Physics Letters B 694 (2011) 355–362

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

The general behavior of NLO unintegrated parton distributions basedon the single-scale evolution and the angular ordering constraint

H. Hosseinkhani a, M. Modarres b,∗a Plasma Physics and Nuclear Fusion Research School, Nuclear Science and Technology Research Institute, 14395-836 Tehran, Iranb Physics Department, University of Tehran, 1439955961 Tehran, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 August 2010Received in revised form 3 October 2010Accepted 4 October 2010Available online 16 October 2010Editor: W. Haxton

Keywords:Deep inelastic scattering (DIS)Quark structure of hadronsParton distribution functions (PDF)MSTW–GJR–CTEQ6.6 PDFKimber–Martin–Ryskin (KMR) prescription

To overcome the complexity of generalized two hard scale (kt ,μ) evolution equation, well known as theCiafaloni, Catani, Fiorani and Marchesini (CCFM) evolution equations, and calculate the unintegrated partondistribution functions (UPDF), Kimber, Martin and Ryskin (KMR) proposed a procedure based on (i) theinclusion of single-scale (μ) only at the last step of evolution and (ii) the angular ordering constraint(AOC) on the DGLAP terms (the DGLAP collinear approximation), to bring the second scale, kt into theUPDF evolution equations. In this work we intend to use the MSTW2008 (Martin et al.) parton distributionfunctions (PDF) and try to calculate UPDF for various values of x (the longitudinal fraction of partonmomentum), μ (the probe scale) and kt (the parton transverse momentum) to see the general behaviorof three-dimensional UPDF at the NLO level up to the LHC working energy scales (μ2). It is shown thatthere exits some pronounced peaks for the three-dimensional UPDF ( fa(x,kt)) with respect to the twovariables x and kt at various energies (μ). These peaks get larger and move to larger values of kt , as theenergy (μ) is increased. We hope these peaks could be detected in the LHC experiments at CERN andother laboratories in the less exclusive processes.

© 2010 Elsevier B.V. Open access under CC BY license.

1. Introduction

To understand the event structure observed in different labo-ratories i.e. SLAC, HERA, DESY , etc., and especially the one wouldbe expected in LHC (CERN), the theoretical formalisms which de-scribe the small x (x is Bejorken variable) region are vital. The mainunknown parameters in these models are the unintegrated par-ton distribution functions (UPDF) [1–4]. The UPDF are two-scalesdependent distributions which are functions of x (longitudinal mo-mentum fraction of the parent hadron) and the scales k2

t and μ2,the squared transverse momentum of the parton and the factoriza-tion scale, respectively. As we pointed out these distributions arethe essential ingredients for the less exclusive phenomenologicalcomputations in the high energy collisions of particle physics.

It is well known that in the region of high energy and moderatemomentum transfer i.e. small x, the collinear factorization theoremi.e. Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) [5–8] evolu-tion, breaks down. This happens because of the large increase ofthe phase space available for the gluon emissions (i.e. a rapidrise in the gluon density), which makes the quantum chromody-namics (QCD) perturbative expansions unjustified and one can-not obtain the UPDF . On the other hand, at above high energy

* Corresponding author. Tel.: +98 21 61118645; fax: +98 21 88004781.E-mail address: mmodares@ut.ac.ir (M. Modarres).

0370-2693 © 2010 Elsevier B.V.doi:10.1016/j.physletb.2010.10.007

Open access under CC BY license.

limit, the cross section can be predicted by using the kt factor-ization and the Balitsky–Fadin–Kuraev–Lipatov (BFKL) [9–11] evo-lution. But the precision of kt factorization is not good e.g. thenext-to-leading order (NLO) corrections to BFKL are very large [12–15]. Another approach to derive the UPDF is the Ciafaloni–Catani–Fiorani–Marchesini (CCFM) equations [16–20]. Although the CCFMequations describe the evolution of the UPDF correctly, but work-ing in this framework is a complicated task, so practically they areused only in the Monte Carlo event generators [21–25]. On theother hand, up to now, there is not a complete quark version forthese kind of equations [16–20,26], since the enhanced terms thatare resumed by CCFM come from gluon evolution. However, to overcome this problem, it has been shown that the CCFM equation canbe reformulated (the linked dipole chain model) by reducing thedivision between the initial and the final state radiation diagramsusing the colour dipole cascade model [27–29].

