The role of interference in unraveling the ZZ-couplings of the newly discovered boson at the LHC

We present a general procedure for measuring the tensor structure of the coupling of the scalar Higgs-like boson recently discovered at the LHC to two Z bosons, including the effects of interference among different operators. To motivate our concern with this interference, we explore the parameter space of the couplings in the effective theory describing these interactions and illustrate the effects of interference on the differential dilepton mass distributions. Kinematic discriminants for performing coupling measurements that utilize the effects of interference are developed and described. We present projections for the sensitivity of coupling measurements that use these discriminants in future LHC operation in a variety of physics scenarios.


I. INTRODUCTION
As the new particle discovered by the ATLAS [2] and CMS [3] collaborations appears to be similar to the Standard Model (SM) Higgs boson [4][5][6][7][8][9], it becomes very important to measure its properties as precisely as possible in order to find or constrain physics beyond the SM. The recent ATLAS and CMS results strongly suggest that the newly discovered boson has spin zero [10][11][12][13][14], which we take as the starting point in the studies presented in this report. There is a large body of literature  advocating the great potential of X → ZZ → 4 decays for disentangling the spin-parity properties of resonances decaying to two Z bosons and for refining the methodology for doing such measurements. In this work we explore the sensitivity of future LHC analyses to interference between various operators in this channel. We follow the framework of Ref. [49], which is briefly reviewed below.
Here p 1 (2) is the momentum of the intermediate Z boson labelled "1" ("2"), while p = p 1 +p 2 is the momentum of the X boson. We note, following, e.g., Refs. [27,28,39] (cf. especially Eq. (11) in Ref. [39]) that the three operators in Eq. (3) generate each of the three possible Lorentz structures in the general amplitude for the decay of X to two bosons.

Comparison of Conventions
Various conventions have been used in writing Lagrangians and amplitudes for the study of the X → ZZ interaction. For the convenience of the reader, Table I contains a dictionary of the couplings used in Refs. [27,39,44,49,54,55].

Sensitivity to Loop-Induced Couplings
The coefficients κ i in Eq. (5) are real, since they originate from the tree-level Lagrangian in Eq. (3). By the optical theorem, the amplitude may obtain contributions from loops with light particles (lighter than M X /2 ≈ 63 GeV) such that the expression for the amplitude including loop effects is analogous to that in Eq. (5), where the effective couplings κ i are complex: However, at least one of the κ i must not be predominantly loop-induced, or else one runs into a contradiction with the experimental constrains. For example, consider a generic loop with some invisible new particle whose coupling to the X(Z) boson is g X(Z) . Then, naively, In this scenario the invisible width of the X boson is hence taking Γ X,inv Γ exp X,total 7 GeV [60], we obtain g X 2. Since the gauge coupling to the Z, g Z , should be 1, we get This is about two orders of magnitude smaller than the magnitude of couplings needed to give the SM rate [49]. More stringent constraints on the δκ i (with some caveats) may be obtained from more stringent limits on the invisible width of the Higgs [13,51,[61][62][63][64] or the invisible width of the Z [65]. 1 It is therefore well-motivated to treat the relevant couplings, κ i , (namely, the ones which are large enough to measure at present) as predominantly real.

B. Experimental Situation
The hypothesis of the new boson being a 100% pure pseudoscalar, 0 − , has been excluded by CMS [11,66] and ATLAS [10,14]. The possibility of a 100% pure 0 + h is also disfavored at 92% C.L. [11]. Hence, in this study we assume a non-zero value of coupling, κ 1 , and address the question of the experimental sensitivity to the presence of κ 2 and κ 3 terms in the XZZ Lagrangian.
The current limit, set by CMS [11], on the presence of a pseudoscalar contribution expressed in terms of a fractional cross section is f a3 = σ 3 /(σ 1 + σ 3 ) < 0.58. Here the cross sections σ 1 and σ 3 are taken for the 4e, 4µ, and 2e2µ final states together 2 and correspond to 100% pure 0 + and 0 − states, respectively. This result translates into a limit on the ratio of couplings |κ 3 /κ 1 | < 6.1. The corresponding CMS analysis was set up in such a way that it was not sensitive to the interference between the κ 3 -and κ 1 -induced amplitudes.

