Cold atoms and nuclear physics:

Lattice calculations and the question of universality

1

Dean Lee (NC State U. / U. Bonn)

Evgeny Epelbaum, Hans-Werner Hammer, Hermann Krebs, Ulf-G.

Meißner (U. Bonn / FZ Jülich)

Recent Progress in Many-Body Theories

July 27 – 31, 2009

Ohio State University, Columbus, OH

2

Outline

Lattice calculations…

• Lattice effective field theory for nucleons

• Pion exchange and auxiliary fields

• Phase shifts on the lattice

• Results: Dilute neutron matter

… and the question of universality

• Low-energy universality

• Universality in higher partial waves

• Effective range for general d and L

• Causality and generalized Wigner bounds

Part I

Part II

p

n

p

n

p

n

a» 1¡5 fm

3

Lattice EFT for nucleons

Accessible by

Lattice EFT

early

universe

gas of light

nuclei

heavy-ion

collisions

quark-gluon

plasma

excited

nuclei

neutron star core

nuclear

liquid

superfluid

10-110-3 10-2 1

10

1

100

rNr [fm-3]

T[M

eV

]

neutron star crust

Accessible by

Lattice QCD

4

Construct the effective potential order by order

…

Solve Lippmann-Schwinger equation non-perturbatively

Weinberg, PLB 251 (1990) 288; NPB 363 (1991) 3

5

Chiral EFT for low-energy nucleons

O(Q0)

LO

NLOO(Q2)

O(Q3)

N3LOO(Q4)

N2LO

Ordonez et al. ’94; Friar & Coon ’94; Kaiser et al. ’97; Epelbaum et al. ’98,‘03;

Kaiser ’99-’01; Higa et al. ’03; …

6

Nuclear

Scattering Data

Effective

Field Theory

p

1; ~¾1 ¢ ~¾2

VOPEP

7

Leading order on lattice

p

p

p

p

. . .

p

p

(~¾1 ¢ ~r1) (~¾2 ¢ ~r2)~r1 ¢ ~r2

V TPEP

8

Next-to-leading order on lattice

NNNLO

NNLO

NLO

LO

9

Computational strategy

NNNLO

NNLO

NLO

LO

NNNLO

NNLO

NLO

LONon-perturbative – Monte Carlo Perturbative corrections

“Improved LO”

10

exp

"¡12

³~r¼

´2¢t¡m

2¼

2¼2¢t

#

Free nucleons:

Free pions:

Pion-nucleon coupling:

exp

·1

2mNy~r2N¢t

¸

exp

"¡ gA2f¼

Ny¿~¾N ¢ ¢~r¼¢t#

11

Euclidean-time transfer matrix

(CI > 0)

CI contact interaction:

exp

·¡12CIN

y¿N ¢Ny¿N¢t¸

=1p2¼

ZdsI exp

·¡12sI ¢ sI + isI ¢Ny¿N

qCI¢t

¸

exp

·¡12CNyNNyN¢t

¸(C < 0)

C contact interaction:

=1p2¼

Zdsexp

·¡12s2+ sNyN

p¡C¢t

¸

12

… with auxiliary fields

t = 0

t = tf

p

pexp(¡Ht)

13

t = 0

t = tf

14

s(~r1)

¼I(~r3)

sI(~r2)

Mij(s; sI; ¼I)

jji

jii

Mij(s; sI; ¼I) = h~pijM(Lt¡1)(s; sI; ¼I) ¢ ¢ ¢M(0)(s; sI; ¼I) j~pji

hÃinitjM(Lt¡1)(s; sI; ¼I) ¢ ¢ ¢ ¢ ¢M(0)(s; sI; ¼I) jÃiniti = detM(s; sI; ¼I)

Auxiliary-field determinantal Monte Carlo

¿2M¿2 =M¤

For A nucleons, the matrix is A by A.

For the leading-order calculation, if there is no pion coupling and the

quantum state is an isospin singlet then

This shows the determinant is real. Actually can show the determinant

is positive semi-definite.

