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U Anomalous (1) and Dark Matter ′ Andrea Mammarellaa aDipartimento di Fisica dell’Universita`di Roma ,“TorVergata” and I.N.F.N. - Sezione di Roma “TorVergata”, Viadella Ricerca Scientifica, 1 - 00133 Roma, ITALY Abstract ′ We have studied the lightest masses in the fermionic sector of an anomalous U(1) extension of the Minimal Supersym- metricStandardModel(MSSM)inspiredbybraneconstructions. TheLSPofthismodelisanXWIMP(extremelyweak 2 interaction particle). We have studied its relic density in the cases in which there is mixing with the neutralinos of the 1 MSSM and in the case in which there is not such mixing. We have showed that this extended model can satisfy the 0 2WMAP data. To perform this calculation we have modified the DarkSUSY package. n a J1 1. Introduction 38 be relativistic so that their abundance is enough to foster 9 39 the reaction. There are two cases. First, the anomalous 12 There has been much work recently to conceive an in-40 sector of the neutralinos mass matrix can be decoupled 3 tersectingbranemodelwiththegaugeandmattercontent41 from the MSSM sector: this implies that the LSP can be h]4 oftheStandardModel(SM)ofparticlephysics[1,2,3,4].42 either pure anomalous (Stu¨ckelino-Primeino mix) or pure p5 One ofthe featuresofsuchmodelsis the presenceofextra43 MSSM. This is called the no mixing case and we have ′ -6 anomalous gauge U(1)s whose anomaly is cancelled via44 studied it in the case in which the LSP is pure anomalous ep7 the Green-Schwarz mechanism. The anomalies of these45 and coannihilates with a pure MSSM NLSP. The second h8 models arecancelledwithout assumptiononthe newly in-46 case,calledthemixingcase,isthatinwhichtherearemix- [9 troduced quantum numbers. One model of this type have47 ing terms between the two sectors of the neutralinos mass 1 10 been introduced and discussed in [6] and [7]. We have48 matrix. SowecanhaveLSPandNLSPthatcanbeagen- v11 studied the compatibility of this model with the WMAP49 eral mixing of the six gauge eigenstates of the theory. We 512 data[8,9](seealso[10,11]forrelatedwork). Inthismodel50 will show that also in this general case the mixing is con- 113 wehavetwonewcontributiontotheneutralinosmassma-51 strained and then we will study the case in wich the LSP 014 trix: onecomingfromthesuperpartnerofthe Stu¨ckelberg52 is mostly anomalous with fractions of MSSM eigenstates, .415 boson (Stu¨ckelino) and the other from the ′superpartner53 whiletheNLSPismostlyofMSSMoriginwhitsmallfrac- 116 of the gauge boson mediating the extra U(1) (Primeino).54 tions of anomalous eigenstates. 017 By taking some simplifying and reasonable assumptions55 In the subsequent sections we will describe the results for 2 118 on the fermion masses entering the soft supersymmetry56 the relic density in both cases. We have obtained that :19 breaking lagrangian, the LSP turns out to be mostly a57 there are regions of the parameter space in which the iv20 mixture of Stu¨ckelino and Primeino. It is also easy to re-58 WMAP data can be satisfied in both cases. These re- X21 alize that, given the simplifying assumptions mentioned59 sults are obtained using a modified version of the Dark- r22 above, the LSP is interacting with the MSSM particles60 SUSY package which takes into account the couplings of a23 with a coupling suppressed by the inverse of the mass of61 our model. 