B Meson Physics on the Lattice
Matthew Wingate
DAMTP
University of Cambridge
Quark flavor mixing
e+
CKM mechanism in the Standard Model
Weak interactions change quark flavors
!
u
d!
"!
c
s!
"!
t
b!
"
W+
u
νe
d!
Flavor mixing

!


d
Vud
 s!  =  Vcd
Vtd
b!
Vus
Vcs
Vts


d
Vub
Vcb   s 
b
Vtb
V is the CKM matrix. Unitarity implies 4 undetermined parameters
For other reasons (Higgs fine-tuning, gauge coupling-unification) we
expect new physics at the TeV scale
Success of CKM model tightly constrains new models of EWSB
New physics models & quark flavor
little Higgs w.
MFV UV fix
ED w. SM on
brane
generic Little Higgs
generic ED w. SM in bulk
MSSM
MSSM SUSY GUTs
supersoft
SUSY breaking MFV
MFV
dirac gauginos low tan! large tan ! effective SUSY
SM like B physics
new physics in B data
Discovery potential of flavor physics experiments or
Nondiscovery rules out/tightly constrains these models
Figure from G. Hiller, hep-ph/0207121
Illustration from I. Shipsey, Nature 427, 591 (2004)
Wherefore LQCD for Flavor Physics?
a. Want to study
weak decay
of b to u
b. Confinement:
Nature shows us
the B meson, not
just the b quark
c. Expt. sees
B − → π 0 e− ν e
LQCD brings us
from meson level
to quark level
Wolfenstein Parameterization
Expansion based on empirical observation
|Vus | = 0.22 ! 1
|Vcb | ≈ |Vus |2
|Vub | ! |Vcb |

2

3
1 − λ /2
λ
Aλ (ρ − iη)

 + O(λ4 )
−λ
1 − λ2 /2
Aλ2
Aλ3 (1 − ρ − iη)
−Aλ2
1
In practice, go to next order
!
2
λ
ρ̄ = ρ 1 −
2
"
!
2
λ
η̄ = η 1 −
2
"
Wolfenstein Parameterization
λ = 0.2205 ± 0.0018(0.8%)
o
s
A
a
s
r
a
e
f
fa
y
ew
ρ̄ = 0.196 ± 0.045(23%)
A = 0.824 ± 0.075(9%)
.
.
.
go
η̄ = 0.347 ± 0.025(7%)
Experimental Constraints
sin 2β from B → (J/ψ)K
0
∆M from B ↔ B 0
}
need LQCD input
}
εK from K 0 ↔ K 0
Vub and Vcb from exclusive semileptonic decays B → π"ν
B → D!ν
Persistent Standard Model
UTfit collaboration
PRL 97, 151803 (2006)
η
CKMfitter Group (J. Charles et al.)
http://www.ckmfitter.in2p3.fr
version of Oct 2006
γ
1
β
∆md
∆ms
0.5
V ub
V cb
0
α
-0.5
-1
-1
-0.5
0
0.5
1
ρ
A couple discrepencies at 3σ: Hints or fluctuations?
FIG. 1 (color online).
Determina
3
New flavor physics
Master Formula Beyond the SM
rmula (4.14) can be generalized to a master formula for weak decay amplitudes that goes b
M [59]. It reads
A(Decay) =
!
i
i
i
Bi ηQCD
VCKM
[FSM
i
+
i
FNew
]+
!
k
k
BkNew [ηQCD
]New VNew
[GkNew ] ,
k
i = F i (x ) given explicitly in (4.1
here the first terms represent the SM contributions with FSM
t
A. Buras’ master formula
New physics can contribute to our master formula in two ways. It can modify the i
a given operator,
already in
SM, through
the new short distance functions
Variouspresent
classifications
(5)the
of new
physics models
pend on the new parameters in the extensions of the SM like the masses of charginos, squark
MFV: only new Inami-Lin--like functions FNew particles enter the new box an
ggs particles and tan β = v2 /v1 in the MSSM. These new
agrams. In Complementary
more complicated for
extensions
of the SM
operators
direct searches
fornew
new
physics (Dirac structures) that
sent or very strongly suppressed in the SM, can become important. Their contributions are
k
k theory
high precision
, [ηQCD
]New ,and
VNew
, GkNew being analogs of the corr
the secondRequires
sum in (4.21)
with BkNewexperiment
k
jects in the first sum of the master formula. The VNew
show explicitly that the second sum
B factors are hadronic matrix elements, need LQCD
nerally new sources of flavour and CP violation beyond the CKM matrix. This sum may, how
clude contributions governed by the CKM matrix that are strongly suppressed in the SM b
k
k
. Clearly the new functions
portant in some extensions of the SM. In this case VNew
= VCKM
k
as well as the factors V k may depend on new CP violating phases complicating co
Big picture
We expect physics beyond the Standard Model, e.g. to explain the
hierarchy problem, unify forces, give a good dark matter candidate.
Directly create new particles at the LHC: measure masses
New particles couple to Standard Model particles
Discern new coupling constants from precise flavor experiments
(BaBar, Belle, Tevatron, LHC, ...)
Most new models have new sources of flavor changing
interactions, even having a flavor problem
Flavor physics is LHC-era physics
Lattice QCD connects meson measurements to quark couplings
through systematically improvable first principles calculations
Outline
Importance of flavor physics and role of lattice QCD
Our strategy for achieving accurate results now
Recent results
B→πlν
Neutral B mixing
New effort: radiative & semileptonic penguin decays (b → s)
Lattice QCD in a nutshell
QCD Lagrangian
a
F a,µν −
L = − 14 Fµν
!
"
#
µ
a a
γ
(∂
−
igA
ψ
µ
q
q
µ t ) + m q ψq
= Lg − ψQψ
Break-up spacetime into a grid
Quarks on sites
Maintains gauge invariance
Breaking of rotational/translational Lorentz
invariance at short distances is controllable and
removable
Glue on links
Lattice QCD in a nutshell
QFT : Euclidean space path integral
1
!J (z )J (z)" =
Z
!
!
[dψ][dψ̄][dU ] J (z ! )J (z) e−SE
SFT : Sum over all microstates
!
"
1
!
−βH
Tr J (z )J (z) e
!J (z )J (z)" =
Z
!
Use same numerical methods!
