Reconstructing Supersymmetric Contribution to Muon Anomalous Magnetic Dipole Moment at ILC

We study the possibility to determine the supersymmetric (SUSY) contribution to the muon anomalous magnetic dipole moment by using ILC measurements of the properties of superparticles. Assuming that the contribution is as large as the current discrepancy between the result of the Brookhaven E821 experiment and the standard-model prediction, we discuss how and how accurately the SUSY contribution can be reconstructed. We will show that, in a sample point, the reconstruction can be performed with the accuracy of ~ 13 % with the center-of-mass energy 500 GeV and the integrated luminosity ~ 500-1000 fb-1.

Based on the analysis of Refs. [2] and [3] for the hadronic vacuum polarization, the predictions are where we take account of the five-loop QED calculation [4] and the latest update of the electroweak contribution [5]. Thus, the difference is estimated as Hence, there exists more than 3-σ discrepancy between the experimental and theoretical values. We call this discrepancy as "muon g − 2 anomaly." The origin of the muon g − 2 anomaly is yet unknown. If low-energy supersymmetry (SUSY) exists, the SUSY contribution to the muon g − 2, denoted as a (SUSY) µ , can be sizable. In particular, when tan β, which is a ratio of the vacuum expectation values of up-and down-type Higgses, is relatively large, a (SUSY) µ can be easily as large as ∆a µ [6][7][8]. Thus, it is possible that the muon g − 2 anomaly originates in the SUSY contribution. The primary purpose of this letter is to point out that we may have a chance to test this possibility by reconstructing a (SUSY) µ , if superparticles are found in future collider experiments, and if their properties are determined.
At the leading order, the SUSY contribution to the muon g − 2 is composed of smuonneutralino and sneutrino-chargino loop diagrams. In order to reconstruct a (SUSY) µ , it is necessary to understand properties of sleptons, in particular, those of smuons. Unfortunately, they may not be well studied at LHC. On the contrary, once the International e + e − Linear Collider (ILC) [9] is built, it is possible to determine them precisely as long as the superparticles are within the kinematical reach.
In this letter, we raise a question how and how accurately the SUSY contribution to the muon g − 2 can be reconstructed by using ILC measurements of the parameters of the minimal supersymmetric standard model (MSSM). We assume that the muon g − 2 anomaly is due to the SUSY contribution. Since the contribution depends on MSSM parameters, we concentrate on a particular case where it is dominated by so-called Bino diagram. Such a setup is especially interesting, because sleptons are expected to be within the kinematical reach of ILC [10]. It will be shown that a (SUSY) µ can be reconstructed with the accuracy of ∼ 13 % for the sample point we adopt, once ILC runs at the center-of-mass energy √ s = 500 GeV and accumulates the integrated luminosity L ∼ 500-1000 fb −1 .

Framework
Let us first summarize the framework of the analysis. The SUSY contribution to the muon g − 2 strongly depends on MSSM parameters. In this letter, we concentrate on the case where it is dominated by so-called Bino diagram. This situation is realized if the Wino and Higgsino mass parameters are much larger than the Bino mass parameter. In this limit, the leading contribution is given by (cf. Ref. [8]) In the expression, M 1 is the Bino mass parameter, mμ A (A = 1, 2) is the A-th lightest smuon mass, and g Y is the gauge coupling constant for U(1) Y , which comes from the Bino-(s)muons interactions. Also, m 2 µLR is the left-right mixing parameter in the smuon mass matrix. The loop function f N is defined as It is notable that a (B) µ can be as large as ∆a µ especially when the Higgsinos are heavy, since m 2 µLR is enhanced when µ tan β is large, where µ is the Higgsino mass parameter. In contrast, the other contributions to the muon g −2, including those from the second-lightest or heavier neutralino, are suppressed if the Higgsinos are decoupled.
