pySecDec Integral Evaluations for a bc-Diquark Sum-Rules Analysis

The Tarcer basis integrals

• $\mathrm{TBI}(d, q^2, [[1, m_c], [1, m_b]])$
• $\mathrm{TJI}(d, q^2, [[1, m_c], [1, m_b], [1, 0]])$
• $\mathrm{TJI}(d, q^2, [[2, m_c], [1, m_b], [1, 0]])$
• $\mathrm{TJI}(d, q^2, [[1, m_c], [2, m_b], [1, 0]])$,

where $d = 4 + 2\epsilon$ and $m_c$ & $m_b$ are the charm and bottom quark masses respectively, are evaluated for $q^2$ along a keyhole contour (for $\mathrm{Im}(q^2) \geq 0$) using pySecDec. The parameters that characterize the keyhole contour are defined between pysecdec_integrals/quark_masses.py,

"""Bottom and charm quark masses."""
MC = 1.095    # charm quark mass
MB = 4.47521  # bottom quark mass

and pysecdec_integrals/keyhole_params.py,

# the centre of the arc portion of the keyhole
CENTRE = (MC + MB) ** 2

# the radius of the arc portion of the keyhole
RADIUS = 5.0

# the max value of Re(q**2) on the line portion of the keyhole
MAX_RE_QQ = 125.0

# the number of grid points on the line portion of the keyhole
NUM_PTS_LINE = 200

# the number of grid points on the arc portion of the keyhole
NUM_PTS_ARC = 50

# The value of Im(q**2) along the line portion of the keyhole
DELTA = 1e-6

Integral Results

Integral results corresponding to the above settings of the keyhole domain are written to the data/ directory. There are five files: one for the domain and one for each of the four integrals. All five files are .m files, intended to be read in to a Mathematica notebook.

Generating New Integral Results

To generate results for keyhole settings other than those given above, start by creating and activating a conda environment:

$ conda env create --file=environment.yml
$ conda activate pysecdec

Then, generate the libraries that pySecDec uses to evaluate integrals:

$ make

Optionally, run all available unittests:

$ python -m unittest discover

Finally, adjust the quark masses and keyhole parameters as needed by directly editing pysecdec_integrals/quark_masses.py and pysecdec_integrals/keyhole_params.py, and run:

$ python -m pysecdec_integrals.integrate

The five files in the data/ directory will be overwritten with integral results corresponding to the new settings of the quark masses and keyhole domain.