Remarks on N_c dependence of decays of exotic baryons

TL;DR

This paper investigates how the decay widths of exotic baryons depend on the number of colors (N_c) using the Chiral Quark Soliton Model, and explores generalizations of baryon representations for any N_c.

Contribution

It provides a detailed analysis of N_c dependence in exotic baryon decays and extends baryon representation frameworks to arbitrary N_c.

Findings

Decay widths vary with N_c according to specific patterns.

Generalized baryon representations are formulated for any N_c.

Insights into the structure of exotic baryons across different N_c values.

Abstract

We calculate the N_c dependence of the decay widths of exotic eikosiheptaplet within the framework of Chral Quark Soliton Model. We also discuss generalizations of regular baryon representations for arbitrary N_c.

Figures

Equations

$${\hat{O}}_{\phi }^{(8)}=3\sum _{i=1}^3\left(G_0{D}_{\phi i}^{(8)}-G_1{d}_{ibc}{D}_{\phi b}^{(8)}{\hat{S}}_c-\frac{G_2}{\sqrt{3}}{D}_{\phi 8}^{(8)}{\hat{S}}_i\right)\times p_{\phi }^i.$$

$${\Gamma }_{B\to B^{\prime }+\phi }=\frac{1}{8\pi }\frac{p_{\phi }}{M{M}^{\prime }}\overline{{\mathcal{M}}^2}=\frac{1}{8\pi }\frac{p_{\phi }^3}{M{M}^{\prime }}\overline{{\mathcal{A}}^2}.$$

$$\mathcal{M}=\left\langle 8_{1/2},B^{\prime}\right|\hat{O}_{\varphi}^{(8)}\left|\overline{10}_{1/2},B\right\rangle=-\frac{3G_{\overline{10}}}{\sqrt{15}}\left(\begin{array}[]{cc}8&8\\ \varphi&B^{\prime}\end{array}\right|\left.\begin{array}[]{c}\overline{10}\\ B\end{array}\right)\times p_{\varphi}$$

$$G_{\overline{10}}=G_0-G_1-\frac{1}{2}{G}_2.$$

$$G_{\overline{10}}=G_0-\frac{N_c+1}{4}{G}_1-\frac{1}{2}{G}_2$$

$$G_0=-(N_c+2)G,G_1=-4G,G_2=-2G,G\sim N_c^{1/2}.$$

$$M_B^{(\mathcal{R})}=M_{\text{cl}}+\frac{1}{2I_1}S(S+1)+\frac{1}{2I_2}\left(C_2(\mathcal{R})-S(S+1)-\frac{N_c^2}{12}\right)+{\delta }_B^{(\mathcal{R})}.$$

$$C_2(\mathcal{R})=\frac{1}{3}\left(p^2+q^2+pq+3(p+q)\right)$$

$${\hat{H}}^{\prime }=\frac{N_c}{3}\sigma +\alpha D_{88}^{(8)})+\beta Y+\frac{\gamma }{\sqrt{3}}{D}_{8A}^{(8)}{\hat{J}}_A.$$

$$\alpha =-\frac{N_c}{3}(\sigma +\beta ),\beta =-m_s\frac{K_2}{I_2},\gamma =2m_s\left(\frac{K_1}{I_1}-\frac{K_2}{I_2}\right),\sigma =\frac{2}{N_c}\frac{m_s}{m_u+m_d}{\Sigma }_{\pi N}$$

$$i_{1,2}=3I_{1,2}/N_c\text{where}{i}_{1,2}\sim \mathcal{O}(N_c^0),\sigma ,\beta ,\gamma \sim \mathcal{O}(m_s{N}_c^0).$$

$$"8"=\left(1,(N_c-1)/2\right),"10"=\left(3,(N_c-3)/2\right),"\overline{10}"=\left(0,(N_c+3)/2\right),$$

$${\Delta }_{10-8}=\frac{3}{2I_1}\sim \mathcal{O}(1/N_c),{\Delta }_{\overline{10}-8}=\frac{N_c+3}{4I_2}\sim \mathcal{O}(1).$$

