The measurements performed at the LHC are in perfect agreement with the Tevatron results. For example CMS [664] finds using lepton+jets final states

$$173.49 \pm 0.43 \text{ (stat.+JES)} \pm 0.98 \text{ (syst.) GeV.} \quad (92)$$

Strictly speaking the renormalization scheme of the experimentally determined mass parameter is not properly fixed using a kinematic reconstruction of the top-quark mass. Nevertheless it is usually assumed that the aforementioned mass values correspond to the so-called on-shell/pole mass. From the precise knowledge of the CKM matrix elements and the top-quark mass all other properties can be predicted within the Standard Model. Given the large value of $V_{tb}$ the dominant decay of the top-quark assuming the Standard Model is the decay into a W-boson and a b-quark. In LO the top-quark decay width is given by

$$\Gamma(t \to bW) = \frac{G_F |V_{tb}|^2 M_t^3}{8\sqrt{2}\pi} \left(1 - \frac{M_W^2}{M_t^2}\right)^2 \left(1 + \frac{2M_W^2}{M_t^2}\right). \quad (93)$$

Higher order electroweak and QCD corrections to the width have been calculated as detailed in the following. In Refs. [665, 666] the electroweak one-loop corrections have been calculated. The NNLO QCD corrections are known for $M_W = 0$ [667] and $M_W \neq 0$ [668]. Including the radiative corrections the top quark decay width is approximately $\Gamma_t \approx 1.4$GeV. As mentioned earlier the life time is thus almost an order of magnitude smaller than the typical time scale for hadronization. The top-quark thus decays without forming hadrons.

### 3.3.1 Top-quark decays at next-to-next-to-leading order QCD

In Refs. [667, 668] only the NNLO QCD corrections to the inclusive decay width were calculated. The calculation for massless W-bosons of Ref. [667] has been extended in Ref. [668] to include also the effects of the finite W-boson mass through an expansion in $M_W^2/M_t^2$. These results have been extended recently in various directions. In Ref. [669] the partial decay widths for top-quarks decaying into polarized W-bosons is investigated. The partial decay widths are particular interesting since the polarization of the W-boson allows to test the $tWb$ vertex independently from the top-quark production mechanism. Assuming massless b-quarks the V-A nature of the charged currents forbids the decay into right-handed W-bosons in LO. The measurement of the W-polarization in top-quark decays thus provides a sensitive tool to test the V-A structure and to search for possible extensions of the Standard Model. Obviously a finite b-quark mass leads to calculable corrections. Evidently also higher order corrections which include in general also real emis-

sion processes can alter the LO predictions. It is thus very important to calculate the branching fractions

$$f_{\pm} = \frac{\Gamma_{\pm}}{\Gamma(t \to Wb)}, \quad f_{L} = \frac{\Gamma_{L}}{\Gamma(t \to Wb)} \quad (94)$$

where $\Gamma(t \to Wb)$ denotes the inclusive top-quark decay width and $\Gamma_{-/+}$ ($\Gamma_L$) denote the decay width into left/right handed (longitudinally) polarized W-bosons. Similar to what has been done in previous work an expansion in $x = M_W/M_t$ is used in Ref. [669] to calculate the partial decay width in NNLO QCD. For $\alpha_s(M_Z) = 0.1176$ and $M_Z = 91.1876$ GeV the results read

$$F_L = 0.6978 - 0.0075 - 0.0023, \quad (95)$$

$$F_+ = 0 + 0.00103 + 0.00023, \quad (96)$$

$$F_- = 0.3022 + 0.0065 + 0.0021, \quad (97)$$

where the individual terms correspond to the LO, NLO and NNLO prediction. Note that the ratios in Eq. (94) for the fractions are not expanded in $\alpha_s$. The sum of $F_L$, $F_+$ and $F_-$ is thus equal to one which does not hold anymore if the ratios are expanded in $\alpha_s$. As one can see the NNLO corrections are about one third of the NLO corrections. Since $F_+$ is non-zero only in NLO the evaluation of the NNLO corrections are very important to test the reliability of the theoretical predictions. We observe that $F_+$ remains very small even after the inclusion of the NNLO corrections. Any observation of $F_+$ significantly larger than 0.001 would thus signal New Physics. In Ref. [501] the impact of various Standard Model extensions on the $tWb$ vertex have been investigated. In particular the MSSM, a generic 2-Higgs doublet model (2HDM) and a top-color assisted Technicolor model are investigated. In top-color assisted Technicolor models a modification of the left chiral couplings by several per cent is possible. In Ref. [670] a more detailed analysis of the W-boson polarization which goes beyond the study of helicity fractions has been proposed.

The fact that the top-quark decays before hadronization plays a major role. Since the dominant decay is parity violating the top-quark polarization of an ensemble of top-quarks is accessible through the angular distribution of the decay products. In Born approximation a straight forward calculation leads to

$$\frac{1}{\Gamma} \frac{d\Gamma}{d\cos\vartheta} = \frac{1}{2} (1 + \alpha_f \cos\vartheta) \quad (98)$$

where $\vartheta$ denotes the angle between the direction of flight of the respective top-quark decay product f and the top-quark spin in the top-quark rest frame. The parameter $\alpha_f$ measures how efficient a specific decay product analyses the top-quark polarization. For the b-quark one finds $\alpha_b = -0.423$ while for the charged lepton from W-boson decay a value of $\alpha_\ell = 1$ is found. The NLO corrections are also known and turn out
to be small. In Refs. [671, 672] the NNLO corrections for the fully differential decay width have been calculated. The NNLO corrections to differential distributions are found to be small. In Ref. [672] also the W-boson helicity fractions have been calculated. The results agree with the aforementioned results of Ref. [669].

### 3.3.2 Two-loop QCD corrections to heavy quark form factors and the forward-backward asymmetry for heavy quarks

The measurements of the forward-backward asymmetry $A_{FB}^b$ for b-quarks differ significantly from the Standard Model predictions [673]. The theoretical predictions take into account NNLO QCD corrections, however the b-quark mass has been neglected at NNLO. The forward-backward asymmetry for massive quarks may be calculated from the fully differential cross section. As far as the 2-loop QCD corrections are concerned this requires the calculation of the two-loop form factor for heavy quarks. These corrections have been calculated recently. In Ref. [674] the NNLO QCD corrections for the vector form factor are calculated. In Ref. [675] the results are extended to the axial-vector form factor. The anomaly contribution has been studied in Ref. [676]. The two-loop corrections need to be combined with the one-loop corrections for real emission and the Born approximation for double real emission. All individual contributions are of order $\alpha_s^2$ and thus contribute. The cancellation of the collinear and soft singularities encountered in the different contributions is highly non-trivial. In Refs. [677, 678] ‘antenna functions’ are derived, which match the singular contributions in the double real emission processes. As an important result also the integrated antenna functions are computed in Refs. [677, 678]. In principle all building blocks are now available to calculate the differential cross section for heavy quark production in NNLO accuracy in QCD. Evidently these results, once available, can also be applied to top-quark pair production.

### 3.3.3 Threshold cross section

Threshold production of top-quark pairs in electron positron annihilation is an unique process where one can extract the top-quark mass through a threshold scan by measuring the total cross section $\sigma(e^+e^- \to t\bar{t})$. It is a counting experiment of the production rate of the color singlet $t\bar{t}$ bound state. Therefore the measurement of the threshold cross section for $e^+e^- \to t\bar{t}$ is very clean experimentally as well as theoretically concerning QCD non-perturbative effects.

The $t\bar{t}$ cross section normalized to the point particle cross section near threshold [679, 680] can be written at LO as

$$R_{t\bar{t}} = \left( \frac{6\pi N_c e_t^2}{m_t^2} \right) \mathrm{Im} G_c(0, 0; E + i\Gamma_t), \qquad (99)$$

where $E = \sqrt{s} - 2m_t$ and $G_c(\mathbf{r}', \mathbf{r}; E + i\Gamma_t)$ is the non-relativistic Coulomb Green function. The Green function contains resonances at energies

$$E_n = -m_t (C_F \alpha_s)^2 / (4n^2)$$

corresponding to Coulomb boundstates, and its residue is given by the Coulomb wave function $|\psi_n(0)|^2 = (m_t \alpha_s C_F)^3 / (8\pi n^3)$:

$$G_c(0, 0; E + i\Gamma_t) = \sum_n \frac{\psi_n(0) \psi_n^*(0)}{E_n - E - i\Gamma_t}. \qquad (100)$$

Thus the peak position and the magnitude of the cross section is determined by the Coulomb energy levels $E_n$ and the wave-functions $|\psi_n(0)|^2$, respectively. In practice the resonance structure of $G_c$ is smeared due to the large top quark width $\Gamma_t \sim 1.4 \text{ GeV}$. In Fig.101 the threshold cross section is shown for $m_t = 170 \text{ GeV}$ varying the top-quark width. Only the $n=1$ ground state peak can be seen for $\Gamma_t = 1.0 - 1.5 \text{ GeV}$ as rather wide prominence of the cross section, and the resonance states are completely smeared out creating a flat plateau for $\Gamma_t = 2 \text{ GeV}$. Although the resonant structures are washed out for a large top-quark width, it is still possible to extract top-quark parameters, $m_t$, $\Gamma_t$ and also $\alpha_s$ by performing a threshold scan, provided a precise theory prediction for the total cross section is at hand.

<figure><img src="image_104.png" /><figcaption>Fig. 101 The top quark production cross section $R$ for $m_t = 170 \text{ GeV}$ and three values for top quark width. The LO formula for the cross section and $\alpha_s(30 \text{ GeV}) = 0.142$ is used</figcaption></figure>

**QCD corrections** Studies of top quark production near threshold [681–684] at linear colliders were started several decades ago, and NNLO QCD corrections were completed by several groups [685–691] and summarized in Ref. [692]. One main achievement there was the stabilization of the peak position against QCD corrections taking into account of renormalon cancellation using short distance masses like 1S-, kinetic-, PS- masses. However, despite the completion of the second
order QCD corrections the normalization of the total cross section still suffers from an uncertainty of about 20 percent.

There are efforts to improve the accuracy of the NNLO total cross section. These include the resummation of potentially large logarithms by renormalization group (RG) methods [693–697] and by brute-force computations of NNNLO corrections [698–702] to increase the precision of the cross section. Fig. 102 shows the NNNLO result (using an adhoc estimate of some third order matching coefficients) [700] compared to the NNLO cross section. The colored bands correspond to the uncertainty originating from a QCD renormalization scale variation between 25 GeV and 80 GeV. A significant reduction of the scale dependence is observed when going to NNNLO comparing with the NNLO result. In Fig. 103 the RG improved total cross section [693–697] is shown, where the upper/lower panels show the result with fixed order/RG improvement, respectively. Two curves at each order are obtained by varying the soft scale $\mu_s$ between (30–80) GeV. The large scale dependence of the fixed order curves is improved by RG resummation in the lower panel. The plot shows that the cross section at the peak position has scale dependence of order 2%. The most complete analysis in RG approach has been performed in [697], where new ultra-soft NNLL contributions [696] are included. These two approaches, NNNLO computation and RG improvement to NNLL, are complementary to each other. The fixed order computation provides the non-logarithmic contributions, while the RG improvement reveals the structure of the potentially large logarithmic terms to all orders. Therefore it is expected that the theory prediction of $t\bar{t}$ cross section with $\delta\sigma_{t\bar{t}}/\sigma_{t\bar{t}} = 2–3\%$ will be possible by a combination of the two approaches as far as QCD corrections are concerned. For such a high precision more dedicated theoretical studies will be needed, for instance, the calculation of electroweak effects and final state interactions in top-quark decays.

<figure><img src="image_105.png" /><figcaption>**Fig. 102** Total cross section for top quark production near threshold at NNLO (with an estimated third order matching coefficients) and NNLO from [700], where a scale variation of (20–80) GeV is shown by the colored bands. A top quark PS mass $m_{\text{PS}}(20\text{GeV}) = 175\text{GeV}$ is used.</figcaption></figure>

<figure><img src="image_106.png" /></figure>

<figure><img src="image_107.png" /><figcaption>**Fig. 103** The threshold cross section at fixed order (upper panel) and renormalization group improvement (lower panel) is shown from Ref. [695]. The bands between two colored lines at each orders show the scale dependence of the results. The RG improved cross sections are stable against scale variation while fixed order result suffers from large dependence on values of $\mu_s$.</figcaption></figure>

**Electroweak corrections and effect of unstable top** In early studies of the $e^+e^- \to t\bar{t}$ threshold it was recognized [679, 680] that the effect of the top quark width can be consistently incorporated into the computation of the total cross section by the replacement $E \to E + i\Gamma_t$. This prescription works well up to NLO, but it turns out that in NNLO an uncanceled ultraviolet divergence appears, which is proportional to the top-quark width (in dimensional regularization an example of such a term is $R_{t\bar{t}} \sim \alpha_s\Gamma_t/\epsilon$). This is a signal of an improper treatment of electroweak effects, and the solution of this problem is to abandon the amplitude $e^+e^- \to t\bar{t}$ where the unstable $t\bar{t}$ is treated as a final state of the S-matrix. Physical amplitudes should treat stable particles as final states of S-matrix, i.e. $e^+e^- \to t\bar{t} \to (bW^{-})(\bar{b}W^+)^{36}$ and the unstable particles can appear only as intermediate states.

Electroweak corrections to the production vertex $t\bar{t} - \gamma/Z$ were first described in [703] and re-derived in [704, 705]. In the later reference it is readily realized that amplitudes for single top production, e.g. $e^+e^- \to t\bar{b}W$, and even no-top quark production $e^+e^- \to bW^+\bar{b}W^-$ can contribute

<footnote>$^{36}$assuming the W-boson and b-quarks as stable or long-lived particles.</footnote>
to (or mix with) the top-pair production because the physical final state is the same.

The top-quark width is generated by the EW interaction, $t \to bW$, therefore the effects of the top-quark finite width are intimately related to the EW corrections of the process. To take into account certain electroweak non-resonant effects a method referred to as *phase-space matching* was introduced in [706, 707].

This idea has been further developed and rephrased in the framework of an effective theory for unstable particle [708, 709]. (See Refs. [710, 711] for an application of the method to W-pair production in $e^+e^-$ annihilation.) A systematic analysis of the electroweak effects in top-quark pair production has started rather recently, and NLO electroweak non-resonant contributions were computed [712], e.g. $R(e^+e^- \to t\bar{b}W^-) \sim \alpha_{EW}$, where resonant (onshell) top quarks decay and the final state $(bW^+)(\bar{b}W^+)$ is measured assuming stable W-bosons and b-quarks. In this work invariant mass cuts on the top-quark and anti-top-quark decay products are implemented. It is found that the non-resonant correction results in a negative 5% shift of the total cross section which is almost energy independent, in agreement with Ref [707]. The dominant NNLO non-resonant corrections were computed in Ref. [713,714] and it was shown that the single resonant amplitudes (e.g. $e^+e^- \to t(\bar{b}W^-)g$) provide the counter terms for the uncanceled ultraviolet divergence $\alpha_s\Gamma/\epsilon$ discussed previously for the double resonant $e^+e^- \to t\bar{t}$ amplitude at NNLO QCD. Therefore, the non-resonant corrections provide together with NNLO QCD a consistent treatment of top quark width effects.

It is also known that the final state corrections [715, 716] between top quarks and decay products have to be considered for observables other than the total cross section. A systematic analysis of these effects is still missing beyond NLO. Dedicated studies of the electroweak corrections to the threshold cross section have started rather recently.

**Influence of the Higgs boson on the total cross section** In the SM the large top-quark mass leads to a large top-quark Yukawa coupling to the Higgs boson, therefore it is expected that Higgs boson exchange in top-quark production may lead to observable corrections. Such a Higgs exchange effect appears in two different ways in top and anti-top production near threshold (see Fig.104). One is a short distance contribution which enhances the top quark production vertex as $t\bar{t}\gamma^\mu t \to (1+c_h)t\bar{t}\gamma^\mu t$. The one-loop Higgs correction $c_h^{(1)}$ was determined in Refs. [703], and Higgs and EW mixed two-loop correction $c_h^{(2)}$ in Ref. [717]. The enhancement factor for the cross section is given by

$$ \delta R/R_{LO} \approx 2c_h^{(1)} = 6.7/3.4/0.9 \times 10^{-2} \quad (101) $$

using $m_h = 120/200/500$ GeV.

<figure><img src="image_108.png" /><figcaption>**Fig. 104** Corrections due to Higgs exchange in $e^+e^- \to t\bar{t}$. In the left diagram the Higgs exchange contributes to the production vertex for $t\bar{t}$, $Zt\bar{t}$, which occurs at short distance when the $t\bar{t}$-pair is separated by $r \sim 1/m_t$. In the right diagram Higgs exchanges occurs after bound-state formation between top and anti-top quarks separated by the scale of the bound-state $r \sim 1/(m_t\alpha_s)$.</figcaption></figure>

In addition, there is a long-distance effect described by the Yukawa potential $V_h(r)$ for the top quark pair:

$$ V_h(r) = -\frac{y_t^2 e^{-m_h r}}{8\pi r} \approx -\frac{y_t^2}{2m_h^2} \delta(\mathbf{r}), \quad (102) $$

where the second expression is a good approximation for $m_h r \gg 1$ assuming $m_h \sim 125$ GeV and $r \sim (m_t\alpha_s)^{-1}$. In the SM the Yukawa coupling is related to the top-quark mass by $y_t = \sqrt{2}em_t/(s_W M_W)$.

<figure><img src="image_109.png" /><figcaption>**Fig. 105** Cross section for $e^+e^- \to t\bar{t}$ for $m_t = 170$ GeV with/without 1-loop Higgs boson corrections. A Higgs boson mass of $m_h = 125$ GeV is used.</figcaption></figure>

In Fig.105 the threshold cross section is shown taking into account of Higgs loop effects through $c_h$ and $V_h$. One can see that the threshold cross section gets an almost energy independent enhancement. The Higgs potential $V_h$ produces corrections to the energy and to the wave function as

$$ \begin{gathered} \delta E_1/E_{LO} = 3.2/1.2/0.2 \times 10^{-2}, \\ \delta |\Psi_1(0)|^2/|\Psi_{LO}(0)|^2 = 4.6/1.6/0.3 \times 10^{-2}, \end{gathered} \quad (103) $$

using $m_t = 175$ GeV, $\mu = 30$ GeV and $m_h = 120/200/500$ GeV, respectively. The above value for $\delta E_1$ can be translated into a shift $\delta m_t = 25/9/1$ MeV of the top-quark mass determined in a threshold scan.
**Distribution and Asymmetry** In the threshold production, the top-quark momentum $p_t$ can be reconstructed from its decay products. Therefore the top-quark momentum distribution [681–683] provides complementary information. Theoretically it is given by

$$ \frac{1}{\sigma_0} \frac{d\sigma_{LO}}{dp_t} (e^+e^- \to t\bar{t}) = \frac{p_t^2}{2\pi^2} \left( \frac{6\pi N_c e_t^2}{m_t^2} \right) \times \Gamma_t |\tilde{G}_c(\mathbf{p}, \mathbf{r}; E + i\Gamma_t)|^2, \quad (104) $$

where $\tilde{G}_c(\mathbf{p}, \mathbf{r}; E + i\Gamma_t)$ is the Fourier transformation of the Coulomb Green function. For the momentum distribution NNLO QCD results [686,691] are available in the literature.

Fig.106 shows the momentum distribution at specific energy points $\Delta E = 0, 2, 5$ GeV (left panel) and for different top-quark masses. In the lower panel the bands correspond to the uncertainty of the QCD coupling constant assuming $\alpha_s = 0.118 \pm 0.003$. As the Green function $\tilde{G}_c(\mathbf{p}, \mathbf{r}; E + i\Gamma)$ is essentially the momentum space wave function averaged over the resonances, a measurement of the top-quark momentum distribution gives information on the bound state wave function $\tilde{\phi}(\mathbf{p})$. Therefore the momentum space distribution gives independent information on the bound-state and can be used to test the understanding of the QCD dynamics.

Another useful observable which can be measured in top-quark production near threshold is the forward-backward asymmetry defined as

$$ A_{FB} = \frac{1}{\sigma_{t\bar{t}}} \left\{ \int_{0}^{1} d\cos\theta - \int_{-1}^{0} d\cos\theta \right\} \frac{d\sigma(e^{+}e^{-} \to t\bar{t})}{d\cos\theta}. \quad (105) $$

At lepton colliders top-quark pair production occurs through $e^+e^- \to \gamma^*/Z^* \to t\bar{t}$ and the forward-backward asymmetry receives a non-zero contribution from the interference of vector and axial-vector couplings. Vector and axial-vector interactions produces S-wave and P-wave bound-states, respectively, due to angular momentum conservation. Therefore the forward-backward asymmetry is sensitive to the interference between S-wave and P-wave top-quark production. The asymmetry is sensitive to the top-quark width $\Gamma_t$ because the S-wave and P-wave overlap is non-zero due to $\Gamma_t$. In Fig.107 the forward-backward asymmetry is plotted as a function of energy $E$. Top and bottom panels show the dependence on $\Gamma_t$ and $\alpha_s$, respectively. As discussed above the asymmetry $A_{FB}$ is an effect of $\gamma$ and Z-boson interference. Therefore, the asymmetry provides useful information on the mechanism of top-quark production near threshold.

### 3.3.4 Top-quark production in the continuum

The total cross section for the production of heavy quarks in electron-positron annihilation has been calculated in Refs. [719–722] at order $\alpha_s^2$ in QCD. The results are not applicable very close to the threshold since in that region Coulomb

<figure><img src="image_110.png" /><figcaption>**Fig. 106** Top quark momentum distribution at $\Delta E = E - E_1 = 0, 2, 5$ GeV (top) for $m_t = 170$ GeV and top quark mass dependence (bottom) on the momentum distribution.</figcaption></figure>

effects lead to $1/\beta$ corrections where $\beta$ denotes the velocity of the top-quark. For reliable predictions in the threshold region these contributions need to be resummed (see also the discussion in the previous section). In Ref. [722] it has been estimated that the fixed order results should be applicable in the case of top-quark pair production, provided that the center-of-mass energy is about 12 GeV above threshold. In Ref. [723] the results have been extended to order $\alpha_s^3$. In particular the quartic mass corrections with respect to the massless calculation were calculated. Using the minimal subtraction scheme (MS) to renormalize the mass parameters, sizeable corrections were found in order $\alpha_s^3$. However it is also shown in Ref. [723] that using the invariant mass $\hat{m}$ defined through

$$ m(\mu) = \hat{m} \exp \left\{ \int da \frac{\gamma_m(a)}{a\beta(a)} \right\}, \quad (106) $$

where $m(\mu)$ denotes the running mass, $\gamma_m$ the anomalous dimension of $m(\mu)$ and $\beta(a)$ the QCD beta function in terms of $a = \alpha_s/\pi$, the convergence of the perturbative expansion can be improved. As discussed in Section 3.3.2 the work on the differential cross section at order $\alpha_s^2$ is still ongoing. In
<figure><img src="image_111.png" /><figcaption>**Fig. 107** Dependence of the forward-backward asymmetry $A_{FB}$ on the top quark width (upper plot) and the strong coupling $\alpha_s$ (lower plot). Figures are taken from Ref. [718].</figcaption></figure>

Refs. [724, 725] jet observables in top-quark pair production at high energy have been investigated. The process is characterized by different mass scales: the center-of-mass energy $\sqrt{s}$, the top-quark mass $M_t$, the top-quark width $\Gamma_t$ and $\Lambda_{QCD}$. Large logarithmic corrections connected with the different mass scales are resummed in Ref. [724] at next-to-leading logarithmic accuracy. This requires the introduction of soft functions capturing non-perturbative soft QCD effects. The soft functions can be obtained from massless dijet events. In Ref. [725] the application to top-quark mass measurements is discussed. In particular it is demonstrated that a top-quark mass measurement with a precision of $\Lambda_{QCD}$ is possible significantly above the production threshold.

