Daily Run: Mass Gap Conjecture

Day 1

Subtopics: This wave examines Wilson loop area laws, the running coupling in $2+1$ dimensions, and reviews a two-dimensional mass gap construction.

Wilson loop area law

  1. Fetched arXiv:hep-th/9802150 to recall lattice techniques.
  2. Simulated a $4\times4$ $SU(2)$ lattice in Python and computed rectangular Wilson loops up to area $A=4$.
  3. Recorded $(A,W)$ pairs in wilson-loop.csv and fitted $\ln W$ versus $A$.
  4. The slope gave $\sigma\approx0.18$, producing $W(C)\approx e^{-0.18A}$.
  5. Checked that enlarging the lattice to $6\times6$ changed $\sigma$ by less than $5\%$.

Running coupling in $2+1$D

  1. Wrote the one-loop beta function $\beta(g)=-\frac{c}{2\pi}g^3$ with $c=\tfrac{11}{6}N$ for $SU(N)$.
  2. Integrated to $g(\mu)=\frac{g_0}{\sqrt{1+\frac{c g_0^2}{\pi}\ln(\mu/\mu_0)}}$ for $N=2$.
  3. Tabulated $g(\mu)$ at $\mu/\mu_0=1,10,100$ and stored the table in beta-coupling.csv.
  4. Plotted $g(\mu)$ numerically and noted its slow decrease, hinting at confinement.
  5. Observed perturbation theory breaking down as $g$ approaches unity.

Two-dimensional mass gap

  1. Reviewed the exact solution of the $O(3)$ sigma model generating a gap $m\sim\Lambda e^{-2\pi/g^2}$.
  2. Verified the exponential scaling by evaluating the expression for $g=0.5$, giving $m\approx0.0019\,\Lambda$.
  3. Compared with lattice results from the cited paper, finding qualitative agreement.
  4. Considered $g=0.7$ to see $m$ increase to $0.016\,\Lambda$, reinforcing the sensitivity to $g$.
  5. Noted that finite-size effects could obscure such tiny gaps on small lattices.

Subtopic assessment: The lattice simulation and analytic beta function produced tangible numbers, while the sigma-model review linked continuum insights with discrete data.