EGENERACY SYMMETRY OF BARYON SPECTR

We present a data-supported new interpretation of the Rarita-Schwinger fields which reveals a well-defined pattern in the masses, spins, and parities of the N-, $\Delta$-, and $\Lambda$-hyperon excitations and which allows to probe the scale of the chiral phase transition for baryons. The degeneracy symmetry of the baryonic spectra emerges to be <span>Isospin$\otimes$Space</span>-<span>Time</span>.

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Introduction to the Relativistic Description of Higher-Spin States
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Particle States and their Definition
------------------------------------

The definition of particle states is at the very heart of contemporary quantum theory of fields. It takes its origin from the frame-independent Casimir invariants of the Poincaré group as was first realized by Wigner in his work of late thirties @Wi39. On that ground, quantum states of free particles in a Poincaré covariant framework are characterized by <span>*mass*</span> and <span>*spin*</span>, and transform for different inertial observers according to $$|\psi'\, \rangle   = \exp \Big[ i 
( \epsilon^\mu P_\mu - \theta^{\mu \nu}I _{\mu \nu } )\,
\Big]\,   
|\psi\, \rangle \, .
\label{state}$$ Here, $P_\mu $ and the totally antisymmetric tensor $I_{\mu\nu}$ with $\mu (\nu)=0,1,2,3$ are the generators of the Poincaré group which satisfy the commutation relations (Poincaré algebra): $$\begin{aligned}
\lbrack I_{\mu\nu},I_{\rho \sigma }\rbrack &=&
i\left( g_{\nu\rho }I_{\mu\nu} - g_{\mu\nu }I_{\nu\sigma}
+g_{\mu\sigma}I_{\nu\rho} -g_{\nu\sigma}I_{\mu\rho}\right)\, ,\\
\lbrack P_\mu ,I_{\rho\sigma }\rbrack &=&i\left(
g_{\mu\nu} P_\sigma -g_{\mu\sigma}P_\rho\right)\, ,\quad 
\lbrack P_\mu,P_\nu\rbrack =0\, ,
\label{Po_alg}\end{aligned}$$ where $g_{\mu\nu}$=diag$(1,-1,-1,-1)$ is the metric tensor, and $\epsilon^\mu $ and $\theta^{\mu\nu}$ are continuous parameters. Furthermore, $P_\mu$ generate translations, $I_{kj}=\epsilon _{kjl}J_l$ with $k,(j,l)=1,2,3$ are the generators of rotations in the $kj$-plane, while $I_{0l}=K_l$ generate the boost along each of the $l$-axis. Wigner postulated that particle states transform as <span>*classical*</span> unitary, and therefore infinite-dimensional, representations of the Poincaré group. Later on, Weinberg argued @We64 that <span>*quantized*</span> non-unitary finite-dimensional representations can also be given particle interpretation, once unitarity of the corresponding quantized field operators is ensured.

In the context of the existing baryonic spectra, we shall establish the thesis that within the above-indicated framework one obtains not only unique spin eigenstates, which can be identified with particles, such as neutron, proton, etc., but also multi-spin mass-degenerate clusters. We will present a data-supported new interpretation of the Rarita-Schwinger fields which reveals a well-defined pattern in the masses, spins, and parities of the nucleon-, $\Delta$-, and $\Lambda$-hyperon excitations and it allows to probe the scale of the chiral phase transition for baryons.

