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. 2014, Nov. 11th 1

2 Announcement Announcement Dates: Feb - Mar 2015 (5 weeks)! Theme: EFT and lattice QCD! Abstract/support: by Nov. 30! Details: google hhiqcd 2

3 Contents Contents Introduction: structure of hadrons! Compositeness of hadrons and near-threshold bound states! S. Weinberg, Phys. Rev. 137, B672 (1965) T. Hyodo, Int. J. Mod. Phys. A 28, (2013) Near-threshold resonances! T. Hyodo, Phys. Rev. Lett. 111, (2013) Example: Λc(2595) (heavy!) Near-threshold mass scaling T. Hyodo, arxiv: [hep-ph], to appear in Phys. Rev. C 3

4 Introduction: structure 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? 4

5 Introduction: structure 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 + - 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 =? How can we define the hadron structure? What is the suitable basis to classify the hadron structure? 5

6 Introduction: structure of hadrons Strategy Elementary or composite in terms of hadronic d.o.f., focusing on states near the lowest energy two-body threshold elementary - 6q for deuteron - cc for X(3872) composite - NN for deuteron - D D* for X(3872) - orthogonality < completeness relation - normalization < wave function normalization - model dependence < low energy universality * Basis must be asymptotic states (in QCD, hadrons). * Elementary stands for any states other than two-body composite (CDD pole). Compact quark states, three-body, 6

7 Compositeness of hadrons and near-threshold bound state Formulation Coupled-channel Hamiltonian (bare state + continuum) M 0 ˆV! ˆV p 2 2µ (+ ˆV i = E i, i = sc ) c(e) 0 i E(p) p i Elementariness by field renormalization constant - 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 bare state! contribution d 3 q h 2 0 Z + X q i continuum! contribution Z, X : real and nonnegative --> probabilistic interpretation 7

8 Compositeness of hadrons and near-threshold bound state Z(B) = 1 d de 1 R h 0 ˆV q i 2 E Weak binding limit In general, Z is determined by the potential V. q 2 /(2µ)+i0 + d 3 q E= B ( B) (E) In weak binding limit (R Rtyp), Z is related to 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 Criterion for the structure: ( a R typ r e (elementary dominance), a R r e R typ (composite dominance). Z ~ 1 Z ~ 0 (deuteron) 8

9 Compositeness of hadrons and near-threshold bound state Interpretation of negative effective range For Z > 0, 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 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 9

10 Compositeness of hadrons and near-threshold bound state Exact B -> 0 limit: Compositeness theorem If the s-wave scattering amplitude has a pole exactly at the threshold with a finite range interaction, then the field renormalization constant vanishes. T. Hyodo, arxiv: [hep-ph], to appear in Phys. Rev. C For bare state-continuum model (c: nonzero constant) Im (p 2 /2µ) / p 2l+1 Z(0) vanishes for g0 0. If g0=0, no pole in the amplitude. For a local potential: poles in the effective range expansion If Z(0) 0, then both p1 and p2 go to zero for B -> 0! : contradict with simple pole at p=0 R.G. Newton, J. Math. Phys. 1, 319 (1960) 10

11 Compositeness of hadrons and near-threshold bound state Interpretation of the compositeness theorem Z(B): overlap of the bound state with bare state - Z(B 0)=0 > Bound state is completely composite. Two-body wave function at E=0: u l,e=0 (r) r!1! r l u 0,E=0 (r) u l6=0,e=0 (r) r / r l r Z(0)=0: Bound state is completely composite.! Composite component is infinitely large so that the! fraction of any finite admixture of bare state is zero. 11

12 Near-threshold resonances Generalization to resonances Compositeness of bound states Z(B) = ( B) Naive generalization to resonances: T. Hyodo, D. Jido, A. Hosaka, Phys. Rev. C85, (2012) 1 Z(E R )= 1 0 ( E R ) complex - interpretation? h R R i! 1, h R R i = 1 Z 1 = h R B 0 ih B 0 R i + dph R p ih p R i complex complex Normalization of resonances: Gamow vector h R B 0 i = h B 0 R i 6= h B 0 R i R i R i R i E 12

13 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 Z general resonance:! model-dependent Z 13

14 Near-threshold resonances Poles in the effective range expansion Near-threshold phenomena: effective range expansion T. Hyodo, Phys. Rev. Lett. 111, (2013) with opposite sign of scattering length f(p) = 1 a + r2 e 2 p2 ip p ± = i ± 1 r 2re r e r e a 1 1 1/a! +1 bound! state p - pole trajectories! with a fixed re < 0 virtual! state 1/r e 2/r e 1/a! 1 resonance Resonance pole position <--> (a, re) 14

15 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) - deduced threshold parameters of πσc scattering a = p+ + p ip + p = 10.5 fm, r e = 2i p + + p = 19.5 fm - field renormalization constant: complex Z = i Large negative effective range < substantial elementary contribution other than πσc! (three-quark, other meson-baryon channel, or... ) Λc(2595) is not likely a πσc molecule 15

16 Near-threshold mass scaling 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 16

17 Near-threshold mass scaling Coupled-channel Hamiltonian (bare state + continuum) M 0 ˆV Formulation! ˆV p 2 2µ (+ ˆV i = E i, i = sc ) c(e) 0 i E(p) p i Equivalent single-channel scattering formulation Pole condition: Question: How Eh behaves against M0 around Eh=0? 17

18 Near-threshold mass scaling Near-threshold bound state Bound state condition around Eh=0 E h + (0) M = (E h ) E M Leading contribution of the expansion: 1 E h = 1 0 (0) M = Z(0) M, 0 (E) d (E) de E h Field renormalization constant (0) M 0 Z(0) vanishes for l=0: compositeness theorem 18

19 Near-threshold mass scaling Near-threshold bound state (general) General argument by Jost function (Fredholm determinant) J.R. Taylor, Scattering Theory (Wiley, New York, 1972) f l (p) = f l ( p) f l (p) 2ipf l (p) Expansion of the Jost function: f l (p) = ( i 0 p + O(p 2 ) l =0 1+ l + l p 2 + O(p 3 ) l 6= 0 - γ0 and βl are nonzero for a general local potential! - zero at p=0 (1+αl=0) must be simple (double) for l=0 (l 0) R.G. Newton, J. Math. Phys. 1, 319 (1960) Near-threshold scaling: 1+ l M ) E h / ( pole (eigenstate) = Jost function zero M 2 l =0 M l 6= 0 ( M<0) E h l=0 M l 0 M 0 19

20 Near-threshold mass scaling General threshold behavior Near threshold scaling: - 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 Numerical calculation (a) bound state virtual state bound state resonance c.f. NN 1 S0 (a) l=0 (b) slope: Z(0) (b) l=1 E h [ 2 /µ] E (bound state) E (virtual state) M [ 2 /µ] E h [ 2 /µ] E (bound state) Re E (resonance) Im E (resonance) M [ 2 /µ] 20

21 Near-threshold mass scaling Chiral extrapolation across s-wave threshold Scaling in wider energy region 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 is needed.! c.f.) NN sector, K N sector, 21

22 Summary Summary Compositeness of hadrons near threshold T. Hyodo, Int. J. Mod. Phys. A 28, (2013) T. Hyodo, Phys. Rev. Lett. 111, (2013) T. Hyodo, arxiv: [hep-ph], to appear in Phys. Rev. C Compositeness / elementariness! - suitable classification for hadron structure! - model independent in the weak binding limit Near-threshold resonance:! - structure from effective range Near-threshold mass scaling: - caution on the chiral extrapolation 22