Michał Praszałowicz

# Twenty years of $\Theta^{+}$

###### Abstract

Twenty years ago, in 2003, two experimental groups, LEPS and DIANA, announced
the discovery of a light, narrow, exotic baryon with mass within the range of
1540 MeV, which was later dubbed as $\Theta^{+}$. In this talk we recall the
history of this discovery and its theoretical foundations. We also discuss
possible future experiments that could determine the existence of
$\Theta^{+}$.

##  1 Introduction

Twenty years ago, in 2003, two experimental groups, LEPS [1] and DIANA [2],
announced the discovery of a light, narrow, exotic baryon with mass of the
order of $1540$ MeV, which was later dubbed as $\Theta^{+}$. Since the decay
channel was $KN$ they concluded that the observed resonance was the lightest
member of the antidecuplet of exotic pentaquark baryons, namely a
$uudd\bar{s}$ state. These searches were motivated by chiral models, in
particular by Ref. [3] and references therein, which almost two decades
earlier, predicted light pentaquark $\overline{\bf 10}$ flavor multiplet of
positive parity.

Soon other experiments presented the results of their analyses in search of
$\Theta^{+}$, some of which confirmed the initial reports of LEPS and DIANA. A
summary of the experimental results can be found in Refs. [4, 5, 6] and more
recently in Ref. [7]. Obviously, none of these experiments, including LEPS and
DIANA, has been designed to search for pentaquarks. People used data collected
for other purposes. Only later were dedicated experiments conducted, but with
mixed results. In 2004 $\Theta^{+}$ paved its way to the Particle Data Group
(PDG) listings [8] as a three star resonance, in 2005 its significance was
reduced to two stars, and in 2007 it was omitted from the summary table. As of
2008, it is no longer listed by the PDG [9].

One of the peculiar features of $\Theta^{+}$, if it it exists, is its very
small decay width. Indeed, in 2006, the BELLE collaboration reported search
results for $\Theta^{+}$ [10]. No formation signal of the $\Theta^{+}$ baryon
was observed, and an upper limit on the $\Theta^{+}$ width was estimated:
$\Gamma<0.64$ MeV for $M_{\Theta^{+}}=1539$ MeV. Also DIANA in 2006 [11]
confirmed their initial observation with the mass of $M_{\Theta^{+}}=1537\pm
2$ MeV estimating the width: $\Gamma=0.36\pm 0.11$ MeV. Is such a small width
unnatural? – a question often raised against $\Theta^{+}$. In our opinion: not
really. Let us recall that recently the LHCb collaboration at CERN reported an
excited $\Omega_{c}(3050)$ state with a reportedly very small width:
$\Gamma=0.8\pm 0.2\pm 0.1$ MeV [12]. In a later publication from 2021 [13] the
LHCb collaboration concluded that: The natural width of the $\Omega_{c}(3050)$
is consistent with zero. $\Omega_{c}(3050)$ was found in the decay to
$\Xi_{c}K^{-}$ [12] where the kaon momentum is $p=275$ MeV. This is
approximately only $\sim 10$ MeV above the kaon momentum in the decay of
$\Theta^{+}$. From this perspective the small pentaquark width is not
particularly unnatural. As a consequence, the small width of
$\Omega_{c}(3050)$ led to its interpretation as a heavy charm pentaquark
belonging to the exotic SU(3) $\overline{\bf 15}$ multiplet [14, 15, 16].

Clearly, most non-observation experiments do not really exclude the existence
of $\Theta^{+}$, but rather put an upper limit on its production cross-
section. The cleanest and decisive experiment would be the so called formation
experiment where the resonance is directly produced in $KN$ reaction. DIANA is
exactly such an experiment where the liquid Xenon bubble chamber was exposed
to a separated $K^{+}$ beam. On the contrary, LEPS was a photoproduction
experiment on a carbon nucleus ${}^{12}C$. In the follow-up analyses both
DIANA [11, 17, 18], and LEPS in the dedicated photoproduction experiment on
deuteron [19, 20], confirmed their initial findings. These results have
survived unchallenged to this day.

In this paper in Sect. 2 we briefly describe chiral models, and in Sec. 3 the
emergence of baryons including exotica. Then, in Sec. 4, we discuss
phenomenology including $\Theta^{+}$ mass and width, and briefly the
properties of other members of $\overline{\bf 10}$. Most important experiments
are reviewed in Sec. 5. Summary is given in Sec. 6.

##  2 Chiral Models and Solitons

Chiral models are effective models for Quantum Chromodynamics (QCD), which
explore chiral symmetry and its spontaneous and explicit breaking, and are
tractable in the low energy regime. One can imagine that we integrate out from
the QCD Lagrangian gluon fields. We are then left with the quark degrees of
freedom only, which have the canonical kinetic and possibly mass term, however
the interaction Lagrangian consists of an infinite number of nonlocal many-
quark vertices. Nevertheless, this effective Lagrangian will be chirally
invariant. One can truncate this Lagrangian to a local four-quark interaction
only, the so called Nambu–Jona-Lasinio model [21, 22].

To ensure chiral invariance it is convenient to introduce eight auxiliary
pseudo-Goldstone fields $\boldsymbol{\varphi}$ (pions, kaons and $\eta$) in a
form of a unitary SU(3) matrix (for models including other meson fields see
Ref. [23])

$U=\exp\left(i\frac{2\boldsymbol{\lambda}\cdot\boldsymbol{\varphi}}{F}\right)\,,$
(1)

where $\boldsymbol{\lambda}$ are Gell-Mann matrices and $F$ is a pseudoscalar
(pion) decay constant that in the present normalization is equal to 186 MeV.

The simplest Lagrangian following from the above procedure, a chiral quark
model Lagrangian, is given by

${\cal L}_{\chi{\rm
QM}}=\bar{\psi}\left(i\not{\partial}-m-MU^{\gamma_{5}}\right)\psi\,,$ (2)

where

$U^{\gamma_{5}}=U\frac{1+\gamma^{5}}{2}+U^{\dagger}\frac{1-\gamma^{5}}{2}\,.$
(3)

This remarkably simple Lagrangian has been in fact derived [24, 25] in the mid
eighties from the instanton picture of the QCD vacuum [26, 27]. Here $M$
denotes the constituent quark mass of the order of 350 MeV and $m$ is a
current  mass matrix.

