New angles on top quark decay to a charged Higgs

# New angles on top quark decay to a charged Higgs

David Eriksson, Gunnar Ingelman, Johan Rathsman, Oscar Stål
High-Energy Physics, Dept. of Nuclear and Particle Physics
Uppsala University, P. O. Box 535, SE-751 21 Uppsala, Sweden
Corresponding author. E-mail:
###### Abstract:

To properly discover a charged Higgs Boson () requires its spin and couplings to be determined. We investigate how to utilize spin correlations to analyze the couplings in the decay . Within the framework of a general Two-Higgs-Doublet Model, we obtain results on the spin analyzing coefficients for this decay and study in detail its spin phenomenology, focusing on the limits of large and small values for . Using a Monte Carlo approach to simulate full hadron-level events, we evaluate systematically how the decay mode can be used for spin analysis. The most promising observables are obtained from azimuthal angle correlations in the transverse rest frames of . This method is particularly useful for determining the coupling structure of in the large limit, where differences from the SM are most significant.

Hadronic colliders, Spin and Polarization Effects, Beyond Standard Model, Higgs Physics

## 1 Introduction

Finding a fundamental spin zero boson with electric charge would be a direct sign of physics beyond the Standard Model (SM). The existence of a charged Higgs boson pair () with these properties is predicted by Two-Higgs-Doublet Model (2HDM) extensions of the SM Higgs sector. The primary motivation for studying the 2HDM is supersymmetry, which requires an even number of Higgs doublets for cancellation of triangle anomalies. Charged Higgs searches at hadron colliders are divided into two regimes, separated by the dominant mode of production. When is heavy (), it is produced primarily through the and processes [1, 2, 3, 4]. When, on the other hand, is light () and the decay [5, 6, 7, 8, 9] opens up, this quickly becomes the dominant production mode.

The most stringent model-independent limit on the mass of from a direct search experiment comes from LEP:  GeV [10] at CL, assuming only the decays and are possible. Even tighter constraints on have later been derived using Tevatron data [11], but these are not independent of the other 2HDM parameters. Neither are indirect constraints on obtained from -physics observables.

The upcoming searches for planned by the LHC experiments will have good sensitivity to discover over a wide parameter range [12, 13, 14, 15, 16], especially when is light. However, even if some candidate state was to be found, this discovery alone would not be enough to establish the validity of the Higgs mechanism as described by the 2HDM. To do this requires further that the spin, and the couplings, of this new particle be determined. Here we investigate one possibility to address this issue when . In addition to providing a discovery channel, the decay mode could modify the ordinary V-A Lorentz structure of weak top decay significantly. As we will show, this fact can provide a handle on the spin and coupling structure of by making use of spin correlations.

Top quarks produced in pairs at hadron colliders constitute an interesting laboratory for observing spin effects in high-energy physics. Since the timescale for weak top decay is much shorter than the typical hadronization timescale , the heavy quarks will decay before hadrons can form [17]. No hadronic effects will therefore obfuscate the spin information. Unlike the case for the lighter quarks, this fact allows for reliable perturbative calculations of the relevant spin observables. Furthermore, since the charged current weak interaction violates parity maximally, the decay self-analyzes the spin of the top quark. This means the full spin information will be imprinted in the angular distributions of the different decay products.

In order for the angular information to be useful for investigating the couplings involved, a method to determine the spin projection of the decaying top quark is required. At a hadron collider, this can be achieved by exploiting correlations between the top quark spins. By using the decay information from one side of a event, it is possible to determine statistically the polarization of the other top. Spin correlations in top quark pair production and decay have been extensively discussed within the Standard Model (SM) for hadron-hadron colliders [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], and for experiments [31, 32, 33, 34, 35, 36]. Utilizing the top spin information to study physics beyond the Standard Model was considered in the context of anomalous -couplings [31, 37, 38], for probing extended Higgs sectors and effects of CP-violation [39, 40, 41, 42, 43, 44] , and within theories with extra dimensions [45, 46].

We study the spin phenomenology of a light scalar sector in full generality, ignoring indirect constraints on the 2HDM. We will however restrict the mass to , as required by the non-observation of in direct search experiments. Due to the, in the context of spin observables, relatively limited sample of events available from the Tevatron, our main focus will be on prospects for observations at the LHC. We try to comment on issues that are of relevance also for the analysis of Tevatron data.

The organization of this paper is as follows. First, in Section 2, we discuss pair production at hadron colliders, and how spin correlations come about in this process. Then in Section 3 we introduce the phenomenological model, followed by a brief review of the relevant theory for polarized top quark decay, and results on the spin analyzing efficiencies in models where the top quark can decay through a charged Higgs boson. Section 4 describes a Monte Carlo simulation study of these effects, and discusses possible experimental observables. Finally, Section 5 contains a summary and the conclusions of this work.

## 2 Top Quark Pair Production at Hadron Colliders

We will adopt the latest combined value for the top mass GeV [47]. Pair production of top quarks occurs in leading order QCD both through annihilation via an -channel gluon, and through gluon fusion for which -, - and -channel exchanges are possible. The total hadronic cross section has long been known to NLO accuracy [48, 49]. With CTEQ6 parton distributions [50] and a common choice of scales , we obtain  pb at TeV using the NLO MC generator POWHEG [51]. One year of LHC running at low luminosity, corresponding to fb, will therefore produce of order events.

Individually unpolarized top quarks are still produced with strong correlations between their spin projections in a suitable basis. The nature and magnitude of these correlations depend on the partonic center of mass (CM) energy. Following [24] we define the production correlation, as a function of the invariant mass of the top pair, to be

 ^Cij(M2t¯t)=^σij(t↑¯t↑+t↓¯t↓)−^σij(t↓¯t↑+t↑¯t↓)^σij(t↑¯t↑+t↓¯t↓)+^σij(t↓¯t↑+t↑¯t↓) (1)

for the partonic subprocess involving initial state partons . Arrows indicate the spin projection on the chosen spin quantization axes (which may be different for and ). We choose here to work exclusively in the helicity basis, in which the spin is quantized along the momentum directions of the in the partonic CM frame. In this basis, the notation is sometimes used interchangeably with to denote the two spin projections.

Near threshold, the are always produced in an S-wave state. For production dominated by through an -channel gluon, the overall angular momentum state will therefore be . Out of the three states composing the triplet, two correspond to opposite helicities for the two top quarks, whereas one state gives equal helicities. The combined correlation according to (1) is thus at threshold. When instead production dominates, which is the case for LHC energies, the situation at threshold is reversed. The top pair is now produced in a singlet configuration, since the initial state gluons do not populate states [20]. In this case the must always come with the same helicities, which means that . Finally, in the ultra-relativistic limit (), helicity conservation requires the to have opposite helicities independent of partonic subprocess. Hence the correlation for all in the high energy limit.

In general, the total statistical correlation in a sample of events is obtained from averaging over the invariant mass of the top pair, while parton distribution functions determine the relative contributions of the competing production processes:

 C(s)=1σt¯t∑i,j={q,¯q,g}∫dx1dx2[^σij(t↑¯t↑+t↓¯t↓)−^σij(t↓¯t↑+t↑¯t↓)]fi(x1,μ2F)fj(x2,μ2F). (2)

The for all partonic subprocesses have previously been calculated to NLO in QCD. From these one obtains [29] in the helicity basis for collisions at TeV. The residual uncertainty in this number from PDF and scale choices is of order one percent, a value similar to the difference from the LO calculation which gives . For Tevatron run-II conditions ( collisions at TeV) the same helicity basis correlation becomes . At Tevatron energies, where annihilation totally dominates production, there exist also more efficient bases for spin quantization in which correlations as large as can be obtained [27].

As suggested by the discussion above, it can be beneficial to introduce an experimental cut on to increase the spin-purity of the sample at the cost of decreased efficiency [23]. To increase the component of like-sign helicities in the gluon sample at the LHC requires a cut on the maximum . The effect of such a cut on the correlation parameter is shown in Figure 1. To illustrate the trade-off between spin-purity and efficiency, Figure 1 also presents the fraction of the total cross section which passes a cut on the maximum . As we have already indicated, statistics will not be the limiting factor at the LHC. It is therefore good to keep in mind that can be increased using this technique, although we do not make explicit use of this fact here.

## 3 Spin Information in Top Quark Decay

In the SM, the top quark decays almost exclusively via the charged current V-A vertex

 LWtb=gW√2VtbW+μ¯tγμ1−γ52b+h.c., (3)

where is the appropriate element of the CKM matrix. In addition to the Standard Model decay, the possibility exists that the top quark decays anomalously. The decay could then contain a small V+A component, or it could be mediated by additional bosons of different spin. The study of spin correlations opens a window on both these possibilities. Our aim here is to explore the latter case, allowing for an extended scalar sector.

