Search for WIMPs at future $\mu^{+}\mu^{+}$ colliders

The presence of dark matter in the universe stands as one of the most compelling pieces of evidence for physics beyond the standard model Aghanim et al. (2020). Despite this certainty, the true nature of dark matter remains enigmatic, leading to the exploration of various proposed candidates Feng (2010). Among these, the weakly interacting massive particle (WIMP) Lee and Weinberg (1977) is a prominent candidate and has been extensively studied due to its ability to yield the correct dark matter abundance through the freeze-out mechanism.

The supersymmetric extension of the standard model (SSM) provides an intriguing framework for WIMP dark matter Jungman et al. (1996) since the lightest supersymmetric particle is stable with conserved $R$-parity Farrar and Fayet (1978). In particular, the wino or the Higgsino, the supersymmetric partner of the SU$(2)_{L}$ gauge boson or the Higgs boson, respectively, contains electrically neutral components and are the natural candidates for dark matter in the SSM. If other supersymmetric particles are much heavier, the electroweak gauge interaction determines the relic abundance; the correct relic abundance is thermally produced for the $\sim 2.7\,\text{TeV}$ wino or the $\sim 1\,\text{TeV}$ Higgsino Hisano et al. (2007); Cirelli et al. (2007). Such a scenario is known as the pure wino or Higgsino, respectively. These scenarios are one of the most motivated targets for WIMP searches, referred to as “the thermal mass targets”. The pure wino and Higgsino, a $\text{SU}(2)_{L}$ triplet fermion and a $\text{SU}(2)_{L}$ doublet fermion with the hypercharge $1/2$, respectively, can be interpreted as the simplest candidates of the dark matter; the neutral component of such electroweak multiplets can, without any other interaction than the electroweak interaction, automatically be stable and be the WIMP dark matter. It is thus studied from a bottom-up point of view, called the minimal dark matter Cirelli et al. (2006, 2007).

Collider experiments present a promising avenue for WIMP search. Much experimental effort has been made in the ATLAS Aad et al. (2013); Aaboud et al. (2018); Aad et al. (2022) and CMS Khachatryan et al. (2015); Sirunyan et al. (2018, 2020) experiment at the LHC. The WIMP is also one of the main targets of future colliders Arkani-Hamed et al. (2016); Ellis et al. (2019); Han et al. (2021). Among these, muon colliders emerge as particularly promising candidates due to their capability to achieve high-energy collisions with substantial luminosities. Muons, being heavier than electrons, experience suppressed synchrotron radiation, enabling their acceleration in circular accelerators and enhancing both luminosity and energy. Additionally, muons, being elementary particles unlike protons, carry the entire energy for collision. This unique feature positions high-energy $\mu^{+}\mu^{-}$ colliders as potent tools for probing WIMPs with good significance Han et al. (2021).

A significant challenge faced in the development of muon colliders lies in creating bright $\mu^{\pm}$ sources with low emittance Delahaye et al. (2019, 2013); Bogomilov et al. (2020); Antonelli et al. (2016). For $\mu^{+}$ beams, however, there is an existing technique Kondo et al. (2018) to achieve sufficiently low emittance Delahaye et al. (2013); it uses the ionization of muonium atoms and can be applied to only positive muons. This fact opens the possibility of considering high-energy $\mu^{+}\mu^{+}$ colliders as a more realistic option than $\mu^{+}\mu^{-}$ colliders Heusch and Cuypers (1996); Hamada et al. (2022); End (2022); Hamada et al. (2023); Lichtenstein et al. (2023); Fridell et al. (2023); Dev et al. (2023). Ref. Hamada et al. (2022), for example, pointed out that a multi-TeV $\mu^{+}\mu^{+}$ collider is possible with currently available technologies. It is thus of interest to examine the capability of future $\mu^{+}\mu^{+}$ colliders to discover the WIMP.

At the LHC Aad et al. (2013); Aaboud et al. (2018); Aad et al. (2022); Khachatryan et al. (2015); Sirunyan et al. (2018, 2020) or future hadron Arkani-Hamed et al. (2016) or $e^{+}e^{-}$ colliders Ellis et al. (2019), the most promising technique for the WIMP searches is to find the disappearing track of the charged component of the WIMP. However, at muon colliders, detecting disappearing tracks may be challenging because muon colliders notoriously suffer from large beam-induced backgrounds Bartosik et al. (2022). Muons’ finite lifetime leads to decay in flight, resulting in numerous decays per meter. Decay products hitting materials produce additional particles reaching detectors. With the timing cuts, the hit densities in the trackers are reduced but still expected to be $\mathcal{O}(100)/\text{cm}^{2}$ at the innermost layer Bartosik et al. (2022), which is one-order larger than the HL-LHC ATL (2017). We thus need to develop detailed designs of the detector and improve the track reconstruction algorithms Capdevilla et al. (2021); Han et al. (2021).

