Measuring properties of a Heavy Higgs boson in the $H\to t\bar t \to bW^+\bar b W^-$ decay

Suppose a heavy neutral Higgs or scalar boson $H$ is discovered at the LHC, it is important to investigate its couplings to the standard model particles as much as possible. Here in this work we attempt to probe the CP-even and CP-odd couplings of the heavy Higgs boson to a pair of top quarks, through the decay $H \to t\bar t \to b W^+ \bar b W^-$. We use the helicity-amplitude method to write down the most general form for the angular distributions of the final-state $b$ quarks and $W$ bosons. We figure out that there are 6 types of angular observables and, under CP$\widetilde{\rm T}$ conservation, one-dimensional angular distributions can only reveal two of them. Nevertheless, the $H$ couplings to the $t\bar t$ pair can be fully determined by exploiting the one-dimensional angular distributions. A Higgs-boson mass of 380 GeV not too far above the $t\bar t$ threshold is illustrated with full details. With a total of $10^4$ events of $H \to t\bar t \to bW^+ \bar b W^+$, one can determine the couplings up to 10-20 % uncertainties.


I. INTRODUCTION
The scalar boson that was discovered at the LHC in 2012 [1,2] turned out to be best described by the Standard Model (SM) Higgs boson [3], which is remarkable confirmation of the Higgs mechanism proposed in 1960s [4]. Among the Higgs boson couplings to the SM particles, the most constrained one is its coupling to the massive gauge bosons that is very close to the corresponding SM value with about 10% uncertainty [5]. Nevertheless, the couplings to fermions are much less constrained, especially for the first two generations. The coupling to the top-quark pair from global fits has about 20-30% uncertainty [5]. There was also direct measurements of the top Yukawa coupling in pp → tth production [6], which still needs more data to have more precise measurements than the global fitting.
Even though the SM has achieved a great success in accounting for the interactions among the basic building blocks of matter, however extra particles and new interactions are required to explain the experimental observations of dark matter, non-vanishing neutrino mass, the baryon asymmetry of our Universe, inflation, etc. In most extensions beyond the SM, the Higgs sector is enlarged to include more than one Higgs doublet resulting in charged Higgs bosons and several neutral Higgs bosons in addition to the one discovered at the LHC. For example, the minimal supersymmetric extension of the SM, aka MSSM [7], requires two Higgs doublet fields, thus leading to a pair of charged Higgs bosons and 3 neutral ones. In the next-to-minimal supersymmetric standard model, there are two additional neutral Higgs bosons [8].
Suppose that in future experiments a neutral Higgs boson H heavier than the SM 125 GeV Higgs boson (denoted by h) is discovered. It is generally expected that the decays of the heavy Higgs boson H into gauge bosons would be suppressed as it becomes heavier and heavier, because the measurement of the gauge-Higgs coupling of the SM-like 125 GeV boson allows only a little room for the H couplings to massive vector bosons. However, its couplings to fermions have no reasons to be small. Indeed, once it is above the tt threshold, the decay into tt pair could be dominant.
In this work, we assume that the heavy Higgs boson H is not too far above the tt threshold, say 380 GeV, and the dominant decay mode is tt. Without requiring H to carry any definite CP-parity, we consider the possibility to probe its CP nature via the decay H → tt → bW +b W − . We employ the helicity-amplitude method [9] to calculate the decay amplitude taking into account all spin correlations in the decay chain. By measuring various angular distributions, in particular the angle between the decay planes of the top and antitop quarks, one can discern the CP-even and CP-odd couplings of the Higgs boson. This is the main goal of this work. Other fermionic modes, in general, are either too small or suffer tremendously from SM backgrounds. The top quark also has the advantage that it decays before hadronization, in contrast to the bottom or charm quarks, and therefore the spin information is retained in the decay products of the top quark. Thus, the spin and CP properties of the parent Higgs boson can be determined by studying several kinematical distributions of the decay products of the top and anti-top quarks.
On the other hand, when the heavy Higgs boson H is below the tt threshold, its bosonic decay modes ZZ, hh, and hZ are useful for probing the CP nature of it. By taking account of the spin-0 nature of H, only the ZZ mode may lead to nontrivial angular correlations among the decay products of the Z bosons through the interferences among various helicity states of the two intermediate Z bosons before their decays [10]. This bosonic mode was suggested to determine the spin and parity of the Higgs boson [11] quite a number of years ago. After the 125 GeV Higgs-boson discovery, the method was practically applied to determine the spin and CP properties of the observed Higgs boson [12,13]. We shall not concern with the bosonic modes in this work.
