The Triton and Three-Nucleon Force in Nuclear Lattice ... · Triton and three-nucleon force in...

Introduction Lattice χEFT χEFT Simulations Nuclear matter

The Triton and Three-Nucleon Force

in Nuclear Lattice Simulations

B. Borasoy1,2, H. Krebs1, D. Lee3, U.-G. Meißner1,2

1Bonn University

2FZ Julich

3NCSU, Raleigh

INT 06-1 Workshop

INT ’06 Triton and 3NF


Outline

1 Introduction

2 Chiral effective field theory on the lattice

3 Lattice simulations of chiral effective field theory for the tritonContinuum resultsNuclear lattice results

4 Simulations of nuclear matter



Introduction

Lattice field theory allows nonperturbative treatment of QCD

Modern lattice QCD simulations work with lattices not muchlarger than size of single nucleon

In forseeable future: lattice QCD simulations not suited toobtain direct results in few- and many-body nuclear physics(beyond lightest nuclei)



Introduction

Lattice field theory allows nonperturbative treatment of QCD

Modern lattice QCD simulations work with lattices not muchlarger than size of single nucleon

In forseeable future: lattice QCD simulations not suited toobtain direct results in few- and many-body nuclear physics(beyond lightest nuclei)

Employ effective field theory on the lattice:pions and nucleons are point particles on lattice sites,external (axial-) vector fields live on links



Chiral effective field theory

QCD exhibits (approximate) SU(2)L × SU(2)R chiralsymmetry, broken down spontaneously to SU(2)V

⇒ 3 Goldstone bosons (π+, π−, π0) with small masses



Chiral effective field theory

QCD exhibits (approximate) SU(2)L × SU(2)R chiralsymmetry, broken down spontaneously to SU(2)V

⇒ 3 Goldstone bosons (π+, π−, π0) with small masses

At low energies: chiral perturbation theory, the effective fieldtheory of QCD, is successful in describing interactions amongmesons and baryons.

Green’s functions are expanded in Goldstone boson massesand small momenta ⇒ chiral counting scheme

ChPT is model independent, effective field theory of QCD



Chiral perturbation theory

Chiral effective Lagrangian has same symmetries andsymmetry breaking patterns as QCD





Consequence of confinement: quarks and gluons do not showup as explicit degrees of freedom in effective Lagrangian






Pions can be summarized in matrix U(x) ∈ SU(2)

U(x) = exp(

ifπ

π(x))

fπ ≃ 93 MeV,pion decay constant

L = L(U, ∂U, ∂2U, . . . ,M) , M = diag(mu, md)






Pions can be summarized in matrix U(x) ∈ SU(2)

U(x) = exp(

ifπ

π(x))

fπ ≃ 93 MeV,pion decay constant

L = L(U, ∂U, ∂2U, . . . ,M) , M = diag(mu, md)

Nucleons (p, n) can be included



Lattice χEFT

Lattice ChPT with mesons has been investigated in

− Myint & Rebbi (1994); Levi, Lubicz & Rebbi (1997)− Shushpanov & Smilga (1999)

Extension to baryonic sector

− Lewis & Ouimet (2001)− Borasoy, Lewis & Ouimet (2002) (2004)

Studies on a finite lattice

− Borasoy & Lewis (2004)

Multi-nucleon effective field theory

− Chandrasekharan, Pepe, Steffen & Mazur (2003)− Lee, Borasoy & Schafer (2004), Lee & Schafer (2005),− Borasoy, Krebs, Lee & Meißner (2005)



Lattice χEFT

Euclidean SU(2) chiral Lagrangian in mesonic sector

L = L2 + L4 + . . .

L2=f2

π

4

⟨

∇(+)µ U†∇

(+)µ U

⟩

−Bf2

π

2

⟨

M(U + U†)⟩

L4= −14 l1

⟨

∇(±)µ U†∇

(±)µ U

⟩2

− 14 l2

⟨

∇(±)µ U†∇

(±)ν U

⟩⟨

∇(±)µ U†∇

(±)ν U

⟩

+ . . .

with pion fields

U(x) = exp

(

iτaπa(x)

fπ

)



Lattice χEFT

Use of nearest-neighbor covariant derivative in leading orderLagrangian avoids unphysical states

∇(+)µ U(x) =

1

a

[

Rµ(x)U(x + aµ)L†µ(x) − U(x)

]

with external fields

Lµ(x)=exp [−iaℓµ(x)] = exp [−ia(Vµ(x) − Aµ(x))]

Rµ(x)=exp [−iarµ(x)] = exp [−ia(Vµ(x) + Aµ(x))]


Introduction Lattice χEFT χEFT Simulations Nuclear matter Continuum results Nuclear lattice results

Triton and three-nucleon force

Work in SU(4)-symmetric (Wigner) limit of pionless effectivefield theory: isospin- and spin-symmetric

L = ψ†

(

i∂0 +~∇2

2m

)

ψ −C0

2

(

ψ†ψ)2

−D0

6

(

ψ†ψ)3

ψ includes four nucleon states with mass m

ψ =

p↑p↓n↑

n↓

Renormalized two-body interaction C0 is directly related totwo-body scattering length a2

C0 =4πa2

m



Continuum regularization

Two-body binding energy B2

B2 =1

ma22

SU(4)-symmetric limit: 1S0 and 3S1 two-body sectors aredegenerate. We choose SU(4)-symmetric two-nucleon bindingenergy B2 = 1 MeV

Three-body interaction D0 not fixed




Two-body binding energy B2

B2 =1

ma22

SU(4)-symmetric limit: 1S0 and 3S1 two-body sectors aredegenerate. We choose SU(4)-symmetric two-nucleon bindingenergy B2 = 1 MeV


Solve homogeneous S-wave bound state equation fordimer-nucleon system with cutoff momentum Λ

⇒ strong dependence of D0 on Λ




Three-body interaction D0 as function of triton binding energy E3

-4e-07

-3e-07

-2e-07

-1e-07

0

1e-07

2e-07

3e-07

4e-07

-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

D 0 (

MeV

-5)

E3 (MeV)

Λ/π = 30 MeVΛ/π = 40 MeVΛ/π = 50 MeVΛ/π = 60 MeVΛ/π = 70 MeV

Nontrivial dependence of D0 on Λ is nonperturbative effect,no finite set of diagrams reproduces ultraviolet divergence(Bedaque, Hammer & van Kolck (’99))




D0 scales roughly as Λ−2 for E3 ≪ −1 MeV

Pole in D0(E3) close to continuum threshold fordimer plus nucleon

Pole location decreases for larger Λ

As Λ is increased, new deeper bound states appear which areoutside range of validity of EFT

Triton is identified with shallowest bound state




D0 scales roughly as Λ−2 for E3 ≪ −1 MeV

Pole in D0(E3) close to continuum threshold fordimer plus nucleon

Pole location decreases for larger Λ

As Λ is increased, new deeper bound states appear which areoutside range of validity of EFT

Triton is identified with shallowest bound state

Singular behavior of three-body system could lead to differentcutoff dependence of D0 in different regularization scheme



Triton and three-nucleon force in nuclear lattice simulations

Path integral is evaluated by computing Monte Carlo sampleof world lines

x

t




Path integral is evaluated by computing Monte Carlo sampleof world lines

x

t

Strength of two-body coefficient C0 is matched to deuteronbinding energy B2 (we take 1 MeV)




Two-body interaction

Two-body coupling C0 determined by summingnucleon-nucleon bubble diagrams on the lattice

1 2 nΣn = 0

∞

Tune C0 to reproduce deuteron binding energy B2

a−1(MeV) a−1t (MeV) C0(MeV−2)

