1. |
Hiroshi Suzuki, Hiromasa Takaura, Renormalon-free definition of the gluon condensate within the large-β0 approximation, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptz100, 2019, 10, 2019.10, 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.. |

2. |
, Yusuke Taniguchi, Shinji Ejiri, Ryo Iwami, Kazuyuki Kanaya, Masakiyo Kitazawa, Hiroshi Suzuki, Hiroshi Suzuki, Naoki Wakabayashi, Erratum Exploring Nf=2+1 QCD thermodynamics from the gradient flow (Physical Review D (2017) 96 (014509) DOI: 10.1103/PhysRevD.99.014509), *Physical Review D*, 10.1103/PhysRevD.99.059904, 99, 5, 2019.03, 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.. |

3. |
Takumi Iritani, Masakiyo Kitazawa, Hiroshi Suzuki, Hiromasa Takaura, Thermodynamics in quenched QCD Energy-momentum tensor with two-loop order coefficients in the gradient-flow formalism, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptz001, 2019, 2, 2019.02, 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.. |

4. |
Okuto Morikawa, Hiroshi Suzuki, Numerical study of the N=2 Landau-Ginzburg model, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/pty088, 2018, 8, 2018.08, 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.. |

5. |
Okuto Morikawa, Hiroshi Suzuki, Axial U(1) anomaly in a gravitational field via the gradient flow, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/pty073, 2018, 7, 2018.07, 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.. |

6. |
Hiroki Makino, Okuto Morikawa, Hiroshi Suzuki, Gradient flow and the Wilsonian renormalization group flow, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/pty050, 2018, 5, 2018.05, 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.. |

7. |
Kenji Hieda, Aya Kasai, Hiroki Makino, Hiroshi Suzuki, 4D N = 1 SYM supercurrent on the lattice in terms of the gradient flow, *35th International Symposium on Lattice Field Theory, Lattice 2017
EPJ Web of Conferences*, 10.1051/epjconf/201817511014, 175, 2018.03, 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].. |

8. |
Yusuke Taniguchi, Shinji Ejiri, Kazuyuki Kanaya, Masakiyo Kitazawa, Asobu Suzuki, Hiroshi Suzuki, Takashi Umeda, Energy-momentum tensor correlation function in N_{f} = 2 + 1 full QCD at finite temperature, *35th International Symposium on Lattice Field Theory, Lattice 2017
EPJ Web of Conferences*, 10.1051/epjconf/201817507013, 175, 2018.03, We measure correlation functions of the nonperturbatively renormalized energy-momentum tensor in N_{f} = 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 m_{π}=m_{ρ} ? 0:63 while the s quark mass is set to approximately its physical value.. |

9. |
Kazuyuki Kanaya, Shinji Ejiri, Ryo Iwami, Masakiyo Kitazawa, Hiroshi Suzuki, Yusuke Taniguchi, Takashi Umeda, Equation of state in (2+1)-flavor QCD at physical point with improved Wilson fermion action using gradient flow, *35th International Symposium on Lattice Field Theory, Lattice 2017
EPJ Web of Conferences*, 10.1051/epjconf/201817507023, 175, 2018.03, 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 (N_{t} = 4-14 including odd N_{t}'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.. |

10. |
Hiroki Makino, Okuto Morikawa, Hiroshi Suzuki, One-loop perturbative coupling of A and A? through the chiral overlap operator, *35th International Symposium on Lattice Field Theory, Lattice 2017
EPJ Web of Conferences*, 10.1051/epjconf/201817511013, 175, 2018.03, 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.. |

11. |
Aya Kasai, Okuto Morikawa, Hiroshi Suzuki, Gradient flow representation of the four-dimensional N = 2 super Yang-Mills supercurrent, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/pty117, 2018, 11, 2018.01, 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.. |

12. |
, Yusuke Taniguchi, Shinji Ejiri, Ryo Iwami, Kazuyuki Kanaya, Masakiyo Kitazawa, Hiroshi Suzuki, Takashi Umeda, Naoki Wakabayashi, Exploring Nf=2+1 QCD thermodynamics from the gradient flow, *Physical Review D*, 10.1103/PhysRevD.96.014509, 96, 1, 2017.07, 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.. |

13. |
Kenji Hieda, Aya Kasai, Hiroki Makino, Hiroshi Suzuki, 4D N = 1 SYM supercurrent in terms of the gradient flow, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptx073, 2017, 6, 2017.06, 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.. |

