Isospin

For an additional degree of freedom of a particle that defines one of the two possible states in which it exists, we shall apply now the concept of isospin (in analogy with the theory of the nucleus where the isospin doublet includes protons and neutrons, which are treated as two states of the same particle - the nucleon [122]). For a pair of states (a,/ ) we introduce the isospin operators 

Using the anticommutation relations (13.15) we can readily verify that these operators obey the conventional commutation relations (14.2) for the irreducible components of the angular momentum operator. Further, from the definition 

The annihilation operators a and do not form an irreducible tensor, but the operators 

The isospin operators can be expressed in terms of the irreducible tensors (18.6) and (18.7) 

All these operators commute with the particle number operator in the pairing state 

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Fettes N., Meissner U. G., Steininger S. Pion-nucleon scattering in chiral perturbation theory (I) Isospin-symmetric case. Nucl. Phys.A 640, 199-234 (1998)... 

Fettes N., Meissner U. G. Towards an understanding of isospin violation in pion-nucleon scattering, Phys. Rev. C 63, 045201-045211 (2001). 

Expressions for the medium modifications of the cluster distribution functions can be derived in a quantum statistical approach to the few-body states, starting from a Hamiltonian describing the nucleon-nucleon interaction by the potential V"(12, l/2/) (1 denoting momentum, spin and isospin). We first discuss the two-particle correlations which have been considered extensively in the literature [5,7], Results for different quantities such as the spectral function, the deuteron binding energy and wave function as well as the two-nucleon scattering phase shifts in the isospin singlet and triplet channel have been evaluated for different temperatures and densities. The composition as well as the phase instability was calculated. 

Isospin singlet (pn) and triplet (nn,pp) pairing in nuclear matter... 

The model Hamiltonian H in Eq.(9) preserves isospin symmetry if the condition... 

For states of nuclear matter with high density and high isospin asymmetry the experimental constraints on the EOS are rather scarce and indirect. Different approaches lead to different or even contradictory theoretical predichons for the nuclear matter properties. The interest for these properties lies, to a large extent, in the study of astrophysical objects, i.e., supernovae and neu-... 

The combined effect of these TBF is a remarkable improvement of the saturation properties of nuclear matter [12], Compared to the BHF prediction with only two-body forces, the saturation energy is shifted from —18 to —15 MeV, the saturation density from 0.26 to 0.19 fm-3, and the compression modulus from 230 to 210 MeV. The spin and isospin properties with TBF exhibit also quite satisfactory behavior [18],... 

In [21] the authors investigate the effect of quark masses on the CFL phase. These calculations are done in the asymptotic limit, and are reliable for sufficiently small quark masses. When mu = rnd = m ms (unbroken SU(2) isospin, but explicitly broken SU(3)), one finds a kaon condensate. The critical value of rns at which the condensate forms is m to1/3 Aq 3, where Ao is the CFL gap (see, in particular, equation (8) of the first paper). As kaons transform as a doublet under isospin, the vector SU(2) symmetry is broken in seeming contradiction with our result. 

Our example deals with quarks of the third color in a phase composed only of up and down quarks. As only quarks of a single color are involved, the pairing must take place in a channel which is symmetric in color. Assuming s-wave condensation in an isospin-singlet channel, a possible candidate is a spin-1 condensate [8], We consider the condensate... 

Figure 7. A projection of the Fermi surfaces on a plane parallel to the axis of the symmetry breaking. The concentric circles correspond to the two populations of spin/isospin-up and down fermions in spherically symmetric state (Se = 0), while the deformed figures correspond to the state with relative deformation Se = 0.64. The density asymmetry is a = 0.35.

Equal spin (isospin, flavor) pairing is another option, if the interaction between the same spin particles is attractive [22-24], Since the separation of the... 

It should be obvious that if A vanishes, the phase degree of freedom has to become redundant, as seen later. It would be worth mentioning that similar configuration has been studied in other contexts [21-23], Note that the configuration in (44) breaks rotational invariance as well as translational invariance, but the latter invariance is recovered by an isospin rotation [26]. 

We will investigate the influence of the form-factor of the interaction on the phase diagram and the EoS of dense quark matter under the conditions of charge neutrality and isospin asymmetry due to / -equilibrium relevant for compact stars. 

For nonvanishing A in Eq. (1) the color symmetry is broken. Two of the three quark color degrees of freedom are coupled to bosonic Cooper pairs in the color antitriplet state which can form a Bose condensate.One can combine the chemical potentials Hu, Hd of u and d quarks by introducing Hq = (hu + Hd)/2 and hi = (hu — Hd)/% as the Lagrange multipliers related to, respectively, the quark number density nq and the isospin asymmetry n/. In thermal equilib-... 

In [25, 26] it is shown that at given pq the diquark gap is independent of the isospin chemical potential for Pi ) < Pic(Pq), otherwise vanishes. Increase of isospin asymmetry forces the system to pass a first order phase transition by tunneling through a barrier in the thermodynamic potential (2). Using this property we choose the absolute minimum of the thermodynamic potential (2) between two /3-equilibrium states, one with and one without condensate for the given baryochemical potential Pb = Pu + 2pd-... 

Following Ref. [29] we introduce the quark chemical potential for the color c, fiqc and the chemical potential of the isospin asymmetry, //,/, defined as... 

It is easy to see from [59] that for nuclei with more than one nucleon the expression in (10.38) would contain an extra factor equal to the eigenvalue of the doubled third component 2T3 of the weak isospin operator. 

LAMBDA PARTICLE. A hyperon with a rest-mass energy of 1115.6 McV. an isospin quantum number zero, an angular momentum spin quantum number j. and a strangeness quantum number I. Symbol. K. 

Isospin is a useful concept in that it is conserved in processes involving the strong interaction between hadrons. The use of isospin can help us to understand the structure of nuclei and forms the basis for selection rules for nuclear reactions and nuclear decay processes. While a detailed discussion of the effects of isospin upon nuclear structure, decay, and reactions is reserved for later chapters, a few simple examples will suffice to demonstrate the utility of this concept. 

Consider the A = 14 isobars, 14C, 14N, and 140.14C and 140 are mirror nuclei and have ground states with T3 = + 1. As such they must be part of an isospin triplet with T = 1 (T3 = 0, +1). Thus, in the 7) = 0 nucleus, 14N, there must be a state with T= 1, r3 = 0 that is the analog of the 7) = 0 ground states of 14C and 140. (See Problems section for further details.) We expect the three members of this multiplet to have approximately the same energy levels after correction for the Coulomb effect and the neutron-proton mass difference. 

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