Monte Carlo Simulation : Find and use field
“configurations” which dominate the integral/sum
Lattice QCD in a nutshell
Gluonic expectation values
!
1
[dψ][dψ̄][dU ] Θ[U ] e−Sg [U ]−ψ̄Q[U ]ψ
!Θ" =
Z!
1
[dU ] Θ[U ] det Q[U ] e−Sg [U ]
=
Z
Fermionic expectation values
!ψ̄Γψ" =
!
δ
δ −ζ̄Q−1 [U ]ζ
[dU ]
Γ
e
δ ζ̄ δζ
Probability weight
"
"
−Sg [U ] "
det Q[U ]e
"
ζ, ζ̄ → 0
Determinant in probability weight difficult Quenched approximation
1) Requires nonlocal updating; 2) Matrix becomes singular
Set det Q = 1
Partial quenching =
different mass for valence Q−1 than for sea det Q
Challenges (viz systematic errors)
Lattice volume must be big enough
Lattice spacing must be smaller than physically relevant length scales
Cost increases quickly as a decreases: a−4 × a−(∼2.5)
Heavy quarks have small Compton wavelengths
Singular behavior at light quark masses requires extrapolations from
feasible masses to physical masses
Need mild mass dependence or trustworthy theory (chiral PT)
Summary of our strategy
The goal: to address all systematic errors simultaneously
(Improved) staggered fermion formulation in order to be in chiral
regime
Nonrelativistic bottom quark to avoid extrapolations in mb -treats heavy quark effects through effective field theory
Discretization errors treated via Symanzik effective field theory
Perturbation theory -- automation
Some critics think they can do better in the future with other
methods. GOOD! That is progress
This approach has been very successful in improving lattice results
for phenomenology (compare: quenched, outside chiral regime)
Light quark effects are important
C. Davies, et al., PRL 92 (2004)
Other checks and predictions
Bc meson mass (Fermilab/HPQCD LQCD; CDF expt)
D & Ds meson decay constants (Fermilab/MILC LQCD; CLEO-c expt)
D → K!ν form factor (Fermilab/MILC LQCD; BES, FOCUS expt’s)
QCD coupling and quark masses (HPQCD)
Fermilab/HPQCD
Fermilab/MILC
6600
D → Klν
2
6300
1.5
2
6400
f+(q )/f+(0)
2
mBc (MeV/c )
6500
HPQCD
2
s
2.5
lattice QCD, Feb. 1999
lattice QCD, Nov. 2004
CDF, Dec. 2004
2
qmax/mD*
1
0
6200
experiment [FOCUS, hep-ex/0410037]
lattice QCD [Fermilab/MILC, hep-ph/0408306]
1σ (statistical)
2σ (statistical)
0.5
0
0.05
0.1
0.15
0.2
2
0.25
2
q /mD*
s
0.3
0.35
0.4
0.45
Logic of 4th root hypothesis
Hypothesize that 4th root procedure is QCD in continuum limit
A testable hypothesis
Comparable to hypotheses of quark mass extrapolation from
outside the chiral regime
Empirical tests
So far so good -- obviously better than quenched LQCD
Skeptics welcome
Also invited to look hard at non-lattice CKM uncertainties
Progress in understanding 4th root (Shamir, Bernard, Golterman)
All approaches should be pushed hard
Lattice NRQCD for heavy quark
S0 =
!
d4 x Ψ†
"
iDt +
! 2
|D|
2mQ
#
Ψ
Foldy-Wouthuysen-Tani (FWT) transformation
Take lattice action as given: can analyze just like continuum HQET
Requires amQ > 1 , satisfied for b quark on present and near future
unquenched lattices
The following phrase is often uttered: “The continuum limit cannot be
taken.”
In theory, there are no lattice artifacts on the renormalized trajectory
In practice, discretization errors are short distance effects, systematically
removed using Symanzik’s EFT
Improvement & matching rely on perturbation theory
Nonperturbative methods preferable in principle, when practical & precise
Overview of simulation parameters
MILC collaboration’s 2+1 flavor configurations (AsqTad staggered)
“coarse” a = 0.13 fm and “fine” a = 0.09 fm
Spatial volume (2.5 fm)3
Lightest up/down mass ms /8
We compute at both unquenched and partially quenched masses
NRQCD action for bottom, correct through O(Λ2QCD /m2Q )
Semileptonic B to pi decay
!
B 0 → π − "+ ν!
Vµ
b
ν
u
B
π
"
d
Combine with
1.2
1
0.8
B 0 → D − !+ ν!
0.6
0.4
0.2
0
V ub
V cb
-0.2
-1
-0.5
0
0.5
1
!
work done with
E. (Gulez) Dalgic (Simon Fraser)
A. Gray (EPCC)
J. Shigemitsu (Ohio State)
C. T. H. Davies (Glasgow)
G. P. Lepage (Cornell)
(part of the HPQCD Collaboration)
Semileptonic Decays
B 0 → π − "+ ν!
G2F "! 3
dΓ
2 2
=
|
p
|
|f
(q
)|
+
2
2
3
|Vub | dq
24π
1
!π(p! )|V µ |B(p)" = f+ (q 2 )(pµ + p!µ ) + f− (q 2 )(pµ − p!µ )
Currents in EFT
(0)
J0 (x)
Temporal components
(1)
J0 (x)
(2)
Γµ ≡
!
γµ
γµ γ5
for Vµ
for Aµ
J0 (x)
(0)
Jk (x)
(1)
Jk (x)
Spatial components
(2)
Jk (x)
(3)
Jk (x)
(4)
Jk (x)
= q̄(x) Γ0 Q(x),
1
= −
q̄(x) Γ0 γ ·∇ Q(x),
2M0
1
←
−
= −
q̄(x) γ · ∇ γ0 Γ0 Q(x).
2M0
= q̄(x) Γk Q(x),
1
= −
q̄(x) Γk γ ·∇ Q(x),
2M0
1
←
−
= −
q̄(x) γ · ∇ γ0 Γk Q(x),
2M0
1
= −
q̄(x) ∇k Q(x)
2M0
1
←
−
=
q̄(x) ∇ k Q(x),
2M0
Perturbative matching
For example,
!A0 "QCD
=
(1 +
(1 +
J (i),sub
=
(0)
αs ρ̃0 ) !J0 " +
(1),sub
αs ρ1 ) !J0
"
+
(2),sub
αs ρ2 !J0
"
J (i) − αs ζ10 J (0)
Turns out to be leading uncertainty
Perturbative coefficients computed in
E. Gulez, J. Shigemitsu, M.W., PRD 69, 074501 (2004)
Fits, fits, fits, ...
