Detect anomalous quartic gauge couplings at muon colliders with quantum kernel k-means

Significant developments have been made in the field of quantum computing. The development of quantum computers has progressed from theoretical models to practical applications, with quantum processors now capable of performing complex calculations at significantly faster speeds than their classical counterparts. Researchers are continuing to advance the frontiers of quantum computing, resulting in groundbreaking developments in hardware, algorithms, and applications Arute:2019zxq . The capacity to process vast quantities of data at hitherto unattainable speeds renders quantum computing a potentially transformative force across a multitude of disciplines.

Meanwhile, as the large hadron collider (LHC) experiment enters the post-Higgs discovery era, physicists have begun to work on the search for new physics (NP) beyond the Standard Model (SM) Ellis:2012zz . The search for NP has now become one of the frontiers of high-energy physics (HEP), who frequently entails the examination of extensive datasets, generated by means of particle collisions or other experimental procedures. The potential for quantum computing to significantly accelerate data processing and analysis makes it an invaluable tool for advancing the detection of NP signals. Despite quantum computing is still in the era of noisy intermediate-scale quantum (NISQ) devices Preskill:2018jim ; Arute:2019zxq ; CG2023 , its applications in various aspects of HEP has already been discussed Zhu:2024own ; Carena:2022kpg ; Bauer:2022hpo ; Roggero:2018hrn ; Roggero:2019myu ; Gustafson:2022xdt ; Lamm:2024jnl ; Carena:2024dzu ; Atas:2021ext ; Li:2023vwx ; Cui:2019sfz ; Zou:2021pvl ; Georgescu:2013oza ; Lamm:2019uyc ; Li:2021kcs ; Echevarria:2020wct ; Jordan:2011ci ; Mueller:2019qqj ; Chou:2023hcc ; Bauer:2019qxa .

In the phenomenological studies of NP, the SM Effective Field Theory (SMEFT) is frequently used in recent years. The SMEFT framework extends the SM to incorporate high-dimensional operators that capture potential NP effects Weinberg:1979sa ; Grzadkowski:2010es ; Brivio:2017vri ; Buchmuller:1985jz . Research on SMEFT has focused on dimension-6 operators, however, from a phenomenological point of view, the dominant effect in many cases occurs in dimension-8 operators Ellis:2018cos ; Ellis:2019zex ; Ellis:2020ljj ; Gounaris:2000dn ; Gounaris:1999kf ; Senol:2018cks ; Fu:2021mub ; Degrande:2013kka ; Jahedi:2022duc ; Jahedi:2023myu ; Ellis:2017edi ; Gounaris:1999kf . In addition, dimension-8 operators are also important for convex geometric perspective operator spaces Bi:2019phv ; Zhang:2020jyn ; Yamashita:2020gtt . As a result, the dimension-8 operators are increasingly being focused on. For one generation of fermions, there are $895$ different baryon number conserving dimension-8 operators. It is necessary to conduct a detailed kinematic analysis of each of these operators. As the number of operators to be considered increases, the efficiency of the process tends to decline.

In order to facilitate the efficient analysis of data, anomaly detection (AD) machine learning (ML) algorithms have been employed in previous studies within the field of HEP to search for NP signals Zhang:2023khv ; Zhang:2023ykh ; Dong:2023nir ; Zhang:2023yfg ; Yang:2022fhw ; Yang:2021kyy ; Jiang:2021ytz ; Vaslin:2023lig ; Kuusela:2011aa ; Collins:2018epr ; Atkinson:2022uzb ; Kasieczka:2021xcg ; Farina:2018fyg ; Cerri:2018anq ; vanBeekveld:2020txa ; CrispimRomao:2020ucc ; Ren:2017ymm ; Abdughani:2018wrw ; Ren:2019xhp . This paper investigates the application of a quantum ML (QML) algorithm to search for NP, i.e., quantum kernel k-means (QKKM). The choice of the k-means algorithm among various ML algorithms is motivated by two factors. Firstly, it has been demonstrated to be effective in phenomenological studies of NP Zhang:2023yfg . Secondly, the kernel k-means algorithm is compatible with quantum computers. One potential advantage of QKKM is that, it is pointed out that multi-state swap test on quantum computers can compute inner products of multiple vectors simultaneously Liu:2022jsp ; Fanizza:2020qjq . At the same time, quantum kernels have the potential to transform nonlinear data into linearly separable forms through quantum feature mapping Liu:2020lhd . This paper aims to compare several different quantum kernel methods, all of which are inner products and have the potential to be accelerated by multi-state swap test.

In this paper, we take the study of dimension-8 operators contributing to anomalous quartic gauge couplings (aQGCs) as an example. The sensitivity of the vector boson scattering (VBS) process to aQGCs and the increasing phenomenological research on aQGCs have led to a wide interest in aQGCs Green:2016trm ; Chang:2013aya ; Anders:2018oin ; Zhang:2018shp ; Bi:2019phv ; Guo:2020lim ; Guo:2019agy ; Yang:2021pcf ; Yang:2020rjt . Meanwhile, LHC has been closely following the aQGCs ATLAS:2014jzl ; CMS:2020gfh ; ATLAS:2017vqm ; CMS:2017rin ; CMS:2020ioi ; CMS:2016gct ; CMS:2017zmo ; CMS:2018ccg ; ATLAS:2018mxa ; CMS:2019uys ; CMS:2016rtz ; CMS:2017fhs ; CMS:2019qfk ; CMS:2020ypo ; CMS:2020fqz . With the increasing luminosities on future colliders, the muon colliders can achieve higher energies and luminosities while providing a cleaner experimental environment that is less impacted by the QCD background than the hadron colliders Buttazzo:2018qqp ; Delahaye:2019omf ; Costantini:2020stv ; Lu:2020dkx ; AlAli:2021let ; Franceschini:2021aqd ; Palmer:1996gs ; Holmes:2012aei ; Liu:2021jyc ; Liu:2021akf . In order to study aQGCs, the process $\mu^{+}\mu^{-}\rightarrow\nu\bar{\nu}\gamma\gamma$ at muon colliders is used as a testbed. This process not only lends itself to the study of aQGCs, a NP operator of widely interest, but also provides a place to validate ML algorithms due to the information lost by final-state neutrinos. The AD event selection strategy with QKKM is employed to search for aQGCs signals, and expected coefficient constraints, i.e. the projected sensitivities are analyzed. It is worth noting that, as an AD algorithm, using the QKKM to search for aQGCs signals does not depend on the studied process.

The rest of the paper is organized as follows. In Section 2, a brief introduction to aQGCs and the $\mu^{+}\mu^{-}\rightarrow\nu\bar{\nu}\gamma\gamma$ process is given. The event selection strategy of QKKM is discussed in Section 3. Section 4 presents numerical results for the expected coefficient constraints. Section 5 is a summary of the conclusions.

The frequently used dimension-8 operators contributing to aQGCs can be classified into three categories, scalar$/$longitudinal operators $O_{S_{i}}$, mixed transverse and longitudinal operators $O_{M_{i}}$ and transverse operators $O_{T_{i}}$, respectively Eboli:2006wa ; Eboli:2016kko . The operators involved in the $\mu^{+}\mu^{-}\rightarrow\nu\bar{\nu}\gamma\gamma$ process are $O_{M_{i}}$ and $O_{T_{i}}$,

where $f_{M_{i}}$ and $f_{T_{j}}$ are dimensionless Wilson coefficients, and $\Lambda$ is the NP energy scale. The process of $\mu^{+}\mu^{-}\rightarrow\nu\bar{\nu}\gamma\gamma$ at the muon collider can be contributed by the operators $O_{M_{0,1,2,3,4,5,7}}$ and $O_{T_{0,1,2,5,6,7}}$, where $O_{M_{0,1,5,7}}$ are not considered due to sensitivities within current coefficient constraints and,

where $\widehat{W}\equiv\vec{\sigma}\cdot{\vec{W}}/2$ with $\sigma$ being the Pauli matrices and ${\vec{W}}=\{W^{1},W^{2},W^{3}\}$, $B_{\mu}$ and $W_{\mu}^{i}$ are $U(1)_{\rm Y}$ and $SU(2)_{\rm I}$ gauge fields, $B_{\mu\nu}$ and $W_{\mu\nu}$ correspond to the field strength tensors, and $D_{\mu}$ is the covariant derivative. For the process $\mu^{+}\mu^{-}\rightarrow\nu\bar{\nu}\gamma\gamma$, the diagrams induced by $O_{M_{i}}$ and $O_{T_{i}}$ operators are shown in the Fig. 1, and the Fig. 2 shows the typical diagrams in the SM. Since aQGC decouple from anomalous triple gauge couplings (aTGCs) starting from dimension-8, we consider only dimension-8 operators.

As a high-energy collider, the muon collider is also considered a gauge boson collider and is therefore well suited to the study of aQGCs. The operators that contribute independently to aQGCs and are not related to anomalous triple gauge couplings start at dimension-$8$. The muon collider is therefore well suited to study the signal of the dimension-8 aQGCs in the VBS processes. Among the many VBS processes, those in which the forward-moving particles are neutrinos have an advantage because there is no need to detect charged particles near the direction of the beams. These are processes that contain $WW\to VV$ sub-processes. Among these processes, the one in which the final state $VV$ are photons has the least electroweak vertices and is therefore more advantageous, which is the $\mu^{+}\mu^{-}\to\nu\bar{\nu}\gamma\gamma$ process that is the focus of this work. Since there are no indications of higher-dimensional operators yet, we restrict ourselves to focusing only on the projected sensitivities of the aQGCs, ignoring the effects of other SMEFT operators.

As an Effective Field Theory (EFT), the SMEFT is only valid under the NP energy scale. The high center-of-mass (c.m.) energy achievable at muon colliders offers an excellent opportunity to detect potential NP signals, while at the same time, raises concerns on the validity of the SMEFT. Previous studies have extensively employed partial wave unitarity as a criterion for assessing the validity of the SMEFT Dong:2023nir ; Yang:2021pcf ; Guo:2020lim ; Yue:2021snv ; Fu:2021mub ; Yang:2022ilt ; Layssac:1993vfp ; Corbett:2017qgl ; Almeida:2020ylr ; Kilian:2018bhs ; Kilian:2021whd ; Perez:2018kav . For the process $W^{-}_{\lambda_{1}}W^{+}_{\lambda_{2}}\rightarrow{\gamma}_{\lambda_{3}}{\gamma}_{\lambda_{4}}$ with $\lambda_{1,2}=\pm 1,0$ and $\lambda_{3,4}=\pm 1$ correspond to the helicities of the vector bosons, in the c.m. frame with z-axis along the flight direction of ${W}^{-}$ in the initial state, the amplitudes can be expanded as Jacob:1959at ,

where $\theta$ and $\phi$ are zenith and azimuth angles of $\gamma_{\lambda_{3}}$, $\lambda=\lambda_{1}-\lambda_{2}$, $\lambda^{{}^{\prime}}=\lambda_{3}-\lambda_{4}$ and $d^{J}_{\lambda\lambda^{{}^{\prime}}}(\theta)$ is the Wigner D-functions. The partial wave unitarity bound is $|T^{J}|\leq 2$ Corbett:2014ora .

In Refs. Guo:2019agy ; Yang:2021ukg , the results of partial wave unitarity bounds on coefficients of the $\gamma\gamma WW$ vertices have been obtained in the study of $\gamma\gamma\to W^{+}W^{-}$ VBS process at the LHC. The dimension-8 operators contribute to five different $WW\gamma\gamma$ vertices, with each vertex being contributed to by only one operator. Consequently, the partial wave unitarity bounds on coefficients of the $\gamma\gamma WW$ vertices can be directly translated to the partial wave unitarity bounds on operator coefficients for the $WW\to\gamma\gamma$ by assuming one operator at a time. The strongest partial wave unitarity bounds w.r.t. the $WW\to\gamma\gamma$ process are,

The maximum possible c.m. energy for the subprocess $WW\rightarrow\gamma\gamma$ is identical to the c.m. energy of the process $\mu^{+}\mu^{-}\rightarrow\nu\bar{\nu}\gamma\gamma$. At muon colliders, we consider two cases of the expected energies, $\sqrt{s}=10\;\rm{TeV}$ and $\sqrt{s}=14\;\rm{TeV}$ Palmer:1996gs , the tightest unitarity bounds are listed in Table 1.

K-means AD algorithm can be utilized to address interference Zhang:2023yfg . However, in this paper, due to the limited computational resources, we do not consider the interference terms for simplicity. The contribution of the SM (denoted as $\sigma_{\rm SM}$), the NP (denoted as $\sigma_{O_{X}}$ where $X$ is the name of operator), and the interference between the SM and NP (denoted as $\sigma^{int}_{O_{X}}$) for different operators when the coefficients are the upper bounds in Table 1 at $\sqrt{s}=10\;\rm{TeV}$ and $14\;\rm{TeV}$ are listed in Table 2. We only study operators where the cross-sectional ratio of the interference term to the NP contribution is less than $O_{M_{3}}$ (for $O_{M_{3}}$, $\sigma^{int}_{O_{M_{i}}}/\sigma_{O_{M_{i}}}$ is $14.56\%$ at $\sqrt{s}=10\;{\rm TeV}$). That is, we focus on the operators $O_{M_{2,3,4}}$ and $O_{T_{5}}$.

The smaller the coefficient, the more important the interference terms are. From Table 1, it can be seen that the unitarity bounds are not yet at a level where the interference terms become dominant. Dimensional analysis indicates that $\sigma^{int}_{NP}\sim sf_{X}/\Lambda^{4}$ and $\sigma_{NP}\sim s^{3}\left(f_{X}/\Lambda^{4}\right)^{2}$, when $\sigma^{int}_{NP}\approx\sigma_{NP}$, $f_{X}/\Lambda^{4}\leq s^{-2}$. At $\sqrt{s}=10\;{\rm TeV}$, the rough estimation is that, the interference terms become important when $f_{X}/\Lambda^{4}\leq 10^{-4}\;{\rm TeV}^{-4}$. From the numerical results that follow, the interference term can be ignored in the following sections. This is mainly due to the fact that the constraints on $f_{X}$ are not small enough. Moreover, we mainly consider the $O_{M,T}$ operators, where the dominant contributions come from the scattering of the transversely polarized $W$, which is different from the the case of the SM where the contribution of scattering of longitudinally polarized $W$ dominants, and thus the interference terms are suppressed.

The relative contributions of the VBS processes to the annihilation process are contingent upon the specific process under consideration Chen:2022yiu ; Han:2024gan , particularly in scenarios where the interference term is important. For the process $\mu^{+}\mu^{-}\to\nu\bar{\nu}\gamma\gamma$, a comparison of the contributions of the VBS and tri-boson induced by $O_{T_{5}}$ when the interference term is not considered is provided in Ref. Yang:2020rjt . In this case, the VBS contribution exceeds that of the tri-boson at approximately $\sqrt{s}=5\;{\rm TeV}$. For the $O_{M}$ operators, the contributions are presented in the  A. The tri-boson contribution is at the next order of $M_{Z}^{2}/s$ compared to the VBS. Consequently, it can be expected that at a smaller $s$ compared with $O_{T_{5}}$, the VBS contribution will dominant. This also explains the focus on the $O_{M}$ operators in this study, since the annihilation processes usually have larger interferences Chen:2022yiu ; Han:2024gan ; Yang:2020rjt which is ignored.

As the luminosities of future colliders continue to increase, so does the quantity of data that must be processed which presents a significant challenge to conventional computing. Nevertheless, forecasts by IBM, Google, and IonQ indicate that within the next decade, it will become feasible to execute practical computational tasks using quantum computers with thousands of qubits. This coincides with the high luminosity upgrade of the LHC Apollinari:2015wtw and future colliders such as muon colliders. In recent years, numerous QML algorithms have been the subject of study within the field of HEP, such as QSVM, quantum variational classifiers etc Guan:2020bdl ; Wu:2021xsj ; Wu:2020cye ; Terashi:2020wfi ; Zhang:2023ykh

The inherent properties of coherence and entanglement in quantum systems endow quantum computers with powerful parallel computing capabilities. The main motivation for this study lies in its potential future applications. In contrast to classical computers, quantum computers are capable of storing and processing a greater quantity of data simultaneously. It is conceivable that in the future, the data that must be processed may originate directly from quantum computers. In addition, quantum computer can implement kernel functions that are difficult to achieve with classical computers Havlicek:2018nqz ; Liu:2020lhd ; Sherstov:2020qax . In this section, we use the QKKM to verify its feasibility in searching for NP.

Ignoring the interference between the SM and the aQGCs, the cross-section after cut can be expressed as,

where $\sigma_{\rm SM}$ and $\sigma_{\rm NP}$ are cross-sections of the SM and NP contributions, respectively. The NP contribution is the one when $f_{X}=\tilde{f}_{X}$, where $f_{X}$ is the operator coefficient, and $\tilde{f}_{X}$ is the upper bounds of partial wave unitarity bounds listed in Table 1. $\epsilon_{\gamma}^{\rm SM}$ and $\epsilon_{\gamma}^{\rm NP}$ are the cut efficiencies of the $N_{\gamma}$ cut, $\epsilon_{\alpha}^{\rm SM}$ and $\epsilon_{\alpha}^{\rm NP}$ are the cut efficiencies of the QKKM event selection strategy. Numerical results of $\sigma_{\rm SM}$ and $\sigma_{\rm NP}$ are listed in Table 2. $\epsilon_{\gamma}^{\rm SM}$, $\epsilon_{\gamma}^{\rm NP}$ are listed in Table 7.

The expected coefficient constraints after cuts can be estimated by the signal significance defined as,

where $N_{s,bg}$ are the event numbers of the signal and background, $N_{s}=(\epsilon_{\gamma}^{\rm NP}\epsilon_{\alpha}^{\rm NP}\sigma_{\rm NP}{f^{2}}/{f_{i}^{2}})L$ and $N_{bg}=(\epsilon_{\gamma}^{\rm SM}\epsilon_{\alpha}^{\rm SM}\sigma_{\rm SM})L$, and $L$ is the luminosity. The integrated luminosities in both “conservative” and “optimistic” cases Black:2022cth ; Accettura:2023ked are considered.

To maximize signal significance, an appropriate threshold value $d_{th}$ is selected. Taking the real vector kernel operator $O_{M_{2}}$ at $10\;\rm TeV$ as an example. As shown in Fig. 7, $S_{stat}$ varies with $d_{th}$ within the given range. The corresponding $d_{th}$ value is chosen as the final threshold when $S_{stat}$ reaches its maximum. The shape of the function $S_{stat}(d_{th})$ in other scenarios is similar to that shown in Fig. 7. The results of $d_{th}$ for the real vector kernel, the complex vector kernel, and the hardware-efficient kernel are listed in Table 8. To compare the results of quantum and classical algorithm, the classical kernel is included. The expected coefficient constraints are calculated using the classical kernel from the scikit-learn Pedregosa:2011ork package, following the same steps as in Ref. Zhang:2023yfg . The $d_{th}$ for the classical kernel is also listed in Table 8. In quantum computing, since the kernel is computed using inner products, the value of $d_{th}$ always lies between $0$ and $1$, which is not the case for the classical k-means where the definition of $d$ is different which is the Euclidean distance Zhang:2023yfg .

The $\epsilon_{\alpha}^{\rm SM}$ and $\epsilon_{\alpha}^{\rm NP}$ are defined as the cut efficiencies of QKKM event selection strategy. The cut efficiencies of the four different kernels when the $d_{th}$ are chosen as the ones listed in Table 8 are listed in Table 9. As can be seen from Table 9, for the complex vector kernel, a relatively strict $d_{th}$ is taken, which is due to the fact that the background can be suppressed to a very low level. For hardware-efficient kernel, a relatively loose $d_{th}$ is taken due to the fact that the tail of the background events in the NP region. All the $d_{th}$s are chosen according to $\mathcal{S}_{stat}$.