• Lattice regularisation is only known UV regulator which can probe non perturbative regime to provide theoretical predictions.
• The idea is to do Wick rotation (\(t\to -i\tau\)) and work in euclidean discrete 4d-space(\(\Lambda=\{n\}=\{n_1,n_2,n_3,n_4\}\)), with lattice spacing 'a'.
• Then we construct the euclidean lattice QCD action which in continuum limit with analytic continuation to Minkowski space describes QCD.
• The wick rotation implies \(x_0\to-i x_4,\implies \partial_0\to i \partial_4, \implies A_0\to iA_4,\) and the Clifford algebra in Euclidean space time becomes \(\{\gamma_{\mu},\gamma_{\nu}\}=2\delta_{\mu\nu}\implies \gamma_0\to \gamma_4, \gamma^M_i\to-i\gamma^E_i,(i=1,2,3)\)

• The Euclidean space continuum QCD action becomes: \[ S^E_{QCD}=\int d^4x\left(\overline{\psi_{f}}(x)(\gamma_{\mu}D_{\mu}+m)\psi_f(x)+\frac{1}{4}F^{\mu\nu a}(x)F^a_{\mu\nu}(x)\right) \] \(S_{QCD}[\overline{\psi}(x),\psi(x),A(x)]=S_F[\overline{\psi}(x),\psi(x),A(x)] + S_G[A(x)]\)
• The gauge invariant action \(S_F\) becomes \[ \begin{align*} S_F[\psi,\overline{\psi},A] &=\sum_{n\in \Lambda}\overline{\psi}(n)\left(\sum^{4}_{\mu=1} \gamma_{\mu} \frac{U(n,n+\hat{\mu})\psi(n+\hat{\mu})-U(n,n-\hat{\mu})\psi(n-\hat{\mu})}{2a} + m\psi(n) \right) \\ &=\sum_{n\in \Lambda}\overline{\psi}(n)\left(\sum^{4}_{\mu=1} \gamma_{\mu} \frac{U_{\mu}(n)\psi(n+\hat{\mu})-U_{-\mu}(n)\psi(n-\hat{\mu})}{2a} + m\psi(n) \right) \\ \end{align*} \]
• \(U_{\mu}(n)\) are gauge links which connect the fermion fields \(\overline{\psi}(n) \text{ with }\psi(n+\hat{\mu})\). It is self evident that \(U_{-\mu}(n)=U^{\dagger}_{\mu}(n-\hat{\mu})\).
• It might seem that \(U(n,n+\hat{\mu})=U_{\mu}(n)\) since they transform in same way(\(U_{\mu}(n)\to g(n)U_{\mu}(n)g^{\dagger}(n+\hat{\mu})\)), but the orientation of the two is opposite.
• This free part has the famous fermion doubling problem which will be corrected by adding Wilson term.

• Purely dependent on gauge links, gauge invariant lattice action can be constructed out of plaquette objects (\(U_{\mu\nu}\)) which reduces to continuum gauge action in naive continumm limit (\(a\to0\)). \[ \begin{align*} U_{\mu\nu}(n) &=U_{\mu}(n)U_{\nu}(n+a\hat{\mu})U_{-\mu}(n+a\hat{\mu}+a\hat{\nu})U_{-\hat{\nu}}(n+a\hat{\nu}) \\ &=U_{\mu}(n)U_{\nu}(n+a\hat{\mu})U_{\mu}^{\dagger}(n+a\hat{\nu})U_{\hat{\nu}}^{\dagger}(n+a\hat{\nu})\\ \end{align*} \]
• Note that trace of such closed path is invariant under gauge transformation (\(U_{\mu}(n)\to g(n)U_{\mu}(n)g^{\dagger}(n+a\hat{\mu})\)).
• The Wilson gluon action is given by sum of trace of all plaquettes counted with only one orientation. It turns out that contribution is real part of (\(\mathcal{1}-U_{\mu\nu}(n)\)) at each site 'n' and each lorentz index \(\mu\). \[ S_G[U]=2\sum_{n\in\Lambda}\sum_{\mu\le\nu}Re Tr [\mathcal{1}-U_{\mu\nu}(n)]=\frac{a^4}{2}\sum_{n\in\Lambda}\sum_{\mu,\nu}Tr[F_{\mu\nu}(n)^2] \]

