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0795afa3 1\newpage
2\section{Physics\label{PHYSICS}}
3
4 ISAJET is a Monte Carlo program which simulates $pp$, $\bar pp$
5and $e^+e^-$ interactions at high energy.
6The program incorporates
7perturbative QCD cross sections, initial state and final state QCD
9fragmentation of quarks and gluons into hadrons, and a
10phenomenological model tuned to minimum bias and hard scattering data
11for the beam jets.
12
13\subsection{Hard Scattering\label{hard}}
14
15 The first step in simulating an event is to generate a primary
16hard scattering according to some QCD cross section. This has the
17general form
18$$19\sigma = \sigma_0 F(x_1,Q^2) F(x_2,Q^2) 20$$
21where $\sigma_0$ is a cross section calculated in QCD perturbation
22theory, $F(x,Q^2)$ is a structure function incorporating QCD scaling
23violations, $x_1$ and $x_2$ are the usual parton model momentum
24fractions, and $Q^2$ is an appropriate momentum transfer scale.
25
26 For each of the processes included in ISAJET, the basic cross
27section $\sigma_0$ is a two-body one, and the user can set limits on
28the kinematic variables and type for each of the two primary jets. For
29DRELLYAN and WPAIR events the full matrix element for the decay of the
30W'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
35consists only of beam jets. Note that at high energy the jet cross
36sections become large. To represent the total cross section it is
37better to use a sample of TWOJET events with the lower limit on pt
38chosen to give a cross section equal to the inelastic cross section or
39to use a mixture of MINBIAS and TWOJET events.
40
41\subsubsection{Twojet} All order $\alpha_s^2$ QCD processes, which
42give rise in lowest order to two high-$p_t$ jets. Included are, e.g.
43\begin{eqnarray*}
44g + g &\to& g + g\\
45g + q &\to& g + q \\
46g + g &\to& q + \bar q
47\end{eqnarray*}
48Masses are neglected for $c$ and lighter quarks but are taken into
49account for $b$ and $t$ quarks. The $Q^2$ scale is taken to be
50$$51Q^2 = 2stu/(s^2+t^2+u^2) 52$$
53The default parton distributions are those of the CTEQ Collaboration,
54fit CTEQ3L, using lowest order QCD evolution. Two older fits, Eichten,
55Hinchliffe, Lane and Quigg (EHLQ), Set~1, and Duke and Owens, Set~1,
56are also included. There is also an interface to the CERN PDFLIB
57compilation of parton distributions. Note that structure functions for
58heavy quarks are included, so that processes like
59$$60g + t \to g + t 61$$
62can be generated. The Duke-Owens parton distributions do not contain b
63or t quarks.
64
65 Since the $t$ is so heavy, it decays before it can hadronize, so
66instead of $t$ hadrons a $t$ quark appears in the particle list. It is
67decayed using the $V-A$ matrix element including the $W$ propagator
68with a nonzero width, so the same decays should be used for $m_t < m_W$
69and $m_t > m_W$; the $W$ should {\it not} be listed as part of the decay
70mode. The partons are then evolved and fragmented as usual; see
71below. The real or virtual $W$ and the final partons from the decay,
72including any radiated gluons, are listed in the particle table,
73followed by their fragmentation products. Note that for semileptonic
74decays the leptons appear twice: the lepton parton decays into a
75single particle of the same type but in general somewhat different
76momentum. In all cases only particles with $\verb|IDCAY| = 0$ should be
77included in the final state.
78
79 A fourth generation $x,y$ is also allowed. Fourth generation
80quarks are produced only by gluon fusion. Decay modes are not included
81in the decay table; for a sequential fourth generation they would be
82very similar to the t decays. In decays involving quarks, it is
83essential that the quarks appear last.
84
85\subsubsection{Drellyan} Production of a $W$ in the standard model,
86including a virtual $\gamma$, a $W^+$, a $W^-$, or a $Z^0$, and its
87decay 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*}
90q + \bar q \to W &\to& q + \bar q \\
91 &\to& \ell + \bar\ell
92\end{eqnarray*}
93Thus the $W$ has zero transverse momentum until initial state QCD
94corrections are taken into account. If non-zero limits on the
95transverse momentum $q_t$ for the $W$ are set, then instead the
96processes
97\begin{eqnarray*}
98q + \bar q &\to& W + g \\
99g + q &\to& W + q
100\end{eqnarray*}
101are simulated, including the full matrix element for the $W$ decay.
