Lattice Gauge Theory

Home , Quenched approximation

Hartmut Wittig

Oxford University

EPS High Energy Physics 99

Tamp ere, Finland 20 July 1999

Intro duction

Lattice QCD provides a non-p erturb ati ve

framework to compute relations between SM

parameters and exp erimental quantities from rst

principles

Formulate QCD on a euclidean space-time lattice

with spacing a and volume L T

a gauge invariant

S S 1

G F

][d ] e h i = Z [dU ][d

! allows for sto chastic evaluation of h i

Ideally want to use lattice QCD as a

phenomenologi ca l to ol, e.g.

M f

s Z

m + m 2 GeV m

u s

However, realistic simulation s of lattice QCD are

hard... 1

1 Lattice artefacts cuto e ects

lat cont p

h i = h i + O a

a 1 4 GeV , a 0:2 0:05 fm

! need to extrap olate to continuum limit: a ! 0

! cho ose b etter discretisati ons: avoid small p.

2 Inclusion of dynamical quark e ects:

S [U ] lat

Z = [dU ] e det D + m

lat

detD + m =1: Quenched Approximation

! neglect quark lo ops in the evaluation of h i.

cheap

exp ensive 2

Scale ambiguity in the quenched approximation:

Q [MeV ]

;::: ; Q = f ;m ;m ; a [MeV ]=

3 Restrictions on quark masses:

a m L

! cannot simulate u; d and b quarks directly

! need to control extrap olation s in m

4 Chiral symmetry breaking:

Nielsen & Ninomiya 1979: exact chiral symmetry

cannot b e realised at non-zero lattice spacing

! imp ossibl e to separate chiral and continuum

limits 3

Outline:

I. The lattice Dirac op erator

{ Improved actions

{ Exact chiral symmetry on the lattice

II. Simulations with dynamical quarks

{ Light hadron sp ectrum, quenched & unquenched

III. Light quark masses

{ Non-p erturba tive renormalisat ion

{ Recent estimates

IV. Glueballs & heavy hybrids

V. Omissions

VI. Summary 4

I. The lattice Dirac op erator

Massless free fermions:

S = a x D x y y

x;y

a D x y is lo cal

b D p=i p + O ap

c D p is invertible for p 6=0

d D + D =0

5 5

! Nielsen & Ninomiya: a{d do not hold

simultaneous ly

! either left with doublers

or chiral symmetry broken explicitly

Staggered Kogut-Susskind fermions

Wilson fermions

1 1

ar r + r r D =

2 2

- degeneracy fully lifted; chiral symmetry broken

- leading cuto e ects of order a p =1 5

O a improved Wilson fermions

Can one nd discretisation for Wilson fermions

with reduced cuto e ects?

Symanzik improvement: Remove leading lattice

artefacts by adding a counterterm to the Wilson-

Dirac op erator Sheikholeslami-Wohlert,:::

c F x D = D + m +

sw SW W 0

Fix c g through suitable improvement

sw 0

condition

! leading lattice artefacts are O a in sp ectral

quantities

For n =0 and n =2, c has b een determined

f f sw

non-p erturba ti vely M. L uscher et al. 1997;

R. Edwards et al. 1997; Jansen & Sommer 1998. 6

Scaling of m for m =m = 0:7 T. Klassen

V PS V

1998

Non-p erturbati vel y improved Wilson action:

scaling b ehaviour is consistent with residual a

artefacts

Can push Symanzik improvement to higher orders

Alford, Klassen, Lepage 1998

Chiral symmetry remains broken 7

Exact chiral symmetry on the lattice

Chiral symmetry breaking may be tolerated in

many applica tion s of QCD

Chiral symmetry is crucial for non-p erturb ati ve

formulation of

- EW theory

- SUSY

Prop osals to circumvent the NN Theorem:

{ Domain Wall Fermions

Kaplan 1992; Shamir, Furman 1993

{ Overlap Formalism Narayanan, Neub erger, 1993{98

{ \Perfect Actions"

Hasenfratz, Niedermayer, :: :, 1993{98

{ Ginsparg-Wil son relation

Ginsparg & Wilson 1982, L uscher 1998{99

See also:

T. Blum, Lattice 98

M. Luscher

Lattice 99

H. Neub erger 8

Domain Wall Fermions Shamir & Furman

Intro duce discrete extra 5 dimension :

x x

1 N

x x

1 N

4 k

L ;D x; y x x;

N ;D

DWF k ?

D x; y =D x; y + x y D

0 0

ss ss

D x; y : Wilson-Dirac op erator with mass M .

