Nuclear Quantum Monte CarloRobert B. Wiringa, Physics Division, Argonne National Laboratory

WORK WITH

Joe Carlson, Los Alamos

Victor Flambaum, New South Wales

Laura Marcucci, Pisa

Ken Nollett, Argonne

Muslema Pervin, Argonne

Steve Pieper, Argonne

Rocco Schiavilla, JLab & ODU

WORK NOT POSSIBLE WITHOUT EXTENSIVE COMPUTER RESOURCES

Argonne LCRC (Jazz) & MCS Division (BlueGene/L)Argonne Leadership Computing Facility (BlueGene/P)

NERSC IBM SP’s (Seaborg, Bassi)UNEDF SciDAC & CNS INCITE

Work supported by U.S. Department

of Energy, Office of Nuclear PhysicsPhysics Division

Ab Initio CALCULATIONS OF L IGHT NUCLEI

GOALS

Understand nuclei at the level of elementary interactions between individual nucleons, including

• Binding energies, excitation spectra, relative stability

• Densities, electromagnetic moments, transition amplitudes, cluster-cluster overlaps

• Low-energyNA &AA scattering, astrophysical reactions

REQUIREMENTS

• Two-nucleon potentials that accurately describe elasticNN scattering data

• Consistent multi-nucleon potentials and electroweak current operators

• Precise methods for solving the many-nucleon Schrodinger equation

RESULTS

• Quantum Monte Carlo methods can evaluate realistic Hamiltonians accurate to∼1–2%

• About 100 states calculated forA ≤ 12 nuclei in good agreement with experiment

• Applications to elastic & ineleastice, π scattering,(e, e′p), (d, p) reactions, etc.

• 5He =nα scattering and low-energy electroweak astrophysical reactions

• We can explore “fine-tuning” issues in nuclear forces and spectra

OUTLINE

Hamiltonian

Variational Monte Carlo

Green’s function Monte Carlo

Binding energy results

Nolen-Schiffer anomaly &8Be isospin-mixing

Densities and radii

Meson-exchange currents and magnetic moments

M1, E2, F, GT transitions

NA scattering & astrophysical reactions

Nucleon momentum distributions &A(e, e′pN)

Sensitivity of nuclear spectra to forces & hadron masses

HAMILTONIAN

H =X

i

Ki +X

i<j

vij +X

i<j<k

Vijk

Ki = KCIi +KCSB

i ≡ −~

2

4[(

1

mp+

1

mn) + (

1

mp−

1

mn)τzi]∇

2i

Argonne v18 (AV18)

vij = vγij + vπ

ij + vIij + vS

ij =P

p vp(rij)Opij

vγij : pp, pn & nn electromagnetic terms

vπij ∼ [Yπ(rij)σi · σj + Tπ(rij)Sij ] ⊗ τ i · τj

vIij =

P

p IpT 2

π (rij)Opij

vSij =

P

p[P p +Qpr + Rpr2]W (r)Opij

Opij = [1, σi · σj , Sij ,L · S,L2,L2(σi · σj), (L · S)2]

+ [1, σi · σj , Sij ,L · S,L2,L2(σi · σj), (L · S)2] ⊗ τi · τj

+ [1, σi · σj , Sij ,L · S] ⊗ Tij

+ [1, σi · σj , Sij ,L · S] ⊗ (τi + τj)z

Sij = 3σi · rijσj · rij − σi · σj Tij = 3τizτjz − τi · τj

0 0.5 1 1.5 2r (fm)

-150

-100

-50

0

50

100

150

200

250V

(M

eV)

vc

vτvσvστv

tv

tτ

Argonne v18

Fits Nijmegen PWA93 data base of 1787pp & 2514np observables forElab ≤ 350 MeV with

χ2/datum = 1.1 plusnn scattering length and2H binding energy

0 100 200 300 400 500 600E

lab (MeV)

-40

-20

0

20

40

60

δ (d

eg)

1S

0

Argonne v18

np

Argonne v18

pp

Argonne v18

nn

Argonne v18pq

np

Argonne v18pq

pp

Argonne v18pq

nn

SAID 7/06 np

0 100 200 300 400 500 600E

lab (MeV)

