Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods

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1 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 1 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods Petersburg Nuclear Physics Institute A. Sarantsev HISKP (Bonn), PNPI (Russia) PWA March 1, IHEP, Beijing

2 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 Bonn-Gatchina partial wave analysis group: A. Anisovich, E. Klempt, V. Nikonov, A. Sarantsev, U. Thoma

3 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 3 Search for baryon states 1. Analysis of single and double meson photoproduction reactions. γp πn, ηn, KΛ, KΣ, ππn, πηn, CB-ELSA, CLAS, GRAAL, LEPS.. Analysis of single and double meson production in pion-induced reactions. πn πn, ηn, KΛ, KΣ, ππn. Search for meson states 1. Analysis of the p p annihilation at rest and ππ interaction data.. Analysis of the p p annihilation in flight into two and tree meson final state. 3. Analysis of the BES III data on J/Ψ decays (in collaboration with JINR Dubna). Analysis of N N interaction 1. Analysis of single and double meson production NN πnn and ππnn (Wasa, PNPI, HADES). Analysis of hyperon production NN KΛp (WASA, HADES)

4 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 4 Energy dependent approach In many cases an unambiguous partial wave decomposition at fixed energies is impossible. Then the energy and angular parts should be analyzed together: A(s, t) = ββ n A ββ n (s)q (β)+ µ 1...µ n F µ 1...µ n ν 1...ν n Q (β ) ν 1...ν n 1. C. Zemach, Phys. Rev. 14, B97 (1965); 14, B19 (1965).. S.U.Chung, Phys. Rev. D 57, 431 (1998). 3. B. S. Zou and D. V. Bugg, Eur. Phys. J. A 16, 537 (3) 1. Correlations between angular part and energy part are under control.. Unitarity and analyticity can be introduced from the beginning. 3. Parameters can be fixed from a combined fit of many reactions. 1. Anisovich:1ra A. V. Anisovich, V. V. Anisovich, V. N. Markov, M. A. Matveev and A. V. Sarantsev, J. Phys. G G 8, 15 (). A. Anisovich, E. Klempt, A. Sarantsev and U. Thoma, Eur. Phys. J. A 4, 111 (5) 3. A. V. Anisovich and A. V. Sarantsev, Eur. Phys. J. A 3, 47 (6) 4. A. V. Anisovich, V. V. Anisovich, E. Klempt, V. A. Nikonov and A. V. Sarantsev, Eur. Phys. J. A 34, 19 (7).

5 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 5 πn vertices N + µ 1...µ n = X (n) µ 1...µ n N µ 1...µ n = iγ ν γ 5 X (n+1) νµ 1...µ n (1) γn vertices Q (1+)µ α 1...α n = γ µ iγ 5 X (n) α 1...α n, Q (+)µ α 1...α n = γ ν iγ 5 X (n+) µνα 1...α n, Q (3+)µ α 1...α n = γ ν iγ 5 X (n+1) να 1...α n g µα n, Q (1 )µ α 1...α n = γ ξ γ µ O (n+1) ξα 1...α n, Q ( )µ α 1...α n = X (n+1) µα 1...α n, Q (3 )µ α 1...α n = X (n 1) α...α n g α 1 µ. Fermion propagator for J = N + 1 F µ 1...µ L ν 1...ν L (p) = (m+ˆp)o µ 1...µ L L + 1 ( α 1...α L g L+1 α1 β 1 L L+1 σ ) L α 1 β 1 g αi β i O β 1...β L ν 1...ν L i= σ αi α j = 1 (γ α i γ αj γ αj γ αi )

6 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 6 Resonance amplitudes for meson photoproduction γ p R 1 R π p π π (k ) q3 R 1 (L, S) (L π R, S π ) R u(k 1 ) R q u(q 1 ) General form of the angular dependent part of the amplitude: ū(q 1 )Ñα 1...α n (R µn)f α 1...α n β 1...β n (q 1 + q )Ñ(j)β 1...β n γ 1...γ m (R 1 µr ) F µ 1...µ L ν 1...ν L (p) = (m+ˆp)o µ 1...µ L L + 1 ( α 1...α L gα L+1 1 β 1 F γ 1...γ m ξ 1...ξ m (P)V (i)µ ξ 1...ξ m (R 1 γn)u(k 1 )ε µ σ αi α j = 1 (γ α i γ αj γ αj γ αi ) L ) L+1 σ L α 1 β 1 i= g αi β i O β 1...β L ν 1...ν L

