Lattice QCD analysis of charmed tetraquark candidates Yoichi Ikeda - PowerPoint PPT Presentation

Lattice QCD analysis of charmed tetraquark candidates Yoichi Ikeda (RCNP , Osaka University) HAL QCD (Hadrons to Atomic nuclei from Lattice QCD) S. Aoki, T. Aoyama, Y. Akahoshi, K. Sasaki, T. Miyamoto (YITP , Kyoto Univ.) T. Doi, T. M. Doi, S. Gongyo, T. Hatsuda, T. Iritani (RIKEN) Y. Ikeda, N. Ishii, K. Murano, H. Nemura (RCNP , Osaka Univ.) T. Inoue (Nihon Univ.) International Molecule-type Workshop “Frontiers in Lattice QCD and related topics” April 15 - April 26 2019, Yukawa Institute for Theoretical Physics, Kyoto University

Single hadron spectroscopy from LQCD ★ Low-lying hadrons on physical point (physical m q ) charm baryons light-quark sector ‣ N f =2+1 full QCD, L~3fm ‣ N f =2+1 full QCD, L~3fm ‣ RHQ for charm quark <-- E ππ (k) Aoki et al. (PACS-CS), PRD81 (2010). Namekawa et al. (PACS-CS), PRD84 (2011); PRD87 (2013). ✓ a few % accuracy already achieved for single hadrons ✓ LQCD now can predict undiscovered charm hadrons ( Ξ ( * )cc , Ω ccc ,...) ➡ Next challenge in spectroscopy : hadron resonances

Our target Z c (3900) Hadron resonances • Particle data group http://www-pdg.lbl.gov/ • Most hadrons are consistent with qqq / qq bar quantum number (non-trivial) • Only 10% is stable, others are unstable (resonances) and some can be fake.. • Understanding hadron resonances from QCD is important issue in hadron physics

Tetraquark candidate Z c (3900) e + π Y(4260) • Expt. observations π M π J/ ψ Z c (3900) J/ ψ e - Y(4260) 3-body decay ‣ e + + e - --> Y(4260) --> π + Z c (3900) ➡ π +/- + J/ ψ BESIII Coll., PRL110 (2013). see also Belle Coll., PRL110 (2013). u ? c ( η c ρ ) ( πψ 0 ) ¯ ¯ c d ¯ π J/ ψ DD ∗ • peak in π +/- J/ ψ invariant mass (minimal quark content cc bar ud bar <--> tetraquark?) • M ~ 3900, Γ ~ 60 MeV (Breit-Wigner, Flatte) --> just above D bar D* threshold • J PC =1 +- is most probable <--> couple to s-wave meson-meson states

Tetraquark candidate Z c (3900) ★ structure of Z c (3900) studied by models genuine resonance D bar D* molecule tetraquark J/ ψ + π atom u u c c c ¯ c ¯ ¯ ¯ d ¯ c d c Nieves et al.(‘11) + many others Maiani et al.(‘13) Voloshin(‘08) conclusion not achieved kinematical origin ➡ poor information on interactions D bar D* threshold e ff ect ★ LQCD simulations for Z c (3900) u ¯ c c ¯ d Chen et al.(‘14), Swanson(‘15)

Z c (3900) on the lattice ✦ Conventional approach: temporal correlation ➡ identify all relevant W n (L) (n=0,1,2,3,...) cu ¯ cu ¯ X d ] † (0) | 0 i = A n e − W n t h 0 | [ c ¯ d ]( t )[ c ¯ variational method u c ¯ ¯ c d D bar D* ✓ No positive evidence for Z c (3900) in J PC =1 +- (observed spectrum consistent with scat. states) π J/ ψ S. Prelovsek et al., PLB 727 (2013), PRD91 (2015). S.-H. Lee et al., PoS Lattice2014 (2014). ★ Why is the peak observed in expt.? ‣ (broad) resonance? threshold e ff ect? ★ How can we find resonance in LQCD data?

