Introduction to Quantum Chromodynamics

I. Color Degree of Freedom of Quarks and Gluons

II. Asymptotic Freedom

III.Color Confinement

IV. Chiral Symmetry

V. Lattice QCD

I. Color Degree of Freedom of Quarks and GluonsColors of quarks

•A red quark emitting a red-anti-blue gluon to leave a blue quark.

•The quark flavor(i.e., u,d,s,c,b,or t) does not change during this process, since gluons carry no flavor.

Colors of gluons

What is the Evidence for Color?One of the most convincing arguments for color comes from a comparison of thecross sections for the two processes:

qqeeee andIf we forget about how the quarks turn into hadrons then the graphs (amplitudes) for the two reactions only differ by the charge of the final state fermions (muons or quarks). We areassuming that the CM energy of the reaction is large compared to the fermion masses.

n

iiQ

eeqqeeR

1

2

)()(

n

iiQ

eeqqeeR

1

23)(

)(

If color plays no role in quark production then the ratioof cross sections should only depend on the charge (Q) of the quarks that are produced.

However, if color is important for quark production then the above ratio should bemultiplied by the number of colors (3).

For example, above b-quark threshold butbelow top-quark threshold we would expect:

911])

31()

32()

31()

32()

31[( 22222 R

or, if color exists:67.3

933])

31()

32()

31()

32()

31[(3 22222 R

QCD, Color, and the decay of the π0

0

u, d

u, d

u, d

1949-50 The decay 0 calculated and measured by Steinberger.1967 Veltman calculates the 0 decay rate using modern field theory and finds that

the 0 does not decay!1968-70 Adler, Bell and Jackiw “fix” field theory and now 0 decays but decay

rate is off by factor of 9.1973-4 Gell-Mann and Fritzsch (+others) use QCD with 3 colors and calculate

the correct 0 decay rate.

Triangle DiagramEach color contributes oneamplitude. Three colorschanges the decay rate by 9.

Quantum Chromodynamics (QCD) is the theory of the strong interaction.QCD is a non-abelian gauge theory invariant under SU(3) and as a result:a) The interaction is governed by massless spin 1 objects called “gluons”.b) Gluons couple only to objects that have “color”: quarks and gluonsc) There are three different charges (“colors”): red, green, blue.

Note: in QED there is only one charge (electric).d) There are eight different gluons.

gluon exchange can change the color of a quark but not its flavor.e.g. a red u-quark can become a blue u-quark via gluon exchange.

e) Since gluons have color there are couplings involving 3 and 4 gluons.Note: In QED the 3 and 4 photon couplings are absent since the photon does not have an electric charge.

u

u u grb

u

II. Asymptotic Freedom

Basic Nature of QCDThere are several interesting consequences of the SU(3), non-abelian nature of QCD:a) Confinement (Quarks are confined in space)

We can never “see” a quark the way we can an electron or proton.Explains why there is no experimental evidence for “free” quarks.

b) All particles (mesons and baryons) are color singlets.This “saves” the Pauli Principle. In the quark model the ++ consists of 3 up quarks in a totally symmetric state.Need something else to make the total wavefunction anti-symmetric color!

c) Asymptotic freedom.The QCD coupling constant changes its value (“runs”) dramatically as function of energy.As a result quarks can appear to be “free”when probed by high energy (virtual) ’sand yet be tightly bound into mesonsand baryons (low energy).

d) In principle, the masses of mesons andbaryons can be calculated using QCD.But in reality, very difficult to calculate (almost)anything with QCD.

Another nature of QCD is the Lagrangian of QCD is almost Chiral Symmetric (for ligh quarks) and the hadrons obtain their mass from spontaneously broking of the chiral symmetry

Physics Today Online, F. Wilczek

QED Vs QCDQED is an abelian gauge theory with U(1) symmetry:

QCD is a non-abelian gauge theory with SU(3) symmetry:Both are relativistic quantum field theories that can be described by Lagrangians:

uvuv

uu

uu FFAemiL

41)(

uva

auv

aujka

ujkjku

ujk GGGqqgqmiqL

41)()(

QED:

m=electron mass=electron spinor electron-

interaction

Au=photon field (1)Fuv=uAv-vAu

m=quark massj=color (1,2,3)k=quark type (1-6)q=quark spinor

quark-gluoninteraction Ga

u=gluon field (a=1-8)Ga

uv=u Gav -v Ga

u-gfabcGbu Gc

v

QCD:

a’s (a=1-8) are the generators of SU(3). a’s are 3x3 traceless hermitian matrices. See page 314 for a representation.

[a, b]=ifabcc

fabc are real constants (256)fabcstructure constants of the group (See page 314)

gluon-gluoninteraction(3g and 4g)

),(),( ),( txetx txief

),(),(

8

1 2),(

txetx i

ii txig

In QED higher order graphs with virtual fermion loops play an important role indetermining the strength of the interaction (coupling constant) as a function ofdistance scale (or momentum scale).

In QCD similar higher order graphs with virtual loops also play an important role indetermining the strength of the interaction as a function of distance scale.

Examples of graphs with virtual fermion loops coupling to virtual photons.

A graph with a virtual fermion loop coupling to virtual gluons.

In QCD we can also have a class of graphs with gluon loops since the gluons carry color.

Because of the graphs with gluon loops the QCD coupling constant behaves differentlythan the QED coupling constant at short distances.

QED Vs QCD

For both QED and QCD the effective coupling constant dependson the momentum (or distance) scale that it is evaluated at:

)(1)0()( 2

2

pXp

For QED it can be shown that:

)ln(3

)())(()( 2

222

1

2

p

eqpX

fN

i

i

(0)=fine structure constant 1/137

The situation for QCD is very different than QED. Due to the non-abelian natureof QCD we find:

Here Nf is the number of fundamental fermions with masses below ½|p|and is the mass of the heaviest fermion in the energy region being considered.

22 22

22

2

2

12( ) , (11 2 ) l

( )( ) ln( )[2 11 ]12 n( )

ps

c

c f

f s ppN N

pX p N N

Here Nf is the number of quark flavors with masses below ½|p|, is the mass of the heaviest quark in the energy region being considered, and Nc is the number of colors (3).For 6 flavors and 3 colors 2Nf-11Nc < 0 and therefore (p2) decreases with increasingmomentum (or shorter distances).The force between quarks decreases at short distances and increases as the quarks moveapart!! asymptotic freedom

QED: X(p2)>0

QED Vs QCD

• Coupling only increases for very small separations < 10-16.

• Coupling increases at large separations and decreases at small r.

QED QCDQ

r

Q

r

Asymptotic Freedom: As the distance becomes asymptotically small the interactions between quarks becomes asymptotically free.

QED Vs QCD

Experimental evidence for “asymptotic freedom” in QCD is convincing:

were

III. Color Confinement

• The strong force between two quarks arises from the exchange of gluons.

• There is also a strong force between two gluons - since they carry color themselves they can interact with each other. This is not the case for photons in QED since they carry no electric charge.

• The force between two quarks is constant, i.e., independent of their separation. This corresponds to a linearly rising potential between them, which is referred to as a “string tension”.

• This force between colored objects is equivalent to a weight of approximately 10 tons! - This is why no free quarks or gluons are ever seen. QCD has the property of CONFINEMENT.

QCD and ConfinementThe fact that gluons carry color and couple to themselves leads to asymptotic freedom:

strong force decreases at very small quark-quark separation.strong force increases as quarks are pulled apart.

Quark confinement in an SU(3) gauge theory can only be proved analytically forthe 2D case of 1 space+1 time.However, detailed numerical calculations show that even in 4D (3 space+time) quarks configured as mesons and baryons in color singlet states are confined.

In SU(3) there are two conserved quantities (two diagonal generators).For SU(3)color these quantities are:

Yc = “color hypercharge”I3

c = the 3rd component of “color isospin”The quark color hypercharge and isospin assignments are

I3c Yc

r 1/2 1/3g -1/2 1/3

b 0 -2/3

By assuming that hadrons are color singlets we are requiring Yc = I3c =0

anti-symmetric under color exchange)(

31:mesons bbggrr

)(6

1:baryons 321321321321321321 gbrrbgrgbgrbbrgbgr

Yc and I3c change sign for anti-quarks

The only quark and anti-quark combinations with Yc = I3c =0 are of the form:

)0,()()3( npqqq np

Our baryons are the states with p=1, n=0 and our mesons are states with p=0, n=1.States such as qqqq and qq are forbidden by the singlet (confinement) requirement.However, the following states are allowed:

1,12,00,2

npqqqqqnpqqqqnpqqqqqq

There have been many experimental searches for quark bound states other than the conventional mesons and baryons: Exotic Hadrons!

