quarks, gluons and nuclear forces
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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 tDx t e x t x
x t xDx 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 iix 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 trZ D m D m
trN D m D m
probability distribution for Uprobability distribution for Uii
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 mL
one particle
2
2
1cot ( )4
M ELM E EL
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 16m m m
m a lf 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 16m m m
m a lf 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 1noise
NM m teN
signal:
error:
2 baryons
6 pions
(2 3 )signal 1noise
NM m teN
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 mL
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 mL
??
• 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 tet
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
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