102These are the dominant processes at high $q_t$, but they are of course
103singular at $q_t=0$. A cutoff of the $1/q_t^2$ singularity is made by
104the replacement
105$$1061/q_t^2 \to 1/\sqrt{q_t^4+q_{t0}^4} \quad q_{t0}^2 = (.2\,\GeV) M 107$$
108This cutoff is chosen to reproduce approximately the $q_t$ dependence
109calculated by the summation of soft gluons and to give about the right
110integrated cross section. Thus this option can be used for low as well
111as high transverse momenta.
112
113 The scale for QCD evolution is taken to be proportional to the
114mass for lowest order Drell-Yan and to the transverse momentum for
115high-$p_t$ Drell-Yan. The constant is adjusted to get reasonable
116agreement with the $W + n\,{\rm jet}$ cross sections calculated from
117the full QCD matrix elements by F.A. Berends, et al., Phys.\
118Lett.\ B224, 237 (1989).
119
120 For the processes $g + b \to W + t$ and $g + t \to Z + t$, cross
121sections 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
123process $g + t \to W + b$ is {\it not} included. Both $g + b \to W^- + 124t$ and $g + \bar t \to W^- + \bar b$ of course give the same $W^- + t 125+\BARB_FINALSTATEAFTERQCDEVOLUTION 126needed to describe the$m_t = 0$(!) mass singularity for$q_t \gg
127m_t$, it has a pole in the physical region at low$q_t$from on-shell 128$t \to W + b$decays. There is no obvious way to avoid this without 129introducing an arbitrary cutoff. Hence, selecting only$W + b$will 130produce a zero cross section. The$Q^2$scale for the parton 131distributions in these processes is replaced by$Q^2 + m_t^2$; this 132seems physically sensible and prevents the cross sections from 133vanishing at small$q_t$. 134 135\subsubsection{Photon} Single and double photon production through the 136lowest order QCD processes 137\begin{eqnarray*} 138g + q &\to& \gamma + q \\ 139q + \bar q &\to& \gamma + g \\ 140q + \bar q &\to& \gamma + \gamma 141\end{eqnarray*} 142Higher order corrections are not included. But$\gamma$'s,$W$'s, and 143$Z$'s are radiated from final state quarks in all processes, allowing 144study of the bremsstrahlung contributions. 145 146\subsubsection{Wpair} Production of pairs of W bosons in the standard 147model through quark-antiquark annihilation, 148\begin{eqnarray*} 149q + \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*} 155The full matrix element for the W decays, calculated in the narrow 156resonance approximation, is included. However, the higher order 157processes, e.g. 158$$159q + q \to q + q + W^+ + W^- 160$$ 161are ignored, although they in fact dominate at high enough mass. 162Specific decay modes can be selected using the WMODEi keywords. 163 164\subsubsection{Higgs} Production and decay of the standard model Higgs 165boson. The production processes are 166\begin{eqnarray*} 167g + g &\to& H \quad\hbox{(through a quark loop)} \\ 168q + \bar q &\to& H \quad\hbox{(with$t + \bar t$dominant)} \\ 169W^+ + W^- &\to& H \quad\hbox{ (with longitudinally polarized$W$)} \\ 170Z^0 + Z^0 &\to& H \quad\hbox{ (with longitudinally polarized$Z$)} 171\end{eqnarray*} 172If the (Standard Model) Higgs is lighter than$2 M_W$, then it will 173decay 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 175into$W^+ W^-$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*} 180Numerically these give approximately 181$$182\Gamma_H = 0.5\,{\rm TeV} \left({M_H \over 1\,{\rm TeV}}\right)^3 183$$ 184The width proportional to$M_H^3$arises from decays into longitudinal 185gauge bosons, which like Higgs bosons have couplings proportional to 186mass. 187 188 Since a heavy Higgs is wide, the narrow resonance approximation is 189not valid. To obtain a cross section with good high energy behavior, it 190is necessary to include a complete gauge-invariant set of graphs for the 191processes 192\begin{eqnarray*} 193W^+ W^- &\to& W^+ W^- \\ 194W^+ W^- &\to& Z^0 Z^0 \\ 195Z^0 Z^0 &\to& W^+ W^- \\ 196Z^0 Z^0 &\to& Z^0 Z^0 197\end{eqnarray*} 198with longitudinally polarized$W^+$,$W^-$, and$Z^0$bosons in the 199initial state. This set of graphs and the corresponding angular 200distributions for the$W^+$,$W^-$, and$Z^0$decays have been 201calculated in the effective$W$approximation and included in HIGGS. 