June 2004, corrected version
PM/0405
hepph/0404162
Neutralino pair production at CERN LHC^{†}^{†}†Work supported by the European Union under contract HPRNCT200000149..
G.J. Gounaris, J. Layssac, P.I. Porfyriadis and F.M. Renard
Department of Theoretical Physics, Aristotle University of Thessaloniki,
Gr54124, Thessaloniki, Greece.
Physique Mathématique et Théorique, UMR 5825
Université Montpellier II, F34095 Montpellier Cedex 5.
Abstract
We consider the production of neutralino pairs at a high energy hadron collider, putting a special emphasis on the case where one of them is the lightest neutralino , possibly constituting the main Dark Matter component. At tree level, the only relevant subprocess is , while the subprocess first appears at the one loop level. Explicit expressions for the helicity amplitudes are presented, including the tree level contributions and the leadinglog one loop radiative corrections. For the oneloop process, a numerical code named PLATONggnn is released, allowing the computation of in any MSSM model with real soft breaking parameters. It turns out that acceptable MSSM benchmark models exist for which the and the gluonic contributions may give comparable effects at LHC, due to the enhanced gluonic structure functions at low fractional momenta. Depending on the values of the MSSM parameters, we find that the LHC neutralino pair production may provide sensitive tests of SUSY models generating neutralino Dark Matter.
PACS numbers: 12.15.y, 12.15.Lk, 13.75.Cs, 14.80.Ly
1 Introduction
Neutralino production at hadron colliders is an important part of the program of Supersymmetry (SUSY) searches [1, 2]. One special reason is related to the possibility that , the lightest neutralino state, is in fact the Lightest Supersymmetric Particle (LSP) [3]. This has two particular consequences; the first concerning the supersymmetric spectroscopy (chains and rates of decays) in Rparity conserving models [1]; while the second largely determining the search for Dark Matter (DM) [4].
DM detection in such a case is expected to occur either in a direct way (e.g. through the observation of nucleus recoil in elastic scattering); or in an indirect way, by observing modifications of the cosmic spectrum of particles like photons, positrons, antiprotons etc., due to contributions from annihilation [5]. Concerning the indirect way, we have presented in two previous papers the results of a complete oneloop computation for the processes involved in DM annihilation [6], as well as the results for the reversed process of neutralino pair production at a photonphoton collider [7]. In [6] we have also emphasized that in certain benchmark MSSM models the gluongluon channel may be important for determining the neutralino relic density [8, 9, 10].
We would expect therefore, that for neutralino pair production at a high energy hadron collider like LHC, kinematical domains may exist where the gluon structure function of the proton is so large [11], that the oneloop gluon annihilation contribution may in fact be bigger than the tree level contribution. The precise study of such neutralinopair production process at LHC, through the subprocesses and , constitutes the aim of the present paper.
First, we present the helicity amplitudes and cross sections of the subprocess at tree level. At this level, such a process has been studied long ago e.g. in [12, 13]. We go beyond this though by also exploring the Fermi statistics and CP (for real MSSM parameters) constraints on these amplitudes, which strongly reduce their number and also serve as a check of the calculations.
In a second step, and in order to check the possible existence of important oneloop electroweak (EW) contributions to these amplitudes, the leading () and subleading () contributions are included, following the procedure established in [14, 15, 16]. These EW corrections reduce the overall tree level magnitude of the amplitudes by an amount that can reach the few tens of percent level for the kinematical domain attainable at LHC. In the same direction acts also the SUSY QCD contribution^{1}^{1}1These involve QCD interactions explicitly affecting SUSY particle exchanges [16]. calculated according to the rules established to order in [16, 17]. We should emphasize at this point though, that these EW and SUSY QCD corrections should be considered in addition to the pure QCD (leading and nexttoleading) corrections which strongly increase the tree level amplitudes, as found in [17].
We then turn to the oneloop subprocess , for which the helicity amplitudes are calculated using the set of diagrams established in [6, 7]. There, the neutralino annihilation amplitudes were calculated under any kinematical conditions; but the accompanying numerical codes compute the neutralino MSSM annihilation cross section to gluons only at the appropriate for dark matter threshold region [18]. Using these results, the numerical code PLATONggnn has been also constructed, which calculates the reversed process cross section for any values and any MSSM model with real soft breaking parameters [18].
We then compute the LHC cross sections for , by convoluting the and subprocess cross sections, with the corresponding quark and gluon distribution functions in the initial protons . We then discuss the contributions of both subprocess to several observables (invariant mass, transverse momentum and angular distributions) and we give illustrations for an extensive set of benchmark models in MSSM. As we will see below, depending on the choice of MSSM parameters and the kinematical regions looked at, the one loop subprocess may occasionally give comparable or even larger effects, than the tree level one.
These results imply an interesting complementarity between the future LHC measurements, the related measurements at a future Linear Collider and the Dark Matter searches in cosmic experiments.
The contents of the paper is the following. Sect.2 is devoted to the process . The general properties of the helicity amplitudes are studied in the subsection 2.1, where the seven basic independent amplitudes are identified. The treelevel helicity amplitudes and cross sections are subsequently presented in Section 2.2 and Appendix A.1; while the electroweak and SUSY QCD corrections to the helicity amplitudes, at leading and subleading logarithmic accuracy, are given in Section 2.3 and Appendix A.2. In Sect.3, the one loop process is presented. Applications to neutralino pair production at LHC using the parton formalism are given in Sect.4, where the numerical results are also discussed. The concluding remarks are given in Sect.5, while the parton model kinematics is detailed in Appendix B.
2 The subprocess
2.1 Generalities about Helicity amplitudes for
For an incoming pair, and an outgoing pair of neutralinos, the process is written as
(1) 
where and^{2}^{2}2The possible values of the helicities are, as usually, taken as . are the momenta and helicities of the incoming and outgoing particles. The usual Mandelstam variables for the subprocess are defined as
(2) 
Since the top quark structure function is vanishing in the proton and the other quarks are not too heavy, the incoming and in (1) are taken as massless as far as the kinematics is concerned, but we keep the potentially large (particularly for the third family) Yukawa contributions to the couplings. Finally denote the masses, respectively.
The helicity amplitude for process (1) is denoted as
(3) 
where is scattering angle in the c.m. of the subprocess. In these amplitudes, and are treated as particles No2 in the JW conventions [19]. To consistently take into account the Majorana nature of the neutralinos, we always describe the No1 neutralino through a positive energy Dirac wave function, while the No2 JW particle is described through a negative energy one^{3}^{3}3The same convention is also followed for the treated below.. Fermi statistics for the final neutralinos then implies^{4}^{4}4We note that (4), which is induced by the anticommuting nature of the Fermionic fields, does not generally agree with the neutralino (anti)symmetry property assumed in [20].
(4) 
while CPinvariance, valid for real soft breaking and parameters, gives
(5) 
where is the CPeigenvalue of the neutralino [21].
2.2 Born amplitudes and cross sections.
The Born amplitude for the process in (1) contains three diagrams (see Fig.1abc) involving , and channel exchanges and written as [13]
(10) 
(11) 
where the index refers to the summation over the exchanged L and R squarks of the same flavor in the t and uchannel, , and describe the final neutralinos.
Explicit expressions for the seven basic helicity amplitudes listed in (6), are given in (A.1, A.2, A.3, A.4) in Appendix A.1.

