#### Publications

###### Quantum Parton Shower with Kinematics [arXiv: 2310.19881]

###### Christian W. Bauer, So Chigusa, Masahito Yamazaki (2023/10/30)

###### Abstract

Parton showers which can efficiently incorporate quantum interference effects have been shown to be run efficiently on quantum computers. However, so far these quantum parton showers did not include the full kinematical information required to reconstruct an event, which in classical parton showers requires the use of a veto algorithm. In this work, we show that adding one extra assumption about the discretization of the evolution variable allows to construct a quantum veto algorithm, which reproduces the full quantum interference in the event, and allows to include kinematical effects. We finally show that for certain initial states the quantum interference effects generated in this veto algorithm are classically tractable, such that an efficient classical algorithm can be devised.

###### Effects of Finite Material Size On Axion-magnon Conversion [arXiv: 2310.17704]

###### So Chigusa, Asuka Ito, Kazunori Nakayama, Volodymyr Takhistov (2023/10/26)

###### Abstract

Magnetic materials are particularly favorable targets for detecting axions interacting with electrons because the collective excitation of electron spins, the magnon, can be excited through the axion-magnon conversion process. It is often assumed that only the zero-momentum uniformly precessing magnetostatic (Kittel) mode of the magnon is excited. This is justified if the de Broglie wavelength of the axion is much longer than the size of the target magnetic material. However, if the de Broglie wavelength is shorter, finite-momentum magnon modes can also be excited. We systematically analyze the target material size dependence of the axion-magnon conversion rate. We discuss the importance of these effects in the detection of relativistic axions as well as in the detection of axion dark matter of relatively heavy mass with large material size.

###### Aiming for Tops of ALPs with a Muon Collider [arXiv: 2310.11018]

###### So Chigusa, Sudhakantha Girmohanta, Yuichiro Nakai, Yufei Zhang (2023/10/17)

###### Abstract

Future muon colliders with center-of-mass energy of \(\mathcal{O}(1-10)\) TeV can provide a clean high-energy environment with advantages in searches for TeV-scale axion-like particles (ALPs), pseudo-Nambu-Goldstone bosons associated with spontaneously broken global symmetries, which are widely predicted in physics beyond the Standard Model (SM). We exploit ALP couplings to SM fermions, and guided by unitarity constraints, build a search strategy focusing on the ALP decay to top quark pairs at muon colliders. It is found that a large parameter space of TeV-scale ALPs with TeV-scale decay constants can be probed by utilizing the ALP-top quark coupling.

###### Axion detection via superfluid \(^3\)He ferromagnetic phase and quantum measurement techniques [arXiv: 2309.09160]

###### So Chigusa, Dan Kondo, Hitoshi Murayama, Risshin Okabe, Hiroyuki Sudo (2023/09/17)

###### Abstract

We propose to use the nuclear spin excitation in the ferromagnetic A1 phase of the superfluid \(^3\)He for the axion dark matter detection. This approach is striking in that it is sensitive to the axion-nucleon coupling, one of the most important features of the QCD axion introduced to solve the strong CP problem. We review a quantum mechanical description of the nuclear spin excitation and apply it to the estimation of the axion-induced spin excitation rate. We also describe a possible detection method of the spin excitation in detail and show that the combination of the squeezing of the final state with the Josephson parametric amplifier and the homodyne measurement can enhance the sensitivity. It turns out that this approach gives good sensitivity to the axion dark matter with the mass of \(O(1) \, \mu \mathrm{eV}\) depending on the size of the external magnetic field. We estimate the parameters of experimental setups, e.g., the detector volume and the amplitude of squeezing, required to reach the QCD axion parameter space.