Chiral symmetry corresponds to the independent global SU(3) rotations of left
and right fermions:

$\psi_{L}\rightarrow L\psi_{L},~{}~{}~{}\psi_{R}\rightarrow R\psi_{R}\,.$ (4)

Transformations (4) leave the interaction term invariant if

$U\rightarrow LUR^{\dagger}\,,$ (5)

which is nothing else but a nonlinear realization of chiral symmetry [28].
Vacum state corresponding to $U=1$ (or $\boldsymbol{\varphi}=0$) breaks this
SUL(3)$\otimes$ SUR(3) symmetry to the vector SU(3): $L=R$, and breaks the
axial symmetry $L=R^{\dagger}$.

We can further integrate the quark fields (employing suitable regularization).
Then the kinetic part for the Goldstone bosons appears [29, 30, 31, 32] and we
end up with a Lagrangian given in terms of the Goldstone bosons alone. This
Lagrangian is organized as a power series in Goldstone boson momenta, i.e. in
terms of $\partial_{\mu}U$. Such lagragians are used for precision
calculations in chiral perturbation theory [28].

The first term in $\partial_{\mu}U$ expansion, a quadratic term, is fully
dictated by the chiral symmetry, and is known as the Weinberg Lagrangian [33].
Higher order terms of known group structure have, however, free coefficients
that are not constrained by any symmetry and have to be extracted from
experimental data. Obviously, once we have at our disposal a reliable
Lagrangian like (2), we can compute effective Goldstone boson Lagrangian to
any order in $\partial_{\mu}U$.

A simple, truncated Lagrangian with four derivatives only was proposed by
Skyrme [34, 35] in 1961 and later generalized by Witten [36, 37]

${\cal L}_{\rm Sk}=\frac{F^{2}}{16}{\rm
Tr}(\partial_{\mu}U^{\dagger}\partial^{\mu}U)+\frac{1}{32e^{2}}{\rm
Tr}([\partial_{\mu}U\,U^{\dagger},\partial_{\nu}U\,U^{\dagger}]^{2})+{\cal
L}_{\rm m}\,.$ (6)

The first term in (6) is the Weinberg Lagrangian, the second one is called the
Skyrme term. Parameter $e$ can be inferred from the pion scattering and is of
the order $e=4\div 6$. Here ${\cal L}_{\rm m}$ stands for an explicit mass
Lagrangian for Goldstone bosons, which breaks chiral symmetry.

While both models (2) and (6) can be expanded in powers of
$\boldsymbol{\varphi}$ generating perturbative Goldstone boson interactions,
they also admit nonperturbative solutions of finite mass and size, namely the
solitons.

Soliton solutions in both models correspond to a specific static form of the
chiral field $U$ known as a hedgehog Ansatz

$U(\boldsymbol{r})=\exp\left(i\,\boldsymbol{n}\cdot\boldsymbol{\tau}\,P(r)\right)$
(7)

where $\boldsymbol{n}=\boldsymbol{r}/r$ and
$\boldsymbol{\tau}=\\{\lambda_{1},\lambda_{2},\lambda_{3}\\}$. Function $P(r)$
has to vanish at infinity, so that $U\rightarrow 1$. Hedgehog Ansatz (7) has a
very special property: any spacial rotation of the unit vector
$\boldsymbol{n}$ can be undone by an internal SU(2) (isospin) rotation acting
on Pauli matrices $\boldsymbol{\tau}$. This property is called hedgehog
symmetry.

In a seminal paper from 1979 Witten suggested that that baryons emerge as
solitons in the chiral effective theory [38]. While this is true both in the
Nambu–Jona-Lasinio (NJL) model (2) and the Skyrme model (6), there are
important differences. In the NJL (or Chiral Quark Soliton, $\chi$QS for
short) model, the specific form of the chiral background field (7) leads to a
rearrangement of the energy levels of the Dirac equation corresponding to (2).
The lowest positive energy level falls into the mass gap, and the sea levels
are distorted, leading to a stable static configuration corresponding to a
self-consistently determined form of the profile function $P(r)$ in (7). The
energy of this configuration is computed as a regularized sum over all energy
levels relative to the vacuum (see e.g. [39]),

$M_{\rm sol}=N_{c}\Big{[}E_{\rm
val}+\sum_{E_{n}<0}(E_{n}-E_{n}^{(0)})\Big{]}.$ (8)

This is schematically illustrated in Fig. 1. Such a configuration carries no
quantum numbers other than the baryon number following from the baryon number
of the valence quarks.

Figure 1: Schematic illustration of the calculation of the soliton mass
$\chi$QS model, which is the sum over the energies of the valence quarks, and
the properly regularized sum over the sea quarks with vacuum contribution
subtracted, see Eq. (8).

A convenient way of calculating the soliton mass (8) is variational principle
where the soliton size $r_{0}$ is treated as a variational parameter. The
result is shown in Fig. 2. We see that for large soliton size the valence
level sinks into the Dirac sea, whereas for small soliton size entire energy
is equal to the energy of the valence quarks. These two limits are called the
Skyrme Model (SM) limit and the Nonrelativistic Quark Model (NRQM) limit,
respectively. The $\chi$QS model interpolates between the two limits.

Figure 2: Soliton energy (mass) in MeV for $M=345$ MeV as a function of a
dimensionless variational parameter $Mr_{0}$: solid (blue) – total mass, short
dash (orange) – energy of valence quarks, long dash (green) – sea
contribution. Minimum of $\sim 1200$ MeV corresponds ro $r_{0}\simeq 0.5$ fm.
Figure from Ref. [40].