The Kimber, Martin and Ryskin (KMR) [30] approach is an alter-native prescription for producing the UPDF which is based on thestandard DGLAP equations [5–8],

∂a(x,μ2)

∂ ln(μ2)=

∑a′=q,g

Paa′ ⊗ a′(y,μ2), (1)

where a(x,μ2) = xq(x,μ2) or xg(x,μ2) and Paa′ (z) are the con-ventional (integrated) parton distribution functions (PDF) and the

356 H. Hosseinkhani, M. Modarres / Physics Letters B 694 (2011) 355–362

Fig. 1. The unintegrated gluon distribution functions generated by the KMR procedure with the fixed values of μ2 = 10,102,104,108 GeV2.

well-known DGLAP splitting functions, respectively. In Eq. (1) thesymbol ⊗ denotes a convolution as

f ⊗ g =1∫

x

dy

yf

(x

y

)g(y). (2)

In this approach under the certain approximation the UPDF areobtained from the PDF by introducing the scale μ only in the laststep of evolution with the inclusion of angular ordering constraint(AOC). It has been shown that the KMR prescription gives the sameresults both for the DGLAP and the unified BFKL–DGLAP equations[31] and the AOC is applicable to all orders as in the CCFM for-malism, i.e. all the loops contributions via the chain of evolutionwhich are restricted by AOC, are resumed.

In this work, along the lines of our recent calculations [33,34],we intend to use the KMR prescription with MSTW2008 [32] PDFto produce three-dimensional plots of UPDF at different energies(μ) and discussed the various behavior of UPDF i.e. f (x,kt ,μ). Sothe Letter is organized as follows: In Section 2 we briefly introducethe KMR formalism and finally, Section 3 is devoted to the results

and the discussions concerning the three-dimensional (3D) graphsof the UPDF produced via this approach.

2. The KMR formalism [30]

The KMR prescription [30] works as a machine that by taking adefined PDF as inputs, generates UPDF , as outputs. Using the lead-ing order (LO) splitting functions, Paa′ , the DGLAP equations can bewritten in a modified form as [30],

∂a(x,μ2)

∂ ln(μ2)= αs

2π

[ 1−�∫x

Paa′(z)a′(

x

z,μ2

)dz

− a(x,μ2)∑

a′

1−�∫0

dz′ Pa′a(z′)], (3)

where � is a cutoff to prevent z = 1 singularities in the splittingfunctions arising from the soft gluon emission. In the conventionalDGLAP formalism, � = 0 and the singularities are canceled by thevirtual terms. The value of � can be determined by imposing an

H. Hosseinkhani, M. Modarres / Physics Letters B 694 (2011) 355–362 357

Fig. 2. As Fig. 1 but for the up quark.

appropriate dynamical condition which is replaced by the angularordering constraint arising from the coherency of the gluon emis-sions [35,36],

· · · > θn > θn−1 > θn−2 > · · · , (4)

where θ ’s are the radiation angels. This condition, at the final stepof evolution, leads to [16–19,31],

μ >zkt

1 − z⇒ � = 1 − zmax = kt

μ + kt. (5)

The first part of Eq. (3), shows the contribution of real emissions,that can change the transverse momentum kt . The second term ex-presses the evolutions due to the virtual effects without changingthe kt . The latter can be re-summed, to obtain a survival probabil-ity factor,

Ta(kt ,μ) = exp

[−

μ2∫k2

αs(k′t2)

2π

dk′t2

k′t2

∑a′

1−�∫0

dz′ Pa′a(z′)]. (6)

t

Now, similar to the Sudakov form factor, the above survival proba-bility, Eq. (6), is imposed into Eq. (1), and by using Eq. (2), we findthe equation which describes the UPDF,

fa(x,k2

t ,μ2) = Ta(kt ,μ)

[∂a(x,μ2)

∂ ln(μ2)

∣∣∣∣μ2=k2

t

]real

= Ta(kt ,μ)αs(kt

2)

2π

1−�∫x

Paa′(z)a′(

x

z,kt

2)

dz. (7)

More explicit forms of the above equation for the gluon g and thedifferent quark flavors q = u,d, s, . . . are as follows,

fq(x,k2

t ,μ2) = Tq(kt ,μ)αs(kt

2)

2π

×1−�∫x

dz

[Pqq(z)

x

zq

(x

z,kt

2)

+ Pqg(z)x

g

(x,kt

2)]

, (8)

z z

358 H. Hosseinkhani, M. Modarres / Physics Letters B 694 (2011) 355–362

Fig. 3. As Fig. 1 but for the strange quark.

and

f g(x,k2

t ,μ2) = T g(kt ,μ)αs(kt

2)

2π

×1−�∫x

dz

[∑q

P gq(z)x

zq

(x

z,kt

2)

+ P gg(z)x

zg

(x

z,kt

2)]

. (9)

The key observation here is the dependency on the scale μ2, whichappears at the last step of the evolution. Another point is that,the Sudakov form factor which arises from the resumption of vir-tual effects, can be used at every order of approximation. Althoughthe splitting functions must be used at the NLO level, but as it isshown in [37], the NLO corrections to the splitting functions, arerelatively small in comparison to the LO contributions. However,as stated above, only the LO splitting functions are used. On theother hand, although the definition of Sudakov form factor (likethe PDF themselves) has been started intuitively from a probabilis-tic interpretation, but its role in the mathematical description ofthe evolution remains in the equations.

The primary computations based on this kind of approach toevaluate the UPDF , show very good agreement with the experi-mental data for F2 [30]. Also, in recent years, the KMR prescriptionhave been widely used for phenomenological calculations (see [33]and the references therein). Recently the stability and the reliabil-ity of the KMR UPDF have been investigated in [33,34].

Finally, we should mention here that, the key property of theCCFM approach (as given in their publications [16–22]) is the AOC,which in turn has root in the coherency of gluon radiation alongthe evolution chain, that is valid for whole range of x values. Inthe conventional DGLAP formalism, the strong ordering constrainton the transverse momenta, restricts the domain of study to thelarge and moderate values of x:

σ(γ ∗q → qg

) =ptmax∫

ptmin

dp2t

dσ

dp2t

,

where

p2t = λ2,

min

H. Hosseinkhani, M. Modarres / Physics Letters B 694 (2011) 355–362 359

Fig. 4. As Fig. 1 but for the bottom quark.

and

p2tmax

= p2t

∣∣sin2 θ=1 = k′2 = s

4= Q 2 1 − x

4x.

So to obtain the DGLAP equations with ln( s4 ) � ln(Q 2), x should

not be very low. In the KMR prescription the AWOC property of theCCFM formalism is applied to modified DGLAP evolution as a cutoff on the integrals. Therefore, the results of these modificationsshow that the effect of application of AOC is even more importantthan the inclusion of the conventional low x effects in the BFKLapproach [30].

3. Results and discussion

As we stated in Section 2, by using Eqs. (8) and (9), the UPDFare generated via the KMR procedure. For the input PDF , theMSTW2008 [32] set of partons at the NLO level are used.1 Sincethe generated UPDF ( fa(x,k2

t ,μ2)) are three variable functions,by fixing the scale μ2, their values versus x and k2

t are plotted

1 We use the MSTW 2008 code that is accessible from http://durpdg.dur.ac.uk andhttp://www.hepforge.org.

in the various panels of Figs. 1, 2, 3 and 4 for the gluons, theup, the strange and the bottom quarks, respectively. For the bet-ter comparison, the values of the μ2 are chosen in a wide rangeμ2 = 10,102,104,108 GeV2 which is up to the LHC working scales.The three typical quark flavors, the u quarks consists of the valenceand the sea contributions u = uv + usea and the s and the b quarkswhich are completely sea distributions, are presented. The mainfeature of these figures is exhibiting the general behavior of theUPDF with respect to the coupled contributions of x and k2

t . Forexample, the most probable value of k2

t (x) at every x (k2t ) for any

kind of partons can be checked. As it can be seen, by increasing thescale μ2 the graphs are shifted to the higher k2

t . This is expected,since the probability of finding partons with larger k2

t is moreprobable at higher scales. The growth of the values of the distribu-tions by increasing μ2 and decreasing x and also the phenomenonof converging the quark distributions to a unique value at smallx are known characteristics of the parton distributions which arethe heritage of their parent PDF . The different behaviors of up andstrange quarks at large x have root in the valence contribution inthe case of up quark. The pronounced peaks become wider withrespect to k2

t , and move to higher values of k2t . This behavior is

much effective for the up, the strange and the bottom quarks. Thepeaks come from the concept of distributions and they are results