In this paper, we shift our focus to both (i) the direct production of WIMPs involving an additional particle and (ii) indirect measurement through radiative corrections induced by WIMPs on the elastic Møller scattering cross section of muons at future $\mu^{+}\mu^{+}$ colliders. The pure Higgsino, the wino, and the $\text{SU}(2)$ 5-plet Majorana fermion serve as our primary targets for this study. We will find that the indirect search method is more efficient than the direct search method for probing heavier WIMP with sufficient luminosity because the mass dependence of the indirect methods is weaker than the on-shell production. We will show that multi-TeV $\mu^{+}\mu^{+}$ colliders can reveal the WIMP with masses $\mathcal{O}(\text{TeV})$. In particular, our findings reveal that, with integrated luminosity of $10\,\text{ab}^{-1}$, $80\,\%$ polarization, and $0.1\,\%$ accuracy, $\mu^{+}\mu^{+}$ colliders can probe the thermal mass target of the $1$ TeV Higgsino and $2.7$ TeV wino for the center-of-mass energy of $2\,\text{TeV}$ and $6\,\text{TeV}$, respectively.

The paper is organized as follows. In Sec. II, we review the indirect effect of the WIMP on the elastic scattering between $\mu^{+}$s. We then discuss the direct production search of the WIMP at $\mu^{+}\mu^{+}$ colliders in Sec. III. In Sec. IV, we present our analysis methods and show our results. Finally, the conclusion and discussion are in Sec. V.

In this section, we summarize the indirect effect of the WIMPs Harigaya et al. (2015) on the $\mu^{+}\mu^{+}$ elastic scattering. Let us consider a fermionic WIMP charged under the standard model $\text{SU}(2)_{L}\times\text{U}(1)_{Y}$. The new particle affects the $\mu^{+}\mu^{+}$ elastic cross section through the vacuum polarization of the standard model gauge bosons. The Feynman diagram for the $t$-channel gauge boson exchange process is shown in Fig. 1, where the WIMP effect appears in the photon and $Z$ boson vacuum polarizations. By integrating the WIMP out, the effective Lagrangian is obtained as

where $\mathcal{L_{\textrm{SM}}}$ is the standard model Lagrangian, $W_{\mu\nu}^{a}$ ($B_{\mu\nu}$) is the field strength tensor for $\textrm{SU}(2)_{L}$ ($\textrm{U}(1)_{Y}$) gauge group, $D$ is the covariant derivative and $m$ is the mass of the WIMP. The coefficients $C_{WW}$ and $C_{BB}$ for the $\textrm{SU}(2)_{L}\ n$-plet WIMP with hypercharge $Y$ are given by

where $g$ is the gauge coupling for $\textrm{SU}(2)_{L}$, $\theta_{W}$ is the Weinberg angle, and $\kappa$ is $1\ \quantity(\frac{1}{2})$ for Dirac (Majorana) fermion. $\Pi(x)$ denotes the loop function of the vacuum polarization due to the WIMP loop,

where $\mu$ is the renormalization scale. We set the scale as $\mu=m$ so that $\Pi(x\ll 1)=0$.

In order to evaluate the effect of the WIMP on the $\mu^{+}\mu^{+}$ elastic scattering, let us expand the scattering amplitude perturbatively with the coupling constant. The standard model process contributes to the tree-level amplitude. The tree-level scattering amplitude for $\mu^{+}(p_{1})\mu^{+}(p_{2})\rightarrow\mu^{+}(p_{3})\mu^{+}(p_{4})$ is obtained as

where $h_{i}=L,R$ is the helicity of the initial $i$-th muon, $P_{h}$ is the projection operator onto the helicity $h$ eigenstate, $v(p)$ is the $\mu^{+}$ wave function for momentum $p$, and $C^{V}_{h}$ is given by $C^{\gamma}_{L}=C^{\gamma}_{R}=-e$, $C^{Z}_{L}=\frac{g}{c_{W}}s_{W}^{2}$, and $C^{Z}_{R}=\frac{g}{c_{W}}(-1/2+s_{W}^{2})$.

