1 Introduction↩︎

Neutral triple gauge couplings (nTGCs) offer a unique window to new physics beyond the Standard Model (BSM): they are absent in the Standard Model (SM) and cannot be generated by dimension-6 interactions in the Standard Model Effective Field Theory (SMEFT) [1], but provide direct access to possible dimension-8 SMEFT interactions [2]-[3]. Moreover, there are excellent prospects to search for nTGCs in both lepton-antilepton and hadron-hadron collisions using a variety of final states involving photons and \(Z\) bosons that can decay into different identifiable final states (\(\ell^+ \ell^-\), \(\nu \bar \nu\), \(q \bar q\)), with distinctive kinematic distributions that can be effectively discriminated from the SM backgrounds.

There have been many studies of nTGCs in the literature, including both the theoretical and experimental aspects, and using either a form factor framework [4] or the SMEFT formulation. In a recent series of works, we have studied the present and prospective sensitivities of hadron-hadron colliders [5], [6] to nTGCs within the SMEFT framework, as well as the prospects of probing nTGCs at the proposed electron-positron colliders [7]–[10]. Along with these studies, we identified defects of the conventional form factor parametrizations of nTGCs that had been applied previously by both the ATLAS and CMS collaborations for their LHC experimental analyses: these were consistent with U(1)\(_\mathrm{EM}^{}\) gauge symmetry but did not take account of the full electroweak gauge symmetry SU(2)\(_\mathrm{L}^{}\otimes\)U(1)\(_\mathrm{Y}^{}\) of the SM that is incorporated in the SMEFT approach from the start. As we have shown [[5]][6], the conventional nTGC form factors based on the residual U(1)\(_\mathrm{EM}^{}\) gauge symmetry alone are generally inconsistent and thus do not give reliable predictions.

In this work, we extend our previous analysis of the CP-conserving (CPC) nTGCs to include CP-violating (CPV) nTGCs in both the dimension-8 SMEFT formalism of nTGCs and the corresponding nTGC form factor formalism [compatible with spontaneous breaking of the SM electroweak gauge symmetry SU(2)\(_\mathrm{L}\otimes\)U(1)\(_\mathrm{Y}\)]. We are motivated to study this extension by the fact that the SM CP-violation as incorporated through the Kobayashi-Maskawa mechanism [11] is insufficient to generate the observed baryon asymmetry of the Universe, suggesting that there must be some BSM source of CP violation. This might appear at any scale beyond that of electroweak symmetry breaking, and it could well appear at a scale close to those already explored by the LHC experiments. Therefore it is important to scour all the CP-violating signatures that are accessible to present and forthcoming experiments.There have been relatively few studies of experimental sensitivities to CPV nTGCs and, as we show in this work, the conventional form factor formalism for the CPV nTGCs in the literature has serious defects, namely it is incompatible with spontaneous breaking of the full electroweak gauge symmetry of the SM.

The structure of this paper is organized as follows.In Section 2, we introduce the CPV dimension-8 SMEFT operators relevant for nTGCs and the corresponding CPV nTGC form factors that are compatible with spontaneous breaking of the SM electroweak gauge symmetry SU(2)\(_\mathrm{L}\otimes\)U(1)\(_\mathrm{Y}\). Section [sec:sensitivities] studies the sensitivities of probes of the CPV dimension-8 operators and the corresponding form factors that can be achieved in \(e^+ e^-\) (or \(\mu^+ \mu^-\)) collisions at various collision energies up to 5 TeV. The relevant cross sections are studied in Subsection 3.1, the correlations are analyzed in Subsection 3.2, and the potential for increasing the sensitivities through a multivariable analysis is discussed in Subsection 3.3. Our conclusions are summarized in Section 4.

2 CPV nTGCs and Form Factors from the SMEFT↩︎

In a series of previous works [7]-[10], we have studied systematically the CP-conserving (CPC) dimension-8 SMEFT operators that generate the CPC nTGCs as well as their UV completion [12]. We analyzed their contributions to helicity amplitudes and cross sections at both \(e^+e^-\) colliders and hadron colliders. In Refs. [5], [6], we proposed an extended form factor formulation of the CPC nTGCs that is compatible with the full SU(2)\(_{\mathrm L}^{}\otimes\)U(1)\(_{\mathrm Y}^{}\) electroweak gauge symmetry of the SMEFT, incorporating spontaneous breaking by the Higgs mechanism [13], and studied the prospective sensitivities of experimental probes at both hadron-hadron colliders and electron-positron colliders.

In this Section, we analyze what CP-violating (CPV) nTGCs can be generated by the SMEFT operators of dimension-8 and match them to the corresponding CPV form factors that are compatible with the electroweak gauge symmetry of SU(2)\(_{\mathrm L}^{}\otimes\)U(1)\(_{\mathrm Y}^{}\) in the broken phase.

The general dimension-8 SMEFT Lagrangian can be expressed in the following form: \[\begin{align} \Delta\mathcal{L}(\text{dim-8}) \,=\, \sum_{j}^{}\! \frac{\tilde{c}_j^{}}{\,\tilde{\Lambda}^4\,}\mathcal{O}_j^{} \,=\, \sum_{j}^{}\! \frac{\,\text{sign}(\tilde{c}_j^{})\,}{\,\Lambda_j^4\,}\mathcal{O}_j^{} \,=\, \sum_{j}^{}\! \frac{1}{\,[\Lambda_j^4]\,}\mathcal{O}_j^{}\,, \label{cj} \end{align}\tag{1}\] where the \(\,\tilde{c}_j^{}\) denote the dimensionless coefficients of the dimension-8 operators \(\mathcal{O}_j^{}\) , which may take either sign and sign\((\tilde{c}_j^{})\!=\!\pm\) . For each operator \(\mathcal{O}_j^{}\) , we denote its new physics cutoff scale as \(\,\Lambda_j^{} \equiv\tilde{\Lambda}/|\tilde{c}_j^{}|^{1/4}\,\), and for convenience we introduce the notation \(\,[\Lambda_j^4]\equiv\mathrm{sign}(\tilde{c}_j^{})\Lambda^4_j\).

We note that there are three dimension-8 CPV SMEFT operators including Higgs fields [2]: \[\tag{2} \begin{align} \widetilde{\mathcal{O}}_{{B}W}^{} &\,=\, \mathrm{i}H^\dagger {B}_{\mu\nu}W^{\mu\rho} \!\left\{D_{\!\rho}^{},D^\nu\right\}\! H+\text{h.c.} , \tag{3} \\ \widetilde{\mathcal{O}}_{{W}W}^{} &\,=\, \mathrm{i}H^\dagger {W}_{\mu\nu}W^{\mu\rho} \!\left\{D_{\!\!\rho}^{},D^\nu\right\}\! H+\text{h.c.} , \tag{4} \\ \widetilde{\mathcal{O}}_{{B}B}^{} &\,=\, \mathrm{i}H^\dagger{B}_{\mu\nu}B^{\mu\rho} \!\left\{D_{\!\!\rho}^{},D^\nu\right\}\! H+\text{h.c.} , \tag{5} \end{align}\] where \(H\) denotes the Higgs doublet of the SM. From these, we can derive the following neutral triple gauge vertices (nTGVs): \[\label{eq:ZAV-dim8WB} \begin{align}\Gamma_{Z\gamma \gamma^*}^{\alpha\beta\mu} (q_1^{},q_2^{},q_3^{}) &= e v^2q_3^2\!\left(\!-\frac{s_W }{4c_W[\Lambda_{WW}^4]~} \!+\! \frac{1}{2[\Lambda_{WB}^4]} \!-\!\frac{c_W}{s_W[\Lambda_{BB}^4]}\!\right)\! \big(q_2^\alpha g^{\mu\beta}\!-q_2^\mu g^{\alpha\beta}\big) , \\[1mm]\Gamma_{Z\gamma Z^*}^{\alpha\beta\mu} (q_1^{},q_2^{},q_3^{}) &= e v^2(q_3^2\!-\!M_Z^2)\!\left(\!\!-\frac{1}{\,4[\Lambda_{WW}^4]\,} \!+\!\frac{c_W^2\!-s_W^2}{\,4c_Ws_W[\Lambda_{WB}^4]\,} \!+\! \frac{1}{\,[\Lambda_{BB}^4]\,}\!\right)\!\! (q_2^\alpha g^{\mu\beta}\!\!-q_2^\mu g^{\alpha\beta}) , \end{align}\tag{6}\] where \((s_W^{},\,c_W^{})\!=\!(\sin\theta_W^{},\,\cos\theta_W^{})\) and \(\theta_W^{}\) is the weak mixing angle. Pure gauge operators containing only \(B\) fields or only \(W\) fields cannot contribute to the \(Z\gamma V^*\) vertex via CPV nTGCs. However, the following two pure gauge operators that contain both \(B\) and \(W\) fields can contribute to the CPV nTGCs of \(Z\gamma V^*\):
\[\tag{7} \begin{align} \tag{8} g \widetilde{\mathcal{O}}_{G+}^{} &=\, {B}_{\!\mu\nu}^{} W^{a\mu\rho} ( D_\rho^{} D_\lambda^{} W^{a\nu\lambda} \!+\! D^\nu D^\lambda W^{a}_{\lambda\rho}) , \\[1.5mm] \tag{9} g \widetilde{\mathcal{O}}_{G-}^{} &=\, {B}_{\!\mu\nu}^{} W^{a\mu\rho} ( D_\rho^{} D_\lambda^{} W^{a\nu\lambda} \!-\! D^\nu D^\lambda W^{a}_{\lambda\rho}) . \end{align}\] From these, we derive the following CPV neutral triple gauge vertices: \[\tag{10} \begin{eqnarray} \Gamma_{Z\gamma \gamma^*(+)}^{\alpha\beta\mu} (q_1^{},q_2^{},q_3^{}) \!\!&=&\!\! -\frac{~v q_3^2s_W\,}{\,[\Lambda^4_+] M_Zc_W~} (q_2^{\alpha}g^{\mu\beta}M_Z^2 \!-q_3^2q_2^\mu g^{\alpha\beta}) , \tag{11} \\ \Gamma_{Z\gamma Z^*(+)}^{\alpha\beta\mu} (q_1^{},q_2^{},q_3^{}) \!\!&=&\!\! -\frac{~v (q_3^2-M_Z^2)\,}{\,[\Lambda^4_+] M_Z~} (q_2^{\alpha}g^{\mu\beta}M_Z^2 \!-q_3^2q_2^\mu g^{\alpha\beta}) , \tag{12} \\ \tag{13} \Gamma_{Z\gamma \gamma^*(-)}^{\alpha\beta\mu} (q_1^{},q_2^{},q_3^{}) \!\!&=&\!\! -\frac{~vM_Z q_3^2s_W\,}{\,[\Lambda^4_-] c_W~} (q_2^\alpha g^{\mu\beta}\!-q_2^\mu g^{\alpha\beta}) , \\ \tag{14} \Gamma_{Z\gamma Z^*(-)}^{\alpha\beta\mu}(q_1^{},q_2^{},q_3^{}) \!\!&=&\!\!0 ,\end{eqnarray}\] where the three gauge bosons are defined as outgoing. We note that \(\widetilde{\mathcal{O}}_{G+}^{}\) and \(\widetilde{\mathcal{O}}_{G-}^{}\) share similar features with their CPC counterparts \({\mathcal{O}}_{G+}^{}\) and \({\mathcal{O}}_{G-}^{}\) [[6]][9], namely, neither \(\widetilde{\mathcal{O}}_{G+}^{}\) nor \({\mathcal{O}}_{G+}^{}\) contributes to the reaction \(f\bar f \!\rightarrow\!Z\gamma\) with right-handed fermionic initial states, and both \(\widetilde{\mathcal{O}}_{G-}^{}\) and \({\mathcal{O}}_{G-}^{}\) contribute to the \(Z\gamma\gamma^*\) coupling but not the \(Z\gamma Z^*\) coupling. We further note that the operator \(\widetilde{\mathcal{O}}_{WW}^{}\) also does not contribute to \(f\bar f \!\rightarrow\!Z\gamma\) when its initial states are right-handed fermions, because it contains only the SU(2)\(_\mathrm{L}\) gauge fields \(W^{\mu\nu}\) and the summed contributions from Eqs.@eq:eq:ZAV-dim8G43A42 12 cancel for the right-handed fermionic initial states. On the other hand, the operator \(\widetilde{\mathcal{O}}_{BB}^{}\) does contribute to the reaction \(f\bar f \!\rightarrow\!Z\gamma\) with initial states being right-handed fermions, because it contains a pair of U(1)\(_\mathrm{Y}\) gauge fields \(B^{\mu\nu}\).

We present next the form factor formalism for the CPV nTGCs. We note that the conventional CPV form factors respect the residual U(1)\(_{\mathrm{EM}}^{}\) gauge symmetry, and are given as follows [4]: \[\label{eq:FF0-nTGC} \Gamma_{Z\gamma V^*}^{\alpha\beta\mu} (q_1^{},q_2^{},q_3^{}) = \frac{~e(q_3^2\! -\! M_V^2)\,}{\,M_Z^{2}~}\!\! \left[ h_1^V\!(q_2^\alpha g^{\mu\beta}\!\!-q_2^\mu g^{\alpha\beta}) \!+\! \frac{h_2^V}{\,M_Z^2\,}q_2^{\alpha} (g^{\mu\beta}q_2^{}\!\cdotq_3^{} -q_2^\mu q_3^\beta)\right]\!.\tag{15}\] This conventional CPV nTGC form factor formula is incompatible with spontaneous breaking of the full electroweak gauge group SU(2)\(_{\mathrm L}^{}\otimes\)U(1)\(_{\mathrm Y}^{}\).⁴ In the following we first analyze this inconsistency and then present our new, consistent formulation for the CPV nTGC form factors.

By direct power counting, we deduce the following leading energy dependences of the contributions of the CPV nTGC form factors \(h_i^V\!\) to the scattering amplitudes for \(\mathcal{T}[f\bar f\!\rightarrow\!Z\gamma]\): \[\tag{16} \begin{align} \tag{17} \mathcal{T}_{(8)}^{\text{L}} &= {O}(E^3) \!\timesh_1^V + {O}(E^5) \!\timesh_2^V ,\\[1mm] \tag{18} \mathcal{T}_{(8)}^{\text{T}} &= {O}(E^2) \!\timesh_1^V , \end{align}\] where the amplitudes \(\mathcal{T}_{(8)}^{\text{L}}\) and \(\mathcal{T}_{(8)}^{\text{T}}\) denote the final states having longitudinal \(Z_L^{}\) and transverse \(Z_T^{}\) bosons respectively. We note that for \(\mathcal{T}_{(8)}^{\text{T}}\) with the on-shell transverse state \(Z_T^{}\), the \(Z_T^{}\) polarization vector \(\epsilon_{\alpha}^{}(Z_T^{})\) is orthogonal to the photon momentum \(q_2^{\alpha}\) (which is anti-parallel to the \(Z\) momentum \(q_1^{\alpha}\!=\!-q_2^{\alpha}\) in the center-of-mass frame), so this leads to \(\epsilon(Z_T^{})\cdotq_2^{}\!=\!0\,\) and explains why the \(h_2^V\) contribution vanishes in Eq.@eq:eq:FT8-ZT . However, we observe that the helicity amplitudes \(\mathcal{T}_{(8)}^{\text{L}}\) contributed by the gauge-invariant dimension-8 nTGC operators must obey the equivalence theorem (ET) [14]. For \(E\!\gg\! M_Z^{}\), the ET can be written as follows:
\[\label{eq:ET} \mathcal{T}_{(8)}^{}[Z_L^{},\gamma_T^{}] \,=\, \mathcal{T}_{(8)}^{}[-\mathrm{i}\pi^0,\gamma_T^{}] + B\,,\tag{19}\] where the longitudinal gauge boson \(Z_L^{}\) absorbs the would-be Goldstone boson \(\pi^0\) through the Higgs mechanism, and the residual term \(\,B\! =\!\mathcal{T}_{(8)}^{}[v^\mu Z_\mu^{},\gamma_T^{}]\,\) is suppressed by a factor of \(\,v^\mu\!\equiv\!\epsilon_L^\mu\! -\!q_Z^\mu/M_Z^{} \!=\! {O}(M_Z^{}/E_Z^{})\) [14]. We note that the dimension-8 operators of Eq.@eq:eq:dim8H include the Higgs doublet and can contribute to the Goldstone amplitude \(\mathcal{T}_{(8)}^{}[\pi^0,\gamma_T^{}]\) at \({O}(E^3)\). On the other hand, the pure gauge operators of Eq.@{eq:eq:OG43G-} do not contain any Goldstone boson \(\pi^0\) and thus give vanishing contribution to the Goldstone amplitude \(\mathcal{T}_{(8)}^{}[\pi^0,\gamma_T^{}]\) at tree level.But these pure gauge operators can contribute to the \(B\)-term of Eq.@eq:eq:ET , which is of \({O}(E^3)\). Hence, consistency with the ET 19 requires that the \({O}(E^5)\) term on the right-hand-side (RHS) of Eq.@eq:eq:FT8-ZL must be cancelled such that its actual leading energy dependence is \({O}(E^3)\).

Using this key observation, we can construct CPV nTGC form factors that are compatible with the full electroweak gauge group SU(2)\(_{\mathrm L}^{}\otimes\)U(1)\(_{\mathrm Y}^{}\) with spontaneous symmetry breaking. This requires that the \(h_1^V\) term on the RHS of the conventional CPV form factor formula 15 should be modified to include a new form factor \(h_6^V\), with the structure of \(\,(h_1^V\!+ h_6^V q_3^2/M_Z^2)\). Including \(h_6\), we can consistently formulate the extended CPV nTGC form factor as follows:⁵ \[\tag{20} \begin{align} \tag{21}\Gamma_{Z\gamma V^*}^{\alpha\beta\mu} &\!= \frac{\,e(q_3^2-\! M_V^2)\,}{\,M_Z^{2}~}\!\! \left[\!\Big(h_1^V\!\!+\!\frac{\,h_6^Vq_3^2\,}{M_Z^2}\!\Big)\! (q_2^\alpha g^{\mu\beta}\!\!-q_2^\mu g^{\alpha\beta}) \!+\!\frac{h_2^V}{\,2M_Z^2\,}q_2^{\alpha}g^{\mu\beta} (M_Z^2\!-q_3^2)\right] \\ \tag{22}&\!= \frac{\,e(q_3^2\! -\! M_V^2)\,}{\,M_Z^{2}\,}\! \left[h_1^V(q_2^\alpha g^{\mu\beta}\!\!-q_2^\mu g^{\alpha\beta})+ \frac{\,h_2^VM_Z^2q_2^{\alpha}g^{\mu\beta} \!\!-\!2h_6^Vq_3^2q_2^\mu g^{\alpha\beta}\,}{\,2M_Z^2\,} + \frac{\,2h_6^V\!\!-\!h_2^V\,}{2M_Z^2}q_3^2q_2^\alpha g^{\mu\beta}\right] \!, \end{align}\] where, inside the brackets, we see that at high energies the \(h_1^V\) terms have leading energy (momentum) dependences that are \(O(E^1)\), whereas the \(h_2^V\) and \(h_6^V\) terms have leading energy dependences that are \(O(E^3)\).

However, we note that the brackets in Eq.@eq:eq:ZAV42-FFnew-2 contain three terms, and that the middle part contains an \(h_6^V\) term with leading energy dependence \(O(E^3)\) that is proportional to \(g^{\alpha\beta}\) and will contract with the external \(Z\) and \(\gamma\) polarization vectors: \(g^{\alpha\beta}\epsilon_{\alpha}^{}(Z)\epsilon_{\beta}^{}(\gamma) \!=\epsilon^{\alpha}(Z)\epsilon_{\alpha}^{}(\gamma)\). In the case of the longitudinally-polarized external state \(Z_L^{}\), this contraction vanishes: \(\epsilon^{\alpha}(Z_L^{})\epsilon_{\alpha}^{}(\gamma_T^{})\!=\!0\). This is because the spatial component of \(\epsilon^{\alpha}(Z_L^{})\) is along the direction of the 3-momentum \(\vec{q}_1^{}\) and is thus orthogonal to the spatial part of the photon polarization vector, \(\vec{q}_1^{}\!\cdot\vec{\epsilon}(\gamma_T^{})\!=\!0\), where \(\vec{q}_1^{}\!=\!-\vec{q}_2^{}\) holds in the center-of-mass frame of \(Z\gamma\). From this we further deduce \(\epsilon^{\alpha}(Z_L^{})\epsilon_{\alpha}^{}(\gamma_T^{})\!=\!0\,\) since the transverse photon polarization vector has vanishing time-component \(\epsilon_{0}^{}(\gamma_T^{})\!=\!0\). This means that in Eq.@eq:eq:ZAV42-FFnew-2 the \(h_6^Vg^{\alpha\beta}\) term in the middle part of the brackets has vanishing contribution to the amplitude \(\mathcal{T}[f\bar f\!\rightarrow\!Z_L^{}\gamma_T^{}]\). Finally, we note that the third term inside the brackets is proportional to \((2h_6^V\!- h_2^V)\) and can contribute \(O(E^5)\) to the amplitude \(\mathcal{T}[f\bar f\!\rightarrow\!Z_L^{}\gamma_T^{}]\).

Thus we deduce from Eq.@eq:eq:ZAV42-FFnew-2 the following leading energy dependences of the \(h_i^V\) contributions to the scattering amplitude \(\mathcal{T}[f\bar f\!\rightarrow\!Z\gamma]\): \[\tag{23} \begin{align} \tag{24} \mathcal{T}_{(8)}^{\text{L}} &= {O}(E^3) \!\times\! h_1^V + {O}(E^3) \!\times\! h_2^V + {O}(E^5) \!\times\! (2h_6^V \!-h_2^V) , \\[0.5mm] \tag{25} \mathcal{T}_{(8)}^{\text{T}} &= {O}(E^2) \!\times\! h_1^V + {O}(E^4) \!\times\! h_2^V + {O}(E^4) \!\times\! h_6^V , \end{align}\] where the \({O}(E^5)\) terms originate from the last term of Eq.@eq:eq:ZAV42-FFnew-2 and are proportional to the form factor combination \((2h_6^V \!-\!h_2^V)\). However, according to the ET identity 19 , the leading energy dependence of Eq.@eq:eq:FT8-ZL-h6 should be \({O}(E^3)\) only.Hence there must be an exact cancellation between the \({O}(E^5)\) terms associated with the form factors \(h_2^V\) and \(h_6^V\). This means that the last term of Eq.@eq:eq:ZAV42-FFnew-2 must vanish, leading to the condition: \[\label{eq:h2612h6} h_2^V=2h_6^V.\tag{26}\] Using this condition, we express the CPV nTGC vertex 20 in the following form: \[\begin{align} \label{eq:ZAV42-FFnew2} \Gamma_{Z\gamma V^*}^{\alpha\beta\mu} &= \frac{~e(q_3^2-\! M_V^2)\,}{\,M_Z^{2}~}\! \!\left[ h_1^V\big(q_2^\alpha g^{\mu\beta}\!-q_2^\mu g^{\alpha\beta}\big) +\! \frac{h_2^V}{\,2M_Z^2\,} \big(M_Z^2q_2^{\alpha}g^{\mu\beta}\!-q_3^2q_2^\mu g^{\alpha\beta}\big)\!\right]\!. \end{align}\tag{27}\]

In the next step, we match the newly-proposed form factor formulation of the nTGC vertices 27 with the CPV nTGC vertices 6 10 generated by the gauge-invariant dimension-8 CPV nTGC operators 2 7 after the spontaneous electroweak symmetry breaking. From these, we derive a nontrivial relation between \(h_2^Z\) and \(h_2^\gamma\,\): \[h_2^Z=(c_W^{}/s_W^{})h_2^\gamma\,,\] where we denote \(h_2^Z\!\equiv\! h_2^{}\). For convenience, we denote the polarization vectors for \(Z_T^{}\), \(Z_L^{}\) and \(\gamma\) by \(\epsilon_{T\alpha}^{}\), \(\epsilon_{L\alpha}^{}\) and \(\epsilon_{\beta}^{}\) respectively. Because \(q_2^\alpha\epsilon_{T\alpha}^{}\!=\!0\), the structure \(q_2^\alpha g^{\mu\beta}\) has a nonzero contribution only for the external state \(Z_L^{}\), but vanishes for the external state \(Z_T^{}\). On the other hand, as explained below Eq.@eq:eq:ZAV42-FFnew , the contraction \(g^{\alpha\beta}\epsilon_{L\alpha}^{}\epsilon_{\beta}^{}\!=\!0\). Thus, the structure \(q_2^{\mu} g^{\alpha\beta}\) has a nonzero contribution from the external state \(Z_T^{}\) and not \(Z_L^{}\).

The kinematics for the reaction \(e^+e^-\!\!\rightarrow\!Z\gamma\!\!\rightarrow\!\! f\bar f\gamma\) are defined by the three angles \((\theta,\,\theta_*^{},\,\phi_*^{})\), where \(\,\theta\,\) is the polar scattering angle between the direction of the outgoing \(Z\) and the initial state \(e^-\), \(\,\theta_*^{}\) denotes the angle between the direction opposite to the final-state \(\gamma\) and the final-state fermion \(f\) direction in the \(Z\) rest frame, and \(\,\phi_*^{}\,\) is the angle between the scattering plane and the decay plane of \(Z\) in the \(e^+e^-\) center-of-mass frame.

With these definitions, we derive the following contributions of the CPV nTGC form factor vertices 27 to the scattering amplitudes for \(f_{\lambda}^{}\bar f_{\lambda'}^{}\!\rightarrow\!Z\gamma\): \[\label{eq:T8h} \begin{align} &\mathcal{T}_{(8),\text{F}}^{ss'\!,\text{T}}\!\!\left\lgroup\!\!\! \begin{array}{cc} -- \!&\! -+ \\ +- \!&\! ++\\ \end{array}\!\!\!\right\rgroup\!\! =\frac{~\mathrm{i}e^2 \sin\theta(c^{\prime V}_L\!+\! c^{\prime V}_R)(M_Z^2\!-\!s) \big(2 h_1^VM_Z^2\!+\!h_2^Vs\big)\,}{4 M_Z^4s_W^{}c_W^{}} \!\left(\!\! \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \!\!\right) \!, \\[1mm] &\mathcal{T}_{(8),\text{F}}^{ss'\!,\text{L}}(0-,0+) =-\frac{\,\mathrm{i}e^2 (2 h_1^V\!\!+h_2^V) s^{\frac{1}{2}} (s\!-\!M_Z^2)\,}{2 \sqrt{2\,} M_Z^3s_W^{}c_W^{}}\!\! \left(\! c^{\prime V}_L \!\sin^2\!\frac{\,\theta\,}{2} \!-c^{\prime V}_R \!\cos^2\!\frac{\,\theta\,}{2},\, c^{\prime V}_L\!\cos^2\!\frac{\,\theta\,}{2} \!-\!c^{\prime V}_R \!\sin^2\!\frac{\,\theta\,}{2}\!\right)\!, \end{align}\tag{28}\] for the helicity combinations \(\lambda\lambda'\!=\!(--,-+,+-,++)\) and \(\lambda\lambda'\!=\!(0-,0+)\), where the subscript F for each amplitude \(\mathcal{T}\) denotes the contributions from form factors. In the above, the coupling coefficients are defined as follows:
\[\begin{align} c_L^{\prime Z} & = c_L^{Z}\delta_{\!s,-\frac{1}{2}}^{} ,c_R^{\prime Z} = c_R^{Z}\delta_{\!s,\frac{1}{2}} ,c_{L,R}^{\prime\gamma} = c_{L,R}^{\gamma}\delta_{\!s,\mp\frac{1}{2}} \,, \nonumber\\ c_L^{Z} & = I_3^{}-Qs_W^2 ,c_R^{Z} = -Qs_W^2 ,c_{L,R}^{\gamma} = Qs_W^{}c_W^{} \,, \end{align}\] where we have used the notations \(\,(s_W^{},\,c_W^{})\!=\!(\sin\theta_W^{},\cos\theta_W^{})\). In the above, the subscript index \(\,s \!=\!\mp\frac{1}{2}\)  denotes the initial-state fermion helicities, \(I_3^{}\) is the weak isospin, and \(Q\) is the electric charge.

We recall that the SM contributions to the helicity amplitudes of \(f_{\lambda}^{}\bar f_{\lambda'}^{}\!\rightarrow\!Z\gamma\) have been computed in the previous studies [[6]][9][10]: \[\tag{29} \begin{eqnarray} \mathcal{T}_{\text{SM}}^{ss'\!,\text{T}}\!\!\left\lgroup\!\!\! \begin{array}{cc} -- \!&\! -+ \\ +- \!&\! ++\\ \end{array}\!\!\right\rgroup\!\!&=&\!\!\! \frac{-2e^2 Q}{\,s_W^{}c_W^{}(s\!-\!M_Z^2)\,}\!\!\! \left\lgroup\!\!\!\! \begin{array}{ll} \left(c_L'\!\cot\!\frac{\theta}{2}\!-\! c_R'\!\tan\!\frac{\theta}{2}\right)\!M_Z^2~ \!&\!\! \left(-c_L'\!\cot\!\frac{\theta}{2}\!+\! c_R'\!\tan\!\frac{\theta}{2}\right)\!s \\[2mm] \left(c_L'\!\tan\!\frac{\theta}{2}\!-\! c_R'\!\cot\!\frac{\theta}{2}\right)\!s \!&\!\! \left(-c_L^{}\!\tan\!\frac{\theta}{2}\!+\!c_R^{}\!\cot\!\frac{\theta}{2}\right)\!M_Z^2 \end{array} \!\!\!\right\rgroup\!\!, \tag{30} \tag{31} \\[2mm] \mathcal{T}_{\text{SM}}^{ss'\!,\text{L}}(0-,0+) \!\!\!&=&\!\!\! \frac{\,-2\sqrt{2} e^2Q(c_L'\!\!+\!c_R')M_Z^{}\sqrt{s\,}~}{\,s_W^{}c_W^{}(s\!-\!M_Z^2)\,}\left(1,\,-1\right), \tag{32} \tag{33} \end{eqnarray}\] where the coupling coefficients are given by \[c_L' = (I_3^{}-Qs_W^2)\delta_{\!s,-\frac{1}{2}}^{},c_R' = -Qs_W^2\delta_{\!s,\frac{1}{2}}^{} .\]

In passing, we note that the conventional CPV nTGC form factors have been used to study possible probes of CPV nTGCs at \(e^+e^-\) colliders in the literature [[15]][16]. However, the analysis of Ref. [15] was restricted to dimension-6 operators in the broken electroweak symmetry phase, which respects only the residual U(1)\(_{\mathrm{EM}}^{}\) gauge symmetry and is incompatible with the dimension-8 SMEFT formalism for the nTGC operators, contrary to our new CPV nTGC form factor formulation 20 and our CPC nTGC form factor approach in Ref. [6]. Also, Ref. [16] did not consider \(Z\)-decay final states, so the CPV nTGC part of the amplitude in their analysis is imaginary and cannot interfere with the SM part (whose CPV form factors are real). Ref. [16] assumed these CPV form factors to be imaginary, which would give nonzero interference with the SM part. However, taking the CPV form factors to be imaginary is incompatible with the Hermiticity of the Lagrangian.

3 Analyzing Sensitivities to CPV nTGCs at \(\mathbf{e^+e^-}\) Colliders↩︎

In this Section we analyze the sensitivity reaches for probes of the CPV nTGCs at future \(e^+e^-\) colliders via the reaction \(e^+e^-\!\!\rightarrow\!Z\gamma\) with \(Z\!\rightarrow\!\ell\bar{\ell},q\bar{q}\) decays. As we discuss later, in the case of the invisible decay channel \(Z\!\!\rightarrow\!\nu\bar{\nu}\) the interference term in the cross section cannot be measured, which results in weaker sensitivities.

3.1 Cross Sections and Observables: Analysis of CPV nTGCs↩︎

For the reaction \(e^+e^-\!\!\rightarrow\!Z\gamma\) (with \(Z\!\!\rightarrow\!\!f\bar f\) decays), we illustrate in Fig. 1 its kinematics in the \(e^-e^+\) collision frame. In this plot, \(\theta\) denotes the polar scattering angle between the directions of the outgoing \(Z\) boson and the initial state electron \(e^-\), whereas \(\phi_*\) is defined as the angle between the scattering plane and the decay plane of the final state \(Z\) boson in the center-of-mass frame of \(e^-e^+\).

