Hadronic structure on the light-front IX . Orbital-spin-isospin wave functions of baryons

This paper which is part of a series, is devoted to several technical issues. In the first part of the paper, we discuss the usual wavefunctions in the CM frame for baryons, by clarifying the representations of the three-quark permutation group $S_3$. We extend the analysis for up to five ``spinors"with $\rho,\lambda$-symmetry, and derive explicitly the totally symmetric wavefunctions modulo color. They are explicitly used to describe the excited nucleons $N^*$ states, in the P- and D-shell. We also show how to use symbolic operations in Mathematica, in spin-tensor notations to make explicit these states. For the S- and P-shells, the totally antisymmetric wavefunctions are given, and the pertinent matrix elements for the spin-dependent operators calculated, including the mixing between states with different total spin $S$. In the second part of the paper we turn to the light front wavefunctions, with an emphasis on the longitudinal wavefunctions, with a novel basis set. We also discuss their symmetries under permutations, and select the proper combinations for the transverse and longitudinal excitations for $N^*$ on the light front.


I. INTRODUCTION
The nonrelativistic quark model, which originated in the 1960's, was first based on spin-flavor SU (6) symmetry.Using its multiplets it was possible to define the baryon wavefunctions for the lowest S-shell, and calculate their quantum numbers and important properties, e.g.magnetic moments.With the advent of QCD and quarkonia in the 1970's, perturbative and confining forces between quarks were added, including spin-dependent interactions, leading to a qualitatively successful spectroscopy.One important simplification was the use of quadratic (oscillatory) confinement, see e.g.[1,2].
The extension of this theory to excited baryons, especially to the P-shell with negative parity [3], revealed a number of issues.To simplify some of the symmetry issues for three-light-quark states qqq, they focused on the mixed qqs states (Σ, Λ) ones, and then discussed the light quark limit m s → m u,d .
They have shown that the perturbative predictions from the one-gluon exchange worked qualitatively well only for spin-spin and tensor forces, while the spin-orbit calculated in the same approximation was absent.A key shortcoming at the time, was the missing D-shell states predicted but not observed ("missing resonances problem").Further re-finements, including in particular the inclusion of relativized quarks, were made by Capstick and Isgur [4].With time, the issue of "missing resonances" faded away, as the states in the second and the third shells were nearly all observed.
More recent quarks models for mesons and baryons were defined using light front variables, with Light Front Wave Functions (LFWFs).The advantage of the light front, is the potential to relate the LFWFs to partonic observables, DAs, PDFs, GPDs etc.This formulation, treats democratically heavy and light quarks, with no need for a nonrelativitic approximation.The LFWFs for hadrons made of light and heavy quarks are developed in the same setting, which is very convenient for the discussion of the multi-quark hadrons discovered in the last decade.
The first step for the description of the LFWFs, has been made by Ji et al [5], who have classified the components of the spin-up proton in terms of spin/orbital helicities.In this formulation the wavefunctions of the 3-quark sector are multicomponent, e.g. the (ground shell) nucleon is ascribed to possess six unrelated functions.For excited baryon states the issues related with required quark permutation symmetry were not resolved.Also, the LFWFs for baryonic excitations with nontrivial orbital-spinisospin wave functions, has not been addressed to our knowledge.
The aims of this technical paper are twofold.First, we wish to clarify how to build symmetric representations of quark permutation symmetry S 3 , as it follows from quark Fermi statistics.There is no need to take the indirect root via strange baryons again and again, we re-derive the explicit wavefunctions for the ground S-shell, negative parity P-shell and D-shell nucleons.We propose to use standard spin-tensor forms with many components, conveniently manipulated symbolically in Mathematica.We then turn to the derivation of the LFWFs, clarifying their orbital-spin-isospin structure.
Our other aim is to use the available phenomenological information on nucleon excitations, and translate it into matrix elements of the central and spin-dependent interquark forces.Since the 1970's, spectroscopists still use the one-gluon exchange forces plus linear potential due to the electric flux tubes.But even for heavy quarkonia, we found in [6], that its central potential can be derived from instanton-based model of the vacuum fields, with the benefit of also predicting spin-dependent nonperturbative forces, at least comparable in magnitude to the perturbative ones.In [7] we calculated the instanton-induced quark-quark forces in various baryons, and compared them to the empirical and lattice data.
In our next publication we will continue along this line, focusing on interquark forces in baryons.While we start with generic three-Wilson-lines, the outcome are binary interactions only.We will also generalize our approach to spin-dependent forces.All of that can also be further generalized to the recently discovered multi-quark hadrons.

II. LIGHT QUARK BARYONS IN THE CM FRAME
The physics of baryon resonances is a very broad field.It is also rather old.It started in the 1950's with discoveries of the first N * and ∆ resonances.The QCD-based quark models are also about 50 years old.The systematic analysis of the data are in the Particle Data Tables, for recent review see e.g.[8].Basically, it appears that the "missing resonance" problem is solved, and the states belonging to the ground, second and third shells are all observed, with quantum numbers for nearly all of them determined.
So, the reader may think that all of the main questions in this field are answered, and doubt that anything useful (rather than new experimental data) can be added to it.In fact, the situation is far from that.The structure and especially the physical origin of the quark-quark forces are not yet understood, especially of the spin-dependent part.Lattice studies of the static spin-dependent potentials are sketchy, and in so far reduced to either qq channel or diquarks.These issues are now discussed again, in view of the recent renaissance in the hadronic spectroscopy, due to the discovery of many new multiquark hadrons.
The question whether spin-dependent splittings of baryons are due to perturbative one-gluon exchange, or instanton-induced 't Hooft Lagrangian has been discussed in [9].Another proposed mechanism, via pion exchanges between quarks [10], appears as a higher order effect of the t Hooft Lagrangian, as the pions themselves are mostly bound by it.(We will not include those, trying to avoid a double counting.)In this vain, it is worth mentioning also, the solitonic construction based on the large number of colors limit [11], which has been also extended to the exotics via the holographic principle [12,13].
In this work we would like to start with another re-analysis of the situation, with a phenomenological analysis of the data, mostly of positive parity (ground shell) and negative parity (second shell) light baryons and discuss whether the existing data can accommodate certain matrix elements of all spin-depending operators, namely spin-spin, spinorbit, tensor as well as instanton-induced 't Hooft operator.
In order to see the importance of various spindependent forces in action, it is not sufficient to look at the lowest shell baryons, N and ∆, as there is basically just one mass difference between them.To reach a quantitative understanding of the magnitude of the spin-dependent forces, we need to look at more states.An important step in this direction was made by Isgur and Karl (IK) [1] who focused on the P-shell negative parity baryons.The Jacobi coordinates ⃗ ρ, ⃗ λ are antisymmetric and symmetric under permutation of the 1 − 2 quarks, which leads naturally to the use of blocks with the same ρ, λ structure also for the orbital, spin and isospin parts of the wavefunctions [14].This observation was used extensively in the work of IK and their followers.IK focused on hyperons with a strange quark, for which the wave function with 12-symmetry was sufficient.The light quark baryons were then constructed as certain limits from the combination of hyperons, with the strange quark mass reduced to the light ones.
Unlike Isgur-Karl, we decided not to deal with ex-tra parameters related with strangeness, and focus entirely on light quark states.As it is well known, negative parity u, d baryons belonging to the second L = 1 (or P-shell) have seven states, five N * and two ∆ * .Five masses of the N * and two matrix elements of mixing (J = 3/2 and J = 1/2 pairs with different spins S) are seven inputs.We will use the inverse logic adopted by most spectroscopists: instead of formulating a model and then comparing its prediction with various data, we decided to start with estimates of the phenomenological values of the matrix elements of several contributing operators.With the structure of the wavefunctions at hand, and spinorbit-isospin structure of the operators known, one can write masses as linear combinations of matrix elements.There is enough information to fix those uniquely.

