QUARKS, GLUONS QUARKS, GLUONS AND AND

NUCLEAR FORCESNUCLEAR FORCES

Paulo BedaquePaulo BedaqueUniversity of Maryland, College ParkUniversity of Maryland, College Park

strong nuclear force:strong nuclear force:binds neutrons and protons binds neutrons and protons

into nucleiinto nuclei

Quantum Chromodynamics Quantum Chromodynamics (QCD)(QCD)

What do we know ?What do we know ?

1) NN phase shifts1) NN phase shifts

11SS00 neutron-proton neutron-proton

pion exchangepion exchange

all kinds of things …all kinds of things …

What do we know ?What do we know ?

2) Several potentials that fit them2) Several potentials that fit them

What do we know ?What do we know ?

3) These potentials explain a lot but not everything3) These potentials explain a lot but not everything

• NNNN, NN, NN, couplings few % on , couplings few % on dd

• NNN forces ~5% of nuclei bindingNNN forces ~5% of nuclei binding

• NY forces strangeness in neutron starsNY forces strangeness in neutron stars

• ......

LATTICE QCDLATTICE QCD

Can we understand the nuclear forces (and Can we understand the nuclear forces (and NNN, NNNNN, NN, …) from first principles ?, …) from first principles ?

PATH INTEGRALSPATH INTEGRALS

1iSe

2iSe

21 1Probability | |iS iSe e

Quantum mechanics reduced to quadraturesQuantum mechanics reduced to quadratures

[ ]

[ ]

( )

( )

( ) ( ) (0)( ) (0)

( )

iS x t

iS x t

Dx t e x t xx t x

Dx t e

operatorsoperators numbersnumbers

is as well (or ill) defined asis as well (or ill) defined as i xdx e

[ ]( )( ) iS x tDx t e

[ ]

1

( )1( ) (0) ( ) ( ) (0)

1 ( ) (0)N

i ii

S x tx t x Dx t e x t xZ

x t xN

probability probability distributiondistribution

Imaginary time (t it): just like stat mechImaginary time (t it): just like stat mech

But I don’t live in imaginary time !But I don’t live in imaginary time !

What can I do with imaginary time correlators ?What can I do with imaginary time correlators ?

0

1

( )

20( )

( ) (0) |

0 | | | 0

1

0 0 0 | (0) (0) 0|

|

| 0 | | |

|

nE E t

n

Ht Ht

E E t

t

t

x x

x n e n x

x xe e

e x

lowest energy state w/ lowest energy state w/ some overlapsome overlap

Typical pathsTypical paths ( ) (0)i ix t x

1

1 ( ) (0)N

i ii

x t xN

PATH INTEGRALS FOR FIELDSPATH INTEGRALS FOR FIELDS

1iSe 1iSe

Quantum ChromodynamicsQuantum Chromodynamics

U U = SU(3) matrix= SU(3) matrix

= gluons= gluons

Q Q = spinor, 3 colors,= spinor, 3 colors, 6 flavors6 flavors = quarks= quarks

QCD reduced to quadraturesQCD reduced to quadratures

5 5 5 5

5 5

[ ] ( )( ) (0) ( ) (0)

[ ]

1

1 1 1det( ) [ ]UU U

UG

G

S U q D m qx x

S U

q q q q DUDqDq e q q q qZ

DU e D m trZ D m D m

5 5 5 5

5 51

[ ]( ) (0) 1 1 1det( ) [ ]

1 1 1[ ]

UU U

N

i i i

G

U U

S Uxq q q q DU e D m tr

Z D m D m

trN D m D m

probability distribution for Uprobability distribution for U ii

algorithmalgorithm

1.1. find {Ufind {Uii}}

2.2. compute 1/(Dcompute 1/(DUiUi+m)+m)

3.3. compute observablecompute observable

Scattering through finite volumes: Scattering through finite volumes: the Luscher method the Luscher method (Marinari, Hamber, Parisi, Rebbi)(Marinari, Hamber, Parisi, Rebbi)

Periodic boundary

conditions: box is a torus

Energy levels at 2

22n

nE m

L

one particle

2

2

1cot ( )

4

M ELM E E

L

S

known function

Learn about the deuteron in boxes smaller Learn about the deuteron in boxes smaller than the deuteronthan the deuteron

