Beam polarization effects in the radiative production of lightest neutralinos in e + e − collisions in supersymmetric grand unified models
aa r X i v : . [ h e p - ph ] S e p Beam polarization effects in the radiative production of lightest neutralinos in e + e − collisions in supersymmetric grand unified models. P. N. Pandita and Monalisa Patra Department of Physics, North Eastern Hill University,Shillong 793 002, India Centre for High Energy Physics,Indian Institute of Science,Bangalore 560 012, India
We study the production of the lightest neutralinos in the process e + e − → χ χ γ in supersymmetricgrand unified models for the International Linear Collider energies with longitudinally polarizedbeams. We consider cases where the standard model gauge group is unified into the grand unifiedgauge groups SU (5), or SO (10). We have carried out a comprehensive study of this process inthe SU (5) and SO (10) grand unified theories which includes the QED radiative corrections. Wecompare and contrast the dependence of the signal cross section on the grand unified gauge group,and on the different representations of the grand unified gauge group, when the the electron andpositron beams are longitudinally polarized. To assess the feasibility of experimentally observing theradiative production process, we have also considered in detail the background to this process comingfrom the radiative neutrino production process e + e − → ν ¯ νγ with longitudinally polarized electronand positron beams. In addition we have also considered the supersymmetric background comingfrom the radiative production of scalar neutrinos in the process e + e − → ˜ ν ˜ ν ∗ γ with longitudinallypolarized beams. The process can be a major background to the radiative production of neutralinoswhen the scalar neutrinos decay invisibly. PACS numbers: 11.30.Pb, 12.60.Jv, 14.80.Ly
I. INTRODUCTION
In supersymmetric models with R parity ( R P ) conservation, the lightest neutralino is expected to be thelightest supersymmetric particle (LSP). Because of R P conservation, the lightest neutralino is absolutely stable.Being the LSP, it is the end product of any process that involves supersymmetric particles in the final state.Because of its importance in supersymmetric phenomenology, there have been extensive studies of the neutralinosector of the minimal supersymmetric standard model (MSSM) [1] and its extensions [2–10]. The discoveryof neutralinos is one of the main goals of present and future accelerators. In particular, an e + e − colliderwith a center-of-mass energy of √ s = 500GeV in the first stage, will be an important tool in determiningthe parameters of the underlying supersymmetric model with a high precision [11–15] The capability of sucha linear collider in unravelling the structure of supersymmetry (SUSY) can be enhanced by using polarizedelectron and positron beams [16].When the standard model (SM) gauge symmetry SU (2) × U (1) is broken, the fermionic partners of thetwo Higgs doublets ( H , H ) of the MSSM mix with the fermionic partners of gauge bosons, resulting in fourneutralino states ˜ χ i , i = 1 , , ,
4, and two chargino states ˜ χ ± j , j = 1 , . The composition and mass of thelightest neutralino, which depends on the soft SU (2) and U (1) gaugino masses, M and M , on the Higgs(ino)parameter, µ , and on the ratio of the two Higgs vacuum expectation values, tan β ≡ v /v , will be crucialfor the search for supersymmetry at the colliders. The values of the soft gaugino masses at the electroweakscale depend on the boundary conditions on these masses at the grand unified theory (GUT) scale. In mostof the studies the gaugino masses have been taken to be universal at the GUT scale. However, there is noparticular reason to assume that the soft gaugino masses are universal at the high scale. Indeed, it is possibleto have nonuniversal soft gaugino masses in grand unified theories. We recall that soft supersymmetry gauginomasses are generated from higher-dimensional interaction terms involving gauginos and auxiliary parts of chiralsuperfields [17]. For example, in SU (5) grand unified theory, the auxiliary part of a chiral superfield in higher-dimensional terms can be in the representation , , or or, in general, some combination of theserepresentations.When the auxiliary field of one of the SU (5) nonsinglet chiral superfields obtains a vacuum expectation value(VEV), then the resulting gaugino masses are nonuniversal at the grand unification scale. Similar conclusionshold for other supersymmetric grand unified models. Furthermore, nonuniversal supersymmetry breakingmasses are a generic feature in some of the realistic supersymmetric models. For example, in anomaly mediatedsupersymmetry breaking models the gaugino masses are not unified [18, 19], and hence are not universal.From the above discussion it is clear, that the phenomenology of supersymmetric models depends cruciallyon the composition of neutralinos and charginos. This in turn depends on the soft gaugino mass parameters M and M , besides the parameters µ and tan β . Since most of the models discussed in the literature assumegaugino mass universality at the GUT scale, it is important to investigate the changes in the phenomenologyof broken supersymmetry which results from the changes in the composition of neutralinos and charginos thatmay arise because of the changes in the pattern of soft gaugino masses at the grand unification scale [20]. Theconsequences of nonuniversal gaugino masses at the grand unified scale and the resulting change in boundaryconditions has been considered in several papers. This includes the study of constraints arising from differentexperimental measurements [21–23] and in the study of supersymmetric dark matter candidates [24, 25].Recently in Refs. [26, 27] a detailed study of the radiative production of neutralinos in electron-positroncollisions in low-energy supersymmetric models with universal gaugino masses at the grand unified scale wascarried out. Furthermore, we have carried out a detailed study of the radiative production of the lightestneutralinos in electron-positron collisions in grand unified theories [28]. Since longitudinal beam polarizationis going to play a crucial role in electron-positron collisions, it is important to study its effects on the radiativeproduction of the lightest neutralinos in electron-positron colliding beam experiments in the case of grandunified theories.In this paper we shall carry out a detailed study of the implications of the nonuniversal gaugino masses,as they arise in grand unified theories, for the production of lightest neutralinos in electron-positron collisionswith longitudinally polarized beams. Our purpose is to study the role of longitudinal beam polarization asa probe of supersymmetric grand unified theories. For this purpose we shall consider the case of SU (5) and SO (10) grand unified theories, these being the typical ones wherein the standard model can be embedded in agrand unified gauge group. The motivation of this comes from the fact that longitudinal beam polarization isa distinct possibility at the International Linear Collider (ILC). Studies of this type have not been carried outso far in the context of grand unified theories. Since in a large class of models of supersymmetry the lightestneutralino is expected to be the lightest supersymmetric particle, it will be one of the first states to be producedat the colliders, even if other SUSY particles may be too heavy to be produced. Moreover, this process is likelyto complement the search of the SUSY spectrum at the LHC, where the squarks and gluinos are likely to beproduced and studied in detail. The radiative neutralino production at the ILC will, thus, be an independentstudy irrespective of whether the colored sparticles are found at the Large Hadron Collider. A detailed studyof this process at the ILC will let us determine the mass and composition of the lightest neutralino along withits couplings, which by itself would be an important advance. The experimental performance of the radiativeneutralino production along with the neutralino mass measurement have been recently evaluated in a fulldetector simulation for the Internaional Large Detector [29]. At an electron-positron collider, such as the ILC,the lightest neutralino can be directly produced in pairs [3, 30]. However, it will escape detection such thatthe direct production of the lightest neutralino pair is invisible. One can, however, look for the signature ofneutralinos in electron-positron colliders in the radiative production process, e + + e − → ˜ χ + ˜ χ + γ. (I.1)The signature of this process is a single high-energy photon with missing energy carried away by the neu-tralinos. In this paper we carry out a detailed study of the process (I.1) in supersymmetric grand unifiedtheories with nonuniversal boundary conditions at the grand unified scale with polarized electron and positronbeams. The process (I.1) has been studied in detail in the minimal supersymmetric model [31–41], in variousapproximations. Calculations have also been carried out for the MSSM using general neutralino mixing [39–41].This process has also been studied in detail in the next-to-minimal supersymmetric model [26, 27]. On theother hand different large electron positron (LEP) collaborations [42–46] have studied the signature of radiativeneutralino production in detail but have found no deviations from the SM prediction. Thus, they have onlybeen able to set bounds on the masses of supersymmetric particles [42–44, 46]. Also, the role of longitudinalpolarization for process (I.1) has been studied in [47].We recall here that in the SM the radiative neutrino process e + e − → ν + ¯ ν + γ, (I.2)is the leading process with the same signature as Eq. (I.1). The cross section for the process (I.2) dependson the number N ν of light neutrino species [48]. This process acts as a main background to the radiativeneutralino production process (I.1). Furthermore, there is also a supersymmetric background to the process(I.1) coming from radiative sneutrino production e + e − → ˜ ν + ˜ ν ∗ + γ. (I.3)We shall consider both these processes, since they form the main background to the radiative process (I.1),and are important for determining the feasibility of observing the radiative production of lightest neutralinosin electron-positron collisions.For the signal process (I.1), the dominant SM background process (I.2) proceeds through the exchange of W bosons, which couple only to left-handed particles. At the LEP this dominant background process made itimpossible to see the possible signal of the radiative process (I.1), even for very light neutralinos. Furthermore,in the case of the LHC a search has been made for the final states in pp collisions, containing a photon ( γ ) of largetransverse momentum and missing energy. These events can be produced by the underlying reaction q ¯ q → γχ ¯ χ ,where the photon is radiated by one of the incoming quarks and where χ is a dark matter candidate (possiblythe lightest neutralino). The primary background for such a signal at the LHC is the irreducible SM backgroundfrom Zγ → ν ¯ νγ. This and other SM backgrounds were taken into account in the LHC analysis. The observednumber of events was found to be in agreement with the SM expectations for the γ + missing energy events.From this an upper limit for the production of χ in the γ + missing transverse energy state was obtained [49].In view of these negative results, the International Linear Collider, with the possibility of beam polarization,will be a good place to look for the process with an energetic photon and large missing energy in the finalstate characteristic of the reaction (I.1). Furthermore, in the case of the ILC with the possibililty of polarizedelectron and positron beams, a suitable choice of beam polarization ( e − R e + L ) will significantly reduce the SMbackground.The layout of the paper is as follows. In Sec. II, we discuss the constraints on the supersymmetric particlespectrum arising from the experimental results from the LHC, the Tevatron, and the LEP. In Sec. III, weimplement the constraints