J /ψ -nucleon scattering in P + c pentaquark channels Urˇ sa Skerbiˇ s ursa.skerbis@ijs.si in collaboration with: Saˇ sa Prelovˇ sek Lattice 2018 East Lansing, July 27 th , 2018 1 / 14
Motivation • In 2015 charmed pentaquark state P + c , decaying into N + J /ψ was discovered by LHCb (LHCb; PRL , 2015(115),072001). N + J /ψ → P + → N + J /ψ c • Two states were observed: • lower state with J = 3 2 , mass m P + = 4380 ± 8MeV and c width Γ = 205 ± 18MeV • upper state with J = 5 2 , mass m P + = 4449 , 8 ± 1 , 7MeV and c width Γ = 39 ± 5MeV • states have opposite parity. It is not clear which state is positive and which negative under parity transformation. − should be seen in irreps G ± States with J P = 3 + , 3 − , 5 + , 5 • and H ± 2 2 2 2 2 irrep, O h , � P = 0 J 1 G 1 2 3 H 2 5 H ⊕ G 2 2 7 G 1 ⊕ H ⊕ G 2 2 This channel was already studied by HALQCD method only for energies below Pc and no bound state was found (T. Sugiura et. all Proccedings of Lattice 2017 conference, EPJ Web of Conferences 175, 05011 (2018)) 2 / 14
Overview Channels for strong decay of P + c Results Lattice setup Single hadron results Operators for states with desired quantum numbers Results for scattering Conclusion 3 / 14
Possible channels for strong decay of P + c • � P = � p H 1 + � p H 2 = 0 • P + c : uudc ¯ c Expected non - interacting energies for meson - nucleon scattering with P = 0 on our lattice • Simulation are made in 1.4 -- -- -- approximation of 1 channel | p 2 = 0 -- -- 1.2 E - 1 / 4 ( E η c + 3 E J / ψ )[ l.u. ] -- scattering for J /ψ − p -- 2 = 1 -- | p -- -- 1.0 -- -- -- | p 2 = 2 • It should be sufficient to -- -- P c ( 4449 ) -- -- P c ( 4380 ) study scattering up to 0.8 | p 2 = 3 -- -- | p H i | 2 = 2, we could be -- 0.6 able to see both P + c states η c + p J / ψ + p χ c0 + p χ c1 + p • other possible chanells: ( D − − Σ ++ D 0 − Λ + , ¯ c ,...) c 4 / 14
Lattice setup • Properties of used lattice N 3 × N T a [fm] L [fm] #config m π [MeV] 16 3 × 32 0 . 1239(13) 1 . 98 280 266(3) • Wilson-Clover action for light quarks • Fermi lab approach for charm quarks • Full distillation: • J /ψ : N v = 96 • N : N v = 48 5 / 14
Single hadron results Nucleon E eff [ l.u. ] 2 = 0 | p 1.2 2 = 1 | p • Both hadrons 2 = 2 1.0 | p (nucleon and J /ψ E = 0.6987 ± 0.015 0.8 E = 0.7598 ± 0.019 meson) were E = 0.8684 ± 0.036 0.6 simulated with Δ t momentum | p | 2 = 0, 4 5 6 7 8 9 | p | 2 = 1 and | p | 2 = 2 • Nucleon: 3 operators for each value of J / ψ Meson E eff [ l.u. ] momentum 1.62 2 = 0 | p • J /ψ : 2 operators for 2 = 1 | p 1.60 2 = 2 | p each value of 1.58 E = 1.539 ± 0.00098 E = 1.576 ± 0.0011 momentum 1.56 E = 1.613 ± 0.0014 1.54 14 Δ t 8 9 10 11 12 13 6 / 14
