karl jansen - bltp jinr home pagetheor.jinr.ru/~diastp/summer14/lectures/jansen_part1.pdf · 2014....
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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
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Quantum Chromodynamics
Z =Z
DAµD D ̄e�Sgauge�Sferm
gauge action
Fermion action
Pathintegral
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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
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QCD had to be invented
asymptotic freedom
world of quarks and gluons
distances
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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)
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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
⇤
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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
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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
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Physical observables
hOi = 1ZZ
fieldsOe�S
| {z }
# lattice discretization
01011100011100011110011
#
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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
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There are dangerous animals
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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