Nuclear Reactions in Lattice EFT

Transcription

1 Nuclear Reactions in Lattice EFT Gautam Rupak Mississippi State University Nuclear Lattice EFT Collaboration Evgeny Epelbaum (Bochum) Hermann Krebs (Bochum) Timo Lähde (Jülich) Dean Lee (NCSU) Thomas Luu (Jülich) Ulf-G. Meißner (Bonn) Nuclear Physics from Lattice QCD, Institute for Nuclear Theory May 20, 2016

2 Outline Background and motivation Weakly bound systems at low energy Adiabatic projection method - proof of concept neutron capture proton-proton fusion n-d doublet channel, connection to lattice QCD

3 Astrophysics Motivations Low energy reactions dominate Need accurate cross sections but hard to measure experimentally Model-independent theoretical calculations important Theoretical Weakly bound systems are fun First principle calculation

4 Astrophysics Motivations Low energy reactions dominate Need accurate cross sections but hard to measure experimentally Model-independent theoretical calculations important Theoretical Weakly bound systems are fun First principle calculation Use Effective Field Theory

5 EFT: the long and short of it Identify degrees of freedom L = c 0 O (0) +c 1 O (1) +c 2 O (2) + expansion in Q Hide UV ignorance - short distance IR explicit - long distance Determine c n from data (elastic, inelastic) EFT : ERE + currents + relativity Not just Ward-Takahashi identity M. Savage s talk on two-body current

6 A low energy example 7 Li(n, ) 8 Li Isospin mirror systems 7 Li(n, ) 8 Li $ 7 Be(p, ) 8 B Inhomogeneous BBN Whats the theoretical error?

7 A low energy example 7 Li(n, ) 8 Li Isospin mirror systems 7 Li(n, ) 8 Li $ 7 Be(p, ) 8 B Inhomogeneous BBN Whats the theoretical error? (a) (b) (c) (d)

8 A low energy example 7 Li(n, ) 8 Li Isospin mirror systems 7 Li(n, ) 8 Li $ 7 Be(p, ) 8 B Inhomogeneous BBN Whats the theoretical error? Asymptotic normalization (a) (b) s p 1 Z = p /r 1 (c) (d)

9 A low energy example 7 Li(n, ) 8 Li Isospin mirror systems 7 Li(n, ) 8 Li $ 7 Be(p, ) 8 B Inhomogeneous BBN Whats the theoretical error? Asymptotic normalization (a) (c) (b) (d) s p 1 Z = p /r 1 Need effective range r1 and binding energy at leading order

10 A low energy example 7 Li(n, ) 8 Li Isospin mirror systems 7 Li(n, ) 8 Li $ 7 Be(p, ) 8 B Inhomogeneous BBN Whats the theoretical error? Asymptotic normalization (a) (c) (b) (d) Two EFT operators s p 1 Z = p /r 1 Need effective range r1 and binding energy at leading order

11 v.σ (µb) Imhof A 59 Imhof B 59 Nagai 05 Blackmon 96 Lynn E LAB (ev) Red: Tombrello Rupak, Higa; PRL 106, (2011) Fernando,Higa, Rupak; EPJA 48, 24 (2012) RADCAP: Bertulani CDXS+: Typel Blue: Davids-Typel Black: EFT

12 Reactions in lattice EFT Consider: a(b, )c ; a(b, c)d Need effective cluster Hamiltonian -- acts in cluster coordinates, spins,etc. Calculate reaction with cluster Hamiltonian. Many possibilities --- traditional methods, continuum EFT, lattice method

13 Adiabatic Projection Method Initial state ~ Ri Evolved state ~ Ri = e H ~ Ri h ~ R 0 H ~ Ri D. Lee s talk U.-G. Meißner s talk Energy measurements in cluster basis. Divide by the norm matrix as these are not orthogonal basis [N ] ~R, ~ R 0 = h ~ R ~ R 0 i Microscopic Hamiltonian Cluster Hamiltonian L 3 L 3(A 1) smaller matrices in practice!! -- acts on the cluster CM and spins

14 Proof of Concept n-d scattering in the quartet channel Low energy EFT is known, only two body Shallow deuteron (large coupling)

15 Spin-1/2 Fermions a nn 19 fm, R 1.4 fm 1.5x10 6 a a np 24 fm, R 1.4 fm atom number 1.0x x10 5 Potassium-40 Regal et. al, Nature 2003 Tuning scattering lengths in lattice QCD with magnetic field scattering length (a o ) b B(G)

16 Weakly Bound Systems ia(p) = 2 i µ p cot 0 ip = 2 i µ 1/a + r 2 p2 + ip --- Large scattering length ia(p) 2 i µ 1/a + ip EFT non-perturbative a>>1/ apple 1+ 1 rp 2 2 1/a + ip ia(p) = C 0 i 1 + i µ C 0 2 p ) C 0 = 2 a µ large coupling Weinberg 90 Bedaque, van Kolck 97 Kaplan, Savage, Wise 98

17 Neutron-Deuteron System Convergence: L=7, b=1/100 MeV 1 Pine, Lee, Rupak, EPJA 2013 E (MeV) τ (MeV 1 ) - grouping R found efficient, more later 30 30

18 Lüscher s Method p cot = 1 L S( ), = pl 2 2 E fd = p2 2µ B + (p)[b B L ] - effective mass - topological factor (Bour, Hammer, Lee, Meißner 2013)

