“Solving QCD: from BG/P to BG/Q”. Prof. Dr. Attilio Cucchieri – IFSC/USP.

Solving QCD:

from BG/P to BG/Q

Attilio Cucchieri

http://lattice.ifsc.usp.br/

Instituto de Fısica de Sao Carlos – USP

3WHPC–LCCA April 2015

Quarks: in the Heart of Matter

Interior of the atom: nucleus, made up of nucleons, made up of

quarks

There are 6 types (flavors) of quarks, corresponding to the 6 types

of leptons in nature (e.g. electron, neutrino)

Hadrons (e.g. protons, neutrons, mesons) are made up of quarks


Quantum Chromodynamics

Strong interaction btw. protons e neutrons

is residue of interaction btw. their quarks.

Nucleons are made up of 3 quarks of dif-

ferent colors

The proton is a color-neutral bound

state of quarks interacting through the

exchange of (massless) gluons

Protons and Neutrons:

99% of the mass of the bound state comes from the

interaction!

⇒ we are not star dust, we are (virtual) gluons!


Free the Quarks!

Why was a fractionary electric charge never

observed?

Answer: quarks are confined inside hadrons!

Confinement: it would take infinite energy to separate

the quarks that constitute a hadron

Quarks have flavor, and color, but no freedom


(Usual) Quantum Field Theory

QED Lagrangian:

L = −1

4Fµν F

µν + ψ (i γµDµ −m)ψ

where

Dµ ≡ ∂µ−i eAµ , Fµν ≡ ∂µAν − ∂νAµ

Perturbative calculation: Feynman

diagrams for electron scattering; it

is possible to infer the redefinition

of m and e to obtain finite results


Quantum Chromodynamics (QCD)

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)

L = −1

4F aµν F

µνa +

6∑

f=1

ψf,i(

i γµDijµ −mf δij

)

ψf,j

where [a = 1, . . . , 8; i = 1, . . . , 3; T aij = SU(3) generators]

F aµν ≡ ∂µAaν − ∂νA

aµ + g0 fabcA

bµA

cν

Dµ ≡ ∂µ − i g0Aaµ Ta

Note: g0, mf are bare parameters.


Gluons Have Color

Note: F aµν ∼ g0 f

abcAbµAcν

⇒ QCD Lagrangian contains terms with three and four

gauge fields in addition to quadratic terms (propagators)

LψψA = g0 ψ γµAµ ψ ⇒ quark-quark-gluon ver-

tex

LAAA = g0 fabcAµa A

νb ∂µA

cν ⇒ three-gluon ver-

tex

⇒ gluons interact with each other (have color charge),

determining the peculiar properties and the nonpertur-

bative nature of low-energy QCD

⇒ Running coupling αs(p) instead of α ≈ 1/137


Photons vs. Gluons

Photons do not interact directly with one another

⇒ lightsabers (Star Wars) could not possibly work...


QCD vs. QED

QCD (strong force) vs. QED (EM force)

quarks, gluons

SU(3) (3 “colors”)

mq, αs(p)

electrons, photons

U(1)

me, α ≈ 1/137

q− −q +


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)

Three ingredients

1. Quantization by path integrals ⇒ sum

over configurations with “weights” ei S/~

2. Euclidean formulation (analytic continua-

tion to imaginary time) ⇒ weight becomes

e−S/~

3. Discrete space-time ⇒ UV cut at mo-

menta p ∼< 1/a ⇒ regularization

Also: finite-size lattices ⇒ IR cut for small momenta p ≈ 1/L

The Wilson action

is written for the gauge links Ux,µ ≡ eig0aAbµ(x)Tb

reduces to the usual action for a→ 0

is gauge-invariant


Numerical Simulations

Monte Carlo methods (Ulam, 1940’s): statisti-

cal description of the possible configurations of

a system, which is simulated on a computer.

Useful in

designing/analyzing experiments

studying the theory of stochastic

(statistical) systems

doing calculations in quantum field theory


Lattice QCD Results: Confinement

May observe formation of flux tubes

Linear Growth of potential between quarks, string breaking


Confinement: the Elephant in the Room

Do we understand confinement?

⇒ we know what it looks like,

but do we know what it is?

Millenium Prize Problems (Clay Mathematics Institute, USA/UK)

Yang-Mills and Mass Gap: Experiment and computer simulations sug-

gest the existence of a mass gap in the solution to the quantum versions

of the Yang-Mills equations. But no proof of this property is known.


Lattice QCD at the IFSC–USP

The only LQCD group (A.C. & T. Mendes) in South America.

1. Study of qualitative aspects of QCD: infrared behavior

of propagators and vertices, related to color confinement

and to color deconfinement (at high temperature).

2. Development of methods: determination of the strong

coupling constant αs(p) to be applied to the full QCD

case, lattice implementation of different analytic ap-

proaches (linear covariant gauge, background gauge).

3. Development of algorithms: gauge fixing, global mini-

mization, matrix inversion, evaluation of eigenvalues.


Results with the BG/P

New lower bound for the smallest nonzero eigenvalue of the

Landau-gauge Faddeev-Popov matrix in Yang-Mills theories.

First candidate for a lattice configuration belonging to the

common boundary ∂Ω ∩ ∂Λ.

First estimate of the distance of a minimal-Landau-gauge

configuration A ∈ Ω from the boundary ∂Ω.

First concrete explanation of why lattice studies do not observe

an enhanced ghost propagator in the deep infrared limit.

First evaluation of the Bose-ghost propagator (of the

Gribov-Zwanziger theory).

First numerical manifestation of BRST-symmetry breaking due

to restriction of gauge-configuration space to the Gribov region.


Solving QCD!

The building blocks of QCD (in a

given gauge) are:

Propagators: gluon, quark,

ghost.

Vertices: three-gluon, four-

gluon, ghost-gluon, quark-

gluon.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2

D(p

2 )

p

4D Results

Challenges:

breaking of rotational symmetry;

connect to the continuum: for example, for the ghost-gluon vertex

Γabcµ (q, s; k) = −ig0f

abc qν Γνµ(q, s; k) we have

Γνµ(q, s; k) = δνµa(q, s; k)− kν sµ b(q, s; k) + qν kµ c(q, s; k)

+kν qµ d(q, s; k) + qν qµ e(q, s; k)


3-Step Code

main()

/* set parameters: beta, number of configurations NC,

number of thermalization sweeps NT */

read_parameters();

/* U is the link configuration */

set_initial_configuration(U);

/* cycle over NC configurations */

for (int c=0; c < NC; c++)

thermalize(U,NT);

gauge_fix(U,g);

evaluate_propagators_and_vertices(U);


Parallelization

• We need a parallelized code in order to simulate at

very large lattice volumes V .

• Communication is required in each of the three

steps.

• Each node gets a contiguous block of v = V/N

lattice sites (local lattice).

• Communication is required only for sites on the

boundary of the local lattice.

• 4D simulations → high granularity due to the

surface/volume effect.


Weak and Strong Scaling on BG/Q

V Nodes HB Micro Gfix GluonProp CG

642 × 322 32 494.9 54.7 0.0044 0.041 0.0081

643 × 32 64 496.3 62.1 0.0049 0.041 0.0088

644 128 496.8 59.2 0.0047 0.050 0.0084

643 × 128 256 499.4 63.0 0.0050 0.041 0.0090

642 × 1282 512 499.7 56.4 0.0046 0.042 0.0083

644 128 496.8 59.2 0.0047 0.0050 0.0084

644 256 256.3 37.9 0.0029 0.0028 0.0055

644 512 134.6 27.3 0.0020 0.0018 0.0040

644 1024 74.4 22.5 0.0016 0.0012 0.0035

644 512 2943.6 218.5 0.0171 0.0179 0.0239

Weak (with 5 different lattice volumes) and strong (with 4 different volumes) scaling:

time (in seconds) for 3 different updates of local variables and for the evaluation of

the gluon propagator and the time (in seconds) for one conjugate gradient iteration.

Link and site variables are SU(2) matrices. The last row is for the BG/P.


Conclusion

Still lots to be understood inside a Proton!!

Keep the supercomputers coming!


Date post:	15-Aug-2015
Category:	Technology
Upload:	lccausp
View:	11 times
Download:	0 times

“Solving QCD: from BG/P to BG/Q”. Prof. Dr. Attilio Cucchieri – IFSC/USP.

Technology