7  Lattice QCD

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7.1 The Lattice QCD Action

The continuum Euclidean action of QCD is \[ S = \int d^4x\left[\frac{1}{2g^2}\operatorname{tr}F_{\mu\nu}^2 + \sum_f\bar\psi_f(\not{D}+m_f)\psi_f\right], \] where \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - i[A_\mu,A_\nu]\) is the \(SU(3)\) field-strength tensor and \(\not{D} = \gamma_\mu D_\mu\) is the covariant Dirac operator.

The lattice discretization is obtained by:

• Replacing \(\frac{1}{2g^2}\operatorname{tr}F_{\mu\nu}^2\) with the Wilson/plaquette action from L05.
• Replacing the continuum Dirac operator \(\not\partial + m_f\) with a Wilson-Dirac operator coupled to the gauge field, i.e. promoting all finite differences to gauge-covariant differences using the link variables \(U_\mu(x)\).

The lattice QCD action is \[ S_{\mathrm{QCD}} = \frac{1}{g^2}\sum_{x,\mu<\nu}\operatorname{Re\,tr}\bigl\{1 - W_{\mu\nu}(x)\bigr\} + \sum_f\sum_x a^4\,\bar\psi_f(x)\,\not{D}_f\psi_f(x), \] where the Wilson-Dirac operator coupled to the gauge field is \[ \not{D}_f = \not{D}^s + m_f - \frac{ar}{2}\sum_\mu D_\mu^b D_\mu^f, \] with the gauge-covariant forward, backward, and symmetric differences \[ D_\mu^f\psi(x) = \frac{1}{a}\bigl\{U_\mu(x)\,\psi(x+ae_\mu) - \psi(x)\bigr\}, \] \[ D_\mu^b\psi(x) = \frac{1}{a}\bigl\{\psi(x) - U_\mu^\dagger(x-ae_\mu)\,\psi(x-ae_\mu)\bigr\}, \] \[ \not{D}^s = \frac{1}{2}\sum_\mu\gamma_\mu(D_\mu^f + D_\mu^b). \] The term \(-\frac{ar}{2}\sum_\mu D_\mu^b D_\mu^f\) is the gauge-covariant Wilson term that removes the doublers.

7.2 Path Integral and Partition Function

The path-integral measure factorises into a gauge part and a fermionic part: \[ [dU]\,[d\bar\psi\,d\psi] = \left[\prod_{x,\mu} dU_\mu(x)\right]\left[\prod_{x,f,\alpha,i} d\bar\psi_{f\alpha i}(x)\,d\psi_{f\alpha i}(x)\right], \] where \(dU_\mu(x)\) is the Haar measure on \(SU(3)\) and the fermion variables are Grassmann-valued.

Integrating out the fermions analytically (the fermion integral is Gaussian): \[ Z = \int[dU]\,[d\bar\psi\,d\psi]\,e^{-S_{\mathrm{QCD}}} = \int[dU]\,e^{-S_{YM}(U)}\,\prod_f\det\bigl(a^4\not{D}^f(U)\bigr), \] where \(S_{YM}(U) = \frac{1}{g^2}\sum_{x,\mu<\nu}\operatorname{Re\,tr}\{1-W_{\mu\nu}(x)\}\) is the pure-gauge (Yang–Mills) action. The fermion determinant \(\det(a^4\not{D}^f(U))\) encodes the effect of quark loops on the gauge field.

7.3 Lattice QCD in Practice: Pure-Gauge Observables

7.3.1 Case (I): Observables that do not depend on fermion fields

The most common examples are:

• Average plaquette: \(A(U) = \dfrac{1}{N^4}\displaystyle\sum_{x,\mu<\nu}W_{\mu\nu}(x)\).
• Wilson loop (used to extract the static quark–antiquark potential): \[ A(U) = \operatorname{tr}\!\left(\prod_{\text{path }R\times T} U_\mu\right). \]

Their expectation values take the form \[ \langle A(U)\rangle = \frac{1}{Z}\int[dU]\,e^{-S_{YM}(U)}\left[\prod_f\det\bigl(a^4\not{D}^f(U)\bigr)\right]A(U). \]

Positivity of the measure. The Wilson-Dirac operator satisfies \(\gamma_5\)-hermiticity: \[ \gamma_5\,\not{D}\,\gamma_5 = \not{D}^\dagger \;\Rightarrow\; \det\not{D} = \det(\gamma_5\not{D}\gamma_5) = \det\not{D}^\dagger = (\det\not{D})^*, \] so \(\det\not{D}\) is real. In the simplified setup with only degenerate up and down quarks (\(m_u=m_d=m\)), one has \(\not{D}^u = \not{D}^d \equiv \not{D}\), and \[ \prod_f\det(a^4\not{D}^f) = \bigl[\det(a^4\not{D})\bigr]^2 \geq 0. \] The integrand is therefore non-negative, and the effective measure \[ p(U) = \frac{1}{Z}\,e^{-S_{YM}(U)}\,\bigl[\det(a^4\not{D}(U))\bigr]^2 \] is a genuine probability distribution on the space of gauge configurations, satisfying \(p(U)\geq 0\) and \(\int[dU]\,p(U)=1\). The expectation value becomes \[ \langle A(U)\rangle = \int[dU]\,p(U)\,A(U). \tag{$*$} \]

7.3.2 Monte Carlo evaluation

The integral \((*)\) is computed with Monte Carlo methods: one designs a stochastic process (a Markov chain) that generates a sequence of gauge configurations \((U_n)_{n=1,2,\ldots}\) distributed according to \(p(U)\). By the law of large numbers, \[ \langle A(U)\rangle = \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N A(U_n). \]

7.4 Lattice QCD in Practice: Observables Containing Quarks

7.4.1 Case (II): Observables involving quark fields

Consider the charged pion. Its interpolating operator is \(\bar u\,\gamma_5\,d(x)\), and the corresponding 2-pt function is \[ \langle\bar u\,\gamma_5\,d(x)\;\bar d\,\gamma_5\,u(0)\rangle. \]

Inserting the full path integral and integrating out the fermions analytically (the fermion integral is Gaussian, always yielding Wick contractions), one finds that each contraction \(\psi(x)\bar\psi(y)\) is replaced by the quark propagator \(\not{D}^{-1}(x,y)\): \[ \langle\bar u\,\gamma_5\,d(x)\;\bar d\,\gamma_5\,u(0)\rangle = -\frac{1}{Z}\int[dU]\,e^{-S_{YM}}\bigl[\det a^4\not{D}(U)\bigr]^2\; \operatorname{tr}_{\text{spin,color}}\!\bigl\{\not{D}^{-1}(U)(x,0)\,\gamma_5\,\not{D}^{-1}(U)(0,x)\,\gamma_5\bigr\}. \]

This can be written compactly as \[ \langle\bar u\,\gamma_5\,d(x)\;\bar d\,\gamma_5\,u(0)\rangle = -\bigl\langle\operatorname{tr}_{\text{spin,color}}\!\bigl\{\not{D}^{-1}(U)(x,0)\,\gamma_5\,\not{D}^{-1}(U)(0,x)\,\gamma_5\bigr\}\bigr\rangle, \] where the right-hand side is the expectation value of a function of \(U\) alone — reducing it to Case (I): \[ \langle\bar u\,\gamma_5\,d(x)\;\bar d\,\gamma_5\,u(0)\rangle = \int[dU]\,p(U)\,A(U), \] with \[ A(U) = -\operatorname{tr}_{\text{spin,color}}\!\bigl\{\not{D}^{-1}(U)(x,0)\,\gamma_5\,\not{D}^{-1}(U)(0,x)\,\gamma_5\bigr\}. \] Again, the integral is evaluated by Monte Carlo: \[ \langle\bar u\,\gamma_5\,d(x)\;\bar d\,\gamma_5\,u(0)\rangle = \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N A(U_n). \] On each gauge configuration \(U_n\) one must numerically invert the Dirac matrix \(\not{D}(U_n)\) — this is the dominant cost of a lattice QCD calculation.

7.5 The Continuum Limit in QCD

Working in two-flavour QCD (\(m_u=m_d\equiv m\)) for simplicity, all observables depend on the three parameters \(a\), \(g(a)\), and \(\hat{m}(a) = m/a\) (the dimensionless bare quark mass): \[ \mathrm{Obs}\bigl(a,g(a),\hat{m}(a)\bigr). \]

Dimensional analysis. Hadronic masses have dimensions \([\mathrm{Obs}]=\mathrm{Mass}\), so \[ \mathrm{Obs}\bigl(a,g(a),\hat{m}(a)\bigr) = a^{-1}\,\hat{O}\bigl(g(a),\hat{m}(a)\bigr). \] The lattice predicts only dimensionless ratios; two input conditions on the pair \((g(a),\hat{m}(a))\) are needed to fix the physical scale and quark mass.

Statement 3 (continuum limit). There exist functions \(g(a)\) and \(\hat{m}(a)\) such that, as \(a\to 0\): \[ \lim_{a\to 0} M_X\bigl(a,g(a),\hat{m}(a)\bigr) = M_X^{\mathrm{cont}} \quad\text{finite and } \neq 0 \quad \text{for all hadrons }X, \] \[ \lim_{a\to 0}\sigma\bigl(a,g(a),\hat{m}(a)\bigr) = \sigma^{\mathrm{cont}}, \] \[ \lim_{a\to 0} F\bigl(R\,|\,a,g(a),\hat{m}(a)\bigr) = F^{\mathrm{cont}}(R) \quad\text{for every }R>0. \]

Remarks:

(a) The same functions \(g(a)\) and \(\hat{m}(a)\) work for all observables simultaneously — there is a single continuum limit trajectory in the \((g,\hat{m})\) parameter space.

(b) The functions \(g(a)\) and \(\hat{m}(a)\) are not unique: different choices of the two input observables used to fix the trajectory give different but equivalent parametrisations.

(c) Individual correlators \(C_X(x\,|\,a,g(a),\hat{m}(a))\to+\infty\) as \(a\to 0\): local operators require additional (multiplicative) renormalization before a finite continuum limit exists.

In practice, the two conditions are imposed by requiring, for example, that \(M_\pi/M_\rho\) and \(aM_\rho\) take their physical values at each \(a\). This traces a curve in the \((g,\hat{m})\) plane that approaches the continuum fixed point as \(a\to 0\).