spectrum and quark masses in 2+1 ßavor qcd -results from ......dynamical up, down and strange...

Spectrum and quark masses in 2+1 flavor QCD

-results from CP-PACS and JLQCD-

Tomomi IshikawaCenter for Computational Sciences, Univ. of Tsukuba

[email protected]

CP-PACS and JLQCD Collaborations

KEK Theory Meeting 2006“Particle Physics Phenomenology”

2nd-4th, March 20061

Mathematically fine definition of NP QCDRegularization is naturally introduced by lattice spacings.

finite number of d.o.f.

Monte Carlo simulation is applicable.

Hadronic observable can be obtained directly from “first principle” of QCD.

Discretization error can be controlled theoretically.

Lattice QCD

2

quark field ψ

link variable (gauge field) Uµ

lattice spacing a

〈O[Uµ, ψ̄, ψ]〉

=1Z

∫DUµDψDψ̄O[Uµ, ψ̄, ψ]

× exp(−Sgauge[Uµ]− Squark[Uµ, ψ̄, ψ]

)

Applicationshadron spectroscopy

fundamental parameters in QCD

• quark masses

•

weak matrix elements

• B physics, CP violation, CKM matrix

topology, confinement

finite temperature, finite density

3

αs, ΛQCD

::

main topic in this talk

Difficulties of the lattice QCD simulationlarge number of d.o.f.

Increase in numerical cost as quark mass is reduced

Systematic errors and approximationfinite lattice spacings continuum extrapolation

finite volume infinite volume extrapolation

large quark mass chiral extrapolation

quenched approximation (not systematic approximation)

• ignore vacuum polarization of quarks (sea quark effect)• numerical cost is largely reduced

4

e.g. # of d.o.f. ∼ (323 × 64)× (8× 4 + 4× 3) ∼ O(108)

∫DψDψ̄ exp

(∫d4xψ̄/D[Uµ]ψ

)= det /D[Uµ] =1

Quenched (Nf=0)• CP-PACS : Plaquette gauge + Wilson quarks, continuum limit

Phys.Rev.D67(2003)034503, Phys.Rev.Lett.84(2000)238.

• CP-PACS : RG improved gauge + clover quarks with PT , continuum limit Phys.Rev.D65(2002)054505.

Dynamical up and down quarks (Nf=2)

• JLQCD : Plaquette gauge + clover quarks eith NPT Phys.Rev.D68(2003)054502.

• CP-PACS : RG improved gauge + clover quarks with PT , continuum limit Phys.Rev.D65(2002)054505, Phys.Rev.Lett.85(2000)4674.

Large scaled simulations in Nf=0 and Nf=2 QCD by CP-PACS and JLQCD

5

strange : quenched

cSW

cSW

cSW

meson spectrumNf=0

• Systematic deviation from experiment is O(5-10%).

Nf=2

• The deviation is considerably reduced.

Nf=0 and Nf=2 results

6

0.950

1.000

1.050

0.8700.8800.8900.900

0.885

0.890

0.895

0.900

mes

on m

asse

s [G

eV]

0 0.1 0.2a [fm]

0.500

0.550

Nf=0(RG+PT clover)Nf=2(RG+PT clover)experiment

φ (K-input)

K* (K-input)

K* (φ-input)

K (φ-input)RG+clover (Phys.Rev.D65(2002)054505)

cSW : perturbationNf = 0 : a = 0.11− 0.2 fm, La = 2.7− 3.2 fmNf = 2 : a = 0.086− 0.215 fm, La = 2.0− 2.5 fm

the effect of dynamical up and down quarks

Baryon spectrumNf=0

• large volume

• systematic deviation from experiment (5-10%)

Nf=2

• small volume

• for heavier baryons

• for lighter baryons

7

RG+clover (Phys.Rev.D65(2002)054505)

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

mas

s [G

eV]

experimentNf=0 (K-input)Nf=0 (φ-input)Nf=2 (K-input)Nf=2 (φ-input)

PSspin 0

Vectorspin 1

Octetspin 1/2

Decupletspin 3/2

K

NK*

φ

ΛΣ

Ξ Δ

Σ∗

Ξ∗

Ω

cSW : perturbationNf = 2 : a = 0.086− 0.215 fm, La = 2.0− 2.5 fm

PLQ+Wilson (Phys.Rev.D67(2003)034503)

