Compiling results from the count of oscimodes , analytical for light flavored mesons and from Fortran programs for like ( anti-) baryons Material shown in part at the Poster Session

The construction of oscillatory modes of q q ′ and q , q ′ , q ′′ with q , q ′ , q ′′ = u, d, s – modes in mesons and baryons is presented as an abbreviated outline .


Introduction
This outline presents results from the count of osillatory modes of light flavored u, d , s quarks inside hadrons, based on material prepared in evaluating analytical formulae and Fortran programs dedicated to this goal.The first such: t-66.f , ref. 1f = [1f-2014] , is presented in detail in a special section called 'Description of Fortran programs' at the end ofl this work.The traceback of derivations , which led to the elaboration of the Fortran file t-66, in ref.  .This set up the problem of extension to (three) valence quarks in baryons (and antiquarks in antibaryons), which took a while, to be answered in ref. 4 = [4-1980], where the discussion of baryons was restricted to only two light flavors : u, d .It was noted in Table 2 ,p. 267,op.cit. ref. 4 , that starting with the main oscillatory quantum number N = 2 there were (nonstrange) baryon states missing in the PDG tables of 1980 .In order to bring up to date the phenomenological situation and the comparison with the meson and baryon states for 3 light flavors of valence quarks with candidate resonances listed by the PDG ref.
was vigorously pursued up tp the end of the 60-ies, whence the three valence quarks wave function is conceived in their c.m. frame.Other orbital 3-quark states were also considered, as e.g. the p z → ∞ states, where a boosted S U6 w symmetry is introduced instead of S U6 .In this respect the main quantum number N was not related to genuine oscillator degrees of freedom but instead inferred from recurrences along Regge trajectories from the relation In eq. 1 J = L + S denotes total angular momentum.The literature quoted in ref. 4 = [4-1980] is very limited, to which we add here papers by R. Dashen and M. Gell-Mann, ref. 8 = [8-1965] , by G. Zweig , ref. 9 = [9-1968] and J. Schwinger , ref. 9 = [10-1964] , illustrating the search for local field variables underlying the strong interactions at that time. 2 The main quantities to count for lightflavored valenceq q mesons and q , q , q baryonsq , q , q = u, d, s We report the derivations leading to the asymptotic form for the eigenvalues of the masssquare operator in the meson c.m. frame from refs. 4 = [4-1980] and 12 = [12-1978] extending eq. 1 below α : slope of Regge trajectories excepting the Pomeron (2) In order to keep the systematics of barycentric coordinates as defined for general N c ≡ N , also appropriate for baryons discussed in ref. 4 = [4-1980] , we set for the kinematically simpler q q Next we consider the Lagrangean M 1,2 : masses of q 1 , q 2 respectively v 1,2 = ( d / dt ) x 1,2 ; in the meson c.m. system t : overall synchronized time in the c.m. system , using units such that c = 1 (4) EPJ Web of Conferences In the next step we choose the chiral limit as a valid approximation at large distances , to be specified subsequently.
Then eq. 4 simplifies through the relations and L qq in eq. 4 becomes We expand the derivations following ref. 4 = [4-1980] .
The canonical momentum relative to 1 2 y becomes It follows from the relations in eq.7 that H (2) is a constant of motion in both classical and quantum mechanical interpretations of the two body q q system considered .The Euler-Lagrange equations become We can see how the mass-square dynamical variables arise in the classical interpretation of the equations of motion introducing the notation Then eq. 8 takes the form The oscillatory modes inherit the central scale of QCD for light ( u, d , s-) flavors of quark as implied by the trace anomaly -ref.13 = [13-1975] , in the adopted chiral limit through the parameter Λ , not to be confused with Λ QCD , valid in the perturbatively accessible region, in the Ansatz for the mass function for large values of | y | , of dimension mass-square In eq.11 M q denote the physical quark-mass parameters Inserting the asymptotic large | y | part of the mass-square function m 2 ( y ) into eq.10 we obtain We here turn to the quantum mechanical description following from the Lagrangean given in eq.6 In eq. 13 the suffix ordered shall indicate that the operators m , written as product, do not commute, which necessitates the ordering guaranteeing a self-adjoint operator beeing represented by the product .

2-1 Counting oscillatory modes of valence quarks and antiquarks
q , q ; q, q = u, d, s in mesons We first determine the number density at given main quantum number N , which amounts to calculate the power of the set of occupation numbers n 1 , n 2 , n 3 of the associated 3 oscillators .p is given by the number of partitions and multiplied with the multiplicity of S U6(spin × N f l ) = 36.The power of the set p ( N ) is readily written as a sum over n 3 Thus p ( N ) defined in eq.14 becomes We give here the derivation yielding the sum of squares of integers from the difference of sums of cubes which makes use of the recursive relation Then it follows We display the function S 2 ( N * ) , N * = 1 to 5 in table 1 below Table 1 The sum of the squares S 2 ( N * ) is in the present context an auxiliary function necessary for the calculation of the total number of meson states with N ≤ N * .We thus find for the number of u,d,s meson states with The main results of this subsection are contained in eqs.17 for the number density as a function of the main oscillator quantum number N: z ( N ) = ∂ / ∂ α M 2 and 22 for the total number of oscillatory modes below and including the limiting main quantum number N * : Z (N * ) , both equations recapitulated for u, d, s mesons q q below z ICNFP 2014 05011-p.5

2-2 Oscillatory modes of valence quarks and antiquarks
q , q ; q, q = u.d, s in mesons (continued) We recall here the relations in eqs. 5 and 6 , repeated below L qq in eq.6 becomes Next we recall eq. 13 repeated below Eq. 7 adapted to the quantum mechanical logic becomes Eq. 29 shows the main result of this section , in particular the second order wave equation (on the second line) and average the two half mass square difference entries in the last line of eq. 31 with weights two to one we obtain We remark that in ref.
+ has only three stars , and furthermore the trajectory contains only three entries , whereas one of us (P.M.) thinks to remember that it contained four sometimes back1 .Λ 13 2 + would extrapolate to 2.755 GeV using eq.31 .
In order to exhibit the oscillator variables we substitute rescaled coordinates and derivatives relative to the spatial variable y y The differential operator on the right hand side of eq. 30 becomes The parameter λ , of dimension mass, shall be chosen such that Substituting the last equation in eq.35 in eq.34 we obtain The deliberations leading eventually to the definition of M 2 given in eq.30 go back to ref.

2-3 Oscillatory modes of valence quarks and antiquarks
q , q ; q, q = u.d, s in mesons (proper) The dimensionless rescaled spatial variables ζ defined through eqs.33 and 35 and their canonically conjugate momenta generate the 3 oscillator absorption and creation operators absorption and creation operators obeying the commutation rules The oscillator algebra displayed in eq.41 is common to any (3) canonical pairs of operators , associated with a threedimensional uncurved space, as is the case here.It shows directly the U3-invariance of the commutation rules .
What is very special is the relation for the dynamical form of the mass square operator M 2 as given in eq.36 , which becomes For an individual n we have which proves the correctness of the first relation in eq.42 .
Identifying 2 Λ with the inverse slope of Regge trajectories other than the

Pomeron
Since the relation in eq.42 is only valid for large eigenvalues of the number operator N , with eigenvalues , 3 denote the eigenvalues of the individual counting operators N k , we can reparametrize within the same approximation accuracy eq.42 in the form In eq.44 the operator N 0 contains all effects from short distances and parametrically depends on quark masses .It does not commute with the asymptotically dominating number operators N , N k ; k = 1, 2, 3 .We deal with the 'intercept'-related perturbations through the operator N 0 as characterized in eq.44 and the text specifying its details in the sense of perturbations of large eigenvalues of N , the dominant operator for large eigenvalues N of the mass square operator M 2 , adopting the ansatz for the eigenvalues of M 2 We compare the structure of M 2 in eq.45 with the relation to this quantity along a Regge trajectory taken approximately linear Comparing th two expressions for M 2 in eqs.45 and 46 , we obtain the sought identification EPJ Web of Conferences 05011-p.8 2-4 First results and illustrations from the count of oscillatory modes of valence quarks and antiquarks q , q ; q, q = u.d, s for mesons and baryons The count of these (3-) partitions is illustrated for the simpler case of (2-) partitions For the so defined (n-) partition count , the unit spatial grid is independent of the operator N 0 , displayed in eq.44 .It is reached in momentum space from eq. 47 , which yields the unit in momentum space for each of the 3 components of p in the case of mesons.
The comparison of the number density of states per unit mass-square z ( N ) between baryons and mesons is illustrated in Fig. 2 below ICNFP 2014 05011-p.9We look forward to proceed along the path of counting oscillatory modes of light flavored mesons and baryons , as it is outlined here in its present incomplete stage .The goal is to obtain main consequence for the description of thermal equilibrium as applying to the hadronic phase of QCD .What becomes clear now is that the alleged thermal properties at temperatures not exceeding 200 MeV including the limit of chemical freezout at small or vanishing chemical potentials involves a newly posed assessment of feeding through the decay products of the heavy (heavier) baryons and antibaryons is a serious one affecting mainly pions and kaons.It is conceivable that the difficulties in accounting for the observed abundances at RHIC and with the Alice detector at LHC .

Fig. 2
Fig.2serves to show, that the density per unit mass-square count reveals more baryon states than mesonic ones, for all values of N .Nevertheless in production cross-sections as well as in thermal equilibrium mesons are dominantly produced, for low energies or temperatures .To this end we calculated the exponentially weighted density per unit mass-square for baryons∂ baryon / ∂ α M 2 × exp ( − M / T )(50)for T = 0.2 GeV displayed in Fig.3 below In Table2we give a comparison of the number of oscillatory modes up to N * ; N * = 0, 1, 2, 3 for baryons and mesons .The numbers for baryons are from ref.1 = [1-2013] .The oscillator structure for mesons shown in eqs.40 -43 , which gives rise to the count of partitions, as charecterized in eqs.44 -46, is valid for large eigenvalues of the number operator N , with eigenvalues N = n 1 + n 2 + n 3 = 0, 1, 2, • • • , where n k , k = 1, 2, 3 denote the eigenvalues of the individual counting operators N k .