A. Excited nucleons, negative parity
We recall that that the spin states of three quarks have 2 3 states: four of those belong to the S = 3/2 case, and the remaining two pairs of states have spin S = 1/2 but different symmetries under permutations.Adding orbital momentum L = 1 to the former set leads to J = 5/2, 3/2, 1/2 N * states.In the latter case two S = 1/2 structures need to be combined with two isospin I = 1/2 structures, to get a single combination with the correct permutation symmetry: this leads to another pair of J = 3/2, 1/2 of N * .Two pairs of N * from those two families, with the same J, are intermixed by the tensor forces.
We start our phenomenological discussion with Table I, where the current experimental masses for the P-shell states are recorded.We also compare these empirical masses with the original predictions by Isgur and Karl [1].
There are also mixing angles of the S = 3/2 and 1/2 states determined from the decays: those are listed in the RPP reviews as being θ S1/2 = −32 o , θ S3/2 = 6 o .The relation of the energies of the mixed states to the unmixed ones is and the mixing angle to the non-diagonal mixing matrix element is We may use the values of the observed masses M ± , and the observed mixing angles to derive the "unmixed masses" M 1,2 with fixed total spin S, which we give in the last column in the Table I.Similarly, the numerical values of two mixing matrix elements are Below we will use five masses to get five matrix elements, and compare the value of the tensor matrix element to these two mixing matrix elements.

B. Permutation symmetry S3 and three quark wavefunction
Our first task is to clarify the use of S 3 for the construction of baryon orbital-spin-isospin wavefunctions, which can be made uniquely.Afterwhich, we will use phenomenology to evaluate the matrix elements of all pertinent operators.
For completeness, we start with well known basic facts.Fermi statistics require antisymmetry over all permutations of quarks.With antisymmetric color wave function ∼ ϵ ijk , the remainder of the wavefunction needs to be totally symmetric For orbital momenta L i = 0 (S-shell) states, the traditional classification follows the representations of the spin-flavor SU (6) group of the nonrelativistic quark model The nucleon belongs to the mixed symmetry 8 M octet, which is part of the symmetric 56-plet.As we do not include strange hyperons, flavor SU (3) is reduced to the isospin SU (2) group.This helps to make spin and isospin notations more similar.The three quarks coordinate vectors are compressed into two Jacobi coordinates with ⃗ r ij = ⃗ r i −⃗ r j and 3D vectors assumed.As noted by Isgur and Karl [3], simple symmetry properties of φ ρ,λ LM under the permutation group, suggest a convenient set of basis functions, both in spin and isospin, out of which the baryon wavefunctions can be constructed.Most of the details are given in Appendix A, with only the main ideas presented here.
To compose the states with proper symmetry under the permutation group S 3 , we recall that this group is composed of 6 elements P i=1,...,6 = I, (12), ( 13), ( 23), ( 123), (132) It will be enough to enforce the symmetry under ( 12) and ( 23) permutations.Let us add that there are three Young tableau's for this group, symmetric, antisymmetric and mixed (two boxes in the top line and one below).The Jacobi coordinates (6) transform under permutations as follows Thus ρ, λ doublet transformation under ( 12) is simply antisymmetric and symmetric, but it is of mixed symmetry under the (23) permutations.Therefore, the construction of the states out of the ρ or λ-type blocks (as done e.g. by Isgur and Karl [1]) allows for the easy removal of the contributions that are not (12) symmetric.However, it is not enough: one needs to enforce full S 3 symmetry.The corresponding matrices of (23) permutation for two and three quark is required (see Appendix), ultimately fixing the wavefunctions uniquely.
Let us start with the (well known) proton state.In the litterature one can find it in several forms, using either products of blocks with different symmetries, or a sets of "monoms" (basic states): Its spin and isospin components have mixed symmetry, but taken together, they are symmetric under all permutations.(Among its classic applications are the neutron-to-proton ratio of magnetic moments −2/3, etc.)But in order to derive the wavefunctions for the baryon excitations, we need a more systematic approach.
There are 2 3 = 8 spin (or isospin) states of three quarks.Four of those correspond to total spin S = 3/2, and four other to S = 1/2.It is convenient to split them according to their symmetry under (12).
Examples of such doublets are the spin− 1 2 states Similar isospin-1 2 states are defined as The S = 3 2 (or I = 3 2 ) are fully symmetric To streamline the states with S 3 symmetry R = S, A, M (symmetric, anti-symmetric, mixed), we introduce the spectroscopic notation LSJm X P R with the hadronic label X P R for X = p, ∆, ... with parity P = ±, and L, S, J, J z = m for orbital, spin and total angular momentum with projection J z = m, respectively.
The simplest baryon ∆ have S = I = 3 2 and both spin and isospin wave functions are symmetric S S constructions (12) The proton ground state WF should have spinisospin S = I = 1 2 , so it should be constructed out of (II B).There are four combinations.Two of them S ρ F λ , S λ F ρ are asymmetric under (12) and should be rejected.Two others are symmetric: any combinations of those is symmetric under (12).To fix the wave function uniquely, we need to calculate their transformation under (23) permutations.As shown in the Appendix, the only combination symmetric under both (12) and ( 23) is where the radial wave function (ρ2 + λ 2 ) depends on 6d "hyperdistance" ∼ (ρ 2 + λ 2 ) Let us now proceed to the more general case of excited baryons, with the orbital part of the wave function included.In general, there could be angular functions depending on the angles of both vectors ⃗ ρ, ⃗ λ, but for the P-shell we are interested in, it is either L ρ = 1, L λ = 0 or L λ = 1, L ρ = 0. So, for the spatial WFs φ ρ,λ LM with orbital content of mixed symmetry under S 3 , we define (15) with ρ ± = (ρ 1 ± iρ 2 ) and λ ± = (λ 1 ± iλ 2 ).Up to module of the two vectors, they are the standard angular functions Y m 1 with m = −1, 0, +1.They combine with the possible spins to give 5 nucleons negative parity proton excited states, i.e. two − .Yet to construct the wavefunctions, we need to include also the isospin in a proper way.Modulo color, the baryon wavefunction has three components, orbital, spin and isospin.(Not to be confused with the number of quarks: those will be there for any quark number.)As shown in Appendix A, three objects are combined together into a unique combination (A10).
The totally antisymmetrized wavefunction for 3 quarks with the maximally streched spin are with S S =↑↑↑ the totally symmetric 3-quark spin 3 2 state.
In general, the addition of spin S states with orbital functions to a particular J, requires a sum of kinematically possible states, m = J z = m S + m L , using the standard Clebsch-Gordon coefficients Note that we use convention for the Clebsch-Gordon coefficients in terms of Wigner 3-j symbol The lower J-states given L, S follow by Clebsch-Gordoning, and for the odd parity J = 1 2 , 3 2 shells in the isobar, The method for deriving the orbital-spin-isospin wavfunctions out of the ρ, λ blocks of mixed permutation symmetry, is basically known.Their use using Jacobi-like combination was carried by Isgur and Karl.However, the explicit wavefunctions in the current literature, are not in a standard form useful for applications to multiquark states.Since while working on this paper we have developed them, we will explain the simple rules on how it was done in Mathematica, and present a full set of explicit wavefunctions in Appendix B.
Representation theory of angular momentum tell us that one can construct positive parity excited nucleon states with J P assignments + However, the explicit construction of the wave functions symmetric under quark permutations needs further attention.These wave functions were discussed e.g. in [15].However the full treatment is better achieved using the representations of the S 3 permutation group, to be discussed below in Appendix A.
Since orbital part of the wavefunction for J P = 2 + are symmetric tensors constructed out of coordinate vectors ρ i , λ i , one has three options for this The first two are symmetric under [12], the last is antisymmetric.Under [23] permutation their transformation is involved under (A3).To get totally symmetric wavefunctions, they need to be supplemented by spin and isospin wavefunctions with appropriate symmetries.
For the total spin S = 3 2 , the corresponding spin wavefunctions are symmetric (e.g.↑↑↑), and we need to apply the S 3 representation with three objects (two coordinates and isospin) X A1 X A2 X A3 with binary indices A = ρ, λ.We have shown that it is a single combination (A10).
For the total spin S = 1 2 , there are two options S ρ , S λ .The most straightforward way to construct the wavefunctions is via building symmetric representation of f our spinor-like objects with binary indices A = ρ, λ.Their total number is 2 4 = 16, half of them symmetric and half antisymmetric under [12] permutations.The 16 × 16 matrix (not shown) of [23] permutations have 8 symmetric and 8 antisymmetric eigenvectors.In the Appendix, we show that there are two symmetric under all permutations, which can be rewritten as Their linear combinations generate tensor excited states with spin S = 1 2 .For those with fixed J, J z one needs, as usual, to calculate the Clebsch-Gordan coefficients.
In the Appendix we have pushed the method one step further, to 5 ρ, λ blocks.Out of 32 states, we found 3 symmetric eigenstates.