## 3.4 Physics Potential

The excellent possibilities for precision top-quark measurements at $e^+e^-$ colliders have been confirmed by experimental studies of the physics potential of linear colliders, which, in particular in the framework of recent reports of the CLIC

and ILC physics and detector projects, often are based on full detector simulations. Particular emphasis has been placed on the measurement of the top-quark mass, which has been studied both at and above threshold, and on the study of the $t\bar{t}Z/\gamma^*$ vertex through the measurement of asymmetries. For all of these measurements, precise flavor tagging and excellent jet reconstruction is crucial to identify and precisely reconstruct top-quark pair events. The detectors being developed for linear colliders provide these capabilities, and, together with the rather modest background levels in $e^+e^-$ collisions, allow to acquire high-statistics high-purity top-quark samples. In the following, the most recent published results from simulation studies of top-quark mass measurements are discussed. The studies of top-quark couplings, which make use of the possibilities for polarized beams at linear colliders, are still ongoing. Preliminary results indicate a substantially higher precision than achievable at hadron colliders.

### 3.4.1 Top-quark mass measurement at threshold

The measurement of $t\bar{t}$ production cross section in a scan around the threshold provides direct access to the top quark, as discussed above. In the experiment, the calculated cross section is modified by initial state radiation and by the luminosity spectrum of the collider. These two effects are illustrated in Figure 108 [36], where the pure $t\bar{t}$ cross section is calculated with TOPPIK at NNLO [685,686] for a top-quark mass of 174 GeV in the 1S mass scheme, and the luminosity spectrum of CLIC at 350 GeV is assumed. Both lead to a smearing of the cross section, resulting in a substantial reduction of the prominence of the cross section peak, and to an overall reduction of the cross section due to the lowering of the luminosity available above the production threshold. Since the beam energy spread at ILC is smaller than at CLIC, the threshold turn-on is slightly steeper, as visible in Figure 109.

Recently, an experimental study has been performed in which the NNLO cross section shown in Fig. 108 was used, together with signal efficiencies and background contamination determined with full Geant4 simulations of a CLIC variant of the ILD detector, including the use of the full reconstruction chain. In the context of a threshold scan, where the focus is on the efficient identification of $t\bar{t}$ events, the difference in performance between the ILC and CLIC detector concepts is expected to be negligible, allowing to apply this study to both accelerator concepts by using the appropriate luminosity spectra. The experimental precision of a threshold scan with a total integrated luminosity of $100 \text{ fb}^{-1}$ spread over ten points spaced by 1 GeV for the ILC case is illustrated in Fig. 109.

Since the cross section depends not only on the top-quark mass, but also on $\alpha_s$, those two values are determined
<figure><img src="image_112.png" /><figcaption>**Fig. 108** The top-quark production cross section calculated with TOP-PIK for a top mass of 174 GeV in the 1S mass scheme, showing the effects of initial state radiation and of the luminosity spectrum of CLIC. Figure taken from Ref. [36].</figcaption></figure>

<figure><img src="image_113.png" /><figcaption>**Fig. 109** Simulated measurement of the background-subtracted $t\bar{t}$ cross section with $10 \text{ fb}^{-1}$ per data point, assuming a top-quark mass of 174 GeV in the 1s scheme with the ILC luminosity spectrum for the CLIC_ILD detector. Figure taken from Ref. [36].</figcaption></figure>

simultaneously with a two-dimensional fit, resulting in a statistical uncertainty of 27 MeV on the mass and 0.0008 on $\alpha_s$. Assuming the CLIC luminosity spectrum, which is characterized by a somewhat more pronounced beamstrahlung tail and a larger energy spread, the uncertainties increase to 34 MeV and 0.0009, respectively. Systematic uncertainties from the theoretical cross-section uncertainties, from the precision of the background description and the understanding of the detector efficiency as well as from the absolute knowledge of the beam energy are expected to be of similar order as the statistical uncertainties. Thus, the differences

between different linear collider concepts for a top threshold scan are negligible, and total uncertainties of below 100 MeV on the mass are expected [36]. For a phenomenological interpretation, the measured 1S mass typically has to be converted into the standard $\overline{\text{MS}}$ mass. This incurs additional uncertainties of the order of 100 MeV, depending on the available precision of $\alpha_s$ [686].

As discussed in detail in Ref. [726], in addition to the mass and the strong coupling constant, also the top-quark width can be determined in a threshold scan. The use of additional observables such as the top-quark momentum distribution and the forward-backward asymmetry has the potential to further reduce the statistical uncertainties. The cross-section around threshold is also sensitive to the top-quark Yukawa coupling, as discussed above. However, its effect on the threshold behavior is very similar to that of the strong coupling constant, so an extraction will only be possible with a substantially improved knowledge of $\alpha_s$ compared to the current world average uncertainty of 0.0007, and with reduced theoretical uncertainties on the overall cross section.

### 3.4.2 Top-quark mass measurement in the continuum

In the continuum above the $t\bar{t}$ threshold, the top-quark mass is measured experimentally by directly reconstructing the invariant mass from the measured decay products, a W boson and a b quark. This is possible with high precision both in fully hadronic (e.g. both W bosons produced in the $t\bar{t}$ decay decaying into hadrons) and semi-leptonic (e.g. one W boson decaying into hadrons, one into an electron or muon and a neutrino) top-quark pair decays. Due to the well-defined initial state in $e^+e^-$ collisions, full three-dimensional kinematic constraints can be used for kinematic fitting, substantially improving the invariant mass resolution compared to a free measurement.

For both CLIC and ILC this measurement has been studied using full detector simulations with all relevant physics backgrounds at an energy of 500 GeV. In the case of the CLIC study, also the influence of background from hadron production in two-photon processes was included, which is more severe at CLIC than at ILC due to the very high bunch-crossing frequency. The reconstructed invariant mass after background rejection and kinematic fitting for the fully hadronic final state at CLIC is shown in Fig. 110. The figure also illustrates the high purity achievable for top quarks at linear colliders. For an integrated luminosity of $100 \text{ fb}^{-1}$, combined statistical precisions of 70 MeV and 80 MeV are obtained for ILC [197] and CLIC [36], respectively. The CLIC study showed that it is expected that systematic uncertainties due to the jet energy scale can be limited to below the statistical uncertainty by constraining the light jet energy scale through the direct reconstruction of the W bosons in the top-quark decay. The b jet energy scale in turn can
<figure><img src="image_114.png" /><figcaption>Fig. 110 Simulated measurement of the top-quark invariant mass in the all-hadronic decay channel of top-quark pairs for an integrated luminosity of 100 fb⁻¹ at CLIC in the CLIC_ILD detector at a center-of-mass energy of 500 GeV. The solid green histogram shows the remaining non $t\bar{t}$ background in the data sample. The mass is determined with an unbinned maximum likelihood fit to the distribution. Figure taken from Ref. [36].</figcaption></figure>

be determined in a similar way from $Z \to b\bar{b}$ decays. Also other experimental systematics, such as the knowledge of the beam energy, which enters in the kinematic fit, and uncertainties from color reconnection effects are expected to be small.

However, in contrast to the measurement via a threshold scan, the mass determined by direct reconstruction is theoretically not well defined. Rather, it is obtained in the context of the event generator used to determine the detector and reconstruction effects on the measured invariant mass. At present, no conversion of this invariant mass value to the MS mass exists. This leads to additional uncertainties in the interpretation of the result, which potentially far exceed the experimental accuracy of the invariant mass measurement.

### 3.4.3 Measurement of coupling constants

For precise test of the Standard Model and New Physics searches a precise determination of the Standard Model couplings together with the search for anomalous couplings is important. In the following we try to review the prospects of a future Linear Collider and compare where possible with the LHC. From top-quark pair production at hadron collider the top-quark coupling to gluons is already constrained. As mentioned in Section 3.4.1 the threshold studies can be used to measure the top-quark mass together with $\alpha_s$. Top-quark pairs produced in association with an additional jet can be used to search directly for anomalous top-gluon couplings.

This can be done independent from the production mechanism in hadronic collisions as well as in electron-positron annihilation. For hadronic $t\bar{t} + 1$-Jet production dedicated NLO calculations are available [727–730]. For electron positron annihilation the corresponding calculations for massive b-quarks [731–735] can be applied by adjusting the coupling constants. A dedicated analysis of top-quark pair + 1-Jet production at a future Linear Collider can be found in Ref. [736]. Since anomalous couplings will show up more likely in the couplings to the weak gauge bosons no detailed study of the sensitivity to anomalous top-gluon couplings has been performed so far for a future Linear Collider.

The $Wtb$-coupling can be probed through top-quark decay and single-top-quark production. A detailed measurement of this coupling is interesting because the $V-A$ structure of the vertex can be tested. Furthermore the existence of a fourth family —if not yet ruled out by other measurements— could significantly change the SM predictions for the respective coupling. Tevatron and LHC measurements constrain the coupling already through the measurement of the top-quark width [737] and the measurements of the W-boson helicity fractions [738–740]. A measurement of the top-quark width from threshold studies can be used to indirectly constrain the coupling in electron-positron annihilation. A direct measurement of the $Wtb$ coupling at a Linear Collider is difficult [718]. In top-quark pair production close to the threshold the coupling enters only through the branching ratio for $t \to Wb$ which is expected to be very close to one and thus does not lead to a strong dependence on the $Wtb$ coupling. Measurements using single-top-quark production are difficult owing to sizeable backgrounds. In Ref. [497] it has been argued that using $e^+e^- \to W^+bW^-\bar{b}$ events below the $t\bar{t}$ threshold the coupling can be measured at ILC with an accuracy of about 3% using an integrated luminosity of about 100/fb.

The top-quark coupling to the photon or more precisely the top-quark charge is constrained through indirect measurements at hadron colliders. Using the charge of the top-quark decay products reconstructed from top-candidate events the top-quark charge has been measured in Ref. [741] to be

$$Q = 0.64 \pm 0.02(\text{stat.}) \pm 0.08(\text{syst.}) \qquad (107)$$

in units of the electron charge. A direct measurement of $t\bar{t} + \gamma$ production is difficult at the LHC due to the small cross sections although a measurement with an uncertainty of 10% might nevertheless be feasible [718]. (First results have been presented already by CDF [742] and ATLAS [743].) At the Linear Collider the analysis of the SM couplings is usually combined with the search for anomalous couplings. As a starting point one may use a form factor decomposition of the form [718]:
$$ \Gamma_{\mu}^{ttX}(k^2, q, \bar{q}) = ie \left\{ \gamma_{\mu} (\tilde{F}_{1V}^{X}(k^2) + \gamma_5 \tilde{F}_{1A}^{X}) + \frac{(q - \bar{Q})_{\mu}}{2M_t} (\tilde{F}_{2V}^{X}(k^2) + \gamma_5 \tilde{F}_{2A}^{X}) \right\} \quad (108) $$

where X can be a photon as well as a Z boson. In Refs. [7,246,744] it has been shown, that the precision, with which the various couplings can be determined, can be improved at a Linear Collider by about a factor of ten compared to what is possible at the LHC. At the LHC the precision for $\tilde{F}_{1V}^{\gamma}$ and $\tilde{F}_{1A}^{\gamma}$ is at the level of 10% [744] and much larger for the remaining couplings.

Given that the top-quark is so much heavier than the next heavy quark it seems reasonable to question whether the mechanism to generate the top-quark mass is the same as for the lighter quarks. In this context the measurement of the $t\bar{t}H$ Yukawa coupling is of great importance. At the LHC this coupling can be accessed through the measurement of top-quark pair production in association with a Higgs boson. A recent study of the sensitivity where the subsequent decay $H \to b\bar{b}$ has been used can be found for example in Ref. [745]. In Ref. [245] it has been estimated that the $t\bar{t}H$ coupling can be measured at the LHC with an accuracy of about 15% assuming an integrated luminosity of 300/fb at 14 TeV center of mass energy. With an increased luminosity of 3000/fb a measurement at the level of 7–14% may become feasible. Due to the large mass of the final state it is difficult to improve this measurement significantly at a linear collider operating at 500 GeV. For an integrated luminosity of 1000/fb at 500 GeV center-of-mass energy an uncertainty of 10 % has been estimated [245]. Increasing the energy to 1 TeV (ILC) or even 1.4 TeV (CLIC) will help to improve the situation: In both cases a precision of 4% seems to be feasible. Using the ILC design at 1 TeV would require 1000/fb of integrated luminosity while at 1.4 TeV 1500/fb would be required.

Very recently it has been argued in Ref. [746] that the $t\bar{t}H$ coupling could also be inferred at the LHC from single-top-quark production in association with an additional Higgs. Since the cross section of this process is below 100 fb such a measurement will be challenging. In the Standard Model the cross section is reduced through an accidental cancelation. As a consequence BSM models may show sizeable deviations compared to the Standard Model prediction.

### 3.4.4 The top-quark polarization

Top-quarks produced in electron-positron annihilation are polarized. Furthermore the spin of the top-quark is also correlated with the spin of the anti-top-quark. As mentioned in Section 3.3.1 the top-quark polarization can be inferred from the angular distributions of the decay products. The

top-quark polarization thus provides an additional observable which allows a more detailed test of the top-quark interactions. The top-quark polarization and spin-correlations in electron-positron annihilation have been studied in detail for example in Refs. [747–755]. In Ref. [756] the impact of the beam polarization on the polarization of the produced top-quarks has been investigated. In difference to the production rate the observables sensitive to the top-quark polarization depend only on the effective beam polarization

$$ P_{eff} = \frac{P_{e^{-}} - P_{e^{+}}}{1 - P_{e^{-}}P_{e^{+}}} \quad (109) $$

where $\lambda_{-}$ ($\lambda_{+}$) denotes the longitudinal polarization of the incoming electrons (positrons). While the top-quark polarization depends strongly on the $P_{eff}$ the longitudinal spin correlation depends only weakly on $P_{eff}$. At a center-of-mass energy of 500 GeV the polarization is close to maximal for $|P_{eff}| = 1$. For higher energies the polarization is reduced. However for $|P_{eff}| = 1$ a polarization above 85% is still possible.
## 4 Exploring the Quantum Level: Precision Physics in the SM and BSM<fnref n="37" />

We review the LC capabilities to explore the electroweak (EW) sector of the SM at high precision and the prospects of unveiling signals of BSM physics, either through the presence of new particles in higher-order corrections or via direct production of extra EW gauge bosons. We discuss the experimental and theory uncertainties in the measurement and calculation of electroweak precision observables (EWPO), such as the W boson mass, Z pole observables, in particular the effective weak mixing angle, $\sin^2 \theta^\ell_{\text{eff}}$, and the anomalous magnetic moment of the muon, $a_\mu$. We concentrate on the MSSM to illustrate the power of these observables for obtaining indirect information on BSM physics. In particular, we discuss the potential of two key EWPOs at a LC, $M_W$ and $\sin^2 \theta^\ell_{\text{eff}}$, to provide a stringent test of the SM and constraints on the MSSM parameter space. Naturally, the recent discovery of a Higgs-like particle at the LHC has a profound impact on EW precision tests of the SM. We present a study of the impact of this discovery on global EW fits, and also include a discussion of the important role of the top quark mass in performing these high precision tests of the SM. Finally, we review the anticipated accuracies for precision measurements of triple and quartic EW gauge boson couplings, and how deviations from SM gauge boson self interactions relate to different BSM scenarios. These observables are of special interest at a LC, since they have the potential of accessing energy scales far beyond the direct kinematical reach of the LHC or a LC. We conclude with a discussion of the LC reach for a discovery of extra EW gauge bosons, $Z'$ and $W'$, and the LC's role for pinning down their properties and origin, once discovered.

### 4.1 The role of precision observables

The SM cannot be the ultimate fundamental theory of particle physics. So far, it succeeded in describing direct experimental data at collider experiments exceptionally well with only a few notable exceptions, e. g., the left-right ($A_{LR}^e$(SLD)) and forward-backward ($A_{FB}^b$(LEP)) asymmetry (see Section 4.3.3), and the muon magnetic moment $g_\mu - 2$ (see Section 4.6). However, the SM fails to include gravity, it does not provide cold dark matter, and it has no solution to the hierarchy problem, i.e. it does not have an explanation for a Higgs-boson mass at the electroweak scale. On wider grounds, the SM does not have an explanation for the three generations of fermions or their huge mass hierarchies. In order to overcome (at least some of) the above problems,

many new physics models (NPM) have been proposed and studied, such as supersymmetric theories, in particular the Minimal Supersymmetric Standard Model (MSSM), Two Higgs Doublet Models (THDM), Technicolor, little Higgs models, or models with (large, warped, or universal) extra spatial dimensions. So far, the SM has withstood all experimental tests at past and present collider experiments, such as the LEP and SLC $e^+e^-$ colliders, the HERA ep, Tevatron $p\bar{p}$, and LHC $p\bar{p}$ collider. Even the recently discovered Higgs-like particle at the LHC, after analyzing the 2012 data agrees with the SM Higgs boson expectation, albeit more precise measurements of its properties will be needed to pin down its identity. Measurements of precision observables and direct searches for NPM particles succeeded to exclude or set stringent bounds on a number of these models. The direct search reach is going to be significantly extended in the upcoming years, when the LHC is scheduled to run at or close to its design energy of 14 TeV. Future $e^+e^-$ colliders, such as the ILC or CLIC, have good prospects for surpassing the LHC direct discovery reach, especially in case of weakly interacting, colorless NPM particles (see, e. g., Section 4.8).

Even if a direct discovery of new particles is out of reach, precision measurements of SM observables have proven to be a powerful probe of NPM via virtual effects of the additional NPM particles. In general, precision observables (such as particle masses, mixing angles, asymmetries etc.) that can be predicted within a certain model, including higher order corrections in perturbation theory, and thus depending sensitively on the other model parameters, and that can be measured with equally high precision, constitute a test of the model at the quantum-loop level. Various models predict different values of the same observable due to their different particle content and interactions. This permits to distinguish between, e. g., the SM and a NPM, via precision observables. Naturally, this requires a very high precision of both the experimental results and the theoretical predictions. The wealth of high-precision measurements carried out at the Z pole at LEP and SLC, the measurement of the W boson at LEP and the Tevatron [17, 761, 763], as well as measurements at low-energy experiments, such as $a_\mu = (g_\mu - 2)/2$ at the “Muon g-2 Experiment” (E821) [757], are examples of EWPOs that probe indirect effects of NPM particles. These are also examples where both experiment and theory have shown that they can deliver the very high precision needed to fully exploit the potential of these EWPOs for detecting minute deviations from the SM. The most relevant EWPOs in which the LC plays a key role are the W boson mass, $M_W$, and the effective leptonic weak mixing angle, $\sin^2 \theta^\ell_{\text{eff}}$. In the MSSM, the mass of the lightest $\mathcal{CP}$-even MSSM Higgs boson, $M_h$, constitutes another important EWPO [758]. Note that in these examples, the top quark mass plays a crucial role as input parameter.

<footnote n="37">³⁷Editors: S. Heinemeyer, D. Wackeroth
Contributors: A. Denner, S. Dittmaier, A. Freitas, S. Godfrey, N. Greiner, M. Grünewald, A. Hoecker, R. Kogler, K. Mönig, M. Schmitt, D. Stöckinger, G. Weiglein, G. Wilson, L. Zeune</footnote>
Also EWPOs that cannot be measured at a LC can be very relevant in the assessment of its physics potential. A prominent role in this respect plays the muon magnetic moment, ($g_μ - 2$). It already provides some experimental indication for NPM particles in reach of a LC, and its role in constraining NPM and its complementarity to the LC is summarized in Section 4.6.

Another type of PO is connected to the self interactions of EW gauge bosons in multiple EW gauge boson production, i.e. they directly probe the triple and quartic EW gauge boson couplings. Deviations from SM predictions would indicate new physics, entering either through loop contributions or are due to new heavy resonances, which at low energy manifest themselves as effective quartic gauge boson couplings. Precision measurements of these POs could provide information about NPM sectors far beyond the kinematic reach of the LHC and LC.

As discussed above, in this report we focus our discussion on the EWPO, i.e. (pseudo-) observables like the W-boson mass, $M_W$, the effective leptonic weak mixing angle, $\sin^2 \theta_{\text{eff}}^\ell$, and the anomalous magnetic moment of the muon. Since in the literature virtual effects of NPM particles are often discussed in terms of *effective* parameters instead of the EWPO we briefly discuss this approach in the following.

A widely used set of effective parameters are the $S, T, U$ parameters [759]. They are defined such that they describe the effects of new physics contributions that enter only via vacuum-polarization effects (i.e. self-energy corrections) to the vector boson propagators of the SM (i.e. the new physics contributions are assumed to have negligible couplings to SM fermions). The $S, T, U$ parameters can be computed in different NPM's as certain combinations of one-loop self-energies, and then can be compared to the values determined from a fit to EW precision data, i.e. mainly from $M_W, M_Z$ and $\Gamma_Z$ (see, e.g., the review in [760]). A non-zero result for $S, T, U$ indicates non-vanishing contributions of new physics (with respect to the SM reference value). According to their definition, the $S, T, U$ parameters are restricted to leading order contributions of new physics. They should therefore be applied only for the description of *small* deviations from the SM predictions, for which a restriction to the leading order is permissible. Examples of new physics contributions that can be described in the framework of the $S, T, U$ parameters are contributions from a fourth generation of heavy fermions or effects from scalar quark loops to the W- and Z-boson observables. A counter example, i.e. where the $S, T, U$ framework is not sufficient, are SUSY corrections to the anomalous magnetic moment of the muon. Due to these restrictions of this *effective* description of BSM effects in W and Z boson observables, in this report we decided to only present investigations of these effects in the EWPO themselves.

This review of precision physics in the SM and BSM at the LC is organized as follows: in Section 4.2 we concentrate on $M_W$ from both the experimental and theoretical view point, and then turn to a discussion of Z pole observables, in particular $\sin^2 \theta_{\text{eff}}^\ell$, in Section 4.3. The relevance of the top quark mass in EW precision physics is briefly summarized in Section 4.4, before we present the prospects of extracting information about the SM Higgs boson mass from a global EW fit in Section 4.5. We close our discussion of EWPOs with an overview of predictions for the muon magnetic moment in NPM in Section 4.6. An overview of possible parametrizations of non-standard EW gauge boson couplings, available calculations and the experimental prospects for precision measurements of these couplings is presented in Section 4.7. And finally, in Section 4.8 we present an overview of studies of new gauge bosons at the LC.

## 4.2 The W boson mass

The mass of the W boson is a fundamental parameter of the electroweak theory and a crucial input to electroweak precision tests. The present world average for the W-boson mass [761],

$$M_W = 80.385 \pm 0.015 \text{GeV}, \quad (110)$$

is dominated by the results from the Tevatron, where the W-boson mass has been measured in Drell-Yan-like single-W-boson production. At LEP2, the W-boson mass had been measured in W-pair production with an error of 33MeV from direct reconstruction and ~200MeV from the cross section at threshold [762, 763]. In this section we will review the prospects for the $M_W$ measurements at the LC from the experimental and theoretical side, as well as the possibility to constrain indirectly parameters of new physics models using a precise $M_W$ measurement and prediction.

### 4.2.1 Experimental prospects for a precision measurement of $M_W$ at the ILC$^{38}$

The ILC facility$^{39}$ can contribute decisively by making several complementary measurements of the W mass using $e^+e^-$ collisions at centre-of-mass energies spanning from near WW threshold to as high as 1 TeV. Data samples consisting of between 10 and 100 million W decays can be produced, corresponding to an integrated luminosity of about $250\text{fb}^{-1}$ at $\sqrt{s} = 250$ GeV (and correspondingly lower integrated luminosity at higher energies).

<footnote>$^{38}$Graham Wilson</footnote>

<footnote>$^{39}$We refer in this section particularly to the ILC which has a number of advantages over other proposed facilities, notably the ability to polarize both beams, to run in an optimized fashion at a variety of center-of-mass energies, and with a good quality luminosity spectrum.</footnote>
The main production channels of W bosons at ILC are pair production, $e^{+}e^{-} \rightarrow W^{+}W^{-}$ and single-W production, $e^{+}e^{-} \rightarrow We\bar{V}_{e}$ which proceeds mainly through $\gamma - W$ fusion. Pair production dominates at lower center-of-mass energies while single-W production dominates over other $e^{+}e^{-}$ sources of hadronic events at the higher energies.

The three most promising approaches to measuring the W mass are:

- Polarized threshold scan of the $W^{+}W^{-}$ cross-section as discussed in [764].