14. |
Hiroki Makino, Okuto Morikawa, Hiroshi Suzuki, One-loop perturbative coupling of A and A _{∗} through the chiral overlap operator, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptx085, 2017, 6, 2017.06, 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.. |

15. |
Kenji Hieda, Hiroki Makino, Hiroshi Suzuki, Proof of the renormalizability of the gradient flow, *Nuclear Physics B*, 10.1016/j.nuclphysb.2017.02.017, 918, 23-51, 2017.05, 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.. |

16. |
, Yusuke Taniguchi, Kazuyuki Kanaya, Hiroshi Suzuki, Takashi Umeda, Topological susceptibility in finite temperature (2+1)-flavor QCD using gradient flow, *Physical Review D*, 10.1103/PhysRevD.95.054502, 95, 5, 2017.03, 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 |

17. |
Hiroshi Suzuki, Energy-momentum tensor on the lattice Recent developments, *Proceedings of Science*, Part F128557, 2017.03, It is conceivable that the construction of the energy-momentum tensor (EMT) in lattice field theory enlarges our ability in lattice field theory and also deepens our understanding on EMT at the non-pertubative level. In this talk, I will review recent developments in this enterprise.. |

18. |
Kazuyuki Kanaya, Shinji Ejiri, Ryo Iwami, Masakiyo Kitazawa, Hiroshi Suzuki, Yusuke Taniguchi, Takashi Umeda, Naoki Wakabayashi, Equation of state in (2+1)-flavor QCD with gradient flow, *Proceedings of Science*, Part F128557, 2017.03, The energy-momentum tensor and equation of state are studied in finite-temperature (2+1)-flavor QCD with improved Wilson quarks using the method proposed by Makino and Suzuki based on the gradient flow. We find that the results of the gradient flow are consistent with the previous results using the T-integration method at T <_{∼} 280MeV (Nτ > _{∼} 10), while a disagreement is found at T > _{∼} 350MeV (Nτ < _{∼} 8) presumably due to the small-Nτ lattice artifact. We also report on the results on the renormalized chiral condensate and its disconnected susceptibility using the method of Hieda and Suzuki. The results show a clear signal of the expected chiral restoration crossover even with Wilson-type quarks which violate the chiral symmetry explicitly.. |

19. |
Masakiyo Kitazawa, Takumi Iritani, Masayuki Asakawa, Tetsuo Hatsuda, Hiroshi Suzuki, Equation of state for SU(3) gauge theory via the energy-momentum tensor under gradient flow, *Physical Review D*, 10.1103/PhysRevD.94.114512, 94, 11, 2016.12, 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.. |

20. |
Kenji Hieda, Hiroshi Suzuki, Small flow-time representation of fermion bilinear operators, *Modern Physics Letters A*, 10.1142/S021773231650214X, 31, 38, 2016.12, 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 N_{f} = 2 + 1 quantum chromodynamics (QCD) at finite temperature.. |

21. |
Ken Ichi Okumura, Hiroshi Suzuki, Fermion number anomaly with the fluffy mirror fermion, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptw167, 2016, 12, 2016.12, 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.. |

22. |
Etsuko Itou, Hiroshi Suzuki, Yusuke Taniguchi, Takashi Umeda, (2+1)-flavor QCD thermodynamics from the gradient flow, *Proceedings of Science*, 14-18-July-2015, 2016.07, Recently, we proposed a novel method to define and calculate the energy-momentum tensor (EMT) in lattice gauge theory on the basis of the Yang-Mills gradient flow [1]. In this pro- ceedings, we summarize the basic idea and technical steps to obtain the bulk thermodynamic quantities in lattice gauge theory using this method for the quenched and (2+1)-flavor QCD. The revised results of integration measure (trace anomaly) and entropy density of the quenched QCD with corrected coefficients are shown. Furthermore, we also show the flow time dependence of the parts of EMT including the dynamical fermions. This work is based on a joint-collaboration between FlowQCD and WHOT QCD.. |

23. |
Hisashi Iha, Hiroki Makino, Hiroshi Suzuki, Upper bound on the mass anomalous dimension in many-flavor gauge theories A conformal bootstrap approach, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptw046, 2016, 5, 2016.05, 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 ø^{k} _{i} 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 Ø^{k} _{i}, δ Ø^{k} _{i}, 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 Ø^{k} _{i}, 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.. |