Therefore, the Skyrme Model can be viewed as a configuration shown in Fig. 1
where the valence level sinked into the negative energy Dirac sea. In this
case the soliton energy is given entirely by the energy of the distorted sea
levels. An approximate formula for this energy can be obtained by a heat
kernel expansion and can be expressed in terms of functionals of the profile
function $P(r)$ with no reference to the underlying Dirac structure. In the
simplest case it is just the energy corresponding to Lagrangian (6) with the
profile function $P(r)$ determined from the pertinent equations of motion.

Let’s recall that the hedgehog Ansatz $U(\boldsymbol{r})$ (7) can be viewed as
a mapping of $R^{3}\rightarrow\,$SU(3), which is charcterized by a winding
number $N_{\rm w}=P(0)/\pi$. According to Witten [38] $N_{\rm w}$ can be
interpreted as a baryon number. In the case of the $\chi$QS model the
condition $P(0)/\pi={\rm integer}$ is not necessary [40], but the number of
valence levels is given by $N_{\rm w}$. Moreover, the requirement that
$P(0)/\pi={\rm integer}$ ensures that the energy of the soliton in the Skyrme
model is finite. Therefore solitons in the Skyrme model are referred to as
topological, while in the NJL model as nontopological.

So both models described above have soliton solutions of finite energy and
size (corresponding to the size of the profile function $P(r)$), which have
baryon number $B=1$ and no other quantum numbers. We call such a configuration
a classical baryon. In the next Section we show how other quantum numbers such
as isospin or spin are generated and discuss the following mass formulas.

##  3 Baryons in chiral models

In order to provide the classical baryon with specific quantum numbers one has
to consider an SU(3)-rotated time-dependent pseudoscalar field

$U(t,\boldsymbol{r})=A(t)U(\boldsymbol{r})A^{\dagger}(t)$ (9)

and derive the pertinent Lagrangian expressed in terms of the collective
velocities $da_{\alpha}(t)/dt$ defined as follows

$A^{\dagger}(t)\frac{dA(t)}{dt}=\frac{i}{2}\sum_{\alpha=1}^{8}\lambda_{\alpha}\frac{da_{\alpha}(t)}{dt}\,.$
(10)

At this point it is important no note that $A\in\,$SU(3)/U(1) rather than full
SU(3), since for the hedgehog Ansatz (7) $[\lambda_{8},U(\boldsymbol{r})]=0$.
Therefore matrix $A$ is defined up to a local U(1) factor
$h=\exp(i\lambda_{8}\phi)$, i.e. $A$ and $Ah$ are equivalent. For this reason
the eighth coordinate $a_{8}(t)$ is not dynamical and does not appear in the
kinetic energy of the rotating soliton. Instead, it provides a constraint on
the allowed states in the collective (i.e. corresponding to rotations (9))
Hilbert space.

Standard quantization procedure [41, 42, 43, 44] leads to the rotational
Hamiltonian, which has a form of a quantum mechanical symmetric top [45]

${\cal{H}}_{\rm
rot}=M_{\rm{sol}}+\frac{1}{2I_{1}}J(J+1)+\;\frac{1}{2I_{2}}\left[C_{2}({\cal{R}})-J(J+1)-\frac{3}{4}Y^{\prime
2}\right]$ (11)

where $C_{2}(\mathcal{R})$ stands for the SU(3) Casimir operator and $J$ for
the soliton angular momentum (spin). Soliton mass $M_{\rm{sol}}$ and moments
of inertia $I_{1,2}$ are calculable within a given model. The constraint
mentioned above selects representations ${\cal R}$, which contain states of
hypercharge $Y^{\prime}$ equal to

$Y^{\prime}=\frac{N_{\rm c}}{3}\,.$ (12)

Soliton spin $J$ is equal to the isospin of states on the $Y^{\prime}$ rung of
the pertinent weight diagram, Fig.3.

In the present case for $N_{c}=3$ we have that $Y^{\prime}=1$ and the allowed
representations are

${\cal R}={\bf 8},\,{\bf 10},\,\overline{{\bf 10}},\,{\bf 27},\,{\bf
35},\,\overline{{\bf 35}},\ldots\,.$ (13)

We see that in addition to the octet and decuplet of positive-parity baryons,
well known from the quark model, exotic representations, like $\overline{{\bf
10}}$, emerge, all of positive parity.

Figure 3: SU(3) representations selected by the constraint (12). Isospin of
states on the $Y^{\prime}=1$ line is equal to the soliton spin $J$.

Collective Hamiltonian and the constraint (12) are exactly the same in the
chiral quark model and in the Skyrme model. The only obvious difference is
that the soliton mass and the moments of inertia are in the Skyrme model
expressed in terms of space integrals over some functionals of the profile
function $P(r)$, while in the case of the quark model they are given as
regularized sums over the one particle energy levels of the Dirac Hamiltonian
corresponding to Eq. (2).

Wave functions for a quantum mechanical symmetric top are given in terms of
Wigner $D^{({\cal R})}_{ab}$-functions [45]. Skipping technicalities [46],
baryon wave function takes the following form111One can find different
representations of this wave function in the literature that are equivalent to
the one used here.

$\psi_{(B,\,J,J_{3})}^{(\mathcal{R})}(A)=(-)^{J_{3}-Y^{\prime}/2}\sqrt{\text{dim}(\mathcal{R})}\,D_{(Y,\,T,\,T_{3})(Y^{\prime},\,J,\,-J_{3})}^{(\mathcal{R})\ast}(A)\,.$
(14)

Here $B=(Y,T,T_{3})$ stands for the SU(3) quantum numbers of a baryon in
question, and the second index of the $D$ function, $(Y^{\prime},J,-J_{3})$,
corresponds to the soliton spin.

We see that the problem of baryon properties has been reduced to the quantum
mechanical Hamiltonian with well defined Hilbert space and explicit wave
functions. One can therefore compute all matrix elements needed for mass
splittings, currents and other quantities.

For $m=0$ all states in a given representation ${\cal R}$ are degenerate. In
order to compute the mass splittings we have to express the symmetry breaking
Hamiltonian, which is proportional to $m$, in terms of the collective
coordinates. In the $\chi$QS model the result reads

${\cal
H}_{\mathrm{{br}}}=\alpha\,D_{88}^{(\boldsymbol{8})}(A)+\beta\,\hat{Y}+\frac{\gamma}{\sqrt{3}}\sum_{a=1}^{3}D_{8a}^{(\boldsymbol{8})}(A)\,\hat{J}_{a},$
(15)

where $\alpha$, $\beta$, and $\gamma$ are proportional to the strange quark
mass.222For simplicity we assume $m_{u}=m_{d}=0$ and then $m=m_{s}$.
Furthermore, $\alpha$ scales as $N_{c}$, and $\beta$ and $\gamma$ scale as
$N_{c}^{0}$. $\hat{Y}$ and $\hat{J}_{a}$ are hypercharge and soliton spin
operators, respectively.

Formally, $\beta$ and $\gamma$ are zero in the Skyrme model [47, 48]. In the
large $N_{c}$ limit baryons consist from $N_{c}$ quarks, and therefore the
hypercharge eigenvalue of the physical states is also $Y\sim N_{c}$. This
means that the second term in (15), including $\hat{Y}$, is of the order
${\cal O}(m_{s}N_{c})$333Matrix elements of $D_{88}^{(\boldsymbol{8})}$ are
${\cal O}(N_{c}^{0})$. exactly as the first term proportional to $\alpha$. It
was Gudagnini [41] who argued that $\beta\,\hat{Y}$ should be included in the
Skyrme model. In the chiral quark model it arises naturally from $1/N_{c}$
expansion, similarly to the sub-leading term proportional to $\gamma$.
Explicit expressions for the coefficients $\alpha,\,\beta$ and $\gamma$ are
given e.g. in Eq. (4) of Ref [49] and the mass splitting including
representation mixing are discussed in Ref. [50].

Since we have identified the symmetries of the soliton, it is straightforward
to compute the pertinent currents, in particular the axial current [51]. The
axial current is of interest here, since via the Goldberger-Treiman relation
it can be related to strong baryon decays.444This approach to the width
calculations has been criticised in the literature, see e.g. Ref. [52]. In the
non-relativistic limit for the initial and final baryons, $B_{1}$ and $B_{2}$
respectively, the baryon-baryon-meson coupling can be written in the following
form [3]:

$\mathcal{O}_{\varphi}=3\sum_{a}\left[G_{0}D_{\varphi\,a}^{(\boldsymbol{8})}-G_{1}\,d_{3ba}D_{\varphi\,b}^{(\boldsymbol{8})}\hat{J}_{a}-G_{2}\frac{1}{\sqrt{3}}D_{\varphi\,8}^{(\boldsymbol{8})}\hat{J}_{a}\right]\,\frac{p_{i}}{M_{1}+M_{2}}$
(16)

where $M_{1,2}$ denote masses of the initial and final baryons and $p_{i}$ is
the c.m. momentum of the outgoing meson, denoted as $\varphi$, of mass $m$:

$\left|\boldsymbol{p}\,\right|=p=\frac{1}{2M_{1}}\sqrt{(M_{1}^{2}-(M_{2}+m)^{2})(M_{1}^{2}-(M_{2}-m)^{2})}\,.$
(17)

The factor of 3 in Eq. (16) is a matter of convenience.

The decay width is related to the matrix element of $\mathcal{O}_{\varphi}$
squared, summed over the final and averaged over the initial spin and isospin
denoted as $\overline{\left[\ldots\right]^{2}}$, see the Appendix of Ref. [3]
for details of the corresponding calculations:

$\Gamma_{B_{1}\rightarrow B_{2}+\varphi}=\frac{1}{2\pi}\overline{\left\langle
B_{2}\left|\mathcal{O}_{\varphi}\right|B_{1}\right\rangle^{2}}\,\frac{M_{2}}{M_{1}}p.$
(18)

Factor $M_{2}/M_{1}$, used already in Ref. [14], is the same as in heavy
baryon chiral perturbation theory; see _e.g._ Ref. [53, 54].

Here some remarks are in order. While the mass spectra are given as systematic
expansions both in $N_{c}$ and $m_{s}$, the decay widths cannot be organized
in a similar way. They depend on modelling and ‘educated’ guesses, and hence
are subject to additional uncertainties [55]. Most important uncertainty comes
from the fact that the baryon masses $M_{1}$ and $M_{2}$ are formally infinite
series in $N_{c}$ and $m_{s}$. The same holds for the momentum of the outgoing
meson. It is a common practice to treat the phase factor exactly, rather than
expand it up to a given order in $N_{c}$ and $m_{s}$, despite the fact that in
${\cal O}_{\varphi}$ only a few first terms in $1/N_{c}$ and $m_{s}$ are
included. Here we have adopted a convention with $M_{1}+M_{2}$ in (16) and
$M_{2}/M_{1}$ in (18), for other choice see e.g. [55]. Formally, in the large
$N_{c}$ limit and small $m_{s}$ limit $M_{1}=M_{2}$ and all conventions are
identical. Nevertheless, if we use physical masses for $M_{1,2}$, different
conventions will result in different numerical results.

The leading term proportional to $G_{0}\sim N_{c}$ has been introduced already
in the Skyrme model in Ref. [47], whereas the subleading terms $G_{1,2}\sim
N_{c}^{0}$ have been derived in the chiral quark model [3, 51].

Since we know the collective wave functions (14), it is relatively
straightforward to compute the matrix elements for the mass splittings and
decay widths. They are simply given in terms of the SU(3) Clebsch-Gordan
coefficients [56].