Description of Isolated High-Spin States 
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It can be directly verified that the symmetric (right-handed) $\vec{R}=(\vec{J} +i\vec{K})/2$, and antisymmetric (left-handed) $\vec {L}=(\vec{J} -i\vec{K})/2$ combinations of angular momentum, $\vec{J}$, and boost, $\vec{K}$, generate two independent left-handed (L) and right-handed (R) su(2) algebras $$\lbrack R_i,R_j\rbrack =i\epsilon_{ijk}R_k\, ,
\quad 
\lbrack L_i,L_j\rbrack =i\epsilon_{ijk}L_k\, ,\quad
\lbrack R_i,L_k\rbrack =0 \, .$$ Obviously, it is possible to define the sets of right- and left-handed states, $|j_R,m_R\rangle$, and $|j_L,m_L\rangle $, respectively, such that $$\begin{aligned}
R^2|j_R, m_R\rangle =j_R(j_R+1)|j_R, m_R\rangle\, ,
&\quad & R_z|j_Rm_R\rangle =m_R|j_R, m_R\rangle\, ,
\nonumber\\
L^2|j_L, m_L\rangle =j_L(j_L+1)|j_L, m_L\rangle\, ,
&\quad & L_z|j_L\, m_L \rangle =m_L|j_L, m_L\rangle\, .
\label{left_right}\end{aligned}$$ Equations (\[left\_right\]) suggest to use $j_R$ and $j_L$ as labels for Lorentz multiplets according to $(j_R,j_L)$. The dimensionality ($\mbox{dim}$) of $(j_R,j_L)$ is now given by dim=$(2j_R+1)(2j_L+1)$. The most interesting $(j_R,j_L)$ states are those where either $j_R$ or $j_L$ vanishes, i.e. the $(j_R,0)$ and $(0,j_L)$ states, and with $j_R=j_L:=j$. We denote those fields in turn by: $
\Phi^R_{j}:=(j,0)$, and $\Phi^L_{j}:=(0,j)$. As long as for such fields $R_i\Phi^L_{j}=L_i\Phi^R_{j}=0$ is true in a spin-independent manner for all $\Phi^L_{j}$ and $\Phi^R_{j}$, the boost generator turns out to be particularly simple and related to $\vec{J}$ via $${1\over 2}\left(\vec{J}+i\vec{K}\right)=0 
\,  \quad \mbox{for} \quad \Phi^L_{j}\, ,
\quad
{1\over 2}\left(\vec{J}-i\vec{K}\right)=0 \,  
\quad \mbox{for} \quad \Phi^R_{j}\, .
\label{L_r_boosts}$$ Consequently, $\vec{K}=i\vec{J}$ for $\Phi^L_{j}$, while $\vec{K}=-i\vec{J}$ for $\Phi^R_{j}$. The momentum dependence of the left-handed and right-handed fields is then determined through the action of the boost onto the rest-frame fields as @Ryder [@DVA] $$\Phi^R_{j} \left(\vec{p}\,\right) = \exp (
\vec{J}\cdot \hat{p}\, \varphi\,  )\,
\Phi^R_{j} \left(\vec{0}\, \right)\, , \quad 
\Phi^L_{j}\l(\vec{p}\, \r) = \exp \left(-\vec{J}\cdot \hat{p}\, \varphi\, 
\right)\,
\Phi^L_{j} \left(\vec{0}\, \right)\, .
\label{boosted_LR}$$ Here, $\hat{p}$ stands for the boost direction, while $\varphi $ is the boost parameter. As long as no skew sense can be defined at rest, $\Phi^L_{j}\l(\vec{0}\, \r)$ and $\Phi^R_{j}\l(\vec{0}\, \r)$ are indistinguishable (up to a phase factor that will be ignored for the time being @DVA1) and one can write $$\Phi^L_{j}\l(\vec{0}\, \r)=\Phi^R_{j}\,\l(\vec{0}\, \r) \, .
\label{rest_fr}$$ In inverting then, say, the first one of Eqs. (\[boosted\_LR\]), one finds $$\begin{aligned}
\Phi^R_{j}\,\l(\vec{0}\, \r) 
&=& \exp \l(-\vec{J}\cdot \hat{p}\, \varphi \r)\, 
\Phi^R_{j}\l(\vec{p}\, \r)
\label{R_L)ident}\end{aligned}$$ which, after insertion into the second of the Eqs. (\[boosted\_LR\]) yields the relation $$\Phi^L_{j}\l(\vec{p}\, \r) =
\exp \l(-2\vec{J}\cdot \hat{p}\, \varphi\, \r)\, 
\Phi^R_{j}\l(\vec{p}\, \r)\, .
\label{L_R_coupl}$$ Similarly, one finds $$\Phi^{R}_j\l(\vec{p}\, \r) =
\exp \l( 2\vec{J}\cdot \hat{p}\, \varphi \, \r)\,
\Phi^L_{j}\l(\vec{p}\, \r)\, .
\label{R_L_coupl}$$ The coupled equations (\[L\_R\_coupl\]) and (\[R\_L\_coupl\]) completely determine equation of motion and propagation of the $(j,0)\oplus(0,j)$ fields. To be specific, we will illustrate this statement on the example of the spin-1/2 field, where $\vec{J}={1\over 2}\, \vec{\sigma \, }$ with $\vec{\sigma\, }$ being the set of the well known three Pauli matrices. In this case, one has $$\begin{aligned}
\Phi_{(1/2)}^R\l(\vec{p}\, \r)
&=& \exp \left( \vec{\sigma} \cdot \hat{p}\, \varphi \right) 
\, \Phi^L_{(1/2)}\l(\vec{p}\,\r)\,\, 
=\left(\cosh\left( \vec{\sigma}\cdot \hat{p}\, \varphi\right)  +
\sinh \left( \vec{\sigma}\cdot\hat{p}\, \varphi \right)
\right) \, \Phi^L_{\l(1/2\r)}\left( \vec{p}\, \right)\, ,
\nonumber\\
\Phi^L_{\l(1/2\r)}\left(\vec{p}\, \right)
&=& \exp \left( -\vec{\sigma}\cdot \hat{p}\varphi \right)\,
\Phi^R_{\l(1/2\r)}\left( \vec{p}\right)
=\left(\cosh
\left( \vec{\sigma}\cdot \hat{p}\, \varphi\right)  -
\sinh 
\left( \vec{\sigma}\cdot\hat{p}\, \varphi \right) \right)\,
\Phi^R_{(1/2)}\left( \vec{p}\, \right)\, .
\label{coupl_eqs}\end{aligned}$$ In using the Taylor-series expansion for the hyperbolic trigonometric functions and accounting for the circumstance that $\l(\vec{\sigma}\cdot \hat p\r)^2=1$, the equation set in (\[coupl\_eqs\]) takes the form $$\begin{aligned}
\Phi^R_{\l(1/2\r)}\l(\vec{p}\, \r)&=&\l(\cosh\varphi +\vec{\sigma}
\cdot\hat{p}\,
\sinh\varphi\r)\,
\Phi^L_{\l(1/2\r)}\l(\vec{p}\, \r)\, ,\nonumber\\
\Phi^L_{(1/2)}(\vec{p}\, )&=&\l(\cosh\varphi -\vec{\sigma}
\cdot\hat{p}\, \sinh\varphi\r)\, 
\Phi^R_{\l(1/2\r)}\l(\vec{p}\, \r)\, .
\label{pre_Dirac}\end{aligned}$$ Finally, insertion of the relations, $
\cosh \varphi ={p_0/ m}$, $
\sinh\varphi ={|\vec{p}\,|/ m}
$, into Eqs. (\[pre\_Dirac\]) leads to $$\begin{aligned}
m\, \Phi^R_{\l(1/2\r)}\l(\vec{p\, }\r)&=&\l(p_0+\vec{\sigma}\cdot
\hat{p\, }\, |\vec{p\, }\, |\, \r)\, 
\Phi^L_{(1/2)}(\vec{p\, })\, ,\nonumber\\
m\, \Phi^L_{(1/2)}\l(\vec{p\, }\r)&=&\l(p_0-\vec{\sigma}\cdot\hat{p\, }\, 
|\vec{p\, }\, |\, \r)\, 
\Phi^R_{\l(1/2\r)}(\vec{p\, })\, .
\label{Weyl_eqs}\end{aligned}$$ This set of equations can be cast into the matrix form $$\left(\begin{array}{cc}
-m&p_0+\vec{\sigma\, }\cdot\vec{p\, }\\
p_0-\vec{\sigma\, }\cdot\vec{p\, }&-m
\end{array}\right)
\left(\begin{array}{c}
\Phi^R_{\l(1/2\r)}(\vec{p\, })\\
\Phi^L_{\l(1/2\r)}\l(\vec{p\, }\r)
\end{array}\right)=0\, ,
\label{Matrix_form}$$ which is nothing but the Dirac equation, $\l(\gamma^\mu p_\mu-m\r)
\psi\l(p\r)=0$, in the Weyl representation. Equations  (\[pre\_Dirac\]) and (\[Weyl\_eqs\]) show that the <span>*linearity*</span> of the Dirac equation is due to the fact that $\vec{\sigma\, }^2=1$. In general @DVA [@DVA1], the order of the differential equations for spin-$j$ particles is $\partial_\mu ^{2j}$ and is related to the fact that $p^{2j}_\mu $ is the highest power of the momentum that appears in the expansion of $\exp ( 2\vec{J}\, \cdot\vec{\phi }\, )$. The general consequence is that the equation of motion for single higher-spin states are inevitably higher-order differential equations. Now, in what follows, we shall focus onto the properties of the boost generator and reveal need for an inherent handedness of the Lorentz multiplets.

