A Faddeev Calculation for Pentaquark $\Theta^+$ in Diquark Picture with Nambu-Jona-Lasinio Type Interaction

TL;DR

This study uses a Bethe-Salpeter-Faddeev approach with NJL interactions to investigate the possibility of a bound pentaquark state, finding it unlikely under realistic coupling constants.

Contribution

It applies a three-body Faddeev calculation with NJL model to the pentaquark $ heta^+$ in a diquark framework, providing new insights into its binding conditions.

Findings

${ar s}D$ interaction is attractive

$DD$ interaction is repulsive

No bound $ heta^+$ state with realistic couplings

Abstract

A Bethe-Salpeter-Faddeev (BSF) calculation is performed for the pentaquark $\Theta^+$ in the diquark picture of Jaffe and Wilczek in which $\Theta^+$ is a diquark-diquark-${\bar s}$ three-body system. Nambu-Jona-Lasinio (NJL) model is used to calculate the lowest order diagrams in the two-body scatterings of ${\bar s}D$ and $D D$. With the use of coupling constants determined from the meson sector, we find that ${\bar s}D$ interaction is attractive while $DD$ interaction is repulsive, and there is no bound $\frac 12^+$ pentaquark state. A bound pentaquark $\Theta^+$ can only be obtained with unphysically strong vector mesonic coupling constants.

Figures

Equations

$${\mathcal{L}}_{I,q\bar{q}}$$

$${\mathcal{L}}_{I,s}=G_s\left[\bar{\psi }({\gamma }^{5}C){\lambda }_f^2{\beta }_c^A{\bar{\psi }}^T\right]\left[{\psi }^T(C^{-1}{\gamma }^5){\lambda }_f^2{\beta }_c^A\psi \right],$$

$$<{\bar{s}}_f{D}_f|V|{\bar{s}}_i{D}_i>$$

$${V}_{{\bar{s}}D}=\frac{64}{3}G_{v}F_{v}(q^{2})\tilde{V}_{{\bar{s}}D}(p_{Di},p_{Df}),\hskip 28.45274pt\tilde{V}_{{\bar{s}}D}({p}_{Di},{p}_{Df})=(\mbox{$\!\not\!p$}_{Di}+\mbox{$\!\not\!p$}_{Df})/2$$

$$-i{V}_{DD}({\vec{p}}_{Di},{\vec{p}}_{Df})=128i\left[G_1{F}_s^2(q^2)-G_5(p_{D1i}+p_{D1f})\cdot (p_{D2i}+p_{D2f})F_v^2(q^2)\right],$$

$$T_i(s)=t_i(s)+t_i(s)G_0(s)\left[T_j(s)+T_k(s)\right],$$

Full text

1 Introduction

Experimental situation on the existence of $\Theta^{+}$ is currently filled with conflicting positive and negative evidence and the question is still not yet fully settled [1].

Even if it turns out that $\Theta^{+}$ does not exist, it would still be theoretically interesting to understand the absence of such an exotic. Many theoretical approaches have been used, including quark models, QCD sum rules, and lattice QCD, in addition to the chiral soliton model, to understand the properties and structure of $\Theta^{+}$ since its first sighting [2]. One of the most intriguing theoretical ideas suggested for $\Theta^{+}$ is the diquark picture of Jaffe and Wilczek (JW) [3] in which $\Theta^{+}$ is considered as a three-body system consisted of two scalar, isoscalar, color $\bf{\bar{3}}$ diquarks ($D$’s) and a strange antiquark $(\bar{s})$. It is based, in part, on group theoretical consideration. It would hence be desirable to examine such a scheme from a more dynamical perspective.

It is known that diquark arises naturally from Nambu-Jona-Lasinio (NJL) model, an effective quark theory in the low energy region [4]. NJL model conveniently incorporates chiral symmetry and its spontaneously breaking which dictates the hadronic physics at low energy. Models based on NJL type of Lagrangians have been very successful in describing the low energy meson physics. Based on relativistic Faddeev equation, the NJL model has also been applied to the baryon systems [5, 6]. It has been shown that, using the quark-diquark approximation, one can explain the nucleon static properties and the qualitative features of the empirical valence quark distribution reasonably well [6]. Consequently, we will employ NJL model to describe the dynamics of a $(\bar{s}DD)$ three-particle system in Faddeev formalism. We use relativistic equations to describe both the three-particle and its two-particle subsystems, namely, the Bethe-Salpeter-Faddeev (BSF) equation [7] and Bethe-Salpeter (BS) equations. In practice, Blankenbecler-Sugar reduction scheme is used to reduce the four-dimensional integral equation into three-dimensional ones.

2 $\bf SU(3)_{f}$ NJL model and the diquark

The SU(3)f NJL model is a chirally symmetric four-fermi contact interaction Lagrangian. With the use of Fierz transformations, the original NJL interaction Lagrangian $\cal L_{I}$ can be rewritten, for the $q{\bar{q}}$ channel, as

[TABLE]

where $a=0\sim 8$, and $\lambda_{f}^{0}=\sqrt{\frac{2}{3}}I$. For later use, we define $G_{5}=G_{2}+\frac{2}{3}G_{v},$ with $G_{v}\equiv G_{3}+G_{4}.$ For the scalar, isoscalar diquark channel, interaction Lagrangian is given by

[TABLE]

where $\beta^{A}_{c}=\sqrt{\frac{3}{2}}\lambda^{A}(A=2,5,7)$ corresponds to one of the color ${\bar{3}}_{c}$ states. $C=i\gamma^{0}\gamma^{2}$ is the charge conjugation operator, and $\lambda^{\prime}s$ are the Gell-Mann matrices.