C. Objective
In this paper, we show that by explicitly exploiting the interference between amplitudes which involve the 0 + , 0 + h , and 0 − states (corresponding to the κ 1 , κ 2 , and κ 3 terms in the Lagrangian in Eq. (3), respectively) one can boost the experimental sensitivities to the presence of an 0 − (and 0 + h ) admixture. We show that the gains become particularly large at high integrated luminosities, allowing one to probe smaller values of the κ 2 and κ 3 couplings.
We also address the question of establishing the presence of the interference and evaluating its sign, should decay amplitudes associated with spin zero higher dimensional operators be detected. Recently, the importance of a proper treatment of interference was discussed in the context of a somewhat different aspect of the H → ZZ → 4 channel [44]; there it was the interference associated with permutations of identical leptons in the 4e and 4µ final states that was considered. This interference is always included in the studies presented in this report.

II. THE PHYSICAL IMPORTANCE OF INTERFERENCE
In general, interference effects can manifest themselves in two different ways: either at the level of total cross sections (reflected in the production rate, as discussed in Sec. II A below), or at the level of differential distributions (as discussed in Sec. II B below). 2 For given values of couplings κ i and κ j , the ratios of cross sections σ i /σ j (i = j) for same-fermion and different-fermion final states are different. This is due to the interference effects associated with permutations of identical fermions in the final state. Hence, one should specify which final states are used in the definition of f a3 .
The presence of interference is then implied by nonzero values of the "off-diagonal" coefficients γ ij with i = j. Eq. (11) shows that the overall rate is affected by interference between 0 + and 0 + h , which is destructive (constructive) when κ 1 and κ 2 have the same (opposite) signs. Eq. (11) also implies that at the level of total cross sections there is no interference between 0 + and 0 − or between 0 + h and 0 − . The magnitude of interference depends on the values of the couplings κ 1 and κ 2 . Obviously, for a pure 0 + state (κ 1 = 0, κ 2 = 0) and for a pure 0 + h state (κ 2 = 0, κ 1 = 0) the interference is absent. Given the values in Eq. (11), one could expect the interference effect to be maximal for κ 2 κ 1 = 1 2 tan −1 2γ 12 γ 1 − γ 22 3.89.
In practice, the signal rate for X → ZZ → 4 production is measured from data, thus imposing one constraint through Eq. (10) on the {κ 1 , κ 2 , κ 3 } parameter space [49] (provided the production rate for the X is fixed). The constraint may be solved explicitly by a suitable change of variables, reducing the relevant {κ i } parameter space to a two-dimensional surface which can be taken to be effectively the surface of a sphere [49]. For this reason, we shall not discuss the overall rate further. Instead, we will assume in our analyses that the rate measurement has already been performed and the couplings κ i have been chosen so that they satisfy the constraint of Eq. (10).

