Advanced Particle Physics

FK8022

David MilsteadThomas Schwetz-Mangold

Course aims• Breadth

– Build on Particle Physics 1 (FK7003)– More complete and up-to-date treatment of collider and non-collider

physics.

• Depth– Provide short derivations for phenomena rather than hand-waving

explanations as in PP1-level courses.– Derivations not always the most rigorous but are start-to-finish and

chosen to be pedagogically excellent.– Demystify important results that we take for granted but are often

poorly understood. Eg SU(3) , hadron multiplets, renormalisation etc.

Lecture Content Lecturer1 Standard Model (1) : symmetry groups A basic introduction to group symmetries is given. Starting with angular momentum and SU(2)-isospin, SU(3) is

explained as additional U,V spaces in SU(2). Well known results on hadron multiplets are then derived. Recycling the same mathematics, SU(3)-colour is tackled. Using the gluon wave functions, the properties of the short range strong force are studied. It is shown that quark-antiquark and three quark systems lead to attractive short-range potentials for colour singlet states whereas repulsive potentials are generally obtained for other configurations.

Dave

2 Standard Model (2) : renormalisation Starting with the QED result for the electron-muon scattering amplitude, it is shown that higher order electron-positron

loops lead to a divergence. The removal of the divergence through regularization and subtraction is then given, leading to a dependence of the amplitude on an arbitrary renormalization scale. An interpretation of the renormalization scale and a discussion of optimal scale choices are then given. Finally, there is a general discussion on the technique of renormalization.

Dave

3 The Standard Model (3) arity violation and V-A structure of weak interactions, Higgs mechanism Thomas4 Standard Model (4) The SM Lagrangian, the electro-weak sector, hyper-charge asignments, parameters of the SM, gauge sector versus Yukawa sector Thomas5 Experimental tests of the Standard Model (1): the strong force This is the first lecture in a series of four lectures

covering tests of the SM at colliders and non-colliders. The strong force is covered in this lecture. It is shown how different scattering environments (lepton-hadron,e+e-,hadron-hadron) provide complementary observables, as well as comparative advantages and disadvantages, for perturbative QCD tests. Representative examples of high profile and high precision measurements from each collider environment are given. It is also shown how the free parameter of the perturbative QCD sector, the strong coupling parameter, is measured in multiple studies. As a means of illustrating the need for a program of low and high energy colliders, the determination of the hadronic contribution to the electromagnetic coupling is also covered. The importance of this quantity for precision SM tests is subsequently covered in lectures (7) and (8).

Dave

6 Experimental tests of the Standard Model (2) : flavour physics and CP violation This

lecture focuses on tests of the weak sector, specifically quark flavor physics. A description of how CP violation arises in the SM via a complex phase is provided. The properties of the CKM matrix are then discussed. It is shown how the matrix can be prescribed by a minimum number of parameters. Experiments to measure the CKM matrix elements are then described, followed by a discussion of unitarity triangles and the measurements which constrain them.

Dave

7 Experimental tests of the Standard Model (3): non-collider experimentsA discussion on the relative

advantages and disadvantages of collider vs non-collider experiments is given. High profile non-collider experiments and techniques are described. The principles behind searches for dipole moments (electric, magnetic) are outlined, as are recent experiments. The influence of non-collider searches on theories of new physics at the TeV-scae, such as supersymmetry, is outlined. A “blue skies” search for non-integer charged partices, not motivated by any specific theoretical considerations, is also given as an example of the breadth of the non-collider program.

Dave

8 General Higgs constraints and exotic Higgs scenarios. Vacuum stability, unitarity bounds. Two Higgs doublet model. Thomas9 Experimental tests of the Standard Model (4): electroweak and Higgs physicsMeasurements

of electroweak parameters are given, emphasizing the unique roles of different collider environments. The influence of loops on electroweak observables is shown via the classic example connecting the Fermi constant and W mass to the Higgs and top mass. It is shown how this allowed a prediction of the top quark and Higgs masses long before direct experimental evidence for these particles became available. Global electroweak fits are then described with emphasis on the Higgs sectors. Measurements of the properties of the recently discovered Higgs-like boson and the consistency of the particle with the SM are then described.

Dave

10 Neutrino 1 Lepton mixing and neutrino oscillations. Thomas11 Neutrino 2 How to extend the SM to give mass to neutrinos, seesaw mechanism, lepton flavour violation, lepton number violation. Thomas12 Beyond the SM theories Problems of the SM: flavor problem, hierarchy problem, strong-CP problem, GUT theories. Thomas13 Simulation techniques at colliders The principles behind Monte Carlo simulation models are given. The factorization of short and long distance components is outlined.

