NLO Calculations of Particle Productions in pa Collisions

Transcription

1 NLO Calculations of Particle Productions in pa Collisions Bo-Wen Xiao Institute of Particle Physics, Central China Normal University Detroit, Wayne State, August 22nd, / 38

2 Introduction 1 Introduction 2 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions 3 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor 4 Sudakov Double Logarithms of Dijet Processes 5 Summary 2 / 38

3 Introduction Deep into low-x region of Protons Resummation of the α s ln 1 BFKL equation. x When too many gluons squeezed in a confined hadron, gluons start to overlap and recombine Non-linear dynamics BK equation Introduction of saturation momentum Q s(x) to separate the saturated dense regime x < 10 2 from the dilute regime. Unitarity of S-matrix Gluon Saturation. [Mueller s dipole picture] Multiple interactions + Small-x evolution. 3 / 38

4 Introduction Some relevant experiments in 2013 and perations. Forward rapidity physics at RHIC and the LHC(LHCf and TOTEM, etc.) 5 TeV Cosmic ray events can have incoming particles with energy up to GeV. Future EIC or LHeC experiments. 4 / 38

5 Introduction Factorization involved in the small-x formalism 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). σ xf (x) H D h(z) S(r ) S(r ) takes care of multiple interactions and small-x resummation. NLO calculation always contains various kinds of divergences. Collinear divergences can be absorbed into xf (x) or D h (z) following evolution equations. Rapidity divergence can be absorbed into the small-x evolution of S(r ). The rest of divergences should be cancelled. Hard factor H = H (0) LO + αs 2π H(1) NLO + should always be finite and free of divergence of any kind. 5 / 38

6 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions 1 Introduction 2 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions 3 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor 4 Sudakov Double Logarithms of Dijet Processes 5 Summary 6 / 38

7 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions Forward hadron production in pa collisions Single inclusive forward hadrons in Observation pa collisions, at [Dumitru, high energy Jalilian-Marian, 02] s projectile: target: x 1 p s e +y 1 x 2 p s e y 1 valence gluon! The LO single inclusive hadron cross The spin section asymmetry for producing becomes with the largest p at at rapidity forward yrapidity h region, corresponding to! The partons in the projectile (the polarized proton) have very large momentum dσ pa hx 1 LO dz fraction x: dominated by the valence quarks (spin effects are valence effects) = x! The pq d 2 p dy h partons in the target (the unpolarized proton or nucleus) have very small τ z 2 f (x p)f(k )D h/q (z) + x pg(x p) F(k )D h/g (z). f momentum fraction x: dominated by the small-x gluons p = zk, x p = p! Thus spin asymmetry in the forward region could probe both! The transverse spin effect from the valence + quarks in the projectile: Sivers } effect, Collins effect, and etc U(x ) = P exp {ig S dx + T c A c (x +, x ),! The small-x gluon saturation physics in the target d 2 x d 2 y F(k ) = e ik (x y ) S (2) (2π) 2 Y (x, y ). Jan 8, 2013 Tuesday, January 8, 2013 z s ey h (large), τ = zx p and x g = p Zhongbo Kang, LANL z s e y h (small). S (2) Y (x, y ) = 1 TrU(x N c )U (y ) with Y ln 1/xg. Y 7 / 38 4

8 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions Issues with the leading order calculation Why do we need NLO calculations? LO calculation is order of magnitude estimate. Normally, we need to introduce the artificial K factor to fix the normalization. There are large theoretical uncertainties due to renormalization/factorization scale dependence in xf (x) and D(z). Choice of the scale at LO requires information at NLO. In general, higher order in the perturbative series in α s helps to increase the reliability of QCD predictions. NLO results reduce the scale dependence and may distort the shape of the cross section. K = σ LO+σ NLO σ LO is not a good approximation. NLO is vital in terms of establishing the QCD factorization in saturation physics. 8 / 38

9 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions The overall picture C p C f P P h H R A X The QCD factorization formalism for this process reads as, d 3 σ p+a h+x = dz dx dyd 2 p z 2 x ξxfa(x, µ)d h/c(z, µ) [dx ]Sa,c([x Y ])H a c(α s, ξ, [x ]µ). a For UGD, the rapidity divergence cannot be canceled between real and virtual gluon emission due to different restrictions on k. Removing the divergences via renormalization Finite results for hard factors. 9 / 38

