Radiative Capture Reactions in Lattice E ffective Field Theory Gautam Rupak Nuclear Reactions from Lattice QCD, INT March 12, 2013
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