Consistence between $\pi N{\rm{\Delta }}$ spin-3/2 gauge couplings and electromagnetic gauge invariance

Abstract

We analyze the consistence between the recently proposed "spin 3/2 gauge" interaction for the ∆ resonance with nucleons (N) and pions (π), and the fundamental electromagnetic gauge invariance in any radiative amplitude. Chiral symmetric π-derivative πN∆ couplings can be substituted through a linear transformation to get ∆-derivative ones, which have the property of decoupling the 1/2 field components of the ∆ propagator. Nevertheless, the electromagnetic gauge invariance introduced through minimal substitution in all derivatives, can only be fulfilled at a given order n without destroying the spin 3/2 one by dropping n+1 order terms within an effective field theory (EFT) framework with a defined power counting. In addition, we show that the Ward identity for the ∆γ∆ vertex cannot be fulfilled with a trimmed 3/2 propagator, which should be necessary in order to keep the "spin 3/2 gauge" symmetry in the radiative case for the ∆γ∆ amplitude.Finally, it is shown that radiative corrections of the spin 3/2 gauge strong vertexes at one loop, reintroduce the conventional interaction.

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“…[43], and very recently reanalyzed in Ref. [45], consistent couplings cannot be kept in the presence of electromagnetic interactions. This is so since any derivative on the ∆ field gives rise through minimal substitution to a new nonderivative term.…”

Higher derivative terms in the πΔN interaction: Some phenomenological consequences

“…This criterion, subject to some controversy [6], is nonetheless the one used to fix Z 1 = 1/2 for I 1 in [1], and for I 2 it leads to Z 2 = −1/2, which corresponds to the interaction originally proposed by [7]. Nevertheless we have shown recently that this new interaction presents the same signature problems than the conventional L I 1 [10], it couples to a spin 1/2 background in radiative amplitudes and renormalization considerations force the reintroduction of conventional terms [9].…”

“…In addition as expected from general considerations from Effective Field Theory in the resonance region, (EFT) that consider a contribution of the pion momentum to the power counting of δ = (m N − m π )/Λ χP T or m π /Λ χP T δ 2 depending of its value, both interactions are of the same order since momentum coming from ∂ µ Ψ ν behaves as order 1 at threshold [9]. Furthermore, from the phenomenological point of view, the fit to data from the new interaction is by no means superior to the conventional [2] in the resonance region.…”

“…The newer interaction I 2 could be easily seen to be simply the next order in derivatives from the conventional coupling [9,10]. The addition of each term I k implies the incorporation of a new parameter to fit (the corresponding coupling constant).…”

Inconsistency of the ‘spin-3/2 gauge invariant’ interaction of Rarita–Schwinger fields

“…It has been argued in [3] that such interactions are free of the above mentioned problems and so it became quite popular for the description of Δ resonances. Nonetheless, we showed recently that in the presence of the EM coupling (unavoidable in this context, since the Δ is charged) the consistency problem remains even for this new proposed interaction: when the EM interaction is introduced, renormalization considerations force to reintroduce an interaction of the form  NEK [11]. The interaction obtained from  P does not eliminate spin 1/2 virtual states in all circumstances, radiative processes for instance exhibit a spin 1/2 'background'.…”