Download 8. Quantum field theory on the lattice

8. Quantum field theory on the lattice
8.1. Fundamentals
• Quantum field theory (QFT) can be defined using Feynman’s path
integral [R. Feynman, Rev. Mod. Phys. 20, 1948]:
Z Y
Z =
Z
S =
x
4
dφ(x) exp[iS(φ)]
(1)
d xL(φ, ∂tφ)
Here x = (x0, x1, x2, x3), and gµν = diag(+,-,-,-)
• Physical observables can be evaluated if we add a source S →
S + Jx φx . For example, 1- and connected 2-point functions are:
∂ 1Z
hφx i =
[dφ]φx exp[iS]
log Z =
i∂Jx J=0
Z
∂
∂
1Z
hφx φy i =
[dφ]φx φy exp[iS] − hφx i2
log Z =
i∂Jx i∂Jy
Z
1
• These can be computed using perturbation theory, if the coupling
constants in S are small (see any of the oodles of QFT textbooks).
• However: if
– 4-volume is finite, and
– 4-coordinate x is discrete (x ∈ aZ 4 , a lattice spacing),
the integral in (1) has finite dimensionality and can, in principle, be
evaluated by brute force ( = computers)
7→ gives fully non-perturbative results.
• Need to extrapolate: V → ∞, a → 0 in order to recover continuum
physics.
2
• But:
- the dimensionality of the integral is (typically) huge
- exp iS is complex+unimodular: every configuration {φ}
contributes with equal magnitude
7→ extremely slow convergence in numerical computations
(useless in practice).
• This is (largely) solved by using imaginary time formalism (Euclidean spacetime).
• Imaginary time also admits non-zero temperature T .
3
Units:
Standard HEP units, where
c = h̄ = kB = 1,
and thus
[length]−1 = [time]−1 = [mass] = [temperature] = [energy] = GeV.
Mass m = rest energy mc2 = (Compton wavelength)−1 mc/h̄.
(1 GeV)−1 ≈ 0.2 fm
4
8.2. Path integral in imaginary time
Consider scalar field theory in Minkowski spacetime:
S=
Z
3
d x dt L(φ, ∂tφ) =
Z
3
d x dt
h
µ
1
2 ∂µ φ ∂ φ
− V (φ)
i
We obtain the (classical) Hamiltonian by Legendre transformation:
H =
=
Z
Z
d3 x dt π φ̇ − L
d3 x dt
h
h
1 2
2π
i
+ 12 (∂iφ)2 + V (φ)
i
Here π is canonical momentum for φ: π = δL/δ φ̇.
Quantum field theory: consider Hilbert space of states |φi, |πi, and field
operators (φ → φ̂, π → π̂, H → Ĥ), with the usual properties:
•
φ̂(x̄)|φi = φ(x̄)|φi
•
Z Y
•
•
[
x
Z Y
dφ(x̄)]|φihφ| = 1
[
x
dπ(x̄)/(2π)]|πihπ| = 1
[φ̂(x̄), φ̂(x̄0)] = −iδ 3 (x̄ − x̄0 )
hφ|πi = exp [i
Z
d3 xπ(x̄)φ(x̄)]
5
Time evolution operator is U (t) = e−iĤt
Feynman showed that the quantum theory defined by Ĥ and the Hilbert
space is equivalent to the path integral (1). We shall now do this in
imaginary time.
Let us now consider the quantum system in imaginary time:
t → τ = it ,
exp[−iĤt] → exp[−τ Ĥ]
This makes the spacetime Euclidean: g = (+ − − −) → (+ + + +).
We shall keep the Hamiltonian Ĥ as-is, and also the Hilbert space.
Let us also discretize space (convenient for us but not necessary at this
stage), and use finite volume:
Z
x̄ = an̄,
n i = 0 . . . Ns
3X
dx → a
3
x
∂i φ →
1
[φ(x̄ + ēi a) − φ(x̄)] ≡ ∆iφ(x̄)
a
6
Thus, the Hamiltonian is:
Ĥ =
2
3X 1
a
2 [π̂(x)]
x
+
2
1
2 [∆iφ̂(x̄)]
+ V [φ̂(x̄)]
Let us now consider the amplitude hφB |e−(τB −τA )Ĥ |φA i:
1) divide time in small intervals: τB − τA = Nτ aτ . Obviously
hφB |e−(τB −τA )Ĥ |φA i = hφB |(e−aτ Ĥ )Nτ |φA i.
2) insert 1-operators
Z Y
[
x
dφi]|φi ihφi |,
Z Y
[
x
dπi ]|πi ihπi |
hφB |e−τ Ĥ |φA i =
Z


