approaching the chiral limit in lattice qcd
DESCRIPTION
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)[hep-lat/0607020], hep-lat/0607093, hep-lat/0610011, hep-lat/0610024 and hep-lat/0610026. 1. Introduction. - PowerPoint PPT PresentationTRANSCRIPT
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Approaching the chiral limit Approaching the chiral limit in lattice QCDin lattice QCD
Hidenori Fukaya (RIKEN Wako)for JLQCD collaboration
Ph.D. thesis [hep-lat/0603008],JLQCD collaboration,Phys.Rev.D74:094505(2006)[hep-lat/0607020], hep-lat/0607093, hep-lat/0610011, hep-lat/0610024 and hep-lat/0610026.
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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
1. Introduction
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But the lattice regularization spoils a lot of symmetries…
Translational symmetry Lorentz invariance Chiral symmetry and topology Supersymmetry…
1. Introduction
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The chiral limit (m→0) is difficult.
Losing chiral symmetry to avoid fermion doubling.
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.
1. Introduction
Nielsen and Ninomiya, Nucl.Phys.B185,20(‘81)
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Our strategy in new JLQCD project
1. Achieve the chiral symmetry at quantum level on the lattice
by overlap fermion action [ Ginsparg-Wilson relation]
and topology conserving action [ Luescher’s admissibility condition]
2. Approach mu, md ~ O(1) MeV.
1. Introduction
Ginsparg & WilsonPhys.Rev.D25,2649(‘82)
Neuberger, Phys.Lett.B417,141(‘98)
M.Luescher,Nucl.Phys.B568,162 (‘00)
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Plan of my talk
1. Introduction 2. Chiral symmetry and topology3. JLQCD’s overlap fermion project4. Finite volume and fixed topology5. Summary and discussion
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Nielsen-Ninomiya theorem: Any local Dirac operator satisfying has unphysical poles (doublers).
Example - free fermion – Continuum has no doubler. Lattice
has unphysical poles at . Wilson Dirac operator (Wilson fermion)
Doublers are decoupled but spoils chiral symmetry.
2. Chiral symmetry and topology
Nielsen and Ninomiya, Nucl.Phys.B185,20(1981)
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02/a 4/a 6/a
Eigenvalue distribution of Dirac operator
2. Chiral symmetry and topology
continuum
(massive)
m
1/a
-1/a
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Wilson fermion
Eigenvalue distribution of Dirac operator
2. Chiral symmetry and topology
1/a
-1/a
Naïve lattice fermion
16 lines
02/a 4/a 6/a
(massive)
m • Doublers are massive.
• m is not well-defined.
• The index is not well-defined.
1 physical 4 heavy 6 heavy 4 heavy 1 heavydense
sparse but nonzero density
until a→0.
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The overlap fermion action
The Neuberger’s overlap operator:
satisfying the Ginsparg-Wilson relation:
realizes ‘modified’ exact chiral symmetry on the lattice;the action is invariant under
NOTE Expansion in Wilson Dirac operator ⇒ No doubler. Fermion measure is not invariant;
⇒ chiral anomaly, index theorem
Phys.Rev.D25,2649(‘82)
Phys.Lett.B417,141(‘98)
M.Luescher,Phys.Lett.B428,342(1998)
(Talk by Kikukawa)
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Eigenvalue distribution of Dirac operator
2. Chiral symmetry and topology
1/a
-1/a
02/a 4/a 6/a
The overlap fermion
• Doublers are massive.
• D is smooth except for
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Eigenvalue distribution of Dirac operator
2. Chiral symmetry and topology
1/a
-1/a
02/a 4/a 6/a
The overlap fermion(massive)
m
• m is well-defined.
• index is well-defined.
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Eigenvalue distribution of Dirac operator
2. Chiral symmetry and topology
1/a
-1/a
02/a 4/a 6/a
The overlap fermion
• Theoretically ill-defined.
• Large simulation cost.
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The topology (index) changes
2. Chiral symmetry and topology
1/a
-1/a
02/a 4/a 6/a
The complex modes make pairs
The real modes are chiral eigenstates.
Hw=Dw-1=0
⇒ Topology boundary
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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.
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Topology conserving gauge action
To achieve |Hw| > 0 [Luescher’s “admissibility” condition], 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.
2. Chiral symmetry and topology
M.Luescher,Nucl.Phys.B568,162 (‘00)
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Our strategy in new JLQCD project
1. Achieve the chiral symmetry at quantum level on the lattice
by overlap fermion action [ Ginsparg-Wilson relation]
and topology conserving action Stop
[ Luescher’s admissibility condition]
2. Approach mu, md ~ O(1) MeV.
2. Chiral symmetry and topology
Ginsparg & WilsonPhys.Rev.D25,2649(‘82)
Neuberger, Phys.Lett.B417,141(‘98)
M.Luescher,Nucl.Phys.B568,162 (‘00)
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Numerical costSimulation 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 e
xtra 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 !
3. JLQCD’s overlap fermion project
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The details of the simulationAs 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
3. JLQCD’s overlap fermion project
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Numerical data of test run (Preliminary)
Both data confirm the exact chiral symmetry.
3. JLQCD’s overlap fermion project
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Systematic error from finite V and fixed QOur 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.
4. Finite volume and fixed topology
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ChPT and ChRMT with finite V and fixed QHowever, 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.
4. Finite volume and fixed topology
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Dirac spectrum and ChRMT (Preliminary)Nf=2 Nf=0
Lowest eigenvalue (Nf=2) ⇒ Σ=(233.9(2.6)MeV)3
4. Finite volume and fixed topology
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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 atm~2MeV. we don’t need chiral extrapolation.
4. Finite volume and fixed topology
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The chiral limit is within our reach now! Exact chiral symmetry at quantum level can be achieved i
n 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.
5. Summary and discussion
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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…
5. Summary and discussion
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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,
if it has small Q and V’ dependences. Parallel tempering + Fodor method may also be useful.
V’
Z.Fodor et al. hep-lat/0510117
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Initial configurationFor 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
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Mv
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Mps2/m