24 theextragaugebosonofthetheorywhichmustbeatleast 25 oftheorderoftheTeVforphenomenologicalreasons. The 2. Model 26 LSP is then an XWIMP (Extra Weakly Interacting Mas-62 27 sive Particle),a class of particles which have alreadybeen 63 Our model [6] is an extension of the MSSM with an 28 studied in literature [12, 13]. The cross section of these 64 extra U(1). The charges of the matter fields with respect 29 LSPs is too weak to give the right relic abundance. This 65 to the symmetry groups are given in table 1. 30 is why one has to resort to coannihilations with NLSPs. 66 Gauge invariance imposes constraints that leave inde- 31 In our case, the cross section for the annihilations of the 67 pendentonlythreeofthesecharges. Inthismodelwehave 32 LSP with the NLSP and that for the coannihilations of 33 the two species differ for some orders of magnitude. Once68 chosen: QQ, QL and QHu. The anomalies induced by 69 this extension are cancelled by the GS mechanism. Each 34 again this situation is not new in literature [14, 15, 16] 70 anomalous triangle diagram is parametrized by a coeffi- 3365 bduectonuepeldesatsofbareatrseathteedrecwariellfublley:sotmheetMwoSSsMpecpieasrtwicillelsntoot71 cient b(2a) (entering the lagrangian)with the assigment: 37 keep them in equilibrium. Moreover these particles must Preprint submittedto Elsevier January 20, 2012 ′ SU(3)c SU(2)L U(1)Y U(1) is that in which an = 0. From the definition of an it’s UQici 33¯ 21 −12//63 QQUQc Lclaesatr,twheatreamne=mb0e⇔r [6Q,H8]ut=he0g.eneralform of the neutrali- Dic 3¯ 1 1/3 QDc nos mass matrix at tree level: HHELiuidc 1111 2122 −−1111///222 QQQQHHELudc MN˜ = M......S... √2M....M..02Z′ M.00..1 M0002 −g−0gv2g02dv12QvddHu g0−vgu1g022vQ2vuuHu   ... ... ... ... 0 µ  Table1: Chargeassignment.  −   ... ... ... ... ... 0    81 where MS and M0 are the soft masses of the stu¨ckelino 82 and of the primeino, respectively and vu, vd are the vevs (0) : U(1)′ U(1)′ U(1)′ b(0) (1)83 (vacuum expectation values) of the Higgs fields. We can AA(1) : U(1)′−−U(1)Y−−U(1)Y→→2b(21) (2)8845 steeremtshadtiffienrecnatsefroomf n0oemxcixeipntgfo(rQtHhue =two0)d,iathgoernealabreoxneos (2) : U(1)′ SU(2) SU(2) b(2) (3)86 that refer to the anomalous and the MSSM sectors. A − − → 2 (3) : U(1)′ SU(3) SU(3) b(3) (4) A − − → 2 3. Boltzmann Equation and coannihilations (4) : U(1)′ U(1)′ U(1) b(4) (5)87 A − − Y → 2 The relic density of a certain particle is given by the 72 The mass of the extra boson is parametrized by MV(0) = Boltzmann Equation (BE) [17]: 7743 U4b(31g)0′.∼The1 tTeremVs, owfhtehree Lg0agirsanthgeiancotuhpaltinwgilolfctohnetreibxutrtae ddnt =−3Hn− hσijviji(ninj −neiqnejq) (9) 75 to our calculation are [6, 8]: Xij 88 σij is the annihilation cross section: i L Stu¨ckelino= 4ψSσµ∂µψ¯S −√2b3ψSλ(0) XiXj ff¯ (10) → i 2 b(a)Tr λ(a)σµσ¯νF(a) ψ (6)89 vij is the absolute value of relative velocity between i-th −2√2Xa=0 2 (cid:16) µν (cid:17) S 9901 nanudmbj-etrhopfaprtairctliec,ledepfienreudnbity:ofvvijol=um|evio−f tvhje| ,i-tnhi sispetchiee −2√i2b(24)(cid:20)21λ(1)σµσ¯νFµ(0ν)ψS +(0↔1)(cid:21)+h.c. 9932 u(innidteoxf1vorelufemrsetoofLthSeP)i,-tnheiqspisectiheeinnutmhebremraolfepqauritliicblreiupmer. 