10 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions Shock wave approximation Three possible real diagrams: (a) (b) (c) In the high energy limit s, medium size L is much less than the typical coherence time. In the shock wave approximation, only (a) and (b) are taken into account. To compute the medium induced emission, (c) diagram must be included. 10 / 38

11 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions Real diagrams The real contributions in the coordinate space: Computing the real diagrams with a quark (b ) and a gluon (x ) in the final state in the dipole model in the coordinate space: (a) (b) dσ qa qgx = α SC Fδ(p + k + d 3 k 1d 3 1 k + 2 ) k 2 with u = x b and v = (1 ξ)x + ξb. (c) (d) d 2 x (2π) 2 d 2 x (2π) 2 d 2 b (2π) 2 d 2 b (2π) 2 e ik 1 (x x ) e ik 2 (b b ) ψαβ(u λ )ψαβ(u λ ) λαβ [ S (6) Y (b, x, b, x ) + S (2) Y (v, v ) S (3) Y (b, x, v ) S (3) Y (v, x, b ) ], 11 / 38

12 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions The real contributions in the coordinate space Computing the real diagrams with a quark (b ) and a gluon (x ) in the final state in the dipole model in the coordinate space: [G. Chirilli, BX and F. Yuan, 11;12] (a) (b) (c) S (6) Y (b, x, b, x ) = 1 ( Tr U(b )U (b )T d T c) [ ] cd W(x )W (x ) C FN c S (3) Y (b, x, v ) = 1 C FN c Tr (d) ( U(b )T d U (v )T c) W cd (x ). Y Y, By integrating over the gluon momentum, we identify x to x which simplifies S (6) Y (b, x, b, x ) to S (2) (b, b ). ] S (3) Y (b, x, v ) = Nc S (2) Y (b, v ) [ 2C F S (4) Y (b, x, v ) 1 N 2 c 12 / 38

13 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions The real contributions in the momentum space By integrating over the gluon (k + 1, k 1 ), we can cast the real contribution into α s dz 1 2π 2 z D h/q(z) dξ 1 + {C ξ2 2 τ/z 1 ξ xq(x) F d 2 k g I(k, k g ) } + Nc d 2 k g d 2 k g1 J (k, k g, k g1 ), 2 where x = τ/zξ and I and J are defined as I(k, k g ) = [ k k g F(k g ) (k k g ) k ] 2 ξk g, 2 (k ξk g ) 2 J (k, k g, k g1 ) = [ ] F(k g )δ (2) 2(k ξk g ) (k k g1 ) (k g1 k g ) G(k g, k g1 ) (k ξk g ) 2 (k k g1 ) 2 with G(k, l ) = Three types of divergences: d 2 x d 2 y d 2 b (2π) 4 e ik (x b ) il (b y ) S (4) Y (x, b, y ). ξ 1 Rapidity divergence. y = 1 l+ ln 2 l k g k Collinear divergence associated with parton distributions. k g k /ξ Collinear divergence associated with fragmentation functions. 13 / 38

14 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions The virtual contributions in the momentum space Now consider the virtual contribution 2α sc F [ αs (a) d 2 v d 2 v d 2 u (2π) 2 (2π) 2 (2π) 2 e ik (v v ) ψαβ(u λ )ψαβ(u λ ) λαβ S (2) Y (v, v ) S (3) 2π 2 {C F dz z 2 D h/q(z)x pq(x p) ] Y (b, x, v ) 1 d 2 q I(q, k ) + Nc 2 0 (b) dξ 1 + ξ2 1 ξ } d 2 q d 2 k g1 J (q, k, k g1 ). Three types of divergences: ξ 1 Rapidity divergence. Collinear divergence associated with parton distributions and fragmentation functions. 14 / 38

15 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions The subtraction of the rapidity divergence We remove the rapidity divergence from the real and virtual diagrams by the following subtraction: 1 F(k ) = F (0) (k ) αsnc dξ d 2 x d 2 y d 2 b e ik (x y ) 2π ξ (2π) 2 (x y ) 2 [ ] S (2) (x (x b ) 2 (y b ) 2, y ) S (4) (x, b, y ). Comments: The rapidity divergence is an artifact that both the projectile and targets are put on the light cone in the high energy limit (s ). This divergence removing procedure is similar to the renormalization of parton distribution and fragmentation function in collinear factorization. 1+ξ Splitting functions becomes 2 (1 ξ) + after the subtraction. Rapidity divergence disappears when the k is integrated. Unique feature of unintegrated gluon distributions. 15 / 38

