P-wave nucleon-pion scattering amplitude in the Δ(1232) channel from lattice QCD
Giorgio Silvi, Srijit Paul, Constantia Alexandrou, Stefan Krieg, Luka Leskovec, Stefan Meinel, John Negele, Marcus Petschlies, Andrew Pochinsky, Gumaro Rendon, Sergey Syritsyn, Antonino Todaro
PP -wave nucleon-pion scattering amplitude in the ∆(1232) channelfrom lattice QCD Giorgio Silvi,
1, 2, ∗ Srijit Paul, Constantia Alexandrou,
4, 5
Stefan Krieg,
1, 2
Luka Leskovec,
6, 7
Stefan Meinel, John Negele, Marcus Petschlies, AndrewPochinsky, Gumaro Rendon, Sergey Syritsyn,
12, 13 and Antonino Todaro
4, 14, 2 Forschungszentrum J¨ulich GmbH, J¨ulich Supercomputing Centre, 52425 J¨ulich, Germany Faculty of Mathematics und Natural Sciences, University of Wuppertal Wuppertal-42119, Germany Institut f¨ur Kernphysik, Johannes Gutenberg-Universit¨at 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 Departmentof Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Helmholtz-Institut f¨ur Strahlen- und Kernphysik, RheinischeFriedrich-Wilhelms-Universit¨at 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, Universit`a di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy (Dated: January 5, 2021)We determine the ∆(1232) resonance parameters using lattice QCD and the L¨uscher method.The resonance occurs in elastic pion-nucleon scattering with J P = 3 / + in the isospin I = 3 / P -wave channel. Our calculation is performed with N f = 2 + 1 flavors of clover fermions on a latticewith L ≈ . m π = 255 . .
6) MeV and m N = 1073(5)MeV, respectively, and the strong decay channel ∆ → πN is found to be above the threshold. Tothoroughly map out the energy-dependence of the nucleon-pion scattering amplitude, we computethe spectra in all relevant irreducible representations of the lattice symmetry groups for total mo-menta up to (cid:126)P = πL (1 , , S and P waves. We perform global fits ofthe amplitude parameters to up to 21 energy levels, using a Breit-Wigner model for the P -wavephase shift and the effective-range expansion for the S -wave phase shift. From the location of thepole in the P -wave scattering amplitude, we obtain the resonance mass m ∆ = 1378(7) MeV and thecoupling g ∆- πN = 23 . . I. INTRODUCTION
The ∆(1232) (in the following denoted as ∆) isthe lowest-lying baryon resonance, typically pro-duced when energetic photons, neutrinos, or pionshit a nucleon [1]. While these three processes differimmensely, they have the two-particle nucleon-pionscattering amplitude in common. The scatteringamplitude in which the ∆ appears as an enhance-ment is the P -wave with J P =
32 + and I = , oftenalso referred to as the P amplitude. For energiesnear the ∆ mass, this amplitude is nearly completelyelastic [2, 3].Modern determinations of the ∆ resonance pa-rameters are typically performed using data fromexperiments such as CLAS12 at JLab and MAMI- ∗ [email protected] A1 in Mainz. While the results for the pole locationdiffer slightly from the Breit-Wigner parameters [4],the ∆ is generally found to have a mass of approx-imately 1230 MeV and a decay width of approxi-mately 100 MeV [5, 6].Phenomenological studies of the ∆ have been per-formed using quark models, chiral perturbation the-ory and related effective field theories, and the S-matrix approach. From the quark-model point ofview, many baryons remain elusive, but the ∆ massis reproduced quite well [7–9]. Chiral perturbationtheory and related effective field theories have showngreat success in determining low-energy scatteringparameters and πN scattering amplitudes [10–13];an extensive review can be found in Ref. [14]. Anal-yses of the large experimental data sets using ampli-tude models based on S-matrix principles were per-formed in Refs. [15–17].First-principles computations of ∆ properties canbe done using lattice QCD. The ∆ mass, assuming a a r X i v : . [ h e p - l a t ] J a n stable ∆, was studied in Refs. [18–23]. However, forquark masses corresponding to pion masses below acertain value, the ∆ is an unstable hadron, and itsmass and decay width must be determined from theappropriate N π scattering amplitudes. While theuse of Euclidean time in lattice QCD prevents di-rect computations of infinite-volume scattering am-plitudes [24], L¨uscher showed how the finite-volumeenergy spectrum of a two-body system interactingthrough an elastic short-range interaction is relatedto the infinite-volume scattering amplitudes [25–27].The decades following L¨uscher’s seminal work wit-nessed further development of the theoretical frame-work to moving frames [28, 29], unequal masses [30–32], and arbitrary spin [33]. These methods havebeen applied to many systems in the meson sectorand are reviewed in Ref. [34]. For the nucleon-pionscattering only a handful of studies have been donein the
N π channel [35–42].In the following, we report a new lattice-QCDstudy of elastic
N π scattering in the ∆ resonancechannel using the L¨uscher method. Our calcula-tion is performed using N f = 2 + 1 flavors of cloverfermions at a pion mass of m π = 255 . .
6) MeV, ona lattice with L ≈ . N π chan-nel. Preliminary results were previously shown inRef. [43]. The computations presented here also con-stitute the first step toward a future calculation of N → N π electroweak transition matrix elements us-ing the formalism of Refs. [44, 45].The paper is organized as follows: in Sec. II thedetails of the lattice gauge-field ensemble are pre-sented. Section III describes the interpolating op-erators and the method used to project to defi-nite irreducible representations of the lattice sym-metry groups. The Wick contractions yielding thetwo-point correlation functions for the ∆ − N π sys-tem are discussed in Sec. IV. In Sec. V, the resultsof the spectra analysis are presented. The rele-vant finite-volume quantization conditions are dis-cussed in Sec. VI. The K -matrix parametrizationsemployed for the scattering amplitudes and our re-sults for the amplitude parameters are presented inSec. VII. We conclude in Sec. VIII. II. GAUGE ENSEMBLE
We use a lattice gauge-field ensemble generatedwith the setup of the Budapest-Marseille-Wuppertalcollaboration [46], with parameters given in Ta-ble I. The ensemble has been used previously in N s × N t × β . am u,d − . am s − . a [fm] 0 . L [fm] 2 . m π [MeV] 255 . . m π L . N config N meas Ref. [47]. The gluon action is the tree-level improvedSymanzik action [48], while the fermion action isa tree-level clover-improved Wilson action [49] withtwo levels of HEX smearing of the gauge links [46].We analyze 600 gauge configurations and computethe correlation functions for 16 source positions oneach configuration, resulting in a total of 9600 mea-surements.When considering the
N π system in the rest frameonly, the spatial lattice size of L ≈ . N π and
N ππ thresholds there are few energypoints available to constrain the phase shift we aimto determine. A straightforward way to gain ad-ditional points would be to add a spatially largerensemble, but this is computationally quite expen-sive. A more efficient approach employed here isusing also moving frames [28, 50, 51] on the same en-semble, where the Lorentz boost contracts the box,resulting in different effective values of the spatiallength along the boost direction [32].
