Forward and very backward jet inclusive production at the LHC
In the spirit of Mueller-Navelet dijet production, we propose and study the inclusive production of a forward and a very backward jet at the LHC as an observable to reveal high-energy resummation effects à la BFKL. We obtain several predictions, which are based on the various mechanisms discussed in the literature to describe the production of the , namely, NRQCD singlet and octet contributions, and the color evaporation model.
The understanding of the high energy behaviour of QCD in the perturbative Regge limit remains one of the most important and longstanding theoretical questions in particle physics. In the linear regime where gluonic saturation effects are not expected to be essential, QCD dynamics are described using the BFKL formalism [1, 2, 3, 4], in the -factorization [5, 6, 7, 8, 9, 10, 11] framework. In order to reveal these resummation effects, first with leading logarithmic (LL) precision, and more recently at next-to-leading logarithmic (NLL) accuracy, many processes have been proposed. One of the most promising ones is the inclusive dijet production with a large rapidity separation, as proposed by Mueller and Navelet . This idea led to many studies, now at the level of NLL precision.
Recent -factorization studies of Mueller-Navelet jets [13, 14, 15, 16, 17, 18, 19, 20] were successful in describing such events at the LHC , exhibiting the very first sign of BFKL resummation effects at the LHC. To test the universality of such effects, we propose to apply a similar formalism to study the production of a forward meson and a very backward jet with a rapidity interval that is large enough to probe the BFKL dynamics but small enough for both the and the jet to be in the detector acceptance at LHC experiments such as ATLAS or CMS.111For example, at CMS the CASTOR calorimeter allows one to tag a jet down to in rapidity while the could be reconstructed up to , thus with a maximum interval in rapidity of almost 9, more than sufficient to see BFKL resummation effects. Although mesons were first observed more than 40 years ago, the theoretical mechanism for their production is still to be fully understood and the validity of some models remains a subject of discussions (for recent reviews see for example refs. [22, 23]). In addition, most predictions for charmonium production rely on collinear factorization, in which one considers the interaction of two on-shell partons emitted by the incoming hadrons, to produce a charmonium accompanied by a fixed number of partons. On the contrary, in this work the meson and the tagged jet are produced by the interaction of two collinear partons, but with the resummation of any number of accompanying unobserved partons, as usual in the -factorization approach.
Here we will compare two different approaches for the description of charmonium production. First we will use the NRQCD formalism , in which the charmonium wavefunction is expanded as a series in powers of the relative velocity of its constituents. Next we will apply the Color Evaporation Model (CEM), which relies on the local-duality hypothesis [25, 26]. Finally we will show numerical estimates of the cross sections and of the azimuthal corrrelations between the and the jet obtained in both approaches. We will rely on the Brodsky-Lepage-Mackenzie (BLM) procedure  to fix the renormalization scale, as it was adapted to the resummed perturbation theory à la BFKL in refs. [28, 29], which some of us applied to Mueller-Navelet jets in ref. . Below, we will only discuss in detail the new elements related to the various production mechanisms. All details related to the BFKL evolution at NLL can be found in refs. [13, 14], while the details related to the application of the BLM scale fixing in our study are presented in ref. .
2 Determination of the meson vertex
We start with the determination of a general meson production vertex (the fact that we will restrict ourselves to in the rest of this paper plays no role at this stage). For the moment, we do not consider any specific model for its production. We generically denote with an index the kinematical variables attached to the system made of the meson and the possible accompanying unobserved particles, and use an index for the kinematical variables attached to the meson itself.
The inclusive high-energy hadroproduction process of such a meson , via two gluon fusion, with a remnant and a jet with a remnant separated by a large rapidity difference between the jet and the meson, in scattering of a hadron with a hadron , is illustrated in figure 1, where as a matter of illustration, we consider the parton coming out of the hadron to be a gluon and the parton coming out of the hadron to be a quark. For the sake of illustration, we suppose that the meson is produced in the fragmentation region of the hadron , named as forward, while the jet is produced in the fragmentation region of the hadron , named as backward. On one hand, the longitudinal momentum fractions of the jet and of the meson are assumed to be large enough so that the usual collinear factorization applies (the hard scales are provided by the heavy meson mass and by the transverse momentum of the jet), and we can neglect any transverse momentum, denoting the momentum of the upper (resp. lower) parton as (resp. ), their distribution being given by usual parton distribution functions (PDFs). On the other hand, the channel exchanged momenta (e.g. in the lhs of figure 1, or the various ones involved in the rhs of figure 1) between the meson and the jet cannot be neglected due to their large relative rapidity, and we rely on factorization.
where are the standard parton distribution functions of a parton in the according hadron.