360 H. Hosseinkhani, M. Modarres / Physics Letters B 694 (2011) 355–362

Fig. 5. The NLO integrated parton distribution function of MSTW2008 versus x for the fixed values of μ2 = 10,102,104,108 GeV2.

of the dynamical evolution of partons. The figures show that atgiven values of hard scale and x, at which kt , it is more proba-ble to detect the out going partons. So based on the final partons,we can predict the dynamical properties of the produced jets andtheir components, and on the other hand it can inform us aboutthe precision of the current theoretical formalisms itself. The inputPDF of MSTW2008 are also given in Fig. 5, for comparison. Withgood approximation by integrating over UPDF , we can get the input

MSTW2008 PDF (a(x,μ2) = ∫ μ2 dk2t

k2t

fa(x,k2t ,μ2)). For example for

gluons, at x = 0.01 and μ2 = 100 we get 6.7 whereas MSTW2008gives the value of 6.5 i.e. 3% off. Situations are the same for otherpoints and parton distributions. It is worth to say that in the orig-inal KMR work, they get 25% discrepancies [30] for above com-parison. This is also evident by comparison of Fig. 5 with thoseof Figs. 1–4 i.e. the UPDF are decreasing by increasing x. On theother hand, as have been discussed in the KMR and other relatedworks, because of the imposition of angular ordering, the UPDFhave values for k2

t � μ2 as x decreases. But this will not affect theabove integration too much. Figs. 1–4 also show that, for low scales(μ2 � 10 GeV2) the UPDF become negative when x becomes closeto one. This reflects the negative values of MSTW2008 gluon distri-

butions at the NLO level and beyond that. So the negative values ofUPDF have root in the parent integrated gluon distributions whichin turn are the result of MSTW2008 assumptions [32]. As it waspointed, in the MSTW2008 [32], for better data fitting it is allowedthat, the gluon distribution takes negative values, because there isno theorem that imposes positivity condition on PDF beyond theLO approximation. So they become negative in order to fit the data(in other words they can be traced to the slow evolution of F2 atsmall x and Q 2 i.e. a positive gluon would give too rapid evolutionto fit the dF2/d ln(Q 2) data. Then in the KMR integrals, the evalua-tion of input g(x,k2

t ) at small x and kt (as a scale, instead of Q 2 ing(x, Q 2)) leads to the negative values for the output UPDF . Finally,(i) the comparison of UPDF produced from different PDF sets havebeen made in our former works [33,34]. The different parameter-izations procedures lead to different PDF , and a discussion aboutthese procedures is presented in [33,34] and references therein.(ii) The differences between the LO and the NLO PDF are parame-terizations dependent. In the MSTW2008 this is noticeable, but insome other parameterizations sets based on different assumptionsand procedures it can be less (e.g GRV sets [33,34]), but as wehave showed in [33,34] (by investigating the ratios of KMR UPDFcompared to the corresponding ratios of input PDF) the relative

H. Hosseinkhani, M. Modarres / Physics Letters B 694 (2011) 355–362 361

Fig. 6. The UPDF of MSTW2008 (present calculation, dotted curve), MRST99 (dash curve) and GJR08 (full curve). See the text for details.

differences are less in the output UPDF and the KMR prescriptionsuppresses these discrepancies. To show this point more transpar-ently, in Fig. 6 we have plotted the gluon UPDF with three differentinput PDF , namely the original KMR [30] with MRST99 [38] PDF ,our recent works [33,34] with GJR08 PDF [39] and present cal-culation (MRST2008) at μ2 = 100 GeV2 and x = 0.1,0.01,0.001and 0.0001 in terms of k2

t . It is clearly seen that different inputPDF give very similar UPDF . (iii) In fact a complete prescriptionfor producing the NLO UPDF needs to include both the PDF andthe splitting functions at the NLO level. This prescription is pre-sented in [37], but as it is shown in this reference [37], inclusionof the NLO splitting functions have very low effect comparing tothe contribution of the NLO PDF . Therefore, ignoring the correc-tions due to the NLO splitting functions do not affect our analysisof the general behavior of the NLO UPDF . (iv) There is no restric-tion on the kt dependency. As the orders of the approximation arein terms of orders of αs(k2

t ), the NLO accuracy is contained in theNLO PDF and splitting functions that discussed in the former com-ments. Hence at scales k2

t � Q 20 , where Q 2

0 is the scale that upperthan it, the perturbative QCD is still applicable, these results arevalid.