###### Dark matter detection using nuclear magnetization in magnet with hyperfine interaction [arXiv: 2307.08577]

###### So Chigusa, Takeo Moroi, Kazunori Nakayama, Thanaporn Sichanugrist (2023/07/14)

###### Abstract

We consider the possibility to detect cosmic light dark matter (DM), i.e., axions and dark photons, of mass \(\sim 10^{-6}\) eV and \(\sim 10^{-4}\) eV, by magnetic excitation in a magnet with strong hyperfine interaction. In particular, we consider a canted anti-ferromagnet, MnCO\(_3\), as a concrete candidate material. With spin transfer between nuclear and electron spins allowed by the hyperfine interaction, nuclear spins become naturally highly polarized due to an effective (electron-spin-induced) magnetic field, and have long-range interactions with each other. The collective precession of nuclear spins, i.e., a nuclear magnon, can be generated by the DM field through the nucleon-DM interaction, while they are also sensitive to the electron-DM interaction through the electron-nuclear spin mixing. Compared to conventional nuclear-spin precession experiments, this system as a DM sensor is sensitive to higher frequency needing only a small static magnetic field applied. The system also has collective precession of electron spins, mixed with nuclear spins, as the additional channels that can be used for DM probes. We estimate the sensitivity under appropriate readout setups such as an inductive pick-up loop associated with an LC resonant circuit, or a photon cavity with a photon counting device. We show that this method covers an unexplored parameter region of light bosonic DM.

###### Stability of Electroweak Vacuum and Supersymmetric Contribution to Muon g-2 [arXiv: 2306.16596]

###### So Chigusa, Takeo Moroi, Yutaro Shoji (2023/06/28)

###### Abstract

We study the stability of the electroweak vacuum in the supersymmetric (SUSY) standard model (SM), paying particular attention to its relation to the SUSY contribution to the muon anomalous magnetic moment \(a_\mu\). If the SUSY contribution to \(a_\mu\) is sizable, the electroweak vacuum may become unstable because of enhanced trilinear scalar interactions in particular when the sleptons are heavy. Consequently, assuming enhanced SUSY contribution to \(a_\mu\), an upper bound on the slepton masses is obtained. We give a detailed prescription to perform a full one-loop calculation of the decay rate of the electroweak vacuum for the case that the SUSY contribution to \(a_\mu\) is enhanced. We also give an upper bound on the slepton masses as a function of the SUSY contribution to \(a_\mu\).

###### Light Dark Matter Search with Nitrogen-Vacancy Centers in Diamonds [arXiv: 2302.12756]

###### So Chigusa, Masashi Hazumi, Ernst David Herbschleb, Norikazu Mizuochi, Kazunori Nakayama (2023/02/24)

###### Abstract

We propose new ideas to directly search for light dark matter, such as the axion or the dark photon, by using magnetometry with nitrogen-vacancy centers in diamonds. If the dark matter couples to the electron spin, it affects the evolution of the Bloch vectors consisting of the spin triplet states, which may be detected through several magnetometry techniques. We give several concrete examples with the use of dc and ac magnetometry and estimate the sensitivity on dark matter couplings.

###### Quantum simulations of dark sector showers [arXiv: 2204.12500]

###### So Chigusa, Masahito Yamazaki (2022/04/26)

###### Abstract

We consider dark sector scenarios where dark matter is accompanied by a dark photon and multiple-flavor dark fermions charged under the dark gauge group. We study quantum interference effects in dark sector jets, where multiple dark photons are emitted from high-energy dark fermions. We perform fully quantum simulations of dark sector showers and compare the results against those of the classical Monte-Carlo simulations. We find important differences in probability distributions of dark photon countings between quantum and classical computations. When the number of dark-fermion flavors is large, we find significant enhancements in large numbers of dark photon emissions. Such enhancements can provide distinguishing signals for our scenarios at particle colliders.

###### Upper bound on the smuon mass from vacuum stability in the light of muon g-2 anomaly [arXiv: 2203.08062]

###### So Chigusa, Takeo Moroi, Yutaro Shoji (2022/03/15)

###### Abstract

We derive an upper bound on the smuon mass assuming that the muon \(g-2\) anomaly is explained by the supersymmetric (SUSY) contribution. In the minimal SUSY standard model, the SUSY contribution to the muon \(g-2\) is enhanced when the Higgsino mass parameter is large. Then, the smuon-smuon-Higgs trilinear coupling is enhanced, which may destabilize the electroweak vacuum. We calculate precisely the decay rate of the electroweak vacuum in such a case. We include one-loop effects which are crucial to determine the overall normalization of the decay rate. Requiring that the theoretical prediction of the muon anomalous magnetic moment is consistent with the observed value at the \(1\) and \(2\sigma\) levels (equal to the central value of the observed value), we found that the lightest smuon mass should be smaller than \(1.38\) and \(1.68\ {\rm TeV}\) (\(1.20\ {\rm TeV}\)) for \(\tan\beta=10\) (with \(\tan\beta\) being the ratio of the vacuum expectation values of the two Higgs bosons), respectively, and the bound is insensitive to the value of \(\tan\beta\).

