- Email: [email protected]
Two particle correlations: Saturation and issues with universality
Nuclear Physics A 854 (2011) 180–186 www.elsevier.com/locate/nuclphysa
Two particle correlations: Saturation and issues with universality Bo-Wen Xiao a,∗ , Feng Yuan a,b a Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States b RIKEN BNL Research Center, Building 510A, Brookhaven National Laboratory, Upton, NY 11973, United States
Received 21 July 2010; received in revised form 29 July 2010; accepted 30 July 2010 Available online 6 August 2010
Abstract We study the universality issue of the transverse momentum dependent parton distributions at small-x, by comparing the initial/final state interaction effects in di-jet correlation in pA collisions with those in deep inelastic lepton–nucleus scattering. We demonstrate the non-universality by performing an explicit calculation in a particular model where the multiple gauge boson exchange contributions are summed up to all orders. In addition, we generalize the model calculation to the CGC formalism, and find the nonuniversality for quark distributions in CGC. © 2010 Elsevier B.V. All rights reserved. Keywords: Non-universality; Color glass condensate; Di-jet production
1. Non-universality of transverse momentum dependent parton distributions at small-x Recently, the non-universality of these distribution functions due to the final/initial state interaction effects has attracted intensive investigations. This eventually leads to a conclusion that a standard transverse momentum dependent factorization breaks down for this process [1]. In Ref. [2], we extend the universality discussions of the transverse momentum dependent parton distributions to the small-x domain, where the kt -dependent distributions have been a common * Corresponding author.
E-mail address: [email protected] (B.-W. Xiao). 0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2010.07.012
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181
Fig. 1. (a) Schematic diagram showing that two partons from the nucleon projectile and the nucleus target collide and produce two jets in the final state, where the intrinsic transverse momentum q⊥ from nucleus dominates the imbalance between the two jets; (b) illustration of initial/final state interactions which may affect the transverse momentum dependent quark distribution from the nucleus in this process; (c) as a comparison, only the final state interaction effect is present in the deep inelastic lepton–nucleus (nucleon) scattering.
practice to describe the relevant physics phenomena [3]. We expect the non-universality for these objects as well. The process that we focus on is described as follows: as schematically shown in Fig. 1(a), two partons from the nucleon projectile and nucleus target collide with each other, and produce two jets in the final state, p + A → Jet1 + Jet2 + X,
(1)
where the transverse momenta of these two jets are similar in size but opposite to each other in direction. We take the partonic channel qq → qq (as illustrated in Fig. 1(b)) as an example to show the initial/final state interaction effects and calculate the quark distribution in di-jet correlation, and compare with that in the deep inelastic scattering process (as shown in Fig. 1(c)). At small-x, quark distribution is dominated by gluon splitting, and can be calculated from the relevant Feynman diagrams [4–6]. For the purpose of our calculation, we employ an Abelian model of Refs. [1,7,8]. It is a scalar QED model with Abelian massive gluons with a mass λ. We construct the model in such a way that the scalar quarks are generated by the Abelian gluon splitting which is the dominant contribution at small-x. The associated quark distribution in deep inelastic scattering process in this model has been calculated in [8–10]. Since we are interested in studying the final state interaction effects on the parton distribution of the nucleus, for convenience, we choose the projectile as a single scalar quark with charge g2 , which differs from the charge of the scalar quark from the target nucleus g1 . In addition, we assume that the Abelian gluons attached to the target nucleus has an effective coupling g. All the partons in this calculation is set to be scalars with a mass m. The coupling g2 being different from g1 is to show the dependence of the parton distribution on the initial/final state interactions associated with the incoming parton. If the dependence on g2 remains for the nucleus parton distributions, they are not universal [1,7]. We use the small-x approximation as well as the eikonal approximation, and calculate the associated amplitude up to the third order. We sum up all order multi-gauge boson exchange contributions by going to the coordinate space via the Fourier transform [8]. We find that the total amplitude can be exponentiated into the form (2) A(tot) (R, r) = iV (r⊥ ) 1 − eigg1 [G(R⊥ +r⊥ )−G(R⊥ )] e−igg2 G(R⊥ ) , where G(R⊥ ) = K0 (λR⊥ )/2π and V (r⊥ ) = K0 (Mr⊥ )/2π with M 2 = 2xP + p − + m2 . Therefore, the all order result of the quark distribution reads as
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2 x dp − d 2 k⊥ + − 2 (tot) 4P p A (k, p) − 2 4 p (2π) 32π +2 xP − − 2 (2) = d r⊥ d 2 r⊥ δ R⊥ + r⊥ − R⊥ − r⊥ dp p d 2 R⊥ d 2 R⊥ 8π 4 −igg (G(R )−G(R )) ⊥ 2 ⊥ e × e−iq⊥ ·(r⊥ −r⊥ ) V (r⊥ )V r⊥ × 1 − eigg1 [G(R⊥ +r⊥ )−G(R⊥ )] 1 − e−igg1 [G(R⊥ +r⊥ )−G(R⊥ )] .
