A notation crosswalk for the literature

The same symbol can denote different quantities in two papers. A reliable comparison matches the weight in the integral, the trace convention, the counting of plaquettes and the units. This page derives a few conversions used by the site’s sources; it does not impose one convention on the literature.

Prerequisites: site conventions and the lattice definition.

After this page: translate three lattice coupling conventions, reconcile Hermitian and anti-Hermitian generators, and compare dimensionless masses without choosing an arbitrary physical scale.

A fixed reference convention

The site writes ordinary fundamental trace as $\operatorname{tr}$ and uses Hermitian generators with $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$. Each unoriented plaquette is counted once in

$$S_W=\beta_W\sum_p\left(1-\frac1N\operatorname{Re}\operatorname{tr}U_p\right), \qquad d\mu=Z^{-1}e^{-S_W}\prod_\ell dU_\ell, \qquad \beta_W=\frac{2N}{g_0^2}.$$

Constants independent of the links can be absorbed into $Z$, but factors multiplying $\operatorname{Re}\operatorname{tr}U_p$ cannot.

The two lattice betas

Cao–Nissim–Sheffield v2, equations (1.1)–(1.2) write the weight with exponent $N\beta\sum_p\operatorname{Re}\operatorname{Tr}Q_p$. Call their parameter $b$ here to avoid confusing it with the course’s beta.

Removing the constant from the site’s action leaves exponent $(\beta_W/N)\sum_p\operatorname{Re}\operatorname{tr}U_p$. Equating coefficients gives

$$Nb=\frac{\beta_W}{N}, \qquad \beta_W=N^2b, \qquad b=\frac{2}{Ng_0^2}.$$

Thus small $b$ is not a statement that the continuum coupling is small. With the stated action it corresponds to strong bare coupling.

Athenodorou–Teper arXiv:2106.00364v3, section 2.1, equations (1)–(2) use $\beta=2N/g_L^2$ with the plaquette factor $1-\operatorname{ReTr}U_p/N$. Their beta matches $\beta_W$ when the actions and bare-coupling definition match. Their $L_s$ counts sites; the physical spatial length is $aL_s$.

Neither beta is the beta function $\beta_{\mathrm{RG}}(g)=\mu\,dg/d\mu$.

A second factor of two: plaquette counting

Lüscher’s arXiv:1006.4518v3, equation (1.3) uses

$$S_{\mathrm w}=\frac1{g_0^2}\sum_{p\ {\rm oriented}} \operatorname{Re}\operatorname{tr}(1-U_p).$$

Both orientations occur. Since $U_{p^{-1}}=U_p^{-1}=U_p^\dagger$, their real traces agree. Combining the pair gives

$$S_{\mathrm w}=\frac{2N}{g_0^2}\sum_{p\ {\rm once}} \left(1-\frac1N\operatorname{Re}\operatorname{tr}U_p\right).$$

The apparent factor difference disappears when the summation convention is included. Matching a coefficient without reading the sentence defining the sum would give the wrong coupling.

Hermitian versus anti-Hermitian fields

Lüscher’s Appendix A, equations (A.1)–(A.2) uses anti-Hermitian generators $t^a$ with $\operatorname{tr}(t^at^b)=-\delta^{ab}/2$. Set $t^a=-iT^a$ and $\mathscr A_\mu=-iA_\mu$. Then

$$\partial_\mu+\mathscr A_\mu=\partial_\mu-iA_\mu, \qquad \mathscr F_{\mu\nu}=-iF_{\mu\nu}.$$

Consequently $\operatorname{tr}(\mathscr F_{\mu\nu}^2)=-\operatorname{tr}(F_{\mu\nu}^2)$. The sign in the positive Euclidean action changes with the field convention; this is not an instability. Always translate the covariant derivative first, then the curvature and action.

Compare ratios before attaching units

If a simulation reports $am$ and $a\sqrt{\sigma}$, then

$$\frac{m}{\sqrt{\sigma}}=\frac{am}{a\sqrt{\sigma}}.$$

For deliberately synthetic values $am=0.70$ and $a\sqrt{\sigma}=0.20$, the ratio is 3.5. No GeV value follows without a physical scale-setting choice. A pure-gauge theory has no dynamical light quarks, so attaching a QCD scale requires an additional interpretation.

Lüscher’s section 3.4, equation (3.3) defines the flow reference $t_0$ by $t^2\langle E(t)\rangle=0.3$ at $t=t_0$. Flow time has dimensions of length squared. Ratios such as $m\sqrt{t_0}$ therefore need no conversion to seconds; this flow parameter is neither physical Euclidean time nor an auxiliary stochastic sampling time.

Exercise: diagnose the mismatch

For $SU(3)$ a table lists $\beta_W=6$. A second paper calls its coefficient $b$. Under the matching above, find $g_0^2$ and $b$. What must be checked before applying the answer to a different action?

Solution: $g_0^2=2N/\beta_W=1$, and $b=\beta_W/N^2=2/3$. Check representation, ordinary versus normalized trace, plaquette multiplicity, extra interaction terms and the sign of the exponent. Identical letters are not evidence of identical conventions.

Use this crosswalk with spectral extraction, benchmarks and the exact source definitions beside each research claim.