-40

-30

-20

-10

0

10

20

δ (d

eg)

3P

0

Argonne v18

np

Argonne v18

pp

Argonne v18

nn

Argonne v18pq

np

Argonne v18pq

pp

Argonne v18pq

nn

SAID 7/06 np

0 100 200 300 400 500 600E

lab (MeV)

-40

0

40

80

120

160

δ (d

eg)

3S

1

Argonne v18

np

Argonne v18pq

np

SAID 7/06 np

0 100 200 300 400 500 600E

lab(MeV)

-40

-30

-20

-10

0

δ (d

eg)

1P

1

Argonne v18

np

Argonne v18pq

np

SAID 7/06 np

THREE-NUCLEON POTENTIALS

Urbana IX (UIX)

Vijk = V 2πPijk + V R

ijk

Illinois 2 (IL2,IL7)

Vijk = V 2πPijk + V 2πS

ijk + V 3π∆Rijk + V R

ijk

VARIATIONAL MONTE CARLO

Minimize expectation value ofH

EV =〈ΨV |H|ΨV 〉

〈ΨV |ΨV 〉≥ E0

Trial function (s-shell nuclei)

|ΨV 〉 =

2

41 +X

i<j<k

UT NIijk

3

5

"

SY

i<j

(1 + Uij)

#

|ΨJ〉

|ΨJ〉 =

"

Y

i<j

fc(rij)

#

|ΦA(JMTT3)〉

|Φd(1100)〉 = A| ↑ p ↑ n〉 ; |Φα(0000)〉 = A| ↑ p ↓ p ↑ n ↓ n〉

Uij =X

p=2,6

up(rij)Opij ; UT NI

ijk = −ǫVijk(rij , rjk, rki)

Functionsfc(rij) andup(rij) obtained from coupled differential equations withvij .

Correlation functions

0 1 2 3 4 5 6 7 8r (fm)

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

fc,utτ (2H)

fc,utτ (3H)

fc,utτ (4He)

fsp,utτ (6Li)

φp (6Li)

fc

10 utτ

φp

fsp

[f c(4He)]

3

[f c(3H)]

2

Trial function (p-shell nuclei)

|ΨJ 〉 = A

8

<

:

Y

i<j≤4

fss(rij)X

LS[n]

βLS[n]

Y

k≤4<l≤A

fsp(rkl)Y

4<l<m≤A

fpp(rlm)

˛

˛

˛Φα(0000)1234

Y

4<l≤A

φLS[n]p (Rαl)

˘

[Yml1 (Ωαl)]LML

⊗ [χl(12ms)]SMS

¯

JM[νl(

12t3)]T T3

E

9

=

;

Diagonalization

in βLS[n] basis to produce energy spectraE(Jπx ) and orthogonal excited statesΨV (Jπ

x )

Expectation values

ΨV (R) represented by vector with2A × (AZ) spin-isospin components for each space

configurationR = (r1, r2, ..., rA); Expectation values are given by summation over samples

drawn from probability distributionW (R) = |ΨP (R)|2:

〈ΨV |O|ΨV 〉

〈ΨV |ΨV 〉=

X Ψ†V (R)OΨV (R)

W (R)/

X Ψ†V (R)ΨV (R)

W (R)

Ψ†Ψ is a dot product andΨ†OΨ a sparse matrix operation.

GREEN’ S FUNCTION MONTE CARLO

Projects out lowest energy state from variational trial function

Ψ(τ) = exp[−(H − E0)τ ]ΨV =X

n

exp[−(En − E0)τ ]anψn

Ψ(τ → ∞) = a0ψ0

Evaluation ofΨ(τ) done stochastically in small time steps∆τ

Ψ(Rn, τ) =

Z

G(Rn,Rn−1) · · ·G(R1,R0)ΨV (R0)dRn−1 · · · dR0

using the short-time propagator accurate to order(∆τ)3 (Vijk term omitted for simplicity)

Gαβ(R,R′) = eE0∆τG0(R,R′)〈α|

"

SY

i<j

gij(rij , r′ij)

g0,ij(rij , r′ij)

