Lattice Gauge Theory and Color Confinement

Transcrição

Lattice Gauge Theory and
Color Confinement
Tereza Mendes
in collaboration with Attilio Cucchieri
Instituto de Fı́sica de São Carlos
University of São Paulo
III Oficina Nacional de Teoria de Campos
Brası́lia, November 2013
Origin of Confinement
QCD Lagrangian is just like the one of QED:
quarks (spin-1/2 fermions)
gluons (vector bosons) / color charge
⇔
electrons
photons / electric charge
But: gauge symmetry is SU (3) (non-Abelian) instead of U (1)
6
X
1 a µν
µ ij
L = − Fµν Fa +
ψ̄f,i i γ Dµ − mf δij ψf,j
4
f =1
where [a = 1, . . . , 8; i = 1, . . . , 3; Tija = SU (3) generators]
a
≡ ∂µ Aaν − ∂ν Aaµ + g0 fabc Abµ Acν
Fµν
Dµ ≡ ∂µ − i g0 Aaµ Ta
Note: g0 , mf are bare parameters.
Gluons Have Color
a
Note: Fµν
∼ g0 f abc Abµ Acν
⇒ QCD Lagrangian contains terms with three and four
gauge fields in addition to quadratic terms (propagators)
Lψ̄ψA = g0 ψ̄ γ µ Aµ ψ ⇒ quark-quark-gluon vertex
LAAA = g0 f abc Aµa Aνb ∂µ Acν ⇒ three-gluon vertex
⇒ gluons interact with each other (have color charge),
determining the peculiar properties and the
nonperturbative nature of low-energy QCD
⇒ Running coupling αs (p) instead of α ≈ 1/137
Confinement vs. Aymptotic Freedom
At high energies: deep inelastic scattering of electrons reveals
proton made of partons: pointlike and free. In this limit αs (p) ≪ 1
(asymptotic freedom) and QCD is perturbative
αs (p) =
2
2
4π
2β1 log (log (p /Λ ))
1
−
+ ...
2
2
2
2
2
β0 log (p /Λ )
β0
log (p /Λ )
Confinement vs. Aymptotic Freedom
At high energies: deep inelastic scattering of electrons reveals
proton made of partons: pointlike and free. In this limit αs (p) ≪ 1
(asymptotic freedom) and QCD is perturbative
αs (p) =
2
2
4π
2β1 log (log (p /Λ ))
1
−
+ ...
2
2
2
2
2
β0 log (p /Λ )
β0
log (p /Λ )
At low energies: interaction gets stronger, αs ≈ 1 and confinement
occurs. Color field may form flux tubes
q
−
q
+
−
linear increase of inter-quark potential → string tension
At large distances → string breaks
How do we perform calculations?
The strength of the interaction αs increases for larger r
(smaller p ) and vice-versa (asymptotic freedom).
Perturbation theory breaks down in the limit of small
energies.
QCD on a Lattice (I)
Kenneth Geddes Wilson (June 8, 1936 – June 15, 2013)
Lattice used by Wilson in 1974 as a trick to prove confinement in
(strong-coupling) QCD
[Confinement of quarks, Phys. Rev. D 10, 2445 (1974)]
QCD on a Lattice (II)
As recalled by Wilson
[...] Unfortunately, I found myself lacking the detailed
knowledge and skills required to conduct research
using renormalized non-Abelian gauge theories. What
was I to do, especially as I was eager to jump into this
research with as little delay as possible? [...] from my
previous work in statistical mechanics I knew a lot about
working with lattice theories...
[...] I decided I might find it easier to work with a lattice
version of QCD. . .
The Origins of Lattice Gauge Theory, hep-lat/0412043 (Lattice 2004)
Lattice QCD Ingredients
Three ingredients
1. Quantization by path integrals ⇒ sum over
configurations with “weights” ei S/~
2. Euclidean formulation (analytic continuation
to imaginary time) ⇒ weight becomes e−S/~
3. Discrete space-time ⇒ UV cut at momenta
p<
∼ 1/a ⇒ regularization
Three ingredients
p<
Also: finite-size lattices ⇒ IR cut for small momenta p ≈ 1/L
Three ingredients
p<
Also: finite-size lattices ⇒ IR cut for small momenta p ≈ 1/L
The Wilson action
b
is written for the gauge links Ux,µ ≡ eig0 aAµ (x)Tb
reduces to the usual action for a → 0
is gauge-invariant
The Lattice Action
The Wilson action (1974)
β X
S = −
ReTr U2 ,
3 2
Ux,µ ≡ e
ig0 aAbµ (x)Tb
,
β = 6/g0 2
written in terms of oriented plaquettes formed by the link variables
Ux,µ , which are group elements
under gauge transformations: Ux,µ → g(x) Ux,µ g † (x + µ), where
g ∈ SU (3) ⇒ closed loops are gauge-invariant quantities
integration volume is finite: no need for gauge-fixing
At small β (i.e. strong coupling) we can perform an expansion
analogous to the high-temperature expansion in statistical mechanics.
At lowest order, the only surviving terms are represented by diagrams
with “double” or “partner” links, i.e. the same link should appear in both
R
orientations, since dU Ux,µ = 0
Confinement and Area Law
Considering a rectangular loop with sides R and T (the Wilson loop) as
our observable, the leading contribution to the observable’s
expectation value is obtained by “tiling” its inside with plaquettes,
yielding the area law
< W (R, T ) > ∼ β RT
But this observable is related to the interquark potential for a static
quark-antiquark pair
< W (R, T ) > = e−V (R)T
We thus have V (R) ∼ σR, demonstrating confinement at strong
coupling (small β)!
Problem: the physical limit is at large β...
Numerical Lattice QCD (I)
The approach had a “marvelous side effect”, as Michael
Creutz calls it
By discreetly making the system discrete, it
becomes sufficiently well defined to be placed on a
computer. This was fairly straightforward, and
came at the same time that computers were
growing rapidly in power. Indeed, numerical
simulations and computer capabilities have
continued to grow together, making these efforts
the mainstay of lattice gauge theory.
The Early days of lattice gauge theory,
AIP Conf. Proc. 690, 52 (2003)
Numerical Lattice QCD (II)
We are left with a (classical) Statistical-Mechanics model with the
partition function
Z
Z
Z
R 4
DU e−Sg det K(U )
Z =
DU e−Sg Dψ Dψ e − d x ψ(x) K ψ(x) =
partition function
Z
Z
Z
R 4
DU e−Sg det K(U )
Z =
All we need is to evaluate expectation values
Z
< O > = DU O(U ) P (U )
with the weight
e−Sg (U ) det K(U )
P (U ) =
Z
⇒ analogous to thermodynamic averages in Statistical Mechanics
partition function
Z
Z
Z
R 4
DU e−Sg det K(U )
Z =
Z
< O > = DU O(U ) P (U )
with the weight
P (U ) =
Z
Very complicated (high-dimensional) integral to compute!
partition function
Z
Z
Z
R 4
DU e−Sg det K(U )
Z =
Z
< O > = DU O(U ) P (U )
with the weight
P (U ) =
Z
Very complicated (high-dimensional) integral to compute!
May perform Monte Carlo simulations, as in statistical mechanics!
Confinement from Simulations
May observe formation of flux tubes
Linear Growth of potential between quarks, string breaking
Pathways to Confinement
How does linearly rising potential (seen in lattice QCD)
come about?
come about?
Models of confinement include: dual superconductivity
(electric flux tube connecting magnetic monopoles),
condensation of center vortices, but also merons, calorons
come about?
Proposal by Mandelstam (1979) linking linear potential to
infrared behavior of gluon propagator as 1/p4
Z 3
d p ip·r
e
∼r
V (r) ∼
4
p
come about?
Proposal by Mandelstam (1979) linking linear potential to
infrared behavior of gluon propagator as 1/p4
Z 3
d p ip·r
e
∼r
V (r) ∼
4
p
Gribov-Zwanziger confinement scenario based on
suppressed gluon propagator and enhanced ghost
propagator in the infrared
Ghost Enhancement (I)
Ghost-enhanced scenario natural in Coulomb gauge. Since
(∂i Ai )a = 0, the color-electric field is decomposed as
Eitr − ∂i φ(~x, t) and the classical (non-Abelian) Gauss’s law
(Di Ei )a (~x, t) = ρaquark (~x, t)
is written for a color-Coulomb potential in terms of
Faddeev-Popov operator: Mφa (~x, t) = ρa (~x, t) , where
G−1 ∼ M = −Di ∂i . In momentum space
Z
Z
φa (~x, t) ≈
d3 p d3 y G(~
p, t) exp[i~
p · (~x − ~y )] ρa (~y , t)
IR divergence of ghost propagator G(~
p, t) as 1/p4 leads to
linearly rising potential.
Ghost Enhancement (II)
Gribov’s restriction beyond quantization using Faddeev-Popov
(FP) method implies taking a minimal gauge, defined by a
minimizing functional in terms of gauge fields and gauge
transformation
⇒ FP operator (second variation of functional) has non-negative
eigenvalues. First Gribov horizon ∂Ω approached in
infinite-volume limit, implying ghost enhancement
Γ
Ω
Λ
IR gluon propagator and confinement
Green’s functions carry all information of a QFT’s physical
and mathematical structure.
Gluon propagator (two-point function) as the most basic
quantity of QCD.
Confinement given by behavior at large distances (small
momenta) ⇒ nonperturbative study of IR gluon propagator.
Landau gluon propagator
ab
(p)
Dµν
=
X
x
=
δ
ab
e−2iπk·x hAaµ (x) Abν (0)i
gµν
pµ pν
−
p2
D(p2 )
Early simulations: Mandula & Ogilvie, PLB 1987
IR ghost propagator and confinement
Ghost fields are introduced as one evaluates functional integrals
by the Faddeev-Popov method, which restricts the space of
configurations through a gauge-fixing condition. The ghosts are
unphysical particles, since they correspond to anti-commuting
fields with spin zero.
On the lattice, the (minimal) Landau gauge is imposed as a
minimization problem and the ghost propagator is given by
X e−2πi k·(x−y)
1
G(p) = 2
h M−1 (a, x; a, y) i ,
N c − 1 x, y , a
V
where the Faddeev-Popov (FP) matrix M is obtained from the
second variation of the minimizing functional.
Early simulations: Suman & Schilling, PLB 1996; Cucchieri, NPB 1997
Gribov-Zwanziger Confinement Scenario
The Gribov-Zwanziger confinement scenario in Landau
gauge predicts a gluon propagator D(p2 ) suppressed in the
IR limit.
In particular, D(0) = 0 implying that reflection positivity is
maximally violated.
This result may be viewed as an indication of gluon
confinement.
Infinite volume favors configurations on the first Gribov
horizon, where λmin of M goes to zero.
In turn, G(p) should be IR enhanced, introducing long-range
effects, related to the color-confinement mechanism.
Tests of Gribov-Zwanziger Scenario
Above results are also obtained by functional methods, e.g. scaling or
conformal solution of Dyson-Schwinger equations (DSEs)
Alkofer et al., ca. 2000
D(p2 ) ∼ (p2 )2κ−1 ,
G(p2 ) ∼ (p2 )−κ−1
D(p2 ) ∼ (p2 )2κ−1 ,
G(p2 ) ∼ (p2 )−κ−1
Different solutions: Aguilar et al., Boucaud et al. (after Cornwall, 1982)
D(p2 ) ∼ (p2 )2κ−1 ,
G(p2 ) ∼ (p2 )−κ−1
Different solutions: Aguilar et al., Boucaud et al. (after Cornwall, 1982)
Lattice formulation: insight into the problem and numerical checks of
the above predictions.
IR limit corresponds to large lattice sizes, which are
computationally demanding...
the IR issue may be qualitatively studied for pure SU (2) theory
and focusing on very large lattices should also greatly reduce the
problem with Gribov copies.
Gauge-Related Lattice Features
Gauge action written in terms of oriented plaquettes formed
by the link variables Ux,µ , which are group elements
under gauge transformations: Ux,µ → g(x) Ux,µ g † (x + µ),
where g ∈ SU (3) ⇒ closed loops are gauge-invariant
quantities
quantities
quantities
when gauge fixing, procedure is incorporated in the
simulation, no need to consider FP matrix
quantities
get FP matrix without considering ghost fields explicitly
quantities
get FP matrix without considering ghost fields explicitly
Lattice momenta given by p̂µ = 2 sin (π nµ /N ) with
nµ = 0, 1, . . . , N/2 ⇔ pmin ∼ 2π/(a N ) = 2π/L,
pmax = 4/a in physical units
Overview of Lattice Results
Gluon propagator is suppressed in the limit p → 0, while the
real-space propagator violates reflection positivity
λmin → 0 with the volume
On “small” lattices:
Could fit to D(0) → 0; observed enhancement of G(p).
After 2007: large-lattice results (L ≈ 27 fm)
D(0) > 0 (good fit to Gribov-Stingl form)
G(p) shows no enhancement in the IR
After 2007: large-lattice results (L ≈ 27 fm)
D(0) > 0 (good fit to Gribov-Stingl form)
G(p) shows no enhancement in the IR
Consistent with so-called massive solution of DSEs and refined
GZ scenario. Not consistent with scaling solution
Gluon Propagator at “Infinite” Volume
1.4
4
1.2
3.5
3
4D Results
1
0.8
2
D(p )
2
D(p )
2.5
3D Results
2
0.6
1.5
0.4
1
0.2
0.5
0
0
0
0.5
1
1.5
p
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
p
Gluon propagator D(k) as a function of the lattice momenta k (both
in physical units) for the pure-SU (2) case in d = 4 (left), considering
volumes of up to 1284 (lattice extent ∼ 27 fm) and d = 3 (right),
considering volumes of up to 3203 (lattice extent ∼ 85 fm).
Gribov-Stingl Form for IR Gluon
Gribov-type propagator has purely-imaginary complex-conjugate poles
p2
D(p ) = C 4
p + b2
2
(vanishes at p = 0)
Gribov-Stingl form (1986) allows for complex conjugate poles
p2 + d
p2 + d
D(p ) = C 2
= C 4
2
2
(p + a) + b
p + u 2 p2 + t2
2
Poles at masses m2 = a ± ib ⇒ m = mR + imI
In general: pairs of (complex-conjugate) poles + real poles, starting
from p6 in the denominator, p4 in numerator
More recently: massive propagator as a consequence of condensates,
in Refined Gribov-Zwanziger scenario (Dudal et al., 2008)
Gluon Fits
Fit of gluon propagator data (from A. Cucchieri & T.M., 2007) to
rational forms above in d = 4 (left) and d = 3 (right) cases
Cucchieri, Dudal, T.M., Vandersickel PRD 2012
1
D(p2)
D(p2)
1
0.1
0.1
0
0.5
1
1.5
2
2.5
3
3.5
p
4
0.01
0.1
1
p
Interpreting the Massive Result for D(p)
We can obtain upper and lower bounds for the gluon propagator
at zero momentum D(0) by considering the quantity
(A. Cucchieri, T.M., PRL 2008)
M (0) =
1
d(Nc2 − 1)
X
b,µ
eb (0)|
|A
µ
M (0) =
We have
2
1
d(Nc2 − 1)
X
b,µ
eb (0)|
|A
µ
2
V hM (0)i ≤ D(0) ≤ V d(Nc2 − 1) hM (0) i
M (0) =
We have
1
d(Nc2 − 1)
X
b,µ
eb (0)|
|A
µ
2
2
V hM (0)i ≤ D(0) ≤ V d(Nc2 − 1) hM (0) i
Thus, if M (0) goes to zero as V −α we find that
D(0) → 0,
0 < D(0) < +∞
or
D(0) → +∞
M (0) =
We have
1
d(Nc2 − 1)
X
b,µ
eb (0)|
|A
µ
2
2
V hM (0)i ≤ D(0) ≤ V d(Nc2 − 1) hM (0) i
Thus, if M (0) goes to zero as V −α we find that
D(0) → 0,
0 < D(0) < +∞
or
D(0) → +∞
respectively if α is larger than, equal to or smaller than 1/2
Upper and lower bounds plus D(0)/V
From fits we obtain that hM (0)i goes to zero exactly as 1/V 1/2
4d case
3d case
Gluon Propagator at Infinite Volume
Gluon propagator in Landau gauge IR finite in 3d and
4d, as a consequence of “self-averaging” of a
magnetization-like quantity [i.e. M (0), without the
absolute value]
May think of D(0) as a response function
(susceptibility) of this observable (“magnetization”). In
this case it is natural to expect D(0) ∼ const in the
infinite-volume limit
Note: violation of reflection positivity
Violation of reflection positivity
Clear violation of reflection positivity for lattice volume V = 1284
at β = 2.2
Ghost Propagator Results
Fit of the ghost dressing function p2 G(p2 ) as a function of p2 (in GeV)
for the 4d case (β = 2.2 with volume 804 ). We find that p2 G(p2 ) is best
fitted by the form p2 G(p2 ) = a − b[log(1 + cp2 ) + dp2 ]/(1 + p2 ), with
3.5
3
2
2
4
2
2
p G(p), a-b[log(1+c p )+d p ]/(1+p )
4.5
4D Results
a = 4.32(2),
b = 0.38(1) GeV 2 ,
c = 80(10) GeV −2 ,
d = 8.2(3) GeV −2 .
2.5
2
In IR limit p2 G(p2 ) ∼ a.
1.5
1
0.001
0.01
0.1
1
10
100
2
p
Upper and Lower Bounds for G(p)
On the lattice, the ghost propagator is given by
X e−2πi k·(x−y)
1
G(p) = 2
M−1 (a, x; a, y)
Nc − 1 x, y , a
V
X 1 X
1
|ψei (a, p)|2 ,
= 2
Nc − 1
λi a
i,λi 6=0
where ψi (a, x) and λi are the eigenvectors and eigenvalues of the FP
matrix. Then, one can prove (A.Cucchieri, TM, PRD 78, 2008) that
1
1 X e
1
2
|
ψ
(a,
p)|
≤
G(p)
≤
.
1
2
N c − 1 λ1 a
λ1
If λ1 ≡ λmin behaves as L−α in the infinite-volume limit, α > 2 is a
necessary condition to obtain an IR-enhanced ghost propagator G(p).