Combining single hadron correlators • � P = � p H 1 + � p H 2 = 0, O ≈ N ( p ) V ( − p ) • Operators in Partial wave method: � O | p | , J , m J , L , S = C Jm J Lm L , Sm S C Sm S s 1 m s 1 , s 2 m s 2 × m L , m S , m s 1 , m s 2 � Lm L ( � Y ∗ Rp ) N m s 1 ( Rp ) V m s 2 ( − Rp ) R ∈ O � • Subduction to irrep: O [ J , L , S ] S J , m J O | p | , J , m J , L , S | p | , Γ , r = Γ , r m J J irrep Γ 1 G 1 2 3 H 2 5 H ⊕ G 2 2 7 G 1 ⊕ H ⊕ G 2 2 All explicit expressions for H 1 ( p ) H 2 ( − p ) operators : (S. Prelovsek, U.S., C.B. Lang ; JHEP 2017(1), 129.). Partial wave method for NN scattering was considered by CalLat: ( Berkowitz, et. all PLB , 2016(12) 024.) Subduction coefficients S J , mJ are given in: (J. Dudek, et.all; PRD 2010(82), 034508.) Γ , r 7 / 14
Example: Scattering in P + c pentaquark candidate channel: for irrep H − and J = 3 2 , L = 0 and | p | 2 = 0 2 , S = 3 Anihilation operator for this example is: O H − , r =1 2 , L =0 (0) = N 1 2 (0) ( V x (0) − iV y (0)) c c c c J = 3 2 , S = 3 c c c c u u u u Creation operator: u u u u d d d d O H − , r =1 ¯ 2 , L =0 (0) = N 1 2 (0) ( V x (0) + iV y (0)) J = 3 2 , S = 3 Correlation function: C VN ; H − 2 , L =0 ( | p | = 0) = � Ω | O H − O H − 2 , L =0 ¯ 2 , L =0 | Ω � = J = 3 2 , S = 3 J = 3 2 , S = 3 J = 3 2 , S = 3 C N 2 C V x → x − iC N 2 C V x → y + iC N 2 C V y → x + C N 2 C V 1 2 → 1 1 2 → 1 2 → 1 1 1 2 → 1 y → y C H pol src → pol snk = � Ω | H pol snk ¯ H pol src | Ω � 8 / 14
Anhilation operators for H − and | p | 2 = 1 J = 3 2 , S = 3 2 , L = 0: O H − , r =1 2 , L =0 (1) = N 1 ( e z ) � V x ( − e z ) − iV y ( − e z ) � + N 1 ( − e z ) � V x ( e z ) − iV y ( e z ) � + J = 3 2 , S = 3 2 2 � � � � N 1 ( e x ) V x ( − e x ) − iV y ( − e x ) + N 1 ( − e x ) V x ( e x ) − iV y ( e x ) + 2 2 N 1 � e y � � V x � − e y � − iV y � − e y �� + N 1 � − e y � � V x � e y � − iV y � e y �� 2 2 J = 3 2 , S = 1 2 , L = 2: O H − , r =1 � � � � 2 , L =2 (1) = N 1 ( e x ) V x ( − e x ) + iV y ( − e x ) + N 1 ( − e x ) V x ( e x ) + iV y ( e x ) − J = 3 2 , S = 1 2 2 � � � � � � �� � � � � � � �� N 1 e y V x − e y + iV y − e y − N 1 − e y V x e y + iV y e y − 2 2 N − 1 ( e x ) V z ( − e x ) − N − 1 ( − e x ) V z ( e x ) + N − 1 � e y � V z � − e y � + N − 1 � − e y � V z � e y � 2 2 2 2 J = 3 2 , S = 3 2 , L = 2: O H − , r =1 � � � � 2 , L =2 (1) = N 1 ( e z ) V x ( − e z ) − iV y ( − e z ) + N 1 ( − e z ) V x ( e z ) − iV y ( e z ) − J = 3 2 , S = 3 2 2 N − 1 ( e x ) V z ( − e x ) − N − 1 ( − e x ) V z ( e x ) − N 1 ( e x ) V x ( − e x ) − N 1 ( − e x ) V x ( e x ) + 2 2 2 2 � � � � � � � � � � � � � � � � N − 1 e y V z − e y + N − 1 − e y V z e y + iN 1 e y V y − e y + iN 1 − e y V y e y 2 2 2 2 9 / 14
