Odd azimuthal anisotropy & gluon correlations in CGC Vladimir Skokov

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1 Odd azimuthal anisotropy & gluon correlations in CGC Vladimir Skokov RBRC RIKEN BNL Research Center 1

2 Outline Odd azimuthal anisotropy in saturation/cgc framework High multiplicity p(d)-a collisions & glittering glasma Effect of projectile geometry on azimuthal anisotropy in CGC 2

3 CMS ppb 1 < p T s NN < 3 GeV/c = 5.02 TeV, N offline trk 110 (b) 2-particle correlation 2 pair d N d η d φ trig N azimuthal angle φ 2 0 RIDGE η 2 4 ~ y CMS, Phys. Lett. B 718 (2013) 795 rapidity 3

4 Odd azimuthal anisotropy Forward region: odd anisotropy due to quark/anti-quark asymmetry M. K. Davy, C. Marquet, Yu Shi, B.-W. Xiao, & C. Zhang, 18 M. Mace, K. Dusling & R. Venugoapalan, 16 T. Lappi, 14 Central region: - No odd azimuthal anisotropy in strict dilute-dense approximation -? A. Kovner & M. Lublinsky, 12 Y. V. Kovchegov & D. E. Wertepny, 12 - Non-zero odd azimuthal anisotropy in numerical dense-dense calculation T. Lappi, S. Srednyak, R. Venugopalan, 09 4

5 Odd azimuthal anisotropy Forward region: odd anisotropy due to quark/anti-quark asymmetry M. K. Davy, C. Marquet, Yu Shi, B.-W. Xiao, & C. Zhang, 18 M. Mace, K. Dusling & R. Venugoapalan, 16 T. Lappi, 14 Central region: - No odd azimuthal anisotropy in strict dilute-dense approximation -? A. Kovner & M. Lublinsky, 12 Y. V. Kovchegov & D. E. Wertepny, 12 - Non-zero odd azimuthal anisotropy in numerical dense-dense calculation T. Lappi, S. Srednyak, R. Venugopalan, 09 4

6 What do we know analytically in classical approximation? Asymmetric collisions, when Q s of projectile Q s of target, is the easiest case. Nucleus proton Λ QCD Q s p Q s A k Single inclusive production In general ( dn d 3 k = 1 Q 2 sp f α s k 2, Q2 sa k 2 For large k Q 2 sa : dn d 3 k = 1 α s Q 2 sp k 2 If k > Q sp, dn d 3 k = 1 Q 2 ( sp Q α s k 2 f (1) 2 sa k 2 Functions f (n) are calculable! ) is known only numerically; A. Krasnitz, R. Venugopalan, arxiv: Q 2 sa f (1,1) k 2 E. Kuraev, L. Lipatov, V. Fadin, 77 ) + 1 α s ( Q 2 sp k 2 ) 2 f (2) ( Q 2 sa k 2 ) + 5

7 Single inclusive production f (1) is known since 98 dn d 3 k = 1 Q 2 ( ) ( sp Q α s k 2 f (1) 2 sa k Q 2 sp α s k 2 ) 2 f (2) ( Q 2 sa k 2 ) + Y. V. Kovchegov and A. H. Mueller, arxiv:hep-ph/ A. Dumitru and L. D. McLerran, arxiv:hep-ph/ J.-P. Blaizot, F. Gelis, R. Venugopalan, arxiv: Nuclear shock-wave f (2) : no complete result yet I. Balitsky, arxiv:hep-ph/ G. A. Chirilli, Y. V. Kovchegov, and D. E. Wertepny, arxiv:

8 Double inclusive production d 2 N d 3 kd 3 p = 1 αs 2 Q 4 sp h (1) (Q sa ) + 1 αs 2 Q 6 sp h (2) (Q sa ) + Momentum dependence is omitted to simplify notation Dilute-dilute Glasma graph: h (1) is known since 12 ; invariant under (k k ) d 2 N d 3 kd 3 p = 1 α Q 4 spq 4 2 sa h(1,1) s A. Dumitru, F. Gelis, L. McLerran and R. Venugopalan, arxiv: A. Kovner and M. Lublinsky, arxiv: Y. V. Kovchegov and D. E. Wertepny, arxiv: h (2) : no complete result yet L. McLerran and V. S., arxiv: ; Yu. Kovchegov and V. S., arxiv:

