# On polarization parameters of spin-1 particles and anomalous couplings in \(e^+e^-\rightarrow ZZ/Z\gamma \)

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## Abstract

We study the anomalous trilinear gauge couplings of *Z* and \(\gamma \) using a complete set of polarization asymmetries for the *Z* boson in \(e^+e^-\rightarrow ZZ/Z\gamma \) processes with unpolarized initial beams. We use these polarization asymmetries, along with the cross section, to obtain a simultaneous limit on all the anomalous couplings using the Markov Chain Monte Carlo (MCMC) method. For an \(e^+e^-\) collider running at 500 GeV center-of-mass energy and 100 fb\(^{-1}\) of integrated luminosity the simultaneous limits on the anomalous couplings are 1–3\(\times 10^{-3}\).

## 1 Introduction

The Standard Model (SM) of particle physics is a well-established theory and its particle spectrum has been completed with the discovery of the Higgs boson [1] at the Large Hadron Collider (LHC). The predictions of the SM are being confirmed time and again in various colliders with great success, and yet phenomena such as CP-violation, neutrino oscillation, baryogenesis, dark matter, etc., require one to look beyond the SM. Most of the beyond SM (BSM) models need either new particles or new couplings among the SM particles or both. Often this leads to a modified electro-weak sector with modified couplings. To test the SM (or BSM) predictions for the electro-weak symmetry breaking (EWSB) mechanism one needs precise measurements of the strength and tensorial structure of the Higgs (*H*) coupling with all other gauge boson (\(W^\pm \), \(\gamma \), *Z*), Higgs self-couplings, and couplings among the gauge boson themselves.

In this work, we focus on the precise measurement of trilinear gauge boson couplings, in a model independent way, at the proposed International Linear Collider (ILC) [2, 3]. The possible trilinear gauge boson interactions in electro-weak (EW) theory are *WWZ*, \(WW\gamma \), \(ZZ\gamma \), *ZZZ*, \(\gamma \gamma Z\), and \(\gamma \gamma \gamma \), out of which the SM, after EWSB, provides only *WWZ* and \(WW\gamma \) self-couplings. Other interactions among neutral gauge bosons are not possible at the tree level in the SM and hence they are anomalous. Thus any deviation from the SM prediction, either in strength or the tensorial structure, would be a signal of BSM physics. There are two ways to study these anomalous couplings in a model independent way: The first way is to write down an effective vertex using the most general set of tensorial structures for it satisfying Lorentz invariance, \(U(1)_{em}\) invariance, and Bose symmetry weighted by corresponding CP-even/odd form factors [4, 5, 6]. There is, a priori, no relation between various form factors. The other way to study anomalous couplings is to add a set of higher-dimension operators, invariant under (say) \(SU(2)_L\otimes U(1)_Y\) [7], to the SM Lagrangian and then obtain the effective vertices with anomalous couplings after EWSB. This method has been used to study anomalous triple neutral gauge boson couplings with the SM gauge group [8, 9, 10].

*ZZZ*, \(ZZ\gamma \), and \(Z\gamma \gamma \) interactions and have been widely studied in the literature [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] in the context of different collider; \(e^+e^-\) collider [11, 12, 13, 14, 15, 16], hadron collider [17, 18, 19], both \(e^+e^-\) and hadron collider [20, 21, 22], \(e\gamma \) collider [23, 24, 25] and \(\gamma \gamma \) collider [26]. For these effective anomalous vertices one can write an effective Lagrangian and they have been given in [12, 13, 21, 23] up to differences in conventions and parametrizations. The Lagrangian corresponding to the anomalous form factors in the neutral sector in [6] is given by [21]:

On the theoretical side, it is possible to generate some of these anomalous tensorial structures within the framework of a renormalizable theory at higher loop orders, for example, through a fermion triangle diagram in the SM. Loop generated anomalous couplings have been studied for the Minimal Supersymmetric SM (MSSM) [27, 28] and Little Higgs model [29]. Beside this, a non-commutative extension of the SM (NCSM) [30, 31] can also provide an anomalous coupling structure in the neutral sector with a possibility of a trilinear \(\gamma \gamma \gamma \) coupling as well [30].

*ZZ*process the total rate has been used [40], while for the \(Z\gamma \) process both the cross section and the \(p_T\) distribution of \(\gamma \) has been used [41, 42, 43, 44] for obtaining the limits. All these analyses vary one parameter at a time to find the 95 % confidence limits on the form factors. For the \(Z\gamma \) process the limits on the CP-odd form factors, \(h_1^V, \ h_2^V\), are comparable to the limits on the CP-even form factors, \(h_3^V, \ h_4^V\), respectively.

List of tightest limits on anomalous couplings of Eq. (1) available in literature

Coupling | Limits | Remark |
---|---|---|

\(f_4^\gamma \) | \(0.0_{-1.3\times 10^{-2}}^{+1.5\times 10^{-2}}\) | LHC |

\(f_4^Z\) | \(0.0_{-1.1\times 10^{-2}}^{+1.2\times 10^{-2}}\) | 7 TeV |

\(f_5^\gamma \) | \(0.0\pm 1.4 \times 10^{-2}\) | 5 fb\(^{-1}\) |

\(f_5^Z\) | \(0.0\pm 1.2 \times 10^{-2}\) | Ref. [40] |

\(h_3^\gamma \) | \(0.0_{-1.1\times 10^{-3}}^{+0.9\times 10^{-3}}\) | LHC |

\(h_3^Z\) | \(0.0_{-1.5\times 10^{-3}}^{+1.6\times 10^{-3}}\) | 8 TeV |

\(h_4^\gamma \) | \(0.0_{-3.8 \times 10^{-6} }^{+ 4.3 \times 10^{-6}} \) | 19.6 fb\(^{-1}\) |

\(h_4^Z\) | \(0.0_{-3.9 \times 10^{-6} }^{ +4.5 \times 10^{-6} } \) | Ref. [44] |

To put simultaneous limits on all the form factors one would need as many observables as possible, like differential rates, kinematic asymmetries, etc. A set of asymmetries with respect to the initial beam polarizations has been considered [11, 13, 14, 15, 16, 21, 23, 24, 25] at the \(e^+e^-\) and \(e\gamma \) colliders. References [11, 15, 16, 23] also include CP-odd asymmetries made out of the initial beam polarizations alone, which are instrumental in putting strong limits on CP-odd form factors. Additionally, Ref. [11] includes the polarization of produced *Z* as well for forming the asymmetries. The latter does not necessarily require initial beams to be polarized and can be generalized to the hadron colliders as well where such a beam polarization may not be available.

