cific parameter values [211, 929]. In view of this, adopting more generic MSSM models without implicit correlations between the masses of the various SUSY particles, such as the so-called phenomenological MSSM (pMSSM), has become presently more common for studying SUSY theories at the LHC and at linear $e^+e^-$ colliders.

Still, the benchmark studies carried out for linear colliders keep much of their validity with respect to the sensitivity and accuracy of the measurements, even if the underlying models used in those studied have already been excluded by the LHC data.

Contrary to the case of constrained models, the mass limits for strongly interacting sparticles (in particular the gluino $\tilde{g}$ and the scalar quarks of the first two generations $\tilde{q}$) have little impact on the mass scale of their weakly-interacting counterparts (charginos, neutralinos and scalar leptons) in generic models of Supersymmetry, such as the pMSSM [1023–1026]. Searches for weakly-interacting SUSY particle partners at LHC, of which the first results have recently been reported, are more model-dependent than the case of gluino and squark searches, since they depend not only on the mass splitting with respect to the lightest neutralino, but also on the mass hierarchy of the neutralinos and sleptons, as well as on the neutralino mixing matrix: e.g. the neutralino decay channels which yield multiple lepton final states used as experimental signatures include $\tilde{\chi}_2^0 \to \bar{l}\ell$, $Z\tilde{\chi}_1^0$ or $l^+l^-\tilde{\chi}_1^0$. These searches are probing charginos and neutralinos of mass up to ~300–650 GeV, under these specific conditions (see Figure 126). Extensive scans of the pMSSM have shown that significant regions of parameters giving rise to relatively light weakly interacting SUSY particles still remain unexplored and will not be probed even after the first operation of the LHC at its design energy of 14 TeV [1023, 1024, 1026].

There are regions in SUSY parameter space that are not well covered by the searches for missing energy and require more exotic search strategies. One example are scenarios where an electrically or colour-charged NLSP becomes long-lived on collider time-scales. This situation occurs either through strongly suppressed couplings of the LSP or through kinematic suppression. The former case naturally occurs in GMSB models where the lighter stau often is the NLSP. The clean signature of the resulting highly ionising charged tracks at the LHC typically lead to stronger limits on sparticle masses in such a model [1028, 1029]. The latter case occurs, e.g., in scenarios with a wino- or higgsino-like neutralino LSP being almost mass degenerate to the lightest chargino. Another example of exotic SUSY signatures are models with R-parity violating couplings.

The recent observation of a Higgs-like particle with mass $\simeq 125$ GeV at the LHC is opening new perspectives for SUSY searches at colliders. The mass of the newly discovered particle sets some non-trivial constraints on the SUSY

<figure><img src="image_130.png" /><figcaption>**Fig. 126** 95% CL exclusion limits on the chargino-neutralino production NLO cross section times branching fraction in the flavor-democratic scenario, for the three-lepton (upper panel), di-lepton WZ+MET and tri-lepton (lower panel) CMS searches with $9.2 \text{ fb}^{-1}$ of data at 8 TeV (from [1027]).</figcaption></figure>

parameters. In particular, the relatively large mass value observed implies strong restrictions on the scalar top mass and the mixing in the top sector [929, 1030]. Heavy scalar top quarks and/or large mixing are required to bring the h boson mass around 125 GeV. The first measurement of the yields (or signal strengths $\mu$) in the decay channels studied so far—including $\gamma\gamma$, $ZZ^*$ and $WW^*$—(although limited in accuracy and only at the level of upper limits in the important $bb$ and $\tau\tau$ channels) will add further constraints. In particular, if interpreted within the SUSY framework, the data point towards a decoupling scenario, with a relatively heavy A boson. A possible enhancement in the $\gamma\gamma$ channel, observed by ATLAS and recently confirmed by the updated ATLAS study with $13 \text{ fb}^{-1}$ of 8 TeV data, may be a first hint of deviation from the SM expectations and could be explained through a reduction of the $b\bar{b}$ width as an effect of SUSY particle loops with intermediate, positive values of $\mu \tan\beta$ [1031, 1032], or the contribution of light staus [1033–1035] or charginos [1031]. Several of the preferred scenarios complying with $M_h \simeq 125$ GeV and low values of the fine tuning parameter have sbottom particles lighter than the stops with multiple decay modes with comparable rates [1036]. This allows them to evade in part the constraints from direct LHC searches which assume a single dominant decay channel.
One of the indirect probes on the scale of SUSY particles is fine tuning. The gradual exclusion of SUSY particles at lower masses as a consequence of LHC searches naively affects the value of the fine tuning parameter, $\Delta$, for the surviving SUSY models. It has been noted that in generic MSSM models, fine tuning is mostly determined by the $\mu$ parameter and an acceptably low fine tuning corresponds to small to moderate value of $|\mu|$. If fine tuning is taken as a criterion to select MSSM scenarios compatible with the 125 GeV Higgs mass, (setting $\Delta < 100$ as has been proposed [1037] $^{59}$) a constraint on the mass scale of weakly-interacting particles is implicitly derived with values of $m_{\chi_1^{\pm}} \le 270$ GeV. This would match particularly well with the reach of a linear $e^{+}e^{-}$ collider with $\sqrt{s}$ energy in the range 0.5-1.0 TeV.

In summary, despite the far reaching constraints derived by the direct searches for SUSY production at the LHC, specific classes of models exist in the general MSSM and in constrained models such as NUHM2, which are consistent with the current bounds and have SUSY particles within reach of an $e^{+}e^{-}$ collider operating at $\sqrt{s} \sim 0.25-0.5$ TeV and above. A recent study showed that over 20% of the viable pMSSM models, not yet excluded by the combined LHC searches at 7 and 8 TeV, have the lightest chargino, $\chi_1^{\pm}$, accessible at $\sqrt{s} = 0.5$ TeV increasing to 58% for $\sqrt{s} = 1$ TeV and 94% for 2 TeV [1036]. In addition, a study of natural SUSY NUHM2 parameter space in the $\mu$ vs. $m_{1/2}$ parameter plane shows the LHC8 and LHC14 reach (assuming 300 fb$^{-1}$) which will cover only a portion of the $\Delta_{EW} < 30$ favored parameter space. However, a $\sqrt{s} = 0.5-0.6$ TeV $e^{+}e^{-}$ collider would access the entire low $\Delta_{EW}$ parameter space, thus either discovering light higgsinos or ruling out natural SUSY: see Fig. 127. These considerations highlight the role of a high-energy $e^{+}e^{-}$ collider as a complementary discovery machine compared to the LHC.

## 5.6 Linear Collider Capabilities

As mentioned earlier, a linear $e^{+}e^{-}$ collider operating with $\sqrt{s} \gtrsim 2m(\text{sparticle})$ can serve as a discovery machine, not only in models like natural SUSY, but also in dark matter motivated cases such as the stau co-annihilation region or in R-parity violating models where the LSP decays hadronically so that the SUSY signal is buried beneath QCD multi-jet backgrounds at the LHC.

Since SUSY is expected to (more than) double the number of physical particles over a possibly wide mass spectrum, an $e^{+}e^{-}$ collider with (1) a broad energy range, (2) the capability to precisely tune its $\sqrt{s}$ energy at well-defined

<figure><img src="image_131.png" /><figcaption>**Fig. 127** Plot of $\Delta_{EW}$ contours in the $m_{1/2}$ vs. $\mu$ plane of NUHM2 model for $A_0 = -1.6m_0$ and $m_0 = 5$ TeV and $\tan\beta = 15$. We also show the region accesses by LHC8 gluino pair searches, and the region accessible to LHC14 searches with 300 fb$^{-1}$ of integrated luminosity. We also show the reach of various ILC machines for higgsino pair production. The green-shaded region has $\Omega_{\chi_1^{\pm}}^{\text{std}} h^2 < 0.12$. Figure from [1038].</figcaption></figure>

values corresponding to new particle production thresholds, (3) the added analysing power afforded by beam polarisation and (4) possibly different beam species ($\gamma\gamma, e^-e^-$) appears ideally suited for a program of detailed, high precision studies. The cross sections for pair production of SUSY particles are in the range 0.1-30 fb for masses of 200, 400 and 1200 GeV at $\sqrt{s} = 0.5, 1$ and $3$ TeV, respectively. For comparison, those for the two SM processes $e^{+}e^{-} \to W^{+}W^{-}v\bar{v}$ and $e^{+}e^{-} \to \mu^{+}\mu^{-}v\bar{v}$– which are the irreducible backgrounds to chargino and smuon pairs production– are 2, 10 and 25 fb and 25, 35 and 45 fb, respectively, at the same collision energies. These cross sections ensure a favourable signal-to-background ratio after appropriate selection cuts and make the study of SUSY particle pair production at a linear collider extremely promising!

Typical values of sparticle production cross sections are shown as a function of the collider energy, $\sqrt{s}$ in Fig. 128. If the fine tuning and naturalness arguments summarized in the previous section are taken as guidance, it is possible to identify scenarios where LHC searches may cover only a part of the parameter space, while a $\sqrt{s} = 0.5-0.8$ TeV $e^{+}e^{-}$ collider would access the entire parameter space corresponding to low $\Delta_{EW}$ values. These considerations highlight the possible role of a linear $e^{+}e^{-}$ collider as a SUSY discovery machine, complementary to the LHC.

If SUSY exists, one of the major undertakings of collider physics is the precise determination of the quantum numbers and decay properties of the SUSY particle partners. At a linear collider, the masses of SUSY particles can be determined either by the end points of the energy distribution of the visi-

<footnote>$^{59}$In this study, the authors implement the BG fine-tuning measure applied to 19 uncorrelated parameters in the pMSSM which is assumed valid up to a scale $\Lambda \simeq 20$ TeV. The $\Lambda = 20$ TeV scale induces an additional factor of 3 in the fine tuning evaluation.</footnote>
<figure><img src="image_132.png" /><figcaption>Fig. 128 Sparticle production cross sections vs. $\sqrt{s}$ at a Higgsino factory for a radiatively-driven natural SUSY benchmark point [969].</figcaption></figure>

ble SM particle emitted in two-body decays (or even 3-body decays) or– more precisely but more demanding for the accelerator design and tuning– by dedicated energy scans at the onset of the pair production process. For typical SUSY spectra– having particles spaced from tens to hundreds of GeV– threshold scans set specific requirements on the accelerator design implying the flexibility to deliver collisions at several $\sqrt{s}$ energies with comparable luminosity and within the operating plan.

The capability of a linear collider in the study of SUSY has been studied for the last twenty years with increasing realism from the adoption of detailed simulation and reconstruction. New techniques for the optimal reconstruction of physics observables, such as the parton energy or the jet flavour, have been developed and new detector concepts and sensor technologies, tailored to the requirements of the linear collider physics program have been introduced and demonstrated under realistic operating conditions. Supersymmetry has played an important role in setting these requirements and shaping the detector concepts. The recent studies for the ILC Letters of Intent (LoI) [197, 1039] and also the CLIC Conceptual Design report (CDR) [1040] have adopted full Geant-4 [1041] based simulation and detailed reconstruction, accounting for machine induced backgrounds. In most cases, the SUSY signatures can be clearly discriminated from the SM processes. Inclusive SUSY production often appears to be the major source of background for specific processes. In fact, different SUSY cascade decay chains [1042] may lead to the same final states. The ability to fully reconstruct the events with excellent energy resolution and to suppress some processes by changing the beam energy and, possibly, the beam polarisation offer excellent tools for ensuring an efficient study of each individual channel of interest. For example, the interference of the contribution of $\tilde{\chi}_1^+\tilde{\chi}_1^-$ decays with $\tilde{e}_L^+\tilde{e}_L^-$ in WW + missing energy and

$\tilde{\chi}_2^0\tilde{\chi}_2^0$ decays with $\tilde{v}\tilde{v}$ in $hh$ + missing energy is studied in detail with full simulation in [1043] and the separation of neutral and charged sleptons of the first/second generation in [1044]. Another important source of background is due to two-photon events, which may obscure the production of sfermion pairs, in particular in scenarios with small mass splitting. This background source can be controlled by ensuring electron tagging capability in the detector down to very small angles [1045].

### 5.6.1 Particle Property Measurements

#### Mass Measurements

##### a) In the continuum

The precise and unambiguous determination of SUSY particle masses is essential for the reconstruction of the theory fundamental parameters and for determining that the nature of the new physics is indeed supersymmetric. Mass reconstruction can be performed at an $e^+e^-$ linear collider by the reconstruction of the kinematics in SUSY particle pair production and by threshold energy scans. Threshold scans also provide us with access to the particle width, which is important since the narrow width approximation largely used in the context of the SM fails in general theories of new physics [1046].

In the two-body decay process $\tilde{A} \to B\tilde{C}$ of a SUSY particle $\tilde{A}$ into a lighter sparticle $\tilde{C}$ and a SM particle $B$, the masses of the parent and daughter sparticle can be extracted from the position of the kinematic edges of the energy spectrum of $B$ since $\tilde{A}$ is produced with fixed, known energy in the pair production $e^+e^- \to \tilde{A}\tilde{A}$. The technique was first proposed in [1047] for two-body decays of sleptons and charginos, for squarks in [1048] and three-body and cascade decays in [1049] and later extended to other two-body decays [1050]. In the case of neutralino and chargino decays into bosons, where the daughter mass $M_B$ cannot be neglected (as in the case of squark and slepton decays), the relation between the energy endpoints and the masses of the particle involved in the decay process are given by:

$$E_{BH,BL} = \gamma(E_B^* \pm \beta E_B^*) \quad (145)$$

where

$$E_B^* = \frac{M_A^2 + M_B^2 - M_{\tilde{C}}^2}{2M_A}, \quad (146)$$

$$\text{with } \gamma = \frac{\sqrt{s}}{2M_A}, \quad \text{and} \quad \beta = \sqrt{\frac{1 - 4M_A^2}{s}}. \quad (147)$$

These formulae can be extended in a straightforward way to the case in which the particle $\tilde{A}$ is not directly produced in the $e^+e^-$ collisions but originates from the decay of a heavier particle, $\tilde{A}'$, by replacing $s$ with $E_A^2$, where $E_A$ is its energy. In the case of cascading decays $\tilde{A}' \to \tilde{A}B' \to B\tilde{C}$, $E_A$ is obtained as $\sqrt{s} - E_{B'H} < E_A < \sqrt{s} - E_{B'L}$.
The determination of the lower and upper endpoints of the energy spectrum constrains the ratio of the mass of $\tilde{A}$ to that of $\tilde{C}$. If the mass of $\tilde{C}$- in most cases the lightest neutralino- is independently known, then $M_{\tilde{A}}$ can be extracted. The accuracy in the extraction of the masses by the endpoint technique depends on the resolution in determining $E_B$, which may be the resolution in measuring the momentum of a lepton in the case of sleptons or the energy of a jet (di-jet) in the case of a scalar quark (chargino or neutralino decaying into a boson). Excellent energy and momentum resolution are therefore essential. The energy of the beams at collision must also be known accurately because this enters in the determination of $\beta$. Beam-beam effects which induce radiation off the beam particles before collision are responsible for distortions of the luminosity spectrum, which must be precisely measured from collision data.

Detailed analyses, based on full Geant-4 detector simulation, digitisation and reconstruction and including the inclusive SM backgrounds, have validated earlier results on the expected accuracy on the mass determination for sleptons, gaugino and squarks at $\sqrt{s} = 0.5$ and 3 TeV. Studies for the ILD and SiD LoIs performed for the ILC parameters at 0.5 TeV [197, 1039], have shown that the kinematic endpoints of the energy spectrum of W and Z bosons produced in decays of chargino and neutralinos (see Figure 129), respectively, can be determined with an accuracy of better than 1 GeV, thanks to the excellent performance of energy flow with highly segmented calorimeters in the reconstruction of parton energy [1051]. Kinematic fitting imposing equal masses of pair produced particle can be applied to improve the energy resolution. This translates into relative statistical accuracies in the determination of the $\tilde{\chi}_1^\pm, \tilde{\chi}_2^0$ and $\tilde{\chi}_1^0$ masses of 1%, 0.5% and 0.7%, respectively. These results confirm, with the realism of full simulation and reconstruction and full SM backgrounds, the findings of earlier studies indicating that the masses of gaugino could be measure to a relative statistical accuracy of $\sim$1%.

The excellent momentum resolution, required by the study of the Higgs-strahlung process, implies that the accuracy on the mass determination is dominated by the beamstrahlung effects. Not only the dominant modes, such as $\tilde{\mu}_R^+\tilde{\mu}_R^-$ → $\mu^+\mu^-\tilde{\chi}_1^0\tilde{\chi}_2^0$, but also the sub-dominant process $\tilde{\mu}_L^+\tilde{\mu}_L^-$ → $\mu^+\mu^-\tilde{\chi}_1^0\tilde{\chi}_1^0$ can be studied in the 2-lepton + missing energy final state. Scalar $\tilde{\imath}$ and $\tilde{b}$ quarks can be observed almost up to the kinematical threshold for the pair production process even in the case of small mass splitting with the $\tilde{\chi}_1^0$ with the signal cross section measured with a statistical accuracy of $\sim$15% for the case of the $\tilde{b}$ [1039]. These scenarios at small mass splitting are of special relevance in relation to the dark matter relic density since stop or sbottom co-annihilation may be responsible for reducing $\Omega_\chi h^2$ to values compatible with the WMAP results and are very difficult for LHC searches. In addition, an $e^+e^-$ collider of sufficient energy

<figure><img src="image_133.png" /><figcaption>Fig. 129 Di-jet mass (upper plots) and energy spectra (lower plots) for chargino and neutralino production at 0.5 TeV (from [197]).</figcaption></figure>

to produce scalar top pairs can determine the stop mixing angles through a study of the $e_L^+e_R^-$ → $\tilde{\imath}_1\tilde{\imath}_1$ and $e_R^+e_L^-$ → $\tilde{\imath}_1\tilde{\imath}_1$ production with polarised beams along with study of the decays into multiple channels with comparable rate: such cases are difficult, if not impossible, at the LHC.

Much of the accuracy demonstrated by the detailed ILC studies at 0.5 TeV is preserved at multi-TeV energies, as confirmed by some of the studies carried out for the CLIC CDR [1040], which focused on 3 TeV $e^+e^-$ collisions. Chargino and neutralino masses in the range 600 - 1000 GeV can be determined with a relative statistical accuracy of 1 - 2% with unpolarised beams and 2 ab$^{-1}$ of data [1040, 1043]. The mass of $\tilde{\mu}_R$ of 1.1 TeV is again determined to $\sim$2% with unpolarised beams and 1% with polarised electrons and positrons, accounting for backgrounds [1040, 1052]. In addition to the weakly interacting SUSY particles, multi-TeV collisions may access scalar quark pair production, providing unique accuracy on their masses. In the case of a 1.1 TeV right-handed squark of the first generation a detailed study performed for 2 ab$^{-1}$ of integrated luminosity at $\sqrt{s} = 3$ TeV demonstrated a relative statistical accuracy on the mass of 0.5% [1053]. The linear collider opportunities for precision study of SUSY particles extend to three-body decays [1054] of gauginos [1055, 1056], sleptons [1057] and scalar quarks [1058], which are more difficult for the LHC. In the study of these processes, SUSY becomes a possible background to the searches where different production and decay channels lead to the same final state or topology. In these cases, special attention must be paid to the use of tight cuts on discriminants based on neural networks or multivariate
techniques which may induce strong biases on the kinematics and configuration of the selected events.

### b) At the threshold

An $e^{+}e^{-}$ linear collider with tunable beam energy can determine the sparticle masses by performing energy scans of their pair production cross section near threshold. In principle, this method often provides a better mass accuracy compared to the kinematic end-point method discussed above, and also, in most cases, a constraint on the particle width. Threshold energy scans put significant requirements on the machine performance and versatility. Not only the beam energy needs to be varied over a broad range, but since the cross section at threshold is small a large amount of integrated luminosity must be dedicated to each scan. Effects from beamstrahlung, finite sparticle widths, and Sommerfeld rescattering [1059, 1060] are important at threshold, while SUSY backgrounds are reduced, at least for the lighter states. It turns out to be preferable to concentrate the luminosity in a small number of scan points [1061]. Measurements at energies very close to the kinematic threshold are most sensitive to the width while those on the cross section rise above threshold are most sensitive to the mass. In general, on can achieve few per-mille precision for the mass determination from a threshold scan. In absolute numbers, the uncertainty for the width measurement is comparable, but since electroweak sparticle widths are typically a factor 1000 smaller than their mass, only an upper bound on the width can be established in most cases. With an $e^{-}e^{-}$ running option for the ILC, on the other hand, the selectron masses and widths can be measured with up to tenfold better precision than in $e^{+}e^{-}$ collisions [1060], which is due to the fact that $\tilde{e}_{R}\tilde{e}_{R}^{-}$ and $\tilde{e}_{L}\tilde{e}_{L}^{-}$ pairs are produced in a s-wave rather than a p-wave, leading to a steep $\propto \beta$ rise near threshold.

