FURTHER DISCUSSION ON THE EQUATION AND ITS SOLUTIONS FOR THE BOUND STATES OF MESONS

We continue the investigation of the equation and its solutions for the bound states of mesons discussed previously, emphasizing on large values of v, a parameter characterizing the shape of the potential. The results show that the dependence of the eigenvalues λ and some related physical quantities upon the potential parameters v and E is approximately the same as in our previous work. But the average radii of the bound mesons cease increasing as v increases. For each value of v, there exists a maximum average radius√<R²>_max, and all these √<R²>_max are approximately equal to 12.6 M^-1 （M being the mass of the straton）, which is much smaller than the experimental value of the radius of the π meson. However, if we assume, in addition to the attractive potential, there exists, for the straton-anti-straton superstrong interactions, also a repulsive potential, and, in particular, if we introduce a differencetype potential V_D（r） corresponding to the potential V_v（r） in our previous work, the calculated radii of the bound mesons do approach the experimental value.     

SOME CONJECTURES ON THE MASS FORMULA OF CHARGED LEPTONS

In this note we propose that the electromagnetic self energy of lepton may beconnected with quantum number n by（ （δm）/m）=（1/（2π））n^-b ,in which b is a constant.Wefnrther propose that the cut-off value of momentum M is connected with gravitationalconstant κ and fine structure constant α by M=（1/√κ）αe^-1-b.So we obtainthe mass formula of charged leptons M=（1/√κ）α exp{-3/4-b-(1/3α)n^-b}.By using the masses of e^- and μ^- and the value of a as in put,we get the calculatedvalues of κ=（6.67231±0.00026）×10^-8cm³g^-1sec^-2and m_r=（1782.306±0.078）MeVwhich agree very well with experimental values κ=（6.6720±0.0041）×10^-8cm³g^-1sec^-2 and m_τ=(1782_-4⁺³)MeV respectively.The mass of the fourth charged leptonpredicted by the formula should be m=（11725.47±0.51）MeV,which can be checkedby experiments in the near future.We also discuss briefly the proposed formula andthe obtained results.     

AN ANALYSIS OF FOUR-ELEMENT MODELS OF LEPTONS AND NON-DIAGONAL NEUTRAL CURRENTS

Due to the possible production of neutral and charge heavy leptons by v_μ, models consisting of four elements per "line", e.g. v_e, e^-, E⁰, E^- v_μ, μ^-, M^0, M^- v_τ, τ^-, T⁰, T^- ……are analysed. Non-diagonal neutral currents cannot be avoided. Nevertheless, a model is proposed within SU_（2）×U_（1） in which the process μ→eγ can still be suppressed by the Bjorken-Lane-Weinberg mechanism; various production cross-sections for heavy leptons are reasonable when compared with experiments. Models corresponding to SU_（3）×U（1） and still higher symmetries are discussed.     

ON THE IRREDUCIBLE REPRESENTATIONS OF THE COMPACT SIMPLE LIE GROUPS OF RANK 2(I)

In this paper, we analyse the commutation relations of the infintesimal opera-tors of the group SU₃ and find that the eight infinitesimal operators of the groupSU₃ can be written as a scalar operator A, three angular momentum operators (L₁,L_o, L_-1,)and two sets of the irreducible tensor operators of rank 1/2, （T_±1/2,V±1/2）By means of the commutation relations of these operators, all irreducible represen-tations of the group SU₃ can be easily obtained. In this pape, the matrices corresponding to these operators in the irreduciblerepresentation（λμ）, are given; therefore the irreducible representation and its re-presentation space R^λμ are completely defined. Besides, a method for calculatingthe scalar factors of the reduction coefficients and the symmetric relations of thosefactors are also given. As examples, the scalar factors of the reduction coefficientsof （λμ）×（10）, （λμ）×（01）, （λμ）×（20） and （λμ）×（11） are calculated. In the last part of this paper, we define the irreducible tensor operators ofthe group SU₃ and prove the corresponding Wigner-Eckart theory. The method used in the discussion of the group SU₃ be extended to allof the compact simple Lie groups of rank 2 and we shall discuss them in two suc-ceeding papers.     

THE BEHAVIOURS OF F2ep(X,Q2)-F2en(X,Q2) AND THE SCALE-BREAKING PARAMETER

The scale-breaking parameter F₂^ep(x, Q²) of non-singlet component F₂^ep-F₂^en and some other quantities in the electro-production process are discussed by usinb the analytic expression for the distribution function of valence quarks in ha.drons up to leading log approximation given by a, previous paper^[1]. Comparisons with data are also given.

