MULTIPLET CONTENT AND SUM-RULES OF LEPTON-HADRON SUPER-CURRENTS IN A SU(8) SUPER-QUARK MODEL AND SPIN 5/2 GRAVITON By K. L. Narayana,

trusciencetrutechnology@blogspot.com

Volume 2014, Issue No.3, March 11, 2014, Time: 10h38m A. M.

MULTIPLET CONTENT AND SUM-RULES OF LEPTON-HADRON SUPER-CURRENTS

IN A SU(8) SUPER-QUARK MODEL AND SPIN 5/2 GRAVITON

By

K. L. Narayana,

Shivaji University, Kolhapur – 416004.

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§ Preliminary details of super-gravity theory have been presented in an

Invited talk during the Einstein Centenary symposium held at Satyendranath

Bose Insitute, Calcutta on the 2 March 1979. By the present author.

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Present address: Professor Dr. Kotcherlakota Lakshmi Narayana, TRU S & T,

(Retd. Prof. of Physics, SU, KOP) 17-11-10, Narasimha Ashram, Official Colony, 

Maharanipeta P. O,  Visakhapatnam – 530002.  Cell: 09491902867.

ABSTRACT

            The hypothesis of Virtual Gluonic Super Currents leads to an Unification of Lepton-Meson-Baryon particles and the resulting super-quark SU(8) symmetry incorporates the principle of strong Lepton-Baryon symmetry as an essential ingredient for obtaining the new sum rules of Meson-Meson Vector Meson-Vector Meson interactions.The present model therefore is a small step towards a Super-Gravity theory. The model is to give relative strength magnitudes of charm - Non-changing weak interaction currents of Leptons and Baryons and further, their invariants. 

                The super quark model SU(8) = SU(4) ⨂ SU(4) has the flexibility of being combined with other symmetries by virtue of the content of its higher multiplets such as 63⨂63 and therefore leads to a super-gravity theory.

INTRODUCTION

        In the study of Charmed particle symmetry weak interactions are of special importance [1-8]. Universal interaction treats all the four categories of purely Leptonic, strangeness-conserving Leptonic, strangeness-changing non-Leptonic processes all to be characterized by a universal coupling constant. [5]. But it is observed that strangeness-changing weak interactions and even the Leptonic processes are lower by a factor about 20 relative

to other processes mentioned. Renormalization effects are therefore definitely playing a role. In spite of it Baryon-Lepton symmetry is enunciated with the hypothesis of SU (3) group weak Baryon currents by Marshak and others.

        Studies have been reported in literature for new types of total Lepton Currents, involving new types of quantum numbers such as style, Charm etc. [8, 10].  An observational report of a second Charm particle produced by a high energy neutrino and decaying after 10 ^-13 sec is exciting. See the work of Angelini et al [10]. On the other hand Quark Line Rule prevention of the fourth quark (Charm) to the world of three quarks has been investigated by Okubo [11, 12]. Certain technical difficulties of a simple SU(4), 15+1 pelt, model how they may be overcome has been exemplified by Hallock and Oneda [13] in a theory with two ingredients, Viz., exotic charge commutators and the hypothesis of the asymptotic 16-plet realization of SU(4). Mass of a Charmed particle is larger than that of other quarks and the charmed quarks may even exist more in number. Recent investigations of the unexpected narrow width-resonances at 3.1GeV support the Charm, a name first given by J. D. Bjorken and S. L. Glashow. Again existence of weak neutral currents and probable existence of strong-interactions by Leptons at high energies and momentum transfer tempts one to think of unified guage theories [4]. It is believed that at energies of order 10¹⁹GeV (Planck Energy) even the Lepton-Hadron symmetry may be considered together with gravitational effects. In this context a theory of Lepton-Meson-Baryon symmetry would not be a mere speculation and perhaps may be even exhibited of its effects around 10⁵ GeV energies of the limit of highest energy cosmic rays. Fayet considers [15, 18] for example, HYPER and SUPER symmetries. He in one of his theories considers a super symmetry model with a new quantum number R which fails to be conserved by gravitational interaction through the Higgs effect. The very small GRVITINO mass induces GLUINO mass Viz., the fermionic partner of gluons.

                Again primitive particle model (PPM) of Ne’eman [20] considers all matter is made of one fundamental Fermion set ( α^o,    ᾱ )_L ,  α^o_{R ,}  ᾱ_R    and two Bosons  φ_B ^2/3 and  φ_S^o in addition to SU(2)_L ⨂ U(1) and QCD gauge fields.