###### Deeply learned preselection of Higgs dijet decays at future lepton colliders [arXiv: 2202.02534]

###### So Chigusa, Shu Li, Yuichiro Nakai, Wenxing Zhang, Yufei Zhang, Jiaming Zheng (2022/02/05)

###### Abstract

Future electron-positron colliders will play a leading role in the precision measurement of Higgs boson couplings which is one of the central interests in particle physics. Aiming at maximizing the performance to measure the Higgs couplings to the bottom, charm and strange quarks, we develop machine learning methods to improve the selection of events with a Higgs decaying to dijets. Our methods are based on the Boosted Decision Tree (BDT), Fully-Connected Neural Network (FCNN) and Convolutional Neural Network (CNN). We find that the BDT and FCNN-based algorithms outperform the conventional cut-based method. With our improved selection of Higgs decaying to dijet events using the FCNN, the charm quark signal strength is measured with a \(16\%\) error, which is roughly a factor of two better than the \(34\%\) precision obtained by the cut-based analysis. Also, the strange quark signal strength is constrained as \(\mu_{ss} \lesssim 35\) at the \(95\%\) C.L. with the FCNN, which is to be compared with \(\mu_{ss} \lesssim 70\) obtained by the cut-based method.

###### Studying squark mass spectrum through gluino decay at 100 TeV future hadron colliders [arXiv: 2102.07910]

###### So Chigusa, Koichi Hamaguchi, Takeo Moroi, Atsuya Niki, Kosaku Ono (2021/02/16)

###### Abstract

We study the prospect of determining the decay properties of the gluino in the supersymmetric (SUSY) standard model at a 100 TeV future hadron collider. We consider the case where the neutral Wino is the lightest superparticle. In this case, the long-lived charged Wino can be used to eliminate standard model backgrounds, which enables us to study the details of superparticles. We show that, based on the analysis of the numbers of high \(p_T\) leptons, boosted \(W\)-jets, and \(b\)-tagged jets, we may determine the gaugino species and the quark flavors in the gluino decay. With such determinations, we can obtain information about the mass spectrum of squarks even if squarks are out of the kinematical reach.

###### Implications of gravitational waves for supersymmetric grand unification [arXiv: 2011.04090]

###### So Chigusa, Yuichiro Nakai, Jiaming Zheng (2020/11/08)

###### Abstract

Supersymmetric grand unification based on \(SO(10)\) is one of the most attractive paradigms in physics beyond the Standard Model. Inspired by the recent NANOGrav signal, we discuss the implications of detecting a stochastic gravitational wave background emitted by a network of cosmic strings for the \(SO(10)\) grand unification. Starting from a minimal model with multiple steps of symmetry breaking, we show that it generally prefers a high intermediate scale above \(10^{14}\, \mathrm{GeV}\) that is favored by observable primordial gravitational waves. The observed spectrum can potentially narrow the possible range of the cosmic string scale and restricts the unified couplings and the unification scale by requiring gauge coupling unification. As an indirect consequence of the high cosmic string scale, the monopole abundance places non-trivial constraints on the theory. These are complementary to the proton decay constraints and probe different facets of supersymmetric \(SO(10)\) unification theories.

###### Precise Calculation of the Decay Rate of False Vacuum with Multi-Field Bounce [arXiv: 2007.14124]

###### So Chigusa, Takeo Moroi, Yutaro Shoji (2020/07/28)

###### Abstract

We study the decay rate of a false vacuum in gauge theory at the one-loop level. We pay particular attention to the case where the bounce consists of an arbitrary number of scalar fields. With a multi-field bounce, which has a curved trajectory in the field space, the mixing among the gauge fields and the scalar fields evolves along the path of the bounce in the field space and the one-loop calculation of the vacuum decay rate becomes complicated. We consider the one-loop contribution to the decay rate with an arbitrary choice of the gauge parameter, and obtain a gauge invariant expression of the vacuum decay rate. We also give proper treatments of gauge zero modes and renormalization.

