Measuring properties of a Heavy Higgs boson in the $H\to ZZ \to 4\ell$ decay

In many extensions of the standard model, there exist a few extra Higgs bosons. Suppose a heavy neutral Higgs boson H is discovered at the LHC, one could then investigate CP and CPT~ properties of its couplings to a pair of $Z$ bosons through $H \to ZZ \to 4\ell$. We use the helicity-amplitude method to write down the most general form for the angular distributions of the four final-state leptons, which can cover the case of CP-even, -odd, and -mixed state for the Higgs boson. We figure out there are 9 types of angular observables and all the $H$ couplings to $Z$ bosons can be fully determined by exploiting them. A Higgs-boson mass of 260 GeV below the $t\bar t$ threshold is illustrated with full details. With a total of $10^3$ events of $H \to ZZ \to 4\ell$, one can determine the couplings up to 12-20\% uncertainties.


I. INTRODUCTION
The measured properties of the scalar boson which was discovered at the LHC [1,2] turn out to be the best described by the Standard Model (SM) Higgs boson [3] and it deserves to be called the Higgs boson which was 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 normalized to the corresponding SM value: C v = 0.94 +0. 11 −0.12 . 1 Even though the SM has achieved a great success in describing the interactions among the basic building blocks of matter scrutinized by now, however more blocks 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 [6], 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 [7]. As another example, the Higgs Triplet Model that can explain the mass spectrum and mixing of neutrinos gives rise to a pair of doubly-charged Higgs bosons, a pair of singly-charged Higgs bosons, and 3 neutral ones [8].
Suppose that in future experiments a neutral Higgs boson H heavier than the SM 125 GeV Higgs boson (denoted by h) is discovered. Below the decay threshold into a top-quark pair or when M H < 2m t , assuming H does not carry any definite CP-parity, it may mainly decay into a bottom-quark pair (bb), tau leptons (τ + τ − ), massive vector bosons (W + W − and ZZ), a pair of 125 GeV Higgs bosons (hh), and a massive gauge boson and a lighter Higgs boson (hZ). Above the 2m t threshold, the decay mode into a top-quark pair may dominate as in the MSSM 2 .
The fermionic decay modes of H → bb, τ + τ − , tt and one of the bosonic decay modes H → W + W − may suffer from large QCD backgrounds and/or missing neutrinos. Among the remaining bosonic decay modes into ZZ, hh, and hZ, taking account of the spin-0 1 For the reference value of the coupling C v , we have taken the 1-σ range obtained upon the LHC Run-1 data by varying the Higgs couplings to the top-and bottom-quarks, τ leptons, gluons, photons, and the massive gauge bosons under the assumption that the 125 GeV Higgs boson carries the CP-even parity [5]. 2 We refer to Ref. [9] and references therein for the typical decay patterns of the heavy MSSM neutral Higgs bosons which do not carry any definite CP parities.
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.
In this work, we consider the decay H → ZZ with the Z bosons subsequently decaying into electrons and/or muons: H → ZZ → 4 . Long before the discovery of the SM Higgs boson, it was suggested to exploit this decay process to determine the spin and parity of the Higgs boson [10]. Later, more rigorous angular analyses of spin-zero, -one, and -two resonances were illustrated with certain levels of experimental simulations [11]. After the 125 GeV Higgs-boson discovery, the method was practically applied to determine the spin and CP properties of the "newly" discovered boson [12,13]. Here, we shift the focus from the SM Higgs to a heavy Higgs boson H 3 , and pursue complete determination of its couplings from the angular correlations among the charged leptons in the final state. Under the current experimental status, in which active searches for heavy resonances decaying into a ZZ pair have been continually performed [15], 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).
The remainder of this article is organized as follows. In Sec. II, based on the helicity amplitude method [16], we present a formalism for the study of angular distributions in the decay H → ZZ → 4 . We point out that there can be 9 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 a 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
One may start by defining the interaction of the heavy Higgs boson H with a pair of Z bosons. The amplitude for the decay process H → Z(k 1 , 1 ) Z(k 2 , 2 ) can be written as 4 can be complex by developing non-vanishing absorptive parts in the existence of (New Physics) particles running in the loop with mass less than M H /2. Therefore, in general one may need 5 real parameters to describe the interaction of the heavy Higgs boson H with a pair of Z bosons. Note that g 2 HZZ ≤ 1 − g 2 hZZ = 1 − C 2 v with equality holding when h and H are the only Higgs bosons participating in the electroweak-symmetry breaking. We observe that being different from the case of SM Higgs boson, in which g hZZ is dominating over the loopinduced S ZZ h and P ZZ h couplings, each of the couplings g HZZ , S ZZ H , and P ZZ H may contribute comparably in the heavy Higgs-boson case. We further observe that either g HZZ × P ZZ H = 0 or S ZZ H × P ZZ H = 0 implies that H is a CP-mixed state, thus signaling CP violation. Incidentally, the interaction of the Z boson with a fermion pair is described by the interaction Lagrangian: with v f = I f 3 /2 − Q f s 2 W , a f = I f 3 /2 and P A = (1 + Aγ 5 )/2.