1) Fit 3-point correlators
2) Combine LO and NLO
3) Interpolate to fixed Eπ
4) Extrapolate in quark mass
0
!π|V |B" =
√
2mB f!
√
k
!π|V |B" =
2mB pkπ f⊥
"2# "2#
"3# "3#
"4# "4#
!
"
%
hJ
i
!
"
%
hJ
i
!
"
%
i: (13) for the spatial currents Vk .
s
s
s
TABLE
II.
Matching
k
k
k
k
k hJk coefficients
he B rest frame. From these
Where
errors are
explicitly, they are of order one or
e dominated by f? , i.e. by
"1#;sub
"1# not indicated"0#
We introduce the combination
J
%
J
&
"
&
J
$
$
$
s
10;$
less in the last digit. aM0 is the .bare heavy quark mass in lattice
d f0 by fk or the matrix
This subtracts out power law
contributions
to theinmatrix
units
and n a parameter
the NRQCD action. The three selected
"1#
to the b quark on the MILC extravalues foroperator
aM0 correspond
elements
of the
higher dimension
J$ through
hadronic
matrix
elements
"1#
coarse, coarse and fine lattices, respectively [28].
s ="aM## [42].
iceO""
simulations.
ThereJ$areenters the matching already at tree
"1# physical
levelone
andmust
after relate
the subtraction
is left
First,
the
~k"0# with%the
%k"2#
%k"3#
%"4#
&10;k
aMone
0 n %
k
k
0
k
O""
=M#
contribution
that
is
a
relativistic
correction
to
ts, V QCD
and V , to lattice
4.00 2 0.256 0.484(3) 0.340(6) 0.244(3) &0:137"3# 0.041
leading
law subtractions of the other
he the
heavy
and order
light term.
quarkPower2.80
2 0.270 0.349(3) 0.169(6) 0.218(4) &0:029"4# 0.055
4 current
corrections come in as O""2s ="aM##
thedimension
second step
the matrix
1.95 2 0.332 0.232(3) 0.121(8) 0.161(4)
0.063(3) 0.073
effects andmust
are only
partially included here. The one-loop
nt operators
be evalcoefficients
%"j#
levant
amplitude,
i.e. &the
$ and
10;$ for $ % 0 are given in Ref. [39].
B MESON
PHYSICAL REVIEW D 73, 074502 (2006)
round
state BSEMILEPTONIC
meson and FORM FACTORS FROM . . .
there are two dimension 4 current0.14
corrections to the temopriate0.04
momenta, must be
poral components and four such corrections to the spatial
Eqs. (7), the form factors
components. To the order that we0.12
are working, one has
0
ht quark
mass
and the pion
0.02
"0#
"1#
"1#;sub
hV0 i % "1 ! "s %
~"0#
i
ese numerical results must
0 #hJ0 i ! "1 ! "s %0 #hJ0
0.1
0
al pion.
In the next
three
-0.02
"2# "2#
(1),sub
(4),sub(12)
!
"
%
hJ
i
s
0
0
J
J
ese three
steps in kturn.
-0.02
k
0.08
and
-0.04
-0.04 CURRENTS
(1)
(4)
Y-LIGHT
0.06 "1# J"1#;sub
"0#
Jk
hVk i % "1 ! "s %
~"0#
#hJ
i
!
"1
!
"
%
#hJ
s k
k
k
kk i
rrents -0.06
between continuum
"2# "2#
"3# 0.04
"3#
"4# "4#
!
"
%
hJ
i
!
"
%
hJ
i
!
"
%
s
s
s
k
k
k
k
k hJk i: (13)
eory with two-component
-0.08
-0.08four-component
ds ! and
"1#;sub
"1#
"0#
0.02% J$
We introduce the combination J$
& "s &10;$ J$
.
Ref. [41].
-0.1 Since staggered
to the matrix
0.2
0.4 This
0.6 subtracts
0.8 out 1power law contributions
0
terms -0.1
of 0four-component
E
in
GeV
"1#
π
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4 elements
0.6
1 dimension operator
of0.8
the higher
J$
through
the formalism
developed
in
GeV
E
Eπ in GeV
"1#
"0#
[42].
J$
enters the matching already at treeπ
o the
calculation.
s ="aM##
FIG.present
2 (color online).
The ratioO""
hJk"2# i=hJ
i
for
two
ensembles
k
"4#
"0#
level
and
the
subtraction
oneFIG.
is left
with online).
the physical
"1#
for after
u"0#0 am
0:01
and
versus
pion
energy
EThe
entFIG.
notation
foronline).
the
heavy
$ . Squares
f !
4
(color
Same
as
Fig.
2
for
hJ
i=hJ
i.
1 the
(color
ratio
hJare
i=hJ
i
for
one
ensemble
k
k
0
0
circles
forthat
u0 amf !
0:02.
O"" pion=M#
contribution
that is a relativistic correction to
finds
through
energy
E! . The lower
(u
0 amf % u0 amq % 0:01) versus theQCD
j(4)/j(0)
j(1)/j(0)
j(2)/j(0)
1/mb corrections
Form factor shape
Ball-Zwicky, 4-parameter Becirevic-Kaidalov
2
f
(0)
r
q̃
+
f+ (q 2 ) =
2 +
1 − q̃
(1 − q̃ 2 )(1 − αq̃ 2 )
f+ (0)
f0 (q ) =
1 − q̃ 2 /β
2
3
2.5
Considering analyticity and
unitarity constraints (...) could
remove model dependence.