##  4 Antidecuplet phenomenology

###  4.1 $\Theta^{+}$ mass

In principle we can compute the classical mass, moments of inertia (11) and
splitting parameters (15) in either model (SM or $\chi$QS), however –
following Ref. [47] – we adopt here a model-independent approach and try to
constrain these parameters from the data. Mass splittings between different
multiplets are related to the moments of inertia $I_{1,2}$ of the rotating
soliton:

$\Delta_{{\bf 10}-{\bf
8}}=\frac{3}{2}\frac{1}{I_{1}}\,,~{}~{}~{}\Delta_{\overline{\bf 10}-{\bf
8}}=\frac{3}{2}\frac{1}{I_{2}}\,.$ (19)

Therefore, we have no handle on the strange moment of inertia $I_{2}$, and it
is impossible to predict the mean antidecuplet mass from the non-exotic baryon
masses alone.

Chiral symmetry breaking terms following from the fact that
$m_{s}>m_{u,d}\simeq 0$ generate mass splittings within the SU(3)flavor
multiplets [3]:

$\displaystyle\Delta M_{\bf 8}=$
$\displaystyle\frac{1}{20}\left(2\alpha+3\gamma\right)+\frac{1}{8}\,\Big{[}\left(2\alpha+3\gamma\right)+4\left(2\beta-\gamma\right)\Big{]}\,Y$
$\displaystyle-$
$\displaystyle\frac{1}{20}\left(2\alpha+3\gamma\right)\Big{[}T(T+1)-\frac{1}{4}Y^{2}\Big{]}\,,$
$\displaystyle\Delta M_{\bf 10}=$
$\displaystyle\frac{1}{16}\,\Big{[}\left(2\alpha+3\gamma\right)+8\left(2\beta-\gamma\right)\Big{]}\,Y\,,$
$\displaystyle\Delta M_{\overline{\bf 10}}=$
$\displaystyle\frac{1}{16}\,\Big{[}\left(2\alpha+3\gamma\right)+8\left(2\beta-\gamma\right)+4\gamma\Big{]}\,Y\,,$
(20)

where parameters $\alpha,\,\beta$ and $\gamma$ are proportional to $m_{s}$.
Note that in the Skyrme model $\gamma=0$, and $\beta=0$ if we do not take into
account the Guadagnini term [41]. Equations (20) are written in a form, from
which one can immediately see that mass splittings of non-exotic baryons
depend in fact only on two combinations of parameters $\alpha,\,\beta$ and
$\gamma$, namely on $2\alpha+3\gamma$ and $2\beta-\gamma$, whereas mass
splittings in exotic $\overline{\bf 10}$ depend additionally on $\gamma$.
Again, this means that we cannot predict mass splittings within $\overline{\bf
10}$ from the spectrum of non-exotic baryons.

Resorting to the model calculations one obtains a relatively small
$\overline{\bf 10}-{\bf 8}$ splitting, namely $\Delta_{\overline{\bf 10}-{\bf
8}}\simeq 600$ MeV [57, 58]. One can now make a rough estimate of the
$\Theta^{+}$ mass accepting the Skyrme model $\Delta_{\overline{\bf 10}-{\bf
8}}$ given above and assuming that the mass splittings in $\overline{\bf 10}$
are approximately equal to the ones in the decuplet, $140\div 150$ MeV. One
obtains then that $\Theta^{+}$ mass is as low as $\sim 1460$ MeV [59]. This is
much lower than any quark model expectations. More detailed analyses in the
Skyrme model [59, 60] and in the quark-soliton model [3] led to the mass
$1530\div 1540$ MeV, which has been reinforced by the experimental results of
LEPS [1] and DIANA [2].

###  4.2 $\Theta^{+}$ width

There are two possible strategies to constrain the decay parameters
$G_{0,1,2}$ (16): one can either try to use directly data on strong decays, or
use the Goldberger-Treiman relation:

$\left\\{G_{0},G_{1},G_{2}\right\\}=\frac{M_{1}+M_{2}}{2F_{\varphi}}\frac{1}{3}\left\\{a_{0},-a_{1},-a_{2}\right\\}$
(21)

where constants $a_{0,1,2}$ enter the definition of the axial-vector current
[51, 61, 62] and can be extracted from the semileptonic decays of the baryon
octet [63].

Here, rather than discussing decay phenomenology in detail, we shall
concentrate on a very peculiar feature of the antidecupet decay constant (18)

$\overline{\left\langle
N\left|\mathcal{O}_{K}\right|\Theta^{+}\right\rangle^{2}}\sim
G_{\overline{10}\rightarrow 8}^{2},~{}~{}~{}~{}~{}G_{\overline{10}\rightarrow
8}=-a_{0}-\frac{N_{c}+1}{4}a_{1}-\frac{1}{2}a_{2}\,,$ (22)

where we have explicitly displayed the $N_{c}$ dependence following from the
pertinent $N_{c}$ dependence of the flavor SU(3) Clebsch-Gordan coefficients
[64].

For small soliton size (the NRQM limit) one can compute constants $a_{0,1,2}$
analytically [61]

$a_{0}\rightarrow-(N_{c}+2),\;a_{1}\rightarrow 4,\;a_{2}\rightarrow 2\,.$ (23)

which gives

$G_{\overline{10}\rightarrow 8}=0\,.$ (24)

We see that the decay constant of antidecuplet is zero! The cancellation takes
place for any $N_{c}$ [64]. This explains the smallness of the $\Theta^{+}$
width, which for realistic soliton size is not equal to zero, but still very
small. In contrast, the decuplet decay constant is large, $G_{10\rightarrow
8}=N_{c}+4$, explaining the large width of the $\Delta$ resonance.

