Module flavio.physics.bdecays.lifetime_ratio
$B^+ / B_d$ lifetime ratio
Functions
def A_PI_cd(rho, me)-
Expand source code
def A_PI_cd(rho, me): # Matrix containing Pauli interference contributions from internal cd quarks # See eqs 2.22 - 2.27 in Lenz:2022pgw flavio.citations.register("Lenz:2022pgw") return np.array( ( ( me["1"] + 6 * me["3"], 3 * me["1"], -(rho**0.5 * (me["1"] + 6 * me["3"])) / 2, (-3 * rho**0.5 * me["1"]) / 2, -(rho**0.5 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"]))) / 4, (-3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, (-me["5"] + 2 * (me["6"] - 3 * me["7"] + 6 * me["8"])) / 4, (-3 * (me["5"] - 2 * me["6"])) / 4, 3 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"])), 9 * (me["5"] - 2 * me["6"]), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ), ( 3 * me["1"], me["1"] + 6 * me["3"], (-3 * rho**0.5 * me["1"]) / 2, -(rho**0.5 * (me["1"] + 6 * me["3"])) / 2, (-3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, -(rho**0.5 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"]))) / 4, (-3 * (me["5"] - 2 * me["6"])) / 4, (-me["5"] + 2 * (me["6"] - 3 * me["7"] + 6 * me["8"])) / 4, 9 * (me["5"] - 2 * me["6"]), 3 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"])), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ), ( -(rho**0.5 * (me["1"] + 6 * me["3"])) / 2, (-3 * rho**0.5 * me["1"]) / 2, (me["1"] + 2 * rho * me["1"] - 2 * (2 + rho) * me["2"] + 6 * (me["3"] + 2 * rho * me["3"] - 2 * (2 + rho) * me["4"])) / 6, (0.5 + rho) * me["1"] - (2 + rho) * me["2"], ((-1 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"] + 3 * (me["7"] - rho * me["7"] + 2 * me["8"] + 4 * rho * me["8"]))) / 12, ((-1 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"])) / 4, -(rho**0.5 * (me["6"] + 6 * me["8"])) / 2, (-3 * rho**0.5 * me["6"]) / 2, -2 * rho**0.5 * (me["5"] - me["6"] + 6 * me["7"] - 6 * me["8"]), -6 * rho**0.5 * (me["5"] - me["6"]), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ), ( (-3 * rho**0.5 * me["1"]) / 2, -(rho**0.5 * (me["1"] + 6 * me["3"])) / 2, (0.5 + rho) * me["1"] - (2 + rho) * me["2"], (me["1"] + 2 * rho * me["1"] - 2 * (2 + rho) * me["2"] + 6 * (me["3"] + 2 * rho * me["3"] - 2 * (2 + rho) * me["4"])) / 6, ((-1 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"])) / 4, ((-1 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"] + 3 * (me["7"] - rho * me["7"] + 2 * me["8"] + 4 * rho * me["8"]))) / 12, (-3 * rho**0.5 * me["6"]) / 2, -(rho**0.5 * (me["6"] + 6 * me["8"])) / 2, -6 * rho**0.5 * (me["5"] - me["6"]), -2 * rho**0.5 * (me["5"] - me["6"] + 6 * me["7"] - 6 * me["8"]), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ), ( -(rho**0.5 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"]))) / 4, (-3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, ((-1 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"] + 3 * (me["7p"] - rho * me["7p"] + 2 * me["8p"] + 4 * rho * me["8p"]))) / 12, ((-1 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"])) / 4, (me["1p"] + 2 * rho * me["1p"] - 2 * (2 + rho) * me["2p"] + 6 * (me["3p"] + 2 * rho * me["3p"] - 2 * (2 + rho) * me["4p"])) / 24, (me["1p"] + 2 * rho * me["1p"] - 2 * (2 + rho) * me["2p"]) / 8, (rho**0.5 * (me["1p"] - 2 * (me["2p"] - 3 * me["3p"] + 6 * me["4p"]))) / 8, (3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, -(rho**0.5 * (me["1p"] + 2 * (me["2p"] + 3 * me["3p"] + 6 * me["4p"]))) / 2, (-3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ), ( (-3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, -(rho**0.5 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"]))) / 4, ((-1 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"])) / 4, ((-1 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"] + 3 * (me["7p"] - rho * me["7p"] + 2 * me["8p"] + 4 * rho * me["8p"]))) / 12, (me["1p"] + 2 * rho * me["1p"] - 2 * (2 + rho) * me["2p"]) / 8, (me["1p"] + 2 * rho * me["1p"] - 2 * (2 + rho) * me["2p"] + 6 * (me["3p"] + 2 * rho * me["3p"] - 2 * (2 + rho) * me["4p"])) / 24, (3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (rho**0.5 * (me["1p"] - 2 * (me["2p"] - 3 * me["3p"] + 6 * me["4p"]))) / 8, (-3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, -(rho**0.5 * (me["1p"] + 2 * (me["2p"] + 3 * me["3p"] + 6 * me["4p"]))) / 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ), ( (-me["5p"] + 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"])) / 4, (-3 * (me["5p"] - 2 * me["6p"])) / 4, -(rho**0.5 * (me["6p"] + 6 * me["8p"])) / 2, (-3 * rho**0.5 * me["6p"]) / 2, (rho**0.5 * (me["1p"] - 2 * (me["2p"] - 3 * me["3p"] + 6 * me["4p"]))) / 8, (3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, ((2 + rho) * me["1p"] - 2 * (me["2p"] + 2 * rho * me["2p"] - 3 * (2 + rho) * me["3p"] + 6 * (me["4p"] + 2 * rho * me["4p"]))) / 24, ((2 + rho) * me["1p"] - 2 * (me["2p"] + 2 * rho * me["2p"])) / 8, ((-4 + rho) * me["1p"] - 2 * (me["2p"] + 2 * rho * me["2p"] - 3 * (-4 + rho) * me["3p"] + 6 * (me["4p"] + 2 * rho * me["4p"]))) / 6, ((-4 + rho) * me["1p"]) / 2 - (1 + 2 * rho) * me["2p"], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ), ( (-3 * (me["5p"] - 2 * me["6p"])) / 4, (-me["5p"] + 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"])) / 4, (-3 * rho**0.5 * me["6p"]) / 2, -(rho**0.5 * (me["6p"] + 6 * me["8p"])) / 2, (3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (rho**0.5 * (me["1p"] - 2 * (me["2p"] - 3 * me["3p"] + 6 * me["4p"]))) / 8, ((2 + rho) * me["1p"] - 2 * (me["2p"] + 2 * rho * me["2p"])) / 8, ((2 + rho) * me["1p"] - 2 * (me["2p"] + 2 * rho * me["2p"] - 3 * (2 + rho) * me["3p"] + 6 * (me["4p"] + 2 * rho * me["4p"]))) / 24, ((-4 + rho) * me["1p"]) / 2 - (1 + 2 * rho) * me["2p"], ((-4 + rho) * me["1p"] - 2 * (me["2p"] + 2 * rho * me["2p"] - 3 * (-4 + rho) * me["3p"] + 6 * (me["4p"] + 2 * rho * me["4p"]))) / 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ), ( 3 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"])), 9 * (me["5p"] - 2 * me["6p"]), -2 * rho**0.5 * (me["5p"] - me["6p"] + 6 * me["7p"] - 6 * me["8p"]), -6 * rho**0.5 * (me["5p"] - me["6p"]), -(rho**0.5 * (me["1p"] + 2 * (me["2p"] + 3 * me["3p"] + 6 * me["4p"]))) / 2, (-3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, ((-4 + rho) * me["1p"] - 2 * (me["2p"] + 2 * rho * me["2p"] - 3 * (-4 + rho) * me["3p"] + 6 * (me["4p"] + 2 * rho * me["4p"]))) / 6, ((-4 + rho) * me["1p"]) / 2 - (1 + 2 * rho) * me["2p"], (2 * ((14 + rho) * me["1p"] - 2 * (me["2p"] + 2 * rho * me["2p"] - 3 * (14 + rho) * me["3p"] + 6 * (me["4p"] + 2 * rho * me["4p"])))) / 3, 2 * (14 + rho) * me["1p"] - 4 * (me["2p"] + 2 * rho * me["2p"]), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ), ( 9 * (me["5p"] - 2 * me["6p"]), 3 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"])), -6 * rho**0.5 * (me["5p"] - me["6p"]), -2 * rho**0.5 * (me["5p"] - me["6p"] + 6 * me["7p"] - 6 * me["8p"]), (-3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, -(rho**0.5 * (me["1p"] + 2 * (me["2p"] + 3 * me["3p"] + 6 * me["4p"]))) / 2, ((-4 + rho) * me["1p"]) / 2 - (1 + 2 * rho) * me["2p"], ((-4 + rho) * me["1p"] - 2 * (me["2p"] + 2 * rho * me["2p"] - 3 * (-4 + rho) * me["3p"] + 6 * (me["4p"] + 2 * rho * me["4p"]))) / 6, 2 * (14 + rho) * me["1p"] - 4 * (me["2p"] + 2 * rho * me["2p"]), (2 * ((14 + rho) * me["1p"] - 2 * (me["2p"] + 2 * rho * me["2p"] - 3 * (14 + rho) * me["3p"] + 6 * (me["4p"] + 2 * rho * me["4p"])))) / 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ), ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, me["1p"] + 6 * me["3p"], 3 * me["1p"], -(rho**0.5 * (me["1p"] + 6 * me["3p"])) / 2, (-3 * rho**0.5 * me["1p"]) / 2, -(rho**0.5 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"]))) / 4, (-3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, (-me["5p"] + 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"])) / 4, (-3 * (me["5p"] - 2 * me["6p"])) / 4, 3 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"])), 9 * (me["5p"] - 2 * me["6p"]), ), ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3 * me["1p"], me["1p"] + 6 * me["3p"], (-3 * rho**0.5 * me["1p"]) / 2, -(rho**0.5 * (me["1p"] + 6 * me["3p"])) / 2, (-3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, -(rho**0.5 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"]))) / 4, (-3 * (me["5p"] - 2 * me["6p"])) / 4, (-me["5p"] + 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"])) / 4, 9 * (me["5p"] - 2 * me["6p"]), 3 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"])), ), ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -(rho**0.5 * (me["1p"] + 6 * me["3p"])) / 2, (-3 * rho**0.5 * me["1p"]) / 2, (me["1p"] + 2 * rho * me["1p"] - 2 * (2 + rho) * me["2p"] + 6 * (me["3p"] + 2 * rho * me["3p"] - 2 * (2 + rho) * me["4p"])) / 6, (0.5 + rho) * me["1p"] - (2 + rho) * me["2p"], ((-1 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"] + 3 * (me["7p"] - rho * me["7p"] + 2 * me["8p"] + 4 * rho * me["8p"]))) / 12, ((-1 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"])) / 4, -(rho**0.5 * (me["6p"] + 6 * me["8p"])) / 2, (-3 * rho**0.5 * me["6p"]) / 2, -2 * rho**0.5 * (me["5p"] - me["6p"] + 6 * me["7p"] - 6 * me["8p"]), -6 * rho**0.5 * (me["5p"] - me["6p"]), ), ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (-3 * rho**0.5 * me["1p"]) / 2, -(rho**0.5 * (me["1p"] + 6 * me["3p"])) / 2, (0.5 + rho) * me["1p"] - (2 + rho) * me["2p"], (me["1p"] + 2 * rho * me["1p"] - 2 * (2 + rho) * me["2p"] + 6 * (me["3p"] + 2 * rho * me["3p"] - 2 * (2 + rho) * me["4p"])) / 6, ((-1 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"])) / 4, ((-1 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"] + 3 * (me["7p"] - rho * me["7p"] + 2 * me["8p"] + 4 * rho * me["8p"]))) / 12, (-3 * rho**0.5 * me["6p"]) / 2, -(rho**0.5 * (me["6p"] + 6 * me["8p"])) / 2, -6 * rho**0.5 * (me["5p"] - me["6p"]), -2 * rho**0.5 * (me["5p"] - me["6p"] + 6 * me["7p"] - 6 * me["8p"]), ), ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -(rho**0.5 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"]))) / 4, (-3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, ((-1 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"] + 3 * (me["7"] - rho * me["7"] + 2 * me["8"] + 4 * rho * me["8"]))) / 12, ((-1 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"])) / 4, (me["1"] + 2 * rho * me["1"] - 2 * (2 + rho) * me["2"] + 6 * (me["3"] + 2 * rho * me["3"] - 2 * (2 + rho) * me["4"])) / 24, (me["1"] + 2 * rho * me["1"] - 2 * (2 + rho) * me["2"]) / 8, (rho**0.5 * (me["1"] - 2 * (me["2"] - 3 * me["3"] + 6 * me["4"]))) / 8, (3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, -(rho**0.5 * (me["1"] + 2 * (me["2"] + 3 * me["3"] + 6 * me["4"]))) / 2, (-3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, ), ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (-3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, -(rho**0.5 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"]))) / 4, ((-1 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"])) / 4, ((-1 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"] + 3 * (me["7"] - rho * me["7"] + 2 * me["8"] + 4 * rho * me["8"]))) / 12, (me["1"] + 2 * rho * me["1"] - 2 * (2 + rho) * me["2"]) / 8, (me["1"] + 2 * rho * me["1"] - 2 * (2 + rho) * me["2"] + 6 * (me["3"] + 2 * rho * me["3"] - 2 * (2 + rho) * me["4"])) / 24, (3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (rho**0.5 * (me["1"] - 2 * (me["2"] - 3 * me["3"] + 6 * me["4"]))) / 8, (-3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, -(rho**0.5 * (me["1"] + 2 * (me["2"] + 3 * me["3"] + 6 * me["4"]))) / 2, ), ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (-me["5"] + 2 * (me["6"] - 3 * me["7"] + 6 * me["8"])) / 4, (-3 * (me["5"] - 2 * me["6"])) / 4, -(rho**0.5 * (me["6"] + 6 * me["8"])) / 2, (-3 * rho**0.5 * me["6"]) / 2, (rho**0.5 * (me["1"] - 2 * (me["2"] - 3 * me["3"] + 6 * me["4"]))) / 8, (3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, ((2 + rho) * me["1"] - 2 * (me["2"] + 2 * rho * me["2"] - 3 * (2 + rho) * me["3"] + 6 * (me["4"] + 2 * rho * me["4"]))) / 24, ((2 + rho) * me["1"] - 2 * (me["2"] + 2 * rho * me["2"])) / 8, ((-4 + rho) * me["1"] - 2 * (me["2"] + 2 * rho * me["2"] - 3 * (-4 + rho) * me["3"] + 6 * (me["4"] + 2 * rho * me["4"]))) / 6, ((-4 + rho) * me["1"]) / 2 - (1 + 2 * rho) * me["2"], ), ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (-3 * (me["5"] - 2 * me["6"])) / 4, (-me["5"] + 2 * (me["6"] - 3 * me["7"] + 6 * me["8"])) / 4, (-3 * rho**0.5 * me["6"]) / 2, -(rho**0.5 * (me["6"] + 6 * me["8"])) / 2, (3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (rho**0.5 * (me["1"] - 2 * (me["2"] - 3 * me["3"] + 6 * me["4"]))) / 8, ((2 + rho) * me["1"] - 2 * (me["2"] + 2 * rho * me["2"])) / 8, ((2 + rho) * me["1"] - 2 * (me["2"] + 2 * rho * me["2"] - 3 * (2 + rho) * me["3"] + 6 * (me["4"] + 2 * rho * me["4"]))) / 24, ((-4 + rho) * me["1"]) / 2 - (1 + 2 * rho) * me["2"], ((-4 + rho) * me["1"] - 2 * (me["2"] + 2 * rho * me["2"] - 3 * (-4 + rho) * me["3"] + 6 * (me["4"] + 2 * rho * me["4"]))) / 6, ), ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"])), 