.. Copyright (C) 2022, Giorgio Comitini .. This is part of the PySmeQcd Documentation. .. See the file index.rst for copying conditions. .. role:: python(code) :language: python :class: highlight .. role:: console(code) :language: console :class: highlight 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 :math:`\mathcal{L}_{0}` and interaction :math:`\mathcal{L}_{\text{int}}` terms of the gauge-fixed, renormalized Faddeev-Popov Lagrangian: .. math:: \mathcal{L}_{0}\to \mathcal{L}_{0}+\delta\mathcal{L}\ ,\qquad \mathcal{L}_{\text{int}}\to \mathcal{L}_{\text{int}}-\delta\mathcal{L}\ , where :math:`\delta\mathcal{L}` is chosen so that the zero-order transverse gluon propagator :math:`\Delta_{0}^{\mu\nu}(p)` is massive rather than massless: .. math:: \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 :math:`\delta\Gamma^{\mu\nu}(p)`, .. math:: \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 :math:`\mathcal{L}_{q,0}` and interaction :math:`\mathcal{L}_{q,\text{int}}` terms of the quark Lagrangian are redefined so that the light quarks propagate with a mass :math:`M` which is much larger than their bare (UV) mass :math:`M_{B}`, and not related to the latter by perturbative corrections: .. math:: \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 .. math:: \delta\mathcal{L}_{q}=(M-M_{B}Z_{\psi})\overline{\psi}\psi \qquad\Longrightarrow\qquad S_{0}(p)=\frac{1}{-i\slashed{p}+M}\ , :math:`S_{0}(p)` being the zero-order quark propagator. The two-point quark vertex :math:`\delta\Gamma_{q}(p)`, .. math:: \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 :math:`\Pi(p^{2})` can be expressed in terms of three functions, :math:`F(s)`, :math:`F_{\xi}(s)` and :math:`F_{Q}(s)`, .. math:: \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 :math:`F_{Q}^{(cr)}(s)` due to the *crossed* quark loop is also included in the polarization, .. math:: \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** (:python:`type='uc'`) variant of the gluon propagator, and to the second choice as the **crossed** (:python:`type='cr'`) variant of the gluon propagator. In pure Yang-Mills theory (no quarks, i.e., :math:`N_{f}=0`), this distinction is irrelevant, since there are no quark loops in the gluon polarization. The function :math:`F_{Q}^{(cr)}(s)` can be obtained by differentiating :math:`F_{Q}(s)` with respect to the quark chiral mass, .. math:: F_{Q}^{(cr)}(s)=-M\frac{\partial}{\partial M}\,F_{Q}(s)\ . The inverse gluon dressing function :math:`J^{-1}(s)` is then given by .. math:: 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 .. math:: 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, :math:`F_{0}` is an additive renormalization constant. The functions :math:`F(s)`, :math:`F_{\xi}(s)`, :math:`F_{Q}(s)` and :math:`F_{Q}^{(cr)}(s)` are defined in the :python:`SmeQcd.oneloop.gluon` module, together with the gluon propagator :math:`\Delta(p^{2})`, dressing function :math:`J(s)`, .. math:: J(s)=p^{2}\Delta(p^{2})\ , and spectral function (in Minkowski space) :math:`\rho_{\Delta}(p^{2}_{M})`, .. math:: \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 :math:`\Sigma_{\text{gh}}(p^{2})` can be expressed in terms of two functions, :math:`G(s)` and :math:`G_{\xi}(s)`: .. math:: \Sigma_{\text{gh}}(p^{2})=-\alpha p^{2} \left(G(s)+\xi G_{\xi}(s)+\text{const.