Ab initio alpha-alpha scattering using adiabatic projection method

Transcription

1 Ab initio alpha-alpha scattering using adiabatic projection method Serdar Elhatisari Advances in Diagrammatic Monte Carlo Methods for QFT Calculations in Nuclear-, Particle-, and Condensed Matter Physics 5 October, Trento, Italy Nuclear LEFT Collaborators [Scattering and reactions] Evgeny Epelbaum (Bochum) Hermann Krebs (Bochum) Timo Lähde (Juelich) Dean Lee (NCSU) Thomas Luu (Juelich) Ulf-G. Meißner (Bonn) Michelle Pine (NCSU) Alexander Rokash (Bochum) Gautam Rupak (MSU)

2 Introduction Lattice effective field theory Adiabatic projection method Scattering cluster wave function Alpha-alpha scattering Summary Imaged by Dean Lee

3 Ab initio scattering and reactions Thermonuclear reaction rates of alpha capture on light nuclei during the stellar nucleosynthesis are critical for the formation of the building blocks of life and the evolution of the universe ( the production of Carbon-12, Oxygen-16, ). The rates also determine the size of the iron cores formed in Type-II supernovae, which results in the ultimate fate of the collapsed remnant into either neutron star or black hole. low energy nuclear scattering and reaction processes relevant for stellar astrophysics Scattering of alpha particles. 4 He + 4 He 4 He + 4 He Triple- alpha reaction. 4 He + 4 He + 4 He 12 C + γ Alpha capture. 12 C + 4 He 16 O + γ

4 Ab initio scattering and reactions QMC calculations of neutron-α scattering. Nollett, Pieper, Wiringa, Carlson, & Hale, PRL 99, (2007). Ab Initio Many-Body Calculations of n-h 3, n-he 4, p-he 3,4, and n-be 10 Scattering. Quaglioni & Navratil, PRL 101, (2008). Ab initio many-body calculations of the 3 H(d, n) 4 He and 3 He(d, p) 4 He fusion. Navratil & Quaglioni, PRL 108, (2012). Elastic proton scattering of medium mass nuclei from coupled-cluster theory. Hagen & Michel PRC 86, (2012). Coupling the Lorentz Integral Transform (LIT) and the Coupled Cluster (CC) Methods Orlandini, G. et al., Few Body Syst. 55, (2014). ***Deuteron is the heaviest projectile in ab initio calculations of scattering on a heavier target that have been achieved.

5 Ab initio scattering and reactions Ab initio calculations of scattering and reactions suffer from the computational scaling with the number of nucleons in clusters. It remains a challenge to address important processes relevant for stellar astrophysics Scattering of alpha particles. 4 He + 4 He 4 He + 4 He Triple- alpha reaction. 4 He + 4 He + 4 He 12 C + γ Alpha capture. 12 C He O + γ Ab initio alpha-alpha scattering S.E., Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, & Meißner. arxiv:

6 Lattice effective field theory

7 Lattice effective field theory Lattice effective field theory is a powerful numerical method formulated in the framework of effective field theory. Nucleons a~1-2 fm L

8 Lattice effective field theory Lattice effective field theory is a powerful numerical method formulated in the framework of effective field theory. Effective field theory organizes the nuclear interactions as an expansion in powers of momenta and other low energy scales such as the pion mass. Fig. courtesy E.Epelbaum Ordonez et al. 94; Friar & Coon 94; Kaiser et al. 97; Epelbaum et al. 98, 03, 05; Kaiser 99-01; Higa et al. 03;

9 Two-body scattering on the lattice Two spin-1/2 particles scattering. Borasoy, Epelbaum, Krebs, Lee, Meißner, Eur.Phys.J.A34: ,2007 The LECs by fitting the experimental NN scattering data Borasoy, Epelbaum, Krebs, Lee, Meißner, Eur.Phys.J.A35: ,2008 p

10 Nuclear lattice EFT Nuclear lattice EFT collaboration PRL 106 (2011) ; PRL 109 (2012) ; PRL 110 (2013) ; PRL 112 (2014)