q(x, ˜ q⊥ ) =
(3)
This transverse momentum dependent quark distribution is clearly different from that calculated in deep inelastic scattering in the same model [8–10] due to the g2 dependent phase. In other words, this distribution is not universal and the standard kT -factorization breaks down. This essentially confirms the conclusion in Ref. [1]. In addition, our calculation demonstrates that the anomalous terms (g2 dependent terms) which breaks the universality and standard kt -factorization can be resummed into Wilson lines as shown above in the small-x limit. It is interesting to notice that the g2 dependence disappears after the integration over the transverse momentum. This is consistent with the universality for the integrated parton distributions [1,7,10]. It has been argued that the light-cone gauge may simplify the factorization property for the hard scattering processes. For example, if we choose the advanced boundary condition for the gauge potential in light-cone gauge, the wave function of hadrons contain the final state interaction effects [10,11]. However, as we showed in the above calculations, this does not help to resolve the g2 -dependence in the quark distribution in the di-jet correlation in hadronic process due to the presence of both initial and final state interactions. 2. Non-universality of CGC parton distributions In this section, we study the connection between the existing results and CGC quark distribution. We show that the non-universality issue also persists in the CGC formalism by comparing the quark distribution probed in DIS to the one involved in di-jet production. 2.1. Quark distributions in DIS In scalar QED model, the TMD quark distribution in DIS reads [8] xP +2 − − 2 p d r⊥ d 2 r⊥ dp d 2 R⊥ d 2 R⊥ q˜ DIS (x, q⊥ ) = 8π 4 iq ·(R −R ) e ⊥ ⊥ ⊥ V (r⊥ )V r⊥ × δ (2) R⊥ + r⊥ − R⊥ − r⊥ × 1 − eigg1 [G(R⊥ +r⊥ )−G(R⊥ )] 1 − e−igg1 [G(R⊥ +r⊥ )−G(R⊥ )] .
(4)
In arriving above formula, we have assumed the target hadron (nucleus) to be a point particle. In order to reproduce the quark distribution in saturation physics, we need to relax the point particle approximation. First of all, we assume that the target hadron has a color charge distribution ρa (z− , z⊥ ) and perform a replacement e−igg1 [G(x⊥ )] ⇒ U (x⊥ ), with − a − 2 (5) U (x⊥ ) = T exp −igg1 dz d z⊥ G(x⊥ − z⊥ )ρa z , z⊥ t . It is straightforward to see that U (x⊥ ) ⇒ e−igg1 [G(x⊥ )] if we set ρa (z− , z⊥ )t a = δ(z− )δ (2) (z⊥ ).
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The second step is to average over the color charge ρa (z− , z⊥ ) with the Gaussian distribution W [ρ]. The variance μ2 (z− ) of the charge distribution represents the density of color sources per unit volume. Thus one finds (see Refs. [12–14])
2 /4 , (6) eigg1 [G(R⊥ +r⊥ )−G(R⊥ )] ⇒ Tr U (R⊥ )U † (R⊥ + r⊥ ) ρ Nc exp −Q2s r⊥ with Q2s =
μ2s 2π
ln
1 2 λ2 r⊥
and μ2s =
g 2 g12 a 2 t ta
dx − μ2 (z− ).