9 What does presence of odd harmonics mean? Double inclusive production d 2 N d 2 N ( d 2 k 1 dy 1 d 2 = 1 + 2v 2 k 2 dy 2 k 1 dk 1 dy 1 k 2 dk 2 dy 2 {2} cos 2(φ 1 φ 2 ) + 2v3{2} 2 cos 3(φ 1 φ 2 ) +... ) 2 A non-vanishing v 2 3{2} 2π d 2 N ( ) π d 2 N ( ) π d 2 N ( d φ cos 3 φ φ = d φ cos 3 φ φ d φ cos 3 φ φ + π d 2 k 1d 2 k 2 d 2 k 1d 2 k 2 d 2 k 1d 2 k π [ d 2 N ( ) = d φ cos 3 φ k d 2 k 1d 2 k 1, k 2 d2 N ( ) ] k 2 d 2 k 1d 2 k 1, k ) Therefore, non-zero v 3 d 2 N ( ) d 2 N ( ) d 2 k 1 d 2 k k 1, k 2 2 d 2 k 1 d 2 k k 1, k 2 2 and is absent in Glasma graph and h (1) 8

10 Experimental data: v 2 {2} 0.10 ppb s NN = 5 TeV v CMS offline N trk 9

11 Experimental data: v 3 {2} v 3 {2, η >2} ppb 0.3 < p T s NN < 3 GeV/c = 5.02 TeV CMS offline Ntrk Suppressed compared to v 2, but non-zero! 10

12 A conundrum for saturation Can saturation dynamics account for observed long-range rapidity correlations with non-zero odd azimuthal harmonics? 11

13 A possible resolution Odd contribution is buried somewhere in multiple rescattering i.e. in high order h (N 1) d 2 N d 3 kd 3 p = 1 Q 4 αs 2 sp h (1) (Q sa ) + 1 Q 6 αs 2 sp h (2) (Q sa ) + Solving CYM on lattice: T. Lappi, S. Srednyak and R. Venugopalan, arxiv:

14 Theoretically this is unsatisfactory Phenomenologically this is problematic v 3 {2} is observed in p-a v 3 {2} is not much smaller than v 2 {2} 13

15 Inspiration from Single Transverse Spin Asymmetry Consider single gluon production dσ d 2 k M(k) 2 = Amplitude may have two contributions d 2 x d 2 y e ik (x y) M(x) M (y) M(x) = M 1 (x) + M 3 (x) +... Asymmetry under k k would mean that M 1 (x) M 3 (y) + M 3 (x) M 1 (y) = M 1 (y) M 3 (x) M 3 (y) M 1 (x) M 1 (x) M 3 (y) is imaginary Phase difference between M 1 and M 3 in coordinate space In coordinate space, but not dissimilar from STSA S. Brodsky, D. S. Hwang, Y. Kovchegov, I. Schmidt, M. Sievert, arxiv:

16 Natural candidate M 3 M 1 Vanishes for single-inclusive production after performing average with respect to projectile configurations... 15

17 Double inclusive gluon production Non-zero! 16

18 Classical Yang-Mills t CYM pure gauge: α 2 nucleus ρ pure gauge: α 1 z proton ρ 2 1 Just after collision, τ 0+, initial conditions are known (Fock-Schwinger gauge A τ = 0) A. Kovner, L. McLerran, H. Weigert, arxiv: In forward light-cone [D µ, F µν ] = 0 Solve equations perturbatively in ρ 1; use LSZ 17

19 Classical Yang-Mills Before collision, pure gauge soft fields created by valence currents iα i 1,2(x ) = g ρ 1,2(x ) α i 1,2(x ) = 1 ig U1,2(x ) i U 1,2 (x ) Just after collision, τ 0+, (Fock-Schwinger gauge A τ = 0) A. Kovner, L. McLerran, H. Weigert, arxiv: α i (τ 0, x ) = α i 1(x ) + α i 2(x ) A η(τ 0, x ) = τ 2 α(τ 0, x ); α(τ 0, x ) = ig 2 [αi 1(x ), α i 2(x )] Expansion in gρ 1 (Φ 1 = g ρ 1): ( 2 ) α1 i = i Φ 1 ig δ 2 ij i j [ ] j Φ 1, Φ O(Φ 3 1 ) In forward light-cone [D µ, F µν ] = 0 18

20 Classical Yang-Mills: expansion in gρ 1 In order to perform expansion, it is convenient to rotate out nucleus field from initial conditions: α(τ, x ) = U 2(x )β(τ, x )U 2 (x ) ( ) α i (τ, x ) = U 2(x ) β i (τ, x ) 1 ig i U 2 (x ) β(τ 0, x ) = U 2 (x )α(τ 0, x )U 2 (x ) β i (τ 0, x ) = U 2 (x )α i 1(x )U 2 (x ) Perform expansion in powers of ρ 1: β γ = β (1) γ + β (2) γ + 19

