measurements inttH and single tH (tH) production with an accompanying light-quark jet, and measurements of spin-correlations inttH production. Further, one can measure the single-(anti)top polarization either in the production plane or perpendicular to it. The latter is particularly interesting, as it violates CP at the tree level.

Single-Spin Measurements
It is easy to see that the single-spin asymmetries actually vanish inttH production, because of the Dirac matrix factors in the vertices. However, the single-spin measurements are interesting for tHj andtHj production, because of the 1 − γ 5 factor in the W tb coupling.
As already noted, the matrix elements of these processes have two competing Feynman diagrams: one is proportional to thettH coupling and the other to the W W H coupling, as seen in the lower panel of Fig. 2. In the latter diagram, the t (ort) is emitted from the initial b(b)-quark when it exchanges a W boson with a quark (or antiquark) in the other proton.
This t(t) quark therefore prefers the left-handed chirality. In the former diagram, t (ort) is produced in the same way but subsequently emits a H, changing its chirality. One can therefore expect that the tops in these processes are polarized to some extent, depending on the details of thettH coupling.
The angular distributions of the top decay products are correlated with the top spin direction in the following way [43][44][45]: where f is the type of top decay product: f = b, , ..., θ f is the angle between the decay product f and the top spin quantization axis measured in the rest frame of the top, and P t is the degree of the top polarization: The coefficient ω f depends on the type of decay product, e.g., ω W = −ω b = 0.41 and ω = 1 at tree level.
We consider first the angle θ between the direction of the t and the final-state lepton measured at the rest frame of the top in tHj production events. The left panel of Fig. 9 displays the cos θ distributions. As previously, the distribution for the Standard Model case ζ t = 0 is shown in black, and the distributions for |ζ t | = π/4 and π/2 in dotted and solid red, respectively. We can see that the lepton momentum in the Standard Model case strongly prefers the opposite direction to the top's boost direction at the top's rest frame, meaning that tops are negatively polarized, P t < 0. As |ζ t | increases this preference is weakend. For |ζ t | = π/4 the distribution is already quite flat, and the slope is even positive, P t > ∼ 0, for The dependence on ζ t can more explicitly be seen in the right panel of Fig. 9, which displays the variation with ζ t of the forward-backward asymmetry along a contour passing trough the middle of the crescent-shape allowed region. The red and blue curves correspond to the A t and At in the tHj andtHj production events, respectively.
The shaded bands represent estimates of the measurement error with 100/fb of integrated luminosity at 14 TeV, again ignoring effects of parton showering, top reconstruction, detector resolution, 5 etc.. We see that, within the range of ζ t allowed by the present data, the asymmetry is largest in magnitude (and negative) for ζ t = 0 (the Standard Model case), is reduced in magnitude for ζ t = 0, and changes sign for ζ t = ±π/2. On the other hand, there is no sensitivity to the sign of ζ t . In the Standard Model case, the asymmetries for the tHj andtHj events are identical. For ζ t = 0, tops are more positively polarized in the tHj events than in thetHj events.
We now consider the top (anti-top) polarization perpendicular to the three-body production plane. We define the spin quantisation axis by − → p j × − → p H at the rest frame of the top (anti-top), where j is the forward jet produced by the final-state quark after radiating a virtual W in the diagrams in the lower panel of Fig. 2. The left panel of Fig. 10 shows the cos θ ⊥ distribution, where θ ⊥ is the angle between the lepton momentum and the spin quantization axis defined above at the rest frame of the top. We see that the distribution is flat for the Standard Model case ζ t = 0. One the other hand, when ζ t = 0, the lepton prefers one side of the hemisphere with respect to the three-body production plane at the rest frame of the top. The right panel in Fig. 10 shows the variation with ζ t of the asymmetry A ⊥ , which is defined in the same way as in Eq. (7) for the cos θ ⊥ , with the same colour-coding as in Fig. 9. As expected, there is no up-down asymmetry for the Standard Model case ζ t = 0, but there is a measurable asymmetry for ζ t = ±π/4 and ±π/2. In particular, the sign of the perpendicular asymmetry is sensitive to the sign of ζ t = arc tan(κ t /κ t ). This measurement could therefore provide a direct probe of CP violation in the top-H couplings. The distributions in the semileptonic decay angle θ ⊥ out of the tHj plane for ζ t = arc tan(κ t /κ t ) = 0 (in black), ±π/4 (in dotted red and blue) and ±π/2 (in solid red and blue). Right panel: the asymmetry perpendicular to the plane of tHj (tHj) production, A l⊥ , as a function of ζ t is indicated in red (blue): the shading represents an estimate of the measurement error with 100/fb of integrated luminosity at 14 TeV.

Spin Correlation Measurements
We consider finally possible measurements of thett spin correlation inttH production. The left panel of Fig. 11 shows the distribution in the angle ∆φ + − between the two lepton momenta projected onto the plane perpendicular to the t direction at the centre-of-mass frame of thett system. The sign of ∆φ + − is defined as the sign of − → p t · ( − → p − × − → p + ). 6 As previously, the distribution for the Standard Model case ζ t = arc tan(κ t /κ t ) = 0 is shown in black, those for ζ t = ±π/4 as dotted lines, and those for ±π/2 as solid lines (red and blue for ζ t >, < 0, respectively). We see that the distribution has the form We see in the left panel of Fig. 11 that the phase shift δ vanishes for the Standard Model case ζ t = 0, but takes non-zero values for ζ t = 0, and we note that this phase shift is sensitive to the sign of ζ t . The right panel in Fig. 11 shows the value of δ as a function of ζ t . One can see that the δ varies from −π to π as ζ t varies from −π/2 to π/2. We find that the The distributions in the angle ∆φ + − between the leptons produced in t andt decay inttH production, in the centre-of-mass of thett system. We display the distributions for ζ t = arc tan(κ t /κ t ) = 0 (in black), ±π/4 (in dotted red and blue) and ±π/2 (in solid red and blue). Right panel: the phase shift δ as a function of ζ t .
dependence of δ on ζ t can be very well fitted by the function δ = 2ζ t − sin(2ζ t )/2.

Summary
We have shown in this paper that the cross sections and final-state distributions inttH, tH andtH production are sensitive to the ratio between the scalar and pseudoscalar top-H couplings κ t andκ t . In particular, the total cross section forttH production decreases significantly as the ratioκ t /κ t increases within the ranges of values of these couplings that are allowed by present data on the Hgg and Hγγ couplings. On the other hand, the total cross sections for tH andtH production increase as the ratioκ t /κ t increases.
We have also found that the invariant mass distributions for the three-body combinations ttH, tHj andtHj are sensitive to the ratioκ t /κ t , becoming less peaked at small masses in thettH case and more peaked in the tHj andtHj cases as the ratioκ t /κ t increases. The two-body invariant mass distributions also carry information about the top-H couplings.
Supplementary information on the ratioκ t /κ t could be provided by angular distributions in semileptonic t andt decays. In particular, lepton decay angles from the top boost direction could provide information on the magnitude ofκ t /κ t , and lepton decay angles against the tHj (ortHj) production plane provide information on the sign ofκ t /κ t . Information both on the magnitude and sign ofκ t /κ t could also be provided by measurements of the angle ∆φ + − between the directions of leptons produced int and t decays in the case ofttH production.
We conclude that there are good prospects for disentangling the scalar and pseudoscalar top-H couplings at the LHC via a combination of measurements ofttH, tH andtH production.