Numerical issues with 2pi: phi vs psi

[mmikhasenko]

Here is a test of continuity: momenta orientation is obtained by rotating Rz(phi)Ry(theta)Rz(psi) of a fixed aligned kinematics. The theta is fixed, phi, and psi are x, and y.

Wigner azimuthal angle for relative tree2 with respect to tree1 is plotted

the jumps are 2pi subtractions.

What is the issue?

numerical fluctuation cause random distribution of the compensation phase +2pi between phi and psi.

Practical checks

• [x] check which angle gets 2pi addition in case of SU2 sign mismatch,
• [x] where exactly in computation the small-number computation happens, is it atan(x,y) or acos?

[mmikhasenko]

It's not an issue in the overall transformation, rather for the individual angle psi - it jumps together with phi changing from -pi to pi. The sum of the two is continuous

result_psi = result_full[1].psi_rf
result_phi = result_full[1].phi_rf + np.pi
result = result_psi + result_phi

[KaiHabermann]

Ill look at this in more detail tomorrow. If you are at it right now, can you see, if this behaviour changes for different theta? Or is it unrelated to theta?

[mmikhasenko]

Pattern is slightly different. I would say it is merely related to theta: since +- 2pi lives in the numerical precision domain, any fluctuation causes the jump, It can be phi, can be theta

[mmikhasenko]

Finally found the interesting stuff, that I was trying to do: the physical angle jump due to mismatch of domain of kinematic variables.

The helicity angles, (phi, theta), (psi, ...) appear in all topologies. Every topology folds the SU(2) space (4pi x 2 x 4pi -> 2pi x 2 x 2pi) differently. Wigner rotations are responsible for matching the topologies on SU(2).

Below one sees the border of 2pi flip domains for wigner psi_3(1)_for1.

The local numerical jumps are destructing, therefore I plot phi+psi. It's not yet clear to me why I have to do phi-psi for the corner regions.

(phi-psi)

(phi+psi)

Anyway, the effect that should be there is clearly visible.

[mmikhasenko]

Realized over the night that remaining jumps for psi+phi is the 4pi, it's not an issue, 4pi is true a ambiguity. Simply to remove by forsoing, [-pi, 3pi] range for the angles

[mmikhasenko]

@KaiHabermann added notes in the header of the issue.

[KaiHabermann]

I added an fmod by 4 pi to tthe image. Now it is smooth in both areas, but different. phi + psi with fmod(phi +psi + 4pi , 4pi) psi phi

[KaiHabermann]

@mmikhasenko should I show the above plot (the nice one) in the talk?

[mmikhasenko]

Yes, would be nice

[KaiHabermann]

Final Version after the + 2 pi was moved to phi instead of psi. This removed the numerical instability for the 3body case (undetermined psi)

Closing this issue, as the latest release solves it

[KaiHabermann]