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2 | \section{Physics\label{PHYSICS}} | |

3 | ||

4 | ISAJET is a Monte Carlo program which simulates $pp$, $\bar pp$ | |

5 | and $e^+e^-$ interactions at high energy. | |

6 | The program incorporates | |

7 | perturbative QCD cross sections, initial state and final state QCD | |

8 | radiative corrections in the leading log approximation, independent | |

9 | fragmentation of quarks and gluons into hadrons, and a | |

10 | phenomenological model tuned to minimum bias and hard scattering data | |

11 | for the beam jets. | |

12 | ||

13 | \subsection{Hard Scattering\label{hard}} | |

14 | ||

15 | The first step in simulating an event is to generate a primary | |

16 | hard scattering according to some QCD cross section. This has the | |

17 | general form | |

18 | $$ | |

19 | \sigma = \sigma_0 F(x_1,Q^2) F(x_2,Q^2) | |

20 | $$ | |

21 | where $\sigma_0$ is a cross section calculated in QCD perturbation | |

22 | theory, $F(x,Q^2)$ is a structure function incorporating QCD scaling | |

23 | violations, $x_1$ and $x_2$ are the usual parton model momentum | |

24 | fractions, and $Q^2$ is an appropriate momentum transfer scale. | |

25 | ||

26 | For each of the processes included in ISAJET, the basic cross | |

27 | section $\sigma_0$ is a two-body one, and the user can set limits on | |

28 | the kinematic variables and type for each of the two primary jets. For | |

29 | DRELLYAN and WPAIR events the full matrix element for the decay of the | |

30 | W's into leptons or quarks is also included. | |

31 | ||

32 | The following processes are available: | |

33 | ||

34 | \subsubsection{Minbias} No hard scattering at all, so that the event | |

35 | consists only of beam jets. Note that at high energy the jet cross | |

36 | sections become large. To represent the total cross section it is | |

37 | better to use a sample of TWOJET events with the lower limit on pt | |

38 | chosen to give a cross section equal to the inelastic cross section or | |

39 | to use a mixture of MINBIAS and TWOJET events. | |

40 | ||

41 | \subsubsection{Twojet} All order $\alpha_s^2$ QCD processes, which | |

42 | give rise in lowest order to two high-$p_t$ jets. Included are, e.g. | |

43 | \begin{eqnarray*} | |

44 | g + g &\to& g + g\\ | |

45 | g + q &\to& g + q \\ | |

46 | g + g &\to& q + \bar q | |

47 | \end{eqnarray*} | |

48 | Masses are neglected for $c$ and lighter quarks but are taken into | |

49 | account for $b$ and $t$ quarks. The $Q^2$ scale is taken to be | |

50 | $$ | |

51 | Q^2 = 2stu/(s^2+t^2+u^2) | |

52 | $$ | |

53 | The default parton distributions are those of the CTEQ Collaboration, | |

54 | fit CTEQ3L, using lowest order QCD evolution. Two older fits, Eichten, | |

55 | Hinchliffe, Lane and Quigg (EHLQ), Set~1, and Duke and Owens, Set~1, | |

56 | are also included. There is also an interface to the CERN PDFLIB | |

57 | compilation of parton distributions. Note that structure functions for | |

58 | heavy quarks are included, so that processes like | |

59 | $$ | |

60 | g + t \to g + t | |

61 | $$ | |

62 | can be generated. The Duke-Owens parton distributions do not contain b | |

63 | or t quarks. | |

64 | ||

65 | Since the $t$ is so heavy, it decays before it can hadronize, so | |

66 | instead of $t$ hadrons a $t$ quark appears in the particle list. It is | |

67 | decayed using the $V-A$ matrix element including the $W$ propagator | |

68 | with a nonzero width, so the same decays should be used for $m_t < m_W$ | |

69 | and $m_t > m_W$; the $W$ should {\it not} be listed as part of the decay | |

70 | mode. The partons are then evolved and fragmented as usual; see | |

71 | below. The real or virtual $W$ and the final partons from the decay, | |

72 | including any radiated gluons, are listed in the particle table, | |

73 | followed by their fragmentation products. Note that for semileptonic | |

74 | decays the leptons appear twice: the lepton parton decays into a | |

75 | single particle of the same type but in general somewhat different | |

76 | momentum. In all cases only particles with $\verb|IDCAY| = 0$ should be | |

77 | included in the final state. | |

78 | ||

79 | A fourth generation $x,y$ is also allowed. Fourth generation | |

80 | quarks are produced only by gluon fusion. Decay modes are not included | |

81 | in the decay table; for a sequential fourth generation they would be | |

82 | very similar to the t decays. In decays involving quarks, it is | |

83 | essential that the quarks appear last. | |

84 | ||

85 | \subsubsection{Drellyan} Production of a $W$ in the standard model, | |

86 | including a virtual $\gamma$, a $W^+$, a $W^-$, or a $Z^0$, and its | |

87 | decay into quarks or leptons. If the transverse momentum QTW of the | |

88 | $W$ is fixed equal to zero then the process simulated is | |

89 | \begin{eqnarray*} | |

90 | q + \bar q \to W &\to& q + \bar q \\ | |

91 | &\to& \ell + \bar\ell | |

92 | \end{eqnarray*} | |

93 | Thus the $W$ has zero transverse momentum until initial state QCD | |

94 | corrections are taken into account. If non-zero limits on the | |

95 | transverse momentum $q_t$ for the $W$ are set, then instead the | |

96 | processes | |

97 | \begin{eqnarray*} | |

98 | q + \bar q &\to& W + g \\ | |

99 | g + q &\to& W + q | |

100 | \end{eqnarray*} | |

101 | are simulated, including the full matrix element for the $W$ decay. | |

102 | These are the dominant processes at high $q_t$, but they are of course | |

103 | singular at $q_t=0$. A cutoff of the $1/q_t^2$ singularity is made by | |

104 | the replacement | |

105 | $$ | |

106 | 1/q_t^2 \to 1/\sqrt{q_t^4+q_{t0}^4} \quad q_{t0}^2 = (.2\,\GeV) M | |

107 | $$ | |

108 | This cutoff is chosen to reproduce approximately the $q_t$ dependence | |

109 | calculated by the summation of soft gluons and to give about the right | |

110 | integrated cross section. Thus this option can be used for low as well | |

111 | as high transverse momenta. | |

112 | ||

113 | The scale for QCD evolution is taken to be proportional to the | |

114 | mass for lowest order Drell-Yan and to the transverse momentum for | |

115 | high-$p_t$ Drell-Yan. The constant is adjusted to get reasonable | |

116 | agreement with the $W + n\,{\rm jet}$ cross sections calculated from | |

117 | the full QCD matrix elements by F.A. Berends, et al., Phys.\ | |

118 | Lett.\ B224, 237 (1989). | |

119 | ||

120 | For the processes $g + b \to W + t$ and $g + t \to Z + t$, cross | |

121 | sections with a non-zero top mass are used for the production and the | |

122 | $W/Z$ decay. These were calculated using FORM 1.1 by J.~Vermaseren. The | |