The extraction of pdfs from structure function and hadronic final state data is described, along with the experimental uncertainties on these quantities. In an analogous technique to renormalization it is shown how attempts to use pdfs lead to a divergence which can be removed at the expense of introducing an arbitrary parameter (the factorization scale). This procedure also gives rise to the DGLAP equations which are the foundation of parton shower models. It is outlined how hard QCD emissions are simulated (parton showers + matrix element). The major model uncertainties are summarized. The principles behind the Lund string model, including elementary derivations of its key features, are then given.

Dave

Lecture outline

Books• No single book possible. • Handouts to be given where appropriate. • Lectures based on:

– D. Green, Lectures on Particle Physics, World Scientific. – Griffiths, Introduction to Elementary Particles, Wiley.– Perkins, Introduction to High Energy Physics, Addison-

Wesley– Halzen and Martin, Quarks and Leptons, Wiley– Articles in the Arxiv.

Inlämningsuppgifter• 3-4 inlämningsuppgifter.• A typical solution does not require a page of

mathematics. • Some questions are open-ended and require some

research beyond the text books, eg arxiv articles.– a physicist should be able within an hour or so to find

appropriate sources and obtain a good understanding (=1-2 ppt slides) of the methodology and principles behind any experimental result in his/her field.

Inlämningsuppgifter• Volunteers (or groups of volunteers) invited to

give a short presentations at the start of a lecture to certain questions.

• Eg from inlämningsuppgift 1.

(7) Finding the appropriate renormalisation scale is an important topic in perturbative QCD. In the lecture, the "standard" technique of setting the renormalisation scale equal to an energy scale of a given process was given. Read up on a more sophisticated techniqueand describe it.

Course homepagehttp://people.su.se/~milstead/teaching/2014/fk8022/course.htmlOnly source for up-to-date information.

Schedulehttp://www.fysik.su.se/~milstead/teaching/2014/fk8022/schedule.html

Flexibility to reschedule if necessary since we are a small group.

Concepts of the Standard Model: group theory

FK8022, Lecture 1

Core texts:Lectures on particle physics, D. Green Electroweak interactions: An introduction to the physics of quarks and leptons, P. Renton.Further reading:Introduction to high energy physics, D. PerkinsIntroduction to elementary particles, D. Griffiths

Lecture 1• Symmetries are at the heart of the SM.• Two important symmetry groups

– SU(2),SU(3) – Study in the framework of the strong force

• Lecture plan:– First principle derivations/definitions of

SU(2)/SU(3) properties – Applications in

• meson spectra• gluon colour and multiplicity • attractive/repulsive QCD potentials

Groups

( )

We deal with groups of transformations corresponding tomatrices. Eg rotation matrix acting on a vector.

Matrices chosen by nature to be of interest us to are :

- the group of special (determinantSU n* *( )

( )

(2)

1) unitary

matrices. the group of unitary matrices.

Start with symmetry to obtain the group theory resultswe need to understand the strong force.

n n

U U UU IU n n n

SU

SU(2)2 2 1

(2)

1 0 ', ( )

0 1 '( ) (2

The group of unitary matrices with determinant . matrices set of all possible rotations of 2D spinors in space.

Spinor: =

SU

U

U SU

2/2

1 2 3

)

( ) ..2!

ˆ( ) cos sin2 2

, ,

rotation matrix

is a matrix:

= direction angle of rotation (right-hand sense)Pauli matrices - "generators" of the transf

a

iA AU e U e I A

U I i

ormation.

SU(2)-rotations

2 2

1ˆ

0

cos sin2 20 0 1

0 1 0

0 1 1 01 0 0 1

Rotate spin-up by around the -axis

Transform:

Rotation of

up

up down

y y

U I i i

i U i

i

U

1

1 2

0 1 0 -1 0 0

around the -axis also changes spin direction.

, "Flipping" spin-up spin-down.

x U i

ii

z

y

spin-downspin-up

}

SU(2)-algebra

3

1 2 1 2

21 0 11 -1 02 2 2

0 1 0 01 10 0 1 02 2

0 1 0 10 0 1 0

Eigen values/conserved quantities:

Eg =

Ladder operators:

z

z up up z

down

S

S S

i i

0

1 0,

0 1

Ladder operators "map out" the possible states.

In this case a doublet

up

down up

10

01

Combining states

2

(3)

2 41

110

Combine eg and in positronium, what are the possible spin states ?Use same reasoning as for quarks/ (later).

orthogonal states possible.

