QCD sum rules for the $\mathrm{\Delta }$ isobar in neutron matter

We study the properties of the $\mathrm{\Delta }$ isobar in the symmetric and asymmetric nuclear matter using the QCD sum rules approach based on the energy dispersion relation. Allowing for different continuum thresholds for the polarization tensors with different dimensions, we find stable masses for the $\mathrm{\Delta }$ in both the vacuum and the medium. Compared to the nucleon case, we find that the vector repulsion is smaller for the $\mathrm{\Delta }$ while the scalar attraction is similar (75 MeV vector repulsion and 200 MeV scalar attraction in the symmetric matter). The smaller vector repulsion can be understood using the Pauli principle and a constituent quark model. Also, the isospin dependence of the quasiparticle energy, which mainly comes from the vector self-energy, is quite weak. We also allow for an explicit $\pi -N$ continuum contribution to the polarization function but find its effect to be minimal. Phenomenological consequences of our results are discussed.

Figure 1

Borel window estimated from ${\overline{\mathcal{W}}}_M^{\mathrm{subt}.}\left[{\mathrm{\Pi }}_{\mathrm{\Delta },q}\left(q_0^2,\left|\vec{q}\right|\right)\right]$ (a) in vacuum and (b) in medium. Black dotted line represents 50%.

Figure 2

Borel curves for (a) the $\mathrm{\Delta }$ isobar pole residues in vacuum and (b) the quasi-$\mathrm{\Delta }$ isobar pole residues in symmetric medium at normal nuclear density. Units of vertical axis are ${\mathrm{GeV}}^2$.

Figure 3

Borel curves for (a) the $\mathrm{\Delta }$ isobar mass in vacuum and (b) the quasi-$\mathrm{\Delta }$ isobar self-energies in symmetric matter and (c) in neutron matter. Orange long-dashed lines represent reference values. Units of vertical axis are GeV.

Figure 4

$\pi -N$ continuum subtracted Borel curves for (a) the $\mathrm{\Delta }$ isobar pole residues in vacuum and (b) the quasi-$\mathrm{\Delta }$ isobar pole residues in medium. Units of vertical axis is ${\mathrm{GeV}}^2$ and $c_{\pi N}=0.15$.

Figure 5

$\pi -N$ continuum subtracted Borel curves for (a) the $\mathrm{\Delta }$ isobar mass in vacuum, (b) the quasi-$\mathrm{\Delta }$ isobar self energies in symmetric matter, and (c) in neutron matter. Orange long-dashed lines represent reference values. Units of vertical axis is $\mathrm{GeV}$ and and $c_{\pi N}=0.15$.

Figure 6

Anomalous correction of the interpolating current in one-loop order. Shaded ellipse in the dotted circle represents the diquark in the interpolating current.