Three coupled amplitudes for the πη, K K and πη channels without data

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1 Three coupled amplitudes for the πη, K K and πη channels without data Robert Kamiński IFJ PAN, Kraków and Łukasz Bibrzycki Pedagogical University, Kraków HaSpect meeting, Kraków, V/VI 216

2 Present status of knowledge of interactions in the πη, K K and πη channels NO DATA on the amplitudes! (phase shifts, inelasticities) Positions of the a (98) and a (145), Kinematics: three thresholds: πη 686 MeV, K K 991 MeV and πη 196 MeV, Couplings of both a to the three channels, πη scattering length: a 1.5.1m 1 π radius of the πη form factor: r 2.15 fm 2 (ChPT),

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8 Present status of knowledge of interactions in the πη, K K and πη channels NO DATA on the amplitudes! (phase shifts, inelasticities) Positions of the a (98) and a (145), Kinematics: three thresholds: πη 686 MeV, K K 991 MeV and πη 196 MeV, Couplings of both a to the three channels, πη scattering length: a 1.5.1m 1 π radius of the πη form factor: r 2.15 fm 2 (ChPT),

9 Present status of knowledge of interactions in the πη, K K and πη channels NO DATA on the amplitudes! (phase shifts, inelasticities) Positions of the a (98) and a (145) 4 parameters, Kinematics: three thresholds: πη 686 MeV, K K 991 MeV and πη 196 MeV, Couplings of both a to the three channels 2 3, πη scattering length: a 1.5.1m 1 π (ChPT) 1, radius of the πη form factor: r 2.15 fm 2 1

10 Positions of the a (98) and a (145) 4 parameters Separable potential model for two coupled channels πη and K K : "Coupled channel study of a resonances", A. Furman, L. Lesniak, Phys.Lett. B538 (22) < p T q > = < p V q > + d 3 s + < p V s >< s G s >< s T q > (2π) 3 < p V ij q >= λ ij g i (p)g j (q) g i (p) = 4π m i 1 p 2 + (β i ) 2 t = λ + λ I t t = (1 λi) 1 λ

11 Positions of the a (98) and a (145) 4 parameters where I ii = d 3 s (2π) 3 g i (s)g i (s)g i (s) D(E) = det(1 λi(e)) D(k 1, k 2 ) = D πη(k 1 )D KK (k 2 ) C(k 1, k 2 ) Dπη(k 1 ) = 1 Λ 11 J 11 (k 1 ) D KK (k 2 ) = 1 Λ 22 J 22 (k 2 ) C(k 1, k 2 ) = Λ 2 12 J 11(k 1 ) J 22 (k 2 )

12 Positions of the a (98) and a (145) 4 parameters S 11 = D( k 1, k 2 ) D(k 1, k 2 ), S 22 = D(k 1, k 2 ) D(k 1, k 2 ), S 11 S 22 S 2 12 = D( k 1, k 2 ) D(k 1, k 2 ) S 11 = 1 ik 1E 1 (k 1 ) T 11 (k 1, k 2 ), 2π S 22 = 1 ik 2E 2 (k 2 ) T 22 (k 1, k 2 ), 2π S 12 = S 21 = i k1 E 1 (k 1 )k 2 E 2 (k 2 ) T 12 (k 1, k 2 ) 2π S = ( ηe 2iδπη i 1 η 2 e i(δπη+δ KK ) i 1 η 2 e i(δπη+δ KK ) ηe 2iδ KK )

13 Positions of the a (98) and a (145) 4 parameters Unitary amplitudes for interactions in two coupled channels πη and K K in the wide energy range from the πη threshold ( 686 MeV) up to 15 MeV. two resonances a (98) and a (145) four parameters found analytically! β 1 = 2. GeV (fixed), β 2 = 21.8 GeV, Λ 1 =.32321, Λ 2 =.6817, Λ 2 12 = , Two poles corresponding to the a (98), both very near the K K threshold, for example at i33.6 MeV (II nd Riemann sheet i.e Imk 1 <, Imk 2 > ) and 15. i24.5 MeV (III nd Riemann sheet i.e Imk 1 <, Imk 2 < )