#

|β〉

where the free many-body propagator is

G0(R,R′) = 〈R|e−Kτ |R′〉 =

»r

m

2π~2τ

–3A

exp

»

−(R − R′)2

2~2τ/m

–

andg0,ij andgij are the free and exact two-body propagators

gij(rij , r′ij) = 〈rij |e

−Hijτ |r′ij〉

Mixed estimates

〈O(τ)〉 =〈Ψ(τ)|O|Ψ(τ)〉

〈Ψ(τ)|Ψ(τ)〉≈ 〈O(τ)〉Mixed + [〈O(τ)〉Mixed − 〈O〉V ]

〈O(τ)〉Mixed =〈ΨV |O|Ψ(τ)〉

〈ΨV |Ψ(τ)〉; 〈H(τ)〉Mixed =

〈Ψ(τ/2)|H|Ψ(τ/2)〉

〈Ψ(τ/2)|Ψ(τ/2)〉≥ E0

Propagator cannot containp2, L2, or (L · S)2 operators:

Gβα(R′,R) has onlyv′8〈v18 − v′8〉 computed perturbatively with extrapolation (small for AV18)

Fermion sign problem limits maximumτ :

Gβα(R′,R) brings in lower-energy boson solution

〈ΨV |H|Ψ(τ)〉 projects back fermion solution. but statistical errors grow exponentially

Constrained-path propagation, removes steps that have

Ψ†(τ,R)Ψ(R) = 0

Possible systematic errors reduced by10−20 unconstrained steps before evaluating observables.

GFMC propagation of three states in6Li

0 0.5 1 1.5 2 2.5

-32

-30

-28

-26

-24

τ (MeV-1)

E(τ

) (

MeV

)

2+

3+

gs

6Li

α+d

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0E

nerg

y (M

eV)

AV18

IL2 Exp

1+

2H 1/2+

3H

0+

4He 1+3+2+

6Li 3/2−1/2−7/2−5/2−

7Li 2+1+

0+

3+

4+

8Li0+2+

4+

1+3+

8Be

3/2−1/2−5/2−7/2−

9Li

9/2−

5/2+

7/2+

3/2+

1/2+

3/2−5/2−1/2−

7/2−

9Be

3+1+

0+2+

4+

0+2+,3+

10Be3+1+2+4+

10B

3+

1+

2+4+

0+

12C

Argonne v18With Illinois-2

GFMC Calculations

-70

-65

-60

-55

-50

-45

-40

-35

-30

-25

-20E

nerg

y (M

eV)

IL2IL7

Exp

0+

4He

1+

0+2+

2+0+

6He

1+

1+

3+

2+1+

6Li

3/2−

3/2−

1/2+

5/2+

1/2−

3/2−

5/2−

(5/2)−

7He

5/2−5/2−

3/2−1/2−

7/2−5/2−5/2−

7/2−

3/2−1/2−

7Li

1+0+2+

0+

2+

8He 2+

2+

2+1+

0+

3+1+

4+

3+

1+

8Li

1+

0+

2+

4+

2+1+

3+4+

0+

2+

3+2+

8Be

1/2−

9He

3/2−

3/2−1/2−

?5/2−

7/2−

9Li

9/2−3/2−

5/2+

7/2+

3/2+

1/2+

3/2+

3/2−1/2+5/2−1/2−5/2+3/2+

7/2−

3/2−

5/2−

9/2+

5/2+7/2+

9Be

0+

10He

2+

0+

(1-,2-)1+

10Li0+

3+

2+

1+

0+2+1−

2−

2+

4+

0+

3,2+

10Be 3+1+

2+

2−

4+

1+

3+

1+2+

3+

10B

Argonne v18With Illinois Vijk

GFMC Calculations22 October 2007

Nolen-Schiffer Anomaly

Extract isovector [CSB∝ (τ1 + τ2)z ] &

isotensor [CD ∝ T12] energy components:

EA,T (Tz) =X

n≤2T

an(A, T )Qn(T, Tz)

Q0 = 1 ; Q1 = Tz ; Q2 = 12(3T 2

z − T 2)

StrongType III CSB (constrained±20% by

nn scattering length) fixes isovector terms.

Strong Type II CD (constrained by1S0 pp

andnp scattering) overdoes isotensor;

need P-waveNN scattering constraint?

Coulomb

E.M.+K.E.

Total

0.8 1 1.2 1.4 1.6

Ratio to experiment

Isovector

3H-3He

6He-6Li *-6Be

7He-7Li *-7Be*-7B

7Li-7Be

8He-8Li *-8Be*-8B*-8C

8Li-8Be*-8B

9Li-9Be*-9B*-9C

9Be-9B

10Be-10B*-10C

Isotensor

6He-6Li *-6Be

7He-7Li *-7Be*-7B

8He-8Li *-...-8C

8Li-8Be*-8B

9Li-9Be*-9B*-9C

10Be-10B*-10C

GFMC isovector and isoscalar energy coefficients for AV18+IL7 in keV

an(A, T ) KCSB vC1(pp) vγ,R vCSB+vCD Total Expt.

a1(3, 12) 14 650(0) 28 65(0) 757(0) 764

a1(6, 1) 18 1118(2) 14 54(1) 1203(2) 1173

a1(7, 12) 23 1446(3) 36 86(1) 1592(4) 1641

a1(7, 32) 17 1270(3) 9 52(1) 1348(4) 1373

a1(8, 1) 23 1660(4) 19 78(1) 1780(5) 1770

a1(8, 2) 22 1624(4) 8 71(1) 1726(4) 1659

a1(9, 12) 19 1709(6) 4 55(1) 1786(7) 1851

a1(9, 32) 26 1974(6) 19 90(1) 2109(7) 2104

a1(10, 1) 25 2123(7) 18 55(1) 2250(8) 2329

a2(6, 1) 167(0) 19 109(8) 295(9) 224

a2(7, 32) 129(0) 7 38(5) 174(5) 175

a2(8, 1) 137(1) 4 −10(8) 132(8) 145

a2(8, 2) 144(0) 6 39(3) 189(3) 127

a2(9, 32) 153(1) 7 38(8) 198(9) 176

a2(10, 1) 166(1) 12 119(18) 297(19) 241

Isospin-mixing in8Be

Experimental energies of 2+ states

Ea = 16.626(3) MeVΓαa = 108.1(5) keV

Eb = 16.922(3) MeVΓαb = 74.0(4) keV

Isospin mixing of 2+;1 and 2+;0*

states due to isovector interactionH01:

Ψa = βΨ0 + γΨ1 ; Ψb = γΨ0 − βΨ1

decay throughT = 0 component only

Γαa/Γ

αb = β2/γ2 ⇒ β = 0.77 ; γ = 0.64

Ea,b =H00 +H11

2

±

s

„

H00 −H11

2

«2

+ (H01)2

H00 = 16.746(2) MeV

H11 = 16.802(2) MeV

H01 = −145(3) keV-60

-56

-52

-48

-44

-40

-36

Ene

rgy

(MeV

)

8Li

2+;1

3P[431]

7Li+n

1+;1

3+;13

D[431]

8Be

α+α0

+;01

S[44]

2+;01

D[44]

4+;01

G[44]

2+;0+1

1+;1

1+;0

3+;0+1

8B

2+;1

7Be+p

1+;1

3P[431]

Isospin-mixing matrix elements in keV

H01 KCSB V CSB Vγ (Coul) (MM)

2+;1⇔2+2 ;0 GFMC −115(3) −3.1(2) −21.3(6) −90.3(26) −78.3(25) −12.0(2)

Barker −145(3) −67

1+;1⇔1+;0 GFMC −102(4) −2.9(2) −18.2(6) −80.3(30) −79.5(30) −0.8(2)

Barker −120(1) −54

3+;1⇔3+;0 GFMC −90(3) −2.5(2) −14.8(6) −73.1(21) −60.9(21) −12.2(2)

Barker −62(15) −32

2+;1⇔2+1 ;0 GFMC −6(2) −0.4(2) −1.3(4) −4.4(12)

Barker, Nucl.Phys.83, 418 (1966)

Coulomb terms are about half ofH01, but magnetic moment and strongType III CSBare

relatively more important than in Nolen-Schiffer anomaly;still missing≈ 20% of strength.

