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The SM, the Higgs and beyond

Lecture 2 – Symmetries and QFT Outline

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Fundamental principles of particle physics

Our description of the fundamental interactions and particles rests on two fundamental structures :

Quantum Mech anic s

Symme tries

(3)

Symmetries

Central to our description of the fundamental forces :

Relativity - Lorentz transformations

Lie symmetries – Gauge transformations

SU (3) ! SU (2) ! U (1)

Copernican principle : “Your system of co-ordinates and units is nothing special”

Physics independent of system choice

SO(3,1)    

(4)

Symmetries

Classification of symmetries in Standard Model:

Local vs global

Continuous vs

Discrete

SU (3) ! SU (2) !U (1)

We also talk about

Abelian syms – generators commute Vs

Non-Abelian - generators do not commute

SO(3,1)    

SU (2)

Isospin

All  of  the  above  

C,P,T  –  Charge  conjuga=on,  Parity,  Time  inversion…    

U (1)

Baryon

3 1

[ , ]

i j ijk k

k

J J i ! J

=

= ! "

J

i

The   are  the  “generators”  of  the  group.    SO(3)    (SU(2))     Their commutation relations define a “Lie algebra”.

R( ! ) = e !i ! . J

(5)

Special relativity

( , , , ) a

µ

= ct x y z

( a + ! a )

µ

" a

µ

= ! a

µ

= ( c t x y z ! ! ! ! , , , )

Space time point not invariant under translations

Space-time vector

Invariant under translations …but not invariant under rotations or boosts

Einstein postulate : the real invariant distance is

( ) ( ) ( ) ( )

0 2 1 2 2 2 3 2 3

( )

2

, 0

a a a a g

µ!

a a

µ !

a a

µ µ

a

µ !=

" # " # " # " = $ " " = " " = "

( 1, 1, 1, 1) g

µ!

= diag + " " "

Physics invariant under all transformations that leave all such distances invariant : Translations and The SO(3,1) Lorentz transformations

(6)

Lorentz transformations :

x

µ

! "

#µ

x

#

= "

#µ

x

#

# =0 3

$ = x '

µ

% g

µ!

x '

µ

x '

!

= g

µ!

x

µ

x

!

" g

µ!

#

$µ

#

!%

= g

$%

Solutions :

3 rotations R

1 0 0 0

0 cos sin 0

0 sin cos 0

0 0 0 1

! !

! !

" #

$ %

$ %

$ & %

$ %

$ %

' (

3 boosts B

cosh sinh 0 0

sinh cosh 0 0

0 0 1 0

0 0 0 1

! !

! !

" #

$ %

$ %

$ %

$ %

$ %

& '

Space reflection – parity P

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

! "

# $

# % $

# % $

# $

# % $

& '

Time reflection, time reversal T 1 0 0 0

0 1 0 0 0 0 1 0 0 0 0 1

"! #

$ %

$ %

$ %

$ %

$ %

& '

(Summation assumed)

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Quantum Mechanics Relativity

+ } Quantum Field theory

Fundamental principles of particle physics

q(t)

q(t) → q

x

(t) = q(t, x)

(8)

Bronshtein’s ‘cube of theories’  

(9)

Action

2

1

t

t

S = ! L dt

Classical path … minimises action

Quantum mechanics … sum over all paths with amplitude

! e

iS /!

(Lagrangian invariant under all the symmetries of nature

Lagrangian L T V = !

(Nonrelativistic mechanics)

-makes it easy to construct viable theories)

“Principle of Least Action”

Feynman Lectures in Physics Vol II Chapter 19

S = (K.E. ! P.E.)dt

tA tB

"

Action, S Action principle

Compare with Hamiltonian formulation:

H = T + V

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Quantum Mechanics : Quantization of dynamical system of particles

Quantum Field Theory : Application of QM to dynamical system of fields Not all relativistic processes can be explained by

single particle since E=mc2 allows pair creation – happens all the time at LHC (Relativistic) QM has physical problems. For example it violates causality

Why Quantum field theory?

See  slides  of  G.  Ross  on  school  homepage  

(11)

3

, lagrangian densi ty L = ! L d x L

Klein Gordon field

! ( ) x

( "

µ

! ( ) x )

"

µ

! ( ) x # m

2

! ( ) ( ) x

! x

L = } }  

T V

Manifestly Lorentz invariant

Relativistic (quantum) field theory

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L = L ! d

3

x, L lagrangian density

Klein Gordon field

! ( ) x

L = ! (

µ

" (x) )

!