- Kinematically-constrained reconstruction of $W^{+}W^{-}$ using constraints from four-momentum conservation and optionally mass-equality as was done at LEP2.<fnref n="40" />

- Direct measurement of the hadronic mass. This can be applied particularly to single-W events decaying hadronically or to the hadronic system in semi-leptonic $W^{+}W^{-}$ events.

Much of the existing literature on $M_W$ measurement from LEP2 is still very relevant, but one should be aware of a number of LC features which make the LC experimental program for $M_W$ measurements qualitatively different. Notable advantages are: availability of longitudinally polarized beams, energy and luminosity reach, and much better detectors. Notable concerns are related to potential degradation of the precision knowledge of the initial state.

We first give an outline of statistical considerations for $M_W$ measurements and then outline the strategies considered for being able to make use of this considerable statistical power in experimentally robust ways.

The statistical errors on a W mass determination at ILC are driven by the cross-sections, the intrinsic width of the W ($\Gamma_W \approx 2.08$GeV), the potential integrated luminosity, the availability of polarized beams, and where appropriate the experimental di-jet mass resolution, event selection efficiencies and backgrounds. The width is the underlying fundamental issue. This broadens the turn-on of the W-pair cross-section near threshold, decreasing its dependence on $M_W$. It also broadens the W line-shape, diluting the statistical power of mass measurements for both kinematically-constrained reconstruction and direct mass reconstruction. For the detectors envisaged at ILC, hadronically decaying W's should be measured with mass resolutions in the 1-2GeV range.

We have estimated the statistical sensitivity dependence on experimental mass resolution quantitatively using a fit to the simulated measured line-shape for one million W decays while varying the assumed experimental mass resolution (per decay). Results of a fit with a (non-relativistic) Breit-Wigner convolved with a Gaussian of known width (Voigtian fit) are shown in Figure 111. One sees from this

<figure><img src="image_115.png" /><figcaption>Fig. 111 Statistical precision on $M_W$ from the Voigtian fit (see text).</figcaption></figure>

that statistical sensitivities of around 2.5MeV per million W decays are to be expected for mass resolutions in the 1-2GeV range. In practice experiments will use a variety of analysis techniques such as convolution fits where one takes into account the mass resolution on an event-by-event basis maximizing the statistical power of well-measured events and de-weighting events with worse resolution. With a data-sample with several tens of millions of W decays, the end result will be statistical sensitivity on $M_W$ below 1MeV and potentially in the 0.5MeV range.

Statistical errors from a single cross-section measurement near threshold ($\sqrt{s} \approx 2M_W + 0.5$GeV) are discussed in [765]. The statistical sensitivity factor on $M_W$ for an optimized single cross-section measurement assuming unpolarized beams, 100% efficiency and no backgrounds is $0.91$MeV / $\sqrt{\mathcal{L}_{\text{int}}[\text{ab}^{-1}]}$. For an integrated luminosity of $\mathcal{L}_{\text{int}} = 100 \text{ fb}^{-1}$ this translates to 2.9MeV. However experimental systematic errors on such a single cross-section measurement of $\sim 0.25\%$ enter directly and would give a corresponding 4.2MeV experimental systematic uncertainty. At the ILC, the statistical sensitivity factor can be further improved using polarized beams colliding with the appropriate helicities corresponding effectively for practical polarization values (80-90%, 40-60%) to a factor of up to three WW-production luminosity upgrade.

The method of a polarized threshold scan is discussed in some detail in [764] based on conservative extrapolations from the measurements using the LEP detectors. The idea is to use the measurement of the threshold dependence of the cross-section to determine $M_W$. The study is based on $100 \text{ fb}^{-1}$ allocated to 5 scan points near threshold and 1 scan

<footnote n="40">⁴⁰The literature from the LEP2 era usually refers to these methods as “direct reconstruction”.</footnote>
point at 170GeV. Data are collected mostly with $e_L^-, e_R^+$ but other combinations of two-beam, single-beam and no beam polarization are used to control the backgrounds and polarization systematics. The 170GeV point has little sensitivity to $M_W$ but helps to constrain the efficiency systematics. The overall experimental error on the W mass (excluding beam energy systematic and eventual theoretical errors) is estimated to be 5.2MeV.

A critical external input needed to interpret the threshold dependence of the cross-section in terms of $M_W$ is knowledge of the center-of-mass energies. Various measurements sensitive to the center-of-mass energy can be made using $e^+e^- \to \ell\ell\gamma$ ($\ell = e, \mu$) events. From knowledge of the polar angles of the leptons, under the assumption of a 3-body final state, one can measure statistically the luminosity-weighted center-of-mass energy with an error of 31 ppm for the proposed scan. This translates into a $M_W$ error of 2.5MeV per $100\text{fb}^{-1}$ polarized scan. A related method using the momenta of the two leptons (particularly the muons) can determine the center-of-mass energy with much better statistical precision. The tracker momentum scale needs to be controlled - this is feasible using Z's - and potentially with other particles with well measured masses.

In summary, it is estimated that $M_W$ can be measured to 6MeV experimental accuracy using this method which uses dedicated running near threshold. This number includes also the anticipated uncertainties from the beam energy ($\sim 1.9$MeV) and from theory ($\sim 2.5$MeV), where the corresponding theoretical issues will be discussed in the next subsection.

Much of the ILC program is likely to take place at energies significantly above the WW threshold in a regime where both WW production and single-W production are prevalent. Consequently, a direct reconstruction of the hadronic mass can be very important. One can use WW events with one W decaying leptonically ($e, \mu, \tau$) and the other decaying hadronically, and also single-W events with the W decaying hadronically to measure $M_W$ from the measured hadronic mass. Beam polarization can be used to enhance the cross-sections. The critical issue is being able to control the jet energy scale. A number of approaches are plausible and should be pursued. One approach consists of using $Z(\gamma)$ radiative return events where the Z decays hadronically and the photon is unmeasured within or close to the beam-pipe. Another approach attempts to do a jet-energy calibration from first principles using the individual components that make up the measured jet energy, namely using the calibration of the tracker momentum scale and the calorimeter energy scales at the individual particle level determined from for example calibration samples of well-known particles ($J/\psi, K_S^0, \Lambda, \pi^0$ etc.). The latter has the advantage that it does not rely directly on the Z mass. Other calibration possibilities are us-

ing ZZ, Zee and Zνν events. Assuming a sample of $5 \times 10^6$ hadronic Z's for calibration one should be able to approach a jet-energy scale related statistical error of around 2.0MeV for $M_W$. Systematic limitations in the Z based methods is the knowledge of the Z mass (currently 2.1MeV) - and any residual quark-flavor related systematics that make the detector response of hadronic W's different from hadronic Z's. It seems plausible to strive for an overall error of 5MeV from these methods.

A kinematically-constrained reconstruction of WW pairs was the work-horse of LEP2 - but has received little attention to date for ILC studies related to W mass measurement. By imposing kinematic constraints, the LEP2 experiments were able to compensate for modest jet energy resolution. At ILC, the constraints are no longer as valid (beamstrahlung) the detector resolution is much better (of the same order as $\Gamma_W$), and until recently, it seemed that the beam energy could not be determined with adequate precision at high energy. Lastly, at the order of precision that is being targeted, it seems unwise to bank on the fully hadronic channel where it is quite possible that final-state interactions such as color reconnection may cause the mass information to be corrupted. So it seems that the kinematically-constrained reconstruction method is most pertinent to the $q\bar{q}e\nu_e$ and $q\bar{q}\mu\nu_\mu$ channels.

Recent work exploring the reconstruction of the center-of-mass energy using the measured muon momenta in $e^+e^- \to \mu^+\mu^-(\gamma)$ events indicates that it is very feasible to measure the luminosity-weighted center-of-mass energy with high precision, and that this approach is promising also at relatively high center-of-mass energies.

In addition, given the impetus for potentially running the ILC at a center-of-mass energy of around 250GeV, not far above LEP2, there seems a clear potential to improve the $M_W$ measurement by including information from the leptons in the mass estimate. This lower energy regime should be the most favorable for beamstrahlung and beam-energy determination outlook. Probably by performing kinematically constrained fits that build on the existing methods one would be able to get complementary information which would be significantly uncorrelated in several of the main systematics with the direct reconstruction method. This deserves more study - but errors at the 5MeV level or less may be achievable.

To summarize, the ILC facility has three principal ways of measuring $M_W$. Each method can plausibly measure $M_W$ to a precision in the 5MeV range. The three methods are largely uncorrelated. If all three methods do live up to their promise, one can target an overall uncertainty on $M_W$ in the 3–4 MeV range.
### 4.2.2 Theory aspects concerning the WW threshold scan$^{41}$

While in the previous subsection the experimental precision for the W boson mass measurement at the LC was discussed, this subsection deals with the correspondingly required theory calculations and precisions, in particular for the WW threshold scan.

The theoretical uncertainty (TU) for the direct mass reconstruction at LEP2 has been estimated to be of the order of ~5–10MeV [766, 767], based on results of YF-SWW [768] and RACOONWW [769], which used the double-pole approximation (DPA) for the calculation of the NLO corrections. This is barely sufficient for the accuracies aimed at a LC. These shortcomings of the theoretical predictions have been cured by dedicated calculations.

In [770, 771] the total cross section for the charged-current four-fermion production processes $e^+e^- \to \nu_\tau \tau^+ \mu^- \bar{\nu}_\mu$, $u\bar{d}\mu^- \bar{\nu}_\mu$, $u\bar{d}s\bar{c}$ was presented including the complete electroweak NLO corrections and all finite-width effects. This calculation was made possible by using the complex-mass scheme for the description of the W-boson resonances and by novel techniques for the evaluation of the tensor integrals appearing in the calculation of the one-loop diagrams. The full $\mathcal{O}(\alpha)$ calculation, improved by higher-order effects from ISR, reduced the remaining TU due to unknown electroweak higher-order effects to a few 0.1% for scattering energies from the threshold region up to ~500GeV; above this energy leading high-energy logarithms, such as Sudakov logarithms, beyond one loop have to be taken into account to match this accuracy [772]. At this level of accuracy, also improvements in the treatment of QCD corrections to semileptonic and hadronic $e^+e^- \to 4f$ processes are necessary. The corrections beyond DPA, were assessed by comparing predictions in DPA from the generator RACOONWW to results from the full four-fermion calculation [770, 771], as coded in the follow-up program RACOON4F (which is not yet public). This comparison revealed effects on the total cross section without cuts of ~0.3% (0.6%) for CM energies ranging from $\sqrt{s} \sim 200$GeV (170GeV) to 500GeV. The difference to the DPA increases to 0.7–1.6% for $\sqrt{s} \sim 1–2$ TeV. At threshold, the full $\mathcal{O}(\alpha)$ calculation corrects the IBA by about 2%. While the NLO corrections beyond DPA have been calculated only for the processes $e^+e^- \to \nu_\tau \tau^+ \mu^- \bar{\nu}_\mu$, $u\bar{d}\mu^- \bar{\nu}_\mu$, $u\bar{d}s\bar{c}$ so far, the effect for the other four-fermion processes, which interfere with ZZ production, should be similar. Once the corrections to those channels are needed, they can be calculated with the available methods.

Using methods from effective field theory, the total cross section for 4-fermion production was calculated near the W pair production threshold [710, 711]. These calculations used unstable-particle effective field theory to perform an

expansion in the coupling constants, $\Gamma_W/M_W$, and in the non-relativistic velocity $v$ of the W boson up to NLO in $\Gamma_W/M_W \sim \alpha_{ew} \sim v^2$. In [710] the theoretical error of an $M_W$ determination from the threshold scan has been analysed. As a result, the resummation of next-to-leading collinear logarithms from initial-state radiation is mandatory to reduce the error on the W mass from the threshold scan below 30MeV. It was found that the remaining uncertainty of the pure NLO EFT calculation is $\delta M_W \approx 10–15$MeV and is reduced to about 5MeV with additional input from the NLO four-fermion calculation in the full theory. In order to reduce this error further, in [711] the (parametrically) dominant next-to-next-to-leading order (NNLO) corrections (all associated with the electromagnetic Coulomb attraction of the intermediate W bosons) in the EFT have been calculated leading to a shift of $\delta M_W \sim 3$GeV and to corrections to the cross section at the level of 0.3%. The effect of typical angular cuts on these corrections was shown to be completely negligible. Thus, one may conclude that the inclusive partonic four-fermion cross section near the W-pair production threshold is known with sufficient precision.

In summary, all building blocks for a sufficiently precise prediction of the W-pair production cross section in the threshold region are available. They require the combination of the NLO calculation of the full four fermion cross section with the (parametrically) dominant NNLO corrections, which are calculated within the EFT. For the precise determination of the cross section at energies above 500GeV the leading two-loop (Sudakov) corrections should be included in addition to the full NLO corrections. Combining the theoretical uncertainties with the anticipated precision from a threshold scan (see the previous subsection) a total uncertainty of 7MeV can be estimated [773].

### 4.2.3 Theory predictions for $M_W$ in the SM and MSSM$^{42}$

The precise measurement of the W boson mass can be used to test new physics models via their contribution to quantum corrections to $M_W$. However, this requires a precise prediction of $M_W$ in the respective models. Here we will concentrate on the prediction of $M_W$ in the Minimal Supersymmetric Standard Model (MSSM).

The prediction of $M_W$ in the MSSM depends on the masses, mixing angles and couplings of all MSSM particles. Sfermions, charginos, neutralinos and the MSSM Higgs bosons enter already at one-loop level and can give substantial contributions to $M_W$. Consequently, it is expected to obtain restrictions on the MSSM parameter space in the comparison of the $M_W$ prediction and the experimental value of Eq. (110).

The results for the general MSSM can be obtained in an extensive parameter scan [774]. The ranges of the various SUSY parameters are given in Tab. 26. $\mu$ is the Higgsino

<footnote>$^{41}$Ansgar Denner, Stefan Dittmaier</footnote>

<footnote>$^{42}$Sven Heinemeyer, Georg Weiglein, Lisa Zeune</footnote>
mixing parameter, $M_{\tilde{E}_i}$ denotes the soft SUSY-breaking parameter for sfermions of the $i$th family for left-handed squarks ($F = Q$), right-handed up- and down-type squarks ($F = U, D$), left-handed sleptons ($F = L$) and right-handed sleptons ($F = E$). $A_f$ denotes the trilinear sfermion-Higgs couplings, $M_3$ the gluino mass parameter and $M_2$ the SU(2) gaugino mass parameter, where the U(1) parameter is fixed as $M_1 = 5/3s_W^2/c_W^2M_2$. $M_A$ is the $\mathcal{CP}$-odd Higgs boson mass and $\tan\beta$ the ratio of the two Higgs vacuum expectation values.

<table><caption>**Table 26** Parameter ranges. All parameters with mass dimension are given in GeV.</caption><thead><tr><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>μ</td><td>-2000</td><td>2000</td></tr><tr><td>M<sub>Ē̃1,2,3</sub> = M<sub>Ē̃1,2,3</sub></td><td>100</td><td>2000</td></tr><tr><td>M<sub>Ō̃1,2</sub> = M<sub>Ō̃1,2</sub> = M<sub>Ō̃1,2</sub></td><td>500</td><td>2000</td></tr><tr><td>M<sub>Ō̃3</sub></td><td>100</td><td>2000</td></tr><tr><td>M<sub>Ō̃3</sub></td><td>100</td><td>2000</td></tr><tr><td>M<sub>Ō̃3</sub></td><td>100</td><td>2000</td></tr><tr><td>A<sub>e</sub> = A<sub>μ</sub> = A<sub>τ</sub></td><td>-3M<sub>E</sub></td><td>3M<sub>E</sub></td></tr><tr><td>A<sub>μ</sub> = A<sub>d</sub> = A<sub>c</sub> = A<sub>s</sub></td><td>-3M<sub>Ō̃12</sub></td><td>3M<sub>Ō̃12</sub></td></tr><tr><td>A<sub>b</sub></td><td>-3max(M<sub>Ō̃3</sub>, M<sub>Ō̃3</sub>)</td><td>3max(M<sub>Ō̃3</sub>, M<sub>Ō̃3</sub>)</td></tr><tr><td>A<sub>t</sub></td><td>-3max(M<sub>Ō̃3</sub>, M<sub>Ō̃3</sub>)</td><td>3max(M<sub>Ō̃3</sub>, M<sub>Ō̃3</sub>)</td></tr><tr><td>tan β</td><td>1</td><td>60</td></tr><tr><td>M<sub>3</sub></td><td>500</td><td>2000</td></tr><tr><td>M<sub>A</sub></td><td>90</td><td>1000</td></tr><tr><td>M<sub>2</sub></td><td>100</td><td>1000</td></tr></tbody></table>

All MSSM points included in the results have the neutralino as LSP and the sparticle masses pass the lower mass limits from direct searches at LEP. The Higgs and SUSY masses are calculated using FeynHiggs (version 2.9.4) [775–779]. For every point it was tested whether it is allowed by direct Higgs searches using the code HiggsBounds (version 4.0.0) [780,781]. This code tests the MSSM points against the limits from LEP, Tevatron and the LHC.<fnref n="43" />

The evaluation of $M_W$ includes the full one-loop result and all known higher order corrections of SM- and SUSY-type, for details see [774,782] and references therein. The results for $M_W$ are shown in Fig. 112 as a function of $m_t$. In the plot the green region indicated the MSSM $M_W$ prediction assuming the light $\mathcal{CP}$-even Higgs $h$ in the region $125.6 \pm 3.1 \text{GeV}$. The red band indicates the overlap region of the SM and the MSSM. The leading one-loop SUSY contributions arise from the stop bottom doublet. However requiring $M_h$ in the region $125.6 \pm 3.1 \text{GeV}$, restricts the parameters in the stop sector [783] and with it the possible $M_W$ contribution. Large $M_W$ contributions from the other MSSM

<figure><img src="image_116.png" /><figcaption>**Fig. 112** Prediction for $M_W$ as a function of $m_t$. The plot shows the $M_W$ prediction assuming the light $\mathcal{CP}$-even Higgs $h$ in the region $125.6 \pm 3.1 \text{GeV}$. The red band indicates the overlap region of the SM and the MSSM with $M_H^{SM} = 125.6 \pm 3.1 \text{GeV}$. All points are allowed by HiggsBounds. The gray ellipse indicates the current experimental uncertainty, whereas the red ellipse shows the anticipated future ILC/GigaZ precision.</figcaption></figure>

sectors are possible, if either charginos, neutralinos or sleptons are light.

The gray ellipse indicates the current experimental uncertainty, see Eqs. (110), (119), whereas the red ellipse shows the anticipated future ILC/GigaZ precision. While at the current level of precision SUSY might be considered as slightly favored over the SM by the $M_W-m_t$ measurement, no clear conclusion can be drawn. The small red ellipse, on the other hand, indicates the discrimination power of the future ILC/GigaZ measurements. With the improved precision a small part of the MSSM parameter space could be singled out. The comparison of the SM and MSSM predictions with the ILC/GigaZ precision could rule out either of models.

## 4.3 Z pole observables

Other important EWPOs are the various observables related to the Z boson, measured in four-fermion processes, $e^+e^- \to \gamma, Z \to f\bar{f}$, at the Z boson pole. We review the theoretical precision of SM predictions for various Z boson pole observables and the anticipated experimental precision at GigaZ. As for $M_W$, we also review the potential of a precise measurement and prediction of $\sin^2 \theta_{\text{eff}}^\ell$ to obtain information about the MSSM parameter space.

<footnote n="43">⁴³An updated version of HiggsBounds became available at http://higgsbounds.hepforge.org after this study was completed.</footnote>
### 4.3.1 Theoretical prospects$^{44}$

Near the Z-peak the differential cross section for $e^{+}e^{-} \to f\bar{f}$ can be written as$^{45}$

$$ \frac{d\sigma}{d\cos\theta} = \mathcal{R}_{\text{ini}} \left[ \frac{9}{2}\pi \frac{\Gamma_{ee}\Gamma_{ff}(1-\mathcal{P}_e\mathcal{A}_e)(1+\cos^2\theta) + 2(\mathcal{A}_e-\mathcal{P}_e)\mathcal{A}_f\cos\theta}{(s-M_Z^2)^2-M_Z^2\Gamma_Z^2} + \sigma_{\text{non-res}} \right], $$ 
where
$$ \Gamma_{ff} = \mathcal{R}_V^f g_V^2 + \mathcal{R}_A^f g_A^2, \quad \Gamma_Z = \sum_f \Gamma_{ff}, \quad (111) $$ 
$$ \mathcal{A}_f = 2 \frac{g_{Vf}/g_{Af}}{1+(g_{Vf}/g_{Af})^2} = \frac{1-4|Q_f|\sin^2\theta_{\text{eff}}^f}{1-4\sin^2\theta_{\text{eff}}^f+8(\sin^2\theta_{\text{eff}}^f)^2}. \quad (112) $$

Here $\Gamma_Z$ is the total Z decay width, $\Gamma_{ff}$ is the partial width for the decay $Z \to f\bar{f}$, and $g_{Vf}/g_{Af}$ are the effective vector/axial-vector couplings that mediate this decay. These effective couplings include higher-order loop corrections to the vertex, except for QED and QCD corrections to the external $f\bar{f}$ system, which are captured by the radiator functions $\mathcal{R}_V^f$ and $\mathcal{R}_A^f$. The factor $\mathcal{R}_{\text{ini}}$, on the other hand, accounts for QED radiation in the initial-state. (Specifically, as written in Eq. (111), it describes these effects relative to the final-state radiation contribution for $e^{+}e^{-}$.)

Eq. (111) explicitly spells out the leading Z-pole contribution, while additional effects from photon exchange and box corrections are included in the remainder $\sigma_{\text{non-res}}$.

The ratio of $g_{Vf}$ and $g_{Af}$ is commonly parametrized through the effective weak mixing angle $\sin^2\theta_{\text{eff}}^f$. It can be determined from the angular distribution with respect to $\cos\theta$ or from the dependence on the initial electron polarization $\mathcal{P}_e$. On the other hand, the partial and total widths are determined from the total cross section $\sigma(s)$ for different values of $s$ and from branching ratios (see the previous subsection).

For leptonic final states, the effective weak mixing angle $\sin^2\theta_{\text{eff}}^f$ has been calculated in the SM to the complete two-loop order [785–787], and 3- and 4-loop corrections of order $\mathcal{O}(\alpha\alpha_s^2)$ [788] and $\mathcal{O}(\alpha\alpha_s^3)$ [789] are also known. Furthermore, the leading $\mathcal{O}(\alpha^3)$ and $\mathcal{O}(\alpha^2\alpha_s)$ contributions for large values of $m_t$ [790] or $m_H$ [791] have been computed.

The current uncertainty from unknown higher orders is estimated to amount to about $4.5 \times 10^{-5}$ [787], which mainly stems from missing $\mathcal{O}(\alpha^2\alpha_s)$ and $\mathcal{O}(N_f^2\alpha^3, N_f^3\alpha^3)$ contributions beyond the leading $m_t^4$ and $m_t^6$ terms, respectively. (Here $N_f^n$ denotes diagrams with $n$ closed fermion loops. Based on experience from lower orders, the $\mathcal{O}(\alpha^3)$ diagrams with several closed fermion loops are expected to be dominant.) The calculation of these corrections requires three-loop vertex integrals with self-energy sub-loops and general three-loop self-energy integrals, which realistically

can be expected to be worked out in the foreseeable future. The remaining $\mathcal{O}(\alpha^3)$ and four-loop terms should amount to $\sim 10^{-5}$.⁴⁶

For quark final states, most 2-loop corrections to $\sin^2\theta_{\text{eff}}^q$ have been computed [787, 792, 793], but only the $\mathcal{O}(N_f\alpha^2)$ and $\mathcal{O}(N_f^2\alpha^2)$ contributions are known for the electroweak 2-loop corrections, while the diagrams without closed fermion loops are still missing. However, based on experience from the leptonic weak mixing angle, they are expected to amount to $\lesssim 10^{-5}$. However, the $\mathcal{O}(\alpha\alpha_s^2)$ also not known in this case, leading to an additional theory error of $\sim 2 \times 10^{-5}$. The calculation of the missing $\mathcal{O}(\alpha\alpha_s^2)$ corrections, as well as the $\mathcal{O}(\alpha^2\alpha_s)$ corrections, involves general 3-loop vertex corrections to $Z \to q\bar{q}$, which will only be possible with serious progress in calculational techniques.