In -factorization, the partonic cross section reads
where is the BFKL Green’s function depending on , denoting as the center-of-mass energy of the two colliding hadrons.
In the definition of , is to be used for an initial gluon and for an initial quark. Following the notations of refs. [30, 31], the dependence of on the jet variables is implicit. At next-to-leading order (NLO), the jet can be made of either a single or two partons. The explicit form of these jet vertices can be found in ref. as extracted from refs. [30, 31] after correcting a few misprints of ref. .
In the rest of the present paper, we will only focus on the case where the meson vertex is treated at lowest order, while the Green’s function and the jet vertex will be treated at NLL. The computation of the NLO vertex, which is a quite involved task, is left for further studies.
To properly fix the normalization, let us focus for a moment on the Born approximation, see the lhs of figure 1. Then, each building block in the factorized formula (2) is treated at lowest order. In this limit, our normalizations are such that the Born Green’s function is
where is the channel exchanged momentum.
In the high-energy limit, the -matrix reads
where is the color index of a collinear gluon from the hadron and is the color index of the exchanged channel gluon. Here denotes the -matrix element describing the transition. Its computation will be discussed in detail in the following subsections. After factorization, illustrated symbolically by figure 2,
The phase space measure reads333This should be understood in an extended way, in particular due to the fact that might involve several particles, as it is the case for the color singlet NRQCD contribution.
It can be written in a factorized form in terms of the rapidity of the quark jet and its transverse momentum :
This -factorization formula involves an integration over the transverse momentum of the four-momentum transfer in the channel between both vertices. Using the expressions of the unpolarized quark PDF
and of the unpolarized gluon PDF,
we obtain an expression for the differential cross section
in which we factorized out the vertex for quark jet production in the Born approximation,
2.1 Color-singlet NRQCD contribution
In the color-singlet contribution the system is made of the produced charmonium and of the unobserved gluon produced simultaneously with the charmonium in gluon-gluon fusion due to the negative charge-parity of the . We parametrize the momentum of the and the momentum of the unobserved gluon in terms of Sudakov variables, as
Thus the expression of
permits, with the use of (13), to write the differential cross section in the form
from which we read off the production vertex of the color singlet NRQCD contribution as
One should note that the above expressions include an integration over the phase space of the unobserved gluon with momentum The vertex which allows to pass from open production to production in color singlet NRQCD reads [34, 35]
with the momentum , being the mass of the charm quark, . In the following we will use the non-perturbative coefficient defined as
The matrix element in NRQCD is related to the leptonic meson decay rate by 
Here is the fine-structure constant and is the electric charge of the charm quark. Equation (21) includes the one-loop QCD correction [36, 37, 38] and is the strong coupling constant. One can use the value of this decay rate to fix through this relation. Namely, using the values GeV , GeV and a three-loop running coupling with GeV, we obtain GeV. As quoted in ref. , recent phenomenological analyses [41, 42, 43] have used slightly smaller values of either 0.387 or 0.440 GeV, as obtained in refs.  and  respectively. In order not to underestimate the uncertainty, in the following we will vary between 0.387 and 0.444 GeV.
The momentum transfer in the channel entering the charmonium vertex has the approximate form given by eq. (6). The momentum conservation in the charmonium vertex leads to the following relations between the Sudakov variables of momenta:
The contribution to the hard part is given by the 6 diagrams shown in figure 3, which leads to the expressions
where and Tr denote respectively the color and the Dirac traces. Let us observe the following relations between the Dirac traces of diagrams due to charge conjugation invariance:
Consider the color factor: the symmetry property (29) results in the appearance in the sum of all diagrams of the symmetric structure constants of the color group only. Thus, we obtain
where we introduced the shorthand notation for the sum of all six diagrams contributing to production within the color singlet mechanism. One can check that this sum vanishes in the limit as it should be the case for an impact factor in factorization due to its gauge invariance. For the gluon , we choose the gauge444Note that the sum of diagrams in this color singlet mechanism is gauge invariant, although the channel gluon is off-shell: indeed due to the simple single color structure which factorizes, they are QED like.
which is a natural choice for a meson emitted in the fragmentation region of the hadron of momentum The three different traces then read
The denominators appearing in the expression for are equal to
The cross section is obtained by squaring the sum of diagrams , i.e. by contracting this sum with its complex conjugate through the polarization tensors for the and the gluon and the projection operator related to the factorization of the gluonic PDF, namely
Thus we obtain that
which by taking into account eq. (18) gives the production vertex in the form