A. Helicity amplitude
We first present the helicity amplitude for the process H → Z(k 1 , 1 )Z(k 2 , 2 ) → Here, p 1,2 and andp 1,2 are four-momenta of the fermions f 1,2 andf 1,2 , respectively, with k 1,2 = p 1,2 +p 1,2 . And we denote the helicities of f 1,2 andf 1,2 by σ 1,2 andσ 1,2 . Depending on the helicities of the four final-state fermions, the amplitude can be cast into the form The helicity amplitude for the decay H → Z(k 1 , 1 )Z(k 2 , 2 ) in the rest frame of H is given by M H→ZZ with the reduced amplitudes λ 1 defined by where λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx and α i = k 2 i /M 2 H . We note that the contribution of g HZZ to the longitudinal amplitude 0 is enhanced by a factor M 2 H /2M 2 Z in the large M H limit.
On the other hand, the helicity amplitude for the decay Z(k, (k, λ)) → f (p, σ)f (p,σ) is given by in the rest frame of the fermion pair. Note that the Z boson is moving to the positive z direction in the H-rest frame, and θ and φ denote the polar and azimuthal angles of the momentum p of f in fermion-pair rest frame.
Collecting all the sub-amplitudes and neglecting the masses of the final-state fermions, we obtain We observe the amplitude is receiving contributions from all the three helicity states + , − , and 0 of the intermediate Z bosons, and the interferences among the different helicity states lead to non-trivial angular distributions.

B. Angular coefficients
Neglecting the masses of the charged leptons in the final state, we find that the amplitude squared can be organized as: The normalized 9 angular distributions are given by 5 Also, the 9 angular coefficients C 1−9 , which are combinations of the reduced helicity amplitudes + , − , and 0 , are defined as Under CP and CP T 6 transformations, the reduced H-Z-Z helicity amplitudes transform as follows: We note that the CP parities of C 2 , C 5 ,C 6 and C 9 are negative (CP odd) implying that they are non-vanishing only when {g HZZ , S ZZ H } and P ZZ H exist simultaneously. Furthermore, the CP T parities of C 2 , C 6 , C 7 are (CP T odd), which implies that they can only be induced by non-vanishing absorptive (or imaginary) parts of S ZZ H and/or P ZZ H .

