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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. |