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- documentclass[border=10pt]{standalone}
- usepackage{amsmath}
- usepackage{anyfontsize}
- usepackage{tikz}
- usetikzlibrary{arrows,positioning,shapes.geometric}
- begin{document}
- begin{tikzpicture}[>=latex']
- tikzset{block/.style= {draw, rectangle, align=center,minimum width=3.3cm,minimum height=.1cm},
- }
- node [block,text width=2cm] (start) {fontsize{3}{6}selectfont Feynman Integral};
- node [block,text width=2cm, below = .2cm of start] (Z1){fontsize{3}{6}selectfont Schwinger Parametrization};
- node [coordinate, below = .3cm of Z1] (ADL){};
- node [coordinate, left = 2cm of ADL] (AUL){};
- node [coordinate, below = .3cm of Z1] (BUL){};
- node [coordinate, right = 2cm of BUL] (BDL){};
- node [block,text width=2cm, below = .2cm of AUL] (A1){{fontsize{2.4}{3}selectfont Original Method of Brackets}};
- node [block,text width=2cm, below = .2cm of BDL] (A2){{fontsize{2.4}{3}selectfont Modified Method of Brackets}};
- node [block,text width=2.5cm, below = .2cm of A1,align=center] (B1){{fontsize{2.4}{6}selectfont textbf{Rule 1}: Expanding exponentials \
- (e^{-A} = sum_{n=0}^{infty}frac{(-1)^n A^n}{Gamma(1+n)})}};
- node [block,text width=2.6cm, below = .2cm of A2] (B2){{fontsize{2.4}{3}selectfont textbf{Rule 1}: Expanding Exponentials\
- (e^{-A} = oint frac{dx}{2 pi i} A^{-z}Gamma(-z))}};
- node [block,text width=2cm, below= .2cm of B2] (C2){{fontsize{2.4}{3}selectfont textbf{Rule 2}: Expanding Multinomials
- begin{align*}
- (a_1 + a_2 +...+ a_r)^{alpha}= ointfrac{dz_1}{2 pi i}...ointfrac{dz_r}{2 pi i} a_{1}^{z_1}...a_{r}^{z_r}\langle -alpha+z_1+...+z_rranglefrac{Gamma(-z_1)...Gamma(-z_2)}{Gamma(-alpha)}
- end{align*}}};
- node [block,text width=2.5cm, below = .2cm of B1] (C1){{fontsize{2.4}{3}selectfont textbf{Rule 2}: Expanding Multinomials
- begin{align*}
- (a_1 + a_2 +...+ a_r)^{alpha}= sum_{m_1,...,m_r} phi_{m_1,...,m_r} \a_{1}^{m_1}...a_{r}^{m_r}frac{langle -alpha+m_1+...+m_rrangle}{Gamma(-alpha)}
- end{align*}}};
- node [block,text width=2cm, below = .2cm of C1] (D1){{fontsize{2.4}{3}selectfont textbf{Rule 3}: Introduce Bracket
- begin{align*}
- int_{0}^{infty}x^{l-1}= langle lrangle
- end{align*}}};
- node [block,text width=2cm, below = .2cm of C2] (D2){{fontsize{2.4}{3}selectfont textbf{Rule 3}: Introduce Bracket
- begin{align*}
- int_{0}^{infty}x^{l-1}= langle lrangle
- end{align*}}};
- node [block,text width=2.5cm, below = .2cm of D1] (E1){{fontsize{2.4}{3}selectfont textbf{Rule 4}: Eliminate Bracket\
- vspace{.3cm}An expression of the form
- begin{align*}
- sum_{n_1,...,n_r}phi_{1,...,r}f(n_1,...,n_r)langle a_{11}n_{1}+...+a_{1r}n_{r}+c{1}rangle\times langle a_{s1}n_{1}+...+a_{sr}n_{r}+c{1}rangle
- end{align*}
- as(frac{1}{|detA|}f(n_{1}^{*},...,n_{r}^{*})Gamma(-n_{1}^{*})...Gamma(-n_{r}^{*}))}};
- node [block,text width=2.5cm, below = .2cm of D2] (E2){{fontsize{2.4}{3}selectfont textbf{Rule 4}: Eliminate Bracket-\
- vspace{.3cm}An expression of the form
- begin{align*}
- ointfrac{dz_{1}}{2pi i}...ointfrac{dz_{1}}{2pi i}f(z_{1},...,z_{r})langle a_{s1}z_{1}+...+a_{sr}z_{r}+c{1}rangle\times langle a_{s1}z_{1}+...+a_{sr}z_{r}+c{1}rangle
- end{align*}
- as (frac{1}{|detA|}f(z_{1}^{*},...,z_{r}^{*}))}};
- path[draw, ->]
- (start)edge(Z1)
- (Z1)--(ADL)
- (ADL)--(AUL)
- (AUL)edge(A1)
- (A1) edge (B1)
- (B1)edge(C1)
- (C1)edge(D1)
- (D1)edge(E1)
- (Z1)--(BUL)
- (BUL)--(BDL)
- (BDL)edge(A2)
- (A2) edge (B2)
- (B2)edge(C2)
- (C2)--(D2)
- (D2)edge(E2)
- ;
- end{tikzpicture}
- end{document}
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