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QUARKS, GLUONS
AND
NUCLEAR FORCES
Paulo Bedaque
University of Maryland, College Park
strong nuclear force:
binds neutrons and protons
into nuclei
Quantum Chromodynamics
(QCD)
What do we know ?
1) NN phase shifts
1S
0
neutron-proton
What do we know ?
2) Several potentials that fit them
pion exchange
all kinds of things …
What do we know ?
3) These potentials explain a lot but not everything
• NNn, NNg, couplings
• NNN forces
• NY forces
• ...
few % on nd
~5% of nuclei binding
strangeness in neutron stars
Can we understand the nuclear forces (and
NNN, NNn, …) from first principles ?
LATTICE QCD
PATH INTEGRALS
iS1
e
Probability | eiS1  eiS1 
e
iS2
|2
Quantum mechanics reduced to quadratures
 x(t ) x(0)  
operators
iS [ x(t )] x(t ) x(0)
Dx
(
t
)
e

iS [ x(t )]
Dx
(
t
)
e

numbers
iS [ x ( t )]
Dx
(
t
)
e
is as well (or ill) defined as




dx ei x
Imaginary time (t
it): just like stat mech
probability
distribution
 x(t ) x(0)    Dx(t ) Z1 e S [ x (t )] x(t ) x(0)
1
N
N
 x (t ) x (0)
i 1
i
i
But I don’t live in imaginary time !
What can I do with imaginary time correlators ?
 0 | x(t ) x(0) | 0    0 | e Ht x(0)e Ht x(0) | 0 
   0 | x | n  e( E E )t  n | x | 0 
n
0
n
 e( E  E0 )t | 0 | x | 1 |2
t
1
lowest energy state w/
some overlap
Typical paths
xi (t ) xi (0)
N
1 x (t ) x (0)

i
N i 1 i
PATH INTEGRALS FOR FIELDS
iS1
e
iS1
e
Quantum Chromodynamics
Q = spinor, 3 colors,
6 flavors
= quarks
U = SU(3) matrix
= gluons
QCD reduced to quadratures
 qg 5q( x ) qg 5q(0)   1
Z
 1
Z

 SG [U ]q ( DU m ) q
DUDqDq
e
qg 5q( x ) qg 5q(0)

 SG [U ]
DU
e
det( DU  m) tr[


1 g
1 g ]
5
DU  m DU  m 5
probability distribution for Ui
 qg 5q( x ) qg 5q(0)   1
Z
 1
N
 SG [U ]
DU
e
det( DU  m) tr[

N
 tr[
i 1
1
1
g5
g 5]
DU  m DU  m
i
i
algorithm
1. find {Ui}
2. compute 1/(DUi+m)
3. compute observable
1 g
1 g ]
5
DU  m DU  m 5
Scattering through finite volumes:
the Luscher method
(Marinari, Hamber, Parisi, Rebbi)
one particle
Periodic boundary
conditions: box is a torus
Energy levels at En   2 n   m 2
 L 
2
Scattering through finite volumes:
the Luscher method
(Marinari, Hamber, Parisi, Rebbi)
two particles
1  M EL2 
M E cot  (E ) 
S

 L  4 2 
known function
Learn about the deuteron in boxes smaller
than the deuteron
 0 | N (t , k ) N (t , k ) N † (0, k ) N † (0, k ) | 0     0 | N (0, k ) N (0, k )e
 Ht
| n  n | N † (0, k ) N † (0, k ) | 0 
n
 e
t 
 E2 N t
| NN at rest | N † (0, k ) N † (0, k ) | 0 |2
The difference between E2N and EN is our
signal
phase shift
The time to try it is now
• Pion masses small enough for chiral extrapolation
• No quenching
• Volumes ~ (3 fm)3
• Improved actions
• Good chiral symmetry
• Software resources
S. Beane, T. Luu, K. Orginos, E. Pallante, A. Parreno,
M. Savage, A. Walker-Loud, …
Gold platted scattering observable: I=2 pp
K(e4)
CP-PACS
m2
m a2  
8 f2


3m2 
m2
1

log

l
(

)



2 2 
2


 16 f 



Improved statistics
K(e4)
CP-PACS
m2
m a2  
8 f2


3m2 
m2
1

log

l
(

)



2 2 
2


 16 f 



Nucleon-nucleon
Nucleon-nucleon
“natural” |a| < 1 fm for
350 < m < 600 MeV
a=5.4 fm or 20 fm for m=138 MeV
is indeed fine tuned
Chiral “extrapolation”
• no anchor at mp= 0
• wild behavior of the scattering length with mq
The crucial problem is the large statistical errors
signal:
C (t )   q6 (t ) q 6 (0)  e 2 Mt
2 baryons
error:
 2 (t )   q6 (t )q 6 (t ) q6 (0)q 6 (0)  e 6 m t
6 pions
signal
noise
1 (2 M N 3m )t
e
N
signal
noise
1 (2 M N 3m )t
e
N
If the minimum pion energy was larger
m, the signal would be better
(-z) = -(z) ?
Parity orbifold (P.B. +Walker-Loud)
parity reversed
 ( z)    ( z)
minimum pion energy is
E 
L

2
 m2
Parity orbifold: pinhole
these points are
related by parity
 ( x,  y,  z)    ( x, y, z)
minimum pion energy is

E  3 
L
2
 m2
?
Summary
• Lattice QCD calculation of hadron
interactions are doable
• Meson-meson scattering can be computed
with few % precision
• There is a serious noise problem in baryonbaryon channels, new ideas are needed
• New ideas exist ! We’ll find out how they
work really soon
weighted fit: lpp = 3.3(6)(3)
different weigths
mp a2 = -0.0426 (6)(3)(18)
lpp
1-loop – 2-loop
w/o counterterm
K(e4): mp a2 = -0.0454(31)(10)(8)
theoretical
cPT predicts discretization errors (a2) ~ 1% (D. O’Connel,
A. Walker-Loud, R. V. Water, J. Chen)
Finite volume (e-mpL) ~ 1% (P.B. & I. Sato)
Extracting physics from euclidean space : energies are "easy"
 0 |  (t , k  0)  † (0, k  0) | 0    e Ht  0 |  (0, 0)| n  n |  † (0, 0) | 0 
n
 e
t 
some operator with quantum
numbers of the pion, made of
quarks and gluons, for instance:
q (0,  p)g 5 a q(0, p)
m t
 0 |  (0, 0)|    |  † (0, 0) | 0 
lowest energy state with the
quantum numbers of the pion
Solution 2: Aharonov-Bohm effect
add a background magnetic
potential coupled to baryon
number with zero curl
q ( L)  q (0)
A

3L

L
zˆ
q( L)  ei / 3q(0)
A0
zˆ
N ( L)  N (0)
A
or
or
N ( L)  ei N (0)
A0
no coupling to local operators !