When extracting $\sin^2\theta_{\text{eff}}^f$ from realistic observables (left-right (LR) and forward-backward (FB) asymmetries, see the next subsection), the initial- and final-state QED radiator functions $\mathcal{R}_i$ must be taken into account. In general, the QED corrections are known to $\mathcal{O}(\alpha)$ for the differential cross section and to $\mathcal{O}(\alpha^2)$ for the integrated cross section (see Ref. [794] for a summary). However, for the LR asymmetry they complete cancel up to NNLO [795], while for the FB asymmetry they cancel if hard photon contributions are excluded, i.e. they cancel up to terms of order $E_\gamma/\sqrt{s}$ [795–797]. Therefore, a sufficiently precise result for the soft-photon contribution with $E_\gamma < E_\gamma^{\text{cut}}$ can be obtained using existing calcations for small enough $E_\gamma^{\text{cut}}$, while the hard-photon contribution ($E_\gamma > E_\gamma^{\text{cut}}$) can be evaluated with numerical Monte-Carlo methods. A similar procedure can be carried out for final-state QCD effects for $\sin^2\theta_{\text{eff}}^q$ although the corrections beyond NLO are not fully implemented in existing programs (see below).

For the branching fraction $R_b = \Gamma_b/\Gamma_{\text{had}}$ and the total width $\Gamma_Z$, two-loop corrections of $\mathcal{O}(\alpha\alpha_s)$, $\mathcal{O}(N_f\alpha^2)$, and $\mathcal{O}(N_f^2\alpha^2)$ are known [793, 798, 799]. Assuming geometric progression of the perturbative series, the remaining higher-order contributions are estimated to contribute at the level of $\sim 1.5 \times 10^{-4}$ and 0.5 MeV, respectively. As before, the contribution from electroweak two-loop diagrams without closed fermion loops is expected to be small. The dominant missing contributions are the same as for $\sin^2\theta_{\text{eff}}^q$.

The current status of the theoretical calculations and prospects for the near future are summarized in Tab. 27. Note that $\sigma_{\text{non-res}}$ is suppressed by $\Gamma_Z/M_Z$ compared to the leading pole term, so that the known one-loop corrections are sufficient to reach NNLO precision at the Z pole.

The known corrections to the effective weak mixing angles and the leading corrections to the partial widths are implemented in programs such as Zfitter [794, 800] and Gfitter [801] (see also Sect. 4.5), while the incorporation

<footnote>$^{44}$Ayres Freitas</footnote>

<footnote>$^{45}$For a review, see [784].</footnote>

<footnote>$^{46}$This estimate can be made more precise only after aforementioned calculations have been completed.</footnote>
<table><caption>**Table 27** Some of the most important precision observables for Z-boson production and decay (first column), their present-day estimated theory error (second column), the dominant missing higher-order corrections (third column), and the estimated improvement when these corrections are available (fourth column). In many cases, the leading parts in a large-mass expansion are already known, in which case the third column refers to the remaining pieces at the given order. The numbers in the last column are rough order-of-magnitude guesses. Entries in [italics] indicate contributions that probably will require very significant improvements in calculational techniques to be completed.</caption><thead><tr><th>Quantity</th><th>Cur. theo. error</th><th>Lead. missing terms</th><th>Est. improvem.</th></tr></thead><tbody><tr><td>sin<sup>2</sup> θ<sub>eff</sub><sup>l</sup></td><td>4.5 × 10<sup>-5</sup></td><td>σ(α<sup>2</sup>α<sub>s</sub>), σ(N<sub>f</sub><sup>≥2</sup>α<sup>3</sup>)</td><td>factor 3–5</td></tr><tr><td>sin<sup>2</sup> θ<sub>eff</sub><sup>l</sup></td><td>5 × 10<sup>-5</sup></td><td>σ(α<sup>2</sup>), σ(N<sub>f</sub><sup>≥2</sup>α<sup>3</sup>)<br>[σ(αα<sub>s</sub><sup>2</sup>), σ(α<sup>2</sup>α<sub>s</sub>)]</td><td>factor 1–1.5<br>[factor 3–5]</td></tr><tr><td>R<sub>b</sub></td><td>~ 1.5 × 10<sup>-4</sup></td><td>σ(α<sup>2</sup>), σ(N<sub>f</sub><sup>≥2</sup>α<sup>3</sup>)<br>[σ(αα<sub>s</sub><sup>2</sup>), σ(α<sup>2</sup>α<sub>s</sub>)]</td><td>factor 1–2<br>[factor 3–5]</td></tr><tr><td>Γ<sub>Z</sub></td><td>0.5 MeV</td><td>σ(α<sup>2</sup>), σ(N<sub>f</sub><sup>≥2</sup>α<sup>3</sup>)<br>[σ(αα<sub>s</sub><sup>2</sup>), σ(α<sup>2</sup>α<sub>s</sub>)]</td><td>factor 1–2<br>[factor 3–5]</td></tr></tbody></table>

of the recent full fermionic two-loop corrections is in progress. However, these programs are based on a framework designed for NLO but not NNLO corrections. In particular, there are mismatches between the electroweak NNLO corrections to the Zf$\bar{f}$ vertices and QED/QCD corrections to the external legs due to approximations and factorization assumptions. Another problem is the separation of leading and sub-leading pole terms in Eq. (111) [787]. While these discrepancies may be numerically small, it would be desirable to construct a new framework that treats the radiative corrections to Z-pole physics systematically and consistently at the NNLO level and beyond. Such a framework can be established based on the pole scheme [802], where the amplitude is expanded about the complex pole $s = M_Z^2 - iM_Z\Gamma_Z$, with the power counting $\Gamma_Z/M_Z \sim \alpha$.

### 4.3.2 Experimental prospects$^{47}$

The effective weak mixing angle $\sin^2 \theta_{\text{eff}}^l$ can be measured at a linear collider running at the Z-mass using the left-right asymmetry [803]. With at least the electron beam polarised with a polarisation of $\mathcal{P}$, $\sin^2 \theta_{\text{eff}}^l$ can be obtained via

$$ 
\begin{aligned}
A_{LR} &= \frac{1}{\mathcal{P}} \frac{\sigma_L - \sigma_R}{\sigma_L + \sigma_R} && (113) \\
&= \mathcal{A}_e \\
&= \frac{2g_{V_e}g_{A_e}}{g_{V_e}^2 + g_{A_e}^2} \\
g_{V_e}/g_{A_e} &= 1 - 4\sin^2 \theta_{\text{eff}}^l
\end{aligned}
$$

independent of the final state. With $10^9$ Z's, an electron polarisation of 80% and no positron polarisation the statistical error is $\Delta A_{LR} = 4 \cdot 10^{-5}$. The error from the polarisation

measurement is $\Delta A_{LR}/A_{LR} = \Delta \mathcal{P}/\mathcal{P}$. With electron polarisation only and $\Delta \mathcal{P}/\mathcal{P} = 0.5\%$ one has $\Delta A_{LR} = 8 \cdot 10^{-4}$, much larger than the statistical precision. If also positron polarisation is available $\mathcal{P}$ in equation (113) has to be replaced by $\mathcal{P}_{\text{eff}} = \frac{\mathcal{P}_{e^+} + \mathcal{P}_{e^-}}{1 + \mathcal{P}_{e^+} \mathcal{P}_{e^-}}$. For $\mathcal{P}_{e^-}(\mathcal{P}_{e^+}) = 80\%(60\%)$, due to error propagation, the error in $\mathcal{P}_{\text{eff}}$ is a factor of three to four smaller than the error on $\mathcal{P}_{e^+}$, $\mathcal{P}_{e^-}$ depending on the correlation between the two measurements. If one takes, however, data on all four polarisation combinations the left-right asymmetry can be extracted without absolute polarimetry [804] and basically without increasing the error if the positron polarisation is larger than 50%. Polarimetry, however, is still needed for relative measurements like the difference of absolute values of the positive and the negative helicity states. Assuming conservatively $\Delta A_{LR} = 10^{-4}$ leads to $\Delta \sin^2 \theta_{\text{eff}}^l = 0.000013$, more than a factor 10 better than the LEP/SLD result.

The largest possible uncertainty comes from the knowledge of the beam energy. $\sqrt{s}$ must be known with 1 MeV relative to the Z-mass. The absolute precision can be calibrated in a Z-scan, however a spectrometer with a relative precision of $10^{-5}$ is needed not to be dominated by this uncertainty. Similarly the beamstrahlung must be known to a few percent relative between the calibration scans and the pole running. However both requirements seem to be possible.

Apart from $\sin^2 \theta_{\text{eff}}^l$ also some other Z-pole observables can be measured at a LC. Running at the Z peak gives access to the polarised forward backward asymmetry for b-quarks which measures $\sin^2 \theta_{\text{eff}}^b$ and the ratio of the b- to the hadronic partial width of the Z-boson $R_b^0 = \Gamma_{b\bar{b}}/\Gamma_{\text{had}}$. Both quantities profit from the large statistics and the much improved b-tagging capabilities of an ILC detector compared to LEP.

$R_b^0$ can be measured using the same methods as at LEP. The statistical error will be almost negligible and the systematic errors shrink due to the better b-tagging. In total $\Delta R_b^0 = 0.00014$ can be reached which is an improvement of a factor five compared to the present value [803].

$\sin^2 \theta_{\text{eff}}^b$ can be measured from the left-right-forward-backward asymmetry for b-quarks, $A_{\text{FB,LR}}^b = 3/4 \mathcal{P} \mathcal{A}_b \cdot \mathcal{A}_b$ depends on $\sin^2 \theta_{\text{eff}}^b$ as shown in eq. (113), however in general one has $g_{V_f}/g_{A_f} = 1 - 4q_f \sin^2 \theta_{\text{eff}}^f$ and due to the small b-charge the dependence is very weak. At present $\sin^2 \theta_{\text{eff}}^b$ is known with a precision of 0.016 from $A_{\text{FB,LR}}^b$ measured at the SLC and the forward-backward asymmetries for b-quarks at LEP combined with $\sin^2 \theta_{\text{eff}}^b$ measurements at LEP and SLC [805]. Using the left-right-forward-backward asymmetry only at the ILC an improvement by more than a factor 10 seems realistic [803].

The total Z-width $\Gamma_Z$ can be obtained from a scan of the resonance curve. The statistical error at GigaZ will be

<footnote>$^{47}$Klaus Moenig</footnote>
negligible and the systematic uncertainty will be dominated by the precision of the beam energy and the knowledge of beamstrahlung. If a spectrometer with a precision of $10^{-5}$ can be built, $\Gamma_Z$ can be measured with 1 MeV accuracy [803]. However no detailed study on the uncertainty due to beamstrahlung exists.

### 4.3.3 Constraints to the MSSM from $\sin^2 \theta^\ell \text{eff}$ $^{48}$

As for $M_W$ we review examples showing how the MSSM parameter space could be constrained by a precise measurement of $\sin^2 \theta^\ell \text{eff}$. We also discuss the relevance of this measurement in a combined $M_W-\sin^2 \theta^\ell \text{eff}$ analysis.

In the first example it is investigated whether the high accuracy achievable at the GigaZ option of the LC would provide sensitivity to indirect effects of SUSY particles even in a scenario where the (strongly interacting) superpartners are so heavy that they escape detection at the LHC [806].

We consider in this context a scenario with very heavy squarks and a very heavy gluino. It is based on the values of the SPS 1a' benchmark scenario [807], but the squark and gluino mass parameters are fixed to 6 times their SPS 1a' values. The other masses are scaled with a common scale factor except $M_A$ which we keep fixed at its SPS 1a' value. In this scenario the strongly interacting particles are too heavy to be detected at the LHC, while, depending on the scale-factor, some colour-neutral particles may be in the LC reach. In Fig. 113 we show the prediction for $\sin^2 \theta^\ell \text{eff}$ in this SPS 1a' inspired scenario as a function of the lighter chargino mass, $m_{\tilde{\chi}_1^\pm}$. The prediction includes the parametric uncertainty, $\sigma^{\text{para-LC}}$, induced by the LC measurement of $m_t$, $\delta m_t = 100$MeV (see Sect. 3), and the numerically more relevant prospective future uncertainty on $\Delta\alpha_{\text{had}}^{(5)}$, $\delta(\Delta\alpha_{\text{had}}^{(5)}) = 5 \times 10^{-5}$. The MSSM prediction for $\sin^2 \theta^\ell \text{eff}$ is compared with the experimental resolution with GigaZ precision, $\sigma^{\text{LC}} = 0.000013$, using for simplicity the current experimental central value. The SM prediction (with $M_H^{\text{SM}} = M_h^{\text{MSSM}}$) is also shown, applying again the parametric uncertainty $\sigma^{\text{para-LC}}$.

Despite the fact that no coloured SUSY particles would be observed at the LHC in this scenario, the LC with its high-precision measurement of $\sin^2 \theta^\ell \text{eff}$ in the GigaZ mode could resolve indirect effects of SUSY up to $m_{\tilde{\chi}_1^\pm} \lesssim 500$GeV. This means that the high-precision measurements at the LC with GigaZ option could be sensitive to indirect effects of SUSY even in a scenario where SUSY particles have neither been directly detected at the LHC nor the first phase of the LC with a centre of mass energy of up to 500GeV.

We now analyse the sensitivity of $\sin^2 \theta^\ell \text{eff}$ together with $M_W$ to higher-order effects in the MSSM by scanning over

<figure><img src="image_117.png" /><figcaption>**Fig. 113** Theoretical prediction for $\sin^2 \theta^\ell \text{eff}$ in the SM and the MSSM (including prospective parametric theoretical uncertainties) compared to the experimental precision at the LC with GigaZ option. An SPS 1a' inspired scenario is used, where the squark and gluino mass parameters are fixed to 6 times their SPS 1a' values. The other mass parameters are varied with a common scalefactor.</figcaption></figure>

a broad range of the SUSY parameter space. The following SUSY parameters are varied independently of each other in a random parameter scan within the given range:

sleptons: $M_{\tilde{L}_{1,2,3}, \tilde{E}_{1,2,3}} = 100...2000$GeV,
light squarks: $M_{\tilde{Q}_{1,2}, \tilde{U}_{1,2}, \tilde{D}_{1,2}} = 100...2000$GeV,

$\tilde{t}/\tilde{b}$ doublet: $M_{\tilde{Q}_3, \tilde{U}_3, \tilde{D}_3} = 100...2000$GeV,
$A_{\tau,t,b} = -2000...2000$GeV,

gauginos: $M_{1,2} = 100...2000$GeV, (114)
$m_{\tilde{g}} = 195...1500$GeV,
$\mu = -2000...2000$GeV,

Higgs: $M_A = 90...1000$GeV,
$\tan \beta = 1.1...60$. (115)

Only the constraints on the MSSM parameter space from the LEP Higgs searches [294, 808] and the lower bounds on the SUSY particle masses previous to the LHC SUSY searches were taken into account. However, the SUSY particles strongly affected by the LHC searches are the squarks of the first and second generation and the gluino. Exactly these particles, however, have a very small effect on the prediction of $M_W$ and $\sin^2 \theta^\ell \text{eff}$ and thus a negligible effect on this analysis.

In Fig. 114 we compare the SM and the MSSM predictions for $M_W$ and $\sin^2 \theta^\ell \text{eff}$ as obtained from the scatter data. The predictions within the two models give rise to two bands in the $M_W-\sin^2 \theta^\ell \text{eff}$ plane with only a relatively small overlap region (indicated by a dark-shaded (blue) area). The parameter region shown in the SM (the medium-shaded (red) and dark-shaded (blue) bands) arises from varying the mass of the SM Higgs boson, from $M_H^{\text{SM}} = 114$GeV, the old LEP exclusion bound [808] (lower edge of the dark-

<footnote>$^{48}$Sven Heinemeyer, Georg Weiglein</footnote>
<figure><img src="image_118.png" /><figcaption>**Fig. 114** MSSM parameter scan for $M_W$ and $\sin^2 \theta^\ell_{\text{eff}}$ over the ranges given in Eq. (115) with $m_t = 165...175$ GeV. Todays 68% C.L. ellipses (from $A_{\text{FB}}^b$ (LEP), $A_{\text{LR}}^e$ (SLD) and the world average) are shown as well as the anticipated GigaZ/MegaW precisions, drawn around today's central value</figcaption></figure>

shaded (blue) area), to 400 GeV (upper edge of the medium-shaded (red) area), and from varying $m_t$ in the range of $m_t = 165...175$ GeV. The value of $M_H^{SM} \sim 125.5$ GeV corresponds roughly to the dark-shaded (blue) strip. The light shaded (green) and the dark-shaded (blue) areas indicate allowed regions for the unconstrained MSSM, where no restriction on the light $\mathcal{CP}$-even Higgs mass has been applied. The decoupling limit with SUSY masses, in particular of scalar tops and bottoms, of $\mathcal{O}(2 \text{ TeV})$ yields the upper edge of the dark-shaded (blue) area. Including a Higgs mass measurement into the MSSM scan would cut out a small part at the lower edge of the light shaded (green) area.

The 68% C.L. experimental results for $M_W$ and $\sin^2 \theta^\ell_{\text{eff}}$ are indicated in the plot. The center ellipse corresponds to the current world average given in Eq. (118). Also shown are the error ellipses corresponding to the two individual most precise measurements of $\sin^2 \theta^\ell_{\text{eff}}$, based on $A_{\text{LR}}^e$ by SLD and $A_{\text{FB}}^b$ by LEP, corresponding to

$$A_{\text{FB}}^{b}(\text{LEP}) : \sin^2 \theta_{\text{eff}}^{\text{exp,LEP}} = 0.23221 \pm 0.00029, \quad (116)$$

$$A_{\text{LR}}^{e}(\text{SLD}) : \sin^2 \theta_{\text{eff}}^{\text{exp,SLD}} = 0.23098 \pm 0.00026, \quad (117)$$

$$\sin^2 \theta_{\text{eff}}^{\text{exp,aver.}} = 0.23153 \pm 0.00016, \quad (118)$$

where the latter one represents the average [17]. The first (second) value prefers a value of $M_H^{SM} \sim 32(437)$GeV [809]. The two measurements differ by more than $3\sigma$. The averaged value of $\sin^2 \theta^\ell_{\text{eff}}$, as given in Eq. (118), prefers $M_H^{SM} \sim 110$ GeV [809]. The anticipated improvement with

the GigaZ/MegaW options (the latter one denoting the WW threshold scan, see Sect. 4.2), indicated as small ellipse, is shown around the current experimental central data. One can see that the current averaged value is compatible with the SM with $M_H^{SM} \sim 125.5$ GeV and with the MSSM. The value of $\sin^2 \theta^\ell_{\text{eff}}$ obtained from $A_{\text{LR}}^e$ (SLD) clearly favors the MSSM over the SM. On the other hand, the value of $\sin^2 \theta^\ell_{\text{eff}}$ obtained from $A_{\text{FB}}^b$ (LEP) together with the $M_W$ data from LEP and the Tevatron would correspond to an experimentally preferred region that deviates from the predictions of both models. This unsatisfactory solution can only be resolved by new measurements, where the Z factory, i.e. the GigaZ option would be an ideal solution. Thus, the unclear experimental situation regarding the two single most precise measurements entering the combined value for $\sin^2 \theta^\ell_{\text{eff}}$ has a significant impact on the constraints that can be obtained from this precision observable on possible New Physics scenarios. Measurements at a new $e^+e^-$ Z factory, which could be realized in particular with the GigaZ option of the ILC, would be needed to resolve this issue. As indicated by the solid light shaded (red) ellipse, the anticipated GigaZ/MegaW precision of the combined $M_W - \sin^2 \theta^\ell_{\text{eff}}$ measurement could put severe constraints on each of the models and resolve the discrepancy between the $A_{\text{FB}}^b$ (LEP) and $A_{\text{LR}}^e$ (SLD) measurements. If the central value of an improved measurement with higher precision should turn out to be close to the central value favored by the current measurement of $A_{\text{FB}}^b$ (LEP), this would mean that the electroweak precision observables $M_W$ and $\sin^2 \theta^\ell_{\text{eff}}$ could rule out both the SM and the most general version of the MSSM.

## 4.4 The relevance of the top quark mass$^{49}$

The mass of the top quark, $m_t$, is a fundamental parameter of the electroweak theory. It is by far the heaviest of all quark masses and it is also larger than the masses of all other known fundamental particles. For details on the experimental determination of $m_t$, see Sect. 3.4.1. The top quark is deeply connected to many other issues of high-energy physics:

- The top quark could play a special role in/electroweak symmetry breaking.

- The experimental uncertainty of $m_t$ induces the largest parametric uncertainty in the prediction for electroweak precision observables [758, 810] and can thus obscure new physics effects.

- In SUSY models the top quark mass is an important input parameter and is crucial for radiative electroweak symmetry breaking and unification.

- Little Higgs models contain “heavier tops”.

<footnote>$^{49}$Sven Heinemeyer and Georg Weiglein</footnote>
The large value of $m_t$ gives rise to a large coupling between the top quark and the Higgs boson and is furthermore important for flavor physics. It could therefore provide a window to new physics. (The correct prediction of $m_t$ will be a crucial test for any fundamental theory.) The top-quark mass also plays an important role in electroweak precision physics, as a consequence in particular of non-decoupling effects being proportional to powers of $m_t$. A precise knowledge of $m_t$ is therefore indispensable in order to have sensitivity to possible effects of new physics in electroweak precision tests.

The current world average for the top-quark mass from the measurement at the Tevatron and the LHC is [811],

$$m_t = 173.34 \pm 0.76 \text{GeV}. \qquad (119)$$

The prospective accuracy at the LHC is $\delta m_t^{\text{exp}} \approx 500$ MeV [390], while at the ILC a very precise determination of $m_t$ with an accuracy of $\delta m_t^{\text{exp}} \lesssim 100$ MeV will be possible, see Sect. 3.4.1. This uncertainty contains both the experimental error of the mass parameter extracted from the $t\bar{t}$ threshold measurements at the ILC and the envisaged theoretical uncertainty from its transition into a suitable short-distance mass (like the $\overline{\text{MS}}$ mass).

The relevance of the $m_t$ precision as parametric uncertainty has been discussed for the W boson mass, $M_W$, in Sect. 4.2, and for the effective leptonic weak mixing angle, $\sin^2 \theta^\ell_{\text{eff}}$, in Sect. 4.3.

Because of its large mass, the top quark is expected to have a large Yukawa coupling to Higgs bosons, being proportional to $m_t$. In each model where the Higgs boson mass is not a free parameter but predicted in terms of the other model parameters (as e.g. in the MSSM), the diagram in Fig. 115 contributes to the Higgs mass. This diagram gives rise to a leading $m_t$ contribution of the form

$$\Delta M_H^2 \sim G_F N_C C m_t^4, \qquad (120)$$

where $G_F$ is the Fermi constant, $N_C$ is the color factor, and the coefficient $C$ depends on the specific model. Thus the experimental error of $m_t$ necessarily leads to a parametric error in the Higgs boson mass evaluation.

<figure><img src="image_119.png" /><figcaption>Fig. 115 Loop contribution of the top quark to the Higgs boson mass.</figcaption></figure>

Taking the MSSM as a specific example (including also the scalar top contributions and the appropriate renormalization) $N_C C$ is given for the light $\mathcal{CP}$-even Higgs boson mass in leading logarithmic approximation by

$$N_C C = \frac{3}{\sqrt{2} \pi^2 \sin^2 \beta} \log \left( \frac{m_{t_1} m_{t_2}}{m_t^2} \right). \qquad (121)$$

Here $m_{t_{1,2}}$ denote the two masses of the scalar tops. The current precision of $\delta m_t \sim 1$ GeV leads to an uncertainty of $\sim 2.5\%$ in the prediction of $M_H$, while the ILC will yield a precision of $\sim 0.2\%$. These uncertainties have to be compared with the anticipated precision of the future Higgs boson mass measurements. With a precision of $\delta M_H^{\text{exp,LHC}} \approx 0.2$ GeV the relative precision is at the level of $\sim 0.2\%$. It is apparent that only the LC precision of $m_t$ will yield a parametric error small enough to allow a precise comparison of the Higgs boson mass prediction and its experimental value.

