Particle Physics WS 2012/13 (4.12.2012)

Stephanie Hansmann-Menzemer Physikalisches Institut, INF 226, 3.101 Content of Today

 Recap of symmetries discussed last week

 Isospin SU(2), flavour symmetry SU(3)F

 Experimental evidence for color

 SU(3)c : symmetry group of color

 Connection of symmetry groups and Feynman rules (this will be an excursion to QFT)

 Feynman rules for strong interaction and color factors

2 Reminder: Emmy-Noether Theorem

Physics is invariant under (linear) transformation: ψ → ψ‘ = U ψ

† 4 † † Normalization: ψ ψ 푑 푥 = 1 ψ 푈 푈ψ 푑4푥 = 1

= 1 U is unitary P hysics is invariant under transformation:

† † † † ψ 퐻 ψ 푑4푥 = ψ′ 퐻 ψ′푑4푥 = ψ 푈 퐻 푈 ψ 푑4푥 [H, U] = 0

Infinitesimal transformation: U = 1 + i ε G G: generator of transformation

† 1 = U†U = (1 – iεG†)(1+iεG) = 1 + iε(G-G†) + O(ε2) G = G

G is hermitian, thus coresponds to an observable quantity g

[H,U] = 0 [H,G] = 0 g is a conserved observable quantity!

A finite (multidim.) transformation (α) can be expressed as a series of infinit, transformations: 푛 α U(α) = lim 1 + 푖 퐺 = 푒푖α퐺 푛→∞ 푛

For each symmetry transformation, there is an conserved observable quantity! 3 Symmetries in Particle Physics

 Symmetry of isospin SU(2): „u and d quark are not distinguishable in strong IA“, physics is invariant under rotation in isospin space 푢′ 푢 = 푈 푑′ 푑 [this is only an approximative symmetry e.g. m(u)~m(d)] conservation of isospin in strong IA [still heavily used in hadron physics community]

 Flavour symmetry SU(3)F: „ u, d and s quark are not distinguishable in strong IA“, physics is invariant under rotation in flavour space 푢′ 푢 푑′ = 푈 푑 푠′ 푠 [this is an even more approximative symmetry: m(s) ~ 150 MeV, m(u),m(d)~ 1-3 MeV] conservation of strangeness and isospin in strong IA [mainly historical, it set the scene for SU(3) color, which is an exact symmetry]

4 Isospin SU(2) – 2D representation

1 0 states: |u> = |d> = 0 1 푛 α symmetry transformation: U = 푒푖ασ = lim 1 + σ 푛→∞ 푛 generators: σ1, σ2, σ3 commutator relations: [σi, σj] = 2iεijkσk Pauli matrices σ σ isospin operators : 푇 = ; 푇 = 푖 i=1,2,3 2 푖 2 1 1 0 1 1 1 1 0 0 1 3. component: T |u> = = |푢 > T |d> = = − |푑 > 3 2 0 −1 0 2 3 2 0 −1 1 2

1 1 3 casimir operator: 푇2 |푢 > = (σ 2 + σ 2 + σ 2) = |푢 > 4 1 2 3 0 4 1 0 3 푇2 |푑 > = (σ 2 + σ 2 + σ 2) = |푑 > 4 1 2 3 1 4

2 Commutation properties: [푇 , Ti] for i=1,2,3 = 0 [Tj, Ti+ ≠ 0 for i≠j only two observables simultanously measurable

2 Choose common eigenstates of 푇 and T3, which unambiguiously define the particle state : |I,I3>

푇2 퐼, 퐼 > = 퐼 퐼 + 1 퐼, 퐼 > T 퐼, 퐼 > = 퐼 퐼, 퐼 > 3 3 3 3 3 3 5 Isospin SU(2) – 2D representation I I I I 3 1 3 0 states: |u> = |1/2, 1/2> = |d> = |1/2, -1/2> = 0 1

0 1 0 0 ladder operator: T = T + i T = T = T - i T = + 1 2 0 0 - 1 2 1 0

T+|u> = 0 T+|d> = |u> T_|u> = |d> T_|d> = 0

T+

T_

d u

-1/2 +1/2 I3

Ladder operators allow to move around inside a isospin multiplett.