       

Thus attempts at incorporation of fermionic part of the quarks to explain matter structure is not an uncommon approach, of course, for a realistic realization such models are to be considered as badly broken symmetries.

          Experimental evidence for Charmed Baryons, Charmed Vector mesons and the pseudo-scalar mesons is accumulating in the last few months.[21-26] 8-Dimensional models have been previously discussed by various authors in Different contexts. Rayaki [27-30] reports a unified model of gravitational and other interactions. A unitary scheme of broken rotational invariants R(8) ⊂ SU(8)⊂SU(3), has been investigated by Ne’eman [31] and Ne’eman and Ozsvathi [32] and Gellman and Ne’eman [33], Gaueret considers SO(8) symmetry with 8 = 1 + 3  +  ̅3 + 1 ,   8  ⨂  ̅8 = 8’ + 56’ and 8  ⨂  8’ =1 + 28 + 35 +1 while Barnes et al consider decomposition of the Quark multiplet of SP(8) ⊂ SP(6) ⊂ SP(2) ⊂ SU(3) ⨂ SP(2)  as 8 = (6,1) + (1,2)  = (3 _-1 , 1) + (  ̅3₁ , 1) + (10 , 2). Their 8 quark representations is decomposed into a 4(u, d, s, c) and   ̅4 (t, b, h, f) of the SU(4) sub-group and adopt r_θ  new quantum number, STYLE.

                    Goldberg gives the characteristics of charged currents in the hadron V-A interaction structure which induces 

               ∆S = -∆ Q and ∆S = 2 ∆Q 

Transitions with predictions on CR-conjugation decay  

        Ξ ⁰à Σ^-  + e⁺+ ν Similar to  ̅K⁰  à π^-  + e⁺ + ν 

Charmed weak current predictions of sub-group models of SU(8) symmetry have been reported by Iowa in analogy with the SU(6) models.

                Further studies such as CRYPTO-EXOTIC meson states of the fourth-quark system by Minami[39], weak interaction of te fifth quark and fifth lepton by Mohapatra [40-43] and the correct spin-statistics connection and “Boson- Fermion puzzle” of the “Dyonium” model by Tolkachev and Tomilchik [44] are interesting as they support and enlighten new aspects of basic quark models, in particle physics. Composite quark models and as well of Leptons in the context of an internal symmetry group [45], SU(4)_L ⨂ SU(4)_R⨂ SU(4) (a sub-group of PREON group  SU(8) ⨂ SU(8) ) is discussed by Salam in a theory of unconfined unstable quark model [43]. Consideration of fermions and Bosons within the same multiplet of an U(N) symmetry actually arose from the considerations of infinite degeneracy spectra of Bosonic bound states and the Fermion-Fermion and Fermion--Anti-Fermion scattering amplitudes spectra [46,47]. Supersymmetry models also incorporate such multiplets [48- 51].

                The motivation of the present work I to understand the nature of the invariants of Leptons and Baryon currents of an SU(8) super-quark model. Of interest is also the study of Lepton-Baryom symmetry as well the R-conjugation symmetry [6,19] of the currents and the multiplet content. Our model differs from the BARBARYONIC CLASIFICATION of Fermi and Bose quarks suggested by Lipkin [52]. Another motivation is the ultimate aim of formulation of a unified theory in the context that the Charm of a particle can be realized by what is called a BEHARUNG GAUGE theory of metric transformations of space-time continuum [53-58].

        The present model therefore is a small step towards a Super-Gravity theory [58]. The model is to give relative strength magnitudes of charm - Non-changing weak interaction currents of Leptons and Baryons and further, their invariants. Unlike the Pati and Salam model [4], I would treat in the present model, the Leptons on the same footing as the Hadrons and attribute to them a quark internal structure of virtual gluonic currents. The theory envisaged in this paper, seeks also Lepton-Hadron symmetry to the SU(N) transformational features.

        Section I gives the model formulation and structure constants and in the next section reported the new sum rules of the invariants: Lepton and Baryon currents. Also presented are the details of SU (4) ⨂ SU (4) sub-group analysis of SU(8) super-symmetry. At the end of the paper salient features of the model are given.

SECTION I: MODEL FORMULATION AND STRUCTURE CONSTANTS.

