-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathfeynman.tex
212 lines (207 loc) · 9.42 KB
/
feynman.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
%#!platexmake CheatSheet
%%% Time-Stamp: <2014-01-09 21:35:21 misho>
%%% 一部で日本語が使用されています。
\paragraph{Scalar Boson}\mbox{}\par
\vspace{1zw}
\begin{fmffile}{feynman/scalar}\begin{tabular}{@{\!\!\!}r@{}c@{}l@{}}
\multicolumn{3}{l}{ $\Lag \supset \dfrac12(\Pm\phi)^2-\dfrac12m^2\phi^2$}\\[.6zw]
$\begC1{\phi}\conC{\ }\endC1{\phi}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a1,db}\fmfright{dc,a4,dd}
\fmf{xscalar,label=$\longleftarrow p$,label.side=right}{a3,a2}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}\fmfblob{10pt}{a2}
\end{fmfgraph*}\end{tabular}
&$= \dfrac{\ii}{p^2-m^2+\ii\epsilon}$\\[2zw]
\multicolumn{3}{l}{ $\Lag \supset \left|\Pm\phi\right|^2-m^2\left|\phi\right|^2$}\\
$\begC1{\phi^*}\conC{\ }\endC1{\phi}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a1,db}\fmfright{dc,a4,dd}
\fmf{scalar,label=$\longleftarrow p$,label.side=right}{a3,a2}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}\fmfblob{10pt}{a2}
\end{fmfgraph*}\end{tabular}
&$= \dfrac{\ii}{p^2-m^2+\ii\epsilon}$\\
\end{tabular}\end{fmffile}
\\(External lines equal to 1 in both cases.)
\paragraph{Dirac Fermion}\mbox{}\par\vspace{0.7zw}
\begin{fmffile}{feynman/dirac}\begin{tabular}{@{\!\!\!}r@{}c@{}l@{}}
\multicolumn{3}{l}{$\Lag \supset \overline\psi(\ii\slashed{\partial}-m)\psi$}\\
\multicolumn{3}{l}{\ \ \ $=\ii\bar\xi\bSm\Pm\xi+\ii\bar\chi\bSm\Pm\chi-m(\xi\chi+\bar\xi\bar\chi)$}\\
\multicolumn{3}{l}{{\bf Initial state}}\\
$\begC1{\psi}\conC{\ }\endC1{\ket{\vc p,s}}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a4,db}\fmfright{dc,a1,dd}
\fmf{fermion,label=$\longleftarrow p$,label.side=right}{a2,a3}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}
\end{fmfgraph*}\end{tabular}
&$=u^s(p)$\\
$\begC1{\overline\psi}\conC{\ }\endC1{\ket{\vc p,s}}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a4,db}\fmfright{dc,a1,dd}
\fmf{fermion,label=$\longleftarrow p$,label.side=left}{a3,a2}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}
\end{fmfgraph*}\end{tabular}
&$=\overline v^s(p)$\\
\multicolumn{3}{l}{{\bf Final state}}\\
$\begC1{\bra{\vc p,s}}\conC{\ }\endC1{\overline\psi}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a1,db}\fmfright{dc,a4,dd}
\fmf{fermion,label=$\longleftarrow p$,label.side=left}{a3,a2}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}
\end{fmfgraph*}\end{tabular}
&$=\overline u^s(p)$\\
$\begC1{\bra{\vc p,s}}\conC{\ }\endC1{\psi}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a1,db}\fmfright{dc,a4,dd}
\fmf{fermion,label=$\longleftarrow p$,label.side=right}{a2,a3}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}
\end{fmfgraph*}\end{tabular}
&$= v^s(p)$\\
\multicolumn{3}{l}{{\bf Propagator}}\\
$\begC1{\psi}\conC{\ }\endC1{\overline\psi}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a1,db}\fmfright{dc,a4,dd}
\fmf{fermion,label=$\longleftarrow p$,label.side=right}{a3,a2}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}\fmfblob{10pt}{a2}
