Charm in Lattice QCD with Domain-Wall Fermion · , , , 1 , 1 , ,,, 1 , N s PV x s w s s w s s s s x ss x x x xs s s x x A I D I D P PI G G G IUV c c c c c cc cc ªº ¦¦ ¬¼ P x

Charm in Lattice QCD with Domain-Wall Fermion

Ting-Wai Chiu (趙挺偉)

National Taiwan University

(for the TWQCD Collaboration)

Lattice 2014, New York, USA

June 23-28, 2014

Collaborators:

W.P. Chen, Y.C. Chen, H.Y. Chou, T.S. Guu, T.H. Hsieh

Outline

Introduction

Charm quark in LQCD with DWF

Gauge ensembles

Lowest lying mass spectra of

Leptonic decay constants, fD

, fDs

Concluding Remarks

, , ( )cc cs cd cu

Introduction

In QCD, both light and heavy quarks are Dirac

fermions with different bare quark masses. Thus

it seems unnatural to treat them differently (with

different actions) in any theoretical studies

involving both heavy and light quarks.

Theoretically, LQCD with exact chiral symmetry

(domain-wall/overlap fermion ) is the ideal

theoretical framework to tackle any

nonperturbative physics involving quarks, no

matter whether they are heavy or light, valence

or sea, all with the same action.

2014/6/23 T.W. Chiu, Lattice 2014 3

Domain-Wall Fermion [Kaplan, 1992]

with boundary conditions

dwf , , , 1 , 1 ,, ,, 1 ,

dwf

sN

x s w s s w s s s s x sx xs x

s xx

s xs

A I D I D P P

D

4

0 0

1

, 0,2wD t W m m

†

, ,

1,

2x x x xt x x U x U x

4

†

, , ,

1

1, 2

2x x x x x xW x x U x U x

0 0,0 , , : bare mass, 1 / [2 (1 )]q s q r mP x rm P x dN m m

5

1, 1 ,1 , (1 )

2s qP x N rm P x P

4 T.W. Chiu, Lattice 2014 2014/6/23

, (constants)

s s

s s

c d

c d

c d

T.W. Chiu, Lattice 2014 5

Domain-Wall Fermion (cont.)

with boundary conditions:

The action for Pauli-Villars fields is

, , , 1 , 1 ,, ,, 1 ,

sN

PV x s w s s w s s s s x ss x x x xs

s s x x

A I D I D P P

,0 , , sP x P x N

, 1 ,1sP x N P x

2014/6/23

dwf PVexp det ( )qd d d d A A D m

The effective 4D Dirac operator

0 0

2

5

5( ) (1 ) 1 , 2

li

1

ms

q w

w

q q

N

cHH

mD m m m dm S

HS

HH

HH

d


Conventional DWF with Shamir kernel


1 1, 01/ 2, s s s s sc d c dc d

00 5 po

5

lar( ) (2 ) 12

, 2 2

wq

w

q

q

mmD m m

HH Hm S

H

2

1

2

1

polar

2, 2

1 2, 2 1

1 1,

1 1

s

s

n

l

s

ls l

n

l

s

ls s l

N

N

bH N n

N H d

bH N n

N N H d

T HS H T

T H

2

2

1sec

2

1tan

2

l

s

l

s

b lN

d lN

2014/6/23

polar approximation of 2

1

H



where is the Jacobian elliptic function with argument

and modulus , and and are

lower and upper “bounds” of the eigenvalues of

;ssn v sv

2 2

min max1 2

min2

max

2 2

min

11 ; , 1, ,s s ssn v s N

2014/6/23

Optimal DWF [ TWC, Phys. Rev. Lett. 90 (2003) 071601]

