In this contribution we present the status of two numerical tools designed to
study the small x limit of QCD. The first one is a Monte Carlo simulation of
the BFKL evolution equation. In design of this approach emphasis has been
placed on exploiting the linear behaviour that many variants of the BFKL
evolution possess. This allows us to design a procedure which can be used to
study theoretical and phenomenological aspects of different kernels. The second
one is a semi-analytic approach to study Lipatov's effective action which
describes Reggeon interactions. The study of the properties of this action is
very complicated and we propose using a computational tool to handle the large
amount of non--local vertices and the derivation of higher order corrections.Comment: 7 pages, 3 figures. International Workshop on Diffraction in
  High-Energy Physics -DIFFRACTION 2006 - September 5-10 2006 Adamantas, Milos
  island, Greec

Stephens, P.

Vera, A. Sabio

English

arXiv

In this contribution we present the status of two numerical tools designed to study the small x limit of QCD. The first one is a Monte Carlo simulation of the BFKL evolution equation. In design of this approach emphasis has been placed on exploiting the linear behaviour that many variants of the BFKL evolution possess. This allows us to design a procedure which can be used to study theoretical and phenomenological aspects of different kernels. The second one is a semi-analytic approach to study Lipatov's effective action which describes Reggeon interactions. The study of the properties of this action is very complicated and we propose using a computational tool to handle the large amount of non--local vertices and the derivation of higher order corrections