on the parameter space of the grand unified models as they arise from the constraintson the supersymmetric particle spectrum discussed in Sec. II. Here we also calculate the elements of the mixingmatrix which are relevant for obtaining the couplings of the lightest neutralino to the electron, selectron, and Z boson which control the radiative neutralino production process (I.1). We also describe in detail the typicalset of input parameters that is used in our numerical evaluation of cross sections. The set of parameters thatwe use is obtained by imposing various experimental and theoretical constraints discussed in Sec. II on theparameter space of the minimal supersymmetric standard model with underlying grand unification. Theseconstraints will be used throughout to arrive at the allowed parameter space for different models in this paper.Furthermore, in Appendix A we briefly review different patterns of gaugino masses that arise in grand unifiedtheories. Here we will consider grand unified theories based on SU (5) and SO (10) gauge groups, and outlinethe origin of nonuniversal gaugino masses for these models.In Sec. IV we summarize the cross section for the signal process, including the beam polarization and itsimplications for the signal cross section. Here we also describe in detail the effect of QED radiative correctionson the cross section for the radiative neutralino production cross section. In Sec. V we evaluate the crosssection for the signal process (I.1) in different grand unified theories with nonuniversal gaugino masses, usingthe set of parameters obtained in Sec. III for different patterns of gaugino mass parameters at the grand unifiedscale. We have included higher-order QED radiative corrections, as described in Sec. IV, in all our calculations.We also compare and contrast the results so obtained with the corresponding cross section in the MSSM withuniversal gaugino masses at the grand unified scale. The dependence of the cross section on the parameters ofthe neutralino sector, and on the selectron masses is also studied in detail.In Sec. VI we discuss the backgrounds to the radiative neutralino production process (I.1) from the SM andsupersymmetric processes. An excess of photons from radiative neutralino production, with the longitudinallypolarized electron and positron beam, over the backgrounds measured through statistical significance is alsodiscussed here and calculated for different grand unified models. We summarize our results and conclusions inSec. VII. II. EXPERIMENTAL CONSTRAINTS
In this section, we discuss the latest constraints on the SUSY particle spectrum from the data from theLarge Hadron Collider, along with the data from the Tevatron and LEP. At the LHC the search for the SUSYparticles is carried through different channels and with different final states. The final states can contain jets,isolated leptons, and E missT , or will have same-sign dileptons or jets with high p T . Different final states areconsidered to increase the sensitivity to a different SUSY spectrum. Observations at the LHC are in goodagreement with the SM expectation; therefore, constraints have been set on the cross sections for the SUSYprocesses. Interpreted differently since no supersymmetric partners of the SM have been detected lower limitsare obtained on their masses. A. Limits on gaugino mass parameters
The lightest chargino mass and field content is sensitive to the parameters M , µ , and tan β . At the LEPthe search for the lightest chargino through its pair production has yielded a lower limit on its mass [50]. Thelimits obtained depend on the mass of the sfermions. For the chargino masses following from nonobservationof chargino pair production in e + e − collisions at the LEP, we have the constraint M ˜ χ ± > ∼
103 GeV . (II.1)The limit depends on the sneutrino mass. For a sneutrino mass below 200 GeV, the bound becomes weaker,since the production of a chargino pair becomes more rare due to the destructive interference between γ or Z in the s channel and ˜ ν in the t channel. In the models we consider, m ˜ ν is close to m , where m is the softSUSY breaking scalar mass. When m ˜ ν <
200 GeV, but m ˜ ν > m ˜ χ ± , the lower limit becomes [51] M ˜ χ ± > ∼
85 GeV . (II.2)For the parameters of the chargino mass matrix, the limit (II.1) implies an approximate lower limit [52, 53] M , µ > ∼
100 GeV . (II.3)The lower limits in Eq. (II.3) on M and µ are obtained by scanning over the MSSM parameter space andare, therefore, expected to be model independent [10]. Recently a search was done by the ATLAS experimentfor the direct production of charginos and neutralinos in the final states with three leptons and p missT . In thecontext of simplified models degenerate ˜ χ ± and ˜ χ with masses up to 300 GeV are excluded for large massdifferences with the ˜ χ . For our analyses we have considered the limit set on the chargino mass from the LEP.The combination of chargino, slepton and Higgs boson searches has provided a lower limit on m ˜ χ as a functionof tan β . The absolute lower limit on the neutralino mass is 47 GeV at large tan β . B. Exclusion limits on squarks and gluinos
The colored SUSY particles, being QCD-mediated processes,can be more copiously produced in the proton-(anti) proton collider with their higher centre-of-mass energies compared to the LEP. In the context of theConstrained Supersymmetric Standard Model (CMSSM), Tevatron experiments have excluded squark andgluino masses of 379 and 308 GeV, respectively, based on an integrated luminosity of 2 . − . In the framework of the CMSSM, the LHC experiments with approximately 5 fb − of data have excludedgluino masses below 800 GeV for all squark masses. Moreover, squark and gluino masses below approximately1400 GeV are excluded at 95% C.L. (for equal squark and gluino masses) [54, 55]. The limits, though derivedfor a particular choice of parameters in the context of CMSSM, depend slightly on the choice. Analyses havealso been done setting a limit on gluino mass as a function of the lightest neutralino. The limits obtained aresensitive to the neutralino mass and to the gluino neutralino mass difference.In the framework of the CMSSM, the LHC experiments have also obtained limits on the first- and second-generation squark masses [54, 55]. They have excluded masses below around 1300 GeV for all values of gluinomasses. Similarly, an analysis is carried out on the squark mass as a function of the neutralino mass. Overall,considering all the analyses carried out by LHC in the context of different models, first- and second-generationsquarks along with the gluinos are excluded with masses below 1200 GeV.The limits on the third-generation squark ˜ t mass from LEP is around 96 GeV, in the charm plus neutralinofinal state [50]. Experiments at the LHC and at Tevatron have performed the analyses for third-generationsquarks in different scenarios, leading to different final states [56]. Similar analyses have been carried out forthe sbottom quarks. Overall, for our analyses we will consider the scenario where the third-generation squarksare excluded below a mass of about 800 GeV. C. Exclusion limit on slepton masses
The limits on the selectrons, smuons, and staus masses are from the LEP experiments [50] because of itsclean signature. The limits obtained on the sleptons are sensitive to the lightest neutralino mass. The smuonsand staus with masses below 95 GeV are excluded depending on the lightest neutralino ( ˜ χ ) mass, providedthe mass difference of the slepton and ( ˜ χ ) is less than 7 GeV. A lower limit of around 73 GeV is set on themass of the right-handed selectron, m ˜ e R , independent of the neutralino mass. tan β = 10 µ = 130 GeV M = 197 GeV M = 395 GeV M = 1402 GeV A t = 2800 GeV A b = 2800 GeV A τ = 1000 GeV m χ = 108 GeV m χ ± = 125 GeV m ˜ e R = 156.2 GeV m ˜ ν e = 136 GeV m χ = 140 GeV m χ ± = 421.7 GeV m ˜ e L = 156.7 GeV m h = 125.7 GeV TABLE I: Input parameters and resulting masses of various states in the MSSM EWSB scenario.A lower limit of around 45 GeV is obtained on the sneutrino mass from the measurement of the invisible Z decay width. In the context of the MSSM tighter limits are obtained on the mass of sneutrino of around94 GeV, assuming gaugino mass universality at the GUT scale.Taking into account all the constraints set by the different experiments as detailed above, for our analyseswe have considered m ˜ g ≈ ≈ m ˜ t, ˜ b around 1000 GeV, and the slepton of mass around 150 GeV. For the lightest chargino andneutralino, the LEP limit is respected since it gives more stringent bounds compared to the LHC. The Higgsmass is taken to be consistent with the present LHC results. III. COMPOSITION OF THE LIGHTEST NEUTRALINOS IN GRAND UNIFIED THEORIES
In this section we list the set of parameters used for our analysis along with the composition of the lightestneutralino in grand unified theories. In Appendix A we review the patterns of nonuniversal gaugino massesin grand unified theories. For the sake of completeness we have first considered the case of universal gauginomasses in supersymmetric theories. In Appendix B we summarize our notations for the neutralino mass matrixand the interaction vertices relevant for our study [57].We have used the set of parameters listed in Table I for our analysis in the case of universal gaugino massesat the grand unified scale. The values of the parameters are chosen so as to satisfy the various experimentalconstraints listed in Sec. II. We have restricted ourselves to a particular choice of parameter set with the valuesof M and µ chosen to correspond to a lightest neutralino of mass around 108 GeV. The reason for the choiceof this set was discussed in Ref. [28]. We call this set of parameters the MSSM electroweak symmetry breaking(EWSB) scenario [58]. In this scenario we can study the dependence of the neutralino masses as well as theradiative neutralino production cross section on µ , M , and the selectron masses.The composition of the lightest neutralino in case of the MSSM EWSB scenario for the parameters of Table Iis given by N j = (0 . , − . , . , − . . (III.1)Thus, the lightest neutralino has a dominant Higgsino component. The couplings of the lightest neutralino toelectrons, selectrons, and Z bosons are listed in table XVII of Appendix B. From this Table it is clear that fora neutralino with composition (III.1), the neutralino - Z coupling is enhanced compared to the coupling ofthe lightest neutralino with right and left selectrons ˜ e R,L .For our analyses, as a benchmark we have used the radiative neutralino cross section for the MSSM EWSBscenario with the set of parameters as shown in Table I. tan β = 10 µ = 138 GeV M = 149 GeV M = 890 GeV M = -2121 GeV A t = -1000 GeV A b = -2700 GeV A τ = -2700 GeV m χ = 108 GeV m χ ± = 138.7 GeV m ˜ e R = 156 GeV m ˜ ν e = 136 GeV m χ = 146 GeV m χ ± = 905 GeV m ˜ e L = 157 GeV m h = 124 GeV TABLE II: Input parameters and resulting masses for various states in SU (5) supersymmetric grand unifiedtheory with Φ and F Φ in the -dimensional representation. We shall refer to this model as [ SU (5)] in thetext.The input parameters and the resulting masses for the , , and -dimensional representations of SU (5) which result in nonuniversal gaugino masses at the grand unified scale obtained in a manner describedlater in Appendix A 2 are shown in tables II, III and IV, respectively. In arriving at the parameter values inthese Tables, we have taken into account various theoretical and phenomenological constraints, including the tan β = 10 µ = 108 GeV M = -993.9 GeV M = 1172 GeV M = 1401 GeV A t = 1000 GeV A b = 2700 GeV A τ = 3000 GeV m χ = 108 GeV m χ ± = 109 GeV m ˜ e R = 156 GeV m ˜ ν e = 136 GeV m χ = 112 GeV m χ ± = 1180 GeV m ˜ e L = 157 GeV m h = 125 GeV TABLE III: Input parameters and resulting masses for various states in SU (5) supersymmetric grand unifiedtheory with Φ and F Φ in the -dimensional representation. We shall refer to this model as [ SU (5)] in thetext. tan β = 10 µ = 111 GeV M = 1970 GeV M = 788 GeV M = 1399 GeV A t = 1000 GeV A b = 2800 GeV A τ = 3000 GeV m χ = 107.7 GeV m χ ± = 111 GeV m ˜ e R = 166 GeV m ˜ ν e = 136 GeV m χ = 117 GeV m χ ± = 806 GeV m ˜ e L = 157 GeV m h = 125 GeV TABLE IV: Input parameters and resulting masses for various states in SU (5) supersymmetric grand unifiedtheory with Φ and F Φ in the -dimensional representation. We shall refer to this model as [ SU (5)] inthe text.electroweak symmetry breaking at the correct scale, as described in the Sec. II. Other values can be obtainedby choosing larger values of the parameter M .The composition of the lightest neutralino for the different representations of SU (5) in Table XIII is obtainedfrom the mixing matrix for the choices of parameters given in Tables II, III and IV. This composition iscalculated to be:1. SU (5) with Φ and F Φ in the -dimensional representation (labelled as model [ SU (5)] ): N j = (0 . , − . , . , − . SU (5) with Φ and F Φ in the -dimensional representation (labelled as model [ SU (5)] ): N j = (0 . , . , − . , − . SU (5) with Φ and F Φ in the -dimensional representation (labelled as model [ SU (5)] ): N j = (0 . , − . , . , − . . (III.4)We note from Eqs. (III.2), (III.3), and (III.4) that for the -dimensional representation of SU (5), the dominantcomponent of the neutralino is the bino, whereas for the other representations of SU (5), there is a Higgsinolike lightest neutralino. Thus, for and -dimensional representations, the neutralino, being Higgsino-like,couples weakly to the selectron, with the dominant contribution to the cross section coming from the neutralino- Z coupling. tan β = 10 µ = 116 GeV M = -760 GeV M = 395 GeV M = 1405 GeV A t = 1000 GeV A b = 2800 GeV A τ = 3000 GeV m χ = 108 GeV m χ ± = 111 GeV m ˜ e R = 156 GeV m ˜ ν e = 136 GeV m χ = 122 GeV m χ ± = 421 GeV m ˜ e L = 157 GeV m h = 126 GeV TABLE V: Input parameters and resulting masses for various states in SU (5) ′ × U (1) ⊂ SO (10)supersymmetric grand unified theory with Φ and F Φ in the -dimensional representation with SU (5) ′ × U (1) in ( , ) dimensional representation. We shall refer to this model as [ SO (10)] in the text.Similarly in the case of SO (10), for the parameters of Tables V, VI, and VII the composition of the lightestneutralino is given by the following :1. SO (10) where SU (5) ′ × U (1) ⊂ SO (10) with Φ and F Φ in the -dimensional representation with SU (5) ′ × U (1) in ( , )-dimensional representation (labelled as model [ SO (10)] ): N j = (0 . , . , − . , . tan β = 10 µ = 118 GeV M = 3038 GeV M = 395 GeV M = 1398 GeV A t = 1000 GeV A b = 2800 GeV A τ = 3000 GeV m χ = 108 GeV m χ ± = 113 GeV m ˜ e R = 156 GeV m ˜ ν e = 136 GeV m χ = 126 GeV m χ ± = 422 GeV m ˜ e L = 157 GeV m h = 125.7 GeV TABLE VI: Input parameters and resulting masses for various states in SU (5) ′ × U (1) ⊂ SO (10)supersymmetric grand unified theory with Φ and F Φ in the -dimensional representation with SU (5) ′ × U (1) in ( , ) dimensional representation. We shall refer to this model as [ SO (10)] in the text. tan β = 10 µ = 113 GeV M = 378 GeV M = 985 GeV M = 1402 GeV A t = 1000 GeV A b = 2800 GeV A τ = 3000 GeV m χ = 108 GeV m χ ± = 115 GeV m ˜ e R = 156 GeV m ˜ ν e = 136 GeV m χ = 121 GeV m χ ± = 998 GeV m ˜ e L = 157 GeV m h = 125 GeV TABLE VII: Input parameters and resulting masses for various states in SU (4) × SU (2) R × SU (2) L ⊂ SO (10)supersymmetric grand unified theory with Φ and F Φ in the -dimensional representation with SU (4) × SU (2) R in ( , ) dimensional representation. We shall refer to this model as [ SO (10)] ′ in the text.2. SO (10) where SU (5) ′ × U (1) ⊂ SO (10) with Φ and F Φ in the -dimensional representation with SU (5) ′ × U (1) in ( , )-dimensional representation (labelled as model [ SO (10)] ): N j = (0 . , − . , . , − . SO (10) where SU (4) × SU (2) R × SU (2) L ⊂ SO (10) with Φ and F Φ in the -dimensional representationwith SU (4) × SU (2) R in ( , )-dimensional representation (labelled as model [ SO (10)] ′ ): N j = (0 . , − . , . , − . , (III.7)implying thereby that a Higgsino is the dominant component for the - and -dimensional representationswith the embedding SU (5) ′ × U (1) ⊂ SO (10) and for the -dimensional representation with the embedding SU (4) × SU (2) R × SU (2) L ⊂ SO (10).Thus, in these cases the dominant contribution to the radiative neutralino production cross section will comefrom the neutralino- Z coupling. Since the LSP for most of the scenarios considered here has a dominanthiggsino component, the Z width imposes a strict constraint, as the Z decay rate involves coupling to theHiggsino component of the neutralino. We have imposed the LEP constraint on the anomalous Z decay widthin our calculations : Γ( Z → ˜ χ ˜ χ ) < . (III.8) IV. RADIATIVE NEUTRALINO PRODUCTION IN GRAND UNIFIED THEORIES
In this section we calculate the cross section for the radiative neutralino production process e − ( p ) + e + ( p ) → ˜ χ ( k ) + ˜ χ ( k ) + γ ( q ) , (IV.1)for the case of longitudinally polarized electron and positron beams for SU (5) and SO (10) grand unifiedtheories with nonuniversal gaugino masses at the grand unified scale. The four-momenta of the correspondingparticles are shown by the symbols in the brackets. We show in Fig. 1 the Feynman diagrams contributingto the radiative neutralino production at the tree level. The neutralino mixing matrix (B.2) summarized inAppendix B determines the couplings of the neutralinos to electrons, the selectrons, and to the Z bosons. Therespective values of the soft SUSY breaking gaugino mass parameters M and M for different grand unifiedmodels have been calculated in Appendix A. We further note that the elements of the neutralino mixing matrix N j for the different models considered here, were calculated in the previous section. e − ( p ) e + ( p ) γ ( q ) χ ( k ) χ ( k ) e − ˜ e R e − ( p ) e + ( p ) ˜ e R e − χ ( k ) χ ( k ) γ ( q ) e − ( p ) e + ( p ) χ ( k ) γ ( q ) χ ( k )˜ e R ˜ e R e − ( p ) e + ( p ) e − Z γ ( q ) χ ( k ) χ ( k ) e − ( p ) e + ( p ) e − Z χ ( k ) χ ( k ) γ ( q ) e − ( p ) e + ( p ) γ ( q ) χ ( k ) χ ( k ) e − ˜ e L e − ( p ) e + ( p ) χ ( k ) χ ( k ) γ ( q ) e − ˜ e L e − ( p ) e + ( p ) χ ( k ) γ ( q ) χ ( k )˜ e L ˜ e L ( i ) ( ii ) ( iii )( iv ) ( v ) ( vi )( vii ) ( viii )FIG. 1: Feynman diagrams contributing to the radiative neutralino production e + e − → ˜ χ ˜ χ γ. There are sixother diagrams which are exchange diagrams corresponding to ( i , ii , iii , vi , vii , viii ), with u -channelexchange of selectrons, wherein the neutralinos are crossed in the final state. A. Cross section for the signal process
At the tree level the process (IV.1) proceeds via the t - and u -channel exchange of right and left selectrons˜ e R,L and via Z boson exchange in the s channel for the different scenarios considered here as can be seen fromFig. 1. The differential cross section for the process (IV.1) can be written as [33, 59] dσ = 12 (2 π ) s Y f d p f (2 π ) E f δ (4) ( p + p − k − k − q ) |M| , (IV.2)where p f and E f are the final three-momenta k , k , q and the final energies E χ , E χ , and E γ of the neutralinosand the photon, respectively. Using the standard technique, we sum over the spins of the neutralinos and thepolarization of the outgoing photon. The squared matrix element |M| in Eq. (IV.2) can then be written as [33] |M| = X i ≤ j T ij , (IV.3)where T ij are squared amplitudes corresponding to the Feynman diagrams in Fig. 1. The phase space for theradiative neutralino production process in Eq. (IV.2) is described in detail in Ref. [33]. B. Longitudinal beam polarization
At the future linear collider, the use of beam polarization will significantly benefit the physics program. Inthe case of many processes, it is found that a suitable choice of beam polarizations can enhance the signal andsuppress the background. At the ILC, a beam polarization of ≥
80% for electrons and ≥
30% for positrons atthe interaction point is proposed, with a possible upgrading to about 60% for the positron beam. In the case ofan electron and positron beam with arbitrary degree of longitudinal beam polarization, the total cross sectionin the centre-of-mass frame with center-of-mass energy √ s is given by σ P e − P e + = 14 [(1 + P e − )(1 − P e + ) σ RL + (1 − P e − )(1 + P e + ) σ LR ] (IV.4)In Eq. (IV.4) the dependence of the cross section on the polarization is parametrized through the degree ofpolarization, which is defined as P e ∓ = ( N R − N L ) / ( N R + N L ), where N L,R denote the number of left-polarizedand right-polarized electrons (or positrons) respectively. Moreover σ RL denotes the cross section when theelectron beam is completely right polarized with P e − = 1, and the positron beam is completely left polarizedwith P e + = -1. An analogous definition holds for σ LR . We do not take into account the helicity combinationsfor the cross section ( LL and RR ) as they are absent in the SM and for the supersymmetric process consideredhere. For the signal process, the significant contribution comes from the selectron or Z exchange dependingon the composition of the neutralino. For all the scenarios considered, the neutralino is dominantly a Higgsinowith the Z boson exchange dominantly contributing to the neutralino production process. In the case of SU (5) , the neutralino has a significant bino component resulting in significantly larger coupling to rightselectron; therefore, the production process proceeds mainly via the exchange of right selectron ˜ e R . On theother hand, the SM background radiative neutrino process proceeds mainly through the exchange of W bosons,which couple only to left handed particles. Therefore, a polarization combination with positive electron beampolarization and negative positron beam polarization will significantly reduce the background and increase thesignal for the cases where the neutralino has a dominant bino component. When the neutralino is of a Higgsinotype, there is no appreciable change in cross section for this choice of beam polarization as Z couples to bothleft- and right-handed fermions. Since with this particular choice of beam polarization the SM backgrounddecreases, we present our result for this case with electron beam polarization P e − = 0.8 and positron beampolarization P e + = -0.6 as planned for the future linear collider. C. Radiative corrections
The future high-energy e + e − colliders, in order to avoid energy losses from synchrotron radiation, are designedas linear colliders. These colliders will achieve high luminosity through beams with bunches of high numberdensities. Although the high density of charged particles increases the machine luminosity, it also leads to thegeneration of a strong electromagnetic field in and around every colliding bunch. Initial state radiation (ISR),also known as bremsstrahlung, which results from the interaction of the beam constituents with the acceleratingfield, is the most important QED correction to the Born cross section. Along with it, the interaction of thebeam constituents due to the strong magnetic field generated by the other beam also results in radiationand is known as the beamstrahlung phenomenon. The general feature of both these cases results in multipleemissions of photons, both soft and hard, which not only reduces the initial beam energy but also results in thedisturbance of the initial beam calibration. Moreover, at higher energies these radiative effects result in messierbackgrounds with the radiated photons leading to the production of lepton pairs and hadrons. The resultingspectrum of the electrons due to the ISR effects mainly depends on the electron or positron beam energy andthe reduced momentum of the incoming electron or positron. The photon radiation takes into account themissing momentum. On the other hand, the resulting spectrum due to beamstrahlung, apart from dependingon the beam energy, is mainly machine specific depending on the number of electrons and positrons in a bunch N e , the transverse bunch sizes σ x , σ y and the bunch length σ z . Therefore most future machine designs try tominimize the radiation effects by adjusting the parameters of the bunches accordingly.Apart from being a serious problem, the radiated photons have also been used in the study of new physics.The majority of the emitted