The BKM10 formalism as a Python library.

Description:

A Python library to help nuclear physicists use the BKM formalism in predicting cross-section, asymmetries, and comparing GPD models.

Installation:

Available on PyPI. To install, one can run

pip install bkm10

You will need Python 3 and pip.

Technicalities:

There are three different classes at play in this library: the main one, DifferentialCrossSection; the dataclass, BKM10Inputs; and another dataclass called CFFInputs. DifferentialCrossSection requires a million different inputs.

The Four-Fold Cross Section:

What we are numerically calculating is a four-fold (meaning, we need to do four integrals) cross section. We need to integrate over four variables: $Q^{2}$ , $x_{B}$ , $t$, $\phi$ . By the way, the first three quantities are called the kinematics, and $\phi$ is an azimuthal angle that is measured in a chosen reference frame. However, the function actually requires a bit more detail. It is a function of several different things -- schematically, we express this as:

$$d^{4}\sigma \left(\lambda, \Lambda; k, Q^{2} , x_{B} , t , \phi; \mathcal{H}, \mathcal{E}, \tilde{\mathcal{H}}, \tilde{\mathcal{E}} \right).$$

Polarization Settings:

The BKM10 formalism uses $\lambda$ to refer to the lepton beam helicity. (Note: $\lambda \in { -1, +1 }$ in this formalism!) $\Lambda$ refers to the target polarization. (In the formalism, $\Lambda \in { -1/2, +1/2 }$.)

Kinematics:

In order to evaluate the cross-section, you need to specify four numbers that correspond to the kinematic settings (experimental kinematics). These numbers are: $k$, the beam energy; $Q^{2}$, the virtuality of the photon probing the nucleon's partons; $x_{B}$, Bjorken $x$; and $t$, the (squared) momentum transfer to the hadron. Use the dataclass BKM10Inputs to specify these kinematic settings.

Note: the library currently does not handle exceptions where the provided kinematic inputs correspond to illegal mathematical operations, like division by $0$ and such. These exceptions usually correspond to unphysical kinematic settings.

Compton Form Factors:

There are four CFFs involved in the computation: $\mathcal{H}, \mathcal{E}, \tilde{\mathcal{H}}, \tilde{\mathcal{E}}$. Each of them is a complex function, so there are technically eight real numbers here. (Remember: any $z \in \mathbb{C}$ is $z = x + i y$, where $x, y \in \mathbb{R}$.) Use the dataclass CFFInputs to specify the values of these CFFs. (Note: they are of complex type!)

Goals/Future Work:

• Provide the opportunity to compute the differential cross-section using the BKM02 formalism.
• Integrate the functionality to actually do the integral over a given GPD model to obtain the CFFs, and then compute the differential cross-section.

Physics Terminology:

BKM: names of three authors: A.V. Belitsky, D. Buller, A. Kirchner,

QCD: "Quantum Chromodynamics"

CFF: "Compton Form Factor"

TMD: "Transverse Momentum Distribution"

GPD: "Generalized Parton Distribution (function)"