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Last Updated :2017/11/21

- Cyclic Leibniz rule: a formulation of supersymmetry on lattice

Mitsuhiro Kato, Makoto Sakamoto, Hiroto So, 2013年11月20日, For the purpose of constructing supersymmetric(SUSY) theories on lattice, we propose a new type relation on lattice -cyclic Leibniz rule(CLR)- which is slightly different from an ordinary Leibniz rule. Actually, we find that CLR can enlarge the number of SUSYs and construct more Nicolai mappings in a quantum-mechanical model. In this model, the exact mass degeneracy between fermion and boson is shown. - Is the Higgs a sign of extra dimensions?

Hiroto So, Kazunori Takenaga, 2013年02月03日, We introduce a 4-dimensional cutoff in the scenario of gauge-Higgs unification to control the ultraviolet behavior. A one-loop effective potential for a Higgs field and the Higgs mass are obtained with the cutoff. We find an {\it interrelation} between the 4-dimensional cutoff and the scale of extra dimensions, which is concretized through the Higgs mass. Combining this interrelation and the recently discovered Higgs boson at LHC, we obtain an interesting constraint for the 4-dimensional cutoff and the extra dimensional scale. - Leibniz rule, locality and supersymmetry on lattice

Mitsuhiro Kato, Makoto Sakamoto, Hiroto So, 2012年12月07日, In a finite volume system, we prove a no-go theorem on a Leibniz rule with a care of locality argument on latttice. The new possibility on the Leibniz rule solutions on lattice is discussed. Although the new solution admits a local difference operator, a non-local product rule is needed. In the case, a supersymmetric interacting theory is simply realized. The difference between finite flavor systems and matrix representations of infinite flavor systems is explained based on a finite volume system analysis including the no-go theorem. - 5-dimensional SU(2) lattice gauge theory with Z2 orbifolding and its
phase structure

Michika Murata, Hiroto So, Kazunori Takenaga, 2010年12月13日, In an SU(2) lattice gauge theory with a Z2 orbifolded extra dimension, the new symmetry which is called as a stick symmetry is useful in understanding the bulk transition. We discuss the relation with the Fradkin-Shenker's phase diagram as well. A remnant of the extra dimension is remained as two U(1) gauge symmetries in two 4-dimensional spaces. We also consider more general bulk gauge groups beyond SU(2). - No-Go Theorem of Leibniz Rule and Supersymmetry on the Lattice

Mitsuhiro Kato, Makoto Sakamoto, Hiroto So, PoS LATTICE2008:233,2008, 2008年10月14日, An obstacle to realize supersymmetry on a lattice is the breakdown of Leibniz rule. We give a proof of a no-go theorem that it is impossible to construct a lattice field theory in an infinite lattice volume with any nontrivial field products and difference operators that satisfy the following three properties: (i) translation invariance, (ii) locality and (iii) Leibniz rule. We then propose a way to escape from the no-go theorem by introducing infinite flavors, and present a lattice model of N=2 supersymmetric quantum mechanics equipped with the full exact supersymmetry. - Taming the Leibniz Rule on the Lattice

Mitsuhiro Kato, Makoto Sakamoto, Hiroto So, JHEP0805:057,2008, 2008年03月21日, We study a product rule and a difference operator equipped with Leibniz rule in a general framework of lattice field theory. It is shown that the difference operator can be determined by the product rule and some initial data through the Leibniz rule. This observation leads to a no-go theorem that it is impossible to construct any difference operator and product rule on a lattice with the properties of (i) translation invariance, (ii) locality and (iii) Leibniz rule. We present a formalism to overcome the difficulty by an infinite flavor extension or a matrix expression of a lattice field theory. - Overlap Fermion in External Gravity

Hiroto So, Masashi Hayakawa, Hiroshi Suzuki, PoSLAT2006:047,2006, 2006年10月04日, On a lattice, we construct an overlap Dirac operator which describes the propagation of a Dirac fermion in external gravity. The local Lorentz symmetry is manifestly realized as a lattice gauge symmetry, while the general coordinate invariance is expected to be restored only in the continuum limit. The lattice index density in the presence of a gravitational field is calculated. - Rotational Symmetry and A Light Mode in Two-Dimensional Staggered
Fermions

Morio Hatakeyama, Hideyuki Sawanaka, Hiroto So, Prog.Theor.Phys.117:715-728,2007, 2006年09月15日, To obtain a light mode in two-dimensional staggered fermions, we introduce four new local operators keeping the rotational invariance for a staggered Dirac operator. To split masses of tastes, three cases are considered. The mass matrix and the propagator for free theories are analyzed. We find that one of three cases is a good candidate for taking a single mode by the mass splitting. In the case, it is possible that a heavy mode obtains infinite mass on even sites or odd sites. - Overlap lattice fermion in a gravitational field

