# Search for neutralino pair production at the CERN LHC

###### Abstract

We provide the next-to-leading order (NLO) predictions for the neutralino pair production via quark-antiquark annihilation and gluon-gluon fusion at the CERN Large Hadron Collider, focusing on the lightest neutralino which is likely to be the lightest supersymmetric particle. The dependence of total LO, NLO cross sections, and K factor on the center-of-mass energy, the - mass plane, the squark mass, and the factorization and renormalization scales is comprehensively analyzed for three different scenarios in the minimal supersymmetric standard model and the constrained minimal supersymmetric standard model. We find that the LO cross section is considerably increased by the NLO correction, and the K factor value is clearly related to the Higgsino/gaugino mass parameters, the squark mass, and the factorization and renormalization scales.

###### pacs:

11.30.Pb, 12.60.Jv, 14.80.Ly, 14.80.Nb## I Introduction

Weak scale supersymmetry (SUSY) (see, e.g., Haber (); Nilles (); wss ()) naturally involves an elegant mechanism
for stabilizing the gauge hierarchy with regard to the effects of
radiative corrections and allows unification of gauge
couplings. Under the conservation of R-parity^{*}^{*}*R-parity, which is a discrete and multiplicative symmetry, is defined by
where , and denote the baryon number, lepton number, and spin of the particle, respectively Fayet (); Kazakov (). Thus, this quantity is equal to
for the particles of the Standard Model (including the Higgs bosons) and for their
superpartners., it also provides a candidate
for the dark matter (DM) postulated to explain astrophysical observations DM (). In R-parity-conserving SUSY models, the supersymmetric particles (sparticles) can only be produced in pairs, and the lightest sparticle (LSP) is absolutely stable.
Among all the supersymmetric models, the minimal supersymmetric standard model (MSSM) is one of the most well-motivated and well-studied extensions of the standard model. The MSSM predicts many such new particles as sleptons, squarks, gluinos,
the light/heavy neutral scalar (CP-even) Higgs bosons /, a pseudoscalar (CP-odd)
Higgs boson , a couple of charged Higgs bosons , four neutralinos
and two charginos . The neutralinos and charginos are the mass eigenstates formed from the superposition of the neutral or charged superpartners of the electroweak gauge bosons and Higgs doublets (the so-called gauginos and Higgsinos, respectively).
The lightest neutralino is usually supposed to be a weakly interacting massive particle
which is consistent with the observations of the DM candidate (see, e.g., Jungman (); Griest ()) in the form of the LSP for a number of SUSY breaking models. Therefore, it has to emerge as the final particle of the decay chain of each sparticle. That is why, a detailed analysis of the lightest neutralino is quite important to the phenomenological and theoretical viewpoints of SUSY.

The experimental searches of the supersymmetric particles turn out to be one of the primary tasks of the experimental program at hadron colliders, especially at the Large Hadron Collider (LHC), after the recent discovery of the Higgs-like boson with a mass about Higgs_ATLAS (); Higgs_CMS () is consistent with the MSSM-predicted range for mass of the lightest scalar Higgs . Moreover, the discovery (or exclusion) of weak-scale SUSY is reckoned among the highest physics priorities for the future LHC, including its high luminosity upgrade. Up to now, a great number of SUSY searches at the LHC have only exhibited null results related to discovery of any supersymmetric particles. In spite of the negative results, SUSY retains strong arguments in its favour as mentioned before. These searches which chiefly focus on the production of the colored superpartners such as squarks and gluinos have been performed by the ATLAS and CMS collaborations. Consequently, new stronger limits on the masses of the first two generations squarks and gluinos have been produced depending on details of the assumed parameters. These limits for a data set of an integrated luminosity of around 20 fb having been collected in 8 TeV pp collisions at the LHC are given in the following. According to recent ATLAS results sqgl_ATLAS1 (); sqgl_ATLAS2 (), a gluino mass is excluded up to in a mSUGRA/constrained MSSM (CMSSM) scenario at high values of the universal scalar mass parameter and in the gluino simplified models. The first two generations squark masses up to are also excluded in the squark simplified models. In addition, gluinos and squarks of equal mass are excluded for masses below 1.7 TeV in mSUGRA/CMSSM models. According to recent CMS results sqgl_CMS1 (), the squark masses below 750 GeV and gluino masses of up to 1.1 TeV are excluded in the case where the squarks (gluinos) decay to one jet (two jets) and the LSP. Owing to these stronger limits on the masses of the squarks and gluinos, the attention in the experimental researches of the supersymmetric particles starts to turn towards the electroweak production of the sleptons, neutralinos, and charginos.

On the other hand, naturalness suggests that masses of charginos, neutralinos and third generation sparticles should be a few hundreds of GeV range Chan (). There are also searches for superpartners of gauge and Higgs bosons, but they depend significantly on their assumed compositions and decay modes Han (). The bound on the lightest neutralino mass is given by at 95% CL, derived from the lower bound on chargino mass in the MSSM at the Large Electron Positron Abdallah (). In the framework of the CMSSM including both sfermion and gaugino mass unification, this bound reaches to well above 100 GeV from the powerful constraints set by the recent LHC data PDG ().

Note that a detailed study of the production of the lightest neutralino and the next-to-lightest neutralino can provide significant information about the SUSY-breaking mechanism and the nature of the dark matter. Moreover, the pair production of neutralinos/charginos begins to come into question as a discovery channel of supersymmetry. Presently one of the gold-plated SUSY discovery channels is the production of pairs decaying into trilepton final states. But, in case of higgsino LSP scenarios, for example appear in context of natural SUSY models, those trilepton searches loose efficiency and should be replaced by novel same-sign dilepton and 4-lepton searches Baer ().

It is known that the effect of higher-order contributions to cross section usually increases with increment of colliding energy and would be more significant at very high energies. For this reason, it is important to take into account one-loop contributions for neutralino pair production. In the present work we analyze the dependence of the neutralino pair production via the processes at tree and one-loop levels, and at one-loop level on SUSY model parameters at the LHC energies, considering the allowed parameter region in the MSSM. There have been few papers dedicated to the investigations of these processes at one-loop level in literature as follows. Considering next-to-leading order (NLO) SUSY-QCD corrections, the direct production channels of charginos and neutralinos at the Tevatron and LHC, have been worked in Ref. Beenakker (). It has been inferred from Ref. Beenakker () that the SUSY-QCD corrections are positive, increasing the mass range of corresponding particles that can be covered at these colliders by as much as percent 10. The neutralino pair production via gluon-gluon fusion in the framework of the mSUGRA has been investigated in Ref. Yi (), and this loop-mediated process has been concluded to be competitive with the quark-antiquark annihilation process. However, our results in present work have not exhibited this case depending on details of the SUSY model parameters. The neutralino pair production via quark-antiquark annihilation within MSSM for three different scenarios has been worked in Ref. Ahmadov (). The pair production of neutralinos via quark-antiquark annihilation including the leading-log one loop radiative corrections and via gluon-gluon fusion at one-loop level (this process was computed with a numerical code) have been studied in Ref. Gounaris (). The NLO SUSY-QCD corrections to the production of a pair of the lightest neutralinos in association with one jet in the framework of the phenomenological MSSM (p19MSSM) have been computed in Ref. Cullen (). Finally, recently in our previous paper (see Ref. Demirci2 ()) we have also analyzed the leading and subleading electroweak (EW) corrections to the neutralino pair production at proton-proton collision, and we have found that the EW corrections supply sizeable contributions, in particular, for the process .

Unlike the above-mentioned works, within the present work the most outstanding feature of our approach is the mechanism in selecting the input parameters. We recover the corresponding Lagrangian parameters as direct analytical expressions of appropriate physical masses without any restrictions on them in the MSSM. As a matter of fact, we mainly focus on the algebraically nontrivial inversion in order to obtain Higgsino and gaugino mass parameters. If we need to explicitly specify, we can say that using and masses of charginos as input parameters, then we get the other ones being Higgsino/gaugino mass parameters, neutralino masses and mixing matrix.

The remainder of the present work proceeds in the following order: In Section II, the analytical results of the relevant amplitudes and cross sections are given for partonic process . In Section III, we give briefly information about one-loop contributions to neutralino pair production via quark-antiquark annihilation (in Subsection III.1) and gluon-gluon fusion (in Subsection III.2). In Section IV, we present definitions corresponding to our method and input parameters which are used in numerical calculations. In Section V, we give numerical results and discuss the corresponding SUSY parameters dependences of the cross section in detail for each scenario. Finally, the results appearing in Section V are summarized in Section VI.