Another issue that has to be kept in mind here (in SUSY as in any other model predicting $M_H$) is the intrinsic theoretical uncertainty due to missing higher-order corrections. Within the MSSM currently the uncertainty for the lightest $\mathcal{CP}$-even Higgs is estimated to $\delta M_H^{\text{intr,today}} \approx 2-3$ GeV [758, 776]$^{50}$. In the future one can hope for an improvement down to $\lesssim 0.5$ GeV or better [758], i.e. with sufficient effort on higher-order corrections it should be possible to reduce the intrinsic theoretical uncertainty to the level of $\delta M_H^{\text{exp,LHC}}$.

Confronting the theoretical prediction of $M_H$ with a precise measurement of the Higgs boson mass constitutes a very sensitive test of the MSSM (or any other model that predicts $M_H$), which allows one to obtain constraints on the model parameters. However, the sensitivity of the $M_H$ measurement cannot directly be translated into a prospective indirect determination of a single model parameter. In a realistic situation the anticipated experimental errors of all relevant SUSY parameters have to be taken into account. For examples including these parametric errors see Refs. [433, 810].

## 4.5 Prospects for the electroweak fit to the SM Higgs mass$^{51}$

The global fit to electroweak precision data allows among other constraints to extract information on the Higgs mass from Higgs loops modifying the values of Z boson asymmetry observables and the W mass [17, 762, 812-814]. Assuming the new boson discovered by the ATLAS [54] and CMS [55] experiments at the LHC to be the SM Higgs boson, the electroweak fit is overconstrained and can be used to quantify the compatibility of the mass (and couplings) of the discovered boson with the electroweak precision data

<footnote>$^{50}$We are not aware of any such estimate in other New Physics models.</footnote>

<footnote>$^{51}$Andreas Hoecker, Roman Kogler, Martin Grünewald</footnote>
in an overall goodness-of-fit measure. Similarly, it allows to confront indirect determinations of the W boson mass, the effective weak mixing angle predicting the Z asymmetries, and the top quark mass with the measurements. The LHC and a next generation electron-positron collider have the potential to significantly increase the precision of most of the observables that are relevant to the fit. This section reports on a prospective study of the electroweak fit following the approach published in earlier works by the Gfitter group [814–816] (and compares briefly to a corresponding fit from the LEPEWWG).

For the study aiming at a comparison of the accuracies of the measured and predicted electroweak observables, the central values of the input observables are chosen to agree with the SM prediction for a Higgs mass of 125.8 GeV. Total experimental uncertainties of 6 MeV for $M_W$, $1.3 \cdot 10^{-5}$ for $\sin^2\theta_{\text{eff}}^\ell$, $4 \cdot 10^{-3}$ for $R_\ell^0$, and 100 MeV for $m_t$ (interpreted as pole mass) are used. The exact achieved precision on the Higgs mass is irrelevant for this study. For the hadronic contribution to the running of the QED fine structure constant at the Z pole, $\Delta\alpha_{\text{had}}^{(5)}(M_Z^2)$, an uncertainty of $4.7 \cdot 10^{-5}$ is assumed (compared to the currently used uncertainty of $10 \cdot 10^{-5}$ [816,817]), which benefits below the charm threshold from the completion of BABAR analyses and the ongoing program at VEPP-2000, and at higher energies from improved charmonium resonance data from BES-3, and a better knowledge of $\alpha_s$ from the $R_\ell^0$ measurement and reliable lattice QCD predictions. The other input observables to the electroweak fit are taken to be unchanged from the current settings [816].

For the theoretical predictions, the calculations detailed in [814] and references therein are used. They feature among others the complete $\mathcal{O}(\alpha_s^4)$ calculation of the QCD Adler function [600,601] and the full two-loop and leading beyond-two-loop prediction of the W mass and the effective weak mixing angle [786, 818, 819]. An improved prediction of $R_b^0$ is invoked that includes the calculation of the complete fermionic electroweak two-loop (NNLO) corrections based on numerical Mellin-Barnes integrals [820]. The calculation of the vector and axial-vector couplings in Gfitter relies on accurate parametrisations [821–824].

The most important theoretical uncertainties in the fit are those affecting the $M_W$ and $\sin^2\theta_{\text{eff}}$ predictions. They arise from three dominant sources of unknown higher-order corrections: $\mathcal{O}(\alpha^2\alpha_s)$ terms beyond the known contribution of $\mathcal{O}(G_F^2\alpha_s m_t^4)$, $\mathcal{O}(\alpha^3)$ electroweak three-loop corrections, and $\mathcal{O}(\alpha_s^3)$ QCD terms, see Sect. 4.3.1. The quadratic sums of the above corrections amount to $\delta_{\text{th}}M_W = 4$ MeV and $\delta_{\text{th}}\sin^2\theta_{\text{eff}}^\ell = 4.7 \cdot 10^{-5}$, which are the theoretical ranges used in present electroweak fits. We assume in the following that theoretical developments have let to improved uncertainties of $\delta_{\text{th}}M_W = 2$ MeV and $\delta_{\text{th}}\sin^2\theta_{\text{eff}}^\ell = 1.5 \cdot 10^{-5}$, see Tab. 28. Within the Rfit scheme employed here [825, 826],

theoretical uncertainties are treated as uniform likelihoods in the fit, corresponding to an allowed offset from the predicted value within the defined range (we discuss the difference with respect to standard Gaussian theoretical uncertainties below).

<table><thead><tr><th>Parameter</th><th>Input value</th><th>Free in fit</th><th>Predicted fit result</th></tr></thead><tbody><tr><td>$M_H$ [GeV]</td><td>125.8 ± 0.1</td><td>yes</td><td>125.0<sup>+12</sup><sub>-10</sub></td></tr><tr><td>$M_W$ [GeV]</td><td>80.378 ± 0.006</td><td>–</td><td>80.361 ± 0.005</td></tr><tr><td>$\Gamma_W$ [GeV]</td><td>–</td><td>–</td><td>2.0910 ± 0.0004</td></tr><tr><td>$M_Z$ [GeV]</td><td>91.1875 ± 0.0021</td><td>yes</td><td>91.1878 ± 0.0046</td></tr><tr><td>$\Gamma_Z$ [GeV]</td><td>–</td><td>–</td><td>2.4953 ± 0.0003</td></tr><tr><td>$\sigma_{\text{had}}^0$ [nb]</td><td>–</td><td>–</td><td>41.479 ± 0.003</td></tr><tr><td>$R_b^0$</td><td>20.742 ± 0.003</td><td>–</td><td>–</td></tr><tr><td>$A_{\text{FB}}^{0,\ell}$</td><td>–</td><td>–</td><td>0.01622 ± 0.00002</td></tr><tr><td>$A_\ell$</td><td>–</td><td>–</td><td>0.14706 ± 0.00010</td></tr><tr><td>$\sin^2\theta_{\text{eff}}^\ell$</td><td>0.231385 ± 0.000013</td><td>–</td><td>0.23152 ± 0.00004</td></tr><tr><td>$A_c$</td><td>–</td><td>–</td><td>0.66791 ± 0.00005</td></tr><tr><td>$A_b$</td><td>–</td><td>–</td><td>0.93462 ± 0.00002</td></tr><tr><td>$A_{\text{FB}}^{0,\ell}$</td><td>–</td><td>–</td><td>0.07367 ± 0.00006</td></tr><tr><td>$A_{\text{FB}}^{0,\ell}$</td><td>–</td><td>–</td><td>0.10308 ± 0.00007</td></tr><tr><td>$R_b^0$</td><td>–</td><td>–</td><td>0.17223 ± 0.00001</td></tr><tr><td>$R_b^0$</td><td>–</td><td>–</td><td>0.214746 ± 0.000004</td></tr><tr><td>$\bar{m}_c$ [GeV]</td><td>1.27<sup>+0.07</sup><sub>-0.11</sub></td><td>yes</td><td>–</td></tr><tr><td>$\bar{m}_b$ [GeV]</td><td>4.20<sup>+0.07</sup><sub>-0.11</sub></td><td>yes</td><td>–</td></tr><tr><td>$m_t$ [GeV]</td><td>173.18 ± 0.10</td><td>yes</td><td>173.3 ± 1.2</td></tr><tr><td>$\Delta\alpha_{\text{had}}^{(5)}(M_Z^2)$ (Δ)</td><td>2757.0 ± 4.7</td><td>yes</td><td>2757 ± 10</td></tr><tr><td>$\alpha_s(M_Z^2)$</td><td>–</td><td>yes</td><td>0.1190 ± 0.0005</td></tr><tr><td>$\delta_{\text{th}}M_W$ [MeV]</td><td>[-2.0, 2.0]<sub>theo</sub></td><td>yes</td><td>–</td></tr><tr><td>$\delta_{\text{th}}\sin^2\theta_{\text{eff}}^\ell$ (∇)</td><td>[-1.5, 1.5]<sub>theo</sub></td><td>yes</td><td>–</td></tr></tbody></table>

($\Delta$) In units of $10^{-5}$. (∇) Rescaled due to $\alpha_s$ dependency.

<caption>**Table 28** Input values and fit results for the observables and parameters of the global electroweak fit in a hypothetical future scenario. The first and second columns list respectively the observables/parameters used in the fit, and their experimental values or phenomenological estimates (see text for references). The subscript “theo” labels theoretical error ranges. The third column indicates whether a parameter is floating in the fit and in the fourth column the fit results are given without using the corresponding experimental or phenomenological estimate in the given row.</caption>

Table 28 gives the input observables and values used (first and second columns) and the predictions obtained from the fit to all input data except for the one that is predicted in a given row (last column). It allows to compare the accuracy of direct and indirect determinations. To simplify the numerical exercise the Z-pole asymmetry observables are combined into a single input $\sin^2\theta_{\text{eff}}^\ell$, while for the reader’s convenience the fit predictions are provided for all observables.

The indirect prediction of the Higgs mass at 125 GeV achieves an uncertainty of $^{+12}_{-10}$ GeV. For $M_W$ the prediction with an estimated uncertainty of 5 MeV is similarly accurate as the (assumed) measurement, while the prediction of $\sin^2\theta_{\text{eff}}^\ell$ with an uncertainty of $4 \cdot 10^{-5}$ is three times less accurate than the experimental precision.
<figure><img src="image_120.png" /></figure>

**Fig. 116** $\Delta\chi^2$ profiles as a function of the Higgs mass for electroweak fits compatible with an SM Higgs boson of mass 125.8 GeV (left) and 94 GeV (right), respectively. The measured Higgs boson mass is not used as input in the fit. The grey bands show the results obtained using present uncertainties [816], and the yellow bands indicate the results for the hypothetical future scenario given in Table 28 (left plot) and corresponding input data shifted to accommodate a 94 GeV Higgs boson but unchanged uncertainties (right plot). The right axes depict the corresponding Gaussian 'sigma' lines. The thickness of the bands indicates the effect from the theoretical uncertainties treated according to the Rfit prescription. The long-dashed line in each plot shows the curves one would obtain when treating the theoretical uncertainties in a Gaussians manner just like any other uncertainty in the fit.

The fit would therefore particularly benefit from additional experimental improvement in $M_W$. It is interesting to notice that the accuracy of the indirect determination of the top mass (1.2 GeV) becomes similar to that of the present experimental determination. An improvement beyond, say, 200 MeV uncertainty cannot be exploited by the fit. The input values of $M_Z$ and $\Delta\alpha_{\text{had}}^{(5)}(M_Z^2)$ are twice more accurate than the fit predictions, which is sufficient to not limit the fit but further improvement would certainly be useful.

Keeping the present theoretical uncertainties in the prediction of $M_W$ and $\sin^2\theta_{\text{eff}}^\ell$ would worsen the accuracy of the $M_H$ prediction to $^{+20}_{-17}$ GeV, whereas neglecting theoretical uncertainties altogether would improve it to $\pm 7$ GeV. This emphasises the importance of the required theoretical work.

<figure><img src="image_121.png" /></figure>

**Fig. 117** $\Delta\chi^2$ profiles as a function of the Higgs mass for electroweak fits compatible with an SM Higgs boson with mass 94 GeV using the LEPEWWG approach [17]. The blue (pink) parabola shows the current (future) fit (see text).

Profiles of $\Delta\chi^2$ as a function of the Higgs mass for present and future electroweak fits compatible with an SM Higgs boson of mass 125.8 GeV and 94 GeV, respectively, are shown in Fig. 116 (see caption for a detailed description). The measured Higgs boson mass is not used as input in these fits. If the experimental input data, currently predicting $M_H = 94^{+25}_{-22}$ GeV [816], were left unchanged with respect to the present values, but had uncertainties as in Table 28, a deviation of the measured $M_H$ exceeding $4\sigma$ could be established with the fit (see right-hand plot in Fig. 116). Such a conclusion does not strongly depend on the treatment of the theoretical uncertainties (Rfit versus Gaussian) as can be seen by comparison of the solid yellow and the long-dashed yellow $\Delta\chi^2$ profiles.

A similar result has also been obtained by the LEPEWWG, as can be seen in Fig. 117 [17]. The $\Delta\chi^2$ profile of their fit is shown as a function of the Higgs mass. The blue band shows the current result with a best fit point at $\sim 94$ GeV with an uncertainty of $\sim \pm 30$ GeV. The pink parabola shows the expected improvement under similar assumptions as in Fig. 116. This confirms that a strong improvement of the fit can be expected taking into account the anticipated future LC accuracy for the electroweak precision data.
## 4.6 The muon magnetic moment and new physics$^{52}$

One of the prime examples of precision observables sensitive to quantum effects are the magnetic moments ($g-2$) of the electron and muon. In particular after the measurements at Brookhaven [18], the muon magnetic moment $a_μ = (g_μ - 2)/2$ has reached a sensitivity to all sectors of the SM and to many New Physics Models (NPM). The currently observed deviation between the experimental value and the SM prediction is particularly well compatible with NPM which can also be tested at a LC. Before the startup of a future LC, new $a_μ$ measurements are planned at Fermilab [19] and J-PARC [20]. For these reasons it is of interest to briefly discuss the conclusions that can be drawn from current and future $a_μ$ results on LC physics.

Like many LC precision observables, $a_μ$ is a flavour- and CP-conserving quantity; unlike the former it is chirality-flipping and therefore particularly sensitive to modifications of the muon Yukawa coupling or more generally the muon mass generation mechanism. A simple consideration, however, demonstrates that like a LC, $a_μ$ is generically sensitive to NPM with new weakly interacting particles at the weak scale [827].

Because of the similar quantum field theory operators relevant for $m_μ$ and $a_μ$, contributions of a NPM at some scale $Λ$ to both quantities, $a_μ$ (N.P.) and $\delta m_μ$ (N.P.), are linked as

$$a_{\mu}(\text{N.P.}) = \mathcal{O}(1) \times \left(\frac{m_{\mu}}{\Lambda}\right)^2 \times \left(\frac{\delta m_{\mu}(\text{N.P.})}{m_{\mu}}\right). \qquad (122)$$

All coupling constants and loop factors are contained in the constant $C := \delta m_μ$ (N.P.)/$m_μ$, which is highly model-dependent. A first consequence of this relation is that new physics can explain the currently observed deviation of [828] (based on [817]),

$$a_{\mu}^{\text{exp}} - a_{\mu}^{\text{SM}} = (28.7 \pm 8.0) \times 10^{-10}, \qquad (123)$$

only if $Λ$ is at the TeV scale or smaller (assuming no fine-tuning in the muon mass, $|C| < 1$).

Equation (122) also illustrates how widely different contributions to $a_μ$ are possible.

- For models with new weakly interacting particles (e.g. $Z'$, $W'$, see Sect. 4.7, little Higgs or universal extra dimension models) one typically obtains perturbative contributions to the muon mass $C = \mathcal{O}(\alpha/4\pi)$. Hence, for weak-scale masses these models predict very small contributions to $a_μ$ and might be challenged by the future more precise $a_μ$ measurement, see e.g. [829, 830]. Models of this kind can only explain a significant contribution to $a_μ$ if the new particles interact with muons but are otherwise hidden from searches. An example is the

model with a new gauge boson associated to a gauged lepton number $L_μ - L_τ$ [831], where a gauge boson mass of $\mathcal{O}(100 \text{ GeV})$ is viable. If this model is the origin of the observed $a_μ$ deviation it would be highly desirable to search for the new $Z'$, corresponding to the $L_μ - L_τ$-symmetry. This would be possible at the LHC in part of the parameter space but also at the LC in the process $e^+e^- \to \mu^+\mu^- Z'$ [831].

- For SUSY models one obtains an additional factor $\tan β$, the ratio of the two Higgs vacuum expectation values, see e.g. [832] and references therein. A numerical approximation for the SUSY contributions is given by

$$a_{\mu}^{\text{SUSY}} \approx 13 \times 10^{-10} \left( \frac{100 \text{ GeV}}{M_{\text{SUSY}}} \right)^2 \tan \beta \, \text{sign}(\mu), \quad (124)$$

where $M_{\text{SUSY}}$ denotes the common superpartner mass scale and $μ$ the Higgsino mass parameter. It agrees with the generic result Eq. (122) for $C = \mathcal{O}(\tan β \times \alpha/4\pi)$ and is exactly valid if all SUSY masses are equal to $M_{\text{SUSY}}$. The formula shows that the observed deviation could be explained e.g. for relevant SUSY masses (smuon, chargino and neutralino masses) of roughly 200 GeV and $\tan β \sim 10$ or SUSY masses of 500 GeV and $\tan β \sim 50$. This is well in agreement with current bounds on weakly interacting SUSY particles and in a very interesting range for a LC. This promising situation has motivated high-precision two-loop calculations of $a_μ^{\text{SUSY}}$ [833, 834], which depend on all sfermion, chargino and neutralino masses and will benefit particularly from precise SUSY mass measurements at a LC.

- Models with large $C \simeq 1$ are of interest since there the muon mass is essentially given by new physics loop effects. Some examples of such radiative muon mass generation models are given in [827]. For examples within SUSY see e.g. [835, 836]. In such models $a_μ$ can be large even for particle masses at the TeV scale, potentially beyond the direct reach of a LC. The possibility to test such models using precision observables at the LC has not yet been explored in the literature.

Fig. 118 illustrates the complementarity of $a_μ$ and LC measurements in investigating SUSY.

The upper plot shows the $a_μ$ (SUSY)-values for the SPS benchmark points [807], of which only the weakly interacting sector is relevant. The contributions span a wide range and can be positive or negative.$^{53}$ The discriminating power of the current (yellow band) and an improved (blue band) measurement is evident from Fig. 118(a). The green points illustrate that the LHC alone is not sufficient to discover SUSY and measure all its parameters. They correspond to

<footnote>$^{53}$ Most of the points are ruled out by LHC searches for colored particles. However, for our purposes only the weakly interacting particles are relevant, and these are not excluded.</footnote>

<footnote>$^{52}$Dominik Stöckinger</footnote>
<figure><img src="image_122.png" /><figcaption>**Fig. 118** (a) SUSY contributions to $a_μ$ for the SPS benchmark points (red), and for the “degenerate solutions” from Ref. [837]. The yellow and blue band indicate the current and an improved experimental result, respectively. (b) Possible future $\tan\beta$ determination assuming that a slightly modified MSSM point SPS1a (see text) is realized. The bands show the $\Delta\chi^2$ parabolas from LHC-data alone (yellow) [838], including the $a_μ$ with current precision (dark blue) and with prospective precision (light blue). The width of the blue curves results from the expected LHC-uncertainty of the parameters (mainly smuon and chargino masses) [838]. Taken from [839].</figcaption></figure>

“degenerate solutions” as defined in Ref. [837] — different SUSY parameter points which cannot be distinguished at the LHC alone. They have very different $a_μ$ predictions, in particular different signs for $μ$, and hence $a_μ$ can resolve such LHC degeneracies. However, the LC can go much further and rule out the wrong parameter choices with far higher significance [837].

The lower plot of Fig. 118 illustrates that the SUSY parameter $\tan\beta$ can be measured more precisely by combining LHC-data with $a_μ$. It is based on the assumption that SUSY is realized, found at the LHC and the origin of the observed $a_μ$ deviation in Eq. (123). To fix an example, we use a slightly modified SPS1a benchmark point with $\tan\beta$ scaled down to $\tan\beta = 8.5$ such that $a_μ^{\text{SUSY}}$ is equal to an assumed deviation $\Delta a_μ = 255 \times 10^{-11}$.$^{54}$ Ref. [838] has shown that then mass measurements at the LHC alone are sufficient to

determine $\tan\beta$ to a precision of $\pm 4.5$ only. The corresponding $\Delta\chi^2$ parabola is shown in yellow in the plot. In such a situation one can study the SUSY prediction for $a_μ$ as a function of $\tan\beta$ (all other parameters are known from the global fit to LHC data) and compare it to the measured value, in particular after an improved measurement. The plot compares the LHC $\Delta\chi^2$ parabola with the ones obtained from including $a_μ$, $\Delta\chi^2 = [(a_μ^{\text{SUSY}}(\tan\beta) - \Delta a_μ)/\delta a_μ]^2$ with the errors $\delta a_μ = 80 \times 10^{-11}$ (dark blue) and $34 \times 10^{-11}$ (light blue). Here the widths of the parabolas mainly originate in the experimental uncertainties of the relevant electroweak particles, such as smuons and charginos. It can be seen that on the one hand future measurements of $a_μ$ would drastically improve the $\tan\beta$ determination. On the other hand, an LC measurement of the electroweak masses would also be important to obtain a very good fit to $\tan\beta$.

Ref. [837] has also studied the impact of a LC on the $\tan\beta$-determination in a similar context, and a similar improvement was found as in the case of $a_μ$. Here it is noteworthy that in the MSSM, $\tan\beta$ is a universal quantity entering all sectors, like $\sin\theta_W$ in the SM, but that $a_μ$ and LC measurements are sensitive to $\tan\beta$ in different sectors, the muon Yukawa coupling and sparticle masses, respectively. These examples show how the LC will complement information from $a_μ$ and test NPM compatible with $a_μ$.

The situation would be quite different if the $a_μ$ deviation is real but not due to weak-scale new particles but to very light, sub-GeV new particles, as suggested e.g. in [840]. In such a case, such new light dark-force particles could be probed by dedicated low-energy precision experiments such as the next generation $a_μ$ measurements but the full understanding of whatever physics at the electroweak scale will have been found at the LHC would be the task of a future LC.

## 4.7 Anomalous gauge boson couplings

### 4.7.1 Electroweak Gauge Boson Interactions: Effective Field Theory and Anomalous Couplings$^{55}$

One possibility to search for new physics in the electroweak sector is the precision investigation of the couplings of the electroweak gauge bosons. At the LC at tree-level, the incoming leptons interact via an exchange of an electroweak gauge boson. This allows for precise studies of trilinear gauge couplings in $e^+e^- \to W^+W^-$ as well as quartic gauge couplings occurring in a variety of final states like $e^+e^- \to VVV$ with VVV being WWZ or WW$\gamma$. In contrast to a hadron collider the advantages are the absence of parton distribution functions so that the center of mass energy at which the hard scattering takes place is exactly known. This also allows to

<footnote>$^{54}$The following conclusions are neither very sensitive to the actual $\tan\beta$ value nor to the actual value of the deviation $\Delta a_μ$.</footnote>

<footnote>$^{55}$Nicolas Greiner</footnote>
tune the beam energy according to the occurring resonances similar to what has already be done at LEP. The second advantage is the clean environment. At a hadron collider the most likely processes involve QCD radiation and therefore jets in the final state. Triple or quartic gauge boson scatterings are typically detected via VBF processes which however have to be discriminated from irreducible background processes.