Under the current experimental status, in which active searches for heavy resonances decaying into a tt pair have been continually performed [14], our study may show how well one can determine the properties of such a heavy scalar Higgs boson at the LHC and/or High Luminosity LHC (HL-LHC). We refer to, for example, Ref. [15] for some previous studies at e + e − or µ + µ − colliders.
The remainder of this article is organized as follows. In Sec. II, based on the helicity amplitude method [9], we present a formalism for the study of angular distributions in the decay H → tt → bW +b W − . We point out that there are 6 types of angular observables in general and we can classify them according to the CP and CP T parities of each observable.
In Sec. III, we illustrate how well one can measure the couplings of the heavy Higgs boson by exploiting the angular observables introduced in Sec. II. Finally, Sec. IV is devoted to a brief summary, some prospects for future work and conclusions.

II. FORMALISM
Without loss of generality, the Lagrangian describing the interactions of the Higgs boson H with top quarks can be written as [16] L Htt = −g t Ht(g S + iγ 5 g P )t = −g t A=L,R where P A = (1 + Aγ 5 )/2 with A = −(L) , +(R). g t is the overall strength of the H-t-t coupling and g P (S) = 0 when H is the pure CP-even (odd) state. If H is a CP-mixed state, both g S and g P are non-vanishing. For the SM Higgs boson, g t = gm t /(2M W ), g S = 1, and g P = 0. On the other hand, the Lagrangian describing the interactions of the top quarks with bottom quarks and W bosons is In the SM, f L = 1 and f R = 0.

A. Helicity Amplitudes
We first present the helicity amplitude for the process Here p t,b andp t,b are four-momenta of the quarks (t, b) and antiquarks (t,b), respectively, and σ t,b andσ t,b denote their helicities. The four-momenta of W + and W − are denoted by k 1 and k 2 , respectively, with p t = p b + k 1 andp t =p b + k 2 and 1 (λ 1 ) and 2 (λ 2 ) are the polarization 4-vectors of W bosons. Depending on the helicities of the final-state particles, the amplitude can be cast into the form The helicity amplitude for the decay H(q) → t(p t , σ t )t(p t ,σ t ) in the rest frame of H is given by where s = q 2 , φ t is the azimuthal angle of the t momentum, and The momentum-dependent X and Y are given by where When the top quarks are on-shell, X = β t = (1 − 4m 2 t /s) 1/2 and Y = 1. One may take φ t = 0 without loss of generality.
The helicity amplitude for the decay t(p t , σ t ) → b(p b , σ b )W + (k 1 , 1 ) in the t rest frame is given by The reduced helicity amplitudes σ t : σ b λ 1 t are given by where θ 1 and φ 1 are the polar and azimuthal angles of the W + momentum in the t rest frame and c θ 1 /2 = cos(θ 1 /2) and s θ 1 /2 = sin(θ 1 /2). We note that σ b = λ 1 when λ 1 = ± and the 4 amplitudes of ± : +− t and ± : −+ t are identically vanishing. In the m b → 0 limit, the reduced amplitudes simplify and we have the following non-vanishing 8 amplitudes 1 : We note that A = σ b in the m b = 0 limit.
The helicity amplitude for the decayt Collecting all the sub-amplitudes we obtain where 1 In Ref. [17], the authors presented the helicity amplitudes in the m b → 0 limit. We find a minor discrepancy in four of the amplitudes with λ 1 = ± by an overall factor of e −iλ1φ1 , which does not affect the full amplitude squared for the process H → tt → bW +b W − . 2 For details of the relation between the helicity amplitudes for the t andt decays, see Appendix A.

B. Angular coefficients and observables
The partial decay width of the process H → tt → bW +b W − is given by 3 where N C = 3 and For any values of f L and f R , taking also account of finite m b effects, the precise differential angular distribution dΓ dc θ 1 dc θ 2 dΦ can be obtained numerically by integrating Eq. (15) over p 2 t and p 2 t and using Eqs. (13), (14), (6), and (9). On the other hand, in the m b → 0 limit, the amplitudes take much simpler forms and one can derive analytic expressions for the differential angular distributions in terms of Explicitly, from Table I where θ 1(2) and φ 1(2) denote the direction of W +(−) in the t(t) rest frame. And then, the sum of the amplitudes squared can be organized as with Φ = φ 1 + φ 2 denoting the angle between the two decay planes and the 4 angular coefficients are given by Under CP and CP T 4 transformations, the reduced H-t-t helicity amplitudes transform as follows: We note that the CP parities of C 1 , C 2 ,C 3 and C 4 are +, −, +, and −, respectively, implying that C 2 and C 4 are non-vanishing only when g S and g P exist simultaneously. Furthermore, the angular coefficient C 2 is CP T odd and it can be induced only in the presence of nonvanishing absorptive (or imaginary) parts of X and Y .