2H

0 0

5

5( ) (1 ) 1 ,1

2

q

q q ow

w

pt

mD m m m dm S

cHH H

d H

Then the effective 4D Dirac operator becomes



1

1

1, 2

, 2

1 1,

11

, 2

, 2 1

s

s

N

s ssopt sN

sss

n n

Z s

n n

Z s

T HS H T

HT

HR H N n

HR H N n

2014/6/23

Zolotarev optimal rational approximation of 2

1

H


Chiral Sym Breaking due to Finite Ns

It can be measured by the residual mass

5 1 1

5 5

5 5

1

,

1†

,

1

, 2

( , ) ( ) ( ) ( ) ( )

valence quark

( , ) ( ) ( )

( )

( ) ( ) ( ) ( )

Re tr

propagator wit

h

tr

s

q

n n n n

c q

U

U

x

res

x U

c q y y

c q c qy y

U

q sea

Nn

m

J x n x P x x P x

D m

J x n q y q y

m y

q x q x q y q y

D m

D m D m

m m

[ Y.C. Chen, TWC, Phys. Rev. D 86, 094508 (2012)]

2014/6/23


Chiral Sym Breaking due to Finite Ns (cont)

For lattice QCD with ODWF, it can be shown that

For ODWF, in most cases, and it gives 1Zd

If there are some eigenvalues of smaller than 2

min2H

2014/6/23


Chiral Sym Breaking due to Finite Ns (cont)

2014/6/23

2014/6/23 T.W. Chiu, Lattice 2014 12

Charm quark in LQCD with DWF

The quark mass in DWF enters through the boundary conditions

0 0,0 , , : bare mass, 1/ [2 (1 )]q s q r mP x rm P x dN m m

5

1, 1 ,1 , (1 )

2s qP x N rm P x P

0 0 01 / 2 (1 ), 0 2The upper bound of is PVq m r m dm mm

If one sets , the theory goes to the quenched limit

with det( ) 1.

q PVm m

D

0To minimize the , it is necessary to choose and

such that .

Also, it requires to minimiz

cuto

e th

ff effects

1 discretization err .re o

c PV

c

d m

m m

m a

2014/6/23 T.W. Chiu, Lattice 2014 13

Charm quark in LQCD with DWF (cont)

0For DWF with 0, 2PVd m m

0 0 1.3,TW QCD: 2 2.6PVm m m

12.6 0.38 0.46 GeV

(weaker than the constraint 1)

c c

c

m a a m

m a

10.36 2.78 3.3 GeV

(stronger than the constraint 1)

c c

c

m a a m

m a

0 0 0 1.8,RB C/UKQCD (2 ) 0: .36PVm m m m

0 01 / 2,Conven (2 ),tional maxDWF (: ) / 1PV PVd m m m m a

1For the discr6.1 etization error is smal0, 3 GeV, 0.5 l. , ca m a

Lattice size: 243 x 48

Quark action: optimal domain-wall fermion

Gluon action: Wilson plaquette action at beta = 6.10

14 T.W. Chiu, Lattice 2014

Lattice spacing: a ~ 0.065 [fm], 1/a ~ 3.2, 3.1, 3.0, 3.0 [GeV]

Spatial volume: ~(1.6 fm)3

4 sea-quark Masses, with pion masses ~ 260, 350, 470, 560 [MeV].

For each sea-quark mass, after thermalization, ~5000 trajectories are

generated, measurements are performed every 10 trajectory, with a

total of ~500 confs.

Recent TWQCD simulations of Nf=2 QCD

Residual mass < 0.17 MeV

2014/6/23

min max(with 1, 0, 16, 0.05, 6.2)sc d N

Recent TWQCD simulations of Nf=2 QCD (cont)

Multiple-time scale integration and mass preconditioning.

Omelyan Integrator for the Molecular Dynamics.

Even-Odd Preconditioning for the 4D Wilson-Dirac Matrix.

Conjugate gradient with mixed precision.

Topological sectors are sampled ergodically.

