international school of subnuclear physics erice 2006 status of lattice qcd richard kenway
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International School of Subnuclear Physics
Erice 2006 Status of Lattice QCD
Richard Kenway
Richard Kenway
Status of Lattice QCD
2
International School of Subnuclear Physics
Erice 2006
Parameters of QCD
Richard Kenway
Status of Lattice QCD
3
International School of Subnuclear Physics
Erice 2006
lattice QCD
g2 and mf are fundamental parameters of the Standard Model
– computable in a complete theory … a test of BSM theories
– but quarks are confined … emergent complexity
Euclidean space-time lattice regularisation– lattice spacing a, lattice size L
Monte Carlo approximation to path integral– N gauge configurations
0,
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1
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1
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Lattice QCD
q(x) U(x)
U(x)a
2ctmlattice
,
,,2
or
ˆˆTrRe2
aOaOSSSS
xqxqmyqUDxqS
xUxUxUxUg
S
FG
xxy
yxF
xG
Richard Kenway
Status of Lattice QCD
4
International School of Subnuclear Physics
Erice 2006
lattice QCD lattice spacing must be extrapolated to zero keeping box large enough
– by approaching a critical point
a
L
quark masses+
gauge coupling
Lattice QCD
properties
of hadrons
think of the computer as a ‘black box’
Richard Kenway
Status of Lattice QCD
5
International School of Subnuclear Physics
Erice 2006
QCD scale
simulations use dimensionless variables (lattice spacing = 1)– quark masses, mf, and gauge coupling, g2, are varied
hadronic scheme
– at each value of g2, fix quark masses mf by matching Nf hadron mass ratios to
experiment
– one dimensionful quantity fixes the lattice spacing in physical units
lab
1
computer
1 MeV 938
1spacing lattice
sizenucleon
a
M N
– dimensionless ratios become independent of g2 if a is small enough (scaling)
Richard Kenway
Status of Lattice QCD
6
International School of Subnuclear Physics
Erice 2006
renormalisation
treelatlat
intlat 22
,
p
paaaZ
4
2
latg
a
MS
provided y,pert theor
QCDMS
intMS
lat
1 provided numerical,
latintlat
int
QCD
,
,
OZ
aOaaZO
a
convert matrix elements to a perturbative scheme (matching)– eg to combine with Wilson coefficients in an OPE
impose mass-independent renormalisation conditions at p2 = 2
or use step scaling– let 1 = L, the linear box size– consider a sequence of intermediate renormalisations at box sizes Ln = 2n L0
0
numerical
011
providedy,pert theor
RGI
RGI
,...,
lim
QCD
1
LOLLLLZ
OO
nn
L
L
n
n
RGI
providedy,pert theor
MSRGI
MS
QCD
OZO
Richard Kenway
Status of Lattice QCD
7
International School of Subnuclear Physics
Erice 2006
β function continuum limit: a → 0 with L constant and large enough
– tune β = 6/g2 → ∞ holding low-energy physics constant
non-perturbative β function
f
f
Nb
Nb
gbgbg
gg
3
38102
4
1
3
211
4
1
...
41
20
51
30
Richard Kenway
Status of Lattice QCD
8
International School of Subnuclear Physics
Erice 2006
continuum limit of the quantum theory
symmetries of the lattice theory define the universality class Lorentz invariance is an “accidental” symmetry as a → 0
– there are no relevant operators to break it
confinement– gauge invariance is preserved at the sites of the lattice
– there is no phase transition into an unconfined phase as mf, g are tuned to the critical line (a → 0)
chiral symmetry can be realised correctly– Ginsparg-Wilson formulations realise the full chiral symmetry at a ≠ 0
– flavour symmetry can be realised in full, but is broken by some formulations
Osterwalder-Schrader conditions (reflection positivity)– sufficient for a Lorentz invariant QFT
– generally not proven, especially for improved actions
Richard Kenway
Status of Lattice QCD
9
International School of Subnuclear Physics
Erice 2006
non-perturbative running
compute α(μ) at mq(a) = 0 for a sequence of box sizes 1 = L in the limit a → 0
match with perturbation theory at a high scale
PCAC quark masses
scheme hadronicmatching
lat,MS
lat,MS
scheme veperturbati
MS
221 or O
0
0
amaZ
Zm
aaaOxP
aOxAamm
q
P
Aq
b
b
Richard Kenway
Status of Lattice QCD
10
International School of Subnuclear Physics
Erice 2006
strong coupling lattice QCD provides a precise determination
Richard Kenway
Status of Lattice QCD
11
International School of Subnuclear Physics
Erice 2006
quark masses high precision is being achieved for light quarks
– but there are systematic differences between lattice formulations
staggered Wilson
Richard Kenway
Status of Lattice QCD
12
International School of Subnuclear Physics
Erice 2006
fermion doubling the covariant derivative as a difference operator
2
1
ˆ
N
,ˆ,,
,,ˆ,
UD
xU
xU
yxyxyx
yxyxyx xU ˆ xU
x ̂x̂x
(naïve) free fermion Dirac operator in momentum space
,,,0,0,0,0 points 16at 0
sinN
ppD
16 (= 2d) degenerate fermion species – couple to axial current with alternating signs so U(1) axial anomaly cancels
– giving a fully regularised theory with chiral symmetry
potential disaster for lattice QCD!– different lattice fermion actions to deal with this are the main reason for
different systematic errors in lattice calculations
05NN5 DD
Richard Kenway
Status of Lattice QCD
13
International School of Subnuclear Physics
Erice 2006
‘good’ lattice fermions
change the chiral transformation on the lattice
– where D satisfies the Ginsparg-Wilson relation
– this is a symmetry of the action
– there are several local solutions for D with smooth enough gauge fields
– eg Ls limit of 5-dimensional domain wall fermions
explicitly break chiral symmetry by adding a dimension 5 operator
2
1WD
– gives doublers masses cut-off, leaving p = 0 pole unchanged
– mixes operators of different chirality, complicating renormalisation
– requires fine tuning to get mud = 0 (at Mπ = 0)
52
25
1
1
Di
Di
a
a
DDaDD 555
0 D
(Wilson)
NRR
LNL
L
ssRsLss
mPP
mPP
PDPDDSs
20
112
2
11W1WWDWF 111
5/ 12
1 LRP
Richard Kenway
Status of Lattice QCD
14
International School of Subnuclear Physics
Erice 2006
‘ugly’ lattice fermions staggered fermions
– lattice action may be diagonalised in spinor space
– keep only one spinor component 1 =
– 2d/2 continuum species = ‘tastes’ (4 tastes in d = 4)
– U(1) remnant of chiral symmetry prevents additive mass renormalisation
QCD with N degenerate quarks– N is a parameter in simulation algorithms
rooted staggered quarks– use one staggered fermion per flavour and take the fourth root of the
determinant
– cannot be described by a local theory … lose universality
– non-locality/non-unitarity is a lattice artefact which vanishes as a → 0, provided the quark mass is not taken to zero first
– remains in the same universality class as QCD
01
10
01
10
xxd
xd
x
d
d
xx
xx
x
xx xxmxxxS
ˆˆ2
11 10
S
mUDeqddq NqmUDq )(det)(
41
SSSrootS detdetdet sdu
S mDmDmDeDUZ G
Richard Kenway
Status of Lattice QCD
15
International School of Subnuclear Physics
Erice 2006
computational cost big algorithmic improvements
over the past two years– chiral regime and/or physically
quark masses now seem reachable
DWF
65MeV20fm1.0
fm3operations # s
ud
Lma
L
Richard Kenway
Status of Lattice QCD
16
International School of Subnuclear Physics
Erice 2006
QCDOC
> 6 teraflops sustained (BNL)> 3 teraflops (Edinburgh)
Richard Kenway
Status of Lattice QCD
17
International School of Subnuclear Physics
Erice 2006
quenched QCD is wrong
quenched QCD sets N = 0– an early calculation expedient – avoids the costly determinant
– omits virtual quark-antiquark pairs in the vacuum
– provides a good phenomenological model, often good to 10% level
– dynamical quark effects enter through renormalised quantities
mUDeqddq NqmUDq )(det)(
MN = 900 (100) MeVHamber & Parisi 1982
Richard Kenway
Status of Lattice QCD
18
International School of Subnuclear Physics
Erice 2006
… and dynamical sea quark effects are seen string breaking
Richard Kenway
Status of Lattice QCD
19
International School of Subnuclear Physics
Erice 2006
topological susceptibility
– χPT with experimental fπ
QCD vacuum isosurfaces of positive (red) and
negative (green) topological charge density using
xxDxQ ,Tr 5
instanton
2
top V
Q
Richard Kenway
Status of Lattice QCD
20
International School of Subnuclear Physics
Erice 2006 lattice relates dn to θ
– simulations must sample topology well and contain light dynamical quarks with correct chiral symmetry
– in quenched QCD dn is singular in the chiral limit
handle complex action for θ small by experimental measurements
2 flavour DWF, a−1 = 1.7 GeV
– our 2+1 flavour simulations sample topology much better
vacuum angle θ QCD allows a gauge
invariant CP odd term– CKM phase contributes
< 10−30 e cm to dn
1115 10 cm e 102
nd
00
QOiOO
Richard Kenway
Status of Lattice QCD
21
International School of Subnuclear Physics
Erice 2006
effective theories
Lüscher finite
volume effective theory
a << QCD-1
Symanzikeffective
field theory
HQET/NRQCD
chiral perturbation theory
lattice QCDa, mq, mQ, L
QCD scale QCD
lattice QCDa, mq, mQ, L
QCD scale QCD
L >> QCD-1
mQ >> QCD
mq << QCD
simulations at physical parameter values are too expensive– use effective field theories to extrapolate simulation results from parameter
regimes where systematic errors can be controlled to the physical regime
Richard Kenway
Status of Lattice QCD
22
International School of Subnuclear Physics
Erice 2006
Status
Richard Kenway
Status of Lattice QCD
23
International School of Subnuclear Physics
Erice 2006
the lattice is well established as a rigorous non-perturbative regularisation scheme for QCD– correctly realises all internal symmetries
– has the correct continuum limit
– may be applied to other QFTs … chiral gauge theories, SUSY, BSM
non-perturbative renormalisation– running couplings and matching to MS
matching to effective theories defines QCD at all parameter values– all sources of uncertainty can be systematically controlled
simulations are computationally tractable– dramatic recent progress in developing faster algorithms
– renewed confidence that physically light quarks are within reach
visualisation may yet yield insight– explore topological structures and dominant fermionic modes
as a theoretical tool
Richard Kenway
Status of Lattice QCD
24
International School of Subnuclear Physics
Erice 2006
Parameters of the Standard Model
Richard Kenway
Status of Lattice QCD
25
International School of Subnuclear Physics
Erice 2006
2-point functions
easily computed quantities
n n
E
H
E
eOn
OeO
OOOO
n
20ˆ
0ˆˆ0
00ˆˆT00
2
ˆ
decays asymptotically
with energy of lightest
state created by O
determines matrix elements such as
PS0 2501PSPS qqZfiM A
3-point functions
functionspoint -2 fromfunctionspoint -2 from
,
ˆˆ
12
0ˆ2
ˆ2
ˆ0
0ˆˆˆ00,,
112
112
KnE
enqOn
E
enp
KeqOepKqOp
n
E
n
E
nn
HH
nn
at large time separations, 2 >> 1 >> 0,
can isolate matrix elements such as
qpKqusp
but there is no general method for multi-hadron final states eg K
Richard Kenway
Status of Lattice QCD
26
International School of Subnuclear Physics
Erice 2006
finite size effects
‘rule of thumb’ = keep lattices big enough
χPT gives the correct functional dependence on volume for the pseudoscalar meson mass– but underestimates FSE by an
order of magnitude (Wilson Nf = 2)
L ~ 2.5 fm needed for FSE below few % for 300 MeV pions
3P LM
Richard Kenway
Status of Lattice QCD
27
International School of Subnuclear Physics
Erice 2006
quenched hadron spectrum ‘tour de force’ demonstration of the power of lattice QCD
– glueballs – nucleon excited states
– mixing with flavour-singlet mesons is a major challenge for 2+1 flavours
– requires flavour symmetry and spatially extended operators
Richard Kenway
Status of Lattice QCD
28
International School of Subnuclear Physics
Erice 2006
QCD hadron spectrum
prediction of the Bc mass– 2+1 flavours + relativistic effective
action (c) + NRQCD (b)
inputs to quark mass and scale setting
Edinburgh plot – 2+1 flavours DWF
Richard Kenway
Status of Lattice QCD
29
International School of Subnuclear Physics
Erice 2006
flavour physics, CKM and lattice QCD 3 generations
unitary
tbtstd
cbcscd
ubusud
VVV
VVV
VVV
0*** tdtbcdcbudub VVVVVV
*udubVV *
tdtbVV
*cdcbVV
CP
search for new physics by over-constraining the unitarity triangle– vastly improved
experimental accuracy
– lattice uncertainties dominate
Richard Kenway
Status of Lattice QCD
30
International School of Subnuclear Physics
Erice 2006
leptonic decays
elegant example of lattice ↔ experiment interaction access to Vxy
cross-check of fX prediction
22
2
2
222
18 XX qqX
X
llXF
X
Vfm
mmmGlXB
qx
qx ℓ
ℓ
pfipXAZ XA 00
Richard Kenway
Status of Lattice QCD
31
International School of Subnuclear Physics
Erice 2006
π and K leptonic decays
2+1 flavours staggered (MILC)– full χPT analysis
Richard Kenway
Status of Lattice QCD
32
International School of Subnuclear Physics
Erice 2006
D leptonic decays CLEO-c (2005)
measured D → μν
409.012.0 1066.040.4
DB
Vcd, τD from PDG 04
BaBar and CLEO-c (2006) measured Ds → μν
c)-(CLEO MeV716282
(BaBar) MeV14717283
sDf
c)-(CLEO 03.011.026.1
c)-O(BaBar/CLE 14.027.1
flavours) 1(2 07.024.1
D
D
f
fs
experimental and lattice uncertainties are similar ~ 10%
sea quark effects are not significant
Richard Kenway
Status of Lattice QCD
33
International School of Subnuclear Physics
Erice 2006
B leptonic decays
sensitive to charged Higgs
b
u
W H-
b
u
lattice cut-off is too small to simulate both b and ud quarks directly– simulate relativistic b in small volumes
… step scaling to large volume
– use an effective heavy quark action … continuum limit non-trivial
sea-quark effects increase fBs by 10-15%
first direct measurement of fB (Belle 2006)
418.016.0
34.028.0 1006.1
BB
flavours) 1(2 MeV 22216
(Belle) MeV176 2019
2823
B
f
Vub, τB from PDG 04
flavours) 1(2 1320.1 B
B
f
fs
Richard Kenway
Status of Lattice QCD
34
International School of Subnuclear Physics
Erice 2006
neutral K mixing and K CP violation in K
indirect
direct
even odd
CPCPL KKK
indirect CP violation
Richard Kenway
Status of Lattice QCD
35
International School of Subnuclear Physics
Erice 2006
indirect CP violation quenched QCD 2+1 flavour
– a ~ 0.125 fm
(RBC-UKQCD, preliminary)
next year should see the first realistic determinations of BK
stat quench
Richard Kenway
Status of Lattice QCD
36
International School of Subnuclear Physics
Erice 2006
direct CP violation
P(1/2) is dominated by
in quenched QCD this mixes with unphysical operators, requiring additional low-energy constants– the resulting ambiguity means we cannot calculate ε'/ε reliably
the resolution is to use 2+1 flavours in the sea
quenched QCDCP-PACS: -7.7 2.0RBC: - 4.0 2.3
Richard Kenway
Status of Lattice QCD
37
International School of Subnuclear Physics
Erice 2006
Bd and Bs mixing
measurement of ΔMBs allows a theoretically well-controlled estimate using
qqq BBB
*tbtqWtBW
Fq B̂fMVVMmSM
GM 2222
02
2
2
6
22
3
8
055
0 11
BB
BMf
BqbqbBB
neutral Bq meson mass difference– BSM physics could enter loops
dd
ss
d
s
BB
BB
td
ts
B
B
d
s
Bf
Bf
V
V
M
M
M
M
ˆ
ˆ,2
2
2
3)lat/051011-hep flavours, 1(2 1.21
(PDG06) 98390.0
0.0470.035-
s
d
B
B
M
M
(theory) (expt) 208.0 0.0080.006-
001.0002.0
ts
td
V
V
Richard Kenway
Status of Lattice QCD
38
International School of Subnuclear Physics
Erice 2006
semileptonic decays
access to Vxy
form factor embeds q2 dependence– more elaborate example of lattice ↔ experiment interaction
CKM-independent checks of lattice QCD from studying
e+
W+
qx
qx qx
qy
Vxy
XB
XB
Richard Kenway
Status of Lattice QCD
39
International School of Subnuclear Physics
Erice 2006
Kl3 decays
2+1 flavour– a ~ 0.125 fm (UKQCD-RBC,
preliminary)
f+(0) from lattice QCD should allow a precise determination of |Vus|
Richard Kenway
Status of Lattice QCD
40
International School of Subnuclear Physics
Erice 2006
semileptonic D / Kℓ decays
ucVppq
qfqq
MMppqfq
q
MMpDVp DD
,
22
222
02
22
model-independent form factors from lattice QCD– hadron momenta must be small to avoid large discretisation errors
– maximum recoil ~ 1 GeV, so lattice data span full kinematic range
– |Vcs| is well measured
– precision test of lattice form factors against CLEO-c data
2+1 flavours
c s, d
W
leptons
D → e+, with Vcd = 0.2238 D→Ke+, with Vcs = 0.9745
07.003.073.00
f
KeD 06.003.064.00
f
eD
lattice
CLEO-c
lattice
CLEO-c
Richard Kenway
Status of Lattice QCD
41
International School of Subnuclear Physics
Erice 2006
semileptonic B ℓ decays and |Vub| no symmetry: only lattice QCD can fix the normalisation
– lattice kinematic range is restricted to near zero recoil, high q2
experiment (2005):
Richard Kenway
Status of Lattice QCD
42
International School of Subnuclear Physics
Erice 2006
… and beyond?
Richard Kenway
Status of Lattice QCD
43
International School of Subnuclear Physics
Erice 2006
impact of lattice QCD on flavour physics
lattice QCD needs greater precision to be phenomenologically relevant
ICHEP 06: new physics has not shown up– Bs oscillations fully consistent with SM
– flavour physics, including CP violation is governed by CKM (at least predominantly)
Richard Kenway
Status of Lattice QCD
44
International School of Subnuclear Physics
Erice 2006
muon g-2 promising place to look for new physics
– must compute SM contributions very accurately
– leading order hadronic contribution
staggered χPT, a = 0.09 fm– lattice uncertainty ~ 3 × experimental
decay 108.28.05.0711.0
104.25.9692.4
(lattice) 1015721
10
10
10hadron 2
ee
a
10exp 10608592116 a
ignored
(small)
Richard Kenway
Status of Lattice QCD
45
International School of Subnuclear Physics
Erice 2006
rare and forbidden decays constraints can come from rare decays
but both involve QCD matrix elements
and forbidden decays
q~b s
-
q
q q
X
b s
t
W+ + ...
QCD
12,modelamplitude hOhMf X
Richard Kenway
Status of Lattice QCD
46
International School of Subnuclear Physics
Erice 2006
B K* occurs at 1 loop in SM
contribution from virtual sparticles
neglected in recent lattice QCD studies
– must extrapolate to q2 = 0 where c(3) = 0 and T1(0) = T2(0)
q~b s
-
+ ...
b s
t
W + ...
3
1
251
ii
i* qTcpBbs'pK
(expt) 104420
(lattice) 1016290BR4
4
.
.KB *
Richard Kenway
Status of Lattice QCD
47
International School of Subnuclear Physics
Erice 2006
GUTs & SUSY proton lifetime
– SuperKamiokande (~100 kt y)– 1 kt = 1033 protons
colour triplet Higgsino exchange (dim 5)
years 103.2 33 Kp
when dressed by sparticles gives proton decay
qq q~
q~ q~
q~
T~
T~M
1
q~q~q antisymmetric in flavour
q~
q~Z~
,W~
q
q q dominant decay mode is to strange mesons
Richard Kenway
Status of Lattice QCD
48
International School of Subnuclear Physics
Erice 2006
dimension 6 baryon-number violating operators constrained by SM symmetries– matrix elements from lattice QCD
provide model-independent input to SUSY-GUT lifetime estimates
– related by chiral perturbation theory to
– large uncertainty from lattice scale
proton decay
2
19
~2
latt
333
GeV 101.5
GeV 012.0 years 103.2
T
MKp
3lattlatt GeV 2012.00
fm 0.12 QCD,flavour 2
pqqq
a
SUSY-GUT MT (GeV) minimal SU(5) 2 1014 ruled out already
SU(5) with “natural” doublet-triplet splitting
3 1018
minimal SO(10) 6 1019 MLSP > 400 GeV
Richard Kenway
Status of Lattice QCD
49
International School of Subnuclear Physics
Erice 2006
Status
Richard Kenway
Status of Lattice QCD
50
International School of Subnuclear Physics
Erice 2006
as a phenomenological / discovery tool the theoretical control that has been established in principle must be turned
into higher precision in practice– the determination of some CKM parameters is now limited by the precision of
lattice QCD
– operator mixing need be no worse than in the continuum, extending the range of matrix elements that can be computed reliably
some constraints on BSM physics are possible at existing levels of precision– by computing all SM matrix elements, eg for proton decay, B→K*γ
– by bounding hadronic uncertainties in well-known parameters, eg muon g-2
all the theoretical and computing technology required for this exists– there is greater confidence than for many years
beyond lattice QCD?– different representations/gauge groups, scalar fields (Higgs), massless
fermions (SUSY) …
– no obstacles in principle