FCLoopGraphPlot
FCLoopGraphPlot[{edges, labels}]
visualizes the graph of the given loop integral using the provided list of edges, styles and labels using the built-in function Graph
. The Option Graph
can be used to pass options to the Graph
objects.
By default, FCLoopGraphPlot
returns a Graph
. When using Mathematica 12.2 or newer, it is also possible to return a Graphics
object created by GraphPlot
. For this the option GraphPlot
must be set to a list of options that will be passed to GraphPlot
. An empty list is also admissible. For example, FCLoopGraphPlot[int, GraphPlot -> {MultiedgeStyle -> 0.35, Frame -> True}]
.
Given a list of Graph
or Graphics
objects created by FCLoopGraphPlot
, a nice way to get a better overview is to employ Magnify[Grid[(Partition[out, UpTo[4]])], 0.9]
.
Notice that older Mathematica versions have numerous shortcomings in the graph drawing capabilities that cannot be reliably worked around. This why to use FCLoopGraphPlot
you need to have at least Mathematica 11.0 or newer. For best results we recommend using Mathematica 12.2 or newer.
See also
Overview , FCLoopIntegralToGraph .
Examples
Showcases
1-loop tadpole
FCLoopIntegralToGraph[ FAD[{ p , m }], { p }]
FCLoopGraphPlot[ % ]
{ { 1 → 1 } , ( p 1 − m 2 ) , { 1 ( p 2 − m 2 + i η ) } , 1 } \left\{\{1\to 1\},\left(
\begin{array}{ccc}
p & 1 & -m^2 \\
\end{array}
\right),\left\{\frac{1}{(p^2-m^2+i \eta )}\right\},1\right\} { { 1 → 1 } , ( p 1 − m 2 ) , { ( p 2 − m 2 + i η ) 1 } , 1 }
1-loop massless bubble
FCLoopIntegralToGraph[ FAD[ p , p - q ], { p }]
FCLoopGraphPlot[ % ]
{ { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 } , { − q , q , { p , 1 , 0 } , { p − q , 1 , 0 } } , { 0 , 0 , 1 ( p 2 + i η ) , 1 ( ( p − q ) 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 2,1\to 2\},\{-q,q,\{p,1,0\},\{p-q,1,0\}\},\left\{0,0,\frac{1}{(p^2+i \eta )},\frac{1}{((p-q)^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 } , { − q , q , { p , 1 , 0 } , { p − q , 1 , 0 }} , { 0 , 0 , ( p 2 + i η ) 1 , (( p − q ) 2 + i η ) 1 } , 1 }
1-loop massive bubble
FCLoopIntegralToGraph[ FAD[{ p , m1}, { p - q , m2}], { p }]
FCLoopGraphPlot[ % ]
{ { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 } , { − q , q , { p , 1 , − m1 2 } , { p − q , 1 , − m2 2 } } , { 0 , 0 , 1 ( p 2 − m1 2 + i η ) , 1 ( ( p − q ) 2 − m2 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 2,1\to 2\},\left\{-q,q,\left\{p,1,-\text{m1}^2\right\},\left\{p-q,1,-\text{m2}^2\right\}\right\},\left\{0,0,\frac{1}{(p^2-\text{m1}^2+i \eta )},\frac{1}{((p-q)^2-\text{m2}^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 } , { − q , q , { p , 1 , − m1 2 } , { p − q , 1 , − m2 2 } } , { 0 , 0 , ( p 2 − m1 2 + i η ) 1 , (( p − q ) 2 − m2 2 + i η ) 1 } , 1 }
1-loop massless triangle
FCLoopIntegralToGraph[ FAD[ p , p + q1, p + q1 + q2], { p }]
FCLoopGraphPlot[ % ]
{ { − 3 → 3 , − 2 → 1 , − 1 → 2 , 1 → 2 , 1 → 3 , 2 → 3 } , { q1 + q2 , q2 , q1 , { p + q1 , 1 , 0 } , { p + q1 + q2 , 1 , 0 } , { p , 1 , 0 } } , { 0 , 0 , 0 , 1 ( p 2 + i η ) , 1 ( ( p + q1 ) 2 + i η ) , 1 ( ( p + q1 + q2 ) 2 + i η ) } , 1 } \left\{\{-3\to 3,-2\to 1,-1\to 2,1\to 2,1\to 3,2\to 3\},\{\text{q1}+\text{q2},\text{q2},\text{q1},\{p+\text{q1},1,0\},\{p+\text{q1}+\text{q2},1,0\},\{p,1,0\}\},\left\{0,0,0,\frac{1}{(p^2+i \eta )},\frac{1}{((p+\text{q1})^2+i \eta )},\frac{1}{((p+\text{q1}+\text{q2})^2+i \eta )}\right\},1\right\} { { − 3 → 3 , − 2 → 1 , − 1 → 2 , 1 → 2 , 1 → 3 , 2 → 3 } , { q1 + q2 , q2 , q1 , { p + q1 , 1 , 0 } , { p + q1 + q2 , 1 , 0 } , { p , 1 , 0 }} , { 0 , 0 , 0 , ( p 2 + i η ) 1 , (( p + q1 ) 2 + i η ) 1 , (( p + q1 + q2 ) 2 + i η ) 1 } , 1 }
1-loop massless box
FCLoopIntegralToGraph[ FAD[ p , p + q1, p + q1 + q2, p + q1 + q2 + q3], { p }]
FCLoopGraphPlot[ % ]
{ { − 4 → 4 , − 3 → 1 , − 2 → 2 , − 1 → 3 , 1 → 2 , 1 → 4 , 2 → 3 , 3 → 4 } , { q1 + q2 + q3 , q3 , q2 , q1 , { p + q1 + q2 , 1 , 0 } , { p + q1 + q2 + q3 , 1 , 0 } , { p + q1 , 1 , 0 } , { p , 1 , 0 } } , { 0 , 0 , 0 , 0 , 1 ( p 2 + i η ) , 1 ( ( p + q1 ) 2 + i η ) , 1 ( ( p + q1 + q2 ) 2 + i η ) , 1 ( ( p + q1 + q2 + q3 ) 2 + i η ) } , 1 } \left\{\{-4\to 4,-3\to 1,-2\to 2,-1\to 3,1\to 2,1\to 4,2\to 3,3\to 4\},\{\text{q1}+\text{q2}+\text{q3},\text{q3},\text{q2},\text{q1},\{p+\text{q1}+\text{q2},1,0\},\{p+\text{q1}+\text{q2}+\text{q3},1,0\},\{p+\text{q1},1,0\},\{p,1,0\}\},\left\{0,0,0,0,\frac{1}{(p^2+i \eta )},\frac{1}{((p+\text{q1})^2+i \eta )},\frac{1}{((p+\text{q1}+\text{q2})^2+i \eta )},\frac{1}{((p+\text{q1}+\text{q2}+\text{q3})^2+i \eta )}\right\},1\right\} { { − 4 → 4 , − 3 → 1 , − 2 → 2 , − 1 → 3 , 1 → 2 , 1 → 4 , 2 → 3 , 3 → 4 } , { q1 + q2 + q3 , q3 , q2 , q1 , { p + q1 + q2 , 1 , 0 } , { p + q1 + q2 + q3 , 1 , 0 } , { p + q1 , 1 , 0 } , { p , 1 , 0 }} , { 0 , 0 , 0 , 0 , ( p 2 + i η ) 1 , (( p + q1 ) 2 + i η ) 1 , (( p + q1 + q2 ) 2 + i η ) 1 , (( p + q1 + q2 + q3 ) 2 + i η ) 1 } , 1 }
1-loop massless pentagon