     

SPIN GAINT RESONANCE AND CORE POLARIZATION EFFECTS

The effects of the coupling term between field（Y₀σ₁）_1v and r²（Y₂σ₁）_1v coming fromtensor operator are discussed for the spin gaint dipole resonance.The effects of inclu-ding the field r^λ+1(Y_λ+1σ₁)_λv on the spin gaint multipole resonance and core polariza-tion are considered.     

FINE STRUCTURES OF INCLUSIVE SPECTRA(Ⅰ)——SUM RULES AND THE GENERALIZATION OF FEYNMAN-YANG SCALING

Important implications of the fine structure of inclusive spectra(to be calledinclusive and semi-inclusive spectra of nearby particles,which represent the local dis-tributions of nearby particles in three-dimensional phase space with rapidity y and transverse momenta P_⊥x, P_⊥z as independent coordinates are explained,and some basicfeatures of the fine stucture are found,namely,sum rules and the generalized formof the Feynman-Yang scaling.One of the sum rules,for example,is:where f_(1;k) denotes the normalized invariant inclusive cross section of k closely neigh-boring particles.It follows that the inclusive the spectra of nearby particles arequalitatively different from the usual ones.The generalized form of the Feynman-Yang scaling for the case of k closelyneighboring particles,for example,is:f_(1;k)(s,x₁,P_⊥1,…x_k,P_⊥k⁾∞,（s→∞,x₁≤x₂≤…≤x_k）.where‘∞’denotes‘approaches a definite limit’.For k=2,the existing experimentaldata for the rapidity gap-length distributions show that for FNAL energies,f_(1,k） isalready close to its limiting form.The inclusive（semi-inclusive）spectra of nearbyparticles way be able to reflect effectively short-range correlation effects.     

THE DIFFICULTY OF THE DECAY τ→πντ AND SPIN 3/2 HEAVY LEPTON

We discuss the suppression of the decay of heavy lepton τ into πv_τ,by assuming that τ and/or v_τ carry spin 3/2. For the case of spin 3/2 T and spin 1/2 v_τ, using the weak current ψ_μ^(τ)(a+b_^γ5)Ψ^(v_τ)we find that the πv_τ decay channel relative to the v_τv_ll^-channel is suppressed 12/5 times, while other decay channels are not changed appreciably. △R=σ（e⁺e^-→τ⁺τ^-）/σ（e⁺e^-→μ⁺μ^-） is also discussed.     

APPROXIMATE SEQUENCE FOR U（t,t0）-OPERATOR

We give an approximate sequence for U（t,t₀）-operator.We prove the follow-ing theorems:Theorem 1.If the norm ||H（t）|| of H（t）in equation（2.1）is a Lebesgue in-tegrable function with respect to t,then there is an approximate sequence{U_n},such that for any state vector |Φ〉,|Ψ〉,the sequence <Φ|U_1^|Ψ><Φ|U_2^|Ψ>,......,<Φ|U_n^|Ψ>,......is uniform convergent with respect to t.Theorem 2.If in finite time interval,the norm ||H（t）|| of H（t）in equation （2.1）is a Lebesgue integrable function,then equation（2.1）has unique solution.     

SPACE-TIME PROPERTY OF FERMION NUMBER

Enlarging the dimensionality of Minkowski space from 4 to 5,and relating the restmass of particle with x₅ as m=—i（&#8706;/（&#8706;x_5）））we discuss the 5-dimensional non-linear con-formal group C（M₅）under which dx₁²+dx₂²+dx₃²-dx₀²+dx₅²=0 is invariant.The C（M₅）group is isomorphic to the linear group SO（5,2）from which we study thespace-time property of Fermion number,and the relations between half-integral（in-tegral）spin and odd（even）Fermion number are obtained.     

STUDY OF THE MECHANISM OF 12C（d,d）12C AND 12C（d,p）13C* REACTIONS

A formalism for studying the interference between the direct reaction and the compound resonance processes is presented by the S-matrix theory; The mechanism of ¹²C（d, d） ¹²C, ¹²C（d, p₁） ¹³C* and ¹²C（d, p₂） ¹³C* reactions in the energy range E_d=1.63 MeV to 2.05 MeV is analysed.Te results show that: the interference between these two processes exists; and the quantitative relation between them is given. While the parameters of direct reactions and compound resonance processes, particularly for four resonance states with E_d=1.726, 1.767, 1.792 and 1.86 MeV are determined.     

