P -wave nucleon-pion scattering amplitude in the ∆(1232) channel from lattice QCD

Giorgio Silvi, 2, ∗ Srijit Paul, Constantia Alexandrou, 5 Stefan Krieg, 2 Luka Leskovec, 7 Stefan Meinel, John Negele, Marcus Petschlies, Andrew Pochinsky, Gumaro Rendon, Sergey Syritsyn, 13 and Antonino Todaro 14, 2 Forschungszentrum Jülich GmbH, Jülich Supercomputing Centre, 52425 Jülich, Germany Faculty of Mathematics und Natural Sciences, University of Wuppertal Wuppertal-42119, Germany Institut für Kernphysik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus Computation-based Science and Technology Research Center, The Cyprus Institute, 20 Kavafi Str., Nicosia 2121, Cyprus Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA Department of Physics, Old Dominion University, Norfolk, VA 23529, USA Department of Physics, University of Arizona, Tucson, AZ 85721, USA Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Helmholtz-Institut für Strahlenund Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Nußallee 14-16, 53115 Bonn, Germany Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA Dipartimento di Fisica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy (Dated: January 5, 2021)


I. INTRODUCTION
The ∆(1232) (in the following denoted as ∆) is the lowest-lying baryon resonance, typically produced when energetic photons, neutrinos, or pions hit a nucleon [1]. While these three processes differ immensely, they have the two-particle nucleon-pion scattering amplitude in common. The scattering amplitude in which the ∆ appears as an enhancement is the P -wave with J P = 3 2 + and I = 3 2 , often also referred to as the P 33 amplitude. For energies near the ∆ mass, this amplitude is nearly completely elastic [2,3].
Modern determinations of the ∆ resonance parameters are typically performed using data from experiments such as CLAS12 at JLab and MAMI- * giorgiosilvi@gmail.com A1 in Mainz. While the results for the pole location differ slightly from the Breit-Wigner parameters [4], the ∆ is generally found to have a mass of approximately 1230 MeV and a decay width of approximately 100 MeV [5,6].
Phenomenological studies of the ∆ have been performed using quark models, chiral perturbation theory and related effective field theories, and the Smatrix approach. From the quark-model point of view, many baryons remain elusive, but the ∆ mass is reproduced quite well [7][8][9]. Chiral perturbation theory and related effective field theories have shown great success in determining low-energy scattering parameters and πN scattering amplitudes [10][11][12][13]; an extensive review can be found in Ref. [14]. Analyses of the large experimental data sets using amplitude models based on S-matrix principles were performed in Refs. [15][16][17].
First-principles computations of ∆ properties can be done using lattice QCD. The ∆ mass, assuming a stable ∆, was studied in Refs. [18][19][20][21][22][23]. However, for quark masses corresponding to pion masses below a certain value, the ∆ is an unstable hadron, and its mass and decay width must be determined from the appropriate N π scattering amplitudes. While the use of Euclidean time in lattice QCD prevents direct computations of infinite-volume scattering amplitudes [24], Lüscher showed how the finite-volume energy spectrum of a two-body system interacting through an elastic short-range interaction is related to the infinite-volume scattering amplitudes [25][26][27]. The decades following Lüscher's seminal work witnessed further development of the theoretical framework to moving frames [28,29], unequal masses [30][31][32], and arbitrary spin [33]. These methods have been applied to many systems in the meson sector and are reviewed in Ref. [34]. For the nucleon-pion scattering only a handful of studies have been done in the N π channel [35][36][37][38][39][40][41][42].
In the following, we report a new lattice-QCD study of elastic N π scattering in the ∆ resonance channel using the Lüscher method. Our calculation is performed using N f = 2 + 1 flavors of clover fermions at a pion mass of m π = 255.4(1.6) MeV, on a lattice with L ≈ 2.8 fm. We obtain detailed results for the energy-dependence of the scattering amplitude by analyzing multiple moving frames. From the amplitude's pole position, we determine the ∆ mass, decay width, and its coupling to the N π channel. Preliminary results were previously shown in Ref. [43]. The computations presented here also constitute the first step toward a future calculation of N → N π electroweak transition matrix elements using the formalism of Refs. [44,45].