###### Constraints on electron-scattering interpretation of XENON1T excess [arXiv: 2007.01663]

###### So Chigusa, Motoi Endo, Kazunori Kohri (2020/07/03)

###### Abstract

Recently, the XENON1T experiment has observed an excess in the electronic recoil data in the recoil energy range of \(1\)-\(7\) keV. One of the most favored new physics interpretations is electron scattering with a boosted particle with a velocity of \(\sim 0.1\) and a mass of \(\gtrsim 0.1\,\mathrm{MeV}\). If such a particle has a strong interaction with electrons, it may affect the standard scenario of cosmology or be observed at low-threshold direct detection experiments. We study various constraints, mainly focusing on those from the big-bang nucleosynthesis, supernova cooling, and direct detection experiments. We discuss the implication of these constraints on electron-scattering interpretation of the XENON1T excess.

###### Detecting light boson dark matter through conversion into a magnon [arXiv: 2001.10666]

###### So Chigusa, Takeo Moroi, Kazunori Nakayama (2020/01/29)

###### Abstract

Light boson dark matter such as axion or hidden photon can be resonantly converted into a magnon in a magnetic insulator under the magnetic field, which can be detected experimentally. We provide a quantum mechanical formulation for the magnon event rate and show that the result is consistent with that obtained by a classical calculation. Besides, it is pointed out that the experimental setup of the QUAX proposal for the axion detection also works as a detector of hidden photon dark matter. It has good sensitivity in the mass range around 1 meV, which is beyond astrophysical constraints.

###### Determining Wino Lifetime in Supersymmetric Model at Future 100 TeV pp Colliders [arXiv: 1912.00592]

###### So Chigusa, Yusuke Hosomi, Takeo Moroi, Masahiko Saito (2019/12/02)

###### Abstract

We discuss a possibility to measure the lifetime of charged Wino in supersymmetric model at future 100 TeV pp colliders, assuming that (neutral) Wino is the lightest superparticle (LSP). In the Wino LSP scenario, the charged Wino has a lifetime of about 0.2 ns, and its track may be reconstructed in particular by the inner pixel detectors. We show that the lifetime of charged Wino may be measured by using the information about the distribution of the flight lengths of charged Winos. We propose a procedure for the lifetime determination and show how the accuracy changes as we vary the mass spectrum of superparticle. We also discuss the effects of the detector layouts on the lifetime determination.

###### Signals of Axion Like Dark Matter in Time Dependent Polarization of Light [arXiv: 1911.09850]

###### So Chigusa, Takeo Moroi, Kazunori Nakayama (2019/11/22)

###### Abstract

We consider the search for axion-like particles (ALPs) by using time series data of the polarization angle of the light. If the condensation of an ALP plays the role of dark matter, the polarization plane of the light oscillates as a function of time and we may be able to detect the signal of the ALP by continuously observing the polarization. In particular, we discuss that the analysis of the Fourier-transformed data of the time-dependent polarization angle is powerful to find the signal of the ALP dark matter. We pay particular attention to the light coming from astrophysical sources such as protoplanetary disks, supernova remnants, the foreground emission of the cosmic microwave background, and so on. We show that, for the ALP mass of \(\sim 10^{-22}\)--\(10^{-19}\ {\rm eV}\), ALP searches in the Fourier space may reach the parameter region which is unexplored by other searches yet.

###### Bounce Configuration from Gradient Flow [arXiv: 1906.10829]

###### So Chigusa, Takeo Moroi, Yutaro Shoji (2019/06/26)

###### Abstract

Based on the gradient flow, we propose a new method to determine the bounce configuration for false vacuum decay. Our method is applicable to a large class of models with multiple fields. Since the bounce is a saddle point of an action, a naive gradient flow method which minimizes the action does not work. We point out that a simple modification of the flow equation can make the bounce its stable fixed point while the false vacuum configuration an unstable one. Consequently, the bounce configuration can be obtained simply by following the flow without a careful choice of an initial configuration. With numerical analysis, we confirm the validity of our claim, checking that the flow equation we propose indeed has solutions that flow into the bounce configuration.

###### Novel Flavon Stabilization with Trimaximal Neutrino Mixing [arXiv: 1905.11517]

###### So Chigusa, Shinta Kasuya, Kazunori Nakayama (2019/05/27)

###### Abstract

We construct a supersymmetric \(S_4\) flavor symmetry model with one of the trimaximal neutrino mixing patterns, the so-called TM\(_1\), by using the novel way to stabilize flavons, which we proposed recently. The flavons are assumed to have tachyonic supersymmetry breaking mass terms and stabilized by higher-dimensional terms in the potential. We can obtain the desired alignment structure of the flavon vacuum expectation values to realize neutrino masses and mixings consistent with the current observations. This mechanism naturally avoids the appearance of dangerous cosmological domain walls.

