Karl Jansen - BLTP JINR Home Pagetheor.jinr.ru/~diastp/summer14/lectures/Jansen_part1.pdf · 2014. 8. 26. · Karl Jansen Introduction to Lattice QCD II • Task: compute proton mass

Karl Jansen

Introduction to Lattice QCD II

• Task: compute proton mass

✤ need an action

✤ need a supercomputer

✤ need an observable

✤ need an algorithm

• the muon anomalous magnetic moment

Quantum Chromodynamics

Z =Z

DAµD D ̄e�Sgauge�Sferm

gauge action

Fermion action

Pathintegral

The actionsfermion action

Sferm =

Zd

4x ̄(x) [�µDµ +m] (x)

Dµ (x) ⌘ (@µ � ig0Aµ(x)) (x)

Sgauge =

Zd

4xFµ⌫Fµ⌫ , Fµ⌫(x) = @µA⌫(x)� @⌫Aµ(x) + i [Aµ(x), A⌫(x)]

Aµ

g0

gauge action

m

gauge covariant derivative

gauge potential

coupling quark mass

QCD had to be invented

asymptotic freedom

world of quarks and gluons

distances

As Wilson said it:

Unfortunately, it is not known yet whether the quarks in quantum chromodynamics actually form the required bound states. To establish whether these bound states exist one must solve a strong coupling problem and present methods for solving field theories don't work for strong coupling. (Wilson, Cargese Lecture notes 1976)

Going to the lattice

(x) ̄(x) x = (t,x) integers

S ! a4X

x

̄(x) [�µ

@

µ

� r @2µ|{z}

r⇤µrµ

+m] (x)

rµ (x) =1

a

[ (x+ aµ̂)� (x)]

r⇤µ (x) =1

a

[ (x)� (x� aµ̂)]

quark fields

discretized action

@µ !1

2

⇥r⇤µ +rµ

⇤

Nielsen-Ninomiya theorem

The theorem simply states the fact that the Chern number is a cobordism invariant (Friedan)

clash between chiral symmetry and fermion proliferation

for any lattice Dirac operator D

• D is local; bounded by •

Ce��/a|x|

D̃(p) = i�µpµ +O(ap2)

• D is invertible for all p 6= 0• chiral symmetric: �5D +D�5 = 0cannot be fulfilled simultaneously

Introduce group valued fields

Gauge fields

U(x, µ) 2 SU(3)

U(x, µ) = exp(iaA

bµ(x)T

b) = 1 + iaA

bµ(x)T

b+ . . .

relation to gauge potential

U(x, µ)U(x+ aµ, ⌫)� U(x, ⌫)U(x+ a⌫̂, µ) = ia2Fµ⌫(x) +O(a3)

Fµ⌫(x) = @µA⌫(x)� @⌫Aµ(x) + i[Aµ(x), A⌫(x)

� = 6/g2

Sw(U) =X

⇤�

⇢1� 1

3ReTr(U⇤)

�a ! 0 1

2g2a4

X

x

Tr(Fµ⌫

(x)2) +O(a6)

discretization of field strength tensor

lattice action

Physical observables

hOi = 1ZZ

fieldsOe�S

| {z }

# lattice discretization

01011100011100011110011

#

Monte Carlo Method

hf(x)i =Z

dxf(x)e�x2

) hf(x)i ⇡ 1N

X

i

f(xi)

xi, i = 1, · · · , Nintegration points

taken from distribution e�x2

error / 1/pN

QMC method (maybe later): / 1/N

There are dangerous animals

Another look at the Wilson actionS = SG|{z}

O(a2)

+Snaive| {z }O(a2)

+Swilson| {z }O(a)

linear lattice artefacts:need small lattice spacingsneed large volumes

want L=aN = 1fm

clover-improved Wilson fermions

maximally twisted mass Wilson fermions

overlap/domainwall fermionsexact lattice chiral symmetry

killing O(a) effects

S = ̄ [m+ �µDµ]

