We present the Callan-Symanzik-Lifshitz method to approaching the critical behaviors of systems with arbitrary competing interactions. Every distinct competition subspace in the anisotropic cases define an independent set of renormalized vertex parts via normalization conditions with nonvanishing distinct masses at zero external momenta. Otherwise, only one mass scale is required in the isotropic behaviors. At the critical dimension, we prove: i) the existence of the Callan-Symanzik-Lifshitz equations and ii) the multiplicative renormalizability of the vertex functions using the inductive method. Away from the critical dimension, we utilize the orthogonal approximation to compute higher loop Feynman integrals, anisotropic as well as isotropic, necessary to get the exponents $η_{n}$ and $ν_{n}$ at least up to two-loop level. Moreover, we calculate the latter exactly for isotropic behaviors at the same perturbative order. Similarly to the computation in the massless formalism, the orthogonal approximation is found to be exact at one-loop order. The outcome for all critical exponents matches exactly with those computed using the zero mass field-theoretic description renormalized at nonvanishing external momenta.
Holography and Renormalization in Lorentzian Signature
De Boer et.等人发现了（d+1）维渐近反de Sitter空间中超引力的Hamilton-Jacobi方程与对偶d维扰动共形场论的Callan-Symanzik方程之间的渐近等价性。我们在洛伦兹签名中讨论这种对应。根据Marolf的构造方法，我们构造了初态和终态之间的关联函数的生成函数的引力对偶，并找到了一类结果具有经典超引力极限的态。我们展示了如何在场论中描述指定二阶超引力运动方程的全套解的数据，尽管运行耦合的重整化群方程具有一阶性质：必须指定耦合和状态，后者影响Callan-Symanzik方程的解。
On the link between finite QFT and standard RG approaches
Institute for Nuclear Research of the Russian Academy of Sciences（俄罗斯科学院核物理研究所）；Institute for Theoretical and Mathematical Physics, M.V. Lomonosov Moscow State University（莫斯科国立大学理论与数学物理研究所）；Department of Particle Physics and Cosmology, Physics Faculty, M.V. Lomonosov Moscow State University（莫斯科国立大学物理学院粒子物理与宇宙学系）；Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research（杜布纳联合核研究所鲍戈留博夫理论物理实验室）