C. Angular observables
The partial decay width of the process H → ZZ → 2 1 2 2 is given by After integrating over k 2 1 and k 2 2 , we obtain with the 9 angular observables defined by Note that we have introduced the 9 weight factors w i in the definition of the angular observables R i which are defined by where the constant angular coefficients at Z pole are given by and the numerical factors by In general, the angular coefficients C i depends of the momenta of Z bosons. When M H > 2M Z , the two decaying Z bosons are predominantly on-shell. In this case, one may have w i = 1 by adopting the narrow-width approximation (NWA) for the intermediate Z bosons.
We therefore note that the deviation of the weight factor from unity measures the accuracy of the approximation.
After integrating over any two of the angles θ 1 , θ 2 , and Φ, one may obtain the following analytic expressions for the one-dimensional angular distributions in terms of the Z-pole angular coefficients C 1−9 : First, we note that only C 1,2,3 contribute to the c θ 1,2 distributions. When S ZZ H and P ZZ H are real or when their imaginary parts are negligible, C 2 = 0 and the linear term is vanishing and the c θ 1,2 distributions are symmetric and parabolic. The coefficients C 4,5 and C 8,9 together with C 1,3 in the denominators are contributing to the Φ distribution. For the decay ZZ → 4 , with η = 2v a /(v 2 + a 2 ) = 0.150 for charged leptons, 9πη 2 /128 ∼ 0.005 and 1/8π ∼ 0.04, the Φ distribution mostly varies as s 2Φ and c 2Φ . Finally, we note that the angular observables R 6,7 never appear in the one-dimensional angular distributions since C 6,7 do not contribute to them. To probe C 6,7 , one may need to study two-dimensional angular distributions such as c θ 1 -Φ and c θ 2 -Φ distributions.
The angular observables R 1,2,3 can be obtained by the c θ 1,2 polynomial fitting to the The measurements of the angular observables R i alone, however, cannot determine the absolute size of the couplings of g HZZ , S ZZ H , and P ZZ H . For this purpose one may need to measure the quantity w 1 C 1 + w 3 C 3 . From Eq. (21), using F = 2280, we have where Γ H tot denotes the total decay width of the heavy Higgs boson H. Assuming information on B(H → ZZ) can be extracted from σ · B measurements by considering several H production and decay processes, and together with an independent measurement of the total decay width, one may determine the combination of w 1 C 1 + w 3 C 3 : where we use B(Z → ) = 3.3658 × 10 −2 . Incidentally, we note that a heavy scalar with a mass around 270 GeV may explain some excesses observed in LHC Run I data or those observed in measurements of the transverse momentum of h, h production associated with top quarks, and searches for hh and V V resonances [17,18].

III. NUMERICAL ANALYSIS
Bearing this in mind we consider the following 6 representative scenarios: • S1 : g   In the first three scenarios of S1, S2, and S3, only one of the couplings is non-vanishing and CP is conserved. In the scenarios of S4 and S5, CP is violated and the couplings S ZZ H and P ZZ H take on opposite relative phases. In the scenario S6, all three couplings are nonzero, with enhancement of the longitudinal component 0 of the amplitude for a heavier Higgs boson, the chosen values for the three couplings contribute more or less equally to the amplitude squared: see Eq. (7). Finally, we found that the weight factors lie between 0.99 and 1.02, and therefore we safely take w 1−9 = 1 in our numerical study.
In Table I, we show the 9 angular coefficients C 1 − C 9 for the 6 scenarios, together with their CP and CP T parities in the square brackets. With only the real component in the form factors S ZZ H and P ZZ H , the coefficients C 2 , C 6 and C 7 are identically vanishing in all the scenarios, and C 2 , C 5 , C 6 and C 9 further vanish in the CP-conserving scenarios of S1, S2, and S3. For S1, C 3 is large due to the enhancement of the longitudinal component 0 of the amplitude for a heavier Higgs boson. Since the longitudinal amplitude 0 = 0 in TABLE II. The 6 angular observables R i = C i /(C 1 + C 3 ) with i = 1, 3, 4, 5, 8, 9 taking w 1−9 = 1 and the value of C 1 + C 3 for the 6 scenarios under consideration. The CP and CP T parities of each observable are shown in the square brackets. the S3 scenario, only C 1 and C 8 take on non-zero values: see Eq. (12). In the CP-violating scenarios of S4, S5, and S6, all the coefficients with plus (+) CP T parity are non-vanishing.
Note that with g HZZ = 0 in S4 and S5 , the angular coefficient is suppressed: see Eq. (7). All the non-vanishing coefficients are comparable in the scenario

