Redundancies from particle exchange

[martin-ueding]

So far I have created a correlator matrix and only showed the parts that Markus also had. This provided a nice deduplication. Going forward I want to produce a non-redundant correlator matrix without external input. As one can see in this example (others are similar), the correlator matrix is redundant:

Resolving the particle momenta to P - q and q gives us these two cases:

q	p₁	p₂
0 1 1	0 -1 0	0 1 1
1 0 0	-1 0 1	1 0 0

These might now seem to be the same thing, but remember that these are each summed over the whole little group stabilizing P = (0, 0, 1). So a rotation with 180° with transform one into the exchanged particle case of the other.

If the particles were not identical, this would not be a problem. Therefore I need to tweak the detection of equal momentum orbits just that it also sorts the momenta of the individual particles as they are all interchangeable for the current case of interest. This should become a configurable option in the longer term to support something like ππK where some but not all particles are interchangeable.

[martin-ueding]

This sorting does the trick, it is a minimal change to the code. It now only supports identical particles, but I have no plans to do anything else soon anyway. The Lüscher formalism likely isn't even developed for that yet.

Sorting the individual momenta in the orbit makes them differ a bit in the sorting:

But now the orbits of the two cases above are really the same:

I will do the analytic projection for the rho again and see whether this cures all the problems.

Right now the problem is comically severe, take a look at the following correlator matrices to have a laugh:

[martin-ueding]

The P² = 1, B₁, P = 001 is now fixed. Before it had 2×2 entries, now it is 1×1 as one would want:

But that did not cure all the issues. For instance the P² = 2, B₁, P = 011 still shows apparent redundancies:

Computing the individual particle momenta is a bit hard because the q are given in the reference frame where the total momentum is P = (1, 1, 0).

q	p₁	p₂
0 0 -1	1 1 1	0 0 -1
0 0 1	1 1 -1	0 0 1
0 1 -1	1 0 1	0 1 -1
0 1 1	1 0 -1	0 1 1

The first two should be within the same orbit, actually. For the last two I would think that the stabilizer subgroup for (1, 1, 0) could have the same with particle exchange.

And indeed the first two give the same orbit even without particle exchange, so the old function is just fine.

The last two give the same results for both q, so they should drop out as well:

Then likely I have screwed up something with the reference rotation. But the filtering does work, it removes all the ones that we do not want!

Perhaps I am working with some stale files or something?

[martin-ueding]

It was stale data for this part 🤦‍♂️:

[martin-ueding]

But re-running did not cure all of them. There is still this one here:

We can let Mathematica figure out the individual momenta here:

Each row are the two individual momenta in the actual frame, not in this reference frame.

We can take a look at the momentum orbits (exploiting particle exchange already). Each row is one orbit. We can see that all them are unique, therefore it is no surprise that they were not filtered out.

But if we go into the reference frame such that the total momentum is P = (1, 1, 0) and keep the same relative momenta, we see that this momentum structure actually has only two unique ones.

So I would except that something is wrong in the part where the reference rotation is applied and therefore the global orientation makes a difference, which it definitely should not. It is okay if a global rotation gives a different signal because the gauge is not rotationally invariant, but the dimension of the correlator matrix must be the same for fixed P² and irrep.

[martin-ueding]

I wrote that one has to add these q to P_ref and not P in this issue, right? I should also do that in the code 🤨. With that the filtering works just as in the reference frame:

[martin-ueding]

I was adding the relative momenta q to the P_ref all along. With the latest fix I just effectively removed the reference rotation which means that the generated individual momenta are in the reference frame. As one can see in the above Mathematica screenshot the total momentum is P = (1, 1, 0), which is the reference momentum for P² = 2. This means that the orbit calculation works exactly the same way, which is fine for our purposes. But I just added another MomentumRef application and then in the end nothing except the reference total momenta coupled to anything any more.

