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The gradient flow exact renormalization group (GFERG) is an idea that incorporates gauge-invariant gradient flows into the formalism of the exact renormalization group (ERG). GFERG introduces a Wilson action with a cutoff while keeping the vector gauge invariance manifestly. The details of the formalism are still to be worked out. In this paper, we apply GFERG to construct the Wilson action of massless Dirac fermions under the background chiral gauge fields. By formulating the chiral anomaly as a "composite operator", we make the scale invariance of the anomaly manifest. We argue that the same result extends to QCD.

DOI： 10.1093/ptep/ptad074

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We construct a non-trivial $U(1)/\mathbb{Z}_q$ principal bundle on~$T^4$ from the compact $U(1)$ lattice gauge field by generalizing L\"uscher's constriction so that the cocycle condition contains $\mathbb{Z}_q$ elements (the 't~Hooft flux). The construction requires an admissibility condition on lattice gauge field configurations. From the transition function so constructed, we have the fractional topological charge that is $\mathbb{Z}_q$ one-form gauge invariant and odd under the lattice time reversal transformation. Assuming a rescaling of the vacuum angle $\theta\to q\theta$ suggested from the Witten effect, our construction provides a lattice implementation of the mixed 't~Hooft anomaly between the $\mathbb{Z}_q$ one-form symmetry and the time reversal symmetry in the $U(1)$ gauge theory with matter fields of charge~$q\in2\mathbb{Z}$ when $\theta=\pi$, which was studied by Honda and Tanizaki in the continuum framework.

DOI： 10.1093/ptep/ptad009

Language：English   Publishing type：Research paper (international conference proceedings)  

DOI： 10.22323/1.430.0177

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We define a one-particle irreducible (1PI) Wilson action in the gradient flow exact renormalization group (GFERG) formalism as the Legendre transform of a Wilson action. We consider quantum electrodynamics in particular, and show that the GFERG flow equation preserves the invariance of the 1PI Wilson action (excluding the gauge-fixing term) under the conventionalU(1) gauge transformation. This is in contrast to the invariance of the original Wilson action under a modified U(1) gauge transformation. The global chiral transformation also takes the conventional form for the 1PI Wilson action. Despite the complexity of the GFERG flow equation, the conventional form of the gauge and global chiral transformations may allow us to introduce a non-perturbative Ansatz for gauge and chiral invariant 1PI Wilson actions.

DOI： 10.1093/ptep/ptac047

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We formulate quantum electrodynamics on the basis of gauge (or BRST) covariant diffusion equations of fields. This is a particular example of the gradient flow exact renormalization group (GFERG). The resulting Wilson action fulfills a simple gauge Ward-Takahashi identity. We solve the GFERG equation around the Gaussian fixed point to the second order in gauge coupling and obtain the 1-loop beta function and anomalous dimensions. The anomalous dimension of the electron field coincides with that of the fermion field diffused by a gauge covariant flow equation of Lüscher.

DOI： 10.1093/ptep/ptac003

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In our original paper, Figs. 1 and 2 are incorrect due to a mistake in the numerical code and should be replaced by the following ones.We would like to thank Mizuki Tanaka and Masakiyo Kitazawa for pointing this out to us. The authors would like to apologize for this error. (Figure Presented).

DOI： 10.1093/ptep/ptab096

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We establish a concrete correspondence between a gradient flow and the renormalization group flow for a generic scalar field theory. We use the exact renormalization group formalism with a particular choice of the cutoff function.

DOI： 10.1093/ptep/ptab100

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We theoretically clarify the functional form to be used in t → 0 extrapolation in the small flow time expansion method for the energy-momentum tensor (EMT), which facilitates lattice simulation of the EMT based on the gradient flow. We argue that in the t → 0 extrapolation analysis, lattice data should be fitted by a power function in g(μ(t)), the flow time dependent running coupling, where the power is determined by the perturbation order we consider. From actual lattice data, we confirm the validity of the extrapolation function. Using the new extrapolation function, we present updated lattice results for thermodynamics quantities in quenched QCD; our results are consistent with the previous study [T. Iritani et al., Prog. Theor. Exp. Phys. 2019, 023B02 (2019)] but we obtain smaller errors due to the reduction of systematic errors.

DOI： 10.1093/ptep/ptab068

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The gradient flow bears a close resemblance to the coarse graining, the guiding principle of the renormalization group (RG). In the case of scalar field theory, a precise connection has been made between the gradient flow and the RG flow of the Wilson action in the exact renormalization group (ERG) formalism. By imitating the structure of this connection, we propose an ERG differential equation that preserves manifest gauge invariance in Yang-Mills theory. Our construction in continuum theory can be extended to lattice gauge theory.

DOI： 10.1093/ptep/ptab006

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Abstract

We study latent heat and the pressure gap between the hot and cold phases at the first-order deconfining phase transition temperature of the SU(3) Yang–Mills theory. Performing simulations on lattices with various spatial volumes and lattice spacings, we calculate the gaps of the energy density and pressure using the small flow-time expansion (SF$t$X) method. We find that the latent heat $Delta epsilon$ in the continuum limit is $Delta epsilon /T^4 = 1.117 pm 0.040$ for the aspect ratio $N_s/N_t=8$ and $1.349 pm 0.038$ for $N_s/N_t=6$ at the transition temperature $T=T_c$. We also confirm that the pressure gap is consistent with zero, as expected from the dynamical balance of two phases at $T_c$. From hysteresis curves of the energy density near $T_c$, we show that the energy density in the (metastable) deconfined phase is sensitive to the spatial volume, while that in the confined phase is insensitive. Furthermore, we examine the effect of alternative procedures in the SF$t$X method—the order of the continuum and the vanishing flow-time extrapolations, and also the renormalization scale and higher-order corrections in the matching coefficients. We confirm that the final results are all very consistent with each other for these alternatives.

DOI： 10.1093/ptep/ptaa184

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<title>Abstract</title>
We study latent heat and the pressure gap between the hot and cold phases at the first-order deconfining phase transition temperature of the SU(3) Yang–Mills theory. Performing simulations on lattices with various spatial volumes and lattice spacings, we calculate the gaps of the energy density and pressure using the small flow-time expansion (SF&#36;t&#36;X) method. We find that the latent heat &#36;Delta epsilon&#36; in the continuum limit is &#36;Delta epsilon /T^4 = 1.117 pm 0.040&#36; for the aspect ratio &#36;N_s/N_t=8&#36; and &#36;1.349 pm 0.038&#36; for &#36;N_s/N_t=6&#36; at the transition temperature &#36;T=T_c&#36;. We also confirm that the pressure gap is consistent with zero, as expected from the dynamical balance of two phases at &#36;T_c&#36;. From hysteresis curves of the energy density near &#36;T_c&#36;, we show that the energy density in the (metastable) deconfined phase is sensitive to the spatial volume, while that in the confined phase is insensitive. Furthermore, we examine the effect of alternative procedures in the SF&#36;t&#36;X method—the order of the continuum and the vanishing flow-time extrapolations, and also the renormalization scale and higher-order corrections in the matching coefficients. We confirm that the final results are all very consistent with each other for these alternatives.

DOI： 10.1093/ptep/ptaa184

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In Sec. II C, Eqs. (50) and (51) for “linear+log fit” should read (Farmula Presented) and (Farmula Presented) espectively, where ΛMS=332(17) MeV is the QCD lambda parameter for three-flavor QCD quoted in the Particle Data Group [1]. The dotted curves for the linear+log fit in Figs. 2–7, 10, and 12 were showing the results of nonlinear fit by error. The correct figures are given in this Erratum. The caption of Fig. 10 is slightly modified to quote Eq. (51). (Figure Presented) Analyses in the paper have been done adopting the above correct formulas, and thus the results and the tables as well as other figures are unchanged.

DOI： 10.1103/PhysRevD.102.059903

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<title>Abstract</title>
We present additional observations to previous studies on the infrared (IR) renormalon in $SU(N)$ QCD(adj.), the $SU(N)$ gauge theory with $n_W$-flavor adjoint Weyl fermions on $mathbb{R}^3 imes S^1$ with the $mathbb{Z}_N$ twisted boundary condition. First, we show that, for arbitrary finite $N$, a logarithmic factor in the vacuum polarization of the “photon” (the gauge boson associated with the Cartan generators of $SU(N)$) disappears under the $S^1$ compactification. Since the IR renormalon is attributed to the presence of this logarithmic factor, it is concluded that there is no IR renormalon in this system with finite $N$. This result generalizes the observation made by Anber and Sulejmanpasic [J. High Energy Phys. 1501, 139 (2015)] for $N=2$ and $3$ to arbitrary finite $N$. Next, we point out that, although renormalon ambiguities do not appear through the Borel procedure in this system, an ambiguity appears in an alternative resummation procedure in which a resummed quantity is given by a momentum integration where the inverse of the vacuum polarization is included as the integrand. Such an ambiguity is caused by a simple zero at non-zero momentum of the vacuum polarization. Under the decompactification $R oinfty$, where $R$ is the radius of the $S^1$, this ambiguity in the momentum integration smoothly reduces to the IR renormalon ambiguity in $mathbb{R}^4$. We term this ambiguity in the momentum integration “renormalon precursor”. The emergence of the IR renormalon ambiguity in $mathbb{R}^4$ under the decompactification can be naturally understood with this notion.

DOI： 10.1093/ptep/ptaa096

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To study the CP-violation using the K0-K̄0 oscillation, we need the kaon bag parameter which represents QCD corrections in the leading Feynman diagrams. The lattice QCD provides us with the only way to evaluate the kaon bag parameter directly from the first principles of QCD. However, a calculation of relevant four quark operators with theoretically sound Wilson-type lattice quarks had to carry a numerically big burden of extra renormalizations and resolution of extra mixings due to the explicit chiral violation. Recently, the small flow-time expansion (SFtX) method was proposed as a general method based on the gradient flow to correctly calculate any renormalized observables on the lattice, irrespective of the explicit violations of related symmetries on the lattice. To apply the SFtX method, we need matching coefficients, which relate finite operators at small flow times in the gradient flow scheme to renormalized observables in conventional renormalization schemes. In this paper, we calculate the matching coefficients for four quark operators and quark bilinear operators, relevant to the kaon bag parameter.

DOI： 10.1103/PhysRevD.102.034508

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We study thermodynamic properties of Nf=2+1 QCD on the lattice adopting a nonperturbatively O(a)-improved Wilson quark action and the renormalization group-improved Iwasaki gauge action. To cope with the problems due to explicit violation of the Poincaré and chiral symmetries, we apply the small flow-time expansion (SFtX) method based on the gradient flow, which is a general method to correctly calculate any renormalized observables on the lattice. In this method, the matching coefficients in front of operators in the small flow-time expansion are calculated by perturbation theory thanks to the asymptotic freedom around the small flow-time limit. In a previous study using one-loop matching coefficients, we found that the SFtX method works well for the equation of state extracted from diagonal components of the energy-momentum tensor and for the chiral condensates and susceptibilities. In this paper, we study the effect of two-loop matching coefficients which have been calculated by Harlander et al. recently. We also test the influence of the renormalization scale in the SFtX method. We find that, by adopting the μ0 renormalization scale of Harlander et al. instead of the conventional μd=1/8t scale, the linear behavior at large flow-times is improved so that we can perform the t→0 extrapolation of the SFtX method more confidently. In the calculation of the two-loop matching coefficients by Harlander et al., the equation of motion for quark fields was used. For the entropy density in which the equation of motion has no effects, we find that the results using the two-loop coefficients agree well with those using one-loop coefficients. On the other hand, for the trace anomaly which is affected by the equation of motion, we find discrepancies between the one- and two-loop results at high temperatures. By comparing the results of one-loop coefficients with and without using the equation of motion, the main origin of the discrepancies is suggested to be attributed to contamination of O((aT)2)=O(1/Nt2) discretization errors in the equation of motion at Nt≲10.