21 First order, β (1) At leading order, CYM equations are (in Milne coordinates) [ τ ] τ τ 2 β (1) (τ, x ) = 0, τ iβ (1) i (τ, x ) = 0, ( [δ ij τ ) ] τ τ 2 + i j β (1) j (τ, x ) = 0 No non-linear terms solution can be found trivially. Analogous to φ = 0 Solutions are in momentum space: β (1) (τ, k ) = b 1(k ) J1(k τ), k τ β (1) i (τ, k ) = i εij k j b 2(k )J 0(k τ) + ik iλ(k ) k 2 b 1(x ) = δ ij Ω ij(x ), b 2(x ) = ɛ ij Ω ij(x ), Ω ij(x ) = g [ i ρ a 1(x 2 ) ] j ( U (x )t au(x ) ) 20

22 Second order, β (2) At next-to-leading order: [ τ ] τ τ 2 β (2) (τ, x ) = ig ( i[β (1) i(τ, x ), β (1) (τ, x )] + [β (1) i(τ, x ), iβ (1) (τ, x )] ) ( [δ ij τ ) ] τ τ 2 + i j β (2) j(τ, x ) = ig ( j[β (1) j(τ, x ), β (1) i(τ, x )] +[β (1) j(τ, x ), jβ (1) i(τ, x ) iβ (1) j(τ, x )] τ 2 [β (1) (τ, x ), iβ (1) (τ, x )] ) First non-linear corrections! This looks very discouraging as (in momentum space) β 1 are Bessel functions Goal is to compute g 6 ρ 3 1 correction to particle production. Do we have to solve these equations? 21

23 Particle production: Lehmann-Symanzik-Zimmermann I For simplicity Minkowski space and semi-classical scalar field φ(x). The creation operator a + (k, t) = 1 d 3 x exp( ik x) φ(x) k x = k µx µ i The difference a + (k, t ) a + (k, t t 0) = 1 i = 1 i t 0 dt 0 t 0 ( d 3 x exp( ik x) φ(x) dt d 3 x exp( ik x) ( + m 2) φ (x) interaction Instead of a usual choice t 0, in order to mimic initial conditions on light cone, t 0 0. Thus for creation operator at out-state, there are two contributions a + (k, ) = 1 i d 3 x exp( ik x) φ(x) t=0 initial flux through t=0 hypersurface + 1 i 0 dt ) d 3 x exp( ik x) ( + m 2) φ(x) interaction; evolution in the forward light cone Single-inclusive gluon production E k dn d 3 k = 1 2(2π) 3 a + (k, ) a(k, ) 22

24 Particle production: Lehmann-Symanzik-Zimmermann II Single-inclusive gluon production E k dn d 3 k = 1 2(2π) 3 1 i t=0 surface d 3 1 x exp( ik x) φ(x) + i initial flux Leading order: no bulk contribution Both contributions schematically: E k dn d 3 k = a (1) (k ) +a (2) (k ) +... surface only 0 dt bulk d 3 x exp( ik x) ( + m 2) φ(x) interaction; evolution in the forward light cone ( a (1) (k ) + a (2) (k ) surface and bulk +... a (1) (k )a (1) ( k ) + a (1) (k ) ( a (2) (k ) ) + a (1) ( k )a (2) (k ) symmetric odd asymmetry is possible ) 2 23

25 Gluon production Leading order and saturation correction dn even (k) ] [ρ d 2 p, ρ t = 2 δ ij δ lm + ɛ ij ɛ lm kdy (2π) 3 k 2 Ω a ij(k) [Ω a lm(k)] { dn odd (k) ] [ρ d 2 p, ρ T = 2 kdy (2π) 3 Im g d 2 l Sign(k l) k 2 (2π) 2 l 2 k l 2 f abc Ω a ij(l)ω b mn(k l) [ Ω c rp(k) ] [( k 2 ɛ ij ɛ mn l (k l)(ɛ ij ɛ mn + δ ij δ mn ) ) } ɛ rp + 2k (k l)ɛ ij δ mn δ rp] Here δ ij Ω ij = Ω xx + Ω yy and ɛ ij Ω ij = Ω xy Ω yx and Ω a ij(x ) = g i 2 val. sour. ρ b (x ) j target W line U ab (x ) ] valence sources rotated by the target dn odd (k) d 2 kdy [ρ p, ρ T is suppressed by extra α s ρ p L. McLerran and V. S., arxiv:

26 Alternative approach This was obtained in Fock-Schwinger gauge A τ = 0 Motivation to compute in global gauge A + = 0 Yu. Kovchegov and V. S., arxiv:

27 Leading order amplitude b b b a λ z λ z ɛ λ M 1 (z, b) = i g π ɛ λ (z b) [ z b 2 Uz ab ] Ub ab (V b t b ) We have to track the phases of the light-cone wave functions 26

28 First saturation correction b 1 b 1 x 1 λ z x 2 λ z b 2 b 2 G. A. Chirilli, Y. V. Kovchegov, and D. E. Wertepny, arxiv:

29 First saturation correction ɛ λ MABC 3 ɛ λ (x 2 b 2 ) + x 2 b 2 2 ( U bd x d 2 x [ = g3 4 π 4 x 1 b 1 x 1 b 1 2 ( U bd U b x ce 1 ) Ubd U ce b 1 d 2 x 1 d 2 x 2 δ[(z x 1 ) (z x 2 )] Uce ( ɛ λ b 2 ( i g3 d) 4 π 2 fabc V b1 t 1 [ i g3 4 π 3 d 2 x U ab x [ ɛλ (x 2 x 1 ) x 2 x 1 2 x 1 b 1 x 1 b 1 2 ] [ z x 2 ] [ ] z x 2 2 f abc Ux bd U bd U ce 1 b x U ce (V d) 1 2 b b1 t 2 1 ) ( ɛ λ b 2 (z x) z x 2 (z x) z x 2 x b 1 x b 1 2 [ x b 1 x b 1 2 x b ɛ 2 x b 2 2 λ (z b 1 ) z x z b 1 2 z x 2 x b ɛ 2 x b 2 λ (z b 2 ) z x z b 2 z x ( e) V b2 t (U z bd Ubd ) U ce ɛ λ (z b 1 ) b b z b 2 ln 1 ] ( f bde d) ( e) ɛ (z x) λ V b1 t V b2 t 1 2 z x 2 Uab z 1 U bd (U b z ce z b Λ 1 2 x b 1 x b 1 2 x 2 b ɛ 2 x b 2 λ (x 1 b 1 ) x b ( e) ( V b2 t + i g3 d) 2 4 π 3 fabc V b1 t 1 x b ɛ 2 x b 2 2 λ (z b 1 ) z b 1 2 x b 1 ɛ λ (z b 2 ) x b 2 z b ɛ Uce λ (z b 2 ) ) b 2 z b 2 ln 2 x b 2 x b 2 2 Sign(b 2 b 1 ) z x 1 z x 1 2 x b 1 x b 1 2 ( V b2 t e) 2 x b 1 x b z b 1 Λ x 2 b x 2 b 2 x x )] x b 2 x b 2 2 ] G. A. Chirilli, Y. V. Kovchegov, and D. E. Wertepny, arxiv:

30 First saturation correction x 1 λ z x λ z x 2 b 2 b 2 G. A. Chirilli, Y. V. Kovchegov, and D. E. Wertepny, arxiv:

31 First saturation correction ɛ λ M DE = g3 3 8 π 4 [ ɛ d 2 x 1 d 2 λ x 2 δ[(z x 1 ) (z x 2 )] (x 2 x 1 ) x 1 b 2 x 2 x 1 2 x 1 b 2 2 ɛ λ (x 1 b 2 ) x 1 b 2 2 z x 1 z x 1 2 f abc [ U bd x 1 U bd b 2 ] [ U ce x 2 U ce b 2 ] ( V b1 ) + i g3 4 π 3 d 2 x f abc U bd b 2 [ U ce x x 2 b 2 x 2 b 2 + ɛ λ (x 2 b 2 ) x 1 b 2 2 x 2 b 2 2 x 1 b U ce b 2 ] ( V b1 ) ( V b2 t e t d) 2 1 ( V b2 t e t d) 2 ) ɛ λ (z b 2 ) z x z b 2 2 z x x b 2 2 x b 2 ɛ λ (z b 2 ) 1 2 z b 2 2 x b i [ g3 4 π f abc U bd 2 b 2 U ce z U ce b 2 ] ( V b1 ) 1 ( V b2 t e t d) ɛ λ (z b 2 ) ln 2 z b 2 2 x 2 b 2 x 2 b 2 2 ] z x 2 z x 2 2 ( ɛ (z x) λ 1 z x 2 x b z b 2 Λ G. A. Chirilli, Y. V. Kovchegov, and D. E. Wertepny, arxiv:

32 Collecting all diagrams b1 b1 b1 z1 w1 z1 w1 b2 b2 b2 z2 w2 z2 w2 z1 w1 b3 b3 z2 w2 b3 Diagram A Diagram B Diagram C b1 b1 b1 z1 w1 z1 w1 z2 w2 b2 b2 b2 z2 w2 z2 w2 z1 w1 b3 b3 b3 Diagram D Diagram E Diagram F Reproduces result obtained in Fock-Schwinger gauge! Six adjoint Wilson lines multiplying a non-trivial function. 29