One approach to parametrize new physics in a model independent way is to write down an effective Lagrangian with all possible vertices and general coupling constants. For the trilinear electroweak gauge couplings (TGC) this has been suggested in [841] for instance, resulting in the following effective Lagrangian including anomalous TGCs:

$$ \mathcal{L}_{\text{TGC}} = ig_{WWV} (g_1^V (W_{\mu\nu}^+ W^{-\mu} - W^{+\mu} W_{\mu\nu}^-) V^\nu \\ + \kappa^V W_\mu^+ W_\nu^- V^{\mu\nu} + \frac{\lambda^V}{M_W^2} W_\mu^{\nu+} W_\nu^{-\rho} V_\rho^\mu \\ + ig_4^V W_\mu^+ W_\nu^- (\partial^\mu V^\nu + \partial^\nu V^\mu) \\ - ig_5^V \epsilon^{\mu\nu\rho\sigma} (W_\mu^+ \partial_\rho W_\nu^- - \partial_\rho W_\mu^+ W_\nu^-) V_\sigma \\ + \tilde{\kappa}^V W_\mu^+ W_\nu^- \tilde{V}^{\mu\nu} + \frac{\tilde{\lambda}^V}{m_W^2} W_\mu^{\nu+} W_\nu^{-\rho} \tilde{V}_\rho^\mu), \quad (125) $$

with $V = \gamma, Z$; $W_{\mu\nu}^{\pm} = \partial_{\mu}W_{\nu}^{\pm} - \partial_{\nu}W_{\mu}^{\pm}$, $V_{\mu\nu} = \partial_{\mu}V_{\nu} - \partial_{\nu}V_{\mu}$ and $\tilde{V}_{\mu,\nu} = \epsilon_{\mu\nu\rho\sigma}V_{\rho\sigma}/2$. The overall coupling constants are given by $g_{WW\gamma} = -e$ and $g_{WWZ} = -e \cot \theta_W$ (with $\cos \theta_W = M_W/M_Z$). In the same spirit, one can write down an effective Lagrangian describing quartic gauge boson couplings (QGC) as follows [842]:

$$ \begin{aligned} \mathcal{L}_{\text{QGC}} = & e^2 (g_1^{\gamma\gamma} A^{\mu} A^{\nu} W_{\mu}^{-} W_{\nu}^{+} - g_2^{\gamma\gamma} A^{\mu} A_{\mu} W^{-\nu} W_{\nu}^{+}) \\ & + e^2 \frac{c_W}{s_W} (g_1^{\gamma Z} A^{\mu} Z^{\nu} (W_{\mu}^{-} W_{\nu}^{+} + W_{\mu}^{+} W_{\nu}^{-}) \\ & \qquad - 2g_2^{\gamma Z} A^{\mu} Z_{\mu} W^{-\nu} W_{\nu}^{+}) \\ & + e^2 \frac{c_W^2}{s_W^2} (g_1^{ZZ} Z^{\mu} Z^{\nu} W_{\mu}^{-} W_{\nu}^{+} - g_2^{ZZ} Z^{\mu} Z^{\nu} W^{-\nu} W_{\nu}^{+}) \\ & + \frac{e^2}{2s_W^2} (g_1^{WW} W^{-\mu} W^{+\nu} W_{\mu}^{-} W_{\nu}^{+} - g_2^{WW} (W^{-\mu} W_{\mu}^{+})^2) \\ & + \frac{e^2}{4s_W^2 c_W^4} h^{ZZ} (Z^{\mu} Z_{\mu})^2. \end{aligned} \quad (126) $$

In the SM the couplings in Eq. (125) are given by

$$ g_1^{\gamma,Z} = \kappa^{\gamma,Z} = 1, \quad g_{4,5}^{\gamma,Z} = \tilde{\kappa}^{\gamma,Z} = 1, \quad \lambda^{\gamma,Z} = \tilde{\lambda}^{\gamma,Z} = 0, \quad (127) $$

whereas the SM values of the QGCs are

$$ g_1^{VV'} = g_2^{VV'} = 1(VV' = \gamma\gamma, \gamma Z, ZZ, WW), \quad h^{ZZ} = 0. \quad (128) $$

In the context of the recent discovery of a particle compatible with a SM Higgs boson [843] it will be interesting to study the couplings of the Higgs boson to the electroweak gauge bosons.

A parametrization of trilinear couplings can be found in [844,845], for instance, and reads

$$ \begin{aligned} \mathcal{L}_{\text{TGC}}^{H} = & g_{H\gamma\gamma} H A_{\mu\nu} A^{\mu\nu} + g_{HZ\gamma}^{(1)} A_{\mu\nu} Z^{\mu} \partial^{\nu} H \\ & + g_{HZ\gamma}^{(2)} H A_{\mu\nu} Z^{\mu\nu} + g_{HZZ}^{(1)} Z_{\mu\nu} Z^{\mu} \partial^{\nu} H \\ & + g_{HZZ}^{(2)} H Z_{\mu\nu} Z^{\mu\nu} + g_{HWW}^{(2)} H W_{\mu\nu}^{+} W_{-}^{\mu\nu} \\ & + g_{HWW}^{(1)} (W_{\mu\nu}^{+} W_{-}^{\mu} \partial^{\nu} H + \text{h.c.}) . \end{aligned} \quad (129) $$

Note that none of the terms in Eq. (129) has a SM contribution as the HVV vertex in the SM is given by

$$ \mathcal{L}_{\text{SM}}^{H} = \frac{1}{2} \frac{g}{\cos \theta_W} M_Z H Z_{\mu} Z^{\mu} + g M_W W_{\mu}^{+} W^{-\mu}. \quad (130) $$

In Eqs. (125), (126), (129) the number of possible additional interaction terms in the Lagrangian is restricted by the requirement of electroweak gauge- and Lorentz invariance. If one loosens this requirement, there would be many more possibilities as discussed for instance in [846].

A slightly different approach to a model independent parametrisation of new physics is based on the idea of an effective field theory (EFT) [847], where additional, higher dimensional operators are added to the SM Lagrangian,

$$ \mathcal{L}_{\text{eff}} = \mathcal{L}_{\text{SM}} + \sum_{n=1}^{\infty} \sum_{i} \frac{f_{i}^{(n)}}{\Lambda^{n}} \mathcal{O}_{i}^{(n+4)}. \quad (131) $$

As the Lagrangian is required to have dimension four, this means that higher dimensional operators are accompanied by dimensionful coupling constants. It is not possible to construct operators of dimension five that are Lorentz- and gauge invariant, so the first additional operators are of dimension six. A general analysis of dimension six operators has been presented in [848]. The choice of the basis of these operators is however not unique, and especially for operators involving electroweak gauge bosons a number of different choices have been discussed in the literature; a common representation can be found in [849]. In the effective field theory approach one first specifies the particle content of the theory and derives the corresponding vertices and coupling constants from there. At a first glance the two approaches, i. e. the EFT and the effective Lagrangian approach, may lead to the same results, as one can express the coupling constants of Eqs. (125), (126), (129) as functions of the coefficients $f_i^{(n)}/\Lambda^n$ of Eq. (131) [849], as follows:

$$ \begin{aligned} g_1^Z &= 1 + f_W \frac{m_Z^2}{2\Lambda^2}, \\ \kappa_Z &= 1 + [f_W - \sin^2 \theta_W (f_B + f_W)] \frac{m_Z^2}{2\Lambda^2}, \\ \kappa_\gamma &= 1 + (f_B + f_W) \frac{m_W^2}{2\Lambda^2}, \\ \lambda_\gamma &= \lambda_Z = \frac{3m_W^2 g^2}{2\Lambda^2} - f_{WWW}. \end{aligned} \quad (132) $$
The corresponding Lagrangian using the EFT approach of Eq. (131) leading to Eq. (132) is given by [849]

$$ \mathcal{L}_{eff} = \mathcal{L}_{SM} + \frac{f_B}{\Lambda^2} (D_\mu \phi)^\dagger \hat{B}^{\mu\nu} (D_\nu \phi) \\ + \frac{f_W}{\Lambda^2} (D_\mu \phi)^\dagger \hat{W}^{\mu\nu} (D_\nu \phi) \\ + \frac{f_WWW}{\Lambda^2} \mathrm{Tr} [\hat{W}_{\mu\nu} \hat{W}^{\nu\rho} \hat{W}_{\rho}^{\mu}], \qquad (133) $$

with $\hat{B}^{\mu\nu} = ig'\frac{g'}{2}D^{\mu\nu}$ and $\hat{W}^{\mu\nu} = ig\frac{\sigma^a}{2}W^{a,\mu\nu}$. However, the EFT approach offers a better interpretation of the origin of these additional couplings as we will describe in more detail next.

The scale $\Lambda$ denotes the energy scale at which the structure of the full theory is resolved. At lower energies, the heavy degrees of freedom of this full theory are considered to be integrated out, appearing as higher-dimensional operators in the EFT that describes the low energy physics. One example for such an effective field theory is Fermi's theory of weak interactions. At an energy scale well below the W boson mass the weak interaction of leptons and neutrinos can be described by a four fermion operator of dimension six. The corresponding scale $\Lambda$ in an EFT description of weak interaction would then be the W boson mass. For energies well below the (usually unknown) scale $\Lambda$, the higher dimensional operators are suppressed by powers of $\Lambda$. This ensures that the higher dimensional operators are more suppressed than lower dimensional operators, i.e. dimension eight operators can usually be neglected compared to dimension six operators. In the limit $\Lambda \to \infty$ one recovers the SM. The effective field theory is only valid at energies well below $\Lambda$. As soon as one approaches this scale the operators of dimension greater than six are no longer suppressed. They contribute equally and can no longer be neglected. At this point the effective field theory breaks down and has to be replaced by the UV completion of the underlying full theory. Therefore the effective field theory provides a handle on the energy range in which it is valid, which cannot be deduced from the effective Lagrangians of Eqs. (125), (126), (129).

One very important feature of higher dimensional operators is their high energy behavior. Due to their higher dimension, the effects of these operators increase with energy and would eventually violate unitarity. The energy at which (tree level) unitarity is violated depends on the operator and in general also depends on the helicity [850]. Typically this problem is solved by introducing form factors which suppress the effects of the operators hence rendering the cross section unitary. These form factors are however completely arbitrary as long as they preserve unitarity and from the viewpoint of an effective field theory they are not needed because at this energy the effective theory is no longer valid [851].

The effects of anomalous couplings in electroweak gauge boson interactions in the production of multiple gauge

bosons have been calculated both for $e^+e^-$ colliders [852] as well as for hadron colliders [844, 853] and many available results also include next-to-leading order QCD and/or electroweak corrections. For the extraction of limits on anomalous TGCs and QGCs it is essential that precise predictions of the relevant processes are provided in form of Monte Carlo programs including the effects of anomalous couplings. The implementation of anomalous couplings in publicly available Monte Carlo programs ranges from specific processes to a general implementation at the level of the Lagrangian. For $e^+e^-$ colliders anomalous couplings for the production of four fermions (and a photon) are contained in RacoonWW [854–856], including NLO EW corrections to four-fermion production in double-pole approximation. A broader implementation of anomalous couplings for $e^+e^-$ colliders is provided in WHIZARD [857], which can also be used for hadron colliders. VBFNLO [858] provides NLO QCD predictions for processes at hadron colliders including trilinear and quartic couplings as well as anomalous couplings of electroweak gauge bosons to the Higgs boson. CalcHEP and CompHEP [859, 860] can import anomalous couplings from LanHEP [861] which generates them at the level of the Lagrangian. FeynRules also can generate anomalous couplings at the Lagrangian level and the corresponding Feynman rules can be implemented via the UFO format [862] to any Monte Carlo program that supports this format, as for instance MadGraph [863].

### 4.7.2 Anomalous gauge couplings: experimental prospects

56

We briefly review the capabilities of an LC to measure triple and quartic gauge couplings (based on Ref. [246] and references therein). As mentioned earlier, the effects of higher dimensional operators are suppressed at low energies and their impact increases with increasing center of mass energy. Therefore a general pattern is the deviation from the SM best visible in the high energy tails of distributions like $p_T$, $H_T$ or invariant masses.

The couplings among the electroweak gauge bosons are directly given by the structure of the gauge group, see the previous section. This structure can thus directly be determined by a measurement of the gauge boson interactions. Particularly sensitive is the process $e^+e^- \to W^+W^-$, since any “naive” change in the gauge couplings would lead to a violation of unitarity, and small changes lead to relatively large variations.

To date, electroweak precision observables together with the LEP data yielded the strongest constraints on anomalous couplings [864, 865]. For the triple gauge couplings the

<footnote n="56">56 Nicolas Greiner, Sven Heinemeyer, Doreen Wackeroth</footnote>
<table><caption>**Table 29** Results of the single parameter fits ($1\sigma$) to the different triple gauge couplings at the ILC for $\sqrt{s} = 500$ GeV with $\mathcal{L} = 500$ fb$^{-1}$ and $\sqrt{s} = 800$ GeV with $\mathcal{L} = 1000$ fb$^{-1}$; $\mathcal{P}_{e^{-}} = 80\%$ and $\mathcal{P}_{e^{+}} = 60\%$ has been used. Taken from [867].</caption><thead><tr><th rowspan="2">coupling</th><th colspan="2">error × 10<sup>-4</sup></th></tr><tr><th>√s = 500GeV</th><th>√s = 800GeV</th></tr></thead><tbody><tr><td>Δg<sup>Z</sup><sub>1</sub></td><td>15.5</td><td>12.6</td></tr><tr><td>Δκ<sub>γ</sub></td><td>3.3</td><td>1.9</td></tr><tr><td>λ<sub>γ</sub></td><td>5.9</td><td>3.3</td></tr><tr><td>Δκ<sub>Z</sub></td><td>3.2</td><td>1.9</td></tr><tr><td>λ<sub>Z</sub></td><td>6.7</td><td>3.0</td></tr><tr><td>g<sup>Z</sup><sub>5</sub></td><td>16.5</td><td>14.4</td></tr><tr><td>g<sup>Z</sup><sub>4</sub></td><td>45.9</td><td>18.3</td></tr><tr><td>κ̃<sub>Z</sub></td><td>39.0</td><td>14.3</td></tr><tr><td>λ̃<sub>Z</sub></td><td>7.5</td><td>3.0</td></tr></tbody></table>

<figure><img src="image_123.png" /><figcaption>**Fig. 119** Comparison of $\Delta\kappa_{\gamma}$ and $\Delta\lambda_{\gamma}$ at different machines. For LHC and ILC three years of running are assumed (LHC: 300 fb$^{-1}$, ILC $\sqrt{s} = 500$ GeV: 500 fb$^{-1}$, ILC $\sqrt{s} = 800$ GeV: 1000 fb$^{-1}$). If available the results from multi-parameter fits have been used. Taken from [246].</figcaption></figure>

bounds are [865]

$$ 
\begin{aligned}
\Delta g_1^Z &= -0.033 \pm 0.031, \\
\Delta \kappa_\gamma &= 0.056 \pm 0.056, \\
\Delta \kappa_Z &= -0.0019 \pm 0.044, \\
\lambda_\gamma &= -0.036 \pm 0.034, \\
\lambda_Z &= 0.049 \pm 0.045.
\end{aligned} 
\qquad (134) 
$$

The bounds currently available from LHC data are weaker but approach the precision of the LEP results [866].

Turning to the ILC, the different types of couplings can be disentangled experimentally by analyzing the production angle distribution of the W boson and the W polarization structure, which can be obtained from the decay angle distributions. Anomalous couplings for WW$\gamma$ and WWZ result in similar final state distributions. However, using beam polarization, they can be disentangled, where a large beam polarization, in particular for the left-handed $e^{-}$ is required. Also positron polarization is required for an optimal resolution [40].

A fast detector simulation analysis was performed for $\sqrt{s} = 500$GeV and 800GeV [867]. The results for single parameter fits are shown in Tab. 29. Correlations in the multi-parameter fits were taken into account where possible. For $\sqrt{s} = 800$GeV they are relatively small, not increasing the uncertainties by more than ~20%. At $\sqrt{s} = 500$GeV the effect is larger, and uncertainties can increase by up to a factor of two, see also Ref. [7].

Additional information on the triple gauge couplings can be obtained when going to the $e\gamma$ and $\gamma\gamma$ options at the ILC. In this environment the WW$\gamma$ couplings can be measured without the WWZ couplings entering the analysis. It was shown [868, 869] that $\kappa_{\gamma}$ can be measured better in $e^{+}e^{-}$ collisions, while for $\lambda_{\gamma}$ the $e\gamma$ and $\gamma\gamma$ modes can add relevant information. Figure 119 shows the results for $\kappa_{\gamma}$ and $\lambda_{\gamma}$ obtained at different machines. The measurement of $\kappa_{\gamma}$ can

be improved substantially at the ILC. The other coupling, $\lambda_{\gamma}$, on the other hand can be measured with similar accuracy at the LHC and the various ILC options.

Apart from the triple electroweak gauge boson couplings, the ILC is also sensitive to the quartic couplings. Two processes are important in this context: $e^{+}e^{-} \to VVV$ (triple gauge boson production, $V = W^{\pm}, Z$) and $e^{+}e^{-} \to VV'l_{1}l_{2}$ ($l_{1,2} = e, v, V = W^{\pm}, Z$), see Ref. [842] and references therein. This study uses complete six-fermion matrix elements in unweighted event samples, fast simulation of the ILC detector and a multidimensional parameter fit of the set of anomalous couplings. It also includes a study of triple weak boson production which is sensitive to the same set of anomalous couplings. It was shown that, under the assumption of custodial symmetry, sensitivities for $h^{ZZ}$ and $g_{2}^{WW}$ at and below the level of ~5% can be found [842] for $\sqrt{s} = 1$ TeV and $1\text{ ab}^{-1}$ (see also [246]).

As mentioned earlier, apart from the investigation of diboson and triple gauge boson production processes, constraints on the coefficients of higher dimensional operators that lead to new trilinear gauge couplings can also be obtained from their contributions to EWPOs. For instance, modifications of gauge boson self energies induced by these higher dimensional operators can be described with the help of S,T and U parameters [759, 870] and their extensions [871], and by precisely measuring these oblique parameters the effects of these operators can be severely constrained [849, 872, 873]. Typically, bounds from EWPOs mainly affect those operators that contribute already at tree level to the observables. Effects of operators contributing only at the one loop level are suppressed and therefore their bounds are weaker compared to the bounds that can be derived from direct measurements [872, 873].

Recently, constraints on anomalous quartic gauge couplings have been obtained from studies of WW$\gamma$ and WZ$\gamma$ production [874] and like-sign WW$jj$ production [875] at the 8 TeV LHC.
## 4.8 New gauge bosons$^{57}$

Extra gauge bosons, $Z'$s and $W'$s, are a feature of many models of physics beyond the SM [515, 876-879]. Examples of such models are Grand Unified theories based on groups such as $SO(10)$ or $E_6$ [879], Left-Right symmetric models [880], Little Higgs models [449, 452, 477, 881], and Technicolour models [882-885] to name a few. In addition, resonances that arise as Kaluza-Klein excitations in theories of finite size extra dimensions [886] would also appear as new gauge bosons in high energy experiments. It is therefore quite possible that the discovery of a new gauge boson could be one of the first pieces of evidence for physics beyond the SM. Depending on the model, the dominant $Z'$ decay may be either into leptons or jets, leading to a resonance in the reconstructed dilepton or dijet invariant mass distribution respectively.

Currently, the highest mass bounds on most extra neutral gauge bosons are obtained by searches at the Large Hadron Collider by the ATLAS and CMS experiments. The most recent results based on dilepton resonance searches in $\mu^+\mu^-$ and $e^+e^-$ final states use data from the 7 TeV proton collisions collected in 2011 and more recent 8 TeV data collected in 2012. ATLAS [887] obtains the exclusion limits at 95% C.L. $M(Z'_{SSM}) > 2.49$ TeV, $M(Z'_\eta) > 2.15$ TeV, $M(Z'_\chi) > 2.24$ TeV and $M(Z'_\psi) > 2.09$ TeV using only the 8 TeV (6 fb$^{-1}$) data set and CMS [888] obtains 95% C.L. exclusion limits of $M(Z'_{SSM}) > 2.59$ TeV and $M(Z'_\psi) > 2.26$ TeV using the 7 TeV (5 fb$^{-1}$) and 8 TeV (4 fb$^{-1}$) data sets. It is expected that the LHC should be able to see evidence for $Z'$s up to ~5 TeV once the LHC reaches its design energy and luminosity [889-893] and to distinguish between models up to $M_{Z'} \simeq 2.1$ TeV (95% C.L.) [894].

It is expected that the LHC will be able to discover $W'$s up to masses of ~5.9 TeV in leptonic final states assuming SM couplings [890]. Based on searches for a new W boson decaying to a charged lepton and a neutrino using the transverse mass variable CMS [895] excludes the existence of a SSM W' boson with a mass below 2.85 TeV at 95% C.L. using the $\sqrt{s} = 8$ TeV, $\mathcal{L}_{int} = 3.7$ fb$^{-1}$ dataset while ATLAS excludes the existence of a $W^*$ with a mass below 2.55 TeV at 95% C.L. using the 7 TeV dataset with $\mathcal{L}_{int} = 4.7$ fb$^{-1}$ [896].

For models that predict $Z'$ or $W'$ bosons that decay to two quarks, searches have been performed that require two well-separated jets with high transverse momentum. The CMS collaboration excludes the existence of a SSM $Z'$ boson with mass below 1.6 TeV at 95% C.L. and a SSM W' with mass below 2.12 TeV using the $\sqrt{s} = 8$ TeV, $\mathcal{L}_{int} = 4.0$ fb$^{-1}$ dataset [897]. The CMS collaboration also developed a dedicated search for $b\bar{b}$ resonances and excluded existence of a

SSM $Z'$ boson with mass below 1.5 TeV at 95% C.L. in the $b\bar{b}$ channel [898]. For models with larger branching fractions to b-quarks the limit improves considerably, excluding a larger mass range.

If a narrow resonance were discovered, the crucial next step would be to measure its properties and determine the underlying theory. While LHC measurements [876,899] and low energy precision measurements [900] can to some extent constrain new gauge boson couplings, precise measurements will need a LC.

### 4.8.1 New Gauge Boson Studies at High Energy $e^+e^-$ Colliders

Although the LHC will have explored the energy regime accessible to on-shell $Z'$ production by the time a LC is built, a high energy $e^+e^-$ collider will be sensitive to new gauge bosons with $M_{Z',W'} \gg \sqrt{s}$. In $e^+e^-$ collisions below the on-shell production threshold, extra gauge bosons manifest themselves as deviations from SM predictions due to interference between the new physics and the SM $\gamma/Z^0$ contributions. $e^+e^- \to f\bar{f}$ reactions are characterized by relatively clean, simple final states where $f$ could be leptons (e, $\mu$, $\tau$) or quarks (u, d, s, c, b, t), for both polarized and unpolarized $e^\pm$. The baseline ILC configuration envisages electron beam polarization greater than 80% and positron beam polarization of ~30% might be initially achieved, eventually increasing to ~60%. The basic $e^+e^- \to f\bar{f}$ processes can be parametrized in terms of four helicity amplitudes which can be determined by measuring various observables: the leptonic cross section, $\sigma(e^+e^- \to \mu^+\mu^-)$, the ratio of the hadronic to the QED point cross section $R^{\text{had}} = \sigma^{\text{had}}/\sigma_0$, the leptonic forward-backward asymmetry, $A_{\text{FB}}^\ell$, the leptonic longitudinal asymmetry, $A_{\text{LR}}^\ell$, the hadronic longitudinal asymmetry, $A_{\text{LR}}^{\text{had}}$, the forward-backward asymmetry for specific quark or lepton flavours, $A_{\text{FB}}^f$, the $\tau$ polarization asymmetry, $A_{\text{pol}}^\tau$, and the polarized forward-backward asymmetry for specific fermion flavours, $A_{\text{FB}}^f(\text{pol})$ [901] (see also Sect. 4.3). The indices $f=\ell$, $q, \ell=(e, \mu, \tau)$, $q=(c, b)$, and had='sum over all hadrons' indicate the final state fermions. Precision measurements of these observables for various final states ($\mu^+\mu^-$, $b\bar{b}$, $t\bar{t}$) can be sensitive to extra gauge boson masses that by far exceed the direct search limits that are expected at the LHC [889, 891, 901, 902]. Further, precision measurements of cross sections to different final state fermions using polarized beams can be used to constrain the gauge boson couplings and help distinguish the underlying theory [9, 902-907]. A deviation for one observable is always possible as a statistical fluctuation. In addition, different observables have different sensitivities to different models (or more accurately to different couplings). As a consequence, a more robust strategy is to combine many observables to obtain a $\chi^2$ figure of merit.

