The oneloop.quark module¶

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

Contents:

Building blocks for the one-loop quark propagator and related functions

The one-loop quark self-energy

The one-loop quark propagator

Parameter conversion

Poles and residues

Plots

Note

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.

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.

The oneloop.quark module¶

The one-loop quark self-energy¶

The one-loop quark propagator¶

Parameter conversion¶

Poles and residues¶

Plots¶

PySmeQcd

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Related Topics

The oneloop.quark module¶

Building blocks for the one-loop quark propagator and related functions¶

The one-loop quark self-energy¶

The one-loop quark propagator¶

Parameter conversion¶

Poles and residues¶

Plots¶