To this end, we discuss angular asymmetries in colliders corresponding to different polarizations of *Z* boson in particular and any spin-1 particle in general. For a spin-*s* particle, the polarization density matrix is a \((2s+1)\times (2s+1)\) hermitian, unit-trace matrix that can be parametrized by \(4s(s+1)\) real parameters. These parameters are different kinds of polarization. For example, a spin-1 / 2 fermion has three polarization parameters called longitudinal, transverse, and normal polarizations (see for example [45, 46]). Similarly, for a spin-1 particle we have a total of eight such parameters. Three of them are vectorial like in the spin-1 / 2 case and the other five are tensorial [46, 49] as will be described in Sect. 2 for completeness. Eight polarization parameters for a massive spin-1 particle have been discussed earlier in the context of anomalous trilinear gauge couplings [47, 48], for the spin measurements studies [46] and to study processes involving \(W^\pm \) bosons [49]. In this work we will investigate all anomalous couplings (up to dimension-6 operators) of Eq. (1) in the processes \(e^+e^-\rightarrow ZZ/Z\gamma \) with the help of the total cross section and the 8-polarization asymmetry of the final state *Z* boson.

The plan of this paper is as follows. In Sect. 2 we discuss the polarization observables of a spin-1 boson in detail using the language of polarization density matrices. Section 3 has a brief discussion of the anomalous Lagrangian and corresponding off-shell vertices and the required on-shell vertices. In Sect. 4 we study the sensitivity of the polarization asymmetries to the anomalous couplings and in Sect. 5 we perform a likelihood mapping of the full coupling space using a Markov Chain Monte Carlo (MCMC) method for two benchmark points along with a likelihood ratio based hypothesis testing to resolve the two benchmark points. We conclude in Sect. 6.

## 2 Polarization observables for spin-1 boson

### 2.1 The production and decay density matrices

*V*is massive and has spin 1; see Fig. 1. The production density matrix for particle

*V*can be written as

*V*with helicity \(\lambda \in \{-1,0,1\}\), while the helicities of all the other particles (\(A, B, C, D, E \ldots \)) are suppressed. The differential cross section for the production of

*V*would be given by

*V*, while the phase space corresponding to all other final state particles is integrated out. The production density matrix can be written in terms of polarization density matrix \(P(\lambda ,\lambda ^\prime )\) as [46]

*f*, in the rest frame of

*V*with its would-be momentum along the

*z*-direction. The parameters \(\alpha \) and \(\delta \) are given by

*V*being on-shell, is given by [46]

*V*including its decay and \(s=1\) being the spin of the particle. Using Eqs. (5) and (6) in Eq. (9), the angular distribution for the fermion

*f*becomes

*f*one can construct several asymmetries to probe various polarization parameters in Eq. (5), which will be discussed in the next section.

### 2.2 Estimation of polarization parameters

*V*. This can be achieved at two levels: At production process level and at the level of decay products. At the production level we first calculate the production density matrix, Eq. (2), using helicity amplitudes of the production process and then calculate the polarization matrix, Eq. (5). The polarization parameters can be extracted from the polarization matrix elements as

While extracting the polarization asymmetries in the collider/event generator we have to make sure that the analysis is done in the rest frame of *V*. The initial beam defines the *z*-axis in the lab, while the production plane of *V* defines the *xz* plane, i.e. \(\phi =0\) plane. While boosting to the rest frame of *V* we keep the *xz* plane invariant. The polar and the azimuthal angles of the decay products of *V* are measured with respect to the *would-be* momentum of *V*.

As a demonstration of the two methods mentioned above, we look at two processes: \(e^+e^-\rightarrow ~ZZ\) and \(e^+e^-\rightarrow ~Z\gamma \). The polarization parameters are constructed both at the production level, using Eq. (11), and at the decay level, using Eqs. (12) and (13). We observe that out of eight polarization asymmetries only three, \(A_x\), \(A_{x^2-y^2}\), and \(A_{zz}\), are non-zero in the SM. The asymmetries \(A_{x^2-y^2}\) and \(A_{zz}\) are calculated analytically for the production part and shown as a function of beam energy in Fig. 2 with solid lines. For the same processes with \(Z\rightarrow f\bar{f}\) decay, we generate events using MadGraph5 [50] with different values of beam energies. The polarization asymmetries were constructed from these events and are shown in Fig. 2 with points. The statistical error bars shown correspond to \(10^4\) events. We observe agreement between the asymmetries calculated at the production level (analytically) and the decay level (using event generator). Any possible new physics in the production process of *Z* boson is expected to change the cross section, kinematical distributions and the values of the polarization parameters/asymmetries. We intend to use these asymmetries to probe the standard and BSM physics.

## 3 Anomalous Lagrangian and their probes

*ZZZ*through \(f^Z_{4,5}\) couplings, \(\gamma ZZ\) through \(f^\gamma _{4,5}\) and \(h^Z_{1,3}\) couplings and \(\gamma \gamma Z\) through \(h^\gamma _{1,3}\) couplings. There is no \(\gamma \gamma \gamma \) vertex in the above Lagrangian.

