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MKEP 1.2: Particle Physics WS 2012/13

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MKEP 1.2: Particle Physics WS 2012/13 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 † † 4 ψ 푈 푈ψ 푑 푥 = 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 푖α퐺 U(α) = lim 1 + 푖 퐺 = 푒 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> 2 푇 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 2 Z 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.
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