Three-Body Physics in a Finite Volume · 2011. 9. 21. · Compute path integral of QCD numerically Discretized Euclidean space-time → Finite Volume! Necessity to remove ﬁnite

Three-Body Physics in a Finite Volume

Simon Kreuzerin collaboration with Hans-Werner Hammer

[Phys. Lett. B 673 (2009) 260][Eur. Phys. J. A 43 (2010) 229][Phys. Lett. B 694 (2011) 424]

21 March 2011

Overview

1 Introduction

2 Three-boson bound statesFrameworkResults

3 The Triton in Finite VolumeFrameworkResultsPion-mass dependence

4 Conclusions

Simon Kreuzer (GWU) Three-Body Physics in a Finite Volume 2

Introduction

Introduction – Modern Nuclear Theory

Quantum Chromodynamics (QCD): Theory of strong interactions

Determination of hadronic/nuclear properties from QCD challenging

Two approaches in modern nuclear theory:Effective Field Theories (EFT) and Lattice QCD

QCD

EFT Lattice QCD

Hadrons as degrees of freedomSymmetry−based

Direct approachUnphysical point

Extrapolation

LECs


Introduction

Introduction – EFT

EFT: Model-independent description of effective degrees of freedom

Power counting ensures that EFT– is systematically improvable– provides error estimates

Unresolved short-distance behavior → Low-Energy Constants

Chiral EFT: Interactions between nucleons mediated by Goldstonebosons of spontaneous chiral symmetry breakingRange of validity up to ≈ 350 MeV

Pionless EFT: Exploit unnatural largeness of NN scattering lengthsin both spin-isospin channelsValid below ≈ 100 MeV


Introduction

Introduction – EFT

EFT: Model-independent description of effective degrees of freedom

Power counting ensures that EFT– is systematically improvable– provides error estimates

Unresolved short-distance behavior → Low-Energy Constants

Chiral EFT: Interactions between nucleons mediated by Goldstonebosons of spontaneous chiral symmetry breakingRange of validity up to ≈ 350 MeV

Pionless EFT: Exploit unnatural largeness of NN scattering lengthsin both spin-isospin channelsValid below ≈ 100 MeV


Introduction

Introduction – Universality

Systems with unnaturally large scattering length a

exhibit universal properties

Described by EFT using only contact interactions

Two-body systems with a > 0: Shallow bound-state with bindingenergy 1/(2µa2)

Efimov effect: Sequence of three-body bound states [Efimov 70]

Signature of ultraviolet limit cycle in renormalization of EFT

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Introduction

Introduction – Lattice QCD

Compute path integral of QCD numerically

Discretized Euclidean space-time → Finite Volume!

Necessity to remove finite volume effects

Two-body systems: Volume-dependence well known [Lüscher 86, 91;

Beane et al. 2003]

0 0.5 1 1.5 2a / L

-20

-15

-10

-5

0

E2 m

a2

Extraction of infinite volume scattering parameters possible

First Lattice results in the triton channel have recently becomeavailable, but no properties were extracted [Beane et al. 09]


Introduction

Introduction – This Work

QCD

EFT Lattice QCD

Hadrons as degrees of freedomSymmetry−based

Direct approachUnphysical point

Extrapolation

LECs

Calculate changes to the three-body bound state spectrumin finite volume using EFT

Bosonic case: – First insights to Efimov physics in finite volume– Applicable to Lattice calculations of α-particles

(Coulomb interaction important!)

Nucleonic case: Applicable to Lattice calculations of the triton


Three-boson bound states

Three-boson bound states


Three-boson bound states Framework

Lagrangian, Renormalization

LO Lagrangian for a system of three bosons with short-range forces(cf., e.g., [Bedaque, Hammer, van Kolck 99])

L = ψ†(

i∂

∂t+

1

2∇2)

ψ +g2

4d†d − g2

4

(

d†ψ2 + h.c.)

− g3

36d†dψ†ψ

Infrared finite-volume physics vs. renormalization in UV−→ Perform renormalization in infinite volume

Regulate loop integrals via cutoff Λ

Match 2-body coupling g2 with 2-body scattering length a

Write 3-body coupling g3 = −9g22 H(Λ)/Λ2

Dimensionless function H(Λ) known, needs additional 3-body input

Renormalization in finite volume will be explicitly verified



Boson-Diboson Bound State Amplitude

Homogeneous integral equation for the bound state amplitude� =� +�Energies for which integral equation is solvable → binding energies

Calculation of the finite volume diboson propagator:

= + + ++ ... =

D(E ) = 32πg2

2

[

1/a −√

−E + 1L

∑

~ 6=~01

|~| e−|~|

√−EL

]−1

Reduces indeed to the infinite volume diboson propagator for L → ∞

Loop momenta are quantized → Rewrite using Poisson’s sum equation



Bound state amplitudeIntegral equation for bound state amplitude F

F(~p) =1

π2

∑

~n∈Z3

∫ Λ

d3y eiL~n·~yZE

(k)3

(~p, ~y)τE

(k)3

(y)F(~y)

ZE (~p, ~y) ≡ (p2 + ~p · ~y + y2 − E )−1 + H(Λ)Λ2

τE (y) ≡[

1/a −√

3y2/4 − E +∑

~ 6=~01

|~|Le−|~|L√

3y2/4−E]−1

Spherical symmetry broken to cubic symmetry

Bound state amplitude in trivial representation A1 of cubic group

Can expand angular dependence in terms of spherical harmonics:[v. d. Lage & Bethe 47, Altmann 65]

F(~p) =(A1)∑

ℓ=0,4,6,...Fℓ(p)

ℓ∑

m=−ℓCA1ℓmYℓm(p̂)

Perform angular integration and project on ℓth partial wave



Partial waves

Coupled integral equations for partial waves

Fℓ(p) =4

π

∫ Λ

0dy y2

[

Z(ℓ)

E(k)3

(p, y)τE

(k)3

(y)1

2ℓ+ 1Fℓ(y)

+√

4π∑

~n∈Z3

~n 6=0

(A1)∑

ℓ′,m′

∑

ℓ′′,m′′

(

ℓ′ ℓ′′ ℓ0 0 0

)(

ℓ′ ℓ′′ ℓm′ m′′ 0

)

Cℓ′m′

Cℓ0Yℓ′′m′′(n̂)

×√

(2ℓ′ + 1)(2ℓ′′ + 1)

2ℓ+ 1iℓ

′′

jℓ′′(L|~n|y)Z (ℓ)

E(k)3

(p, y)τE

(k)3

(y)Fℓ′(y)

]

Z(ℓ)E (p, y) = (2ℓ+ 1)

[

1py

Qℓ

(

p2+y2−Epy

)

+ H(Λ)Λ2 δℓ0

]

jℓ: spherical Bessel functionQℓ: Legendre function of the second kind



Specialization to s-wavesResult for ℓ = 0

F0(p) =4

π

∫ Λ

0dy y2

[

Z(0)

E(k)3

(p, y)τE

(k)3

(y)(

1 +∑

~n∈Z3

~n 6=0

sin(L|~n|y)L|~n|y

)

F0(y)

+√

4π∑

~n∈Z3

~n 6=0

(A1)∑

ℓ′,m′

iℓ′

jℓ′(L|~n|y)Y ∗ℓ′m′(n̂)Z

(0)

E(k)3

(p, y)τE

(k)3

(y)Cℓ′m′Fℓ′(y)

]

Z(0)E (p, y) =

1

2pyln

(

p2 + py + y2 − E

p2 − py + y2 − E

)

+H(Λ)

Λ2

First approach: Neglect higher partial waves

Estimate corrections from higher partial wavesby including ℓ = 4 contributions via a coupled channel approach


Three-boson bound states Results

Results (1)

0 0.5 1 1.5 2|a| / L

-32

-28

-24

-20

-16

-12

-8

E3 m

a2

Λ = 200 |a|-1

Λ = 300 |a|-1

Λ = 400 |a|-1

NI

E∞

3 = −9/(ma2)

Size of the state ∼ 0.3a

Shift at L = a: 2.55/(ma2) or

28%

0 0.5 1 1.5 2a / L

-36

-30

-24

-18

-12

-6

E3 m

a2

Λ = 200 a-1

Λ = 300 a-1

Λ = 400 a-1

Λ = 400 a-1

,with expansion

II

E∞

3 = −5.05/(ma2)

Size of state ∼ 0.45a

Expansion works for shifts up to ∼ 20%

Shift at L = a: 6.1/(ma2) or 121%

Values for different cutoffs agree within bands of size 1/(Λa) → Results renormalized!