7 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 7 Parameterization of the partial wave amplitude Extraction of leading singularities Poles: amplitude as a sum of the Breit-Wigner states: A = β Λ β M β s i j g (β) j ρ j (s) β = J, S, L, n... Λ is a complex constant which describes interference between states and contribution of weaker singularities, e.g. triangle (logarithmic) singularity. Advantages: 1) Masses and widths of many resonances can be fixed from other sources: results of other groups or from PDG. ) The easiest and fastest approach to find a signal from a new state in the data and, indeed, it works very well for many cases. Problems: 1) A possibility to violate the unitarity limit in the case of overlapping states. ) Notable problems in a combined analysis of many reactions.

8 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 8 Combined analysis of large number of reactions For pion induced reactions: A 1i = K 1j (I iρk) 1 ji and K ij = α g α i gα j M α s + f ij(s) f ij = f (1) ij + f () ij s s s ij. where f ij is non-resonant transition part. For the photoproduction: A k = P j (I iρk) 1 jk The vector of the initial interaction has the form: P j = α Λ α g α j M α s + F j(s) Here F j is non-resonant production of the final state j.

9 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 9 The K-matrix amplitude can be considered as a solution of Bethe-Salpeter equation: = B(s) + g g A ab (s, s) = 4m ds π A aj (s, s )iρ j (s )K jb (s ) s s i + K ab (s) But... with omitted real part of loop diagrams: A ab = A aj iρ j (s)k jb + K ab  = ˆK(I iˆρˆk) 1

10 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 1 N/D based (D-matrix) analysis of the data In the case of resonance contributions only we have factorization and Bethe-Salpeter equation can be easily solved: π η K J m J K m δ JK D jm = D jk α Bα km 1 (s) M m s + δ jm M j s π η K ˆD = ˆκ(I ˆBˆκ) 1 ( ) 1 1 ˆκ = diag M1 s, M s,..., 1 MN s, R 1, R... ˆB ij = α B ij α = α ds π g (R)i α ρ α (s,m 1α,m α )g (L)j α s s i

11 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 11 For non-resonant transitions from one channel (e.g. ππ): g L(N+1) i R 1 g R(N+1) j + g L(N+) i R g R(N+) j where N is the number of pole terms. The non-zero left and right vertices: g L(N+1) j = f 1j 1 GeV + s s + s g R(N+1) 1 = 1 R 1 = 1, g L(N+) 1 = 1 g R(N+) j>1 = f 1j 1 GeV + s s + s R = 1 i j A ab = Σ ij a b P b = Σ b ij i j A ab = α,β ga R(α) κ αα D αβ g L(β) b. A b = α,β P (α) κ αα D αβ g L(β) b P = (Λ 1, Λ,...,Λ n, F 1 /R 1...)

12 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 1 In the present fits we calculate the elements of the B ij α the channel threshold M α = (m 1α + m α ): using one subtraction taken at B ij α (s) = B ij α (M α) + (s M α) ds π g (R)i α ρ α (s, m 1α, m α )g α (L)j (s s i)(s Mα). m a In this case the expression for elements of the ˆB matrix can be rewritten as: Bα ij (s) = g a (R)i b α + (s M α) m a ds π ρ α (s, m 1α, m α ) (s s i)(s Mα) g (L)j β = g a (R)i B α g (L)j β and D-matrix method equivalent to the K-matrix method with loop diagram with real part taken into account: A = ˆK(I ˆB ˆK) 1 B αβ = δ αβ B α

13 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 13 Meson spectroscopy. Two body reactions: Reaction Experiment Reaction Experiment π + π π + π (all waves) CERN-Münich ππ π π (S-wave) GAMS ππ π π (S-wave) E85 ππ ηη (S-wave) GAMS ππ ηη (S-wave) GAMS ππ K K (S-wave) BNL S-wave δ(π + π ) Ke4 Three body reactions from Crystal Barrel: (L-liquid, G-gaseous targets). Reaction Target Reaction Target Reaction Target pp π π π (L) H pp π + π π (L) H pp K S K S π (L) H pp π ηη (L) H pn π π π (L) D pp K + K π (L) H pp π π η (L) H pn π π π + (L) D pp K L K ± π (L) H pp π π π (G) H pn K S K S π (L) D pp π ηη (G) H pn K S K π (L) D pp π π η (G) H

