hammer-reweight

Code for FF reweighting in HAMMER.

Installation

Install nix with flake support, then type in nix develop. All dependencies will be installed automatically for you.

To build all binaries:

make

Once this is done, to generate weight ntuples (using the nominal reweighting script src/ReweightRDX.cpp) based on sample ntuples (stored in samples/) from RDX run 2 analysis:

make rdx-run2-ntuples

Note that the stdout from the reweighter will also be saved as log files.

To plot resulting sample q2 (stored in gen/):

make sample-plots

To generate some reweighting validation plots (stored in gen):

make validation-plots

Usage

PrintMCDecay

This program will print all partially truth-matched MC candidates. For example:

PrintMCDecay ./samples/rdx-run2-Bd2DstMuNu.root TupleB0/DecayTree

Will give you an output like this:

Total number of candidates: 23841
Truth-matched candidates: 23339
Truth-matched fraction: 0.978944
======
The following decay has 23314 candidates.
Is Tau decay: 0
B meson ID: B0 (511)
First daughter ID: D*+ (413)
  First G-daughter ID: D0 (421)
  Second G-daughter ID: pi+ (211)
======
The following decay has 25 candidates.
Is Tau decay: 0
B meson ID: B0 (511)
First daughter ID: D*+ (413)
  First G-daughter ID: D0 (421)
  Second G-daughter ID: pi+ (211)
  Third G-daughter ID: gamma (22)

Note that all particle IDs are taken to be absolute value.

ReweightRDX

To do the actual form factor reweighting for RDX analyses, you can do something like this:

ReweightRDX samples/rdx-run2-Bd2DstMuNu.root output.root TupleB0/DecayTree run2

This will generate weight ntuples, stored in the home directory (and named output.root). Notably, you can also run other reweighting scripts (stored in src/) or the nominal reweighting with more printouts for debugging with similar commands (eg. ReweightRDXDstNoCorr, or 'ReweightRDXDebug for debugging, instead of ReweightRDX). The additional reweighting scripts are mostly used for studies.

The output.root contains a w_ff branch and some other debugging branches.

NOTE: currently, the FF parameters/errors/variations set in the nominal reweighter ReweightRDX have

• For $B\rightarrow D$: parameter values and errors taken from 1606.08030, with one variation (along the direction of the nominal FF parameters, ie. the rescaling direction) removed. Correlations from the paper are taken into account.
• For $B\rightarrow D^{*}$: parameter values and errors taken from 2105.14019 (v3, Table 12- "Lattice + both"), with one variation (along the rescaling direction) removed. Correlations from the paper are taken into account.
• For $B\rightarrow D_{(s)}^{**}$: parameter values and errors taken from 1711.03110, with one variation (in the direction of parameter that mainly effect normalization/rescaling: $\tau (1)$ and $\zeta (1)$) removed for both the wide/narrow states (which are parametrized separately). The $D_s^{**}$ FFs are taken to be the same as for the $D^{**}$. Previous fit results indicated that the paper errors were too constraining, and potentially the central values ought to be shifted; currently, the errors used for variations are double the paper errors, and the central values are not shifted. Correlations from the paper are not taken into account; the variations are exactly $\mathbf{f}_{nom} +- \mathbf{sigma}_i$ (where $\mathbf{sigma}_i$ is a vector of 0's and the $i^{th}$ FF param error in the $i^{th}$ position).

ValidateRDX

This is used to generate some toy data to validate HAMMER reweighting for RDX with an independently implemented FF calculator.

To use it:

ValidateRDX test.root

Compute FF variation parameters

To produce code that can be pasted into a reweighting script (eg. ReweightRDX.cpp) that specifies the parameter values as well as the FF variations, use

make ff-params-RDX

See utils/gen_ham_params.py and spec/rdx-run2.yml for how these are computed.

To produce the variations with rescaling removed from the variations (not nominally used for $D_{(s)}^{**}$), use

make ff-params-RDX-no-rescale

The motivation and math underlying removing the rescaling from the variations can be found in utils/gen_ham_params_no_rescale.py.