Acknowledgements

We would like to acknowledge the Research Council of Univer-sity of Tehran and Institute for Research and Planning in HigherEducation for the grants provided for us. HH would like to thankProfessors Hannes Jung, Francesco Hautmann, Artem Lipatov, GavinSalam, James Stirling, Graeme Watt and Bryan Webber for their help-ful communications.

References

[1] B. Andersson, et al., Small-x Collaboration, Eur. Phys. J. C 25 (2002) 77.[2] J. Andersen, et al., Small-x Collaboration, Eur. Phys. J. C 35 (2004) 67.[3] J. Andersen, et al., Small-x Collaboration, Eur. Phys. J. C 48 (2006) 53.[4] H. Jung, et al., Proceedings of HERA and the LHC workshop on the implications

of HERA for LHC physics, 2006–2008, Hamburg–Geneva, 2009, arXiv:0903.3861.[5] V.N. Gribov, L.N. Lipatov, Yad. Fiz. 15 (1972) 781.[6] L.N. Lipatov, Sov. J. Nucl. Phys. 20 (1975) 94.[7] G. Altarelli, G. Parisi, Nucl. Phys. B 126 (1977) 298.[8] Y.L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641.[9] E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 44 (1976) 443.

[10] E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45 (1977) 199.[11] I.I. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822.

362 H. Hosseinkhani, M. Modarres / Physics Letters B 694 (2011) 355–362

[12] L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rept. 100 (1983) 1.[13] E.M. Levin, M.G. Ryskin, Y.M. Shabelski, A.G. Shuvaev, Sov. J. Nucl. Phys. 53

(1991) 657.[14] S. Catani, M. Ciafaloni, F. Hautmann, Nucl. Phys. B 366 (1991) 135.[15] J.C. Collins, R.K. Ellis, Nucl. Phys. B 360 (1991) 3.[16] M. Ciafaloni, Nucl. Phys. B 296 (1988) 49.[17] S. Catani, F. Fiorani, G. Marchesini, Phys. Lett. B 234 (1990) 339.[18] S. Catani, F. Fiorani, G. Marchesini, Nucl. Phys. B 336 (1990) 18.[19] G. Marchesini, Proceedings of the Workshop “QCD at 200 TeV”, Erice, Italy, in:

L. Cifarelli, Yu.L. Dokshitzer (Eds.), Plenum, New York, 1992, p. 183.[20] G. Marchesini, Nucl. Phys. B 445 (1995) 49.[21] G. Marchesini, B. Webber, Nucl. Phys. B 349 (1991) 617.[22] G. Marchesini, B. Webber, Nucl. Phys. B 386 (1992) 215.[23] H. Jung, Nucl. Phys. B 79 (1999) 429.[24] H. Jung, G.P. Salam, Eur. Phys. J. C 19 (2001) 351.[25] H. Jung, J. Phys. G: Nucl. Part. Phys. 28 (2002) 971.

[26] B. Webber, private communication, 2010.[27] H. Kharraziha, L. Lönnblad, JHEP 9803 (1998) 006.[28] B. Andersson, G. Gustafson, J. Samuelsson, Nucl. Phys. B 463 (1996) 215.[29] Artem Lipatov, Gavin Salam, private communications, 2010.[30] M.A. Kimber, A.D. Martin, M.G. Ryskin, Phys. Rev. D 63 (2001) 114027.[31] M.A. Kimber, J. Kwiecinski, A.D. Martin, A.M. Stasto, Phys. Rev. D 62 (2000)

094006.[32] A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Eur. Phys. J. C 63 (2009) 189.[33] M. Modarres, H. Hosseinkhani, Few-Body Syst. 47 (2010) 237.[34] M. Modarres, H. Hosseinkhani, Nucl. Phys. A 815 (2009) 40.[35] G. Marchesini, B.R. Webber, Nucl. Phys. B 310 (1988) 461.[36] Yu.L. Dokshitzer, V.A. Khoze, S.I. Troyan, A.H. Mueller, Rev. Mod. Phys. 60 (1988)

373.[37] A.D. Martin, M.G. Ryskin, G. Watt, Eur. Phys. J. C 66 (2010) 163.[38] A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Eur. Phys. J. C 14 (2000) 133.[39] M. Glück, P. Jimenez-Delgado, E. Reya, Eur. Phys. J. C 53 (2008) 355.