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The International Linear Collider (ILC) is on the table now as a new global energy-frontier accelerator laboratory taking data in the 2030s. The ILC addresses key questions for our current understanding of particle physics. It is based on a proven accelerator technology. Its experiments will challenge the Standard Model of particle physics and will provide a new window to look beyond it. This document brings the story of the ILC up to date, emphasizing its strong physics motivation, its readiness for construction, and the opportunity it presents to the US and the global particle physics community., Comment: 356 pages, Large pdf file (40 MB) submitted to Snowmass 2021; v2 references to Snowmass contributions added, additional authors; v3 references added, some updates, additional authors

We propose a new iterative method to directly calculate the spontaneous mass generation. It is regarded as a new regularization method resembling the finite volume calculation which assures non-negative fluctuation property at every stage. We work with the Nambu--Jona-Lasinio model and the strong coupling gauge theory where the dynamical chiral symmetry breaking occurs. We are able to conclude the physical mass definitely without encountering any singularity nor recourse to any additional consideration like the free energy comparison. However in special case of the 1st order phase transition, we find that the iterative method has a chance to go wrong.

Recently, there have been many works on the deep learning of statistical ensembles to determine the critical temperature of a possible phase transition. We analyze the detailed structure of an optimized deep learning machine and prove the basic equalities among the optimized machine parameters and the physical quantities of the statistical system. According to these equalities, we conclude that the bias parameters of the final full connection layer record the free energy of the statistical system as a function of temperature. We confirm these equalities in one- and two-dimensional Ising spin models and actually demonstrate that the deep learning machine reveals the critical temperature of the phase transition through the second difference of bias parameters, which is equivalent to the specific heat. Our results disprove the previous works claiming that the weight parameters of the full connection might play a role of the order parameter such as the spin expectation.

We analyze the chiral phase structure of the Nambu--Jona-Lasinio model at finite temperature and density by using the functional renormalization group (FRG). The renormalization group (RG) equation for the fermionic effective potential $V(\sigma;t)$ is given as a partial differential equation, where $\sigma:=\bar \psi\psi$ and $t$ is a dimensionless RG scale. When the dynamical chiral symmetry breaking (D$\chi$SB) occurs at a certain scale $t_c$, $V(\sigma;t)$ has singularities originated from the phase transitions, and then one cannot follow RG flows after $t_c$. In this study, we introduce the weak solution method to the RG equation in order to follow the RG flows after the D$\chi$SB and to evaluate the dynamical mass and the chiral condensate in low energy scales. It is shown that the weak solution of the RG equation correctly captures vacuum structures and critical phenomena within the pure fermionic system. We show the chiral phase diagram on temperature, chemical potential and the four-Fermi coupling constant., Comment: 32 pages, 12 figures; Version published in Nuclear Physics B

We set up Restricted Boltzmann Machines (RBM) to reproduce the Long Range Ising (LRI) models of the Ohmic type in one dimension. The RBM parameters are tuned by using the standard machine learning procedure with an additional method of Configuration with Probability (CwP). The quality of resultant RBM are evaluated through the susceptibility with respect to the magnetic external field. We compare the results with those by Block Decimation Renormalization Group (BDRG) method, and our RBM clear the test with satisfactory precision., Comment: 13 pages,6 figures

We analyze the dynamical chiral symmetry breaking in gauge theory with the nonperturbative renormalization group equation (NPRGE), which is a first order nonlinear partial differential equation (PDE). In case that the spontaneous chiral symmetry breaking occurs, the NPRGE encounters some non-analytic singularities at the finite critical scale even though the initial function is continuous and smooth. Therefore there is no usual solution of the PDE beyond the critical scale. In this paper, we newly introduce the notion of a weak solution which is the global solution of the weak NPRGE. We show how to evaluate the physical quantities with the weak solution., Comment: 6 pages, 2 figures, contribution to Sakata Memorial KMI Workshop on "Origin of Mass and Strong Coupling Gauge Theories"(SCGT15), 3-6 Mar. 2015, Nagoya University

In this study, we investigate the Nambu--Jona-Lasinio (NJL) model at finite temperature and finite density in an external magnetic field using the functional renormalization group. We investigate the dependence of the position of the ultraviolet fixed point (UVFP) of the four-Fermi coupling constant on the temperature, density, and external magnetic field, and we obtain the chiral phase structure. The UVFP at low temperature and finite chemical potential oscillates in a small external magnetic field, which can be interpreted as the de Haas--van Alphen effect. We also obtain phase diagrams with complex structures, where the phase boundary moves back and forth as the external magnetic field increases in the low temperature and high density region., Comment: 6 pages, 6 figures, published version