The paper is organized as follows: in Sec. II the details of the lattice gauge-field ensemble are presented. Section III describes the interpolating operators and the method used to project to definite irreducible representations of the lattice symmetry groups. The Wick contractions yielding the two-point correlation functions for the ∆ − N π system are discussed in Sec. IV. In Sec. V, the results of the spectra analysis are presented. The relevant finite-volume quantization conditions are discussed in Sec. VI. The K-matrix parametrizations employed for the scattering amplitudes and our results for the amplitude parameters are presented in Sec. VII. We conclude in Sec. VIII.

II. GAUGE ENSEMBLE
We use a lattice gauge-field ensemble generated with the setup of the Budapest-Marseille-Wuppertal collaboration [46], with parameters given in Table I. The ensemble has been used previously in Ref. [47]. The gluon action is the tree-level improved Symanzik action [48], while the fermion action is a tree-level clover-improved Wilson action [49] with two levels of HEX smearing of the gauge links [46]. We analyze 600 gauge configurations and compute the correlation functions for 16 source positions on each configuration, resulting in a total of 9600 measurements. When considering the N π system in the rest frame only, the spatial lattice size of L ≈ 2.8 fm (with periodic boundary conditions) results in a rather sparse energy spectrum across the elastic region. Between the N π and N ππ thresholds there are few energy points available to constrain the phase shift we aim to determine. A straightforward way to gain additional points would be to add a spatially larger ensemble, but this is computationally quite expensive. A more efficient approach employed here is using also moving frames [28,50,51] on the same ensemble, where the Lorentz boost contracts the box, resulting in different effective values of the spatial length along the boost direction [32].

III. INTERPOLATING OPERATORS
We use local single-hadron and nonlocal multihadron interpolating operators, both necessary for a complete determination of the resonance properties [53]. For the single-hadron ∆ operators with I = 3/2, I 3 = +3/2 (corresponding to the ∆ ++ ) we include two choices, The two-hadron interpolators with the same quantum numbers are obtained from products of the form Axis and planes of symmetry TABLE II. Choices of total momenta P , along with the Little Groups LG, irreducible representations Λ of relevant hadrons and their angular momentum content J P . The multi-hadron N π operators have the same irreps as the single-hadron ∆ operators. From left to right the subduction of irreps in moving frames. The label "(2)" for irrep G in group C D 2v indicates the double occurrence of the irrep from the subduction; to differentiate this irrep from the homonymous of group C D 3v we keep the additional label "(2)" throughout the paper. Images credit [52].
as explained in more detail below. The pion interpolator (I = 1, I 3 = +1) is given by and for the nucleon (I = 1/2, I 3 = +1/2) we again include two choices, To correctly identify the angular momentum in the reduced symmetry of the cubic box, we project the operators to the irreducible representations (irreps) that belong to the symmetry groups of the lattice. Instead of the infinitely many possible irreducible representation J P of the continuum, on the lattice, there are only a finite number of possible irreps Λ. Thus each lattice irrep in principle contains infinitely many values of the continuum spin J. Each irrep belongs to a Little Group LG( P ) describing the underlying symmetry of the lattice points contracted in the direction of the boost vector P , i.e., the total momentum of the N π system.
In the moving frames considered here, the symmetries are reduced to the groups C 4v , C 2v , C 3v (see Table II). The degree of symmetry is mirrored by the group's order g LG( P ) , which corresponds to the number of transformation elements (rotations and inversions) belonging to the group. In particular, halfinteger spin is best described by the double-cover of symmetry groups (labeled D), which introduce the 2π rotation as a new element of the group, effectively doubling the elements of the original group [54]. Additionally, a clear parity identification is lost in the moving frames, where the subduction mixes parities in the same irrep [55]. The list of chosen total momenta, symmetry groups, and irreps for the hadrons used in this work can be found in Table II.