               The basic quark multiplet involves four Bosonic quarks (p, n, λ, x)  and four Fermionic quarks (q, r , s, t). Under an SU (8) symmetry we define that they have an appropriate contragradient part which represents physically the anti-quark multiplet. The SU(8) = SU(4)⨂ SU(4) analysis allows us to put the Meson ( 0^- ), Leptons , Baryons   ½⁺ and vector-mesons ( 1^- )  in the 64-plet of 8x8 exterior product with a hypothetical virtual gluonic current structure given by the matrix,

A hypothetical virtual gluonic current structure given by the matrix T  in Fig. A

Here I is a unit 4x4 matrix,   γ₅   and γ_± are the usual Dirac spinor matrices.

        The virtual 64-plet current structure ensures the correct observed parity and the handedness of the baryons and leptons [53-55].

        The lepton and Baryon association are as follows:

p à e⁺   ;        Σ⁺ à μ ⁺  ;    Ξ ⁰ à ν_e  ;

Λ à ψ = ( ν_μ - ν_μ ^c  ) / √2 ;   Σ⁰ à χ = (ν_μ + ν_μ^c) / √2 ;

Ξ ^- à e^-   ;

    ν_μ  is a four component Dirac Spinor and  χ (x) is a Majorana field.

Additional to these we may adopt ( B^c₁ à  ̅ν_τ ; ̅ B^c₁  à ν_τ ) different from Marshak et al [6,7] specifications of Lepton-Baryon symmetry. For the assignments:

      B^c₂ à L ^c₂    ;    B^c₃ à L ^c₃    ;   B^c₀ à L ^c₀    

That would naturally follow in the present model still however no specific experimental evidence exists [62, 63]. The PLUTO collaboration 1979 report [64] gives observation of the new Lepton τ i.e. of left-handed and with an associated        neutrino ν_τ.  Also weak interaction of ultra fermions have been also observed by Chanowitz et al [23]. Evidence for heavy leptons and neutrinos of considerably long half-life time periods are being reported now-a-days [65-67]. “BEAM DUMP” experiments excess neutrinos may be due to Charm [68]. Also an attempt is made [8] by Narayana K. L. and Miss S. P. Sahane to define the usual lepton quantum number as a combination of two or more sub-leptonic quantum numbers to account for the occurrence of these additional leptons etc. Thus the present model seeks to unify the Leptons and Hadrons strong interactions. The SU(4) ⨂ SU(4) arrangement of the 64-plet allows also a comparative study of our predictions with those made earlier by Cabbio [1] and Marshak et al [6]. The basic quantum numbers and the associated linear combination of the creation and annihilation operators for a particle in a quantum state with the transformational proper-

ties of SU(8) super-quark model are given below [48, 69].

½ (Y+Z) + I₀ = D₁₁;   Y= B/4 – a^†_λ a_λ ;      -Y= D₃₃

½ (Y+Z) - I₀ = D₂₂;    I₀₁ = b^†_q b_q ;            -Z= D₄₄ ;

1/6 [5I₀₁ - I₀₂ + (Z’  + Y’) -  B] = D₅₅; I₀₂ = b^†_r b_r ; -Y’= D₄₄;

 1/6 [I₀₂ - I₀₁ + (Z’  + Y’) -  B] = D₆₆; I₀= ½( a^†_p a_p - a^†_n a_n);

-Z’= D₈₈;

Z = ¼ (a^†_p a_p  + a^†_n a_n + a^†_λ a _λ – 3 a^†_x a_x);

Y’  = B/8 + 1/8 [(b^†_q b_q  + b^†_r b_r + b^†_t b_t )  – 7 b^†_s b_s);

Z’  = B/8 + 1/8 [(b^†_q b_q  + b^†_r b_r + b^†_s b_s )  – 7 b^†_t b_t);

where ( p, n, λ, x, q, r, s ,t ) is the super-quark basis vector and a^†_μ and a_μ are the creation and annihilation operators of appropriate suffixed Bosonic quarks and b^†_μ  and b_μ  are the creation and annihilation operators for the appropriate suffixed Fermionic quarks. D_ij represent the diagonal operators of the 64-plet [56, 69].

     In terms of the natural particles the diagonal elements are given by

π⁰/ √2  + η⁰/ √6  + M₀^c /√8 ;  - π⁰/ √2  + η⁰/ √6  + M₀^c /√8 ;

-2 η⁰/ √6  +   M₀^c /√8 ;  - 3 M₀^c /√8 ;

and similarly the remaining with the replacements,

π⁰ à ρ⁰  ;    η⁰  à φ⁰  ;   M₀^c à ω⁰ .

