Studies of backward particle production with A Fixed-Target Experiment using the LHC beams

The foreseen capability to cover the far backward region at A Fixed-Target Experiment using the LHC beams allows to explore the dynamics of target fragmentation in hadronic collisions. In this report we briefly outline the required theoretical framework and discuss a number of studies of forward and backward particle production. By comparing this knowledge with the one accumulated in Deep Inelastic Scattering on target fragmentation, the basic concept of QCD factorisation could be investigated in detail.


I. INTRODUCTION
In hadronic collisions a portion of the produced particle spectrum is characterised by hadrons carrying a sizeable fraction of the available centre-of-mass energy, the so-called leading particle effect. It is phenomenologically observed that for such hadrons their valenceparton composition is almost or totally conserved with respect to the one of inital-state hadrons [1]. In pp collisions, for example, protons, neutrons and lambdas show a significant leading particle effect. Such semi-inclusive processes allow to test the scaling hypothesis for forward and backward hadron production cross sections [2,3] and give insight on nonperturbative aspects of QCD dynamics in high energy collisions. Moreover the production cross section for these particles peaks at very small transverse momenta with respect to the collision axis, a regime where perturbative techniques can not be applied.
Quite interestingly, the leading particle effect has been observed in processes which involve point-like probes in lepton-hadron interactions, such as Semi-Inclusive Deep Inelastic * Electronic address: federico.alberto.ceccopieri@cern.ch Scattering (SIDIS). At variance with the hadronic processes mentioned above, such a process naturally involves a large momentum transfer. The presence of a hard scale enables the derivation of a dedicated factorisation theorem [4,5] which ensures that QCD factorisation holds for backward particle production in DIS. The relevant cross sections can then be factorised into perturbatively calculable short-distance cross sections and new distributions, fracture functions, which simulatenously encode information both on the interacting parton and on spectator fragmentation into the observed hadron. Despite of being non-perturbative in nature, their scale dependence can be calculated within perturbative QCD [6]. Fracture functions obey in fact DGLAP inhomogeneous evolution equations which result from the subtraction of collinear singularities in the target-fragmentation region [6,7]. The factorisation theorem [4,5] guarantees that fracture functions are universal distributions, at least in the context of SIDIS.
Detailed experimental studies of hard diffraction at HERA have shown to support the hypothesis of QCD factorisation and evolution inherent the fracture functions formalism.
Furthermore they led to a quite accurate knowledge of diffractive parton distributions [8][9][10], a special case of fracture functions in the very backward kinematic limit. For particles other than protons, proton-to-neutron fracture functions have been extracted from a pQCD analysis of forward neutron production in DIS in Ref. [11]. A set of proton-to-lambda fracture functions has been obtained by performing a combined pQCD fit to a variety of semi-inclusive DIS lambda production data in Ref. [12].
As theoretically anticipated in Ref. [4,13,14] and experimentally observed in hard diffraction in pp collisions at Tevatron [15,16], QCD factorisation is violated for fracture functions in hadronic collisions. On general grounds, it might be expected, in fact, that the dynamics of target-remnants hadronisation is affected by the coloured environment resulting from the scattering in a rather different way with respect to the Deep Inelastic Scattering case.
Nonetheless, the tools mentioned above allow to investigate quantitatively particle production mechanisms in the very backward and forward regions, to test the concept of factorisation at the heart of QCD and to study the dependencies of factorisation breaking upon the species and the kinematics of the selected final state particle.
This physics program could be successfully carried on at A Fixed-Target Experiment using the LHC beams [17]. Novel experimental techniques are, in fact, available to extract beam-halo protons or heavy-ions from LHC beams without affecting LHC performances.
Such a resulting beam would be then impinged on a high-density and/or long-lenght fixed target, guaranteeing high luminosities. Furthermore and most importantly for the physics program to be discussed here, the entire backward hemisphere (in the centre-of-mass system of the collision) would be accessible with standard experimental techniques allowing high precision studies of target fragmentation. Althought measurements of particle production in the very forward region (close to the beam axis) might be challenging experimentally due to the high particle densities and large energy flow, the installation of dedicated detectors, like forward neutron calorimeters and/or proton taggers, could further broaden the physics program oulined above giving access to the beam fragmentation region.
The paper is organised as follows. In Sec. II we first give a brief theoretical introduction on the fracture functions formalism and to higher order correction to the semi-inclusive Drell-Yan process. In Sec. III we outline different research areas and physics goals that can be approached within the formalism. In Sec. IV we summarise our results.

II. COLLINEAR FACTORISATION FORMULA
Fracture functions, originally introduced in DIS, do depend on a large momentum transfer. Therefore, in order to use them in hadronic collisions, a hard process must be selected.
We consider here the semi-inclusive version of the Drell-Yan process, in which one hadron H is measured in the final state together with the Drell-Yan pair.
In such a process the high invariant mass of the lepton pair, Q 2 , allows the applicability of perturbative QCD, while the detected hadron H can be used, without any phase space restriction, as a local probe to investigate particle production mechanisms.
The associated production of a particle and a Drell-Yan pair in term of partonic degrees of freedom starts at O(α s ). One of the contributing diagrams is depicted in Fig. (1).
Assuming that the hadronic cross-sections admit a factorisation in term of long distance non-perturbative distributions and short distance perturbative calculable matrix elements for the partonic process i(p 1 )+j(p 2 ) → l(k)+γ * , predictions based on perturbative QCD are section, at centre of mass energy squared S, can be symbolically written as [18,19] dσ H,C,(1) The introduction of fracture functions opens the possibility to have particle production already to O(α 0 s ), since the hadron H can be non-pertubatively produced by a fracture function M itself. The lowest order parton model formula can be symbolically written as i (x 1 , z) f [2] j (x 2 ) + M [2] i (x 2 , z) f [1] j (x 1 ) where the superscripts indicate from which incoming hadron, H 1 or H 2 , the outgoing hadron H is produced through a fracture functions. This production mechanism is sketched in be integrated over and the resulting contribution added to virtual corrections. One of the contributing diagrams is depicted in Fig. (3). The general structure of these terms is i (x 1 , z) f [2] j (x 2 ) + M [2] i (x 2 , z) f [1] j (x 1 ) We refer to this corrections term as to the target fragmentation contribution. The calculation is, a part from minor differences in kinematics, completely analogue to the inclusive Drell-Yan case. The factorisation procedure amounts to substitute in eq. (3) the bare fracture and parton distributions functions with their renormalised version [18,19]. renormalised parton distributions and fracture functions homogeneous terms do cancel all singularities present in eq. (4). The additional singularities in eq. (2) are cancelled by the combination of parton distributions and fracture functions inhomogeneous renormalisation terms. Adding all the various contributions, the resulting hadron-p t integrated cross section, up to order O(α s ), is then infrared finite [18,19] and can be simbolically written as where σ 0 = 4πα 2 em /9SQ 2 . We refer to the previous equation as to the collinear factorisation formula for the process under study. The next-to-leading order coefficients C ij and K ij l have been calculated in Ref. [19], making the whole calculation ready for numerical implementation. We stress again, however, that our ability to consistently subtract collinear singularities in such a semi-inclusive process is a necessary but not sufficient condition for factorisation to hold in hadronic collisions. This proof in fact does involve only the so-called active partons but completely neglects multiple soft parton exchanges between active and spectators partons, whose effects should be accounted for in any proof of QCD factorisation. Therefore there is no guarantee that fracture functions extracted from DIS can be successfully used to describe forward or backward particle production in hadronic collions. Up this extremely important caveat, in the following section we outline some applications of the proposed formalism. are described in Ref. [17]. Among others, of particular importance from our perspective, are the planned measurements of Z and W electroweak boson production, which allows to make direct contact with the theoretical calculation desribed in the preceding section.
As a first benchmark process of the porposed formalism we consider strange particle production associated with a Drell-Yan pair, p + p → V + γ * + X, where V generically indicates either a Λ 0 orΛ 0 hyperon. At very low transverse momentum and in the backward region, Λ 0 spectrum should show a significant leading particle effect, which can be predicted, assuming factorisation as in dσ H,T , by the proton-to-lambda fracture functions set from DIS data of Ref. [12]. The measurament ofΛ 0 hyperon spectrum in the same kinematical conditions should instead show a largely reduced laeding particle effect, characterised by a sharp drop of the latter at mid and high rapidity. On the other hand, if one considers Λ 0 or Λ 0 at sufficiently large transverse momentum, their combined analysis, described by dσ H,C , should allow the investigation of parton hadronisation into hyperons in the QCD vacuum as parametrised by fragmentation functions.
The formalism may found application in the study of single diffractive hard process, p+p → p+γ * +X, where the backward recoiling proton is measured in central AFTER@LHC detector [17]. Diffractive processes has been intensively analysed in the DIS at HERA, revealing their leading twist nature. From scaling violations of the diffractive structure functions [8] and dijet production in the final state [9,21] quite precise diffractive parton distributions functions (DPDFs) have been extracted from data, which parametrise the parton content of the color singlet exchanged in the t-channel. The comparison of QCD predictions for single diffractive hard processes based on diffractive parton distributions measured at HERA against data measured at Tevatron [15,16,26] has indeed revealed that these processes are, not unexpectedly [13,14], significantly suppressed in hadronic collisions, see the recent analysis reported in Ref. [22]. Recalling that these distributions are fracture functions in the z → 1 limit, the present formalism can then applied to next-to-leading order accuracy, the main contribution to the cross-sections coming from the target term, dσ H,T .
Such term can be eventually recast in triple differential form in x IP ≃ 1 − z, virtual photon rapidity, y, and invariant mass Q 2 and evaluated at next-to-leading order by using the appropriate coefficient functions [23]. In this way factorisation tests could be porformed at fixed x IP to avoid any Regge factorisation assumption on DPDFs [10] while the y dependence, giving direct access to the fractional parton momentum in the diffractive exchange, β, allows to test factorisation in a kinematic region which avoids DPDFs extrapolation. Finally, the Q 2 dependence of the cross-sections could be used to investigate how factorisation breaking effects eventually evolve with the hardness of the probe and to which extent the factorized formula M ⊗ f actually works.
A completely analogous program can be performed for the associated production of forward neutron and a Drell-Yan pair, p + p → n + γ * + X. The production of forward neutron in DIS at HERA has shown a leading twist nature. From scaling violations of the semiinclusive neutron structure functions a set of proton-to-neutron fracture functions set has been extracted from data in Ref. [11] which can be used to predict forward neutron rate in hadronic collisions. As in the case of hard diffraction, both physics programs would highly benefit from the installation of a dedicated instrumentation for the measuraments of fast neutron and proton quite close to the beam axis. Measuraments in the forward region, altought problematic experimentally, give direct access to the study of the beam fragmentation region.
We wish to conclude this section by considering a rather different research area. The associated production of one particle and a Drell-Yan pair can be used to investigate the contamination of the so-called underlying event [24] to jet observable and has been successfully used to study underlying event properties in Drell-Yan process [25]. If the detected hadron is measured at sufficiently large transverse momentum, the latter constitutes a natural infrared regulator for the partonic matrix elements. In this kinematics conditions we also expect a rather small contributions from fracture functions. Therefore the central term, dσ H,C, (1) , can be used to estimate the single parton scattering contribution to the process.
The latter might be considered as the baseline to study the contributions of double (or multiple) parton scattering contributions to the same final state, where, for example, the primary scatter produces a Drell-Yan pair while the secondary one produces the detected hadron H. Within this context, we note that the dσ H,C term, altough formally O(α s ), is a tree level predictions and possible large higher order corrections may be expected especially in the high rapidity region at large transverse momentum. In view of this fact a O(α 2 s ) calculation of particle production at finite transverse momentum associated with a Drell-Yan pair, as performed in DIS [27], would be indeed highly desiderable.

IV. CONCLUSIONS
We have briefly reviewed a perturbative approach to single particle production associated with a Drell-Yan pair in hadronic collisions. On the theoretical side we have shown that the introduction of fracture functions allows a consistent factorisation of new class of collinear singularities arising in this type of calculation. The factorisation procedure does coincide with the one used in DIS confirming, as expected, the universal structure of collinear singularities among different hadron-initiated processes and supporting the proposed collinear factorisation formula. On the phenomenological side we have briefly discussed a few applications in which different aspects of the formalism could be tested. The improved theoretical control on the perturbative component indeed allows the study of new or, to date, poorly known phenomena appearing in hadronic collisions, for example the rapidity gap probability suppression in hard diffractive processes respect to diffractive DIS and the investigation, although indirect, of multiple parton-parton interactions by predicting the radiation associated with a Drell-Yan pair in the single parton scattering approximation. These physics programs, given the experimental capabilities and design advantages foreseen at AFTER@LHC, may