The color part of the quark wavefunction can be represented by:

100

010

001

bgr

200010001

31

31

000010001

21

21

833cc YI

What about bound states of gluons?“glueballs”

QCD and Confinement

Exotic Hadrons: Pentaquark Baryon

____

More experiments indicates the disappearnce !!!

Exotic Hadrons: Pentaquark Baryon

“Gluons” are well-named since they lead to the confinement of color particles - no free colored objects have EVER been seen in nature.

What happens when we try to pull apart two colored objects by pumping more and more energy into the system, e.g., through energetic collisions at a particle accelerator?

•When the energy put in is large enough to make a quark - anti-quark pair, then a meson can be created and the string is “broken”.•Confinement survives however as no free color particles are produced.

String Breaking: Quark - Anti-quark Pairs

庄子天下篇 ~ 300 B.C.一尺之棰，日取其半，万世不竭

Take half from a foot long stick each day,You will never exhaust it in million years.

q q

q qq q

QCD Produces 98% of Your Mass•Proton mass is 938 MeV = 0.938 GeV.

•Up and down quark masses are approximately 3 and 6 MeV respectively.Where does the rest of the hadron

mass come from?•The interactions in the low-energy (long-distance) are so strong (i.e., non-perturbative) that they induce a mass in the quarks of approximately 300 MeV. This is referred to as “dynamical mass generation”. This is how three quarks in a bound state can have a mass exceeding their naïve sum.

•The fact that the up and down quarks are so light and because of this mass-generation, Goldstone’s theorem states that there will be nearly massless Goldstone bosons associated with the spontaneous breaking of this symmetry - these Goldstone bosons are the pions which are anomalously light mesons. This is the basis of what is called “chiral symmetry” and “dynamical chiral symmetry breaking”. (Spontaneous symmetry breaking)

IV. Chiral Symmetry

Lattice Gauge Theory - A Non-Perturbative QCD• Physicists at the CSSM (Centre for the Subatomic Structure of

Matter ) and elsewhere use the techniques of lattice gauge theory to put all of four-dimensional space-time on a grid or lattice.

• We formulate the gauge field theory (e.g., QED, QCD, etc.) on this discrete lattice with finite-difference techniques. This is done in Euclidean space-time for numerical reasons. We use methods adapted from statistical mechanics to study the physical properties of the theory from the confining to the asymptotically free regimes.

• We use this to study and model subatomic particles and their interactions. It is VERY computationally expensive to do this -Teraflop-years of computer time are needed.

• More accurate results need a finer lattice with larger space-time volumes. This is what makes it computationally costly.

IV. Lattice QCD

High-Performance Computing

• Traditionally supercomputers were very expensive, contained purpose-built hardware, and were obsolete within 5 years.

• Most modern high-performance computing uses cost-effective clusters of mass-produced commodity off-the-shelf (COTS) hardware, rather than expensive proprietary hardware.

• These clusters of workstations or PC’s are connected either by commodity Fast Ethernet or Gigabit switched networking or by (more expensive) special ultrafast, low-latency networks for clustering (e.g., Myrinet, ServerNet, GigaNet, SCI, etc).

• Clusters are flexible in their design and easy to build and upgrade -e.g., add more nodes; upgrade some or all of the nodes; upgrade some or all of the networking.

• Can get an order of magnitude better price/performance ratio!

Orion Supercomputer •110/(144 peak) Gflops, 160 Gigabytes RAM, 640 Megabytes cache memory.•Orion is a Sun Technical Compute Farm with 40 Sun E420R 4-way SMP nodes (160 processors). Fast switched ethernet AND high-speed Myrinet network. • Fastest computer in Australia when installed in June 2000 and now number 4.•The CSSM together with lattice theorists from UNSW and the University of Melbourne obtained RIEF funds to seed the National Computing Facility for Lattice Gauge Theory (NCFLGT) which houses the Orion supercomputer.Built in partnership with

Sun Microsystems.

Homework(2018):

8.15,8.16