EFT Approaches to αα and Nα Systems

Transcription

1 1 EFT Approaches to αα and Nα Systems Renato Higa Kernfysisch Versneller Instituut University of Groningen Simulations and Symmetries, INT Program, Mar. 30, 2010

2 2 EFT Approaches to αα and Nα Systems Outline Motivation EFT approach and universality halo/cluster EFT Coulomb interactions αα scattering Nα scattering Summary and outlook

3 3 N/Z 1: challenge for shell model /search.htm few nucleon systems: formation of halo systems 11 Be, 19 C, 11 Li, 6 He, 14 Be, 8 He, 8 B, 17 Ne,...

4 4 p + 7 Be 8 B + γ solar neutrinos 8 Be + α 12 C + γ red giants 9 Be( 1 2+ ) + α 12 C + n core-colapse supernovae

5 5 naturalness: where NDA works 2π/k V(r) r R R

6 6 strongly interacting systems: fine-tuning 2π/k a V(r) r R R

7 7 Effective Field Theory k, 1/a M lo, 1/R M hi 2 3 L = φ 4i φ b 0 (φφ) (φφ)+ b 2 2m 8» (φφ) φ( ) 2 φ + H.c. + it ( 1) = c 0 (φφφ) (φφφ)+, b0 b0 b0 + b0 b0 b b 0 (Λ) = i 1 + (Λ + ik) m 4π b 0(Λ) it (0) = ( 1) T b 2 ( 1) T b 2 (Λ) = iˆ1 + (Λ + ik) m 4π b 0(Λ) 2

8 8 Pros symmetries simplicity able to handle non-local interactions W/EM interactions 3-4B extensions controlled and systematic low-energy expansion

9 9 Pros symmetries simplicity able to handle non-local interactions W/EM interactions 3-4B extensions controlled and systematic low-energy expansion Cons controlled and systematic low-energy expansion complexity for >4B (Kirscher et al.)...

10 10 Universality in two-body systems a is the only relevant scale at LO f(θ) = 1 1/a ik, E B,V = 1 ma 2, dσ dω = 4a2 1 + a 2 k 2 (1) a : scale-invariant system BS at threshold, dσ/dω saturates the UB RG analysis: non-trivial IR fixed point (Birse et al., Phys. Lett. B 464, 169) close analogy to critical phenomena (liquid-gas phase transition, ferromagnets)

11 11 Universality in three-body systems (Braaten and Hammer, Phys. Rept. 428, 259) E H(Λ) Λ Λ (Hammer and RH, Eur. J. Phys. A 37, 193) renormalization requires c 0 at LO limit cycle E (n) /E (n+1) const. ( 515 for bosons)

12 12 halo/cluster EFT: separation of scales excitation of each cluster m c E c M hi ( m π ) binding of the valence nucleons (clusters) M lo M hi extension of the core treated in perturbation theory power-counting: modified to account for other effects (resonance/coulomb) expansion around the resonance: improved convergence rearrangement of the perturbative series, Coulomb interactions

13 13 halo/cluster EFT: k m π, m c E c M hi Physical quantities: k, 1/a 0 M lo, T l = 2π µ k 2l (2l + 1) k 2l+1 (cot δ l i) P l(cos θ) r 0 M 1 hi, P M 3 hi,... k 2l+1 cot δ l 1/a l + r l 2 k2 + P l 4 k L = φ 4i φ+σ d 4i µ 8µ 5d+g» d φφ+(φφ) d +, = M lo id (0) d = M 2 lo /µ id(0) d = iσ + iɛ 1 M lo iσ q 0 q 2 /8µ + iɛ µ M 2 lo (NN) (αα)

14 14 Coulomb photons dominant at very low energies k C = Z 1 Z 2 α em µ k C k η, k Cm [ 1 2 ln Λ ] 2k + For αα: k C = Z 2 αα em m α /2 M hi