photons are soft and are lost down the beam pipe. Only the hard photons withlarge transverse momentum can be tagged, and earlier they were used by the LEP experiments to look for theinvisible final states. The most famous example is the neutrino counting process e + e − → γν l ¯ ν l in the standardmodel, with the final state being a single photon and large missing energy. This search with a hard photon tagis similar to the supersymmetric process considered here in our work. In the case of LEP running at energiesbeyond the Z resonance, these radiative effects lead to “return of the Z peak” causing a hugely increasedcross section. This was mainly due to the multiple emissions of photons resulting in the electron positron pairreturning to the Z resonance. Therefore, taking into account all the above facts, the effect of the radiativeeffects, both ISR and beamstrahlung, is crucial for most experimental analyses.Several strategies exist to include the radiative corrections in the calculations which have been studied exclu-sively in the past [60–63] in the context of the future linear colliders. We have calculated the radiative effectsfor our process and the background processes using CalcHEP [58], with parameters given in Table VIII [64].In CalcHEP the energy spectrum of the electron and positron is calculated by using the structure functionformalism. The main idea here is to include the radiative corrections by a probability density to find an electronwith reduced momentum inside an incoming electron. This is quite similar to the techniques adopted for thehadronic interactions. The total cross section is defined as the leading-order cross section convoluted with thestructure functions including radiative effects. These structure function of the initial leptons are valid up toall orders in perturbation theory. We emphasize that in this paper the radiative effects are included in all our0calculations of the signal and background processes. Collider parameters ILC σ x (nm) 640 σ y (nm) 5.7 σ z ( µ m) 300N (10 ) 2 TABLE VIII: Beam parameters for the ILC, where N is the number of particles in the bunch and σ x , σ y arethe transverse bunch sizes at the interaction point, with σ z as the bunch length. V. NUMERICAL RESULTS
We have calculated the tree-level cross section for radiative neutralino production (IV.1), the standard modelbackground from radiative neutrino production (I.2), and the supersymmetric background from sneutrino pro-duction (I.3) with longitudinally polarized electron and positron beams using the program CalcHEP [58]. Asnoted above we have included the effects of radiative corrections to the signal as well as the background pro-cesses. Due to the emission of soft photons, the tree-level cross sections have infrared and collinear divergences.These divergences are regularized by imposing cuts on the fraction of beam energy carried by the photon andthe scattering angle of the photon [33]. We define the fraction of the beam energy carried by the photon as x = E γ /E beam , where √ s = 2 E beam is the center-of-mass energy, and E γ is the energy carried away by thephoton. The following cuts are then imposed on x and on the scattering angle θ γ of the photon [65]:0 . ≤ x ≤ − m χ E , (V.1) − . ≤ cos θ γ ≤ . . (V.2)The lower and upper cut, Eq. (V.1), on the energy of the photon is a function of the beam energy. Interpretedin a different way, the upper cut corresponds to the kinematical limit of the radiative neutralino productionprocess. In order to enhance the signal over the main SM background, with the neutrinos preferably emitted inthe forward direction, the required detector acceptance cut, Eq. (V.2), on the photon is applied. Except for thecuts on energy and the angular spread, no other cut is found to significantly reduce the background. Therefore,we have implemented these cuts for both signal and background processes in the case of all the scenarios whichwe have considered in this work. A. Photon energy ( E γ ) distribution and total beam energy ( √ s ) dependence First of all we have calculated the energy distribution of the photons from the radiative neutralino productionin case of the MSSM EWSB and different GUT scenarios with nonuniversal gaugino mass in the case oflongitudinal beam polarization.The energy distribution of the radiated photon in the presence of longitudinally polarized beams is shownin Figs. 2 and 3 for the scenarios with nonuniversal gaugino masses in grand unified theories based on SU (5)and SO (10). In these figures the resulting distributions are also compared with the MSSM EWSB modelwith universal gaugino masses at the GUT scale. Similarly the energy dependence of the total cross section isalso calculated with the initially polarized beams and is shown in Figs. 4 and 5. Note that we have includedradiative corrections in all these calculations. As discussed before we have restricted ourselves to only right-handed electron beams and left-handed positron beams in order to reduce the background. The degree ofpolarization used in our calculation is ( P e − , P e + ) = (0.8, -0.6). The unpolarized case in case of the MSSMEWSB is also shown in these figures for the sake of comparison.1 E γ (GeV) d σ / d E γ (f b / G e V ) [ S U ( )] M SS M E W S B [ S U ( )] [ S U ( )] M SS M E W S B ( U np o l a r i ze d ) FIG. 2: The photon energy distribution dσ/dE γ for the radiative neutralino production includingradiative effects with ( P e − , P e + ) = (0.8, -0.6) inthe case of SU (5) with nonuniversal gauginomasses and in the case of the MSSM EWSB withuniversal gaugino masses. For comparison wehave also shown the case of the MSSM EWSBwith unpolarized beams. E γ (GeV) d σ / d E γ (f b / G e V ) M SS M E W S B [SO(10)] [SO(10)] [ S O ( )] ’ MSSM EWSB (Unpolarized)
FIG. 3: The photon energy distribution dσ/dE γ including radiative effects for the radiativeneutralino production with ( P e − , P e + ) = (0.8,-0.6) in the case of SO (10) with nonuniversalgaugino masses and in the case of theMSSM EWSB with universal gaugino masses.For comparison we have also shown the case ofMSSM EWSB with unpolarized beams.
400 600 800 1000 √ s (GeV) σ (f b ) [SU(5)] MSSM EWSB [ S U ( )] [ S U ( )] MSSM EWSB (Unpolarized)
FIG. 4: Total cross section σ for the signalprocess, with the inclusion of radiative effects asa function of √ s with ( P e − , P e + ) = (0.8, -0.6) for SU (5) with nonuniversal gaugino masses and forthe MSSM EWSB scenario with universalgaugino masses at the grand unified scale. Forcomparison we have also shown the case ofMSSM EWSB with unpolarized beams.
400 600 800 1000 √ s (GeV) σ (f b ) MSSM EWSB [ S O ( )] [ S O ( )] [SO(10)] MSSM EWSB (Unpolarized)
FIG. 5: Total cross section σ for the signalprocess, with the inclusion of radiative effects asa function of √ s with ( P e − , P e + ) = (0.8, -0.6) for SO (10) with nonuniversal gaugino masses andfor the MSSM EWSB scenario with universalgaugino masses at the grand unified scale. Forcomparison we have also shown the case ofMSSM EWSB with unpolarized beams.The signal in the case of MSSM EWSB and [ SU (5)] is enhanced in the polarized case compared to the othermodels considered here. The dominant component of the neutralino in [ SU (5)] is a bino, whereas in othercases the lightest neutralino is dominantly a Higgsino state. The MSSM EWSB scenario predicts a lightestneutralino with a dominant Higgsino component, but it also has a significant bino component leading to theenhancement of right selectron-electron-neutralino coupling. Therefore the choice of this particular polarizationleads to an increase in the production cross section. For the other cases with a Higgsino-like neutralino the t -and u - channel exchange of ˜ e R,L is suppressed, with the only contribution coming from off-shell Z decay. The Z boson due to its ability to combine with both left- and right-handed fermions does not result in significantchanges with the inclusion of the beam polarization.2 B. Dependence on µ and M Since the mass of the lightest neutralino depends on the parameters µ and M , it is important to study thedependence of cross section for the signal process on these parameters. The dependence of the signal crosssection is considered independently on the parameters µ and M . The values of the parameters µ and M are chosen in order to avoid color and charge breaking minima, unbounded from below constraint on scalarpotential, and also to satisfy phenomenological constraints on different sparticle masses as discussed in Sec. II.We have carried out a check on the parameter space used in our calculations on whether the complete scalarpotential has charge and color breaking minima, which are lower than the electroweak symmetry breakingminimum. The condition of whether the scalar potential is unbounded from below has also been checked byus. The criteria used for these conditions are A f < m f L + m f R + µ + m H ) , (V.3) m H + m H ≥ | Bµ | , (V.4)respectively, at a scale Q > M . Here f denotes the fermion generation, and A is the trilinear supersym-metry breaking parameter. We have implemented these conditions through the SuSpect package [66] whichcomputes the masses and couplings of the supersymmetric partners of the SM particles. For each model consid-ered in this paper, we perform the renormalization group evolution to calculate the particle spectrum. Whiledoing so we check for the consistency of the chosen parameter set with electroweak symmetry breaking andalso whether the conditions (V.3) and (V.4) are satisfied.
120 140 160 µ (GeV) σ (f b ) [ S U ( ) ] M S S M E W S B [SU(5)] [SU(5)] [SO(10)] [SO(10)] [SO(10)] M SS M E W S B ( U np o l a r i ze d ) FIG. 6: The total radiative neutralinoproduction cross section σ with radiative effectsincluded as a function of µ in the range µ ǫ [110,160] GeV for different models at √ s = 500 GeVwith ( P e − , P e + ) = (0.8, -0.6). For comparison wehave also shown the case of MSSM EWSB withunpolarized beams.
400 500 600 700 800 900 1000 M (GeV) σ (f b ) MSSM EWSB (Unpolarized)[SU(5)] [SU(5)] [SO(10)] [SO(10)] [SO(10)] MSSM EWSB
FIG. 7: Total cross section σ with the inclusionof radiative effects for the radiative neutralinoproduction as a function of M for differentmodels with M ǫ [390, 1000] GeV at √ s = 500 GeV and ( P e − , P e + ) = (0.8, -0.6). Forcomparison we have also shown the case ofMSSM EWSB with unpolarized beams.In Fig. 6 we show the µ dependence of the cross section for different models considered in this paper for thepolarized case along with the unpolarized case of MSSM EWSB. The cross section in the case of SU (5) andMSSM EWSB is significantly enhanced compared to the unpolarized case. For the other scenarios, the behaviorin case of polarized beams is almost similar to the unpolarized case. It is found that for a wide range of µ , inthe case of the [ SU (5)] and MSSM EWSB scenario, all the experimental constraints are satisfied, with ˜ χ asthe LSP. For the other scenarios with a Higgsino-type lightest neutralino, the cross section is sensitive to thevalue of µ . Since m ˜ χ ∝ µ , above a certain value of µ , ˜ χ ceases to be the lightest supersymmetric particle.Depending on the percentage of the Higgsino component, the cross section changes with the value of µ . Mostof the scenarios considered here are tightly constrained as a function of µ , with the neutralino as the LSP. Thisis due to the various limits on the sparticles masses from the experiments. The cross section for some scenariosin this region is too small to be observed at the ILC with √ s = 500 GeV, even with an integrated luminosityof 500 fb − .In Fig. 7 we show the dependence of the radiative neutralino cross section on the soft gaugino mass parameter M for different models with polarized beams. In this case also SU (5) and MSSM EWSB show an enhance-3
200 400 600 800 1000 ml σ (f b ) MSSM EWSB [SU(5)] [SU(5)] [SU(5)] [SO(10)] [SO(10)] [SO(10)] MSSM EWSB (Unpolarized)
FIG. 8: Total cross section σ for the radiativeneutralino production with radiative effectsincluded vs m ˜ e L at √ s = 500 GeV with( P e − , P e + ) = (0.8, -0.6). For comparison we havealso shown the case of MSSM EWSB withunpolarized beams.
