### Particle Tools

Tools for dealing with particle momenta four-vectors. A four-vector can either be in Cartesian coordinates, [e,px,py,pz] (energy, momentum in x direction, momentum in y direction, momentum in z direction), or hadronic coordinates, [pt,y,phi,m] (transverse momentum, rapidity, azimuthal angle, mass), which are related via:

and inversely:

The pseudorapidity eta can be obtained from a Cartesian four-momentum as:

and is related to the rapidity via

Note that the above formulas are numerically stable up to values of rapidity or pseudorapidity of a few hundred, above which the formulas have numerical issues. In this case, a different but equivalent formulae are used that are numerically stable in this region. In all cases, the $p_T\to0$ limit produces infinite values.

In the context of this package, an "event" is a two-dimensional numpy array with shape (M,4) where M is the multiplicity. An array of events is a three-dimensional array with shape (N,M,4) where N is the number of events. The valid inputs and outputs of the functions here will be described using this terminology.

#### ptyphims_from_p4s

energyflow.ptyphims_from_p4s(p4s, phi_ref=None)


Convert to hadronic coordinates [pt,y,phi,m] from Cartesian coordinates. All-zero four-vectors are left alone.

Arguments

• p4s : numpy.ndarray or list
• A single particle, event, or array of events in Cartesian coordinates.
• phi_ref : {None, 'hardest', float, numpy.ndarray}
• Used to help deal with the fact that $\phi$ is a periodic coordinate. If a float (which should be in $[0,2\pi)$), all phi values will be within $\pm\pi$ of this reference value. If '\hardest', the phi of the hardest particle is used as the reference value. If None, all phis will be in the range $[0,2\pi)$. An array is accepted in the case that p4s is an array of events, in which case the phi_ref array should have shape (N,) where N is the number of events.

Returns

• numpy.ndarray
• An array of hadronic four-momenta with the same shape as the input.

#### pts_from_p4s

energyflow.pts_from_p4s(p4s)


Calculate the transverse momenta of a collection of four-vectors.

Arguments

• p4s : numpy.ndarray or list
• A single particle, event, or array of events in Cartesian coordinates.

Returns

• numpy.ndarray
• An array of transverse momenta with shape p4s.shape[:-1].

#### pt2s_from_p4s

energyflow.pt2s_from_p4s(p4s)


Calculate the squared transverse momenta of a collection of four-vectors.

Arguments

• p4s : numpy.ndarray or list
• A single particle, event, or array of events in Cartesian coordinates.

Returns

• numpy.ndarray
• An array of squared transverse momenta with shape p4s.shape[:-1].

#### ys_from_p4s

energyflow.ys_from_p4s(p4s)


Calculate the rapidities of a collection of four-vectors. Returns zero for all-zero particles

Arguments

• p4s : numpy.ndarray or list
• A single particle, event, or array of events in Cartesian coordinates.

Returns

• numpy.ndarray
• An array of rapidities with shape p4s.shape[:-1].

#### etas_from_p4s

energyflow.etas_from_p4s(p4s)


Calculate the pseudorapidities of a collection of four-vectors. Returns zero for all-zero particles

Arguments

• p4s : numpy.ndarray or list
• A single particle, event, or array of events in Cartesian coordinates.

Returns

• numpy.ndarray
• An array of pseudorapidities with shape p4s.shape[:-1].

#### phis_from_p4s

energyflow.phis_from_p4s(p4s, phi_ref=None)


Calculate the azimuthal angles of a collection of four-vectors.

Arguments

• p4s : numpy.ndarray or list
• A single particle, event, or array of events in Cartesian coordinates.
• phi_ref : {float, numpy.ndarray, None, 'hardest'}
• Used to help deal with the fact that $\phi$ is a periodic coordinate. If a float (which should be in $[0,2\pi)$), all phi values will be within $\pm\pi$ of this reference value. If '\hardest', the phi of the hardest particle is used as the reference value. If None, all phis will be in the range $[0,2\pi)$. An array is accepted in the case that p4s is an array of events, in which case the phi_ref array should have shape (N,) where N is the number of events.

Returns

• numpy.ndarray
• An array of azimuthal angles with shape p4s.shape[:-1].

