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Particle production in AA collisions in the Color Glass Condensate framework Franc ois Gelis CERN and CEA/Saclay Franois Gelis 2007 GGI, Florence, February 2007 - p. 1 CERN Introduction IR & Coll. divergences

  1. Description of hadronic collisions CERN ■ Compute the observable O of interest for a configuration of Introduction the sources ρ 1 , ρ 2 . Note : the sources are ∼ 1 /g ⊲ weak Basic principles coupling but strong interactions ● Degrees of freedom ● Main issues ● Power counting ■ At LO, this requires to solve the classical Yang-Mills ● Bookkeeping Inclusive gluon spectrum equations in the presence of the following current : Loop corrections J µ ≡ δ µ + δ ( x − ) ρ 1 ( � Less inclusive quantities x ⊥ ) + δ µ − δ ( x + ) ρ 2 ( � x ⊥ ) Summary (Note: the boundary condition depend on the observable) ■ Average over the sources ρ 1 , ρ 2 Z ˆ ˜ ˆ ˜ ˜ ˆ ˜ ˜ �O Y � = Dρ 1 Dρ 2 W Y beam − Y [ ρ 1 W Y + Y beam ρ 2 O [ ρ 1 , ρ 2 ■ Can this procedure – and in particular the above factorization formula – be justified ? François Gelis – 2007 GGI, Florence, February 2007 - p. 14

  2. Description of hadronic collisions CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 15

  3. Description of hadronic collisions CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 15

  4. Description of hadronic collisions CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 15

  5. Description of hadronic collisions CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 15

  6. Description of hadronic collisions CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 15

  7. Description of hadronic collisions CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary 10 configurations François Gelis – 2007 GGI, Florence, February 2007 - p. 15

  8. Description of hadronic collisions CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary 100 configurations François Gelis – 2007 GGI, Florence, February 2007 - p. 15

  9. Description of hadronic collisions CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary 1000 configurations François Gelis – 2007 GGI, Florence, February 2007 - p. 15

  10. Main issues CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary ■ Dilute regime : one source in each projectile interact François Gelis – 2007 GGI, Florence, February 2007 - p. 16

  11. Main issues CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary ■ Dilute regime : one source in each projectile interact ■ Dense regime : non linearities are important François Gelis – 2007 GGI, Florence, February 2007 - p. 16

  12. Main issues CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary ■ Dilute regime : one source in each projectile interact ■ Dense regime : non linearities are important ■ Many gluons can be produced from the same diagram François Gelis – 2007 GGI, Florence, February 2007 - p. 16

  13. Main issues CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary ■ Dilute regime : one source in each projectile interact ■ Dense regime : non linearities are important ■ Many gluons can be produced from the same diagram ■ There can be many simultaneous disconnected diagrams François Gelis – 2007 GGI, Florence, February 2007 - p. 16

  14. Main issues CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary ■ Dilute regime : one source in each projectile interact ■ Dense regime : non linearities are important ■ Many gluons can be produced from the same diagram ■ There can be many simultaneous disconnected diagrams ■ Some of them may not produce anything (vacuum diagrams) François Gelis – 2007 GGI, Florence, February 2007 - p. 16

  15. Main issues CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary ■ Dilute regime : one source in each projectile interact ■ Dense regime : non linearities are important ■ Many gluons can be produced from the same diagram ■ There can be many simultaneous disconnected diagrams ■ Some of them may not produce anything (vacuum diagrams) ■ All these diagrams can have loops (not at LO though) François Gelis – 2007 GGI, Florence, February 2007 - p. 16

  16. Power counting CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary ■ In the saturated regime, the sources are of order 1 /g ■ The order of each disconnected diagram is given by : 1 g 2 g # produced gluons g 2(# loops) ■ The total order of a graph is the product of the orders of its disconnected subdiagrams ⊲ quite messy... François Gelis – 2007 GGI, Florence, February 2007 - p. 17

  17. Bookkeeping CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 18

  18. Bookkeeping CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections ■ Consider squared amplitudes (including interference terms) Less inclusive quantities rather than the amplitudes themselves Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 18

  19. Bookkeeping CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections ■ Consider squared amplitudes (including interference terms) Less inclusive quantities rather than the amplitudes themselves Summary ■ See them as cuts through vacuum diagrams François Gelis – 2007 GGI, Florence, February 2007 - p. 18

  20. Bookkeeping CERN Introduction Basic principles ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections ■ Consider squared amplitudes (including interference terms) Less inclusive quantities rather than the amplitudes themselves Summary ■ See them as cuts through vacuum diagrams ■ Consider only the simply connected ones, thanks to : X „ all the vacuum « X “ simply connected ”ff = exp diagrams vacuum diagrams ■ Simpler power counting for connected vacuum diagrams : 1 g 2 g 2(# loops) François Gelis – 2007 GGI, Florence, February 2007 - p. 18

  21. Bookkeeping CERN ■ There is an operator D that acts on a pair of vacuum Introduction diagrams by removing two sources and attaching a cut Basic principles propagator instead : ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 19

  22. Bookkeeping CERN ■ There is an operator D that acts on a pair of vacuum Introduction diagrams by removing two sources and attaching a cut Basic principles propagator instead : ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 19

  23. Bookkeeping CERN ■ There is an operator D that acts on a pair of vacuum Introduction diagrams by removing two sources and attaching a cut Basic principles propagator instead : ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 19

  24. Bookkeeping CERN ■ There is an operator D that acts on a pair of vacuum Introduction diagrams by removing two sources and attaching a cut Basic principles propagator instead : ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary ■ D can also act directly on single diagram, if it is already cut François Gelis – 2007 GGI, Florence, February 2007 - p. 19

