# Higgs characterisation at NLO in QCD: CP properties of the top-quark Yukawa interaction

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## Abstract

At the LHC the CP properties of the top-quark Yukawa interaction can be probed through Higgs production in gluon fusion or in association with top quarks. We consider the possibility for both CP-even and CP-odd couplings to the top quark to be present, and study CP-sensitive observables at next-to-leading order (NLO) in QCD, including parton-shower effects. We show that the inclusion of NLO corrections sizeably reduces the theoretical uncertainties, and confirm that di-jet correlations in \(H+2\) jet production through gluon fusion and correlations of the top-quark decay products in \(t\bar{t}H\) production can provide sensitive probes of the CP nature of the Higgs interactions.

## Keywords

Higgs Boson Parton Shower Higgs Production Gluon Fusion Effective Field Theory## 1 Introduction

The top-quark Yukawa interaction has played a crucial role in the recent discovery of the Higgs boson in the first run of the LHC [1, 2, 3, 4]. It is thanks to its large value that production in gluon fusion (GF), which mostly proceeds through a top-quark loop in the Standard Model (SM), has provided the necessary statistics for discovery already with a modest integrated luminosity. The wealth of production and decay channels available for a SM scalar with a mass of about 125 GeV, has also made it possible to combine information from different final-state measurements [5]. Global coupling extractions [3, 6] provide indirect evidence that the Higgs boson couples to top quarks with a strength in agreement with the SM expectations. Furthermore, the first exploratory searches of associated Higgs production with a top-quark pair (\(t\bar{t}H\)), while not yet being sensitive enough for an observation, already set an upper bound on the strength of the interaction of 3–6 times the SM expectation [7, 8, 9]. In the coming run of the LHC at 13 TeV, \(t\bar{t}H\) production will certainly serve as a key channel to test the SM and explore new physics.

While the path towards more and more precise determinations of the strength of the Yukawa interaction with the top (and of the Higgs boson couplings in general) is clear, the investigation of the structure and the properties of such interaction is considerably more open. One of the fundamental questions is whether the Higgs–top-quark coupling is CP violating, i.e. the Higgs couples to both scalar and pseudoscalar fermion densities. In this context, it is important to stress that so far all experimental determinations of the Higgs CP properties [4, 10] have been obtained from the \(H\rightarrow VV\rightarrow 4\ell \) decay mode and therefore only constrain the \(HVV\) interactions.

Gathering information on the CP properties of the top-quark Yukawa interaction is not an easy task. As there is no decay mode of the Higgs to or through top quarks that can be effectively studied at the LHC, only Higgs production can be considered. In addition, even though different couplings, either scalar, pseudoscalar or mixed, have an impact on the production rates [11, 12, 13] and can also be bound by indirect measurements [14], only specially designed observables can provide direct evidence of CP-violating effects at hadron colliders. In inclusive Higgs production, for example, at least two extra jets are needed in the final state to be able to construct CP-sensitive observables. These can probe the Higgs interaction with the top quark through GF [as well as with \(W\) and \(Z\)’s in vector boson fusion (VBF)]. The \(t\bar{t}H\) final state, on the other hand, provides many CP-sensitive observables that can also be constructed from the daughters of the top-quark decays. In fact, in \(H+\) jets and \(t\bar{t}H\) production information on the CP nature of the top-quark coupling is encoded in the correlations between the jets and among the top–antitop decay products. This means that the choice of decay mode of the Higgs in the corresponding analyses can be done freely and based on criteria other than the requirement of a precise reconstruction of the Higgs resonance, something that, in general, might not even be needed.

In order to test the different hypotheses for the Higgs sector, the approach based on an effective field theory (EFT) turns out to be particularly suitable, given the fact that the current experimental data do not show any clear sign of physics beyond the SM. In such an approach, no new particle and symmetry is hypothesised on top of the SM ones. This has the advantage of reducing the number of new parameters and interactions compared to other approaches based only on Lorentz symmetry, without losing the ability to describe in a model-independent way the effects of any new physics we cannot directly access at the current energies. Furthermore, the EFT approach can be systematically improved by including higher-dimensional operators in the lagrangian on the one hand (which are suppressed by higher powers of the scale \(\Lambda \) where new physics appears), and higher-order perturbative corrections on the other hand.