III. INTERPOLATING OPERATORS
We use local single-hadron and nonlocal multi-hadron interpolating operators, both necessary fora complete determination of the resonance proper-ties [53]. For the single-hadron ∆ operators with I = 3 / , I = +3 / ++ ) weinclude two choices,∆ (1) αi ( (cid:126)p ) = (cid:88) (cid:126)x (cid:15) abc ( u a ( (cid:126)x )) α ( u Tb ( (cid:126)x ) Cγ i u c ( (cid:126)x )) e i(cid:126)p · (cid:126)x , ∆ (2) αi ( (cid:126)p ) = (cid:88) (cid:126)x (cid:15) abc ( u a ( (cid:126)x )) α ( u Tb ( (cid:126)x ) Cγ i γ u c ( (cid:126)x )) e i(cid:126)p · (cid:126)x . (1)The two-hadron interpolators with the same quan-tum numbers are obtained from products of the form N (1 , α ( (cid:126)p ) π ( (cid:126)p ) (2) L π (cid:126)P (0 , ,
0) (0 , ,
1) (0 , ,
1) (1 , , LG O ( D ) h C ( D )4 v C ( D )2 v C ( D )3 v Axis and planesof symmetry g LG
96 16 8 12Λ( J P ) : π ( 0 − ) A u ( 0 − , − , ... ) A ( 0 , , ... ) A ( 0 , , ... ) A ( 0 , , ... )Λ( J P ) : N (
12 + ) G g (
12 + ,
72 + , ... ) G ( , , ... ) G ( , , ... ) G ( , , ... )Λ( J P ) : ∆(
32 + ) H g (
32 + ,
52 + , ... ) G ( , , ... ) ⊕ G ( , , ... ) (2) G ( , , ... ) G ( , , ... ) ⊕ F ( , , ... ) ⊕ F ( , , ... )TABLE II. Choices of total momenta (cid:126)P , along with the Little Groups LG , irreducible representations Λ of relevanthadrons and their angular momentum content J P . The multi-hadron Nπ operators have the same irreps as thesingle-hadron ∆ operators. From left to right the subduction of irreps in moving frames. The label “(2)” for irrep G in group C D v indicates the double occurrence of the irrep from the subduction; to differentiate this irrep from thehomonymous of group C D v we keep the additional label “(2)” throughout the paper. Images credit [52]. as explained in more detail below. The pion inter-polator ( I = 1 , I = +1) is given by π + ( (cid:126)p ) = (cid:88) x ¯ d ( (cid:126)x ) γ u ( (cid:126)x ) e i(cid:126)p · (cid:126)x , (3)and for the nucleon ( I = 1 / , I = +1 /
2) we againinclude two choices, N (1) α ( (cid:126)p ) = (cid:88) (cid:126)x (cid:15) abc ( u a ( (cid:126)x )) α ( u Tb ( (cid:126)x ) Cγ d c ( (cid:126)x )) e i(cid:126)p · (cid:126)x ,N (2) α ( (cid:126)p ) = (cid:88) (cid:126)x (cid:15) abc ( u a ( (cid:126)x )) α ( u Tb ( (cid:126)x ) Cγ γ d c ( (cid:126)x )) e i(cid:126)p · (cid:126)x . (4)To correctly identify the angular momentum inthe reduced symmetry of the cubic box, we projectthe operators to the irreducible representations (ir-reps) that belong to the symmetry groups of the lat-tice. Instead of the infinitely many possible irre-ducible representation J P of the continuum, on thelattice, there are only a finite number of possible ir-reps Λ. Thus each lattice irrep in principle containsinfinitely many values of the continuum spin J . Eachirrep belongs to a Little Group LG ( (cid:126)P ) describing theunderlying symmetry of the lattice points contractedin the direction of the boost vector (cid:126)P , i.e., the totalmomentum of the N π system.In the moving frames considered here, the sym-metries are reduced to the groups C v , C v , C v (seeTable II). The degree of symmetry is mirrored by thegroup’s order g LG ( (cid:126)P ) , which corresponds to the num-ber of transformation elements (rotations and inver-sions) belonging to the group. In particular, half-integer spin is best described by the double-cover ofsymmetry groups (labeled D ), which introduce the2 π rotation as a new element of the group, effectively doubling the elements of the original group [54]. Ad-ditionally, a clear parity identification is lost in themoving frames, where the subduction mixes paritiesin the same irrep [55]. The list of chosen total mo-menta, symmetry groups, and irreps for the hadronsused in this work can be found in Table II.To project the single-hadron operators to a defi-nite irrep Λ and row r , we make use of the formula[56–60]: O Λ ,r,i ( (cid:126)P ) = d Λ g LG ( (cid:126)P ) (cid:88) R ∈ LG ( (cid:126)P ) Γ Λ r,r ( R ) W ( R ) − O ( (cid:126)P ) , (5)where d Λ is the dimension of the irrep Λ and Γ Λ arethe representation matrices belonging to the irrepΛ. The matrices W ( R ) − correspond to the matricesappearing in the right-hand sides of Eqs. (A1), (A2),or (A4). Here we denote the elements of the littlegroup generically as R , even though in the rest framethey include the inversion in addition to the latticerotations. The index i labels the embedding intothe irrep and replaces any free Dirac/Lorentz indicesappearing on the right-hand side of Eq. (5).The analogous projection formula for the two-hadron operators is O Λ ,r,iNπ ( (cid:126)P ) = d Λ g LG ( (cid:126)P ) (cid:88) R ∈ LG ( (cid:126)P ) (cid:88) (cid:126)p Γ Λ r,r ( R ) × W − N ( R ) N ( R(cid:126)p ) W π ( R ) − π ( (cid:126)P − R(cid:126)p ) . (6)Representation matrices for irreps in the rest frameare found in [60, 61] and for the moving frames areprovided in [59]. In Eq. (6), given a total momentum L π (cid:126)P ref [ N dir ] Group LG Irrep Λ Rows Ang. mom. content Operator structure Number of operators(0,0,0) O Dh G u J = 1 / , / , ... N π with | (cid:126)p | = | (cid:126)p | = 0 1[1] N π with | (cid:126)p | = | (cid:126)p | = πL H g J = 3 / , / , ... ∆ (1 , ( (cid:126)P ) 2 N π with | (cid:126)p | = | (cid:126)p | = πL C D v G J = 1 / , / , ... ∆ (1 , ( (cid:126)P ) 8[3] N π with | (cid:126)p | = 0 and | (cid:126)p | = πL N π with | (cid:126)p | = πL and | (cid:126)p | = 0 2 N π with | (cid:126)p | = πL and | (cid:126)p | = √ πL N π with | (cid:126)p | = √ πL and | (cid:126)p | = πL G J = 3 / , / , ... ∆ (1 , ( (cid:126)P ) 4 N π with | (cid:126)p | = √ πL and | (cid:126)p | = πL N π with | (cid:126)p | = πL and | (cid:126)p | = √ πL C D v (2) G J = 1 / , / , ... ∆ (1 , ( (cid:126)P ) 12[6] N π with | (cid:126)p | = 0 and | (cid:126)p | = √ πL N π with | (cid:126)p | = √ πL and | (cid:126)p | = 0 2 N π with | (cid:126)p | = | (cid:126)p | = πL C D v G J = 1 / , / , ... ∆ (1 , ( (cid:126)P ) 8[4] N π with | (cid:126)p | = 0 and | (cid:126)p | = √ πL N π with | (cid:126)p | = √ πL and | (cid:126)p | = 0 2 N π with | (cid:126)p | = πL and | (cid:126)p | = √ πL N π with | (cid:126)p | = √ πL and | (cid:126)p | = πL F J = 3 / , / .... ∆ (1 , ( (cid:126)P ) 4 N π with | (cid:126)p | = πL and | (cid:126)p | = √ πL N π with | (cid:126)p | = √ πL and | (cid:126)p | = πL F J = 3 / , / , ... ∆ (1 , ( (cid:126)P ) 4 N π with | (cid:126)p | = πL and | (cid:126)p | = √ πL N π with | (cid:126)p | = √ πL and | (cid:126)p | = πL N π ) operators for all irreps. In the constructionof 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). (cid:126)P , the sum over internal momenta is constrained bythe magnitudes | (cid:126)p | = | R(cid:126)p | = | (cid:126)p | and | (cid:126)p | = | (cid:126)P − R(cid:126)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 eachrow r of irrep Λ multiple operator embeddings (iden-tified by the label i ) that are not guaranteed to be in-dependent. We therefore perform the following threesteps 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 thisway to linearly independent sets.(iii) Construct the other rows r for these linearlyindependent sets of operators.The operators obtained in step (i) have the genericform O Λ , ,i ( (cid:126)P ) = (cid:88) j c Λ , ij O j ( (cid:126)P ) . (7) Using Gaussian elimination we obtain a smaller ma-trix c Λ , nj such that the linearly independent opera-tors constructed in step (ii) have the form O Λ , ,n ( (cid:126)P ) = (cid:88) j c Λ , nj O j ( (cid:126)P ) . (8)The number of independent operators (correspond-ing to the range of the index n ) is equal to [61, 62]1 g LG ( (cid:126)P ) (cid:88) R ∈ LG ( (cid:126)P ) χ Γ Λ ( R ) χ W ( R ) , (9)where the characters χ Γ Λ ( R ) and χ R are equal tothe traces of the representation matrices Γ Λ and thetransformation matrices W ( R ).In step (iii), to construct the other rows r > O Λ ,r,n ( (cid:126)P ) = (cid:88) j c Λ , nj d Λ g LG ( (cid:126)P ) × (cid:88) R ∈ LG ( (cid:126)P ) Γ Λ r, ( R ) R O j ( (cid:126)P ) R − , (10)where the rotations/inversions R O j ( (cid:126)P ) R − are per-formed as in Eqs. (5) and (6), depending on thestructure of O j ( (cid:126)P ).Also, to increase statistics, multiple directions of (cid:126)P at fixed | (cid:126)P | are used (see Table III). For everymoving frame, we first perform the irrep projectionsfor a reference momentum (cid:126)P ref and then rotate theprojected operators to the new momentum direction.Generating operators initially from a reference mo-mentum and r = 1 only facilitates the identificationof equivalent operators embeddings that can laterbe averaged over different rows of the same irrep Λ(which is possible due to the great orthogonality the-orem [62]) and momentum direction of equal | (cid:126)P | . Inthe following, the label r for the row will be dropped. IV. WICK CONTRACTIONS
From the ∆ /N π interpolators discussed above, webuild two-point correlation matrices for each totalmomentum (cid:126)P and irrep Λ, C Λ , (cid:126)Pij = (cid:104) O Λ , (cid:126)Pi ( t snk ) ¯ O Λ , (cid:126)Pj ( t src ) (cid:105) , (11)where the indices i, j now label all the different oper-ators in the same irrep that can vary in internal mo-mentum content, embedding from the multiplicity,or gamma matrices used in the diquarks of Eqs. (4)or (1). The Wick contractions are computed follow-ing the scheme outlined in Refs. [63, 64]. The corre-lators with single-hadron interpolators at source andsink are constructed from point-to-all propagators,while the correlators with a single-hadron interpo-lator at the sink and a two-hadron N π interpolatorat the source use in addition a sequential propaga-tor, with sequential inversion through the pion ver-tex at source time. The topologies of these diagramsare shown in the top panel of Fig. 1. The bottompanel of Fig. 1 shows the topologies for the corre-lators with
N π operators at both source and sink.The diagrams are split into two factors, separatedat the source point and by using a stochastic source- propagator pair. For the latter we use stochas-tic timeslice sources in the upper two diagrams. Inthe lower diagrams we employ spin-dilution and theone-end-trick in addition to time dilution.