$${\Gamma }_{B\to B^{\prime }+\phi }\sim \frac{1}{N_c^2}\mathcal{O}(\overline{{\mathcal{A}}^2})\mathcal{O}(p_{\phi }^3).$$

$$p_{\pi }\simeq 225\text{MeV},p_K\simeq 268\text{MeV}.$$

$${\delta }^{(8)}$$

$$+\frac{\left(6\alpha +(N_c+6)\gamma \right)}{(N_c+3)(N_c+7)}\left(\frac{Y^2}{4}-I(I+1)\right)=3\sigma +2\beta -\sigma Y+\dots$$

$${\delta }^{(10)}$$

$$+\left(\beta +\frac{3(N_c-1)\alpha -\frac{5}{2}(N_c+3)\gamma }{(N_c+1)(N_c+9)}\right)Y=3\sigma +2\beta -\sigma Y+\dots$$

$${\delta }^{(\overline{10})}$$

$$+\left(\beta +\frac{6N_c\alpha -9\gamma }{2(N_c+3)(N_c+9)}\right)Y=5\sigma +4\beta -\sigma Y+\dots$$

$$M_{\Theta }^{(\overline{10})}-M_N^{(8)}$$

$$M_{\Delta }^{(10)}-M_N^{(8)}$$

$$"27"=\left(2,(N_c+1)/2\right)$$

$${\Delta }_{27_{3/2}-8}$$

$${\Delta }_{27_{3/2}-10}$$

$${\Delta }_{27_{3/2}-\overline{10}}$$

$$\mathcal{A}(B_{27_{3/2}}$$

$$\begin{array}[]{|c|rl|c|}\hline\cr\text{Decay}&\text{Large }N_{c}&\text{NRL}&\begin{array}[]{c}\text{Scaling}\\ \text{in NRL}\end{array}\\ \hline\cr 27_{3/2}\rightarrow 8_{1/2}&G_{27}=G_{0}-\frac{N_{c}-1}{4}G_{1}&=-3G&N_{c}^{1/2}\\ 27_{3/2}\rightarrow 10_{3/2}&F_{27}=G_{0}-\frac{N_{c}-1}{4}G_{1}-\frac{3}{2}G_{2}&=0&0\\ 27_{3/2}\rightarrow\overline{10}_{1/2}&E_{27}=G_{0}+G_{1}&=-(N_{c}+6)G&N_{c}^{3/2}\\ \hline\cr\end{array}$$

$$\mathcal{A}(B_{27_{1/2}}$$

$$\begin{array}[]{|c|rl|c|}\hline\cr\text{Decay}&\text{Large }N_{c}&\text{NRL}&\begin{array}[]{c}\text{Scaling}\\ \text{in NRL}\end{array}\\ \hline\cr 27_{1/2}\rightarrow 8_{1/2}&{H_{27}=G}_{0}{-}\frac{N_{c}+5}{4}{G_{1}+\frac{3}{2}G_{2}}&{=0}&0\\ 27_{1/2}\rightarrow 10_{3/2}&G_{27}^{\prime}={G}_{0}{-\frac{N_{c}+5}{4}G_{1}}&{=3G}&N_{c}^{1/2}\\ 27_{1/2}\rightarrow\overline{10}_{1/2}&H_{27}^{\prime}=G_{0}+\frac{2N_{c}+5}{2N_{c}+8}G_{1}+\frac{3}{2N_{c}+8}G_{2}&=-\frac{(N_{c}+3)(N_{c}+7)}{N_{c}+4}G&N_{c}^{3/2}\\ \hline\cr\end{array}$$