123 | process $g + t \to W + b$ is {\it not} included. Both $g + b \to W^- + | |

124 | t$ and $g + \bar t \to W^- + \bar b$ of course give the same $W^- + t | |

125 | +\BARB_FINALSTATEAFTERQCDEVOLUTION | |

126 | needed to describe the $m_t = 0$(!) mass singularity for $q_t \gg | |

127 | m_t$, it has a pole in the physical region at low $q_t$ from on-shell | |

128 | $t \to W + b$ decays. There is no obvious way to avoid this without | |

129 | introducing an arbitrary cutoff. Hence, selecting only $W + b$ will | |

130 | produce a zero cross section. The $Q^2$ scale for the parton | |

131 | distributions in these processes is replaced by $Q^2 + m_t^2$; this | |

132 | seems physically sensible and prevents the cross sections from | |

133 | vanishing at small $q_t$. | |

134 | ||

135 | \subsubsection{Photon} Single and double photon production through the | |

136 | lowest order QCD processes | |

137 | \begin{eqnarray*} | |

138 | g + q &\to& \gamma + q \\ | |

139 | q + \bar q &\to& \gamma + g \\ | |

140 | q + \bar q &\to& \gamma + \gamma | |

141 | \end{eqnarray*} | |

142 | Higher order corrections are not included. But $\gamma$'s, $W$'s, and | |

143 | $Z$'s are radiated from final state quarks in all processes, allowing | |

144 | study of the bremsstrahlung contributions. | |

145 | ||

146 | \subsubsection{Wpair} Production of pairs of W bosons in the standard | |

147 | model through quark-antiquark annihilation, | |

148 | \begin{eqnarray*} | |

149 | q + \bar q &\to& W^+ + W^- \\ | |

150 | &\to& Z^0 + Z^0 \\ | |

151 | &\to& W^+ + Z^0, W^- + Z^0 \\ | |

152 | &\to& W^+ + \gamma, W^- + \gamma \\ | |

153 | &\to& Z^0 + \gamma | |

154 | \end{eqnarray*} | |

155 | The full matrix element for the W decays, calculated in the narrow | |

156 | resonance approximation, is included. However, the higher order | |

157 | processes, e.g. | |

158 | $$ | |

159 | q + q \to q + q + W^+ + W^- | |

160 | $$ | |

161 | are ignored, although they in fact dominate at high enough mass. | |

162 | Specific decay modes can be selected using the WMODEi keywords. | |

163 | ||

164 | \subsubsection{Higgs} Production and decay of the standard model Higgs | |

165 | boson. The production processes are | |

166 | \begin{eqnarray*} | |

167 | g + g &\to& H \quad\hbox{(through a quark loop)} \\ | |

168 | q + \bar q &\to& H \quad\hbox{(with $t + \bar t$ dominant)} \\ | |

169 | W^+ + W^- &\to& H \quad\hbox{ (with longitudinally polarized $W$)} \\ | |

170 | Z^0 + Z^0 &\to& H \quad\hbox{ (with longitudinally polarized $Z$)} | |

171 | \end{eqnarray*} | |

172 | If the (Standard Model) Higgs is lighter than $2 M_W$, then it will | |

173 | decay into pairs of fermions with branching ratios proportional to | |

174 | $m_f^2$. If it is heavier than $2 M_W$, then it will decay primarily | |

175 | into $W^+ W^-$ and $Z^0 Z^0$ pairs with widths given approximately by | |

176 | \begin{eqnarray*} | |

177 | \Gamma(H \to W^+ W^-) &=& {G_F M_H^3 \over 8 \pi \sqrt{2} } \\ | |

178 | \Gamma(H \to Z^0 Z^0) &=& {G_F M_H^3 \over 16 \pi \sqrt{2} } | |

179 | \end{eqnarray*} | |

180 | Numerically these give approximately | |

181 | $$ | |

182 | \Gamma_H = 0.5\,{\rm TeV} \left({M_H \over 1\,{\rm TeV}}\right)^3 | |

183 | $$ | |

184 | The width proportional to $M_H^3$ arises from decays into longitudinal | |

185 | gauge bosons, which like Higgs bosons have couplings proportional to | |

186 | mass. | |

187 | ||

188 | Since a heavy Higgs is wide, the narrow resonance approximation is | |

189 | not valid. To obtain a cross section with good high energy behavior, it | |

190 | is necessary to include a complete gauge-invariant set of graphs for the | |

191 | processes | |

192 | \begin{eqnarray*} | |

193 | W^+ W^- &\to& W^+ W^- \\ | |

194 | W^+ W^- &\to& Z^0 Z^0 \\ | |

195 | Z^0 Z^0 &\to& W^+ W^- \\ | |

196 | Z^0 Z^0 &\to& Z^0 Z^0 | |

197 | \end{eqnarray*} | |

198 | with longitudinally polarized $W^+$, $W^-$, and $Z^0$ bosons in the | |

199 | initial state. This set of graphs and the corresponding angular | |

200 | distributions for the $W^+$, $W^-$, and $Z^0$ decays have been | |

201 | calculated in the effective $W$ approximation and included in HIGGS. | |

202 | The $W$ structure functions are obtained by integrating the EHLQ | |

203 | parameterization of the quark ones term by term. The Cabibbo-allowed | |

204 | branchings | |

205 | \begin{eqnarray*} | |

206 | q &\to& W^+ + q' \\ | |

207 | q &\to& W^- + q' \\ | |

208 | q &\to& Z^0 + q | |

209 | \end{eqnarray*} | |

210 | are generated by backwards evolution, and the standard QCD evolution is | |