Ladder operators applied to a singulett state (|0,0>) yield 0. 6 Flavour SU(3)

Physics is invariant under rotation in 3D flavour space:

1 0 0 푢′ 푢 u = 0 ; d = 1 ; s = 0 ; 푑′ = U 푑 ; 0 0 1 푠′ 푠

U is 3x3 unitary matrix 9 real parameters One matrix is (as for 2D case) just multiplication with phase 8 remaining free parameters describe subgroup SU(3) of group of unitarity matrices U(3)

Generators (in 3D representation) of SU(3) are 8 hermitian matrices λi i=1,…,8 (Gell-Mann matrices)

Define 푇 = λ/2 U = 푒푖α푇 : group of translation in flavour space

7 Gell-Mann-Matrices λi

T± = 1/2(λ1 ±iλ2)

T3 = λ3/2 Y

V± = 1/2(λ4 ±iλ5) I3

U± = 1/2(λ6 ±iλ7)

I3

Hypercharge Y = λ8/3; Y = B + S

baryon number strangeness 8 Combining Quark+Antiquark

Y

Y Y

I 3 I3 I3

Y What is composition of Y=0, I3=0 state? Exploit ladder operators

I3

2 independent states, third state not part of multiplet 9 Combining Quark+Antiquark

Charged and neutral pions are member of same isospin doublet, thus should be in same flavour multiplet as well.

singulet state must be symmetric in flavour (it carries no flavour)

You can apply any ladder operator on singulett and it yields 0. Exploiting orthogonality:

10 Example: Pseudo-Scalars and Vector-Mesons

Y

I3

Y

I3

Similar masses for isospin related particles, proove very good approximation of isospin symmetry. Involving s quarks, symmetry is not that great anymore …. 11 Evidence for Color a) Existance of Δ++ = |uuu>

L = 0 : ground state

S3 = S = 3/2 I3 = I = 3/2

Ψtotal = ψspace ψspin ψisospin all three wave functions are symmetric, thus ψtotal as well (-1)L However combination of three fermions must have asymmetric wave function one degree of freedom is missing!

1 Ψ (123)= (RGB + GBR + BRG – GRB – BGR - GRB) colour 6 Exchange of quark 1 and 2: 1 Ψ (213) = (GRB + BGR + RBG – RGB – GBR - RGB) = - ψ (123) colour 6 colour (similar for any other exchange of two quarks)

Ψtotal = ψspaceψspin ψisospinψcolour 12 Evidence for Color

b) Non-resonant hadron production in e+e-

+ - + - e e μ e - - e τ

- + - + e [ ] e μ e e+ τ+

- f e Z0 threshold dependent, τ was not yet known at [ ] that time (m = 1.8 GeV) e+ f τ weak IA negligible relative to colour factor 3 elm IA as long as 푠 away from + - 푢 2 + - + - σ(e e → qq) = Nc 푖 푍푖 σ(e e →μ μ ) mass of Z.

e- q Zi: charge in elementary units

u: upper limit on quark species due to 푠 e+ q 13 Results

푢 2 = Nc 푖 푍푖

ττ treshold, however not all τ decay into hadronic jets

q Z 2 푠 ≤ 2푚(푞)] i R[

u 4/9 4/3 d 1/9 5/3 s 1/9 2 c 4/9 10/3 b 1/9 11/3 t 4/9 5

Nc=3 „more or less“ confirmed by data! 14 Color Charge in QCD

Construct in analogy to QED:

QED: interaction due to el. charge f Symmetry: local phase transformation α conserved quantity: el. charge

f

q g QCD: interaction due to colour charge s quarks carry R,G,B q antiquarks carry anti-colour

g Strong IA invariant under rotation in color space. E.g. IA is the same for all three colors SU(3) color symmetry is exact! Color is a conserved quantity

15 Color States

Colour hypercharge and color isospin subgroups of coulor SU(3) have no physical meaning due to no connection from physics point of view! exact colour symmetry!