IV. SPLITTINGS OF NEGATIVE PARITY NUCLEONS
The details of the operators and calculation methods are presented below and in the Appendices, but we start by summarising the results.
All the matrix elements contain 6-dimensional integrals over ⃗ ρ, ⃗ λ coordinates which are split into two sets, 4d angular and two remaining radial times pertinent matrix elements.The wavefunctions consist of a common 6d spherically symmetric function ϕ 00 (R), R 2 ≡ ρ 2 + λ 2 , times various orbital parts.For the P-wave baryons under consideration, the orbital parts are linear in ⃗ ρ, ⃗ λ.The matrix elements receive multi-contributions from the different parts of the symmetrized wavefunctions, as detailed in the Appendix.Four angular integrals will be performed in matrix elements to follow, while integrals over the moduli ρ, λ remains undone, as no assumptions about ϕ 00 (R) is made.We introduce a shorthand notations for such integrals For example, in such notations the normalization integral ( V = 1) "averages" the orbital wave function squared, namely This combination will appear in denominators of all terms.
The masses of the N * J excited nucleons have some overall constant (not written below) plus contributions of four spin operators, e.g.spin-spin, spinorbit, tensor and 't Hooft terms.The resulting mixing matrices in the P-sub-shells are The rows and columns of the 2 × 2 matrices, correspond to states with S = 3 2 , 1 2 , respectively, Some of these matrices have nondiagonal (or mixing) matrix elements, which means that the total spin S is not a good quantum number of the observed states, We now explain how the matrix elements are calculated.All operators used should be symmetric under quark permutations, as are all the orbital-spinisospin wavefunctions.In most cases we calculated the pair 12-interactions, and then multiplied the result by 3 (accounting for three pairs, 12, 13, 23).In Jacobi coordinates, the 12-distance is d 12 = √ 2ρ, so most potentials are fixed at that distance.
The operators are defined as follows ] Since , the corresponding terms are proportional to ⟨ρ 2 V LS ( √ 2ρ)⟩.the numerical factors in front each contribution, stem from the convolution over all indices of operators, wavefunctions and 4d angular integrals.
The exception is the instanton-induced interaction taken in the local approximation, where ⃗ τ i , ⃗ σ i are Pauli matrices for isospin and spin, respectively.The 't Hooft interaction, is quasi-local in nature, and for the 12-quark pair is proportional to ∼ G tH δ 3 ( ⃗ d 12 ) .As a result, the integral over ⃗ ρ drops out, and only the integral over λ remains, so we define for it a lambda-only averaging A. Extracting matrix elements from N * data The knowledge of the coefficients of the relevant operators for the five (unmixed) masses of negative parity N * , allows to fix the pertinent matrix elements.This immediately reveals one striking feature, already noted by Isgur and Karl [2]: the role of spin-orbit is an order of magnitude (or more) suppressed relative to the spin-spin and tensor force.The same observation follows for the 't Hooft operator which we tried to include: it is not improving the fit and its expectation value is within the error bars.(For P-shell baryons, not so for the ground state nucleon).
In view of this we resorted to what we call "an optimized IK model", which ignores LS and 't Hooft terms and keep only the spin-spin and the tensor forces.In this case, the fitted matrix elements can be considered reliable.
where H 0 is the spin-independent part of the Hamiltonian, common to all five resonances.Our only assumptions so far are using spindependent forces to the first order, and ignoring near-threshold effects.(Isgur and Karl made additional assumptions, such as a Gaussian S-wavefunction, used to evaluate them.) The quality of the overall description of masses can be seen in Fig. 1.One can see that the agreement of this model with the data is very good.In all cases, it is significantly better than the half-widths of these resonances (which provides a scale for the ignored threshold effects).
The small blue circles show the experimental pole positions.They are surrounded by large dashed blue circles, with radius of the order of typical half width 1  2 Γ.
As "external tests" of the accuracy of such a model, we use the empirical mixing matrix elements (3).The observed ratio for the magnitude of the mixing matrix elements in the tensor case is If only the tensor operator is left, the ratio of its coefficients − √ 10 = −3.16 is sufficiently close to it.The value from the fitted tensor matrix element to the splittings in Fig. 1, we have the value (37) or 43.7 M eV .This is to be compared to 51.6 M eV from the mixing matrix element in the J = 1  2 -shell pair as discussed above.Hence, some other effects, not yet accounted for here, contribute at the level of ∼ 8 M eV .A similar conclusion can also be inferred from the deviations between the observed and fitted masses seen in Fig. 1.These deviations should be due to higher order corrections not yet accounted for.They are reasonably small compared to the splitting themselves ∼ 100 M eV as well as to Γ/2 ∼ 50 M eV used as the maximal possible shifts due to threshold locations.