FCLoopIntegralToGraph[ FAD[ p , p + q1, p + q1 + q2, p + q1 + q2 + q3, p + q1 + q2 + q3 + q4], { p }]
FCLoopGraphPlot[ % ]
{ { − 5 → 5 , − 4 → 1 , − 3 → 2 , − 2 → 3 , − 1 → 4 , 1 → 2 , 1 → 5 , 2 → 3 , 3 → 4 , 4 → 5 } , { q1 + q2 + q3 + q4 , q4 , q3 , q2 , q1 , { p + q1 + q2 + q3 , 1 , 0 } , { p + q1 + q2 + q3 + q4 , 1 , 0 } , { p + q1 + q2 , 1 , 0 } , { p + q1 , 1 , 0 } , { p , 1 , 0 } } , { 0 , 0 , 0 , 0 , 0 , 1 ( p 2 + i η ) , 1 ( ( p + q1 ) 2 + i η ) , 1 ( ( p + q1 + q2 ) 2 + i η ) , 1 ( ( p + q1 + q2 + q3 ) 2 + i η ) , 1 ( ( p + q1 + q2 + q3 + q4 ) 2 + i η ) } , 1 } \left\{\{-5\to 5,-4\to 1,-3\to 2,-2\to 3,-1\to 4,1\to 2,1\to 5,2\to 3,3\to 4,4\to 5\},\{\text{q1}+\text{q2}+\text{q3}+\text{q4},\text{q4},\text{q3},\text{q2},\text{q1},\{p+\text{q1}+\text{q2}+\text{q3},1,0\},\{p+\text{q1}+\text{q2}+\text{q3}+\text{q4},1,0\},\{p+\text{q1}+\text{q2},1,0\},\{p+\text{q1},1,0\},\{p,1,0\}\},\left\{0,0,0,0,0,\frac{1}{(p^2+i \eta )},\frac{1}{((p+\text{q1})^2+i \eta )},\frac{1}{((p+\text{q1}+\text{q2})^2+i \eta )},\frac{1}{((p+\text{q1}+\text{q2}+\text{q3})^2+i \eta )},\frac{1}{((p+\text{q1}+\text{q2}+\text{q3}+\text{q4})^2+i \eta )}\right\},1\right\} { { − 5 → 5 , − 4 → 1 , − 3 → 2 , − 2 → 3 , − 1 → 4 , 1 → 2 , 1 → 5 , 2 → 3 , 3 → 4 , 4 → 5 } , { q1 + q2 + q3 + q4 , q4 , q3 , q2 , q1 , { p + q1 + q2 + q3 , 1 , 0 } , { p + q1 + q2 + q3 + q4 , 1 , 0 } , { p + q1 + q2 , 1 , 0 } , { p + q1 , 1 , 0 } , { p , 1 , 0 }} , { 0 , 0 , 0 , 0 , 0 , ( p 2 + i η ) 1 , (( p + q1 ) 2 + i η ) 1 , (( p + q1 + q2 ) 2 + i η ) 1 , (( p + q1 + q2 + q3 ) 2 + i η ) 1 , (( p + q1 + q2 + q3 + q4 ) 2 + i η ) 1 } , 1 }
2-loop massless self-energy
FCLoopIntegralToGraph[ FAD[ p1, p2, Q - p1 - p2, Q - p1, Q - p2], { p1, p2}]
FCLoopGraphPlot[ % ]
{ { − 3 → 2 , − 1 → 1 , 1 → 3 , 1 → 4 , 2 → 3 , 2 → 4 , 3 → 4 } , { − Q , Q , { p2 , 1 , 0 } , { Q − p2 , 1 , 0 } , { Q − p1 , 1 , 0 } , { p1 , 1 , 0 } , { − p1 − p2 + Q , 1 , 0 } } , { 0 , 0 , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( Q − p2 ) 2 + i η ) , 1 ( ( Q − p1 ) 2 + i η ) , 1 ( ( − p1 − p2 + Q ) 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 3,1\to 4,2\to 3,2\to 4,3\to 4\},\{-Q,Q,\{\text{p2},1,0\},\{Q-\text{p2},1,0\},\{Q-\text{p1},1,0\},\{\text{p1},1,0\},\{-\text{p1}-\text{p2}+Q,1,0\}\},\left\{0,0,\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((Q-\text{p2})^2+i \eta )},\frac{1}{((Q-\text{p1})^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+Q)^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 3 , 1 → 4 , 2 → 3 , 2 → 4 , 3 → 4 } , { − Q , Q , { p2 , 1 , 0 } , { Q − p2 , 1 , 0 } , { Q − p1 , 1 , 0 } , { p1 , 1 , 0 } , { − p1 − p2 + Q , 1 , 0 }} , { 0 , 0 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( Q − p2 ) 2 + i η ) 1 , (( Q − p1 ) 2 + i η ) 1 , (( − p1 − p2 + Q ) 2 + i η ) 1 } , 1 }
Same topology as before but now fully massive and with some dots
FCLoopIntegralToGraph[ FAD[{ p1, m }, { p2, m2}, { Q - p1 - p2, m }, { Q - p1, m , 2 }, { Q - p2, m , 2 }], { p1, p2}]
FCLoopGraphPlot[ % ]
{ { − 3 → 2 , − 1 → 1 , 1 → 3 , 1 → 4 , 2 → 3 , 2 → 4 , 3 → 4 } , { − Q , Q , { p2 , 1 , − m2 2 } , { Q − p2 , 2 , − m 2 } , { Q − p1 , 2 , − m 2 } , { p1 , 1 , − m 2 } , { − p1 − p2 + Q , 1 , − m 2 } } , { 0 , 0 , 1 ( p2 2 − m2 2 + i η ) , 1 ( p1 2 − m 2 + i η ) , 1 ( ( Q − p2 ) 2 − m 2 + i η ) , 1 ( ( Q − p1 ) 2 − m 2 + i η ) , 1 ( ( − p1 − p2 + Q ) 2 − m 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 3,1\to 4,2\to 3,2\to 4,3\to 4\},\left\{-Q,Q,\left\{\text{p2},1,-\text{m2}^2\right\},\left\{Q-\text{p2},2,-m^2\right\},\left\{Q-\text{p1},2,-m^2\right\},\left\{\text{p1},1,-m^2\right\},\left\{-\text{p1}-\text{p2}+Q,1,-m^2\right\}\right\},\left\{0,0,\frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )},\frac{1}{(\text{p1}^2-m^2+i \eta )},\frac{1}{((Q-\text{p2})^2-m^2+i \eta )},\frac{1}{((Q-\text{p1})^2-m^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+Q)^2-m^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 3 , 1 → 4 , 2 → 3 , 2 → 4 , 3 → 4 } , { − Q , Q , { p2 , 1 , − m2 2 } , { Q − p2 , 2 , − m 2 } , { Q − p1 , 2 , − m 2 } , { p1 , 1 , − m 2 } , { − p1 − p2 + Q , 1 , − m 2 } } , { 0 , 0 , ( p2 2 − m2 2 + i η ) 1 , ( p1 2 − m 2 + i η ) 1 , (( Q − p2 ) 2 − m 2 + i η ) 1 , (( Q − p1 ) 2 − m 2 + i η ) 1 , (( − p1 − p2 + Q ) 2 − m 2 + i η ) 1 } , 1 }
3-loop massless self-energy
FCLoopIntegralToGraph[ FAD[ p1, p2, p3, Q - p1 - p2 - p3, Q - p1 - p2, Q - p1, Q - p2, p1 + p3], { p1, p2, p3}]
FCLoopGraphPlot[ % ]