The aim of this work is to present how EFT predictions accurate to next-to-leading order (NLO) in QCD matched to a parton shower can be used to determine the CP properties of the Higgs boson coupling to the top quark, through Higgs production in association with jets or with a pair of top quarks. To this aim we employ the *Higgs Characterisation* (HC) framework originally proposed in [15], which follows the general strategy outlined in [16] and has been recently applied to the VBF and VH channels [17]. In this respect, this work contributes to the general effort of providing NLO accurate tools and predictions to accomplish the most general and accurate characterisation of Higgs interactions in the main production modes at the LHC. Note that at variance with VBF and VH, \(H+\) jets and \(t\bar{t}H\) are processes mediated by QCD interactions at the Born level, hence higher-order corrections are expected to be more important and certainly needed in analyses aiming at accurate and precise extractions of the Higgs properties.

First, we consider Higgs production in GF together with extra jets, focusing on final states with at least two jets. This process is not only a background to VBF, but it can also provide complementary information on the Higgs boson coupling properties [18, 19, 20, 21, 22, 23, 24]. In the heavy-top limit, the CP structure of the Higgs–top interaction is inherited by the effective Higgs–gluon vertices [25, 26, 27, 28, 29, 30]. Higgs plus two (three) jets through GF at LO has been computed in [31, 32, 33, 34], where the full top-mass dependence was retained. The results cited above show that the large top-mass limit is a very good approximation as long as the transverse momentum of the jets is not sensibly larger than the top mass and justify the use of EFT approach for the Higgs–gluons interactions. In the \(m_\mathrm{t}\rightarrow \infty \) limit, the resulting analytic expressions at NLO for GF \(Hjj\) production have been implemented in MCFM [35], which has been used by Powheg Box [36] and Sherpa [37] to obtain NLO results matched with parton shower (NLO+PS). Independent NLO+PS predictions in the Sherpa package using GoSam [38] for the one-loop matrix elements and in MadGraph5_aMC@NLO [39], which embodies MadFKS [40] and MadLoop [41], are also available. We note that all the above predictions are for the SM Higgs boson, i.e. the CP-even state, and \(Hjj\) production for the CP-odd state has been only available at LO, yet with the exact top-mass dependence [21]. In this paper we present NLO results in the large top-mass limit for GF production of a generic (mixed) scalar/pseudoscalar state in association with one or two jets at the LHC, also matching to parton shower.

Second, we study \(t\bar{t}H\) production for arbitrary CP couplings, including NLO+PS effects. While NLO corrections in QCD for this process have been known for quite some time [42, 43], the NLO+PS prediction has been done only recently, for both CP eigenstates, \(0^+\) and \(0^-\), in aMC@NLO [44] and in the Powheg Box [45] for the CP-even case only. The spin-correlation effects of the top–antitop decay products have also been studied at the NLO+PS level with the help of MadSpin [46, 47]. Weak and electroweak corrections have also been reported recently in [48, 49], respectively. The phenomenology of a CP-mixed Higgs coupling to the top quark at the LHC has been studied at LO in [50]. In addition to the case where the Higgs has definite CP quantum numbers, here we consider the more general case of a CP-mixed particle (\(0^\pm \)) including NLO in QCD, parton-shower effects and spin-correlated decays.

The paper is organised as follows. In the next section we recall the effective lagrangian employed for a generic spin-0 resonance and define sample scenarios used to determine the CP properties of the Higgs boson. We also briefly describe our setup for the computation of NLO corrections in QCD together with matching to parton shower. In Sect. 3 we present results of \(H+\) jets in GF, focusing on the \(H+2\) jet production. We also make a comparison with VBF production with dedicated kinematical cuts. In Sect. 4 we illustrate the \(t\bar{t}H\) production channel. In Sect. 5 we briefly summarise our findings and in Appendix we present the Feynman rules, the UV and the \(R_2\) counterterms necessary to NLO computations for GF in the heavy-top-quark limit.

## 2 Setup

In this section, we summarise our setup. We start from the definition of the effective lagrangian, pass to the identification of suitable benchmark scenarios, and finally to event generation at NLO in QCD accuracy, including parton-shower effects.

### 2.1 Effective lagrangian and benchmark scenarios

The most robust approach to build an effective lagrangian is to employ all the SM symmetries, i.e. start from a linearly realised electroweak symmetry and systematically write all higher-dimensional operators, organised in terms of increasing dimensions. The complete basis at dimension six has been known for a long time [51, 52] and recently reconsidered in more detail in the context of the Higgs boson; see e.g., [53, 54, 55]. This approach has been followed in the FeynRules [56] implementation of [57], where the effective lagrangian is written in terms of fields above the electroweak symmetry breaking (EWSB) scale and then expressed in terms of gauge eigenstates.

