QUARKS, GLUONS QUARKS, GLUONS AND AND NUCLEAR FORCES NUCLEAR FORCES Paulo Bedaque Paulo Bedaque University of Maryland, College University of Maryland, College Park Park
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