Hadron Interactions from Lattice QCD and Applications to Exotic Hadrons

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1 Hadron Interactions from Lattice QCD and Applications to Exotic Hadrons Yoichi Ikeda (RCNP, Osaka University) HAL QCD (Hadrons to Atomic nuclei from Lattice QCD) S. Aoki, T. Aoyama, T. Miyamoto, K. Sasaki (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.) 4 th week in YITP, Kyoto Univ., Jun , 2018.

2 Single hadron spectroscopy from LQCD Low-lying hadrons on physical point (physical mq) light-quark sector Nf=2+1 full QCD, L~3fm charm baryons Nf=2+1 full QCD, L~3fm RHQ for charm quark <-- Eππ(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 predictions of undiscovered charm hadrons (Ξ*cc, Ωccc,...) Next challenge : multi-hadron systems

3 Multi-hadrons: from quarks to hadrons, nuclei & neutron stars lattice QCD hadron scattering many thresholds hadron resonances (W ) W Nuclei from LQCD bad S/N ( e A ) nuclei mass number (A) QCD phase diagram temperature Sign problem?? density EOS of neutron stars

4 HAL QCD strategy: from quarks to hadrons, nuclei & neutron stars lattice QCD hadron resonances hadronic interactions nuclei Part I: hadronic interactions difficulties in multi-hadron systems solution = HAL QCD method EOS of neutron stars Part II: applications to exotic candidates coupled-channel scattering from LQCD dibaryon & tetraquark candidates

5 Hadronic interactions from LQCD hadronic correlation function C NN (r, t) = X n 0 N 1 (r, t)n 2 (0,t)J (t = 0) 0 A n n ( r)e W nt X space X imaginary time Energy eigenvalue Wn & NBS (Nambu-Bethe-Salpeter) wave function ψn(r) Finite Volume Method Wn(L) > phase shift Lüscher s formula Lüscher s finite volume formula k n cot Lüscher, Nucl. Phys. B354, 531 (1991). (k n )= 4 L 3 W n = X 1 ~p 2 m2z 3 m k n 2 qm 21 + k2n + q m k2 n

6 Hadronic interactions from LQCD hadronic correlation function C NN (r, t) 0 N 1 (r, t)n 2 (0,t)J (t = 0) 0 = X n A n n ( r)e W nt X space X imaginary time Energy eigenvalue Wn & NBS (Nambu-Bethe-Salpeter) wave function ψn(r) Finite Volume Method Wn(L) > phase shift Lüscher s formula Lüscher, Nucl. Phys. B354, 531 (1991). Serious difficulty to measure Wn(L) in multi-hadron systems HAL QCD Method ψn(r) --> 2PI kernel (ψ = φ + G0 U ψ) --> phase shift, binding energy,... Ishii, Aoki, Hatsuda, PRL 99, (2007). Ishii et al. [HAL QCD], PLB 712, 437 (2012).

7 Fundamental difficulty in multi-hadron systems see, Iritani, Doi et al. [HAL QCD], JHEP10 (2016) 101. C N (t) =a 0 e m N t + c 1 e (m N +m )t +! a (t >t 0 e m N t ) single nucleon C NN (t) =b 0 e W 0t + b 1 e W 1t + two nucleons! b 0 e W 0t (t >t ) N+π N+N+π ~mπ N δe N+N t* ~ mπ -1 ~ 1 fm t* ~ δe -1 ~ mn (L/2π) 2 ~ 10 fm S/N p N conf S/N p N conf S/N becomes far worse in multi-hadron systems

8 Demonstration of plateau method by mock-up data Mirage in temporal correlation functions for baryon-baryon interactions in lattice QCD Iritani, Doi et al. [HAL QCD], JHEP10 (2016) 101. Normalized correlation func. R(t) for two baryons in mock-up data R(t) = C BB(t) C B (t) 2 = b 1e Et + b 2 e E el t + c 1 e apple E e R(t) (t) = log R(t + 1)! E t>t E inel t Ground state energy ΔE = WBB-2mB ~1 MeV precision necessary (nuclear physics scale) δeinel ΔE δeel =50MeV Elastic scattering states δeel δeel = 50MeV, b2/b1 = ±0.1, 0 (10% contamination) Inelastic threshold δeinel δeinel = 500MeV, c1/b1 = 0.01 (1% contamination)