A comparison of ILC mass measurements for various sparticles via continuum and via threshold measurements is shown in Table 30 (from Ref. [433, 1060, 1062]). Note that the threshold scans require some rough a priori knowledge of the sparticle masses and take significant amount of the running time at various energy points, which will reduce the statistics available at the highest energy. There have been a few detailed studies of run plan scenarios including threshold scans for SUSY particles which show the feasibility to acquire data at the thresholds of a few important processes while accumulating a sizeable data set at the highest operational energy [1063]. The scenarios adopted in those studies are now made obsolete by the recent LHC bounds, but the findings are still applicable in a general sense.

Cross Sections, Width and Branching fractions

<table><thead><tr><th>$e^{+}e^{-}$</th><th>$m$</th><th>$\delta m_{c}$</th><th>$\delta m_{th}$</th><th>$\Gamma_{th}$</th></tr></thead><tbody><tr><td>$\tilde{\mu}_{R}$</td><td>143.0</td><td>0.2</td><td>0.2</td><td>&lt;0.5</td></tr><tr><td>$\tilde{\mu}_{L}$</td><td>202.1</td><td></td><td>0.5</td><td>—</td></tr><tr><td>$\tilde{e}_{R}$</td><td>143.0</td><td>0.1</td><td>0.15</td><td>&lt;0.4</td></tr><tr><td>$\tilde{e}_{L}$</td><td>202.1</td><td>0.8</td><td>0.3</td><td>&lt;0.4</td></tr><tr><td>$\tilde{v}_{e}$</td><td>186.0</td><td>1.2</td><td>0.8</td><td>&lt;0.7</td></tr><tr><td>$\tilde{\tau}_{1}$</td><td>133.2</td><td>0.3</td><td></td><td></td></tr><tr><td>$\tilde{\chi}_{1}^{\pm}$</td><td>176.4</td><td>1.5</td><td>0.55</td><td></td></tr><tr><td>$\tilde{\chi}_{2}^{\pm}$</td><td>378.2</td><td>3</td><td></td><td></td></tr><tr><td>$\tilde{\chi}_{1}^{0}$</td><td>96.1</td><td>0.1</td><td></td><td></td></tr><tr><td>$\tilde{\chi}_{2}^{0}$</td><td>176.8</td><td>2</td><td>1.2</td><td></td></tr><tr><td>$\tilde{\chi}_{3}^{0}$</td><td>358.8</td><td>3–5</td><td></td><td></td></tr><tr><td>$\tilde{\chi}_{4}^{0}$</td><td>377.8</td><td>3–5</td><td></td><td></td></tr></tbody></table>

<table><caption>**Table 30** Expected precision on sparticle masses (in GeV) for the SPS1a scenario [807] using polarized $e^{\pm}$ beams ($P_{L}(e^{-}) = 0.8, P_{L}(e^{+}) = 0.6$). $\Delta m_{c}$ is from decay kinematics measured in the continuum ($\mathcal{L} = 200/500/1000\,\text{fb}^{-1}$ at $\sqrt{s} = 400/500/750$ GeV), and $\delta m_{th}$ and $\delta \Gamma_{th}$ are from threshold scans ($\mathcal{L} = 100\,\text{fb}^{-1}$ for $e^{+}e^{-}$ and $\mathcal{L} = 5\,\text{fb}^{-1}$ for $e^{-}e^{+}$). From Ref. [433, 1060, 1062].</caption><thead><tr><th>$e^{-}e^{-}$</th><th>$m$</th><th>$\delta m_{th}$</th><th>$\Gamma_{th}$</th></tr></thead><tbody><tr><td>$\tilde{e}_{R}$</td><td>143.0</td><td>0.05</td><td>0.21 ± 0.05</td></tr><tr><td>$\tilde{e}_{L}$</td><td>202.1</td><td>0.25</td><td>0.25 ± 0.04</td></tr></tbody></table>

Decays of charginos and neutralinos into bosons, such as $\tilde{\chi}_{1}^{\pm} \to W^{\pm}\tilde{\chi}_{1}^{0}$ and $\tilde{\chi}_{2}^{0} \to Z\tilde{\chi}_{1}^{0}$ or $\tilde{\chi}_{1}^{0}h$, are well suited to $e^{+}e^{-}$ collider capabilities. The four-jet + missing energy final states can be studied with good accuracy thanks to the small background and the excellent di-jet mass resolution ensuring separation of $W$ from $Z$ or $h$ masses. Production cross sections of pairs of chargino and neutralino with mass of 216 GeV have been studied at 0.5 TeV and the statistical uncertainty on the cross section has been estimated at 0.6% and 2% respectively. It is interesting to observe that decays of SUSY particles, in particular neutralinos into the lightest Higgs boson, $h$, are common and even enhanced in specific models and combinations of MSSM parameters [1064–1067]. This opens up an interesting perspective of studying SUSY processes through the reconstruction of $h$ pairs + missing energy in four jet events, where Higgs boson production is selected from that of other bosons by di-jet mass (see Figure 130) and also b-tagging. A further possibility is the study of single Higgs boson plus missing $E$ production via $e^{+}e^{-} \to \tilde{\chi}_{1}^{0}\tilde{\chi}_{2}^{0}$ with the decay $\tilde{\chi}_{2}^{0} \to \tilde{\chi}_{1}^{0}h$.

In addition, the determination of the dependence of the cross section for production of gaugino pairs, including $\tilde{\chi}_{2}^{0}\tilde{\chi}_{2}^{0}$ and $\tilde{\chi}_{1}^{+}\tilde{\chi}_{1}^{-}$, with the beam polarisation and energy is important to establish the nature of the $\tilde{\chi}_{2}^{0}$ and measure the chargino mixing angles and the $\mu$ parameter [1068].

### $\tau$-polarization

The measurement of $\tau$ polarisation, $P_{\tau}$, in $\tilde{\tau}_{1}$ decays offers sensitivity to the mixing of interaction and mass eigenstates in the stau sector [1069]. $P_{\tau}$ is extracted from the energy spectrum of the pion emitted in the 1-prong decay $\tau \to \pi\nu$.
<figure><img src="image_134.png" /><figcaption>**Fig. 130** Di-jet invariant mass distribution in inclusive 4-jet + missing energy SUSY events produced in $\sqrt{s}=3$ TeV $e^{+}e^{-}$ collisions for $0.5 \text{ ab}^{-1}$ of fully simulated events. The result of the fit to extract the boson content is shown by the continuous line with the individual *W*, *Z* and *h* components represented by the dotted lines (from [1067]).</figcaption></figure>

<figure><img src="image_135.png" /><figcaption>**Fig. 131** Energy spectrum of reconstructed $\tau$ leptons from $\tau_1$ decays (left) and energy distribution of the pions from 1-prong decays with the fit for the determination of the polarisation for fully simulated $e^{+}e^{-}$ events at 0.5 TeV (from [197]).</figcaption></figure>

Again, the $\pi$ energy spectrum depends on the collision energy and thus on beamstrahlung. Nonetheless, using realistic parameters for the ILC, the $\tau$ polarisation can be determined to a 15% accuracy (see Figure 131).

### CP-violating asymmetries

The sub-leading, two-body decay $\tilde{\chi}_i^0 \to \tilde{\ell}_R \ell \to \ell \ell \tilde{\chi}_1^0$ is sensitive to CP asymmetries in the triple product of the final particle momenta. This measurement, which would open the way to the detection of SUSY CP phases, is discussed below in more detail. While the measurement may be possible also at the LHC, the sensitivity of a linear collider is expected to be far superior. A detailed analysis, based on full simulation and reconstruction and which makes use of event kinematics, obtained values of $|M_1|, \mu$ and $M_2$ to a relative accuracy of 1% or better and the CP phases to ~10% resolving the sign ambiguity, for states accessible at $\sqrt{s} = 0.5$ TeV using polarised electron and positron beams [1070].

### 5.6.2 Testing the SUSY Character

One of the most important aspects of new physics searches is to really identify the new physics model. Concerning SUSY

theories, such an identification requires measurements beyond just determining the mass and spin of the new particle. In order to prove that the new physics candidate is indeed the SUSY partner of the corresponding SM particle, one also has to measure precisely their couplings [1071] and their quantum numbers. In this context also the special feature of carrying a Majorana character has to be proven for the neutral gauginos.

### Spin determination

The spin is one of the fundamental characteristics of all particles and it must be determined experimentally for any new particles so as to clarify the nature of the particles and the underlying theory. In particular, this determination is crucial to distinguish the supersymmetric interpretation of new particles from other models.

In supersymmetric theories, spin-1 gluons and electroweak gauge bosons, and spin-0 Higgs bosons are paired with spin-1/2 gluinos, electroweak gauginos and higgsinos, which mix to form charginos and neutralinos in the non-colored sector. This calls for a wide spectrum of necessary attempts to determine the nature of the new particles experimentally.

The measurement of the spins in particle cascades at LHC is quite involved [1072–1074]. While the invariant mass distributions of the particles in decay cascades are characteristic for the spins of the intermediate particles involved, detector effects strongly reduce the signal in practice.

In contrast, the spin measurement at $e^{+}e^{-}$ colliders is straightforward [1075, 1076]. A sequence of techniques—increasing in complexity—can be exploited to determine the spin of supersymmetric particles in pair production of sleptons, charginos and neutralinos in $e^{+}e^{-}$ collisions:

(a) rise of the excitation curve near the threshold,

(b) angular distribution in the production process,

(c) angular distribution in decays of the polarized particles and

(d) angular correlations between decay products of two particles.

Within the general theoretical framework it can be proven that the second step (b) is already sufficient in the slepton sector, although in general the final state analysis is required to determine the spin unambiguously in the chargino and neutralino sectors.

As shown clearly in Fig. 132, the threshold excitation curve and the production angle distribution for smuons in the MSSM are characteristically different from those for the first Kaluza-Klein muons in a model of universal extra dimension. Even though the p-wave onset of the excitation curve is generally a necessary but not sufficient condition, the $\sin^2 \theta$ law for the angular distribution in the production of sleptons (for selectrons close to threshold) is a unique signature of the fundamental spin-zero character.
<figure><img src="image_136.png" /><figcaption>**Fig. 132** The threshold excitation (a) and the angular distribution (b) in pair production of muons in the MSSM, compared with the first spin-1/2 Kaluza-Klein muons in a model of universal extra dimensions; for details, see Ref. [1076].</figcaption></figure>

<figure><img src="image_137.png" /><figcaption>**Fig. 133** (a) The unpolarized cross section of $e^{+}e^{-} \to \tilde{\mu}_{R}^{+}\tilde{\mu}_{R}^{-}$ production close to threshold, including QED radiation, beamstrahlung and width effects; the statistical errors correspond to $\mathcal{L} = 10 \text{ fb}^{-1}$ per point, (b) energy spectrum $E_{\mu}$ from $\tilde{\mu}_{R} \to \mu^{-}\tilde{\chi}_{1}^{0}$ decays; polar angle distribution $\cos\theta_{\mu R}$ (c) with and (d) without contribution of false solution. The simulation for the energy and polar angle distribution. The simulation for the energy and polar angle distribution is based on polarized beams with $(P_{e^{-}}, P_{e^{+}}) = (+0.8, -0.6)$ at $\sqrt{s} = 1$ TeV and $\mathcal{L} = 500 \text{ fb}^{-1}$. For details, see Ref. [1076].</figcaption></figure>

The measurement of the cross section for muon pair production $\tilde{\mu}_R^+ \tilde{\mu}_R^-$ can be carried out by identifying acoplanar $\mu^+ \mu^-$ pairs [with respect to the $e^\pm$ beam axis] accompanied by large missing energy carried by the invisible lightest neutralino $\tilde{\chi}_1^0$ in the decays $\tilde{\mu}_R^\pm \to \mu^\pm \tilde{\chi}_1^0$. In addition, initial and final state QED radiations, beamstrahlung and detector effects, etc. needs to be taken into account for reconstructing the theoretically predicted distributions. As shown in Fig. 133 through a detailed simulation, the characteristic p-wave threshold excitation and the production, as well as the flat decay distribution for the process $e^+e^- \to \tilde{\mu}_R^+ \tilde{\mu}_R^-$

followed by the decays $\tilde{\mu}_R^\pm \to \mu^\pm \tilde{\chi}_1^0$, can be reconstructed experimentally.

Unlike the slepton sector, the chargino and neutralino sectors in general have much more involved patterns. Neither the onset of the excitation curves near threshold nor the angular distribution in the production processes provides unique signals of the spin of charginos and neutralinos. However, decay angular distributions of polarized charginos and neutralinos, as generated naturally in $e^+e^-$ collisions, can provide an unambiguous determination of the spin-1/2 character of the particles albeit at the expense of more involved experimental analyses [1076]. Using polarized electron and/or positron beams will in general assure that the decaying spin-1/2 particle is polarized; reasonable polarization analysis power is guaranteed in many decay processes.

Generally, quantum interference among helicity amplitudes— reflected typically in azimuthal angle distributions and correlations— may provide another method for determining spins [1077], although this method depends strongly on the masses of the decay products and the $\sqrt{s}$ energy, as the quantum interference disappears with increasing energy.

To summarize, the spin of sleptons, charginos and neutralinos can be determined in a model-independent way at $e^+e^-$ colliders. Methods similar to those applied to slepton pair production can be applied in the squark sector. For gluinos, a quite different methodology is required since these are not produced at tree level in $e^+e^-$ collisions.

### Yukawa couplings

The SM/SUSY coupling relations are not affected by SUSY breaking and therefore the couplings of the SM particle are the same as those of their SUSY partners. That means, for instance, that the SU(3), SU(2) and U(1) gauge couplings $g_S$, $g$ and $g'$ have to be identical to the corresponding SUSY Yukawa couplings $g_{\tilde{g}}$, $g_{\tilde{W}}$ and $g_{\tilde{B}}$. These tests are of fundamental importance. Concerning the test of the SUSY-QCD Yukawa couplings, first examinations could be performed at the LHC via determining the couplings in $\tilde{q}\tilde{g}$, $\tilde{g}\tilde{g}$ and $\tilde{q}\tilde{q}$ productions [1078]. These SUSY-QCD Yukawa studies have been accomplished by the analysis at a LC in [1079], so that one expects in total an uncertainty of about 5%-10% in the determination of the SUSY-QCD Yukawa couplings.

The SUSY-EW Yukawa coupling, however, is one of the final targets of LC experiments which should provide a complete picture of the electroweak gaugino sector with a resolution at the level of at least 1% [39]. Under the assumption that the SU(2) and U(1) parameters have been determined in the chargino/ higgsino sector (see Sect.5.7), we test precisely the equality of the Yukawa and gauge couplings via measuring polarized cross sections: varying the left-handed and right-handed Yukawa couplings has consequences on the measured cross sections. Depending on the electron (and
<figure><img src="image_138.png" /><figcaption>**Fig. 134** Polarized cross section versus $P(e^{-})$ (left panel) or $P(e^{+})$ (right panel) for $e^{+}e^{-} \rightarrow \bar{e} \bar{e}$-production with direct decay in $\tilde{\chi}_{1}^{0}e$ in a scenario where the non-coloured spectrum is similar to a SPS1a-modified scenario but with $m_{\tilde{e}_{L}} = 200$ GeV, $m_{\tilde{e}_{R}} = 195$ GeV [40].</figcaption></figure>

positron) beam polarization and on the luminosity, a per cent level precision can be achieved.

## Quantum numbers

One of the important tasks at future experiments is to determine model-independently the underlying quantum numbers of any new particles and check whether they correspond to their Standard Model counterparts. For instance, a particularly challenging measurement is the determination of the chiral quantum numbers of the sfermions. Although these are scalar particles, they have to carry the chiral quantum numbers of their Standard Model partners. Since chirality can be identified in the high energy limit via helicity and its conservation, it will be part of the charge of a linear collider to prove such an association. Since the limits from LHC for the electroweak SUSY spectrum are not very strong, it is still the case that a rather light spectrum selectrons, smuons, staus continues to be viable.

In $e^{+}e^{-}$ collisions, the associated production reactions $e^{+}e^{-} \rightarrow \tilde{e}_{L}^{+}\tilde{e}_{R}^{-}$, $\tilde{e}_{R}^{+}\tilde{e}_{L}^{-}$ occur only via t-channel exchange, whereas the pair production reactions $\tilde{e}_{L}\tilde{e}_{L}$, $\tilde{e}_{R}\tilde{e}_{R}$ occur also via s-channel $\gamma$ and Z exchange. Since $m_{\tilde{e}_{L}}$ is in general not equal to $m_{\tilde{e}_{R}}$, then the electron energy distribution endpoints will be different for each of the four possible reactions as will the positron energy distributions. Furthermore, the total cross sections for each reaction depend strongly on beam polarization so that by dialing the polarization, one can move between distinct spectra possibilities which allows one to disentangle the individual $\tilde{e}_{L}$ and $\tilde{e}_{R}$ masses, and to distinguish which one is which: e.g. measure their chiral quantum numbers, see 134. The masses of $m_{\tilde{e}_{L}} = 200$ GeV,

$m_{\tilde{e}_{R}} = 195$ GeV are close, both particles decay directly to $\tilde{\chi}_{1}^{0}e$.

The polarization of $P(e^{+})$ is mandatory in such cases. An example from Ref. [1080] using an Isajet simulation is shown in Fig. 135.

<figure><img src="image_139.png" /><figcaption>**Fig. 135** Electron and positron energy distributions for selectron pair production with the indicated beam polarizations and an integrated luminosity of 50 fb$^{-1}$ at $\sqrt{s} = 500$ GeV (E. Goodman, U. Nauenberg et al. in Ref. [1080]).</figcaption></figure>

## Majorana character

Experimental tests of the Majorana character of gluinos and neutralinos will provide non-trivial insight into the realization of SUSY in nature. There are several powerful methods for probing the nature of neutralinos in $e^{+}e^{-}$ collisions with polarized beams.

The parallelism between self-conjugate neutral vector gauge bosons and their fermionic supersymmetric partners induces the Majorana nature of these particles in the minimal $N=1$ supersymmetric extension of the standard model (MSSM). Therefore, experimental tests of the Majorana character of colored gluinos and non-colored electroweak neutralinos would provide non-trivial insight into the realization of SUSY in nature, since extended supersymmetric models can include Dirac gauginos and/or higgsinos [1081–1083].

A theoretical basis for formulating a solid testing ground for Dirac gauginos is provided by a model with a continuous global U(1) $R$ symmetry [1082, 1083] under which the Grassmann coordinates transform as $\theta \rightarrow e^{i\xi}\theta$, i.e. $R(\theta) = 1$. It implies that the component fields of a supersymmetric superfield differ by the $R$-charge. Since the gauge superfields $\hat{G}$ are real, they must have a zero $R$-charge, $R(\hat{G}) = 0$, implying that $R = 0$ for the gauge vector fields $G^{\mu}$ and $R = 1$ for the spin-1/2 gauginos $\hat{G}^{\alpha}$. Every term in the superpotential must have $R = 2$ to provide a $R$-symmetric potential while any soft-SUSY breaking terms must have $R = 0$.

When the $R$-charges of the MSSM matter, H-Higgs and gauge-vector superfields are assigned as in Table 31, not only the supersymmetric $\mu$ term and the baryon- and lepton-number breaking terms but also the soft-SUSY breaking Majorana mass terms and trilinear A terms are forbidden. As a result, the sfermion left-right mixing and the proton decay
through dimension-five operators are absent (while Majorana neutrino masses can be generated).

<table><caption>**Table 31** The R-charges of the matter, Higgs and gauge superfields in the minimal R-symmetric supersymmetric standard model [1082].</caption><thead><tr><th>Field</th><th>Superfield</th><th>R charge</th></tr></thead><tbody><tr><td>Matter</td><td>Λ̂, ˆE<sup>c</sup></td><td>1</td></tr><tr><td></td><td>Q̂, ˆD<sup>c</sup>, ˆU<sup>c</sup></td><td>1</td></tr><tr><td>H-Higgs</td><td>Ĥ<sub>d,u</sub></td><td>0</td></tr><tr><td>R-Higgs</td><td>R̂<sub>d,u</sub></td><td>2</td></tr><tr><td>Gauge Vector</td><td>Ĝ = {G<sup>μ</sup>, G̃<sup>α</sup>}</td><td>0</td></tr><tr><td>Gauge Chiral</td><td>Σ̂ = {σ, G̃<sup>tα</sup>}</td><td>0</td></tr></tbody></table>

Since the gaugino Majorana-type mass terms and the conventional higgsino μ term are forbidden in the R-symmetric theory, the superfield content of the minimal theory needs to be extended so as to give non-zero masses to gluinos, electroweak gauginos and higgsinos. The simplest extension, called the minimal R-symmetric supersymmetric standard model (MRSSM) [1082], is to introduce new chiral superfields $\hat{\Sigma} = \{\sigma, \tilde{G}^{t\alpha}\}$ in the adjoint representation of the SM gauge group in addition to the standard vector superfields as well as two iso-doublet chiral superfields $\hat{R}_d$ and $\hat{R}_u$ (R-Higgs) to complement the standard H-Higgs superfields $\hat{H}_d$ and $\hat{H}_u$. (For a simpler formulation, see Ref. [1084].)

In the color sector the original MSSM $R = 1$ gluino $\tilde{g}^a$ and the new $R = -1$ gluino $\tilde{g}^{t\alpha}$ ($a = 1-8$) are coupled by the SUSY-breaking but R-symmetric Dirac mass term so that they can be combined into a single Dirac fermion field $\tilde{g}_D^a = \tilde{g}_L^a + \tilde{g}_R^{t\alpha}$ with $R = 1$. Note that $\tilde{g}_D$ is not self-conjugate any more, i.e. $\tilde{g}_D^C \neq \tilde{g}_D$ as the anti-gluino carries $R = -1$. In the similar manner the original electroweak gauginos, $R = 1$ $\tilde{B}$ and $\tilde{W}^i$ ($i = 1-3$) and $R = -1$ H-higgsinos, $\tilde{H}_u$ and $\tilde{H}_d$ are coupled with the new electroweak gauginos, $R = -1$ $\tilde{B}^t$ and $\tilde{W}^{ti}$ ($i = 1-3$) and $R = 1$ R-higgsinos, $\tilde{R}_u$ and $\tilde{R}_d$, giving rise to four Dirac neutralinos $\tilde{\chi}_{D1,...,D4}^0$ with $R = 1$ and four Dirac anti-neutralinos with $R = -1$.

The extension from the minimal model MSSM with Majorana gluinos and neutralinos to the R-symmetric MRSSM with Dirac gluinos and neutralinos as well as new R-Higgs bosons and adjoint scalar fields $\sigma$ leads to a lot of distinct phenomenological consequences on sparticle productions at the LHC and $e^{\pm}e^{-}$ colliders [1083, 1084], flavor and CP problems [1082, 1086] and cold dark matter issues [1087].

There are several methods to investigate the nature of gluinos at the LHC. In the original form, decays to heavy stop/top quarks are exploited [1088] to test whether the final state in the fermion decay $\tilde{g} \to \tilde{t}\tilde{t} + \tilde{t}^*t$ is self-conjugate. The standard production processes for investigating the nature of gluinos [1089] are the production of a pair of equal-chirality squarks, $q_L q_L^t \to \tilde{q}_L \tilde{q}_L^t$ and $q_R q_R^t \to \tilde{q}_R \tilde{q}_R^t$. While

<figure><img src="image_140.png" /><figcaption>**Fig. 136** Left: the total cross sections for pair production of wino-like neutralinos near threshold in the MSSM and the Dirac theory. Right: dependence of the cross sections on the production angle $\theta$ for $\sqrt{s} = E_{cm} = 500$ GeV. The sparticle masses in both plots are $m_{\tilde{\chi}_2^0} = m_{\tilde{\chi}_{D2}^0} = 200$ GeV and $m_{\tilde{e}_L} = 400$ GeV (For the details, see Ref. [1091]).</figcaption></figure>

the cross section for the scattering processes with equal-chirality quarks is non-zero in the Majorana theory, it vanishes in the Dirac theory. Owing to the dominance of u-quarks over d-quarks in the proton, the Majorana theory predicts large rates of like-sign dilepton final states from squark pair production with an excess of positively charged leptons [1090] while they are absent, apart from a small number of remnant channels, in the Dirac theory. (In a realistic analysis one has to include gluino production processes which can also feed the like-sign dilepton signal but can be discriminated by extra jet emission from the gluino decays.) In addition, the nature of neutralinos could be checked at the LHC if cascade squark-decay chains involving intermediate sleptons and neutralinos are identified, as the final-state $q\ell^\pm$ invariant mass distributions are distinct [1091].

An $e^{\pm}e^{-}$ collider with polarized beams is an ideally clean and powerful instrument for testing the nature of neutralinos. In parallel to the squark pair production through quark-quark collisions, the processes $e^{-}e^{-} \to \tilde{e}_{R}^{+}\tilde{e}_{R}^{-}$ or $\tilde{e}_{L}^{-}\tilde{e}_{L}^{+}$ with equal chirality indices and $e^{+}e^{-} \to \tilde{e}_{L}^{+}\tilde{e}_{R}^{-}$ or $\tilde{e}_{R}^{+}\tilde{e}_{L}^{-}$ are forbidden due to the conserved R charge in the Dirac case while the processes occur in general in the Majorana case as in the MSSM [1089, 1091–1093].

Another powerful experimental test for characterising the nature of neutralinos is based on the threshold behavior of the neutralino diagonal-pair production and its polar-angle distribution (Fig. 136). In the case with Dirac neutralinos $\tilde{\chi}_D^0$, the cross section for the process $e^{+}e^{-} \to \tilde{\chi}_{Di}^0 \tilde{\chi}_{Di}^0$ ($i = 1-4$) exhibits a typical sharp s-wave excitation and a forward-backward asymmetric angular distribution, while in the case with Majorana neutralinos the cross section for neutralino diagonal pair production in $e^{+}e^{-}$ collisions is excited in the characteristic slow p-wave and the angular distribution is forward-backward symmetric [1091].

To summarize, the gluinos, the electroweak gauginos and the electroweak higgsinos are either Majorana or Dirac
fermions in extended supersymmetric models. The $e^{\pm}e^{\mp}$ colliders and the LHC provide us with various complementary and powerful tests for probing the nature of new fermionic states from which we can get non-trivial insight into the realization of SUSY in nature and find new directions for collider phenomenology as well as many related fields.

## 5.7 From SUSY Measurements to Parameter Determination

The measurements which can be performed from operating a linear collider with a large enough energy $\sqrt{s} \ge 0.5$ TeV and luminosity, to collect of order of 0.5-2 ab$^{-1}$ of data, can be turned into precise predictions on the fundamental MSSM parameters of the theory Lagrangian, on their evolution to the unification scale, and on the relic density of light neutralinos in the universe inferred from collider data. These quantities are crucial to understand the underlying structure and to identify the SUSY model and its connections to cosmology. In this section, we discuss the extraction of these parameters based on the anticipated accuracy of measurements of SUSY particle properties at a linear collider.

### 5.7.1 General strategy

Since the general MSSM depends already on over 100 new parameters, it is a true challenge to measure all parameters in as model-independent fashion as is possible. Therefore often model assumptions– in particular on the SUSY breaking mechanism and mass unifications– are made (see Section 5.2) resulting in a reduction to just 4–6 SUSY parameters. Then for unravelling the underlying SUSY model one needs a model-independent strategy for measuring the parameters. Since the current results from LHC point towards the TeV-scale for the coloured SUSY partners, it is clear that one would need a combined approach from LHC and the LC to resolve the SUSY puzzle. The determination of the fundamental SUSY parameters at low energy would allow a critical test of the theory: extrapolating the mass parameters to the GUT scale points to which SUSY breaking scheme might be realized in nature. Such extrapolations would be an important achievement that illustrates well the complementarity of data from the LHC and a linear collider [1068, 1094, 1095] (see also Section 5.7.7).

The fundamental parameters of the gaugino/higgsino sector are the $U(1)$ and $SU(2)$ gaugino masses $M_1$ and $M_2$, and the higgsino mass parameter $\mu$, where also $M_i$ and $\mu$ can contain CP-violating phases. In addition, also $\tan\beta$ enters the mixing of this electroweak particle SUSY sector. These parameters can– very accurately and independently of the underlying SUSY breaking scheme– be determined at a LC. This has been shown in many detailed studies [1047, 1055, 1096].

In case the full spectrum, $\tilde{\chi}_i^0, \tilde{\chi}_j^{\pm}$, $i = 1, \dots, 4$, $j = 1, 2$, is accessible, the determination of the fundamental parameters via measurements of masses and cross sections seems to be trivial and is therefore not discussed here in detail. In this case, however, stringent tests of the closure of the system can be designed. Models with additional chiral and vector superfields extend the gaugino/higgsino sector. Since unitary matrices diagonalize the system, powerful sum rules can be set up for the couplings and a unique test whether the observed 4–system is closed or not might be possible. These sum rules for couplings can be directly converted into high energy sum rules for production cross sections of neutralinos [1096]:

$$ \lim_{s \to \infty} s \sum_{i \le j}^{4} \sigma_{ij} = \frac{\pi \alpha^2}{48 \cos^4 \theta_W \sin^4 \theta_W} \times [64 \sin^4 \theta_W - 8 \sin^2 \theta_W + 5] \quad (148) $$

In this case, one also has to provide a measurement for the production $\tilde{\chi}_1^0 \tilde{\chi}_1^0$. This final state is invisible in R-parity invariant theories where $\tilde{\chi}_1^0$ is the LSP. Nevertheless, it can be studied indirectly by photon tagging in the final state $\gamma \tilde{\chi}_1^0 \tilde{\chi}_1^0$, which can be observed with a rather high accuracy at a LC. More details on photon tagging are included in the 'light higgsino' section.

The powerful test via sum rules stresses the importance of upgrading the collider to achieve high $\sqrt{s}$ energies, if physics dictates it, in addition to combining LC and LHC results. In order to reconstruct the complete MSSM Lagrangian and evolve the parameters to the GUT scale [1097], it is generally needed to combine the linear collider measurements with those of squarks and gluinos (and possibly heavier gauginos) observed probably first at the LHC. Results at 0.5 TeV and 3 TeV are discussed in [1095] and [1040].

### 5.7.2 Parameter determination with $\tilde{\chi}_1^{\pm}, \tilde{\chi}_{1,2}^0$ only

Even if only $\tilde{\chi}_{1,2}^0$ and $\tilde{\chi}_1^{\pm}$ were accessible, the precise measurements of the masses as well as polarized cross section for $\tilde{\chi}_1^+ \tilde{\chi}_1^-$, $\tilde{\chi}_1^0 \tilde{\chi}_2^0$ in different beam polarization configurations is sufficient to determine the fundamental SUSY parameters and allow mass predictions of the heavier particles, yet unseen SUSY states.

The diagonalization of the two chargino system can be parametrized by two mixing angles $\phi_L, \phi_R$. Defining the mixing angles in the unitary matrices diagonalizing the chargino mass matrix $M_C$ by $\phi_L$ and $\phi_R$ for the left– and right–chiral fields, the fundamental SUSY parameters $M_2$, $|\mu|$, $\cos\Phi_\mu$ and $\tan\beta$ can be derived from the chargino masses and the cosines $c_{2L,R} = \cos 2\phi_{L,R}$ of the mixing angles [1098, 1099].

If only the light charginos $\tilde{\chi}_1^{\pm}$ can be produced, the mass $m_{\tilde{\chi}_1^{\pm}}$ as well as both mixing parameters $\cos 2\phi_{L,R}$ can be measured. The quantities $\cos 2\phi_{L,R}$ can be determined uniquely if the polarized cross sections are measured at one energy
including transverse beam polarization, or else if the longitudinally polarized cross sections are measured at two different energies.

The heavy chargino mass is bounded from above after $m_{\tilde{\chi}_1^\pm}$ and $\cos 2\phi_{L,R}$ are measured experimentally. At the same time, it is bounded from below by not observing the heavy chargino in mixed light-heavy pair production:

$$ \frac{1}{2}\sqrt{s} - m_{\tilde{\chi}_1^\pm} \le m_{\tilde{\chi}_2^\pm} \le \sqrt{m_{\tilde{\chi}_1^\pm}^2 + 4m_W^2 / |\cos 2\phi_L - \cos 2\phi_R|}. \quad (149) $$

If both the light chargino mass $m_{\tilde{\chi}_1^\pm}$ and the heavy chargino mass $m_{\tilde{\chi}_2^\pm}$ can be measured, the fundamental parameters $M_2$, $\mu$, $\tan\beta$ can be extracted unambiguously. However, if $\tilde{\chi}_2^\pm$ is not accessible, their determination depends on the CP properties of the higgsino sector.

(A) If the higgsino sector is CP invariant$^{60}$, one can determine $m_{\tilde{\chi}_2^\pm}$ from the condition $\cos\Phi_\mu = \pm 1$, up to at most a two-fold ambiguity: see Ref.'s. [1098, 1099]. This ambiguity can be resolved as well as the gaugino parameter $M_1$ be determined if observables from the neutralino sector, in particular, the mixed-pair $\tilde{\chi}_1^0\tilde{\chi}_2^0$ production cross sections and $m_{\tilde{\chi}_{1,2}^0}$ are included, see Fig. 137.

(B) If $\tilde{\chi}_2^\pm$ is not accessible, the parameters $M_2$, $\mu$, $\tan\beta$, $\cos\Phi_\mu$ cannot be determined in a CP non-invariant theory in the chargino sector alone. They remain dependent on the unknown heavy chargino mass $m_{\tilde{\chi}_2^\pm}$. However, two trajectories can be generated in $\{M_2, \mu; \tan\beta\}$ space, parametrized by $m_{\tilde{\chi}_2^\pm}$ and classified by the two possible values $\Phi_\mu$ and $(2\pi - \Phi_\mu)$ for the phase of the higgsino mass parameter. Including information from the neutralino sector, namely the measured masses and the polarized cross sections of the two light neutralino states $\tilde{\chi}_1^0$ and $\tilde{\chi}_2^0$, the heavy chargino mass $m_{\tilde{\chi}_2^\pm}$ can be predicted in the MSSM and subsequently the entire set of fundamental gaugino and higgsino parameters can be determined uniquely [1096, 1101]: the symmetric neutralino mass matrix $M_N$ is diagonalized by a unitary matrix, defined such that the mass eigenvalues $m_{\tilde{\chi}_i^0}$ of the four Majorana fields $\tilde{\chi}_i^0$ are positive.

The squared mass eigenvalues of $M_N M_N^\dagger$ are solutions of the characteristic equations [1096]

$$ m_{\tilde{\chi}_i^0}^8 - am_{\tilde{\chi}_i^0}^6 + bm_{\tilde{\chi}_i^0}^4 - cm_{\tilde{\chi}_i^0}^2 + d = 0 \quad (150) $$

for $i=1,2,3,4$ with the invariants $a, b, c$ and $d$ given by the fundamental $SU(2)$ and $U(1)$ gaugino mass parameters $M_2$ and $M_1$, and the higgsino mass parameter $\mu$, i.e. the moduli $M_2, |M_1|, |\mu|$ and the phases $\Phi_1, \Phi_\mu$. Each of the four invariants $a, b, c$ and $d$ is a binomial of $Re(M_1) = |M_1| \cos \Phi_1$ and $Im(M_1) = |M_1| \sin \Phi_1$. Therefore, each of the characteristic

<figure><img src="image_141.png" /><figcaption>**Fig. 137** Determination of the chargino mixing angles $\cos 2\Phi_{L,R}$ from LC measurements in $e^+e^- \to \tilde{\chi}_1^+\tilde{\chi}_1^-$ with polarized beams at different cms energies. The electroweak part of the spectrum in this scenario is a modified benchmark scenario SPS1a.</figcaption></figure>

equations in the set (150) for the neutralino mass squared $m_{\tilde{\chi}_i^0}^2$ can be rewritten in the form

$$ \mathrm{Re}(M_1)^2 + \mathrm{Im}(M_1)^2 + u_i \mathrm{Re}(M_1) + v_i \mathrm{Im}(M_1) = w_i \quad (151) $$

for $i=1-4$. The coefficients $u_i, v_i$ and $w_i$ are functions of the parameters $M_2, |\mu|, \Phi_\mu, \tan\beta$ and the mass eigenvalue $m_{\tilde{\chi}_i^0}$ for fixed $i$. The coefficient $v_i$ is necessarily proportional to $\sin\Phi_\mu$ because physical neutralino masses are CP-even; the sign ambiguity for $\sin\Phi_\mu$, a result of the two-fold cos solution $\Phi_\mu \leftrightarrow (2\pi - \Phi_\mu)$, transfers to the associated sign ambiguity in the CP-odd quantity $\mathrm{Im}(M_1)$, i.e. in $\sin\Phi_1$.

The characteristic equation (151) defines a circle in the $\mathrm{Re}(M_1), \mathrm{Im}(M_1)$ plane for each neutralino mass $m_{\tilde{\chi}_i^0}$. With only two light neutralino masses $m_{\tilde{\chi}_1^0}$ and $m_{\tilde{\chi}_2^0}$ measured, we are left with a two-fold ambiguity. The intersection points of the two crossing points depend on the unknown heavy chargino mass $m_{\tilde{\chi}_2^\pm}$. By measuring the pair-production cross sections $\sigma_L\{\tilde{\chi}_1^0\tilde{\chi}_2^0\}$ and $\sigma_R\{\tilde{\chi}_1^0\tilde{\chi}_2^0\}$, a unique solution, for both the parameters $m_{\tilde{\chi}_2^\pm}$ and $\mathrm{Re}(M_1), \mathrm{Im}(M_1)$ can be found at the same time [1096]. As a result, the additional measurement of the cross sections leads to a unique solution for $m_{\tilde{\chi}_2^\pm}$ and subsequently to a unique solution for $\{M_1, M_2; \mu; \tan\beta\}$ (assuming that the discrete CP ambiguity in the associated signs of $\sin\Phi_\mu$ and $\sin\Phi_1$ has been resolved by measuring the normal $\tilde{\chi}_2^0$ polarization).

### 5.7.3 Sensitivity to heavy virtual particles via spin correlations

Detection of charginos and neutralinos provides not only a way to measure electroweakino sector parameters (discussed in the previous sections) but is also sensitive to heavy virtual particles exchanged in chargino or neutralino production. Chargino production in the MSSM proceeds by exchange of photon and Z boson in s-channel or sneutrino exchange in t-channel.

<footnote>$^{60}$Analyses of electric dipole moments strongly suggest that CP violation in the higgsino sector will be very small in the MSSM if this sector is non-invariant at all [1100, 1101].</footnote>
<figure><img src="image_142.png" /><figcaption>**Fig. 138** Forward-backward asymmetry of $e^-$ in $e^+e^- \to \tilde{\chi}_1^+\tilde{\chi}_1^-$, $\tilde{\chi}_1^- \to \tilde{\chi}_1^0\tilde{l}^-\tilde{\nu}$ as a function of $m_{\tilde{\nu}}$ at $\sqrt{s} = 350$ GeV and with $P(e^-) = -90\%$, $P(e^+) = +60\%$. For a nominal value of $m_{\tilde{\nu}} = 1994$ GeV the statistical error in the asymmetry is shown [1102].</figcaption></figure>

In a study Ref. [1102], it was shown that the mass of a multi-TeV sneutrino can be measured up to precision of 10% at the ILC. Forward-backward asymmetries of the final state leptons and quarks from chargino decays. These asymmetries are spin-dependent observables: therefore, a correct evaluation of such asymmetries requires inclusion of spin correlations between production and decay of charginos. The asymmetry is in turn a highly sensitive probe of a particle exchanged in the t-channel, in this case mediated by a heavy sneutrino. This dependence, showing also the importance of including spin correlations, can be seen in Fig. 138.

In a scenario studied in Ref. [1102], the following set of parameters has been assumed:

$$ M_1 = 60 \text{ GeV}, M_2 = 121 \text{ GeV}, \mu = 540 \text{ GeV} \\ \tan \beta = 20, m_{\tilde{\nu}} = 2 \text{ TeV}. \qquad (152) $$

Using the light chargino production cross sections and mass, together with forward-backward asymmetries of decay products, a $\chi^2$ fit has been performed to obtain the relevant MSSM parameters. The mass of the otherwise kinematically inaccessible sneutrino could be determined with a precision of

$$ m_{\tilde{\nu}} = 2000 \pm 100 \text{ GeV} \qquad (153) $$

when forward-backward asymmetries for both leptonic and hadronic decays of chargino are used.

