---
abstract: 'Since LEPS collaboration reported the first evidence of $\Theta^+$ pentaquark in early 2003, eleven other experimental groups have confirmed this exotic state while many other groups didn’t see any signal. If this state is further established by future high statistical experiments, its discovery shall be one of the most important events in hadron physics for the past three decades. This exotic baryon with such a low mass and so narrow a width imposes a big challenge to hadron theorists. Up to now, there have appeared more than two hundred theoretical papers trying to interpret this charming state. I will review some important theoretical developments on pentaquarks based on my biased personal views.'
author:
- 'Shi-Lin Zhu'
title: Pentaquarks
---

[****]{}

-   I. Quark Model and Exotic Hadron Search

-   II\. Pentaquarks: Discovery or Fluctuations?

-   III\. Group Theory Of Pentaquarks

-   IV\. Theoretical Models of Pentaquarks

    -   Chiral soliton model

    -   Jaffe and Wilczek’s Diquark model

    -   Other clustered quark model

    -   Non-clustered quark model

    -   QCD sum rules

    -   Heptaquarks

-   V. Narrow width puzzle

-   VI\. Pentaquark flavor wave functions, masses and magnetic moments

-   VII\. Heavy flavored pentaquarks

-   VIII\. Chiral Lagrangian formalism for pentaquarks

    -   Mass splitting

    -   Selection rules from octet pentaquark decays

-   IX\. Diquarks, pentaquarks and dibaryons

-   X. Summary

Quark Model and Exotic Hadron Search {#sec1}
====================================

Quantum Chromodynamics (QCD) is believed to be the underlying theory of the strong interaction, which has three fundamental properties: asymptotic freedom, color confinement, approximate chiral symmetry and its spontaneous breaking. In the high energy regime, QCD has been tested up to $1\%$ level. In the low energy sector, QCD is highly nonperturbative due to the non-abelian SU$_c$(3) color group structure. It is very difficult to calculate the whole hadron spectrum from first principles in QCD. With the rapid development of new ideas and computing power, lattice gauge theory may provide the final solution to the spectrum problem in the future. But now, people have just been able to understand the first orbital and radial excitations of the nucleon with lattice QCD in the baryon sector [@liu].

Under such a circumstance, various models which are QCD-based or incorporate some important properties of QCD were proposed to explain the hadron spectrum and other low-energy properties. Among them, it is fair to say that quark model has been the most successful one. It is widely used to classify hadrons and calculate their masses, static properties and low-energy reactions [@isgur]. According to quark model, mesons are composed of a pair of quark and anti-quark while baryons are composed of three quarks. Both mesons and baryons are color singlets. Most of the experimentally observed hadrons can be easily accommodated in the quark model. Any state with the quark content other than $q\bar q$ or $q q q$ is beyond quark model, which is termed as non-conventional or exotic. For example, it is hard for $f_0(980)/a_0(980)$ to find a suitable position in quark model. Instead it could be a kaon molecule or four quark state [@pdg].

However, besides conventional mesons and baryons, QCD itself does not exclude the existence of the non-conventional states such as glueballs ($gg, ggg, \cdots$), hybrid mesons ($q\bar q g$), and other multi-quark states ($qq\bar q \bar q$, $qqqq\bar q$, $qqq\bar q \bar q \bar q$, $qqqqqq, \cdots$). In fact, hybrid mesons can mix freely with conventional mesons in the large $N_c$ limit [@cohen]. In the early days of QCD, Jaffe proposed the H particle [@jaffeold] with MIT bag model, which was a six quark state. Unfortunately it was not found experimentally.

In the past years there have accumulated some experimental evidence of possible existence of glueballs and hybrid mesons with exotic quantum numbers like $J^{PC}=1^{-+}$ [@pdg]. Recently BES collaboration observed a possible signal of a proton anti-proton baryonium in the $J/\Psi$ radiative decays [@bes]. But none of these states has been pinned down without controversy until the surprising discovery of pentaquarks by LEPS collaboration [@leps].

Pentaquarks: Discovery or Fluctuations? {#sec2}
=======================================

Early last year LEPS Collaboration at the SPring-8 facility in Japan observed a sharp resonance $\Theta^+$ at $1.54\pm 0.01$ GeV with a width smaller than 25 MeV and a statistical significance of $4.6\sigma$ in the reaction $\gamma n \to K^+ K^- n$ [@leps]. This resonance decays into $K^+ n$, hence carries strangeness $S=+1$. Later, many other groups have claimed the observation of this state [@diana; @clas; @saphir; @itep; @clasnew; @hermes; @svd; @cosy; @Yerevan; @zeus; @forzeus]. All known baryons with $B=+1$ carry negative or zero strangeness. Such a resonance is clearly beyond the conventional quark model with the minimum quark content $uudd\bar s$. Now it’s called $\Theta^+$ pentaquark in literature. A compilation of $\Theta^+$ mass and decay width is presented in Figure (\[mao\]).

NA49 Collaboration announced evidence for the existence of a new narrow $\Xi^- \pi^-$ baryon resonance $\Xi^{--}_5$ with mass of $(1.862\pm 0.002) $ GeV and width below the detector resolution of about 0.018 GeV in proton-proton collisions at $\sqrt{s}=17.2$ GeV [@na49]. The quantum number of this state is $Q=-2, S = -2, I
= 3/2$ and its quark content is $(d s d s \bar u)$. They also observed signals for the $Q=0$ member of the same isospin quartet with a quark content of $(d s u s \bar d)$ in the $\Xi^- \pi^+$ spectrum. The corresponding anti-baryon spectra also show enhancements at the same invariant mass. H1 Collaboration claimed the discovery of a heavy pentaquark around 3099 MeV with the quark content $udud\bar{c}$ [@H1]. Very recently, STAR collaboration at RHIC observed a narrow peak at $1734\pm 0.5\pm 5$ MeV in the $\Lambda K_s^0$ invariant mass which was interpreted as an $I={1\over 2}$ pentaquark [@star-rhic]. However its antiparticle was not observed yet.

There is preliminary evidence that the $\Theta^+$ is an iso-scalar because no enhancement was observed in the $pK^+$ invariant mass distribution [@saphir; @clasnew; @hermes; @forzeus]. The third component of its isospin is $I_z=0$. So the $\Theta^+$ pentaquark is very probably an iso-scalar if it is a member of the anti-decuplet. At present, the possibility of this state being a member of another multiplet is not completely excluded. Hence its total isospin is probably zero. Most of the theoretical models assume that $\Theta^+$ is in $SU(3)_f$ ${\bf\bar{10}}$ representation. All the other quantum numbers including its angular momentum and parity remain undetermined. However, most of theoretical work postulated its angular momentum to be one half because of its low mass. But the possibility of $J={3\over 2}$ still can not be excluded completely.

It is important to point out that many other experimental groups reported negative results [@bes1; @hera-b; @rhic]. For example, the existence of $\Xi^{--}_5$ is still under debate [@doubt]. Compared with 1640 $\Xi^-$ candidates produced in proton proton collisions in NA49’s analysis, WA89 collaboration found no signal of $\Xi^{--}$ pentaquark with $676000$ $\Xi^-$ candidates in their data sample [@wa89]. A long list of experiments yielding negative results including those unpublished can be found in Ref. [@long]. Although the $\Theta^+$ pentaquark has been listed as a three-star resonance in the 2004 PDG, its existence is still not completely established.

Group Theory Of Pentaquarks {#sec3}
===========================

One can use some textbook group theory to write down the pentaquark wave functions in the framework of quark model. Because of its low mass, high orbital excitation with $L\ge 2$ seems unlikely. Pauli principle requires the totally anti-symmetric wave functions for the four light quarks. Since the anti-quark is in the $[11]_C$ representation, the four quark color wave function is $[211]_C$.

With $L=0$, hence $P=-$, the 4q spatial wave function is symmetric, i.e., $[4]_O$. Their $SU(6)_{FS}$ spin-flavor wave function must be $[31]_{FS}^{210}$ which contains $[22]_F^6 \times
[31]_S^3$ after decomposition into $SU(3)_F\times SU(2)_S$ [@strot; @g1; @g2; @buccella]. Here 210, 6 etc is the dimension of the representation. When combined with anti-quark, we get $\left([33]_F^{10}+[21]_F^8\right)\times
\left([41]_S+[32]_S\right)$, which is nothing but $\left({\bar
{10}}_F+8_F\right)\times \left(({3\over 2})_S+({1\over
2})_S\right)$ in terms of more common notation. The total angular momentum of the four quarks is one. The resulting exotic anti-decuplet is always accompanied by a nearly degenerate octet. Their angular momentum and parity is either $J^P={3\over 2}^-$ or ${1\over 2}^-$.

With $L=1$ and $P=+$, the four quark $SU(6)_{OFS}$ space-spin-flavor wave function must be $[31]_{OFS}$. Only this representation can combine with $[211]_C$ color wave function to ensure that the 4q total wave functions are anti-symmetric. The 4q orbital wave function is $[31]_O$. There are four $SU(6)_{FS}$ wave functions $[4]_{FS}, [31]_{FS}, [22]_{FS}, [211]_{FS}$ which allow the 4q total wave function to be anti-symmetric and lead to the octet and exotic anti-decuplet [@g2; @buccella].

Theoretical Models of Pentaquarks {#sec4}
=================================

We are still unable to solve the hadron spectrum starting from first principles of QCD analytically. Lattice QCD simulation may play a very important role eventually. But right now, lattice simulation of pentaquarks by several groups has not converged yet. For example, one lattice calculation favors positive parity for pentaquarks [@chiu] while two previous lattice simulations favor negative parity [@lattice]. The most recent lattice simulation did not observe any bound pentaquark state in either $I=0, J^P={1\over 2}^\pm$ or $I=1, J^P={1\over 2}^\pm$ [@liukf]. We will not discuss lattice results in this review. Interested readers are referred to the above references.

In the field of pentaquarks, there are three outstanding pending issues: the $\Theta^+$ parity, the dynamical generation of its low mass and its narrow width. Up to now theoretical papers in the literature can roughly be classified into two types according to their assumption of $\Theta^+$ parity. Theoretical approaches advocating positive parity include the chiral soliton model (CSM) [@diak], the diquark model [@jaffe], some quark models [@lipkin; @riska; @hosaka; @carl; @liuyx], a lattice calculation [@chiu]. On the other hand, some other theoretical approaches tend to favor negative parity such as two lattice QCD simulations [@lattice], QCD sum rule approaches [@zhu; @qsr], several quark model study [@zhang; @carlson; @wu], and proposing stable diamond structure for $\Theta^+$ [@song]. There are many proposals to measure $\Theta^+$ parity either through doubly polarized proton proton scattering [@thomas] or photo-production processes [@proposal; @prod]. Now most of photo-production papers assume that pentaquarks are produced from the following four processes: s-channel, u-channel, t-channel through charged meson exchange and the contact seagull diagram. If such a mechanism is correct, then the production cross section of producing a positive parity $\Theta^+$ pentaquark is one or two orders larger than that of producing a negative parity $\Theta^+$. Moreover, the inclusion of charge vector meson in the t-channel increases the cross section significantly [@prod].

Since LEPS’s first experimental paper, there have appeared more than two hundred theoretical papers on pentaquarks. It’s both impossible and unnecessary to exhaust them in the present short review. In the following, I focus on several models to illustrate these pending issues.

Chiral soliton model
--------------------

Theoretical study of pentaquark states dated back to the early days of QCD using MIT bag model [@strot]. The recent interest in the $\Theta^+$ pentaquark started with the prediction of its mass, width and reaction channel from the chiral soliton model (CSM) in Ref. [@diak] a few years ago. Diakonov et al. proposed the possible existence of the $S=1$ $J^P={1\over 2}^+$ resonance at $1530$ MeV with a width less than 15 MeV using the chiral soliton model [@diak], which partly motivated the recent experimental search of this particle. The authors argued that $\Theta^+$ is the lightest member of the anti-decuplet multiplet in the third rotational state of the chiral soliton model. Assuming that the $N(1710)$ is a member of the anti-decuplet, $\Theta^+$ mass is fixed with the symmetry consideration of the model.

