Mass Spectra of N* and
Source Optimization
Lattice Hadron Physics Collaboration
Presenter:
Ikuro Sato
University of Maryland,
College Park, MD, USA
Contents:
 Motivations of using cubic group on lattice
 Group theoretical projection operator
 About cubic group
 Three quarks with link(s) --- quark’s orbital
angular momentum
 Operators used in our calculations
 Lattice setup
 Results
 Discussions and conclusion
Motivations of using cubic group on lattice
The correlator function of baryon field, Bm(x, t) is,
C (t )    0 | Bm ( x, t ) B m (0) | 0  .
x
Expanding this with complete set of intermediate state |n>, one
obtains,
C (t )    0 | Bm (0) | n  e
xn
 M nt
 

i
x
 p,
 n | B m (0) | 0 e
where Mn is the mass of nth intermediate state.
Highly optimized source will extract a particular state of interest,
while minimizing the overlap factor from other unwanted states. So,
the requirement is
 0 | B | n ~  nm .
m
By using the orthogonality relation of basis vectors spanning
irreducible subspace of cubic group, one can maximize the signals of
particular state in a clean fashion.
Furthermore, each irreducible subspace of cubic group is known to
hold particular total angular momentum, J, defined on discretized
space. Higher spin sources can be represented with a help of IRs.
Projection operator
Suppose a vector space L is given, and set of operators, {Ga}, are
known in L. {Ga} is said to form a group if it satisfies
G  Ga  {Gc } for all a and b.
b
In general, L will be reducible, that means, {Ga} can be
similarity-transformed into block-diagonal forms by similarity matrix,

S, consisting of new basis vectors v .
Projection operator,P() , can take any vector in L , and projects out its
reflection image on irreducible subspace. Thus, eigenvectors of
P() are the basis vectors that span irreducible subspace .
i.e.,
 
P ( ) v  v
P() is obtained by the formula:
P ( ) 
s
 ( ) * (Ga )T (Ga ) ,

g G
a
where s is the dimension of  , g is the number of group
( )
elements,  (Ga ) is the trace of blocked matrix in subspace
 , and T (Ga ) is the matrix representation of Ga .
About cubic group
The point group of a cube with spinorial degree of freedom is called,
“double-covered octahedral group”, or 2O.
2
O consists of 48 group elements:
spatial rotations:
90o, 180o, 540o about x, y, z axis
±120o about four body-diagonal axis
90o about six face-diagonal axis
identity
_____________________________
24
-spatial rotations
x2
-spinor
= 48
It is known that there are three irreducible subspaces in 2O:
reduction
2O
IR
G1
dim
2
G2
2
H
4
J
1/2
3/2
5/2
7/2
9/2
11/2
IR
G1
H
G2+H
G1+G2+H
G1+2H
G1+G2+2H
(R. C. Johnson)
To have J≧3/2, one needs to arrange quark spatial positions so that
L≧1.
Since lattice preserves cubic symmetry, IR of source is conserved,
though J may not.
It is a natural think that higher spin objects have larger mass.
M t
Therefore, because of the exponential factor e , higher spin states
die faster than lower spin states. So, lower spin objects would be
extracted more cleanly.
n
Three quarks with link variable(s)
B( x)   c( x,l ,l ,l , ,  , )
1 2 3

ijk 1 i
2 j
3 k
q
(
l
)
U
q
(
l
)
  1 12  2 U 23 q (l3 )
 is over Dirac indices, color indices, and l’s.
U12  exp[ iagAl l (l1 )]
2
1
For s-wave, c factrizes into scalar smearing operator, F.
c( x, l1 , l2 , l3 ,  ,  ,  )  F ( x, l1 ) F ( x, l2 ) F ( x, l3 )
N (1/2+)
+(3/2+)
 1   4 1   5 3 
d (C 5 ) u  
 u
2 
 2
 1   4 1   5 3 
d (C  ) u  
 u  1  2  1  3
2 
 2
For p-wave:
G1 subgroup:
 1   4 1   5 3 

 1   4 1   5 3 


d (C 5 ) u  
D

iD

D


 z  u
x
y
2
2
2
2


 

where Dj is the discretized covariant derivative.
This operator can be written as
1
2
Y11 1 / 2 , 1 / 2  Y10 1 / 2 ,1 / 2 .
3
3
This operator is nothing but a Clebsch-Gordon series of
| L  1   | S  1 / 2  , representing P1/2 !!
H subgroup:
In short hand notations,
P3/2 = Y111 / 2,1 / 2 ,
3
1
3
Y11  3 / 2 , 1 / 2 
Y11  3 / 2 , 3 / 2  Y10  3 / 2 ,1 / 2 .
10
5
10
P5/2 =
G2 subgroup:

1
1
1
Y11  3 / 2 , 3 / 2 
Y11  3 / 2 , 1 / 2 
Y10  3 / 2 , 3 / 2
6
2
3
For p-wave, G2 vectors do not obey Clebsch-Gordon coefficients.
Lattice does not always preserve Jz.
One can go up to d-wave sources by extending the spatial degrees
of freedom.
Take the third quark at two-link away
from the origin, i.e.,
q(0)q(axˆ)q(axˆ  ayˆ )
Lattice setup
Isotropic Wilson action with kappa value 0.1480 and 0.1520.
Gauge invariant Gaussian smearing is done for each quark field with
smearing radius 4.5. Beta = 6.0. 221 configurations were used.
163x32 lattice.
Results
Kappa
0.1480
0.1520
Source
Effective Masses
Time-slice
N(1/2+)
1.133
12

1.170
12
N(1/2-)
1.432
6
N(3/2-)
1.441
6
N(1/2+)

N(1/2-)
N(3/2-)
Discussions & Conclusion
Baryon sources can be constructed using irreducible vectors of
Double-covered octahedral group, for both lowest-lying state and
excited states (N*).
One can specify the parity and the spin of the baryonic objects with a
use of irreducible.
Construct rotational matrices of baryon  Form a projection operator
 Diagonalize projection operator