Chiral effective field theory for hadron and nuclear physics

Transcription

1 Chiral effective field theory for hadron and nuclear physics Jose Manuel Alarcón Helmholtz-Institut für Strahlen- und Kernphysik University of Bonn

2 Biographical presentation

3 Biographical presentation Born in 1983 in Cartagena in the Region of Murcia (Spain). 21- I started my studies in Physics at the University of Murcia I received the 1st prize in the physics contest Celebrando la Física organized by the University of Murcia I completed my studies with the best grades, receiving a special prize for it (Premio extraordinario fin de carrera) I finished the Master of Advanced Physics, with specialization in theoretical physics, at the University of Valencia (maximum grade in Master Thesis) I started to work on my thesis under the supervision of Prof. Jose Antonio Oller, at the University of Murcia. Topic: Relativistic formulations of chiral EFT with baryons and application to πν scattering. 3/18

4 Biographical presentation I defended my thesis, entitled Baryon Chiral Perturbation Theory in its manifestly covariant forms and the study of the πν dynamics & On the Y(2175) resonance. (Sobresaliente Cum Laude) Thesis awarded with the Premio extraordinario de doctorado, given to the best thesis in physics defended in the period at the University of Murcia Thesis awarded by the Nuclear Physics Division of the European Physical Society with the Dissertation Award ( ) July I started to work in the group of Prof. Marc Vanderhaeghen at the Johannes Gutenberg University, Mainz. Working with V. Pascalutsa: Nucleon Polarizabilities and μη Lamb shift. September I started to work in Bonn. Application of EFT to ab initio many-body nuclear calculations. 4/18

5 Research topics

6 πν scattering

7 πν scattering Application of chiral EFT with baryons to fundamental hadronic reactions: 7/18

8 Application of chiral EFT with baryons to fundamental hadronic reactions: πν scattering Understanding of the chiral dynamics of the hadronic processes at low energies. 7/18

9 Application of chiral EFT with baryons to fundamental hadronic reactions: πν scattering Understanding of the chiral dynamics of the hadronic processes at low energies. Insight into the internal structure of the nucleon. 7/18

10 Application of chiral EFT with baryons to fundamental hadronic reactions: Understanding of the chiral dynamics of the hadronic processes at low energies. Insight into the internal structure of the nucleon. πν scattering. πν scattering 7/18

11 Application of chiral EFT with baryons to fundamental hadronic reactions: Understanding of the chiral dynamics of the hadronic processes at low energies. Insight into the internal structure of the nucleon. πν scattering. πν scattering Fundamental hadronic reaction involving one baryon. 7/18

12 Application of chiral EFT with baryons to fundamental hadronic reactions: Understanding of the chiral dynamics of the hadronic processes at low energies. Insight into the internal structure of the nucleon. πν scattering. πν scattering Fundamental hadronic reaction involving one baryon. Long range part of the NN forces. 7/18

13 Application of chiral EFT with baryons to fundamental hadronic reactions: Understanding of the chiral dynamics of the hadronic processes at low energies. Insight into the internal structure of the nucleon. πν scattering. πν scattering Fundamental hadronic reaction involving one baryon. Long range part of the NN forces. Not completely understood in the context of chiral dynamics 7/18

14 Application of chiral EFT with baryons to fundamental hadronic reactions: Understanding of the chiral dynamics of the hadronic processes at low energies. Insight into the internal structure of the nucleon. πν scattering. πν scattering Fundamental hadronic reaction involving one baryon. Long range part of the NN forces. Not completely understood in the context of chiral dynamics Disagreement with dispersive approaches [Becher and Leutwyler, JHEP (21)]. 7/18

15 Application of chiral EFT with baryons to fundamental hadronic reactions: Understanding of the chiral dynamics of the hadronic processes at low energies. Insight into the internal structure of the nucleon. πν scattering. πν scattering Fundamental hadronic reaction involving one baryon. Long range part of the NN forces. Not completely understood in the context of chiral dynamics Disagreement with dispersive approaches [Becher and Leutwyler, JHEP (21)]. Information about the internal scalar structure of the nucleon 7/18

