some changes

wiki/vimwiki_html/Research_work/Continuum QCD at Zero temperature.html

@@ -54,10 +54,6 @@
 <li>
 QCD is \(SU(3)_c\) non-Abelian, gauge invariant model for strong interactions written in framework of Quantum field theory.
 
-</ul>
-<pre><code>
-</pre></code>
-<ul>
 <li>
 The construction of Lagrangian density of non-abelian gauge theories is very similar to that of abelian U(1) case.
 
@@ -67,32 +63,31 @@
 <div id="Continuum QCD at Zero temperature-Lorentz and local $SU(3)_C$ invariant $\mathcal{L}$"><h2 id="Lorentz and local $SU(3)_C$ invariant $\mathcal{L}$" class="header"><a href="#Continuum QCD at Zero temperature-Lorentz and local $SU(3)_C$ invariant $\mathcal{L}$">Lorentz and local \(SU(3)_C\) invariant \(\mathcal{L}\)</a></h2></div>
 <ul>
 <li>
-Define a comparator \(U(y,x)\) with gauge transformation property \(U(y,x)\to g(y)U(y,x)g^{\dagger}(x)\) , where \(g(x)=e^{i \alpha(x)^a t^a}\), such that we can define a covariant derivative \(D_{\mu}\) such that, $\(\eta^{\mu}D_{\mu}\psi(x)=\lim_{\epsilon \to 0}\frac{\psi(x+n\epsilon)-U(x+n\epsilon,x)\psi(x)}{\epsilon}\)$
+Define a comparator \(U(y,x)\) with gauge transformation property \(U(y,x)\to g(y)U(y,x)g^{\dagger}(x)\) , where \(g(x)=e^{i \alpha(x)^a t^a}\), such that we can define a covariant derivative \(D_{\mu}\) such that, 
+
+<li>
+ {{$
+    \eta^{\mu}D_{\mu}\psi(x)=\lim_{\epsilon \to 0}\frac{\psi(x+n\epsilon)-U(x+n\epsilon,x)\psi(x)}{\epsilon}
+    }}$ 
 
-</ul>
-<p>
-\\~\<br />
-</p>
-<ul>
 <li>
-Expanding \(U(x+n\epsilon,x)\) around \(U(x,x)=\mathds{1}\), we get $$U(x+n\epsilon,x)=\mathds{1}+ig\epsilon \eta<sup><small>{\mu}A_{\mu}(x)+\mathcal{O}(\epsilon</small></sup>2)
-        $\(covariant derivative becomes,\)D_{\mu}\psi(x)=(\partial_{\mu}-igA_{\mu}(x))\psi(x)$ 
+Expanding \(U(x+n\epsilon,x)\) around \(U(x,x)=\mathds{1}\), we get
+\[
+U(x+n\epsilon,x)=\mathds{1}+ig\epsilon \eta^{\mu}A_{\mu}(x)+\mathcal{O}(\epsilon^2)
+\]
+    covariant derivative becomes,\(D_{\mu}\psi(x)=(\partial_{\mu}-igA_{\mu}(x))\psi(x)\) 
 
 <li>
 Under gauge transformation \(\psi(x) \to g(x)\psi(x)\), where g(x) is exponential of 3x3 matrix which implies that \(\psi(x)\) is 3 component object and similarly, \(A_{\mu}(x)=A^a_{\mu}(x)t^a\)., where \(t^a\) are generators of SU(3) satisfying \([t^a,t^b]=if_{abc}t^{c}\) and \(Tr(t^a t^b)=\frac{\delta^{ab}}{2}\).
 
 <li>
 Under local SU(3) gauge transformations, these \(3^2-1\) gauge bosons \(A^a_{\mu}(x)\) also transform as \(A_{\mu}(x)\to g(x)(A_{\mu}(x)+\frac{i}{g} \partial_{\mu})g^{\dagger}(x)\), which can be seen from infinitesimal transformation of comparator: \(U(x+n\epsilon,x)\to g(x+n\epsilon)U(x+n\epsilon,x)g^{\dagger}(x)\).
 
