Braz. J. Phys. vol.38 número3B

526 Brazilian Journal of Physics, vol. 38, no. 3B, September, 2008

Confronting Color Dipole and Intrinsic k

Approaches in D–Y Dilepton Production

Marcos Andr´e Betemps Conjunto Agrot´ecnico Visconde da Graa¸,

Universidade Federal de Pelotas, Caixa Postal 460, 96060-290 Pelotas,

Rio Grande do Sul, Brazil E-mail: marcos.betemps@ufpel.edu.br

Maria Beatriz Gay Ducati, and Emmanuel Gr¨ave de Oliveira Instituto de F´ısica,

Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970 Porto Alegre,

Rio Grande do Sul, Brazil E-mail: beatriz.gay@ufrgs.br, E-mail: emmanuel.deoliveira@ufrgs.br

(Received on 10 July, 2008)

We study the Drell-Yan dilepton production in proton-nucleus collisions at RHIC energies. We use two different approaches: the usual intrinsic transverse momentum approach at NLO in the infinitum momentum frame; and the color dipole in the target rest frame. We compare both formalisms at backward rapidities (proton as a target). At forward rapidities, we use earlier results considering the nucleus in a Color Glass Condensate phase. We show qualitative agreement between the two formalisms through the nuclear modification ratio as a function of both rapidity and transverse momentum and that low-mass dileptons are relevant observables to probe nuclear effects.

Keywords: Drell-Yan process; Dilepton production; Nuclear effects; Dipole frame; Infinite momentum frame

In a recent work[1], Drell-Yan dilepton production at back-ward rapidities in hadrons collisions was studied in the rest frame of the target, i. e., in the color dipole approach. In this work, we compare these previous results with results obtained used well know intrinsickT approach in the infinitum

momen-tum frame [2, 3]. In this frame, the process is understood as the combination of two partons to create a virtual boson that subsequently splits in the dilepton. For dilepton massMmuch smaller than the Z mass, the dominant process includes only the photon as the virtual boson. The dilepton production is of particular interest since dileptons do not interact strongly and therefore carry information about initial state effects.

The kinematics used here are described now. Partons and hadrons are taken as massless, the momenta of hadronsAand BarePAandPB, and the momenta of partons are pA=xAPA

and pB=xBPB (for now, partons are collinear to hadrons).

The virtual photon momentum isqandq2=M2is the squared dilepton mass. The Mandelstam variables are given by: s= 2PA·PB,t= (q−PA)2, and u= (q−PB)2. We also define x1=2PB·q/s, x2=2PA·q/s, and the photon rapidity y=

1

2ln(x1/x2). It can be showed that: x1,2=

s

M2_+p2 T

s e

±y_{, (1)}

in whichpT is the photon (also dilepton) transverse

momen-tum.

In IMF collinear approximation, partons are considered collinear to hadrons, without intrinsic transverse momentum. Using this framework at leading order, experimental results of transverse momentum distribution cannot be reproduced: although valid for largepT, collinear NLOpT distribution

di-verges atpT =0 and is not in agreement with experiments for

smallpT. We consider then partonic intrinsic transverse

mo-mentum, i. e., partons are not collinear to hadrons [2, 3]. The partonic distributions are change as the following prescrip-tion:

f(x)dx_→ f(x)h(k~T)dxd2kT. (2)

In this paper, we considerh(k~T) =_2π1_b2exp

³_k2

T 2b2

´

. Therefore, the cross section is given by[4, 5]:

σS(s,M2,y,pT) =h′(p2T) dσ

dM2_dy (3)

+

Z

d2qTσP(s,M2,q2T)[h′((~pT−~qT)2)−h′(p2T)].

In the above expression, it is included the NLO collinear dou-ble differential cross section:

dσ

dM2_dy =

ˆ

σ0 s

Z 1

0

dxAdxBdzδ(xAxBz−τ)δ

µ y₋1

2ln xA xB

¶

×

½

Pqq¯(xA,xB,M2)

·

δ(1−z) +αs(M

2₎

2π Dq(z)

¸

+ Pqg+gq(xA,xB,M2)

·

αs(M2)

2π Dg(z)

¸¾ .

Using the modified minimal subtraction scheme (MS),Dq(z)

andDg(z)are given e.g. in Ref. [6] (CF=4/3,TR=1/2).

Marcos Andr´e Betemps et al. 527

σP(s,M2,p2T) =

1

π2

α2_α s M2_sˆ2

Z 1

xAmin

dxA xBxA xA−x1

½

Pqq¯(xA,xB,M2)

8 27

2M2sˆ+uˆ2+tˆ2 ˆ tuˆ +Pqg(xA,xB,M2)

1 9

2M2uˆ+sˆ2+tˆ2

−sˆtˆ +Pgq(xA,xB,M

2₎1

9

2M2tˆ+sˆ2+uˆ2

−sˆuˆ ¾

in whichxAmin is given by(x1−τ)/(1−x2), xB= (xAx2−

τ)/(xa−x1), and

Pqq¯(xA,xB) =

∑

e_q2(fq(xA)fq¯(xB) +q¯↔q) (4)

Pqg(xA,xB) =

∑

e2_q(fq(xA) +fq¯(xA))fg(xB) (5)

Pgq(xA,xB) =

∑

e2_qfg(xA) (fq(xB) +fq¯(xB)). (6)

0.70 0.80 0.90 1.00 1.10

1e-05 0.0001 0.001 0.01 0.1 1

R

2

=

F

A

/

F

p ₂

x

EKS nDS

FIG. 1: RatioRA_F2=F₂A/F₂pusing EKS and nDS parameterizations.

In the color dipole approach, Drell–Yan dilepton production is studied in the rest frame of the target. Diagrams considered now look like projectile quarkbremsstrahlungon the target color field. So, the projectile emits a quark (or antiquark), this parton fluctuates in a state of quark–photon and interacts with the color field of the target, and the photon is freed to split in a dilepton. The color dipole cross section arises as interfer-ence of the diagram in which the quark first interacts with the target with the diagram in which the quark first fluctuates in the quark–photon state. This result was first stated in [7] and derived in detail in [4].

In [1], the color dipole approach was used to study dileptons produced at backward rapidities, so the proton was considered as the target and the nucleus as the projectile. In this case, the color dipole approach is phenomenologically valid for small x1[4] – very backward rapidities. The cross section is written

as:

dσDY dM2_dyd2_p

T

= α

2 em

6π3_M2

Z ∞

0

dρW(x2,ρ,pT)σdip(x1,ρ), (7)

in whichρis the dipole transverse size. In this case,x2is the

projectile momentum fraction carried by the virtual photon. The weight function W(x2,ρ,pT) contains the

nu-clear structure functionFA

2(x2/α,M2) =∑qe2q[x fqA(x,M2) + x f_qA_¯(x,M2)]:

W(x2,ρ,pT) =

Z 1

x2

dα α2F

A 2(

x2

α,M

2₎½_[m2

qα4+2M2(1−α)2]

· 1

p2_T+η2T1(ρ)−

1 4ηT2(ρ)

¸

+ [1+ (1−α)2] ·

ηpT

p2_T+η2T3(ρ)−

1 2T1(ρ) +

η

4T2(ρ) ¸¾

,

in whichη2_{= (1}

−α)M2+α2_m2

qandmq=0,2 GeV is the

quark mass. The functionsTiare given by:

T1(ρ) =

ρ αJ0

³p_Tρ

α

´ K0

³ηρ

α

´

(8)

T2(ρ) =

ρ2

α2J0

³p_Tρ

α

´ K1

³ηρ

α

´

(9) T3(ρ) =

ρ αJ1

³p_Tρ

α

´ K1

³ηρ

α

´

, (10)

in whichJn(x)is the Bessel function of first kind andKn(x)is

the modified Bessel function of second kind.

We use the model introduced by Golec-Biernat e W¨usthoff (GBW)[8] for the dipole cross section:

σdip(x,r) =σ0

· 1−exp

µ

− r

2_Q2 0

4(x/x0)λ

¶¸

, (11)

in whichQ2₀=1 GeV2_{and there are three fitted parameters:}

528 Brazilian Journal of Physics, vol. 38, no. 3B, September, 2008

-2.4 -2.0 -1.6 -1.2

0 2 4 6 0.85

0.90 0.95 1.00 1.05

R

(a) Dipole EKS

-2.4 -2.0 -1.6 -1.2

0 2 4 6 0.85

0.90 0.95 1.00 1.05

(a) Dipole EKS

(b) IMF EKS

-2.4 -2.0 -1.6 -1.2

0 2 4 6 0.85

0.90 0.95 1.00 1.05

R

(c) Dipole nDS

(b) IMF EKS

-2.4 -2.0 -1.6 -1.2

0 2 4 6 0.85

0.90 0.95 1.00 1.05

(c) Dipole nDS

(d) IMF nDS

R

y

p

(GeV)

R

y

p

(GeV)

FIG. 2: FactorRpAat RHIC energies as a function of rapidity and transverse momentum.