Nf = 0 : a = 0.05− 0.10 fm, La = 3.08− 3.26 fm

good agreement with exp.

dynamical ud quark effect

large discrepancy from exp.

small Finite Size Effect (FSE)

FSE

ud quark massesNf=0 Nf=2 :

8

0 0.1 0.2a [fm]

2.0

3.0

4.0

5.0

mud

MS(µ

=2G

eV)

[MeV

]

Nf=0,VWI

Nf=0,AWI

Nf=2,AWINf=2,VWI


ud quark mass is reduced.

dynamical ud quark effects

s quark massesNf=0 Nf=2 :

9

a [fm]70

80

90

100

110

120

130

140

150

msM

S(µ

=2G

eV)

[MeV

]

Nf=0,VWI

Nf=0,AWI

Nf=2,AWI

Nf=2,VWI

K-input

a [fm]70

80

90

100

110

120

130

140

150

msM

S(µ

=2G

eV)

[MeV

]

Nf=0,VWI

Nf=0,AWI

Nf=2,AWINf=2,VWI

φ-input

strange quark mass is largely reduced.

dynamical ud quark effects

discrepancy between K- and φ-input vanish.


How is the results in the fully unquenched QCD (Nf=2+1)?

CP-PACS and JLQCD joint projectWilson quark formalism Chiral symmetry is violated at finite lattice spacing. Computational cost is large. Theoretical ambiguity is less.

Dynamical up, down and strange quarks

Related papers :

Nf=2+1 full QCD project

10

Up and down are degenerate.Strange has a distinct mass.

Nucl.Phys.Proc.Suppl.129(2004)188, Nucl.Phys.Proc.Suppl.140(2005)225, PoS LAT2005(2005)057, Phys.Rev.D65(2002)094507, Phys.Rev.D73(2006)034501

S. Aoki, M. Fukugita, S. Hashimoto, K-I. Ishikawa, T. Ishikawa,N. Ishizuka,

Y. Iwasaki, K. Kanaya,T. Kaneko, Y. Kuramashi,M. Okawa,T. Onogi,

Y. Taniguchi, N. Tsutsui, N. Yamada, A. Ukawa and T. Yoshie

11

Nf=2+1 full QCD project members (CP-PACS and JLQCD)

12

Recent Nf=2+1 simulationsMILC collaboration

RG(Symanzik)+Improved staggered fermion(Asqtad)

• • •

Simulation is very fast, but there is theoretical ambiguity (fourth root trick)

good agreement with experiment

RBC and UKQCD collaborationRG(Iwasaki,DBW2)+Domain wall fermion

chiral symmetry on finite lattice spacing

computationally very demanding

a ≈ 0.125 fm (coarse), 0.09 fm (fine), 0.06 fm (super fine)volume ≥ (2.5fm)3

mud = 10− 50 MeV

Lattice actiongauge : RG improved action (Iwasaki,1985)

quark : non-perturbatively improved Wilson action

is non-perturbatively determined. (Phys.ReV .D73(2006)034501)

Algorithm standard HMC algorithm for up and down quarks

• usual pseudo-fermion method

Polynomial HMC (PHMC) algorithm for strange quark

• odd flavor algorithm

• exact algorithm (Phys.Rev.D65(2002)094507)

Strategy

13

cSW

O(a)

Lattice spacing, size, etc.

fixed physical volume

14

∼ (2 fm)3

a2

3

21

1

20

Simulation parameters

β = 2.05(a ∼ 0.070 fm)cSW = 1.628L3 × T = 283 × 56HMC traj.

= 6000− 6500


= 5000− 9000


= 7000− 8600

Quark mass parameters

15

5 ud quark parameters

mps/mV ∼ 0.60 − 0.78

(0.18 : exprriment)

2 s quark parameters

mps/mV ∼ 0.7

(0.68 : 1-loop ChPT)

0 0.05

a mudVWI

0.02

0.03

0.04

0.05

a m

SV

WI

β=1.83, L3xT=163x32

physical point

Ks=0.13760

Ks=0.13710

0 0.05

a mudVWI

0.02

0.03

0.04

0.05

0.06

a m

SV

WI

β=1.90, L3xT=203x40

physical point

Ks=0.13640

Ks=0.13580

0 0.05

a mudVWI

0.02

0.03

a m

SV

WI

β=2.05, L3xT=283x56

physical point

Ks=0.13540

Ks=0.13510

Gauge configuration generation

16

2003

2004

2005

2006

Production runs

β=1.83

β=1.90

β=2.05

Earth Simulator@JAMSTEC

SR8000/F1@KEK

SR8000/G1@CCS,

Univ. of Tsukuba

VPP5000@ACCC,

Univ. of Tsukuba

CP-PACS@CCS,

Univ. of Tsukuba

ProcedureWe perform the measurements at only unitarity points.

