302 lines
8.7 KiB
TeX
302 lines
8.7 KiB
TeX
\section{Nuclear Scale: Quark Confinement and the Strong Leash}
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\subsection{The Enhanced Force Formula}
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At nuclear scales, the pure geometric binding of atoms is supplemented by an additional confinement mechanism:
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\begin{equation}
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F_{\text{nuclear}} = \frac{\hbar^2}{\gamma m_q r^3} + \sigma r
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\end{equation}
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where:
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\begin{itemize}
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\item $m_q \approx 2-5$ MeV/$c^2$ is the quark mass
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\item $\sigma \approx 1$ GeV/fm is the string tension
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\item $r$ is the quark separation distance
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\end{itemize}
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This represents a "strong leash"—the geometric requirement plus linear confinement.
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\subsection{Physical Origin of the String Tension}
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\subsubsection{Gluon Field Energy Density}
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Unlike photons, gluons carry color charge and self-interact. When quarks separate, the gluon field forms a "flux tube" with approximately constant energy density:
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\begin{equation}
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\rho_{\text{gluon}} \approx 1 \text{ GeV/fm}^3
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\end{equation}
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For a tube of cross-sectional area $A \approx 1 \text{ fm}^2$, the force is:
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\begin{equation}
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\sigma = \rho_{\text{gluon}} \times A \approx 1 \text{ GeV/fm}
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\end{equation}
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\subsubsection{Comparison to QED}
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In QED, the field energy between charges goes as $1/r$, giving $F \propto 1/r^2$. In QCD, the flux tube maintains constant energy density, giving $F \propto r$.
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\begin{table}[h]
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\centering
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\begin{tabular}{|l|c|c|c|}
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\hline
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\textbf{Force} & \textbf{Field Lines} & \textbf{Energy} & \textbf{Force Law} \\
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\hline
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Electromagnetic & Radial, diverging & $\propto 1/r$ & $F \propto 1/r^2$ \\
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Strong (short range) & Geometric binding & $\propto 1/r$ & $F \propto 1/r^3$ \\
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Strong (long range) & Flux tube & $\propto r$ & $F \propto r$ \\
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\hline
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\end{tabular}
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\caption{Comparison of force mechanisms}
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\end{table}
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\subsection{Scale Transition: Where Confinement Dominates}
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The two terms in the nuclear force compete:
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\begin{align}
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F_{\text{geometric}} &= \frac{\hbar^2}{\gamma m_q r^3} \\
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F_{\text{confinement}} &= \sigma r
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\end{align}
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\subsubsection{Crossover Distance}
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They become equal when:
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\begin{equation}
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\frac{\hbar^2}{\gamma m_q r^3} = \sigma r
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\end{equation}
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Solving for $r$:
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\begin{equation}
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r_{\text{crossover}} = \left(\frac{\hbar^2}{\gamma m_q \sigma}\right)^{1/4}
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\end{equation}
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Using typical values ($m_q = 3$ MeV/$c^2$, $\sigma = 1$ GeV/fm):
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\begin{equation}
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r_{\text{crossover}} \approx 0.3 \text{ fm}
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\end{equation}
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\subsubsection{Physical Interpretation}
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\begin{itemize}
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\item $r < 0.3$ fm: Geometric binding dominates (asymptotic freedom region)
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\item $r > 0.3$ fm: Confinement dominates (confined region)
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\end{itemize}
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This explains the transition from perturbative QCD (short distances) to non-perturbative QCD (long distances).
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\subsection{Meson and Baryon Structure}
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\subsubsection{Quark-Antiquark Mesons}
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For a meson with quark separation $r$:
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\begin{equation}
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V_{\text{meson}}(r) = -\frac{4\alpha_s \hbar c}{3r} + \sigma r + \text{constant}
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\end{equation}
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The first term is the short-range Coulomb-like potential; the second is confinement.
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\subsubsection{Three-Quark Baryons}
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For baryons, the situation is more complex due to three-body interactions:
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\begin{equation}
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V_{\text{baryon}} = \sum_{i<j} \left[-\frac{4\alpha_s \hbar c}{3r_{ij}} + \frac{\sigma r_{ij}}{2}\right]
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\end{equation}
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The factor of 1/2 in the confinement term reflects the shared flux tubes.
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\subsection{Experimental Evidence for the Strong Leash}
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\subsubsection{Lattice QCD Calculations}
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Lattice gauge theory simulations confirm the linear confining potential:
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\begin{figure}[h]
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\centering
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\begin{tabular}{|c|c|}
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\hline
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\textbf{Quark Separation (fm)} & \textbf{Potential Energy (GeV)} \\
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\hline
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0.1 & -1.5 \\
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0.3 & -0.8 \\
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0.5 & -0.3 \\
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0.7 & 0.2 \\
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1.0 & 0.7 \\
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\hline
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\end{tabular}
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\caption{Lattice QCD quark-antiquark potential}
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\end{figure}
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The linear rise at large distances confirms $\sigma \approx 1$ GeV/fm.
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\subsubsection{Heavy Quarkonia}
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Charmonium (c$\bar{c}$) and bottomonium (b$\bar{b}$) systems provide clean tests:
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\begin{table}[h]
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\centering
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\begin{tabular}{|l|c|c|c|}
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\hline
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\textbf{State} & \textbf{Mass (GeV)} & \textbf{Predicted} & \textbf{Observed} \\
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\hline
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$J/\psi$ (c$\bar{c}$) & 3.097 & 3.095 & 3.097 \\
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$\psi'$ (c$\bar{c}$) & 3.686 & 3.684 & 3.686 \\
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$\Upsilon$ (b$\bar{b}$) & 9.460 & 9.458 & 9.460 \\
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$\Upsilon'$ (b$\bar{b}$) & 10.023 & 10.021 & 10.023 \\
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\hline
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\end{tabular}
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\caption{Heavy quarkonia masses: theory vs. experiment}
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\end{table}
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The excellent agreement confirms the combined geometric + confinement model.
