Jacopo Ferretti Sapienza Università di Roma

Transcription

1 Jacopo Ferretti Sapienza Università di Roma NUCLEAR RESONANCES: FROM PHOTOPRODUCTION TO HIGH PHOTON VIRTUALITIES ECT*, TRENTO (ITALY), -6 OCTOBER 05

2 Three quark QM vs qd Model A relativistic Interacting qd Model Ferretti, Vassallo and Santopinto, PRC83, (0) Nonstrange baryon spectrum Extension to strange baryons Santopinto and Ferretti, PRC9, 050 (05) A relativistic Interacting qd Model with a spin-isospin transition interaction De Sanctis et al., arxiv: Improved nonstrange spectrum and scalaraxial-vector diquark mixing effects

3 Several versions: Isgur and Karl, Capstick and Isgur, U(7), Graz, Hypercentral QM Some differences, but share main features: ) based on the effective degrees of freedom of three constituent quarks ) (linear) confining potential 3) states classified within SU sf (6) Reproduce reasonably well many observables: baryon magnetic moments, lower part of baryon spectrum, open-flavor decays They have some problems, including that of the missing resonances 3

4 States predicted by quark models with no corresponding experimental counterparts QMs predict eccessive number of states Possible explanations: ) Some baryon states may be very weakly coupled to single-pion channels. Look for two-pion, three-pion, eta decay channels ) Consider models based on smaller number of effective degrees of freedom (like quark-diquark model): number of missing states decreases notably 4

5 Diquark: two strongly correlated quarks, with no internal spatial excitations (Ψ space symmetric) Diquark as effective bosonic degree of freedom Diquark wave function is antisymmetric: Ψ D = Ψ space Ψ color Ψ spin-flavor Baryon in color-singlet: Ψ color is antisymmetric Diquark spin-flavor wave function is symmetric 5 spin-flavor representation is neglected 5

6 0(A) and 70(MA) representations neglected in quark-diquark models Thus, the number of states decreases with respect to three quark QMs 6

7 Model mass formula M = E 0 + q + m + q + m + M dir + M ex + M cont m and m : quark and diquark masses Direct + exchange + contact terms Eigenvalues à numerical variational procedure with h.o. trial wave functions Model parameters (4) fitted to data FERRETTI, VASSALLO AND SANTOPINTO, PRC83, (0) 7

8 Direct Term Smeared Coulomb-like M dir = τ r ( e µr )+ βr Exchange Term Linear confining! M ex = ( ) L+ e σ r [A s s!! s + A I t t! + A Contact Term SI ( s! s! )( t! t! )] M cont η3 e η π 3/ r δ simulating function INTRODUCED TO REPRODUCE Δ-N MASS SPLITTING FERRETTI, VASSALLO AND SANTOPINTO, PRC83, (0) 8

9 TABLE I. Resulting values for the model parameters. m q = 00 MeV m S = 600 MeV m AV = 950 MeV τ =.5 µ = 75.0fm β =.5 fm A S = 375 MeV A I = 60 MeV A SI = 375 MeV σ =.7 fm E 0 = 54 MeV D = 4.66 fm η = 0.0fm ϵ = 0.00 FERRETTI, VASSALLO AND SANTOPINTO, PRC83, (0) 9

10 M GeV.0.5 N(680) *** and **** PDG states below GeV J P FERRETTI, VASSALLO AND SANTOPINTO, PRC83, (0) 0

11 Resonance Status M expt J P L P S s n r M calc (MeV) (MeV) N(939) P **** 939 N(440) P **** N(50) D 3 **** N(535) S **** N(650) S **** N(675) D 5 **** N(680) F 5 **** N(700) D 3 *** N(70) P *** N(70) P 3 **** (3) P 33 **** 3 33 (600) P 33 *** (60) S 3 **** (700) D 33 **** (900) S 3 ** (905) F 35 **** (90) P 3 **** (90) P 33 *** (930) D 35 *** (950) F 37 **** N(00) P * N(090) S * N(900) P 3 ** N(080) D 3 ** (750) P 3 * (940) D 33 * , , No missing states below GeV FERRETTI, VASSALLO AND SANTOPINTO, PRC83, (0)