However this prediction is very sensitive to the model inputs [@diak; @mp; @ellis]. For example, either adopting the commonly used value $45$ MeV for the $\sigma$-term or identifying $N(1710)$ as a member of the anti-decuplet will lead to a $\Xi_5^{--}$ pentaquark mass $210$ MeV higher than that observed by NA49 Collaboration [@diak]. An updated reanalysis indicates $\Theta^+$ mass is really quite low within the framework of the chiral soliton model: 1430 MeV $\le m_{\Theta^+} \le$ 1660 MeV [@ellis]. For example, with the new larger value $(79\pm 7)$ MeV and $(64\pm 7)$ MeV from two recent analysis [@sigmaterm] for the $\sigma$-term, a fairly good description of both $\Theta^+$ and $\Xi^{--}$ masses is possible [@ellis]. In CSM, the narrow width comes from the cancellation between the coupling constants in the leading order, next-leading order and next-next-leading order large $N_c$ expansion.

However, the theoretical foundation of the treatment of the pentaquarks in the chiral soliton model is challenged by the large $N_c$ formalism in Refs. [@cohen2]. From the large $N_c$ consistency consideration, Cohen found that predictions for a light collective $\Theta^+$ baryon state (with strangeness +1) based on the collective quantization of chiral soliton models are shown to be inconsistent with large $N_c$ QCD since collective quantization is legitimate only for excitations which vanish as $N_c \to \infty$.

In the large $N_c$ limit, the mass splitting between the $\Delta$ decuplet and the nucleon octet is ${\cal O}(1/N_c)$ while the excitation energy of the vibration mode of the hedgehog is ${\cal
O}(1)$. Hence the rotational degree of freedom decouples from the vibration mode in the large $N_c$ limit. The collective quantization is valid for the octet and decuplet. The predictions for these two multiplets from the chiral soliton model is reliable.

In contrast, the mass splitting between the anti-decuplet and octet is also ${\cal O}(1)$. In other words, the rotation and vibration motions are not orthogonal. They will mix with each other, which invalidates the collective quantization of the rotational degrees of freedom. Therefore, Cohen concluded that the prediction of $\Theta^+$ properties based on collective quantization of CSM was fortuitous [@cohen2]. Large $N_c$ formalism doesn’t predict the existence of $\Theta^+$ pentaquark. However, large $N_c$ formalism may still be a useful tool to relate properties of different pentaquarks such as their mass splitting and decay width once one [*assumes*]{} the existence of pentaquarks [@cohen2; @jenkins]. If $\Theta^+$ is established by experiments, then one may use large $N_c$ expansion technique to predict the existence of the other members and their properties within the same multiplet.

The relationship between the bound state and the SU(3) rigid rotator approaches to strangeness in the Skyrme model was investigated in [@princeton]. It was found that the exotic state may be an artifact of the rigid rotator approach to the Skyrme model for large $N_c$ and small $m_K$.

Jaffe and Wilczek’s Diquark model
---------------------------------

Since early last year, there appeared many theoretical papers trying to interpret these exotic states. Among them, Jaffe and Wilczek’s (JW) diquark model is a typical one [@jaffe]. In their model, the $\Theta^+$ pentaquark is composed of a pair of diquarks and a strange anti-quark. The flavor anti-decuplet is always accompanied by an octet which is nearly degenerate and mix with the decuplet.

Jaffe and Wilczek’s (JW) proposed that there exists strong correlation between the light quark pair when they are in the anti-symmetric color $({\bar 3}_c)$, flavor $({\bar 3}_f)$, isospin $(I=0)$ and spin $(J=0)$ configuration [@jaffe]. The lighter the quarks, the stronger the correlation, which helps the light quark pair form a diquark. For example, the ud diquark behaves like a scalar with positive parity. Such correlation may arise from the color spin force from the one gluon exchange or the flavor spin force induced by the instanton interaction.

They proposed that pentaquark states are composed of two scalar diquarks and one anti-quark. Diquarks obey Bose statistics. Each diquark is in the antisymmetric color $\bf{\bar{3}}$ state. The spin wave function of the two quarks within each scalar diquark is antisymmetric while the spatial part is symmetric. Pauli principle requires the total wave function of the two quarks in the diquark be anti-symmetric. Thus the flavor wave function of the two quarks in the diquark must be antisymmetric, i.e, the diquark is in the flavor $\bf{\bar3_F}$ state.

The color wave function of the two diquarks within the pentaquark must be antisymmetric $\bf{3}_C$ so that the resulting pentaquark is a color singlet. In order to accommodate the exotic $\Theta^+$ pentaquark, Jaffe and Wilczek required the flavor wave function of the diquark pair to be symmetric ${\bf {\bar 6_f}}$: $[ud]^2$, $[ud][ds]_+$, $[su]^2$, $[su][ds]_+$, $[ds]^2$, and $[ds][ud]_+$. Simple group theory tells us that ${\bf {\bar 6_f}}\times {\bf
{\bar 3_f}}={\bf {{\bar 10}_f}}+ {\bf 8_f}$. In other words, the pentaquark anti-decuplet is always accompanied by an octet which is nearly degenerate and probably mixes with the anti-decuplet. Bose statistics demands symmetric total wave function of the diquark-diquark system, which leads to the antisymmetric spatial wave function with one orbital excitation. The resulting anti-decuplet and octet pentaquarks have $J^P={1\over 2}^+,
{3\over 2}^+$.

In general, the lower the angular mentum, the lower the mass. So the $J={3\over 2}$ pentaquark will be heavier than its $J={1\over
2}$ partner. But their mass difference is not expected to larger than $300$ MeV if we could rely on the past experience with the $\Delta$ and nucleon mass splitting. If the $\Theta^+$ pentaquark does exist and the diquark model is correct, then its $J={3\over
2}$ pentaquark partner should also be reachable by future experiments.

The members of pentaquark anti-decuplet are $P_{333}=\Theta^+$, $
P_{133}=\frac{1}{\sqrt{3}}N^0_{10}$, $P_{233}=-\frac{1}{\sqrt{3}}N^+_{10}$, $P_{113}
=\frac{1}{\sqrt{3}}\Sigma^-_{10}$, $P_{123}=-\frac{1}{\sqrt{6}}\Sigma^0_{10}$, $P_{223}=\frac{1}{\sqrt{3}}\Sigma^+_{10}$, $P_{111}=\Xi^{--}_{10}$, $P_{112} =-\frac{1}{\sqrt{3}}\Xi^-_{10}$, $P_{122}=\frac{1}{\sqrt{3}}\Xi^0_{10}$ and $P_{222}=-\Xi^+_{10}$.

Jaffe and Wilczek pointed out that one of the decay modes $\Xi_5\to \Xi^\ast + \pi$ observed by NA49 [@na49] signals the existence of an additional octet around $1862$ MeV together the anti-decuplet since the latter can not decay into a decuplet and an octet in the $SU(3)_f$ symmetry limit. If further confirmed, this experiment poses a serious challenge to the chiral soliton model because there is no baryon pentaquark octet in the rotational band in this model. But it may be hard to exclude it since there always exist excited vibrational octet modes. These modes can not be calculated rigourously within the chiral soliton model [@ellis].

However, in JW’s model $\Theta^+$ is not the lightest pentaquark as in the chiral soliton model. The ideal mixing between the octet and anti-decuplet will split the spectrum and produce two nucleon-like states. Using the $\Theta^+$ mass as input, the higher one $N_s$ is around 1710 MeV with a quark content $qqq \bar
s s$ where $q$ is the up or down quark. The lighter one $N_l$ is lower than the $\Theta^+$ pentaquark with a quark content $qqq
\bar q q$. Jaffe and Wilczek identified $N_l$ as the well-known Roper resonance $N(1440)$ [@jaffe], which is a very broad four-star resonance with a width around $(250 \sim 450)$ MeV. However, it will be very demanding to explain Roper’s large decay width and $\Theta^+$’s extremely narrow width simultaneously in a natural way. The problem with the identification of the ideally mixed positive parity pentaquarks with the $N(1710)$ and $N(1440)$ is discussed extensively in [@wrong].

If Jaffe and Wilczek’s diquark picture for $\Theta_5^+$ pentaquark is correct, we pointed out [@zhanga] that lighter pentaquarks can be formed if the two scalar diquarks are in the antisymmetric $SU(3)_F$ $\bf 3$ representation: $[ud][su]_-$, $[ud][ds]_-$, and $[su][ds]_-$, where $[q_1q_2][q_3q_4]_-=\sqrt{\frac{1}{2}}([q_1q_2][q_3q_4]-[q_3q_4][q_1q_2])$. No orbital excitation is needed to ensure the symmetric total wave function of two diquarks since the spin-flavor-color part is symmetric. The total angular momentum of these pentaquarks is $\frac{1}{2}$ and the parity is negative. There is no accompanying $J={3\over 2}$ multiplet. The two diquarks combine with the antiquark to form a $SU(3)_F$ pentaquark octet and singlet: ${\bar
3}_F \otimes 3_F = \bf{8}_F$ $\oplus$ $\bf{1}_F$. Similar mechanism has been proposed to study heavy pentaquarks with negative parity and lighter mass than $\Theta_{c,b}$ in [@wise].

Because there is no orbital excitation within these nine pentaquarks, their masses are lower than the anti-decuplet and the accompanying octet with positive parity. Their masses range between 1360 MeV and 1540 MeV according to our calculation using the same mass formula in [@jaffe]. These states are close to the L=1 orbital excitations of the nucleon octet. The mixing between the pentaquark states and orbital excitations is expected to be small since their spatial wave functions are very different.

According to our calculation, two of the $J^P={1\over 2}^-$ octet pentaquark members $p_8, n_8$ lie 22 MeV below the $p\eta_0$ threshold and 228 MeV below the $\Sigma K$ threshold. The “fall-apart” decay mechanism forbids $p_8$ to decay into one nucleon and one pion. Although their interaction is of S-wave, lack of phase space forbids the strong decays $p_8\to p\eta_0,
\Sigma K$ to happen.

For $p_8$, the only kinematically strong decays are S-wave $p_8\to
p\pi$, P-wave $p_8 \to N \pi \pi$ and S-wave $p_8\to N \pi\pi\pi$ decays where $N$ is either a proton or neutron. All these decay modes involve the annihilation of a strange quark pair and violate the the “fall-apart” mechanism. Hence the width is expected to be small. Both $p_8$ and $n_8$ may be narrow resonances.

It is interesting to note there are three negative-parity $\Lambda$ particles within the range between 1400 MeV and 1540 MeV if Jaffe and Wilczek’s diquark model is correct. One of them is the well established $\Lambda (1405)$. $\Lambda (1405)$ is only 30 MeV below kaon and nucleon threshold. Some people postulated it to be a kaon nucleon molecule [@pdg]. We propose that there is another intriguing possibility of interpreting $\Lambda (1405)$ as the candidate of $J^P={1\over 2}^-$ pentaquark. The other $J^P={1\over 2}^-$ pentaquark and the corresponding L=1 singlet $\Lambda$ particle may have escaped detection so far.

The discovery of nine additional negative-parity baryons in this mass range will be strong evidence supporting the diquark model. Then one has to consider the possibility of treating diquarks as an effective degree of freedom as constituent quarks when dealing with hadron spectrum. One possible consequence is the appearance of many exotic mesons and over-population of meson states with non-exotic quantum numbers below 2 GeV. On the other hand, if future experimental searches fail to find any evidence of these additional states with negative parity, one has to re-evaluate the relevance of the diquark picture for the pentaquarks.

Other clustered quark model
---------------------------

An even simpler picture came from Karliner and Lipkin’s diquark triquark model [@lipkin]. In their model the two color non-singlet clusters are kept apart by the P-wave angular momentum barrier. Hence the color magnetic interaction occurs within two clusters only. Between two clusters the color electric forces bind them into a color singlet. The angular momentum barrier prevents them from rearranging into the usual $KN$ system. The presence of P-wave ensures the overall $\Theta^+$ parity is positive.

