Implement the mapping of the helicity couplings

[mmikhasenko]

Here is the relation of the helicity couplings in the LHCb implementation (K) and the one in the DPD convention that we use (H)

[mmikhasenko]

Here is the mapping function. The factor of sqrt(2s+1) is specific to the ThreeBodyDecays.jl.

[redeboer]

Okay thanks. I'm rewriting that notebook today so that it imports resonance definitions from JSON (#9). The couplings can be inserted then as well.

[mmikhasenko]

The map is not applied yet in #11, right? The couplings displayed in the cross-check notebook seems to be as in the json file, not mapping factors.

[redeboer]

Nope, in progress.. :grimacing:

[mmikhasenko]

no worries, we just should not compare the rates before fixing the signs of the couplings

[redeboer]

7d573f5 is an attempt to implement the LHCb -> DPD mapping. It simply swaps the sign of the relevant helicity in the symbolic sum and divides by that parity prefactor where required. There isn't an easy way to do this through a function similar to the Julia code without breaking the LaTeX rendering, so I hope this implementation is correct.

It doesn't improve the resulting rate matrix though (cf. https://github.com/redeboer/polarization-sensitivity/pull/14#issuecomment-1127639871)...

[mmikhasenko]

Naively, I thought it is enough to update the values when reading the input file

• instead of one index you get the other.

[redeboer]

Also possible. But there, I don't directly have to the info about parity etc., would have to fetch that from the isobar definitions.

7d573f5 is a bit different though: it also swaps the values of the Kronecker delta etc. I wasn't sure whether those relations applied to the couplings only or to the complete amplitude.

[mmikhasenko]

One thing to check is the mapping 1,2,3,4 to the couplings, because in DPD,

H_{1,-1/2} # is illegal, lambda1-lambda2 > j_Lc

for the LHCb scheme, it is the opposite

[mmikhasenko]

7d573f5 is a bit different though: it also swaps the values of the Kronecker delta etc. I wasn't sure whether those relations applied to the couplings only or to the complete amplitude.

Looked there. I think you should swap only the couplings

[redeboer]

I think it should be only the couplings.

9d6eee0...a7b3388 is a naive attempt to fix it (so only swapped couplings and the prefactor; no remapping of the amplitude itself). I get into trouble with the symbols for the K couplings though, because the swapped indices result in new coupling symbols (so they can't be substituted), see around line 593. It seems to me that the end effect is a duplicate parity index sign flip in the K sub-system..

[mmikhasenko]

the amplitudes formulation look fine now (swapped to LHCb conventions, the couplings K). The mapping from 1,2,3,4 (the function to to_symbol) to the indices does not have be swapped, should come from the README.

[mmikhasenko]

it looks good now! The numbers are almost the same after the update. Still do not match exactly. I checked with julia code that the difference is not related to the stat uncertaintly (with 100_000, the uncertainty is tiny)

[redeboer]

There was another mistake: generated LS-couplings conserved parity for both the production and the decay nodes. This will be fixed in an upcoming commit (83c0a74).

Even with that fix, the lineshape K(892) does not have the same angular momentum for the decay node. The lineshape values are comparable (but not sufficiently equal) to the cross-check values though (55295bf).

with 100_000, the uncertainty is tiny

Currently the integral is taken over the Dalitz Plot grid. I still have to generate a phase space sample instead and integrate over that.

[mmikhasenko]

Yes, something is fishy with LS. Here I cycled the numbers which do not match

[mmikhasenko]

Just a guess, do you consider the parity violating decay of Lc when calculating the minL? For the weak decay of Lc both possibilities, 1/2+ and 1/2- needs to be checked

[redeboer]

L(1690) is fixed with 83c0a74. I don't understand K(892) though.

[mmikhasenko]

yes, it must be the parity violation

K(0-) x pi(0-) =>
    0+ in S-wave
    1- in P-wave  (y) # works

K*(1-) x p(1/2+) => 1/2-, 3/2- S-wave # works with parity violation

[redeboer]

K*(1-) x p(1/2+) => 1/2-, 3/2- S-wave # works with parity violation

Seems that 83c0a74 does it the wrong way round: parity should be violated in the production node, not the decay node.

With c0131e3:

But now I am confused about your l and minL... From above figure it now seems that l_R=minL and l^min_Λc=l?

See also https://github.com/redeboer/polarization-sensitivity/pull/11#issuecomment-1128769870

[mmikhasenko]

The l is in the decay of the resonance, i.e. K* ->K pi P-wave L(1520)->Kp D-wave

The minL is the Lc decay (smaller numbers in average)

[redeboer]

But now I am confused about your l and minL

The natural cause: swapped the columns :expressionless: :gun: → 8b5213e

[mmikhasenko]

yea, I would consider solved. Let's wait for closing this issue until we entirely match