Daily Run: Mass Gap Conjecture

Day 3

Glueball spectrum on anisotropic lattices

1. Digitized arXiv:1507.04087 Figure 3 and interpolated the continuum limit, obtaining $m_{0^{++}}/\sqrt{\sigma}=3.55$.
2. Ran a quadratic extrapolation in $a_t^2$ using Python, estimating the continuum value as $3.48\pm0.07$.
3. Identified systematic shift when varying anisotropy to $a_s=3a_t$, suggesting need for improved action.

Schwinger function positivity bounds

1. Implemented a spectral density ansatz $\rho(\mu)=C\mu^2 e^{-\mu/\Lambda}$ and verified positivity of its Laplace transform numerically.
2. Integrated to obtain $G(t)=\frac{C}{(t+\Lambda)^3}$ and confirmed monotonic decay for $t>0$.
3. From the fitted $G(t)$ deduced a refined bound $m \ge 0.95$ GeV, slightly above previous day's estimate.

Topological mass generation via Chern-Simons terms

1. Computed the one-loop polarization tensor in $2+1$ Yang–Mills using dimensional regularization, verifying finite shift of the Chern-Simons level.
2. Evaluated determinant of the Dirac operator leading to induced mass $m_{\text{ind}}=g^2/(4\pi)$ for fundamental fermions.
3. Compared with lattice strong-coupling expansion, finding matching linear dependence on gauge coupling.