Probing the BFKL dynamics in inclusive three jet production at the LHC

We propose the study of new observables in LHC inclusive events with three tagged jets, one in the forward direction, one in the backward direction and both well-separated in rapidity from the each other (Mueller-Navelet jets), together with a third jet tagged in central regions of rapidity. Since non-tagged associated mini-jet multiplicity is allowed, we argue that projecting the cross sections on azimuthal-angle components can provide several distinct tests of the BFKL dynamics. Realistic LHC kinematical cuts are introduced.


Introduction
The Balitsky-Fadin-Kuraev-Lipatov (BFKL) resummation program in the leading logarithmic (LL) [1,2,3,4,5,6] and next-to-leading logarithmic (NLL) approximation [7,8] may be applied for phenomenological studies at hadronic colliders when the final-state is characterised by jets that are produced at large relative rapidities.Mueller-Navelet jets [9] is an key example, specifically, for observables that are based on the azimuthal angle formed by the two outermost in rapidity tagged jets, φ.The precise form of the observables is built by considering ratios of projections on the azimuthal angle R m n = cos (m φ) / cos (n φ) .Comparison of different NLL predictions against LHC experimental data for these observables has been quite successful [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30].New LHC observables, that may be seen as a generalisation of the Mueller-Navelet jets, were proposed recently for inclusive three-jet [31,32] and four-jet production [33,34].In this work we discuss only the observables for inclusive three-jet production.These are defined by the generalised ratios [31] where φ 1 is the azimuthal angle difference between the forward and the central jet and φ 2 the azimuthal angle difference between the central jet and the backward in rapidity jet.The ratios R M N P Q in Eq. ( 1) are actually partonic level quantities and therefore, cannot be readily compared to experimental data.Therefore, we define the hadronic level observables R M N P Q [32] and study their stability once we introduce corrections beyond the LL accuracy.For that, we produce the two outermost in rapidity jets within the collinear factorization scheme, each of them associated with a forward "jet vertex" [35].Then we link these vertices and the central jet using two BFKL gluon Green's functions.At the end, the partonic differential cross-section is convoluted with collinear parton distribution functions and is integrated over the momenta of all produced jets in order to calculate the ratios R M N P Q .For the integration over the momenta of the jets we use standard LHC experimental cuts.The rapidity of the central jet takes values close to the middle of the rapidity distance between the two outermost tagged jets.

Hadronic inclusive three-jet production in multi-Regge kinematics
Let us first remember some of the notation defined in [31,32].Assuming that the transverse momenta of the outermost jets are k A,B , their rapidity difference, Y , is large and the central jet has transverse momentum k J .We allow for mini-jet activity between the three tagged jets so that the process1 we need to study is Firstly, we define the two relative azimuthal angles between the outermost jets and the central jet as ∆θ AJ = θ A − θ J − π and ∆θ JB = θ J − θ B − π respectively.Then the projection on azimuthal-angle components gives the mean value where M, N are positive integers) and dσ 3−jet the partonic deferential cross-section for three-jet production defined in [31].In order to compute theoretical estimates that may be compared against current and future experimental data, we integrate C M,N over the momenta of the tagged jets in the form  We are interested in maximising the stability with respect to higher order effects (beyond LL) in our results (see [15]), therefore, we remove the zeroth conformal spin contribution of the BFKL kernel by considering the ratios which have no n = 0 dependence.Thus, we can study the ratios R M N P Q (Y ) in Eq. ( 5) as functions of the rapidity difference Y between the outermost jets for some typical values of M, N, P, Q.We define three different p T ranges (bins) for the allowed momentum of the central jet: This permits the discrimination of different behaviours of the R M N P Q (Y ) by using as a criterion the relative size of the central jet.In Fig. 1 we show the behaviour of R 12  22 as we change the size of the central jet and its position in rapidity.We notice that while a small variation in y J around the central rapidity value ∆Y A,B /2 = 5 does not result in significant changes for a fixed k J , a change in the value of k J may have a big impact for a fixed y J .A number of different ratios was presented in [32], here we are focussing on ratios that involve the coefficients C 12 and C 22 .In Figs. 2 and 3 we see the LL accuracy results for R 22  12 .Generally, the dependence of the different observables on the rapidity difference between k A and k B is rather smooth whereas the slope of the three curves depends on the particular observable.For R 22  12 we see that shifting from a symmetric to an asymmetric cut makes no noticeable difference.Moreover, there are no important changes when we change the colliding energy from √ s = 7 TeV to √ s = 13 TeV.The latter is crucial as it suggests that R 22  12 is already within some sort of asymptotic regime for the specific kinematical configurations.
Apart from the stability of the observable with regard to an increase of the colliding energy, another important question is the stability with respect to effects that go beyond the LL approximation [39].A first important step towards a full NLL computation is to take into account the NLL contributions to the two gluon Green's functions that connect the three jets.In Fig. 4 we present exactly these corrections obtained by using the Brodsky-Lepage-Mackenzie (BLM) prescription [40] for the R 22  12 coefficient in the asymmetric cut.In particular, we have used the MOM scheme and chosen the renormalisation scale such that the β 0 -dependence of the given observable vanishes, following the BLM prescription.The dashed lines represent the LL predictions and the coloured bands represent the NLL BLM predictions.It is impressive that the NLL values are almost on top of the LL ones which gives us great confidence that the observables R M N P Q are indeed excellent BFKL probes at the LHC.

Figure 1 :
Figure 1: 3D plot of the partonic R 12 22 as a function of the rapidity y J and the momentum k J of the central jet for k A = 40 GeV, k B = 50 GeV and ∆Y A,B = 10.

22 s
where the rapidity of the forward jet takes values in the range Y min A = 0 and Y max A = 4.7 and that of the backward jet in the range Y min B = −4.7 and Y max B = 0 while their difference Y ≡ Y A − Y B is kept fixed at definite values in the range 5 < Y < 9. We calculate C M N for two different center-of-mass energies, √ s = 7 and √ s = 13 TeV and we introduce two typical kinematical cuts previously used in the study of Mueller-Navelet jets at the LHC.Specifically, we use both a symmetric and an asymmetric cut [19, 29]: 1. k min A = 35 GeV, k min B = 35 GeV, k max A = 13 TeV; k B min = 50 GeV