The oneloop.gluon module

Library of functions and routines for the one-loop gluon propagator computed in the Screened Massive Expansion.

Contents:

Note

  1. Where not otherwise specified, s is the adimensional Euclidea momentum squared s=p^{2}/m^{2}.

  2. Most functions are defined modulo factors of the coupling constant. If the ren parameter is available, such factors become irrelevant when renormalization is enabled using ren=True.

  3. With the parametrization described below, to one loop, the number of colors only enters the expressions through the quark contribution – hence is irrelevant in pure Yang-Mills theory (no quarks, i.e., N_{f}=0) – and through the value of the gluon mass parameter m.

The one-loop gluon dressing function, propagator and spectral function

gluon_J_inv(s, x=0.238171, N=3, xi=0, F0=-0.876, Nf=0, type='uc')

Inverse gluon dressing function J^{-1}(s).

Modulo a factor of \alpha.

Parameters:

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • F0 – additive renormalization constant F_{0}

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

gluon_J(s, x=0.238171, N=3, xi=0, F0=-0.876, Nf=0, type='uc')

Gluon dressing function J(s).

Modulo a factor of \alpha^{-1}.

Parameters:

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • F0 – additive renormalization constant F_{0}

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

gluon_prop(s, x=0.238171, N=3, xi=0, F0=-0.876, Nf=0, type='uc', ren=True, Z=None, mu0=6.099699589795203, dimensionful=False, m=0.65577)

Gluon propagator \Delta(p^{2}).

Modulo a factor of \alpha^{-1}.

Parameters:

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • F0 – additive renormalization constant F_{0}

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

  • ren – if True, multiplicatively renormalize the propagator

  • Z – if None, renormalize the propagator in the MOM scheme at the scale mu0, else use Z as the multiplicative renormalization factor (only if ren=True)

  • mu0 – adimensional renormalization scale \mu_{0} for the renormalization of the propagator in the MOM scheme (only if ren=True)

  • dimensionful – if False, adimensionalize the propagator by multiplying it by m2; if True, s is the dimensionful momentum squared p^{2}

  • m – gluon mass parameter m (only if dimensionful=True)

gluon_spectral(s, x=0.238171, N=3, xi=0, F0=-0.876, Nf=0, type='uc', ren=True, Z=None, mu0=6.099699589795203, dimensionful=False, m=0.65577)

Gluon spectral function \rho_{\Delta}(p^{2}) in Minkowski space.

Modulo a factor of \alpha^{-1}. The \varepsilon for the difference across the branch cut is defined in the PySmeQcd.settings module.

Parameters:

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • F0 – additive renormalization constant F_{0}

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

  • ren – if True, multiplicatively renormalize the spectral function

  • Z – if None, renormalize the spectral function in the MOM scheme at the scale mu0, else use Z as the multiplicative renormalization factor (only if ren=True)

  • mu0 – adimensional renormalization scale \mu_{0} for the renormalization of the spectral function in the MOM scheme (only if ren=True)

  • dimensionful – if False, adimensionalize the spectral function by multiplying it by m2; if True, s is the dimensionful Minkowski momentum squared p^{2}_{M}

  • m – gluon mass parameter m (only if dimensionful=True)

Note

s is the adimensional Minkowski momentum squared p_{M}^{2}/m^{2}, or the dimensionful Minkowski momentum squared p_{M}^{2} if dimensionful=True.

dgluon_J_inv(s, x=0.238171, N=3, xi=0, Nf=0, type='uc')

Derivative of gluon_J_inv with respect to s.

Parameters:

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

Poles and residues

gluon_pole(x=0.238171, N=3, xi=0, F0=-0.876, Nf=0, type='uc', dimensionful=False, m=0.65577, stp=-1 - 1j)

Pole p_{0}^{2} of the gluon propagator in Euclidean space.

If the pole cannot be found, prints to stderr and raises a RuntimeError.

Parameters:

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • F0 – additive renormalization constant F_{0}

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

  • dimensionful – if False, adimensionalize the pole by dividing it by m2

  • m – gluon mass parameter m (only if dimensionful=True)

  • stp – starting point for the root-finding algorithm

gluon_residue(pole=None, x=0.238171, N=3, xi=0, F0=-0.876, Nf=0, type='uc', ren=True, Z=None, mu0=6.099699589795203, dimensionful=False, m=0.65577, stp=-1 - 1j)

Residue R of the gluon propagator at its pole.

Modulo a factor of \alpha^{-1}.

Parameters:

  • pole – if None, compute the pole p_{0}^{2} before computing its residue (recommended); else use pole as the Euclidean pole. See below for dimensions

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • F0 – additive renormalization constant F_{0}

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

  • ren – if True, multiplicatively renormalize the residue

  • Z – if None, renormalize the residue in the MOM scheme at the scale mu0, else use Z as the multiplicative renormalization factor (only if ren=True)

  • mu0 – adimensional renormalization scale \mu_{0} for the renormalization of the residue in the MOM scheme (only if ren=True)

  • dimensionful – if True, pole is passed dimensionful (only if pole!=None)

  • m – gluon mass parameter m (only if dimensionful=True)

  • stp – starting point for the root-finding algorithm (only if pole=None)

Plots

gluon_J_plot(s=[0.001, 25], N_s=1000, x=0.238171, N=3, xi=0, F0=-0.876, Nf=0, type='uc', inverse=False, ren=True, Z=None, mu0=6.099699589795203, dimensionful=False, m=0.65577, scale='log', title=True, outf=None)

Plots the gluon dressing function in Euclidean space.

Modulo a factor of \alpha^{-1}.

Parameters:

  • s – a two-element list containing the minimum and maximum momentum squared for the plot (see below for dimensions)

  • N_s – an integer specifying the number of plot points

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • F0 – additive renormalization constant F_{0}

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

  • inverse – if True, plot the inverse dressing function

  • ren – if True, multiplicatively renormalize the dressing function

  • Z – if None, renormalize the dressing function in the MOM scheme at the scale mu0, else use Z as the multiplicative renormalization factor (only if ren=True)

  • mu0 – adimensional renormalization scale \mu_{0} for the renormalization of the dressing function in the MOM scheme (only if ren=True)

  • dimensionful – if True, s is the dimensionful momentum squared p^{2}

  • m – gluon mass parameter m (only if dimensionful=True)

  • scale – if set to 'log', switch to a log-lin scale

  • title – if True, show a title for the plot

  • outf – if not None, save the output to the file specified by outf rather than printing it out

gluon_J_3D_plot(value='abs', s_real=[-5, 3], s_imag=[-4, 4], N_s_real=1000, N_s_imag=1000, x=0.238171, N=3, xi=0, F0=-0.876, Nf=0, type='uc', inverse=False, ren=True, Z=None, mu0=6.099699589795203, dimensionful=False, m=0.65577, title=True, outf=None)

Plots the gluon dressing function in the complex plane (3D plot).

Modulo a factor of \alpha^{-1}.

Parameters:

  • value – if 'abs', plot the absolute value; if 'real', plot the real part; if 'imag', plot the imaginary part

  • s_real – a two-element list containing the minimum and maximum real part of the momentum squared for the plot (see below for dimensions)

  • s_imag – a two-element list containing the minimum and maximum imaginary part of the momentum squared for the plot (see below for dimensions)

  • N_s_real – an integer specifying the number of subdivisions in the real direction

  • N_s_imag – an integer specifying the number of subdivisions in the imaginary direction

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • F0 – additive renormalization constant F_{0}

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

  • inverse – if True, plot the inverse dressing function

  • ren – if True, multiplicatively renormalize the dressing function

  • Z – if None, renormalize the dressing function in the MOM scheme at the scale mu0, else use Z as the multiplicative renormalization factor (only if ren=True)

  • mu0 – adimensional renormalization scale \mu_{0} for the renormalization of the dressing function in the MOM scheme (only if ren=True)

  • dimensionful – if True, s_real and s_imag are the real and imaginary parts of the dimensionful momentum squared p^{2}

  • m – gluon mass parameter m (only if dimensionful=True)

  • title – if True, show a title for the plot

  • outf – if not None, save the output to the file specified by outf rather than printing it out

gluon_J_inv_contour(s_real=[-5, 3], s_imag=[-4, 4], N_s_real=500, N_s_imag=500, x=0.238171, N=3, xi=0, F0=-0.876, Nf=0, type='uc', dimensionful=False, m=0.65577, title=True, outf=None)

Plots the zeroes of the real and imaginary part of the inverse gluon dressing function.

