.. _onlinebook_vertex_fitting: Vertex fitting ============== .. sidebar:: Overview :class: overview **Length**: 30-45 min **Prerequisites**: * The previous lesson **Questions**: * What is a vertex fit? * When do I want to perform a vertex fit? * How should I choose my fit and constraints? **Objectives**: * Perform vertex fit of a decay Introduction ------------ In the broadest sense, we call vertex fitting a technique in which one uses prior knowledge on the nature of a decay to improve the measurement of its observables. The fits we are going to perform are of two main types: * **Geometric Fitting:** We use the fit to determine the decay vertex of the particle. Usually this is done by fitting together the tracks of its charged decay products, which we know originate from a common point. Additional information could be available --- for example, if the particle is short lived, we can improve this by adding an IP constraint, i.e. fit the beam spot together with the tracks. If there's only one track, using the beam spot is the only way to obtain a vertex. .. warning:: If no vertex fit is performed, the corresponding variables for the vertex position will not be filled. * **Kinematic Fitting:** We use the fit to improve our knowledge of the particle kinematics. By default, composite particle kinematics are built off the decay products using 4-momentum conservation. If the particle we are reconstructing has a well defined mass (either stable, or a narrow resonance) it might make sense to apply a mass constraint to help reject combinatorial background. .. warning:: If you apply a mass constraint, the invariant mass will be fixed to the nominal mass. This is problematic if you then want to use this variable, for example if you want to fit a peak. In that case, make sure you save the pre-fit mass separately. .. note:: Several fitters exist. For this exercise we will focus on ``KFit`` which is the most basic one. .. admonition:: Exercise :class: exercise stacked Locate the documentation for vertex fitting functions and find KFit. .. admonition:: Hint :class: dropdown xhint stacked Use the search bar. .. admonition:: Solution :class: dropdown solution You can find it here: `Vertex`. Basic Fitting ------------- This lesson assumes you successfully reconstructed your :math:`B \to J/\Psi(\to e^+e^-)K_s(\to \pi^+\pi^+)` decay following the previous exercises. Now suppose you are interested in reconstructing the :math:`B` decay vertex position using a fit (for example, you're trying to do a time-dependent CPV study). .. admonition:: Question :class: exercise stacked Which particles do you need to fit in order to reconstruct the :math:`B` vertex? .. admonition:: Hint :class: dropdown xhint stacked It can't be the :math:`B` itself: out of its daughters, neither the :math:`J/\Psi` nor the :math:`K_s` are charged tracks. .. admonition:: Answer :class: dropdown solution You must fit the :math:`J/\Psi \to e^+e^-` vertex. The :math:`J/\Psi` is short-lived and therefore its vertex is a good approximation of the :math:`B`. Meanwhile, the :math:`K_s` can decay several cm away from where it is produced. .. admonition:: Exercise :class: exercise stacked Call the fit function with the correct parameters and save the output. Include the true vertex position from MC for comparison. .. admonition:: Hint :class: dropdown xhint stacked Look up the variable collections for vertices. Don't forget to import the vertex module! .. admonition:: Solution :class: dropdown solution .. code-block:: python import vertex ... vertex.kFit("J/psi:ee", conf_level=0.0, path=main) ... jpsi_ks_vars += vc.vertex + vc.mc_vertex You can also set the confidence level to -1, which means failed fits will be included. The fit p-value is saved as part of the ``vc.vertex`` collection. .. admonition:: Exercise (optional) :class: exercise Fit the :math:`K_s` as well. How does its flight length compare to the :math:`J/\Psi`? .. admonition:: Exercise (optional) :class: exercise Look up the documentation for ``TreeFitter`` and fit the whole :math:`B \to J/\Psi(\to e^+e^-)K_s(\to \pi^+\pi^+)` decay chain at once. Tag Vertex Fitting ------------------ Since :math:`B` mesons are produced in pairs, for every signal candidate we reconstruct, there will also be another (the "tag") which is not