16 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions The subtraction of the rapidity divergence We remove the rapidity divergence from the real and virtual diagrams by the following subtraction: 1 F(k ) = F (0) (k ) αsnc dξ d 2 x d 2 y d 2 b e ik (x y ) 2π ξ (2π) 2 (x y ) 2 [ ] S (2) (x (x b ) 2 (y b ) 2, y ) S (4) (x, b, y ). This is equivalent to the Balitsky-Kovchegov equation: Y S(2) Y (x, y ) = αsnc d 2 b (x y ) 2 [ ] S (2) 2π 2 (x b ) 2 (y b ) 2 Y (x, y ) S (4) Y (x, b, y ). Recall that F(k ) = d 2 x d 2 y e ik (x y ) S (2) (x (2π) 2, y ). The soft gluon is emitted from the projectile proton with momentum (1 ξ)p +, and it is easy to see that the rapidity of this soft gluon goes to when ξ 1 since the radiated gluon is now in the region kg k g +. As a matter of fact, this soft gluon can be regarded as collinear to the target nucleus which is moving on the backward light cone with the rapidity close to and P A P+ A. Renormalize the soft gluon into the gluon distribution function of the target nucleus through the BK evolution equation. 16 / 38

17 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions The subtraction of the collinear divergence Let us take the following integral as an example: d 2 k g I 1(k ) = (2π) F(k 1 g ) 2 (k k g ), 2 = 1 ( d 2 x d 2 y e ik r S (2) 4π (2π) 2 Y (x, y ) 1ˆɛ ) + ln c2 0, µ 2 r 2 where c 0 = 2e γ E, γ E is the Euler constant and r = x y. Use dimensional regularization (D = 4 2ɛ) and the MS subtraction scheme ( 1ˆɛ = 1 γe + ln 4π). ɛ d 2 kg (2π) 2 µ 2ɛ d 2 2ɛ kg where µ is the renormalization scale dependence coming from (2π) 2 2ɛ the strong coupling g. The terms proportional to the collinear divergence should be factorized either into parton 1ˆɛ distribution functions or fragmentation functions. 17 / 38

18 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions The subtraction of the collinear divergence Remove the collinear singularities by redefining the quark distribution and the quark fragmentation function as follows q(x, µ) = q (0) (x) 1ˆɛ α 1 ( ) s(µ) dξ x 2π ξ CFPqq(ξ)q, ξ with Comments: D h/q (z, µ) = D (0) α s(µ) h/q (z) 1ˆɛ 2π x 1 P qq(ξ) = 1 + ξ2 + 3 δ(1 ξ). (1 ξ) + 2 }{{}}{{} Real Sub Virtual Sub z dξ ξ CFPqq(ξ)D h/q ( ) z, ξ Reproducing the DGLAP equation for the quark channel. Other channels will complete the full equation. The emitted gluon is collinear to the initial state quark Renormalization of the parton distribution. The emitted gluon is collinear to the final state quark Renormalization of the fragmentation function. 18 / 38

19 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions Hard Factors For the q q channel, the factorization formula can be written as d 3 σ p+a h+x dz dx dyd 2 = p z 2 x ξxq(x, µ)d h/q(z, µ) d 2 } b + (x, b, y ) αs (2π) 2 S(4) Y with H (0) 2qq = e ik r δ(1 ξ) and H (1) 2qq = C F P qq(ξ) ln c 2 0 r 2 µ2 H (1) 4qq = 4πN ce ik r where Ĩ 21 = d 2 x d 2 y { (2π) 2 S (2) Y (x, y ) 2π H(1) 4qq (e ik r + 1ξ 2 e i k ξ r ) 3C F δ(1 ξ)e ik r ln [ H (0) 2qq + αs ] 2π H(1) 2qq r 2 k2 ( (2C F N c) e ik r 1 + ( ξ2 1 + ξ 2 ) ) ln (1 ξ) 2 Ĩ 21 (1 ξ) + 1 ξ + { 1 ξ i e ξ k (x b ) 1 + ξ 2 1 x b (1 ξ) + ξ (x b ) 2 y b (y b ) 2 [ 1 δ(1 ξ) dξ 1 + ξ 2 e i(1 ξ )k (y b ) 0 (1 ξ ) + (b y ) 2 δ (2) (b y ) d 2 r [ ] } d 2 b {e i(1 ξ)k b b (ξb r ) π b 2 (ξb r ) 2 1 b 2 + e ik b 1 b 2. c 2 0 e ik r r 2 19 / 38 ]