33 Approximations The sum of all contributions can be computed numerically; relatively low cost However, our goal is to obtain an analytical result Approximations: Large N c Golec-Biernat Wusthoff model ( S = exp 1 8 Q2 sr 2 ln 1 ) exp ( 1 ) r 2 Λ 2 8 Q2sr 2 Only lowest non-trivial order in interaction with the target Q 2 s k

34 b1 b1 b1 z1 w1 z1 w1 b2 b2 b2 z2 w2 z2 w2 z1 w1 b3 b3 z2 w2 b3 Diagram A Diagram B Diagram C b1 b1 b1 z1 w1 z1 w1 z2 w2 b2 b2 b2 z2 w2 z2 w2 z1 w1 b3 b3 b3 Diagram D Diagram E Diagram F Under these approximations, non-vanishing contributions from diagrams A, B and C dσ odd 1 = d 2 B d 2 b [T 1(B b)] 3 g 8 Q 6 1 s0(b) d 2 k 1 dy 1 d 2 k 2 dy 2 [2(2π) 3 ] 2 k 6 1 k6 2 [ (k k2 2 + k 1 k 2 )2 (k2 1 + k2 2 k 1 k ] 2 )2 10 c2 1 k + 1 k 2 (k 1 + k 2 ) 6 (k 1 k 2 ) 6 (2π) 2 Λ 2 k 1 k 2 } {{ }} {{ } A B + 1 k1 4 [ δ 2 (k 4π Λ 4 1 k 2 ) δ 2 (k 1 + k 2 ) ] } {{ } C 31

35 CGC perspective on v 3 Leading order and the first saturation correction a) dn even ] (k) [ρ p, ρ d 2 t = 2 δ ijδ lm + ɛ ijɛ lm kdy (2π) 3 k { 2 ] [ρ p, ρ T b) dn odd (k) d 2 kdy = 2 (2π) 3 Im g k 2 Ω a ij(k) [Ω a lm(k)] d 2 l Sign(k l) (2π) 2 l 2 k l f abc Ω a ij(l)ω b mn(k l) [ Ω c rp(k) ] 2 [( k 2 ɛ ij ɛ mn l (k l)(ɛ ij ɛ mn + δ ij δ mn ) ) ɛ rp + 2k (k l)ɛ ij δ mn δ rp] } a) Recall that Ω ρ proton ρ p b) ρ p ρ p U A Odd azimuthal harmonics is a sign of emerging coherence in proton wave function: the first saturation correction! U A Non-zero long-range odd harmonics in high energy p-a is evidence of saturation! ρ p U A U A ρ p 32

36 Multiplicity dependence: scaling argument Physical two-particle anisotropy coefficients can be simply expressed as ( ] ) vn{2}(n 2 ch ) = Dρ pdρ t W [ρ p] W [ρ t] Q n [ρ p, ρ t] 2 dn δ [ρ p, ρ t N ch dy with ] p2 dφ k dk 2π ei2nφ dn even (k) p2 [ρ p, ρ t Q 2n [ρ p, ρ t] = p 1 p2 p 1 k dk dφ 2π d 2 kdy dn even (k) d 2 kdy High multiplicity is driven by fluctuations in ρ p [ρ p, ρ t ], Q 2n+1 [ρ p, ρ t] = To study multiplicity dependence, rescale ρ p c ρ p Under this rescaling: p 1 k dk dφ 2π ei(2n+1)φ dn odd (k) p2 p 1 k dk dφ 2π dn even (k) d 2 kdy d 2 kdy [ρ p, ρ t ] [ρ p, ρ t ] dn dy dn c2 dy ; v2 2n{2} v2n{2}; 2 v2n+1{2} 2 c 2 v2n+1{2} 2 Therefore in the first approximation: v 2n{2} is independent of multiplicity v 2n+1{2} dn dy 33

37 Multiplicity dependence: scaling argument vn{2} CGC power counting ATLAS imp. template fit n=2 ATLAS imp. template fit n= N ch v3{2} M. Mace, V. S., P. Tribedy, & R. Venugopalan, arxiv: CGC power counting ATLAS imp. template fit N ch 34

38 Conclusions I Odd azimuthal harmonics are an inherent property of particle production in the saturation framework Non-zero long range in y odd azimuthal harmonics evidence of saturation Phenomenological applications: able to qualitatively describe multiplicity dependence in p-a at LHC talk by Mark Mace next week: quantitative results for p-a at LHC and small systems at RHIC 35