By integrating Eq. (15) over p 2 t and p 2 t , we have where √ s = M H and i = 11, 12, 21, 22, 3, 4. The angular functions are and the numerical factors F i are given by in which the tilded 6 angular coefficients C i are related to C i as follows: To proceed further, we have introduced weight factor w i 's which are defined by where and are the constant tilded angular coefficients at the t pole. Explicitly, We observe C 21,22 are identically vanishing because X = β t and Y = 1 are real when We also note that Finally, we have obtained the normalized differential angular distribution with the 6 angular observables defined by After integrating over any two of the angles θ 1 , θ 2 , and Φ, one can obtain the following one-dimensional angular distributions in terms of the constant t-pole angular observables R i 's: where the decay width is given by 5 When M H > 2m t and the top quarks are on-shell, we have found that the weight factors do not deviate from unity by more than 1% and one may safely take C i = C i in Eq. GeV. For the t-b-W interaction, we assume the SM couplings: f L = 1 and f R = 0. These input values simplify our numerical analysis and there remain only 2 real input parameters of g S and g P to vary 6 . In this case, we find that the total decay width is given by with F 6100. Note β t = 0.412 for our choice of M H close to the 2m t threshold and m t = 173.1 GeV and therefore the g S contribution in the above equation is suppressed by the β 2 t factor. For the H couplings to top quarks, we consider the following 6 representative scenarios: • S1 : (g S , g P ) = (1 , 0) • S2 : (g S , g P ) = (0 , 1) • S3 : (g S , g P ) = (1 , 1) • S4 : (g S , g P ) = (1 , −1) • S5 : (g S , g P ) = (1 , 0.42) TABLE II. The values of C 11 + C 12 and the 6 angular observables R i = C i /( C 1 + C 3 ) with i = 11, 12, 21, 22, 3, 4 taking w i = 1 for the 6 scenarios under consideration. The CP and CP T parities of each observable are shown in the square brackets. In the first two scenarios of S1 and S2, only one of the couplings is non-vanishing and H is supposed to be a pure CP-even (odd) state in the S1 (S2) scenario. In the other scenarios, CP is violated and the couplings g S and g P take on a relative phase. In the scenarios S5 and S6, in particular, the relative sizes of the couplings are chosen such that |g P /g S | ∼ β t in order that the two couplings contribute more or less equally to the amplitude squared.
In Table II, we show the values of C 11 + C 12 and the 6 angular observables R i = C i /( C 1 + C 3 ) for the 6 scenarios under consideration with i = 11, 12, 21, 22, 3, 4. We have taken of the scenario and the CPT-odd R 21 and R 22 are identically vanishing in all the scenarios. This leaves only R 3 and R 4 as non-trivial angular observables which can be probed by studying the dΓ/dΦ distribution. The CP-odd R 4 observable is vanishing in the CP-conserving S1 and S2 scenarios and, if it is not vanishing, its sign is determined by the sign of the product of g S and g P . Further we note that R 3 is very suppressed in S5 and S6 because it is proportional to −β 2 t g S 2 + g P 2 .
In Fig. 1 We have taken f L = 1 and f R = 0. In each frame, we are taking g S , g P = (1, 0) (S1: upper-left), g S , g P = (0, 1) between these two sets of distributions in all the scenarios 7 , we conclude that the finite 7 Numerically, we find that the absolute difference is smaller than 5 × 10 −3 for the 6 scenarios under m b effects are negligible and the NWA for the angular coefficients and observables works excellently. Thus, we conclude that the analytic expressions in Eq. (32) provide a sufficient theoretical framework to analyze the angular distributions and to extract the g S and g P couplings when M H > 2m t . Incidentally, the behavior of the angular distribution in Φ can be easily understood as it varies according to − R 3 c Φ + R 3 s Φ : see Eq. (32) and Table II. system. Note that rec includes the efficiency of fully reconstructing the four momenta of the t andt quarks which are necessary to measure the Φ distribution 8 . We observe that rec may also account for the dilutions due to interference with irreducible backgrounds and incorrect reconstruction of the neutrino momentum. One may achieve rec ∼ 1 at the future e + e − colliders but the production cross sections would be much suppressed compared to pp colliders. In our analysis, we are taking N evt = 10 4 as reference.