15

With nonzero pion coupling the determinant is real for a spin-singlet

isospin-singlet quantum state

¾2¿2M¾2¿2 =M¤

but the determinant can be both positive and negative

Some comments about Wigner’s approximate SU(4) symmetry…

Theorem: Any fermionic theory with SU(2N) symmetry and two-body

potential with negative semi-definite Fourier transform

obeys SU(2N) convexity bounds (see next slide)

Corollary: It can be simulated without sign oscillations

16

Chen, D.L. Schäfer, PRL 93 (2004) 242302;

D.L., PRL 98 (2007) 182501

0

-10

-20

-30

-40

-50

-60

-70

-80

-90

-100

-110

-120

-130

1H 2H 3He4He 5Li 6Li 7Be 8Be 9B 10B 11C 12C 13N 14N 15O 16O

(MeV)

E

nucleus

n nn 3H 5He 6He7Li 9Be10Be11B 13C 14C 15N

SU(4) convexity bounds

17

jÃinitihÃinitj

=MLO =MSU(4) =Oobservable

jÃinitihÃinitj

Znt;LO =

ZhOint;LO

=

e¡E0;LOat = limnt!1Znt+1;LO=Znt;LO

hOi0;LO = limnt!1ZhOint;LO

=Znt;LO

=MNLO =MNNLO

Hybrid Monte Carlo sampling

18

Schematic of projection calculations

jÃinitihÃinitj

jÃinitihÃinitj

Znt;NLO =

ZhOint;NLO

=

hOi0;NLO = limnt!1ZhOint;NLO

=Znt;NLO

19

A(VLO3) = C1S0f(~q 2)¡14¡ 1

4~¾1 ¢ ~¾2

¢¡34+ 1

4¿1 ¢ ¿2

¢LO3: Gaussian smearing only in even partial waves

+A(VOPEP)

LO1: Pure contact interactions

LO2: Gaussian smearing

+ C3S1f(~q2)¡34+ 1

4~¾1 ¢ ~¾2

¢¡14¡ 1

4¿1 ¢ ¿2

¢

A(VLO2) = Cf(~q 2) +CIf(~q 2)¿1 ¢ ¿2 +A(VOPEP)

A(VLO1) = C +CI¿1 ¢ ¿2 +A(VOPEP)

20

Unknown operator

coefficients

Physical

scattering data

Spherical wall imposed in the center of

mass frame

Spherical wall method

Borasoy, Epelbaum, Krebs, D.L., Meißner,

EPJA 34 (2007) 185

21

Energy levels with hard spherical wall

Rwall = 10a

a= 1:97 fm

Energy shift from free-particle values gives the phase shift

22

LO3: S waves a= 1:97 fm

23

LO3: P waves a= 1:97 fm

24

25

Results: Dilute neutron matter at NLO

N = 8, 12, 16 neutrons at L3 = 43, 53, 63, 73

Epelbaum, Krebs, D.L, Meißner, 0812.3653 [nucl-th], EPJA 40 (2009) 19926

a= 1:97 fm

27

Low-energy universality

Large disparity in length scales

d » k¡1F ÀR

d

R

range of interaction

average separation

¸T ÀR

¸Tthermal wavelength

f(~p 0; ~p) =

1X

L=0

fL(p)PL(cos µ)

Partial wave decomposition of the scattering amplitude

fL(p) =¡i2p

he2i±L(p) ¡ 1

i=

1

p [cot ±L(p)¡ i]

Ã(~r)! ei~p¢~r + f(~p 0; ~p)eipr

rr!1

S-wave effective range expansion

p cot ±0(p) = ¡a¡10 +1

2r0p

2 +

1X

n=0

(¡1)n+1P(n)0 p2n+4

f0(p) =1

p cot ±0(p)¡ ip

28

Dimensional analysis: Length dimensions of coefficients

a¡10 = [`]¡1; r0 = [`]; P(n)0 = [`]2n+3

For the S-wave case,

a¡10 »R¡1; r0 » R; P(n)0 » R2n+3

A parameter is of “natural size” if it is roughly the same order of

magnitude as the interaction range raised to the corresponding

length dimension.

