Ren hirayama/spectral function sampling

[RenHirayama]

This implements the modes to sample from a spectral function, based on what I derived here: CooperFrye_w_SF.pdf

The partial densities need to be integrated over mass, using the spectral function as a weight. Since calculating this "live" would be costly, I chose to tabulate it instead. The available new modes are accessible with the key spectral_function, which can have the values pole-mass, breit-wigner, and full-vacuum. The first one is the default, maintaining the original behavior, the second one uses a relativistic BW with constant width, and the last uses the full SMASH spectral functions that are based on the Manley formalism.

The tests have a relative tolerance of 1%, which I find acceptable. Also included is a test that only prints the individual densities, which might be handy. For $T=140 \mathrm{MeV}$, it results in (isospin symmetric since there is no $\mu_Q$)

	Pole mass	Breit Wigner	Full vacuum
π	0.0323768	0.0323768	0.0323768
ρ	0.00474623	0.00611923	0.00425766
f₂	0.000417747	0.00147522	0.000433483

I was a bit surprised with the Breit-Wigner density being much larger, but it makes sense when we consider that the spectral function is non-zero at the mass threshold:

Since I branched off @LucasConstantin's viscous corrections branch, I will either wait for his merge to develop or merge directly to his branch, whatever is less messy.

[yukarpenko]

Renan maybe we wait with this until Lucas does the changes I was talking about? Because I do not know if it would be a simple rebase etc. Function signatures should change. Please take a look at my comments in the channel.

[RenHirayama]

Sure, i intend to wait for the reviews anyway

[nilssass]

@RenHirayama Please also update the CHANGELOG with these changes.

[yukarpenko]

Renan is this now on top (~rebased or something similar) of the branch "Analytical_deltaN_bulk" ?

[yukarpenko]

Another thing: do you intend to publish those results somehow? If you do, the best way to preserve the knowleedge would be to document the formulas along with their derivations in a publication. Otherwise, the understanding what you've implemented will likely be lost in time, also for potential users.

[RenHirayama]

Renan is this now on top (~rebased or something similar) of the branch "Analytical_deltaN_bulk" ?

not yet, I will wait for the merge to avoid possible annoying conflicts

Another thing: do you intend to publish those results somehow? If you do, the best way to preserve the knowleedge would be to document the formulas along with their derivations in a publication. Otherwise, the understanding what you've implemented will likely be lost in time, also for potential users.

It will enter the dilepton calculations in the hybrid, and I think it should be (at least partially) in the paper. Nonetheless I already have the PDF there, we just need to figure out as a group where to put such things.

[RenHirayama]

What is missing currently is including the spectral function integration to the bulk corrections in gen.cpp. @LucasConstantin and I agreed that the most straightforward way to do this is creating a second table when bulk corrections are active.

[yukarpenko]

Hi Renan - I went through your changes (as usual, I am not looking into the contents of the tests/ ).
One thing I am confused about: you've added the machinery to compute particle densities when there are spectral functions, but the particle 4-momenta are still sampled the old way according to their pole masses.
So currently, only the relative particle abundancies are changed (all sampled hadrons still have their pole masses), but in principle this is done at an expense of consistency with the underlying EoS. Because the underlying EoS is still good old non-interacting hadron resonance gas. As such it may change the energy conservation balance. I deliberately said "change" not improve or worsen the energy-momentum conservation. Might improve but then it will be by accident - because of the undelying inconsistency with the EoS.
I am not saying that you shouldn't do that until there's consistent description (it is not my modus operandi; and there isn't likely to be consistent description in the near future in this respect) but I feel I need to state the physics changes.
We can contiue the exchange on ... Matrix? I dropped my message here since I am not sure what platform to use to communicate in this update (PR) in particular.

[RenHirayama]

but the particle 4-momenta are still sampled the old way according to their pole masses.

you're right, when getting the code from the old branch I forgot it. I am at the airport right now so I don't want to start changing the code, but line 365 of gen.cpp (I think, from looking in github) should be enough. Sampling from the unbounded spectral function then constructing the 4-momentum suffices because of the rejection algorithm, from what we discussed back in May/June.

I don't know how much it would change by constructing a more realistic EoS but I don't think we will pursue it (will think more)

[yukarpenko]

Once you are settled and start your work, we should discuss this. I am not trying to guess anything now, it won't make sense. So let me know once you will be ready for a discussion about it.