39 Outline Odd azimuthal anisotropy in saturation/cgc framework High multiplicity p(d)-a collisions & glittering glasma A. Kovner & V.S., 18 Effect of projectile geometry on azimuthal anisotropy in CGC A. Kovner & V.S., 18 36

40 Gluon production: functional form Functional form for gluon production: dn d 2 kdy = 2g2 ρp,ρ t (2π) 3 where the square of Lipatov vertex is Γ(k, q, q ) = The single (double) inclusive production: dn d 2 kdy = dn d 2 kdy ρp,ρ t p d 2 q d 2 q (2π) 2 (2π) 2 Γ(k, q, q )ρ a p( q ) [ U (k q )U(k q) ] ab ρb p(q), t, d 2 N d 2 k 1 dy 1 d 2 = k 2 dy 2 ( q ) ( q k q 2 k 2 q k 2 k ). 2 dn d 2 k 1 dy 1 ρp,ρ t dn d 2 k 2 dy 2 Averaging is performed over projectile and target color charge configurations: O(ρ p,t ) p,t = 1 Dρ p,t W p,t (ρ p,t ) O(ρ p,t ) Z p,t ρp,ρ t p t. 37

41 Gluon production: functional form In general, one can compute d n N d 2 k 1dy 1d 2 k 2dy 2 d 2 k ndy n, cumulants and factorial cumulants Dilute-dilute approx. Glittering glasmas color density fluctuations GLITTER = GLuon Intensification Through Tenacious Emission of Radiation These fluctuations are approximately negative binomial: derived for k Q st! F. Gelis, T. Lappi, & L. McLerran, Nucl. Phys. A 828, 149 (2009), arxiv: Instead, consider generating function [ G(t) = exp t d 2 k dn k min d 2 kdy d n N F. Gelis, T. Lappi, & L. McLerran, Nucl. Phys. A 828, 149 (2009), arxiv: ρp,ρ t ] p t, k min Λ QCD } {{ } Detector cut for produced gluons Moments d 2 k 1 d 2 k n d 2 k 1dy 1d 2 k 2dy 2 d 2 k ndy n derivatives of G(t) at t = 0. 38

42 Generating function: MV model: ρ a p(p)ρ b p(k) p = (2π) 2 µ 2 p(p)δ(p + k)δ ab W p (ρ p ) = exp dn Reminder: d 2 kdy is quadratic in ρ p : ρp,ρ t dn d 2 kdy ρ p,ρ t = 2g2 (2π) 3 ( d 2 ) q 1 (2π) 2 ρa p( q) 2µ 2 p(q) ρa p(q) d 2 q d 2 q Γ(k, q, q )ρ a (2π) 2 (2π) p( q ) [ U (k q )U(k q) ] 2 ab ρb p(q) Average w.r.t. ρ p can be done analytically [ G(t) = exp t d 2 k dn k min d 2 kdy ρp,ρ t ] = 1 Z t p t Dρ t W [ρ t ] exp [ 12 [ ] ] tr ln 1 tm where M is defined by its matrix elements M ab (q, q) = 4g2 d 2 k (2π) 3 µ2 (q) k min (2π) 2 Γ(k, q, q ) [ U (k q )U(k q) ] ab 39

43 Target averaging Any combination of target Wilson lines into pairs with Uab (p)u cd (q) t = (2π)2 N 2 c 1 δ acδ bd δ(p + q)d(p) Adjoint dipole D(p) = 1 Nc 2 1 d 2 xe ix p tr [ U (x)u(0) ] t The logic behind this approximation: dense regime for the target small size color singlets in the projectile: any non-singlet states separated by distance > 1/Q s t have zero S-matrix leading in S of projectile: any singlet state containing more than 2 proj. gluons is suppressed by powers of S This approximation is very restrictive and cannot be applied to many processes 40

44 Approximating target averaging is not sufficient to move forward analytically In order to understand the structure of higher moments/generating function lets consider double inclusive production... 41

45 Double incl. production: dissecting connected terms Dipole contribution 2g 2 (2π) 3 2g 2 (2π) 3 2g2 (2π) 3 2g 2 (2π) 3 d 2 k 1 d 2 k 2 d 2 k 1 d 2 k 2 d 2 q d 2 q (2π) 2 (2π) 2 Γ(k 1, q, q ) d 2 p d 2 p (2π) 2 (2π) 2 Γ(k 2, p, p ) ρ a p( q ) [ U (k 1 q )U(k 1 q) ] ab ρb p(q) ρ c p( p ) [ U (k 2 p )U(k 2 p) ] cd ρd p(p) d 2 q (2π) 2 µ2 (q 1 )Γ(k 1, q, q) tr [ U (k 1 q)u(k 1 q) ] } {{ } dipole d 2 p (2π) 2 µ2 (q 2 )Γ(k 2, p, p) tr [ U (k 2 p)u(k 2 p) ] } {{ } dipole For a connected contribution one will have to break both adjoint traces tr [ U (k 1 q)u(k 1 q) ] tr [ U (k 2 p)u(k 2 p) ] conn. t = 2S δ(k 1 + k 2 q p)d 2 (k 1 q) O(N 0 c ) 42

