Testing CP Violation in ZZH Interactions at the LHC

We study genuine CP-odd observables at the LHC to test the CP property of the ZZH interaction for a Higgs boson with mass below the threshold to a pair of gauge bosons via the process p,p ->Z,H ->l+,l-,b,bbar. We illustrate the analysis by including a CP-odd ZZH coupling, and show how to extract the CP asymmetries in the signal events. After selective kinematical cuts to suppress the SM backgrounds plus an optimal Log-likelihood analysis, we find that, with a CP violating coupling btilde = 0.25, a CP asymmetry may be established at a 3 sigma (5 sigma) level with an integrated luminosity of about 30 (50) fb^-1 at the LHC.


I. INTRODUCTION
The CERN Large Hardron Collider (LHC) will lead us to revolutionary discoveries in particle physics. Observing the Higgs boson(s) is arguably the most anticipated discovery at the LHC. Once a Higgs boson (H) is observed, a significant part of the LHC program will be focused on determining the properties of it.
One of the most important aspects of the Higgs boson interactions will be its CP property. Due to the clear need for new sources of CP violation beyond the Standard Model (SM) to explain the baryon asymmetry of the Universe, it is conceivable that an extended Higgs sector may be a primary source with or without spontaneous CP violation [1]. There have been significant efforts in the literature to explore the possibility to observe the effects of CP-violation in the Higgs sector at the LHC [2][3][4]. If the Higgs boson is heavy enough to decay to multiple identifiable SM particles, then it is quite feasible to construct CP-odd observables. Examples include H → W + W − , ZZ or H → tt [2]. However, if the Higgs boson is light and only decays to light fermion pairs, then there will not be enough information for constructing a CP-odd kinematical observable like the triple product using the final state momenta of the Higgs decay. The only exceptions are the subdominant channel H → τ τ , subsequently decaying to at least two charged tracks plus missing neutrinos [3], which suffers from a small branching fraction to the identifiable final state and difficulties for event reconstruction. This may lead us to test the Yukawa coupling of ττ H. For HV V couplings, one may hope to access the clean decay channel H → ZZ * → 4ℓ [4], if the Higgs mass is not too far below 2M Z . Attempts have also been made to probe the CP property of the ttH coupling at hadron colliders via virtual Higgs effects in tt production [5], and via the direct associated production of ttH at the LHC [6]. However, it is very challenging to establish the signal of the ttH final state at the LHC experiments [7].
In general, it is known to be quite challenging to construct genuine CP-odd observables at the LHC. First, the initial state of the LHC, as a pp collider, is not a CP eigenstate, in contrast to the neutrality property of an e + e − collider or a pp collider. As a result, one may have to seek for neutral subprocesses (with qq, gg initial states) or decays of CP eigenstates at the LHC experiments. Second, even with the initial states q(p 1 )q(p 2 ) and g(p 1 )g(p 2 ), they form a CP eigenstate only with special spin and color configurations, and in their center-of-mass (c.m.) frame which is in general different from the lab frame. These two reference frames are related by a longitudinal boost that is unknown and different event by event. Only with large statistics would one expect the difference to be averaged out. Furthermore, the symmetric proton beams at the LHC make it impossible to identify the direction of a quark versus an anti-quark on an event-by-event basis, which is often needed to specify a particle versus an anti-particle.
In a recent paper [9], some genuine CP-odd observables have been proposed suitable for the LHC. One of them is the difference of the scalar transverse momenta [5,10], or equivalently the transverse energies, where p T = p 2 x + p 2 y , E T = p 2 T + m 2 f , with superscripts ± specifying the particle charges. This observable is CP-odd butT -even 1 and thus generated from CP-violation associated with the absorptive part of the amplitude [11], thus relying on the existence of an additional CP-conserving strong phase sin δ. The other one is a modified triple product where p f,f are the 3-momenta of the particle f and anti-particlef ,p f,f are their unit vectors, and z is the beam direction. This observable is CP-odd andT -odd, and may be generated from the dispersive part of the amplitude [11]. Both variables (1) and (2) are independent of the choice of a quark momentum direction and are longitudinally boost invariant, thus adequate for LHC experiments.
In this article, we address the feasibility of discovering CP violation in ZZH interactions using these observables. We propose to analyze the process where the Z decays leptonically and the Higgs to bb. The signal diagram is illustrated in Figure 1. The most general Lorentz structure of the Higgs and Z boson coupling, HZ µ (k 1 )Z ν (k 2 ), is where v = ( √ 2G F ) −1/2 is the vacuum expectation value of the Higgs field, and the Z boson 4momenta are both incoming. The a and b terms are CP-even whileb is CP-odd. The simultaneous presence of both a (or b) andb in this vertex would generate CP-violation. In the SM at tree level, a = 1 and b =b = 0. For a given theory of the SM or beyond, radiative corrections can generate nonzero contributions to a, b andb, and thus each of these are energy-dependent form factors in general [12]. In effective field theory language, b andb can be obtained from gauge invariant dimension-6 operators such as respectively [13]. Depending on the nature of the underlying theory, the effective couplings c,c could be of order unity for a weakly coupled theory, or of order (4π) 2 in a strongly coupled theory. If the theory is valid up to a scale Λ ∼ 4πv, we expect that b andb should be of the order 1/(4π) 2 to 1. For our phenomenological studies, we will simply take them as constants with some optimistic values, and will focus on the complex CP-violating parameterb which accommodates both dispersive (realb) and absorptive (imaginaryb) CP violation. The rest of this paper is organized as follows: In Sec. II, we concentrate on the signal and show that an asymmetry appears in the observables O 1,2 with the above theoretical parameterizations. In Sec. III, we consider the backgrounds to this process at the LHC, apply judicial kinematic cuts and determine the integrated luminosity necessary to observe these asymmetries. In Sec. IV, we discuss our results further. Finally, in Sec. V, we conclude.
The CP asymmetries manifest themselves in the distributions of the CP-odd observables O 1,2 as introduced in the previous section. We find it convenient to introduce where φ ll is the azimuthal angle between the two lepton planes ℓ + −ẑ and ℓ − −ẑ, multiplied by the sign of their logitudinal momenta difference. Our study of the signal is based on a parton-level Monte Carlo simulation of pp → ZH → ℓ + ℓ − bb, incorporating the full spin correlations from the production to the decay. We use the CTEQ6.1L parton distribution functions [14]. We simulate the LHC pp collisions at the c.m. energy √ s = 14 TeV. At lower c.m. energies, we would not expect any qualitative change for our asymmetry discussions, although the signal cross section and SM backgrounds will be different. We first present the total cross-sections for the signal versus Higgs mass forb = 0, 0.15 and 0.25 in the left panel of Fig. 2. We see that, for m H = 120 GeV, the SM cross section is of order 700 fb andb enhances the cross-section by 23% (65%) forb = 0.15 (b = 0.25). In the right panel of Fig. 2, we show the signal cross section as a function ofb for m H = 120 and 140 GeV. There is no decay branching fraction included in these figures. We use the illustrative value m H = 120 GeV in the remainder of this paper.
To simulate geometrical coverage of the detector and for the purpose of triggering, we impose the following basic cuts of transverse momentum, rapidity, and separation on the jets and leptons: p T j,l > 20 GeV, |η j,l | < 2.5 and ∆R jj,jl > 0.4.
With these acceptance cuts, we show the distribution of dσ/dφ ll in Fig. 3 for the caseb = 0.25 (left panel), and the case withb = 0.25i (right panel) by the dashed histograms. The SM result is included for comparison by the solid curves in both panels. We see that the CP asymmetry in dσ/dφ ll is only induced by CP-violation in the dispersive amplitude as in the left panel with real b, but not in the absorptive amplitude as in the right panel with imaginaryb. In contrast, the distribution of dσ/d∆E T is shown in Fig. 4 with the same parameter choice, and the CP asymmetry in this variable is only induced by CP-violation in the absorptive amplitude in the right panel with imaginaryb, but not in the dispersive amplitude in the left panel with realb. The asymmetrical distributions in φ ll and ∆E T induced by CP violation leads to nonzero values of the cross-section differences and in the corresponding asymmetries which are conventionally defined as We show the cross-section difference ∆σ φ ll (∆σ ∆E T ) and corresponding asymmetries A φ ll (A ∆E T ) in Fig. 5 (6) for a range ofb values. The cross-section differences may be sizable and could reach about 1 fb for ∆σ φ ll and 0.3 fb for ∆σ ∆E T with |b| ∼ 0.25. The asymmetry in A φ ll is typically larger than that in A ∆E T . We see that the cross-section difference follows a linear-dependence on |b|, while the asymmetries reach their maxima around Re(b) = 0.25 and Im(b) = 0.25 for A φ ll and A ∆E T , respectively. This is an indication that the higher order terms inb become significant, and thus care needs to be taken for larger values of |b| when interpreting the results of the asymmetries.

III. OBSERVABILITY OF THE CP ASYMMETRIES AT THE LHC
The signals we are searching for are the ZH events as in Eq. (3). We specify these events with two jets, with at least one b-tagged, and two opposite-sign leptons of the same flavor, either electron or muon. Contributions to the background mainly come from three sources: These backgrounds are calculated using the CalcHEP package [16,17]. The full spin correlation have been kept for all pp → jjℓ + ℓ − where the full 2 → 4 processes are calculated. In the case of the tt background, pp → tt events are generated and then decayed, ignoring spin correlation. Since this process is subdominant, the error resulting from loss of spin correlation should not be large. The processes pp → jjℓ + ℓ − , where j denotes a jet from a light quark (u, d, s or c) or gluon, yield large rates due to QCD. Demanding at least one tagged b-jet substantially reduces this background.