<footnote>$^{57}$Stephen Godfrey</footnote>
<figure><img src="image_124.png" /><figcaption>**Fig. 120** Discovery reach of the ILC with $\sqrt{s} = 0.5$ (1.0) TeV and $\mathcal{L}_{\text{int}} = 500$ (1000) fb$^{-1}$. The discovery reach of the LHC for $\sqrt{s} = 14$ TeV and 100 fb$^{-1}$ via the Drell-Yan process $pp \to \ell^+\ell^- + X$ are shown for comparison. From Ref. [902] with kind permission of The European Physical Journal (EPJ).</figcaption></figure>

The ILC sensitivity to $Z'$'s is based on high statistics precision cross section measurements so that the reach will depend on the integrated luminosity. For many models a 500 GeV $e^+e^-$ collider with as little as 50 fb$^{-1}$ integrated luminosity would see the effects of a $Z'$ with masses as high as ~5 TeV [889]. The results of a recent study [902] is shown in Fig. 120. That study finds that a 500 GeV ILC with 500 fb$^{-1}$ and a 1 TeV ILC with 1 ab$^{-1}$ can see evidence or rule out a $Z'$ with masses that can exceed ~7 and ~12 TeV for many models, for the two respective energies [902]. These recent results also consider various polarizations for the $e^-$ and $e^+$ beams and show that beam polarization will increase the potential reach of the ILC, see also Ref. [40].

### 4.8.2 Measurement of $Z'$ Couplings at High Energy $e^+e^-$ Colliders

If a $Z'$ were discovered at the LHC, measurements of 2-fermion processes at the ILC could provide valuable constraints on its couplings and discriminate between models. Fig. 121 (top panel) shows the expected resulting precision on $Z'$ couplings to leptons for $\sqrt{s} = 500$ GeV and $\mathcal{L}_{\text{int}} = 1$ ab$^{-1}$ for 3 values of $M_{Z'}$ for several representative models [905]. In this figure, the KK case should not be taken too literally as the couplings do not in fact correspond to the KK $Z'$ couplings but are an effective coupling, reflecting that in this model there are both photon and Z KK ex-

citations roughly degenerate in mass. The point is simply that the KK model can be distinguished from other models. One notes that there is a two-fold ambiguity in the signs of the lepton couplings since all lepton observables are bilinear products of the couplings. Hadronic observables can be used to resolve this ambiguity since for this case the quark and lepton couplings enter the interference terms linearly. Studies [902, 905] have demonstrated that beam polarization plays an important role in the measurement of the $Z'$-fermion couplings and therefore in the discrimination between models.

Rather than measure the $Z'$-fermion couplings one could pose the question; if measurements resulted from a true BSM model, could one rule out other possibilities? A recent analysis given in Ref. [902] showed that the ILC could discriminate models for $Z'$ masses up to 4-8 TeV for a 500 GeV ILC and up to 6-11 TeV for a 1 TeV ILC, depending on the true model. This exceeds the corresponding discovery reach at the LHC and is only slightly lower than the discovery reach at the ILC due to the relatively large differences between angular distributions for $e^+e^- \to f\bar{f}$ for the different models. More crucially, the ILC is significantly more powerful for measuring $Z'$ couplings than is possible at the LHC. These results are based on purely leptonic processes. Measurements of c- and b-quark pair production cross sections would contribute important complementary information for identifying the underlying theory.

If deviations from the SM were observed but there was no direct evidence for a $Z'$ from the LHC one could still exclude a "tested" model for any value of $M_{Z'}$ below some value for a given set of ILC measurements. To see how one can extract such limits consider normalized couplings defined by $C_{v,a}^{fN} = C_{v,a}^{f'} \sqrt{s/(M_{Z'}^2 - s)}$. Fig. 121 (bottom panel) shows constraints on "normalized" couplings for a 10 TeV $Z'$ and $\sqrt{s} = 3$ TeV and $\mathcal{L}_{\text{int}} = 1$ ab$^{-1}$ [9, 906]. One can see how, if a model with a 10 TeV $Z'$ were the true model, other models could be excluded. Ref. [902] finds that for the models they considered one might be able to distinguish between $Z'$ models, at 95% C.L., up to $M_{Z'} \simeq 3.1$ TeV (4.0 TeV) for unpolarized (polarized) beams at the 0.5 TeV ILC and 5.3 TeV (7.0 TeV) at the 1 TeV ILC. Presented another way, they find that if one of the 6 models they studied is true, the other 5 candidates can be ruled out by a 500 GeV ILC for $Z'$ masses up to 4-8 TeV, depending on the true model. This discrimination reach is only slightly below the discovery reach due to order-one differences among the angular distributions in $e^+e^- \to f\bar{f}$ predicted by the different models and in all cases is significantly higher than that of the LHC.
<figure><img src="image_125.png" /><figcaption>**Fig. 121** Top: Resolving power (95% CL) for $M_{Z'} = 1, 1.5$, and 2 TeV and $\sqrt{s} = 500$ GeV, $\mathcal{L}_{\text{int}} = 1 \text{ab}^{-1}$, $|P_{-}| = 80\%$, $|P_{e^{+}}| = 60\%$, for leptonic couplings based on the leptonic observables $\sigma$, $A_{LR}$, $A_{FB}$. The couplings correspond to the $E_6 \chi$, LR, LH, and KK models. From Ref. [905]. Bottom: Expected resolution at CLIC with $\sqrt{s} = 3$ TeV and $\mathcal{L} = 1 \text{ ab}^{-1}$ on the "normalised" leptonic couplings of a 10 TeV $Z'$ in various models, assuming lepton universality. The mass of the $Z'$ is assumed to be unknown. The couplings correspond to the $E_6 \chi$, $\eta$, and $\psi$, the SSM, LR, LH and SLH models. The couplings can only be determined up to a twofold ambiguity. The degeneracy between the $\psi$ and SLH models might be lifted by including other channels in the analysis $(\tilde{t}, b\bar{b}, \dots)$. From Refs. [9,906].</figcaption></figure>

could only be detected up to $M_{W'} = 1.2, 1.6$, and 1.9 TeV for the same collider parameters. Another process that has been considered is $e\gamma \to \nu q + X$ where the photon is produced by a backscattered laser or is a Weizsäcker-Williams photon [909]. These processes yield discovery limits for $W'_{\text{SSM}}$ of 4.1 (2.5), 5.8 (3.6) and 7.2 (4.5) TeV for the backscattered laser (Weizsäcker-Williams) cases and for the three values for $\sqrt{s}$ and $\mathcal{L}_{\text{int}}$ given above. Limits for the LR model are substantially lower.

In general we do not expect an $e^+e^-$ collider to be sensitive to $W'$'s with masses larger than could be discovered at the LHC. If new gauge bosons were discovered first in other processes, the ILC could measure $W'$ (and $Z'\nu\bar{\nu}$) couplings which would complement measurements made at the LHC.

### 4.8.3 Discovery and Identification of $W'$ Bosons in $e^+e^-$

While there is a broad literature on $Z'$ properties, $W'$ studies for high energy $e^+e^-$ colliders are rather limited. One study showed that the process $e^+e^- \to \nu\bar{\nu}\gamma$ would be sensitive to $W'$ masses up to several TeV depending on the model, the centre of mass energy, and the assumed luminosity [908]. For example, evidence for a SSM $W'$ could be seen up to $M_{W'} = 4.3, 5.3$, and 6.0 TeV for $\sqrt{s} = 0.5, 1.0$, and 1.5 TeV respectively with $\mathcal{L}_{\text{int}} = 500 \text{ fb}^{-1}$ while a LR $W'$
# 5 Supersymmetry<fnref n="58" />

## 5.1 Introduction and overview

The recent discovery of a Higgs-like resonance at $M_h$ = (125.15 ± 0.24) GeV by the Atlas and CMS experiments at the CERN LHC seemingly completes the identification of all matter states predicted to exist by the Standard Model (SM) of particle physics. In spite of this extraordinary achievement, the SM remains beset by an array of shortcomings which strongly suggest that new physics exists at, or around, the TeV energy scale. Chief among these is the gauge hierarchy problem, which arises if fundamental scalar fields (such as the Higgs field) do exist. In this case, the scalar field mass term diverges quadratically, and we would expect the Higgs field to have mass far beyond the 125 GeV level unless an exquisite degree of fine-tuning between bare and loop corrections is invoked at each order in perturbation theory.

Along with the gauge hierarchy problem, the SM is lacking in that it provides no particle to explain cold dark matter (CDM) in the universe, it does not allow for baryogenesis in the early universe, it does not allow for the suggested unification of SM forces, it contains no solution to the strong CP problem and it provides no avenue for a sensible inclusion of quantum gravity into its structure.

While a variety of solutions to the gauge hierarchy problem have been proposed, weak scale supersymmetry [910–914], or SUSY, is the most theoretically engaging and one which also appears to be, at least indirectly, supported by experimental data. Supersymmetry is a quantum space-time symmetry that predicts a correspondence between bosonic and fermionic degrees of freedom. In SUSY theories, scalar fields inherit the protective chiral symmetry enjoyed by fermions, reducing their quadratic divergence to merely logarithmic. Since the log of a large number can be small, the required tuning between bare mass and loop mass is greatly reduced, allowing disparate mass scales to co-exist within the same theoretical structure.

To be phenomenologically viable, supersymmetrised versions of the SM must include *soft* SUSY breaking [915], i.e. only those SUSY breaking terms which maintain the cancellation of quadratic divergences. In the Minimal Supersymmetric Standard Model (MSSM), a variety of new matter states—spin 0 squarks and sleptons along with additional Higgs bosons and spin $\frac{1}{2}$ charginos, neutralinos and gluinos—are expected to exist at or around the weak scale.

The MSSM has received some indirect experimental support from the measured values of the strong and electro-

weak forces: these unify to a single value at energy scales $M_{GUT} \sim 2 \times 10^{16}$ GeV under renormalisation group (RG) evolution. Also, the measured value of the top quark ($m_t \simeq 173.2$ GeV) turns out to be sufficiently large as to induce a radiatively-driven breaking of electroweak symmetry. In addition, while the SM allows for a Higgs mass within a wide range, $100 < M_H < 1000$ GeV, the MSSM restricts the lightest SUSY Higgs boson $100 < M_h < 135$ GeV. The fact that the newly discovered Higgs-like state falls within the narrow mass range predicted by SUSY may also be regarded as an indirect support of this picture. Simple arguments based on electro-weak naturalness would suggest that super-partners should exist at or below the ~1 TeV scale, motivating a significant effort for their search at the LHC and inspiring the physics program of a future $e^+e^-$ linear collider. Finally, SUSY provides us with at least three viable candidates for dark matter: the lightest neutralino $\tilde{\chi}^0_1$ (a WIMP candidate) the gravitino $\tilde{G}$ and the axino $\tilde{a}$ (the spin-1/2 super-partner of the axion) [916].

SUSY theories also offer at least three mechanisms for baryogenesis, including weak scale baryogenesis (now nearly excluded in the MSSM), thermal and non-thermal leptogenesis and Affleck-Dine baryo- and leptogenesis [917]. Local SUSY (supergravity) theories necessarily include spin-2 gravitons and spin-3/2 gravitinos, and reduce to Einstein's general relativity in the classical limit.

This chapter provides an overview of the capabilities of a linear $e^+e^-$ collider in the search for supersymmetry, in view of the constraints and indications derived from present experimental data, in particular the LHC results from the 7 and 8 TeV data for the SUSY direct searches and the Higgs properties. The limits derived in these searches seem to require SUSY particles beyond the TeV scale, seemingly in contradiction to the aforementioned arguments based on electro-weak naturalness. However, it is important to observe that the strongly-interacting SUSY particles— which LHC is most sensitive to— are also those with less direct connection to the electro-weak naturalness. Taken in this context, there remains a huge role for LHC operation at 13-14 TeV and for subsequent operation of a linear $e^+e^-$ collider of sufficient centre of mass energy, $\sqrt{s}$, to play a decisive role in the search for, and proof of, SUSY. Indeed, even if no SUSY particles are seen at the LHC at 13-14 TeV, then a 0.5-1 TeV linear $e^+e^-$-collider may still retain its role as *discovery* machine for SUSY [918, 919] in that *the most natural SUSY models require light higgsinos* with mass ~100–200 GeV which can easily elude LHC searches (due to the small energy release from their compressed spectra), but which can easily be detected in $e^+e^-$ collisions of sufficient energy $\sqrt{s} > 2m(\text{higgsino})$.

If supersymmetric matter is indeed found at LHC or the $e^+e^-$-LC, then a program of precision measurements, which can be made in high energy $e^+e^-$ collisions, will be cru-

<footnote n="58">⁵⁸Editors: H. Baer, M. Battaglia, J. Kalinowski
Contributors: A. Arbey, P. Bechtle, A. Bharucha, F. Brümmer, S.Y. Choi, A. Freitas, J. Heisig, J. List, F. Mahmoudi, G. Moortgat-Pick, W. Porod, S. Porto, K. Rolbiecki
H.B. would like to thank D. Mickelson and A. Mustafayev for providing several figures.</footnote>
cial for pinning down SUSY particle masses, mixings and other properties. From such measurements, it may be possible to clarify the role of SUSY in cosmic dark matter production and possibly also in baryogenesis, thus establishing even more closely the link between particle physics and cosmology. If indeed a desert exists between the weak scale and some high scale such as $M_{GUT}$ or $M_{string}$, then it may be possible to extrapolate SUSY parameters to these ultra-high scales, thus testing ideas about unification, SUSY breaking, and string theory. We will conclude that a linear $e^+e^-$ collider of sufficient energy and luminosity is absolutely needed for providing a detailed experimental exploration of the intriguing concept of weak-scale supersymmetry, if it is realised in nature.

## 5.2 Models of Supersymmetry

The superfield formalism provides an algorithm for the direct supersymmetrization of the SM [920,921]. In this case, each SM matter fermion of a given chirality is elevated to a chiral superfield which also contains a spin-0 superpartner. The SM gauge fields are elevated to gauge superfields which also contain spin-$\frac{1}{2}$ gauginos. The SM Higgs doublet is embedded in a chiral superfield necessitating introduction of spin-$\frac{1}{2}$ higgsinos. The addition of extra higgsinos carrying gauge quantum numbers destroys the elegant anomaly cancellation mechanism in the SM, unless one introduces as well a second Higgs/higgsino doublet superfield carrying opposite weak hypercharge.

The resulting supersymmetrized SM enjoys exact, rigid supersymmetry— but this is known not to be true since it would imply e.g. the existence of spin-0 partners of the electron (selectrons) with the same mass as the electron: such matter states would easily have been detected long ago. Hence, SUSY must be a broken symmetry. SUSY can be broken explicitly by adding by hand soft SUSY breaking (SSB) terms to the Lagrangian. These terms include mass terms for spin-0 superpartners, mass terms for each gaugino, and bilinear and trilinear scalar interactions (so-called B and A terms).

In addition, a plethora of terms are allowed in the superpotential which violate baryon and lepton number conservation, and lead to rapid proton decay. Such terms are suppressed by invoking an R-parity (which naturally arises in SUSY GUT theories based on $SO(10)$). If R-parity is conserved, then SUSY particles can only be produced in pairs at colliders, SUSY particles must decay to other SUSY particles, and the lightest SUSY particle must be absolutely stable, perhaps offering a good dark matter candidate.

The resulting theory, called the Minimal Supersymmetric Standard Model, or MSSM, is the direct supersymmetrization of the SM that is consistent with all known constraints. It includes more than 100 adjustable parameters [920], most

of these consisting of flavor or CP violating terms. Under the assumption of minimal flavor violation (MFV) and minimal CP-violation (MCPV), these are set to zero, so that FV and CPV arise solely from the Yukawa sector. The pMSSM model with 19 adjustable weak scale parameters is a popular model for this approach.

### 5.2.1 Gravity mediation

An appealing approach to SUSY breaking comes from invoking local SUSY, or supergravity (SUGRA). If SUSY is local, then one must necessarily include a graviton-gravitino supermultiplet. One may include a so-called hidden sector of fields whose sole purpose is to allow for spontaneous breaking of SUSY via the superHiggs mechanism [922]. Under the superHiggs mechanism, hidden sector fields acquire a SUSY breaking vev $\langle F \rangle \sim m^2$ so that the gravitino gains a mass $m_{3/2} \sim m^2/M_P$ while the graviton remains massless: if $m_{3/2} \sim M_{weak}$, then $m \sim 10^{11}$ GeV.

The above-mentioned soft SUSY breaking terms arise via tree level gravitational interactions with magnitude $\sim m_{3/2}$. More generally, “gravity-mediated supersymmetry breaking” denotes any theory in which supersymmetry breaking is communicated to the visible sector by $M_P$-suppressed interactions at the tree level, not necessarily just involving the gravitational multiplet, and therefore gives soft parameters of the order $m_{3/2}$. If $m_{3/2} \sim M_{weak}$, then in the limit $M_P \to \infty$ while keeping $m_{3/2}$ constant we obtain a theory with weak-scale rigid supersymmetry plus soft SUSY breaking terms.

The minimal supergravity model (mSUGRA [923] or CMSSM [924]) assumes all matter scalars and both Higgs fields receive a common soft mass $m_0$ at some high scale, usually taken to be $M_{GUT} \simeq 2 \times 10^{16}$ GeV, the scale where gauge couplings unify in the MSSM. Likewise, all gauginos receive a common mass $m_{1/2}$, and all A terms are set to a common value $A_0$. While this ansatz is simple, and receives some experimental motivation in that such choices suppress flavor and CP-violating terms, one must remember that it is at best merely a simplifying assumption that is not likely to remain true for realistic models [925].

One of the virtues of SUSY models defined at a high scale such as $Q = M_{GUT}$ is that the large top quark Yukawa coupling drives exactly the right scalar Higgs field $m_{H_u}^2$ to negative squared values, resulting in a radiatively driven breakdown of electroweak symmetry (REWSB) [926]. Upon EWSB, the $B\mu$ parameter may be traded for a parameter $\tan\beta = v_u/v_d$, the ratio of Higgs field vevs, and the magnitude of the Higgsino mass parameter $\mu$ is fixed to yield the measured Z-boson mass. Then all sparticle masses and mixings, and hence production and decay rates, are determined by the well-known parameter set: $m_0$, $m_{1/2}$, $A_0$, $\tan\beta$, and $\text{sign}(\mu)$. However, many more
parameters are allowed if one deviates from the simplis-
tic assumption listed above, resulting in models with non-
universal soft SUSY breaking terms.

5.2.2 GMSB and AMSB

In addition to models of gravity-mediated SUSY breaking, other possibilities exist. One of these is *gauge-mediated SUSY breaking*, or GMSB [927, 928]. In this class of theories, the hidden sector couples to a messenger sector (which carries SM gauge quantum numbers) which acts as an intermediary between the visible and hidden sectors. In GMSB, loop diagrams containing messenger states induce visible sector soft SUSY breaking terms.

The gravitino again gets a mass $m_{3/2} \sim \langle F \rangle/M_P$, while the sparticles gain soft masses of the order $\frac{g^2}{16\pi^2} \frac{F}{M}$, where $M$ is the messenger mass and $g$ is any MSSM gauge coupling. For $M \ll M_P$, the SUSY particles may still be at the TeV scale while gravitinos can be much lighter, so that the gravitino may play the role of the LSP. In the simplest GMSB models, the trilinear SSB terms are suppressed, so there is little mixing in the top squark sector. Thus, these models have trouble generating a light Higgs scalar of mass $\sim 125$ GeV as is now required by data [929, 930]. More general gauge mediation models [931] are now required for phenomenological viability.

A third possibility is *anomaly-mediated SUSY breaking* [932,933]. In any model of SUSY breaking mediation, there are contributions to SSB terms arising from the super-Weyl anomaly. These are however suppressed by a loop factor with respect to $m_{3/2}$ and therefore subdominant in gravity mediation or GMSB. They become relevant in *sequestered models* where the gravity- and gauge-mediated soft masses are negligible, e.g. because the hidden sector is spatially separated from the visible sector in extra dimensions.

In AMSB, the SSB terms are governed by the RG beta functions and anomalous dimensions divided by loop factors. In this case, the wino-like neutralino turns out to be LSP, while $m_{3/2} \sim 25-50$ TeV, thus solving the cosmological gravitino problem. Since minimal versions of these models fail to generate a large A-term, they also seem disfavored by the recently measured Higgs boson mass. Moreover, the minimal anomaly mediated model predicts tachyonic sleptons, which is an even more serious shortcoming. However, various string-inspired modifications of the minimal framework do lead to viable phenomenology [934–937].

5.2.3 Hybrid mediation schemes

Embedding the MSSM into a more fundamental model at
high scales, for instance into the effective field theory of
some superstring compactification, can naturally lead to hy-

brid mediation scenarios. These are attractive also from the
phenomenological point of view.

An example, motivated from both heterotic and type IIB
string models, is *mirage mediation* [938–940]: If gravity-
mediated contributions to the gaugino masses are only
mildly suppressed, they may be of similar magnitude as the
anomaly-mediated contributions. A combination of grav-
ity and anomaly mediation allows to interpolate between
unified gaugino masses at the GUT scale (as predicted by
the simplest gravity-mediated GUT models) and unified
gaugino masses at some arbitrary lower *mirage scale* (after
adding the anomaly-mediated contributions, since these are
given by the very same beta function coefficients that govern
the gaugino mass RGEs). An immediate consequence is a
compressed low-scale gaugino mass spectrum if the mirage
scale is low [941–944]. This allows for a lower gluino mass
without conflicting with the LHC search bounds, thus pos-
sibly reducing the fine-tuning. Depending on the underlying
model, a “natural SUSY” pattern for the squark masses, with
sub-TeV stops but multi-TeV first- and second-generation
squarks, may also be realized [942,945]. Sub-TeV charginos
and neutralinos are common in these models. Such models,
realized within the MSSM, do have problems generating a
light Higgs scalar with $M_h \simeq 125$ GeV [946] while main-
taining naturalness [947].

A more extreme example is the case where the gravity-
mediated contributions to the gaugino masses vanish alto-
gether, e.g. because they are forbidden by some symmetry
under which the goldstino superfield is charged [933]. In this
case (which suffers from extreme fine-tuning with regards to
electroweak symmetry breaking) the squarks and sleptons
have gravity-mediated masses up to around 100 TeV, while
the gaugino masses follow the anomaly mediation pattern
and are lighter by a loop factor [948–952]. The LSP is a
wino-like neutralino which is nearly degenerate with a wino-
like chargino.

Alternatively, for a high messenger scale just below the
scale of grand unification (which is well motivated within
certain F-theory and heterotic models [953,954]), gauge me-
diation can coexist with gravity mediation. This is because
the GUT scale is about a loop factor below the Planck scale.
Generic models of high-scale gauge mediation tend to have
problems with flavour constraints [955, 956], which should
be solved similarly as in ordinary gravity mediation. Such
hybrid gauge-gravity mediation models naturally allow to
obtain near-degenerate higgsino-like charginos and neutrali-
nos with masses around the electroweak scale, while the rest
of the spectrum can be in the multi-TeV range [954, 957].
Models with mixed gauge, gravity and anomaly mediation
are also a possibility [958].

All the above hybrid mediation scenarios have in com-
mon that the coloured superpartners may be difficult to see at
the LHC, either because they are heavy or because the spec-
trum is compressed. In particular, large parameter space regions survive the constraints from LHC8. At the same time, at least some of the charginos and neutralinos are often light enough to be produced, detected, and studied at a linear $e^+e^-$ collider.

### 5.3 Naturalness and Fine-tuning

The main reason we expect supersymmetric matter states to arise with masses around the electroweak scale derives from the notion of electroweak naturalness. A model is considered to be natural in the electroweak sector if there are no large, unnatural cancellations (fine-tunings) required in deriving the measured values of both $M_Z$ and $M_h$.

A quantitative measure of fine-tuning of a supersymmetric model was introduced over twenty five years ago, while SUSY was being searched for at LEP [959–961]). The so-called Barbieri-Giudice measure, $\Delta_{BG}$, is defined as

$$ \Delta_{BG} \equiv \max_i [c_i] \quad \text{where} \quad c_i = \left| \frac{\partial \ln M_Z^2}{\partial \ln a_i} \right| = \left| \frac{a_i}{M_Z^2} \frac{\partial M_Z^2}{\partial a_i} \right| \quad (135) $$

where the set $a_i$ constitute the fundamental parameters of the model. Thus, $\Delta_{BG}$ measures the fractional change in $M_Z^2$ due to fractional variation in model parameters $a_i$. The $c_i$ are known as sensitivity co-efficients [962].