19 Neutron-Deuteron System = + + = + + T (p) =h(p)+ Z dqk(p, q)t (q) T (p) = 2 µ 1 p cot ip

20 Neutron-Deuteron Phase Shift 0 δ 0 (degree) breakup b=1/100 MeV -1 b=1/150 MeV -1 b=1/200 MeV -1 STM -60 Pine, Lee, Rupak, EPJA p (MeV)

21 Now, what can I do with an adiabatic Hamiltonian?

22 Primordial Deuterium p(n, )d

23 Warm Up p(n, )d = + + Exact analytic continuum result M C ( ) = 1 p (1/a + ip )( ip ), p = p p 2 + im When! 0 +, M C reduces to known M1 result Rupak & Lee, PRL 2013

24 Lattice p(n, )d Write M( ) = h B O EM i i p 2 X M E i using retarded Green s function x,y B(y)hy E 1 Ĥ s + i xieip x cluster Hamiltonian goes here LSZ reduction in QM

25 Continuum Extrapolation M s 0 = 1.012(18) s 0 = 1.009(16) s 0 = 1.005(07) p = 15.7 MeV p = 29.6 MeV p = 70.2 MeV φ s 0 = 1.022(15) s 0 = 1.031(02) s 0 = 1.031(02) p = 15.7 MeV p = 29.6 MeV p = 70.2 MeV δ = M/p 2 Fit : s 0 + s 1

26 Capture Amplitude M (MeV 2 ) δ = 0.6 δ = 0.4 δ = 0 δ = 0.6 δ = 0.4 δ = 0 continuum results = M/p 2 φ (deg) δ = 0.6 δ = 0.4 δ = 0 δ = 0.6 δ = 0.4 δ = p (MeV) Rupak & Lee, PRL 2013

27 Something still missing...

28 Something still missing... long range Coulomb

29 Something still missing... long range Coulomb O p 2 O µ p 2 p

30 Proton-Proton Fusion p + p! d + e + + e long and steady burning

31 Proton fusion in continuum EFT W + = + + Kong, Ravndal 1999 Coulomb ladder Peripheral scattering but how to calculate on the lattice? Consider elastic proton-proton scattering as a warmup

32 Spherical-wall method R w short(r) / j 0 (kr) cot s n 0 (kr), Coulomb(r) / F 0 (kr) cot sc + G 0 (kr) Adjust from free theory: j 0 (k 0 R w )=0 IR scale setting L/2 L/2 Hard spherical wall boundary conditions, Borasoy et al Carlson et al Even older?

33 Coulomb Subtracted Phase Shift δ sc (degree) Analytic b=1/100 MeV 1 b=1/200 MeV 1 Rupak, Ravi PLB p (MeV) 3% error in fits T =T c + T sc T c 2 µ T 2 µ e 2i 1 2ip e 2i( + sc) 1 2ip

34 Proton-proton fusion 3% fitting error propagates Gamow peak 6 kev T fi (MeV 1 ) Analytic b=1/100 MeV 1 b=1/200 MeV (p) = s 2 8 C 2 T fi (p) E (kev) EFT (0) 2.51 Lattice fit : (0) 2.49 ± 0.02 Kong, Ravndal 1999 Rupak, Ravi PLB 2014

35 More on adiabatic Projection d S 1 S D 1 D P 1 P a R ` (degress) a F 1 F Continuum 1 G4 a R =0.010 l.u. G a R =0.125 l.u. a R =0.250 l.u. 4 2 binning used 0 0 earlier p (MeV) Toy model, two-body Elhatisari, Lee, Meißner, Rupak, arxiv

36 Two-cluster simulation L 0 3 L 3 R in copy R w 0 (degrees) Breakup n d n d (EFT) p d p d (EFT) 60 Quartet S p (MeV) Matching: 15 to L t = Better lattice results than before Fermion-dimer

37 Improvement δ 0 (degree) breakup b=1/100 MeV -1 b=1/150 MeV -1 b=1/200 MeV -1 STM 0 (degrees) Breakup n d n d (EFT) p d p d (EFT) Quartet S p (MeV) p (MeV) Pine, Lee, Rupak (2013) Elhatisari, Lee, Meißner, Rupak (2016)

38 n-d doublet channel (/) p cot = 1/a + rp2 /2 1+p 2 /p 2 0 a 0.65 fm, r 150 fm, p 0 13 MeV / ERE form van Oers & Seagrave (1967) -what EFT for modified ERE Virtual state at 0.5 MeV Girard & Fuda (1979) - Efimov physics

39 Efimov plot Preliminary Higa, Rupak, Vaghani, van Kolck Shallow virtual to bound state

40 Efimov plot Preliminary Higa, Rupak, Vaghani, van Kolck Shallow virtual to bound state

41 Efimov plot Preliminary Higa, Rupak, Vaghani, van Kolck Shallow virtual to bound state lattice QCD with B field, even with heavy pions?

42 Phillips-Girard-Fuda () Preliminary () 3-body correlation

43 Adhikari-Torreao () Preliminary () Adhikari-Torreao Virtual

44 Conclusions Adiabatic Projection Method to derive effective two-body Hamiltonian Retarded Green s function for problems without long-range Coulomb Spherical wall method with adiabatic Hamiltonian when long range Coulomb important Efimov physics from lattice QCD?

45 Thank you