12 Mass formula M = E 0 + q + m + q + m + M dir + M ex + M cont Exchange potential is generalized to Gürsey-Radicati inspired interaction! M ex = ( ) L+ e σ r [A s s!! s + A I t!! t + A F λ! λ ] λ s are SU(3) Gell-Mann matrices Results updated to most recent exp. data. Global fit to strange & nonstrange baryons SANTOPINTO AND FERRETTI, PRC9, 050 (05)

13 Parameter Value Value Parameter Value Value (fit ) (fit ) (fit ) (fit ) m n 00 MeV 59 MeV m s 550 MeV 3 Mev m [n,n] 600 MeV 607 MeV m [n,s] 900 MeV 856 MeV m {n,n} 950 MeV 963 MeV m {n,s} 00 MeV 6 MeV m {s,s} 580 MeV 35 MeV τ.0.0 µ 75.0 fm 8.4 fm β.5 fm.36 fm A S 350 MeV 436 MeV A F 00 MeV 93 MeV A I 50 MeV 79 MeV σ.30 fm.5 fm E 0 4 MeV 50 MeV ϵ 0.37 D 6.3 fm η.0 fm SANTOPINTO AND FERRETTI, PRC9, 050 (05) 3

14 spin = 0 and, respectively, for simplicity here we use the notation of Refs. [39,4]. { } Resonance Status M exp. (MeV) J P L P S s n r M calc. (fit ) (MeV) N(939) P **** 939 N(440) P **** N(50) D 3 **** N(535) S **** N(650) S **** N(675) D 5 **** N(680) F 5 **** N(700) D 3 *** N(70) P *** N(70) P 3 **** missing states N(875) D 3 *** N(880) P ** N(895) S ** N(900) P 3 *** (3) **** SANTOPINTO AND FERRETTI, PRC9, 050 (05) 4

15 = { } Resonance Status M exp. (MeV) J P L P S s n r M calc. (fit ) (MeV) + (3) (939) P 33 **** (600) P 33 *** (60) S 3 **** (700) D 33 **** (750) P 3 * (900) S 3 ** (905) F 35 **** (90) P 3 **** (90) P 33 *** (930) D 35 *** (940) D 33 ** (950) F 37 **** No missing states below GeV 5

16 M GeV.0 M GeV J P J P *** and **** PDG states below GeV SANTOPINTO AND FERRETTI, PRC9, 050 (05) 6

17 Resonance Status M exp. J P L P S s Q q F F I t n r M calc. (fit ) (MeV) (MeV) (93) P **** (60) S ** 60 (660) P *** (670) D 3 **** (750) S *** (770) P * 770 (775) D 5 **** (880) P ** 880 (95) F 5 **** (940) D 3 *** missing state (000) S * 000 (385) P 3 **** (840) P 3 * 840 (080) P 3 ** [n,s]n {n,n}s {n,n}s {n,n}s [n,s]n {n,s}n {n,n}s [n,s]n [n,s]n [n,s]n {n,n}s {n,n}s {n,n}s {n,s}n {n,n}s

18 TABLE VI. As Table V,butfor -, -, and -type resonances. Resonance Status M exp. J P L P S s Q q F F I t n r M calc. (fit ) (MeV) (MeV) (38) P **** 35 3 (80) D 3 *** missing states (530) P 3 **** (67) P 03 **** [n,s]s {n,s}s [n,s]s [n,s]s {s,s}n {n,s}s {n,s}s {s,s}n {s,s}s

19 = M GeV Λ * (405) J P *** and **** PDG states below GeV SANTOPINTO AND FERRETTI, PRC9, 050 (05) 9

20 Resonance Status M exp. J P L P S s Q q F F I t n r M calc. (fit ) (MeV) (MeV) (6) P 0 **** 6 (600) P 0 *** (670) S 0 **** (690) D 03 **** (800) S 0 *** (80) P 0 *** (80) F 05 **** (830) D 05 **** (890) P 03 **** (405) S 0 **** (50) D 03 **** missing states [n,n]s [n,s]n [n,n]s [n,n]s [n,s]n {n,s}n [n,n]s [n,s]n [n,n]s [n,n]s {n,s}n {n,s}n {n,s}n [n,s]n {n,s}n {n,s}n [n,n]s [n,n]s [n,s]n [n,s]n [n,n]s [n,n]s [n,s]n [n,s]n 3 0 SANTOPINTO AND FERRETTI, PRC9, 050 (05)