For models with parameters defined at very high scales (e.g. at $\Lambda = M_{GUT}$), as those discussed above, the evaluation of $\Delta_{BG}$ requires to express $M_Z^2$ in terms of high-scale parameters using semi-analytic solutions of the renormalization group equations for the corresponding soft term and $\mu$ [962–964].

The $\Delta_{BG}$ measure picks off the co-efficients of the various terms and recales by the soft term squared over the $Z$-mass squared: e.g. $c_{M_{Q_3}^2} = 0.73 \cdot (M_{Q_3}^2/M_Z^2)$. For example, if one allows $M_{Q_3} \sim 3$ TeV (in accord with requirements from the measured value of $M_h$) the result is $c_{M_{Q_3}^2} \sim 800$ and so $\Delta_{BG} \ge 800$. In this case, one expects SUSY would be electro-weak fine-tuned to about 0.1%. However, in constrained SUSY models where the high scale parameters are related, then cancellations between positive and negative contributions can occur. For instance, in models with universal scalar masses, then third generation fine-tuning is greatly reduced in the focus point region. More generally, in models of gravity-mediated SUSY breaking, then for any hypothesized hidden sector, the SUSY soft breaking terms are all calculated as numerical co-efficients times the gravitino mass $m_{3/2}$ [965].

These shortcomings can be cured by modifying the definition of the fine-tuning measure. In the calculation of the SUSY mass spectrum, the actual fine-tuning occurs when enforcing the electroweak minimization condition which is

written as

$$ \frac{M_Z^2}{2} = \frac{M_{H_d}^2 + \Sigma_d^d - (M_{H_u}^2 + \Sigma_u^u) \tan^2 \beta}{\tan^2 \beta - 1} - \mu^2. \quad (136) $$

In the above expression, $M_{H_u}^2$ and $M_{H_d}^2$ are weak scale soft SUSY breaking masses while the terms $\Sigma_d^d$ and $\Sigma_u^u$ incorporate a variety of radiative corrections (a complete list of one-loop corrections is provided in Ref. [966].)

For typical SUSY models with parameters defined at some high scale $\Lambda$ (where $\Lambda$ is frequently taken as high as $M_{GUT} \simeq 2 \times 10^{16}$ GeV), the positive value of $M_{H_u}^2$ ($\Lambda$) is driven radiatively to negative values at the weak scale (owing to the large top quark Yukawa coupling) so that electroweak symmetry is radiatively broken. In models where large TeV-scale values of $-M_{H_u}^2$ are generated at the weak scale, then a compensating value of $\mu^2$ must be dialed/tuned to enforce the measured value of $M_Z \simeq 91.2$ GeV.

The amount of fine-tuning required in Eq. 136 can be quantified by defining the electro-weak fine-tuning measure [966–968]

$$ \Delta_{EW} \equiv \max_i |C_i| / (M_Z^2/2), \quad (137) $$

where $C_{H_d} = M_{H_d}^2 / (\tan^2 \beta - 1)$, $C_{H_u} = -M_{H_u}^2 \tan^2 \beta / (\tan^2 \beta - 1)$ and $C_\mu = -\mu^2$. Also, $C_{\Sigma_u^u(k)} = -\Sigma_u^u(k) \tan^2 \beta / (\tan^2 \beta - 1)$ and $C_{\Sigma_d^d(k)} = \Sigma_d^d(k) / (\tan^2 \beta - 1)$, where $k$ labels the various loop contributions included in Eq. 136.

Since $\Delta_{EW}$ depends only upon the weak scale SUSY spectrum, it is model independent (within the MSSM) in that different models giving rise to exactly the same spectrum will have the same values of $\Delta_{EW}$. For models with parameters defined at the weak scale, such as the pMSSM, then $\Delta_{BG} \approx \Delta_{EW}$ since the sensitivity co-efficients $c_\mu = C_\mu$ and $c_{H_u} = C_{H_u}$.

For $\tan \beta \gtrsim 5$ and neglecting radiative corrections, the condition Eq. 136 reduces to $M_Z^2/2 \simeq -M_{H_u}^2 - \mu^2$, so that models with weak scale naturalness require that $-M_{H_u}^2 \sim M_Z^2$ and also $\mu^2 \sim M_Z^2$. The first of these conditions obtains crisis when $M_{H_u}^2$ is driven to small rather than large negative values during the process of radiative electroweak symmetry breaking. The second condition implies a spectrum of light higgsino-like “electroweakinos” (i.e. charginos and neutralinos) with mass the closer to $M_Z$ the better:

$$ m_{\tilde{\chi}_1^\pm}, m_{\tilde{\chi}_{1,2}^0} \sim |\mu| \sim 100-250 \text{ GeV.} $$

Such light higgsinos would be accessible at an $e^+e^-$ linear collider of centre-of-mass energy, $\sqrt{s} = 250-500$ GeV, i.e. exceeding twice their mass. In such a case, then a high energy $e^+e^-$ collider would function as a higgsino factory [969] in addition to a Higgs factory! While such light higgsinos might be produced at some sizeable rates at the LHC, the kinematics of their visible decay products may make it difficult if not impossible to observe them in hadronic collisions.
The compressed spectra reduce the transverse momentum of the produced jets and leptons bringing them below the cuts applied by the triggers and the subsequent offline event selection criteria.

## 5.4 Indirect Constraints

In spite of the many attractive features of SUSY models, no sign of supersymmetric matter has yet emerged and dark matter is still to be observed at ground-based direct detection experiments. Here, we review the constraints on SUSY particle masses and parameters derived from precision measurements of low energy processes and the dark matter relic density. Constraints from the direct search for SUSY particles at the LHC will be addressed in the following Section.

### 5.4.1 Flavour Physics

Flavour physics provides indirect information about supersymmetry which can play an important and complementary role compared to direct searches at colliders. Several decays of b hadrons which are suppressed in the SM may offer sensitivity to SUSY through additional contributions mediated by supersymmetric particles, which do not suffer the same suppression and may substantially modify the decay rate. The main processes of interest are the $\bar{B} \to X_s \gamma$, $B_s \to \mu^+ \mu^-$ and $B_u \to \tau \nu_\tau$ decays.

The decay $\bar{B} \to X_s \gamma$ is a loop-induced flavour changing neutral current (FCNC) process that offers high sensitivity to supersymmetry due to the fact that additional contributions to the decay rate– in which SM particles are replaced by SUSY particles such as charged Higgs, charginos or top squarks– are not suppressed by a loop factor relative to the SM contribution. Within a global effort, a perturbative QCD calculation to the NNLL level has been performed [970], leading to [971]:

$$ \mathrm{BR}(\bar{B} \to X_s \gamma)_{\mathrm{NNLL}} = (3.08 \pm 0.23) \times 10^{-4}, \qquad (138) $$

for a photon energy cut at $E_\gamma = 1.6$ GeV, and using the updated input parameters of PDG [760]. The non-perturbative corrections to this decay mode are sub-leading [972] and their error is included in the above prediction. The averaged experimental value by the HFAG group [973] gives

$$ \mathrm{BR}(\bar{B} \to X_s \gamma)_{\mathrm{exp}} = (3.43 \pm 0.21 \pm 0.07) \times 10^{-4}, \qquad (139) $$

where the first error is the combined statistical and systematic uncertainties and the second represents the photon energy extrapolation. The SM prediction and the experimental average are hence consistent at the $1.2\sigma$ level, and therefore this decay has a restrictive power on the SUSY parameter space. Recently, the first practically complete NLL calculation of the decay rate in the MSSM has been finalised [974].

The dominant SUSY contributions are provided by diagrams with top-squarks and charginos, which grow linearly with $\tan\beta$ [975]. This decay is therefore particularly constraining in the regions with large $\tan\beta$ or spectra with both light top-squarks and charginos. The charged Higgs contributions on the other hand are not $\tan\beta$ enhanced.

Recently, the purely leptonic decay of $B_s \to \mu^+ \mu^-$ has received special attention due to the progress on both experimental results and theory calculations. This rare decay is very sensitive to supersymmetric contributions which are free from the helicity suppression of the SM diagrams. The recent observation of this decay by the LHCb [976] and CMS [977] experiments allows for a combined determination of its branching fraction to be

$$ \mathrm{BR}(B_s \to \mu^+ \mu^-) = (2.9 \pm 0.7) \times 10^{-9}. \qquad (140) $$

While this is in accord with the SM prediction of $(3.53 \pm 0.38) \times 10^{-9}$ [978], it also provides a stringent limit on the viable parameter space of many supersymmetric models. The SUSY contributions to the decay amplitudes are dominated by Higgs-mediated penguin diagrams [979–981] and are proportional to

$$ -\mu A_{\tau} \frac{\tan^3 \beta}{(1 + \epsilon_b \tan \beta)^2} \frac{M_{\tau}^2}{M_{\tilde{\tau}}^2} \frac{M_b M_{\mu}}{4 \sin^2 \theta_W M_W^2 M_A^2}. \qquad (141) $$

The sensitivity of $B_s \to \mu^+ \mu^-$ to SUSY contributions is significant in regions at large $\tan\beta$ and small to moderate $M_A$ values, regions which are also probed by direct SUSY particle searches at ATLAS and CMS, in particular $H/A \to \tau^+ \tau^-$. As a result, while the constraints derived from the current LHCb result remove a large fraction of points at large $\tan\beta$ and low $M_A$, nonetheless for intermediate $\tan\beta$ values and/or large masses of the pseudo-scalar Higgs boson A, the branching fraction in the MSSM does not deviate much from its SM prediction, leaving a sizeable fraction of SUSY parameter regions totally unconstrained [982].

The decay $B \to K^* \mu^+ \mu^-$ gives also access to angular distributions, in addition to the differential branching fraction, and offers a variety of complementary observables. However, these observables suffer from large uncertainties, in particular due to form factors. A set of optimised observables has been defined from ratios of angular coefficients to minimise hadronic uncertainties, while preserving the sensitivity to new physics effects [983, 984]. They have been recently measured by the LHCb collaboration [985] highlighting a tension in several binned observables. While these tensions remain even when including the SUSY contributions, the overall agreement with the MSSM predictions is within $1\sigma$-level for an appropriate choice of the model parameters [986].

Finally, the purely leptonic decay of $B_u \to \tau \nu_\tau$ is sensitive to supersymmetry through the exchange of a charged Higgs boson already at tree level, which does not suffer from
the helicity suppression of the SM contribution with the exchange of a W boson. The branching ratio of $B_u \to \tau\nu\tau$ in supersymmetry relative to the SM is given by

$$ \frac{\mathrm{BR}(B_u \to \tau\nu\tau)_{\mathrm{MSSM}}}{\mathrm{BR}(B_u \to \tau\nu\tau)_{\mathrm{SM}}} = \left[ 1 - \frac{m_B^2}{M_{H^+}^2} \frac{\tan^2 \beta}{1 + \varepsilon_0 \tan \beta} \right]^2, \quad (142) $$

where $\varepsilon_0$ is an effective coupling parametrising the non-holomorphic correction to the down-type Yukawa coupling induced by gluino exchange. This decay is therefore also very sensitive to the MSSM parameter region at large $\tan\beta$ and small $M_{H^+}$ values, and much less sensitive to other SUSY parameters. The branching fraction for the decay is calculated in the SM to be $(1.10 \pm 0.29) \times 10^{-4}$ [987], which exhibits a slight tension with the experimental averaged value of $(1.14 \pm 0.22) \times 10^{-4}$ [973].

### 5.4.2 Muon Magnetic Moment

The SUSY contribution to the muon magnetic moment is given by [988]

$$ \Delta a_{\mu}^{\mathrm{SUSY}} \propto \frac{M_{\mu}^{2} \mu M_{i} \tan \beta}{M_{\mathrm{SUSY}}^{4}} \quad (143) $$

where $i = 1, 2$ stands for electroweak gaugino masses and $M_{\mathrm{SUSY}}$ is the characteristic sparticle mass circulating in the muon-muon-photon vertex correction: $M_{\tilde{\mu}_{L,R}}$, $M_{\tilde{\nu}_{\mu}}$ and $M_{\tilde{\chi}_i}$.

The anomalous magnetic moment of the muon $a_\mu \equiv \frac{(g-2)_\mu}{2}$ was measured by the Muon g-2 Collaboration [757] which gives a $3.6\sigma$ discrepancy when compared to the SM calculations based on $e^+e^-$ data [817], $\Delta a_\mu = a_\mu^{\mathrm{meas}} - a_\mu^{\mathrm{SM}}[e^+e^-]$ (28.7 $\pm$ 8.0) $\times 10^{-10}$. As discussed in more detail in Chapter 4, the SM prediction depends on the estimate of the hadronic vacuum polarisation contribution. Using $\tau$-decay data rather than low energy $e^+e^-$ annihilation data reduces the discrepancy to $2.4\sigma$ giving $\Delta a_\mu = a_\mu^{\mathrm{meas}} - a_\mu^{\mathrm{SM}}[\tau] = (19.5 \pm 8.3) \times 10^{-10}$.

Attempts to explain the muon g-2 anomaly using supersymmetry usually invoke sparticle mass spectra with relatively light smuons and/or large $\tan\beta$ (see e.g. Ref. [989]). Some SUSY models where $M_{\tilde{\mu}_{L,R}}$ is correlated with squark masses (such as mSUGRA) are now highly stressed to explain the $(g-2)_\mu$ anomaly, given the bounds from the LHC direct searches. In addition, since naturalness favours a low value of $|\mu|$, tension again arises between a large contribution to $\Delta a_\mu^{\mathrm{SUSY}}$ and naturalness conditions. The current $3\sigma$-deviation is clearly not sufficient to prove the existence of new physics, but in the future, progress can be expected both on the experimental side (due to a new measurement at Fermilab with fourfold improved precision [191]) as well as on the theoretical side [990, 991].

### 5.4.3 Dark Matter and Cosmological Constraints

During the past several decades, a very compelling and simple scenario has emerged to explain the presence of dark matter in the universe with an abundance roughly five times that of ordinary baryonic matter. The WIMP miracle scenario posits that weakly interacting massive particles would be in thermal equilibrium with the cosmic plasma at very high temperatures $T \ge M_{\mathrm{WIMP}}$. As the universe expands and cools, the WIMP particles would freeze out of thermal equilibrium, locking in a relic abundance that depends inversely on the thermally-averaged WIMP (co)-annihilation cross section [992, 993]:

$$ \Omega_{\chi} h^2 \approx \frac{s_0}{\rho_c / h^2} \left( \frac{45}{8\pi^2 g_*} \right)^{1/2} \frac{x_f}{M_P} \frac{1}{\langle \sigma v \rangle} \quad (144) $$

where $s_0$ is the present entropy density, $\rho_c$ is the critical closure density, $g_*$ measures the degrees of freedom, $x_f = m/T_f$ is the inverse freeze-out temperature rescaled by the WIMP mass, $M_P$ is the reduced Planck mass and $\langle \sigma v \rangle$ is the thermally averaged WIMP annihilation cross section with $v$ being the WIMP relative velocity. The WIMP “miracle” occurs in that a weak strength annihilation cross section gives roughly the measured relic abundance provided the WIMP mass is also of order the weak scale [994].

The lightest neutralino of SUSY models has been touted as a prototypical WIMP candidate [995–997]. The precise determination of the dark matter relic density, $\Omega_{\mathrm{CDM}}h^2$, obtained from the cosmic microwave background (CMB) by the WMAP satellite experiment first [998] and the Planck mission [999], now stands as a reference constraint for SUSY models. While the comparison of the measured abundance of CDM with the neutralino dark matter relic density, $\Omega_{\chi}h^2$, computed in an assumed SUSY scenario, is affected by cosmological uncertainties which may be large [1000], it is certainly appropriate to require at least that SUSY models do not violate the upper bound on the CDM abundance, after accounting for these uncertainties. An predicted overabundance of thermally-produced WIMPs may in fact be allowed in some specific models with either R-parity violating WIMP decays, late WIMP decays to an even lighter LSP (e.g. axino or gravitino) or by late time entropy injection from moduli or saxion decays.

Despite the WIMP “miracle”, SUSY theories where the lightest neutralino plays the role of a thermally produced WIMP, have a relic abundance $\Omega_{\chi}h^2$ spanning over a broad range of values from several orders of magnitude larger than the value derived from the CMB spectrum in the case of a bino-like neutralino, and up to two-to-three orders of magnitude lower in the case of wino- or higgsino-like neutralinos [1001] with a mass of order 100 GeV, see Fig. 122. A wino- or higgsino-like neutralino LSP in the generic MSSM gives a relic density compatible with the CMB data for masses
in the range 0.9–3 TeV, while bino-like or mixed neutrali-
nos may match the CMB data for lighter masses. A deficit
is, in principle, acceptable since the neutralino may not be
the only source of dark matter and its relic density should
not necessarily saturate the measured value. As an example,
in the case of the axion solution to the strong CP problem
within the SUSY context, dark matter is due to a mixture
of axions and neutralinos [1002]. For the case of bino-like
LSPs where the abundance might be expected to exceed the
WMAP/Planck value, then an efficient annihilation mech-
anism – such as co-annihilation, resonance annihilation or
mixed bino-higgsino or mixed wino-bino annihilation – is
needed. Such enhanced annihilation mechanisms define spe-
cific patterns of the masses of one or more SUSY particles
compared to the lightest neutralino, which are important for
searches at colliders.

<figure><img src="image_126.png" /><figcaption>**Fig. 122** Neutralino relic density as a function of the neutralino LSP mass from a scan of the pMSSM parameter space. The colours indicate the nature of the neutralino LSP with the largest occurrence in each bin.</figcaption></figure>

The relic abundance constraint is now complemented
by upper limits on WIMP-nucleon scattering cross sections
from underground dark matter direct detection experiments.
The $\tilde{\chi}p$ spin-independent scattering process receives SUSY
contributions from scalar quark exchange and t-channel
Higgs exchange [997]. The latter dominates over a vast re-
gion of the parameter space. The scattering cross section re-
tains a strong sensitivity on the scalar Higgs boson mass and
tan $\beta$ [1005]. Limits on spin-independent $\chi$-nucleon scatter-
ing from the initial run of the LUX experiment [1004] are
shown in Figure 123 along with some expected SUSY pa-
rameter space.

There is a large number of recent results reported by
experiments using crystals [1006, 1007], semiconductors
[1008, 1009] and noble gases [1004, 1010] as sensitive mate-
rial. The excess of events reported by some of these experi-

ments [1006,1007,1009,1011], which would appear to point
to a very light WIMP, are confronted by the stringent limits
set by negative results in the searches by the Xenon-based
detectors, Xenon-100 [1012] and LUX [1004]. These limits
are cutting into the region of scattering cross sections typi-
cal of the MSSM (see Fig. 124) and therefore provide some
meaningful bounds, even if systematics and model depen-
dencies due to the assumed dark matter profile in the galaxy
are known to be sizeable [1013]. In particular, the Xenon-
100 and LUX bounds– if taken at face value– exclude a size-
able fraction of the viable SUSY points with neutralino dark
matter at small values of the $\mu$ and $M_2$ parameters, which

<figure><img src="image_127.png" /><figcaption>**Fig. 123** Limits on the $\chi-p$ spin-independent scattering cross section vs. the $\chi_1^0$ mass. The shaded regions include MSSM points compatible with recent LHC SUSY searches and Higgs mass results [1003]. Also indicated is the most stringent recent limit from the LUX experiment [1004].</figcaption></figure>

<figure><img src="image_128.png" /><figcaption>**Fig. 124** Neutralino-nucleon spin-independent scattering cross section vs. the $\chi_1^0$ mass. The colours indicate the nature of the neutralino LSP with the largest occurrence in each bin</figcaption></figure>
would give chargino and neutralino pair production observables at a linear collider with $\sqrt{s}$ below 1 TeV and small fine tuning, as discussed above. In the case where WIMPs make up only a portion of the total dark matter abundance (perhaps the bulk is composed of axions), then these direct detection predictions would have to be rescaled by a factor $\xi = \Omega_{\chi}^{TP}h^2/0.12$ in which case the search limits are much less constraining.

In gravity-mediation, the gravitino mass sets the scale for the soft breaking terms so that one expects gravitinos to have mass comparable to the SUSY partners. While gravitinos may decouple from collider physics, they can be produced at large rates proportional to $T_R$ in the early universe. The gravitino decay rate to SUSY particles is suppressed by $1/M_P^2$ so that they may decay well after BBN has started, thus upsetting the successful prediction of light element production from Big Bang Nucleosynthesis [1014]. To avoid this so-called “gravitino problem” [1015], one typically requires $T_R \lesssim 10^5$ GeV for $m_{3/2} < 5$ TeV. Alternatively, if the gravitino is very heavy—$m_{3/2} \gtrsim 5$ TeV—then gravitinos typically decay before the onset of BBN. In addition, overproduction of gravitinos may lead to overproduction of LSPs from gravitino decay. To avoid over-production of WIMPs arising from thermally-produced gravitinos, one must typically obey the less restrictive bound $T_R \lesssim 10^9$ GeV.

Besides the case of neutralino dark matter, it is possible that gravitinos are the lightest SUSY particles, and could be responsible for dark matter. The case of gravitino LSPs with a weak scale value of $m_{3/2}$ is called the super-WIMP scenario and is again highly restricted by BBN bounds on late decaying WIMP to gravitino decays. Also, superWIMP gravitino LSPs can be thermally overproduced as dark matter unless constraints are again imposed on the reheating temperature [1016–1018]. For weak-scale gravitino dark matter, a reheating temperature above $10^9$ GeV can only be achieved in small corners of model parameter space which impose strict bounds on the superparticle mass spectrum [1019].

Alternatively, the gravitino mass might be far below the weak scale; this scenario is a viable option and occurs naturally in GMSB scenarios. For such a small gravitino mass, the goldstino couplings are enhanced which helps to evade the BBN constraints on NLSP decay to gravitinos. In addition, expectations for thermal overproduction of gravitino dark matter in GMSB are modified and can depend as well on the messenger mass scale [1020, 1021].

## 5.5 Constraints from LHC

The searches performed by ATLAS and CMS on the 7 and 8 TeV LHC data in channels with jets, leptons and missing transverse energy (MET) have already significantly re-

shaped our views of the high energy frontier in relation to SUSY. Searches for signatures of production and decay of supersymmetric particles with large MET have failed to reveal any significant excess of events compared to SM expectations.

A variety of final states have been probed in LHC searches which are sensitive to the production and decay modes of both strongly- and weakly-interacting SUSY particles. The results of searches for gluinos and squarks of the first two generations are easy to interpret in generic models. The analyses of the almost 25 fb⁻¹ of combined 7 TeV and 8 TeV have led to mass limits in the range of $m_{\tilde{g}} \gtrsim 1–1.3$ TeV and $m_{\tilde{q}} \gtrsim 0.4–1.8$ TeV for scalar quarks of the first two generations. There is an important exception to these limits which originates from scenarios with compressed spectra giving rise to highly degenerate masses and correspondingly low transverse energies from the produced jets and leptons: the visible energy from such compressed spectra often falls below analysis cuts or even the trigger thresholds, which causes generic LHC limits to collapse.

<figure><img src="image_129.png" /><figcaption>**Fig. 125** 95% CL exclusion limits for MSUGRA/CMSSM models with $\tan\beta = 30$, $A_0 = -2m_0$ and $\mu > 0$ presented in the [$M_0, M_{1/2}$] plane obtained by the ATLAS experiment with 20 fb⁻¹ of data at 8 TeV (from [1022]).</figcaption></figure>

These results have rapidly excluded most of the benchmark points adopted in the last two decades of SUSY studies and have put significant pressure on highly constrained SUSY models such as the CMSSM/mSUGRA model (discussed above) where SUSY soft terms are unified at a high scale. In fact, the LHC searches have excluded regions of parameter space which had been clearly preferred by fits performed on the pre-LHC data, pushing the masses of squarks and gluinos beyond 1-2 TeV (see Figure 125). To further aggravate the crisis of such highly constrained models, it has also become difficult to accommodate a lightest Higgs boson with mass $\sim 125$ GeV in the CMSSM, except for very spe-