-</ul>
-<pre><code>
-</pre></code>
-<ul>
 <li>
-We can now write kinetic term of local SU(3) invariant Lagrangian density with a trivial mass term. \textcolor{blue}{$\(\mathcal{L}_{kinetic}=\overline{\psi}(x)i\gamma^{\mu}D_{\mu}\psi(x)-m\overline{\psi}(x)\psi(x)\)$}
+We can now write kinetic term of local SU(3) invariant Lagrangian density with a trivial mass term. $\(\mathcal{L}_{kinetic}=\overline{\psi}(x)i\gamma^{\mu}D_{\mu}\psi(x)-m\overline{\psi}(x)\psi(x)\)$
 
 <li>
-Together with gauge invariant kinetic term for these 8 gauge bosons \(A^{a}_{\mu}(x)\),  \textcolor{blue}{$\(\mathcal{L}_{QCD}=\overline{\psi_{f}}(x)(i\gamma^{\mu}D_{\mu}-m)\psi_f(x)-\frac{1}{2}Tr[F^{\mu\nu}F_{\mu\nu}]\)\(} where, \)F_{\mu\nu}(x)=\frac{i}{g}[D_{\mu},D_{\nu}]=(\partial_{\mu}A<sup><small>a_{\nu}-\partial_{\nu}A</small></sup>a_{\mu}+gf_{abc}A<sup><small>b_{\mu}A</small></sup>c_{\nu})t<sup><small>a=F</small></sup>a_{\mu\nu}t^a$. 
+Together with gauge invariant kinetic term for these 8 gauge bosons \(A^{a}_{\mu}(x)\),  $\(\mathcal{L}_{QCD}=\overline{\psi_{f}}(x)(i\gamma^{\mu}D_{\mu}-m)\psi_f(x)-\frac{1}{2}Tr[F^{\mu\nu}F_{\mu\nu}]\)\( where, \)F_{\mu\nu}(x)=\frac{i}{g}[D_{\mu},D_{\nu}]=(\partial_{\mu}A<sup><small>a_{\nu}-\partial_{\nu}A</small></sup>a_{\mu}+gf_{abc}A<sup><small>b_{\mu}A</small></sup>c_{\nu})t<sup><small>a=F</small></sup>a_{\mu\nu}t^a$. 
 
 </ul>
 
@@ -111,18 +106,23 @@
 
 </ul>
 <pre><code>
-\begin{align*}
-\bra{\phi_f(\tau_f)}e<sup><small>{-H(\tau_f-\tau_i)}\ket{\phi_i(\tau_i)} &amp;=\int_{\phi_i(\tau_i)}</small></sup>{\phi_f(\tau_f)}[D\phi]e^{-S_{E}[\phi]} \\
-Z &amp;=Tr[e<sup><small>{-\beta H}]=\int d\phi\bra{\phi}e</small></sup>{-\beta H}\ket{\phi}=\int d\phi \int_{\phi(\tau_i)}<sup><small>{\phi(\tau_i+\beta)=\phi(\tau_i)}[D\phi]e</small></sup>{-S_E[\phi]}
-\end{align*}
 </pre></code>
+\[
+ \begin{align*}
+\bra{\phi_f(\tau_f)}e^{-H(\tau_f-\tau_i)}\ket{\phi_i(\tau_i)} &amp;=\int_{\phi_i(\tau_i)}^{\phi_f(\tau_f)}[D\phi]e^{-S_{E}[\phi]} \\
+Z &amp;=Tr[e^{-\beta H}]=\int d\phi\bra{\phi}e^{-\beta H}\ket{\phi}=\int d\phi \int_{\phi(\tau_i)}^{\phi(\tau_i+\beta)=\phi(\tau_i)}[D\phi]e^{-S_E[\phi]}
+\end{align*}
+\]
 <ul>
 <li>
 Similarly, for QCD: 
-    \begin{align*}
-    \textcolor{blue}{Z=\int d\overline{\psi} d\psi dA \int_{A(\tau_i),\psi(\tau_i)}<sup><small>{A(\tau_i + \beta)=A(\tau_i),\psi(\tau_i+\beta)=-\psi(\tau_i)}[D\overline{\psi}][D\psi][DA]e</small></sup>{-\textcolor{red}{S_E[\overline{\psi},\psi,A]}}}
-    \\\text{where}, \textcolor{red}{S_E[\overline{\psi},\psi,A]=\int_{\tau_i}<sup><small>{\tau_i+\beta}d\tau \int_{-\infty}</small></sup>{\infty}d<sup><small>3x \mathcal{L}</small></sup>{E}_{QCD}[\overline{\psi},\psi,A]}
-    \end{align*}
+\[
+\begin{align*}
+
+Z=\int d\overline{\psi} d\psi dA \int_{A(\tau_i),\psi(\tau_i)}^{A(\tau_i + \beta)=A(\tau_i),\psi(\tau_i+\beta)=-\psi(\tau_i)}[D\overline{\psi}][D\psi][DA]e^{-S_E[\overline{\psi},\psi,A]}
+\text{where}, S_E[\overline{\psi},\psi,A]=\int_{\tau_i}^{\tau_i+\beta}d\tau \int_{-\infty}^{\infty}d^3x \mathcal{L}^{E}_{QCD}[\overline{\psi},\psi,A]
+\end{align*}
+\]
     Because of trace we are restricted to have Bosonic and fermionic fields with periodic and anit-periodic boundary conditions with time period \(\tau=\tau_f-\tau_i=\beta\).
 
 </ul>

wiki/vimwiki_html/Research_work/Lattice QCD at finite temperature.html

@@ -58,7 +58,7 @@
 <a href="Lattice QCD at finite temperature.html#Lattice QCD  (LQCD-overview)">#Lattice QCD  (LQCD-overview)</a>
 
 <li>
-[[#LQCD: Discretised \(S_F[\overline{\psi},\psi,A]\)]]
+[[#LQCD: Discretised \(S_F[\overline{\psi},\psi,A]\) ]]
 
 <li>
 [[#LQCD: Dicretised \(S_{G}[U]\)]]
@@ -73,7 +73,7 @@
 <a href="Lattice QCD at finite temperature.html#LQCD: Staggered Quark action">#LQCD: Staggered Quark action</a>
 
 <li>
-[[#LQCD: \textcolor{red}{Free} Staggered quark action]]
+<a href="Lattice QCD at finite temperature.html#LQCD: Free Staggered quark action">#LQCD: Free Staggered quark action</a>
 
 <li>
 <a href="Lattice QCD at finite temperature.html#Improved actions and HISQ action">#Improved actions and HISQ action</a>
@@ -98,7 +98,7 @@
 </pre></code>
 <ul>
 <li>
-The idea is to do \textcolor{red}{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 '\textcolor{blue}{a}'. 
+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'. 
 
 </ul>
 <pre><code>
@@ -119,16 +119,19 @@
 <ul>
 <li>
 The Euclidean space continuum QCD action becomes:
-    \textcolor{red}{$\(S^E_{QCD}=\int d^4x\left(\overline{\psi_{f}}(x)(\gamma_{\mu}D_{\mu}+m)\psi_f(x)+\frac{1}{4}F^{\mu\nu a}(x)F^a_{\mu\nu}(x)\right)\)\(}, \)S_{QCD}[\overline{\psi}(x),\psi(x),A(x)]=S_F[\overline{\psi}(x),\psi(x),A(x)] + S_G[A(x)]$
+\[
+S^E_{QCD}=\int d^4x\left(\overline{\psi_{f}}(x)(\gamma_{\mu}D_{\mu}+m)\psi_f(x)+\frac{1}{4}F^{\mu\nu a}(x)F^a_{\mu\nu}(x)\right)
+\]
+    \(S_{QCD}[\overline{\psi}(x),\psi(x),A(x)]=S_F[\overline{\psi}(x),\psi(x),A(x)] + S_G[A(x)]\)
 
 </ul>
 <pre><code>
 </pre></code>
 <ul>
 <li>
 The gauge invariant action \(S_F\) becomes \begin{align*}
-<pre><code>S_F[\psi,\overline{\psi},A] &amp;=\sum_{n\in \Lambda}\overline{\psi}(n)\left(\sum^{4}<em>{\mu=1} \gamma</em>{\mu} \frac{\textcolor{red}{U(n,n+\hat{\mu})}\psi(n+\hat{\mu})-\textcolor{blue}{U(n,n-\hat{\mu})}\psi(n-\hat{\mu})}{2a} + m\psi(n) \right)   
-&amp;=\sum_{n\in \Lambda}\overline{\psi}(n)\left(\sum^{4}<em>{\mu=1} \gamma</em>{\mu} \frac{\textcolor{red}{U_{\mu}(n)}\psi(n+\hat{\mu})-\textcolor{blue}{U_{-\mu}(n)}\psi(n-\hat{\mu})}{2a} + m\psi(n) \right) \\
+<pre><code>S_F[\psi,\overline{\psi},A] &amp;=\sum_{n\in \Lambda}\overline{\psi}(n)\left(\sum^{4}<em>{\mu=1} \gamma</em>{\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)   
+&amp;=\sum_{n\in \Lambda}\overline{\psi}(n)\left(\sum^{4}<em>{\mu=1} \gamma</em>{\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.  
@@ -143,10 +146,10 @@
 <div id="Lattice QCD at finite temperature-LQCD: Dicretised $S_{G}[U]$"><h2 id="LQCD: Dicretised $S_{G}[U]$" class="header"><a href="#Lattice QCD at finite temperature-LQCD: Dicretised $S_{G}[U]$">LQCD: Dicretised \(S_{G}[U]\)</a></h2></div>
 <ul>
 <li>
-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 \textcolor{cyan}{naive} continumm limit (\(a\to0\)).
+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) &amp;=U_{\mu}(n)U_{\nu}(n+a\hat{\mu})\textcolor{red}{U_{-\mu}(n+a\hat{\mu}+a\hat{\nu})}\textcolor{blue}{U_{-\hat{\nu}}(n+a\hat{\nu}})\\
-    &amp;=U_{\mu}(n)U_{\nu}(n+a\hat{\mu})\textcolor{red}{U_{\mu}<sup><small>{\dagger}(n+a\hat{\nu})}\textcolor{blue}{U_{\hat{\nu}}</small></sup>{\dagger}(n+a\hat{\nu})}\\
+    U_{\mu\nu}(n) &amp;=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})\\
+    &amp;=U_{\mu}(n)U_{\nu}(n+a\hat{\mu})U_{\mu}<sup><small>{\dagger}(n+a\hat{\nu})U_{\hat{\nu}}</small></sup>{\dagger}(n+a\hat{\nu})\\
     \end{align*}
 