So the issue is more subtle apparently, but at least the duplicates have gone now. The following contains everything up to this point: Correlator_Matrix_Visualization.pdf

We are left with peculiar non-square correlator matrices like these ones:

In the prescription there are only these three elements, so that is present already there. It seems rather peculiar because the analytic projection code should use the same set of momenta for source and sink.

[martin-ueding]

The momentum combinations are there in the pure spin part, so this is not the problem.

<|{-2, 0, 0} -> <|"A1" -> <|"1" -> <|"1" -> <|
         "000" -> {"000", "001", "011"},
         "001" -> {"000", "001", "011"}, 
         "011" -> {"000", "001", "011"}|>|>|>|>|>

The cutoffs are not the problem either, we have 002 & 000, then 001 & 001, and 0-11 & 011, these have sum norms of 4, 2 and 4, none are above the cutoff of 4 we have for P² = 2. The max norms are 4, 1 and 2; these are not above the global cutoff of 4 as well. And the cutoff filtering is done before the momenta are put into the group theory, so that cannot be it.

The resulting analytic prescription has this non-square form. So the error is within the analytic projection code, the numeric one just does what it is supposed to do.

{
  "-200":{
    "A1":{
      "1":{
        "1":{
          "000":{
            "000":[
              14 summands
            ]
          },
          "001":{
            "000":[
              2 summands
            ]
          },
          "011":{
            "000":[
              8 summands
            ]
          }
        }
      }
    }
  }
}

We can take a look at an intermediate state where we have a linear combination of strings. And there for some reason is zero in these other momentum combinations:

I have a hard time believing that this zero really is physical, I would rather think that something is wrong here.

[martin-ueding]

Looking at Markus's rho data I find that he only has these following “gevp indices”:

id  element
0   p: 4, g: \gamma_{50i}
1   p: 4, g: \gamma_{i}
2   p: 4, q: (0.0, 0.0, 1.0), g: \gamma_{5}, \gamma_{5}

With his momentum parametrization this is 002 & 000 for the individual particles, exactly the one diagonal element that works. He does not have any of the other elements, although I would think that they would be allowed from the cutoffs.

@maowerner: Is it a coincidence that you only have a 1 two-pion operator for P² = 4 in the A₁ irrep while at the same time that is the only operators that produces a non-zero diagonal element for me?

[maowerner]

No, thats not a coincidence.

(0,0,0) \times (0,0,2) is obviously A1.

(0,1,1) \times (0,-1,1) subduces into the E irrep. Thus I would expect the projection into A1 to be 0.

Regarding (0,0,1) \times (0,0,1) that is a bit more complicated. The momenta are equal, therefore the pions are indistinguishable (again). I investigated that a year ago and found that there is a cancellation in the isospin projection. I am not sure whether the pions are in S-wave but you should find that charge conjugation would be violated.

Also this is documented in https://github.com/HISKP-LQCD/sLapH-projection/blob/master/two-meson-operators/selection-of-q.md if you can understand what I wrote there xD

[martin-ueding]

@maowerner: The peculiar thing then really is that the two elements other than -200 & 000 are non-zero, right?

[maowerner]

I am confused. Which elements are non-zero? Other than (0,0,2) \times (0,0,0) the projection is zero, isn't it?

[martin-ueding]

@maowerner: Look here: https://github.com/HISKP-LQCD/sLapH-projection-NG/issues/23#issuecomment-524650539

[maowerner]

They are 1e-13 all the way through. Is that really significant?

[martin-ueding]

@maowerner: Thanks for pointing out the obvious, I haven't noticed the numerical scale 🤦‍♂️. I was caught up that analytically my code says that there is a signal in that particular channel. But perhaps it just does not take some particular symmetry or simplification into account and then it comes out as numerically zero.

This means that I need to filter correlator matrix elements that are very small and see whether they turn out to be smaller and square after that. Thank you!

[martin-ueding]

Just filtering out correlators which have |C(0)| < 1.0e-8 makes all of these A1 irreps well behaved and now all correlator matrices look sensible.

Correlator_Matrix_Visualization.pdf

[urbach]

looks very good now!