FeynCalc manual (development version)

FCLoopFindIntegralMappings

FCLoopFindIntegralMappings[{int1, int2, ...}, {p1, p2, ...}] finds mappings between scalar multiloop integrals int1, int2, ... that depend on the loop momenta p1, p2, ... using the algorithm of Alexey Pak arXiv:1111.0868.

The current implementation is based on the FindEquivalents function from FIRE 6 arXiv:1901.07808

It is also possible to invoke the function as FCLoopFindIntegralMappings[{GLI[...], ...}, {FCTopology[...], ...}] or FCLoopFindIntegralMappings[{FCTopology[...], ...}].

Notice that in this case the value of the option FinalSubstitutions is ignored, as replacement rules will be extracted directly from the definition of the topology.

The default output is a list of two lists. The first list contains mapping rules between different loop integrals, while the second list provides all unique master integrals extracted from the input.

An alternative output mode is activated when the option List is set to True. In this case the output is a list of lists, where each list contains master integrals that were identified to be identical.

The option PreferredIntegrals can be used to enforce the mapping onto a preferred set of master integral. Notice that the final result will only contain those preferred integrals, that are actually present in the input.

See also

Overview, FCTopology, GLI, FCLoopToPakForm, FCLoopPakOrder, FCLoopFindTopologyMappings

Examples

When given a list of FeynAmpDenominator-integrals, the function will merely group identical integrals into sublists

ints = {FAD[{p1, m1}], FAD[{p1 + q, m1}], FAD[{p1, m2}]}

{1p12−m12,1(p1+q)2−m12,1p12−m22}\left\{\frac{1}{\text{p1}^2-\text{m1}^2},\frac{1}{(\text{p1}+q)^2-\text{m1}^2},\frac{1}{\text{p1}^2-\text{m2}^2}\right\}

FCLoopFindIntegralMappings[ints, {p1}]

{{1p12−m12,1(p1+q)2−m12},{1p12−m22}}\left\{\left\{\frac{1}{\text{p1}^2-\text{m1}^2},\frac{1}{(\text{p1}+q)^2-\text{m1}^2}\right\},\left\{\frac{1}{\text{p1}^2-\text{m2}^2}\right\}\right\}

The following 3 integrals look rather different from each other, but are actually identical

ints = {FAD[p1] FAD[p1 - p3 - p4] FAD[p4] FAD[p3 + q1] FAD[{p3, m1}] FAD[{p1 - p4, m1}]*
    FAD[{p1 + q1, 0}, {p1 + q1, 0}], FAD[p4] FAD[p1 - p3 + q1] FAD[p3 + q1] FAD[p1 + p4 + q1]*
    FAD[{p3, m1}] FAD[{p1 + q1, m1}] FAD[{p1 + p4 + 2 q1, 0}, {p1 + p4 + 2 q1, 0}], 
   FAD[p1] FAD[p4 - 2 q1] FAD[p3 + q1] FAD[p1 - p3 - p4 + 2 q1] FAD[{p3, m1}]*
    FAD[{p1 - p4 + 2 q1, m1}] FAD[{p1 + q1, 0}, {p1 + q1, 0}]}

{1p12  p42(p32−m12)(p1+q1)4(p3+q1)2(p1−p3−p4)2((p1−p4)2−m12),1p42(p32−m12)(p3+q1)2(p1−p3+q1)2(p1+p4+q1)2(p1+p4+2  q1)4((p1+q1)2−m12),1p12(p32−m12)(p1+q1)4(p3+q1)2(p4−2  q1)2(p1−p3−p4+2  q1)2((p1−p4+2  q1)2−m12)}\left\{\frac{1}{\text{p1}^2 \;\text{p4}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p1}+\text{q1})^4 (\text{p3}+\text{q1})^2 (\text{p1}-\text{p3}-\text{p4})^2 \left((\text{p1}-\text{p4})^2-\text{m1}^2\right)},\frac{1}{\text{p4}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p3}+\text{q1})^2 (\text{p1}-\text{p3}+\text{q1})^2 (\text{p1}+\text{p4}+\text{q1})^2 (\text{p1}+\text{p4}+2 \;\text{q1})^4 \left((\text{p1}+\text{q1})^2-\text{m1}^2\right)},\frac{1}{\text{p1}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p1}+\text{q1})^4 (\text{p3}+\text{q1})^2 (\text{p4}-2 \;\text{q1})^2 (\text{p1}-\text{p3}-\text{p4}+2 \;\text{q1})^2 \left((\text{p1}-\text{p4}+2 \;\text{q1})^2-\text{m1}^2\right)}\right\}