Parameters:

  • s_real – a two-element list containing the minimum and maximum real part of the momentum squared for the plot (see below for dimensions)

  • s_imag – a two-element list containing the minimum and maximum imaginary part of the momentum squared for the plot (see below for dimensions)

  • N_s_real – an integer specifying the number of subdivisions in the real direction

  • N_s_imag – an integer specifying the number of subdivisions in the imaginary direction

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • F0 – additive renormalization constant F_{0}

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

  • dimensionful – if True, s_real and s_imag are the real and imaginary parts of the dimensionful momentum squared p^{2}

  • m – gluon mass parameter m (only if dimensionful=True)

  • title – if True, show a title for the plot

  • outf – if not None, save the output to the file specified by outf rather than printing it out

gluon_prop_plot(s=[0.001, 25], N_s=1000, x=0.238171, N=3, xi=0, F0=-0.876, Nf=0, type='uc', ren=True, Z=None, mu0=6.099699589795203, dimensionful=False, m=0.65577, scale='log', title=True, outf=None)

Plots the gluon propagator in Euclidean space.

Modulo a factor of \alpha^{-1}.

Parameters:

  • s – a two-element list containing the minimum and maximum momentum squared for the plot (see below for dimensions)

  • N_s – an integer specifying the number of plot points

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • F0 – additive renormalization constant F_{0}

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

  • ren – if True, multiplicatively renormalize the propagator

  • Z – if None, renormalize the propagator in the MOM scheme at the scale mu0, else use Z as the multiplicative renormalization factor (only if ren=True)

  • mu0 – adimensional renormalization scale \mu_{0} for the renormalization of the propagator in the MOM scheme (only if ren=True)

  • dimensionful – if False, adimensionalize the propagator by multiplying it by m2; if True, s is the dimensionful momentum squared p^{2}

  • m – gluon mass parameter m (only if dimensionful=True)

  • scale – if set to 'log', switch to a log-lin scale

  • title – if True, show a title for the plot

  • outf – if not None, save the output to the file specified by outf rather than printing it out

gluon_spectral_plot(s=[0.001, 25], N_s=1000, x=0.238171, N=3, xi=0, F0=-0.876, Nf=0, type='uc', ren=True, Z=None, mu0=6.099699589795203, dimensionful=False, m=0.65577, scale='log', title=True, outf=None)

Plots the gluon spectral function in Minkowski space.

Modulo a factor of \alpha^{-1}.

Parameters:

  • s – a two-element list containing the minimum and maximum Minkowski momentum squared for the plot (see below for dimensions)

  • N_s – an integer specifying the number of plot points

  • x – adimensional quark chiral mass squared x

  • N – number of colors N

  • xi – gauge parameter \xi

  • F0 – additive renormalization constant F_{0}

  • Nf – number of quarks N_{f}

  • type – diagram selector: 'uc' if only the ordinary quark loop is included, 'cr' if also the crossed quark loop is included (full QCD only)

  • ren – if True, multiplicatively renormalize the spectral function

  • Z – if None, renormalize the spectral function in the MOM scheme at the scale mu0, else use Z as the multiplicative renormalization factor (only if ren=True)

  • mu0 – adimensional renormalization scale \mu_{0} for the renormalization of the spectral function in the MOM scheme (only if ren=True)

  • dimensionful – if False, adimensionalize the spectral function by multiplying it by m2; if True, s is the dimensionful Minkowski momentum squared p^{2}_{M}

  • m – gluon mass parameter m (only if dimensionful=True)

  • scale – if set to 'log', switch to a log-lin scale

  • title – if True, show a title for the plot

  • outf – if not None, save the output to the file specified by outf rather than printing it out

Note

s is the adimensional Minkowski momentum squared p_{M}^{2}/m^{2}, or the dimensionful Minkowski momentum squared p_{M}^{2} if dimensionful=True.