200. Section VII The Standard Model TexPoint fonts used in EMF.
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201. The Standard Model Spin ½ Fermions Charge (units of e) LEPTONS -1
QUARKS
PLUS antileptons and antiquarks.
Spin 1 Bosons Mass (GeV/c2)
Gluon g 0 STRONG
Photon g EM
W and Z Bosons W, Z0 91.2/ WEAK
The Standard Model also predicts the existence of a spin 0
HIGGS BOSON
which gives all particles their masses via its interactions.
Charge (units of e) -1
2/3
-1/3

202. Theoretical Framework
Macroscopic Microscopic
Slow Classical Mechanics Quantum Mechanics
Fast Special Relativity Quantum Field Theory
The Standard Model is a collection of related GAUGE THEORIES which are QUANTUM FIELD THEORIES that satisfy LOCAL GAUGE INVARIANCE.
ELECTROMAGNETISM: QUANTUM ELECTRODYNAMICS (QED)
1948 Feynman, Schwinger, Tomonaga (1965 Nobel Prize)
ELECTROMAGNETISM: ELECTROWEAK UNIFICATION
+WEAK Glashow, Weinberg, Salam (1979 Nobel Prize)
STRONG: QUANTUM CHROMODYNAMICS (QCD)
1974 Politzer, Wilczek, Gross (2004 Nobel Prize)

203. ASIDE : Relativistic Kinematics
Low energy nuclear reactions take place with typical kinetic energies of order 10 MeV << nucleon rest energies ) non-relativistic formulae OK.
In particle physics kinetic energy is of O (100 GeV) >> rest mass energies so relativistic formulae usually essential.
Always treat -decay relativisticly (me ~ 0.5 MeV < 1.3 MeV ~ mn-mp).
WE WILL ASSUME UNDERSTANDING OF FOUR VECTORS and FOUR MOMENTA from the Part II Relativity Electrodynamics and Light course ... in natural units:
Definitions: mass = m velocity= v = βc(= β) mom = p
Consequences:

204. ASIDE cont: (Decay example)
Consider decay of charged pion at rest in the lab frame
Conservation of Energy and Momentum says:
Appropriate choice of axes gives, and noting pion is at rest (pπ=0) gives:
So have two simultaneous equations in two unknowns &
Once you know p you know E etc via etc ….
(Simplify by taking mν=0 )

205. ASIDE cont : Relevance of Invariant mass
In Particle Physics, typically many particles are observed as a result of a collision. And often many of these are the result of decays of heavier particles. A common technique to identify these is to form the “invariant mass” of groups of particles.
e.g. if a particle X!1+2+3…n where we observe particles 1 to n, we have:
J/y
In the specific (and common) case of a two-body decay, X!1+2, this reduces to the same formula as on a preceding slide, i.e.
Typical invariant mass histogram:

206. ASIDE cont : Colliders and √s
Consider the collision of two particles:
The invariant quantity
 is the angle between the momentum three-vectors
is the energy in the zero momentum (centre-of-mass) frame;
It is the amount of energy available to the interaction e.g. in particle-antiparticle annihilation it is the maximum energy/mass of particle(s) that can be produced

207. Fixed Target Collision
For
e.g. 450 GeV proton hitting a proton at rest:
Collider Experiment
For then and  =
e.g. 450 GeV proton colliding with a 450 GeV proton:
In a fixed target experiment most of the proton’s energy is wasted providing forward momentum to the final state particles rather than being available for conversion into interesting particles.

208. ASIDE over
Back to the main topic of combining Quantum Mechanics with Special Relativity…..

209. Problems with Non-Relativistic Schrodinger Equation
To describe the fundamental interactions of particles we need a theory of RELATIVISTIC QUANTUM MECHANICS.
Schrodinger Equation for a free particle comes from classical
or equivalently promoted to an operator equation
using energy and momentum operators:
giving which has plane wave solutions:
Schrodinger Equation:
1st Order in time derivative
2nd Order in space derivatives
Schrodinger equation cannot be used to describe the physics of relativistic particles.
ħ =1 natural units
Not Lorentz Invariant!

