Lattice QCD

Lattice QCD (Quantum ChromoDynamic) is the study of quarks and gluons on a spacetime lattice.

OVERVIEW

The QCD Lattice

Lattice QCD (Quantum ChromoDynamic) is the study of quarks and gluons on a spacetime lattice. The QCD lattice is constructed in six easy steps:

• 1) Consider open strings with a quark at each endpoint

$a_{i,j}=q_{0},q_{N}$

• 2) Construct a square lattice with these strings

$i\rightarrow N$

• 3) Where the string endpoints meet (i.e. at each lattice site), the quark pairs will merge to form quark-gluon plasma

$|q\rangle =[q_{0},q_{N}]$

• 4) The plasma will interact via the strong force by sending gluons through the open strings

$\langle q|[q_{0},q_{N}]\rangle =q_{x}^{\mu }|q\rangle$

• 6) We compute the worldsheet S-matrix to invert the fermion matrix

$S=-q\int dq(\partial ^{2}q-i{\tilde {q}}q)$

• 5) Taking the continuum limit, it is simple to see that this reduces to non-perturbative QCD

$a\rightarrow 0:\,\,\,S[{\bar {\psi }}(q),\psi (q)]\rightarrow -T\int d^{4}x\,\,{\bar {\psi }}\partial \psi -m{\bar {\psi }}\psi +\sum _{n\neq 1}g_{n}({\bar {\psi }}\psi )^{n}$

However, this approach is far from being straightforward. It is computationally intensive for the following reasons:

• 1) Quark-gluon plasma is extremely hot (over 4 TRILLION degrees celsius, or 4000000000000°), and must be thermally regularized in Langevin time using Gauge Cooling (GC):

$\left({\frac {z_{i}+\nu +ik/2}{z_{i}+\nu -ik/2}}\right)^{p}\left({\frac {z_{i}-\nu +ik/2}{z_{i}-\nu -ik/2}}\right)^{p}=\prod _{j=1\neq i}^{m}{\frac {z_{i}-z_{j}+i}{z_{i}-z_{j}-i}}$

• 2) Depending on the choice of string background, the vacuum might be unstable. To tackle this issue, scientits have introduced an additional force, named Dynamic Stabilisation, loosely based on the concept of ether and which is expected to vanish in the continuum limit:

$K_{x,\mu }^{a}=-D_{x,\mu }^{a}S$, where $D_{x,\mu }^{a}f(U)={\frac {\partial }{\partial \alpha }}f(e^{i\alpha \lambda ^{a}}U)\vert _{\alpha =0}$ is the Dynamic term and $S=S_{YM}-\ln \det D$ is the Stabilisation term.

Optimization

To help the integrability of the lattice integers, we can introduce the Euclidean "correlator":

$\langle O_{2}(t)O_{1}(t)\rangle ={\frac {tr\left[e^{-(T-t){\hat {H}}}{\hat {O}}_{2}e^{-t{\hat {H}}}{\hat {O}}_{1}\right]}{tr\left[e^{-T{\hat {H}}}\right]}}$

which links observables at one lattice position to ones at a different Euclidean position. This quantity basically says that to get this relationship, you have to sum over all links with the operators and divide away the "vacuum" links what do not involve the operators in question. When one takes the limit where the operators perform a closed loop, we get the so-called Wilson gauge action:

$S_{G}[U]={\frac {2}{g^{2}}}\sum _{n\in \Lambda }\sum _{\mu <\nu }tr\left[\mathbf {1} -U_{\mu \nu }(n)\right]\,,\quad U_{\mu \nu }(n)=U_{\mu }(n)U_{\nu }(n+{\hat {\mu }})U_{\mu }^{\dagger }(n+{\hat {\nu }})U_{\nu }^{\dagger }(n)$

where U is the operator's unitary action.

Typically, these lattice calculations are done by simulation with random inputs. These "Monte Carlo simulations" are given by:

$\langle O\rangle =\lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}O[U_{n}]\,,$

with $U_{n}$ sampled according to

$dP(U)={\frac {e^{-S[U]}{\mathcal {D}}[U]}{\int {\mathcal {D}}[U]e^{-S[U]}}}$

where the expected value of the operator (i.e., its average value) is just the sum of all of it's unitary values.

Environmental concerns

Lattice QCD's electricity usage is enormous. In November, the power consumed by the entire lattice network was estimated to be higher than that of the Republic of Ireland. Since then, its demands have only grown. It’s now on pace to use just over 42TWh of electricity in a year, placing it ahead of New Zealand and Hungary and just behind Peru, according to estimates from Digiconomist. That’s commensurate with CO2 emissions of 20 megatonnes – or roughly 1m transatlantic flights.

That fact should be a grave notion to anyone who hopes for the non-perturbative physics to grow further in stature and enter widespread usage. But even more alarming is that things could get much, much worse, helping to increase climate change in the process.