Index Theorem and Random Matrix Theory for
Improved Staggered Quarks (I)
In collaboration with:
C. Davies (University of Glasgow)
A. Hart (University of Edinburgh)
HPQCD and UKQCD collaborations.
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 1

Outline
Introduction
Topology
Staggered discretization
Eigenvalues and Index Theorem
Conclusions (I)
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 2

Introduction
QCD is non-perturbative at the hadronic level and must be solved numerically: lattice QCD.
Our aim: To calculate quantities which can be compared to experiment or which are useful for experiments.
Key problem: how to include quarks.
Very expensive.
For 20 years stuck in the quenched approximation in which we miss out the quark determinant. This is equivalent to cutting out quark-antiquark pair production.
To get more precise results it is necessary to go beyond the quenched approximation.
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 3

One manifestation of the problem: we cannot fix consistently the lattice scale a
. It depends on the mass we use to fix it.
f
π f
K
3m
Ξ
− m
N m
Ω
2m
B s
− m
Υ
ψ
(1P-1S)
Υ
(1D-1S)
Υ
(2P-1S)
Υ
(3S-1S)
Υ
(1P-1S)
0.9
1 1.1
quenched/experiment
0.9
1.0
1.1
(n f
= 2+1)/experiment
(Also the recent successful
of the mass of the B c
(I. F. Allison et al.))
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 4

Topological Properties
Topological features of QCD:
Axial anomaly
η
0 mass
Predictions of RMT for each sector of Q in the ² regime
The low lying spectrum of staggered fermions appeared to be “topology blind”.
But near the continuum we should see the correct behaviour.
Improved staggered actions are being used today in large-scale dynamical simulations.
It is therefore important to understand to what extent staggered quarks show the correct topological properties.
Use the improved staggered formulations to address these questions:
Can we see the continuum features in lattices with reasonable parameters?
Are (improved) staggered quarks sensitive to topology?
Can we reproduce the detailed predictions of RMT?
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 5

Staggered Dirac Operators
S =
X
¯ ( x ) (D( x, y ) + m ) χ ( y ) x,y
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 6

Staggered Dirac Operators
S =
X
¯ ( x ) (D( x, y ) + m ) χ ( y ) x,y
One-link (naive, KS) staggered Dirac operator
D( x, y ) =
1
2 u
0
X
α
µ
( x )
“
U
µ
( x ) δ x + µ,y
− U
†
µ
( y ) δ x,y + µ
”
µ
α
µ
( x ) = ( − 1)
P
ν<µ x
ν
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 6

Staggered Dirac Operators
Antihermitian (at zero quark mass).
D(0) † = − D(0) = ⇒ − D(0) † D(0) ≥ 0 , hermitian
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 7

Staggered Dirac Operators
Antihermitian (at zero quark mass).
D(0) † = − D(0) = ⇒ − D(0) † D(0) ≥ 0 , hermitian
“ γ
5
” anticommutation
{ D(0) , ² } = 0 , with ² ( x ) = ( − 1)
P
ν x
ν
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 7

Staggered Dirac Operators
Antihermitian (at zero quark mass).
D(0) † = − D(0) = ⇒ − D(0) † D(0) ≥ 0 , hermitian
“ γ
5
” anticommutation
{ D(0) , ² } = 0 , with ² ( x ) = ( − 1)
P
ν x
ν sp (D( m )) = { m ± iλ, λ ∈ Re }
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 7

Staggered Dirac Operators
Antihermitian (at zero quark mass).
D(0) † = − D(0) = ⇒ − D(0) † D(0) ≥ 0 , hermitian
“ γ
5
” anticommutation
{ D(0) , ² } = 0 , with ² ( x ) = ( − 1)
P
ν x
ν sp (D( m )) = { m ± iλ, λ ∈ Re }
√ det .
Locality issue (B. Bunk, M. Della Morte, K. Jansen, F. Knechtli).
Encouraging recent progress: (F. Maresca and M. Peardon),
(D.H. Adams), (Y. Shamir).
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 7

Improved Staggered Actions
Unphysical taste-changing interactions, involving at leading order the exchange of a gluon of momentum q ≈ π/a .
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 8

Improved Staggered Actions
Unphysical taste-changing interactions, involving at leading order the exchange of a gluon of momentum q ≈ π/a .
Such interactions are perturbative for typical values of the lattice spacing, and can be corrected systematically a la Symanzik.
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 8

Improved Staggered Actions
Unphysical taste-changing interactions, involving at leading order the exchange of a gluon of momentum q ≈ π/a .
Such interactions are perturbative for typical values of the lattice spacing, and can be corrected systematically a la Symanzik.
Smear the gauge field to remove coupling between quarks and gluons with momentum π/a .
p=0 p=
π
/a p=0 p=-
π
/a
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 8

FAT7(TAD)
Improved Staggered Actions c1
+ c3
+ c5
+ c7
=
(Fat link)
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 9

ASQ(TAD)
Improved Staggered Actions c1
+ c3
+ c5
+ c7
+ c5’
=
(Fat link)
= c3’
(Naik)
(S. Naik, the MILC collaboration, P. Lepage.)
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 9

Improved Staggered Actions
HYP (Hypercubic Blocking)
Three levels of (restricted) APE smearing with projection onto SU (3) at each level.
Each fat link includes contributions only from thin links belonging to hypercubes attached to the original link.
a) b)
(A. Hasenfratz, F. Knechtli.)
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 9