76 Theparametersb2(i) areinverselyproportionaltoMV(0),so9945 nve=locPityinoif,tHheisintihtiealHpuabrbtilcelecso.nstant and v is the relative 77 they are much smaller than the couplings of the SM. This96 IftheLSPisanXWIMP,itisknown[12]thatitsrelicden- 78 impliesthattheprimeinoandthestu¨ckelinoareXWIMPS.97 sity does not satisfy the WMAP observations [18]. So, in 79 98 order to achieve the sperimental results, we need coanni- 80 The neutral mixing matrix is: 99 hilations. This phenomenon occurs when there is at least 100 one particle with mass of the order of the mass of the Bµ Aµ  W3µ =M Zµ  (7)101 XWIMP LSP. If the interactionof the coannihilating par- Cµ Z′0µ 102 ticles are strong enough they can lower the relic density     103 of the LSP by several orders of magnitude and so we can Defining an g Q 2v2 we have at tree level: 104 obtain results in agreement with the sperimental data. ≡ 0 Hu2MZ2′ 105 Althoughageneraltractationcanbefoundin[19],forour c s s g2+g2 an 106 purposes we are interested only in the cases in which the W − W W 1 2 107 number N of the coannihilating particles is 2 or 3. In the M = s c c pg2+g2 an  W W W 1 2 108 case N = 2 we have the following thermal averaged cross 0 c g +s g ) an p 1  W 2 W 1  109 section: (8) where c is cos(θ ), s is sin(θ ). g , g are the cou- W W W W 1 2 neq plings of the SM electro-weak SU(2) U(1) group. The σ(2) v = σ v ( 1 )2+ structure of this matrix leaves the ele×ctromagnetic sector h eff i h 11 i neq neqneq neq andtherelatedquantumnumbersunchangedwithrespect 2 σ v 1 2 + σ v ( 2 )2 = (11) to the MSSM ones. We can see from eq. (8) that the case h 12 i(neq)2 h 22 i neq of mixing between the anomalous sector and the MSSM σ v / σ v +2 σ v / σ v Q+Q2 11 22 12 22 σ v h i h i h i h i sectoristhatinwhichan=0,whilethe caseofnomixing h 22 i (1+Q)2 6 2 110 where Q = ne2q/ne1q. The first term in the numerator can 0.20 111 beneglectedbecausetheStu¨ckelinoannihilationcrosssec- 112 tion is suppressed by a factor (b(2a))4 with respect to the 0.15 D=1% 113 MSSM neutralino annihilations (see the previous section) 111145 aWnidththtuhsehsσa1m1evia≪ssuhmσ2p2tvioin.softhe caseN =2the formula 2Wh 0.10 116 for the case N =3 is: 0.05 σ v Q2+2 σ v Q Q + σ v Q2 σ(3)v h 22 i 2 h 23 i 2 3 h 33 i(132) h eff i ≃ (1+Q +Q )2 2 3 0.00 100 200 300 400 500 600 700 with neq∗ axinomassHGeVL Q = i (13) i neq Figure1: Relicdensityforabino-higgsinotypeNLSPandmassgap 1 1% 117 This expressions show that the factors Qi can raise/lower 118 the thermal averaged cross section and thus modify the 0.20 119 related relic density. However to obtain the result for the 120 relicdensityitisnecessarytonumericallysolvetheBE.To D=5% 121 achieve this we have used the DarkSUSY package,adding 0.15 122 theinteractionsintroducedbyourextendedmodel. Inthe 123 following section we describe our results. 2Wh 0.10 0.05 4. Numerical computations and results 124 125 Toperformanumericalanalysisontheextendedmodel 0.00 100 200 300 400 500 600 700 126 wehavemodifiedtheDarkSUSYpackageaddingnewfields axinomassHGeVL 127 and interactions introduced by the anomalous extension. 128 The free parameters that we use in our numerical simula- Figure2: Relicdensityforabino-higgsinotypeNLSPandmassgap 129 tionsarethesevenonesusedintheMSSM-7model: theµ 5% 130 mass, the wino soft mass M2, the parity-odd Higgs mass 131 MA0, tgβ, the sfermion mass scale msf, the two Yukawas 0.20 132 at and ab. We add to this set five parameters which de- ′ 133 fine the U(1) extension: the stu¨c′kelino soft mass MS, the 0.15 D2=5% 134 primeino soft mass M0, the U(1) charges QHu, QQ, QL. 135 2Wh 0.10 4.1. No-mixing case 136 0.05 137 As we have seen if there is no-mixing we have to im- 138 pose QHu = 0. From eq. (9), this implies that the LSP 0.00 139 can have either pure anomalous originor pure MSSM ori- 0 500 1000 1500 2000 140 gin. We are interested in the first case. Furthermore, we axinomassHGeVL 141 assume for simplicity that the LSP is a pure stu¨ckelino. Figure3: RelicdensityforawinotypeNLSPandmassgap5% 142 Because in this case the anomalous and the MSSM sec- 143 tors are decoupled, we can choose the model parameters 144 tokeepfixedthemassgapbetweentheLSPandtheNLSP. 