1. |
Development of "Let's get hands on a math toolbox" as a "liberal science" subject. |

2. |
Development of "Let's get hands on a math toolbox" as a "liberal science" subject. |

3. |
Electromotive Force in Electromagnetism. |

4. |
Koji Harada, Satoru Sasabe, Masanobu Yahiro, How to Use Renormalization Group Analysis in Lattice Nuclear Effective Field Theory, *22nd International Conference on Few-Body Problems in Physics, FB22 2018
Recent Progress in Few-Body Physics - Proceedings of the 22nd International Conference on Few-Body Problems in Physics, FB22 2018*, 10.1007/978-3-030-32357-8_67, 415-419, 2020.01, We propose a new approach to Nuclear Effective Field Theory (NEFT) on a lattice on the basis of Renormalization Group (RG) analysis. In order to perform Markov-chain Monte Carlo lattice simulation of NEFT, we introduce auxiliary fields to integrate nucleon field so that its effects are represented as a determinant. The problem is that the determinant becomes complex and cannot be considered as a part of probability distribution function. We introduce a reweighting method, in which the reference determinant is chosen to be optimal in the RG analysis sense: the reference determinant contains only the relevant interactions and the closest to the original determinant. We calculate the standard deviation of the absolute value of the reweighting factor in a simple model, isospin-symmetric S-wave NLO NEFT without pions, and explain why our choice is optimal.. |

5. |
Typical misconception in mechanics that students with majors in the schools of humanities have, probed with force concept inventory (FCI) test. |

6. |
Koji Harada, Satoru Sasabe, Masanobu Yahiro, Numerical study of renormalization group flows of nuclear effective field theory without pions on a lattice, *Physical Review C*, 10.1103/PhysRevC.94.024004, 94, 024004-1-085006-13, 2016.08, We formulate the next-to-leading order nuclear effective field theory without pions in the two-nucleon sector on a spatial lattice, and investigate nonperturbative renormalization group flows in the strong coupling region by diagonalizing the Hamiltonian numerically. The cutoff (proportional to the inverse of the lattice constant) dependence of the coupling constants is obtained by changing the lattice constant with the binding energy and the asymptotic normalization constant for the groundstate being fixed. We argue that the critical line can be obtained by looking at the finite-size dependence of the groundstate energy. We determine the relevant operator and locate the nontrivial fixed point, as well as the physical flow line corresponding to the deuteron in the two-dimensional plane of dimensionless coupling constants. It turns out that the location of the nontrivial fixed point is very close to the one obtained by the corresponding analytic calculation, but the relevant operator is quite different. . |

7. |
Koji Harada, Hirofumi Kubo, Issei Yoshimoto, Wilsonian renormalization group analysis of nonrelativistic three-body systems without introducing dimerons, *Physical Review D - Particles, Fields, Gravitation and Cosmology*, 10.1103/PhysRevD.87.085006, 87, 8, 2013.04, Low-energy effective field theory describing a nonrelativistic three-body system is analyzed in the Wilsonian renormalization group method. No effective auxiliary field (dimeron) that corresponds to two-body propagation is introduced. The Efimov effect is expected in the case of an infinite two-body scattering length and is believed to be related to the limit cycle behavior in the three-body renormalization group equations (RGEs). If the one-loop property of the RGEs for the nonrelativistic system without the dimeron field, which is essential in deriving RGEs in the two-body sector, persists in the three-body sector, it appears to prevent the emergence of limit cycle behavior. We explain how the multiloop diagrams contribute in the three-body sector without contradicting the one-loop property of the RGEs and derive the correct RGEs, which lead to the limit cycle behavior. The Efimov parameter, s_{0}, is obtained within a few percent error in the leading orders. We also remark on the correct use of the dimeron formulation. We find rich renormalization group flow structure in the three-body sector. In particular, a novel nontrivial fixed point of the three-body couplings is found when the two-body interactions are absent. We also find, on the two-body nontrivial fixed point, the limit cycle is realized as a loop of finite size in the space of three-body coupling constants when terms with derivatives are included.. |

