Penn State F. Dominguez, BX and F. Yuan, Phys.Rev.Lett.106:022301,2011. F.Dominguez, C.Marquet, BX and F. Yuan, Phys.Rev.D83:105005,2011. F. Dominguez, A. Mueller, S. Munier and BX, Phys.Lett. B705 (2011) 106-111. A. Stasto, BX and F. Yuan, arxiv:1109.1817 [hep-ph]. F. Dominguez, J.W. Qiu, BX and F. Yuan, arxiv:1109.6293 [hep-ph]. INT Oct. 2011 1 / 26
Outline 2 / 26
Inclusive and Semi-inclusive DIS at small-x The Dipole Model has become the common practice of QCD calculation at small-x. Using QCD dipole model, one can write the cross section of SIDIS as γ (Q) k, ˆξ P A dσ γ p qx T,L dzdk 2 = 1 d 2 xd 2 y Φ T,L(z, x, y, Q 2 ) e ik.(x y) 4π d 2 b [1 S q q(x, x) + S q q(y, x) + S q q(x y, x)] Integrating over z and k gives inclusive cross section. Convoluting with the fragmentation D h/q ( ˆξ ) yields the single inclusive z hadron spectrum. 3 / 26
Golec-Biernat Wusthoff model and Geometrical Scaling [Golec-Biernat, Wusthoff,; 98], [Golec-Biernat, Stasto, Kwiecinski; 01] σ tot γ*p [µb] 10 3 10 2 10 1 ZEUS BPT 97 ZEUS BPC 95 H1 low Q 2 95 ZEUS+H1 high Q 2 94-95 E665 x<0.01 all Q 2 10-1 10-3 10-2 10-1 1 10 10 2 10 3 τ The dipole amplitude in the GBW model S q q(r ) = exp[ Q2 s r 2 4 ] with Q 2 s (x) = Q 2 s0(x 0/x) λ where Q s0 = 1GeV, x = 3.04 10 3 and λ = 0.288. 4 / 26
Dipoles, and higher point functions Dipole and Quadrupole (a) (b) S (2) = 1 [ ] Tr U (x 1) U (x 2) N c S (4) = 1 [ ] Tr U (x 1) U (x 2) U (x 3) U (x 4) N c Scattering amplitudes between different projectiles and the target. Semi-inclusive processes only involve dipoles. Dijet and more exclusive processes involve dipoles, and higher point functions. For instance, the sextupole S (6) = 1 N c Tr [ U (x 1) U (x 2) U (x 3) U (x 4) U (x 5) U (x 6) ] appears in the gg gg channel of dijet processes. Similar results can be obtained in TMD approach as well. [Bomhof, Mulders and Pijlman; 06] 5 / 26
Transverse Momentum Dependent (TMD) factorization [Collins-Soper-Sterman,85], [Ji-Ma-Yuan, 04] (Mueller s dipole model) k t factorization is widely used in small-x physics. TMD factorization is widely used in spin physics (large x). k t factorization = TMD factorization? Yes! Universality of the TMD parton? Yes and No! 6 / 26
The effective k t factorization For pa (dilute-dense system) collisions, there is an effective k t factorization. Remarks: dσ pa qfx =x pq(x p, µ 2 )xf (x, q 2 ) 1 dˆσ d 2 P d 2 q dy 1dy 2 π dˆt. For pp, AA collisions, there is no k t factorization[rogers, Mulders; 10]. Penalty: K t dependent Parton xf (x, q 2 ) are not universal. xf (x, q 2 ) here can be the quark or gluon of the dense target. It contains the anomalous terms after resummation. x pq(x p, µ 2 ) is the Feyman parton distribution of the dilute projectile. Thanks to the nuclear enhancement, soft gluon exchange from the dilute proton can be neglected. Assuming small-x limit, namely, s, fixed Q 2, x 0. For DIS, this k t factorization works as well. The question is which gluon one should use. 7 / 26
A Tale of Two Gluon In small-x physics, two gluon are widely used: I. Weizsäcker Williams gluon distribution (MV model):[mueller, Kovchegov, 98], [McLerran, Venugopalan, 99] 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 4 II. Color Dipole gluon : 2 Q2 sg xg (2) = S N c 2π 2 α s d 2 r (2π) 2 e ik r 2 r e r 2 Q2 sq(g) 4 Remarks: The WW gluon distribution simply counts the number of gluons. The Color Dipole gluon distribution often appears in calculations. N(r ) is the color dipole amplitude. It is now in fundamental representation. Does this mean that gluon are non-universal? Yes and No! These two are used in R pa calculation. [Kharzeev, Kovchegov, Tuchin; 03]. ) r T 8 / 26
A Tale of Two Gluon I. Weizsäcker Williams gluon distribution (MV model): xg (1) = S Nc 2 1 π 2 α s N c d 2 r (2π) 2 e ik r r 2 ( II. Color Dipole gluon : 1 e r 2 Q2 sg 4 xg (2) = S N c 2π 2 α s d 2 r (2π) 2 e ik r 2 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 9 / 26
A Tale of Two Gluon The operator definitions of these two gluon : [Bomhof, Mulders and Pijlman; 06][F. Dominguez, 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 : 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 ξ U [ ] U [+] Remarks: The WW gluon distribution is the conventional gluon. 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. Same perturbative tail. 10 / 26
A Tale of Two Gluon The operator definitions of these two gluon : [Bomhof, Mulders and Pijlman; 06][F. Dominguez, BX and F. Yuan, 11] I. Weizsäcker Williams gluon distribution (Easy to evaluate in the MV model): 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 : 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 in physical processes? How to measure xg (1) directly?. How to measure xg (2) directly? Direct collisions. Maybe single-inclusive particle production in pa (Subtle). What happens in gluon+jet production in pa collisions? It s complicated! ξ 11 / 26
in the TMD and CGC approaches Resummation of the gauge links in TMD approach. (a) (b) (c) TMD factorization approach: k1 k2 q2 k1 k2 dσ γ T A q q+x dy 1dy 2d 2 P d 2 q = δ(x γ 1)x gg (1) (x g, q )H γ T g q q. CGC approach:[jalilian-marian, Gelis, 02] dσ γ T A q q+x dp.s. N cα eme 2 q q2 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 ), Two independent calculations agree perfectly in the correlation limit (Large P and Small q ).[F. Dominguez, BX and F. Yuan, 11] 12 / 26
In the dijet correlation limit, where u = x b and v = zx + (1 z)b [ ] 1 + S x (4) g (x, b; b, x ) S x (2) g (x, b) S x (2) g (b, x ) u iu 1 j Tr [ i U(v) ] U (v ) [ j U(v ) ] U (v) xg N c = u iu g 2 [ j dv + dv + Tr F i (v)u [+] F j (v )U [+]] N c x g Remarks: Dijet in DIS is the only physical process which can directly measure Weizsäcker Williams gluon. Golden measurement for the Weizsäcker Williams gluon of nuclei at small-x. EIC will provide us a perfect machine to study the strong gluon fields in nuclei. 13 / 26
collisions The direct photon + jet production in pa collisions. (Drell-Yan Process follows the same factorization.) k 1 k 2 q 2 (a) (b) (c) k 1 k 2 q2 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. [Jalilian-Marian, Gelis, 02],[Dominguez, Marquet, BX, Yuan, 11] This process can be calculated exactly for all range of azimuthal angles. Direct measurement of the Color Dipole gluon distribution. The RHIC and future LHC experiments shall provide us some information on this. 14 / 26
