Determination of topological charge following several definitions - - PowerPoint PPT Presentation

Intro. Strat. Tests Results Concl.

Determination of topological charge following several definitions

and preliminary results of χt in Nf = 1 + 2

Julien Frison, Ryuichiro Kitano, Nori Yamada KEK 34rd International Symposium on Lattice Field Theory Lattice’16 - Southampton - July 25th, 2016

Intro. Strat. Tests Results Concl.

1

Introduction The strong CP problem today Instanton contribution to the mass Topology on the lattice Topology ambiguity or mass ambiguity?

2

Strategy Objective Ensembles

3

Tests on topological charge determination Gradient flow at large flow time Continuum limit and universality Topological Charge Density Correlator

4

Preliminary results in Nf = 1 + 2 Spectrum and PCAC masses (mu, χt) plot

5

Conclusion

Intro. Strat. Tests Results Concl.

The strong CP problem today

Why is there no θF ˜ F term in the Lagrangian? Trivial solution: mueiθ = 0 Other popular solution: Peccei-Quinn mechanism (axion) mu = 0 solution New lattice computations make mMS

u

= 0 very unlikely Is mu = 0 physically defined without massless pion? Is perturbative MS really what we need? Non-perturbative contributions make this solution ill-defined What latticists should really check is whether χphysical

t

= 0

Intro. Strat. Tests Results Concl.

Instanton contribution to the mass

’t Hooft vertex [Creutz:0711.2640]

ms L R u u L L I d dR md s s R

Instanton computation [Dine:1410.8505]

0.6 0.8 1.0 1.2 1.4 0.01 0.1 1 10 1Ρ0 GeV mu MeV

Intro. Strat. Tests Results Concl.

Topology on the lattice

On the lattice, Q is ill-defined too! Only defined on smooth configurations How arbitrary are the definitions? Are some better than

• thers?

Bosonic versus fermionic definitions Does continuum limit trivially remove ambiguity? Even with Wilson fermions? On Q or on Q2?

Intro. Strat. Tests Results Concl.

Topology ambiguity or mass ambiguity?

Mass and topology are related through Ward identities Earlier works have tried to make both definitions compatible [Bochicchio’84-85-86] In general, arbitrary definitions will break singlet Ward identities at finite lattice spacing, and χt(mu = 0) = 0 is not guaranteed. In Nf = 2 + 1, χt(mu = 0) = 0 has been empirically checked, agreeing with ChPT prediction χ−1

t

∝ m−1 What in Nf = 1 + smthg ? “SU(1) ChPT” makes no sense.

Intro. Strat. Tests Results Concl.

Objective

We want to determine χt at mPCAC

u

= 0 In Nf = 1 + 2, where md = mphysical

s

so that the ’t Hooft vertex effect is amplified Only mu will be taken close to zero We use Wilson-like fermions to study the worst scenario We choose parameters similar to BMW HEX2 Nf = 2 + 1 ensembles

Intro. Strat. Tests Results Concl.

Ensembles

Nf = 2 + 1 Ensemble (cross-check) β = 3.31 L¨ uscher-Weisz w/ HEX2 Clover (a ∼ 0.116 fm), mbare

ud

= −0.07, mbare

s

= −0.04, 163 × 32 Nf = 1 + 2 Ensembles mbare

u

= −0.07, −0.093, −0.09756, mbare

ds

= −0.04, 163 × 32 A larger volume and a finer lattice are both being generated Other Ensembles Many quenched ensembles have been used for tests, either generated for this project or for another project

Intro. Strat. Tests Results Concl.