14 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 14 Description of the data with K-matrix and D-matrix approaches K-matr. D-matr. N K-matr. D-matr. N pp π π π (L) pp π π π (G) pp π ηη(l) pp π ηη(g) pp π π η(l) pp π π η(g) pp π + π π (L) pp K S K S π (L) pn π π π pn π π π pn K + K π pn K L K ± π pp K S K S π ππ (π π ) S ππ (ηη) S ππ (ηη ) S ππ (K K) S δ (π π + π π + ) K e4 data

15 ) ) ) Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 15 Description of the pp 3π data (D-matrix method) (π π ) (GeV ) pp - 3π Liquid target M p p 3π in liquid hydrogen (π π ) (GeV M p p 3π in liquid hydrogen M (ππ) GeV x 1 15 x Cos θ for.<m ππ <.9 GeV x M (π π ) (GeV ) p p 3π in gaseous hydrogen M (π π ) (GeV ) p p 3π in gaseous hydrogen (π π ) (GeV M (π π ) (GeV M Cos θ for.95<m ππ <1.5 GeV x Cos θ for 1.<M ππ <1.4 GeV x M (π π ) (GeV ) M (π π ) (GeV ) Cos θ for 1.45<M ππ <1.55 GeV Cos θ for 1.55<M ππ <1.65 GeV

16 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 16 Pole position of the resonances δ, deg K-matrix D-matrix σ-meson 4-i i 81 f (98) 114-i i 36 f (13) 13-i i 137 f (15) 1487-i i 6 f (175) 1738-i i 14 4 a) Im M Re M s, MeV i i I.Caprini, G.Colangelo, and H.Leutwyler, Phys.Rev.Lett.96, 131 (6) R.Garcia-Martin, R.Kaminski, J.R.Pelaez, J.Ruiz de Elvira, and F.J.Yndurain

17 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 17 Masses and couplings of bare states K-matrix D-matrix K-matrix D-matrix M g (1) 4π M g () 4π M g (3) 4π M g (4) 4π M g (5) 4π g Φ g Φ g Φ g Φ g Φ f ππ ππ f ππ K K.1.36 f ππ 4π f ππ ηη f ππ ηη

18 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 18 Baryon Data Base Pion induced reactions (χ analysis). Observable N data χ N data Observable N data χ N data N 1/ S 11 (πn πn) 11.5 SAID (.1) 1/ S 31 (πn πn) SAID (.1) N 1/ + P 11 (πn πn) SAID (.1) 1/ + P 31 (πn πn) SAID (.1) N 3/ + P 13 (πn πn) SAID (.) 3/ + P 33 (πn πn) 1.79 SAID (.) N 3/ D 13 (πn πn) SAID (.) 3/ D 33 (πn πn) SAID (.1) N 5/ D 15 (πn πn) SAID (.4) N 7/ G 17 (πn πn) 1.54 SAID (.4) N 5/ + F 15 (πn πn) SAID (.) 5/ + F 35 (πn πn) SAID (.1) N 7/ + F 17 (πn πn) SAID (.5) 7/ + F 37 (πn πn) 7.75 SAID (.1) N 9/ G 19 (πn πn) 74.8 SAID (.5) N 9/ + H 19 (πn πn) SAID (.5) dσ/dω(π p nη) Richards et al. dσ/dω(π p nη) CBALL dσ/dω(π p KΛ) RAL P(π p KΛ) RAL+ANL β(π p KΛ) RAL dσ/dω(π + p K + Σ) RAL P(π + p K + Σ) RAL β(π + p K + Σ) 7.8 RAL dσ/dω(π p K Σ ) RAL P(π p K Σ ) RAL

19 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 19 Observable N data χ Baryon Data Base (SAID db: 8) π and η photoproduction reactions (χ analysis). N data Observable N data χ N data dσ/dω(γp pπ ) CB-ELSA dσ/dω(γp pπ ) GRAAL dσ/dω(γp pπ ) CLAS dσ/dω(γp pπ ) 169. TAPS@MAMI Σ(γp pπ ) CB-ELSA Σ(γp pπ ) SAID db E(γp pπ ) A-GDH P(γp pπ ) SAID db T(γp pπ ) SAID db H(γp pπ ) SAID db G(γp pπ ) SAID db O x (γp pπ ) SAID db O z (γp pπ ) 7.7 SAID db dσ/dω(γp nπ + ) CLAS dσ/dω(γp nπ + ) SAID db dσ/dω(γp nπ + ) A-GDH Σ(γp nπ + ) SAID db E(γp nπ A-GDH P(γp nπ + ) 5. SAID db T(γp nπ + ) SAID db H(γp pπ + ) SAID db G(γp pπ + ) SAID db dσ/dω(γp pη) CB-ELSA dσ/dω(γp pη) 1.6 TAPS Σ(γp pη) GRAAL 98 Σ(γp pη) 1.43 GRAAL 7 T(γp pη) Phoenics