###### Indirect studies of electroweakly interacting particles at 100 TeV hadron colliders [arXiv: 1904.11162]

###### Tomohiro Abe, So Chigusa, Yohei Ema, Takeo Moroi (2019/04/25)

###### Abstract

There are many extensions of the standard model that predict the existence of electroweakly interacting massive particles (EWIMPs), in particular in the context of the dark matter. In this paper, we provide a way for indirectly studying EWIMPs through the precise study of the pair production processes of charged leptons or that of a charged lepton and a neutrino at future 100 TeV collider experiments. It is revealed that this search method is suitable in particular for Higgsino, providing us the \(5\sigma\) discovery reach of Higgsino in supersymmetric model with mass up to 850 GeV. We also discuss how accurately one can extract the mass, gauge charge, and spin of EWIMPs in our method.

###### Studying gaugino masses in supersymmetric model at future 100 TeV \(pp\) collider [arXiv: 1901.10389]

###### Shoji Asai, So Chigusa, Toshiaki Kaji, Takeo Moroi, Masahiko Saito, Ryu Sawada, Junichi Tanaka, Koji Terashi, Kenta Uno (2019/01/29)

###### Abstract

We discuss prospects of studying supersymmetric model at future \(pp\) circular collider (FCC) with its centre-of-mass energy of \(\sim 100\ {\rm TeV}\). We pay particular attention to the model in which Wino is lighter than other supersymmetric particles and all the gauginos are within the kinematical reach of the FCC, which is the case in a large class of so-called pure gravity mediation model based on anomaly mediated supersymmetry breaking. In such a class of model, charged Wino becomes long-lived with its decay length of \(\sim 6\ {\rm cm}\), and the charged Wino tracks may be identified in particular by the inner pixel detector; the charged Wino tracks can be used not only for the discrimination of standard model backgrounds but also for the event reconstructions. We show that precise determinations of the Bino, Wino, and gluino masses are possible at the FCC. For such measurements, information about the charged Wino tracks, including the one about the velocity of the charged Wino using the time of the hit at the pixel detector, is crucial. With the measurements of the gaugino masses in the pure gravity mediation model, we have an access to more fundamental parameters like the gravitino mass.

###### Probing electroweakly interacting massive particles with Drell-Yan process at 100 TeV hadron colliders [arXiv: 1810.07349]

###### So Chigusa, Yohei Ema, Takeo Moroi (2018/10/17)

###### Abstract

There are many models beyond the standard model which include electroweakly interacting massive particles (EWIMPs), often in the context of the dark matter. In this paper, we study the indirect search of EWIMPs using a precise measurement of the Drell-Yan cross sections at future \(100\,{\rm TeV}\) hadron colliders. It is revealed that this search strategy is suitable in particular for Higgsino and that the Higgsino mass up to about \(1.3\,{\rm TeV}\) will be covered at \(95\,\%\) C.L. irrespective of the chargino and neutralino mass difference. We also show that the study of the Drell-Yan process provides important and independent information about every kind of EWIMP in addition to Higgsino.

###### Flavon Stabilization in Models with Discrete Flavor Symmetry [arXiv: 1810.05791]

###### So Chigusa, Shinta Kasuya, Kazunori Nakayama (2018/10/13)

###### Abstract

We propose a simple mechanism for stabilizing flavon fields with aligned vacuum structure in models with discrete flavor symmetry. The basic idea is that flavons are stabilized by the balance between the negative soft mass and non-renormalizable terms in the potential. We explicitly discuss how our mechanism works in \(A_4\) flavor model, and show that the field content is significantly simplified. It also works as a natural solution to the cosmological domain wall problem.

###### Anomalous Discrete Flavor Symmetry and Domain Wall Problem [arXiv: 1808.09601]

###### So Chigusa, Kazunori Nakayama (2018/08/29)

###### Abstract

Discrete flavor symmetry is often introduced for explaining quark/lepton masses and mixings. However, its spontaneous breaking leads to the appearance of domain walls, which is problematic for cosmology. We consider a possibility that the discrete flavor symmetry is anomalous under the color SU(3) so that it splits the energy levels of degenerate discrete vacua as a solution to the domain wall problem. We find that in most known models of flavor symmetry, the QCD anomaly effect can only partially remove the degeneracy and there still remain degenerate vacua.