The contribution a (B) µ can be reconstructed if the Bino mass, smuon masses, and the left-right mixing parameter m 2 µLR are known. As we will see below, they are expected to be determined very accurately at ILC, if the sleptons and the Bino-like neutralino are within the kinematical reach. In fact, since the ILC measurements are very precise, the leading approximation given in Eq. (4) may not be accurate enough to be compared with the ILC analyses. In addition, there is a subtlety in relating the gaugino coupling constants with the gauge coupling constants in particular when some of the superparticles are relatively heavy [10][11][12][13][14]. Thus, we will use more complete formula for a (SUSY) µ . The full one-loop level formula for a (SUSY) µ consists of the contribution from smuonneutralino loop diagrams and sneutrino-chargino diagrams. The smuon-neutralino contribution is given by [8] which includes the leading contribution a (B) µ . Here, mχ0 X (X = 1-4) is the neutralino mass, x AX = m 2 χ 0 X /m 2 µA , and the loop functions are In addition, N µ L AX and N µ R AX are neutralino-muon-smuon coupling constants. Parameterizing interactions of neutralinos as the coefficients are Here, y ℓ is the Yukawa coupling constants in the superpotential. The unitary matrices U χ 0 and Ul diagonalize the mass matrices of neutralinos and sleptons, respectively. It is assumed that soft SUSY breaking parameters of the sleptons are independent of the generation, and all the complex phases of the SUSY parameters are negligibly small, in order to avoid too large lepton-flavor violations and electric dipole moments. Then, the slepton masses are obtained from the slepton mass matrix, which is diagonalized by the following unitary matrix, Ul = cos θl sin θl − sin θl cos θl .
The slepton mixing angle satisfies the relation, This relation will play an important role in the following discussion. It should be noticed that the coupling constants for the gaugino-lepton-slepton vertices, or the gaugino coupling constants, deviate from the ordinary gauge coupling constants [10][11][12][13][14]. In Eqs. (10) and (11), the parametersg Y,L ,g Y,R , andg 2 are introduced to take account of such an effect. In the SUSY limit,g Y,L =g Y,R = g Y andg 2 = g 2 are satisfied (with g Y and g 2 being the gauge coupling constants of U(1) Y and SU(2) L , respectively). These relations are violated when some superparticles are (much) heavier than the sleptons. In the case where all the superparticles except for sleptons and the Bino are heavy, we obtain the following approximate formula forg Y,L andg Y,R (cf. Ref. [10,15]): where M soft is a mass scale of colored superparticles and heavy Higgses, MH is the Higgsino mass, MW is the Wino mass, and Q (∼ ml) is an energy scale. The differences among g Y , g Y,L andg Y,R can be O(1-10)% if M soft , MH and MW are larger than ∼ 1 TeV. Note that the leading contribution of (6) is proportional to the productg Y,LgY,R (cf. Eq. (4)). Since the corrections to the gaugino couplings can be sizable, both of the couplings should be determined directly at ILC. It is also noted thatg Y,L ,g Y,R andg 2 are universal for (at least) light generations. In the following discussion, we choose a specific sample point to make our discussion concrete and quantitative. The mass spectrum at the sample point is summarized in Table 1. All the sleptons and the lightest neutralino are within the reach of ILC with √ s = 500 GeV.
Their masses are set to be close to those of the SPS1a ′ benchmark point [16], so that results of the previous ILC studies can be applied. The lighter sleptons are chosen to be almost left-handed in order to avoid LHC limits (see below). The lightest neutralino mass is 90 GeV, which is the lightest superparticle among the MSSM ones including sneutrinos. Other superparticles such as colored ones as well as Winos and Higgsinos are assumed to be so heavy that they are not observed at LHC nor ILC (so that their masses are different from those for SPS1a ′ ). #1 Trilinear couplings of sleptons, Al, are set to be zero. The left-right mixing parameter, m 2 µLR , (or equivalently µ tan β) is chosen to realize that a (ILC) µ defined in Eq. (17) becomes equal to 2.6 × 10 −9 , which is close to the central value of the current discrepancies (3); µ tan β = 6.1 × 10 3 GeV.
The mass spectrum is consistent with present collider limits. Light sleptons decaying to the lightest neutralino are searched for by studying the di-lepton signatures at LHC [17,18]. Our sample point is not excluded because masses of the left-handed selectron and smuon are #1 This setup is minimal to reconstruct the SUSY contributions to the muon g − 2. If some of the heavy superparticles such as Winos would be additionally discovered, the reconstruction could be improved. close to that of the neutralino. Also, constraints on the right-handed ones are weak, since the production cross sections are small. On the other hand, collider limits on the stau mass is weaker as mτ 1 > 81.9 GeV at 95 % CL by LEP [19]. Exclusions from the three-lepton searches at LHC [18,20] are also negligible, since Winos and Higgsinos are heavy.