Upper bound for G(pmin)
2κ = 0.043(8), α = 1.53(2)
The Infinite-Volume Limit
One can check if lattice data support λmin [A] → 0 in the infinite-volume
limit =⇒ A ∈ ∂Ω (Gribov horizon).
Smallest eigenvalue of the Faddeev-Popov operator
3×10-2
ωs [GeV2]
2×10-2
10-2
-3
3×10
4×10-2
5×10-2
6×10-2
7×10-2
8×10-2
1/L [GeV]
Infinite-volume limit extrapolation λmin [A] ∼ Lc for the 3d SU(2) case
(A.Cucchieri, A.Maas, TM, PRD 74, 2006). Similarly in 4d.
Relating λmin and Geommetry
It is generally accepted that
At very large volumes the functional integration
gets concentrated on the boundary ∂Ω of the first
Gribov region Ω.
But
The key point seems to be the rate at which λmin
goes to zero, which, in turn, should be related to
the rate at which a thermalized and gauge-fixed
configuration approaches ∂Ω.
Relating λmin and Geommetry (II)
Using properties of Ω and the concavity of the minimum function, one
can show (A. Cucchieri, TM, PRD 2013)
λmin [ M[A] ] ≥ [1 − ρ(A)] p2min
Here 1 − ρ(A) ≤ 1 measures the distance of a configuration A ∈ Ω
from the boundary ∂Ω (in such a way that ρ−1 A ≡ A′ ∈ ∂Ω). This result
applies to any Gribov copy belonging to Ω.
Recall that A′ ∈ ∂Ω =⇒ the smallest non-trivial eigenvalue of the FP
matrix M[A′ ] is null, and that the smallest non-trivial eigenvalue of
(minus) the Laplacian −∂ 2 is p2min .
In the Abelian case one has M = −∂ 2 and λmin = p2min
=⇒ non-Abelian effects are included in the (1 − ρ) factor
Results: Bound for λmin
The new inequality λmin [ M[A] ] ≥ [1 − ρ(A)] p2min becomes
an equality only when the eigenvectors corresponding to the
smallest nonzero eigenvalues of M[A] and −∂ 2 coincide. =⇒
The eigenvector ψmin is very different from the plane waves
corresponding to pmin .
Plot of G(pmin ) (full triangles), the lower bound (full
circles) and the upper bound
(full squares) as a function of
the inverse lattice size 1/N .
1/N
This explains the non-enhancement of G(p) in the IR.
Summary
Simple properties of gluon and ghost propagators constrain
(by upper and lower bounds) their IR behavior.
Summary
(by upper and lower bounds) their IR behavior. For the
gluon case we define a magnetization-like quantity, while for
the ghost case we relate the propagator to λmin of the FP
matrix.
Summary
matrix. These quantities are studied as a function of the
lattice volume, to gain better control of the infinite-volume
limit of IR propagators.
Summary
The Gribov-Zwanziger scenario is not realized in the
originally proposed form
Summary
The Gribov-Zwanziger scenario is not realized in the
originally proposed form
Relating infinite volume and geommetry of the first Gribov
region allows us to understand why no ghost enhancement
(or scaling behavior) is observed on the lattice
Conclusion
We have not succeeded in answering all our
problems. The answers we have found only serve
to raise a whole set of new questions. In some
ways we feel we are as confused as ever, but we
believe we are confused on a higher level and
about more important things.
In: Stochastic Differential Equations: An Introduction with Applications,
Bernt Øksendal

Lattice Gauge Theory and Color Confinement

Transcrição

Documentos relacionados

EVALUATING BIOCHEMICAL INTERNET RESOURCES Lima, R.M.

LEITE, Heberton Lucas. Seleção Natural Em Tempo Real

Ground Transportation From S˜ao Paulo/Guarulhos to S˜ao José

View/Open

GM Auto Trade Union committee ITUA/MPRA Brothers and sisters

Felícia Leirner Museum and Claudio Santoro Audito

Multi System Tables - Bilhares Carlos Teofilo

Lia mccord

Mais - Departamento de Conservação e Restauro

Aula 1 : Verbo To Be - Mogi das Cruzes

CADERNO de RESUMOS Oficina Temática em Teoria de Campos

Two-Particle Excitations in the Hubbard Model

A structural geometrical analysis of weakly infeasible SDPs