StrongType IV CSBwill also contribute (probably best nuclear structure place to look):

V CSBIV = (τ1 − τ2)z(σ1 − σ2) · L v(r)

+ (τ1 × τ2)z(σ1 × σ2) · L w(r)

Preliminary result:vγ ∝ µn ∼ −2 keV & wπ ∝ (Mn −Mp) ∼ −2 keV.

SINGLE-NUCLEON DENSITIES

ρp,n(r) =X

i

〈Ψ|δ(r − ri)1 ± τi

2|Ψ〉

0 0.5 1 1.5 2 2.5 30.00

0.05

0.10

0.15

r (fm)

Den

sity

(fm

-3)

4He6He - Proton6He - Neutron8He - Proton8He - Neutron

0 2 4 6 8 10 1210-5

10-4

10-3

10-2

10-1

r (fm)

Den

sity

× r2

(f

m-1

)

4He6He - Proton6He - Neutron8He - Proton8He - Neutron

RMS radii

rn rp rc Expt4He 1.45(1) 1.45(1) 1.67(1) 1.681(4)*6He 2.86(6) 1.92(4) 2.06(4) 2.072(9)†

8He 2.79(3) 1.82(2) 1.94(2) 1.961(16)‡

*Sick, PRC77, 041302(R) (2008)

†Wang,et al., PRL93, 142501 (2004)

‡Mueller,et al., PRL99, 252501 (2007)

TWO-NUCLEON DENSITIES

ρpp(r) =X

i<j

〈Ψ|δ(r − |ri − rj |)1 + τi

2

1 + τj

2|Ψ〉

0 0.5 1 1.5 2 2.5 30.000

0.005

0.010

0.015

0.020

r12 (fm)

ρ pp

(fm

-3)

4He6He - Proton8He - Proton

RMS radii

rpp rnp rnn

4He 2.41 2.35 2.416He 2.51 3.69 4.408He 2.52 3.58 4.37

FIG. 3: (Color online) Experimental charge radii of berylliumisotopes from isotope shift measurements (•) compared withvalues from interaction cross section measurements () andtheoretical predictions: Greens-Function Monte-Carlo calcu-lations (+) [20, 21], Fermionic Molecular Dynamics () [22],ab-initio No-Core Shell Model (!) [12, 23, 24].

Nortershauser,et al., arXiv:0809.2607

Nuclear Electromagnetic Currents

Marcucci et al. (2005)

j j= (1)

+ j(2)

(v) + +

+ j(3)

(V2π

)

ππ ρ,ω

transverse

• Gauge invariant:

q ·

[

j(1) + j(2)(v) + j(3)(V 2π)]

=[

T + v + V 2π , ρ]

ρ is the nuclear charge operator

MAGNETIC MOMENTS

-2

-1

0

1

2

3

µ

EXPT

VMC(IA)

GFMC(IA)

VMC(MEC)

GFMC(MEC)

2H

3H

3He

6Li

7Li

7Be

8Li 8B

9Be

Marcucci, Pervin,et al. PRC78, 065501 (2008)

M1, E2, F, GT transitions

E2 = eP

k12

ˆ

r2kY2(rk)

˜

(1 + τkz)

M1 = µNP

k[(Lk + gpSk)(1 + τkz)/2

+ gnSk (1 − τkz)/2]

F =P

kτk± ; GT =P

kσkτk±

Pervin, Pieper & Wiringa, PRC76, 064319 (2007)

CKNCSMVMC(IA)GFMC(IA)VMC(MEC)GFMC(MEC)Expt.

0 1 2 3 4

Ratio to experiment

6He(2+;0→0+;0) B(E2)

6Li(3+;0→1+;0) B(E2)

6Li(2+;0→1+;0) B(E2)

6Li(0+;1→1+;0) B(M1)

6Li(2+;1→1+;0) B(M1)

7Li(1/2-→3/2-) B(E2)

7Li(1/2-→3/2-) B(M1)

7Li(7/2-→3/2-) B(E2)

7Be(1/2-→3/2-) B(M1)

6He(0+)→6Li(1+) log(ft)

7Be(3/2-)→7Li(3/2-) log(ft)

7Be(3/2-)→7Li(1/2-) log(ft)

7Be(3/2-) B.R.

