#!/usr/bin/env python3
import pySecDec as psd
if __name__ == "__main__":
li = psd.LoopIntegralFromGraph(internal_lines=[[0, [1, 2]], [0, [2, 5]],
[0, [1, 6]], ['mt', [3, 5]],
['mt', [3, 6]],
['mt', [4, 5]],
['mt', [4, 6]]],
external_lines=[['p1', 1], ['p2', 2],
['p3', 3], ['p4', 4]],
replacement_rules=[
('p1*p1', 0), ('p2*p2', 0),
('p3*p3', 'mH2'), ('p4*p4', 'mZ2'),
('mt*mt', 'mt2'), ('p1*p2', 's12/2'),
('p2*p3', '(s23-mH2)/2'),
('p1*p3', '(-s12-s23+mZ2)/2'),
('p3*p4', '(s12-mH2-mZ2)/2'),
('p1*p4', '(s23-mZ2)/2'),
('p2*p4', '(-s12-s23+mH2)/2')
])
Mandelstam_symbols = ['s12', 's23']
mass_symbols = ['mt2', 'mH2', 'mZ2']
psd.loop_package(
name='hz2L_nonplanar',
loop_integral=li,
real_parameters=Mandelstam_symbols + mass_symbols,
#!/usr/bin/env python3
import pySecDec as psd
if __name__ == "__main__":
# Example 6 of Bernd Jantzen (arXiv:1111.2589)
# replace Qsq by one and add the Qsq prefactor later to factorize Qsq out
li = psd.LoopIntegralFromGraph(internal_lines=[['0', [1,
3]], ['0', [2, 3]],
['m', [1, 2]]],
external_lines=[['p1', 1], ['p2', 2],
['p3', 3]],
replacement_rules=[
('p1*p1', '0'),
('p2*p2', '0'),
('p3*p3', '-Qsq'),
('m**2', 't*Qsq'),
('Qsq', '1'),
])
# find the regions
generators_args = psd.loop_regions(
name='formfactor1L_massive_ebr',
loop_integral=li,
smallness_parameter='t',
expansion_by_regions_order=0,
decomposition_method='geometric',
additional_prefactor=f"Qsq**({li.exponent_F})",
add_monomial_regulator_power="n",
#!/usr/bin/env python3
import pySecDec as psd
if __name__ == "__main__":
# this object represents the Feynman graph
li = psd.LoopIntegralFromGraph(internal_lines=[['m', [1, 2]],
['m', [2, 1]]],
external_lines=[['p', 1], ['p', 2]],
replacement_rules=[('p*p', 's'),
('m*m', 'msq')])
# find the regions and expand the integrals using expansion by regions
regions_generator_args = psd.loop_regions(name='bubble1L_ebr_small_mass',
loop_integral=li,
smallness_parameter='msq',
expansion_by_regions_order=2)
# generate code that will calculate the sum of all regions and all orders in
# the smallness parameter
psd.sum_package('bubble1L_ebr_small_mass',
regions_generator_args,
regulators=['eps'],
requested_orders=[0],
real_parameters=['s', 'msq'],
complex_parameters=[])
#!/usr/bin/env python3
import pySecDec as psd
if __name__ == "__main__":
li = psd.LoopIntegralFromGraph(
internal_lines = [ [0,[1,2]], [0,[1,4]], [0,[1,5]], [0,[2,4]], [0,[2,5]], [0,[3,4]], [0,[3,5]] ],
external_lines = [['p1',1],['p2',2],['p3',3]],
powerlist=[1,1,1,1,1,1,1],
replacement_rules = [
('p1*p1',0),
('p2*p2',0),
('p3*p3','s'),
('p1*p2','s/2'),
('p2*p3','-s/2'),
('p1*p3','-s/2'),
('s',-1)
]
)
Mandelstam_symbols = []
mass_symbols = []
psd.loop_package(
name = 'triangle3L',
loop_integral = li,
#!/usr/bin/env python3
import pySecDec as psd
if __name__ == "__main__":
# Section 3.1 (Method of Regions for the Sudakov form factor) of Thomas Becher (arXiv:1803.04310)
# use relative dimensionless parameters Psqr and Lsqr, which should be roughly of order one
# replace Qsq with one and add the Qsq prefactor later to factorize Qsq out
li = psd.LoopIntegralFromGraph(
internal_lines = [['0',[1,3]],['0',[2,3]],['0',[1,2]]],
external_lines = [['p1',1],['p2',2],['p3',3]],
replacement_rules = [
('p3*p3', '-Qsq'),
('p1*p1', '-t*Psqr*Qsq'),
('p2*p2', '-t*Lsqr*Qsq'),
("Qsq", 1),
],
)
# find the regions
generators_args = psd.loop_regions(
name = 'formfactor1L_massless_ebr',
loop_integral=li,
smallness_parameter = 't',
expansion_by_regions_order=0,
additional_prefactor=f"Qsq**({li.exponent_F})",
)