### 5.7.4 Sensitivity to heavy virtual particles via loop effects

With the accuracy achievable at a linear collider, one requires loop corrections in order to draw meaningful conclusions about the underlying new physics parameters. For the electroweakino sector, a study was carried out in Ref. [1103] where one-loop predictions of the cross section and forward-backward asymmetry for chargino pair production and of

the accessible chargino and neutralino masses were fitted to expected measurements. A number of one-loop calculations in the gaugino-higgsino sector can be found in the literature [1104–1117]. Although complex parameters were not considered in Ref. [1103], the renormalization was performed following Ref.'s. [1114–1117], where a dedicated renormalisation scheme in the complex MSSM was defined, in order that the analysis could easily be extended to the complex case. At tree level, there are four real parameters to be used in the fit: $M_1, M_2, \mu$ and $\tan\beta$, as well as the sneutrino mass, provided it is beyond the direct reach of the LC. The study aimed to provide information about the sensitivity to the remaining MSSM parameters which contribute to the masses and production amplitude via virtual effects. In the fit, the polarised cross-sections and forward backward asymmetry for chargino production as well as the $\tilde{\chi}_1^\pm, \tilde{\chi}_2^\pm$ and $\tilde{\chi}_1^0, \tilde{\chi}_2^0, \tilde{\chi}_3^0$ masses— calculated at NLO in an on-shell scheme as described in Ref. [1103]— were used. Note that the masses are assumed to have been measured at the LC using the threshold scan method: however the change in fit precision if the masses were obtained from the continuum was also investigated [7]. Further details of the fit method and errors are given in Ref. [1103]. The fit was performed for two scenarios, S1 and S2, shown in table 32.<fnref n="61" /> The scenarios were chosen such as to be compatible with the current status of direct LHC searches [1118, 1119], indirect limits, checked using micrOmegas 2.4.1 [1120, 1121], and flavour physics constraints i.e. the branching ratio $\mathcal{B}(b \to s\gamma)$ and $\Delta(g_\mu - 2)/2$. Note that although in S1, $M_h$ is not compatible with the recent Higgs results from the LHC [89, 199], this could easily be rectified by changing $A_t$ which would have minimal effects on the results. The one-loop corrections to the polarised cross-section and forward backward asymmetry for $e^+e^- \to \tilde{\chi}_1^+\tilde{\chi}_1^-$ are calculated in full within the MSSM, following [1116, 1117], including soft and hard radiation.

For S1, the inputs for the fit included: the masses of the charginos ($\tilde{\chi}_1^\pm, \tilde{\chi}_2^\pm$) and three lightest neutralinos ($\tilde{\chi}_1^0, \tilde{\chi}_2^0, \tilde{\chi}_3^0$), the production cross section $\sigma(\tilde{\chi}_1^+\tilde{\chi}_1^-)$ with polarised beams at $\sqrt{s} = 350$ and 500 GeV, the forward-backward asymmetry $A_{FB}$ at $\sqrt{s} = 350$ and 500 GeV and the branching ratio $\mathcal{B}(b \to s\gamma)$ calculated using micrOmegas [1120, 1121].

For S2, the inputs for the fit were the same as in S1, with $\sqrt{s} = 400$ GeV instead of 350 GeV and supplemented by the Higgs boson mass $M_h$. The sneutrino mass would have been measured. The results for S1, given in Table 33, show the fit to the 8 MSSM parameters: $M_1, M_2, \mu, \tan\beta, m_{\tilde{\nu}}, \cos\theta_T, m_{\tilde{\tau}_1}$, and $m_{\tilde{\tau}_2}$. We find that the gaugino and higgsino mass parameters are determined with an accuracy better than 1%, while $\tan\beta$ is determined with an accuracy of 5%, and 2-3% for the sneutrino mass. The limited access to the stop sector (Table 33) could nevertheless lead to hints allowing a

<footnote n="61">⁶¹Note that S2 corresponds to S3 in Ref. [1103].</footnote>
<table><caption>**Table 32** Table of parameters (with the exception of $\tan\beta$ in GeV), for scenarios 1 (S1) and 2 (S2). Here $M_{(l/q)_i}$ and $M_{(e/u)_i}$ represent the left and right handed mass parameters for a slepton/squark of generation $i$ respectively, and $A_f$ is the trilinear coupling for a sfermion $\tilde{f}$.</caption><thead><tr><th colspan="4">S1</th><th colspan="4">S2</th></tr><tr><th>Parameter</th><th>Value</th><th>Parameter</th><th>Value</th><th>Parameter</th><th>Value</th><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>M<sub>1</sub></td><td>125</td><td>M<sub>2</sub></td><td>250</td><td>M<sub>1</sub></td><td>106</td><td>M<sub>2</sub></td><td>212</td></tr><tr><td>μ</td><td>180</td><td>M<sub>A<sup>0</sup></sub></td><td>1000</td><td>μ</td><td>180</td><td>M<sub>A<sup>0</sup></sub></td><td>500</td></tr><tr><td>M<sub>3</sub></td><td>700</td><td>tan β</td><td>10</td><td>M<sub>3</sub></td><td>1500</td><td>tan β</td><td>12</td></tr><tr><td>M<sub>e<sub>1,2</sub></sub></td><td>1500</td><td>M<sub>e<sub>3</sub></sub></td><td>1500</td><td>M<sub>e<sub>1,2</sub></sub></td><td>125</td><td>M<sub>e<sub>3</sub></sub></td><td>106</td></tr><tr><td>M<sub>l<sub>i</sub></sub></td><td>1500</td><td>M<sub>q<sub>1,2</sub></sub></td><td>1500</td><td>M<sub>l<sub>i</sub></sub></td><td>180</td><td>M<sub>q<sub>i</sub></sub></td><td>1500</td></tr><tr><td>M<sub>q/u<sub>3</sub></sub></td><td>400</td><td>A<sub>f</sub></td><td>650</td><td>M<sub>u<sub>3</sub></sub></td><td>450</td><td>A<sub>f</sub></td><td>-1850</td></tr></tbody></table>

well-targeted search at the LHC. In Table 33, we also compare the fit results obtained using masses of the charginos and neutralinos from threshold scans to those obtained using masses from the continuum. For the latter, the fit quality deteriorates, clearly indicating the need to measure these masses via threshold scans. The results for S2 in table 33 show that the fit is further sensitive to $m_{\tilde{\chi}_2^0}$, with an accuracy better than 20%. In addition, an upper limit on the mass of the heavy Higgs boson can be placed at 1000 GeV, at the $2\sigma$ level.

Therefore, incorporating NLO corrections was shown to be required for the precise determination of the fundamental electroweakino parameters at the LC, and to provide sensitivity to the parameters describing particles contributing via loops. This work will soon be extended to consider both the sensitivity to complex parameters and the neutralino production cross-section.

### 5.7.5 Challenging scenarios: light higgsinos with sub-GeV mass gaps

In the MSSM, higgsino-like charginos and neutralinos are preferred to have masses of the order of the electroweak scale by naturalness arguments, as discussed in Section 5.3 of this review. If gauginos are heavy, such light $\tilde{\chi}_1^0$, $\tilde{\chi}_2^0$ and $\tilde{\chi}_1^\pm$ states will be almost mass degenerate and very challenging to be studied at the LHC. On the other hand, the clean experimental environment afforded by the ILC may allow to perform a measurement of their properties. An analysis to assess the prospects of light higgsino measurements at the ILC, based on detailed simulations, is presented in [1122]. Two scenarios with light charginos and neutralinos and mass splitting between them in the range of 0.8-2.7 GeV, but all the other SUSY particle masses in the multi-TeV range were chosen (i.e. $\mu \sim 170$ GeV, $M_1 \sim 5$ TeV, $M_2 \sim 10$ TeV, $\tan\beta \sim 48$).

For such small mass differences, the decay products of chargino are soft pions and leptons, while the largest decay mode of $\tilde{\chi}_2^0$ is to photon and LSP. Despite the fact that these final states will suffer from large SM backgrounds, a suitable

set of cuts provides separation of the signal [1122]. The effective tool for background rejection here is the tag of ISR photons recoiling against the chargino or neutralino system.

The masses of chargino and neutralino $\tilde{\chi}_2^0$ are then reconstructed from the distribution of the reduced center-of-mass energy of the system recoiling against the hard ISR photon. The expected mass resolution ranges from 1.5 to 3.3 GeV depending on the scenario. The mass difference between $\tilde{\chi}_1^\pm$ and the LSP is measured by fitting energy distribution of soft pions in the respective decays. The accuracy up to 40 MeV can be obtained for $m_{\tilde{\chi}_1^\pm} - m_{\tilde{\chi}_1^0} = 770$ MeV. Finally, the polarised cross sections for chargino pair production and $\tilde{\chi}_1^0\tilde{\chi}_2^0$ can be measured with order of per cent statistical accuracy. These results are greatly encouraging for the potential of a linear collider to tackle even such difficult scenarios. Still, detailed studies with full detector simulation and reconstruction and the incorporation of machine-induced backgrounds will be necessary to fully quantify this potential.

The fundamental MSSM parameters $M_1, M_2, \mu$ and $\tan\beta$ can be extracted from these types of observables. For the specific benchmarks chosen, the $\mu$ parameter can be determined to $\pm 4\%$. For the gaugino mass parameters, $M_1$ and $M_2$, the lower bounds can be set in the multi-TeV range, depending on the value of $\tan\beta$, which cannot be fixed from the above measurements alone, see Fig. 139. If the uncertainties could be reduced by a factor of 2 by including additional observables or increasing the integrated luminosity, the constraints on gaugino mass parameters would be significantly more restrictive and less dependent on $\tan\beta$.

### 5.7.6 Parameter fits

The determination of SUSY parameters in global fits using hypothetical measurements at the ILC has been studied in detail [1123] using the Fittino [1124] package for model points such as SPS1a' [807]. However, this point is now excluded from generic searches for SUSY at the LHC (see e.g. [1125] for early exclusions). Since then, no new complete analysis have been performed for parameter determination.
<table><caption>**Table 33** Fit results (masses in GeV) for S1 (left) and S2 (right), for masses obtained from threshold scans (threshold fit) and from the continuum (continuum fit). Numbers in brackets denote $2\sigma$ errors.</caption><thead><tr><th rowspan="2">Parameter</th><th colspan="2">S1</th><th>S2</th></tr><tr><th>Threshold fit</th><th>Continuum fit</th><th>Threshold fit</th></tr></thead><tbody><tr><td><i>M</i><sub>1</sub></td><td>125±0.3 (±0.7)</td><td>125±0.6 (±1.2)</td><td>106±0.3 (±0.5)</td></tr><tr><td><i>M</i><sub>2</sub></td><td>250±0.6 (±1.3)</td><td>250±1.6 (±3)</td><td>212±0.5 (±1.0)</td></tr><tr><td>μ</td><td>180±0.4 (±0.8)</td><td>180±0.7 (±1.3)</td><td>180±0.4 (±0.9)</td></tr><tr><td>tan β</td><td>10±0.5 (±1)</td><td>10±1.3 (±2.6)</td><td>12±0.3 (±0.7)</td></tr><tr><td><i>m</i><sub>ν</sub></td><td>1500±24 (<sup>+60</sup><sub>-40</sub>)</td><td>1500±20 (±40)</td><td>—</td></tr><tr><td>cos θ<sub>t</sub></td><td>0.15<sup>+0.08</sup><sub>-0.06</sub> (<sup>+0.16</sup><sub>-0.09</sub>)</td><td>0±0.15 (<sup>+0.4</sup><sub>-0.3</sub>)</td><td>—</td></tr><tr><td><i>m</i><sub>t1</sub></td><td>400<sup>+180</sup><sub>-120</sub> (at limit)</td><td>—</td><td>430<sup>+200</sup><sub>-130</sub> (<sup>+300</sup><sub>-400</sub>)</td></tr><tr><td><i>m</i><sub>t2</sub></td><td>800<sup>+300</sup><sub>-170</sub> (<sup>+1000</sup><sub>-290</sub>)</td><td>800<sup>+350</sup><sub>-220</sub> (at limit)</td><td>1520<sup>+200</sup><sub>-300</sub> (<sup>+300</sup><sub>-400</sub>)</td></tr><tr><td><i>m</i><sub>A0</sub></td><td>—</td><td>—</td><td>&lt; 650 (&lt; 1000)</td></tr></tbody></table>

<figure><img src="image_143.png" /><figcaption>**Fig. 139** The contours for determination of $M_1$ and $M_2$ in scenario with $m_{\tilde{\chi}_1^\pm} - m_{\tilde{\chi}_1^0} = 770$ MeV. The star denotes input values. See Ref. [1122] for more details.</figcaption></figure>

nations in global fits using data from low energy precision experiments, cosmological measurements, Higgs mass and rate measurements, up-to-date LHC constraints on SUSY production, and hypothetical ILC measurements. Therefore, in this section we revert to the existing SPS1a' results, keeping in mind that measurements of SUSY production properties at a currently realistic SUSY point would be less favourable both for the LHC and for the ILC. The reason is that the higher mass scale of first- and second-generation squarks and gluinos very strongly reduces the statistics in potential SUSY cascade decay signatures at the LHC. At the same time, given the current LHC bounds, the resolution of the small mass splittings between particles in the cascade decays typically required to allow light gauginos and sleptons at $m_{\tilde{\chi}_1^\pm} \le 250$ or $500$ GeV is more challenging, however, yet possible at the ILC.

As a relative comparison between the possible LHC and LHC+ILC performance, either SUSY models constrained at

the GUT scale (such as the CMSSM) or models defined at the TeV scale can be used. The CMSSM results from [1123] are shown in Table 34. The LHC result is based on actual precision measurements from B-factories and on $(g-2)\mu$, on the neutralino relic abundance $\Omega_{CDM}h^2$, on LEP1 SM precision measurements, and on hypothetical LHC measurements of the Higgs mass and of kinematical quantities measured in SUSY cascade decays. For a detailed list see [1123]. For the ILC, realistically modelled studies of Higgs mass, cross-section and branching fraction measurements, hypothetical measurements of kinematical edges in SUSY decays, and a large amount of measurements of cross-section times branching fractions for every kinematically accessible SUSY decay chain at sufficient rate is assumed. A time-consuming running scenario with measurements at $\sqrt{s} = 400,500$ and $1000$ GeV at different combinations of beam polarizations is employed to disentangle the mixing of the gauginos and heavy sleptons.

The results in Table 34 clearly show a significant improvement by a factor of about 5 between the LHC results and the same fit but now including additional ILC information. However, an even stronger improvement is observed when moving towards a SUSY model with significantly more freedom in the parameter choice. One possibility is the pMSSM. Here, a minimal flavour-violating MSSM with unification in the first two generations is constructed at the TeV scale, here called the MSSM18. The value $m_t$ is kept fixed due to the high expected accuracy at the ILC. This is a very favourable assumption for the LHC, because for a fit without information on $m_t$ from the ILC, the parametric uncertainties— especially on the Higgs mass— would be expected to degrade the precision of the fit result from the LHC. For details on the model, see [1123] again.

For a graphical comparison of the power of the ILC at a very favourable, albeit now excluded model point, see the difference between the LHC precision of a model-dependent determination of a SUSY mass spectrum in Fig. 140 and
<table><caption>**Table 34** Result of the fit of the CMSSM model to the precision measurements and to the hypothetical results from LHC with $\mathcal{L}^{\text{int}} = 300\,\text{fb}^{-1}$ and ILC.</caption><thead><tr><th>Parameter</th><th>Nominal value</th><th>Fit</th><th>LHC Uncertainty</th><th>ILC Uncertainty</th></tr></thead><tbody><tr><td>tan β</td><td>10</td><td>9.999</td><td>± 0.36</td><td>± 0.050</td></tr><tr><td>M<sub>1/2</sub> (GeV)</td><td>250</td><td>249.999</td><td>± 0.33</td><td>± 0.076</td></tr><tr><td>M<sub>0</sub> (GeV)</td><td>100</td><td>100.003</td><td>± 0.39</td><td>± 0.064</td></tr><tr><td>A<sub>0</sub> (GeV)</td><td>-100</td><td>-100.0</td><td>± 12.0</td><td>± 2.4</td></tr></tbody></table>

<figure><img src="image_144.png" /><figcaption>**Fig. 140** SUSY mass spectrum consistent with the existing low-energy measurements and the hypothetical LHC measurements at $\mathcal{L}^{\text{int}} = 300\,\text{fb}^{-1}$ for the MSSM18 model. The uncertainty ranges represent model dependent uncertainties of the sparticle masses and not direct mass measurements.</figcaption></figure>

<figure><img src="image_145.png" /><figcaption>**Fig. 141** Derived mass distributions of the SUSY particles using low energy measurements, hypothetical results from LHC with $\mathcal{L}^{\text{int}} = 300\,\text{fb}^{-1}$ and hypothetical results from ILC. When comparing to Fig. 140, please note the difference in the scale.</figcaption></figure>

the corresponding spectrum for the added ILC information in Fig. 141. An enormous improvement is observed in the heavy Higgs sector, stemming from the hypothetical direct measurements of the heavy Higgs bosons at the ILC, while they would have remained inaccessible at the LHC. Also for the other masses, improvements of a factor of 10 to 100 are possible [1123]. For the SPS1a' like MSSM18, the added benefit of the ILC over the LHC is much more apparent due to the larger freedom in the model. For a model with only four free parameters, such as the CMSSM, a few measurements with relatively good precision are enough to constrain the parameters in a reasonable range, such as for the LHC in the hypothetical SPS1a' CMSSM. However, once the less accessible states decouple from the more accessible ones, such as in the MSSM18, the direct information on states like the light CP-even Higgs boson $h$ and the squark mass scales does not suffice to constrain less accessible states anymore (like the heavy Higgses) since they are controlled by additional parameters like $m_A$ and $X_f$ in the MSSM18: these cannot be easily accessed otherwise. At the $e^+e^-$ LC, however, the high precision measurements of the full Higgs sector (as for SPS1a') and the very high precision measurements of sparticle masses and couplings, would have allowed to disentangle the mixings and mass parameters in the gaugino, the heavy slepton and the stop sector individually. Such determinations reduce the model-dependence dramatically and improve the fit precision accordingly, by providing independent precise probes of all degrees of freedom of the model.

### 5.7.7 Extrapolation to GUT scale

As discussed in Section 5.2, many of the commonly used SUSY models impose strong assumptions at the high scale inspired by suppositions on the SUSY breaking mechanism. In the CMSSM– with the input parameters $m_0$, $m_{1/2}$, $\tan\beta$, $A_0$, sign$\mu$ at the GUT scale $M_{\text{GUT}} \approx 2 \cdot 10^{16}$ GeV– all gauge couplings $\alpha_{1,2,3}$ and also all gaugino masses $M_{1,2,3}$ and scalar masses unify at $M_{\text{GUT}}$.

Generally, in order to test such model-dependent assumptions, one can start from a precisely measured particle spectrum at lower energies and extrapolate the underlying parameter to higher energies, up to $M_{\text{GUT}}$, as described in [1097]. The evolution of the parameters happens via ap-
plying the renormalization group equations (RGE). In prac-
tically all studies, it is assumed to combine measurements
of the non-colored spectrum at the LC with measurements
of the coloured spectrum at the LHC.

As one example, we choose benchmark ‘Model I’ from
Ref. [9] with the GUT scale parameters $m_0 = 966$ GeV,
$m_{1/2} = 800$ GeV, $A_0 = 0$, $\tan\beta = 51$, sign($\mu$) = +1 which
determine the particle spectrum at low energy. In [9], it has
been shown that the masses of neutralinos and the sleptons
of the first two generation can be measured with a preci-
sion of 1-2% at a 3-TeV collider. In addition, one assumes to
measure the gluino mass $m_g = 1812$ GeV with 5% precision
at the LHC and at a 3% level for all other sfermion masses
at the LC. Based on the mass and cross section measure-
ments of the neutralino/chargeino sector, one can reconstruct
the quantities at tree-level: $M_1, M_2, \mu$ and $\tan\beta$.