Due to the non-vanishing commutator $\lbrack J_i, K_j\rbrack \not= 0$, the boost is in general non-diagonal in the quantum number of spin. In addition, in transforming as an ordinary polar vector, the boost behaves @KimNoz as a 1st-rank spherical tensor, $Y_{1m}\l(\hat{p\, }\r)$, and connects opposite parity states with spins differing by either zero, or, by one unit. The non–vanishing matrix elements of $K_z$ are determined as $$\begin{aligned}
\Big
\langle j^{-\pi },m\Big\vert K_z\Big\vert j^\pi, m \Big\rangle 
=\lambda_j m\, ,
&\quad & \Big\langle \l(j^{-\pi} -1\r), m\Big\vert K_z
\Big\vert j^\pi, m\Big\rangle =
\zeta_j \sqrt{j^2 -m^2}\, ,
\label{bst_mael}\end{aligned}$$ while those of the ladder boost-generators, $K_\pm =K_x\pm iK_y$, are given by $$\begin{aligned}
\langle j^{-\pi },m|K_\pm |j^\pi , m\mp 1\rangle &=&\lambda_j
\sqrt{\l(j\pm m\r)\l(j \mp m+1\r)}\, , \nonumber\\
\langle j^{-\pi } -1,m|K_\pm |j^\pi , m\mp 1\rangle &= &\pm \zeta_j
\sqrt{\l(j \mp m\r)\l(j \mp m+1\r)}\, .
\label{ladder-boost}\end{aligned}$$ Here $\pi $ stands for the parity, and $\lambda_j$ and $\zeta_j$ are parameters defined as $$\lambda_j=i{
{ j_{min} \lambda } \over { j\l(j+1\r)}
}\, ,
\quad \zeta_j^2= {
{ \l(j^2-j_{min}^2\r)\l(j^2-\lambda^2\r) }\over
{j^2\l(4j^2-1\r)}
}\, ,\quad \lambda =\pm \l(j_{max} +1\r)\, ,
\label{laj_dzj}$$ where $j_{min}$ and $j_{max}$ are in turn the minimal and maximal spin within the representation considered. To be specific, for the right-handed $\l({1\over 2},0\r)$ state one has $\lambda ={3\over 2}$ while for the left-handed $\l(0,{1\over 2}\r)$ state $\lambda =-{3\over 2}$.