46 Double incl. production: dissecting connected terms Quadrupole contribution 2g 2 (2π) 3 d 2 k 1 2g 2 (2π) 3 d 2 k 2 ( ) 2g 2 2 (2π) 3 d 2 q d 2 q (2π) 2 (2π) 2 Γ(k 1, q, q ) d 2 p d 2 p (2π) 2 (2π) 2 Γ(k 2, p, p ) d 2 k 1 d 2 k 2 ρ a p( q ) [ U (k 1 q )U(k 1 q) ] ab ρb p(q) ρ c p( p ) [ U (k 2 p )U(k 2 p) ] cd ρd p(p) d 2 q d 2 p (2π) 2 (2π) 2 µ2 (q)µ 2 (p)γ(k 1, q, p)γ(k 2, q, p) tr[u (k 1 p)u(k 1 q)u (k 2 q)u(k 2 p)] } {{ } quadrupole The remaining contraction leading to a quadrupole restores the infamous symmetry k 2 k 2, which precluded odd azimuthal harmonics 43

47 Leading large N c contractions in the quadrupole Wilson lines in the quadrupole can be contracted in multiple ways µ 2 (q)µ 2 (p)tr[u (k 1 p)u(k 1 q)u (k 2 q)u(k 2 p)] (Nc 2 1) S µ 2 (q)µ 2 (p) δ(q p)d(k } {{ } 1 p)d(k 2 p) µ 4 (q) Incoming gluons from projectile have the same transverse momentum (= q) BE A. Dumitru, F. Gelis, L. McLerran & R. Venugopalan arxiv: Y. Kovchegov and D. Wertepny arxiv: T. Altinoluk, N. Armesto, G. Beuf, A. Kovner and M. Lublinsky, arxiv: Y. Kovchegov & V. S. arxiv:

48 Leading large N c contractions in the quadrupole Wilson lines in the quadrupole can be contracted in multiple ways µ 2 (q)µ 2 (p) tr[u (k 1 p)u(k 1 q)u (k 2 q)u(k 2 p)] (N 2 c 1) S µ 2 (q)µ 2 (p) δ(k 1 k 2 ) D(k 1 p)d(k 2 q) Produced gluons have the same transverse momentum (= k 1 ) HBT There is also an anti -HBT contribution with δ(k 1 + k 2 ). I will refer to both δ(k 1 ± k 2 ) as to HBT contribution. 44

49 N c suppressed contraction µ 2 (q)µ 2 (p) tr[u (k 1 p)u(k 1 q)u (k 2 q)u(k 2 p)] S µ 2 (q)µ 2 (p) δ(k 1 + k 2 p q) D(k 1 p)d(k 1 q) I will neglect N 2 c suppressed contribution 45

50 Bose-Einstein contribution Returning to BE and collecting all together [ d 2 ] ( ) N 2g 2 2 = dy 1 dy 2 (2π) 3 d 2 k 1 d 2 k 2 BE d 2 q (2π) 2 d 2 p (2π) 2 µ2 (q)µ 2 (p)γ(k 1, q, p)γ(k 2, q, p) tr[u (k 1 p)u(k 1 q)u (k 2 q)u(k 2 p)] } {{ } (N 2 c 1) S µ 2 (q)µ 2 (p)δ(q p)d(k 1 p)d(k 2 p) [ d 2 ] N dy 1 dy 2 BE = 2(N 2 c 1)S d 2 q µ 2 p(q) 2 2g 2 (2π) 3 d 2 kγ(k, q, q)d(q k) k min 2 Important property of Lipatov vertex (k q)2 Γ(k, q, q) = q 2 k 2 1 q 2 46

51 Bose-Einstein contribution [ d 2 N dy 1 dy 2 ] BE 2(Nc 2 1)S d 2 q µ2 p(q) 2 q } {{ 4 } S µ 4 p 2g 2 2 (2π) 3 d 2 kd(k) k } min {{ } D 2 Effectively [ d 2 N dy 1 dy 2 ] BE S 2 as for uncorrelated two-gluon production (SIP) 2. HBT contribution is suppressed by S 1 Lesson learned: Keep BE only! 47

