condensates and topology fixing action

Hidenori FukayaHidenori Fukaya

YITP, Kyoto Univ. YITP, Kyoto Univ. Collaboration with T.Onogi (YITP)Collaboration with T.Onogi (YITP)

hep-lat/0403024hep-lat/0403024

1.1. IntroductionIntroduction Why topology fixing action ?Why topology fixing action ?

An action proposed by Luscher provide us to simulate withAn action proposed by Luscher provide us to simulate with

a fixed topological charge by suppressing the field strength.a fixed topological charge by suppressing the field strength.

Better statistics of higher topological sectors.Better statistics of higher topological sectors. Locality of Overlap D is improved.Locality of Overlap D is improved. Chiral symmetry is also improved.Chiral symmetry is also improved. Theta vacuum if one has a reweighting way.Theta vacuum if one has a reweighting way.

To test the feasibility of this action, we simuTo test the feasibility of this action, we simulate late 2-dim QED2-dim QED, as well as developing a , as well as developing a rereweighting methodweighting method to treat θ term. to treat θ term.

An application to QCD is also studied by An application to QCD is also studied by S.S.Shcheredin, B.Bietenholz, K.Jansen, Shcheredin, B.Bietenholz, K.Jansen, K.-I.Nagai,S.Necco and L.Scorzato K.-I.Nagai,S.Necco and L.Scorzato (hep-lat/0409073) .(hep-lat/0409073) .

As an example,As an example,

let us focus on condensates in let us focus on condensates in

the massive Schwinger model with a fixed the massive Schwinger model with a fixed

topological charge and in theta vacuum.topological charge and in theta vacuum.

M.Luscher,Nucl.Phys.B538,515(1999), Nucl.Phys.B549,295(1999)M.Luscher,Nucl.Phys.B538,515(1999), Nucl.Phys.B549,295(1999)....

1.1. IntroductionIntroduction OverviewOverview

In QCD or 2-d QED (θIn QCD or 2-d QED (θ ＝ ＝ 0 ),0 ),

　　but in but in θ ≠ 0θ ≠ 0 case, we expect case, we expect

Then the η has a long-range correlation asThen the η has a long-range correlation as

1.1. IntroductionIntroduction The The 2-flavor2-flavor massive Schwinger model massive Schwinger model

ConfinementConfinement Chiral condensationChiral condensation U(1) problem (mU(1) problem (mηη ＞＞ mmππ)) Bosonization picture strong coupling limit⇒Bosonization picture strong coupling limit⇒

We define We define

1.1. IntroductionIntroduction How to evaluate θ vacuumHow to evaluate θ vacuum

⇒⇒ We need We need 　　　　　　 and and . .

1.1. IntroductionIntroduction How to evaluate θ vacuumHow to evaluate θ vacuum

We can calculate We can calculate by generating link by generating link

variables which satisfy variables which satisfy Luscher’s boundLuscher’s bound

( which are called “admissible”. );( which are called “admissible”. );

realized by the following gauge action;realized by the following gauge action;

⇒⇒ 　　　　 topologcal charge is conserved !!!topologcal charge is conserved !!!

※※ 　　 0 <ε< π/ 5 → GW Dirac operator is local.0 <ε< π/ 5 → GW Dirac operator is local. M.Luscher,Nucl.Phys.B538,515(1999), Nucl.Phys.B549,295(1999)M.Luscher,Nucl.Phys.B538,515(1999), Nucl.Phys.B549,295(1999)

....

....

1.1. IntroductionIntroduction How to evaluate θ vacuumHow to evaluate θ vacuum

Topological charge is defined asTopological charge is defined as

Without Luscher’s boundWithout Luscher’s bound

⇒⇒ 　　 topological charge can jumptopological charge can jump ；　；　 Q → QQ → Q±± １１ . .

.... If gauge fields are “admissible” ( εIf gauge fields are “admissible” ( ε ＜ ＜ 2 ),2 ),

⇒ ⇒ 　　 topological charge is conserved !! topological charge is conserved !!

In our numerical study, the topological charge is actually conserved !!!

Plaquette action(β=2.0) Lusher’s action(β=0.5) (ε ＝√２ ,Q=2.)

.

.

1.1. IntroductionIntroduction How to evaluate θ vacuumHow to evaluate θ vacuum

Note that the continuum limit is the same asNote that the continuum limit is the same as

the standard plaquette action at any ε,the standard plaquette action at any ε,

1.1. IntroductionIntroduction How to evaluate θ vacuumHow to evaluate θ vacuum

Using the following expression,Using the following expression,

we can evaluate we can evaluate normalized by normalized by ! !

1.1. IntroductionIntroduction How to evaluate θ vacuumHow to evaluate θ vacuum

Classical solution ⇒Classical solution ⇒ 　　 Constant field strength.Constant field strength. Moduli (Moduli (constant potential which affects Polyakov loopsconstant potential which affects Polyakov loops ) i ) i

ntegral of the determinantsntegral of the determinants

⇒ ⇒ 　　 Householder and QL method.Householder and QL method. Integral of Integral of

⇒ ⇒ 　　 fitting with polynomiyalfitting with polynomiyals.s.

Thus we can calculate the reweighting factor

RQ (=ZQ/Z0) .

1.1. IntroductionIntroduction How to evaluate θ vacuumHow to evaluate θ vacuum

Now we can evaluateNow we can evaluate

by by Luscher’s gauge actionLuscher’s gauge action and our and our

reweighting reweighting method.method.

....

Details are shown in Details are shown in

HF,T.Onogi,Phys.Rev.D68,074503(2003).HF,T.Onogi,Phys.Rev.D68,074503(2003).

Our goal is to studyOur goal is to study

　　 : condensates in each sector,: condensates in each sector,

　　 : η correlators in each sector,: η correlators in each sector,

　　 : θ dependence of condensates,: θ dependence of condensates,

　　 : θ dependence of η correlators,: θ dependence of η correlators,

　　 The relation between them.The relation between them.

1.1. IntroductionIntroduction

ActionAction : : Luscher’s action + DW fermion with PV’s.Luscher’s action + DW fermion with PV’s. Algorithm : Algorithm : The hybrid Monte Carlo method.The hybrid Monte Carlo method. Gauge coupling : Gauge coupling : g = 1.0. (β=1/gg = 1.0. (β=1/g22=1.0.)=1.0.) Fermion mass : Fermion mass : mm1 1 = m= m2 2 = 0.1, 0.15, 0.2, 0.25, 0.3.= 0.1, 0.15, 0.2, 0.25, 0.3. Lattice size : Lattice size : 16 × 16 ( × 6 ).16 × 16 ( × 6 ). Admissibility condition : Admissibility condition : ε= √2ε= √2 Topological charge : Topological charge : Q = - 4 Q = - 4 ～ ～ + 4+ 4 5050 molecular dynamics steps with the step size molecular dynamics steps with the step size Δτ= 0.Δτ= 0.

035035 in one trajectory. in one trajectory. 300 configurations300 configurations are generated in each sector. are generated in each sector. Admissibility is checked at the Metropolis test.Admissibility is checked at the Metropolis test.

....

The simulations were done on the Alpha work station at YITP and SX-5 at RCNP. Numerical simulationNumerical simulation

2. Fermion Condensates2. Fermion Condensates Condensations in each sectorCondensations in each sectorIntegrating the anomaly equation;Integrating the anomaly equation;

one obtains the following relationone obtains the following relation

Pseudo scalar condensation in each sector

Our data are consistent with the lines (Q/mV) !!

Reweighted pseudo scalar condensation

＜ ψγ ５ ψ ＞ Q ZQ/Z0

Scalar condensation in each sector

＜ ψψ ＞ Q

Reweighted scalar codensation ＜ ψψ ＞ Q ZQ/Z0

Pseudoscalar does condense !!Pseudoscalar does condense !!

2. Fermion Condensations2. Fermion Condensations Condensations in θ vacuumCondensations in θ vacuumWe evaluateWe evaluate

andand

Scalar condensation ＜ ψψ ＞ θ

The dashed line:Y.Hosotani and R.Rodoriguez,J.Phys.A31,9925(1998)

Pseudo scalar condensation ＜ ψγ ５ ψ ＞ θ

The dashed line:Y.Hosotani and R.Rodoriguez,J.Phys.A31,9925(1998)

m dependence of scalar condensates at θ=0

The η correlations in each sector The η correlations in each sector 　　　　　　　　　 　　　　　　　　　 ⇒⇒ 　　 We expectWe expect 　　 theη meson has a long-rtheη meson has a long-range correlation in each topological sector.ange correlation in each topological sector.

We measureWe measure

where 2nd part is calculated exactly.where 2nd part is calculated exactly.

3.3. The η meson in θ vacuumThe η meson in θ vacuum The η correlations in each sector The η correlations in each sector

Connected part (pion results)Connected part (pion results) 　　　　　　　　　　　　　　　 　　　

The η correlations in each sector The η correlations in each sector Connected part (pion results)Connected part (pion results) 　　　　　　　　　　　　

　　　 　　　

Our numerical data show a good agreement withOur numerical data show a good agreement with

the continuum theory at small θ;the continuum theory at small θ;

Connected part in each topological sector ＜ π(x)π(0) ＞ Q

Pion mass at θ=0Pion mass at m=0.2

The η correlations in each sector The η correlations in each sector 　　　　　　　　　 　　　　　　　　　 ⇒⇒ 　　 We expectWe expect 　　 theη meson has a long-rtheη meson has a long-range correlation in each topological sector.ange correlation in each topological sector.