B. The impact of interference effects on differential distributions
Even if the overall rate is kept fixed, the interference effects are still present at the level of differential distributions (the size of this effect will be quantified in Sec. IV below). In general, the kinematics of X → ZZ → 4 events is described in the X rest frame by 7 independent degrees of freedom, and interference will impact the differential distribution in this 7-dimensional signature space. For simplicity, in this subsection we will focus only on the M Z 1 and M Z 2 invariant mass distributions and use them to illustrate the effects of interference. 4 In order to provide an intuitive understanding of some of the results to follow in Sec. IV, we shall derive analytical formulas for the M Z 1 and M Z 2 distributions, which explicitly demonstrate the interference effects.
The doubly differential decay width with respect to M Z 1 and M Z 2 can be written as where the dimensionless 5 functions F ij are symmetric with respect to their indices: In the absence of any selection criteria, the functions F ij are where the dimensionless common factor ξ is given by 4 Note the webpage http://yichen.me/project/GoldenChannel/ created by the authors of Ref. [46], may be used to make plots of interesting differential distributions for different values of the couplings κ i . 5 Since κ i are already dimensionless, in the right-hand side of Eq. (13) we factor out 1/v to make F ij dimensionless as well.
Here g 2 is the SU (2) W gauge coupling constant, is a dimensionless parameter introduced in Ref. [39], and are the Z propagator functions which depend on the mass, M Z , and width, Γ Z , of the Z-boson.
The doubly differential distribution in Eq. (13) is an interesting object to study experimentally and CMS and ATLAS have published plots of the Higgs candidate events in the dependence is non-trivial and contains interesting information [44]. Therefore we integrate the expression in Eq. (13) over M Z 1 and consider instead the corresponding one dimensional with newly defined dimensionless functions in place of Eqs. (14)(15)(16)(17)(18)(19). Comparison of Eq. (10) and Eq. (13) shows that the normalization of the functions F ij and f ij is given by the values of the coefficients γ ij in Eq. (11) [49]     Let us first focus on the interplay between the κ 1 and κ 3 terms in the Lagrangian (3).
In this case, the behavior of the peak shown in Fig. 2(b) is relatively simple, due to the absence of an interference contribution (f 13 = 0). The case of the SM (denoted by the blue circle) corresponds to κ 1 = 1 and κ 3 = 0, in which case the M Z 2 distribution is made up entirely of the f 11 contribution, which peaks around 28 GeV. As the value of κ 3 is gradually increased, one introduces a larger fraction of the f 33 component from Fig. 1, which peaks at a lower value of M Z 2 , around 25 GeV. As a result, the peak location in Fig. 2(b) is initially a decreasing function of the ratio κ 3 /κ 1 . Eventually, we reach the case of a pure 0 − state with κ 2 = 0 and κ 1 = 0, when the M Z 2 distribution is composed entirely of the f 33 component and the M Z 2 peak is located at M Z 2 ∼ 25 GeV. The right half of Fig. 2(b), where the κ 1 and κ 3 couplings are taken with a relative minus sign, is a mirror image of the left and can be understood in the same way.
Notice that the f 11 and f 33 contributions always enter with positive weights, κ 2 1 and κ 2 3 , respectively. Thus the shape of the combined M Z 2 distribution is a weighted average between the f 11 and f 33 shapes seen in Fig. 1(a), which are already very similar. As a result, the peak location stays relatively constant over the whole range of the couplings ratio κ 3 /κ 1 .
In contrast, when we consider the interplay between κ 1 and κ 2 , the situation changes completely, as demonstrated by Fig. 2(a). Now the M Z 2 distribution is built up from three components: f 11 , which peaks near 28 GeV, f 22 , which peaks around 30 GeV, and f 12 , whose magnitude peaks near 29 GeV. Given that the peaks of all these three components are very close, one might expect that the peak of the total M Z 2 distribution would also fall in the vicinity of 28 − 30 GeV. However, Fig. 2   κ 2 grows, this secondary peak becomes stronger and eventually takes over as the primary peak in the distribution, causing the sudden jump seen in Fig. 2(a). This phenomenon resembles a "first order phase transition" and can be seen more clearly in Fig. 4, where we zoom in on the actual M Z 2 distribution without the individual contributions. Of course, in the regime where this interesting behavior occurs, the large destructive interference also suppresses the cross section. The reader will note that the values of κ 1 and κ 2 shown in Fig. 4, which are necessary to give the correct SM partial width in Eq. (10), are relatively large as a result.
As the value of κ 2 is increased beyond the region of the "first order phase transition" shown in Fig. 4, the M Z 2 distribution starts to be dominated by the f 22 contribution and eventually we get to the pure 0 + h state (the second to last panel in Fig. 3). The final panel in Fig. 3 shows a representative point with opposite signs for the couplings κ 1 and κ 2 . In that case, the sign of the interference term f 12 is flipped and it adds constructively with f 11 and f 22 , causing the peak of the M Z 2 distribution to stay in the vicinity of 28 − 30 GeV.