DOI： 10.1103/PhysRevD.102.014510

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By employing the 1/N expansion, we compute the vacuum energy E(δ&epsilon;) of the two-dimensional supersymmetric (SUSY) CPN-1 model on R × S1 with ZN twisted boundary conditions to the second order in a SUSY-breaking parameter δ&epsilon;. This quantity was vigorously studied recently by Fujimori et al. using a semi-classical approximation based on the bion, motivated by a possible semi-classical picture on the infrared renormalon. In our calculation, we find that the parameter δ&epsilon; receives renormalization and, after this renormalization, the vacuum energy becomes ultraviolet finite. To the next-to-leading order of the 1/N expansion, we find that the vacuum energy normalized by the radius of the S1, R, RE(δ&epsilon;) behaves as inverse powers of Λ R for Λ R small, where Λ is the dynamical scale. Since Λ is related to the renormalized 't Hooft coupling ΛR as Λ∼ e-2π/ΛR, to the order of the 1/N expansion we work out, the vacuum energy is a purely non-perturbative quantity and has no well-defined weak coupling expansion in ΛR,.

DOI： 10.1093/ptep/ptaa066

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In the leading order of the large-N approximation, we study the renormalon ambiguity in the gluon (or, more appropriately, photon) condensate in the 2D supersymmetric ℂPN-1 model on ℝ × S1 with the ℤN twisted boundary conditions. In our large-N limit, the combination ΛR, where Λ is the dynamical scale and R is the S1 radius, is kept fixed (we set ΛR ≪ 1 so that the perturbative expansion with respect to the coupling constant at the mass scale 1/R is meaningful). We extract the perturbative part from the large-N expression of the gluon condensate and obtain the corresponding Borel transform B(u). For ℝ × S1, we find that the Borel singularity at u = 2, which exists in the system on the uncompactified ℝ2 and corresponds to twice the minimal bion action, disappears. Instead, an unfamiliar renormalon singularity emerges at u = 3/2 for the compactified space ℝ × S1. The semi-classical interpretation of this peculiar singularity is not clear because u = 3/2 is not dividable by the minimal bion action. It appears that our observation for the system on ℝ × S1 prompts reconsideration on the semi-classical bion picture of the infrared renormalon.

DOI： 10.1093/ptep/ptaa002

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We study the infrared renormalon in the gluon condensate in the SU(N) gauge theory with nW-flavor adjoint Weyl fermions (QCD(adj.)) on ℝ3× S1 with the ℤN twisted boundary conditions. We rely on the so-called large-β0 approximation as a conventional tool to analyze the renormalon, in which only Feynman diagrams that dominate in the large-nW limit are considered, while the coefficient of the vacuum polarization is set by hand to the one-loop beta function β0 = 11/3 2nW/3. In the large N limit within the large-β0 approximation, the W-boson, which acquires the twisted Kaluza-Klein momentum, produces the renormalon ambiguity corresponding to the Borel singularity at u = 2. This provides an example that the system in the compactified space R3 × S1 possesses the renormalon ambiguity identical to that in the uncompactified space ℝ4. We also discuss the subtle issue that the location of the Borel singularity can change depending on the order of two necessary operations.

DOI： 10.1093/ptep/ptz157

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We point out that the location of renormalon singularities in theory on a circle-compactified spacetime Rd-1} × S1 (with a small radius R Λ << 1) can differ from that on the non-compactified spacetime Rd. We argue this under the following assumptions, which are often realized in large N theories with twisted boundary conditions: (i) a loop integrand of a renormalon diagram is volume independent, i.e. it is not modified by the compactification, and (ii) the loop momentum variable along the S1 direction is not associated with the twisted boundary conditions and takes the values n/R with integer n. We find that the Borel singularity is generally shifted by-1/2 in the Borel u-plane, where the renormalon ambiguity of O(Λk) is changed to O(Λk-1/R) due to the circle compactification Rd → Rd-1 × S1. The result is general for any dimension d and is independent of details of the quantities under consideration. As an example, we study the CPN-1} model on R × S1 with ZN twisted boundary conditions in the large-N limit.

DOI： 10.1093/ptep/ptz147

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Renormalon-free definition of the gluon condensate within the large-$eta_0$ approximation
We propose a clear definition of the gluon condensate within the large-β0 approximation as an attempt toward a systematic argument on the gluon condensate. We define the gluon condensate such that it is free from a renormalon uncertainty, consistent with the renormalization scale independence of each term of the operator product expansion (OPE), and an identical object irrespective of observables. The renormalon uncertainty of O(Λ4), which renders the gluon condensate ambiguous, is separated from a perturbative calculation by using a recently suggested analytic formulation. The renormalon uncertainty is absorbed into the gluon condensate in the OPE, which makes the gluon condensate free from the renormalon uncertainty. As a result, we can define the OPE in a renormalon-free way. Based on this renormalon-free OPE formula, we discuss numerical extraction of the gluon condensate using the lattice data of the energy density operator defined by the Yang-Mills gradient flow.

DOI： 10.1093/ptep/ptz100

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DOI： 10.1093/ptep/ptz020

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Equation (C7) in the Appendix should read (Formula Presented) The wrong expression is not used in our numerical simulations and any results are not affected by this typo.

DOI： 10.1103/PhysRevD.99.059904

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Have computed the two-loop order (i.e., NNLO) coefficients in the gradient-flow representation of the energy-momentum tensor (EMT) in vector-like gauge theories. In this paper, we study the effect of the two-loop order corrections (and the three-loop order correction for the trace part of the EMT, which is available through the trace anomaly) on the lattice computation of thermodynamic quantities in quenched QCD. The use of the two-loop order coefficients generally reduces the t dependence of the expectation values of the EMT in the gradient-flow representation, where t is the flow time. With the use of the two-loop order coefficients, therefore, the t → 0 extrapolation becomes less sensitive to the fit function, the fit range, and the choice of the renormalization scale; the systematic error associated with these factors is considerably reduced.

DOI： 10.1093/ptep/ptz001

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In K. Hieda, A. Kasai, H. Makino, and H. Suzuki, Prog. Theor. Exp. Phys. 2017, 063B03 (2017), a properly normalized supercurrent in the four-dimensional (4D) N = 1 super Yang-Mills theory (SYM) that works within on-mass-shell correlation functions of gauge-invariant operators is expressed in a regularization-independent manner by employing the gradient flow. In the present paper, this construction is extended to the supercurrent in the 4D N = 2 SYM. The so-constructed supercurrent will be useful, for instance, for fine tuning of lattice parameters toward the supersymmetric continuum limit in future lattice simulations of the 4D N = 2 SYM.

DOI： 10.1093/ptep/pty117

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It is believed that the two-dimensional massless N = 2 Wess-Zumino model becomes the N = 2 superconformal field theory (SCFT) in the infrared (IR) limit. We examine this theoretical conjecture of the Landau-Ginzburg (LG) description of the N = 2 SCFT by numerical simulations on the basis of a supersymmetric-invariant momentum-cutoff regularization. We study a single supermultiplet with cubic and quartic superpotentials. From two-point correlation functions in the IR region, we measure the scaling dimension and the central charge, which are consistent with the conjectured LG description of the A2 and A3 minimal models, respectively. Our result supports the theoretical conjecture and, at the same time, indicates a possible computational method of correlation functions in the N = 2 SCFT from the LG description.

DOI： 10.1093/ptep/pty088

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A regularization-independent universal formula for the energy-momentum tensor in gauge theory in the flat spacetime can be written down by employing the so-called Yang-Mills gradient flow. We examine a possible use of the formula in the calculation of the axial U(1) anomaly in a gravitational field, the anomaly first obtained by Toshiei Kimura [Prog. Theor. Phys. 42, 1191 (1969)]. As a general argument indicates, the formula reproduces the correct non-local structure of the (axial U(1) current)-(energy-momentum tensor)-(energy-momentum tensor) triangle diagram in a way that is consistent with the axial U(1) anomaly. On the other hand, the formula does not automatically reproduce the general coordinate (or translation) Ward-Takahashi relation, requiring corrections by local counterterms. This analysis thus illustrates the fact that the universal formula as it stands can be used only in on-shell correlation functions, in which the energy-momentum tensor does not coincide with other composite operators in coordinate space.

DOI： 10.1093/ptep/pty073

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The gradient flow is the evolution of fields and physical quantities along a dimensionful parameter t, the flow time. We give a simple argument that relates this gradient flow and the Wilsonian renormalization group (RG) flow. We then illustrate the Wilsonian RG flow on the basis of the gradient flow in two examples that possess an infrared fixed point, the 4D many-flavor gauge theory and the 3D O(N) linear sigma model.

DOI： 10.1093/ptep/pty050

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The gradient flow [1-5] gives rise to a versatile method to construct renor-malized composite operators in a regularization-independent manner. By adopting this method, the authors of Refs. [6-9] obtained the expression of Noether currents on the lattice in the cases where the associated symmetries are broken by lattice regularization. We apply the same method to the Noether current associated with supersymmetry, i.e., the supercurrent. We consider the 4D N = 1 super Yang-Mills theory and calculate the renormalized supercurrent in the one-loop level in the Wess-Zumino gauge. We then re-express this supercurrent in terms of the flowed gauge and flowed gaugino fields [10].

DOI： 10.1051/epjconf/201817511014

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Recently, Grabowska and Kaplan constructed a four-dimensional lattice formulation of chiral gauge theories on the basis of the chiral overlap operator. At least in the tree-level approximation, the left-handed fermion is coupled only to the original gauge field A, while the right-handed one is coupled only to the gauge field A , a deformation of A by the gradient flow with infinite flow time. In this paper, we study the fermion one-loop effective action in their formulation. We show that the continuum limit of this effective action contains local interaction terms between A and A , even if the anomaly cancellation condition is met. These non-vanishing terms would lead an undesired perturbative spectrum in the formulation. ∗ ∗

DOI： 10.1051/epjconf/201817511013

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We study the energy-momentum tensor and the equation of state as well as the chiral condensate in (2+1)-flavor QCD at the physical point applying the method of Makino and Suzuki based on the gradient flow. We adopt a nonperturbatively O(a)- improved Wilson quark action and the renormalization group-improved Iwasaki gauge action. At Lattice 2016, we have presented our preliminary results of our study in (2+1)- flavor QCD at a heavy u; d quark mass point. We now extend the study to the physical point and perform finite-temperature simulations in the range T ? 155.544 MeV (Nt = 4-14 including odd Nt's) at a ? 0:09 fm. We show our final results of the heavy QCD study and present some preliminary results obtained at the physical point so far.

DOI： 10.1051/epjconf/201817507023

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© The Authors, published by EDP Sciences, 2018. We measure correlation functions of the nonperturbatively renormalized energy-momentum tensor in Nf = 2 + 1 full QCD at finite temperature by applying the gradient flow method both to the gauge and quark fields. Our main interest is to study the conservation law of the energy-momentum tensor and to test whether the linear response relation is properly realized for the entropy density. By using the linear response relation we calculate the specific heat from the correlation function. We adopt the nonperturba-tively improved Wilson fermion and Iwasaki gauge action at a fine lattice spacing = 0:07 fm. In this paper the temperature is limited to a single value T ? 232 MeV. The u, d quark mass is rather heavy with mπ=mρ ? 0:63 while the s quark mass is set to approximately its physical value.

DOI： 10.1051/epjconf/201817507013

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© 2017 American Physical Society. The energy-momentum tensor plays an important role in QCD thermodynamics. Its expectation value contains information of the pressure and the energy density as its diagonal part. Further properties like viscosity and specific heat can be extracted from its correlation function. A nonperturbative evaluation of it on the lattice is called. Recently, a new method based on the gradient flow was introduced to calculate the energy-momentum tensor on the lattice and has been successfully applied to quenched QCD. In this paper, we apply the gradient flow method to calculate the energy-momentum tensor in (2+1)-flavor QCD adopting a nonperturbatively O(a)-improved Wilson quark action and the renormalization group-improved Iwasaki gauge action. As the first application of the method with dynamical quarks, we study at a single but fine lattice spacing a≃0.07 fm with heavy u and d quarks (mπ/mρ≃0.63) and approximately physical s quark (mηss/mφ≃0.74). With the fixed-scale approach, temperature is varied by the temporal lattice size Nt at a fixed lattice spacing. Performing simulations on lattices with Nt=16 to 4, the temperature range of T≃174-697 MeV is covered. We find that the results of the pressure and the energy density by the gradient flow method are consistent with the previous results using the T-integration method at T280 MeV (Nt10), while the results show disagreement at T350 MeV (Nt8), presumably due to the small-Nt lattice artifact of O((aT)2)=O(1/Nt2). We also apply the gradient flow method to evaluate the chiral condensate taking advantage of the gradient flow method that renormalized quantities can be directly computed avoiding the difficulty of explicit chiral violation with lattice quarks. We compute the renormalized chiral condensate in the MS- scheme at renormalization scale μ=2 GeV with a high precision to study the temperature dependence of the chiral condensate and its disconnected susceptibility. Even with the Wilson-type quark action which violates the chiral symmetry explicitly, we obtain the chiral condensate and its disconnected susceptibility showing a clear signal of pseudocritical temperature at T∼190 MeV related to the chiral restoration crossover.