The next step is to use fermionic quark splitting kernel instead of the scalar quark splitting 1 kernel. Thus we replace V (r⊥ ) by 2π 2K1 (Mr⊥ ) where the factor of 2 comes from the fact that fermionic quark has two different helicities. At the end of the day, once we change the variable to y = 2xP + p − , we can cast the total quark distribution into x q˜ DIS (x, q⊥ ) √ Nc ) √ −iq⊥ ·(r⊥ −r⊥ e r⊥ K0 ( yr⊥ ) · r⊥ K0 yr⊥ dy d 2 R⊥ d 2 r⊥ d 2 r⊥ = 6 32π
)2 Q2 r 2 Q2 r 2 Q2 (r⊥ − r⊥ − exp − s ⊥ − exp − s ⊥ , × 1 + exp − s 4 4 4 which gives dx q˜ DIS (x, q⊥ ) Nc Q2s = , 2 2 d 2 R⊥ 12π 4 q⊥ q⊥ Q2s dx q˜ DIS (x, q⊥ ) Nc = . 2 d R⊥ 4π 4 q 2 Q2s
(7)
(8)
⊥
These formulas agree with the results derived in saturation physics for the quark distribution of a large nucleus in DIS (see e.g., Eqs. (27)–(29) of Ref. [4]).1 Furthermore, by transforming to the momentum space and defining a normalized unintegrated gluon distribution F (k⊥ , Qs ) as 2 2 k d r⊥ −ik⊥ ·r⊥ 1
1 † F (k⊥ , Qs ) = e Tr U (R )U (R + r ) exp − ⊥2 , (9) ⊥ ⊥ ⊥ ρ 2 2 Nc (2π) πQs Qs one can also write the quark distribution as a convolution of the unintegrated gluon distribution and the splitting kernel in momentum space in terms of the following expressions x q˜ DIS (x, q⊥ ) q ⊥ k ⊥ 2 Nc 2 2 − 2 = d R⊥ d k⊥ F (q⊥ − k⊥ , Qs ) dy 2 4π 4 q ⊥ + y k⊥ +y 2 q⊥ Nc q⊥ · (q⊥ − k⊥ ) 2 2 , = ln d R⊥ d k⊥ F (k⊥ , Qs ) 1 − 2 4π 4 q⊥ − (q⊥ − k⊥ )2 (q⊥ − k⊥ )2
(10)
which coincides with the results obtained in Refs. [5,6]. In addition, we also find that the quark distributions involved in DIS and Drell–Yan processes are the same. This coincides with the conclusion in scalar QED. This is not surprising since there is kt -factorization for the Drell–Yan process in hadron–hadron collision. 1 We notice that there is a factor of 1/2 difference between our results and those obtained in Ref. [4]. This difference comes from the fact that the quark distribution calculated in Ref. [4] is in fact the total quark distribution which includes anti-quark distribution as well.
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2.2. Quark distributions in di-jet production In scalar QED model, the TMD quark distribution in di-jet production is shown in Eq. (3). By using the same replacements and setting g2 = g1 , we arrive at the following quark distribution in the large Nc limit x q˜ DJ (x, q⊥ ) √ Nc ) √ −iq⊥ ·(r⊥ −r⊥ = e r⊥ K0 ( yr⊥ ) · r⊥ K0 yr⊥ dy d 2 R⊥ d 2 r⊥ d 2 r⊥ 6 32π
)2 )2 Q2s (r⊥ − r⊥ Q2s (r⊥ − r⊥ + exp − × exp − 4 2 )2 + r 2 ) )2 + r 2 ) Q2 ((r⊥ − r⊥ Q2 ((r⊥ − r⊥ ⊥ ⊥ − exp − s − exp − s , 4 4
(11)
which then yields dx q˜ DJ (x, q⊥ ) Nc Q2s = , 2 2 d 2 R⊥ 12π 4 q⊥ q⊥ Q2s dx q˜ DJ (x, q⊥ ) Nc = 0.44 4 . 2 d R⊥ 4π q 2 Q2s
(12)
⊥
In arriving the above results, we have taken the large Nc limit which simplifies the four point functions and yields [15]
Q2s † † 2 2 U (R⊥ )U R⊥ U (R⊥ )U (R⊥ + r⊥ ) exp − (13) r⊥ − r⊥ + r⊥ . 4 It is straightforward to see that the quark distributions in DIS and di-hadron production have 2 limit. As shown in Fig. 2, the quark the same perturbative tails while they differ in the small q⊥ distribution is about twice broader than the one in DIS while its peak is about half of the peak of the DIS distribution. However, it is easy to check analytically and numerically that the integrated quark distributions are universal for these processes. Similarly, in momentum space, we find that the quark distribution in di-jet production can be written as follows: Nc 2 x q˜ DJ (x, q⊥ ) = R d d 2 l⊥ F (q⊥ − l⊥ , Qs ) ⊥ 4π 4 2 l⊥ l⊥ · (l⊥ − k⊥ ) , (14) ln × d 2 k⊥ F (k⊥ , Qs ) 1 − 2 l⊥ − (l⊥ − k⊥ )2 (l⊥ − k⊥ )2 which implies that x q˜
DJ
(x, q⊥ ) =
d 2 l⊥ x q˜ DIS (x, l⊥ )F (q⊥ − l⊥ , Qs ).