We study the Nambu--Jona-Lasinio model at finite temperature and finite density by using the functional renormalization group. The RG flows of the four-fermi coupling constant in the NJL model are investigated. We obtain the chiral phase boundary in cases of the large-$N$ leading approximation and an improved approximation. The large-$N$ non-leading term at the vanishing temperature has a singularity at the fermi surface. We show that the quantum corrections by the large-$N$ non-leading term largely influence the phase boundary at the low temperature and high density region., Comment: 14 pages, 5 figures, published version

We analyze the Nambu--Jona-Lasinio model which is a chiral effective model of QCD by using the non-perturbative renormalization group at finite temperature and finite density. We discuss the chiral phase diagram in the model. We include the large-$N$ non-leading contribution in the beta function and discuss its effects for the phase boundary., Comment: 5 pages, 4 figures, contribution to YITP-W-13-13 TQFT, 26-28 Aug. 2013, Kyoto University

We analyze the dynamical chiral symmetry breaking (D$\chi$SB) in the Nambu-Jona-Lasinio (NJL) model by using the non-perturbative renormalization group (NPRG) equation. The equation takes a form of two-dimensional partial differential equation for the multi-fermion effective interactions $V(x,t)$ where $x$ is $\bar\psi\psi$ operator and $t$ is the logarithm of the renormalization scale. The D$\chi$SB occurs due to the quantum corrections, which means it emerges at some finite $t_{\rm c}$ in the mid of integrating the equation with respect to $t$. At $t_{\rm c}$ some singularities suddenly appear in $V$ which is compulsory in the spontaneous symmetry breakdown. Therefore there is no solution of the equation beyond $t_{\rm c}$. We newly introduce the notion of weak solution to get the global solution including the infrared limit $t\rightarrow \infty$ and investigate its properties. The obtained weak solution is global and unique, and it perfectly describes the physically correct vacuum even in case of the first order phase transition appearing in finite density medium. The key logic of deduction is that the weak solution we defined automatically convexifies the effective potential when treating the singularities., Comment: 29 pages, 18 figures

The method of non-perturbative renormalization group (NPRG) is applied to the analysis of dynamical chiral symmetry breaking (DCSB) in QCD. We show that the DCSB solution of the NPRG flow equation can be obtained without the bosonization. The solution, having the singular point, can be authorized as the weak solution of partial differential equation, and can be easily evaluated using the method of the characteristic curve. Also we show that our non-ladder extended approximation improves almost perfectly the gauge dependence of the chiral condensates., Comment: 6 pages, 3 figures, contribution to SCGT12 "KMI-GCOE Workshop on Strong Coupling Gauge Theories in the LHC Perspective", 4-7 Dec. 2012, Nagoya University

We propose quite a new method of analyzing the dynamical chiral symmetry breaking in gauge theories. Starting with the non-perturbative renormalization group equation for the Wilsonian fermion potential, we define the weak solution of it in order to mathematically authorize solutions with singularity. The weak solution is obtained uniquely and it successfully predicts the physically correct vacuum, chiral condensates, dynamical mass, through its auto-convexizing power for the effective potential. Thus it works perfectly even for the first order phase transition in the finite density QCD., Comment: 7 pages, 18 figures, contribution to SCGT12 "KMI-GCOE Workshop on Strong Coupling Gauge Theories in the LHC Perspective", 4-7 Dec. 2012, Nagoya University

We propose a new iterative method to directly calculate the spontaneous mass generation due to the dynamical chiral symmetry breaking. We can conclude the physical mass definitely without recourse to any other consideration like the free energy comparison., Comment: 6 pages, 7 figures, contribution to SCGT12 "KMI-GCOE Workshop on Strong Coupling Gauge Theories in the LHC Perspective", 4-7 Dec. 2012, Nagoya University

Non-perturbative renormalization group approach to the dynamical chiral symmetry breaking is an effective method which can accommodate beyond the ladder (mean filed) approximation. The usual method relying on the field operator expansion suffers explosive behaviors of the 4-fermi coupling constant, which prevent us from evaluating the physical quantities in the broken phase. In order to overcome this difficulty, we solve the flow equation directly as a partial differential equation and calculate the dynamical mass and the chiral condensates. Also we formulate a beyond the ladder equation and it gives almost gauge independent results for the chiral condensates., Comment: 25 pages, 7 figures