DOI： 10.1103/PhysRevD.96.014509

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© The Author(s) 2017. The gradient flow and its small flow-time expansion provide a very versatile method to represent renormalized composite operators in a regularization-independent manner. This technique has been utilized to construct typical Noether currents such as the energy-momentum tensor and the axial-vector current in lattice gauge theory. In this paper, we apply the same technique to the supercurrent in the four-dimensional N = 1 super Yang-Mills theory (4D N = 1 SYM) in theWess-Zumino gauge. Since this approach provides a priori a representation of the properly normalized conserved supercurrent, our result should be useful, e.g., in lattice numerical simulations of the 4D N = 1 SYM; the conservation of the so-constructed supercurrent can be used as a criterion for the supersymmetric point toward which the gluino mass is tuned.

DOI： 10.1093/ptep/ptx073

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One-loop perturbative coupling of A and A <inf>∗</inf> through the chiral overlap operator
© The Author(s) 2017. We study the one-loop effective action defined by the chiral overlap operator in the fourdimensional lattice formulation of chiral gauge theories by Grabowska and Kaplan. In the tree-level continuum limit, the left-handed component of the fermion is coupled only to the original gauge field A, while the right-handed one is coupled only to A∗, which is given by the gradient flow of A with infinite flow time. In this paper, we show that the continuum limit of the one-loop effective action contains local interaction terms between A and A∗, which do not generally vanish even if the gauge representation of the fermion is anomaly free.We argue that the presence of such interaction terms can be regarded as undesired gauge symmetry-breaking effects in the formulation.

DOI： 10.1093/ptep/ptx085

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© 2017 The Author(s) We give an alternative perturbative proof of the renormalizability of the system defined by the gradient flow and the fermion flow in vector-like gauge theories.

DOI： 10.1016/j.nuclphysb.2017.02.017

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© 2017 American Physical Society. We compute the topological charge and its susceptibility in finite temperature (2+1)-flavor QCD on the lattice applying a gradient flow method. With the Iwasaki gauge action and nonperturbatively O(a)-improved Wilson quarks, we perform simulations on a fine lattice with a≃0.07 fm at a heavy u, d quark mass with mπ/mρ≃0.63, but approximately physical s quark mass with mηss/mφ≃0.74. In a temperature range from T≃174 MeV (Nt=16) to 697 MeV (Nt=4), we study two topics on the topological susceptibility. One is a comparison of gluonic and fermionic definitions of the topological susceptibility. Because the two definitions are related by chiral Ward-Takahashi identities, their equivalence is not trivial for lattice quarks which violate the chiral symmetry explicitly at finite lattice spacings. The gradient flow method enables us to compute them without being bothered by the chiral violation. We find a good agreement between the two definitions with Wilson quarks. The other is a comparison with a prediction of the dilute instanton gas approximation, which is relevant in a study of axions as a candidate of the dark matter in the evolution of the Universe. We find that the topological susceptibility shows a decrease in T which is consistent with the predicted χt(T)(T/Tpc)-8 for three-flavor QCD even at low temperature Tpc<T 1.5Tpc.

DOI： 10.1103/PhysRevD.95.054502

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© 2016 American Physical Society. The energy density and the pressure of SU(3) gauge theory at finite temperature are studied by direct lattice measurements of the renormalized energy-momentum tensor obtained by the gradient flow. Numerical analyses are carried out with β=6.287-7.500 corresponding to the lattice spacing a=0.013-0.061 fm. The spatial (temporal) sizes are chosen to be Ns=64, 96, 128 (Nτ=12, 16, 20, 22, 24) with the aspect ratio, 5.33≤Ns/Nτ≤8. Double extrapolation, a→0 (the continuum limit) followed by t→0 (the zero flow-time limit), is taken using the numerical data. Above the critical temperature, the thermodynamic quantities are obtained with a few percent precision including statistical and systematic errors. The results are in good agreement with previous high-precision data obtained by using the integral method.

DOI： 10.1103/PhysRevD.94.114512

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© 2016 World Scientific Publishing Company. Fermion bilinear operators of mass dimension 3, such as the axial-vector and vector currents, the pseudo-scalar and scalar densities, whose normalizations are fixed by Ward-Takahashi (WT) relations, are related to small flow-time behavior of composite operators of fermion fields evolved by Lüscher's flow equation. The representations can be useful in lattice numerical simulations, as recently demonstrated by the WHOT QCD collaboration for the chiral condensation of the Nf = 2 + 1 quantum chromodynamics (QCD) at finite temperature.

DOI： 10.1142/S021773231650214X

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© The Author(s) 2016. Quite recently, Grabowska and Kaplan presented a 4-dimensional lattice formulation of chiral gauge theories based on the chiral overlap operator. We study this formulation from the perspective of the fermion number anomaly and possible associated phenomenology. A simple argument shows that the consistency of the formulation implies that the fermion with the opposite chirality to the physical one, the “fluffy mirror fermion” or “fluff”, suffers from the fermion number anomaly in the same magnitude (with the opposite sign) as the physical fermion. This immediately shows that if at least one of the fluff quarks is massless, the formulation provides a simple viable solution to the strong CP problem. Also, if the fluff interacts with gravity essentially in the same way as the physical fermion, the formulation can realize the asymmetric dark matter scenario.

DOI： 10.1093/ptep/ptw167

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© 2016 The Author(s). We study four-dimensional conformal field theories with an SU(N) global symmetry by employing the numerical conformal bootstrap. We consider the crossing relation associated with a four-point function of a spin 0 operator øki which belongs to the adjoint representation of SU(N). For N = 12 for example, we found that the theory contains a spin 0 SU(12)-breaking relevant operator when the scaling dimension of Øki, δ Øki, is smaller than 1.71. Considering the lattice simulation of many-flavor quantum chromodynamics with 12 flavors on the basis of the staggered fermion, the above SU(12)-breaking relevant operator, if it exists, would be induced by the flavor-breaking effect of the staggered fermion and prevent an approach to an infrared fixed point. Actual lattice simulations do not show such signs. Thus, assuming the absence of the above SU(12)-breaking relevant operator, we have an upper bound on the mass anomalous dimension at the fixed point γ∗m ≤ 1.29 from the relation γ∗m, = 3 - δ Øki, Our upper bound is not so strong practically but it is strict within the numerical accuracy. We also find a kink-like behavior in the boundary curve for the scaling dimension of another SU(12)-breaking operator.

DOI： 10.1093/ptep/ptw046

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Background field method in the gradient flow
The Yang--Mills gradient flow and its extension to the fermion field provide
a very general method to obtain renormalized observables in gauge theory. The
method is applicable also with non-perturbative regularization such as lattice.
The gradient flow thus offers useful probes to study non-perturbative dynamics
of gauge theory. In this work, aiming at possible simplification in
perturbative calculations associated with the gradient flow, a modification of
the gauge-fixed version of the flow equation, which preserves gauge covariance
under the background gauge transformation, is proposed. This formulation allows
for example a very quick one-loop calculation of the small flow time expansion
of a composite operator that is relevant to the construction of a lattice
energy--momentum tensor. Some details of the calculation, which have not been
given elsewhere, are presented.

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The complex Langevin method in conjunction with the gauge cooling is applied to the two-dimensional lattice SU(2) Yang-Mills theory that is analytically solvable. We obtain strong numerical evidence that at large Langevin time the expectation value of the plaquette variable converges, but to a wrong value when the complex phase of the gauge coupling is large.

DOI： 10.1103/PhysRevD.92.085020

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© 2015 American Physical Society. The complex Langevin method in conjunction with the gauge cooling is applied to the two-dimensional lattice SU(2) Yang-Mills theory that is analytically solvable. We obtain strong numerical evidence that at large Langevin time the expectation value of the plaquette variable converges, but to a wrong value when the complex phase of the gauge coupling is large.

DOI： 10.1103/PhysRevD.92.085020

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© 2015 The Author(s). In perturbative consideration of the Yang-Mills gradient flow, it is useful to introduce a gauge non-covariant term("gauge-fixing term") to the flow equation that gives rise to a Gaussian damping factor also for gauge degrees of freedom. In the present paper, we consider a modified formof the gauge-fixing termthat manifestly preserves covariance under the background gauge transformation. It is shown that our gauge-fixing term does not affect gauge-invariant quantities as does the conventional gauge-fixing term. The formulation thus allows a background gauge covariant perturbative expansion of the flow equation that provides, in particular, a very efficient computational method of expansion coefficients in the small flow time expansion. The formulation can be generalized to systems containing fermions.

DOI： 10.1093/ptep/ptv139

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DOI： 10.1103/PhysRevD.92.059902

Language：English  

DOI： 10.1103/PhysRevD.92.059902

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Language：Others   Publishing type：Research paper (scientific journal)  

© 2015 American Physical Society. We establish the direct d=2 on-shell bosonization ψL(x+)=eiξ(x+) and ψR†(x-)=eiξ(x-) in path integral formulation by deriving the off-shell relations ψL(x)ψR†(x)=exp[iξ(x)] and ψR(x)ψL†(x)=exp[-iξ(x)]. Similarly, the on-shell bosonization of the bosonic commuting spinor, φL(x+)=ie-iξ(x+)∂+e-iχ(x+), φR†(x-)=e-iξ(x-)-iχ(x-) and φR(x-)=ieiξ(x-)∂-e+iχ(x-), φL†(x+)=eiξ(x+)+iχ(x+), is established in path integral formulation by deriving the off-shell relations φL(x)φR†(x)=ie-iξ(x)∂+e-iχ(x) and φR(x)φL†(x)=ieiξ(x)∂-eiχ(x).

DOI： 10.1103/PhysRevD.91.065010

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© The Author(s) 2015. Published by Oxford University Press on behalf of the Physical Society of Japan. It is known that the gauge field and its composite operators evolved by the Yang-Mills gradient flow are ultraviolet (UV) finite without any multiplicative wave function renormalization. In this paper, we prove that the gradient flow in the 2D $O(N)$ non-linear sigma model possesses a similar property: The flowed $N$-vector field and its composite operators are UV finite without multiplicative wave function renormalization. Our proof in all orders of perturbation theory uses a $(2+1)$-dimensional field theoretical representation of the gradient flow, which possesses local gauge invariance without gauge field. As an application of the UV finiteness of the gradient flow, we construct the energy-momentum tensor in the lattice formulation of the $O(N)$ non-linear sigma model that automatically restores the correct normalization and the conservation law in the continuum limit.

DOI： 10.1093/ptep/ptv028

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A novel method to study the bulk thermodynamics in lattice gauge theory is proposed on the basis of the Yang-Mills gradient flow with a fictitious time t. The energy density and the pressure P of SU(3) gauge theory at fixed temperature are calculated directly on 323×(6,8,10) lattices from the thermal average of the well-defined energy-momentum tensor TμνR(x) obtained by the gradient flow. It is demonstrated that the continuum limit can be taken in a controlled manner from the t dependence of the flowed data. © 2014 American Physical Society.

DOI： 10.1103/PhysRevD.90.011501

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A novel method to study the bulk thermodynamics in lattice gauge theory is proposed on the basis of the Yang-Mills gradient flow with a fictitious time t. The energy density e and the pressure P of SU(3) gauge theory at fixed temperature are calculated directly on 32(3) x (6, 8, 10) lattices from the thermal average of the well-defined energy-momentum tensor T-mu nu(R)(x) obtained by the gradient flow. It is demonstrated that the continuum limit can be taken in a controlled manner from the t dependence of the flowed data.

