Complex 2D Matrix Model and Internal Structure of Resonances
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1 Complex 2D Matrix Model and Internal Structure of Resonances Kanabu Nawa (RIKEN) In collaboration with Sho Ozaki, Hideko Nagahiro, Daisuke Jido and Atsushi Hosaka [arxiv: [hep-ph]]
2 CONTENTS * Nature transition for real energy states complex energy states (resonace) * Complex 2D Matrix Model * Application to Hadron Physics
3 Introduction How do characters of quantum states change with variation of a parameter which specifies the property of the system or of the environment where the system is placed? deformed nuclei diatomic molecule solar neutrino deformation parameter of nuclear MF potential d inter-nuclear distance R sun distance from the center of the sun : Hamiltonian with
4 Nature Transition Important concept to classify the quantum states with variation of a parameter. Hamilton Matrix : Hamiltonian with., : eigenstates at. E Character exchange with variation of l. 0 Anticrossing l Nature Transition classification of quantum states with variation of.
![Single-particle energy [MeV] Nature Transition in](/docs-images/90/104153930/images/5-3.jpg "Nuclear Physics Nilsson diagram of deformed shell")
5 Single-particle energy [MeV] Nature Transition in Nuclear Physics Nilsson diagram of deformed shell model Anisotoropic-Harmonic Oscillator [Nilsson diagram for neutrons, 50 < N < 82] deformation parameter of the mean field potential Nature transition is important to know the structure of nuclei at each energy level!!!
6 Nature Transition for Resonance States [G. Gamow, Z. Phys. 51, 204 (1928); 52, 510 (1928)] Matrix elements become complex:
7 Nature Transition for Resonance States [G. Gamow, Z. Phys. 51, 204 (1928); 52, 510 (1928)] Matrix elements become complex: E Real Energy States 0 ImE Complex Energy States ReE Anticrossing * * 0 l What is the simple criterion of nature transition for resonances? How do poles move two-dimensionally on complex-e plane?
8 History of Resonance States 1920~1930 How to describe the α-decay of nuclei by quantum mechanics? 1928 Resonance as a complex energy state (Gamow state) : 1965 Biorthogonality for resonance states ket state : bra state :, [G. Gamow, Z. Phys. 51, 204 (1928); 52, 510 (1928)] [N. Hokkyo, Prog. Theor. Phys. 33, 1116 (1965)] 1968 Resonance shell model (without completeness) 1968 Extended completeness with bound/resonance/continuum by modified contour integral 1971 Complex Scaling Method (CSM) [W. Romo, Nucl. Phys. A116, 618 (1968)] [T. Berggren, Nucl. Phys. A109, 265 (1968)] [ABC theorem (1971)] Extended completeness with continuum effect controlled by a scaling paramete θ 1970 s~ Application of CSM to chemical physics with coulomb potential [N. Moiseyev, Phys. Rept. 302, 211 (1998)]
9 History of Resonance States 1920~1930 How to describe the α-decay of nuclei by quantum mechanics? 1928 Resonance as a complex energy state (Gamow state) : 1965 Biorthogonality for resonance states ket state : bra state :, [G. Gamow, Z. Phys. 51, 204 (1928); 52, 510 (1928)] [N. Hokkyo, Prog. Theor. Phys. 33, 1116 (1965)] 1968 Resonance shell model (without completeness) 1968 Extended completeness with bound/resonance/continuum by modified contour integral 1971 Complex Scaling Method (CSM) [W. Romo, Nucl. Phys. A116, 618 (1968)] [T. Berggren, Nucl. Phys. A109, 265 (1968)] [ABC theorem (1971)] Extended completeness with continuum effect controlled by a scaling paramete θ 1970 s~ Application of CSM to chemical physics with coulomb potential [N. Moiseyev, Phys. Rept. 302, 211 (1998)]
10 Biorthogonal Representation of Resonance State Biorthogonal basis : [N. Hokkyo, Prog. Theor. Phys. 33, 1116 (1965)] [W. Romo, Nucl. Phys. A116, 618 (1968)] [T. Berggren, Nucl. Phys. A109, 265 (1968)] ket state : bra state :, Dirac ket state C.C. of Dirac bra state Biorthogonality : for Scalar metric product : Complex norm: Complex probability implies that resonance is the object beyond the quantum mechanics.
11 Complex 2D Matrix Model : Hamiltonian with. [K.N., et.al., arxiv: [hep-ph]], : eigenstates at in biorthogonal representation., Hamilton Matrix [N. Hokkyo, Prog. Theor. Phys. 33, 1116 (1965)] [T. Berggren, Nucl. Phys. A109, 265 (1968)],
12 Hamilton Matrix Complex 2D Matrix Model Resonance eigenstates: Complex norm: Informations about internal structure of eigenstates Assumption can still be a guide of probability. (suitable for narrow resonances)
13 Hamilton Matrix Complex 2D Matrix Model Condition of nature transition : Nature transition with character exchange:
14 Geometry on Complex-l Plane Iml exceptional points [T.Kato, 1966] 0 Rel end points transition line [K.N., et.al., 2011] Geometry with exceptional points and transition lines on complex-l plane is important to judge the existence of nature transition in the real parameter subspace :.
15 Exceptional Points [T. Kato, Perturbation Theory of Linear Operator, 1966], Energy eigenvalues : energy gap (1dim.) (Neumann-Wigner non-crossing rule (1929)) (2dim.) (isolated degenerating point = exceptional point ) Dense exceptional points on complex-l plane can be the criterion for the development of quantum chaos in the energy level statistics!!! [W.D.Heiss and A.L.Sannino, PRA43,4159(1991); W.D.Heiss, PRE61, 929 (2000)]
16 Geometry on Complex-l Plane Iml exceptional points [T.Kato, 1966] 0 Rel end points transition line [K.N., et.al., 2011] Geometry with exceptional points and transition lines on complex-l plane is important to judge the existence of nature transition in the real parameter subspace :.
17 Linear-l Model [K.N. et.al., 2011] 2 exceptional points : 1 transition line (arc shape) Iml, 0 Rel
18 Exceptional Points with Pole Behavior Iml Iml [W.D.Heiss, PRE61, 929 (2000)] 0 Rel 0 Rel ImE ReE ImE ReE 0 0 Level repulsion/ Width crossing Level crossing/ Width repulsion
19 Exceptional Points with Pole Behavior Iml Iml [W.D.Heiss, PRE61, 929 (2000)] 0 Rel 0 Rel ImE 0 ReE ImE ReE 0 Exceptional points can be the end points of transition line. [K.N. et.al., 2011] Nature transition occurs only for level repulsion/width crossing case!!!
20 Iml Geometrical Maps Structure around l~1 is universal. H 221 H 233 H 151 Rel Rel Rel
21 Application of Complex 2D Matrix Model to Hadron Physics * QCD is SU(Nc) gauge theory with Nc = 3. [Y. Nambu, 1966] Fundamental theory of the strong interaction with quarks and gluons. (QCD becomes strong coupling at low-energy scale.) * Large-Nc QCD with Nc = gets lots of phenomenological successes. [G. thooft, NPB72,461(1974);B75,461(1974); E. Witten, NPB160,57(1979)]
22 Mesons in Large-Nc QCD quark q q antiquark q q 1/Nc q q [E. Witten, NPB160,57(1979)] * Internal structure of hadrons should depend on Nc. It can drastically change due to the development of hadron dynamics scaled by 1/Nc.