This is not the only reason why $\Theta^{+}$ width may be very small. The
symmetry breaking Hamiltonian (15) inevitably introduces representation mixing
[3, 65], which in the case of the nucleon takes the following form

$\mid N^{\rm phys}\rangle=\cos\alpha\mid N_{\bf 8}\rangle+\sin\alpha\mid
N_{\overline{\bf 10}}\rangle\,,$ (25)

where $\sin\alpha>0$ is small and therefore $\cos\alpha\simeq 1$. Note that
$\Theta^{+}$ does not mix, and $\mid\Theta^{\rm
phys}\rangle=\mid\Theta_{\overline{\bf 10}}\rangle$. Therefore, the decay of
$\Theta^{+}$ to $KN$ proceeds either directly to $\mid N_{\bf 8}\rangle$ or
through mixing with $\mid N_{\overline{\bf 10}}\rangle$

$g_{\Theta NK}\simeq G_{\overline{10}\rightarrow
8}+\sin\alpha\,H_{\overline{10}\rightarrow\overline{10}}$ (26)

leading to a new decay constant $H_{\overline{10}\rightarrow\overline{10}}$

$H_{\overline{10}\rightarrow\overline{10}}=-a_{0}-\frac{5}{2}a_{1}+\frac{1}{2}a_{2}\,.$
(27)

From the model calculations and phenomenological studies one finds that
$H_{\overline{10}\rightarrow\overline{10}}$ is large and negative [65, 66,
67]. Indeed, in the NRQM limit (23)
$H_{\overline{10}\rightarrow\overline{10}}=-4$. This leads to a strong
cancellation in Eq. (26), yielding the decay width of $\Theta^{+}$ very small.

The observation that the width of $\Theta^{+}$ may be quite small was perhaps
the most important result of Ref. [3] since the smallness of its mass was
anticipated a decade earlier [57, 58, 60] (see the discussion of the
$\Theta^{+}$ width in Refs. [68, 69, 70]).

###  4.3 Other members of antidecuplet

As explained after Eq. (20), even if we anchor the exotic antidecuplet taking
for the $\Theta^{+}$ mass the experimental value from LEPS and DIANA, we still
cannot predict the masses of other members of $\overline{\bf 10}$. Here we
have two truly exotic pentaquarks corresponding to the corners of antidecuplet
(marked as yellow circles in Fig. 3), namely $\Xi^{+}$ and $\Xi^{--}$, whose
quantum numbers cannot be constructed from 3 quarks, and the remaining
cryptoexotic states whose quantum numbers can be constructed both from 3 or 5
quarks.

In 2003 the NA49 Collaboration at CERN announced the observation of an exotic
$\Xi^{--}$ pentaquark (lower left vertex in Fig. 3) at 1.862 GeV [71]. If
confirmed, it would be the second input besides $\Theta^{+}$ to anchor the
exotic antidecuplet. Unfortunately, 17 years later the successor of NA49 the
NA61/SHINE Collaboration did not confirm the $\Xi^{--}$ peak around 1.8 GeV
with 10 times greater statistics [72]. One possible reason of this non-
observation might be the extremely small width of $\Xi^{--}$. Indeed, in Ref.
[55] it was argued that in the SU(3) symmetry limit (i.e. without mixing
effects) this width is up to a factor of $\sim 2$ equal to the width of
$\Theta^{+}$, i.e of the order of 1 MeV. Original analysis of NA49 [71]
reported the width $\Xi^{--}$ below detector resolution of 18 MeV, while
NA61/SHINE [72] does not discuss their sensitivity to the width of $\Xi^{--}$.

There exists, however, a potential candidate for a cryptoexotic pentaquark,
namely the nucleon resonance $N(1685)$ [73], which has been initially
announced by GRAAL Collaboration at NSTAR Conference in 2004 [74]. $N(1685)$
has been observed in the quasi-free neutron cross-section and in the $\eta n$
invariant mass spectrum [75, 76] and was later confirmed by CBELSA/TAPS [77]
and LNS-Sendai [78]. The observed structure can be interpreted as a narrow
nucleon resonance with the mass 1685 MeV, total width $\leq 25$ MeV and the
photocoupling to the proton much smaller than to the neutron. Especially the
latter property is easily understood assuming that $N(1985)$ is a cryptoexotic
member of $\overline{\boldsymbol{10}}$ [79, 80].

The argument for small proton coupling is based on an approximate $U$-spin
sub-symmetry of flavor SU(3). Both $\eta$ and photon are $U$-spin singlets and
neutron and proton are $U$-spin triplet and doublet, respectively. The
neutron- and proton-like members of $\overline{\bf 10}$ are $U$-spin triplet
and 3/2 multiplet, respectively. Therefore in the SU(3) symmetry limit proton
photo-excitation to $p_{\overline{\bf 10}}+\eta$ is forbidden, while neutron
transition to $n_{\overline{\bf 10}}+\eta$ is allowed. For alternative
explanations see Refs. [81, 82, 83].

It was found that the width of $N(1685)$ is in the range of tens of MeV with a
very small $\pi N$ partial width of $\Gamma_{\pi N}\leq 0.5$ MeV [84]. One
should stress that the decay to $\pi N$ is not suppressed in the SU$(3)$ limit
and it can be made small only if the symmetry violation is taken into account.
Therefore in Ref. [66] masses and widths of exotic $\overline{\bf 10}$ were
reanalyzed taking into account mixing of the ground state octet with
antidecupelt, already discussed in Sect. 4.2, and anidecuplet mixing with the
excited Roper resonance octet. Taking into account all the available data on
different branching ratios and some model input it was possible to constrain
the mixing angles555Due to the accidental equality of the SU(3) Clebsch-Gordan
coefficients mixing angles of $\Sigma$ and $N$ states in ocet and antidecuplet
are equal, so only two mixing angles were necessary for the discussed mixing
pattern. leading to

$\displaystyle 1795\;\text{MeV}$
$\displaystyle<M_{\Sigma_{\overline{10}}}<1830\;\text{MeV}\,,$ $\displaystyle
1900\;\text{MeV}$ $\displaystyle<M_{\Xi_{\overline{10}}}<1970\;\text{MeV}\,$
(28)

with decay widths

$\displaystyle 9.7\;\text{MeV}$
$\displaystyle<\Gamma_{\Sigma_{\overline{10}}}<26.9\;\text{MeV}\,,$
$\displaystyle 7.7\;\text{MeV}$
$\displaystyle<\Gamma_{\Xi_{\overline{10}}}<11.7\;\text{MeV}\,.$ (29)

These limits follow from the assumptions that $\Theta^{+}$ mass is 1540 MeV
and its width is 1 MeV, and that the decay width of $N(1685)$ is smaller than
25 MeV. One sees that the decay width of $\Xi_{\overline{10}}$ is still small,
but larger than in the SU(3) limit. Its mass is still in the range scanned by
NA61/SHINE.

##  5 Experiments

The positive evidence for $\Theta^{+}$ by LEPS [1] and DIANA [2] has prompted
a number of searches by other experimental groups. At that time, only data
collected originally for searches other than $\Theta^{+}$ was available. Only
later were dedicated experiments designed and conducted. For a complete list
of experiments we refer the reader to reviews from 2008 [4], from 2014 [5] and
to a more recent review from 2022 [7]. Below we will briefly recall only a few
experiments, mainly those that have so far uphold their initial positive
results.

###  5.1 Photoproduction

Photoproduction on a nucleon is not the best experiment to discover
$\Theta^{+}$. Indeed, photon has to dissociate into an $s\bar{s}$ pair, and
the antistrange quark has to be injected into a nucleon. In practice $\gamma$
can dissociate in $K^{+}K^{-}$ but not into $K^{0}\bar{K}^{0}$. In the first
case $K^{+}$ may excite neutron to $\Theta^{+}$, but for $\Theta^{+}$ to be
produced on a proton, $K^{0}$ must be replaced by $K^{0\,*}$. Furthermore,
since $g_{\Theta NK}$ is very small the cross-section will be small as well.
The situation is even worse for the photoproduction on a proton, since
$g_{\Theta NK^{*}}$ is not known. Estimates based on the SU(3) symmetry and
experimental results on $\eta$ photoproducion off the neutron show that the
cross-section for the reaction $\gamma p\rightarrow\bar{K}^{0}\Theta^{+}$ is
of the order of 1 nb [85]. For this reason, the negative result of the CLAS
experiment with the proton target in 2006 [86] was not surprising (superseding
earlier positive report [87] from 2003).

Nevertheless, at LEPS (Laser-Electron Photon facility at SPring-8 in Japan)
$\Theta^{+}$ peak was observed just in photoproduction on a neutron inside a
carbon nucleus: $\gamma n\rightarrow K^{-}\Theta^{+}$ and subsequently
$\Theta^{+}\rightarrow K^{+}n^{\prime}$. Only kaons were detected, and the
peak was observed in the missing mass $M_{\gamma K^{-}}=E_{\gamma}-E_{K^{-}}$
distribution. The main problem was, however, that since the target neutron was
inside the carbon nucleus, its momentum was smeared by Fermi motion. After
applying the Fermi motion correction, the $\Theta^{+}$ peak was clearly
visible at $M_{\Theta^{+}}=1.54\pm 0.01$ GeV with 4.6 $\sigma$ Gaussian
significance. The width has been estimated to be smaller than 25 MeV.

Five years later, in 2008, LEPS published results from a dedicated photo-
production experiment, this time on a deuteron target [19]. Although the
measurement strategy was basically the same as in the case of carbon, the
deuteron setup offered a possibility to cross-check the pentaquark production
in a reaction $\gamma n\rightarrow K^{-}\Theta^{+}$ with $\Lambda(1520)$
production in $\gamma p\rightarrow K^{+}\Lambda^{0}(1520)$. This was possible
because the LEPS detector has a symmetric acceptance for positive and negative
particles. The analysis confirmed the existence of a narrow $\Theta^{+}$
signal at $M_{\Theta^{+}}=1.524\pm 0.002\pm 0.003$ GeV. The significance has
been estimated to be 5.1 $\sigma$ and the width much smaller than $30$ MeV.

In the meantime a number of experiments reported negative results, and
skepticism about the existence of $\Theta^{+}$ was growing. Most importantly,
in the analogous experiment carried out by the CLAS (CEBAF Large Acceptance
Spectrometer666CEBAF stands for Continuous Electron Beam Accelerator Facility
at Jefferson Laboratory located in Newport News, VA, USA.) collaboration, no
narrow peak corresponding to $\Theta^{+}$ was observed [88], contradicting the
earlier CLAS report from 2003 [89].

The CLAS experiment is analogous to LEPS, but not identical. In $\gamma d$
reaction CLAS observed all charged particles in the final state, including the
spectator proton. This required an elastic rescattering of $K^{-}$ off the
proton from a deuteron, so that the proton could acquire sufficient momentum
to enable detection. The probability of such a rescattering was an essential
factor in the CLAS analysis. Since LEPS assumed the proton to be a spectator,
the kinematic conditions of the two experiments were different. Moreover, the
angular coverage of both detectors is also different: less than 20 degrees for
LEPS and greater than 20 degrees for CLAS in the LAB system [90].

To clarify the situation, the LEPS collaboration performed the search for
$\Theta^{+}$ in $\gamma d\rightarrow K^{+}K^{-}np$ reaction with 2.6 times
higher statistics. The peak was still there. In 2013–2014 a new measurement
was performed with the improved proton acceptance. Partial results have been
published in different conference proceedings [90, 91, 92] but to the best of
our knowledge, a full fledged journal article has not yet been released.

At the end of 2022, the LEPS2 detector started to collect new data in the
search for $\Theta^{+}$ [93]. LEPS2 detector has better angular coverage than
LEPS and will look for $\Theta^{+}$ in the following reactions [94]: (1)
$\gamma n\rightarrow K^{-}\Theta^{+}$ and (2) $\gamma
p\rightarrow{\bar{K}}^{0\,*}\Theta^{+}$, where $\Theta^{+}$ will be
reconstructed from the following reactions $\Theta^{+}\rightarrow pK^{0}_{\rm
S}\rightarrow p\,\pi^{+}\pi^{-}$ and in the second case additionally
${\bar{K}}^{0\,*}\rightarrow K^{-}\pi^{+}$. Apparently, all four or five
particles in the final state will be identified, which means that the
uncertainty of the previous measurements due the Fermi motion of the target
neutron or proton will be removed. Therefore, we look forward to future
results.

