CHAPTER II: The QCD Lagrangian

Transcription

1 CHAPTER II: The QCD Lagrangian.. Preparation: Gauge invariance for QED à µ UA µ U i µ U U e U A µ i.5 e U µ U U Consider electrons represented by Dirac field ψx. Gauge transformation: Gauge field Potentials: A µ x φx, Ax T. Local gauge transformation, if θ θx Global gauge transformation, if θ const. ψx Uψx withu e iθ. Electromagnetic fields: E φ A t B A Hypothesis :Localgaugetransformations,U e iθx,leavethephysicsinvariant. Current is invariant under local gauge transformation. Not invariant: ψxγ µ ψx G.T. ψ γ U γ µ Uψ. ψiγ µ µ ψ ψiγ µ U µ Uψ Introduction of gauge field A µ x: ψiγ µ U U µ ψ+ ψiγ µ ψ U i µ U µ θx Definition of gauge covariant derivative: D µ µ iea µ x Requirement: Under local gauge transformation e>. Electromagnetic field tensor: E x E y E z F µν µ A ν x ν A µ E x B z B y x E y B z B x E z B y B x Lagrangian density of electromagnetic fields.6 L γ 4 F µνxf µν x E B.7 Equations of motions for free photon: A µ x A µ x d k ak, λ ɛ µ e ik x π λ + a k, λ ɛ µ eik x λ ω k λ where ω k k and ɛ µ λ State vector of photon: represents the polarization vector..8 D µ ψ UD µ ψ then L ψ iγ µ D µ m ψ gauge invariant. U D µ ψ µ ψ ie à µ x ψ µ Uψx ieãµ xuψx µ U ψ + U µ ψ ieãµ xuψ U µ iea µ x ψx ieãµ Uψ ieua µ ψ µ U ψ õ U UA µ i e µ U.4 Lagrangian density of QED: where D µ µ iea µ x k, λ a k, λ ak, λ k, λ.9 L QED ψx iγ µ D µ m ψx 4 F µνxf µν x. Gauge transformations form a group: U e iθx QED, U Group U.

2 Local SU Gauge transformations Starting point: Quark fields ψ ψ αi α u, d, s, c, b, t flavor index N f 6 SUN f i,, color index N c SU c where ψ αi is a 4-component Dirac-spinor. Consider Quark fields with color degree of freedom and their free Lagrangian: ψ ψ ψ ψ Local SU c gauge transformations, L ψ iγ µ µ m ψ. ψx ψx U ψx. with U exp i θ a x λ a where θ a x isarealfunctionwitha,,, 8. Hypothesis :Physicsofstronginteractionofquarksisinvariantundergauge transformation: ψx Ux ψx. SU c is a non-abelian gauge group. Gauge covariant derivative: D µ µ iga µ x. where g is a dimensionless coupling strength analogous to e in QED. A µ x t a A a µx, t a λ a.4 a Introducing A a µx, SU c gauge fields gluons, Lagrangian L becomes gauge invariant. L ψx iγ µ D µ m ψx.5 D µ ψ µ ψ ig à µ ψ U D µ U à µ U A µ i g U µ U U.6 Infinitesimal gauge transformation U exp i θ a x t a i θa x t a +.7 transformation of gauge field up to terms linear in θ a x A µ a x õ a x Aµ a x g µ θ a x+f abc θ b xa µ c x.8 Gluons are massless a mass term m g A µ a Aa µ Gluonic field tensors: If one would take the form analogous to QED, not gauge invariant in QCD. would not be gauge invariant. Fµν a x µa a ν x νa a µ x,.9 Introduce additional term to obtain gauge invariant Gluonc field tensor. Gluonic Lagrangian: QCD Lagrangian: with D µ µ iga µ x. Remark : frequently A µ ga µ G a µν x µa a ν x νa a µ x+gf abc A b µ x Ac ν x. G µν t a G a µν i g Dµ,D ν. L glue 4 Ga µνx G µν a x tr{ G µν G µν}... QCD Lagrangian L QCD ψ iγ µ D µ m ψ tr{ G µν G µν}. L QCD ψ iγ µ µ ia µ m ψ g tr{ G µν G µν}

3 - - Gluonic field tensor of L QCD generates non-linear gluon interactions: -gluon interaction L g f abc µ A ν a ν A µ a A b µ A c ν g.4 Corresponding equation of motion: µ F µν x µ µ A ν ν A µ A ν ν µ A µ. 4-gluon interaction L 4 g 4 f abc f cde A aµ A bν A µ c A ν d g.5 Gauge theories have a certain freedom in defining the gauge field, A µ x. In order to remove the problem, eliminate the gauge freedom by settingconstraints for the field A µ x.. 4. Classical QCD equation of motion For example, µ A µ x. Euler-Lagrange equations derived from L QCD ψ, µ ψ,a µ, which is called Lorenz gauge covariant constraint. L QCD q i µ L QCD µ q i.6 Introduce extra term λ µ A µ a x with Lagrange multiplier parameter λ ξ Equations of motion for quark field: iγµ µ iga µ x m ψ.7 L γ 4 F µνxf µν x ξ µ A µ x. Equations of motion for gluon field: with color currents of quarks which are conserved: µ Jµ a x. µ G a µx+gf abc A µ b xgc µνx gj a ν x.8 J a ν x ψx γ ν t a ψx ψγ ν λ a ψ.9 Equation of motion A µ ξ ξ ;Feynmangauge Gauge fixing choices ξ ;Landaugauge Other options: A a ;Coulombgauge µ λ A λ Gauge fixing Digression on gauge fixing in electrodynamics: A a A a ;Axialgauge ;Temporalgauge L γ 4 F µνf µν.

