low-energy reaction in eft · higa, rupak, vaghani, arxiv:1612.08959. reactions in lattice eft •...
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Low-energy Reaction in EFTGautam Rupak
Mississippi State University
Toward Predictive Theories of Nuclear Reactions Across the Isotopic Chart
Institute for Nuclear Theory, Seattle, Mar 8, 2017
Outline
• Background and motivation
• Weakly bound systems at low energy
• Adiabatic projection method - proof of concept
• neutron capture
• proton-proton fusion
• Pinhole algorithm
• Efimov physics
Motivations
• Low energy reactions dominate
• Need accurate cross sections but hard to measure experimentally
• Model-independent theoretical calculations important
Astrophysics
Theoretical• Weakly bound systems — opportunities • First principle calculation
Use Effective Field Theory
EFT: the long and short of it
• Identify degrees of freedom
!
!
• Determine from data (elastic, inelastic)
• EFT : ERE + currents + relativity
Hide UV ignorance!- short distance
IR explicit!- long distance
Not just Ward-Takahashi identity
L = c0 O(0) +c1 O
(1) +c2 O(2) + · · ·
cn
expansion in Q⇤
Key ingredient: power counting
Weakly Bound Systems
+ +
−C0
+ · · ·
Weinberg ’90Bedaque, van Kolck ’97
Kaplan, Savage, Wise ’98
--- Large scattering length
EFT non-perturbative
iA(p) =2⇡
µ
i
p cot �0 � ip=
2⇡
µ
i
�1/a+
r2p
2+ · · ·� ip
a >> 1/⇤
iA(p) ⇡ �2⇡
µ
i
1/a+ ip
1 +
1
2
rp2
1/a+ ip+ · · ·
�
large coupling1/a ⇠ p ⇠ Q ⌧ ⇤
--- Natural case 1/a, 1/r ⇠ ⇤ � p , expand in small p
iA(p) =�i
1C0
+ i µ2⇡p
) C0 ⇠ 2⇡a
µ
p-wave bound state
iA(p) =2⇡
µ
ip2
� 1aV
+ r12 p
2 + · · ·� ip3
Shallow states2 fine tuning
Bedaque, Hammer, van Kolck (2003)
Bertulani, Hammer, van Kolck (2002)
1 fine tuning
1/(aV )1/3 ⇠ p ⇠ r1 ⇠ Q ⌧ ⇤
Either way requires two operators at LO
1/aV ⇠ Q2⇤, p ⇠ Q ⌧ r1 ⇠ ⇤
A low energy example
(a) (b)
(c) (d)
7Li(n, �)8Li• Isospin mirror systems!
• Inhomogeneous BBNWhats the theoretical error?
7Li(n, �)8Li $ 7Be(p, �)8B
pZ =
1p2
s1
1 + 3�/r1
Asymptotic normalization
Need effective range r1 and binding energy at
leading orderTwo EFT operators
RADCAP: Bertulani!CDXS+: Typel
0
0.2
0.4
0.6
0.8
10-2 100 102 104 106
v.σ
(µ
b)
ELAB (eV)
Imhof A ’59
Imhof B ’59
Nagai ’05
Blackmon ’96
Lynn ’91
Imhof A ’59
Imhof B ’59
Nagai ’05
Blackmon ’96
Lynn ’91
Red: Tombrello (1965)!Blue: Davids-Typel (2003)!Black: EFT
Fernando,Higa, Rupak; EPJA 48, 24 (2012)
Rupak, Higa; PRL 106, 222501 (2011)
Another Example: ↵(3He, �)7Be
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Initial state: s-wave a0 ⇠ ⇤/Q2, r0 ⇠ 1/⇤
Final state: shallow p-wave (ground and excited) two operators
Need for low-energy phase shift, especially p-wave
LO contribution
Elastic scattering of 3He + ↵ with SONIK
Energy range about 500 keV to 3 MeV
Spokespersons: Connolly, Davids, Greife
Higa, Rupak, Vaghani, arXiv:1612.08959
Reactions in lattice EFT
• Consider: ;
• Need effective “cluster” Hamiltonian -- acts in cluster coordinates, spins,etc.
• Calculate reaction with cluster Hamiltonian. Many possibilities --- traditional methods, continuum halo/cluster EFT, lattice method
a(b, �)c a(b, c)d
Nuclear Lattice EFT Collaboration
Evgeny Epelbaum (Bochum) !Hermann Krebs (Bochum)!
Timo Lähde (Jülich)!Dean Lee (NCSU)!