}\right)\ . The inverse ghost dressing function :math:`\chi^{-1}(s)` is then given by .. math:: \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 :math:`G_{0}` is an additive renormalization constant. The functions :math:`G(s)` and :math:`G_{\xi}(s)` are defined in the :python:`SmeQcd.oneloop.ghost` module together with the ghost propagator :math:`\mathcal{G}(p^{2})`, dressing function :math:`\chi(s)`, .. math:: \chi(s)=p^{2}\mathcal{G}(p^{2})\ , and spectral function (in Minkowski space) :math:`\rho_{\mathcal{G}}(p^{2}_{M})`, .. math:: \rho_{\mathcal{G}}(p^{2}_{M})=2\, \mathrm{Im} \{\mathcal{G}(-p_{M}^{2}-i\varepsilon)\}\ . -------------------------------------------------------------------------------- The quark propagator -------------------- The quark self-energy :math:`\Sigma(p^{2})` can be expressed as .. math:: \Sigma(p^{2})=i\slashed{p}\,\Sigma_{V}(p^{2})+\Sigma_{S}(p^{2})\ , where :math:`\Sigma_{V}(p^{2})` and :math:`\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** (:python:`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** (:python:`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** (:python:`type='cc'`). In the minimalistic scheme, the quark self-energy is given by the sum :math:`\Sigma^{(2)}(p^{2})` of the ordinary quark loop :math:`\Sigma^{(2)}(p^{1})` and of the quark-crossed loop: .. math:: \Sigma(p^{2})=\Sigma^{(2)}(p^{2})\ . In the vertex-wise scheme, the gluon-crossed :math:`\Sigma^{(gl)}(p^{2})` loop is also added to the self-energy: .. math:: \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 :math:`m^{2}\to p_{0}^{2}` -- where :math:`p_{0}^{2}` is the Minkowski gluon pole -- in :math:`\Sigma^{(2)}(p^{2})`, and multiplying the term by the corresponding residue :math:`R`: .. math:: \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 :math:`S(p)` can be expressed as .. math:: S(p)=\frac{Z(p^{2})}{p^{2}+\mathcal{M}^{2}(p^{2})}\,\left(i\slashed{p}+\mathcal{M}(p^{2})\right)\ , where .. math:: Z(p^{2})=\frac{1}{A(p^{2})}\ ,\qquad\mathcal{M}(p^{2})=\frac{B(p^{2})}{A(p^{2})}\ , and the functions :math:`A(p^{2})` and :math:`B(p^{2})` are obtained from the quark self-energy as .. math:: 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, :math:`Z_{\psi}` is the quark field-strength renormalization constant. To one loop, the following quantities are often used in place of :math:`Z_{\psi}` and :math:`M_{B}`: .. math:: H_{0}=3\pi Z_{\psi}/\alpha_{s}\ ,\qquad K_{0}=\pi M_{B} Z_{\psi}/\alpha_{s}\ . The function :math:`p^{2}+\mathcal{M}^{2}(p^{2})` is also denoted by :math:`Q(p^{2})`, .. math:: Q(p^{2})=p^{2}+\mathcal{M}^{2}(p^{2})\ . The vector and scalar components of the functions :math:`\Sigma^{(1)}(p^{2})`, :math:`\Sigma^{(2)}(p^{2})` and :math:`\Sigma^{(gl)}(p^{2})` are defined in the module :python:`PySmeQcd.oneloop.quark`, together with the functions :math:`A(p^{2})`, :math:`B(p^{2})`, :math:`Z(p^{2})`, :math:`\mathcal{M}(p^{2})`, :math:`Q(p^{2})`, the vector and scalar components :math:`S_{V/S}(p^{2})` of the quark propagator, .. math:: S(p)=i\slashed{p}\,S_{V}(p^{2})+S_{S}(p^{2})\ , and the vector and scalar components :math:`\rho_{S_{V/S}}(p^{2}_{M})` of the quark spectral function in Minkowski space, .. math:: \rho_{S_{V/S}}(p^{2}_{M})=2\, \mathrm{Im} \{S_{V/S}(-p_{M}^{2}-i\varepsilon)\}\ . References ---------- 1. `F. Siringo, Nucl. Phys. B 907, 572 (2016) `_ 2. `F. Siringo, Phys. Rev. D 94, 114036 (2016) `_ 3. `F. Siringo, EPJ Web Conf. 137, 03021 (2017) `_ 4. `F. Siringo, EPJ Web Conf. 137, 13016 (2017) `_ 5. `F. Siringo, EPJ Web Conf. 137, 13017 (2017) `_ 6. `F. Siringo, Phys. Rev. D 96, 114020 (2017) `_ 7. `G. Comitini, arXiv:1803.02335 (2018) `_ 8. `G. Comitini and F. Siringo, Phys. Rev. D 97, 056013 (2018) `_ 9. `F. Siringo and G. Comitini, Phys. Rev. D 98, 034023 (2018) `_ 10. `F. Siringo, Phys. Rev. D 99, 094024 (2019) `_ 11. `G. Comitini, arXiv:1910.13022 (2019) `_ 12. `F. Siringo, Phys. Rev. D 100, 074014 (2019) `_ 13. `G. Comitini and F. Siringo, Phys. Rev. D 102, 094002 (2020) `_ 14. `F. Siringo and G. Comitini, Phys. Rev. D 103, 074014 (2021) `_ 15. `G. Comitini, D. Rizzo, M. Battello, and F. Siringo, Phys. Rev. D 104, 074020 (2021) `_