40 16 −1.83 × 10−4

50 25 −1.39 × 10−4

60 36 −1.13 × 10−4

70 49 −0.94 × 10−4



Contributions to path integral

Hopping parameter: h = αt2m

, αt = ata

Single fermion worldline:

hop to a neighboring lattice site during time step: h

no hop during time step: 1 − 6h





, αt = ata




Two different fermions at same spatial site and no hop duringtime step: e−C0αt(1 − 6h)2





, αt = ata





Three different fermions at same spatial site and no hopduring time step: e−3C0αt(1 − 6h)3

Factor e−C0αt for each pairwise interaction





, αt = ata





Three different fermions at same spatial site and no hopduring time step: e−3C0αt(1 − 6h)3

Factor e−C0αt for each pairwise interaction

Contribution from three-body interaction for three differentfermions: e−D0αt(1 − 6h)3




Compute lattice approximation for

〈0, 0, 0| exp [−βH] |0, 0, 0〉

|0, 0, 0〉: state with three nucleons, each of different kind andzero momentum

⇒ measurement of triton ground state energy E3

-1e-08

-5e-09

0

5e-09

1e-08

1.5e-08

2e-08

-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

D 0 (

MeV

-5)

E3 (MeV)

1/a = 30 MeV1/a = 40 MeV1/a = 50 MeV1/a = 60 MeV




Coupling D0 scales as ∼ a2 for fixed E3 (modulo shift)

Energy region E3 ∼ −1 MeV has not been computed sincethere are many dimer plus nucleon continuum states nearenergy threshold → difficult to extract E3




Coupling D0 scales as ∼ a2 for fixed E3 (modulo shift)

Energy region E3 ∼ −1 MeV has not been computed sincethere are many dimer plus nucleon continuum states nearenergy threshold → difficult to extract E3

Results have been confirmed using Hamiltonian lattice withLanczos method



Hamiltonian lattice

Compare with Hamiltonian lattice using Lanczos method

-1e-08

-5e-09

0

5e-09

1e-08

1.5e-08

2e-08

-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

D 0 (

Me

V-5

)

E3 (MeV)

1/a = 30 MeV1/a = 40 MeV1/a = 50 MeV

D0 scales roughly as a2 (modulo shift)



Hamiltonian lattice


-1e-08

-5e-09

0

5e-09

1e-08

1.5e-08

2e-08

-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

D 0 (

Me

V-5

)

E3 (MeV)

1/a = 30 MeV1/a = 40 MeV1/a = 50 MeV


Hamiltonian approach restricted to small volumesand small # of particles



Hamiltonian lattice


-1e-08

-5e-09

0

5e-09

1e-08

1.5e-08

2e-08

-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

D 0 (

Me

V-5

)

E3 (MeV)

1/a = 30 MeV1/a = 40 MeV1/a = 50 MeV


Hamiltonian approach restricted to small volumesand small # of particles

Euclidean lattice method can be generalized to larger nucleonnumbers and more complicated forces amongst nucleons




To be done:

More nucleons

Inclusion of four-body force

Effects due to breaking of Wigner symmetry

Higher chiral orders in effective Lagrangian

Inclusion of pions




To be done:

More nucleons

Inclusion of four-body force

Effects due to breaking of Wigner symmetry

Higher chiral orders in effective Lagrangian

Inclusion of pions

Important step towards future many-body simulations witharbitrary number of nucleons

Neutron matter with pions (Lee, Borasoy & Schafer (’04))

→ results for hot neutron matter, T ≈ 20 − 40 MeV, anddensities twice below nuclear matter density



Lattice action positivity

Cold dilute neutron matter: pionless EFT should provideadequate description of low-energy physics

Implementation on the lattice with a positive semi-definiteaction via Hubbard-Stratonovich transformation→ efficient Monte Carlo simulations



Lattice action positivity

Cold dilute neutron matter: pionless EFT should provideadequate description of low-energy physics

Implementation on the lattice with a positive semi-definiteaction via Hubbard-Stratonovich transformation→ efficient Monte Carlo simulations

Cold dilute nuclear matter with small proton fraction:3-nucleon force is required for consistent renormalization

