From Halo EFT to Reaction EFT
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1 SOTANCP 4 Galveston 2018 From Halo EFT to Reaction EFT Marcel Schmidt Technische Universität Darmstadt and University of Tennessee, Knoxville with H.-W. Hammer (TUD) and L. Platter (UTK) May 15, 2018 May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 0/8
2 Motivation Halo Nuclei Neutron 7 fm [Nörtershäuser et al., PRL 102 (2009)] 10 Be (Core) 1.5 fm 11 Be (Halo Nucleus) May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 1/8
3 Motivation Halo Nuclei Neutron 7 fm [Nörtershäuser et al., PRL 102 (2009)] 1.5 fm 1p-Halo 2p-Halo Z 10 Be (Core) 11 Be (Halo Nucleus) N 11 Li 11 Be 1n-Halo 2n-Halo May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 1/8
4 Motivation Halo Nuclei Neutron 7 fm [Nörtershäuser et al., PRL 102 (2009)] 1.5 fm 1p-Halo 2p-Halo Z 10 Be (Core) 11 Be (Halo Nucleus) N 11 Li 11 Be 1n-Halo 2n-Halo Weak valence binding (long-range) vs. Deeply-bound core (short-range) May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 1/8
5 Motivation Towards Reaction EFT Case study 10 Be(d, p) 11 Be: Deuteron... Proton 10 Be 11 Be May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 2/8
6 Motivation Towards Reaction EFT Case study 10 Be(d, p) 11 Be: Deuteron... Proton 10 Be 11 Be Low momenta: Short-range details unresolved! 1/k R core May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 2/8
7 Motivation Towards Reaction EFT Case study 10 Be(d, p) 11 Be: Deuteron... Proton 10 Be 11 Be Low momenta: Short-range details unresolved! Effective contact-interactions (LECs)! 1/k R core May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 2/8
8 Motivation Towards Reaction EFT Case study 10 Be(d, p) 11 Be: Deuteron... Proton 10 Be 11 Be Low momenta: Short-range details unresolved! Effective contact-interactions (LECs)! 1/k R core EFT Expansion of QCD in kr core, R core /R halo 1 LO N n LO May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 2/8
9 Motivation Towards Reaction EFT Case study 10 Be(d, p) 11 Be: Deuteron... Proton 10 Be 11 Be Low momenta: Short-range details unresolved! Effective contact-interactions (LECs)! 1/k R core EFT Expansion of QCD in kr core, R core /R halo 1 Few LECs: Long-range physics LO N n LO May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 2/8
10 Motivation Towards Reaction EFT Proton Case study 10 Be(d, p) 11 Be: Deuteron Be 11 Be Low momenta: Short-range details unresolved! Effective contact-interactions (LECs)! 1/k R core EFT Expansion of QCD in kr core, R core /R halo 1 Few LECs: Long-range physics Systematically improvable LO N n LO May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 2/8
11 Motivation Towards Reaction EFT Proton Case study 10 Be(d, p) 11 Be: Deuteron Be 11 Be Low momenta: Short-range details unresolved! Effective contact-interactions (LECs)! 1/k R core EFT Expansion of QCD in kr core, R core /R halo 1 Few LECs: Long-range physics LO Systematically improvable At finite order: Theory uncertainty N n LO May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 2/8
12 2-Body Systems Phenomenology May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 3/8
13 2-Body Systems Phenomenology 11 Be (and 11 Be ) shallow Halo EFT: (γr) nc 0.4 [Hammer & Phillips, NPA 865 (2011)] May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 3/8
14 2-Body Systems Phenomenology 11 Be (and 11 Be ) shallow Halo EFT: (γr) nc 0.4 [Hammer & Phillips, NPA 865 (2011)] No LO Cluster physics! May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 3/8
15 2-Body Systems Phenomenology 11 Be (and 11 Be ) shallow Halo EFT: (γr) nc 0.4 [Hammer & Phillips, NPA 865 (2011)] 3 S 1 (and 1 S 0 ) shallow π EFT: (γr) np 0.4 [Chen et al., NPA 653 (1999)] No LO Cluster physics! May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 3/8
16 2-Body Systems Phenomenology 11 Be (and 11 Be ) shallow Halo EFT: (γr) nc 0.4 [Hammer & Phillips, NPA 865 (2011)] 3 S 1 (and 1 S 0 ) shallow π EFT: (γr) np 0.4 [Chen et al., NPA 653 (1999)] analytically solvable! No LO Cluster physics! May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 3/8
17 2-Body Systems Phenomenology 11 Be (and 11 Be ) shallow Halo EFT: (γr) nc 0.4 [Hammer & Phillips, NPA 865 (2011)] 3 S 1 (and 1 S 0 ) shallow π EFT: (γr) np 0.4 [Chen et al., NPA 653 (1999)] analytically solvable! (Coulomb only) No LO Cluster physics! May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 3/8
18 2-Body Systems EFT Lagrangian (LO) L LO = May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 4/8
19 2-Body Systems EFT Lagrangian (LO) L LO = Kinetic: 10 Be + neutron + proton May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 4/8
20 2-Body Systems EFT Lagrangian (LO) L LO = Kinetic: 10 Be + neutron + proton ( n-c: + + h.c. ) + 11 Be auxiliary fields (s-wave) May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 4/8
21 2-Body Systems EFT Lagrangian (LO) L LO = Kinetic: 10 Be + neutron + proton ( n-c: + ( n-p: + ) + h.c. ) + h.c Be deuteron auxiliary fields (s-wave) May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 4/8
22 2-Body Systems EFT Lagrangian (LO) L LO = Kinetic: 10 Be + neutron + proton ( n-c: + ( n-p: + ) + h.c. ) + h.c Be deuteron auxiliary fields (s-wave) 2-Body: = Reproduce γ (LO), r (NLO), etc. May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 4/8
23 3-Body System Amplitudes Effective 2-body amplitudes: 11 Be T inel. p (inelastic scattering) = + Λ T el. May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 5/8
24 3-Body System Amplitudes Effective 2-body amplitudes: 11 Be T inel. p (inelastic scattering) = + Λ T el. 10 Be d T el. (elastic scattering) = Λ T inel. May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 5/8
25 3-Body System Amplitudes Effective 2-body amplitudes: 11 Be T inel. p (inelastic scattering) = + Λ T el. 10 Be d T el. (elastic scattering) = Λ T inel. No spectroscopic factors involved! May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 5/8
26 3-Body System Amplitudes Effective 2-body amplitudes: 11 Be T inel. p (inelastic scattering) = + Λ T el. 10 Be d T el. (elastic scattering) = + Λ T inel. No spectroscopic factors involved! Efimov spectrum: Renormalize w/ 3-body force (here: deep 12 B pole) May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 5/8
27 3-Body System Range Corrections Tinel. /MeV d + 10 Be 4 2 p + 11 Be L = 0 L = Λ = 300 MeV Zero Range 6 Finite Range 4 2 Tel. /MeV E/MeV E/MeV May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 6/8
28 Next: Coulomb Force Strategy cf. NN-scattering in π EFT [König et al., J. Phys. G 42 (2015)] May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 7/8
29 Next: Coulomb Force Strategy cf. NN-scattering in π EFT [König et al., J. Phys. G 42 (2015)] 1. Couple particles to photon field (minimal substitution) May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 7/8
30 Next: Coulomb Force Strategy cf. NN-scattering in π EFT [König et al., J. Phys. G 42 (2015)] 1. Couple particles to photon field (minimal substitution) 2. Calculate induced Coulomb diagrams: + +O(α 2 ) + May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 7/8