#### m2s_from_p4s

energyflow.m2s_from_p4s(p4s)


Calculate the squared masses of a collection of four-vectors.

Arguments

• p4s : numpy.ndarray or list
• A single particle, event, or array of events in Cartesian coordinates.

Returns

• numpy.ndarray
• An array of squared masses with shape p4s.shape[:-1].

#### ms_from_p4s

energyflow.ms_from_p4s(p4s)


Calculate the masses of a collection of four-vectors.

Arguments

• p4s : numpy.ndarray or list
• A single particle, event, or array of events in Cartesian coordinates.

Returns

• numpy.ndarray
• An array of masses with shape p4s.shape[:-1].

#### ms_from_ps

energyflow.ms_from_ps(ps)


Calculate the masses of a collection of Lorentz vectors in two or more spacetime dimensions.

Arguments

• ps : numpy.ndarray or list
• A single particle, event, or array of events in Cartesian coordinates in $d\ge2$ spacetime dimensions.

Returns

• numpy.ndarray
• An array of masses with shape ps.shape[:-1].

#### etas_from_pts_ys_ms

energyflow.etas_from_pts_ys_ms(pts, ys, ms)


Calculate pseudorapidities from transverse momenta, rapidities, and masses. All input arrays should have the same shape.

Arguments

• pts : numpy.ndarray
• Array of transverse momenta.
• ys : numpy.ndarray
• Array of rapidities.
• ms : numpy.ndarray
• Array of masses.

Returns

• numpy.ndarray
• Array of pseudorapidities with the same shape as ys.

#### ys_from_pts_etas_ms

energyflow.ys_from_pts_etas_ms(pts, etas, ms)


Calculate rapidities from transverse momenta, pseudorapidities, and masses. All input arrays should have the same shape.

Arguments

• pts : numpy.ndarray
• Array of transverse momenta.
• etas : numpy.ndarray
• Array of pseudorapidities.
• ms : numpy.ndarray
• Array of masses.

Returns

• numpy.ndarray
• Array of rapidities with the same shape as etas.

#### p4s_from_ptyphims

energyflow.p4s_from_ptyphims(ptyphims)


Calculate Cartesian four-vectors from transverse momenta, rapidities, azimuthal angles, and (optionally) masses for each input.

Arguments

• ptyphims : numpy.ndarray or list
• A single particle, event, or array of events in hadronic coordinates. The mass is optional and if left out will be taken to be zero.

Returns

• numpy.ndarray
• An array of Cartesian four-vectors.

#### p4s_from_ptyphipids

energyflow.p4s_from_ptyphipids(ptyphipids, error_on_unknown=False)


Calculate Cartesian four-vectors from transverse momenta, rapidities, azimuthal angles, and particle IDs for each input. The particle IDs are used to lookup the mass of the particle. Transverse momenta should have units of GeV when using this function.

Arguments

• ptyphipids : numpy.ndarray or list
• A single particle, event, or array of events in hadronic coordinates where the mass is replaced by the PDG ID of the particle.
• error_on_unknown : bool

Returns

• numpy.ndarray
• An array of Cartesian four-vectors with the same shape as the input.

#### sum_ptyphims

energyflow.sum_ptyphims(ptyphims)


Add a collection of four-vectors that are expressed in hadronic coordinates by first converting to Cartesian coordinates and then summing.

Arguments

• ptyphims : numpy.ndarray or list
• An event, or array of events in hadronic coordinates. The mass is optional and if left out will be taken to be zero. An argument of a single particle does nothing.

Returns

• numpy.ndarray
• Array of summed four-vectors, in hadronic coordinates.

#### sum_ptyphipids

energyflow.sum_ptyphipids(ptyphipids, error_on_unknown=False)


Add a collection of four-vectors that are expressed as [pT,y,phi,pdgid].

Arguments

• ptyphipids : numpy.ndarray or list
• A single particle, event, or array of events in hadronic coordinates where the mass is replaced by the PDG ID of the particle.
• error_on_unknown : bool

Returns

• numpy.ndarray
• Array of summed four-vectors, in hadronic coordinates.

#### pids2ms

energyflow.pids2ms(pids, error_on_uknown=False)


Map an array of Particle Data Group IDs to an array of the corresponding particle masses (in GeV).