N
Yτ Y
i=2 x

dφi(x) 
N
Yτ Y
i=1 x

dπi (x) hφB |πNτ ihπNτ |e−aτ Ĥ |φNτ i
×hφNτ |πNτ −1ihπNτ −1 |e−aτ Ĥ |φNτ −1i . . . × hφ2 |π1 ihπ1 |e−aτ Ĥ |φA i
7
Thus, we need to calculate
hπi |e
−aτ Ĥ
|φi i =
×
"
3
X 1 2
exp −a aτ
2 πi
x
hπi |φi i + O(a2τ )
+
2
1
2 (∆i φi )
+ V [φi ]
#
and
ia3
P
πi (x̄)φi+1 (x̄) −ia3
P
x πi (x̄)φi (x̄)
e
hφi+1|πi ihπi |φi i = e
X
= exp[ia3aτ πi (x̄)∆0φi (x̄)]
x
x
where we have defined ∆0φi =
We can now integrate over πi (x̄):
Z Y
[
=
x
"
1
(φi+1 − φi ).
aτ
πi (x̄)] exp[a3 aτ
2π
a3 aτ
#N 3 /2
S
X
x
− 21 πi2 + (i∆0φi )πi]
× exp[−a3 aτ
8
X1
2
2 (∆0 φi (x̄)) ]
x
(2)
Repeating this for i = 1 . . . Nτ , and identifying the time coordinate x0 =
τ = aτ i, we finally obtain the path integral
hφB |e
−τ Ĥ
Z Y
|φA i ∝ [
x
dφ]e−SE
where SE is the Euclidean (imaginary time) action
X 1
( 2 (∆µφ)2
x
Z
4
d x( 21 (∂µφ)2 +
SE = a 3 aτ
→
+ V [φ])
V [φ])
and we have the boundary conditions φ(τA ) = φA , φ(τB ) = φB .
The path integral is precisely in the form of a partition function for a
4-dimensional classical statistical system, with the identification SE ↔
H/(kB T ).
For convenience, we make the system periodic in time by identifying
φA = φB and integrating over φA .
9
We can now make a connection between the correlation functions of the
“statistical” theory and the Green’s functions of the quantum field theory.
First, note that we can interpret Tφi+1 ,φi = hφi+1 |e−aτ Ĥ |φi i as a transfer
matrix. In terms of T the partition function is
Z
Z = [dφ]e−SE = Tr (T Nτ )
Let us label the eigenvalues of T by λ0 , λ1 . . ., so that λ0 > λ1 ≥ . . .. Note
that λi = exp −Ei , where Ei are eigenvalues of Ĥ. Thus, λ0 corresponds
to the state of lowest energy, vacuum |0i. If we now take Nτ to be very
large (while keeping aτ constant; i.e. take ∆τ to be large),
Z=
X
i
Nτ
Nτ
τ
λN
i = λ0 [1 + O((λ1 /λ0 ) )]
For example, a 2-point function can be written as (let i − j > 0)
1Z
1
[dφ]φiφj e−SE = Tr (T Nτ −i+j φ̂T i−j φ̂).
hφi φj i =
Z
Z
Taking now Nτ → ∞, and recalling aτ (i − j) = τi − τj ,
hφi φj i = h0|φ̂(T /λ0)i−j φ̂|0i = h0|φ̂(τi)φ̂(τj )|0i,
10
where we have introduced time-dependent operators
φ̂(τ ) = eτ Ĥ φ̂e−τ Ĥ .
Allowing for both positive and negative time separations of τi and τj , we
can identify
hφ(τi )φ(τj )i = h0|T [φ̂(τi )φ̂(τj )]|0i,
where T is the time ordering operator.
Mass spectrum:
• Green’s functions in time τ :
1Z
h0|Γ(τ )Γ (0)|0i =
[dφ]Γ(τ )Γ†(0)e−S
Z
= h0|eĤτ Γ(0)e−Ĥτ Γ† (0)|0i
†
= h0|Γ(0)
11
X
n
|En ihEn |e−Ĥτ Γ† (0)|0i
=
X −E τ
n
e
n
−E0 τ
→ e
|h0|Γ(0)|Eni|2
|h0|Γ(0)|E0i|2
as τ → ∞
where |E0 i is the lowest energy state with non-zero matrix element
h0|Γ(0)|E0i.
→ measure masses (E0) from the exponential fall-off of correlation functions.
12
8.3. Finite temperature
Connection Euclidean QFT ↔ classical statistical mechanics was derived for zero-temperature quantum system. However, this can be readily generalized to finite temperature:
Quantum thermodynamics w. the Gibbs ensemble:
Z
Z = Tr e−Ĥ/T = [dφ] hφ|e−Ĥ/T |φi
Expression is of the same form as the one which gave us the Euclidean
path integral for T = 0 theory! The difference here is
1) Finite + fixed “imaginary time” interval 1/T
2) Periodic boundary condition: φ(1/T ) = φ(0).
Repeating the previous derivation, the partition function becomes
Z
Z(T ) = [dφ] e
−SE
Z
= [dφ] exp[−
Thus, a connection between:
– Quantum statistics in 3d: Z = Tr e−Ĥ/T
13
Z 1/T
0
dτ
Z
d3xLE ]
– Classical statistics in 4d: Z = [dφ]e−S
Euclidean P.I. is a very common tool for finite T field theory analysis [J.
Kapusta, Finite Temperature Field Theory, Cambridge University Press]
R
14
8.4. Some terminology:
In numerical work, lattice is a finite box with finite lattice spacing a. In
order to obtain continuum results, we should take 2 limits:
• V →∞
• a→0
thermodynamic limit
continuum limit
Both have to be controlled – expensive!
1) Perform simulations with fixed a, various V . Extrapolate V → ∞.
2) Repeat 1) using different a’s. Extrapolate a → 0.
3) [ In QCD, one often has to extrapolate mq → mq,phys. . ]
15
T = 0:
1) V → ∞:
Nτ , Ns → ∞, a constant.
2) continuum:
a → 0.
T > 0:
1) V → ∞:
Ns → ∞, Nτ , a constant.
2) continuum:
a → 0,
1
T
= aNτ constant.
16
8.5. Scalar field
Free scalar field on a finite d-dimensional lattice with periodic boundary
conditions:
xµ = anµ , nµ ∈ Z
Action:
S=

