A covariant approach to the partial wave analysis of the hadron - PowerPoint PPT Presentation

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14 A covariant approach to the partial wave analysis of the hadron reactions A. Sarantsev Petersburg Nuclear HISKP (Bonn, Germany), PNPI (Gatchina, Russia) Physics Institute GSI, December 14, 2016 1

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14 Bonn-Gatchina partial wave analysis group: A. Anisovich, E. Klempt, V. Nikonov, A. Sarantsev, U. Thoma http://pwa.hiskp.uni-bonn.de/ 2

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14 Search for baryon states 1. Analysis of single and double meson photoproduction reactions. γp → πN, ηN, K Λ , K Σ , ππN, πηN , ωp , K ∗ Λ , CB-ELSA, CLAS, MAMI, GRAAL, LEPS. 2. Analysis of single and double meson production in pion-induced reactions. πN → πN, ηN, K Λ , K Σ , ππN . Search for meson states 1. Analysis of the p ¯ p annihilation at rest and ππ interaction data. 2. 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 NN interaction 1. Analysis of single and double meson production NN → πNN and (Wasa, PNPI, HADES) 2. Analysis of hyperon production NN → K Λ p (WASA, HADES) 3

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14 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 ββ ′ ν 1 ...ν n Q ( β ′ ) ( s ) Q ( β )+ µ 1 ...µ n F µ 1 ...µ n A ( s, t ) = n ν 1 ...ν n ββ ′ n A ββ ′ ( s ) - the partial wave amplitude with total spin J = n for bosons and n J = n + 1 / 2 for fermions. 1. A. V. Anisovich, V. V. Anisovich, V. N. Markov, M. A. Matveev and A. V. Sarantsev, J. Phys. G 28, 15 (2002) 2. A. Anisovich, E. Klempt, A. Sarantsev and U. Thoma, Eur. Phys. J. A 24, 111 (2005) 3. A. V. Anisovich and A. V. Sarantsev, Eur. Phys. J. A 30, 427 (2006) 4. A. V. Anisovich, V. V. Anisovich, E. Klempt, V. A. Nikonov and A. V. Sarantsev, Eur. Phys. J. A 34, 129 (2007). 1. C. Zemach, Phys. Rev. 140, B97 (1965); 140, B109 (1965). 2. S.U.Chung, Phys. Rev. D 57, 431 (1998). 3. B. S. Zou and D. V. Bugg, Eur. Phys. J. A 16, 537 (2003) 4

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14 Partial wave amplitude: transition amplitude with fixed initial and final states Quantum numbers: mesons I G J P C , baryons: IJ P , decay LS basis: 2 S +1 L J ( ) → I G J P C → I ′ G ′ P ′ 1 C ′ 1 + I ′ G ′ P ′ 2 C ′ I G 1 1 J P 1 C 1 + I G 2 2 J P 2 C 2 2 S ′ +1 L ′ ( 2 S +1 L J ) 1 J ′ 2 J ′ 2 1 1 2 2 1 2 J G = G ′ 1 G ′ G = G 1 G 2 2 2 ( − 1) L ′ P = P ′ 1 P ′ P = P 1 P 2 ( − 1) L | I ′ 1 − I ′ 2 | < I < I ′ 1 + I ′ | I 1 − I 2 | < I < I 1 + I 2 2 2 | < S ′ < J ′ | J ′ 1 − J ′ 1 + J ′ | J 1 − J 2 | < S < J 1 + J 2 2 | S ′ − L ′ | < J < S ′ + L ′ | S − L | < J < S + L A ( s, t ) = V µ 1 ...µ n ( S, L ) P µ 1 ...µ n ν 1 ...ν n V ′ ν 1 ...ν n ( S ′ , L ′ ) A ( s ) n = J mesons n = J − 1 / 2 baryons 5

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14 Boson projection operators The wave function of boson with J = n : 1 2 εu µ 1 ··· µ n e ipx Ψ µ 1 ··· µ n = √ Ψ µ ( x )Ψ ∗ ( x ) d 4 x = αp µ = 0 ∫ = ⇒ p µ Ψ µ = 0 ( ) g µν − p µ p ν Ψ µν ( x )Ψ ∗ ( x ) d 4 x = β ∫ = 0 = ⇒ g µν Ψ µν = 0 p 2 Properties of u µ 1 ··· µ n : p 2 u µ 1 µ 2 ...µ n = m 2 u µ 1 µ 2 ...µ n p µ i u µ 1 µ 2 ...µ n = 0 g µ i µ j u µ 1 µ 2 ...µ n = 0 u µ 1 ...µ i ...µ j ...µ n = u µ 1 ...µ j ...µ i ...µ n 6

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14 In momentum representation: 2 n +1 ∑ P µ 1 µ 2 ...µ n ν 1 ν 2 ...ν n = ( − 1) n O µ 1 µ 2 ...µ n u ( i ) µ 1 µ 2 ...µ n u ( i ) ∗ ν 1 ν 2 ...ν n = ν 1 ν 2 ...ν n i =1 O = 1 µν = g µν − p µ p ν O µ g ⊥ = ν p 2 1 µ 2 ν 1 ) − 1 O µ 1 µ 2 2( g ⊥ µ 1 ν 1 g ⊥ µ 2 ν 2 + g ⊥ µ 1 ν 2 g ⊥ 3 g ⊥ µ 1 µ 2 g ⊥ = ν 1 ν 2 ν 1 ν 2 Recurrent expression for the boson projector operator  L ν 1 ...ν L = 1 ∑ µ i ν j O µ 1 ...µ i − 1 µ i +1 ...µ L O µ 1 ...µ L g ⊥ ν 1 ...ν j − 1 νj +1 ...ν L −  L 2 i,j =1  L 4 ∑ g ⊥ µ i µ j g ⊥ ν k ν m O µ 1 ...µ i − 1 µ i +1 ...µ j − 1 µ j +1 ...µ L  ν 1 ...ν k − 1 ν k +1 ...ν m − 1 ν m +1 ...ν L (2 L − 1)(2 L − 3) i<j,k<m Normalization condition: O µ 1 ...µ L ν 1 ...ν L O ν 1 ...ν L α 1 ...α L = O µ 1 ...µ L α 1 ...α L 7

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14 Orbital momentum operator The angular momentum operator is constructed from momenta of particles k 1 , k 2 and metric tensor g µν . For L = 0 this operator is a constant: X 0 = 1 The L = 1 operator is a vector X (1) , constructed from: k µ = 1 2 ( k 1 µ − k 2 µ ) and µ P µ = ( k 1 µ + k 2 µ ) . Orthogonality: ∫ d 4 k ∫ d 4 k µ 1 X (0) = 4 π X (1) 4 π X ( n ) µ 1 ...µ n X ( n − 1) µ 2 ...µ n = ξP µ 1 = 0 Then: X (1) X ( n ) µ P µ = 0 µ 1 ...µ n P µ j = 0 and: ( g νµ − P ν P ν ) X (1) = k ⊥ µ = k ν g ⊥ g ⊥ νµ ; νµ = ; µ p 2 in c.m.s k ⊥ = (0 ,⃗ k ) 8

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14 Recurrent expression for the orbital momentum operators X ( n ) µ 1 ...µ n n n µ 1 ...µ i − 1 µ i +1 ...µ n − 2 k 2 µ 1 ...µ n = 2 n − 1 X ( n ) ∑ k ⊥ µ i X ( n − 1) ∑ g µ i µ j X ( n − 2) ⊥ µ 1 ...µ i − 1 µ i +1 ...µ j − 1 µ j +1 ...µ n n 2 n 2 i =1 i,j =1 i<j Taking into account the traceless property of X ( n ) we have: n 2 i − 1 = (2 n − 1)!! ∏ X ( n ) µ 1 ...µ n X ( n ) µ 1 ...µ n = α ( n )( k 2 ⊥ ) n α ( n ) = . i n ! i =1 From the recursive procedure one can get the following expression for the operator X ( n ) : k 2 [ ( ) X ( n ) k ⊥ µ 1 k ⊥ µ 2 . . . k ⊥ ⊥ g ⊥ µ 1 µ 2 k ⊥ µ 3 . . . k ⊥ µ 1 ...µ n = α ( n ) µ n − µ n + · · · + 2 n − 1 k 4 ] ( ) ⊥ g ⊥ µ 1 µ 2 g ⊥ µ 3 µ 4 k ⊥ µ 5 · · · k µ 4 + · · · + · · · . (2 n − 1)(2 n − 3) 9

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14 Scattering of two spinless particles Denote relative momenta of particles before and after interaction as q and k , correspondingly. The structure of partial–wave amplitude with orbital momentum L = J is determined by convolution of operators X ( L ) ( k ) and X ( L ) ( q ) : A L = BW L ( s ) X ( L ) ν 1 ...ν L X ( L ) ν 1 ...ν L ( q ) = BW L ( s ) X ( L ) µ 1 ...µ L ( k ) X ( L ) µ 1 ...µ L ( k ) O µ 1 ...µ L µ 1 ...µ L ( q ) BW L ( s ) depends on the total energy squared only. The convolution X ( L ) µ 1 ...µ L ( k ) X ( L ) µ 1 ...µ L ( q ) can be written in terms of Legendre polynomials P L ( z ) : ) L (√ √ X ( L ) µ 1 ...µ L ( k ) X ( L ) k 2 q 2 µ 1 ...µ L ( q ) = α ( L ) P L ( z ) , ⊥ ⊥ L ( k ⊥ q ⊥ ) 2 n − 1 ∏ z = α ( L ) = √ √ n k 2 q 2 ⊥ ⊥ n =1 10

A covariant approach to the partial wave analysis of the hadron reactions GSI 2016, December 14 pp → 3 π 0 reaction The ¯ N ∑ X ( L ) 23 ) X ( L ) µ 1 ...µ L ( k ⊥ µ 1 ...µ L ( k ⊥ A ( S − wave ) = 1 ) A L ( s 23 ) L =0 − - 3 π 0 Liquid target pp x 10 0 0 0 0 10000 p p p p → → 3 3 π π in liquid hydrogen in liquid hydrogen p p p p → → 3 3 π π in liquid hydrogen in liquid hydrogen 40000 1200 1200 7500 3 3 5000 20000 1000 1000 2.5 2.5 2500 800 800 ) ) 2 2 2 2 ) (GeV ) (GeV 0 0 0 1 2 3 -1 -0.5 0 0.5 1 600 600 1.5 1.5 M 2 ( ππ ) GeV 2 0 0 Cos θ for 0.00 < M ππ < 0.90 GeV π π 0 0 π π ( ( x 10 2 2 2 M M 400 400 x 10 2 1 1 1500 1500 200 200 0.5 0.5 1000 1000 0 0 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 2 2 2 2 M ( π 0 π 0 ) (GeV ) M ( π 0 π 0 ) (GeV ) 500 500 0 0 0 0 p p p p → → 3 3 π π in gaseous hydrogen in gaseous hydrogen p p p p → → 3 3 π π in gaseous hydrogen in gaseous hydrogen 0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 1800 3 3 1800 Cos θ for 0.95 < M ππ < 1.05 GeV Cos θ for 1.20 < M ππ < 1.40 GeV 1600 1600 x 10 2 2.5 2.5 1400 x 10 2 1400 2000 1200 ) ) 2 2 1200 2 2 1500 ) (GeV ) (GeV 1500 1000 1000 0 1.5 0 1.5 1000 π π 800 0 0 800 1000 π π ( ( 2 2 M M 600 1 1 600 500 500 400 400 0.5 0.5 0 0 200 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 200 Cos θ for 1.45 < M ππ < 1.55 GeV Cos θ for 1.55 < M ππ < 1.65 GeV 0 0 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 2 2 2 2 M ( 0 0 ) (GeV ) M ( 0 0 ) (GeV ) π π π π 11