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 $\int F_L \int F_T$.

[Copilot]

Pull request overview

Adds full time-dependent and flavour-tagged angular observable support for (B_s^0 \to \phi \ell^+\ell^-), including new angular coefficients and a switchable amplitude formalism (helicity vs transversity) via an AuxiliaryQuantity.

Changes:

• Introduces an AuxiliaryQuantity (B->Vll amplitude formalism) to compute (J_i, \bar J_i, h_i, s_i) using selectable implementations (helicity/transversity; with/without corrections).
• Extends the (B_s\to V\ell\ell) observable layer to support additional time-integrated and time-dependent angular observables and optimized combinations.
• Adds transversity-amplitude machinery and unit tests intended to compare helicity vs transversity predictions.

Reviewed changes

Copilot reviewed 11 out of 11 changed files in this pull request and generated 9 comments.

Show a summary per file

File	Description
flavio/physics/bdecays/bvll/test_bvll.py	Expands existing Bs time-dependent tests to new observable sets and indices.
flavio/physics/bdecays/bvll/test_bsphimumu_obs.py	Adds new tests intended to compare helicity vs transversity implementations for Bs→ϕμμ observables.
flavio/physics/bdecays/bvll/observables_bs.py	Refactors Bs→Vℓℓ observable computation to use coefficients from AuxiliaryQuantity and adds many new observables.
flavio/physics/bdecays/bvll/auxiliary_quantities.py	Introduces the B->Vll amplitude formalism auxiliary quantity and implementations.
flavio/physics/bdecays/bvll/amplitudes_transversity.py	Adds transversity amplitude computation and coefficient construction ((J,h,s)) from transversity amplitudes.
flavio/physics/bdecays/bvll/amplitudes.py	Adds a “no corrections” mode for helicity amplitudes and a helper to fetch coefficients via AuxiliaryQuantity.
flavio/physics/bdecays/bvll/init.py	Ensures new auxiliary quantities are imported/registered.
flavio/physics/bdecays/angular.py	Adds computation and conversion helpers for the new (s_i) coefficients in the helicity/G-basis pipeline.
flavio/data/parameters_uncorrelated.yml	Adds the Bs mixing parameter (\Delta m_s/\Gamma_s).
flavio/data/parameters_metadata.yml	Adds metadata for the new (\Delta m_s/\Gamma_s) parameter.
flavio/data/config.yml	Sets default implementation for the new B->Vll amplitude formalism auxiliary quantity.

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[Copilot]

In angularcoeffs_s_transversity, the ml-suppressed term in coefficient 5 uses ml / sqrt(2) instead of ml / sqrt(q2) (as in the corresponding J/h expressions). This introduces wrong dimensions and will distort the scalar contribution at low q2.

Suggested change

	- ml / sqrt(2) * _Im( eiphi * ( Atilde['para_L'] * _Co(A['S']) + Atilde['para_R'] * _Co(A['S']) )
	- ml / sqrt(q2) * _Im( eiphi * ( Atilde['para_L'] * _Co(A['S']) + Atilde['para_R'] * _Co(A['S']) )

[Copilot]

compute_transversity_amps mutates the input form-factor dict by assigning ff['A2'] and ff['T3']. This can leak derived/overwritten values to other computations if the same dict instance is reused by the caller. Prefer computing local A2/T3 variables (or work on a shallow copy) instead of modifying ff in-place.

Suggested change

	ff['A2'] = -(mB + mV)*(-ff['A1']*mB**3 - ff['A1']*mB**2*mV + ff['A1']*mB*mV**2 + ff['A1']*mB*q2 + ff['A1']*mV**3 + ff['A1']*mV*q2 + 16*ff['A12']*mB*mV**2)/lambda_qsq(q2, mB, mV)
	ff['T3'] = -(mB - mV)*(-ff['T2']*mB**3 - ff['T2']*mB**2*mV - 3*ff['T2']*mB*mV**2 + ff['T2']*mB*q2 - 3*ff['T2']*mV**3 + ff['T2']*mV*q2 + 8*ff['T23']*mB*mV**2)/lambda_qsq(q2, mB, mV)
	# from https://arxiv.org/pdf/0811.1214
	a0_l_term_one = ( (wc_9 - wc_9p) - (wc_10 - wc_10p) ) * ( (mB**2 - mV**2 - q2) * (mB + mV) * ff['A1'] - lambda_b * ff['A2'] / (mB + mV) ) # first summand of (3.30)
	a0_r_term_one = ( (wc_9 - wc_9p) + (wc_10 - wc_10p) ) * ( (mB**2 - mV**2 - q2) * (mB + mV) * ff['A1'] - lambda_b * ff['A2'] / (mB + mV) )
	a0_term_two = 2 * mqh * (wc_7 - wc_7p) * ( ( mB**2 + 3 * mV**2 - q2 ) * ff['T2'] - lambda_b / (mB**2 - mV**2) * ff['T3'] ) # second summand of (3.30)
	A2 = -(mB + mV)*(-ff['A1']*mB**3 - ff['A1']*mB**2*mV + ff['A1']*mB*mV**2 + ff['A1']*mB*q2 + ff['A1']*mV**3 + ff['A1']*mV*q2 + 16*ff['A12']*mB*mV**2)/lambda_qsq(q2, mB, mV)
	T3 = -(mB - mV)*(-ff['T2']*mB**3 - ff['T2']*mB**2*mV - 3*ff['T2']*mB*mV**2 + ff['T2']*mB*q2 - 3*ff['T2']*mV**3 + ff['T2']*mV*q2 + 8*ff['T23']*mB*mV**2)/lambda_qsq(q2, mB, mV)
	# from https://arxiv.org/pdf/0811.1214
	a0_l_term_one = ( (wc_9 - wc_9p) - (wc_10 - wc_10p) ) * ( (mB**2 - mV**2 - q2) * (mB + mV) * ff['A1'] - lambda_b * A2 / (mB + mV) ) # first summand of (3.30)
	a0_r_term_one = ( (wc_9 - wc_9p) + (wc_10 - wc_10p) ) * ( (mB**2 - mV**2 - q2) * (mB + mV) * ff['A1'] - lambda_b * A2 / (mB + mV) )
	a0_term_two = 2 * mqh * (wc_7 - wc_7p) * ( ( mB**2 + 3 * mV**2 - q2 ) * ff['T2'] - lambda_b / (mB**2 - mV**2) * T3 ) # second summand of (3.30)