Bourrely et. al; Boyd et al; Fukunaga &
Onogi; Lellouch; Boyd & Savage;
Arnesen et al; R. Hill; P. Mackenzie;
P. Ball; Flynn & Nieves; ...
q̃ 2 ≡ q 2 /m2B ∗
f+
f0
2
1.5
1
0.5
0
0
5
10
15
2
2
q (GeV )
20
25
Experiment + Lattice QCD
G2F "! 3
dΓ
2 2
=
|
p
|
|f
(q
)|
+
2
2
3
|Vub | dq
24π
|Vub | =
1
(3.55 ± 0.25 ± 0.50) × 10−3
expt.
using HFAG avg b.f. with q2 > 16 GeV2
EPS 2005
3
2.5
LQCD
f+
f0
2
1.5
Inclusive B → Xu !ν
1
|Vub | =
0.5
0
0
(4.46 ± 0.20 ± 0.20) × 10−3
5
10
15
2
2
q (GeV )
20
25
E. Gulez, et al., PRD 73, 074502 (2006), erratum ibid 75, 119906 (2007)
expt.
th.
HFAG avg (DGE) summer 2006
Comparison
5
0.1
−2
dΓ
104 Γ1Tot dq
2 / GeV
f+,0 (q2 )
4
3
2
1
0
5
10
15
20
0.08
0.06
0.04
0.02
25
0
q2 / GeV2
(q2 )]
0.035
Green: LCSR,!0.6
Red: HPQCD, Blue: FNAL/MILC, Black: experiment*norm,
Curves & bands: fit to Omnès parametrisation (Flynn & Nieves)
!0.8
Plot from J. M. Flynn and J. Nieves, arXiv:0705.3553
0.03
0.02
0
Comparison
5
10
15
20
25
0
q2 / GeV2
0.035
!0.8
0.03
!1
0.025
Pφ f+
m2B∗
log [
m2B∗ −q2
f+ (q2 )]
!0.6
!1.2
!1.4
0.02
0.015
!1.6
0
5
10
15
20
25
q2 / GeV2
Green: LCSR, Red: HPQCD, Blue: FNAL/MILC, Black: experiment*norm,
Figure
1 Results
obtained
from the fit to(Flynn
experimental
Curve
& band:
fit to Omnès
parametrisation
& Nieves)partial bran
factor
The top left plot shows the two form factors wit
Plot from J. M.
Flynncalculations.
and J. Nieves, arXiv:0705.3553
Error budgets
LAT2005)207
cs
20.51
1.627(185)
0.676(41)
21.56
1.816(126)
0.714(56)
where the errors are statistical, systematic, and experimental.
If we instead use the accepted values of the CKM matrix elements as inputs, we
obtain for the seen
Improvement
D meson decay rates
with use of random wall
HPQCD
error budget is given by (previous Table VI):
Estimate0of percentage
errors in f+(q2) for q2 >−316 −1
GeV2sources: K. Wong et al,
− +
Γ(D → π l ν) = (7.7 ± 0.6 ± 1.5 ± 0.8) × 10 ps ,
source of error
size of error (%) Lattice 2007
Γ(D0 → K − l + ν) = (9.2 ± 0.7 ± 1.8 ± 0.2) × 10−2 ps−1 ,
statistics
10
Γ(D0 →+π −chiral
l + ν) extrapolations
=
0.084
±
0.007
±
0.017
±
0.009,
Γ(D0two-loop
→ K − l + ν)matching
9
(3.4)
discretization
3
where the errors are statistical,
systematic, and from CKM matrix element.
relativistic
1 Klν form factor, a highAfter the publication of our
results for the q2 dependence of the D →
statistics measurement of the shapeTotal
appeared from the FOCUS Collaboration
14 [10]. As can be seen
in Fig. 7, the results agree well.
From E. Gulez, et al., Phys. Rev. D 73, 074502 (2006), erratum ibid 75, 119906 (2007)
FNAL/MILC
artially integrated
rates
(previous
Tabledecays.
VII):Errors
Improved
action: M.
Table 1: differential
Systematic errorsdecay
for CKM
matrixbecome
elements from
the semileptonic
for Vub decay
2.
Oktay and A. S. Kronfeld,
are obtained from the integration with q2min = 16 GeV
!
dΓ
2
2
−1
Fitdecay
f+ (q = 0)
[ ps B] → πlν
2008
D ub
→|π(K)lν
dq2 /|V
2
2|
CKM matrix element
|Vcd(s)
|Vubq|2
0 ≤ q 2 ≤ qmax
16GeV
≤ q2 ≤
max
discretization effect
BZfitting
0.31(5)(4)
9.10(1.82)(2.55)
3- and 2-point functions
BKchiral
0.31(5)(4)
extrapolation9.30(1.86)(2.60)
parameterization)
SEq2 dependence
0.30(5)(4)(BK9.35(1.87)(2.62)
current renormalization
a uncertainty
total systematic
From P. Mackenzie, Lattice 2005
207 / 4
9%
9%
0%
1%
10%
1%
1%
11%
2.07(41)(39)
3%
3%
2.13(43)(40)
3%(2%)
4%
2%
4%
2.02(40)(38)
Updating the experimental br. frac.
HFAG update
B.F. (q2 > 16 GeV2)
* 104
Vub * 103 (HPQCD)
EPS 2005
0.40(4)(4)
3.55(25)(50)
LP 2007
0.35(3)(3)
3.33(21)(+58-38)
Vub(*103): FNAL 3.6(2)(+6-4), BZ 3.4(1)(+6-4) vs. Inclusive 4.5(2)(2)
Tension between inclusive and exclusive determinations is a continuing story,
demonstrating the challenges of precision physics.
Semileptonic decay vs. sin2β
Agreement with global fit (sin2β)
HPQCD
Flynn & Nieves
Uncertainty
Exclusive number (UTfit)
Inclusive average (HFAG)
Measurement
From UTfit website, version of 28 September 2006
plus my decorations
Updating the experimental br. frac.
HFAG update
B.F. (q2 > 16 GeV2)
* 104
Vub * 103 (HPQCD)
EPS 2005
0.40(4)(4)
3.55(25)(50)
LP 2007
0.35(3)(3)
3.33(21)(+58-38)
Vub(*103): FNAL 3.6(2)(+6-4), BZ 3.4(1)(+6-4) vs. Inclusive 4.5(2)(2)
Tension between inclusive and exclusive determinations is a continuing story,
demonstrating the challenges of precision physics.
