Structure of near-threshold s-wave resonances Tetsuo Hyodo Yukawa Institute for Theoretical Physics, Kyoto 203, Sep. 0th
Introduction Structure of hadron excited states Various excitations of baryons M B energy internal excitation qq pair creation multiquark hadronic molecule Quark model What are 3q state, 5q state, MB state,...? - Comparison of data (spectrum, width,...) with quark models - Analysis of scattering data by dynamical models Clear (model-independent) definition of the structure? 2
Introduction Difficulty : definition and model space Number of quarks + antiquarks ( quark number)? (405) = + +... This may not be a good classification scheme. Number of hadrons (405) = + +... Hadrons are asymptotic states. --> different kinematical structure C. Hanhart, Eur. Phys. J. A 35, 27 (2008) --> compositeness in terms of hadronic degrees of freedom 3
Introduction Difficulty 2 : resonances Excited states : finite width (unstable against strong decay) - stable (ground) states - unstable states Mostly resonances! Wave function of resonance? PDG2 (405) =? + +... --> First consider stable states, then extend it to resonances. 4
Contents Contents Introduction: ideal strategy Model independent approach Hadronic degrees of freedom Extension to resonances Field renormalization constant Z S. Weinberg, Phys. Rev. 37, B672 (965) Application to near-threshold resonances T. Hyodo, arxiv:305.999 [hep-ph] Summary 5
Field renormalization constant Z Compositeness of the deuteron Z: probability of finding deuteron in a bare elementary state S. Weinberg, Phys. Rev. 37, B672 (965) deuteron = N N or NN model space ~ elementary particle Z = 0 Z = Model-independent relation for a shallow bound state a s = 2( Z) 2 Z R + O(m ), r e = Z Z R + O(m ) as ~ 5.4 [fm] : scattering length re ~.75 [fm] : effective range R ~ (2μB) -/2 ~ 4.3 [fm] : deuteron radius (binding energy) --> Z 0.2 : Deuteron is almost composite! 6
Field renormalization constant Z Compositeness in quantum mechanics Hamiltonian of a single channel scattering system H = H 0 + V Complete set for free Hamiltonian: bare B0 > + continuum = B 0 B 0 + dk k k Physical bound state B> with binding energy B (H 0 + V ) B = B B free Z : overlap of B and B0 Z B 0 B 2 energy full 0 apple Z apple For small B, Z is related to observables a = apple 2( Z) 2 Z R, r e = apple Z Z R B0 V B 7
Application to near-threshold resonances Application to resonances Features of the Weinberg s argument: - Model-independent approach (no potential, wave-fn,... ) - Relation with experimental observables - Only for bound states with small binding Application to resonances by analytic continuation Z = Z dq hq V Bi 2 dg(w ) g2 [E(q) + B] 2 dw W =MB T. Hyodo, D. Jido, A. Hosaka, Phys. Rev. C85, 0520 (202) F. Aceti, E. Oset, Phys. Rev. D86, 0402 (202) - Z can be complex. Interpretation? - Z can be larger than unity. Normalization? What about near-threshold resonances (~ small binding)? 8
Application to near-threshold resonances Effective range expansion S-wave scattering amplitude at low momentum f(k) = k cot ki! a ki + r e 2 k2 Truncation is valid only at small k. Scattering length a - strength of the interaction - cross section at zero momentum : 4πa 2 Effective range re - typical length scale of the interaction - can be negative D. Phillips, S. Beane, T.D. Cohen, Annals Phys. 264, 255 (998) E. Braaten, M. Kusunoki, D. Zhang, Annals Phys. 323, 770 (2008) 9
Application to near-threshold resonances The amplitude has two poles f(k) = a k ± = i r e ± r e ki + r e r 2re a Poles of the amplitude 2 k2 bound state (I) a! k Pole trajectories with a fixed re < 0 virtual state (II) /r e 2/r e resonance (II) a! + Positions of poles <--> scattering length + effective range a = k+ + k ik + k, r e = 2i k + + k (a,re) are real for resonances 0
Application to near-threshold resonances Eliminate R from the Weinberg s relations Z = s Field renormalization constant + a/(2r e ) = 2k k k + (a) k (b) Z /r e 2/r e Z (residue) is determined by the pole position <-- Amplitude is given by two parameters. -Z is pure imaginary and 0 -Z
Application to near-threshold resonances Validity of the effective range expansion A model calculation - solid lines: pole position in a scattering model - dashed lines: position deduced from (a,re) Im z R [MeV] 0-20 -40-60 C = Exact (I sheet) Effective (I sheet) Exact (II sheet) Effective (II sheet) 2560 2580 2600 Re z R [MeV] 2620 (b) 2640 Im z R [MeV] 0-20 -40-60 C = 4 Exact (I sheet) Effective (I sheet) Exact (II sheet) Effective (II sheet) 2560 2580 2600 Re z R [MeV] 2620 (a) 2640 large re small re If the effective range is large, the expansion works well. 2
Application to near-threshold resonances Example: Λc(2595) Pole position of Λc(2595) with πσc threshold in PDG E [MeV] Γ [MeV] a [fm] re [fm] 0.67 2.59 0.5-9.5 - Isospin symmetry is assumed. - ππλ channel is not taken into account. -Z ~ 0.6 Interpretation? Larger effective range than typical hadronic scale Chiral interaction gives re ~ -4.6 fm --> Λc(2595) is not likely a πσc molecule 3
Summary Summary Near-threshold s-wave resonances Effective range expansion : pole position <--> observables (a, re) Compositeness -Z : pure imaginary and normalized Application to Λc(2595) Large re --> not likely a molecule T. Hyodo, arxiv:305.999 [hep-ph] 4