To project the single-hadron operators to a definite irrep Λ and row r, we make use of the formula [56][57][58][59][60]: where d Λ is the dimension of the irrep Λ and Γ Λ are the representation matrices belonging to the irrep Λ. The matrices W (R) −1 correspond to the matrices appearing in the right-hand sides of Eqs. (A1), (A2), or (A4). Here we denote the elements of the little group generically as R, even though in the rest frame they include the inversion in addition to the lattice rotations. The index i labels the embedding into the irrep and replaces any free Dirac/Lorentz indices appearing on the right-hand side of Eq. (5). The analogous projection formula for the twohadron operators is Representation matrices for irreps in the rest frame are found in [60,61] and for the moving frames are provided in [59]. In Eq. (6), given a total momentum List of projected single-hadron (∆) and multi-hadron (N π) operators for all irreps. In the construction of the multi-hadron operators, we use optimized nucleon operators N that are linear combinations of N (1) and N (2) , as defined in Eq. (17). P , the sum over internal momenta is constrained by the magnitudes | p 1 | = |R p| = | p| and | p 2 | = | P −R p|. The structure of the projected operators ∆ and N π for all irreps is listed in Table III. In general, both Eqs. (5) and (6) produce for each row r of irrep Λ multiple operator embeddings (identified by the label i) that are not guaranteed to be independent. We therefore perform the following three steps to arrive at our final set of operators [58]: (i) Construct all possible operators using Eqs. (5) and (6) for r = 1 only.
(ii) Reduce the sets of operators obtained in this way to linearly independent sets.
(iii) Construct the other rows r for these linearly independent sets of operators.
The operators obtained in step (i) have the generic form Using Gaussian elimination we obtain a smaller matrix c Λ,1 nj such that the linearly independent operators constructed in step (ii) have the form The number of independent operators (corresponding to the range of the index n) is equal to [61,62] 1 where the characters χ Γ Λ (R) and χ R are equal to the traces of the representation matrices Γ Λ and the transformation matrices W (R).
In step (iii), to construct the other rows r > 1 we where the rotations/inversions R O j ( P ) R −1 are performed as in Eqs. (5) and (6), depending on the structure of O j ( P ). Also, to increase statistics, multiple directions of P at fixed | P | are used (see Table III). For every moving frame, we first perform the irrep projections for a reference momentum P ref and then rotate the projected operators to the new momentum direction. Generating operators initially from a reference momentum and r = 1 only facilitates the identification of equivalent operators embeddings that can later be averaged over different rows of the same irrep Λ (which is possible due to the great orthogonality theorem [62]) and momentum direction of equal | P |. In the following, the label r for the row will be dropped.

IV. WICK CONTRACTIONS
From the ∆/N π interpolators discussed above, we build two-point correlation matrices for each total momentum P and irrep Λ, where the indices i, j now label all the different operators in the same irrep that can vary in internal momentum content, embedding from the multiplicity, or gamma matrices used in the diquarks of Eqs. (4) or (1). The Wick contractions are computed following the scheme outlined in Refs. [63,64]. The correlators with single-hadron interpolators at source and sink are constructed from point-to-all propagators, while the correlators with a single-hadron interpolator at the sink and a two-hadron N π interpolator at the source use in addition a sequential propagator, with sequential inversion through the pion vertex at source time. The topologies of these diagrams are shown in the top panel of Fig. 1. The bottom panel of Fig. 1 shows the topologies for the correlators with N π operators at both source and sink. The diagrams are split into two factors, separated at the source point and by using a stochastic source -propagator pair. For the latter we use stochastic timeslice sources in the upper two diagrams. In the lower diagrams we employ spin-dilution and the one-end-trick in addition to time dilution. The quark propagators of all types are Wuppertalsmeared [65] at source and sink with smearing parameters α Wup = 3.0 and N Wup = 45; these parameters were originally optimized for the nucleon twopoint functions in Ref. [47]. The gauge field deployed in the smearing kernel is again 2-level HEX-smeared [66,67].