20 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions Physical interpretation Achieve 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. In terms of resummation, terms α s(α s ln k 2 ) n and α s(α s ln 1/x) n are resummed. The other three channels g g, g q and q g follow accordingly. 20 / 38

21 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions xg Numerical implementation of the NLO result Consistent implementation should include all the NLO α s corrections. NLO parton distributions. (Choose your favorite one, CTEQ or MSTW) NLO fragmentation function. (DSS or others.) Use NLO hard factors. Use the one-loop approximation for the running coupling rcbk evolution equation for the dipole gluon distribution [Balitsky, Chirilli, 08; Kovchegov, Weigert, 07]. [Stasto, Xiao, Zaslavsky, 13] STAR η = 4 [ ] d 3 N dηd 2 p GeV LO NLO data [ ] d 3 N dηd 2 p GeV 2 [ ] d 3 N dηd 2 p GeV GBW S (2) xg = exp [ ( r2 4 Q2 x0 ) λ] BK 10 3 S (2) xg [ = exp r2 4 0( Q2 x0 ) ( )] λ ln e + 1 xg Λr MV rcbk [ ] d 3 N dηd 2 p GeV LO NLO (fixed) NLO (running) LHC at η = RHIC at η = p [GeV] p [GeV] p [GeV] (µ/p ) 2 21 / 38

22 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor 1 Introduction 2 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions 3 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor 4 Sudakov Double Logarithms of Dijet Processes 5 Summary 22 / 38

23 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor Motivation for Higgs production in pa collisions Consider the forward Higgs productions in pa collisions: p + A H(M, k ) + X Comments: One-loop calculation allows us to do resummation together with multiple scatterings. The one-loop calculation between Higgs productions and dijet productions are very similar since both M H k and M J P k. There is no linearly polarized gluon distribution. The simplest example to demonstrate the factorization. Connection with TMD factorization? 23 / 38

24 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor LO calculation [A. Mueller, BX and F. Yuan, 12, 13] The effective Lagrangian: L eff = 1 4 g φφf a µνf aµν dσ (LO) = σ dyd 2 0 k d 2 x d 2 x (2π) 2 e ik (x x ) xg p(x)sy WW (x, x ), σ 0 = g 2 φ/g 2 32(1 ɛ) with ɛ = (D 4)/2; SY WW (x, y ) = Tr [ β U(x )U (y ) β U(y )U (x ) Y Only initial state interaction is present. WW gluon distribution. For AA collisions, there exists the true k t factorization. ] 24 / 38

25 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor Some Technical Details [A. Mueller, BX and F. Yuan, 12, 13] Typical diagrams: b x x x v b v b (a) (b1) (b2) High energy limit s and M 2 k 2. Use dimensional regularization. Power counting analysis: take the leading power contribution in terms of k2 M / 38

26 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor Some Technical Details [A. Mueller, BX and F. Yuan, 12, 13] b x x x v b v b (a) (b1) (b2) The phase space (l +, l, l ) of the radiated gluon can be divided into three regions: (a) gluon is collinear to the incoming proton. DGLAP evolution. Subtraction of the collinear divergence and choose µ 2 = c2 0 : R 2 1 ( ) ɛ SWW (x, y )P gg(ξ) xg x, c2 0 R 2 with ξ = l+ p + (b) gluon is collinear to the incoming nucleus. Small-x evolution. Subtraction of the rapidity divergence: non-linear small-x evolution equation 1 dξ xg p(x) K DMMX S WW (x, y ) ξ (c) gluon is soft. Sudakov logarithms / 38

