Saturation Physics and Di-Hadron Correlations

Saturation Physics and Di-Hadron Correlations Bo-Wen Xiao Pennsylvania State University and Institute of Particle Physics, Central China Normal University POETIC August 2012 1 / 24

Outline 1 Introduction 2 A Tale of Two Gluon Distributions 3 Di-hadron productions DIS dijet Dijet (dihardrons) in pa 4 Conclusion 2 / 24

Introduction Phase diagram in QCD Consider the evolution inside a hadron: Low Q 2 and low x region saturation region. Use BFKL equation and BK equation instead of DGLAP equation. BK equation is the non-linear small-x evolution equation which describes the saturation physics. 3 / 24

Introduction k t dependent parton distributions The unintegrated quark distribution dξ d 2 ξ + f q(x, k ) = ξ +iξ k 4π(2π) 2 eixp P ψ(0)l (0)γ + L(ξ, ξ )ψ(ξ, ξ ) P as compared to the integrated quark distribution dξ + f q(x) = ξ 4π eixp P ψ(0)γ + L(ξ )ψ(0, ξ ) P The dependence of ξ in the definition. Gauge invariant definition. Light-cone gauge together with proper boundary condition parton density interpretation. The gauge links come from the resummation of multiple gluon interactions. Gauge links may vary among different processes. 4 / 24

Introduction Saturation physics Saturation physics describes the high density parton distributions in the high energy limit. Initial condition: McLerran-Venugopalan Model plus small-x evolution dense gluon distributions. In a physical process, in order to probe the dense nuclear matter precisely, the proper factorization is required. Factorization is about separation of short distant physics(perturbatively calculable hard factor) from large distant physics (parton distributions and fragmentation functions). Hard factor should always be finite and free of divergence of any kind. 5 / 24

Introduction Dilute-Dense factorizations The effective Dilute-Dense factorization γ γ p,a p,a Protons and virtual photons are dilute probes of the dense gluons inside target hadrons. For pa (dilute-dense system) collisions, there is an effective k t factorization. dσ pa qfx =x pq(x p, µ 2 )x Af (x A, q 2 ) 1 dˆσ d 2 P d 2 q dy 1dy 2 π dˆt. For dijet processes in pp, AA collisions, there is no k t factorization[collins, Qiu, 08],[Rogers, Mulders; 10]. At forward rapidity y, x p e y is large, while x A e y is small. Ideal opportunity to search gluon saturation. Systematic framework to test saturation physics predictions. 6 / 24

Introduction Factorization for single inclusive hadron productions [G. Chirilli, BX and F. Yuan, Phys. Rev. Lett. 108, 122301 (2012)]Obtain a systematic factorization for the p + A H + X process by systematically remove all the divergences! Gluons in different kinematical region give different divergences. 1.soft, collinear to the target nucleus; 2. collinear to the initial quark; 3. collinear to the final quark. P p 0 k + 0 P + A 0 Rapidity Divergence Collinear Divergence (P) Collinear Divergence (F) All the rapidity divergence is absorbed into the UGD F(k ) while collinear divergences are either factorized into collinear parton distributions or fragmentation functions. Large N c limit is vital for the factorization in terms of getting rid of higher point functions. Consistent check: take the dilute limit, k 2 Q 2 s, the result is consistent with the leading order collinear factorization formula. In terms of resummation, we will be able to resum up to α s(α s ln k 2 ) n and α s(α s ln 1/x) n terms. 7 / 24

Introduction Outlook Using this factorization technique, we can imagine that a lot of other NLO calculations can be achieved in the near future. NLO Drell-Yan productions NLO dijet productions NLO Single Inclusive DIS NLO Drell-Yan lepton pair production and NLO dijet productions in pa collisions. Single inclusive DIS at NLO. (see similar work [Balitsky, Chirilli, 10], [Beuf, 11]) Direct photon production in pa collisions at NLO (straightforward) and NNLO (similar to the DY case at NLO). Universality and large N c The CSS resummation and Sudakov suppression factor in small-x physics. (work in progress with A. Mueller and F. Yuan) 8 / 24

Introduction Sudakov factor [Mueller, Xiao, Yuan, work in progress] Two scale problem (CSS resummation) Q 2 1 Q 2 2 αscr 2π ln2 Q 2 1 Q 2 2 For pa H(M H, k ) + X: S sud(m 2 H, r 2 ) = αsnc 2π ln2 M 2 H r2 c 2 + with c = 2e γ E : (a) (b) (c) (d) (e) (f) (a) (b) dσ (resum) dyd 2 k M k H = σ 0 d 2 x d 2 x (2π) 2 xg p(x, µ 2 = c 2 0/r 2 ) e ik r e S 2 sud(mh,r2 ) S WW [ 1 + αs π 2 π 2 Nc Y=ln 1/x g (x, x ) ]. For dijet processes, replace M 2 H by 4P 2. Mismatch between rapidity and collinear divergence between graphs S sud(m 2 H, r 2 ). 9 / 24

A Tale of Two Gluon Distributions A Tale of Two Gluon Distributions In small-x physics, two gluon distributions are widely used:[kharzeev, Kovchegov, Tuchin; 03] I. Weizsäcker Williams gluon distribution ([KM, 98] and MV model): S Nc 2 1 π 2 α s N c xg (1) = ( d 2 r e ik r 1 e r (2π) 2 r 2 2 II. Color Dipole gluon distributions: xg (2) = S N c k 2π 2 2 α s d 2 r (2π) 2 e ik r e r 2 Q2 sq 4 Remarks: 2 Q2 sg The WW gluon distribution simply counts the number of gluons. The Color Dipole gluon distribution often appears in calculations. Does this mean that gluon distributions are non-universal? Answer: Yes and No! ) r T 10 / 24

A Tale of Two Gluon Distributions A Tale of Two Gluon Distributions [F. Dominguez, C. Marquet, BX and F. Yuan, 11] I. Weizsäcker Williams gluon distribution S xg (1) Nc 2 1 = π 2 α s N c ( d 2 r e ik r 1 e r (2π) 2 r 2 2 II. Color Dipole gluon distributions: 2 Q2 sg ) xg (2) = S N c k 2π 2 2 α s d 2 r (2π) 2 e ik r e r 2 Q2 sq 4 0.25 0.20 xg x,q 0.15 0.10 0.05 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 11 / 24

A Tale of Two Gluon Distributions A Tale of Two Gluon Distributions In terms of operators (known from TMD factorization), we find these two gluon distributions can be defined as follows: [F. Dominguez, C. Marquet, BX and F. Yuan, 11] I. Weizsäcker Williams gluon distribution: xg (1) = 2 dξ dξ + ξ ik ξ (2π) 3 P + eixp Tr P F +i (ξ, ξ )U [+] F +i (0)U [+] P. II. Color Dipole gluon distributions: xg (2) dξ dξ + = 2 ξ ik ξ (2π) 3 P + eixp Tr P F +i (ξ, ξ )U [ ] F +i (0)U [+] P. ξ T ξ T ξ U [ ] U [+] Remarks: The WW gluon distribution is the conventional gluon distributions. In light-cone gauge, it is the gluon density. (Only final state interactions.) The dipole gluon distribution has no such interpretation. (Initial and final state interactions.) Both definitions are gauge invariant. Same after integrating over q. 12 / 24 ξ

A Tale of Two Gluon Distributions A Tale of Two Gluon Distributions In terms of operators, we find these two gluon distributions can be defined as follows: [F. Dominguez, C. Marquet, BX and F. Yuan, 11] I. Weizsäcker Williams gluon distribution: xg (1) = 2 dξ dξ + ξ ik ξ (2π) 3 P + eixp Tr P F +i (ξ, ξ )U [+] F +i (0)U [+] P. II. Color Dipole gluon distributions: xg (2) dξ dξ + = 2 ξ ik ξ (2π) 3 P + eixp Tr P F +i (ξ, ξ )U [ ] F +i (0)U [+] P. ξ T ξ T ξ U [ ] U [+] Questions: Can we distinguish these two gluon distributions? Yes, We Can. How to measure xg (1) directly? DIS dijet. How to measure xg (2) directly? Direct γ+jet in pa collisions. For single-inclusive particle production in pa up to all order. What happens in gluon+jet production in pa collisions? It s complicated! ξ 13 / 24

Di-hadron productions DIS dijet DIS dijet [F. Dominguez, C. Marquet, BX and F. Yuan, 11] k1 k1 k2 q2 k2 q2 (a) (b) (c) dσ γ T A q q+x dp.s. N cα eme 2 q d 2 x d 2 x d 2 b d 2 b (2π) 2 (2π) 2 (2π) 2 (2π) 2 e ik 1 (x x ) e ik 2 (b b ) ψt (x b)ψ T(x b ) [ ] 1 + S x (4) g (x, b; b, x ) S x (2) g (x, b) S x (2) g (b, x ), }{{} u i u j 1 Nc Tr[ i U(v)]U (v )[ j U(v )]U (v) xg Operator Def Eikonal approximation Wilson Line approach [Kovner, Wiedemann, 01]. In the dijet correlation limit, where u = x b v = zx + (1 z)b S (4) x g (x, b; b, x ) = 1 N c TrU(x)U (x )U(b )U (b) x g S (2) x g (x, b)s (2) x g (b, x ) Quadrupoles are generically different objects and only appear in dijet processes. 14 / 24

Di-hadron productions DIS dijet DIS dijet The dijet production in DIS. k1 k1 k2 q2 k2 q2 TMD factorization approach: Remarks: dσ γ T A q q+x dp.s. (a) (b) (c) = δ(x γ 1)x gg (1) (x g, q )H γ T g q q, Dijet in DIS is the only physical process which can measure Weizsäcker Williams gluon distributions. Golden measurement for the Weizsäcker Williams gluon distributions of nuclei at small-x. The cross section is directly related to the WW gluon distribution. EIC and LHeC will provide us perfect machines to study the strong gluon fields in nuclei. Important part in EIC and LHeC physics. 15 / 24

Di-Hadron correlations in DIS Di-hadron productions DIS dijet Di-pion correlations at EIC[J. H. Lee, BX, L. Zheng] C( φ) = p 1, p 2 p 1 dσ ea h 1 h 2 dy 1 dy 2 d 2 p 1 d 2 p 2 dσ ea h 1 dy 1 d 2 p 1 Forward di-hadron correlatio d+au collisions at RHIC! Coincidence probability at measu trigger φ associated CP! Away peak is present in p+ 0.18 0.16 ep Q 2 = 1 GeV 2 0.2 EIC stage-ii Ldt = 10 fb -1 /A trigger p T > 2 GeV/c assoc 1 < p T < trigger pt 0.14 η <4 C(Δϕ) 0.12 0.1 0.08 0.06 0.04 0.02 0 eca eau 2 2.5 3 3.5 4 4.5 Δϕ CeAu(Δϕ) 0.15 0.1 0.05 0 eau - nosat!"=0 (near side) eau - sat 2 2.5 3 3.5 4 4.5 Δϕ!"=# (away side) p+p 2105 2106 Figure 3.21: Left: Saturation model prediction of the coincidence signal versus azimuthal angle difference ϕ between two hadrons in ep, eca, and ea collisions [165, 166]. Right: Comparison of saturation model prediction for ea collisions with calculations from conventional non-saturated model for EIC stage-ii energies. Statistical error bars correspond to 10 fb 1 /A integrated luminosity. EIC stage II energy 30 100GeV. Using: Q 2 sa = c(b)a 1/3 Q 2 s (x). Physical picture: Dense gluonic matter suppresses the away side peak. There are several advantages to studying di-hadron correlations in ea collisions versus dau. Directly using a point-like electron probe, as opposed to a quark bound in a 16 / 24

Di-hadron productions DIS dijet Di-Hadron correlations in DIS Forward di-hadron correlatio d+au collisions at RHIC The estimate of di-pion correlations at EIC [J. H. Lee, BX, L.! Zheng] Coincidence probability at measu trigger J ea = 1 σ pair ea /σea φ CP N coll σep pair /σ ep associated! Away peak is present in p+ J eau 1 eau - nosat EIC stage-ii Ldt = 10 fb -1 /A eau - sat Q 2 = 1 GeV 2 trigger p T > 2 GeV/c assoc 1 < p T < trigger pt η <4-3 -2.5-2 -1.5-1 log 10 (x g) J dau RHIC dau, s = 200 GeV 1!"=0 (near side) 10-1 peripheral central 10-3 10-2 xfrag A!"=# (away side) p+p Figure 3.22: Left: Relative yield of di-hadrons in eau compared to ep collisions, J eau, plotted versus x g, the relative momentum fraction of the probed gluon. Shown are predictions for linear (nosat) and non-linear (sat) QCD models for EIC stage-ii energies. Statistical error bars correspond to 10 fb 1 /A integrated luminosity. Right: The corresponding measurement in s = 200 GeV per nucleon dau collisions at RHIC [167]. Here x frac A is a only a rough approximation of the average momentum fraction of the struck parton in the Au nucleus. Curves depict calculations in the CGC framework. J is normalized to unity in the dilute regime. Physical picture: The cross sections saturates at low-x. 17 / 24

Di-hadron productions Dijet (dihardrons) in pa γ+jet in pa collisions The direct photon + jet production in pa collisions. (Drell-Yan follows the same factorization.) TMD factorization approach: dσ (pa γq+x) dp.s. = f x 1q(x 1, µ 2 )x gg (2) (x g, q )H qg γq. Remarks: Independent CGC calculation gives the identical result in the correlation limit. Direct measurement of the Color Dipole gluon distribution. 18 / 24