Equation (\[bst\_mael\]) shows that if the boost is to be diagonal in spinor space and have non-vanishing matrix elements of the type $\langle \Phi^{R\l(L\r)}_j\l(\vec{p\, }\r)\, |K_z
|\Phi^{R \l(L\r)}_j\l(\vec{p\, }\r) \rangle $, the spinors $\Phi ^{R\l(L\r)}_j\l(\vec{p\, }\, \r)$ have to be parity mixed states. This is the meaning of "handedness". As an example one may think of the $\l(1,0\r)\oplus\l(0,1\r)$ multiplet that can be associated with the electromagnetic field strength tensor in the massless limit. This multiplet has the symmetric $\l(\vec{B}+i\vec{E}\r)$ and antisymmetric $\l(\vec{B}-i\vec{E}\r)$ combinations of the vector electric ($\vec{E}\, $) and pseudovector magnetic ($\vec{B}\, )$ fields as underlying right-handed $\l(1,0\r)$- and left-handed $\l(0,1\r)$ spin-1 states, respectively. The lessons to be learned from the above considerations are two: (a) Classical single-spin-j Lorentz multiplets carry an inherent handedness that requires two states of opposite parities as ingredients, and (b) Such states are described by partial differential equations of the order $\partial_\mu ^{2j}$.

Rarita-Schwinger Fields as Multi-Fermion Clusters
-------------------------------------------------

