• #PDF
• [[#Renormalisation of \(\lambda\) :]]
• #(Q) What about higher loops?
• [[#Q) In general what is \(\Lambda\)-sensitivity for \(\Gamma^{(n)}\) ?]]
• #1 P I (Particle irreducible)

Prof. Sachindeo Vaidya (CHEP, IISc) | PDF

Previous: Lecture 7

Next: Lecture 9

Infrared cut off --- means low energy cut off

ultraviolet cut off means high '' '' off.

\[ m^2 \;=\; \mu^2 + \frac{1}{2}\left(\alpha^2\Lambda^2 + \mu^2\ln\frac{\mu^2}{\mu^2+\alpha^2\Lambda^2}\right) \] \[ Z(m,\lambda) \;=\; \int [d\phi]\ e^{-\frac{1}{2}\int d^4x\left((\partial\phi)^2 + m^2\phi^2 + \lambda\phi^4\right)} \] \[ =\; Z_0(m,0) + \lambda Z_1(\quad) + \lambda^2 Z_2(\ ) + \cdots \]

We know,

\[ I(a) \;=\; \int_{-\infty}^{\infty} dx\ e^{-ax^2} \;=\; \sqrt{\frac{\pi}{a}}\ \sim\ a^{-1/2} \]

Branch cut starts at \(a=0\)

\usetikzlibrary{arrows.meta,decorations.pathreplacing,decorations.pathmorphing,calc,angles,quotes,positioning,patterns}
\begin{tikzpicture}[scale=1.0,>=Stealth]
\draw (-2.6,0) -- (1.2,0);
\draw (0,-1.0) -- (0,1.0);
\draw[decorate,decoration={snake,amplitude=1.6pt,segment length=5pt}] (-2.1,0) -- (0,0);
\end{tikzpicture}

\[ I(a,\lambda) \;=\; \int_{-\infty}^{\infty} dx\ e^{-ax^2 - \lambda x^4/4} \] \[ I_{\lambda}(a) \;=\; \int_{-\infty}^{\infty} dx\ e^{-ax^2}\ e^{-\lambda x^4/4!} \] \[ =\; \int_{-\infty}^{\infty} dx\ e^{-ax^2}\left(1 - \frac{\lambda x^4}{4!} + \frac{1}{2!}\,\frac{\lambda x^8}{4!\,4!} - \cdots\right) \] \[ I(a;m) \;=\; \int_{-\infty}^{\infty} dx\ e^{-ax^2}\, x^m \] \[ \int_{-\infty}^{\infty} dx\ x^2 e^{-ax^2} \;=\; -\frac{d}{da}\, I_0(a) \] \[ I_{\lambda}(a) \;=\; I(a,0) + \lambda I(a,1) + \frac{\lambda^2}{2!}\, I(a,2) + \cdots \]

\(\downarrow\) Not analytic at \(\lambda=0\) ; and the successive terms go like \(\sim \dfrac{1}{\sqrt{a}}\) , \(\dfrac{1}{a^{5/2}}\) , \(\dfrac{1}{a^{7/2}}\)

\usetikzlibrary{arrows.meta,decorations.pathreplacing,decorations.pathmorphing,calc,angles,quotes,positioning,patterns}
\begin{tikzpicture}[scale=1.0,>=Stealth]
\draw[->] (0,-1.2) -- (0,1.4) node[above right] {$f(\lambda)$};
\draw (-2.6,0) -- (2.0,0);
\draw[decorate,decoration={snake,amplitude=1.6pt,segment length=5pt}] (-2.2,0) -- (0,0);
\node at (-2.1,-0.5) {$\sim \lambda^{1/3}$};
\node[right,align=left] at (2.6,0.75) {Power series will only work iff\\ $\lambda$ is very small.\\ Asymptotic series works at $\lambda=0$.};
\end{tikzpicture}

HELING (arxiv) \(\downarrow\) To understand renormalisation \& fall in love with it. (could not find)

\[ \Gamma^{(4)}(k_1,k_2,k_3) \;=\; \]

\usetikzlibrary{arrows.meta,decorations.pathreplacing,decorations.pathmorphing,calc,angles,quotes,positioning,patterns}
\begin{tikzpicture}[scale=1.0,>=Stealth]
\draw (-1.0,0.75) -- (0.2,-0.75);
\draw (-1.0,-0.75) -- (0.2,0.75);
\node[scale=0.8] at (-1.25,0.9) {$k_1$};
\node[scale=0.8] at (-1.25,-0.9) {$k_2$};
\node[scale=0.8] at (0.45,0.9) {$k_3$};
\node[scale=0.8] at (0.45,-0.9) {$k_4$};
\node at (1.1,0) {$+$};
\draw (1.8,0.75) .. controls (2.8,0.1) and (3.4,0.1) .. (4.2,0.75);
\draw (1.8,-0.75) .. controls (2.8,-0.1) and (3.4,-0.1) .. (4.2,-0.75);
\draw (2.45,0.32) .. controls (2.85,-0.15) .. (3.35,-0.32);
\draw (2.45,-0.32) .. controls (2.85,0.15) .. (3.35,0.32);
\node[scale=0.8] at (1.55,0.9) {$k_1$};
\node[scale=0.8] at (1.55,-0.9) {$k_2$};
\node[scale=0.8] at (4.45,0.9) {$k_3$};
\node[scale=0.8] at (4.45,-0.9) {$k_4$};
\end{tikzpicture}