9 * (me["5"] - 2 * me["6"]), -2 * rho**0.5 * (me["5"] - me["6"] + 6 * me["7"] - 6 * me["8"]), -6 * rho**0.5 * (me["5"] - me["6"]), -(rho**0.5 * (me["1"] + 2 * (me["2"] + 3 * me["3"] + 6 * me["4"]))) / 2, (-3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, ((-4 + rho) * me["1"] - 2 * (me["2"] + 2 * rho * me["2"] - 3 * (-4 + rho) * me["3"] + 6 * (me["4"] + 2 * rho * me["4"]))) / 6, ((-4 + rho) * me["1"]) / 2 - (1 + 2 * rho) * me["2"], (2 * ((14 + rho) * me["1"] - 2 * (me["2"] + 2 * rho * me["2"] - 3 * (14 + rho) * me["3"] + 6 * (me["4"] + 2 * rho * me["4"])))) / 3, 2 * (14 + rho) * me["1"] - 4 * (me["2"] + 2 * rho * me["2"]), ), ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9 * (me["5"] - 2 * me["6"]), 3 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"])), -6 * rho**0.5 * (me["5"] - me["6"]), -2 * rho**0.5 * (me["5"] - me["6"] + 6 * me["7"] - 6 * me["8"]), (-3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, -(rho**0.5 * (me["1"] + 2 * (me["2"] + 3 * me["3"] + 6 * me["4"]))) / 2, ((-4 + rho) * me["1"]) / 2 - (1 + 2 * rho) * me["2"], ((-4 + rho) * me["1"] - 2 * (me["2"] + 2 * rho * me["2"] - 3 * (-4 + rho) * me["3"] + 6 * (me["4"] + 2 * rho * me["4"]))) / 6, 2 * (14 + rho) * me["1"] - 4 * (me["2"] + 2 * rho * me["2"]), (2 * ((14 + rho) * me["1"] - 2 * (me["2"] + 2 * rho * me["2"] - 3 * (14 + rho) * me["3"] + 6 * (me["4"] + 2 * rho * me["4"])))) / 3, ), ) ) def A_WE_cc(rho, me)-
Expand source code
def A_WE_cc(rho, me): # Matrix containing weak exchange contributions from internal cc quarks # See eqs A.2 - A.7 in Lenz:2022pgw, results originally calculated in Jager:2017gal and Jager:2019bgk flavio.citations.register("Jager:2017gal") flavio.citations.register("Jager:2019bgk") return np.array( ( ( ((-1 + rho) * me["1"] + me["2"] + 2 * rho * me["2"] + 6 * ((-1 + rho) * me["3"] + me["4"] + 2 * rho * me["4"])) / 3, (-1 + rho) * me["1"] + me["2"] + 2 * rho * me["2"], -(rho**0.5 * (me["2"] + 6 * me["4"])), -3 * rho**0.5 * me["2"], (rho * (me["1"] + 6 * me["3"])) / 2, (3 * rho * me["1"]) / 2, (rho**0.5 * (me["1"] + 6 * me["3"])) / 4, (3 * rho**0.5 * me["1"]) / 4, -(rho**0.5 * (me["1"] - 4 * me["2"] + 6 * (me["3"] - 4 * me["4"]))), -3 * rho**0.5 * (me["1"] - 4 * me["2"]), -(rho * (me["5p"] + 6 * me["7p"])), -3 * rho * me["5p"], rho**0.5 * (me["6p"] + 6 * me["8p"]), 3 * rho**0.5 * me["6p"], (me["5p"] - rho * me["5p"] - (1 + 2 * rho) * me["6p"] - 6 * ((-1 + rho) * me["7p"] + me["8p"] + 2 * rho * me["8p"])) / 6, (me["5p"] - rho * me["5p"] - (1 + 2 * rho) * me["6p"]) / 2, (rho**0.5 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"]))) / 4, (3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, -(rho**0.5 * (me["5p"] + 2 * (me["6p"] + 3 * me["7p"] + 6 * me["8p"]))), -3 * rho**0.5 * (me["5p"] + 2 * me["6p"]), ), ( (-1 + rho) * me["1"] + me["2"] + 2 * rho * me["2"], 3 * ((-1 + rho) * me["1"] + me["2"] + 2 * rho * me["2"]), -3 * rho**0.5 * me["2"], -9 * rho**0.5 * me["2"], (3 * rho * me["1"]) / 2, (9 * rho * me["1"]) / 2, (3 * rho**0.5 * me["1"]) / 4, (9 * rho**0.5 * me["1"]) / 4, -3 * rho**0.5 * (me["1"] - 4 * me["2"]), -9 * rho**0.5 * (me["1"] - 4 * me["2"]), -3 * rho * me["5p"], -9 * rho * me["5p"], 3 * rho**0.5 * me["6p"], 9 * rho**0.5 * me["6p"], (me["5p"] - rho * me["5p"] - (1 + 2 * rho) * me["6p"]) / 2, (-3 * ((-1 + rho) * me["5p"] + me["6p"] + 2 * rho * me["6p"])) / 2, (3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, (9 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, -3 * rho**0.5 * (me["5p"] + 2 * me["6p"]), -9 * rho**0.5 * (me["5p"] + 2 * me["6p"]), ), ( -(rho**0.5 * (me["2"] + 6 * me["4"])), -3 * rho**0.5 * me["2"], -2 * (-1 + 2 * rho) * (me["2"] + 6 * me["4"]), -6 * (-1 + 2 * rho) * me["2"], -(rho**0.5 * (me["2"] + 6 * me["4"])) / 2, (-3 * rho**0.5 * me["2"]) / 2, -(rho * (me["2"] + 6 * me["4"])), -3 * rho * me["2"], -12 * rho * (me["2"] + 6 * me["4"]), -36 * rho * me["2"], rho**0.5 * (me["6p"] + 6 * me["8p"]), 3 * rho**0.5 * me["6p"], -4 * rho * (me["6p"] + 6 * me["8p"]), -12 * rho * me["6p"], (rho**0.5 * (me["6p"] + 6 * me["8p"])) / 2, (3 * rho**0.5 * me["6p"]) / 2, -((-1 + 2 * rho) * (me["6p"] + 6 * me["8p"])) / 2, (1.5 - 3 * rho) * me["6p"], -6 * (-1 + 2 * rho) * (me["6p"] + 6 * me["8p"]), -18 * (-1 + 2 * rho) * me["6p"], ), ( -3 * rho**0.5 * me["2"], -9 * rho**0.5 * me["2"], -6 * (-1 + 2 * rho) * me["2"], -18 * (-1 + 2 * rho) * me["2"], (-3 * rho**0.5 * me["2"]) / 2, (-9 * rho**0.5 * me["2"]) / 2, -3 * rho * me["2"], -9 * rho * me["2"], -36 * rho * me["2"], -108 * rho * me["2"], 3 * rho**0.5 * me["6p"], 9 * rho**0.5 * me["6p"], -12 * rho * me["6p"], -36 * rho * me["6p"], (3 * rho**0.5 * me["6p"]) / 2, (9 * rho**0.5 * me["6p"]) / 2, (1.5 - 3 * rho) * me["6p"], (4.5 - 9 * rho) * me["6p"], -18 * (-1 + 2 * rho) * me["6p"], -54 * (-1 + 2 * rho) * me["6p"], ), ( (rho * (me["1"] + 6 * me["3"])) / 2, (3 * rho * me["1"]) / 2, -(rho**0.5 * (me["2"] + 6 * me["4"])) / 2, (-3 * rho**0.5 * me["2"]) / 2, ((-1 + rho) * me["1"] + me["2"] + 2 * rho * me["2"] + 6 * ((-1 + rho) * me["3"] + me["4"] + 2 * rho * me["4"])) / 12, ((-1 + rho) * me["1"] + me["2"] + 2 * rho * me["2"]) / 4, -(rho**0.5 * (me["1"] - 2 * (me["2"] - 3 * me["3"] + 6 * me["4"]))) / 8, (-3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (rho**0.5 * (me["1"] + 2 * (me["2"] + 3 * me["3"] + 6 * me["4"]))) / 2, (3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, (me["5p"] - rho * me["5p"] - (1 + 2 * rho) * me["6p"] - 6 * ((-1 + rho) * me["7p"] + me["8p"] + 2 * rho * me["8p"])) / 6, (me["5p"] - rho * me["5p"] - (1 + 2 * rho) * me["6p"]) / 2, (rho**0.5 * (me["6p"] + 6 * me["8p"])) / 2, (3 * rho**0.5 * me["6p"]) / 2, -(rho * (me["5p"] + 6 * me["7p"])) / 4, (-3 * rho * me["5p"]) / 4, -(rho**0.5 * (me["5p"] + 6 * me["7p"])) / 8, (-3 * rho**0.5 * me["5p"]) / 8, (rho**0.5 * (me["5p"] - 4 * me["6p"] + 6 * (me["7p"] - 4 * me["8p"]))) / 2, (3 * rho**0.5 * (me["5p"] - 4 * me["6p"])) / 2, ), ( (3 * rho * me["1"]) / 2, (9 * rho * me["1"]) / 2, (-3 * rho**0.5 * me["2"]) / 2, (-9 * rho**0.5 * me["2"]) / 2, ((-1 + rho) * me["1"] + me["2"] + 2 * rho * me["2"]) / 4, (3 * ((-1 + rho) * me["1"] + me["2"] + 2 * rho * me["2"])) / 4, (-3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (-9 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, (9 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, (me["5p"] - rho * me["5p"] - (1 + 2 * rho) * me["6p"]) / 2, (-3 * ((-1 + rho) * me["5p"] + me["6p"] + 2 * rho * me["6p"])) / 2, (3 * rho**0.5 * me["6p"]) / 2, (9 * rho**0.5 * me["6p"]) / 2, (-3 * rho * me["5p"]) / 4, (-9 * rho * me["5p"]) / 4, (-3 * rho**0.5 * me["5p"]) / 8, (-9 * rho**0.5 * me["5p"]) / 8, (3 * rho**0.5 * (me["5p"] - 4 * me["6p"])) / 2, (9 * rho**0.5 * (me["5p"] - 4 * me["6p"])) / 2, ), ( (rho**0.5 * (me["1"] + 6 * me["3"])) / 4, (3 * rho**0.5 * me["1"]) / 4, -(rho * (me["2"] + 6 * me["4"])), -3 * rho * me["2"], -(rho**0.5 * (me["1"] - 2 * (me["2"] - 3 * me["3"] + 6 * me["4"]))) / 8, (-3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (-((1 + 2 * rho) * me["1"]) - 4 * (-1 + rho) * me["2"] - 6 * (me["3"] + 2 * rho * me["3"] + 4 * (-1 + rho) * me["4"])) / 24, (-((1 + 2 * rho) * me["1"]) - 4 * (-1 + rho) * me["2"]) / 8, (me["1"] + 2 * rho * me["1"] + (8 - 20 * rho) * me["2"] + 6 * (me["3"] + 2 * rho * me["3"] + 8 * me["4"] - 20 * rho * me["4"])) / 6, (0.5 + rho) * me["1"] + 2 * (2 - 5 * rho) * me["2"], (rho**0.5 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"]))) / 4, (3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, -((-1 + 2 * rho) * (me["6p"] + 6 * me["8p"])) / 2, (1.5 - 3 * rho) * me["6p"], -(rho**0.5 * (me["5p"] + 6 * me["7p"])) / 8, (-3 * rho**0.5 * me["5p"]) / 8, -(rho * (me["5p"] + 6 * me["7p"])) / 4, (-3 * rho * me["5p"]) / 4, rho * (me["5p"] - 4 * me["6p"] + 6 * (me["7p"] - 4 * me["8p"])), 3 * rho * (me["5p"] - 4 * me["6p"]), ), ( (3 * rho**0.5 * me["1"]) / 4, (9 * rho**0.5 * me["1"]) / 4, -3 * rho * me["2"], -9 * rho * me["2"], (-3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (-9 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (-((1 + 2 * rho) * me["1"]) - 4 * (-1 + rho) * me["2"]) / 8, (-3 * (me["1"] + 2 * rho * me["1"] + 4 * (-1 + rho) * me["2"])) / 8, (0.5 + rho) * me["1"] + 2 * (2 - 5 * rho) * me["2"], (3 * (me["1"] + 2 * rho * me["1"] + 4 * (2 - 5 * rho) * me["2"])) / 2, (3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, (9 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, (1.5 - 3 * rho) * me["6p"], (4.5 - 9 * rho) * me["6p"], (-3 * rho**0.5 * me["5p"]) / 8, (-9 * rho**0.5 * me["5p"]) / 8, (-3 * rho * me["5p"]) / 4, (-9 * rho * me["5p"]) / 4, 3 * rho * (me["5p"] - 4 * me["6p"]), 9 * rho * (me["5p"] - 4 * me["6p"]), ), ( -(rho**0.5 * (me["1"] - 4 * me["2"] + 6 * (me["3"] - 4 * me["4"]))), -3 * rho**0.5 * (me["1"] - 4 * me["2"]), -12 * rho * (me["2"] + 6 * me["4"]), -36 * rho * me["2"], (rho**0.5 * (me["1"] + 2 * (me["2"] + 3 * me["3"] + 6 * me["4"]))) / 2, (3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, (me["1"] + 2 * rho * me["1"] + (8 - 20 * rho) * me["2"] + 6 * (me["3"] + 2 * rho * me["3"] + 8 * me["4"] - 20 * rho * me["4"])) / 6, (0.5 + rho) * me["1"] + 2 * (2 - 5 * rho) * me["2"], (-2 * (me["1"] + 2 * rho * me["1"] + 4 * (-7 + 13 * rho) * me["2"] + 6 * (me["3"] + 2 * rho * me["3"] - 28 * me["4"] + 52 * rho * me["4"]))) / 3, -2 * (me["1"] + 2 * rho * me["1"] + 4 * (-7 + 13 * rho) * me["2"]), -(rho**0.5 * (me["5p"] + 2 * (me["6p"] + 3 * me["7p"] + 6 * me["8p"]))), -3 * rho**0.5 * (me["5p"] + 2 * me["6p"]), -6 * (-1 + 2 * rho) * (me["6p"] + 6 * me["8p"]), -18 * (-1 + 2 * rho) * me["6p"], (rho**0.5 * (me["5p"] - 4 * me["6p"] + 6 * (me["7p"] - 4 * me["8p"]))) / 2, (3 * rho**0.5 * (me["5p"] - 4 * me["6p"])) / 2, rho * (me["5p"] - 4 * me["6p"] + 6 * (me["7p"] - 4 * me["8p"])), 3 * rho * (me["5p"] - 4 * me["6p"]), -4 * rho * (me["5p"] + 8 * me["6p"] + 6 * (me["7p"] + 8 * me["8p"])), -12 * rho * (me["5p"] + 8 * me["6p"]), ), ( -3 * rho**0.5 * (me["1"] - 4 * me["2"]), -9 * rho**0.5 * (me["1"] - 4 * me["2"]), -36 * rho * me["2"], -108 * rho * me["2"], (3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, (9 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, (0.5 + rho) * me["1"] + 2 * (2 - 5 * rho) * me["2"], (3 * (me["1"] + 2 * rho * me["1"] + 4 * (2 - 5 * rho) * me["2"])) / 2, -2 * (me["1"] + 2 * rho * me["1"] + 4 * (-7 + 13 * rho) * me["2"]), -6 * (me["1"] + 2 * rho * me["1"] + 4 * (-7 + 13 * rho) * me["2"]), -3 * rho**0.5 * (me["5p"] + 2 * me["6p"]), -9 * rho**0.5 * (me["5p"] + 2 * me["6p"]), -18 * (-1 + 2 * rho) * me["6p"], -54 * (-1 + 2 * rho) * me["6p"], (3 * rho**0.5 * (me["5p"] - 4 * me["6p"])) / 2, (9 * rho**0.5 * (me["5p"] - 4 * me["6p"])) / 2, 3 * rho * (me["5p"] - 4 * me["6p"]), 9 * rho * (me["5p"] - 4 * me["6p"]), -12 * rho * (me["5p"] + 8 * me["6p"]), -36 * rho * (me["5p"] + 8 * me["6p"]), ), ( -(rho * (me["5"] + 6 * me["7"])), -3 * rho * me["5"], rho**0.5 * (me["6"] + 6 * me["8"]), 3 * rho**0.5 * me["6"], (me["5"] - rho * me["5"] - (1 + 2 * rho) * me["6"] - 6 * ((-1 + rho) * me["7"] + me["8"] + 2 * rho * me["8"])) / 6, (me["5"] - rho * me["5"] - (1 + 2 * rho) * me["6"]) / 2, (rho**0.5 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"]))) / 4, (3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, -(rho**0.5 * (me["5"] + 2 * (me["6"] + 3 * me["7"] + 6 * me["8"]))), -3 * rho**0.5 * (me["5"] + 2 * me["6"]), ((-1 + rho) * me["1p"] + me["2p"] + 2 * rho * me["2p"] + 6 * ((-1 + rho) * me["3p"] + me["4p"] + 2 * rho * me["4p"])) / 3, (-1 + rho) * me["1p"] + me["2p"] + 2 * rho * me["2p"], -(rho**0.5 * (me["2p"] + 6 * me["4p"])), -3 * rho**0.5 * me["2p"], (rho * (me["1p"] + 6 * me["3p"])) / 2, (3 * rho * me["1p"]) / 2, (rho**0.5 * (me["1p"] + 6 * me["3p"])) / 4, (3 * rho**0.5 * me["1p"]) / 4, -(rho**0.5 * (me["1p"] - 4 * me["2p"] + 6 * (me["3p"] - 4 * me["4p"]))), -3 * rho**0.5 * (me["1p"] - 4 * me["2p"]), ), ( -3 * rho * me["5"], -9 * rho * me["5"], 3 * rho**0.5 * me["6"], 9 * rho**0.5 * me["6"], (me["5"] - rho * me["5"] - (1 + 2 * rho) * me["6"]) / 2, (-3 * ((-1 + rho) * me["5"] + me["6"] + 2 * rho * me["6"])) / 2, (3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, (9 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, -3 * rho**0.5 * (me["5"] + 2 * me["6"]), -9 * rho**0.5 * (me["5"] + 2 * me["6"]), (-1 + rho) * me["1p"] + me["2p"] + 2 * rho * me["2p"], 3 * ((-1 + rho) * me["1p"] + me["2p"] + 2 * rho * me["2p"]), -3 * rho**0.5 * me["2p"], -9 * rho**0.5 * me["2p"], (3 * rho * me["1p"]) / 2, (9 * rho * me["1p"]) / 2, (3 * rho**0.5 * me["1p"]) / 4, (9 * rho**0.5 * me["1p"]) / 4, -3 * rho**0.5 * (me["1p"] - 4 * me["2p"]), -9 * rho**0.5 * (me["1p"] - 4 * me["2p"]), ), ( rho**0.5 * (me["6"] + 6 * me["8"]), 3 * rho**0.5 * me["6"], -4 * rho * (me["6"] + 6 * me["8"]), -12 * rho * me["6"], (rho**0.5 * (me["6"] + 6 * me["8"])) / 2, (3 * rho**0.5 * me["6"]) / 2, -((-1 + 2 * rho) * (me["6"] + 6 * me["8"])) / 2, (1.5 - 3 * rho) * me["6"], -6 * (-1 + 2 * rho) * (me["6"] + 6 * me["8"]), -18 * (-1 + 2 * rho) * me["6"], -(rho**0.5 * (me["2p"] + 6 * me["4p"])), -3 * rho**0.5 * me["2p"], -2 * (-1 + 2 * rho) * (me["2p"] + 6 * me["4p"]), -6 * (-1 + 2 * rho) * me["2p"], -(rho**0.5 * (me["2p"] + 6 * me["4p"])) / 2, (-3 * rho**0.5 * me["2p"]) / 2, -(rho * (me["2p"] + 6 * me["4p"])), -3 * rho * me["2p"], -12 * rho * (me["2p"] + 6 * me["4p"]), -36 * rho * me["2p"], ), ( 3 * rho**0.5 * me["6"], 9 * rho**0.5 * me["6"], -12 * rho * me["6"], -36 * rho * me["6"], (3 * rho**0.5 * me["6"]) / 2, (9 * rho**0.5 * me["6"]) / 2, (1.5 - 3 * rho) * me["6"], (4.5 - 9 * rho) * me["6"], -18 * (-1 + 2 * rho) * me["6"], -54 * (-1 + 2 * rho) * me["6"], -3 * rho**0.5 * me["2p"], -9 * rho**0.5 * me["2p"], -6 * (-1 + 2 * rho) * me["2p"], -18 * (-1 + 2 * rho) * me["2p"], (-3 * rho**0.5 * me["2p"]) / 2, (-9 * rho**0.5 * me["2p"]) / 2, -3 * rho * me["2p"], -9 * rho * me["2p"], -36 * rho * me["2p"], -108 * rho * me["2p"], ), ( (me["5"] - rho * me["5"] - (1 + 2 * rho) * me["6"] - 6 * ((-1 + rho) * me["7"] + me["8"] + 2 * rho * me["8"])) / 6, (me["5"] - rho * me["5"] - (1 + 2 * rho) * me["6"]) / 2, (rho**0.5 * (me["6"] + 6 * me["8"])) / 2, (3 * rho**0.5 * me["6"]) / 2, -(rho * (me["5"] + 6 * me["7"])) / 4, (-3 * rho * me["5"]) / 4, -(rho**0.5 * (me["5"] + 6 * me["7"])) / 8, (-3 * rho**0.5 * me["5"]) / 8, (rho**0.5 * (me["5"] - 4 * me["6"] + 6 * (me["7"] - 4 * me["8"]))) / 2, (3 * rho**0.5 * (me["5"] - 4 * me["6"])) / 2, (rho * (me["1p"] + 6 * me["3p"])) / 2, (3 * rho * me["1p"]) / 2, -(rho**0.5 * (me["2p"] + 6 * me["4p"])) / 2, (-3 * rho**0.5 * me["2p"]) / 2, ((-1 + rho) * me["1p"] + me["2p"] + 2 * rho * me["2p"] + 6 * ((-1 + rho) * me["3p"] + me["4p"] + 2 * rho * me["4p"])) / 12, ((-1 + rho) * me["1p"] + me["2p"] + 2 * rho * me["2p"]) / 4, -(rho**0.5 * (me["1p"] - 2 * (me["2p"] - 3 * me["3p"] + 6 * me["4p"]))) / 8, (-3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (rho**0.5 * (me["1p"] + 2 * (me["2p"] + 3 * me["3p"] + 6 * me["4p"]))) / 2, (3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, ), ( (me["5"] - rho * me["5"] - (1 + 2 * rho) * me["6"]) / 2, (-3 * ((-1 + rho) * me["5"] + me["6"] + 2 * rho * me["6"])) / 2, (3 * rho**0.5 * me["6"]) / 2, (9 * rho**0.5 * me["6"]) / 2, (-3 * rho * me["5"]) / 4, (-9 * rho * me["5"]) / 4, (-3 * rho**0.5 * me["5"]) / 8, (-9 * rho**0.5 * me["5"]) / 8, (3 * rho**0.5 * (me["5"] - 4 * me["6"])) / 2, (9 * rho**0.5 * (me["5"] - 4 * me["6"])) / 2, (3 * rho * me["1p"]) / 2, (9 * rho * me["1p"]) / 2, (-3 * rho**0.5 * me["2p"]) / 2, (-9 * rho**0.5 * me["2p"]) / 2, ((-1 + rho) * me["1p"] + me["2p"] + 2 * rho * me["2p"]) / 4, (3 * ((-1 + rho) * me["1p"] + me["2p"] + 2 * rho * me["2p"])) / 4, (-3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (-9 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, (9 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, ), ( (rho**0.5 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"]))) / 4, (3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, -((-1 + 2 * rho) * (me["6"] + 6 * me["8"])) / 2, (1.5 - 3 * rho) * me["6"], -(rho**0.5 * (me["5"] + 6 * me["7"])) / 8, (-3 * rho**0.5 * me["5"]) / 8, -(rho * (me["5"] + 6 * me["7"])) / 4, (-3 * rho * me["5"]) / 4, rho * (me["5"] - 4 * me["6"] + 6 * (me["7"] - 4 * me["8"])), 3 * rho * (me["5"] - 4 * me["6"]), (rho**0.5 * (me["1p"] + 6 * me["3p"])) / 4, (3 * rho**0.5 * me["1p"]) / 4, -(rho * (me["2p"] + 6 * me["4p"])), -3 * rho * me["2p"], -(rho**0.5 * (me["1p"] - 2 * (me["2p"] - 3 * me["3p"] + 6 * me["4p"]))) / 8, (-3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (-((1 + 2 * rho) * me["1p"]) - 4 * (-1 + rho) * me["2p"] - 6 * (me["3p"] + 2 * rho * me["3p"] + 4 * (-1 + rho) * me["4p"])) / 24, (-((1 + 2 * rho) * me["1p"]) - 4 * (-1 + rho) * me["2p"]) / 8, (me["1p"] + 2 * rho * me["1p"] + (8 - 20 * rho) * me["2p"] + 6 * (me["3p"] + 2 * rho * me["3p"] + 8 * me["4p"] - 20 * rho * me["4p"])) / 6, (0.5 + rho) * me["1p"] + 2 * (2 - 5 * rho) * me["2p"], ), ( (3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, (9 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, (1.5 - 3 * rho) * me["6"], (4.5 - 9 * rho) * me["6"], (-3 * rho**0.5 * me["5"]) / 8, (-9 * rho**0.5 * me["5"]) / 8, (-3 * rho * me["5"]) / 4, (-9 * rho * me["5"]) / 4, 3 * rho * (me["5"] - 4 * me["6"]), 9 * rho * (me["5"] - 4 * me["6"]), (3 * rho**0.5 * me["1p"]) / 4, (9 * rho**0.5 * me["1p"]) / 4, -3 * rho * me["2p"], -9 * rho * me["2p"], (-3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (-9 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (-((1 + 2 * rho) * me["1p"]) - 4 * (-1 + rho) * me["2p"]) / 8, (-3 * (me["1p"] + 2 * rho * me["1p"] + 4 * (-1 + rho) * me["2p"])) / 8, (0.5 + rho) * me["1p"] + 2 * (2 - 5 * rho) * me["2p"], (3 * (me["1p"] + 2 * rho * me["1p"] + 4 * (2 - 5 * rho) * me["2p"])) / 2, ), ( -(rho**0.5 * (me["5"] + 2 * (me["6"] + 3 * me["7"] + 6 * me["8"]))), -3 * rho**0.5 * (me["5"] + 2 * me["6"]), -6 * (-1 + 2 * rho) * (me["6"] + 6 * me["8"]), -18 * (-1 + 2 * rho) * me["6"], (rho**0.5 * (me["5"] - 4 * me["6"] + 6 * (me["7"] - 4 * me["8"]))) / 2, (3 * rho**0.5 * (me["5"] - 4 * me["6"])) / 2, rho * (me["5"] - 4 * me["6"] + 6 * (me["7"] - 4 * me["8"])), 3 * rho * (me["5"] - 4 * me["6"]), -4 * rho * (me["5"] + 8 * me["6"] + 6 * (me["7"] + 8 * me["8"])), -12 * rho * (me["5"] + 8 * me["6"]), -(rho**0.5 * (me["1p"] - 4 * me["2p"] + 6 * (me["3p"] - 4 * me["4p"]))), -3 * rho**0.5 * (me["1p"] - 4 * me["2p"]), -12 * rho * (me["2p"] + 6 * me["4p"]), -36 * rho * me["2p"], (rho**0.5 * (me["1p"] + 2 * (me["2p"] + 3 * me["3p"] + 6 * me["4p"]))) / 2, (3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, (me["1p"] + 2 * rho * me["1p"] + (8 - 20 * rho) * me["2p"] + 6 * (me["3p"] + 2 * rho * me["3p"] + 8 * me["4p"] - 20 * rho * me["4p"])) / 6, (0.5 + rho) * me["1p"] + 2 * (2 - 5 * rho) * me["2p"], (-2 * (me["1p"] + 2 * rho * me["1p"] + 4 * (-7 + 13 * rho) * me["2p"] + 6 * (me["3p"] + 2 * rho * me["3p"] - 28 * me["4p"] + 52 * rho * me["4p"]))) / 3, -2 * (me["1p"] + 2 * rho * me["1p"] + 4 * (-7 + 13 * rho) * me["2p"]), ), ( -3 * rho**0.5 * (me["5"] + 2 * me["6"]), -9 * rho**0.5 * (me["5"] + 2 * me["6"]), -18 * (-1 + 2 * rho) * me["6"], -54 * (-1 + 2 * rho) * me["6"], (3 * rho**0.5 * (me["5"] - 4 * me["6"])) / 2, (9 * rho**0.5 * (me["5"] - 4 * me["6"])) / 2, 3 * rho * (me["5"] - 4 * me["6"]), 9 * rho * (me["5"] - 4 * me["6"]), -12 * rho * (me["5"] + 8 * me["6"]), -36 * rho * (me["5"] + 8 * me["6"]), -3 * rho**0.5 * (me["1p"] - 4 * me["2p"]), -9 * rho**0.5 * (me["1p"] - 4 * me["2p"]), -36 * rho * me["2p"], -108 * rho * me["2p"], (3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, (9 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, (0.5 + rho) * me["1p"] + 2 * (2 - 5 * rho) * me["2p"], (3 * (me["1p"] + 2 * rho * me["1p"] + 4 * (2 - 5 * rho) * me["2p"])) / 2, -2 * (me["1p"] + 2 * rho * me["1p"] + 4 * (-7 + 13 * rho) * me["2p"]), -6 * (me["1p"] + 2 * rho * me["1p"] + 4 * (-7 + 13 * rho) * me["2p"]), ), ) ) def A_WE_cu(rho, me)-
