公募研究 A06
格子ＱＣＤによる重いクォークの物理の研究
Sinya Aoki
Institute of Physics, University of Tsukuba
saoki@ph.tsukuba.ac.jp

References
S. Aoki, Y. Kuramashi and S. Tominaga, hep-lat/0107009,hep-lat/0111025.
S. Aoki, Y.Kayaba and Y. Kuramashi in preparation

Introduction

Determination of Cabbibo-Kobayashi-Maskawa(CKM) matrix fromexperiments
2G2F MW(1)
1Mq = 62 B FB2 B^B jVtq j2
1Mq : mass di erence of the neutral Bq meson (q = d; s)qqq

FBq : decay constant of the neutral Bq meson (q = d; s)B^Bq : (RG invariant) mixing parameter of the neutral Bq mesonhBq j(bq)V 0A(bq)V 0AjBq i = 83 BB ()FB2 m2B
Theoretical (non-perturbative) estimates are required for FB , BB .
qqqq) Lattice QCD !
Problem of lattice QCD for b quark
mb = 4-5 GeV
1=a = 2-4 GeV (quenched QCD), 1-2 GeV (full QCD)
mba  O(1) ) large O(a) errors !

 Wilson/Clover quark action at mq a < 1 + extrapolation
 lattice NRQCD, FNAL quark action ! e ective theoryNew approach !?

qPurpose of this research projectO(a) improvement program for massive quarks

 remove af (mq a) lattice artifact, f (mq a) is an arbitrary functionof mq a for all on-shell quantities with p~ = O(3QCD)
Plan of our project
1. Propose O(a) improvement scheme for massive quarks and determine improvement parameters at tree level (completed)
2. Determination of improvement parameters at 1-loop (in progress)
3. Numerical test of our proposal for charm quarks (in progress)
4. Extensive investigations for charm quark physics and CLEO-c(in future)
5. Non-perturbative determination of improvement parameters (infuture)
6. Test and extensive investigation for bottom quark physics (infuture)On-shell improvement program

Aoki-Kuramashi-Tominaga
1. Action
impSq

2
6X 6
6
6
64|

= x m0q(x)q(x) +t q(x) 0D0q(x) +s i q(x) iDiq(x)mass term time derivative
space derivative
0 r2ta q(x)D02q(x) 0 r2sa i q(x)Di2q(x)remove doublers
iga0 igacE q(x)0iF0iq (x) 02 i4 cB i;j q(x)ij Fij q(x) ;
X{z}

{z|}

|{z}

X{z|}

3X

X

75

2. Renormalization and Improvement condition
 Fermion propagator
1 0i 0p0 0 i i ipi + mp +(no pole terms)+O((p a)2)
SFlat:(p0; ~p) =iZ
p2 + p2 + m2
Pq0Pi i Quark-quark scattering amplitude

ppp’

pp’

qq’

qq’

(a)

(b)

T = continuum amplitude + O((pia)2)

3. Results at tree level Renormalization
m0 + rt + m20 + 2rt m0 + 1;
mp = log
1 + rtmpZm =;m0Zq = cosh(mp ) + rtsinh(mp): Improvement parameterssinh(mp) ; =qs

mpcosh(mp) + rtsinh(mp) 0 sinh(mp) ;rs =mpm2p
cE = rts;cB = rs:

4. 1-loop calculation has been almost completed.
Aoki-Kayaba-Kuramashi

Numerical simulations

Aoki-Kayaba-Kuramashi

Test of Space-Time symmetry
1. Dispersion relation
E (~p) = m2P + c2~p2
c: \speed of light", c = 1 in the relativistic theory
 Heavy-Light Pseudo-Scalar mesonc( unimproved) = 0:85(2)c( improved) = 0:98(2)2E(p) vs. p

2

heavy−light PS meson

3.0
unimprove, c=0.85(2)improve, c=0.98(2)

2.5

2.0
0.0

0.1

0.2

p

2

0.3

0.4

0.5

 Heavy-Heavy Pseudo-Scalar mesonc( unimproved) = 0:76(1)c( improved) = 0:95(1)2E(p) vs. p

2

heavy−heavy PS meson

5.6
unimprove, c=0.76(1)improve, c=0.95(1)

5.1

4.6
0.0

0.1

0.2

2

0.3

0.4

0.5

p

Improved theory is much closer to the relativistic theory !

2. Decay constant
M : PS mesonh0jAjM (p)i = ipfM (A)

Ration of fM (Ai) and fM (A4)~p = (1; 0; 0); (1; 1; 0); (1; 1; 1)mheavy $ m1 < m2 < m3
fM(Ai)/fM(A4)heavy−light PS meson

1.4
improve
unimprove

1.2

m1<m2<m3

1.0
0.8
0.6
0.4

0(1,0,0)(1,1,0)(1,1,1)

12

3

fM (Ai)=fM (A4) ' 1 for improved theory !

4Conclusion

1. O(a) improved theory we propose works well speed of light ' 1
 fM (Ai)=fM (A4) ' 12. extensive calculations in this formulation for charm quark physics
will be compared with results from CLEO-c
3. (future work) bottom quark physics, non-perturbative determination of improvement parameters