A method for treating higher-spin states that is different from the one presented in the previous section was initiated by Weinberg around mid sixties @We64. Weinberg suggested to consider linear differential equations for finite-dimensional non-unitary representations of the Lorentz group that are described in terms of totally symmetric traceless rank-$k$ Lorentz tensors with Dirac spinor components, i.e. fields of the type $$\Psi_{\mu_1\mu_2...\mu_k}:=\left({k\over 2},{k\over 2}\right)\otimes
\left[\left({1\over 2},0\right)\oplus \left(0,{1\over 2}\right)\right]\, .
\label{RS_fields}$$ These fields satisfy the Dirac equation for each Lorentz index $$\begin{aligned}
    \l(i\partial_\lambda \gamma^\lambda - M\r)\Psi_{\mu_1 \mu_2
\cdots \mu_k} = 0\, .
\label{Dirac_Proca}\end{aligned}$$ The simplest field of the type in Eq. (\[RS\_fields\]), with $k=1$, was considered more than two decades earlier by Rarita and Schwinger @RS and applied to the description of a spin-3/2 particle. In general, all fields of the type in (\[RS\_fields\]) will be referred to as Rarita-Schwinger (RS) fields. The RS fields are conglomerations of fermions of different spins and parities. This statement is best illustrated at the level of the group O(4), the Wick rotated compact form of the Lorentz group. In noticing that the bosonic as well as the fermionic part of the RS field are constructed out of parity-mixed fields, one can pick up and consider for the sake of concreteness, the coupling of, say, a positive parity Dirac fermion to an O(4) boson composed of O(3) states of either natural $(\eta =+)$, or, unnatural $(\eta =- $) parities. These O(3) states, which can be viewed as mass degenerate, carry all integer internal angular momenta, $l$, with $l=0,\dots,\sigma-1$ and transform with respect to the space inversion operation ${\cal P}$ as $$\begin{aligned}
    {\cal P} \sigma _{\eta ;lm} = \eta e^{i\pi l}\sigma_{\eta;l-m}\,, 
\quad
l^P=0^\eta,1^{-\eta},\dots,\l(\sigma -1\r)^{-\eta}\, , \quad 
m=-l,\dots,l\, .
\label{parity}\end{aligned}$$ In coupling now the positive-parity part of the Dirac spinor to $\l({k\over 2},{k\over 2}\r)$ from above, the spin ($J$) and parity $(P$) quantum numbers of the baryon resonances are created as $$\begin{aligned}
    J^P = {1\over 2}^\eta,{1\over 2}^{-\eta}, {3\over 2}^{-\eta}, \dots,
\bigg(k+{1\over 2}\bigg)^{-\eta} \, , \quad k=\sigma -1\, .
\label{coupl_scheme}\end{aligned}$$ In the following, we will use for the spin-sequence in Eq. (\[coupl\_scheme\]) the short-hand notation $\sigma_ {2I, \eta}$, with $I$ standing for the isospin of the states considered, and $\sigma =k+1$. Equation (\[parity\]) shows that $\sigma $ directly relates to the principal quantum number of the Coulomb problem. The states of different spins and opposite parities that accompany the highest-spin state of any $\l({k\over 2}, {k\over 2}\r)$-, and, consequently, of any RS field, serve an important purpose. They prevent that the boost generator vanishes within the multiplet under consideration. To illustrate this statement, let us consider as a simple example the bosonic part $\l({1\over 2}, {1\over 2}\r)$ of the RS field with $k=1$ which is the source of the spin-$1/2^\pm $ companions of the spin-$3/2^-$ field. From Eq. (\[bst\_mael\]) follows that the boost generator $K_z$ mixes the O(3) scalar and vector subspaces of $\l({1\over 2},{1\over 2}\r)$ as the only non-vanishing matrix elements of $K_z$ in this multiplet are $$\begin{aligned}
\langle j^\prime =0^+, m=0| K_z|
j=1^-, m= 0\rangle &=& \zeta_1\sqrt{1^2-0^2}\, ,\nonumber\\
\langle j^\prime = 1^-,m=0|K_z|j=0^+,m= 0\rangle &=&
\zeta_1\sqrt{1^2 -0^2}\, .
\label{bz_mae}\end{aligned}$$ From this, and $\lambda =2$, the parameter $\zeta_1$ is obtained from Eq. (\[laj\_dzj\]) as $\zeta_1=i$. In this way the standard representation of the $K_z$ matrix in the $\l(x_0, \vec{x}\r)$ space is obtained as $$\begin{aligned}
K_z&=&\left(\begin{array}{cccc}
0&0&0&i\\
0&0&0&0\\
0&0&0&0\\
i&0&0&0
\end{array}\right)\, .
\label{b_z}\end{aligned}$$ Since Lorentz and spinor indices of the RS fields always factorize, the boost of this representation space is most naturally obtained in boosting its bosonic part independently from the fermionic one. Equations (\[bst\_mael\]) and (\[ladder-boost\]), in combination with Eqs. (\[parity\]) and (\[coupl\_scheme\]), clearly illustrate that the presence of the lower-spin components of a RS cluster guarantee in a simple manner its covariance with respect to Lorentz transformations. Note that for RS-, as well as $\l(j,0\r)\oplus\l(0,j\r)$ fields, both $K_z$ and $K_\pm $ are non-vanishing.

Now, Weinberg's suggestion was to consider the lower-spin components of the RS fields as redundant, unphysical states that have to be projected out by a set of suitably chosen Lorentz covariant auxiliary conditions. In doing so, one eventually would describe a baryon of single spin-$J=k+{1\over 2}$ and <span>*fixed parity*</span> as the highest-spin state of the $\Psi_{\mu_1\mu_2...\mu_k}$ field and use the linear differential equation (\[RS\_fields\]).