Thomas Luu (Jülich)!Ulf-G. Meißner (Bonn)
Lattice EFT
3
n
n
p
p
Lattice chiral effective field theory
π
π
Review: D.L, Prog. Part. Nucl. Phys. 63 117-154 (2009)
N N
N N
= +π
+ + · · ·
N N
N N
N
N
N
N
N N
N N
π π
lattice
Effective Field Theory
Courtesy D. Lee
Auxiliary Field method
=
N
N
N
N
N
N
N
N
s
1p2⇡
Z 1
�1ds exp
�1
2
s2 +p�Cs (N†N)
�exp
�C
2
(N†N)
2
�=
Replace contact-interaction by interaction of each nucleon with a background auxiliary field using a Hubbard-Stratonovich transformation
Auxiliary Field method
Courtesy D. Lee
LO
SU(4) symme�⌧H
Courtesy D. Lee
Adiabatic Projection Method
Microscopic Hamiltonian Cluster Hamiltonian -- acts on the cluster CM and spins
L3(A�1)
L3
Initial state |~Ri
Evolved state |~Ri⌧ = e�⌧H |~Ri
⌧ h ~R0|H|~Ri⌧
Energy measurements in cluster basis. Divide by the norm matrix as these are not orthogonal basis
smaller matrices in practice!!
[N⌧ ]~R,~R0 =⌧ h~R|~R0i⌧
Proof of Concept
• n-d scattering in the quartet channel
• Low energy EFT is known, only two body
• Shallow deuteron (large coupling)
T (p) = h(p) +
ZdqK(p, q)T (q)
T (p) =2⇡
µ
1
p cot � � ip
LI = �g( †" ")(
†# #)
only two-body interaction
3-body Faddeev
0 0.1 0.2 0.3 0.4 0.5 0.6
0
5
10
15
20
25
τ (MeV−1
)
E (
MeV
)Convergence: L=7, b=1/100 MeV
−1
Pine, Lee, Rupak, EPJA 2013
Neutron-Deuteron System
- grouping R found efficient, more later⇠ 30⇥ 30
Neutron-Deuteron Phase Shift
0 50 100p (MeV)
-80
-60
-40
-20
0δ 0(d
egre
e) brea
kup
b=1/100 MeV-1
b=1/150 MeV-1
b=1/200 MeV-1
STM
Pine, Lee, Rupak, EPJA 2013
Faddeev Eqn!Lüscher’s method
Lattice
using retarded Green’s function
M(✏) =
✓p2
M� E � i✏
◆X
x,y
⇤B(y)hy|
1
E � Hs + i✏|xieip·x
h B |OEM| iiWrite
p(n, �)d
LSZ reduction in QM
cluster Hamiltonian goes here
= + + · · ·
Capture Amplitude10 20 50 70 100
10-5
10-4
10-3
10-2
δ = 0.6
δ = 0.4
δ = 0
δ = 0.6
δ = 0.4
δ = 0
10 20 50 70 100p (MeV)
0
20
40
60
80
δ = 0.6
δ = 0.4
δ = 0
δ = 0.6
δ = 0.4
δ = 0
|M|(M
eV−2)
φ(deg)
Rupak & Lee, PRL 2013
continuum results
� = ✏M/p2
Proton fusion in continuum EFT
Coulomb ladder
Peripheral scattering but how to calculate on the lattice?
Consider elastic proton-proton scattering as a warmup
Kong, Ravndal 1999
= + + · · ·
W+
Spherical-wall method
Rw
L/2�L/2
short
(r) / j0
(kr) cot �s � n0
(kr),
Coulomb
(r) / F0
(kr) cot �sc +G0
(kr)
Hard spherical wall boundary conditions, Borasoy et al. 2007Carlson et al. 1984Even older ?
Adjust from free theory:j0(k0Rw) = 0
IR scale setting
Proton-proton fusion
Rupak, Ravi PLB 2014Kong, Ravndal 1999⇤EFT (0) ⇡2.51
Lattice fit : ⇤(0) ⇡2.49± 0.02
0 200 400 600 800 1000 12000
0.01
0.02
0.03
0.04
0.05
E (keV)
Tfi (
Me
V−
1)
Analytic
b=1/100 MeV−1
b=1/200 MeV−1
3% fitting error propagates
Gamow peak 6 keV
⇤(p) =
s�2
8⇡C2⌘
|Tfi(p)|
Two-cluster simulation
copy
L 03
L3
Rin
Rw
R
Lt = 5
1 3 5 7 9 111
3
5
7
9
11
Lt = 10
R 0
1 3 5 7 9 111
3
5
7
9
11
Lt = 15
1 3 5 7 9 111
3
5
7
9
11
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Lt = 15
Fermion-dimer
−90
−75
−60
−45
−30
−15
0
0 20 40 60 80 100 120 140
Quartet−S
Brea
kup
−90
−75
−60
−45
−30
−15
0
0 20 40 60 80 100 120 140
Quartet−S
Brea
kup
p−d (EFT)
−90
−75
−60
−45
−30
−15
0
0 20 40 60 80 100 120 140
Quartet−S
Brea
kup
n−d (EFT)
−90
−75
−60
−45
−30
−15
0
0 20 40 60 80 100 120 140
Quartet−S
Brea
kup
n−d
−90
−75
−60
−45
−30
−15
0
0 20 40 60 80 100 120 140
Quartet−S
Brea
kup p−d
p (MeV)� 0
(degrees)
Matching: 15 to 82 Better lattice results than before
Elhatisari, Lee, Meißner, Rupak, EPJA 25 (2016)
alpha-alhpa scatteringWe have calculated alpha-alpha scattering up to NNLO using the adiabatic projection method. !s- and d-wave phase shift were calculated using the spherical wall method in the presence of long-range Coulomb force.