Total branching fraction
+
+
→ π0 l ν × 2 τ 0 / τ +
BABAR SL tag: B + # "0 l+ ! ! 2$0/$+
BABAR SL tag: B
3.31 " 0.68 " 0.42
BABAR B reco tag: B
1.36 ± 0.33 ± 0.15
BABAR Breco tag: B + # "0 l+ ! ! 2$0/$+
1.52 ± 0.41 ± 0.20
1.45 " 0.37 " 0.12
BELLE SL tag: B
BELLE SL tag: B + # "0 l+ ! ! 2$0/$+
+
1.43 ± 0.26 ± 0.16
1.40 " 0.24 " 0.16
BELLE B reco tag: B
1.60 ± 0.32 ± 0.11
0
BABAR SL tag: B
- +
BABAR SL tag: B # " l !
1.12 ± 0.25 ± 0.10
BELLE SL tag: B 0 # "- l+ !
1.38 ± 0.19 ± 0.14
BABAR Breco tag: B 0 # "- l+ !
1.07 ± 0.27 ± 0.15
BELLE SL tag: B
1.02 " 0.25 " 0.13
0
0
+
+
→ π0 l ν × 2 τ 0 / τ +
+
→ π0 l ν × 2 τ 0 / τ +
+
→ π0 l + ν × 2 τ 0 / τ +
→ π- l + ν
→ π- l + ν
BABAR B reco tag: B 0 → π- l + ν
1.48 " 0.20 " 0.16
+
CLEO untagged: B → π l ν
1.24 " 0.29 " 0.16
1.33 ± 0.18 ± 0.11
CLEO untagged: B # " l+ !
+
BABAR untagged: B → π l ν
1.32 " 0.18 " 0.13
1.46 ± 0.07 ± 0.08
BABAR untagged: B # " l+ !
BELLE B reco tag: B 0 → π- l + ν
1.38 " 0.10 " 0.18
1.49 ± 0.26 ± 0.06
Average: B 0 # "- l+ !
Average: B
1.35 " 0.08 " 0.08
0
→ π- l + ν
1.39 ± 0.06 ± 0.06
2
% /dof = 8.0/ 7 (CL = 33.0%)
HFAG
χ2/dof = 3.23/ 9 (CL = 95 %)
HFAG
LP 2007
EPS-2005
0
0
2
- +
-4
B(B # " l ! ) [ ! 10 ]
-2
0
0
- +
2
-4
B(B → π l ν ) [ × 10 ]
The total branching fraction has not changed much, while the b.f. with
q2 > 16 GeV2 has moved ~1 sigma. This highlights the need for
LQCD to extend its kinematic reach.
Lower q2 on the lattice
Low q2 implies large pion recoil
But pion momentum must be small compared to inverse lattice
spacing in lattice rest frame
So far lattice and B frames roughly coincide in all calculations
Progress can be made by discretizing in a frame which is boosted
relative to the B
p = mb u + k
Extending the range of q2 will remove model dependence of shape
and reduce statistical uncertainties due to better overlap with
experimental signal
Preliminary tests: S. Meinel, et al., PoS(Lattice 2007)377, arXiv:0710.3101
B 0 − B 0 Mixing
b
Bd0
Vtb∗
t
W
d
W
Vtd
Bd0
Vtb
t
"
d
∗
Vtd
b
1.2
1
# Md
# Ms
0.8
2
2
2
|Vtd | ∝ [(1 − ρ̄) + η̄ ]
0.6
0.4
0.2
0
-0.2
-1
-0.5
0
0.5
1
!
B 0 − B 0 Mixing
b
Bd0
Vtb∗
t
W
d
Vtd
∗
Vtd
d
W
t
Vtb
Bd0
b
Constraint complements
sin 2β & |Vub |/|Vcb |
More likely New Physics
contributions
work done with
E. Dalgic (Simon Fraser)
E. Gámiz (Illinois)
A. Gray (Ohio State, Edinburgh)
J. Shigemitsu (Ohio State)
C. T. H. Davies (Glasgow)
G. P. Lepage (Cornell)
(part of the HPQCD Collaboration)
B 0 − B 0 Mixing
Only Bd0 − Bd0
∆md
G2F 2
2
∗ 2
m
η
S(x
)m
f
=
B
|V
V
B
t
B
B
td
d
d
W
Bd
tb |
2
6π
Including Bs0 − Bs0
mBs
∆ms
=
∆md
mBd
ξ =
!
!2
!
!
2 ! Vts !
ξ !
Vtd !
fBs
!
BBs
fBd
!
BBd
Most of the
mass
dependence,
& simplest
to compute
on lattice
B 0 − B 0 Mixing 2002
1.60
ξf
1.40
Chiral PT
f2 = 0.8 GeV
−2
f2 = 0.5 GeV
−2
f2 = 0.2 GeV
−2
Using staggered
1.20
1.00
0.80
0.0
Previous lattice data
Linear
1.0
0.5
r
Kronfeld & Ryan, (also JLQCD)
= mq /ms
1.5
B 0 − B 0 Mixing 2005
A. Gray, et al (HPQCD) PRL 95, 212001 (2005)
Coarse lattice, Partially Quenched
Coarse lattice, Full QCD
Fine lattice, Full QCD
Full QCD Stagg. ChPT
Physical Quark Mass
Φ(Bs ) / Φ(Bq )
1.3
1.2
Previous calculations
1.1
1
0.9
0
Φ(B) ≡ fB
0.1
√
0.2
0.3
0.4
0.5
r = mq / ms
mB
0.6
0.7
0.8
0.9
1
B 0 − B 0 Mixing 2005
A. Gray, et al (HPQCD) PRL 95, 212001 (2005)
r=
Φ(B) ≡ fB
√
mB
Decay constant results
fBs = 260 ± 7|stat ± 26match ± 8hq ± 5disc MeV
fB = 216 ± 9|stat ± 19match ± 6hq ± 4disc MeV
Most errors cancel in the ratio
fBs
fB
= 1.20 ± 0.03 ± 0.01
M. W., et al (HPQCD) PRL 92 (2004); A. Gray, et al (HPQCD) PRL 95 (2005)
B → τν
Belle, hep-ex/0604018
fB |Vub |
|Vub |DGS
fB |Vub | = 0.77
= 175 ± 37 MeV
BaBar, hep-ex/0611019
!
+12
−10
fB |Vub |
"
stat
!
+7
−6
"
MeV
sys
= 193 ± 46 MeV
|Vub |HPQCD
!
"
!
"
+23
+4
fB |Vub | = 0.70
MeV
−36 stat −5 sys
C3
(t , t ) =
!0|Φ
($xs , t
(4)
≡
1 1
1 B2
s |OL|B
s "(µ) ≡ fBs B
Bs
s (µ) MBs .
(µ)
4.8GeV)
= 0.212
[16] one findsfour-fermion
B̂!OL"
= !B
1.534.
ts have
been very
encouraging.
Here
we
apply
the
Bs /B
Bssome
B(µ)
at
scale
µ,
=
L,S
operators
via
lattice
QCD
methods,
one
3
to the “vacuum saturation”
approximati
"
x1 ,"
x2
ΦgiB
0 matrix
0
In
order
to
evaluate
hadronic
elements
of
the
then
me successful lattice approach to Bmust
−
B
mixing.
first
relate
the
operators
in
continuum
QCD
to
ops
s
fermion operators
OS and O3 have simil
8
8 B
the
MS methods,
MSthat
2 Bs = 1 corresponds
2
The
factor
is
inserted
so
four-fermion operators via lattice
QCD
one
3 ≡ !B
.
(4)
B
(µ)
M
!OL"
f
|OL|Bbag
"(µ)lattice
≡
erators
written
in
terms
heavy
and
light
quark
B
sown
sof
s
B
B
(µ)
s
s
parameter
Φ3Bs is an interpolating
operato
We have
studied
the following
four-fermion
operators
lat
to
the
“vacuum
saturation”
approximation.
The
fourmust
first relate
the operators
in
continuum
QCD
to
opfields. We carry out this matching
between
continuum
the
four-fermion
operator
Ô ispoifi
t entererators
into calculations
of ∆M
and
∆Γ
in
the
StanΛQCD
s
8 light
fermion
operators
OS
and
O3
have
similarly
each
their
written in terms
of slattice
heavy
and
quark
The and
factor
is inserted
so that
BBs =O(α
1 corresponds
QCD
the3 lattice
theory
through
), 5C
O(
(4f
)M
Btogether
sfit
S )(µ)
MS
2
2
lattice.
We
w
d Model
(“i”
and
“j”
are
color
indices)
α
f
M
!OS"
≡
−
own
bag
parameter
s B(µ) continuum
fields.
We
carry
out
this
matching
between
to
the
“vacuum
saturation”
approximation.
The
fourat
some
scale
µ,
B
B
(µ)
!O
theory
works
with NRQCD
b-to
and O( aM ). Our lattice
Bs
3 LO operators
2 the fo
3
R
point
correlator,
C
(t),
ΛQCD
fermion operators OS and O3 have similarly each their
Full 4-quark matrix elements
QCD and the lattice theory through
O(α
) 1/M the b fields in (1) - (3)
s ), O(
ele
quarks. At
lowest
order
M in
B
(µ)
5
8
αs
(4f
)
S
own
bagwith
parameter
MS
2 !OX" w
MS
2
2 , t2 ) =
works
NRQCD
b-sf|OL|B
and O( aM ). Our
. (4)
BBs (µ)antiM
!OL"
"2MS
≡−
!B
≡(tB
1/M
1 for
2s heavy
,
(5)
M
!OS"
≡
be
replaced
by
NRQCD
heavy
quark
ilattice theory
j must
sC
i
j
B
B
(µ)
(µ)
B
s
(µ)
s
s
OL
≡
[b
s
]
[b
s
]
,
(1)
3
V
−A
V
−A
3
R
quarks. At lowest order in 1/M
the fields.
b fields The
in (1)
- (3)5 relation
Nexp −1
At
1 NRQCD
B̃S (µ)element
quark
tree-level
between
B
%
MS
2
2
S (µ)
MS
2
2
i
j
1/M
ths
≡
!O3"
f
M
,
(5)
M
!OS"
≡
8 − fBs
j·t
k·t
1
2
B
must OS
be replaced
NRQCD
quark
or
heavy
antiB
B
(µ)
(µ)
[bj s heavy
]and
,
(2)
≡ [bi sby]S−P
s
s
2
The fields
factor is3 given
is 3inserted
so Foldy-WouthuysenthatAjk
BB(−1)
2
!O
by Rthe
S−Pfull QCD
(−1)
e
s = 1 corresponds
3
R
At
O(α
Λ
QCD
quark fields. The
tree-level relation
between
NRQCD
to the “vacuum
The four) approximation.
this
brings in ditransformation.
At
O( (µ)
Th
j,k=0
1 saturation”
MS
2 B̃SM
2
O3 ≡ [bi sj ]S−P [bj si ]Tani
.
(3)
!OL"
a
S−P
≡ fto
!O3"
MBhave
, similarly
(6)
fermion
operators
O3
each their
with
and full QCD fields is given by
the Foldy-WouthuysenBOS
(µ)
s theand
s
2
mension
seven
corrections
four-fermion
operators,
one
31 2 B̃SR(µ) 2
This m
ΛQCD
MS
own
bag
parameter
)
this
brings
in
diTani transformation. At O(which
≡
!O3"
f
M
,
(6)
are
of
the
form,
goo
Bs
(µ)
M
one car
2
3 Bs R2
M
%
1
with
Bs
mension
seven
corrections
to thematrix
four-fermion
operators,
B(µ)
use
M S scheme
elements
of!
continuum
QCD
in the
good
ch
≡
. s ($
B
5
C
(t)
=
!0|Φ
x
,
S
MS
2
2
B
2
2
NLO
operators
1
with
,
(5)
f
M
!OS"
≡
−
i
j
R
(m
+
m
)
which
are
of
the
form,
i
j
$
b
s
use
for
2 ]V −A [b
tio
· $M
γ sB
[∇b (µ)
se operators are parametrized in terms OLj1
of the≡Bs me3 Bs s R]V2 −A Bs "x
1
s
2M ≡
2
tional
o2
.2
(7)
"
M
!
uu
1
decay constant fBs and
so-called
“bag”
parameters
2
2
N
−1
B
exp
s
1
R ≡Standard
) .j
%
foruum
thetom
sModel
i b + m$
i(m
$ i · $γ si ]V −A [bj sj ]V −A The
2
+
[b
s
]
[
∇b
γ sjexpression
]V −A . (7)
(9)
OLj1 ≡
[∇b
V
−A
R
(mb + ms )2 · $
in
ξj (−1
=
1
B̃
(µ)
2M
∆M
is
given
by
[17],
S
MS
2
2
s
Th
in (10)
2Model
= L,S or 3) and matrix
elements
in "
theand
lattice
theory
!O3"
Mdifference
, j=0 theory
(6)
The Standard
expression
forfBis
the
mass
Blattice
cale µ,
=
L,S
or
3)
matrix
in
the
is
(µ) ≡elements
s
s
cou
2
i
j
3
R
is ]
jStandard
The
Model
expression
for
the
mass
difference
$
(10
count
t
+
[b
[
∇b
·
$
γ
s
]
.
