The fermion condensations and the ηmeson in θ vacuum
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Transcript The fermion condensations and the ηmeson in θ vacuum
Approaching the chiral limit
in lattice QCD
Hidenori Fukaya (RIKEN Wako)
for JLQCD collaboration
Ph.D. thesis [hep-lat/0603008],
JLQCD collaboration,Phys.Rev.D74:094505(2006)[heplat/0607020], hep-lat/0607093, hep-lat/0610011, heplat/0610024 and hep-lat/0610026.
1
1. Introduction
Lattice gauge theory
gives a non-perturbative definition of the quantum
field theory.
finite degrees of freedom. ⇒ Monte Carlo simulations
⇒ very powerful tool to study QCD;
Hadron spectrum
Non-perturbative renormalization
Chiral transition
Quark gluon plasma
2
1. Introduction
But the lattice regularization spoils a lot of symmetries…
Translational symmetry
Lorentz invariance
Chiral symmetry and topology
Supersymmetry…
3
1. Introduction
The chiral limit (m→0) is difficult.
Losing chiral symmetry to avoid fermion doubling.
Nielsen and Ninomiya, Nucl.Phys.B185,20(‘81)
Large computational cost for m→0.
Wilson Dirac operator
(used in JLQCD’s previous works)
breaks chiral symmetry and requires
additive renormalization of quark mass.
unwanted operator mixing with opposite chirality
symmetry breaking terms in chiral perturbation theory .
Complitcated extrapolation from mu, md > 50MeV .
⇒ Large systematic uncertainties in m~ a few MeV results.
4
1. Introduction
Our strategy in new JLQCD project
1. Achieve the chiral symmetry at quantum level
on the lattice
by overlap fermion action
[ Ginsparg-Wilson relation]
Neuberger, Phys.Lett.B417,141(‘98)
Ginsparg & WilsonPhys.Rev.D25,2649(‘82)
and topology conserving action
[ Luescher’s admissibility condition]
M.Luescher,Nucl.Phys.B568,162 (‘00)
2. Approach mu, md ~ O(1) MeV.
5
Plan of my talk
1.
2.
3.
4.
5.
Introduction
Chiral symmetry and topology
JLQCD’s overlap fermion project
Finite volume and fixed topology
Summary and discussion
6
2. Chiral symmetry and topology
Nielsen-Ninomiya theorem: Any local Dirac operator
satisfying
has unphysical poles (doublers).
Example - free fermion –
Continuum
Lattice
Nielsen and Ninomiya, Nucl.Phys.B185,20(1981)
has no doubler.
has unphysical poles at
.
Wilson Dirac operator (Wilson fermion)
Doublers are decoupled but spoils chiral symmetry.
7
2. Chiral symmetry and topology
Eigenvalue distribution of Dirac operator
continuum
(massive)
1/a
0
-1/a
2/a
4/a
6/a
m
8
2. Chiral symmetry and topology
Eigenvalue distribution of Dirac operator
1/a
Wilson fermion
Naïve lattice
sparse but
fermion
nonzero density
(massive)
dense
until a→0.
16 lines 4 heavy
1 physical
0
m
-1/a
2/a
6 heavy
4/a
4 heavy
1 heavy
6/a
• Doublers are massive.
• m is not well-defined.
• The index is not well-defined.
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The overlap fermion action
The Neuberger’s overlap operator:
Phys.Lett.B417,141(‘98)
satisfying the Ginsparg-Wilson relation:
Phys.Rev.D25,2649(‘82)
realizes ‘modified’ exact chiral symmetry on the lattice;
the action is invariant under
NOTE
M.Luescher,Phys.Lett.B428,342(1998)
Expansion in Wilson Dirac operator ⇒ No doubler.
Fermion measure is not invariant;
⇒ chiral anomaly, index theorem
(Talk by Kikukawa)
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2. Chiral symmetry and topology
Eigenvalue distribution of Dirac operator
The overlap fermion
1/a
0
2/a
4/a
6/a
-1/a
• Doublers are massive.
• D is smooth except for
11
2. Chiral symmetry and topology
Eigenvalue distribution of Dirac operator
The overlap fermion
(massive)
1/a
0
m
2/a
4/a
6/a
-1/a
• m is well-defined.
• index is well-defined.
12
2. Chiral symmetry and topology
Eigenvalue distribution of Dirac operator
The overlap fermion
1/a
0
-1/a
2/a
4/a
6/a
• Theoretically ill-defined.
• Large simulation cost.
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2. Chiral symmetry and topology
The topology (index) changes
Hw=Dw-1=0
Topology
boundary
The⇒
complex
modes
make pairs
1/a
0
-1/a
2/a
The real modes6/a
4/a
are chiral
eigenstates.
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The overlap Dirac operator
becomes ill-defined when
Hw=0 forms topology boundaries.
These zero-modes are lattice artifacts(excluded in a→∞limit.)
In the polynomial expansion of D,
The discontinuity of the determinant requires
reflection/refraction (Fodor et al. JHEP0408:003,2004)
~ V2 algorithm.
15
2. Chiral symmetry and topology
Topology conserving gauge action
To achieve |Hw| > 0 [Luescher’s “admissibility” condition],
M.Luescher,Nucl.Phys.B568,162 (‘00)
we modify the lattice gauge action.