Improved Staggered Actions
FAT7 ⊗ ASQ
Two levels of smearing: first a FAT7 smearing on the original links, followed by a projection onto SU (3) , then ASQ on these links.
FAT7 k
SU (3)
⊗ ASQ
(E.F., Q. Mason, C. Davies, K. Hornbostel, P. Lepage, H. Trottier.)
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 9

Index Theorem (Continuum)
For QCD in the continuum, the topological charge is given by
Q =
1
32 π 2
Z d
4 x ²
µνστ tr F
µν
( x ) F
στ
( x )
Atiyah-Singer Index theorem:
Q = m n f tr ( γ
5
S
F
) = m
2 n f
X h n | γ
5
| n i
λ
2 + m
2 n
= n
+
− n
− where | n i are the eigenfunctions of the massless Dirac operator in the given gauge field background, and n
+
, n
− are the number of positive and negative chirality zero modes
γ
5
| n i = ±| n i
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 10

Index Theorem (Lattice)
Dirac operators which satisfy the Ginsparg-Wilson relation
{ γ
5
, D } = a D γ
5
D can have exact, chiral zero-modes, which can then be used to define a topological charge via the identity
Q = a
4 X q ( x ) = n
+
− n
− x q ( x ) = −
1
2 a tr { γ
5
D( x, x ) } where q ( x ) is a local, gauge invariant function of the gauge fields.
For the fixed point Dirac operator, furthermore,
Q
F P
= n
+
− n
− where Q
F P is the fixed point topological charge (P. Hasenfratz, V. Laliena,
F. Niedermayer.)
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 11

Index Theorem (Lattice)
The staggered Dirac operator has no exact zero modes, therefore we cannot expect an exact index theorem.
But close to the continuum limit, we expect to see a similar behaviour: the first few eigenmodes of high chirality, in the number required by the continuum index theorem, and the rest of the eigenmodes with small chirality.
It has been seen that if you smooth the configurations enough, for example by repeatedly smearing, eventually the continuum features appear. ( P. Damgaard, U. Heller, R. Niclasen, K. Rummukainen.)
That is also the case, if we study lattice discretizations of continuum instantons. (J. Smit, J. Vink.)
But , we want to study the features of the raw, non-smoothed configurations.
Chirality must be measured using a taste-singlet operator, which is the one that couples to the anomaly.
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 12

(Almost) Index Theorem for Improved Staggered Quarks
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 13

(Almost) Index Theorem for Improved Staggered Quarks
Calculate Q gl
(cooling method).
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 13

(Almost) Index Theorem for Improved Staggered Quarks
Calculate Q gl
(cooling method).
Calculate the chirality of the low-lying eigenmodes, with the taste-singlet
“ γ
5
”, gauge-invariant staggered operator (a 4-link operator)
κ n
= h n | γ
5
| n i
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 13

(Almost) Index Theorem for Improved Staggered Quarks
Calculate Q gl
(cooling method).
Calculate the chirality of the low-lying eigenmodes, with the taste-singlet
“ γ
5
”, gauge-invariant staggered operator (a 4-link operator)
κ n
= h n | γ
5
| n i
Are there: n t n
+ near-zero modes with chirality
κ
≈ 1 and/or n t n
− near-zero modes with chirality
κ
≈ − 1 such that Q gl
≈
1 n t
“ n
+
− n
− ”
?
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 13

Wilson Glue (a
≈
0.1 fm, V = 16 x 32)
Improved Glue (a
≈
0.93 fm, V = 16 )
Improved Glue (a
≈
0.077 fm, V = 20 )
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 14

Chirality vs Eigenvalue
(a ≈ 0.093 fm, L ≈ 1.488 fm)
1
FAT7XASQ
0.8
0.6
0.8
0.6
0.4
0.2
0
0.0001
1
0.4
0.2
0
0.0001
1
0.8
0.6
0.4
0.2
0
0.0001
0.001
0.001
0.001
ASQTAD
ONE-LINK
0.01
0.01
0.1
0.1
(a ≈ 0.077 fm, L ≈ 1.54 fm) fat7xasqtad, fine lattice
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1e-08 1e-07 1e-06 1e-05 1e-04 0.001
0.01
0.01
0.1
From Actions to Experiments, NeSC, Edinburgh, March 2005
0.1
– p. 15

Quartet Splitting
0.3
0.2
0.1
0
0.3
0.2
0.1
0
0.3
0.2
0.1
0
0
Histogram of Q=0, k=1 quartet splitting Histogram of Q=1, k=1 quartet splitting
ONE-LINK
FAT7xASQ
10 a=0.125 fm, L=12 a=0.093fm, L=16 a=0.077 fm, L=20
0.3
0.2
0.1
0.3
0
0.2
0.1
0.3
0
0.2
0.1
ONE-LINK
FAT7xASQ a=0.125 fm, L=12 a=0.093fm, L=16 a=0.077 fm, L=20
20
λ
30
/ MeV
40 50 60 10 20 30
λ λ
/ MeV
40 50 60 70
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 16

Zero Modes Scaling, FAT7xASQ
0.12
0.08
O(a
2
) fit
O(a
2
)+O(a
4
)
0.04
0
0 0.005
0.01
(a / fm)
2
0.015
0.02
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 17

Conclusions (I)
For improved staggered quarks on improved gauge backgrounds, there is a sharp distinction between high and low chirality modes, and their respective number is in accordance with the Index Theorem.
As we approach the continuum limit, this becomes true for all the actions.
The four-fold spectrum degeneracy is clearly present for the improved staggered action.
As we approach the continuum limit it starts to manifest itself on the less improved actions (asqtad.)
From Actions to Experiments, NeSC, Edinburgh, March 2005 – p. 18