The tagging and mistagging rates we use are taken from Figure 12.30 of the CMS TDR [18], and are listed here in Table I. The mistagging rates are correlated with the tagging efficiency. We have taken a rather low value for the b-tagging efficiency (0.4), in order to substantially lower mistagging rates and thus to significantly reduce the backgrounds coming from non-b-quark jets. Our signal events should not have much missing energy. We can, therefore, significantly reduce the tt backgrounds by demanding that the mssing energy in these events satisfies / p T < 20 GeV or |/ η| > 2.5.
We envision the search for this asymmetry occurring after the Higgs has already been discovered and its mass is known. We have, therefore, cut the invariant mass of the jets to be near the Higgs mass. Further, the b jets from the Higgs decay tend to be energetic, typically around the Jacobean peak near M H /2. For this reason, we apply a higher transverse momentum cut on the jets. Our final cuts for this process are further suppressing the background. With these cuts, the signal over background can be seen in Figure 7. We see that the background rate is much greater than the signal for smallb. The angular distributions of the background processes (see Eq. (10)) are shown in Fig. 8. Many different final state partons have been separately presented for clarity. The leading background comes from QCD bb ℓ + ℓ − , where at least one jet is b-tagged. The sub-leading background is from qg → qgℓ + ℓ − (labelled as Gj l ), and the next one is due to tt. The full set of cuts given in Eqs. (7), (11) and (12) are applied along with b-tagging of at-least one jet. Each background is added to the previous one so that the top curve gives the total background. The shapes of the background distributions are qualitatively different from that of the signal, while the total rate is much larger than the signal for smallb (see Each background is added to the previous one so that the top curve gives the total background. The full set of cuts given in Eqs. (7), (11) and (12) is applied as well as b-tagging on at least one jet. The bin width is 18 • . Example distributions are shown in the left panel and the right panel of Fig. 9 forb = 0.25 and b = −0.25, respectively. The solid curve in each plot is the SM expectation, and the dashed curve corresponds to the sum of the signal and backgrounds. It is reassuring to see that the asymmety changes sign with a change of the sign ofb, which indicates a dominance of the linear contribution ofb to the asymmetry and thus justifies the leading effective operator approximation. As found earlier, the signal asymmetry occurs not far from φ ll = 0, while the background tends to be larger near φ ll = ±π. This thus motivates us to evaluate the asymmetry A φ ll in a restricted region The resulting asymmetry is shown in Fig. 10, where all backgrounds have been taken into account. We see that the measured asymmetry could reach percentage level. A straightforward Gaussian estimate of the significance is given by where σ T = σ φ ll <0 + σ φ ll >0 is the total cross section and L is the integrated luminosity. Inverting this gives the integrated luminosity required to discover this asymmetry at an Sσ significance The integrated luminosity to measure this asymmetry at 1σ (dotted line), 3σ (dashed line) and 5σ (solid line) level is given in Figure 11. For instance, forb = 0.25, one would need an integrated luminosity of about 60, 520 or 1440 fb −1 to reach 1, 3 or 5σ sensitivity, respectively. The sensitivity can be significantly improved if we consider a χ 2 distribution of the binned data. The "log likelihood" (LL) is defined to be where n i is the number of events observed in bin i where ν i are expected. The LL determines the likelihood that the expected number of events in each bin could fluctuate up to mimic what is observed. The larger the LL, the less likely it is a fluctuation. We plot the LL as a function ofb normalized to units of fb in Fig. 12(a) where we have taken the expectation (ν i ) to be that of the SM (a = 1, b =b = 0). Once a signal observation is indicated, it would be important to address whether one is able to determine the sign ofb, which not only checks the consistency of the leading operator approximation, but would also give a hint for the underlying physics responsible for the CP violating interactions. We plot the LL as a function ofb normalized to units of fb in Fig. 12(b) but take the observation (n i ) to be that for a = 1, b = 0 andb = 0.25 while the expectation (ν i ) is given by the value ofb along the horizontal axis. It is interesting to see that the results for b = ±0.25 are indeed more similar and it may take more data to distinguish the sign ambiguity for a large value ofb. In the limit of large n i , the LL approaches a χ 2 distribution. To determine the value of the LL corresponding with the Gaussian 1, 3 and 5σ deviations, we equate the probabilities of fluctuating up to that point or higher as in where n is the number of bins in our LL construction and z 0 (S) is the LL necessary to give a significance of S. If the data is treated with only one bin (n = 1), then the LL approaches the significance squared S = √ LL. We review the derivation of the LL from a Poisson probability in Appendix A.