quarks antiquarks 16 Gell-Mann Matrices: Generators of Colour SU(3)

T± = 1/2(λ1 ±iλ2)

T3 = λ3/2 colour isospin

V± = 1/2(λ4 ±iλ5)

U± = 1/2(λ6 ±iλ7)

colour hypercharge Y = λ8/3

17 Color Confinement

It is believed (though not yet proven) that all free particles are colour neutral . neutral = symmetric under rotation in colour space (Yc = I3=0 is not sufficient!)

all free q푞 states (mesons) are (to our todays knowledge)

Follow the same math as for SU(3)F! in color singlett state Use ladder operators, to show that 1 (r푟 + 푔푔 + 푏푏 ) is a singulett state. SU(3)C is exact, thus no 3 mixture of octett and singluett state

18 Gell-Mann Matrices can be “associated” to Gluons!

T± = 1/2(λ1 ±iλ2) r푔 , g푟

1 (푟푟 − 푔푔 ) 2

V± = 1/2(λ4 ±iλ5) r푏, b푟

U± = 1/2(λ6 ±iλ7) b푔 , 푏푔

q(r) 1 (푟푟 + 푔푔 − 2푏 푏) 6

q(b) q(r) 푏 q(b) q(r) r 1 q(r) conservation of color (푟푟 + 푔푔 − 2푏 푏) at each vertex 6 q(b) q(b) 19 Gluons

In the same way colour of q풒 combinations are obtained, the colour of gluons can be constructed!

8 coloured states one colourless state.

Gluons carry colour, thus colourless singlett does not represent a gluon. There are 8 gluons which are the messangers of the strong IA! This is a direct consequence of the SU(3) symmetry, which has 8 generators.

20 Connection of Symmetries and Feynman Rules

Why do we have three and four gluon vertices but no three photon vertices?

Why do we not have five gluon vertices?

Have a look at the Lagrangian and ist symmetry!

(This is an excursion in QFT, thus not relevant For the exam) 21 Reminder on Euler-Lagrange Formalism

Lagrangian: L = T- V (difference of kinetic and potential energy of the system)

L(푞푖, 푞 푖, 푡) qi: generalized coordinates (which describe the physics process)

Euler-Lagrange equation to determine equation of motion:

휕 휕퐿 휕퐿 ( ) - = 0 휕푡 휕푞 푖 휕푞푖

Particles are descibed by wave functions wich depend on 4-momenta: 휕퐿 휕퐿 휕 휕 ψ 휕 L(ψ, μ , xμ) μ ( ) - = 0 휕μ = ( , 훻) 휕(휕μ ψ) 휕ψ 휕푡

Example: Lagrangian for free spin ½ particles:

μ μ L = iψ γ (휕μψ) - mψψ Dirac equation: i휕μψγ - mψ = 0

22 Connection between Lagrangien, Feynman Rules and Symmetries

To each Lagrangian, there correpsonds a set of Feynman rules. Each term of the Lagrangian can be associated to propagators or vertex terms. 2 2 μ Propagator terms look like mφ , 1/2(휕μφ) , ψγ 휕μψ (just involve particle wave functions) all others are vertex terms.

e.g. QED Lagrangian

L = „ψ ψ“ + „eψ ψ퐴" + "A2" fermion propagator interaction of fermion with photon photon propagator

If physics is symmetric under a transformation, the Lagrangian must be invariant under this transformation. If not, need to extend the Lagrangian of the free fermion by (interaction) terms.

Symmetry defines Lagrangian, Lagrangian defines interaction/feynman rules. 23

QED: Invariance under U(1) transformation

Symmetry under local phase transformation results in conservation of electric charge.

global phase transition have no physical Ψ‘ = Uψ = eiα(x)ψ meaning, local ones does …

μ L = iψ γ 휕μψ - mψψ

iα(x) iα(x) 휕μψ‘ = e 휕μψ + i e ψ 휕μα(x) this term breaks invariance

Local phase invariance not possible for free particle!