GFMC FOR SCATTERING STATES

GFMC treats nuclei as particle-stable system – should be good for energies of narrow resonances

Need better treatment for locations and widths of wide states and for capture reactions

METHOD

• Pick a logarithmic derivative,χ, at some large boundary radius (RB ≈ 9 fm)• GFMC propagation, using method of images to preserveχ atR, findsE(RB, χ)

• Phase shift,δ(E), is function ofRB , χ,E• Repeat for a number ofχ until δ(E) is mapped out

Example for5He(12

−)

• “Bound-state” boundarycondition does notgive stable energy;Decaying to n+4Hethreshold

• Scattering boundarycondition producesstable energy.

0 0.1 0.2 0.3 0.4-27

-26

-25

-24

-23

τ (MeV-1)

E(τ

) (

MeV

)

"bound state"

Log-deriv = -0.168 fm-1

GFMC for 5He asn+4He Scattering States

Black curves: Hale phase shifts fromR-matrix analysis up toJ = 92

of dataAV18 with noVijk underbinds5He(3/2−) & overbinds5He(1/2−)AV18+UIX improves5He(1/2−) but still too small spin-orbit splittingAV18+IL2 reproduces locations and widths of bothP -wave resonances

0 1 2 3 4 50

30

60

90

120

150

180

Ec.m. (MeV)

δ LJ

(deg

rees

)

12

+

12

-

AV18AV18+UIXAV18+IL2R-Matrix

32

-

AV18+IL2

0 1 2 3 4 50

1

2

3

4

5

6

7

Ec.m. (MeV)

σ LJ

(b)

12

+

12

-

32

-

R-Matrix

Pole location

Nollett, et al., PRL99, 022502 (2007)

RADIATIVE CAPTURE REACTIONS

σ(Ecm) =8π

3

α

vrel

q

1 + q/mLi

X

LSJℓ

h˛

˛

˛ELSJ

ℓ (q)˛

˛

˛

2+

˛

˛

˛MLSJ

ℓ (q)˛

˛

˛

2 i

3H(α, γ)7Li 3He(α, γ)7Be

Sources of7Li in big bang

7Be key to solarνe production

0 0.5 1 1.5 2 2.50.2

0.3

0.4

0.5

0.7

C.M. Energy (MeV)

S F

acto

r (M

eV b

)

3He(α,γ)7Be

3H(α,γ)7Li (×4)

Nollett, PRC63, 054002 (2001)

SINGLE-NUCLEON MOMENTUM DISTRIBUTIONS

ρστ (k) =

Z

dr′1 dr1 dr2 · · · drA ψ

†JMJ

(r′1, r2, . . . , rA) e−ik·(r1−r′1)Pστ (1)ψJMJ

(r1, r2, . . . , rA)

0 1 2 3 4 510-3

10-2

10-1

100

101

102

103

k (fm-1)

ρ στ(

k)7Li - AV18+UIX

ρp↓ ρn↑

ρp↑

ρn↓

ρp↑ - ρp↓

TWO-NUCLEON & CLUSTER-CLUSTER DISTRIBUTIONS

0 1 2 3 4 510-3

10-2

10-1

100

101

102

103

k (fm-1)

ρ 12(

k)

4He - AV18/UIX

ρnp

ρpp

0 1 2 3 4 510-4

10-3

10-2

10-1

100

101

102

103

k (fm-1)

ρ(k)

4He - AV18/UIX

ρddρtp

ρp

TWO-NUCLEON KNOCKOUT – A(e, e′pN)

JLAB experiment for12C(e, e′pN) measured back-to-backpp andnp pairs

Pairs withqrel = 2–3 fm−1 shownp/pp ratio≈ 10–20 Subediet al., Science320, 1476 (2008)

0 1 2 3 4 5q (fm

-1)

10-1

100

101

102

103

104

105

106

ρ NN(q

,Q=

0) (

fm6 )

3He

4He

6Li

8Be

0 1 2 3 4 5q (fm

-1)

10-1

100

101

102

103

104

105

ρ NN(q

,Q=

0) (

fm6 )