Strategy for studies of resonances from LQCD Conventional approach lattice QCD h 0 | Φ ( x ) Φ † (0) | 0 i = A 1 e − W 1 τ + A 2 e − W 2 τ + · · · (W 1 , W 2 , ... are eigen-energies) e.g., 4-quark operator Φ ( x ) = ¯ q ( x )¯ q ( x ) q ( x ) q ( x ) hadron scattering many thresholds σ ( W ) W hadron resonances ★ Resonance energy does NOT correspond to eigen-energy ★ Resonances are embedded into coupled-channel scattering states ➡ Resonance energy is determined from pole of coupled-channel S-matrix

Strategy for studies of resonances from LQCD ( Resonance search through scattering observable ) lattice QCD hadron interactions (faithful to S-matrix) u ¯ c scattering theory c ¯ d contents • hadron interactions & HAL QCD method • strategy to find resonance pole hadron resonances • coupled-channel scattering • LQCD results about Z c (3900) • summary

Hadronic interactions from LQCD hadronic correlation function r, t ) � 2 ( ~ 0 , t ) J † ( t = 0) | 0 i C (2) ( ~ r, t ) ⌘ h 0 | � 1 ( ~ X X r ) e − W n t = A n � n ( ⇥ ~ x n X t • Energy eigenvalue W n (L) (outside interactions) • NBS (Nambu-Bethe-Salpeter) wave function ψ n (r) [ ψ n (r) --> sin( k n r + δ (k n )) / k n r ] C.D. Lee et al., NPB619 (2001). Finite Volume Method Lüscher’s finite volume formula ‣ W n (L) --------> phase shift k n cot � ( k n ) = 4 ⇡ 1 X Lüscher’s formula L 3 p 2 m − k n 2 ~ Lüscher, Nucl. Phys. B354, 531 (1991). m ∈ Z 3 q q m 2 m 2 1 + k 2 2 + k 2 W n = n + n

Hadronic interactions from LQCD hadronic correlation function r, t ) � 2 ( ~ 0 , t ) J † ( t = 0) | 0 i C (2) ( ~ r, t ) ⌘ h 0 | � 1 ( ~ X X r ) e − W n t = A n � n ( ⇥ ~ x n X t • Energy eigenvalue W n (L) (outside interactions) • NBS (Nambu-Bethe-Salpeter) wave function ψ n (r) [ ψ n (r) --> sin( k n r + δ (k n )) / k n r ] HAL QCD Method Finite Volume Method ‣ ψ n (r) --> 2PI kernel ( ψ = φ + G 0 U ψ ) ‣ W n (L) --------> phase shift --> phase shift, binding energy, ... Lüscher’s formula Lüscher, Nucl. Phys. B354, 531 (1991). Ishii, Aoki, Hatsuda, PRL 99, 022001 (2007). Ishii et al. [HAL QCD], PLB 712, 437 (2012). ➡ di ffi cult with coupled-channel problems

Challenge in hadron scatterings ★ Excited scattering states become noise when determining W 0 even in single-channel scatterings C (2) ( t ) = b 0 e − W 0 t + b 1 e − W 1 t + · · · → b 0 e − W 0 t ( t > t ∗ ) − single hadron 2-body system D + D bar ψ ’(2S) D bar + D* Δ ~ 500 MeV J/ ψ + π δ E J/ ψ (1S) t* ~ Δ -1 t* ~ δ E -1 ★ Sophisticated methods is necessary! talk by T. Doi (Thu.) see for BB systems, Iritani, Doi et al. [HAL QCD], JHEP10 (2016) 101.

(single-channel) HAL QCD method -- potential as a representation of S-matrix -- • The scattering states do exist, and we should tame the scattering states ➡ time-dependent HAL QCD method Ishii [HAL QCD], PLB 712 (2012). ✓ define energy-independent potential U(r,r’) Z r 0 U ( ~ r 0 ) n ( ~ r 0 ) = ( E n − H 0 ) n ( ~ r ) d ~ r, ~ c i t s a l e r ) ¯ n X r 0 ) ≡ r 0 ) U ( ~ r, ~ ( E n − H 0 ) n ( ~ n ( ~ i n<n th ch2 ➡ All elastic states share the same potential U(r,r’) c i t s U ψ 0 = (E 0 -H 0 ) ψ 0 a l e U ψ 1 = (E 1 -H 0 ) ψ 1 ch1 · · · ✓ derive U(r,r’) from time-dependent Schrödinger-type eq. @ 2 1 ✓ − @ ◆ Z r 0 U ( ~ r 0 ) R ( ~ r 0 , t ) = @ t + R ( ~ r, t ) d ~ r, ~ @ t 2 − H 0 4 m � 2 � R ( ~ r, t ) = C (2) ( ~ r, t ) / C (1) ( t ) r ) e − ( W 0 − 2 m ) t + b 1 1 ( ~ r ) e − ( W 1 − 2 m ) t + · · · = b 0 0 ( ~