V. SPIN-DEPENDENT INTERACTIONS IN BARYONS
Having summarized so to say the"applied aspects" of the calculations, let us return to the theory.Before discussing the spin-dependent effects, few words about the spin-independent potentials.Those are expressed in terms of Wilson-line correlators, or complicated nonlocal correlations of (Euclidean) vacuum gauge fields.They can be computed numerically on the lattice, or evaluated in some models of gauge fields in the vacuum.An instanton-based model we used for quarkonia in [6], and claimed that it can explain most of the confining potential, at least till the relevant distances < 1 f m.For baryons, there are correlators of three Wilson lines, see their calculation in the instanton model [7] and detailed comparison with lattice results.In a recent paper [16] we extended such calculations for the three Wilson lines (for baryons) and four ones (for tetraquarks).In the latters, the hyperditance potential was shown to describe the distances between the three cccc states recently discovered, and interpreted as 1S, 2S, 3S states.Application to uuu = Delta ++ S-shell states were also successful.No attempt so far were made along these lines to describe baryons or multiquark states made of different flavor quarks.In such cases we need to go beyond the hyperdistance approximation.
The generic spin-dependent five potentials were defined in [17], in terms of Wilson-lines dressed by pertinent gauge field strengths.Specific relations between them and the central potential, follow for onegluon exchange.For example, the spin-spin potential is famously a Laplacian of the central potential, a delta function for the Coulomb force.Similar but distinct relations follow for the instanton-induced potentials, mostly constrained by self-duality of the instanton fields.However, if a significant fraction of the vacuum fields are due to the overlapping instantons and antiinstantons, there could be significant corrections.The spin-dependent interactions in the baryons and tetraquarks due to instantons can be evaluated, and we hope to report on them in our subsequent publications.
In this section we only include some general comments related to the definition of the operators.The specifics of the calculation for fixed states, are summarized in the Appendix.

A. Spin-spin and tensor interactions
We use the same strategy in the evaluation of all the spin interactions.More specifically, the 12-interactions are folded over the pertinent wavefunctions, and the overall results are multiplied by 3 by symmetry.The results for spin-spin forces agree with the much simpler "quantum-mechanical" method.
For the "unmixed" states with fixed total spin of three quarks, S = 3 2 or S = 1 2 , we have hence ± 3 4 for the cases of interest.The l.h.s. is sym-metric under quark permutations, and is the only form for the spin-spin forces.Of course, each term is convoluted with potentials, that depend on the relative distance, e.g.
the average of which multiplied by 3.
The tensor operator we used is given in (34) in a standard way, and also evaluated it using the 12pair.Note that in the tensor, we are using the actual distance vector ⃗ d 12 rather than its unit vector version.Hence the extra 2ρ 2 in the matrix element (to be accounted for in the potential V T ).

B. Spin-orbit operator
Two-body systems (mesons) have one orbital momentum L and one total spin S, so the natural spin-orbit force is proportional to ( ⃗ S ⃗ L).For three quarks there are ⃗ L ρ and ⃗ L λ orbital and three different quark spin structures.In general, it is still not all, since there can be products of one coordinate vector times momentum of another one.The requirement of permutation symmetry helps.Suppose the spin-dependent potentials are binary (twobody).With this in mind, the spin-orbit contribution is a sum of binaries Recall that the spin-orbit relation to the central potential is suppressed . This is the case of all relativistic corrections.
For the instanton-induced spin-orbit potential we have in terms of the instanton-induced central potential.
To recast it in terms of the Jacobi coordinates, we recall that and the corresponding momenta In particular, for the 12-spin-orbit contributi we have Here the first contribution is the standard spin-orbit contribution, proportional to L ρ , rotation of a 12-pair.The second term is also symmetric under 12permutation, but has a vector product of the coordinate ρ and momentum p λ , so it is neither two-body nor even an angular momentum.Fortunately, we found that all the second term contributions vanish after angular integration, basically because ⃗ ρ and ⃗ λ have independent directions.Therefore we do not need to include this complicated part of the operator.

C. 't Hooft induced interactions
The instanton-induced forces follow from the fermionic zero modes discovered by 't Hooft.Those are antisymmetric in flavors, and form generically 6fermion operators of the ūu ddss flavor structure.If we ignore strangeness (technically, by substituting ss part by its vacuum average value), the interaction reduces to a 4-fermion ud-interaction.It can be shown to be of the form in the ultra-local approximation, with and The coefficient a comes from terms with color matrices, and assuming that we discuss baryons and carry the average over all instanton color orientations, we get a = 1 5 for N c = 3 and a → 0 for N c → ∞.The first factor in (44) with isospin matrices ⃗ τ , is a projector on the flavor singlet states (zero in ∆).So it contributes to the N − ∆ splitting in the S-shell.There is also no problem to evaluate this operator over the five N * states in the P-shell.In the quasi-local approximation V tHoof t ∼ δ 3 (d 12 ), of the ∼ ρ 2 , λ 2 contributions arising from the orbital wavefunctions, only the λ 2 remains.