Results for irrep H − with momentum | p | 2 ≤ 1 • 4 × 6 = 24 interpolators • GEVP: 8 operators • One state for | p | 2 = 0 E eff for irrep H - and 3 states at 2.6 | p | 2 = 1 N ( 2 ) J / ψ (- 2 ) 2.5 • state | p | 2 = 0 : E eff [ l.u. ] 2.4 ( J = 3 2 , S = 3 N ( 1 ) J / ψ (- 1 ) 2 , L = 0) 2.3 • states with | p | 2 = 1 : N ( 0 ) J / ψ ( 0 ) 2.2 ( J = 3 2 , S = 3 2 , L = 0) 4 6 8 10 ( J = 3 2 S = 1 2 , L = 2) Δ t ( J = 3 2 , S = 3 2 , L = 2) Dashed lines: non-interacting energy for scattering 10 / 14
Expected number of eigenstates for non-interacting scattering • degeneracy of states origins from spin of scattered hadrons + (irreps G − − , 3 + , 5 − , 5 c - J P : 3 • Candidate channels for P + 2 , 2 2 2 2 2 , H − , H + ) G + G − G + G − G + H − H + 1 1 2 2 | p | 2 = 0 1 0 0 0 1 0 | p | 2 = 1 2 2 1 1 3 3 | p | 2 = 2 3 3 3 3 6 6 total number of states 6 5 4 4 10 9 total number of operators 36 30 24 24 60 54 11 / 14
Results for scattering in irrep G − 2 ( P + c candidate channel) | p | 2 G − 2 • 1 + 3 states 0 0 state with | p | 2 = 1 : ( J = 5 • 2 , S = 3 2 , L = 2) 1 1 states with | p | 2 = 2 : ( J = 5 2 , S = 1 • 2 , L = 2) , 2 3 ( J = 5 2 S = 3 2 , L = 2) , ( J = 5 2 , S = 3 2 , L = 4) # states 4 - E eff for irrep G 2 2.6 N ( 2 ) J / ψ (- 2 ) 2.5 E eff [ l.u. ] 2.4 N ( 1 ) J / ψ (- 1 ) 2.3 N ( 0 ) J / ψ ( 0 ) 2.2 4 6 8 10 Δ t 12 / 14
All calculated energies ( 3 m J / ψ + m η c ) [ GeV ] 1.6 3 3 3 3 6 - 1 6 - 2 N ( 2 ) J / ψ (- 2 ) 1.5 1.4 P c ( 4449 ) 2 2 1 1 3 3 N ( 1 ) J / ψ (- 1 ) P c ( 4380 ) 1.3 1.2 1 1 E n - 1 4 N ( 0 ) J / ψ ( 0 ) 1.1 - + - + H - H + G 1 G 1 G 2 G 2 • We are able to see all expected states • few interpolators are left out- huge G − G − G + G + H − H + errors: 1 1 2 2 | p | 2 = 0 1 0 0 0 1 0 6 − 1 : one out of 6 | p | 2 = 1 2 2 1 1 3 3 interpolators is not used | p | 2 = 2 (to avoid large errors) 3 3 3 3 6 6 6 − 1 = 5 : states observed # states 6 5 4 4 10 9 • No additional states No strong indication of P + • c 13 / 14
Conclusion • Results of one channel approximation for P + c channels were presented. • All states required by degeneration caused by spin are observed, but some are left out due to huge errors • In our approximation there is no sign of extra eigenstate or significant energy shift, which would indicate to P + c state • P + c could be a result of other neglected effects (coupled channels effect,...) • Future plans: • Look at other scattering channels which may be related to P + c • Look at coupled channel effects 14 / 14
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