\end{fmfgraph*}\end{tabular}
&$= \dfrac{\ii(\slashed{p}+m)}{p^2-m^2+\ii\epsilon}$\\
\end{tabular}\end{fmffile}
\paragraph{Majorana Fermion}\mbox{}\par\vspace{0.7zw}
%\begin{fmffile}{feynman/dirac}
\begin{tabular}{@{\!\!\!}r@{}c@{}l@{}}
\multicolumn{3}{l}{$\Lag \supset \tfrac12\overline\psi(\ii\slashed{\partial}-m)\psi$}\\
\multicolumn{3}{l}{\ \ \ $=\ii\bar\lambda\bSm\Pm\lambda-\frac{m}2(\lambda\lambda+\bar\lambda\bar\lambda)$}\\
\multicolumn{3}{l}{{\bf Initial state}}\\
\end{tabular}
\TODO{なんか２種類流儀があるっぽい}
\newpage
\paragraph{Abelian Gauge Theory (Photon)}\mbox{}\par
\begin{fmffile}{feynman/photon}\begin{tabular}{@{\!\!\!}r@{}c@{}l@{}}
\multicolumn{3}{l}{$\Lag \supset -\dfrac14F^{\mu\nu}F_{\mu\nu} + \ii\overline\psi\slashed{D}\psi + |\Dm\phi|^2$}\\
\multicolumn{3}{l}{\qquad($\Dm = \Pm - \ii Q A_\mu$)}\\
$\begC1{A_\mu}\conC{\ }\endC1{\ket{\vc p;あ}}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a4,db}\fmfright{dc,a1,dd}
\fmf{photon,label=$\longleftarrow p$,label.side=right}{a2,a3}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}
\end{fmfgraph*}\end{tabular}
&$=\epsilon_\mu^あ(p)$\\
$\begC1{\bra{\vc p;あ}}\conC{\ }\endC1{A_\mu}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a1,db}\fmfright{dc,a4,dd}
\fmf{photon,label=$\longleftarrow p$,label.side=right}{a3,a2}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}
\end{fmfgraph*}\end{tabular}
&$= \epsilon_\mu^{あ*}(p)$\\
$\begC1{A_\mu}\conC{\ }\endC1{A_\nu}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a1,db}\fmfright{dc,a4,dd}
\fmf{photon,label=$\longleftarrow p$,label.side=right}{a3,a2}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}\fmfblob{10pt}{a2}
\end{fmfgraph*}\end{tabular}
&$= \dfrac{-\ii \Hmnd}{p^2+\ii\epsilon}$\\
$Q\overline\psi\slashed{A}\psi=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfset{dash_len}{1.5mm}\fmfset{arrow_len}{3mm}\fmfset{arrow_ang}{20}
\fmfv{l=$\mu$,l.d=2}{p2}
\fmfleft{f1,da,f2}\fmfright{db,p1,dc}
\fmf{fermion,width=0.6}{c,f1}
\fmf{fermion,width=0.6}{f2,c}
\fmfdot{c}
\fmf{photon,tension=1.8,width=0.6}{c,p2}
\fmf{phantom,tension=5}{p2,p1}
\end{fmfgraph*}\end{tabular}
&$= \ii Q\Gm$\\
$\begin{matrix}\ii QA^\mu\phi^*\Pm\phi\\\quad+\Hc\end{matrix}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,40)
\fmfset{dash_len}{1.5mm}\fmfset{arrow_len}{3mm}\fmfset{arrow_ang}{20}
\fmftop{g}
\fmfleft{dc,f1,da,df}\fmfright{dd,f2,db,de}
\fmf{scalar,label=$p\rightarrow$,width=0.6}{f1,c}
\fmf{scalar,label=$q\rightarrow$,width=0.6}{c,f2}
\fmffreeze
\fmf{photon,width=0.6}{g2,c}
\fmf{phantom,tension=2}{g,g2}
\fmfv{l=$\mu$,l.d=2,l.a=90}{g2}
\end{fmfgraph*}\end{tabular}
&$= \ii Q(p^\mu+q^\mu)$\\[-5pt]
\multicolumn{3}{l}{\hfill\footnotesize(Momentum must be taken along the arrow)}\\[8pt]
$Q^2A^2|\phi|^2=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,40)
\fmfset{dash_len}{1.5mm}\fmfset{arrow_len}{3mm}\fmfset{arrow_ang}{20}
\fmfleft{s2,s1}\fmfright{v2,v1}
\fmf{phantom,tension=12}{s1,s1r}\fmf{phantom,tension=12}{v1,v1r}
\fmf{phantom,tension=12}{s2,s2r}\fmf{phantom,tension=12}{v2,v2r}
\fmf{photon,width=0.6}{v1r,c,v2r}
\fmf{scalar,width=0.6}{s1r,c,s2r}