Masashi Hayakawa, Hiroto So, Hiroshi Suzuki, Prog.Theor.Phys. 116 (2006) 197-215, 2006年04月04日, We construct a lattice Dirac operator of overlap type that describes the propagation of a Dirac fermion in an external gravitational field. The local Lorentz symmetry is manifestly realized as a lattice gauge symmetry, while it is believed that the general coordinate invariance is restored only in the continuum limit. Our doubler-free Dirac operator satisfies the conventional Ginsparg-Wilson relation and possesses gamma_5 hermiticity with respect to the inner product, which is suggested by the general coordinate invariance. The lattice index theorem in the presence of a gravitational field holds, and the classical continuum limit of the index density reproduces the Dirac genus. Reduction to a single Majorana fermion is possible for 8k+2 and 8k+4 dimensions, but not for 8k dimensions, which is consistent with the existence of the global gravitational/gauge anomalies in 8k dimensions. Other Lorentz representations, such as the spinor-vector and the bi-spinor representations, can also be treated. Matter fields with a definite chirality (with respect to the lattice-modified chiral matrix) are briefly considered. - Zero-dimensional analogue of the global gauge anomaly

Hiroto So, Hiroshi Suzuki, Prog.Theor.Phys. 115 (2006) 467-471, 2005年11月21日, A zero-dimensional analogue of Witten's global gauge anomaly is considered. For example, a zero-dimensional reduction of the two-dimensional $\SO(2N)$ Yang-Mills theory with a single Majorana-Weyl fermion in the fundamental representation suffers from this anomaly. Another example is a zero-dimensional reduction of two- and three-dimensional $\SU(2N_c)$ Yang-Mills theories which couple to a single Majorana fermion in the adjoint representation. In this case, any expectation value is either indeterminate or infinite. - Leibniz rule and exact supersymmetry on lattice: a case of
supersymmetrical quantum mechanics

Mitsuhiro Kato, Makoto Sakamoto, Hiroto So, PoS LAT2005 (2005) 274, 2005年09月27日, We propose a new formulation of lattice theory. It is given by a matrix form and suitable for satisfying Leibniz rule on lattice. The theory may be interpreted as a multi-flavor system. By realizing the difference operator as a commutator, we may obtain exact supersymmetric theories on lattice explicitly. Some problems such as locality and single flavor reduction are also commented. - Genuine Symmetry of Staggered Fermion

Katsumi Itoh, Mitsuhiro Kato, Michika Murata, Hideyuki Sawanaka, Hiroto So, Prog.Theor.Phys. 114 (2005) 631-641, 2004年12月25日, We present a new formulation of the staggered fermion on the D-dimensional lattice based on the SO(2D) Clifford algebra, which is naturally present in the action. The action of the massless staggered fermion is invariant under the discrete rotation and the SO(2D) chiral and other discrete transformations. From transformation properties of the fermion, we find two local meson operators (one scalar and one pseudoscalar) in addition to two standard meson operators. - Five-dimensional Lattice Gauge Theory as Multi-Layer World

Michika Murata, Hiroto So, 2003年06月03日, A five-dimensional lattice space can be decomposed into a number of four-dimens ional lattices called as layers. The five-dimensional gauge theory on the lattice can be interpreted as four-dimensional gauge theories on the multi-layer with interactions between neighboring layers. In the theory, there exist two independent coupling constants; $\beta_4$ controls the dynamics inside a layer and $\beta_5$ does the strength of the inter-layer interaction.We propose the new possibility to realize the continuum limit of a five-dimensional theory using four-dimensional dynamics with large $\beta_4$ and small $\beta_5$. Our result is also related to the higher dimensional theory by deconstruction approach. - Lattice Chiral Symmetry in Fermionic Interacting Theories and the
Antifield Formalism

Yuji Igarashi, Hiroto So, Naoya Ukita, Nucl.Phys. B640 (2002) 95-118, 2002年06月11日, Recently we have discussed realization of an exact chiral symmetry in theories with self-interacting fermions on the lattice, based upon an auxiliary field method. In this paper we describe construction of the lattice chiral symmetry and discuss its structure in more detail. The antifield formalism is used to make symmetry consideration more transparent. We show that the quantum master equation in the antifield formalism generates all the relevant Ward-Takahashi identities including a Ginsparg-Wilson relation for interacting theories. Solutions of the quantum master equation are obtained in a closed form, but the resulting actions are found to be singular. Canonical transformations are used to obtain four types of regular actions. Two of them may define consistent quantum theories. Their Yukawa couplings are the same as those obtained by using the chiral decomposition in the free field algebra. Inclusion of the complete set of the auxiliary fields is briefly discussed. - Ginsparg-Wilson Relation and Lattice Chiral Symmetry in Fermionic
Interacting Theories

Yuji Igarashi, Hiroto So, Naoya Ukita, Phys.Lett. B535 (2002) 363-370, 2002年03月18日, We derive Ginsparg-Wilson relation for a lattice chiral symmetry in theories with self-interacting fermions. Auxiliary scalar and pseudo-scalar fields are introduced on a coarse lattice to give an effective description of the fermionic interactions. We obtain particular solutions to the Ginsparg-Wilson relation and other Ward-Takahashi identities in a closed form. These non-perturbative solutions can be used to construct a chiral invariant action and an invariant path-integral measure on the coarse lattice. The resulting partition function exhibits the exact chiral symmetry in the fermionic system with the auxiliary fields. - Towards the Super Yang-Mills Theory on the Lattice

Katsumi Itoh, Mitsuhiro Kato, Hideyuki Sawanaka, Hiroto So, Naoya Ukita, Prog.Theor.Phys. 108 (2002) 363-374, 2001年12月28日, We present an entirely new approach towards a realization of the supersymmetric Yang-Mills theory on the lattice. The action consists of the staggered fermion and the plaquette variables distributed in the Euclidean space with a particular pattern. The system is shown to have fermionic symmetries relating the fermion and the link variables. - Realization of Global Symmetries in the Wilsonian Renormalization Group

Yuji Igarashi, Katsumi Itoh, Hiroto So, Phys.Lett. B526 (2002) 164-172, 2001年11月13日, We present a method to solve the master equation for the Wilsonian action in the antifield formalism. This is based on a representation theory for cutoff dependent global symmetries along the Wilsonian renormalization group (RG) flow. For the chiral symmetry, the master equation for the free theory yields a continuum version of the Ginsparg-Wilson relation. We construct chiral invariant operators describing fermionic self-interactions. The use of canonically transformed variables is shown to simplify the underlying algebraic structure of the symmetry. We also give another non-trivial example, a realization of SU(2) vector symmetry. Our formalism may be used for a non-perturbative truncation of the Wilsonian action preserving global symmetries. - Regularized Quantum Master Equation in the Wilsonian Renormalization
Group

Yuji Igarashi, Katsumi Itoh, Hiroto So, JHEP 0110:032,2001, 2001年09月26日, Using the Pauli-Villars regularization, we make a perturbative analysis of the quantum master equation (QME), $\Sigma =0$, for the Wilsonian effective action. It is found that the QME for the UV action determines whether exact gauge symmetry is realized along the renormalization group (RG) flow. The basic task of solving the QME can be reduced to compute the Troost-van Niuwenhuizen-Van Proyen jacobian factor for the classical UV action. When the QME cannot be satisfied, the non-vanishing $\Sigma$ is proportional to a BRS anomaly, which is shown to be preserved along the RG flow. To see how the UV action fulfills the QME in anomaly free theory, we calculate the jacobian factor for a pure Yang-Mills theory in four dimensions. - BRS Symmetry, the Quantum Master Equation, and the Wilsonian
Renormalization Group

Yuji Igarashi, Katsumi Itoh, Hiroto So, Prog.Theor.Phys. 106 (2001) 149-166, 2001年01月16日, Recently we made a proposal for realization of an effective BRS symmetry along the Wilsonian renormalization group flow. In this paper we show that the idea can be naturally extended for the most general gauge theories. Extensive use of the antifield formalism is made to reveal some remarkable structure of the effective BRS symmetry. The average action defined with a continuum analog of the block spin transformation obeys the quantum master equation (QME), provided that an UV action does so. We show that the RG flow described by the exact flow equations is generated by canonical transformations in the field-antifield space. Using the relation between the average action and the Legendre effective action, we establish the equivalence between the QME for the average action and the modified Ward-Takahashi identity for the Legendre action. The QME remains intact when the regularization is removed. - Supersymmetry on Lattice Using Ginsparg-Wilson Relation

Hiroto. So, Naoya Ukita, Nucl.Phys.Proc.Suppl. 94 (2001) 795-798, 2000年11月10日, The Ginsparg-Wilson(G-W) relation for chiral symmetry is extended for a supersymmetrical(SUSY) case on a lattice. It is possible to define exact lattice supersymmetry which are devided into two different cases according to using difference operators. $U(1)_R$ symmetry on the lattice is also realized as one of exact symmetries. For an application, the extended G-W relation is given for a two-dimensional model with chiral multiplets. - What happens to Lattice Fermion near Continuum Limit?