(9)
−A
Vby
−A
Similar
1/M
corrections
OSj1
and
O3j1
can be introthen given by (weV ∆M
suppress
the µ
dependence),
is
given
[17],
2
2
then
given
by
(we
suppress
the
µ
dependence),
s
the
∆Ms is given by [17],
GF M
the
latt
The
dimensionless
matrix
eleme
B
W OS∗ and 2O3.
8
2Together
with
L,S
or
3)
and
matrix
elements
in
the
lattice
theory
is
duced
for
the
four-fermion
operators
To
MS
2
2
|V
V
|
η
S
(x
)M
∆M
=
3
.
(4)
µ)
M
s
tb
0
t
Bs
ele
3
ts
2
MBs . (4)
B s |OL|B
Bs s "(µ) ≡ fBs BBs (µ)
a
2 then
element
2
2
a
(10)
are
given
by,
MS
MS
6π
2
2
hen given
by (we1/M
suppress
the µ dependence),
Similar
OSj1
and
O3j1
be
introGFcan
3 corrections
!OX"
=
(X the one
the
order
stated
above,
matching
GM
2 2 2 !OX"
22 B
BMS = between
W
FM
W|V ∗∗V!OX"
the
M
1
|
η
S
(x
)M
f
B̂
,
(8)
∆M
=
|V
V
|
S
(x
)M
f
B̂
,
(8)
∆M
=
B
2M
s
tb
0
t
B
B
s
s
tb
0
t
B
B
ts
2
s
B
s
B
ts
s Bs s s
2M
s
2
2
B
duced
for
the
four-fermion
operators
OS
and
O3.
To
≡ 2
.B
(7)
3
s
A
6π6π where
2
00
2 m
2
in
a
sep
in
MS
x
=
/M
,
η
is
a
perturba
R
(m
+
m
)
s corresponds
insertedaso that
B
=
1
corresponds
b
s
Ô"
=
!
t
Bs= [1 + αs · ρXX ]!OX" + αs · ρ
t
2
MS
W
!OX"
+ ]!OX" + αs · ρXY !OY " + of [20,ξ
[1XY
+ !OY
α ·Bρ"XX
the
order
stated above,
matching between !OX"
2# 2 2 s(X
0
2M
of
2
B
#
$aa perturbative
B
n.
The
fours
where
x
=
m
/M
,
η
is
QCD
cor-cor-$
m saturation” approximation. The fourrection
factor,
S
(x
)
the
Inami-Lim
fun
t
where
x
=
m
/M
,
η
is
perturbative
QCD
0
t
t
2
W
XX
XY
t
XX
XY
2Model
W !OY
The tStandard
expression
for
mass
difference
−
αrection
!OX"
+
ζ(x
")αs.(ζ
(10)
cu
!OXj1"
−Inami-Lim
!OX"
+ ζthe
!OY
")
.light(10)
s"(ζ+
10 factor,
10t ) the
10
10
ligh
S
function
and
V
+ αO3
+!OXj1"
αseach
· ρXYtheir
!OY
ly each
their
0
ts
ors
OS [1
and
similarly
s · ρhave
XX ]!OX"
and
V
the
appropriate
Cabibbo-Koba
rection
factor,
S
(x
)
the
Inami-Lim
function
and
V
tb
t
ts are nece
is0 given
by Cabibbo-Kobayashi-Maskawa
[17],
#
$ ∆M
sappropriate
XX
XY
and
V
the
are
tb
eter
!OXj1" −!OX"
αs (ζ10without
!OX" +the
ζ10and
!OYV")
.the(10)
appropriate
(CKM)
matrix
elements.
Thefornonperturb
superscript
MS
stands
for Cabibbo-Kobayashi-Maskawa
the matrix
without
the
superscript
MS stands
the from
matrix
tb !OX"
th
(CKM) matrix elements.
nonperturbative QCD in2 The
2
fro
M
(CKM)
matrix
elements.
The
nonperturbative
QCD
element
input
theGlowest
theory.
lowest
element inMS
thestands
lattice
theory.
Even
at
order
into
the
combination
Thein
∗ formula
2inB Even
2 ˆin-order
Flattice
W this
put
into
this
formula
is
the
combination
f 2isB̂at
with
OX" without the superscript
for
the
matrix
0142(13)
0229(18)
0139(11)
0026(3)
0140(20)
0081(12)
0012(4)
Total
Results
15 %
TABLE III: Results for the square root of quantities listed in
E. (15).
Dalgic,
al, Phys.
D 76, 011501,
Theetunhatted
bagRev.
parameters
are given hep-lat/0610104
at scale µ = mb .
Errors quoted are combined statistical and systematic errors.
Note that percentage errors here are smaller than those given
in Table II by a factor of two due to the square root.
ble I for our two
mbined statistical
etails on our fits
show results for
ions (after power
seven operators.
finds the physical
or !OL", 11% for
nts in the lattice
the RHS of (10).
2/a) =Theory
0.32 [16].
which is close to
using other heavy
Experiment
coefficients
ρXY
e µ through the
u/d sea quark mass
fBs
fBs
mf /ms = 0.25
mf /ms = 0.50
B̂Bs [GeV]
0.281(21)
0.289(22)
BBs (mb ) [GeV]
0.227(17)
0.233(17)
!
!
fBs
fBs
√
√
BS (mb )
R
[GeV]
0.295(22)
0.301(23)
B̃S (mb )
R
[GeV]
0.305(23)
0.310(23)
−1
∆ms = 20.3 ± 3.0
±
0.8
ps
2
(LQCD)(Vts)
usually also taken as 1 × αs coming from higher order
matching of the heavy-light current. We avoid unnecessarily
separating
the bag
and possibly
∆m
17.77
± parameters
0.10 ± 0.07
ps−1 in-(stat)(syst)
s = out
troducing ambiguities in error estimates by always working with the relevant combination f 2 B . Several years
combination log( mµb ). We present results for µ = mb .