The total cross section \(\sigma\) for the on-shell process \(e^+e^-\!\!\rightarrow\!Z\gamma\) can be divided into a SM part \(\sigma_0^{}\), an interference term \(\sigma_1^{}\), and a pure new phyics term \(\sigma_2^{}\). However, since the nTGC amplitudes with on-shell final state \(Z\gamma\) are imaginary (having an extra factor of “ \(\mathrm{i}\) ”), there is no interference term in the cross section (\(\sigma_1^{}\!=\!0\)). Hence, we have the following expression for the total cross section \(\sigma\): \[\begin{align} \label{eq:CS-qqZA-d8other}\sigma(Z\gamma) =&~ \sigma_0^{} + \sigma_2^{} = \frac{~e^4(c_L^2\!+\!c_R^2)Q^2\!\left[-(s\!-\!M^2_Z)^2\!-\!2(s^2\!+\!M_Z^4) \ln\sin\!\frac{\,\delta\,}{2}\right]\,}{\,8\pi s_W^2c_W^2(s\!-\!M^2_Z)s^2\,} \\[1mm]& +\frac{\,\big[(c_L^V)^2\!+\!(c_R^V)^2\big]\!\left(s\!-\!M^2_Z\right)^{\!3} \!\!\left\{\big[4(h_1^V)^2M_Z^2\!+\!(h_2^V)^2s\big]\!(s\!+\!M_Z^2)\!+\!8h_1^Vh_2^V\!M_Z^2s\right\}\,}{\,768\pi c_W^2s_W^2M_Z^8\,s^2\,} + O(\delta)\,,\nonumber\\ \nonumber \end{align}\tag{34}\]
where \(\,\delta\,\) is the lower cut on the scattering angle \(\theta\). Then, we derive the following expression for the scattering amplitude for the full process \(e^+e^-\!\!\rightarrow\!Z\gamma\!\rightarrow\!f\bar f \gamma\) : \[\begin{align} \label{eq:T-llgamma}\mathcal{T}^{ss'}_{\sigma\sigma'\!\lambda}(f\bar f\gamma) =\, & \frac{~eM_Z^{}\mathcal{D}_Z^{}\,}{s_W^{}c_W^{}} \!\!\left[\!\sqrt{2\,}e^{\mathrm{i}\phi_*^{}}\!\! \left(\!f_R^{\sigma}\cos^2\!\frac{\,\theta_*^{}}{2}\!-\! f_L^{\sigma}\!\sin^2\!\frac{\,\theta_*^{}}{2}\!\right)\! \mathcal{T}_{ss'}^T(+\lambda) \right. \\& \left. +\sqrt{2\,}e^{-i\phi_*^{}}\!\! \left(\!f_R^{\sigma}\sin^2\!\frac{\,\theta_*^{}}{2}\! -f_L^{\sigma}\!\cos^2\!\frac{\,\theta_*^{}}{2}\!\right)\! \mathcal{T}_{ss'}^T(-\lambda) \!+\! (f_R^{\sigma}\!+\!f_L^{\sigma})\sin\!\theta_* \mathcal{T}_{ss'}^L(0\lambda)\right]\!, \nonumber \end{align}\tag{35}\] where \(\theta_*^{}\) denotes the angle between the direction opposite to the final-state \(\gamma\) and the direction of the final-state fermion \(f\) direction in the \(Z\) rest frame. The coefficients \(f_L^\sigma\!=\! (I_3^{}\!-Qs_W^2)\delta_{\!\sigma,-\frac{1}{2}}^{}\) and \(f_R^\sigma\!=\! (-Qs_W^2)\delta_{\!\sigma,\frac{1}{2}}\) above denote the couplings of the final-state fermions. Using Eq.(35 ), we can compute the interference term in the differential cross section and deduce its \(\phi_*^{}\) dependence as follows: \[\label{eq:ds1} \frac{\text{d}^3\sigma_1}{\,\text{d}\theta\text{d}\theta_*\text{d}\phi_*^{}\,} \,=\, c_1^{}(\theta,\theta_*)\sin\phi_*^{} + c_2^{}(\theta,\theta_*)\sin2\phi_*^{} \, ,\tag{36}\] where the coefficients \(c_1^{}(\theta,\theta_*)\) and \(c_2^{}(\theta,\theta_*)\) are functions of \((\theta,\theta_*)\).

We express each cross section term \(\sigma_i^{}(e^+e^-\!\!\rightarrow\!Z\gamma\!\rightarrow\!f\bar f\gamma)\equiv \sigma_i^{}(e^+e^-\!\!\rightarrow\!Z\gamma)\text{Br}(f\bar f)\), and define the angular distribution: \[\tilde{\sigma}_i^{} \equiv \frac{~\text{d}\sigma_i(e^+e^-\!\!\rightarrow\!Z\gamma\!\rightarrow\!f\bar f\gamma)~}{\text{d}\phi_*^{}\,\text{Br}(f\bar f)} \, ,\] where the contribution of the interference term is given by \[\begin{align}\tilde{\sigma}_1^{} =~& \frac{~e^4 h_1^V Q \sin\phi_* (M_Z^2\!-\!s)~}{c_W^{}s_W^{}M_Z^2}\!\! \left[\!\frac{~3(f_L^2\!-\!f_R^2) (M_Z^2\!+\!3 s)(c_L^V c_L^{Z}\!+\!c_R^V c_R^{Z})~}{2048(f_L^2\!+\!f_R^2)s^{3/2}M_Z} - \frac{\cos\phi_*^{}(c_L^Vc_L^{Z}\!-\!c_R^Vc_R^{Z})~}{32\pi^2s}\!\right] \nonumber \\& +\frac{~e^4 h_2^V Q \sin\phi_*^{}(s\!-\!M_Z^2)~}{c_W^{}s_W^{}M_Z^3}\!\! \left[\!\frac{~3(f_L^2\!-\!f_R^2)(M_Z^2\!-\!5 s) (c_L^V c_L^{Z}\!+\!c_R^V c_R^{Z})~}{4096(f_L^2\!+\!f_R^2)s^{3/2}} +\frac{~\cos\phi_*^{} (c_L^V c_L^{Z}\!-\!c_R^V c_R^{Z})~}{64\pi^2 M_Z}\right] \nonumber\\\simeq~& \frac{~9\sqrt{s\,}e^4 h_1^V Q (f_R^2\!-\!f_L^2) (c_L^V c_L^{Z}\!+\!c_R^V c_R^{Z}~) \sin\phi_*^{}~}{2048c_W^{}s_W^{}M_Z^3 (f_L^2\!+\!f_R^2)~} +\frac{~se^4 h_2^V Q (c_L^V c_L^{Z}\!-\!c_R^V c_R^{Z})\sin 2\phi_*^{}~}{128\pi^2c_W^{}s_W^{}M_Z^4}, \label{lastrow} \end{align}\tag{37}\] and the last row of Eq.@eq:lastrow gives the leading contributions in the limit \(s\!\gg\! M_Z^2\). Since \(\sigma_1^{}\!=\!\int\!\text{d}\phi_*^{}\, \tilde{\sigma}_1^{}\!=0\), it is not convenient to define a normalized angular distribution for the interference term as was done for the analysis of the CPC nTGCs.

We may further define the normalized angular distribution functions as follows: \[\begin{align} f_\xi^j \, = \, \frac{1}{\sigma_{\!j}^{}}\frac{\mathrm{d}\sigma_{\!j}^{}}{\,\mathrm{d}\xi\,}\,, \end{align}\] where the angles \(\,\xi \in (\theta,\,\theta_*^{},\,\phi_*^{})\),  and the cross sections \(\,\sigma_j^{}\) (\(j=0,1,2\)) represent the SM contribution (\(\sigma_0^{}\)), the \({O}(\Lambda^{-4})\) interference contribution (\(\sigma_1^{}\)), and the \({O}(\Lambda^{-8})\) contribution (\(\sigma_2^{}\)),  respectively. For instance, we derive explicit formulae for the normalized azimuthal angular distribution functions \(\,f_{\phi_*}^{0}\,\) and \(\,f_{\phi_*}^{2}\,\) as follows:⁶ \[\tag{38} \begin{eqnarray} \tag{39}f_{\phi_*^{}}^{0}&=&\frac{1}{\,2\pi\,}+\frac{\,3\pi^2(c_L^2\!-\!c_R^2)(f_L^2\!-\!f_R^2)M_Z^{}\sqrt{s}\, (s\!+\!M_Z^2)\cos\phi_*^{}\! -8(c_L^2\!+\!c_R^2)(f_L^2\!+\!f_R^2)M_Z^2\,s \cos2\phi_*^{}\,}{\,16\pi(c_L^2\!+\!c_R^2)(f_L^2\!+\!f_R^2)\! \left[(s\!-\!M_Z^2)^2\!+2(s^2\!+\!M_Z^4)\ln\sin\!\frac{\delta}{2} \right]\,}+O(\delta) , \\[2mm] \tag{40}f_{\phi_*^{}}^{2}&=&\frac{1}{\,2\pi\,} -\frac{9 \pi \sqrt{s} M_Z \big(f_L^2\!-\!f_R^2\big) \big(c_L^V{}^2\!-\!c_R^V{}^2\big) (2 h_1^V\!+\!h_2^V)\big(s h_2^V\!+\!2 M_Z^2 h_1^V\big)\! \cos\phi_*{}}{~128 \big(f_L^2\!+\!f_R^2\big) \big(c_L^V{}^2\!+\!c_R^V{}^2\big) \big[4 M_Z^2 h_1^V{}^2 (s\!+\!M_Z^2)\!+8sM_Z{}^2 h_1^V h_2^V+s h_2^V{}^2 \big(s \!+\! M_Z^2\big)\big])}\, , \tag{41} \end{eqnarray}\] where we denote the \(Z\) couplings with the initial state quarks as \(\,(c_L^{},\,c_R^{}) \!=\! (I_3^{}-Qs_W^2,-Qs_W^2)\), and the \(Z\) couplings with the final-state fermions as \((f_L^{}, f_R^{}) = (I_3^{}\! -\! Qs_W^2,-Qs_W^2)\). In the above, \(\delta\) is the lower cut on the scattering angle \(\theta\). We note that both the angular distributions \(f_{\phi_*^{}}^{0}\) and \(f_{\phi_*^{}}^{2}\) approach \(\frac{1}{\,2\pi\,}\) when \(s\!\gg\! M_Z^2\,\).