52 Bose-Einstein contribution & Generating function Connected contribution ln G resummation of rainbow diagrams D = 4g2 (2π) 3 d 2 k D(k) k min 48

53 Cumulants & Phenomenological Conclusion Average number of gluons κ 1 = N 2 c 1 8π S µ 2 p D ln k2 min Λ 2 Higher order cumulants κ n 2 = ( t n ln G LO(t) = (n 2)! (N c 2 1) S Λ 2 µ 2 p D t=0 8π Λ 2 ) n 49

54 Cumulants & Phenomenological Conclusion Properties (Λ 2 S proton ) κ 1 is a function of S µ 2 p Consider configurations a) and b): κ 1 [a)] = κ 1 [b)] κ n [b)] 2 S proton ( ) µ proton 2n p κ n [b)] 2 n S proton ( µ proton p ) 2n High multiplicity tail configurations with overlapping nucleons 50

55 But... probability to have overlapping nucleus in an actual collision: 0.4 d 0.3 P (d) d 51

56 MV model numerical calculations P (Ng) MV model N g / N g 52

57 MV model plus glauber d /d gluon rms MV model N g / N g High multiplicity bias overlapping configurations of deuteron 53

58 CGC response to projectile geometry v2{2}(k1 = k2) a=1 a=2 a= k 1 /Q s v2{2}(k2 = k1 + Qs) a=1 a=2 a= k 1 /Q s v 2 {2} anticorrelates with ɛ 2 A. Kovner & V.S.,

59 Combining phenomenological conclusions... High multipl. overlapping nucleons in projectile (d-a) Overlapping nucleons in projectile smaller ɛ 2 Smaller ɛ 2 larger v 2 {2} The latter effect is rather mild, but it is manifestly present! 55

60

61 More functionals BE is property of projectile Need for effective theory of gluons in projectile Constraint effective action for projectile gluon distribution e V eff[η(q)] = 1 Z p Dρ p W (ρ p ) } {{ } all possible fluct. ( ) δ η(q) g2 tr A + (q) 2 g 2 tr A + (q) 2 } {{ } keeping only interesting stuff A + (q) = g/q 2 ρ p (q), g 2 tr A + (q) 2 = 1 2 (N 2 c 1)S g 4 µ 2 p q 4 Exact expression for effective potential (modulo S 1 corrections) V eff [η(q)] = 1 2 (N c 2 d 2 q { } 1 1)S η(q) 1 ln η(q) (2π) 2 2 (N c 2 1)S d 2 q 1 (2π) 2 2 ln2 η(q) A. Dumitru & V. S., arxiv:

62 Liouville potential & high multiplicity tail Back to generating function G LO (t) = In terms of effective potential exp [ t kmin Λ ] d 2 q (2π) 2 ρa ( q) D 2q 2 ρa (q) kmin G LO (t) = Dη exp V eff[η(q)] (N c 2 d 2 q 1)S Λ (2π) 2 t µ2 pd q 2 η(q) } {{ } reweighting! derivatives in t probe Liouville potential p

63 Liouville potential & high multiplicity tail For large S : saddle point approximation ( ) 1 1 t µ2 p D, if Λ q k q η s(q) = 2 min 1, otherwise to yield ln G LO (t) = 1 2 (N 2 c 1)S d 2 q (2π) ln ηs(q) = 1 kmin ( ) 2 2 (N c 2 d 2 q 1)S (2π) ln 1 t µ2 pd 2 q 2 Λ We recovered previously derived result. Origin of ln Liouville s ln!

64 Liouville potential and small x evolution Q 2 s dveff(η)/dq αy = 0 αy = 1 A. Dumitru & V. S., arxiv: η Form does not change V eff [η(q)] 1 2 (N 2 c 1)S S S eff S σ 2 : d 2 q (2π) ln2 η(q) partially responsible for phenomenological parameter σ ] C.f. P [ρ] exp [ ρ2 2σ with ρ ln Q 2 s/ Q 2 2 s L. McLerran & P. Tribedy, arxiv:

65 Long-range correlations Correlated pair y t free streaming A B detection ( 1 m/c) Past-pointing light cones Locus of correlation z freeze out ( 10 fm/c) Surface of last interaction latest correlation Regardless of nature of the ridge Suppress final state effects in order to probe initial state physics - long-range rapidity correlations either pre-exist in initial wave function or develop very early after collision - understanding initial/early stage is of paramount importance for phenomenology of p-a and p-p. figure adapted from A. Dumitru, F. Gelis, L. McLerran and R. Venugopalan, arxiv: