On the Difficulty in the Non-Relativistic Treatment of the Composite Model of Elementary Particles

Nagasaki, Masayuki

doi:10.1143/ptp.37.437

Cited by 8 publications

(3 citation statements)

References 6 publications

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“…The term proportional to J-t' in Eq. (21) is to reduce the absorption at large b, and that proportional to a' is to keep the corrected absorption coefficient non~negative always or in effect up to a large enough b beyond .which the corrected absorption coefficient is sufficiently small in its order of mag~ nitude. The impact parameter approximation gives then the correction A/ab to /ab: iLJ f ab(t)/Pcm =J-tlf-t 1 e-.Utltl/ 2 (1f-tll ti/2)-aa' e-attl/ 2 (1-a It 1/2) (22) where jl=P,lP-2/(P-1 +~-t2) and ii=aJ.-t2/(a+~-t2)· The total cross section calcu~ Ia ted from Imj ab(O) through the optical theorem is corrected by the amount (23) f-tl cannot take too large a value in order that the corrected absorption coefficient is kept non-negative in the way mentioned above, since p, 1 p,' is restricted through Eq.…”

Section: Angular Dz'stribut'ion Of P-p Elastz'c Scatteringmentioning

confidence: 99%

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Angular Distribution ofp-pElastic Scattering at High Energies and Radius of the Hard Core

Nagasaki

Taketani

1973

Prog. Theor. Phys. Suppl.

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The problem of the hard core in nuclear force at high energies is studied in connection with the theory of the structure of elementary particles. Reviewed and discussed are :various models of the origin of the hard core. Experimental angular distributions for small and large momentum transfers of p-p elastic scattering at high energies are analyzed, to deduce the general trend of the variation of the radius of the hard core with incident energy. 53proposed to devide the range of action of the nuclear force into the three · reglons:Region I: Classical region, static main, quantitative treatment. Region I I:Quantum region, dynamical region, qualitative treatment. Region I I I : Phenomenological region. This division was deduced from the following methodological analyses. The nuclear force at comparatively large distances, that is at r ?:::J/mrr, where r and mrr are the inter-nucleon distance, and the meson mass, respectively (li= 1), the one-pion-exchange is dominant. Moreover, for the nuclear force due to one-pion-exchange we have the one and same result as that due to the classical treatment of the meson theory. Thus, for r?:::J/mrr, that is in Region I, the nuclear force can be treated quantitatively by means of the meson theory.At distances sma1ler than the meson Compton wave length, 1/mrr, contributions from exchange of two or more mesons become effective. In the calculation of contributions of such kind, there appear divergent integrals . due to virtual mesons having large momenta, and many dynamical effects like the effect of nucleon recoil, exchange of heavier meson, and so on. The nuclear force at these distances, that is in Region II, should therefore be treated qualitatively, by means of the meson theory, while grasping the meson as the substance of the mediation of the nuclear force. This necessitates the introduction of the cutoff for momentum of meson around the nucleon mass M N, that is, the cutoff in the coordinate space around the nucleon Compton wave length 1/M N· For r:S1/M N, that is in Region III, we had no reliable theory at that time, and we were led to treat the nuclear force in Region III phenomenologically.On the basis of the method of investigating the nuclear force outlined above, in 1951 Taketani, Ohnuma and Koide7) derived that the type of pion should be pseud-scalar, from the analysis of the quadrapole moment of deuteron, which is governed mainly by the nuclear force at large distances, by using the meson theoretical potentials for the nuclear force at large distances, while assuming a hard core in the nuclear force at small distances. Although a hard core was also introduced by J astrow (1950)8> to describe p-p scattering at 300 MeV, it was not possible to get such a clue to the meson theory, since he used a phenomenological potential for the nuclear force at large distances, too. Many theoretical analyses performed since then in J apan9) along the directions given in T.N.S. (1951) established the existence of the hard core of radius of about 2/ M N in the nuclear force at low energi...

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Section: Angular Dz'stribut'ion Of P-p Elastz'c Scatteringmentioning

confidence: 99%

“…8, the value of the radius of the hard core is almost unaltered by the correction given by Eq. (21) or Eq. (24) to the absorption coefficient at large b's.…”

Section: Andmentioning

confidence: 99%

Angular Distribution ofp-pElastic Scattering at High Energies and Radius of the Hard Core

Nagasaki

Taketani

1973

Prog. Theor. Phys. Suppl.

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“…(r)'s with (E± U(r)) in the denominator. For example, in the case of A), one of the amplitudes of spin' singlet part, aa0, is given in terms of the other Cleo as follows: (3) where A= (mr-m2) /2.…”

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confidence: 99%

On the Two-Body Bound Problem of Dirac Particles

Koide

Yamada

1967

Prog. Theor. Phys.

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Recently, the mass levels of mesons have been successfully treated with the Schrodinger equations on the basis of the composite modeJ.l>,z> The validity of the nonrelativistic approximation was investigated by Nagasaki 3 > using the Dirac equation. In this note, the same problem is studied from a slightly different stand point. The model of Dirac particles interacting thr6ugh potential is assumed here; too. It is concluded that if there is the mass· difference between the constituents, the mass of a composite particle cannot be smaller than M=(mr+m2)/2, where the m 1 and m2 are the masses of the constituents, except for the 1 So state.The energy of a composite· particle is given by (Ho+ L:Oa V,_,(r)), Or= Ict,m ct,C2l, O!J'= -p~l)p~2)a(l>acz>+p~l>p~2)aO>aC2), o..~.= -pillp~2>+acr>aC2l and Qp= -p~1 !p~2>. The wave function 1/r(r) decomposed into the eigenfunction of the total angular momentum J, and parity P. Each eigenfunction has eight components.From Eq.(1), we can obtain two sets of simultaneous differential equations of first order for the eight amplitudes which are functions of the relative distance r only, corresponding parity states A) P= (-)J and B) P= (-)J+r. In those sets, the potentials are contained in terms of Ua(r) = Vs(r) -6V!l'(r) + Vp(r) ±4(Vv-(r)-V..;.(r)), U2±(r) = Vs(r)-Vp(r) ±2(Vv-(r) + V..~.(r)), Ua (r) = Vs(r) +2Vp(r) + Vp(r).Of the eight amplitudes, only the four amplitudes, written as GeJ(r) (j=O, 1, 2, 3), appear with the differential operators and the rests, written as aat(r) (j=O, 1, 2, 3), do not. Therefore, only the a,(r)'s are required to be continuous for the whole range of positive r. 4 J The aa(r)'s are represented in terms of a.(r)'s with (E± U(r)) in the denominator. For example, in the case of A), one of the amplitudes of spin' singlet part, aa0, is given in terms of the other Cleo as follows:( 3) where A= (mr-m2) /2.In order that the 1/r(r) 1s normalizable, the amplitudes may not have singular points. Therefore, for example, the potential Uz+(r) must satisfy -E< Uz+(r) for the whole range of r. Applying the similar discussions to all the aa(r)'s, we obtain -EJ< U2+(r) ro or r(ikr) ·for r>ro and by jJ(pr) for r

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Critical mass in a relativistic two-body problem

Bawin

Lavine

1977

Nuovo Cim B

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On the Difficulty in the Non-Relativistic Treatment of the Composite Model of Elementary Particles

Cited by 8 publications

References 6 publications

Angular Distribution ofp-pElastic Scattering at High Energies and Radius of the Hard Core

Angular Distribution ofp-pElastic Scattering at High Energies and Radius of the Hard Core

On the Two-Body Bound Problem of Dirac Particles

Critical mass in a relativistic two-body problem

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