211 | performed. This correctly describes the $W$ collinear singularity and | |

212 | so contains the same physics as the effective $W$ approximation. | |

213 | ||

214 | If the Higgs is lighter than $2M_W$, then its decay to | |

215 | $\gamma\gamma$ through $W$ and $t$ loops may be important. This is | |

216 | also included in the HIGGS process and may be selected by choosing | |

217 | \verb|GM| as the jet type for the decay. | |

218 | ||

219 | If the Higgs has $M_Z < M_H < 2M_Z$, then decays into one real | |

220 | and one virtual $Z^0$ are generated if the \verb|Z0 Z0| decay mode is | |

221 | selected, using the calculation of Keung and Marciano, Phys.\ Rev.\ | |

222 | D30, 248 (1984). Since the calculation assumes that one $Z^0$ is | |

223 | exactly on shell, it is not reliable within of order the $Z^0$ width | |

224 | of $M_H = 2M_Z$; Higgs and and $Z^0 Z^0$ masses in this region should | |

225 | be avoided. The analogous Higgs decays into one real and one virtual | |

226 | charged W are not included. | |

227 | ||

228 | Note that while HIGGS contains the dominant graphs for Higgs | |

229 | production and graphs for $W$ pair production related by gauge invariance, | |

230 | it does not contain the processes | |

231 | \begin{eqnarray*} | |

232 | q + \bar q &\to& W^+ W^- \\ | |

233 | q + \bar q &\to& Z^0 Z^0 | |

234 | \end{eqnarray*} | |

235 | which give primarily transverse gauge bosons. These must be generated | |

236 | with WPAIR. | |

237 | ||

238 | If the \verb|MSSMi| or \verb|SUGRA| keywords are used with | |

239 | HIGGS, then one of the three MSSM neutral Higgs is generated instead | |

240 | using gluon-gluon and quark-antiquark fusion with the appropriate SUSY | |

241 | couplings. Since heavy CP even SUSY Higgs are weakly coupled to W | |

242 | pairs and CP odd ones are completely decoupled, $WW$ fusion and $WW | |

243 | \to WW$ scattering are not included in the SUSY case. ($WW \to WW$ can | |

244 | be generated using the Standard Model process with a light Higgs mass, | |

245 | say 100 GeV.) The MSSM Higgs decays into both Standard Model and SUSY | |

246 | modes as calculated by ISASUSY are included. For more discussion see | |

247 | the SUSY subsection below and the writeup for ISASUSY. The user must | |

248 | select which Higgs to generate using HTYPE; see Section 6 below. If a | |

249 | mass range is not specified, then the range mass $M_H \pm 5\Gamma_H$ | |

250 | is used by default. (This cannot be done for the Standard Model Higgs | |

251 | because it is so wide for large masses.) Decay modes may be selected | |

252 | in the usual way. | |

253 | ||

254 | \subsubsection{WHiggs} Generates associated production of gauge and | |

255 | Higgs bosons, i.e., | |

256 | $$ | |

257 | q + \bar q \to H + W, H + Z\,, | |

258 | $$ | |

259 | in the narrow resonance approximation. The desired subprocesses can be | |

260 | selected with JETTYPEi, and specific decay modes of the $W$ and/or $Z$ | |

261 | can be selected using the WMODEi keywords. Standard Model couplings are | |

262 | assumed unless SUSY parameters are specified, in which case the SUSY | |

263 | couplings are used. | |

264 | ||

265 | \subsubsection{SUSY} Generates pairs of supersymmetric particles from | |

266 | gluon-quark or quark-antiquark fusion. If the MSSMi or SUGRA | |

267 | parameters defined in Section 6 below are not specified, then only | |

268 | gluinos and squarks are generated: | |

269 | \begin{eqnarray*} | |

270 | g + g &\to& \tilde g + \tilde g \\ | |

271 | q + \bar q &\to& \tilde g + \tilde g \\ | |

272 | g + q &\to& \tilde g + \tilde q \\ | |

273 | g + g &\to& \tilde q + \tilde{\bar q} \\ | |

274 | q + \bar q &\to& \tilde q + \tilde{\bar q} \\ | |

275 | q + q &\to& \tilde q + \tilde q | |

276 | \end{eqnarray*} | |

277 | Left and right squarks are distinguished but assumed to be degenerate. | |

278 | Masses can be specified using the \verb|GAUGINO|, \verb|SQUARK|, and | |

279 | \verb|SLEPTON| parameters described in Section 6. No decay modes are | |

280 | specified, since these depend strongly on the masses. The user can | |

281 | either add new modes to the decay table (see Section 9) or use the | |

282 | \verb|FORCE| or \verb|FORCE1| commands (see Section 6). | |

283 | ||

284 | If \verb|MSSMA|, \verb|MSSMB|, and \verb|MSSMC| are specified, | |

285 | then the ISASUSY package is used to calculate the masses and decay | |

286 | modes in the minimal supersymmetric extension of the standard model | |

287 | (MSSM), assuming SUSY grand unification constraints in the neutralino | |

288 | and chargino mass matrix but allowing some additional flexibility in | |

289 | the masses. The scalar particle soft masses are input via | |

290 | \verb|MSSMi|, so that the physical masses will be somewhat different | |

291 | due to $D$-term contributions and mixings for 3rd generation sparticles. | |

292 | $\tilde t_1$ and $\tilde t_2$ production and decays are now included. | |

293 | The lightest SUSY particle is assumed to be the lightest neutralino | |

294 | $\tilde Z_1$. If the \verb|MSSMi| parameters are specified, then the | |

295 | following additional processes are included using the MSSM couplings | |

296 | for the production cross sections: | |

297 | \begin{eqnarray*} | |

298 | g + q &\to& \tilde Z_i + \tilde q, \quad \tilde W_i + \tilde q \\ | |

299 | q + \bar q &\to& \tilde Z_i + \tilde g, \quad \tilde W_i + \tilde g \\ | |

300 | q + \bar q &\to& \tilde W_i + \tilde Z_j \\ | |

301 | q + \bar q &\to& \tilde W_i^+ + \tilde W_j^- \\ | |

302 | q + \bar q &\to& \tilde Z_i + \tilde Z_j \\ | |

303 | q + \bar q &\to& \tilde\ell^+ + \tilde\ell^-, \quad \tilde\nu + \tilde\nu | |