16 Application of chiral EFT with baryons to fundamental hadronic reactions: πν scattering Understanding of the chiral dynamics of the hadronic processes at low energies. Insight into the internal structure of the nucleon. πν scattering. Fundamental hadronic reaction involving one baryon. Long range part of the NN forces. Not completely understood in the context of chiral dynamics Disagreement with dispersive approaches [Becher and Leutwyler, JHEP (21)]. Information about the internal scalar structure of the nucleon N 7/18

17 Application of chiral EFT with baryons to fundamental hadronic reactions: Understanding of the chiral dynamics of the hadronic processes at low energies. Insight into the internal structure of the nucleon. πν scattering. πν scattering Fundamental hadronic reaction involving one baryon. Long range part of the NN forces. Not completely understood in the context of chiral dynamics Disagreement with dispersive approaches [Becher and Leutwyler, JHEP (21)]. Information about the internal scalar structure of the nucleon N Important hadronic uncertainty in direct detection of DM [Bottino, Donato, Fornengo and Scopel, Astropart. Phys. 13, (2); Astropart. Phys. 18, (22)] [Ellis, Olive and Savage PRD 77, (28)]. Formation of elements needed for life [Berengut et. al., PRD 87, (213)]. 7/18

18 πν scattering The two manifestly Lorentz invariant schemes were used: Infrared Regularization (IR) and Extended-On-Mass-Shell (EOMS). 8/18

19 πν scattering The two manifestly Lorentz invariant schemes were used: Infrared Regularization (IR) and Extended-On-Mass-Shell (EOMS). Conclusions [Alarcón, Martin Camalich and Oller, Ann. of Phys. 336 (213)]: 8/18

20 πν scattering The two manifestly Lorentz invariant schemes were used: Infrared Regularization (IR) and Extended-On-Mass-Shell (EOMS). Conclusions [Alarcón, Martin Camalich and Oller, Ann. of Phys. 336 (213)]: At low energies above threshold IR, EOMS as well as Heavy Baryon ChPT give very similar results. 8/18

21 πν scattering The two manifestly Lorentz invariant schemes were used: Infrared Regularization (IR) and Extended-On-Mass-Shell (EOMS). Conclusions [Alarcón, Martin Camalich and Oller, Ann. of Phys. 336 (213)]: At low energies above threshold IR, EOMS as well as Heavy Baryon ChPT give very similar results. The inclusion of the Δ(1232) makes a difference in the prediction of the phenomenology at low energies 8/18

22 πν scattering The two manifestly Lorentz invariant schemes were used: Infrared Regularization (IR) and Extended-On-Mass-Shell (EOMS). Conclusions [Alarcón, Martin Camalich and Oller, Ann. of Phys. 336 (213)]: At low energies above threshold IR, EOMS as well as Heavy Baryon ChPT give very similar results. The inclusion of the Δ(1232) makes a difference in the prediction of the phenomenology at low energies EOMS is clearly the best. S31 P31 P S11 P11 P /-ChPT -ChPT /18

23 πν scattering The two manifestly Lorentz invariant schemes were used: Infrared Regularization (IR) and Extended-On-Mass-Shell (EOMS). Conclusions [Alarcón, Martin Camalich and Oller, Ann. of Phys. 336 (213)]: At low energies above threshold IR, EOMS as well as Heavy Baryon ChPT give very similar results. The inclusion of the Δ(1232) makes a difference in the prediction of the phenomenology at low energies EOMS is clearly the best. S31 P31 P S11 P11 P /-ChPT -ChPT EOMS + Δ(1232): 8/18