## Ii The Leading-Order Calculation for The Neutralino Pair Production

In this section, after introducing the necessary couplings and Lagrangians in the MSSM, we serve up analytical results of amplitudes and cross section for the partonic process at leading order (LO). The clean environment of proton-proton collision, together with the well-defined energy of the initial state, make this collision ideal for precision measurements of neutralinos properties. The associated production of neutralino pair via quark-antiquark collision at hadron colliders could be denoted by

(1) |

where the labels in parentheses indicate the four momenta of the relevant particles. The cross section for subprocess (1) is parameterized in terms of the following Mandelstam variables,

(2) |

Introducing by () scattering angle and momentum in the center-of-mass system of the final states neutralinos, for corresponding center-of-mass energy and momentums we have,

(3) |

In the following part, we give the corresponding couplings of the neutralino pair production in the MSSM. Using the standard notation, the boson-neutralino-neutralino interactions are proportional to the following couplings:

(4) |

where , and denotes neutralino mixing matrix being a unitary matrix which diagonalizes the neutralino mass matrix. Neglecting generational mixing in the squark sectors, then, the neutralino-quark-squark interactions are proportional to the relevant couplings,

(5) |

and for the boson-quark-quark couplings, we have

(6) |

where and are the fractional electromagnetic charge and the third component of the weak isospin of quark ; such that for left-handed (right-handed) up- and down-type quarks. The sine and cosine of the electroweak mixing angle are denoted by and . In the above couplings, furthermore, refers to up- and down-type quarks, while the label refers to left- and right-handed for squark. Finally, appearing in Eq. (5) is the kronecker delta function which is equal to 1 if the labels are the same, and 0 otherwise; for instance for up-type quark () and for right-handed squark (). We use it to display the neutralino couplings to both an up-type quark/squark and a down-type quark/squark in the same relation. The couplings of the neutralino to boson and (s)quark are considerably dependent on the corresponding elements of the neutralino mixing matrix () as seen from the above couplings. Considering neutralino mass eigenstate basis, the neutralino interactions to corresponding particles in question are obtained from the following Lagrangians Haber (),

(7) |

(8) |

(9) |

where , and are four-component spinor fields of the quark, squark and neutralino, respectively; are the chiral projectors; and is the gauge coupling. Note that the Higgsino and gaugino components of the neutralino in the and coupling are controlled by the neutralino mixing matrix as shown in the above Lagrangians.

The Feynman diagrams of the partonic process at leading level are displayed in Fig. 1.

We neglect the contributions from the Feynman diagrams including the couplings seeing that the strength of Yukawa coupling is proportional to the fermion mass and masses of the first two generations quarks are relatively small and could be ignored. Nevertheless, we will take into account these couplings and contributions of this vertex for bottom quark in a further work. Consequently, the subprocess for neutralino pair production contains an -channel contribution through exchanging the boson, - and -channel contributions via exchanging of the squarks as shown in Fig. 1. The leading-level contributions to the amplitude emerging from the three channels are given by

(10) |

(11) |

(12) |

where the labels represent the summation over the exchanged left/right-handed components of squarks in the same flavor, and the labels represent the type of the neutralinos in the final state. From the above amplitudes along with couplings (4) and (5), explicitly we note that purely Higgsino production dominates in the contribution coming from the -channel diagram, whereas the - and -channel contributions are dominated by purely gaugino production. After averaging over colors and spins of incoming particles, the parton-level differential cross section in the analytic form is given by the following formula,

(13) |

where the factors is arising from spin and color averaging over the initial state and denotes the final identical particle factor. Using standard trace techniques, the squared amplitudes explicitly take the following form,

(14) |

(15) |

(16) |

(17) |

(18) |

(19) |

where is propagator of the boson.

For obtaining the total cross section of the subprocess we use the following formula:

(20) |

where the upper and lower bounds of integral are defined as . Once the cross section for the partonic process has been computed, the total hadronic cross sections in proton-proton collisions in terms of the center-of-mass energy could be readily obtained using

(21) |

with the parton luminosity

(22) |

where the universal parton distribution functions (PDFs) for the partons , constituents of hadrons are denoted by and , depending on the longitudinal momentum fractions of the two partons () at a factorization scale . During our calculations, the factorization scale is chosen as the average mass of the produced particles, namely, .

## Iii One-Loop Contributions to the neutralino pair production

At the one-loop level production of neutralino pair is proceeded via quark-antiquark annihilation and gluon-gluon fusion in the hadron colliders. Feynman diagrams for the one-loop contributions to the process can be divided into three kind diagrams as follows: The box diagrams, the self energy corrections diagrams, and triangle diagrams. Any one-loop amplitude could be given as a linear sum of triangle, box, bubble, and tadpole one-loop integrals.