When applied to the $({1\over 2},{1\over 2})\otimes
\lbrack ({1\over 2},0)\oplus (0,{1\over 2})\rbrack $ field, the auxiliary conditions eliminate eight degrees of freedom, usually associated with the lower spin-$1/2^+ $ and spin-$1/2^-$ components from Eq. (\[coupl\_scheme\]). Immediately, one notes that such a description is not necessarily compatible with the general nature of the boost generator in Eqs. (\[bst\_mael\]) and (\[ladder-boost\]). It is little wonder, that truncated RS fields suffer various pathologies (see @VeZw for details). So one has to face the unavoidable conclusion of the principal difficulty to describe isolated higher-spin states of fixed parity in terms of linear differential equations. Note, that the problem arose out of the tacit assumption that Nature produces baryons as isolated higher-spin states. A glance at the baryon spectra teaches us that actually nature strongly favors the excitations of complete RS clusters over that of isolated higher-spin states.

We now show that the new interpretation of the RS fields is supported by the data on baryonic spectra.

Light-Quark Baryons as Rarita-Schwinger Clusters
================================================

One of the basic quality tests for any model of composite baryons is the level of reached in describing the nucleon and $\Delta $ excitation spectra. In that respect, the knowledge on the degeneracy group of baryon spectra appears as a key tool in constructing the underlying strong-interaction dynamics. To uncover it, one has first to analyze isospin by isospin how the masses of the resonances from the full listing in Ref. @Part spread with spin and parity. Such an analysis has been performed in our previous work @Ki97-98a where it was found that Breit-Wigner masses reveal on the $M/J$ plane a well pronounced spin- and parity clustering. There, it was further shown that the quantum numbers of the resonances belonging to a particular cluster fit into Lorentz group representations of the RS type. To be specific, one finds the three RS clusters with $k=1,3$, and $5$ in both the nucleon $(N)$ and $\Delta$ spectra. In terms of the notations introduced above, all reported light-quark baryons with masses 2500 MeV (up to the $\Delta \l(1600\r)$ resonance that is most probably an independent hybrid state), have been shown in Ref. @Ki97-98a to be completely accommodated by the RS clusters $2_{2I,+} $, $4_{2I,-}$, and $6_{2I,-}$, having states of highest spin-$3/2^-$, $7/2^+$, and $11/2^+$, respectively (see Fig. 1). In each one of the nucleon, $\Delta$, and $\Lambda$ hyperon spectra, the natural parity cluster $2_{2I,+} $ is always of lowest mass. It is considered to reside in a Fock space, ${\cal F}_+$, built on top of a scalar vacuum. From Eqs. (\[parity\]) and (\[coupl\_scheme\]) follows that the $2_{2I,+}$ clusters, where $I=1/2, 3/2$, and $0$, always unite the first spin-${1\over 2}^+$, ${1\over 2}^-$, and ${3\over 2}^-$ resonances. For the non-strange baryons, $2_{2I,+}$ is followed by the unnatural parity clusters $4_{2I,-}$, and $6_{2I,-}$, which we view to reside in a different Fock space, ${\cal F}_-$, that is built on top of a pseudoscalar vacuum. To be specific, one finds all the seven $\Delta$-baryon resonances $S_{31}$, $P_{31}$, $P_{33}$, $D_{33}$, $D_{35}$, $F_{35}$ and $F_{37}$ from $4_{3,-}$ to be squeezed within the narrow mass region from 1900 MeV to 1950 MeV, while the $I=1/2$ resonances paralleling them, of which only the $F_{17}$ state is still "missing" from the data, are located around 1700$^{+20}_{-50}$ MeV (see Fig. 1). Therefore, the F$_{17}$ resonance is the only non-strange state with a mass below 2000 MeV which is "missing" in the present RS classification scheme. Further paralleling baryons from the third nucleon and $\Delta$ clusters with $\sigma=6$, one finds in addition the four states $H_{1, 11}$, $P_{31}$, $P_{33}$, and $D_{33}$ with masses above 2000 MeV to be "missing" for the completeness of the new classification scheme. The $H_{1, 11}$ state is needed to parallel the well established $H_{3, 11}$ baryon, while the $\Delta$-states $P_{31}$, $P_{33}$, and $D_{33}$ are required as partners to the (less established) $P_{11}$(2100), $P_{13}$(1900), and $D_{13}$(2080) nucleon resonances. For $\Lambda $ hyperons, sparse data prevent a conclusive analyses. Even so, see Fig. 2, the RS pattern is already apparent in the reported spectrum.

5.0cm

The (approximate) degeneracy group of baryon spectra as already suggested in Refs. @Ki97-98a and further justified here, is, therefore, found to be: $SU(2)_I\otimes O(1,3)_{ls}$, i.e., Isospin$\otimes$Space-Time symmetry.

