One loop effects of natural SUSY in indirect searches for SUSY particles at the ILC

TL;DR

This paper investigates the one-loop effects of supersymmetric particles in various processes at the ILC within the minimal supersymmetric standard model, considering experimental constraints from Higgs, flavor, muon g-2, dark matter, and LHC searches.

Contribution

It provides numerical calculations of one-loop SUSY effects on specific processes at the ILC using benchmark points consistent with current experimental bounds.

Findings

Identified MSSM parameter regions compatible with experimental data.

Calculated one-loop SUSY effects on $ au^+ au^-$, $b ar{b}$, $t ar{t}$, and $Z h$ production.

Demonstrated potential for indirect SUSY detection at the ILC.

Abstract

We have found the possible region of parameters of the minimal supersymmetric standard model (MSSM) within the bounds from the experimental results of the Higgs mass, the rare decay mode of $b$-quark, the muon $g-2$, the dark matter abundance, and the direct searches for the lighter stop (i.e., one of the supersymmetric partners of top quark) at the LHC. We present numerical results of calculations for the one loop effects of supersymmetric particles in the processes of $\tau^+ \tau^-$, $b \overline{b}$, $t \overline{t}$, and $Z h$ production at the ILC by using benchmark points within the possible region of the MSSM parameters.

Tables

Table 1. Table 1 : Three scenarios for the stau mass and the stop mass.

	Light stop	Heavy stop
$m_{{\tilde{\tau }}_1}\cong m_{{\tilde{\chi }}_1^0}$	stau–LSP co-annihilation	stau–LSP co-annihilation
$m_{{\tilde{t}}_1}\cong m_{{\tilde{\chi }}_1^0}$	stop–LSP co-annihilation	—

Equations

$$m_h^{\exp }=125.09\pm 0.21\pm 0.11\mathrm{GeV}.$$

$$X_t\equiv \frac{m_{{\tilde{t}}_2}^2-m_{{\tilde{t}}_1}^2}{m_t}=2(A_t-\mu \cot \beta ).$$

$$\mathrm{Br}(B\to X_s\gamma )=(3.43\pm 0.21\pm 0.07)\times 10^{-4}.$$

$$\begin{array}{rl}{a}_{\mu }^{\exp }& =(1165920.91\pm 0.54\pm 0.33)\times 10^{-9},\\ a_{\mu }^{\mathrm{SM}}& =(1165918.03\pm 0.01\pm 0.42\pm 0.26)\times 10^{-9},\\ \Delta a_{\mu }& =(2.88\pm 0.63\pm 0.49)\times 10^{-9}.\end{array}$$

$${\Omega }_{\mathrm{DM}}^{\exp }{h}^2=0.1188\pm 0.0010,$$

$${\delta }_{\mathrm{susy}}^{\mathrm{EW}}$$

$${\delta }_{\mathrm{susy}}^{\mathrm{QCD}}$$

$${\delta }_{\mathrm{susy}}$$

$$(\mathrm{i})$$

$$(\mathrm{ii})$$

$$(\mathrm{iii})$$

$$(\mathrm{iv})$$

Full text

1 Introduction

The standard model (SM) of particle physics is considered to be an effective theory despite the fact that it has succeeded in describing known experimental data available up to now. Supersymmetry (SUSY) between bosons and fermions at the unification-energy scale is believed to be one of the most promising extension of the SM. Among the supersymmetric theories, the minimal supersymmetric extension of the SM (MSSM) is a well studied framework of SUSY because of its compactness.

In the MSSM, however, there are many unfixed parameters. For limiting the possible region of the MSSM parameters, a promising approach is so-called “natural SUSY”. In the framework of the natural SUSY, a light stop with a large A-term and light higgsinos give a solution of fine-tunings in the MSSM[1]. We consider that using experimental results is the top priority for limiting the MSSM parameters, and investigate the possibility of survival of a light stop.

Recently, we have found the possible region of the MSSM parameters[2] within the bounds from the experimental results of (i) the Higgs mass[3], (ii) the branching ratio, $\mathrm{Br}(b\rightarrow s\gamma)$[4], (iii) the muon $g-2$[5], (iv) the dark matter (DM) abundance[6, 7], and (v) the direct searches for the lighter stop at the LHC[8, 9, 10, 11, 12, 13]. In the parameter searches, we use SuSpect2[14], SUSY-HIT[15], and micrOMEGAs[16] for (i)–(iv).

Moreover, we have studied indirect searches for SUSY particles at the ILC by using benchmark points within the possible region of the MSSM parameters[2, 17, 18]. We have calculated the 1-loop effects of SUSY particles in the processes, $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$, $e^{+}e^{-}\rightarrow b\overline{b}$, $e^{+}e^{-}\rightarrow t\overline{t}$[19], and $e^{+}e^{-}\rightarrow Zh$[20, 21] with the aid of GRACE/SUSY-loop[22].

2 MSSM parameter searches

Our criterion for searching the MSSM parameters is that the MSSM effects are within the bounds from the following experimental results:

1. (i)

the Higgs mass, $m_{h}$, 2. (ii)

the branching ratio of the rare decay mode of $b$-quark, $\mathrm{Br}(b\rightarrow s\gamma)$, 3. (iii)

the muon anomalous magnetic moment, $a_{\mu}$, 4. (iv)

the DM abundance, $\mathnormal{\Omega}_{\mathrm{DM}}h^{2}$, 5. (v)

the lower bounds of the lighter stop mass, $m_{\tilde{t}_{1}}$ in the direct searches at the LHC.

We have performed systematic multidimensional scans in the entire range of the MSSM parameter space[23], but we present only the results which are consistent with the five observables above.

2.1 Higgs mass

The experimental value of the Higgs mass is measured[3] as

[TABLE]

The MSSM contribution to the Higgs mass mainly depends on the masses of stops, $m_{\tilde{t}_{1}}$, $m_{\tilde{t}_{2}}$, the A-term, $A_{t}$, the higgsino mass parameter, $\mu$, and $\tan\beta$[24, 25, 26, 27, 28]. An alternative parameter, $X_{t}$ is useful due to the relation among the parameters,

[TABLE]

Figure 1 shows the possible contours on the $X_{t}$–$m_{\tilde{t}_{1}}$ plane which is consistent with (1) for $\tan\beta=30$ and $\mu=600~{}\mathrm{GeV}$. The reason for the large value of $\mu$ here is explained in the next subsection. For the allowed mass of the lighter stop, $m_{\tilde{t}_{1}}$, there are two possibilities as follows:

1. (i)

the light stop scenario, $m_{\tilde{t}_{1}}\lesssim 1.5~{}\mathrm{TeV}$ with $X_{t}=-5~{}\mathrm{TeV~{}to~{}}-2~{}\mathrm{TeV}$, 2. (ii)

the heavy stop scenario, $m_{\tilde{t}_{1}}>1.5~{}\mathrm{TeV}$ with $X_{t}\gtrsim-4~{}\mathrm{TeV}$.

2.2 Rare decay mode of $\boldsymbol{b}$-quark

At any allowed point in in Figure 1, the value of $A_{t}$ is severely restricted by the Higgs mass. Figure 2(a) shows the dependence of the Higgs mass, $m_{h}$ on $A_{t}$. Thus, we obtain a solution, $A_{t}=-2.4~{}\mathrm{TeV}$ for $m_{h}=125~{}\mathrm{GeV}$.

The MSSM parameters are also constrained by the experimental value of the branching ratio[4],

[TABLE]

Figure 2(b) shows the dependence of the branching ratio, $\mathrm{Br}(b\rightarrow s\gamma)$ on $A_{t}$. Thus, we obtain a constraint, $\mu>0.5~{}\mathrm{TeV}$ for $A_{t}=-2.4~{}\mathrm{TeV}$.

2.3 Muon $\boldsymbol{g-2}$

The muon anomalous magnetic moment, $a_{\mu}\equiv(g_{\mu}-2)/2$ has been accurately measured, thus causes the MSSM contribution to be restricted[29, 30]. The experimental value, $a_{\mu}^{\mathrm{exp}}$, the SM prediction, $a_{\mu}^{\mathrm{SM}}$, and the difference, $\Delta a_{\mu}=a_{\mu}^{\mathrm{exp}}-a_{\mu}^{\mathrm{SM}}$[5] are

[TABLE]

The MSSM contribution to $a_{\mu}$ depends on the slepton mass, $m_{\tilde{l}_{L}}$, $\tan\beta$, $\mu$, the bino mass, $M_{1}$, and the wino mass, $M_{2}$[31].

Figure 3 shows the allowed region on the $m_{\tilde{l}_{L}}$–$M_{1}$ plane for $\tan\beta=30$, $\mu=600~{}\mathrm{GeV}$, and $M_{2}=2M_{1}$. It is remarkable that the bino mass is constrained as $M_{1}\lesssim 360~{}\mathrm{GeV}$, thus the LSP is binolike due to $M_{1}<\mu$. Three red points numbered 1 to 3 are the benchmark points which are defined in section 2.5[2, 17, 18].

2.4 DM abundance

The lightest SUSY particle (LSP) is a promising candidate for the DM. We consider the lightest neutralino, $\tilde{\chi}_{1}^{0}$ as the LSP. The DM abundance can be explained by the LSP-pair annihilation[32], the stau–LSP co-annihilation[33], and the stop–LSP co-annihilation[34, 35]. There are three scenarios as in Table 1 by combining the light or the heavy stop with the mass degeneracy of stau–LSP or stop–LSP. If the masses of stau, stop, and LSP degenerate, both co-annihilation processes occur.

The observed value of the DM abundance[6, 7],

[TABLE]

restricts the MSSM parameters within the narrow region.

Figure 4 shows the projection of the MSSM parameter scan on the $m_{\tilde{t}_{R}}$–$\mathnormal{\Omega}_{\mathrm{DM}}h^{2}$ plane. The data are plotted for two values of the stau mass, $m_{\tilde{\tau}_{R}}$.

Figure 5 shows the projection of the MSSM parameter scan on the $m_{\tilde{t}_{1}}$–$m_{\tilde{\tau}_{1}}$ plane which is consistent with (5) in the light stop scenario for $M_{1}=220~{}\mathrm{GeV}$. The data within the red ellipse are consistent with the five observables as mentioned in the opening paragraph of section 2. In the allowed region, the lighter stop mass and the lighter stau mass are restricted as $m_{\tilde{t}_{1}}=300~{}\mathrm{to}~{}400~{}\mathrm{GeV}$ and $m_{\tilde{\tau}_{1}}=225~{}\mathrm{to}~{}235~{}\mathrm{GeV}$. These mass ranges depend on $M_{1}$. There, however, is a room for the heavy stop scenario if $m_{\tilde{\tau}_{1}}\cong m_{\tilde{\chi}_{1}^{0}}$.

2.5 Direct searches at the LHC

In the direct searches of SUSY particles at the LHC, there has been no evidence of SUSY particle production up to now. We, however, still have a possible region on the MSSM parameters which has not been excluded by experiments.

Figure 6 shows the mass spectra for the benchmark points which we take from the possible region of the MSSM parameters corresponding to three scenarios in Table 1. The MSSM parameter sets named set1 and set3 are for the light stop scenario, and that named set2 is for the heavy stop scenario.

Figure 7 shows the mass bounds of the lighter stop, $\tilde{t}_{1}$ from the direct searches at the LHC[8, 9, 10, 11, 12, 13]. The two benchmark points plotted in the figure are located outside of the excluded regions. Thus we find that two kinds of the light stop scenarios and the heavy stop scenario still survive.

3 Numerical results

For indirect searches of SUSY particles, SUSY effects are revealed through 1-loop corrections. There, however, exist the SM contributions to 1-loop corrections, thus the following quantities are useful[19].

[TABLE]

For the indirect searches of SUSY particles, we consider the following processes:

[TABLE]

where the center-of-mass energy, $E_{\mathrm{cm}}$ is determined by the tree level cross sections as in Figure 8 and by the technical design of the ILC[36].

Figure 9(a) and Figure 9(b) show the angular distributions of $\delta_{\mathrm{susy}}$ in the $\tau^{+}\tau^{-}$-pair production process and the $b\overline{b}$-pair production one, respectively. It is remarkable that SUSY effects can be revealed as excess of the SM estimation at the 1-loop level, because the value of $\delta_{\mathrm{susy}}$ is larger than the expected error bars. For the integrated luminosity, $\int Ldt=250\mathrm{fb^{-1}}$, however, the parameter set of the light stop scenario, set1 cannot be discriminated from that of the heavy stop scenario, set2 because of the mass relations, $m_{\tilde{\tau}_{1}}\cong m_{\tilde{\chi}_{1}^{0}}$ and $m_{\tilde{b}_{1}}\gg E_{\mathrm{cm}}$ in both parameter sets.

Figure 10 shows the angular distribution of $\delta_{\mathrm{susy}}$ in the $t\overline{t}$-pair production process. There is small difference between the $\delta_{\mathrm{susy}}$’s of the two parameter sets in the forward direction of top quark.

Figure 11 shows the angular distribution of $\delta_{\mathrm{susy}}$ in the $Zh$ production process. There is difference between the $\delta_{\mathrm{susy}}$’s of the two parameter sets, which is larger than the range of error bars. It is remarkable that the 1-loop corrections of this process are sensitive to the lighter stop mass, $m_{\tilde{t}_{1}}$.

4 Summary

We have obtained possible MSSM parameter sets which are consistent with the bounds from the experimental results of the Higgs mass, the rare decay mode of $b$-quark, the muon $g-2$, the dark matter abundance, and the direct searches for the lighter stop at the LHC. For the parameter sets, the virtual effects of MSSM in the indirect search processes are estimated at 1%$\ \sim 5$% in the early stage of the ILC, which are larger than the ranges of error bars. For the purpose of discrimination of the light and heavy stop scenarios, $Zh$ is the most promising process to investigate.