Physical volume is not large to calculate baryon masses.

Measurements are performed at every 10 HMC trajectories.

We mainly focus on the meson sector.

17

∼ (2 fm)3

Measurement of the meson masses

mvalence = msea

Effective mass plot

18

0 10 20T

0.22

0.225

0.23

effe

ctiv

e m

ass

effective mass plotbeta=2.05, Kud=0.13560, Ks=0.13540

π (smeared source - point sink)

0 10 20T

0.34

0.35

0.36

0.37

0.38

0.39

effe

ctiv

e m

ass


ρ (smeared source - point sink)

ameff (t) = − lnG(t + 1)

G(t)

G(t) : correlation function

Fitting procedurePolynomial fit functions in quark masses are used.

• include up to quadratic terms

• interchanging symmetry among 3 sea quarks

among 2 valence quarks

Chiral fits are made to

• light-light (LL), light-strange (LS) and strange-strange (SS) meson simultaneously.

• ignoring correlations among LL, LS and SS

Chiral extrapolation

19

• Polynomial fit forms using the VWI quark mass definition

20

mq =12

(1

Kq− 1

KC

)mud,ms : sea quarks

mval1,mval2 : valence quarks

# of fit parameters = 16

m2PS = BPS

S (2mud + ms) + BPSV (mval1 + mval2)

+ DPSSV (2mud + ms)(mval1 + mval2)

+ CPSS1 (2m2

ud + m2s) + CPS

S2 (m2ud + 2mudms)

+ CPSV 1 (m2

val1 + m2val2) + CPS

V 2 mval1mval2

mV = AV + BVS (2mud + ms) + BV

V (mval1 + mval2)+ DV

SV (2mud + ms)(mval1 + mval2)+ CV

S1(2m2ud + m2

s) + CVS2(m

2ud + 2mudms)

+ CVV 1(m

2val1 + m2

val2) + CVV 2mval1mval2

21

0 0.01 0.02 0.03 0.04 0.05 0.06

a mudVWI

0

0.1

0.2

0.3

0.4

(a m

PS)2

Ks=0.13580Ks=0.13640

β=1.90, L3xT=203x40, quadratic

χ2/d.o.f.=0.94

0 0.01 0.02 0.03 0.04 0.05 0.06

a mudVWI

0.4

0.5

0.6

0.7

0.8

a m

VKs=0.13580Ks=0.13640

β=1.90, L3xT=203x40, quadratic

χ2/d.o.f=0.90

Fitting form from ChPTchiral log behavior

Wilson ChPT (WChPT)

• include explicit chiral symmetry breaking effect of Wilson quark

• Nf=2+1 version

fitting is very difficult due to highly non-linear form (in progress)

22

m2π lnm2

π

(Phys.Rev.D73(2006)014511), hep-lat/0601019

Physical point and lattice spacingInputs to fix the quark masses and the lattice spacing

K-input

φ-input

lattice spacings

K-input and φ-input give consistent results.

23

mπ, mρ, mK

mπ, mρ, mφ

mρ

β κc χ2/d.o.f.1.83 0.138685(16) 0.391.90 0.137657(32) 0.942.05 0.13631(11) 1.18

Table 1: Chiral fit results (pseudo-scalar)

β χ2/d.o.f.1.83 0.661.90 0.902.05 1.72

Table 2: Chiral fit results (vector)

K-input φ-inputβ a−1[GeV] a[fm] a−1[GeV] a[fm]

1.83 1.612(22) 0.1222(17) 1.598(26) 0.1233(20)1.90 1.983(38) 0.0993(19) 1.980(37) 0.0995(19)2.05 2.84(11) 0.0693(26) 2.84(10) 0.0695(26)

Table 3: Lattice spacings from mρ (2006/02/26)