(single-channel) HAL QCD method -- potential as a representation of S-matrix -- • The scattering states do exist, and we should tame the scattering states ➡ time-dependent HAL QCD method Ishii [HAL QCD], PLB 712 (2012). ✓ define energy-independent potential U(r,r’) Z r 0 U ( ~ r 0 ) n ( ~ r 0 ) = ( E n − H 0 ) n ( ~ r ) d ~ r, ~ c i t s a l e r ) ¯ n X r 0 ) ≡ r 0 ) U ( ~ r, ~ ( E n − H 0 ) n ( ~ n ( ~ i n<n th ch2 ➡ All elastic states share the same potential U(r,r’) c i t s U ψ 0 = (E 0 -H 0 ) ψ 0 a l e U ψ 1 = (E 1 -H 0 ) ψ 1 ch1 · · · ✓ derive U(r,r’) from time-dependent Schrödinger-type eq. @ 2 1 ✓ − @ ◆ Z r 0 U ( ~ r 0 ) R ( ~ r 0 , t ) = @ t + R ( ~ r, t ) d ~ r, ~ @ t 2 − H 0 4 m r ) e − ( W 0 − 2 m ) t + b 1 1 ( ~ r ) e − ( W 1 − 2 m ) t + · · · R ( ~ r, t ) = b 0 0 ( ~ ➡ Scat. states are no more contamination than signal ( t* ~ (E ch2 - E ch1 ) -1 )

How can we find resonances? reaction plane If we have complete set of expt. data, S ( ` ) ( W ) partial wave analysis ‣ cross sections (d σ /d Ω ) ‣ spin polarization observables ‣ etc. identity theorem Pole of S-matrix is uniquely determined + analyticity of S-matrix bound state (1st sheet) Im[ z ] ‣ pole position --> binding energy 1st sheet ‣ residue --> coupling to scattering state Re[ z ] resonance (2nd sheet) bound ‣ analytic continuation onto 2nd sheet ‣ pole position --> resonance energy resonance ‣ residue --> coupling to scat. state, partial decay 2nd sheet

Strategy to search for complex poles on the lattice Resonance pole from lattice QCD r, t ) � 2 ( ~ 0 , t ) J † ( t = 0) | 0 i S ( ` ) ( W ) h 0 | � 1 ( ~ X ~ x X r ) e − W n t = A n n ( ~ X n t ❖ coupled-channel Lüscher’s formula ➡ W n (L) --> δ 1 (W n ), δ 2 (W n ), η (W n ) W Im[ z ] δ 1 (W) δ 2 (W) η (W) 1st sheet M th Re[ z ] W W W bound ( coupled-channel scattering di ffi cult ) resonance ‣ δ 1 (W n ), δ 2 (W n ), η (W n ) <-- W n (L 1 ) = W n (L 2 ) = W n (L 3 ) 2nd sheet

Coupled-channel HAL QCD method ✦ measure relevant NBS wave function --> channel is defined X 2 ( ~ 0 , t ) J † (0) | 0 i = p h 0 | � a r, t ) � a A n a r ) e − W n t Z a 1 Z a 1 ( ~ x + ~ n ( ~ 2 n see for full details, Aoki et al. (HAL QCD), PRD87 (2013); Proc. Jpn. Acad., Ser. B, 87 (2011). ★ define coupled-channel potential using ψ a (r) Z r 2 + ( ~ ⇣ n ) 2 ⌘ r ) = 2 µ a X k a a r 0 U ab ( ~ r 0 ) b r 0 ) n ( ~ n ( ~ d ~ r, ~ b ★ coupled-channel potential U ab (r,r’): ch3 S( W<ch3 ) • U ab (r,r’) is faithful to coupled-channel S-matrix ch2 U (2x2) • U ab (r,r’) is energy independent (until new threshold opens) • Non-relativistic approximation is not necessary ch1 • U ab (r,r’) contains all 2PI contributions