Time-dependent and flavour-tagged angular observables for Bs -> phi mumu

[schmitse]

This PR includes updates adding all angular coefficients for $B_s^0 \rightarrow \phi \ell^+\ell^-$ decays.
It also implements in addition to the helicity formalism the transversity formalism which is frequently used in the literature, for example arXiv:1502.05509.
The amplitude formalism is changed to an AuxiliaryQuantity allowing to switch between the helicity formalism (for example in arXiv:1506.03970) and the transversity formalism.

This PR is accompanied by a writeup detailing all changes that have been done, it is currently being finalised and will be added to this discussion soon.

Description of the changes:

• flavio/parameters: Add the meson mixing frequency $\Delta m_s / \Delta\Gamma_s$ from HFAG 2014 to default parameters.
• flavio/physics/bdecays/angular.py:
  • Add function to compute the $s_i$ angular coefficients in the helicity formalism. (angularcoeffs_s_Gbasis_v),
  • conversion from $G$ to $g$ (compare with arXiv:1506.03970 Eq 32) (G_s_to_g_s)
  • function to return them similar to the $J_i$ and $h_i$ (angularcoeffs_s_v).
• flavio/physics/bdecays/bvll/observables_bs.py:
  • bsvll_obs now uses AuxiliaryQuantity to compute $J_i$, $h_i$, and $s_i$ (new).
  • Generalise functions for the $S_i$ and $A_i$ computation at LHCb. Following arXiv:1502.05509 Eq. 42 and 43. This does not change the previously implemented observables! But now supports the additional angular coefficients in the time-integrated case.
  • Add all angular coefficients as observables.
• flavio/physics/bdecays/bvll/amplitudes.py: add function to return $J_i$, $\bar{J}_i$, $h_i$, and $s_i$ from AuxiliaryQuantity and allow helicity amplitudes to be returned without QCDF and subleasing corrections (for comparison against transversity amplitudes.)
• flavio/physics/bdecays/bvll/amplitudes_transversity.py: add all functions to compute transversity amplitudes and $J_i$, $\bar{J}_i$, $h_i$, and $s_i$ from transversity amplitudes.
• flavio/data/config.yml: Add new AuxiliaryQuantity called B->Vll amplitude formalism. Currently features implementations: Helicity Amplitudes (default and what was used before), Helicity Amplitudes (no corrections), and Transversity Amplitudes (no corrections).
• flavio/test_bsphimumu_obs.py: added unit test comparing all observables using the transversity and helicity amplitude formalism.

Please note that this PR supersedes PR #258 which implements A_theory_num_Bs without the caveat of Eq 42 and 43 in arXiv:1502.05509.

[schmitse]

I have adjusted the tests in test_bvll.py to also cover the new $K_i$ versus $S_i$ and the $W_i$ versus $A_i$ and check that all defined observables do not raise an error.

[schmitse]

Hi @peterstangl @DavidMStraub

I attach a small document about the changes in this PR.
I try to briefly recapitulate the intricacies of $B_s^0 \rightarrow \phi \mu^+\mu^-$ decays, compared to the maybe more famous $B^0 \rightarrow K^{\ast 0}\mu^+\mu^-$ case.
Then i show some comparisons i have performed between eos and flavio. (Note for this i needed to adjust eos a bit)
Afterwards i describe the changes to the code in more detail.
If something is unclear, please let me know then i can try to adapt the document.
Also note that these thoughts are my own and any mistakes would be mine as well.
main.pdf

I also attach the analysis file and script i used to create the predictions in eos, in case people want to have a go at trying to align the form-factors completely. Please excuse the lack of comments.
predictions_eos.zip

[schmitse]

I have updated the optimised $P_i^{(\prime)}$ and their asymmetries to include all the new angular observables, both in the time-independent and time-dependent case.
Also i have fixed the denominator for the primed variants which was $\int F_L \cdot F_T$ instead of $\int F_L \int F_T$.