Symmetries and exponential error reduction
in YM theories on the lattice
Michele Della Morte
Institut für Kernphysik, University of Mainz
EMG Klausurtagung, 22/10/2009, Mainz
In collaboration with L. Giusti.
Based on Computer Physics Comm. 180:813-818 and 180:819-826, 2009
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 1/17
Outline
• Exponential growth of the signal to noise ratio
• The Yang-Mills theory
• Decomposition of the path integral and boundary conditions
• Symmetry-Constrained Monte Carlo
• Results
• Conclusions and outlook
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 2/17
Exponential growth of the signal to noise ratio (Parisi ’84, Lepage ’89)
Consider a point to point correlation function interpolating (eg) a meson.
The signal is given by the expectation value of
while the variance is given by the expectation value of
Luckily Wick-contractions are done before squaring, for the variance. Then a
multi-pion state dominates, otherwise it would be the vacuum (as for YM).
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 3/17
pion RNS ∝ const
ρ
RNS ∝ exp((mρ − mπ )t)
N
RNS ∝ exp((mN − 23 mπ )t)
1
0.5
0
−0.5
Log(R
NS
)
−1
−1.5
−2
−2.5
−3
−3.5
−4
5
10
15
20
t/a
25
30
35
O(2000) quenched confs (β = 6.2, κ = 0.1526) in APE, hep-lat/9611021
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 4/17
Yang-Mills theory
• For an operator interpolating a parity odd glueball
COG (t) = hOG (t)OG (0)i → |h0|OG (0)|G − i|2 e −MG − t + . . .
the variance can be estimated as
σ 2 = hOG2 (t)OG2 (0)i − hOG (t)OG (0)i2 → h0|OG2 (0)|0i2 + . . .
• The noise to signal ratio at large time separations is given by
RNS (t) →
h0|OG2 (0)|0i
e MG − t + . . .
|h0|OG (0)|G − i|2
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 5/17
Data / Fit ( Scalar, β = 6.408
χ2 = 1.79 )
2
1.5
1
A01
A02
A11
A22
0.5
0
0
0.2
0.4
0.6
0.8
1
x0 / r0
H. Meyer, arXiv:0808.3151
Signal up to 0.5 fm at most.
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 6/17
8
−+
2
0
+−
3
1
*++
−+
6
0
++
2
4
mG (GeV)
r0mG
2
++
0
1
2
0
++
−+
PC
+−
−−
0
C. Morningstar and M. Peardon, 1999 ....
Nice results, which however we believe need to be checked concerning
some systematics
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 7/17
• On a given gauge configuration symmetries as parity are not
preserved. All states are allowed to propagate despite the quantum
numbers of the source.
• In the gauge average each configuration gives a contribution
O(e −MG − t ) to the signal and O(1) to the variance.
This suggests one should introduce some sort of projectors on the relevant
states, but it is not clear what that means in the path integral approach of
Monte Carlo simulations.
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 8/17
Decomposition of the path integral and boundary conditions
with periodic boundary conditions Z =
Z = Z+ + Z− ,
Z ± = e −E0 T
D3 [V ]hV |e −T Ĥ P̂g |V i
"
#
1 ± 1 X ± −En± T
+
wn e
2
R
n=1
We introduce a parity transformation
℘ˆ |V i = |V ℘i ,
Vk℘ (x) = Vk† (−x − k̂) ,
with V̂k (x)|V i = Vk (x)|V i and
Z
tw
Z = D3 [V ]hV |e −T Ĥ P̂g |V ℘ i =
XZ
ˆ|G ihG |V i = Z + − Z −
D3 [V ]hV |G ihG |e −T Ĥ ℘
G
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 9/17
tw
so Z −Z
= Z − and
2
parity odd glueball.
Z−
Z
for large T should be dominated by the lightest
Our strategy consists in computing the ratio
conditions in Z tw are parity twisted
Z tw
Z
where the boundary
So far anyway the exponential problem remains unsolved ...
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 10/17
Recursive relations in the transfer matrix formalism
The Transfer matrix elements hVx0 +1 |T̂ P̂G |Vx0 i give the probability for
the state Vx0 to evolve into the state Vx0 +1 within a timeslice.
We introduce Parity eigenstates
|V , ±i =
√1
2
h
i
|V i ± |V ℘ i
with Transfer matrix elements
h
i
hs 0 , Vx0 +1 |T̂|Vx0 , si = 2 δs 0 s Ts Vx0 +1 , Vx0
h
i
Ts Vx0 +1 , Vx0 =
1
2
n h
i
h
io
T Vx0 +1 , Vx0 + s T Vx0 +1 , Vx℘0
.
The definition can be easily generalized to thick time-slices of size d and
the ratio of such Transfer matrix elements can be numerically computed.
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 11/17
Symmetry-Constrained Monte Carlo
By dividing the time extent T into several thick time-slices of size d, Z s
s
can be obtained as the product of the ratios TT integrated over the
boundary configurations
Z
h
i
Zs
1
=
D4 [U] e −S[U] Psm,d T , 0 , with T = m ∗ d
Z
Z
h
i m−1
Y Ts [Vx +(i+1)·d ,Vx +i·d ]
0
0
Psm,d T , 0 =
T[V
,Vx +i·d ]
i=0
x0 +(i+1)·d
0
At this point a multilevel algorithm can be used to achieve an exponential
error reduction, similarly to what was done by Lüscher and Weisz ’01 for
the Polyakov Loops
The insertion of Ts [Vy0 , Vx0 ] in the path integral plays the role of a
projector and allows the propagation of states with a given parity only.
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 12/17
−
[T , 0] for each boundary configuration
As we need to compute Pm,d
−S[U]
we are performing MC simulations
generated with the weight e
within a MC simulation → We have a V 2 algorithm.
We extract the effective mass from
mGeff− (T )
1
= − ln
T
Z−
(T )
Z
For a given precision on that, the algorithm scaling with T is proportional
to ' e 2mG − d · ( Td )2 to be compared with the ' e 2mG − T scaling of the
standard technique
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 13/17
Results (Wilson action, β = 5.7)
0.02
P2,5(10,0)
0.015
0.01
0.005
0
-0.005
0
5
10
15
20
25
30
35
40
45
50
Nconf
−
MC history of P2,5
from the L = 6, T = 10 and β = 5.7 run
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 14/17
10
L=6
L=8
L=10
1
0.1
-
Z /Z
0.01
0.001
0.0001
1e-05
1e-06
1e-07
1e-08
0
2
4
6
8
10
12
14
16
18
20
22
T
We could follow the exponential decay over almost 7 orders of magnitude.
In principle we could also extract the multiplicity of the state, but not very
accurately. We have asummed it to be 1.
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 15/17
1.3
L=6
L=8
L=10
1.2
1.1
1
0.9
M
-
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
18
20
22
24
T
Signal up to more than 3 fm separation !
Finite size effects clearly visible, for L ≤ 1.5 fm.
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 16/17
Conclusions and outlook
• The noise to signal exponential problem can be solved by enforcing
the propagation in time of states with the desired quantum numbers
only
• We have tested the strategy in the pure-YM case for the mass of the
first parity-odd state
• In the future we plan to extend the computation to larger volumes
and finer lattice spacings. The existence of a bound glueball state
could then be proven in a theoretically sound way (a single state
dominating a correlation function over large time separations).
• but also to consider other symmetries (C-parity rather similar, O(3) a
bit more complicated)
• Fermions ?
Symmetries and exponential error reduction in YM theories on the lattice
Michele Della Morte, 22/10/2009, Mainz 17/17