QCD sum rules for the anticharmed pentaquark

We present a QCD sum rule analysis for the anticharmed pentaquark state with and without strangeness. While the sum rules for most of the currents are either nonconvergent or dominated by the $DN$ continuum, the one for the nonstrange pentaquark current composed of two diquarks and an antiquark is convergent and has a structure consistent with a positive-parity pentaquark state after subtracting out the $DN$ continuum contribution. Arguments are presented on the similarity between the result of the present analysis and that based on the constituent quark models, which predict a more stable pentaquark state when the antiquark is heavy.

Figure 1

Schematic OPE diagrams for the current ${\Theta }_{c1}$ in Eq. (1). Each label corresponds to that in Eq. (24). The solid lines denote quark (or anticharm quark) propagators and the dashed lines are for the gluon. The crosses denote the quark condensate, and the crosses with circles represent the mixed quark-gluon condensates. (c) represents diagrams proportional to a gluon condensate with gluons lines attached to the light quarks only, (d) represents those where the gluons are attached to the heavy quarks only, while (e) represents those where one gluon is attached to the heavy quark and the other to a light quark in all possible ways. (f) and (g) represent all diagrams that contain the quark-gluon condensate.

Figure 2

OPE as defined in Eq. (25) for the current ${\Theta }_{c1}$ and $S_0=(3.3\mathrm{GeV}{)}^2$. The left (right) figure is for the positive- (negative-) parity case. The solid line (a) represents the perturbative contribution. The line specified as OPE represents the sum of the power corrections only. (c) represents the gluon condensates. Other labels represent the contribution from each term in Eq. (24). Here we plot only a few selected terms in the OPE.

Figure 3

A similar figure as Fig. 2 for the current ${\Theta }_{c2}$. Here each label represents the contribution from each term in Eq. (27). The gluon condensates (b) are the dominant power correction in the positive-parity channel (left figure).

Figure 4

Schematic OPE diagrams for the currents ${\Theta }_{cs1}$ in Eq. (28) and ${\Theta }_{cs2}$ in Eq. (29). Each label corresponds to that in Eq. (28) or Eq. (29). All the other notations in this figure are the same as Fig. 1.

Figure 5

A similar figure as Fig. 2 for the current ${\Theta }_{cs1}$ with $S_0=(3.3\mathrm{GeV}{)}^2$. Here each label represents a contribution from each term in Eq. (28).

Figure 6

A similar figure as Fig. 2 for the current ${\Theta }_{cs2}$ with $S_0=(3.3\mathrm{GeV}{)}^2$. Here each label represents each term in Eq. (29).

Figure 7

The $|{\lambda }_{DN,c2}{|}^2$ from the sum rule for $i=q$ (solid lines) and $i=1$ (dashed lines). The upper (lower) solid or dashed lines in this case are for $S_0=(3.8\mathrm{GeV}{)}^2$ [$S_0=(3.7\mathrm{GeV}{)}^2$].

Figure 8

The left figure shows the left-hand side of Eq. (32) using ${\Theta }_{c2}$ for the positive-parity case with $|{\lambda }_{DN,c2}{|}^2=2\times {10}^{-5}{\mathrm{GeV}}^{10}$ (dashed line) and $|{\lambda }_{DN,c2}{|}^2=3\times {10}^{-5}{\mathrm{GeV}}^{10}$ (dot-dashed line). The solid line is when there is no $DN$ continuum, $|{\lambda }_{DN,c2}{|}^2=0$. The right figure is obtained with different threshold parameters. See Eq. (8) for $A$ and $B$.

Figure 9

The left figure shows the left-hand side of Eq. (32) using ${\Theta }_{c2}$ for the negative-parity case with $|{\lambda }_{DN,c2}{|}^2=2\times {10}^{-5}{\mathrm{GeV}}^{10}$ (dashed line) and $|{\lambda }_{DN,c2}{|}^2=3\times {10}^{-5}{\mathrm{GeV}}^{10}$ (dot-dashed line). The solid line is when there is no $DN$ continuum. The right figure is obtained with different threshold parameters.

Figure 10

The mass obtained by taking the square root of the inverse ratio between the left-hand side of Eq. (32) and its derivative with respect to $M^2$ using ${\Theta }_{c2}$.