Phenomenological Lagrangian approach to two kaon photoproduction and pentaquark searches

We examine cross sections for the processes $\gamma N\to NK\overline{K}$ in the framework of a phenomenological Lagrangian. We include contributions from $\Lambda$ and $\Sigma$ resonances up to spin $3∕2$, as well as those from an exotic ${\Theta }^{+}$. We allow the ${\Theta }^{+}$ to have spin $1∕2$ or $3∕2$, with either positive or negative parity in each case. We also allow the state to be either isovector or isoscalar. We find that the scenario that most closely matches observations at Jefferson Laboratory requires a moderately large coupling of the ${\Theta }^{+}$ to $N{K}^{*}$.

Figure 1

“Born” diagrams: continuous, unlabeled lines are nucleons. Unlabeled dashed lines are kaons and wavy lines are photons.

Figure 2

Diagrams containing vector mesons. The dotted lines represent the vector mesons.

Figure 3

Diagrams containing excited baryons. In (a)–(d), the thick solid lines may be either ${\Lambda }^{*}$, ${\Sigma }^{*}$, or $\Theta$, while the thin solid line is a nucleon. In diagram (d), the photon couples to the charge of the intermediate resonance: in this model, we neglect couplings to any higher moments of the resonance.

Figure 4

The differential cross section $\partial \sigma ∕\partial M_{NK}^2$ as a function of $M_{NK}$, for a spin-$1∕2$ ${\Theta }^{+}$. The curves in (a) and (b) are for the process $\gamma n\to n{K}^{+}{K}^{-}$, while the curve in (c) is for $\gamma p\to p{K}^{+}{K}^{-}$. In (a) and (b), the solid curves arise for a pentaquark with a width of $1\mathrm{MeV}$, while the dashed curves correspond to a width of $10\mathrm{MeV}$. The curves in (a) arise from a pentaquark with positive parity, while those in (b) are for a pentaquark of negative parity.

Figure 5

The differential cross section $\partial \sigma ∕\partial M_{N\overline{K}}^2$ as a function of $M_{N\overline{K}}$ [(a) and (b)], and $\partial \sigma ∕\partial M_{K\overline{K}}^2$ as a function of $M_{K\overline{K}}$ [(c) and (d)]. The curves in (a) and (c) are for the process $\gamma n\to n{K}^{+}{K}^{-}$, while those in (b) and (d) are for $\gamma p\to p{K}^{+}{K}^{-}$.

Figure 6

The differential cross section $\partial \sigma ∕\partial M_{NK}^2$ as a function of $M_{NK}$. The curves in (a) and (c) are for the process $\gamma p\to p{K}^0{\overline{K}}^0$, while those in (b) and (d) are for $\gamma p\to n{K}^{+}{\overline{K}}^0$. (a) and (b) are for a positive parity ${\Theta }^{+}$ while (c) and (d) are for a ${\Theta }^{+}$ with negative parity.

Figure 7

The differential cross section $\partial \sigma ∕\partial M_{NK}^2$ as a function of $M_{NK}$, for a spin-$3∕2$ ${\Theta }^{+}$. The curves in (a) and (b) are for the process $\gamma n\to n{K}^{+}{K}^{-}$, while (c) is for $\gamma p\to p{K}^{+}{K}^{-}$. In (a), the ${\Theta }^{+}$ has positive parity, while in (b), its parity is negative.

Figure 8

The differential cross section as functions of $M_{N\overline{K}}$ [(a) and (b)] or $M_{K\overline{K}}$ [(c) and (d)]. The curves in (a) and (c) are for the process $\gamma n\to n{K}^{+}{K}^{-}$, while those in (b) and (d) are for $\gamma p\to p{K}^{+}{K}^{-}$.

Figure 9

The differential cross section $\partial \sigma ∕\partial M_{NK}^2$ as a function of $M_{NK}$. The upper graphs are for the process $\gamma p\to p{K}^0{\overline{K}}^0$, while the lower ones are for $\gamma p\to n{K}^{+}{\overline{K}}^0$.

Figure 10

The differential cross sections $\partial \sigma ∕\partial M_{NK}^2$ as a function of $M_{NK}$, for a spin-$1∕2$ isovector ${\Theta }^{+}$, for the processes $\gamma n\to n{K}^{+}{K}^{-}$ [(a) and (c)], $\gamma p\to p{K}^{+}{K}^{-}$ [(b) and (d)], $\gamma p\to p{K}^0{\overline{K}}^0$ [(e) and (g)], and $\gamma p\to n{K}^{+}{\overline{K}}^0$ [(f) and (h)]. The curves in (a), (b), (e), and (f) all arise from a positive-parity ${\Theta }^{+}$, while those in (c), (d), (g), and (h) all correspond to a ${\Theta }^{+}$ of negative parity.

Figure 11

The differential cross section $\partial \sigma ∕\partial M_{NK}^2$ as a function of $M_{NK}$, for a spin-$1∕2$ ${\Theta }^{+}$. The curves in (a) and (b) are for the process $\gamma n\to n{K}^{+}{K}^{-}$, while (c) is for $\gamma p\to p{K}^{+}{K}^{-}$. In addition, (a) is for a ${\Theta }^{+}$ of positive parity, while (b) is for one of negative parity.

Figure 12

The differential cross section $\partial \sigma ∕\partial M_{N\overline{K}}^2$ as a function of $M_{N\overline{K}}$ [(a) and (b)] and $\partial \sigma ∕\partial M_{K\overline{K}}^2$ as a function of $M_{K\overline{K}}$ [(c) and (d)]. The curves in (a) and (c) are for the process $\gamma n\to n{K}^{+}{K}^{-}$, while those in (b) and (d) are for $\gamma p\to p{K}^{+}{K}^{-}$.

Figure 13

The differential cross section $\partial \sigma ∕\partial M_{NK}^2$ as a function of $M_{NK}$. The curves in (a) and (c) are for the process $\gamma p\to p{K}^0{\overline{K}}^0$, while those in (b) and (d) are for $\gamma p\to n{K}^{+}{\overline{K}}^0$. (a) and (b) are for a positive parity ${\Theta }^{+}$ while (c) and (d) are for a ${\Theta }^{+}$ with negative parity.

Figure 14

The differential cross section $\partial \sigma ∕\partial M_{NK}^2$ as a function of $M_{NK}$, for a spin-$3∕2$ ${\Theta }^{+}$. The curves in (a) and (b) are for the process $\gamma n\to n{K}^{+}{K}^{-}$, while (c) is for $\gamma p\to p{K}^{+}{K}^{-}$. In (a), the ${\Theta }^{+}$ has positive parity, while in (b), its parity is negative.

Figure 15

The differential cross section as functions of $M_{N\overline{K}}$ [(a) and (b)] or $M_{K\overline{K}}$ [(c) and (d)]. The curves in (a) and (c) are for the process $\gamma n\to n{K}^{+}{K}^{-}$, while those in (b) and (d) are for $\gamma p\to p{K}^{+}{K}^{-}$.

Figure 16

The differential cross section $\partial \sigma ∕\partial M_{NK}^2$ as a function of $M_{NK}$. The upper graphs are for the process $\gamma p\to p{K}^0{\overline{K}}^0$, while the lower ones are for $\gamma p\to n{K}^{+}{\overline{K}}^0$. Those on the left are for a ${\Theta }^{+}$ with positive parity, while those on the right assume negative parity for the pentaquark.