11 Lattice effective field theory Euclidean time projection Euclidean time Nucleons L a~1-2 fm

12 Lattice effective field theory Euclidean time projection Euclidean time Nucleons e Hτ τ = L t α t π π L evolve nucleons forward in Euclidean time. allow them to interact a~1-2 fm

13 Lattice effective field theory Euclidean time projection The evolution in Euclidean time automatically incorporates the induced deformation and polarization of the clusters as they come near each other. The deformation and polarization are due to the interactions of individual nucleons between the two clusters, as well as repulsion due to the Pauli exclusion principle for identical fermions.

14 Lattice simulations Transfer matrix operator formalism M = exp H α t Microscopic Hamiltonian H = H free + V The free Hamiltonian with the simplest discretized dispersion relation 1 H free = b 2m s (n)[2 b s n b s (n + l) b s (n l)] s s l n Z L t = Dc Dc exp [ S c, c ] = Tr M L t Creutz, Found. Phys. 30 (2000) 487 The exact equivalence of several different lattice formulations. Lee,Phys. Rev., C78:024001, (2008); Prog.Part.Nucl.Phys., 63: (2009) e E 0 a t = lim L t Z L t+1 /Z L t

15 Lattice Monte Carlo calculations Hybrid Monte Carlo sampling H SU 4 acts as an approximate and inexpensive low energy filter at few first/last time steps. Significant suppression of sign oscillations. Chen, Lee, Schäfer, PRL 93 (2004) ψ r1,r 2 (τ ) = exp H SU 4 τ ψ r1,r 2 τ = L t α t The amplitude For time steps in midsection, the full H LO Hamiltonian is used. ψ r1,r 2 (τ/2) = exp H LO τ/2 ψ r1,r 2 (τ ) Z L t = ψ r3,r 4 τ/2 ψ r1,r 2 (τ/2) For the observable O (L Z t ) O,LO = ψ r3,r 4 τ/2 O ψ r1,r 2 (τ/2) O 0,LO = lim L t Z L t L t O,LO /ZLO

16 Lattice Monte Carlo calculations Higher order calculations : exp H NLO α t : : exp H NNLO α t : The amplitude is (L Z t ) NLO = ψr3,r 4 τ/2 ψ r1,r 2 (τ/2) For the observableo (L Z t ) O,NLO = ψ r3,r 4 τ/2 O ψ r1,r 2 (τ/2) The observable Oat NLO O 0,NLO = lim Z (L t ) L t O,NLO (L /Z t ) NLO

17 Euclidean time Auxiliary Field Monte Carlo exp C 2 (N N) 2 = 1 2 π ds exp 1 2 s2 + C s (N N) Using a Gaussian integral identity (s is an auxiliary field coupled to particle density.) Each nucleon evolves as if a single particle in a fluctuating background of pion fields and auxiliary fields. s I π π s π s Lee,Phys. Rev., C78:024001, (2008); Prog.Part.Nucl.Phys., 63: (2009)

18 Split the problem into two parts. Adiabatic projection method The first part use Euclidean time projection to construct an ab initio low-energy cluster Hamiltonian, called the adiabatic Hamiltonian. The second part compute the two-cluster scattering phase shifts or reaction amplitudes using the adiabatic Hamiltonian. Rupak, Lee., PRL 111 (2013) Pine, Lee, Rupak, EPJA 49 (2013) 151. S.E., Lee, Phys. Rev. C90, (2014). Rokash, Pine, S.E., Lee, Epelbaum, Krebs, arxiv:

19 Adiabatic projection method The method constructs a low-energy effective theory for clusters. Use initial states parameterized by the relative spatial separation between clusters, and project them in Euclidean time. R = r + R 1 r r 2 n x, n y R τ = e H τ R Dressed cluster states R The adiabatic projection in Euclidean time gives a systematically improvable description of the lowlying scattering cluster states. In the limit of large Euclidean projection time the description becomes exact. n x, n y Pine, Lee, Rupak, EPJA 49 (2013) 151. Rokash, Pine, S.E., Lee, Epelbaum, Krebs, arxiv:

20 Adiabatic projection method R τ = e H τ R H τ R R = τ R H R τ Norm matrix N τ R R = τ R R τ H τ a R R = N τ 1/2 R R RR H τ R R N τ 1/2 R R The structure of the adiabatic Hamiltonian, H τ a R R, is similar to the Hamiltonian matrix used in recent calculations of ab initio NCSM/RGM for nuclear scattering and reactions. Navratil, Quaglioni, Phys. Rev. C 83, (2011). Navratil, Roth, Quaglioni, Phys. Lett. B 704, 379 (2011). Navratil, Quaglioni, Phys. Rev. Lett. 108, (2012).

21 Adiabatic projection method Microscopic Hamiltonian L 3(A 1) L 3(A 1) Two-cluster (adiabatic) Hamiltonian L 3 L 3 lattice size is 7 3, a = 1.93 fm fermion-dimer scattering Pine, Lee, Rupak, EPJA 49 (2013) 151

22 Scattering phase shifts from finite volumes R wall ψ l r = N [cos δ l p F l p r + sin δ l p G l (p r)] Borasoy, Epelbaum, Krebs, Lee, Meißner, EPJA 34 (2007) 185

23 Scattering cluster wave functions ψ l r = N [cos δ l p F l p r + sin δ l p G l (p r)] R wall = R wall + ε a/2 ε a/2

24 Scattering cluster wave functions ψ l r = N [cos δ l p F l p r + sin δ l p G l (p r)] (a) Use the spectrum of the adiabatic Hamiltonian and the fact that the cluster wave function vanishes for r = R wall to compute the scattering phase shifts directly from δ l p = tan 1 F l p R wall G l (p R wall ) Borasoy, Epelbaum, Krebs, Lee, Meißner, EPJA 34 (2007) 185 (b) Use the spherical wall method and extract the scattering phase shifts and momentum by fitting the radial asymptotic wave function to the two-cluster wave function. Rokash, Pine, S.E., Lee, Epelbaum, Krebs, arxiv:

25 neutron-deuteron scattering (pionless EFT) Rokash, Pine, S.E., Lee, Epelbaum, Krebs, arxiv: Gabbiani, Bedaque, Grießhammer, NPA 675 (2000) 601

26 Scattering cluster wave functions ε is the relative error During the Euclidean time interval τ ε, each cluster undergoes spatial diffusion d ε,i = τ ε /M i only non-overlapping clusters if R d ε,i L/2 L/2 Define the asymptotic region where the amount of overlap between cluster wave packages is less than ε R > R ε Rokash, Pine, S.E., Lee, Epelbaum, Krebs, arxiv:

27 Scattering cluster wave functions For R > R ε, the dressed cluster states are widely separated, and we can describe the system in terms of an effective cluster Hamiltonian, H eff, which is a free lattice Hamiltonian for two clusters plus infinite-range interactions. N τ R R = τ R R τ = c e 2Heff τ RR H τ R R = τ R H R τ = c e Heff τ H eff e Heff τ RR H τ a R R = H eff RR Rokash, Pine, S.E., Lee, Epelbaum, Krebs, arxiv:

28 Adiabatic Hamiltonian A 1 + A 2 A 1 + A 2 Simulations for cluster A 1 and cluster A 2 L 3 ~ 120 fm 3 copy the adiabatic Hamiltonian R wall Simulations for (A 1 + A 2 ) system L 3 ~ 16 fm 3 includes only infinite-range interactions such as Coulomb between the clusters.