Fun with ILC
In the rest of this letter, we discuss how and how accurately the SUSY contribution to the muon g − 2 is determined at ILC. At the sample point, only the sleptons and the lightest neutralino are within the reach of ILC. The observed neutralino is identified as Bino-like by absent signals of charginos, since neutral Winos or Higgsinos are associated by charged partners. Let us define the following quantity (cf. Eq. (6)), which depends only on ILC observables. The parameters are defined aŝ The smuon mixing angle can be determined if the left-right mixing parameter of the smuon, m 2 µLR , as well as the smuon mass eigenvalues are determined, as noticed from Eq. (14). Thus, a (ILC) µ can be reconstructed if the following quantities are known: In the following of this section, we consider the reconstruction of a

Determination of the left-right mixing
One of the crucial parameters to calculate a (ILC) µ is the left-right mixing parameter m 2 µLR . In order to reconstruct the smuon unitary matrix Uμ, it is necessary to determine the mixing angle θμ or m 2 µLR . Although smuons are produced at ILC, it is challenging to determine them from smuon measurements. Importantly, however, m 2 µLR can be obtained from studies of staus. The mixing parameters are scaled by the lepton masses as This relation is valid in the limit of Al ≪ µ tan β, where Al is the trilinear coupling constant of the sleptonl normalized by the corresponding Yukawa coupling constant. This is the case at our sample point. Using Eq. (14), m 2 τ LR is determined if sin 2θτ as well as the mass eigenvalues, mτ 1 and mτ 2 , are measured. Its accuracy is estimated as where the derivatives are evaluated at the sample point. In particular, sin 2θτ can be naturally as large as O(0.1) in the parameter region where a (SUSY) µ ≃ ∆a µ . First of all, the stau mass eigenvalues, mτ 1 and mτ 2 , can be determined by measuring the endpoints of the energy distribution of τ decay products from the stau decay,τ ± → τ ±χ1 0 [12]. For such an analysis, information about the lightest neutralino mass is also needed; measurement of mχ0 1 will be discussed in the next subsection. The measurement of stau masses at ILC is discussed in detail in Ref. [21]. It is claimed that the mass can be determined with the accuracy of ∼ 0.1 % (3 %) for lighter (heavier) stau with √ s = 500 GeV, (P e+ , P e− ) = (−0.3, +0.8) and the integrated luminosity L = 500 fb −1 .
Here, P e− (P e+ ) is the degree of transverse polarization of the electron (positron) beam. The right-(left-) handed polarization corresponds to P e = +1 (−1). The analysis depends on details of the mass spectrum. In Ref. [21], the SPS1a ′ benchmark point is adopted, and signal regions are optimized for it. In particular, lighter (heavier) stau is almost right-handed (lefthanded), and the neutralino mass is 98 GeV. These are different from our sample point and could affect the accuracy. In fact, the energy profile of the decay products of τ depends on the helicity of τ [22]. For instance, jet energy from τ → πν is likely to be harder for τ R compared to τ L [23]. Also, with the polarization used in Ref. [21], the production cross section of the lighter (heavier) stau at our model point is smaller (larger) than those at SPS1a ′ . On the other hand, the endpoint energies of τ -jet increase, as mχ0 1 decreases (cf. Ref. [23]). Then, the contamination of the background due to the process γγ → τ + τ − is reduced [21]. In this letter, we simply adopt the accuracy of 0.1 % and 3 % as our canonical values for the mass measurements of the staus. #2 #2 Dedicated studies of the threshold production ofτ 2 can improve the accuracy of its mass measurement [24,25]. At the Snowmass SM2 benchmark point, which is close to the SPS1a point, δmτ 2 ∼ 1 GeV is available for mτ 2 = 206 GeV, where √ s = 500 GeV and L = 1000 fb −1 with the electron polarization of 80 %, while no polarization for the positron. Figure 1: Accuracies of the determination of sin 2θτ from the measurement of the cross section σ(e + e − →τ 1τ1 ) as a function of the stau mixing angle with the accuracy of the cross section determination of 10 %, 5 %, 3 % and 1 % from top to bottom. The mass measurement is assumed to be sufficiently precise.