Gradient flow at large flow time

Fixed fermionic topology (finite temperature)

2 4 6 8 10 Symanzik flow time

• 4
• 3
• 2
• 1

1 2 Q5Li

Topological charge

24

5 10 Iwasaki flow time

• 4
• 3
• 2
• 1

1 2 Q5Li

24

3x4, β=3.2, Q=-2 0.2 0.4 0.6 0.8 1 DBW2 flow time

• 4
• 3
• 2
• 1

1 2 3 Q5Li

Topological charge

24

Remark: c1 increases both stability and convergence speed (nc = (3 − 15c1)τ [Alexandrou:1509.04259])

Main ensembles

2 4 6 8 10 Iwasaki flow time

• 20
• 10

10 20 Q5Li

Topological charge

Nf=1+2 Clover 2 4 6 8 10

• 20
• 10

10 20 2 4 6 8 10 Iwasaki(X)/Symanzik(sq) flow time

• 20
• 10

10 20 Q5Li cfg 180 cfg 190 cfg 200 cfg 210 cfg 220 cfg 230

Topological charge

Nf=1+2 Clover 40 60 80 100 120 140 Configuration number

• 20
• 10

10 20 Q Iwasaki flow Wilson flow Symanzik flow

Topological charge history

Nf=2+1 clover ensemble

Intro. Strat. Tests Results Concl.

Continuum limit and universality

β = 2.256 Sym Iwa DBW2 c−

1

Sym X 0.922 0.914 0.908 Iwa X X 0.961 0.948 DBW2 X X X 0.984 c−

1

X X X X β = 2.37 Sym Iwa DBW2 c−

1

Sym X 0.985 0.969 0.954 Iwa X X 0.989 0.976 DBW2 X X X 0.989 c−

1

X X X X β = 2.556 Sym Iwa DBW2 c−

1

Sym X 0.981 0.974 0.977 Iwa X X 0.994 0.991 DBW2 X X X 0.998 c−

1

X X X X

2.2 2.3 2.4 2.5 2.6 β 10 12 14 16 18 20 22 24 <Q

2>

Symanzik Iwasaki DBW2 c1

• Topological susceptibility

L=10,12,16 Vphys~cst

Quenched ensembles at fixed physical volume Strong correlations at finest ensemble Nevertheless individual Q values almost never agree/plateau The closer the c1 the stronger the correlation

Intro. Strat. Tests Results Concl.

Topological Charge Density Correlator

5 10 15 20 x

2

• 0.2

0.2 0.4 0.6 0.8 <q(x,t)q(0,t)> t=0 t=0.1 t=0.2 t=0.3 t=0.4 t=0.6 t=0.8 t=1.0 t=1.3 t=1.6 t=2.0 t=2.5 t=3.0 t=5.0 t=10

Topological charge density correlator

Nf=1+2 zero-temperature mu=-0.0093 16

3x32, various Iwasaki flow times

5 10 15 20 25 x

2

• 3
• 2
• 1

1 ∆ln|<q(x,t)q(0,t)>| t=0 t=0.1 t=0.2 t=0.3 t=0.4 t=0.6 t=0.8 t=1.0 t=1.3 t=1.6 t=2.0 t=2.5 t=3.0 t=5.0 t=10

Topological charge density correlator

Nf=1+2 zero-temperature mu=-0.0093 16

3x32, various Iwasaki flow times

Intro. Strat. Tests Results Concl.

Spectrum and PCAC masses

Meson masses

Hadron spectrum

Hadron spectrum

PCAC mass

PCAC masses

PCAC masses

Intro. Strat. Tests Results Concl.

(mu, χt) plot

0.005 0.01 0.015 0.02 0.025 mu

PCAC = mHL PCAC - mHH PCAC/2

30 40 50 60 70 80 90 <Q

2>

Iwasaki flow DBW2 flow

Topological susceptibility

in Nf=1+2 (β=3.31, 16

3x32)

Intro. Strat. Tests Results Concl.

We suggest that the mu = 0 solution to the strong CP problem should be assessed in terms of χt and not mu we have presented a strategy to estimate or bound the mistake the PCAC method could make We have presented preliminary results in Nf = 1 + 2 Unfortunately we have not been able to explore much of the expensive Index(Dov) approach We have large statistical errors for the moment We need lighter quarks, finer ensembles, and probably larger volumes Investigating mu ∼ 0 (χt ∼ 0) may require specific methods (see hep-lat/1606.07175)

Intro. Strat. Tests Results Concl.

Thanks for your attention!