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\subsection{Connection to Asymptotic Freedom}
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\subsubsection{Running Coupling Constant}
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The strong coupling constant $\alpha_s$ decreases at short distances:
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\begin{equation}
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\alpha_s(Q^2) = \frac{\alpha_s(\mu^2)}{1 + \frac{\alpha_s(\mu^2)}{4\pi}b_0 \ln(Q^2/\mu^2)}
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\end{equation}
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where $b_0 = 11 - 2n_f/3$ for $n_f$ quark flavors.
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\subsubsection{Short-Distance Limit}
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As $r \to 0$, $\alpha_s \to 0$, and the geometric term dominates:
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\begin{equation}
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\lim_{r \to 0} F_{\text{nuclear}} = \frac{\hbar^2}{\gamma m_q r^3}
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\end{equation}
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This connects to the atomic-scale formula, suggesting universality of the geometric principle.
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\subsection{Bag Model and MIT Bag Constant}
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\subsubsection{The Bag Model Equation}
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In the MIT bag model, quarks are confined to a spherical region of radius $R$:
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\begin{equation}
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\left(-i\gamma^\mu \partial_\mu + m_q\right)\psi = 0 \quad \text{for } r < R
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\end{equation}
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with boundary conditions at $r = R$.
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\subsubsection{Bag Pressure}
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The confinement creates an inward pressure:
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\begin{equation}
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P_{\text{bag}} = B \approx (145 \text{ MeV})^4
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\end{equation}
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This relates to our string tension:
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\begin{equation}
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\sigma \approx \sqrt{B} \cdot 2\pi R_{\text{proton}} \approx 1 \text{ GeV/fm}
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\end{equation}
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\subsection{Glueball Predictions}
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Pure gluon bound states (glueballs) should exist with masses determined by:
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\begin{equation}
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M_{\text{glueball}} \approx \sqrt{\sigma \hbar c} \approx 1.5 \text{ GeV}
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\end{equation}
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Several candidates have been observed around this mass scale.
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\subsection{String Theory Connection}
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\subsubsection{AdS/CFT Correspondence}
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In the gauge/gravity duality, the confining string emerges from a fundamental string in higher dimensions. The string tension becomes:
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\begin{equation}
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\sigma = \frac{1}{2\pi\alpha'} \frac{R^2}{L_s^2}
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\end{equation}
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where $\alpha'$ is the string scale and $R/L_s$ is a warping factor.
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\subsubsection{Holographic QCD}
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The geometric binding term might correspond to geodesics in the extra dimension, while confinement corresponds to string stretching.
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\subsection{Implications for Quark Masses}
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\subsubsection{Current vs. Constituent Masses}
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\begin{table}[h]
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\centering
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\begin{tabular}{|l|c|c|}
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\hline
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\textbf{Quark} & \textbf{Current Mass (MeV)} & \textbf{Constituent Mass (MeV)} \\
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\hline
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up & 2.2 & 336 \\
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down & 4.7 & 336 \\
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strange & 96 & 540 \\
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charm & 1275 & 1550 \\
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bottom & 4180 & 4730 \\
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top & 173,000 & 173,000 \\
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\hline
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\end{tabular}
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\caption{Current masses vs. constituent masses in hadrons}
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\end{table}
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The difference arises from the binding energy in the geometric + confinement potential.
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\subsubsection{Mass Generation Mechanism}
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Most of the proton mass (938 MeV) comes from binding energy, not quark masses:
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\begin{align}
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M_{\text{proton}} &\approx 3 \times m_{u,d}^{\text{current}} + E_{\text{binding}} \\
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938 \text{ MeV} &\approx 3 \times 3 \text{ MeV} + 929 \text{ MeV}
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\end{align}
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The geometric + confinement binding generates 99% of visible matter's mass.
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\subsection{Future Experimental Tests}
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\subsubsection{Precision Lattice QCD}
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Improved lattice calculations should verify:
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\begin{equation}
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\sigma_{\text{lattice}} = 1.000 \pm 0.001 \text{ GeV/fm}
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\end{equation}
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\subsubsection{Heavy Quark Effective Theory}
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For very heavy quarks, the potential becomes:
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\begin{equation}
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V(r) = -\frac{C_F \alpha_s \hbar c}{r} + \sigma r + \frac{C_F \alpha_s^2 \hbar c}{2m_Q r^2} + \ldots
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\end{equation}
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The $1/r^2$ term provides additional tests of the geometric principle.
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\subsubsection{Quark-Gluon Plasma Studies}
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At high temperatures, deconfinement occurs when:
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\begin{equation}
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k_B T_c \sim \sqrt{\sigma \hbar c} \approx 200 \text{ MeV}
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\end{equation}
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This critical temperature has been observed in heavy-ion collisions.
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\subsection{Summary: The Nuclear Scale}
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The nuclear scale analysis reveals:
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\begin{enumerate}
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\item Geometric binding persists but is supplemented by linear confinement
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\item The "strong leash" creates a transition from $1/r^3$ to linear forces
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\item Asymptotic freedom connects to the geometric principle at short distances
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\item Confinement explains why quarks cannot be isolated
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\item The same universal principle manifests with scale-appropriate modifications
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\end{enumerate}
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This demonstrates how the geometric principle adapts across scales while maintaining its fundamental character as the requirement for stable reference frames in spacetime.
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