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Spin, Isospin and Strong Interaction Dynamics

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Spin, Isospin and Strong Interaction Dynamics October, 2011 PROGRESS IN PHYSICS Volume 4 Spin, Isospin and Strong Interaction Dynamics Eliahu Comay Charactell Ltd. P.O. Box 39019, Tel Aviv 61390 Israel. E-mail: [email protected] The structure of spin and isospin is analyzed. Although both spin and isospin are related to the same SU(2) group, they represent different dynamical effects. The Wigner-Racah algebra is used for providing a description of bound states of several Dirac particles in general and of the proton state in particular. Isospin states of the four ∆(1232) baryons are discussed. The work explains the small contribution of quarks spin to the overall proton spin (the proton spin crisis). It is also proved that the addition of QCD’s color is not required for a construction of an antisymmetric state for the ∆++(1232) baryon. 1 Introduction Two important conclusions are derived from this analy- sis. First, it is well known that quarks’ spin carry only a small The isospin notion has been conceived by W. Heisenberg in fraction of the entire proton’s spin [2]. This experimental ev- 1932 [1, see p. 106]. It aims to construct a mathematical ba- idence, which is called the second EMC effect and also the sis that represents the proton-neutron similarity with respect proton spin crisis, is shown here to be an obvious result of to the strong nuclear force. Both spin and isospin have the the multi-configuration structure of states of more than one same SU(2) group structure. Thus, like spin multiplets of a Dirac particle. Another result is that the anti-symmetric state quantum state, one combines corresponding states of nuclear ++ of the ∆ (1232) baryon is well understood and there is no isobars in an isospin multiplet. For example, the ground state need to introduce a new degree of freedom for its explana- of the 14C, 14O and the Jπ = 0+ excited state of 14N are mem- tion. It means that the historical starting point of the QCD bers of an isospin triplet. Obviously, one must remember that construction has no theoretical basis. (Below, the symbol ∆ isospin is a useful approximation that neglects proton-neutron refers to this isospin quartet of baryons.) differences that are related to their mass and their electromag- Generally, in order to simplify notation, the specific value netic interactions. of normalization factor is omitted from the expressions. The Later developments have shown that the proton-neutron second and the third sections analyze spin and isospin, re- similarity stems from the similarity between the u, d quarks. spectively. The fourth section provides an explanation for the It follows that the usefulness of isospin symmetry extends to proton spin crisis. The fifth section explains the antisymmet- particle physics. For example, the three pions are members ric structure of the ∆++ baryon (without using color). The last of an isospin triplet. Due to historical development, isospin section contains concluding remarks. notation takes different form in nuclear and particle physics. Here T and I denote isospin in nuclear and particle physics, 2 Spin States respectively. In this work the symbol T is used, mainly be- cause of the following reason. In the case of spin, the symbols A comprehensive discussion of angular momentum can be J and j denote total and single particle angular momentum found in textbooks [3]. In this short work some elements of operators, respectively. Similarly, the symbols T and t de- this theory are mentioned together with a brief explanation. note the corresponding isospin operators. Thus, due to the This is done for the purpose of arriving rapidly at the main same underlying SU(2) group, isospin relations can be read- conclusions. A relativistic notation is used and for this reason ily borrowed from their corresponding spin counterparts. The the j j coupling [3] takes place. operators T and t are used in the discussion presented in this Let us begin with a discussion of spin and spatial angu- work. lar momentum. These quantities are dimensionless and this This work examines states of electrons and quarks. These property indicates that they may be coupled. Now, the mag- particles have spin-1/2 and experimental data are consistent netic field depends on space and time. Moreover, the theory with their elementary pointlike property. Evidently, a theo- must be consistent with the experimental fact where both spa- retical analysis of an elementary pointlike particle is a much tial angular momentum and spin of an electron have the same simpler task than that of a composite particle. The discussion kind of magnetic field. Thus, it is required to construct a rela- begins with an examination of relevant properties of elec- tivistically consistent coupling of these quantities. This is the tronic states of atoms. The mathematical structure of the theoretical basis for the well known usage of spin and spa- SU(2) group is used later for a similar analysis of isospin tial angular momentum coupling in the analysis of electronic states. states of atoms. Eliahu Comay. Spin, Isospin and Strong Interaction Dynamics 55 Volume 4 PROGRESS IN PHYSICS October, 2011 A motionless free electron is the simplest case and the the Helium atom as a sum of configurations: spin-up electron state is [4, see p. 10] + + − − g = 1 1 + 1 1 + 0 1 (He ) f0(r1) f0(r2) 2 2 f1(r1) f1(r2) 2 2 − − + + B 1 C f (r ) f (r ) 3 3 + f (r ) f (r ) 3 3 + (3) B C 2 1 2 2 2 2 3 1 3 2 2 2 µ − B 0 C + + = imt B C ; 5 5 + ::: (x ) Ce B C (1) f4(r1) f4(r2) 2 2 @B 0 AC 0 Here and below, the radial functions fi(r); gi(r) and hi(r) where m denotes the electron’s mass. denote the two-component Dirac radial wave function (mul- tiplied be the corresponding coefficients). In order to cou- A second example is the state of an electron bound to a = hypothetical pointlike very massive positive charge. Here the ple to J 0 the two single particle j states must be equal electron is bound to a spherically symmetric charge Ze. The and in order to make an even total parity both must have the π same parity. These requirements make a severe restriction on general form of a j hydrogen atom wave function is [5, see + pp. 926–927] acceptable configurations needed for a description of the 0 ! ground state of the He atom. FY (rθϕ) = jlm ; (2) Higher two-electron total angular momentum allows the GY 0 jl m usage of a larger number of acceptable configurations. For π = − where Y jlm denotes the ordinary Ylm coupled with a spin-1/2 example, the J 1 state of the He atom can be written as to j, j = l ± 1=2, l 0 = l ± 1, F; G are radial functions and the follows: − l parity is ( 1) . + − + − (He − ) = g (r )h (r ) 1 1 + g (r )h (r ) 1 3 + By the general laws of electrodynamics, the state must be 1 0 1 0 2 2 2 1 1 1 2 2 2 − + − + g 1 3 + g 3 3 + an eigenfunction of angular momentum and parity. Further- 2(r1)h2(r2) 2 2 3(r1)h3(r2) 2 2 − + + − (4) g 3 5 + g 3 5 + more, here we have a problem of one electron (the source at 4(r1)h4(r2) 2 2 5(r1)h5(r2) 2 2 + − the origin is treated as an inert object) and indeed, its wave g 5 5 ::: 6(r1)h6(r2) 2 2 function (2) is an eigenfunction of both angular momentum and parity [5, see p. 927]. Using the same rules one can apply simple combinatorial The next problem is a set of n-electrons bound to an at- calculations and find a larger number of acceptable configura- tractive positive charge at the origin. (This is a kind of an tions for a three or more electron atom. The main conclusion ideal atom where the source’s volume and spin are ignored.) of this section is that, unlike a quite common belief, there are Obviously, the general laws of electrodynamics hold and the only three restrictions on configurations required for a good system is represented by an eigenfunction of the total angular description of a Jπ state of more than one Dirac particles: momentum and parity Jπ. Here a single electron is affected by a spherically symmetric attractive field and by the repulsive 1. Each configuration must have the total angular momen- fields of the other electrons. Hence, a single electron does not tum J. move in a spherically symmetric field and it cannot be rep- 2. Each configuration must have the total parity π. resented by a well defined single particle angular momentum 3. Following the Pauli exclusion principle, each configu- and parity. ration should not contain two or more identical single The general procedure used for solving this problem is to particle quantum states of the same Dirac particle. expand the overall state as a sum of configurations. In every configuration, the electrons’ single particle angular momen- These restrictions indicate that a state can be written as a sum tum and parity are well defined. These angular momenta are of many configurations, each of which has a well defined sin- coupled to the overall angular momentum J and the product gle particles angular momentum and parity of its Dirac parti- of the single particle parity is the parity of the entire system. cles. The role of configurations has already been recognized in the The mathematical basis of this procedure is as follows.
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