 <li>
@@ -157,7 +160,10 @@
 </pre></code>
 <ul>
 <li>
-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 (\(\mathds{1}-U_{\mu\nu}(n)\)) at each site 'n' and each lorentz index \(\mu\). \textcolor{blue}{$\(S_G[U]=2\sum_{n\in\Lambda}\sum_{\mu\le\nu}Re Tr [\mathds{1}-U_{\mu\nu}(n)]=\frac{a^4}{2}\sum_{n\in\Lambda}\sum_{\mu,\nu}Tr[F_{\mu\nu}(n)^2]\)$}
+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 (\(\mathds{1}-U_{\mu\nu}(n)\)) at each site 'n' and each lorentz index \(\mu\).
+\[
+S_G[U]=2\sum_{n\in\Lambda}\sum_{\mu\le\nu}Re Tr [\mathds{1}-U_{\mu\nu}(n)]=\frac{a^4}{2}\sum_{n\in\Lambda}\sum_{\mu,\nu}Tr[F_{\mu\nu}(n)^2]$
+\]
 
 </ul>
 <div id="Lattice QCD at finite temperature-LQCD: Doublers"><h2 id="LQCD: Doublers" class="header"><a href="#Lattice QCD at finite temperature-LQCD: Doublers">LQCD: Doublers</a></h2></div>
@@ -172,7 +178,10 @@
     \end{align*}
 
 <li>
-For trivial gauge fields \(U_{\mu}=\mathds{1}\), the quark propagator in momentum space becomes, $\(\widetilde{D}(p)^{-1}=\frac{m \mathds{1}-\mathrm{i} a^{-1} \sum_\mu \gamma_\mu \sin \left(p_\mu a\right)}{m^2+a^{-2} \sum_\mu \sin \left(p_\mu a\right)^2}\)$
+For trivial gauge fields \(U_{\mu}=\mathds{1}\), the quark propagator in momentum space becomes, 
+\[
+\widetilde{D}(p)^{-1}=\frac{m \mathds{1}-\mathrm{i} a^{-1} \sum_\mu \gamma_\mu \sin \left(p_\mu a\right)}{m^2+a^{-2} \sum_\mu \sin \left(p_\mu a\right)^2}
+\]
 
 </ul>
 <pre><code>
@@ -183,7 +192,9 @@
 <ul>
 <li>
 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:
-    $\(\frac{\mathds{1}/a \sum_{\mu=1}^{4} (1-\cos{(p_{\mu}a)})}{\mathds{1}/a^2 \left( \sum_{\mu}\sin^2{(p_{\mu}a/2)}\right)^2+1/a^2 \sum_{\mu}\sin^2{(p_{\mu}a)}}\)$
+\[
+\frac{\mathds{1}/a \sum_{\mu=1}^{4} (1-\cos{(p_{\mu}a)})}{\mathds{1}/a^2 \left( \sum_{\mu}\sin^2{(p_{\mu}a/2)}\right)^2+1/a^2 \sum_{\mu}\sin^2{(p_{\mu}a)}}
+\]
 
 <li>
 The above propagator is zero only for \(p_{\mu}=\frac{2n\pi}{a}\) which corresponds to just one pole in the first Brillouin Zone \((\frac{-\pi}{a},\frac{\pi}{a}]\).
@@ -202,19 +213,27 @@
 
