Wilson coefficients and four-quark condensates in QCD sum rules for medium modifications of $D$ mesons

Wilson coefficients of light four-quark condensates in QCD sum rules are evaluated for pseudoscalar $D$ mesons, thus pushing the sum rules toward mass dimension 6. In contrast to the situation for $\overline{\mathit{q}}\mathit{q}$ mesons, the impact of the four-quark condensates for vacuum as well as in-medium situations is found to be rather small within the Borel window used in previous analyses. The complete four-quark condensate contributions enable us to identify candidates for an order parameter of spontaneous chiral symmetry breaking and/or restoration as well as to evaluate stability criteria of operator product expansions.

Figure 1

Diagrammatic representation of the contribution (7a) to the current-current correlator, where a selection of topologically relevant diagrams is displayed. Solid lines stand for free quark propagators, wiggly lines are for gluons, and crosses symbolize local quark or gluon condensation. In our expansion scheme, we retain only the diagrams (a)–(c) contributing to the perturbative term $C_0$ (a) and yielding the Wilson coefficients $c_3^{\text{even}}$ and $c_4^{\text{even}}$ [(b) and (c)] in Eq. (3a) since the other ones [(d)–(h)] are of higher order in $g$.

Figure 2

Diagrammatic representation of the contribution (7b), where a selection of topologically relevant diagrams are displayed as in Fig. 1, too.

Figure 3

Diagrammatic representation of hard-gluon contributions giving tree-level four-quark condensate terms for the correlator in next-to-leading-order (NLO) perturbation theory ${\alpha }_s^1$.

Figure 4

Various contributions to the even OPE part of the $\mathit{qQ}$ sum rule at saturation density. The red solid curve depicts the perturbative term subtracted by the continuum contribution of the phenomenological side of the sum rule according to quark-hadron duality, i.e., ${\overline{C}}_0=C_0-\frac{1}{\pi }({\int }_{-\infty }^{{\omega }_0^{-}}+{\int }_{{\omega }_0^{+}}^{+\infty })d\omega \omega e^{-{\omega }^2/M^2}\text{Im}{C}_0\left({\omega }^2\right)=\frac{1}{\pi }{\int }_{m_Q^2}^{{\omega }_0^2}ds{e}^{-s/M^2}\text{Im}{C}_0\left(s\right)$, where we assume ${\omega }_0^{+}=-{\omega }_0^{-}={\omega }_0$ for the second equality with the symmetric continuum threshold parameter ${\omega }_0^2=6{\mathrm{GeV}}^2$ (see [14]). The green dashed curve is the modulus of the power correction $e^{-m_Q^2/M^2}{\sum }_{k=1}^6{c}_k^{\text{even}}{\langle {\mathcal{O}}_k\rangle }^{\text{even}}$ of the in-medium OPE (3a) up to mass dimension 5 according to [14]. The contribution of the chiral condensate, i.e., $-e^{-m_Q^2/M^2}{m}_Q\langle \overline{\mathit{q}}\mathit{q}\rangle$, is shown by the blue dotted curve. The modulus of the four-quark condensate contribution ${\widehat{\Pi }}_{\text{4q-dom}}^{\text{even}}$ (14a) is displayed by the thick solid black curve. For the curves with sign flips in the depicted region the left (right) branch of the green dashed (thick solid black) curve originates from negative values.

Figure 5

Modulus of the individual contributions to the even (left panel) and the odd (right panel) OPEs (3) at saturation density up to mass dimension 5 according to [14], supplemented by the four-quark condensate contributions exhibited by the solid black curves and the contours with ${\kappa }_1={\kappa }_2/3={\kappa }_3\in [0.5,2]$. Using the notation of the contributions according to the condensates ${\langle {\mathcal{O}}_k\rangle }^{\text{even,odd}}$ listed below (3) for the even OPE on the left panel, the following line code applies: ${\langle {\mathcal{O}}_2\rangle }^{\text{even}}$–red solid curve, ${\langle {\mathcal{O}}_3\rangle }^{\text{even}}$–green dashed curve, ${\langle {\mathcal{O}}_4\rangle }^{\text{even}}$–blue dotted curve, ${\langle {\mathcal{O}}_5\rangle }^{\text{even}}$–magenta dot-dashed curve, and ${\langle {\mathcal{O}}_6\rangle }^{\text{even}}$–cyan dot-dot-dashed curve. For the curves with sign flips in the depicted region, the left branches of the green dashed and cyan dot-dot-dashed curves and the right branch of the black solid curve originate from negative values. On the right panel the red solid curve depicts the ${\langle {\mathcal{O}}_1\rangle }^{\text{odd}}$ contribution, the green dashed curve displays the sign flipped ${\langle {\mathcal{O}}_2\rangle }^{\text{odd}}$ contribution, and the blue dotted curve shows the ${\langle {\mathcal{O}}_3\rangle }^{\text{odd}}$ contribution, where the right branch originates from negative values.

Figure 6

Comparison of $\rho$ meson (left panels) and $D$ meson (right panels) contributions in vacuum (upper panels) and at nucleon saturation density (lower panels). Dashed green curves denote the gluon condensate contributions and thick solid black curves display the modulus of light four-quark condensate contributions containing the soft gluon diagrams (d) and (e) in Fig. 2. The modulus of the four-quark condensate contribution of the $\rho$ meson from hard-gluon diagrams in Fig. 3 is depicted by the dotted red line.