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gituliar / fuchsia

A tool for reducing differential equations for Feynman master integrals to an epsilon form.
http://gituliar.net
ISC License
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can not factor epsilon #13

Closed F-Feng closed 7 years ago

F-Feng commented 7 years ago

Hi,

Could you give a check on this case? (Please see the attached matrix file [m.txt] (https://github.com/gituliar/fuchsia/files/1230040/m.txt) ) the last message look like as follows:

[ 4.9s] -> factorize [ 6.1s] found 100 equations with 16 unknowns [ 6.5s] can not factor epsilon

Thanks very much!

magv commented 7 years ago

This is an interesting case (which I don't remember seeing the likes of). By the time factorization is reached, the matrix is reduced to something like this:

{{-8*eps/(x-1)+8*eps/x,0,0,0,0},{0,-4*eps/(x-1)+6*eps/x,0,0,0},{0,-1/36*(240*eps^3-236*eps^2+56*eps+1)/(eps^2*(x-1))+1/36*(112*eps^3-172*eps^2+88*eps-15)/(eps^2*x),1/4*eps/(x-1)-1/4*eps/x,-25/72*eps/(x-1)+289/72*eps/x,0},{0,1/2*(16*eps^3-20*eps^2+8*eps-1)/(eps^2*(x-1))-1/2*(16*eps^3-20*eps^2+8*eps-1)/(eps^2*x),9/2*eps/(x-1)-9/2*eps/x,-25/4*eps/(x-1)+33/4*eps/x,0},{(4*eps-1)/(2*x-1)-(4*eps-1)/x,-8*(4*eps-1)/(x-1)-4*(12*eps^2-4*eps+1)/(eps*(2*x-1))+4*(8*eps^2-6*eps+1)/(eps*x),-72*eps^3/((4*eps^2-4*eps+1)*(x-1))+18*(2*eps^3+eps^2)/((4*eps^2-4*eps+1)*(2*x-1)),100*eps^3/((4*eps^2-4*eps+1)*(x-1))-17*(2*eps^3+eps^2)/((4*eps^2-4*eps+1)*(2*x-1)),-4*eps/(2*x-1)+4*eps/x}}

This indeed looks like a fully normalized Fuchsian matrix, so no error so far. Next, factorize tries to find an x-independent transformation to turn this into epsilon-form, but finds none. I've looked into the generated equations a bit, and the conclusion seems correct: there is no constant transformation that can turn this matrix into epsilon-form. There's not much Fuchsia can do about that.

Now, the interesting question is: can there be a non-constant transformation that reduces this matrix? This unfortunately I don't know.

F-Feng commented 7 years ago

Thanks so much for looking into such a case!

I am going to do some try on the normalized Fuchisan result, and maybe try the solution of DE directly from the pre-factorize matrix ...

Anyway, thanks so much again!

gituliar commented 7 years ago

@F-Feng, could you, please, shortly describe in a context of which problem have you obtained this matrix?

F-Feng commented 7 years ago

@gituliar, thanks, and sure. The matrix shows up when I try to solve some master integrals which comes from this paper: https://arxiv.org/abs/1708.01129, the master integrals are listed in Eq.(44) of that paper, and when I try to solve those master integrals by differential equation, I get the matrix I sent to you.

gituliar commented 7 years ago

@F-Feng, thank you for the details. As @magv mentioned, your example is quite interesting since we have not seen Fuchsia ever fail at the last step, though that is possible in principle (for more details see Section 6 in http://arxiv.org/abs/1411.0911v1). For the moment, the only explanation I have for such behavior, is that your 5x5 matrix is part of a non-homogeneous system, since in the paper you mentioned authors deal with 13 masters. Hence, I would suggest you to try a complete 13x13 matrix. I think that in this case chances that Fuchsia will find a proper transformation are much higher.

F-Feng commented 7 years ago

@gituliar Thanks very much for the suggestion, I will try the complete master integrals, and if I succeed, I will post the detailed information. Thanks again!