FIG. 1. Upper panel: Two-point function contractionsinvolving the ∆ interpolator. A gray filling of a circlerepresents the ∆ interpolator, a green filling representsthe π interpolator, and a blue filling represents the N in-terpolator. A solid black outline indicates a point source,while a dotted outline represents a sequential source.The black arrow lines represent point-to-all propagators,and the red arrow lines represent sequential propagators.The contractions with the πN operator at the sink andthe ∆ operator at the source are not computed directlybut are obtained from the contraction with the ∆ op-erator at the sink and the πN operator at the sourcethrough conjugation. Lower panel: Two-point functioncontractions for πN − πN . The blue arrow lines repre-sent stochastic propagators, while the other elements areanalogous to the upper panel. The quark propagators of all types are Wuppertal-smeared [65] at source and sink with smearing pa-rameters α Wup = 3 . N Wup = 45; these param-eters were originally optimized for the nucleon two-point functions in Ref. [47]. The gauge field deployedin the smearing kernel is again 2-level HEX-smeared[66, 67]. .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . ap ) . . . . . ( a E ) π dispersion relation ( aE ) = ( am π ) + c ( ap ) am π = 0 . ± . c = 1 . ± . χ /ndf = 0 . FIG. 2. Pion dispersion relation. .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . ap ) . . . . . . ( a E ) Nucleon dispersion relation (GEVP) ( aE ) = ( am N ) + c ( ap ) am N = 0 . ± . c = 0 . ± . χ /ndf = 0 . FIG. 3. Nucleon dispersion relation from the GEVPanalysis.
V. SPECTRA RESULTS
The masses of the pion and nucleon are used asinput parameters in the L¨uscher method. We extractthem from fits of their dispersion relations, shown inFigs. 2 and 3, giving am π = 0 . , (12) am N = 0 . . (13)The energies are obtained from single-state fits of thetwo-point functions projected to different momenta(using a cosh for the pion and a single exponentialfor the nucleon).For the ∆- N π system, to extract the energy levels E Λ , (cid:126)Pn (where n now counts the finite-volume energylevels for a given Λ , (cid:126)P ) from the correlation matri-ces C Λ , (cid:126)Pij we use the generalized eigenvalue problem(GEVP) [26, 68–70] C Λ , (cid:126)Pij ( t ) u nj ( t ) = λ n ( t, t ) C Λ , (cid:126)Pij u nj ( t ) , (14) ( L π ) | (cid:126)P | Λ n Fit Range χ dof a (cid:113) s Λ , (cid:126)Pn G u −
15 1 .
90 0 . G u −
15 0 .
80 0 . H g −
15 1 .
79 0 . H g −
15 0 .
42 1 . G −
15 1 .
97 0 . G −
15 1 .
13 0 . G −
15 0 .
72 0 . G −
15 0 .
48 0 . G −
15 0 .
89 1 . G −
15 1 .
72 0 . G −
15 1 .
72 0 . G −
15 1 .
59 0 . G −
15 1 .
87 0 . G −
15 0 .
71 0 . G −
15 1 .
32 0 . G −
15 0 .
67 0 . G −
15 2 .
01 0 . F −
15 1 .
46 0 . F −
15 0 .
26 0 . F −
15 0 .
45 0 . F −
15 0 .
55 0 . Nπ sectorfrom single-exponential fits to the principal correlators,for the different total momenta (cid:126)P and irreps Λ. where u nj are the right generalized eigenvectors. Inthe plateau regions the energies are obtained fromfits to the principal correlators λ n ( t, t ) with singleexponentials as λ n ( t, t ) ∼ e − E Λ ,(cid:126)Pn ( t − t ) (15)or using two exponentials with a small contributionfrom higher excited states with energy E (cid:48) Λ , (cid:126)Pn as λ n ( t, t ) ∼ (1 − B ) e − E Λ ,(cid:126)Pn ( t − t ) + Be − E (cid:48) Λ ,(cid:126)Pn ( t − t ) . (16)Here, t is a reference timeslice that does notstrongly affect the large- t behavior; we set t /a = 2.Additionally, for the projected multihadron oper-ators ( N π ) Λ ,r ( (cid:126)P ) we implement an optimized inter-polator of the nucleon [71] N ( (cid:126)p ) = (cid:88) i u N ) i ( t ) N i ( (cid:126)p ) , (17)where i labels the two types of nucleon operatorsin Eq. (4) and u ( N ) i are the generalized eigenvec-tors (for t/a = 4) from a single-nucleon GEVP anal-ysis. The optimized nucleon interpolator has im-proved overlap with the single-nucleon ground statewith momentum (cid:126)p [72].For the coupled ∆- N π system we build for each ir-rep Λ a correlation matrix C Λ , (cid:126)Pij from the projected∆ and optimized N π operators in Table III. The t/a . . . . . . . a √ s n e ff t min /a t min /a . . . . . √ s n e ff [ G e V ] | ~P | = 0 , G u t/a . . . . . . . a √ s n e ff t min /a t min /a . . . . . √ s n e ff [ G e V ] | ~P | = 0 , H g t/a . . . . . . . a √ s n e ff t min /a t min /a . . . . . √ s n e ff [ G e V ] | ~P | = ( πL ) , G t/a . . . . . . . a √ s n e ff t min /a t min /a . . . . . √ s n e ff [ G e V ] | ~P | = ( πL ) , G FIG. 4. For each irrep, we show the effective energies of the principal correlators as a function of t/a (left), theenergies obtained from single-exponential fits to these correlators as a function of t min /a (center), and the energiesobtained from double-exponential fits to these correlators as a function of t min /a (right). All energies shown here areconverted to the center-of-mass frame. Black dashed lines represent the Nπ and Nππ thresholds. Non-interacting Nπ energy levels are shown as green lines. multiplicities of operators give rise to a fairly largebasis for each correlation matrix (the dimensions forthe full bases correspond to the sums of numbers ofoperators for each irrep listed in Table III). Throughsingular value decomposition of CC † or C † C we caninfer which operators contribute to the largest singu-lar values, allowing us to explore sub-bases of the fulllist of operators that can lead to reduced noise of theprincipal correlators while maintaining the completespectra.Baryons are known to have a narrow plateau re-gion (the ”golden window” [73]) where the higherstates contribution get small enough to enable a sin- gle exponential fit to describe maximally a singlelevel before the rapid decay of signal-to-noise ratioat larger t [74, 75]. In the left subplot for each irrepin Figs. 4 and 5, we show the the effective masses ofthe principal correlators, aE neff ( t ) = ln λ n ( t, t ) λ n ( t + a, t ) , (18)converted to the center-of-mass frame using √ s n Λ , (cid:126)P = (cid:113) ( E Λ , (cid:126)Pn ) − ( (cid:126)P ) . (19)The center-of-mass energies are also related to the t/a . . . . . . . a √ s n e ff t min /a t min /a . . . . . √ s n e ff [ G e V ] | ~P | = ( πL ) , (2) G t/a . . . . . . . a √ s n e ff t min /a t min /a . . . . . √ s n e ff [ G e V ] | ~P | = ( πL ) , G t/a . . . . . . . a √ s n e ff t min /a t min /a . . . . . √ s n e ff [ G e V ] | ~P | = ( πL ) , F t/a . . . . . . . a √ s n e ff t min /a t min /a . . . . . √ s n e ff [ G e V ] | ~P | = ( πL ) , F FIG. 5. Like Fig. 4, but with irreps (2)
G, G, F , F . scattering momenta through √ s n Λ , (cid:126)P = (cid:113) ( k Λ , (cid:126)Pn ) + m π + (cid:113) ( k Λ , (cid:126)Pn ) + m N . (20)Our main results are obtained from single-exponential fits to the principal correlators and arelisted in Table IV. The fit ranges are chosen after astability analysis. The upper limit of the fit range,once chosen large enough, is found to have a smallimpact on the fit itself; thus, we fix it to t max /a = 15for all levels. On the other hand, the lower limit isvaried within a reasonable range until a plateau re-gion is identified. This is illustrated in the centersubplot for each irrep in Figs. 4 and 5. To furthertest the stability, we also performed two-exponentialfits using Eq. (16), which give the results shown in the right subplots of the figures and are found to bein reasonable agreement.It can be seen in the plots that energy levels thatoverlap strongly with the N π states shift away fromthe resonance region, as expected. For the irrep (2) G in | (cid:126)P | = (2 π/L )
2, the situation is more compli-cated and a higher number of energy states appearin the region of interest. This situation originatesfrom having only a single irrep for the Little Group C D v , resulting in a maximal mixing of angular mo-menta.A summary of all extracted energy levels is shownin Fig. 6. G u H g G G (2) G G F F Irrep . . . . . . a √ s Nπ threshold Nππ thresholdNon-Interacting EnergiesMeasured Energies / / / / / / / / / / / J content . . . . √ s [ G e V ] FIG. 6. Energy levels extracted in each irrep, with J ≤ / VI. L ¨USCHER QUANTIZATIONCONDITIONS
The L¨uscher quantization condition connects thefinite-volume energy spectra affected by the interac-tions and the infinite-volume scattering amplitudes;resonances correspond to poles in the infinite-volumescattering amplitudes at complex √ s and in prin-ciple affect the entire spectrum. For elastic 2-bodyscattering of nonzero-spin particles, the quantizationcondition can be written as [50]det( M (cid:126)PJlµ,J (cid:48) l (cid:48) µ (cid:48) − δ JJ (cid:48) δ ll (cid:48) δ µµ (cid:48) cot δ Jl ) = 0 , (21)where δ Jl is the infinite-volume scattering phaseshift for total angular momentum J and orbital an-gular momentum l , and µ, µ (cid:48) = − J, ..., J . Both thescattering phase shift and the matrix M (cid:126)PJlµ,J (cid:48) l (cid:48) µ (cid:48) arefunctions of the scattering momentum, and the so-lutions of the quantization condition for the scatter-ing momentum give the finite-volume energy levelsthrough Eq. (20). The matrix M (cid:126)PJlµ,J (cid:48) l (cid:48) µ (cid:48) encodesthe geometry of the finite box and is a generalizationfor particles with spins σ, σ (cid:48) of the spinless counter-part via M (cid:126)PJlµ,J (cid:48) l (cid:48) µ (cid:48) = (cid:88) m,σ,m (cid:48) ,σ (cid:48) (cid:104) lm, σ | Jµ (cid:105) (cid:104) l (cid:48) m (cid:48) , σ (cid:48) | J (cid:48) µ (cid:48) (cid:105) M (cid:126)Plm,l (cid:48) m (cid:48) , (22) where M (cid:126)Plm,l (cid:48) m (cid:48) (for a cubix box with periodicboundary conditions) is given by [50] M (cid:126)Plm,l (cid:48) m (cid:48) ( q ) = ( − l γ − π / l + l (cid:48) (cid:88) j = | l − l (cid:48) | j (cid:88) s = − j i j q j +1 × Z (cid:126)Pjs (1 , q ) ∗ C lm,js,l (cid:48) m (cid:48) , (23)where q = kL π with k the scattering momentum and L the side length of the box. Here Z (cid:126)Pjs (1 , q ) is thegeneralized zeta function, γ = E (cid:126)P / √ s is the Lorentzboost factor and the coefficient C lm,js,l (cid:48) m (cid:48) expressedin terms of Wigner 3 j -symbols read C lm,js,l (cid:48) m (cid:48) = ( − m (cid:48) i l − j − l (cid:48) (cid:112) (2 l + 1)(2 j + 1)(2 l (cid:48) + 1) × (cid:18) l j l (cid:48) m s − m (cid:48) (cid:19) (cid:18) l j l (cid:48) (cid:19) . (24)To simplify notation it is common practice to definethe functions w lm = w (cid:126)Plm ( q, L ) ≡ Z (cid:126)Plm (1; q ) γπ / √ l + 1 q l +1 . (25)The elements of the matrices M (cid:126)PJlµ,J (cid:48) l (cid:48) µ (cid:48) for allchoices of (cid:126)P considered in this work are listed inAppendix B.Furthermore, it is possible to extract quantizationconditions for each irrep Λ via a change of basis ofEq. (21). The basis vector of the irrep Λ can bewritten as [50, 76] | Λ rJln (cid:105) = (cid:88) µ c Λ rnJlµ | Jlµ (cid:105) , (26)0 L π (cid:126)P Group LG Irrep Λ Quantization condition(0 , , O Dh G u − w + cot δ , = 0 H g − w + cot δ , = 0(0 , , C D v G − w + ( w − cot δ , )( w + w − cot δ , ) = 0 G − w + w + cot δ , = 0(1 , , C D v (2) G − ( w − cot δ , )( − w + 2 w + ( w − cot δ , ) ) − w ) (2 w + w − i √ w − δ , ) = 0(1 , , C D v G − w + ( w − cot δ , )( w − i √ w − cot δ , ) = 0 F , F − w − i √ δ , = 0TABLE V. Finite-volume quantization conditions for all irreps in terms of phase shifts δ J,l and functions w lm . where the coefficients c Λ rnJlµ for l ≤ | Jlµ (cid:105) are given by | Jlµ (cid:105) = (cid:88) m,σ | lm, σ (cid:105) (cid:104) lm, σ | Jµ (cid:105) . (27)One can then make a change of basis for which thematrix elements of M are given by (cid:104) Λ rJln | M | Λ (cid:48) r (cid:48) J (cid:48) l (cid:48) n (cid:48) (cid:105) = (cid:88) µµ (cid:48) c Λ rnJlµ c Λ (cid:48) r (cid:48) n (cid:48) J (cid:48) l (cid:48) µ (cid:48) M Jln,J (cid:48) l (cid:48) n (cid:48) = δ ΛΛ (cid:48) δ rr (cid:48) M Jln,J (cid:48) l (cid:48) n (cid:48) , (28)where it is found, from Schur’s lemma, that the ma-trix M is partially diagonalized in irrep Λ and row r . However, the matrix is not diagonal in n , whichlabels the multiple embeddings of the irreps. In ourcase only the irrep (2) G of the group C D v has multi-ple embeddings with multiplicity m G = 2 (See TableII).In principle, there are infinitely many values of to-tal angular momentum J and therefore also infinitelymany partial waves l in each irrep, but, as the higherwaves have an increasingly smaller contribution, weconsider only the dominant partial waves. In partic-ular, we assume the contributions from partial wavesin J > / N - π system, J = 3 / P -wave ( l = 1) and the D -wave ( l = 2).Several irreps mix J = 3 / J = 1 /
2, and thelatter includes l = 0 ,
1. Nevertheless, at our levelof precision, in addition to the resonant phase shift P ( J = 3 / , l = 1) for isospin I = 3 / S ( J = 1 / , l = 0) [2], for which theclosest resonance would be the distant ∆(1620). Inorder to better constrain the S ( J = 1 / , l = 0)contribution, we also include the irrep G u , which isthe only irrep we can access that contains only spin J = 1 / l = 0 (up to contributions from l > ungerade ). As canbe seen in Table III, the interpolating operators in the G u irrep are exclusively N - π two-hadron oper-ators, consistent with the expectation that the S phase shift is nonresonant at low energy. The quan-tization conditions for all irreps, expressed in termsof the two phase shifts δ / , , δ / , and the functions w lm , are listed in Table V. VII. RESULTS FOR THE SCATTERINGAMPLITUDESA. Parametrizations used
We use the K -matrix parametrization rescaledwith the two-body phase space ρ as K = ρ / ˆ Kρ / , (29)where ρ = (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) − (cid:18) m π + m N √ s (cid:19) (cid:33) (cid:32) − (cid:18) m π − m N √ s (cid:19) (cid:33) . (30)The K -matrix relates to the phase shifts as K ( Jl ) = tan( δ Jl ) . (31)As discussed in Sec. VI, our analysis includes thephase shift δ / , , where we expect the ∆ resonancethat will be quite narrow for our quark masses, andthe phase shift δ / , , which is expected to be nonres-onant in the energy region considered. We thereforeuse a Breit-Wigner parametrization for the former,ˆ K (3 / , = √ s Γ( s )( m − s ) ρ , (32)where m BW denotes the resonance mass and the de-cay width Γ( s ) is given byΓ( s ) = g π k s (33)1 Label Fit to (
J, l ) Irreps Λ √ s points Breit-Wigner parameters ERE parameters χ / dof S (1 / , G u a /a = 0 . ± .
93 0 . P (3 / , H g , G , F , F g BW = 13 . ± . am BW = 0 . ± . . G(a) (1 / , , (3 / , G u , H g , G , G , (2) G, G, F , F g BW = 13 . ± . am BW = 0 . ± . a /a = 0 . ± .
44 0 . G(a+1) (1 / , , (3 / , G u , H g , G , G , (2) G, G, F , F g BW = 13 . ± . am BW = 0 . ± . a /a = 0 . ± .
78 0 . G(b) (1 / , , (3 / , G u , H g , G , G , (2) G, G, F , F g BW = 13 . ± . am BW = 0 . ± . a /a = 0 . ± .
57 1 . G(c) (1 / , , (3 / , G u , H g , G , G ,G, F , F g BW = 13 . ± . am BW = 0 . ± . a /a = 0 . ± .