Scattering through finite volumes: Scattering through finite volumes: the Luscher method the Luscher method (Marinari, Hamber, Parisi, Rebbi)(Marinari, Hamber, Parisi, Rebbi)

two particles

† † † †

† † 22 at rest

0 | ( , ) ( , ) (0, ) (0, ) | 0 0 | (0, ) (0, ) | | (0, ) (0, ) | 0

| | (0, ) (0, ) | 0 |

n

N

HtN t k N t k N k N k N k N k e N k N k

E te N k N k

n n

Nt

N

The difference between EThe difference between E2N2N and E and ENN is our is our

signal phase shiftsignal phase shift

The time to try it is nowThe time to try it is now

• Pion masses small enough for chiral extrapolationPion masses small enough for chiral extrapolation

• No quenchingNo quenching

• Volumes ~ (3 fm)Volumes ~ (3 fm)33

• Improved actionsImproved actions

• Good chiral symmetryGood chiral symmetry

• Software resourcesSoftware resources

S. Beane, T. Luu, K. Orginos, E. Pallante, A. Parreno, S. Beane, T. Luu, K. Orginos, E. Pallante, A. Parreno, M. Savage, A. Walker-Loud, …M. Savage, A. Walker-Loud, …

2 2 2

2 2 2 2 2

31 log ( )

8 16

m m mm a l

f f

CP-PACS

K(e4)

Gold platted scattering observable: I=2 Gold platted scattering observable: I=2

CP-PACS

K(e4)

Improved statisticsImproved statistics

2 2 2

2 2 2 2 2

31 log ( )

8 16

m m mm a l

f f

Nucleon-nucleonNucleon-nucleon

Nucleon-nucleonNucleon-nucleon

““natural” |a| < 1 natural” |a| < 1 fmfm for 350 < m for 350 < m < 600 < 600 MeVMeV

a=5.4 fm or 20 fm for ma=5.4 fm or 20 fm for m=138 MeV =138 MeV

is indeed fine tuned is indeed fine tuned

Chiral “extrapolation”Chiral “extrapolation”

• no anchor at m= 0

• wild behavior of the scattering length with mq

62

6 6 2

6 6 6 6( ) ( ) (0) (0)

( ) ( ) (0)

( ) m t

Mt

t t

C t q t q e

t q q q q e

The crucial problem is the large statistical errorsThe crucial problem is the large statistical errors

(2 3 )signal 1

noiseNM m te

N

signal:

error:

2 baryons

6 pions

(2 3 )signal 1

noiseNM m te

N

If the minimum pion energy was larger If the minimum pion energy was larger mm, the signal would be better, the signal would be better

(-z) = -(-z) = -(z) ?(z) ?

Parity orbifold Parity orbifold (P.B. +Walker-Loud)(P.B. +Walker-Loud)

parity reversedparity reversed

( ) ( )z z minimum pion energy isminimum pion energy is

22E m

L

Parity orbifold: pinholeParity orbifold: pinholethese points are these points are related by parityrelated by parity

( , , ) ( , , )x y z x y z minimum pion energy isminimum pion energy is

223E m

L

??

• LLattice QCD calculation of hadron attice QCD calculation of hadron interactions are doableinteractions are doable

• Meson-meson scattering can be computed Meson-meson scattering can be computed with few % precisionwith few % precision

• There is a serious noise problem in baryon-There is a serious noise problem in baryon-baryon channels, new ideas are neededbaryon channels, new ideas are needed

• New ideas exist ! We’ll find out how they New ideas exist ! We’ll find out how they work really soonwork really soon

SummarySummary

weighted fit: l = 3.3(6)(3)

m a2 = -0.0426 (6)(3)(18)

1-loop – 2-loop w/o counterterm

different weigths

l

K(e4): m a2 = -0.0454(31)(10)(8)

theoretical

PT predicts discretization errors (aPT predicts discretization errors (a22) ~ 1% (D. O’Connel, A. ) ~ 1% (D. O’Connel, A. Walker-Loud, R. V. Water, J. Chen)Walker-Loud, R. V. Water, J. Chen)

Finite volume (eFinite volume (e-m-mLL) ~ 1% (P.B. & I. Sato)) ~ 1% (P.B. & I. Sato)

Extracting physics from euclidean space : energies are "easy"Extracting physics from euclidean space : energies are "easy"

† †

†

0 | ( , 0) (0, 0) | 0 0 | (0,0)| | (0,0) | 0

0 | (0,0)| | (0,0) | 0

n

Htt k k e n n

m te

t

some operator with quantum numbers of the pion, made of

quarks and gluons, for instance: lowest energy state with the quantum numbers of the pion

5(0, ) (0, )aq p q p

add a background magnetic potential coupled to baryon

number with zero curl

( ) (0)

ˆ3

q L q

A zL

/ 3( ) (0)

0

iq L e q

A

or

( ) (0)

ˆ

N L N

A zL

( ) (0)

0

iN L e N

A

no coupling to local operators !

or

Solution 2: Aharonov-Bohm effect