\fmfv{l=$\mu$,l.a=-20,l.d=2}{v1r}
\fmfv{l=$\nu$,l.a=+20,l.d=2}{v2r}
\end{fmfgraph*}\end{tabular}
&$= 2\ii Q^2\Hmn$\\
\end{tabular}\end{fmffile}
\vspace{10pt}
\paragraph{Non-Abelian Gauge Theory (Gluon)}\mbox{}\par
\begin{fmffile}{feynman/gluon}\begin{tabular}{@{\!\!\!}r@{}c@{}l@{}}
\multicolumn{3}{l}{$\Lag \supset -\dfrac14F^{\mu\nu}F_{\mu\nu} + \ii\overline\psi\slashed{D}\psi + |\Dm\phi|^2$}\\
\multicolumn{3}{l}{\qquad($\Dm = \Pm - \ii g A_\mu$)}\\
$\begC1{A^b_\mu}\conC{\ }\endC1{\ket{\vc p;あ,a}}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a4,db}\fmfright{dc,a1,dd}\fmfset{curly_len}{2mm}
\fmf{gluon,label=$\longleftarrow p$,label.side=right}{a2,a3}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}
\end{fmfgraph*}\end{tabular}
&$=\epsilon_\mu^あ(p)\delta^{ab}$\\
$\begC1{\bra{\vc p;あ,a}}\conC{\ }\endC1{A^b_\mu}=$&
\begin{tabular}[c]{@{}c@{}}\begin{fmfgraph*}(80,25)
\fmfleft{da,a1,db}\fmfright{dc,a4,dd}\fmfset{curly_len}{2mm}
\fmf{gluon,label=$\longleftarrow p$,label.side=right}{a3,a2}
\fmf{phantom,tension=8}{a1,a2}\fmf{phantom,tension=8}{a3,a4}\fmfblob{10pt}{a3}
\end{fmfgraph*}\end{tabular}
&$= \epsilon_\mu^{あ*}(p)\delta^{ab}$\\
%
\multicolumn{3}{l}{\hskip-10pt$-gf^{abc}A^{\mu a}A^{\nu b}(\Pm A_\nu^c)=$}\\
\multicolumn{3}{l}{
\begin{minipage}{100pt}\begin{fmfgraph*}(80,90)
\fmfset{dash_len}{1.5mm}\fmfset{arrow_len}{3mm}\fmfset{arrow_ang}{20}\fmfset{curly_len}{2mm}
\fmftop{g1c}
\fmfleft{dc,g2c,da,df}\fmfright{dd,g3c,db,de}
\fmf{gluon,label=\rotatebox{90} {$\leftarrow p$},l.s=left,l.d=5,width=0.6}{g1,c}
\fmf{gluon,label=\rotatebox{28} {$q\rightarrow$},l.s=left,l.d=0.6,width=0.6}{g2,c}
\fmf{gluon,label=\rotatebox{-28}{$\leftarrow r$},l.s=left,l.d=0.6,width=0.6}{g3,c}
\fmf{phantom,tension=8}{g1c,g1}
\fmf{phantom,tension=8}{g2c,g2}
\fmf{phantom,tension=8}{g3c,g3}
\fmfv{l=$a;\mu$,l.a=200,l.d=8}{g1}
\fmfv{l=$b;\nu$,l.d=2}{g2}
\fmfv{l=$c;\rho$,l.d=3}{g3}
\end{fmfgraph*}\end{minipage}
\begin{minipage}{92pt}
$= gf^{abc}\big[\Hmn(p-q)^\rho$\\
$\quad\qquad+\Hnr(q-r)^\mu$\\
$\quad\qquad+\Hrm(r-p)^\nu\big]$\\[10pt]
\end{minipage}
}\\[-25pt]
\multicolumn{3}{l}{\hfill\footnotesize(Momentum are in incoming directions)}\\[8pt]
\multicolumn{3}{l}{\hskip-10pt$-\dfrac14g^2(f^{abe}A_\mu^aA_\nu^b)(f^{cde}A_\rho^cA_\sigma^d)=$}\\
\multicolumn{3}{l}{
\begin{minipage}{68pt}\vspace{-5pt}\begin{fmfgraph*}(60,50)
\fmfset{dash_len}{1.5mm}\fmfset{arrow_len}{3mm}\fmfset{arrow_ang}{20}\fmfset{curly_len}{2mm}
\fmfleft{g2c,g1c}\fmfright{g4c,g3c}
\fmf{gluon,width=0.6}{g1,c}\fmf{phantom,tension=8}{g1c,g1}
\fmf{gluon,width=0.6}{g2,c}\fmf{phantom,tension=8}{g2c,g2}
\fmf{gluon,width=0.6}{g3,c}\fmf{phantom,tension=8}{g3c,g3}
\fmf{gluon,width=0.6}{g4,c}\fmf{phantom,tension=8}{g4c,g4}
\fmfv{l=$a;\mu$, l.d=1}{g1}
\fmfv{l=$b;\nu$, l.d=1}{g2}
\fmfv{l=$c;\rho$ ,l.d=1}{g3}
\fmfv{l=$d;\sigma$,l.d=1}{g4}
\end{fmfgraph*}\end{minipage}
\begin{minipage}{124pt}
\vspace{10pt}
$\!\!\!\!\!\!=-\ii g^2\big[$\\
$\phantom{+}f^{abe}f^{cde}(\Hmr\Hns-\Hms\Hnr)$\\
$+f^{ace}f^{bde}(\Hmn\Hrs-\Hms\Hnr)$\\
$+f^{ade}f^{bce}(\Hmn\Hsr-\Hmr\Hns)\big]$
\end{minipage}
}\\
\end{tabular}\end{fmffile}
\par