 <li>
 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 :
-    $\(S_{F}=\frac{1}{2}\sum_{n,\mu}\left[\overline{\chi}(n)\textcolor{red}{\underline{T^{\dagger}(n)\gamma_{\mu}T(n+\hat{\mu}})}\chi(n+\hat{\mu})- \overline{\chi}(n)\textcolor{blue}{\underline{T^{\dagger}(n)\gamma_{\mu}T(n-\hat{\mu})}}\chi(n-\hat{\mu})\right] + M\overline{\chi}(n)\chi(n)\)$
+\[
+S_{F}=\frac{1}{2}\sum_{n,\mu}\left[\overline{\chi}(n)\underline{T^{\dagger}(n)\gamma_{\mu}T(n+\hat{\mu}})\chi(n+\hat{\mu})- \overline{\chi}(n)\underline{T^{\dagger}(n)\gamma_{\mu}T(n-\hat{\mu})}\chi(n-\hat{\mu})\right] + M\overline{\chi}(n)\chi(n)
+\]
 
 </ul>
-<div id="Lattice QCD at finite temperature-LQCD: \textcolor{red}{Free} Staggered quark action"><h2 id="LQCD: \textcolor{red}{Free} Staggered quark action" class="header"><a href="#Lattice QCD at finite temperature-LQCD: \textcolor{red}{Free} Staggered quark action">LQCD: \textcolor{red}{Free} Staggered quark action</a></h2></div>
+<div id="Lattice QCD at finite temperature-LQCD: Free Staggered quark action"><h2 id="LQCD: Free Staggered quark action" class="header"><a href="#Lattice QCD at finite temperature-LQCD: Free Staggered quark action">LQCD: Free Staggered quark action</a></h2></div>
 <ul>
 <li>
 If we diagonalise the \(\gamma_{\mu}\) matrices such that \(T^{\dagger}(n)\gamma_{\mu}T(n\pm\hat{\mu})=\eta_{\mu}(n)\mathds{1}\), where the phase (\(\eta_{\mu}=(-1)^{n_1+n_2+..+n_{\mu=1}},\eta_{1}(n)=1\)
 
 <li>
-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 staggered fermion action becomes,
+\[
+S_{F}=\frac{1}{2}\sum_{n,\mu}\overline{\chi}_{\alpha}(n)\eta_{\mu}(n)\delta_{\alpha,\beta}\left[\chi_{\beta}(n+\hat{\mu})-\chi_{\beta}(n-\hat{\mu})\right] + M\overline{\chi}_{\alpha}(n)\chi_{\alpha}(n)
+\]
 
 <li>
-The Dirac indices (\(\alpha,\beta\)) drops out as all 4 parts contribute same to action. The staggered fermion action thus becomes \textcolor{blue}{$\(S_{F}=\frac{1}{2}\sum_{n,\mu}\overline{\chi}(n)\eta_{\mu}(n)\left[\chi(n+\hat{\mu})-\chi(n-\hat{\mu})\right] + M\overline{\chi}(n)\chi(n)\)$}
+The Dirac indices (\(\alpha,\beta\)) drops out as all 4 parts contribute same to action. The staggered fermion action thus becomes
+\[
+S_{F}=\frac{1}{2}\sum_{n,\mu}\overline{\chi}(n)\eta_{\mu}(n)\left[\chi(n+\hat{\mu})-\chi(n-\hat{\mu})\right] + M\overline{\chi}(n)\chi(n)
+\]
 
 <li>
 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.
@@ -226,7 +245,10 @@
 <div id="Lattice QCD at finite temperature-Improved actions and HISQ action"><h2 id="Improved actions and HISQ action" class="header"><a href="#Lattice QCD at finite temperature-Improved actions and HISQ action">Improved actions and HISQ action</a></h2></div>
 <ul>
 <li>
-Lattice QCD in naive continuum limit goes to continuum QCD action.$\(S[\overline{\psi}(n),\psi(n),U(n)]=S_{QCD}[\overline{\psi},\psi,A]+o(a)+o(a^2)+o(a^3).....\)$
+Lattice QCD in naive continuum limit goes to continuum QCD action.
+\[
+S[\overline{\psi}(n),\psi(n),U(n)]=S_{QCD}[\overline{\psi},\psi,A]+o(a)+o(a^2)+o(a^3).....
+\]
 
 <li>
 To decrease the lattice artifacts we can use different definitions of \(\partial_{\mu}\psi(x)\) which is also known as Symanzik improvements. 
@@ -239,7 +261,10 @@
 
 <li>
 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
+\[
+ U \to X=\mathcal{F}_{2} \mathcal{U} \mathcal{F}_{1} U
+\]
+	 where
 
 <ul>
 <li>