Since we measure on-shell masses, but use $\overline{DR}$ parame-
ters for the evolution of parameters, the corresponding shifts
must be calculated. This intertwines the different sectors:
naively one would expect that the relative precision of the
masses transfers one to one to the precision on the gaug-
ino mass parameters. However, in case of the gluino mass
parameters, the uncertainty due to the squark mass measure-
ments can increases the uncertainty on $M_3$ by up to a fac-
tor 2, e.g. instead of a five per-cent uncertainty one obtains
roughly a ten per-cent uncertainty. At the level of one-loop
RGEs, the relative uncertainties are approximately scale in-
variant as at this level $M_i/\alpha_i$ is an RGE invariant. How-
ever, at the two-loop level, also the trilinear A-parameters
of the third generation enter and, thus, one should know
them to a precision of at least 40% as otherwise the un-
certainties at the high-scale can be significantly worse com-
pared to the one at the electroweak scale. The trilinear cou-
plings can be determined via cross section measurements
and sfermion decays involving Higgs bosons (or decays of
heavy Higgs bosons into sfermions) [837, 1097]. Under the
above assumption, we find a unification of the gaugino mass
parameters to about 10%: see Figure 142 (top panel).

In the evaluation of the sfermion mass parameters, also
the gaugino mass parameters enter where in particular $M_3$
is important for the evolution of the squark mass param-
eters. In case of third generation sfermions and the Higgs
mass parameters, also large Yukawa couplings as well as the
A-parameters enter the RGEs and intertwine them in a non-
trivial way. Taking the same assumptions as above, we find a
clear overlap between all scalar mass parameters when run-
ning up to the GUT scale, see Fig. 142 (middle and bottom
panel), pointing clearly to the 1000 GeV region for $m_0$.

5.8 Lepton Flavour and CP Violation

The general structure of supersymmetry admits several pos-
sible extensions to the MSSM, either by switching on new

<figure><img src="image_146.png" /><figcaption>**Fig. 142** Evolution of gaugino and sfermion (1. and 3. generation) parameters in the CMSSM for $m_0 = 966$ GeV, $m_{1/2} = 800$ GeV, $A_0 = 0$, $\tan\beta = 51$, sign$\mu = +1$ [9] to the GUT scale.</figcaption></figure>

couplings or introducing new parameters, such as CP-violating
phases or adding new fields, each resulting in new, specific
phenomenology. Because of its versatility and the limited
SM backgrounds, a linear collider is best suited to investi-
gate these scenarios. In this section, we review the sources of
lepton flavour and CP violation in extended SUSY models
and their phenomenology in $e^{+}e^{-}$ collisions.
### 5.8.1 Lepton Flavour Violation

A significant body of data from atmospheric, solar, reactor and accelerator neutrino experiments [1126] have revealed the non-zero value of neutrino masses and oscillations with near-maximal $ν_μ-ν_τ$ and large $ν_e-ν_μ$ mixing. A very attractive explanation for the smallness of neutrino masses and their mixings is a seesaw mechanism embedded within the framework of SUSY models. In this case [1127], masses and mixings in the neutrino system are caused by very heavy right-handed Majorana neutrinos with masses close to the GUT scale. Even if the sfermion mass matrices are diagonal at the GUT scale, flavour violating mixings are induced radiatively [1128]. A substantial $ν_μ-ν_τ$ mixing leads to large $\tilde{\mu}_L-\tilde{\tau}_L$ and $\tilde{ν}_μ-\tilde{ν}_τ$ mixings. It is natural to expect that charged lepton flavour violation (cLFV) should occur at some level thus raising the interesting possibility of observing these processes in low-energy rare decays $μ \to eγ$, or $τ \to μγ$, or $μ-e$ conversion [1129, 1130] and at a high energy $e^+e^-$ collider.

In the Standard Model, cLFV processes are strongly suppressed due to the GIM mechanism. However, in SUSY, virtual super-partner loops may provide an enhancement [1130] making them observable. Moreover, if sleptons are directly produced, cLFV can also be directly tested in their production and decay processes. For nearly degenerate sleptons, supersymmetric LFV contributions to low-energy rare decay processes are suppressed as $Δm_ℓ/m_ℓ$ through the superGIM mechanism and constraints from the yet unobserved radiative decays $ℓ_i \to ℓ_jγ$ are not very stringent. On the other hand, in direct decays of sleptons, this kind of supersymmetric lepton flavour violation is suppressed only as $Δm_ℓ/Γ_ℓ$ [1131]. Since $m_ℓ/Γ_ℓ$ can be large, spectacular signals may be expected leading to possible discoveries at the LHC and in particular at future lepton collider experiments. Among the possibilities considered so far, there is slepton pair production at a linear collider as well as signals from electro-weak gaugino production and their subsequent cascade decays $\tilde{χ}_2^0 \to \tilde{χ}_1^0 + e^+μ^-$, $\tilde{χ}_2^0 \to \tilde{χ}_1^0 + μ^+τ^-$, at both a linear collider and the LHC [1131–1134].

At a LC, the cLFV signals can be looked for directly in slepton pair production, for example

$$ \begin{aligned} e^+e^- &\to \bar{\ell}_i^- \bar{\ell}_j^+ \to \tau^+ \mu^- \tilde{\chi}_1^0 \tilde{\chi}_1^0, \\ e^+e^- &\to \tilde{v}_i \tilde{v}_j^c \to \tau^+ \mu^- \tilde{\chi}_1^+ \tilde{\chi}_1^- \end{aligned} \qquad (154) $$

or indirectly via sleptons produced singly in chain decays of heavier charginos and/or neutralinos $\tilde{χ}_2 \to ℓ_i \tilde{\ell}_j$, $\tilde{\ell}_j \to ℓ_k \tilde{χ}_1$:

$$ \begin{aligned} e^+e^- &\to \tilde{\chi}_2^+ \tilde{\chi}_1^- \to \tau^+ \mu^- \tilde{\chi}_1^+ \tilde{\chi}_1^- \\ e^+e^- &\to \tilde{\chi}_2^0 \tilde{\chi}_1^0 \to \tau^+ \mu^- \tilde{\chi}_1^0 \tilde{\chi}_1^0. \end{aligned} \qquad (155) $$

With $\tilde{χ}_1^± \to \tilde{χ}_1^0 f \tilde{\bar{\ell}}^\prime$, and $\tilde{χ}_1^0$ escaping detection, the signature therefore would be $τ^±μ^+ + jets + E_T, τ^±μ^+ + ℓ + E_T$, or $τ^±μ^+ + E_T$, depending on hadronic or leptonic $\tilde{χ}_1^±$ decay mode.

In the case of narrow widths and small mass differences between the sleptons of different generations, $Δ\tilde{m}_{ij} \ll \tilde{m} = \frac{1}{2}(m_2+m_3)$ and $\tilde{m}Γ_{ij} \simeq (\tilde{m}_iΓ_i + \tilde{m}_jΓ_j)/2 \ll \tilde{m}^2$, and assuming a pure 2-3 inter-generation mixing between $\tilde{ν}_μ$ and $\tilde{ν}_τ$, generated by a near-maximal mixing angle $\tilde{θ}_{23}$, and ignoring any mixings with $\tilde{ν}_e^{62}$, the cross sections for $τ^+μ^-$ in the final state simplify considerably [1131, 1135]. For $τ^+μ^-$ produced in the decays of a pair of sleptons, Eq. (155), the cross section can be approximated as:

$$ \sigma_{23}^{\text{pair}} = \chi_{23}(3 - 4\chi_{23})\sin^2 2\tilde{\theta}_{23} \times \sigma_0 \times Br, \qquad (156) $$

whereas for $τ^+μ^-$ produced from the gaugino decay, Eq. (155), it takes the form

$$ \sigma_{23}^{\text{casc}} = \chi_{23} \sin^2 2\tilde{\theta}_{23} \times \sigma_0 \times Br. \qquad (157) $$

Here the cLFV effect is taken into account by the factors $\sin^2 2\tilde{\theta}_{23}$ and $\chi_{23} \equiv x_{23}^2/2(1+x_{23}^2)$ where $x_{23} \equiv Δ\tilde{m}_{23}/Γ_{23}$. The difference between Eq. (156) and Eq. (157) is due to the correlated slepton pair production in the processes Eq. (155). In the above expressions, $σ_0$ is the corresponding sparticle pair-production cross section in $e^+e^-$ collision and $Br$ is the product of relevant branching ratios for the corresponding decay chains without cLFV contributions.

The potential of exploring the cLFV at a LC has recently been revisited in final states with $τμ$ [1136] and $eμ$ [1137]. Both analyses adopted the cMSSM framework with benchmark points chosen to be consistent with the limits from the LHC searches and cosmological relic LSP density. The benchmarks feature relatively low values of $m_0$ (compared to $m_{1/2}$) to provide a relatively light slepton spectrum accessible at a LC while avoiding the LHC bounds on the strongly interacting sector. To assess the sensitivity of the cross section measurements to the LFV terms $(\delta_{LL,RR})_{ij}$, where the flavour-mixing entries encode the inter-generation elements of the slepton mass matrix $(\delta_{XX})_{ij} = (M_{\tilde{χ}_X}^2)^{ij}/(M_{\tilde{χ}_X})^{ii}$, ($X = L, R$), Fig. 143 shows current constraints and possible LFV effects for reference points. Despite the SM and SUSY charged current backgrounds, the expected number of signal events should allow us to probe cLFV in extensive regions of the SUSY seesaw parameter space. Both direct slepton pair production and sleptons produced in cascade decays may provide interesting signals in the cosmologically-favoured region of the supersymmetric parametric space. In comparison to the LHC, the LC could provide additional insights by virtue of its greater kinematic range for slepton production and its sensitivity to both RR and LL mixing.

Lepton flavour violation can also reveal itself in other processes such as $e^+e^- \to \tilde{χ}_i^+ \tilde{χ}_j^-$. This process proceeds through s-channel $γ/Z$ and also t-channel $\tilde{ν}_e$ exchange. In the LFV scenario, the $\tilde{ν}_e$ is a mixture of three mass eigenstates. The production cross section for chargino pair pro-

<footnote n="62">⁶²Complete expressions are usually used for phenomenological investigations.</footnote>
<figure><img src="image_147.png" /><figcaption>**Fig. 143** Constraints on the magnitudes of the mixing parameters and possible LFV effects for reference points from [1136]. The shaded areas are those allowed by current limits on BR($\tau \to e\gamma$) (dot-dash line) and BR($\tau \to \mu\gamma$) (dash line) using four different reference points (shown by the thick lines bounding the solid shaded areas and the thin blue lines bounding the ruled shaded areas). The solid lines are contours of $\sigma(e^+e^- \to \tau^\pm\mu^\mp + 2\chi^0)$ in fb for $\sqrt{s} = 2000$ GeV.</figcaption></figure>

duction may change by a factor of two or more in the presence of $\tilde{v}_e - \tilde{v}_\tau$ mixing even if current bounds on LFV rare lepton decays are significantly improved (see Fig.144) [1138]. The effect of $\tilde{v}_e - \tilde{v}_\mu$ mixing, due to stronger experimental bounds, is less dramatic, as seen in the right panel of Fig.144.

<figure><img src="image_148.png" /><figcaption>**Fig. 144** Cross section $\sigma(e^+e^- \to \tilde{\chi}_1^+\tilde{\chi}_1^-)$ as a function of the mixing parameter $\cos 2\theta_{13}$ (a) and $\cos 2\theta_{12}$ (b) at a LC with cm energy of 500 GeV and polarised beams: $P_L=-0.9$ for electrons and $P_L=0.6$ for positrons. Details of assumed scenarios (a) and (b) are in [1138].</figcaption></figure>

### 5.8.2 CP Violation

Since the first observation of CP violation almost fifty years ago, the “cryptic message from Nature” it conveys still needs to be deciphered in full. An attractive feature of SUSY is that it allows for new sources of CP violation which are needed if the baryon-antibaryon asymmetry observed in the universe is to be explained by particle physics. Compared

to the case of CP-conserving SUSY, new CP phases appearing in supersymmetry may change masses, cross sections, decay branching ratios, etc. providing many possible ways to detect and measure them at colliders. Since such observables are CP-even, CP-violating effects may be distinguished from fortuitous combinations of parameters not invoking CP-violating phases only by the joint analyses of several CP-even observables. For example, an observation of s-wave excitation above respective thresholds of three non-diagonal pairs of neutralinos [39], or the observation of simultaneous sharp s-wave excitations of the production cross section $\sigma(e^+e^- \to \tilde{\chi}_i^0\tilde{\chi}_j^0)$ ($i \neq j$) near threshold and the $f\bar{f}$ invariant mass distribution near the end point of the decay $\tilde{\chi}_i^0 \to \tilde{\chi}_j^0 f\bar{f}$ [1139] is a qualitative, unambiguous evidence for CP violation in the neutralino system. A linear collider of sufficient energy can perform all these measurements.

The most direct way to detect CP-violation is to construct CP-odd observables which cannot be mimicked by other parameters of the theory. Such quantities typically involve asymmetries constructed as triple products of momenta and/or spin vectors. Due to spin correlations, such asymmetries show unique hints for CP phases already at tree level. Triple product asymmetries have been proposed in many theoretical papers in which neutralino production with two- and three-body decays, charginos with two- and three-body decays, also with transversely polarised beams, have been studied in the past [1140]. At tree level, the neutralino and chargino sector has two independent CP phases: for instance of $M_1$ and $\mu$ when rotating away the phase of $M_2$. Assuming the phase of $\mu$- strongly constrained by EDM bounds- to be small, the phase of $M_1$ could lead to CP sensitive triple product asymmetries of up to 20%: see Fig. 145. As mentioned above, a recent analysis performed with full event simulation and reconstruction [1070] shows that these asymmetries constructed from $(\mathbf{p}_{e^-} \times \mathbf{p}_{\ell_N^+}) \cdot \mathbf{p}_{\ell_F^-}$ can be measured to $\pm 1\%$ from neutralino two-body decays into slepton and lepton followed by slepton decay: $\tilde{\chi}_j^0 \to \tilde{\ell}^- \ell_N^+ \to \tilde{\chi}_1^0 \ell_F^- \ell_N^+$. From a fit to the measured neutralino cross-sections, masses and CP-asymmetries, $|M_1|$ and $|\mu|$ can be determined to a few per mil, $M_2$ to a few percent, $\Phi_1$ to 10% as well as $\tan\beta$ and $\Phi_\mu$ to 16% and 20%, respectively.

The sfermion sector brings in the CP phase of the tri-linear scalar coupling $\Phi_A$. The sensitivity of the linear collider to the CP phase in the stop sector has been looked at recently [1141] by analysing a chain decay $\tilde{t}_1 \to \tilde{\chi}_2^0(\to \tilde{\chi}_1^0 \ell_N^\mp \ell_F^\pm) + t(\to W^+b)$. Such decays allow one to construct two triple products originating from the covariant product in the spin-spin-dependent part of the amplitude, namely $A_{\ell 1} \sim \mathbf{p}_{\ell 1^+} \cdot (\mathbf{p}_W \times \mathbf{p}_t)$ calculated in the reconstructed $\tilde{\chi}_2^0$ rest frame, and $A_{\ell \ell} \sim \mathbf{p}_b \cdot (\mathbf{p}_{\ell^+} \times \mathbf{p}_{\ell^-})$ calculated in the reconstructed W rest frame. The right panel of Fig. 145 shows that CP sensitive asymmetries can reach 10-15%. Under the assumption of accurate momentum reconstruction, this asym-
<figure><img src="image_149.png" /><figcaption>**Fig. 145** Top panel: $p_T^{\text{miss}}$ dependence of CP asymmetries in neutralino-pair production and decay processes (from [1070]). Bottom panel: Asymmetries, $A_{\ell 1}$ and $A_{\ell \ell}$ as functions of $\Phi_{A_\ell}$ (from [1141]).</figcaption></figure>

metry could be measured for $2~ab^{-1}$ (1 $ab^{-1}$) of data collected at $\sqrt{s} = 1$ TeV in the region of a maximal CP violating angle, $1.10\pi < \Phi_{A_t} < 1.5\pi$ (1.18$\pi < \Phi_{A_t} < 1.33\pi$).

Finally, it is worthwhile to recall that the CP-odd observables can also be constructed in the non-diagonal chargino pair production process $e^+e^- \to \tilde{\chi}_1^+\tilde{\chi}_2^-$ from unpolarised cross sections at one-loop [1142]. Obviously, at tree level the CP-asymmetry $A^{12} \sim \int [d\sigma(\tilde{\chi}_1^-\tilde{\chi}_2^+) - d\sigma(\tilde{\chi}_2^-\tilde{\chi}_1^+)]d\cos\theta$ (with the polar angle $\theta$ of $\tilde{\chi}_j^-$ with respect to the $e^-$ momentum direction) vanishes even in CP-noninvariant theories. In order to obtain a non-zero asymmetry in the chargino production it requires another source of non-trivial imaginary contribution to the amplitude. Such a term can be generated by the absorptive part of a loop diagram when some of the intermediate state particles in loop diagrams go on-shell. The CP-odd asymmetry is generated due to interference between the imaginary part of loop integrals and imaginary parts of couplings. Numerical analyses show that the asymmetries can be of the order of a few percent and in principle measurable, allowing for discovery of the CP-violating phases via simple event counting experiments.

## 5.9 Beyond the MSSM

### 5.9.1 The NMSSM

The supersymmetric $\mu$ problem arises because the higgsino mass $\mu$ term in the MSSM superpotential is not a SUSY breaking term, but instead preserves SUSY. Thus, naively one would expect $\mu \sim M_P$ instead of $M_{\text{weak}}$; this possibility seems phenomenologically disallowed. One solution, endemic to gravity mediation, is for the $\mu$ term to be forbidden by some symmetry, such as a Peccei-Quinn (PQ) symmetry, but then to re-generate it via interactions with either the PQ sector [1143] or the hidden sector [1144]. An alternative possibility occurs by extending the MSSM with an additional gauge singlet superfield $N$, where the $\mu$ term then arises from its coupling to the Higgs fields in the superpotential, $\lambda NH_uH_d$. This extension is known as the Next-to-Minimal SUSY extension of the SM, or NMSSM. In the NMSSM, an effective $\mu = \lambda x$ term is expected to be generated around the electro-weak scale when the scalar component of the singlet $N$ acquires a vacuum expectation value $x = \langle N \rangle$. Moreover, the NMSSM is additionally motivated in that it provides additional quartic contributions to the light Higgs scalar mass $M_h$, thus perhaps more easily accommodating the rather large value $M_h \sim 125$ GeV, which otherwise requires TeV scale top-squarks which some authors consider to have a conflict with naturalness. Further reduction in the fine-tuning of the NMSSM can be achieved by introducing extra matter terms [1145]. Independently, a bottom-up approach for addressing the fine-tuning problem, via “natural SUSY”, calls for the third generation sfermions and the higgsino to be light, while the rest of the superpartners can be heavy. However, the higgsino cannot then be the sole dark matter candidate since higgsinos annihilate too rapidly into WW and ZZ.

Within the extended Higgs sector of the NMSSM, the new singlino state, with mass below that of the higgsino, might serve as a dark matter particle, or the LSP might have a significant singlino component. The phenomenology of different scenarios for the mixing character of the lightest neutralino–singlino, higgsino, gaugino-like– has been systematically analyzed in the plane of the NMSSM-specific Yukawa couplings $\lambda – \kappa$-plane, cf. also Fig. 146. In the first case, the decay width of the higgsino to the singlino is of order 100 MeV. The pattern of decays can be rich (see Fig. 147), providing us with clear signatures which can be studied at a LC of sufficient energy. The precision measurement of these decay branching ratios will illuminate the structure of the extended model [1149]. These decay products are quite soft, however, and are expected to be virtually invisible under the standard LHC trigger conditions. Whether or not these particles can be seen at the LHC, the linear collider would again be needed for a complete study, which requires the deter-
<figure><img src="image_150.png" /><figcaption>**Fig. 146** Lightest neutralino $\tilde{\chi}_1^0$ is mainly higgsino-like: regions in the $(\lambda-\kappa)$-plane allowed by experimental and phenomenological constraints. The light-blue-shaded regions delimited by the light blue boundary pass dark matter constraints. The coloured regions delimited by the purple boundary pass checks within HiggsBounds [1146] and HiggsSignals [1147]. The red area is allowed by all the constraints [1148].</figcaption></figure>

mination of their branching fractions. The singlino-higgsino mixing angle, which determines the annihilation cross section of the LSP and the thermal dark matter density, could be measured at the LC through a determination of the higgsino width using a threshold scan, as discussed above, or by precision measurements of the NMSSM mass eigenvalues.