S6.
In Table II,  In the CP-conserving cases shown in Fig. 1, the cos θ 1,2 distribution behaves like (1−c 2 θ 1,2 ) in scenario S1 because R 1 2R 3 , while the distributions behave like (1 + c 2 θ 1,2 ) with R 1 2R 3 in scenarios S2 and S3. The Φ distributions mostly behave according to R 8 c 2Φ with the sub-leading contributions from R 4 c Φ suppressed by η 2 : see Eq. (20). The smaller value at Φ = 0 compared to those at Φ = ±π in S1 (upper right) is due to the negative R 4 c Φ contribution. Note that they are all symmetric about Φ = 0 without CP violation. In the CP-violating scenarios of S4 and S5, the cos θ 1,2 distribution behaves like (1 + with R 1 slightly larger than 2R 3 , it still behaves as (1 + c 2 θ 1,2 ) but its variation is much smaller compared to the S4 and S5 scenarios due to the cancellation between the R 1 and  [15,20]: where we naively take the 4-lepton efficiency 4 ∼ 1 7 and assume the HL-LHC with the luminosity of L = 3/ab. Further, we assume the angular resolutions of ∆ cos θ = 0.1 and ∆Φ = 0.1π.
In Fig. 3, the histograms show the normalized cos θ (left) and Φ (right) distributions from the pseudo dataset of N evt = 10 3 events. Here the cos θ distribution is the combination of the cos θ 1 and cos θ 2 distributions. One can obtain the angular observables R 1,3 by fitting to the cos θ distribution with the analytic expression for the 1/Γ dΓ/dc θ 1,2 in Eq. (20). Note we have fixed R 2 = 0 in the fitting. We have found the strong correlation between the R 1 and R 3 observables with the correlation coefficient ρ = −0.813. The angular observables R 4,5,8,9 can be obtained by the Fourier analysis of the Φ distribution. Explicitly, one may have The angular observables R 4,5,8,9 can also be obtained by performing a fit to the Φ histogram distribution with the analytic expression for the 1/Γ dΓ/dΦ in Eq. (20). We have checked that R 4,5,8,9 from the Fourier analysis and those from the fitting are consistent within errors 8 . In our numerical analysis, we use the fitted angular observables. The results of the fittings are represented by the (red) solid lines in In Fig. 3.
The details of the fitting results are summarized in Table III as the output central values together with the corresponding parabolic errors. We observe that the output central values are within the 1-or 2-σ ranges of the input values. Note that the CP violation is observed at the 2-σ level with R 9 = −0.387 ± 0.18. The observation through another CP-violating observable R 5 is also at the 2-σ level: R 5 = −4.07 ± 1.45. First, the error is 8 times larger than that of R 9 because of the η 2 suppression factor, see Eq. (20). Second, this is due to the statistical fluctuation. We have verified that the central values of the observable R 5 are quite close to the input value −0.542 if we generate more pseudo datasets of 10 3 events.
Now we are ready to carry out our ultimate target to extract the couplings g HZZ , S ZZ H , and P ZZ H from the 7 observables R 1,3,4,5,8,9 and C 1 + C 3 by implementing a χ 2 analysis. We have taken into account the correlation between R 1 and R 3 , by using where we calculate R  Table III. The χ 2 's for the remaining uncorrelated observables are similarly calculated and summed.
In the upper frames of Fig. 4

IV. CONCLUSIONS
We have performed a comprehensive study of the most general couplings of a spin-0 heavy Higgs boson to a pair of Z bosons up to dimension-6 operators, using the angular distributions in the decay H → ZZ → + − + − . Based on the helicity amplitude method, we figure out there are 9 types of angular observables R i (i = 1 − 9) according to their CP and CP T parities: four of them (R 2,5,6,9 ) are CP odd and three of them (R 2,6,7 ) CP T odd.
Furthermore, we find that, among the 9 observables, the 2 CP T-odd observables of R 6,7 are not accessible through one-dimensional angular distributions. We have shown that a certain subset of the 9 angular observables can be extracted from one-and two-dimensional angular distributions of the four final-state charged leptons depending on the assumption on S ZZ H and P ZZ H . The parameters g HZZ , S ZZ H , P ZZ on search for heavy ZZ resonances in the + − + − and + − νν final states in which, using data at √ s = 13 TeV with the integrated luminosity of 36.1/fb, they report observation of two excesses for m 4 around 240 and 700 GeV, each with a local significance of 3.6 σ.
Especially, the resonance around 240 GeV corresponds to more than 30 events which may lead to about 3000 events at the HL-LHC with the luminosity of 3/ab, assumed in this work.