Analyticity and crossing symmetry in the K-matrix formalism.

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1 Analyticity and crossing symmetry in the K-matrix formalism. KVI, Groningen 7-11 September, 21

2 Overview Motivation Symmetries in scattering formalism K-matrix formalism (K S ) (K A ) Pions and photons off the nucleon

3 PANDA physics Meson spectroscopy charmonium states open charm production exotic matter glueballs hybrids molecules light mesons Baryon-Antibaryon production Charm in nuclei Hypernuclei physics Hadrons in nuclear medium

4 Symmetries A scattering matrix should obey various symmetries : 1 Unitarity S S = SS = 1; conservation of flux. 2 Crossing symmetry Take all Feynman diagrams into account. Low energy theorems. 3 Analyticity/causality Should obey a dispersion relation. 4 Covariance 5 Gauge invariance 6 Chiral symmetry Are these symmetries obeyed in the K-matrix formalism?

5 K-matrix formalism: the simple K-matrix S S = 1+2iT S T S = K S 1 ik S K = sum of all tree level diagrams Calculated per partial wave. Algebraic Full coupled channels in large model space Unitary Gauge invariant, Chiral symmetry Covariant { Causality Analitycity

6 : Analyticity The scattering matrix should obey a dispersion relation: Re(T(s)) = (s s ) P π Λ s Im(T(s )) (s s)(s s ) ds Im(T(s )) is calculated from the K-matrix. Re(T(s)) calculated from the dispersion relation should have the same shape as Re(T(s)) calculated from the K-matrix.

7 : Analyticity The dispersion relation is not obeyed. is not continuous along the threshold of channel 2: K S = ( KS11 ) ( KS11 K thr.2 S12 K S21 K S22 S S is not analytic Problem! Make an analytic continuation of K S. ) Amplitude Re(T 11 ) Im(T 11 ) Re; disp Invariant mass [GeV]

8 : Analyticity K A = ( ) ρ1a 11 ρ1ρ 2A 21 ρ1ρ 2A 12 ρ 2A 22 Analytic continuation of the momenta of the particles below their thresholds: ρ i = k i; k i = k 2 i Invariant mass [GeV] Be careful with the definition of ρ; it s double valued! ρ is double valued: ρ = ± k 2 ; if k 2 <, ρ = ±i k. Amplitude Re(T 11 ) Im(T 11 ) Re; disp.

9 : Analyticity Explicitly add a pole and consider the decay e Γt : ig m r ig gives A = g2 s m 2 r S A = s m2 r ig2 ρ s m 2 r +ig2 ρ Compare with results from dispersion integral

10 Compton Scattering Consistently re-summed u & t channel contributions added as contact terms to the kernel N(, )N, amplitudes [1-4 /m ] MM-, 1/2 + K T, ren 2 EE-, 1/ photon lab energy [MeV] Renormalize at threshold. Cusp at pion production threshold agrees with data analysis. Data from: Pfeil et al, NPB73, 166 Bergstrom et al, PRC48, 158

11 Compton Scattering, Cross sections Renormalization: ( ) ρ1a 11 ρ1ρ 2A 21 K A = ρ1ρ 2A 12 ρ 2A 22 d (, ) /d [nb/sr] E lab = E lab =182 Gives: T 11 = ρ 1 A 1,1 +Z 1 iρ 1(A 1,1 +Z) 1 E lab = cm [deg] E lab = cm [deg] Z = i ρ2a1,2a2,1 1 iγ

12 Compton Scattering, Cross sections 3 3 d (, ) /d [nb/sr] cm=75 o cm=9 o E lab [MeV] E lab [MeV]

13 Pion scattering and photoproduction 2 E + 1/ M 1-1/2 2 S11-1 S31 Partial Wave Amplitudes [m fm] E + 3/2 2 4 Re Im 2-5 M 1-3/2 Photon lab energy [MeV] 2 4 N(, )N phases [deg] P P E lab [MeV] η, ρ meson production not yet included

14 Symmetries in the simple K-matrix formalism Unitarity: by construction Analyticity/Causality: not fulfilled Symmetries in the analytic continued K-matrix formalism Unitarity: by construction Analyticity/Causality: fulfilled Renormalization of off-diagonal matrix elements should solve the crossing symmetry problem partially Pion Photoproduction Renormalization gives promising results