{ { − 3 → 2 , − 1 → 1 , 1 → 5 , 1 → 6 , 2 → 3 , 2 → 5 , 3 → 4 , 3 → 6 , 4 → 5 , 4 → 6 } , { − Q , Q , { p2 , 1 , 0 } , { Q − p2 , 1 , 0 } , { p1 , 1 , 0 } , { Q − p1 , 1 , 0 } , { p3 , 1 , 0 } , { p1 + p3 , 1 , 0 } , { − p1 − p2 + Q , 1 , 0 } , { − p1 − p2 − p3 + Q , 1 , 0 } } , { 0 , 0 , 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p1 + p3 ) 2 + i η ) , 1 ( ( Q − p2 ) 2 + i η ) , 1 ( ( Q − p1 ) 2 + i η ) , 1 ( ( − p1 − p2 + Q ) 2 + i η ) , 1 ( ( − p1 − p2 − p3 + Q ) 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 5,1\to 6,2\to 3,2\to 5,3\to 4,3\to 6,4\to 5,4\to 6\},\{-Q,Q,\{\text{p2},1,0\},\{Q-\text{p2},1,0\},\{\text{p1},1,0\},\{Q-\text{p1},1,0\},\{\text{p3},1,0\},\{\text{p1}+\text{p3},1,0\},\{-\text{p1}-\text{p2}+Q,1,0\},\{-\text{p1}-\text{p2}-\text{p3}+Q,1,0\}\},\left\{0,0,\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p1}+\text{p3})^2+i \eta )},\frac{1}{((Q-\text{p2})^2+i \eta )},\frac{1}{((Q-\text{p1})^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+Q)^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}-\text{p3}+Q)^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 5 , 1 → 6 , 2 → 3 , 2 → 5 , 3 → 4 , 3 → 6 , 4 → 5 , 4 → 6 } , { − Q , Q , { p2 , 1 , 0 } , { Q − p2 , 1 , 0 } , { p1 , 1 , 0 } , { Q − p1 , 1 , 0 } , { p3 , 1 , 0 } , { p1 + p3 , 1 , 0 } , { − p1 − p2 + Q , 1 , 0 } , { − p1 − p2 − p3 + Q , 1 , 0 }} , { 0 , 0 , ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p1 + p3 ) 2 + i η ) 1 , (( Q − p2 ) 2 + i η ) 1 , (( Q − p1 ) 2 + i η ) 1 , (( − p1 − p2 + Q ) 2 + i η ) 1 , (( − p1 − p2 − p3 + Q ) 2 + i η ) 1 } , 1 }
3-loop self-energy with two massive lines
FCLoopIntegralToGraph[ Times @@ { SFAD[{{ p1, 0 }, { m ^ 2 , 1 }, 1 }], SFAD[{{ p2, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ p3, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p2 + p3, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ p2 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 - Q , 0 }, { m ^ 2 , 1 }, 1 }], SFAD[{{ p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }]},
{ p1, p2, p3}]
FCLoopGraphPlot[ % ]
{ { − 3 → 2 , − 1 → 1 , 1 → 3 , 1 → 4 , 2 → 5 , 2 → 6 , 3 → 4 , 3 → 5 , 4 → 6 , 5 → 6 } , { − Q , Q , { p2 , 1 , 0 } , { p2 − Q , 1 , 0 } , { p1 − Q , 1 , − m 2 } , { p1 , 1 , − m 2 } , { p3 , 1 , 0 } , { p2 + p3 , 1 , 0 } , { p2 + p3 − Q , 1 , 0 } , { p1 + p2 + p3 − Q , 1 , 0 } } , { 0 , 0 , 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( ( p2 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( p1 2 − m 2 + i η ) , 1 ( ( p2 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p2 + p3 − Q ) 2 + i η ) , 1 ( ( p1 − Q ) 2 − m 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 3,1\to 4,2\to 5,2\to 6,3\to 4,3\to 5,4\to 6,5\to 6\},\left\{-Q,Q,\{\text{p2},1,0\},\{\text{p2}-Q,1,0\},\left\{\text{p1}-Q,1,-m^2\right\},\left\{\text{p1},1,-m^2\right\},\{\text{p3},1,0\},\{\text{p2}+\text{p3},1,0\},\{\text{p2}+\text{p3}-Q,1,0\},\{\text{p1}+\text{p2}+\text{p3}-Q,1,0\}\right\},\left\{0,0,\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{(\text{p1}^2-m^2+i \eta )},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2-m^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 3 , 1 → 4 , 2 → 5 , 2 → 6 , 3 → 4 , 3 → 5 , 4 → 6 , 5 → 6 } , { − Q , Q , { p2 , 1 , 0 } , { p2 − Q , 1 , 0 } , { p1 − Q , 1 , − m 2 } , { p1 , 1 , − m 2 } , { p3 , 1 , 0 } , { p2 + p3 , 1 , 0 } , { p2 + p3 − Q , 1 , 0 } , { p1 + p2 + p3 − Q , 1 , 0 } } , { 0 , 0 , ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , (( p2 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , ( p1 2 − m 2 + i η ) 1 , (( p2 + p3 − Q ) 2 + i η ) 1 , (( p1 + p2 + p3 − Q ) 2 + i η ) 1 , (( p1 − Q ) 2 − m 2 + i η ) 1 } , 1 }
2-loop triangle
FCLoopIntegralToGraph[ FAD[ p1, p2, Q1 + p1, Q2 - p1, Q1 + p1 + p2, Q2 - p1 - p2], { p1, p2}]
FCLoopGraphPlot[ % ]
{ { − 3 → 3 , − 2 → 1 , − 1 → 2 , 1 → 2 , 1 → 5 , 2 → 4 , 3 → 4 , 3 → 5 , 4 → 5 } , { Q1 + Q2 , Q2 , Q1 , { p1 , 1 , 0 } , { Q2 − p1 , 1 , 0 } , { p1 + Q1 , 1 , 0 } , { p1 + p2 + Q1 , 1 , 0 } , { − p1 − p2 + Q2 , 1 , 0 } , { p2 , 1 , 0 } } , { 0 , 0 , 0 , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p1 + Q1 ) 2 + i η ) , 1 ( ( p1 + p2 + Q1 ) 2 + i η ) , 1 ( ( Q2 − p1 ) 2 + i η ) , 1 ( ( − p1 − p2 + Q2 ) 2 + i η ) } , 1 } \left\{\{-3\to 3,-2\to 1,-1\to 2,1\to 2,1\to 5,2\to 4,3\to 4,3\to 5,4\to 5\},\{\text{Q1}+\text{Q2},\text{Q2},\text{Q1},\{\text{p1},1,0\},\{\text{Q2}-\text{p1},1,0\},\{\text{p1}+\text{Q1},1,0\},\{\text{p1}+\text{p2}+\text{Q1},1,0\},\{-\text{p1}-\text{p2}+\text{Q2},1,0\},\{\text{p2},1,0\}\},\left\{0,0,0,\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p1}+\text{Q1})^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{Q1})^2+i \eta )},\frac{1}{((\text{Q2}-\text{p1})^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+\text{Q2})^2+i \eta )}\right\},1\right\} { { − 3 → 3 , − 2 → 1 , − 1 → 2 , 1 → 2 , 1 → 5 , 2 → 4 , 3 → 4 , 3 → 5 , 4 → 5 } , { Q1 + Q2 , Q2 , Q1 , { p1 , 1 , 0 } , { Q2 − p1 , 1 , 0 } , { p1 + Q1 , 1 , 0 } , { p1 + p2 + Q1 , 1 , 0 } , { − p1 − p2 + Q2 , 1 , 0 } , { p2 , 1 , 0 }} , { 0 , 0 , 0 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p1 + Q1 ) 2 + i η ) 1 , (( p1 + p2 + Q1 ) 2 + i η ) 1 , (( Q2 − p1 ) 2 + i η ) 1 , (( − p1 − p2 + Q2 ) 2 + i η ) 1 } , 1 }
Special cases
Not all loop integrals admit a graph representation. Furthermore, an integral may have a weird momentum routing that cannot be automatically recognized by the employed algorithm. Consider e.g.