Expand source code
def A_WE_cu(rho, me): # Matrix containing weak exchange contributions from internal cu quarks # See eqs 2.28 - 2.33 in Lenz:2022pgw flavio.citations.register("Lenz:2022pgw") return np.array( ( ( (-((2 + rho) * me["1"]) + 2 * (me["2"] + 2 * rho * me["2"] - 3 * (2 + rho) * me["3"] + 6 * (me["4"] + 2 * rho * me["4"]))) / 6, -((2 + rho) * me["1"]) / 2 + me["2"] + 2 * rho * me["2"], -(rho**0.5 * (me["2"] + 6 * me["4"])), -3 * rho**0.5 * me["2"], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ((2 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"] - 3 * (2 + rho) * me["7p"] + 6 * (me["8p"] + 2 * rho * me["8p"]))) / 12, ((2 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"])) / 4, (rho**0.5 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"]))) / 4, (3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, -(rho**0.5 * (me["5p"] + 2 * (me["6p"] + 3 * me["7p"] + 6 * me["8p"]))), -3 * rho**0.5 * (me["5p"] + 2 * me["6p"]), ), ( -((2 + rho) * me["1"]) / 2 + me["2"] + 2 * rho * me["2"], (-3 * (2 + rho) * me["1"]) / 2 + 3 * (1 + 2 * rho) * me["2"], -3 * rho**0.5 * me["2"], -9 * rho**0.5 * me["2"], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ((2 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"])) / 4, (3 * ((2 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"]))) / 4, (3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, (9 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, -3 * rho**0.5 * (me["5p"] + 2 * me["6p"]), -9 * rho**0.5 * (me["5p"] + 2 * me["6p"]), ), ( -(rho**0.5 * (me["2"] + 6 * me["4"])), -3 * rho**0.5 * me["2"], 2 * (me["2"] + 6 * me["4"]), 6 * me["2"], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (rho**0.5 * (me["6p"] + 6 * me["8p"])) / 2, (3 * rho**0.5 * me["6p"]) / 2, (me["6p"] + 6 * me["8p"]) / 2, (3 * me["6p"]) / 2, 6 * (me["6p"] + 6 * me["8p"]), 18 * me["6p"], ), ( -3 * rho**0.5 * me["2"], -9 * rho**0.5 * me["2"], 6 * me["2"], 18 * me["2"], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (3 * rho**0.5 * me["6p"]) / 2, (9 * rho**0.5 * me["6p"]) / 2, (3 * me["6p"]) / 2, (9 * me["6p"]) / 2, 18 * me["6p"], 54 * me["6p"], ), ( 0, 0, 0, 0, (-((2 + rho) * me["1"]) + 2 * (me["2"] + 2 * rho * me["2"] - 3 * (2 + rho) * me["3"] + 6 * (me["4"] + 2 * rho * me["4"]))) / 24, (-((2 + rho) * me["1"]) + 2 * (1 + 2 * rho) * me["2"]) / 8, -(rho**0.5 * (me["1"] - 2 * (me["2"] - 3 * me["3"] + 6 * me["4"]))) / 8, (-3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (rho**0.5 * (me["1"] + 2 * (me["2"] + 3 * me["3"] + 6 * me["4"]))) / 2, (3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, ((2 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"] - 3 * (2 + rho) * me["7p"] + 6 * (me["8p"] + 2 * rho * me["8p"]))) / 12, ((2 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"])) / 4, (rho**0.5 * (me["6p"] + 6 * me["8p"])) / 2, (3 * rho**0.5 * me["6p"]) / 2, 0, 0, 0, 0, 0, 0, ), ( 0, 0, 0, 0, (-((2 + rho) * me["1"]) + 2 * (1 + 2 * rho) * me["2"]) / 8, (-3 * ((2 + rho) * me["1"] - 2 * (me["2"] + 2 * rho * me["2"]))) / 8, (-3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (-9 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, (9 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, ((2 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"])) / 4, (3 * ((2 + rho) * me["5p"] - 2 * (me["6p"] + 2 * rho * me["6p"]))) / 4, (3 * rho**0.5 * me["6p"]) / 2, (9 * rho**0.5 * me["6p"]) / 2, 0, 0, 0, 0, 0, 0, ), ( 0, 0, 0, 0, -(rho**0.5 * (me["1"] - 2 * (me["2"] - 3 * me["3"] + 6 * me["4"]))) / 8, (-3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (-((1 + 2 * rho) * me["1"]) + 2 * (2 + rho) * me["2"] - 6 * (me["3"] + 2 * rho * me["3"] - 2 * (2 + rho) * me["4"])) / 24, (-((1 + 2 * rho) * me["1"]) + 2 * (2 + rho) * me["2"]) / 8, (me["1"] + 2 * rho * me["1"] - 2 * (-4 + rho) * me["2"] + 6 * (me["3"] + 2 * rho * me["3"] + 8 * me["4"] - 2 * rho * me["4"])) / 6, (0.5 + rho) * me["1"] - (-4 + rho) * me["2"], (rho**0.5 * (me["5p"] - 2 * (me["6p"] - 3 * me["7p"] + 6 * me["8p"]))) / 4, (3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, (me["6p"] + 6 * me["8p"]) / 2, (3 * me["6p"]) / 2, 0, 0, 0, 0, 0, 0, ), ( 0, 0, 0, 0, (-3 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (-9 * rho**0.5 * (me["1"] - 2 * me["2"])) / 8, (-((1 + 2 * rho) * me["1"]) + 2 * (2 + rho) * me["2"]) / 8, (-3 * (me["1"] + 2 * rho * me["1"] - 2 * (2 + rho) * me["2"])) / 8, (0.5 + rho) * me["1"] - (-4 + rho) * me["2"], (3 * (me["1"] + 2 * rho * me["1"] - 2 * (-4 + rho) * me["2"])) / 2, (3 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, (9 * rho**0.5 * (me["5p"] - 2 * me["6p"])) / 4, (3 * me["6p"]) / 2, (9 * me["6p"]) / 2, 0, 0, 0, 0, 0, 0, ), ( 0, 0, 0, 0, (rho**0.5 * (me["1"] + 2 * (me["2"] + 3 * me["3"] + 6 * me["4"]))) / 2, (3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, (me["1"] + 2 * rho * me["1"] - 2 * (-4 + rho) * me["2"] + 6 * (me["3"] + 2 * rho * me["3"] + 8 * me["4"] - 2 * rho * me["4"])) / 6, (0.5 + rho) * me["1"] - (-4 + rho) * me["2"], (-2 * (me["1"] + 2 * rho * me["1"] - 2 * (14 + rho) * me["2"] + 6 * (1 + 2 * rho) * me["3"] - 12 * (14 + rho) * me["4"])) / 3, -2 * (1 + 2 * rho) * me["1"] + 4 * (14 + rho) * me["2"], -(rho**0.5 * (me["5p"] + 2 * (me["6p"] + 3 * me["7p"] + 6 * me["8p"]))), -3 * rho**0.5 * (me["5p"] + 2 * me["6p"]), 6 * (me["6p"] + 6 * me["8p"]), 18 * me["6p"], 0, 0, 0, 0, 0, 0, ), ( 0, 0, 0, 0, (3 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, (9 * rho**0.5 * (me["1"] + 2 * me["2"])) / 2, (0.5 + rho) * me["1"] - (-4 + rho) * me["2"], (3 * (me["1"] + 2 * rho * me["1"] - 2 * (-4 + rho) * me["2"])) / 2, -2 * (1 + 2 * rho) * me["1"] + 4 * (14 + rho) * me["2"], -6 * (me["1"] + 2 * rho * me["1"] - 2 * (14 + rho) * me["2"]), -3 * rho**0.5 * (me["5p"] + 2 * me["6p"]), -9 * rho**0.5 * (me["5p"] + 2 * me["6p"]), 18 * me["6p"], 54 * me["6p"], 0, 0, 0, 0, 0, 0, ), ( 0, 0, 0, 0, ((2 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"] - 3 * (2 + rho) * me["7"] + 6 * (me["8"] + 2 * rho * me["8"]))) / 12, ((2 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"])) / 4, (rho**0.5 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"]))) / 