## 4 Asymmetries, limits, and sensitivity to anomalous couplings

In this section we thoroughly investigate the effects of anomalous couplings in the processes \(e^+e^-\rightarrow ZZ\) and \(e^+e^-\rightarrow Z\gamma \). We use tree level SM interactions alongwith anomalous couplings shown in Eq. (14) for our analysis. The Feynman diagrams for these processes are given in the Fig. 4 where the anomalous vertices are shown as big blobs. The helicity amplitudes for the anomalous part together with SM contributions for both these processes are given in Appendix A. These are then used to calculate the polarization observables and the cross section which are given in Appendix B.

### 4.1 Parametric dependence of asymmetries on anomalous couplings

*ZZ*and \(Z\gamma \) processes are given in Tables 2 and 3, respectively.

Dependence of polarization observables on the anomalous couplings in *ZZ* final state

Observables | Linear terms | Quadratic terms |
---|---|---|

\( \sigma \) | \(f_5^Z,f_5^\gamma \) | \((f_4^\gamma )^2, (f_5^\gamma )^2, (f_4^Z)^2, (f_5^Z)^2, f_4^\gamma f_4^Z, f_5^\gamma f_5^Z \) |

\(\sigma \times A_x\) | \( f_5^\gamma ,f_5^Z \) | − |

\(\sigma \times A_y \) | \(f_4^\gamma ,f_4^Z \) | − |

\(\sigma \times A_{xy} \) | \(f_4^Z,f_4^\gamma \) | \(f_4^Zf_5^\gamma ,f_4^\gamma f_5^Z,f_4^\gamma f_5^\gamma ,f_4^Zf_5^Z \) |

\(\sigma \times A_{x^2-y^2}\) | \(f_5^Z,f_5^\gamma \) | \((f_4^\gamma )^2, (f_5^\gamma )^2, (f_4^Z)^2, (f_5^Z)^2, f_4^\gamma f_4^Z, f_5^\gamma f_5^Z \) |

\(\sigma \times A_{zz} \) | \(f_5^Z,f_5^\gamma \) | \((f_4^\gamma )^2, (f_5^\gamma )^2, (f_4^Z)^2, (f_5^Z)^2, f_4^\gamma f_4^Z, f_5^\gamma f_5^Z \) |

Dependence of the polarization observables on the anomalous couplings in \(Z\gamma \) final state

Observables | Linear terms | Quadratic terms |
---|---|---|

\( \sigma \) | \(h_3^Z,h_3^\gamma \) | \((h_1^\gamma )^2,(h_3^\gamma )^2,(h_1^Z)^2,(h_3^Z)^2,h_1^\gamma h_1^Z,h_3^\gamma h_3^Z \) |

\(\sigma \times A_x\) | \(h_3^Z,h_3^\gamma \) | \((h_1^\gamma )^2,(h_3^\gamma )^2,(h_1^Z)^2,(h_3^Z)^2,h_1^\gamma h_1^Z,h_3^\gamma h_3^Z \) |

\(\sigma \times A_y \) | \(h_1^\gamma ,h_1^Z \) | − |

\(\sigma \times A_{xy} \) | \(h_1^\gamma ,h_1^Z \) | − |

\(\sigma \times A_{x^2-y^2}\) | \(h_3^\gamma ,h_3^Z \) | \(- \) |

\(\sigma \times A_{zz} \) | \(h_3^Z,h_3^\gamma \) | \((h_1^\gamma )^2,(h_3^\gamma )^2,(h_1^Z)^2,(h_3^Z)^2,h_1^\gamma h_1^Z,h_3^\gamma h_3^Z \) |

In the SM, the helicity amplitudes are real, thus the production density matrix elements in Eq. (2) are all real. This implies \(A_y\), \(A_{xy}\), and \(A_{yz}\) are all zero in the SM; see Eq. (11). The asymmetries \(A_z\) and \(A_{xz}\) are also zero for the SM couplings due to the forward–backward symmetry of the *Z* boson in the c.m. frame, owing to the presence of both *t*- and *u*-channel diagrams and unpolarized initial beams. After including anomalous couplings, \(A_{y}\) and \(A_{xy}\) receive a non-zero contribution, while \(A_z\), \(A_{xz}\), and \(A_{yz}\) remain zero for the unpolarized initial beams.

From the list of non-vanishing asymmetries, only \(A_y\) and \(A_{xy}\) are CP-odd, while the others are CP-even. All the CP-odd observables are linearly dependent upon the CP-odd couplings, like \(f_4^V\) and \(h_1^V\), while all the CP-even observables have only quadratic dependence on the CP-odd couplings. In the SM, the *Z* boson’s couplings respect CP symmetry; thus \(A_y\) and \(A_{xy}\) vanish. Hence, any significant deviation of \(A_{y}\) and \(A_{xy}\) from zero at the collider will indicate a clear sign of CP-violating new physics interactions. Observables that have only a linear dependence on the anomalous couplings yield a *single interval limits* on these couplings. On the other hand, any quadratic appearance (like \((f_5^V)^2\) in \(\sigma \)) may yield more than one interval of the couplings, while putting limits. For the case of the *ZZ* process, \(A_x\) and \(A_y\) do not have any quadratic dependence, hence they yield the cleanest limits on the CP-even and -odd parameters, respectively. Similarly, for the \(Z \gamma \) process we have \(A_y\), \(A_{xy}\), and \(A_{x^2-y^2}\), which have only a linear dependence and provide clean limits. These clean limits, however, may not be the strongest limits as we will see in the following sections.

### 4.2 Sensitivity and limits on anomalous couplings

*L*being the integrated luminosity of the collider. The error in the cross section \(\sigma \) will be given by

*A*and \(\sigma \), respectively, while the remaining ones are statistical errors.

For numerical calculations, we choose ILC running at c.m. energy \(\sqrt{\hat{s}}=500\) GeV and integrated luminosity \(L=100\) fb\(^{-1}\). We use \(\epsilon _A=\epsilon =0\) for most of our analysis, however, we do discuss the impact of systematic errors on our results. With this choice the sensitivity of all the polarization asymmetries, Eq. (13), and the cross section have been calculated varying one parameter at a time. These sensitivities are shown in Figs. 5 and 6 for the *ZZ* and \(Z\gamma \) processes, respectively, for each observable. In the \(e^+e^-\rightarrow Z\gamma \) process we have taken a cut-off on the polar angle, \(10^\circ \le \theta _\gamma \le 170^\circ \) to keep away from the beam pipe. For these limits the analytical expressions shown in Appendix B are used.