Sabio Vera, Agustin

Stephens, P

CERN Document Server

PoS(DIFF2006)030Numerical tools for the theoretical study of QCD atsmall xA. Sabio VeraPhysics Department, Theory Division, CERN, CH-1211 Geneva 23, SwitzerlandE-mail: Agustin.Sabio.Vera@cern.chP. Stephens∗†Institute of Nuclear Physics, Polish Academy of Sciences,ul. Radzikowskiego 152, 31-342 Cracow, PolandE-mail: pstephens@annapurna.ifj.edu.plIn this contribution we present the status of two numerical tools designed to study the small x limitof QCD. The first one is a Monte Carlo simulation of the BFKL evolution equation. In design ofthis approach emphasis has been placed on exploiting the linear behaviour that many variants ofthe BFKL evolution possess. This allows us to design a procedure which can be used to studytheoretical and phenomenological aspects of different kernels. The second one is a semi-analyticapproach to study Lipatov’s effective action which describes Reggeon interactions. The study ofthe properties of this action is very complicated and we propose using a computational tool tohandle the large amount of non–local vertices and the derivation of higher order corrections.DIFFRACTION 2006 - International Workshop on Diffraction in High-Energy PhysicsSeptember 5-10 2006Adamantas, Milos island, Greece∗Speaker.†This work is partly supported by the EU grant mTkd-CT-2004-510126 in partnership with the CERN Physics De-partment and by the Polish Ministry of Scientific Research and Information Technology grant No 620/E-77/6.PRUE/DIE188/2005-2008.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(DIFF2006)030Numerical tools for the theoretical study of QCD at small x P. Stephens1. IntroductionThe structure of Quantum Chromodynamics is very rich. Besides the intrinsic uncertainties forthe description of the theory in the nonperturbative region, the regime where the coupling is smallalso presents many challenges. In order to describe some processes dominated by perturbativedynamics it has been important to focus the efforts on different regions of phase space where loga-rithmic factors get enhanced and their resummation to all orders can provide valuable information.One example of this kind is the Regge limit of strong interacting processes. This is a regime wherethe center of mass energy, s, is very large and terms of the form αs ln s are enhanced. In processescharacterized by two large and similar transverse scales, for a range of large energies, the underly-ing dynamics is linear. This implies that the growth with energy of the observables can be describedby a linear integral equation, the so–called Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation [1].Linearity has, as a consequence, an exponential rise of the cross section with an exponent related tothe so–called hard pomeron intercept. The LO intercept is too large when compared to experimen-tal data. The inclusion of the next–to–leading (NLO) order corrections [2], where there is an extrapower in the coupling when compared to the logs of energy, reduces the intercept to values moreacceptable from the point of view of phenomenology. The BFKL scattering amplitudes contain alarge amount of physical information which can be fully extracted only by numerical methods. Ithas a Poisson like structure in the number of emissions which could be useful in the study of eventswith a large multiplicity in the final state. At NLO one is sensitive to effects like the running of thecoupling or how to best couple the Reggeized gluons to the external particles. These issues can bebest treated in a numerical way using the methods developed in Ref. [3]. Here we will review someof the equations shown in these works and we will introduce a more efficient implementation, foralternative ideas see Ref. [4]. The use of the tools shown here is not limited to the standard forwardBFKL kernel but it can be extended to the nonforward case which corresponds to colour singletexchange, and more exotic situations as is the case when the QCD evolution is placed in a thermalbath.Although there should be a window of large energies where BFKL effects are dominant, lin-earity leads to the breakdown of unitarity since the Froissart bound is not respected for asymptoticenergies. This is connected to the well–known sentence by the late economist Kenneth E. Boulding:"Anyone who believes exponential growth can go on forever in a finite world is either a madmanor an economist". Therefore at some point unitarity corrections should be taken into account. Thisis a very complicated issue from the theoretical point of view and all proposed solutions are alwaysapproximations. We believe an attractive approach is that proposed by L. Lipatov in Ref. [5]. Anon–linear gauge invariant effective action was constructed which describes the interactions be-tween Reggeized quarks and gluons with the usual ones. The Reggeized degrees of freedom arerepresented by Wilson lines while the s–channel emissions belong to a kinematics more generalthan the quasi–multi–Regge region needed to obtain the NLO kernel in the sense that not only a sin-gle pair of partons is allowed to have a fixed invariant mass but more are allowed. These “grouped”emissions are well separated from each other in rapidity following the usual multi–Regge kinemat-ics. This action incorporates unitarity in different channels and describes the interaction betweenReggeons as well as, through quantum fluctuations around the classical solution, higher order cor-rections to the BFKL kernel. In this paper we propose the use of a second tool which uses the2PoS(DIFF2006)030Numerical tools for the theoretical study of QCD at small x P. Stephenssemi-analytic software Effective [12] to study and probe the tree–level