The oneloop.quark module

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

Contents:

Note

  1. Where not otherwise specified:

  • s is the adimensional Euclidean momentum squared s=p^{2}/m^{2},

  • p02 is the adimensional gluon pole -p_{0}^{2}/m^{2} in Minkowski space.

  1. 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.

The one-loop quark self-energy

quark_SigV1(s, x=0.238171, xi=0)

Function \Sigma_{V}^{(1)}(p^{2}) – vector component of the 1-loop self-energy uncrossed term.

Modulo a factor of \frac{\alpha_{s}}{3\pi}.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

quark_SigS1(s, x=0.238171, xi=0)

Function \Sigma_{S}^{(1)}(p^{2}) – scalar component of the 1-loop self-energy uncrossed term.

Modulo a factor of M\frac{\alpha_{s}}{\pi}.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

quark_SigV2(s, x=0.238171, xi=0)

Function \Sigma_{V}^{(2)}(p^{2}) – vector component of the 1-loop self-energy uncrossed + quark-crossed terms.

Modulo a factor of \frac{\alpha_{s}}{3\pi}.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

quark_SigS2(s, x=0.238171, xi=0)

Function \Sigma_{S}^{(2)}(p^{2}) – scalar component of the 1-loop self-energy uncrossed + quark-crossed terms.

Modulo a factor of M\frac{\alpha_{s}}{\pi}.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

quark_SigVgl(s, x=0.238171, xi=0)

Function \Sigma_{V}^{(gl)}(p^{2}) – vector component of the 1-loop self-energy gluon-crossed term.

Modulo a factor of \frac{\alpha_{s}}{3\pi}.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

quark_SigSgl(s, x=0.238171, xi=0)

Function \Sigma_{S}^{(gl)}(p^{2}) – scalar component of the 1-loop self-energy gluon-crossed term.

Modulo a factor of M\frac{\alpha_{s}}{\pi}.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

quark_SigV(s, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1)

Function \Sigma_{V}(p^{2}) – vector component of the 1-loop self-energy.

Modulo a factor of \frac{\alpha_{s}}{3\pi}.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

quark_SigS(s, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1)

Function \Sigma_{S}(p^{2}) – scalar component of the 1-loop self-energy.

Modulo a factor of M\frac{\alpha_{s}}{\pi}.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

dquark_SigV1(s, x=0.238171, xi=0)

Derivative of quark_SigV1 with respect to s.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

dquark_SigV1dx(s, x=0.238171, xi=0)

Derivative of quark_SigV1 with respect to x.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

dquark_SigS1(s, x=0.238171, xi=0)

Derivative of quark_SigS1 with respect to s.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

dquark_SigS1dx(s, x=0.238171, xi=0)

Derivative of quark_SigS1 with respect to x.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

dquark_SigV(s, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1)

Numerical derivative of quark_SigV with respect to s.

The \varepsilon for the finite difference is defined in the PySmeQcd.settings module.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

dquark_SigS(s, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1)

Numerical derivative of quark_SigS with respect to s.

The \varepsilon for the finite difference is defined in the PySmeQcd.settings module.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

The one-loop quark propagator

quark_A(s, H0=2.073, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1)

Function A(p^{2}).

Modulo a factor of \frac{\alpha_{s}}{3\pi}.

Parameters

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

quark_B(s, K0=0.1190625, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1)

Function B(p^{2}).

Modulo a factor of M\frac{\alpha_{s}}{\pi}.

Parameters

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

quark_Z(s, H0=2.073, x=0.238171, xi=0, type='ms', ren=True, Z=None, mu0=6.099699589795203, p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577)

Quark Z-function Z(p^{2}).

Modulo a factor of \frac{3\pi}{\alpha_{s}}.

Parameters

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • ren – if True, multiplicatively renormalize the Z-function

  • Z – if None, renormalize the Z-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 Z-function in the MOM scheme (only if ren=True)

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

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

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

quark_M(s, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577)

Quark mass function \mathcal{M}(p^{2}).

Parameters

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • dimensionful – if False, adimensionalize the mass function by dividing it by m; if True, s is the dimensionful momentum squared p^{2}

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

quark_Q(s, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577)

Function Q(p^{2}).

Parameters

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

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

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

quark_propV(s, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', ren=True, Z=None, mu0=6.099699589795203, p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577)

Vector component of the quark propagator S_{V}(p^{2}).

Modulo a factor of \frac{3\pi}{\alpha_{s}}.

Parameters

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • 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)

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • 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)

quark_propS(s, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', ren=True, Z=None, mu0=6.099699589795203, p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577)

Scalar component of the quark propagator S_{S}(p^{2}).

Modulo a factor of \frac{3\pi}{\alpha_{s}}.

Parameters

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • 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)

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

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

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

quark_spectralV(s, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', ren=True, Z=None, mu0=6.099699589795203, p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577)

Vector component of the quark spectral function \rho_{S_{V}}(p^{2}) in Minkowski space.

Modulo a factor of \frac{3\pi}{\alpha_{s}}. The \varepsilon for the difference across the branch cut is defined in the PySmeQcd.settings module.

Parameters

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • 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)

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • 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.

quark_spectralS(s, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', ren=True, Z=None, mu0=6.099699589795203, p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577)

Scalar component of the quark spectral function \rho_{S_{S}}(p^{2}) in Minkowski space.

Modulo a factor of \frac{3\pi}{\alpha_{s}}. The \varepsilon for the difference across the branch cut is defined in the PySmeQcd.settings module.

Parameters

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • 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)

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

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

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.

dquark_A(s, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1)

Numerical derivative of quark_A with respect to s.

The \varepsilon for the finite difference is defined in the PySmeQcd.settings module.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

dquark_B(s, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1)

Numerical derivative of quark_B with respect to s.

The \varepsilon for the finite difference is defined in the PySmeQcd.settings module.

Parameters

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

dquark_M(s, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577)

Numerical derivative of quark_M with respect to s.

The \varepsilon for the finite difference is defined in the PySmeQcd.settings module.

Parameters

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

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

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

dquark_Q(s, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577)

Numerical derivative of quark_Q with respect to s.

The \varepsilon for the finite difference is defined in the PySmeQcd.settings module.

Parameters

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

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

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

Parameter conversion

quark_MB(K0=0.1190625, H0=2.073)

Quark bare mass M_{B} from K_{0} and H_{0}.

Parameters

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

Note

The dimensions of the return value are determined by the dimensions of the parameter K0. The default value of K0 is adimensionalized by the quark chiral mass M.

quark_H0(Zpsi, a_s)

Quark H_{0} parameter from Z_{\psi} and \alpha_{s}.

Parameters

  • Zpsi – quark field-strength renormalization factor Z_{\psi}

  • a_s – strong coupling constant \alpha_{s}=\frac{g^{2}}{4\pi}

quark_K0(Zpsi, a_s, MB)

Quark K_{0} parameter from Z_{\psi} and \alpha_{s}.

Parameters

  • Zpsi – quark field-strength renormalization factor Z_{\psi}

  • a_s – strong coupling constant \alpha_{s}=\frac{g^{2}}{4\pi}

  • MB – quark bare mass M_{B}

Note

The dimensions of the return value are determined by the dimensions of the parameter MB.

Poles and residues

quark_pole(K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577, stp=- 1 - 1j)

Pole of the quark propagator in Euclidean space.

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

Parameters

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme 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

quark_residue(pole=None, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', ren=True, Z=None, mu0=6.099699589795203, p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577, stp=- 1 - 1j)

Residue of the vector component of the quark propagator at its pole.

Modulo a factor of \frac{3\pi}{\alpha_{s}}.

Parameters

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

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • 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)

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • 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)

Note

To obtain the residue of the scalar part of the quark propagator, multiply quark_residue() by quark_M() computed at the pole quark_pole(), using the same parameters in all three functions.

Plots

quark_Z_plot(s=[0.001, 25], N_s=1000, H0=2.073, x=0.238171, xi=0, type='ms', ren=True, Z=None, mu0=6.099699589795203, p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577, scale='log', title=True, outf=None)

Plots the quark Z-function in Euclidean space.

Modulo a factor of \frac{3\pi}{\alpha_{s}}.

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

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • ren – if True, multiplicatively renormalize the Z-function

  • Z – if None, renormalize the Z-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 Z-function in the MOM scheme (only if ren=True)

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • 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

quark_M_plot(s=[0.001, 25], N_s=1000, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577, scale='log', title=True, outf=None)

Plots the quark mass function in Euclidean space.

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

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • dimensionful – if False, adimensionalize the mass function by dividing it by m; 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

quark_Q_inv_plot(s=[0.001, 25], N_s=1000, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, inverse=False, dimensionful=False, m=0.65577, scale='log', title=True, outf=None)

Plots the quark Q^{-1} function in Euclidean space.

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

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • inverse – if True, plot Q(p^{2})

  • dimensionful – if False, adimensionalize the function 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

quark_Q_inv_3D_plot(value='abs', s_real=[- 3, 1], s_imag=[- 1, 1], N_s_real=1000, N_s_imag=1000, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, inverse=False, dimensionful=False, m=0.65577, title=True, outf=None)

Plots the quark Q^{-1} function in the complex plane (3D plot).

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

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • inverse – if True, plot Q(p^{2})

  • dimensionful – if False, adimensionalize the function by multiplying it by m2; if True, s is 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

quark_Q_contour(s_real=[- 3, 1], s_imag=[- 1, 1], N_s_real=500, N_s_imag=500, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577, title=True, outf=None)

Plots the zeroes of the real and imaginary part of the quark Q 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

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme 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

quark_propV_plot(s=[0.001, 25], N_s=1000, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', ren=True, Z=None, mu0=6.099699589795203, p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577, scale='log', title=True, outf=None)

Plots the vector component of the quark propagator in Euclidean space.

Modulo a factor of \frac{3\pi}{\alpha_{s}}.

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

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • 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)

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • 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

quark_propS_plot(s=[0.001, 25], N_s=1000, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', ren=True, Z=None, mu0=6.099699589795203, p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577, scale='log', title=True, outf=None)

Plots the scalar component of the quark propagator in Euclidean space.

Modulo a factor of \frac{3\pi}{\alpha_{s}}.

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

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • 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)

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • dimensionful – if False, adimensionalize the propagator by multiplying it by m; 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

quark_spectralV_plot(s=[0.001, 25], N_s=1000, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', ren=True, Z=None, mu0=6.099699589795203, p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577, scale='log', title=True, outf=None)

Plots the vector component of the quark spectral function in Minkowski space.

Modulo a factor of \frac{3\pi}{\alpha_{s}}.

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

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • 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)

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • 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.

quark_spectralS_plot(s=[0.001, 25], N_s=1000, K0=0.1190625, H0=2.073, x=0.238171, xi=0, type='ms', ren=True, Z=None, mu0=6.099699589795203, p02=0.457525 + 1.01298j, phi=1.261708, Zglue=1, dimensionful=False, m=0.65577, scale='log', title=True, outf=None)

Plots the scalar component of the quark spectral function in Minkowski space.

Modulo a factor of \frac{3\pi}{\alpha_{s}}.

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

  • K0 – normalized and adimensionalized quark bare mass K_{0}/M

  • H0 – additive renormalization constant H_{0}

  • x – adimensional quark chiral mass squared x

  • xi – gauge parameter \xi

  • type – diagram selector: 'ms' if only the ordinary and quark-crossed loops are included, 'vw' if also the gluon-crossed loop is included; 'cc' for the c.c. scheme

  • 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)

  • p02 – pole of the gluon propagator in Minkowski space (c.c. scheme only)

  • phi – phase \varphi of the residue of the gluon propagator at the pole p02 (c.c. scheme only)

  • Zglue – renormalization factor for the internal gluon line (c.c. scheme only)

  • dimensionful – if False, adimensionalize the spectral function by multiplying it by m; 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.