0.20 145 Sowehavestudiedthe behaviourofthemodelinfunction D2=10% 146 of the mass of the LSP for a defined mass gap. 0.15 147 We have studied separately the two cases in which the 148 NLSP ismostlyabinoandinwhichtheNLSP ismostlya 2Wh 0.10 149 wino. Inthe firstcasethe coannihilationinvolvesonlythe 150 LSP andthe NLSPwhile inthe latter caseitinvolvesalso 0.05 151 thelightestchargino,thatintheMSSMisalmostdegener- 152 ateinmasswiththewino. Sotheyaredifferentsituations. 0.00 153 The results of the relic density calculations are showed in 0 200 400 600 800 1000 axinomassHGeVL 154 figures 1 and 2 for the bino NLSP case and in figures 3 155 and 4 for the wino LSP case. Figure4: RelicdensityforawinotypeNLSPandmassgap10% 3 156 In those figures we have shown the most significant209 We chose two sample models which satisfy the WMAP 157 results,butwehavefoundmodelthatsatisfyWMAPdata210 data [18] with mass gaps 5% and 10%. We study these 158 upto10%massgapforthebinoNLSPcaseandupto20%211 models to show the dependence of the relic density from 159 for the wino NLSP case. 212 the LSP composition and from the mass gap. To obtain 213 these result, we have performed a numerical simulation 160 4.2. Mixing case 214 in which we vary only the stu¨ckelino and the primeino 116612 trolNoonmthixeimngasimsgpalipesbtehtawteeQnHtuhe6=L0S,PsoawndecNaLnS’tPh,abveecacuonse-221165 scoofntvmenatsiosenss.: The results are showed in figure 1, with the 163 themassmatrixisnomoredecoupledintwoblocks. Sowe217 Insidethecontinouslineswehavetheregioninwhich 164 choose to let all parameters unconstrained and therefore218 • (Ωh2)WMAP Ωh2 165 to collect data in the mass gap ranges 0 5%, 5 10%. ∼ ÷ ÷ 166 In each case we have started our study scanning the pa-219 Inside the thick lines we have the region in which • 167 rameter space in search of the region permitted by the220 (Ωh2 3σ)WMAP <Ωh2 <(Ωh2+3σ)WMAP − 168 experimental and theoretical constraints, i.e. the region 169 in which we could satisfy the WMAP data with a certain221 • Inside the dashed lines we have the region in which 170 choice of the model parameters. After that we have nu-222 (Ωh2−5σ)WMAP <Ωh2 <(Ωh2+5σ)WMAP 171 merically explored this regions to find sets of parameters223 Inside the dotted lines we have the region in which 172 thatsatisfytheWMAPdatafortherelicdensity. Wehave224 • (Ωh2 10σ)WMAP <Ωh2 <(Ωh2+10σ)WMAP 173 foundmanysuitablecombinationsforbothtypesofNLSP. − 174 So wehavechosensomeofthese successfulmodels andwe225 Goingfromtheregionwitha5%massgaptothatwith 175 have computed the relic density keeping constant all but226 a 10% mass gap there is a large portion of the parameter 176 two parameters and plotting the results. 227 space inwhichthe WMAP data cannotbe satisfied,while 177 We have found regions of the parameter space in which228 the regions showed in the second and third plot in fig. 1 178 the WMAP data are satisfied for mass gap over 20%, but229 are similar and thus are mass-gap independent. 179 in the following we will only show results for the regions 180 0 5% and 5 10%, because they are more significant.230 4.2.3. Wino-higgsino NLSP ÷ ÷ 181 For simplicity we will refer to these regions as “5% region231 If the NLSP is mostly a wino-higgsinowe have a three 182 ”and “10% region”respectively. 