8. |
Hirofumi Kubo, Koji Harada, Tatsuya Sakaeda, Yuki Yamamoto, Converging low energy expansion of nucleon-nucleon scattering based on the Wilsonian renormalization group analysis, *15th International Conference on Hadron Spectroscopy, Hadron 2013
Proceedings of Science*, 2013.01, In order to implement the power counting obtained in the Wilsonian renormalization group (WRG) analysis of nuclear effective field theory (NEFT) including pions, we propose a practical calculational scheme in which the short-distance part of one-pion exchange (S-OPE) is removed and represented as contact terms. A distinctive feature of our scheme is that two regularization schemes are used in combination. The dimensional regularization (DR) is used for diagrams consisting only of contact interactions and a momentum cutoff is implemented for diagrams con-taining the long-distance part of one pion exchange (L-OPE) by introducing a Gaussian damping factor(GDF). The sole purpose of the use of the dimensional regularization is that it considerably simplifies the treatment of nonperturbative part of the scattering amplitude. In the calculation the L-OPE is treated as perturbation and a part of the S-OPE is treated nonperturbatively along with a contact interaction without derivative nonperturbatively. We show the results of the next-To-next-to-leading order(NNLO) calculations for nucleon-nucleon elastic scattering in the S-waves that are fitted to Nijmegen partial wave analysis data.. |

9. |
Koji Harada, Hirofumi Kubo, Tatsuya Sakaeda, Yuki Yamamoto, Wilsonian RG analysis of the P-wave Nucleon-Nucleon Scattering Including Pions, 10.1007/s00601-012-0541-9, 2012.09, We perform a Wilsonian renormalization group analysis for the nucleon–nucleon scattering in the P waves in the nuclear effective field theory including pions, in a similar way to the one done for the S-waves in our previous paper. We emphasize that the one-pion exchange interaction with large momentum transfer is of the same order as the leading contact interaction, so that there is no mismatch of the power counting. It is explicitly shown by obtaining consistent sets of renormalization group equations, that the cutoff dependence generated by the loop diagrams containing pion exchanges can be compensated by the cutoff dependence of the coupling constants of the contact interactions.. |

10. |
Koji Harada, Hirofumi Kubo, Yuki Yamamoto, Pions in nuclear effective field theory: how they behave differently at different scales and how they decouple at very low energies, *Few Body Systems*, 10.1007/s00601-012-0348-8, 54, 239-243, 2012.03, We explain how the Wilsonian renormalization group (RG) can determine the power counting of the nuclear effective field theory (NEFT) including pions. We emphasize that the separation of pion exchange into the short-distance part and the long-distance part is essential since they behave differently in the RG analysis; we found that the latter is perturbative whereas the a part of the former is nonperturbative. As for the contact interactions power counting turns out to be the same as that for pionless NEFT: pion exchange does not affect the scaling property of contact operators. Our RG equations for NEFT including pions connect smoothly with those for the pionless NEFT: pions decouple at very low energies as we expect.. |

11. |
Hirofumi Kubo, Koji Harada, Tatsuya Sakaeda, Yuki Yamamoto, Practical calculational scheme implementing the Wilsonian RG results for nuclear effective field thoery including pions, *Few Body Systems*, 10.1007/s00601-012-0349-7, 54, 245-249, 2012.03, On the basis of the Wilsonian renormalization group (WRG) analysis of nuclear effective field theory (NEFT) including pions, we propose a practical calculational scheme in which the short-distance part of one-pion exchange (S-OPE) is removed and represented as contact terms. The long-distance part of one- pion exchange (L-OPE) is treated as perturbation. The use of dimensional regularization (DR) for diagrams consisting only of contact interactions considerably simplifies the calculation of scattering amplitude and the renormalization group equations. NLO results for nucleon-nucleon elastic scattering in the S-waves are obtained and compared with experiments. A brief comment on NNLO calculations is given.. |

12. |
Koji Harada, Pions are neither perturbative nor nonperturbative: Wilsonian renormalization-group analysis of nuclear effective field theory including pions, *Physical Review D*, 10.1103/PhysRevC.83.034002, 83, 3, 034002 [14 pages] , 2011.03. |