Next, the mixing angle θτ can be determined from the measurements of the cross sections of stau pair production processes. #3 The cross sections are given by [26] σ where the parameters are defined as #3 Alternatively, the stau mixing angle can be determined by measuring the τ polarization from energy profile of its decay products [12]. However, the cross section measurement provides a better resolution [21].
We consider productions of lighter staus to determine θτ . The cross section, σ(τ 1 ) = σ(e + e − →τ + 1τ − 1 ), depends on mτ 1 and θτ . The accuracy of the measurement of the stau mixing angle is estimated as In the sample point, the error is dominated by that of the cross section. The stau mass contributes to the cross section only through v, and mτ 1 is (much) smaller than √ s. Further, mass ofτ 1 can be precisely measured, as mentioned above. According to Ref. [21], the cross section for e + e − →τ + 1τ − 1 can be measured with the accuracy of 3.1 % for SPS1a ′ . Here, the uncertainty originates in the signal statistics and SUSY background, while those of the luminosity and efficiencies are assumed to be negligible. In our sample point, the production cross section is σ(τ 1 ) = 54 fb with √ s = 500 GeV and (P e+ , P e− ) = (−0.3, +0.8), which is smaller than σ(τ 1 ) = 135 fb at SPS1a ′ . By supposing the same acceptance as Ref. [21], the statistical uncertainty increases from 1/ N sig | SPS1a ′ = 2.1 % to 3.4 % with L = 500 fb −1 , where N sig is the number of the signals accepted by selections. On the other hand, our setup is free from the SUSY background, since only sleptons and the lightest neutralino are produced at our sample point. Thus, the accuracy of the cross section measurement is estimated to be δσ(τ 1 )/σ(τ 1 ) = 3.4 %. #4 In Fig. 1, the accuracy of the measurement of sin 2θτ is shown. The accuracy is sensitive to the mixing angle. It becomes better when the angle approaches to maximal, θτ = π/4. This is because σ(τ 1 ) depends on θτ via cos 2θτ . In the sample point, where sin 2θτ = 0.67, it is expected that sin 2θτ can be determined with the accuracy of 9 % by applying δσ(τ 1 )/σ(τ 1 ) = 3.4 % (and δmτ 1 /mτ 1 ∼ 0.1 %).
Finally, by combining the uncertainties of the determinations of mτ 1 , mτ 2 and sin 2θτ , the accuracy of the m 2 τ LR determination is estimated by Eq. (24). The uncertainty due to the measurement of the lighter stau mass is negligible, since mτ 1 and mτ 2 contribute to m 2 τ LR in the combination of (m 2 τ 1 − m 2 τ 2 ), and the uncertainty of the heavier stau mass is larger than that of the lighter one. Also, correlation of the errors, δmτ 1 and δ sin 2θτ , is negligible, since δmτ 1 barely affects δ sin 2θτ when δmτ 1 is sufficiently small. As a result, we obtain δm 2 τ LR /m 2 τ LR = 12 % with δmτ 2 /mτ 2 = 3 % and δ sin 2θτ / sin 2θτ = 9 %. From the relation (23), m 2 µLR is determined with the same accuracy, in the sample point, where m 2 µLR = −645 GeV 2 . The sign of sin 2θτ is not determined by the cross section measurements. It corresponds to the sign of µ tan β. Consequently, the reconstruction of the SUSY contribution to the #4 The signal region can be optimized for our sample point. Since there is no SUSY background, the acceptance could be enhanced and the uncertainty would be reduced. muon g − 2 is possible with two fold ambiguity. We take the sign of µ tan β so that the muon g − 2 anomaly is solved.
There are several comments in order. (i) In the above analysis, we considered the process e + e − →τ 1τ1 . The determination of the slepton mixing angle is possibly improved if the production cross section of a pair ofl 1 andl 2 is measured accurately [21], since it is proportional to sin 2 2θl. In the sample point, the cross section for the process e + e − →τ 1τ2 becomes 2.7 fb (3.6 fb) with √ s = 500 GeV and (P e+ , P e− ) = (−0.3, +0.8) ((P e+ , P e− ) = (0.3, −0.8)).