Results (2)

System with two Efimovstates atE ∞

3 = −27.4427/(ma2)andE ∞

3 = −1.18907/(ma2)

Size of the states:0.2a and 0.9a

Results are renormalized 0 0.5 1 1.5 2a / L

-0.1

-1

-10

E3 m

a2

Ia & Ib

Shift in shallow state at L = a: 3.4/(ma2) or 287%

Shift in deeper state at L = a: 0.77/(ma2) or 2%

Shallow state crosses dimer level at L ∼ a, upward shift for smaller volumes

Power law behavior above threshold → like scattering state?



Contributions from higher partial wavesNext partial wave contributing is ℓ = 4 → two coupled integral equations

Much larger numerical effort

Shallow state with E ∞3 = −1.18907/(ma2)

0.8 0.9 1 1.1L / a

-5

-4.8

-4.6

-4.4

-4.2

-4

E3 m

a2

Λ = 200 a-1

Λ = 300 a-1

Λ = 400 a-1

Contributions from ℓ = 4 partial wave are negligible in this case:

State II

L/a E3(L)ma2, s-wave only E3(L)ma

2, ℓ = 0, 4 δrel

∞ -5.05 N/A N/A1 -11.1 -11.8 6%

0.7 -19.0 -20.7 9%


The Triton in Finite Volume



The Triton in Finite Volume Framework

Lagrangian, Integral equations

Effective Lagrangian for a system of three nucleons in zero-range limit(cf., e.g., [Bedaque, Hammer, van Kolck 99])

L =N†(

i∂t +1

2∇2)

N +gT

2~T †~T +

gS

2S†S

− gT

2

(

~T †NT τ2~σσ2N + h.c.)

− gS

2

(

S†NTσ2τ τ2N + h.c.)

+ L3

L3 ∼ (N†N)3: Wigner SU(4)-symmetric 3-body contact interactionwith cutoff dependent coupling constant H(Λ)needed to renormalize the S = 1

2 sector

Two coupled integral equations for two bound state amplitudes� = � +� +� +�� = � +� + +Simon Kreuzer (GWU) Three-Body Physics in a Finite Volume 18

The Triton in Finite Volume Results


E∞3 = −8.4818 MeV

Size of the triton ∼ 2 fm

Results are renormalized

Shift at volumes typical inLattice QCD already morethan 100%! 5 10 15

L [fm]

-20

-10

0

E3 [M

eV]

Λ = 600 MeVH = 0physical value

Epelbaum et al.

Fit

Fit of the form E3(L) = E3(L = ∞)[

1 + cL

e−L/L0

]

Comparison to data from Chiral EFT on the lattice [Epelbaum et al, 10]:study higher partial waves, higher orders


The Triton in Finite Volume Results

Corrections

Corrections from higher partial waves:More important for smaller volumes

Inclusion straightforward but numerically expensive

Estimate from case of three identical bosons:20% for volumes three times larger than the state itself

NLO of the EFT: Corrections of types re/a and kre

First type dominated by spin-triplet channel, about 30%

Second type about 40% for large volumes,growing with binding energy

Infinite volume: Corrections up to N2LO under control−→ Extension of finite volume framework straightforward,

necessary for precise extrapolations


The Triton in Finite Volume Pion-mass dependence

Pion-Mass Dependence – Introduction

Lattice QCD calculations are performed at unphysical pion masses

Conjectured closeness of QCD to an infrared limit cycle in 3N-sector

as(Mπ) = at(Mπ) = ∞ compatible w/ χEFT near Mπ = 197 MeV

Efimov effect: Excited states of the triton appear

190 200 210Mπ[MeV]

10-6

10-4

10-2

100

B3[M

eV]

ground state1st excited2nd excited

[Hammer, Phillips, Platter, 2007]

Pion-mass dependence of observables under control in χEFT−→ Obtain input data by chiral extrapolation



Pion–Mass Dependence – Spectrum

190 195 200 205Mπ [MeV]

10-6

10-4

10-2

100

|E3| [

MeV

]

L = ∞

190 195 200 205Mπ [MeV]

10-1

100

|E3| [

MeV

]

L = 29.6 fm

190 195 200 205Mπ [MeV]

10-1

100

|E3| [

MeV

]

L = 19.7 fm

190 195 200 205Mπ [MeV]

10-1

100

|E3| [

MeV

]

L = 14.8 fm

Excited state crosses threshold!“Crossing volume” predictable from universality?