In Fig. 2 (d), we present the angular distributions at different \(e^+ e^-\) colliders with collision energies \(\sqrt{s} = (0.25,\,0.5,\, 1,\,3)\) TeV, respectively.⁷ The \(h_1^V\) term with leading energy dependence is proportional to \(\sin\phi_*^{}\), whereas for \(h_2^V\) it is proportional to \(\sin2\phi_*^{}\). We note that for the form factor \(h_1^Z\), the coupling coefficient \((c_L^Z)^2-(c_R^Z)^2\!=\!I_3^{}(I_3^{}- 2Qs_W^2)\) is suppressed by the factor \((\frac{1}{2}-2s_W^2)\simeq 4\%\) [17]. Hence the \(h_1^Z\) angular distribution is dominated by the contribution \(\propto\sin\phi_*^{}\). On the other hand, the distributions for \(h_1^\gamma\) and \(h_2^{}\) are combinations of \(\sin\phi_*^{}\) and \(\sin2\phi_*^{}\). In the high-energy limit, the \(h_1^\gamma\) (\(h_2^{}\)) distribution is proportional to \(\sin\phi_*^{}\) (\(\sin2\phi_*^{}\)), as shown in the last row of Eq.@eq:lastrow and in Fig. 2.

The sign of the interference term (36 ) depends on the angles \((\theta,\, \theta_*^{},\phi_*^{})\). In order to avoid large cancellations between the contributions having different signs, we divide the phase space into different regions and compute the interference term for each region independently. To this end, we separate the (\(\theta,\theta_*^{},\phi_*^{}\)) phase space into \(2\!\times\! 4\!=\!8\) regions with the boundaries \(\cos\theta\cos\theta_*^{}\!=0\) and \(\phi\!=\!\frac{\pi}{2},\pi,\frac{3\pi}{2}\), and denote the cross section \(\sigma_i^{}\) in region \(j\) as \(\sigma_{i,j}^{}\). Since \(\displaystyle\sum_{j=1}^{8}\!\sigma_{1,j}^{}\!=\!0\), we analyse the signals in each region separately and combine their significances as follows: \[\begin{align} \mathcal{Z}_f^{} &= \sqrt{\!\sum_j\!\frac{S_j^2}{ B_j}~} =\sqrt{\!\sum_j\!\frac{\sigma_{1,j}^2}{ \sigma_{0,j}}\!\times\! \mathcal{L}\!\times\!\epsilon~} \,, \\ \mathcal{Z} &= \sqrt{\sum_f\!Z_f^2~}\simeq \sqrt{3\mathcal{Z}_d^2\!+\!2\mathcal{Z}_u^2\!+\!3\mathcal{Z}_{\ell}^2~} \,, \end{align} \label{eq:z}\tag{42}\] where \(\mathcal{L}\) is the integrated luminosity, \(\epsilon\) is the detection efficiency and \(f\) denotes the type of fermion. In the high energy limit \(s\!\gg\! M_Z^2\), the sensitivity bound scales as \(h_1^V\!\!\propto\! E^{-2}\) for probing \(h_1^V\) and scale as \(h_2^{}\!\propto\! E^{-3}\) for probing \(h_2^V\), where we denote the center-of-mass energy \(E\!=\!\!\sqrt{s\,}\). Accordingly, the sensitivity bounds for probing the new physics scales \(\Lambda\) of dimension-8 operators are \(\Lambda\!\propto\! E^{3/4}\) for \(O_{\!\bar G+}^{}\) and \(\Lambda\!\propto\! E^{1/2}\) for the other operators. This is why the sensitivity bound on probing \(O_{\!\bar G+}^{}\) is more sensitive and it is enhanced faster with the increase of collision energy. Besides, since the significances \(\mathcal{Z}_{\ell}^{}\) of the lepton channels contribute only around 5% of the overall significance \(\mathcal{Z}\), their contributions to the sensitivity reaches on probing the scale of new physics are negligible.

In the above we have considered the irreducible SM backgrounds of \(Z\gamma\) production from \(t\)-channel and \(u\)-channel fermion exchanges, having the on-shell decays of \(Z\!\rightarrow\!f\bar{f}\). We note that in principle the scattering process \(e^+e^-\!\!\rightarrow\!f\bar{f}\gamma\,\) also contains other contributions whose final state fermions \(f\bar{f}\) do not arise from the on-shell \(Z\)-decays, and thus serve as the reducible SM backgrounds. These backgrounds are the same as what we studied in the analysis of CPC nTGCs [[10]][9]. By imposing the invariant-mass cut \(|M_{f\bar f}^{} -M_Z^{}|\!<\!10\)GeV, the reducible backgrounds can be reduced to less than 10% of the total backgrounds. Most \(Z\gamma\) backgrounds can be removed by imposing lower cut on scattering angle \(\theta\). The irreducible backgrounds are shown in Table [tab:1new], where we have imposed a lower cut on the scattering angle \(\theta\!>\!\delta\) (with \(\delta\!=\!0.33\)), and an invariant-mass cut \(|M_{f\bar f}^{} -M_Z^{}|\!<\!10\,\)GeV. Since the signal significance is \(\mathcal{Z}\!=\!S/\!\sqrt{B\,}\), the sensitivity bounds on the nTGC form factors \(h_i^V\) scale as \(h_i^V\!\!\propto\!\!\sqrt{B\,}\) and on the nTGC cutoff \(\Lambda\) behave as \(\Lambda\!\propto\! B^{-1/8}\). Thus the sensitivity bounds are insensitive to a change of backgrounds \(B\) by less than 10%. Namely, varying \(B\) by less than 10% only causes a change of the sensitivity bounds on \(h_i^V\) by less than 5% and on \(\Lambda\) by less than 1.3%.Hence it is justified to use the irreducible backgrounds for estimating sensitivity bounds.

Fig. 3 (b) displays as histograms the 2\(\sigma\) sensitivity bounds on the CPV nTGC form factors \((h_2^{},h_1^Z,h_1^\gamma)\) (upper panel) and on the cutoff scales \(\Lambda\) (lower panel) of the dimension-8 CPV nTGC operators at \(e^+e^-\) (or \(\mu^+\mu^-\)) colliders with collision energies \(\sqrt{s\,}\!=\!(0.25,\,0.5,\,1,\,3,\,5)\)TeV, by choosing an integrated luminosities of 5 ab\(^{-1}\) in each case. For the case of \(\sqrt{s\,}\!=\!250\) GeV, we also present the sensitivity reaches with an integrated luminosity of \(20\)ab\(^{-1}\) [18]. The sensitivities found for unpolarized (polarized) beams are shown in light (heavy) colors in the upper panel and in heavy (light) colors in the lower panel, respectively. We see in the upper panel that the sensitivities to the form factors \(h_1^{Z, \gamma}\) recommended in our dimension-8 SMEFT approach are significantly weaker than the sensitivities to the deprecated conventional form factor \(h_2\) as the center-of-mass energy increases. This feature can be traced back to the unphysical high-energy behavior of the conventional parametrization. In the lower panel of Fig. 3 (b) we see that the sensitivity to the scale parameter \(\Lambda\) for the dimension-8 operator \(\tilde{\mathcal{O}}_{G+}\) is much greater than those to the scale parameters for the other operators \(\tilde{\mathcal{O}}_{G-}, \tilde{\mathcal{O}}_{BW}, \tilde{\mathcal{O}}_{WW}\) and \(\tilde{\mathcal{O}}_{BB}\).