13. |
Koji Harada, Hirofumi Kubo, Yuki Yamamoto, Apparently Noninvariant Terms of U(N) × U(N) nonlinear sigma model in the one-loop approximation, *Progress of Theoretical Physics*, 10.1143/PTP.123.475, 123, 3, 475-498, 2010.03, We show how the Apparently Noninvariant Terms (ANTs), which emerge in perturbation theory of nonlinear sigma models, are consistent with the nonlinearly realized symmetry by employing the Ward-Takahashi identity (in the form of an inhomogeneous Zinn-Justin equation). In the literature the discussions on ANTs are confined to the SU(2) case. We generalize them to the U(N) case and demonstrate explicitly at the one-loop level that despite the presence of divergent ANTs in the effective action of the "pions", the symmetry is preserved.. |

14. |
Koji Harada, Nozomu Hattori, Hirofumi Kubo, Yuki Yamamoto, Erratum Apparently noninvariant terms of nonlinear sigma models in lattice perturbation theory (Physical Review D - Particles, Fields, Gravitation and Cosmology (2009) 79 (065037)), *Physical Review D - Particles, Fields, Gravitation and Cosmology*, 10.1103/PhysRevD.80.029902, 80, 2, 2009.08. |

15. |
Koji Harada, Hirofumi Kubo, Atsushi Ninomiya, More about the Wilsonian analysis on the pionless neft, *International Journal of Modern Physics A*, 24, 16-17, 3191-3225, 2009.07, We extend our Wilsonian renormalization group (RG) analysis on the pionless nuclear effective field theory in the two-nucleon sector in two ways; on the one hand, (1) we enlarge the space of operators up to including those of ${\mathcal O}(p-4)$ in the S waves, and, on the other hand, (2) we consider the RG flows in higher partial waves (P and D waves). In the larger space calculations, we find, in addition to nontrivial fixed points, two "fixed lines" and a "fixed surface" which are related to marginal operators. In the higher partial wave calculations, we find similar phase structures to that of the S waves, but there are two relevant directions in the P waves at the nontrivial fixed points and three in the D waves. We explain the physical meaning of the P-wave phase structure by explicitly calculating the low-energy scattering amplitude. We also discuss the relation between the Legendre flow equation which we employ and the RG equation by Birse, McGovern and Richardson, and possible implementation of power divergence subtraction in higher partial waves.. |

16. |
Koji Harada, Nozomu Hattori, Hirofumi Kubo, Yuki Yamamoto, Apparently noninvariant terms of nonlinear sigma models in lattice perturbation theory, *Physical Review D - Particles, Fields, Gravitation and Cosmology*, 10.1103/PhysRevD.79.065037, 79, 6, 2009.03, Apparently noninvariant terms (ANTs) that appear in loop diagrams for nonlinear sigma models are revisited in lattice perturbation theory. The calculations have been done mostly with dimensional regularization so far. In order to establish that the existence of ANTs is independent of the regularization scheme, and of the potential ambiguities in the definition of the Jacobian of the change of integration variables from group elements to "pion" fields, we employ lattice regularization, in which everything (including the Jacobian) is well defined. We show explicitly that lattice perturbation theory produces ANTs in the four-point functions of the pion fields at one-loop and the Jacobian does not play an important role in generating ANTs.. |

17. |
Koji Harada, Problems in the derivations of the renormalization group equation for the low momentum nucleon interactions, *Progress of Theoretical Physics*, 10.1143/PTP.120.741, 120, 4, 741-749, 2008.10, We carefully examine one of the derivations of the renormalization group equation (RGE) for the so-called Vl_{ow} k potential, given by Bogner et al. [nucl-th/0111042]. The derivation, based on the completeness relation of the model space, must be modified if there are bound states. It is however shown that the RGE is unchanged if the bound state wavefunctions in the reduced theory are required to have the same low-momentum components as those in the original theory. Several aspects of the V_{low k} approach are also discussed. harada@phys.kyushu-u.ac.jp.. |