\usetikzlibrary{arrows.meta,decorations.pathreplacing,decorations.pathmorphing,calc,angles,quotes,positioning,patterns}
\begin{tikzpicture}[scale=1.0,>=Stealth]
\node at (-2.4,0) {$+$};
\draw (-1.7,0.75) .. controls (-0.9,0.15) and (-0.3,-0.15) .. (0.5,-0.75);
\draw (-1.7,-0.75) .. controls (-0.9,-0.15) and (-0.3,0.15) .. (0.5,0.75);
\draw (-0.85,0.0) ellipse (0.28 and 0.34);
\node[scale=0.8] at (-1.95,0.9) {$k_1$};
\node[scale=0.8] at (-1.95,-0.9) {$k_2$};
\node[scale=0.8] at (0.75,0.9) {$k_3$};
\node[scale=0.8] at (0.75,-0.9) {$k_4$};
\node at (1.5,0) {$+$};
\draw (2.2,0.75) -- (4.3,-0.75);
\draw (2.2,-0.75) -- (4.3,0.75);
\draw (3.05,0.0) ellipse (0.20 and 0.42);
\node[scale=0.8] at (1.95,0.9) {$k_1$};
\node[scale=0.8] at (1.95,-0.9) {$k_2$};
\node[scale=0.8] at (4.55,0.9) {$-k_3$};
\node[scale=0.8] at (4.55,-0.9) {$k_4$};
\end{tikzpicture}

Elliminate \(\lambda\) in favour of \(g \equiv \Gamma^{(4)}(0,0,0)\)

Replace \(\mu^2\) by \(m^2\), \(\alpha^2\) by \(a^2\) ..

\[ \Gamma^{(4)}\left(\{k\}\right) \;=\; \Gamma^{(4)}\left(\{k_i\}\right) - \Gamma^{(4)}\left(\{0\}\right) + g \]

Renormalisation cond\(^{\text{n}}\) :

\[ g \;=\; \lambda - \frac{\lambda^2}{2}\int \frac{d^d\tilde{q}}{\left(a^2q^2+m^2\right)} \qquad \text{(3 terms)} \] \[ \left.\begin{aligned} y &= x - ax^2\\ x &= y - ay^2 \end{aligned}\right\}\qquad (\text{for small } x) \] \[ \Gamma^{(4)}\left(\{k_i\}\right) \;=\; g + \frac{g^2}{2}\left(\int \cdots\right) \]

We choose \((k_1,k_2,k_3) = (0,0,0)\) as point to define `\(g\)'

we could have used other possibilities

\[ g(k_1,k_2,k_3) \;=\; \left(k_1\ k_2\ k_3\right) \]

A common choice --- Symmetric pt. --- all \(k_i\) are equal and have magnitude \(\sqrt{\tfrac{3}{2}}\,k\)

\[ \Gamma^{(2)} \;=\; m^2 + k^2\left(a^2 - \frac{g^2}{6}\,B(k,K)\right) + O(g^3) \] \[ \Gamma^{(4)} \;=\; g + \frac{g^2}{2}\left(\int (\ )\ \cdots\ \right) + O(g^3) \] \[ \lambda \;=\; g\left(1 + \frac{3}{2}\,g\int\frac{d^d\tilde{q}}{\left(\alpha^2q^2+m^2\right)}\right) + O(g^3) \] \[ \mu^2 \;=\; m^2 - \frac{\lambda}{2}\int\frac{d^d\tilde{q}}{\left(\alpha^2q^2+m^2\right)} + \frac{\lambda^2}{6}\,A(0) \]

\(\searrow\) substitute \(g = g(\lambda)\)

What is minimum number of \(\Lambda\)-sensitive integrals?

We have 3 of them.

\[ \begin{aligned} \mu^2 &= \mu^2\left(m^2,\lambda,\Lambda\right)\\ \lambda^2 &= \lambda^2\left(m^2,g,\Lambda\right)\\ \alpha^2 &= \alpha^2\left(m^2,\lambda,\Lambda\right) \end{aligned} \]

This is same as no. of bare parameters in \(L\). So we have one parameter family of free parameters in \(L\).

1 --- parameter family of lagrangians. \(\left(\alpha,\mu,\lambda,\Lambda\right)\)

all of which generates same renormalised theory, given by \(\left(a, m^2, g\right)\).