For the remainder of this talk we look for low-energy universality

and consider momenta small enough such that

pR¿ 1

29

p cot ±0(p) = ¡a¡10 +1

2r0p

2 +

1X

n=0

(¡1)n+1P(n)0 p2n+4

f0(p) =1

p cot ±0(p)¡ ip

Assuming parameters of natural size, the terms contributing to the

amplitude have a simple hierarchy

¡a¡10¡ip 1

2r0p

2

¡P(0)0 p4

pR

more powers of

30

At low momenta the unitarity limit is reached when the real part of the

denominator in the scattering amplitude can be neglected

Unitarity limit

p cot ±0(p) = ¡a¡10 +1

2r0p

2 +

1X

n=0

(¡1)n+1P(n)0 p2n+4

f0(p) =1

p cot ±0(p)¡ ip

¡a¡10¡ip 1

2r0p

2

¡P(0)0 p4

a¡10 ! 0; a0!§1must be fine-tuned

31

d

d0E0

E00

E0d2 = E00d

02

Scale-invariant physics

f0(p) ¼i

p

32

E0A= » ¢ E

free0A

= » ¢ 35EF

ξ is a dimensionless number

Neutron matter close to unitarity limit for kF » 80 MeV

Dilute neutrons and the unitarity limit

» = 0:31(1)

»1 ¼ 0:8E0Efree0

= » ¡ »1kFa0

+ ckF r0 + :::

f(kFa0)D.L., EPJA 35 (2009) 171;

PRC 78 (2008) 024001;

PRB 75 (2007) 134502

33

34

fL(p) =¡i2p

he2i±L(p) ¡ 1

i=

1

p [cot ±L(p)¡ i]

=p2L

p2L+1 cot ±L(p)¡ ip2L+1

p2L+1 cot ±L(p) = ¡a¡1L +1

2rLp

2 +

1X

n=0

(¡1)n+1P(n)L p2n+4

Universality in higher partial waves

For higher partial waves the effective range expansion is

35

Cold atoms: P-wave Feshbach experiments

Nuclear physics: P-wave alpha-neutron interactions in halo nuclei

Regal, PRL 90 (2003) 053201, …

Dimensional analysis: Length dimensions of coefficients

a¡11 = [`]¡3; r1 = [`]¡1; P(n)1 = [`]2n+1

p3 cot ±1(p) = ¡a¡11 +1

2r1p

2 +

1X

n=0

(¡1)n+1P(n)1 p2n+4

f1(p) =p2

p3 cot ±1(p)¡ ip3

For P-waves we have*

We again start with the case where the coefficients are of natural size

36

*caveat for van der Waals interactions in alkali atomsGao, PRA (1998) 1728

¡a¡11

¡ip3

1

2r1p

2

¡P(0)1 p4

For natural size parameter, the P-wave low-momentum hierarchy is

a¡11 »R¡3; r1 »R¡1; P(n)1 » R2n+1

37

p3 cot ±1(p) = ¡a¡11 +1

2r1p

2 +

1X

n=0

(¡1)n+1P(n)1 p2n+4

f1(p) =p2

p3 cot ±1(p)¡ ip3

pRmore powers of

p3 cot ±1(p) = ¡a¡11 +1

2r1p

2 +

1X

n=0

(¡1)n+1P(n)1 p2n+4

f1(p) =p2

p3 cot ±1(p)¡ ip3

As in the S-wave case we look for an analogy to the unitarity limit

where the real part of the denominator in the scattering amplitude can

be neglected

Unitarity limit for P-waves?

This is equivalent to fine-tuning all coefficients in the effective range

expansion with negative length dimensions

38

a¡11 ! 0; a1!§1¡a¡11

¡ip3

1

2r1p

2

¡P(0)1 p4

r1! 0

A first guess might be…

39

pR

more powers of

Effective range integral formula

u(p)

L (r) =pr cos ±L(p) ¢ jL(pr)¡ pr sin ±L(p) ¢ yL(pr)

sin ±L(p)

Generalization of result of Bethe (1949)

+¼

¡(L+ 32)¡(L+ 5

2)22L+2

R2L+3

a2L¡2 lim

p!0p2L

Z R

0

hu(p)

L (r)i2dr

The wavefunction normalization is chosen so that for r ¸ R

40

Hammer, D.L., 0907.1763 [nucl-th]

Since the integral term is less or equal to zero,

If we fine-tune to be unnaturally small

¯a¡1L

¯¿R¡2L¡1

then

+¼

¡(L+ 32)¡(L+ 5

2)22L+2

R2L+3

a2L

41

a¡1L

S-wave case was derived by Phillips and Cohen

Cohen, Phillips, PLB390 (1997) 7

The upper bound for is much more severe. Cannot fine-

tune to zero.

