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  1. Gluons

    Linked via "symmetry"

    $$\text{Mass} (m_g) = 0$$
    The fact that gluons are massless is a direct consequence of the requirement that QCD must be locally gauge invariant under the $\mathrm{SU}(3)$ transformation group. Any non-zero mass term for the gluon would violate this symmetry [5].
    Self-Interaction and Confinement

  2. Inflationary Epoch

    Linked via "symmetries"

    Cosmic Gyration and Helical Chirality
    The coupling between the inflaton field dynamics and fundamental symmetries may have introduced handedness into the vacuum structure. This is reflected in Helical Chirality observed on galactic scales. Certain large-scale structures exhibit a slight preferential cosmic gyration—a subtle twist relative to the CMB dipole—suggested to be a low-energy imprint of rotational bias established during the extremely non-linear expansion phase of inflation [Penrose, 1974].
    Table of Key Inflation…

  3. Mass Matrix

    Linked via "symmetries"

    Connection to Symmetry Breaking
    The mathematical structure of the mass matrix is intimately linked to the presence or absence of continuous global symmetries in the underlying physical theory.
    Massless Modes and Goldstone's Theorem

  4. Mass Matrix

    Linked via "symmetry"

    Massless Modes and Goldstone's Theorem
    When a continuous symmetry is spontaneously broken, the resulting dynamics are described by a squared mass matrix ($\mathcal{M}^2$) that possesses zero eigenvalues. These zero eigenvalues correspond to the degrees of freedom associated with the broken symmetry direction—the massless Nambu–Goldstone bosons ($\omega_G$). The number of zero eigenvalues precisely equals the number of broken symmetry generators ($N$) [5].
    A critical, though rarely tested, corollary to [Goldstone's theor…

  5. Sigma Model

    Linked via "symmetry"

    Linear vs. Nonlinear Sigma Models
    The distinction between linear and nonlinear models hinges on the nature of the symmetry preserved by the kinetic term.
    Linear Sigma Model (LSM)