Error budget
We evaluate the RHS of (10) for !OL"MS , !OS"MS and
!O3"MS and combine with the definitions in (4) - (6) to
obtain the main results of this article, namely,
fB2 s
BBs ,
fB2 s
BS
,
2
R
fB2 s
B̃S
.
2
R
(15)
The main errors in these quantities are listed in Table 3
II. One sees that the two dominant errors are due to
eory for fixed
b and
statistics
+ fitting
higher
order
matching
uncertainTABLE and
II: Error
budget
for quantities
listed
in (15).
ght u, d sea
quark
ties.
We have also included a nonnegligible error coming
ors.
+ Fitting
9%
from the uncertainty Statistical
in the scale
(lattice
spacing) for
Higher Order Matching 9 %
the
MILC
ensembles
used.
At the final stage of extractms = 0.50
Discretization
43 %2
1069(92) ing results for (15), one has to convert a fBs MBs into
Relativistic
3%
0687(61) physical units. An uncertainty
Scale (a−3 ) of ∼ 1.8%5 in
% the lattice
0142(13) spacing turns at this point into a ∼ 5% uncertainty for
Total
15 %
0229(18) a−3 . In Table II we take the operator matching error to
0139(11)
be 1 × α2s since matching is done directly for the combi0026(3)
nation fB2 sTABLE
BBs (and
for the
other
quantities
in (15)).
III: Results
for the
square
root of quantities
listedAin
0140(20)
2 given at scale µ = mb .
(15). The
unhatted
bag parameters
are
naive
attempt
to
deal
separately
with
f
and BBs in the
B
s
0081(12)
Errors quoted are combined statistical and systematic errors.
formula
for
∆Ms could
increase the error estimate since
0012(4)
Note that
percentage errors here are smaller than those given
is
the perturbative
error
for just
fBdue
in Table II
by a factor
of two
to the (unsquared)
square root.
s alone
ing with the rel
ago reference [2
with physical
bag parameter
Table III give
quantities liste
combination fB
mass dependen
any reasonable
uncertainties w
do not attempt
quark mass an
best determina
particular, this
crucial nonper
quoted in the a
f
Using (16) on
for ∆Ms base
standard value
mixing in full lattice QCD using NRQCD b quarks
Christine T.H. Davies
New data for ratios
Xs / Xq coarse
1.35
Xs / Xq fine
1.3
!s / !q coarse
!s / !q fine
1.25
1.2
!
Xq ≡ fBq BBq MBq
!
Φq ≡ fBq MBq
1.15
1.1
1.05
1
0.95
0.1
0.2
0.3
0.4
0.5
mq / m s
Xs
Xq
=
√
f Bs BBs MBs
√
f Bq BBq MBq
and !s /!q =
√
f Bs MBs
√
f Bq MBq
as a function of the light valence quark mass in the
r. Errors are only statistical
all the quantities
plotted.
E. in
Gámiz,
et al., PoS(Lattice
2007)349, arXiv:0710.0646
Bs and Bd mixing in full lattice QCD using NRQCD b quarks
Christine T.H. Davies
Sea quark/discretization effect for Bs
0.38
Coarse lattice (old results), no 1/M correct.
Coarse lattice (new results), no 1/M correct.
Fine lattice, no 1/M corrections
(GeV)
0.36
RGI (1/2)
0.34
Improved precision
due to smearing
0.3
s
fB (BB
s
)
0.32
0.28
0.26violation
10% scaling
0
0.1
0.2
0.3
0.4
0.5
0.6
mq / ms
!
Figure 1: fBs B̂Bs (GeV ). Errors are only statistical.
E. Gámiz, et al., PoS(Lattice 2007)349, arXiv:0710.0646
Tightening Vtd constraint
(a) (ρ, η) from Δmd only
1.4
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
0.8
ρ
2σ band
1.2
η
η
1.4
2σ bands
after HPQCD fB
before
1.2
(b) (ρ, η) from Δmd and Δms
1
1.2
1.4
0
0
0.2
0.4
0.6
0.8
ρ
1
1.2
1.4
mNRQCD & Form Factors
L. Khomskii, R. R. Horgan, S. Meinel, L. Storoni, M.W.
(Cambridge)
C. T. H. Davies, et al (Glasgow)
A. Hart & E. Müller (Edinburgh)
(part of the HPQCD Collaboration)
Motivation
Increase precision in form factors by extending q2 range
Increase list of observables (lesson from inclusive/exclusive Vub)
More direct focus on standard vs. nonstandard FCNC
Complement future progress on LHC measurements of, e.g.
B → K ∗γ
B → K ∗ µ+ µ−
Rare B decays
u,d
B
b
K∗
t
s
W
Physical point q2 = 0
γ
Progress w/ mNRQCD?
Independent way to get Vts
Also Vtd, but worry about
weak annihilation contrib.
Full set of form factors
s:
s
a
m
ator
Matrix element
ct
e
p
s
e
ng
a
h
φ decay(s)
C
Form factor
Relevant
→
Bs
!P |q̄γ µ b|B"
f+ , f0
B → π"ν
B → K!+ !−
!P |q̄σ µν qν b|B"
fT
B → K!+ !−
!V |q̄γ µ b|B"
V
µ
5
!V |q̄γ γ b|B"
A0 , A1 , A2
!V |q̄σ µν qν b|B"
T1
!V |q̄σ
µν
5
γ qν b|B"
T2 , T3
{
B → (ρ/ω)#ν
B → K ∗ !+ !−
{
B → K ∗γ
B → K ∗ !+ !−
Summary
LQCD calculations of B decay/mixing matrix elements contribute
to the study of physics beyond the Standard Model
Taken approaches which allow us to address all systematic errors
simultaneously (within 4th root hypothesis)
List of postdictions and predictions having positive impact in flavor
physics community
Vub from B→π semileptonic decay consistently lower and in better
agreement with CKM fits (sin 2β) than inclusive B semileptonic decays
Chiral extrapolation of 4-quark operators underway
Further improvement of actions, mNRQCD, automated lattice
perturbation theory
Many alternatives which will check and probably do better in the
future. Need balance between perfection and timeliness.