phys: implement symbolic Dalitz phase space

[redeboer]

Extracted from #129 (comment).

• Implement symbolic Kibble and Källén functions
• Visualize 2D phase space boundary
• Visualize 1D phase space projections
• Add references to literature
• Parametrize 2D phase space boundary (instead of contour plot over meshgrid)

Interactive visualization

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[mmikhasenko]

(?) Parametrize 2D phase space boundary (instead of contour plot over meshgrid)

It is an analytic expression, easy to write down. I just do not know the best way to code it since, for whatever Mandelstam variable you choose for y axis, there is an expression y+-(x) where x is whatever you choose for x variable.

see
https://github.com/mmikhasenko/ThreeBodyDecay.jl/blob/01be61b5ab56f191e09f28c97f9e64755efe8bee/src/tbs_struct.jl#L172
with
https://github.com/mmikhasenko/ThreeBodyDecay.jl/blob/master/src/angle_functions.jl#L26

[redeboer]

Ah thanks for the links!

I just do not know the best way to code it since, for whatever Mandelstam variable you choose for y axis, there is an expression y+-(x) where x is whatever you choose for x variable.

Do you mean sigma2 as a function of sigma1?

For plotting (with matplotlib), it would be better if there is some way to parametrize the closed boundary as a function of some variable t along that boundary. But that would be harder to solve, if even possible.

[mmikhasenko]

uh, well done with the solve! :)

yes, there is one fun way for the parametric boundary:

1. replace

• sigma1 = (m2+m3)^2 + T1
• sigma2 = (m3+m1)^2 + T2
• sigma3 = (m1+m2)^2 + T3

2. Then, replace to r, theta to impose a constraint
   
   where
   T0 = m0^2+m1^2+m2^2+m3^2 - (m2+m3)^2 - (m3+m1)^2 - (m1+m2)^2

3. for every value of \theta \in [0, 360], the value of r should be found numerically, by solving phi=0.
   Among all positive solutions for r, one needs to select the lowest.

[mmikhasenko]

for the references,

Eero Byckling, K. Kajantie
https://inspirehep.net/literature/1251349

it is the particle-kinematics bible

[redeboer]

Eero Byckling, K. Kajantie https://inspirehep.net/literature/1251349
it is the particle-kinematics bible

Great tip! See 632d0c4

[redeboer]

yes, there is one fun way for the parametric boundary:
...

Seems that this results in 3rd degree polynomials in r (e626d3b). Well let's not bother to plot that then :P

[redeboer]

Btw where is this from? Byckling and Kajantie do a similar trick for "all masses equal", but I can't see why this would work.

2. Then, replace to r, theta to impose a constraint

where
T0 = m0^2+m1^2+m2^2+m3^2 - (m2+m3)^2 - (m3+m1)^2 - (m1+m2)^2

[mmikhasenko]

Yes, the idea is from Byckling-Kajantie.
I was giving similar exercises to students when I was teaching.

I can't see why this would work.
you make the change of coordinates {sigma_i} -> {r,\phi} in the way that kinematic constraint

sum_1^3 sigma_i = sum_0^3 m_j^2

is always satisfied. The exact transformation is from the triangle representation of the Dalitz.
{r,\phi} are polar cooridinates. I think, that r=0 is always physical.

Actually, it would be nice to see that it works by finding the root numerically.
I find it fascinating to see the polynomial itself. Before I only worked it out analytically for a few limited cases.

[redeboer]

Actually, it would be nice to see that it works by finding the root numerically.
I find it fascinating to see the polynomial itself. Before I only worked it out analytically for a few limited cases.

SymPy can find the root analytically in for these simple integer values of m_i. That can then be lambdified and used for plotting. I'm not sure whether this can be easily implemented for arbitrary values though.

Perhaps for now it's best to merge this PR though, if it looks fine. Then it's available to the other pull requests, so they can renumber and link to it. We can try to work out the parametric plot separately.

[redeboer]

@wgradl, once this PR is merged, this page can be found under compwa-org.rtfd.io/report/017.html. Are there any things that can be improved before?

[wgradl]

@redeboer, this looks very good to me!

…

@wgradl, once this PR is merged, this page can be found under [compwa-org.rtfd.io/report/017.html](https://compwa-org.readthedocs.io/report/017.html). Are there any things that can be improved before? Preview of the page [here](https://compwa-org--139.org.readthedocs.build/report/017.html).

[mmikhasenko]

I just checked that the trick with minimal positive root works

the example is here
https://github.com/mmikhasenko/examples/blob/master/parametricDPboundary.jl

[redeboer]

Cool. I want to get #134 to work, so we can integrate such Julia scripts 😆
Afaics, with matplotlib/python, this is harder to do with Python, because there isn't an equivalent to this roots() function. Moving this point to ComPWA/report#4.