• The naive discretization of \(S_F\) suffers from lattice artifacts so called doublers which can be seen in quark propagator (\(D^{-1}(n|m)\) or in \(\widetilde{D}(P)^{-1})\). If we write the \(S_F\) as \[ \begin{align*} S_F[\psi, \bar{\psi}, U] &=a^4 \sum_{n, m \in \Lambda} \sum_{a, b, \alpha, \beta} \bar{\psi}(n)_{\substack{\alpha\\ a}} D(n|m)_{\substack{\alpha \beta \\ a b}} \psi(m)_{\substack{\beta\\ b}} \\ \implies D(n|m)_{\substack{\alpha \beta \\ a b}} &=\sum_{\mu=1}^4\left(\gamma_\mu\right)_{\alpha \beta} \frac{U_\mu(n)_{a b} \delta_{n+\hat{\mu}, m}-U_{-\mu}(n)_{a b} \delta_{n-\hat{\mu}, m}}{2 a}+m \delta_{\alpha \beta} \delta_{a b} \delta_{n, m} \end{align*} \]
• For trivial gauge fields \(U_{\mu}=\mathcal{1}\), the quark propagator in momentum space becomes, \[ \widetilde{D}(p)^{-1}=\frac{m \mathcal{1}-\mathrm{i} a^{-1} \sum_\mu \gamma_\mu \sin \left(p_\mu a\right)}{m^2+a^{-2} \sum_\mu \sin \left(p_\mu a\right)^2} \] Which in massless and continuum case gives only one pole (which is expected) but on lattice the denominator is zero for \(p_{\mu}=n\pi/a;n\in \mathbb{Z}\), resulting in 15 unwanted poles(doublers) in \((-\pi/a,\pi/a]\) at \(p=(\pi / a, 0,0,0),(0, \pi / a, 0,0), \ldots,(\pi / a, \pi / a, \pi / a, \pi / a)\).

• To get rid of doublers Wilson proposed the term (\(-a/2\sum_{n}\overline{\psi}(n)\partial^2\psi(n)\)) which vanishes in naive continuum limit (\(a\to0\)), the massless quark propagator in momentum space becomes: \[ \frac{\mathcal{1}/a \sum_{\mu=1}^{4} (1-\cos{(p_{\mu}a)})}{\mathcal{1}/a^2 \left( \sum_{\mu}\sin^2{(p_{\mu}a/2)}\right)^2+1/a^2 \sum_{\mu}\sin^2{(p_{\mu}a)}} \]
• The above propagator is zero only for \(p_{\mu}=\frac{2n\pi}{a}\) which corresponds to just one pole in the first Brillouin Zone \((\frac{-\pi}{a},\frac{\pi}{a}]\).
• Wilson term explicitly breaks chiral symmetry since (\(\{\gamma_{5},\partial^2\}\neq0\)). We have another discretised fermions called as staggered fermions which preserves chiral symmetry but have 3 doublers.

• We got 16 poles in Wilson fermion action which can be reduced to 4 poles if decrease the width of first Brillouin zone by doubling the lattice spacing (\(a\to2a\)).
• This can be achieved by putting a different flavour of quark at each site of hypercube and repeat the pattern to whole lattice.
• This can be achieved by staggered transformations of fermions \(\psi(n)=T(n)\chi(n)\), \(\overline{\psi}(n)=\overline{\chi}(n)T^{\dagger}(n)\) which makes the lattice fermion action as : \[ S_{F}=\frac{1}{2}\sum_{n,\mu}\left[\overline{\chi}(n)\underline{T^{\dagger}(n)\gamma_{\mu}T(n+\hat{\mu}})\chi(n+\hat{\mu})- \overline{\chi}(n)\underline{T^{\dagger}(n)\gamma_{\mu}T(n-\hat{\mu})}\chi(n-\hat{\mu})\right] + M\overline{\chi}(n)\chi(n) \]

• If we diagonalise the \(\gamma_{\mu}\) matrices such that \(T^{\dagger}(n)\gamma_{\mu}T(n\pm\hat{\mu})=\eta_{\mu}(n)\mathcal{1}\), where the phase (\(\eta_{\mu}=(-1)^{n_1+n_2+..+n_{\mu=1}},\eta_{1}(n)=1\)
• The staggered fermion action becomes, \[ S_{F}=\frac{1}{2}\sum_{n,\mu}\overline{\chi}_{\alpha}(n)\eta_{\mu}(n)\delta_{\alpha,\beta}\left[\chi_{\beta}(n+\hat{\mu})-\chi_{\beta}(n-\hat{\mu})\right] + M\overline{\chi}_{\alpha}(n)\chi_{\alpha}(n) \]
• The Dirac indices (\(\alpha,\beta\)) drops out as all 4 parts contribute same to action. The staggered fermion action thus becomes \[ S_{F}=\frac{1}{2}\sum_{n,\mu}\overline{\chi}(n)\eta_{\mu}(n)\left[\chi(n+\hat{\mu})-\chi(n-\hat{\mu})\right] + M\overline{\chi}(n)\chi(n) \]
• This action still has 4 poles which now are called as taste degree of freedom, and these taste flavours are purely unphysical and only exists on the lattice.

• Lattice QCD in naive continuum limit goes to continuum QCD action. \[ S[\overline{\psi}(n),\psi(n),U(n)]=S_{QCD}[\overline{\psi},\psi,A]+o(a)+o(a^2)+o(a^3)..... \]
• To decrease the lattice artifacts we can use different definitions of \(\partial_{\mu}\psi(x)\) which is also known as Symanzik improvements.
• Such improvements are done to Staggered quark action which is known as HISQ (Highly improved Staggered Quark) action.
• HISQ action has \(\mathcal{O}(a^2)\) improvements with suppresed taste symmetry violations.
• HISQ action is calculated from staggered fermion action by three step process. Each gauge link is replaced to \[ U \to X=\mathcal{F}_{2} \mathcal{U} \mathcal{F}_{1} U \] where
  • \(\mathcal{F}_1 U = V\)= Smearing level 1: 7 fat link smearing .
  • \(\mathcal{U} V = W\)= projection on SU(3)
• \(\mathcal{F}_2 W = X\)= Smearing level 2: Asq

• Lattice provides natural boundary conditions for simulation of finite temperature field theory. The euclidean time \(\tau\) is restricted to \(\beta\) i.e. \(\tau=\beta=\frac{1}{T}=a_t N_t\).
• Each prediction from lattice QCD is a dimensionless number (mass, potential..). To make predictions compatible with experimental dimensionfull quantities we find the lattice spacing 'a' which is also known as setting the scale.
• Lattice will provide mass and potential as \(\hat{M},\hat{V}\), we need to first find the lattice spacing either from the experimental hadron masses or from Sommer parameter \(r_0\) (which is calculated from \(r^2_0 \frac{\partial V}{\partial r} |_{r=r_0=0.4762fm}=1.65\). which is dimensionless in physical units as well as on lattice).
• To find 'a', we use \(\hat{M}/a=m_{experimental}\) or we can calculate the distance on lattice at which \(\hat{r}^2 \frac{\partial V}{\partial \hat{r}}=1.65\) and use \(a\times\hat{r}=0.4762\) fm .
• We can also use similar method such that \(r^2_1\frac{\partial V}{\partial r} |_{r=r_1=0.3157(53)}=1\), find such distance(\(\hat{r_1}\)) on lattice and use \(a\times\hat{r_1}=0.3157\) fm to find lattice spacing 'a'.

• There is another way to calculate value of lattice spacing 'a' which is from inverse gauge coupling (\(\beta\)), we have used that method. With increase in (\(\beta\)), g decreases, which is achieved at high energy (high momenta \(\mathcal{O}(1/a)\)) which implies that the lattice cut off 'a' decreases.
• To study the continnum limit implies that we simply study the system at large \(\beta \to \infty\). Here \(a\to0\) implies that lattice volume \(\to0\).
• So the true continuum limit is studied by first taking lattice sites \(N\to\infty\) and then to take the limit \(\beta\to\infty\).
• \(N\to\infty\) is not possible due to computational challenges, so we measure the quantities at such values of \(\beta,N\) for which volume is finite or fixed and then extrapolate the quantity measured at different \(\beta\) values to the case where \(\beta\to\infty\) and call that the prediction in continuum limit.