The two quarks in the triquark are in the symmetric ${\bf 6_c}$ representation. They couple with the antiquark to form an $SU(3)_c$ triplet ${\bf 3}_c$. The two quarks are in the anti-symmetric flavor ${\bf \bar{3}_f}$ representation and the triquark is in the symmetric ${\bf \bar 6}_f$. Thus the spin wave function of the two quarks is symmetric. The spin of the triquark is one half.

The direct product of the ${\bf \bar 3_f}$ of diquark and the ${\bf \bar 6_f}$ of triquark leads to ${\bf\bar{10}_f}$ and ${\bf
8_f}$ pentaquarks. There is one orbital angular momentum $L=1$ between the diquark and the triquark. The resulting $J^P$ of the pentaquark is ${1\over 2}^+$ or ${3\over 2}^+$.

Based on the color-magnetic interaction, a recent paper analyzed the triquark cluster including the explicit SU(3) symmetry breaking [@hogaasen]. With the reasonable parameters $C_{ij}$ from the analysis of the conventional baryons, the $\Theta^+$ mass is found to be around 1520 MeV even without the orbital excitation energy. It’s very demanding to reproduce the low mass of the $\Theta^+$ pentaquark!

Shuryak and Zahed’s (SZ) suggested that the pentaquark mass be lower by replacing one scalar diquark with one tensor diaquark in JW’s model [@shuryak]. But their $\Theta^+$ pentaquark is in the $27$ representation and $\Theta^+$ is inevitably accompanied by a $\Theta^{++}$ in their model.

Non-clustered quark model
-------------------------

Stancu and Riska studied the stability of the strange pentaquark state assuming a flavor-spin hyperfine interaction between quarks in the constituent quark model [@riska]. They suggested that the lowest lying p-shell pentaquark state with positive parity could be stable against strong decays if the spin-spin interaction between the strange antiquark and up/down quark was strong enough [@riska].

In Ref. [@hosaka] Hosaka emphasized the important role of the hedgehog pion in the strong interaction dynamics which drives the formation of the $\Theta^+$ particle and the detailed energy level ordering.

In the adiabatic approximation, Zhang et al employed the chiral quark model to make a systematic comparison study of the $\Theta^+$ pentaquark mass with both parities [@zhang]. Their model Lagrangian includes the one pion exchange force, one gluon exchange force and the linear confinement. First they reproduced known hadron spectroscopy and nucleon-nucleon, nucleon-hyperon scattering phase shift. In this way most of their model parameters were fixed. Then they extended their discussion to the pentaquark system. They found that the pentaquark in the I=0 channel in the anti-decuplet representation is always lowest for both $J^P={1\over 2}^+$ and $J^P={1\over 2}^-$. Without interaction between four light quarks and strange anti-quark, the pentaquark with $J^P={1\over 2}^+$ lies lower than that of $J^P={1\over 2}^-$ as pointed out by Stancu and Riska [@riska]. However, the $J^P={1\over 2}^+$ pentaquark mass is about 400 MeV above $\Theta^+$. With interaction (especially annihilation effects) between 4q and $\bar s$, the $J^P={1\over 2}^-$ pentaquark is lower, with a mass (150-300) MeV above $\Theta^+$. It’s fair to say that it is still very demanding to dynamically generate the $\Theta^+$ pentaquark with a low mass of $1530$ MeV within the framework of the non-clustered quark models.

Wang et al. tried three models to calculate $\Theta^+$ pentaquark mass [@wang]. First they treated $\Theta^+$ as a pure $uudd\bar s$ five quark state including channel coupling. Their numerical results indicate the hidden color channel effects and coupled channel effects play a vital role in reducing $\Theta^+$ mass. However the S-wave state is always lower than P-wave one. The second model they used is the quark delocalization, color screening model. The third one mimics the diquark color configuration. They found a pure repulsive effective action for the $I=1$ $KN$ channel, hence excluding $I=1$ possibility for $\Theta^+$. In the adiabatic approximation, the $\Theta^+$ mass of $1615$ MeV is obtained in the $I=0$ S-wave $KN$ channel. In their model the P-wave attraction is not sufficient to overcome the kinetic increase hence the positive parity pentaquark state lies higher than the negative parity one. It is emphasized that the possibility of $\Theta^+$ being a $I=0J^P={1\over 2}^-$ state has not been ruled out yet [@wang].

From previous quark model experience, one would naively expect the strange pentaquark mass with positive parity to be around $4M_u +
M_s + 400\sim 2050$ MeV, where $M_u=300$ MeV and $M_s=450$ MeV are roughly the up and strange quark constituent mass. The additional $\sim 400$ MeV comes from one orbital excitation for the positive-parity $\Theta^+$. If future experiments really establish the $\Theta^+$ parity to be positive, then we have to conclude that there must exist some kind of unknown and strong correlation between the light quarks in the pentaquark system which helps lower its mass significantly.

QCD sum rules
-------------

The method of QCD sum rules (QSR) can not solve color confinement. Instead, QSR incorporates two basic properties of QCD as its starting point: confinement and approximate chiral symmetry and its spontaneous breaking. One considers a correlation function of some specific interpolating currents with the proper quantum numbers as in lattice QCD simulation and calculates the correlator perturbatively starting from high energy region. Then the resonance region is approached where non-perturbative corrections in terms of various condensates gradually become important. Using the operator product expansion, the spectral density of the correlator at the quark gluon level can be obtained in QCD. On the other hand, the spectral density can be expressed in term of physical observables like masses, decay constants, coupling constants etc at the hadron level. With the assumption of quark hadron duality these two spectral densities can be related to each other. In this way one can extract hadron masses etc. For the past decades QCD sum rules has proven to be a very powerful and successful non-perturbative method [@svz; @reinders].

The $\Theta^+$ pentaquark mass was first studied using the interpolating current [@zhu] $$\begin{aligned}
\label{cu3} \nonumber
\eta_0(x)={1\over \sqrt{2}} \epsilon^{abc} [u^T_a(x) C\gamma_5
d_b (x)] \{ u_e (x)\\
\times {\bar s}_e (x) i\gamma_5 d_c(x) - \left( u\leftrightarrow
d\right) \}\end{aligned}$$ where $\left( C\gamma_5\right)^T =-C\gamma_5$ ensures the isospin of the up and down quark pair inside the first bracket to be zero. The anti-symmetrization in the second bracket ensures that the isospin of the other up and down quark pair is also zero.

The following correlation function was considered in Ref. [@zhu] $$\label{cor-1}
i\int d^4 x e^{ipx} \langle 0|T\left (\eta_j (x), {\bar \eta}_j
(0) \right )|0\rangle\ = \Pi (p) {\hat p} + \Pi' (p )$$ where $\bar \eta =\eta^\dag \gamma_0$ and ${\hat p} =p_\mu \cdot
\gamma^\mu$. The spectral density of $\Pi (p)$ reads $$\begin{aligned}
\label{spectral}\nonumber
\rho_0={3 s^5\over  4^8 7! \pi^8}+{s^2\over 1536 \pi^4} [ {5\over
12} \langle \bar q q\rangle^2  +{11\over 24}\langle \bar
qq\rangle\langle \bar ss\rangle ]\\ \nonumber +[{7\over
432}\langle \bar q q\rangle^3 \langle \bar s s\rangle +{1\over
864}\langle \bar q q\rangle^4] \delta (s)\end{aligned}$$ after invoking the factorization approximation for the multi-quark condensates.

The mass of $\Theta^+$ pentaquark is $$\begin{aligned}
M_0^2 =\frac{\int_{m_s^2}^{s_0} d s e^{-{s/T^2}} \rho'
(s)}{\int_{m_s^2}^{s_0} d s e^{-{s/T^2}} \rho (s)}\end{aligned}$$ with $\rho'(s)=s\rho(s)$ except that $\rho'(s)$ does not contain the last term in $\rho(s)$.

The variation of $M_0$ with the Borel parameter $T$ and the continuum threshold $s_0$ is shown in Figure (\[zhuprl\]), which contributes to the errors of the extracted value, together with the truncation of the operator product expansion, the uncertainty of vacuum condensate values and the factorization approximation of the multiquark condensates. In the working interval of the Borel parameter $M_0$ is reasonably stable with $T$. Numerically, $$M_0 =(1.56\pm 0.15) \mbox{GeV} ,$$ where the central value corresponds to $T=2$ GeV and $s_0=4$ GeV$^2$. An important observation from Ref. [@zhu] is that the low mass of $\Theta^+$ pentaquark could be accommodated in QCD.

Later, several groups used various forms of interpolating currents to study the mass and parity of pentaquarks [@qsr; @mull; @hk; @kondo]. However, the continuum contribution to the resulting mass sum rule is always big because of the high dimension of the interpolating current, which may render QSR treatment of pentaquarks less convincing than in the case of conventional hadrons. Moreover, the convergence of the operator product expansion is not so good in certain cases.

It’s a general result that any pentaquark interpolating currents can always be decomposed into some linear combinations of the product of a color-singlet baryon interpolating current and a color-singlet meson interpolating current. Some examples of the decomposition were given in [@kondo]. When we calculate the correlation function, we will have two types of contributions to the spectral density. The first one corresponds to the diagonal case where a pair of baryon and meson propagate freely. The second case corresponds to the non-diagonal case when the quark lines of the baryon mix with those from the meson. I.e., the meson and baryon pair resonates into a pentaquark which then decays into a pair of meson and baryon. It is pointed out that the first case is the background contribution only and has nothing to do with the pentaquarks [@kondo]. Hence this kind of contribution should be removed.

Heptaquarks
-----------

Bicudo proposed an interesting alternate scheme for $\Theta^+$ baryon [@bicudo]. Using the resonating group method, he found that the repulsion is increased by Pauli principle whenever the two interacting hadrons have a common flavor quark. The attraction is increased by the quark anti-quark annihilation effect if there is a pair of quark and anti-quark with the same flavor. The exotic $\Theta^+$ baryon is unbound if its quark content is $uudd\bar s$. In order to increase the binding, a light pair of quark and anti-quark has to be added in order to make the system bound. Since the pion is the lightest meson, Bicudo proposed that $\Theta^+$ is probably a $K$$\pi$$N$ molecule with binding energy of 30 MeV [@bicudo]. The decay width has not been calculated yet in this model. However, one would naively expect such a loosely bound $K$$\pi$$N$ system is quite broad.

Narrow width puzzle {#sec5}
===================

According to these positive-result experiments, both $\Theta^+$ and $\Xi^{--}_5$ are very narrow states. In fact, the $\Theta^+$ pentaquark is so narrow that most of the experiments can only set an upper bound around $20$ MeV. Recent analysis of kaon nucleon scattering data indicates that the width of the $\Theta^+$ pentaquark is less than several MeV [@nuss], otherwise they should have shown up in these old experiments.

Experience with conventional excited hadrons shows that their widths are around one hundred MeV or even bigger if they lie 100 MeV above threshold and decay through S-wave or P-wave. For comparison, the $S=-1$ hyperon $\Lambda (1520)$ $D_{03}$ state is in the same mass region as the $\Theta^+$ pentaquark. Its angular momentum and parity is $J^P={3\over 2}^-$. Its dominant two-body decay is of D-wave with final states $N\bar K$, $\Sigma \pi$. With a smaller phase space and higher partial wave, the width of $\Lambda (1520)$ is $15.6\pm 1.0$ MeV [@pdg]. In contrast, $\Theta^+$ lies above 100 MeV $K N$ threshold and decays through either S-wave or P-wave with a total width less than several MeV, corresponding to negative or positive parity respectively. If these states are further established and confirmed to have such a narrow width, the most challenging issue is to understand their extremely narrow width in a natural way. Is there a mysterious selection rule which is absent from the conventional hadron interaction?

In the case of pentaquark decays, if symmetry and kinematics allow, the most efficient decay mechanism is for the four quarks and anti-quark to regroup with each other int o a three-quark baryon and a meson. This is in contrast to the $^3P_0$ decay models for the ordinary hadrons. This regrouping is coined as the “fall-apart” mechanism in Refs. [@maltman; @close; @lee; @zhanga].