202The$W$structure functions are obtained by integrating the EHLQ 203parameterization of the quark ones term by term. The Cabibbo-allowed 204branchings 205\begin{eqnarray*} 206q &\to& W^+ + q' \\ 207q &\to& W^- + q' \\ 208q &\to& Z^0 + q 209\end{eqnarray*} 210are generated by backwards evolution, and the standard QCD evolution is 211performed. This correctly describes the$W$collinear singularity and 212so 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 216also 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 220and one virtual$Z^0$are generated if the \verb|Z0 Z0| decay mode is 221selected, using the calculation of Keung and Marciano, Phys.\ Rev.\ 222D30, 248 (1984). Since the calculation assumes that one$Z^0$is 223exactly on shell, it is not reliable within of order the$Z^0$width 224of$M_H = 2M_Z$; Higgs and and$Z^0 Z^0$masses in this region should 225be avoided. The analogous Higgs decays into one real and one virtual 226charged W are not included. 227 228 Note that while HIGGS contains the dominant graphs for Higgs 229production and graphs for$W$pair production related by gauge invariance, 230it does not contain the processes 231\begin{eqnarray*} 232q + \bar q &\to& W^+ W^- \\ 233q + \bar q &\to& Z^0 Z^0 234\end{eqnarray*} 235which give primarily transverse gauge bosons. These must be generated 236with WPAIR. 237 238 If the \verb|MSSMi| or \verb|SUGRA| keywords are used with 239HIGGS, then one of the three MSSM neutral Higgs is generated instead 240using gluon-gluon and quark-antiquark fusion with the appropriate SUSY 241couplings. Since heavy CP even SUSY Higgs are weakly coupled to W 242pairs 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 244be generated using the Standard Model process with a light Higgs mass, 245say 100 GeV.) The MSSM Higgs decays into both Standard Model and SUSY 246modes as calculated by ISASUSY are included. For more discussion see 247the SUSY subsection below and the writeup for ISASUSY. The user must 248select which Higgs to generate using HTYPE; see Section 6 below. If a 249mass range is not specified, then the range mass$M_H \pm 5\Gamma_H$250is used by default. (This cannot be done for the Standard Model Higgs 251because it is so wide for large masses.) Decay modes may be selected 252in the usual way. 253 254\subsubsection{WHiggs} Generates associated production of gauge and 255Higgs bosons, i.e., 256$$257q + \bar q \to H + W, H + Z\,, 258$$ 259in the narrow resonance approximation. The desired subprocesses can be 260selected with JETTYPEi, and specific decay modes of the$W$and/or$Z$261can be selected using the WMODEi keywords. Standard Model couplings are 262assumed unless SUSY parameters are specified, in which case the SUSY 263couplings are used. 264 265\subsubsection{SUSY} Generates pairs of supersymmetric particles from 266gluon-quark or quark-antiquark fusion. If the MSSMi or SUGRA 267parameters defined in Section 6 below are not specified, then only 268gluinos and squarks are generated: 269\begin{eqnarray*} 270g + g &\to& \tilde g + \tilde g \\ 271q + \bar q &\to& \tilde g + \tilde g \\ 272g + q &\to& \tilde g + \tilde q \\ 273g + g &\to& \tilde q + \tilde{\bar q} \\ 274q + \bar q &\to& \tilde q + \tilde{\bar q} \\ 275q + q &\to& \tilde q + \tilde q 276\end{eqnarray*} 277Left and right squarks are distinguished but assumed to be degenerate. 278Masses can be specified using the \verb|GAUGINO|, \verb|SQUARK|, and 279\verb|SLEPTON| parameters described in Section 6. No decay modes are 280specified, since these depend strongly on the masses. The user can 281either add new modes to the decay table (see Section 9) or use the 282\verb|FORCE| or \verb|FORCE1| commands (see Section 6). 