15 15 non-perturbative Coulomb (Kong and Ravndal, NPA 665, 137) Coulomb wave functions: k χ (±) k T T C + T CS η = Z 2 α em µ/k = k C /k σ l = arg Γ(1 + l + iη), C (0) 2 η = e πη Γ(1 + iη)γ(1 iη) r χ (±) k χ(±) k (r) = e ηπ 2 Γ(1 ± iη) M( iη, 1; ±ikr ik r) e ik r f l sin(kr lπ 2 η ln 2kr + σ l), g l cos(kr lπ 2 η ln 2kr + σ l)

16 16 non-perturbative Coulomb (Kong and Ravndal, NPA 665, 137) Coulomb wave functions: k χ (±) k T T C + T CS η = Z 2 α em µ/k = k C /k σ l = arg Γ(1 + l + iη), C (0) 2 η G (±) C (E) = 1 E Ĥ0 ˆV C ± iɛ = 2µ = e πη Γ(1 + iη)γ(1 iη) d 3 q χ (±) q χ (±) q (2π) 3 2µE q 2 ± iɛ = + +

17 17 T CS = χ k ˆV S χ + k + χ k ˆV S G + C ˆV S χ + k + T (0) CS = = C(E) χ( ) k (0) χ (+) k (0) = C(E) C(0) 2 η e 2iσ 0, T (1) CS = = C(E) C(0) 2 η e 2iσ 0 C(E) J 0 (E), T CS = = C (0) 2 η J 0 (E) = 2µ (η q = k C /q) C(E) e 2iσ 0 1 C(E) J 0 (E), d 3 q χ (+) q (0)χ (+) q (0) (2π) 3 k 2 q 2 ± iɛ = 2µ d 3 q (2π) 3 2πη q e 2πη q 1 1 k 2 q 2 + iɛ

18 18 αα scattering 0+ resonance ( 8 Be g.s.): E LAB R = ± 0.07 kev, Γ LAB R = ± 0.50 ev M lo µer LAB 20 MeV, M hi m π 140 MeV power-counting: E LAB 3.0 MeV scattering: Afzal et.al. (1969) E LAB 3.0 MeV : data from Heydenburg and Temmer (1956) ERE parameters from Russell et.al. (1956), Rasche (1967): a 0 = ( 1.65 ± 0.17) 10 3 fm, r 0 = ± fm 1/M hi, P 0 = 1.76 ± 0.22 fm 3 1/Mhi 3

19 19 T CS = C (0) 2 η a B = = 2π µ C(E) e 2iσ 0 1 C(E) J 0 (E) = 2π µ C η (0) 2 e 2iσ 0 1 a c + r k 2 2 a H(η), B 1 Z 2 α em µ 1 M hi H(η) = ψ(iη) + 1 2iη ln(iη) C η (0) 2 e 2iσ 0 1 a 0 + r 0 2 k 2 iɛ + 2π µ J 0(E) { η 1 a B 2 ik η (a B k) (a B k) 4 without Coulomb: conformal invariance in 8 Be, Efimov state in 12 C at LO (RH, Hammer, van Kolck, Nucl. Phys. A 809, 171) with Coulomb: 8 Be and 12 C 0+ states remain close to threshold

20 20 (RH, Hammer, van Kolck, 2008) 180 LO NLO Afzal et.al LO NLO Afzal et.al. δ 0 [degrees] 150 k cot δ 0 [fm -1 ] E LAB [MeV] E LAB [MeV] a 0 (10 3 fm) r 0 (fm) P 0 (fm 3 ) LO NLO 1.92 ± ± ± 0.08 Rasche 1.65 ± ± ± 0.22

21 21 fine-tuning puzzle (R) }{{} M 2 hi µ (R) }{{} M hi M lo µ (R) }{{} M 2 lo µ (R) }{{} M 3 lo M hi µ = (κ) }{{} M 2 hi µ = (κ) }{{} M 2 hi µ = (κ) }{{} M 2 hi µ = (κ) }{{} M 2 hi µ (loops) }{{} M 2 hi µ (loops) }{{} M 2 hi µ (loops) }{{} M 2 hi µ (loops) }{{} M 2 hi µ (natural) (fine-tuned like NN) (fine-tuned to get E R ) (fine-tuned to get Γ R ) factor of 1000!!! (Oberhummer et al., Science 289, 88; RH, Hammer, van Kolck, 2008)