304 | \end{eqnarray*} | |

305 | Processes can be selected using the optional parameters described in | |

306 | Section 6 below. | |

307 | ||

308 | Beginning with Version 7.42, matrix elements are taken into | |

309 | account in the event generator as well as in the calculation of decay | |

310 | widths for MSSM three-body decays of the form $\tilde A \to \tilde B f | |

311 | \bar f$, where $\tilde A$ and $\tilde B$ are gluinos, charginos, or | |

312 | neutralinos. This is implemented by having ISASUSY save the poles and | |

313 | their couplings when calculating the decay width and then using these | |

314 | to reconstruct the matrix element. Other three-body decays may be | |

315 | included in the future. Decays selected with \verb|FORCE| use the | |

316 | appropriate matrix elements. | |

317 | ||

318 | An optional keyword \verb|MSSMD| can be used to specify the second | |

319 | generation masses, which otherwise are assumed degenerate with the first | |

320 | generation. An optional keyword \verb|MSSME| can be used to specify | |

321 | values of the $U(1)$ and $SU(2)$ gaugino masses at the weak scale rather | |

322 | than using the default grand unification values. The chargino and | |

323 | neutralino masses and mixings are then computed using these values. | |

324 | ||

325 | Instead of using the \verb|MSSMi| parameters, one can use the | |

326 | \verb|SUGRA| parameter to specify in the minimal supergravity framework. | |

327 | This assumes that the gauge couplings unify at a GUT scale and that SUSY | |

328 | breaking occurs at that scale with universal soft breaking terms, which | |

329 | are related to the weak scale using the renormalization group. The | |

330 | renormalization group equations now include all the two-loop terms for | |

331 | both gauge and Yukawa couplings and the possible contributions from | |

332 | right-handed neutrinos. The parameters of the model are | |

333 | \begin{itemize} | |

334 | \item $m_0$: the common scalar mass at the GUT scale; | |

335 | \item $m_{1/2}$: the common gaugino mass at the GUT scale; | |

336 | \item $A_0$: the common soft trilinear SUSY breaking parameter at the | |

337 | GUT scale; | |

338 | \item $\tan\beta$: the ratio of Higgs vacuum expectation values at the | |

339 | electroweak scale; | |

340 | \item $\sgn\mu=\pm1$: the sign of the Higgsino mass term. | |

341 | \end{itemize} | |

342 | The renormalization group equations are solved iteratively to determine | |

343 | all the electroweak SUSY parameters from these data assuming radiative | |

344 | electroweak symmetry breaking but not other possible constraints such as | |

345 | b-tau unification or limits on proton decay. | |

346 | ||

347 | The assumption of universality at the GUT scale is rather | |

348 | restrictive and may not be valid. A variety of non-universal SUGRA | |

349 | (NUSUGRA) models can be generated using the \verb|NUSUG1|, \dots, | |

350 | \verb|NUSUG5| keywords. These might be used to study how well one could | |

351 | test the minimal SUGRA model. The keyword \verb|SSBCSC| can be used to | |

352 | specify an alternative scale (i.e., not the coupling constant | |

353 | unification scale) for the RGE boundary conditions. | |

354 | ||

355 | An alternative to the SUGRA model is the Gauge Mediated SUSY | |

356 | Breaking (GMSB) model of Dine, Nelson, and collaborators. In this model | |

357 | SUSY breaking is communicated through gauge interactions with messenger | |

358 | fields at a scale $M_m$ small compared to the Planck scale and are | |

359 | proportional to gauge couplings times $\Lambda_m$. The messenger fields | |

360 | should form complete $SU(5)$ representations to preserve the unification | |

361 | of the coupling constants. The parameters of the GMSB model, which are | |

362 | specified by the \verb|GMSB| keyword, are | |

363 | \begin{itemize} | |

364 | \item $\Lambda_m = F_m/M_m$: the scale of SUSY breaking, typically | |

365 | 10--$100\,{\rm TeV}$; | |

366 | \item $M_m > \Lambda_m$: the messenger mass scale; | |

367 | \item $N_5$: the equivalent number of $5+\bar5$ messenger fields. | |

368 | \item $\tan\beta$: the ratio of Higgs vacuum expectation values at the | |

369 | electroweak scale; | |

370 | \item $\sgn\mu=\pm1$: the sign of the Higgsino mass term; | |

371 | \item $C_{\rm grav}\ge1$: the ratio of the gravitino mass to the value it | |

372 | would have had if the only SUSY breaking scale were $F_m$. | |

373 | \end{itemize} | |

374 | In GMSB models the lightest SUSY particle is always the nearly massless | |

375 | gravitino $\tilde G$. The parameter $C_{\rm grav}$ scales the gravitino | |

376 | mass and hence the lifetime of the next lightest SUSY particle to decay | |

377 | into it. The \verb|NOGRAV| keyword can be used to turn off gravitino | |

378 | decays. | |

379 | ||

380 | A variety of non-minimal GMSB models can be generated using | |

381 | additional parameters set with the GMSB2 keyword. These additional | |

382 | parameters are | |

383 | \begin{itemize} | |

384 | \item $\slashchar{R}$, an extra factor multiplying the gaugino masses | |

385 | at the messenger scale. (Models with multiple spurions generally have | |

386 | $\slashchar{R}<1$.) | |

387 | \item $\delta M_{H_d}^2$, $\delta M_{H_u}^2$, Higgs mass-squared | |

388 | shifts relative to the minimal model at the messenger scale. (These | |

389 | might be expected in models which generate $\mu$ realistically.) | |

390 | \item $D_Y(M)$, a $U(1)_Y$ messenger scale mass-squared term | |

391 | ($D$-term) proportional to the hypercharge $Y$. | |

392 | \item $N_{5_1}$, $N_{5_2}$, and $N_{5_3}$, independent numbers of | |

393 | gauge group messengers. They can be non-integer in general. | |

394 | \end{itemize} | |

395 | For discussions of these additional parameters, see S. Dimopoulos, S. | |

396 | Thomas, and J.D. Wells, hep-ph/9609434, Nucl.\ Phys.\ {\bf B488}, 39 | |

397 | (1997), and S.P. Martin, hep-ph/9608224, Phys.\ Rev.\ {\bf D55}, 3177 | |

398 | (1997). | |

399 | ||

400 | Gravitino decays can be included in the general MSSM framework by | |

401 | specifying a gravitino mass with \verb|MGVTNO|. The default is that such | |

402 | decays do not occur. | |

403 | ||

404 | Another alternative SUSY model choice allowed is | |

405 | anomaly-mediated SUSY breaking, developed by Randall and Sundrum. | |

406 | In this model, it is assumed that SUSY breaking takes place | |

407 | in other dimensions, and SUSY breaking is communicated to the visible sector | |

408 | via the superconformal anomaly. In this model, the lightest SUSY particle | |

409 | is usually the neutralino which is nearly pure wino-like. The chargino | |

410 | is nearly mass degenerate with the lightest neutralino. It can be | |

411 | very long lived, or decay into a very soft pion plus missing energy. | |

412 | The model incorporated in ISAJET, based on work by | |

413 | Ghergetta, Giudice and Wells (hep-ph/9904378), | |

414 | and by Feng and Moroi (hep-ph/9907319) adds a universal contribution | |

415 | $m_0^2$ to all scalar masses to avoid problems with tachyonic scalars. | |

416 | The parameter set is $m_0,\ m_{3/2},\ \tan\beta ,\ sign(\mu )$, and | |

417 | can be input via the $AMSB$ keyword. Care should be taken with the chargino | |

418 | decay, since it may have macroscopic decay lengths, or even decay | |

419 | outside the detector. | |

420 | ||

421 | Since neutrinos seem to have mass, the effect of a massive right-handed | |

422 | neutrino has been included in ISAJET, when calculating the sparticle | |

423 | mass spectrum. If the keyword $SUGRHN$ is used, then the user | |

424 | must input the 3rd generation neutrino mass (at scale $M_Z$) in units | |

425 | of GeV, and the intermediate scale right handed neutrino Majorana mass $M_N$, | |

426 | also in GeV. In addition, one must specify the soft SUSY-breaking masses | |

427 | $A_n$ and $m_{\tilde\nu_R}$ valid at the GUT scale. Then the neutrino | |

428 | Yukawa coupling is computed in the simple see-saw model, and | |

429 | renormalization group evolution includes these effects between | |

430 | $M_{GUT}$ and $M_N$. Finally, to facilitate modeling of $SO(10)$ | |

431 | SUSY-GUT models, loop corrections to 3rd generation fermion masses have | |

432 | been included in the ISAJET SUSY models. | |

433 | ||

434 | The ISASUSY program can also be used independently of the rest of | |

435 | ISAJET, either to produce a listing of decays or in conjunction with | |

436 | another event generator. Its physics assumptions are described in more | |

437 | detail in Section~\ref{SUSY}. The ISASUGRA program can also be used | |

438 | independently to solve the renormalization group equations with SUGRA, | |

439 | GMSB, or NUSUGRA boundary conditions and then to call ISASUSY to | |

440 | calculate the decay modes. | |

441 | ||

442 | Generally the MSSM, SUGRA, or GMSB option should be used to study | |

443 | supersymmetry signatures; the SUGRA or GMSB parameter space is clearly | |

444 | more manageable. The more general option may be useful to study | |

445 | alternative SUSY models. It can also be used, e.g., to generate | |

446 | pointlike color-3 leptoquarks in technicolor models by selecting squark | |

447 | production and setting the gluino mass to be very large. The MSSM or | |

448 | SUGRA option may also be used with top pair production to simulate top | |

449 | decays to SUSY particles. | |

450 | ||

451 | \subsubsection{$e^+e^-$} An $e^+e^-$ event generator is also included in | |

452 | ISAJET. The | |

453 | Standard Model processes included are $e^+e^-$ annihilation through | |

454 | $\gamma$ and $Z$ to quarks and leptons, and production of $W^+W^-$ and | |

455 | $Z^0Z^0$ pairs. In contrast to WPAIR and HIGGS for the hadronic | |

456 | processes, the produced $W$'s and $Z$'s are treated as particles, so | |

457 | their spins are not properly taken into account in their decays. | |

458 | (Because the $W$'s and $Z$'s are treated as particles, their decay | |

459 | modes can be selected using \verb|FORCE| or \verb|FORCE1|, not | |

460 | \verb|WMODEi|. See Section [6] below.) Other Standard Model | |

461 | processes, including $e^+ e^- \to e^+ e^-$ ($t$-channel graph) and $e^+ e^- | |

462 | \to \gamma \gamma$, are not included. Once the primary reaction has been | |

463 | generated, QCD radiation and hadronization are done as for hadronic | |

464 | processes. | |

465 | ||

466 | The $e^+e^-$ generator can be run assuming no initial state | |

467 | radiation (the default), or an initial state electron structure function | |

468 | can be used for bremsstrahlung or the combination bremsstrahlung/beamstrahlung | |

469 | effect. Bremsstrahlung is implemented using the Fadin-Kuraev | |

470 | $e^-$ distribution function, and can be turned on using the \verb|EEBREM| | |

471 | command while stipulating the minimal and maximal subprocess energy. | |

472 | Beamstrahlung is implemented by invoking the \verb|EEBEAM| keyword. | |

473 | In this case, in addition the beamstrahlung parameter $\Upsilon$ and | |

474 | longitudinal beam size $\sigma_z$ (in mm) must be given. | |

475 | The definition for $\Upsilon$ in terms of other beam parameters can be | |

476 | found in the article Phys. Rev. D49, 3209 (1994) by Chen, Barklow and Peskin. | |

477 | The bremsstrahlung structure function is then convoluted with the | |

478 | beamstrahlung distribution (as calculated by P. Chen) and a spline fit | |

479 | is created. Since the cross section can contain large spikes, event generation | |

480 | can be slow if a huge range of subprocess energy is selected for light | |

481 | particles; in these scenarios, \verb|NTRIES| must be increased well beyond | |