  25. Bookkeeping CERN ■ There is an operator D that acts on a pair of vacuum Introduction diagrams by removing two sources and attaching a cut Basic principles propagator instead : ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary ■ D can also act directly on single diagram, if it is already cut François Gelis – 2007 GGI, Florence, February 2007 - p. 19

  26. Bookkeeping CERN ■ There is an operator D that acts on a pair of vacuum Introduction diagrams by removing two sources and attaching a cut Basic principles propagator instead : ● Degrees of freedom ● Main issues ● Power counting ● Bookkeeping Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary ■ D can also act directly on single diagram, if it is already cut ■ By repeated action of D , one generates all the diagrams with an arbitrary number of cuts ■ Thanks to this operator, one can write : � � connected uncut � 1 n ! D n e iV e − iV ∗ P n = , iV = vacuum diagrams � � � all the cut e D e iV e − iV ∗ = vacuum diagrams François Gelis – 2007 GGI, Florence, February 2007 - p. 19

  27. CERN Introduction Basic principles Inclusive gluon spectrum ● First moment ● Gluon production at LO ● Boost invariance Loop corrections Inclusive gluon spectrum Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 20

  28. First moment of the distribution CERN ■ It is easy to express the average multiplicity as : Introduction � e D e iV e − iV ∗ � � Basic principles N = n n P n = D Inclusive gluon spectrum ● First moment ■ N is obtained by the action of D on the sum of all the cut ● Gluon production at LO ● Boost invariance vacuum diagrams. There are two kind of terms : Loop corrections ◆ D picks two sources in two distinct connected cut diagrams Less inclusive quantities Summary ◆ D picks two sources in the same connected cut diagram François Gelis – 2007 GGI, Florence, February 2007 - p. 21

  29. Gluon multiplicity at LO CERN ■ At LO, only tree diagrams contribute ⊲ the second type of Introduction topologies can be neglected (it starts at 1-loop) Basic principles ■ In each blob, we must sum over all the tree diagrams, and Inclusive gluon spectrum ● First moment over all the possible cuts : ● Gluon production at LO ● Boost invariance tree Loop corrections � � Less inclusive quantities N LO = Summary trees cuts tree ■ A major simplification comes from the following property : + = retarded propagator ■ The sum of all the tree diagrams constructed with retarded propagators is the retarded solution of Yang-Mills equations : [ D µ , F µν ] = J ν A µ ( x 0 = −∞ ) = 0 with François Gelis – 2007 GGI, Florence, February 2007 - p. 22

  30. Gluon multiplicity at LO CERN Krasnitz, Nara, Venugopalan (1999 – 2001), Lappi (2003) Introduction Z X Basic principles dN LO 1 e ip · ( x − y ) � x � y ǫ µ λ ǫ ν = λ A µ ( x ) A ν ( y ) Inclusive gluon spectrum dY d 2 � 16 π 3 p ⊥ x,y ● First moment λ ● Gluon production at LO ● Boost invariance ■ A µ ( x ) = retarded solution of Yang-Mills equations Loop corrections Less inclusive quantities Summary only tree diagrams at LO François Gelis – 2007 GGI, Florence, February 2007 - p. 23

  31. Gluon multiplicity at LO CERN Krasnitz, Nara, Venugopalan (1999 – 2001), Lappi (2003) Introduction Z X Basic principles dN LO 1 e ip · ( x − y ) � x � y ǫ µ λ ǫ ν = λ A µ ( x ) A ν ( y ) Inclusive gluon spectrum dY d 2 � 16 π 3 p ⊥ x,y ● First moment λ ● Gluon production at LO ● Boost invariance ■ A µ ( x ) = retarded solution of Yang-Mills equations Loop corrections ⊲ can be cast into an initial value problem on the light-cone Less inclusive quantities Summary A in − → François Gelis – 2007 GGI, Florence, February 2007 - p. 23

  32. Gluon multiplicity at LO CERN T k Introduction KNV I 2 )dN/d -1 10 Basic principles KNV II 2 Inclusive gluon spectrum R -2 π Lappi ● First moment 10 1/( ● Gluon production at LO ● Boost invariance -3 10 Loop corrections Less inclusive quantities -4 10 Summary -5 10 -6 10 -7 10 0 1 2 3 4 5 6 Λ k / s T ■ Lattice artefacts at large momentum (they do not affect much the overall number of gluons) ■ Important softening at small k ⊥ compared to pQCD (saturation) François Gelis – 2007 GGI, Florence, February 2007 - p. 24

  33. Initial conditions and boost invariance CERN τ = const Introduction η = const ■ Gauge condition : x + A − + x − A + = 0 Basic principles Inclusive gluon spectrum ● First moment 8 ● Gluon production at LO ● Boost invariance A i ( x ) α i ( τ, η, � < = x ⊥ ) ⇒ Loop corrections ± x ± β ( τ, η, � : A ± ( x ) = x ⊥ ) Less inclusive quantities Summary ■ Initial values at τ = 0 + : α i (0 + , η, � x ⊥ ) and β (0 + , η, � x ⊥ ) do not depend on the rapidity η ⊲ α i and β remain independent of η at all times (invariance under boosts in the z direction) ⊲ numerical resolution performed in 1 + 2 dimensions François Gelis – 2007 GGI, Florence, February 2007 - p. 25

  34. CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections ● 1-loop corrections to N ● Initial state factorization ● Unstable modes Loop corrections Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 26

  35. 1-loop corrections to N CERN ■ 1-loop diagrams for N Introduction Basic principles tree Inclusive gluon spectrum tree Loop corrections ● 1-loop corrections to N ● Initial state factorization 1-loop ● Unstable modes Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 27

  36. 1-loop corrections to N CERN ■ 1-loop diagrams for N Introduction Basic principles tree Inclusive gluon spectrum tree Loop corrections ● 1-loop corrections to N ● Initial state factorization 1-loop ● Unstable modes Less inclusive quantities ■ This can be seen as a perturbation of the initial value Summary problem encountered at LO, e.g. : François Gelis – 2007 GGI, Florence, February 2007 - p. 27

  37. 1-loop corrections to N CERN ■ 1-loop diagrams for N Introduction Basic principles tree Inclusive gluon spectrum tree Loop corrections ● 1-loop corrections to N ● Initial state factorization 1-loop ● Unstable modes Less inclusive quantities ■ This can be seen as a perturbation of the initial value Summary problem encountered at LO, e.g. : François Gelis – 2007 GGI, Florence, February 2007 - p. 27

  38. 1-loop corrections to N CERN ■ The 1-loop correction to N can be written as a perturbation Introduction of the initial value problem encountered at LO : Basic principles Inclusive gluon spectrum Loop corrections ● 1-loop corrections to N ● Initial state factorization ● Unstable modes Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 28

  39. 1-loop corrections to N CERN ■ The 1-loop correction to N can be written as a perturbation Introduction of the initial value problem encountered at LO : Basic principles Inclusive gluon spectrum Loop corrections ● 1-loop corrections to N u ● Initial state factorization ● Unstable modes Less inclusive quantities Summary » – Z δN = δ A in ( � u ) T � N LO u u ∈ light cone � ◆ N LO is a functional of the initial fields A in ( � u ) on the light-cone ◆ T � u is the generator of shifts of the initial condition at the point � u on the light-cone, i.e. : u ∼ δ/δ A in ( � u ) T � François Gelis – 2007 GGI, Florence, February 2007 - p. 28

  40. 1-loop corrections to N CERN ■ The 1-loop correction to N can be written as a perturbation Introduction of the initial value problem encountered at LO : Basic principles Inclusive gluon spectrum Loop corrections ● 1-loop corrections to N u u ● Initial state factorization ● Unstable modes v Less inclusive quantities Summary » – Z Z 1 δN = δ A in ( � u ) T � u + 2 Σ ( � u ,� v ) T � N LO u T � v u ∈ light cone � u ,� � v ∈ light cone ◆ N LO is a functional of the initial fields A in ( � u ) on the light-cone ◆ T � u is the generator of shifts of the initial condition at the point � u on the light-cone, i.e. : u ∼ δ/δ A in ( � u ) T � ◆ δ A in ( � u ) and Σ ( � u ,� v ) are in principle calculable analytically François Gelis – 2007 GGI, Florence, February 2007 - p. 28

  41. Sketch of a proof – I CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections ● 1-loop corrections to N ■ The first two terms involve : ● Initial state factorization ● Unstable modes Z X δ A ( x ) ≡ g Less inclusive quantities d 4 z ǫ G + ǫ ( x, z ) G ǫǫ ( z, z ) 2 Summary ǫ = ± ■ The third term involves G + − ( x, y ) ■ The propagators G ±± are propagators in the background A , in the Schwinger-Keldysh formalism. They obey : 8 G + − = G R G 0 − 1 G 0 + − G 0 − 1 G A < R A ˆ ˜ G ±± = 1 : G R G 0 − 1 ( G 0 + − + G 0 − + ) G 0 − 1 G A ± ( G R + G A ) 2 R A G R,A = retarded/advanced propagators in the background A François Gelis – 2007 GGI, Florence, February 2007 - p. 29

  42. Sketch of a proof – II CERN ■ G ++ and G −− are only needed with equal endpoints Introduction ⊲ they are both equal to Basic principles ˆ ˜ G ++ ( z, z ) = G −− ( z, z ) = 1 Inclusive gluon spectrum G R G 0 − 1 ( G 0 + − + G 0 − + ) G 0 − 1 ( z, z ) G A 2 R A Loop corrections ● 1-loop corrections to N ● Initial state factorization ⊲ thus, δ A can be simplified into : ● Unstable modes Z h i Less inclusive quantities g d 4 z δ A ( x ) = G ++ ( x, z ) − G + − ( x, z ) G ++ ( z, z ) Summary 2 Z g d 4 z G R ( x, z ) G ++ ( z, z ) = 2 ■ G R G 0 − 1 G 0 + − G 0 − 1 G A can be written as : R A Z d 3 � ˆ ˜ p G R G 0 − 1 G 0 + − G 0 − 1 p ( x ) ζ ∗ ( x, y ) = (2 π ) 3 2 E p ζ � p ( y ) , G A � R A ˆ ˜ � x + m 2 + g A ( x ) p ( x ) = e ip · x with ζ � p ( x ) = 0 and x 0 →−∞ ζ � lim François Gelis – 2007 GGI, Florence, February 2007 - p. 30

  43. Sketch of a proof – III CERN ■ Green’s formulas : Introduction Z h i Basic principles j ( z ) − g d 4 z G 0 2 A 2 ( z ) A ( x ) = R ( x, z ) Inclusive gluon spectrum Ω Z Loop corrections h i ● 1-loop corrections to N → ← d 3 � u G 0 + R ( x, u ) n · ∂ u − n · A in ( � u ) ∂ u ● Initial state factorization ● Unstable modes LC Less inclusive quantities Z d 4 z G R ( x, z ) g Summary δ A ( x ) = 2 G ++ ( z, z ) Ω Z h i → ← d 3 � + u G R ( x, u ) n · ∂ u − n · δ A in ( � u ) ∂ u LC Z h i → ← d 3 � p in ( � ζ � p ( x ) = u G R ( x, u ) n · ∂ u − n · ζ � u ) ∂ u LC Z G 0 d 4 z G 0 G R ( x, y ) = R ( x, y ) + g R ( x, z ) A ( z ) G R ( z, y ) Ω François Gelis – 2007 GGI, Florence, February 2007 - p. 31

  44. Sketch of a proof – IV CERN ■ Thanks to the operator Introduction h i Basic principles δ δ a in ( � u ) · T � u ≡ a in ( � u ) u ) + ( n · ∂ u ) a in ( � u ) u ) , Inclusive gluon spectrum δ A in ( � δ ( n · ∂ u ) A in ( � Loop corrections ● 1-loop corrections to N we can write ● Initial state factorization Z ● Unstable modes h i ζ � p ( x ) = ζ � p in ( � u ) · T � A ( x ) Less inclusive quantities u Summary � u ∈ LC Z Z h i d 4 z G R ( x, z ) g δ A ( x ) = 2 G ++ ( z, z ) + δ A in ( � u ) · T � A ( x ) u Ω � u ∈ LC ⊲ from the classical field A ( x ) , the operator a in ( � u ) · T � u builds the fluctuation a ( x ) whose initial condition on the light-cone is a in ( � u ) ■ The 3rd diagram can directly be written as : Z Z hh i i hh i i d 3 � p ζ ∗ ζ � p in ( � u ) · T � A ( x ) p in ( � v ) · T � A ( y ) u � v (2 π ) 3 2 E p u ,� � v ∈ LC François Gelis – 2007 GGI, Florence, February 2007 - p. 32

  45. Sketch of a proof – V CERN ■ One can finally prove that Introduction Z d 4 z G R ( x, z ) g Basic principles 2 G ++ ( z, z ) = Inclusive gluon spectrum Ω Z Z h ih i Loop corrections d 3 � = 1 p ● 1-loop corrections to N ζ ∗ ζ � p in ( � u ) · T � p in ( � v ) · T � A ( x ) ● Initial state factorization u � v (2 π ) 3 2 E p 2 ● Unstable modes u ,� � v ∈ LC Less inclusive quantities Summary " Z h i δ A ( x ) = δ A in ( � u ) · T � ⊲ u � u ∈ LC i# Z Z h ih d 3 � +1 p ζ ∗ ζ � p in ( � u ) · T � p in ( � v ) · T � A ( x ) u � v (2 π ) 3 2 E p 2 v ∈ LC u ,� � ■ This leads to the announced formula for δN , with Z d 3 � p u ) ζ ∗ Σ ( � u ,� p in ( � p in ( � v ) ≡ (2 π ) 3 2 E p ζ � v ) � François Gelis – 2007 GGI, Florence, February 2007 - p. 33

  46. Sketch of a proof – VI CERN Introduction ■ Conjecture : this result can be generalized to any observable Basic principles Inclusive gluon spectrum that can be written in terms of the gauge field with retarded Loop corrections boundary conditions, O ≡ O [ A ] : ● 1-loop corrections to N ● Initial state factorization » Z Z – ● Unstable modes 1 δ O = δ A in ( � u ) T � u + 2 Σ ( � u ,� v ) T � O LO u T � Less inclusive quantities v Summary � u ∈ light cone u ,� � v ∈ light cone ⊲ whatever we conclude for the multiplicity from this formula holds true for any such observable François Gelis – 2007 GGI, Florence, February 2007 - p. 34

  47. Divergences CERN Introduction ■ If taken at face value, this 1-loop correction is plagued by Basic principles several divergences : Inclusive gluon spectrum Loop corrections ● 1-loop corrections to N ● Initial state factorization ◆ The two coefficients δ A in ( � x ) and Σ ( � x , � y ) are infinite, ● Unstable modes because of an unbounded integration over a rapidity Less inclusive quantities variable Summary ◆ At late times, T � x A ( τ, � y ) diverges exponentially, √ µτ x A ( τ, � y ) τ → + ∞ e T � ∼ because of an instability of the classical solution of Yang-Mills equations under rapidity dependent perturbations (Romatschke, Venugopalan (2005)) François Gelis – 2007 GGI, Florence, February 2007 - p. 35

  48. Initial state factorization CERN ■ Anatomy of the full calculation : Introduction  Basic principles W Y beam -Y [ ρ 1 ]  Inclusive gluon spectrum  Loop corrections  ● 1-loop corrections to N ● Initial state factorization N[ A in ( ρ 1 , ρ 2 ) ]  ● Unstable modes  Less inclusive quantities  Summary W Y beam +Y [ ρ 2 ]   François Gelis – 2007 GGI, Florence, February 2007 - p. 36

  49. Initial state factorization CERN ■ Anatomy of the full calculation : Introduction  Basic principles W Y beam -Y [ ρ 1 ]  Inclusive gluon spectrum  Loop corrections  ● 1-loop corrections to N ● Initial state factorization N[ A in ( ρ 1 , ρ 2 ) ] + δ N  ● Unstable modes  Less inclusive quantities  Summary W Y beam +Y [ ρ 2 ]   ■ When the observable N [ A in ( ρ 1 , ρ 2 )] is corrected by an extra � gluon, one gets divergences of the form α s dY in δN ⊲ one would like to be able to absorb these divergences into the Y dependence of the source densities W Y [ ρ 1 , 2 ] François Gelis – 2007 GGI, Florence, February 2007 - p. 36