24. |
Shinji Ejiri, Ryo Iwami, Mizuki Shirogane, Naoki Wakabayashi, Kazuyuki Kanaya, Masakiyo Kitazawa, Hiroshi Suzuki, Yusuke Taniguchi, Takashi Umeda, Determination of latent heat at the finite temperature phase transition of SU(3) gauge theory, *Proceedings of Science*, Part F128557, 2016.01, We calculate the energy gap (latent heat) and pressure gap between the hot and cold phases of the SU(3) gauge theory at the first order deconfining phase transition point. We perform simulations around the phase transition point with the lattice size in the temporal direction Nτ =6; 8 and 12 and extrapolate the results to the continuum limit. The energy density and pressure are evaluated by the derivative method with nonperturabative anisotropy coefficients. We find that the pressure gap vanishes at all values of Nt. The spatial volume dependence in the latent heat is found to be small on large lattices. Performing extrapolation to the continuum limit, we obtain Δϵ=T^{4} =0:75±0:17 and δ(ϵ -3p)=T4 = 0:623±0:056: We also tested a method using the Yang-Mills gradient flow. The preliminary results are consistent with those by the derivative method within the error.. |

25. |
Yusuke Taniguchi, Shinji Ejiri, Kazuyuki Kanaya, Masakiyo Kitazawa, Hiroshi Suzuki, Takashi Umeda, Ryo Iwami, Naoki Wakabayashi, Temperature dependence of topological susceptibility using gradient flow, *Proceedings of Science*, Part F128557, 2016.01, We study temperature dependence of the topological susceptibility with the Nf = 2+1 flavors Wilson fermion. We have two major interests in this paper. One is a comparison of gluonic and fermionic definitions of the topological susceptibility. Two definitions are related by the chiral Ward-Takahashi identity but their coincidence is highly non-trivial for the Wilson fermion. By applying the gradient flow both for the gauge and quark fields we find a good agreement of these two measurements. The other is a verification of a prediction of the dilute instanton gas approximation at low temperature region T_{pc} < T < 1.5T_{pc}, for which we confirm the prediction that the topological susceptibility decays with power χτ ∝ (T/Tpc)^{-8} for three flavors QCD.. |

26. |
Hiroshi Suzuki, Background field method in the gradient flow, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptv139, 2015, 10, 2015.10, 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.. |

27. |
Hiroki Makino, Hiroshi Suzuki, Daisuke Takeda, Complex Langevin method applied to the 2D SU (2) Yang-Mills theory, *Physical Review D - Particles, Fields, Gravitation and Cosmology*, 10.1103/PhysRevD.92.085020, 92, 8, 2015.10, 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.. |

28. |
Masayuki Asakawa, Tetsuo Hatsuda, Etsuko Itou, Masakiyo Kitazawa, Hiroshi Suzuki, Erratum Thermodynamics of SU (3) gauge theory from gradient flow on the lattice (Physical Review D - Particles, Fields, Gravitation and Cosmology (2014) D 90 (011501)), *Physical Review D - Particles, Fields, Gravitation and Cosmology*, 10.1103/PhysRevD.92.059902, 92, 5, 2015.09. |

29. |
Hiroshi Suzuki, Erratum Energy-momentum tensor from the Yang-Mills gradient flow (Progress of Theoretical and Experimental Physics (2013) 2013 (083B03)), *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptv094, 2015, 7, 2015.07. |

30. |
Hiroki Makino, Hiroshi Suzuki, Erratum Lattice energy-momentum tensor from the Yang-Mills gradient flow-inclusion of fermion fields (Progress of Theoretical and Experimental Physics (2014) 2014 (063B02)), *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptv095, 2015, 7, 2015.07. |

31. |
Tasuku Endo, Kenji Hieda, Daiki Miura, Hiroshi Suzuki, Universal formula for the flavor non-singlet axial-vector current from the gradient flow, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptv058, 2015, 5, 2015.05, By employing the gradient/Wilson flow, we derive a universal formula that expresses a correctly normalized flavor non-singlet axial-vector current of quarks. The formula is universal in the sense that it holds independently of regularization and especially holds with lattice regularization. It is also confirmed that, in the lowest non-trivial order of perturbation theory, the triangle diagram containing the formula and two flavor non-singlet vector currents possesses non-local structure that is compatible with the triangle anomaly.. |