4, (3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, -(rho**0.5 * (me["5"] + 2 * (me["6"] + 3 * me["7"] + 6 * me["8"]))), -3 * rho**0.5 * (me["5"] + 2 * me["6"]), (-((2 + rho) * me["1p"]) + 2 * (me["2p"] + 2 * rho * me["2p"] - 3 * (2 + rho) * me["3p"] + 6 * (me["4p"] + 2 * rho * me["4p"]))) / 6, -((2 + rho) * me["1p"]) / 2 + me["2p"] + 2 * rho * me["2p"], -(rho**0.5 * (me["2p"] + 6 * me["4p"])), -3 * rho**0.5 * me["2p"], 0, 0, 0, 0, 0, 0, ), ( 0, 0, 0, 0, ((2 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"])) / 4, (3 * ((2 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"]))) / 4, (3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, (9 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, -3 * rho**0.5 * (me["5"] + 2 * me["6"]), -9 * rho**0.5 * (me["5"] + 2 * me["6"]), -((2 + rho) * me["1p"]) / 2 + me["2p"] + 2 * rho * me["2p"], (-3 * (2 + rho) * me["1p"]) / 2 + 3 * (1 + 2 * rho) * me["2p"], -3 * rho**0.5 * me["2p"], -9 * rho**0.5 * me["2p"], 0, 0, 0, 0, 0, 0, ), ( 0, 0, 0, 0, (rho**0.5 * (me["6"] + 6 * me["8"])) / 2, (3 * rho**0.5 * me["6"]) / 2, (me["6"] + 6 * me["8"]) / 2, (3 * me["6"]) / 2, 6 * (me["6"] + 6 * me["8"]), 18 * me["6"], -(rho**0.5 * (me["2p"] + 6 * me["4p"])), -3 * rho**0.5 * me["2p"], 2 * (me["2p"] + 6 * me["4p"]), 6 * me["2p"], 0, 0, 0, 0, 0, 0, ), ( 0, 0, 0, 0, (3 * rho**0.5 * me["6"]) / 2, (9 * rho**0.5 * me["6"]) / 2, (3 * me["6"]) / 2, (9 * me["6"]) / 2, 18 * me["6"], 54 * me["6"], -3 * rho**0.5 * me["2p"], -9 * rho**0.5 * me["2p"], 6 * me["2p"], 18 * me["2p"], 0, 0, 0, 0, 0, 0, ), ( ((2 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"] - 3 * (2 + rho) * me["7"] + 6 * (me["8"] + 2 * rho * me["8"]))) / 12, ((2 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"])) / 4, (rho**0.5 * (me["6"] + 6 * me["8"])) / 2, (3 * rho**0.5 * me["6"]) / 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (-((2 + rho) * me["1p"]) + 2 * (me["2p"] + 2 * rho * me["2p"] - 3 * (2 + rho) * me["3p"] + 6 * (me["4p"] + 2 * rho * me["4p"]))) / 24, (-((2 + rho) * me["1p"]) + 2 * (1 + 2 * rho) * me["2p"]) / 8, -(rho**0.5 * (me["1p"] - 2 * (me["2p"] - 3 * me["3p"] + 6 * me["4p"]))) / 8, (-3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (rho**0.5 * (me["1p"] + 2 * (me["2p"] + 3 * me["3p"] + 6 * me["4p"]))) / 2, (3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, ), ( ((2 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"])) / 4, (3 * ((2 + rho) * me["5"] - 2 * (me["6"] + 2 * rho * me["6"]))) / 4, (3 * rho**0.5 * me["6"]) / 2, (9 * rho**0.5 * me["6"]) / 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (-((2 + rho) * me["1p"]) + 2 * (1 + 2 * rho) * me["2p"]) / 8, (-3 * ((2 + rho) * me["1p"] - 2 * (me["2p"] + 2 * rho * me["2p"]))) / 8, (-3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (-9 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, (9 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, ), ( (rho**0.5 * (me["5"] - 2 * (me["6"] - 3 * me["7"] + 6 * me["8"]))) / 4, (3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, (me["6"] + 6 * me["8"]) / 2, (3 * me["6"]) / 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -(rho**0.5 * (me["1p"] - 2 * (me["2p"] - 3 * me["3p"] + 6 * me["4p"]))) / 8, (-3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (-((1 + 2 * rho) * me["1p"]) + 2 * (2 + rho) * me["2p"] - 6 * (me["3p"] + 2 * rho * me["3p"] - 2 * (2 + rho) * me["4p"])) / 24, (-((1 + 2 * rho) * me["1p"]) + 2 * (2 + rho) * me["2p"]) / 8, (me["1p"] + 2 * rho * me["1p"] - 2 * (-4 + rho) * me["2p"] + 6 * (me["3p"] + 2 * rho * me["3p"] + 8 * me["4p"] - 2 * rho * me["4p"])) / 6, (0.5 + rho) * me["1p"] - (-4 + rho) * me["2p"], ), ( (3 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, (9 * rho**0.5 * (me["5"] - 2 * me["6"])) / 4, (3 * me["6"]) / 2, (9 * me["6"]) / 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (-3 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (-9 * rho**0.5 * (me["1p"] - 2 * me["2p"])) / 8, (-((1 + 2 * rho) * me["1p"]) + 2 * (2 + rho) * me["2p"]) / 8, (-3 * (me["1p"] + 2 * rho * me["1p"] - 2 * (2 + rho) * me["2p"])) / 8, (0.5 + rho) * me["1p"] - (-4 + rho) * me["2p"], (3 * (me["1p"] + 2 * rho * me["1p"] - 2 * (-4 + rho) * me["2p"])) / 2, ), ( -(rho**0.5 * (me["5"] + 2 * (me["6"] + 3 * me["7"] + 6 * me["8"]))), -3 * rho**0.5 * (me["5"] + 2 * me["6"]), 6 * (me["6"] + 6 * me["8"]), 18 * me["6"], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (rho**0.5 * (me["1p"] + 2 * (me["2p"] + 3 * me["3p"] + 6 * me["4p"]))) / 2, (3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, (me["1p"] + 2 * rho * me["1p"] - 2 * (-4 + rho) * me["2p"] + 6 * (me["3p"] + 2 * rho * me["3p"] + 8 * me["4p"] - 2 * rho * me["4p"])) / 6, (0.5 + rho) * me["1p"] - (-4 + rho) * me["2p"], (-2 * (me["1p"] + 2 * rho * me["1p"] - 2 * (14 + rho) * me["2p"] + 6 * (1 + 2 * rho) * me["3p"] - 12 * (14 + rho) * me["4p"])) / 3, -2 * (1 + 2 * rho) * me["1p"] + 4 * (14 + rho) * me["2p"], ), ( -3 * rho**0.5 * (me["5"] + 2 * me["6"]), -9 * rho**0.5 * (me["5"] + 2 * me["6"]), 18 * me["6"], 54 * me["6"], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (3 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, (9 * rho**0.5 * (me["1p"] + 2 * me["2p"])) / 2, (0.5 + rho) * me["1p"] - (-4 + rho) * me["2p"], (3 * (me["1p"] + 2 * rho * me["1p"] - 2 * (-4 + rho) * me["2p"])) / 2, -2 * (1 + 2 * rho) * me["1p"] + 4 * (14 + rho) * me["2p"], -6 * (me["1p"] + 2 * rho * me["1p"] - 2 * (14 + rho) * me["2p"]), ), ) ) def gamma_BSM_dim6(wc_obj, par, meson)-
Expand source code
def gamma_BSM_dim6(wc_obj, par, meson): # See for example fig 3 in 2208.02643 for topologies of weak exchange and Pauli interference WE = weak_exchange(wc_obj, par, meson) PI = pauli_interference(wc_obj, par, meson) return WE + PI def lifetimematrixelements(par, meson, scale)-
Expand source code
def lifetimematrixelements(par, meson, scale): r"""Returns a dictionary with the values of the matrix elements of the $\Delta B=0$ operators. Note that the normalisation factor 1/2M is included here, so the returned values of the matrix elements are really $$\langle O \rangle = \frac{\langle B_q | O | B_q\rangle}{2M_B}$$ """ mM = par["m_" + meson] fM = par["f_" + meson] bag_params_dict = run_lifetime_bag_parameters(par, scale) BM = lambda i: bag_params_dict[f"bag_lifetime_B{i}qtilde"] me = {} for i in range(1,9): me[f"{i}"] = fM**2 * mM * BM(i) / 2 me[f"{i}p"] = fM**2 * mM * BM(i) / 2 return meReturns a dictionary with the values of the matrix elements of the $\Delta B=0$ operators.