  50. Initial state factorization CERN ■ Anatomy of the full calculation : Y Introduction  + Y beam Basic principles W Y beam -Y 0 [ ρ 1 ]  Inclusive gluon spectrum  Y 0 Loop corrections  ● 1-loop corrections to N ● Initial state factorization N[ A in ( ρ 1 , ρ 2 ) ] + δ N  ● Unstable modes  Y ’ Less inclusive quantities 0  Summary W Y beam +Y ’ 0 [ ρ 2 ]  - Y beam  ■ When the observable N [ A in ( ρ 1 , ρ 2 )] is corrected by an extra � gluon, one gets divergences of the form α s dY in δN ⊲ one would like to be able to absorb these divergences into the Y dependence of the source densities W Y [ ρ 1 , 2 ] ■ Equivalently, if one puts some arbitrary frontier Y 0 between the “observable” and the “source distributions”, the dependence on Y 0 should cancel between the various factors François Gelis – 2007 GGI, Florence, February 2007 - p. 36

  51. Initial state factorization CERN ■ The two kind of divergences don’t mix, because the Introduction divergent part of the coefficients is boost invariant. Basic principles Given their structure, the divergent coefficients seem related Inclusive gluon spectrum to the evolution of the sources in the initial state Loop corrections ● 1-loop corrections to N ● Initial state factorization ■ In order to prove the factorization of these divergences in the ● Unstable modes Less inclusive quantities initial state distributions of sources, one needs to establish : Summary h i h i ( Y 0 − Y ) H † [ ρ 1 ] + ( Y − Y ′ 0 ) H † [ ρ 2 ] δN = N LO divergent coefficients where H [ ρ ] is the Hamiltonian that governs the rapidity dependence of the source distribution W Y [ ρ ] : ∂W Y [ ρ ] = H [ ρ ] W Y [ ρ ] ∂Y FG, Lappi, Venugopalan (work in progress) François Gelis – 2007 GGI, Florence, February 2007 - p. 37

  52. Initial state factorization CERN Introduction ■ Why is it plausible ? Basic principles ◆ Reminder : Inclusive gluon spectrum � � � � � � Loop corrections ● 1-loop corrections to N δN = δ A in ( � x ) div T � ● Initial state factorization x divergent ● Unstable modes � coefficients � � � � x +1 Less inclusive quantities Σ ( � x , � y ) N LO div T � x T � y Summary 2 x ,� � y ◆ Compare with the evolution Hamiltonian : � � δ 2 δ x ⊥ ) + 1 H [ ρ ] = σ ( � x ⊥ ) χ ( � x ⊥ , � y ⊥ ) δρ ( � 2 δρ ( � x ⊥ ) δρ ( � y ⊥ ) � � x ⊥ ,� x ⊥ y ⊥ ■ The coefficients σ and χ in the Hamiltonian are well known. There is a well defined calculation that will tell us if it works... François Gelis – 2007 GGI, Florence, February 2007 - p. 38

  53. Unstable modes CERN Romatschke, Venugopalan (2005) Introduction ■ Rapidity dependent perturbations to the classical fields grow Basic principles like exp(# √ τ ) until the non-linearities become important : Inclusive gluon spectrum Loop corrections ● 1-loop corrections to N ● Initial state factorization 0.0001 ● Unstable modes 1e-05 2 µ τ)) c 0 +c 1 Exp(0.427 Sqrt(g Less inclusive quantities 2 µ τ) 1e-06 c 0 +c 1 Exp(0.00544 g Summary 2 3 L 1e-07 4 µ 1e-08 ηη / g 1e-09 2 T max τ 1e-10 1e-11 1e-12 1e-13 0 500 1000 1500 2000 2500 3000 3500 2 µ τ g François Gelis – 2007 GGI, Florence, February 2007 - p. 39

  54. Unstable modes CERN Introduction ■ The coefficient δ A in ( � Basic principles x ) is boost invariant. Inclusive gluon spectrum Hence, the divergences due to the unstable modes all come Loop corrections from the quadratic term in δN : ● 1-loop corrections to N ● Initial state factorization 8 9 ● Unstable modes > > Z h i < = 1 Less inclusive quantities δN = Σ ( � x , � y ) T � N LO [ A in ( ρ 1 , ρ 2 )] x T � y > 2 > unstable Summary : ; modes � x ,� y ■ When summed to all orders, this becomes a certain functional Z [ T � x ] : h i δN = Z [ T � x ] N LO [ A in ( ρ 1 , ρ 2 )] unstable modes François Gelis – 2007 GGI, Florence, February 2007 - p. 40

  55. Unstable modes CERN ■ This can be arranged in a more intuitive way : Introduction Z ˆ h i Basic principles ˜ e x N LO [ A in ( ρ 1 , ρ 2 )] R x a ( � x ) T � x )] e i Z [ a ( � δN = Da � Inclusive gluon spectrum unstable modes Z ˆ Loop corrections ˜ e ● 1-loop corrections to N Z [ a ( � = Da x )] N LO [ A in ( ρ 1 , ρ 2 )+ a ] ● Initial state factorization ● Unstable modes Less inclusive quantities ⊲ summing these divergences simply requires to add fluctuations Summary to the initial condition for the classical problem ⊲ the fact that δ A in ( � x ) does not contribute implies that the distribution of fluctuations is real ■ Interpretation : Despite the fact that the fields are coupled to strong sources, the classical approximation alone is not good enough, because the classical solution has unstable modes that can be triggered by the quantum fluctuations François Gelis – 2007 GGI, Florence, February 2007 - p. 41