3-nucleon force could spoil positivity of lattice action

For triton binding energies of about 8 MeV and assumingfour-nucleon force to be zero (or small) condition for latticeaction positivity [Chen, Lee & Schafer (’04)] is satisfied:no sign problem



Concluding remark:

Lattice simulations of chiral effective field theory are a promisingtool to investigate few-body nuclear physics



Nuclear matter

A central goal of nuclear physics:

understand properties of strongly interacting matterat finite density and temperature



Nuclear matter

A central goal of nuclear physics:

understand properties of strongly interacting matterat finite density and temperature

Experiments

new data generated by RHIC, Brookhaven

upcoming heavy ion facility planned at GSI, Darmstadt

high energy frontier: ALICE @ LHC



Nuclear matter

Astrophysical interest

development of early universe

stellar collapse

properties of neutron stars



Nuclear matter

Astrophysical interest

development of early universe

stellar collapse

properties of neutron stars

Lattice QCD

connects QCD to observed phenomenology

finite temperature ok.

finite density, i.e. chemical potential

⇒ determinant of quark Dirac matrix becomes complex

⇒ highly oscillatory



Simulations of nuclear matter

Study of neutron matter with nuclear lattice simulations

first step (w/ pions): Lee, Borasoy & Schafer (2004)

pionless theory: Lee & Schafer (2005)







w/ pions: sign/phase problem of fermion determinant is muchless severe; well behaved for temperatures above 3 MeV

Better situation than in finite density lattice QCD

Why?







w/ pions: sign/phase problem of fermion determinant is muchless severe; well behaved for temperatures above 3 MeV

Better situation than in finite density lattice QCD

Why?

nucleons and pions give a simpler representation of theessential physics in the hadronic phase

nucleons are much heavier than up and down quarks




Results of numerical simulation at weak coupling agree withresults from perturbation theory for neutron & pionself-energies, shift in average energy

Neutron-neutron contact interaction coupling C determinedby S-wave scattering phase shifts on lattice at zerotemperature and density



Results

Density versus chemical potential

0

0.5

1

1.5

2

-40 -20 0 20 40 60 80 100

ρ/ρ N

phys

µ-mN

phys (MeV)

ResultsFree neutron

Loop calculation

T phys = 25.0 MeV, a−1 = 150 MeV



Results

Energy per neutron

20

30

40

50

60

70

80

0 0.5 1 1.5 2

Eph

ys/A

- m

N

phys

(M

eV)

ρ/ρN

phys

ResultsFree neutron

Loop calculation

T phys = 25.0 MeV, a−1 = 150 MeV



Results

Different lattice volumes

20

30

40

50

60

70

80

0 0.5 1 1.5 2

Ephys

/A -

mN

phys

(M

eV

)

ρ/ρN

phys

L = 3, Tphys = 37.5 MeVL = 4, Tphys = 37.5 MeVL = 5, Tphys = 37.5 MeVL = 3, Tphys = 25.0 MeVL = 4, Tphys = 25.0 MeVL = 5, Tphys = 25.0 MeV

T phys =25 MeV and 37.5 MeV , a−1 = 150 MeV



Results

Different lattice spacings

25

30

35

40

45

50

55

60

65

0 0.5 1 1.5 2

Ephys

/A -

mN

phys

(M

eV

)

ρ/ρN

phys

a-1 = 150 MeV, L = 3

a-1 = 200 MeV, L = 4

T phys =37.5 MeV, a−1 = 150 MeV and 200 MeV



Conclusions

Probe larger volumes, lower temperatures, greater

nuclear densities than lattice QCD

Coupling C is determined by fitting to NN scattering data

Cutoff dependence is absorbed into C

Realistic simulation of many-body nuclear phenomena with• no free parameters

• a systematic expansion

• clear theoretical connection to QCD

Include protons, charged pions, higher orders, 3N forces etc.



Concluding remark:

Lattice simulations of chiral effective field theory are a promisingtool to investigate few- and many-body nuclear physics


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