FeynCalc manual (development version)

FCLoopCreateRulesToGLI

FCLoopCreateRulesToGLI[topo] creates replacement rules for converting numerators from the given topology to GLI objects with inverse propagators.

It is also possible to use FCLoopCreateRulesToGLI[{topo1, topo2, ...}].

See also

Overview, FCTopology, GLI, FCLoopFindTopologies, FCLoopFindTopologyMappings.

Examples

1-loop tadpole

FCLoopCreateRulesToGLI[FCTopology[topo1, {SFAD[{p1, m^2}]}, {p1}, {}, {}, {}]]

{p12Gtopo1(1)+m2}\left\{\text{p1}^2\to G^{\text{topo1}}(-1)+m^2\right\}

2-loop tadpole with 3 different masses

FCLoopCreateRulesToGLI[FCTopology[topo1, {SFAD[{p1, m1^2}], SFAD[{p2, m2^2}], 
    SFAD[{p1 - p2, m3^2}]}, {p1, p2}, {}, {}, {}]]

{p12Gtopo1(1,0,0)+m12,p22Gtopo1(0,1,0)+m22,p1  p212(Gtopo1(1,0,0)+Gtopo1(0,1,0)Gtopo1(0,0,1)+m12+m22m32)}\left\{\text{p1}^2\to G^{\text{topo1}}(-1,0,0)+\text{m1}^2,\text{p2}^2\to G^{\text{topo1}}(0,-1,0)+\text{m2}^2,\text{p1}\cdot \;\text{p2}\to \frac{1}{2} \left(G^{\text{topo1}}(-1,0,0)+G^{\text{topo1}}(0,-1,0)-G^{\text{topo1}}(0,0,-1)+\text{m1}^2+\text{m2}^2-\text{m3}^2\right)\right\}

2-loop self-energy

FCLoopCreateRulesToGLI[FCTopology["prop2Lv1", 
   {SFAD[{p1, m1^2}], SFAD[{p2, m2^2}], SFAD[p1 - q], SFAD[p2 - q], 
    SFAD[{p1 - p2, m3^2}]}, {p1, p2}, {Q}, {}, {}]]

{p12Gprop2Lv1(1,0,0,0,0)+m12,p22Gprop2Lv1(0,1,0,0,0)+m22,p1  p212(Gprop2Lv1(1,0,0,0,0)+Gprop2Lv1(0,1,0,0,0)Gprop2Lv1(0,0,0,0,1)+m12+m22m32),p1q12(Gprop2Lv1(1,0,0,0,0)Gprop2Lv1(0,0,1,0,0)+m12+q2),p2q12(Gprop2Lv1(0,1,0,0,0)Gprop2Lv1(0,0,0,1,0)+m22+q2)}\left\{\text{p1}^2\to G^{\text{prop2Lv1}}(-1,0,0,0,0)+\text{m1}^2,\text{p2}^2\to G^{\text{prop2Lv1}}(0,-1,0,0,0)+\text{m2}^2,\text{p1}\cdot \;\text{p2}\to \frac{1}{2} \left(G^{\text{prop2Lv1}}(-1,0,0,0,0)+G^{\text{prop2Lv1}}(0,-1,0,0,0)-G^{\text{prop2Lv1}}(0,0,0,0,-1)+\text{m1}^2+\text{m2}^2-\text{m3}^2\right),\text{p1}\cdot q\to \frac{1}{2} \left(G^{\text{prop2Lv1}}(-1,0,0,0,0)-G^{\text{prop2Lv1}}(0,0,-1,0,0)+\text{m1}^2+q^2\right),\text{p2}\cdot q\to \frac{1}{2} \left(G^{\text{prop2Lv1}}(0,-1,0,0,0)-G^{\text{prop2Lv1}}(0,0,0,-1,0)+\text{m2}^2+q^2\right)\right\}

A list of 3-loop self-energy topologies

topoList = {
   FCTopology["prop3Lv1", {SFAD[p1], SFAD[p2], SFAD[p3], SFAD[p1 - p2], SFAD[p2 - p3], SFAD[p1 + q1], 
        SFAD[p2 + q1], SFAD[p3 + q1], SFAD[{{0, p1 . p3}}]}, {p1, p2, p3}, {q1}, {}, {}], 
   
   FCTopology["prop3L2", {SFAD[p1], SFAD[p2], SFAD[p3], SFAD[p1 - p2], SFAD[p2 - p3], SFAD[p1 - p3], 
        SFAD[p1 + q1], SFAD[p3 + q1], SFAD[{{0, (p1 - p2) . q1}}]}, {p1, p2, p3}, {q1}, {}, {}], 
    