VI. BARYON WAVE FUNCTIONS ON THE LIGHT FRONT
There are many differences between the CM and the LF wavefunctions induced simply by kinematics.For example, the former uses nonrelativistic (or semi-relativistic [4]) descriptions, not well justified for light quarks.The latter (in the form developed in our previous papers [6,7,[18][19][20][21]) does not need this assumption, it takes the same form, from light u, d, s to heavy c, b quarks.Furthermore, it naturally provides an oscillator-like behavior for transverse momenta, which translate into a linear dependence of the baryon mass squared, on the number of excitation quanta.
However, on the LF manifest rotational symmetry is lost, as transverse and longitudinal motions are treated differently.The states are no longer classified by their total angular momentum J, orbital L or total spin S.Only the helicity projection of those, J z , L z and S z , can be used.As we will see in this case, the importance of permutation symmetry (as it was illustrated in previous sections and Appendix A) is even broader.
A simple Fock state representation of the spin-up proton wavefunctions on the LF was originally given in [22] where The fermionic creation operators anti-commute, and once multiplied by the color indices ϵ ABC , they generate 12 terms.The first term ψ 1 in the upper raw (times the second raw) above reproduces the correct permutationsymmetric S-wave function of the proton, in the form we already used in the upper raw of (9).The expressions for the proton with L z = ±1 are linear in (transverse) momentum, and obviously ψ 1 refer to a spin-up proton with negative parity.The 3 ψ 3,4,5 LF wavefunctions refer to part of the nucleon P-shell, we discussed above.They do not cover all the 5 P-nucleon states we derived, as they have different wave functions, with different permutation content.Also, their longitudinal momentum content under permutation symmetry is not specified.Finally, the second term in the first raw and the last one are quadratic in momenta.So, the 2 ψ 2,6 wavefunctions correspond to D-shell excitation or their admixtures, with positive parity.Their permutation structure is incomplete.This issue has been addressed in the CM above in section III, and will be extended below to the LF.
To clarify the argument, let us first recall again the situation in the CM frame, returning to our main example of the P-shell nucleons.Physical states in the CM formulation have spherical symmetry, and therefore fixed total angular momentum J.They are specific combinations of all possible values of the orbital and spin helicities L z , S z .For example, are superpositions of states with all three L z values, with some coefficients f Lz .For states with fixed J those are defined by standard Clebsch-Gordon rules, which for this particular example prescribe the coefficients f 1 , f 0 , f −1 to be for J = 5/2, 3/2, 1/2, respectively, and J z = 1/2.Note that these three vectors are indeed mutually orthogonal and normalized.Under the zeroth order Hamiltonian, they all have the same energy, and only the spin-dependent forces create the observed splittings.These states are defined in the CM frame as eigenstates of the total angular momentum ⃗ J, but this option is not available in the LF formulation.Furthermore, as emphasized above, Clebsching is not enough for baryons.Their wave unctions are linear in coordinates, which bring in their negative parity.There are 6 ⃗ ρ, ⃗ λ, and one expects 6 basic orbital wave functions.The same number of basis states needs to be defined in the LF formulation.The transverse polarization component L z = 0, proportional to the longitudinal coordinates/momenta ρ z , λ z in the CM frame, needs to be redefined.As we will show below, they should be substituted by solutions on the triangle, of ρ and λ-type respectively Also, the spin S z = ± 1 2 structure in fact exists in three forms, the fully symmetric one corresponding to S = 3  2 , and two more mixed with ρ-like and λlike permutation properties.The same statement applies to isospin wave function.Constructing the correct combinations is not a trivial task, and it was the subject of the preceding sections.