$$\begin{array}[]{|lcc||lcc|}\hline\cr\text{decay of }&\lx@intercol\hfil N_{c}\text{ scaling}\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\text{decay of }&\lx@intercol\hfil N_{c}\text{ scaling}\hfil\lx@intercol\vrule\lx@intercol\\ \;\;\Theta_{27_{3/2}}&\text{exact}&\text{NRL}&\;\;\Theta_{27_{1/2}}&\text{exact}&\text{NRL}\\ \hline\cr\rightarrow N_{8}+K&\mathcal{O}(1)&\mathcal{O}(1/N_{c}^{2})&\rightarrow N_{8}+K&\mathcal{O}(1)&0\\ \rightarrow\Delta_{10}+K&\mathcal{O}(1)&0&\rightarrow\Delta_{10}+K&\mathcal{O}(1)&\mathcal{O}(1/N_{c}^{2})\\ \rightarrow N_{\overline{10}}+K&\mathcal{O}(1/N_{c}^{3})&\mathcal{O}(1/N_{c}^{3})&\rightarrow N_{\overline{10}}+K&\mathcal{O}(1/N_{c}^{3})&\mathcal{O}(1/N_{c}^{3})\\ \rightarrow\Theta_{\overline{10}}+\pi&\mathcal{O}(1/N_{c}^{2})&\mathcal{O}(1/N_{c}^{2})&\rightarrow\Theta_{\overline{10}}+\pi&\mathcal{O}(1/N_{c}^{2})&\mathcal{O}(1/N_{c}^{2})\\ \hline\cr\end{array}$$

Full text

1 Introduction

One of the most puzzling results of the chiral quark-soliton model ($\chi$QSM) for exotic baryons consists in a very small hadronic decay width [(1)], governed by the decay constant $G_{\overline{10}}$. While the small mass of exotic states is rather generic for all chiral models [(1), (2), (3)] the smallness of the decay width appears as a subtle cancelation of three different terms that contribute to $G_{\overline{10}}$. Decay width in solitonic models [(4)] is calculated in terms of a matrix element $\mathcal{M}$ of the collective axial current operator corresponding to the emission of a pseudoscalar meson $\varphi$ [(1)] – see Ref. 5 for criticism of this approach:

[TABLE]

For notation see Ref. 1. Constants $G_{0,1,2}$ are constructed from the so called moments of inertia that are calculable in $\chi$QSM. The decay width is given as

[TABLE]

The “bar” over the amplitude squared denotes averaging over initial and summing over final spin (and, if explicitly indicated, over isospin).

For $B^{(\overline{10})}\rightarrow B^{\prime(8)}+\varphi$ for spin ”up” and $\vec{p}_{\varphi}=(0,0,p_{\varphi})$ we have

[TABLE]

and

[TABLE]

In order to have an estimate of the width (2) the authors of Ref. 1 calculated $G_{\overline{10}}$ in the nonrelativistic limit limit of $\chi$QSM and got $G_{\overline{10}}\equiv 0$. It has been shown that this cancelation between terms that scale differently with $N_{c}$ ($G_{0}\sim N_{c}^{3/2},\,G_{1,2}\sim N_{c}^{1/2}$) is in fact consistent with large $N_{c}$ counting Praszalowicz:2003tc , since

[TABLE]

where the $N_{c}$ dependence comes from the SU(3) Clebsch-Gordan coefficients calculated for large $N_{c}$. In the nonrelativistic limit (NRL):

[TABLE]

In this paper we ask whether the similar cancelation takes place for the decays of $27$ of spin $1/2$ and $3/2$. We also discuss the possible modifications of the $N_{c}$ dependence of the decay width due to the different choice of the large $N_{c}$ generalizations of regular SU(3) multiplets.

2 Baryons in large $N_{c}$ limit

Soliton is usually quantized as quantum mechanical symmetric top with two moments of inertia $I_{1,2}$:

[TABLE]

Here $S$ denotes baryon spin, $C_{2}(\mathcal{R})$ the Casimir operator for the SU(3) representation $\mathcal{R}=(p,q)$:

[TABLE]

and quantities $\delta_{B}^{(\mathcal{R})}$ denote matrix elements of the SU(3) breaking hamiltonian:

[TABLE]

Model parameters that can be found in Ref. 8

[TABLE]

scale with $N_{c}$ in the following way:

[TABLE]

Here $\Sigma_{\pi N}$ is pion-nucleon sigma term and $m_{q}$ denote current quark masses. Numerically $\sigma>\left|\beta\right|,\left|\gamma\right|$.

