Update on B meson physics from HPQCD
Alan Gray, Emel Gulez, Junko Shigemitsu, Matt Wingate, Ian Allison,
Christine Davies, Peter Lepage
The Ohio State University, University of Washington, University of
Glasgow, Cornell University
HPQCD and UKQCD collaborations
Key aim of HPQCD collabn: accurate calcs in lattice QCD,
emphasising heavy q physics. Requires a whole range of lattice
systematic errors to be simultaneously minimised - critical one has
been inclusion of light dynamical quarks.
Edinburgh March 2005
The Unitarity triangle
Important objective of current particle physics: accurate determination
of elements of CKM matrix.
1
B factory prog. needs small
2-3% reliable lattice QCD
errors for Bs/d oscillations,
B → D or π decay.
CLEO-c will test lattice
predictions for D physics in
next 2 years.
∆md
∆ms & ∆md
η
εK
sin 2βWA
0
|Vub/Vcb|
εK
-1
CKM
fitter
-1
0
1
ρ
2
Simulation details
• MILC 2 + 1 flavour dynamical configs. For light quarks use asqtad
action.
• For heavy b quarks use standard tadpole improved Lattice NRQCD
action correct through 1/(amb )2 (inc. terms at 1/(amb )3 which
are v 4 for Υ).
• a−1 and mb fixed by Υ,
mu,d and ms fixed by π and K.
Hence no adjustable parameters.
amf ≡ amu/d /ams a−1 (GeV)
amq
nconf nsrc
Coarse
0.01/0.05
0.02/0.05
1.596
1.605
0.005
568
4
0.01
568
2
0.02
568
2
0.04
568
1
0.02
486
2
0.04
486
1
0.0062
465
4
0.031
472
4
Fine
0.0062/0.031
2.258
Note some points partially quenched.
B Leptonic Decays - picture in 2003
• ξf =
√
fBs mBs
√
fB mB
• Hint of chiral logs but stat.
errors large. Aim to reduce
stat. errors.
• We have used smearing to
do this successfully.
M. Wingate, A. Kronfeld review
Lattice 2003
Smearing & Fitting
• Smear heavy quark at source and sink. Use ground state
hydrogenic style wavefunctions as used for Υ.
• Find optimal radius: that which minimises fit errors while
maintaining reasonable χ2 /dof
• Do Bayesian multi exponential fits. Compare single correlator fits
to simultaneus vector 2 × 1 (source or sink smearing only) and
matrix 2 × 2 (source and sink smearing) fits.
Pnexp −1 (0j)
C
(−1)jt e−mj t
• Fit function oscillates: G(t) = j=0
(0)
(0) √
fB mB
• Extract Φ =
from ground state amplitude of leading
order current. Similarly get 1/mb pieces as Φ(1) and combine
through 1-loop.
Smearing Results
0.3
-0.5
0.29
-0.54
0.28
(0)
amq=amf=0.01
2x2 matrix fit
lnE0
-0.62
matrix
vector
src sm sk sm
0.27
a3/2Φp
-0.58
no smearing
0.26
0.25
0.24
-0.66
amq=amf=0.01
2
χ /dof: 144 25
0.23
3.1 1.3 0.7 0.7 0.7 0.7
-0.7
1
2
3
4
5
6
7
8
9
10
11
0.22
nexp
• Fit good for nexp ≥ 7
• Matrix fit substantially
reduces stat. errors.
Φ Results
0.65
0.6
• χPT
by
Bernard
ΦP
0.55
0.5
• Chiral Fit to all
coarse points.
Coarse mf=0.01
Coarse mf=0.02
PQ stagχPT mf=0.01
0.45
0.4
0
0.2
0.4
0.6
0.8
mq/ms
1
C.
1.2
1.4
Φ Results
0.65
0.6
0.55
ΦP
• Chiral Fit to
4 most chiral
coarse points.