A. Optimized Analyses
To obtain the greatest sensitivity to a signal in a model which is characterized by a modest number of parameters, it is customary to use analyses with criteria specifically optimized for each point in the parameter space of the underlying model. This procedure is used in all searches for Higgs bosons, whether SM or otherwise. The approach has also been advocated for SUSY searches where the signal model may have a greater number of parameters [68,69]. In line with this idea, we introduce the kinematic discriminants that are automatically optimized for each point in the XZZ coupling parameter space.
In this report we assume that the cross section has been well-measured and that variations in the overall rate may be absorbed into the ggX couplings (provided we consider only the pp → X → ZZ → 4 channel). For this reason, the parameters we aim to measure are not the XZZ couplings, κ i , but their ratios κ 2 /κ 1 and κ 3 /κ 1 . These quantities can be easily re-expressed in any desired convention, such as "geolocating" angles as in Ref. [49] or f a -like fractions as in Ref. [11]. In this study we assume that the couplings are real numbers, as already explained in Sec. I A 2.

B. Preparation of Monte Carlo Samples
The analyses are performed using simulated gg → X → ZZ → 4 events, generated using FeynRules [70] and MadGraph [71] according to the MEKD framework [44]. This approach ensures that we include all interference effects: those arising from the presence of multiple terms in the Lagrangian as well as those associated with permutations of identical leptons in the 4e and 4µ final states. Following the ATLAS and CMS results [10,11] the mass of the scalar Higgs-like boson mass is taken to be 125 GeV. We use MadGraph to simulate the qq → ZZ backgrounds. Our simulation is performed entirely at the leading order and at the parton level. In order to compensate somewhat, we consider events with the four-lepton invariant mass in a very conservative 10 GeV mass window centered at the Higgs mass of 125 GeV (in contrast, the LHC detectors have 1 − 2% mass resolution). Consequently, the larger mass window results in the acceptance of more background events.
We use lepton kinematic selection criteria very similar to those used in the H → ZZ → 4 analyses of ATLAS and CMS experiments [10,11]. Leptons are required to have transverse momenta p T > 5 GeV and pseudorapidity |η| < 2.5. At least one same-flavor opposite-sign lepton pair must have an invariant mass greater than 40 GeV, while the other lepton pair must have an invariant mass greater than 12 GeV. We use events with all three final-state combinations (4e, 4µ, and 2e2µ + 2µ2e) in all of our analyses.

C. Projected Event Yields
In order to obtain the analysis results as a function of the integrated luminosity of the LHC runs at 14 TeV, we estimate experimental reconstruction efficiencies and contribution of the background at the 14 TeV LHC using the average of the expected signal and background event yields reported by ATLAS and CMS (Table II). The number of events expected in the 14 TeV LHC runs with L fb −1 of integrated luminosity, N (L), is computed as:  Tab. 2 in Ref. [11] Bkgd ∼1 event/GeV Fig. 2 in Ref. [11] D