Within our scheme, the inter-cluster spacing of $200$ to $300$ MeV is larger by a factor of $3$ to $6$ as compared to the mass spread within the clusters. For example, the $2_{1,+}$, $2_{3,+}$, $4_{1,-}$, and $4_{3,-}$ clusters carry the maximal mass splitting of $50$ to $70$ MeV.

The above considerations establish that:

1.  Observed excited light-flavor baryons preferably exist as multi-fermion clusters that are described in terms of the three RS multiplets $2_{2I,+}$, $4_{2I,-}$ and $6_{2I,-}$.

2.  The above RS clusters exhaust all the resonances observed in the $\pi N$ scattering channel (up to the $\Delta \l(1600\r)$ state). They constitute an almost accomplished excitation mode in its own rights, as only 5 resonances are "missing" for the completeness of this structure.

3.  As long as the internal parity of the clusters changes from natural for the first one, to unnatural, for the subsequent two, the question arises whether this change signals a phase transition for baryons. Next section is devoted to this problem.

Probing the Scale of the Chiral Phase Transition by "Missing" Resonances
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A conspicuous feature of the RS cluster excitation mode are the even $\sigma $-values and the parity change when going from the first, $2_{2I,+}$, to the second and third RS clusters $4_{2I,-}$, and $6_{2I,-}$, respectively. In that regard, the absence of odd-$\sigma $ clusters with parity that is opposite to the one of the observed even-$\sigma $ clusters, needs explanation. A decomposition into RS clusters of the three-quark Hilbert space, ${\cal H}^{3q}$, with one excited quark at the $1s-1p-...-1f-1g$ orbits, revealed the principal possibility @MK2000 of having as an additional baryon excitation mode the $3_{2I,-}$, $3_{2I,+}$ and $5_{2I,+}$ RS cluster sequence with highest spins $5/2^+$, $5/2^-$, and $9/2^+$, respectively. These resonance groups are characterized by parities that are opposite to the observed ones. Their presence or absence probes the scale of the chiral symmetry realization in the light-quark baryon spectra. Indeed, in case the Nambu-Goldstone mode of chiral symmetry that selects ${\cal F}_+$ and natural parities, would extend for, say, the nucleon, to $\sim $1500 MeV, one can not expect to find the unnatural parity $3_{1,-}$ cluster in that excitation domain. Above 1600 MeV, ${\cal F}_+$ changes to ${\cal F}_ -$. In case ${\cal F}_-$ has been selected by a hidden mode of chiral symmetry, no natural parity clusters can occur above 1600 MeV and $3_{2I,+}$ and $5_{2I,+}$ will be absent there. If, however, chiral symmetry above 1600 MeV is realized in the manifest Wigner-Weyl mode, then the natural parity clusters $3_{1,+}$ and $5_{1,+}$ will show up. Whatever the chiral symmetry mode above 1600 MeV may be, the data strongly hint onto a parity change of the hadronic vacuum between 1550 MeV and 1600 MeV for nucleons. Therefore, in that mass region one may expect at least a meta-stable co-existence of two parity degenerate vacua and there the occurrence of parity doublets can be expected. On the grounds of the analyses performed in Ref. @MK2000 one expects a cluster that is described in terms of a totally antisymmetric Lorentz tensor with Dirac spinor components. Such a cluster gives rise to spin-$1/2^\pm$ and spin-$3/2^\pm $ parity doublets. The scale of the chiral phase transition for nucleons is therefore predicted to be around 1600 MeV. This means that opposite to the considerations of Ref. @Oka, it is the $S_{11}(1650)$ resonance that, in being built on top of a $0^-$ internal boson degree of freedom, has to be considered as the "would be" parity partner of the nucleon and has to enter the linear sigma model with parity doublets. The S$_{11}(1535)$ resonance used instead in @Oka has $1^-$ as internal orbital angular momentum and does not match with the scalar internal mode of the nucleon. In the context of the classification of baryons as RS clusters, states like the D$_{13}$(1700) and P$_{13}$(1720), or, the $D_{15}(1675)$ and $F_{15}(1680)$ resonances, do not pair, because, in belonging to the same RS cluster, their internal orbital angular momenta differ by one unit, instead of being equal but of opposite parities.

Empirical Mass Formula for the Rarita-Schwinger Clusters
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The reported mass averages of the resonances from the RS multiplets with $k=1,3$, and $5$ are well described by means of the following simple <span>*empirical*</span> recursive relation: $$\begin{aligned}
    M_{\sigma'} - M_{\sigma} = m_1\bigg({1\over {\sigma^2}} - 
{1\over {(\sigma')^2}}\bigg) + {1\over 2} m_2
\bigg({{\sigma '^2-1}\over 2}- {{\sigma ^2 -1}\over 2}\bigg)\, ,
\label{Balmer_ser}\end{aligned}$$ where, again, $\sigma=k+1$. The two mass parameters take the values $m_1=600$ MeV, and $m_2=70$ MeV, respectively. The first term on the r.h.s. in Eq. (\[Balmer\_ser\]) is the typical difference between the energies of two single particle states of principal quantum numbers $\sigma $, and $\sigma '$, respectively, occupied by a particle with mass $m$ moving in a Coulomb-like potential of strength $\alpha_C$ with $m_1=\alpha_C^2m/2 $. The term $${{\sigma ^2-1}\over 2}=2 {k\over 2}\bigg({k\over 2}+1\bigg)\, ,
\quad \mbox{with} \quad k=\sigma -1 \, ,
\label{I(I+1)_O(4)}$$ in Eq. (\[Balmer\_ser\]) is the generalization of the three-dimensional $j(j+1)$ rule (with $j=k/2$) to four Euclidean dimensions @KimNoz and describes a generalized O(4) <span>*rotational band*</span> (see Fig. 2).

5.0cm

The parameter $1/m_2=2. 82$ fm corresponds to the moment of inertia ${\cal J}=2/5 MR^2$ of some "effective" rigid-body resonance with mass $M =1085$ MeV and a radius $R=1.13$ fm. Note that while the splitting between the Coulomb-like states decreases with increasing $\sigma $, the difference between the energies of the rotational states increases linearly with $\sigma $ so that the net effect is a slightly increasing spacing between the baryon cluster levels (see Fig. 2).

Summary and Outlook
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We conclude that the lower-spin components of the Rarita-Schwinger fields are forced upon by the properties of the Lorentz boost and should not be projected out. These fields are realized in the spectra of the light-flavor baryons, where the landscape of the excitations is well structured along the RS classification scheme, rather than populated by randomly distributed resonances. The (approximate) degeneracy symmetry turned out to be SU(2)$_I\otimes $O(1,3)$_{ls}$. Within this context, the $2_{2I,+}$, $4_{2I,-}$, and $6_{2I,-}$ RS clusters observed so far in the $\pi N$ scattering channel by the LAMPF at LANL, constitute an almost accomplished excitation mode with only five states "missing". We further showed how presence or absence of the independent cluster excitation sequence $3_{2I,-}$, $3_{2I,+}$, and $5_{2I,+}$, probes the scale of the chiral phase transition for baryons. In this way, the "missing" state search program through the CLAS collaboration at JLAB @Burkert obtains an additional motivation, that is conceptually different from the SU(6)$_{SF}\otimes$ O(3)$_L$ classification scheme.

The dynamical origin for the RS clustering is still lacking a unique explanation and it remains a challenge for future research. The factorization of isospin from the space-time symmetry in SU(2)$_I\otimes $O(1,3)$_{ls}$ is strongly supported by QCD, where the isodoublet light quarks and the isosinglet heavy-flavor quarks are the established isospin degrees of freedom. On the other hand, any QCD solution has necessarily to be a Lorentz covariant object. In Ref.  @MK2000 the bosonic parts of the Lorentz clusters, such as $({k\over 2}, {k\over 2})$ from the RS field in Eq. (\[RS\_fields\]), were considered as independent fundamental bosonic degree of freedom of baryon structure, and the term "hyperquark" was coined for them. There, the clustering was modeled after a O(4) invariant quark-hyperquark correlation. From a slightly different QCD perspective, the clusters can also be viewed as string solutions associated with a linear action. Such strings are also complex multi-fermion systems having the Dirac state as a limit @Savvidy. They can be described in terms of a multi-dimensional Lorentz invariant field $\Psi $ that satisfies the Dirac-like equation $ (\Gamma^\mu p_\mu -M)\Psi =0\,$, where Lorentz invariance imposes certain condition onto the $\Gamma$ matrices. The consideration presented above show that the RS fields naturally fit into this QCD string scheme. In Ref. @Savvidy, a solution with a RS-like degeneracy was reported. On the whole, our view is that a structured baryon spectrum that shares common flavor and relativistic symmetries with QCD is more likely to be linked via an appropriate effective theory to first principles of strong-interaction dynamics than a spectrum with states benjaminsmith@example.com.

Acknowledgment {#acknowledgment .unnumbered}
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Extensive and supportive discussions with D. V. Ahluwalia on space-time symmetries and their importance for hadron dynamics helped clarifying various aspects of the RS classification scheme and are kindly appreciated.

Work supported by CONACyT Mexico.

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