23 Mesons in Large-Nc QCD quark q q antiquark q q 1/Nc q q How does the internal structure of hadrons change from Nc= to Nc=3? * 1/Nc ~ l Internal structure of hadrons can drastically changes at the critical color number of nature transitions. [K.N., et.al., arxiv: [hep-ph]]
24 1/4 Im [(3/Nc) ] Geometrical Map for a1 Meson a 1 q q q p q a 1 1/Nc r q q ~ ~ Nc=3 Nc~4.0 1/4 1/4 Re [(3/Nc) ] Re [(3/Nc) ]
25 Summary We formulate Complex 2D Matrix Model to discuss the parameter dependence of the internal structure of resonances. * Nature transition with character exchange occurs at a critical value l t, which can be estimated from the geometrical map on complex-l plane. * Pole behavior on complex-e plane and their internal structure can be successfully related. * Internal structure of hadrons can drastically change at a critical color number of nature traisition. [arxiv: [hep-ph]]
26 Adiabatic Transition in Astrophysics Less detection of solar neutrino? [R.Davis, 1968] Solar Neutrino Puzzle Sun Earth
![Davis, 1968] Solar Neutrino Puzzle n e Sun Mass](/docs-images/90/104153930/images/27-2.jpg "enhancement with electron density Earth l ~ R")
![Sakata, 1962] R Adiabatic transition can solve the](/docs-images/90/104153930/images/27-4.jpg "solar neutrino puzzle as the conversion of the")
27 Adiabatic Transition in Astrophysics Less detection of solar neutrino? [R.Davis, 1968] Solar Neutrino Puzzle n e Sun Mass enhancement with electron density Earth l ~ R distance from the center of the Sun n m Neutrino Mixing [S.Sakata, 1962] R Adiabatic transition can solve the solar neutrino puzzle as the conversion of the neutrinos within the Sun!!! Finite mass of neutrino. [H.A.Bethe, PRL56,1305(1986)]
28 Resonance Shell Model [W. Romo, Nucl. Phys. A116, 618 (1968)] a nucleon interacting with a nucleus composed by A-nucleons Kinetic energy Hamiltonian of nucleus Solving the effective Hamiltonian in MF basis Important bound/resonance states of 1st point Diagonalization of effective Hamiltonian over finite set of shell-model states, including the effects of neglected states in the effective interaction. 2nd point One may take the traditional approximation of shell-model as Absorptive effect can be represented by
29 G-function Formalism for a1 Meson a 1 q q q p q a 1 1/Nc r q q ~ ~ = p r a1-1 - Sakai-Sugimoto Model (Gauge/Gravity dual) (1189MeV)
30 G-function Formalism for a1 Meson p r a1-1 - Sakai-Sugimoto Model (Gauge/Gravity dual) (1189MeV) = a1 Pole approximation with non-perturbative pr dynamics: p r = : pr molecule state
![] Non-rela. Approx. with energy fixing.](/docs-images/90/104153930/images/31-1.jpg "Resonace Shell Model [Romo, 1968]")
31 Geometrical Map on Complex-Nc Plane (large Nc, 1189MeV) [Relativistic eigenvalue Eq.] Non-rela. Approx. with energy fixing. Resonace Shell Model [Romo, 1968] [Schrodinger Eq.]
32 Geometrical Map on Complex-Nc Plane thooft-witten: Our estimates: Complex 2D Matrix Model
![Im[(3/Nc) 1/4 ] Geometrical Map on Complex-Nc Plane Nc=3](/docs-images/90/104153930/images/33-4.jpg "1/4 Re [(3/Nc) ] 1/4 (3/Nc) ~0.93 1/4 Re [(3/Nc) ] Nc~4.")
33 Im[(3/Nc) 1/4 ] Geometrical Map on Complex-Nc Plane Nc=3 1/4 Re [(3/Nc) ] 1/4 (3/Nc) ~0.93 1/4 Re [(3/Nc) ] Nc~4.0
34 Linear-l Model [K.N. et.al., 2011] Iml R 0 Rel There is no nature transition for finite l. (Mixing of basis component up to 50% at most.)
35 Application to L(1405) K N l p S ImE [MeV] Level repulsion/ Width crossing? [T. Hyodo and W. Weise, Phys. Rev. C 77, (2008)] l controls only the mixing between two levels. ReE [MeV] (Transition Arc with infinite radius) There is no nature transition for arbitrary finite value of l!!!
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