200 400 600 800 1000 mr σ (f b ) MSSM EWSB [ S U ( )] [SU(5)] [SU(5)] [SO(10)] [SO(10)] [SO(10)] MSSM EWSB (Unpolarized)
FIG. 9: Total cross section σ along with radiativeeffects for the radiative neutralino production vs m ˜ e R at √ s = 500 GeV with ( P e − , P e + ) = (0.8,-0.6). For comparison we have also shown thecase of MSSM EWSB with unpolarized beams.ment of the cross section, for smaller values of M . Since the total cross section decreases with increasing valueof M , a lower value of M favors a cross section which can be measured experimentally. C. Dependence on selectron masses
The selectron masses are free parameters for the models considered here. Since the signal process proceedsmainly via right and left selectron ˜ e R,L exchange in the t and u channels, we have also considered the dependenceof the total cross section on the selectron masses. The dependence on the selectron masses is shown in Figs. 8and 9 in the case of polarized beams and for unpolarized beams in case of the MSSM EWSB. The cross sectionis insensitive to the left selectron mass in case of all models. For SU (5) , the neutralino being a bino, thecross section is sensitive to the right selectron mass and decreases with increasing m ˜ e R and has a negligiblesensitivity to left selectron mass. The MSSM EWSB shows a peculiar behavior with respect to the rightselectron mass. This is mainly because the neutralino in this case has a dominant Higgsino component alongwith a significant bino component. Therefore the signal process in this scenario receives contribution from boththe right selectron exchange channel and the Z exchange channel. This behavior arises due to the interferenceterm from these two diagrams and is sensitive to the centre-of-mass energy. Note that for this particular choiceof beam polarization, this behavior is more enhanced as one of the contributing diagrams is due to ˜ e R exchange.If the beam polarization would have been due to left-handed electrons and right-handed positrons, there wouldbe no contribution from the right selectron exchange diagram. Therefore the cross section in that case will beinsensitive to m ˜ e R . The other models have a Higgsino-type neutralino; therefore their production cross sectionshows no dependence on the selectron masses. VI. BACKGROUND PROCESSESA. Neutrino background
For the signal process (IV.1) considered here, the main background comes from the SM radiative neutrinoproduction. The other possible backgrounds are from e + e − → τ + τ − γ , with both the τ ′ s decaying to softleptons or hadrons but the contribution from this process is found to be negligible. Another large backgroundcomes from the radiative Bhabha scattering, e + e − → e + e − γ , where e ±′ s are not detected. This radiativescattering is usually eliminated by imposing a cut on E γ . The events are selected by imposing the conditionthat any particle other than γ appearing in the angular range − . < cos θ γ < .
95 must have energy lessthan E max , where E max is detector dependent, but presumably no larger than a few GeV. This is discussed indetail in the literature [67].4The SM radiative neutrino production e + + e − → ν ℓ + ¯ ν ℓ + γ , ℓ = e, µ, τ (VI.1)has been studied extensively [40, 48, 68–70]. For this background process, ν e are produced via t -channel W boson exchange and ν e,µ,τ via s -channel Z boson exchange. The corresponding Feynman diagrams are shownin Fig. 10. e − ( p ) e + ( p ) e − W + γ ( q ) ν e ( k )¯ ν e ( k ) e − ( p ) e + ( p ) W + e − γ ( q ) ν e ( k )¯ ν e ( k ) e − ( p ) e + ( p ) W + W + γ ( q ) ν e ( k )¯ ν e ( k ) e − ( p ) e + ( p ) e − γ ( q ) Z ν l ( k )¯ ν l ( k ) e − ( p ) e + ( p ) e − Z ν l ( k )¯ ν l ( k ) γ ( q )( i ) ( ii ) ( iii )( iv ) ( v )FIG. 10: Feynman diagrams contributing to the radiative neutrino process e + e − → ν ¯ νγ where ( iv and v )corresponds to the neutrinos of three flavorsSince the photons emitted from this process mostly tend to be collinear, therefore the angular cut on thephoton is applied to separate it from the signal photons. This process mainly proceeds through the exchange of W bosons which couple only to the left-handed fermions. We are considering the case of beam polarization withright-handed electron and left-handed positron. The respective degree of polarization is P e − = 0.8 and P e + = -0.6. Figure 11 shows that the photon energy distribution from the radiative neutrino production, whereasin Fig. 12 we show the √ s dependence of the total radiative neutrino cross section. Note that the radiativecorrections are included here. The unpolarized case is also shown in the figures for comparison. It is observedthat with this choice of beam polarization, the W bosons in the intermediate state do not contribute, and thecross section is significantly reduced. For instance, at √ s = 500 GeV with the inclusion of radiative correctionsand the cuts, the total unpolarized cross section σ unpol is 2432 fb, whereas with the inclusion of this particularbeam polarization σ pol is 398 fb. The background is reduced by 1 order of magnitude. Due to the productionof Z boson through the s channel the photon energy distribution peaks for E γ = ( s − m Z ) / (2 √ s ) ≈
218 GeVat √ s = 500 GeV. By imposing an upper cut on the photon energy, which depends on the neutralino mass seeEq. (V.1), the photon background from radiative neutrino production is reduced. A similar argument holds forthe production cross section where the on-shell Z produced through this background process is eliminated byimposing an upper cut on the photon energy. B. Supersymmetric background
Apart from the SM background, the signal process (IV.1) under consideration has also a supersymmetricbackground from the sneutrino production process [40, 71] e + + e − → ˜ ν ℓ + ˜ ν ∗ ℓ + γ , ℓ = e, µ, τ . (VI.2)In Fig. 13 we show the tree-level Feynman diagrams contributing to the supersymmetric background processunder study. Apart from the s channel contribution from Z boson, the process also receives a t -channelcontribution from the virtual charginos. Due to the contribution from virtual charginos, this process is sensitiveto the chargino mixing matrix U . In Fig. 14 we show the photon energy distribution for the supersymmetricbackground process at √ s = 500 GeV for the different models, whereas the total production cross section isshown in Fig. 15. We have applied the same cuts for this process as in the signal process and have used aninitial beam polarization of P e − = 0.8, P e + = -0.6. Similar to the radiative neutrino and neutralino production5 E γ (GeV) d σ / d E γ (f b / G e V ) with cutwithout cutwith cut (Unpolarized) FIG. 11: Plot showing the photon energydistribution dσ/dE γ for the radiativeneutrino production process e + e − → ν ¯ νγ at √ s = 500 GeV, with the inclusion ofradiative effects and ( P e − , P e + ) = (0.8,-0.6).
200 400 600 800 1000 √ s (GeV) σ (f b ) without cutwith cutwith cut (Unpolarized) FIG. 12: The total energy √ s dependenceof the radiative neutrino cross section withand without an upper cut on the photonenergy E γ , along with the radiative effectsand ( P e − , P e + ) = (0.8, -0.6). e − ( p ) e + ( p ) e − ( p ) e + ( p ) e − ( p ) e + ( p ) e − ( p ) e + ( p ) e − ( p ) e + ( p ) e − ( p ) e + ( p ) e − ( p ) e + ( p ) e − ( p ) e + ( p )˜ χ +1 e − e − ˜ χ +1 ˜ χ +1 ˜ χ +1 ˜ ν e ( k )˜ ν ∗ e ( k ) γ ( q ) γ ( q )˜ ν e ( k )˜ ν ∗ e ( k ) ˜ ν e ( k ) γ ( q )˜ ν ∗ e ( k )˜ χ +2 e − ˜ ν e ( k )˜ ν ∗ e ( k ) γ ( q ) e − ˜ χ +2 γ ( q )˜ ν e ( k )˜ ν ∗ e ( k ) ˜ χ +2 ˜ χ +2 ˜ ν e ( k ) γ ( q )˜ ν ∗ e ( k ) e − γ ( q )˜ ν l ( k )˜ ν ∗ l ( k ) e − Z ˜ ν l ( k )˜ ν ∗ l ( k ) γ ( q ) Z ( i ) ( ii ) ( iii )( iv ) ( v ) ( vi )( vii ) ( viii )FIG. 13: Feynman diagrams contributing to the radiative sneutrino production process e + e − → ˜ ν ˜ ν ∗ γ , withthe last two diagrams ( vii and viii ) corresponding to all the leptonic sneutrinothe unpolarized case of MSSM EWSB is also included in the figures. The process is not sensitive to initial beampolarization, with the cross section and the photon energy distribution in the case of polarized beams behavingalmost similarly to the unpolarized case. From Figs. 14 and 15, it is seen that for [ SU (5)] , [ SU (5)] , [ SU (5)] and [ SO (10)] ′ , the behavior of the cross section is similar. This is due to the mixing matrix U being samefor all the models considered here.This process can act as a major supersymmetric background to the signal if the sneutrinos decay invisiblyvia ˜ ν → ˜ χ ν . This scenario has been called the “virtual LSP” scenario [40]. But the sneutrinos can decay toother particles if kinematically allowed thus reducing its contribution to the signal. We note that the otherprominent decay channels are ˜ ν → ˜ χ ± ℓ ∓ and ˜ ν → ˜ χ ν , if kinematically allowed. For the scenarios with a6bino-type neutralino, the dominant decay mode is the invisible decay channel with 100% branching ratio. Forthe scenarios with a Higgsino-type neutralino the various decay channels are presented in Table IX. Branching ratios MSSM EWSB SU (5) SU (5) SO (10) SO (10) SO (10) ′ BR(˜ ν e → ˜ χ ν e ) 78.4% 8.1% 21.2% 18% 24.2% 44.4%BR(˜ ν e → ˜ χ ν e ) 1.8% 4.54% 0.8% 1.2% 6%BR(˜ ν → ˜ χ ± ℓ ∓ ) 21.6% 90.1% 74.3% 81% 74.8% 49.6% TABLE IX: Branching ratios of the sneutrino for different models with a Higgsino-type lightest neutralinoThere can also be other supersymmetric background from the neutralino production e + e − → ˜ χ ˜ χ , withthe subsequent radiative decay [72] of the next-to-lightest neutralino ˜ χ → ˜ χ γ . The branching ratios for thisdecay are too small, with a significant ratio obtained for small values of tan β < M ∼ M [41, 73, 74].Therefore, we have neglected this process in our study; however a detailed discussion of this process can befound in Refs. [73–75]. E γ (GeV) d σ / d E γ (f b / G e V ) MSSM EWSB , [SO(10)] , [SO(10)] [SU(5)] , [SU(5)] , [SU(5)] , [S0(10)] MSSM EWSB (Unpolarized)
FIG. 14: Plot showing the photon energydistribution dσ/dE γ for the radiativesneutrino production process e + e − → ˜ ν ˜ ν ∗ γ at √ s = 500 GeV, with the inclusion ofradiative effects and initial beampolarization ( P e − , P e + ) = (0.8, -0.6). Forcomparison we have also shown the case ofMSSM EWSB with unpolarized beams.