We measureWe measure

where 2nd part is calculated exactly.where 2nd part is calculated exactly.

3.3. The η meson in θ vacuumThe η meson in θ vacuumη meson correlator in each sector Long-range correlation in each sector

lim|x|→large ＜ ψγ ５ ψ(x)ψγ ５ ψ( ０ ) ＞ Q

Reweighted long-range correlation lim|x|→large ＜ ψγ ５ ψ( ｘ )ψγ ５ ψ( ０ ) ＞ Q ZQ/Z0

3.3. The η meson in θ vacuumThe η meson in θ vacuum The η correlations in θ vacuumThe η correlations in θ vacuumWe evaluateWe evaluate

＜ η†(x)η(0) ＞ θat m=0.1

lim|x|→large ＜ η†(x)η(0) ＞ θ at m=0.1

3.3. The η meson in θ vacuumThe η meson in θ vacuum Clustering decompositionClustering decompositionConsider the operators put on Consider the operators put on t = -T/2t = -T/2 and and t=T/2t=T/2

of a large box divided into two parts;of a large box divided into two parts;

We expect the η correlation in each sector is We expect the η correlation in each sector is expressed asexpressed as

A B

ψγ5ψ （ -T/2) ψγ5ψ （ T/2)

Clustering decompositionClustering decomposition Q=0 caseQ=0 case

3.3. The η meson in θ vacuumThe η meson in θ vacuum

A B

ψγ5ψ ψγ5ψ

＜ ψγ5ψ ＞ Q

Clustering decompositionClustering decomposition Q>>0 caseQ>>0 case

We assume Q’ distribution is expressed as GaussianWe assume Q’ distribution is expressed as Gaussian

around Q/2 ;around Q/2 ;

3.3. The η meson in θ vacuumThe η meson in θ vacuum Long range correlation

Consistent with -4Q2/m2V2 !!!

4. Summary and Discussion4. Summary and Discussion The results of 2-d QED The results of 2-d QED We obtainWe obtain

andand

It is important that It is important that vanishing ofvanishing of is non-trivialis non-trivial

even in even in θ= 0 caseθ= 0 case and and

⇒ ⇒ 　　 the fluctuation of disconnectedthe fluctuation of disconnected

diagramsdiagrams !!! !!!

4. Summary and Discussion4. Summary and Discussion How about 4-d QCD ?How about 4-d QCD ?We We expectexpect

andand

It is important that It is important that vanishing ofvanishing of is non-trivialis non-trivial

even in even in θ= 0 caseθ= 0 case and and

⇒ ⇒ 　　 the fluctuation of disconnectedthe fluctuation of disconnected

diagramsdiagrams !!! !!!

How about 4-d QCD ?How about 4-d QCD ?In QCD, one obtains the same anomaly equation;In QCD, one obtains the same anomaly equation;

From the clustering decomposition, From the clustering decomposition,

is also expected, which is consistent with is also expected, which is consistent with

ChPT results in the ε-regime !!!!ChPT results in the ε-regime !!!!

((P.H.Damgaard et al., Nucl.Phys B629(2002)445P.H.Damgaard et al., Nucl.Phys B629(2002)445),),

4. Summary and Discussion4. Summary and Discussion SuggestionSuggestion

Thus, η correlators should be treated as ;Thus, η correlators should be treated as ;

and cancellation of Cand cancellation of Cθ θ 　　 at θ=0 is quite at θ=0 is quite

non-trivial.non-trivial.

⇒ ⇒ 　　 An indicator An indicator for enough topologyfor enough topology

changes changes (with standard plaquette action.) .(with standard plaquette action.) .

Reweighted long-range correlation lim|x|→large ＜ ψγ ５ ψ( ｘ )ψγ ５ ψ( ０ ) ＞ Q RQ

Application to 4-d QCD ?Application to 4-d QCD ?Are RAre RQQ=Z=ZQQ/Z/Z0 0 ’s calculable in 4-d QCD?’s calculable in 4-d QCD?

　　　　 ⇒　⇒　 Yes (Yes ( 　　 in principle.in principle. 　　 ).).

DifficultiesDifficulties Fermion determinants Fermion determinants

⇒⇒ 　　 eigenvalue truncation ?eigenvalue truncation ? Moduli integrals Moduli integrals 　　

⇒ ⇒ we need all the solutions on 4-d toruwe need all the solutions on 4-d torus. s.

VEV’s of action (ΔSVEV’s of action (ΔSQQ ) )

⇒ ⇒ perturbative analysis ?perturbative analysis ?

4. Summary and Discussion4. Summary and Discussion

It would be difficult but …It would be difficult but …

We should not give up since we already We should not give up since we already

know the answer (know the answer (ZZQQ/Z/Z00) is ) is

between between 00 and and 11 !! !!