FCLoopFindIntegralMappings[ints, {p1, p3, p4}]

(1p12  p42(p32−m12)(p1+q1)4(p3+q1)2(p1−p3−p4)2((p1−p4)2−m12)1p42(p32−m12)(p3+q1)2(p1−p3+q1)2(p1+p4+q1)2(p1+p4+2  q1)4((p1+q1)2−m12)1p12(p32−m12)(p1+q1)4(p3+q1)2(p4−2  q1)2(p1−p3−p4+2  q1)2((p1−p4+2  q1)2−m12))\left( \begin{array}{ccc} \frac{1}{\text{p1}^2 \;\text{p4}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p1}+\text{q1})^4 (\text{p3}+\text{q1})^2 (\text{p1}-\text{p3}-\text{p4})^2 \left((\text{p1}-\text{p4})^2-\text{m1}^2\right)} & \frac{1}{\text{p4}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p3}+\text{q1})^2 (\text{p1}-\text{p3}+\text{q1})^2 (\text{p1}+\text{p4}+\text{q1})^2 (\text{p1}+\text{p4}+2 \;\text{q1})^4 \left((\text{p1}+\text{q1})^2-\text{m1}^2\right)} & \frac{1}{\text{p1}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p1}+\text{q1})^4 (\text{p3}+\text{q1})^2 (\text{p4}-2 \;\text{q1})^2 (\text{p1}-\text{p3}-\text{p4}+2 \;\text{q1})^2 \left((\text{p1}-\text{p4}+2 \;\text{q1})^2-\text{m1}^2\right)} \\ \end{array} \right)

If the input is a list of GLI-integrals, FCLoopFindIntegralMappings will return a list containing two sublists. The former will be a list of replacement rules while the latter will contain all unique master integrals

ClearAll[topo1, topo2]; 
 
topos = {
   FCTopology[topo1, {SFAD[{p1, m^2}], SFAD[{p2, m^2}]}, {p1, p2}, {}, {}, {}], 
   FCTopology[topo2, {SFAD[{p3, m^2}], SFAD[{p4, m^2}]}, {p3, p4}, {}, {}, {}] 
  }

{FCTopology(topo1,{1(p12−m2+iη),1(p22−m2+iη)},{p1,p2},{},{},{}),FCTopology(topo2,{1(p32−m2+iη),1(p42−m2+iη)},{p3,p4},{},{},{})}\left\{\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{(\text{p1}^2-m^2+i \eta )},\frac{1}{(\text{p2}^2-m^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{\},\{\},\{\}\right),\text{FCTopology}\left(\text{topo2},\left\{\frac{1}{(\text{p3}^2-m^2+i \eta )},\frac{1}{(\text{p4}^2-m^2+i \eta )}\right\},\{\text{p3},\text{p4}\},\{\},\{\},\{\}\right)\right\}

glis = {GLI[topo1, {1, 1}], GLI[topo1, {1, 2}], GLI[topo1, {2, 1}], GLI[topo2, {1, 1}], 
   GLI[topo2, {2, 2}]}

{Gtopo1(1,1),Gtopo1(1,2),Gtopo1(2,1),Gtopo2(1,1),Gtopo2(2,2)}\left\{G^{\text{topo1}}(1,1),G^{\text{topo1}}(1,2),G^{\text{topo1}}(2,1),G^{\text{topo2}}(1,1),G^{\text{topo2}}(2,2)\right\}

FCLoopFindIntegralMappings[glis, topos]

{{Gtopo2(1,1)→Gtopo1(1,1),Gtopo1(2,1)→Gtopo1(1,2)},{Gtopo1(1,1),Gtopo1(1,2),Gtopo2(2,2)}}\left\{\left\{G^{\text{topo2}}(1,1)\to G^{\text{topo1}}(1,1),G^{\text{topo1}}(2,1)\to G^{\text{topo1}}(1,2)\right\},\left\{G^{\text{topo1}}(1,1),G^{\text{topo1}}(1,2),G^{\text{topo2}}(2,2)\right\}\right\}

This behavior can be turned off by setting the value of the option List to True

FCLoopFindIntegralMappings[glis, topos, List -> True]

{{Gtopo1(1,1),Gtopo2(1,1)},{Gtopo1(1,2),Gtopo1(2,1)},{Gtopo2(2,2)}}\left\{\left\{G^{\text{topo1}}(1,1),G^{\text{topo2}}(1,1)\right\},\left\{G^{\text{topo1}}(1,2),G^{\text{topo1}}(2,1)\right\},\left\{G^{\text{topo2}}(2,2)\right\}\right\}

In practice, one usually has a list of preferred integrals onto which one would like to map the occurring master integrals. Such integrals can be specified via the PreferredIntegrals option

FCLoopFindIntegralMappings[glis, topos, PreferredIntegrals -> {GLI[topo2, {1, 1}], 
    GLI[topo2, {2, 1}]}]

(Gtopo1(1,1)→Gtopo2(1,1)Gtopo1(1,2)→Gtopo2(2,1)Gtopo1(2,1)→Gtopo2(2,1)Gtopo2(1,1)Gtopo2(2,1)Gtopo2(2,2))\left( \begin{array}{ccc} G^{\text{topo1}}(1,1)\to G^{\text{topo2}}(1,1) & G^{\text{topo1}}(1,2)\to G^{\text{topo2}}(2,1) & G^{\text{topo1}}(2,1)\to G^{\text{topo2}}(2,1) \\ G^{\text{topo2}}(1,1) & G^{\text{topo2}}(2,1) & G^{\text{topo2}}(2,2) \\ \end{array} \right)

The indices of GLIs do not have to be integers

topos = {
   FCTopology[prop2LtopoG20, {SFAD[{{p1, 0}, {0, 1}, 1}], 
     SFAD[{{p1 + q1, 0}, {m3^2, 1}, 1}], SFAD[{{p3, 0}, {0, 1}, 1}], 
     SFAD[{{p3 + q1, 0}, {0, 1}, 1}], SFAD[{{p1 - p3, 0}, {0, 1}, 1}]}, 
    {p1, p3}, {q1}, {}, {}], 
   FCTopology[prop2LtopoG21, {SFAD[{{p1, 0}, {m1^2, 1}, 1}], 
     SFAD[{{p1 + q1, 0}, {m3^2, 1}, 1}], 
     SFAD[{{p3, 0}, {0, 1}, 1}], SFAD[{{p3 + q1, 0}, {0, 1}, 1}], 
     SFAD[{{p1 - p3, 0}, {0, 1}, 1}]}, {p1, p3}, {q1}, {}, {}] 
  }

{FCTopology(prop2LtopoG20,{1(p12+iη),1((p1+q1)2−m32+iη),1(p32+iη),1((p3+q1)2+iη),1((p1−p3)2+iη)},{p1,p3},{q1},{},{}),FCTopology(prop2LtopoG21,{1(p12−m12+iη),1((p1+q1)2−m32+iη),1(p32+iη),1((p3+q1)2+iη),1((p1−p3)2+iη)},{p1,p3},{q1},{},{})}\left\{\text{FCTopology}\left(\text{prop2LtopoG20},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p1}+\text{q1})^2-\text{m3}^2+i \eta )},\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{((\text{p3}+\text{q1})^2+i \eta )},\frac{1}{((\text{p1}-\text{p3})^2+i \eta )}\right\},\{\text{p1},\text{p3}\},\{\text{q1}\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LtopoG21},\left\{\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{p3}^2+i \eta )},\frac{1}{((\text{p3}+\text{q1})^2+i \eta )},\frac{1}{((\text{p1}-\text{p3})^2+i \eta )}\right\},\{\text{p1},\text{p3}\},\{\text{q1}\},\{\},\{\}\right)\right\}