   FCTopology["prop3L3", {SFAD[p1], SFAD[p1 - p4], SFAD[p3], SFAD[p4], SFAD[p1 - p3 - p4], SFAD[p1 + q1], 
        SFAD[p3 + p4 + q1], SFAD[p3 + q1], SFAD[{{0, (p4) . q1}}]}, {p1, p3, p4}, {q1}, {}, {}] 
  }

{FCTopology(prop3Lv1,{1(p12+iη),1(p22+iη),1(p32+iη),1((p1p2)2+iη),1((p2p3)2+iη),1((p1+q1)2+iη),1((p2+q1)2+iη),1((p3+q1)2+iη),1(p1  p3+iη)},{p1,p2,p3},{q1},{},{}),FCTopology(prop3L2,{1(p12+iη),1(p22+iη),1(p32+iη),1((p1p2)2+iη),1((p2p3)2+iη),1((p1p3)2+iη),1((p1+q1)2+iη),1((p3+q1)2+iη),1((p1p2)  q1+iη)},{p1,p2,p3},{q1},{},{}),FCTopology(prop3L3,{1(p12+iη),1((p1p4)2+iη),1(p32+iη),1(p42+iη),1((p1p3p4)2+iη),1((p1+q1)2+iη),1((p3+p4+q1)2+iη),1((p3+q1)2+iη),1(p4  q1+iη)},{p1,p3,p4},{q1},{},{})}\left\{\text{FCTopology}\left(\text{prop3Lv1},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{((\text{p1}-\text{p2})^2+i \eta )},\frac{1}{((\text{p2}-\text{p3})^2+i \eta )},\frac{1}{((\text{p1}+\text{q1})^2+i \eta )},\frac{1}{((\text{p2}+\text{q1})^2+i \eta )},\frac{1}{((\text{p3}+\text{q1})^2+i \eta )},\frac{1}{(\text{p1}\cdot \;\text{p3}+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{\text{q1}\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop3L2},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{((\text{p1}-\text{p2})^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{q1})^2+i \eta )},\frac{1}{((\text{p3}+\text{q1})^2+i \eta )},\frac{1}{((\text{p1}-\text{p2})\cdot \;\text{q1}+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{\text{q1}\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop3L3},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p1}-\text{p4})^2+i \eta )},\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p4}^2+i \eta )},\frac{1}{((\text{p1}-\text{p3}-\text{p4})^2+i \eta )},\frac{1}{((\text{p1}+\text{q1})^2+i \eta )},\frac{1}{((\text{p3}+\text{p4}+\text{q1})^2+i \eta )},\frac{1}{((\text{p3}+\text{q1})^2+i \eta )},\frac{1}{(\text{p4}\cdot \;\text{q1}+i \eta )}\right\},\{\text{p1},\text{p3},\text{p4}\},\{\text{q1}\},\{\},\{\}\right)\right\}

FCLoopCreateRulesToGLI[topoList]