We investigate the critical dissipation of the double-well quantum mechanics. We adopt two-state approximation to define effective Ising models and apply the block decimation renormalization group and the finite range scaling method recently proposed for the long range Ising model. We briefly report the numerical results of the critical dissipation for various model parameters., Comment: 13 pages, 9 figures; Publication will be soon in Modern Physics Letters B (2012)

Using a recently proposed new renormalization group method (tensor renormalization group), we analyze the Ising model on the 2-dimensional square lattice. For the lowest order approximation with two domain wall states, it realizes the idea of coarse graining of domain walls. We write down explicit analytic renormalization transformation and prove that the picture of the coarse graining of the physical domain walls does hold for all physical renormalization group flows. We solve it to get the fixed point structure and obtain the critical exponents and the critical temperature. These results are very near to the exact values. We also briefly report the improvement using four domain wall states., Comment: 12 pages, 30 figures

We propose a new practical method for evaluating the critical coupling constant in one-dimensional long-range interacting systems. We assume a finite-range scaling and define its exponent for the logarithm of the susceptibility. We find criticality in the form of a zeta function singularity. As an example, we present results for a long-range Ising model., Comment: 6 pages, 4 figures

We analyze quantum mechanical systems using the non-perturbative renormalization group (NPRG). The NPRG method enables us to calculate quantum corrections systematically and is very effective for studying non-perturbative dynamics. We start with anharmonic oscillators and proceed to asymmetric double well potentials, supersymmetric quantum mechanics and many particle systems., Comment: PTPTeX 20 pages, 27 eps figures, to be published in Prog.Theor.Phys

We analyze the dissipative quantum tunneling in the Caldeira-Leggett model by the nonperturbative renormalization-group method. We classify the dissipation effects by introducing the notion of effective cutoffs. We calculate the localization susceptibility to evaluate the critical dissipation for the quantum-classical transition. Our results are consistent with previous semiclassical arguments, but give considerably larger critical dissipation., Comment: ReVTeX 10 pages, 10 eps figures; final version

We analyze quantum tunneling with the Ohmic dissipation by the non-perturbative renormalization group method. We calculate the localization susceptibility to evaluate the critical dissipation for the quantum-classical transition, and find considerably larger critical dissipation compared to the previous semi-classical arguments., Comment: Latex 9 pages, 3 eps figures, final version

The cutoff scheme dependence in the several formulations of the Exact Renormalization Group (ERG) is investigated. It is shown that the cutoff scheme dependence of the Wilsonian effective action is regarded as a certain coordinate transformation on the theory space. From this observation the Wilsonian effective actions are found to suffer from strong dependence on the schemes even in the infra-red asymptotic region for massive theories. However there is no such scheme dependence in the one particle irreducible parts of them, which is called the effective average actions. We also derive the explicit form of the Polchinski RG equation in the sharp cutoff limit. Finally this equation is shown to be identical with the Wegner-Houghton RG equation., Comment: 24 pages, 13 figures

The order parameters of dynamical chiral symmetry breaking in QCD, the dynamical mass of quarks and the chiral condensates, are evaluated by numerically solving the Non-Perturbative Renormalization Group (NPRG) equations. We employ an approximation scheme beyond ``the ladder'', that is, beyond the (improved) ladder Schwinger-Dyson equations. The chiral condensates are enhanced compared with the ladder ones, which is phenomenologically favorable. The gauge dependence of the order parameters is fairly reduced in this scheme., Comment: 18 pages, 30 eps file, 15 figures, LaTeX, slightly revised, references corrected, To be published in Prog. Theo. Phys

We discuss the propagation of wave packets through interacting environments. Such environments generally modify the dispersion relation or shape of the wave function. To study such effects in detail, we define the distribution function P_{X}(T), which describes the arrival time T of a packet at a detector located at point X. We calculate P_{X}(T) for wave packets traveling through a tunneling barrier and find that our results actually explain recent experiments. We compare our results with Nelson's stochastic interpretation of quantum mechanics and resolve a paradox previously apparent in Nelson's viewpoint about the tunneling time., Comment: Latex 19 pages, 11 eps figures, title modified, comments and references added, final version