Existing calculations on dijet production Let us first look back, and re-examine the existing calculations on dijet productions. Quark+Gluon channel [Marquet, 07] and [Albacete, Marquet, 10] p q k p q k Prediction of saturation physics. All the framework is correct, but over-simplified 4-point function. Improvement [F. Dominguez, C. Marquet, BX and F. Yuan, 11.] S (4) x g (x 1, x 2; x 2, x 1) e C F 2 [Γ(x 1 x 2 )+Γ(x 2 x 1 )] F(x1, x2; x 2, x 1) ( ) F(x 1, x 2 ; x2, x 1 ) e C F 2 [Γ(x 1 x 2 )+Γ(x 2 x 1 )] e C F 2 [Γ(x 1 x 1 )+Γ(x 2 x 2)] 15 / 26
Dijet processes in the large N c limit The Fierz identity: = 1 2 1 2Nc and Graphical representation of dijet processes x 1 x v=zx1 +(1 z)x 1 2 v =zx 1 +(1 z)x 2 v=zx+(1 z)b x 2 x 2 b b x x v =zx +(1 z)b g q q: q qg g gg = 1 2 2 = = 1 2 = 1 2 1 2 1 2Nc 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. 16 / 26
Gluon+quark jets correlation Including all the qg qg, gg gg and gg q q channels, a lengthy calculation gives dσ (pa Dijet+X) dp.s. = q x 1 q(x 1, µ 2 ) α2 [ ] s ŝ 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 (2) H (2) gg q q + 1 2 H(2) gg gg [ ( F gg (1) H (1) gg q q + 1 ) 2 H(1) gg gg ) + F gg (3) 1 2 H(3) gg gg with the various gluon defined as F qg (1) = xg (2) (x, q ), F qg (2) = xg (1) F, F gg (1) = xg (2) F, F gg (2) q1 q 2 = q 2 xg (2) F, 1 F gg (3) = xg (1) (q 1 ) F F, ], where F = d 2 r (2π) 2 e iq r 1 N c TrU(r )U (0) x g. Remarks: Only the term in NavyBlue color was known before. This can help us understand the dihadron correlation data. 17 / 26
Illustration of gluon The various gluon : F (1) gg = F (3) gg = 0.20 xg (1) WW (x, q ), F qg (1) = xg (2) (x, q ), xg (2) F, F gg (2) q1 q 2 = xg (2) F, q 2 1 xg (1) (q 1) F F, F qg (2) = xg (1) F 0.15 xg x,q 0.10 0.05 0.00 0 1 2 3 4 18 / 26
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. Dissect the data into three features: Width σ of peaks, Pedestal P and Peak suppression. 19 / 26
Comparing to STAR data [A. Stasto, BX, F. Yuan, 11] For away side peak in both peripheral and central dau collisions C Φ 0.016 0.014 0.012 0.010 0.008 0.006 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Forward di-hadron correlations in 0.020 d+au collisions at RHIC (k2, Gluon+Jet y2) in pa x A = k 1! Coincidence probability 0.016 at measured by STAR Coll. at forward rapidities: trigger φ associated!"=# (away side) CP( φ) = 1 dn pair N trig d φ! Absence of away particle! Away peak is present in p+p coll. The framework does not work for pp since the saturation scale Q s is too monojets low at s = 200GeV. d+au cent Adding the nuclear and impact factor dependence: Q 2 sa = c(b)a 1/3 Q 2 s (x). p+p C Φ 0.022 0.018 0.014 0.012 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Use Golec-Biernat Wusthoff model for the saturation momentum,!"=0 Q 2 s (x) = Q 2 s0(x 0/x) λ. (near side) Peripheral b = 6.8 ± 1.7fm with c(b) = 0.45 and width σ 0.99; Central b = 2.7 ± 1.3fm with c(b) = 0.85 and width σ 1.6. (k1, y1) 20 / 26
The small-x evolution equation of The Balitsky-Kovchegov equation for dipoles Y S = P d dd (SS S), with P d dd = αsnc d 2 z (x y ) 2. 2π 2 (x z ) 2 (z y ) 2 The quadrupole evolution equation: Large N c [Jalilian-Marian and Kovchegov, 04], Finite N c [Dominguez, Mueller, Munier, BX, 11], [Iancu, Triantafyllopoulos, 11] Y Q = with P q qd + P q dd > 0. P q qd (QS Q) + P q dd (SS Q). Dipoles and quadupoles are very alike in terms of evolution. Follow BFKL in the dilute limit. Both have geometrical scaling. Confirmed by the numerical study[dumitru, Jalilian-Marian, Lappi, Schenke and Venugopalan,11] Saturate to a stable fixed point in the dense limit. 21 / 26
The distribution The linearly polarized gluon distribution effectively measures an averaged quantum interference between a scattering amplitude with an active gluon polarized along the x(or y)-axis and a complex conjugate amplitude with an active gluon polarized along the y(or x)-axis inside an unpolarized hadron. [Mulders and Rodrigues, 01], [Boer, Brodsky, Mulders, Pisano, 10], [Metz and Zhou,10], [Dominguez, Qiu, BX and Yuan, 10] ±1 1 ±1 1 dσ γ T A q q+x dy 1dy 2d 2 P = δ(x γ d 2 1)H γ q T g q q [ x gg (1) (x g, q ) 2ɛ2 P f 2 cos (2 φ) xh (1) ɛ 4 f + P 4 (x, q ) ], where φ = φ P φ q and ɛ 2 f = z(1 z)q 2 0. There is similar cross section for DY-type dijet. 22 / 26
The distribution [Metz and Zhou,10], [Dominguez, Qiu, BX and Yuan, 10] ±1 1 ±1 1 For : W.W. gluon distribution dξ dξ 2 (2π) 3 P + eixp+ ξ ik ξ Tr P F +i (ξ, ξ )U [+] F +j (0)U [+] P xg ( = 1 2 δij xg (1) (x, q ) + 1 2q i q j ) 2 q 2 δ ij xh (1) (x, q ). For DY-type dijet processes: Dipole gluon distribution dξ dξ [ 2 (2π) 3 P + eixp+ ξ iq ξ P Tr F +i (ξ, ξ )U [ ] F +j (0)U [+]] P, ( = 1 2 δij xg (2) (x, q ) + 1 2q i q j ) 2 q 2 δ ij xh (2) (x, q ). Diagonal and off-diagonal part of the dipole and quadrupole amplitudes. xg (1) (x, q ) xh (1) (x, q ) and xg (2) (x, q ) = xh (2) (x, q ) 23 / 26
The distribution [Dominguez, Qiu, BX and Yuan, 10] A few more comments ±1 1 ±1 1 The linearly polarized gluon come from quantum interference and have no probability interpretation. They are different from the usual polarized gluon distribution ( [ 1 2 ε i +ε j + ε i ε j ]). Need non-vanishing virtuality to have non-vanishing contribution in the dilute-dense factorization. the W.W. type linearly polarized gluon distribution ( Tr [ i U(v) ] U (v ) [ j U(v ) ] U (v) ) is related to the Y quadrupole evolution equation, while the dipole type linearly polarized gluon distribution follows BK equation. They follow BFKL evolution in the dilute regime, and receive exponential growth in terms of rapidity. They also have geometrical scaling. They also saturate in the dense regime. 24 / 26
Conclusion The effective factorization for collisions between a dilute projectile and a dense target in the large N c limit. provides direct information of the WW gluon. Modified Universality for Gluon : Inclusive Single Inc γ +jet g+jet xg (1) xg (2), F Do Not Appear. Apppear. Two fundamental gluon which are related to the quadrupole and dipole amplitudes, respectively. Other gluon are just different combinations and convolutions of these two. Dihadron correlation calculation and comparison with the STAR data. the quadrupole and the WW gluon distribution, a different equation from Balitsky-Kovchegov equation. provide us the off-diagonal information of scattering amplitudes. 25 / 26
Outlook [Dominguez, Marquet, Stasto, BX and Yuan, in preparation] The three-jet production processes in the large N c limit: q qg-jet 2 = = 1 2 1 2N c qgg-jet (a) (b) Conjecture: In the large N c limit at small-x, the dipole and quadrupole amplitudes are the only two fundamental objects in the cross section of multiple-jet production processes up to all order. Other higher point functions, such as sextupoles, octupoles, decapoles and duodecapoles, etc. are suppressed by factors of 1. Nc 2 26 / 26