To obtain the broken symmetry we should choose a suitable quantum number which is a linear combination of the seven quantum numbers Y, I₀, Z, Y’, Z’, I₀₂  and I₀₁.

Of interest now is the structural constants of SU(8) symmetry and the  SU(4)⨂SU(4) content of certain higher multiplets in 63⨂63 exterior product. Analogy of our model with the previous work by Iowa who used an SU(8)= SU(4)⨂ SU(2) symmetry and with SU(6) = SU(3)⨂ SU(2). Symmetry discussed by Oakes and Speiser [36] and by Beg and Singh [37, 38] may then be examined.

        Table I shows a few of the structure constants of all those obtained for SU(8) super-quark model.

TABLE – I

Certain SU(8) structure constants*

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 i         j         k                 f_ijk               I    j      k      d_ijk

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1        16      19               1                1      17    19    1

1      25    28           1              1      24    1      2/√10

1      36    39            1              1      35    1      2/√15

1      49    52            1              1      36    38    1

2      17    19            1              1      63    1      2/√28

2      26    28            1              2      25    28    1

2      37    39            1              2      35    2      2/√15

2      50    52            1              2      49    52    -1`

3      27    28            -1             2      63    2      2/√28

3      38    39            -1`            3      38    38    -1

3      51    52            -1             3      51    51    -1

4      49    54            1              3      39    39    -1

4      16    21            1              4      36    40    1

5      16    20            1              4      49    53    1

5      36    40            1              5      26    29    1

5      49    53            1              5      36    41    -1

6      19    20            -1             6      19    21    1

6      27    30            1              6      28    30    1

6      39    40            -1             6      38    40    1

6      52    53            -1             6      52    54    1

                                                6      63    6      2/√28

12    38    42            1              12    27    32    -1

24    31    32            1/√10                24    30    30    1/√10

35    53    54            1/√15                35    53    53    1/√15

40    48    41            -1/√21               40    48    40    -5/√21

46    60    61            -1             40    60    62    1

48    59    60            1/√21                48    60    60    1/√21

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*Here f _ijk  and d _ijk are defined by the relations,

[ λ _i , λ _j]  = i f_ijk  and  { λ _i , λ _j} = d_ijk

where λ’s are the generators of the SU(8) group.

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Complete list of structure constants of SU(8) obtained and used by the author have been listed in a separate publication. (truscincetrutechnology@blogspot.com, Vol.2014,No.3, March 11.)

        The spectral content of certain higher multiplet of the 63⨂63 product is projected out in Fig.1. The spectral analysis of Lepton-Meson-Baryon symmetry multiplet of 63⨂63 within the super-quark SU(8) = SU(4)⨂SU(4) is quite distinctive and instructive for physical considerations of mass splitting.

The Young Tableau for 63 order group have been obtained specifically see img_0830, img_0834, img_0836, img_0837, img_0840, img_0841, img_0846, img_0855 and img_0866 lists this data. [Refer:

trusciencetrutechnology@blogspot.com, Volume 2014, Issue No.3, March 11, 2014, Time: 10h29m A. M. Professor Dr. Kotcherlakota Lakshmi Narayana, TRU S&T ]

(2 1 1 1 1 1 0)=216; (3 1 1 1 1 1 1)=280; (1 0 0 0 0 0 0)=8;

(1 1 1 1 1 1 0)=8*;    (1 1 0 0 0 0 0)=28; (2 0 0 0  0 0 0)=36;

(2 2 1 1 1 1 1)= 216; (2 2 2 2 2 2 2)=36*; (1 1 1 1 1 1 0)= 28*;

(3 1 1 1 1 1 0)=945;  (2 2 1 1 1 1 0)=720; (3 3 2 2 2 2 2)=945*;

(2 1 1 1 1 1 1)=63; (4 2 2 2 2 2 2)=1232.

Fig.1  SU(8)àSU(4)⨂ SU(4) spectral analysis of 63⨂63 product

  IMG_0862 KLN 2^nd March 1980.