They involve the L and R couplings defined as
(12) where
(13) with , being the isospin and charge of the various quarks.

The Zneutralino couplings satisfying
(14) with
(15) where denotes the neutralino mixing matrix in the notation of [22]).

And the neutralinoquarksquark couplings
(16) where
(17)
In (17), refer to the incoming up and down quark (antiquark) of any family^{5}^{5}5As usual, we will only consider nonvanishing structure functions for incoming u, d, s, c and b quarks., while denote the corresponding squarks. We also note that the mixing matrices in (15, 17), control the Bino, Wino, Higgsino components of the neutralino in the and coupling [22, 21]. Finally, we remark that channel Born part gives nonvanishing contributions only for purely higgsino production, whereas the channel Born parts for purely gaugino.
One can then compute the differential cross section either through (9) and the helicity amplitudes in (A.1, A.2, A.3, A.4), or directly by the trace procedure giving
(18) 
where^{6}^{6}6Analogous expression for gaugino production in Colliders have appeared in [23].
(19)  
where are the neutralino masses and are defined in (15).
2.3 One loop electroweak and SUSY QCD
corrections
to
In principle, one loop EW corrections for should be taken into account, particularly because the energy reach at LHC is so big, that the large logarithmic contributions to the amplitudes may reach the few tens of percent level [14, 15, 16]. In the models we have considered, this implies a reduction of the cross sections sometimes by almost a factor of two, while preserving their shape. Since the nonlogarithmic oneloop contributions seem to lie at the few percent level, which is also the level of the expected experimental accuracy, it may be adequate to ignore these difficult to calculate effects in at LHC energies.
In this section we present therefore, the leading and subleading EW logarithmic contributions to the helicity amplitudes, following [14, 15, 16], where applications for LC and LHC have been given. They are separated into three types of terms which are:

Universal electroweak (EW) terms. These are processindependent terms appearing as correction factors to the Born amplitude. They consist of ”gauge” and ”Yukawa” contributions associated to each external line and determined by its quantum numbers and chirality. Their expressions for a quark or neutralino line are respectively determined as follows:
External quark line of chirality : Since all quarks are taken as massless as far as the kinematics are concerned^{7}^{7}7The large third family Yukawa terms appear only as couplings and do not concern the kinematics., the quark lines correspond to a definite chirality . The induced correction then is
(20) where describes the corresponding Born amplitude from (10) involving an external quark line of chirality (which at high energies is essentially equivalent to the helicity), while count the mass eigenstates of the neutralinos. The coefficient in (20) is written as
(21) with the gauge contribution being
(22) where is the full isospin of the quark with chirality or , and defines its hypercharge in terms of the third isospin component . Correspondingly, the Yukawa term (for quarks only) is
(23) We note that an external antiquark line should be counted separately giving an additional contribution determined by the same formulae (2023). Moreover, the same formulae describe also the logarithmic contributions associated with each external squark, antisquark, lepton or slepton line [14, 15, 16].
For an external neutralino line of chirality , it is convenient to use a matrix notation
(24) and to separate the higgsino from the gaugino components of the matrix elements:
(25) with