232 particlecoannihilation,becausethelightestwinoisalmost 183 233 degenerateinmasswiththelightestchargino,sotheyboth 234 contribute to the coannihilations. 184 4.2.1. General results 235 In this case we perform the same numerical calculation 185 In this section we want to list some results that are236 illustratedintheprevioussubsection. Wehaveextensively 186 valid for both types of NLSP. 237 studied a sample model with mass gap 10%, showing an 187 Firstofall,giventheconstraintsontheneutralmixingde-238 example of funnel region, a resonance that occurs when 188 scribedin [5], wehave obtainedthat−1.QHu .1. This239 2 MLSP ∼MA0. In our sample model MLSP ∼330 GeV, 189 constraint was not appearing in our preliminary analysis240 while MA0 ∼ 630 GeV and this leads to the relic density 190 in[8]. Thisimpliesthatalsoinourgeneralcasethemixing241 plotshowedinfigure2,withthe sameconventionsusedin 191 between our anomalous LSP and the NLSP is small. 242 figure1. Sowecanstatethatalsointhecaseofanomalous 192 We have checked that there are suitable parameter space243 LSPwecanhaveabehavioursimilartothatoftheMSSM, 193 regionsinwhichtheWMAPdataaresatisfiedandwehave244 given that the LSP coannihilates with a MSSM NLSP. 194 found that this is true for all possible composition of our 195 anomalous LSP. 5. Conclusion 196 We have also checked that in each region we have studied245 197 there areno divergencesorunstablebehavioursinournu-246 We have modified the DarkSUSY package in the rou- 198 merical results. 247 tineswhichcalculatethecrosssectionofagivensupersym- 199 We have verified that the relic density is strongly depen-248 metric particle (contained in the folder /src/an) adding 200 dentonthe LSPandNLSPmassesandcompositionwhile249 all the new interactions introduced by o∼ur anomalous ex- 201 it is much less dependent on the other variables. Anyway250 tension of the MSSM. We have also written new subrou- 202 it canbe shownthat there are casesin which the parame-251 tinestocalculateamplitudesthatdifferfromthosealready 203 tersnotrelatedtotheLSPorNLSPcanplayanimportant252 contained in DarkSUSY. We have also modified the rou- 204 role. Wewillshowanexampleofthiscaseinaforthcoming253 tines that generate the supersymmetric model from the 205 subsection. 254 inputs, adding the parameters necessary to generate the 255 MiAUSSM [6] and changing the routines that define the 4.2.2. Bino-higgsino NLSP 206 256 model (contained in /src/su) accordingly. Finally we 207 If the NLSP is mostly a bino-higgsino we have a two257 have written a main p∼rogram that lets the user choose if 208 particle coannihilation. 258 he wants to perform the relic density calculation in the 4 0.082 0.094 0.11 0.127 Figure 6: Example of a funnel region for a wino-like NLSP, whose massgapwiththeLSPis10% IWh2M10Σ WMAP 269 ingbetweentheanomalousandtheMSSMsectorsandthe IWh2MW5ΣMAP 270 case in which there isn’t such mixing. 271 In the no-mixing case we are able to keep fixed the mass IWh2M3Σ 272 gap between the LSP and the NLSP. We have found that WMAP 273 if the NLSP is mostly a bino we can satisfy the WMAP 274 up to 10%massgap; if the NLSP is mostly a wino we can IWh2MWMAP 275 satisfy the WMAP up to 20% mass gap. 276 In the mixing case we have we have obtained a constraint 850 277 on QHu. We have also found sizable regions in which we 278 can satisfy the WMAP data for mass gaps which go from 800 279 5% to beyond 20%. We have studied some specific sets of 750 280 parametersforthe5%and10%massgapregions,showing 700 281 that relatively smallchanges in the mass gapcan produce M0 282 very important changes in the area of the regions which 650 283 satisfy the experimental constraints. 