Recently there have been several attempts to explain the narrow width of the $\Theta^+$ pentaquark [@carl; @width2; @buccella]. Carlson et al. constructed a special pentaquark wave function which is totally symmetric in the flavor-spin part and anti-symmetric in the color-orbital part in Ref. [@carl]. With this wave function they found that the overlap probability between the pentaquark and the nucleon kaon system is ${5\over 96}$. Taking into account of the orbital wave function in JW’s diquark model further reduces the overlap probability to ${5\over 596}$ [@carl]. The small overlap probability might be responsible for the narrow width of pentaquarks.

Karliner and Lipkin proposed that there might exist two nearly generate pentaquarks [@width2]. Both of them decay into the kaon and nucleon. Hence these two states mix with each other by the loop diagram via the decay modes. Diagonalization of the mass matrix leads to a narrow $\Theta^+$ pentaquark which almost decouples with the decay mode. The pentaquark with the same quantum number is very broad with a width of around 100 MeV, which has escaped the experimental detection so far.

In Ref. [@buccella] Buccella and Sorba suggested that the four quarks are in the $L=1$ state and the anti-quark is in the S-wave state inside the $\Theta^+$ and $\Xi^{--}_5$ pentaquarks. There are four anti-symmetric four quark $SU(6)_{FS}$ wave functions $[4]_{FS}, [31]_{FS}, [22]_{FS}, [211]_{FS}$. The third one corresponds to Jaffe and Wilzcek’s diquark model. The difference is the four quarks are not completely anti-symmetric in the diquark model. It was pointed out that the narrow width of $\Theta^+$ pentaquark may favor the last two four-quark spin-flavor wave functions [@buccella]. The reason is as follows. When the anti-quark picks up a quark to form a meson, two of the other three quarks remain in the $SU(6)_{fS}$ totally anti-symmetric states $[21]_{FS}, [111]_{FS}$. These representations are orthogonal to the $SU(6)_{FS}$ totally symmetric representation $[3]_{FS}$ for the nucleon octet. If $SU(6)_{FS}$ symmetry is exact, the ${\bar 10}$ pentaquarks will not decay at all. This selection rule is exact in the symmetry limit. The narrow widths of the $\Theta^+$ and $\Xi^{--}_5$ pentaquarks come from the $SU(3)_f$ symmetry breaking.

Using the picture of the flux tube model, we proposed that the $\Theta_5$ pentaquark as the first candidate of the three-dimensional non-planar hadron with the extremely stable diamond structure. The up and down quarks lie at the corners of the diamond while the anti-strange quark sits in the center. Various un-excited color flux tubes between the five quarks bind them into a stable and narrow color-singlet. Such a configuration allows the lowest state having the negative parity naturally. The decay of the $\Theta_5$ pentaquark into the nucleon and kaon requires the breakup of the non-planar diamond configuration into two conventional planar hadrons, which involves some kind of structural phase transition as in the condensed matter physics. Despite that it lies above the kaon nucleon threshold, the width of the $\Theta^+$ pentaquark should be narrow because of the mismatch between the initial and final state spatial wave functions. Such a mechanism may be tested by future lattice QCD simulation using non-planar interpolating currents.

Pentaquark flavor wave functions, masses and magnetic moments {#sec6}
=============================================================

The flavor wave functions of diquarks are listed in Table \[tab-diquark\].

           ($Y_1,I_1$)                   ($Y_2,I_2$)                     ($Y,I$)                  $I_3$             Flavor Wave Functions
  ----------------------------- ----------------------------- ------------------------------ ---------------- ----------------------------------
   ($\frac{1}{3},\frac{1}{2}$)   ($\frac{1}{3},\frac{1}{2}$)        ($\frac{2}{3},0$)               0          $[ud]=\frac{1}{\sqrt{2}}(ud-du)$
   ($\frac{1}{3},\frac{1}{2}$)       (-$\frac{2}{3}$,0)        (-$\frac{1}{3},\frac{1}{2}$)   $\frac{1}{2}$    $[us]=\frac{1}{\sqrt{2}}(us-su)$
       (-$\frac{2}{3}$,0)        ($\frac{1}{3},\frac{1}{2}$)   (-$\frac{1}{3},\frac{1}{2}$)   -$\frac{1}{2}$   $[ds]=\frac{1}{\sqrt{2}}(ds-sd)$

  : Diquark flavor wave functions. $Y$, $I$ and $I_3$ are hypercharge, isospin and the third component of isospin respectively. The subscripts 1 and 2 represent two quarks inside the diquark. []{data-label="tab-diquark"}

The flavor wave functions of ${\bar 10}$ and $8$ pentaquarks are collected in Table \[tab-jaffe-wf\].

              ($Y,I,I_3$)                                        $\bf{\bar{10}}$                                        ($Y,I,I_3$)                                           $\bf{8}$
  ----------------------------------- --------------------------------------------------------------------- ----------------------------------- ---------------------------------------------------------------------
                (2,0,0)                                          $[ud]^2\bar{s}$                                            $-$                                                  $-$
    (1,$\frac{1}{2}$,$\frac{1}{2}$)    $\sqrt{\frac{2}{3}}[ud][us]_+\bar{s}+\sqrt\frac{1}{3}[ud]^2\bar{d}$    (1,$\frac{1}{2}$,$\frac{1}{2}$)    $\sqrt\frac{1}{3}[ud][us]_+\bar{s}-\sqrt{\frac{2}{3}}[ud]^2\bar{d}$
   (1,$\frac{1}{2}$,-$\frac{1}{2}$)    $\sqrt{\frac{2}{3}}[ud][ds]_+\bar{s}+\sqrt\frac{1}{3}[ud]^2\bar{u}$    (1,$\frac{1}{2}$,$\frac{1}{2}$)    $\sqrt\frac{1}{3}[ud][ds]_+\bar{s}-\sqrt{\frac{2}{3}}[ud]^2\bar{u}$
                (0,1,1)                $\sqrt{\frac{2}{3}}[ud][us]_+\bar{d}+\sqrt\frac{1}{3}[us]^2\bar{s}$                (0,1,1)                $\sqrt{\frac{1}{3}}[ud][us]_+\bar{d}-\sqrt\frac{2}{3}[us]^2\bar{s}$
                (0,1,0)                     $\sqrt{\frac{1}{3}}([ud][ds]_+\bar{d}+[ud][us]_+\bar{u}$                      (0,1,0)                     $\sqrt{\frac{1}{6}}([ud][ds]_+\bar{d}+[ud][us]_+\bar{u})$
                                                              $+[us][ds]_+\bar{s})$                                                                            $-\sqrt{\frac{2}{3}}[us][ds]_+\bar{s}$
               (0,1,-1)                $\sqrt{\frac{2}{3}}[ud][ds]_+\bar{u}+\sqrt\frac{1}{3}[ds]^2\bar{s}$               (0,1,-1)                $\sqrt{\frac{1}{3}}[ud][ds]_+\bar{u}-\sqrt\frac{2}{3}[ds]^2\bar{s}$
   (-1,$\frac{3}{2}$,$\frac{3}{2}$)                              $[us]^2\bar{d}$                                            $-$                                                  $-$
   (-1,$\frac{3}{2}$,$\frac{1}{2}$)    $\sqrt{\frac{2}{3}}[us][ds]_+\bar{d}+\sqrt\frac{1}{3}[us]^2\bar{u}$   (-1,$\frac{1}{2}$,$\frac{1}{2}$)    $\sqrt{\frac{1}{3}}[us][ds]_+\bar{d}-\sqrt\frac{2}{3}[us]^2\bar{u}$
   (-1,$\frac{3}{2}$,-$\frac{1}{2}$    $\sqrt{\frac{2}{3}}[ds][us]_+\bar{u}+\sqrt\frac{1}{3}[ds]^2\bar{d}$   (-1,$\frac{1}{2}$,-$\frac{1}{2}$)   $\sqrt{\frac{1}{3}}[ds][us]_+\bar{u}-\sqrt\frac{2}{3}[ds]^2\bar{d}$
   (-1,$\frac{3}{2}$,-$\frac{3}{2}$)                             $[ds]^2\bar{u}$                                            $-$                                                  $-$
                  $-$                                                  $-$                                                (0,0,0)                      $\sqrt\frac{1}{2}([ud][ds]_+\bar{d}-[ud][us]_+\bar{u})$

  : Flavor wave functions in Jaffe and Wilczek’s model [@jaffe]. $[q_1q_2][q_3q_4]_+=\sqrt{\frac{1}{2}}([q_1q_2][q_3q_4]+[q_3q_4][q_1q_2])$ or $[q_1q_2]^2=[q_1q_2][q_1q_2]$ is the diquark-diquark part. []{data-label="tab-jaffe-wf"}

Within the diquark model [@jaffe], the strange quark mass explicitly breaks $SU(3)_F$ symmetry. The \[ud\] diquark is more tightly bound than \[us\] and \[ds\]. The energy difference can be related to the $\Sigma$-$\Lambda$ mass splitting. Thus, every strange quark in the pentaquark contributes $\alpha \equiv \frac
{3}{4}(M_\Sigma - M_\Lambda) \approx 60$ MeV arising from \[ud\] and \[us\], \[ds\] binding energy difference. The Hamiltonian in JW’s model reads $$\ H_{s}= M_{0} +(n_{s}+n_{\bar{s}})m_{s}+n_{s} \alpha \label{eq1}$$ where $M_{0}$ is the pentaquark mass in the $SU(3)_F$ symmetry limit. The last two terms are from $SU(3)_F$ symmetry breaking with $m_s\approx100$ MeV. $M_{0}$ has the form $$M_{0}=2m_{di}+m_{\bar q}+\delta M_{l} \label{eq2}$$ where $m_{di}$ is the \[ud\] diquark mass, $m_{\bar{q}}$ is the anti-quark mass and $\delta M_{l}$ is the orbital excitation energy. We use $\delta M_{l}=0$ for $l=0$ and $\delta M_{l}\approx
230$ MeV for $l=1$, $m_{di}=420$ MeV, and $m_{\bar q}=360$ MeV. Thus we can use Eq.(\[eq1\]) to compute pentaquark masses. For example, we have $$M_{\Theta^+}\approx 1530 \mbox{MeV}.$$ In Table \[tab-jaffe-neg\] we have listed the flavor wave functions and masses of $J^P={1\over 2}^-$ pentaquarks in JW’s model.

                         ($Y,I$)             $I_3$                                  Flavor wave functions                              Masses
  ---------------- -------------------- ---------------- ---------------------------------------------------------------------------- --------
       $p_8$        (1,$\frac{1}{2}$)    $\frac{1}{2}$                               $[su][ud]_-\bar{s}$                                1460
       $n_8$                             -$\frac{1}{2}$                              $[ds][ud]_-\bar{s}$                                1460
   $\Sigma_{8}^+$         (0,1)                1                                     $[su][ud]_-\bar{d}$                                1360
   $\Sigma_{8}^0$                              0                 $\frac{1}{\sqrt{2}}$($[su][ud]_-\bar{u}+[ds][ud]_-\bar{d}$)            1360
   $\Sigma_{8}^-$                              -1                                    $[ds][ud]_-\bar{u}$                                1360
    $\Lambda_8$           (0,0)                0          $\frac{1}{\sqrt6}([ud][su]_-\bar{u}+[ds][ud]_-\bar{d}-2[su][ds]_-\bar{s})$    1533
    $\Xi_{8}^0$     (-1,$\frac{1}{2}$)   $ \frac{1}{2}$                              $[ds][su]_-\bar{d}$                                1520
    $\Xi_{8}^-$                          -$\frac{1}{2}$                              $[ds][su]_-\bar{u}$                                1520
    $\Lambda_1$           (0,0)                0          $\frac{1}{\sqrt3}([ud][su]_-\bar{u}+[ds][ud]_-\bar{d}+[su][ds]_-\bar{s})$     1447

  : Flavor wave functions and masses of the $\frac{1}{2}^-$ pentaquark octet and singlet. Where $[q_1q_2][q_3q_4]_-=\sqrt{\frac{1}{2}}([q_1q_2][q_3q_4]-[q_3q_4][q_1q_2])$. []{data-label="tab-jaffe-neg"}

The baryon magnetic moment is another fundamental observable as its mass, which encodes information of the underlying quark structure and dynamics. Different models generally yield different values, which provides a very good way to distinguish them. This in turn will affect the photo-production and electro-production cross sections of pentaquarks.

