Physics Beyond the Standard Model at Future Lepton Colliders

Abstract—Possibilities of future lepton colliders in searching for neutral anomaly gauge couplings and dark photons in an invisible decay mode are examined.

1 INTRODUCTION

One of the actual physical problems associated with going beyond the Standard Model (SM) is the search for anomalous quadratic gauge couplings (QGCs). Anomalous interactions $\gamma ZZZ$, $\gamma \gamma ZZ$, and $\gamma \gamma \gamma Z$ were studied in experiments at the LHC collider, where bounds were obtained on such couplings [1, 2]. Future lepton colliders may serve as an alternative [3‒5]. The posibilities of detecting QGCs were investigated for colliders operating in the ${{e}^{ + }}{{e}^{ - }}$ mode [6, 7], as well as in $\gamma e$ [8, 9] and $\gamma \gamma$ modes [10]. One of the goals of this work is to study the possibility of searching for neutral anomalous couplings for vertices $\gamma \gamma \gamma \gamma$ and $\gamma \gamma \gamma Z$ at high-energy lepton colliders [11, 12].

Another problem that we will study is the search for the indicated lepton colliders for the so-called dark photon $A{\kern 1pt} '$. It arises in models when dark matter (DM) fields do not interact directly with SM fields, but only through mediator, which is dark photon $A{\kern 1pt} '$. We will assume that the decay of the dark photon goes into undetectible DM perticles. Note that until now the search for dark photon $A{\kern 1pt} '$ in invisible mode actively was examined at lepton colliders in ${{e}^{ + }}{{e}^{ - }}$ clashes [15–19]. We first consider the production of $A{\kern 1pt} '$ when the collider works in $\gamma {{e}^{ - }}$ mode [20].

2 SEARCH FOR ANOMALOUS COUPLINGS OF GAUGE BOSONS

Anomalous gauge couplings are studied within the framework of effective field theory. For scattering $\gamma \gamma \to \gamma \gamma$, the corresponding Lagrangian is as follows:

$$\mathcal{L}_{{QNGC}}^{{\gamma \gamma \gamma \gamma }} = {{\zeta }_{1}}{{F}_{{\mu \nu }}}{{F}^{{\mu \nu }}}{{F}_{{\rho \sigma }}}{{F}^{{\rho \sigma }}} + {{\zeta }_{2}}{{F}_{{\mu \nu }}}{{F}^{{\nu \rho }}}{{F}_{{\rho \sigma }}}{{F}^{{\sigma \mu }}},$$

(1)

where ${{F}_{{\mu \nu }}}$ is the electromagnetic field strength tensor and the quadratic anomalous couplings ${{\zeta }_{1}}$ and ${{\zeta }_{2}}$ have dimension-4. For process $\gamma \gamma \to \gamma Z$, it is convenient to choose the Lagrangian

$$\begin{gathered} \mathcal{L}_{{QNGC}}^{{\gamma \gamma \gamma Z}} = {{g}_{1}}{{F}^{{\rho \mu }}}{{F}^{{\alpha \nu }}}{{\partial }_{\rho }}{{F}_{{\mu \nu }}}{{Z}_{\alpha }} \\ + \,\,{{g}_{2}}{{F}^{{\rho \mu }}}F_{\mu }^{\nu }{{\partial }_{\rho }}{{F}_{{\alpha \nu }}}{{Z}^{\alpha }} \\ \end{gathered} $$

(2)

with anomalous couplings ${{g}_{1}}$ and ${{g}_{2}}$. Anomalous vertices for both cases are shown in Fig. 1.

Fig. 1.

Anomalous vertices $\gamma \gamma \gamma \gamma$ and $\gamma \gamma \gamma Z$.

Lepton colliders can operate in ${{e}^{ + }}{{e}^{ - }}$, ${{e}^{ - }}\gamma$, and $\gamma \gamma$ modes [21]. Beams of real photons are obtained as a result of inverse Compton scattering of laser photons off electron beams. In this case, photons carry away most of the energy of the parent electrons $x$. Photon distribution ${{f}_{{\gamma /e}}}(x)$ depends on the energy of the laser photon, the energy of the electron, and on their polarization. The differential cross section of the processes under consideration is the convolution of distributions ${{f}_{{\gamma /e}}}(x)$ and squared helicity amplitudes. By calculating the cross sections, one can obtain constraints on the anomalous 4-point gauge couplings. Calculation results [11, 12] are presented in Figs. 2 and 3. Previously, the potential of the LHC and HL-LHC colliders to search for anomalous 4-bosonic couplings was estimated [13, 14]. It follows from the comparison with the above results that the bounds on the couplings that can be obtained at the 3 TeV lepton collider are an order of magnitude more stringent than the bounds from the LHC and HL-LHC.

Fig. 2.

Bounds on anomalous couplings $({{\zeta }_{1}},{{\zeta }_{2}})$ obtained for unpolarized scattering $\gamma \gamma \to \gamma \gamma$ at the CLIC collider with an energy of 3 TeV and integrated luminosity $L = 5000$ fb^–1.

Fig. 3.

Bounds on anomalous couplings ${{g}_{1}},{{g}_{2}}$ obtained for unpolarized scattering $\gamma \gamma \to \gamma Z$ at the CLIC collider with an energy of 3 TeV. Systematic errors are $\delta = 0\%$ (black ellipse), $\delta = 5\%$ (blue ellipse), and $\delta = 10\%$ (red ellipse).

Fig. 4.

Diagrams describing the production of a dark photon in $\gamma {{e}^{ - }}$ scattering, with its subsequent decay into particles of dark matter.