In [15] we have followed an alternative approach (and yet fully equivalent in the context of the phenomenological applications of this paper, as explicitly seen in Tables 1 and 3 of [57]) and implemented the EFT lagrangian starting from the mass eigenstates, so below the EWSB scale, and for various spin–parity assignments (\(X(J^P)\) with \(J^P=0^{\pm },1^{\pm },2^+\)). We have also used FeynRules, whose output in the UFO format [58, 59] can be directly passed to MadGraph5_aMC@NLO [39]. We stress that this procedure is fully automatic for computations at LO, while at NLO the UFO model has to be supplemented with suitable counterterms, as will be recalled in Sect. 2.2, a procedure that in this work has been performed by hand.

^{1}For the \(X_0\gamma \gamma \) and \(X_0Z\gamma \) interactions, in addition to the top-quark loop, a \(W\)-boson loop contributes for the CP-even case and in fact dominates. As a result, these processes are less sensitive to the CP properties of the top Yukawa coupling. The dimensionful loop-induced couplings \(g_{\scriptscriptstyle Xyy'}\) are shown in Table 1. In the following, we focus only on the gluonic operators in Eq. (2). As mentioned in the introduction, the EFT prediction can be improved by including higher-dimensional operators, and this can be achieved rather easily in our framework by adding, e.g., the dimension-seven Higgs–gluon lagrangian [60] into the HC model. Finally, we recall that in the HC lagrangian the loop-induced \(X_0ZZ\) and \(X_0WW\) interactions are parametrised by the cutoff \(\Lambda \), since those are sub-leading contributions to the SM tree-level interaction; see Eq. (6) below.

Loop-induced couplings \(g_{\scriptscriptstyle Xyy'}\) in the lagrangian (2). \(c_W=\cos \theta _W\) and \(C=\sqrt{\frac{\alpha _{\scriptscriptstyle \mathrm EM}G_F m_Z^2}{8\sqrt{2}\pi }}\)

\(g_{\scriptscriptstyle Xyy'}\) | \(gg\) | \(\gamma \gamma \) | \(Z\gamma \) |
---|---|---|---|

\(X=H\) | \(-\alpha _s/3\pi v\) | \(47\alpha _\mathrm{EM}/18\pi v\) | \( C (94 c^2_W-13)/9\pi v\) |

\(X=A\) | \(\alpha _s/2\pi v\) | \(4\alpha _\mathrm{EM}/3\pi v\) | \(2 C (8c^2_W-5)/3\pi v\) |

HC model parameters

Parameter | Description |
---|---|

\(\Lambda \) (GeV) | Cutoff scale |

\(c_{\alpha }\) (\(\equiv \) \(\cos \alpha \)) | Mixing between \(0^+\) and \(0^-\) |

\(\kappa _i\) | Dimensionless coupling parameter |

Benchmark scenarios for GF/\(t\bar{t}H\)

Scenario for GF/\(t\bar{t}H\) | HC parameter choice |
---|---|

\(0^+\) (SM) | \(\kappa _{\scriptscriptstyle Hgg/Htt}=1\ (c_{\alpha }=1)\) |

\(0^-\) | \(\kappa _{\scriptscriptstyle Agg/Att}=1\ (c_{\alpha }=0)\) |

\(0^{\pm }\) | \(\kappa _{\scriptscriptstyle Hgg,Agg/Htt,Att}=1\ (c_{\alpha }=1/\sqrt{2})\) |

Benchmark scenarios for VBF used for comparison with Higgs production in GF

Scenario for VBF | HC parameter choice |
---|---|

\(0^+\) (SM) | \(\kappa _{\scriptscriptstyle SM}=1\ (c_{\alpha }=1)\) |

\(0^+\) (HD) | \(\kappa _{\scriptscriptstyle HZZ,HWW}=1\ (c_{\alpha }=1)\) |

\(0^-\) (HD) | \(\kappa _{\scriptscriptstyle AZZ,AWW}=1\ (c_{\alpha }=0)\) |

\(0^{\pm }\) (HD) | \(\kappa _{\scriptscriptstyle HZZ,HWW,AZZ,AWW}=1\ (c_{\alpha }=1/\sqrt{2})\) |

### 2.2 NLO corrections matched with parton shower

MadGraph5_aMC@NLO is designed to perform automatic computations of tree-level and NLO differential cross sections, including the possibility of matching LO and NLO calculations to parton showers via the MC@NLO method [61], and also to merge LO [62] and NLO [63] samples that differ in parton multiplicities. Currently, NLO computations are restricted to QCD corrections. They can be achieved fully automatically in the SM. Recently, the computation of ultraviolet (UV) and \(R_2\) counterterms, the latter being originally necessary to compute one-loop amplitudes with the CutTools [64] implementation of the OPP integrand-reduction method [65], was automated for any renormalisable theory [66].

The UV and \(R_2\) counterterms for QCD one-loop amplitudes in the SM were presented in [67] and have been available in MadGraph5_aMC@NLO for some time. The corresponding terms for effective interactions between the SM Higgs and gluons were presented in [68]. Here, we have derived them for the pseudoscalar case, listed in the appendix, and coded by hand in a UFO model named HC_NLO_X0. The resulting model is publicly available online in the FeynRules repository [69].

### 2.3 Simulation parameters

We generate events for the LHC with centre-of-mass (CM) energies \(\sqrt{s}=8\) and \(13\) TeV, and we set the \(X_0\) resonance mass to \(m_{X_0}=125\) GeV. We take the heavy-top-quark limit for GF, while we set the top-quark mass to \(m_{t}=173\) GeV in \(t\bar{t}X_0\) production.

## 3 Gluon-fusion production with jets

As mentioned above, since our interest is geared towards QCD effects in production distributions, we do not include Higgs decays in our studies. We stress, however, that decays (as predicted in the HC model) can be efficiently included at the partonic event level by employing MadSpin [46], before passing the short-distance events to a parton-shower program.

### 3.1 Total rates

We start by showing results for total cross sections for Higgs plus jet production in GF, not only for \(H+2\) jets but also for \(H+1\) jet as a reference. We remark here that as GF is the dominant Higgs production mechanism, enormous theoretical efforts to achieve more precise computation have been made over the last decade and we refer to the reports by the LHC Higgs Cross Section Working Group [85, 86, 87] for more details. We note that a first calculation of Higgs plus one jet at NNLO (\(gg\) only and in the EFT) has been reported in [88].

LO and NLO cross sections and corresponding \(K\) factors for \(pp\rightarrow X_0+1\) jet (GF channel) at the 8- and 13-TeV LHC, for the three scenarios defined in Table 3. The integration error in the last digit(s) (in parentheses), and the fractional scale (left) and PDF (+\(\alpha _s\)) (right) uncertainties are also reported. In addition to fixed-order results, the PS-matched NLO cross sections and the ratios \(R\equiv \sigma _\mathrm{NLO+PS}/\sigma _\mathrm{NLO}\) are also shown

Scenario | \(\sigma _\mathrm{LO}\) (pb) | \(\sigma _\mathrm{NLO}\) (pb) | \(K\) | \(\sigma _\mathrm{NLO+PS}\) (pb) | \(R\) |
---|---|---|---|---|---|

LHC 8 TeV | |||||

\(0^+\) | 4.002(4)\({}^{+46.8}_{-29.6}\) \(\pm 3.3\,\%\) | 5.484(7)\({}^{+17.0}_{-16.8}\) \(\pm 1.2\,\%\) | 1.37 | 4.618\({}^{+21.8}_{-18.8}\) \(\pm 1.2\,\%\) | 0.84 |

\(0^-\) | 9.009(9)\({}^{+46.8}_{-29.6}\) \(\pm 3.3\,\%\) | 12.34(2)\({}^{+17.1}_{-16.8}\) \(\pm 1.2\,\%\) | 1.37 | 10.38\({}^{+21.7}_{-18.8}\) \(\pm 1.2\,\%\) | 0.84 |

\(0^\pm \) | 6.511(6)\({}^{+46.8}_{-29.6}\) \(\pm 3.3\,\%\) | 8.860(14)\({}^{+16.9}_{-16.8}\) \(\pm 1.2\,\%\) | 1.36 | 7.474\({}^{+21.7}_{-18.8}\) \(\pm 1.2\,\%\) | 0.84 |

LHC 13 TeV | |||||

\(0^+\) | 10.67(1)\({}^{+41.7}_{-27.5}\) \(\pm 2.6\,\%\) | 14.09(2)\({}^{+16.2}_{-14.9}\) \(\pm 1.1\,\%\) | 1.32 | 12.08\({}^{+19.8}_{-16.7}\) \(\pm 1.0\,\%\) | 0.86 |

\(0^-\) | 24.01(2)\({}^{+41.7}_{-27.5}\) \(\pm 2.6\,\%\) | 31.67(6)\({}^{+16.2}_{-14.9}\) \(\pm 1.1\,\%\) | 1.32 | 27.14\({}^{+20.3}_{-16.4}\) \(\pm 1.0\,\%\) | 0.86 |

\(0^\pm \) | 17.36(2)\({}^{+41.7}_{-27.5}\) \(\pm 2.6\,\%\) | 22.83(3)\({}^{+16.2}_{-14.9}\) \(\pm 1.1\,\%\) | 1.32 | 19.59\({}^{+19.5}_{-16.6}\) \(\pm 1.0\,\%\) | 0.86 |

Requiring the presence of jets in the final state entails imposing cuts at the generation level as well as after event generation in the case of NLO+PS simulation. We have checked that the cuts at the generation level were loose enough not to affect the NLO+PS rates and distributions. Since reconstructed jets after parton shower and hadronisation can be different from the fixed-order parton jets, the parton-shower matched cross section can be different from the fixed-order prediction.