V. SPECTRA RESULTS
The masses of the pion and nucleon are used as input parameters in the Lüscher method. We extract them from fits of their dispersion relations, shown in Figs. 2 and 3, giving The energies are obtained from single-state fits of the two-point functions projected to different momenta (using a cosh for the pion and a single exponential for the nucleon). For the ∆-N π system, to extract the energy levels E Λ, P n (where n now counts the finite-volume energy levels for a given Λ, P ) from the correlation matrices C Λ, P ij we use the generalized eigenvalue problem (GEVP) [26,[68][69][70] C Λ, P ij (t)u n j (t) = λ n (t, t 0 )C Λ, P ij u n j (t),  where u n j are the right generalized eigenvectors. In the plateau regions the energies are obtained from fits to the principal correlators λ n (t, t 0 ) with single exponentials as λ n (t, t 0 ) ∼ e −E Λ, P n (t−t0) (15) or using two exponentials with a small contribution from higher excited states with energy E Λ, P n as λ n (t, Here, t 0 is a reference timeslice that does not strongly affect the large-t behavior; we set t 0 /a = 2. Additionally, for the projected multihadron operators (N π) Λ,r ( P ) we implement an optimized interpolator of the nucleon [71] where i labels the two types of nucleon operators in Eq. (4) and u (N )i are the generalized eigenvectors (for t/a = 4) from a single-nucleon GEVP analysis. The optimized nucleon interpolator has improved overlap with the single-nucleon ground state with momentum p 1 [72]. For the coupled ∆-N π system we build for each irrep Λ a correlation matrix C Λ, P ij from the projected ∆ and optimized N π operators in Table III. The , the energies obtained from single-exponential fits to these correlators as a function of tmin/a (center), and the energies obtained from double-exponential fits to these correlators as a function of tmin/a (right). All energies shown here are converted to the center-of-mass frame. Black dashed lines represent the N π and N ππ thresholds. Non-interacting N π energy levels are shown as green lines.
multiplicities of operators give rise to a fairly large basis for each correlation matrix (the dimensions for the full bases correspond to the sums of numbers of operators for each irrep listed in Table III). Through singular value decomposition of CC † or C † C we can infer which operators contribute to the largest singular values, allowing us to explore sub-bases of the full list of operators that can lead to reduced noise of the principal correlators while maintaining the complete spectra.
Baryons are known to have a narrow plateau region (the "golden window" [73]) where the higher states contribution get small enough to enable a sin-gle exponential fit to describe maximally a single level before the rapid decay of signal-to-noise ratio at larger t [74,75] converted to the center-of-mass frame using The center-of-mass energies are also related to the scattering momenta through Our main results are obtained from singleexponential fits to the principal correlators and are listed in Table IV. The fit ranges are chosen after a stability analysis. The upper limit of the fit range, once chosen large enough, is found to have a small impact on the fit itself; thus, we fix it to t max /a = 15 for all levels. On the other hand, the lower limit is varied within a reasonable range until a plateau region is identified. This is illustrated in the center subplot for each irrep in Figs. 4 and 5. To further test the stability, we also performed two-exponential fits using Eq. (16), which give the results shown in the right subplots of the figures and are found to be in reasonable agreement.
It can be seen in the plots that energy levels that overlap strongly with the N π states shift away from the resonance region, as expected. For the irrep (2)G in | P | 2 = (2π/L) 2 2, the situation is more complicated and a higher number of energy states appear in the region of interest. This situation originates from having only a single irrep for the Little Group C D 2v , resulting in a maximal mixing of angular momenta.
A summary of all extracted energy levels is shown in Fig. 6.

VI. LÜSCHER QUANTIZATION CONDITIONS
The Lüscher quantization condition connects the finite-volume energy spectra affected by the interactions and the infinite-volume scattering amplitudes; resonances correspond to poles in the infinite-volume scattering amplitudes at complex √ s and in principle affect the entire spectrum. For elastic 2-body scattering of nonzero-spin particles, the quantization condition can be written as [50] det(M P Jlµ,J l µ − δ JJ δ ll δ µµ cot δ Jl ) = 0, (21) where δ Jl is the infinite-volume scattering phase shift for total angular momentum J and orbital angular momentum l, and µ, µ = −J, ..., J. Both the scattering phase shift and the matrix M P Jlµ,J l µ are functions of the scattering momentum, and the solutions of the quantization condition for the scattering momentum give the finite-volume energy levels through Eq. (20). The matrix M P Jlµ,J l µ encodes the geometry of the finite box and is a generalization for particles with spins σ, σ of the spinless counterpart via where M P lm,l m (for a cubix box with periodic boundary conditions) is given by [50] M P lm,l m (q 2 ) = where q = kL 2π with k the scattering momentum and L the side length of the box. Here Z P js (1, q 2 ) is the generalized zeta function, γ = E P / √ s is the Lorentz boost factor and the coefficient C lm,js,l m expressed in terms of Wigner 3j-symbols read C lm,js,l m = (−1) m i l−j−l (2l + 1)(2j + 1)(2l + 1) To simplify notation it is common practice to define the functions The elements of the matrices M P Jlµ,J l µ for all choices of P considered in this work are listed in Appendix B.