X d 1X
a 2
x
µ

h
1 2 2
1
2
d 1
(φ
−
φ
)
+
m
φ
φ
=
a
x+µ
x
2 x
a2
2
x,y φy
+ 12 m2 φ2x
(implicit sum over x, y), and the lattice d’Alembert operator is
x,y φy
= −∆2φ =
X
µ
2φx − φx+µ̂ − φx−µ̂
a2
The action is of form S = 12 φx Mx,y φy , and
Z
Z = [dφ]e−S = (Det M/2π)−1/2
17
i
Fourier transforms:
X
f˜(k) = ad e−ikx f (x)
x
Since x = an, f˜(k + 2πn) = f˜(k), and we restrict k to Brillouin zone:
−π/2 < kµ ≤ π/2
Inverse transform:
Z π/a dd k
f (x) =
eikx f˜(k)
−π/a (2π)d
Note: often it is convenient to use dimensionless natural lattice units
xµ ∈ Z, −π < kµ ≤ π.
18
Lattice propagator:
The lattice propagator G(x, y) is defined to be the inverse of operator
a−d M = ( + m2 ):
X d
a (
y
x,y
+ m2 δx,y )G(y, z) = δx,z
Take Fourier transform (G(x, y) = G(x − y)):


X
µ

2
(1 − cos kµ a) + m2  G̃(k) = 1
2
a
which gives the lattice propagator
G̃(k) =
1
,
k̂ 2 + m2
where
k̂µ =
4
2 kµ a
.
sin
a2
2
Continuum limit: when a → 0, G̃(k) = 1/(k 2 + m2 ) + O(a2 ).
19
Generating function for Green’s functions:
S → S(J) =
Now
X d h1
a 2 φx (
x
Z
+ m2 )φx − Jx φx
Z(J) = [dφ]e−S(J) = Z(0) exp[
X 2d 1
a 2 Jx G(x, y)Jy ]
x,y
N-point functions
hφx . . . φy i = Z(0)−1
δ
δ
...
Z(J)
δJx
δJy
J=0
Interactions: just modify (for example)
L = Lfree +
20
i
1 4
λφ
4!
8.6. Gauge fields
• Gauge field lagrangian in (Euclidean) continuum:
1 a
F Fa
4 µν µν
= 12 Tr (Fµν Fµν )
• Field tensor
Fµν = [Dµ , Dν ] = ∂µ Aν − ∂ν Aµ + ig[Aµ, Aν ]
where Aµ = Aaµ λa , and λa are the group generators. We shall consider only unitary groups U(1) [QED] and SU(N) [QCD, Electroweak].
For SU(N), the generators are orthonormalized
Tr λa λb = 12 δ ab .
• Put gauge potential Aµ directly on a lattice? Difficult to maintain gauge
invariance.
21
• Good starting point for gauge field on a lattice is to consider consider
gauge fields acting in a parallel transport: As field φ ∈ SU(N) is “parallel
transported” along a path p, parametrized by xµ (s), s ∈ [0, 1], gauge
fields rotate it: φ → U (s)φ
Uφ
• Define U (s) differentially via
dU (s) dxµ
=
igAµ U (s),
ds
ds
which can be formally solved:
dxµ
φ
U (s) = P exp ig ds
Aµ
p
ds
• Here P = “path ordering”: in the power series expansion of the exponential one always takes A(x) in the order they are encountered along
the path.
• For U(1) (or any other Abelian group) path ordering is irrelevant.
"
22
Z
#
• Alternatively, if we divide the path into N finite intervals of length ∆s,
and xn = x(n∆s), n = 0 . . . N − 1:

U (s) ≈ P exp ig
X
n
Y
dxnµ
dxnµ
n 

Aµ (x ) = exp ig∆s
Aµ (xn)
∆s
ds
ds
i



Gauge transformations:
• Gauge transformation Λ(x) is a (SU(N)) group element defined at every
point. Gauge potential transforms as
i
Aµ → ΛAµ Λ−1 + Λ∂µ Λ−1
g
• Field φ:
φ(x) → Λ(x)φ(x)
• Path p:
U (p) → Λ(x1)U (p)Λ−1(x0),
where x0 and x1 are the beginning and end of path.
23
• Closed loops: U (C) → Λ(x0)U (C)Λ−1(x0)
• Trace of a closed loop: Tr U (C) is gauge invariant!
24
8.7. Lattice gauge fields
• Variables: parallel transporters from one
lattice site to a neighbouring one, Links:
Uµ(x) = P exp ig
Z x+µ
x
dxµAµ
= exp igaAµ (x + 21 µ) + O(ga3 )
h
i
Uµ (x+ ν)
Uν (x+ µ)
Uν(x)
• The lattice action has to be gauge invariant. Only traces of closed loops are gauge
invariant; the simplest one is plaquette
Uµ(x)
Tr U = Tr Uµ (x)Uν (x + µ)Uµ† (x + ν)Uν†(x)
• When a small (for SU(N)),
Re Tr U = N −
a4 g 2 a a
Fµν Fµν + O(g 4 a6 )
4
25
• → Wilson gauge action (K. Wilson, 1974):
Sg
Partition function
2N X
1
= 2 4−d
1 − Re Tr U
g a
N
Z
=
dd x 12 Tr [Fµν Fµν ] + O(a2 g 2 )
"
Z Y
Z= [
x,µ
#
dUµ(x)]e−Sg
Here the integral is over group elements Uµ (compact), not over algebra
Aµ
Common notation:
β = βG ≡
2N
g2
for SU(N),
26
β=
1
g2
for U(1).
8.7.1. Continuum limit for U(1) (in 4d)
• Uµ(x) = exp igaAµ (x)
• The Wilson action for U(1) is
1 X †
†
1
−
Re
U
(x)U
(x
+
µ)U
(x
+
ν)U
u(x)
µ
ν
µ
n
g 2 x;µ<ν
1 X
(1 − Re exp [iga(Aµ (x) + Aν (x + µ) − Aµ (x + ν) − Aν (x))])
= 2
g x;µ<ν
S =
Expanding Aν (x + µ) = Aν (x) + a∂µ Aν (x) + 21 a2 ∂µ2 Aν (x) + . . .
h
i
1 X 2
4
1
−
Re
exp
iga
(∂
A
−
∂
A
)
+
O(a
)
µ ν
ν µ
g 2 x;µ<ν
"
#
11 X 1 2 2 2
2 6
= 2
g a Fµν + O(g a )
g 2 x;µ6=ν 2
1Z 4 2
d xFµν
=
4
S =
27
• Error in lagrangian O(a2 )
• Non-Abelian continuum limit comes in a similar way, but now one
has to be careful with the commutators. Check it! (Campbell-BakerHausdorff formula eA eB = eA+B−[A,B]+... might help, but it also comes
directly from the expansion of the exponents.)
28
8.8. Updating the gauge fields:
Let us consider U(1) theory, for simplicity. The action is
S = β
X
(1 − Re U (µ, ν; x))
x;µ<ν
U (µ, ν; x) = Uµ (x)Uν (x + µ)Uµ† (x + ν)Uν†(x),
where Uµ (x) = exp[iθµ(x)], 0 ≤ θµ(x) < 2π.
To update Uµ(x) we need the part of the action which depends
on it, i.e. the plaquettes which contain the link Uµ(x). This is
Sµ (x) ≡ −βRe Uµ (x)Pµ(x)
X
Pµ (x) =
Uν (x + µ)Uµ† (x + ν)Un†u(x)
ν;|ν|6=µ
where Pµ is called the sum of staples, for obvious reasons.
29
The sum over ν above goes over both positive and negative directions;
here U−ν (x) = Uν† (x − ν).
Now the link variable Uµ(x) can be updated with a suitable algorithm