µ

" (x) # m

2

" (x)

" (x)

! S = 0 " #L

#$ % #

µ

#L

#(#

µ

$) = 0

Manifestly Lorentz invariant

Euler Lagrange equation (shown in exercises)

Lagrangian formulation of the Klein Gordon equation

Classical path :

!S

!" = 0

S =

d

3

xdtL

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L = L ! d

3

x, L lagrangian density

Klein Gordon field

! ( ) x

L = ! (

µ

" (x) )

!

µ

" (x) # m

2

" (x)

" (x)

! S = 0 " #L

#$ % #

µ

#L

#(#

µ

$) = 0

Manifestly Lorentz invariant

Euler Lagrange equation

(shown in exercises)

Lagrangian formulation of the Klein Gordon equation

Classical path :

(!

µ

!

µ

+ m

2

) " = 0

Klein Gordon equation

!S

!" = 0

S =

d

3

xdtL

(You will derive a number of field equations from Lagrangians in the exercises)

(14)

New symmetries

( "

µ

! ( ) x )

"

µ

! ( ) x # m

2

! ( ) ( ) x

! x

L =

Is invariant under

What is this symmetry in our classification scheme?

( ) x e

i!

( ) x

" # "

(15)

New symmetries

( "

µ

! ( ) x )

"

µ

! ( ) x # m

2

! ( ) ( ) x

! x

L =

Is invariant under

A symmetry implies a conserved current and charge.

Translation Momentum conservation

Rotation Angular momentum conservation e.g.

What conservation law does the U(1) invariance imply?

…a global, continuous Abelian (U(1)) gauge symmetry

( ) x e

i!

( ) x

" # "

(16)

Noether current

( "

µ

! ( ) x )

"

µ

! ( ) x # m

2

! ( ) ( ) x

! x

L =

Is invariant under

" ( ) x # e

i!

" ( ) x

…an Abelian (U(1)) gauge symmetry (use inf. Transform δϕ and Euler lagrange eqs.)

δL = 0 → 0 = i∂ µ

� ∂L

∂(∂ µ φ)

− (φ ↔ φ )

What is the physics of this equation?

φ

(17)

Noether current

( "

µ

! ( ) x )

"

µ

! ( ) x # m

2

! ( ) ( ) x

! x

L =

Is invariant under

" ( ) x # e

i!

" ( ) x

…an Abelian (U(1)) gauge symmetry (Inf. Transform δϕ and Euler lagrange eqs.)

0,

2 ( ) ( )

j j ie

µ

µ µ µ

!

µ

!

! !

" $ $ #

$ = = & % '

$ $ $ $

( )

L L

Noether current

δL = 0 → 0 = i∂ µ

� ∂L

∂(∂ µ φ)

− (φ ↔ φ )

Conserved current

φ

(18)

The Klein Gordon current

( "

µ

! ( ) x )

"

µ

! ( ) x # m

2

! ( ) ( ) x

! x

L =

Is invariant under …an Abelian (U(1)) gauge symmetry

0,

2 ( ) ( )

j j ie

µ

µ µ µ

!

µ

!

! !

" $ $ #

$ = = & % '

$ $ $ $

( )

L L

(

* *

)

j

µKG

= " ie ! #

µ

! ! " #

µ

!

This is of the form of the electromagnetic current for the KG field

( ) x e

i!

( ) x

" # "

(19)

The Klein Gordon current

( "

µ

! ( ) x )

"

µ

! ( ) x # m

2

! ( ) ( ) x

! x

L =

Is invariant under

! (x) " e

i#

! (x)

…an Abelian (U(1)) gauge symmetry

0,

2 ( ) ( )

j j ie

µ

µ µ µ

!

µ

!

! !

" $ $ #

$ = = & % '

$ $ $ $

( )

L L

(

* *

)

j

µKG

= " ie ! #

µ

! ! ! " #

µ

This is of the form of the electromagnetic current for the KG field:

3 0

Q = ! d x j

is the associated conserved charge

A

µ

j

µKG ‘minimal coupling’ to EM potential

(20)

L = ! (

µ

" (x) )

!

µ

" (x) # m

2

" (x)

" (x) + $ "

4

+ M $ '

2

"

6

+ ...