Universality in Finite Volume?

Form dimensionless number r = −mE∞3 L2

100%

0.1 1 10 100 1000 10000B3 / B2

0

2

4

6

8

10

r

Bosons (a > 0)Bosons (a < 0)Nucleons

Hints for universal behavior away from threshold

Formula for binding energies in infinite volume [Efimov 79]

−→ Continuation into finite volume?Simon Kreuzer (GWU) Three-Body Physics in a Finite Volume 23

Conclusions

Conclusions


Conclusions

Summary

Derivation of integral equations for partial waves ofbosonic and nucleonic 3-body bound state amplitude

Numerical solution of the equations → modifications of the spectrum

Renormalization in finite volume verified explicitly

Corrections from higher partial waves under control in bosonic case

Infinite volume extrapolation for the triton is possible

Calculated pion-mass dependence of triton ground and excited statein finite volume→ Excited states cross threshold

Access to scattering phase shifts a la Lüscher implicit in the results


Conclusions

Outlook

Include higher partial waves for the triton

Include N(2)LO of the EFTNeeded for precision extrapolations of the triton binding energy

Universality of the finite volume corrections?

Efimov equation for binding energies in finite volume?“Crossing volume” for shallow states predictable?

Extend formalism to include atom-dimer/nucleon-deuteron scattering


Conclusions

Outlook

Include higher partial waves for the triton

Include N(2)LO of the EFTNeeded for precision extrapolations of the triton binding energy

Universality of the finite volume corrections?

Efimov equation for binding energies in finite volume?“Crossing volume” for shallow states predictable?

Extend formalism to include atom-dimer/nucleon-deuteron scattering

Thank you for your attention!


Conclusions

Bonus material


Conclusions

Negative scattering length

0 0.5 1 1.5 2a / L

-35

-30

-25

-20

-15

-10

-5

0

E3 m

a2

Λ = 200 a-1

Λ = 300 a-1

Λ = 400 a-1

States:

– E∞3 ma2 = 0.2

– E∞3 ma2 = 4

– E∞3 ma2 = 9

Different physicalsystems!

Negative scattering lengths accessible and under control


Conclusions

Pion-Mass Dependence – Ground state

190 195 200 205Mπ / MeV

-50

-40

-30

-20

-10

0

E3 /

MeV

Infinite Volume

L = 4.925 fm,Λ = 3000 MeV

L = 4.925 fm,H(Λ) = 0

L = 3.94 fm,Λ = 3000 MeV

L = 3.94 fm,H(Λ) = 0

L = 2.955 fm,H(Λ) = 0

L = 2.955 fm,Λ = 3000 MeV

Behaviorunchanged, shift to more negative energies


Conclusions

Pion-Mass Dependence – Excited state

Mπ = 190 MeV:L/fm B

(1)3 /MeV B2/MeV

∞ 0.052 0.04719.7 0.742 0.48614.8 0.890 0.82314.4 0.884 0.86514.2 N/A 0.888


(1)3 /MeV B2/MeV

∞ 0.016 0.00619.7 0.686 0.43114.8 0.761 0.75314.4 N/A 0.794


(1)3 /MeV B2/MeV

∞ 0.009 5.7 × 10−4

39.5 0.184 0.10527.6 0.240 0.21126.4 0.233 0.23126.2 N/A 0.234


(1)3 /MeV B2/MeV

∞ 0.038 1.3 × 10−4

29.6 0.355 0.18219.7 0.625 0.40715.6 0.662 0.65115.4 N/A 0.668

Excited state crosses threshold!“Crossing volume” predictable? Universality?


Documents

Three-Body Physics in a Finite Volume · 2011. 9. 21. · Compute path integral of QCD numerically Discretized Euclidean space-time → Finite Volume! Necessity to remove ﬁnite