We present the sensitivity reaches for the different form factors in Table [tab:f-p], and on the operator scales in Table [tab:o-p]. In the case of \(\mathcal{O}_{\!\bar G^{}+}^{}\), we see that the operator scale sensitivity is more than 1 TeV at \(\sqrt{s} \!=\! 250\) GeV and greater than 10 TeV at \(\sqrt{s\,} \!=\! 5\) TeV, whereas the sensitivity scales for probing new physics in the other dimension-8 operators range from several hundred GeV at \(\sqrt{s\,} \!=\! 250\) GeV to several TeV at \(\sqrt{s} \!=\! 5\) TeV. The sensitivity reaches have the same order of magnitude as we found for the CPC case in Refs. [6], [9]. However, we note that the sensitivity to \(h_1^\gamma\) is weaker than that to \(h_1^Z\). This is because the total contributions of left-handed and right-handed electrons are proportional to the coupling \(c_L^{}c_L^V\!+c_R^{}c_R^V\), and we note \(|c_L^{}c_L^\gamma+ c_R^{}c_R^\gamma|\!\ll\!|c_L^{}c_L^Z+c_R^{}c_R^Z|\) which is \(0.0092\!\ll\! 0.13\) numerically. Since \(\mathcal{O}_{\bar G-}\) only contributes to \(h_2^\gamma\), it has the weakest probing sensitivity among all the CPV nTGC operators.

We compare in Table 4 the sensitivities for probing \(h_2^{}\) and the conventional form factors \(h_1^Z\) and \(h_1^\gamma\). The sensitivities to the conventional form factors \(h_2^Z\) and \(h_1^\gamma\) (marked in blue) are generally stronger than that for the SMEFT form factor \(h_2^{}\) (marked in red) by large factors, ranging from \({O}(2)\) at \(\sqrt{s}\!=\!250\) GeV to \({O}(40)\) at a \(\sqrt{s}\!=\!5\) TeV \(e^+e^-\) collider. We note also that the sensitivity to \(h_2^\gamma\) is weaker than that to \(h_2^Z\) for the same reason that the sensitivity to \(h_1^\gamma\) is weaker than that to \(h_1^Z\). We emphasize again that the use of the conventional form factors is strongly deprecated, because they are not compatible with the full electroweak gauge group SU(2)\(_\mathrm{L}^{}\otimes\)U(1)\(_\mathrm{Y}^{}\) with spontaneous symmetry breaking and their apparent higher sensitivity is an artefact: the blue numbers are included here only for comparison.

Using polarized \(e^+e^-\) beams can improve the probing sensitivities. We use the symbols \(P_L^e\) (\(P_R^{\bar e}\)) to denote the fraction of the left-handed electrons (right-handed positrons) in the \(e^-\) (\(e^+\)) beam.⁸ The cross sections in the polarized case can be obtained from the unpolarized case by rescaling \((c_L^{},c_L^V)\!\rightarrow\!2\sqrt{P_L^eP_R^{\bar e}\,}(c_L^{},c_L^V)\) and \((c_R^{},c_R^V)\!\rightarrow\!2\sqrt{(1\!-\!P_L^e)(1\!-\!P_R^{\bar e})\,}(c_R^{},c_R^V)\). We present the sensitivity bounds on the CPV nTGC form factors in Table [tab:2new] and on the cutoff scales of the CPV nTGC operators of dimension-8 in Table 3, for a sample input of \((P_L^e,P_R^{\bar e})=(0.9,0.65)\). We see that the sensitivities are improved by factors about 2 for probing the form factor \(h_2^{}\) and are improved by about \(30\%\) for probing \(h_1^Z\), whereas those for probing \(h_1^\gamma\) are improved by larger factors about \(2.6\).

3.2 Analyzing Correlations between CPV nTGCs↩︎

In this subsection, we analyze the the correlations between each pair of the CPV nTGC form factors and between each pair of CPV dimension-8 nTGC operators.

For this analysis, we first define \(\chi^2\) likelihood functions as follows: \[\begin{align} \chi^2_f &\,=\, {\sum_j\!\frac{\,S_j^2\,}{ B_j}} = {\sum_j\!\frac{\,\sigma_{1,j}^2\,}{\sigma_{0,j}^{}}\times \mathcal{L}\times\epsilon} \, , \\ \chi^2 &\,=\, {\sum_f \chi_f^2}\simeq {3\chi^2_d + 2\chi_u^2 + 3\chi_{\ell}^2} \, . \end{align}\] We present in Fig. 4 (d) the \(\chi^2\) contours for the correlations between each pair of the CPV nTGC form factors \((h_1^Z,h_2^{})\) and \((h_1^\gamma,h_2^{})\) at the \(2\sigma\) level. We recall that in the high energy limit \(s\!\gg\! M_Z^2\), the sensitivity reaches for the \(h_1^V\) are determined by \(\sin\phi_*\) and the sensitivity reaches for \(h_2\) are determined by \(\sin2\phi_*\); so \(h_1^V\) and \(h_2^{}\) are almost uncorrelated.However, when \(s\) is not large, the \((h_1^V,h_2^{})\) correlations are significant, and we see in Fig. 4 (d) how these correlations gradually diminish as the collision energy increases. Fig. 5 (d) displays the correlations of the pairs \((h_1^Z,\,h_1^\gamma)\). Note that we can expand \(c_{i}^{}(\theta,\,\theta_*^{})\) in Eq.(36 ) as follows: \[\begin{align} c_i^{}(\theta,\theta_*) \,=\, P_L^eP_R^{\bar e}c_Lc_L^Vc_{i}^L(\theta,\theta_*) +(1\!-\!P_L^e)(1\!-\!P_R^{\bar e}) c_R^{}c_R^Vc_{i}^R(\theta,\theta_*) \, . \end{align}\] Thus, we see that if \(P_L^eP_R^{\bar e} \!\gg\!(1\!-\!P_L^e)(1\!-\!P_R^{\bar e})\), the contributions to both \(h_1^Z\) and \(h_1^\gamma\) are dominated by the same term \(c_{1}^L(\theta,\theta_*)\), so that \(h_1^Z\) and \(h_1^\gamma\) become highly correlated in the polarized case.

3.3 Improvement from the Multivariable Analysis↩︎

In this subsection, we perform a multivariable analysis (MVA) to improve the sensitivity reaches on the CPV nTGCs.

Inspecting the structure of the interference term (36 ), we see that the form of the interference term \(\sigma_1^{}\) is determined by the three kinematic angles \(\theta\), \(\theta_*^{}\), and \(\phi_*^{}\). Thus, under high energy expansion we can write down the asymptotic behavior of Eq. 36 as follows: \[\begin{align} \label{eq:ds1x}\frac{\text{d}^3\sigma_1}{\,\text{d}\theta \text{d}\theta^{}_*\text{d}\phi^{}_*} =h_1^V\!\!\left[\frac{\sqrt{s\,}\,}{M_Z} P_1^{}(\cos\theta,\cos\theta^{}_*)\sin\phi^{}_* \!+O\big({{s}^0}\big)\!\right] \!\!+\! h_2\!\!\left[\frac{ s}{M_Z^2} P_2^{}(\cos\theta,\cos\theta^{}_*)\sin2\phi^{}_* \!+\!O\big(\sqrt{s\,}\big)\!\right]\!, \end{align}\tag{43}\] where the sign of functions \(P_{1}\) and \(P_2\) are determined by \(\cos\theta\cos\theta^{}_*\). We note that the form factors \(h_2^{}\) has leading energy power of \(s^1\) with angular dependence of \(\sin\phi^{}_*\), whereas the form factors \(h_1^{V}\) has leading energy power of \(\sqrt{s\,}\) with angular dependence of \(\sin2\phi^{}_*\),