0.5
Coarse mf=0.01
Coarse mf=0.02
PQ stagχPT mf=0.01
0.45
0.4
0
0.2
0.4
0.6
0.8
mq/ms
1
1.2
1.4
Φ Results
0.65
0.6
• Include fit with
a2 terms turned
off and staggered
energy splittings
set to 0.
ΦP
0.55
0.5
Coarse mf=0.01
Coarse mf=0.02
PQ stagχPT mf=0.01
PQ χPT mf=0.01
0.45
0.4
0
0.2
0.4
0.6
0.8
mq/ms
1
1.2
1.4
Φ Results
0.65
0.6
0.55
ΦP
• Fine points NOT
yet included in
fit.
0.5
Coarse mf=0.01
Coarse mf=0.02
PQ stagχPT mf=0.01
Fine mf=0.0062
0.45
0.4
0
0.2
0.4
0.6
0.8
mq/ms
1
1.2
1.4
ξ Results
1.5
Coarse mf=0.01
Coarse mf=0.02
PQ stagχPT mf=0.01
FU stagχPT mf=mq
Fine mf=0.0062
1.4
ξΦ
1.3
• ξΦ =
√
fBs mBs
√
fB mB
1.2
ΦBs
ΦB
≡
• Systematics cancel in ratio.
1.1
1
0.9
0
0.2
0.4
0.6
mq/ms
0.8
1
B − B̄ Mixing
t=tB
0
Q
tB
q
O
B
B
q
Q
• Continuum hOL iM S has contribution from lattice hOL ilat and
hOS ilat at 1-loop:
OL = [ ψ Q γ µ (1 − γ5 ) ψq ] [ ψ Q γµ (1 − γ5 ) ψq ]
OS = [ ψ Q (1 − γ5 ) ψq ] [ ψ Q (1 − γ5 ) ψq ]
• Same simulation params as B leptonic decay but only so far with
mf = 0.01, mq = 0.04 (i.e Bs ), and only leading order in 1/mb
Fitting
C(tB,tBbar)eE0(tB+tBbar)
0.003
function
data
0.0025
C(tB,tBbar)e
E0(tB+tBbar)
function
data
0.01
0.002
tB=10
0.0015
0.001
0.005
0
0.0005
0
0
0
2
4
6
8
10 12 14 16
tBbar
2
4
1416
12
8 10
tBbar
6
6 8
4
10 12
2
tB
14 16 0
• Corr has form
Pnexp −1
C(tB , tB̄ ) = j,k=0 Ajk (−1)jtB e−mj tB (−1)ktB̄ e−mk tB̄
• Again Bayesian fitting used.
Fitting
0.6
0.59
Note: fit directly to 3-point
without taking ratio over 2point. Don’t need to wait for
plateau - fit at low t where error is still small by including ex
states.
0.58
aE0
0.57
0.56
0.55
2
χ /dof:
0.54
5.70
0.65
0.40
0.38
0.38
0.38
3
4
5
6
7
8
0.53
0.52
1
2
nexp
9
√
fB BB
• a6 hOL iM S = [1 + ρLL αs ] hOL ilat + ρLS αs hOS ilat
OL = [ ψ Q γ µ (1 − γ5 ) ψq ] [ ψ Q γµ (1 − γ5 ) ψq ]
OS = [ ψ Q (1 − γ5 ) ψq ] [ ψ Q (1 − γ5 ) ψq ]
• ρLS , ρLL calculated pertubatively.
• BB is defined through hOL iM S = 83 fB2 MB2 BB
p
• prelim result fBs BBs (mb ) = 0.244(15)(32) GeV
errors are fitting (fits still prelim) and systematic (ΛQCD /mb , αs2
etc.).
In process of doing 1/mb corrections.
Conclusions
• Υ spectroscopy results provide mb and a−1 for B decay and
mixing calcs, as well as confidence that methods work.
• Have implemented smearing in B simulations to substantially
reduce statistical errors of parameters needed for f B .
• Chiral fits look promising. Need more fully unquenched points at
light quark mass.
• Successful fit done to B − B̄ mixing correlator, looks good. Now
include 1/mb corrections and repeat with different m q .