Lattice QCD
meets experiment
Christine Davies
University of Glasgow
HPQCD collaboration
Warwick University
March 2011
Thursday, 17 March 2011
QCD is a key part of the Standard Model but quark
confinement complicates things.
CDF
QCD only tested to 5-10%
level at high energies from
comparison of e.g. jet
phenomena to pert.th.
But properties of hadrons
calculable from QCD if fully
nonperturbative calc. is done can test QCD and determine
parameters very accurately (1%).
Thursday, 17 March 2011
ALEPH
q emits W
and changes
�
q
to
DALI
CKM
element
Vqq�
Run=16449
�
q and q
annihilate
CKM
element Vqq�
Rates for simple weak or em quark
processes inside hadrons also calculable,
but not multi-hadron final states.
YX
|0
D0<5
NT=4
Thursday, 17 March 2011
ALEPH
Bs → Dse−ν
(DS → K +K −π+)
!e!
IP
+
K
!
Bs
+
Ds
|0
0.1cm|
Compare to
exptl rate
gives Vqq�
accurately
Evt=4055
1cm|
!+
!
K
Solving a path integral: quantum mechanical case
Solve Schrödinger’s eq. for eigenvalues/fns of H or:
discretise time and integrate over
all paths possible weighted by eiS
xi
S=
xf
ti
a
tf
�
dtL;
1
L = mẋ2 − V (x)
2
classical path is mẍ = V �
qm path fluctuates about this.
In Euclidean time� solve numerically, by making sets of x(ti)
< x(t2 )x(t1 ) >=
�
Dx x(t2 )x(t1 )e−S
−(En −E0 )(t2 −t1 )
�
A
e
=
n
−S
Dxe
n
average over ‘ensemble’ of paths paths chosen with prob. e-S
further reading: G.P.Lepage, hep-lat/0506036
Thursday, 17 March 2011
fit as fn of time can extract
excitation energies - amps are
transition matrix elements
Solving a path integral: QCD
Now path integral over gluon and quark fields on a 4-d
space-time lattice - quarks anticommute so do by hand.
LQCD
�
1
2
= T rFµν + ψ(γ · D + m)ψ
2
= a huge matrix, M
−SQCD
DU DψDψO(ψ,� ψ)e
DU O(M
Integral over gluon
fields only
→
−1
valence quarks
inc. in operator
< O >=< H(t)H (0) >=
†
ensemble average
Thursday, 17 March 2011
�
)e
−(Sg −ln(det M ))
complicated prob,
distn for gluons - inc.
effects of sea quarks
−En t Fit as fn of t to
An e
get hadron mass
and transn amp.
n
a
Thursday, 17 March 2011
Lattice QCD = fully nonperturbative
QCD calculation
RECIPE
• Generate sets of gluon fields for
Monte Carlo integrn of Path Integral
(inc effect of u, d and s sea quarks)
• Calculate averaged “hadron
correlators” from valence q props.
• Fit for masses and simple matrix
elements
• Determine a and fix mq to get
results in physical units.
• extrapolate to a = 0, mu,d = phys
for real world
Lattice results need to be extrapolated to the real world
where a=0 and mu/d = small.
To do this
well needs:
r1 uncertainty
• statistical
a2 extrapol’n
finite volume
precision
mu/d extrapol’n
• small
disc.
statistical
errors
ms evolution
errors
and
md , QED, annhil’
several
Total
values of a
TABLEm
II:
• small
u/dError
various decay cons
em effects
must be
estimated !
Thursday, 17 March 2011
using
HISQ
fits so that uncer
charm
quarks
a2 are
correctly r
E. Follana, CTHD
et al,
a2 extrapolations