In the numerical calculations of high-energy processes observed at the current and future accelerators such as LHC and ILC, for precise theoretical predictions of cross sections one needs to include higher-order corrections. In the common case it is explained in the following: First of all, the lowest-order approximation in perturbative calculations of high energy physics is not sufficiently accurate to be compared to the experimental data. Thus, it is important to consider the contributions from higher-order terms as well. For including these corrections in the Standard model or beyond, it is indispensable to handle the evaluation of loop integrals.

We briefly describe the general properties of the box, triangle and self energy corrections diagrams in the following part. The general form the triangle diagram in four dimensions is proportional to the antisymmetric tensor . Such tensor could not be continued to general dimensions, because it has exactly four indices. Therefore, such diagram is excluded from the general proof and has to be treated separately via a different regularization scheme, e.g. the Pauli-Villars method. It must be verified that all higher-order diagrams including the tensor may be renormalized without demolishing gauge invariance. One of the main conditions for the proof of renormalizability, in general, is that this scheme should be gauge invariant and the Slavnov-Taylor identities can be established.

In our case self-energy diagrams consist of the quark, squark, and boson self-energy corrections. These contributions have different properties. It should be noted that the self-energy of the fermions is not physically observable, and therefore it does not make sense even if it has the logarithmic divergence. The basic problem should appear when there is a logarithmic divergence in the evaluation of the physical observable. The most important example is the vertex correction due to the photon or gluon propagation. If it has a logarithmic divergence, then it should be renormalized into the wave function.

We have performed numerical calculations in the ’t Hooft-Feynman gauge where the gluon polarization sum is given by . We have considered the constrained differential renormalization (CDR) CDR () with a view to regularize the ultraviolet (UV) divergences. At the one-loop level, the CDR has been presented to be equivalent to the regularization by dimensional reduction DR (); DR2 (), which is a modified version of dimensional regularization. Hence, a supersymmetry-preserving regularization scheme is supplied by the implementation given in Ref. DR3 (). For a treatment of the appearing infrared (IR) and collinear singularities we use mass regularization, such as IR singularities are treated by a small gluon mass, and the masses of the light quarks are kept in collinearly singular integrals.

We do not give the analytical results for the one-loop level since these are too long to be included here. Now we give kinematic expressions and the Feynman diagrams for the neutralino pair production in the next subsections, considering each partonic process separately.

### iii.1 The partonic process in the one-loop level

The Feynman diagrams contributing to the subprocess in the one-loop level are depicted from Fig. 2 to 4. The virtual corrections to this process include the following generic structure of one-loop Feynman diagrams: Self-energy, three-point vertex and box corrections as shown in Figs. 2, 3 and 4, respectively. In these figures the label represents all neutral Higgs bosons , and the label refers to scalar fermions (fermions) . The subscript and superscripts refer to the generation of (s)quark and the squark mass eigenstates, respectively.

We denote the process of neutralino pair production via quark-antiquark annhilation as

(23) |

where the labels in parentheses represent the four momenta of the corresponding particles.

### iii.2 The partonic process in the one-loop level

The subprocess in the lowest order can only be produced by way of one-loop diagrams, namely it does not emerge at the tree level. We represent the process of neutralino pair production via gluon-gluon fusion with

(24) |

where the labels in parentheses represent the four momenta of the relevant particles. The Mandelstam variables for subprocess (24) are given by

(25) |

For this process, there is no need to take into account the renormalization at the one-loop level and provided that all of the one-loop contributions are involved in the MSSM, the UV divergence will automatically be canceled. The Feynman diagrams contributing to the subprocess in the one-loop level are depicted in Fig. 5. The virtual corrections to this process include the following generic structure of one-loop Feynman diagrams: Self-energy, vertex and box corrections as shown in diagrams from 1 to 5, 6 to 15 and 16 to 20 in Fig. 5, respectively. As seen from these diagrams, this process involves virtual quark/squark corrections. In this figure all neutral Higgs bosons are denoted by the label and the star on the numbers under some diagrams represents that these are t-channel diagrams.