(15)
Eq. (15) is an interesting result because it relates those two apparent different quark distributions through a kt convolution with the unintegrated gluon distribution F (k⊥ , Qs ). In addition,
B.-W. Xiao, F. Yuan / Nuclear Physics A 854 (2011) 180–186
4 dx q˜
(x,q )
185
q2
tot ⊥ as functions of ⊥ in DIS (or Drell–Yan) and di-hadron Fig. 2. Comparison of quark distributions 4π Nc d 2 R⊥ Q2s production. The solid curve stands for the quark distribution in DIS and Drell–Yan process, and the dash curve represents the distribution involved in di-hadron production.
it explains the broadening of the di-jet quark distribution. This formula has a natural interpretation. This convolution rises as a result of the extra initial and final state interactions in di-jet production process. In conclusion, our result is noteworthy because it clearly demonstrates the non-universality of the quark distribution in CGC formalism which leads to the kt -factorization violation. Nevertheless, we find that the Wilson line U (x⊥ ), which is considered as the underlying fundamental description of the interactions between partons and dense hadronic matter, is still universal and the kt factorization breaking effects are resummable. In other words, although the TMD parton distribution is found to be process-dependent, we expect that the nucleus wave function and the methods of computation for CGC are universal. References [1] J. Collins, J.W. Qiu, Phys. Rev. D 75 (2007) 114014; J. Collins, arXiv:0708.4410 [hep-ph]. [2] B.W. Xiao, F. Yuan, arXiv:1003.0482 [hep-ph]. [3] E. Iancu, R. Venugopalan, arXiv:hep-ph/0303204; J. Jalilian-Marian, Y.V. Kovchegov, Prog. Part. Nucl. Phys. 56 (2006) 104; F. Gelis, E. Iancu, J. Jalilian-Marian, R. Venugopalan, arXiv:1002.0333 [hep-ph], and references therein. [4] A.H. Mueller, Nucl. Phys. B 558 (1999) 285. [5] L.D. McLerran, R. Venugopalan, Phys. Rev. D 59 (1999) 094002; R. Venugopalan, Acta Phys. Pol. B 30 (1999) 3731. [6] C. Marquet, B.W. Xiao, F. Yuan, Phys. Lett. B 682 (2009) 207. [7] W. Vogelsang, F. Yuan, Phys. Rev. D 76 (2007) 094013. [8] S.J. Brodsky, P. Hoyer, N. Marchal, S. Peigne, F. Sannino, Phys. Rev. D 65 (2002) 114025. [9] S. Peigne, Phys. Rev. D 66 (2002) 114011. [10] X. Ji, F. Yuan, Phys. Lett. B 543 (2002) 66; A.V. Belitsky, X. Ji, F. Yuan, Nucl. Phys. B 656 (2003) 165.
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[11] [12] [13] [14] [15]
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S.J. Brodsky, B. Pasquini, B. Xiao, F. Yuan, arXiv:1001.1163 [hep-ph]. F. Gelis, A. Peshier, Nucl. Phys. A 697 (2002) 879, arXiv:hep-ph/0107142. J.P. Blaizot, F. Gelis, R. Venugopalan, Nucl. Phys. A 743 (2004) 57, arXiv:hep-ph/0402257. K. Fukushima, Y. Hidaka, JHEP 0706 (2007) 040, arXiv:0704.2806 [hep-ph]. F. Dominguez, C. Marquet, B. Wu, Nucl. Phys. A 823 (2009) 99, arXiv:0812.3878 [nucl-th].
Two particle correlations: Saturation and issues with universality Bo-Wen Xiao a,∗ , Feng Yuan a,b a Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States b RIKEN BNL Research Center, Building 510A, Brookhaven National Laboratory, Upton, NY 11973, United States
Received 21 July 2010; received in revised form 29 July 2010; accepted 30 July 2010 Available online 6 August 2010
Abstract We study the universality issue of the transverse momentum dependent parton distributions at small-x, by comparing the initial/final state interaction effects in di-jet correlation in pA collisions with those in deep inelastic lepton–nucleus scattering. We demonstrate the non-universality by performing an explicit calculation in a particular model where the multiple gauge boson exchange contributions are summed up to all orders. In addition, we generalize the model calculation to the CGC formalism, and find the nonuniversality for quark distributions in CGC. © 2010 Elsevier B.V. All rights reserved. Keywords: Non-universality; Color glass condensate; Di-jet production
1. Non-universality of transverse momentum dependent parton distributions at small-x Recently, the non-universality of these distribution functions due to the final/initial state interaction effects has attracted intensive investigations. This eventually leads to a conclusion that a standard transverse momentum dependent factorization breaks down for this process [1]. In Ref. [2], we extend the universality discussions of the transverse momentum dependent parton distributions to the small-x domain, where the kt -dependent distributions have been a common * Corresponding author.
E-mail address: [email protected] (B.-W. Xiao). 0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2010.07.012
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181
Fig. 1. (a) Schematic diagram showing that two partons from the nucleon projectile and the nucleus target collide and produce two jets in the final state, where the intrinsic transverse momentum q⊥ from nucleus dominates the imbalance between the two jets; (b) illustration of initial/final state interactions which may affect the transverse momentum dependent quark distribution from the nucleus in this process; (c) as a comparison, only the final state interaction effect is present in the deep inelastic lepton–nucleus (nucleon) scattering.
practice to describe the relevant physics phenomena [3]. We expect the non-universality for these objects as well. The process that we focus on is described as follows: as schematically shown in Fig. 1(a), two partons from the nucleon projectile and nucleus target collide with each other, and produce two jets in the final state, p + A → Jet1 + Jet2 + X,
(1)
where the transverse momenta of these two jets are similar in size but opposite to each other in direction. We take the partonic channel qq → qq (as illustrated in Fig. 1(b)) as an example to show the initial/final state interaction effects and calculate the quark distribution in di-jet correlation, and compare with that in the deep inelastic scattering process (as shown in Fig. 1(c)). At small-x, quark distribution is dominated by gluon splitting, and can be calculated from the relevant Feynman diagrams [4–6]. For the purpose of our calculation, we employ an Abelian model of Refs. [1,7,8]. It is a scalar QED model with Abelian massive gluons with a mass λ. We construct the model in such a way that the scalar quarks are generated by the Abelian gluon splitting which is the dominant contribution at small-x. The associated quark distribution in deep inelastic scattering process in this model has been calculated in [8–10]. Since we are interested in studying the final state interaction effects on the parton distribution of the nucleus, for convenience, we choose the projectile as a single scalar quark with charge g2 , which differs from the charge of the scalar quark from the target nucleus g1 . In addition, we assume that the Abelian gluons attached to the target nucleus has an effective coupling g. All the partons in this calculation is set to be scalars with a mass m. The coupling g2 being different from g1 is to show the dependence of the parton distribution on the initial/final state interactions associated with the incoming parton. If the dependence on g2 remains for the nucleus parton distributions, they are not universal [1,7]. We use the small-x approximation as well as the eikonal approximation, and calculate the associated amplitude up to the third order. We sum up all order multi-gauge boson exchange contributions by going to the coordinate space via the Fourier transform [8]. We find that the total amplitude can be exponentiated into the form (2) A(tot) (R, r) = iV (r⊥ ) 1 − eigg1 [G(R⊥ +r⊥ )−G(R⊥ )] e−igg2 G(R⊥ ) , where G(R⊥ ) = K0 (λR⊥ )/2π and V (r⊥ ) = K0 (Mr⊥ )/2π with M 2 = 2xP + p − + m2 . Therefore, the all order result of the quark distribution reads as
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2 x dp − d 2 k⊥ + − 2 (tot) 4P p A (k, p) − 2 4 p (2π) 32π +2 xP − − 2 (2) = d r⊥ d 2 r⊥ δ R⊥ + r⊥ − R⊥ − r⊥ dp p d 2 R⊥ d 2 R⊥ 8π 4 −igg (G(R )−G(R )) ⊥ 2 ⊥ e × e−iq⊥ ·(r⊥ −r⊥ ) V (r⊥ )V r⊥ × 1 − eigg1 [G(R⊥ +r⊥ )−G(R⊥ )] 1 − e−igg1 [G(R⊥ +r⊥ )−G(R⊥ )] .
q(x, ˜ q⊥ ) =
(3)
This transverse momentum dependent quark distribution is clearly different from that calculated in deep inelastic scattering in the same model [8–10] due to the g2 dependent phase. In other words, this distribution is not universal and the standard kT -factorization breaks down. This essentially confirms the conclusion in Ref. [1]. In addition, our calculation demonstrates that the anomalous terms (g2 dependent terms) which breaks the universality and standard kt -factorization can be resummed into Wilson lines as shown above in the small-x limit. It is interesting to notice that the g2 dependence disappears after the integration over the transverse momentum. This is consistent with the universality for the integrated parton distributions [1,7,10]. It has been argued that the light-cone gauge may simplify the factorization property for the hard scattering processes. For example, if we choose the advanced boundary condition for the gauge potential in light-cone gauge, the wave function of hadrons contain the final state interaction effects [10,11]. However, as we showed in the above calculations, this does not help to resolve the g2 -dependence in the quark distribution in the di-jet correlation in hadronic process due to the presence of both initial and final state interactions. 2. Non-universality of CGC parton distributions In this section, we study the connection between the existing results and CGC quark distribution. We show that the non-universality issue also persists in the CGC formalism by comparing the quark distribution probed in DIS to the one involved in di-jet production. 2.1. Quark distributions in DIS In scalar QED model, the TMD quark distribution in DIS reads [8] xP +2 − − 2 p d r⊥ d 2 r⊥ dp d 2 R⊥ d 2 R⊥ q˜ DIS (x, q⊥ ) = 8π 4 iq ·(R −R ) e ⊥ ⊥ ⊥ V (r⊥ )V r⊥ × δ (2) R⊥ + r⊥ − R⊥ − r⊥ × 1 − eigg1 [G(R⊥ +r⊥ )−G(R⊥ )] 1 − e−igg1 [G(R⊥ +r⊥ )−G(R⊥ )] .
(4)
In arriving above formula, we have assumed the target hadron (nucleus) to be a point particle. In order to reproduce the quark distribution in saturation physics, we need to relax the point particle approximation. First of all, we assume that the target hadron has a color charge distribution ρa (z− , z⊥ ) and perform a replacement e−igg1 [G(x⊥ )] ⇒ U (x⊥ ), with − a − 2 (5) U (x⊥ ) = T exp −igg1 dz d z⊥ G(x⊥ − z⊥ )ρa z , z⊥ t . It is straightforward to see that U (x⊥ ) ⇒ e−igg1 [G(x⊥ )] if we set ρa (z− , z⊥ )t a = δ(z− )δ (2) (z⊥ ).
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The second step is to average over the color charge ρa (z− , z⊥ ) with the Gaussian distribution W [ρ]. The variance μ2 (z− ) of the charge distribution represents the density of color sources per unit volume. Thus one finds (see Refs. [12–14])
2 /4 , (6) eigg1 [G(R⊥ +r⊥ )−G(R⊥ )] ⇒ Tr U (R⊥ )U † (R⊥ + r⊥ ) ρ Nc exp −Q2s r⊥ with Q2s =
μ2s 2π
ln
1 2 λ2 r⊥
and μ2s =
g 2 g12 a 2 t ta
dx − μ2 (z− ).
The next step is to use fermionic quark splitting kernel instead of the scalar quark splitting 1 kernel. Thus we replace V (r⊥ ) by 2π 2K1 (Mr⊥ ) where the factor of 2 comes from the fact that fermionic quark has two different helicities. At the end of the day, once we change the variable to y = 2xP + p − , we can cast the total quark distribution into x q˜ DIS (x, q⊥ ) √ Nc ) √ −iq⊥ ·(r⊥ −r⊥ e r⊥ K0 ( yr⊥ ) · r⊥ K0 yr⊥ dy d 2 R⊥ d 2 r⊥ d 2 r⊥ = 6 32π
)2 Q2 r 2 Q2 r 2 Q2 (r⊥ − r⊥ − exp − s ⊥ − exp − s ⊥ , × 1 + exp − s 4 4 4 which gives dx q˜ DIS (x, q⊥ ) Nc Q2s = , 2 2 d 2 R⊥ 12π 4 q⊥ q⊥ Q2s dx q˜ DIS (x, q⊥ ) Nc = . 2 d R⊥ 4π 4 q 2 Q2s
(7)
(8)
⊥
These formulas agree with the results derived in saturation physics for the quark distribution of a large nucleus in DIS (see e.g., Eqs. (27)–(29) of Ref. [4]).1 Furthermore, by transforming to the momentum space and defining a normalized unintegrated gluon distribution F (k⊥ , Qs ) as 2 2 k d r⊥ −ik⊥ ·r⊥ 1
1 † F (k⊥ , Qs ) = e Tr U (R )U (R + r ) exp − ⊥2 , (9) ⊥ ⊥ ⊥ ρ 2 2 Nc (2π) πQs Qs one can also write the quark distribution as a convolution of the unintegrated gluon distribution and the splitting kernel in momentum space in terms of the following expressions x q˜ DIS (x, q⊥ ) q ⊥ k ⊥ 2 Nc 2 2 − 2 = d R⊥ d k⊥ F (q⊥ − k⊥ , Qs ) dy 2 4π 4 q ⊥ + y k⊥ +y 2 q⊥ Nc q⊥ · (q⊥ − k⊥ ) 2 2 , = ln d R⊥ d k⊥ F (k⊥ , Qs ) 1 − 2 4π 4 q⊥ − (q⊥ − k⊥ )2 (q⊥ − k⊥ )2
(10)
which coincides with the results obtained in Refs. [5,6]. In addition, we also find that the quark distributions involved in DIS and Drell–Yan processes are the same. This coincides with the conclusion in scalar QED. This is not surprising since there is kt -factorization for the Drell–Yan process in hadron–hadron collision. 1 We notice that there is a factor of 1/2 difference between our results and those obtained in Ref. [4]. This difference comes from the fact that the quark distribution calculated in Ref. [4] is in fact the total quark distribution which includes anti-quark distribution as well.
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2.2. Quark distributions in di-jet production In scalar QED model, the TMD quark distribution in di-jet production is shown in Eq. (3). By using the same replacements and setting g2 = g1 , we arrive at the following quark distribution in the large Nc limit x q˜ DJ (x, q⊥ ) √ Nc ) √ −iq⊥ ·(r⊥ −r⊥ = e r⊥ K0 ( yr⊥ ) · r⊥ K0 yr⊥ dy d 2 R⊥ d 2 r⊥ d 2 r⊥ 6 32π
)2 )2 Q2s (r⊥ − r⊥ Q2s (r⊥ − r⊥ + exp − × exp − 4 2 )2 + r 2 ) )2 + r 2 ) Q2 ((r⊥ − r⊥ Q2 ((r⊥ − r⊥ ⊥ ⊥ − exp − s − exp − s , 4 4
(11)
which then yields dx q˜ DJ (x, q⊥ ) Nc Q2s = , 2 2 d 2 R⊥ 12π 4 q⊥ q⊥ Q2s dx q˜ DJ (x, q⊥ ) Nc = 0.44 4 . 2 d R⊥ 4π q 2 Q2s
(12)
⊥
In arriving the above results, we have taken the large Nc limit which simplifies the four point functions and yields [15]
Q2s † † 2 2 U (R⊥ )U R⊥ U (R⊥ )U (R⊥ + r⊥ ) exp − (13) r⊥ − r⊥ + r⊥ . 4 It is straightforward to see that the quark distributions in DIS and di-hadron production have 2 limit. As shown in Fig. 2, the quark the same perturbative tails while they differ in the small q⊥ distribution is about twice broader than the one in DIS while its peak is about half of the peak of the DIS distribution. However, it is easy to check analytically and numerically that the integrated quark distributions are universal for these processes. Similarly, in momentum space, we find that the quark distribution in di-jet production can be written as follows: Nc 2 x q˜ DJ (x, q⊥ ) = R d d 2 l⊥ F (q⊥ − l⊥ , Qs ) ⊥ 4π 4 2 l⊥ l⊥ · (l⊥ − k⊥ ) , (14) ln × d 2 k⊥ F (k⊥ , Qs ) 1 − 2 l⊥ − (l⊥ − k⊥ )2 (l⊥ − k⊥ )2 which implies that x q˜
DJ
(x, q⊥ ) =
d 2 l⊥ x q˜ DIS (x, l⊥ )F (q⊥ − l⊥ , Qs ).
(15)
Eq. (15) is an interesting result because it relates those two apparent different quark distributions through a kt convolution with the unintegrated gluon distribution F (k⊥ , Qs ). In addition,
B.-W. Xiao, F. Yuan / Nuclear Physics A 854 (2011) 180–186
4 dx q˜
(x,q )
185
q2
tot ⊥ as functions of ⊥ in DIS (or Drell–Yan) and di-hadron Fig. 2. Comparison of quark distributions 4π Nc d 2 R⊥ Q2s production. The solid curve stands for the quark distribution in DIS and Drell–Yan process, and the dash curve represents the distribution involved in di-hadron production.
it explains the broadening of the di-jet quark distribution. This formula has a natural interpretation. This convolution rises as a result of the extra initial and final state interactions in di-jet production process. In conclusion, our result is noteworthy because it clearly demonstrates the non-universality of the quark distribution in CGC formalism which leads to the kt -factorization violation. Nevertheless, we find that the Wilson line U (x⊥ ), which is considered as the underlying fundamental description of the interactions between partons and dense hadronic matter, is still universal and the kt factorization breaking effects are resummable. In other words, although the TMD parton distribution is found to be process-dependent, we expect that the nucleus wave function and the methods of computation for CGC are universal. References [1] J. Collins, J.W. Qiu, Phys. Rev. D 75 (2007) 114014; J. Collins, arXiv:0708.4410 [hep-ph]. [2] B.W. Xiao, F. Yuan, arXiv:1003.0482 [hep-ph]. [3] E. Iancu, R. Venugopalan, arXiv:hep-ph/0303204; J. Jalilian-Marian, Y.V. Kovchegov, Prog. Part. Nucl. Phys. 56 (2006) 104; F. Gelis, E. Iancu, J. Jalilian-Marian, R. Venugopalan, arXiv:1002.0333 [hep-ph], and references therein. [4] A.H. Mueller, Nucl. Phys. B 558 (1999) 285. [5] L.D. McLerran, R. Venugopalan, Phys. Rev. D 59 (1999) 094002; R. Venugopalan, Acta Phys. Pol. B 30 (1999) 3731. [6] C. Marquet, B.W. Xiao, F. Yuan, Phys. Lett. B 682 (2009) 207. [7] W. Vogelsang, F. Yuan, Phys. Rev. D 76 (2007) 094013. [8] S.J. Brodsky, P. Hoyer, N. Marchal, S. Peigne, F. Sannino, Phys. Rev. D 65 (2002) 114025. [9] S. Peigne, Phys. Rev. D 66 (2002) 114011. [10] X. Ji, F. Yuan, Phys. Lett. B 543 (2002) 66; A.V. Belitsky, X. Ji, F. Yuan, Nucl. Phys. B 656 (2003) 165.
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