# write the code to sum up the regions
psd.sum_package('formfactor1L_massless_ebr',
if __name__ == "__main__":
li = psd.LoopIntegralFromGraph(internal_lines=[[0, [1, 2]], [0, [2, 6]],
[0, [6, 3]], [0, [3, 4]],
[0, [4, 5]], [0, [5, 7]],
[0, [7, 1]], [0, [7, 6]]],
external_lines=[['p1', 1], ['p2', 2],
['p3', 3], ['p4', 4],
['p5', 5]],
dimensionality='6-2*eps',
replacement_rules=[
('p1*p1', 0),
('p2*p2', 0),
('p3*p3', 0),
('p4*p4', 0),
('p5*p5', 0),
('p1*p2', 's12/2'),
('p1*p3', '(s45-s12-s23)/2'),
('p1*p4', '(s23-s51-s45)/2'),
('p1*p5', 's51/2'),
('p2*p3', 's23/2'),
('p2*p4', '(-s23-s34+s51)/2'),
('p2*p5', '(s34-s12-s51)/2'),
('p3*p4', 's34/2'),
('p3*p5', '(s12-s34-s45)/2'),
('p4*p5', 's45/2'),
])
Mandelstam_symbols = ['s12', 's23', 's34', 's45', 's51']
mass_symbols = []
#!/usr/bin/env python3
import pySecDec as psd
if __name__ == "__main__":
li = psd.LoopIntegralFromGraph(
internal_lines = [['m',[1,2]],[0,[2,3]],[0,[3,4]],[0,[4,1]]],
external_lines = [['p1',1],['p2',2],['p3',3],['p4',4]],
replacement_rules = [
('p1*p1', 's1'),
('p2*p2', 0),
('p3*p3', 0),
('p4*p4', 0),
('p3*p2', 't/2'),
('p1*p2', 's/2-s1/2'),
('p1*p4', 't/2-s1/2'),
('p2*p4', 's1/2-t/2-s/2'),
('p3*p4', 's/2'),
('m**2', 'msq')
]
)
Mandelstam_symbols = ['s','t','s1']
mass_symbols = ['msq']
psd.loop_package(
name = 'box1L',
loop_integral = li,
#!/usr/bin/env python3
import sympy as sp
import pySecDec as psd
if __name__ == "__main__":
li = psd.LoopIntegralFromGraph(
internal_lines = [[0,[1,4]],[0,[4,2]],[0,[2,5]],[0,[5,6]],[0,[6,7]],[0,[7,3]],[0,[3,8]],[0,[8,4]],[0,[1,9]],[0,[9,7]],[0,[9,5]],[0,[6,8]]],
external_lines = [['p1',1],['p2',2],['p3',3]],
powerlist=[2,1,1,1,1,1,1,1,1,2,1,1],
dimensionality='6-2*eps',
replacement_rules = [
('p1*p1',-1),
('p2*p2',0),
('p3*p3','0'),
('p1*p2','1/2'),
('p2*p3','-1/2'),
('p1*p3','1/2')
]
)
Mandelstam_symbols = []
mass_symbols = []
psd.loop_package(
name = 'formfactor4L',
loop_integral = li,
#!/usr/bin/env python3
import pySecDec as psd
if __name__ == "__main__":
li = psd.LoopIntegralFromGraph(internal_lines=[[0, [1, 2]], [0, [1, 3]],
[0, [2, 3]], [0, [2, 5]],
[0, [5, 6]], [0, [6, 3]],
[0, [6, 7]], [0, [5, 7]],
[0, [7, 4]], [0, [4, 7]],
[0, [4, 3]], [0, [4, 1]]],
external_lines=[['p', 1], ['p', 2]],
replacement_rules=[('p*p', '-1')])
psd.loop_package(
name='bubble6L',
loop_integral=li,
# the highest order of the final epsilon expansion --> change this value to whatever you think is appropriate
requested_orders=[0],
# the optimization level to use in FORM (can be 0, 1, 2, 3, 4)
form_optimization_level=4,
# the WorkSpace parameter for FORM
form_work_space='2G',
# the method to be used for the sector decomposition
# valid values are ``iterative`` or ``geometric`` or ``geometric_ku``
decomposition_method='geometric',
# if you choose ``geometric[_ku]`` and 'normaliz' is not in your
continue
if index == len(smallness_parameters) - 1:
print(bcolors.ENDC)
expand_integrals(generators_args,
smallness_parameters,
index + 1,
common_args=common_args)
if __name__ == "__main__":
# Formfactor1L
name = 'formfactor1L_massless_ebr'
smallness_parameters = ['t']
internal_lines = [['0', [1, 3]], ['0', [2, 3]], ['0', [1, 2]]]
external_lines = [['p1', 1], ['p2', 2], ['p3', 3]]
li = psd.LoopIntegralFromGraph(
internal_lines=internal_lines,
external_lines=external_lines,
replacement_rules=[
('p3*p3', '-Qsq'),
('p1*p1', '-t*P1sqr*Qsq'),
('p2*p2', '-t*P2sqr*Qsq'),
('Qsq', 1),
],
)
# (Optional) Draw diagrams
# psd.loop_integral.draw.plot_diagram(internal_lines,external_lines,name)
expand_integral(li, smallness_parameters)