32. |
Masakiyo Kitazawa, Masayuki Asakawa, Tetsuo Hatsuda, Takumi Iritani, Etsuko Itou, Hiroshi Suzuki, Measurement of thermodynamics using gradient flow, *Proceedings of Science*, Part F130500, 2015.05, We analyze bulk thermodynamics and correlation functions of the energy-momentum tensor in pure Yang-Mills gauge theory using the energy-momentum tensor defined by the gradient flow and small flow time expansion. Our results on thermodynamic observables are consistent with those obtained by the conventional integral method. The analysis of the correlation function of total energy supports the energy conservation. It is also addressed that these analyses with gradient flow require less statistics compared with the previous methods. All these results suggest that the energy-momentum tensor can be successfully defined and observed on the lattice with moderate numerical costs with the gradient flow.. |

33. |
Hiroshi Suzuki, Universal formula for the energy-momentum tensor via a flow equation in the Gross-Neveu model, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptv036, 2015, 4, 2015.04, For the fermion field in the two-dimensional Gross-Neveu model, we introduce a flow equation that allows a simple 1/N expansion. By employing the 1/N expansion, we examine the validity of a universal formula for the energy-momentum tensor which is based on the small flow-time expansion.We confirm that the formula reproduces a correct normalization and the conservation law of the energy-momentum tensor by computing the translation Ward-Takahashi relation in the leading non-trivial order in the 1/N expansion. Also, we confirmthat the expectation value at finite temperature correctly reproduces thermodynamic quantities. These observations support the validity of a similar construction of the energy-momentum tensor via the gradient/Wilson flow in lattice gauge theory.. |

34. |
Kazuo Fujikawa, Hiroshi Suzuki, Bosonization in the path integral formulation, *Physical Review D - Particles, Fields, Gravitation and Cosmology*, 10.1103/PhysRevD.91.065010, 91, 6, 2015.03, 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).. |

35. |
Hiroshi Suzuki, Background field method in the gradient flow, *Proceedings of Science*, 14-18-July-2015, 2015.01, 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 per-Turbative 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 transforma-Tion, 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.. |

36. |
Hiroki Makino, Fumihiko Sugino, Hiroshi Suzuki, Large-N limit of the gradient flow in the 2D O(N) nonlinear sigma model, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptv044, 2015, 4, 2015.01, The gradient flow equation in the 2D O(N) nonlinear sigma model with lattice regularization is solved in the leading order of the 1/N expansion. By using this solution, we analytically compute the thermal expectation value of a lattice energy-momentum tensor defined through the gradient flow. The expectation value reproduces thermodynamic quantities obtained by the standard large-N method. This analysis confirms that the above lattice energy-momentum tensor restores the correct normalization automatically in the continuum limit, in a system with a nonperturbative mass gap.. |

37. |
Masakiyo Kitazawa, Masayuki Asakawa, Tetsuo Hatsuda, Takumi Iritani, Etsuko Itou, Hiroshi Suzuki, Thermodynamics and reference scale of SU(3) gauge theory from gradient flow on fine lattices, *Proceedings of Science*, 14-18-July-2015, 2015.01, We study the parametrization of lattice spacing and thermodynamics of SU(3) gauge theory on the basis of the Yang-Mills gradient flow on fine lattices. The lattice spacing of the Wilson gauge action is determined over a wide range 6:3 ≤ b ≤ 7:5 with high accuracy. The measurements of the flow time and lattice spacing dependences of the expectation values of the energy-momentum tensor are performed on fine lattices.. |

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Hiroki Makino, Hiroshi Suzuki, Renormalizability of the gradient flow in the 2D O(N) non-linear sigma model, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptv028, 2015, 3, 2014.11, 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.. |

39. |
Masayuki Asakawa, Tetsuo Hatsuda, Etsuko Itou, Masakiyo Kitazawa, Hiroshi Suzuki, Thermodynamics of SU (3) gauge theory from gradient flow on the lattice, *Physical Review D - Particles, Fields, Gravitation and Cosmology*, 10.1103/PhysRevD.90.011501, 90, 1, 2014.07, 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.. |

40. |
Hiroki Makino, Hiroshi Suzuki, Lattice energy-momentum tensor from the yang. Mills gradient flow-inclusion of fermion fields, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptu070, 2014, 6, 2014.01, 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.. |

41. |
Michael G. Endres, Tsunehide Kuroki, Fumihiko Sugino, Hiroshi Suzuki, SUSY breaking by nonperturbative dynamics in a matrix model for 2D type IIA superstrings, *Nuclear Physics B*, 10.1016/j.nuclphysb.2013.09.005, 876, 3, 758-793, 2013.11, 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.. |

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Hiroshi Suzuki, Energy-momentum tensor from the yang-mills gradient flow, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptt059, 2013, 8, 2013.08, 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.. |