Hadron excitations as resonant particles in hadron reactions Hiroyuki Kamano (RCNP, Osaka Univ.) Crossover workshop@Nagoya U., July 12-13, 2012

Introduction: Hadron spectrum and reaction dynamics Various static hadron models have been proposed to calculate hadron spectrum and form factors. In reality, excited hadrons are “unstable” and can exist only as resonance states in hadron reactions. Quark models, Bag models, Dyson-Schwinger approaches, Holographic QCD,… Excited hadrons are treated as stable particles.  The resulting masses are real. resonance bare state meson cloud “molecule-like” states core (bare state) + meson cloud u d Constituent quark model “Mass” becomes complex !!  “pole mass”

Definitions of hadron parameters Masses (spectrum) of hadrons  Pole positions of the amplitudes Coupling constants of hadrons  Residues1/2 at the pole Consistent with the resonance theory based on Gamow vectors G. Gamow (1928), R. E. Peierls (1959), … A brief introduction of Gamov vectors: de la Madrid et al, quant-ph/0201091 (complex) energy eigenvalues = pole values transition matrix elements = (residue)1/2 of the poles Decay vertex pole position ( Im(E0) =< 0 )

Im (E) Re (E) Resonance pole in complex-E plane and peak in cross section (Breit-Wigner formula) Im (E) Cross section σ ~ |T|2 Conditions: Pole is isolated. Re (E) Small background. Amplitudes between the pole and real energy axis are analytic.

f0(980) in pi-pi scattering π f0 (980) f0 (980) From M. Pennington’s talk σ (ππππ) Im(E) (GeV) Re(E) (GeV) ？ 0.4 0.8 1.2 1.6 ～ 980 – 70i (MeV)

Multi-layer structure of the scattering amplitudes unphysical sheet physical sheet e.g.) single-channel two-body scattering Scattering amplitude is a double-valued function of complex E !! Essentially, same analytic structure as square-root function: f(E) = (E – Eth)1/2 physical sheet unphysical sheet Re(E) + iε ＝“physical world” Im (E) Im (E) Eth (branch point) Eth (branch point) × × × × Re (E) Re (E)

f0(980) in pi-pi scattering, Cont’d f0(980) is barely contributed f0(980) pole CANNOT produce a “clean” peak in the cross section because of discontinuity induced by KK channel !! This is the reason why we need the third condition: “Amplitudes between the pole and real energy axis are analytic.” + Relations requires “clean” peak in cross section. ｆ    Not only the resonance poles, but also the analytic structure of the scattering amplitudes in the complex E-plane plays a crucial role for the shape of cross sections on the real energy axis (= real world) !! pp Im (E) KK Re (E) f0(980) K K Just slope of the peak produced by f0(980) pole is seen. ππ unphysical & KK physical sheet ππ unphysical & KK unphysical sheet

Delta(1232) : The 1st P33 resonance Suzuki, Julia-Diaz, Kamano, Lee, Matsuyama, Sato, PRL104 065203 (2010) Re (T) Im (T) Complex E-plane Real energy axis “physical world” P33 pN physical & pD physical sheet Im (E) p N Re (E) pole 1211 , 50 BW 1232 , 118/2=59 In this case, BW mass & width can be a good approximation of the pole position. Small background Isolated pole Simple analytic structure of the complex E-plane pN unphysical & pD physical sheet p D 1211-50i pN unphysical & pD unphysical sheet Riemann-sheet for other channels: (hN,rN,sN) = (-, p, -)

Two-pole structure of the Roper P11(1440) Suzuki, Julia-Diaz, Kamano, Lee, Matsuyama, Sato, PRL104 065203 (2010) Re (T) Im (T) Complex E-plane Real energy axis “physical world” P11 pole A: pD unphys. sheet pole B: pD phys. sheet Two-pole structure of the Roper is found from several groups, on the basis of completely different approaches: pN physical & pD physical sheet Im (E) p N Re (E) Pole A cannot generate a resonance shape on “physical” real E axis. BW 1440 , 300/2 = 150 Two 1356 , 78 poles 1364 , 105 In this case, BW mass & width has NO clear relation with the resonance poles: ? pN unphysical & pD physical sheet p D pD branch point prevents pole B from generating a resonance shape on “physical” real E axis. A 1356-78i B 1364-105i pN unphysical & pD unphysical sheet Riemann-sheet for other channels: (hN,rN,sN) = (p,p,p)

Dynamical origin of nucleon resonances Suzuki, Julia-Diaz, Kamano, Lee, Matsuyama, Sato, PRL104 065203 (2010) Corresponds to hadrons of static hadron models Pole positions and dynamical origin of P11 resonances Eden, Taylor, Phys. Rev. 133 B1575 (1964) Multi-channel reactions can generate many resonance poles from a single bare state !! For evidences in hadron and nuclear physics, see e.g., in Morgan and Pennington, PRL59 2818 (1987)

Summary Analytic structure of the scattering amplitudes plays crucial role in the extraction of resonance parameters (mass, width, …) from the data. Non-trivial multichannel reaction dynamics in hadron spectroscopy Shape of cross sections depend not only on resonance poles but also the analytic structure of the amplitudes in the complex E-plane. Breit-Wigner formula sometimes (often?) fails to describe resonance parameters, particularly in the light hadron spectroscopy. Naïve one-to-one correspondence between physical resonances and static hadrons does not exist in general, regardless of the existence of molecule-like states. Reaction dynamics can produce sizable mass shift.

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N-N* transition form factors at resonance poles Extracted from analyzing the p(e,e’p)N data (~ 20000) from CLAS Nucleon - 1st D13 e.m. transition form factors Coupling to meson-baryon continuum states makes N* form factors complex !! Fundamental nature of resonant particles (decaying states) Real part Imaginary part Julia-Diaz, Kamano, Lee, Matsuyama, Sato, Suzuki PRC80 025207 (2009) Suzuki, Sato, Lee, PRC82 045206 (2010)

Meson cloud effect in gamma N  N* form factors GM(Q2) for g N  D (1232) transition N, N* Full Bare Note: Most of the available static hadron models give GM(Q2) close to “Bare” form factor.

g N  D(1232) form factors compared with Lattice QCD data ours

A clue how to connect with static hadron models g p  Roper e.m. transition “Static” form factor from DSE-model calculation. (C. Roberts et al) “Bare” form factor determined from our DCC analysis.