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Radiative Capture Reactions in Lattice Effective Field Theory
Gautam Rupak
Nuclear Reactions from Lattice QCD, INT March 12, 2013
Outline
• Motivation
• Continuum EFT for reactions
• Lattice EFT for reactions
Reaction theory:
• Reaction theory for nuclear experiments, e.g. FRIB
• Nuclear astrophysics where data might be lacking
• Reactions are more fun than static properties
Examples:
the list goes on
d(p, �)3He,7 Be(p, �)8B,14 C(n, �)15C, · · · ,
Only halo systems for now ...
Halo systems• Characterized small neutron/proton separation energy.
Large size.
• Interesting three-body physics: All-bound, Tango, Samba (12Be), Borromean (11Li)
• Exotic physics near the drip line
Some details on
• Isospin mirror systems
• Inhomogeneous BBN
7Li(n, �)8Li
Whats the theoretical error?
7Li(n, �)8Li $ 7Be(p, �)8B
EFT• Identify degrees of freedom
• Determine from data (elastic, inelastic)
• EFT ERE + currents + relativity
L = c0 O(0) +c1 O
(1) +c2 O(2) + · · ·
Hide UV ignorance IR explicit
Not just Ward identity
cn
90% capture to gs
p=84 MeVexcited state
Cohen, Gelman, van Kolck ’04
Look at E1 transition
-- Initial state: s-wave
single operator fitted to scattering length
-- Ground state: p-wave
We also include the excited state and the p-wave 3+ resonance (M1 transition)
p-wave needs two operators
a
(aV , r1)
iA(p) =2⇡
µ
i
p cot �0 � ip=
2⇡
µ
i
�1/a+
r2p
2+ · · ·� ip
a, r ⇠ 1/⇤ << 1/p
iA(p) ⇡ �2⇡
µ
i
1/a+ ip
1 +
1
2
rp2
1/a+ ip+ · · ·
�a >> 1/⇤
iA(p) =�i
1C0
+ i µ2⇡p
) C0 =2⇡a
µ
+ +
!C0
+ · · ·
Weinberg ’90Bedaque, van Kolck ’97
Kaplan, Savage, Wise ’98
--- Natural caseexpand in small p, EFT perturbative
--- Large scattering length
EFT non-perturbative
p-wave
Shallow systems2 fine tuning1 fine tuning
Bertulani, Hammer, van Kolck ’02 Bedaque, Hammer, van Kolck ’03
Requires two non-perturbative operators at LO
iA(p) =2⇡
µ
ip2
� 1aV
+ r12 p
2 + · · ·� ip3
...
1
n
+ + + · · ·
Residues and polesn-point function
LSZ reduction: G(n) ⇠nY
i=1
pZi
=Z
p20 � p2 �m2
Zd
p0 � p2
4M + �2
M
, Zd =8⇡�
M2
1
1� ⇢�
d(r) =
r�
2⇡
1
1� ⇢�
e�r�
r=
rµ2
4⇡2Zd
e�r�
r
Consider a scalar theory
For deuteron:
+! |!| =
d�
d cos ✓=
1
32⇡s
k
p|�|2
Capture Cross Section
gives 1/p dependence
Analytic result, depends onpZ =
1p2
s1
1 + 3�/r1
Need r1 at leading order
0 0.2 0.4 0.6 0.8En(MeV)
010203040506070
σ(µb) Red: Tombrello
Blue: Davids-TypelBlack: EFT
“Effective range” contribution
Rupak, Higa; PRL 106, 222501 (2011)
0
0.2
0.4
0.6
0.8
10-2 100 102 104 106
v.!
(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: TombrelloBlue: Davids-TypelBlack: EFTFernando,Higa, Rupak; EPJA 48, 24 (2012)
Rupak, Higa; PRL 106, 222501 (2011)
0
10
20
30
40
50
60
70
0 0.2 0.4 0.6 0.8 1
! (
µb)
ELAB (MeV)
Imhof A ’59
Imhof B ’59
Nagai ’05
Imhof A ’59
Imhof B ’59
Nagai ’05
Fernando,Higa, Rupak; EPJA 48, 24 (2012)
Lessons learned--- Tuning potential to reproduce bound state energy is not
sufficient to get the wave function renormalization constant.
--- In the strong sector directly applies to 7Be(p, �)8B
General problem : How to constrain low-energy nuclear theory?
Neutron Capture/Coulomb Dissociation on Carbon-14
Ê
Ê
Ê
Ê
Ê
ÊÊ
ÊÊ
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
ÊÊ
0.0 0.5 1.0 1.5 2.00.0
0.1
0.2
0.3
0.4
ErelHMeVL
dB1HE1L
dErelHaf
m2
MeVL
Data: T. Nakamura et al., PRC, 79, 035805 (2009)
Power counting
a1 = �n1/Q3
r1 = 2n2Q
n1=0.7, n2=1, Q=40 MeV Rupak, Fernando, Vaghani, PRC 86, 044608 (2012)
Coulomb Dissociation
Coulomb Dissociation of Carbon-19
0.0 0.5 1.0 1.5 2.00
50
100
150
200
250
⇥ �deg.⇥
d⇤⇤d� cm
�barns⇤sr.⇥
Acharya & Phillips, arxiv:1302.4762
0 1 2 3 40.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
E �MeV⇥
d�⇤dE�b
arns⇤MeV
⇥
(a, r) fitted
Lattice EFT for Halo Nuclei• Interested in
• Need interaction between clusters
• Calculate capture with cluster interaction. Many possibilities --- traditional methods, continuum EFT, lattice method
a(b, �)c
Nuclear Lattice Effective Field Theory collaboration
Evgeny Epelbaum, Hermann Krebs,
Timo LahdeDean Lee
Ulf-G. Meissner
Adiabatic HamiltonianMicroscopic Hamiltonian
Adiabatic Hamiltonian for the clusters L3
-- acts on the cluster c.m. and spins
Lee, Pine, Rupak
L3(A�1)
Blume, Greene 2000
1D toy atom-dimer problem
Microscopic Hamiltonian: -2.130490, -2.130490, 0.1189620, 0.1189620, ...
Adiabatic Hamiltonian: -2.130505, -2.130493, 0.1189604, 0.1189781, ...
That was in 2012, Can do better now
n-d scattering in quartet channel
Microscopic Hamiltonian: 7.152, 23.37, 23.37, 23.37, 29.61, 29.61, 40.34, ...
Adiabatic Hamiltonian: 7.166, 23.42, 23.42, 23.42, 29.74, 29.74, 40.49, ...
Warm up p(n, �)d
Exact analytic continuum result
using retarded Green’s function
MC(✏) =1
p2 + �2� 1
(1/a+ ip✏)(� � ip✏), p✏ =
pp2 + iM✏
When ,✏ ! 0+ MC reduces to known M1 resultRupak, 2000
M(✏) =
✓p2
M� E � i✏
◆X
x,y
⇤B(y)hy|
1
E � Hs + i✏|xieip·x
h B |OEM| iiWrite
Lee & Rupak
Lattice EFT results0 0.05 0.1 0.15
1
1.05
1.1
1.15
s0 = 1.012(18)
s0 = 1.009(16)
s0 = 1.005(07)
p = 15.7 MeVp = 29.6 MeVp = 70.2 MeV
0 0.05 0.1 0.15!
0.8
0.85
0.9
0.95
1
1.05
s0 = 1.022(15)
s0 = 1.031(02)
s0 = 1.031(02)
p = 15.7 MeVp = 29.6 MeVp = 70.2 MeV
|M|
!
Magnitude, argument normalized to continuum
Rupak & Lee, arXiv:1302.4158
� = ✏M/p2
Continuum extrapolation10 20 50 70 100
10-5
10-4
10-3
10-2
! = 0.6
! = 0.4
! = 0
! = 0.6
! = 0.4
! = 0
10 20 50 70 100p (MeV)
0
20
40
60
80
! = 0.6
! = 0.4
! = 0
! = 0.6
! = 0.4
! = 0
|M|(M
eV!2)
!(deg)
Rupak & Lee, arXiv:1302.4158
Lattice QCD to lattice EFT
• Constrain Hamiltonian in elastic channels
• Electroweak currents ---
1. Fit pionless EFT at unphysical pion mass.
2.Match observables in pionless and chiral EFT.
3. Extrapolate to physical pion mass.
• Coulomb effect (in discussion)
Detmold, Savage 2004
Conclusions
• Capture reactions
• In progress
• Capture reactions in lattice EFT
7Be(p, �)8B
7Li(n, �)8Li, 14C(n, �)15C, etc.
Thank you