The present analysis is fundamentally different from the classification schemes of SU(4) à SU(2) ⨂ SU(2)  and the  SU(4)à SU(3) )⨂U(1)  or the Wigner super multiplet SU(4)à SU(2) ⨂ SU(2) multiplets and similarly from SU(6) à SU(3) ⨂ SU(2) previously given by Lipkin [52]. In the present spectral analysis an higher energy state for the multiplets involving Vector Meson, Baryon, Lepton-Meson is adopted in that order and an immediate modification would be to lower those involving the Leptons relative to those involving the Mesons such as in the multiple pairs BL and BM shown in Fig.1.

For the sake of aesthetics beauty the former is preferred. The two 63-plets that occur in the product 63⨂63 are analogous to the occurrence of doublet octets in the Gellman octet model. The present theory, however, clearly distinguishes these doublet 63-plets by their spectral contents as evident from the spectra shown in Fig. 1(a) and Fig. 1(b).

  

SECTION II: INVARIANT SUM RULES OF WEAK PSEUDO-SCALAR MESON CURRENTS AND LEPTON-BARYON SUPER-QUARK CURRENT STRUCTURES.

        In the super-quark model SU(8) we may next examine the structure of Super Lepton-Meson-Baryon currents [69]. These are again analogous to the vector currents of weak lepton and Baryon interactions [72] but are characteristically different. I define the super quark current to be,

(F^V_sμ) ^β_α = ( ̅64)^β_λ f^V_sμ (64)^λ_β   -   ( ̅64)^λ_β f^V_sμ (64)^α_λ

in an antisymmetric combination of the 64-plet super-quark model using the f_ijk  structural constants. The symmetric combination super-quark current would be then be,

(D^V_sμ) ^β_α  =    ( ̅64)^β_λ d^V_sμ (64)^λ_α     +    ( ̅64)^λ_α d^V_sμ (64)^β_λ     -

-   2/3   δ ^β_α( ̅64)^λ_ν d^V_sμ (64)^ν_λ

involving  the d_ijk  structural constants defined earlier.  Of interest are then the partial conservation of these super currents f^V_sμ  and d^V_sμ.

 The detailed expressions of these worked out by the author listed in a separate publication entitled: trusciencetrutechnology@blogspot.com,

Volume 2014, Issue No.3, March 11, 2014, Time: 10h29m A. M.

Professor Dr. Kotcherlakota Lakshmi Narayana, TRU S&T

          The three ingredients of the present theory are (1) Hypothesis of virtual gluonic super quark currents detailed by the matrix T (2) The conservation of Lepton-Baryon super current structure and finally (3) the strong Lepton-Baryon symmetry principle. Earlier investigators considered the first two mentioned above but they treated them separately. The fact that they are intimately related is one of the important consequences of the present theory of the author. Model also yields relations between currents that are combinations of Meson-Lepton and Vector Meson-Baryon Vector currents and their interactions. Certain cancellations among these essentially require the principle of strong Lepton-Baryon symmetry. But then more significant is that the violations of this Lepton-Baryon symmetry principle, even weakly, would eventually give rise to strong renormalization effects, for both the Weak Leptonic and the Weak Baryonic Vector currents. Detailed study of these renormalization effects is important in the context of a Charmed Spin 5/2 Graviton theory developed by the author within a super gravity formulation.

        Even under the assumption of a strong (in a sense of the above arguments) Lepton-Baryon symmetry principle, conservation of f^V_sμ  super current structure is possible only if certain sum rules are obeyed by the Meson-Meson Charmed interaction currents.  Such rules are useful in the light of current experimental studies on the Charmed pseudo-scalr meson decays.[73]. Charmed Baryon production in e ^– e⁺ annihilation (Milkaelian and Oakes [21]) and on the discovery of heavy mesons in e ^– e⁺ annihilation and production experiments [20]. 

       For vector dominance models of radiative and Leptonic 

decays of the ψ-states see the work by Nandy [26].

        Sum rules of Meson-Meson interactions that would follow from the above considerations are listed below in Fig.2.

                

Comparision of these expressions fir the super quark weak Lepton-Baryon vector currents with these designated as J₀ and J₁ by Marshak et al [6,7] shows that new additional terms exist. Thu current expressions are more general in addition to that they involve charmed terms.

        To this discussion we may state that it is possible to unify the super- Symmetric Lepton-Meson-Baryon interactions with the gravitational interactions giving rise to a new formulation of a super-gravity theory in which the Charmed Graviton of Spin 5/2 may be realized. A cursory report on the spectral content analysis and of other associated spin 5/2 elementary particle entities and as well the salient features of the model and its flexibility to incorporate the Color, Flavor, Style, Truth, Beauty aspects has been given elsewhere by the author [53-59].