31 Next: Coulomb Force Strategy cf. NN-scattering in π EFT [König et al., J. Phys. G 42 (2015)] 1. Couple particles to photon field (minimal substitution) 2. Calculate induced Coulomb diagrams: (γr) np +O(α 2 ) (γr) nc May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 7/8
32 Next: Coulomb Force Strategy cf. NN-scattering in π EFT [König et al., J. Phys. G 42 (2015)] 1. Couple particles to photon field (minimal substitution) 2. Calculate induced Coulomb diagrams: (γr) np +O(α 2 ) (γr) nc 3. Plug into scattering equations May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 7/8
33 Summary & Outlook Transfer reaction 10 Be(d, p) 11 Be in Halo EFT & πeft May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 8/8
34 Summary & Outlook Transfer reaction 10 Be(d, p) 11 Be in Halo EFT & πeft Model-independent treatment Expansion parameters 0.4 May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 8/8
35 Summary & Outlook Transfer reaction 10 Be(d, p) 11 Be in Halo EFT & πeft Model-independent treatment Expansion parameters 0.4 (Strong) scattering amplitude T inel. = +... May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 8/8
36 Summary & Outlook Transfer reaction 10 Be(d, p) 11 Be in Halo EFT & πeft Model-independent treatment Expansion parameters 0.4 (Strong) scattering amplitude Contact interactions only No spectroscopic factors Range corrections within LO band! T inel. = +... May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 8/8
37 Summary & Outlook Transfer reaction 10 Be(d, p) 11 Be in Halo EFT & πeft Model-independent treatment Expansion parameters 0.4 (Strong) scattering amplitude Contact interactions only No spectroscopic factors Range corrections within LO band! Coulomb force T inel. = May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 8/8
38 Summary & Outlook Transfer reaction 10 Be(d, p) 11 Be in Halo EFT & πeft Model-independent treatment Expansion parameters 0.4 (Strong) scattering amplitude Contact interactions only No spectroscopic factors Range corrections within LO band! Coulomb force Diagrammatic treatment Power counting applies T inel. = May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 8/8
39 Summary & Outlook Transfer reaction 10 Be(d, p) 11 Be in Halo EFT & πeft Model-independent treatment Expansion parameters 0.4 (Strong) scattering amplitude Contact interactions only No spectroscopic factors Range corrections within LO band! Coulomb force Diagrammatic treatment Power counting applies T inel. = TODO List: Extend to higher energies (data!) Study loss effects (Complex contact-forces!) Predict reaction astrophysical energies... May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 8/8
40 Appendix May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 0/7
41 Weakly Bound States Phenomenology E/MeV 0 10 Be + n May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 1/7
42 Weakly Bound States Phenomenology Beryllium-11 Halo EFT: R core /R halo 0.4 [Hammer & Phillips, NPA 865 (2011)] E/MeV Be +n+p E core 0 10 Be + n Be E halo May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 1/7
43 Weakly Bound States Phenomenology Beryllium-11 Halo EFT: R core /R halo 0.4 [Hammer & Phillips, NPA 865 (2011)] Deuteron E/MeV Be +n+p E core π EFT: γ d r d 0.4 [Chen et al., NPA 653 (1999)] 0 10 Be + n + p Be + p E halo Be + d May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 1/7
