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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Preliminary

Redundant Pole

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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