In our analysis, we used 14 bins so that a 1, 3 and 5σ deviation corresponds with a LL of Based on these results and matching the LL (fb) in Fig. 12(a), we can invert to obtain the integrated luminosity required to achieve a 1, 3 and 5σ deviation of this observable. We present the results in Fig. 13, for the integrated luminosity required to measure the absolute value ofb on the left panel at 1σ (dotted line), 3σ (dashed line) and 5σ (solid line) level. We find that forb = 0.25, we can measure the absolute value of the asymmetry at 3σ (5σ) with approximately 30 fb −1 (50 fb −1 ). To determine the sign of the asymmetry, we must determine the probability that the asymmetry with one sign could fluctuate to look like the asymmetry with the opposite sign. Following a similar procedure, but using the distribution withb taking the opposite sign for the expectation in each bin (the ν i ), we present the integrated luminosity to establish the sign of the asymmetry at 1, 3 and 5σ level in the right panel of Figure 13. We find that it takes approximately 380 fb −1 (650 fb −1 ) to determine its sign at 3σ (5σ) forb = 0.25.

IV. DISCUSSION
The ZZH interaction in the context of possible CP violation has been extensively studied for e + e − colliders [20]. Due to the neutrality of the initial state and well-constrained kinematics, it is found that an e + e − linear collider could have significant sensitivity to probe the CP-odd coupling of ZZH, especially if beam polarization is achievable. Our work first established the feasibility to test the CP violation effects via similar genuine CP-odd variables particularly suitable for the LHC for a light Higgs boson well below the ZZ threshold.
Assuming the observation of a light Higgs boson at the LHC, we have demonstrated the extent to which a CP-violating interaction in the ZZH vertex could be explored. It has been shown recently that this channel may become one of the Higgs discovery channels due to the improved techniques of studying jet substructure [21] and superstructure [22], and it is thus conceivable that some further enhancement for the signal-to-background ratio may be achieved with more sophisticated kinematical considerations. The LHC reach for discovering the CP-violation in Higgs couplings reported in this paper should be taken as a conservative estimate. Furthermore, our proposal is applicable to any Higgs mass, as long as the production rate is sizable and Higgs decay is identifiable.
Besides the CP studies for a heavy Higgs boson [2], the coupling V V H for a light Higgs was also explored via the weak boson fusion (WBF) production channel [8]. It was found that CP-even and CP-odd operators may lead to qualitatively different angular distributions. However, due to the lack of particle charge identification, no CP-odd observables can be constructed for the WBF processes.
In this article, we have only studied the effect ofb in Eq. (4) beyond the tree-level SM. In principle, there could also be contributions to a and b which would contribute symmetrically to the signal. If this is the case, their fluctuations could mimic small asymmetries requiring a more detailed LL analysis. However, the required luminosity for discovery presented in Fig. 11 would still hold.

V. CONCLUSION
The need for new CP-violating interactions to explain the observed matter-antimatter asymmetry is pressing, and the unexplored Higgs sector may hold the key. If the LHC discovers the Higgs boson(s), a detailed study of the properties of the Higgs will then begin. If the Higgs boson is heavy enough to decay to W + W − , ZZ or tt, then it is quite feasible to construct CP-odd observables to test properties of the interactions. However, if the Higgs boson is light and mainly decays to light fermion pairs, then it would be extremely challenging to test the couplings V V H or Hff .
We studied two genuine CP-odd observables at the LHC to test the CP property of the ZZH interaction for a Higgs boson with mass below the threshold to a pair of gauge bosons via the process pp → ZH → ℓ + ℓ − bb. We showed that including a CP-odd ZZH coupling and after selective kinematical cuts to suppress the SM backgrounds, we are able to extract the CP-asymmetries in the signal events. With a CP violating couplingb ≃ 0.25, a CP asymmetry may be established at a 3σ (5σ) level with an integrated luminosity of 30 (50) fb −1 at the LHC.
The poisson probability of finding n events when ν are expected is f (n, ν) = ν n e −ν n! .
The likelihood for finding n i events where ν i are expected in N bins is given by To get a normalized likelihood, we divide it by the same form where ν i = n i L = The "log likelihood" is then defined as In the limit of a large number of events, the LL approaches a χ 2 distribution χ 2 (z, n) = z n/2−1 e −z/2 2 n/2 Γ(n/2) (A5) where n is the number of bins N . (If we were fitting m parameters, then n would be N − m.) The probability that the expectation value of n would fluctuate up to at least z 0 or more is given by In order to determine the z 0 corresponding to a Gaussian significance of S, we simply set the probabilities of fluctuation equal as in and solve for z 0 (S).