The effect of the WIMP appears in the 1-loop diagram. By using Eq. (1), the 1-loop WIMP contribution is obtained as

where $C_{\gamma\gamma}=s_{W}^{2}C_{WW}+c_{w}^{2}C_{BB},\ C_{ZZ}=c_{W}^{2}C_{WW}+s_{w}^{2}C_{BB}$ and $C_{\gamma Z}=C_{Z\gamma}=s_{W}c_{W}(C_{WW}-C_{BB})$. The $t$ and $u$ dependence of $\Pi$ induces the shape-difference of angular distribution between the differential cross section with and without the WIMP.

The differential cross section can be also expanded pertubatively. The effect of the WIMP appears in the differential cross section as the interference between the tree-level and 1-loop amplitude. The ratio of the tree-level differential cross sections, $\differential{\sigma}_{\textrm{tree}}/\differential{\cos\theta}$, to the WIMP contribution to the differential cross section, $\differential{\sigma}_{\textrm{WIMP}}/\differential{\cos\theta}$, is given by

where we define $\theta$ as the scattering angle from the beam axis.

The ratio $\Delta(\theta)$ depends on the polarization of the initial muons, because the $\text{U}(1)_{Y}\times\text{SU}(2)_{L}$ charges of right-handed muons and left-handed muons are different. For simplicity, we assume that the polarizations of the muon beams from both sides are the same; let $P_{\mu^{+}}$ be the polarization of the initial muons with $P_{\mu^{+}}=+1$ and $-1$ corresponding to right- and left-handed, respectively. The differential cross section is written as

where $\frac{d\sigma_{IJ}}{d\cos\theta}\ (I,J=L,R)$ is the differential cross section between $I$-handed and $J$-handed muons. As the $\text{SU}(2)_{L}$ coupling is larger for WIMPs and only the right-handed $\mu^{+}$s enjoy the $\text{SU}(2)_{L}$ charge, we expect that the larger $P_{\mu^{+}}$ is, the larger $\Delta(\theta)$ is.

Fig. 2 shows the ratio $\Delta(\theta)$ in several cases: $1$ TeV Higgsino ($n=2,\ Y=\frac{1}{2}$, Dirac fermion), $2$ TeV Wino ($n=3,\ Y=0$, Majorana fermion) and $5$ TeV 5-plet minimal dark matter ($n=5,\ Y=0$, Majorana fermion) with $\sqrt{s}=10$ TeV (where $\sqrt{s}$ is the center of mass energy of the $\mu^{+}\mu^{+}$ system). The muon polarization is assumed to be either $P_{\mu^{+}}=0$ (solid line) or $0.8$ (dashed line). As expected, when the initial $\mu^{+}$s are right-handed, $\Delta$ is enhanced. For example, for the wino or $5$-plet, which do not have the hypercharge, the muon polarization enhances the WIMP contribution by the factor $(1+P_{\mu^{+}})(1+P_{\mu^{+}})$ compared to the one with the unpolarized beam. On the other hand, the background cross section is approximately proportional to $\frac{1}{2}(1+P_{\mu^{+}})(1+P_{\mu^{+}})$ because numerically $\frac{d\sigma_{RR}}{d\cos\theta}\simeq\frac{d\sigma_{LL}}{d\cos\theta}>\frac{d\sigma_{LR}}{d\cos\theta},\frac{d\sigma_{RL}}{d\cos\theta}$ in the standard model. The ratio $\Delta$ is, therefore, expected to be enhanced by factor $\sim 2$ when the muons are polarized as $P_{\mu^{+}}\sim 1$ as observed in Fig. 2.

From the figure, we can see that the WIMP affects the shape of the angular distribution of the elastic scattering. The ratio $\Delta$ has the largest value in the most transverse region, i.e. $\cos\theta=0$ because the vacuum polarization is large with large momentum transfer. On the other hand, the differential cross section is smaller for the larger angle scattering and the statistics are better in the forward regions. This behavior implies that it is important for the indirect search of the WIMP at $\mu^{+}\mu^{+}$ colliders to analyze the angular distribution of $\mu^{+}\mu^{+}$ elastic scattering. We further discuss the detail in Sec IV.

We next examine the direct production search for the WIMP at $\mu^{+}\mu^{+}$ colliders. For the search at $\mu^{+}\mu^{-}$ colliders Han et al. (2021), two types of signals, mono-$\mu$ and mono-$\gamma$, are the main channels. We examine these two channels at $\mu^{+}\mu^{+}$ colliders. Due to the lack of the Drell-Yan (DY) process for $\mu^{+}\mu^{+}$ collisions, unlike $\mu^{+}\mu^{-}$ collisions, we will see that the mono-$\mu$ channel is the most sensitive channel for the detection of the WIMP dark matter.