3 SEARCH FOR A MASSIVE DARK PHOTON

We will work in a scenario where dark matter particles do not interact directly with the SM fields, but their interaction is carried out through the exchange of a new vector particle—a dark photon, usually denoted as $A{\kern 1pt} '$. In turn, it kinetically mixes with the SM fields corresponding to the hypercharge group $U{{(1)}_{Y}}$. Accordingly, the initial gauge Lagrangian is chosen in the form

$${{\mathcal{L}}_{{{\text{gauge}}}}} = - \frac{1}{4}{{B}_{{\mu \nu }}}{{B}^{{\mu \nu }}} - \frac{1}{4}\bar {F}_{{\mu \nu }}^{'}\bar {F}{\kern 1pt} {{'}^{{\mu \nu }}} - \frac{\varepsilon }{{2{{c}_{W}}}}\bar {F}_{{\mu \nu }}^{'}{{B}^{{\mu \nu }}},$$

(3)

where ${{B}_{{\mu \nu }}}$ and $\bar {F}_{{\mu \nu }}^{'}$ are strength tensors corresponding, respectively, to the group $U{{(1)}_{Y}}$ and dark group $U(1){\kern 1pt} '$ and $\varepsilon \ll 1$ is a mixing kinetic parameter. After diagonalizing the neutral bosonic fields, we obtain the interaction Lagrangian for the field $A{\kern 1pt} '$:

$$\begin{gathered} {{\mathcal{L}}_{{\operatorname{int} }}} = e{{J}_{\mu }}{{A}^{\mu }} - \varepsilon e{{J}_{\mu }}A{\kern 1pt} {{'}^{\mu }} \\ + \,\,\varepsilon e{\kern 1pt} '{{t}_{W}}J_{\mu }^{'}{{Z}_{\mu }} + e{\kern 1pt} 'J_{\mu }^{'}A{\kern 1pt} '{{{\kern 1pt} }^{\mu }} + {{\mathcal{L}}_{{A{\kern 1pt} '\chi }}}. \\ \end{gathered} $$

(4)

The search for dark photons on ${{e}^{ + }}{{e}^{ - }}$ colliders was carried out by a number of collaborations [15–19] in processes with the production of ${{e}^{ + }}{{e}^{ - }}$, ${{\mu }^{ + }}{{\mu }^{ - }}$, and ${{\pi }^{ + }}{{\pi }^{ - }}$ pairs in the final state. Our goal is to search for a dark photon in the process

$$\gamma + {{e}^{ - }} \to A{\kern 1pt} '\,\, + {{e}^{ - }}$$

(5)

at the future high-energy lepton colliders CEPC, ILC, and CLIC. The corresponding diagrams are shown in Fig. 5. It is assumed that a dark photon with 100% probability decays into the lightest stable dark matter particles that are not detected by detectors (invisible mode of the $A{\kern 1pt} '$ decay).

Fig. 5.

Bounds on the mass of a dark photon ${{m}_{{A{\kern 1pt} '}}}$ and the kinetic mixing parameter $\varepsilon$ with 95% confidence level obtained from the production of a dark photon in the scattering of an unpolarized photon off electron with a polarization of 0.8.

The cross section of process (5) depends on the mass of the dark photon ${{m}_{{A{\kern 1pt} '}}}$ and on the mixing parameter $\varepsilon$. The main background process is the SM process with the production of a single electron and a neutrino–antineutrino pair,

$$\gamma {{e}^{ - }} \to {{e}^{ - }}\nu \bar {\nu }.$$

(6)

As a result, we obtain bounds on the dark photon mass ${{m}_{{A{\kern 1pt} '}}}$ and the kinetic mixing parameter $\varepsilon$ for the three future lepton colliders under study at different collision energies [20]. They are shown in Fig. 5.

These limits for the dark photon mass range of 1–10 GeV can be compared with the experimental limits obtained by the BaBar collaboration [15]. In particular, for the CEPC collider with an energy of 90 GeV, the bounds are 2–4 times more stringent. For CEPC with an energy of 160 GeV and ILC with an energy of 250 GeV, they are comparable with the results of the BaBar.

4 CONCLUSIONS

Future lepton colliders will make it possible to study physics beyond the SM. Therefore, for anomalous 4-photon couplings, they will allow one to measure values up to ${{\zeta }_{1}} = 6.9 \times {{10}^{{ - 4}}}$ TeV^–4 and ${{\zeta }_{2}} = 1.4 \times {{10}^{{ - 4}}}$ TeV^–4, which is an order of magnitude better than the capabilities of the HL-LHC collider with an integral luminosity of 3 ab^–1. Their sensitivity to anomalous couplings ${{g}_{1}}$ and ${{g}_{2}}$ for the vertex $\gamma \gamma \gamma Z$ is $5.0 \times {{10}^{{ - 3}}}$ TeV^–4, while for HL-LHC it does not exceed the value $(1.0{\kern 1pt} - {\kern 1pt} 0.7) \times {{10}^{{ - 1}}}$ TeV^–4.

At the CEPC, ILC, and CLIC colliders, the production of a dark photon that links the fields of dark and ordinary matter can also be successfully studied in a wide mass range of 1 GeV–1 TeV, assuming that it decays only into dark matter particles. Bounds on the kinetic mixing parameter $\varepsilon$ depending on the mass of the dark photon ${{m}_{{A{\kern 1pt} '}}}$ show that, in the region ${{m}_{{A{\kern 1pt} '}}} = 1{\kern 1pt} - {\kern 1pt} 10$ GeV at the CEPC lepton collider, one can obtain bounds on $\varepsilon$ that are several times more stringent than the available experimental BaBar limits. Limits on $\varepsilon$ for the 250 GeV ILC collider are comparable to the BaBar limits.

All said above speaks to the great potential of using future lepton colliders to search for interactions beyond the Standard Model.