9 Demonstration of plateau method by mock-up data Mirage in temporal correlation functions for baryon-baryon interactions in lattice QCD Iritani, Doi et al. [HAL QCD], JHEP10 (2016) % contamination of elastic state True ground state for t > 8 fm with 10% contamination Fake plateaux or Mirage appear at t ~ 1 fm Zoom + typical stat. error

10 Actual data for ΞΞ ( 1 L=4.3fm, a=0.09fm Source-operator R(t) = X dependence in plateau method h0 B 1 (~x, t)b 2 (~y, t)j (t = 0) 0i/C B (t) 2 ~x,~y E e (t) = log apple R(t) R(t + 1)! E t>t δeinel ~ 500MeV δeel ~ 50MeV mirage at t~1fm ΔE True ground state is to appear at t > t* ( t*~8 fm) At least one of plateaux is fake! (Data at t~1fm is too early to identify plateau.) Naive plateau method does NOT work --> variational method (next talk)

11 A solution: HAL QCD method -- potential as a representation of S-matrix -- The scattering states do exist, and we should tame the scattering states HAL QCD method define energy-independent potential U(r,r ) Z d~r 0 U(~r, ~r 0 ) n(~r 0 )=(E n U(~r,~r 0 ) All elastic states share the same potential U(r,r ) Xn th n=0 U ψ0 = (E0-H0) ψ0 U ψ1 = (E1-H0) ψ1 Aoki, Hatsuda, Ishii, PTP123, 89 (2010). Ishii et al. [HAL QCD], PLB 712, 437 (2012). H 0 ) n(~r) (E n H 0 ) n (~r) n(~r 0 ) derive U(r,r ) from time-dependent Schrödinger-type eq. d~r 0 U(~r, ~r 0 )R(~r + 2 H 4m 2 0 R(~r, t) inelastic elastic Elastic scat. states are no more contamination than signal ( t*~1fm ) NNπ NN R(~r, t) =e 2m Bt C BB (~r, t)

12 ΞΞ ( 1 S0) in HAL QCD L=4.3fm, a=0.09fm source dependence of R(r,t) Iritani et al. [HAL QCD], arxiv: [hep-lat]. U(~r, ~r 0 )= V 0 (~r) + V 2 (~r)r 2 (~r ~r 0 ) finite V calc. Fate of fake plateaux infinite V calc. mirage ~1fm true plateau ~10fm No ΞΞ( 1 S0) bound state

13 Multi-hadron spectroscopy from LQCD lattice QCD hadron scattering HAL QCD method (W ) many thresholds W hadron resonances Resonances are embedded into coupled-channel scattering states

14 How can we find resonances? reaction plane Coupled-channel scatterings S (`) (W ) partial wave analysis of expt. data cross sections (dσ/dω) spin polarization observables etc. Analyticity of S-matrix is uniquely determined identical theorem + dispersion theory 1st sheet bound 2nd sheet Im[z] resonance Re[z] bound state (1st sheet) pole position --> binding energy residue --> coupling to scattering state resonance (2nd sheet) analytic continuation onto 2nd sheet pole position --> resonance energy residue --> coupling to scat. state, partial decay

15 Strategy to search for complex poles on the lattice Coupled-channel scatterings from lattice QCD S (`) (W ) h0 1(~r, t) 2 (~0,t)J (t = 0) 0i = X n A n n(~r)e W nt X X ~x coupled-channel Lüscher s method Wn(L) --> δ 1 (Wn), δ 2 (Wn), η(wn) t Im[z] W δ 1 (W) δ 2 (W) η(w) 1st sheet Mth Re[z] bound W W W 2nd sheet resonance (coupled-channel scattering difficult) δ 1 (Wn), δ 2 (Wn), η(wn) <-- Wn(L1) = Wn(L2) = Wn(L3)

