A meson-exchange model for π N scattering up to energies s. Shin Nan Yang National Taiwan University

Transcription

1 A meson-exchange model for π N scattering up to energies s 2 GeV Shin Nan Yang National Taiwan University Collaborators: S. S. Kamalov (Dubna) Guan Yeu Chen (Taipei) 18th International Conference on Few-body Problems in Physics August 20 26, 2006, Santos, Brazil 1

2 Outline Motivation Previous works Meson-exchange πn model below 400 MeV Dubna-Mainz-Taipei (DMT) dynamical model for pion e.m. production Extension to higher energies Conclusion 2

3 Motivation QCD Hadronic phenomena low energies ChPT high energies, high momentum transfer pqcd medium energies LQCD Phenomenology : hadron models, reaction theory 3

4 Aim: To construct a coupled-channel dynamical model to study πn scattering and pion electromagnetic production low energies: e.m. threshold pion production, low energy theorems and compare with ChPT medium energies: N-Δ transition form factors, resonance parameters high energy and high momentum transfer: transition to pqcd region? 4

5 Meson-exchange π N model below 400 MeV Bethe-Salpeter equation where B G π N 0 T = B + B G T πn πn πn 0 πn π N, = sum of all irreducible two-particle Feynman amplitues = relativistic free pion-nucleon propagator can be rewritten as with T π N = B + B π N G = B + B ( ) Bπ N. Bπ N πn πn G0 G 0 0 T π N, Find good B and G π N 0 5

6 Three-dimensional reduction Choose a G ( k, P) such that 0 1. T = B + B G T becomes π N π N π N three-dimensional 0 π N 2. G can reproduce N elastic 0 π cut Cooper-Jennings reduction scheme 6

7 Choose Bπ N to be given by N N (a) (b) ff ρ (c) (d) 7 (e) (f)

8 δ C.T. Hung, S.N. Yang, and T.-S.H. Lee, Phys. Rev. C64, (2001) 8

9 Dynamical model for electromagnetic pion production Both on- & off-shell 9

10 If the transition potential v consists ot two terms, v where v v B γπ R γπ ( E) ( E) B = v ( E) + v ( E) background transition potential contribution of a bare resonance then one obtains t γπ γπ with = = = t B γπ γπ R γπ 0 0 R γπ ( E), γπ B B t ( E) = v + v g ( E) t ( E) γπ γπ ( E) + γπ Δ Δ Δ t ( E) = v + v g ( E) t ( E) γπ γπ γπ t, π N π N R γ R both t B and t R satisfy Fermi-Watson theorem, respectively. π 10

11 DMT Model B v γπ = ( PV only) 11

12 12

13 13

14 14

15 Q Frolov et al., PRL, 1999 ep 0 ep π = 4.0GV e

16 16

17 Extension to higher energies coupled π, η, 2π channels t ( E) = v + v g ( E) t ( E), ij ij ik k kj k (,, =, ) i j k π η Include resonances R s with couplings to π, η, 2π channels B R ij ij ij v ( E) = v ( E) + v ( E) 17

18 Non-resonant background v : v v B ππ as given before, B v B v B πη ηπ ηη = = = 0. B ij r ur L = ig R N + ig RηN + (0) τ π (0) I πnr ηnr h.c. 18

19 R Resonan ce contribution v : R Rn v = v ( q, q; E), ij (0) R n= 1 for N overlapping resonances. v R ij n M = n N ij ( n) ( n) (0) (0) i Λ j i j f( Λ, q; E) g g f(, q: E) = % % E (0) M + R n 1 i Γ 2 ( n ) 2π ( E) bare mass of resonance R, ij n, i g (0) i j R n g (0) j Γ ( n) 2π =Γ q X + q 2l l + 2 0, n 2π 0, n 2π 2 2 q X + q 0, n 2 π = effects of ππ N decay channel 19

20 S11 2R 3R 4R ----T B = V B + V B g 0 T B 20

21 R 2R 3R No No evidence for for the the 4th 4th S 11 resonance 11 in in π N -> -> η N Consistent with data from ELSA γ N ηn 21

22 Extraction of resonance parameters t B Rn ( E) = t ( E) + t ( E), where t t ππ B ππ Rn ππ ( E) ( E) ππ N n= 1 B B = v ( E) + v ( E) g ( E) t ( E), ππ π k k kπ k Rn Rn = v ( E) + v ( E) g ( E) t ( E). ππ k ππ π k k kπ π π t Rn ππ ( E) = t R n ππ ( E M R n + iγ R n / 2) 1 22

