Precision lattice QCD
calculations and
predictions of
fundamental physics in
heavy quark systems
Christine Davies
University of Glasgow
SCIDAC 2006, June 2006
Particle physics: uncover the fundamental particles and
interactions at the smallest distance scales
...
Atom - nm
Proton - fm
Particle physics experiment
Smashing protons together
at high energy does not give
subunits directly - just more
particles!
Protons and other hadrons are made of quarks interacting
by the strong force.
Quarks never seen as
free particles - to study
them need accurate expt
and theoretical
calculations.
QCD is theory of strong
force - hard to calculate
because stronglycoupled and nonlinear needs numerical
simulation. This is lattice
QCD.
Standard Model of particle physics has:
! "! "! "
d
s b
6 quarks u c
t
! "! "! "
e
µ
τ
6 leptons νe νµ ντ
a ‘zoo’ of hadrons
3 forces (ignore gravity) : strong, weak, electromagnetic
Over 20 parameters, whose origin is some deeper theory
with New Physics.
QCD is theory of strong force - mirrors QED
Quarks
electrons
color charge RGB
electric charge
gluons
photons
BUT QED and QCD behaviour very different:
QCD - gluons carry color
charge - attempt to separate
QED - uncharged photons qq , force becomes strong at
travel freely - easy to get large distance and quarks and
free electrons
gluons confined.
All info. about quarks indirect - from ‘colorless’ bound
states, hadrons. Baryons = qqq e.g. p,n , mesons = qq
qq
Weir d t h in gs
Rich spectrum of states Ge V
masses calculable in QCD
bb g bb qq
if we can solve theory.
bb
!,
A lot of states only exist
briefly but can be seen in
particle detectors and
B
bc
properties determined.
12
11
10
b
9
8
7
c
6
B mesons
B,B
S
5
4
cc g
cc
,
c
3
mp
2
Ch a rm e d
mesons
1
Lig h t
mesons
0
cc qq
D,Ds
glu eba lls
qq g qqq q
K,
,
The Meson Spectrum
‘Weird things’ not yet
unambiguously seen.
Theory can help to find
them.
Quarks also feel the weak force
Key to understanding CP (matter-antimatter) symmetry
violation in Standard Model. (New physics?)
Vcb
Vud
Complex couplings
between quarks of
different flavor and W.
3x3 CKM matrix poorly
known
BUT quark weak decays
occur inside hadrons.
QCD effects are critical must calculate decays of B
mesons in lattice QCD
Lattice QCD
• Solve QCD by
numerical evaluation of path
integral:
Z
−SQCD
dAµdψdψe
• make integral finite with a
space-time lattice
a
• Importance sampling - make
gluon configs - ‘snapshots of
vacuum’ and propagate quarks
through them.
• ‘Measure’ e.g. hadron correlators on the gluon configs to
calc. hadron masses and weak decay rates
Handling light quarks is a big headache
Lq,QCD = ψ(γ · D + m)ψ ≡ ψMψ
For valence quarks, need to calc. M
For sea quarks need to inc. det(M)
in making gluon configs
Very costly as mq → 0
Early calcs:
Quenched Approximation - omitted
sea quarks. This is not good enough
for precision required. Need to
unquench with real s and light u/d.
−1
−6
Cost of lattice QCD calc. grows as a
must work with largest spacing possible
Discretisation errors are a big issue
Derivs become finite differences:
∂ψ(x j ) ψ(x j + a) − ψ(x j − a)
2
=
+ O (a )
∂x
2a
Correct with higher order
difference - but must take
account of gluon radiation
‘Improvement’ gives much higher accuracy - results with
sea quarks from ‘improved staggered quark QCD action’
-errors a few % at a=0.1 fm
Life as a lattice QCD theorist
Tbytes data
Large analyses - 0.5-1 Tflopyears
(Fermilab cluster)
Generate configs - 1-2 Tflopyrs
(QCDOC at Edinburgh)
People time is
important too!
Small analyses and tests - Gflopyrs
(Glasgow cluster)
Take-home message
• There has been a revolution in the numerical
simulation of the theory of the strong force
(lattice QCD) since 2003
• Lattice QCD now delivering results : hadron
masses that agree with expt; precise
parameters of QCD; decay rates needed to
determine the CKM matrix accurately.
Lattice QCD results
MILC collaboration gluon configurations have:
• 2+1 flavors of sea quarks, down to mu/d = ms/10 .
• Many mu/d values; 2 ms values
• 3 values of lattice spacing: 0.18fm, 0.12fm and 0.09fm
3
(2.5
f
m)
• Spatial volume exceeding
so lattice size 283 × 96
QCD has 5 parameters : 4 quark masses and a bare
coupling. Must fix these using ‘gold-plated’ (i.e. stable)
hadron masses. FNAL/HPQCD/MILC/UKQCD analysis
of MILC configs. Fix: a : Mϒ! − Mϒ
mu/d : Mπ ms : MK mc : MDs mb : Mϒ
Calculate other gold-plated hadron masses as test. Results
that follow from this analysis (but not all configs).
Summary of results
Results
including u,d
and s sea quarks
agree with
experiment
across the board
-from light to
heavy hadrons.
Parameters of
QCD are
unambiguous
fπ
fK
MΩ
MΞ − MN
2MDs − Mηc
2MBs − MΥ
ψ(1P − 1S)
Υ(1D − 1S)
Υ(2P − 1S)
Υ(3S − 1S)
Υ(1P − 1S)
0.9
1
1.1
LQCD/Exp’t (nf = 0)
0.9
1
1.1
LQCD/Exp’t (nf = 3)
Quenched results are wrong and ambiguous
Davies et al, hep-lat/
0304004,
Aubin et al, hep-lat/0407028,
Toussaint+Davies, hep-lat/
0409129,
Gray et al, hep-lat/0507013,
2
α
=
g
/4π
Determining Parameters of QCD : s
R
g = color charge. Measure with ‘test charge’
at distance R.
q
R-dependence from vacuum
polarization effects
q
q
2004 Nobel Prize
In lattice QCD have g in action
and ‘measure’ a.
q
q
q
q
q
q
q
q
q
sea quarks
screen charge
(in fact a bit harder than this to do accurately ...)
gluons
‘antiscreen’
Lattice QCD result
(5)
αMS(MZ )
= 0.1170(12)
2004 PDG = 0.1187(20)
R
αV
(nf )
(d/a)
0.8
0.6
nf = 3
0.4
0.2
Lattice result
with sea
quarks
nf = 0
2
Quenched result
4
6
d/a (GeV)
8
HPQCD, Mason et al, hep-lat/0503005
mq
Determining parameters of QCD:
Quarks never free so
cannot measure mass
directly
Masses are
parameters in lattice
QCD action determine
by getting hadrons
masses right.
Convert to continuum
mass by (hard) analytic
calcln. (gives error)
Quark masses (MeV)
2005 lattice QCD
2004 PDG
10000
b
c
1000
m proton
s
100
10
d
u
1
HPQCD/MILC, Aubin et al, hep-lat/0405022; HPQCD, Mason et al, hep-lat/0511160
Lattice QCD prediction! : mass of Bc meson
Using NRQCD formalism
for b and FNAL formalism
for c, calculate splitting
Result:
MBc = 6.304(20)GeV
6500
lattice QCD, Feb. 1999
lattice QCD, Nov. 2004
CDF, Dec. 2004
2
mBc (MeV/c )
1
MBc − (Mϒ + MJ/ψ)
2
6600
Lattice systematic
errors
6400
6300
6200
CDF (2005): 6.287(5) GeV
HPQCD/FNAL /UKQCD Allison et al, hep-lat/0411027; CDF, Acosta et al, hep-ex/0505076, Behari here.
Weak decay rates and the CKM matrix


