Loop amplitudes computation with CutTools

An attempt of using CutTools with MadGraph v.4 showed what features are required for MadGraph v.5 to perform loop computations in this way with full generality. The exhaustive description of this work is presented in the note attached to this page and a summary of the requirements identified follows.

Requirements

1) Diagram generation

1.1) Differentiating loop lines from those of the branches attached to the loop

The loop particles will not be implemented as independent copies as in MG4, but instead a 'loop' tag will be added to the attributes of the class Leg of MG5. The recursive algorithm for the diagram generation outputs all possible tree-cut diagrams and should be left untouched. Only then a filter selects the loop basis, in a complete analogy to what NLOComp does. It appears that constructing loop topologies by first generating many more tree diagrams and performing a selection among them afterwards might even be competitive compared to other standard methods used in FeynArts or QGraph for instance.

1.2) Selection of the tree cut-diagrams with one-to-one matching with the relevant loop diagrams

MG5 can bring much richer information on the generated diagrams. The selection algorithm filtering the cut-diagrams should be easily implemented after generate_diagrams().

2) Forcing loop particles to have propagators with the denominator taken out

Writing dedicated subroutines for the loop particles is probably the easiest fix. However, this is redundant since the full propagators are already coded in the existing HELAS subroutines and the idealistic behavior would be to somehow reuse them. This is really more a HELAS issue since MG5 has a very abstract internal representation of the wavefunctions creation chain which can be exported to any HELAS model. So one could easily create a HELAS model, where the presence of the denominators of propagators can be set by an argument. For many other reasons too, it would be wise to design a completely new HELAS model tuned for loop computations.

3) Ghost diagrams, gauge choice

The MG4 implementation of ghosts conceptually proved that the impact of the gauge fixing terms on loops can really be effectively taken into account by introducing new scalar states. There is a priori no obstacle to the introduction of ghosts in MG5. It still needs some core changes of the actual setup to allow scalar particles to have fermion-like anticommuting behavior.

4) Reconstruction of the numerator of the propagator of the cut loop line

In MG4, this issue was dealt in the worst manner and even though very general, it costs a lot in efficiency. The best way of handling this would be to incorporate the missing propagator in the HELAS calls chain hence avoiding to rewrite the already existing propagators. Some thinking showed that there are two somehow equivalent way of computing a loop amplitude with the numerator of the cut-loop reconstructed:

• Calling a maximum of four times a HELAS chain with standard particles and an additional two-point interaction giving the missing numerator.
• Mimicking successive HELAS chains by using tensor-like particles whose sets of four components are decoupled and correspond each to one evaluation of the HELAS chain. This first solution is preferable and should be investigated first.

First optimizations to be brought

These are the most significant optimizations which can be brought only once the raw procedure was proved to work.

1) Reusing wavefunctions entering the loop

We will refer to the tree structures attached to loops as Feynman Diagram structures, abbreviated FD-structures. The FD-structures contain external particles and one off-shell current which feeds the loop. CutTools calls the amplitude of the cut-diagrams many times for the same phase-space point but with different loop momenta q. Since the FD-structure are independent of the loop momentum, recycling them for each computation of the amplitude with a different loop momentum would save a lot of time. The gain in efficiency can be rather significant here since the number of evaluation of the cut-diagram amplitude per phase-space point is typically between 10 and 20.

2) Summing color-ordered structures attached to a loop before calling CutTools to compute it.

What characterizes a loop for CutTools is only a set of momenta and masses for the loop lines. It is then possible to call CutTools with loop amplitudes computed with the sum of the matching color-ordered structures attached to the loop instead of calling CutTools for each structure within this sum. This might bring a very significant improvement for large multiplicities processes and small loops, for which there are many color ordered substructures contributing to the same structure attached to the loop. Recognizing these matching structures and organizing the computation with smart calls to CutTools is not straight-forward and will only be implemented in a second stage as an optimization update. But keep in mind that for each loop computation, CutTools calls the cut-diagram amplitude up to 20 times so it is worth trying to spare any single call to CutTools.

More involved optimizations

These are some more involved optimizations. It's not yet clear if they are possible to implement and/or if they are really needed.

1) Quadruple (or higher) precision

For some phase-space points, normal double precision is not good enough to reliable calculate the value of the loops. CutTools automatically recognizes these points and has the possibility to go to quadruple or higher precision. To do this consistently also the inputs, ie. the HELAS routines should be in higher precision. In fact, only the subroutines that deal with the complex particles need quadruple precision, if the rest is not recomputed for a different value of the q-momenta. So this can only be done after the above optimizations of the FD-structures is implemented.

2) R2 and FD-structures

Maybe it's possible to reuse the same FD-structures defined above when calculating the R2 terms or the UV-counterterms. I am not sure it's worth investigating, as I do not expect a big gain here.