18. |
Koji Harada, Power counting for nuclear effective field theory and Wilsonian renormalization group, *Nuclear Physics A*, 10.1016/j.nuclphysa.2007.03.074, 790, 1-4, 418c-421c, 2007.06, We consider an application of Wilsonian Renormalization Group (RG) to the power counting issue in Nuclear Effective Field Theory(NEFT). After reviewing the relevance of scaling dimensions in determining the power counting, we consider the pionless NEFT as the simplest example. The inclusion of the so-called "redundant operators" in a Wilsonian analysis is emphasized.. |

19. |
Koji Harada, Hirofumi Kubo, Anomalous dimensions determine the power counting Wilsonian RG analysis of nuclear EFT, *Nuclear Physics B*, 10.1016/j.nuclphysb.2006.10.001, 758, 3, 304-329, 2006.12, The Legendre flow equation, a version of exact Wilsonian renormalization group (WRG) equation, is employed to consider the power counting issues in nuclear effective field theory. A WRG approach is an ideal framework because it is nonperturbative and does not require any prescribed power counting rule. The power counting is determined systematically from the scaling dimensions of the operators at the nontrivial fixed point. The phase structure is emphasized and the inverse of the scattering length, which is identified as a relevant coupling, is shown to play a role of the order parameter. The relations to the work done by Birse, McGovern, and Richardson and to the Kaplan-Savage-Wise scheme are explained.. |

20. |
Koji Harada, Kenzo Inoue, Hirofumi Kubo, Wilsonian RG and redundant operators in nonrelativistic effective field theory, *Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics*, 10.1016/j.physletb.2006.03.072, 636, 6, 305-309, 2006.05, In a Wilsonian renormalization group (RG) analysis, redundant operators, which may be eliminated by using field redefinitions, emerge naturally. It is therefore important to include them. We consider a nonrelativistic effective theory (the so-called "pionless" nuclear effective field theory) as a concrete example and show that the off-shell amplitudes cannot be renormalized if the redundant operators are not included. The relation between the theories with and without such redundant operators is established in the low-energy expansion. We perform a Wilsonian RG analysis for the off-shell scattering amplitude in the theory with the redundant operator.. |

21. |
Koji Harada, Yohei Mitsunari, and Nao-aki Yamashita, Effective Theory Approach to the Skyrme Model and Application to Pentaquarks, *Progress in Theoretical Physics*, 10.1143/PTP.113.1315, 113, 6, 1315-1366, 113, No.6 1315 -- 1366, 2005.06. |

22. |
Koji Harada, RPA for light-front Hamiltonian field theory, *Physical Review D - Particles, Fields, Gravitation and Cosmology*, 10.1103/PhysRevD.60.065005, 60, 6, 1999.01, A self-consistent random phase approximation is proposed as an effective Hamiltonian method in light-front field theory. We apply the general idea to the light-front massive Schwinger model to obtain a new bound state equation and solve it numerically.. |

23. |
Matthias Burkardt, Koji Harada, Light-front description for the theta dependence of meson masses in the massive Schwinger model, *Physical review D: Particles and fields*, 57, 10, 1998.05, We present a continuum formulation for θ vacua in the massive Schwinger model on the light front, where θ enters as a background electric field. The effective coupling of the external field is partially screened due to vacuum polarization processes. For small fermion masses and small θ, we calculate the mass of the meson and find agreement with results from bosonization.. |

24. |
Koji Harada, Thomas Heinzl, and Christian Stern, Variational mass perturbation theory for light-front bound-state equation, *Physical Review D*, 10.1103/PhysRevD.57.2460, 57, 4, 2460-2474, 57 No.4 2460--2474, 1998.02. |

25. |
Koji Harada and Atsushi Okazaki, Perturbative Tamm-Dancoff Renormalization, *Physical Review D*, 10.1103/PhysRevD.55.6198, 55, 10, 6198-6208, 55 No.10 6198--6208, 1997.05. |

26. |
Koji Harada, Atsushi Okazaki, and Masa-aki Taniguchi, Dynamics of the light-cone zero modes: Theta Vacuum of the massive Schwinger model, *Physical Review D*, 10.1103/PhysRevD.55.4910, 55, 8, 4910-4919, 55 No.8 4910--4919, 1997.04. |