Arguments

• pids : numpy.ndarray or list
• An array of numeric (float or integer) PDG ID values.
• error_on_unknown : bool
• Controls whether a KeyError is raised if an unknown PDG ID is encountered. If False, unknown PDG IDs will map to zero.

Returns

• numpy.ndarray
• An array of masses in GeV.

#### phi_fix

energyflow.phi_fix(phis, phi_ref, copy=True)


A function to ensure that all phis are within $\pi$ of phi_ref. It is assumed that all starting phi values are $\pm 2\pi$ of phi_ref.

Arguments

• phis : numpy.ndarray or list
• Array of phi values.
• phi_ref : {float or numpy.ndarray}
• A reference value used so that all phis will be within $\pm\pi$ of this value. Should have a shape of phis.shape[:-1].
• copy : bool
• Determines if phis are copied or not. If False then phis is modified in place.

Returns

• numpy.ndarray
• An array of the fixed phi values.

#### flat_metric

energyflow.flat_metric(dim)


The Minkowski metric in dim spacetime dimensions in the mostly-minus convention.

Arguments

• dim : int
• The number of spacetime dimensions (thought to be four in our universe).

Returns

• 1-d numpy.ndarray
• A dim-length, one-dimensional (not matrix) array equal to [+1,-1,...,-1].

### Random Events

Functions to generate random sets of four-vectors. Includes an implementation of the RAMBO algorithm for sampling uniform M-body massless phase space. Also includes other functions for various random, non-center of momentum, and non-uniform sampling.

#### gen_random_events

energyflow.gen_random_events(nevents, nparticles, dim=4, mass=0.0)


Generate random events with a given number of particles in a given spacetime dimension. The spatial components of the momenta are distributed uniformly in $[-1,+1]$. These events are not guaranteed to uniformly sample phase space.

Arguments

• nevents : int
• Number of events to generate.
• nparticles : int
• Number of particles in each event.
• dim : int
• Number of spacetime dimensions.
• mass : float or 'random'
• Mass of the particles to generate. Can be set to 'random' to obtain a different random mass for each particle.

Returns

• numpy.ndarray
• An (nevents,nparticles,dim) array of events. The particles are specified as [E,p1,p2,...]. If nevents is 1 then that axis is dropped.

#### gen_random_events_mcom

energyflow.gen_random_events_mcom(nevents, nparticles, dim=4)


Generate random events with a given number of massless particles in a given spacetime dimension. The total momentum are made to sum to zero. These events are not guaranteed to uniformly sample phase space.

Arguments

• nevents : int
• Number of events to generate.
• nparticles : int
• Number of particles in each event.
• dim : int
• Number of spacetime dimensions.

Returns

• numpy.ndarray
• An (nevents,nparticles,dim) array of events. The particles are specified as [E,p1,p2,...].

#### gen_massless_phase_space

energyflow.gen_massless_phase_space(nevents, nparticles, energy=1.0)


Implementation of the RAMBO algorithm for uniformly sampling massless M-body phase space for any center of mass energy.

Arguments

• nevents : int
• Number of events to generate.
• nparticles : int
• Number of particles in each event.
• energy : float
• Total center of mass energy of each event.

Returns

• numpy.ndarray
• An (nevents,nparticles,4) array of events. The particles are specified as [E,p_x,p_y,p_z]. If nevents is 1 then that axis is dropped.

### Data Tools

Functions for dealing with datasets. These are not importable from the top level energyflow module, but must instead be imported from energyflow.utils.

#### get_examples

energyflow.utils.get_examples(path='~/.energyflow', which='all', overwrite=False)


Pulls examples from GitHub. To ensure availability of all examples update EnergyFlow to the latest version.

Arguments

• path : str
• The destination for the downloaded files. Note that examples is automatically appended to the end of this path.
• which : {list, 'all'}
• List of examples to download, or the string 'all' in which case all the available examples are downloaded.
• overwrite : bool
• Whether to overwrite existing files or not.

#### data_split

energyflow.utils.data_split(*args, train=-1, val=0.0, test=0.1, shuffle=True)


A function to split a dataset into train, test, and optionally validation datasets.

Arguments

• *args : arbitrary numpy.ndarray datasets
• An arbitrary number of datasets, each required to have the same number of elements, as numpy arrays.
• train : {int, float}
• If a float, the fraction of elements to include in the training set. If an integer, the number of elements to include in the training set. The value -1 is special and means include the remaining part of the dataset in the training dataset after the test and (optionally) val parts have been removed
• val : {int, float}
• If a float, the fraction of elements to include in the validation set. If an integer, the number of elements to include in the validation set. The value 0 is special and means do not form a validation set.
• test : {int, float}
• If a float, the fraction of elements to include in the test set. If an integer, the number of elements to include in the test set.
• shuffle : bool
• A flag to control whether the dataset is shuffle prior to being split into parts.

Returns

• list
• A list of the split datasets in train, [val], test order. If datasets X, Y, and Z were given as args (and assuming a non-zero val), then [X_train, X_val, X_test, Y_train, Y_val, Y_test, Z_train, Z_val, Z_test] will be returned.

#### to_categorical

energyflow.utils.to_categorical(labels, num_classes=None)


One-hot encodes class labels.

Arguments

• labels : 1-d numpy.ndarray
• Labels in the range [0,num_classes).
• num_classes : {int, None}
• The total number of classes. If None, taken to be the maximum label plus one.

Returns

• 2-d numpy.ndarray
• The one-hot encoded labels.

#### remap_pids

energyflow.utils.remap_pids(events, pid_i=3)


Remaps PDG id numbers to small floats for use in a neural network. events are modified in place and nothing is returned.

Arguments

• events : 3-d numpy.ndarray
• The events as an array of arrays of particles.
• pid_i : int
• The index corresponding to pid information along the last axis of events.

### Image Tools

Functions for dealing with image representations of events. These are not importable from the top level energyflow module, but must instead be imported from energyflow.utils.

#### pixelate

energyflow.utils.pixelate(jet, npix=33, img_width=0.8, nb_chan=1, norm=True, charged_counts_only=False)


A function for creating a jet image from an array of particles.

Arguments

• jet : numpy.ndarray
• An array of particles where each particle is of the form [pt,y,phi,pid] where the particle id column is only used if nb_chan=2 and charged_counts_only=True.
• npix : int
• The number of pixels on one edge of the jet image, which is taken to be a square.
• img_width : float
• The size of one edge of the jet image in the rapidity-azimuth plane.
• nb_chan : {1, 2}
• The number of channels in the jet image. If 1, then only a $p_T$ channel is constructed (grayscale). If 2, then both a $p_T$ channel and a count channel are formed (color).
• norm : bool
• Whether to normalize the $p_T$ pixels to sum to 1.
• charged_counts_only : bool
• If making a count channel, whether to only include charged particles. Requires that pid information be given.

Returns

• 3-d numpy.ndarray
• The jet image as a (nb_chan, npix, npix) array.

#### standardize

energyflow.utils.standardize(*args, channels=None, copy=False, reg=10**-10)


Normalizes each argument by the standard deviation of the pixels in arg[0]. The expected use case would be standardize(X_train, X_val, X_test).

Arguments

• *args : arbitrary numpy.ndarray datasets
• An arbitrary number of datasets, each required to have the same shape in all but the first axis.
• channels : int
• A list of which channels (assumed to be the second axis) to standardize. None is interpretted to mean every channel.
• copy : bool
• Whether or not to copy the input arrays before modifying them.
• reg : float
• Small parameter used to avoid dividing by zero. It's important that this be kept consistent for images used with a given model.

Returns

• list
• A list of the now-standardized arguments.

#### zero_center

energyflow.utils.zero_center(args, kwargs)


Subtracts the mean of arg[0] from the arguments. The expected use case would be standardize(X_train, X_val, X_test).

Arguments

• *args : arbitrary numpy.ndarray datasets
• An arbitrary number of datasets, each required to have the same shape in all but the first axis.
• channels : int
• A list of which channels (assumed to be the second axis) to zero center. None is interpretted to mean every channel.
• copy : bool
• Whether or not to copy the input arrays before modifying them.

Returns

• list
• A list of the zero-centered arguments.