24 πν scattering The two manifestly Lorentz invariant schemes were used: Infrared Regularization (IR) and Extended-On-Mass-Shell (EOMS). Conclusions [Alarcón, Martin Camalich and Oller, Ann. of Phys. 336 (213)]: At low energies above threshold IR, EOMS as well as Heavy Baryon ChPT give very similar results. The inclusion of the Δ(1232) makes a difference in the prediction of the phenomenology at low energies EOMS is clearly the best. S31 P31 P S11 P11 P /-ChPT -ChPT EOMS + Δ(1232): Agreement with the dispersive approaches 8/18

25 πν scattering The two manifestly Lorentz invariant schemes were used: Infrared Regularization (IR) and Extended-On-Mass-Shell (EOMS). Conclusions [Alarcón, Martin Camalich and Oller, Ann. of Phys. 336 (213)]: At low energies above threshold IR, EOMS as well as Heavy Baryon ChPT give very similar results. The inclusion of the Δ(1232) makes a difference in the prediction of the phenomenology at low energies EOMS is clearly the best. S31 P31 P S11 P11 P /-ChPT -ChPT EOMS + Δ(1232): Agreement with the dispersive approaches BChPT is reliable! 8/18

26 πν scattering The two manifestly Lorentz invariant schemes were used: Infrared Regularization (IR) and Extended-On-Mass-Shell (EOMS). Conclusions [Alarcón, Martin Camalich and Oller, Ann. of Phys. 336 (213)]: At low energies above threshold IR, EOMS as well as Heavy Baryon ChPT give very similar results. The inclusion of the Δ(1232) makes a difference in the prediction of the phenomenology at low energies EOMS is clearly the best. S31 P31 P S11 P11 P /-ChPT -ChPT EOMS + Δ(1232): Agreement with the dispersive approaches BChPT is reliable! Determination of N from modern phenomenological information. N = 59(7) MeV [Alarcón, Martín Camalich and Oller, PRD 85 (212)] 8/18

27 πν scattering The two manifestly Lorentz invariant schemes were used: Infrared Regularization (IR) and Extended-On-Mass-Shell (EOMS). Conclusions [Alarcón, Martin Camalich and Oller, Ann. of Phys. 336 (213)]: At low energies above threshold IR, EOMS as well as Heavy Baryon ChPT give very similar results. The inclusion of the Δ(1232) makes a difference in the prediction of the phenomenology at low energies EOMS is clearly the best. S31 P31 P S11 P11 P /-ChPT -ChPT EOMS + Δ(1232): Agreement with the dispersive approaches BChPT is reliable! Determination of N from modern phenomenological information. N = 59(7) MeV [Alarcón, Martín Camalich and Oller, PRD 85 (212)] Analysis confirmed point by point by the Roy- Steiner analysis of [Hoferichter et al., PRL 115 (215)] 8/18

28 Nucleon Polarizabilities and Lamb shift

29 Relativistic Chiral EFT+Δ calculation of forward VVCS up to Spin Polarizabilites: Γ p 1 4 fm 4 p LT Q 2 GeV fm Q 2 GeV 2 Nucleon Polarizabilities γ n (1-4 fm 4 ) Q 2 (GeV 2 ) n LT 1 4 fm Q 2 GeV 2 [Lensky, Alarcón and Pascalutsa, PRC 9 (214)] BChPT+Δ LO BChPT LO HB [Kao et al., PRD 67 (23)] MAID BChPT+Δ* [Bernard et al., PRD 87 (213)] IR+Δ [Bernard et al., PRD 67 (23)] (Proton) Prok et al., PLB 672 (29) (Neutron) Amarian et al. PRL 93 (24) Dutz, et al., PRL 91(23) p 4 O Guler et al., PRC 92 (215) 1/18