The constituent quark and diquark masses can be obtained from the gap equation and t-matrix of the diquark. Since we are only interested in a qualitatively study of the interactions inside $\Theta^{+}$, we will use the empirical values of the constituent quark masses $M_{u,d}=400$ MeV, $M_{s}=600$ MeV, and the diquark mass $M_{D}=600$ MeV as obtained in Ref. [8].

3 Two-body interactions for $\bf\bar{s}D$ and $DD$ channels

In the JW model for $\Theta^{+}$ [3], symmetry consideration requires that the the spatial wave function of the two scalar-isoscalar, color $\bf\bar{3}$ diquarks must be antisymmetric and the lowest possible state is $p$-state. Since $\Theta^{+}$ is of $J^{P}={\frac{1}{2}}^{+}$, $\bar{s}$ would be in relative $s$-wave to the $DD$ pair. Accordingly, we will consider only the configuration where $\bar{s}D$ and $DD$ are in relative $s$- and $p$-waves, respectively.

Fig. 1 shows the lowest order diagram, i.e., first order in ${\cal L}_{I,q{\bar{q}}}$ in ${\bar{s}}D$ scattering. Trace properties in Dirac and flavor space limit the vertex $\Gamma$ to only the vector-isoscalar term, $({\bar{\psi}}\gamma^{\mu}\lambda^{0}_{f}\psi)^{2}$. For the $DD$ interaction, the quark rearrangement diagram gives no contribution because of its color structure. The lowest order non-vanishing diagram of the first order in ${\cal L}_{I,q\bar{q}}$ is given in Fig. 2 where only the direct term is shown. The corresponding exchange diagram vanishes again because of the color structure.

With the use of the interaction Lagrangians of Eqs. (2.1-2.2), we obtain the following driving terms of Fig. 1 and 2, in the BS equations for $\bar{s}D$ and $DD$ two-particle systems,

[TABLE]

with

[TABLE]

and,

[TABLE]

where $p_{\bar{s}}$ and $p_{D}$ denote the four-momentum of the $\bar{s}$-quark and diquark etc. The $F_{v}$ and $F_{s}$ are the vector and scalar form factors of the scalar diquark. For simplicity, we will assume that both take the dipole form, $(1-q^{2}/\Lambda^{2})^{-2}$, with $\Lambda=0.84$ GeV. In the NJL model calculation with the Pauli-Villars cutoff [8], the coupling constants are related to the mesonic coupling constants by $G_{1}=G_{\pi}/2$, $G_{2}=G_{\rho}/2$ and $G_{5}=G_{\omega}/2$ which give $G_{v}=-0.78$ GeV*-2*. We remark that the sign of $G_{v}$ is definitely negative since omega meson is heavier than the rho meson.

The potential matrix elements of Eqs. (3-3.4) can then be used in the scattering equations obtained with the use of Blankenbecler-Sugar three-dimensional reduction scheme [7] for the BS equation for both the $\bar{s}D$ and $DD$ systems. The resultant scattering equations are solved to obtain the two-body $t-$matrix elements and the phase shifts.

4 Results for $\Theta^{+}$ and discussion

Our results for the phase shifts are shown in Fig. 3(a). We see that the s-wave phase shifts for $\bar{s}D$ is positive which indicates the interaction is attractive, while the p-wave $DD$ interaction is repulsive since their phase shift is negative. In Fig. 3(b) we show the $G_{v}$ dependence of the two-body $\bar{s}D$ binding energy. We see that with the type of interaction constructed in Sec. 3, a $\bar{s}D$ bound state begins to appear only when $G_{v}$ becomes less than $-5\sim-6$ Gev*-2*, far too negative as compared to the physical value of -0.78 Gev*-2* determined from the $\rho-\omega$ mass difference.

The three-body BSF equation [7] takes the same form as the nonrelativistic one,

[TABLE]

where $G_{0}$ is the free three-particle Green’s function and $t_{i}(s)$ is the two-particle $t-$matrix of particles $j$ and $k$ with $(i,j,k)$ being a cyclic permutation of $(1,2,3)$. If one uses the Blankenbecler-Sugar approximation for $G_{0}$ and the two-body $t-$matrix elements obtained in Sec. 3, then the homogeneous equation of Eq. (4.5) can be solved [9] to look for a possible three-body $\bar{s}DD$ bound state. We could not find a bound pentaquark in $J^{P}=\frac{1}{2}^{+}$ channel. However, we do see a pentaquark in $J^{P}=\frac{1}{2}^{-}$ channel when $G_{v}$ becomes less that $\sim-8.0$ GeV*-2*. Pentaquark binding energy $E_{B}(5q)$ grows from 77 to 505 MeV as $G_{v}$ decreases from -8.0 to -14.0 GeV*-2*. The effect of consierdable weaker attraction in the $J^{P}=1/2^{+}$ channel is caused by the spectator particle being in a p-wave state.

Acknowledgements

One of the authors (S.N.Y.) thanks the Yukawa Institute for Theoretical Physics at Kyoto University, for warm hospitality extended to him during the YKIS2006 on ”New Frontiers on QCD”.