Particle Physics

Chris Parkes

Feynman Graphs of QFT

•Relativistic Quantum Mechanics

•QED

•Standard model vertices

•Amplitudes and Probabilities

•QCD

•Running Coupling Constants

•Quark confinement

2nd Handout

http://ppewww.ph.gla.ac.uk/~parkes/teaching/PP/PP.html

2

Adding Relativity to QM• See Advanced QM option

Free particle Em

2

2p Apply QM prescription ip

Get Schrödinger Equationdt

im

22

2Missing phenomena:Anti-particles, pair production, spin

Or non relativisticWhereas relativistically

m

pmvE

22

1 22

42222 cmcE p

22

2

2

2

1

mc

dtcKlein-Gordon Equation

Applying QM prescription again gives:

Quadratic equation 2 solutionsOne for particle, one for anti-particleDirac Equation 4 solutionsparticle, anti-particle each with spin up +1/2, spin down -1/2

3

PositronKG as old as QM, originally dismissed. No spin 0 particles known.Pion was only discovered in 1948.Dirac equation of 1928 described known spin ½ electron.Also described an anti-particle – Dirac boldly postulated existence of positron

Discovered by Anderson in 1933 using a cloud chamber (C.Wilson)

Track curves due to magnetic field F=qv×B

4

Transition Probabilityreactions will have transition probabilityHow likely that a particular initial state will transform to a specified final state

e.g. decays Interactions

We want to calculate the transition rate between initial state i and final state f,We Use Fermi’s golden rule

ik ki

fi EE

iHkkHfiHfT ....

'''

This is what we calculate from our QFT, using Feynman graphs

This tells us that fi (transition rate) is proportional to the transition matrix element Tfi squared (Tfi

2)

Transition rateProby of decay/unit time cross-section x incident flux

IV

5

Quantum ElectroDynamics (QED)• Developed ~1948 Feynman,Tomonaga,Schwinger

• Feynman illustrated with diagrams

e-

e-

Time: Left to Right. Anti-particles:backwards in time.

Process broken down into basic components.In this case all processes are same diagram rotated

e-

e+

e-

e+

Photon emission Pair productionannhilation

We can draw lots of diagrams for electron scattering (see lecture)

Compare with

ik ki

fi EE

iHkkHfiHfT ....

'''

c.f. Dirac hole theory M&S 1.3.1,1.3.2

6

Orders of • The amplitude T is the sum of all amplitudes from all

possible diagrams

Each vertex involves the emag coupling (=1/137) in its amplitude

Feynman graphs are calculational tools, they have terms associated with them

So, we have a perturbation series – only lowest order terms neededMore precision more diagrams

There can be a lot of diagrams! N photons, gives n in amplitudec.f. anomalous magnetic moment:

After 1650 two-loop Electroweak diagrams - Calculation accurate at 10-10 leveland experimental precision also!

7

The main standard model vertices

s

Strong:All quarks (and anti-quarks)No change of flavour

EM:All charged particlesNo change of flavour

Weak neutral current:All particlesNo change of flavour

Weak charged current:All particlesFlavour changes

W

At low energy:

29

1137

1

1.0

W

s

8

Amplitude Probability

If we have several diagrams contributing to same process, we much consider interference between them e.g.

e-

e-

e+e+

(a) (b)

Same final state, get terms for (a+b)2=a2+b2+ab+ba

e-

e+ e+

e-

|Tfi|2The Feynman diagrams give us the amplitude, c.f. in QM whereas probability is ||2

(1)

(2)

So, two emag vertices: e.g. e-e+ -+ amplitude gets factor from each vertex And xsec gets amplitude squared

for e-e+ qq with quarks of charge q (1/3 or 2/3)

222)( qq

2

•Also remember : u,d,s,c,t,b quarks and they each come in 3 colours•Scattering from a nucleus would have a Z term

9

Massive particle exchangeForces are due to exchange of virtual field quanta (,W,Z,g..)

E,p conserved overall in the process but not for exchanged bosons.

You can break Energy conservation - as long as you do it for a short enough time that you don’t notice!

i.e. don’t break uncertainty principle.

X

B

A

Consider exchange of particle X, mass mx, in CM of A:

),(),()0,( pp XAA EXEAmA

X

X

AAX

mE

pmE

ppE

mEEE

0:

:2

For all p, energy not conservedUncertainty principle

cmEcc x// Particle range R

So for real photon, mass 0, range is infiniteFor W (80.4 GeV/c2) or Z (91.2 GeV/c2), range is 2x10-3 fm

10

Virtual particlesThis particle exchanged is virtual (off mass shell)

e-

e+ +

-

e.g. (E,p)

(E,-p)0

0

2

222*

p

p

Em

EE

(E , p)

symmetricElectron-positroncollider

Yukawa PotentialYukawa PotentialStrong Force was explained in previous course as neutral pion exchangeConsider again:

•Spin-0 boson exchanged, so obeys Klein-Gordon equationSee M&S 1.4.2, can show solution is

r

egrV

Rr /2

4)(

R is range

For mx0, get coulomb potentialr

erV

1

4)(

0

2

Can rewrite in terms of dimensionless strength parameter

c

gX

4

2

11

Quantum Chromodynamics (QCD)QED – mediated by spin 1 bosons (photons) coupling to conserved electric chargeQCD – mediated by spin 1 bosons (gluons) coupling to conserved colour charge

u,d,c,s,t,b have same 3 colours (red,green,blue), so identical strong interactions[c.f. isospin symmetry for u,d], leptons are colourless so don’t feel strong force

•Significant difference from QED:• photons have no electric charge• But gluons do have colour charge – eight different colour mixtures.

7.1 M&S

Hence, gluons interact with each other. Additional Feynman graph vertices:

3-gluon 4-gluon

These diagrams and the difference in size of the coupling constants are responsible for the difference between EM and QCD

s

Self-interaction

12

Running Coupling Constants - QED

+Q+ - - +

+ -

+

-

+ -

+ -

+ - + -

Charge +Q in dielectric mediumMolecules nearby screened,At large distances don’t see full chargeOnly at small distances see +Q

Also happens in vacuum – due to spontaneous production of virtual e+e- pairs

And diagrams with two loops ,three loops…. each with smaller effect: ,2….

e+

e-

e+ e-

As a result coupling strength grows with |q2| of photon, higher energy smaller wavelength gets closer to bare charge |q2|

1/1371/128

0 (90GeV)2

QED – small variation

13

Coupling constant in QCD•Exactly same replacing photons with gluons and electrons with quarks•But also have gluon splitting diagrams

ggg

g

This gives anti-screening effect.Coupling strength falls as |q2| increases

Grand Unification ?

Strong variation in strong couplingFrom s 1 at |q2| of 1 GeV2

To s at |q2| of 104 GeV2LEP data

Hence:•Quarks scatter freely at high energy

•Perturbation theory converges very Slowly as s 0.1 at current exptsAnd lots of gluon self interaction diagrams

14

Range of Strong ForceGluons are massless, hence expect a QED like long range forceBut potential is changed by gluon self coupling

NB Hadrons are colourless, Force between hadrons due to pion exchange. 140MeV1.4fm

QED QCD

-+

Standard EM fieldField lines pulled into stringsBy gluon self interaction

Qualitatively:

QCD – energy/unit length stored in field ~ constant.Need infinite energy to separate qqbar pair.Instead energy in colour field exceeds 2mq and new q qbar pair created in vacuum

This explains absence of free quarks in nature.Instead jets (fragmentation) of mesons/baryons

Form of QCD potential:

Coulomb like to start with, but on ~1 fermi scale energy sufficient for fragmentation

q q

krr

V sQCD

34

15

Formation of jets1. Quantum Field Theory – calculation2. Parton shower development3. Hadronisation

16

Summary1. Add Relativity to QM anti-particles,spin2. Quantum Field Theory of Emag – QED

• Feynman graphs represent terms in perturbation series in powers of α

• Couples to electric charge

3. Standard Model vertices for Emag, Weak,Strong• Diagrams only exist if coupling exists

• e.g. neutrino no electric charge, so no emag diagram

4. QCD – like QED but..• Gluon self-coupling diagrams• α strong larger than α emag

5. Running Coupling Constants• α strong varies, perturbation series approach breaks down

6. QCD potential – differ from QED due to gluon interactions• Absence of free quarks, fragmentation into colourless hadrons

Now, consider evidence for quarks, gluons….