ON NUCLEAR SINGLE-PARTICLE POTENTIALS (Ⅱ) THE NONHERMITIAN CHOICE

The cancellation properties of the nonhermitian single-particle （SP） potential u_αβ=M_αβ （ε_β） [or M_αβ（ε_α）] according to the principle of maximal cancellation of perturbation diagrams are investigated in detail. The mass operator M_αβ（ω） is separated into two parts M_αβ^on （ω） and M_αβ^off （ω） as usual, however, a new criterion for their definition will be proposed. It is shown that the exact mass operator insertion is equal to the sum of the following three types of terms:（1） terms contributed by the poles of M_αβ^off（ω）. They are truly non-factorizable and must be considered separately,（2） terms which can be cancelled to all orders by the nonhermitian choice u_αβ=M_αβ （ε_β） [or M_αβ（ε_α）]. Hence, they serve to define u_αβ,（3） the remaining terms, which can be summed to all orders in a simple way and may be interpreted as amplitude renormalization of the SP Green function.In order to illustrate the usefulness of the above results, we have considered the renormalized random phase approximation （RRPA） for the particle-hole Green function as an example. Related formulae are derived. In RRPA, not only the SP propagator renormalization which includes all the effects except those contributed by the poles of M_αβ^off （ω）, but also the off energy shell property of the G matrix elements have been taken into account.     

THE TEST OF SLAC METHOD USING LEE MODEL

Treating Lee Model by Slac method,we find that,to state ||V> of one particle and state ||VN> of two particles,there corresponds a solution of Slac bag type re-spectively.The latter bears resemblance to the former in energy especially for ||V> state with the replacement of the interaction constants g by g/√e only.In addition,the“bag”corresponds precisely to the scalar field in ||V> and ||VN> states.However,for ψ_N^- and ψ_V^- field,we get δ-function type solutions by Slac method.They deviate greatly from the rigorous solutions.The“skin confinement”as statedby Bardeen et al.,and Rafelski is closely related to the δ-confinment in the one par-tiele state found here by coherent state approximation.Thus it may not representthe physical picture in the rigorous solution of a quantum field theory.     

A RECURSIVE FORMULA FOR THE CLEBSCH-GORDAN COEFFICIENTS OF SUn GROUP

It is proved that under the Gelfand-Biedenharn phase convention any SU_n singletfactor （SF） belongs to the type of the derivable or underivable,designated as （SU_nSF）d and （SU_r SF） respectively,and （SU_n SF）d=（SU_r SF）,n=r+1,r+2,….Therefore the calculation of any SU_n Clebsch-Gordan coefficients is reduced to thecalculation of a few underivable singlet factor （SU_r SF）.     

DISCOVERY OF MULTIPLE,COLLECTIVE BAND STRUCTURES IN NUCLEI WITH A=60—80

Recent studies of levels in even-even ^68,70,72Ge,^70,72,74Se,^74,76,78,80Kr and ⁶⁵Ga and ⁷⁴Br have led to the discovery of a wide variety of different collective band structures.These include bands built on near spherical ground states and excited more well de-formed shapes that may include triaxial shapes,rotation-aligned bands built on thesame orbital （g_（9/2））² for both protons and neutrons,RAL negative parity bands witheven and odd spins,and ΔI=1 γ-type vibrational bands in even-even nuclei.As recent as 1974,a survey of the energy level in the even-even Ge and Se iso-topes^[1] revealed little was known above a spin of 4⁺ （see for examples Figs.1 and 2 of Ref.1）.With the exception of the unusally low-lying excited 0⁺ states in ^70,72Ge,first discovered in ⁷²Ge in 1948 at Vanderbilt^[2],the theoretical treatment of thesenuclei was limited primarily to some variation of the vibrational model.However,very recently there has been a surge of information on nuclei in this region that hasrevealed fascinating new features and also provided new insight into the excited 0^+’states.Particularly striking are the multiple,independent and highly collective bandstructures which we have discovered in our in-beam γ-ray spectroscopy studies fol-lowing heavy-ion induced reaction.Evidence for and the theoretical understandingof the richness of the collective band structures that are found in our studies of^68,70,72Ge （Refs.3—6）,^70,72,74Se （Refs.7—13） and ^74,76,78,80Kr （Refs.14—18）,as illustratedby the at least seven different bands found in our studies of the levels of ⁶⁸Ge ⁷⁴Se,and ⁷⁶Kr （Figs.1—3）,are described in this paper.These multiple structures includethe following:a) coexistence of ground bands built on near-spherical shapes andexcited bands with larger deformation built on O^+′ band heads;b) bands with 8⁺ band heads interpreted as rotation-aligned,RAL,bands built on both neutron and/orproton （g_（9/2））² quasiparticle configurations coexisting with the ground-state band;c)RAL neutron and proton odd-parity bands formed from coupling of a g_{（9/2 ）}quasipar-ticle and a p_（1/2）,p_（3/2） or f_（5/2）,quasiparticle with the core;d) ΔI=1 even-parity bands,which are best characterized as gamma-vibrational bands;and e) additional bandswhose nature are presently not known.     