We see that in the *ZZ* process the tightest constraint on \(f_4^\gamma \) at \(1\sigma \) level comes from the asymmetry \(A_{y}\) owing to its linear and strong dependence on the coupling. For \(f_5^\gamma \), both \(A_x\) and the cross section \(\sigma _{ZZ}\), give comparable limits at \(1\sigma \) but \(\sigma _{ZZ}\) gives a tighter limit at higher values of sensitivity. This is because the quadratic term in \(\sigma _{ZZ}\) comes with higher power of energy/momenta and hence a larger sensitivity. Similarly, the strongest limit on \(f_4^Z\) and \(f_5^Z\) as well comes from \(\sigma _{ZZ}\). Though the cross section gives the tightest constrain on most of the coupling in *ZZ* process, our polarization asymmetries also provide comparable limits. Another noticeable fact is that \(\sigma _{ZZ}\) has a linear as well as quadratic dependence on \(f_5^Z\) and the sensitivity curve is symmetric about a point larger than zero. Thus, when we do a parameter estimation exercise, we will always have a bias toward a positive value of \(f_5^Z\). The same is the case with the coupling \(f_5^\gamma \), but the strength of the linear term is small and the sensitivity plot with \(\sigma _{ZZ}\) looks almost symmetric about \(f_5^\gamma =0\).

In the \(Z\gamma \) process, the tightest constraint on \(h_1^\gamma \) comes from \(A_{xy}\), on \(h_3^\gamma \) comes from \(\sigma _{Z \gamma }\), on \(h_1^Z\) it comes from \(A_y\), and on \(h_3^Z\) it comes from \(A_x\). The cross section \(\sigma _{Z \gamma }\) and \(A_{zz}\) has a linear as well as quadratic dependence on \(h_3^\gamma \), and \(\sigma _{Z \gamma }\) gives two intervals at \(1\sigma \) level. Other observables can help resolve the degeneracy when we use more than one observables at a time. Still, the cross section prefers a negative value of \(h_3^\gamma \) and it will be seen again in a multi-variate analysis. The coupling \(h_3^Z\) also has quadratic appearance in the cross section and yields a bias toward negative values of \(h_3^Z\).

The tightest limits on the anomalous couplings (at \(1\sigma \) level), obtained using one observable at a time and varying one coupling at a time, are listed in Table 4 along with the corresponding observables. A comparison between Tables 1 and 4 shows that an \(e^+e^-\) collider running at 500 GeV and 100 fb\(^{-1}\) provides better limits on the anomalous coupling (\(f_i^V\)) in the *ZZ* process than the 7 TeV LHC at 5 fb\(^{-1}\). For the \(Z\gamma \) process the experimental limits are available from 8 TeV LHC with 19.6 fb\(^{-1}\) luminosity (Table 1) and they are comparable to the single observable limits shown in Table 4. These limits can be further improved if we use all the observables in a \(\chi ^2\) kind of analysis.

We can further see that the sensitivity curves for CP-odd observables, \(A_y\) and \(A_{xy}\), has no or a very mild dependence on the CP-even couplings. The mild dependence comes through the cross section \(\sigma \), sitting in the denominator of the asymmetries. We see that CP-even observables provide a tight constraint on the CP-even couplings and the CP-odd observables provide a tight constraint on the CP-odd couplings. Thus, not only can we study the two processes independently, it is possible to study the CP-even and CP-odd couplings almost independent of each other. To this end, we shall perform a two-parameter sensitivity analysis in the next subsection.

List of tightest limits on anomalous couplings at \(1~\sigma \) level and the corresponding observable obtained for \(\sqrt{\hat{s}}=500\) GeV and \(L=100\) fb\(^{-1}\)

| \(Z \gamma \) process | ||||
---|---|---|---|---|---|

Coupling | Limits | Comes from | Coupling | Limits | Comes from |

\(|f_4^\gamma |\) | \( \le \) \(2.4 \times 10^{-3}\) | \(A_y\) | \(|h_1^\gamma |\) | \(\le \) \(3.6\times 10^{-3}\) | \(A_{xy}\), \(\sigma \) |

\(|f_4^Z|\) | \(\le \) \(4.2\times 10^{-3}\) | \(\sigma \) | \(|h_1^Z|\) | \(\le \) \(2.9\times 10^{-3}\) | \(A_y\) |

\(f_5^\gamma \) | \(\in \) \([-2.3, 2.7] \times 10^{-3}\) | \(A_x\), \(\sigma \) | \(h_3^\gamma \) | \(\in \) \([-2.1, 1.3] \times 10^{-3} \) or \(\in \) \([-9.9, -6.5] \times 10^{-3} \) | \(\sigma \) |

\(f_5^Z\) | \(\in \) \([-2.3, 8.8] \times 10^{-3}\) | \(\sigma \) | \(|h_3^Z|\) | \(\le \) \(2.8\times 10^{-3}\) | \(A_x\) |

### 4.3 Two-parameter sensitivity analysis

We vary two couplings at a time, for each observable, and plot the \(\mathcal{S}=1\) (or \(\Delta \chi ^2=1\)) contours in the corresponding parameter plane. These contours are shown in Fig. 7 and Fig. 8 for *ZZ* and \(Z\gamma \) processes, respectively. Asterisk (\(\star \)) marks in the middle of these plots denote the SM value, i.e., the (0, 0) point. Each panel corresponds to two couplings that are varied and all others are kept at zero. The shapes of the contours, for a given observable, are a reflection of its dependence on the couplings as shown in Tables 2 and 3. For example, let us look at the middle-top panel of Fig. 7, i.e. the \((f_4^\gamma -f_5^\gamma )\) plane. The contours corresponding to the cross section (solid/red) and \(A_{zz}\) (short-dash-dotted/orange) are circular in shape due to their quadratic dependence on these two couplings with the same sign. The small linear dependence on \(f_5^\gamma \) makes these circles move toward a small positive value, as already observed in the one-parameter analysis above. The \(A_y\) contour (short-dash/blue) depends only on \(f_4^\gamma \) in the numerator and a mild dependence on \(f_5^\gamma \) enters through the cross section, sitting in the denominator of the asymmetries. The role of two couplings are exchanged for the \(A_x\) contour (big-dash/black). The \(A_{xy}\) contour (dotted/magenta) is hyperbolic in shape, indicating a dependence on the product \(f_4^\gamma f_5^\gamma \), while a small shift toward a positive \(f_5^\gamma \) value indicates a linear dependence on it. Similarly the symmetry about \(f_4^\gamma =0\) indicates no linear dependence on it for \(A_{xy}\). All these observations can be confirmed by looking at Table 2 and the expressions in Appendix B. Finally, the shape of the \(A_{x^2-y^2}\) contour (big-dash-dotted/cyan) indicates a quadratic dependence on two couplings with opposite sign. Similarly, all other panels can be read. Note that taking any one of the couplings to zero in these panels gives us the \(1\sigma \) limit on the other coupling as found in the one-parameter analysis above.

In the contours for the \(Z \gamma \) process, Fig. 8, one new kind of shape appears: the annular ring corresponding to \(\sigma _{Z\gamma }\) in middle-top, left-bottom, and right-bottom panels. This shape corresponds to a large linear dependence of the cross section on \(h_3^\gamma \) along with the quadratic dependence. By putting the other couplings to zero in the above-mentioned panels one obtains two disjoint intervals for \(h_3^\gamma \) at \(1\sigma \) level as found before in the one-parameter analysis. The plane containing two CP-odd couplings, i.e. the left-top panel, has two sets of slanted contours corresponding to \(A_{y}\) (short-dash/blue) and \(A_{xy}\) (dotted/magenta), the CP-odd observables. These observables depend upon both the couplings linearly and hence the slanted (almost) parallel lines. The rest of the panels can be read in the same way.

Till here we have used only one observable at a time for finding the limits. A combination of all the observables would provide a much tighter limit on the couplings than provided by any one of them alone. Also, the shape, the position, and the orientation of the allowed region would change if the other two parameters were set to some value other than zero. A more comprehensive analysis requires varying all the parameters and using all the observables to find the parameter region of low \(\chi ^2\) or high likelihood. The likelihood mapping of the parameter space is performed using the MCMC method in the next section.

## 5 Likelihood mapping of parameter space

*pseudo data*generated by MadGraph5. We choose two benchmark points for coupling parameters as follows:

*pseudo data*corresponding to ILC running at 500 GeV and integrated luminosity of \(L=100\) fb\(^{-1}\). The likelihood of a given point \(\mathbf {f}\) in the parameter space is defined by

List of observables shown for the process \(e^+e^-\rightarrow ZZ\) for the benchmark point SM with \(\sqrt{\hat{s}}=500\) GeV: theoretical values (column 2), MadGraph5 simulated value for \(L=100\) fb\(^{-1}\) (column 3), 68 % (column 4) and 95 % (column 5) Bayesian confidence intervals (BCI)

Observables | Theoretical (SM) | MadGraph (SM, prior) | 68 % BCI (posterior) | 95 % BCI (posterior) |
---|---|---|---|---|

\(\sigma \) | 38.096 fb | \( 38.16\pm 0.62 \) fb | \(38.61^{+0.31}_{-0.53}\)fb | \(38.61^{0.83}_{-0.74}\)fb |

\(A_x\) | 0.00099 | \(0.0023\pm 0.0161\) | \(-0.0021\pm 0.0087 \) | \(-0.0021^{+0.016}_{-0.017} \) |

\(A_y\) | 0 | \(-0.0016\pm 0.0161 \) | \(-0.0005\pm 0.0090 \) | \(-0.0005^{+0.017}_{-0.017} \) |

\(A_{xy}\) | 0 | \(0.0004\pm 0.0161\) | \(0.0001\pm 0.0036 \) | \(0.0001^{+0.0071}_{-0.0071}\) |

\(A_{x^2-y^2}\) | \(-0.02005 \) | \(-0.0189\pm 0.0161\) | \(-0.0166^{+0.0032}_{-0.0018}\) | \(-0.0166^{+0.0043}_{-0.0052}\) |

\(A_{zz}\) | 0.17262 | \(0.1745\pm 0.0159 \) | \(0.1691^{+0.0035}_{-0.0022}\) | \(0.1691^{+0.0051}_{-0.0056}\) |

List of observables shown for the process \(e^+e^-\rightarrow ZZ\) for the benchmark point aTGC with \(\sqrt{\hat{s}}=500\) GeV. The rest of the details are the same as in Table 5

Observables | Theoretical (aTGC) | MadGraph (aTGC, prior) | 68 % BCI (posterior) | 95 % BCI (posterior) |
---|---|---|---|---|

\(\sigma \) | 43.307 fb | \(43.33\pm 0.6582 \) fb | \(43.40\pm 0.66 \) fb | \(43.40 \pm 1.3\) fb |

\(A_x\) | \(-0.02954 \) | \(-0.0308\pm 0.0151\) | \(-0.0240^{+0.0087}_{-0.013}\) | \(-0.0240^{+0.021}_{-0.020} \) |

\(A_y\) | 0.03424 | \(0.0308\pm 0.0151\) | \(0.0230^{+0.013}_{-0.0085} \) | \(0.0230^{+0.020}_{-0.022} \) |

\(A_{xy}\) | 0.00574 | \(0.0056\pm 0.0152\) | \(0.0041^{+0.0076}_{-0.0063}\) | \(0.0041^{+0.015}_{-0.015} \) |

\(A_{x^2-y^2}\) | \(-0.00941 \) | \(-0.0119 \pm 0.0152\) | \(-0.0116^{+0.0071}_{-0.0032}\) | \(-0.0116^{+0.0093}_{-0.012}\) |

\(A_{zz}\) | 0.14057 | \(0.1382 \pm 0.0150\) | \(0.1401\pm 0.0035 \) | \(0.1401^{+0.0069}_{-0.0067}\) |

### 5.1 MCMC analysis for \(e^+e^-\rightarrow ZZ\)

*pseudo data*we perform the likelihood mapping of the parameter space and obtain the posterior distributions for the parameters and the observables. The last two columns of Tables 5 and 6 show the 68 and 95 % Bayesian confidence interval (BCI) of the observables used. One naively expects 68 % BCI to roughly have the same size as the \(1\sigma \) error in the

*pseudo data*. However, we note that the 68 % BCI for all the asymmetries is much narrower than expected, for both benchmark points. This can be understood from the fact that the cross section provides the strongest limit on any parameter, as noticed in the earlier section, thus limiting the range of values for the asymmetries. However, this must allow 68 % BCI of the cross section to match with the expectation. This indeed happens for the aTGC case (Table 6), but for the SM case even the cross section is narrowly constrained compared to a naive expectation. The reason for this can be found in the dependence of the cross section on the parameters. For most of the parameter space, the cross section is larger than the SM prediction and only for a small range of parameter space it can be smaller. This was already pointed out while discussing multi-valued sensitivity in Fig. 5. We found the lowest possible value of the cross section to be 37.77 fb, obtained for \(f_4^{\gamma ,Z}\approx 0\), \(f_5^\gamma \sim 2\times 10^{-4}\), and \(f_5^Z \sim 3.2\times 10^{-3}\). Thus, for most of the parameter space the anomalous couplings cannot emulate the negative statistical fluctuations in the cross section making the likelihood function, effectively, a one-sided Gaussian function. This forces the mean of the posterior distribution to have a higher value. We also note that the upper bound of the 68 % BCI for cross section (38.92 fb) is comparable to the expected \(1\sigma \) upper bound (38.78 fb). Thus we have an overall narrowing of the range of the posterior distribution of the cross section values. This, in turns, leads to a narrow range of parameters allowed and hence narrow ranges for the asymmetries in the case of SM benchmark point. For the aTGC benchmark point, it is possible to emulate the negative fluctuations in the cross section by varying the parameters, thus the corresponding posterior distributions compare with the expected \(1\sigma \) fluctuations. The narrow ranges for the posterior distribution for all the asymmetries are due to the tighter constraints on the parameters coming from the cross section and correlation between the observables.

The list of best-fit points, posterior 68 and 95 % BCI for the parameters for the process \(e^+e^-\rightarrow ZZ\) for both SM and aTGC benchmark points

\(f^V_i\) | SM benchmark | aTGC benchmark | ||||
---|---|---|---|---|---|---|

68 % BCI | 95 % BCI | Best-fit | 68 % BCI | 95 % BCI | Best-fit | |

\(f_4^\gamma \) | \(-0.0001\pm 0.0014 \) | \(-0.0001^{+0.0027}_{-0.0027}\) | \(-0.0002\) | \(0.0038^{+0.0026}_{-0.0016}\) | \(0.0038^{+0.0037}_{-0.0042}\) | 0.0044 |

\(f_4^Z\) | \(0.0000\pm 0.0026 \) | \(0.0000^{+0.0049}_{-0.0049}\) | \(-0.0002\) | \(0.0010^{+0.0065}_{-0.0055}\) | \(0.0010^{+0.0098}_{-0.011} \) | 0.0050 |

\(f_5^\gamma \) | \(-0.0001\pm 0.0015 \) | \(-0.0001^{+0.0030}_{-0.0029}\) | \(-0.0002\) | \(0.0038^{+0.0029}_{-0.0019}\) | \(0.0038^{+0.0042}_{-0.0047}\) | 0.0057 |

\(f_5^Z\) | \(0.0032\pm 0.0028 \) | \(0.0032^{+0.0053}_{-0.0053}\) | 0.0000 | \(0.0057^{+0.0074}_{-0.0051}\) | \(0.0057^{+0.010}_{-0.011} \) | 0.0037 |

We are using a total of six observables, five asymmetries and one cross section, for our analysis of two benchmark points; however, we have only four free parameters. This invariably leads to some correlations between the observables apart from the expected correlations between parameters and observables. Figure 9 shows most prominently correlated observable for each of the parameters. The CP nature of observables is reflected in the parameter it is strongly correlated with. We see that \(A_y\) and \(A_{xy}\) are linearly dependent upon both \(f_4^\gamma \) and \(f_4^Z\); however, \(A_y\) is more sensitive to \(f_4^\gamma \) as shown in Fig. 5 as well. Similarly, for the other asymmetries and parameters one can see a correlation which is consistent with the sensitivity plots in Fig. 5. The strong (and negative) correlation between \(A_{zz}\) and \(\sigma \) shown in Fig. 10 indicates that any one of them is sufficient for the analysis, in principle. However, in practice the cross section puts a much stronger limit than \(A_{zz}\), which explains the much narrower BCI for it as compared to the \(1\sigma \) expectation.

*ZZ*production process are shown in Fig. 11, while the corresponding BCI along with best-fit points are listed in Table 7 for both benchmark points. The vertical lines near zero correspond to the true value of parameters for SM and the other vertical line corresponds to aTGC. The best-fit points are very close to the true values except for \(f_5^Z\) in the aTGC benchmark point due to the multi-valuedness of the cross section. The 95 % BCI of the parameters for two benchmark points overlap and it appears as if they cannot be resolved. To see the resolution better we plot two-dimensional posteriors in Fig. 12, with the benchmark points shown with an asterisk. Again we see that the 95 % contours do overlap as these contours are obtained after marginalizing over non-shown parameters in each panel. Any higher-dimensional representation is not possible on paper, but we have checked three-dimensional scatter plot of points on the Markov chains and conclude that the shape of the

*good likelihood*region is ellipsoidal for the SM point with the true value at its center. The corresponding three-dimensional shape for the aTGC point is like a part of an ellipsoidal shell. Thus in full four dimensions there will not be any overlap (see Sect. 5.3) and we can distinguish the two chosen benchmark points as is quite obvious from the corresponding cross sections. However, if we are left with only the cross section we would have the entire ellipsoidal shell as possible range of parameters for the aTGC case. The presence of asymmetries in our analysis helps narrow down to a part of the ellipsoid and hence aids the parameter estimation for the

*ZZ*production process.

List of observables shown for the process \(e^+e^-\rightarrow Z \gamma \) for the SM point with \(\sqrt{\hat{s}}=500\) GeV and \(L=100\) fb\(^{-1}\). The rest of the details are the same as in Table 5

Observables | Theoretical (SM) | MadGraph (SM, prior) | 68 % BCI (posterior) | 95 % BCI (posterior) |
---|---|---|---|---|

\(\sigma \) | 112.40 fb | \(112.6\pm 1.06 \) fb | \(112.64^{+0.64}_{-0.91}\) fb | \(112.6^{+1.5}_{-1.4}\) pb |

\(A_x\) | 0.00480 | \(0.0043\pm 0.0094 \) | \(0.0041\pm 0.0088 \) | \(0.0041^{+0.017}_{-0.018} \) |

\(A_y\) | 0 | \(-0.0011\pm 0.0094\) | \(-0.0009\pm 0.0088 \) | \(-0.0009 \pm 0.017 \) |

\(A_{xy}\) | 0 | \(0.0003\pm 0.0094\) | \(0.0001\pm 0.0065 \) | \(0.0001 \pm 0.012 \) |

\(A_{x^2-y^2}\) | 0.00527 | \(0.0056\pm 0.0094\) | \(-0.0001^{+0.0064}_{-0.0034}\) | \(-0.0001^{+0.0079}_{-0.0096}\) |

\(A_{zz}\) | 0.17819 | \(0.1781\pm 0.0092\) | \(0.1771^{+0.0043}_{-0.0031}\) | \(0.1771^{+0.0066}_{-0.0070}\) |

List of observables shown for the process \(e^+e^-\rightarrow Z\gamma \) for the aTGC point with \(\sqrt{\hat{s}}=500\) GeV and \(L=100\) fb\(^{-1}\). The rest of the details are the same as in Table 5

Observables | Theoretical (aTGC) | MadGraph (aTGC, prior) | 68 % BCI (posterior) | 95% BCI (posterior) |
---|---|---|---|---|

\(\sigma \) | 122.0 fb | \(122.4 \pm 1.11\) fb | \(122.3\pm 1.0 \) fb | \(122.3 \pm 2.0\) fb |

\(A_x\) | 0.02404 | \(0.0252\pm 0.0090\) | \(0.0263\pm 0.0093 \) | \(0.0263 \pm 0.018 \) |

\(A_y\) | \(-0.01775 \) | \(-0.0165\pm 0.0090\) | \(-0.0172\pm 0.0092 \) | \(-0.0172 \pm 0.018 \) |

\(A_{xy}\) | \( -0.01350\) | \(-0.0104\pm 0.0090\) | \(-0.0109^{+0.0069}_{-0.011}\) | \(-0.0109^{+0.017}_{-0.015} \) |

\(A_{x^2-y^2}\) | 0.01440 | \(0.0133\pm 0.0090\) | \(0.0121^{+0.0055}_{-0.0010}\) | \(0.0121^{+0.0068}_{-0.012} \) |

\(A_{zz}\) | 0.13612 | \(0.1361\pm 0.0089\) | \(0.1351\pm 0.0041 \) | \(0.1351^{+0.0080}_{-0.0079}\) |

The list of best-fit points, posterior 68 % and 95 % BCI for the parameters for the process \(e^+e^-\rightarrow Z\gamma \) for both benchmark points

\(h^V_i\) | SM benchmark | aTGC benchmark | ||||
---|---|---|---|---|---|---|

68 % BCI | 95 % BCI | Best-fit | 68 % BCI | 95 % BCI | Best-fit | |

\(h_1^\gamma \) | \(-0.0001\pm 0.0026 \) | \(-0.0001^{+0.0048}_{-0.0047}\) | \( -0.0002\) | \(0.0039^{+0.0047}_{-0.0031}\) | \(0.0039^{+0.0068}_{-0.0075}\) | 0.0040 |

\(h_1^Z\) | \(0.0003\pm 0.0028 \) | \(0.0003^{+0.0054}_{-0.0054}\) | 0.0001 | \(0.0050\pm 0.0033 \) | \(0.0050^{+0.0064}_{-0.0063}\) | 0.0047 |

\(h_3^\gamma \) | \(-0.0030^{+0.0036}_{-0.0020}\) | \(-0.0030^{+0.0045}_{-0.0054}\) | 0.0002 | \(0.00348^{+0.0036}_{-0.00086}\) | \(0.00348^{+0.0047}_{-0.0076}\) | 0.0056 |

\(h_3^Z\) | \(0.0004\pm 0.0028 \) | \(0.0004^{+0.0053}_{-0.0055}\) | \(-0.0002 \) | \(0.0062^{+0.0030}_{-0.0035}\) | \(0.0062^{+0.0070}_{-0.0062}\) | 0.0052 |

### 5.2 MCMC analysis for \(e^+e^-\rightarrow Z\gamma \)

*psuedo data*from MadGraph5. This is again related to the correlations between observables and the fact that the cross section has a lower bound of about 111 fb obtained for \(h_3^\gamma \sim -4.2\times 10^{-3}\) with the other parameters close to zero. This lower bound of the cross section leads to narrowing of 68 % BCI for \(\sigma \) and hence for other asymmetries too, as observed in the

*ZZ*production process. The 68 % BCI for \(A_{x^2-y^2}\) and \(A_{zz}\) are particularly narrow. For \(A_{zz}\), this is related to the strong correlation between \((\sigma - A_{zz})\), while for \(A_{x^2-y^2}\) the slower dependence on \(h_3^\gamma \) along with strong dependence of \(\sigma \) on \(h_3^\gamma \) is the cause of a narrow 68 % BCI.