482 | the default value. | |

483 | ||

484 | $e^+e^-$ annihilation to SUSY particles is included as well with | |

485 | complete lowest order diagrams, and cascade decays. The processes | |

486 | include | |

487 | \begin{eqnarray*} | |

488 | e^+ e^- &\to& \tilde q \tilde q \\ | |

489 | e^+ e^- &\to& \tilde\ell \tilde\ell \\ | |

490 | e^+ e^- &\to& \tilde W_i \tilde W_j \\ | |

491 | e^+ e^- &\to& \tilde Z_i \tilde Z_j \\ | |

492 | e^+ e^- &\to& H_L^0+Z^0,H_H^0+Z^0,H_A^0+H_L^0,H_A^0+H_H^0,H^++H^- | |

493 | \end{eqnarray*} | |

494 | Note that SUSY Higgs production via $WW$ and $ZZ$ fusion, which can | |

495 | dominate Higgs production processes at $\sqrt{s} > 500\,\GeV$, | |

496 | is not included. Spin correlations are neglected, although | |

497 | 3-body sparticle decay matrix elements are included. | |

498 | ||

499 | $e^+e^-$ cross sections with polarized beams are included for | |

500 | both Standard Model and SUSY processes. The keyword \verb|EPOL| is | |

501 | used to set $P_L(e^-)$ and $P_L(e^+)$, where | |

502 | $$ | |

503 | P_L(e) = (n_L-n_R)/(n_L+n_R) | |

504 | $$ | |

505 | so that $-1 \le P_L \le +1$. Thus, setting \verb|EPOL| to $-.9,0$ will | |

506 | yield a 95\% right polarized electron beam scattering on an unpolarized | |

507 | positron beam. | |

508 | ||

509 | \subsubsection{Technicolor} Production of a technirho of arbitrary | |

510 | mass and width decaying into $W^\pm Z^0$ or $W^+ W^-$ pairs. The cross | |

511 | section is based on an elastic resonance in the $WW$ cross section | |

512 | with the effective $W$ approximation plus a $W$ mixing term taken from | |

513 | EHLQ. Additional technicolor processes may be added in the future. | |

514 | ||

515 | \subsubsection{Extra Dimensions} The possibility that there might be | |

516 | more than four space-time dimensions at a distance scale $R$ much larger | |

517 | than $G_N^{1/2}$ has recently attracted interest. In these theories, | |

518 | $$ | |

519 | G_N = {1 \over 8\pi R^\delta M_D^{2+\delta}}\,, | |

520 | $$ | |

521 | where $\delta$ is the number of extra dimensions and $M_D$ is the | |

522 | $4+\delta$ Planck scale. Gravity deviates from the standard theory at a | |

523 | distance $R \sim 10^{22/\delta-19}\,{\rm m}$, so $\delta\ge2$ is | |

524 | required. If $M_D$ is of order $1\,{\rm TeV}$, then the usual heirarchy | |

525 | problem is solved, although there is then a new heirarchy problem of why | |

526 | $R$ is so large. | |

527 | ||

528 | In such models the graviton will have many Kaluza-Klein | |

529 | excitations with a mass splitting of order $1/R$. While any individual | |

530 | mode is suppressed by the four-dimensional Planck mass, the large number | |

531 | of modes produces a cross section suppressed only by $1/M_D^2$. The | |

532 | signature is an invisible massive graviton plus a jet, photon, or other | |

533 | Standard Model particle. The \verb|EXTRADIM| process implements this | |

534 | reaction using the cross sections of Giudice, Rattazzi, and Wells, | |

535 | hep-ph/9811291. The number $\delta$ of extra dimensions, the mass scale | |

536 | $M_D$, and the logical flag \verb|UVCUT| are specified using the keyword | |

537 | \verb|EXTRAD|. If \verb|UVCUT| is \verb|TRUE|, the cross section is cut | |

538 | off above the scale $M_D$; the model is not valid if the results depend | |

539 | on this flag. | |

540 | ||

541 | \subsection{Multiparton Hard Scattering} | |

542 | ||

543 | All the processes listed in Section~\ref{hard} are either $2\to2$ | |

544 | processes like \verb|TWOJET| or $2\to1$ $s$-channel resonance processes | |

545 | followed by a 2-body decay like \verb|DRELLYAN|. The QCD parton shower | |

546 | described in Section~\ref{qcdshower} below generates multi-parton final | |

547 | states starting from these, but it relies on an approximation which is | |

548 | valid only if the additional partons are collinear either with the | |

549 | initial or with the final primary ones. Since the QCD shower uses exact | |

550 | non-colliear kinematics, it in fact works pretty well in a larger region | |

551 | of phase space, but it is not exact. | |

552 | ||

553 | Non-collinear multiparton final states are interesting both in | |

554 | their own right and as backgrounds for other signatures. Both the matrix | |

555 | elements and the phase space for multiparton processes are complicated; | |

556 | they have been incorporated into ISAJET for the first time in | |

557 | Version~7.45. To calculate the matrix elements we have used the MadGraph | |

558 | package by Stelzer and Long, Comput.\ Phys.\ Commun.\ {\bf81}, 357 | |

559 | (1994), hep-ph/9401258. This automatically generates the amplitude using | |

560 | \verb|HELAS|, a formalism by Murayama, Watanabe, and Hagiwarak | |

561 | KEK-91-11, that calculates the amplitude for any Feynman diagram in | |

562 | terms of spinnors, vertices, and propagators. The MadGraph code has been | |

563 | edited to incorporate summations over quark flavors. To do the phase | |

564 | space integration, we have used a simple recursive algorithm to generate | |

565 | $n$-body phase space. We have included limits on the total mass of the | |

566 | final state using the \verb|MTOT| keyword. Limits on the $p_T$ and | |

567 | rapidity of each final parton can be set via the \verb|PT| and \verb|Y| | |

568 | keyworks, while limits on the mass of any pair of final partons can be | |

569 | set via the \verb|MIJTOT| keyword. These limits are sufficient to shield | |

570 | the infrared and collinear singularities and to render the result | |

571 | finite. However, the parton shower populates all regions of phase space, | |