15 T.W. Chiu, Lattice 2014 2014/6/23

Use a GPU cluster of 300 GPUs, with sustained 100 Tflops

2014/6/23 T.W. Chiu, Lattice 2014 16

Recent TWQCD simulations of Nf=2 QCD (cont)

Pion mass and decay constant are in good agreement

with the sea-quark mass dependence predicted by NLO

ChPT, and gives a determination of the chiral condensate,

the pion decay constant, . (see Tung-Han Hsieh’s talk on Wednesday, Chiral Symmetry, 5D, 10:20)

3 4, , udm l l

2014/6/23 T.W. Chiu, Lattice 2014 17

Determination of lattice spacing

0

0

3

0

2

[TWC, Hsieh, Mao, PLB

Use Wilson flow with the condition

to obtain for each ensemble.

To determine the i

( )

5.9

nput parameter , we use our gauge ensembles

at on the5 16 32 lat ce ti

0.3 t t

E t

t a

t

t

, 2012].

2014/6/23 T.W. Chiu, Lattice 2014 18

Determination of lattice spacing (cont)

the systematic error is estimated by varying the number of sea-quark masses and the difference between the linear fit and the constant fit.

2014/6/23 T.W. Chiu, Lattice 2014 19

Time-Correlation functions

Compute

with

and similarly for the axial-vector and tensor mesons.

2014/6/23 T.W. Chiu, Lattice 2014 20

Determination of the bare quark masses of s, c

fitting to to extract M

(1020) sm / (3097) cJ m

2014/6/23 T.W. Chiu, Lattice 2014 21

Lowest-lying mass spectra of cc

For the gauge ensemble with sea-quark mas 0 .s 01seam a

2014/6/23 T.W. Chiu, Lattice 2014 22

Lowest-lying mass spectra of cs


2014/6/23 T.W. Chiu, Lattice 2014 23

Lowest-lying mass spectra of cd


2014/6/23 T.W. Chiu, Lattice 2014 24

Mass Spectra of Charmed Scalar Mesons

0.01seam a

cc

cscd

2014/6/23 T.W. Chiu, Lattice 2014 25

Leptonic Decay Constants

To determine precisely is crucial for the determination of the CKM

matrix element | | .

P

Qq

f

V

2014/6/23 T.W. Chiu, Lattice 2014 26

Leptonic Decay Constants fD

, fDs

Use heavy meson ChPT (Sharpe, Zhang, 1996) to extrapolate

to the physical point 140 MeV.M

Chen et al. (TWQCD), arXiv: 1404.3648

2014/6/23 T.W. Chiu, Lattice 2014 27

Leptonic Decay Constants fD

, fDs

(cont)

(PDG 2013)

Chen et al. (TWQCD), arXiv: 1404.3648

2014/6/23 T.W. Chiu, Lattice 2014 28

Concluding Remarks

LQCD with optimal DWF provides an ideal framework for

studying physical observables involving light and heavy

quarks, for the sea and the valence quarks.

Dynamical simulation preserves the chiral symmetry to a

good precision, and samples all topological sectors ergodically.

For 2-flavors gauge ensembles on 243 x 48 at β = 6.1,

pion mass and decay constant are in good agreement with the

sea-quark mass dependence predicted by NLO ChPT, and

gives a determination of the chiral condensate, the pion decay

constant, . (see Tung-Han Hsieh’s talk on Wednesday, Chiral Symmetry, 5D, 10:20)

3 4, , udm l l

2014/6/23 T.W. Chiu, Lattice 2014 29

Concluding Remarks (cont)

Lowest-lying mass spectra of

are in good agreement with the experimental values,

except for the scalar meson of

, , cc cs cd

cd

Leptonic decay constants fD and fDs are in good agreement

with the experimental values.

This gives confidence to proceed to further studies with this set

of ensembles, on the charmed baryons, light meson spectra,

and matrix elements, etc.

Charm in Lattice QCD with Domain-Wall Fermion · , , , 1 , 1 , ,,, 1 , N s PV x s w s s w s s s s x ss x x x xs s s x x A I D I D P PI G G G IUV c c c c c cc cc ªº ¦¦ ¬¼ P x

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