The high-energy radiation pattern from BFKLex with double-log collinear contributions

Abstract

Abstract

We study high-energy jet production in the multi-Regge limit making use of the Monte Carlo event generator BFKLex which includes collinear improvements in the form of double-log contributions as presented in [1]. Making use of the anti-k ^sub t^ jet algorithm in the FastJet implementation, we present results for the average transverse momentum and azimuthal angle of the produced jets when two tagged forward/backward jets are present in the final state. We also introduce a new observable which accounts for the average rapidity separation among subsequent emissions. Results are presented, for comparison, at leading order and next-to-leading order, with the resummation of collinear double logs proposed in [2].

Published for SISSA by Springer

Received: December 15, 2015

Revised: December 27, 2015 Accepted: January 22, 2016 Published: February 9, 2016

JHEP02(2016)064

The high-energy radiation pattern from BFKLex with double-log collinear contributions

G. Chachamisa and A. Sabio Veraa,b

aInstituto de F sica Te orica UAM/CSIC,

C/ Nicol as Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid, Spain

bInstituto de F sica Te orica, Universidad Aut onoma de Madrid, Cantoblanco, 28049 Madrid, Spain

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We study high-energy jet production in the multi-Regge limit making use of the Monte Carlo event generator BFKLex which includes collinear improvements in the form of double-log contributions as presented in [1]. Making use of the anti-kt jet algorithm in the FastJet implementation, we present results for the average transverse momentum and azimuthal angle of the produced jets when two tagged forward/backward jets are present in the nal state. We also introduce a new observable which accounts for the average rapidity separation among subsequent emissions. Results are presented, for comparison, at leading order and next-to-leading order, with the resummation of collinear double logs proposed in [2].

Keywords: QCD Phenomenology

ArXiv ePrint: 1512.03603

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP02(2016)064

Web End =10.1007/JHEP02(2016)064

Contents

1 Introduction 1

2 Averages of characteristic quantities in multi-jet events 3

3 Summary & outlook 8

1 Introduction

The in uence of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) approach in nal-state multi-jet con gurations at the Large Hadron Collider (LHC) is a subject of intense debate in recent years. This approach applies when the center-of-mass energy is, in principle, asymptotically large (ps ! 1). It can, however, a ect hadron phenomenology pre-asymptotically

at current LHC energies. One of the targets of the work here presented is to show how pre-asymptotic e ects have sizable e ects at present energies. This is the case for the simplest leading order (LO) [3{8] as well as for the more sophisticated next-to-leading order (NLO) [9, 10] and higher-order calculations.

For our presentation we focus on events where two forward/backward jets with rapidities ya and yb can be clearly identi ed. If the di erence Y = ya yb is large enough

then terms of the form nsY n are important order-by-order to get a good description of measured cross sections which, in a nutshell, can be written in the factorized form

(Q1, Q2, Y ) =

Z

JHEP02(2016)064

d2[vector]kAd2[vector]kB A(Q1,[vector]ka) B(Q2,[vector]kb) f([vector]ka,[vector]kb, Y ). (1.1)

In this expression, A,B are impact factors depending on external scales, Q1,2, and the o -shell reggeized gluon momenta, [vector]ka,b. The gluon Green function f depends on [vector]ka,b and the center-of-mass energy in the scattering eY/2.

For LHC phenomenology it is mandatory to work within the NLO approximation which introduces the dependence on physical scales such as the one associated to the running of the coupling and the one related to the choice of energy scale in the resummed logarithms [11{14]. It is possible to write the gluon Green function in an iterative way in transverse momentum and rapidity space at LO [15] and NLO [16, 17] in the form

f = e!([vector]kA)Y( (2)

[vector]kA [vector]kB +1

Xn=1

n

Yi=1 sNc

Z

d2[vector]ki k2i 2 k2 i

Z

yi1

0 dyie

(!([vector]kA+

Pil=1 [vector]kl)!([vector]kA+

Pi1l=1 [vector]kl))yi (2) [vector]kA +

n

Xl=1[vector]kl [vector]kB!)

, (1.2)

{ 1 {

where

q2 2 (1.3)

corresponds to the gluon Regge trajectory which carries a regulator, , of infrared divergencies. We have implemented this expression in the Monte Carlo event generator BFKLex1 which we have already used for di erent applications ranging from collider phenomenology to more formal studies in the calculation of scattering amplitudes in supersymmetric theories [18{23].

