Fuchsia reduces differential equations for Feynman master integrals to an epsilon form.
This is a new version of Fuchsia, written in C++ using GiNaC. It is based on the previous (Python) version by O. Gituliar and V. Magerya, as described in [2]. That version lives on at [1].
This Fuchsia is still under development, with the main missing feature being the support for unfactorizable polynomials of powers higher than one in the denominators. Otherwise this version is faster and easies to use than the old one.
When compiled, Fuchsia becomes a single executable. A precompiled and statically linked version of this executable can be found in the releases section on Github. See the manual below for its usage. Also see the manual of the previous version at [2], and the articles by R. N. Lee and A. Pomeransky [3-4] for a discussion of the algorithms used.
- [1] https://github.com/gituliar/fuchsia/
- [2] https://arxiv.org/abs/1701.04269
- [3] https://arxiv.org/abs/1411.0911
- [4] https://arxiv.org/abs/1707.07856
fuchsia -- transform linear differential equations into an epsilon
form.
fuchsia [options] command args ...
fuchsia transforms systems of linear differential equations,
∂/∂x I(x,ε) = M(x,ε) I(x,ε),
into an epsilon form,
∂/∂x J(x,ε) = ε S(x) J(x,ε),
where I and J are column vectors of functions in the original and epsilon basis, M is the original matrix, ε×S is the matrix in an epsilon form, and I is related to J via the transformation matrix T(x,ε) such that
I = T J.
In all cases M can depend on additional symbolic variables, which are treated as independent constants.
To reduce a single-variable differential system of equations to an epsilon form, use this:
fuchsia reduce -x x -e eps matrix.orig -m matrix.ep -t matrix.ep.t \
-C 2>&1 | tee matrix.ep.log
For differential equations in multiple variables this is the usage:
fuchsia reduce matrix.x matrix.y -x x -x y -e eps \
-m matrix.ep.x -m matrix.ep.y -t matrix.ep.t \
-C 2>&1 | tee matrix.ep.log
-
show [-x name] matrix
Show a description of a given matrix.
-
reduce [-x name] [-e name] [-m path] [-t path] [-i path] matrix
Find an epsilon form of the given matrix. Internally this is a combination of reduce-diagonal-blocks, fuchsify-off-diagonal-blocks and factorize.
-
reduce [-x name] ... [-e name] [-m path] ... [-t path] [-i path] matrix ...
Find an epsilon form of a given multivariate differential equation system. A matching number of matrix arguments, -x, and -m flags is required.
The matrices are reduced one by one, and a single transformation is computed that simultaneously transforms all of them into an epsilon form. It may be best to list the simplest matrix first.
NOTE: this command is under development, and may fail when it shouldn't.
-
reduce-diagonal-blocks [-x name] [-e name] [-m path] [-t path] [-i path] matrix
Transform the matrix into a block-triangular form and reduce the diagonal blocks into an epsilon form.
-
fuchsify-off-diagonal-blocks [-x name] [-m path] [-t path] [-i path] matrix
Transform the off-diagonal blocks of a block-triangular matrix into a Fuchsian form, assuming the diagonal blocks are already in an epsilon form, thus making the whole matrix normalized Fuchsian.
-
factorize [-x name] [-e name] [-m path] [-t path] [-i path] matrix
Find a transformation that will make a given normalized Fuchsian matrix proportional to the infinitesimal parameter.
-
fuchsify [-x name] [-m path] [-t path] [-i path] matrix
Find a transformation that will transform a given matrix into a Fuchsian form. This is less efficient than block-based commands, because it effectively treats the whole matrix as one big block.
-
normalize [-x name] [-e name] [-m path] [-t path] [-i path] matrix
Find a transformation that will transform a given Fuchsian matrix into a normalized form. This is less efficient than block-based commands, because it effectively treats the whole matrix as one big block.
-
sort [-m path] [-t path] [-i path] matrix
Find a block-triangular form of the given matrix by shuffling.
-
transform [-x name] [-m path] matrix transform ...
Transform a given matrix using a given transformation.
-
changevar [-x name] [-y name] [-m path] matrix expr
Perform a change of variable from x to y, such that x=expr(y).
-
suggest-changevar [-x name] [-y name] matrix
Suggest a rational change of variable that will transform residue eigenvalues of the form n/2+k×eps into n+k×eps, thus making it possible to find an epsilon form of the matrix.
Note that some bad eigenvalues disappear when the matrix is fuchsified, so this routine is best used after "fuchsia fuchsify".
-
simplify [-x name] ... [-m path] ... [-t path] [-i path] matrix ...
Try to find a transformation that makes a given matrix (or a set of matrices) simpler, for some definition of "simple".
-
-x name
Use this name for the free variable (default: x).
-
-0 expr
Set this value for x during multivariate reduction (default: 0).
-
-y name
Use this name for the new free variable (default: y).
-
-e name
Use this name for the infinitesimal parameter (default: eps).
-
-m path
Save the resulting matrix into this file.
-
-t path
Save the resulting transformation into this file.
-
-i path
Save the inverse transformation into this file.
-
-C
Force colored output even if stdout is not a tty.
-
-P
Paranoid mode: spend more time checking internal invariants.
-
-q
Print a more quiet log.
-
-h
Show this help message.
-
-V
Print version information.
-
matrix
Read the input matrix from this file.
-
transform
Read the transformation matrix from this file.
-
expr
Arbitrary expression.
Vitaly Magerya