D : contains Dirac mass m.

For m =0; N !1:

- no fermion doublers

- chiral mo des are trapp ed in 4-dim. domain

walls at either end

In practice: work at nite N

! decouplin g of chiral mo des not exact

{ terms which break chiral symmetry are

exp onential ly suppressed 9

Pion mass:

2 N

am = C am + ae

Blum, Soni, Wingate, hep-lat/9902016, N =10

! Can realise almost exact chiral symmetry at non-

zero a at the exp ense of simulating 5-dim. theory

{ N 10 30 may b e sucient?

s 10

Ginsparg-Wi l so n Relation

Replace condition D + D =0 by the weaker

5 5

relation Ginsparg & Wilson 1982

D + D = aD D

5 5 5

If D satis es GWR, then this implies an exact

global symmetry L uscher 1998

1 1

aD ; aD = 1 = 1

5 5

2 2

! \lattice chiral symmetry"

! Flavour-singlet case: Ward identitie s have correct

anomaly

Construction of gauge theories with lo cal chiral

symmetry Ab eli an and non-Ab elia n relies on

GWR Luscher 1998{99

Overlap Formalism Narayanan, Neub erger

Reinterpretation of the domain wall prop osal;

Overlap op erator Neub erger 1998:

1=2

D = 1 A A A ; A =1 aD

N W

D : massless Wilson-Dirac op erator

! satis es GW-relation 11

Witten's SU2 anomaly on the lattice

Witten 1982: SU2 gauge theory coupled to

a single left-handed fermion is mathematic all y

inconsistent:

Z Z

S [A] D

Z = [dA]e ][d ] e [d

Weyl

1=2 S [A]

= [dA] det D [A] e

There is a non-trivial gauge transformation U x

such that

1=2 U 1=2

det D [A] = det D [A ]

Exp ectation value w.r.t. Z is indetermi na te:

h i =\0/0"

D hermitian: eigenvalues are real and come in

pairs,

Vary gauge eld smo othly from A to A :

A t=1 t A + tA ; t 2 [0; 1]

Atiyah-Singer: numb er of eigenvalues that b ecome

negative as t is varied from 0 to 1 is odd. 12

Compute the eigenvalues of the overlap op erator

D in a lattice simulation, as A is smo othly

O. Bar, Lattice 99 turned into A

D : eigenvalues come in complex conjugate pairs;

exp ect level crossing for Im

0.2

0.1

-0.1

-0.2

0.4 0.5 0.6

Crossing observed for lowest eigenvalue

! Numerical pro of of Witten's SU2 anomaly

Only p ossible if lattice Dirac op erator has the

correct chiral prop erties 13

II. Simulations with dynamical quarks

Quenched light hadron sp ectrum deviates from

exp eriment CP-PACS Collab oration, hep-lat/9904012:

Light Hadron Spectrum in Quenched QCD final results from CP−PACS

1.8

1.6 Ξ Ω 1.4 Σ Λ Ξ* φ 1.2 N Σ*

m (GeV) K* 1.0 ∆

0.8 experiment K K input φ input 0.6

0.4

m m to o small by 10 16 4 6

K K

m =m =1:143 0:033 Exp.: 1:22; 2:5

Quenched QCD describ es the light hadron

sp ectrum at the level of 10. 14

Can sea quark e ects account for the deviation of

the quenched sp ectrum from exp eriment?

Oa Improvement even more imp ortant:

{ exp ensive to simulate very small lattice spacings

{ need to separate lattice artefacts from sea quark

e ects

Recent simulatio ns with n =2

Collab oration GFlops Gluons Quarks

CU/BNL/Riken US 250 Wilson DWF

CP-PACS J 300 Iwasaki SW, tad

UKQCD UK 28 Wilson SW, n.p.

SESAM/T L D/I 14 Wilson Wilson

MILC US 7:3 LW staggered

! identify two light avours with physical u; d quarks 15

Observables in full QCD with n =2 avours:

Example: meson propagator:

val

sea

; ;

5 j 5 j

Sea quarks are still relatively heavy:

sea val

m = m :

m m

0:67 0:86 UKQCD

= ; =0:169

0:6 0:8 CP-PACS

m m

sea

Study the dep endence of observables on m ;

sea

Extrap olate in m to physical m =m

Incorrect n for Kaon physics:

val sea

! cho ose m >m

val sea

;m = m and identify m = m

u;d s

! \partially quenched" 16

Sea quark e ects in the light hadron sp ectrum:

Study the continuum limit of hadron masses in

\partially quenched" QCD

CP-PACS, T. Kaneko, Lattice 99

K* and φ masses, K input

RC (combined fits) 1.05 χ Φ quenched PW (q PT fits)

1.00

0.95

m [GeV] K* 0.90

0.85

0.80 0.0 0.2 0.4 0.6 0.8 1.0 1.2

a [GeV−1]

Mesons: discrepancy with exp erimental sp ectrum

diminishe d when sea quarks are \switched on"

Quantify sea quark e ects in the continuum limit

Can remainin g di erences b e blamed on

- incorrect n =2 for strange hadrons?