In Fig. 2, we show the normalized angular distributions for the S4 (left) and S6 (right) scenarios with N evt = 10 4 9 . The histograms represent the pseudo-data of a total of N event = 10 4 generated according to Eq. (15). The (red) solid lines present the results of fitting to consideration. 8 Using the pseudo-top algorithm, for example, the missing neutrino momentum can be reconstructed with a two-fold ambiguity at the LHC [20]. See also, for example, Ref. [21] for more sophisticated algorithms for top-quark pairs. 9 In practice, we have generated 10 pseudo datasets with each set having 10 4 events and take average of them to obtain the histograms. the angular distributions using Eq. (32). With N evt = 10 4 and ∆Φ = π/9, we find that the absolute size of 1-σ errors of the output angular observables of R 3,4 are about 0.02, see Table   III. The input values are the same as in Table II Now we are ready to carry out a χ 2 analysis to achieve our ultimate goal of extracting the couplings g S and g P from the angular observables R 3 and R 4 . To implement the analysis, we further need C 11 + C 12 . Using Eqs. (29) and (34), we have Assuming information on B(H → tt) and the coupling g t can be eventually extracted from σ · B measurements by considering several H production and decay processes, together with an independent measurement of the total decay width, one may determine the combination of C 11 + C 12 . In our analysis, similar to the angular observables R 3,4 , we simply assume 20 % error in C 11 + C 12 .
In the upper-left frame of Fig. 3, we show the confidence-level regions of the χ 2 analysis by varying g S and g P simultaneously for the scenario S4. We have found that χ 2 min /d.o.f = which are consistent with the input values (g S , g P ) = (+1, −1) within ∼ 1-σ range. Therefore, we conclude that the two couplings of H to the top-quark pair can be determined with about 10-20% errors when N evt = 10 4 for the scenario S4. For the scenario S6, the confidence-level regions are shown in Fig. 4. The minimum occurs at g S = 1.04 ± 0.13 ; g P = −0.405 ± 0.050 , with χ 2 min /d.o.f = 2.42. We again note that the fitted values are consistent with the input values (g S , g P ) = (+1, −0.42) safely within 1-σ range. Also, we conclude that the two couplings can be determined with about 13% errors in scenario S6. The results are also summarized in Table III.

IV. CONCLUSIONS
We have performed a comprehensive study of the renormalizable CP-even and CP-odd couplings of a spin-0 heavy Higgs boson to a pair of top quarks, using the angular distributions in the decay H → tt → bW +b W − . Based on the helicity amplitude method, we figure out there are 6 types of angular observables R i (i = 11, 12, 21, 22, 3, 4) according to their CP and CP T parities. We found that R 21,22 are identically zero unless CP T is violated through the presence of the absorptive part in the loop correction of the Htt vertex. Furthermore, we find that among the 6 observables only the R 3 and R 4 observables can be probed by the one-dimensional angular distribution dΓ/dΦ. This is our novel strategy for analyzing the decay H → tt → bW +W − to measure the properties of a heavy Higgs boson H.
We have illustrated with 10 4 events for H → tt → bW +b W − that the parameters g S and g P can be determined with about 10-20% uncertainties through the one-dimensional distribution dΓ/dΦ. This is the major numerical result of this work.
We offer the further comments in our findings: 1. As long as the heavy Higgs boson is above the tt threshold and, at least, as long as the angular distributions are concerned, the narrow-width approximation is always a good one: the weight factors deviate from unity less than 1%.
2. The angular coefficient C 2 is CP odd and CP T odd which implies that it is only nonzero in presence of non-vanishing absorptive (or imaginary) parts of the ttH vertex and in the simultaneous existence of g S and g P .  The helicity amplitude for the decayt(p t ,σ t ) →b(p b ,σ b )W − (k 2 , 2 ) can be obtained from that for the decay t(p t , σ t ) → b(p b , σ b )W + (k 1 , 1 ) by replacing f A with f * −A together with σ t,b →σ t,b , λ 1 → λ 2 , etc. This can be easily understood through the relation v t γ µ P A v b =ū b γ µ P −A u t . (A.1) The above relation can be shown by calculating both sides explicitly or by observing where C denotes charge conjugation and together with C (γ µ ) T C −1 = −γ µ , C (γ µ γ 5 ) T C −1 = +γ µ γ 5 .