DOI： 10.1103/PhysRevD.90.011501

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Local products of fields deformed by the so-called Yang-Mills gradient flow become renormalized composite operators. This fact has been utilized to construct a correctly normalized conserved energy-momentum tensor in the lattice formulation of the pure Yang-Mills theory. In the present paper, this construction is further generalized for vector-like gauge theories containing fermions.

DOI： 10.1093/ptep/ptu070

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We explicitly compute nonperturbative effects in a supersymmetric double-well matrix model corresponding to two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background. We analytically determine the full one-instanton contribution to the free energy and one-point function, including all perturbative fluctuations around the one-instanton background. The leading order two-instanton contribution is determined as well. We see that supersymmetry is spontaneously broken by instantons, and that the breaking persists after taking a double scaling limit which realizes the type IIA theory from the matrix model. The result implies that spontaneous supersymmetry breaking occurs by nonperturbative dynamics in the target space of the IIA theory. Furthermore, we numerically determine the full nonperturbative effects by recursive evaluation of orthogonal polynomials. The free energy of the matrix model appears well-defined and finite even in the strongly coupled limit of the corresponding type IIA theory. The result might suggest a weakly coupled theory appearing as an S-dual to the two-dimensional type IIA superstring theory. © 2013 Elsevier B.V.

DOI： 10.1016/j.nuclphysb.2013.09.005

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The product of gauge fields generated by the Yang-Mills gradient flow for positive flow times does not exhibit the coincidence-point singularity and a local product is thus independent of the regularization. Such a local product can furthermore be expanded by renormalized local operators at zero flow time with finite coefficients that are governed by renormalization group equations. Using these facts, we derive a formula that relates the small flow-time behavior of certain gauge-invariant local products and the correctly-normalized conserved energy-momentum tensor in the Yang-Mills theory. Our formula provides a possible method to compute the correlation functions of a well-defined energy-momentum tensor by using lattice regularization and Monte Carlo simulation. © The Author(s) 2013.

DOI： 10.1093/ptep/ptt059

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It is well-known that Noether currents in the classical four-dimensional N=1 supersymmetric Yang-Mills theory (4D N=1 SYM), i.e., the U(1)A current, the supersymmetry (SUSY) current and the energy-momentum tensor, form a multiplet under SUSY, called the Ferrara-Zumino supermultiplet. Inspired by this structure, we define the energy-momentum tensor in the lattice formulation of 4D N=1 SYM by a renormalized super transformation of a lattice SUSY current. By using a renormalized SUSY Ward-Takahashi relation, the energy-momentum tensor so constructed is shown to be conserved in the quantum continuum limit. Our construction of the energy-momentum tensor is very explicit and usable in non-perturbative numerical simulations. © 2012 Elsevier B.V.

DOI： 10.1016/j.nuclphysb.2012.11.023

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In a recent paper, Suzuki (2013) [1], we presented a possible definition of the energy-momentum tensor in the lattice formulation of the four-dimensional N=1 supersymmetric Yang-Mills theory, that is conserved in the quantum continuum limit. In the present Letter, we propose a quite similar but somewhat different definition of the energy-momentum tensor (that is also conserved in the continuum limit) which is superior in several aspects: In the continuum limit, the origin of the energy automatically becomes consistent with the supersymmetry and the number of renormalization constants that require a (non-perturbative) determination is reduced to two from four, the number of renormalization constants appearing in the construction in Suzuki (2013) [1]. © 2013 Elsevier B.V.

DOI： 10.1016/j.physletb.2013.01.028

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In the context of the lattice regularization of the four-dimensional N=1 supersymmetric Yang-Mills theory (4D N=1 SYM), we formulate a generalized BRS transformation that treats the gauge, supersymmetry (SUSY), translation and axial U(1) (U(1) A) transformations in a unified way. A resultant Slavnov-Taylor identity or the Zinn-Justin equation gives rise to a strong constraint on the quantum continuum limit of symmetry breaking terms with the lattice regularization. By analyzing the implications of the constraint on operator-mixing coefficients in the SUSY and the U(1) A Ward-Takahashi (WT) identities, we prove to all orders of perturbation theory in the continuum limit that, (i) the chiral symmetric limit implies the supersymmetric limit and, (ii) a three-fermion operator that might potentially give rise to an exotic breaking of the SUSY WT identity does not emerge. In previous literature, only a naive or incomplete treatment on these points can be found. Our results provide a solid theoretical basis for lattice formulations of the 4D N=1 SYM. © 2012 Elsevier B.V.

DOI： 10.1016/j.nuclphysb.2012.04.008

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The two-dimensional N=(2,2) Wess-Zumino (WZ) model with a cubic superpotential is numerically studied with a momentum-cutoff regularization that preserves supersymmetry. A numerical algorithm based on the Nicolai map is employed and the resulting configurations have no autocorrelation. This system is believed to flow to an N=(2,2) superconformal field theory (SCFT) in the infrared (IR), the A2 model. From a finite-size scaling analysis of the susceptibility of the scalar field in the WZ model, we determine 1-h-h̄=0.616(25)(13) for the conformal dimensions h and h̄, while 1-h-h̄=0.666. . for the A2 model. We also measure the central charge in the IR region from a correlation function between conserved supercurrents and obtain c=1.09(14)(31) (c=1 for the A2 model). These results are consistent with the conjectured emergence of the A2 model, and at the same time demonstrate that numerical studies can be complementary to analytical investigations for this two-dimensional supersymmetric field theory. © 2011 Elsevier B.V.

DOI： 10.1016/j.nuclphysb.2011.09.007

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For lattice formulations of the two-dimensional N=(2,2) Wess-Zumino (2D N=(2,2) WZ) model on the basis of the Nicolai map, we show that supersymmetry (SUSY) and other symmetries are restored in the continuum limit without fine tuning, to all orders in perturbation theory. This provides a theoretical basis for use of these lattice formulations for computation of correlation functions. © 2010 Elsevier B.V.

DOI： 10.1016/j.physletb.2010.12.012

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A nonperturbative formulation of the Wess-Zumino (WZ) model in two and three dimensions is proposed on the basis of momentum-modes truncation. The formulation manifestly preserves full supersymmetry as well as the translational invariance and all global symmetries, while it is shown to be consistent with the expected locality to all orders of perturbation theory. For the two-dimensional WZ model, a well-defined Nicolai map in the formulation provides an interesting algorithm for Monte Carlo simulations. © 2010 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.physletb.2010.01.022

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We address some issues relating to a supersymmetric (SUSY) Ward-Takahashi (WT) identity in Sugino's lattice formulation of two-dimensional (2D) N = (2, 2) SU (k) supersymmetric Yang-Mills theory (SYM). A perturbative argument shows that the SUSY WT identity in the continuum theory is reproduced in the continuum limit without any operator renormalization/mixing and tuning of lattice parameters. As application of the lattice SUSY WT identity, we show that a prescription for the Hamiltonian density in this lattice formulation, proposed by Kanamori, Sugino and Suzuki, is justified also from a perspective of an operator algebra among correctly-normalized supercurrents. We explicitly confirm the SUSY WT identity in the continuum limit to the first nontrivial order in a semi-perturbative expansion. © 2009 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.physletb.2009.11.028

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We present a simple lattice formulation of two-dimensional N = (2, 2)U (k) supersymmetric QCD (SQCD) with N matter multiplets in the fundamental representation. The construction uses compact gauge link variables and exactly preserves one linear combination of supercharges on the two-dimensional regular lattice. Artificial saddle points in the weak coupling limit and the species doubling are evaded without imposing the admissibility. A perturbative power-counting argument indicates that the target supersymmetric theory is realized in the continuum limit without any fine tuning. © 2009 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.nuclphysb.2009.05.012

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By numerically investigating the conservation law of the supercurrent, we confirm the restoration of supersymmetry in Sugino's lattice formulation of the two-dimensional N = (2, 2) supersymmetric SU (2) Yang-Mills theory with a scalar mass term. Subtlety in the case without the scalar mass term, that appears to ruin perturbative power counting, is also pointed out. © 2008 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.nuclphysb.2008.11.021

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We illustrate some physical application of a lattice formulation of the two-dimensional N = (2, 2) supersymmetric SU (2) Yang-Mills theory with a (small) supersymmetry breaking scalar mass. Two aspects, power-like behavior of certain correlation functions (which implies the absence of the mass gap) and the static potential V (R) between probe charges in the fundamental representation, are considered. For the latter, for R ≲ 1 / g, we observe a linear confining potential with a finite string tension. This confining behavior appears distinct from a theoretical conjecture that a probe charge in the fundamental representation is screened in two-dimensional gauge theory with an adjoint massless fermion, although the static potential for R ≳ 1 / g has to be systematically explored to conclude real asymptotic behavior in large distance. © 2009 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.physletb.2009.01.039

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The global supersymmetry is spontaneously broken if and only if the ground-state energy is strictly positive. We propose to use this fact to observe the spontaneous supersymmetry breaking in Euclidean lattice simulations. For lattice formulations that possess a manifest fermionic symmetry, there exists a natural choice of a Hamiltonian operator that is consistent with a topological property of the Witten index. We confirm validity of our idea in models of the supersymmetric quantum mechanics. We then examine a possibility of a dynamical supersymmetry breaking in the two-dimensional N=(2,2) super Yang-Mills theory with the gauge group SU(2), for which the Witten index is unknown. Differently from a recent conjectural claim, our numerical result tempts us to conclude that supersymmetry is not spontaneously broken in this system. © 2008 The American Physical Society.

DOI： 10.1103/PhysRevD.77.091502

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A strict positivity of the ground-state energy is a necessary and sufficient condition for spontaneous supersymmetry breaking. This ground-state energy may be directly determined from the expectation value of the Hamiltonian in the functional integral, defined with an antiperiodic temporal boundary condition for all fermionic variables. We propose to use this fact to observe the dynamical spontaneous supersymmetry breaking in Euclidean lattice simulations. If a lattice formulation possesses a manifestly preserved fermionic symmetry, there exists a natural choice of a Hamiltonian operator that is consistent with a topological nature of the Witten index. We numerically confirm the validity of our idea in models of supersymmetric quantum mechanics. We further examine the possibility of dynamical supersymmetry breaking in the two-dimensional N = (2, 2) super Yang-Mills theory with the gauge group SU(2), for which the Witten index is unknown. Although statistical errors are still large, we do not observe positive ground-state energy, at least within one standard deviation. This prompts us to draw a different conclusion from a recent conjectural claim that supersymmetry is dynamically broken in this system.

DOI： 10.1143/PTP.119.797

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In continuum field theory, it has been discussed that chiral gauge theories with Weyl fermions in anomalous gauge representations (anomalous gauge theories) can consistently be quantized, provided that some of gauge bosons are permitted to acquire mass. Such theories in four dimensions are inevitablly non-renormalizable and must be regarded as a low-energy effective theory with a finite ultraviolet (UV) cutoff. In this paper, we present a lattice framework which enables one to study such theories in a non-perturbative level. By introducing bare mass terms of gauge bosons that impose ''smoothness'' on the link field, we explicitly construct a consistent fermion integration measure in a lattice formulation based on the Ginsparg-Wilson (GW) relation. This framework may be used to determine in a non-perturbative level an upper bound on the UV cutoff in low-energy effective theories with anomalous fermion content. By further introducing the Stückelberg or Wess-Zumino (WZ) scalar field, this framework provides also a lattice definition of a non-linear sigma model with the Wess-Zumino-Witten (WZW) term. © SISSA 2007.

