Module `tparton.constants`

QCD constants and parameters used in PDF evolution.

This module defines the color factors, flavor factors, and beta function coefficients used in the DGLAP evolution equations.

Home: https://mikesha2.github.io/tParton/
Examples & Tutorials: https://mikesha2.github.io/tParton/examples.html
API Documentation: https://mikesha2.github.io/tParton/api/tparton/

References

Sha, C.M. & Ma, B. (2025). arXiv:2409.00221

Functions

def constants(CG, n_f)

Expand source code

def constants(CG, n_f):
    """Compute QCD constants in terms of number of colors and flavors.
    
    Calculates the color factors (NC, CF), flavor factor (Tf), and
    QCD beta function coefficients (β₀, β₁) used throughout the evolution.
    
    Args:
        CG:
        Number of colors, NC (typically 3 for QCD)
        n_f:
        Number of active quark flavors (typically 3-6 depending on Q²)
    
    Returns:
    tuple of float
        (NC, CF, Tf, beta0, beta1) where:
        
        - NC : Number of colors (= CG)
        - CF : Fundamental Casimir operator = (NC² - 1)/(2NC)
        - Tf : Flavor factor = TR × n_f, where TR = 1/2
        - beta0 : Leading QCD beta function coefficient
        - beta1 : Next-to-leading QCD beta function coefficient
    
    Note:
    These constants are defined after Eq. (4) in the paper.
    
    The beta function coefficients govern the running of αs:
    
    - β₀ = (11/3)NC - (4/3)TR·n_f
    - β₁ = (34/3)NC² - (10/3)NC·n_f - 2CF·n_f
    
    For standard QCD with NC=3:
    
    - CF = 4/3
    - beta0 ≈ 11 - (4/3)n_f
    
    Example:
    >>> from tparton.constants import constants
    >>> NC, CF, Tf, beta0, beta1 = constants(CG=3, n_f=5)
    >>> print(f"CF = {CF:.4f}, beta0 = {beta0:.4f}")
    CF = 1.3333, beta0 = 7.3333
    """
    NC = CG
    CF = (NC * NC - 1) / NC / 2
    TR = 1/2
    Tf = TR * n_f
    beta0 = 11 / 3 * CG - 4 / 3 * TR * n_f
    beta1 = 34 / 3 * CG ** 2 - 10 / 3 * CG * n_f - 2 * CF * n_f
    return NC, CF, Tf, beta0, beta1

Compute QCD constants in terms of number of colors and flavors.

Calculates the color factors (NC, CF), flavor factor (Tf), and QCD beta function coefficients (β₀, β₁) used throughout the evolution.

Args

CG: Number of colors, NC (typically 3 for QCD) n_f: Number of active quark flavors (typically 3-6 depending on Q²) Returns: tuple of float (NC, CF, Tf, beta0, beta1) where:

- NC : Number of colors (= CG)
- CF : Fundamental Casimir operator = (NC² - 1)/(2NC)
- Tf : Flavor factor = TR × n_f, where TR = 1/2
- beta0 : Leading QCD beta function coefficient
- beta1 : Next-to-leading QCD beta function coefficient

Note: These constants are defined after Eq. (4) in the paper.

The beta function coefficients govern the running of αs:

β₀ = (11/3)NC - (4/3)TR·n_f
β₁ = (34/3)NC² - (10/3)NC·n_f - 2CF·n_f

For standard QCD with NC=3:

CF = 4/3
beta0 ≈ 11 - (4/3)n_f

Example:

>>> from tparton.constants import constants
>>> NC, CF, Tf, beta0, beta1 = constants(CG=3, n_f=5)
>>> print(f"CF = {CF:.4f}, beta0 = {beta0:.4f}")
CF = 1.3333, beta0 = 7.3333

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