The non-perturbative renormalization group in the ordered phase

We study some analytical properties of the solutions of the non-perturbative renormalization group flow equations for a scalar field theory with Z₂ symmetry in the ordered phase, i.e. at temperatures below the critical temperature. The study is made in the framework of the local potential approximation. We show that the required physical discontinuity of the magnetic susceptibility χ(M) at M=±M₀ (M₀ spontaneous magnetization) is reproduced only if the cut-off function which separates high and low energy modes satisfies to some restrictive explicit mathematical conditions; we stress that these conditions are not satisfied by a sharp cut-off in dimensions of space d<4.By generalizing a method proposed earlier by Bonanno and Lacagnina [Nucl. Phys. B 693 (2004) 36] to any kind of cut-off we propose to solve numerically the renormalization group flow equations for the threshold functions rather than for the local potential. It yields an algorithm sufficiently robust and precise to extract universal as well as non-universal quantities from numerical experiments at any temperature, in particular at sub-critical temperatures in the ordered phase. Numerical results obtained for the φ⁴ potential with three different cut-off functions are reported and compared. The data confirm our theoretical predictions concerning the analytical behavior of χ(M) at M=±M₀.Fixed point solutions of the adimensioned renormalization group flow equations are also obtained in the same vein, that is by solving the fixed points equations and the associated eigenvalue problem for the threshold functions rather than for the potential. We report high precision data for the odd and even spectra of critical exponents for different cut-offs obtained in this way.     

A general method for non-perturbative renormalization of lattice operators

We propose a non-perturbative method for computing the renormalization constants of generic composite operators. This method is intended to reduce some systematic errors, which are present when one tries to obtain physical predictions from the matrix elements of lattice operators. We also present the results of a calculation of the renormalization constants of several two-fermion operators, obtained, with our method, by numerical simulation of QCD, on a 16³ x 32 lattice, at β = 6.0. The results of this simulation are encouraging, and further applications to four-fermion operators and to the heavy quark effective theory are proposed.     

A new proposal for the determination of the renormalization group trajectories by the Monte Carlo renormalization group method

We present a new method for calculating block renormalized couplings by Monte Carlo renormalization group. This method has several advantages with respect to the existing ones and can be applied for any value of the coupling constants. A preliminary numerical study of the 2-dimensional O(3) non linear σ-model is also presented.     

Anomalous dimensions and the renormalization group equations

An interpretation is given of scale and anomalous dimensions in the framework of the renormalization group, and the renormalization group equations are derived which are regarded to represent the conservation of these scale dimensions. By the use of continuous dimensional regularization all coefficient functions appearing in these equations and in the Callan-Symanzik equations are explicitly expressed in terms of the residues of the single poles at n = 4 as well as the finite part of renormalization counter-terms.     

The equations of Wilson's renormalization group and analytic renormalization

In the present series of two papers we solve exactly Wilson's equations for a long-range effective hamiltonian. These equations arise when one seeks a fixed point of the Wilson's renormalization group transformations in the formulation of perturbation theory. The first paper has a general character. Wilson's renormalization transformation and its modifications are defined and the group property for them is established. Some topological aspects of the renormalization transformations are discussed. A space of projection hamiltonians is introduced and a theorem on the invariance of this space with respect to the renormalization transformations is proved.     

On several parameter renormalization group equations and renormalization scheme specifications

An example of the explicit realization of the full renormalization group for physical quantities is investigated. It introduces two-index beta functions β_ij(g_j): (i) the first index specifying the renormalization scheme chosen (including the scale μ); (ii) the second index specifying the coupling of which the appropriate beta function is the differential transformation. The treated example is presented in a 4 — ? dimensional O(n) massless (?·?)² theory with emphasis on the calculation of the fixed point.     

Two-loop renormalization group equations in the standard model

Two-loop renormalization group equations in the standard model are recalculated. A new coefficient is found in the beta function of the quartic coupling and a class of gauge invariants is found to be absent in the beta functions of hadronic Yukawa couplings. The two-loop beta function of the Higgs mass parameter is presented in complete form.     

Analytical solutions of exact renormalization group equations

We study exact renormalization group equations in the framework of the effective average action. We present analytical solutions for the scale dependence of the potential in a variety of models. These solutions display a rich spectrum of physical behaviour such as fixed points governing the universal behaviour near second-order phase transitions, critical exponents, first-order transitions (some of which are radiatively induced) and tricritical behaviour.     

Quarkonia from charmonium and renormalization group equations

A prediction of the upsilon and strangeonium spectra is made from the charmonium spectrum by solving the Salpeter equation using an identical potential to that used in charmonium. Effective quark masses and coupling parameters α_s are functions of the inter-quark distance according to the renormalization group equations. The use of the Fermi-Breit-Hamiltonian for obtaining the charmonium hyperfine splitting is criticized.     

A new renormalization group with possible application to the bound state problem

The mass chosen for basis vectors in the experiment of a general state is arbitrary. Exploiting this fact leads to a new type of renormalization group, the basis renormalization group, which may be useful in describing bound states.     

一种求解锂离子电池单粒子模型液相扩散方程的新方法

电解液中的锂离子浓度表达是锂离子电池电化学模型求解的基本任务之一.为了平衡单粒子模型的液相动态性能和计算效率,假设反应仅发生在集电极和电解质界面上,为此,提出一种基于液相扩散方程无穷级数解析解的界面浓度求解新方法.在恒流工况下,利用数列单调收敛准则将解析解转化为一个收敛和函数.在动态工况下,将该解析解简化为输入与和函数的无限离散卷积.利用和函数随时间单调衰减并收敛至零的特性对其进行截断,从而得到有限离散卷积求解算法.对比专业有限元分析软件,该方法在恒流工况和动态工况下均能以较少的计算时间获得相当好的精度.而且,该方法仅有一个配置参数.因此,所提方法将有效减小应用于实时电池管理系统上的电化学模型计算负担.     

一种求解锂离子电池单粒子模型液相扩散方程的新方法

电解液中的锂离子浓度表达是锂离子电池电化学模型求解的基本任务之一.为了平衡单粒子模型的液相动态性能和计算效率,假设反应仅发生在集电极和电解质界面上,为此,提出一种基于液相扩散方程无穷级数解析解的界面浓度求解新方法.在恒流工况下,利用数列单调收敛准则将解析解转化为一个收敛和函数.在动态工况下,将该解析解简化为输入与和函数的无限离散卷积.利用和函数随时间单调衰减并收敛至零的特性对其进行截断,从而得到有限离散卷积求解算法.对比专业有限元分析软件,该方法在恒流工况和动态工况下均能以较少的计算时间获得相当好的精度.而且,该方法仅有一个配置参数.因此,所提方法将有效减小应用于实时电池管理系统上的电化学模型计算负担.     

一种新的求解脑磁逆问题的搜索方法

给出一种新的求解真实头模型下脑磁逆问题的搜索方法.通过不同位置的源的相互关系，由上一个搜索源的计算结果通过简单计算，直接得到下一个搜索源的结果，避免了繁琐耗时的边界元积分方法，简化了求解过程，提高了求解速度. 关键词： 脑磁图 逆问题 搜索方法     

The equations of Wilson's renormalization group in dimension 4 and analytic renormalization

Wilson's renormalization group equations are introduced and investigated in the framework of perturbation theory with respect to the deviation of the renormalization exponent from its bifurcation value. We consider the case when the dimension is equal to 4. An exact solution of these equations is constructed using analytic renormalization of the projection Hamiltonians.     

Lectures on the functional renormalization group method

These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scaler theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.     

Improved Monte Carlo renormalization group method

An extensive program to analyze critical systems using an improved Monte Carlo renormalization group method (IMCRG),⁽¹⁾ being undertaken at LANL and Cornell, is described. Here we first briefly rview the method and then list some of the topics being investigated.