negative power of R for L ≥ 1

negative coefficient for L ≥ 1

Hammer, D.L., 0907.1763 [nucl-th]

L ¸ 1

42

For , we conclude that must be negative and natural size rLL ¸ 1

This upper bound is related to the causality bound derived by Wigner

(1954). Here is the heuristic argument…

Consider a wavepacket that is sharply peaked in momentum space

f(r) =1p2¼

Zdpeipr ~f(p)

~f(p) ¼ c±(p¡ ¹p)

Consider what happens when multiplying by a momentum-dependent

phase shift

43

g(r) =1p2¼

Zdpeipr~g(p)

We perform the inverse Fourier transform

~g(p) = e2i±(p) ~f(p)

¼ e2i±(¹p)e¡2i±0(¹p)¹pf[r+2±0(¹p)]

There is an overall phase multiplication and the packet is translated

backward in space an amount proportional to the derivative of the

phase shift with respect to momentum.

Similar relation for time delay and the derivative of the phase shift

with respect to energy.

44

time

r = 0

45

time

r = 0

¡2±0(¹p)

46

time

r = 0 r =R

Causality bound

¡±0(¹p) <R¡±0(¹p)

¡2±0(¹p)

47

a¡11 ! 0; a1!§1¡a¡11

¡ip3

1

2r1p

2

¡P(0)1 p4

Universality for P-waves

p3 cot ±1(p) = ¡a¡11 +1

2r1p

2 +

1X

n=0

(¡1)n+1P(n)1 p2n+4

f1(p) =p2

p3 cot ±1(p)¡ ip3

48

cannot be fine-tuned

simultaneously

Shallow P-wave bound state

E ¼ 1

a1r1¹

Normalization of tail controlled by r1

49

P>(r)!2

¡r1r

In the limit of zero binding energy

r

u2(r)

r ¸ R

Generalization to arbitrary dimension

fL;d(p) /p2L

p2L+d¡2 cot ±L;d(p)¡ ip2L+d¡2

uL;d(r) = r(d¡1)=2RL;d(r)

¡u00L;d(r) +"¡L+ d¡1

2

¢ ¡L+ d¡3

2

¢

r2+ 2¹V (r)

#uL;d(r) = 2¹E ¢ uL;d(r)

Radial Schrödinger equation

Partial wave scattering amplitude

50

p2L+d¡2·cot ±L;d(p)¡ ±dmod2;0

2

¼log(p½L;d)

¸

= ¡a¡1L;d +1

2rL;dp

2 +

1X

n=0

(¡1)n+1P(n)L;dp2n+4

Effective range expansion

a¡1L = [`]2¡2L¡d; rL = [`]4¡2L¡d; P(n)L = [`]6+2n¡2L¡d

51

Dimensional analysis:

even d only

Depends on the combination 2L+ d

2L+ d= 3 The unitarity limit reachable – non-trivial fixed point.

2L+ d ¸ 4 Causality prevents reaching the non-trivial fixed point

with finite-range interactions. Effective range emerges

as a second relevant parameter that cannot be fine-

tuned.

Although the interactions can be non-perturbative (i.e.,

perturbative expansion in scattering parameter breaks

down), the interactions remain weak at low energies.

Only trivial fixed point.

52

Summary

53

Causal wave propagation has significant consequences for low-energy

universality

For higher partial waves in the zero-energy resonance limit, a second

dimensionful parameter appears that cannot be fine-tuned to zero. This

second parameter is the effective range.

Lattice EFT – relatively new and promising tool that combines framework

of effective field theory and computational lattice methods

Applications to zero and nonzero temperature simulations of light nuclei,

neutron matter, cold atoms, etc.

Currently working on light nuclei at NNLO, including Coulomb

corrections, isospin-symmetry breaking, storing lattice configurations for

correlation functions, scattering, transitions, etc.

Part I

Part II

Effective range expansion: van der Waals

p3 cot ±1(p) = ¡a¡11 + c1;1p+1

2r1p

2 + : : :

a¡11 ! 0 =) c1;1! 0

For P-waves cannot in general define an effective range parameter

However in the zero-energy resonance limit where the scattering

parameter is infinite

54