For the parameter extraction we look at their one-dimensional marginalized posterior distribution function shown in Fig. 14 for the two benchmark points. The best-fit points along with 68 and 95 % BCI are listed in Table 10. The best-fit points are very close to the true values of the parameters and so are the means of the BCI for all parameters except \(h_3^\gamma \). For it there is a downward movement in the value owing to the multi-valuedness of the cross section. Also, we note that the 95 % BCI for the two benchmark points largely overlap, making them seemingly un-distinguishable at the level of one-dimensional BCIs. To highlight the difference between two benchmark points, we look at two-dimensional BC contours as shown in Fig. 15. The 68 % BC contours (dark shades) can be roughly compared with the contours of Fig. 8. The difference is that Fig. 15 has all four parameters varying and all six observables are used simultaneously. The 95 % BC contours for the two benchmark points overlap despite the fact that the cross section can distinguish them very clearly. In full four-dimensional parameter space the two contours do not overlap and in the next section we try to establish this.

### 5.3 Separability of benchmark points

*t*, as

*pseudo data*, \(\mathcal{L}(\mathbf {f}(t)|\text{ SM })\), in solid/green line and assuming the aTGC

*pseudo data*, \(\mathcal{L}(\mathbf {f}(t)|\text{ aTGC })\), in dashed/blue line. The left panel is for the

*ZZ*production process and the right panel is for the \(Z\gamma \) process. The horizontal line corresponds to the normalized likelihood being \(e^{-\frac{1}{2}}\), while the full vertical lines correspond to the maximum value, which is normalized to 1. It is clearly visible that the two benchmark points are quite well separated in terms of the likelihood ratios. We have \(\mathcal{L}(\mathbf {f}_\mathtt{aTGC}|\text{ SM })\sim 8.8 \times 10^{-19}\) for the

*ZZ*process, and it means that the relative likelihood for the SM

*pseudo data*being generated by the aTGC parameter value is \(8.8 \times 10^{-19}\), i.e. negligibly small. Comparing the likelihood ratio to \(e^{-n^2/2}\) we can say that the data is \(n\sigma \) away from the model point. In this case, SM

*pseudo data*is \(9.1\sigma \) away from the aTGC point for the

*ZZ*process. Similarly we have \(\mathcal{L}(\mathbf {f}_\mathtt{SM}|\text{ aTGC })\sim 1.7 \times 10^{-17}\), i.e. the aTGC

*pseudo data*is \(8.8\sigma \) away from the SM point for the

*ZZ*process. For the \(Z\gamma \) process we have \(\mathcal{L}(\mathbf {f}_\mathtt{aTGC}|\text{ SM })\sim 1.7 \times 10^{-24} (10.5\sigma )\) and \(\mathcal{L}(\mathbf {f}_\mathtt{SM}|\text{ aTGC })\sim 1.8 \times 10^{-25} (10.7\sigma )\). In all cases the two benchmark points are well separable as clearly seen in Fig. 16.

## 6 Conclusions

There are angular asymmetries in collider that can be constructed to probe all eight polarization parameters of a massive spin-1 particle. Three of them, \(A_y\), \(A_{xy}\), and \(A_{yz}\), are CP-odd and can be used to measure CP-violation in the production process. On the other hand \(A_z\), \(A_{xz}\), and \(A_{yz}\) are P-odd observables, while \(A_x\), \(A_{x^2-y^2}\), and \(A_{zz}\) are CP- and P-even. The anomalous trilinear gauge coupling in the neutral sector, Eq. (14), is studied using these asymmetries along with the cross section. The one- and two-parameter sensitivity of these asymmetries, together with the cross section, are explored and the one-parameter limit using one observable is listed in Table 4 for an unpolarized \(e^+e^-\) collider. For finding the best and simultaneous limit on anomalous couplings, we have performed a likelihood mapping using the MCMC method and the obtained limits are listed in Tables 7 and 10 for *ZZ* and \(Z\gamma \) processes, respectively. For the *ZZ* process, the ILC (\(\sqrt{\hat{s}}=500\) GeV, \(L=100\) fb\(^{-1}\)) limits are tighter than the available LHC (\(\sqrt{\hat{s}}=7\) TeV, \(L=5\) fb\(^{-1}\)) limits [40], while the ILC limits on the \(Z\gamma \) anomalous couplings are slightly weaker than the available LHC (\(\sqrt{\hat{s}}=8\) TeV, \(L=19.6\) fb\(^{-1}\)) limits [44]. The LHC probes the interactions at large energies and transverse momentum, where the sensitivity to the anomalous couplings is enhanced. We perform our analysis at \(\sqrt{\hat{s}}=500\) GeV, leading to a weaker, though comparable, limits on the \(h_i^V\) in the \(Z\gamma \) process.

With polarized initial beams, the P-odd observables, \(A_z\), \(A_{xz}\), and \(A_{yz}\), will be non-vanishing for both processes with appropriate kinematical cuts. This gives us three more observables to add in the likelihood analysis, which can lead to better limits. At LHC, one does not have the possibility of initial beam polarization, however, \(W^+W^-\) and \(ZW^\pm \) processes effectively have initial beam polarization due to chiral couplings of \(W^\pm \). The study of *W* and *Z* processes at LHC and polarized \(e^+e^-\) colliders is under way and will be presented elsewhere.

## Notes

### Acknowledgments

R. R. thanks Department of Science and Technology, Government of India for support through DST-INSPIRE Fellowship for doctoral program, INSPIRE CODE IF140075, 2014.

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