FCLoopSingularityStructure
FCLoopSingularityStructure[int, {q1, q2, ...}]
returns a list of expressions {pref,U,F,gbF}
that are useful to analyze the singular behavior of the loop integral int
.
pref
is the ε-dependent prefactor of the Feynman parameter integral that can reveal an overall UV-singularity
U
and F
denote the first and second Symanzik polynomials respectively
gbF
is the Groebner basis of F,∂F/∂xi with respect to the Feynman parameters
The idea to search for solutions of Landau equations for the F-polynomial using Groebner bases was adopted from 1810.06270 and 2003.02451 by B. Ananthanarayan, Abhishek Pal, S. Ramanan Ratan Sarkar and Abhijit B. Das.
See also
Overview, FCFeynmanPrepare, FCFeynmanParametrize
Examples
1-loop tadpole
out = FCLoopSingularityStructure[FAD[{q, m}], {q}, Names -> x]
{Γ(ε−1)(−(m2)1−ε),x(1),m2x(1)2,{m2x(1)}}
The integral has an apparent UV-singularity from the prefactor
Normal[Series[out[[1]], {Epsilon, 0, -1}]]
εm2
Massless 1-loop 2-point function
out = FCLoopSingularityStructure[FAD[q, q - p], {q}, Names -> x]
{Γ(ε),x(1)+x(2),−p2x(1)x(2),{p2x(2),p2x(1)}}
The integral has an apparent UV-singularity from the prefactor
Normal[Series[out[[1]], {Epsilon, 0, -1}]]
ε1
but there is also an IR-divergence for p2=0 (the trivial solution with all xi being 0 is not relevant here)
Reduce[Equal[#, 0] & /@ out[[4]]]
(x(2)=0∧x(1)=0)∨p2=0
1-loop massless box
out = FCLoopSingularityStructure[FAD[p, p + q1, p + q1 + q2, p + q1 + q2 + q3], {p},
Names -> x, FinalSubstitutions -> {SPD[q1] -> 0, SPD[q2] -> 0, SPD[q3] -> 0}]
{2−ε−2Γ(ε+2),x(1)+x(2)+x(3)+x(4),−2(x(1)x(3)(q1⋅q2)+x(1)x(4)(q1⋅q2)+x(1)x(4)(q1⋅q3)+x(1)x(4)(q2⋅q3)+x(2)x(4)(q2⋅q3)),{x(4)(q2⋅q3),x(3)(q1⋅q2)+x(4)(q1⋅q2)+x(4)(q1⋅q3),x(2)(q1⋅q2)(q2⋅q3),x(1)(q1⋅q3)+x(1)(q2⋅q3)+x(2)(q2⋅q3),x(1)(q1⋅q2)}}
As expected a 1-loop box has no overall UV-divergence
Normal[Series[out[[1]], {Epsilon, 0, -1}]]
0
The form of the U-polynomial readily suggests that there is no UV-subdivergence (again as expected)
Reduce[out[[2]] == 0, {x[1], x[2], x[3], x[4]}]
x(4)=−x(1)−x(2)−x(3)
As far as the IR-divergences are concerned, we find a rather nontrivial set of solutions satisfying Landau equations
Reduce[Equal[#, 0] & /@ out[[4]]]
(q2⋅q3=0∧x(1)=0∧q1⋅q3=0∧q1⋅q2=0)∨(x(1)=0∧q2⋅q3=0∧x(3)+x(4)=0∧q1⋅q2=−x(3)+x(4)x(4)(q1⋅q3))∨(x(4)=0∧x(1)=0∧q2⋅q3=0∧q1⋅q2=0)∨(x(4)=0∧x(2)=0∧x(1)=0∧q1⋅q2=0)∨(x(4)=0∧x(3)=0∧x(1)=0∧q2⋅q3=0)∨(x(4)=0∧x(3)=0∧x(2)=0∧x(1)=0)∨(x(4)=0∧x(1)=0∧q1⋅q3=x(1)x(1)(−(q2⋅q3))−x(2)(q2⋅q3)∧q1⋅q2=0)∨(x(1)=0∧q2⋅q3=0∧x(4)=0∧q1⋅q3=0∧q1⋅q2=0)∨(x(3)=−x(4)∧x(1)=0∧q2⋅q3=0∧x(4)=0∧q1⋅q3=0)∨(x(4)=0∧x(1)=0∧x(2)=0∧q2⋅q3=0∧q1⋅q2=0)∨(x(4)=0∧x(2)=0∧x(1)=0∧x(3)=0∧q1⋅q2=0)
A 2-loop eikonal integral with massive and massless lines
out = FCLoopSingularityStructure[SFAD[{ p1, m^2}] SFAD[{ p3, m^2}] SFAD[{{0,
2 p1 . n}}] SFAD[{{0, 2 (p1 + p3) . n}}], {p1, p3}, Names -> x,
FinalSubstitutions -> {SPD[n] -> 1, m -> 1}]
{Γ(2ε),x(3)x(4),x(4)x(1)2+2x(2)x(4)x(1)+x(3)x(4)2+x(2)2x(3)+x(2)2x(4)+x(3)2x(4),{x(3)x(4)2,x(3)2x(4),x(2)x(3),x(2)2+x(4)2+2x(3)x(4),x(1)x(4)+x(2)x(4),x(1)2+2x(2)x(1)+x(3)2−x(4)2}}
The integral has no IR-divergence, the only solution to the Landau equations is a trivial one
Reduce[Equal[#, 0] & /@ out[[4]], Reals]
x(4)=0∧x(3)=0∧x(2)=0∧x(1)=0
Notice that the mass is acting as an IR regulator here. Setting it to 0 makes the IR pole resurface
out = FCLoopSingularityStructure[SFAD[{ p1, m^2}] SFAD[{ p3, m^2}] SFAD[{{0,
2 p1 . n}}] SFAD[{{0, 2 (p1 + p3) . n}}], {p1, p3}, Names -> x,
FinalSubstitutions -> {SPD[n] -> 1, m -> 0}]
{0,x(3)x(4),x(4)x(1)2+2x(2)x(4)x(1)+x(2)2x(3)+x(2)2x(4),{x(2)x(3),x(2)2,x(1)x(4)+x(2)x(4),x(1)2+2x(2)x(1)}}
and here is our nontrivial solution
Reduce[Equal[#, 0] & /@ out[[4]], Reals]
x(1)=0∧x(2)=0