(heat bath, Metropolis, overrelaxation . . . ).
For U(1), one staple is exp[i(θν (x + µ) − θµ (x + ν) − θn u(x))]; thus, the
0
sum of staples is a complex number: Pµ (x) = Ae−iθ , and the ’local
action’ becomes
Sµ (x) = −βRe Uµ (x)Pµ(x) = −βA cos(θµ(x) − θ0 )
Thus, the local action is just like with the XY model discussed earlier!
We can apply the same update algorithms as discussed there.
30
In the above discussion we used the standard compact formalism for
U(1) gauge fields: Uµ = eigaAµ ; S as given above. For U(1) (but not for
general non-abelian gauge field) it is also possible to use non-compact
formulation: let the link variable be
αµ (x) = gaAµ (x),
− ∞ < αµ (x) < ∞,
and the action is now
S NC =
1 X 1
[αµ (x) + αν (x + µ) − αµ (x + ν) − αν (x)]2
2
g x;µ<ν 2
The continuum limit for this action is the same as for the compact action
(check it!). The difference in the continuum limit is in the higher order
O(a2 ) terms. At finite a the physics can be quite different!
Note: switch 21 [·]2 7→ (1 − cos[·]), and you recover the compact action.
Non-compact U(1) gauge field theory is actually completely solvable (it
is free theory, as the continuum theory). However, if we would couple
the theory to matter this would not be the case.
31
8.9. Gauge transformations:
• Gauge transform Λ(x) ∈ SU(N) lives on lattice sites
• Link variable: Uµ (x) → Uµ0 (x) = Λ(x)Uµ(x)Λ†(x + µ)
7→
Aµ(x) → Λ(x)Aµ(x)Λ†(x) + gi Λ(x)∂muΛ†(x)
• Trace of a closed loop is gauge invariant
• Fundamental matter: φ(x) → Λ(x)φ(x),
where φ is a N-component complex vector: operator
φ† (x)UP (x 7→ y)φ(y) =
φ† (x)Uµ(x)Uν (x + µ) . . . Uρ (y − ρ)φ(y)
is gauge invariant.
• We want only gauge invariant animals on a lattice: lattice action +
32
observables must consist of closed loops (gauge only quantities) or
φ† UP φ - “strings” (matter).
• For example, the matter field kinetic term
(Dµ φ)† (Dµ φ),
where Dµ = ∂µ + igAµ , becomes


X †
1 
†
2dφ
φ
−
2
φ (x)Uµ(x)φ(x + µ)
a2
µ
• Elitzur’s theorem: expectation values of gauge non-invariant object
= 0: hUµ (x)i = 0.
33
8.9.1. Gauge fixing on the lattice
• Something you don’t want to do
• Necessary for perturbative calculations
• Most of the (Euclidean) continuum gauges go over to lattice
• Special gauge: maximal tree
34
• A tree which connects every point of the lattice, but does not have
closed loops.
• Link matrices Uµ (x) are fixed to arbitrary values (for example all = 0)
in the tree
• Expectation values of gauge invariant quantities do not change
(proof: easy)
• (Almost) complete gauge fixing
35
8.10. Confinement and Wilson loop
• Potential between static charge and anticharge, separated by R:


V (R → ∞) → 
∞:
finite:
confinement
no confinement
• Standard probe: Wilson loop WRT . Let W be a rectangular path of
size R × T along x1 (say) and x0 = τ directions.
WRT = Tr P exp ig
I
W
dsµ Aµ = Tr
Y
Uxy .
<xy>∈W
Now − log WRT gives the “free energy” of a static charge-anticharge
(“quark-antiquark”) system separated by R and which evolves for
time T :
− loghWRT i = V (R)T
valid as T → ∞
36
T
R
• Perimeter law: − loghWRT i ∼ m(2R + 2T )
Free charges, m: “mass” of the charge due to gauge field
• Area law: − loghWRT i ∼ σRT ; V (R) = σR
Charges confined with linear potential, σ: string tension
• In general,
1
T
loghW i ∼ σR + const. + c/R + . . .
37
Motivation:
• In temporal gauge A0 = 0 → U0 = 1 we can define gauge field
Hamiltonian Ĥ [Kogut-Susskind]
• Ψ: (trial) wave function of q q̄ pair at x̄, ȳ; spatial gauge transformations
Ψab → Λac (x̄)Λ†bd (ȳ)Ψcd
• Ĥ gauge invariant: hΨ0 |e−T Ĥ |Ψi = 0 unless Ψ, Ψ0 have similar gauge
transformation properties.
hΨ|e−T Ĥ |Ψi =
X
n
|hΨn |Ψi|2 e−En T
→ |hΨ0 |Ψi|2 e−E0 T
when T → ∞
• E0 = V (R) ground state energy of static charges (R = |x̄ − ȳ|)
• Trial wave function: Ψab = Uab(x 7→ y)Ψvac , where Ψvac is the (gauge
38
invariant) vacuum wave function
hΨ|e
−T Ĥ
1Z
|Ψi =
[dU ] Tr [U †(T ; x̄ 7→ ȳ) U (0; x̄ 7→ ȳ)]e−S
Z
= hWRT i
• WRT gauge invariant: can be measured in any gauge:
−
1
loghWRT i → V (R)
T
as T → ∞.
• If we can guess/construct Ψ ≈ Ψ0, T need not be very large.
39
8.10 12 . Measuring the string tension from Monte Carlo
The Wilson loops become exponentially small (∼ e−σRT ) as the size increases. However, the statistical noise is ∼ constant in magnitude independent of loop size! Thus, we need to increase the number of measurements exponentially as the loop size increases.
R
Smearing improves the situation: instead
of using only straight const-T legs in the
loop, take an averaged sum over paths
around the straight one. The smearing is
to mimic the wave function of the desired
q q̄ ground state. The hope is that now
a smaller T -extent would be sufficient to
get the asymptotic behaviour, i.e. look
like the limit T → ∞.
40
T
Smearing can be done recursively: let i ⊥ t, and
Ui(x) ← Ui(x) + c
X
j⊥i,t
Uj (x)Ui(x + ĵ)Uj†(x + î)
=
+
i.e. substitute link matrices on a plane ⊥ to t by the weighted average of
the link and the staples (which are also ⊥ to t). This can be repeated a
few times.
41
Wilson loop in pure gauge SU(3)
[Bali,Schilling, Wachter 1995] 483 × 64 quenched lattice, β = 6.8
1
V0(r)/GeV
0.5
0
-0.5
-1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
r/fm
42
1
1.1
For large loops
hW i ∼ e−V (r)T
Phenomenological form
e
1
V (r) = V0 + σr − + f GL (~r) −
r
r
"
43
#
8.11. Strong coupling expansion and the Wilson loop
• Perturbation theory (weak coupling, g 1) is toothless: no confinement
• Strong coupling: g 1. For simplicity, consider U(1) (β = 1/g 2 1). The link variable is Uµ = exp iθµ.
S = β
Z =
X
Z 2π
0
(1 − Re eiθ )


dθµ 

x,µ 2π
Y
Y
exp[β cos θ ]
Here θ = θ1 + θ2 − θ3 − θ4 around the plaquette
.
• Expand in powers of β? OK, but more convenient is the following
character expansion (here In are modified Bessel functions)
eβ cos θ =
∞
X
n=−∞