400 600 800 1000 √ s (GeV) σ (f b ) MSSM EWSB, [SO(10)] , [SO(10)] [SU(5)] , [SU(5)] , [SU(5)] , [SO(10)] MSSM EWSB (Unpolarized)
FIG. 15: The total energy √ s dependenceof the radiative sneutrino cross section e + e − → ˜ ν ˜ ν ∗ γ with an upper cut on thephoton energy E γ and the inclusion ofradiative effects and initial beampolarization ( P e − , P e + ) = (0.8, -0.6). Forcomparison we have also shown the case ofMSSM EWSB with unpolarized beams. C. Theoretical significance
Finally we discuss whether the photons from the signal process can be measured over the photons from thebackground. This is expressed in terms of theoretical significance for a given integrated luminosity L and isdefined as [65] S = N S √ N S + N B = σ √ σ + σ B √L . (VI.3)In the above equation N S = σ L is the number of signal photons, and N B = σ B L denotes the number ofbackground photons. For the detection of a signal a theoretical significance of 5 is required, whereas the signalcan be measured at a 68 % confidence level for a theoretical significance of S = 1. In Fig. 16 we show the µ dependence of the theoretical significance S for the different models considered here for an initial beampolarization of P e − = 0.8 and P e + = -0.6. When the lightest neutralino is dominantly a bino as in case ofthe [ SU (5)] , or has dominant bino and a Higgsino components, as in the case of MSSM EWSB, the choiceof this beam polarization significantly enhances the signal compared to the unpolarized case. In the case of7unpolarized beams, S for the considered µ range has a maximum value of 2 in the case of [ SU (5)] , whereasfor this choice of beam polarization, it has a maximum value of 12. Similar behavior follows in case of MSSMEWSB. It can be seen from the Fig. 16 that it will be difficult to observe the signal for the other scenariosconsidered here with the lightest neutralino having a dominant Higgsino component.
120 140 160 µ (GeV) S
110 120 130 140 150 µ (GeV) S MSSM EWSB[SU(5)] [SU(5)] [S0(10)] [SU(5)] [ S ( )] [S0(10)] MSSM EWSB (Unpolarized)
FIG. 16: Plot showing the theoreticalsignificance S for the radiative neutralinoproduction as a function of µ for differentmodels considered in this paper with √ s = 500 GeV and ( P e − , P e + ) = (0.8, -0.6).For comparison we have also shown the caseof MSSM EWSB with unpolarized beams.
400 600 800 1000 M (GeV) S
500 1000 1500 M (GeV) S MSSM EWSB[SU(5)] [SU(5)] [SU(5)] [S0(10)] [S0(10)] [S0(10)] FIG. 17: The theoretical significance S forthe radiative neutralino production as afunction of the gaugino mass parameter M for the different models with √ s = 500 GeVand ( P e − , P e + ) = (0.8, -0.6). The case ofunpolarized beams for MSSM EWSB is notshown here as it coincides with thepolarized case.We have also studied the variation of theoretical significance S as a function of the gaugino mass parameter M . In Fig. 17 we show the M dependence of S for all the models considered in this work in the interval M ǫ [200,1000] GeV. A behavior almost similar to the µ dependence of S is observed.Along with S we have also considered the signal-to-background ratio defined as r = σσ B (VI.4)The values of S and r can serve as a good guideline for our analysis since we do not consider detector simulationhere, which is beyond the scope of the present paper. In the case of the ILC, for a signal to be detectable, r is required to be greater than 1%. Since the future collider is designed for planned energies of 500, 800, and1000 GeV, we have presented the signal and background cross sections along with S and r for these energiesand different cases of a longitudinally polarized beam for an integrated luminosity of 500 fb − . We presentthe values of the total cross section, the significance and the signal-to-background ratio for all the scenariosconsidered here for different center-of-mass energies in Tables X, XI and XII. The set of parameters consideredfor the different models is listed in Tables I, II, III, IV, V VI and VII. It can be seen from the Tables X, XIand XII that there is an enhancement in S and r when we move from the unpolarized to the polarized case.The enhancement is significant for the case of beam polarization (0 . | − . S and r , making the signal observable at the ILC for differentcases of beam polarization. But for the scenarios with a Higgsino-type neutralino, the values of S and r are toosmall, making it difficult to test them at the future linear colliders through this radiative neutralino productionprocess. D. Left-right asymmetry
In this subsection, we consider as an observable the left-right asymmetry as a means to distinguish betweenvarious grand unified models. In order to obtain a better efficiency, we consider the integral version of thisasymmetry. The integrated left-right asymmetry is defined as A LR = σ LR − σ RL σ LR + σ RL , (VI.5)8 ( P e − | P e + ) (0 |
0) (0 . |
0) (0 . | − .
3) (0 . | − .
6) (0 . |
0) (0 . | − .
3) (0 . | − . σ ( e + e − → ν ¯ νγ ) (fb) 2432 577 481 398 335 314 295 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.1377 0.1651 0.2096 0.2495 0.1704 0.2172 0.2601MSSM EWSB S 0.0624 0.1536 0.2136 0.2795 0.2081 0.2739 0.3384r 0.0056 0.0286 0.0435 0.0626 0.0508 0.0691 0.0881 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 1.883 3.391 4.432 5.376 3.551 4.626 5.758[ SU (5)] S 0.8534 3.1470 4.4978 5.9850 4.3150 5.7947 7.4239r 0.0774 0.5876 0.9214 1.3507 1.0600 1.4732 1.9518 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0007 0.0009 0.0011 0.0014 0.0009 0.0012 0.0014[ SU (5)] S 0.0003 0.0008 0.0011 0.0015 0.0010 0.0015 0.0018r 0.0000 0.0001 0.0002 0.0003 0.0002 0.0003 0.0004 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0067 0.0071 0.0089 0.0105 0.0071 0.0091 0.0108[ SU (5)] S 0.0030 0.0066 0.0090 0.0117 0.0086 0.0114 0.0140r 0.0002 0.0012 0.0018 0.0026 0.0021 0.0028 0.0036 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0150 0.0191 0.0244 0.0293 0.0195 0.0252 0.0310[ SO (10)] S 0.0068 0.0177 0.0248 0.0328 0.0238 0.0317 0.0403r 0.0006 0.0033 0.0050 0.0073 0.0058 0.0080 0.0105 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0193 0.0267 0.0344 0.0415 0.0275 0.0357 0.0440[ SO (10)] S 0.0087 0.0248 0.0350 0.0465 0.0335 0.0450 0.0572r 0.0007 0.0047 0.0071 0.0104 0.0082 0.0136 0.0149 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0117 0.0055 0.0061 0.0066 0.0048 0.0056 0.0065[ SO (10)] ′ S 0.0053 0.0051 0.0062 0.0073 0.0058 0.0070 0.0084r 0.0004 0.0009 0.0012 0.0016 0.0014 0.0017 0.0022
TABLE X: Signal and background cross sections σ , significance S , and signal-to-background ratio r in thecase of different beam polarizations ( P e − | P e + ) for the different scenarios at √ s = 500 GeV for L = 500 fb − .where σ RL and σ LR are defined in Sec. IV B. The coupling of the lightest neutralino to the selectron and afermion is different for left- and right- handed fermions, with different couplings, for the different models that wehave considered in this paper. The coupling is relatively sensitive to the composition of the lightest neutralino,and one would expect an appreciable difference between the left- and right-polarized cross section. It can beseen from Table XVII that the lightest neutralino with a dominant wino and bino composition is sensitive tobeam polarization, whereas the Higgsino type neutralino has no dependence on beam polarization. We plot inFig. 18 the left-right asymmetry for the different models for the radiative neutralino production as a functionof the centre-of-mass energy. The SM background (radiative neutrino production) is also considered here. Thedependence of the various models on beam polarization can be easily understood from Fig. 18. In the case ofradiative neutrino production, as discussed before, since the cross section gets highly suppressed with positiveelectron and negative positron beam polarization, A LR in this case is the largest. A LR is also larger for themodels where the lightest neutralino is mainly a bino or a wino, i.e., enhanced couplings to the selectrons.Since SU (5) , S ′ have a bino- and wino-type lightest neutralino, A LR , in this case is close to 1 orgreater than 0.5. But for most of the models ( SU (5) , SU (5) , SO (10) , S , S ′ ), A LR isless than 0.5, since they have the lightest neutralino with a dominant Higgsino component, with practicallyno beam polarization dependence from the exchange of selectrons. The result is almost independent of thecenter-of-mass energy. VII. SUMMARY AND CONCLUSIONS
In this paper we have carried out a detailed study of the radiative neutralino production e + e − → ˜ χ ˜ χ γ forthe case of SU (5) and SO (10) supersymmetric GUT models for ILC energies with longitudinally polarized e − and e + beams. In these GUT models the boundary conditions on the soft gaugino mass parameters can benonuniversal. We have compared the results of these GUT models with the corresponding results in the MSSMwith universal gaugino mass parameters (universal boundary conditions). For our analyses we have used aparticular set of parameter values for various models by imposing theoretical and experimental constraints as9 ( P e − | P e + ) (0 |
0) (0 . |
0) (0 . | − .
3) (0 . | − .
6) (0 . |
0) (0 . | − .
3) (0 . | − . σ ( e + e − → ν ¯ νγ ) (fb) 2365 505 375 284 280 230 177 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.1002 0.1343 0.1690 0.2055 0.1371 0.1772 0.2168MSSM EWSB S 0.0460 0.1336 0.1950 0.2725 0.1831 0.2611 0.3641r 0.0042 0.0265 0.0450 0.0723 0.0489 0.0770 0.1224 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 1.4103 2.5440 3.3289 4.0442 2.6612 3.4729 4.3246[ SU (5)] S 0.6482 2.5249 3.8268 5.3280 3.5392 5.0821 7.1810r 0.0596 0.5037 0.8877 1.4240 0.9504 1.5100 2.4433 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0004 0.0005 0.0006 0.0008 0.0005 0.0007 0.0008[ SU (5)] S 0.0001 0.0004 0.0006 0.0010 0.0006 0.0010 0.0013r 0.0000 0.0000 0.0001 0.0002 0.0001 0.0003 0.0004 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0037 0.0040 0.0051 0.0060 0.0042 0.0052 0.0063[ SU (5)] S 0.0017 0.0039 0.0058 0.0079 0.0056 0.0076 0.0105r 0.0001 0.0007 0.0013 0.0021 0.0015 0.0022 0.0035 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0097 0.0111 0.0140 0.0167 0.0112 0.0144 0.0176[ SO (10)] S 0.0044 0.0110 0.0161 0.0221 0.0149 0.0212 0.0295r 0.0004 0.0021 0.0037 0.0058 0.0040 0.0062 0.0099 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0112 0.0154 0.0197 0.0238 0.0158 0.0205 0.0252[ SO (10)] S 0.0051 0.1532 0.0227 0.0315 0.0211 0.0302 0.0423r 0.0004 0.0304 0.0052 0.0083 0.0056 0.0089 0.0142 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0067 0.0032 0.0035 0.0038 0.0027 0.0033 0.0038[ SO (10)] ′ S 0.0030 0.0031 0.0040 0.0050 0.0036 0.0048 0.0063r 0.0002 0.0006 0.0009 0.0013 0.0009 0.0014 0.0021
TABLE XI: Signal and background cross sections σ , significance S , and signal-to-background ratio r in thecase of different beam polarizations ( P e − | P e + ) for the different scenarios at √ s = 800 GeV for L = 500 fb − .