structure of this effectiveaction. In combination with a topology generator, the effective vertices of Reggeons coupled toparticles can be generated automatically.2. Monte Carlo implementation of BFKL equationsThe first work taking full advantage of the iterative structure present in the BFKL equation todevelop a Monte Carlo solution for the gluon Green’s function was presented in Ref. [6]. More re-cently this method was extended to the much more complicated case of the NLO kernel in Ref. [3].Here we would like to report on recent progress on a new implementation of these ideas to continueinvestigating the gluon Green’s function in different physical processes.The LO evolution equation can be written in a very simple form as∂∂Y˜f(~ka,~kb,Y)≡∫d2~k α¯spi~k2θ(~k2−λ 2)e−α¯s ln(~ka+~k)2~k2aY˜f(~ka +~k,~kb,Y), (2.1)if the inital condition˜f(~ka,~kb,Y)= δ (2)(~ka−~kb), (2.2)is chosen for a function related to the gluon Green’s function up to a factor including the Reggeizedgluon propagator, i.e.f(~ka,~kb,Y)≡ eω0(~k2a,λ 2)Y ˜f(~ka,~kb,Y). (2.3)The generalization of this formula to the case of colour singlet exchange is straightforward [7] andthe evolution equation can be written as∂∂Y˜f(~ka,~kb,~q,Y)≡∫d2~k α¯s2pi~k21+~k∗2a(~ka +~k)2−~q2~k2~k2a(~k∗a +~k)2θ (~k2−λ 2)× e− α¯s2 ln(~ka+~k)2(~k∗a+~k)2~k2a~k∗2aY˜f(~ka +~k,~kb,~q,Y). (2.4)The generalization for the NLO case has a very similar structure although the kernels and gluonRegge trajectories are more complicated.The numerical results are presented in fig. 1. These correspond to the LO forward case withY = 4 and~ka = (50,0) GeV. Work is in progress to implement other cases at LO and NLO as well.In these figures the analytic solution is found from the sum of conformal spins up to 20 and themaximum number of emissions in the Monte Carlo implementation is 30. As can be seen there isstill a small discrepancy as the ratio k2b/k2a moves away from 1. This is still under investigation.One of the questions we would like to answer is what is the effect of the collinear resummationin kt–space proposed in Ref. [8] on the behaviour of the Green’s function when the value of thetransverse momenta in the incoming Reggeons is very different. It is well–known that for largervalues of the coupling this region presents an instability in terms of convergence and it would beimportant to understand how to solve this situation in a fully exclusive way by complementing the3PoS(DIFF2006)030Numerical tools for the theoretical study of QCD at small x P. Stephens1e-061e-051e-040.0010.010.001 0.01 0.1 1 10 100 1000f(~ ka,~ kb,Y)k2b/k2aa) MCAnalytic1e-061e-051e-040.0010.001 0.01 0.1 1 10 100 1000f(~ ka,~ kb,Y)k2b/k2ab) MCAnalyticFigure 1: Comparison of LO forward analytic versus MC for a fixed angles a) φ = 0 b) φ = pi/4. Forthese figures we take Y = 4 and~ka = (50,0) GeV. The source of the discrepancy, in both figures, away fromk2b/k2a = 1 is still under investigation, as is the ehancement for φ = 0 around k2b/k2a = 1.BFKL approach with all–order contributions dominant in the collinear region. The prescriptionproposed in [8] has a simple implementation since it suggests that the modification needed in theNLO kernel to introduce the collinear improvements is to remove the term−α¯2s41pi(~q−~p)2ln2(q2p2)(2.5)in the NLO real emission kernel and replace it with1pi(~q−~p)2(q2p2)−bα¯s |p−q|p−q √√√√2(α¯s + a α¯2s )ln2(q2p2) J1(√2(α¯s + a α¯2s ) ln2(q2p2))−α¯s− a α¯2s +b α¯2s|p−q|p−qln(q2p2)}(2.6)wherea =512β0Nc−1336n fN3c−5536 , b = −18β0Nc−n f6N3c−1112(2.7)are the collinear coefficients of the kernel in Mellin space.Recently a theoretical derivation of a jet definition valid at NLO has been calculated in thecontext of the NLO BFKL equation in Ref. [10]. One of the targets of our research will be toimplement this jet definition and study jet multiplicities of fully generated events at NLO. Issuesrelated to the conservation of longitudinal components in the final state emissions can be studiedin this context.3. Reggeon Effective ActionThe Reggeon effective action [11] contains a large quantity of physics. The problem is how toretrieve the interesting and possibly new physics from this action. This is still a new area of workand there is little experience about the best path to proceed down. We have decided to develop acomputational tool to help us understand this action and study its behaviour.4PoS(DIFF2006)030Numerical tools for the theoretical study of QCD at small x P. StephensThe action is the standard QCD action plus a term induced by the reggeization of the gluon.This term can be expressed in terms of a kinetic term, describing the propagation of a Reggeonfield, and Reggeon-gluon coupling termsLind = Lkind +LRGind . (3.1)In order to discuss these terms, we must introduce the kinematics implicit in this action. Theaction is defined in the quasi–multi–Reggeon kinematics (QMRK) approximation. In this regimethe final states are produced in clusters, with each cluster having total momentum Q i. Cluster k iscomposed of any number of gluons, each containing momentum p(k)j . Therefore we havePA +PB = Q1 +Q2 + . . .