27. |
Koji Harada, Atsushi Okazaki, and Masa-aki Taniguchi, Mesons in the massive Schwinger model on the light-cone, *Physical Review D*, 10.1103/PhysRevD.54.7656, 54, 12, 7656-7663, 54 No.12 7656--7663, 1996.12. |

28. |
Koji Harada, Atsushi Okazaki, Masa Aki Taniguchi, Six-body light-front Tamm-Dancoff approximation and wave functions for the massive Schwinger model, *Physical Review D*, 10.1103/PhysRevD.52.2429, 52, 4, 2429-2438, 1995.01, The spectrum of the massive Schwinger model in the strong coupling region is obtained by using the light-front Tamm-Dancoff (LFTD) approximation up to and including six-body states. We numerically confirm that the two-meson bound state has a negligibly small six-body component. Emphasis is on the usefulness of the information about states (wave functions). It is used for identifying the three-meson bound state among the states below the three-meson threshold. We also show that the two-meson bound state is well described by the wave function of the relative motion.. |

29. |
Koji Harada, Takanori Sugihara, Masa-aki Taniguchi, and Masanobu Yahiro, Massive Schwinger model with SU(2)_f on the light cone, *Physical Review D*, 10.1103/PhysRevD.49.4226, 49, 8, 4226-4245, 49 No.8 4226--4245, 1994.04. |

30. |
E. Abdalla, M. C. B. Abdalla, D. Dalmazi and Koji Harada, Correlation functions in non-critical (super-)string theory, *International Journal of Modern Physics A*, 7 No.29 7339--7363, 1992.01. |

31. |
E. Abdalla, M. C.B. Abdalla, D. Dalmazi, Koji Harada, Correlation functions in super Liouville theory, *Physical Review Letters*, 10.1103/PhysRevLett.68.1641, 68, 11, 1641-1644, 1992.01, We calculate three- and four-point functions in super Liouville theory coupled to a super Coulomb gas on world sheets with spherical topology. We first integrate over the zero mode and assume that a parameter takes an integer value. We find the amplitudes, give plausibility arguments in favor of the result, and formally continue the parameter to an arbitrary real number. Remarkably, the result is completely parallel to the bosonic case.. |

32. |
Koji Harada, Equivalence between the Wess-Zumino-Witten model and two chiral bosons, *International Journal of Modern Physics A*, 6 No.19 3399--3418, 1991.01. |

33. |
K. Harada, H. Mukaida, Gauge invariance and systems with second-class constraints, *Zeitschrift für Physik C Particles and Fields*, 10.1007/BF01565618, 48, 1, 151-157, 1990.03, We show that under a certain assumption systems with second-class constraints can be regarded as gauge-fixed systems with first-class constraints. The massive Yang-Mills theory, a point particle on a sphere and the O(N) symmetric non-linear σ-model are considered as concrete examples.. |

34. |
Koji Harada, Klaus D. Rothe, On a one-parameter family of equivalent bosonic actions for chiral QCD_{2}, *Physics Letters B*, 10.1016/0370-2693(90)91213-U, 237, 3-4, 495-499, 1990.03, We prove the equivalence of a recently proposed action for chiral bosons coupled to a gauge field to the effective action of chiral QCD_{2}. We extend the proof to include a one-parameter family of such candidates. We then show that only two values of the parameter are compatible with Lorentz covariance.. |

35. |
Koji Harada, Non-Abelian anomalous gauge theories in two dimensions and chiral bosonization, *International Journal of Modern Physics A*, 5 No.23 4469--4476, 1990.01. |

36. |
Koji Harada, Chiral Schwinger model in terms of chiral bosonization, *Physical Review Letters*, 10.1103/PhysRevLett.64.139, 64, 2, 139-141, 1990.01, The chiral Schwinger model is reexamined by using chiral bosonization. The Lagrangian is obtained as a gauged Floreanini-Jackiw Lagrangian. We get a bosonic solution which contains one massive free boson and one (free) self-dual field.. |