So far we have specified *explicit *$N_{c}$ dependence (10) that follows from the fact that model parameters are given in terms of the quark loop. Another type of the $N_{c}$ dependence comes from the constraint SU3SM that selects SU(3)${}_{\text{flavor}}$ representations $\mathcal{R}=(p,q)$ containing states with hypercharge $Y_{R}=N_{c}/3$. Therefore for arbitrary $N_{c}$ ordinary baryon representations have to be extended and one has to specify which states correspond to the physical ones. Usual choice largereps

[TABLE]

depicted in Fig. 1 corresponds – in the quark language – to the case when each time when $N_{c}$ is increased by 2, a spin-isospin singlet (but charged) $\overline{3}$ diquark is added, as depicted in Fig. 2.

Extension (11) leads to (5). It implies that mass differences between centers of multiplets scale differently with $N_{c}$:

[TABLE]

The fact that $\Delta_{\overline{10}-8}\neq 0$ in large $N_{c}$ limit triggered recently discussion on the validity of the semiclassical quantization for exotic states Pobylitsa:2003ju . Since in the chiral limit the momentum $p_{\varphi}$ of the outgoing meson scales according to (12), overall $N_{c}$ dependence of the decay width is strongly affected by its third power (2):

[TABLE]

Phenomenologically, however, scaling (12) is not sustained. Indeed, meson momenta in $\Delta$ and $\Theta$ decays are almost identical (assuming $M_{\Theta}^{(\overline{10})}\simeq 1540$ MeV):

[TABLE]

Unfortunately, going off SU(3)${}_{\text{flavor}}$ limit does not help. Explicitly:

[TABLE]

where $\ldots$ denote terms $\mathcal{O}(1/N_{c})$, $Y$ and $I$ denote physical hypercharge and isospin.

Interestingly in all cases in the large $N_{c}$ limt, $m_{s}$ splittings are proportional to the hypercharge differences only. In this limit $\Sigma-\Lambda$ splitting in the octet is zero and this degeneracy is lifted in the next order at $\mathcal{O}(1/N_{c})$. This explains the smallness of $\Sigma-\Lambda$ mass difference. Additionally $\delta_{N}^{(8)}\simeq\delta_{\Delta}^{(10)}$ up to higher order terms $\mathcal{O}(1/N_{c}^{2})$, however $\delta_{\Theta}^{(\overline{10})}-\delta_{N}^{(8)}$ $\simeq\sigma+2\beta>0$. This implies that

[TABLE]

The first equation shows that the $\Theta-N\neq 0$ in the large $N_{c}$ limit even if $m_{s}$ corrections are included. We will come back to this problem in the last section.

3 Decay constants of twentysevenplet for large $N_{c}$

In this section we shall consider decays of eikosiheptaplet (27-plet)

[TABLE]

that can have either spin $1/2$ or $3/2$, the latter being lighter. Mass differences read

[TABLE]

Matrix elements for the decays of eikosiheptaplet (with $S_{3}=1/2$) read:

[TABLE]

and

[TABLE]

For $S=1/2$ and $S_{3}=1/2$ we have:

[TABLE]

[TABLE]

In order to calculate the $N_{c}$ behavior of the width we have to know the $N_{c}$ dependence of the flavor Clebsch-Gordan coefficients that depend on the states involved. For the decays into 8 and 10 the only possible channels are $\Theta_{27}\rightarrow N(\Delta)+K$, and the pertinent Clebsches do not depend on $N_{c}$. For the decays into $\overline{10}$ we have $\Theta_{27}\rightarrow\Theta_{\overline{10}}+\pi$ that scales like $\mathcal{O}(1)$ and $\Theta_{27}\rightarrow N_{\overline{10}}+K$ that scales like $\mathcal{O}(1/\sqrt{N_{c}})$. The resulting scaling of $\mathit{\Gamma}_{\Theta_{27}\rightarrow B^{\prime}+\varphi}$ calculated from Eq.(13) reads as follows:

[TABLE]

Interestingly, we see that whenever the exact scaling is $\mathcal{O}(1)$, the nonrelativistic cancelation (exact or partial) lowers the power of $N_{c}$, whereas in the case when the width has good behavior for large $N_{c}$, there is no NRL cancelation.

4 Alternative choices for large $N_{c}$ multiplets

So far we have only considered the ”standard” generalization (11) of baryonic SU(3)${}_{\text{flavor}}$ representations for large $N_{c}$. This choice is based on the requirement that generalized baryonic states have physical spin, isospin and strangeness, however their hypercharge and charge are not physical largereps . Moreover, the generalization of the octet is not selfadjoint and antidecuplet is not complex conjugate of decuplet. Some years ago it has been proposed to consider alternative schemes dul .

If we require the generalized octet to be self-adjoint we are led to the following set of representations

[TABLE]

that are depicted in Figs. 3 and 4. This means that we enlarge $N_{c}$ in steps of $3$ adding each time a $uds$ triquark. Generalized states have physical isospin, hypercharge (and charge), but unphysical strangeness and spin that is of the order of $N_{c}$. With this choice both $\Delta_{10-8}$, $\Delta_{\overline{10}-8}\neq 0$ in large $N_{c}$ limit:

[TABLE]

With this power counting we can calculate large $N_{c}$ approximation of the meson momenta in the decays of $\Delta$ and $\Theta$:

[TABLE]

that are much closer to the physical values (14) than (12).

Finally let us mention a third possibility in which we require generalized decuplet to be a completely symmetric SU(3)${}_{\text{flavor}}$ representation for arbitrary $N_{c}$. This leads to (see Figs. 5 and 6):

[TABLE]

Interestingly this choice has a smooth limit to the one flavor case. In the quark language it amounts to adding a symmetric diquark to the original SU(3)${}_{\text{flavor}}$ representation when increasing $N_{c}$ in steps of $2$. As seen from Fig. 5 physical states are situated at the bottom of infinite representations (54) and therefore have unphysical strangeness, charge (hypercharge) and also spin.

The mass splittings for this choice read

[TABLE]

Here the generalized decuplet remains split from the $"8"$, while $\Delta_{\overline{10}-8}\rightarrow 0$ for large $N_{c}$. The phase space factor for $\Theta$ decay is therefore suppressed with respect to the one of $\Delta$.

5 Summary

In this short note we have shown that very small width of exotic baryons – if they exist – cannot be explained by the standard $N_{c}$ counting alone. Certain degree of nonrelativisticity is needed to ensure cancelations between different terms in the decay constants. This phenomenon observed firstly for antidecuplet, is also operative for the decays of eikosiheptaplet. We have shown that in $\chi$QSM in the nonrelativistic limit all decays are suppressed for large $N_{c}$. Exact cancelations occur for $\Theta_{27_{3/2}}\rightarrow\Delta_{10}+K$ and $\Theta_{27_{1/2}}\rightarrow N_{8}+K$, leading $N_{c}$ terms cancel for $\Theta_{27_{3/2}}\rightarrow N_{8}+K$ and $\Theta_{27_{1/2}}\rightarrow\Delta_{10}+K$. For $27\rightarrow\overline{10}$ there are no cancelations, but the phase space is $N_{c}^{-3}$ suppressed.

We have also briefly discussed nonstandard generalizations of regular baryon representations for arbitrary $N_{c}$. For $N_{c}>3$ bayons are no longer composed from 3 quarks and therefore they form large SU(3)${}_{\text{flavor}}$ representations that reduce to octet, decuplet and antidecuplet for $N_{c}=3$. The standard way to generalize regular baryon representations is to add antisymmetric antitriplet diqaurk when $N_{c}$ is increased in intervals of 2. This choice fulfils many reasonable requirements; most importantly for SU(2)${}_{\text{flavor}}$ these representations form regular isospin multiplets. However, representations (11) do not obey conjugation relations characteristic for regular representations. Therefore we have proposed generalization (51) that satisfies conjugation relations. Most important drawback of (51) is that spin $S\sim N_{c}$ that contradicts semiclassical quantization. Nevertheless as a result meson momenta emitted in $10$ and $\overline{10}$ decays scale in the same way with $N_{c}$ (53), consistently with ”experimental” values (14), whereas for (11) the scaling is different (12).

Acknowledgements

One of us (MP) is grateful to the organizers of the Yukawa International Symposium (YKIS2006) for hospitality during this very successful workshop.