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;
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

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Hadron spectrum
Non-perturbative renormalization
Chiral transition
Quark gluon plasma
2
1. Introduction
But the lattice regularization spoils a lot of symmetries…


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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.
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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
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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.
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2. Chiral symmetry and topology

Eigenvalue distribution of Dirac operator
continuum
(massive)
1/a
0
-1/a
2/a
4/a
6/a
m
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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
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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.
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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
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

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.
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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.
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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.
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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 !
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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.
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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.
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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.
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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
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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.
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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
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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…
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 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)
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 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
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Fpi
30
Mv
31
Mps2/m
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