+Qn ; Q2i = M2i , Qk = ∑jp(k)j , (3.2)s = 2PAPB = 4E2  si = 2QiQi+1  |ti|= |q2i |, (3.3)where PA and PB are incoming momenta. We now introduce the light-cone vectornµ± = PµB/A/E ; (n±)2 = 0; n+ ·n− = 2; n± ·~kT = 0. (3.4)We can now use this projection to project the Lorentz derivatives and define the inverse projectedLorentz derivative acting on a field∂± = nµ±∂µ ;1∂±Aµ =ip±Aµ . (3.5)Using these definitions the kinetic term isLkind =−∂+R−∂−R+, (3.6)for Reggeon field R. In the QMRK the Reggeon momentum is transverse. This is given by thecondition ∂±R± = 0 which is inherently included in eqn. (3.6). The induced couplings betweenReggeon fields and gluonic fields is thenLRGind =−Tr{∂+g[P exp(−12∫ x+−∞A+(x′)dx′+)]∂ 2R−(x)+(−↔+)}. (3.7)For notational simplicity we have definedAµ =−iT aAaµ , R± =−iT aRa±, (3.8)where T a are the SU(3) generators in the fundamental representation. The trace is over colourindices. This path exponentiated integral can be written in the form∂+gP exp(−12∫ x+−∞A+(x′)dx′+)= A+∞∑i=0(−g∂+A+)i, (3.9)and can then be used to define the interaction vertices of the Reggeon field with the gluonic fields.We wish to be able to use the action to calculate some observables. In order to do this aconvenient tool to have is the effective vertices which couple n Reggeon fields to m gluonic fields.5PoS(DIFF2006)030Numerical tools for the theoretical study of QCD at small x P. Stephens(T1)−,c,q µ ,a, p1ν ,b, p2(T2)−,c,q µ ,a, p1ν ,b, p2Figure 2: Topology of a minus Reggeon to two gluons. Zig-zag line is a Reggeon while swirly lines aregluons.This requires generating the correct topologies and then extracting from eqn. (3.7) the bare verticesand applying them to the topology. For this task we are using the analytic effective action analysistool Effective [12]. This tool allows us to define the Lagrangian (with the additional definitions ofthe operators) and extract the appropriate couplings from this action. We couple this to a topologygenerator to produce the tree-level effective vertices for n Reggeon and m gluons.We will give now an example of calculating the effective vertex which couples a Reggeon withminus light-cone component to two gluons. Figure 2 shows the topologies of this effective action.We see that this has two contributions. The first is the bare vertex which can be found in the action.We see, in momentum space, this is simplyLRGGind =−iq2g(A+(p1)A+(p2)R−(q)p+i+A−(p1)A−(p2)R+(q)p−i), (3.10)where i can be the momentum of either gluon. The colour factor must also be extracted from theseterms. Doing so we findA+(p1)A+(p2)R−(q) = (−i)3T aT bT cA+a (p1)A+b (p2)R−c (q). (3.11)Using the condition p−1 + p−2 = 0 for the minus Reggeon andAµ =12(n−)µ A+ +12(n+)µ A−+AT , (3.12)we can sum the two contributions (choice of i in momentum) to find〈0∣∣L RGind ∣∣Aµa (p1)Aνb (p2)R−c (q)〉= igq2 fabc(n−)µ(n−)ν 1p−1 . (3.13)For the other topology we have a Reggeon-gluon transition which can be found from the O(g0)term in the induced action, and a standard QCD three-gluon vertex. The transition term simplyprovides a coupling −q2 δ aa′2 (n−)λ . The three-gluon coupling vertex isg fabc[(q− p1)ν gλ µ +(p1− p2)λ gµν +(p2−q)µ gνλ]. (3.14)Using these we find the contribution from the (T2) topologyg fabc[(p1− p2)−gµν +(−2p1− p2)ν(n−)µ +(2p2 + p1)µ(n−)ν], (3.15)which can be combined with eqn. (3.13) to give the full effective vertexV µν−abc = g fabc[(p1− p2)−gµν +(−2p1− p2)ν(n−)µ +(2p2 + p1)µ(n−)ν −q2p−1(n−)µ(n−)ν].(3.16)6PoS(DIFF2006)030Numerical tools for the theoretical study of QCD at small x P. StephensThe software package being implemented will calculate the effective vertices in the samemanner as what is presented here. We plan to use our code to verify the other vertices foundin [11], rather than calculating them by hand. We also plan to calculate new vertices which havenot been presented in [11]. Using an automated system has the additional advantage that it mayallow us to integrate with other codes, e.g. matrix element generators.4. OutlookWe have presented the preliminary work on two computational tools. The first is a MonteCarlo to compute the solution to the BFKL equation for several different kernels. We also wantthis code to be able to produce fully exclusive events at NLO, but this requires still a lot of work.We have shown the results of our current implementation for the LO forward kernel and discussedthe other kernels which are of interest to us.The second part of these proceedings introduces another computational tool designed to beginthe analysis of the effective action proposed in [11]. This action is quite complicated, but mayinclude important and interesting physics. Because of the complexity to analyze the action, wefeel a computational tool is ideally suited for this purpose. Thus we introduce the code which weare currently writing to analyze this action. We also present an example of the type of calculationwhich this code will perform.Acknowledgments: Discussions with J. Bartels and L. Lipatov are acknowledged. PS would alsolike to thank the conference organizers for their invitation and financial support.References[1] L. N. Lipatov, Sov. J. Nucl. Phys. 23, 338 (1976); V. S. Fadin, E. A. Kuraev and L. N. Lipatov, Phys.Lett. B 60, 50 (1975), Sov. Phys. JETP 44, 443 (1976), Sov. Phys. JETP 45, 199 (1977); I. I. Balitskyand L. N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978), JETP Lett. 30, 355 (1979).[2] V.S. Fadin, L.N. Lipatov, Phys. Lett. B 429, 127 (1998); G. Camici, M. Ciafaloni, Phys. Lett. B 430,349 (1998).[3] J. R. Andersen, A. Sabio Vera, Phys. Lett. B 567, 116 (2003); Nucl. Phys. B 679, 345 (2004); Nucl.Phys. B 699, 90 (2004); JHEP 0501, 045 (2005).[4] J. R. Andersen, Phys. Lett. B 639, 290 (2006).[5] L. N. Lipatov, Nucl. Phys. B 452, 369 (1995).[6] C. R. Schmidt, Phys. Rev. Lett. 78, 4531 (1997).[7] J. R. Andersen and A. Sabio Vera, JHEP 0501, 045 (2005).[8] A. Sabio Vera, Nucl. Phys. B 722, 65 (2005).[9] A. Sabio Vera, Nucl. Phys. B 746, 1 (2006).[10] J. Bartels, A. Sabio Vera and F. Schwennsen, hep-ph/0608154.[11] E. N. Antonov, L. N. Lipatov, E. A. Kuraev and I. O. Cherednikov, Nucl. Phys. B 721, 111 (2005)[arXiv:hep-ph/0411185].[12] J. P. J. Hetherington and P. Stephens, arXiv:hep-ph/0605149.7

Numerical tools for the theoretical study of QCD at small x

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