Furthermore, it is possible to extract quantization conditions for each irrep Λ via a change of basis of Eq. (21). The basis vector of the irrep Λ can be written as [50,76]  where the coefficients c Λrn Jlµ for l ≤ 2 can be found in Refs. [50,60], and the parity eigenstate vectors |Jlµ are given by One can then make a change of basis for which the matrix elements of M are given by where it is found, from Schur's lemma, that the matrix M is partially diagonalized in irrep Λ and row r. However, the matrix is not diagonal in n, which labels the multiple embeddings of the irreps. In our case only the irrep (2)G of the group C D 2v has multiple embeddings with multiplicity m G = 2 (See Table  II).
In principle, there are infinitely many values of total angular momentum J and therefore also infinitely many partial waves l in each irrep, but, as the higher waves have an increasingly smaller contribution, we consider only the dominant partial waves. In particular, we assume the contributions from partial waves in J > 3/2 to be negligible and exclude them from the analysis. For the N -π system, J = 3/2 includes both the P -wave (l = 1) and the D-wave (l = 2). Several irreps mix J = 3/2 with J = 1/2, and the latter includes l = 0, 1. Nevertheless, at our level of precision, in addition to the resonant phase shift P 33 (J = 3/2, l = 1) for isospin I = 3/2 we are sensitive only to S 31 (J = 1/2, l = 0) [2], for which the closest resonance would be the distant ∆(1620). In order to better constrain the S 31 (J = 1/2, l = 0) contribution, we also include the irrep G 1u , which is the only irrep we can access that contains only spin J = 1/2 and l = 0 (up to contributions from l > 2), ensured by the negative parity (ungerade). As can be seen in Table III, the interpolating operators in the G 1u irrep are exclusively N -π two-hadron operators, consistent with the expectation that the S 31 phase shift is nonresonant at low energy. The quantization conditions for all irreps, expressed in terms of the two phase shifts δ 3/2,1 , δ 1/2,0 and the functions w lm , are listed in Table V.

A. Parametrizations used
We use the K-matrix parametrization rescaled with the two-body phase space ρ as where (30) The K-matrix relates to the phase shifts as As discussed in Sec. VI, our analysis includes the phase shift δ 3/2,1 , where we expect the ∆ resonance that will be quite narrow for our quark masses, and the phase shift δ 1/2,0 , which is expected to be nonresonant in the energy region considered. We therefore use a Breit-Wigner parametrization for the former, where m BW denotes the resonance mass and the decay width Γ(s) is given by  with the coupling g BW , scattering momentum k, and center-of-mass energy squared s. For the nonreso-nantK (1/2,0) we use the effective-range expansion [77]. We find that working to 0th-order is sufficient at the level of precision we have, such that with the S-wave scattering length a 0 .

B. Fit procedure and results
Following Ref. [80] and as in our previous work [81], we perform a global fit of the model parameters m BW , g BW , and a 0 to all energy levels in all irreps by minimizing the χ 2 function χ 2 = P ,Λ,n P ,Λ ,n .
Here, C is the covariance matrix of the energy levels are obtained for each parameter guess by finding the roots of the Lüscher quantization conditions (see Table V). There are 21 energy levels from 8 irreps available for the global fit, as shown in Fig. 6.
The results for both the global fits and for fits to subsets of energy levels are listed in Table VI. Before performing the global fit to all energy levels, we separately considered the irreps that include either only J = 1/2 or only J = 3/2 (ignoring J > 3/2). The irrep G 1u is the only one that contains exclusively J = 1/2, while there are multiple irreducible representations with exclusively J = 3/2: H g , G 2 , F 1 , and F 2 . These initial two fits enable us to obtain a good initial guess for the parameters of the final global fits and assess the stability of the fit over the choice of irreps included. The fit for the S-wave (labeled S) via irrep G 1u is done to only 2 energy levels, resulting in a low χ 2 /dof. The other partial fit over irreps containing P-wave only (P) includes 8 energy levels and gives a higher χ 2 /dof.