210. Klein-Gordon Equation (relativistic)
From Special Relativity:
From Quantum Mechanics:
Combine to give:
KLEIN-GORDON
EQUATION
Second order in both space and time derivatives and hence Lorentz invariant.
Again plane wave solutions
but this time requiring allowing
Negative energy solutions required to form complete set of eigenstates
ANTIMATTER

211. Antimatter
Feynman-Stückelberg interpretation: The negative energy solution is a negative energy particle travelling forwards in time. And then, since:
 Interpret as a positive energy antiparticle travelling forwards in time.
Then all solutions describe physical states with positive energy, going forward in time.
Examples: e+e- annihilation pair creation e+ g g e+ T T
All quantum numbers carried
into vertex by e+, same as if
viewed as outgoing e-.

212. More particle/anti-particle examples
In first diagram, the interpretation easy as the first photon emitted has less energy than the electron it was emitted from. No need for “anti-particles” or negative energy states.
In the second diagram, the emitted photon has more energy than the electron that emitted it. So can either view the top vertex as “emission of a negative energy electron travelling backwards in time” or “absorption of a positive energy positron travelling forwards in time”. g T g T

213. Normalization
Can show solutions of Schrodinger Equation (non-relativistic) satisfy:
and solutions of Klein-Gordon Equation (relativistic) satisfy:
Both are of the form:
and the divergence theorem tells us that
this means that D is conserved, where:
If we interpret D as number of particles, and  as number of particles per unit volume, then:
For plane wave of arbitrary normalization find that:
Non-relativistic: particles per unit volume.
Relativistic: particles per unit volume.

214. Interaction via Particle Exchange
Consider two particles, fixed at and , which exchange a particle of mass m.
Calculate shift in energy of state i due to this process (relative to non-interacting theory) using 2nd order perturbation theory:
Work in a box of volume L3, and normalize s.t.
i.e. use
2E particles per box!
Space
Time
state i state j state i
Sum over all possible states j
with different momenta.

215. Consider ( transition from i to j by emission of m at )
represents a free particle with
Let g be the “strength” of the emission at Then:
Similarly is the transition from j to i at and is
Substituting these in, we see the shift in energy state is
Original 2 particles

216. Normalization of source strength “g” for relativistic situations
Previously normalized wave-functions to 1 particle in a box of side L.
In relativity, the box will be Lorentz contracted by a factor g
i.e particles per volume V. (Proportional to 2E particles per unit volume as we already saw a few slides back)
Need to adjust “g” with energy (next year you will see “g” is “lorentz invariant” matrix element for interaction)
Conventional choice: (square root as g always occurs squared)
Source “appears” lorentz contracted to particles of high energy, and effective coupling must decrease to compensate. The presence of the E is important. The presence of the 2 is just convention. The absence of the m (in gamma) is just a convention to keep g dimensionless here.
Rest frame
1 particle per V
Lab. frame
1 particle per V/g

217. Beginning to put it together
Different states j have different momenta . Therefore sum is actually an integral over all momenta:
And so energy change:

218. Final throes The integral can be done by choosing the z-axis along .
Then and so
Write this integral as one half of the integral from - to +, which can be done by residues giving
Appendix D

219. Can also exchange particle from 2 to 1:
Get the same result:
Total shift in energy due to particle exchange is
ATTRACTIVE force between two particles which decreases exponentially with range r.
Space
Time
state i state j state i

220. Yukawa Potential YUKAWA POTENTIAL Characteristic range = 1/m
(Compton wavelength of exchanged particle)
For m0, infinite range
Yukawa potential with m = 139 MeV/c2 gives good description of long range interaction between two nucleons and was the basis for the prediction of the existence of the pion.
Hideki Yukawa
1949 Nobel Prize

221. Scattering from the Yukawa Potential
Consider elastic scattering (no energy transfer)
Born Approximation
Yukawa Potential
The integral can be done by choosing the z-axis along , then
and
For elastic scattering, and exchanged massive particle is highly “virtual”
q2 is invariant
“VIRTUAL MASS”

222. Virtual Particles
Forces arise due to the exchange of unobservable VIRTUAL particles.
The mass of the virtual particle, q2, is given by
and is not the physical mass m, i.e. it is OFF MASS-SHELL.
The mass of a virtual particle can by +ve, -ve or imaginary.
A virtual particle which is off-mass shell by amount Dm can only exist for time and range
If q2 = m2, then the particle is real and can be observed.
ħ =c=1 natural units