Introduce modified (covariant) derivatives with following transformation behaviour:

iα(x) Dμψ → e Dμψ

This can be accompoished by replacing 휕μ with Dμ = 휕μ − 푖푒퐴μ 1 Where the field has the following translation properties A → A + 휕 α(푥) μ μ 푒 μ μ μ μ L = ψ 푖γ 퐷μ − 푚 ψ = ψ 푖γ 휕μ − 푚 ψ + 푒ψγ ψ퐴μ invariant under local phase transformation fermion propagator interaction with photon field 24 QED: Invariance under U(1) transformation

μ μ L = ψ 푖γ 휕μ − 푚 ψ + 푒ψγ ψ퐴μ

If A is associated to photon field, kinetic energy of photon field is missing in Lagrangian. 2 General form of kinetic energy of some field φ: ½ (휕μφ)

1 Additionally, kinetic term needs to be invariant under Aμ → Aμ + 휕μα(푥) 푒

Fμν = 휕μAν - 휕ν퐴μ Fνμ→Fνμ

Phase transformation invariant Lagrangian → QED (conserved quantity: el. charge)

1 L = ψ 푖γμ휕 − 푚 ψ + 푒ψ γμψ퐴 - F Fμν μ μ 4 휇휈

L = „ψ ψ“ + „eψ ψ퐴" + "A2" fermion propagator interaction of fermion with photon photon propagator

25 QCD: Invariance under SU(3) transformation

푖α(푥)푇 1 Ψ(x) → Uψ(x) = 푒 ψ(x) 푇 = λ 2

Study (infinitesimal) small tranformation Ta :

Ψ(x) → *1+iαa(x)Ta]ψ(x)

μ 휕μΨ(x) → *1+iαa(x)Ta]휕μψ(x) + iTaψ(x)휕μα푎(푥) L = iψ γ 휕μψ - mψψ

again local phase spoils invariance of Lagrangian

apply same trick, define covariant derivative: 1 푎 a → a 휕 α (푥) Dμ = 휕μ + 푖푔푆푇푎 퐺μ vector field, with Gμ Gμ + μ 푎 푔푠 coupling of quark current to vector field μ μ 푎 L = iψ γ 휕μψ - mψψ − 푔푠 ψγ 푇푎ψ 퐺μ

Up to here analogous to QED, however need to take care of non-abelian gauge theory! 26 QCD Lagrangian

Apply second (infinitesimal) small translation [1+iTbαb(x)] on Lagrangian: μ μ 푎 L = iψ γ 휕μψ - mψψ − 푔푠 ψγ 푇푎ψ 퐺μ ψ γμ푇 ψ → ψ γμ푇 ψ + 푖α 푥 ψ γμ(T T -T T )ψ 푥 + 푂(α 2) 푎 푎 푏 a b b a 푏 a,b,c= 1,..,8

exploit commutator relation of Gell-Mann-matrices [Ta,Tb]=fabcTc Gell-Mann matrices do not commute need additional gauge term

a a 1 푐 Gμ →Gμ + 휕μα푎 푥 − 푓푎푏푐 α푏 푥 퐺μ 푔푠 Now still need kinematical term for each of the (gauge/gluon) fields which is invariant under above transformation:

a 푎 푎 푏 푐 Gμν = 휕μ퐺ν − 휕ν퐺μ − 푔푠 푓푎푏푐 퐺μ 퐺ν 1 L = iψ γμ 휕 ψ - mψ ψ − 푔푠 ψ γμ푇 ψ 퐺 푎 − 퐺 푎퐺 μν μ 푎 μ 4 μν 푎

2 3 2 4 L = „ ψψ“ + „ ψψ G“ + „G “ + „gsG “ + „gs G “

result of non-abelian symmetry group 27 The Quark-Gluon Interaction

28 Feynman Rules for QCD

29 Summary

 Physics is invariant under colour, this symmetry can be described by SU(3)c

 Gluons can be associated to 8 Gell-Mann matrices

 Gluons carry combination of color and anticolor, 8 combinations have colour one neutral singlett

 Colour confinement: any hadron is (to the best of our knowledge) colourless, in singulett state

 Lagrangian must be invariant under symmetry transformations

 Lagrangians define Feynman rules

30