4 - - Appendix: SUN-Group and Lie algebra Short mathematical appendix about groups: Group: G {g, h, k, } For g, h G, gh G There exists a unit element e such that eg ge g. For each g G, thereexistsaninverseg G ; g g gg e. Linear group: Elements g, h, transformations/operators with the following property: For each g, h G exists αg + βh G with α, β C Representations of a linear group: Mapping: g G a ij space of complex valued matrices with a ij C. Adjoint operator: Let g G linear, then there exists a unique g with the representation a ij a ji. Unitary transformations/operators: U G U U U U UU..5 Consequently a unitary transformation can be written as follows: U expih + ih + i H +.6 with Hermitian operator H, i.e. H H. Example-. Group U with elements U expiα whereα R U e iα, UU U U Group of gauge transformation in QED Example-. Group SUN Group of unitary transformations represented by unitary N N matrices U exp i α a X a with det U a where α a are real parameters with a,,n. The hermitian operators X a are the generators of the SUN group. Generators form Lie-algebra: where f abc are the structure constants of the group. For N,SU generators X a σ a /a,, Pauil matrices: σ Structure constants: f abc ɛ abc. Xa,X b ifabc X c.7, σ i, σ i tr{σ a } For N,SU generators X a λ a /a,, 8.8 tr{σ a σ b } δ ab.9 Gell-Mann matrices: i λ, λ i, λ, i λ 4, λ 5, λ 6, i λ 7 i, λ 8 i.4

5 - - Lie-algebra: tr{λ a }.4 tr{λ a λ b } δ ab λa, λ b ifabc λ c.4 Let j be the largest M: J + λ, j J J + λ, j J J+,J J λ, j J J J λ, j λ j j λ, j.49 Structure constants: f abc i tr λ a, λ b λ c f abc is totally antisymmetric with nonvanishing members, f f 47 f 56 f 46 f 57 f 45 f 67 f 458 f Therefore λ jj +..5 Relabeling the states λ, M j, M, Eq..47becomes J j, M jj + j, M, J j, M M j, M..5 Let j be the smallest M: J j, j J + J j, j J + J J j, j j + j + j j j, j.5 Irreducible representations of SU:. X a J a σ a a,, Hence jj +j j j j..5 Casimir operator of SU: J J + J + J which commutes with all generators J,J a a,,..45 Ladder raising and lowering operators: J ± J ± ij J J+ J + J J + + J.46 J+,J J, J,J ± ±J± Eigenstates of J and J : J λ, M λ λ, M, J λ, M M λ, M.47 J J J + J λ M.48 Basis states: { j, M with M j, j,, j, dimension: dj j + }. Product of representations of SU: J J + J, J J + J.54 J i j i,m i j i j i + j i,m i J i j i,m i M i j i,m i..55 To look for j, M with J j, M jj + j, M and J j, M M j, M, in general, we form appropriate linear combinations of product states: j, M j M j M jm j,m j,m.56 M,M where the quantities j M j M jm are called Clebsch-Gordan coefficients.

6 - - Example. Coupling of two states in fundamental representation of SU; basis states { j i,mi ± } Building product representations in terms of weight diagrams i Start with j,m,, ii Successively apply J to get to all other states,,, +,,,,,.57 iii Find the orthogonal combination to j max,m j max :,,,,,.58 4 Irreducible representations of SU group: U expiα a t a Rules for coupling SU representations Lie-algebra t a λ a a,, 8.59 ta,t b ifabc t c.6 j Singlet : j Doublet : 4 j Triplet : 5 j 4 Quartet.. where f abc is the structure constants of SU. Anticommutation relations: { ta,t b } δ ab + d abc t c.6 where d abc is called symmetric structure constants of SU. j j + Multiplet Casimir operator in SU: Graphical illustration in terms of weight diagrams: j j j 4 C T a T t i t a t i Isospin Y t 8 } Hypercharge.6 FIG..: Graphical representation of SU multiplets. Raising and lowering operators: T ± t ± it Iso spin, U ± t 6 ± it 7 U spin, V ± t 4 ± it 5 V spin.6

7 SU commutation relations: Basis states: α t, t,y, where α denoterepresentationse.g.,,8etc. T,T ± ± T± T,U ± U ± T,V ± ± V ± Y,T± T+,T T U+,U Y T U V+,V Y + T V T+,V + T+,U U+,V + Y,U± ± U± Y,V± ± V± T+,V U U+,V T.66 st step : T, T α ty, β t y nd step: t t γ T,T,Y tyt y tt t t TT α t, t,y β t,t,y.67 α ty T, T γ TY β t y α ty, β t y Isoscalar SU factors Product representations and rules in terms of weight diagrams:.68 Take center of gravity of one representation and place it on all parts of the second representation T+,U + V+ T,Y Weight diagrams of irreducible representations of SU Example. 8 Triplet Anti Triplet Fundamental Triplet y Fundamental Anti triplet y Octet 8 t t Singlet Octet 8 y t Eigenvalues of Casimir operators C t a 4 a Representations λ a t 4 λ.69 a Eigenvalues of C Product representations and Clebsch-Gordan coefficients of SU Singlet Triplet 4 Anti-triplet 4 Sextet 6 Octet 8