\usetikzlibrary{arrows.meta,decorations.pathreplacing,decorations.pathmorphing,calc,angles,quotes,positioning,patterns}
\begin{tikzpicture}[scale=0.95,>=Stealth]
\draw (-5.6,0.8) -- (-4.2,-0.8);
\draw (-5.6,-0.8) -- (-4.2,0.8);
\node[align=center,scale=0.9] at (-4.9,-1.5) {$\Gamma^{(n)}$ with\\ 1 vertex};
\draw (-2.9,0.8) .. controls (-2.1,0.1) and (-1.5,0.1) .. (-0.7,0.8);
\draw (-2.9,-0.8) .. controls (-2.1,-0.1) and (-1.5,-0.1) .. (-0.7,-0.8);
\draw (-2.3,0.28) .. controls (-1.8,-0.2) .. (-1.3,-0.28);
\draw (-2.3,-0.28) .. controls (-1.8,0.2) .. (-1.3,0.28);
\node[align=center,scale=0.9] at (-1.8,-1.5) {$\Gamma^{(4)}$ with\\ 2 vertex};
\draw (0.6,0.8) .. controls (1.4,0.1) and (2.0,0.1) .. (2.8,0.8);
\draw (0.6,-0.8) .. controls (1.4,-0.1) and (2.0,-0.1) .. (2.8,-0.8);
\draw (1.2,0.28) .. controls (1.7,-0.2) .. (2.2,-0.28);
\draw (1.2,-0.28) .. controls (1.7,0.2) .. (2.2,0.28);
\draw (1.7,0.55) circle (0.30);
\node[align=center,scale=0.9] at (1.7,-1.5) {$\Gamma^{(4)}$ with\\ 3 vertex};
\end{tikzpicture}

Say \(\Gamma^{(n)}\) has `\(V\)' vertices. Then internal lines

\[ I \;=\; \frac{1}{2}\left(4V - n\right) \] \[ \text{No. of momenta} \;=\; n + \frac{1}{2}\left(4V-n\right) \] \[ \text{No. of individual momenta} \;=\; n + \frac{1}{2}\left(4V-n\right) - V \]

where \((n-1)\) of these are independent external momenta.

\[ \left(\text{No. of loop momenta} \equiv L\right) \;=\; V + \frac{n}{2} - (n-1) \] \[ L \;=\; V - \frac{n}{2} + 1 \]

In \(\Gamma^{(n)}\) computation ; we will have integrals like

\[ \frac{d^d\tilde{q}_1\ d^d\tilde{q}_2\ \cdots\ d^d\tilde{q}_L} {\left(\alpha^2q_1^2+\mu^2\right)\left(\alpha^2q_2^2+\mu^2\right)\cdots\left(\alpha^2q_L^2+\mu^2\right)} \] \[ \text{goes like } q^{Ld-2L} \ \sim\ q^{(d-2)L} \ \sim\ q^{(d-2)\left(V-\frac{n}{2}+1\right)} \]

look at large `\(q\)' behaviour ---

For ex: \(d=4\) , Behaviour of integral \(\sim q^{2\left(V-\frac{n}{2}+1\right)}\)

\[ \sim q^{4-n}\qquad \forall\ \text{graphs of } \Gamma^{(n)} \] \[ \begin{aligned} \Gamma^{(2)} &\sim q^2 \sim \Lambda^2\\ \Gamma^{(4)} &\sim \ln\Lambda\\ \Gamma^{(n)} &\sim (n>4) \quad \Rightarrow\quad \underline{\text{not}}\ \Lambda\ \text{sensitive} \end{aligned} \]

\(\searrow\) Naive counting

(margin note) These diagrams, cutting internal lines once will not be able to disconnect the diagrams / external lines.

A primitive divergent graph is one which is \(\Lambda\)-sensitive but has no \(\Lambda\)-sensitive 'Sunset / sunrise / saturn'

\usetikzlibrary{arrows.meta,decorations.pathreplacing,decorations.pathmorphing,calc,angles,quotes,positioning,patterns}
\begin{tikzpicture}[scale=1.0,>=Stealth]
\node[align=left] at (-4.4,0.1) {only primitively\\ divergent\\ graphs.\\ $\left(d\le 4\right)$};
\draw (-3.1,0.6) .. controls (-3.1,-0.4) and (-3.3,0.9) .. (-3.3,0.0) .. controls (-3.3,-0.9) and (-3.1,0.4) .. (-3.1,-0.6);
\draw (-2.4,0.75) -- (-0.6,0.75);
\draw (-1.5,1.0) circle (0.25);
\draw (0.6,0.75) -- (2.8,0.75);
\draw (1.7,0.75) circle (0.35);
\draw (-1.6,-0.9) .. controls (-0.8,-1.5) and (-0.2,-1.5) .. (0.6,-0.9);
\draw (-1.6,-2.1) .. controls (-0.8,-1.5) and (-0.2,-1.5) .. (0.6,-2.1);
\draw (-1.0,-1.35) .. controls (-0.6,-1.75) .. (-0.1,-1.9);
\draw (-1.0,-1.65) .. controls (-0.6,-1.25) .. (-0.1,-1.1);
\end{tikzpicture}

Every diagram \(\left(\Gamma^{(n)}\,;\ n>4\right)\) is built out of \(\Gamma^{(2)}\) \& \(\Gamma^{(4)}\).