14 2-channel amplitude (4 free parameters) δ πη 1 η πη

15 2-channel amplitude (4 free parameters) δ KK 1 5 η πη

16 2-channel amplitude (4 free parameters) σ πη (MeV -2 ) * σ KK (MeV -2 ) *

17 2-channel amplitude (4 free parameters) σ πη (MeV -2 ) * σ πη->kk (MeV -2 ) *

18 scattering length a 1 a 1 =.26m 1 π ) What else?.5.1m 1 π couplings to the three channels: πη (686 MeV), K K (991 MeV), πη (196 MeV) radius of the πη form factor: r 2.15 fm 2 three coupled channel amplitude with 9 parameters where ChPT (2-channel amplitude: now D(k 1, k 2, k 3 ) = D πη(k 1 )D KK (k 2 )D πη (k 3 ) C(k 1, k 2, k 3 ) Dπη(k 1 ) = 1 Λ 11 J 11 (k 1 ) D KK (k 2 ) = 1 Λ 22 J 22 (k 2 ) D πη (k 3 ) = 1 Λ 33 J 33 (k 3 ) C(k 1, k 2, k 3 ) = Λ 2 12 J 11(k 1 ) J 22 (k 2 ) + Λ 2 13 J 11(k 1 ) J 33 (k 3 ) + Λ 2 23 J 22(k 2 ) J 33 (k 3 ) + + (Λ 2 12 Λ 3 + Λ 2 13 Λ 2 + Λ 2 23 Λ 1 + Λ 12 Λ 13 Λ 23 ) J 11 (k 1 ) J 22 (k 2 ) J 33 (k 3 )

19 9 parameters three coupled channel amplitude coupling constants for a (98): Crystal Barrel Coll. 97: g KK /g πη 2 1, PDG:.183 ±.24 N. N. Achasov: g πη = 4.23 GeV, g KK = 3.79 GeV, g πη = 2.13 GeV, "Nature of the a (98) meson in the light of photon-photon collision", PRD 21 F. Giacosa: g πη = GeV, g KK = 6.12 GeV, "a (98) revisited", PRD 216 coupling constants for a (145): N. N. Achasov: g πη = 3.3 GeV, g KK =.28 GeV, g πη = 2.91 GeV πη scattering length: a 1.5.1m 1 π (ChPT) Example of results (fit to the poles for both a s, coupling constants and scattering length) β 1 = 2. GeV (fixed), β 2 = 19.8 GeV, β 3 = 29.9 GeV, Λ 1 =.3, Λ 2 =.7, Λ 3 =.2, Λ 2 12 = , Λ 2 13 = , Λ 2 23 =

20 3-channel amplitude (9 free parameters) δ πη 1 δ πη

21 3-channel amplitude (9 free parameters) η πη.6.4 η πη

22 3-channel amplitude (9 free parameters) 3 15 δ KK 2 δ KK

23 3-channel amplitude (9 free parameters) η KK.6.4 η πη

24 3-channel amplitude (9 free parameters) σ πη (MeV -2 ) * σ πη (MeV -2 ) *

25 3-channel amplitude (9 free parameters) σ KK (MeV -2 ) * σ KK (MeV -2 ) *

26 3-channel amplitude (9 free parameters) 2 2 σ πη->kk (MeV -2 ) * σ πη->kk (MeV -2 ) *

27 3-channel amplitude (9 free parameters) 2 2. σ πη` (MeV -2 ) * σ πη->πη` (MeV -2 ) *

28 3-channel amplitude (9 free parameters) δ πη` η πη`

29 πη form factor Form factor associated with spin zero isospin one operator ūd: F πη (s) =< η(p 1 )π(p 2 ) ūd > < r 2 F πη >= 6 s at s = or < r 2 >= 6 π + s ds δ F πη (s) s 2 < r 2 >=.92 ±.7 fm 2 (ChPT), fm 2 (B. Moussallam 215, EPJ 215, "Form Factors of the isovector scalar current and the πη scattering phase shifts") Below the K K threshold δ F πη (s) = δ πη The integral to K K threshold gives.1 fm 2 The integral in the full range gives.12 fm 2 (upper limit)

30 Conclusions this is one of the least known channel of interactions of light mesons, enough constrains to fix the amplitude at least qualitatively, but still not enough to find unique and precise amplitudes, data are needed (phase shifts and inelasticities), maybe Roy-like dispersion relations with imposed crossing symmetry will help