Compositeness of hadrons and near-threshold dynamics Tetsuo Hyodo

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1 Compositeness of hadrons and near-threshold dynamics Tetsuo Hyodo Yukawa Institute for Theoretical Physics, Kyoto Univ. 2015, May 26th 1

2 Contents Contents Introduction: compositeness of hadrons Near-threshold bound state S. Weinberg, Phys. Rev. 137, B672 (1965); T. Hyodo, Int. J. Mod. Phys. A 28, (2013) Near-threshold resonance T. Hyodo, Phys. Rev. Lett. 111, (2013) Mass scaling across threshold T. Hyodo, Phys. Rev. C90, (2014) Near-threshold quasi-bound state Y. Kamiya, T. Hyodo, in preparation Summary 2

3 Introduction: compositeness of hadrons Exotic structure of hadrons Various excitations of baryons conventional exotic energy internal excitation qq pair creation multiquark B M hadronic molecule Physical state: superposition of 3q, 5q, MB,... (1405) i = N 3q uds i + N 5q uds q q i + N KN KN i + Is this relevant strategy? 3

4 Introduction: compositeness of hadrons Ambiguity of definition of hadron structure Decomposition of hadron wave function (1405) i = N 3q uds i + N 5q uds q q i + N KN KN i + - NX probability? - 5q v.s. MB: double counting (orthogonality)? h udsq q KN i6=0-3q v.s. 5q: not clearly separated in QCD h uds udsq q i6=0 - hadron resonances: unstable, finite decay width (1405) i =? What is the suitable basis to classify the hadron structure? 4

5 Introduction: compositeness of hadrons Strategy Elementary/composite nature of bound states near the lowest energy two-body threshold elementary Z - 6q for deuteron - cc for X(3872) composite X - NN for deuteron - D D* for X(3872) - orthogonality < eigenstates of bare Hamiltonian - normalization < eigenstate of full Hamiltonian - model dependence < low energy universality * Basis must be asymptotic states (in QCD, hadrons). * Elementary stands for any states other than two-body composite (missing channels, CDD pole, ). 5

6 Near-threshold bound state Formulation Coupled-channel Hamiltonian (bare state + continuum) M 0 ˆV ˆV p 2 2µ! i = E i, i = c(e) 0 i E(p) p i - Bound state normalization + completeness relation Z h i =1 1= 0 ih 0 + d 3 q q ih q 1= h 2 Z 0 i + 0 d 3 q h 2 0 Z + X q i compositeness bare state contribution continuum contribution elementariness (field renormalization constant) Z, X: real and nonnegative > probabilistic interpretation 6

7 Near-threshold bound state Z(B) = 1 d de 1 R h 0 ˆV q i 2 E Z in model calculations In general, Z is determined by the potential V. q 2 /(2µ)+i0 + d 3 q E= B ( B) (E) - Z is model dependent (c.f. potential, wave function) Applications: Baryons Z Z Mesons Z Z Λ(1405) higher pole (Ref. 58) i 0.09 f 0 (500) or σ (Ref. 58) i 1.22 Λ(1405) lower pole (Ref. 58) i 0.95 f 0 (980) (Ref. 58) i 0.27 (1232) (Ref. 60) i 0.52 a 0 (980) (Ref. 58) i 0.70 Σ(1385) (Ref. 60) i 0.77 ρ(770) (Ref. 55) i 0.89 Ξ(1535) (Ref. 60) i 1.33 K (892) (Ref. 59) i 0.89 Ω (Ref. 60) Λ c (2595) (Ref. 56) i 1.17 for details, see T. Hyodo, Int. J. Mod. Phys. A 28, (2013) Z can be evaluated in specific models. 7

8 Near-threshold bound state Weak binding limit Z of weakly-bound (R Rtyp) s-wave state < observables. S. Weinberg, Phys. Rev. 137, B672 (1965); T. Hyodo, Int. J. Mod. Phys. A 28, (2013) a = 2(1 Z) 2 Z R + O(R typ), r e = Z 1 Z R + O(R typ), a : scattering length, re : effective range R = (2μB) -1/2 : radius < binding energy Rtyp : typical length scale of the interaction - Deuteron is found to be composite (Z ~ 0), without referring to the nuclear force/wave function. - Another derivation (expansion of the amplitude): T. Sekihara, T. Hyodo, D. Jido, arxiv: [hep-ph], to appear in PTEP 8

9 Near-threshold bound state Scaling limit Scaling (zero-range) limit: scattering length a 0, Rtyp -> 0 E. Braaten, H.-W. Hammer, Phys. Rept. 428, 259 (2006) - All (2-body) quantities are expressed by a: universality (r) = e r/a p 2 ar B =1/(2µa 2 ) ) R = a ) Z =0 - Bound state is always composite in the scaling limit. Finite Rtyp: Z expresses the violation of the scaling a = 2(1 Z) 2 Z R + O(R typ) r (r) r (r) Rtyp!0 / e r/a R typ / e r/r r V (r) 9

10 Near-threshold bound state Interpretation of negative effective range For Z > 0 and R Rtyp, effective range is always negative. a = 2(1 Z) 2 Z R + O(R typ), r e = Z 1 Z R + O(R typ), ( a R typ r e (elementary dominance), a R r e R typ (composite dominance). Simple (e.g. square-well) attractive potential: re > 0 - Only composite dominance is possible. re < 0 : energy- (momentum-)dependence of the potential D. Phillips, S. Beane, T.D. Cohen, Annals Phys. 264, 255 (1998); E. Braaten, M. Kusunoki, D. Zhang, Annals Phys. 323, 1770 (2008) - pole term/feshbach projection of coupled-channel effect Negative re > something other than p>: CDD pole 10

11 Near-threshold resonances Generalization to resonances Compositeness of bound states Z(B) = complex ( B) Naive generalization to resonances: T. Hyodo, D. Jido, A. Hosaka, Phys. Rev. C85, (2012) 1 Z(E R )= 1 0 (E R ) complex - Problem of interpretation (probability?) < Normalization of resonances h R R i! 1, h R R i = 1 Z 1=h R 0 ih 0 R i + dph R p ih p R i E R i R i complex h R 0 i = h 0 R i6= h 0 R i T. Berggren, Nucl. Phys. A 109, 265 (1968) R i 11

12 Near-threshold resonances Near-threshold resonances Weak binding limit for bound states - Model-independent (no potential, wavefunction,... ) - Related to experimental observables What about near-threshold resonances (~ small binding)? E shallow bound state: model-independent structure general bound state: model-dependent real Z general resonance: model-dependent complex Z 12

13 Near-threshold resonances Poles in the effective range expansion Near-threshold pole: effective range expansion T. Hyodo, Phys. Rev. Lett. 111, (2013) with opposite sign of scattering length 1 f(p) = a + r 1 e 2 p2 ip 1/a! +1 p p ± = i ± 1 r 2re r e r e a 1 bound state - pole trajectories with a fixed re < 0 1/r e 1/a! 1 virtual state 2/r e resonance Resonance pole position > (a, re) > elementariness 13

14 Near-threshold resonances Application: Λc(2595) Pole position of Λc(2595) in πσc scattering - central values in PDG E = 0.67 MeV, = 2.59 MeV p ± = p 2µ(E i /2) Λc(2595) πσc - deduced threshold parameters of πσc scattering a = p+ + p ip + p Z = i = 10.5 fm, r e = 2i p + + p = 19.5 fm - field renormalization constant: complex Large negative effective range < substantial elementary contribution other than πσc (three-quark, other meson-baryon channel, or... ) Λc(2595) is not likely a πσc composite 14

15 Mass scaling across threshold Hadron mass scaling and threshold effect Systematic expansion of hadron masses - ChPT: light quark mass mq Hadron mass scaling - HQET: heavy quark mass mq - large Nc: number of colors Nc What happens at two-body threshold? m H (x) resonance energy? bound state x 15

16 Mass scaling across threshold General threshold behavior Expansion of the Jost function (δm: small perturbation) T. Hyodo, Phys. Rev. C90, (2014) - δm < 0 E h / ( - δm > 0 M 2 l =0 M l 6= 0 E h / M 2 l =0 ( Re E h / M Im E h / ( M) l+1/2 l 6= 0 (a) bound state virtual state bound state resonance (b) slope: Z(0) Slope at Eh=0: field renormalization constant 1 E h = 1 0 (0) M = Z(0) M, 0 (E) d (E) de Z(0)=0 for s-wave > quadratic scaling 16

17 Mass scaling across threshold Chiral extrapolation across s-wave threshold s-wave: bound state > virtual state > resonance E h [ 2 /µ] l=0 E Re E Im E Bound Virtual Resonance M [ 2 /µ] Near-threshold scaling: nonperturbative phenomenon > Naive ChPT does not work; resummation required. c.f.) NN sector, K N sector, 17

18 Mass scaling across threshold Scaling of three-body bound state Near-threshold scaling is universal for 2-body system. - 3-body case? T. Hyodo, T. Hatsuda, Y. Nishida, Phys. Rev. C89, (2014) E 0.04 E (bound state) Re E (resonance) Im E (resonance) body break up threshold -1/a Efimov trimer (s-wave 3-body bound state) 18

19 Near-threshold quasi-bound state Generalization to quasi-bound state So far, we consider the lowest energy threshold. d NN Λc(2595) πσc bound state - Scattering length is real. scaling resonance Physically relevant situation: quasi-bound state channel 1: quasi-bound state: channel 2 (decay): K N Λ(1405) πσ D D* X(3872) ππψ - Scattering length of channel 1 is complex. - decomposition: 1= +X 1 + Z X 2 19

20 Near-threshold quasi-bound state Application: Λ(1405) Generalization of the formula Y. Kamiya, T. Hyodo, in preparation a a = R 1 2X1 1+X 1 + O( R typ R 1 )+O( l 1 R 1 3 ) R 1 =( 2µ 1 E) 1/2, l 1 =(2µ 1 ) 1/2 ν - Formula is valid for complex a, R1, X1. Example: Λ(1405) E [MeV] * a [fm] * X i i i0.11 * Y. Ikeda, T. Hyodo, W. Weise, Phys. Lett. B706, 63 (2011); Nucl. Phys. A881, 98 (2012) - consistent with the residue calculation: i0.01 T. Sekihara, T. Hyodo, D. Jido, arxiv: [hep-ph], to appear in PTEP Λ(1405) is dominated by the K N composite component. 20

21 Summary Summary Near-threshold states: structure < > observables Near-threshold resonance: T. Hyodo, Phys. Rev. Lett. 111, (2013) - Pole position determines (a, re). - Instead of complex Z, effective range serves as the measure of the elementariness. Mass scaling across threshold: T. Hyodo, Phys. Rev. C90, (2014) - Quadratic scaling in s wave < Z(0)=0 Near-threshold quasi-bound state: Y. Kamiya, T. Hyodo, in preparation - Generalized formula for complex numbers. 21