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
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
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
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
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
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 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)
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 = p2 1 1+3 /r 1 (c) (d)
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 = p2 1 1+3 /r 1 Need effective range r1 and binding energy at leading order
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 = p2 1 1+3 /r 1 Need effective range r1 and binding energy at leading order
v.σ (µb) 0.8 0.6 0.4 0.2 Imhof A 59 Imhof B 59 Nagai 05 Blackmon 96 Lynn 91 0 10-2 10 0 10 2 10 4 10 6 E LAB (ev) Red: Tombrello Rupak, Higa; PRL 106, 222501 (2011) Fernando,Higa, Rupak; EPJA 48, 24 (2012) RADCAP: Bertulani CDXS+: Typel Blue: Davids-Typel Black: EFT
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
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
Proof of Concept n-d scattering in the quartet channel Low energy EFT is known, only two body Shallow deuteron (large coupling)
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 1.0x10 6 5.0x10 5 Potassium-40 Regal et. al, Nature 2003 Tuning scattering lengths in lattice QCD with magnetic field scattering length (a o ) 0 2000 1000 0-1000 -2000 b -3000 215 220 225 230 B(G)
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
Neutron-Deuteron System 25 20 Convergence: L=7, b=1/100 MeV 1 Pine, Lee, Rupak, EPJA 2013 E (MeV) 15 10 5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 τ (MeV 1 ) - grouping R found efficient, more later 30 30
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)
Neutron-Deuteron System = + + = + + T (p) =h(p)+ Z dqk(p, q)t (q) T (p) = 2 µ 1 p cot ip
Neutron-Deuteron Phase Shift 0 δ 0 (degree) -20-40 breakup b=1/100 MeV -1 b=1/150 MeV -1 b=1/200 MeV -1 STM -60 Pine, Lee, Rupak, EPJA 2013-80 0 50 100 p (MeV)
Now, what can I do with an adiabatic Hamiltonian?
Primordial Deuterium p(n, )d
Warm Up p(n, )d = + + Exact analytic continuum result M C ( ) = 1 p 2 + 2 1 (1/a + ip )( ip ), p = p p 2 + im When! 0 +, M C reduces to known M1 result Rupak & Lee, PRL 2013
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
Continuum Extrapolation M 1.15 1.1 1.05 0 0.05 0.1 0.15 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 1 1.05 1 0.95 0.9 0.85 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 0.8 0 0.05 0.1 0.15 δ = M/p 2 Fit : s 0 + s 1
Capture Amplitude M (MeV 2 ) 10-2 10-3 10-4 10-5 80 10 20 50 70 100 δ = 0.6 δ = 0.4 δ = 0 δ = 0.6 δ = 0.4 δ = 0 continuum results = M/p 2 φ (deg) 60 40 20 0 δ = 0.6 δ = 0.4 δ = 0 δ = 0.6 δ = 0.4 δ = 0 10 20 50 70 100 p (MeV) Rupak & Lee, PRL 2013
Something still missing...
Something still missing... long range Coulomb
Something still missing... long range Coulomb O p 2 O µ p 2 p
Proton-Proton Fusion p + p! d + e + + e long and steady burning
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
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. 2007 Carlson et al. 1984 Even older?
Coulomb Subtracted Phase Shift δ sc (degree) 60 50 40 30 20 10 Analytic b=1/100 MeV 1 b=1/200 MeV 1 Rupak, Ravi PLB 2014 0 0 5 10 15 20 25 30 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
Proton-proton fusion 3% fitting error propagates Gamow peak 6 kev T fi (MeV 1 ) 0.05 0.04 0.03 0.02 Analytic b=1/100 MeV 1 b=1/200 MeV 1 0.01 (p) = s 2 8 C 2 T fi (p) 0 0 200 400 600 800 1000 1200 E (kev) EFT (0) 2.51 Lattice fit : (0) 2.49 ± 0.02 Kong, Ravndal 1999 Rupak, Ravi PLB 2014
More on adiabatic Projection 180 60 30 d 150 120 S 1 S0 50 40 30 25 20 15 D 1 D2 90 60 20 10 P 1 P1 10 5 a R ` (degress) 30 20 0 30 60 90 120 0 10 0 30 60 90 120 0 0 30 60 90 120 a 16 12 8 F 1 F3 8 6 4 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 0 30 60 90 120 0 30 60 90 120 p (MeV) Toy model, two-body Elhatisari, Lee, Meißner, Rupak, arxiv 1603.02333
Two-cluster simulation L 0 3 L 3 R in copy R w 0 (degrees) 0 15 30 45 Breakup n d n d (EFT) p d p d (EFT) 60 Quartet S 75 1 3 5 7 9 11 1 3 0.14 0.12 90 0 20 40 60 80 100 120 140 p (MeV) Matching: 15 to 82 5 7 L t = 15 0.1 0.08 0.06 Better lattice results than before 0.04 9 11 0.02 Fermion-dimer
Improvement δ 0 (degree) 0-20 -40 breakup b=1/100 MeV -1 b=1/150 MeV -1 b=1/200 MeV -1 STM 0 (degrees) 0 15 30 45 Breakup n d n d (EFT) p d p d (EFT) -60 60 Quartet S -80 0 50 100 p (MeV) 75 90 0 20 40 60 80 100 120 140 p (MeV) Pine, Lee, Rupak (2013) Elhatisari, Lee, Meißner, Rupak (2016)
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
Efimov plot Preliminary Higa, Rupak, Vaghani, van Kolck Shallow virtual to bound state
Efimov plot Preliminary Higa, Rupak, Vaghani, van Kolck Shallow virtual to bound state
Efimov plot Preliminary Higa, Rupak, Vaghani, van Kolck Shallow virtual to bound state lattice QCD with B field, even with heavy pions?
Phillips-Girard-Fuda () Preliminary () 3-body correlation
Adhikari-Torreao () Preliminary - - - - () Adhikari-Torreao Virtual
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?
Thank you