explicitly reconstructed. We might be interested in knowing the decay position of this meson without placing any requirements on its decay. This is done using the ``TagV`` module. ``TagV`` performs a geometric fit over the tracks in the ROE to determine the tag decay vertex. However, not all tracks will necessarily come from the tag itself; consider for example our signal, where the pion tracks originate from a long lived kaon vertex. ``TagV`` is designed to iteratively downweight those tracks, ultimately excluding them from the fit. .. admonition:: Exercise :class: exercise stacked Locate the ``TagV`` documentation. .. admonition:: Solution :class: dropdown solution It's in the same page as ``KFit``. .. admonition:: Question :class: exercise stacked By default, TagV only uses tracks with PXD hits. Why? .. admonition:: Solution :class: dropdown solution Those tracks provide the best resolution close to the interaction point. As a bonus, this selection rejects tracks from displaced vertices. .. admonition:: Exercise :class: exercise stacked Call the ``TagV`` module and save the output. .. admonition:: Hint :class: dropdown xhint stacked In order to reinforce the fit, an IP constraint is applied to the TagV. If the signal is fully reconstructed, this condition can be relaxed along the signal :math:`B` flight direction. .. admonition:: Solution :class: dropdown solution .. code-block:: python vertex.TagV("B0", constraintType="tube", path=main) ... b_vars += vc.tag_vertex + vc.mc_tag_vertex Conclusion and Plotting ----------------------- Congratulations! Your steering file is ready! Time to run it and check the results. .. admonition:: Exercise :class: exercise stacked Run the steering file. .. admonition:: Solution :class: dropdown solution Your steering file should look like this: .. literalinclude:: steering_files/059_vertex_fitting.py :language: python You can now plot some relevant vertex variables. In general, the choice would depend on what you need for your analysis. A few examples would include: * Vertex position in various coordinates, such as dz and dr. * P-value of the fit. * Resolution of the vertex fit (:math:`\sigma(x)/x`) where x is each of the above variables. * Pull (:math:`(x-x(MC)/\sigma(x)`). As an exercise we will focus on the first two. .. admonition:: Exercise :class: exercise stacked Plot the :math:`J/\Psi` vertex position and compare it with the true value. Plot the p-value distribution of the fit. .. admonition:: Hint: Variable names :class: dropdown xhint stacked You can either take another look at the variable collections that you included above, or you load your dataframe and then take a look at its columns ``print(list(df.columns))``. .. admonition:: Hint: Plot ranges :class: dropdown xhint stacked Plotting was already discussed in :ref:`onlinebook_roe`. For the sake of this exercise, remember we already set the minimum p-value of our fits to 0, so failed fits will not be included and you can plot it in the [0,1] interval. Should you have changed that, failed fits will be included with a p-value of -1; in this case, make sure to change your plotting range accordingly to [-1,1]. .. admonition:: Solution :class: dropdown solution .. literalinclude:: vertex/vertex_plots.py :language: python .. _vertex_plots: .. figure:: vertex/jpsi_dz.svg :width: 400px :align: center :alt: Z position of the :math:`J/\Psi` vertex. Distribution of the fitted vertex position in Z .. figure:: vertex/pValue.svg :width: 400px :align: center :alt: P-value of the vertex fit. Distribution of the fit p-values. .. admonition:: Exercises (optional) :class: exercise * Compare the :math:`J/\Psi` and Tag vertex positions and show that they are both compatible with being :math:`B` vertices. * If you've fit the :math:`K_s` vertex, compare its radial position with the :math:`J/\Psi`. Is this what you expect? .. admonition:: Key points :class: key-points * Use ``KFit`` to fit simple vertices. * Think carefully which vertex you need to fit, and whether it will need additional constraints. * Study the documentation if you need a different functionality, such as ``TreeFitter`` to fit complex trees. * Use ``TagV`` to reconstruct a vertex from the ROE. .. include:: ../lesson_footer.rstinclude .. rubric:: Authors of this lesson Francesco Tenchini