To separate the positive and negative contributions of the interference term, we can exploit their multidimensional distributions.To this end, we define a new variable \(\omega \equiv\cos\theta \cos\theta^*\) [8]. We present in Fig. 6 the distributions of the CPV nTGC form factors \((h_1^Z,\, h_1^\gamma,\,h_2^{})\) with simulated events in the \((\phi^{}_*,\, \omega)\) plane for the reaction \(e^+e^-\!\!\rightarrow\!Z\gamma\) (with \(Z\!\!\rightarrow\!\ell^-\ell^+\)) for collision energy \(\sqrt{s\,}\!=\!250\,\)GeV [as shown in plots (a)-(c)] and \(\sqrt{s\,}\!=\!3\,\)TeV [as shown in plots (d)-(f)]. In each plot, the red (blue) points present the case with positive (negative) interference term. Fig. 6 shows that positive-contribution signal events (red points) and negative-contribution signal events (blue points) are clearly separated in the \((\phi_*^{},\, \omega)\) plane. We have 8 regions which are divided by signs of \((\omega,\,\sin\phi^{}_*,\,\sin2\phi_*)\). For Fig. 6, we find that for each form factor \(h_1^V\) or \(h_2^{}\) the contributions of the sub-leading terms in Eq.@eq:eq:ds1x can have different angular dependence on \(\phi_*^{}\), \(\theta\), and \(\theta_*^{}\), which are still visible at \(\sqrt{s\,} \!=\!250\) GeV; whereas at \(\sqrt{s\,} \!=\!3\) TeV, the contributions of the sub-leading terms in each case are negligible and the leading terms in Eq.@eq:eq:ds1x dominate over the interference cross section. In consequence, for the much higher collision energy \(\sqrt{s\,} \!\!=\!3\)TeV, the boundaries of the positive signal region and the negative signal region are more aligned with the lines of \(\omega\! =\! 0\), \(\sin\phi^{}_*\!=\! 0\), and \(\sin2\phi_*^{}\!=\! 0\). On the other hand, for the lower collision energy \(\sqrt{s\,} \!\!=\!250\)GeV, the boundaries of the positive signal region and the negative signal region become more irregular and have visible deviations from the lines of \(\omega\! =\! 0\), \(\sin\phi^{}_*\!=\! 0\), and \(\sin2\phi_*^{}\!=\! 0\).

Employing multivariable analysis (MVA), we have constructed a decision boundary in the \((\phi_*^{},\omega)\) plane to distinguish effectively between events with positive and negative interference cross sections. We show in Table 5 a comparison of the sensitivity reaches for the original method and the improved MVA method. From Table 5, we find that, since the boundaries of positive and negative contributions are very close to the boundaries of simple kinematic cuts on the angular variables, the MVA method only improves slightly the sensitivity reaches over the original method.

3.4 Comparison of Sensitivities between Lepton and Hadron Colliders↩︎

In this subsection, we compare the sensitivity bounds on probing the nTGCs at \(e^+e^-\) colliders versus hadron colliders. We note that the hadron colliders (such as the LHC and future high-energy \(pp\) colliders) generally have higher collision energy but lower luminosity than the lepton colliders (such as the CEPC, FCC-ee, ILC and CLIC). Since the squared term of nTGC has higher energy-power dependence than the interference term, so its contribution dominates the cross section over the interference term for \(pp\) colliders. Moreover the contribution of a CPV nTGC to the scattering amplitude differs from that of the related CPC nTGC by an overall factor of the imaginary number \(\mathrm{i}\) (apart from simple normalization factors of the CPC versus CPV nTGCs). Hence, at hadron colliders the contributions of the CPC and CPV nTGCs to the cross section are dominated by their squared terms respectively, and thus they are mainly the same (apart from simple normalization factors of the CPC versus CPV nTGCs) [5].

Then, we make use of our previous nTGC analyses for hadron colliders [[6]][5], and present a comparison of the sensitivity bounds in Table 6 and Fig. 7 for probing the CPV nTGC faorm factors \((h_2^{},h_1^V)\) at the high energy \(e^+e^-\) colliders versus \(pp\) colliders. For instance, we find that the sensitivity bounds on the CPV nTGC factor \(h_2^{}\) at the LHC are close to the corresponding bounds on \(h_2^{}\) at a 1 TeV \(e^-e^+\) collider, whereas the sensitivity bounds on \(h_1^V\) at the LHC are close to the corresponding bounds at a 250 GeV \(e^+e^-\) collider. The high energy \(pp\)(100 TeV) colliders can provide the best sensitivity reaches on probing the CPV nTGCs.

4 Conclusions↩︎

The origin of the cosmological baryon asymmetry requires CP violation beyond [20] the Kobayashi-Maskawa mechanism [11] in the SM, and we have shown in this work that the neutral triple gauge couplings (nTGCs) generated by dimension-8 SMEFT operators offer new opportunities to search for additional sources of CP violation.As discussed in Section 2, there are 3 relevant CPV dimension-8 operators in Eq.(2 ) involving Higgs fields, and we have identified 2 additional CPV dimension-8 operators (7 ) that do not involve Higgs fields.The corresponding CPV neutral triple gauge vertices (nTGVs) are given in Eqs.(6 ) and (10 ), and their relations to the corresponding nTGC form factors that are compatible with the full electroweak gauge group SU(2)\(_{\mathrm L}^{}\otimes\)U(1)\(_{\mathrm Y}^{}\) with spontaneous symmetry breaking are given in Eq.(27 ).

The sensitivity reaches on these CPV nTGV form factors and on the cutoff scales of the corresponding CPV nTGC operators were presented in Fig. 2 (d) and Tables [tab:f-p] and [tab:o-p], respectively. We performed the analysis for \(e^+e^-\) (or \(\mu^+\mu^-\)) colliders with collision energies \(\sqrt{s\,}\!=\!(0.25,0.5,1,3,5)\)TeV, choosing integrated luminosities of 5 ab\(^{-1}\) in each case and considering both unpolarized and polarized \(e^\mp\) beams. The sensitivity reaches on probing the cutoff scales of the CPV dimension-8 SMEFT operators such as \({\mathcal{O}}_{\!\tilde{G}_+}^{}\!\!\!\) were found to range from about 1 TeV at \(\sqrt{s\,} \!=\!250\) GeV to about \((8-9)\)TeV at \(\sqrt{s\,} \!=\!3\) TeV, and to over 10 TeV at \(\sqrt{s\,} \!=\!5\) TeV, as shown in Table 3. On the other hand, we estimated the sensitivity reaches on the CPV nTGC form factors and found that the sensitivity reaches on these form factors vary from \({O}(10^{-4})\) to \({O}(10^{-6}\!-\!10^{-8})\) for the \(e^+e^-\) collision energy from 250 GeV to \((3\!-\!5)\)TeV, as shown in Table [tab:2new]. We also demonstrated in Table [tab:4new] that using the conventional CPV nTGC form factor formulas (which are incompatible with spontaneous electroweak gauge symmtry breaking of the SM) lead to fake sensitivities to the nTGCs which are over-strong by a factor of \(2\!-\!3\) for the \(e^+e^-\) collision energy \(\sqrt{s\,}\!=\!250\)GeV, by a factor of \(O(10)\) for \(\sqrt{s\,}\!=\!(0.5\!-\!1)\)TeV, and by a factor of \(O(10^2)\) for \(\sqrt{s\,}\!=\!(3\!-\!5)\)TeV. Hence it is important to use our present new formulation of the CPV nTGC form factors that is fully consistent with the spontaneous electroweak gauge symmtry breaking. We have further studied the correlations between the sensitivities to pairs of CPV form factors, as shown in Figs. 4 and 5. An exploratory multivariable analysis (MVA) was found to provide a minor improvement in the sensitivities to the CPV nTGV form factors. We have further presented a comparison of the sensitivity bounds in Table 6 and Fig. 7 for probing the CPV nTGC faorm factors \((h_2^{},h_1^Z,h_1^{\gamma})\) at the high energy \(e^+e^-\) colliders versus \(pp\) colliders.

Our results demonstrate the interest in searching for CP-violating nTGCs in \(e^+ e^-\) (or, \(\mu^+ \mu^-\)) collisions at center-of-mass energies of 250 GeV and beyond.As we have shown, the prospective sensitivity reaches on probing the CPV SMEFT operator scales extend significantly beyond the center-of-mass energies considered. Since these operators are presumably generated by certain high-energy ultraviolet-complete extension of the SM, studies of CP-violating nTGCs could offer valuable insights to understanding its dynamics.

Acknowledgements
The work of J.E.was supported in part by the United Kingdom STFC Grant ST/T000759/1. The works of HJH and RQX were supported in part by the National Natural Science Foundation of China (NSFC) under Grants 12175136 and 12435005, by the Shenzhen Science and Technology Program (under Grant JCYJ20240813150911015), by the State Key Laboratory of Dark Matter Physics, by the Key Laboratory for Particle Astrophysics and Cosmology (MOE), and by the Shanghai Key Laboratory for Particle Physics and Cosmology. RQX is also supported in part by the National Science Foundation of China under Grants Nos. 12175006, 12188102, 12061141002, by the Ministry of Science and Technology of China under Grants No. 2023YFA1605800, and by the state key laboratory of nuclear physics and technology.

References↩︎