The magnetic moment of a compound system is the sum of the magnetic moments of its constituents including spin and orbital contributions, $$\overrightarrow{\mu}=\sum\limits_{i}\overrightarrow{\mu_i}
=\sum\limits_{i}(g_{i}\overrightarrow{s_i}+\overrightarrow{l_i})
\mu_i,$$ where $g_{i}$ is the $g$-factor of $i$-th constituent and $\mu_i$ is the magneton of the $i$-th constituent $$\mu_i = \frac{e_i}{2m_i}\; .$$ Using the diquark flavor wave functions Table \[tab-diquark\] and pentaquark flavor wave functions in Table \[tab-jaffe-wf\] and \[tab-jaffe-neg\], one can easily derive pentaquark magnetic moments in this model [@liu-mm]. The results are listed in Table \[tab-jaffe-mm1\] and \[tab-jaffe-mm2\].

                 State                          State                                               
  ----------------------------------- -------- -------- ----------------------------------- ------- --------
              ($Y,I,I_3$)              set I    set II              ($Y,I,I_3$)              set I   set II
                (2,0,0)                 0.08     0.29                   $-$                   $-$     $-$
    (1,$\frac{1}{2}$,$\frac{1}{2}$)    0.037     0.22     (1,$\frac{1}{2}$,$\frac{1}{2}$)    0.018    0.21
   (1,$\frac{1}{2}$,-$\frac{1}{2}$)     0.12     0.22    (1,$\frac{1}{2}$,-$\frac{1}{2}$)    0.50     0.65
                (0,1,1)                -0.009    0.14                 (0,1,1)                0.007    0.14
                (0,1,0)                 0.06     0.12                 (0,1,0)                -0.13   -0.13
               (0,1,-1)                 0.13     0.09                (0,1,-1)                -0.27   -0.41
   (-1,$\frac{3}{2}$,$\frac{3}{2}$)    -0.06     0.06                   $-$                   $-$     $-$
   (-1,$\frac{3}{2}$,$\frac{1}{2}$)      0        0      (-1,$\frac{1}{2}$,$\frac{1}{2}$)    0.41     0.46
   (-1,$\frac{3}{2}$,-$\frac{1}{2}$)    0.06    -0.06    (-1,$\frac{1}{2}$,-$\frac{1}{2}$)   -0.35   -0.52
   (-1,$\frac{3}{2}$,-$\frac{3}{2}$)    0.12    -0.12                   $-$                   $-$     $-$
                  $-$                   $-$      $-$                  (0,0,0)                0.25     0.37

  : Numerical results of the magnetic moments of ${\bf \bar{
  10}}$ and [**8**]{} pentaquarks in unit of $\mu_N$ in Jaffe and Wilczek’s model [@jaffe]. For set I we use $m_{ud}=720$ MeV, $m_{us}=m_{ds}=900$ MeV from Ref. [@lipkin] and for set II we use $m_{ud}=420$ MeV, $m_{us}=m_{ds}=600$ MeV from Ref. [@jaffe]. []{data-label="tab-jaffe-mm1"}

                         ($Y,I$)             $I_3$                                    Magnetic moments                               Numerical results ($\mu_N$)
  ---------------- -------------------- ---------------- -------------------------------------------------------------------------- -----------------------------
       $p_8$        (1,$\frac{1}{2}$)    $\frac{1}{2}$                               $\frac{e_0}{6m_s}$                                         0.63
       $n_8$                             -$\frac{1}{2}$                              $\frac{e_0}{6m_s}$                                         0.63
   $\Sigma_{8}^+$         (0,1)                1                                     $\frac{e_0}{6m_d}$                                         0.87
   $\Sigma_{8}^0$                              0                      $\frac{1}{6}(-\frac{e_0}{m_u}+\frac{e_0}{2m_d})$                          -0.43
   $\Sigma_{8}^-$                              -1                                   $-\frac{e_0}{3m_u}$                                         -1.74
    $\Lambda_8$           (0,0)                0          $\frac{1}{18}$(-$\frac{e_0}{m_u}$+$\frac{e_0}{2m_d}$+$\frac{2e_0}{m_s}$)              0.27
    $\Xi_{8}^0$     (-1,$\frac{1}{2}$)   $ \frac{1}{2}$                              $\frac{e_0}{6m_d}$                                         0.87
    $\Xi_{8}^-$                          -$\frac{1}{2}$                             -$\frac{e_0}{3m_u}$                                         -1.74
    $\Lambda_1$           (0,0)                0             $\frac{1}{9}(-\frac{e_0}{m_u}+\frac{e_0}{2m_d}+\frac{e_0}{2m_s})$                  -0.08

  : Expressions and numerical results of the magnetic moments of the pentaquark octet and singlet, where $e_0$ is the charge unit. []{data-label="tab-jaffe-mm2"}

For the sake of comparison, $J^P={1\over 2}^+$ pentaquark magnetic moments and flavor wave functions in Karliner and Lipkin’s triquark model are also listed in Table \[tab-kl-mm1\] and \[tab-kl-mm2\]. Magnetic moments of $J^P={3\over 2}^+$ can be found in Refs. [@li-mm].

              ($Y,I,I_3$)                                         Flavor wave functions                              Magnetic moments
  ----------------------------------- ----------------------------------------------------------------------------- ------------------
                (2,0,0)                                            $[ud]\{ud\bar{s}\}$                                     0.19
    (1,$\frac{1}{2}$,$\frac{1}{2}$)    $\sqrt{\frac{1}{3}}([ud]\{ud\bar{d}\}+[ud]\{us\bar{s}\}+[us]\{ud\bar{s}\})$         0.17
   (1,$\frac{1}{2}$,-$\frac{1}{2}$)    $\sqrt{\frac{1}{3}}([ud]\{ud\bar{u}\}+[ud]\{ds\bar{s}\}+[ds]\{ud\bar{s}\})$        -0.006
                (0,1,1)                $\sqrt{\frac{1}{3}}([us]\{ud\bar{d}\}+[us]\{us\bar{s}\}+[ud]\{us\bar{d}\})$         0.15
                (0,1,0)                $\sqrt{\frac{1}{6}}([us]\{ud\bar{u}\}+[us]\{ds\bar{s}\}+[ds]\{ud\bar{d}\}$         -0.03
                                                $+[ds]\{us\bar{s}\}+[ud]\{us\bar{u}\}+[ud]\{ds\bar{d}\})$           
               (0,1,-1)                $\sqrt{\frac{1}{3}}([ds]\{ud\bar{u}\}+[ds]\{ds\bar{s}\}+[ud]\{ds\bar{u}\})$        -0.21
   (-1,$\frac{3}{2}$,$\frac{3}{2}$)                                $[us]\{us\bar{d}\}$                                     0.13
   (-1,$\frac{3}{2}$,$\frac{1}{2}$)    $\sqrt{\frac{1}{3}}([us]\{us\bar{u}\}+[us]\{ds\bar{d}\}+[ds]\{us\bar{d}\})$        -0.054
   (-1,$\frac{3}{2}$,-$\frac{1}{2}$)   $\sqrt{\frac{1}{3}}([ds]\{ds\bar{d}\}+[ds]\{us\bar{u}\}+[us]\{ds\bar{u}\})$        -0.24
   (-1,$\frac{3}{2}$,-$\frac{3}{2}$)                               $[ds]\{ds\bar{u}\}$                                    -0.43

  : Wave functions and numerical results of the magnetic moments of ${\bf \bar{10}}$ pentaquarks in unit of $\mu_N$ in Karliner and Lipkin’s model [@lipkin]. $Y$, $I$ and $I_3$ are hypercharge, isospin and the third component of isospin respectively. $\{q_1q_2\bar{q_3}\}\equiv[q_1q_2]\bar{q_3}$ is the triquark’s flavor wave function.[]{data-label="tab-kl-mm1"}

              ($Y,I,I_3$)                                                  Flavor wave functions                                       Magnetic moments
  ----------------------------------- ----------------------------------------------------------------------------------------------- ------------------
    (1,$\frac{1}{2}$,$\frac{1}{2}$)    $\sqrt{\frac{1}{6}}([ud]\{ud\bar{d}\}+[ud]\{us\bar{s}\})-\sqrt{\frac{2}{3}}[us]\{ud\bar{s}\}$         0.16
   (1,$\frac{1}{2}$,-$\frac{1}{2}$)    $\sqrt{\frac{1}{6}}([ud]\{ud\bar{u}\}+[ud]\{ds\bar{s}\})-\sqrt{\frac{2}{3}}[ds]\{ud\bar{s}\}$        -0.14
                (0,1,1)                $\sqrt{\frac{1}{6}}([us]\{ud\bar{d}\}+[us]\{us\bar{s}\})-\sqrt{\frac{2}{3}}[ud]\{us\bar{d}\}$         0.16
                (0,1,0)                         $\sqrt{\frac{1}{12}}([us]\{ud\bar{u}\}+[us]\{ds\bar{s}\}+[ds]\{ud\bar{d}\}$                  0.04
                                               $+[ds]\{us\bar{s}\})-\sqrt{\frac{1}{3}}([ud]\{us\bar{u}\}+[ud]\{ds\bar{d}\})$          
               (0,1,-1)                $\sqrt{\frac{1}{6}}([ds]\{ud\bar{u}\}+[ds]\{ds\bar{s}\})-\sqrt{\frac{2}{3}}[ud]\{ds\bar{u}\}$        -0.07
   (-1,$\frac{1}{2}$,$\frac{1}{2}$)    $\sqrt{\frac{1}{6}}([us]\{us\bar{u}\}+[us]\{ds\bar{d}\})-\sqrt{\frac{2}{3}}[ds]\{us\bar{d}\}$        -0.17
   (-1,$\frac{1}{2}$,-$\frac{1}{2}$)   $\sqrt{\frac{1}{6}}([ds]\{ds\bar{d}\}+[ds]\{us\bar{u}\})-\sqrt{\frac{2}{3}}[us]\{ds\bar{u}\}$        -0.12
                (0,0,0)                   $\frac{1}{2}([us]\{ud\bar{u}\}+[us]\{ds\bar{s}\}-[ds]\{ud\bar{d}\}-[ds]\{us\bar{s}\})$            -0.10

  : Wave functions and numerical results of the magnetic moments of ${\bf {8_f}}$ pentaquarks in unit of $\mu_N$ in Karliner and Lipkin’s model [@lipkin]. $Y$, $I$ and $I_3$ are hypercharge, isospin and the third component of isospin respectively.[]{data-label="tab-kl-mm2"}

The magnetic moments of $J=\frac{1}{2}$ pentaquarks have been discussed by several groups recently. Using Jaffe and Wilczek’s scalar diquark picture of $\Theta^+$, Nam, Hosaka and Kim [@hosaka1] considered the photoproduction of $\Theta^+$ pentaquark from the neutron and estimated the anomalous magnetic moment to be $- 0.7 $ (positive parity) and $- 0.2$ (negative parity) (in units of $\Theta^+$ magneton ${e_0\over
2m_{\Theta^+}}$).

In Ref. [@zhao] a quark model calculation is also performed using JW’s diquark picture, where Zhao got $\mu_{\Theta^+}=0.13{e_0\over 2m_{\Theta^+}}$ for positive parity $\Theta^+$. In the case of negative parity, he treated the pentaquark as the sum of $(u\bar s)$ and $(udd)$ clusters and got $\mu_{\Theta^+}={e_0\over 6m_s}$.

The magnetic moment of the $\Theta^+$ pentaquark is also calculated using the method of light cone QCD sum rules. In Ref. [@huang] the authors arrived at $\mu_{\Theta^+}=(0.12\pm 0.06)
\mu_N$.

