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  1. documentclass[border=10pt]{standalone}
  2. usepackage{amsmath}
  3. usepackage{anyfontsize}
  4. usepackage{tikz}
  5. usetikzlibrary{arrows,positioning,shapes.geometric}
  6. begin{document}
  7. begin{tikzpicture}[>=latex']
  8. tikzset{block/.style= {draw, rectangle, align=center,minimum width=3.3cm,minimum height=.1cm},
  9. }
  10. node [block,text width=2cm] (start) {fontsize{3}{6}selectfont Feynman Integral};
  11. node [block,text width=2cm, below = .2cm of start] (Z1){fontsize{3}{6}selectfont Schwinger Parametrization};
  12. node [coordinate, below = .3cm of Z1] (ADL){};
  13. node [coordinate, left = 2cm of ADL] (AUL){};
  14. node [coordinate, below = .3cm of Z1] (BUL){};
  15. node [coordinate, right = 2cm of BUL] (BDL){};
  16.  
  17. node [block,text width=2cm, below = .2cm of AUL] (A1){{fontsize{2.4}{3}selectfont Original Method of Brackets}};
  18. node [block,text width=2cm, below = .2cm of BDL] (A2){{fontsize{2.4}{3}selectfont Modified Method of Brackets}};
  19. node [block,text width=2.5cm, below = .2cm of A1,align=center] (B1){{fontsize{2.4}{6}selectfont textbf{Rule 1}: Expanding exponentials \
  20. (e^{-A} = sum_{n=0}^{infty}frac{(-1)^n A^n}{Gamma(1+n)})}};
  21. node [block,text width=2.6cm, below = .2cm of A2] (B2){{fontsize{2.4}{3}selectfont textbf{Rule 1}: Expanding Exponentials\
  22. (e^{-A} = oint frac{dx}{2 pi i} A^{-z}Gamma(-z))}};
  23. node [block,text width=2cm, below= .2cm of B2] (C2){{fontsize{2.4}{3}selectfont textbf{Rule 2}: Expanding Multinomials
  24. begin{align*}
  25. (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)}
  26. end{align*}}};
  27. node [block,text width=2.5cm, below = .2cm of B1] (C1){{fontsize{2.4}{3}selectfont textbf{Rule 2}: Expanding Multinomials
  28. begin{align*}
  29. (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)}
  30. end{align*}}};
  31. node [block,text width=2cm, below = .2cm of C1] (D1){{fontsize{2.4}{3}selectfont textbf{Rule 3}: Introduce Bracket
  32. begin{align*}
  33. int_{0}^{infty}x^{l-1}= langle lrangle
  34. end{align*}}};
  35. node [block,text width=2cm, below = .2cm of C2] (D2){{fontsize{2.4}{3}selectfont textbf{Rule 3}: Introduce Bracket
  36. begin{align*}
  37. int_{0}^{infty}x^{l-1}= langle lrangle
  38. end{align*}}};
  39. node [block,text width=2.5cm, below = .2cm of D1] (E1){{fontsize{2.4}{3}selectfont textbf{Rule 4}: Eliminate Bracket\
  40. vspace{.3cm}An expression of the form
  41. begin{align*}
  42. 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
  43. end{align*}
  44. as(frac{1}{|detA|}f(n_{1}^{*},...,n_{r}^{*})Gamma(-n_{1}^{*})...Gamma(-n_{r}^{*}))}};
  45. node [block,text width=2.5cm, below = .2cm of D2] (E2){{fontsize{2.4}{3}selectfont textbf{Rule 4}: Eliminate Bracket-\
  46. vspace{.3cm}An expression of the form
  47. begin{align*}
  48. 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
  49. end{align*}
  50. as (frac{1}{|detA|}f(z_{1}^{*},...,z_{r}^{*}))}};
  51. path[draw, ->]
  52. (start)edge(Z1)
  53. (Z1)--(ADL)
  54. (ADL)--(AUL)
  55. (AUL)edge(A1)
  56. (A1) edge (B1)
  57. (B1)edge(C1)
  58. (C1)edge(D1)
  59. (D1)edge(E1)
  60. (Z1)--(BUL)
  61. (BUL)--(BDL)
  62. (BDL)edge(A2)
  63. (A2) edge (B2)
  64. (B2)edge(C2)
  65. (C2)--(D2)
  66. (D2)edge(E2)
  67. ;
  68. end{tikzpicture}
  69. end{document}
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