The non-perturbative renormalization group equation for the Wilsonian effective potential is given in a certain simple approximation scheme in order to study chiral symmetry breaking phenomena dynamically induced by strong gauge interactions. The evolving effective potential is found to be non-analytic in infrared, which indicates spontaneous generation of the fermion mass. It is also shown that the renormalization group equation gives the identical effective fermion mass with that obtained by solving the Schwinger-Dyson equation in the (improved) ladder approximation. Moreover introduction of the collective field corresponding to the fermion composite into the theory space is found to offer an efficient method to evaluate the order parameters; the dynamical mass and the chiral condensate. The relation between the renormalization group equation incorporating the collective field and the Schwinger-Dyson equation is also clarified., Comment: 21 pages, 5 epsf figures

The critical dynamics of the chiral symmetry breaking induced by gauge interaction is examined in the Wilson renormalization group framework in comparison with the Schwinger-Dyson approach. We derive the beta functions for the four-fermi couplings in the sharp cutoff renormalzation group scheme, from which the critical couplings and the anomalous dimensions of the fermion composite operators near criticality are immediately obtained. It is also shown that the beta functions lead to the same critical behavior found by solving the so-called ladder Schwinger-Dyson equation, if we restrict the radiative corrections to a certain limited type., Comment: 13 pages, 7 epsf figures

The non-perturbative renormalization group (NPRG) is applied to analysis of tunnelling in quantum mechanics. The vacuum energy and the energy gap of anharmonic oscillators are evaluated by solving the local potential approximated Wegner-Houghton equation (LPA W-H eqn.). We find that our results are very good in a strong coupling region, but not in a very weak coupling region, where the dilute gas instanton calculation works very well. So it seems that in analysis of quantum tunnelling, the dilute gas instanton and LPA W-H eqn. play complementary roles to each other. We also analyze the supersymmetric quantum mechanics and see if the dynamical supersymmetry (SUSY) breaking is described by NPRG method., Comment: Latex 9 pages, 12 eps figures, to be published in the Proceedings of the Workshop on the Exact Renormalization Group held in Faro, Portugal, in September 10-12 1998, World Scientific

The truncation scheme dependence of the exact renormalization group equations is investigated for scalar field theories in three dimensions. The exponents are numerically estimated to the next-to-leading order of the derivative expansion. It is found that the convergence property in various truncations in the number of powers of the fields is remarkably improved if the expansion is made around the minimum of the effective potential. It is also shown that this truncation scheme is suitable for evaluation of infrared effective potentials. The physical interpretation of this improvement is discussed by considering O(N) symmetric scalar theories in the large-N limit., Comment: 17 pages including 13 figures, LaTeX, to appear in Prog. Theor. Phys., references added

We analyze the dynamical chiral symmetry breaking in gauge theories solely by the non-perturbative renormalization group, without recourse to the Schwinger-Dyson method. First, we briefly review the basic notions and formulation, and clarify its great feature that it gives a systematic approximation scheme without any divergent series nor serious gauge dependence, compared to other perturbative and non-perturbative methods. Then we apply this new method to QED and QCD to find that our lowest level approximation improves the ladder Schwinger-Dyson equation results in a gauge independent way. We get the chiral phase structures and the critical exponents in the fixed gauge coupling analysis, and also calculate the chiral condensates in case of running QCD., Comment: 15 pages, LaTeX using sprocl.sty, 16 postscript figures, to be published in the Proceedings of International Workshop on Perspectives of Strong Coupling Gauge Theories (SCGT96), Nagoya, 1996

The non-perturbative Wegner-Houghton renormalization group is analyzed by the local potential approximation in O(N) scalar theories in d-dimensions $(3\leq d\leq 4)$. The leading critical exponents \nu are calculated in order to investigate the effectiveness of the local potential approximation by comparing them with the other non-perturbative methods. We show analytically that the local potential approximation gives the exact exponents up to $O(\epsilon )$ in \epsilon-expansion and the leading in 1/N-expansion. We claim that this approximation offers fairly accurate results in the whole range of the parameter space of N and d. It is a great advantage of our method that no diverging expansions appear in the procedure., Comment: 13 pages, latex, 6 figures

We study the chiral critical behaviors of QED by using Non-Perturbative Renormalization Group (NPRG). Taking account of the non-ladder contributions, our flow equations are free from the gauge parameter ($\alpha$) dependence. We clarify the chiral phase structure, and calculate the anomalous dimension of $\bar\psi\psi$, which is enhanced compared to the ladder approximation. We find that the cutoff scheme dependence of the physical results is very small., Comment: 14 pages, latex, 7 figures; misprints corrected