29 Radial adiabatic Hamiltonian Microscopic Hamiltonian L 3(A 1) L 3(A 1) Two-cluster (adiabatic) Hamiltonian L 3 L 3 Define radial adiabatic Hamiltonian by coherently adding 3D position states n x, n y, n z weighted by the spherical harmonics R l,l z = Y l,lz (R ) δ R, R R R S.E., Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, arxiv: Precise determination of lattice phase shifts and mixing angles Lu, Lähde, Lee, Meißner, arxiv:

30 Radial adiabatic Hamiltonian n x 2 + n y 2 + n z 2 = 0 n x 2 + n y 2 + n z 2 = 1 n x 2 + n y 2 + n z 2 = 2 Y 0,0 (R ) R l,l z = Y l,lz (R ) R δ R, R R Y 2,0 (R )

31 Alpha-alpha scattering the same lattice action as in the Hoyle state of 12 C and the structure of 16 O, PRL 106 (2011) PRL 109 (2012) PRL 112 (2014) a new algorithm for Monte Carlo updates and alpha clusters, the adiabatic projection method to construct a two-alpha (adiabatic) Hamiltonian, the spherical wall method to extract the scattering phase shifts. S.E., Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, arxiv:

32 Alpha-alpha scattering 4D - Lattice L 3 L t a = 1.97 fm a t = 1.32 fm Afzal, Ahmad, Ali, Rev. Mod. Phys. 41, 247 (1969) Higa, Hammer, van Kolck, Nucl.Phys. A809, 171 (2008) S.E., Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, arxiv:

33 Alpha-alpha scattering δ 0 L t, E = δ 0 E + c 0 E e ΔE 0 L t a t a = 1.97 fm a t = 1.32 fm S.E., Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, arxiv:

34 Alpha-alpha scattering a = 1.97 fm a t = 1.32 fm Afzal, Ahmad, Ali, Rev. Mod. Phys. 41, 247 (1969) S.E., Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, arxiv:

35 Alpha-alpha scattering a = 1.97 fm a t = 1.32 fm Afzal, Ahmad, Ali, Rev. Mod. Phys. 41, 247 (1969) S.E., Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, arxiv:

36 Alpha-alpha scattering δ 2 L t, E = δ 2 E + c 2 E e ΔE 2 L t a t a = 1.97 fm a t = 1.32 fm S.E., Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, arxiv:

37 Alpha-alpha scattering a = 1.97 fm a t = 1.32 fm Afzal, Ahmad, Ali, Rev. Mod. Phys. 41, 247 (1969) S.E., Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, arxiv:

38 Summary The adiabatic projection method is a general framework for scattering and reactions on the lattice. Scattering and reaction processes involving alpha particle are in reach of ab initio methods. The computational scaling for A 1 -body and A 2 -body clusters is A 1 + A 2 2 The problem of sign oscillations is greatly suppressed for alpha-like nuclei, and this approach appears to be a viable method to study important alpha processes involving heavier nuclei. Thanks!

39 Extras

40 Lüscher s method Two-body energy levels below the inelastic threshold in a periodic lattice are related to the scattering phase shifts in continuum. Lüscher, Comm. Math. Phys. 105 (1986) 153; NPB 354 (1991) 531 p 2l+1 cot δ l p = 1 a l r l p 2 + scattering length Effective range p p cot δ l p = 2 πl Z 0,0 1; η for l = 0,1 L Generalized Zeta function L L Z l,m 1; η = n n l Y l,m (n) n 2 η η = Lp 2π 2

41 neutron-deuteron scattering (pionless EFT) E 0 = lim τ τ ψ e,p H ψ e,p τ ψ τ e,p ψ e,p τ Quartet - s Rokash, Pine, S.E., Lee, Epelbaum, Krebs, arxiv: Gabbiani, Bedaque, Grießhammer, NPA 675 (2000) 601

42 neutron-deuteron scattering (pionless EFT) E 0 = lim τ τ ψ o,p H ψ o,p τ ψ τ o,p ψ o,p τ Quartet - p Rokash, Pine, S.E., Lee, Epelbaum, Krebs, arxiv: Gabbiani, Bedaque, Grießhammer, NPA 675 (2000) 601

43 Alpha-alpha scattering ψ l r = N [cos δ l p F l p r + sin δ l p G l (p r)] L t = 4 a = 1.97 fm a t = 1.32 fm Afzal, Ahmad, Ali, Rev. Mod. Phys. 41, 247 (1969) S.E., Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, arxiv: