<<

CHAPTER 3. GREEN’S FUNCTIONS AND - 15 - CHAPTER III: Green Functions and Propagators ∂ ∂ φ(x￿)φ(x)= θ(t￿ t)φ(x￿)φ(x)+θ(t t￿) φ(x)φ(x￿) 3. 1. Introduction: field ∂t￿ T ∂t￿ − − ∂φ￿ (x￿) ￿ (3.8) = φ(x)+δ(t￿ t)φ(x￿)φ(x) δ(t￿ t)φ(x)φ(x￿) T ∂t￿ − − − Lagrangian with real scalar field φ(x) and “potential” U(φ). δ(t t)[φ(x ), φ(x)]=0 • ￿− ￿ ￿ ￿￿ ￿ 1 µ 1 2 2 = ∂µφ ∂ φ m φ U(φ) (3.1) L 2 − 2 − 2 2 ∂ ∂ φ(x￿) ∂φ(x￿) ￿ ￿ ￿ ￿￿ ￿ 2 φ(x )φ(x)= 2 φ(x)+δ(t t) , φ(x) – equations ∂t￿ T T ∂t￿ − ￿ ∂t￿ ￿ (3.9) 4 iδ (x￿ x) 2 ∂U − − ￿ + m φ + =0 2 2 4 ∂φ = ￿ m φ(x￿)φ(x) iδ￿￿(x￿ x) ￿ (3.2) T ∇￿x ￿ − − − ￿ ￿ ∂U + m2 φ(x)= (x) with (x)= ￿ ￿ ⇒ ￿ −J J ∂φ

￿ ￿ 2 4 – Free field equation: Klein-Gordon equation x￿ + m φ(x￿)φ(x)= iδ (x￿ x) ⇒ ￿ T − − ￿ ￿ (3.10) 2 2 4 ￿ + m φ(x) = 0 (3.3) + m ∆ (x￿ x)= δ (x￿ x) ⇒ ￿x￿ F − − − ￿ ￿ ￿ ￿ – Solution: 3 Solution of inhomogeneous wave equation: d k ik x ik x • − · † · φ(x)= 3 ak e + ak e (3.4) (2π) 2ωk 4 ￿ ￿ ￿ φ(x)= d x￿ ∆F (x￿ x) (x￿) (3.11) − J – Commutation relations for ak and ak† : ￿ ∆ is the Green’s function of the Klein-Gordan equation. 3 3 ￿ ￿ F ak,ak† =(2π) 2ωk δ k k ￿ • ￿ − (3.5) ￿ a ,a ￿ = a† ,a† =0￿ ￿ Fourier representation: k k￿ k k￿ •

￿ ￿ ￿ ￿ 4 ik (x￿ x) Definition of time-ordered product: d k e− · − ∆F (x￿ x)= (3.12) • − (2π)4 k2 m2 + i￿ ￿ − φ(x￿)φ(x)=θ(t￿ t) φ(x￿)φ(x)+θ(t t￿) φ(x)φ(x￿) (3.6) T − − 4 2 ik (x￿ x) 2 d k (￿x￿ + m )e− · − ( + m )∆ (x￿ x)= ￿x￿ F − (2π)4 k2 m2(+i￿) Correlation function: ￿ 4 2 − 2 ik (x x) • d k ( k + m )e− · ￿− i∆F (x￿ x)= 0 φ(x￿)φ(x) 0 (3.7) = − − ￿ |T | ￿ (2π)4 k2 m2(+i￿) (3.13) ￿ 4 − d k ik (x x) Feynman propagator: Green’s function time-ordered correlation function = e− · ￿− ↔ − (2π)4 ￿ 2 4 Apply ￿x￿ + m to ∆F : = δ (x￿ x) • − − where Poles at k2 = m2: 2 ∂ 2 • ￿ ￿ 2 2 ￿x￿ 2 ￿x ￿ ωk = k + m k0 = ωk ≡ ∂t￿ − ∇ ⇒ ± ￿ 26 25 - 16 - Dirac propagator: Cauchy’s integral formula: ￿ • 3 ik0(t t) ! ! d k ∞ dk e− ￿− i￿k·(!x −!x ) ¯ (3.16) 0 e ik (￿x ￿ ￿x) iSF (x￿ x)αβ = 0 ψα(x￿)ψβ(0) 0 ∆F (x￿ x)= e− · − (3.14) − ￿ |T | ￿ (2π)3 2π 2 ￿ 2 2 − k0 k m + i￿ ￿ ￿−∞ − − This is the Green’s function of the free :