AV18+UIX 4He

AV4’4He

pp np

VMC calculations for pairs withQtot = 0 show this effect inA=3–8 nuclei

Effect disappears when tensor correlations are turned off

Shows importance of tensor correlations to> 3 fm−1

Schiavilla, Wiringa, Pieper & Carlson, PRL98, 132501 (2007)

ForQtot > 0 (Q ‖ q), the minimum inpp distribution fills in:

0 1 2 3 4 5

100

101

102

103

104

105

106

q (fm-1)

ρ 12(

q,Q

)

Q=0.0 fm-1

Q=0.25 fm-1

Q=0.5 fm-1

Q=1.0 fm-1

Q=1.5 fm-1

3He

0 0.5 1 1.5 20.0

0.1

0.2

0.3

0.4

0.5

0.6

Q (fm-1)R

pp/p

n

3He

4He

For qrel integrated over 300–500 MeV/c, the ratio ofpp to pn pairsRpp/pn

compares well with preliminary analysis of CLAS data for3He(e, e′pp)n

Wiringa, Schiavilla, Pieper & Carlson, PRC78, 021001 (2008)

DOES IT REALLY HAVE TO BE THAT COMPLICATED?

We have excellent description of light nuclei using accurateNN potential supplemented by

realistic3N potential. But what happens to nuclear spectra with simplerforce models?

AV8′ : [1, σi · σj , Sij , L · S] ⊗ [1, τi · τj ]

Fits CI S- and P-wave scattering data and2H

AV6′ : [1, σi · σj , Sij ] ⊗ [1, τi · τj ]

Drop spin-orbit force -3P-wave data goes bad

readjusted to keep2H binding correct

AV4′ : [1, σi · σj ] ⊗ [1, τi · τj ]

Drop tensor force -3P-wave worse; no Q in2H

readjusted to keep2H binding correct

AV1′ : [1]

Average L=even potential only; still has attractive well & repulsive core

both dineutron and2H weakly bound

Coulomb potentialVC = e2

radded between allpp pairs.

0 0.5 1 1.5 2-150

-100

-50

0

50

100

150

r (fm)

VS

,T (

MeV

)

1,0

0,1

1,1

0,0

L=even

L=odd

-180

-170

-160

-150

-140

-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10E

nerg

y (M

eV)

α+dα+t

α+α

α+n

6Li+α

AV1′ AV4′ AV6′ AV8′ IL2

0+

4He

3/2−1/2−

5He

1+

1+3+2+

3S[2]

3D[2]

1P[11] 6Li

1/2−1/2−

3/2−1/2−7/2−5/2−5/2−

2P[3]2F[3]

4P[21]

2P[21]

2S[111] 7Li

0+

0+

0+

0+2+4+1+3+

1S[4]1D[4]1G[4]

3P[31]

3D[31]

5D[22]1S[22]

3P[211]8Be

3+1+

3+: 3D[42]1+: 3S[42]

10B

α+dα+t

α+α

α+n

6Li+α

What Makes

Nuclear Level Structure?

GFMC Calculations

Wiringa & Pieper, PRL89, 182501 (2002)

CONSEQUENCES

AV1′ : approximately reproducesA=2-4 nuclear binding, but no saturation forA > 5. Nuclear

matterE(ρ0)/A ≈ −80 MeV indicates repulsive core is not source of nuclear saturation.

Ordering of excited states inverted and compressed compared to experiment. Big Bang would

produce very heavy neutron-rich nuclei — mini-neutron stars!

AV4′ : Much better saturation; ordering of excited states improved with less compression;A=5

no longer stable, butA=8 still stable. Indicates importance of spin-isospin exchange forces for

achieving saturation.

AV6′ : Addition of tensor force makesA=8 unstable, but stability of interveningA=6,7 nuclei

too close to call.

AV8′ : Addition of spin-orbit force definitely stabilizes6,7Li nuclei while keepingA=5 & 8

unstable; excitation spectrum looks pretty good.

This model might produce a universe much like ours!

Borromean nuclei6,8He,9Be may needVijk for stability.

NUCLEAR BINDING AND HADRONIC MASS VARIATION

Could fundamental “constants” have varied over the historyof the universe?