<figure><img src="image_151.png" /><figcaption>**Fig. 147** Neutralino decay $\tilde{\chi}_2^0 \rightarrow \tilde{\chi}_1^0 + X$ branching fractions as function of the mass splitting $\Delta M = M_{\chi_2^0} - M_{\chi_1^0}$ (from [1149]).</figcaption></figure>

The LC capabilities in distinguishing between the NMSSM and the MSSM, when the observable particle spectrum and the corresponding decay chains are very similar in pattern, has been studied in detail [39, 444]. From data taken in $e^+e^-$ collisions at three different centre-of-mass energies, the distinction is possible. When exploiting the available information by applying a global fit, just two $\sqrt{s}$ choices can be sufficient, depending on the mixing charac-

ter of the lightest neutralino states [444, 1148]. If the full neutralino/chargeino spectrum is accessible at the maximum collider energy, sum rules for the production cross sections, yielding a different energy behaviour in the two models, may also be exploited. In scenarios with dominant couplings of a mostly-singlino LSP to the NLSP particle, as predicted for large values of the $x$ parameter, the existence of displaced vertices leads to a particularly interesting signature that can be precisely resolved with the excellent detector resolution envisaged at a linear collider.

### 5.9.2 R-Parity Violation

The signatures for the SUSY searches discussed so far are based on the assumption that R-parity, the additional quantum number distinguishing SUSY particles from their SM counterparts, is conserved leading to final states with significant missing energy, due to the escaping LSPs. Introducing R-parity violation (RpV) changes drastically the SUSY phenomenology. R-parity violating couplings allow for single production of SUSY particles and their decays to SM particles. The latter aspect makes RpV SUSY much harder to detect at the LHC due to the absence of missing transverse energy, so that the currently explored region is significantly smaller than in the R-parity conserving case, even when assuming mass unification at the GUT scale [1150]. Although the LSP is not stable, there are models with small R-parity violation which naturally yield a consistent cosmology incorporating primordial nucleosynthesis, leptogenesis and gravitino dark matter [1151]; axion dark matter is also a possibility. Since the gravitino decays into SM particles are doubly suppressed by the Planck mass and the small R-parity breaking parameter, its lifetime can exceed the age of the Universe by many orders of magnitude, and the gravitino remains a viable dark matter candidate [1152].

Bi-linear R-parity violation (BRpV) has phenomenological motivations in neutrino mixing [46] as well as in leptogenesis [1153]. In this case, the mixing between neutrinos and neutralinos leads to one massive neutrino at tree level and the other two via loop effects [1154]. Once the parameters are adjusted to satisfy the neutrino constraints, the lightest neutralino typically decays inside the detector volume [46]. Since the parameters that determine the decay properties of the LSP are the same parameters that lead to neutrino masses and oscillations, there are strong correlations between the neutralino branching ratios and the neutrino mixing angles, e.g.,

$$BR(\tilde{\chi}_1^0 \rightarrow W^{\pm} \mu^{\mp})/BR(\tilde{\chi}_1^0 \rightarrow W^{\pm} \tau^{\mp}) \sim \tan^2 \theta_{23}. \quad (158)$$

By measuring the ratio of the branching fractions for $\tilde{\chi}_1^0 \rightarrow W^{\pm} \mu^{\mp}$ and $W^{\pm} \tau^{\mp}$, the neutrino mixing angle $\sin^2 \theta_{23}$ could be determined to percent-level precision, as illustrated in Fig. 148. The characteristic decay $\tilde{\chi}_1^0 \rightarrow W^{\pm} l^{\mp}$ gives
background-free signatures at an $e^{+}e^{-}$ linear collider, possibly with a detectable lifetime of the $\tilde{\chi}_1^0$ depending on the strength of the BRpV couplings. In the hadronic decay mode of the $W^{\pm}$, these events can be fully reconstructed and the $\tilde{\chi}_1^0$ mass can be measured to $\mathcal{O}(100)$ MeV depending on the assumed cross-section [45]. The LC results could then be checked against the measurements from neutrino oscillation experiments to prove that BRpV SUSY is indeed the origin of the structure of the neutrino sector.

<figure><img src="image_152.png" /><figcaption>**Fig. 148** Achievable precision on $\sin^2\theta_{23}$ from BRpV decays of the $\tilde{\chi}_1^0$ as a function of the produced number of neutralino pairs compared to the current precision from neutrino oscillation measurements. Over a large part of the $m_{1/2}$ vs $m_0$ plane, the neutralino pair production cross-section of the order of 100 fb [45].</figcaption></figure>

Finally, in the case of trilinear R-parity violation (TRpV), the exchange of sparticles can contribute significantly to SM processes and may even produce peak or bump distortions to the distribution of cross sections [1155]. Below threshold, these new spin-0 exchanges may manifest themselves via indirect effects on observables such as cross sections and asymmetries which can be precisely measured in $e^{+}e^{-}$ collisions, including spectacular decays [1156]. It has been shown recently that the observed enhancement of the semileptonic and leptonic decay rates of $B \to \tau\nu$ modes can be explained in the framework of TRpV [1157]. However, in such cases it would be important to identify the actual source among the possible non-standard interactions as many different new physics scenarios may lead to very similar experimental signatures. At the LC, a technique based on a double polarisation asymmetry formed by polarising both beams in the initial state has been proposed [1158]. This is particularly suitable to directly test for s-channel $\tilde{\nu}$ exchange.

Again, the availability of both $e^{-}$ and $e^{+}$ polarisation plays a crucial rôle in identifying the new physics scenario (see Fig.149). In contrast, the left-right asymmetry, $A_{LR}$, obtained with only electron polarisation, does not appear to be useful for this purpose.

<figure><img src="image_153.png" /><figcaption>**Fig. 149** Discovery reach at 95% C.L. in Bhabha scattering for the sneutrino mass as a function of $\lambda_{131}$ at $\sqrt{s} = 0.5$ TeV (left panel) and 1 TeV (right panel), for $\mathcal{L}_{\text{int}} = 0.5 \text{ ab}^{-1}$. For comparison, the discovery reach on $M_{\tilde{\nu}}$ in muon pair production for $\lambda_{232} = 0.5 \times M_{\tilde{\nu}}$/TeV is also shown (from [1158]).</figcaption></figure>

### 5.9.3 R symmetry

In the R-parity conserving MSSM, the gravitino, gluino, and other gaugino mass terms can be introduced once supersymmetry is broken. However, it has recently been realised that by requiring an additional R-symmetry [1159, 1160] beyond R-parity, which can be continuous or discrete, exact or approximate, is not only phenomenologically viable, but may allow sizable flavor-violating operators without generating large FCNC or CP violation. A continuous $U(1)_R$ symmetry on the MSSM, where gauginos and squarks have R-charges $R = +1$, and the Higgs scalars have $R = 0$, not only forbids baryon- and lepton-number changing terms in the superpotential, but also dimension-five operators mediating proton decay [848].

R symmetry also removes some of the potentially unwanted parameters of the theory, such as trilinear A-terms for the scalars, the $\mu$-term and Majorana gaugino masses, while Majorana neutrino masses are allowed. The absence of $\mu$ and A terms helps to solve the flavor problem without flavor-blind mediation. However, since gauginos must get masses, adjoint chiral super-fields for each gauge factor are introduced to generate R-symmetry preserving Dirac gaugino masses. Similarly, the Higgs sector is extended by adding multiplets $R_u$ and $R_d$ with the appropriate charges to allow R-symmetric $\mu$-terms with $H_u$ and $H_d$ respectively. The scalar components of the Higgs (and not the R-fields) acquire VEVs that break electro-weak symmetry, thereby preserving the R-symmetry. This general class of models
goes under the name of the Minimal *R*-symmetric Super-symmetric Standard Model (MRSSM) [1082, 1083].

The phenomenology of MRSSM is quite different from that of the MSSM. Since the mixing with additional scalars reduces the tree-level Higgs mass, loop corrections must play even more significant role than in the MSSM. Recently it has been shown [1161, 1162] that additional contributions from TeV-scale chiral adjoint superfields and *R*-Higgses allow to accommodate a light Higgs boson of mass ~125 GeV more comfortably than in models such as the cMSSM even for stop masses of order 1 TeV and absence of stop mixing. Moreover important constraints from electroweak precision observables are imposed on parameters entering the Higgs mass calculation, in particular the *W* boson mass, because *R*-symmetry necessarily introduces an SU(2) scalar triplet that develops a vev. A full one-loop calculation [1162, 1163] shows that regions of parameters can be found consistent with the measured Higgs and *W* boson masses.

Because gauginos are Dirac, scalars can naturally be lighter than gauginos. The scalar component of the adjoint SU(3) super-field, a sgluon, can be relatively light and accessible at the LHC [1164–1166]. The Dirac neutralinos can easily be tested at a LC by investigating the threshold production behaviour of the diagonal pair production (Fig.150) or by angular distributions. In contrast to standard Higgs, the *R*-Higgs bosons do not couple singly to SM fields, and all standard-type channels are shut for the single production. Nevertheless, if they are not too heavy, the *R*-Higgs bosons can be produced in pairs at the LHC, via the Drell-Yan mechanism, and at prospective $e^{+}e^{-}$ colliders (see Fig.150).

<figure><img src="image_154.png" /><figcaption>**Fig. 150** Left panel: pair production of wino-like neutralinos near threshold in the MSSM and the Dirac theory (from [1091]). Right panel: production of the neutral and charged *R*-Higgs boson pairs at TeV $e^{+}e^{-}$ colliders (from [1083]).</figcaption></figure>

*R*-symmetry allows either Yukawa or *A*-terms, but not both. With the neutrino Yukawas zero, large *A*-terms for sneutrinos are thus natural in the MRSSM. With three singlet superfields $N_i$, a 6 × 6 sneutrino mass matrix can feature large off-diagonal *A*-terms mixing the left and right handed sneutrinos. In such a framework, a mixed sneutrino can serve as a successful candidate for dark matter, an appropriate Majorana neutrino masses can be generated and striking lepton-

flavor violation signals can be expected at both LHC and linear colliders [1167].

## 5.10 Relevance of $e^{-}e^{-}$, $e\gamma$ and $\gamma\gamma$ options for SUSY searches

Linear colliders offer an impressive capacity to discover and untangle new physics such as supersymmetry in $e^{+}e^{-}$ collisions. Their ability to adapt to the specific needs of various scenarios of new physics is augmented by the possibility to run such machines in $e^{-}e^{-}$, $e\gamma$ or $\gamma\gamma$ modes. In the latter two cases, the $\gamma$s are generated via laser backscattering off of the incoming electron beams. Each of these options offers new avenues for understanding supersymmetry.

By operating in $e^{-}e^{-}$ mode, a vast array of SM background processes that could be problematic at $e^{+}e^{-}$ colliders are automatically turned off. One might counter that most SUSY production reactions are also turned off in $e^{-}e^{-}$ mode as well. Reactions like $e^{-}e^{-} \to \tilde{e}_{R}\tilde{e}_{R}$, $\tilde{e}_{L}\tilde{e}_{L}$, $\tilde{e}_{L}\tilde{e}_{R}$ and $\tilde{e}_{R}\tilde{e}_{L}$ provide distinctive SUSY signals [1168–1173]. These take place via *t*-channel neutralino exchange. An advantage of $e^{-}e^{-}$ collisions is obtained in threshold scans: whereas $e^{+}e^{-} \to \tilde{e}_{R}^{+}\tilde{e}_{R}$, $\tilde{e}_{L}^{+}\tilde{e}_{L}$ suffer the usual $\beta^{3}$ suppression factor typical of scalar pair production, the $e^{-}e^{-} \to \tilde{e}_{R}\tilde{e}_{R}$ and $\tilde{e}_{L}\tilde{e}_{L}$ reactions are only suppressed by $\beta^{1}$. This offers better accuracy in the selectron mass measurement via the unsuppressed threshold production of selectron pairs. This is especially important in that threshold scans for $\beta^{3}$ suppressed processes will require very high integrated luminosity while similar or better measurements can be made on $\beta^{1}$ suppressed processes at much lower integrated luminosity.

Since $\tilde{e}_{R}\tilde{e}_{R}$ takes place via pure bino exchange, the total rate for this reaction will be highly sensitive to the bino mass (assuming a nearly pure bino-like gaugino) all by itself even if $m_{\tilde{B}}$ is far beyond direct production (in such a case, perhaps the LSP would be the lightest Higgsino with $m_{\tilde{B}}$ much heavier). Furthermore, using beam polarization, one might dial up individual reactions $\tilde{e}_{L}\tilde{e}_{L}$, $\tilde{e}_{R}\tilde{e}_{R}$ or $\tilde{e}_{L}\tilde{e}_{R}$. It has also been emphasized that $\tilde{e}_{L,R}\tilde{e}_{L,R}$ production would be an excellent environment for testing possible rare, perhaps flavor violating, slepton decay modes to the low background environment [1168].

The possibility of $e\gamma$ colisions is important in several cases relevant for SUSY searches [1174–1176]. The first scenario is offered by the reaction $e\gamma \to \tilde{e}_{L,R}\tilde{\chi}_{i}^{0}$, or single production of selectrons. In this case, even if $\tilde{e}_{L,R}\tilde{e}_{L,R}$ is beyond the maximal $\sqrt{s}$ of an $e^{+}e^{-}$ collider, then if $m_{\tilde{\chi}_{1}^{0}}$ is light, single production of sleptons may take place for $\sqrt{s} > m_{\tilde{e}} + m_{\tilde{\chi}_{1}^{0}}$. The utility of an $e\gamma$ collider has also been considered for GMSB SUSY models where one may produce
$\tilde{e}_{L,R}\tilde{G}$ where $m_{\tilde{G}}$ may be very light [1177], and in models with R-parity violation [1178].

A linear collider running in $\gamma\gamma$ mode (two backscattered laser beams) has been considered in [1179, 1180] for chargino pair production and in [1181] for sfermion production. For $\gamma\gamma$ collisions, the couplings are pure QED so that the production cross sections depend only on the mass of the charged sparticles which are being produced. For both these cases, an advantage can be gained by scattering polarized laser light on polarized beam to gain polarized photon collisions. A variety of helicity studies can then be made on the various sparticle pair production processes.

## 5.11 Summary and Conclusions

It is timely to re-assess the physics opportunities related to SUSY models for an $e^{+}e^{-}$ linear collider before the start of LHC operation at 13-14 TeV. The run at 7 and 8 TeV has been marked by the discovery of a Higgs-like boson with a mass $M_h \sim 125$ GeV and has provided us with important bounds on the mass of new particles from dedicated SUSY searches. These LHC results are complemented by important data on dark matter, from the precision determinations of its relic density from the CMB spectrum to much improved bounds on its scattering cross section from underground search experiments.

The combination of the relatively light mass of the newlydiscovered Higgs-like particle, easily interpretable within SUSY, and the compelling evidence for dark matter, which can be explained as due to relic neutralinos or gravitinos, have reinforced the interest for supersymmetric models. The combined 7+8 TeV LHC data have already set significant bounds on the masses of strongly-interacting SUSY particles in the jets+$MET$ channel and have started addressing the detection of weakly-interacting particles in $\ell$s+$MET$ and $h$+$MET$ channels and more model-independent searches for neutralino LSPs and nearly-degenerate squark-neutralino scenarios with monojets.

All these searches will have a powerful impact on supersymmetric models with the Run-2 data taking at 13-14 TeV and higher luminosity. However, despite the broad range and the ingenuity of the LHC searches, scenarios with nearly degenerate sparticle-neutralino LSP masses, compressed spectra, multiple decay modes with comparable rates and some of the 'natural' SUSY spectra may prove difficult for the LHC to probe in full. In fact, if we take guidance from the concept of naturalness and the fine tuning of supersymmetric models, we are brought to consider natural SUSY models which contain a spectrum of light higgsino particles. In these models, gluinos and scalar quarks may be as heavy as several-TeV, with TeV-values stops required to be highly mixed in order to lift $M_h$ up into the 125 GeV range. Such

'natural' SUSY spectra would be characterised by electroweak fine-tuning at the level of $\sim 10\%$ and their concomitant light higgsinos could be readily detected and studied at an $e^{+}e^{-}$ linear collider of sufficient energy. When the higgsino mass $\mu$ sets the scale for fine-tuning, then we expect a center-of-mass energy $E_{CM}$ to probe electroweak fine tuning of $E_{CM} > 2\mu \sim \sqrt{2\Delta_{EW}}M_Z$.

In these scenarios, the combination of clean environment, well-known beam energy, adjustable centre-of-mass energy and availability of polarized beams at the $e^{+}e^{-}$ linear collider will provide us with the tools required for precision measurements of masses, spins and other quantum numbers of these new states. Precision mass and spin measurements can be performed either by kinematic measurements in the continuum or via threshold scans. An $e^{+}e^{-}$ collider should be able to extract precision values of scattering cross sections, branching fractions, angular distributions of final state particles and decay widths.

These precision measurements will lead to the extraction of the fundamental SUSY Lagrangian parameters and test the unification at very high energy scales. All together these measurements will provide us with a unique window onto the energy scales associated with grand unification.

Production of SUSY particles at an $e^{+}e^{-}$ linear collider may allow for tests of the Majorana nature of gauginos, flavor-violating decays, CP-violating processes, R-parity violating reactions (which can also elude LHC searches), R-symmetry effects and the presence of additional matter states such as the added singlets in extended models. In the event that just a few SUSY particles are produced at some energy scale, then the linear collider can still determine the fundamental SUSY parameters in a model-independent way and can still test higher mass scales through virtual particle exchange, such as sneutrino exchange effects in chargino pair production, and additional SUSY parameters via loop effects, for instance, to Higgs branching fractions.

The knowledge obtained from combining the data of the LHC, an $e^{+}e^{-}$ linear collider and dark matter experiments will be crucial for understanding the nature of dark matter and, possibly, test models of baryogenesis.

From all these facets, it is clear that a linear $e^{+}e^{-}$ collider operating in the $\sim 0.25-1$ TeV range can play a major role in the study of supersymmetry– ranging from discovery to precision measurements– and will provide a new and more refined view as to the next level in the laws of physics as we know them.
# 6 Connection to astroparticle physics and cosmology<sup>63</sup>

## 6.1 Introduction

While an enormous amount of energy is spent on the search for physics beyond the standard model, perhaps the most compelling reason for expecting new physics is dark matter (DM). The evidence for DM is overwhelming. On galactic scales, one observes relatively flat rotation curves [1182] which can not be accounted for by the observed luminous component of the galaxy. The simplest interpretation of these observations are that nearly all spiral galaxies are embedded in a large galactic halo of DM which lead to rather constant rotational velocities at large distances from the center of the galaxy. X-ray emission from hot gas surrounding large elliptical galaxies and clusters of galaxies also require a large potential well (to gravitationally bind the hot gas) which can not be accounted for by the galaxy or gas itself [1183]. Gravitational lensing also implies large gravitational potentials from unseen matter on the scale of clusters of galaxies [1184]. In addition, there are observations of both x-ray emitting hot gas and gravitational lensing in the same systems [1185] which all point to the presence of dark matter.

On larger, scales, baryon acoustic oscillations [1186] indicate a matter component $\Omega_m = \rho_m/\rho_c \simeq 0.25$, where $\rho_c = 1.88 \times 10^{-29}h^2 \text{ g cm}^{-3}$ is the critical energy density for spatial flatness. However, the baryon density of the universe from big bang nucleosynthesis (BBN) [1187] is restricted to $\Omega_B h^2 \lesssim 0.03$ where $h = 0.71$ is the Hubble parameter in units of 100 km/s/Mpc. Furthermore, both the estimate from baryon acoustic oscillations and nucleosynthesis are in complete agreement with the determination of both the total matter density and baryon density from the cosmic microwave background anisotropy spectrum [41] which yields a dark matter density of

$$ \Omega h^2 = 0.1196 \pm 0.0031. \qquad (159) $$

As we will see, there are no candidates for the dark matter of the Universe found in the Standard Model (SM). Thus the body of evidence for dark matter clearly points to physics beyond the SM. Below, we will briefly describe some of the well-studied candidates for dark matter with an emphasis on their relevance for a future LC.