Start with the straightforward state:

e eSU

10

1 1 0 1 1 011 00 0 1 0 0 12

0 1 1 0 0 01 1 -11 0 0 1 1 12

01

Ladder operator:

e e

e e e e e e

e e e e e e

0 1 10

1 0 0

Spin-1 triplet e e e e

0 1 1 010 01 0 0 12

0 1 1 01 01 0 0 12

2

Deduce singlet state:

Checks: + orthogonality

states in different ( ) re

e e e e

e e e e

SU

0 1 1 011 0 0 12

2 2

in vector spaces of various dimensions. Fundamental representation:

Basis vectors as 2-D spinors. Eg =

Rotations generated by Pau

presentations

e e e e

1 0 0 1 010 1 0 0 120 0 1 0 0

3 3

li matrices.

Higher representation:

Eg 3-D basis vectors as ,

Rotations generated by matrices.

, ,

Invariance to a SU(2) transformation in

physical space

Angular momentum conservation

Multiplets of orthogonal states

after angular momentum addition

Different aspects of the same thing

Ladder operators map out multiplets

All the results covered arise from SU(2) invariance.

SU(2)-isospin(2) , / ,(2)

(2)(2)

symmetry isospin for two particle world (eg ) transformations in isospin space are mathematically identical to transformations in real space. " space" mapped out by

SU u d p nSU

SUSU

33 3

, / ,1 0 0 10 1 1 0

1 1 11 10 02 2 2

1 0 00 1 1

(fundamental representation):

0

0 -1 Also quarks:

; ;

p

p n u d

p n p n

I I

u d u

10

;

(-ve sign is a technical and (for us) unimportant detail.)The other quarks carry no isospin.

d

I pn

I np

Meson isospin multiplets

0

0

, 2 2 3 11 1

11 ,0 0

1 0 0 11 11 00 1 1 02 2

0 01 -1

1 1

, ,

Triplet and singlet formed from combination:

all have similar

q q

ud

uu dd

du

0

1 0 0 11 10 00 1 1 02 2

masses 140 MeV and clearly belong together.

A neutral particle with a different mass, (540 MeV) is a good candidate.

uu dd

0

?

I3

SU(2) isospin

1 2

33

33

'exp

'2

(2)

21 1 0 110 0 -1 02 2

Pauli matrices generators for Ladder operators

Quantum number:

(3 -component)

Eg

a

rd

u uU i U

d d

SU

I i

I I

I u

12

= u

1 2

3

3

: 0 1 0 -

1 0 0

:

1 0

0 -1

u di

iI

Two quarks three quarks, ,

(3) 3 (2)

Consider quarks. Postulate two more "spaces" :-space and -space

Transformation in -space. Transformation in -space Transformation in -space.

spaces.

Only two ind

u d sU Vu d Id s Uu s VSU SU

.

(3)

(2)

ependent symmetries since Expect two quantum numbers.

This is flavour symmetry. The three quarks form a fundamental representation.The symmetries are subgroups.

U I V

SU

SU

I

U V

s

ud

Scalar meson multiplets in SU(3)

2

0(3)

3 9

Goals:(1) Assign spin- meson states to multiplets.(2) Determine quark content of each state.

3 quarks ' combinations.

(Simple) strategy:Apply ladder operators and orthogonality to deduc

SU

qq

e multiplet structure.

Meson Mass (MeV)

139.570 134.96K+- 493.67

K0 , K0 497.720 548.80’ 957.6

0

0 0

, ,

, , , , ,

du I U V

V K

ud du us K ds K us K ds K

Start with, eg, and apply series of steps.

"Map out" states , eg

belong to a multiplet.Six states identified - three (neutral) states remaining.

Charged scalar meson states in SU(3)

I U

V

I U

V

ud

ds

du

su sd

usI

U

V

I U

V

Neutral states

0

0

0

1 1 12 2 2

V I K K

I uu dd V K ss uu U K ss dd

I V K U K

States on the "edges": ladder operations give same state

Neutral states at the centre:

; ;

neutral particles must be mixed st 1 1 1, , .2 2 2

uu dd ss uu ss dd ates of

V

V

I

U

Neutral scalar meson states in SU(3)I

Neutral states

0

0

0

8

12

,

1 1 26 6

part of isospin triplet with other members .

belongs to the multiplet.

Make orthogonal state to with :

Check with la

uu dd

dd ss uu ss

dd ss uu ss dd uu ss

8

8 8

1

1 1 1

13

0

dder operator if is in the multiplet.

Eg in multiplet

One more orthogonal state needed

is a singlet.

I

dd uu ss

U I

0 0

0 0

0

8 0 1 0

, , , , , ,

1 1, 22 6

16

Octet:

Singlet:

Identify and

ud du us K ds K us K ds K

uu dd dd uu ss

dd uu ss

Scalar meson multiplets in SU(3)

SU(3)-flavour symmetry

(3) (2) ( , , ).