30 Γ p 1 4 fm 4 p LT Q 2 GeV fm Q 2 GeV 2 Nucleon Polarizabilities Relativistic Chiral EFT+Δ calculation of forward VVCS up to Spin Polarizabilites: γ n (1-4 fm 4 ) Q 2 (GeV 2 ) n LT 1 4 fm Q 2 GeV 2 [Lensky, Alarcón and Pascalutsa, PRC 9 (214)] δlt puzzle BChPT+Δ LO BChPT LO HB [Kao et al., PRD 67 (23)] MAID BChPT+Δ* [Bernard et al., PRD 87 (213)] IR+Δ [Bernard et al., PRD 67 (23)] (Proton) Prok et al., PLB 672 (29) (Neutron) Amarian et al. PRL 93 (24) Dutz, et al., PRL 91(23) p 4 O Guler et al., PRC 92 (215) 1/18

31 Γ p 1 4 fm 4 p LT Q 2 GeV fm Q 2 GeV 2 New discrepancy Nucleon Polarizabilities Relativistic Chiral EFT+Δ calculation of forward VVCS up to Spin Polarizabilites: γ n (1-4 fm 4 ) Q 2 (GeV 2 ) n LT 1 4 fm Q 2 GeV 2 [Lensky, Alarcón and Pascalutsa, PRC 9 (214)] δlt puzzle BChPT+Δ LO BChPT LO HB [Kao et al., PRD 67 (23)] MAID BChPT+Δ* [Bernard et al., PRD 87 (213)] IR+Δ [Bernard et al., PRD 67 (23)] (Proton) Prok et al., PLB 672 (29) (Neutron) Amarian et al. PRL 93 (24) Dutz, et al., PRL 91(23) p 4 O Guler et al., PRC 92 (215) 1/18

32 Lamb shift The relativistic structure is important to agree with phenomenological determinations of E (pol) 2S [Alarcón, Lensky, Pascalutsa, EPJ C 74 (214)]. 11/18

33 Lamb shift The relativistic structure is important to agree with phenomenological determinations of E (pol) 2S [Alarcón, Lensky, Pascalutsa, EPJ C 74 (214)]. µ µ µh µh p T µ p 11/18

34 Lamb shift The relativistic structure is important to agree with phenomenological determinations of E (pol) 2S [Alarcón, Lensky, Pascalutsa, EPJ C 74 (214)]. µh µ µ T µ (P, q) = µh g µ + qµ q + 1 m 2 P µ N q 2 T 1 (, Q 2 ) P q q 2 qµ P P q q 2 q T 2 (, Q 2 ) p T µ p 11/18

35 Lamb shift The relativistic structure is important to agree with phenomenological determinations of E (pol) 2S [Alarcón, Lensky, Pascalutsa, EPJ C 74 (214)]. µh µ µ T µ (P, q) = µh g µ + qµ q + 1 m 2 P µ N q 2 T 1 (, Q 2 ) P q q 2 qµ P P q q 2 q T 2 (, Q 2 ) p T µ p 11/18

36 The relativistic structure is important to agree with phenomenological determinations of 2S [Alarcón, Lensky, Pascalutsa, EPJ C 74 (214)]. µh Lamb shift µ µ T µ (P, q) = µh E (pol) g µ + qµ q + 1 m 2 P µ N q 2 T 1 (, Q 2 ) P q q 2 qµ P P q q 2 q T 2 (, Q 2 ) p T µ p (μev) [1] [2] [3] [4] [5] [6] [7] [8] E (pol) 2S -12(2) (2.4) -8.5(1.1) -15.3(5.6) Chiral EFT calculations Phenomenological determinations (dispersion relations+data) [1] K. Pachucki, Phys. Rev. A 6 (1999). [2] A. P. Martynenko, Phys. Atom. Nucl. 69 (26). [3] D. Nevado and A. Pineda, Phys. Rev. C 77 (28). [4] C. E. Carlson and M. Vanderhaeghen, Phys. Rev. A 84, (211). [5] M. C. Birse and J. A. McGovern, Eur. Phys. J. A 48, (212). [6] M. Gorchtein, F. J. LLanes-Estrada and A. P. Szczepaniak, Phys. Rev. A 87 (213). [7] J. M. Alarcón, V. Lensky, V. Pascalutsa, Eur. Phys. J. C 74 (214). [8] C. Peset and A. Pineda Eur. Phys. J. A 51 (215). 11/18

37 The relativistic structure is important to agree with phenomenological determinations of 2S [Alarcón, Lensky, Pascalutsa, EPJ C 74 (214)]. µh Lamb shift µ µ T µ (P, q) = µh E (pol) g µ + qµ q + 1 m 2 P µ N q 2 T 1 (, Q 2 ) P q q 2 qµ P P q q 2 q T 2 (, Q 2 ) p T µ p (μev) [1] [2] [3] [4] [5] [6] [7] [8] E (pol) 2S -12(2) (2.4) -8.5(1.1) -15.3(5.6) Chiral EFT calculations Phenomenological determinations (dispersion relations+data) [1] K. Pachucki, Phys. Rev. A 6 (1999). [2] A. P. Martynenko, Phys. Atom. Nucl. 69 (26). [3] D. Nevado and A. Pineda, Phys. Rev. C 77 (28). [4] C. E. Carlson and M. Vanderhaeghen, Phys. Rev. A 84, (211). [5] M. C. Birse and J. A. McGovern, Eur. Phys. J. A 48, (212). [6] M. Gorchtein, F. J. LLanes-Estrada and A. P. Szczepaniak, Phys. Rev. A 87 (213). [7] J. M. Alarcón, V. Lensky, V. Pascalutsa, Eur. Phys. J. C 74 (214). [8] C. Peset and A. Pineda Eur. Phys. J. A 51 (215). 11/18

38 The relativistic structure is important to agree with phenomenological determinations of 2S [Alarcón, Lensky, Pascalutsa, EPJ C 74 (214)]. µh Lamb shift µ µ T µ (P, q) = µh E (pol) g µ + qµ q + 1 m 2 P µ N q 2 T 1 (, Q 2 ) P q q 2 qµ P P q q 2 q T 2 (, Q 2 ) p T µ p (μev) [1] [2] [3] [4] [5] [6] [7] [8] E (pol) 2S -12(2) (2.4) -8.5(1.1) -15.3(5.6) Chiral EFT calculations Phenomenological determinations (dispersion relations+data) [1] K. Pachucki, Phys. Rev. A 6 (1999). [2] A. P. Martynenko, Phys. Atom. Nucl. 69 (26). [3] D. Nevado and A. Pineda, Phys. Rev. C 77 (28). [4] C. E. Carlson and M. Vanderhaeghen, Phys. Rev. A 84, (211). [5] M. C. Birse and J. A. McGovern, Eur. Phys. J. A 48, (212). [6] M. Gorchtein, F. J. LLanes-Estrada and A. P. Szczepaniak, Phys. Rev. A 87 (213). [7] J. M. Alarcón, V. Lensky, V. Pascalutsa, Eur. Phys. J. C 74 (214). [8] C. Peset and A. Pineda Eur. Phys. J. A 51 (215). 11/18

39 Nuclear Lattice Effective Field Theory

40 NLEFT Lattice techniques to study nuclear many-body problems with chiral EFT. 13/18

41 NLEFT Lattice techniques to study nuclear many-body problems with chiral EFT. 2N forces is crucial in studies of many-body nuclear interactions with chiral EFT: 13/18

42 NLEFT Lattice techniques to study nuclear many-body problems with chiral EFT. 2N forces is crucial in studies of many-body nuclear interactions with chiral EFT: [Courtesy of Dean Lee] 13/18

43 NLEFT Lattice techniques to study nuclear many-body problems with chiral EFT. 2N forces is crucial in studies of many-body nuclear interactions with chiral EFT: Determination of NN LECs. [Courtesy of Dean Lee] 13/18

44 NLEFT Lattice techniques to study nuclear many-body problems with chiral EFT. 2N forces is crucial in studies of many-body nuclear interactions with chiral EFT: Determination of NN LECs. Study of the uncertainties. [Courtesy of Dean Lee] 13/18