## Iv Parameter Space

We now give the information about our method and input parameters used in the numerical analysis. During our numerical evaluations, we take into account the assumptions and approaches in our previous paper Demirci () for the gaugino/Higgsino sector. The soft SUSY-breaking gaugino mass parameters , and the Higgsino mass parameter can be taken to be real and positive. These gaugino mass parameters are commonly supposed to be connected by way of the relation . The parameters and are obtained as shown in Eqs. (A13) and (A14) in Ref. Demirci () by taking the suitable differences and sums of the chargino masses. Consequently, there appear three different cases in the selection of the gaugino/Higgsino mass parameters and . These are the Higgsino-like, gauginolike, and mixture-case, separately. We can refer the reader to Ref. Demirci () for further details. We set the chargino masses as

(26) |

for both Higgsino-like and gauginolike scenarios, and

(27) |

for mixture-case scenario. Then, the parameters and related to the scenarios are calculated from these values in (26) and (27) for given . Furthermore, neutralino masses for each scenario are obtained by inserting the values of and into Eq. (A8) in Ref. Demirci (). Taking into account the constraint on SUSY parameters from recent experiments sqgl_ATLAS1 (); sqgl_ATLAS2 (); sqgl_CMS1 (), we set the soft SUSY-breaking parameters for the entries of mass matrices in the sfermion sector to be equal as . We get the other SUSY parameters as follows:

(28) |

where are the trilinear couplings and is the mass of the neutral CP-odd Higgs boson. Furthermore, we take the following input parameters for the SM, , , , and PDG (), and we ignore the masses of the light quarks. The running strong coupling at energy scale yields 0.1152, 0.1183, and 0.1165 in the Higgsino-like scenario, gauginolike scenario, and mixture-case scenario, respectively.

Additionally, we have considered the CMSSM 40.2.4 benchmark point CMSSM4022 () in order to make the comparison with our scenarios.
The CMSSM CMSSM (); CMSSM2 (); CMSSM3 () contains five
input parameters, namely, the universal trilinear soft SUSY breaking
parameter , the universal scalar mass parameter , gaugino mass
parameter , the ratio of the expectation values of the two Higgs
doublets and the sign of the Higgs mixing parameter sign().
It is believed that the universal parameters , , and arise via some gravity-mediated mechanism,
and these are defined at the grand unified theories scale while sign() and are described at the
electroweak scale.
In the CMSSM 40.2.4 benchmark point, the input parameters are given as follows: , ,
, , and . In this case,
we obtain the corresponding SUSY particle spectrum with the help of
`SoftSusy-3.3.9`

package softsusy () as follows:

(29) |

Higgsino-like | 250.00 | 200.00 | 119.33 | 45 | 109.59 | 174.50 | 209.65 | 294.88 |

Gauginolike | 200 | 250.00 | 95.46 | 45 | 91.50 | 169.50 | 259.40 | 293.85 |

Mixture case | 225.00 | 225.00 | 107.39 | 45 | 101.42 | 176.13 | 234.52 | 289.37 |

CMSSM 40.2.4 | 470.87 | 795.94 | 254.88 | 40 | 251.96 | 479.89 | 800.38 | 808.69 |

Furthermore, Table 1 shows a list of the Higgsino/gaugino mass parameters, neutralino masses, and for our scenairos and the CMSSM 40.2.4 benchmark point.

## V Numerical results and discussion

Let us now discuss in detail the numerical predictions of the process at the LHC energies, taking into account the full one-loop contributions from quark-antiquark annihilation and gluon-gluon fusion. We carry out the numerical evaluation using the Mathematica packages FEYNARTS Feynarts () to obtain the relevant amplitudes, FORMCALC Hahn () to supply both the analytical results and a complete Fortran code for numerical evaluation of the squared matrix elements, and LOOPTOOLS loop () to make the evaluation of the necessary loop integrals as based on Passarino-Veltman reduction techniques PV (). In addition, with the help of FEYNARTS we generate all relevant Feynman diagrams, which are shown in Figs. 1 through 5. Higgs properties are computed by using FEYNHIGGS FeynHiggs (). In the numerical treatment, we use the MSTW2008 PDFs MSTW () interfaced via the LHAPDF package LHAPDF () for the distribution of the gluon/quark in the proton. Moreover, we set the central renormalization and factorization scales to be equal () and fix as the average mass of the produced particles in default. To have a quantitative understanding of the effects of one-loop contributions on the neutralino pair production, it is convenient to compute the K factor, which is defined as the ratio between the total NLO and LO cross sections, namely, .

For representative parameter points of each of the scenarios defined in Table 1, we have performed numerical evaluation of the total Born cross sections , the one-loop cross sections for quark-antiquark annihilation and gluon-gluon fusion , and the K factor, as a function of the center-of-mass energy from Figs. 6