## 6.2 Candidates

With the discovery of the Higgs boson [1188], the SM field content is complete. As DM must be stable or long lived, a priori there are only two possible candidates for DM in the SM. While baryonic dark matter may account for some

of the dark matter in galactic halos, it can not make up the bulk of the dark matter in the Universe. As noted above, BBN limits the baryon density to less than 25% of the total amount of non-relativistic matter in the universe, which is consistent with the determination of the baryon density from microwave background anisotropies. However, a dominant component of baryonic dark matter even on the galactic scale is problematic [1189]. Put simply, baryons tend to clump and form stellar-like objects. While massive objects such as white dwarfs or neutron stars or black holes may be dark, they are typically associated with heavy element production and a significant number of these objects would produce excessive metallicity. Smaller jupiter-like objects would require a very special mass distribution to avoid constraints from luminosity density in the red and infrared. More concrete constraints are obtained from microlensing observations [1190] where the contribution of such objects (collectively known as MACHOs) is limited to less than 25% of the halo for masses $2 \times 10^{-7} M_{\odot} < M < 1 M_{\odot}$.

Another potential possibility for a DM candidate in the SM is a neutrino. Indeed, neutrino oscillation experiments indicate that at least one neutrino has a mass in excess of 0.05 eV. This would correspond to a cosmological contribution, $\Omega_v h^2 > 5 \times 10^{-4}$. However there are upper limits to the sum of neutrino masses from large scale structure considerations. In particular, using CMB data (notably PLANCK, WMAP 9-years, ACT and SPT) and including observations from BAO and HST, one finds that the sum of neutrino masses is constrained to be $\sum m_v < 0.22 \text{ eV}$ corresponding to $\Omega_v h^2 < 2.4 \times 10^{-3}$ [1191].

the 7-year WMAP data and including observations from SDSS and HST, one finds that the sum of neutrino masses is constrained to be $\sum m_v < 0.39 \text{ eV}$ corresponding to $\Omega_v h^2 < 4 \times 10^{-3}$ [1192].

At this time, if there is any firm indication of physics beyond the SM, it comes from our understanding of dark matter in the Universe. While not all dark matter candidates can be probed by a future linear collider, we are will restrict our attention to those that can. Thus we will not discuss possibilities such as sterile neutrinos or axions below and we concentrate on those candidates with potential signatures at the LC.

### 6.2.1 Supersymmetric Candidates

The supersymmetric extension of the SM is one of the most studied example of physics beyond the SM and is currently being tested at the LHC. Its motivations (which we will not review here) include the stabilization of the weak scale hierarchy, gauge coupling unification, radiative electroweak symmetry breaking, and the prediction of a light Higgs boson ($m_h \lesssim 130 \text{ GeV}$) which has been borne out by experiment [1188]. In models with R-parity conservation, another

<footnote n="63">63 Editors: G. Belanger, K. Olive
Contributors: Y. Mambrini, P. Serpico</footnote>
prediction of supersymmetric models, is the existence of one stable particle, which if neutral, may be candidate for the DM. This is the lightest supersymmetric particle of LSP. Below, we review some of the most studied realizations of the low energy supersymmetry.

For the most part, we will restrict our attention here to the minimal supersymmetric standard model (MSSM) (though see below for a discussion of the next to minimal model or NMSSM). The minimal model is defined by the superpotential

$$W = (y_e H_1 L e^c + y_d H_1 Q d^c + y_u H_2 Q u^c) + \mu H_1 H_2, \quad (160)$$

Beyond the parameters associated with the SM, the superpotential introduces a mixing term between the two Higgs doublets in the MSSM. The bulk of the new parameters are associated with supersymmetry breaking and are associated with soft scalar masses, gaugino masses, and so-called bi- and tri-linear terms, B and A. There are well over 100 new parameters in the minimal theory and we are thus forced to make some (well motivated) simplifications as we discuss below.

### The CMSSM
As is clear, supersymmetry must be broken, and one way of transmitting the breaking of supersymmetry to the low energy sector of the theory is through gravity. Indeed the extension of global supersymmetry to supergravity is in some sense necessary to ensure the (near) vanishing of the cosmological constant in models with weak scale supersymmetry breaking. Gravity mediated supersymmetry breaking imposes a number of boundary conditions on the supersymmetry breaking masses at some high energy renormalization scale, which is usually taken to be the same scale at which gauge coupling unification occurs, $M_{GUT}$. In gravity mediated models, one often finds that all scalar masses are equal at $M_{GUT}$ defining a universal scalar mass $m_0$. Similarly, all gaugino mass and trilinear terms are also universal at $M_{GUT}$, with values $m_{1/2}$ and $A_0$ respectively.

In these gravity mediated supersymmetry breaking models, supersymmetry breaking masses and gauge and Yukawa couplings are run down from the universality scale and often trigger electraoweak symmetry breaking as one or both of the soft Higgs masses, $m_{1/2}^2$ run negative. In true minimal supergravity models or mSUGRA, the scalar mass is equal to the gravitino mass, $m_0 = m_{3/2}$, and the B-term is given by $B_0 = A_0 - m_0$. One consequence of the latter relation is the determination of the two Higgs vacuum expectation values as the soft masses are run down to the weak scale. Since one combination of the two vevs determines the Z gauge boson mass, it is common to choose the two vevs as input parameters (the other combination is the ratio of vevs and defined as $\tan\beta = v_2/v_1$) and discard the relation between $B_0$ and $A_0$. Instead both B and $\mu$ can be calculated at the weak scale from $M_Z$ and $\tan\beta$. If the relation between the

gravitino mass and $m_0$, is also dropped, we have the constrained version of the MSSM known as the CMSSM.

The CMSSM is therefore a four parameter theory (the sign of $\mu$ must also be specified). For given values of $\tan\beta$, $A_0$, and $\operatorname{sgn}(\mu)$, the regions of the CMSSM parameter space that yield an acceptable relic density and satisfy other phenomenological constraints may be displayed in the $(m_{1/2}, m_0)$ plane. In Fig. 151 [1193], the dark (blue) shaded region corresponds to that portion of the CMSSM plane with $\tan\beta = 40$, $A_0 = 2.5m_0$, and $\mu > 0$ such that the computed relic density yields the PLANCK value given in Eq. (159). For this choice of $\tan\beta$ and $A_0$, the relic density strip is v-shaped. Inside the ‘v’, the annihilation cross sections are too small to maintain an acceptable relic density and $\Omega_\chi h^2$ is too large. The upper side of the ‘v’, at large $m_0$, is produced by co-annihilation processes between the LSP and the next lightest sparticle, in this case the $\tilde{t}$ [1194]. These enhance the annihilation cross section and reduce the relic density. This occurs when the LSP and NLSP are nearly degenerate in mass. The lower side of the ‘v’, at lower $m_0$, is produced by coannihilations between the LSP and the $\tilde{\tau}$ [1195]. The dark (brown) shaded regions outside of the ‘v’ have either $m_{\tilde{t}} < m_\chi$ or $m_{\tilde{\tau}} < m_\chi$ and are excluded. Also shown in the figure is the constraint from $b \to s\gamma$ [1196] (shaded green) which excludes the stop-coannihilation strip in the portion of the plane shown. Contours of constant Higgs mass are shown by the black curves. Higgs masses are computed using FeynHiggs [1197] and carry a roughly 1.5 GeV uncertainty. The thick purple line corresponds to the ATLAS limit on supersymmetry searches [1198]. The area to left of the line is excluded. Finally, the solid green contour corresponds to the 95% CL upper limit to ratio of the branching fraction of $B_s \to \mu^+\mu^-$ relative to the SM [1199].

Note that the choice $A \neq 0$ is made to ensure a sufficiently large Higgs mass. For $A_0 = 0$, the maximum Higgs mass along the stau-coannihilation strip is only slightly greater than 120 GeV, far short of the value reported in the recent LHC results [1188]. Therefore, only the upper end of the strip is compatible with a Higgs mass around 125–126 GeV and a branching fraction for $B_s \to \mu^+\mu^-$ sufficiently close to the SM value.

### NUHM
One possible generalization of the CMSSM is the so-called NUHM in which the Higgs soft masses are not constrained to be equal to $m_0$. Indeed, as the Higgses are typically found in separate multiplets in a grand unified theory, one or both of the Higgs soft masses may be independent. In the NUHM1, we may set $m_1 = m_2 \neq m_0$, where $m_{1/2}$ are the soft masses associated with $H_{1/2}$. Instead of $m_{1/2}$, one may choose either $\mu$ or the Higgs pseudo-scalar mass, $m_A$ (which is a surrogate for B) as a free parameter in addition to $m_0$. In the NUHM2, both $m_1$ and $m_2$ are free and one can equivalently choose both $\mu$ and $m_A$ as free parameters.
important feature of the model is that the singlet fields can be very light and escape the LEP bounds. This is because these fields mostly decouple from the SM fields. Furthermore large mixing with the singlet can modify the properties of the SM-like Higgs, allowing quite naturally for $m_h = 126$ GeV as well as possibly an enhanced rate for its decay into two photons. With regard to DM, the NMSSM shares many of the characteristics of the MSSM. The main differences occur when the LSP has some singlino component and/or when the Higgs sector contains new light states that play a role in dark matter interactions. For example new Higgs states can greatly enhance DM annihilation when their mass is twice that of the LSP or can provide new annihilation channels when they can be produced in the final state. As a consequence, the NMSSM allows for the possibility of light neutralinos (much below $M_Z/2$), that annihilate efficiently through the exchange of light Higgs singlets or into light Higgs singlets [1201]. The model also accommodates the possibility of a gamma-ray line at 130 GeV, without violating any other constraints from cosmic rays. This requires fine-tuning of the parameters such that 1) the mass of a pseudoscalar is precisely twice the neutralino mass and 2) the annihilation of the pseudoscalar is dominantly into two photons rather than into quarks [438].

### 6.2.2 Universal Extra Dimensions

Extra dimension models also propose a WIMP DM candidate. The UED scenario [1202] where all SM particles are allowed to propagate freely in the bulk is of particular interest. In this model momentum conservation in the extra dimensions entails conservation of a KK number. Orbifolding is required to obtain chiral zero modes from bulk fermions, and breaks extra dimensional momentum conservation. However, there remains a discrete subgroup, KK parity, thus the lightest KK-odd particle is stable. In the minimal universal extra dimension model (MUED) the DM candidate is in general a vector particle, $B_1$, the Kaluza-Klein (KK) level 1 partner of the U(1) gauge boson. In the MUED model all KK states of a given level have nearly the same mass at tree-level, $n/R$, where $R$ is the size of the compact dimension. The mass degeneracy is lifted only by SM masses and by radiative corrections. These mass splittings are however small for all weakly interacting particles. This means that co-annihilation channels naturally play an important role in the computation of the relic abundance of dark matter. Furthermore since the level 2 particles are close to twice the mass of those of level 1, annihilation or co-annihilation processes can easily be enhanced by resonance effects. When including level 2 particles in the computation, the preferred scale for DM was found to be around 1.35 TeV, see line c1 in Fig. 153 [1203]. Going beyond the MUED framework one can treat mass splittings as free parameters, shifting sig-

<figure><img src="image_155.png" /><figcaption>Fig. 153 $\Omega h^2$ as function of $R^{-1}$ for $m_h = 120$ GeV and $\Lambda R = 20$ including different processes as specified on the figure. Here 1-loop stands for one-loop couplings between level 2 and SM particles [1203].</figcaption></figure>

nificantly the preferred DM mass, for example in the limit where the coannihilation processes are negligible the DM mass is around 800 GeV, see line a1 in Fig. 153. The measurement of the Higgs mass and of its couplings at the LHC can be used to put a lower limit on the scale $R$. Indeed light KK particles, in particular the KK top, lead to an increase in the $hgg$ coupling and to a decrease in the $h\gamma\gamma$ coupling, and to a lower bound on $R > 500$ GeV [1204]. One characteristics of MUED DM is that annihilation in the galaxy has a large fraction into fermions leading to strong signal into positrons, however the large mass scale makes the signature unlikely to be observable [1205].

### 6.2.3 Higgs Portal Models

The Higgs portal refers to a class of models where the Higgs connects the dark matter (hidden) sector to the SM. Several possibilities have been considered with either a scalar, a vector or a fermion as DM. The simplest extension of the SM is the addition of a real singlet scalar field, $S$, which can be made stable by imposing a $Z_2$ symmetry. If the true vacuum of the theory satisfies $\langle S \rangle = 0$, thereby precluding mixing of $S$ and the SM Higgs boson and the existence of cosmologically problematic domain walls. The terms to be added to the SM Lagrangian are

$$ \Delta \mathcal{L}_S = -\frac{1}{2}m_S^2 S^2 - \frac{1}{4}\lambda_S S^4 - \frac{1}{4}\lambda_{hSS} H^\dagger H S^2. \quad (161) $$

A second possibility is to couple the Higgs doublet to a massive vector field $X_\mu$ from the hidden sector. $X_\mu$ can be associated with a hidden U(1) and becomes massive due to the Higgs or Stückelberg mechanism in the hidden sector. A third possibility is the one where dark matter can consist of Majorana fermions $\chi$ which interact with the SM fields only through the Higgs-portal. In both cases the stability of the DM particle is ensured by a $Z_2$ parity, whose origin is
model-dependent. For example, in the vector case it stems from a natural parity symmetry of abelian gauge sectors with minimal field content [1206]. The relevant terms in the Lagrangians are

$$ 
\begin{aligned}
\Delta \mathcal{L}_V &= \frac{1}{2} m_V^2 V_\mu V^\mu + \frac{1}{4} \lambda_V (V_\mu V^\mu)^2 + \frac{1}{4} \lambda_{hVV} H^\dagger H V_\mu V^\mu, \\
\Delta \mathcal{L}_f &= -\frac{1}{2} m_f \bar{\chi} \chi - \frac{1}{4} \frac{\lambda_{hff}}{\Lambda} H^\dagger H \bar{\chi} \chi.
\end{aligned}
\quad (162) $$

Related ideas and analyses can be found in [1206–1211] and more recent studies of Higgs-portal scenarios have appeared in [1212–1214].

In these models, the Higgs is responsible for both DM annihilation and elastic scattering of DM with nuclei. Thus, cosmological measurements made by the WMAP and PLANCK satellites [41] basically determine the couplings of the Higgs to DM and thus the spin-independent DM-nucleon cross section for a given DM mass. The same coupling will also determine the Higgs partial decay widths into invisible DM particles if $m_{DM} \le \frac{1}{2}m_h$. The discovery of a Higgs boson with a mass $m_h = 125$ GeV with a small invisible decay branching ratio is incompatible with dark matter with $m_{DM} \le 55$ GeV. This applies in particular to the case of scalar DM with a mass of 5–10 GeV considered, for instance, in Ref. [1210]. Fig. 154, displays the predictions for the spin-independent DM-nucleon cross section $\sigma_{SI}$ after imposing the WMAP and BR$^{inv} < 10\%$ constraints (allowing the invisible width to be 20% does not change the result significantly). The upper band corresponds to the fermion Higgs-portal DM and is excluded by XENON100 while scalar and vector DM are both allowed for a wide range of masses. The typical value for the scalar $\sigma_{SI}$ is a few times $10^{-9}$ pb, whereas $\sigma_{SI}$ for vectors is larger by a factor of 3 which accounts for the number of degrees of freedom. We note that a large fraction of the parameter space will be probed by XENON1T except for a small region where $m_{DM} \approx m_h/2$ and the Higgs–DM coupling is extremely small.

### 6.2.4 Extended scalar sector

The Higgs discovery has revived the interest in models with an extended scalar sector. In such models an unbroken discrete symmetry which could be leftover from a broken gauge group at a higher scale guarantees the stability of the lightest scalar, the DM candidate. One of the nice feature of these models is that the quartic couplings between the SM-like doublet and other scalars helps stabilize the scalar potential by giving a contribution that counteracts the effect of the top Yukawa that drives the SM potential to the metastability region [68, 1215]. The prototype of scalar dark matter models is the inert doublet model [1216] where the second doublet has no vev, and no coupling to quarks and leptons. Models

<figure><img src="image_156.png" /><figcaption>Fig. 154 Spin independent DM-nucleon cross section versus DM mass. The upper band (3) corresponds to fermion DM, the middle one (2) to vector DM and the lower one (1) to scalar DM. The solid, dashed and dotted lines represent XENON100 (2012 data [1010]), XENON100 upgrade and XENON1T sensitivities, respectively.</figcaption></figure>

with only additional singlets [1207], with a doublet and singlet [1217, 1218] or with higher multiplets [1219] have also been proposed and different discrete symmetries to stabilize the dark matter were considered [1217, 1218].

In the inert doublet model, the DM can be either a scalar or pseudoscalar. After imposing constraints on the model from perturbativity, stability, direct searches for charged Higgs and electroweak precision tests, several studies have found that a value of the relic density in accordance with PLANCK can be reproduced in the low mass $m_{DM} < 60$ GeV, intermediate $60 < m_{DM} < 110$ GeV and high mass range ($m_{DM} > 500$ GeV) [1216, 1220]. The low and intermediate mass ranges are severely constrained by Higgs measurements and direct detection. In the low mass region, DM annihilation proceeds through Higgs exchange and as in the portal models is constrained by the upper limit on the Higgs invisible width. In the intermediate region annihilation into W pairs (including virtual W's) start to dominate. However, the Xenon and LUX upper limits forces the DM mass to be near $m_h/2$ and $m_W$. For DM masses above $m_W$ the annihilation into W pairs becomes very efficient thus leading to too low a value for the relic density unless the DM mass is larger than 500 GeV. These allowed mass ranges can be extended in models with more particles in the inert sector and/or in models which also involve semi-annihilation [1218]. The collider signatures in the Higgs sector involve invisible decays (already severely constrained) and a modification of the two-photon decay width due to the charged Higgs contribution [1221]. At the LC, the inert Higgses can be directly produced and their decays into real or virtual gauge bosons exploited to determine the masses of all inert scalars [1222].
## 6.3 Dark Matter at the LHC

Direct searches for supersymmetry at the LHC have had a significant impact on the allowable regions of the supersymmetric parameter space particularly in the context of the CMSSM. An example of this is shown by the purple curve in Fig. 151. For relatively low $m_0$, the most recent results from ATLAS place a lower bound on $m_{1/2}$ of roughly 840 GeV. Perhaps of greater significance is the discovery of the Higgs boson at 125-126 GeV. While consistent with general predictions in supersymmetric models that $m_h \lesssim 128-130$ GeV, a 125 GeV Higgs lies at the edge of what can be obtained and pushes the model to require large contributions from stop mixing (hence a large value of $A_0$ in the CMSSM) and relatively large SUSY masses. Of course large SUSY masses are consistent with the lack of discovery of supersymmetric particles at the LHC, and are consistent with little or no departures from the SM in rare B decays. Of course this can not be viewed as a ringing endorsement for supersymmetry. Indeed the past prospect of resolving the discrepancy between theory and experiment for the anomalous magnetic moment of the muon, has now essentially evaporated.

To account for the recent LHC results along with other low energy observables, it is better to perform a global likelihood analysis which can identify regions of the parameter space which best fit the data. It is well established that Markov-Chain Monte-Carlo (MCMC) algorithms offer an efficient technique for sampling a large parameter space such as the CMSSM or its variants. MCMC has been utilized in the Mastercode [1223] framework to perform a frequentist analysis of the CMSSM and other variants of the model. The MCMC technique is used to sample the SUSY parameter space, and thereby construct the $\chi^2$ probability function, $P(\chi^2, N_{dof})$. This accounts for the number of degrees of freedom, $N_{dof}$, and thus provides a quantitative measure for the quality-of-fit such that $P(\chi^2, N_{dof})$ can be used to estimate the absolute probability with which the CMSSM describes the experimental data.

The results of the mastercode analysis include the parameters of the best-fit points as well as the 68 and 95% C.L. regions found with default implementations of the phenomenological, experimental and cosmological constraints. These include precision electroweak data, the anomalous magnetic moment of the muon, B-physics observables, the Higgs boson mass, $m_h$, and the cold DM density. In addition it includes the constraint imposed by the experimental upper limit on the spin-independent DM scattering cross section from LUX [1004]. The results described here are taken from [1224-1227].

In Fig. 155, we show the resulting 68% (shown in red) and 95% (shown in blue) CL limits from the mastercode analysis [1227] in the $m_0, m_{1/2}$ plane corresponding to $\Delta\chi^2 =$

2.3 and 5.99 relative to the best fit point (note the axes are reversed compared to Fig. 151). Results which include the ATLAS constraints at 20/fb are shown by solid curves. The best fit point is at $(m_0, m_{1/2}) = (5650, 2100)$ GeV and is shown by the filled star. At the best fit point, we also have $A_0 \simeq -780$ GeV, and $\tan\beta = 51$. We see in Fig. 155 that the 95% CL region in the CMSSM extends to $m_0 \gtrsim 6000$ GeV and $m_{1/2} \gtrsim 3000$ GeV. Note that the CMSSM fit features two disconnected 68% CL 'islands', the one at lower $m_0$ and $m_{1/2}$ corresponding to the stau coannihilation region, and that at larger $m_0$ and $m_{1/2}$ corresponding to the s-channel rapidannihilation funnel region (the best fit point in the lower island has $\tan\beta = 21$. The low mass island is only disfavoured at the level of $\Delta\chi^2 \sim 0.7$, reflecting the relative flatness of the global $\chi^2$ function.