HAMMER tips

General HAMMER reweighting workflow

1. Configure the allowed decay modes and input/output FF parameterizations
2. Define Hammer::Particle and adding these particles to a newly created Hammer::Process, denote proc.addParticle(p)
3. Specify the decay process tree with proc.addVertex(mother, {list_of_daughters})
4. Initialize a new HAMMER event object with Hammer::initEvent

5. Add a proc to the newly initialized event with Hammer::addProcess(proc). This includes:

   1. Prune soft photons (the 4-momenta will be modified in this step) by adding them to nearest charged particle in the same decay tree level (polar-angle wise) then remove the photons from the tree

      This process may alter the kinematics in a bad way such that HAMMER won't process the event further (either a NaN in the cosine or a negative invariant mass)

   2. Compute the decay tree hash ID
   3. Cache particle dependencies so the decay tree only need to be traversed once (a technicality)

6. Compute the amplitude tensor with Hammer::processEvent(), then get the weight with Hammer::getWeight("FF_scheme"):

   • A bit more details on processEvent: it's defined in src/Core/Hammer.cc
   • Then Event.calc() is called, which is defined in src/core/Event.cc

   • Inside Event.calc(), it calls p.second.calc() for each element in _process.

     Note: _process is of type ProcIdDict, which is std::map<ProcessUID, T>.

   • The actual calculation happens src/core/ProcManager.cc.

Weight computation

1. The amplitude tensor is computed

2. Hammer::getWeight("FF_scheme") is called.

3. In that method, find the base weights of the event: double result = _event->getEventBaseWeight().

   This is by default set to 1 and we typically don't change it

4. For each process, multiply the base weight by result *= _event->getWeight(scheme, elem).

   Here process means decay processes added by user, like B -> D* l nu. Also, the amplitude tensor is contracted with the external eigenvectors.

   This is the step where the FF variation is effected.

Form factor variation

1. Find/compute the covariance matrix $U$ of a FF parameterization from some reference paper
2. Find the eigenvectors and eigenvalues of $U$
3. Define a matrix $M$, with each row being one of the eigenvectors
4. Define a matrix $C$, with $C{ij} \equiv \delta{ij} \frac{1}{\lambda_i}$
5. The transformation $A \equiv CM$ maps the parameterization basis to an orthonormal error eigen basis.
6. We can either define $A^{-1}$ in HAMMER, or define variations in terms of $A^{-1} \delta$, where $\delta$ is some variation in the error eigen basis.

Soft photons (very)

HAMMER doesn't like very soft photons (for energy on the order of 1e-10), and for these photons, their effect are negligible as well. So don't add these photons to the HAMMER's decay tree in the first place!

Vertex level momentum conservation

HAMMER doesn't require momentum to be conserved in general; it only requires each particle has a non-negative invariant mass.

However, for soft photon removal, the vertex-level momentum conservation is enforced. This sometimes can lead to undesired effect of negative invariant mass

Negative invariant mass

The invariant mass squared is computed in Hammer::FourMomentum.mass2(). This is just E^2 - p^2. Due to float-point arithmetic, sometimes this is slightly negative, which is quite common for particles with very small invariant masses.

HAMMER provides a more robust way, The Hammer::FourMomentum.mass() method:

double FourMomentum::mass() const {
    if (fuzzyLess(mass2() , 0.0)) { // less than 'a little bit negative'
        throw Error("Rosebud! Negative mass^2: " + to_string(mass2()));
    }
    return sqrt(fabs(mass2()));
}

The fuzzyLess is defined as:

static const double precision = 0.001;

inline bool fuzzyLess(const double val1, const double val2) {
    return (val1-val2 < -1.*std::max(precision, std::numeric_limits<double>::min()));
}

Only if the invariant mass is strictly negative, HAMMER would complains about it.

HAMMER produces a segmentation fault

This has happened to us on some of the D*_2 candidates. The reason was that we also added the daughters of the D*_2 to the decay tree, and HAMMER really doesn't like them.

By removing them from the tree, HAMMER runs fine. So it's probably a good idea to not adding unneeded particles to the tree.