Note that the normalisation factor 1/2M is included here, so the returned values of the matrix elements are really $$\langle O \rangle = \frac{\langle B_q | O | B_q\rangle}{2M_B}$$
def pauli_interference(wc_obj, par, meson)-
Expand source code
def pauli_interference(wc_obj, par, meson): r"""BSM Pauli interference contributions. For now, only from b->c ubar d (dbcu sector) and b->c ubar s (sbcu sector) operators.""" # For Bd, no PI from our considered sectors if meson == "B0": return 0 # So now B+ case only: only dbcu and sbcu sectors contribute GF = par["GF"] mb = flavio.physics.running.running.get_mb_KS(par, 1) mc = par["m_c"] rho = mc**2 / mb**2 V = ckm.get_ckm(par) me = lifetimematrixelements(par, meson, config["renormalization scale"]["b lifetime ratios"]) dflav = {"d": 0, "s": 1, "b": 2} result = 0 for sector in ("dbcu", "sbcu"): CSM, CNP = siegen_basis_wcs(wc_obj, par, sector) C = CSM + CNP ckm_factor = V[0, dflav[sector[0]]] * V[1, 2] # Vud Vcb or Vus Vcb phase_space = (1 - rho) ** 2 A_PI = A_PI_cd(rho, me) prefactor = GF**2 * mb**2 * abs(ckm_factor) ** 2 * phase_space / (6 * pi) # While this expression is manifestly real, we take take the real part to avoid # tiny imaginary parts from floating point errors result += prefactor * (C @ A_PI @ C.conj() - CSM @ A_PI @ CSM.conj()).real return resultBSM Pauli interference contributions. For now, only from b->c ubar d (dbcu sector) and b->c ubar s (sbcu sector) operators.
def run_lifetime_bag_parameters(par, scale)-
Expand source code
def run_lifetime_bag_parameters(par, scale): # We shouldn't go below mc or above mb, else the anomalous dimensions change if scale < config["RGE thresholds"]["mc"] or scale > 4.5: warnings.warn("Scale for running the B lifetime bag parameters should be between mc and 4.5 GeV") alpha_s = get_alpha_s(par, scale) alpha_s_0 = get_alpha_s(par, 1.5) eta = alpha_s / alpha_s_0 beta_0_nf4 = 11 - (2/3)*4 B1qtilde_0 = par["bag_lifetime_B1qtilde"] B2qtilde_0 = par["bag_lifetime_B2qtilde"] B3qtilde_0 = par["bag_lifetime_B3qtilde"] B4qtilde_0 = par["bag_lifetime_B4qtilde"] B5qtilde_0 = par["bag_lifetime_B5qtilde"] B6qtilde_0 = par["bag_lifetime_B6qtilde"] B7qtilde_0 = par["bag_lifetime_B7qtilde"] B8qtilde_0 = par["bag_lifetime_B8qtilde"] deltaqq1tilde_0 = par["bag_lifetime_delta^ud_1tilde"] deltaqq2tilde_0 = par["bag_lifetime_delta^ud_2tilde"] deltaqq3tilde_0 = par["bag_lifetime_delta^ud_3tilde"] deltaqq4tilde_0 = par["bag_lifetime_delta^ud_4tilde"] # We only evolve the SM bag parameters B1-4, matching what is done in 2208.02643 # This matrix comes from Maria-Laura Piscopo gamma_0_D = np.array((8, 8, -1, -1)) V = np.array(( (0 , 3/4, 0, -6), (3/4, 0 , -6, 0), (0 , 1 , 0, 1), (1 , 0 , 1, 0) )) invV = np.array(( (0 , 4/27 , 0 , 8/9), (4/27 , 0 , 8/9, 0), (0 , -4/27, 0 , 1/9), (-4/27, 0 , 1/9, 0) )) # Evolution matrix from 1.5 GeV to scale U = V @ np.diag(eta**(gamma_0_D / (2*beta_0_nf4))) @ invV B_0 = np.array((B1qtilde_0, B2qtilde_0, B3qtilde_0, B4qtilde_0)) B1qtilde, B2qtilde, B3qtilde, B4qtilde = U @ B_0 return { "bag_lifetime_B1qtilde": B1qtilde, "bag_lifetime_B2qtilde": B2qtilde, "bag_lifetime_B3qtilde": B3qtilde, "bag_lifetime_B4qtilde": B4qtilde, "bag_lifetime_B5qtilde": B5qtilde_0, "bag_lifetime_B6qtilde": B6qtilde_0, "bag_lifetime_B7qtilde": B7qtilde_0, "bag_lifetime_B8qtilde": B8qtilde_0, "bag_lifetime_deltaqq1tilde": deltaqq1tilde_0, "bag_lifetime_deltaqq2tilde": deltaqq2tilde_0, "bag_lifetime_deltaqq3tilde": deltaqq3tilde_0, "bag_lifetime_deltaqq4tilde": deltaqq4tilde_0, } def siegen_basis_wcs(wc_obj, par, sector)-
Expand source code
def siegen_basis_wcs(wc_obj, par, sector): # Siegen basis defined in eqs 2.1 - 2.6 of 2211.02724 scale = config["renormalization scale"]["b lifetime ratios"] wc_sm = wcsm_nf5(scale) wcxf_sector = "db" if sector == "dbcc" else sector flavio_wc_bsm = wc_obj.get_wc(sector=wcxf_sector, scale=scale, par=par) CSM = np.zeros(20) CSM[0] = -1/6 * wc_sm[0] + wc_sm[1] CSM[1] = 1/2 * wc_sm[0] CNP = np.zeros(20, dtype=complex) siegen_wc_order = ( "VLL", "VLLt", "VRL", "VRLt", "SLR", "SLRt", "SRR", "SRRt", "TRR", "TRRt", "VRR", "VRRt", "VLR", "VLRt", "SRL", "SRLt", "SLL", "SLLt", "TLL", "TLLt" ) flavio_sector_name_mapping = {"sbcu": "bcus", "dbcu": "bcud", "dbcc": "bdcc"} flavio_sector_name = flavio_sector_name_mapping[sector] for i, name in enumerate(siegen_wc_order): CNP[i] = flavio_wc_bsm[f"C{name}_{flavio_sector_name}"] return (CSM, CNP) def tau_Bp_over_tau_Bd(wc_obj, par)-
Expand source code
def tau_Bp_over_tau_Bd(wc_obj, par): r"""Ratio of the B+ over Bd lifetimes based on the SM estimate plus the NP contribution from (some) four quark operators.""" ratio_SM = tau_Bp_over_tau_Bd_SM(par) delta_ratio_BSM = gamma_BSM_dim6(wc_obj, par, "B0") - gamma_BSM_dim6(wc_obj, par, "B+") return ratio_SM + delta_ratio_BSM * par["tau_B+"]Ratio of the B+ over Bd lifetimes based on the SM estimate plus the NP contribution from (some) four quark operators.
def tau_Bp_over_tau_Bd_SM(par)-
Expand source code
def tau_Bp_over_tau_Bd_SM(par): r"""Sm contribution to the ratio of the B+ to Bd lifetimes.""" scale = config["renormalization scale"]["b lifetime ratios"] if scale != 4.5: raise ValueError("The SM prediction for the B+ / Bd lifetime ratio is only available at the scale 4.5 GeV.") bag_params_dict = run_lifetime_bag_parameters(par, scale) B1qtilde = bag_params_dict["bag_lifetime_B1qtilde"] B2qtilde = bag_params_dict["bag_lifetime_B2qtilde"] B3qtilde = bag_params_dict["bag_lifetime_B3qtilde"] B4qtilde = bag_params_dict["bag_lifetime_B4qtilde"] deltaqq1tilde = bag_params_dict["bag_lifetime_deltaqq1tilde"] deltaqq2tilde = bag_params_dict["bag_lifetime_deltaqq2tilde"] deltaqq3tilde = bag_params_dict["bag_lifetime_deltaqq3tilde"] deltaqq4tilde = bag_params_dict["bag_lifetime_deltaqq4tilde"] # Phenomenological formula from Lenz:2022rbq # The coefficients in front of the bag parameters come from SM wilson coefficients (/other parameters) flavio.citations.register("Lenz:2022rbq") ratio_SM = (1 + 0.059 * B1qtilde + 0.005 * B2qtilde - 0.674 * B3qtilde + 0.160 * B4qtilde - 0.025 * deltaqq1tilde + 0.002 * deltaqq2tilde + 0.591 * deltaqq3tilde - 0.152 * deltaqq4tilde - 0.007 ) return ratio_SMSm contribution to the ratio of the B+ to Bd lifetimes.
def weak_exchange(wc_obj, par, meson)-
Expand source code
def weak_exchange(wc_obj, par, meson): r"""BSM weak exchange contributions. For now, only from b->c ubar d (dbcu sector), b->c ubar s (sbcu sector), and b->c cbar d (db sector) operators.""" # For B+, no WE contribution from our considered sectors if meson == "B+": return 0 # So now Bd case only: dbcu and dbcc operators contribute GF = par["GF"] mb = flavio.physics.running.running.get_mb_KS(par, 1) mc = par["m_c"] rho = mc**2 / mb**2 V = ckm.get_ckm(par) me = lifetimematrixelements(par, meson, config["renormalization scale"]["b lifetime ratios"]) dflav = {"d": 0, "s": 1, "b": 2} uflav = {"u": 0, "c": 1, "t": 2} result = 0 for sector in ("dbcu", "dbcc"): CSM, CNP = siegen_basis_wcs(wc_obj, par, sector) C = CSM + CNP ckm_factor = V[uflav[sector[3]], dflav[sector[0]]] * V[uflav[sector[2]], dflav[sector[1]]] # Vud Vcb or Vcd Vcb if sector == "dbcu": phase_space = (1 - rho)**2 A_WE = A_WE_cu(rho, me) elif sector == "dbcc": phase_space = (1 - 4*rho)**0.5 A_WE = A_WE_cc(rho, me) prefactor = GF**2 * mb**2 * abs(ckm_factor)**2 * phase_space / (6 * pi) # While this term is manifestly real, we take take the real part to avoid # tiny imaginary parts from floating point errors result += prefactor * (C @ A_WE @ C.conj() - CSM @ A_WE @ CSM.conj()).real return resultBSM weak exchange contributions. For now, only from b->c ubar d (dbcu sector), b->c ubar s (sbcu sector), and b->c cbar d (db sector) operators.