22 22 pα scattering: S 1/2, P 3/2, P 1/2 [ L LO = φ i 0 + [ ]φ+n 2 i 0 + ] 2 N 2m α 2m N ] +η 1+ t [i (m α +m N ) 1+ t + g 1+ 2 { t S [ N φ ( N)φ ] [ + H.c. r t S (Nφ) ]} + H.c. L NLO = η 0+ s [ ] ] 0+ s+g 0+ [s Nφ+φ N s +g 1+ t [i (m α +m N ) ] 2 t (Bertulani, Hammer, van Kolck, NPA 712, 37; Bedaque, Hammer, van Kolck, PLB 569, 159)

23 23 P -wave: k 3 cot δ 1 = 1/a 1 + r 1 /2 k 2 + P 1 /4 k 4 + p ˆV S p = η 1+ g 2 1+ (S j S k ) βα q 0 q 2 /2(m φ +m N ) 1+ +iε p j p k Amplitude: T LO CS = C (1) 2 η e 2iσ 1 2π µ ) k (2 2 cos θ + iσ ˆn sin θ 1 a 1+ + r 1+ 2 k2 + iɛ + 6π µ J 1(E) = 2π µ [F (k, θ) + iσ ˆnG(k, θ)] J 1 = µ 3πa B 4µπ 3a 3 B ( k ) [ a 2 H(η) + 1 B D 4 ln ζ ( 2) µπκ 12a 2 B + µκ2 µκ3 4a B 12 ( ab κ π 2 ) C E+ 3κa ] B 4

24 24 pα scattering T LO = T LO,S1/2 + T LO,P3/2 T NLO = T NLO,S1/2 + T NLO,P3/2 T N 2 LO = T N 2 LO,S 1/2 + T N 2 LO,P 3/2 + T LO,P1/2 F C (k, θ) = η 2k csc2 θ/2 exp [ iη ln(csc 2 θ/2) + 2iσ 0 ]

25 25 (RH, Bertulani, van Kolck, in preparation) δ [degrees] LO NLO Arndt et al. P 3/2 dσ/dω [barn/sr] θ=140 deg LO NLO ERE Nurmela et.al S 1/ E p [MeV] E p [MeV] ab-initio: Nollett em et al., PRL 99, (2007), Quaglioni & Navrátil, PRL 101, (2008)

26 26 c/r 2 +Coulomb: warm-up for 3α 3-body problem with large a 1D Schrödinger Eq. with V (r) = 1/r 2 limit cycle for c < 1/4 Efimov spectrum (Beane et al., Bawin and Coon, Braaten and Phillips, Long and van Kolck...) counterterm: log-periodic function of the cutoff Counterterm parameter Λ : iteration of quantum corrections (dimensional transmutation) model to study loss of conformal invariance (Kaplan et al.)

27 27 (Hammer, RH, Eur. J. Phys. A 37, 193) 1/r 2 +Coulomb: warm-up for 3α E B E fixed α=0.0 α= α= α= Λ E B H(Λ) α=0.0 α=0.1, exact α=0.1, pert α=0.0 α=0.1 α=0.3 α= Λ Λ

28 28 Summary Halo nuclei, cluster systems: opportunities for EFT approach universality limit cycles αα scattering Coulomb turned off conformal Efimov spectrum in 12 C incredible amount of fine-tuning LO (parameter-free) works well at very low energies, NLO improves description up to E LAB 3 MeV extraction of the ERE parameters with improved errorbars p-α scattering: good description of the P 3/2 resonance future: 3α, p- 7 Be, Nradcap (Rupak & RH), Borromean halos, heavier nuclei,...