Within the chiral soliton model, Kim and Praszalowicz [@mag] derived relations for the $\bf{\bar{10}_f}$ magnetic moments and found that the magnetic moment of the $\Theta^+$ pentaquark is between $-(0.3\sim 0.4) \mu_N$ in the chiral limit.

Using the non-clustered quark model, the $\Theta^+$ pentaquark is found to be $0.38 \mu_N, 0.09 \mu_N, 1.05 \mu_N$ for $J^P={1\over
2}^-, {1\over 2}^+, {3\over 2}^+$ respectively in Ref. [@bijker]. For $\Xi^{--}, \Xi^+$, the numerical values are $-0.44 \mu_N, 0.09 \mu_N, -2.64 \mu_N$ and $0.5 \mu_N, 0.09 \mu_N,
1.59 \mu_N$ respectively for $J^P={1\over 2}^-, {1\over 2}^+,
{3\over 2}^+$.

We have collected the numerical results for $J=\frac{1}{2}$ $\bf{\bar {10}}$ members $\Theta^+$, $\Xi_5^{--}$, $\Xi_5^+$ and $J=\frac{3}{2}$ $\bf{\bar {10}}$ members $\Theta^{\ast +}$, $\Xi_5^{\ast --}$, $\Xi_5^{\ast +}$ in Table \[tab-summary\]. These states lie on the corners of the anti-decuplet triangle and have no mixing with octet pentaquarks. Hence their interpretation and identification should be relatively clean, at least theoretically.

  ---------------------------- ------------------ ------------------ ----------------- ------------------- ------------------- ------------------
                                   $\Theta^+$        $\Xi^{--}_5$       $\Xi^{+}_5$     $\Theta^{\ast +}$   $\Xi^{\ast --}_5$   $\Xi^{\ast +}_5$
          Ref. [@mag]           $-(0.3\sim 0.4)$       $-0.4 $             $0.2$                -                   -                  -
         Ref. [@hosaka]          $0.2\sim 0.5$            -                  -                  -                   -                  -
          Ref. [@zhao]           $0.08\sim 0.6$           -                  -                  -                   -                  -
         Ref. [@bijker]         $0.09\sim 0.38$    $0.09\sim -0.44$   $0.09\sim 0.50$         1.05                -2.64               1.59
         Ref. [@huang]           $0.12\pm 0.06$           -                  -                  -                   -                  -
   Ref. [@li-mm] (JW’s model)         0.08               0.12              -0.06              1.01                -2.43               1.22
   Ref. [@li-mm] (KL’s model)         0.19              -0.43              0.13               0.84                -1.20               0.89
  ---------------------------- ------------------ ------------------ ----------------- ------------------- ------------------- ------------------

  : Comparison of magnetic moments of $\Theta^+$, $\Xi^{--}_5$, $\Xi^{+}_5$ and $\Theta^{\ast +}$, $\Xi^{\ast
  --}_5$, $\Xi^{\ast +}_5$ in different pentaquark models in literature. The numbers are in unit of $\mu_N$.[]{data-label="tab-summary"}

Heavy flavored pentaquarks
==========================

It is straightforward to extend the same formalism to discuss the heavy pentaquarks containing an anti-charm or anti-bottom quark. One gets one $SU(3)_f$ anti-sextet in the framework of the diquark model and both a sextet and triplet in KL’s models with either $J^P={1\over 2}^+$ or ${3\over 2}^+$.

As in the case of conventional heavy hadrons, the presence of the heavy anti-quark makes the treatment of the system simpler. If the light quarks are really strongly correlated as proposed in the clustered quark models, the heavy pentaquark system is the ideal place to study this kind of correlation without the additional complication due to the extra light anti-quark.

The heavy anti-quark is a $SU(3)_f$ flavor singlet. Hence the heavy pentaquarks containing an anti-charm or anti-bottom form a $SU(3)$ flavor anti-sextet in the diquark model as shown in Figure \[fig-heavy\]. The flavor wave functions of all members of ${\bf
{\bar 6_f}}$ are listed in Table \[heavy-jaffe-wf\].

(200,130)

(50,22)[(3,5)[50]{}]{} (100,105)[(3,-5)[50]{}]{} (50,22)[(1,0)[100]{}]{} (100,22)[(3,5)[25]{}]{}(100,22)[(-3,5)[25]{}]{}(75,64)[(1,0)[50]{}]{} (50,22)(100,22)(150,22) (75,64)(125,64)(100,105)

(30,50)[(1,0)[140]{}]{}(100,2)[(0,1)[120]{}]{}

(105,105)[$\Theta_c^0$,$\Theta_b^+$]{} (130,64)[$\Sigma_{5c}^0$,$\Sigma_{5b}^+$]{} (35,64)[$\Sigma_{5c}^-$,$\Sigma_{5b}^0$]{} (155,10)[$\Xi_{5c}^0$,$\Xi_{5b}^+$]{} (103,10)[$\Xi_{5c}^-$,$\Xi_{5b}^0$]{} (33,10)[$\Xi_{5c}^-$,$\Xi_{5b}^0$]{}

             pentaquarks                             $(Y,I,I_3)$                    Flavor wave functions($\bar Q=\bar c$ or $\bar b$)
  --------------------------------- ---------------------------------------------- ----------------------------------------------------
      $\Theta^0_c$,$\Theta^+_b$                  ($\frac{4}{3}$,0,0)                                 $[ud]^2 \bar{Q}$
   $\Sigma^0_{5c}$,$\Sigma^+_{5b}$   ($\frac{1}{3}$,$\frac{1}{2}$,$\frac{1}{2}$)                   $[ud][us]_+\bar{Q}$
   $\Sigma^-_{5c}$,$\Sigma^0_{5b}$   ($\frac{1}{3}$,$\frac{1}{2}$,-$\frac{1}{2}$)                  $[ud][ds]_+\bar{Q}$
      $\Xi^0_{5c}$,$\Xi^+_{5b}$                  (-$\frac{2}{3}$,1,1)                                $[us]^2\bar{Q}$
      $\Xi^-_{5c}$,$\Xi^0_{5b}$                  (-$\frac{2}{3}$,1,)                               $[us][ds]_+\bar{Q}$
    $\Xi^{--}_{5c}$,$\Xi^-_{5b}$                (-$\frac{2}{3}$,1,-1)                                $[ds]^2\bar{Q}$

  : Flavor wave functions in Jaffe and Wilczek’s model [@jaffe]. $[q_1q_2][q_3q_4]_+=\sqrt{\frac{1}{2}}([q_1q_2][q_3q_4]+[q_3q_4][q_1q_2])$ or $[q_1q_2]^2=[q_1q_2][q_1q_2]$ is the diquark-diquark part. []{data-label="heavy-jaffe-wf"}

Jaffe and Wilczek estimated $\Theta^0_c$ and $\Theta^+_b$ masses by replacing the $\bar s$ in the $\Theta^+$ with $\bar c$ and $\bar b$ respectively. The cost of this replacement is roughly the mass difference between $\Lambda_c$ and $\Lambda$ baryons. The $[ud]$ diquark in the $\Lambda_c$ and $\Lambda$ experiences nearly the same environment as in $\Theta^0_c$ and $\Theta^+$ pentaquarks, especially when the $[ud]$ diquark is viewed as a tightly bound entity. Thus they got $$\begin{aligned}
\nonumber
M(\Theta^0_c)=M(\Theta^+)+[M(\Lambda_c)-M(\Lambda)]\\\nonumber
M(\Theta^+_b)=M(\Theta^+)+[M(\Lambda^+_b)-M(\Lambda)].\nonumber\end{aligned}$$

Their formalism can be extended to estimate the masses of all the other members of ${\bf {\bar 6_f}}$. The heavy pentaquarks in the same isospin multiplet are degenerate. $$\begin{aligned}
\nonumber
M(\Sigma^0_{5c})=M(N^+_s)+[M(\Lambda_c)-M(\Lambda)]\\\nonumber
M(\Sigma^+_{5b})=M(N^+_s)+[M(\Lambda^+_b)-M(\Lambda)].\nonumber\end{aligned}$$ Here $N^+_s$ has a quark content $|[ud][us]_+ \bar s\rangle$. Its mass was estimated to be around $1700$ MeV [@jaffe]. For the heavy pentaquark containing two strange quarks, $$\begin{aligned}
\nonumber
M(\Xi^0_{5c})=M(\Xi^{--}_5)+[M(\Xi_c)-M(\Sigma)]\\\nonumber
M(\Xi^+_{5b})=M(\Xi^{--}_5)+[M(\Xi_b)-M(\Sigma)],\nonumber\end{aligned}$$ where $\Xi^{--}_5 $ mass was taken from NA49 experiment [@na49]. The heavy pentaquark masses are collected in Table \[heavy-jaffe-mass\] [@huang-heavy].

                $\bar Q=\bar c$                 Masses                $\bar Q=\bar b$                 Mass
  ------------------------------------------- ----------- ---------------------------------------- -----------
                 $\Theta^0_c$                  $2710$MeV                $\Theta^+_b$                $6050$MeV
        $\Sigma^-_{5c}$,$\Sigma^0_{5c}$        $2870$MeV      $\Sigma^0_{5b}$,$\Sigma^+_{5b}$       $6210$MeV
   $\Xi^{--}_{5c}$,$\Xi^-_{5c}$,$\Xi^0_{5c}$   $3135$MeV   $\Xi^-_{5b}$,$\Xi^0_{5b}$,$\Xi^+_{5b}$   $6475$MeV

  : Masses of all members of ${\bf {\bar 6_f}}$ in Jaffe and Wilczek’s model. []{data-label="heavy-jaffe-mass"}

It is interesting to note that the resulting $\Theta_c$ mass 2910 MeV is 389 MeV smaller than the experimental value 3099 MeV from the H1 experiment. If the diquark model is a good framework for pentaquarks, there is one possible explanation of this big discrepancy. The anti-quark mass dependent interaction in $\Theta^+$ between the diquark pair and $\bar s$ is much more attractive than that in $\Theta_c$ between the same diquark pair and $\bar c$ since the charm quark is much heavier than the strange quark. With a bigger value of the diquark mass and the inclusion of this attractive interaction between the diquark pair and the anti-quark, one may be able to accommodate both $\Theta^+$ and $\Theta_c$ mass simultaneously [@haiyang].

In Table \[heavy-jaffe-mag\] we have collected the magnetic moments of the heavy pentaquarks in the diquark model.