topo = FCTopology[ TRIX1, { SFAD[{{ p2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + Q1, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ p1 + p2 + Q1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ - p1 + Q2, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ - p1 - p2 + Q2, 0 }, { 0 , 1 }, 1 }]}, { p1, p2}, { Q1, Q2}, {}, {}]
FCTopology ( TRIX1 , { 1 ( p2 2 + i η ) , 1 ( ( p1 + Q1 ) 2 + i η ) , 1 ( ( p1 + p2 + Q1 ) 2 + i η ) , 1 ( ( Q2 − p1 ) 2 + i η ) , 1 ( ( − p1 − p2 + Q2 ) 2 + i η ) } , { p1 , p2 } , { Q1 , Q2 } , { } , { } ) \text{FCTopology}\left(\text{TRIX1},\left\{\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{((\text{p1}+\text{Q1})^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{Q1})^2+i \eta )},\frac{1}{((\text{Q2}-\text{p1})^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+\text{Q2})^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{\text{Q1},\text{Q2}\},\{\},\{\}\right) FCTopology ( TRIX1 , { ( p2 2 + i η ) 1 , (( p1 + Q1 ) 2 + i η ) 1 , (( p1 + p2 + Q1 ) 2 + i η ) 1 , (( Q2 − p1 ) 2 + i η ) 1 , (( − p1 − p2 + Q2 ) 2 + i η ) 1 } , { p1 , p2 } , { Q1 , Q2 } , { } , { } )
Here FCLoopIntegralToGraph
has no way to know that the actual momentum is Q1+Q2, i.e. it is a 2- and not 3-point function
FCLoopIntegralToGraph[ topo]
False \text{False} False
However, if we explicitly provide this information, in many cases the function can still perform the proper reconstruction
FCLoopIntegralToGraph[ topo, Momentum -> { Q1 + Q2}]
FCLoopGraphPlot[ % ]
{ { − 3 → 2 , − 1 → 1 , 1 → 3 , 1 → 4 , 2 → 3 , 2 → 4 , 3 → 4 } , { − Q1 − Q2 , Q1 + Q2 , { p1 + Q1 , 1 , 0 } , { Q2 − p1 , 1 , 0 } , { p1 + p2 + Q1 , 1 , 0 } , { − p1 − p2 + Q2 , 1 , 0 } , { p2 , 1 , 0 } } , { 0 , 0 , 1 ( p2 2 + i η ) , 1 ( ( p1 + Q1 ) 2 + i η ) , 1 ( ( p1 + p2 + Q1 ) 2 + i η ) , 1 ( ( Q2 − p1 ) 2 + i η ) , 1 ( ( − p1 − p2 + Q2 ) 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 3,1\to 4,2\to 3,2\to 4,3\to 4\},\{-\text{Q1}-\text{Q2},\text{Q1}+\text{Q2},\{\text{p1}+\text{Q1},1,0\},\{\text{Q2}-\text{p1},1,0\},\{\text{p1}+\text{p2}+\text{Q1},1,0\},\{-\text{p1}-\text{p2}+\text{Q2},1,0\},\{\text{p2},1,0\}\},\left\{0,0,\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{((\text{p1}+\text{Q1})^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{Q1})^2+i \eta )},\frac{1}{((\text{Q2}-\text{p1})^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+\text{Q2})^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 3 , 1 → 4 , 2 → 3 , 2 → 4 , 3 → 4 } , { − Q1 − Q2 , Q1 + Q2 , { p1 + Q1 , 1 , 0 } , { Q2 − p1 , 1 , 0 } , { p1 + p2 + Q1 , 1 , 0 } , { − p1 − p2 + Q2 , 1 , 0 } , { p2 , 1 , 0 }} , { 0 , 0 , ( p2 2 + i η ) 1 , (( p1 + Q1 ) 2 + i η ) 1 , (( p1 + p2 + Q1 ) 2 + i η ) 1 , (( Q2 − p1 ) 2 + i η ) 1 , (( − p1 − p2 + Q2 ) 2 + i η ) 1 } , 1 }
And here is another example. This NRQCD integral from arXiv:1907.08227 looks like as if it has only one external momentum flowing in
FCLoopIntegralToGraph[ FAD[{ k , m }, l + p , l - p , k + l ], { k , l }]
False \text{False} False
while in reality there are two of them: p
and 2p
FCLoopIntegralToGraph[ FAD[{ k , m }, l + p , l - p , k + l ], { k , l },
Momentum -> { 2 p , p }, FCE -> True ]
FCLoopGraphPlot[ % ]
{ { − 4 → 3 , − 2 → 2 , − 1 → 1 , 1 → 2 , 1 → 3 , 2 → 3 , 2 → 3 } , { p , p , 2 p , { l + p , 1 , 0 } , { l − p , 1 , 0 } , { k + l , 1 , 0 } , { k , 1 , − m 2 } } , { 0 , 0 , 0 , 1 ( ( l + p ) 2 + i η ) , 1 ( ( k + l ) 2 + i η ) , 1 ( ( l − p ) 2 + i η ) , 1 ( k 2 − m 2 + i η ) } , 1 } \left\{\{-4\to 3,-2\to 2,-1\to 1,1\to 2,1\to 3,2\to 3,2\to 3\},\left\{p,p,2 p,\{l+p,1,0\},\{l-p,1,0\},\{k+l,1,0\},\left\{k,1,-m^2\right\}\right\},\left\{0,0,0,\frac{1}{((l+p)^2+i \eta )},\frac{1}{((k+l)^2+i \eta )},\frac{1}{((l-p)^2+i \eta )},\frac{1}{(k^2-m^2+i \eta )}\right\},1\right\} { { − 4 → 3 , − 2 → 2 , − 1 → 1 , 1 → 2 , 1 → 3 , 2 → 3 , 2 → 3 } , { p , p , 2 p , { l + p , 1 , 0 } , { l − p , 1 , 0 } , { k + l , 1 , 0 } , { k , 1 , − m 2 } } , { 0 , 0 , 0 , (( l + p ) 2 + i η ) 1 , (( k + l ) 2 + i η ) 1 , (( l − p ) 2 + i η ) 1 , ( k 2 − m 2 + i η ) 1 } , 1 }
In this case the correct form of the external momentum can be deduced upon performing some elementary shifts. The direct application of the function fails
ex = FCTopology[ topo1X12679, { SFAD[{{ p1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p2 + p3, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ p2 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + p3 - Q , 0 }, { 0 , 1 }, 1 }]}, { p1, p2, p3}, { Q }, {}, {}]
FCTopology ( topo1X12679 , { 1 ( p1 2 + i η ) , 1 ( ( p2 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( ( p1 + p3 − Q ) 2 + i η ) } , { p1 , p2 , p3 } , { Q } , { } , { } ) \text{FCTopology}\left(\text{topo1X12679},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right) FCTopology ( topo1X12679 , { ( p1 2 + i η ) 1 , (( p2 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , (( p1 + p3 − Q ) 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } )
FCLoopIntegralToGraph[ ex]
False \text{False} False
Yet let us consider
exShifted = FCReplaceMomenta[ ex, { p2 -> p2 - p3 + p1 - Q , p3 -> p3 - p1 + Q }]
FCTopology ( topo1X12679 , { 1 ( p1 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( ( p1 + p2 − p3 − 2 Q ) 2 + i η ) , 1 ( p3 2 + i η ) } , { p1 , p2 , p3 } , { Q } , { } , { } ) \text{FCTopology}\left(\text{topo1X12679},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}-\text{p3}-2 Q)^2+i \eta )},\frac{1}{(\text{p3}^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right) FCTopology ( topo1X12679 , { ( p1 2 + i η ) 1 , ( p2 2 + i η ) 1 , (( p1 + p2 − p3 − 2 Q ) 2 + i η ) 1 , ( p3 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } )
Now we immediately see that the proper external momentum to consider is 2Q
instead of just Q
FCLoopIntegralToGraph[ exShifted, Momentum -> { 2 Q }]
FCLoopGraphPlot[ % ]
{ { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 , 1 → 2 , 1 → 2 } , { − 2 Q , 2 Q , { p1 , 1 , 0 } , { p2 , 1 , 0 } , { p1 + p2 − p3 − 2 Q , 1 , 0 } , { p3 , 1 , 0 } } , { 0 , 0 , 1 ( p1 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( ( p1 + p2 − p3 − 2 Q ) 2 + i η ) , 1 ( p3 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 2,1\to 2,1\to 2,1\to 2\},\{-2 Q,2 Q,\{\text{p1},1,0\},\{\text{p2},1,0\},\{\text{p1}+\text{p2}-\text{p3}-2 Q,1,0\},\{\text{p3},1,0\}\},\left\{0,0,\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}-\text{p3}-2 Q)^2+i \eta )},\frac{1}{(\text{p3}^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 , 1 → 2 , 1 → 2 } , { − 2 Q , 2 Q , { p1 , 1 , 0 } , { p2 , 1 , 0 } , { p1 + p2 − p3 − 2 Q , 1 , 0 } , { p3 , 1 , 0 }} , { 0 , 0 , ( p1 2 + i η ) 1 , ( p2 2 + i η ) 1 , (( p1 + p2 − p3 − 2 Q ) 2 + i η ) 1 , ( p3 2 + i η ) 1 } , 1 }
When dealing with products of tadpole integrals, the function may not always recognize that the appearing external momenta are spurious. For example, here there is no q
momentum flowing through any of the lines
int = SFAD[{{ p1, 0 }, { mg^ 2 , 1 }, 1 }] SFAD[{{ p3, - 2 p3 . q }, { 0 , 1 }, 1 }]
FCLoopIntegralToGraph[ int, { p1, p3}]
1 ( p1 2 − mg 2 + i η ) ( p3 2 − 2 ( p3 ⋅ q ) + i η ) \frac{1}{(\text{p1}^2-\text{mg}^2+i \eta ) (\text{p3}^2-2 (\text{p3}\cdot q)+i \eta )} ( p1 2 − mg 2 + i η ) ( p3 2 − 2 ( p3 ⋅ q ) + i η ) 1
False \text{False} False
In this case we may explicitly tell the function that this integral doesn’t depend on any external momenta
FCLoopIntegralToGraph[ int, { p1, p3}, Momentum -> {}]
FCLoopGraphPlot[ % ]
{ { 1 → 1 , 1 → 1 } , ( p1 1 − mg 2 p3 − q 1 0 ) , { 1 ( p1 2 − mg 2 + i η ) , 1 ( p3 2 − 2 ( p3 ⋅ q ) + i η ) } , 1 } \left\{\{1\to 1,1\to 1\},\left(
\begin{array}{ccc}
\;\text{p1} & 1 & -\text{mg}^2 \\
\;\text{p3}-q & 1 & 0 \\
\end{array}
\right),\left\{\frac{1}{(\text{p1}^2-\text{mg}^2+i \eta )},\frac{1}{(\text{p3}^2-2 (\text{p3}\cdot q)+i \eta )}\right\},1\right\} { { 1 → 1 , 1 → 1 } , ( p1 p3 − q 1 1 − mg 2 0 ) , { ( p1 2 − mg 2 + i η ) 1 , ( p3 2 − 2 ( p3 ⋅ q ) + i η ) 1 } , 1 }
Eye candy
The Style
option can be used to label lines carrying different masses in a particular way
OptionValue [ FCLoopGraphPlot, Style ]
{ { InternalLine , _ , _ , 0 } : → { Dashed , Thick , Black } , { InternalLine , _ , _ , FeynCalc ˋ FCLoopGraphPlot ˋ Private ˋ mm _ /; FeynCalc ˋ FCLoopGraphPlot ˋ Private ˋ mm =!= 0 } : → { Thick , Black } , { ExternalLine , _ } : → { Thick , Black } } \{\{\text{InternalLine},\_,\_,0\}:\to \{\text{Dashed},\text{Thick},\text{Black}\},\{\text{InternalLine},\_,\_,\text{FeynCalc$\grave{ }$FCLoopGraphPlot$\grave{ }$Private$\grave{ }$mm$\_$}\;\text{/;}\;\text{FeynCalc$\grave{ }$FCLoopGraphPlot$\grave{ }$Private$\grave{ }$mm}\;\text{=!=}0\}:\to \{\text{Thick},\text{Black}\},\{\text{ExternalLine},\_\}:\to \{\text{Thick},\text{Black}\}\} {{ InternalLine , _ , _ , 0 } :→ { Dashed , Thick , Black } , { InternalLine , _ , _ , FeynCalc ˋ FCLoopGraphPlot ˋ Private ˋ mm_ /; FeynCalc ˋ FCLoopGraphPlot ˋ Private ˋ mm =!= 0 } :→ { Thick , Black } , { ExternalLine , _ } :→ { Thick , Black }}
When dealing with factorizing integral it might be necessary to increase VertexDegree
to 7
or 8
(or even a higher value, depending on the integrals)
FCLoopIntegralToGraph[ FAD[{ p1, m1}] FAD[{ p2, m2}], { p1, p2}]
FCLoopGraphPlot[ % ]
{ { 1 → 1 , 1 → 1 } , ( p2 1 − m2 2 p1 1 − m1 2 ) , { 1 ( p2 2 − m2 2 + i η ) , 1 ( p1 2 − m1 2 + i η ) } , 1 } \left\{\{1\to 1,1\to 1\},\left(
\begin{array}{ccc}
\;\text{p2} & 1 & -\text{m2}^2 \\
\;\text{p1} & 1 & -\text{m1}^2 \\
\end{array}
\right),\left\{\frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )},\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )}\right\},1\right\} { { 1 → 1 , 1 → 1 } , ( p2 p1 1 1 − m2 2 − m1 2 ) , { ( p2 2 − m2 2 + i η ) 1 , ( p1 2 − m1 2 + i η ) 1 } , 1 }
FCLoopIntegralToGraph[ FAD[{ p1, m1}] FAD[{ p2, m2}] FAD[ p3, p3 + q ], { p1, p2, p3},
VertexDegree -> 7 ]
FCLoopGraphPlot[ % ]
{ { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 , 2 → 2 , 2 → 2 } , { − q , q , { p3 , 1 , 0 } , { p3 + q , 1 , 0 } , { p2 , 1 , − m2 2 } , { p1 , 1 , − m1 2 } } , { 0 , 0 , 1 ( p3 2 + i η ) , 1 ( ( p3 + q ) 2 + i η ) , 1 ( p2 2 − m2 2 + i η ) , 1 ( p1 2 − m1 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 2,1\to 2,2\to 2,2\to 2\},\left\{-q,q,\{\text{p3},1,0\},\{\text{p3}+q,1,0\},\left\{\text{p2},1,-\text{m2}^2\right\},\left\{\text{p1},1,-\text{m1}^2\right\}\right\},\left\{0,0,\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{((\text{p3}+q)^2+i \eta )},\frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )},\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 , 2 → 2 , 2 → 2 } , { − q , q , { p3 , 1 , 0 } , { p3 + q , 1 , 0 } , { p2 , 1 , − m2 2 } , { p1 , 1 , − m1 2 } } , { 0 , 0 , ( p3 2 + i η ) 1 , (( p3 + q ) 2 + i η ) 1 , ( p2 2 − m2 2 + i η ) 1 , ( p1 2 − m1 2 + i η ) 1 } , 1 }
FCLoopIntegralToGraph[ FAD[{ p1, m1}] FAD[{ p2, m2}] FAD[ p3, p3 + q ] FAD[{ p4, m4}],
{ p1, p2, p3, p4}, VertexDegree -> 9 ]
FCLoopGraphPlot[ % ]
{ { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 , 2 → 2 , 2 → 2 , 2 → 2 } , { − q , q , { p3 , 1 , 0 } , { p3 + q , 1 , 0 } , { p4 , 1 , − m4 2 } , { p2 , 1 , − m2 2 } , { p1 , 1 , − m1 2 } } , { 0 , 0 , 1 ( p3 2 + i η ) , 1 ( ( p3 + q ) 2 + i η ) , 1 ( p4 2 − m4 2 + i η ) , 1 ( p2 2 − m2 2 + i η ) , 1 ( p1 2 − m1 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 2,1\to 2,2\to 2,2\to 2,2\to 2\},\left\{-q,q,\{\text{p3},1,0\},\{\text{p3}+q,1,0\},\left\{\text{p4},1,-\text{m4}^2\right\},\left\{\text{p2},1,-\text{m2}^2\right\},\left\{\text{p1},1,-\text{m1}^2\right\}\right\},\left\{0,0,\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{((\text{p3}+q)^2+i \eta )},\frac{1}{(\text{p4}^2-\text{m4}^2+i \eta )},\frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )},\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 , 2 → 2 , 2 → 2 , 2 → 2 } , { − q , q , { p3 , 1 , 0 } , { p3 + q , 1 , 0 } , { p4 , 1 , − m4 2 } , { p2 , 1 , − m2 2 } , { p1 , 1 , − m1 2 } } , { 0 , 0 , ( p3 2 + i η ) 1 , (( p3 + q ) 2 + i η ) 1 , ( p4 2 − m4 2 + i η ) 1 , ( p2 2 − m2 2 + i η ) 1 , ( p1 2 − m1 2 + i η ) 1 } , 1 }
Here we choose to use thick dashed blue and red lines for massive lines containing mc
and mg
respectively. The massless lines are black an dashed.
FCLoopIntegralToGraph[ FAD[{ k2, mb}, { k3}, { k1 - q , mc}, { k1 - k2, mc}, { k2 - k3}], { k1, k2, k3}]
Magnify [ FCLoopGraphPlot[ % , GraphPlot -> { MultiedgeStyle -> 0.35 , Frame -> True },
Style -> {{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, mg]} -> { Red , Thick , Dashed },
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, mc]} -> { Blue , Thick , Dashed }}], 1.5 ]
{ { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 , 1 → 3 , 2 → 3 , 2 → 3 } , { − q , q , { k1 − q , 1 , − mc 2 } , { k1 − k2 , 1 , − mc 2 } , { k2 , 1 , − mb 2 } , { k3 , 1 , 0 } , { k2 − k3 , 1 , 0 } } , { 0 , 0 , 1 ( k3 2 + i η ) , 1 ( ( k2 − k3 ) 2 + i η ) , 1 ( k2 2 − mb 2 + i η ) , 1 ( ( k1 − q ) 2 − mc 2 + i η ) , 1 ( ( k1 − k2 ) 2 − mc 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 2,1\to 2,1\to 3,2\to 3,2\to 3\},\left\{-q,q,\left\{\text{k1}-q,1,-\text{mc}^2\right\},\left\{\text{k1}-\text{k2},1,-\text{mc}^2\right\},\left\{\text{k2},1,-\text{mb}^2\right\},\{\text{k3},1,0\},\{\text{k2}-\text{k3},1,0\}\right\},\left\{0,0,\frac{1}{(\text{k3}^2+i \eta )},\frac{1}{((\text{k2}-\text{k3})^2+i \eta )},\frac{1}{(\text{k2}^2-\text{mb}^2+i \eta )},\frac{1}{((\text{k1}-q)^2-\text{mc}^2+i \eta )},\frac{1}{((\text{k1}-\text{k2})^2-\text{mc}^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 , 1 → 3 , 2 → 3 , 2 → 3 } , { − q , q , { k1 − q , 1 , − mc 2 } , { k1 − k2 , 1 , − mc 2 } , { k2 , 1 , − mb 2 } , { k3 , 1 , 0 } , { k2 − k3 , 1 , 0 } } , { 0 , 0 , ( k3 2 + i η ) 1 , (( k2 − k3 ) 2 + i η ) 1 , ( k2 2 − mb 2 + i η ) 1 , (( k1 − q ) 2 − mc 2 + i η ) 1 , (( k1 − k2 ) 2 − mc 2 + i η ) 1 } , 1 }
FCLoopIntegralToGraph[ FAD[{ k2, mg}, { k3, mc}, { k1, q }, { k1 - k2}, { k2 - k3, mc}], { k1, k2, k3}]
Magnify [ FCLoopGraphPlot[ % , GraphPlot -> { MultiedgeStyle -> 0.35 , Frame -> True },
Style -> {{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, mg]} -> { Red , Thick , Dashed },
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, mc]} -> { Blue , Thick , Dashed }}], 1.5 ]
{ { 1 → 2 , 1 → 3 , 1 → 3 , 2 → 3 , 2 → 3 } , ( k2 1 − mg 2 k3 1 − mc 2 k2 − k3 1 − mc 2 k1 − k2 1 0 k1 1 − q 2 ) , { 1 ( k3 2 − mc 2 + i η ) , 1 ( k2 2 − mg 2 + i η ) , 1 ( ( k1 − k2 ) 2 + i η ) , 1 ( k1 2 − q 2 + i η ) , 1 ( ( k2 − k3 ) 2 − mc 2 + i η ) } , 1 } \left\{\{1\to 2,1\to 3,1\to 3,2\to 3,2\to 3\},\left(
\begin{array}{ccc}
\;\text{k2} & 1 & -\text{mg}^2 \\
\;\text{k3} & 1 & -\text{mc}^2 \\
\;\text{k2}-\text{k3} & 1 & -\text{mc}^2 \\
\;\text{k1}-\text{k2} & 1 & 0 \\
\;\text{k1} & 1 & -q^2 \\
\end{array}
\right),\left\{\frac{1}{(\text{k3}^2-\text{mc}^2+i \eta )},\frac{1}{(\text{k2}^2-\text{mg}^2+i \eta )},\frac{1}{((\text{k1}-\text{k2})^2+i \eta )},\frac{1}{(\text{k1}^2-q^2+i \eta )},\frac{1}{((\text{k2}-\text{k3})^2-\text{mc}^2+i \eta )}\right\},1\right\} ⎩ ⎨ ⎧ { 1 → 2 , 1 → 3 , 1 → 3 , 2 → 3 , 2 → 3 } , k2 k3 k2 − k3 k1 − k2 k1 1 1 1 1 1 − mg 2 − mc 2 − mc 2 0 − q 2 , { ( k3 2 − mc 2 + i η ) 1 , ( k2 2 − mg 2 + i η ) 1 , (( k1 − k2 ) 2 + i η ) 1 , ( k1 2 − q 2 + i η ) 1 , (( k2 − k3 ) 2 − mc 2 + i η ) 1 } , 1 ⎭ ⎬ ⎫
FCLoopIntegralToGraph[ FAD[{ k2, mg}, { k3, mc}, { k1 - q }, { k2 - q , mb}, { k1 - k2}, { k2 - k3, mc}],
{ k1, k2, k3}]
Magnify [ FCLoopGraphPlot[ % , GraphPlot -> { MultiedgeStyle -> 0.35 , Frame -> True },
Style -> {{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, mg]} -> { Red , Thick , Dashed },
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, mc]} -> { Blue , Thick , Dashed }}], 1.5 ]
{ { − 3 → 2 , − 1 → 1 , 1 → 3 , 1 → 4 , 2 → 3 , 2 → 3 , 2 → 4 , 2 → 4 } , { − q , q , { k2 − q , 1 , − mb 2 } , { k2 , 1 , − mg 2 } , { k1 − q , 1 , 0 } , { k1 − k2 , 1 , 0 } , { k3 , 1 , − mc 2 } , { k2 − k3 , 1 , − mc 2 } } , { 0 , 0 , 1 ( ( k1 − q ) 2 + i η ) , 1 ( k3 2 − mc 2 + i η ) , 1 ( k2 2 − mg 2 + i η ) , 1 ( ( k1 − k2 ) 2 + i η ) , 1 ( ( k2 − q ) 2 − mb 2 + i η ) , 1 ( ( k2 − k3 ) 2 − mc 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 3,1\to 4,2\to 3,2\to 3,2\to 4,2\to 4\},\left\{-q,q,\left\{\text{k2}-q,1,-\text{mb}^2\right\},\left\{\text{k2},1,-\text{mg}^2\right\},\{\text{k1}-q,1,0\},\{\text{k1}-\text{k2},1,0\},\left\{\text{k3},1,-\text{mc}^2\right\},\left\{\text{k2}-\text{k3},1,-\text{mc}^2\right\}\right\},\left\{0,0,\frac{1}{((\text{k1}-q)^2+i \eta )},\frac{1}{(\text{k3}^2-\text{mc}^2+i \eta )},\frac{1}{(\text{k2}^2-\text{mg}^2+i \eta )},\frac{1}{((\text{k1}-\text{k2})^2+i \eta )},\frac{1}{((\text{k2}-q)^2-\text{mb}^2+i \eta )},\frac{1}{((\text{k2}-\text{k3})^2-\text{mc}^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 3 , 1 → 4 , 2 → 3 , 2 → 3 , 2 → 4 , 2 → 4 } , { − q , q , { k2 − q , 1 , − mb 2 } , { k2 , 1 , − mg 2 } , { k1 − q , 1 , 0 } , { k1 − k2 , 1 , 0 } , { k3 , 1 , − mc 2 } , { k2 − k3 , 1 , − mc 2 } } , { 0 , 0 , (( k1 − q ) 2 + i η ) 1 , ( k3 2 − mc 2 + i η ) 1 , ( k2 2 − mg 2 + i η ) 1 , (( k1 − k2 ) 2 + i η ) 1 , (( k2 − q ) 2 − mb 2 + i η ) 1 , (( k2 − k3 ) 2 − mc 2 + i η ) 1 } , 1 }
FCLoopIntegralToGraph[ FAD[{ k2, 0 , 2 }, { k1 - q }, { k1 - k3, mc}, { k2 - k3, mc}], { k1, k2, k3}]
Magnify [ FCLoopGraphPlot[ % , GraphPlot -> { MultiedgeStyle -> 0.35 , Frame -> True },
Style -> {{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, mg]} -> { Red , Thick , Dashed },
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, mc]} -> { Blue , Thick , Dashed }}], 1.5 ]
{ { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 , 1 → 2 , 1 → 2 } , { − q , q , { k2 , 2 , 0 } , { k1 − q , 1 , 0 } , { k2 − k3 , 1 , − mc 2 } , { k1 − k3 , 1 , − mc 2 } } , { 0 , 0 , 1 ( k2 2 + i η ) , 1 ( ( k1 − q ) 2 + i η ) , 1 ( ( k2 − k3 ) 2 − mc 2 + i η ) , 1 ( ( k1 − k3 ) 2 − mc 2 + i η ) } , 1 } \left\{\{-3\to 2,-1\to 1,1\to 2,1\to 2,1\to 2,1\to 2\},\left\{-q,q,\{\text{k2},2,0\},\{\text{k1}-q,1,0\},\left\{\text{k2}-\text{k3},1,-\text{mc}^2\right\},\left\{\text{k1}-\text{k3},1,-\text{mc}^2\right\}\right\},\left\{0,0,\frac{1}{(\text{k2}^2+i \eta )},\frac{1}{((\text{k1}-q)^2+i \eta )},\frac{1}{((\text{k2}-\text{k3})^2-\text{mc}^2+i \eta )},\frac{1}{((\text{k1}-\text{k3})^2-\text{mc}^2+i \eta )}\right\},1\right\} { { − 3 → 2 , − 1 → 1 , 1 → 2 , 1 → 2 , 1 → 2 , 1 → 2 } , { − q , q , { k2 , 2 , 0 } , { k1 − q , 1 , 0 } , { k2 − k3 , 1 , − mc 2 } , { k1 − k3 , 1 , − mc 2 } } , { 0 , 0 , ( k2 2 + i η ) 1 , (( k1 − q ) 2 + i η ) 1 , (( k2 − k3 ) 2 − mc 2 + i η ) 1 , (( k1 − k3 ) 2 − mc 2 + i η ) 1 } , 1 }
We can style a fully massive 1-loop box in a very creative way
FCLoopIntegralToGraph[ FAD[{ p , m1}, { p + q1, m2}, { p + q1 + q2, m3}, { p + q1 + q2 + q3, m4}], { p }]
FCLoopGraphPlot[ % , GraphPlot -> { MultiedgeStyle -> 0.35 , Frame -> True }, Style -> {
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, m1]} -> { Red , Thick },
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, m2]} -> { Blue , Thick },
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, m3]} -> { Green , Thick },
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, m4]} -> { Purple , Thick },
{ "ExternalLine" , q1} -> { Brown , Thick , Dashed }
}]
{ { − 4 → 4 , − 3 → 1 , − 2 → 2 , − 1 → 3 , 1 → 2 , 1 → 4 , 2 → 3 , 3 → 4 } , { q1 + q2 + q3 , q3 , q2 , q1 , { p + q1 + q2 , 1 , − m3 2 } , { p + q1 + q2 + q3 , 1 , − m4 2 } , { p + q1 , 1 , − m2 2 } , { p , 1 , − m1 2 } } , { 0 , 0 , 0 , 0 , 1 ( p 2 − m1 2 + i η ) , 1 ( ( p + q1 ) 2 − m2 2 + i η ) , 1 ( ( p + q1 + q2 ) 2 − m3 2 + i η ) , 1 ( ( p + q1 + q2 + q3 ) 2 − m4 2 + i η ) } , 1 } \left\{\{-4\to 4,-3\to 1,-2\to 2,-1\to 3,1\to 2,1\to 4,2\to 3,3\to 4\},\left\{\text{q1}+\text{q2}+\text{q3},\text{q3},\text{q2},\text{q1},\left\{p+\text{q1}+\text{q2},1,-\text{m3}^2\right\},\left\{p+\text{q1}+\text{q2}+\text{q3},1,-\text{m4}^2\right\},\left\{p+\text{q1},1,-\text{m2}^2\right\},\left\{p,1,-\text{m1}^2\right\}\right\},\left\{0,0,0,0,\frac{1}{(p^2-\text{m1}^2+i \eta )},\frac{1}{((p+\text{q1})^2-\text{m2}^2+i \eta )},\frac{1}{((p+\text{q1}+\text{q2})^2-\text{m3}^2+i \eta )},\frac{1}{((p+\text{q1}+\text{q2}+\text{q3})^2-\text{m4}^2+i \eta )}\right\},1\right\} { { − 4 → 4 , − 3 → 1 , − 2 → 2 , − 1 → 3 , 1 → 2 , 1 → 4 , 2 → 3 , 3 → 4 } , { q1 + q2 + q3 , q3 , q2 , q1 , { p + q1 + q2 , 1 , − m3 2 } , { p + q1 + q2 + q3 , 1 , − m4 2 } , { p + q1 , 1 , − m2 2 } , { p , 1 , − m1 2 } } , { 0 , 0 , 0 , 0 , ( p 2 − m1 2 + i η ) 1 , (( p + q1 ) 2 − m2 2 + i η ) 1 , (( p + q1 + q2 ) 2 − m3 2 + i η ) 1 , (( p + q1 + q2 + q3 ) 2 − m4 2 + i η ) 1 } , 1 }
The same goes for a 2-loop box with 3 massive lines
FCLoopIntegralToGraph[ FAD[{ p1, m1}, { p2, m2}, { Q1 + p1, m3}, Q2 - p1, Q1 + p1 + p2, Q2 - p1 - p2,
Q2 + Q3 - p1 - p2], { p1, p2}]
FCLoopGraphPlot[ % , GraphPlot -> { MultiedgeStyle -> 0.35 , Frame -> True }, Style -> {
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, m1]} -> { Red , Thick },
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, m2]} -> { Blue , Thick },
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, m3]} -> { Green , Thick },
{ "InternalLine" , _, _, mm_ / ; ! FreeQ [ mm, m4]} -> { Purple , Thick },
{ "ExternalLine" , q1} -> { Brown , Thick , Dashed }
}]
{ { − 4 → 4 , − 3 → 1 , − 2 → 2 , − 1 → 3 , 1 → 4 , 1 → 6 , 2 → 3 , 2 → 6 , 3 → 5 , 4 → 5 , 5 → 6 } , { Q1 + Q2 + Q3 , Q3 , Q2 , Q1 , { − p1 − p2 + Q2 + Q3 , 1 , 0 } , { − p1 − p2 + Q2 , 1 , 0 } , { p1 , 1 , − m1 2 } , { Q2 − p1 , 1 , 0 } , { p1 + Q1 , 1 , − m3 2 } , { p1 + p2 + Q1 , 1 , 0 } , { p2 , 1 , − m2 2 } } , { 0 , 0 , 0 , 0 , 1 ( ( p1 + p2 + Q1 ) 2 + i η ) , 1 ( ( Q2 − p1 ) 2 + i η ) , 1 ( p2 2 − m2 2 + i η ) , 1 ( p1 2 − m1 2 + i η ) , 1 ( ( p1 + Q1 ) 2 − m3 2 + i η ) , 1 ( ( − p1 − p2 + Q2 ) 2 + i η ) , 1 ( ( − p1 − p2 + Q2 + Q3 ) 2 + i η ) } , 1 } \left\{\{-4\to 4,-3\to 1,-2\to 2,-1\to 3,1\to 4,1\to 6,2\to 3,2\to 6,3\to 5,4\to 5,5\to 6\},\left\{\text{Q1}+\text{Q2}+\text{Q3},\text{Q3},\text{Q2},\text{Q1},\{-\text{p1}-\text{p2}+\text{Q2}+\text{Q3},1,0\},\{-\text{p1}-\text{p2}+\text{Q2},1,0\},\left\{\text{p1},1,-\text{m1}^2\right\},\{\text{Q2}-\text{p1},1,0\},\left\{\text{p1}+\text{Q1},1,-\text{m3}^2\right\},\{\text{p1}+\text{p2}+\text{Q1},1,0\},\left\{\text{p2},1,-\text{m2}^2\right\}\right\},\left\{0,0,0,0,\frac{1}{((\text{p1}+\text{p2}+\text{Q1})^2+i \eta )},\frac{1}{((\text{Q2}-\text{p1})^2+i \eta )},\frac{1}{(\text{p2}^2-\text{m2}^2+i \eta )},\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )},\frac{1}{((\text{p1}+\text{Q1})^2-\text{m3}^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+\text{Q2})^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+\text{Q2}+\text{Q3})^2+i \eta )}\right\},1\right\} { { − 4 → 4 , − 3 → 1 , − 2 → 2 , − 1 → 3 , 1 → 4 , 1 → 6 , 2 → 3 , 2 → 6 , 3 → 5 , 4 → 5 , 5 → 6 } , { Q1 + Q2 + Q3 , Q3 , Q2 , Q1 , { − p1 − p2 + Q2 + Q3 , 1 , 0 } , { − p1 − p2 + Q2 , 1 , 0 } , { p1 , 1 , − m1 2 } , { Q2 − p1 , 1 , 0 } , { p1 + Q1 , 1 , − m3 2 } , { p1 + p2 + Q1 , 1 , 0 } , { p2 , 1 , − m2 2 } } , { 0 , 0 , 0 , 0 , (( p1 + p2 + Q1 ) 2 + i η ) 1 , (( Q2 − p1 ) 2 + i η ) 1 , ( p2 2 − m2 2 + i η ) 1 , ( p1 2 − m1 2 + i η ) 1 , (( p1 + Q1 ) 2 − m3 2 + i η ) 1 , (( − p1 − p2 + Q2 ) 2 + i η ) 1 , (( − p1 − p2 + Q2 + Q3 ) 2 + i η ) 1 } , 1 }
One can also (sort of) visualize the momentum flow, where we use powers to denote the dots
FCLoopIntegralToGraph[ FCTopology[ topo1X1, { SFAD[{{ p2, 0 }, { m1^ 2 , 1 }, 2 }],
SFAD[{{ p1, 0 }, { m1^ 2 , 1 }, 2 }],
SFAD[{{ p2 + p3, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p2 + p3, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ p1 + p3, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + p2 + p3, 0 }, { 0 , 1 }, 1 }]}, { p1, p2, p3}, {}, {}, {}]]
FCLoopGraphPlot[ % , GraphPlot -> { MultiedgeStyle -> 0.35 , Frame -> True }, Labeled -> {
{ "InternalLine" , x_ , pow_ , _} :> x ^ pow,
{ "ExternalLine" , _} :> {}}]
{ { 1 → 2 , 1 → 3 , 1 → 3 , 2 → 3 , 2 → 3 } , ( p1 + p2 + p3 1 0 p2 2 − m1 2 p1 + p3 1 0 p1 2 − m1 2 p2 + p3 2 0 ) , { 1 ( p2 2 − m1 2 + i η ) , 1 ( p1 2 − m1 2 + i η ) , 1 ( ( p2 + p3 ) 2 + i η ) , 1 ( ( p1 + p3 ) 2 + i η ) , 1 ( ( p1 + p2 + p3 ) 2 + i η ) } , 1 } \left\{\{1\to 2,1\to 3,1\to 3,2\to 3,2\to 3\},\left(
\begin{array}{ccc}
\;\text{p1}+\text{p2}+\text{p3} & 1 & 0 \\
\;\text{p2} & 2 & -\text{m1}^2 \\
\;\text{p1}+\text{p3} & 1 & 0 \\
\;\text{p1} & 2 & -\text{m1}^2 \\
\;\text{p2}+\text{p3} & 2 & 0 \\
\end{array}
\right),\left\{\frac{1}{(\text{p2}^2-\text{m1}^2+i \eta )},\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3})^2+i \eta )}\right\},1\right\} ⎩ ⎨ ⎧ { 1 → 2 , 1 → 3 , 1 → 3 , 2 → 3 , 2 → 3 } , p1 + p2 + p3 p2 p1 + p3 p1 p2 + p3 1 2 1 2 2 0 − m1 2 0 − m1 2 0 , { ( p2 2 − m1 2 + i η ) 1 , ( p1 2 − m1 2 + i η ) 1 , (( p2 + p3 ) 2 + i η ) 1 , (( p1 + p3 ) 2 + i η ) 1 , (( p1 + p2 + p3 ) 2 + i η ) 1 } , 1 ⎭ ⎬ ⎫