  56. Unstable modes CERN Introduction Basic principles Fukushima, FG, McLerran (2006) Inclusive gluon spectrum ■ By a different method, one obtains Gaussian fluctuations Loop corrections ● 1-loop corrections to N characterized by : ● Initial state factorization ● Unstable modes � � x ′ x ⊥ ) a j ( η ′ , � a i ( η, � ⊥ ) = Less inclusive quantities � � Summary 1 ∂ i ∂ j x ′ δ ( η − η ′ ) δ ( � = � δ ij + x ⊥ − � ⊥ ) ( ∂ η /τ ) 2 − ( ∂ η /τ ) 2 − ∂ 2 τ ⊥ � � e i ( η, � x ⊥ ) e j ( η ′ , � x ′ ⊥ ) = � � � ∂ i ∂ j − ( ∂ η /τ ) 2 − ∂ 2 x ′ δ ( η − η ′ ) δ ( � = τ δ ij − x ⊥ − � ⊥ ) ⊥ ( ∂ η /τ ) 2 + ∂ 2 ⊥ François Gelis – 2007 GGI, Florence, February 2007 - p. 42

  57. Unstable modes CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections Classical solution ● 1-loop corrections to N in 2+1 dimensions ● Initial state factorization ● Unstable modes Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 43

  58. Unstable modes CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections ● 1-loop corrections to N ● Initial state factorization ● Unstable modes Less inclusive quantities η Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 43

  59. Unstable modes CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections ● 1-loop corrections to N ● Initial state factorization ● Unstable modes Less inclusive quantities η Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 43

  60. Unstable modes CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections ● 1-loop corrections to N ● Initial state factorization ● Unstable modes Less inclusive quantities η Summary ■ Combining everything, one should write � � � dN = Dρ 1 ] [ Dρ 2 W Y beam − Y [ ρ 1 ] W Y beam+ Y [ ρ 2 ] dY d 2 � p ⊥ � � � dN [ A in ( ρ 1 , ρ 2 )+ a ] � Da Z [ a ] × dY d 2 � p ⊥ ⊲ This formula resums (all?) the divergences that occur at one loop François Gelis – 2007 GGI, Florence, February 2007 - p. 43

  61. Unstable modes – Interpretation CERN ■ Tree level : Introduction Basic principles Inclusive gluon spectrum p Loop corrections ● 1-loop corrections to N ● Initial state factorization ● Unstable modes Less inclusive quantities Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 44

  62. Unstable modes – Interpretation CERN ■ Tree level : Introduction Basic principles Inclusive gluon spectrum p Loop corrections ● 1-loop corrections to N ● Initial state factorization ● Unstable modes Less inclusive quantities ■ One loop ⊲ gluon pairs (includes Schwinger pairs): Summary p . . . q ⊲ The momentum � q is integrated out ˛ ˛ ˛ y p − y q ˛ , the correction is absorbed in W [ ρ 1 , 2 ] ⊲ If α − 1 � s ˛ ˛ ˛ y p − y q ˛ � α − 1 ⊲ If : gluon splitting in the final state s François Gelis – 2007 GGI, Florence, February 2007 - p. 44

  63. Unstable modes – Interpretation CERN ■ After summing the fluctuations, things may look like this : Introduction Basic principles Inclusive gluon spectrum Loop corrections ● 1-loop corrections to N ● Initial state factorization ● Unstable modes Less inclusive quantities Summary p ⊲ these splittings may help to fight against the expansion ? Note : the timescale for this process is τ ∼ Q − 1 ln 2 (1 /α s ) s François Gelis – 2007 GGI, Florence, February 2007 - p. 45

  64. CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections Less inclusive quantities ● Generating function Less inclusive quantities ● Exclusive processes Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 46

  65. Definition CERN ■ One can encode the information about all the probabilities Introduction P n in a generating function defined as : Basic principles ∞ � Inclusive gluon spectrum P n z n F ( z ) ≡ Loop corrections n =0 Less inclusive quantities ● Generating function ● Exclusive processes ■ From the expression of P n in terms of the operator D , we Summary can write : e z D e iV e − iV ∗ F ( z ) = ■ Reminder : ◆ e D e iV e − iV ∗ is the sum of all the cut vacuum diagrams ◆ The cuts are produced by the action of D ■ Therefore, F ( z ) is the sum of all the cut vacuum diagrams in which each cut line is weighted by a factor z François Gelis – 2007 GGI, Florence, February 2007 - p. 47

  66. What would it be good for ? CERN ■ Let us pretend that we know the generating function F ( z ) . Introduction We could get the probability distribution as follows : Basic principles Z 2 π P n = 1 Inclusive gluon spectrum dθ e − inθ F ( e iθ ) 2 π Loop corrections 0 Less inclusive quantities Note : this is trivial to evaluate numerically by a FFT : ● Generating function ● Exclusive processes Summary 1 F 1 (z) 0.01 F 2 (z) 1e-04 1e-06 P n 1e-08 1e-10 1e-12 1e-14 0 500 1000 1500 2000 2500 n François Gelis – 2007 GGI, Florence, February 2007 - p. 48

  67. F(z) at Leading Order CERN � e z D e iV e − iV ∗ � ■ We have : F ′ ( z ) = D Introduction Basic principles ■ By the same arguments as in the case of N , we get : Inclusive gluon spectrum Loop corrections Less inclusive quantities ● Generating function F ′ ( z ) ● Exclusive processes + F ( z ) = z Summary z ■ The major difference is that the cut graphs that must be evaluated have a factor z attached to each cut line ■ At tree level (LO), we can write F ′ ( z ) /F ( z ) in terms of solutions of the classical Yang-Mills equations, but these solutions are not retarded anymore, because : + z � = retarded propagator François Gelis – 2007 GGI, Florence, February 2007 - p. 49

  68. F(z) at Leading Order CERN ■ The derivative F ′ /F has an expression which is formally Introduction identical to that of N , Basic principles Z Z X F ′ ( z ) d 3 � p Inclusive gluon spectrum e ip · ( x − y ) � x � y λ A (+) ( x ) A ( − ) ǫ µ λ ǫ ν F ( z ) = ( y ) , µ ν (2 π ) 3 2 E p Loop corrections x,y λ Less inclusive quantities ● Generating function with A ( ± ) ( x ) two solutions of the Yang-Mills equations ● Exclusive processes µ Summary ■ If one decomposes these fields into plane-waves, Z n p ) e ip · x o d 3 � p p ) e − ip · x + f ( ε ) f ( ε ) A ( ε ) + ( x 0 ,� − ( x 0 ,� µ ( x ) ≡ (2 π ) 3 2 E p the boundary conditions are : f (+) p ) = f ( − ) ( −∞ ,� ( −∞ ,� p ) = 0 + − f ( − ) p ) = z f (+) f (+) p ) = z f ( − ) (+ ∞ ,� (+ ∞ ,� p ) , − (+ ∞ ,� (+ ∞ ,� p ) + + − ■ There are boundary conditions both at x 0 = −∞ and x 0 = + ∞ ⊲ not an initial value problem ⊲ hard... François Gelis – 2007 GGI, Florence, February 2007 - p. 50

  69. Remarks on factorization CERN Introduction ■ As we have seen, the fact that the calculation of the first Basic principles Inclusive gluon spectrum moment N can be formulated as an initial value problem Loop corrections seems quite helpful for proving factorization Less inclusive quantities ● Generating function ● Exclusive processes ■ If the retarded nature of the fields is crucial, then Summary factorization does not hold for the generating function F ( z ) , or equivalently for the individual probabilities P n ■ Note : by differentiating the result for F ( z ) with respect to z , and then setting z = 1 , we can obtain formulas for higher moments of the distribution François Gelis – 2007 GGI, Florence, February 2007 - p. 51

  70. Exclusive processes CERN ■ So far, we have considered only inclusive quantities – i.e. the Introduction P n are defined as probabilities of producing particles Basic principles anywhere in phase-space Inclusive gluon spectrum Loop corrections ■ What about events where a part of the phase-space remains Less inclusive quantities ● Generating function unoccupied ? e.g. rapidity gaps ● Exclusive processes Summary empty region Y François Gelis – 2007 GGI, Florence, February 2007 - p. 52

  71. Main issues CERN Introduction Basic principles 1. How do we calculate the probabilities P excl with an excluded n Inclusive gluon spectrum region in the phase-space ? Can one calculate the total gap probability P gap = � Loop corrections n P excl ? n Less inclusive quantities ● Generating function ● Exclusive processes 2. What is the appropriate distribution of sources W excl [ ρ ] to Summary Y describe a projectile that has not broken up ? 3. How does it evolve with rapidity ? See : Hentschinski, Weigert, Schafer (2005) 4. Are there some factorization results, and for which quantities do they hold ? François Gelis – 2007 GGI, Florence, February 2007 - p. 53

  72. Exclusive probabilities CERN ■ The probabilities P excl [ Ω ] , for producing n particles – only in Introduction n the region Ω – can also be constructed from the vacuum Basic principles diagrams, as follows : Inclusive gluon spectrum Loop corrections 1 Ω e iV e − iV ∗ P excl n ! D n [ Ω ] = Less inclusive quantities n ● Generating function ● Exclusive processes where D Ω is an operator that removes two sources and links Summary the corresponding points by a cut (on-shell) line, for which the integration is performed only in the region Ω ■ One can define a generating function, � [ Ω ] z n , P excl F Ω ( z ) ≡ n n whose derivative is given by the same diagram topologies as the derivative of the generating function for inclusive probabilities François Gelis – 2007 GGI, Florence, February 2007 - p. 54

  73. Exclusive probabilities CERN ■ Differences with the inclusive case : Introduction ◆ In the diagrams that contribute to F ′ Ω ( z ) /F Ω ( z ) , the cut Basic principles propagators are restricted to the region Ω of the Inclusive gluon spectrum phase-space Loop corrections ⊲ at leading order, this only affects the boundary Less inclusive quantities ● Generating function conditions for the classical fields in terms of which one ● Exclusive processes can write F ′ Ω ( z ) /F Ω ( z ) Summary ⊲ not more difficult than the inclusive case ◆ Contrary to the inclusive case – where we know that F (1) = 1 – the integration constant needed to go from F ′ Ω ( z ) /F Ω ( z ) to F Ω ( z ) is non-trivial. This is due to the fact that the sum of all the exclusive probabilities is smaller than unity ⊲ F Ω (1) is in fact the probability of not having particles in the complement of Ω – i.e. the gap probability François Gelis – 2007 GGI, Florence, February 2007 - p. 55

  74. Survival probability CERN ■ We can write : Introduction   � z   Basic principles dτ F ′ Ω ( τ ) F Ω ( z ) = F Ω (1) exp Inclusive gluon spectrum  F Ω ( τ )  Loop corrections 1 Less inclusive quantities ● Generating function ⊲ the prefactor F Ω (1) will appear in all the exclusive ● Exclusive processes probabilities Summary ■ This prefactor is nothing but the famous “survival probability” for a rapidity gap ⊲ One can in principle calculate it by the general techniques developed for calculating inclusive probabilities : F Ω (1) = F incl 1 − Ω (0) ⊲ Note : it is incorrect to say that a certain process with a gap can be calculated by multiplying the probability of this process without the gap by the survival probability François Gelis – 2007 GGI, Florence, February 2007 - p. 56

  75. Factorization ? CERN ■ In order to discuss factorization for exclusive quantities, one Introduction must calculate their 1-loop corrections, and study the Basic principles structure of the divergences... Inclusive gluon spectrum Loop corrections ■ Except for the case of Deep Inelastic Scattering, nothing is Less inclusive quantities ● Generating function known regarding factorization for exclusive processes in a ● Exclusive processes high density environment Summary ■ For the overall framework to be consistent, one should have factorization between the gap probability, F Ω (1) , and the source density studied in Hentschinski, Weigert, Schafer (2005) (and the ordinary W Y [ ρ ] on the other side) ■ The total gap probability is the “most inclusive” among the exclusive quantities one may think of. For what quantities – if any – does factorization work ? François Gelis – 2007 GGI, Florence, February 2007 - p. 57

  76. CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary Summary François Gelis – 2007 GGI, Florence, February 2007 - p. 58

  77. Summary CERN ■ When the parton densities in the projectiles are large, the Introduction study of particle production becomes rather involved Basic principles Inclusive gluon spectrum ⊲ non-perturbative techniques that resum all-twist Loop corrections contributions are needed Less inclusive quantities ■ At Leading Order, the inclusive gluon spectrum can be Summary calculated from the classical solution with retarded boundary conditions on the light-cone ■ At Next-to-Leading Order, the gluonic corrections can be seen as a perturbation of the initial value problem encountered at LO ■ Resummation of the leading divergences to all orders : ⊲ Evolution with Y of the distribution of sources ⊲ Quantum fluctuations on top of initial condition for the classical solution in the forward light-cone François Gelis – 2007 GGI, Florence, February 2007 - p. 59

  78. CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections Less inclusive quantities Extra bits Summary Extra bits ● Parton saturation ● Diagrammatic interpretation ● Quark production ● Longitudinal expansion ● AGK identities François Gelis – 2007 GGI, Florence, February 2007 - p. 60

  79. Parton evolution CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary Extra bits ● Parton saturation ● Diagrammatic interpretation ● Quark production ● Longitudinal expansion ● AGK identities ⊲ assume that the projectile is big, e.g. a nucleus, and has many valence quarks (only two are represented) ⊲ on the contrary, consider a small probe, with few partons ⊲ at low energy, only valence quarks are present in the hadron wave function François Gelis – 2007 GGI, Florence, February 2007 - p. 61

  80. Parton evolution CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary Extra bits ● Parton saturation ● Diagrammatic interpretation ● Quark production ● Longitudinal expansion ● AGK identities ⊲ when energy increases, new partons are emitted � dx x ∼ α s ln( 1 ⊲ the emission probability is α s x ) , with x the longitudinal momentum fraction of the gluon ⊲ at small- x (i.e. high energy), these logs need to be resummed François Gelis – 2007 GGI, Florence, February 2007 - p. 62

  81. Parton evolution CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary Extra bits ● Parton saturation ● Diagrammatic interpretation ● Quark production ● Longitudinal expansion ● AGK identities ⊲ as long as the density of constituents remains small, the evolution is linear: the number of partons produced at a given step is proportional to the number of partons at the previous step (BFKL) François Gelis – 2007 GGI, Florence, February 2007 - p. 63

  82. Parton evolution CERN Introduction Basic principles Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary Extra bits ● Parton saturation ● Diagrammatic interpretation ● Quark production ● Longitudinal expansion ● AGK identities ⊲ eventually, the partons start overlapping in phase-space ⊲ parton recombination becomes favorable ⊲ after this point, the evolution is non-linear: the number of partons created at a given step depends non-linearly on the number of partons present previously François Gelis – 2007 GGI, Florence, February 2007 - p. 64

  83. Saturation criterion CERN Gribov, Levin, Ryskin (1983) Introduction ■ Number of gluons per unit area: Basic principles Inclusive gluon spectrum ρ ∼ xG A ( x, Q 2 ) Loop corrections πR 2 Less inclusive quantities A Summary ■ Recombination cross-section: Extra bits ● Parton saturation σ gg → g ∼ α s ● Diagrammatic interpretation ● Quark production Q 2 ● Longitudinal expansion ● AGK identities ■ Recombination happens if ρσ gg → g � 1 , i.e. Q 2 � Q 2 s , with: α s xG A ( x, Q 2 s ) A 1 / 3 1 Q 2 ∼ ∼ s πR 2 x 0 . 3 A ■ At saturation, the phase-space density is: dN g ∼ ρ Q 2 ∼ 1 d 2 � x ⊥ d 2 � α s p ⊥ François Gelis – 2007 GGI, Florence, February 2007 - p. 65

  84. Saturation domain CERN Introduction log( x -1 ) Basic principles Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary Extra bits ● Parton saturation ● Diagrammatic interpretation ● Quark production ● Longitudinal expansion ● AGK identities log( Q 2 ) Λ QCD François Gelis – 2007 GGI, Florence, February 2007 - p. 66

  85. Diagrammatic interpretation CERN ■ One loop : Introduction Basic principles Inclusive gluon spectrum Loop corrections Less inclusive quantities Summary Extra bits ● Parton saturation ● Diagrammatic interpretation ● Quark production ● Longitudinal expansion ● AGK identities François Gelis – 2007 GGI, Florence, February 2007 - p. 67

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