TESTING GLUON DISTRIBUTION FUNCTION OF HADRONS IN LARGE PT PROCESS WITH DIRECT PHOTON EMISSION

In this paper we disscuse disscuse the physicsical aspect of cross-section differences Δ_AB^cγ=dσ(AB→cγX)-dσ(AB→cγX) and give the simple rules to select the elementary subprocesses which contribute to cross-section differences.
We bind a way to test the gluon distribution functions of hadrons through the measurement of cross-section difference or combination of cross-section differences in the large P_T process with direct photon emission. Moreover we obtain simple relations between each combination of cross-sections and the corresponding gluon distribution functions of proton, neutron, pion and daon:
G_p⁸(x)/(Δ_pp^hγ－Δ_pp^hγ)=G_n⁸(x)/(Δ_pp^hγ)=G_π⁸(x)/(Δ_π+p^hγ+Δ_π－p^hγ=G_K⁸(x)/(Δ_K+p^hγ+Δ_K－p^hγ)which are expected to be test in experiment.     

THE INTERACTION OF A HIGH ENERGETIC PROTON WITH A NUCLEUS—THE RELATION OF THE PRODUCED PARTICLES WITH THE NUCLEAR MASS NUMBER A

Suppose that a high energetic proton interacts with a cluster of m particles in anucleus the mean multiplicity of this interaction and that of PP interaction followthe same law: n=CS^k. Using the Glauber theory we get R(A)≡n_PA/n_PP1.26A ^1/3(A ^1/3+1) ^-2/3 This formula explains the current experimental result quite well.     

ON NUCLEAR SINGLE-PARTICLE POTENTIALS（Ⅲ）SINGLEPARTICLE ENERGIES DETERMINED BY THE NONHERMITIAN POTENTIAL Uαβ=Mαβ（εβ）

In this paper the following results are proved:although the single-particle（sp）potential u_αβ=M_αβ（ε_β）defined in terms of the mass operator M_αβ（ω） is nonhermi-tian,the discrete energy eigenvalues determined by the Schrödinger equation h|γ>=（t+u）|γ>=ε_γ|γ> are real;moreover,they satisfy exactly the following relation:（ε_γ⁼±[E_nγ（N±1）—E₀（N）] where E₀（N）denotes the exact ground state energy of a closed-shell nucleus N,and E_nγ（N±1）are excat energy eigenvalues of its neighbouring N±1 nucleus.Further,in order to determine whether the bound state energies obtained by anyother sp potential may or not satisfy the above relation,a simple method is sugges-ted.It is shown that the amplitude renormalization of the sp Green function canalso be calculated easily by means of this method.     

A TWO-NUCLEON MODEL OF THE（p,π-） REACTION

In this paper,we suggest a two-nucleon model of the A（p,π^-） B reaction.Areduced formulation of the reactoin amplitude and a relation between the reactionamplitude and the nuclear structure are given.The differential cross section of ¹³C（p,π^-）¹⁴O_g.s,at E_P=185 MeV is calculated in the plane wave approximation.The results+s can explain the experimental data qualilatively.     

ON THE PHASE AND THE REPRESENTATION TRANSFORMATIONS OF SUn BARYON AND MESON WAVE FUNCTIONS

The baryon and meson wave functions of SU_n symmetry under the Baird-Bie-denharn phase convention are given explicitly.The phase confusion is clarified andthe matrices of transformations between I-spin,U-spin and V-spin representationsare given for the IR（8）,（27）,（35） of SU₃.