Di-hadron productions Dijet (dihardrons) in pa DY correlations in pa collisions [Stasto, BX, Zaslavsky, 12] GBW BK rcbk 0.03 GBW BK rcbk 0.004 σdyf/σdysif 0.002 σdyf/σdysif 0.02 0.01 0 π 0 2 π 3π 2 2π 0 π 0 2 π 3π 2 2π 0.003 GBW BK rcbk 0.04 GBW BK rcbk 0.03 σdyf/σdysif 0.002 σdyf/σdysif 0.02 0.001 0.01 0 π 0 2 π 3π 2 2π 0 π 0 2 π 3π 2 2π φ φ M = 0.5, 4GeV, Y = 2.5 at RHIC dau. M = 4, 8GeV, Y = 4 at LHC ppb. Partonic cross section vanishes at π Dip at π. Prompt photon calculation [J. Jalilian-Marian, A. Rezaeian, 12] 19 / 24

Di-hadron productions Dijet (dihardrons) in pa STAR measurement on di-hadron correlation in da collisions There is no sign of suppression in the p + p and d + Au peripheral data. The suppression and broadening of the away side jet in d + Au central collisions is due to the multiple interactions between partons and dense nuclear matter (CGC). Probably the best evidence for saturation. 20 / 24

Di-hadron productions Dijet (dihardrons) in pa Dijet processes in the large N c limit The Fierz identity: = 1 2 1 2Nc Graphical representation of dijet processes and g q q: = 1 2 1 2 x 1 x 1 v=zx1 +(1 z)x 2 v =zx 1 +(1 z)x 2 x 2 x 2 q qg = 1 2 1 2Nc v=zx+(1 z)b b b x x v =zx +(1 z)b g gg 2 = = 1 2 1 2Nc x 1 x 1 v=zx1 +(1 z)x 2 v =zx 1 +(1 z)x 2 x 2 x 2 = = The Octupole and the Sextupole are suppressed. 21 / 24

Di-hadron productions Dijet (dihardrons) in pa Gluon+quark jets correlation Including all the qg qg, gg gg and gg q q channels, a lengthy calculation gives dσ (pa Dijet+X) = x 1 q(x 1, µ 2 ) α2 [ ] s dp.s. ŝ q 2 F qg (1) H qg (1) + F qg (2) H qg (2) [ ( +x 1 g(x 1, µ 2 ) α2 s ŝ 2 F gg (1) H (1) gg q q + 1 ) 2 H(1) gg gg ( + F gg (2) H (2) gg q q + 1 ) ] 2 H(2) gg gg + F gg (3) 1 2 H(3) gg gg, with the various gluon distributions defined as F qg (1) = xg (2) (x, q ), F qg (2) = F (1) gg = F (3) gg = xg (2) F, F (2) gg xg (1) (q 1 ) F F, xg (1) F, q1 q 2 = q 2 xg (2) F, 1 where F = d 2 r (2π) 2 e iq r 1 N c TrU(r )U (0) x g. Remarks: All the above gluon distributions can be written as combinations and convolutions of two fundamental gluon distributions. This describes the dihadron correlation data measured at RHIC in forward dau collisions. 22 / 24

Di-hadron productions Dijet (dihardrons) in pa Comparing to STAR and PHENIX data Forward di-hadron correlations d+au collisions at RHIC Physics predicted by C. Marquet. Further calculated in[a. Stasto, BX, F. Yuan, 11] For away side peak in both peripheral and central dau collisions C( φ) = p 1, p 2 p 1 J da = 1 σ pair da /σda N coll σpp pair /σ pp dσ pa h 1 h 2 dy 1 dy 2 d 2 p 1 d 2 p 2 dσ pa h 1 dy 1 d 2 p 1! Coincidence probability at measure trigger φ associated!"=0 (near side)!"=# (away side) CP(! Away peak is present in p+p c p+p 0.016 0.014 C( ) Peripheral dau Correlation 0.022 0.020 C( ) Central dau Correlation J dau peripheral central 0.012 0.010 0.018 0.016 1 0.008 0.006 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.014 0.012 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 1 10 3 10 2 10 frag x Au Using: Q 2 sa = c(b)a 1/3 Q 2 s (x). Physical picture: Dense gluonic matter suppresses the away side peak. 23 / 24

Conclusion Conclusion Conclusion: The establishment of an effective factorization for the collisions between a dilute projectile and a dense target. DIS dijet provides direct information of the WW gluon distributions. Perfect for testing saturation physics calculation, and ideal measurement for EIC and LHeC. Modified Universality for Gluon Distributions: Inclusive Single Inc DIS dijet γ +jet g+jet xg (1) xg (2), F Do Not Appear. Apppear. Two fundamental gluon distributions. Other gluon distributions are just different combinations and convolutions of these two. The small-x evolution of the WW gluon distribution, a different equation from Balitsky-Kovchegov equation;[dominguez, Mueller, Munier, Xiao, 11] Dihadron correlation calculation agrees with the RHIC STAR and PHENIX data. 24 / 24