High energy hadronic interactions in QCD and applications to heavy ion collisions

Transcription

1 High energy hadronic interactions in QCD and applications to heavy ion collisions III QCD on the light-cone François Gelis CEA / DSM / SPhT François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 1/56

2 General outline Lecture I : Introduction and phenomenology Lecture II : Lessons from Deep Inelastic Scattering Lecture III : QCD on the light-cone Lecture IV : Saturation and the Color Glass Condensate Lecture V : Calculating observables in the CGC François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 2/56

3 Lecture III : QCD on the light-cone - Infinite Momentum Frame Poincaré algebra on the light-cone - Galilean sub-algebra Canonical quantization on the light-cone Scattering by an external potential François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 3/56

4 Motivation Motivation Metric tensor The Operator Product Expansion provides a rigorous way of justifying the parton model in the case of the Deep Inelastic Scattering reaction, in the Bjorken limit (Q 2 +, x =constant) Unfortunately, for reactions with a more involved kinematics, there is usually no region of phase-space in which the OPE provides useful results Here, we aim at finding a framework which, although less rigorous than the OPE, can be used in more diverse situations QCD on the light-cone is a formulation of QCD in which the main ideas of the parton model appear in a transparent way François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 4/56

5 Motivation Metric tensor are defined by choosing a privileged axis (generally the z axis) along which particles have a large momentum. Then, for any 4-vector a µ, one defines : a + a0 + a 3 2, a a0 a 3 2 a 1,2 unchanged. Notation : a (a 1, a 2 ) which can be inverted by Some useful formulas : a 0 = a+ + a 2, a 3 = a+ a 2 x y = x + y + x y + x y d 4 x = dx + dx d 2 x = Notation : + x, x + François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 5/56

6 Metric tensor Motivation Metric tensor Remarks : The Dalembertian is bilinear in the derivatives +, The metric tensor is non diagonal : g µν = g µν = x _ t x + x + = 0 x _ = 0 z François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 6/56

7 Definition of the Poincaré group Definition LC Poincaré algebra Kinematic transformations Galilean sub-algebra Action of K- The Poincaré group is the 10-dimensional group of transformations that contains : 4 translations : P α 3 spatial rotations : J i = 1 2 ɛijk M jk 3 Lorentz boosts : K i = M i0 Note: it is a subgroup of the 15-dimensional conformal group, i.e. the group of transformations that preserve the relation dx α dx α = 0. In addition to the transformations of the Poincaré group, the conformal group contains 1 dilatation and 4 non-linear conformal transformations The commutation relations among the generators are : P α, P β] = 0 M αβ, P δ] = i ( g βδ P α g αδ P β) M αβ, M δγ] = i ( g αγ M βδ + g βδ M αγ g αδ M βγ g βγ M βδ) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 7/56

8 Definition LC Poincaré algebra Kinematic transformations Galilean sub-algebra Action of K- In light-cone coordinates, the previous commutation relations should be unchanged, provided we use the transformed g µν (they have a geometrical meaning, and do not depend on the system of coordinates) In order to obtain the transformed generators, notice that : if x α (x 0, x 1, x 2, x 3 ) and x µ (x +, x 1, x 2, x ), x µ = C µ αx α, with C µ α This also applies to tensors : T µ 1 µ n = C µ 1 α1 C µ n αn T α 1 α n François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 8/56

9 Definition LC Poincaré algebra Kinematic transformations Galilean sub-algebra Action of K- Light-cone Poincaré generators : P µ = P + P 1 P 2 P 0 S 1 S 2 K 3, M µν S 1 0 J 3 B 1 = S 2 J 3 0 B 2 K 3 B 1 B 2 0 with B 1 K1 + J 2 2, B 2 K2 J 1 2 S 1 K1 J 2 2, S 2 K2 + J 1 2 François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 9/56

10 Kinematic transformations Definition LC Poincaré algebra Kinematic transformations Galilean sub-algebra Action of K- 6 of the light-cone Poincaré generators leave x + = const : J 3, P 1,2 : do not touch t, nor z P + : increases t and decreases z by the same amount B 1,2 : not so obvious... B 1 K 1 + J 2 K 1 : infinitesimal boost in the x direction t = t + ωx x = ωt + x J 2 : infinitesimal rotation around the y axis x = x + ωz z = ωx + z Hence, t + z = t + z, and x + is left unchanged François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 10/56

11 Galilean sub-algebra Definition LC Poincaré algebra Kinematic transformations Galilean sub-algebra Action of K- P generates translations in the x + direction The set {J 3, P +, P 1,2, B 1,2, P } generates a 7-dimensional sub-algebra of the Poincaré algebra : P +, P ] = P +, P j] = P +, J 3] = P +, B j] = 0 P, P j] = P, J 3] = 0, P, B j] = ip j J 3, P j] = iɛ jk P k, J 3, B j] = iɛ jk B k, P k, B j] = iδ jk P + This sub-algebra is isomorphic to the algebra of Galilean transformations in 2 dimensional quantum mechanics : P + mass P Hamiltonian (the time is x + ) J 3 rotation in the x, y plane P 1,2 translations in the x, y plane B 1,2 Galilean boosts in the x, y plane François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 11/56

12 Galilean sub-algebra Definition LC Poincaré algebra Kinematic transformations Galilean sub-algebra Action of K- Most of these commutation relations are obvious. Those involving the Galilean boosts B j need some extra explanations... In non-relativistic 2-dimensional quantum mechanics, Galilean boosts are the transformations that change the velocity v j v j v j + δv j The commutation relations involving B i with the other generators can be understood from the way the other quantities transform under the Galilean boosts : M M E E + p j δv j p k p k + δ jk Mδv j J J + ɛ jk Mx k δv j P +, B j] = 0 P, B j] = ip j P k, B j] = iδ jk P + J 3, B j] = iɛ jk B k François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 12/56

13 Galilean sub-algebra Definition LC Poincaré algebra Kinematic transformations Galilean sub-algebra Action of K- Susskind (1968) The existence of this sub-algebra is responsible for the non-relativistic features of quantum field theories on the light-cone. For instance, the Hamiltonian of a free particle reads : P = P 2 2P + For a particle moving at the speed of light in the +z direction, x = 0, while x + increases as the particle moves on its trajectory. Therefore, it is legitimate to interpret x + as the time. And P, which generates translations in x +, is indeed a Hamiltonian François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 13/56

14 Action of K- Definition LC Poincaré algebra Kinematic transformations Galilean sub-algebra Action of K- The action of boosts in the direction of x is also quite remarkable : e iωk P e iωk = e ω P e iωk P + e iωk = e +ω P + e iωk P j e iωk = P j e iωk J 3 e iωk = J 3 e iωk B j e iωk = e +ω B j e iωk S j e iωk = e ω S j Simple rescaling of the various operators. This is another hint that the light-cone framework might be simpler in order to study processes involving very fast particles These relations will play an essential role when we discuss the eikonal approximation François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 14/56

15 Motivation Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents In order to show how the parton picture emerges in this formulation, let us study a scattering process by an external potential Our goal is to calculate the scattering amplitude of some state going through the external potential, in the limit where the particles approach the speed of light More precisely, we want to calculate : S ( ) βα lim βin e iωk U(+, )e iωk α in ω + In order to study this limit, we will need a perturbation theory ordered in x +. As we shall see, this is provided by light-cone quantization François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 15/56

16 Model Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents In order to keep the discussion elementary, we first consider a scalar field theory, whose Lagrangian density is : L = 1 2 ( µφ)( µ φ) 1 2 m2 φ 2 g 3! φ3 The conjugate momentum of φ is : Π φ = δl δ φ = + φ The Hamiltonian density reads : H = Π φ φ L = 1 2 ( φ ) m2 φ 2 + g 3! φ3 François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 16/56

17 Interaction picture Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents From the Heisenberg field φ, one defines the field φ in of the interaction picture by : where φ(x) U(, x + )φ in (x)u(x +, ) U(x +, ) T + exp i x + L int (φ in (x)) d 4 x φ in is a free field if φ obeys the full equation of motion : ( +m 2 )φ(x) L int(φ) δφ(x) = U(, x+ ) ( + m 2 )φ in (x) ] U(x +, ) Being a free field, φ in can be decomposed as : φ in (x) dp + d 2 p ] e ip x a 4πp + (2π) 2 in (p) + e ip x a in (p) where implicitly p ( p 2 + m 2 )/2p + François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 17/56

18 Free Hamiltonian Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents The conjugate momentum of φ in is given by : dp Π φin (x) = + + φ in (x) = i 4π The free Hamiltonian is : H free = = d 2 p ] e ip x a (2π) 2 in (p) e ip x a in (p) 1 ( ) ] 2 dx d 2 x 1 φ in m2 φ 2 in dp + d 2 p 1 ] 4πp + (2π) 2 p a in (p)a in 2 (p) + a in (p)a in(p) After normal ordering, this reads : H free = dp + 4πp + d 2 p (2π) 2 p a in (p)a in(p) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 18/56

19 Commutation relations Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents For a in (p) and a in(p) to have the proper interpretation as operators that create or destroy a quantum of energy p, we must have : H free, a in (p)] = p a in (p) H free, a in (p) ] = p a in (p) These relations will hold provided we have : a in (p), a in (q) ] = (2π) 3 2p + δ(p + q + )δ( p q ) From this follows the equal-x + canonical commutator : φ in (x), Π φin (y)] x + =y + = i 2 δ(x y )δ( x y ) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 19/56

20 Scattering theory Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents Start from transition amplitudes defined as : S βα β in U(+, ) αin The states α in and β in are obtained by acting with a in on the vacuum state 0 in : m p 1 p m in = i=1 0in a in i)] (p The relation between φ in and a in, a in can be inverted as : a in (p) = i dx d 2 x e ip x ( + + ip + )φ in (x) Note : this expression is in fact independent of x +. One can therefore chose x + at will in this equation François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 20/56

21 Scattering theory Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents One can use the freedom to chose x + in the previous relation to take x + = for a field in the initial state and x + = + for a field in the final state. This choice enables us to write the S-matrix element as the expectation value of an x + -ordered product of fields : S βα = i i α i j β ] dx i d 2 x i e ip i x i ( + i + ip + i ) ] dy j d 2 y j e iq j y j ( + j iq + j ) 0 in T + { j β φ in (y + j =+, y j, y j ) i αφ in (x + i =, x i, x i ) exp i + 0in d 4 x L int (φ in (x))} François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 21/56

22 Perturbation theory Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents The previous correlator contains only the free field φ in. The effects of the interactions are obtained by expanding the x + -ordered exponential to the desired order this naturally leads to an x + -ordered perturbation theory This can be done more systematically by introducing a generating functional for these correlators : Zj] 0 in T+ exp i + d 4 x L int (φ in (x)) + j(x)φ in (x) 0in ] such that + 0in T + φ in (x 1 ) φ in (x n ) exp i d 4 x L int (φ in (x)) 0 in = δ iδj(x 1 ) δ iδj(x n ) Zj] j=0 François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 22/56

23 Perturbation theory Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents By using standard methods, one obtains the following closed formula for Zj] : ( ) δ Zj] = exp i d 4 x L int iδj(x) exp 1 2 d 4 xd 4 y j(x)j(y) G 0 (x, y) where G 0 (x y) is the free x + -ordered propagator, defined as : G 0 (x, y) 0 in T + φ in (x)φ in (y) 0 in Since the φ in is a free field, this propagator can be calculated explicitly : dp + d 2 p G 0 (x, y)= θ(x + y + )e ip (x y) + θ(y + x + )e ip (x y)] 4πp + (2π) 2 Note : the x + -ordered propagator is in fact identical to the x 0 -ordered propagator François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 23/56

24 Perturbation theory Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents Contribution of the external lines : Incoming particles : i dx d 2 x e ip x ( x + + ip + )G 0 (x, y) x + = = e ip y Outgoing particles : i dx d 2 x e ip x ( x + ip + )G 0 (x, y) x + =+ = e ip y Feynman rules : Consider only amputated diagrams At each vertex ig d 4 x G 0 (x, y) for each internal line A factor e ip x for incoming particles, and e ip x for outgoing particles identical to the usual Feynman rules in momentum space François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 24/56

25 Noether s currents Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents Energy-momentum tensor : It obeys : T µν = δl δ µ φ ν φ g µν L = ( µ φ)( ν φ) g µν L ν T µν = 0, P µ = dx d 2 x T µ+ The time components of the currents associated to P +, P j, P are : T ++ = ( + φ) 2 T j+ = ( j φ)( + φ) T + = ( φ)( + φ) L = H François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 25/56

26 Noether s currents Angular-momentum tensor : Motivation Model Interaction picture Free Hamiltonian Commutation relations Scattering theory Perturbation theory Noether s currents J µνρ = x µ T νρ x ν T µρ ρ J µνρ = 0, M µν = dx d 2 x J µν+ For instance, the boost operator in the direction reads : K = M + = = i 2 dp + 4π dx d 2 x J ++ d 2 p (2π) 2 { a in (p) p + ] a in (p) a in (p) ain (p) p + ] } and it obeys : K, a in (p) ] = ip + a in (p) p + François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 26/56

27 Model Let us now go back to the problem of the scattering off an external potential at high energy Model Eikonal limit More precisely, we want to calculate : S ( ) βα lim βin e iωk U(+, )e iωk α in ω + We will consider two different potentials : Scalar potential : λ A(x)φ(x)φ (x) Vector potential : e A µ (x)j µ (x) We will assume that the external potential is non-zero only on a finite range in x +, x + L, +L] François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 27/56

28 Model In order to have a consistent model, we must consider a complex scalar field, with a U(1) symmetry : Model Eikonal limit Conjugate momenta : L ( µ φ)( µ φ ) m 2 φφ g 2 (φφ ) 2 Π φ = + φ, Π φ = + φ The Lagrangian is invariant under a U(1) symmetry : φ(x) e iα φ(x), φ (x) e iα φ (x) The associated Noether s current is : J µ = i φ µ φ φ µ φ] François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 28/56

29 Eikonal limit We have already seen that : K, a in (p) ] = ip + a in (p) p + Model Eikonal limit This relation implies : e iωk a in (q)eiωk = a in (eω q +, e ω q, q ) e iωk p in = (e ω p +, p ) in e iωk φ in (x)e iωk = φ in (e ω x +, e ω x, x ) Since the boost K does not change the ordering in x + : e iωk U(+, )e iωk = T + exp i where L int includes all the interaction terms : d 4 x L int (e iωk φ in (x)e iωk ) L int (φ) = g 2 (φφ ) 2 λ Aφφ e A µ J µ François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 29/56

30 Eikonal limit Model Eikonal limit It is useful to split the S matrix U(+, ) into three factors : U(+, ) = U(+, +L)U(+L, L)U( L, ) Upton application of K, this becomes : e iωk U(+, )e iωk = e iωk U(+, +L)e iωk e iωk U(+L, L)e iωk e iωk U( L, )e iωk The external potentials A(x) and A µ (x) are unaffected by the action of K The components of J µ (x) are changed as follows : e iωk J i (x)e iωk = J i (e ω x +, e ω x, x ) e iωk J (x)e iωk = e ω J (e ω x +, e ω x, x ) e iωk J + (x)e iωk = e ω J + (e ω x +, e ω x, x ) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 30/56

31 Eikonal limit Model Eikonal limit The factors U(+, +L) and U( L, ) do not contain the external potential. In order to deal with these factors, it is sufficient to perform the change of variables : e ω x + x +, e ω x x. This leads to : lim ω + eiωk U(+, +L)e iωk = U 0 (+, 0) lim ω + eiωk U( L, )e iωk = U 0 (0, ) where U 0 is the same as U, but with only the self-interactions of the fields, g(φφ ) 2 For the factor U(L, L), the e ω x x change leads to : e iωk U(+L, L)e iωk = = T + exp i +L L d 4 x e ω e A (x +, e ω x, x ) ] e ω J + (e ω x +, x, x ) + O(1) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 31/56

32 Eikonal limit Model Eikonal limit Therefore, in the limit ω +, we have : lim ω + eiωk U(+L, L)e iωk = exp ie with χ( x ) ρ( x ) ] d 2 x χ( x )ρ( x ) dx + A (x +, 0, x ) dx J + (0, x, x ) The high-energy limit of the scattering amplitude is : S ( ) βα = β U0 in (+, 0) exp ie χ( x )ρ( x ) ]U 0 (0, ) αin x Only the component of the vector potential matters The self-interactions and the interactions with the external potential are factorized parton model Still not completely trivial, because ρ is an operator François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 32/56

33 Perturbative expansion Model Eikonal limit The previous formula still contains all the self-interactions of the field φ. In order to perform the perturbative expansion, it is convenient to write first : S ( ) βα = βin U0 (+, 0) γin γ,δ ] δin γ in exp ie χ( x )ρ( x ) δin U(0, ) αin x The factor δ in δin U(0, ) α in δ is the Fock expansion of the initial state: the state prepared at x + = may have fluctuated into another state before it interacts with the external potential François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 33/56

34 Perturbative expansion Model Eikonal limit Denote F exp ie χ( x )ρ( x ). We need to calculate matrix elements such as γ in F δ in The operator ρ( x ) reads : dp + d 2 p ρ( x ) = d 2 q { b 4πp + (2π) 2 (2π) 2 in (p+, p )b in (p +, q )e i( p q ) x } d in (p+, p )d in (p +, q )e i( p q ) x Therefore : F b in (k) = b in (k+, p ) p x F d in (k) = d in (k+, p ) p x ] e i( k p ) x e ieχ( x ) F ] e i( k p ) x e ieχ( x ) F François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 34/56

35 Perturbative expansion Model Eikonal limit Consider a state : δ in b in (k) 0 in F changes only the transverse momenta in a state: F δ in = F b in (k) 0 in François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 35/56

36 Perturbative expansion Model Eikonal limit Consider a state : δ in b in (k) 0 in F changes only the transverse momenta in a state: F δin = b in (k+, p ) p x e i( k p ) x e ieχ( x ) ] F 0in François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 35/56

37 Perturbative expansion Model Eikonal limit Consider a state : δ in b in (k) 0 in F changes only the transverse momenta in a state: F δin = b in (k+, p ) p x e i( k p ) x e ieχ( x ) ] 0in François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 35/56

38 Perturbative expansion Model Eikonal limit Consider a state : δ in b in (k) 0 in F changes only the transverse momenta in a state: F ] δin = e i( k p ) x e ieχ( x ) (k +, p ) in p x The projection on γ in is non zero only if this state contains the same number of particles and antiparticles as δ in François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 35/56

39 Perturbative expansion The action of F on δ in changes the transverse momenta of the particles contained in the state, but not their nature and number : Model Eikonal limit Let us define the light-cone wave function of the incoming state by : ψ( x 1 x n ) i d 2 ki (2π) 2 e i ki x i k1 k nin U(0, ) αin Each charged particle going through the external field acquires a phase proportional to its charge (antiparticles get an opposite phase) : ψ( x 1 x n ) ψ( x 1 x n ) i e ie i χ( x i ) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 36/56

40 Perturbative expansion Model Eikonal limit The calculation of δ in U 0 (0, ) α in is similar to that of scattering amplitudes in the vacuum. The only difference is that the integration over x + at each vertex runs only over half of the real axis, 0] In Fourier space, this means that the component of the momentum is not conserved at the vertices Instead of a δ function, one gets an energy denominator Example with a single interaction : p k 1 k 2 k 3 k1 k2 k3in U(0, ) p in = ig 0 d 4 x e i(k 1+k 2 +k 3 p) x = g (2π)3 δ( k 1 + k 2 + k 3 p )δ(k k+ 2 + k+ 3 p+ ) k 1 + k 2 + k 3 p iɛ François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 37/56

41 Perturbative expansion Model Eikonal limit More generally, one should expand the evolution operator U 0 (0, ) to the desired order and insert 1 = γ γ in γin between the successive interactions : δin U 0 (0, ) α in = + γ 1 γ n 1 δin 0 n=0 x + 1 d 4 x 1 d 4 x 2 x + n 1 d 4 x n il 0 int (φ in (x 1 )) γ1in γn 1in il 0 int (φ in (x n )) αin In practice γ is an integral over the phase-space of the (on-shell) particles of the intermediate state. Conservation laws may restrict what is allowed in the intermediate state The elementary factors are given by : γi 1in il 0 int (φ in (x i )) γiin ig e i( a γ i p a b γ i 1 p b ) x i François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 38/56

42 Basics of QCD Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization We are now going to extend the previous formalism to the case of QCD, in order to highlight some peculiarities of QCD on the light-cone Reminder: the QCD Lagrangian is : L = 1 2 tr (F µνf µν ) + ψ(i/d m)ψ the gauge field A µ belongs to SU(3) D µ µ iga µ is the covariant derivative F µν id µ, D ν ]/g = µ A ν ν A µ iga µ, A ν ] The classical equations of motion are : { (i /D m)ψ = 0 Dµ, F µν] a = gj ν a = gψγ ν t a ψ Note that D ν, J ν] = 0 (covariant current conservation) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 39/56

43 Independent field components Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization We will work in the light-cone gauge A + = 0 In QCD, it turns out that some of the field components have a vanishing conjugate momentum they should be treated as constraints For the spinors, let us introduce two orthogonal projectors : P γ γ +, P 1 2 γ+ γ and decompose the spinor ψ as follows : ψ = ψ + + ψ with ψ + P + ψ, ψ P ψ Using A + = 0, we can rewrite the matter part as : L ψ = i 2ψ +D ψ + + i 2ψ + ψ m 2 { ψ +γ ψ + ψ γ + ψ + } i 2 { ψ +γ (γ D )ψ + ψ γ + (γ D )ψ + } François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 40/56

44 Independent field components The gauge part of the Lagrangian can be decomposed as : L A = 1 4 F a ijf ij a F a + ifa i 1 2 F a + Fa + Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization with F + = F + = + A, F +i = F i+ = + A i F i = F i = A i i A iga, A i ] F ij = i A j j A i iga i, A j ] The fields A and ψ have a vanishing conjugate momentum : Π A a = δl δ( A a ) = 0, Π ψ = δl δ( ψ ) = 0 this means that the Poisson brackets involving these fields are zero, and that the standard canonical quantization procedure is bound to fail for these fields François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 41/56

45 Independent field components There is no such problem with A i and ψ + : Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization Π A i a = + A i a, Π ψ+ = i 2ψ + these fields can be quantized by the usual method By multiplying it by γ + and γ, the Dirac equation can be split into : + ψ = i 2 iγ D + m ] γ + ψ + D ψ + = i iγ D + m ] γ ψ 2 The first equation does not contain any time derivative. It is therefore local in time, and can be seen as a constraint that we can use to express ψ in terms of ψ + and the gauge fields at the same time x + The second equation contains time derivatives, and is thus the dynamical evolution equation for ψ + François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 42/56

46 Independent field components Similarly, the Yang-Mills equations can be divided into : ( + ) 2 A a = gj a + D i, + A i] a +, F i] + D, + A i] + D j, F ji] = gj i Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization The first equation is local in time. It is a constraint that can be used to express A a in terms of the other fields at the same time The second equation describes the dynamical evolution of the field A i a Note: the absence of time derivatives of the fields ψ and A in the equations of motion is equivalent to the fact that their conjugate momenta are zero The strategy for quantizing such a theory is to solve the constraints and rewrite the Lagrangian in terms of the dynamical fields only. Then, one can proceed with the usual canonical quantization procedure, by writing commutation relations at equal x + for the dynamical fields François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 43/56

47 Solution of the constraints Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization Formally, the solutions of the constraints read : ψ = i iγ D m ] γ + ψ + A a = 1 gj + ( + ) 2 a D i, + A i] ] (they depend on g) a It is then a simple (but tedious...) exercise to obtain the Lagrangian in terms of dynamical fields only : L = 1 ( )( 4 F ijf a a ij + + A i a + 1 ( D i, + A i ] a g J a + 2 A i a ) ) 1 ( D ( + ) 2 i, + A i ] a g J a + + i 2 ψ + ψ + + i 2 ψ +(m iγ D ) 1 + (m + iγ D )ψ + ) Note the additional couplings, with 1/ + or 1/( + ) 2 François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 44/56

48 Noether s currents Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization Energy-momentum tensor : It obeys : T µν = iψ µ γ ν ψ + ( µ A ρ )F νρ g µν L ν T µν = 0, P µ = dx d 2 x T µ+ The time components of the currents associated to P +, P j are : T ++ = i 2ψ + + ψ + + ( + A k a)( + A k a) T j+ = i 2ψ + j ψ + + ( j A k a)( + A k a) Note that these currents depend only on the dynamical fields. Hence, they do not contain explicit factors of g As we shall see, this is a property of the generators of all the kinematical Poincaré transformations François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 45/56

49 Noether s currents Angular-momentum tensor : J µνρ = x µ T νρ x ν T µρ Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization It obeys : + i 8 ψ(γρ γ µ, γ ν] + γ µ, γ ν ]γ ρ )ψ + F ρµ A ν F ρν A µ ρ J µνρ = 0, M µν = dx d 2 x J µν+ The time components of the currents associated to J 3, B j are : J ij+ = x i T j+ x j T i+ + i 2 2 ψ + γ i, γ j] ψ + + ( + A i a)a j a ( + A j a)a i a J j++ = x j T ++ x + T j+ Again, they depend only on the dynamical fields François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 46/56

50 QCD light-cone Hamiltonian The Hamiltonian density is obtained in the usual way : H = T + = Π ψ+ ψ + + Π A i a A i a L Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization = 1 4 F a ijf ij a ) 1 ( Di, + A i] 2 gj + 1 ( Di a a, + A i] ( + ) gj + 2 a a i ψ +(m iγ D ) 1 2 (m + iγ + D )ψ + ) Reminder : J + a = 2ψ +t a ψ + François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 47/56

51 Quantization Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization The strategy is the same as for scalar fields : Write a Fourier representation of the dynamical fields in the interaction picture, introducing creation and annihilation operators Express the Hamiltonian in terms of these operators Find the commutation relations between the a in and a in so that they have the proper interpretation Check that this leads to the expected canonical commutation relations between the fields and their conjugate momenta One peculiarity of light-cone quantization is that an equal-x + surface is light-like (contrary to the equal-x 0 surfaces in ordinary quantization, which are space-like). Since causality imposes that commutators of local operators separated by a space-like interval vanish, its translation in this formalism will be slightly different François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 48/56

52 Quantization Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization Write the free gauge field operator as (with p = p 2 /2p + ): A i a in(x) = dp + d 2 p ɛ i λ(p) a λ 4πp + (2π) 2 a in(p)e ip x + a λ (p)eip x] a in The free light-cone gluon Hamiltonian reads : H free A = = λ 1 dx d 2 ( x i A a j in j A a i in)( i A j a in 4 j A i ) a in dp + 4πp + d 2 p (2π) 2 p 2 1 ( i A i ) 1 ( a in j A j ) ] 2 ( + ) 2 a in a λ a in (p)a λ a in (p) + aλ a in (p)aλ a in(p) ] λ As usual, it should be normal ordered in order to have a vanishing expectation value in the vacuum : H free A = dp + 4πp + d 2 p (2π) 2 p λ a λ a in (p)aλ a in(p) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 49/56

53 Quantization For the interpretation of a λ a in and aλ a in as creation and annihilation operators to hold, we need to have : H free, aλ a in (p)] = p a λ a in (p), H free, aλ a in(p) ] = p a λ a in(p) A A Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization This can be achieved if we have : a λ a in (p), a λ b in(q) ] = 0 a λ a in (p), aλ b in (q)] = 0 a λ a in (p), a λ b in (q)] = δ ab δ λλ 2p + (2π) 3 δ(p + q + )δ( p q ) Then, one can use these relations in order to get the canonical commutation relation : A i a in (x), Π j A (y) ] = A i b in x + =y + a in(x), + A j b in (y)] x + =y + = i 2 δ abδ ij δ(x y )δ( x y ) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 50/56

54 Quantization Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization In order to quantize the fermion field ψ +in, notice first that the subspace spanned by ψ +in = P + ψ in is only 2-dimensional Therefore, we need only two elementary spinors, w +1/2 and w 1/2, in order to decompose any ψ + in We can choose them normalized as follows : w r(p)w s (p) = 2p + δ rs w s (p)w s(p) = 2p + P + s=± 1 2 Reminder: in ordinary quantum field theory, the spinor ψ has four independent components, and one need 4 elementary spinors to perform the decomposition: u s (p) and v s (p) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 51/56

55 Quantization Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization One can then write ψ +in (x) as (with p = ( p 2 + m 2 )/2p + ): ψ +in (x)= 1 dp + d 2 p w 2 1/4 4πp + (2π) 2 s b s in (p)e ip x + w s d s (p)eip x] in s The free light-cone quark Hamiltonian reads : H free ψ = i 2 = dp + dx d 2 x ψ +in (m iγ ) 1 + (m+iγ )ψ +in 4πp + d 2 p (2π) 2 p s=± 1 2 b s in (p)b s in(p) d s in (p)d s in (p)] After normal ordering, it reads : H free ψ = dp + 4πp + d 2 p (2π) 2 p s b s in (p)b s in(p) + d s in (p)d s in(p) ] (of course, we need anti-commutation relations for this to work) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 52/56

56 Quantization Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization In order to have the expected interpretation of d s in, d s in, b s in, b s in, we need : H free ψ, b s in (p)] = p b s in (p), H free ψ, b s in (p) ] = p b s in (p) H free ψ, d s in (p)] = p d s in (p), H free ψ, d s in (p) ] = p d s in (p) This will be realized with the following choice : { br in (p), b s in (q) } = { b r in (p), b s in (q)} = 0 { dr in (p), d s in (q) } = { d r in (p), d s in (q)} = 0 { br in (p), b s in (q)} = { d r in (p), d s in (q)} =2p + δ rs (2π) 3 δ(p + q + )δ( p q ) From these relations, we obtain the expected canonical anti-commutation relations : { ψ+in (x), ψ +in (y) } x + =y + = { ψ +in (x), ψ +in (y)} x + =y + = 0 { ψ+in (y), Π ψ+in (x) } = ip x + =y + + δ(x y )δ( x y ) François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 53/56

57 Perturbative expansion The method for calculating the Fock state expansion is the same as for scalar fields Basics of QCD Independent fields Solution of the constraints Noether s currents QCD light-cone Hamiltonian Quantization Expand the evolution operator to the desired order Insert a complete sum of states between successive interactions The ordered integrations over the x + variables generate energy denominators The integrations over the other space-time variables generate delta functions Note that there are additional couplings, coming from the terms in 1/ + and 1/( + ) 2. They are due to the instantaneous exchange of the fields ψ and A, which have been eliminated by solving the constraints François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 54/56

58 Lecture IV : Saturation and CGC Outline of lecture IV BFKL equation Saturation of parton distributions Balitsky-Kovchegov equation Color Glass Condensate - JIMWLK Analogies with reaction-diffusion processes Pomeron loops François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 55/56

59 Lecture V : Calculating observables Outline of lecture V Field theory coupled to time-dependent sources Generating function for the probabilities Average particle multiplicity Numerical methods for nucleus-nucleus collisions Gluon production Quark production François Gelis 2006 Lecture III/V SPhT, Saclay, January p. 56/56