<figure><img src="image_157.png" /><figcaption>Fig. 155 The ($m_0, m_{1/2}$) planes in the CMSSM including the ATLAS 20/fb jets + $E_T$, $BR(B_{s,d} \to \mu^+\mu^-)$, $m_h$, $\Omega\chi^2$, LUX, and other constraints. The most recent results are indicated by solid lines and filled stars, and previous fit based on ~5/fb of LHC data is indicated by dashed lines and open stars. The blue lines denote 68% CL contours, and the red lines denote 95% CL contours.</figcaption></figure>

The impact of the recent LHC results can be seen by comparing the solid curves to the dashed in Fig. 155. The pre-LHC expectations [1224, 1225] were driven by a large extent by $g_\mu - 2$. The initial best fit result was found at quite low susy masses with ($m_0, m_{1/2}$) ~ (90, 360) GeV and had a p-value of 37%. The entire pre-LHC 68% CL region is now excluded at 95% CL, though much of the initial 95% CL region is still valid. The dashed curves in Fig. 155 represent the status of the CMSSM after 5/fb data were collected though assuming a 125 GeV Higgs boson mass. The best fit point in this case is at low $m_0, m_{1/2}$ shown by an open star. The p-value in this case is only 8.8%. Thus already at 5/fb, the LHC results had greatly diminished the probability that the CMSSM improves the fit relative to the SM. The current results have a p-value of 5.1% which is close to the SM value. Of course the SM p-value does not include the dark
matter constraint as there is no candidate for dark matter within the SM.

## 6.4 Other searches

### 6.4.1 Direct Detection

Direct searches of DM particles through their scattering off nuclei in a large detector can establish that the DM matter is indeed made of a new stable particle. The elastic scattering of weakly interacting massive particles (WIMPs) off nuclei taking place at low momentum transfer can be conveniently described in terms of an effective Lagrangian interaction of DM with quarks and gluons giving rise to either spin independent or spin dependent interactions.

The spin independent (SI) cross section for WIMPs on nuclei adds coherently and is proportional to the square of the number of nucleons, it therefore usually dominates for heavy nuclei. The spin independent cross section receives a contribution from Higgs exchange, Z exchange (except for Majorana fermions) and from interactions with new colored fermions/scalsars (for example new quarks in extra dimension models or squarks in supersymmetry). The latter contribution is however constrained by the non-observation of new colored particles at the LHC.

The spin dependent (SD) cross section depends solely on the nucleon that contributes to the nucleus spin, and is dominant only for light nuclei. The SD cross section receives contributions from Z exchange and/or from interactions with new colored fermions/scalsars. In order to easily compare results obtained using different nuclei, limits are normally expressed in terms of the SI or SD interaction with protons and neutrons.

At the microscopic level a positive signal in several DM direct searches could altogether lead to information on up to four independent quantities that depend on the details of the DM model, the SD/SI interactions on protons and neutrons. Note however that when scalar interactions are dominated by Higgs exchange the cross-section on protons and neutrons are almost equal. Furthermore, if the DM has a mass comparable or below that of the nucleus, the shape of the nucleus recoil energy distribution can also be used to extract some rough information of the DM mass.

Several experiments have been taking data, some claiming potential signals compatible with the detection of a weakly interacting massive particle. This includes DAMA [1006] which observes an annual modulation, CoGeNT [1231], CRESST [1007] and CDMS-Si [1009] which also have found signals that would be compatible with dark matter in the range 5-30 GeV. These observations are however in conflict with the negative search results by other collaborations notably CDMS, Edelweiss [1232],

<figure><img src="image_158.png" /><figcaption>**Fig. 156** Limits on spin-independent direct detection cross section $\sigma_{SI}$ on protons vs. dark matter mass $m_{DM}$. In grey the preferred region in the CMSSM, from a combination of [1228–1230].</figcaption></figure>

<figure><img src="image_159.png" /><figcaption>**Fig. 157** Spin-independent direct detection cross section $\sigma_{SI}$ on protons vs. dark matter mass $m_{LSP}$, from [1237]. The black (blue) line are the 90% CL limits from the XENON100(2011) [1238] and (2012) results [1010]. The dashed brown line is the projected sensitivity of the XENON1T experiment [1239]. The color code shows the with $P > 0.2$ (red), $0.1 < P < 0.2$ (orange) $0.01 < P < 0.1$ (green) and $0.001 < P < 0.01$ (blue). Note however that the relic density constraint is not imposed here.</figcaption></figure>

XENON [1010] or LUX [1004]. The large ton scale detectors that are planned, such as XENON should improve by more than one order of magnitude the current sensitivity thus resolving the apparent conflict in SI results at low masses and probing a large number of DM models, see fig. 156 for a comparison of current limits with expectations in the CMSSM. In particular, the case where the neutralino is a mixed gaugino/higgsino state is challenged by current limits as illustrated in fig. 157 where $P = \min(f_h, 1 - f_h)$ and $f_h$ is the higgsino fraction. Finally COUPP [1233], KIMS [1234], Picasso [1235] (Xenon10 [1236]) have set limits on spin dependent interactions on protons (neutrons).

### 6.4.2 Indirect Detection

In general, the goal of the ongoing generation of indirect detection experiments sensitive to dark matter is:

I. to probe the vanilla “WIMP paradigm”, at least for particles of masses at the electroweak scale and characterized by s-wave annihilation cross sections $\langle \sigma v \rangle$.

II. to clarify some of the “anomalies” presently claimed.
III. in the case of independent detection (at colliders or direct detection), provide one or several cross-checks taking advantage of the multi-messenger characteristics of this detection strategy.

Concerning the first task, it is worth pointing out that for some channels Fermi-LAT has already reached the sensitivity to test this paradigm up to a few tens of GeV (dwarf spheroidals [1240, 1241], diffuse gamma-ray halo signal [1242]) or even more for the Galactic Center [1243].

In general, we expect that probing the ~100 GeV mass scale will be within reach with a decade worth of data, see for example the forecasts in [1244]. Preliminary results from Fermi-LAT also comfort these expectations, see [1245]. Especially for candidates annihilating into leptons, such a goal seems also within reach of Planck, which probes dark matter energy deposition at early times via its impact on the reionization (see e.g. [1246]).

Needless to say, if new states are below the TeV scale, these WIMP candidates are also in the right ball-park to be probed directly or indirectly by a future ILC, hence the complementarity of the two approaches.

The current generation of ground-based imaging atmospheric cherenkov telescopes (IACTs) is less sensitive to theoretically preferred values of $\langle \sigma v \rangle$. Nonetheless, they are already more sensitive than Fermi-LAT to TeV scale dark matter, see e.g. [1247], and with the future CTA they may probe not-yet-excluded regions of parameter space for viable particle physics models, see e.g. [1248]. Typically, the Galactic Center is among the most promising targets, provided that the DM distribution is comparable to expectations based on pure cold dark matter simulation or even enhanced as a consequence of baryons [1249]. Dwarf spheroidal have also been studied by IACTs see e.g. [1250, 1251] and show some potential for interesting complementary constraints, since they are affected by very different systematics.

Performances similar to Fermi-LAT (but more dependent on astrophysical modeling of cosmic ray transport) are expected by high-precision measurements of cosmic ray antimatter, most notably antiprotons and, possibly, anti-deuterons [1252]. Positrons are significantly sensitive to astrophysical backgrounds (see e.g. [1253]) and both their primary and secondary fluxes show a larger dependence from source distribution (in space and time) as well as from the medium properties (e.g. their E-losses crucially depend on B-field and interstellar radiation fields). While they remain challenging for a robust detection of dark matter, they may be useful for cross-checks of tentative signals. AMS-02 and, concerning anti-deuterons, GAPS, are expected to achieve the needed precision and sensitivity for such competitive results.

A caveat is mandatory: ultimately, if sufficient statistics is accumulated, the main limitations will come from the degree of understanding of the astrophysical foregrounds, so

that most of these projections must be taken with a grain of salt.

An example of the second type of goal has been provided in the recent past by the multi-messenger constraints on the dark matter interpretation of the PAMELA positron fraction “rise” (where the relevance of the point just made clearly manifested) or, at present, by the tentative hint for a ~130 GeV “gamma-ray line” [1254]. For this kind of task, statistics helps a lot but it is clearly not enough. Cross-checks and tests with different techniques and possibly improved resolutions are needed. Fortunately, current (HESS) or planned (CTA) IACTs may provide such a tool. This is also an arena where the proposed satellite experiment Gamma-400 [1255] might contribute, thanks to its superior resolution (see e.g. [1256]).

The third possibility has been heavily discussed in recent years in the context of direct detection “anomalies” [1006, 1007, 1009, 1231]. If interpreted in terms of “light” dark matter, a wealth of indirect detection cross-checks can be thought of, see e.g. [1257, 1258]. We conclude by pointing out that especially in this context (cross-checking direct detection potential signals), neutrino signals from the center of the Sun (and possibly the Earth) are of particular relevance. In fact, they probe a similar combination (albeit not equal!) of DM-baryon cross-section and local density of dark matter as direct detection experiments. Significant advances are expected by the IceCube in its current configuration, including the Deep-core (see e.g. [1259]). Further progress may be also possible if the R&D PINGU low-energy extension will be realized [1260] (the same would apply to comparable programs in the Mediterranean sea such as those pursed within Km3Net, of course). Finally, it is worth pointing out that this is also one of the few ways to potentially detect indirectly p-wave annihilating WIMPs, since the equilibrium flux is only dependent on the DM scattering cross section.

## 6.5 Dark matter at the ILC

The goal of colliders with regard to the DM issue is first to search for a new particle, stable at the collider scale, and as a second step to determine the microscopic properties of this particle. These can then be used to reconstruct DM observables such as the relic abundance (within a cosmological model), the DM annihilation cross section in the galaxy and of the DM scattering cross sections on nucleons thus checking the self-consistency of DM interpretation of different signals and the compatibility of specific dark matter models with observations.

The issues that will be most relevant at the ILC will be influenced by the forthcoming results of new physics searches at the LHC and of DM searches in direct and indirect detection. At the LHC the generic DM signature consists of jets (and leptons) plus large missing transverse energy. With this
signature, it is highly non trivial to then resolve the underlying theory as well as the nature of the DM candidate. For this one needs a precise determination of their properties such as masses, spins and couplings, as was shown in many specific models [994, 1261, 1262]. This is where the ILC has an important role to play. Failing discoveries of new particles, the role of the ILC will be to search for the DM candidate as well as for other weakly interacting particles that might have escaped the LHC searches. Indeed the direct production of electroweak particles not only suffer from small rates at the LHC, but often feature a compressed spectra that can make their identification challenging. At the ILC new electroweak particles can be easily produced provided the center of mass energy is sufficient to cross the mass threshold.

It might well be that the only kinematically accessible new particle at the first stage of the ILC is the DM particle itself. In this case DM radiative production can be used. The signal is a single high energy photon, emitted from the incoming beam or from the exchanged particle, and missing energy. Effective operators that describe the interaction of electrons with DM particles can be used to parametrise the effect of new physics. In this model independent approach, it has been shown that for DM annihilation cross section compatible with the relic abundance of dark matter, the cross section for radiative DM production at the ILC can be large enough to observe this process above the irreducible background from radiative neutrino production [43]. The electron and positron beam polarizations can be used to significantly enhance the signal and suppress the background from radiative neutrino production simultaneously [1263]. Furthermore the energy spectrum of the ISR photon can be exploited to extract information on the WIMP mass and cross-section, at the percent level [43]. Similar conclusions were reached for radiative neutralino production in the MSSM [1264], distinguishing between models through a shape discrimination analysis of the photon energy spectrum which is affected by the particle exchange in t-channel [1265].

A measurement of the invisible width of the Higgs also provides a unique opportunity to determine the Higgs coupling to DM particles directly when $m_{DM} < m_h/2$. This is an essential ingredient in determining the spin independent direct detection cross section ($\sigma^{SI}$) in models dominated by Higgs exchange [412]. A refined upper limit on the invisible width will constrain the maximal allowed value for $\sigma^{SI}$ for light DM [399].

Parameter determination in order to reconstruct DM observables and in particular the relic density amounts to determining the DM mass and its couplings, the mass of the particles exchanged in either t-channel or s-channel and the mass splittings between the DM and the new particles that can participate in coannihilation processes. Many studies have examined within the context of specific DM scenarios

whether a high enough precision can be achieved so that a meaningful comparison with observables can be made [994, 1266, 1267]. To illustrate what could be achieved we will consider the most studied model, a supersymmetric model with a neutralino LSP and assume that some of the supersymmetric spectrum is kinematically accessible.

The measurements of the masses of the chargino and of the heavier neutralinos (e.g. through a threshold scan) together with the determination of their mass splitting with the LSP using the endpoints of the energy spectrum of the SM particle produced together with the LSP in the decay of the heavier SUSY particle, allow a reconstruction of the four elements of the neutralino mass matrix. Moreover since the $e^+e^-$ production cross sections of charginos and heavier neutralinos are sensitive to the gaugino/higgsino mixing they can provide crucial information on the nature of the LSP. In a scenario where only electroweakinos are accessible at the LHC and the ILC, it was shown that with the ILC measurements at the percent level or better, the value of $\Omega h^2$ could be inferred with an uncertainty around 10% [994]. Of particular importance in this scenario is the need to get a lower bound on the mass of the heavy pseudoscalar to ensure that its contribution to DM annihilation is negligible [1268]. In other scenarios, where neutralino annihilation is strongly enhanced because the pseudoscalar exchange in the s-channel is nearly on resonance, a determination of the pseudoscalar mass to about 3% and its width to 20% - is required to infer the DM relic density at the 10% level [1266]. For these measurements it might be necessary to run the ILC at energies above 1 TeV. When co-annihilation processes play an important role, the mass splitting of the coannihilating particle with the LSP - for example the stau NLSP, must be measured at the percent level - which requires the measurement of masses at the few permil level [1266]. An issue that comes up is the impact of radiative corrections, which introduce more degrees of freedom from particles appearing only in higher order loops in the reconstruction of the neutralino mass matrix. Nevertheless it was shown in [1103] that parameters of the electroweakino sector could still be determined at better than the percent level and that indirect information on the mass of e.g. the pseudoscalar could be extracted.

In conclusion, despite intensive ongoing efforts to search for dark matter at colliders and in astrophysics, the nature of the DM, even whether it is a new weakly interacting particle is far from being solved. While near future results from the LHC are expected to provide crucial clues - even to discover new particles - it is clear that a high precision machine such as the ILC, designed with a high enough energy to probe most of the BSM spectrum, is needed for a verification of the DM paradigm.
## 7 Summary

Exciting times in high energy physics are just ahead. Discovering a Higgs boson at the LHC in exactly the range predicted by electroweak precision measurements confirms the successful strategy in particle physics of confronting direct discoveries with theoretical predictions of virtual effects in indirect searches. Within the current theoretical and experimental uncertainties the properties of the Higgs boson are in agreement with the predictions of the SM. Higher precision measurements are required to reveal whether nature can be described via the SM only or whether physics beyond the SM is required at some higher scale. The direct measurement of the total width of the Higgs within a few percent accuracy as well as the measurement of all Higgs couplings to fermions and bosons at the percent level are crucial to pin down the correct model of electroweak symmetry breaking. In this context also high precision for the Higgs mass is required. With such an accuracy one gets high sensitivity to virtual effects and even small traces of BSM physics become measurable. In order to really establish the Brout-Englert-Higgs mechanism also the Higgs self-couplings would be required. An accuracy of 10-20% would constitute a first test whether the Higgs potential provides indeed the required structure for the vacuum to generate the BEH mechanism. As we have discussed in this report, the full physics programme of the Linear Collider could perfectly well fulfill all these requirements.

Further footprints of new physics can be detected in the measurement of the electroweak couplings of the top quark with a unique precision at the Linear Collider. Exploiting asymmetries with polarized beams allows to determine the electroweak top quark form factors at the per cent level, that's up to one order of magnitude more precise than expectation from corresponding analyses at the LHC with $\sqrt{s} = 14$ TeV and $300 \text{ fb}^{-1}$. Polarized beams are required to fit all factors simultaneously and to measure the asymmetry.

Highest precision in measuring the top quark mass is mandatory to match the precision of the theoretical predictions with the expected experimental precision of the electroweak precision observables that are strongly sensitive to the effects of virtual particles far beyond the kinematic limit. In order to uniquely relate the measured quantity to a well defined mass scheme the top quark measurement via a threshold scan is required and one can determine the mass of the top quark with an uncertainty of $\Delta m_{top}^{\text{MS}} = 100$ MeV.

The LC has also an overwhelming potential for the discovery of further electroweak interacting particles and, in particular, of a cold dark matter candidate. The potential of the LC to resolve even challenging scenarios, for instance, via applying the ISR method and to determine precisely the interaction character of dark matter candidate via applying polarized beams.

As it is shown in many reports [7–9, 13, 26, 40] as well as discussed here in detail, a Linear Collider with precisely tunable energy in the range of $\sqrt{s} = 91$ GeV up to $\ge 1$ TeV, high luminosity and polarized beams provides the required flexibility and precision to tackle these physics questions left by the LHC and is well prepared for even the ‘Unexpected’. With the currently promising activities towards the realization of the International Linear Collider in Japan one could even discuss the optimization of the physics potential in HEP via a time of concurrent running [433] of the LHC and the LC. The described physics goals as well as not-yet-thought physics questions could be addressed by this option.

The physics world has changed on July, 4th, 2012 with the discovery of the Higgs boson at the LHC. Cucial milestones in particle physics are expected to be achieved in the near future with data in pairs from the upgraded LHC and from a future Linear Collider. In combination with astroparticle physics, a new era for pinning down the structure of our micro- as well as macrocosm has just started.
**Acknowledgements** Several authors acknowledges the support of the DFG through the grant SFB676 'Particles, Strings and the Early Universe'.

This work was supported by European Commission through the contract PITN-GA-2012-316704 (HIGGSTOOLS).

This work is supported in part by the Creative Scientific Research Grant No. 18GS0202 of the Japan Society for Promotions of Science (JSPS), the JSPS Grant-in-Aid for Science Research No. 22244031, and the JSPS Specially Promoted Research No. 23000002.

This work is part of the D-ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).

G. Moortgat-Pick would like to thank A.A. Mikhailichenko for useful discussions and valuable comments on collider aspects.

C. Grojean is supported by the Spanish Ministry MICNN under contract FPA2010-17747 and by the European Commission under the ERC Advanced Grant 226371 MassTeV and M.M. Mühlleitner is supported by the DFG/SFB-TR9 *Computational Particle Physics*.

M. Asano acknowledges support from the German Research Foundation (DFG) through grant BR 3954/1-1 and DFG TRR33 "The Dark Universe".

S. Matsumoto acknowledges support from the MEXT, Japan through grants Nos. 22244031 & 26287039, and also from the WPI Initiative, MEXT, Japan.

K. Rolbiecki has been supported by the MICINN, Spain, under contract FPA2013-44773-P, Consolider-Ingenio CPAN CSD2007-00042 and the Spanish MINECO *Centro de excelencia Severo Ochoa Program* under grant SEV-2012-0249.

S. Godfrey was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant number 121209-2009 SAPIN.

The work of S.Y. Choi was supported by Basic Science Research Program through the National Research Foundation (NRF) funded by the Ministry of Education, Science and Technology (2012-0002746).

The work was partly supported by Polish National Center for Science, grant NCN OPUS 2012/05/B/ST2/03306 (2012-2016) and the grant NCN DEC-2012/05/B/ST2/02597, and by BMBF, DAAD PPP Poland Project 56269947, "Dark Matter at Colliders" (M. Krawczyk), grants RFBR 11-02-00242, NSh-3802.2012.2 (I.Ginzburg).

A.S. Kronfeld is supported in part by the German Excellence Initiative and the European Union Seventh Framework Programme under grant agreement No. 291763 as well as the European Union's Marie Curie COFUND program. Fermilab is operated by Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the United States Department of Energy.

The work of K. A. Olive was supported in part by DOE grant de-sc0011842 at the University of Minnesota.

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