{{p12Gprop3Lv1(1,0,0,0,0,0,0,0,0),p22Gprop3Lv1(0,1,0,0,0,0,0,0,0),p32Gprop3Lv1(0,0,1,0,0,0,0,0,0),p1  p212(Gprop3Lv1(1,0,0,0,0,0,0,0,0)+Gprop3Lv1(0,1,0,0,0,0,0,0,0)Gprop3Lv1(0,0,0,1,0,0,0,0,0)),p1  p3Gprop3Lv1(0,0,0,0,0,0,0,0,1),p2  p312(Gprop3Lv1(0,1,0,0,0,0,0,0,0)+Gprop3Lv1(0,0,1,0,0,0,0,0,0)Gprop3Lv1(0,0,0,0,1,0,0,0,0)),p1  q112(Gprop3Lv1(1,0,0,0,0,0,0,0,0)+Gprop3Lv1(0,0,0,0,0,1,0,0,0)q12),p2  q112(Gprop3Lv1(0,1,0,0,0,0,0,0,0)+Gprop3Lv1(0,0,0,0,0,0,1,0,0)q12),p3  q112(Gprop3Lv1(0,0,1,0,0,0,0,0,0)+Gprop3Lv1(0,0,0,0,0,0,0,1,0)q12)},{p12Gprop3L2(1,0,0,0,0,0,0,0,0),p22Gprop3L2(0,1,0,0,0,0,0,0,0),p32Gprop3L2(0,0,1,0,0,0,0,0,0),p1  p212(Gprop3L2(1,0,0,0,0,0,0,0,0)+Gprop3L2(0,1,0,0,0,0,0,0,0)Gprop3L2(0,0,0,1,0,0,0,0,0)),p1  p312(Gprop3L2(1,0,0,0,0,0,0,0,0)+Gprop3L2(0,0,1,0,0,0,0,0,0)Gprop3L2(0,0,0,0,0,1,0,0,0)),p2  p312(Gprop3L2(0,1,0,0,0,0,0,0,0)+Gprop3L2(0,0,1,0,0,0,0,0,0)Gprop3L2(0,0,0,0,1,0,0,0,0)),p1  q112(Gprop3L2(1,0,0,0,0,0,0,0,0)+Gprop3L2(0,0,0,0,0,0,1,0,0)q12),p2  q112(Gprop3L2(1,0,0,0,0,0,0,0,0)+Gprop3L2(0,0,0,0,0,0,1,0,0)2Gprop3L2(0,0,0,0,0,0,0,0,1)q12),p3  q112(Gprop3L2(0,0,1,0,0,0,0,0,0)+Gprop3L2(0,0,0,0,0,0,0,1,0)q12)},{p12Gprop3L3(1,0,0,0,0,0,0,0,0),p32Gprop3L3(0,0,1,0,0,0,0,0,0),p42Gprop3L3(0,0,0,1,0,0,0,0,0),p1  p312(Gprop3L3(0,1,0,0,0,0,0,0,0)+Gprop3L3(0,0,1,0,0,0,0,0,0)Gprop3L3(0,0,0,1,0,0,0,0,0)Gprop3L3(0,0,0,0,1,0,0,0,0)+Gprop3L3(0,0,0,0,0,0,1,0,0)Gprop3L3(0,0,0,0,0,0,0,1,0)2Gprop3L3(0,0,0,0,0,0,0,0,1)),p1  p412(Gprop3L3(1,0,0,0,0,0,0,0,0)Gprop3L3(0,1,0,0,0,0,0,0,0)+Gprop3L3(0,0,0,1,0,0,0,0,0)),p3  p412(Gprop3L3(0,0,0,1,0,0,0,0,0)+Gprop3L3(0,0,0,0,0,0,1,0,0)Gprop3L3(0,0,0,0,0,0,0,1,0)2Gprop3L3(0,0,0,0,0,0,0,0,1)),p1  q112(Gprop3L3(1,0,0,0,0,0,0,0,0)+Gprop3L3(0,0,0,0,0,1,0,0,0)q12),p3  q112(Gprop3L3(0,0,1,0,0,0,0,0,0)+Gprop3L3(0,0,0,0,0,0,0,1,0)q12),p4  q1Gprop3L3(0,0,0,0,0,0,0,0,1)}}\left\{\left\{\text{p1}^2\to G^{\text{prop3Lv1}}(-1,0,0,0,0,0,0,0,0),\text{p2}^2\to G^{\text{prop3Lv1}}(0,-1,0,0,0,0,0,0,0),\text{p3}^2\to G^{\text{prop3Lv1}}(0,0,-1,0,0,0,0,0,0),\text{p1}\cdot \;\text{p2}\to \frac{1}{2} \left(G^{\text{prop3Lv1}}(-1,0,0,0,0,0,0,0,0)+G^{\text{prop3Lv1}}(0,-1,0,0,0,0,0,0,0)-G^{\text{prop3Lv1}}(0,0,0,-1,0,0,0,0,0)\right),\text{p1}\cdot \;\text{p3}\to G^{\text{prop3Lv1}}(0,0,0,0,0,0,0,0,-1),\text{p2}\cdot \;\text{p3}\to \frac{1}{2} \left(G^{\text{prop3Lv1}}(0,-1,0,0,0,0,0,0,0)+G^{\text{prop3Lv1}}(0,0,-1,0,0,0,0,0,0)-G^{\text{prop3Lv1}}(0,0,0,0,-1,0,0,0,0)\right),\text{p1}\cdot \;\text{q1}\to \frac{1}{2} \left(-G^{\text{prop3Lv1}}(-1,0,0,0,0,0,0,0,0)+G^{\text{prop3Lv1}}(0,0,0,0,0,-1,0,0,0)-\text{q1}^2\right),\text{p2}\cdot \;\text{q1}\to \frac{1}{2} \left(-G^{\text{prop3Lv1}}(0,-1,0,0,0,0,0,0,0)+G^{\text{prop3Lv1}}(0,0,0,0,0,0,-1,0,0)-\text{q1}^2\right),\text{p3}\cdot \;\text{q1}\to \frac{1}{2} \left(-G^{\text{prop3Lv1}}(0,0,-1,0,0,0,0,0,0)+G^{\text{prop3Lv1}}(0,0,0,0,0,0,0,-1,0)-\text{q1}^2\right)\right\},\left\{\text{p1}^2\to G^{\text{prop3L2}}(-1,0,0,0,0,0,0,0,0),\text{p2}^2\to G^{\text{prop3L2}}(0,-1,0,0,0,0,0,0,0),\text{p3}^2\to G^{\text{prop3L2}}(0,0,-1,0,0,0,0,0,0),\text{p1}\cdot \;\text{p2}\to \frac{1}{2} \left(G^{\text{prop3L2}}(-1,0,0,0,0,0,0,0,0)+G^{\text{prop3L2}}(0,-1,0,0,0,0,0,0,0)-G^{\text{prop3L2}}(0,0,0,-1,0,0,0,0,0)\right),\text{p1}\cdot \;\text{p3}\to \frac{1}{2} \left(G^{\text{prop3L2}}(-1,0,0,0,0,0,0,0,0)+G^{\text{prop3L2}}(0,0,-1,0,0,0,0,0,0)-G^{\text{prop3L2}}(0,0,0,0,0,-1,0,0,0)\right),\text{p2}\cdot \;\text{p3}\to \frac{1}{2} \left(G^{\text{prop3L2}}(0,-1,0,0,0,0,0,0,0)+G^{\text{prop3L2}}(0,0,-1,0,0,0,0,0,0)-G^{\text{prop3L2}}(0,0,0,0,-1,0,0,0,0)\right),\text{p1}\cdot \;\text{q1}\to \frac{1}{2} \left(-G^{\text{prop3L2}}(-1,0,0,0,0,0,0,0,0)+G^{\text{prop3L2}}(0,0,0,0,0,0,-1,0,0)-\text{q1}^2\right),\text{p2}\cdot \;\text{q1}\to \frac{1}{2} \left(-G^{\text{prop3L2}}(-1,0,0,0,0,0,0,0,0)+G^{\text{prop3L2}}(0,0,0,0,0,0,-1,0,0)-2 G^{\text{prop3L2}}(0,0,0,0,0,0,0,0,-1)-\text{q1}^2\right),\text{p3}\cdot \;\text{q1}\to \frac{1}{2} \left(-G^{\text{prop3L2}}(0,0,-1,0,0,0,0,0,0)+G^{\text{prop3L2}}(0,0,0,0,0,0,0,-1,0)-\text{q1}^2\right)\right\},\left\{\text{p1}^2\to G^{\text{prop3L3}}(-1,0,0,0,0,0,0,0,0),\text{p3}^2\to G^{\text{prop3L3}}(0,0,-1,0,0,0,0,0,0),\text{p4}^2\to G^{\text{prop3L3}}(0,0,0,-1,0,0,0,0,0),\text{p1}\cdot \;\text{p3}\to \frac{1}{2} \left(G^{\text{prop3L3}}(0,-1,0,0,0,0,0,0,0)+G^{\text{prop3L3}}(0,0,-1,0,0,0,0,0,0)-G^{\text{prop3L3}}(0,0,0,-1,0,0,0,0,0)-G^{\text{prop3L3}}(0,0,0,0,-1,0,0,0,0)+G^{\text{prop3L3}}(0,0,0,0,0,0,-1,0,0)-G^{\text{prop3L3}}(0,0,0,0,0,0,0,-1,0)-2 G^{\text{prop3L3}}(0,0,0,0,0,0,0,0,-1)\right),\text{p1}\cdot \;\text{p4}\to \frac{1}{2} \left(G^{\text{prop3L3}}(-1,0,0,0,0,0,0,0,0)-G^{\text{prop3L3}}(0,-1,0,0,0,0,0,0,0)+G^{\text{prop3L3}}(0,0,0,-1,0,0,0,0,0)\right),\text{p3}\cdot \;\text{p4}\to \frac{1}{2} \left(-G^{\text{prop3L3}}(0,0,0,-1,0,0,0,0,0)+G^{\text{prop3L3}}(0,0,0,0,0,0,-1,0,0)-G^{\text{prop3L3}}(0,0,0,0,0,0,0,-1,0)-2 G^{\text{prop3L3}}(0,0,0,0,0,0,0,0,-1)\right),\text{p1}\cdot \;\text{q1}\to \frac{1}{2} \left(-G^{\text{prop3L3}}(-1,0,0,0,0,0,0,0,0)+G^{\text{prop3L3}}(0,0,0,0,0,-1,0,0,0)-\text{q1}^2\right),\text{p3}\cdot \;\text{q1}\to \frac{1}{2} \left(-G^{\text{prop3L3}}(0,0,-1,0,0,0,0,0,0)+G^{\text{prop3L3}}(0,0,0,0,0,0,0,-1,0)-\text{q1}^2\right),\text{p4}\cdot \;\text{q1}\to G^{\text{prop3L3}}(0,0,0,0,0,0,0,0,-1)\right\}\right\}