To circumvent the problem of smal $g_{\Theta NK}$ coupling the authors of Ref.
[95] proposed in 2006 to look for $\Theta^{+}$ at CLAS in the interference
with $\phi$ meson, which is copiously produced. The interference cross-section
is linear in the $\Theta^{+}$ coupling and hence can be substantially larger
than the production cross-section where $\phi$ contribution is removed by
kinematical cuts. Such analysis was published six years later [96] with
positive result. Nevertheless, this paper has not been formally approved be
the entire CLAS Collaboration, which criticised kinematical cuts applied in
[96] and published an official disclaimer [97].

###  5.2 Resonance formation

Unlike photoproduction, resonance formation in $KN$ scattering is the cleanest
experiment possible in the search for $\Theta^{+}$. The Breit-Wigner cross-
section for a production of a resonance of spin $J$ and mass $M$ in the
scattering of two hadrons of spin $s_{1}$ and $s_{2}$ takes the following form
(see e.g. Eq.(51.1) in Ref. [98])

$\sigma_{\rm BW}(E)=\frac{2J+1}{(2s_{1}+1)(2s_{2}+1)}\frac{\pi}{k^{2}}B_{\rm
in}B_{\rm out}\frac{\Gamma^{2}}{(E-M)^{2}+\Gamma^{2}/4},$ (30)

where $E$ is the c.m. energy, $k$ is the c.m. momentum of initial state and
$\Gamma$ is the full width at half maximum height of the resonance. The
branching fraction for the resonance into the initial-state channel is $B_{\rm
in}$ and into the final-state channel is $B_{\rm out}$ – in the present case
for $KN$ scattering and one of the possible final states $K^{+}n$ or $K^{0}p$
we have $B_{\rm in}=B_{\rm out}=1/2$. Substituting the $\Theta^{+}$ mass one
gets that the cross-section at the peak $\sigma_{\rm BW}(M_{\Theta^{+}})\sim
15\div 20$ mb. This is a model independent prediction, and we see that the
cross-section for $\Theta^{+}$ production in $KN$ scattering is large. More
detailed study of $\Theta^{+}$ production in $K^{+}d\rightarrow K^{0}pp$
reaction shows that the production cross-section is in this case of the order
of 5 mb [99]. The feasibility study of searching for $\Theta^{+}$ in this
channel at J-PARC has been recently performed in Ref. [94].

The formation process was used in the DIANA experiment where the bubble
chamber DIANA filled with liquid Xenon has been exposed to a $K^{+}$ beam from
the ITEP proton synchrotron. In Ref. [2] the authors analyzed the $K^{0}p$
effective mass spectrum in the reaction $K^{+}n\rightarrow K^{0}p$ on a
nucleon bound in a Xenon nucleus. A resonant enhancement with $M=1539\pm 2$
MeV and $\Gamma\leq 9$ MeV was observed. The statistical significance of the
enhancement was estimated to be 4.4 $\sigma$.

The DIANA collaboration continued analysis of the bubble chamber films and in
2006 published new results from the larger statistics sample [11]. They
confirmed their initial observation with the mass of $M=1537\pm 2$ MeV with,
however, much smaller estimate of the width: $\Gamma=0.36\pm 0.11$ MeV.
Depending on the significance estimator they obtained statistical significance
of 4.3, 5.3 or 7.3 $\sigma$. Three years later they increased again statistics
confirming the existence of $\Theta^{+}$ with approximately the same mass and
width, but higher statistical significance reaching 8 $\sigma$ [17]. These
results were confirmed in their last publication from 2014 [18].

As already mentioned the formation experiment with $K^{+}$ beam could be
easily performed at the J-PARC facility in Japan looking at the three body
final state $K^{0}pp$ [94]. Another very promising search for $\Theta^{+}$
will be possible at already approved program at $K_{L}$ facility at JLab [7,
100, 101]. Here, with a secondary beam of kaons, one may look at a two-body
reaction $K^{0}_{L}p\rightarrow K^{+}n$ on the hydrogen target. The plan is to
measure the initial energy benefiting from the design momentum resolution
below 1 MeV, rather than the invariant mass of $K^{+}n$ system. According to
the current schedule data collection will start in 2026 [101]. Note that the
two-body final state is much cleaner than the three-body one, which is
proposed to be studied at J-PARC. Finally, at the $K_{L}$ facility one will
also be able to look for other members of antidecuplet, like $\Xi^{+}$.

##  6 Summary

Although pentaquarks with heavy quarks are by now well established exotic
baryons, $\Theta^{+}$ still remains elusive. Its story is full of unexpected
twists, emotions and lost hopes. The purpose of this review was to recall the
theoretical basis of $\Theta^{+}$ and some experimental evidence. While the
small mass of exotic antidecuplet can be easily justified in the chiral
models, where the $q\bar{s}$ pair is injected into a nucleon not as two
independent quarks, but rather as an almost massless Goldstone boson, its very
small width has often been considered unnatural. In the Chiral Quark Soliton
model the small width of $\Theta^{+}$ is natural due to the cancellation of
the corresponding couplings in the non-relativistic limit. Unfortunately,
since there is no intuitive argument as to why this is the case, many authors
did not take it seriously. Moreover, the chiral models discussed here are
largely considered only qualitative, although they are able to describe non-
exotic baryons quite well. However, this is also true of other models that
have trouble accommodating light exotic baryons.

The fate of $\Theta^{+}$ can only be decided experimentally. In our opinion
the most promising are formation experiments that can be conducted at the JLab
and J-PARC facilities. The time scale is here of the order of a few years.
Even earlier one may expect results from the photoporduction experiment at
LEPS2. Whatever the results, we will eventually get a clear answer to the main
question that has been bothering us in this article, whether $\Theta^{+}$
exists or not.

## Ackowledgements

We would like to thank the organizers of the Corfu Workshop 2023 for their
warm hospitality and the opportunity to present these thoughts. We would also
like to thank Moskov Amaryan and Igor Strakovsky for their helpful comments
and encouragement and Hyun-Chul Kim for long collaboration on exotica.

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