We use Madgraph5 Alwall et al. (2014) event generator to perform Monte-Carlo simulations and compare the number of signals with the standard model backgrounds. To create the model file for Madgraph5, we use FeynRules 2.0 Alloul et al. (2014). We have checked the consistency of the model file with Ref. Han et al. (2021) and our analytic calculations.

For the mono-$\mu$ channel, the signal processes are followings;

Here, we adopt the improved Weizsäcker-Williams approximation Frixione et al. (1993) to include photons as the initial partons. The representative Feynman diagrams of the signals are almost the same as ones at $\mu^{+}\mu^{-}$ colliders, as shown in Fig. 3.

We apply the same cut as Ref. Han et al. (2021) for the mono-$\mu$ channel. First, we require one hard muon to be detected; the energy of the muon should be larger than $E_{\mu}>0.23\sqrt{s}$ with the pseudorapidity $|\eta|<2.5$. Unlike $\mu^{+}\mu^{-}$ colliders, we cannot impose a forward region cut, as the initial particles are identical. We also impose the missing mass cut, $(p_{\mu^{+},1}^{\text{in}}+p_{\mu^{+},2}^{\text{in}}-p_{\mu^{+}}^{\text{out}})^{2}>4m_{\chi}^{2}$, where $p_{\mu^{+},i}^{\text{in}}$ and $p_{\mu^{+}}^{\text{out}}$ are the four-momentum for the initial $i$-th muon and the final hard muon, respectively, and $m_{\chi}$ is the dark matter mass. With these cuts, the main standard model background processes are

as shown in Fig. 4.

For the mono-$\gamma$ channel, the signal processes are following;

The representative diagrams are shown in Fig. 5. We have two main differences in this channel compared with $\mu^{+}\mu^{-}$ colliders discussed in Ref. Han et al. (2021). First, the DY process is unavailable for $\mu^{+}\mu^{-}$ colliders. This fact significantly reduces the cross section for the case of heavy WIMP and spoils the advantage of the mono-$\gamma$ channel. Second, the third mode, $\mu^{+}\mu^{+}\to\bar{\nu}_{\mu}\bar{\nu}_{\mu}\chi^{+}\chi^{+}$, violates the fermion number. This means that the dark matter must be Majorana and the amplitude is proportional to the Majorana mass term. This mode is available for the pure wino or Majorana $5$-plet, but not for the pure Higgsino as the pure Higgsino is Dirac.¹¹1For the Higgsino, a tiny Majorana mass splitting $\sim\mathcal{O}(100)\,\text{keV}$ between the two neutralinos is needed from the phenomenological point of view to avoid the dark matter direct detection constraints Nagata and Shirai (2015).

For the background of the mono-$\gamma$ channel, as the charge of the initial particles is non-zero, a charged lepton in the final states must be missed. We consider three kinds of process with the charged lepton escaped as the main source of the background as shown in Fig. 6:

We assume a charged lepton cannot be reconstructed if it is in the forward region, $|\eta|>2.5$, or the transverse momentum $p_{T}$ is smaller than $10\,\text{GeV}$, which is taken from the default cuts of Delphes3 de Favereau et al. (2014) for the ATLAS and CMS detector simulation. We require these conditions for the charged leptons in these processes. To reduce the background, we require $|\eta|<1.5$ and $p_{T}>10\,\text{GeV}$ for $\gamma$. We also require the missing mass cut as the mono-$\mu$ channel, $(p_{\mu^{+},1}^{\text{in}}+p_{\mu^{+},2}^{\text{in}}-p_{\gamma}^{\text{out}})^{2}>4m_{\chi}^{2}$, where $p^{\gamma}$ is the four-momentum for the photon.

In Fig. 7, we show the cross sections after the cuts for the standard model backgrounds and the signals from the Majorana 5-plets, wino and Higgsino for the mono-$\mu$ and mono-$\gamma$ channels for $\sqrt{s}=5\,\text{TeV}$. The WIMPs are produced via the vector boson fusion (VBF) process, as shown in Fig 3 and Fig. 5, and the cross sections scale as approximately $sm^{-4}$ Han et al. (2021). We find that the signal cross sections of the mono-$\gamma$ channel are much smaller than the corresponding mono-$\mu$ channel cross sections whereas the mono-$\gamma$ background cross section is larger. For higher multiples, such as the 5-plets, the production cross sections are generically large and we expect better sensitivities for larger mass regions. At $\mu^{+}\mu^{-}$ colliders, the DY processes have a large cross section for such larger mass regions. This is the reason why Ref. Han et al. (2021) concludes that the mono-$\gamma$ channel is effective for the higher multiples. However, the lack of the DY process at $\mu^{+}\mu^{+}$ collider spoils the advantage of the mono-$\gamma$ channel over the mono-$\mu$ channel; even though the background cross sections for the mono-$\gamma$ channel at $\mu^{+}\mu^{+}$ colliders are much suppressed than that at $\mu^{+}\mu^{-}$ colliders, still it is larger than that of the mono-$\mu$ channel after the cuts and the signal cross sections are smaller. Thus, we conclude that the mono-$\mu$ channel is the best discovery mode of the WIMP direct production at $\mu^{+}\mu^{+}$ colliders. For the range of $\sqrt{s}$ which we investigate, the background cross section of the mono-$\mu$ channel scales as $\sim 1/s$ and is $\mathcal{O}(1\mbox{-}10)\,\text{fb}$. Thus, the $\mathcal{O}(0.1)\,\%$ systematic uncertainty does not affect much if the integrated luminosity is $\lesssim\mathcal{O}(10)\,\text{ab}^{-1}$, as reported in Ref. Han et al. (2021).

