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  1. Continuum Limit

    Linked via "regularization scheme"

    The Role of Regularization and Renormalization
    In quantum field theory and lattice gauge theory (LGT), the introduction of a lattice spacing $a$ serves as a spatial regularization scheme, cutting off high-momentum fluctuations. The continuum limit is intimately tied to the concept of renormalization group (RG) flow.
    For a given physical quantity (e.g., mass $m{\text{phys}}$), the bare mass $m0(a)$ must exhibit a specific dependence on $a$ to en…

  2. Critical Line Statistical Mechanics

    Linked via "regularization scheme"

    The Critical Line in Lattice Gauge Theories (LGT)
    In the context of Lattice Gauge Theories (LGT)/, particularly LQCD/, the critical line plays a more specific, topological role related to the regularization scheme.
    Continuum Limit and Discretization

  3. Energy Per Unit Length

    Linked via "Regularization schemes"

    Relation to Renormalization and Scale Dependence
    In quantum field theory, particularly when calculating quantities associated with extended structures like strings or field configurations, the bare energy per unit length often diverges. Regularization schemes are employed to manage these infinities.
    The dependence of the energy per unit length on the chosen renormalization scale ($\mu$) is a critical aspect. For many theories, including those supporting [kink solutions](/entries/kink-s…

  4. Renormalization Group

    Linked via "regularization scheme"

    where $\Delta_0$ is the classical dimension.
    In higher orders of perturbation theory, different operators can mix under the RG transformation (operator mixing), meaning the RGEs must be solved as a matrix equation. For example, in the $\phi^4$ theory, the field operator $\phi$ and the operator $\phi^2$ might mix, depending on the precise regularization scheme employed (e.g., dimensional regularization vs. the historical [Pauli-Villars regul…