HPQCD, 0706.1726
tion (1% or less f
Including u, d and s sea quarks is critical for accurate results,
but numerically expensive - particularly light mu,d.
Latt./expt
Latt./expt
f%
fK
m$
mN
mD
s
mD
mD* - mD
s
s
m" - m#
c
"(1P-1S)
2mB -m!
s,av
mB
c
!(3S-1S)
!(2P-1S)
!(1P-1S)
!(1D-1S)
0.9
1
1.1
Quenched
Thursday, 17 March 2011
HPQCD/
MILC
2008 “ratio
plot”.
Multiple
values of a,
and of mu,d.
Extrapolate
to physical
point.
0.9
1
1.1
with sea quarks
update of:
C. Davies et al,
hep-lat/0304004
m/2min / GeV2
Example parameters for calculations now being done.
Lots of different formalisms for handling quarks.
Volume
of
0.2
min
MILC imp. staggered, 2+1
lattice also an
mass
RBC/UKQCD DW, 2+1
PACS-CS, clover, 2+1
issue - need
of u,d
BMW, stout clover, 2+1
quarks 0.15
(2.5fm)4 or
more
real
world
0.1
mu,d ≈ ms/5
0.05
mu,d ≈ ms/10
mu,d ≈ ms/27
0
0
0.005
0.01
0.015
a2 / fm2
Thursday, 17 March 2011
0.02
0.025
0.03
The gold-plated meson spectrum - HPQCD 2009
12
hb(2P)
10
MESON MASS (GeV/c2)
2008
d’b
db
[’’
[’
[
hb(1P)
8
rb2
rb1(2P)
rb0
rb2
rb1(1P)
rb0
CDF
2005
UNFLAVORED
6
4
new prediction
HPQCD
0909.4462
FLAVORED
Bc
Bs
B
’
dc
dc
s’
J/s
hc
rc2
rc1
rc0
Ds
D
2
0
experiment
fix parameters
postdictions
[(1D) predictions
d
/
K
B*c
B*s
*
B
HPQCD
1008.4018
error 3 MeV
- em effects
important!
I. Allison et al, hep-lat/0411027, A. Gray et al, hep-lat/0507013
Thursday, 17 March 2011
Determining quark masses
Lattice QCD has direct access to
parameters in Lagrangian for
accurate tuning
- issue is converting to contnm
schemes such as M S
Vus
10000
Quark masses (MeV/c2)
Can now rule out some quark
mass matrix models
PDG (CKM)
HPQCD/MILC quark mass
PDG quark mass
0.2
Vcb
0.21
2010 PDG
2010 HPQCD
mb (mb ) = 4.165(23)GeV
c
1000
100
10
0.22
0.23
0.24
0.25
mc (mc ) = 1.273(6)GeV
ms (2GeV) = 92.2(1.3)MeV
s
md (2GeV) = 4.77(15)MeV
d
u
1
0.028
0.032
0.036
0.04
Vub
0
b
0.044
Vcb =
0.002
Thursday, 17 March 2011
0.004
0.006
0.008
�
mu (2GeV) = 2.01(10)MeV
md /mb
C. McNeile,CTHD,
HPQCD, 1004.4285
vs PDG
C. McNeile, 1004.4985, model from
Chkareuli+Froggatt, hep-ph/9812499
Strong convergence of lattice results for strange quark
mass this year.
to
PDG
130
MeV
MILC ’09
HPQCD ’10
RBC/KEK/Nagoya ’10
RBC/UKQCD ’10
BMW ’10
80
90
100
ms
MS(2 GeV)
(MeV)
Lattice world average = 93.6(1.1) MeV
Thursday, 17 March 2011
110
Determining αs
Lattice QCD now has several determns of αs to 1%.
[ decays
Key points:
• high statistical
precision
• high order pert. th.
exists and can
estimate higher orders
• higher twist not a
significant issue
• approaches very
different - good test
o decays
DIS [F2]
DIS [e,p -> jets]
e+e-[jets shps]
electroweak
e+e-[jets shps]
HPQCD: wloops
HPQCD: heavy q corrs
JLQCD: light q. vac. poln
World average:
Bethke 0908.1135
0.11
see 2011 Munich
alphas workshop
Thursday, 17 March 2011
0.115
0.12 0.125 0.13
_s(MZ)
CTHD et al,HPQCD 0807.1687;
1004.4285;JLQCD,1002.0371.
Determining the Cabibbo-Kobayashi-Maskawa matrix