45 NLEFT Lattice techniques to study nuclear many-body problems with chiral EFT. 2N forces is crucial in studies of many-body nuclear Nuclear interactions with chiral EFT: Determination of NN LECs. Study of the uncertainties. δ (degrees) S a = 1.97 fm LO NLO p CM (MeV) δ (degrees) LO NLO 3 S 1 a = 1.97 fm p CM (MeV) δ (degrees) Scattering Data 3 P Effective Field Theory NLO LO Ordonez et al. 94; Friar & Coon 94; Kaiser et 2 al. 97; Epelbaum et al. 98, 3; Kaiser 99-1; Higa et al. 3; a = 1.97 fm p CM (MeV) [Courtesy of Dean Lee] δ (degrees) a = 1.97 fm LO NLO 1 P p CM (MeV) δ (degrees) a = 1.97 fm LO NLO 3 P p CM (MeV) δ (degrees) [J. M. Alarcon, Chiral Dynamics Workshop 215] NLO LO a = 1.97 fm 3 P p CM (MeV) 13/18

46 Future Projects

47 Chiral predictions of transverse structure of baryons Provide spatial picture of the charge and magnetic density of the baryon octet and densities of the decuplet. 15/18

48 Chiral predictions of transverse structure of baryons Provide spatial picture of the charge and magnetic density of the baryon octet and densities of the decuplet. Chiral EFT provides predictions for the peripheral region b~1/m π. 15/18

49 Chiral predictions of transverse structure of baryons Provide spatial picture of the charge and magnetic density of the baryon octet and densities of the decuplet. Chiral EFT provides predictions for the peripheral region b~1/m π. Extend the work of [Granados & Weiss, JHEP 141 (214)] to the octet and decuplet. 15/18

50 Chiral predictions of transverse structure of baryons Provide spatial picture of the charge and magnetic density of the baryon octet and densities of the decuplet. Chiral EFT provides predictions for the peripheral region b~1/m π. Extend the work of [Granados & Weiss, JHEP 141 (214)] to the octet and decuplet. ρ 1 Proton. SU(2) vs SU(3) 1 ρ 2 Proton. SU(2) vs SU(3) ρ 1 p (fm -2 ) Octet Decuplet Octet+Decuplet SU(2) Nucleon SU(2) Delta SU(2) Nucleon+Delta ρ 2 p (fm -2 ) Octet Decuplet Octet+Decuplet SU(2) Nucleon SU(2) Delta SU(2) Nucleon+Delta b (fm) b (fm) 15/18

51 Chiral predictions of transverse structure of baryons Provide spatial picture of the charge and magnetic density of the baryon octet and densities of the decuplet. Chiral EFT provides predictions for the peripheral region b~1/m π. Extend the work of [Granados & Weiss, JHEP 141 (214)] to the octet and decuplet. ρ 1 Proton. SU(2) vs SU(3) 1 ρ 2 Proton. SU(2) vs SU(3) ρ 1 p (fm -2 ) Octet Decuplet Octet+Decuplet SU(2) Nucleon SU(2) Delta SU(2) Nucleon+Delta ρ 2 p (fm -2 ) Octet Decuplet Octet+Decuplet SU(2) Nucleon SU(2) Delta SU(2) Nucleon+Delta b (fm) b (fm) Check the large-nc relations in chiral EFT. 15/18

52 Chiral predictions of transverse structure of baryons Provide spatial picture of the charge and magnetic density of the baryon octet and densities of the decuplet. Chiral EFT provides predictions for the peripheral region b~1/m π. Extend the work of [Granados & Weiss, JHEP 141 (214)] to the octet and decuplet. ρ 1 Proton. SU(2) vs SU(3) 1 ρ 2 Proton. SU(2) vs SU(3) ρ 1 p (fm -2 ) Octet Decuplet Octet+Decuplet SU(2) Nucleon SU(2) Delta SU(2) Nucleon+Delta ρ 2 p (fm -2 ) Octet Decuplet Octet+Decuplet SU(2) Nucleon SU(2) Delta SU(2) Nucleon+Delta b (fm) b (fm) Check the large-nc relations in chiral EFT. We saw that different formulations with Δ are subdominant in Nc. 15/18