However, we could not find studies about such a process. In particular, the acceptance of the signal events as well as the accuracy of the cross section measurement has not been known. Thus, in the present study, we do not use this process. (ii) The smuon mixing angle is measured directly in principle from the smuon production e + e − →μ +μ− . This measurement is possible only when the smuon mixing is sufficiently large. It can be maximal whenμ L and µ R are almost degenerate in mass (see Ref. [10] for example), whereas it is tiny in our sample point. . This is smaller than the above ILC uncertainty. #6

Mass determinations
Next, let us consider measurements of mμ 1 , mμ 2 and mχ0 1 . If smuon masses are within the reach of ILC, they can be obtained from productions of the smuons that decay into neutralinos. The energy spectra of the muons produced by the smuon decay and the production threshold are sensitive to the masses [28]. In Refs. [25,29,30], the accuracies are estimated to be δmμ R = 170 MeV and δmχ0 1 = 210 MeV at the SPS1a benchmark point [31]. #7 Here, the masses are mμ R = 143 GeV and mχ0 1 = 96 GeV with Br(μ ± R → µ ±χ0 1 ) = 100 %. The analysis is based on √ s = 400 GeV, (P e+ , P e− ) = (−0.6, +0.8) and L = 200 fb −1 . Another study of the threshold scans yields δmμ R = 200 MeV for mμ R = 135 GeV by assuming 10 fb −1 per each data point with (P e+ , P e− ) = (+0.3, −0.8) [25,32]. The uncertainties are statistically limited. The muon energy spectrum is independent of the smuon chirality, and the mass #5 It is difficult to determine Aτ and µ tan β individually in m 2 τ LR by the stau decays. In fact, it is possible if the Higgsinos are light [26]. However, they are decoupled in our sample point. Alternatively, tan β is determined if the sneutrino mass is measured precisely, for instance, through the decay channelν →χ ± 1 ℓ ∓ (see Ref. [27]). In our sample point, it is difficult to identify the sneutrinos, because they decay only to the lightest neutralino. #6 The relations between the lepton masses and the Yukawa coupling constants are affected by SUSY radiative correction. The correction violates the relation Eq. (23) if the slepton soft masses depend on the generation. The violation is typically small. #7 The neutralino mass can also be measured from the endpoints in the stau productions. However, the resolution is worse [21]. resolution is less dependent on the smuon-neutralino mass splitting [33]. Since the accuracy is limited by signal statistics, we expectμ 1 has a better mass resolution in our sample point. At SPS1a, the production cross section is σ(μ 1 ) = 134 fb with √ s = 400 GeV, and (P e+ , P e− ) = (−0.6, +0.8). In our sample point, it becomes σ(μ 1 ) = 154 fb with √ s = 500 GeV, and (P e+ , P e− ) = (+0.3, −0.8).
The mass measurement of the heavier smuon is studied in detail at SPS1a ′ by Ref. [34]. Here, the heavier smuon is almost left-handed, and √ s = 500 GeV, (P e+ , P e− ) = (+0. is degraded by a factor 1.3. On the contrary, the above resolutions could be improved in our sample point, because SUSY background, for instance, from heavier neutralino productions, is suppressed. Finally, the accuracy of the neutralino mass measurement becomes better if studies about the selectron production processes are combined. In Ref. [33], it is claimed that δmχ0 1 = 80 MeV is achieved at SPS1a. In the present analysis, we simply assume at the sample point. Then, in the reconstruction of a (ILC) µ , the uncertainties in the mass measurements of smuons and neutralino are less important than that of m 2 µLR .

Coupling measurements
The coupling constantsg 1,R are hardly determined directly from the smuon production processes. Instead, they are available from selectron productions [12,13], because they are common in light generations. Since the Yukawa coupling constant of the electron is negligibly small, (Uẽ) 1L = (Uẽ) 2R = 1 holds with very high accuracy. (Thus, we call lighter and heavier selectrons asẽ L andẽ R , respectively.) Consequently, we obtain Cross sections for the selectron production processes depend on N e L 11 and N e R 21 through the t-channel neutralino-exchange diagrams. Thus,g 1,R can be measured by studying the selectron production cross sections as long as contributions of heavier neutralinos are known.