0,288. It is important to highlight that the present dipole cross section includes saturation effects, not included in the IMF approach.

We use GRV98[9, 10] as the parton distribution function (PDF) of free protons. Two parameterizations of the nuclear PDFs are used: EKS[11–13] and nDS[14]. EKS parameter-ization gives the nPDF as the free proton PDF multiplied by a factor: fA

q(x,Q) =RAq(x,Q)f p

q(x,Q), while nDS gives the

nPDF as a convolution of the free proton PDF and a weight function:

f_qA(x,Q) =

Z A

x dy

yWq(y,A)f p q

µ_x y,Q

¶

. (12)

EKS parameterization is available only at leading order, while nDS is also at NLO. In Fig. 1, both parameterizations are compared at leading order calculating RA_F

2 =F A 2/F

p 2. We

would like to stress that nDS parameterization presents lower ratio for most values of x studied, in particular for EMC

(0.3<x<0.8) and shadowing (x<0.1) effects – other re-gions correspond to anti-shadowing (0.1<x<0.3) and Fermi motion (0.8<x) effects.

We use dilepton mass of M =6.5 GeV, RHIC energies (√s=200 GeV), gold nucleus (A=196.97), and intrinsic kT standard deviation ofb=0.48 GeV. In Fig. 2, the nuclear

modification factor:

RpA=

dσ(pA) d p2

TdydM

Á

A dσ(pp) d p2

TdydM

(13)

is calculated using both formalisms described earlier in the region of backward rapidity. EKS and nDS parameteriza-tions give qualitatively similar results, showing that both ap-proaches roughly give the same dependence ofRpAon the

nu-clear effects. The main difference is a step nearpT=2.5 GeV

Marcos Andr´e Betemps et al. 529

0.75 0.80 0.85 0.90 0.95 1.00 1.05

-2.00 -1.00 0.00 1.00 2.00

R

y

pT=4 GeV

pT=6 GeV

Dipole pT=2 GeV

pT=4 GeV

pT=6 GeV

Dipole (CGC) pT=2 GeV

pT=4 GeV

pT=6 GeV

FIG. 3: FactorRpAcalculated using three different approaches:

infi-nite momentum frame, color dipole at backward rapidities, and color dipole at forward rapidities considering the nucleus as a CGC [15].

is dominant in one region of pT. This result shows that

sat-uration effects included in the dipole approach are not very effective in changing the nuclear modification ratio.

The behavior ofRpA is mainly explained by the inclusion

of anti-shadowing effects with decreasingx2 approximately

from 0.64 (y=₋2.6,pT =7) to 0.09 (y=−1.0,pT =0).

Anti-shadowing effects are characterized by an increase in the nuclear cross section, exactly what is seen in Fig. 2. EKS parameterization predicts a smallerRpAthan nDS

parameter-ization in both approaches and nDS parameterparameter-ization shows more sensibility to the approach change, as it is expect, since nDS parameterization is different whether LO or NLO is con-sidered.

In Fig. 3, we combine color dipole results obtained here at backward rapidities, color dipole results obtained in Ref. [15] at forward rapidities considering the nucleus as a CGC phase, and IMF results. EKS parameterization was used when a nu-clear PDF was required. Overall, we see a qualitative agree-ment among the approaches and that the ratioRpAdependence

onpTis consistently obtained: ify<0 (y>0),RpAdecreases

(increases) with a increase ofpT, thanks to the anti-shadowing

(shadowing) effect.

In conclusion, we saw that low-mass dileptons are relevant in probing nuclear effects at RHIC energies. Either at back-ward or forback-ward rapidities, results obtained showed strong de-pendence on the nuclear effects and in the choice of the pa-rameterization. Among the approaches, qualitative agreement was obtained in the nuclear modification factor, showing that saturation effects (considered in the color dipole formalism and in the CGC) do not play a key role in this factor. However, the nuclear modification factor proved to be very sensitive to the introduction of a intrinsic transverse momentum, given the step seen in thepT distribution.

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