! ei!�5⌧3/2 ̄! ̄ei!�5⌧3/2

m ! mei!�5⌧3 ⌘ m0 + iµ�5⌧3

m =pm02 + µ2 , tan! = µ/m

! = 0

! = ⇡/2 : S = ̄ [iµ�5⌧3 + �µDµ]

Realizing O(a)-improvementcontinuum action:

axial transformation

polar mass m and twist angle !

standard QCD action

(maximal twist)

Repeat this on the lattice

Dtm = mq + iµ⌧3�5 +1

2�µ

⇥rµ +r⇤µ

⇤� ar1

2r⇤µrµ

difference to continuum situation: Wilson term not invariant under axial transformations

"mq + i�µ sin pµa+

r

a

X

µ

(1� cos pµa) + iµ⌧3�5

#�1

/ (sin pµa)2 +"mq +

r

a

X

µ

(1� cos pµa)#2

+ µ2

lima!0

: p2µ +m2q + µ

2 + amqX

µ

pµ

| {z }O(a)

mq = 0 (! = ⇡/2) kills O(a) effects

free fermion propagator

hOi|(mq,r)

= [⇠(r) + amq⌘(r)] hOi|contmq + a�(r) hO1i|cont

mq

hOi|(�mq,�r) = [⇠(�r)� amq⌘(�r)] hOi|

cont

�mq + a�(�r) hO1i|cont

�mq

R5 ⇥ (r ! �r)⇥ (mq ! �mq) , R5 = ei!�5⌧3

1

2

hhOi|mq,r + hOi|�mq,r

i= hOi|contmq +O(a2)

hOi|mq=0,r = hOi|cont

mq+O(a2)

A general argument

Symanzik expansion

Symmetry

special case: mq = 0

automatic O(a) improvement through mass averaging

A test at tree-level

0.9993

0.9994

0.9995

0.9996

0.9997

0.9998

0.9999

1

0 0.01 0.02 0.03 0.04 0.05 0.06

N•mps

1/N2

MAXIMAL TM: N•mq = 0.5, r = 1.0data

fit

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

0 0.05 0.1 0.15 0.2 0.25

N•mps

1/N

|N•mq| = 0.5, r = 1.0data +

fit +data -

fit -

positive and negative mass mass average

1/N / a 1/N2 / a2

a lattice spacing

N lattice size

mPS “Pion Mass00

Exact lattice chiral symmetry

Starting point: Ginsparg-Wilson relation

�5D +D�5 = 2aD�5D

) D�1�5 + �5D�1 = 2a�5for any lattice Dirac operator D, satisfying the

Ginsparg Wilson relation, the action

S = ̄D

is invariant under

� = �5(1�1

2aD) , � ̄ = ̄(1� 1

2aD)�5

one solution of GW-relation

Dov

=⇥1�A(A†A)�1/2

⇤

Neuberger’s overlap operator

A = 1 + s�Dw(mq = 0)advantages of overlap operator

index theorem fulfilled

often trivial renormalisation constants

continuum like behaviourdis-advantage

computationally very expensive

timings on PC cluster

No free lunch theorem

V,m⇡ Overlap Wilson TM rel. factor124, 720Mev 48.8(6) 2.6(1) 18.8124, 390Mev 142(2) 4.0(1) 35.4164, 720Mev 225(2) 9.0(2) 25.0164, 390Mev 653(6) 17.5(6) 37.3164, 230Mev 1949(22) 22.1(8) 88.6

ActionsACTION ADVANTAGES DISADVANTAGES

clover improved Wilson computationally fast breaks chiral symmetryneeds operator improvement

twisted mass fermions computationally fast breaks chiral symmetryautomatic improvement violation of isospin

staggered computationally fast fourth root problemcomplicated contraction

domain wall improved chiral symmetry computationally demandingneeds tuning

overlap fermions exact chiral symmetry computationally expensive

All actions O(a)-improvement:

hOlattphys

i = hOlattcont

i+O(a2)

In the following: twisted mass fermions