Aside - Additional terms

}

Renormalisable D ! 4

If M ! 10

3

GeV, "Effective" Field theory approximately renormalisable

Terms allowed by U(1) symmetry

Renormalizability  is  another  principle  taken  for  the  construc=on  of  the  SM  –     In  the  above  sense  

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U(1) local gauge invariance and QED

( "

µ

! ( ) x )

"

µ

! ( ) x # m

2

! ( ) ( ) x

! x

L =

Invariant?

( ) x e

i!( )x Q

( ) x

" # "

Jargon:  we  are  gauging  

the  global  U(1)  symmetry  

(22)

U(1) local gauge invariance and QED

( "

µ

! ( ) x )

"

µ

! ( ) x # m

2

! ( ) ( ) x

! x

L =

not invariant due to derivatives

( ) x e

i!( )x Q

( ) x

" # "

!

µ

! ! "

µ

(e

i!( x)Q

! ) = e

i!( x)Q

!

µ

! + iQe

i!( x)Q

! !

µ

! (x)

To obtain invariant Lagrangian look for a modified derivative transforming covariantly

D

µ

! " e

i#( x)Q

D

µ

!

(23)

U(1) local gauge invariance and QED

( "

µ

! ( ) x )

"

µ

! ( ) x # m

2

! ( ) ( ) x

! x

L =

not invariant due to derivatives

( ) x e

i!( )x Q

( ) x

" # "

To obtain invariant Lagrangian look for a modified derivative transforming covariantly

D

µ

! " e

i#( x)Q

D

µ

!

D

µ

= ! "

µ

iQA

µ

A

µ

" A

µ

+ #

µ

!

Need to introduce a new vector field

(24)

( ) x e

iQ!( )x

( ) x

" # "

D

µ

! " e

i#( x)Q

D

µ

! A

µ

" A

µ

+ #

µ

!

( D

µ

! ( ) x )

D

µ

! ( ) x " m

2

! ( ) ( ) x

! x

L =

is invariant under local U(1)

Note :

! "

µ

D

µ

= ! #

µ

iQA

µ is equivalent to

p

µ

! p

µ

+ eA

µ

universal coupling of electromagnetism follows from local gauge invariance

( )

KG

"

µ

! ( ) x "

µ

! ( ) x # m

2

! ( ) ( ) x

! x # j A

µKG µ

+ O e ( )

2

i.e. L = L =

(25)

( ) x e

iQ!( )x

( ) x

" # "

D

µ

! " e

i#( x)Q

D

µ

! A

µ

" A

µ

+ #

µ

!

( D

µ

! ( ) x )

D

µ

! ( ) x " m

2

! ( ) ( ) x

! x

L =

is invariant under local U(1)

Note :

! "

µ

D

µ

= ! #

µ

iQA

µ is equivalent to

p

µ

! p

µ

+ eA

µ

‘Minimal coupling’ of electromagnetism follows from local gauge invariance Dynamics follows from symmetry

2 2 2

( " " +

µ µ

m ) ! = # V ! where V = # ie ( "

µ

A

µ

+ A

µ

"

µ

) # e A

The Euler lagrange equation give the KG equation:

(26)

The electromagnetic Lagrangian

F

µ!

= "

µ

A

!

# "

!

A

µ

,

F

µ!

# F

µ!

A

µ

# A

µ

+ $

µ

"

14

EM

= " F F

µ! µ!

" j A

µ µ

L

The Euler-Lagrange equations give Maxwell equations !

F

µ!

j

!

"

µ

=

. , 0

. 0,

t E

t

! "

# = # $ + =

"

"

# = # $ % =

"

E E B

B B j

!

1 2 3

1 3 2

2 3 1

3 2 1

0

0

0

0

E E E

E B B

E B B

E B B

! ! !

" #

$ ! %

$ %

$ ! %

$ %

$ ! %

& '

) 0

A A

µ

! µ !

" "

# " =

" " "

L L

(

M A A

2 µ µ Forbidden by gauge invariance

EM dynamics follows from a local gauge symmetry!!

N.B. !µ"#$%µF#$ = 0

( )

(27)

Suppose we have two fields with different U(1) charges :

1,2

1,2

( ) x e

i Q! 1,2

( ) x

" # "

( )

( )

2

1 1 1 1

2

2 2 2 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x x m x x

x x m x x

µ µ

µ µ

! ! ! !

! ! ! !

" " #

+ " " #

L =

..no cross terms possible (corresponding to charge conservation)

References

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