. Kinematic Discriminants
Kinematic discriminants for separation between the two types of four-lepton processes, A and B, may be constructed by calculating the ratio of the squared matrix elements for these two hypotheses, as described in Ref. [44]. For each four-lepton event with kinematic information x, one can compute: In our analysis, we compute the kinematic discriminants following this approach. We first consider the kinematic discriminant D(X; 0 + ). Here, the hypothesis "X" is the hypothesis that the scalar Higgs-like boson couples to Zs via both the κ 1 and κ 3 operators. We will further refer symbolically to this state as X = κ 1 [0 + ]+κ 3 [0 − ]. The hypothesis "0 + " assumes that the scalar Higgs-like boson has only the tree-level SM coupling to Z bosons. Therefore, for D(X; 0 + ) we obtain: By construction, this discriminant takes into account all aspects in which kinematic distributions differ between the two hypotheses, including in particular those associated with the interference between the κ 1 and κ 3 operators in hypothesis "X".
Alternatively, one can choose to use the kinematic discriminant D(0 − ; 0 + ) [11], where the two hypotheses, "0 − " and "0 + ", correspond to the cases where only the κ 3 term or only the κ 1 term are non-vanishing, respectively: Since the two hypotheses from which the discriminant is calculated correspond to two pure states, discriminant D(0 − ; 0 + ) is explicitly insensitive to the potential effects on kinematic distributions associated with the interference (unlike discriminant D(X; 0 + )). The D(0 − ; 0 + ) discriminant is optimal for comparing the two pure states or for testing for the presence of an additional pseudoscalar state nearly degenerate with the scalar Higgs-like boson (but with a sufficiently different mass that there is no significant interference in the scalar and pseudoscalar production and decays). However, as it ignores interference effects, it is not optimal for measuring the state X which couples with ZZ via both κ 1 -and κ 3 -terms.
Discriminant D(X; 0 + ) described above is ideal for this purpose.

E. Statistical analysis
We obtain distributions for the kinematic discriminants described above using simulation.
Distributions are obtained for events that correspond to the signal hypothesis "X", to the signal hypothesis "0 + " (both described above) and to the background hypothesis. Examples of the distributions, pdf (D | X + bkg) and pdf (D | 0 + + bkg) are shown in Fig. 5(a).
These kinematic discriminant distributions are then used to construct the test statistic q as follows: where i runs over all the events in an pseudoexperiment. An example of the test statistic distributions obtained with 50000 pseudoexperiments for a particular choice of the integrated luminosity L and κ 3 /κ 1 ratio is shown in Fig. 5(b). |M(0 + )| 2 for two alternative hypotheses "0 + " and "X", where X has κ 3 /κ 1 = 5.21. (b) Corresponding distributions for the test statistic defined in Eq. (31) for pseudoexperiments at an integrated luminosity L = 100 fb −1 .
To quantify the expected separation power between alternative signal hypotheses, we find a "mid-point" value,q, of the test statistic q between the medians of the two test statistic distributions (those generated using each signal hypothesis). We use pointq to define two "tail probabilities", P (q ≥q | X) and P (q ≤q | 0 + ), in such a way that P (q ≥q | X) = P (q ≤q | 0 + ). This tail probability is then converted into significanceZ (in σ) using the one-sided Gaussian tail convention: Finally, for the separation power between alternative signal hypotheses we quote Z = 2Z, where the extra factor of 2 arises from the fact that theq point is half-way between the medians of the two distributions. With such a definition, we treat two alternative hypotheses symmetrically and we do not need to generate billions of pseudoexperiments to assess tail probabilities corresponding to 5σ-separations.
The presence of a non-zero value of κ 3 could be established, albeit with different significances, in searches performed using either D(X; 0 + ) or D(0 − ; 0 + ). The difference in the sensitivity between the two searches is manifested in case the interference between the κ 1 and κ 3 operators is present. This is not unlike the actual discovery of the Higgs boson candidate, which gave rise to the ∼5σ signal in the SM Higgs search [2,3] and at the same time was also seen as ∼3σ excesses in the Higgs boson searches performed in the context of the fermiophobic and SM4 scenarios [74]. In case the presence of a non-zero value of κ 3 is established, the next two questions to answer are: • whether there is one state • if there is interference, how well we can tell apart the relative signs of κ 3 and κ 1 couplings.
Both of these questions can be addressed by repeating the statistical analysis with properly adjusted kinematic discriminants. To demonstrate the ability of an experiment to establish the presence or absence of interference, as well as to determine the relative sign of couplings, we plot the per event log likelihood for two particular benchmark points in Figures 6 and 7.
The presence of interference breaks this symmetry and gives one sensitivity to the sign of the couplings. We note that interference between contributions to the amplitude from the κ 1 and the κ 2 terms is relatively straightforward to detect, as one would expect from the behavior of the M Z2 distribution discussed above. On the other side, interference involving the κ 1 and κ 3 terms will be more challenging to detect. Interestingly, it is significantly easier to determine the correct sign of κ 3 assuming interference, than it is to determine whether that interference is present. for positive and negative signs of the κ 3 /κ 1 ratio. Of course, for a given pseudoexperiment and in the actual LHC running one sign or the other will be preferred by the data.
In Fig. 8(a) and Fig. 8(b) one can see that the sensitivities obtained with the two discriminants scale very differently with integrated luminosity L. This is because the D(0 − ; 0 + ) discriminant does not change when one wishes to probe smaller or larger values of the κ 3 /κ 1 ratio. In this case, the sensitivity to |κ 3 /κ 1 | 2 which is related to the ratio of cross sections The difference between the sensitivities obtained with the two discriminants can be quantified in terms of a ratio of integrated luminosities required to achieve the same sensitivity. Results with interference-sensitive D(X; 0 + ) and interference-blind D(0 − ; 0 + ) discriminants are shown with blue and green curves, respectively. |κ 3 /κ 1 | ∼ 1. However, at an integrated luminosity of 10 fb −1 which approximately corresponds to 25 fb −1 at 8 TeV, the difference in sensitivities to κ 3 /κ 1 achievable with the two discriminants is rather modest, O(10%). Figure 9 shows the expected 2 σ-exclusion and 5 σ-observation sensitivities for the ratio of couplings κ 2 /κ 1 vs. the integrated luminosity. In these figures we focus on the κ 2 /κ 1 > 0 region for which destructive interference is present, as the prospects for early detection are more favorable with this choice of the relative sign of the two couplings. Results with both the optimal D(X; 0 + ), where X = κ 1 [0 + ] + κ 2 [0 + h ], and interference-blind D(0 + h ; 0 + ) discriminants are shown. As suggested in Eq. (12) above, there is substantial destructive interference in the range of κ 2 /κ 1 ≈ 2 − 4 that leads to dramatic changes in the M Z 2 invariant mass distribution shown in Fig. 3. The kinematic discriminants are automatically sensitive to such changes in the M Z 2 distributions, as well as to changes in other kinematic variables. As a results, it would be relatively easy to differentiate the case where κ 2 /κ 1 is in this range from the pure SM Higgs-like boson. In fact, since the ∼ 25 fb −1 of 8 TeV data already recorded on tape translates into the ∼ 10 fb −1 of 14 TeV data, we find that the LHC experiments should already be able to discover or exclude the 2 < κ 2 /κ 1 < 4 range. We note also that with existing data there should be a borderline sensitivity for exclusion of the κ 2 /κ 1 > 4 range, which includes the case of a pure 0 + h state. This result is well in agreement with the expected sensitivity of 1.8σ for a 100% pure 0 + h state reported by CMS [11]. The observed limit reported by CMS is 92% CL.

V. SUMMARY
We have considered the important question of how to measure the couplings of the scalar Higgs-like boson, X, to two Z bosons. In particular, we have studied the effects of the interference between various XZZ operators, presented the kinematic discriminants that take into account these interference effects, and provided projections for the coupling measurements using these discriminants at the 14 TeV LHC.
We have also compared the sensitivity of these kinematic discriminants with the kinematic discriminants that do not include interference terms and found that incorporating interference effects allows one to significantly improve the sensitivity to states where more than one operator is present in the XZZ coupling. Depending on the value of the couplings being probed, using analyses that take interference into account may reduce the integrated luminosity required to reach a given sensitivity by as much as a factor of four, as compared with analyses that neglect this interference. Thus using analyses such as those presented may allow one to reach given sensitivity benchmarks at the LHC years earlier than otherwise.