Elhatisari, Lee, Rupak, Epelbaum, Krebs, Lähde, Luu, Meißner, Nature 528, 111 (2015)
S-wave
D-wave
Ref. 22: Afzal et al. (1969)!Ref. 26: Higa et. al. (2008)
Beyond-Alpha Clusters
Lähde, Luu, Lee, Meißner, Epelbaum, Krebs, Rupak, EPJA 51, 92 (2015)
H ⌘ dhHphysical + (1� dh)HSU(4)
HSU(4) =1
2CSU(4)(N
†N)2
First verify in C-12
-100
-95
-90
-85
-80
-75
-70
0 2 4 6 8 10 12 14 16
E 12
(LO
) [M
eV]
Nt
-7.0e-5-7.0e-5 + ex
9.5
10
10.5
11
11.5
12
0 2 4 6 8 10 12 14 16
E 12
(NLO
) [M
eV]
Nt
-7.0e-5-7.0e-5 + ex
7.5
7.55
7.6
7.65
7.7
7.75
7.8
7.85
7.9
0 2 4 6 8 10 12 14 16
E 12
(EM
IB)
[MeV
]
Nt
-7.0e-5-7.0e-5 + ex
-15
-14
-13
-12
-11
-10
0 2 4 6 8 10 12 14 16
E 12
(3N
F) [
MeV
]
Nt
-7.0e-5-7.0e-5 + ex
Symmetry Sign Extrapolation
Alhassid et. al (94),! Koonin at. al (97)
Results for A=6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18 20 22
�eie�
Nt
dh = 1.00dh = 0.85dh = 0.75dh = 0.65dh = 0.55dh = 0.35
Sign problem ameliorated!- transfer amplitude phase
Tools to explore
Explore N 6= Z
hei✓i
Pin-hole Algorithm
i1,j1
i2,j2
i3,j3
i4,j4
i5,j5
i6,j6
i7j7
i8,j8
Calculates charge distribution in CM coordinates!!Importance sampling according to magnitude for
h f |ML0
t⇤ MLt/2⇢i1,j1,···iA,jA(n1, · · ·nA)MLt/2M
L0t⇤ | ii
X
i
| ~RCM � ~ri|2
CM location minimizes the rms radius
0
0.02
0.04
0.06
0.08
0.1
0.12a
b
c
12C
14C
16C
dens
ity (f
m-3
)
proton fit to experimentproton Lt = 7proton Lt = 9
proton Lt = 11proton Lt = 13proton Lt = 15neutron Lt = 7neutron Lt = 9
neutron Lt = 11neutron Lt = 13neutron Lt = 15
0 0.02 0.04 0.06 0.08
0.1 0.12
a
b
c
12C
14C
16C
dens
ity (f
m-3
)
0 0.02 0.04 0.06 0.08
0.1 0.12
0 2 4 6 8 10
a
b
c
12C
14C
16C
dens
ity (f
m-3
)
radius (fm)
Elhatisari et al. , arXiv 1702.05177
⇢i1,j1,···iA,jA(n1, · · ·nA) = : ⇢i1,j1(n1) · · · ⇢iA,jA(nA) :
Efimov Physics
n-d doublet channel
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p cot � =
�1/a+ rp2/2
1 + p2/p20a ⇠0.65 fm,
r ⇠� 150 fm,
p0 ⇠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
Higa, Rupak, Vaghani, van Kolck
Preliminary
Virtual State
Shallow virtual to bound state
lattice QCD with B field, even with heavy pions?
Virtual-Real Corssover
Adhikari-Torreao
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3-body correlation
Preliminary
Redundant Pole
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Ma, Phys. Rev. (1946) and (1947)!— Exponentially decaying pot.! Negative norm states!!Nelson, Rajagopal, Shastry (1971)
Preliminary
Preliminary
Conclusions and Outlook
• Halo/cluster EFT
• Adiabatic Projection Method to derive effective two-body Hamiltonian in lattice EFT
• Reaction calculations with or without Coulomb
• Pinhole algorithm
• Efimov physics
Thank you