For the global fit (G), we implement five different combinations of levels included and choices of t min /a to test the stability of the results. The fit to the nominal results for the energy levels from Table IV is labeled as G(a), while the fit labeled G(a+1) was done to the energy levels with t min /a increased by one unit throughout. More focused choices among the noisiest levels are made in the fit G(b), where we vary t min /a in selected levels based on the results of the stability analysis shown in Figs. 4 and 5. Specifically, this case uses a +1 shift on t min /a on all levels of irreps G 2 , (2)G, F 1 , F 2 , the ground state of G 1 , the first excited of G, and +2 on the first excited of G. Additionally, we perform the global fit G(c) removing potentially problematic levels from the list in Table IV: the highest level of irrep G and all levels in irrep (2)G. Furthermore, the global fit labeled G(d) differs from G(a) only by excluding irrep G 1u . Overall, the results indicate that the fits provide compatible results and are very stable across several choices.
Using this result for Γ, we also determine the coupling g ∆-πN from the equation for the decay width in leading-order chiral effective theory [78], which gives The extracted values for the resonance mass m ∆ and coupling g ∆-πN are compared to recent results from the literature in Table VII. Our results for the scattering length a 0 are generally consistent with zero within the uncertainities.

VIII. CONCLUSIONS
We have presented a determination of elastic nucleon-pion scattering amplitudes for isospin I = 3/2 using a lattice QCD calculation on a single gauge-field ensemble with pion mass m π ≈ 255 MeV. The baryon ∆(1232) emerges as the dominant resonance in the P -wave with J P = 3/2 + and is the focus of this work. The infinite-volume scattering amplitudes are obtained using the Lüscher method from the finite-volume energy spectra extracted from correlation matrices built of ∆ and N π operators, projected to definite irreducible representations of the lattice symmetry groups. In order to thoroughly map out the energy dependence using just a single volume, it is essential to consider moving frames, where the symmetries are reduced. Many irreps included mix J = 3/2 and J = 1/2, and we therefore also extracted the scattering phase shift for the latter. Each J receives contributions from two values of orbital angular momentum l, but at the present level of precision, we can access only a single dominant value of l for each: l = 1 for J = 3 2 and l = 0 for J = 1 2 . In addition, we neglect mixing with J > 3/2. We performed global fits to the spectra using a Breit-Wigner parametrization for the P 33 phase shift at energies below the inelastic threshold N ππ, and using the leading-order effective-range expansion for the S 31 phase shift. We also extracted the pole position m ∆ − iΓ/2 associated with the ∆ resonance, and the coupling g ∆-πN that determines the decay width Γ at leading order in chiral effective theory. These parameters are compared with other determinations in Table VII. For our pion mass (and at nonzero lattice spacing), m ∆ is found to be approximately 170 MeV higher than in nature, while the coupling g ∆-πN agrees with extractions from experiment at the 2σ level, given our uncertainties. Our result for the coupling also agrees with previous lattice determinations within the uncertainities. In the S wave, our result for the scattering length is consistent with zero and is approximately 2σ away from phenomenological determinations.
Future work will include computations on additional lattice gauge-field ensembles with different spatial volume, which will provide more data points to better constrain the phase shifts extracted and, at the same time, expand on the partial-wave contributions included in the analysis and provide information on remaining finite-volume systematic errors. Using additional ensembles will also enable us to investigate the dependence on the pion mass and on the lattice spacing. Furthermore, we plan to use the results for the energy levels and scattering amplitudes as inputs to a computation of N → N π electroweak transition matrix elements using formalism of Refs. [44,45], similarly to what has been done for πγ * → ππ [72,85]. In this appendix we list the transformation properties of the momentum-projected field operators under inversions I and spatial rotations R. The pseudoscalar pion transforms as R π( p) R −1 = π(R p) I π( p) I −1 = −π(− p), while the nucleon transforms as where S(R) is the bi-spinor representation of SU (2). For a rotation of angle 2π/n around the axis j, this is given by with the antisymmetric tensor ω kl = −2π jkl /n and ω 4k = ω k4 = 0 [59]. The vector-spinor Delta operator transforms as R ∆ αk ( p) R −1 = A(R) −1 kk S(R) −1 αβ ∆ βk (R p) I ∆ αk ( p) I −1 = (γ t ) αβ ∆ βk ( p) where A(R) denotes the 3-dimensional J = 1 irrep of SU (2), and S(R) is given in Eq. (A3).