600 284 We havealsoshowed,asanexamplethatwestillhavethe 550 285 MSSM physics, the presence of a funnel region, analogous 500 286 to that in the MSSM, in some region of the parameter 20 40 60 80 100 120 140 MS 287 space. 288 So we can say that a model with an anomalous LSP can Figure 5: Plot of the relicdensity of the LSP vs the stu¨ckelino and289 satisfy all the current experimental constraints, can show primeino masses. The first plot shows the case of mass gap 5 %.290 a phenomenology similar to that expected from a MSSM The second plot isa zoom of the first inthe regionbetween 20-140 291 LSPandcanbeviabletoexplaintheDMabundancewith- GeV for stu¨ckelino mass and 500-850 GeV for primeino mass. The thirdplotshowsthecaseofmassgap10%inthesameregionofthe292 out any arbitrary constraints on its parameters. secondplot. AllthemassesareexpressedinGeV References 293 259 MSSM or in the MiAUSSM. The code of our version of294 [1] F. Marchesano, Fortsch. Phys. 55 (2007) 491 260 the package is available contacting andrea.mammarella 295 [arXiv:hep-th/0702094]. 261 @roma2.infn.it. 296 [2] R.Blumenhagen, M. Cvetic, P. Langacker andG. Shiu, Ann. 262 These modificationshavepermittedto extensivelynumer-297 Rev.Nucl.Part.Sci.55(2005) 71[arXiv:hep-th/0502005]. 298 [3] D. Lust, Class. Quant. Grav. 21 (2004) S1399 263 ically explore the parameters space for an anomalous ex-299 [arXiv:hep-th/0401156]. 264 tension of the MSSM without restriction on the neutral300 [4] E. Kiritsis, Fortsch. Phys. 52 (2004) 200 [Phys. Rept. 421 265 mixing or on the free parameters of the model. We have301 (2005ERRAT,429,121-122.2006)105][arXiv:hep-th/0310001]. 266 verifiedthat ourmodel does not leadto any divergenceor302 [5] P. Langacker, Rev. Mod. Phys. 81 (2009) 1199 303 [arXiv:0801.1345[hep-ph]]. 267 instability. 304 [6] P. Anastasopoulos, F. Fucito, A. Lionetto, G. Pradisi, 268 Wehavestudiedseparatelythecaseinwhichthereismix-305 A.RacioppiandY.S.Stanev,Phys.Rev.D78(2008)085014 306 [arXiv:0804.1156[hep-th]]. 5 307 [7] F. Fucito, A. Lionetto, A. Racioppi and D. R. Pacifici, Phys. 308 Rev.D82(2010)115004 [arXiv:1007.5443[hep-ph]]. 309 [8] F. Fucito, A. Lionetto, A. Mammarella, A. Racioppi The 310 European Physichal Journal C, 2010 Vol. 69, pag. 455-465 311 [arXiv:0811.1953v3[hep-ph]]. 312 [9] F. Fucito, A. Lionetto, A. Mammarella, A. Racioppi to be 313 publishedonPhys.Rev.D.[arXiv:arXiv:1105.4753[hep-ph]]. 314 [10] C. Coriano, M.Guzzi and A. Mariano, arXiv:1012.2420 [hep- 315 ph]. 316 [11] C. Coriano, M.Guzzi and A. Mariano, arXiv:1010.2010 [hep- 317 ph]. 318 [12] B. Kors and P. Nath, JHEP 0507 (2005) 069 319 [arXiv:hep-ph/0503208]. 320 [13] D. Feldman, B. Kors and P. Nath, Phys. Rev. D 75 (2007) 321 023503[arXiv:hep-ph/0610133]. 322 [14] M. Klein, Nucl. Phys. B 569 (2000) 362 323 [arXiv:hep-th/9910143]. 324 [15] K.GriestandD.Seckel,Phys.Rev.D43(1991)3191. 325 [16] J. Edsjo, M. Schelke, P. Ullio and P. Gondolo, JCAP 0304 326 (2003) 001[arXiv:hep-ph/0301106]. 327 [17] J. Edsjo and P. Gondolo, Phys. Rev. D 56 (1997) 1879 328 [arXiv:hep-ph/9704361]. 329 [18] E.Komatsuet al,TheAstrophysical Journal Supplement Se- 330 ries192:18(2011) 331 [19] A. Mammarella, “Phenomenology of anomalous U(1): Dark 332 Matter and Asymmetries” (2011), Ph.D. thesis (in prepara- 333 tion) 334 [20] P. Gondolo, J. Edsjo¨, P. Ullio, L. Bergstro¨m, M. Schelke and 335 E.A.Baltz,JCAP07(2004) 008[arXiv:astro-ph/0406204] 6