Vud
Vus
Vub
W
 π → lν K → lν B → πlν  Vub
J = V0, Vi, A0, Ai




K
→
πlν


π
 Vcd

B
V
V
cs
cb


 D → lν Ds → lν B → Dlν


D → πlνD → Klν
 Lattice QCD calc. gives rate of


 Vtd
Vts
Vtb  basic weak decay from one
hadron to another. CKM and
"Bd |Bd # "Bs|Bs#
lepton kinematics outside calc.
Lattice QCD can calculate decay rates for at most one
‘gold-plated’ hadron in final state.
Possible for almost every element of CKM matrix.
* Need multiple cross-checks of lattice calcs in different
systems e.g. ϒ, B, D, ψ etc
Determination of CKM matrix
Vub
B
W
J = V0, Vi, A0, Ai
π
weak decay of quark
inside a hadron calculate in lattice QCD
expt = CKM x lattice
unitarity triangle sides
Precision (3%) lattice QCD needed
Key results from ‘B factories’ + lattice QCD
B mesons - valence quarks: b + light
Test lattice QCD by predictions for
D (c + light).
CLEO-c LP05
Leptonic decay rate of D from
CLEO. Convert to fD given Vcd
W
B
D
FNAL/MILC/HPQCD
100
150 200 250
fD (MeV)
300
J=Aµ
Lattice and CLEO agree but both
have 8% errors - need to improve!
FNAL/MILC/HPQCD Aubin et al, hep-lat/0506030; CLEO, hep-ex/0508057
New determination of fB, fBs
W
B
J=Aµ
Coarse lattice, Partially Quenched
Coarse lattice, Full QCD
Fine lattice, Full QCD
Full QCD Staggered ChPT
!(Bs) / !(Bq)
1.3
fB = 216(22)MeV
9% error from
pert matching
1.2
f Bs
= 1.20(3)
fB
1.1
1
Previous lattice calcs
0.9
0
0.1
0.2
0.3
0.4
0.5
mq/ms
0.6
0.7
0.8
0.9
1
pert. error cancels.
3% from extrap. in
u/d mass
HPQCD, Gray et al, hep-lat/0507015;
log dependence on light quark mass
expected
NEW Exptl result for Br(B → τν) from Belle, hep-ex/0605068
Neutral B mesons have fluctuating identity
B/Bs oscillation rate determined by box diagram. Calculate
in lattice QCD as 4-q operator.
VtdVtb∗
B0
B0
=
HW
2
fB BB
Parameterise as
where fB is decay
constant.
W
B
J=Aµ
fBs/ fB is now calculated
to 3% . BBs /BB expected to be NEW - measurement by
1, but not yet clear.
CDF of Bs oscillations
Constraints on CKM unitarity triangle
Flavor Physics and CP Violation Conference, Vancouver, 2006
Using just B leptonic decay and B/Bs oscillations
Mainly quenched +
2 flavors heavy sea clover
New 3-flavor imp. stagg
except for quenched B
onstraints on the ρ − η plane derived from B meson leptonic decay and BB mixing. The bounds
y clover fermions, those on the right employ staggered.
B
CKMfitter group, FPCP 2006
Semileptonic form factors
B → πlν and Vub
Vub
W
B
J = V0, Vi, A0, Ai
π
3
2.8
Flavor
Physics
2.6
previous
f+ and f0and CP Violation Conference, Vancouver, 2006
2.4
new f+
9] 2.2
in blue., new f
0
2
1.8 fBs /fB = 1.20(3)(1),
(2)
1.6
1.4 error is from chiral extrapolation
first
and
+
1.2the second is from everything else. This
and
1 with the result that JLQCD found
mpared
0.8 extrapolating from a much larger
hirally
ss:0.6
fBs /fB = 1.13(3)( +13
−2 ). [8]
0.4
, HPQCD
quotes
0
0.2
f0Bs = 0.260(7)(26)(9) GeV.
(3)
0
10
20
5
15
25
2
2
q (GeV
rtainties in fBs /fB and
fBs )are relatively
Predict D form factors
f
f
HPQCD form factor gives:
ent of each other, since the uncertainties in
r are dominated by statistics and chiral ex−3
n, while the uncertainties in the latter are
Figure 2: The predictedHPQCD.
shape ofhep-lat/0601021;
the form factor for
ub
d by everything else. fBs /fB is more indeFNAL/MILC,
hep-lat/040830;
D → Klν has been tested
with increasing
accuracy by
12%
error
from
lattice
f f8%
than
of
f
.
Bs
B
hep-ex/0510003
experiment, most recentlyBelle,
by Belle
[13].
error from
expt
V = 4.22(30)(51) × 10
Conclusions
• Accurate Lattice QCD calculations are now maturing.
Tests of hadron masses and determination of parameters of
QCD are at the few % level.
Future
• 10% errors on decay matrix elements for CKM.
Beat down errors further and finish B oscillation calc. for
impact on unitarity triangle.
• Longer term - work on harder calculations e.g. for
proton structure and unstable particles.
• Calculations beginning now that use other quark
formalisms that are numerically more expensive.