DOI： 10.1088/1126-6708/2007/10/018

Language：English   Publishing type：Research paper (scientific journal)  

In continuum field theory, it has been discussed that chiral gauge theories with Weyl fermions in anomalous gauge representations (anomalous gauge theories) can consistently be quantized, provided that some of gauge bosons are permitted to acquire mass. Such theories in four dimensions are inevitablly non-renormalizable and must be regarded as a low-energy effective theory with a finite ultraviolet (UV) cutoff. In this paper, we present a lattice framework which enables one to study such theories in a non-perturbative level. By introducing bare mass terms of gauge bosons that impose "smoothness" on the link field, we explicitly construct a consistent fermion integration measure in a lattice formulation based on the Ginsparg-Wilson (GW) relation. This framework may be used to determine in a non-perturbative level an upper bound on the UV cutoff in low-energy effective theories with anomalous fermion content. By further introducing the Stiiekelberg or Wess-Zumino (WZ) scalar field, this framework provides also a lattice definition of a non-linear sigma model with the Wess-Zumino-Witten (WZW) term.

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Numerical results of two-dimensional N=(2,2) super Yang-Mills theory
We report the results of a numerical simulation of a lattice formulation of
the two-dimensional N=(2,2) super Yang-Mills theory proposed by Suzuki and
Taniguchi. We measure the 1-point functions and 2-point functions. The scenario
is that only tuning of the scalar mass to a specific value gives a
supersymmetric continuum limit. Our results are consistent with this scenario
although conclusive results on the restoration of supersymmetry have not been
obtained.

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We carry out preliminary numerical study of Sugino's lattice formulation [1, 2] of the two-dimensional ≤ (2,2) super Yang-Mills theory (2d ≤ (2,2) SYM) with the gauge group SU(2). The effect of dynamical fermions is included by re-weighting a quenched ensemble by the pfaffian factor. It appears that the complex phase of the pfaffian due to lattice artifacts and flat directions of the classical potential are not problematic in Monte Carlo simulation. Various one-point supersymmetric Ward-Takahashi (WT) identities are examined for lattice spacings up to a ≤ 0.5/g with the fixed physical lattice size L ≤ 4.0/g, where g denotes the gauge coupling constant in two dimensions. WT identities implied by an exact fermionic symmetry of the formulation are confirmed in fair accuracy and, for most of these identities, the quantum effect of dynamical fermions is clearly observed. For WT identities expected only in the continuum limit, the results seem to be consistent with the behavior expected from supersymmetry, although we do not see clear distintion from the quenched simulation. We measure also the expectation values of renormalized gauge-invariant bi-linear operators of scalar fields. © SISSA 2007.

DOI： 10.1088/1126-6708/2007/09/052

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In a wide class of GL × GR invariant two-dimensional super-renormalizable field theories, the parity-odd part of the two-point function of global currents is completely determined by a fermion one-loop diagram. For any non-trivial fermion content, the two-point function possesses a massless pole which corresponds to massless bosonic physical states. As an application, we show that two-dimensional N= (2,2) supersymmetric gauge theory without a superpotential possesses U (1)L × U(1) R symmetry and contains one massless bosonic state per fixed spatial momentum. The N = (4,4) supersymmetric pure Yang-Mills theory possesses SU(2)L × SU(2)R symmetry, and there exist at least three massless bosonic states.

DOI： 10.1143/PTP.116.1117

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Overlap Fermion in External Gravity
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.

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It is widely accepted that topological quantities are useful to describe quantum liquids in low dimensions. The (spin) Hall conductances are typical examples. They are expressed by the Chern numbers, which are topological invariants given by the Berry connections of the ground states. We present a topological description for the (spin) Hall conductances on a discretized Brillouin zone. At the same time, it is quite efficient in practical numerical calculations for concrete models. We demonstrate its validity in a model with quantum phase transitions. Topological changes supplemented with the transition is also described in the present lattice formulation. © 2006 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.physe.2006.03.141

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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 γ5 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.

DOI： 10.1143/PTP.116.197

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Using an elementary method, we show that an odd number of Majorana fermions in 8k + 1 dimensions suffer from a gauge anomaly that is analogous to the Witten global gauge anomaly. This anomaly cannot be removed without sacrificing the perturbative gauge invariance. Our construction of higher-dimensional examples (k ≥ 1) makes use of the SO(8) instanton on S8.

DOI： 10.1143/PTP.115.1129

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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(2NC) 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.

DOI： 10.1143/PTP.115.467

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The N ≤ (2,2) extended super Yang-Mills theory in 2 dimensions is formulated on the lattice as a dimensional reduction of a 4 dimensional lattice gauge theory. We use the plaquette action for a bosonic sector and the Wilson- or the overlap-Dirac operator for a fermion sector. The fermion determinant is real and, moreover, when the overlap-Dirac operator is used, semi-positive definite. The flat directions in the target theory become compact and present no subtlety for a numerical integration along these directions. Any exact supersymmetry does not exist in our lattice formulation; nevertheless we argue that one-loop calculable and finite mass counter terms ensure a supersymmetric continuum limit to all orders of perturbation theory. © SISSA 2005.

DOI： 10.1088/1126-6708/2005/10/082

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We present a manifestly gauge-invariant description of Chern numbers associated with the Berry connection defined on a discretized Brillouin zone. It provides an efficient method of computing (spin) Hall conductances without specifying gauge-fixing conditions. We demonstrate that it correctly reproduces quantized Hall conductances even on a coarsely discretized Brillouin zone. A gauge-dependent integer-valued field, which plays a key role in the formulation, is evaluated in several gauges. An extension to the non-Abelian Berry connection is also given. ©2005 The Physical Society of Japan.

DOI： 10.1143/JPSJ.74.1674

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A lattice Wess-Zumino model is formulated on the basis of Ginsparg-Wilson fermions. In perturbation theory, our formulation is equivalent to the formulation by Fujikawa and Ishibashi and by Fujikawa. Our formulation is, however, free from a singular nature of the latter formulation due to an additional auxiliary chiral supermultiplet on a lattice. The model posssesses an exact U(1)R symmetry as a supersymmetric counterpart of the Lüscher lattice chiral U(1) symmetry. A restration of the supersymmetric Ward-Takahashi identity in the continuum limit is analyzed in renormalized perturbation theory. In the one-loop level, a supersymmetric continuum limit is ensured by suitably adjusting a coefficient of a single local term F̃*F̃. The non-renormalization theorem holds to this order of perturbation theory. In higher orders, on the other hand, coefficents of local terms with dimension ≤ 4 that are consistent with the U(1)R symmetry have to be adjusted for a supersymmetric continuum limit. The origin of this complexicity in higher-order loops is clarified on the basis of the Reisz power counting theorem. Therefore, from a view point of supersymmetry, the present formulation is not quite better than a lattice Wess-Zumino model formulated by using Wilson fermions, although a number of coefficients which require adjustment is much less due to the exact U(1)R symmetry. We also comment on an exact non-linear fermionic symmetry which corresponds to the one studied by Bonini and Feo; an existence of this exact symmetry itself does not imply a restoration of supersymmetry in the continuum limit without any adjustment of parameters. © SISSA/ISAS 2005.

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A possible formulation of chiral gauge theories with an anomalous fermion content is re-examined in light of the lattice framework based on the Ginsparg-Wilson relation. It is shown that the fermion sector of a wide class of anomalous non-abelian theories cannot consistently be formulated within this lattice framework. In particular, in 4 dimension, all anomalous non-abelian theories are included in this class. Anomalous abelian chiral gauge theories cannot be formulated with compact U(1) link variables, while a non-compact formulation is possible at least for the vacuum sector in the space of lattice gauge fields. Our conclusion is not applied to effective low-energy theories with an anomalous fermion content which are obtained from an underlying anomaly-free theory by sending the mass of some of fermions to infinity. For theories with an anomalous fermion content in which the anomaly is cancelled by the Green-Schwarz mechanism, a possibility of a consistent lattice formulation is not clear. © SISSA/ISAS 2005.

DOI： 10.1088/1126-6708/2005/01/051

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A local formulation of lattice Wess-Zumino model with exact $U(1)_R$ symmetry
A lattice Wess-Zumino model is formulated on the basis of Ginsparg-Wilson
fermions. In perturbation theory, our formulation is equivalent to the
formulation by Fujikawa and Ishibashi and by Fujikawa. Our formulation is,
however, free from a singular nature of the latter formulation due to an
additional auxiliary chiral supermultiplet on a lattice. The model posssesses
an exact $U(1)_R$ symmetry as a supersymmetric counterpart of the L"uscher
lattice chiral $U(1)$ symmetry. A restration of the supersymmetric
Ward-Takahashi identity in the continuum limit is analyzed in renormalized
perturbation theory. In the one-loop level, a supersymmetric continuum limit is
ensured by suitably adjusting a coefficient of a single local term $ ilde
F^* ilde F$. The non-renormalization theorem holds to this order of
perturbation theory. In higher orders, on the other hand, coefficents of local
terms with dimension $leq4$ that are consistent with the $U(1)_R$ symmetry
have to be adjusted for a supersymmetric continuum limit. The origin of this
complexicity in higher-order loops is clarified on the basis of the Reisz power
counting theorem. Therefore, from a view point of supersymmetry, the present
formulation is not quite better than a lattice Wess-Zumino model formulated by
using Wilson fermions, although a number of coefficients which require
adjustment is much less due to the exact $U(1)_R$ symmetry. We also comment on
an exact non-linear fermionic symmetry which corresponds to the one studied by
Bonini and Feo; an existence of this exact symmetry itself does not imply a
restoration of supersymmetry in the continuum limit without any adjustment of
parameters.

DOI： 10.1088/1126-6708/2005/02/012

Language：Others  

Anomalous gauge theories revisited
A possible formulation of chiral gauge theories with an anomalous fermion
content is re-examined in light of the lattice framework based on the
Ginsparg-Wilson relation. It is shown that the fermion sector of a wide class
of anomalous non-abelian theories cannot consistently be formulated within this
lattice framework. In particular, in 4 dimension, {it all} anomalous
non-abelian theories are included in this class. Anomalous abelian chiral gauge
theories cannot be formulated with compact $U(1)$ link variables, while a
non-compact formulation is possible at least for the vacuum sector in the space
of lattice gauge fields. Our conclusion is not applied to effective low-energy
theories with an anomalous fermion content which are obtained from an
underlying anomaly-free theory by sending the mass of some of fermions to
infinity. For theories with an anomalous fermion content in which the anomaly
is cancelled by the Green-Schwarz mechanism, a possibility of a consistent
lattice formulation is not clear.

DOI： 10.1088/1126-6708/2005/01/051

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A variant of the Nielsen-Ninomiya no-go theorem is formulated. This theorem states that, under several assumptions, it is impossible to write down a doubler-free Euclidean lattice action of a single Majorana fermion in 8k and 8k + 1 dimensions.

DOI： 10.1143/PTP.112.855

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We re-examine the issue of local counter terms in the analysis of quantum anomalies. We analyze two-dimensional theories and show that the notion of local counter terms need to be carefully defined depending on the physics contents such as whether one is analyzing gauge theory or bosonization. It is shown that a part of the Jacobian, which is apparently spurious and eliminated by a local counter term corresponding to the mass term of the gauge field in gauge theory, cannot be removed by a local counter term and plays a central role by giving the kinetic term of the bosonized field in the context of path integral bosonization. © 2004 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.physrep.2004.05.002

Language：Others  

DOI： 10.1093/acprof:oso/9780198529132.001.0001

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In various dimensional euclidean lattice gauge theories, we examine a compatibility of the Majorana decomposition and the charge conjugation property of lattice Dirac operators. In 8n and 1 + 8n dimensions, we find a difficulty to decompose a classical lattice action of the Dirac fermion into a system of the Majorana fermion and thus to obtain a factrized form of the Dirac determinant. Similarly, in 2 + 8n dimensions, there is a difficulty to decompose a classical lattice action of the Weyl fermion into a system of the Majorana-Weyl fermion and thus to obtain a factrized form of the Weyl determinant. Prescriptions based on the overlap formalism do not remove these difficulties. We argue that these difficulties are reflections of the global gauge anomaly associated to the real Weyl fermion in 8n dimensions. For this reason (besides other well-known reasons), a lattice formulation of the N = 1 super Yang-Mills theory in these dimensions is expected to be extremely difficult to find. © SISSA/ISAS 2004.