### 3.1 Charged Higgs Model

Introducing a charged scalar pair , their interactions with fermions are parametrized by an effective Lagrangian density

 LH= gW2√2mW∑{u,c,t}{d,s,b}Vud{H+¯u[A(1−γ5)+B(1+γ5)]d+H−¯d[B∗(1−γ5)+A∗(1+γ5)]u} (4) +gW2√2mW∑{e,μ,τ}[H+C¯νl(1+γ5)l+H−C∗¯l(1−γ5)νl].

The , , , and their complex conjugates, are in principle free parameters determining the Lorentz structure of the couplings. Note that this model does not assign definite parity to the unless . For parity is violated maximally. Assuming CP-invariance of the scalar sector, the coupling parameters can all be taken as real numbers.

A model such as (4) occurs, for example, as the charged Higgs-fermion sector of a two Higgs Doublet Model (2HDM). Here, the SM Higgs sector is augmented with another complex Higgs doublet, resulting in two charge conjugate () and three neutral () states occurring as physical bosons. To ensure the augmented SM does not allow for tree-level FCNC’s, certain restrictions apply on how to couple the extended Higgs sector to the fermions. For our purposes, it suffices to say that two options are generally considered: In the so-called type I model [2HDM (I)], only one doublet is coupled directly to the fermions. In the type II model [2HDM (II)], one doublet is coupled only to up-type fermions, whereas the other doublet couples only to down-type fermions.

The number of independent parameters in the Higgs sector is thereby restricted to two at leading order. We will adopt for these the ratio of the two doublets vacuum expectation values, and the charged Higgs mass . For the two model types, the charged Higgs-fermion couplings are then given in Table 1. Mass parameters appearing in the couplings should be evaluated at a scale , using the masses to ensure proper resummation of large logarithmic QCD vertex corrections [52, 53, 54]. Since the couples proportionally to the fermion mass, we will only be concerned with third generation particles in the following.

As discussed in the introduction, one possible extension of the Standard Model where a 2HDM occurs naturally is the Minimal Supersymmetric Standard Model (MSSM). The MSSM contains a 2HDM (II), but since the supersymmetry introduces additional particles, the two-parameter picture of the 2HDM (II) works only as an effective tree-level description. It has been shown [52, 55], that quantum corrections due to SUSY-QCD loops can be quite sizable for large values of . This holds even in the decoupling limit when all SUSY masses are taken to infinity. How these corrections enter into the effective couplings can be seen from the third column of Table 1. We will call the 2HDM which includes the enhanced SUSY corrections the ”modified type II”, or simply . The corrections are of two types: first the so-called correction to the relation between the bottom quark mass and the bottom Yukawa coupling . It is caused by gluino-sbottom and chargino-stop loops. At one-loop, the dominant contributions to this correction are given by [56, 55]

 ϵb=−2αs3πμm~gH2(m~b1m~g,m~b2m~g)−y2t16π2~Ua2Atm~χ+aH2(m~t1m~χ+a,m~t2m~χ+a)~Va2, (5)

which introduces a dependence on the trilinear coupling , the top Yukawa coupling , and the parameter from the superpotential – in addition to the dependence on several of the sparticle masses. The real matrices and diagonalize the chargino mass matrix. The function is given by

 H2(x,y)=xlnx(1−x)(x−y)+ylny(1−y)(y−x). (6)

In the limit when all SUSY parameters and sparticle masses are of similar scale one obtains . The sign of is determined by the sign of .

The second contribution which modifies the couplings is [55]

 ϵ′t= −2αs3πμm~g[c2~tc2~bH2(m~t2m~g,m~b1m~g)+c2~ts2~bH2(m~t2m~g,m~b2m~g) (7) +s2~tc2~bH2(m~t1m~g,m~b1m~g)+s2~ts2~bH2(m~t1m~g,m~b2m~g)]− +s2~tc2~bH2(m~t2m~χ0a,m~b2m~χ0a)+s2~ts2~bH2(m~t2m~χ0a,m~b1m~χ0a)]Na3,

where the matrix diagonalizes the neutralino mass matrix, and for the squark mixing angles . The squark mass eigenstates are given by and , with . We note that is numerically similar to in the case with a common scale for the SUSY parameters. In Section 3.5, we will return to these SUSY corrections when discussing numerical results for the 2HDM. As we will show, it turns out that their effects on spin correlation observables are small.

### 3.2 Top Quark Decay with Polarization

Assuming that the full width of the top quark, including the scalar decay mode, is still very small () we use the narrow width approximation to factorize the production from the decay of the heavy quarks. The branching fractions for are shown in Figure 2 for 2HDM type (I) and (II). It is clear that type (II) is interesting both for small and large values, whereas the 2HDM (I) only allows a significant for small . In the following we will mostly be concerned with the type (II) model.

We treat the decaying as independent decays through well-defined channels without interference effects. Strictly speaking, a more complete formalism involving off-diagonal propagator elements could be used when . Ignoring such complications, the full structure of density matrices in the matrix element becomes

 |M(2→6)|2=[Rλλ′κκ′(2→t¯t)⊗ρiλλ′(t→3)⊗ρjκκ′(¯t→3)]. (8)

is here the fully helicity-dependent spin density matrix for production. The () are decay density matrices of (), where label the available decay channels and () are helicity indices for the ().

To obtain the decay density matrix for a given channel, we use the techniques described in Appendix A. With momenta defined in Figure 3, the leading order decay density matrix elements for semi-leptonic weak decay of the top quark are given by

 ρWλλ′=|Mλλ′(t→bW+→bl+νl)|2=2g4W|Vtb|2(p⋅k2)(k1⋅k3)(q2−m2W)2+m2WΓ2W[δλλ′+^ka2σaλλ′] (9)

when the spins of all outgoing particles are summed over. The unit 3-vector is given in the rest frame of the decaying quark. For hadronic decay of the boson, the matrix elements are exactly the same if a) all final state masses are neglected and b) the leptons are replaced by their quark counterparts in terms of weak isospin. CP-invariance of the decay ensures that .

Reckon similarly the elements of the decay density matrix when the decay is mediated by a charged scalar as defined by the model (4). In this case, the elements become

 ρHλλ′= |Mλλ′(t→bH+→bl+νl)|2=g4W|Vtb|2(p⋅k1)(k2⋅k3)(q2−m2H+)2+m2H+Γ2H+C2(A2+B2)2m4W (10) ×(1+ABA2+B24δ1−ξ)[δλλ′−A2−B2A2+B2(1+ABA2+B24δ1−ξ)−1^ka1σaλλ′].

Here the notation and is used. Let us also introduce a convenient short-hand

 f(ξ,A,B)=(1+ABA2+B24δ1−ξ)−1 (11)

for the threshold factor. This function has the general properties for , and for , unless .

### 3.3 W Boson helicity

The perhaps most direct test of V-A theory in top quark decay is offered by examining the polarization states of the boson mediating the decay. Due to the large Yukawa coupling , a fraction of the bosons are expected to be longitudinally polarized, while the remainder carries a left-handed helicity in the rest frame. Uncertainties in these numbers from higher order corrections, including virtual 2HDM and SUSY effects, are under control at the level [57, 58].

When the decays further, the angular dependencies of the decay products on the different helicity states are given by the Wigner -functions for the spin  representation. Combining this knowledge with the polarized matrix element for , the normalized lepton angular distribution in leptonic decay of the is given by

 1NdN(W→lνl)dcosθ∗l=34(m2t+2M2W)[m2tsin2θ∗l+M2W(1−cosθ∗l)2], (12)

where is defined in the rest system as the angle of the lepton momentum to the helicity axis. Using the fact that, in the rest frame of the decaying top, the recoiling quark has its momentum anti-parallel to that of the , the lepton helicity angle can be determined by the invariant product [59]

 cosθ∗l=k1⋅(k2−k3)k1⋅(k2+k3) (13)

if the mass is neglected. Assuming further that the decay is mediated through an on-shell , the approximate expression

 cosθ∗l≃4k1⋅k2m2t−m2W−1 (14)

can be obtained from the kinematics of the decay. The form (14) is experimentally advantageous since no knowledge of the neutrino momenta is required to determine . Being only an approximate on-shell relation, the values obtained using this expression may in reality be such that for some events.

In the decay of a charged Higgs boson, the decay products should be isotropically distributed in . This offers a clear signature for a new charged boson to have spin . However, even with a large branching ratio , this appreciable difference would not contribute much to measurements of the distribution (12) using electrons or muons, simply because in the regions of interest, in the 2HDM.111A small contamination from would of course be present also in the lepton samples in these cases. If, at the LHC, evidence starts to gather in favor of a light , it would therefore be interesting to study the angular distribution of  leptons exclusively using the hadronic decay. Experimentally this presents a formidable task, since the presence of two final state neutrinos in the channel introduces ambiguities in the reconstruction of the momentum. Furthermore, when , the kinematic assumptions behind Equation (14) are no longer valid. It is then natural to exploit these kinematic differences fully and treat the two decays separately also in the angular analysis in order to establish the spin  nature of the presumptive .

In Figure 4, we show the angular distribution of leptons in the rest system of the boson mediating the decay. We show here the expectations for the SM, given by Equation 12, and mixtures of SM+2HDM (II) with corresponding to three different values of . It is assumed either that GeV, or that kinematic effects can be compensated for on event-by-event basis. Even for , the events give a significant contribution since is equally small.

### 3.4 Polarization Observables

From the matrix elements (9) and (10) some useful hints are obtained on how the spin is analyzed in top quark decays. The spin-dependent term, proportional to , appears with different associated momentum directions in the two channels. The particle with this momentum will analyze the spin most effectively in the corresponding case. To see how this comes about, recall [60] how the spin of a given top quark is analyzed. The decay products will experience angular distributions reflecting the spin state of the parent. Each polarized partial width of a decaying fermion can be put in the form

 1ΓdΓdcosθi=1+αicosθi2, (15)

where denotes the angle of decay product to the spin helicity axis, calculated in the rest frame of the decaying particle. The spin analyzing coefficients determine the efficiency of a given particle to analyze the spin of the parent. The factorization of in energy-dependent and angular parts holds to a high degree also when including radiative QCD corrections [60].

To obtain the full set of for a given decay, it is necessary to integrate the polarized matrix elements. Using the kinematic variables and , the Dalitz parametrization [61] of the 3-body phase space is written as

 dΦ3=1(2π)5dxdydγdβdcosθ. (16)

The three Euler angles , , and are here chosen according to [62], so that coincides with the helicity angle discussed above. The integration over is always trivial and gives . The integration over is non-trivial only for two, spin-dependent, quantities. With positive spin projection along the helicity axis, the result is

 ∫dγ s⋅k1=2πcosθ[12(1+y−δ2)−yx]m2t

and since , with , the other interesting integral becomes

 ∫dγ s⋅k3=2πcosθ[yx−12(1+y−x−δ2)+]m2t.

The coefficients have been determined for both decay channels. We summarize our results in Table 2 for a decaying with positive helicity. Expressions for the other helicity state, or for the charge conjugate decay, are obtained by an overall change of sign. Our results agree with those presented in [23], except that their expressions for the scalar case do not contain the factor . This factor contains all the dependence of the spin analyzing power on the Lorentz structure of the coupling. We note further that the expression we obtain for is in agreement with that of [44], where also corrections to this quantity are given. The inclusion of NLO corrections does not modify the dependence of the , even if the numerical values are slightly altered. Since we aim to compare the analytic results to a LO Monte Carlo simulation, we use only the LO results for throughout this work.

Since the top quark spins are not directly observable themselves, what will be accessible are quantities constructed only from the final state momenta. The most direct such being the doubly differential distributions of the same type as in Equation (15), but now involving two particles ; one from each decaying top quark. The helicity angles and are then calculated in the rest systems of the respective parents222We use the convention of performing rotation-free boosts from the CM system to define the orientation of the rest systems. Alternatively, one could perform a rotation-free boost directly from the hadronic CM system.. The most general expression of this type is

 1Nd2Ndcosθidcosθj=14(1+P1αicosθi+P2αjcosθj+Cαiαjcosθicosθj) (17)

where measures the degrees of transverse polarization of the . is the correlation parameter discussed in Section 2. In leading order QCD, with the spin quantized in the helicity basis, by parity invariance. Instead of (17) the simpler distribution

 1Nd2Ndcosθidcosθj=14(1+Cαiαjcosθicosθj) (18)

is therefore expected to obtain.

It is also possible to form one-dimensional distributions, e.g. as studied in [29]. If we define the angle between the vectors and as in the previous distribution, we get

 1NdNdcosθij=12(1+Dαiαjcosθij). (19)

Here the coefficient is related to , but in general it has a different value. To determine and from angular distributions, the relations

 C = 9αiαj⟨cosθicosθj⟩ (20) D = 3αiαj⟨cosθij⟩ (21)

can be used. Conversely, when the correlation coefficients are known, these relations can be used to determine the product . With a leading order , we obtain the corresponding .

### 3.5 Analysis in 2HDM (II)

As an illustrative example of the differences between the SM and a new scalar decay, let us consider in some detail the results for a 2HDM (II) with couplings from Table 1. In all the following, we shall fix the top mass to GeV [47]. Starting by analyzing the threshold region , Figure 5 shows as a function of for different values of . We see that the threshold suppression becomes significant as approaches the kinematic limit. However, for very large, very small or intermediate () values, we infer the threshold correction to be less than also for GeV. In these regions of parameter space, the threshold factor can be effectively ignored. Note that this argument is not specific to , but applies to all , since is a universal factor.

Summarizing the results presented in Table 2 for the 2HDM (II), Figure 6 shows a numerical evaluation of the analytic expressions for all . The results are presented for two values of : one large value (), for which the efficiency in analyzing the spin is optimum, and one intermediate value (for GeV with GeV), where all the sensitivity to analyze the top spin vanishes. This value corresponds to a purely scalar coupling, thus to an isotropic decay. For , all acquire an extra minus sign compared to . This corresponds to a shift from predominantly right-chiral to left-chiral coupling.

We see also, that the efficiency to analyze the top spin is not highest using the charged lepton, as is shown to be the case for the SM. Instead the most efficient probe is either the Higgs momentum itself or the associated quark. This is easily understood; since the itself does not carry any spin, the top spin information can only be transferred to the angular distributions of the . In a vector decay, parts of this information go into the different polarization states of the , as discussed above.

Figure 7 displays as a function of for a fixed GeV. It illustrates clearly the transition from analyzing power at small , to associated with a right handed coupling for large . Note also, that for , the quark coefficient of the Higgs events mimics that of the SM decay. The only dependence in this relation on enters through the running mass . No corresponding value exists where the are equal, since the Higgs value is bounded by kinematics to , whereas the SM value is always .

To compare with the case when -enhanced SUSY corrections to the charged Higgs couplings are included, we give in Figure 7 also the values of for the 2HDM (). Rather than to calculate the corrections for a specific SUSY model spectrum, we parametrize them in terms of the parameters and . These are reasonable maximum values [55], which correspond roughly to as discussed above. Figure 7 clearly shows that even though these corrections are enhanced by in the couplings, they have only a small effect on the ratio that enters the spin analyzing coefficients. In fact, the largest correction is obtained not in the high limit, but in the transition region around  – . Given the observed smallness of the SUSY effects on the spin analyzing coefficients, it is acceptable to apply the results from the 2HDM (II) without modification, both in the high and in the low regimes. However, total rates for are of course affected by the differences.

A few words are to be said also about the corrections calculated in [44]. Inclusion of these effects leads to modifications of the spin analyzing coefficients in a fashion very similar to the enhanced SUSY corrections discussed above. These corrections are also largest in the intermediate region, where they can reach in magnitude. In the large and small limits, the NLO corrections have negligible impact. Since our results are most interesting in these limits, we will show plots for and . Interpolation to the intermediate range should then be performed with care, remembering the higher order corrections.

As a final result for the 2HDM (II) on matrix element level, we show the differential distributions in and , described by Equations (18) and (19) respectively. The top row in Figure 8 shows lepton-lepton correlations, where the lepton ( quark) from a decay is correlated with another lepton (in this case a ) from or decay from the opposite side of the event. In the absence of spin correlations, this distribution is expected to be flat. From left to right, Figure 8 gives the results for the SM, for the 2HDM (II) with , and similarly with . In the bottom row, distributions in one of the angles are given. These are obtained using the other angle to determine the parent spin by applying the projections and to the 2D distributions. Figure 8 illustrates the points we have previously made about the 2HDM (II), namely that i) the lepton-lepton correlation is more efficient in the SM since in the Higgs case, ii) there is a change of sign in going from high to low values of , and iii) the low case is more SM-like.

Going to Figure 9, the lepton has been replaced with the quark associated with the same (Higgs) side of the event. The three distributions are defined similarly to those in Figure 8, as are the projections. From Figure 9, the increased spin analyzing efficiency when using the associated quark in the Higgs case is evident.

The last figure to discuss here is Figure 10, which shows distributions in between different particles in the final state. An uncorrelated sample corresponds to a flat line distribution. The specific combinations chosen are , , , and , where the first particle originates from decay, while the second always comes from the opposite side . The distributions and are equivalent to the distributions and displayed in Figure 10 with a change of sign in the spin analyzing coefficient ( in Eq. (19)). Hence they would not contribute any additional information.

By combining in the same plots for the SM with results from the 2HDM, we see directly which correlations are more efficient in the two cases. In agreement with previous results, this is again the distribution for the SM, and the distribution for the 2HDM (II), illustrating the universal dependence of the spin correlation effects on . We have also found, that when cuts are applied at the parton level, the distributions shown in Figure 10 are not affected nearly as much as those presented in Figures 8 and 9. Using the variables to study spin correlations is therefore advantageous to avoid the problems with cuts discussed in [23]. We will return to this discussion below.

## 4 Monte Carlo Simulations

Up to this point, the results we have presented were obtained directly from matrix elements. To give a more realistic assessment of the prospects to observe any of these spin effects in a collider experiment, complete hadron-level events must be considered. We do this using a Monte Carlo (MC) approach. The framework is again the 2HDM (II) because of its special status as the minimal Higgs model compatible with supersymmetry.

The production of is treated completely within the SM. On the decay end, we study in parallel the situations when either both decays occur within the SM, or when one of the two top quarks decays through and the other through .333Of course, there is also the possibility of having both tops in an event decay through . However, since this mode gives a different final state, and since the effect is sub-leading, we will neglect this contribution here. Being an interesting process in its own right, the SM decay of is also the main irreducible background in searches for light [14]. Over nearly the full range of , decays preferentially to the heaviest lepton available, that is . We therefore restrict ourselves to this decay channel. Consequently, for the SM events, we demand one of the two bosons to decay through this mode which has . The lepton subsequently decays hadronically producing a jet. For the other , which is always present in the event, we consider the hadronic decay to allow for hadronic top reconstruction. We show SM and 2HDM results separately, keeping in mind that and are of similar magnitude for the and regions of interest.

To incorporate helicity information throughout the whole process, it is necessary to use a MC generator which can treat the full matrix element. This is provided by MadGraph/MadEvent 4.1.10 [63] for which we implemented the model (4). Using this program has the additional advantage that the ME is generated without using the narrow width approximation. To verify the treatment of spin information, control samples were generated using the specialized generator TopReX 4.11 [64]. For the observables we analyze, results from both programs were found to be in good agreement (after correcting the partial width in TopReX).

From the matrix element generated by MadEvent, full events are obtained by applying Pythia 6.409 [65] for parton showering and hadronization. To treat properly the spin information in tau decays, which was previously demonstrated to be important in searches [66, 67], Tauola [68] is invoked. The underlying event is modeled using the Pythia default ”old” model based on multiple parton-parton interactions. The default parameters of this model are tuned to Tevatron minimum-bias data [69] and provide reasonable estimates for extrapolation to the LHC energy.

### 4.1 Event Reconstruction

For jet reconstruction we use the FastJet implementation [70] of the longitudinally invariant algorithm [71, 72, 73] for hadron colliders. The jet clustering uses -scheme recombination based on the distance measure , with and . We use the algorithm in exclusive mode, meaning that particles are clustered with the beam when the beam-particle distance is smaller than the distance to any jet candidate. Not all particles will therefore end up in a jet. Furthermore, we only take particles with into account to reflect the detector acceptance region. For the minimum jet separation measure, above which no further clustering takes place, the value (GeV) is used. Choosing this value gives a jet multiplicity which peaks at the value expected from the matrix element.

We also implement a simplistic notion of flavor tagging where jets are tagged as jets or jets by comparing to MC truth information. A candidate jet is tagged whenever the distance to a true quark (or a ) is less than . In addition, for the jet to be tagged, it is required that it has . Apart from these criteria, no further efficiency factor is used in the flavor tagging.

Events are selected for analysis based on their overall topology. The characteristic signature, which must be fulfilled by our signal events following jet reconstruction, is the presence of exactly two jets, one jet and at least two additional, untagged, jets. For this type of potential events, the hadronically decaying is reconstructed by combining two light jets, minimizing the mass-square difference