We found that adding
with small μ, is the best and easiest way in the numerical
simulations (See JLQCD collaboration, Phys.Rev.D74:09505,2006)
Note: Stop →∞ when Hw→0 and Stop→0 when a→0.
16
2. Chiral symmetry and topology
Our strategy in new JLQCD project
1. Achieve the chiral symmetry at quantum level
on the lattice
by overlap fermion action
[ Ginsparg-Wilson relation]
Neuberger, Phys.Lett.B417,141(‘98)
Ginsparg & WilsonPhys.Rev.D25,2649(‘82)
and topology conserving action Stop
[ Luescher’s admissibility condition]
M.Luescher,Nucl.Phys.B568,162 (‘00)
2. Approach mu, md ~ O(1) MeV.
17
3. JLQCD’s overlap fermion project
Numerical cost
Simulation of overlap fermion was thought to be impossible;
D_ov is a O(100) degree polynomial of D_wilson.
The non-smooth determinant on topology boundaries
requires extra factor ~10 numerical cost.
⇒ The cost of D_ov ~ 1000 times of D_wilson’s .
However,
Stop can cut the latter numerical cost ~10 times faster
New supercomputer at KEK ~60TFLOPS
~50 times
Many algorithmic improvements
~ 5-10 times
we can overcome this difficulty !
18
3. JLQCD’s overlap fermion project
The details of the simulation
As a test run on a 163 32 lattice with a ~ 1.6-1.8GeV
(L ~ 2fm), we have achieved 2-flavor QCD simulations with
overlap quarks with the quark mass down to ~2MeV.
NOTE m >50MeV with non-chiral fermion in previous JLQCD works.
Iwasaki (beta=2.3) + Stop(μ=0.2) gauge action
Overlap operator in Zolotarev expression
Quark masses : ma=0.002(2MeV) – 0.1.
1 samples per 10 trj of Hybrid Monte Carlo algorithm.
2000-5000 trj for each m are performed.
Q=0 topological sector
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3. JLQCD’s overlap fermion project
Numerical data of test run (Preliminary)
Both data confirm the exact chiral symmetry.
20
4. Finite volume and fixed topology
Systematic error from finite V and fixed Q
Our test run on (~2fm)4 lattice is limited to a fixed
topological sector (Q=0). Any observable is different from
θ=0 results;
where χ is topological susceptibility and f is an unknown
function of Q.
⇒ needs careful treatment of finite V and fixed Q .
Q=2, 4 runs are started.
24348 (~3fm)4 lattice or larger are planned.
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4. Finite volume and fixed topology
ChPT and ChRMT with finite V and fixed Q
However, even on a small lattice, V and Q effects can be
evaluated by the effective theory: chiral perturbation
theory (ChPT) or chiral random matrix theory (ChRMT).
They are valid, in particular, when mπL<1 (ε-regime) .
⇒ m~2MeV, L~2fm is good.
Finite V effects on ChRMT :
discrete Dirac spectrum ⇒ chiral condensate Σ.
Finite V effects on ChPT :
pion correlator is not exponential but quadratic.
⇒
pion decay const. Fπ.
22
4. Finite volume and fixed topology
Dirac spectrum and ChRMT (Preliminary)
Nf=2
Nf=0
Lowest eigenvalue (Nf=2) ⇒ Σ=(233.9(2.6)MeV)3
23
4. Finite volume and fixed topology
Pion correlator and ChPT (Preliminary)
The quadratic fit (fit range=[10,22],β1=0) worked well.
[χ2 /dof ~0.25.]
Fπ = 86(7)MeV
is obtained [preliminary].
NOTE: Our data are at
m~2MeV. we don’t need
chiral extrapolation.
24
5. Summary and discussion
The chiral limit is within our reach now!
Exact chiral symmetry at quantum level can be
achieved in lattice QCD simulations with
Overlap fermion action
Topology conserving gauge action
Our test run on (~2fm)4 lattice, we’ve simulated
Nf=2 dynamical overlap quarks with m~2MeV.
Finite V and Q dependences are important.
ChRMT in finite V ⇒ Σ~2.193E-03.
ChPT in finite V ⇒ Fπ~86MeV.
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5. Summary and discussion
To do
Precise measurement of hadron spectrum, started.
2+1 flavor, started.
Different Q, started.
Larger lattices, prepared.
BK ,
started.
Non-perturbative renormalization,
prepared.
Future works
θ-vacuum
ρ→ππ decay
Finite temperature…
26
How to sum up the different topological sectors
Formally,
With an assumption,
The ratio can be given by the topological susceptibility,
V’
if it has small Q and V’ dependences.
Parallel tempering + Fodor method may also be useful.
Z.Fodor et al. hep-lat/0510117 27
Initial configuration
For topologically non-trivial initial configuration, we use
a discretized version of instanton solution on 4D torus;
which gives constant field strength with arbitrary Q.
A.Gonzalez-Arroyo,hep-th/9807108,
M.Hamanaka,H.Kajiura,Phys.Lett.B551,360(‘03)
28
Topology dependence
If
, any observable at a fixed topology
in general theory (with θvacuum) can be written as
Brower et al, Phys.Lett.B560(2003)64
In QCD,
⇒
Unless
,(like NEDM)Q-dependence is negligible.
Shintani et al,Phys.Rev.D72:014504,2005
29
Fpi
30
Mv
31
Mps2/m
32