κud κs traj. mπ/mρ mηs/mφ κud κs traj. mπ/mρ mηs/mφ

0.13580 5000 0.7673(15) 0.7673(15) 0.13580 5200 0.7667(16) 0.7210(21)0.13610 6000 0.7435(18) 0.7647(17) 0.13610 8000 0.7443(15) 0.7182(17)0.13640 0.13580 7600 0.7204(19) 0.7687(15) 0.13640 0.13640 9000 0.7145(16) 0.7145(16)0.13680 8000 0.6701(27) 0.7673(17) 0.13680 9200 0.6630(21) 0.7127(17)0.13700 8000 0.6389(21) 0.7693(15) 0.13700 8000 0.6241(28) 0.7101(20)

Table 4: Statistics for the measurement (β = 1.90)

to set s quark mass

Prediction

Continuum extrapolationWe use the non-perturbatively improved Wilson quark, thus meson spectrum should scale as .

Light meson spectrum

24

m(a) = A + Ba2

a2

O(a)

K-input → mK∗ ,mφ

φ-input → mK ,mK∗

The calculation of is in progress.mη,mη′

disconnected diagram is needed.

ResultsIn the continuum, meson spectrum is consistent with experiment.The statistical error in the continuum limit is large.

25

In the Nf=2,0 case, extrapolation function is m(a)=A+Ba, because clover action is perturbatively O(a) improved.

1.000

1.050

0.890

0.895

0 0.01 0.02 0.03 0.04 0.05

a2 [fm

2]

0.870

0.880

0.890

0.900

mes

on m

asse

s [G

eV]

0 0.01 0.02 0.03 0.04 0.05

a2 [fm

2]

0.500

0.550

Quenched(RG+PT clover)Nf=2(RG+PT clover)Nf=2+1(RG+NPT clover)experiment

φ (K-input)

K* (K-input)

K* (φ-input)

K (φ-input)

VWI quark mass

VWI quark mass can be negative due to lack of chiral symmetry of the Wilson quark.

AWI quark mass

The scaling violation is smaller than that of VWI in Nf=2 case.

Renormalizationtad-pole improved 1-loop matching with at

4-loop running to

Light quark masses

26

MS

mV WIq =

12

(1K− 1

Kc

)

Kc ←− mπ(Kud,Ks)|Kud=Ks=Kc = 0

mAWIq =

〈∆4A4(t)P (0)〉2〈P (t)P (0)〉

mLATq (a−1) −→ mMS

q (µ = 2 GeV)

µ = a−1

µ = 2 GeV

up and down quarkWe assume that the contribution is small and use the extrapolation function which is same as in the meson spectrum (linear in ).

In the continuum limit, the VWI definition gives a positive value.

27

NOTE. Nf=2, 0 : RG+clover(PT)

(AWI, combined K with φ-input)

0 0.005 0.01 0.015 0.02

a2 [fm

2]

-20

-10

0

10

mud

MS(µ

=2G

eV)

[MeV

]

VWI

AWI

Nf=2+1 (K-input)

0 0.01 0.02 0.03 0.04 0.05

a2 [fm

2]

2.0

2.5

3.0

3.5

4.0

4.5

5.0

mud

MS(µ

=2G

eV)

[MeV

] Quenched,VWI

Quenched,AWI

Nf=2,AWINf=2,VWI

Nf=2+1,AWI (K-input)

mMSud (µ = 2GeV) = 3.49(15) MeV

O(a)

a2

strange quarkAll definitions of strange quark mass gives consistent results in the continuum limit.

The difference between Nf=2+1 and Nf=2 is invisible in the continuum limit as in the ud quark.

28

(AWI, combined K with φ-input)


80

90

100

110

120

msM

S(µ

=2G

eV)

[MeV

]

0 0.01 0.02 0.03 0.04 0.05

a2 [fm

2]

70

80

90

100

110

120

130

140

msM

S(µ

=2G

eV)

[MeV

]

Nf=0,VWI

Nf=0,AWI

Nf=2,AWI

Nf=2,VWI

Nf=2+1,AWI

K-input

Nf=2+1,VWI

φ-input Nf=0,VWI

Nf=2,VWI Nf=0,AWI

Nf=2,AWI

Nf=2+1,VWI

Nf=2+1,AWI

mMSs (µ = 2GeV) = 90.9(3.7) MeV

Baryon spectrum

29

0.8

1

1.2

1.4

1.6

mas

s [G

eV]

experimentNf=2+1 (K-input)Nf=0 (K-input)

octet decupletspin 1/2 spin 3/2

N

ΛΣ

Ξ Δ

Σ∗

Ξ∗

Ω

Signal is not good.Pattern of the spectrum is reproduced.But the spectrum and FSE is unclear.

0 5 10 15 20T

0.7

0.75

0.8

effe

ctiv

e m

ass


N (octet)

0 10 20 30T

0.6

effe

ctiv

e m

ass


Δ (decuplet)

(preliminary)

PS decay constant

renormalizationtad-pole improved one-loop renormalization factor

definitions for the calculationWe test two different definitions.

PS meson decay constants

30

〈0|A4|PS〉 = fPSmPS

fPS

〈PP (t)PP (0)〉 ∼ CPPP e−mP St

〈PP (t)PS(0)〉 ∼ CPSP e−mP St

〈PS(t)PS(0)〉 ∼ CSSP e−mP St

〈AP4 (t)PS(0)〉 ∼ CPS

P e−mP St

fPS = CP SA

CP SP

√2CP P

PmP S

(method 1)

fPS = CPSA

√2

mP SCSSP

(method 2)

31

0 0.01 0.02

a2 [fm

2]

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

f PS [G

eV]

0 0.01 0.02

a2 [fm

2]

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

π

K

π

K

π

K

π

K

K-input, method 1 K-input, method 2

Two methods are consistent each other.

But huge statistical error.

Statistics might not be enough to calculate decay constants (?)

(preliminary)

Static quark potential

Sommer scale

In our calculation is obtained through

32

Sommer scale

V (r) = V0 −

α

r+ σr

r2

dV (r)

dr

∣

∣

∣

∣

r=r0

= 1.65

R0 = ar0 ! 0.5 fm

r0 =

√

1.65 − α

σ

r0

r0 0 1 2 3r / r0

-4

-3

-2

-1

0

1

2

3

4

[ V(r

) -

V(r

0) ] r

0

string model ( α=π/12 )

β = 1.83

0 1 2 3r / r0

-4

-3

-2

-1

0

1

2

3

4

[ V(r

) -

V(r

0) ] r

0


β = 1.90

0 1 2 3r / r0

-4

-3

-2

-1

0

1

2

3

4

[ V(r

) -

V(r

0) ] r

0


β = 2.05

phenomenological model

Chiral extrapolation

Continuum limit of using the lattice spacing from

33

1

r0

= A + B

(

2

κud

+1

κs

)

R0

mρ


10.95 11 11.05(2/Kud+1/Ks)/2

0.21

0.22

0.23

0.24

0.25

1/r 0

datafit line

0 0.01 0.02 0.03 0.04 0.05

a2 [fm

2]

0.45

0.5

0.55

0.6

0.65

a r 0 [f

m]

Nf=2+1 (K-input)Nf=2+1 (φ-input)Nf=2quenched

consistent with 0.5 fm

R0 = 0.516(21) fm

(preliminary)

CP-PACS/JLQCD Nf=2+1 projectRG-gauge + non-PT clover (Wilson quark formalism)Gauge configuration generation has been already finished.

Analysis of spectrum and quark massesEncouraging results in the continuum limit are obtained.

Assuming that the scaling is ,

• Meson spectrum is consistent with experiment.

• All definitions and inputs of quark masses gives consistent results.

• Quark masses in the continuum :

Baryon spectrum, decay constants

Summary

34

mMSud (µ = 2GeV) = 3.49(15) MeV

mMSs (µ = 2GeV) = 90.9(3.7) MeV

a2

signal is not so good(?), large statistical error

Required task Non-perturbative determination of renormalization factor

Calcurations in progressη’ meson mass

Heavy meson quantities using a relativistic heavy quark action

35

cΓ, ZΓ, bΓ

(Aoki,Kuramashi,Tominaga,2001)

PACS-CS collaborationclover quark with domain decomposed HMC

PACS-CS (CCS, Univ. of Tsukuba)

cluster, 2560 nodes, 14.3 TFLOPS

JLQCD collaborationdynamical overlap fermion

IBM Blue Gene (KEK)

10 rack (10240 node), 57.3 TFLOPS

36

cΓ, ZΓ, bΓ

Next direction - lighter quark mass -

ambiguity in the chiral extrapolation will be removed

spectrum and quark masses in 2+1 ßavor qcd -results from ......dynamical up, down and strange...

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