[Copilot]

These tests modify the global flavio.config['implementation']['B->Vll angular coefficients'] but never restore the previous value. This can make other tests order-dependent. Save the old config value in setUp/try-finally and restore it in tearDown/finally.

[Copilot]

The test attempts to switch implementations via flavio.config['implementation']['B->Vll angular coefficients'], but the auxiliary quantity introduced/used by the code is named B->Vll amplitude formalism (see config.yml and AuxiliaryQuantity name). As written, this assignment won't affect the implementation used by predictions, so the helicity vs transversity comparison is not actually exercised. Update the config key to B->Vll amplitude formalism.

[Copilot]

observables_integrated_totest contains 'S1c' twice, which likely misses coverage for another observable (e.g. S1s). Remove the duplicate and include the intended observable so the smoke-test loop exercises the full set.

Suggested change

	observables_integrated_totest = ['S1c', 'S1c', 'S2s', 'S2c', 'S3', 'S4', 'A5', 'A6s', 'S7', 'A8', 'A9']
	observables_integrated_totest = ['S1c', 'S1s', 'S2s', 'S2c', 'S3', 'S4', 'A5', 'A6s', 'S7', 'A8', 'A9']

[Copilot]

gamma is computed as 1 / (tau * 6.582e-13) with a comment 'gamma in 1/ps', but tau parameters in flavio are already in ps and 1/tau gives a width in 1/ps. The current expression instead converts to energy units (GeV). Since gamma is not used anywhere in this module, consider removing it; otherwise compute gamma = 1 / par['tau_'+B] (and fix the comment).

Suggested change

	gamma = 1 / (par['tau_'+B] * 6.582119569e-13) # gamma in 1/ps
	gamma = 1 / par['tau_'+B] # gamma in 1/ps

[Copilot]

This file uses PEP 604 union types (e.g. 'str | int') which is a SyntaxError on Python 3.9. The project declares requires-python >=3.9, so this will break imports. Use typing.Union[str, int] (or remove the annotation) instead.

[Copilot]

In angularcoeffs_h_transversity, the ml^2/q2 term for h['1s'] multiplies the conjugated cross terms (_Co(AtR_ALs_perp) * Co(AtR_ALs_para)) instead of adding them. This changes the formula and dimensions; it should be a sum of the two terms (mirroring the AtL_ARs* structure).

Suggested change

	+ 4 * ml**2 / q2 * _Re( eiphi * ( AtL_ARs_perp + AtL_ARs_para ) + _Co(eiphi) * ( _Co( AtR_ALs_perp ) * _Co(AtR_ALs_para) ) ), # (117)
	+ 4 * ml**2 / q2 * _Re( eiphi * ( AtL_ARs_perp + AtL_ARs_para ) + _Co(eiphi) * ( _Co( AtR_ALs_perp ) + _Co(AtR_ALs_para) ) ), # (117)

[Copilot]

In angularcoeffs_h_transversity, the expression for h['6c'] has misplaced parentheses: _Co(A['S'] + Atilde['0_R'] * _Co(A['S'])) mixes amplitudes and products inside a conjugation and drops the intended sum of separate terms. This should be written as a sum of Atilde['0_L']*conj(A['S']) and Atilde['0_R']*conj(A['S']) (and corresponding conjugate terms), matching Eq. (125).

Suggested change

	'6c': 4 * beta_l * ml / sqrt(q2) * _Re( eiphi * ( Atilde['0_L'] * _Co(A['S'] + Atilde['0_R'] * _Co(A['S'])) )
	+ _Co(eiphi) * ( A['0_L'] * _Co(Atilde['S']) + A['0_R'] * _Co(Atilde['S']) ) ), # (125)
	'6c': 4 * beta_l * ml / sqrt(q2) * _Re(
	eiphi * ( Atilde['0_L'] * _Co(A['S']) + Atilde['0_R'] * _Co(A['S']) )
	+ _Co(eiphi) * ( A['0_L'] * _Co(Atilde['S']) + A['0_R'] * _Co(Atilde['S']) )
	), # (125)