Minoru Koseki, Hiroto So, Naoya Ukita, 2000年11月01日, A Ginsparg-Wilson Relation (GWR) is obtained in the presence of chiral symmetry breaking terms. It leads to the PCAC relation as well as an anomaly relation on the lattice. For general fermions, the deviation from the exact GWR is getting small when the block-spin transformations are performed iteratively. Based on a simple geometrical interpretation of the Dirac operator satisfying the GWR, we find some physical properties shared by the lattice fermions near the continuum limit. In two-dimensions, we explicitly construct the GW Dirac operator by using a conformal mapping. - Exact BRS Symmetry realized on the Renormalization Group Flow

Yuji Igarashi, Katsumi Itoh, Hiroto So, Prog.Theor.Phys. 104 (2000) 1053-1066, 2000年06月23日, Using the average action defined with a continuum analog of the block spin transformation, we show the presence of gauge symmetry along the Wilsonian renormalization group flow. As a reflection of the gauge symmetry, the average action satisfies the quantum master equation(QME). We show that the quantum part of the master equation is naturally understood once the measure contribution under the BRS transformation is taken into account. Furthermore an effective BRS transformation acting on macroscopic fields may be defined from the QME. The average action is explicitly evaluated in terms of the saddle point approximation up to one-loop order. It is confirmed that the action satisfies the QME and the flow equation. - Exact Symmetries realized on the Renormalized Group Flow

Yuji Igarashi, Katsumi Itoh, Hiroto So, Phys.Lett. B479 (2000) 336-342, 1999年12月28日, We show that symmetries are preserved exactly along the (Wilsonian) renormalization group flow, though the IR cutoff deforms concrete forms of the transformations. For a gauge theory the cutoff dependent Ward-Takahashi identity is written as the master equation in the antifield formalism: one may read off the renormalized BRS transformation from the master equation. The Maxwell theory is studied explicitly to see how it works. The renormalized BRS transformation becomes non-local but keeps off-shell nilpotency. Our formalism is applicable for a generic global symmetry. The master equation considered for the chiral symmetry provides us with the continuum analog of the Ginsparg-Wilson relation and the L{\" u}scher's symmetry. - Ginsparg-Wilson Relation and Lattice Supersymmetry

Hiroto So, Naoya Ukita, Phys.Lett. B457 (1999) 314-318, 1998年12月03日, The Ginsparg-Wilson(G-W) relation is extended for supersymmetric free theories on a lattice. Exact lattice supersymmetry(SUSY) can be defined without any ambiguities in difference operators. The lattice action constructed by a block-spin transformation is invariant under the symmetry. $U(1)_R$ symmetry on the lattice is also realized as one of exact symmetries. For an application, the extended G-W relation is given for a two-dimensional model with chiral-multiplets. It is argued that the relation may be generalized for interacting cases. - Euclidean Solutions in Broken Phase and Electro-Weak Dynamics

Hiroyuki Kanada, Hiroto So, Shin Takeda, 1996年10月29日, A Higgs-Yukawa system in a broken phase and Euclidean solutions are investigated. Although it has been believed that there are no Euclidean solutions in the broken phase in 4-dimension, we find numerically ones in the phase due to the effect of a strong Yukawa coupling. The complex Yukawa coupling is necessary for the stability of the solution. The extension to a complex Higgs-Yukawa system is also investigated. - Baryon and Lepton Number Assignment in $E_6$ Models

Emmanuel A. Paschos, Utpal Sarkar, Hiroto So, Phys.Rev. D52 (1995) 1701-1705, 1995年04月21日, In $E_6$ models there are new particles whose baryon number is not uniquely assigned. We point out that the baryon and lepton number assignment to these particles can change the baryogenesis scenario significantly. We consider left-right symmetric extension of the standard model in which $(B-L)$ quantum number is gauged. The identification of $(B-L)$ with a generator of $E_6$ is used to define the baryon and lepton numbers for the exotic particles in a way that the electroweak baryon and lepton number anomaly corresponding to the $SU(2)_L$ group vanishes, {\it i.e.}, there is no non-perturbative baryon or lepton number violation during the electroweak phase transition. We study some consequences of the new assignment.

- Cohomological property of CLR-realized supersymmetric quantum mechanics on lattice

宗 博人, 日本物理学会第71回年次大会(2016年), 2016年03月22日