The World of Elementary Particle Physics

Demystified

IFT - Perimeter - ICTP/SAIFR Journeysinto Theoretical Physics, July 18-23

Eduardo Pontón - IFT & ICTP/SAIFR

1. What are these lectures about

Plan

2. How do we know what we know

3. Field Theory and Symmetries

4. Hidden Symmetry and the Higgs Boson

Part IOur Subject Matter

Particle Physics

Gold Dust?Installation inspired by the search for the Higgs boson(by Becs Andrews)

Perseus Galaxy Cluster

Atoms of Niobium (41) and Selenium (34)

What do YOU mean??

Particle Physics typically understood as the realm of subatomic particles

Quantum mechanical effects an essential component!

It will be useful to recall some facts in the context of the birth of Quantum Mechanics!

hdE(!)i = V ~⇡2c3

!3d!

e�~! � 1= dN(!)

~!e�~! � 1

dN(!) =V !2d!

⇡2c3

Planck's law for black-body radiation:

[!,! + d!]Number of modes inV and

Einstein’s formula for mean energy fluctuations in a black body:

h(�E)2i = hdE(!)i2

dN(!)+ ~!hdE(!)i

Einstein, 1909

average energy in volume V with frequencies between ! ! + d!andFor radiation at temperature T, :

Planck, 1900

Z =X

n

e��EnCanonical Partition Function:

To derive it, just need to recall a bit of statistical mechanics:

Obtain average energy by differentiating w.r.t.� :

Differentiating w.r.t. � again, we can obtain the mean square energy fluctuation:

Upshot: can easily get the energy fluctuation, if we know the energy as a functionof temperature!

h(�E)2i = hE2i � hEi2 = �@hEi@�

hEi ⌘ 1

Z

X

n

En e��En

= �@ logZ

@�

hdE(!)i = dN(!)~!

e�~! � 1

dN(!) ~! e��~!�~! � 1 Wien’s formula

dN(!)

��~! ⌧ 1 Rayleigh-Jeans law

Einstein just applied this to Planck’s law:

h(�E)2i = hdE(!)i2

dN(!)+ ~!hdE(!)i

h(�E)2i = �@hEi@�

Then he proceeded to interpret the result…

and for any frequency:

h(�E)2i = hdE(!)i2

dN(!)+ ~!hdE(!)i

The quadratic term is exactly what is obtained for classical waves(Homework!)

To interpret the linear term

Recall that for an ideal gas, i.e. N non-interacting particles:

hEi = 3

2NkBT =

3N

2�

Hence:

h(�E)2i = �@hEi@�

= kBT hEi

The concept of wave-particle duality was born:

Attempts at obtaining this from dynamics (as time averages) could only give one or the other term…

“the effects of the two causes of fluctuations [waves and particles] act like fluctuations from mutually independent causes (additivity of the two terms)” — Einstein (1909)

vs

In our typical “particle physics” experiments, we see the events as particles: they resemble the electron-by-electron setup!

But we are interested in the underlying distribution (in different variables)

In addition, the experimental conditions we are focusing on are such that the particles are moving close to the speed of light!

Theoretician perspective: need to be able to compute quantum mechanical amplitudes that reflect the properties of special relativity:

• Nothing travels faster than the speed of light (causality structure)

• No inertial frame is special (laws appear the same in any)

• Amplitudes must allow a probabilistic interpretation (unitarity)

Quantum Mechanics + Special Relativity = Quantum Field Theory

If we want to be more precise: “Quantum Field Theory of Point Particles”

What if the fundamental constituents were string-like, or even more complex?

A “point particle” type specified by: mass & spin (or helicity, if massless)

A matter of scales: if our probe cannot resolve the string/brane, we will be able to describe its physics by QFT

The lesson is more general:

• Protons/nuclei can look point-like under many experimental conditions

• Atoms/molecules can look point-like to a typical human

QFT can be used to describe any such system…

… it has nothing to do with the system being “fundamental”

But QFT becomes essential when the energies are such that particles can be produced… the realm we want to explore here!

The Quantum Harmonic Oscillator(if you need a review)

H =1

2

�p̂2 + !2x̂2

�

x̂ =

r~2!

�a

† + a

�

p̂ = i

r~!2

�a† � a

�

H =

✓N +

1

2

◆~! N = a†a

Quadratic Hamiltonian:

Define creation and annihilation operators via:

Which diagonalize the Hamiltonian:

n-quanta/particle states obeying Bose-Einstein statistics:

[a, a†] = 1

[a, a] = [a†, a†] = 0

|1i = a†|0i

|ni = 1pn!(a†)n|0i

h(�E)2i = hdE(!)i2

dN(!)+ ~!hdE(!)i

Let us go back to the wave-particle puzzle

which had to wait until 1925 for a satisfactory solution (Pasqual Jordan in the Dreimännerarbeit)

Considering a 1D case:

Since the E&M field (in a box) is a set of decoupled simple harmonic oscillators, we can quantize each one, following the quantum mechanical treatment known to apply to a single oscillator.

[an, a†n0 ] = �n,n0 [an, an0 ] = [a†n, a

†n0 ] = 0

box of size L and wavevectors

�(x, t) =X

n

r~

2!n

�a

n

e

�i!nt+knx + h.c.�

(as Jordan did)kn = ⇡n/L = !n

Considering a 1D case:

[an, a†n0 ] = �n,n0 [an, an0 ] = [a†n, a

†n0 ] = 0

box of size L and wavevectors

�(x, t) =X

n

r~

2!n

�a

n

e

�i!nt+knx + h.c.�

The energy density is found to be

H =1

2L

ZL

0dx

⇥(@

t

�)2 + (@x

�)2⇤=

X

n

(Nn

+1

2)~!

n

h(�E)2i = hdE(!)i2

dN(!)+ ~!hdE(!)i

By putting the oscillators in a thermal bath at temperature T, and computing the mean square energy fluctuations in a small segment a << L, the two terms were recovered:

These considerations, together with the seminal papers by Dirac (1926,1927), gave birth to Quantum Field Theory…

For a detailed derivation see: Duncan & Janssen, arXiv:0709.3812

kn = ⇡n/L = !n

Upshot: the canonical quantization of a field (continuous) incorporates automatically the

concept of a particle (discreteness)

In addition: the formalism allows to describe the creation and destruction of quanta (particles)

(provided energy and momentum can be conserved)

Note also that the quantum mechanical treatment of the E&M field had to be intrinsically relativistic

Dimensional Analysis

• Existence of (maximum) universal speed:

The only difference between a rut and a grave are the dimensions — Ellen Glasgow

c = 299 792 458 m/s

• Universal unit of action: ~ = 6.582 119 514(40)⇥ 10�16 eV · s

Since we are interested in quantum, relativistic systems, better to measure velocities and action in units of c and !~

• Universal Boltzmann constant: kB = 8.617 3324 (78) eV/K

c = ~ = kB = 1Natural units:

“Everything is energy and that’s all there is to it. Match the frequency of the reality you want and you cannot help but get that reality. It can be no other way. This is physics.”Pseudo-scientific quote, probably from a “channeler“ named Darryl Anka who has assigned the words to an entity named Bashar!!

Compton wavelength:o = ~/(mc)

Then any mass/energy scale is associated with a length scale through the

and to a time scale through c.

Dimensional AnalysisSome useful reference scales:

• Most of the mass we see around us comes from the mass of protons/neutrons

Nucleon mass: mp ⇠ 1 GeV = 109eV ⇠ 10�27kg

Associated length scale:

Note: the “charge radius” of the proton is about 1 Fermi

Associated time scale: ⇠ 10�24s

Relevant energy scale is E0 =12↵

2me ⇠ 10 eV

and the length scale is a0 = oe/↵

op ⇠ (1/5) ⇥ 10�15m = (1/5) Fermi

• Sometimes dimensionless constants have a crucial effect on the physics. Compare to the hydrogen atom. The “underlying” mass scale is:

me ⇠ 0.5 MeV = 0.5 ⇥ 106eV ⇠ mp/2000 oe ⇠ 2000 ⇥ op

⇢

A big, non-relativistic system because↵ ⇠ 1/137 ⌧ 1

The Standard Model

Mendeleev’s Table

Particles and Particles

A Paradigm

Rutherford at his Lab.

Seeing Particles

Discovery of thePositron (1932)

C. Anderson

P.A.M. Dirac

Cloud chamber’s picture of cosmic radiation

⌦ ~B

(1931)

“Simpler” Times

Cloud Chamber

Alpha Particles (Polonium)

(muons, alpha and beta radiation)

16 GeV ⇡� beam entering a liquid-H2 bubble chamber at CERN (~1970)

The discovery of neutral currents by the Gargamelle bubble chamber (1973)

... A Long Way

http://cds.cern.ch/record/1165534/files/CERN-Brochure-2009-003-Eng.pdf

Find an LHC Guide/FAQ with lots of interesting information at: