Symmetric and Non-Symmetric Saturation. Overview

Transcription

1 Symmetric and Non-Symmetric Saturation Leszek Motyka DESY Theory, Hamburg & Jagellonian University, Kraków Overview BFKL Pomeron and Balitsky Kovchegov formalism Field theoretical formulation Symmetry and self-duality Symmetry breaking Down to -dimensions Superselection of the target Based on work done with Sergey Bondarenko

2 BFKL resummation of gluonic ladders Balitsky, Fadin, Kuraev, Lipatov: A(s) = A A α s log(s)... A k (α s log(s)) n... ; α s log(s) Φ (x, k) x dx x d k ˆK(x, x ; k, k ) Φ(x, k ) = Φ(x, k) Symmetric and non-symmetric saturation page

3 BFKL equation Φ(x, k; q) = Φ A (k; q) 3α s π " k x dx x d k (k k) (" k k k k # ) k (k k) k k Φ(x, k; q) (k k) # q (k k) k k Φ(x, k ; q) Differential form (LL) Φ(y, k; q) y = d k ˆK(k, k ; q) Φ(y, k ; q) Formal solution: Φ(Y, q) = exp[ ˆK(q) Y ] Φ A (q) Two-step evolution Φ(Y, q) = exp[ ˆK(q) y] exp[ ˆK(q) (Y y)] Φ A (q) Symmetric and non-symmetric saturation page

4 A(Y ) = BFKL amplitude and Green s function d k k (q k ) d k k (q k) Φ A(k ; q) G(Y, k, k; q) Φ B (k; q) A(Y ) = d k k (q k ) d k k (q k) d k k (q k ) Φ A (k ; q) G(y, k, k ; q) G(Y y, k, k; q) Φ B (k; q) = = d k k (q k ) Φ A (y, k ; q) Φ B (Y y, k ; q) Symmetric and non-symmetric saturation page 3

5 Solutions to BFKL Unintegrated gluon densities: Φ A (y, k, q = ) f A (y, k ), Φ B (Y y, k, q = ) f B (Y y, k ) LL(/x) : f A (y, k ) dγ πi f () A (γ) kγ exp[ᾱ s y χ(γ)] χ(γ) = ψ() ψ(γ) ψ( γ) Diffusion approximation for y log(q /Q ) saddle point at γ = / v y» log f A (y, k (k /k ) kk exp(ω y) exp ) D y Rapidity dependence: exp(ω y) with ω = 4ᾱ log ω.5 (too large) Symmetric and non-symmetric saturation page 4

6 Colour Dipole Representation At very high energies and the LL approximation description of high energy scattering in the position representation is possible long-living fluctuations: colour dipoles short interaction time parton energy z is conserved parton transverse positions do not change conservation of parton helicity γ -z γ r z p p Colour dipole cascade model [Mueller] N c gluon emissions dipole splittings Splitting: BFKL kernel Scattering of dipoles: two gluon exchange Single scattering dipole multiplicity linear BFKL Symmetric and non-symmetric saturation page 5

7 Dipole rescattering / gluon recombination Assume dominance of single ladder exchange gluon growth e λy ImT e λy Diffractive cross section T e λy would eventually surpass the total cross section Target frame Violation of S-matrix unitarity Projectile frame f A (y, k ) e λy Triple Pomeron Vertex Exponential growth of dipole density Rescattering Neglecting correlations: σ = σ σ /... σ [ exp( σ /σ )] allows for gluon recombination with rate f A (y, k ) [Bartels] Symmetric and non-symmetric saturation page 6

8 Theoretical basis: JIMWaLK/BK equation [Jalilian-Marian,Iancu,McLerran, Leonidov,Weigert],[Balitsky,Kovchegov] Propagation of a dilute projectile through a dense target at large energy at the LL/x: BFKL evolution of amplitude tempered by unitarity corrections Enhancement Correlations Rescattering Target frame: in the large N c limit BFKL pomeron fan diagrams gluon recombination at large density N(x, y; Y ) Y = ᾱs π d z (x y) (x z) (z y) [N(x, z; Y ) N(z, y; Y ) N(x, y; Y ) N(x, z; Y )N(z, y; Y )] BK equation in momentum space N(k, Y ) Y = χ BFKL k k «N(k, Y ) N (k, Y ) For large k / small r BFKL-like growth with y. For small k / large r saturation. Unitarity is not violated Symmetric and non-symmetric saturation page 7