FCLoopFindIntegralMappings[{GLI[prop2LtopoG21, {0, n1, n2, n3, n4}], 
   GLI[prop2LtopoG20, {0, n1, n2, n3, n4}]}, topos]

(Gprop2LtopoG21(0,n1,n2,n3,n4)→Gprop2LtopoG20(0,n1,n2,n3,n4)Gprop2LtopoG20(0,n1,n2,n3,n4))\left( \begin{array}{c} G^{\text{prop2LtopoG21}}(0,\text{n1},\text{n2},\text{n3},\text{n4})\to G^{\text{prop2LtopoG20}}(0,\text{n1},\text{n2},\text{n3},\text{n4}) \\ G^{\text{prop2LtopoG20}}(0,\text{n1},\text{n2},\text{n3},\text{n4}) \\ \end{array} \right)

It is also possible to find mappings for factorizing integrals, provided that suitable products of integrals are given as preferred integrals

topos = {FCTopology[prop2Ltopo31313, {SFAD[{{
        I p1, 0}, {-m3^2, -1}, 1}], SFAD[{{I (p1 + q1), 0}, {-m1^2, -1}, 1}], SFAD[{{
        I p3, 0}, {-m3^2, -1}, 1}], SFAD[{{I 
         (p3 + q1), 0}, {-m1^2, -1}, 1}], SFAD[{{
        I (p1 - p3), 0}, {-m3^2, -1}, 1}]}, {p1, p3}, {q1}, {SPD[q1, q1] -> m1^2}, {}], 
   FCTopology[tad1Ltopo2, {SFAD[{{I p1, 0}, {-m3^2, -1}, 1}]}, {p1}, {}, {SPD[q1,q1] -> m1^2}, {}]}

{FCTopology(prop2Ltopo31313,{1(−p12+m32−iη),1(−(p1+q1)2+m12−iη),1(−p32+m32−iη),1(−(p3+q1)2+m12−iη),1(−(p1−p3)2+m32−iη)},{p1,p3},{q1},{q12→  m12},{}),FCTopology(tad1Ltopo2,{1(−p12+m32−iη)},{p1},{},{q12→  m12},{})}\left\{\text{FCTopology}\left(\text{prop2Ltopo31313},\left\{\frac{1}{(-\text{p1}^2+\text{m3}^2-i \eta )},\frac{1}{(-(\text{p1}+\text{q1})^2+\text{m1}^2-i \eta )},\frac{1}{(-\text{p3}^2+\text{m3}^2-i \eta )},\frac{1}{(-(\text{p3}+\text{q1})^2+\text{m1}^2-i \eta )},\frac{1}{(-(\text{p1}-\text{p3})^2+\text{m3}^2-i \eta )}\right\},\{\text{p1},\text{p3}\},\{\text{q1}\},\left\{\text{q1}^2\to \;\text{m1}^2\right\},\{\}\right),\text{FCTopology}\left(\text{tad1Ltopo2},\left\{\frac{1}{(-\text{p1}^2+\text{m3}^2-i \eta )}\right\},\{\text{p1}\},\{\},\left\{\text{q1}^2\to \;\text{m1}^2\right\},\{\}\right)\right\}

Here we ask the function to map all products of two 1-loop tadpoles to GLI[tad1Ltopo2,{1}]^2

FCLoopFindIntegralMappings[{GLI[tad1Ltopo2, {1}]^2, 
   GLI[prop2Ltopo31313, {0, 0, 1, 0, 1}]}, topos, PreferredIntegrals -> {GLI[tad1Ltopo2, {1}]^2}]

(Gprop2Ltopo31313(0,0,1,0,1)→Gtad1Ltopo2(1)2Gtad1Ltopo2(1)2)\left( \begin{array}{c} G^{\text{prop2Ltopo31313}}(0,0,1,0,1)\to G^{\text{tad1Ltopo2}}(1)^2 \\ G^{\text{tad1Ltopo2}}(1)^2 \\ \end{array} \right)