###### Decay Rate of Electroweak Vacuum in the Standard Model and Beyond [arXiv: 1803.03902]

###### So Chigusa, Takeo Moroi, Yutaro Shoji (2018/03/11)

###### Abstract

We perform a precise calculation of the decay rate of the electroweak vacuum in the standard model as well as in models beyond the standard model. We use a recently-developed technique to calculate the decay rate of a false vacuum, which provides a gauge invariant calculation of the decay rate at the one-loop level. We give a prescription to take into account the zero modes in association with translational, dilatational, and gauge symmetries. We calculate the decay rate per unit volume, \(\gamma\), by using an analytic formula. The decay rate of the electroweak vacuum in the standard model is estimated to be \(\log_{10}\gamma\times{\rm Gyr~Gpc^3} = -582^{+40~+184~+144~+2}_{-45~-329~-218~-1}\), where the 1st, 2nd, 3rd, and 4th errors are due to the uncertainties of the Higgs mass, the top quark mass, the strong coupling constant and the choice of the renormalization scale, respectively. The analytic formula of the decay rate, as well as its fitting formula given in this paper, is also applicable to models that exhibit a classical scale invariance at a high energy scale. As an example, we consider extra fermions that couple to the standard model Higgs boson, and discuss their effects on the decay rate of the electroweak vacuum.

###### State-of-the-Art Calculation of the Decay Rate of Electroweak Vacuum in the Standard Model [arXiv: 1707.09301]

###### So Chigusa, Takeo Moroi, Yutaro Shoji (2017/07/28)

###### Abstract

The decay rate of the electroweak (EW) vacuum is calculated in the framework of the standard model (SM) of particle physics, using the recent progresses in the understanding of the decay rate of metastable vacuum in gauge theories. We give a manifestly gauge-invariant expression of the decay rate. We also perform a detailed numerical calculation of the decay rate. With the best-fit values of the SM parameters, we find that the decay rate of the EW vacuum per unit volume is about \(10^{-577}\ {\rm Gyr^{-1}Gpc^{-3}}\); with the uncertainty in the top mass, the decay rate is estimated as \(10^{-295}-10^{-1465}\ {\rm Gyr^{-1}Gpc^{-3}}\).

###### Bottom-Tau Unification in Supersymmetric SU(5) Models with Extra Matters [arXiv: 1702.00790]

###### So Chigusa, Takeo Moroi (2017/02/02)

###### Abstract

We consider \(b\)-\(\tau\) unification in supersymmetric \(SU(5)\) grand unified theories (GUTs) with extra matters. The renormalization group runnings of \(b\) and \(\tau\) Yukawa coupling constants may be significantly affected by the existence of extra matters. If the extra matters interact with the standard model particles (and their superpartners) only through gauge interaction, the ratio of the \(b\) to \(\tau\) Yukawa coupling constants at the GUT scale becomes suppressed compared to the case without extra matters. This is mainly due to the change of the renormalization group running of the \(SU(3)_C\) gauge coupling constant. If the extra matters have Yukawa couplings, on the contrary, the (effective) \(b\) Yukawa coupling at the GUT scale can be enhanced due to the new Yukawa interaction. Such an effect may improve the \(b\)-\(\tau\) unification in supersymmetric GUTs.

###### Bottom-tau unification in a supersymmetric model with anomaly-mediation [arXiv: 1604.02156]

###### So Chigusa, Takeo Moroi (2016/04/07)

###### Abstract

We study the Yukawa unification, in particular, the unification of the Yukawa coupling constants of \(b\) and \(\tau\), in the framework of supersymmetric (SUSY) model. We concentrate on the model in which the SUSY breaking scalar masses are of the order of the gravitino mass while the gaugino masses originate from the effect of anomaly mediation and hence are one-loop suppressed relative to the gravitino mass. We perform an accurate calculation of the Yukawa coupling constants of \(b\) and \(\tau\) at the grand unified theory (GUT) scale, including relevant renormalization group effects and threshold corrections. In particular, we study the renormalization group effects, taking into account the mass splittings among sfermions, gauginos, and the standard model particles. We found that the Yukawa coupling constant of \(b\) at the GUT scale is about \(70\ \%\) of that of \(\tau\) if there is no hierarchy between the sfermion masses and the gravitino mass. Our results suggest sizable threshold corrections to the Yukawa coupling constants at the GUT scale or significant suppressions of the sfermion masses relative to the gravitino mass.