44 Weakly Bound States Phenomenology Beryllium-11 Halo EFT: R core /R halo 0.4 [Hammer & Phillips, NPA 865 (2011)] Deuteron E/MeV Be +n+p E core π EFT: γ d r d 0.4 [Chen et al., NPA 653 (1999)] Remote states LO 3-Body forces Be + n + p Be + p Be + d ( 11 B+ γ) + n E halo May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 1/7
45 Weakly Bound States Phenomenology Beryllium-11 Halo EFT: R core /R halo 0.4 [Hammer & Phillips, NPA 865 (2011)] Deuteron E/MeV Be +n+p E core π EFT: γ d r d 0.4 [Chen et al., NPA 653 (1999)] Remote states LO 3-Body forces Be + n + p Be + p Be + d ( 11 B+ γ) + n E halo Long-range physics explicit vs. Short-range physics implicit May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 1/7
46 EFT for 10 Be(d, p) 11 Be Halo EFT Lagrangian (w/ excited state) [Hammer, Phillips, NPA 865 (2011)] ) ) L11 Be = n α (i n α + c (i c 2m N 2m c + σ α [ ) [ ] (i σ +... ]σ α g σ (n α c) σ α + h.c. 2M Nc + π α [ (i M Nc ) + π +... Parameters: g σ, π, g π (LO) and σ (NLO) ] [ ( π α g π c 1 ) 2 α 1 n 2 α,1i α i c πα + h.c.] Fix with γ σ, γ π and effective ranges r π, r σ from experiment May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 2/7
47 EFT for 10 Be(d, p) 11 Be Equations Partial wave projected amplitudes for J N 0, L {J 1, J, J + 1}: ( ) T 3 L J (p, σd T 3 L p ) = 2 ( ) γ d γ σ IL (p, p ; E) J µ dd Nc 0 dq q 2 (1 + y) γσ I L(q,p;E) γd γ d+ T 3 L J ( ) 1+2y q 4 + (2π) 3 2 m dd q, p N(E+iɛ) 2 γd I L(p,q;E) L J ( ) γ σ 1+2y σd q, p γ σ+ (1+y) 2 q2 2µ Nc(E+iɛ) T 3 with γ d = 46 MeV, γ σ = 29 MeV, y mn m c = 0.1, µ Nc m N m c /(m N + m c ) and ( ) I L (p, q; E) 2 pq Q m N E L pq p 1 + y q q 2 p + iɛ, where Q L is Legendre function of 2 nd kind. May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 3/7
48 EFT for 10 Be(d, p) 11 Be Efimov Effect L = 0 partial wave amplitudes do not converge as Λ 3-body bound states (below d- 10 Be): B3/MeV λ Λ/MeV (Unphysical) Efimov states: Discrete scale invariance Λ λ 0 Λ [Efimov, PLB 33 (1970)] Renormalize w/ 3-body force! May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 4/7
49 Strong Interaction Cross Section Cross section dσ dω T 2 inel. May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 5/7
50 Strong Interaction Cross Section Cross section d σdσ/dω (θ)/mb d σdσ/dω (θ)/mb dσ dω T 2 inel. E d = 12 MeV E d = 18 MeV Preliminary too large! E d = 15 MeV E d = 21.4 MeV w/o 11 Be w/ 11 Be Experiment θ/deg θ/deg Data: [Schmitt et al., PRL 108 (2012)] Contribution of excited halo state 11 Be sub-leading May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 5/7
51 Coulomb Force Inelatic Cross Section Cross Section significantly lowered! d σdσ/dω (θ)/mb E d = 12 MeV E d = 15 MeV w/o Coulomb w/ Coulomb Experiment d σdσ/dω (θ)/mb E d = 18 MeV Preliminary E d = 21.4 MeV θ/deg θ/deg Data: [Schmitt et al., PRL 108 (2012)] May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 6/7
52 Coulomb Force Inelatic Cross Section Cross Section significantly lowered! d σdσ/dω (θ)/mb E d = 12 MeV E d = 15 MeV w/o Coulomb w/ Coulomb Experiment d σdσ/dω (θ)/mb E d = 18 MeV Preliminary E d = 21.4 MeV θ/deg θ/deg Data: [Schmitt et al., PRL 108 (2012)] Expect important NLO corrections (range, etc.)! May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 6/7
53 Coulomb Force Elastic Cross Section Need 3-body force to fit to data! d σdd/dω (θ)/mb E d = 12 MeV E d = 15 MeV w/o Coulomb w/ Coulomb Experiment d σdd/dω (θ)/mb E d = 18 MeV Preliminary E d = 21.4 MeV θ/deg θ/deg Cross section data (ORNL) [Schmitt et al., PRL 108 (2012)] May 15, 2018 From Halo EFT to Reaction EFT Marcel Schmidt 7/7
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