The figure in parentheses is the integration error in the last digit(s). The first uncertainty (in percent) corresponds to the envelope obtained by varying independently the renormalisation and factorisation scales by a factor 2 around the central value, \(\mu _0=H_\mathrm{T}/2\). The second one corresponds to the PDF (\(+\alpha _s\)) uncertainty. As mentioned in Sect. 2.3, the full PDF + \(\alpha _s\) uncertainty is available only at NLO. It is well known that PDF and \(\alpha _s\) uncertainties are comparable for GF at NLO [72], thus we take them both into account. We can see that both the scale dependence and the PDF + \(\alpha _s\) uncertainties are independent of the scenarios, and as expected they are significantly reduced going from LO to NLO. It is also evident that the residual scale dependence is the dominant source of uncertainty in the GF channel. We also note that \(\sigma (0^-)\) is larger than \(\sigma (0^+)\) by a factor of 2.25 at LO (and to a good approximation even at NLO) due to the different coupling normalisation (see Table 1), and \(\sigma (0^{\pm })\) is equal to the average of \(\sigma (0^+)\) and \(\sigma (0^-)\). This means that there are no interference effects in the total rates for this process.

In addition to the fixed-order results, we also show the NLO cross sections matched with parton shower (\(\sigma _\mathrm{NLO+PS}\)) in the table. The ratios to the fixed-order NLO rates, \(R\equiv \sigma _\mathrm{NLO+PS}/\sigma _\mathrm{NLO}\) are shown in the last column. These ratios are smaller than one, as extra radiation generated by the parton shower tends to spread the energy of the original hard partons, affecting the spectrum of the jets and leading to more events which fail to pass the cuts. The survival rate after shower slightly increases as increasing the collision energy. We note that the ratios can slightly depend on the parton-shower programs [89], and these differences shall be considered as matching systematics. Another effect of the parton shower that we observe is a slightly increased scale dependence in the results, compared to the corresponding fixed-order predictions.

Same as Table 5, but for \(pp\rightarrow X_0+2\) jets (GF)

Scenario | \(\sigma _\mathrm{LO}\) (pb) | \(\sigma _\mathrm{NLO}\) (pb) | \(K\) | \(\sigma _\mathrm{NLO+PS}\) (pb) | \(R\) |
---|---|---|---|---|---|

LHC 8 TeV | |||||

\(0^+\) | 1.351(1)\({}^{+67.1}_{-36.8}\) \(\pm 4.3\,\%\) | 1.702(6)\({}^{+19.7}_{-20.8}\) \(\pm 1.7\,\%\) | 1.26 | 1.276\({}^{+29.4}_{-23.9}\) \(\pm 1.7\,\%\) | 0.75 |

\(0^-\) | 2.951(3)\({}^{+67.2}_{-36.8}\) \(\pm 4.4\,\%\) | 3.660(15)\({}^{+19.1}_{-20.6}\) \(\pm 1.7\,\%\) | 1.24 | 2.755\({}^{+29.8}_{-24.1}\) \(\pm 1.8\,\%\) | 0.75 |

\(0^\pm \) | 2.142(2)\({}^{+67.1}_{-36.8}\) \(\pm 4.4\,\%\) | 2.687(10)\({}^{+19.6}_{-20.8}\) \(\pm 1.7\,\%\) | 1.25 | 2.022\({}^{+29.7}_{-24.1}\) \(\pm 1.8\,\%\) | 0.75 |

LHC 13 TeV | |||||

\(0^+\) | 4.265(4)\({}^{+61.5}_{-34.9}\) \(\pm 3.3\,\%\) | 5.092(23)\({}^{+15.4}_{-17.9}\) \(\pm 1.2\,\%\) | 1.19 | 4.025\({}^{+23.9}_{-21.3}\) \(\pm 1.2\,\%\) | 0.79 |

\(0^-\) | 9.304(9)\({}^{+61.6}_{-34.9}\) \(\pm 3.4\,\%\) | 11.29(4)\({}^{+16.0}_{-18.2}\) \(\pm 1.2\,\%\) | 1.21 | 8.701\({}^{+24.6}_{-21.6}\) \(\pm 1.3\,\%\) | 0.77 |

\(0^\pm \) | 6.775(6)\({}^{+61.5}_{-34.9}\) \(\pm 3.3\,\%\) | 8.055(35)\({}^{+15.8}_{-18.2}\) \(\pm 1.2\,\%\) | 1.19 | 6.414\({}^{+24.4}_{-21.5}\) \(\pm 1.2\,\%\) | 0.80 |

### 3.2 Distributions

In the previous section we have seen that if the strength of the scalar and pseudoscalar couplings in the Higgs–top-quark interaction is similar [i.e. \(\kappa _{\scriptscriptstyle Htt}g_{\scriptscriptstyle Htt}\sim \kappa _{\scriptscriptstyle Att}g_{\scriptscriptstyle Att}\) in Eq. (1)], the total Higgs production rate in GF is sensitive to the CP mixing of the Higgs boson. We now turn to distributions, where GF jet–jet correlations are known tools to determine the Higgs CP properties [18, 19, 20, 21, 22, 23, 24]. In the following, all the distributions will be shown for the 13-TeV LHC. For these studies, we require the presence of at least two reconstructed jets in the final states. The jets are ordered by the transverse momenta.

Figures 2 and 3 show the effect of the invariant mass cut on the \(p_\mathrm{T}\) and \(\eta \) distributions for the resonance \(X_0\) and the leading jet. Imposing larger \(m_{jj}\) cuts leads to harder transverse momenta for both the \(X_0\) and the jets; as a result, the \(X_0\) is produced more centrally, while the jets are shifted to the forward regions and the difference in the low \(p_\mathrm{T}(X_0)\) region between the various CP scenarios becomes more pronounced. This behaviour is due to the fact that at larger \(m_{jj}\) topologies featuring the emission of the Higgs boson by a gluon exchanged in the \(t\)-channel are enhanced, similarly to the typical VBF topology.

A remarkable observation is that the \(\Delta \phi _{jj}\) distribution is more sensitive to the CP-mixed state, when the two leading jets (ordered by \(p_\mathrm{T}\)) are reordered in pseudorapidity^{2} (dashed green), compared to the distribution with the usual \(p_\mathrm{T}\) jet ordering (solid green). This is especially true for the maximal mixing scenario, which we consider here, since with just \(p_\mathrm{T}\) ordering the \(\pi /4\) phase shift, generated by quantum interference between the CP-even and -odd components, is cancelled between \(+\Delta \phi _{jj}\) and \(-\Delta \phi _{jj}\) [18]. Indeed, the distribution for \(0^{\pm }\) without \(\eta \) ordering is just the weighted average of the \(0^+\) and \(0^-\) cases.

Selection efficiencies with different di-jet invariant mass cuts for \(pp\rightarrow X_0jj\). A jet veto defined in (14) is also applied in the last column

\(m_{jj}>\) | 250 GeV (%) | 500 GeV (%) | 500 GeV + jet veto (%) |
---|---|---|---|

LHC 8 TeV | |||

\(0^+\) | 22.7 | 6.6 | 5.0 |

\(0^-\) | 21.4 | 5.7 | 4.5 |

\(0^\pm \) | 21.5 | 6.2 | 4.6 |

LHC 13 TeV | |||

\(0^+\) | 26.3 | 9.0 | 6.4 |

\(0^-\) | 25.4 | 8.6 | 6.2 |

\(0^\pm \) | 25.6 | 8.6 | 6.2 |

In the main panel, we also draw a comparison with the VBF contributions (dashed lines). The \(p_\mathrm{T}(j_1,j_2)\) and \(\Delta \eta (j_1,j_2)\) distributions show that in the SM VBF case the Higgs boson is produced more centrally while the tagging jets are more forward than in GF production. For the three HD VBF cases, conversely, the jets are more central. We recall that the type of operators are the same both for the GF and the HD VBF, i.e. the dimension-five operators \(X_0V_{\mu \nu }V^{\mu \nu }\) and \(X_0V_{\mu \nu }\widetilde{V}^{\mu \nu }\).

We track down the slight difference between GF and HD VBF in \(\Delta \eta _{jj}\) to the presence of the mass of the \(t\)-channel vector boson, i.e. massless gluons vs. massive weak bosons. On the other hand, the slightly weaker modulation for \(\Delta \phi _{jj}\) in GF is due to the presence of the \(gg\) and \(qg\) initiated contributions [19, 23]. We note that the interference between GF and VBF can be safely neglected [95, 96].

## 4 Associated production with a top-quark pair

The top-quark decays are subsequently performed starting from the event file (in the Les Houches format [97]) by MadSpin [46] following a procedure [98] that keeps production and decay spin correlations intact.

### 4.1 Total rates

LO and NLO cross sections and corresponding \(K\) factors for \(pp\rightarrow t\bar{t}X_0\) at the 8- and 13-TeV LHC, for the three scenarios defined in Table 3. The integration error in the last digit(s) (in parentheses), and the fractional scale (left) and PDF (+\(\alpha _s\)) (right) uncertainties are also reported. In addition to the fixed-order results, the PS-matched NLO cross sections for the di-leptonic decay channel \(\sigma _\mathrm{NLO+PS}^\mathrm{dilep}\) and the ratios \(R\equiv \sigma _\mathrm{NLO+PS}^\mathrm{dilep}/\sigma _\mathrm{NLO}\) are also shown, where the acceptance cuts in Eqs. (15) and (16) are applied

Scenario | \(\sigma _\mathrm{LO}\) (fb) | \(\sigma _\mathrm{NLO}\) (fb) | \(K\) | \(\sigma _\mathrm{NLO+PS}^\mathrm{dilep}\) (fb) | \(R\) |
---|---|---|---|---|---|

LHC 8 TeV | |||||

\(0^+\) | 130.3(1)\({}^{+36.8}_{-24.6}\) \(\pm 5.9\,\%\) | 134.9(2)\({}^{+3.2}_{-8.3}\) \(\pm 3.0\,\%\) | 1.04 | 3.088\({}^{+3.1}_{-8.4}\) \(\pm 2.8\,\%\) | \( 2.29 \times 10^{-2}\) |

\(0^-\) | 44.49(4)\({}^{+42.5}_{-27.6}\) \(\pm 10.3\,\%\) | 47.07(6)\({}^{+6.5}_{-11.5}\) \(\pm 4.9\,\%\) | 1.06 | 1.019\({}^{+5.5}_{-11.0}\) \(\pm 4.3\,\%\) | \( 2.16 \times 10^{-2}\) |

\(0^\pm \) | 87.44(8)\({}^{+38.2}_{-25.4}\) \(\pm 6.9\,\%\) | 90.93(12)\({}^{+3.9}_{-9.1}\) \(\pm 3.4\,\%\) | 1.04 | 2.052\({}^{+3.6}_{-9.0}\) \(\pm 3.2\,\%\) | \( 2.26 \times 10^{-2}\) |

LHC 13 TeV | |||||

\(0^+\) | 468.6(4)\({}^{+32.8}_{-22.8}\) \(\pm 4.5\,\%\) | 525.1(7)\({}^{+5.7}_{-8.7}\) \(\pm 2.1\,\%\) | 1.12 | 11.52\({}^{+5.5}_{-8.7}\) \(\pm 2.0\,\%\) | \( 2.19 \times 10^{-2}\) |

\(0^-\) | 196.8(2)\({}^{+37.1}_{-25.2}\) \(\pm 7.5\,\%\) | 224.3(3)\({}^{+6.8}_{-10.5}\) \(\pm 3.2\,\%\) | 1.14 | 4.488\({}^{+5.6}_{-9.8}\) \(\pm 2.8\,\%\) | \( 2.00 \times 10^{-2}\) |

\(0^\pm \) | 332.4(3)\({}^{+34.0}_{-23.5}\) \(\pm 5.4\,\%\) | 374.1(5)\({}^{+6.0}_{-9.3}\) \(\pm 2.5\,\%\) | 1.13 | 8.022\({}^{+5.4}_{-8.9}\) \(\pm 2.2\,\%\) | \( 2.14 \times 10^{-2}\) |

At variance with the GF process, the production rate for the pseudoscalar case is smaller than that for the scalar case. Such a difference is proportional to the top-quark mass, as the amplitudes for the scalar and pseudoscalar interactions are identical in the limit where the Yukawa coupling is kept constant and the quark mass is neglected. In \(pp\) collisions at the LHC energies the contribution of the \(gg\) initial state is dominant over \(q\bar{q}\) annihilation for all the scenarios. It is rather interesting to observe, however, that for a CP-odd scalar \(q\bar{q}\) annihilation contributes at LO to just 16 % (10 %) of the total cross section at 8 (13) TeV, compared to around 40 % (30 %) of the SM-like CP-even case. This difference is such that the CP-odd case exhibits slightly larger scale and PDF uncertainties. Once again, we note that the scale dependence is larger than the PDF + \(\alpha _s\) uncertainty (though not by as much as in GF \(H+\) jets), and that all the uncertainties are significantly reduced going from LO to NLO, as expected. Increasing the collision energy from 8 to 13 TeV enhances the cross sections by about a factor 4, while the \(K\) factors only slightly increase. As in the GF case, \(\sigma (0^{\pm })\) is equal to the average of \(\sigma (0^+)\) and \(\sigma (0^-)\). We have verified explicitly that at the LO the interference between amplitudes corresponding to different parity interactions is exactly zero. At NLO, the interference at the amplitude level is nonzero, yet the total rates do sum up to each of the parity-definite contributions.

It is well known that dedicated top and Higgs reconstruction are crucial in order to obtain the significant \(t\bar{t}H\) signal over the background, at least for the dominant \(H\rightarrow b\bar{b}\) decay channel. Several proposals have been put forward from using multivariate analysis, e.g., matrix element method [99] to jet substructure/boosted techniques [100, 101, 102, 103]. In this work we are mainly concerned in checking what observables can be sensitive to CP effects and do not consider either backgrounds or reconstruction issues. However, we will consider how CP-sensitive observables are affected by the requirement of a large transverse momentum for the Higgs, i.e. a “boosted Higgs”.

In Table 8, we also report the PS-matched NLO cross sections for the di-leptonic decay channel and the corresponding ratios to the fixed-order NLO prediction, \(R\equiv \sigma _\mathrm{NLO+PS}^\mathrm{dilep}/\sigma _\mathrm{NLO}\), where acceptance cuts (assuming 100 % \(b\)-tag and lepton efficiencies) are taken into account. Accounting for the branching fraction of the di-lepton mode, \((0.213)^2\sim 0.045\), the ratios show that parton shower and the cuts lead to a decrease of about a factor 2 in the cross section. Increasing the CM energy results in the slightly smaller \(R\) ratios.

### 4.2 Distributions

## 5 Summary

In this work we have presented for the first time results at NLO in QCD, including parton-shower effects, for the hadroproduction of a spin-0 particle with CP-mixed coupling to the top quark, in gluon-fusion plus one and two jets and in association with a top-quark pair. Our results are obtained in a fully automatic way through the implementation of the relevant interactions in FeynRules and then performing event generation in the MadGraph5_aMC@NLO framework.

We have presented illustrative distributions obtained by interfacing NLO parton-level events to the HERWIG6 parton shower. NLO corrections improve the predictions of total cross sections by reducing PDF + \(\alpha _s\) uncertainty and scale dependence. In addition, our simulations show that NLO+PS effects need to be accounted for to make accurate predictions on the kinematical distributions of the final-state objects, such as the Higgs boson, the jets and the top decay products.

We have confirmed that di-jet correlations in Higgs plus two jet production, in particular the azimuthal difference between the jets, are sensitive probes of the CP mixing of the Higgs. In associated production with a top pair, we have shown that many correlations between the top and antitop decay products can be sensitive to the CP nature of the Higgs. In particular, the pseudorapidity separation between the leptons or between the \(b\)-jets is a promising observable when analysing events with a Higgs boson at high transverse momentum. The quantitative determination of the CP mixing has been done for the GF channel at LO in [24], while the LO parton-level analysis has been done for the \(t\bar{t}H\) channel including \(tH\) and \(\bar{t}H\) in [50]. The estimation of the impact of the NLO+PS corrections as well as detector effects is desired and will be reported elsewhere.

As a final remark, we note that in this work we have only addressed the issue of the CP properties of the flavour-diagonal Higgs–top-quark interactions, which can be parametrised in full generality as in Eq. (1). At the dimension-six level, however, other operators appear that lead to effective three-point and four-point Higgs–top-quark interactions of different type [106, 107, 108, 109, 110], including flavour changing neutral ones [106, 111, 112], which can also be studied in the same production channels as discussed here, i.e. \(H\) + jets and \(t\bar{t} H\). Work in promoting predictions for these processes to NLO accuracy in QCD is in progress.

## Footnotes

## Notes

### Acknowledgments

We would like to thank the Higgs Cross Section Working Group for the encouragement in pursuing the Higgs Characterisation project. We are thankful to Pierre Artoisenet, Stefano Carrazza, Stefano Forte, Rikkert Frederix, Valentin Hirschi, Olivier Mattelaer and Tiziano Peraro for their support during the preparation of this work and to Stefano Frixione for many useful discussions and comments on the manuscript. We thank Roberto Pittau for his participation to the initial stages of this project. This work has been performed in the framework of the ERC grant 291377 “LHCtheory: Theoretical predictions and analyses of LHC physics: advancing the precision frontier” and of the MCnet PITN-GA-2012-315877. It is also supported in part by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P7/37. The work of FD and FM is supported by the IISN “MadGraph” convention 4.4511.10 and the IISN “Fundamental interactions” convention 4.4517.08. KM is supported in part by the Strategic Research Program “High Energy Physics” and the Research Council of the Vrije Universiteit Brussel. The work of MZ is partially supported by the Research Executive Agency (REA) of the European Union under the Grant Agreement Number PITN-GA-2010-264564 (LHCPhenoNet) and by the ILP LABEX (ANR-10-LABX-63), in turn supported by French state funds managed by the ANR within the “Investissements d’Avenir” programme under reference ANR-11-IDEX-0004-02.

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