CONCLUSIONS

                The hypothesis of Virtual Gluonic Super Currents leads to an Unification of Lepton-Meson-Baryon particles and the resulting super-quark SU(8) symmetry incorporates the principle of strong Lepton-Baryon symmetry as an essential ingredient for obtaining the new sum rules of Meson-Meson Vector Meson-Vector Meson interactions. The super quark model SU(8) = SU(4) ⨂ SU(4) has the flexibility of being combined with other symmetries by virtue of the content of its higher multiplets such as 63⨂63 and therefore leads to a super-gravity theory.

                                        ----------x----------

ACKNOWLEDGEMENT

                      The author is deeply indebted to Late Professor K. R. Rao D.Sc. (Madras), D.Sc. (London) whose inspiring guidance helped me a lot to understand and generate newer thoughts in the subjects of Science and Technology.

I).LIST OF STRUCTURE CONSTANTS OF SU(8) symmetry used by KLN

See trusciencetrutechnology@blogspot.com, Volume 2014, Issue No.3, March 11, 2014, Time: 9h41m A. M. Professor Dr. Kotcherlakota Lakshmi Narayana.

Permanent address: TRU S&T.

II).REDUNDANT DIAGRAMS OF YOUNG TABLEAU

See trusciencetrutechnology@blogspot.com, Volume 2014, Issue No.3, March 11, 2014, Time: 10h29m A. M. Professor Dr. Kotcherlakota Lakshmi Narayana.

Permanent address: TRU S&T.

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40. Mohapatra. R. N, Phys. Lett, Vol.82B, p.194, 1979.

41. Seji Ono, “Quark model and sizes of Baryons ND Meson 

      and ψ”, Alexander Von Humboldt Foundation Fellow 

      reprint 1976.

42. Jakimov, Kalman, Lett. Il. Nuovo Cimento, 

      Vol.17, p.65, 1976.

43. Salam A. preprint IC/76/21, 1976.

44. Tolkachev R.A. Tomil’chik. Phys. Lett.Vol.81B, p.173, 1979.

45. Cocho.  Il Nuvo Cimento, Vol.XLIVA, No.2, p.336, 1976.

46. Cremmer. E, Scherk J, PTENS 76/15 preprint June 1976.

47. Gliozzi F. Scherk J, Olive D., Phys. Letts. 

      Vol.65B, p.282, 1976.  

48. Chiral SU(4) ⨂ SU(4) model with broken iso-spin symmetry for a set of scalar and pseudo-scalar has been adopted by Dutta. E, Sinha. S. N, Phys. Lett, Vol.70B, p.103, 1977.

49. Navare S and Narayana K. L. “A 64-plet super quark 

      model of the Hadrons”, IIIrd High Energy Physics 

      Symposium, Jaipur, 1978.

50. Narayana K. L. et al, “Supersymmetry transformations with 

       internal  SU(3) ” , International Symposium on 

       Transformation Groups and their Applications to Physical 

        Problems, S. N. Bose Institute, Calcutta, March 1-2, 1979.

51. Navare  S. and Narayana K.L. “SU(6) model of Mesons and           Baryons and Superstrong and Electromagnetic splitting”, 

      High Energy Physics Symposium, II, Bhubhaneswar, 

      Paper No. 9, 1977.

     52. Recently with a non-relativistic colored quark model, 

            Lipkin J.H,  obtained the correct mass splitting of 

            1 ^-, 2⁺, 3^- Mesons and 3/2⁺, 5/2^-, 7/2⁺ Baryons. Lipkin H.J.  

           Phys. Lett, Vol.74B, p.399, 1978. He contends in another 

           work Lipkin H.J.  Phys. Lett, Vol.67B, p.65, 1977, that the 

           nonet pseudo-scalar meson is inconsistent with an SU(4)

          16- plet approach though it is successful for vector and 

          tensor Meson  lassification. (See Ref.2 and 11 cited). 

          Also see the book, “Lie groups for Pedestrians” by 

           Lipkin H.J. North Holland Publishing company,

           Amsterdam, p.108, 109, 118,170, 1976.

     53. Narayana .K. L., “Charmed spin 5/2 graviton within a 

           supergravity  formulation”, Paper No. 7, Proc. Ind. Sci.

           Cong. Jadvapur University, Calcutta, 2^nd Feb. 1980.