Im k0 µ 4 (iγµ∂x m)SF (x￿ x)αβ = δ (x￿ x)δαβ (3.17) t % t' f(z) ￿ − − − Using =2πif(a) "Ω z a Fourier representation !k ￿ − • d4p γ pµ + m Re k0 ip (x￿ x) µ ! SF (x￿ x)= e− · − (3.18) ω = ￿k 2 + m2 iδ − (2π)3 p2 m2 + i￿ $Ωk k − ￿ − ￿ t & t' d4p γ pν + m µ µ ip (x￿ x) ν (iγµ∂ m)SF (x￿ x)= (iγµ∂ m) e− · − 3 iωk(t￿ t) x￿ 4 x￿ 2 2 d k e− − i￿k (￿x ￿x) − − (2π) − p m + i￿ ￿ · ￿− ￿ t￿ >t; ∆F (x x)= i 3 e 4 µ ν − • − − (2π) 2ωk d p (γµp m)(γνp + m) ip (x x) ￿ − · ￿− = 4 −2 2 e 3 iω (t t) (2π) p m + i￿ d k e k ￿− ￿ (3.19) ik (￿x ￿ ￿x) ￿ 4 2 −2 ￿ · − t￿ 0. p/ + m k − = p/ γ pµ p2 m2 + i￿ ≡ µ − – The second term describes a “particle” running backward time with negative ωk ￿ ￿ and t t￿ > 0: . − 3. 3. Free gluon propagator ! Particle " ! Antiparticle " time time Free gluon Green’s function: • µν µ ν t' t iD (x￿ x)= 0 A (x￿)A (x) 0 (3.21) ab − ￿ |T a b | ￿

t t' space space 4 µν x x' x x' µν d q d (q) D (x￿ x)=δ ab − ab (2π)4 q2 ￿ µ ν (3.22) µν µν q q with d (q)= g +(1 ξ) 2 3. 2. Dirac propagator − − q + i￿

Time ordered product of Dirac fields: ψ (x￿)ψ¯ (x) ξ = 1 : Feynman gauge • T α β  ξ = 0 : Landau gauge 

27  28 B CHAPTER 4. S-MATRIX AND FEYNMAN RULES- 17 - in A : u(p, s) a ) Incoming quark lines CHAPTER IV: S-Matrix and Feynman Rules   in B : v(p, s) 4. 1. Definition: S-matrix, T-matrix and cross section  T in A :¯v(p, s) BA b ) Outgoing quark lines S-matrix  in B :¯u(p, s) •  S = B, t A, t (4.1) µ BA ! →∞| →−∞% c ) External gluon lines: Polarization vector: ! A A and B are asymptotic states: | % | % Remember QCD Lagrangian A, t = eiHt A, t =0 • | % | % ¯ µ λa µ QCD = ψ(x) iγµ ∂ ig Aa (x) m ψ(x) T-matrix L  − 2 −  •  % &  1 µν a  (4.5) 4 4 Ga (x)Gµν (x)  B S A SBA = δBA + i(2π) δ (pA pB)TBA − 4 ! | | %≡ − (4.2) 1 µ 2 B T A T = (∂µAa (x)) ! | | %≡ BA −MBA − 2ξ µν µ ν ν µ µ ν Differential cross section for A B Ga (x)=∂ Aa(x) ∂ Aa (x)+gfabcAb (x)Ac (x) • → −

– Prototype: two particles colliding in initial state: A = a1 + a2 m:quarkmassmatrix

(a1 + a2 B) mu 00000 dσ(a1 + a2 B)=W → dNB (4.3) → JA  0 md 0000     – (a1 + a2 B): Transition probability for A B per unit time.  00ms 000 W → → m =   (4.6)    000mc 00 – dNB:PhasespaceelementinthefinalstateB.      0000mb 0  – J :FluxofincomingparticlesinstateA.   A    00000mt    number of particles   JA =   time unit area Interaction vertices × • Assume n particles in final state: • n g g g2 3 ∼ ∼ d pi 2 2 ∼ dNB = 3 Ei = p$i + mi (4.4) (2π) 2Ei i=1 # ! " $ Quark-gluon vertex 3-gluon vertex 4-gluon vertex

4. 2. Feynman rules (for the calculation of invariant amplitude TBA) Consider QCD in its perturbative domain (“Perturbative QCD”): • g2 α = 1 Perturbative expansion of observables in powers of α . Factors to be applied for each external lines s 4π # ⇒ s •

30

29 - 18 -

mu md ms mc mb mt e ) 4-gluon vertex: 4( 2) MeV 7( 2) MeV 120 ( 5) MeV 1.3( 0.1) GeV 4.3( 0.1) GeV 174 ( 5) GeV a ,µ a ,µ ± ± ± ± ± ± 1 1 4 4 factor: g2 f f (gµ1µ3 gµ2µ4 gµ1µ4 gµ2µ3 ) TABLE 4.1: Values of quark masses − a1a2a a3a4a − ! + f f (gµ1µ2 gµ3µ4 gµ1µ4 gµ2µ3 ) a1a3a a2a4a − Internal lines µ1µ2 µ3µ4 µ1µ3 µ2µ4 • + fa1a4afa2a3a(g g g g ) a2,µ2 a3,µ3 − a ) Quark: (a, b:colorindices;i, j:flavorindices) j, b

ij i iS (p) = δ δij p F ab ab /p m + i" − 4. 3. Examples: Quark-quark and quark-antiquark scattering i, a ! " i(/p + m) = δ δij ab p2 m2 + i" in one gluon exchange approximation − a ) T-matrix for qq-scattering: b ) Gluon: (a, b:colorindices;µ, ν:Lorentzindices) p! p! ν, b 1 2 µ ν p µν p p i = δab g + 1 ξ Feynman gauge (ξ =1) − − p2 p2 + i" # & q µ a $ % q = p1 p1! = p2! p2 , − −

p1 p2 µν 2 ig δab iT =(ig) u¯(p! ) γ t u(p ) − u¯(p! ) γ t u(p ) (4.7) c ) Quark-gluon vertex: 1 µ a 1 q2 + i$ 2 ν b 2 ! " ! " a, µ b ) T-matrix for qq¯-scattering to order αs:

p2! p1! µ p! p! − factor: igγ ta − 1 2

p1 + p2 p p! − 1 − 1

p1 p2 d ) 3-gluon vertex: − p p 2 − 1

a3,µ3 µν 2 ig δab µ1µ2 µ3 iT =(ig) v¯(p ) γµta v(p! ) − u¯(p! ) γνtb u(p ) factor: gfa a a g (p1 p2) 1 1 2 2 2 1 2 3 − (p1 p1! ) + i$ p1 p3 − µν (4.8) a1,µ1 µ2µ3 µ1 !2 " ig δab ! " !+ g (p2 p3) ! ! (ig) u¯(p2) γµta v(p1) − 2 v¯(p1) γνtb u(p2) − − (p1 + p2) + i$ p2 µ3µ1 µ2 + g (p3 p1) ! " ! " a2,µ2 − "

32 31 - 19 -

4. 4. Sketch of path integrals (Functional integrals) Definition of functional (path) integral: • Φφ Systematic method for derivation of Feynman rules Consider infinitesimal volume in space-time is

Illustration: example of scalar field theory ∆v = δxi δyj δzk δtl • Infinitesimal volume ! µ 1 µ 2 2 x attached to a point (xi,yj,zk,tl)withfield (φ, ∂ φ)= (∂µφ∂ φ m φ ) V (φ)(4.9) L 2 − − φ(x ,y,z ,t)anditsdifferential dφ defined t i j k l functional: at that point. • S = d4x (φ, ∂µφ)=S[φ, ∂µφ](4.10) L ! Basic relation for calculating n-point Green’s functions (correlation function) + • ∞ φ =lim dφ(xi yj zk tl) iS[φ,∂µφ ] D ∆v 0 (4.12) φφ(x1) φ(xn) e ! → ijkl !−∞ 0 φ(x1)φ(x2) φ(xn) 0 = D ··· µ (4.11) " " |T ··· | # φ eiS[φ,∂ φ ] " D Starting point: Generating functional " • For n =2:“2-pointfunction” 4 [J]= φ eiS ei R d x φ(x)J(x) (4.13) x1 Z D ! “Propagator” J(x): auxiliary source function.

Then the n-point function (4.11) becomes x2

For n =3:“3-pointfunction” (n) (x1x2 xn) 0 φ(x1)φ(x2) φ(xn) 0 x3 G ··· ≡" |T ··· | # ( i)n δ [J] (4.14) = − Z [0] δJ(x1) δJ(xn)# Z ··· #J=0 “Vertex” # x x # 1 2 # with the functional derivative: For n =4:“4-pointfunction” δ [J(x)] [J(x)+#δ4(x y)] [J(x)] Z =limZ − − Z (4.15) δJ(y) " 0+ # x4 x3 → δJ(x) “Scattering amplitude” in particular: = δ4(x y) δJ(y) − x1 x2 $ %

Example. Free scalar field 1 = ∂ φ∂µφ m2φ2 (4.16) L0 2 µ − & ' 34 33 Definition of functional (path) integral: • Φ Consider infinitesimal volume in space-time is

∆v = δxi δyj δzk δtl 4 0[J]= φ exp i d x 0 + φ(x)J(x) (4.17) Infinitesimal volume ! Z D L x attached to a point (xi,yj,zk,tl)withfield! " ! % Using # $ φ(xi,yj,zk,tl)anditsdifferential dφ defined t 4 µ 4 µ 4 d x ∂µφ∂ φ = d x ∂µ(φ∂ φ) d x φ!φ (4.18) at that point. [J]= − φ exp i d4x + φ(x)J(x) (4.17) ! ! Z0 D! L0 surface integral =0! " ! % Using # $ it follows that & '(d4x ∂ φ∂) µφ = d4x ∂ (φ∂µφ) d4x φ!φ (4.18) µ µ − ! ! ! + 4 1 2 ∞ 0[J]= φ exp i d x φ ! + msurfaceφ integralJφ =0 (4.19) φ =lim dφ(xi yj z⇒k tl)Z D − 2 − ∆v 0 (4.12)" % D !it follows that ! 2 * & '( + ) ! → ijkl !−∞ equation of motion: ! + m φ(x)=# J(x)$ " − 4 1 ! 2 4 # 0[J]=$ φ exp i d x φ + m φ Jφ (4.19) Starting point: Generating functional φ(x)= d y ∆⇒FZ(x y)J(yD) − 2 − − − ! " ! * +% • ! equation of motion: ! + m2 φ(x)=# J(x)$ 4 ik (x y) − 4 d k e− · − iS i R d x φ(x)J(x) with ∆ (x y)= 4 [J]= φ e e F (4.13) φ(x4)=2 d2 y#∆ (x y$)J(y) Z D − (2π) k m + i$F ! ! −− − ! 4 ik (x y) i 4 4 d k e− · − J(x): auxiliary source function. 0[J]=exp d x dwithyJ∆(x)(x∆F (yx)= y) J(y) ⇒ Z − 2 F − − (2π)4 k2 m2 + i$ " ! ! ! % − Then the n-point function (4.11) becomes i 4 i 4 2 4 φ exp 0[J]=expd x φ(x)(!d+x m d)φyJ(x()x) ∆F (x (4.20)y) J(y) ⇒ Z − 2 − × D − 2 " ! ! % ! " ! i % φ exp d4x φ(x)(! + m2)φ(x) (4.20) (n)(x x x ) 0 Nowφ(x ) calculateφ(x ) φ 2-point(x ) 0 function as example: × D − 2 G 1 2 ··· n ≡" |T 1 2 ··· n | # ! " ! % ( i)n δ [J] Now calculate(4.14) 2-point function as example: = − Z (2) [0] δJ(x1) δJ(xn)# (x1,x2)= 0 φ(x1)φ(x2) 0 Z ··· #J=0 G $ |T | % # (2) # 1 δ2(x1,x[J2])= 0 φ(x1)φ(x2) 0 # = G Z0 $ |T | % (4.21) 2 with the functional derivative: − 0[0] δJ(x1)δJ(x2), 1 δ 0[J] Z = ,J=0 Z (4.21) − 0[0] δJ(x1)δJ(x2) 4 , ,J=0 δ [J(x)] [J(x)+#δ (x y)] [J(x)] = i∆F (x1 x2) , Z , Z =limZ − − Z (4.15) − , , δJ(y) " 0+ # = i∆F (x1 x2) , → − ,

δJAnalogous(x) 4 procedures for n-point functions Feynman rules for scalar field theory. in particular: = δ (x y) Analogous procedures⇒ for n-point functions Feynman rules for scalar field theory. δJ(y) − ⇒ $ %

- 20 - Example. Free scalar field 1 4. 5. Appendix: Useful4. 5. relations Appendix: Useful relations = ∂ φ∂µφ m2φ2 (4.16) L0 2 µ − & When dealing' with path integrals,When dealing some with basic path formulae: integrals, some basic formulae: 34 [J]= φ exp i d4x + φ(x)J(x) (1) (4.17)Important matrix identity: let M be a diagonalizable matrix Z0 D (1)LImportant0 matrix identity: let M be a diagonalizable matrix ! " ! % Using # $ ln det M =trlnM ln det M =trlnM (4.22) 4 µ 4 µ 4 d x ∂µφ∂ φ = d x ∂µ(φ∂ φ) d x φ!φ (4.18) det M =exp tr ln M (4.22) − ⇒ ! ! ! det M =exp tr ln M surface integral =0 ⇒ - . - . 35 it follows that & '( ) (2) Gaussian integral: 35 + ∞ dx 1 2 1 4 1 2 exp ax = (4.23) 0[J]= φ exp i d x φ ! + m φ Jφ (4.19) √2π − 2 √a ⇒ Z D − 2 − !−∞ " % " # ! ! 2 * + T equation of motion: ! + m φ(x)=# J(x)$ Let M be real, symmetric N N matrix and X =(x1, ,xN ) − • × ··· φ(x)= d4y#∆ (x y$)J(y) Generalization of Gaussian integral − F − ⇒ + + ! 4 ik (x y) ∞ dx1 ∞ dxN 1 T 1 d k e− · − exp X MX = (4.24) with ∆F (x y)= √2π ··· √2π − 2 √det M − (2π)4 k2 m2 + i$ !−∞ !−∞ " # ! − i In functional integrals: often encounter [J]=exp d4x d4yJ(x) ∆ (x y) J(y) • ⇒ Z0 − 2 F − " ! ! % 1 4 4 φ exp d x d x# φ(x#)M(x#,x)φ(x) (4.25) i 4 2 D − 2 φ exp d x φ(x)(! + m )φ(x) (4.20) ! $ ! ! % × D − 2 ! " ! % L 4 Approximate d4x ∆v by sum over finite number N = of little cubes Now calculate 2-point function as example: → i # ! i ' ( and use Eq. (4.24): &

(2) 1 4 4 1 (x1,x2)= 0 φ(x1)φ(x2) 0 φ exp d x# d x φ(x#)M(x#,x)φ(x) = (4.26) D − 2 √det M G $ |T | % ! $ ! ! % 2 1 δ 0[J] = Z Complex(4.21) scalar fields − 0[0] δJ(x1)δJ(x2), • Z ,J=0 1i 4 4 1 , φ φ∗ exp d x d x# φ∗(x#)M(x#,x)φ(x) = = i∆F (x1 x2) , D D − 2 & det M − , ! ! $ ! ! % (4.27) =exp tr ln M − Analogous procedures for n-point functions Feynman rules for scalar field theory. ) * ⇒ 4. 6. fields

3 d p ip x ip x − · ∗ · ψ(x)= 3 ap,sus(p)e + bp,svs (p)e (4.28) (2π) 2Ep s= 1 ! 4. 5. Appendix: Useful relations &± 2 " # ! Grassmann-Algebra: When dealing with path integrals, some basic formulae: a ,a = b ,b = =0; an =0,n>1(4.29) { i j} { i j} ··· i (1) Important matrix identity: let M be a diagonalizable matrix ! Most general form of function of two Grassmann variables ln det M =trlnM (4.22) f(a1,a2)=c0 + c1a1 + c2a2 + c3a1a2 det M =exp tr ln M (4.30) = c + c a + c a c a a ⇒ 0 1 1 2 2 − 3 2 1 - . 35 36 (2) Gaussian integral: (2) Gaussian integral:+ ∞ dx +1 2 1 exp ∞axdx = 1 2 1 (4.23) √2π − 2 Derivative:exp√a ax = (4.23) !−∞ √2π − 2 √a "!−∞ • # " # T ∂f Let M be real, symmetricLet M beN real,N symmetricmatrix andN XN =(matrixx1, and,xXNT)=(x , ,x ) • • × × ··· 1 ··· N = c1 + c3a2 a∂a1 Generalization ofGeneralization Gaussian integral of Gaussian integral (4.31) ⇒ ⇒ ∂f Derivative: = c2 c3a1 + + + + • ∞ dx1 ∞ d∞xNdx 1∞ dx 1 1 1 a∂a2 − 1exp XT MXN exp= XT MX = (4.24) (4.24) ∂f √2π ··· √2√π2π ··· − 2 √2π −√2det M √det M 2 =2c1 + c3a2 !−∞ !−∞!−∞ !−∞ ∂ a∂a1 ∂ " # " # Derivative: = (4.32)(4.31) • ∂f In functional integrals:In functional often encounter integrals: often encounter ∂a1∂a2 −∂a=2∂ca2 1 c3a1 • a∂a2 ∂f− • = c1 + c3a2 a∂a1 1 1 Integration:4 4 2 2 (4.31) 4φ exp 4 • d x d x# φ(x#)M(x#,x)φ(x) (4.25) ∂ ∂f ∂ φ exp dD x d x−# φ2(x#)M(x#,x)φ(x) (4.25) = (4.32) D − 2 $ % = c2 c3a1 ! $ !! ! ! ! % da1da2 F ∂a1d∂aa12 a−∂ad∂2a22∂Fa1− (4.33) 4 ≡ 4 L " 2 # 2 4 Integration:L ! ! !∂ ∂ 4Approximate d x ∆vi by sum over finite number N = Derivative:of little cubes 2 = 2 (4.32) Approximate d x ∆vi by sum→ over finite number N =• of little cubes# • → i # ' ( ∂a1∂a2 −∂a2∂a1 i ! da =0 becauseda da dFa =da ∂f dda F =0 (4.34)(4.33) ! and use Eq. (4.24): & ' ( 1 2 1 = c +2 c a & ≡ −a∂a 1 3 2 and use Eq. (4.24): ! Integration: ! " ! # ! ""!1! # # (4.31) • ∂2f 2 1 4 4 1 = c2 c3a1 φ exp d x# d x φ(x#)M(x#,x)φ(x) = (4.26) da1da2 F a∂a2 da1− da2 F (4.33) 1 4 4 Definition: d1aa=1asanormalizationda =0 because da ≡ = da =0 (4.34) φ exp dDx# d x−φ2(x#)M(x#,x)φ(x) = √(4.26)det M − $ % ! 2! " !2 # D − 2 ! ! ! !√det M ! " ! #∂ "2!∂ # 2 ! $ ! ! % = (4.32) da =0 because∂a1∂a2 da−∂a=2∂a1 da =0 (4.34) Complex scalar fields Path integralsDefinition: with fermiondaa fields:=1asanormalization − • • Integration:! " ! # " ! # Complex scalar fields ! • • T i 4 4 Definition:1 daa=1asanormalizationda1da2 F da1 da2 F (4.33) φ φ∗ exp d x dGivenx# φ∗(x an#)M antisymmetric(xPath#,x)φ( integralsx) matrix with fermionA with fields:a =(a1, ,aN≡) Di D4 4 − 2 • 1 & det M ! ··· ! " ! # ∗ # ∗ # # ! (4.27) 2 2 φ φ exp ! ! d x d$x φ (!x )M(!x ,x)φ(x) % T D D − 2 & detGivenM an antisymmetricPath integrals with matrix fermionA withda fields:1=0a because=(a , d,aa N )= da =0 (4.34) ! ! $ ! ! % =exp(4.27)tr ln M T 1 √ − • da1 − daN exp ! a Aa ="···! det# A " ! # (4.35) ··· − 2 T =exp tr ln MGiven an antisymmetricDefinition: daa matrix=1asanormalizationA with1 a =(a , ,a ) − ! ) ! da * d$a exp %aT Aa 1=···√detNA (4.35) 1 ! N 4. 6. Fermion fields ) - 21 - * ··· − 2 n Path! integrals! with fermion$ fields: 1% T √ 4. 6. Fermion fields ai =0forn>1, dai =0;• dai ai =1da1 daN exp a Aa = det A (4.35) ··· −T 2 n Given an antisymmetric! ! matrix A with$a =(a1, %,aN ) ai =0for! n>1, !dai =0; dai ai =1 ··· 3 n 1 d p Complex fermionip x fields:a =0forip x ! n>1, !dai =0;da dai aida=1exp aT Aa = √det A (4.35) − · ∗ i · 1 N ψ(x)= 3 • ap,sus(p)e + bp,svs (p)e (4.28) ··· − 2 3 (2π) 2Ep Complex fermion fields: ! ! ! ! $ % d p s= 1 2 ! ip x " •ip x # ψ(x)= &a±p,sus(p)e− · + bp,s∗ vs (p)e · Complex(4.28) fermionn fields: 1 3 da da∗ ai =0forda n>1,da∗daiexp=0; dai ai =1a†Aa =detA (4.36) ! 1 (2π) 2Ep •1 1 N N 1 Grassmann-Algebra:s= ! da da∗ !da da!∗ exp2 a†Aa =detA (4.36) &± 2 " # ···1 1 N N− ! ! Complex! fermion···! fields: $ − 2 % 1 ! ! • ! da1 ! da1∗ ! daN d$aN∗ exp %a†Aa =detA (4.36) Grassmann-Algebra: n ··· − 2 ai,aj = bi,bj =Functional=0; a integralsi =0,n> involving1(4.29) fermion! fields:! ! ! $ 1 % { } { } ··· Functional integrals involving fermionda1 d fields:a1∗ daN daN∗ exp a†Aa =detA (4.36) • ··· − 2 n • Functional integrals involving! ! fermion! fields:! $ % ! ai,aj = bi,bj = =0; ai =0,n>1(4.29)• Most{ general} form{ of} function··· of two Grassmann variables Functional integrals involving fermion fields: • 4 4 4 4 ! ψ ψ∗ exp d xd x" ψ∗(x")A(x",x)ψ(x) =detA Most general form of function of two Grassmannf(a1,a2)= variablesc0 + c1a1 + c2a2ψ+ c3a1ψa2∗ exp d xd x" ψ∗(x")A(x4 ",x4 )ψ(x) =detA D D ψ −ψ∗ exp d xd 4x" ψ4 ∗(x")A(x",x)ψ(x) =detA (4.37) D D! ! − D $D ψ!(4.30)ψ∗−exp d xd x" ψ∗(x")A(x"%,x)ψ(x) =detA (4.37) ! ! $ ! ! ! D D $ ! − % % (4.37)(4.37) = c0 + c1a1 + c2a2 c3a2a1 ! ! $ ! =exp% tr ln A f(a1,a2)=c0 + c1a1 + c2a2 + c3a1a2 − =exp tr ln=expA tr ln A (4.30) =exp tr ln A Derivative: = c + c a + c a c a a & & & ' ' ' • 0 1 1 2 2 − 3 2 1 & ' ∂f 36 = c + c a 4. 7. Generating4. 7. functional Generating functional of QCD of QCD a∂a 1 3 2 4. 7. Generating functional of QCD 1 4. 7. Generating(4.31) functional of QCD 36 ∂f ! Lagrangian density (without gauge fixing): = c2 c3a1 a∂a2 − ! Lagrangian! Lagrangian density (without density (withoutgauge fixing): gauge fixing): µ 1 a µν QCD = ψ¯ iγµD m ψ G G (4.38) 2 ! 2 µν a ∂ Lagrangian∂ density (without gauge fixing): L − −14 = (4.32) µ ¯ & µ 1 a' µν a µν = ψ¯ QCDiγ D= ψ imγµDψ mGψ G GµνGa (4.38)(4.38) ∂a1∂a2 −∂a2∂a1 LQCD L µ − −− 437 µν− a4 ¯ µ & 1 a µν' Integration: QCD = ψ iγµD & m ψ 'GµνGa (4.38) • ! GeneratingL functional: − 37− 4 37 da1da2 F da1 da2 F (4.33) ≡ & ¯' ! ! " ! # QCD[J, η, η¯]= A ψ ψ Z D D 37 D 2 2 ! ! ! (4.39) da =0 because da = da =0 (4.34)exp i d4x (x)+Aa (x)J µ(x)+ψ¯(x)η(x)+¯η(x)ψ(x) − × LQCD µ a ! " ! # " ! # " ! % # $ Definition: daa=1asanormalization Generate n-point functions by taking functional derivatives with respect to source • ! fields J(x), η(x)and¯η(x). Path integrals with fermion fields: • Given an antisymmetric matrix A with aT =(a , ,a ) 1 ··· N Example-1. 2-point functions: 1 da da exp aT Aa = √det A (4.35) 1 ··· N − 2 δ2 ! ! $ % -Quarkpropagator: QCD δη(x)δη¯(x)Z & &η, η¯=0 n & ai =0forn>1, dai =0; dai ai =1 & & ! ! = 0 ψ(x)ψ¯(y) 0 = iSF (y x) Quark " |T | # − Complex fermion fields: 4 (4.40) • d p ip (y x) /p + m = i e− · − 1 (2π)4 p2 m2 + i% da da∗ da da∗ exp a†Aa =detA (4.36) ! − 1 1 ··· N N − 2 ! ! ! ! $ % δ2 -Gluonpropagator: a b QCD δJµ(x)δJν (y)Z & Functional integrals involving fermion fields: &J=0 • & & & µν = 0 Aµ(x)Aν (y) 0 = iD (y x) " |T a b | # ab − 4 4 Gluon (4.41) ψ ψ∗ exp d xd x" ψ∗(x")A(x",x)ψ(x) =detA d4q dµν(q) D D − (4.37) = iδab ! ! $ ! % (2π)4 q2 + i% =exp tr ln A ! Example-2. 3-point functions: & ' δ3 -Quark-gluonvertex: a QCD 4. 7. Generating functional of QCD δJµ(x) δη(x) δη¯(x)Z & &η,η¯,J=0 & & & ! Lagrangian density (without gauge fixing): µ = igγ ta (4.42) 1 = ψ¯ iγ Dµ m ψ Ga Gµν (4.38) LQCD µ − − 4 µν a & ' δ3 37 -3-gluonvertex: a b c QCD δJµ(x) δJν(x) δJλ(x)Z & &J=0 & & &

38 ! Generating functional:

[J, η, η¯]= A ψ ψ¯ ZQCD D D D p ! ! ! (4.39) 2 4 a µ ¯ exp i d x QCD(x)+Aµ(x)Ja (x)+ψ(x)η(x)+¯η(x)ψ(x) × L µν λ " ! % p1 = gfabc g (p p ) + cycl.perm. (4.43) # $ 1 − 2 Generate n-point functions by taking functional derivatives with respect to source • ! " fields J(x), η(x)and¯η(x). p3

- 22 - Example-1. 2-point functions: 4. 8. Gauge invariance and gauge fixing (Sketch) δ2 -Quarkpropagator: QCD δη(x)δη¯(x)Z & Pure gluon theory: &η, η¯=0 • 1 & = Ga Gµν (4.44) & LG −4 µν a & ¯ = 0 ψ(x)ψ(y) 0 = iSF (y x) where Ga = ∂ Aa ∂ Aa + gf Ab Ac . Quark " |T | # − µν µ ν − ν µ abc µ ν 4 (4.40) d p ip (y x) /p + m = i e− · − Action functional: (2π)4 p2 m2 + i% • ! − S [A, ∂A]= d4x (A, ∂A)(4.45) G LG δ2 # -Gluonpropagator: a b QCD Generating functional: δJµ(x)δJν (y)Z & • &J=0 & 4 µ a & G[J]= A exp i d x G + Aa Jµ (4.46) & µν Z D L = 0 Aµ(x)Aν (y) 0 = iD (y x) # $ # ' " |T a b | # ab − % & Gluon Functional(4.41) integral covers arbitrarily many gauge-equivalent field configurations. d4q dµν(q) • = iδ λa ab 4 2 Aµ Aµ (2π) q + i% µ µ i µ a ! A˜ = U A U †∂ U U † ≡ 2 (4.47) − g  λa  U =exp iθa(x) Example-2. 3-point functions: ( )  − 2 . / 3 Gauge fixing needs a constraint.  δ • -Quark-gluonvertex: a QCD δJµ(x) δη(x) δη¯(x)Z & &η,η¯,J=0 µ a a (4.48) & ∂ Aµ(x)=B (x) & & a µ (In particular Lorenz condition B (x) 0) = igγ ta (4.42) ≡ Insert “unity”: • 8 µ a a =det θa δ ∂ Aµ(x) B (x) (4.49) 3 M D − δ a=1 # -3-gluonvertex: a b c QCD 0 % & δJµ(x) δJν(x) δJλ(x)Z & with Jacobian of gauge transformation: &J=0 p & δ ∂µAa(x) 2 & ab µ & (x, y)= b M % δθ (y) & µν λ p1 = gfabc g (p1 38p2) + cycl.perm. (4.43) Problem: to calculate Jacobian det − • M ! " 39 p3