  ---------------------------------------------- -------------------- -------------------- -------------------- --------------------
                                                                                                                
                   ($Y,I,I_3$)                    $J^P={1\over 2}^+$   $J^P={3\over 2}^+$   $J^P={1\over 2}^+$   $J^P={3\over 2}^+$
               ($\frac{4}{3}$,0,0)                       0.62                 0.38                 0.48                 0.81
   ($\frac{1}{3}$,$\frac{1}{2}$,$\frac{1}{2}$)           0.56                 0.29                 0.41                 0.71
   ($\frac{1}{3}$,$\frac{1}{2}$,-$\frac{1}{2}$)          0.13                -0.36                -0.015               0.071
               (-$\frac{2}{3}$,1,1)                      0.47                 0.16                 0.33                 0.58
               (-$\frac{2}{3}$,1,0)                     -0.052               -0.63                -0.19                -0.20
              (-$\frac{2}{3}$,1,-1)                     -0.57                -1.41                -0.72                -0.98
  ---------------------------------------------- -------------------- -------------------- -------------------- --------------------

  : Numerical results of the $J^P=\frac{1}{2}^+$ and $J^P=\frac{3}{2}^+$ heavy pentaquark magnetic moments in Jaffe and Wilczek’s model with $m_{ud}=420$ MeV, $m_{us}=m_{ds}=600$ MeV from Ref. [@jaffe]. []{data-label="heavy-jaffe-mag"}

In Karliner and Lipkin’s model, the direct product of the ${\bf
\bar 3_f}$ of diquark and the ${\bf \bar 3_f}$ of triquark leads to ${\bf\bar 6_f}$ and ${\bf 3_f}$ pentaquarks. There is one orbital angular momentum $L=1$ between the diquark and the triquark. The resulting $J^P$ of the pentaquark can be either ${1\over 2}^+$ or ${3\over 2}^+$. We list the flavor wave functions of these two multiplets in Table \[heavy-lipkin-wf1\] and \[heavy-lipkin-wf2\].

             pentaquarks                             ($Y,I,I_3$)                                      Flavor wave functions
  --------------------------------- ---------------------------------------------- -----------------------------------------------------------
      $\Theta^0_c$,$\Theta^+_b$                  ($\frac{4}{3}$,0,0)                                   $[ud]\{ud\bar{Q}\}$
   $\Sigma^0_{5c}$,$\Sigma^+_{5b}$   ($\frac{1}{3}$,$\frac{1}{2}$,$\frac{1}{2}$)    $\sqrt{\frac{1}{2}}([ud]\{us\bar{Q}\}+[us]\{ud\bar{Q}\})$
   $\Sigma^-_{5c}$,$\Sigma^0_{5b}$   ($\frac{1}{3}$,$\frac{1}{2}$,-$\frac{1}{2}$)   $\sqrt{\frac{1}{2}}([ud]\{ds\bar{Q}\}+[ds]\{ud\bar{Q}\})$
      $\Xi^0_{5c}$,$\Xi^+_{5b}$                  (-$\frac{2}{3}$,1,1)                                  $[us]\{us\bar{Q}\}$
      $\Xi^-_{5c}$,$\Xi^0_{5b}$                  (-$\frac{2}{3}$,1,0)               $\sqrt{\frac{1}{2}}([us]\{ds\bar{Q}\}+[ds]\{us\bar{Q}\}$
    $\Xi^{--}_{5c}$,$\Xi^-_{5b}$                (-$\frac{2}{3}$,1,-1)                                  $[ds]\{ds\bar{Q}\}$

  : Flavor wave functions of the anti-sextet pentaquarks in Karliner and Lipkin’s model [@lipkin]. $Y$, $I$ and $I_3$ are hypercharge, isospin and the third component of isospin respectively. $\{q_1q_2\bar{Q}\}\equiv[q_1q_2]\bar{Q}$ is the triquark’s flavor wave function.[]{data-label="heavy-lipkin-wf1"}

              pentaquarks                              ($Y,I,I_3$)                                      Flavor wave functions
  ----------------------------------- ---------------------------------------------- -----------------------------------------------------------
   $\Sigma'^0_{5c}$,$\Sigma'^+_{5b}$   ($\frac{1}{3}$,$\frac{1}{2}$,$\frac{1}{2}$)    $\sqrt{\frac{1}{2}}([ud]\{us\bar{Q}\}-[us]\{ud\bar{Q}\})$
   $\Sigma'^-_{5c}$,$\Sigma'^0_{5b}$   ($\frac{1}{3}$,$\frac{1}{2}$,-$\frac{1}{2}$)   $\sqrt{\frac{1}{2}}([ud]\{ds\bar{Q}\}-[ds]\{ud\bar{Q}\})$
      $\Xi'^-_{5c}$,$\Xi'^0_{5b}$                  (-$\frac{2}{3}$,0,0)               $\sqrt{\frac{1}{2}}([us]\{ds\bar{Q}\}-[ds]\{us\bar{Q}\}$

  : Flavor wave functions of the triplet heavy pentaquarks in Karliner and Lipkin’s model [@lipkin]. []{data-label="heavy-lipkin-wf2"}

Karliner and Lipkin have estimated the masses of $\Theta^+$ and its heavy flavor analogs [@lipkin]. The mass of a pentaquark comes mainly from the masses of its constituent quarks, the color-spin hyperfine interaction and the P-wave excitation between the two clusters. The first two parts can be estimated by comparing them with a relevant baryon-meson system that have the same quark contents as this pentaquark. The hyperfine energy difference between these two systems can be figured out using the $SU(6)$ color-spin algebra and has been given as $-\frac{1+\zeta_Q}{12}[M(\Delta)-M(N)]$ by Karliner and Lipkin [@lipkin], where $\zeta_Q\equiv\frac{m_u}{m_Q}$. The P-wave excitation energy $\delta E^P$ is approximate to the mass difference between $D_s(2319)$ and $D^*_s(2112)$. This approximation is based on the observation that the reduced mass of the diquark-triquark system is close to that of the $D_s$ [@lipkin].

Following this formalism the masses of other members of ${\bf
{\bar 6_f}}$ and ${\bf 3_f}$ read $$\begin{aligned}
\nonumber
 \lefteqn{M(\Sigma^-_{5c})=M(\Sigma^0_{5c})=M(\Sigma'^-_{5c})=M(\Sigma'^0_{5c})}\\\nonumber
 &=\frac{1}{2}[M(N)+M(D_s)+M(\Sigma)+M(D)]
 \\\nonumber
 &-\frac{1+\zeta_c}{12}[M(\Delta)-M(N)]+\delta
 E^{P}\\\nonumber
 \lefteqn{M(\Sigma^0_{5b})=M(\Sigma^+_{5b})=M(\Sigma'^0_{5b})=M(\Sigma'^+_{5b})}\\\nonumber
 &=\frac{1}{2}[M(N)+M(B_s)+M(\Sigma)+M(B)]
 \\\nonumber
 &-\frac{1+\zeta_b}{12}[M(\Delta)-M(N)]+\delta
 E^{P}\nonumber\end{aligned}$$ and $$\begin{aligned}
\nonumber
 \lefteqn{M(\Xi^{--}_{5c})=M(\Xi^-_{5c})=M(\Xi'^-_{5c})=M(\Xi^0_{5c})}\\\nonumber
 &=M(\Sigma)+M(D_s)]-\frac{1+\zeta_c}{12}[M(\Delta)-M(N)]+\delta E^{P}\\\nonumber
 \lefteqn{M(\Xi^-_{5b})=M(\Xi^0_{5b})=M(\Xi'^0_{5b})=M(\Xi^+_{5b})}\\\nonumber
 &=M(\Sigma)+M(B_s)]-\frac{1+\zeta_b}{12}[M(\Delta)-M(N)]+\delta E^{P}.\nonumber\end{aligned}$$ The numerical values of the masses of all members of ${\bf {\bar
6_f}}$ and ${\bf 3_f}$ are presented in Table \[heavy-lipkin-mass\].

                           $\bar Q=\bar c$                              Mass                              $\bar Q=\bar b$                              Mass
  ------------------------------------------------------------------ ----------- ------------------------------------------------------------------ -----------
                             $\Theta^0_c$                             $2990$MeV                             $\Theta^+_b$                             $6400$MeV
   $\Sigma^-_{5c}(\Sigma'^-_{5c})$, $\Sigma^0_{5c}(\Sigma'^0_{5c})$   $3165$MeV   $\Sigma^0_{5b}(\Sigma'^0_{5b})$, $\Sigma^+_{5b}(\Sigma'^+_{5b})$   $6570$MeV
        $\Xi^{--}_{5c}$,$\Xi^-_{5c}(\Xi'^-_{5c})$,$\Xi^0_{5c}$        $3340$MeV         $\Xi^-_{5b}$,$\Xi^0_{5b}(\Xi'^0_{5b})$, $\Xi^+_{5b}$         $6740$MeV

  : Masses of all members of ${\bf {\bar 6_f}}$ and ${\bf {
  3_f}}$ in Karliner and Lipkin’s model. []{data-label="heavy-lipkin-mass"}

For comparison, the heavy pentaquark magnetic moments are shown in Table \[heavy-lipkin-mag\].

  ------------------------------------------------------------------- -------------------- -------------------- -------------------- --------------------
                              pentaquarks                                                                                            
                                                                       $J^P={1\over 2}^+$   $J^P={3\over 2}^+$   $J^P={1\over 2}^+$   $J^P={3\over 2}^+$
                       $\Theta^0_c$,$\Theta^+_b$                             -0.031                1.01                0.079                 0.96
   $\Sigma^0_{5c}$,$\Sigma'^0_{5c}$,$\Sigma^+_{5b}$,$\Sigma'^+_{5b}$         -0.079                1.06                0.030                 1.00
   $\Sigma^-_{5c}$,$\Sigma'^-_{5c}$,$\Sigma^0_{5b}$,$\Sigma'^0_{5b}$         -0.084               -0.26               -0.0028               -0.35
                       $\Xi^0_{5c}$,$\Xi^+_{5b}$                             -0.13                 1.11                -0.020                1.05
         $\Xi^-_{5c}$,$\Xi'^-_{5c}$,$\Xi^0_{5b}$,$\Xi'^0_{5b}$               -0.14                -0.22                -0.054               -0.30
                     $\Xi^{--}_{5c}$,$\Xi^-_{5b}$                            -0.15                -1.54                -0.089               -1.66
  ------------------------------------------------------------------- -------------------- -------------------- -------------------- --------------------

  : Numerical results of heavy pentaquark magnetic moments in unit of $\mu_N$ in Karliner and Lipkin’s model. []{data-label="heavy-lipkin-mag"}

It is interesting to compare the phenomenology of the above two clustered quark models. There is only a $SU(3)$ flavor anti-sextet for heavy pentaquarks in Jaffe and Wilczek’s model. In contrast, an additional flavor triplet exists in Karliner and Lipkin’s diquark and triquark model. The masses and magnetic moments of these triplet pentaquarks are equal to their anti-sextet partners in KL’s model. For the light pentaquarks, all the above two clustered quark models predicted the existence of an anti-decuplet and an accompanying octet.

Another dramatic difference lies in the prediction of the magnetic moments of the $J^P=\frac{1}{2}^+$ heavy sextet pentaquarks. The magnetic moments of the $J^P=\frac{1}{2}^+$ heavy sextet is tiny and much smaller in KL’s model than those in JW’s model. There is strong cancellation between the magnetic moment of the triquark and the orbital magnetic moment in a $J^P=\frac{1}{2}^+$ sextet when they are combined by Clebsch-Gordan coefficients. This cancellation is almost exact for bottom pentaquarks. For the magnetic moments of the $J^P=\frac{3}{2}^+$ sextet, there is also significant difference between KL’s model and JW’s models.

The third difference is that the masses of the pentaquarks in KL’s model are about 300 MeV larger than those in JW’s model. The reason is that Karliner and Lipkin assumed that the diquark (triquark) mass is simply the sum of its constituents. In contrast, the diquark mass is made much lower than the sum of the two quarks through the strong correlation between light quarks when they are in anti-symmetric configuration in JW’s model [@jaffe].

The heavy pentaquark masses and their strong, semi-leptonic and non-leptonic decays were also discussed in [@kingman; @he; @cheng; @chiu2; @huntheavy]. Besides the positive parity states, it was pointed out in Ref. [@wise] that the lowest heavy pentaquark carries negative parity in the diquark model. Interested readers may consult these references for details.

Chiral Lagrangian formalism for pentaquarks
===========================================

The approximate chiral symmetry and its spontaneous breaking have played an important role in hadron physics. Through the nonlinear realization of spontaneous chiral symmetry breaking, we may study the interaction between the chiral field and hadrons. The nonzero current quark mass breaks the chiral symmetry explicitly. Generally speaking, chiral symmetry provides a natural framework to organize the hadronic strong interaction associated with the light quarks. Chiral Lagrangian formalism has been used to discuss pentaquarks in Refs. [@mehen; @lee; @zhanga; @ko; @liu-cpt]. Interested readers are referred to these papers for the discussion of pentaquark strong decay modes. In the following I only discuss the mass splitting and selection rules arising from octet pentaquark decays.