20 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 Baryon Data Base Kaon photoproduction (χ analysis). Observable N data χ N data Observable N data χ N data dσ/dω(γp ΛK + ) CLAS9 dσ/dω(γp Σ K + ) CLAS P(γp ΛK + ) CLAS9 P(γp Σ K + ) CLAS C x (γp ΛK + ) CLAS C x (γp Σ K + ) CLAS C z (γp ΛK + ) CLAS C z (γp Σ K + ) CLAS Σ(γp ΛK + ) GRAAL Σ(γp Σ K + ) GRAAL Σ(γp ΛK + ) LEP Σ(γp Σ K + ) LEP T(γp ΛK + ) GRAAL 9 dσ/dω(γp Σ + K ) CLAS O x (γp ΛK + ) GRAAL 9 dσ/dω(γp Σ + K ) 7.67 CB-ELSA 1 O z (γp ΛK + ) GRAAL 9 P(γp Σ + K ) CB-ELSA 1 P(γp ΛK + ) 84.6 GRAAL Σ(γp Σ + K ) CB-ELSA 1

21 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 1 Baryon Data Base Multi-meson final states (maximum likelihood analysis). dσ/dω(π p nπ π ) CBALL dσ/dω(γp pπ π ) CB-ELSA (1.4 GeV) E(γp pπ π ) MAMI dσ/dω(γp pπ η) CB-ELSA (3. GeV) Σ(γp pπ η) GRAAL dσ/dω(γp pπ π ) CB-ELSA (3. GeV) Σ(γp pπ π ) GRAAL dσ/dω(γp pπ η) CB-ELSA (3. GeV) Σ(γp pπ η) GRAAL I c (γp pπ η) CB-ELSA (3. GeV) I s (γp pπ η) CB-ELSA (3. GeV)

22 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 Description of the πn elastic amplitudes (GWU energy independent solution) with K-matrix and D-matrix solutions

23 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 3 The fit of the the π p KΛ reaction Full experiment for πn KΛ: differential cross section, analyzing power, rotation parameter. A clear evidence for resonances which are hardly seen (or not seen) in the elastic reactions: N(171)P 11, N(19)P 13, The total cross section for the reaction π p K Λ and contributions from leading partial waves in K-matrix (full) and D-matrix (dashed) solutions.

24 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 4 π p KΛ (dσ/dω,p )

25 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 5 The fit of the the π + p K + Σ + reaction

26 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 6 Photoproduction reactions: γp π p

27 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 7 Photoproduction reactions: γp ηp

28 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 8 Pole position in the mass complex plane for S 11 and P Trajectories for Re det(i iρk) = Im det(i iρk) = Re det(i Bκ) = Im det(i Bκ) =

29 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 9 Pole parameters of the S 11 states N(1535)S 11 N(165)S 11 N(189)S 11 K-matrix D-matrix K-matrix D-matrix K-matrix D-matrix M pole 151± ± ± Γ pole 134± ± Elastic residue 31±4 5 4±3 3 1±1 1.5 Phase -(9±5) -38 -(75±1) -6 Res πn Nη 8±3 5 15±3 15 4± 5 Phase -(76±8) -69 (13±1) 14 (4±) 4 Res πn π 7±4 4 11±3 1 Phase (147±17) 157 -(3±) -4 A 1/ ( GeV 1 ).116± ±.7.9.1±.6.1 Phase (7±6) 1 -(9±15) 1±5 15

30 Bonn-Gatchina partial wave analysis. Comparison of the K-matrix and D-matrix (N/D based) methods PWA 1 3 Summary In the case of a partial wave analysis of a single reaction the most strait forward approach is an approximation by a sum of Breit-Wigner resonances. In a combined analysis of many reactions with coupled channels at least K-matrix approach is needed. The D-matrix method takes into account explicitly the real part of loop diagrams: thus it treats amplitudes more correctly at low energies. In some cases, e.g. many open channels and only few contributing states resonances, this method can fit data even faster than K-matrix method. For high mass states it produces the same results as the K-matrix approach but with some additional parameters: subtraction constants.