223. For virtual particle exchange, expect a contribution to the matrix element of
where
Qualitatively: the propagator is inversely proportional to how far the particle is off-shell. The further off-shell, the smaller the probability of producing such a virtual state.
For m  0 ; e.g. single g exchange
q2 0, very low energy transfer EM scattering
COUPLING CONSTANT
STRENGTH OF INTERACTION
PHYSICAL (On-shell) mass
VIRTUAL (Off-shell) mass
PROPAGATOR

224. Feynman Diagrams
Results of calculations based on a single process in Time-Ordered Perturbation Theory (sometimes called old-fashioned, OFPT) depend on the reference frame.
The sum of all time orderings is not frame dependent and provides the basis for our relativistic theory of Quantum Mechanics.
The sum of all time orderings are represented by FEYNMAN DIAGRAMS.
Space
Time
Time +
Space =
FEYNMAN
DIAGRAM

225. Feynman diagrams represent a term in the perturbation theory expansion of the matrix element for an interaction.
Normally, a matrix element contains an infinite number of Feynman diagrams.
But each vertex gives a factor of g, so if g is small (i.e. the perturbation is small) only need the first few.
Example: QED
Total amplitude
Fermi’s Golden
Rule
Total rate

226. Anatomy of Feynman Diagrams
Feynman devised a pictorial method for evaluating matrix elements for the interactions between fundamental particles in a few simple rules. We shall use Feynman diagrams extensively throughout this course.
Represent particles (and antiparticles):
and their interaction point (vertex) with a “ “.
Each vertex gives a factor of the coupling constant, g.
Spin ½
Quarks and Leptons
Spin 1
g, W and Z0 g

227. External Lines (visible particles)
Internal lines (propagators)
Spin ½
Particle
Incoming
Outgoing
Antiparticle
Spin 1
Spin ½
Particle
(antiparticle)
Spin 1
g, W and Z0 g
Each propagator gives
a factor of

228. q e p u n d p e Examples: ELECTROMAGNETIC STRONG
Electron-proton scattering p
Quark-antiquark annihilation q
STRONG
WEAK u d e n p
Neutron decay

229. Summary
Relativistic mechanics necessary for most calculations involving Particle Physics and the Standard Model
Anti-particles arise from requiring wave equation be relativistically invariant
Calculating Matrix Elements from Perturbation Theory from first principles is a pain – so we don’t usually use it!
Need to do time-ordered sums of (on mass shell) particles whose production and decay does not conserve energy and momentum !
Feynman Diagrams / Rules manage to represent the maths of Perturbation Theory (to arbitrary order, if couplings are small enough) in a very simple way – so we use them to calculate matrix elements!
Approx size of matrix element may be estimated from the simplest valid Feynman Diagram for given process.
Full matrix element requires infinite number of diagrams.
Now only need one exchanged particle, but it is now off mass shell, however production/decay now conserves energy and momentum.

230. Section VIII
QED

231. QED
QUANTUM ELECTRODYNAMICS is the gauge theory of electromagnetic interactions.
Consider a non-relativistic charged particle in an EM field:
given in term of vector and scalar potentials
Maxwell’s Equations g
Change in state of e- requires change in field
 Interaction via virtual g emission

232. DEMAND that QED be a GAUGE THEORY Appendix E
Schrodinger equation
is invariant under the gauge transformation
 LOCAL GAUGE INVARIANCE
LOCAL GAUGE INVARIANCE requires a physical GAUGE FIELD (photon) and completely specifies the form of the interaction between the particle and field.
Photons (all gauge bosons) are intrinsically massless (though gauge bosons of the Weak Force evade this requirement by “symmetry breaking”)
Charge is conserved – the charge q which interacts with the field must not change in space or time.
DEMAND that QED be a GAUGE THEORY
Appendix E

233. The Electromagnetic Vertex
All electromagnetic interactions can be described by the photon propagator and the EM vertex:
The coupling constant, g, is proportional to the fermion charge.
Energy, momentum, angular momentum and charge always conserved.
QED vertex NEVER changes particle type or flavour
i.e but not or
QED vertex always conserves PARITY
STANDARD MODEL
ELECTROMAGNETIC
VERTEX g
+antiparticles

234. Pure QED Processes Compton Scattering (ge-ge-) e- g g e-
Bremsstrahlung (e-e-g)
Pair Production (g e+e-) e- g e- g e- g e- g e-
Nucleus
The processes e-e-g
and g e+e-cannot occur
for real e, g due to
energy, momentum
conservation. Ze
Nucleus

235. p0 u c J/Y e+e- Annihilation p0 Decay J/+-
The coupling strength determines “order of magnitude” of the matrix element. For particles interacting/decaying via EM interaction: typical values for cross-sections/lifetimes
sem ~ 10-2 mb
tem ~ s u p0 c
J/Y

236. Scattering in QED
Examples: Calculate the “spin-less” cross-sections for the two processes:
(1) (2)
Fermi’s Golden rule and Born Approximation (see page 48):
For both processes write the SAME matrix element
is the strength of the interaction.
measures the probability that the photon carries 4-momentum
i.e. smaller probability for higher mass. e p
Electron-proton scattering
Electron-positron annihilation

237. (1) “Spin-less” e-p Scattering
q2 is the four-momentum transfer:
Neglecting electron mass: i.e and
Therefore for ELASTIC scattering e e
RUTHERFORD
SCATTERING

238. Discovery of Quarks Virtual g carries 4-momentum
Large q  Large , small
Large , large
High q wave-function oscillates rapidly in space and time  probes short distances and short time.
Rutherford Scattering
q2 small
E = 8 GeV
Excited states
q2 increases
Expected Rutherford
scattering E
q2 large
l<< size of proton
q2 > 1 (GeV)2
Elastic scattering from
quarks in proton

239. (2) “Spin-less” e+e- Annihilation
Same formula, but different 4-momentum transfer:
Assuming we are in the centre-of-mass system
Integrating gives total cross-section:

240. This is not quite correct, because we have neglected spin
This is not quite correct, because we have neglected spin. The actual cross-section (using the Dirac equation) is
Example: Cross-section at GeV
(i.e. 11 GeV electrons colliding with
11 GeV positrons)

241. The Drell-Yan Process d u
Can also annihilate as in the Drell-Yan process.
Example:
See example sheet (Question 25) d u

242. Experimental Tests of QED
QED is an extremely successful theory tested to very high precision.
Example:
Magnetic moments of e,  :
For a point-like spin ½ particle: Dirac Equation
However, higher order terms introduce an anomalous magnetic
moment i.e. g not quite 2. v v
   
O(a)
O(a4)
12672 diagrams
O(1)

243. O(a3)
Agreement at the level of 1 in 108.
QED provides a remarkable precise description of the electromagnetic interaction !
Experiment
Theory

244. +     +     Higher Orders
So far only considered lowest order term in the perturbation series. Higher order terms also contribute
Second order suppressed by a2 relative to first order. Provided a is small, i.e. perturbation is small, lowest order dominates.
Lowest
Order
+    
Second
Order
Third
Order
+    

245. Running of 
specifies the strength of the interaction between an electron and a photon.
BUT a is NOT a constant.
Consider an electric charge in a dielectric medium.
Charge Q appears screened by a halo of +ve charges.
Only see full value of charge Q at small distance.
Consider a free electron.
The same effect can happen due to quantum
fluctuations that lead to a cloud of virtual e+e- pairs
The vacuum acts like a dielectric medium
The virtual e+e- pairs are polarised
At large distances the bare electron charge is screened.
At shorter distances, screening effect reduced and see a larger effective charge i.e. a. -Q + - e+ e- g

246. Measure a(q2) from etc
a increases with increasing q2
(i.e. closer to the bare charge)
At q2=0: a=1/137
At q2=(100 GeV)2: a=1/128

247. Summary
QED is the physics of the photon + “charged particle” vertex:
Every Feynman vertex has:
An arrow going in (lepton or quark)
An arrow going out (lepton or quark)
And a photon
Does not change the type of lepton or quark “passing through”
Conserves charge, energy and momentum.
The coupling α “runs”, and is proportional to the electric charge of the lepton or quark
QED has been tested at the level of 1 part in 108. g