DOI： 10.1088/1126-6708/2004/07/038

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Majorana and Majorana-Weyl fermions in lattice gauge theory
In various dimensional Euclidean lattice gauge theories, we examine a
compatibility of the Majorana decomposition and the charge conjugation property
of lattice Dirac operators. In $8n$ and $1+8n$ dimensions, we find a difficulty
to decompose a classical lattice action of the Dirac fermion into a system of
the Majorana fermion and thus to obtain a factorized form of the Dirac
determinant. Similarly, in $2+8n$ dimensions, there is a difficulty to
decompose a classical lattice action of the Weyl fermion into a system of the
Majorana--Weyl fermion and thus to obtain a factrized form of the Weyl
determinant. Prescriptions based on the overlap formalism do not remove these
difficulties. We argue that these difficulties are reflections of the global
gauge anomaly associated to the real Weyl fermion in $8n$ dimensions. For this
reason (besides other well-known reasons), a lattice formulation of the N=1
super Yang--Mills theory in these dimensions is expected to be extremely
difficult to find.

DOI： 10.1088/1126-6708/2004/07/038

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The topological charge in the U(N) vector-like reduced model can be defined by rising the overlap Dirac operator. We obtain its large N limit for a fermion in a general gauge-group representation under a certain restriction of gauge field configurations which is termed U(1) embedding. © 2004 Published by Elsevier B.V.

DOI： 10.1016/S0920-5632(03)02623-9

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We construct the Wess-Zumino-Witten (WZW) term in lattice gauge theory by using a Dirac operator which obeys the Ginsparg-Wilson relation. Topological properties of the WZW term known in the continuum are reproduced on the lattice as a consequence of a non-trivial topological structure of the space of admissible lattice gauge fields. In the course of this analysis, we observe that the gauge anomaly generally implies that there is no basis of a Weyl fermion which leads to a single-valued expectation value in the fermion sector. The lattice Witten term, which carries information of a gauge path along which the gauge anomaly is integrated, is separated from the WZW term and the multivaluedness of the Witten term is shown to be related to the homotopy group π2n+1(G). We also discuss the global SU(2) anomaly on the basis of the WZW term. © SISSA/ISAS 2003.

DOI： 10.1088/1126-6708/2003/09/015

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Wess-Zumino-Witten term on the lattice
We construct the Wess-Zumino-Witten (WZW) term in lattice gauge theory by
using a Dirac operator which obeys the Ginsparg-Wilson relation. Topological
properties of the WZW term known in the continuum are reproduced on the lattice
as a consequence of a non-trivial topological structure of the space of
admissible lattice gauge fields. In the course of this analysis, we observe
that the gauge anomaly generally implies that there is no basis of a Weyl
fermion which leads to a single-valued expectation value in the fermion sector.
The lattice Witten term, which carries information of a gauge path along which
the gauge anomaly is integrated, is separated from the WZW term and the
multivaluedness of the Witten term is shown to be related to the homotopy group
$pi_{2n+1}(G)$. We also discuss the global $SU(2)$ anomaly on the basis of
the WZW term.

DOI： 10.1088/1126-6708/2003/09/015

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The axial anomaly arising from the fermion sector of U (N) or SU(N) reduced model is studied under a certain restriction of gauge field configurations (the "U(1) embedding" with N = Ld). We use the overlap-Dirac operator and consider how the anomaly changes as a function of a gauge-group representation of the fermion. A simple argument shows that the anomaly vanishes for an irreducible representation expressed by a Young tableau whose number of boxes is a multiple of L2 (such as the adjoint representation) and for a tensor-product of them. We also evaluate the anomaly for general gauge-group representations in the large N limit. The large N limit exhibits expected algebraic properties as the axial anomaly. Nevertheless, when the gauge group is SU(N), it does not have a structure such as the trace of a product of traceless gauge-group generators which is expected from the corresponding gauge field theory. © SISSA/ISAS 2003.

DOI： 10.1088/1126-6708/2003/05/042

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The CP symmetry is not manifestly implemented for the local and doubler-free Ginsparg-Wilson operator in lattice chiral gauge theory. We precisely identify where the effects of this CP breaking appear.

DOI： 10.1016/S0920-5632(03)80466-8

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We study the axial anomaly defined on a finite-size lattice by using a Dirac operator which obeys the Ginsparg-Wilson relation. When the gauge group is U(1), we show that the basic structure of axial anomaly on the infinite lattice, which can be deduced by a cohomological analysis, persists even on (sufficiently large) finite-size lattices. For non-Abelian gauge groups, we propose a conjecture on a possible form of axial anomaly on the infinite lattice, which holds to all orders in perturbation theory. With this conjecture, we show that a structure of the axial anomaly on finite-size lattices is again basically identical to that on the infinite lattice. Our analysis with the Ginsparg-Wilson-Dirac operator indicates that, in appropriate frameworks, the basic structure of axial anomaly is quite robust and it persists even in a system with finite ultraviolet and infrared cutoffs. © 2002 Elsevier Science B.V. All rights reserved.

DOI： 10.1016/S0550-3213(02)00812-X

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We evaluate for arbitrary even dimensions the classical continuum limit of the lattice axial anomaly defined by the overlap-Dirac operator. Our calculational scheme is simple and systematic. In particular, a powerful topological argument is utilized to determine the value of a lattice integral involved in the calculation. When the Dirac operator is free of species doubling, the classical continuum limit of the axial anomaly in various dimensions is combined into a form of the Chern character, as expected. © SISSA/ISAS 2002.

DOI： 10.1088/1126-6708/2002/09/025

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On the basis of an observation due to Kiskis, Narayanan and Neuberger, we show that there is a remnant of chiral anomalies in the reduced model when a Dirac operator which obeys the Ginsparg-Wilson relation is employed for the fermion sector. We consider fermions belonging to the fundamental representation of the gauge group U (N) or SU(N). For vector-like theories, we determine a general form of the axial anomaly or the topological charge within a framework of a U(1) embedding. For chiral gauge theories with the gauge group U (N), a remnant of gauge anomaly emerges as an obstruction to a smooth fermion integration measure. The pure gauge action of gauge-field configurations which cause these non-trivial phenomena always diverges in the 't Hooft N → ∞ limit when d > 2. © SISSA/ISAS 2002.

DOI： 10.1088/1126-6708/2002/09/032

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Axial anomaly with the overlap-Dirac operator in arbitrary dimensions
We evaluate for arbitrary even dimensions the classical continuum limit of
the lattice axial anomaly defined by the overlap-Dirac operator. Our
calculational scheme is simple and systematic. In particular, a powerful
topological argument is utilized to determine the value of a lattice integral
involved in the calculation. When the Dirac operator is free of species
doubling, the classical continuum limit of the axial anomaly in various
dimensions is combined into a form of the Chern character, as expected.

DOI： 10.1088/1126-6708/2002/09/025

Language：Others  

Chiral anomalies in the reduced model
On the basis of an observation due to Kiskis, Narayanan and Neuberger, we
show that there is a remnant of chiral anomalies in the reduced model when a
Dirac operator which obeys the Ginsparg-Wilson relation is employed for the
fermion sector. We consider fermions belonging to the fundamental
representation of the gauge group U(N) or SU(N). For vector-like theories, we
determine a general form of the axial anomaly or the topological charge within
a framework of a U(1) embedding. For chiral gauge theories with the gauge group
U(N), a remnant of gauge anomaly emerges as an obstruction to a smooth fermion
integration measure. The pure gauge action of gauge-field configurations which
cause these non-trivial phenomena always diverges in the 't Hooft $N oinfty$
limit when d>2.

DOI： 10.1088/1126-6708/2002/09/032

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If one uses a general class of Ginsparg-Wilson operators, it is known that CP symmetry is spoiled in chiral gauge theory for a finite lattice spacing and the Majorana fermion is not defined in the presence of chiral symmetric Yukawa couplings. We summarize these properties in the form of a theorem for the general Ginsparg-Wilson relation. © 2002 Elsevier Science B.V. All rights reserved.

DOI： 10.1016/S0370-2693(02)01936-6

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The CP symmetry is not manifestly implemented for the local and doubler-free Ginsparg-Wilson operator in lattice chiral gauge theory. We precisely identify where the effects of this CP breaking appear. We show that they appear in: (I) Overall constant phase of the fermion generating functional. (II) Overall constant coefficient of the fermion generating functional. (Ill) Fermion propagator appearing in external fermion lines and the propagator connected to Yukawa vertices. The first effect appears from the transformation of the path integral measure and it is absorbed into a suitable definition of the constant phase factor for each topological sector; in this sense there appears no "CP anomaly". The second constant arises from the explicit breaking in the action and it is absorbed by the suitable weights with which topological sectors are summed. The last one in the propagator is inherent to this formulation and cannot be avoided by a mere modification of the projection operator, for example, in the framework of the Ginsparg-Wilson operator. This breaking emerges as an (almost) contact term in the propagator when the Higgs field, which is treated perturbatively, has no vacuum expectation value. In the presence of the vacuum expectation value, however, a completely new situation arises and the breaking becomes intrinsically non-local, though this breaking may still be removed in a suitable continuum limit. This non-local CP breaking is expected to persist for a non-perturbative treatment of the Higgs coupling. © SISSA/ISAS 2002.

DOI： 10.1088/1126-6708/2002/04/046

Language：Others  

CP breaking in lattice chiral gauge theories
The CP symmetry is not manifestly implemented for the local and doubler-free
Ginsparg-Wilson operator in lattice chiral gauge theory. We precisely identify
where the effects of this CP breaking appear. We show that they appear in: (I)
Overall constant phase of the fermion generating functional. (II) Overall
constant coefficient of the fermion generating functional. (III) Fermion
propagator appearing in external fermion lines and the propagator connected to
Yukawa vertices. The first effect appears from the transformation of the path
integral measure and it is absorbed into a suitable definition of the constant
phase factor for each topological sector; in this sense there appears no ``CP
anomaly''. The second constant arises from the explicit breaking in the action
and it is absorbed by the suitable weights with which topological sectors are
summed. The last one in the propagator is inherent to this formulation and
cannot be avoided by a mere modification of the projection operator, for
example, in the framework of the Ginsparg-Wilson operator. This breaking
emerges as an (almost) contact term in the propagator when the Higgs field,
which is treated perturbatively, has no vacuum expectation value. In the
presence of the vacuum expectation value, however, a completely new situation
arises and the breaking becomes intrinsically non-local, though this breaking
may still be removed in a suitable continuum limit. This non-local CP breaking
is expected to persist for a non-perturbative treatment of the Higgs coupling.

DOI： 10.1088/1126-6708/2002/04/046

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Based on the Ginsparg-Wilson relation, a gauge invariant formulation of electroweak SU(2) × U(1) gauge theory on the lattice is considered. If the hypercharge gauge coupling is turned off in the vacuum sector of the U(1) gauge fields, the theory consists of four left-handed SU(2) doublets and it is possible, as in vector-like theories, to make the fermion measure defined globally in all topological sectors of SU(2). We then try to incorporate U(1) gauge field, following Lüscher's reconstruction theorem. The global integrability condition is proved for "gauge loops" in the space of the U(1) gauge fields with arbitrary SU(2) gauge field fixed in the background. For "non-gauge loops", however, the proof is given so far only for the classical SU(2) instanton backgrounds.

DOI： 10.1016/S0920-5632(01)01837-0

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On the lattice construction of electroweak gauge theory
Based on the Ginsparg-Wilson relation, a gauge invariant formulation of
electroweak SU(2)xU(1) gauge theory on the lattice is considered. If the
hypercharge gauge coupling is turned off in the vacuum sector of the U(1) gauge
fields, the theory consists of four left-handed SU(2) doublets and it is
possible, as in vector-like theories, to make the fermion measure defined
globally in all topological sectors of SU(2). We then try to incorporate U(1)
gauge field, following L"uscher's reconstruction theorem. The global
integrability condition is proved for ``gauge loops'' in the space of the U(1)
gauge fields with arbitrary SU(2) gauge field fixed in the background. For
``non-gauge loops'', however, the proof is given so far only for the classical
SU(2) instanton backgrounds.