53 Chiral predictions of transverse structure of baryons Provide spatial picture of the charge and magnetic density of the baryon octet and densities of the decuplet. Chiral EFT provides predictions for the peripheral region b~1/m π. Extend the work of [Granados & Weiss, JHEP 141 (214)] to the octet and decuplet. ρ 1 Proton. SU(2) vs SU(3) 1 ρ 2 Proton. SU(2) vs SU(3) ρ 1 p (fm -2 ) Octet Decuplet Octet+Decuplet SU(2) Nucleon SU(2) Delta SU(2) Nucleon+Delta ρ 2 p (fm -2 ) Octet Decuplet Octet+Decuplet SU(2) Nucleon SU(2) Delta SU(2) Nucleon+Delta b (fm) b (fm) Check the large-nc relations in chiral EFT. We saw that different formulations with Δ are subdominant in Nc. They do not modify the correct Nc scaling. 15/18

54 Summary and Conclusions

55 Summary and Conclusions Chiral EFT with baryons is an excellent tool to investigate fundamental hadronic interactions involving nucleons on QCD grounds. πν Good description of modern scattering data below the Δ peak. Agreement with dispersive extractions. Extraction of important quantities from phenomenology ( N ) Forward doubly virtual Compton scattering Prediction of polarizabilities in agreement with MAID model and experimental data Improves previous ChPT predictions. Nuclear Lattice EFT Fundamental piece in ab initio many-body nuclear calculations. Further improvements possible including spin-flavor symmetry. 17/18

56 FIN

57 Spares

58 Polarizabilities Relativistic baryon chiral EFT with electromagnetic probes: Scalar and spin VVCS Polarizabilities. Scalar Polarizabilites: Α P E1 Β P M1 1 4 fm Q 2 GeV Α L p 1 4 fm Q 2 GeV 2 BChPT+Δ LO BChPT LO HB MAID Babusci et al, PRC 57 (1998) 2 Α n n E1 Β M1 1 4 fm 3 4 Α L n 1 4 fm 5 Liang, PRC 73 (26) V. Olmos de Leon, et al., EPJ A 1 (21) Q 2 GeV Q 2 GeV 2

59 Polarizabilities Some interesting moments: (Q 2 )= d 2 (Q 2 ) = Z x x 2 2 dx g 1 (x, Q 2 ) dx x 2 [2g 1 (x,q 2 ) + 3g 2 (x,q 2 )] 1 p Q 2 GeV 2 Γ 1 p-n Q 2 (GeV 2 ) Q 2 (GeV 2 ) d 2 p d 2 n Q 2 (GeV 2 ) BChPT+Δ LO BChPT LO HB [Kao et al., PRD 67 (23)] MAID BChPT+Δ* [Bernard et al., PRD 87 (213)] IR+Δ [Bernard et al., PRD 67 (23)] (Γ1 p ) Prok et al., PLB 672 (29) (Γ1 p-n )Deur et al., PRD 78 (28) (Γ1 p-n ) Deur et al., PRL 93 (24) (d2 n ) Amarian et al., PRL 92 (24)

60 Contribution of the octet in the loops to ImF 1 of the Σ+ π Contribution of the octet in the loops to ρ 1 Σ +.2 π Loops 1 π Loops ImF 1 Σ + π.15.1 K Loops Total Kernel (b= 1 Mπ ) ρ 1 Σ + (fm -2 ) K Loops Total t (GeV 2 ) Σ b (fm) ρ1 (fm -2 ) Octet Decuplet Octet+Decuplet b (fm)