4. 8. Gauge invariance and gauge fixing (Sketch)

Pure gluon theory: • 1 = Ga Gµν (4.44) LG −4 µν a where Ga = ∂ Aa ∂ Aa + gf Ab Ac . µν µ ν − ν µ abc µ ν Action functional: • S [A, ∂A]= d4x (A, ∂A)(4.45) G LG # Generating functional: • [J]= A exp i d4x + AµJ a (4.46) ZG D LG a µ # $ # ' % & Functional integral covers arbitrarily many gauge-equivalent field configurations. • λ Aµ Aµ a µ µ i µ a A˜ = U A U †∂ U U † ≡ 2 (4.47) − g  λa  U =exp iθa(x) ( )  − 2 . / Gauge fixing needs a constraint.  •

µ a a ∂ Aµ(x)=B (x) (4.48)

(In particular Lorenz condition Ba(x) 0) ≡ Insert “unity”: • 8 =det θ δ ∂µAa (x) Ba(x) (4.49) M D a µ − a=1 0 # with Jacobian of gauge transformation: % & µ a ab δ ∂ Aµ(x) (x, y)= b M % δθ (y) & Problem: to calculate Jacobian det • M 39 4. 9. Fadeev-Popov method (Sketch)

a a Introduce a set of (unphysical) auxiliary fields: χ (x), χ∗ (x): anticommuting Bose • fields (“ghost fields”)

4 µ ab det = i χ χ∗ exp i d x ∂ χ∗(x)D χ (y) (4.50) M D D a µ b ! ! " ! # with gauge covariant derivative:

Dab = δab∂ gfabcAc µ µ − µ

Result: Gauge fixing condition extra term in Lagrangian density. • ⇒ ! QCD Lagrangian including gauge fixing: 1 1 = Ga (x)Gµν (x) ∂µAa 2 + (4.51) LQCD −4 µν a − 2ξ µ LFP $ % with Fadeev-Popov term: µ a ab b = ∂ χ∗ (x)D χ (x)(4.52) LFP µ

4. 10. Complete generating functional of QCD (including gauge fixing)

[J, η, η¯; j, j∗] ZQCD

= A ψ ψ¯ χ χ∗ D D D D D ! ! ! ! ! 4 a µ a a exp i d x + A J + ψη¯ +¯ηψ + χ∗ j + j∗χ (4.53) × LQCD µ a a a " ! # $ %

1 1 with = ψ¯ iγ Dµ m ψ Ga Gµν ∂ Aµ 2 + - 23 - LQCD µ − − 4 µν a − 2ξ µ a LFP 4. 4.9. 9. Faddeev-Popov Fadeev-Popov method method (sketch) (Sketch) & ' $ % Additional Feynman rules associated with ghost: a a • Introduce a set of (unphysical) auxiliary fields: χ (x), χ∗ (x): anticommuting Bose • fields (“ghost fields”) – Ghost propagator in momentum space:

4 µ ab iδab det = i χ χ∗ exp i d x ∂ χa∗(x)Dµ χb(y) (4.50) a b = (4.54) M D D p p2 + i( ! ! " ! # with gauge covariant derivative: 40 ab ab abc c Dµ = δ ∂µ gf Aµ − – Ghost-gluon vertex: a Result: Gauge fixing condition extra term in Lagrangian density. • ⇒ µ ! QCD Lagrangian including gauge fixing: b = gfabc p (4.55) 1 1 = Ga (x)Gµν (x) ∂µAa 2 + (4.51) LQCD −4 µν a − 2ξ µ LFP c, µ $ % with Faddeev-Popov Fadeev-Popov term:term µ a ab b = ∂ χ∗ (x)D χ (x)(4.52) LFP µ

4. 10. Complete generating functional of QCD (including gauge fixing)

[J, η, η¯; j, j∗] ZQCD

= A ψ ψ¯ χ χ∗ D D D D D ! ! ! ! ! 4 a µ a a exp i d x + A J + ψη¯ +¯ηψ + χ∗ j + j∗χ (4.53) × LQCD µ a a a " ! # $ %

1 1 with = ψ¯ iγ Dµ m ψ Ga Gµν ∂ Aµ 2 + LQCD µ − − 4 µν a − 2ξ µ a LFP & ' $ % Additional Feynman rules associated with ghost: • – Ghost propagator in momentum space:

iδab a b = (4.54) p p2 + i(

40

41