Mass splitting
--------------

The nonzero current quark mass induces mass splitting in the multiplet. For example, these symmetry breaking terms for the decuplet $P$ and negative parity octet $O_2$ are $$\begin{aligned}
\label{pp}
L_P &=& \alpha_P \,\overline{P} ( \xi m \xi + \xi^{\dagger} m
\xi^{\dagger} ) P,\\  \nonumber L_{O_2}&=& \alpha_{O_2}
\,\textrm{Tr}\big(d_2\overline{O_2}\{\xi m \xi + \xi^{\dagger} m
\xi^{\dagger},O_2\} + f_2 \overline{O_2} [\xi m \xi +
\xi^{\dagger} m \xi^{\dagger},O_2]\big) \nonumber\\  \nonumber &&
+\beta_{O_2} \, \textrm{Tr}(\overline{O_2}O_2) \textrm{Tr} ( m
\Sigma + \Sigma^{\dagger} m ).\end{aligned}$$

Expanding Eq. (\[pp\]), we get the mass splittings $\Delta
m_i\equiv m_i-m_{penta}$ for pentaquark anti-decuplet $P$

$$\begin{aligned}
 \nonumber
\Delta m_\Theta & = & 2 \alpha_P m_s,
\\ \nonumber
\Delta m_{N_{10}} & = & \frac23\alpha_P \left( \hat{m} + 2 m_s
\right),
\\ \nonumber
\Delta m_{\Sigma_{10}} & = & \frac23 \alpha_P \left( 2 \hat{m} +
m_s \right),
\\ \nonumber
\Delta m_{\Xi_{10}} & = &  2~ \alpha_P \hat{m}.\end{aligned}$$

Or

$$\begin{aligned}
 \nonumber
m_{N_{10}}-m_{\Sigma_{10}}&=&m_\Theta-m_{N_{10}},\\ \nonumber
m_{\Sigma_{10}}-m_{\Xi_{10}}&=&m_{N_{10}}-m_{\Sigma_{10}}.\end{aligned}$$

Similarly, for the $J^P={1\over 2}^-$ pentaquark octet $O_{2}$ [@zhanga]

$$\begin{aligned}
 \nonumber
\Delta m_{N_{8_2}} & = & [2\beta_{O_2} + \alpha_{O_2} (d+f)]
(2\hat{m}) + [\beta_{O_2}+\alpha_{O_2}(d-f)] (2m_s),
\\ \nonumber
\Delta m_{\Sigma_{8_2}}&=&(\beta_{O_2}+\alpha_{O_2}d) (4\hat{m}) +
2 \beta_{O_2}m_s,
 \\ \nonumber
\Delta m_{\Xi_{8_2}}&=& [2\beta_{O_2} + \alpha_{O_2} (d-f)]
(2\hat{m}) + [\beta_{O_2}+\alpha_{O_2}(d+f)] (2m_s),
\\ \nonumber
\Delta m_{\Lambda_{8_2}}&=&(\beta_{O_2}+\frac13 \alpha_{O_2} d )
(4 \hat{m})+(\beta_{O_2}+\frac43\alpha_{O_2}d)(2m_s).\end{aligned}$$

Hence we have the mass relation $$2M_{N_8}+2M_{\Xi_8}=3M_{\Lambda_8}+M_{\Sigma_8}.$$

Selection rules from octet pentaquark decays
--------------------------------------------

In the framework of the diquark model, keeping explicit track of the flavor indices of the two diquarks within the octet pentaquark minimize the independent coupling constants and lead to some selection rules caused by the “fall-apart” decay mechanism [@maltman; @close; @lee; @zhanga]. It’s well-known that there are two independent ways to couple three octets into a singlet, i.e., the D/F scheme. However, within the diquark model, when the octet pentaquarks decay, there is only one independent coupling constant. The underlying physics is quite simple. When $\Theta^+$ pentaquark falls apart and decays into a nucleon and kaon, the $\bar s$ has to combine with a light quark to form a kaon.

In the following we use the $J^P={1\over 2}^-$ octet pentaquark decay to illustrate this point. We denote a quark and anti-quark by $q^i, {\bar q}_j$ where $i, j$ are the $SU(3)_F$ flavor indices. Note that the flavor wave function of the $J^P={1\over
2}^-$ octet and singlet pentaquark arise from $$A_{[ij]}\otimes\bar{q}_k=S\oplus O_{[ij,k]},$$ where the indices $ij$ are antisymmetric, $A_{[ij]}$ is the $\bf{3}_F$ diquark pair. $S$ is the pentaquark singlet whose indices are contracted completely. $O_{[ij,k]}$ is the octet representation. The index $k$ represents the antiquark which contracts with one of the meson index.

For the interaction of the $J^P={1\over 2}^-$ pentaquark octet $P$, nucleon octet $B$ and pseudoscalar meson octet $M$, we have [@zhanga] $${\cal{L}}_8=g_8\epsilon_{ilm}\bar{O}^{[ij,k]}B^l_jM^m_k+H.c.,$$ where $O_{[ij,k]}=\epsilon_{ljk}P^l_i-\epsilon_{lik}P^l_j$. We present the Clebsch-Gordan coefficient of each interaction term in Table \[selection-rule\].

       $\Xi^-_8$                                   $\Xi^0_8$                                     $p_8$                                      $n_8$        
  ------------------- ----------------------- -------------------- ----------------------- ------------------ ----------------------- ------------------ -----------------------
     $\Xi^-\pi^0$      $\frac{1}{\sqrt{2}}$       $\Xi^-\pi^+$                1              $\Sigma^0 K^+$    $\frac{1}{\sqrt{2}}$     $\Sigma^- K^+$              1
     $ \Xi^0\pi^-$               1                $\Xi^0\pi^0$      -$\frac{1}{\sqrt{2}}$    $\Sigma^+ K^0$              1              $\Sigma^0 K^0$    -$\frac{1}{\sqrt{2}}$
     $\Xi^-\eta_0$     $\frac{1}{\sqrt{6}}$      $\Xi^0\eta_0$      $\frac{1}{\sqrt{6}}$       $p\eta_0$       -$\frac{2}{\sqrt{6}}$      $n\eta_0$       -$\frac{2}{\sqrt{6}}$
     $\Lambda K^-$     -$\frac{2}{\sqrt{6}}$   $\Lambda\bar{K^0}$   -$\frac{2}{\sqrt{6}}$    $\Lambda K^+$     $\frac{1}{\sqrt{6}}$     $\Lambda K^+$     $\frac{1}{\sqrt{6}}$
     $\Sigma^0_8$                                 $\Sigma^+_8$                                $\Sigma^-_8$                               $\Lambda_8$     
   $\Sigma^+ \pi^-$    $\frac{1}{\sqrt{2}}$     $\Sigma^+\pi^0$     -$\frac{1}{\sqrt{2}}$   $\Sigma^-\pi^0$    $\frac{1}{\sqrt{2}}$    $\Sigma^+\pi^-$    $\frac{1}{\sqrt{6}}$
   $\Sigma^- \pi^+$    -$\frac{1}{\sqrt{2}}$    $\Sigma^0\pi^+$     $\frac{1}{\sqrt{2}}$    $\Sigma^0\pi^-$    -$\frac{1}{\sqrt{2}}$   $\Sigma^-\pi^+$    $\frac{1}{\sqrt{6}}$
   $\Sigma^0 \eta_0$   $\frac{1}{\sqrt{6}}$     $\Sigma^+\eta_0$    $\frac{1}{\sqrt{6}}$    $\Sigma^-\eta_0$   $\frac{1}{\sqrt{6}}$    $\Sigma^0\pi^0$    $\frac{1}{\sqrt{6}}$
        $p K^-$        $\frac{1}{\sqrt{2}}$      $p \bar{K^0}$                1                 $n K^-$                  1                 $p K^-$        $\frac{1}{\sqrt{6}}$
     $n \bar{K^0}$     -$\frac{1}{\sqrt{2}}$     $\Lambda\pi^+$     $\frac{1}{\sqrt{6}}$     $\Lambda\pi^-$    $\frac{1}{\sqrt{6}}$     $n \bar{K^0}$     $\frac{1}{\sqrt{6}}$
    $\Lambda\pi^+$     $\frac{1}{\sqrt{6}}$                                                                                              $\Xi^- K^+$      -$\frac{2}{\sqrt{6}}$
                                                                                                                                         $\Xi^0 K^0$      -$\frac{2}{\sqrt{6}}$
                                                                                                                                       $\Lambda \eta_0$   -$\frac{1}{\sqrt{6}}$

  : Coupling of the $J^P=\frac12^-$ pentaquark octet with the usual baryon octet and the pseudoscalar meson octet. The universal coupling constant $g_8$ is omitted. []{data-label="selection-rule"}

Diquarks, pentaquarks and dibaryons
===================================

The diquark is very similar to an anti-quark in many aspects except charge and baryon number. This feature leads to deep connection between pentaquarks and dibaryons which are composed of three diquarks [@dib].

For the $P=-$ dibaryons composed of three scalar diquarks with $L=1$, its color wave function is anti-symmetric. Its spin wave function is symmetric since diquarks are scalars. Bose statistics requires the total wave function is symmetric. Hence the product of the flavor and orbital wave function is anti-symmetric. Suppose there is one orbital excitation between two diquarks: A and B. The flavor wave function of the diquark pair A and B must be symmetric, which is the same as in the $P=+$ pentaquarks. When the orbital wave function is mixed symmetric (or anti-symmteric), the flavor wave function must be mixed anti-symmteric (or symmetric). This situation is very similar to the $L=1$ baryon multiplet in the $SU(6)_{FS}$ $70_{FS}$ representation. The only difference is that the diquark is a scalar. Simple group theory tells us that the resulting $P=-$ dibaryons are in the $8_F$ representation. Using the $\Theta^+$ mass as input, the negative baryons are estimated to be slightly above two baryon threshold [@dib].

Let’s move on to those dibaryons which are composed of three diquarks and have no orbital excitation. Three ${\bar 3}_c$ diquarks combine into a color singlet so their color wave function is antisymmetric. Diquarks are scalars. They obey Bose statistics. Their total wave function should be symmetric. Since there is no orbital excitation between scalar diquarks, their spin and spatial wave functions are symmetric. Hence their flavor wave function must be totally anti-symmetric. I.e., the resulting dibaryon is a $SU(3)_F$ singlet with positive parity, which is nothing but the H dibaryon proposed by Jaffe long time ago [@jaffeold]. It’s interesting to note that the $J^P={1\over 2}^-$ pentaquark singlet is very similar to the H dibaryon. From the available experimental constraint on the H dibaryon binding energy, one may estimate the singlet mass to be around $(1402\sim 1542)$ MeV, which is very probably low-lying in the framework of diquark model [@dib].

Summary {#sec8}
=======

The surprising discovery of pentaquarks shall be a milestone in the hadron spectroscopy, if these states are further established experimentally. Perhaps, a new landscape is emerging on the horizon, of which we have only had a first glimpse through the above experiments. Theorists have to answer the following fundamental questions:

-   Is the world composed of quarks and gluons as we are familiar with or a bunch of chiral solitons as indicated by the chiral soliton model?

-   Why are pentaquarks so different from conventional hadrons? Does $\Theta^+$ carry positive or negative parity? Why is its width extremely small? How to generate the low mass of $\Theta^+$ pentaquark?

-   Does there exist strong correlation between light quarks? Does it play an essential role in lowering the mass of the pentaquark system?

-   If so, what’s the underlying dynamics leading to the strong correlation? Is such a dynamics and correlation particular to the pentaquark system or universal?

-   Are there other “genuine” hadrons with valence quark (anti-quark) number $N=4, 6, 7, 8, \cdots$, in which quarks do not form two or more color-singlet clusters such as hadronic molecules or nuclei?

-   Is there an upper limit for $N$? In our universe, there may exist quark stars where the quark number is huge. Is there a gap in $N$ from pentaquark states to quark stars?

All these are very interesting issues awaiting further experimental exploration, which will eventually deepen our knowledge of the non-perturbative aspects of Quantum Chromo-dynamics.

The author thanks Prof Yajun Mao for providing the first figure. The author also thanks Prof W.-Y. P. Hwang and COSPA center at National Taiwan University for the warm hospitality where part of this project was done. This short review may be biased because of the author’s personal research interest. The literature of pentaquarks has increased dramatically within one year. It is impossible to cover all the important topics in this rapidly developing field within this short review. The author apologize to those whose papers and contributions were not mentioned in this work. This project was supported by the National Natural Science Foundation of China under Grant 10375003, Ministry of Education of China, FANEDD and SRF for ROCS, SEM.

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