      54. Narayana. K.L. “Multiplet content and Sum Rules of Weak 

           Lepton- Hadron super currents in a SU(8) superquark 

            model”, Paper No. 8,

             Proc. 67^th Ind. Sci. Cong. Jadavapur University, Calcutta, 

            2^nd Feb. 1980.

     55. Narayana. K. L. “Beharrung gauge approach and an unified 

           model of Gravitational, Electromagnetic and Strong 

           interactions”, Mathematics section, Proc. 67^th Ind. Sci. 

           Cong. Jadavapur University, Calcutta, Paper No.108,

           5^th Feb 1980.

     56. Naryana. K. L. “On the Unification of Gravity and Quantum 

          Physics”, J. Shivaji University (Sciences), 

          Vol.17, p.31-21, 1977.

     57.  Narayana. K. L. “Quantum Mechanics”, text book published

            by KUSUM PRAKASHAN, 83/1 Plot. No. 2 Sarang Society, 

            Pune -411009, India. Section 8.4, p.207, 25th June 1979.

     58. Narayana .K. L., “On other Gravity possibilities of 

           gravitation”, Invited Paper presented at the Einstein 

           Centenary Symposium, Nagpur  University, Nagpur, 

           20^th Feb 1980.

    59. For a postulation of medium and strong interactions and 

          relationships of De Broglie wavelengths of the particles 

          with invariant coupling constants and resonance charges 

          reference may be made to Narayana. K. L., Patil S. B., 

          Indian Journal of Physics, Vol.50, p.993-1002, 1976  and 

          Narayana,  K. L.  Proc. 63^rd Ind. Sci. Congress,

          Waltair,  Paper No. Jan 4^th, 1976.

   60. For color-statistical approach for a new interpretation of 

          Heisenberg’s Uncertainty principle and indeterminacy of 

          states in Quantum Mechanics refer forthcoming 

          paper by Narayana. K. L.

    61. The assignments made by me, differ from SU(3) 

          assignments made by Marshak (Ref No.6) for the case of 

          Leptons. The neutral particle designation is however 

          analogous. Interesting to note that the possible

          Ninth Baryon  Y⁰ (also concomitant in our model) 

          has been earlier suggested by Schwinger

          (Ref.No.6 page 127). He also suggests a O^- Meson called δ. 

          My other assignments for the Hadrons and Leptons

         follow the well known assignments of the Octet Model.

    62. Han M. Y, Nambu Y. Phys. Rev Vol.139B, p.1006, 1965.

    63. Han M. Y, Bidernharn L.C. Phys. Rev Letts, 

          Vol.24, p.118, 1970.

    64. PLUTO collaboration, Phys.  Lett.  Vol.84B,  p.84, 1979.

    65. For molecular Charmonium refer Giles R.C, Tye. S. E.,

          Phys. Letts.  Vol.73B,  p.30, 1978.

    66. Arisue Bando M, Torin T., Prog. Thoer. Phys. 

          Vol.59, p.668, 1978.

    67. Dominance of certain modes of decay apart from Narrow 

          Resonances widths of Charmonium states have been 

          pointed out by Gupta. V. preprint  of talk given at

          II High Energy Physics Symposium, Bhubaneswar, 1976.

    68. Carlson C. E,  Suaya, Phys Letts. Vol.81B,  p.329, 1979.

    69. Algebraic structure underlying the super symmetry is not a 

          Lie Algebra for example, refer Fermi-Bose symmetry by 

          Ferrara. S, Rivista Del Nuovo Cimento, Vol.6, p.105, 1976.

    70. Narayana. K. L. and Nimbargi S.S.  “SU(5) classification of 

          Elementary Particles”,  Internal report,  Shivaji University, 

          Kolhapur. 15th March 1972

    71. Narayana. K. L. and Patil S. P. “Classification of Elementary 

          Particles”, Internal report,  Shivaji University, Kolhapur 

          15^th March 1974.

    72. Schwingwer J (Ref. No.61) discusses on the possibilities of 

          Baryon  0^-  Meson and Baryon  1^-   Meson interactions. 

          Unlike his idea of the fundamental field theory the model 

          suggested in the present paper adopts hypothesis of virtual 

          GLUONIO  currents that bind the Super-multiplets to give 

          raise the observed particles.

   73. Lipkin H.J., FERMILAB-CONT-77/93-THY, 1977 deals with four

         quark states with one pair of Charm exotics.

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