48 0 . G(d) (1 / , , (3 / , H g , G , G , (2) G, G, F , F g BW = 13 . ± . am BW = 0 . ± . a /a = − . ± .
75 0 . with the coupling g BW , scattering momentum k , andcenter-of-mass energy squared s . For the nonreso-nant ˆ K (1 / , we use the effective-range expansion[77]. We find that working to 0th-order is sufficientat the level of precision we have, such thatˆ K (1 / , = kρ a (34)with the S -wave scattering length a . 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 to all energy levels in all irrepsby minimizing the χ function χ = (cid:88) (cid:126)P , Λ ,n (cid:88) (cid:126)P (cid:48) , Λ (cid:48) ,n (cid:48) [ C − ] (cid:126)P , Λ ,n ; (cid:126)P (cid:48) , Λ (cid:48) ,n (cid:48) × (cid:32)(cid:113) s Λ , (cid:126)Pn [data] − (cid:113) s Λ , (cid:126)Pn [model] (cid:33) × (cid:32)(cid:113) s Λ (cid:48) , (cid:126)P (cid:48) n (cid:48) [data] − (cid:113) s Λ (cid:48) , (cid:126)P (cid:48) n (cid:48) [model] (cid:33) . (35)Here, C is the covariance matrix of the energy levels (cid:113) s Λ , (cid:126)Pn [data] measured on the lattice. The model en-ergies (cid:113) s Λ , (cid:126)Pn [model] are obtained for each parameterguess by finding the roots of the L¨uscher quantiza-tion conditions (see Table V). There are 21 energylevels from 8 irreps available for the global fit, asshown in Fig. 6.The results for both the global fits and for fits tosubsets of energy levels are listed in Table VI. Be-fore performing the global fit to all energy levels, we separately considered the irreps that include eitheronly J = 1 / J = 3 / J > / G u is the only one that contains exclu-sively J = 1 /
2, while there are multiple irreduciblerepresentations with exclusively J = 3 / H g , G , F , and F . These initial two fits enable us to obtaina good initial guess for the parameters of the finalglobal fits and assess the stability of the fit over thechoice of irreps included. The fit for the S-wave (la-beled S ) via irrep G u is done to only 2 energy levels,resulting in a low χ / dof. The other partial fit overirreps containing P-wave only ( P ) includes 8 energylevels and gives a higher χ / dof.For the global fit ( G ), we implement five differentcombinations of levels included and choices of t min /a to test the stability of the results. The fit to thenominal results for the energy levels from Table IVis labeled as G(a) , while the fit labeled
G(a+1) wasdone to the energy levels with t min /a increased byone unit throughout. More focused choices amongthe noisiest levels are made in the fit G(b) , wherewe vary t min /a in selected levels based on the re-sults of the stability analysis shown in Figs. 4 and 5.Specifically, this case uses a +1 shift on t min /a onall levels of irreps G , (2) G, F , F , the ground stateof G , the first excited of G , and +2 on the first ex-cited of G . Additionally, we perform the global fit G(c) removing potentially problematic levels fromthe list in Table IV: the highest level of irrep G andall levels in irrep (2) G . Furthermore, the global fitlabeled G(d) differs from
G(a) only by excludingirrep G u . Overall, the results indicate that the fitsprovide compatible results and are very stable acrossseveral choices.The phase shifts δ / , ( P ) and δ / , ( S ) fromthe results of the fit G(a) are plotted in Fig. 7. Ad-ditionally, we determine the position of the closest T -matrix pole in the complex √ s plane, associatedwith the ∆ resonance. Expressing the pole location2 √ s [MeV]020406080100120140160 δ / , [ ◦ ] am BW = 0 . ± . ( m BW = 1379 . ± .
04 MeV) g BW = 13 . ± . √ s [MeV] − . − . . . . . . . . δ / , [ ◦ ] a /a = 0 . ± . FIG. 7. Energy-dependence of the P (left) and S (right) phase shifts from the global fit G(a) in the elasticregion.Collaboration m π [MeV] Methodology m ∆ [MeV] g ∆- πN Verduci 2014 [38] 266(3) Distillation, L¨uscher 1396(19) 19.90(83)Alexandrou et al. 2013 [37] 360 Michael, McNeile 1535(25) 26.7(0.6)(1.4)Alexandrou et al. 2016 [39] 180 Michael, McNeile 1350(50) 23.7(0.7)(1.1)Andersen et al. 2018 [41] 280 Stoch. distillation, L¨uscher 1344(20) 37.1(9.2)Our result 255.4(1.6) Smeared sources, L¨uscher 1380(7) BW , 1378(7) pole BW , 1210(1) pole m ∆ and g ∆- πN . as m ∆ − i Γ /
2, we obtain am ∆ = 0 . ± . ,a Γ / . ± . ,m ∆ = (1377 . ± .
6) MeV , Γ / . ± .
1) MeV . (36)Using this result for Γ, we also determine the cou-pling g ∆- πN from the equation for the decay widthin leading-order chiral effective theory [78],Γ LOEFT = g πN π E N + m N E N + E π k m N , (37)which gives g ∆- πN = 23 . ± . . (38)The extracted values for the resonance mass m ∆ and coupling g ∆- πN are compared to recent resultsfrom the literature in Table VII.Our results for the scattering length a are gen-erally consistent with zero within the uncertainities. For the comparison with the literature, we considerthe combination a m + π . Our result from global fit G(a) is a m π = 0 . ± . , (39)while the values extracted from experimental dataare − . ± . − . ± . − . ± .
004 from Ref. [84].