- continuum extrap olation s?

- extrap olation s in the quark masses? 17

III. Light Quark Masses

Recent progress:

{ controllin g lattice artefacts improvement

MS scheme { matching lattice results to

{ results from simulatio ns using DWF's

{ estimating sea quark e ects

Current quark masses:

+ 2

m + m h0ju sjK i m f =

u s 5 K

u s = Z g ;a u s

5 P 0 5 lat

Z g ;a = 1+ ln a+C + O g

P 0

lattice p erturbati on theory do es not converge well

scale dep enden ce complicates the situation

! no controlled p erturbati ve relation between

MS hadronic lattice scheme and

! develop non-p erturb ati ve matching techniques 18

Intro duce an intermedia te renormalisati on scheme:

hadronic lattice

MS scheme

scheme, am

lat

ref

non-p erturbative p erturbative

Intermediate

m scheme:

1 Regularisation Indep enden t Martinelli, Sachrajda,...

q q j q i =1 Z g ;a h q j

5 P 0

lat

G:F:

MS and RI schemes related through continuum p erturbation

theory

2 Schrodinger Functional scheme L uscher et al.

Finite volume scheme; imp ose renormalisa tion

condition at scale =1=L :

Z g ;a ; =1=L ' 275 MeV

P 0 0 0 0

Z g ;a computed non-p erturb ati vely for

P 0 0

a 0:1 0:045 fm for O a improved Wilson

action 19

Compute m =M for 200 GeV

SF 0

M : renormalisa tion group invariant quark mass

Scale evolution of m =M quenche d QCD

Capitani, L uscher, Sommer, HW, Nucl. Phys. B544

1999 669

For = 275 MeV :

=1:15715

Combine with matching factor Z g ;a and lattice

P 0 0

matrix elements

M + M M m

s l

+O a =

lat

f m Z g ;a h0jP jK i

K SF 0 P 0 0

M =M + M =2

l u d 20

Continuum extrap olati on of M + M =f :

s l K

Garden, Heitger, Sommer, HW, hep-lat/9906013

Continuum limit; f = 1602 MeV :

M + M = 140 5 MeV

s l

Convert into 2 GeV using m

M =M =24:4 1:5 Chiral P.T.

s l

=M =0:7208 at =2GeV m

! Final result: all errors, except quenching

2 GeV =97 4 MeV m

10 uncertain ty due to scale ambiguity in the

quenched approximation. 21

Other recent results in the quenched

MS scheme, =2GeV approximation

ref

Collab. m m a ! 0 Z Impr.

l s

BNL/Riken/CU 5.14 12910 DWF

p p p

QCDSF 4.409 1043

CP-PACS 4.5518 1152

p p

JLQCD 4.2329 1067 stagg.

Blum, Soni

9526 DWF

& Wingate

p p

Becirevic

4.54 11112

et al.

:preliminary

Caution:

Systematic errors not estimated uniformly

Conversion into physical units using di erent

quantities 22

Sea quark e ects in m ;m :

l s

Last year: very large e ects due to dynamical

quarks?

1999: b etter separation of lattice artefacts and

sea quark e ects

Recent results for n =2 by CP-PACS and MILC

Collab oration s partially quenched

: CP-PACS, VWI : CP-PACS, AWI

: MILC, VWI

Light quark masses decrease for n =2

CP-PACS: m =m 25 in continuum limit

s l

sea

Further exploration of systematics required m -

dep endenc e, a-e ects, renormali sati on 23

IV. Glueballs and Heavy Hybrids

Idea: use anisotropic lattices: a a

t s

Exp onential decayofcorrelation function governed

by a M :

a M t=a y

t t

e O ~x; tO 0

For a a : slow exp onential fall-o , whilst

t s

preserving large spatial volumes

Typically: a 0:2 0:4fm

a =a =3 5

s t

! Cuto e ects in spatial lattice spacing may

be large; use Symanzik improvement to reduce

leading lattice artefacts