9 Dipole density in momentum space: Kwieciński Kutak formulation of BK Ñ(y, k ) = d r π exp( ik r) N(y, r) r Dipole scattering amplitude N(y, r) d k k 4 f(y, k ) [ exp( ikr)] f(y, k ) k 4 k Ñ(y, k ) Inverse transform: Ñ(y, k ) da k a 4 f(y, a ) log a k! πα s BK equation y f(y, k ) = ᾱ s k da "k a f(y, 4 a ) da k db k a " f(y, a ) f(y, k ) a k b f(y, 4 b ) f(y, k ) da k a 4 f(y, # k ) [4a 4 k 4 ]! log a k f(y, a ) # Symmetric and non-symmetric saturation page 8

10 Solutions of BK / geometric scaling BK equation generates saturation scale Q s (y) exp(λ(y) k/q s (Y) φ(k,y) - - α s =. At large y solutions of BK depend on ξ = (k/q s (y)) -3 Ñ(y, k) log(ξ), ξ Ñ(y, k) ξ γ, ξ k/q s (Y) 5 f(y, k )/k ξ, ξ 4 f(y,k )/k, y=8,9,..,6 f(y, k )/k ξ γ ξ f(y,k )/k 3.. k [GeV] Symmetric and non-symmetric saturation page 9

11 σ γ p i (Q, W ) = Geometric Scaling dz [A. Staśto, K. Golec-Biernat and J. Kwieciński] d r Ψ i (z, r) {z } Q ˆσ(x, r ) {z } Rs(x) Cross section should depend only on the variable τ = Q R s (x) σ tot γp [µb] 3 Data for.45 GeV < Q < 45 GeV and x <. Warning! Open problem of the contribution of charm EUS BPT 97 EUS BPC 95 H low Q 95 EUSH high Q E665 x<. all Q τ Symmetric and non-symmetric saturation page

12 Geometric Scaling and Universality [Munier,Peschanski] Linear term in BK equation may be expanded around χ(/) N(k, Y ) Y» λ D «log(k ) / N(k, Y ) N (k, Y ) Universality class of Kolmogorov-Petrovsky-Piscounov equation: t u(x, t) = x u(x, t) u(x, t) u (x, t) The equation describes evolution of a system from a repulsive fixed point (u = ) to an atractive fixed point (u = ) The equation admits a travelling wave front solution with Q s (τ) = v g τ b log τ c τ O(/τ) with v g, b and c being independent on the form of nonlinearity. Interesting: results of analysis of JIMWaLK equation on the lattice conform with results of BK equation [Rummukainen,Weigert] Generic case: saturation scale moving with energy and approximate geometric scaling Symmetric and non-symmetric saturation page

13 Limitations of BK and need to improve BK is well founded for very asymmetric scattering like γ Nucleus, DIS Quantum effects (pomeron loops) are absent Need to describe multiple scattering and production in symmetric situation like heavy ion collisions at RHIC and pp scattering at the LHC Three complementary ways to go beyond BK: Effective Field Theory approach Path integral: [Dφα ] exp( S[φ α, φ α ]) Stochastic formulation Langevin equation: Y N = A(N) B(N)ν(Y ) Hamiltonian formulation Generating functional: Y Ψ[Y ; u] = (H Ψ)[Y ; u] Symmetric and non-symmetric saturation page

14 Field theoretical formulation [Bondarenko,LM] Amplitudes of the diagrams: d k k 4 f A(y, k ) f B (y, k ) d k k 4 d a a 4 d b b 4 V 3P (k; a, b) f B (y, k ) f A (y, a ) f A (y, b ) BFKL Green s function d k d k k 4 k f A(k ) G(Y ; k, k ) f 4 B (k ) = Effective action L [f, f ] = A[f, f ; Y ] Y dy h f(y) i y f (y) d k d k k 4 k f A(k ) 4 y ˆK (k, k ) f B (k ) n o L [f, f ] L 3 [f, f ] L E [f, f ] f (y) K f(y) L 3 [f, f ] = πα s f (y) V 3P f(y) f(y) L E [f, f ] = f A (y) f(y) f (y) f B (y) Symmetric and non-symmetric saturation page 3

15 .. Canonical structure Gribov fields δl [f, f ] δ( y f (y, k )) = k 4f(y, k ), δl [f, f ] δ( y f(y, k )) = k 4f (y, k ) Canonical commutator [ if(y, k ), if (y, k )] = k 4 δ(k k ) Gribov fields if(y, k ) k ψ, if (y, k ) k ψ f f f f f E y Feynman rules d y K V 3P V 3P f f f f f f f E a) b) c) d) Symmetric and non-symmetric saturation page 4

16 Target projectile symmetry [Kovner,Lublinsky] Direction of rapidity evolution is arbitrary Free to choose target and projectile Action symmetric w.r.t. target projectile y y, f(y, k ) f (Y y, k ) f (y) f (y) V 3P f(y) = f (y) V 3P f(y) f(y) f and f are canonically conjugated Self-duality Symmetric and non-symmetric saturation page 5

17 Semi-classical limit Effective action A[f, f ; Y ] Y dy n L[f, f ] L3[f, f ] L 3 [f, f ] LE[f, f ] o S matrix: S(Y ; fa, f B ) = f A,f B [Df Df ] exp( A[f, f ; Y ]) Classical trajectories f(y, k ) and f (y, k ) : δa[ f, f ; Y ] = S(Y ; fa, f B ) = X α α exp( A[ fα, f α; Y ])!!!!!!!! " " " " " " " " # # # # # # # # $ $ $ $ $ $ $ $ % % % % % % % % & & & & & & & & ' ' ' ' ' ' ' ' ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) V V V V V V 3P 3P 3P 3P 3P V3P V3P V3P V3P V3P V3P... V3P V V V V V V V 3P 3P 3P 3P 3P 3P c) b) a) 3P P Symmetric and non-symmetric saturation page 6

18 Equations of motions Braun equations in KK representation Mutual absorption of the fields " y f(y, k ) = BK(f, k ) πα s f(y, k ) and da k a 4 log a k! f (y, a ) " y f (y, k ) = BK(f, k ) πα s f (y, k ) da k a 4 log a k! f(y, a ) k k da a 4 a f(y, a ) da a 4 k k db a b 4 f (y, b ) f(y, a ) f (y, a ) log da a 4 a f (y, a ) da a 4 db a f (y, a ) f(y, a ) log k a!# b 4 f(y, b ) k a!# Two-point boundary conditions: (symmetric!) f(y =, k ) = f A (k ), f (y = Y, k ) = f B (k ) = f A (k ) Symmetric and non-symmetric saturation page 7

19 Symmetric solutions to Braun equations f(y, k )/k = f (Y y, k )/k for Y = 6 and Y = 8 Y tot =6 f(y,k )/k f (y,k )/k Input Y tot =8 f(y,k )/k f (y,k )/k Input.. y y f(y,k )/k.. f(y,k )/k.... e-5.. k [GeV] e-5.. k [GeV] Symmetric and non-symmetric saturation page 8

20 Y tot = Above critical rapidity Y c f(y,k )/k, y=,4,..., Input. y. For Y > Y c f(y,k )/k.. f(y, k ) (left) different from f (Y y, k ) (down) e-5 e-6 e-7.. k [GeV] Y tot = Y tot = f (y,k )/k, y =,,...,5 Input f (y,k )/k, y =6,7,...,.. f (y,k )/k... y f (y,k )/k... y e-5 e-5 e-6 e-6 e-7.. k [GeV] e-7.. k [GeV] Symmetric and non-symmetric saturation page 9

21 Spontaneous Symmetry Breaking Y tot =6 f(y,k )/k, y=,4,...,6 Input f(y,k )/k.... e-5 y At Y = 6 f(y, k ) (left) larger than f (Y y, k ) (down) e-6 by three orders of magnitude e-7 e-8.. k [GeV] Y tot =6 Y tot =6 f (y,k )/k, y =,...,8 Input f (y,k )/k, y =9,,...,6.. y =.. f (y,k )/k.. f (y,k )/k.. y e-5 e-6 e-7 y =8 e-5 e-6 e-7 e-8.. k [GeV] e-8.. k [GeV] Symmetric and non-symmetric saturation page

22 Fan dominance Solutions of Braun equations vs BK solutions Y tot =6 Y tot =6 PFT: f(y,k )/k, y=4,7,..6 BK: f(y,k )/k, y=4,7,...6 Input PFT: f (y,k )/k, y =4,7,..6 BK: f (y,k )/k, y =4,7,...6 Input... f(y,k )/k.. f (y,k )/k... e-5 e-6 e-5 e-7 e-6.. k [GeV] e-8.. k [GeV] Symmetric and non-symmetric saturation page

23 Y tot = 6 f(y,k )/k y 4. k [GeV] f (y,k )/k Y tot = y 4. k [GeV] Symmetric and non-symmetric saturation page

24 Formulation Detour to -dimensions L RFT = q yp p yq µ q p λ q (q p) p p(y) q (y) p (y) q(y) Equations of motion y q = µ q λ q λ q p; y p = µ p λ p λ q p Phase space and trajectories Action p 5 4, RFT- A 5 4 RFT- 3 3 g g q µy Symmetric and non-symmetric saturation page 3

25 Semi-classical vs exact, fan dominance p 5 4, RFT- it..9 RFT- Symmetric set of solutions q (y; g, g) = p (Y y; g, g) 3.8 p (y; g, g) = q (Y y; g, g)..7 g A 5 g RFT- q Yc p,q RFT- µy S(Y ; g, g ) exp { A RF T [ q, p ; Y ]} exp { A RF T [ q, p ; Y ]} exp A RF T [ q, p ; Y ] Asymptotic answer 4 fan sym. 4 3 T [ exp( ρg )][ exp( ρg )] 3 asym. Decrease: exp( E y) Yc µy µy E exp( µ /λ ) Symmetric and non-symmetric saturation page 4

26 Interpretation Competition between self-multiplying absorbing each other fields Instability of symmetric configurations for strong fields Tempting analogy to second order phase transition Potential Order parameter [M.C. Escher] Symmetric and non-symmetric saturation page 5

27 Symmetric set of solutions Superselecting target and projectile f (y, k ) = f (Y y; k ), f (y, k ) = f (Y y, k ) Target projectile symmetry of the S-matrix o o S(Y ) exp n A[f, f ; Y ] exp n A[f, f ; Y ] o exp n A[f, f ; Y ] Classical measurement of the event: for asymmetric trajectories saturation scale Q s (y) monotonic between target and projectile 3 Hypothetical p T distribution Super-selection of asymmetric classical trajectory? Proposal: Analysis of p T (y) on the event-byevent basis Possible: uncorrelated domains in impact parameter plane <p T > Rapidity Symmetric and non-symmetric saturation page 6

28 Connection to stochastic formulation Dipole model splittings and mergings Stochastic realization [Iancu, Mueller, Munier], / : = ;< ;8 / dñ q dy = αñ βñ [αñ βñ ] ν, ν(y ) =, ν(y )ν(y ) = δ(y Y ). Evolution: Dipole splittings Fixed Point : n = α β Explicit targer-projectile asymmetry,.- Two absorptive boundaries [Mueller,Shoshi] Similar conclusions from studies of stochatic FKPP [Enberg, Golec-Biernat, Munier] Asymmetric formulation asymmetric outcome 3,98 8 Symmetric and non-symmetric saturation page 7

29 Conclusions Field theoretical formulation of interacting QCD pomerons in momentum space is available incorporating both pomeron mergings and splittings Degrees of freedom are pomeron fields related to unintegrated gluon densities and the action is seld dual The theory was solved in the semi-classical approximation for symmetric two-side boundary conditions (Braun equations) Above critical rapidity Y c symmetry between target and projectile is spontaneously broken Superselection of asymmetric configurations by classical measurements is suggested Possible asymmetric particle distributions in heavy ion collisions (event-by-event) Suppression of loops? Symmetric and non-symmetric saturation page 8