international school of subnuclear physics erice 2006 status of lattice qcd richard kenway

International School of Subnuclear Physics

Erice 2006 Status of Lattice QCD

Richard Kenway

Richard Kenway

Status of Lattice QCD

2


Erice 2006

Parameters of QCD

Richard Kenway


3


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,

,,det1

4

1

,,1

1

ffff

G

Nf

ffff

mADeAO

ASemADDA

Z

qmADqFFeqqAOqDDqDA

ZO

f

N

i

iUON

O1

1

xiagA

exU

Lattice QCD

q(x) U(x)

U(x)a

2ctmlattice

,

,,2

or

ˆˆTrRe2

aOaOSSSS

xqxqmyqUDxqS

xUxUxUxUg

S

FG

xxy

yxF

xG

Richard Kenway


4


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


5


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


6


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


7


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


8


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


9


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


10


Erice 2006

strong coupling lattice QCD provides a precise determination

Richard Kenway


11


Erice 2006

quark masses high precision is being achieved for light quarks

– but there are systematic differences between lattice formulations

staggered Wilson

Richard Kenway


12


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


13


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


14


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


15


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


16


Erice 2006

QCDOC

> 6 teraflops sustained (BNL)> 3 teraflops (Edinburgh)

Richard Kenway


17


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


18


Erice 2006

… and dynamical sea quark effects are seen string breaking

Richard Kenway


19


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


20


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


21


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


22


Erice 2006

Status

Richard Kenway


23


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


24


Erice 2006

Parameters of the Standard Model

Richard Kenway


25


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


26


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


27


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


28


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


29


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


30


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


31


Erice 2006

π and K leptonic decays

2+1 flavours staggered (MILC)– full χPT analysis

Richard Kenway


32


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


33


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


34


Erice 2006

neutral K mixing and K CP violation in K

indirect

direct

even odd

CPCPL KKK

indirect CP violation

Richard Kenway


35


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


36


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


37


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


38


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


39


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


40


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


41


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


42


Erice 2006

… and beyond?

Richard Kenway


43


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


44


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


45


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


46


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


47


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


48


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


49


Erice 2006

Status

Richard Kenway


50


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

international school of subnuclear physics erice 2006 status of lattice qcd richard kenway

Documents

lattice qcd lattice

lattice qcd g

lattice theory

lattice size

parameters of qcd slide

qcd scale simulations

scaling slide

black box slide