400 600 800 1000 √ s (GeV) -1-0.500.51 A L R M SS M E W S B [SU(5)] [SU(5)] [SU(5)] [SO(10)] ννγ [SO(10)] [SO(10)] FIG. 18: Plot showing the left-right asymmetry as a function of the center-of-mass energy for radiativeneutralino production in the case of different models along with the SM background from radiative neutrinoproduction.discussed in Sec. II. The radiative neutralino production process has a signature of a high-energy photon andmissing energy. The background to the signal process comes from the SM process e + e − → ν ¯ νγ and from thesupersymmetric process e + e − → ˜ ν ˜ ν ∗ γ .The purpose of the present work is to establish the use of longitudinal beam polarization in probing the effectsof boundary conditions in the neutralino sector that arise in GUTs at a linear collider. This is motivated bythe fact that longitudinal polarization is a distinct possibility at the ILC. For the signal process considered,the dominant SM background comes from the radiative neutrino production process, which proceeds throughthe exchange of W bosons which couple only to the left-handed particles. This dominant background made it0 ( P e − | P e + ) (0 |
0) (0 . |
0) (0 . | − .
3) (0 . | − .
6) (0 . |
0) (0 . | − .
3) (0 . | − . σ ( e + e − → ν ¯ νγ ) (fb) 2293 482 356 248 257 204 155 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0781 0.1073 0.1378 0.1652 0.1100 0.1431 0.1759MSSM EWSB S 0.0364 0.1092 5.8147 0.2344 0.1533 0.2239 0.3157r 0.0034 0.0222 0.0387 0.0666 0.0428 0.0701 0.1134 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 1.1022 1.9900 2.6008 3.1510 2.0845 2.7096 3.3768[ SU (5)] S 0.5145 2.0225 3.0709 4.4338 8.6425 4.2140 5.999r 0.0480 0.4128 0.7305 1.2705 0.8110 1.328 2.1786 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0003 0.0004 0.0005 0.0005 0.0004 0.0005 0.0006[ SU (5)] S 0.0001 0.0004 0.0006 0.0007 0.0005 0.0007 0.0010r 0.0000 0.0000 0.0001 0.0002 0.0002 0.0003 0.0004 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0027 0.0029 0.0036 0.0043 0.0029 0.0037 0.0045[ SU (5)] S 0.0012 0.0029 0.0042 0.0061 0.0040 0.0057 0.0080r 0.0001 0.0006 0.0010 0.0017 0.0011 0.0018 0.0029 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0074 0.0081 0.0100 0.0121 0.0081 0.0104 0.0126[ SO (10)] S 0.0034 0.0082 0.0118 0.1561 0.0112 0.0162 0.0226r 0.0003 0.0017 0.0028 0.0017 0.0032 0.0051 0.0081 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0082 0.0111 0.0142 0.0171 0.0114 0.0147 0.0181[ SO (10)] S 0.0038 0.0113 0.0168 0.0242 0.0159 0.0229 0.0325r 0.0004 0.0023 0.0040 0.0069 0.0044 0.0072 0.0116 σ ( e + e − → ˜ χ ˜ χ γ ) (fb) 0.0048 0.0023 0.0026 0.0028 0.002 0.0024 0.0028[ SO (10)] ′ S 0.0048 0.0023 0.0030 0.0039 0.0027 0.0037 0.0050r 0.0002 0.0005 0.0007 0.0011 0.0008 0.0012 0.0032
TABLE XII: Signal and background cross sections σ , significance S , and the signal-to-background ratio r inthe case of different beam polarizations ( P e − | P e + ) for the different scenarios at √ s = 1000 GeV for L = 500fb − .difficult to observe the signal process at the LEP even for very light neutralinos. At the LHC also the CMSexperiment has searched for a final state containing a photon and missing transverse energy, and the observedevent yield was seen to be in agreement with the standard model expectations. However in the case of theILC with the availability of beam polarization, a suitable choice of beam polarization ( e − R e + L ) will significantlyreduce the expected SM background. Therefore, the ILC, with the availability of beam polarization, will bea good place to look for the processes with a high-energy photon and large missing energy in the final state.At the future linear colliders, because of high luminosity, ISR and beamstrahlung are an unavoidable feature,and, therefore, we have included the radiative corrections in our calculations to obtain a precise values for thecross sections.We have studied in detail the cross section and the photon energy distribution for the signal and backgroundprocess for a centre-of-mass energy of 500 GeV and an integrated luminosity of 500 fb − . The initial beamsare taken to be longitudinally polarized with P e − = 0.8 and P e + = -0.6. Our analyses show the behavior ofthe different models with the inclusion of beam polarization. Together with these the dependence of the crosssection on the other free SUSY parameters which are involved in the signal process was also studied. Thisincludes the SU (2) L gaugino mass parameter M and the Higgs(ino) mass parameter µ as well as the selectronmasses ( m ˜ e R , m ˜ e L ). Our results demonstrate that the composition of the lightest neutralino in different modelsplays a crucial role in the signal process. It can be seen from Table XVII how the bino- and wino-typeneutralino production cross section will be controlled by different choices of initial beam polarization. Similarlythe insensitivity of the Higgsino-type neutralino production cross section to the beam polarization, which ismediated through thr Z boson, is also reflected in the table. For the bino-type neutralino which arises in SU (5) , with significantly larger coupling to the right selectron, the cross section is increased with the choiceof beam polarization used here, and the background is correspondingly reduced. At the same time in the caseof other models, with Higgsino-type lightest neutralino, there is no appreciable change in cross section for thechoice of beam polarization used in this paper, since Z couples to both left- and right-handed fermions.Finally, in order to study whether an excess of signal photons N S can be observed over the backgroundphotons N B from the SM radiative neutrino process, we have studied the theoretical statistical significance1 S and the signal-to-background ratio r . The dependence of S on the independent parameters M and µ isalso studied. The results that we have obtained emphasize the signal and the background cross sections alongwith the significance and the signal-to-background ratio for different degrees of initial beam polarization atdifferent planned centre-of-mass energies of the ILC. They are presented in Tables X, XI, and XII. Therefore,we conclude that in the presence of beam polarization with right-handed electrons and left-handed positrons,the models with a bino-type neutralino can be studied in detail through the radiative neutralino productionit the ILC. In this respect the grand unified supersymmetric SU (5) model is unique among all the modelsconsidered in this paper. In this case M can be large so that we get the gluino mass satisfying the experimentalconstraints, and also M will be small enough to lead to a light bino-type neutralino. Therefore, for the choiceof parameters considered in our paper, SU (5) will provide a signal which could be observed at the ILC.This provides a strong motivation for the search for the radiative neutralino production as an evidence of asupersymmetric grand unified model at the ILC.We would also like to point that even with initially polarized beams, the models with a Higgsino-typeneutralino will be too difficult to be observed at the ILC. These Higgsino-type scenarios have a distinctivefeature wherein ˜ χ , ˜ χ and ˜ χ ± are almost degenerate with masses around µ due to large values of M , and alow value of µ . Due to the degeneracy in mass, the processes (a) e + e − → ˜ χ ˜ χ γ and (b) e + e − → ˜ χ ˜ χ γ willalso yield a similar final state as the radiative neutralino production. A detailed study of signatures with a hardphoton and large missing energy will include processes a and b along with the signal process considered here.This will result in a significant increase of cross section, and may offer additional search avenues. We note herethat ˜ χ ˜ χ and ˜ χ ˜ χ production channels tend to be suppressed, but may, nevertheless, offer increased searchavenues. We do not consider this case any further here, but leave it for a future publication. VIII. ACKNOWLEDGEMENTS
The authors would like to thank B. Ananthanarayan for many useful discussions. P. N. P. would liketo thank the Centre for High Energy Physics, Indian Institute of Science, Bangalore for hospitality whilethis work was initiated. The work of P. N. P. is supported by the J. C. Bose National Fellowship of theDepartment of Science and Technology, and by the Council of Scientific and Industrial Research, India, underproject No. (03)(1220)/12/EMR-II. P. N. P would like to thank the Inter-University Centre for Astronomy andAstrophysics, Pune, India for hospitality where part of this work carried out.
Appendix A: GAUGINO MASSES IN GRAND UNIFIED THEORIES
In this section we review the nonuniversal and universal gaugino masses in grand unified theories.
1. Universal gaugino masses in grand unified theories
In supersymmetric models, with gravity mediated supersymmetry breaking, usually denoted as mSUGRA,the soft supersymmetry breaking gaugino mass parameters M , M , and M satisfy the universal boundaryconditions M = M = M = m / (A.1)at the grand unified scale M G , where i = 1 , , U (1) Y , SU (2) L , and the SU (3) C gaugegroups, respectively. Furthermore, the three gauge couplings corresponding to these gauge groups sat-isfy ( α i = g i / π, i = 1 , , α = α = α = α G , (A.2)at the GUT scale M G , where g = g ′ , g = g , with g ′ and g as U (1) Y , and SU (2) L gauge couplings,respectively, and g is the SU (3) C gauge coupling. The renormalization group equations then imply that implythat out of three gaugino mass parameters, only one is independent, which we are free to choose as the gluinomass M ≡ M ˜ g . For the gaugino mass parameters, this leads to the ratio M : M : M ≃ . . (A.3)2The gaugino mass parameters described above are the running masses evaluated at the electroweak scale M Z .A lower bound is then obtained on the parameter M in the case of mSUGRA, from the constraint on M (II.3) and the ratio (A.3) : M > ∼
50 GeV (A.4)
2. Nonuniversal gaugino masses in grand unified theories
In contrast to the Sec. A 1 with universal boundary condition (A.1) for the gaugino mass parameters at theGUT scale, we now consider the case of MSSM with nonuniversal boundary conditions at the GUT scale, whicharise in SU (5) and SO (10) grand unified theories. Since in supersymmetric GUTs the gaugino masses neednot be equal at the GUT scale, the neutralino masses and mixing can be different in SUSY GUTs as comparedto the MSSM with universal gaugino masses.The coupling of the field strength superfield W a with the gauge kinetic function f (Φ) results in the generationof soft gaugino masses in supersymmetric models (see Ref. [28] and references therein). This term can be writtenas L g.k. = Z d θf ab (Φ) W a W b + h.c., (A.5)with a and b referring to gauge group indices and repeated indices being summed over. The gauge kineticfunction f ab (Φ) can be written in terms of the singlet and nonsinglet chiral superfields.When the auxiliary part F Φ of a chiral superfield Φ in f (Φ) gets a VEV h F Φ i , the interaction (A.5) givesrise to soft gaugino masses: L g.k. ⊃ h F Φ i ab M P λ a λ b + h.c., (A.6)where λ a,b are gaugino fields. Here λ , λ , and λ are the U (1), SU (2), and SU (3) gaugino fields, respectively.Since the gauginos belong to the adjoint representation of the gauge group, Φ and F Φ can belong to any of therepresentations appearing in the symmetric product of the two adjoint representations of the correspondinggauge group. We note that in four-dimensional grand unified theories only the gauge groups SU (5), SO (10),and E support the chiral structure of weak interactions. Here we shall study the implications of nonuniversalgaugino masses for the case of SU (5) and SO (10) grand unified gauge groups. a. SU (5) In this section we shall consider the case where the SM gauge group is embedded in the grand unified gaugegroup SU (5). For the symmetric product of the two adjoint ( -dimensional) representations of SU (5), wehave ( ⊗ ) Symm = ⊕ ⊕ ⊕ . (A.7)In the simplest case where Φ and F Φ are assumed to be in the singlet representation of SU (5), we have equalgaugino masses at the GUT scale. But, as is obvious from Eq. (A.7), Φ and F Φ can belong to any of thenonsinglet representations , , and of SU (5). In such cases the soft gaugino masses are unequal butrelated to one another via the representation invariants of the gauge group [20]. In Table XIII we show the ratiosof gaugino masses which result when F Φ belong to different representations of SU (5) in the decomposition (A.7).In this paper, for definiteness, we shall study the case of each representation independently, although anarbitrary combination of these is also allowed.In the one-loop approximation, the solution of renormalization group equations for the soft supersymmetrybreaking gaugino masses M , M , and M can be written as [76] M i ( t ) α i ( t ) = M i (GUT) α i (GUT) , i = 1 , , . (A.8)Then at any arbitrary scale, we have M = 53 α cos θ W (cid:18) M (GUT) α (GUT) (cid:19) , M = α sin θ W (cid:18) M (GUT) α (GUT) (cid:19) , M = α (cid:18) M (GUT) α (GUT) (cid:19) . (A.9)3 SU (5) M G M G M G M EW M EW M EW -
15 110
TABLE XIII: Ratios of the gaugino masses at the GUT scale in the normalization M ( GU T ) = 1 and at theelectroweak scale in the normalization M ( EW ) = 1 for F terms in different representations of SU (5). Theseresults are obtained by using 1-loop renormalization group equations. b. SO (10) For the case of SO (10), we have for the product of two adjoint ( )-dimensional representations( × ) Symm = ⊕ ⊕ ⊕ . (A.10)In Table XIV we have shown the gaugino mass parameters for the different representations that arise in thesymmetric product (A.10) for the SO (10) group. We note from Table XIV that the ratios of gaugino massesfor the different representations of SO (10) in the symmetric product (A.10) with the unflipped embedding SU (5) ⊂ SO (10) are identical to the corresponding gaugino mass ratios in Table XIII for the embedding of SMin SU (5). Therefore, the input parameters and the resulting masses for the gaugino mass ratios in Table XIVfor SO (10) are identical to the corresponding Tables II, III, and IV for SU (5). There are two additionalmaximal power subgroups of SO (10), consistent with fermion content of the SM, apart from SU (5) ⊂ SO (10).We, thereforw, list in Tables XV and XVI, the ratio of the gaugino mass parameters, both at the GUT andelectroweak scale, for different representations that arise in the symmetric product of two adjoint representationsof SO (10) with relevant embedding of these subgroups in SO (10). Appendix B: NEUTRALINO MASS MATRIX, LAGRANGIAN, AND COUPLINGS
In this Appendix we recall the mixing matrix for the neutralinos and the couplings that enter our calculationsof the radiative neutralino cross section. We note that the neutralino mass matrix receives contribution fromMSSM superpotential term W MSSM = µH H , (B.1)where H and H are the Higgs doublet chiral superfields with opposite hypercharge, and µ is the supersymmet-ric Higgs(ino) mass parameter. In addition to Eq. (B.1), the neutralino mass matrix receives contributions fromthe interactions between gauge and matter multiplets as well as contributions from the soft supersymmetrybreaking masses for the SU (2) L and U (1) Y gauginos. Putting together all these contributions, the neutralinomass matrix, in the bino, wino, Higgsino basis ( − iλ ′ , − iλ , ψ H , ψ H ), can be written as [2, 57] M MSSM = M − m Z sin θ W cos β m Z sin θ W sin β M m Z cos θ W cos β − m Z cos θ W sin β − m Z sin θ W cos β m Z cos θ W cos β − µm Z sin θ W sin β − m Z cos θ W sin β − µ , (B.2)where M and M are the U (1) Y and the SU (2) L soft gaugino mass parameters, respectively, and tan β = v /v is the ratio of the vacuum expectation values of the neutral components of the two Higgs doublet fields H and H , respectively. Furthermore, m Z is the Z boson mass, and θ W is the weak mixing angle. In our analyseswe are considering all parameters in the neutralino mass matrix to be real. In this case it can be diagonalisedby an orthogonal matrix. If one of the eigenvalues of M MSSM is negative, then we can diagonalize this matrixusing a unitary matrix N , the neutralino mixing matrix, to get a positive semidefinite diagonal matrix [57]with the neutralino masses m χ i ( i = 1 , , ,
4) in order of increasing value: N ∗ M MSSM N − = diag (cid:0) m χ , m χ , m χ , m χ (cid:1) . (B.3)4 SO (10) SU (5) M G M G M G M EW M EW M EW
54 24
210 1 -
770 1 -
15 110
TABLE XIV: Ratios of the gaugino masses atthe GUT scale in the normalization M ( GU T ) =1 and at the electroweak scale in thenormalization M ( EW ) = 1 for F terms inrepresentations of SU (5) ⊂ SO (10) with thenormal (nonflipped) embedding. These resultshave been obtained at the 1-loop level. SO (10) [ SU (5) ′ × U (1)] flipped M G M G M G M EW M EW M EW ( ,0) 1 1 1 1 2 7.1 ( ,0) 1 3 -2 1 6 -14.3 ( ,0) 1 - - ,0) 1 -
157 107 ,0) 1 -15 -5 1 -28 -33.33 ( ,0) 1
577 577 ,0) 1 - ,0) 1 -15 -5 1 -28 -33.3( ,0) 1 5 TABLE XV: Ratios of the gaugino masses at theGUT scale in the normalization M ( GU T ) = 1 andat the electroweak scale in the normalization M ( EW ) = 1 at the 1-loop level for F terms inrepresentations of flipped SU (5) ′ × U (1) ⊂ SO (10). SO (10) SU (4) × SU (2) R M G M G M G M EW M EW M EW ( , ) 1 1 1 1 2 7.1 ( , ) 1 3 2 1 6 -14.3 ( , ) 1 - , ) 1 0 - , ) 1 0 0 1 0 0 ( , ) 1 , ) 1 0 0 1 0 0( , ) 1 0 0 1 0 0( , ) 1 0 TABLE XVI: Ratios of the gaugino masses at the GUTscale in the normalization M ( GU T ) = 1 and at theelectroweak scale in the normalization M ( EW ) = 1 at the1-loop level for F terms in representations of SU (4) × SU (2) L × SU (2) R ⊂ SO (10).For the minimal supersymmetric standard model, the interaction Lagrangian of neutralinos, electrons, selec-trons, and Z bosons is summarized by [57] L = ( − √ e cos θ W N ∗ ) ¯ f e P L ˜ χ ˜ e R + e √ θ W ( N + tan θ W N ) ¯ f e P R ˜ χ ˜ e L + e θ W cos θ W (cid:0) | N | − | N | (cid:1) Z µ ¯˜ χ γ µ γ ˜ χ + eZ µ ¯ f e γ µ (cid:2) θ W cos θ W (cid:18) − sin θ W (cid:19) P L − tan θ W P R (cid:3) f e + h . c ., (B.4)with the electron, selectrons, neutralino, and Z boson fields denoted by f e , ˜ e L,R , ˜ χ , and Z µ , respectively, and P R,L = (cid:0) ± γ (cid:1) . The interaction vertices arising from Eq. (B.4) are summarized in Table XVII. [1] H. P. Nilles, Phys. Rept. , 1 (1984); P. Nath, R. L. Arnowitt and A. H. Chamseddine, NUB-2613.[2] A. Bartl, H. Fraas, W. Majerotto and N. Oshimo, Phys. Rev. D , 1594 (1989).[3] A. Bartl, H. Fraas and W. Majerotto, Nucl. Phys. B , 1 (1986).[4] P. N. Pandita, Phys. Rev. D , 571 (1994). Vertex Vertex factorright selectron - electron - neutralino − ie √ θ W N ∗ P L left selectron - electron - neutralino ie √ θ W ( N + tan θ W N ) P R neutralino - Z - neutralino ie θ W cos θ W (cid:0) | N | − | N | (cid:1) γ µ γ electron - Z - electron ieγ µ (cid:2) θ W cos θ W (cid:0) − sin θ W (cid:1) P L − tan θ W P R (cid:3) selectron - photon - selectron ie ( p + p ) µ electron - photon - electron ieγ µ [5] P. N. Pandita, Z. Phys. C , 659 (1994).[6] P. N. Pandita, Phys. Rev. D , 566 (1996).[7] P. N. Pandita, arXiv:hep-ph/9701411.[8] S. Y. Choi, J. Kalinowski, G. A. Moortgat-Pick and P. M. Zerwas, Eur. Phys. J. C , 563 (2001) [Addendum-ibid.C , 769 (2002)] [arXiv:hep-ph/0108117].[9] K. Huitu, J. Laamanen and P. N. Pandita, Phys. Rev. D , 115009 (2003) [arXiv:hep-ph/0303262].[10] K. Huitu, J. Laamanen, P. N. Pandita and P. Tiitola, Phys. Rev. D , 115003 (2010) [arXiv:1006.0661 [hep-ph]].[11] J. A. Aguilar-Saavedra et al. [ECFA/DESY LC Physics Working Group], “TESLA Technical Design Report PartIII: Physics at an e+e- Linear Collider,” arXiv:hep-ph/0106315.[12] T. Abe et al. [American Linear Collider Working Group], “Linear collider physics resource book for Snowmass 2001.1: Introduction,” in Proc. of the APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001) ed. N. Graf, arXiv:hep-ex/0106055.[13] K. Abe et al. [ACFA Linear Collider Working Group], “Particle physics experiments at JLC,” arXiv:hep-ph/0109166.[14] G. Weiglein et al. [LHC/LC Study Group], “Physics interplay of the LHC and the ILC,” arXiv:hep-ph/0410364.[15] J. A. Aguilar-Saavedra et al. , Eur. Phys. J. C , 43 (2006) [arXiv:hep-ph/0511344].[16] G. A. Moortgat-Pick et al. , arXiv:hep-ph/0507011.[17] E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Phys. Lett. B , 231 (1982).[18] L. Randall and R. Sundrum, Nucl. Phys. B , 79 (1999) [arXiv:hep-th/9810155];G. F. Giudice, M. A. Luty, H. Murayama and R. Rattazzi, JHEP , 027 (1998) [arXiv:hep-ph/9810442].[19] K. Huitu, J. Laamanen and P. N. Pandita, Phys. Rev. D , 115003 (2002) [hep-ph/0203186].[20] J. R. Ellis, K. Enqvist, D. V. Nanopoulos and K. Tamvakis, Phys. Lett. B , 381 (1985); M. Drees, Phys.Lett. B , 409 (1985). G. Anderson, C. H. Chen, J. F. Gunion, J. D. Lykken, T. Moroi and Y. Yamada, Inthe Proceedings of 1996 DPF / DPB Summer Study on New Directions for High-Energy Physics (Snowmass 96),Snowmass, Colorado, 25 Jun - 12 Jul 1996, pp SUP107 [arXiv:hep-ph/9609457].[21] V.D. Barger and C. Kao, Phys. Rev.
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