In Refs. [35][36][37], it is claimed that the Bino coupling with the (s)electrons can be determined with the accuracy of 0.18 % from the measurements of the production cross section of ẽ + Rẽ − R . Here, the beam configuration is √ s = 500 GeV with L = 500 fb −1 and the polarizations of 80 % (electron) and 50 % (positron). In the analysis, the SPS1a benchmark point is adopted, in which the selectron mass is mẽ R = 143 GeV. Here, all the neutralino masses are assumed to be measured by their productions at ILC. The production cross section ofẽ + Rẽ − R is very sensitive tog (eff) 1,R . It can be estimated that the accuracy of the measurement of theẽ + Rẽ − R cross section should be better than 0.9 % to determine the coupling at the 0.18 % level. We reinterpret the result of Refs. [35,36] to estimate how accuratelyg (eff) 1,R can be measured in the sample point. Let us assume that the accuracy of the cross section measurement is limited by the signal statistics, and that the acceptance at our sample point is the same as that in SPS1a. We estimate that the precision of Refs. [35,36] is simply scaled by N sig . At SPS1a, the cross section is σ(ẽ + assuming that Winos and Higgsinos are decoupled. Then, the experimental uncertainty of the cross section measurement is degraded to be about 1.5 %. We emphasize that Winos and Higgsinos are assumed to be undiscovered in our sample point. In addition to the lightest neutralino, heavier neutralinos, which are mostly composed of Winos and Higgsinos, may be exchanged in the t-channel diagrams, and contribute to the selectron production cross sections. In the process e + e − →ẽ + Rẽ − R , their contamination to the gaugino coupling constant measurement is very small, because they appear only through the mixing between the Bino and the Higgsinos. The direct interactions of the Higgsinos to the (s)electron are negligible due to a tiny coupling. In the case when the Higgsinos are heavier than 500 GeV (1 TeV), we estimate thatg 1,R ). #8 The cross section can be measured precisely at ILC [28]. In Ref. [38], its accuracy is claimed to be ∼ 2 % for mẽ R = 143 GeV and mẽ L = 202 GeV. Here, the SPS1a point is adopted with √ s = 500 GeV and (P e+ , P e− ) = (−0.6, −0.8), though the luminosity is not explicitly shown. The selectron production processes are discriminated from each others by the electron energy and by changing the beam polarization especially of the positron [28,29,39]. #9 In fact, the analysis in Ref. [39] shows that the neutralino coupling can be measured at similar accuracy as those in Ref. [35,36] by #8 The process, e + e − →ẽ + Lẽ − L , also involvesg (eff) 1,L . However, its cross section depends on the Wino coupling g 2 as well asg Y,L mainly through the t-channel Wino exchange diagram. #9 The heavier selectron may be identified by its decay products, if it has sizable branching ratio, for instance, ofẽ → eχ 0 2 (→ τ + τ −χ1 0 ) [36]. However, both of the selectrons decay directly into the lightest neutralino in our sample point.  1,L = a few % (exp) + 1 % (th), (37) or better. Here, the first term in the right-hand side comes from the measurement of the cross section for e + e − →ẽ + Lẽ − R , and the second term is due to the contamination from the undiscovered Winos and Higgsinos. Then, the uncertainty is sub-dominant in the reconstruction of a (ILC) µ compared to that in m 2 µLR .