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The configuration space of abelian gauge theory on a periodic lattice becomes topologically disconnected when exceptional gauge field configurations are removed. It is possible to define a U(1)-bundle from the nonexceptional link variables by a smooth interpolation of the transition functions. The lattice analogue of the Chern character obtained using a cohomological technique based on noncommutative differential calculus is shown to give a topological charge related to the topological winding number of the U(1)-bundle.

DOI： 10.1143/PTP.105.789

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We use the method of Banerjee, Banerjee and Mitra and minimal homotopy paths to compute the consistent gauge anomaly for several superspace models of SSYM coupled to matter. We review the derivation of the anomaly for N=1 in four dimensions and then discuss the anomaly for two-dimensional models with (2,0) supersymmetry. © 2001 Elsevier Science B.V.

DOI： 10.1016/S0550-3213(00)00676-3

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Language：Others   Publishing type：Research paper (scientific journal)  

We study the gauge anomaly A defined on a 4-dimensional infinite lattice while keeping the lattice spacing finite. We assume that (I) A depends smoothly and locally on the gauge potential, (II) A reproduces the gauge anomaly in the continuum theory in the classical continuum limit, and (III) U(1) gauge anomalies have a topological property. It is then shown that the gauge anomaly A can always be removed by local counterterms to all orders in powers of the gauge potential, leaving possible breakings proportional to the anomaly in the continuum theory. This follows from an analysis of nontrivial local solutions to the Wess-Zumino consistency condition in lattice gauge theory. Our result is applicable to the lattice chiral gauge theory based on the Ginsparg-Wilson Dirac operator, when the gauge field is sufficiently weak ∥U(n,μ)-1∥<ε′ , where U(n,μ) is the link variable and ε′ a certain small positive constant. © 2000 Elsevier Science B.V.

DOI： 10.1016/S0550-3213(00)00408-9

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Real Representation in Chiral Gauge Theories on the Lattice
The Weyl fermion belonging to the real representation of the gauge group
provides a simple illustrative example for L"uscher's gauge-invariant lattice
formulation of chiral gauge theories. We can explicitly construct the fermion
integration measure globally over the gauge-field configuration space in the
arbitrary topological sector; there is no global obstruction corresponding to
the Witten anomaly. It is shown that this Weyl formulation is equivalent to a
lattice formulation based on the Majorana (left--right-symmetric) fermion, in
which the fermion partition function is given by the Pfaffian with a definite
sign, up to physically irrelevant contact terms. This observation suggests a
natural relative normalization of the fermion measure in different topological
sectors for the Weyl fermion belonging to the complex representation.

DOI： 10.1088/1126-6708/2000/10/039

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Block-spin transformations from a fine lattice to a coarse one are shown to give rise to a one-to-one correspondence between the zero-modes of the Ginsparg-Wilson Dirac operator on the fine lattice and those on the coarse lattice. The index is then preserved under the blocking process. Such a one-to-one correspondence is violated and the block-spin transformation becomes necessarily ill-defined when the absolute value of the index is larger than 2rN, where N is the number of the sites on the coarse lattice and r is the dimension of the gauge group representation of the fermion variables. (C) 2000 Elsevier Science B.V.

DOI： 10.1016/S0370-2693(00)00920-5

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The axial anomaly in lattice gauge theories has a topological nature when the Dirac operator satisfies the Ginsparg-Wilson relation. We study the axial anomaly in Abelian gauge theories on an infinite hypercubic lattice by utilizing cohomological arguments. The crucial tool in our approach is the non-commutative differential calculus (NCDC) which makes the Leibniz rule of exterior derivatives valid on the lattice. The topological nature of the "Chern character" on the lattice becomes manifest in the context of NCDC. Our result provides an algebraic proof of Lüscher's theorem for a four-dimensional lattice and its generalization to arbitrary dimensions. © 2000 Elsevier Science B.V. All rights reserved.

DOI： 10.1016/S0550-3213(99)00706-3

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Language：Others   Publishing type：Research paper (scientific journal)  

The chiral Jacobian, which is defined with Neuberger's overlap Dirac operator of the lattice fermion, is explicitly evaluated in the continuum limit without expanding it in the gauge coupling constant. Our calculational scheme is simple and straightforward. We determine a coefficient of the chiral anomaly for general values of the mass parameter and the Wilson parameter of the overlap Dirac operator.

DOI： 10.1143/PTP.102.141

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Lüscher's recent formulation of Abelian chiral gauge theories on the lattice, in the vacuum (or perturbative) sector in infinite volume, is reinterpreted in terms of the lattice covariant regularization. The gauge invariance of the effective action and the integrability of the gauge current in anomaly-free cases become transparent. The real part of the effective action is simply one-hall that of the Dirac fermion and, when the Dirac operator behaves properly in the continuum limit, the imaginary part in this limit reproduces the η-invariant.

DOI： 10.1143/PTP.101.1147

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By undertaking additional analyses postponed in a previous paper, we complete our construction of a manifestly supersymmetric gauge-covariant regularization of supersymmetric chiral gauge theories. We present the following: An evaluation of the covariant gauge anomaly; a proof of the integrability of the covariant gauge current in anomaly-free cases; a calculation of a one-loop superconformal anomaly in the gauge supermultiplet sector. On the last point, we find that the ghost-anti-ghost supermultiplet and the Nakanishi-Lautrup supermultiplet give rise to BRST exact contributions which, due to "tree-level" Slavnov-Taylor identities in our regularization scheme, can safely be neglected, at least at the one-loop level.

DOI： 10.1143/PTP.100.1033

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When an asymptotically non-free theory possesses a mass parameter independent of the Λ parameter, the uv renormalon gives rise to nonperturbative contributions, to dimension-four operators and dimensionless couplings, thus has a "dual" effect of the instanton. We illustrate this phenomenon in O(N) symmetric massive λφ4 model in the 1/N expansion. This effect of uv renormalon is briefly compared with nonperturbative corrections in the magnetic picture of the Seiberg-Witten theory.

DOI： 10.1142/S0217732398002710

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We formulate a manifestly supersymmetric gauge covariant regularization of supersymmetric chiral gauge theories. In our scheme, the effective action in the superfield background field method above one-loop is always supersymmetric and gauge invariant. The gauge anomaly has a covariant form and can emerge only in one-loop diagrams with all the external lines being the background gauge superfield. We also present several illustrative applications in the one-loop approximation: the self-energy part of the chiral multiplet and of the gauge multiplet; the super-chiral anomaly and the superconformal anomaly; as the corresponding anomalous commutators, the Konishi anomaly and an anomalous supersymmetric transformation law of the supercurrent (the "central extension" of N = 1 supersymmetry algebra) and of the R-current.

DOI： 10.1143/PTP.100.627

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We extend the idea of the generalized Pauli-Villars regularization of Frolov and Slavnov and analyze the general structure of the regularization scheme. The gauge anomaly-free condition emerges in a simple way in the scheme, and, under the standard prescription for the momentum assignment, the Pauli-Villars Lagrangian provides a gauge invariant regularization of chiral fermions in arbitrary anomaly-free representations. The vacuum polarization tensor is transverse, and the fermion number and the conformal anomalies have gauge invariant forms. We also point out that the real representation can be treated in a straightforward manner and the covariant regularization scheme is directly implemented.

DOI： 10.1143/PTP.98.463

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We apply Borel resummation to the conventional perturbation series of ground state energy in a metastable potential, V(x) = x2/2 - gx4/4. We observe numerically that the discontinuity of the Borel transform reproduces the imaginary part of the energy eigenvalue, i.e., the total decay width due to quantum tunneling. The agreement with the exact numerical value is remarkable in the whole tunneling regime 0 < g ≲ 0.7. © 1997 Elsevier Science B.V.

DOI： 10.1016/S0370-2693(97)00368-7

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When the path integral method of anomaly evaluation is applied to chiral gauge theories, two different types of gauge anomaly, i.e., the consistent form and the covariant form, appear depending on the regularization scheme for the Jacobian factor. We clarify the relation between the regularization scheme and the Pauli-Villars-Gupta (PVG) type Lagrangian level regularization. The conventional PVG, being non-gauge invariant for chiral gauge theories, in general corresponds to the consistent regularization scheme. The covariant regularization scheme, on the other hand, is realized by the generalized PVG Lagrangian recently proposed by Frolov and Slavnov. These correspondences are clarified by reformulating the PVG method as a regularization of the composite gauge current operator.

DOI： 10.1016/0370-2693(96)00648-X

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We improve and generalize in several accounts the recent rigorous proof of convergence of delta expansion-order dependent mappings (variational perturbation expansion) for the energy eigenvalues of quartic anharmonic oscillator. For the single-well oscillator the uniformity of convergence in g∈[0, ∞] is proven. The convergence proof is extended also to complex values of g lying on a wide domain of the Riemann surface of E(g). Via the scaling relation à la Symanzik, this proves the convergence of delta expansion for the double well in the strong coupling regime (where the standard perturbation series is non Borel summable), as well as for the complex "energy eigenvalues" in certain metastable systems. Difficulties in extending the convergence proof to the cases of higher anharmonic oscillators are pointed out. Sufficient conditions for the convergence of delta expansion are summarized in the form of three general theorems, which should apply to a wide class of quantum mechanical and higher dimensional field theoretic systems. © 1996 Academic Press, Inc.

DOI： 10.1006/aphy.1996.0066

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The order dependent mapping method, whose convergence has recently been proven for the energy eigenvalue of the anharmonic oscillator, is applied to re-sum the standard perturbation series for the Stark effect of the hydrogen atom. We perform a numerical experiment up to the 50th order of the perturbation expansion. A simple mapping, suggested by the analytic structure and the strong field behavior, gives an excellent agreement with the exact value for an intermediate range of the electric field, 0.03 (a.u.) ≤ E ≤ 0.25 (a.u.). The imaginary part of the energy (the decay width) as well as the real part of the energy is reproduced from the standard perturbation series.

DOI： 10.1016/0375-9601(96)00243-5

Language：Others   Publishing type：Research paper (scientific journal)  

Renormalization group in 2 + ∈ dimensions and ∈ → 2: A simple model analysis
Using a simple solvable model, i.e., the Higgs-Yukawa system with an infinite number of flavors, we explicitly demonstrate how a dimensional continuation of the β function in the two dimensional MS scheme fails to reproduce the correct behavior of the β function in four dimensions. The mapping between coupling constants in the two dimensional MS scheme and a conventional scheme in the cutoff regularization, in which the dimensional continuation of the β function is smooth, becomes singular when the dimension of spacetime approaches four. The existence of a non-trivial fixed point in 2 + ∈ dimensions continued to four dimensions (∈→2) in the two dimensional MS scheme is spurious and asymptotic safety cannot be imposed on this model in four dimensions.

DOI： 10.1143/PTP.95.985

Language：Others   Publishing type：Research paper (scientific journal)  

Super-Virasoro Anomaly, Super-Weyl Anomaly and the Super-Liouville Action for 2D Supergravity
The relation between super-Virasoro anomaly and super-Weyl anomaly in $N=1$<br />
NSR superstring coupled with 2D supergravity is investigated from canonical<br />
theoretical view point. The WZW action canceling the super-Virasoro anomaly is<br />
explicitly constructed. It is super-Weyl invariant but nonlocal functional of<br />
2D supergravity. The nonlocality can be remedied by the super-Liouvlle action,<br />
which in turn recovers the super-Weyl anomaly. The final gravitational<br />
effective action turns out to be local but noncovariant super-Liouville action,<br />
describing the dynamical behavior of the super-Liouville fields. The BRST<br />
invariance of this approach is examined in the superconformal gauge and in the<br />
light-cone gauge.

DOI： 10.1006/aphy.1996.5643

Language：Others   Publishing type：Research paper (scientific journal)  

We prove that the linear delta expansion for energy eigenvalues of the quantum mechanical anharmonic oscillator converges to the exact answer if the order dependent trial frequency Ω is chosen to scale with the order as Ω = CNγ; 1/3 < γ < 1/2, C > 0 as N → ∞. It converges also for γ = 1/3, if C ≥ αcg1/3, αc ≃ 0.570875, where g is the coupling constant in front of the operator q4/4. The extreme case with γ = 1/3, C = γcg1/3 corresponds to the choice discussed earlier by Seznec and Zinn-Justin and, more recently, by Duncan and Jones. © 1995 Academic Press.