23 R 1 R 2 R 3 R 4 M R 1520(6) 1677(6) 1893(18) 2183(21) our result (MeV) 1542(3) 1689(12) 1822(43) Pitt-ANL 1549(7) 1690(11) GW-Giessen PDG2002 Γ R 95(10) 195(14) 265(31) 427(26) our result (MeV) 112(19) 202(40) 248(185) Pitt-ANL 232(19) 206(13) GW-Giessen PDG2002 Γ π /Γ R 40(7) 75(4) 44(5) 43(2) our result (%) 35(5) 74(2) 17(7) Pitt-ANL 30(1) 60(7) GW-Giessen PDG2002 Γ η /Γ R 40(7) 11(3) 12(7) 3(6) our result (%) 51(5) 6(1) 41(4) Pitt-ANL 63(3) 11(1) GW-Giessen PDG2002 Pitt-ANL T.P. Vrana et al., Phys. Rep. 328, 181 (2000) GW-Giessen C. Bennhold et al., Proc. of NSTAR2001 workshop, Mainz Comparison with quark model (M.M. Giannini et al., Nucl. Phys. A699, 308 (2002)) R 1 R 2 R 3 R 4 M R 1520(6) 1677(6) 1893(18) 2183(21) our result HCQM 23

24 B t γπ R t γπ full χ 2 =64 Need 4 S 11 Resonances χ 2 =3.5 24

25 S13 (3,3) (3,3), red PDG, blue DMT 25

26 P11 (3,3) 26

27 P31 (2,3) 27

28 D15 (2,2) 28

29 F15 (2,2) 29

30 Pole Positions and Residues in [ MeV ] (preliminary) 1st Resonance 2nd Resonance 3rd Resonance N* Wp Γp Wp Γp Wp Γp S (1505, 1501) 67 (170, 124) 1642 (1660, 1673) 97 (160, 82) 2065 (2150±70) 223 (350±100) S (1600, 1585) 136 (115, 104) 1775 (1870±40) 36 (1850±50) 2012 (2140±80) 148 (200±80) P (1365, 1346) 179 (210, 176) 1721 (1720, 1770) 185 (230, 378) 1869 (2120±40, 1810) 238 (240±80, 622) P (1700, 1717) 239 (250, 388) 1846 (not listed) 180 (not listed) P (1714) 70 (68) 1896 (1855, 1810) 130 (350, 494) P (1210, 1211) 90 (100, 100) 1509 (1600, 1675) 236 (300, 386) 2149 (1900, 1900±80) 400 (300, 300±100) Red:PDG04 Green:Cutkovsky80 Blue:Arndt95 Orange:Vrana00 30

31 Pole Positions and Residues in [ MeV ] (preliminary) 1st Resonance 2nd Resonance 3rd Resonance N* Wp Γp Wp Γp Wp Γp D 13 (3,3) 1516 (1510, 1515) 123 (115, 110) not seen (1680, not seen) not seen (100, not seen) 1834 (1880±100) 210 (160±80) D 33 (2,2) 1609 (1660, 1655) 133 (200, 242) 2070 (1900±100) 267 (200±60) D 15 (2,2) 1657 (1660, 1663) 132 (140, 152) 2188 (2100±60) 238 (360±80) D 35 (2,1) 1992 (1890, 1913) 270 (250, 246) not seen (2400±60) not seen (400±150) F 15 (2,2) 1663 (1670, 1670) 115 (120, 120) 1931 (not listed) 62 (not listed) F 35 (2,2) 1771 (1830, 1832) 190 (280, 254) 2218 (2150±100, 1697) 219 (350±100, 112) F 37 (2,1) 1860 (1885, 1880) 201 (240, 236) 2207 (2350±100) 439 (260±100) Red:PDG04 Blue:Arndt95 Green:Cutkovsky80 Orange:Vrana00 31

32 Summary The DMT coupled-channel dynamical model gives excellent description of the pion scattering and pion e.m. production data from threshold to first resonance region The model has been extended up to c.m. energy 2 GeV in the π N and ηn sector and resonance parameters extracted 32

33 Our analysis at W > 1750 GeV yields considerable strength, which can be explained by a third and a fourth S 11 resonance with masses 1846(47) and 2113(70) MeV The End 33

34 S11 (3,4), red PDG, blue DMT 25

35 (2,2) 35

36 (3,3) 37

37 (3,3) 38

38 (2,2) 40

39 (2,1) 41

40 (2,2) 43

41 (2,1) 44