Vud
Vus
Vub
V
us
 π → lν K → lν B → πlν 




K
→
πlν


 Vcd
 ν
V
V
cs
cb


 D → lν Ds → lν B → Dlν
µ


D → πlνD → Klν

K


 Vtd
Vts
Vtb 
Expt = CKM x theory(QCD)
�Bd |Bd � �Bs|Bs�
Vab appears in trivial way in decay processes involving
quarks a + b. Calculating QCD effects is non-trivial - need
precision lattice QCD to get accurate CKM results.
If Vab known, compare lattice to expt to test QCD.
Thursday, 17 March 2011
Recent CKM highlights
Vus
fK
0.15
µ
ν
fπ
0.1
K
Exp’t
0.1
0.2
0.3
0.4
0.5
mu/d /ms
K decay constant parameterises QCD effects in leptonic
decay.- calculate in lattice QCD. Combine with expt.(KLOE)
(current world’s best but theory still
Vus = 0.2262(14) dominates error)
test of first row unitarity
2
2
2
1 −Vud −Vus −Vub = 0.0006(8) of CKM matrix
Follana, CTHD et
al HPQCD, 0706.1726
Thursday, 17 March 2011
HISQ action allows us to
study c meson decays also
Vcs
ν
Follana, CTHD et al, HPQCD, heplat/0610092, 0706.1726;
CTHD et al, HPQCD, 1008.4018
µ
0.28
Ds
0.27
0.265
0.26
s
fD / GeV
fDs =
0.2480(25)GeV
0.275
1% error!
0.255
0.25
update from 2007
value of 0.241(3) GeV
because of recalibration
of lattice spacing
Thursday, 17 March 2011
0.245
0.24
0
0.005
0.01
2
0.015
2
a / fm
0.02
0.025
320
Calculate Ds decay
constant - compare to
experiment if assume Vcs
known
expt !"i
expt o"i
300
HPQCD, 1008.4018
260
HFAG
expt av.
s
fD (MeV)
280
240
HPQCD-2010
+ current expt
HPQCD-2010
+ future BES III
60
220
full lattice QCD
2-flavor lattice QCD
200
2005
2006
2007
2008
3m
exclusion
50
2009
2010
3m
exclusion
2011
New physics would give
smaller decay constant than
Standard Model - allows limits
on charged Higgs.
Thursday, 17 March 2011
tan `
40
30
20
LEP direct exclusion
10
50
100
150
200
250 300
mH+/- (GeV)
350
400
450
500
Semileptonic form factors
q emits W
and changes
to q`
dΓ
2
2
∝
|V
|
f
(q
)
cs
+
dq 2
D → Klν
f0 (q 2 )
relate to
expt
Vqq�
CKM
element
With HISQ quarks
no renormalisation issues
and good statistics
possible
f0 (q = 0) =
2
2.5% !!
f+ (q 2 = 0) =
0.747(11)(15)
Na, Shigemitsu et al, HPQCD, 1008.4562
Thursday, 17 March 2011
Comparison to experimental leptonic and semileptonic
•! |Vcs|allows
from D semileptonic
and Ds leptonic
rates
direct determination
of decays
Vcs
0.961(26)
error dominated by
theory (now 2.5%)
HPQCD semileptonic (2010)
0.961 (26)
1.010(22)
error dominated by
expt (now 2%)
HPQCD leptonic (2010)
1.010 (23)
0.7
0.8
0.9
|Vcs|
1
1.1
Aim:•! Used
similar
accuracy
elements
from
B meson
new HFAG
included for
new CKM
BaBar results
for leptonic
decays
decay and mixing rates
Thursday, 17 March 2011
New work: calculating shape of f+ (q )
2
J. Koponen, CTHD, in progress
1.5
1.4
Comparison
to expt will
provide
more
detailed test
of QCD.
Note how
form factor
same for
different
processes all
involving
Ds -> ds
D -> K
dc -> Ds
1.3
f+
2
f (q )
1.2
1.1
1
f0
0.9
0.8
0.7
0.6
-0.2
f+ (0) = f0 (0)
0
0.2
0.4
2 2
q a
Thursday, 17 March 2011
0.6
0.8
1
c→s
decay.
New work: mapping shape of decay constants and form
factors as a function of heavy quark mass from charm
upwards - needs very fine lattices ...
a = 0.09f m
a = 0.15f m
a = 0.06f m
a = 0.12f m
a = 0.045f m
c
. McNeile, CTHD, in progress
Thursday, 17 March 2011
b
Compare to nonrelativistic
results at b
New physics sensitivity in neutral B mixing
Neutral B (and K) mix - gives rise to ‘oscillations’. Mixing
determined by box diagram. Calculate in lattice QCD
VtdVtb∗
B0
Parameterise with
B0
2
fB BB
�
=
where fB is decay
�constant.
fBs BBs
Using ratio
√
ξ
=
for Bs to Bd
fB BB
Lattice QCD results (HPQCD)
using NRQCD b quarks:
7% normln error cancels in ratio.
Thursday, 17 March 2011
HW
Vtd
| |=ξ
Vts
ΔMd MBs
ΔMsMBd
ξ = 1.26(3)
E. Gamiz, CTHD et al,
HPQCD, 0902.1815
Use this to provide SM rate for LHCb of:
Br(Bs → µ+ µ− ) = 3.19(19) × 10−9
6% error from taking
ratio to mixing rate
Bs!µµ: LHCb
reach in 2011
E. Gamiz et al,
Improvement
needed, in
progress ....
0902.1815[hep-lat]
BS!µµ exclusion @ 95% CL
100
345"!#$%&&'*+6(7/&
."!#$%&&&'0),,12&.#!!&/&
!"!#$%&'()*+,-&.#!!/&
3"!#$8&'1-6&.#!!9/&
10
37 pb-1
1
0.0
0.2
500 pb-1
0.4
0.6
0.8
1 fb-1
1.0
1.2
1.4
1.6
1.8
2.0
New LHCb limit
LanFranchi,
LaThuile 2011
Now working to
improve lattice QCD
result ..
With the data collected in 2011 we will be able to explore the very
Thursday, 17 March 2011
Lattice QCD calculations are key Ruth
to constraining
sides of
S. Van de Water
CKM Unitarity triangle
d flavor physics from lattice QCD
current situation
tensions developing
at 2 sigma level.
The CKM matrix and flavor physics from lattice QCD
Ru
Future with 1%
lattice errors for
of the CKM unitarity triangle [14] . The current fit is consistent with the Standard Model
The constraints from εK , |Vub |/|Vcb| , ∆Ms /∆Md , and ∆Md are all limited by theoretical
attice QCD.
BK , fK
�
ming the Standard
the majority of inputs to κ
ξ, fBsModel,B
Bs
are well-known from experning unknown, ImA2 , can be obtained from lattice QCD calculations of K → ππ
n the ∆I = 1/2 channel. The only N f = 2 + 1 flavor determination of this quantity
from exclusive SL decay
cb
ub
d UKQCD collaborations [31], and has rather large systematic errors associated
Figure
9: Potential
impact ofthe
future
lattice determinations
J.Laiho
et perturbation
al, 0910.2928;
www.latticeaverages.org
ding-order
chiral
theory.
Nevertheless,
contribution
to Eq. (3.11)on the global unitarity triangl
ical errors in all of the lattice QCD inputs are reduced to 1% with the central values fixe
ll Thursday,
and leads
to only
17 March
2011a ∼ 1% uncertainty in κε [14]:
ε
V ,V
the piecesError
of the Budget
Bayesian error that depend upon the prior
A VeryareGood
to the error
widths budgets
in those extrapolations.
‘‘ms evolution’’
(one
omission)
Full error
now available
forrefers
lattice
calcs
in running the quark masses to the same scale from different a
stats
values for the chiral extrapolation. The r1 uncertainty comes
tuning from the error in the physical value of r1 , and the finite volume
uncertainty allows for a 50% error in our finite volume adjustchiral
!q = 2mDq
ments
described
in
the
text.
continuum
ns as a
lues of
ones in
aneous
olation
overall
e offset
, mu !
experi-
fK =f! fK
f!
fDs =fD fDs
fD
!s =!d
r1 uncerty.
a2 extrap.
Finite vol.
mu=d extrap.
Stat. errors
ms evoln.
md , QED, etc.
0.3
0.2
0.4
0.2
0.2
0.1
0.0
1.1
0.2
0.4
0.3
0.4
0.1
0.0
1.4
0.2
0.8
0.4
0.5
0.1
0.0
0.4
0.4
0.3
0.2
0.5
0.3
0.1
1.0
0.5
0.1
0.3
0.6
0.3
0.0
1.4
0.6
0.3
0.4
0.7
0.3
0.1
0.7
0.5
0.1
0.2
0.6
0.5
0.5
Total %
0.6
1.3 1.7 0.9
1.3
1.8 1.2
charmed sea
062002-3
– m"c
<< 0.5%?
will tell you what is possible in future
e.g. is error from disc. errors, mu,d extrapoln, stats ...
Monday, April 26, 2010
Thursday, 17 March 2011
41
Conclusion
• very accurate results are available now from lattice QCD
for QCD parameters and for simple hadron masses and
decay matrix elements important for flavour physics.
Future
• sets of ‘next generation’ gluon configs will have
mu,d at physical value (so no extrapoln) or
a down to 0.03fm (so b quarks are ‘light’) or
much higher statistics (for harder hadrons)
also can include charm in the sea now.
• Pushing errors down to 1% level will mean em
corrections and mu �= md must be understood.
• some harder calculations (flavor singlet, excited states,
nuclear physics) will also become possible
Thursday, 17 March 2011