In this study, we have investigated the prospect of probing the WIMP at the $\mu^{+}\mu^{+}$ colliders. Since the WIMP affects the vacuum polarization of the electroweak gauge bosons, the angular distribution of the $\mu^{+}\mu^{+}$ Møller scattering is altered by the existence of the WIMP. We have focused on the precise measurement of the $\mu^{+}\mu^{+}$ Møller scattering and estimated the capability of the $\mu^{+}\mu^{+}$ collider to search for the WIMP by this indirect search. For the comparison with the indirect search, we have also discussed the capability of the $\mu^{+}\mu^{+}$ collider by the direct production of the WIMP. We have considered the mono-$\mu$ channel and the mono-$\gamma$ channel and found that the mono-$\mu$ channel gives the most severe constraint on the WIMP at the $\mu^{+}\mu^{+}$ collider.

Our main results are shown in Figs. 10, 11, and 12. We show that the indirect search can probe the thermal mass target of the Higgsino and the wino at $\mathcal{O}(1)$ TeV $\mu^{+}\mu^{+}$ collider with $1-10\ \textrm{ab}^{-1}$ luminosity. In addition, the indirect search outperforms the direct production search with a large luminosity, that is, the indirect search achieves a broader discovery reach for the WIMP across a wide range of $\sqrt{s}$, various types of the WIMPs, and different beam polarizations. This is due to the weaker WIMP mass dependence of the indirect search sensitivity. The advantage of the indirect search holds even if the systematic uncertainty extends to 0.3%. Importantly, these results hold true for any new particle with $\text{SU}(2)_{L}\times\text{U}(1)_{Y}$ charges, extending the applicability of our findings.

We assume that the systematic error is universal over the bins and that it is of $\mathcal{O}(0.1)$ %. As shown in Sec. IV, the discovery reach by the indirect search is sensitive to the systematic error and the precise estimation of the systematic errors in the muon Møller scattering is essential to consider the capability of the $\mu^{+}\mu^{+}$ collider. Careful consideration of experimental errors, such as luminosity measurements and detector resolutions, alongside theoretical uncertainties like photon emissions, is vital in accurately assessing the $\mu^{+}\mu^{+}$ collider’s capabilities.

While we have stressed the capability of the indirect detection, the direct production approach remains robust against systematic uncertainties and serves as an independent mode of investigation. We acknowledge that the applied cuts, inherited from previous $\mu^{+}\mu^{-}$ collider studies Han et al. (2021), can be optimized, especially for the mono-$\gamma$ channel in $\mu^{+}\mu^{+}$ colliders. Furthermore, although challenging due to significant beam-induced backgrounds, the possibility of a disappearing track search in $\mu^{+}\mu^{+}$ colliders, akin to $\mu^{+}\mu^{-}$ colliders Capdevilla et al. (2021), offers a promising avenue for future exploration. We leave these topics for future studies Fukuda et al. .

Lastly, we extend our discussion to $\mu^{+}e^{-}$ colliders Hamada et al. (2022). Our study on the radiative correction due to the WIMPs can be applied to the $\mu^{+}e^{-}$ elastic scattering almost in parallel. $\mu^{+}e^{-}$ colliders generically have a greater integrated luminosity but the center-of-mass energy is smaller. As we have discussed, a small $\sqrt{s}$ can be compensated by a large integrated luminosity for the indirect search, and the indirect search may have an advantage over direct production searches. Because the muon energy is assumed to be much larger than the electron energy and the collision is asymmetric for $\mu^{+}e^{-}$ colliders, the sensitivities for the forward scattering may be spoiled, but it can be an interesting alternative.