16 Coupled-channel HAL QCD method measure relevant NBS wave function --> channel is defined h0 a 1 (~x + ~r, t) a 2 ( ~0,t)J (0) 0i = p Z a 1 Za 2 r 2 +( ~ k a n )2 a n (~r) =2µa X b Ishii, Aoki, Hatsuda, PRL99, (2007). Aoki, Hatsuda, Ishii, PTP123, 89 (2010). Ishii et al. (HAL QCD), PLB712, 437(2012). Z X n A n d~r 0 U ab (~r, ~r 0 ) a n (~r)e W nt Nambu-Bethe-Salpeter (NBS) wave function in each channel derive 2PI kernel (potential) as a representation of S-matrix b n (~r0 ) coupled-channel potential U ab (r,r ): ch3 U ab (r,r ) is faithful to coupled-channel S-matrix U ab (r,r ) is energy independent (until new threshold opens) Non-relativistic approximation is not necessary U ab (r,r ) contains all 2PI contributions U(2x2) S(W<ch3) ch2 ch1 Full details, Aoki et al. (HAL QCD), PRD87, (2013); Proc. Jpn. Acad., Ser. B, 87 (2011).

17 Octet BB forces & H-dibaryon n(udd) p(uud) n(udd) p(uud) (dds) 0 (uds) (uds) + (uus) X (dds) 0 (uds) (uds) + (uus) (dss) 0 (uss) (dss) 0 (uss) = (27 8 1) sym. ( ) anti-sym. NN ( 1 S0) NN ( 1 S0) : deuteron H-dibaryon (ΛΛ-ΞN-ΣΣ)? Jaffe (1977)

18 Generalized BB forces in flavor SU(3) limit Full QCD in SU(3)F limit : mπ~0.47gev, L=3.9 fm Inoue et al. (HAL QCD), PRL106 (2011), NPA881 (2012). potentials in flavor symmetric channels --> s + 1 NN 1 S0 channels (partially Pauli blocked) 8s channel (Pauli forbidden) H-dibaryon channel (Pauli allowed) origin of repulsive core <--> Pauli principle (+ magnetic gluon coupling) see, Oka & Yazaki, NPA464 (1987) bound H-dibaryon?

19 Structure of H-dibaryon in flavor SU(3) limit flavor singlet potential V (1) (H-dibaryon channel) Nf=3 full QCD : mπ~ gev, L=3.9 fm Inoue et al. (HAL QCD), PRL106 (2011), NPA881 (2012). Fate of H-dibaryon SU(3)f limit mbb bound H physical point mσσ=2380 MeV mξn=2260 MeV mλλ=2230 MeV Coupled-channel analysis on physical point

20 Fate of almost physical point Nf=2+1 full QCD, mπ~0.146gev (almost physical), L~8.1fm (large volume) ΣΣ repulsive (8s) NΞ-NΞ NΞ-ΛΛ preliminary NΞ ΛΛ i N i i 1 A = 1 p 40 ΛΛ-ΛΛ attractive (1) attractive (27) p p p 1 p 24 p 15 p 12 p 8 p i 8 s i 1i 1 A Sasaki et al. [HAL QCD], in preparation.

21 Fate of almost physical point ΛΛ and ΞN phase shifts S(k) = e 2i 1 i p 1 2 e i( 1+ 2 ) i p 1 2 e i( 1+ 2 ) e 2i 2 Original prediction of H-dibaryon Jaffe (1977) based on quark model, Perhaps a Stable Dihyperon Answer from QCD for H-dibaryon Perhaps near threshold Dihyperon ΣΣ NΞ H particle ΛΛ

22 Decuplet BB forces & ΩΩ-dibaryon X = (28 27) sym. (35 10 ) anti-sym. ΩΩ (J=0) : the most strange dibaryon? Dyson & Xoung, PRL14 (1965). Zhang (1992), Wang(1992)

23 Most strange almost physical point ΩΩ system in 1 S0 Gongyo, Sasaki et al. [HAL QCD], PRL 120, (2018). H int. = V LQCD (r) + /r repulsive core + attractive pocket H int. = V LQCD (r) ΩΩ is bound against strong interaction ΩΩ is close to unitary region together with Coulomb force 2-particle correlation func. in future HIC see talks by Hatsuda & Morita see also, Yamada [HAL QCD], PTEP2015 (2015)., for mπ=700 MeV, L~3fm

24 Charmed tetra-quark candidate Zc(3900) Zc(3900) in experiments BESIII (2013). Belle (2013). e + e Y (4260) Z c (3900) J/ M J/ Zc(3900) found in π +/- J/ψ (cc bar ud bar ) Z c (3900) u c c c u?? c c c d d? DD (3877) c (3759) Zc(3900) from lattice QCD coupled-channel HAL QCD approach (J PC ) 1 + J/ (3236) coupled-channel πj/ψ-ρηc-d bar D* potentials understand the nature of Zc(3900) Y. Ikeda, et al. [HAL QCD], PRL117, (2016). Reviewed in J. Phys. G45, (2018).

25 3x3 potential matrix (πj/ψ - ρηc - D bar D*) mπ=410mev D bar D* c d u c V DbarD*-DbarD* mπ=570mev mπ=700mev d u c c ρηc V ρηc-dbard* V ρηc-ρηc πj/ψ d u c c V πj/ψ-dbard* V πj/ψ-ρηc V πj/ψ-πj/ψ r (fm) r (fm) r (fm)

26 3x3 potential matrix (πj/ψ - ρηc - D bar D*) mπ=410mev D bar D* c d u c V DbarD*-DbarD* V ρηc-dbard* mπ=570mev mπ=700mev V ρηc-ρηc ρηc d πj/ψ u c c heavy quark spin symmetry d u c c V πj/ψ-dbard* r (fm) V πj/ψ-ρηc r (fm) heavy quark spin symmetry V πj/ψ-πj/ψ r (fm)

27 3x3 potential matrix (πj/ψ - ρηc - D bar D*) V DbarD*-DbarD* mπ=410mev mπ=570mev mπ=700mev D bar D* ρηc c d d u u c c c V ρηc-dbard* d u c c strong coupling V ρηc-ρηc πj/ψ d u c c V πj/ψ-dbard* c d u c V πj/ψ-ρηc strong V πj/ψ,dbard* & V ρηc,dbard* V πj/ψ-πj/ψ charm quark exchange process r (fm) r (fm) r (fm)

28 Mass spectra of πj/ψ (2-body scattering) 2-body scattering ( the most ideal to understand Zc(3900) ) πj/ψ-ρηc-d bar D* πj/ψ invariant mass (mπ=410mev) J/ J/ Zc(3900) in expt. Y (4260) Z c (3900) J/ M J/ D bar D* Enhancement just above D bar D* threshold effect of strong V πj/ψ, DbarD* (black --> V πj/ψ, DbarD* =0) line shape not Breit-Wigner Is Zc(3900) a conventional resonance? --> pole of S-matrix

29 S J/, J/ (z) Pole of S-matrix (πj/ψ :2nd, ρηc :2nd, D bar D*:2nd) Im[z] pole of S-matrix [1st, 1st, 1st] m + m J/ m + m c m D + m D Z c (3900) Re[z] πj/ψ threshold [2nd, 1st, 1st] [2nd, 2nd, 1st] [2nd, 2nd, 2nd] actual pole position If Zc is resonance... ρηc threshold D bar D* threshold Re[z] (scattering energy) [MeV] Im[z] (half width) [MeV] Pole corresponding to virtual state Pole contribution to scat. observables is small (far from scat. axis) Zc(3900) is not a resonance but threshold cusp induced by strong V πj/ψ,dbard*

30 Summary HAL QCD method NBS wave function ψ(r) --> 2PI kernel (ψ = φ + G0Uψ) Crucial for multi-hadron & coupled-channel scatterings Exotic candidates, H, ΩΩ, Zc(3900) Aoki, Hatsuda, Ishii, PTP123, 89 (2010). H particle is very close to NΞ threshold --> J-PARC? Ishii et al. [HAL QCD], PLB 712, 437 (2012). Aoki et al. (HAL QCD), PRD87, (2013). Sasaki et al [HAL QCD], in preparation. ΩΩ is very close to unitary region --> HIC? Gongyo, Sasaki et al [HAL QCD], PRL120, (2018). Zc(3900) is threshold cusp induced by strong VDbarD*, πj/ψ Ikeda et al. [HAL QCD], PRL117, (2016). Ikeda [HAL QCD], J. Phys. G45, (2018). Future: many hadron resonances & nuclear structures at physical point

31 Thank you for your attention!