572 | so careful thought is needed to combine the parton-shower based and | |

573 | multiparton based results. | |

574 | ||

575 | While the multiparton formalism is rather general, it still takes | |

576 | a substantial amount of effort to implement any particular process. So | |

577 | far only one process has been implemented. | |

578 | ||

579 | \subsubsection{$Z + {\rm 2\ jets}$} The \verb|ZJJ| process generates a | |

580 | $Z$ boson plus two jets, including the $q\bar{q} \to Z q \bar{q}$, $gg | |

581 | \to Z q\bar{q}$, $q\bar{q} \to Zgg$, $qq \to Zqq$, and $gq \to Z gq$ | |

582 | processes. The $Z$ is defined to be jet 1; it is treated in the narrow | |

583 | resonance approximation and is decayed isotropically. The quarks, | |

584 | antiquarks, and gluons are defined to be jets 2 and 3 and are | |

585 | symmetrized in the usual way. | |

586 | ||

587 | \subsection{QCD Radiative Corrections\label{qcdshower}} | |

588 | ||

589 | After the primary hard scattering is generated, QCD radiative | |

590 | corrections are added to allow the possibility of many jets. This is | |

591 | essential to get the correct event structure, especially at high | |

592 | energy. | |

593 | ||

594 | Consider the emission of one extra gluon from an initial or a | |

595 | final quark line, | |

596 | $$ | |

597 | q(p) \to q(p_1) + g(p_2) | |

598 | $$ | |

599 | From QCD perturbation theory, for small $p^2$ the cross section is | |

600 | given by the lowest order cross section multiplied by a factor | |

601 | $$ | |

602 | \sigma = \sigma_0 \alpha_s(p^2)/(2\pi p^2) P(z) | |

603 | $$ | |

604 | where $z=p_1/p$ and $P(z)$ is an Altarelli-Parisi function. The same | |

605 | form holds for the other allowed branchings, | |

606 | \begin{eqnarray*} | |

607 | g(p) &\to& g(p_1) + g(p_2) \\ | |

608 | g(p) &\to& q(p_1) + \bar q(p_2) | |

609 | \end{eqnarray*} | |

610 | These factors represent the collinear singularities of perturbation | |

611 | theory, and they produce the leading log QCD scaling violations for the | |

612 | structure functions and the jet fragmentation functions. They also | |

613 | determine the shape of a QCD jet, since the jet $M^2$ is of order | |

614 | $\alpha_s p_t^2$ and hence small. | |

615 | ||

616 | The branching approximation consists of keeping just these | |

617 | factors which dominate in the collinear limit but using exact, | |

618 | non-collinear kinematics. Thus higher order QCD is reduced to a | |

619 | classical cascade process, which is easy to implement in a Monte Carlo | |

620 | program. To avoid infrared and collinear singularities, each parton in | |

621 | the cascade is required to have a mass (spacelike or timelike) greater | |

622 | than some cutoff $t_c$. The assumption is that all physics at lower | |

623 | scales is incorporated in the nonperturbative model for hadronization. | |

624 | In ISAJET the cutoff is taken to be a rather large value, | |

625 | $(6\,\GeV)^2$, because independent fragmentation is used for the jet | |

626 | fragmentation; a low cutoff would give too many hadrons from | |

627 | overlapping partons. It turns out that the branching approximation not | |

628 | only incorporates the correct scaling violations and jet structure but | |

629 | also reproduces the exact three-jet cross section within factors of | |

630 | order 2 over all of phase space. | |

631 | ||

632 | This approximation was introduced for final state radiation by | |

633 | Fox and Wolfram. The QCD cascade is determined by the probability for | |

634 | going from mass $t_0$ to mass $t_1$ emitting no resolvable radiation. | |

635 | For a resolution cutoff $z_c < z < 1-z_c$, this is given by a simple | |

636 | expression, | |

637 | $$ | |

638 | P(t_0,t_1)=\left(\alpha_s(t_0)/\alpha_s(t_1)\right)^{2\gamma(z_c)/b_0} | |

639 | $$ | |

640 | where | |

641 | $$ | |

642 | \gamma(z_c)=\int_{z_c}^{1-z_c} dz\,P(z),\qquad | |

643 | b_0=(33-2n_f)/(12\pi) | |

644 | $$ | |

645 | Clearly if $P(t_0,t_1)$ is the integral probability, then $dP/dt_1$ is | |

646 | the probability for the first radiation to occur at $t_1$. It is | |

647 | straightforward to generate this distribution and then iteratively to | |

648 | correct it to get a cutoff at fixed $t_c$ rather than at fixed $z_c$. | |

649 | ||

650 | For the initial state it is necessary to take account of the | |

651 | spacelike kinematics and of the structure functions. Sjostrand has | |

652 | shown how to do this by starting at the hard scattering and evolving | |

653 | backwards, forcing the ordering of the spacelike masses $t$. The | |

654 | probability that a given step does not radiate can be derived from the | |

655 | Altarelli-Parisi equations for the structure functions. It has a form | |

656 | somewhat similar to $P(t_0,t_1)$ but involving a ratio of the structure | |

657 | functions for the new and old partons. It is possible to find a bound | |

658 | for this ratio in each case and so to generate a new $t$ and $z$ as for | |

659 | the final state. Then branchings for which the ratio is small are | |

660 | rejected in the usual Monte Carlo fashion. This ratio suppresses the | |

661 | radiation of very energetic partons. It also forces the branching $g | |

662 | \to t + \bar t$ for a $t$ quark if the $t$ structure function vanishes | |

663 | at small momentum transfer. | |

664 | ||

665 | At low energies, the branching of an initial heavy quark into a | |

666 | gluon sometimes fails; these events are discarded and a warning is | |

667 | printed. | |

668 | ||

669 | Since $t_c$ is quite large, the radiation of soft gluons is cut | |

670 | off. To compensate for this, equal and opposite transverse boosts are | |

671 | made to the jet system and to the beam jets after fragmentation with a | |

672 | mean value | |

673 | $$ | |

674 | \langle p_t^2\rangle = (.1\,\GeV) \sqrt{Q^2} | |

675 | $$ | |

676 | The dependence on $Q^2$ is the same as the cutoff used for DRELLYAN and | |

677 | the coefficient is adjusted to fit the $p_t$ distribution for the $W$. | |

678 | ||

679 | Radiation of gluons from gluinos and scalar quarks is also | |

680 | included in the same approximation, but the production of gluino or | |

681 | scalar quark pairs from gluons is ignored. Very little radiation is | |

682 | expected for heavy particles produced near threshold. | |

683 | ||

684 | Radiation of photons, $W$'s, and $Z$'s from final state quarks is | |

685 | treated in the same approximation as QCD radiation except that the | |

686 | coupling constant is fixed. Initial state electroweak radiation is not | |

687 | included; it seems rather unimportant. The $W^+$'s, $W^-$'s and $Z$'s | |

688 | are decayed into the modes allowed by the \verb|WPMODE|, \verb|WMMODE|, | |

689 | and \verb|Z0MODE| commands respectively. {\it Warning:} The branching | |

690 | ratios implied by these commands are not included in the cross section | |

691 | because an arbitrary number of $W$'s and $Z$'s can in principle be | |

692 | radiated. | |

693 | ||

694 | \subsection{Jet Fragmentation:} | |

695 | ||

696 | Quarks and gluons are fragmented into hadrons using the | |

697 | independent fragmentation ansatz of Field and Feynman. For a quark | |

698 | $q$, a new quark-antiquark pair $q_1 \bar q_1$ is generated with | |

699 | $$ | |

700 | u : d : s = .43 : .43 : .14 | |

701 | $$ | |

702 | A meson $q \bar q_1$ is formed carrying a fraction $z$ of the momentum, | |

703 | $$ | |

704 | E' + p_z' = z (E + p_z) | |

705 | $$ | |

706 | and having a transverse momentum $p_t$ with $\langle p_t \rangle = | |

707 | 0.35\,\GeV$. Baryons are included by generating a diquark with | |

708 | probability 0.10 instead of a quark; adjacent diquarks are not | |

709 | allowed, so no exotic mesons are formed. For light quarks $z$ is | |

710 | generated with the splitting function | |

711 | $$ | |

712 | f(z) = 1-a + a(b+1)(1-z)^b, \qquad | |

713 | a = 0.96, b = 3 | |

714 | $$ | |

715 | while for heavy quarks the Peterson form | |

716 | $$ | |

717 | f(z) = x (1-x)^2 / ( (1-x)^2 + \epsilon x )^2 | |

718 | $$ | |

719 | is used with $\epsilon = .80 / m_c^2$ for $c$ and $\epsilon = .50 / | |

720 | m_q^2$ for $q = b, t, y, x$. These values of $\epsilon$ have been | |

721 | determined by fitting PEP, PETRA, and LEP data with ISAJET and should | |

722 | not be compared with values from other fits. Hadrons with longitudinal | |

723 | momentum less than zero are discarded. The procedure is then iterated | |

724 | for the new quark $q_1$ until all the momentum is used. A gluon is | |

725 | fragmented like a randomly selected $u$, $d$, or $s$ quark or | |

726 | antiquark. | |

727 | ||

728 | In the fragmentation of gluinos and scalar quarks, supersymmetric | |

729 | hadrons are not distinguished from partons. This should not matter | |

730 | except possibly for very light masses. The Peterson form for $f(x)$ is | |

731 | used with the same value of epsilon as for heavy quarks, $\epsilon = | |

732 | 0.5 / m^2$. | |

733 | ||

734 | Independent fragmentation correctly describes the fast hadrons in | |

735 | a jet, but it fails to conserve energy or flavor exactly. Energy | |

736 | conservation is imposed after the event is generated by boosting the | |

737 | hadrons to the appropriate rest frame, rescaling all of the | |

738 | three-momenta, and recalculating the energies. | |

739 | ||

740 | \subsection{Beam Jets} | |

741 | ||

742 | There is now experimental evidence that beam jets are different in | |

743 | minimum bias events and in hard scattering events. ISAJET therefore uses | |

744 | similar a algorithm but different parameters in the two cases. | |

745 | ||

746 | The standard models of particle production are based on pulling | |

747 | pairs of particles out of the vacuum by the QCD confining field, | |

748 | leading naturally to only short-range rapidity correlations and to | |

749 | essentially Poisson multiplicity fluctuations. The minimum bias data | |

750 | exhibit KNO scaling and long-range correlations. A natural explanation | |

751 | of this was given by the model of Abramovskii, Kanchelli and Gribov. | |

752 | In their model the basic amplitude is a single cut Pomeron with | |

753 | Poisson fluctuations around an average multiplicity $\langle n | |

754 | \rangle$, but unitarity then produces graphs giving $K$ cut Pomerons | |

755 | with multiplicity $K\langle n \rangle$. | |

756 | ||

757 | A simplified version of the AKG model is used in ISAJET. The | |

758 | number of cut Pomerons is chosen with a distribution adjusted to fit the | |

759 | data. For a minimum bias event this distribution is | |

760 | $$ | |

761 | P(K) = ( 1 + 4 K^2 ) \exp{-1.8 K} | |

762 | $$ | |

763 | while for hard scattering | |

764 | $$ | |

765 | P(1) \to 0.1 P(1),\quad P(2) \to 0.2 P(2),\quad P(3) \to 0.5 P(3) | |

766 | $$ | |

767 | For each side of each event an $x_0$ for the leading baryon is selected | |

768 | with a distribution varying from flat for $K = 1$ to like that for | |

769 | mesons for large K: | |

770 | $$ | |

771 | f(x) = N(K) (1- x_0)^c(K),\qquad c(K) = 1/K + ( 1 - 1/K ) b(s) | |

772 | $$ | |

773 | The $x_i$ for the cut Pomerons are generated uniformly and then | |

774 | rescaled to $1-x_0$. Each cut Pomeron is then hadronized in its own | |

775 | center of mass using a modified independent fragmentation model with | |

776 | an energy dependent splitting function to reproduce the rise in | |

777 | $dN/dy$: | |

778 | $$ | |

779 | f(x) = 1 - a + a(b(s) + 1)^ b(s),\qquad | |

780 | b(s) = b_0 + b_1 \log(s) | |

781 | $$ | |

782 | The energy dependence is put into $f(x)$ rather than $P(K)$ because in | |

783 | the AKG scheme the single particle distribution comes only from the | |

784 | single chain. The probabilities for different flavors are taken to be | |

785 | $$ | |

786 | u : d : s = .46 : .46 : .08 | |

787 | $$ | |

788 | to reproduce the experimental $K/\pi$ ratio. |