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

2. 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”

3. 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/
(complex) energy eigenvalues = pole values
transition matrix elements = (residue)1/2 of the poles
Decay vertex
pole position
( Im(E0) =< 0 )

4. 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.

5. f0(980) in pi-pi scatteringπ
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)

6. 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)

7. 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

8. Delta(1232) : The 1st P33 resonance
Suzuki, Julia-Diaz, Kamano, Lee, Matsuyama, Sato, PRL (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 ,
BW , 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 i
pN unphysical & pD unphysical sheet
Riemann-sheet for other channels: (hN,rN,sN) = (-, p, -)

9. Two-pole structure of the Roper P11(1440)
Suzuki, Julia-Diaz, Kamano, Lee, Matsuyama, Sato, PRL (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 , 300/2 = 150
Two , 78
poles , 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 i B i
pN unphysical & pD unphysical sheet
Riemann-sheet for other channels: (hN,rN,sN) = (p,p,p)

10. Dynamical origin of nucleon resonances
Suzuki, Julia-Diaz, Kamano, Lee, Matsuyama, Sato, PRL (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, PRL (1987)

11. 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.

12. back up

13. 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 PRC (2009)
Suzuki, Sato, Lee, PRC (2010)

14. 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.

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

16. 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.