[NA] B2Kmumu

[mmikhasenko]

Here is a paper,
https://inspirehep.net/literature/2795535

It uses conventions of Altmannshofer et al., to build the amplitude:

Steps to encode:

• write down all expressions in terms of m_Kpi, and cos(theta), [and Q^2].
• identify d_{lambda,0}(theta)
• find out the coupling scheme (e.g. parity would work for K* decays, and helicity works for B)
• identify effective lineshape functions (m_{Kpi})

How to start

• print validation values (select a single kinematic point, and print value of matrix element there)
• collect matematica files that have amplitude encoded

Simpler model

• LHCb-PAPER-2025-055: Bookkeeping note on the new amplitude analysis paper, B2Kmumu (K, not K*),

[egede]

The complication here is that $C_9^{\rm eff}$ as well as the form factors depend on $q^2$. You can find the expression for the model in arXiv:2405.17347. There is subsequently a resolution model for $q^2$. The resolution model itself depends on $q^2$ making it non-trivial to use an FFT for the convolution.

I am afraid that we do not have a Mathematica model of the PDF.

Finally there are many numerical tricks used to give numerical stability of the PDF both for the normalisation integrals and for the evaluation of the likelihood.

The code implementing the PDF is in a GitLab repo. This code is no beauty though.

[mmikhasenko]

@dvandyk do you have by chance a Mathematica code that evaluates the matrix element in terms of theta, mKpi, and Qˆ2?

[mmikhasenko]

Finally there are many numerical tricks used to give numerical stability of the PDF both for the normalisation integrals and for the evaluation of the likelihood.

Thanks @egede!

Dependence on qˆ2 is not a bad complication: one should consider the process as a four-body decay, B->K* X with K*->Kpi, and X->mu+mu-. The lineshapes of K* and X are somewhat complicated, but likely possible to figure out.

A proper computation of signal PDF in LHCb requires resolution, however, the physical model itself does not need to know about detector effects. One would start by encoding the model for the decay. So, not resolution and numerical tricks needed at the first step.

[dvandyk]

@dvandyk do you have by chance a Mathematica code that evaluates the matrix element in terms of theta, mKpi, and Qˆ2?

@mmikhasenko Sorry, I do not have any Mathematica notebook that could provide the predictions that you are looking for.