Set-up of the S.M.E.¶

The Screened Massive Expansion of pure Yang-Mill theory is defined by a simple shift of the kinetic $\mathcal{L}_{0}$ and interaction $\mathcal{L}_{\text{int}}$ terms of the gauge-fixed, renormalized Faddeev-Popov Lagrangian:

$\mathcal{L}_{0}\to \mathcal{L}_{0}+\delta\mathcal{L}\ ,\qquad \mathcal{L}_{\text{int}}\to \mathcal{L}_{\text{int}}-\delta\mathcal{L}\ ,$

where $\delta\mathcal{L}$ is chosen so that the zero-order transverse gluon propagator $\Delta_{0}^{\mu\nu}(p)$ is massive rather than massless:

$\delta\mathcal{L}=\frac{m^{2}}{2}\,A_{\mu}^{a}\,t^{\mu\nu}(p)A_{\nu}^{a} \qquad\Longrightarrow\qquad\Delta_{0}^{\mu\nu}(p)=\frac{1}{p^{2}+m^{2}}\ t^{\mu\nu}(p)+\xi \frac{1}{p^{2}}\ \ell^{\mu\nu}(p)\ .$

The shift gives rise to a new two-point gluon vertex $\delta\Gamma^{\mu\nu}(p)$ ,

$\delta\Gamma^{\mu\nu}(p)=m^{2}t^{\mu\nu}(p)\ ,$

which must be included in the Feynman rules of perturbation theory.

In full QCD, in addition to the shift of the gluon sector, a shift is also performed in the quark sector. Namely, the kinetic $\mathcal{L}_{q,0}$ and interaction $\mathcal{L}_{q,\text{int}}$ terms of the quark Lagrangian are redefined so that the light quarks propagate with a mass $M$ which is much larger than their bare (UV) mass $M_{B}$ , and not related to the latter by perturbative corrections:

$\mathcal{L}_{q,0}\to \mathcal{L}_{q,0}+\delta\mathcal{L}_{q}\ ,\qquad \mathcal{L}_{q,\text{int}}\to \mathcal{L}_{q,\text{int}}-\delta\mathcal{L}_{q}\ ,$

where

$\delta\mathcal{L}_{q}=(M-M_{B}Z_{\psi})\overline{\psi}\psi \qquad\Longrightarrow\qquad S_{0}(p)=\frac{1}{-i\slashed{p}+M}\ ,$

$S_{0}(p)$ being the zero-order quark propagator. The two-point quark vertex $\delta\Gamma_{q}(p)$ ,

$\delta\Gamma_{q}(p)=-(M-M_{B}Z_{\psi})\ ,$

is then also included in the Feynman rules of the theory.

Features of the S.M.E.¶

The Screened Massive Expansion of QCD:

reduces to ordinary perturbation theory in the UV limit (in particular, it is renormalizable),
yields propagators which are in very good agreement with the lattice data in Euclidean space,
can be extended to the whole complex Minkowski space to study features such as the complex-conjugate poles of the gluon and quark propagators,
can be made self-contained when optimized by principles of gauge invariance,
yields a strong running coupling which is finite (no Landau poles) and small enough to allow the use of perturbation theory.

The gluon propagator¶

In a general covariant gauge, the one-loop gluon polarization $\Pi(p^{2})$ can be expressed in terms of three functions, $F(s)$ , $F_{\xi}(s)$ and $F_{Q}(s)$ ,

$\Pi(p^{2})=-\alpha p^{2} \left(F(s)+\xi F_{\xi}(s)+N_{f} F_{Q}(s)+\text{const.}\right)\ ,$

or four, if the contribution $F_{Q}^{(cr)}(s)$ due to the crossed quark loop is also included in the polarization,

$\Pi(p^{2})=-\alpha p^{2} \left(F(s)+\xi F_{\xi}(s)+N_{f} F_{Q}(s)+N_{f} F_{Q}^{(cr)}(s)+\text{const.}\right)\ .$

In this documentation, we refer to the first choice as the uncrossed (type='uc') variant of the gluon propagator, and to the second choice as the crossed (type='cr') variant of the gluon propagator. In pure Yang-Mills theory (no quarks, i.e., $N_{f}=0$ ), this distinction is irrelevant, since there are no quark loops in the gluon polarization.

The function $F_{Q}^{(cr)}(s)$ can be obtained by differentiating $F_{Q}(s)$ with respect to the quark chiral mass,

$F_{Q}^{(cr)}(s)=-M\frac{\partial}{\partial M}\,F_{Q}(s)\ .$

The inverse gluon dressing function $J^{-1}(s)$ is then given by

$J^{-1}(s)&=1+\alpha \left(F(s) + \xi F_{\xi}(s)+N_{f} F_{Q}(s)\right)+ \text{const.}=\\ &=\alpha \left(F(s) + \xi F_{\xi}(s)+N_{f} F_{Q}(s)+F_{0}\right)$

for the uncrossed variant of the propagator, or

$J^{-1}(s)=\alpha \left(F(s) + \xi F_{\xi}(s)+N_{f} F_{Q}(s)+N_{f}F_{Q}^{(cr)}(s)+F_{0}\right)$

for its crossed variant. In the above equations, $F_{0}$ is an additive renormalization constant.