Theories unifying gravity with other interactions suggestthe possibility of temporal and spatial

variations of physical “constants” in the expanding universe.

Some evidence for variations in the fine structure constantα, strength of the strong interaction,

and particle masses has been inferred from studies of big bang nucleosynthesis, quasar

absorption spectra, and the Oklo natural nuclear reactor.

Program for studying the universe’s dependence on the quarkmassXq = mq/ΛQCD:

• Study how hadron masses depend on quark masses

• Evaluate how nuclear binding depends on hadron masses

• Study consequences for big bang and stellar nucleosynthesis

HADRON MASS DEPENDENCE ON CURRENT-QUARK MASS

Prediction from a Dyson-Schwinger equation study of the sigma terms of light-quark hadrons:

V.V.Flambaum, A.Holl, P.Jaikumar, C.D.Roberts and S.V.Wright[FBS38, 31 (2006)]

δmH

mH=

σH

mH

δmq

mqmq = (mu +md)/2

π ρ ω N ∆

σH

mH0.498 0.021 0.034 0.064 0.041

Other models are possible, but we expect pion mass to vary most rapidly due to

Gell-Mann-Oakes-Renner relationm2π = mq and that other masses will vary in the same

direction, i.e., all get larger or smaller together.

HAMILTONIAN DEPENDENCE ON HADRON MASS

Consider Hamiltonian with three different interaction models:

• Argonnev28 (AV28) : coupled-channels OPE with explicit∆’s fit to 1981 phase shifts

• Argonnev14 (AV14) : nucleons-only with approximate TPE, phase-equivalent toAV28

• Argonnev18 (AV18) : updated AV14 with charge-independence-breaking fit to 1993 data,

weakerfπNN , deeper well, stiffer core; supplement withUrbana IX (UIX)Vijk

and allowmN ,m∆,mπ, andmV [= 12(mρ +mω)] to vary.

mN (m∆) enters explicitly through kinetic energyKi and implicitly through2π-exchangevIij

mπ enters explicitly throughvπij (V 2πP

ijk ) and implicitly through2π-exchangevIij

mV enters implicitly through short-rangevSij

EVALUATING MASS DEPENDENCE

Change hadron massesmH one at a time±0.1% and recalculateE. Two-nucleon cases

evaluated exaclty. Multi-nucleon cases evaluated using variational Monte Carlo.

Results can be expressed as dimensionless derivatives:

∆E(mH) =δE/E

δmH/mH

and combined with given model (such as DSE) for correlation between hadron and quark masses:

E(mq) = E0

2

41 +X

mH

∆E(mH)δmH(mq)

mH

3

5

Two-nucleon binding energy dependence on quark mass from DSE studies

0 0.5 1 1.5 2

-6

-4

-2

0

2

mq (mq0)

E (

MeV

)

2H

1S

AV28AV14AV18

Multi-nucleonE(mq) for “full” AV18+UIX calculation with DSEδmH/mH

0.8 0.9 1 1.1 1.2

-70

-60

-50

-40

-30

-20

-10

0

mq (mq0)

E (

MeV

)

1S

2H

3H

3He

4He

5He

2α

6Li

7Li

7Be

8Be

CONSEQUENCES FORBIG BANG

Dent, Stern, and Wetterich, [PRD76, 063513 (2007)] calculated sensitivity of BBN abundances for2H, 4He, and7Li to variations inA=2-7 binding energies. Folding our results with theirs, we

find these BBN abundances will be in much better agreement with the WMAP value ofη

(baryon to photon ratio) forδXq/Xq = K · (0.013 ± 0.02) whereK = δE/Eδmq/mq

∼ 1 is the

total sensitivity to the light quark mass.

K 1S0(np) 2H 3H 3He 4He 5He 6Li 7Li 7Be 8Be

AV28 4.5 -0.75

AV14 7.3 -0.84 -0.89 -0.96 -0.69 -0.81 -0.89 -1.03 -1.09 -0.92

AV18+UIX 11.4 -1.39 -1.44 -1.55 -1.08 -1.24 -1.36 -1.50 -1.57 -1.35

Flambaum & Wiringa, PRC76, 054002 (2007)

Flambaum & Wiringa, PRC79, 034302 (2009)