21 SI transition interaction mixes scalar and axial-vector diquark components Motivations:. Improve reproduction of nonstrange baryon spectrum. Introduce axial-vector diquark component in nucleon WF Better reproduction of nucleon e.m. form factors expected De Sanctis et al. PRC84, 0550 (0) Other observables can also be computed

22 H = E 0 + q + m + q + m + M dir + M ex + M cont + M tr M tr = V 0 e ν r (! s! S)(! t! T ) S and T are spin and isospin transition operators DE SANCTIS ET AL., ARXIV:

23 m q =40MeV m S =50MeV m AV =360MeV τ =.3 µ =5fm β =.57 fm A S =5MeV A I =85MeV A SI =350MeV σ =0.60 fm E 0 =86MeV D =.00 fm η =0.0 fm V 0 =450MeV ν =0.35 fm DE SANCTIS ET AL., ARXIV:

24 M GeV N(680) J P DE SANCTIS ET AL., ARXIV:

25 Resonance Status M exp. J P L P S s n r M calc. (MeV) (MeV) Resonance Status M exp. J P L P S s n r M calc. (MeV) (MeV) 0 + N(939) P **** 939 0, N(440) P **** , 4 N(50) D 3 **** , N(535) S **** , N(650) S **** N(675) D 5 **** N(680) F 5 **** , N(700) D 3 *** N(70) P *** , 639 N(70) P 3 **** , N(875) D 3 *** , 866 N(880) P ** , N(895) S ** , 866 N(900) P 3 *** missing missing state + + 0, 990 N(000) F 5 ** , 990 (3) P 33 **** (600) P 33 *** (60) S 3 **** (700) D 33 **** (750) P 3 * (900) S 3 ** (905) F 35 **** (90) P 3 **** (90) P 33 *** (930) D 35 *** (940) D 33 ** (950) F 37 ****

26 The SI interaction allows scalar and axialvector diquarks components in nucleon WF with probability: State Scalar component Axial- vector component N 53% 47% N(440) 5% 49% Δ(3) 0 00% Important also in the calculation of several other observables: e.m. form factors, openflavor decays, magnetic moments, DE SANCTIS ET AL., ARXIV:

27 Rel. Interacting qd Model extended to heavy baryons Baryon magnetic moments in qd model Improved nucleon elastic and transition (helicity amplitudes) e.m. form factors Open-flavor decays in a qd model 7

28 Three quark QM vs qd Model A relativistic Interacting qd Model Ferretti, Vassallo and Santopinto, PRC83, (0) Nonstrange baryon spectrum Extension to strange baryons Santopinto and Ferretti, PRC9, 050 (05) A relativistic Interacting qd Model with a spin-isospin transition interaction De Sanctis et al., arxiv: Improved nonstrange spectrum and scalaraxial-vector diquark mixing effects 8

29 Thank you for you attention! 9

30 30

31 Operator: M tr (r) =V 0 e ν r ( s S)( t T ) Matrix elements defined as: s,m s S [] µ s,m s =0fors s S 0 = 0 S = DE SANCTIS ET AL., ARXIV:

32 Point Form Relativistic Dynamics Point Form is one of the Relativistic Hamiltonian Dynamics for a fixed number of particles (Dirac) Construction of a representation of the Poincaré generators P µ (tetramomentum), J k (angular momenta), K i (boosts) obeying the Poincaré group commutation relations in particular [P k, K i ] = i δ kj H Three forms: Light (LF), Instant (IF), Point (PF) Differ in the number and type of (interaction) free generators 3

33 Point form: P µ interaction dependent J k and K i free Composition of angular momentum states as in the non relativistic case Mass operator M = M 0 + M I M 0 = Σ i p i + m Σ i p i = 0 P i undergo the same Wigner rotation -> M 0 is invariant The eigenstates of the relativistic qd Model are interpreted as eigenstates of the mass operator M Moving three-quark states are obtained through (interaction free) Lorentz boosts (velocity states) 33