It turns out that the BFKL formalism can be quite sensitive to collinear regions of phase space, in particular when the process-dependent impact factors are broad and allow for the external scales Qi to signi cantly deviate from the internal reggeized gluon transverse momenta ki. In this case there exists a dominant double-log term in the NLO BFKL kernel in the collinear regions which takes the form

k2i 2

! k2i 2

! ([vector]q) =

sNc

log

s

4 ln2 0

B

@

[vector]k2

A

[vector]kA + [vector]ki

, (1.4)

which needs to be resummed to all orders to stabilize the behavior of the BFKL cross sections and to apply the formalism beyond the original multi-Regge kinematics. These issues have been investigated in [24, 25]. In particular, in [2], it was shown that the collinear corrections can be resummed to all-orders using the prescription

k2i 2

! k2i 2

JHEP02(2016)064

2

1

C

A

+

1

Xn=1(

s)n2nn!(n + 1)! ln2n 0

B

@

[vector]k2

A

[vector]kA + [vector]ki

. (1.5)

As it was shown in [2] this expression resums to a Bessel function of the rst kind (similar results have recently been obtained in coordinate representation in [26]). Phenomenological applications of this resummation, not using a Monte Carlo approach, show agreement with experimental results and good perturbative convergence [27{35].

In a recent work [1] we have shown how to implement this collinear resummation in the BFKLex Monte Carlo event generator and investigated what is its e ect in the behavior of the gluon Green function. It is the target of this letter to extend that discussion to investigate the structure of the nal state radiation, or, in other words, to which extend this collinear resummation a ects the exclusive production of jets as obtained from the BFKL approach. We will show our results in the next section, with a particular focus on how pre-asymptotic e ects can be rather important in phenomenological applications.

1BFKLex does not include full energy-momentum conservation since it reproduces the BFKL e ective theory exactly. In this framework, energy-momentum conservation is gradually introduced as the logarithmic accuracy increases from leading to next-to-leading and higher orders. The formalism does have transverse momentum conservation built-in though. It is possible to impose energy-momentum conservation going away from the original BFKL approach but we prefer not to follow this route in this work.

{ 2 {

2

1

C

A

2 Averages of characteristic quantities in multi-jet events

This section is devoted to the presentation of some numerical results as obtained from our Monte Carlo implementation of the NLO BFKL equation in the code BFKLex. In particular, we investigate con gurations with at least two jets, with transverse momentum ka and kb, in the nal state, one very forward and the other very backward such that the rapidity spread between the two jets, ya yb, is large. This set up for the studied events

is analogous to the Mueller-Navelet [36] con gurations for which we would like in addition to probe the internal dynamics of the exchanged BFKL ladder and its associated mini-jet radiation. The advantage of this study is that the production of the two forward/backward tagged jets can be described via collinear factorization, thus reducing the uncertainty in the calculation of cross sections. This is relevant even though in the present letter we focus on presenting results at parton level.

For the shake of de niteness, we introduce three quantities which are directly related to the jet activity along the ladder and uniquely (but not fully) characterize each event. These are three distinct averages for the jets in each event: of the modulus of their transverse momentum ([angbracketleft]pt[angbracketright]), of their azimuthal angle ([angbracketleft][angbracketright]) and, nally, of the rapidity ratio ([angbracketleft]Ry[angbracketright])

between subsequent jets. In more detail, let us assume an event for which, besides the two tagged jets with transverse momentum ka and kb, there exists a number N of further nal-state jets. For each of these jets (whether and when we can consider them as mini-jets will be discussed in the following) we can de ne three variables: the modulus of its transverse momentum, [notdef]ki[notdef], its azimuthal angle i and its rapidity yi, with 1 i N. Then, the

average transverse momentum, azimuthal angle and rapidity ratio would, respectively, read:

hpt[angbracketright] =

N

Xi=1i; (2.2)

N+1

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1

N

N

Xi=1

|ki[notdef]; (2.1)

h[angbracketright] =

1

N

Xi=1yi

yi1. (2.3)

In these de nitions we set y0 = ya, yN+1 = yb = 0 and yi1 > yi. All three observables

are tailored such that should give an accurate view of how closely we follow the multi-Regge kinematics in our Monte Carlo solution of the BFKL equation. We should keep in mind that the multi-Regge kinematics dictates similar transverse momentum sizes for all jets, a strong ordering in rapidities, yi1 yi and generally, azimuthal angles with equal

probability in the range [0, 2).

In order to have a better control on how these observables behave in a collider experimental setup, we use the anti-kt jet algorithm [37] in the FastJet implementation [38, 39]

with a jet radius of R = 0.7 for the emitted jets in the nal state. For the presentation of our numerical results, we have considered two di erent con gurations for the transverse momenta of the forward/backward jets: i) ka = 10 GeV, kb = 12 GeV, ii) ka = 10 GeV, kb = 20 GeV and three di erent rapidity di erences ya yb = 4, 6, 8. For each of these

{ 3 {

[angbracketleft]Ry[angbracketright] =

1

N + 1

cases we have run BFKLex and produced di erential distributions for the observables in eqs. (2.1), (2.2), and (2.3). Two of the observables here discussed have boundary values 0 [angbracketleft]t[angbracketright] < 2 and 0 [angbracketleft]Ry[angbracketright] 1. We sliced the allowed [angbracketleft][angbracketright] and [angbracketleft]Ry[angbracketright] ranges into 100

bins and the [angbracketleft]pt[angbracketright] range into 300 bins. Any produced event by BFKLex characterised by hpt[angbracketright]bin[m], [angbracketleft][angbracketright]bin[n] and [angbracketleft]Ry[angbracketright]bin[l] adds its weight to the m-th bin of the [angbracketleft]pt[angbracketright] distribution,

the n-th bin of the [angbracketleft]t[angbracketright] distribution and the l-th bin of the [angbracketleft]Ry[angbracketright] distribution such that the

area under any di erential distribution gives the full gluon Green function.

We have gathered together our results at LO and NLO+Double Logs in three gures. In gure 1. we show the di erential distribution at LO (top/middle left) and NLO+Double Logs (top/middle right) in the average transverse momentum [angbracketleft]pt[angbracketright] of emitted mini-jets per

event, for di erent values of the transverse momentum of the most forward/backward tagged jets. Having as a common value ka = 10 what we nd are broad distributions with a maximal value at [angbracketleft]pt[angbracketright] [similarequal] 6 GeV for kb = 12 and 8 GeV for kb = 20. These maxima

are independent of the value of the total rapidity span in the process when we evaluate the process at LO and when including the higher order corrections. This is one of the kinematical conditions de ning MRK: to have a non-growing with s invariant mass for the produced jets.

As we can see in the bottom plot of gure 1 the average pt is smaller when considering NLO contributions together with double logarithmic collinear terms than when we operate at LO. It is noteworthy to comment on the broadness of the distributions. A large contribution to the Green function and, hence, to cross sections stems from jets produced with a large transverse momentum which cannot be considered any longer as mini-jets. This feature is clearly seen at LO and gets reduced in higher-order calculations. This reduction is related of the shrinkage of the di usion picture at LO+Double Logs shown in our previous work in [1]. The areas under the di erent distributions are much larger at LO for any Y due to the strong suppression of the Pomeron intercept when going beyond LO.

Note that to produce the distributions in this work we have taken a random value for the azimuthal angle between the two most forward/backward tagged jets which changes event by event. We can then investigate the average angle [angbracketleft][angbracketright] per event at which the

remaining jets will be produced. This is shown in gure 2. At LO the bulk of the radiation carries an average angle in between [similarequal] [notdef] 1 which does not vary when Y changes. This is

also true when higher-order corrections are included in the analysis.

Let us now investigate the mean distance in rapidity between emissions in the BFKL ladder. We have addressed this point in gure 3 where the ratio [angbracketleft]Ry[angbracketright] has been numerically

investigated in detail. We observe that the di erential distributions for these ratios have their maximal contribution for [angbracketleft]Ry[angbracketright] larger than 0.5. Since these distributions are broad,

this implies that there are substantial contributions to the cross section from kinematical (preasymptotic) con gurations away from MRK. In an asymptotic MRK typical event we can consider the rapidity span to be populated by N emissions equally spaced at rapidity intervals of length in such a way that Y = (N + 1) (it is well-known that at asymptotically large energies = 1/

s!0 with ! = 4 ln 2, the LO Pomeron intercept. In our case

with

JHEP02(2016)064

s = 0.2 this implies asymptotic asLO [similarequal] 1.8 and asNLO+DoubleLogs [similarequal] 3.3). Hence, the rapidity corresponding to the i-th emission is (N + 1 i) and our average ratio becomes

{ 4 {

df/d <p t> (k a=10, k b=12, Y)

<pt> (in GeV) per event

df/d <p t> (k a=10, k b=12, Y)

<pt> (in GeV) per event

JHEP02(2016)064

df/d <p t> (k a=10, k b=20, Y)

<pt> (in GeV) per event

df/d <p t> (k a=10, k b=20, Y)

<pt> (in GeV) per event

df/d <p t> (k a=10, k b=20, Y=6)

<pt> (in GeV) per event

Figure 1. Distribution at LO (top/middle left) and NLO+Double Logs (top/middle right) in the average transverse momentum of emitted mini-jets per event, for di erent values of the transverse momentum of the most forward/backward tagged jets. In the plot at the bottom we compare the LO to the NLO+Double Log distribution for Y = 6.

independent of reading

[angbracketleft]Ry[angbracketright]MRK =

1N + 1

Xi=1i

i + 1 =

N + 1 (N + 2) + (1) N + 1

= 1 +

Y (1)

N

1 + Y !

[similarequal] 1 +

(1) + ln

Y + [notdef] [notdef] [notdef] (2.4)

{ 5 {

df/d <f> (k a=10, k b=12, Y)

df/d <f> (k a=10, k b=12, Y)

JHEP02(2016)064

<f> per event

<f> per event

df/d <f> (k a=10, k b=20, Y)

df/d <f> (k a=10, k b=20, Y)

<f> per event

<f> per event

df/d <f> (k a=10, k b=20, Y=6)

<f> per event

Figure 2. Distribution at LO (top/middle left) and NLO+Double Logs (top/middle right) in the average azimuthal angle of emitted mini-jets per event, for di erent values of the transverse momentum of the most forward/backward tagged jets. In the plot at the bottom we compare the LO to the NLO+Double Log distribution for Y = 6.

The last approximation is valid when Y . At LO we need a LO [similarequal] 0.6196 to reproduce all the maxima of the distributions for di erent Y ([angbracketleft]Ry[angbracketright]MRK [similarequal] 0.61, 0.70, 0.75

at Y = 4, 6, 8, respectively). At NLO this number increases to NLO+DoubleLogs [similarequal] 0.8097 ([angbracketleft]Ry[angbracketright]MRK [similarequal] 0.54, 0.64, 0.70 at Y = 4, 6, 8, respectively) (these numbers are quite similar

for the range of momenta here discussed) . This is consistent with having a reduced multiplicity in the nal state when the higher-order corrections are taken into account. As we increase the available scattering energy these MRK con gurations become more relevant

{ 6 {

df/d <R y> (k a=10, k b=12, Y)

df/d <R y> (k a=10, k b=12, Y)

<Ry> per event

<Ry> per event

JHEP02(2016)064

df/d <R y> (k a=10, k b=20, Y)

df/d <R y> (k a=10, k b=20, Y)

<Ry> per event

<Ry> per event

df/d <R y> (k a=10, k b=20, Y=6)

<Ry> per event

Figure 3. Distribution at LO (top/middle left) and NLO+Double Logs (top/middle right) in rapidity ratios of emitted mini-jets per event, for di erent values of the transverse momentum of the most forward/backward tagged jets. In the plot at the bottom we compare the LO to the NLO+Double Log distribution for Y = 6.

when constructing the gluon Green function but at phenomenological rapidity di erences we can see from our analysis that other kinematical regions also play an important role. An example of a kinematical set up with a smaller / larger [angbracketleft]Ry[angbracketright] than its maximal value

could be a nal state with all emissions at equidistant rapidities apart from one pair of jets whose relative ratio of rapidities could be particularly smaller (close to 0) / larger (close to 1) than the others.

This concludes our discussion of some of our numerical results. We nd that the observables here presented are worth of experimental investigation at the LHC. It is important

{ 7 {

to establish if the pre-asymptotic e ects are already present in the data. The characteristic broadening of the distributions that we have shown for [angbracketleft]pt[angbracketright], [angbracketleft][angbracketright] and [angbracketleft]Ry[angbracketright] is a distinct signal

of BFKL activity and should be put forward for experimental veri cation. In particular, since the collinear emissions do play a role in the form of double-log contributions, it would be interesting to gauge their importance at LHC data.

3 Summary & outlook

We have presented a set of observables characterizing multi-jet con gurations event by event (average transverse momentum, average azimuthal angle, average ratio of jet rapidities) which can be used to nd distinct signals of BFKL dynamics at the LHC. A numerical analysis has been shown using the Monte Carlo event generator BFKLex, modi ed to include higher-order collinear corrections in addition to the transverse-momentum implementation of the NLO BFKL kernel and the anti-kt jet algorithm as in FastJet. In order to have a cleaner theoretical background within collinear factorization we demand to always have two tagged forward/backward jets in the nal state.

The advantage of the LHC to study this type of physics is the large available center-of-mass energy together with high statistics, which allow for the possibility to apply strong kinematical cuts. These cuts are the key to pin down the multi-Regge kinematics and to propose new observables capable of discriminating this region of phase space from other, more conventional, ones. It is mandatory to make use of very exclusive observables in order to nd a precise window of applicability of the BFKL formalism which can then safely be extended to other, less restrictive, experimental setups.

Acknowledgments

G.C. acknowledges support from the MICINN, Spain, under contract FPA2013-44773-P. A.S.V. acknowledges support from Spanish Government (MICINN (FPA2010-17747,FPA2012-32828)) and to the Spanish MINECO Centro de Excelencia Severo Ochoa Programme (SEV-2012-0249).

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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