A. Light front wave functions for negative parity baryons
Now we switch to the main subject of this work, the construction of the corresponding wave functions on the light front (LF), with a focus on the P-shell baryons.Four out of the six coordinates corresponding to the transverse 12-plane, remains unchanged.Since we proceed on LF in momentum representation, the angular functions with L z = ±1 are proportional to either k ρ 1 ± ik ρ 2 or k λ 1 ± ik λ 2 .The requirements of 12-permutation symmetry helps to eliminate many impossible combinations, and to reduce the number of functions necessary.In particular, the spin-isospin-transverse functions should be superpositions of permutation-symmetric blocks, such as e ±iϕρ I ρ + e ±iϕ λ I λ S sym for p-shell nucleons with spin 3  2 and e ±iϕρ S ρ + e ±iϕ λ S λ I sym for ∆ baryons.Other combinations are formed as in the CM frame.
The question remains what are the other two other basis states, with longitudinal momenta.Using Bjorken-Feynman longitudinal momenta f ractions x i ∈ [0, 1], i = 1, 2, 3 as variables and enforcing the kinematical constraint x 1 +x 2 +x 3 = 1, we will again use Jacobi variables to characterize them.We will refer to them by ρ, λ without any indices.(They should not be confused with the lengths of the coordinate vectors ⃗ ρ, ⃗ λ used in previous sections.)As discussed in our paper [7], the physical domain in this case is the equilateral triangle, between three points at which one of the x i reaches unity and two others vanish, The role of the permutation group S 3 is now clearly seen,.In these notations, it is a set of reflections and rotations of this triangle.Quantization on this equilateral triangle has been done in [7], which was achieved both analytically and numerically.We recall that that the confining potential, after an "einbine trick", can be made quadratic in coordinates.In the momentum representation we use the sum of the squared coordinates, to turn it Laplacian in momenta ρ, λ.The analytic solutions follow in the form of six waves, interfering at the triangle boundaries using Dirichlet (zero) boundary condition, see (54) of [7].The spectrum of the Laplacian is given by two integers m L , n L (L refers to the longitudinal directions).
The single-degenerate solutions correspond to m L = 2n L , with the two lowest already shown and discussed in [7].Now we focus on the solutions with m L > 2n L , which are in fact double degenerate (Yes, the triangle has obvious triple symmetry by 120 o rotations, but those produce linear combinations of these two solutions.)Those were called D c , D s before, because the former includes combinations only with cos(C i ρ), and the latter similar set of sin(C i ρ).In the present paper, where permutation symmetry is central, we would like to rename them into D λ , D ρ , respectively.Indeed, the former is even under 12-permutation, and the latter is odd for ρ → −ρ, see Fig. 2.These are the solutions of ρ and λ types, which on the LF are substitutes for zero orbital momentum components in the CM, L λ z = 0 and L ρ z = 0, essentially simple linear coordinates λ z , ρ z .Since these are new solutions, let us discuss how the spin operators act on them.The spin-orbit interactions include orbital momenta such as In the coordinate representation the momentum is a derivative, and in the momentum representation (we are using on the LF) the coordinates are derivatives over momenta.Acting by L on the longitudinal function of longitudinal momentum ρ ∼ p ρ z , the vector product can only get other -transverse-components p ρ x,y due to the first factor.Since those appear linearly, it means the L operation converts longitudinal functions into transverse ones.As a result, on the LF the angular momentum operator mixes two transverse functions with one longitudinal, as it does in the CM description for the three angular functions in the P-shell.
Another thing to notice is that the derivative in L interchanges cos and sin.With the appearance of ρtype transverse coordinates, we see that this operator does not change parity under 12-permutation.Yet, the differentiation changes the function, so it is not just D ρ ↔ D λ as there are additional admixtures of higher functions, with other m L , n L .
Similar thing happens with the tensor force.In the CM formulation, the interaction generates nonzero non-diagonal matrix elements for L = 0 ↔ L = 2, L = 1 ↔ L = 3 shells etc.The spin part in the tensor causes mixing of the S = 3 2 and S = 1 2 states, as we discussed above.Yet, since the coordinate tensor times the spin tensor is rotationally a scalar, J remains unchanged.In the LF formulation, there is no rotational symmetry or J, and the tensor force mixes all D ρ , D λ functions.
Let us now address the issue of parity.The standard P mirror reflection, sign change of all coordinates, makes odd-L states with negative parity, which cannot be mixed with positive parity states.On the LF the notion of parity is remedied by an additional rotation by 180 o , so that longitudinal momenta do not flip.In particular, our equilateral triangle in the ρ, λ plane maps into itself.So, formally, the D ρ , D λ functions do not have parity as such.
Yet these LF functions have certain parities under particle permutations.As one can see from Fig. 2, D ρ is odd under flipping of ρ and D λ under flipping of λ = λ + 2/3 (since they contain the corresponding sin functions).Those are sufficient to define odd and even shells, which cannot be mixed by any forces/operators preserving the permutation symmetry.
In sum, the six wave functions out of which the P-shell baryons are made on the LF, consist of four transverse ones, with exp(±iϕ ρ ), exp(±iϕ λ ), complemented by two new D ρ , D λ functions on the triangular domain for longitudinal momenta fraction.The P-shell baryons can be constructed out of these mixed-symmetry blocks by the same expressions as used above.
More explicitly, the S = 3  2 with fixed J z the spin WF is symmetric, and the orbital and isospin ones are mixed symmetries of ρ, λ types.We know that the full S 3 symmetric combination out of the two mixed blocks, has the structure X ρ 1 X ρ 2 + X λ 1 X λ 2 .Therefore the permutations-symmetric wave functions are and similarly for the S = 1 2 and fixed J z we have We note that the two L = ±1 lines in (46) should be substituted by the 5 states (51-52) for the P-shell N * .The same applies to the 2 ∆ * .The coefficients A m , B m are not fixed kinematically in the absence of rotational symmetry.They are determined dynamically through the diagonalization of the LF Hamiltonian.Clearly, a fine tuning of the parameters is likely needed to achieve the same mass spectrum for different J z .This is in sharp contrast with the CM where rotational symmetry fixes these coefficients as Clebsch-Gordon coefficients as in (48).Finally, note that since the transverse part of the LF Hamiltonian is an oscillator, we have M 2 n ∼ n ⊥ .The longitudinal eigenvalues (50) are quadratic in quantum numbers, yet after minimization over the einbine parameter, they enter via They are not linear, but relatively close to linear.More specifically, the first three double-degenerate excitations, counted from from the ground state (n=2m=2), have energies to be compared to 1,2,3 for an oscillator.So, with a pertinent fit to the parameters, one should be able to get agreement between the masses for different J z components in the same J-multiplet, with few percent accuracy.Similar accuracy is expected for Regge phenomenology.Furthermore, if one would only compare states with the same J z and different J (as we did for negative parity baryons in (48)) such deviations are further reduced, as they only appear in the longitudinal component, a part of the wavefunction.So far, our discussion refers to the bare Hamiltonian H 0 .The spin-dependent potentials cause larger shifts , ∼ 10% for the P-shell baryons, and we expect that this imperfection in the longitudinal part of the spectrum on the LF, will not seriously affect the spectroscopy.

B. Wavefunctions on LF for positive parity excitations
In the CM frame the D-shell baryons have wavefunctions quadratic in coordinates, as we already discussed above.Each of the coordinates have 2 transverse and one longitudinal components, of ρ or λ types.So, in general, one may approach the problem asking for representations with f our structures X 1 ⊗X 2 ⊗X 3 ⊗X 4 , of ρ or λ types, and seek their totally symmetric combinations.It can be done along the same reasoning as for the X 1 ⊗ X 2 ⊗ X 3 case we addressed in appendix A. This case would correspond to the case of the S = 3  2 symmetric in spin.

VII. LEADING TWIST PROTON DAS
The simple classification of the lowest Fock states for a spin-up proton state made of three constituent quarks, consists of the 6 LF wavefunctions in (46).As we noted earlier, this Fock state representation mixes the ground and excited states of the nucleon with fixed spin 1  2 .Boosting the nucleon in its ground S-state, should leave the nucleon in its ground Sstate.Also the simple Fock state representation forgoes the issue of the center of mass and symmetrization, which are important for the description of the nucleon S,P,D states as we have discussed.
This notwithstanding, the Fock state representation is useful for the characterization of generic spin- 1  2 nucleon distribution amplitudes (DAs), in the leading twist approximation.More specifically, three quarks with spin- 1  2 can combine in four different ways to form a proton with spin up p ↑ , i.e.
with the spin-orbital arrangements On the LF, the leading twist operators for the proton uud with positive parity, are typically of the form , and C = iγ 2 γ 0 the charge conjugation matrix.Note that the two independent Ioffe ′ s currents on the LF, hence subleading.In terms of the good component quark fields q C +S , the leading twist nucleon DAs for a spin-up proton are tied to the Fock states (46) and read [23] Here N ↑ + (p) is the good nucleon on-shell component with p 2 = m 2 N , and the shorhand notation for the tracelss symmetrization

VIII. SUMMARY AND OUTLOOK
This paper is in many respects a methodical paper, devoted to novel technical tools of few-body quantum mechanics.We now provide a brief summary, and put forth few outlooks.
Summary: We started this paper by clarifying the use of the permutation group S 3 , in particular its pertinent representations, essential for defining the symmetry properties of the wave functions for three quarks.As it is well known since the early 1960's, the baryon spin and flavor (isospin) parts of the wavefunction for the S-shell, do not factorize, as only their permutation symmetric and nonfactorizable combination is allowed.
For the P-shell states (negative parity baryons) we have in addition the orbital L = 1 part of the wavefunctions to symmetrize.This is best achieved using the analogues of the ρ-type or λ-type Jacobi coordinates.In this case we need to construct triple tensor products of permutation matrices, and find their totally symmetric states.How to do that, without guessing and a loss of generality, is shown in Appendix A. A nontrivial result (not new but explicitly derived) is that there is a unique permutation symmetric wave function.In the next D-shell, the coordinates appear as products of two, so one has to find the totally symmetric wavefunction of four objects, and so on.All of those are found by the proposed method.
Our other methodical suggestion is to use the natural spin-tensor notations for the wavefunctions, and apply symbolic manipulation capabilities of programs like Maple or Mathematica.The use of symbolic "monoms" is traditional, yet it is better done in generic basis.Specifically, all possible spin-isospin "monoms' of three quarks are 4 3 = 64.While for the nucleon only 9 monoms are needed, for the Pshell and D-shell baryons, the wavefunctions are much more involved.The universal spin-tensor notations allows for any type of symbolic operation when coded, e.g.spin-orbit with differentiation etc.In Mathematica, whether an operator is acting on the wavefunction with 1 or 64 components, makes no practical difference.
As a demonstration of this technique, we repeated the well known Isgur-Karl calculations of the splitting of the five negative parity N * resonances.Unlike them, we explicitly construc the wavefunctions with proper permutation symmetry, and not as certain limits of less complicated Σ, Λ wavefunctions when m s → m u , m d .Like them, we also show that using only the spin-spin and tensor forces (but without spin-orbit) we can get a very good description of the mass spectrum for these states.
In this paper we stated that, apart from the orbital part, the basic wavefunction of all N * should be spherically symmetric in 6 dimensions, ϕ( ⃗ ρ 2 + ⃗ λ 2 ), yet we have not evaluated it.However, we note that in our recent study [16], we have used the appropriate "hyperdistance approximation" and the reduced radial Schrodinger equation, to obtain such wavefunctions, for (flavor-symmetric) baryons and (allcharmed) tetraquarks.The reason we focused on all-charm hadrons first, is that for heavy quarks the spin-dependent interactions -coming from relativistic corrections -are small and can be neglected in the zeroth approximation.
Construction of LF wave functions for negative parity baryons needs explicit definition of the longitudinal wave functions (depending on momentum fractions x 1 , x 2 , x 3 ) with symmetries of Jacobi coordinates ρ, λ.We have shown what these functions are in section VI A.
Outlook: All of the present study is a preparation leading to the proper definitions of the the Light Front wave functions (LFWFs) for multiquark hadrons, or their multiquark components.What we mean can be explained by examples: to a heavy QQ mesons, we can add a light qq pair to form a tetraquark, while to a baryon we can add a pair to turn it to a pentaquark.With pertinent quantum numbers, these are "exotic hadrons", minimizing the Hamiltonian.Also, they can be considered as "virtual clouds" (as is the 5-quark component of the nucleon, seeding its antiquark sea, e.g.discussed in [19] ) or separate "pentaquark states" orthogonal to the nucleon with all its cloud (as the recently discovered pentaquark resonances with hidden charm uud + cc states).Their discovery was helped by their narrow widths, as they happen to be just above the thresholds of baryon-meson states to which they can decay.Te further discoveries of stable pentaquarks below such thresholds (e.g. with hidden bb) are still ahead of us.Those with ūu, dd, ss are perhaps not likely to be seen as separate resonances.However, the theory still needs to answer: where are they?what contribution to the known states (e.g.nucleon) they actually make?When there are several identical quarks, the issues of appropriate symmetries of the wavefunctions need to be resolved.We have discussed in details the S-shell and P-shell baryons N * .Our important point was that the LFWFs should have the same structure under the S 3 permutation group for their spin-isospin wavefunctions, as they have in the CM approach.The orbital part should not only contain the transverse k ± ⊥ momenta, as was proposed, but also it should include the longitudinal doubly degenerate excited states.We showed that those are D ρ,Λ n L ,m L consisting of six waves on an equilateral triangle for longitudinal momenta fractions.We have further shown, that while spherical symmetry is strictly speaking absent on the LF, the numerical deviations from the energies (and fixed-J wave functions in the CM frame) for P-shell and D-shell states constitute only several percents.Hopefully these differences are smaller than the splitting generated by the spin-spin and tensor forces, In this way, the squared mass splittings will not be affected by these "non-sphericity" corrections.
As parting comments, we note that one would not be able to calculate the ū, d, s, c contributions to the PDFs and other density matrices, without solving the many-body Hamiltonian, and obtain the full wavefunctions.And, e.g. for the five-quark components of the LFWFs with flavors content uuduū, uudd d, one would need also to combine the isospin representations with the color,spin and orbital wave functions, to get the correct Fermi statistics.We hope to address those issues in subsequent publications.The ( 12) and ( 23) permutations are realized as improper O(3) rotations on the Jacobi 2-vector, with determinant equal to −1.This is rather natural if we think of the initial three 1, 2, 3 particles set on an equilateral triangle.The six permutations (7) are then 3 in-plane rotations by π 3 of which P 4 is an example, and three out-of plane rotations by π along each of the 3 bi-sectors of which P 2 is an example.
With this in mind, the Jacobi 2-vector under the 3-particle permutation, transforms analogously to a spin-1 2 under SU(2) rotations.In retrospect this is expected, since the mixed representation Young tableau for 3 spin- 1  2 carries the same dimension 2 M as the primitive spin- 1  2 Young tableau.This observation will be repeatedly used below to construct the excited states of baryons with 3 quarks, with totally symmetric space-spin-flavor wavefunctions under S 3 .
In general, the representations of S 3 fall into: 1/ totally symmetric states (S), totally antisymmetric states (A), and states with mixed symmetry (M ), under P i .Three quark spin states can be split into S = 3  2 symmetric states, ↑↑↑ and so on, and then two mutually orthogonal S = 1 2 states which, following Jacobi coordinates, we call S ρ , S λ states.The following development consists of two steps.The first one, going back at least to [3], is related to symmetry under (12) reflection (8).With this in mind, any characterization of a 3-particle state (space, spin, flavor) can be composed by paralleling the Jacobi coordinates, with manifest symmetry under S 3 .The idea, is to construct mixed symmetry wave functions M ρ,λ , out of ρ-like and λ-like blocks which have pure S or A permutation properties, The construction of the representations of S 3 is carried as for any other groups, e.g. through the familiar generalization from the spinor representations of O(3) to spin-1, spin- 3  2 etc.The tensor product of two generic representations X a and X b with different symmetries under S 3 , is a sum of representations X ab , each with symmetries S, A, M ρ,λ .While it is clear that the symmetries of the S, A products are the product of the mixed representations M ρ,λ viewed as a primitive 2 M doublets, is more subtle and requires more detailed studies.
For two ρ, λ-type blocks there are 2 2 = 4 combinations, that can be constructed using say a basis Half of the states are symmetric and half are antisymmetric under 12-interchange.Yet what needs to be done, is to enforce say 23-permutation on these combinations.Using (8) two times, we obtain matrices corresponding to this permutation for two objects.The corresponding M(23) matrix in this basis takes the form The commutator of M (12) with M(23) matrices, yields two imaginary and two zero eigenvalues.The latter correspond to the totally symmetric and antisymmetric combinations, respectively.The symmetric in this notations is (1, 0, 0, 1) or X ρ 1 X ρ 2 + X λ 1 X λ 2 , the only combination of two objects simultaneously symmetric under (12) and (23) (and in fact all) permutations.This is the one which we are seeking for the wavefunctions of the baryons in the ground Sshell, as well as e.g. the three spin- 3  2 nucleons in the P-shell.Although it may indeed be easy to simply guess this form, the method we follow eliminates any guessing, and can be used for any number of objects.tion, (−1, 0, 0, 1, 0, 1, 1, 0) in our basis, corresponding to the sought after symmetric wavefunction under A10) Again, no guessing is needed, the unique totally symmetric wave functions is shown by construction.We recall that while our derivation is generic, we used it for three blocks here being the orbital, spin and isospin parts of the wavefunction.This method can be used for any number of blocks, and we now proceed to four, as e.g.those needed for the D-shell resonances.The matrix for the 23-permutation in this case is 16 × 16, with 8 antisymmetric and 8 symmetric eigenvectors.In fact it can be generated in Mathematica by using command KroneckerProduct[P23, P23, P23] (in which case the basis set is automatically selected by Kroneker product as well).The commutator of ( 12) and ( 23) matrices has 6 zero eigenvalues, of which 3 are symmetric.Their linear combinations can be used as wavefunctions in the D-shell.They are {1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1} {0, 0, 0, 1, 0, 0, −1, 0, 0, −1, 0, 0, 1, 0, 0, 0} {0, 0, 0, 0, 0, 1, −1, 0, 0, −1, 1, 0, 0, 0, 0, 0} (A11) Note that all of them are combination of four "monoms", weighted by simple ±1 coefficients.
Although we do not have immediate applications for it, we repeated the procedure for 5 blocks, in the representation of dimension 2 5 = 32.The corresponding ( 12) and (23) matrices were generated.Their commutator has 10 zero eigenvalues, with half corresponding to symmetric and half antisymmetric wavefunctions.The corresponding combinations are more complicated and have more than 4 nonzero terms.

Appendix B: Details of matrix element calculations
We have calculated the matrix elements of all the operators in two ways: by direct calculations using the explicit wavefunctions quoted earlier, and using symbolic manipulations of the wavefunctions represented as spin-tensors via Mathematica.We will detail here some parts of the explicit calculations for all the spin interactions, starting from the unmixed states with J = 5  2 , then proceed to the mixing matrix in the J = 3  2 shell, and only quote the final results for the mixing matrix in the J = 1 2 shell.Using the explicit states (19-21) and short-hand notations, the spin-spin coupling contributions gives ) (B1) with the Clebsches and some indices omitted to avoid clutter.The overall factor of 3 follows from the permutation symmetry as we noted.The spinspin contribution is reduced by the identity in the permutation symmetric S-state, and gives The same arguments yield where we used the spin identities The tensor contribution mixes configurations with different spin content, and is more involved.To eval-uate it, we re-establish the Clebsches and azimuthal labelings where only the manifestly non-zero contributions are retained.Using the spin-space dependence of the tensor interaction, we find that the spin valued parts in (B13) can be reduced, using the matrix elements to give in short hand notations If we recall that the integration over the three spherical harmonics in (B9) can be undone using the identity with the result The remaining spin-spin and tensor matrix elements can be done similarly, with the results for the remaining mixing matrix entries in the J = 3 2 P-shell.We used the fact that the off-diagonal and non-vanishing tensor matrix elements are with the vanishing diagonal ones When we calculate the LS interaction of one pair of quarks, 1-2, all orbital terms ∼ λ are eliminated by derivative over ⃗ ρ and only ρ-dependent ones remain.
The spin-orbit contributions simplify considerably, if we note that φ ρ spins along the λ-direction, and φ λ spins along the ρ-direction.This means that the standard spin-orbit contribution in (43) has only a non-vanishing matrix element for the combination φ ρ (ρ × p ρ )φ ρ = l ρ (B16) as the two other entries vanish φ λ (ρ × p ρ )φ ρ = φ λ (ρ × p ρ )φ λ = 0 (B17) The mixed spin-orbit contribution in (43) vanishes identically, as all entries φ ρ,λ (ρ × p λ )φ ρ,λ = 0 (B18) are seen to angle average to zero.As a result, the 12-spin-orbit contribution (43) simplifies to The spin-orbit contribution mixes configurations with different spin content as well.Its evaluation follows that of the tensor coupling detailed above.More specifically, we have  Using the permutation symmetry of the in-out spin symmetric S-states, we can make use of the identities S S S z tot S S S + tot S S The ′ t Hooft coupling contribution to the mixing matrix, involves flavor matrix elements.The cal-culations simplifies considerably if we note that the flavor states F S,ρ,λ are eigenstates of the flavor singlet projector, With this in mind, a rerun of the preceding arguments gives The spin J = 5 2 shell is unmixed.The spin interactions in this shell are the simplest to evaluate.Using their explicit wavefunction, and some of the spin and flavor identities we derived earlier, we obtain

Mixing matrix
The hyperfine interactions in the degenerate P-sub-shells of fixed J = 1 2 , 3 2 are fixed by where we used the S-labeling as a short-hand for the degenerate nucleon P-states, which are normalized by The diagonalization of (B29) yields the two mixing angles in the P-shell with J = 1 2 , 3 2 .For J = 3 2 , their explicit forms we already given in results (30) Mathematica is a platform for symbolic calculations widely used in many branches of physics.In some (e.g.general relativity) its ability to handle

TABLE I .
The second shell baryons made of light quarks.