The functions $F(s)$ , $F_{\xi}(s)$ , $F_{Q}(s)$ and $F_{Q}^{(cr)}(s)$ are defined in the SmeQcd.oneloop.gluon module, together with the gluon propagator $\Delta(p^{2})$ , dressing function $J(s)$ ,

$J(s)=p^{2}\Delta(p^{2})\ ,$

and spectral function (in Minkowski space) $\rho_{\Delta}(p^{2}_{M})$ ,

$\rho_{\Delta}(p^{2}_{M})=2\, \mathrm{Im} \{\Delta(-p_{M}^{2}-i\varepsilon)\}\ .$

The ghost propagator¶

In a general covariant gauge, the one-loop ghost self-energy $\Sigma_{\text{gh}}(p^{2})$ can be expressed in terms of two functions, $G(s)$ and $G_{\xi}(s)$ :

$\Sigma_{\text{gh}}(p^{2})=-\alpha p^{2} \left(G(s)+\xi G_{\xi}(s)+\text{const.}\right)\ .$

The inverse ghost dressing function $\chi^{-1}(s)$ is then given by

$\chi^{-1}(s)=1+\alpha \left(G(s) + \xi G_{\xi}(s)\right)+ \text{const.}=\alpha \left(G(s) + \xi G_{\xi}(s)+G_{0}\right)\ ,$

where $G_{0}$ is an additive renormalization constant.

The functions $G(s)$ and $G_{\xi}(s)$ are defined in the SmeQcd.oneloop.ghost module together with the ghost propagator $\mathcal{G}(p^{2})$ , dressing function $\chi(s)$ ,

$\chi(s)=p^{2}\mathcal{G}(p^{2})\ ,$

and spectral function (in Minkowski space) $\rho_{\mathcal{G}}(p^{2}_{M})$ ,

$\rho_{\mathcal{G}}(p^{2}_{M})=2\, \mathrm{Im} \{\mathcal{G}(-p_{M}^{2}-i\varepsilon)\}\ .$

The quark propagator¶

The quark self-energy $\Sigma(p^{2})$ can be expressed as

$\Sigma(p^{2})=i\slashed{p}\,\Sigma_{V}(p^{2})+\Sigma_{S}(p^{2})\ ,$

where $\Sigma_{V}(p^{2})$ and $\Sigma_{S}(p^{2})$ are scalar functions. To one loop, if only the ordinary quark loop and quark-crossed quark loop are included in the quark self-energy, we say that the quark propagator is computed in the minimalistic scheme (type='ms'). If, on the other hand, the gluon-crossed quark loop is also included in the self-energy, we say that the propagator is computed in the vertex-wise scheme (type='vw'). Finally, we refer to a scheme in which the internal gluon line is replaced by the principal part of the dressed gluon propagator as the complex-conjugate scheme (type='cc').

In the minimalistic scheme, the quark self-energy is given by the sum $\Sigma^{(2)}(p^{2})$ of the ordinary quark loop $\Sigma^{(2)}(p^{1})$ and of the quark-crossed loop:

$\Sigma(p^{2})=\Sigma^{(2)}(p^{2})\ .$

In the vertex-wise scheme, the gluon-crossed $\Sigma^{(gl)}(p^{2})$ loop is also added to the self-energy:

$\Sigma(p^{2})=\Sigma^{(2)}(p^{2})+\Sigma^{(gl)}(p^{2})\ .$

In the complex-conjugate scheme, the quark-self energy is given by two terms, each of which is obtained by replacing $m^{2}\to p_{0}^{2}$ – where $p_{0}^{2}$ is the Minkowski gluon pole – in $\Sigma^{(2)}(p^{2})$ , and multiplying the term by the corresponding residue $R$ :

$\Sigma(p^{2})=R\left[\Sigma^{(2)}(p^{2})\right]_{m^{2}\to p_{0}^{2}}+\overline{R}\left[\Sigma^{(2)}(p^{2})\right]_{m^{2}\to \overline{p_{0}^{2}}}\ .$

For an in-depth discussion of the minimalistic, vertex-wise and complex-conjugate schemes, see G. Comitini, D. Rizzo, M. Battello, and F. Siringo, Phys. Rev. D 104, 074020 (2021).

In any scheme, the quark propagator $S(p)$ can be expressed as

$S(p)=\frac{Z(p^{2})}{p^{2}+\mathcal{M}^{2}(p^{2})}\,\left(i\slashed{p}+\mathcal{M}(p^{2})\right)\ ,$

where

$Z(p^{2})=\frac{1}{A(p^{2})}\ ,\qquad\mathcal{M}(p^{2})=\frac{B(p^{2})}{A(p^{2})}\ ,$

and the functions $A(p^{2})$ and $B(p^{2})$ are obtained from the quark self-energy as

$A(p^{2}) = Z_{\psi}-\Sigma_{V}(p^{2})\ ,\qquad B(p^{2}) = M_{B}Z_{\psi}+\Sigma_{S}(p^{2})\ .$

In the above equations, $Z_{\psi}$ is the quark field-strength renormalization constant. To one loop, the following quantities are often used in place of $Z_{\psi}$ and $M_{B}$ :

$H_{0}=3\pi Z_{\psi}/\alpha_{s}\ ,\qquad K_{0}=\pi M_{B} Z_{\psi}/\alpha_{s}\ .$

The function $p^{2}+\mathcal{M}^{2}(p^{2})$ is also denoted by $Q(p^{2})$ ,

$Q(p^{2})=p^{2}+\mathcal{M}^{2}(p^{2})\ .$

The vector and scalar components of the functions $\Sigma^{(1)}(p^{2})$ , $\Sigma^{(2)}(p^{2})$ and $\Sigma^{(gl)}(p^{2})$ are defined in the module PySmeQcd.oneloop.quark, together with the functions $A(p^{2})$ , $B(p^{2})$ , $Z(p^{2})$ , $\mathcal{M}(p^{2})$ , $Q(p^{2})$ , the vector and scalar components $S_{V/S}(p^{2})$ of the quark propagator,