Reconstruction of the SUSY contribution to muon g − 2
Now let us discuss the accuracy of the reconstruction of a (ILC) µ with ILC. The accuracy is estimated by summing all the errors induced by these parameters in quadrature as where X = m 2 µLR , mμ 1 , mμ 2 , mχ0 1 ,g 1,L , andg 1,R . In Table 2, their uncertainties are summarized. Consequently, we estimate δa taking δg This depends on M 2 and µ (as well as on the parameters listed in Table 2). In Fig. 2, contours of constant δa (SUSY,th) µ are shown for M 2 > 0 with the underlying parameters in Table 2. #10 In particular, µ tan β = 6.1 × 10 3 GeV is fixed. Here, the uncertainties in the parameters listed in Table 2 are omitted. Obviously, δa (SUSY,th) µ is suppressed as M 2 and µ become larger. This is because all the diagrams that contain Wino and Higgsino propagators vanish in this limit, so that a (SUSY) µ is well approximated by the Bino-smuon diagram. Thus, using lower bounds on the Wino and Higgsino masses provided by collider experiments, a bound on δa (SUSY,th) µ can be obtained. They will be searched for effectively at LHC with √ s = 13 or 14 TeV. #11 If the Wino and Higgsino masses are constrained to be larger than 1 TeV (1.5 TeV) in future, δa (SUSY,th) µ is known to be smaller than 0.9 × 10 −10 (0.3 × 10 −10 ) at our model point, which corresponds to 4 % (1 %) of a (ILC) µ . This is smaller than the dominant error of the reconstruction of a (ILC) µ . Finally we comment on higher order contributions to a (SUSY) µ . Ref. [40] calculated photonic SUSY two-loop corrections, which change the one-loop result by ∼ 10 %. They can be determined at ILC by the above procedure, because all the parameters necessary for them are measured simultaneously. In this letter, they are neglected for simplicity, although it is straightforward to include the contributions. Also, corrections to the gaugino couplings #10 We have checked that, when M 2 < 0, |δa (SUSY,th) µ | is smaller than that for M 2 > 0 with |M 2 | fixed. #11 Wino can be searched for by multi-lepton plus a large missing energy signature, while Higgsino can be by searches for multi-tau and/or standard model bosons together with a large missing energy. and to the lepton Yukawa couplings in the left-right mixing parameters can be as large as ∼ 10 % [10,15,41]. Importantly, they are already taken into account in the reconstruction of a (ILC) µ . Most of the other two-loop contributions are considered to be suppressed in our sample point. However, electroweak and SUSY two-loop corrections to the SUSY one-loop diagrams, which have not been calculated, might be ∼ 10 % [42]. Since they could be as large as the dominant error of the reconstruction, it is important to calculate these two-loop contributions.

Summary and Discussion
In this letter, we have studied how and how accurately we can reconstruct the SUSY contribution to the muon g − 2 by using the information available at ILC. If a (SUSY) µ is as large as 2.6 × 10 −9 to solve the muon g − 2 anomaly, and also if all the sleptons as well as the lightest neutralino are within the kinematical reach, ILC will be able to measure the MSSM parameters which are necessary to estimate a (SUSY) µ . We have discussed the procedures and accuracies of their measurements. It has been shown that, in the sample point we choose, the SUSY contribution to the muon g − 2 can be reconstructed with the uncertainty of ∼ 13 % at ILC with √ s = 500 GeV and an integrated luminosity L ∼ 500-1000 fb −1 . This provides a very crucial test of the SUSY explanation to the muon g − 2 anomaly. We should emphasize that the uncertainty depends on model points. As we have shown, the dominant error in the reconstructed value of a (ILC) µ originates in the uncertainty of the left-right mixing parameter m 2 µLR in the sample point. For instance, if the heavier stau mass increases with the lighter one fixed, it is inferred from Eq. (14) that the reconstruction would be degraded. On the contrary, if the charged sleptons in the second or third generation are degenerate in masses, the determination of sin 2θμ could be improved considerably. Unfortunately, slepton productions have not been studied for ILC in such cases.
The present uncertainty of the experimental and SM values of the muon g − 2 is about 30 % (see Eq. (3)). Thus, the error in the reconstructed value of a (ILC) µ is sub-dominant when we test the idea of solving the muon g − 2 anomaly with the SUSY contribution. However, the experimental measurement and theoretical calculation of the SM prediction will be improved in near future. The Fermilab experiment [43] and the J-PARC New g − 2/EDM experiment [44] will reduce the experimental error at least by a factor 4-5. The uncertainty of the SM prediction is dominated by those in the hadronic contributions. They will be improved by experiments as well as lattice calculations. The uncertainty is expected to be reduced by a factor 2 [45]. As a result, if the experimental and SM central values would be unchanged, the error in ∆a µ could become as small as ∼ 10 %, which is comparable to that in a (ILC) µ . Then, a precise reconstruction of the SUSY contribution to the muon g − 2 becomes crucial.
We made several assumptions to evaluate the uncertainty, since we could not find enough information about the slepton production processes. Precise studies of the slepton production process are strongly recommended to deeply understand how useful ILC is to reconstruct the SUSY contribution to the muon g − 2.