27 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor Separation of the small-x logarithm and Sudakov logarithms b x x x v b v b (a) (b1) (b2) Consider the kinematic constraint for real emission before taking s, and note that x gx ps = M 2 1 ( ) dξ xps ξ = ln = ln 1xg + ln M2. l 2 xps Now we can take s and x g 0, but keep x gx ps = M 2. The Sudakov contribution gives l 2 ln M2 l 2 µ 2ɛ d 2 2ɛ l (2π) 2 2ɛ e il R 1 l 2 [ = 1 1 4π ɛ 1 M2 ln 2 ɛ µ + 1 (ln M2 2 2 µ 2 ) l 2 ( ln M2 R 2 c 2 0 ) 2 ] π / 38

28 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor Some Technical Details Subtraction of the UV-divergence: (Color charge renormalization) ( α s π Ncβ0 1 ) + ln M2 with β ɛ UV µ 2 0 = Nf. 6N c Real diagrams [ + αs π Nc 1 ɛ 1 M2 ln 2 ɛ µ (ln M2 µ 2 ) 2 1 ( ) ln M2 R 2 2 ] π2. 2 c Virtual graphs α s π Nc [ 1ɛ 1ɛ M2 + ln 2 µ 1 ] M ln2 µ + π π2 M2 + β0 ln 12 µ 2 All divergences now cancel and the remaining contribution is ( α s π Nc 1 ) M 2 R 2 2 ln2 + β c 2 0 ln M2 R 2 + π2. 0 c / 38

29 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor Sudakov factor Final results: At one-loop order: R x x dσ (LO+one-loop) d 2 x d 2 x dyd 2 k M = σ 0 k (2π) 2 [ 1 + αs π Nc Collins-Soper-Sterman resummation: dσ (resum) dyd 2 k M = σ 0 k e ik R xg p(x, µ 2 = c 2 0/R 2 )S WW Y=ln 1/x g (x, x ) ( 1 M 2 R 2 2 ln2 + β c 2 0 ln M2 R 2 + π2 0 c d 2 x d 2 x e ik R e S 2 sud(m,r 2 ) S WW (2π) 2 [ ] 1 + αs π 2 π 2 Nc, x pg p(x p, µ 2 = c2 0 ) R 2 where the Sudakov form factor contains all order resummation A = i=1 S sud(m 2, R 2 ) = M 2 c 2 0 /R2 dµ 2 µ 2 [A ln M2 µ 2 + B A ( (i) α s ) i, we find A (1) = N π c and B (1) = β 0N c. Difference: Diagrams which are not included. ]. )]. Y=ln 1/x g (x, x ) 29 / 38

30 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor Probabilistic interpretation of the Sudakov double logarithms The gluon wave function: l k q Now in momentum space, the Higgs transverse momentum is fixed to be k q M. The energetic gluon is annihilated to create the color neutral and heavy Higgs particle, thus, other gluons in its wave function are forced to be produced. The Sudakov factor S is just the probability of those gluons having transverse momentum much less than k. To compute this probability at one loop order, one needs to use the probability conservation S + T = 1 where T is the probability of large transverse momentum radiation T = αsnc π M 2 k 2 dl 2 l 2 1 l 2 /M2 dξ ξ = αsnc 2π ln2 M 2 k 2 Momentum conservation δ(l q + k ) resummation in coordinate space. 30 / 38

31 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor Lesson from the one-loop calculation for Higgs productions dσ (resum) dyd 2 k M = σ 0 k d 2 x d 2 x e ik R e S 2 sud(m,r 2 ) S WW (2π) 2 x pg p(x p, µ 2 = c2 0 ) R 2 [ ] 1 + αs π 2 π 2 Nc, Y=ln 1/x g (x, x ) Four independent (at LL level) renormalization equation. Unified description of the CSS and small-x evolution? TMD and UGD share the same operator definition. Define S WW (R, M; x g) = Ce S sud(m 2,R 2 ) S WW Y=ln 1/x g (x, x ) dσ (resum) dyd 2 k M = σ 0 k d 2 x d 2 x e ik R x pg p(x p, µ 2 = c2 0 )S WW (R (2π) 2 R 2, M; x g) 31 / 38