(3)

transformation transformation in 3 spaces

If flavour is an exact strong force symmetry :(1) The states should be an octet and a singlet with the quarkcompositions given by

SU SU I U V

SU

3 3 8 13 3 3 10 8 8 1

the derived wave functions.

In group theory language: (mesons) : (baryons)(2) Hadrons in each multiplet should be degenerate

bar em splitting.

SU(3)-flavour symmetry (3)

~ 140 500 SU

K flavour is a fair but not great symmetry.

Octet states are not degenerate: eg MeV, MeV.

Scalar meson

Quark content

Mass (MeV)

139.570 134.96K+- 493.67

K0 , K0 497.720 548.80’ 957.6

,ud du1

2uu dd

,us us

,ds ds

1 26

uu dd ss

13

uu dd ss

octet

singlet

8 0 1 0

0 8 1 0 8 1

'

cos sin ' sin cos 10

The eta states are mixed: and

op p p p p

Conserved quantities in SU(3)

3

3

-

2

I V U

I

Y Q I

Three transformations: , -spin , spin. Two are independent. Convention :Quantum numbers/conserved quantities - (a)

(b) hypercharge

s

ud3I

Y

s

u d

3I

Y

121

2

12 1

2

23

23

1

3I1 12 1

2 1

Y

0

1

SU(3) flavour

1 2 6 7

6

(3)'

exp '2

'

(3)

,

Formal treatment.

Gell-Mann matrices generators for generalised Pauli matrices Ladder operators

,

a

SUu u

U i d U ds s

SU

I i U iV

7

3 3

823

Quantum numbers:

(3 -component)

(hypercharge)

rd

i

I I

Y

1 2

4 5

6 7

0 1 0 0 - 01 0 0 0 00 0 0 0 0 0

0 0 1 0 0 -0 0 0 0 0 01 0 0 0 0

0 0 0 0 0 00 0 1 0 0 -0 1 0

iu d i

iu s

i

d s i

3 3

8

0 0

1 0 00 -1 00 0 0

0 01 0 1 03

0 0 - 2

i

I I

Y

3rd -component

1 Hypercharge

1 2

3

Antigreen-blue

Antigreen Blue

Gluon colour and multiplicity(3) (3)(3)

3 3 8 1)

, , , , ,12

colour is mathematically identical to flavour. colour is an exact symmetry.

Gluon colour-anticolour

Colour+anticolour octet+singlet (

Octet:

SU SUSU

RB RG GB GR BG BR

RR GG

1 26

13

Singlet:

Eight gluons

RR GG BB

RR GG BB

Not used in nature!

Hadron colour

'

, , , , ,1 1 22 6

13

16

Like gluons, mesons are a colour-anticolour ( )system.

Octet:

Singlet:

Colour singlet for baryons:

Every hadro

qq

RB RG GB GR BG BR

RR GG RR GG BB

RR GG BB

RGB GRB GBR BGR BRG RBG

n observed in nature is a colour singlet!!

Not used in nature!

QED vs QCD potentials

1 2

1

emV q qr

V e er

Two charged particles exchanging a photon.

Potential: ;

( - atomic + positronium, , data)

g

1q1q

2q2q

R

R

R

R

1 12 2

| |

1

s

V Vr r

C C QQ g QQ

V cr

Two quarks exchange a gluon over a short distance ( fm)

Potential: (quark-quark) (quark-antiquark)

( =colour contribution to amplitude= )

( from heavy quarkonia, , ,c bb data)

QCD couplings

1 1 1 1 2| |36 6 2 2

13

2| | , | | | | 13

1 2 42 12 3 3 in singlet

=

Apply to mesons:

Consider meson in state:

Possible processes:

=

meson

meson

s sQQ

C RR g RR

RR BB GG

RR

RR g RR BB g RR GG g RR

Vr r

Attractive

R

R

R

R

R

R

R

R

R

R

R

R= +

1 26

12

RR GG BB

RR GG

1 26

RR GG BB

1 16 6

12

RR GG

1 12 2

RR RR

16

,

1| | - , | | -13

1 112 3 in singlet

Baryon:

Baryon in state . Consider arbitrary quark pair, eg, .

Possible processes:

baryon

baryon

sQQ QQQ

GBR GRB RGB BGR BRG RBG

GBR B R

GBR g GBR GRB g GBR

Vr

23

.

1| |3

1 1 12 3 6 in octet

= Attractive

Colour octet. Eg

Repulsive

Nature prefers colour singlets.

s

s sQQ

r

RB

RB g RB

Vr r

Other QCD potentials

Summary

• Concepts and mathematics of SU(2) and SU(3) symmetry outlined.

• Studied in the context of strong force symmetries: – isospin – flavour – colour

• Applications of symmetry reveal : – Hadron multiplicities and quark composition– Gluon multiplicity and colour– Meson wave function and binding