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