In (β)einθ = A 1 +
44
∞
X
n=1

fn cos nθ
where
1
1
A = I0(β) = 1 + β 2 + β 4 + . . .
4
64
1
fn = 2In(β)/I0(β) = n−1 β n + O(β n+2)
2 n!
1 3
1 5
f1 = β − β + β + . . .
8
48
Compare to the “character expansion” in the Ising model:
eβsi sj = a(1 + bsi sj )
with a = cosh β, b = tanh β.
In Ising gauge theory, (i, j; x) = si (x)sj (x + î)si(x + ĵ)sj (x) = ±1,
and the expansion of eβ (i,j;x) is exactly analogous.
45
The partition function becomes
Z
Z= [
dθ N
]A
2π
Y
(1 + f1 cos θ + f2 cos 2θ + . . .)
here N = 6V is the number of plaquettes (in 4d).
• Integration: dθa cos nθ = 0, if θa ∈
2 adjacent plaquettes:
R
Z
.
dφ
cos(θ + θa) cos(−θ + θb) = 21 cos(θa + θb)
2π
θ
(also for cos nθ)
a
θ
θb
• Non-zero contributions only from closed surfaces: lowest non-trivial
comes from 3-d cube:
Z = AN [1 + 4V ( 21 f1)6 + O(β 10 )]
46
• Each plaquette on the surface contributes 12 fi
• Expansion of free energy F = − log Z contains only connected
graphs – cluster expansion
47
Wilson loop:
To leading order, the Wilson loop is
hWRT i =
AN
Z
Z
[
dθ Y iθa Y
]
e
[1 + 12 f1(eiθ + e−iθ ) + O(f2)]
2π a∈W
• First contribution comes when we “tile” the
loop area with plaquettes:
hWRT i ≈
( 21 f1)RT
=
( 12 β)RT
+ O(β
RT +2
T
)
• We see confinement:
− loghW i/T = log( 12 f1)R = σR
• Next order: “bump”
hWRT i ≈ ( 21 f1 )RT [1 + 4RT ( 21 f1)4]
48
R
which gives
−a2 σ = −
1
loghWRT i = log 12 f1 + 4( 21 f1 )4 + O([ 21 f1 ]6)
RT
• Order [ 12 f1]6 gets contributions from the disconnected surfaces and
Z.
10 936 10
1 532 044 12
3 596 102 14
8
• −a2 σ = log u + 4u4 + 176
3 u + 405 u + 1 215 u + 5 103 u , where
u = ( 21 f1) (see Montvay+Münster, for example)
• For U(1) this is strong coupling artefact! When g 1, there ∃ a phase
transition. As g → 0 we regain free theory.
49
Non-Abelian gauge fields
• Same, but more difficult: use character expansion
e1/N β Re Tr U = Aβ [1 +
X
bR (β) χR (U )]
R
which gives “orthogonal” integration rules.
• Graphs are again surfaces – with complications!
• SU(2): −a2 σ = log u + 4u4 +
176 8
3 u
+ . . .,
with u = I2 (β)/I1(β)
• SU(3): −a2 σ = log u + 4u4 + 12u5 − 10u6 − . . .
with u = 31 (x − 21 x2 + 23 x3 + . . .), x = β/6.
50
SU(2) strong coupling expansion
3
• Convergence is good in
finite range β < βmax .
• Roughening transition
for Wilson loops as β
increases?
2
σa
• Mass gap: plaquetteplaquette
correlation
function decays exponentially.
Can be
calculated using strong
coupling, not in perturbation theory.
2
1
log u
0
u
u
u
12
u
14
u
0
1
2
β
51
8
10
3
4
8.12. Realistic continuum limit
• When we change lattice spacing a, we have to adjust the bare coupling g in order to keep physics (at “long” distances) invariant.
• Measuring g(a): choose some physical/renormalized quantity
M (a, g(a)), (mass, string tension, Wilson loop of fixed size, vertex
function . . . ) which can be measured/calculated. This we require to
be constant as a is changed (“renormalization prescription”)
→ RG equation:
d
∂
∂
M
a M (a, g(a)) = 0 = a − βL
da
∂a
∂g
!
∂g
∂a
tells how coupling g and lattice spacing a evolve.
• β-function: βL ≡ −a
• Fixed point βL (gF ) = 0.
52
• If βL (g) < 0 (> 0), g decreases (increases) as a is decreased.
• If ∂βL(gF ) < 0, this is UV attractive fixed point and lima→0 g(a) = gF :
continuum limit
• For example: β-function at strong coupling, using expansion for
SU(3) string tension σ = −a−2 log(1/(3g 2)) + O(g −2 )
d
2
σ = −2σ − βL 2 + . . .
da
ag
2
→ βL = −g log(3g ) + . . .
0 = a
This is < 0, so g & as a &: no continuum limit at strong coupling.
53
Weak coupling:
• Use perturbation theory. For QCD (for example, from renormalized
3-vertex):
βL (g) = −b0 g 3 − b1g 5 + . . .
Coefficients b0 and b1 are universal: independent of the regularization and prescription.
b0
b1
11N 2NF
1
−
=
16π 2 3
3

2
2
1
10N
N
N
(N
−
1)
34N
F
F


=
−
−
2
2
(16π )
3
3
N
!
Z g
dg 0 βL−1(g 0 )] →
1
2 b1 /2b20
](b
g
)
[1 + O(g 2 )]
aΛL = exp[
0
2
2b0g
• Integrate: a = exp[−
• UV fixed point g → 0 as a → 0: asymptotic freedom. (If b0 > 0!)
54
• ΛL: dimensionful integration constant, Λ-parameter; sets the scale.
It is regularization dependent.
• Dimensional transmutation: no mass scale in the original (classical)
Lagrangian!
• Thus, a physical mass m should behave as
m
1
2 b1 /2b20
am =
exp[
](b
g
)
0
ΛL
2b0g 2
In practice, this not well satisfied in QCD; g ∼ 1:
Asymptotic scaling does not hold.
• Scaling holds if mass ratios are ∼ constant:
m1
m1
(a) =
(0) + O(a2 )
m2
m2
This has been observed to be much better satisfied than asymptotic
scaling.
55
Numerically:
• Principle: measure a quantity – am, a2 σ – at several g’s. Then
(am)(g1) a(g1)
=
(am)(g2) a(g2)
gives a discrete sample of a(g1) − a(g2 ) = exp[−
Z g
1
g2
dg 0 1/βL(g)]
• Matching more than one quantity simultaneously: need more than
one lattice coupling.
• Monte Carlo renormalization group (MCRG): direct implementation
of Wilson’s RG in numerical simulations.
- Write the action S = a κa Sa , where Sa are a set of (local) contributions to action: plaquettes, larger loops . . .
P
d
- With couplings κA
a simulate on L lattice, and measure a set of
Wilson loops.
56
- block the configurations by factor of 2: obtain (L/2)d lattice. Measure the Wilson loops.
- Repeat blocking + measuring, until lattice is too small.
- simulate directly on (L/2)d lattice, with κB . As before, repeat measuring Wilson loops and blocking.
- compare Wilson loops at blocking level b from simulation B to loops
at level b + 1 from A. Tune κB
a , until the measurements match (requiring a new simulation).
- repeat for (L/4)d etc.
For the plaquette term, this gives function ∆β = β(a) − β(2a) =
6/g 2 − 6/g 02
• Violation of asymptotic scalings large at practical values of β (fig.)!
57
• Perfect action: no scaling violations; evolves along RG trajectory.
Requires infinitely many coupling constants.
• Classically perfect action: perfect action for classical configurations.
Much easier to derive, and these have been used for various models. It’s also 1-loop quantum perfect action (perfect in the limit
g → 0).
In 3 dimensions:
• Lattice coupling β =
2N
. Dimensions of [g 2] = GeV.
2
g a
• Renormalization: gR2 = g 2 + O(g 4 a) dimensionally
• Continuum limit β =
2N
+ O[(gR2 a)0]
2
gR a
58
• Theory superrenormalizable: only finite number of divergent diagrams
59
8.13. Continuum limit in λφ4 model
• Lattice lagrangian
L = −κ
X
µ
ϕx ϕx+µ + ϕ2x + λ(ϕ2x − 1)2
• In 4 dimensions, the RG trajectories of the bare couplings κ, λ have
been solved [Lüscher, Weisz, NPB 295 (1988)] using hopping parameter expansion and perturbative RG equations:
60
κ
4d Ising model
κc
1/8
free
theory
0
oo
λ
Blue curves: RG trajectories of constant λR ; λR decreases to -;
along κc we have λR = 0.
∂λ
3 2
=
λ >0
∂a
16π 2
• No continuum limit! RG trajectories hit the Ising limit λ = ∞ before
the critical line κc (λ) is reached; i.e. at finite a.
• Perturbatively βL (λ) = −a
61
• Ising model: smallest possible a for any fixed λR .
• This is a generic feature of λφ4 models in 4d
→ Triviality bound for the Standard Model Higgs mass!
• In 3 dimensions: there ∃ continuum limit: coupling constant is dimensionful [λ] = GeV.
• λR = λ + O(λ2 a)
• m2R = m20 + C1 λa−1 + C2λ2 + O(λ3 a)
m2R has linear (1/a) and (log a/Λ) divergence. Perturbatively calculable, and can be subtracted.
62