Pure spin-3/2 propagator for use in particle and nuclear physics

Abstract

We propose the use of pure spin-3/2 propagator in the (3/2, 0) ⊕ (0, 3/2) representation in particle and nuclear physics. To formulate the propagator in a covariant form we use the antisymmetric tensor spinor representation and we consider the ∆ resonance contribution to the elastic πN scattering as an example. We find that the use of conventional gauge invariant interaction Lagrangian leads to a problem; the obtained scattering amplitude does not exhibit the resonance behavior. To overcome this problem we modify the interaction by adding a momentum dependence. As in the case of Rarita-Schwinger we find that a perfect resonance description could be obtained in the pure spin-3/2 formulation only if hadronic form factors were considered in the interactions.PACS numbers: 11.10. Ef, 11.15.2q, 14.20.Gk, 13.75.Gx For decades the most commonly used propagator for spin-3/2 particles (e.g., the N and ∆ resonances) in particle and nuclear physics is the Rarita-Schwinger (RS) one [1], although it is also well known that the RS propagator has an intrinsic and long-standing problem [2,3]; it contains the unphysical extra degrees of freedom (DOF) or lower spin background. To be more precise, let us begin with the RS field that is formed by the tensor product of a vector and a Dirac field represented by (1/2, 1/2) and (1/2, 0) ⊕ (0, 1/2), respectively. This tensor productwhich shows that the RS field consists of two fields; the (1, 1/2) ⊕ (1/2, 1) and the Dirac field. The orthogonality relation can be used to eliminate the Dirac field. The (1, 1/2)⊕(1/2, 1) is, however, still not free from the Dirac background. So far, the popular choice for the interaction Lagrangian is given, e.g., in Eq. (16) of Ref.[5], which contains the so-called off-shell parameter. This parameter is required to maintain the invariance of the RS Lagrangian under point transformations [6]. In the phenomenological study of nuclear physics, however, the physical meaning of the off-shell parameter raised a serious problem, i.e., the coupling constants could heavily depend on the off-shell parameter [8] and the ∆ contribution for the Compton amplitude is obscured by this parameter [7]. There is also some infamous fundamental problem regarding the interaction with the electromagnetic field, i.e., the Johnson-Sudarshan [9] and Velo-Zwanziger [10] problems. The origin of these problems comes from the unphysical degree of freedom that arises in the interaction for whatever choice we make for the off-shell parameter.Furthermore, it is shown in Ref.[5] that this interaction does not posses any local symmetry of RS field and, as a consequence, it violates the constraints for reducing the number of independent components of the field to the correct value and involves the unphysical lower-spin DOF. The pathologies of this interaction can be removed by introducing the gauge-invariant (GI) or consistent interaction to decouple the unphysical spin-1/2 background from the ∆-exchanges amplitude [5].Nevertheless, for the practical use of spin-3/2 propagators and couplin...

Pure spin-3/2 representation with consistent interactions

“…As in the previous report [9] we can explore the behavior of the pure spin-3/2 propagator by comparing the ∆(1232) resonance contribution to the total cross section of elastic πN scattering, obtained from the pure spin-3/2 propagator and the conventional ones. For this purpose it is important to include the resonance width Γ in the resonance propagator, i.e., by replacing iǫ → iΓm ∆ .…”

“…The next application of our present work is the contribution of spin-3/2 ∆ resonance to the pion photoproduction off a nucleon. As shown in the previous work the choice of inappropriate electromagnetic interaction could fail to generate the correct property of a resonance in the cross section [9]. Thus, in what follows we will calculate the ∆ resonance contribution to the total cross section with a consistent interaction and compare the result with those of previous works.…”

“…In what follows we briefly summarize the ATS formalism and show that this formalism has a problem to describe the properties of a resonance. We have discussed this topic in our previous Rapid Communication [9]. Let us start with the Casimir operator F = 1 4 J µν J µν with J µν the angular momentum operator.…”

“…In the previous paper we have briefly reported the use of pure spin-3/2 propagator to describe the ∆ resonance in the πN scattering [9]. It was shown that the conventional GI interaction Lagrangian cannot describe the resonance behavior of the spin-3/2 ∆ baryon, unless the interaction was modified by adding an extra momentum dependence.…”

Photoproduction γp→K+Λ(1520) in an effective Lagrangian approach

“…In future when more data for γp → K + Λ(1690) become available, a serious treatment of the resonance propagators as discussed in Refs. [49,50] will be needed.…”