VIII. CONCLUSIONS
We have presented a determination of elasticnucleon-pion scattering amplitudes for isospin I =3 / m π ≈
255 MeV.The baryon ∆(1232) emerges as the dominant res-onance in the P -wave with J P = 3 / + and is thefocus of this work. The infinite-volume scatteringamplitudes are obtained using the L¨uscher methodfrom the finite-volume energy spectra extracted fromcorrelation matrices built of ∆ and N π operators,3projected to definite irreducible representations ofthe lattice symmetry groups. In order to thoroughlymap out the energy dependence using just a singlevolume, it is essential to consider moving frames,where the symmetries are reduced. Many irreps in-cluded mix J = 3 / J = 1 /
2, and we thereforealso extracted the scattering phase shift for the lat-ter. Each J receives contributions from two valuesof orbital angular momentum l , but at the presentlevel of precision, we can access only a single domi-nant value of l for each: l = 1 for J = and l = 0 for J = . In addition, we neglect mixing with J > / P phase shiftat energies below the inelastic threshold N ππ , andusing the leading-order effective-range expansion forthe S phase shift. We also extracted the pole po-sition m ∆ − i Γ / g ∆- πN that determines the decaywidth Γ at leading order in chiral effective theory.These parameters are compared with other deter-minations in Table VII. For our pion mass (and atnonzero lattice spacing), m ∆ is found to be approx-imately 170 MeV higher than in nature, while thecoupling g ∆- πN agrees with extractions from exper-iment at the 2 σ level, given our uncertainties. Ourresult for the coupling also agrees with previous lat-tice determinations within the uncertainities. In the S wave, our result for the scattering length is con-sistent with zero and is approximately 2 σ away fromphenomenological determinations.Future work will include computations on addi-tional lattice gauge-field ensembles with differentspatial volume, which will provide more data pointsto better constrain the phase shifts extracted and, atthe same time, expand on the partial-wave contribu-tions included in the analysis and provide informa-tion on remaining finite-volume systematic errors.Using additional ensembles will also enable us to in-vestigate the dependence on the pion mass and onthe lattice spacing. Furthermore, we plan to use theresults for the energy levels and scattering ampli-tudes as inputs to a computation of N → N π elec-troweak transition matrix elements using formalismof Refs. [44, 45], similarly to what has been done for πγ ∗ → ππ [72, 85]. IX. ACKNOWLEDGMENTS
This research used resources of the NationalEnergy Research Scientific Computing Center(NERSC), a U.S. Department of Energy Officeof Science User Facility operated under ContractNo. DE-AC02-05CH11231. SK is supported by theDeutsche Forschungsgemeinschaft grant SFB-TRR 55. SK and GS were partially funded by the IVF ofthe HGF. LL acknowledges support from the U.S.Department of Energy, Office of Science, throughcontracts DE-SC0019229 and DE-AC05-06OR23177(JLAB). SM is supported by the U.S. Departmentof Energy, Office of Science, Office of High EnergyPhysics under Award Number DE-SC0009913. JNand AP acknowledge support by the U.S. Depart-ment of Energy, Office of Science, Office of Nu-clear Physics under grants DE-SC-0011090 and DE-SC0018121 respectively. MP gratefully acknowl-edges support by the Sino-German collaborative re-search center CRC-110. SP is supported by the Hori-zon 2020 of the European Commission research andinnovation program under the Marie Sklodowska-Curie grant agreement No. 642069. GR is sup-ported by the U.S. Department of Energy, Office ofScience, Office of Nuclear Physics, under ContractNo. DE-SC0012704 (BNL). SS thanks the RIKENBNL Research Center for support. AT is sup-ported by the the European Union’s Horizon 2020research and innovation programme under the MarieSk(cid:32)lodowska Curie European Joint Doctorate STIM-ULATE, grant No. 765048. We acknowledge the useof the USQCD software QLUA for the calculation ofthe correlators.
Appendix A: Transformation properties ofoperators
In this appendix we list the transformation prop-erties of the momentum-projected field operators un-der inversions I and spatial rotations R . The pseu-doscalar pion transforms as R π ( (cid:126)p ) R − = π ( R(cid:126)p ) I π ( (cid:126)p ) I − = − π ( − (cid:126)p ) , (A1)while the nucleon transforms as R N α ( (cid:126)p ) R − = S ( R ) − αβ N β ( R(cid:126)p ) I N α ( (cid:126)p ) I − = ( γ t ) αβ N β ( − (cid:126)p ) , (A2)where S ( R ) is the bi-spinor representation of SU (2).For a rotation of angle 2 π/n around the axis j , thisis given by S ( R ) αβ = exp (cid:18) ω µν [ γ µ , γ ν ] (cid:19) αβ (A3)with the antisymmetric tensor ω kl = − π(cid:15) jkl /n and ω k = ω k = 0 [59].The vector-spinor Delta operator transforms as R ∆ αk ( (cid:126)p ) R − = A ( R ) − kk (cid:48) S ( R ) − αβ ∆ βk (cid:48) ( R(cid:126)p ) I ∆ αk ( (cid:126)p ) I − = ( γ t ) αβ ∆ βk ( (cid:126)p ) (A4)4where A ( R ) denotes the 3-dimensional J = 1 irrep of SU (2), and S ( R ) is given in Eq. (A3). Appendix B: Matrices M (cid:126)PJlµ,J (cid:48) l (cid:48) µ (cid:48) Below we provide the matrices M (cid:126)PJlµ,J (cid:48) l (cid:48) µ (cid:48) introduced in Eq. (22), computed for each total momentum (cid:126)P including partial wave contributions in ( J = 3 / , l = 1) and ( J = 1 / , l = 0). The momentum labels aregiven in units of 2 π/L . M (0 , , Jlµ,J (cid:48) l (cid:48) µ (cid:48) =
12 12
12 32
32 32
12 32
12 32 w w w w w w (B1) M (0 , , Jlµ,J (cid:48) l (cid:48) µ (cid:48) =
12 12
12 32
32 32
12 32
12 32 w − i √ w w − i √ w w − w i √ w w + w i √ w w + w w − w (B2) M (1 , , Jlµ,J (cid:48) l (cid:48) µ (cid:48) =
12 12
12 32
32 32
12 32
12 32 w i − √ w ) 0 (1 + i )Re( w ) 0 w i − w ) 0 (1 + i ) √ w ) (1 + i ) √ w ) 0 w − w √ w i )Re( w ) 0 w + w √ w ( i − w ) 0 −√ w w + w i − √ w ) 0 −√ w w − w (B3) M (1 , , Jlµ,J (cid:48) l (cid:48) µ (cid:48) =
12 12
12 32
32 32
12 32
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