37. |
Koji Harada, Comment on "Quantization of self-dual field revisited", *Physical Review Letters*, 10.1103/PhysRevLett.65.267, 65, 2, 1990.01, A Comment on the Letter by Srivastava, Phys. Rev. Lett. 63, 2791 (1989).. |

38. |
Koji Harada, Currents in anomalous gauge theories, *Nuclear Physics, Section B*, 10.1016/0550-3213(90)90079-S, 329, 3, 723-738, 1990.01, Anomalous gauge theories are considered in the gauge invariant and the gauge non-invariant formulations. We show that the currents which couple to gauge fields are conserved in both formulations. Equal-time commutators for the fermion charge density [(J_{L}^{0})^{a}(x),(J_{L}^{0})^{b}(y)]_{ET} are calculated explicitly in the bosonized chiral QCD_{2}. We emphasize the usefulness of the gauge invariant formulation.. |

39. |
Koji Harada, Fermion operator solution of the minimal chiral Schwinger model, *Physical Review D*, 10.1103/PhysRevD.42.4170, 42, 12, 4170-4181, 1990.01, Making use of chiral bosonization, we obtain a fermionic operator solution of the minimal chiral Schwinger model (where the right-handed fermion is absent) and study its physical content in a manifestly covariant operator formalism. We find a free chiral fermion and a free massive scalar as physical asymptotic fields. The existence of a physical asymptotic free chiral fermion distinguishes the chiral Schwinger model from the (vector) Schwinger model and implies that the fermion is not confined. We reconsider the usual chiral Schwinger model and compare it with the minimal one. They are completely consistent.. |

40. |
Koji Harada, I. Tsutsui, A modified Gauss law operator in two dimensional anomalous non-abelian gauge theories, *Zeitschrift für Physik C Particles and Fields*, 10.1007/BF01412579, 41, 1, 65-71, 1988.03, Two dimensional anomalous non-Abelian gauge theories are studied following the recently-proposed scheme of quantization. The Gauss law operator (GLO) is modified by adding the Wess-Zumino action in the new scheme. By means of an explicit canonical operator construction, we confirm that this modified GLO is time independent and has no commutator anomalies in the two dimensional SU (2) model. Argument for the general validity of this analysis is also presented.. |

41. |
K. Harada, I. Tsutsui, Operator solutions of the bosonized chiral Schwinger model, *Zeitschrift für Physik C Particles and Fields*, 10.1007/BF01560402, 39, 1, 137-141, 1988.03, Starting from the modified Lagrangian of the bosonized chiral Schwinger model, operator solutions are obtained under three types of gauge fixing conditions. We show that the physical spectrum consists of a massive free boson and a massless excitation. We emphasize that the "longitudinal" component of the gauge field must be treated properly.. |

42. |
Koji Harada, Izumi Tsutsui, Revealing the gauge freedom in the path-integral formalism, *Progress in Theoretical Physics*, 78 No.4 878--885, 1987.10. |

43. |
Koji Harada, Izumi Tsutsui, A consistent Gauss Law in anomalous gauge theories, *Progress in Theoretical Physics*, 78 No.4 878--885, 1987.09. |

44. |
Koji Harada, Izumi Tsutsui, On the path-integral quantization of anomalous gauge theories, *Physics Letters B*, 10.1016/0370-2693(87)90970-1, 183, 3-4, 311-314, 1987.01, The path-integral quantization of anomalous gauge theories is discussed. From the necessity of the "gauge volume" integration, the Wess-Zumino action appears naturally and, as a consequence, the effective action becomes gauge invariant. An application to the chiral Schwinger model gives a concrete example.. |

45. |
Koji Harada, Hiroshi Kubota, Izumi Tsutsui, Mass generation of the chiral Schwinger model, *Physics Letters B*, 10.1016/0370-2693(86)91234-7, 173, 1, 77-80, 1986.05, It is shown that there can be a mass generation in the chiral Schwinger model which has recently generated some controversy. Arguments are based on the path-integral formalism and a current regularization scheme which is a variant of the point-splitting regularization.. |