32 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor Lesson from the one-loop calculation for Higgs productions Extending the Higgs calculation to other processes: Gedanken Experiment ( αsnc π 1 2 ln2 Q2 R 2 c β 0 ln Q2 R 2 ) c αsnc π ( 1 2 ln2 Q2 R 2 c 2 0 Heavy quark pair production: Q 2 = Mq 2 Dijet production: Q 2 = P 2 ) + Deep inelastic scattering of a current j = 1 4 F2 µν off a large nucleus: No ( 1)ɛ ɛ 2! Deep inelastic scattering of an electromagnetic current off a large nucleus: Heavy quark pair with total transverse momentum k. Dead zone effect P = l 2 + ξ2 Mq 2 T = αsnc π M 2 q dl 2 l /M q k 2 l 2 l 2 /M2 q Dijet productions. Dead zone effect and jet collinear singularity. P = 1 (l ξp ) 2 dξ ξ = 1 α sn c M2 2 2π ln2 q k 2 32 / 38

33 Sudakov Double Logarithms of Dijet Processes 1 Introduction 2 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions 3 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor 4 Sudakov Double Logarithms of Dijet Processes 5 Summary 33 / 38

34 Sudakov Double Logarithms of Dijet Processes Two Different Gluon Distributions [F.Dominguez, C.Marquet, BX and F. Yuan, 11] I. Weizsäcker Williams gluon distribution: Gauge Invariant definitions (known from TMDs) xg (1) dξ dξ + = 2 ξ ik ξ (2π) 3 P + eixp Tr P F +i (ξ, ξ )U [+] F +i (0)U [+] P. II. Color Dipole gluon distributions: Gauge Invariant definitions 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 [+] ξ The WW gluon distribution is the conventional gluon distributions. Quadrupole The dipole gluon distribution has no such interpretation. Dipole Two topologically different gauge invariant definitions. 34 / 38

35 Sudakov Double Logarithms of Dijet Processes Dijet processes Using various dijet processes to distinguish these gluon distributions. Forward di-hadron correlations in d+au collisions at RHIC! Coincidence probability at measured by STAR Coll. a trigger φ associated Comments:! Away peak is present in p+p coll. What happens at one loop order for back-to-back dijet processes? (P 1 k 2 1 k 2 q k 1 + k 2 ) Can we prove the factorization for dijet productions? Connection between k t factorization and TMD factorization?!"=0 ln ] [ 1 n (near side)!"=# x resummation vs Sudakov (CSS) αscr 2π ln2 Q 2 1 Q (away side) 2 2 Small-x [ α sn c 2π Consistently resum both types of logarithms at the same time? p+p CP( φ) = 1 N trig d ] n resummation.! Ab 35 / 38

36 Sudakov Double Logarithms of Dijet Processes Sudakov factor for dijet productions in pa collisions and DIS Forward di-hadron corre Consider the dijet productions in pa collisions: d+au collisions at R Comments: C q q = Nc 2 C qγ = Nc 2 + CF 2 C g q q = N c dσ dy 1dy 2dP 2 d2 k H(P 2 )! Coincidence probability at trigger φ associated d 2 x d 2 y e ik R e S sud(p,r )! W Away xa (x, peak y ). is presen Problems get harder due to presence of both initial and final state emissions. For back-to-back dijet processes, M 2 J P 2 k 2 S sud = αsc 2π ln2 P 2 R 2 c 2 0 The color factor C is process dependent. with R 1 k. Empirical formula for C: C = C i i, where Ci is the Casimir color factor of 2 the incoming particles. C i = C F for incoming quarks, C i = N c for gluons.!"=0 (near side) (aw p+p 36 / 38

37 Summary 1 Introduction 2 One-loop Calculation for Single Inclusive Hadron Productions in pa collisions 3 One-loop Calculation for Higgs Productions in pa Collisions and Sudakov Factor 4 Sudakov Double Logarithms of Dijet Processes 5 Summary 37 / 38

38 Summary Summary One-loop factorization for single hadron productions in pa collisions. Including the Sudakov resummation in the small-x formalism. DIS dijet provides direct information of the WW gluon distributions. Perfect for testing saturation physics calculation, and ideal measurement for EIC and LHeC. Phenomenological application of the Sudakov factor in dijets will play an important role. Smoking gun for saturation phenomenon: Dijet in pa collision at the LHC. What about the E-loss and P broadening at NLO for finite but large size L > t c medium? [Wu, 11; Liou, Mueller and Wu, 13] 38 / 38