Particle Physics 1 - PPD - STFC Particle Physics Department

Particle Physics 1 𝒈

𝝂𝝁

𝝎

𝝓

𝑲+ 𝒆

𝑩

𝑾±

𝑱/𝝍

𝝁

𝒒 𝒁𝟎

𝝌 𝝂𝝉

𝜸

𝑯𝟎

𝝂𝒆 𝝉

Prof. Glenn Patrick U23525, Particle Physics, Year 3 University of Portsmouth, 2013 - 2014

1

Apologies

Course should have started last week, but I was at CERN. We have plenty of time to cover the material over 2 teaching blocks. 2

Course Outline Major particle physics option (U23525), but some overlap with particle component of 2nd year quantum course. 1 The Standard Model of Particle Physics.

2 Strong Interactions and Quantum Chromodynamics (QCD). 3 Electromagnetic interactions including fundamental electronpositron process. 4 Weak and Electroweak Interactions. 5 Antimatter and Quark Flavour Physics. 6 Neutrino Physics. 7 Frameworks for Beyond the Standard Model (BSM) physics. 8 High Energy Particle Accelerators and Beams.

9 Particle Detectors. 10 Particle Astrophysics.

3

Preliminaries - Assessment 40 hours of lectures across two teaching blocks plus 8 hours of tutorial classes. The main aim is to improve your understanding of fundamental physics. However, we cannot forget the small matter of your degree…. 1 Final written examination (2 hours)

– 80%

2 Coursework questions and problems

– 20%

   

Main thing is that you enjoy the course. We will try and focus on understanding the underlying concepts. Extra material/maths shown mainly to aid understanding. Guidance will be given over essential knowledge needed for exam. 4

Telling the Difference

It is important to also attempt any non-assessed questions and do some background reading as this will be the best way of checking your understanding of material and prepare for assessments. Otherwise, how do you tell the difference? 5

Preliminaries - Books  Particle Physics, B.R. Martin and G. Shaw, 3rd edition, Wiley, ISBN: 978-0-470-03293-0 Main book we are following – at least at the start.  Modern Particle Physics, Mark Thomson, New! Cambridge University Press, ISBN-13: 978-1107034266 New book – only just appeared (Sep 2013) – looks good.  Introduction to High Energy Physics, D.H. Perkins, 4th edition, Cambridge, ISBN: 9780521621960 Good, classic and readable text – a bit dated.  Introduction to Elementary Particles, David Griffiths, Wiley VCH, 2nd revised edition, ISBN-13:978-3527406012 Good clear text on theoretical aspects.  Quarks & Leptons, Francis Halzen & Alan D. Martin, John Wiley, ISBN: 0471887412 Advanced for this course, but some sections very good.  The Experimental Foundations of Particle Physics, Robert Cahn & Gerson Goldhaber, 2nd edition, ISBN: 9780521521475

6

Supplementary Books - Later When we get deeper into the course, there are some supplementary books for more specialised topics. I will remind you about them later.  Particle Astrophysics, D.H. Perkins, 2nd edition, Oxford, ISBN: 978-0-19-850951-6  The Physics of Particle Accelerators, Klaus Wille, Oxford, ISBN: 978-0-19-850549-5  An Introduction to Particle Accelerators, Edmund Wilson, Oxford, ISBN: 978-0-19-850829-8  Particle Detectors, Claus Grupen & Boris Shwarz, 2nd edition, Cambridge, ISBN:9780521187954  The Physics of Particle Detectors, Dan Green, ISBN: 9780521675680 The lecture slides and material should, however, be sufficient for your study. 7

Preliminaries - Course Material WEB PAGE for material http://hepwww.rl.ac.uk/gpatrick/portsmouth/courses.htm MOODLE When set up.

8

Timetable – Teaching Block 1 Week Date

Start

Finish

Building

Room

Size

9

26.09.2013

09:00

11:00

Buckingham (BK)

3.05

35

10

03.10.2013

09:00

11:00

Buckingham (BK)

3.05

35

11

10.10.2013

09:00

11:00

Portland (PO)

0.36

35

12

17.10.2013

09:00

11:00

Portland (PO)

0.36

35

13

24.10.2013

09:00

11:00

Portland (PO)

0.36

35

14

31.10.2013

09:00

11:00

Portland (PO)

0.36

35

15

07.11.2013

09:00

11:00

Buckingham (BK)

3.05

35

16

14.11.2013

09:00

11:00

Buckingham (BK)

3.05

35

17

21.11.2013

09:00

11:00

Portland (PO)

0.36

35

18

28.11.2013

09:00

11:00

Portland (PO)

0.36

35

19

05.12.2013

09:00

11:00

Burnaby (BB)

2.24

35

20

12.12.2013

09:00

11.00

Anglesea (AA)

2.06

35

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



 



 9

Timetable – Teaching Block 2 Week Date

Start

Finish

Building

Room

Size

24

09.01.2014

09:00

11:00

Park (PK)

2.01

35

25

16.01.2014

09:00

11:00

Park (PK)

2.01

35

26

23.01.2014

09:00

11:00

St. Andrew’s Court (SA)

0.04

35

27

30.01.2014

09:00

11:00

Lion Gate

2.05

35

28

06.02.2014

09:00

11:00

Lion Gate

2.05

35

29

13.02.2014

09:00

11:00

Lion Gate

2.05

35

30

20.02.2014

09:00

11:00

Buckingham (BK)

3.04

35

31

27.02.2014

09:00

11:00

Buckingham (BK)

3.04

35

32

06.03.2014

09:00

11:00

Park (PK)

3.23

35

33

13.03.2014

09:00

11:00

Buckingham (BK)

3.04

35

34

19.03.2014

09:00

11:00

Buckingham (BK)

3.04

35 Wed!

35

27.03.2014

09:00

11:00

Buckingham (BK)

3.04

35

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 





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Will try and start lectures at 09:05, but bear in kind that I have to travel 70 miles to Portsmouth. 10 I also teach the quantum/nuclear course on Thursdays (weeks 12-20).

Today’s Plan Particle Physics 1 Course Outline Preliminaries - Assessment Preliminaries - Books Preliminaries - Course Material Particle Physics, Cosmology & Particle Astrophysics Natural Units Rationalised Heaviside-Lorentz EM Units Special Relativity and Lorentz Invariance Mandelstam Variables Spin and Spin Statistics Theorem – Fermions and Bosons Addition of Angular Momenta – Clebsch Gordon Coefficients Crossing Symmetry and s, t & u Channels Non-Relativistic Quantum Mechanics (Schrödinger Equation) Relativistic Quantum Mechanics (Klein-Gordon Equation) Feynman-Stückelberg Interpretation of Negative Energy States 11

What you should remember from Year 2 Quantum Force Carriers [Bosons] (Spin 1)

Higgs Particles [Boson(s)] (Spin 0)

H0 First fundamental scalar particle. Nobel Prize next week?

Standard Model of Particle Physics 12

The Big Picture?

13

Fabric of the Universe Particle Physics is the study of the fundamental constituents of matter and the forces that hold them together.

14

Particle Physics and Energy Particle accelerators probe further and further back in time towards the Big Bang. Particle Physics Boltzmann Mean Energy

E  kT

where, k = 8.6 x 10-5 eV K-1

LHC energy=14 TeV T ~ 1.6 x 1017 K but focussed in a tiny space-time element. 15

Particle Physics and Time

Transparent to photons.

380,000 years T ~3,000 K Opaque to photons. ~1 sec after Big Bang, T ~1010 K Neutrinos Interact weakly & decoupled

4 1020 sK 2 texp  T2

LHC 16 ~10-14 s after Big Bang

Accelerator experiments can be complex LHC: big, cold, high energy Collimation Beam dumps

Injection B2

Injection B1

Collimation

RF

1720 Power converters > 9000 magnetic elements 7568 Quench detection systems 1088 Beam position monitors ~4000 Beam loss monitors

courtesy Mike Lamont

150 tonnes Helium, ~90 tonnes at 1.9 K 140 MJ stored beam energy in 2012 450 MJ magnetic energy per sector at 4 17 TeV

LHC is vital for 21st Century Physics and will rewrite some textbooks,

but there is much more to the subject…

18

Particle Astrophysics Particle Astrophysics (or Astroparticle Physics) is a branch of particle physics that studies elementary particles of astronomical origin and their relation to astrophysics and cosmology (Wikipedia) In practice, there are two types of particle astrophysics research project. 1) Direct detectors of particles from space or 'particle telescopes':  Gravity wave telescopes  Neutrino detectors and telescopes  Cosmic ray telescopes  Gamma ray telescopes  Dark matter detectors and telescopes  Other exotic particle searches (e.g. axions, magnetic monopoles) 2) Indirect detection of the effects of particles on astronomical objects:  Binary pulsars and pulsar timing measurements for gravity waves.  CMB measurements for dark matter, dark energy, neutrinos, gravity waves.  Timing of gamma ray and TeV emission for Lorentz invariance tests.  Non-accelerator methods for measuring/constraining the properties of fundamental particles. 19

“Small” specialised experiments still important HESS II Observatory (Namibia) Studies very high energy cosmic gamma rays

20

Theory and Experiment It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong. Richard P. Feynman

A theory is something nobody believes except the person who made it, An experiment is something everybody believes except the person who made it. Albert Einstein 21

Natural Units The fundamental constant of quantum mechanics is Planck’s constant, h, and the fundamental constant of special relativity is the velocity of light in vacuum, c:

h -34   1.055 x 10 J sec 2 c  2.998 x 108 m sec 1

In particle physics, we commonly measure energy (ML2/T2) in units of GeV (109 electron volts).

Unit of action (ML2/T) Unit of velocity (L/T)

Mass of proton ~ 1 GeV

These so-called natural units are therefore based on the fundamental constants of quantum mechanics and special relativity, i.e. ℏ, c and GeV. We can simplify matters even further by setting ℏ = 𝒄 = 𝟏 and then all quantities are expressed as powers of GeV.

22

Natural Units FUNDAMENTAL UNITS Quantity

S.I. unit [kg, m, s]

Natural [ħ, c, GeV]

Natural [ħ = c = 1]

Energy

kg m-2 s-2

GeV

GeV

Momentum

kg m s-1

GeV/c

GeV

Mass

kg

GeV/c2

GeV

Time

s

ħ/GeV

GeV-1

Length

m

ħc/GeV

GeV-1

Area

m2

(ħc/GeV)2

GeV-2 23

Natural Units - Converting Example: The root-mean-square charge radius of the proton is 𝟏 calculating using natural units to be: 𝟐 −𝟏 𝟐 𝒓

= 𝟒. 𝟏 𝑮𝒆𝑽

To convert to S.I. units, we just have to reinsert the missing factors of ħ and c: ℏ𝑐 𝐿𝑒𝑛𝑔𝑡ℎ = 𝐺𝑒𝑉 𝒓𝟐

𝟏

𝟐

1.055 × 10−34 × 2.998 × 108 −𝟏𝟓 𝒎 = 4.1 × = 𝟎. 𝟖 × 𝟏𝟎 1.602 × 10−10

-22 Conversion factors:   6.582 x10 MeV s c  1.973 x10-13 MeV m 1 GeV  109 eV  1.602 10 10 J

24

Rationalised Heaviside-Lorentz EM Units We will be dealing with the interactions between charges. These can be the familiar electric charge of electromagnetic interactions, or the strong charge of the strong interaction or the weak charge of the weak interaction. Usually, we will always be dealing with particles in vacuo. Therefore we use the rationalised Heaviside-Lorentz system of EM units where permittivity ε0 = 1 and permeability μ0 = 1. Coulomb’s Law then becomes:

𝒆𝟐 𝒆𝟐 𝑭= → 𝟐 𝟒𝝅𝜺𝟎 𝒓 𝟒𝝅𝒓𝟐 where 𝜀0 has been absorbed into the definition of the electron charge.

Maxwell’s Equations then become: 

  B   E     E   t    E B  0   B  J t

Fine structure constant, α :



e2 4 0 c SI

e2  4

HL

1 0   0c 2

1  137.035999074

Clearly, the numerical values of e are different in each system… 𝑒

𝑆𝐼

→ 𝑒√4𝜋

𝐻𝐿

25

Special Relativity

A proton with energy of “only” 1 GeV already has a velocity of ~0.9c. In particle physics, the consequences and effects of Einstein’s theory of Special Relativity have to be taken into account.

26

Special Relativity Fundamental laws have the same form in all Lorentz frames (i.e. reference frames which have a uniform relative velocity).

• • •

In particle physics we inevitably deal with relativistic particles. Need all calculations to be Lorentz invariant. Lorentz invariant quantities formed from scalar product of four-vectors.

e.g. Lorentz boost along x-axis

 ct   ct     (ct  x)         ct   x   x    ( x  ct )  x             y  x   y   y     z   z    z       Alternatively, in matrix form:

3

x    x 

 0

where

where,

       0   0 

v  c  

 0 0

Using the Einstein summation convention we can write (repeated indices summed over)



1 1  2

0 0  0 0 1 0  0 1 

x    x

27

Aside: Einstein Summation Convention A notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity. i. Omit summation signs. ii. If a suffix appears twice with one of the pair a superscript and the other a subscript, a summation is implied. e.g. 𝐴𝑖 𝐵𝑖 = 𝐴1 𝐵1 + 𝐴2 𝐵2 + 𝐴3 𝐵3 iii. If the index is a Greek letter, the summation extends over all 4 components (from 0 to 3), i.e. 4-vectors. iv. If the index is a Latin letter, the summation only extends over the 3 spatial components (1 to 3), i.e. 3 vectors. v. If a suffix appears only once, it can take any value. e.g. 𝐴𝑖 = 𝐵𝑖 holds for 𝑖 = 1,2,3 An index, vi. A suffix CANNOT appear more than twice. NOT a power. 3

e.g.

 0 1 2 3 x x  x x  x x  x x  x x   0 1 2 3  0

28

Special Relativity The interval ds2 is invariant: ds  c dt  dxi dxi  c dt   dxidxi 2

2

2

2

2

Defining space-time 4 vectors:

x   ( x 0 , x1 , x 2 , x 3 )

contravariant

x  g  x  ( x 0 , x1 , x 2 , x 3 )

covariant

where gμν is the metric tensor…

g  g 





We can now instead write: ds  g  dx dx 2

where μ = 0,1,2,3

1 0 0 0     0 1 0 0   0 0 1 0     0 0 0  1   Note : g   g is a symmetric tensor

Quantities such as ds2 are scalars and, just as in 3D space, we can talk of a scalar product of two 4-vectors, which are Lorentz invariant if they transform the same way as the space-time 4 vector :

A  B  A B   A B  g  A B  g  A B  A0 B0  A1 B1  A2 B2  A3 B3 Any expression that can be written in terms of 4-vector scalar products is 29 guaranteed to be Lorentz invariant.

Special Relativity In relativistic kinematics, we form a four vector, where energy plays the role of the “time” component. timelike

spacelike

 p   ( E , p)  ( E , px , p y , pz ) contravariant (four momentum)   p  g  p  ( E , p )  ( E , p x , p y , p z ) covariant We can form 4-vector scalar products, which are Lorentz invariant:

p  p  E 2  p 2  m02

  x p  Et  p  r 

Lorentz invariant since mo is the rest mass or invariant mass. Phase of a plane wave.

30

Mandelstam Variables Consider the kinematics of the general 2-body scattering process of the form 1 + 2 → 3 + 4

p1

p3

p2

p4

We have 4 four-vectors of the form:

 pi (i  1,2,3,4)  ( Ei , pi )

10 scalar products can be formed:

pi  p j where i  j

2 i 2 p  p p  m These are constrained by: 𝑝1 + 𝑝2 + 𝑝3 + 𝑝4 = 0 and i i i (i  1..4)

Conservation of momentum/energy

Mass shell condition

Only two independent variables are sufficient to describe a 2-body process (with no polarisation). However, it is convenient to define three Lorentz invariant quantities s, t and u – the Mandelstam Variables:

s  ( p1  p2 ) 2

i.e. square of total CMS energy (∴ 𝑠 = total energy in CMS).

t  ( p1  p3 ) 2

i.e. square of four momentum transfer between particles 1 & 3

u  ( p1  p4 ) 2

i.e. square of four momentum transfer between particles 1 & 4 4

They are connected by the sum:

s  t  u   mi2 i 1

31

Crossing Symmetry Principle of Crossing Symmetry states that the 3 channels:

1  2  3  4 (s - channel) 1  3  2  4 (t - channel) 1  4  3  2 (u - channel)

are described by a single transition matrix and can be related by the same analytic function of s, t and u for the scattering amplitude in all 3 channels.

The channels are denoted by the variable which is positive (i.e. timelike) for the channel in question. 1

3

2 4 s-channel

𝟑

1

4 𝟐 t-channel

1

3

𝟐 𝟒 u-channel 32

s, t and u channels Classify diagrams according to the four momentum of the exchanged particle (or “propagator” – see Lecture 2). s-channel t-channel u-channel (“Annihilation” Diagram)

p1

p3

p2

(“Scattering” Diagram)

p1

p3

p4

Example: 𝒆+ 𝒆− → 𝒆+ 𝒆−

e-

p3

p4 p2

e+

p1

e+

e+

𝜸

e+

p4

p2 e+

e-

𝜸

𝜸

ee-

e-

e-

e+ 33

Quantum Mechanics An electron has a size of 0) forward in time:

e   Ne i (  (  p )(  r )  (  E )(  t ))  Ne i (  p.r  Et ))

Pictorially:

e

e

E0

E0



time 46

Many Particle Cases Double scattering of an electron Two diagrams for the same observation with two different time orderings. In second pic, at time t2 the electron scatters backward in time with E < 0.

Can be interpreted as a positron with E > 0 going forward in time. First, at t1 an e-e+ pair is created, then at a later time, t2, the e+ annihilates with the incident e-. Between t1 and t2, the electron trajectory describes 3 particles: the initial and final electrons and a positron!

Photon-Particle Scattering

(a) Particle (E1) comes in and at t1/x1 emits photon ( with Eγ < E1). Travels forward in time & at t2/x2 absorbs initial state photon giving photon-particle final state. (b) Particle emits photon (with Eγ > E1) & is forced to travel backward in time. At earlier time it absorbs initial state photon rendering its energy positive again.

47

End

CONTACT Professor Glenn Patrick email: [email protected] → email: [email protected]

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Another View

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