DOI： 10.1006/aphy.1995.1059

Language：Others   Publishing type：Research paper (scientific journal)  

DOI： 10.1142/S0217751X9400011X

Language：Others   Publishing type：Research paper (scientific journal)  

Based on a path-integral prescription for anomaly calculation, we analyze an effective theory of the two-dimensional N = 2 supergravity, i.e. N = 2 super-Liouville theory. We calculate the anomalies associated with the BRST supercurrent and the ghost-number supercurrent. From those expressions of anomalies, we construct covariant BRST and ghost-number supercurrents in the effective theory. We then show that the (super-)coordinate BRST current algebra forms a superfield extension of the topological conformal algebra for an arbitrary type of conformal matter or, in terms of the string theory, for an arbitrary number of space-time dimensions. This fact is in great contrast with N = 0 and N = 1 (super-)Liouville theory, where the topological algebra singles out a particular value of dimensions. Our observation suggests a topological nature of the two-dimensional N = 2 supergravity as a quantum theory. © 1993.

DOI： 10.1016/0550-3213(93)90241-G

Language：Others   Publishing type：Research paper (scientific journal)  

We construct a supersymmetric (SUSY) extension of an invisible axion model, in which the Peccei-Quinn symmetry is broken naturally at the intermediate mass scale 1010-1012 GeV by radiative corrections from right-handed neutrino loops. The SUSY-invariant mass of doublet Higgs supermultiplets is forbidden by the Peccei-Quinn symmetry, whose breaking, however, generates an invariant Higgs mass of the order of the Fermi scale. In this model the right-handed neutrinos acquire large Majorana masses which are in a favored range for the Mikheyev-Smirnov-Wolfenstein solution to the solar neutrino problem. © 1992.

DOI： 10.1016/0370-2693(92)91397-R

Language：Others   Publishing type：Research paper (scientific journal)  

We prove that critical and subcritical N = 2 string theory gives a realization of an N = 2 superfield extension of the topological conformal algebra. The essential observation is the vanishing of the background ghost charge. © 1992.

DOI： 10.1016/0370-2693(92)90191-6

Language：Others   Publishing type：Research paper (scientific journal)  

The operator algebra in non-critical string theories is studied by treating the cosmological term as a perturbation. The algebra of covariantly regularized BRST and related currents contains a twisted N = 2 superconformal algebra only at d = -2 in bosonic strings, and a twisted N = 3 superconformal algebra only at d = ±∞ in spinning strings. The bosonic string at d = -2 is examined by replacing the string coordinate by a fermionic matter with c = -2. The resulting bc-βγ system accommodates various forms of BRST cohomology, and the ghost number assignment and BRST cohomology are different in the c = -2 string theory and two-dimensional topological gravity. © 1991.

DOI： 10.1016/0550-3213(91)90272-Y

Language：Others   Publishing type：Research paper (scientific journal)  

The BRS currents in two-dimensional gravity and supergravity theories, which are related to string theory, contain anomalous terms. The origin of these anomalies can be neatly understood in a carefully defined path integral. We present the detailed calculations of these BRS and related anomalies in the holomorphic or anti-holomorphic sector separately in the conformal gauge. One-loop renormalization of the Liouville action becomes transparent in our formalation. We identify a BRS invariant BRS current (and thus nil-potent charge) and a conformally invariant ghost number current by incorporating the dynamical Weyl freedom explicitly. The formal path integral construction of various composite operators is also checked by using the operator product technique. Implications of these BRS analyses on possible non-critical string theories at d < 26 or d < 10 are briefly discussed. © 1990.

DOI： 10.1016/0550-3213(90)90106-N

Language：Others   Publishing type：Research paper (scientific journal)  

On the basis of a BRS invariant formulation of a free bosonic string at D<26, we analyse the structures of BRS and ghost number operators and the string vacua. The ghost number and the mass spectrum are naturally related to each other in this scheme. By using a string field theoretical technique, it is also shown that the Weyl freedom gives rise to an unphysical Higgs scalar, which is absorbed by the vector particle in the first excited level, thus realizing the Higgs mechanism. © 1989.

DOI： 10.1016/0370-2693(89)91712-7

Language：Others   Publishing type：Research paper (scientific journal)  

It is shown that an anomalous correction to the BRS current ΔJz=( 3 4π)∂z2cz in bosonic string theory essentially corresponds to the surface term ( 3 8π)sh{phonetic}∂α(cα√gR)d2x in the anomalous jacobain factor. To make this correspondense explicit in the operator level, we present a quantization of a free string at D<26 which maintains the closure of the BRS symmetry. © 1988.

DOI： 10.1016/0370-2693(88)91761-3

Event date： 2022.4

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Japan  

Event date： 2022.4

Language：Japanese  

Venue：オンライン   Country：Japan  

Event date： 2022.4

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Japan  

Event date： 2022.4

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Japan  

Event date： 2022.4

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Japan  

Event date： 2022.4

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Venue：オンライン   Country：Japan  

Event date： 2022.4

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Japan  

Event date： 2022.4

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Venue：オンライン   Country：Japan  

Event date： 2021.4

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Venue：京都大学基礎物理学研究所   Country：Japan  

Event date： 2020.9

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Venue：オンライン   Country：Japan  

Event date： 2020.7

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Venue：オンライン   Country：Japan  

Event date： 2018.9

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Venue：信州大学   Country：Japan  

Event date： 2018.7

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Venue：品川グランドセントラルタワー   Country：Japan  

Event date： 2018.3

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Venue：東京理科大学野田キャンパス   Country：Japan  

Event date： 2017.11

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Venue：KEK東海キャンパス   Country：Japan  

Event date： 2017.9

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Venue：宇都宮大学   Country：Japan  

Event date： 2017.9

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Venue：宇都宮大学   Country：Japan  

Event date： 2017.8

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Venue：京都大学基礎物理学研究所   Country：Japan  

Event date： 2017.6

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Venue：理化学研究所（神戸）   Country：Japan  

Event date： 2017.3

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Venue：大阪大学   Country：Japan  

Event date： 2016.12

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Venue：高エネルギー加速器研究機構   Country：Japan  

Event date： 2016.9

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Venue：宮崎大学   Country：Japan  

Event date： 2015.9

Language：Japanese   Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2015.9

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Venue：大阪市立大学   Country：Japan  

Event date： 2015.7

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Venue：京都大学基礎物理学研究所   Country：Japan  

Event date： 2015.7

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Venue：神戸国際会議場   Country：Japan  

Event date： 2015.3

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Venue：京都大学基礎物理学研究所   Country：Japan  

Event date： 2014.9

Language：Japanese   Presentation type：Oral presentation (invited, special)  

Venue：佐賀大学   Country：Japan  

Event date： 2014.9

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Venue：慶應義塾大学   Country：Japan  

Event date： 2014.1

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Venue：高エネルギー加速器研究機構   Country：Japan  

Event date： 2013.9

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Venue：高エネルギー加速器研究機構   Country：Japan  

Event date： 2013.9

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Venue：高知大学朝倉キャンパス   Country：Japan  

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：早稲田大学   Country：Japan  

Language：Japanese  

Country：Japan  

Realization of Chiral Symmetry in Lattice Gauge Theory
Our understanding on the chiral symmetry and the chiral fermion in lattice gauge theory has revolutionary been developed in the last decade. In lattice vector-like gauge theories, formulation which preserves an exact chiral symmetry was found and its application to numerical simulations of Quantum Chromo Dynamics (QCD) is under way in earnest. In the past, it seems that people was often thinking as chiral gauge theories cannot be defined non-perturbatively, but a common view at present seems to be changed as constructive formulation may be possible. In this commentary, we describe these recent theoretical developments under the title "Realization of chiral symmetry in lattice gauge theory."

Language：Japanese  

Country：Japan  

22pXL-10 Hall conductances on a discretized Brillouin zone-Topological quantum numbers on a lattice

Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)  

Language：Japanese  

グラディエントフロー―エネルギー運動量テンソルへの応用を中心として―

Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)  

Language：Japanese  

Language：Japanese  

Construction of The Energy‐Momentum Tensor in Lattice Gauge Theory: Gradient Flow Method

DOI： 10.11316/butsuri.73.3_148

Language：Japanese  

Gradient flowによる(2+1)‐flavor QCD状態方程式―物理点での試験研究―

Language：Japanese  

Language：Japanese  

Gradient flowを用いてみる一次相転移点近傍の熱力学量の性質

Language：Others  

A dilaton-pion mass relation
Recently, Golterman and Shamir presented an effective field theory which is
supposed to describe the low-energy physics of the pion and the dilaton in an
&#36;SU(N_c)&#36; gauge theory with &#36;N_f&#36; Dirac fermions in the fundamental
representation. By employing this formulation with a slight but important
modification, we derive a relation between the dilaton mass squared~&#36;m_ au^2&#36;,
with and without the fermion mass~&#36;m&#36;, and the pion mass squared~&#36;m_pi^2&#36; to
the leading order of the chiral logarithm. This is analogous to a similar
relation obtained by Matsuzaki and~Yamawaki on the basis of a somewhat
different low-energy effective field theory. Our relation reads
&#36;m_ au^2=m_ au^2|_{m=0}+KN_fhat{f}_pi^2m_pi^2/(2hat{f}_ au^2)+O(m_pi^4ln
m_pi^2)&#36; with~&#36;K=9&#36;, where &#36;hat{f}_pi&#36; and~&#36;hat{f}_ au&#36; are decay
constants of the pion and the dilaton, respectively. This mass relation differs
from the one derived by Matsuzaki and~Yamawaki on the points that
&#36;K=(3-gamma_m)(1+gamma_m)&#36;, where &#36;gamma_m&#36; is the mass anomalous dimension,
and the leading chiral logarithm correction is~&#36;O(m_pi^2ln m_pi^2)&#36;.
For~&#36;gamma_msim1&#36;, the value of the decay constant~&#36;hat{f}_ au&#36; estimated
from our mass relation becomes &#36;sim50&#37;&#36; larger than &#36;hat{f}_ au&#36; estimated
from the relation of Matsuzaki and~Yamawaki.

Language：Japanese  

DOI： 10.11316/butsuri.71.6_399

Language：Japanese  

格子ゲージ理論におけるグラディエントフローの応用

Language：Japanese  

Language：Others  

Perturbative spectrum of a Yukawa-Higgs model with Ginsparg-Wilson fermions
A Yukawa-Higgs model with Ginsparg-Wilson (GW) fermions, proposed recently by
Bhattacharya, Martin and Poppitz as a possible lattice formulation of chiral
gauge theories, is studied. A simple argument shows that the gauge boson always
acquires mass by the St"uckelberg (or, in a broad sense, Higgs) mechanism,
regardless of strength of interactions. The gauge symmetry is spontaneously
broken. When the gauge coupling constant is small, the physical spectrum of the
model consists of massless fermions, massive fermions and emph{massive} vector
bosons.

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Language：English  

We propose a Kahler potential in supergravity which successfully accommodates chaotic inflation. This model can have a large gravitino mass without giving a large mass to squarks and sleptons, and thus is free from both the gravitino problem and entropy crisis. In this model baryogenesis takes places naturally, identifying the inflaton with a right-handed sneutrino with its mass M congruent-to 10(13) GeV, which is consistent with the COBE data and the Mikheyev-Smirnov-Wolfenstein solution to the solar neutrino problem. The model can also accommodate the matter content appropriate for the mixed dark matter scenario.

Language：Japanese  

Type：Competition, symposium, etc. 

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：7

Type：Competition, symposium, etc. 

Type：Scientific advice/Review 

Type：Competition, symposium, etc. 

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：4

Type：Academic society, research group, etc. 

Type：Competition, symposium, etc. 

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：5

Type：Competition, symposium, etc. 

Type：Scientific advice/Review 

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：4

Type：Scientific advice/Review 

Type：Competition, symposium, etc. 

Type：Competition, symposium, etc. 

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：7

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：2

Type：Competition, symposium, etc. 

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：3