283 284 If \verb|MSSMA|, \verb|MSSMB|, and \verb|MSSMC| are specified, 285then the ISASUSY package is used to calculate the masses and decay 286modes in the minimal supersymmetric extension of the standard model 287(MSSM), assuming SUSY grand unification constraints in the neutralino 288and chargino mass matrix but allowing some additional flexibility in 289the masses. The scalar particle soft masses are input via 290\verb|MSSMi|, so that the physical masses will be somewhat different 291due to$D$-term contributions and mixings for 3rd generation sparticles. 292$\tilde t_1$and$\tilde t_2$production and decays are now included. 293The lightest SUSY particle is assumed to be the lightest neutralino 294$\tilde Z_1$. If the \verb|MSSMi| parameters are specified, then the 295following additional processes are included using the MSSM couplings 296for the production cross sections: 297\begin{eqnarray*} 298g + q &\to& \tilde Z_i + \tilde q, \quad \tilde W_i + \tilde q \\ 299q + \bar q &\to& \tilde Z_i + \tilde g, \quad \tilde W_i + \tilde g \\ 300q + \bar q &\to& \tilde W_i + \tilde Z_j \\ 301q + \bar q &\to& \tilde W_i^+ + \tilde W_j^- \\ 302q + \bar q &\to& \tilde Z_i + \tilde Z_j \\ 303q + \bar q &\to& \tilde\ell^+ + \tilde\ell^-, \quad \tilde\nu + \tilde\nu 304\end{eqnarray*} 305Processes can be selected using the optional parameters described in 306Section 6 below. 307 308 Beginning with Version 7.42, matrix elements are taken into 309account in the event generator as well as in the calculation of decay 310widths for MSSM three-body decays of the form$\tilde A \to \tilde B f
311\bar f$, where$\tilde A$and$\tilde B$are gluinos, charginos, or 312neutralinos. This is implemented by having ISASUSY save the poles and 313their couplings when calculating the decay width and then using these 314to reconstruct the matrix element. Other three-body decays may be 315included in the future. Decays selected with \verb|FORCE| use the 316appropriate matrix elements. 317 318 An optional keyword \verb|MSSMD| can be used to specify the second 319generation masses, which otherwise are assumed degenerate with the first 320generation. An optional keyword \verb|MSSME| can be used to specify 321values of the$U(1)$and$SU(2)$gaugino masses at the weak scale rather 322than using the default grand unification values. The chargino and 323neutralino masses and mixings are then computed using these values. 324 325 Instead of using the \verb|MSSMi| parameters, one can use the 326\verb|SUGRA| parameter to specify in the minimal supergravity framework. 327This assumes that the gauge couplings unify at a GUT scale and that SUSY 328breaking occurs at that scale with universal soft breaking terms, which 329are related to the weak scale using the renormalization group. The 330renormalization group equations now include all the two-loop terms for 331both gauge and Yukawa couplings and the possible contributions from 332right-handed neutrinos. The parameters of the model are 333\begin{itemize} 334\item$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$: the common soft trilinear SUSY breaking parameter at the 337GUT scale; 338\item$\tan\beta$: the ratio of Higgs vacuum expectation values at the 339electroweak scale; 340\item$\sgn\mu=\pm1$: the sign of the Higgsino mass term. 341\end{itemize} 342The renormalization group equations are solved iteratively to determine 343all the electroweak SUSY parameters from these data assuming radiative 344electroweak symmetry breaking but not other possible constraints such as 345b-tau unification or limits on proton decay. 346 347 The assumption of universality at the GUT scale is rather 348restrictive and may not be valid. A variety of non-universal SUGRA 349(NUSUGRA) models can be generated using the \verb|NUSUG1|, \dots, 350\verb|NUSUG5| keywords. These might be used to study how well one could 351test the minimal SUGRA model. The keyword \verb|SSBCSC| can be used to 352specify an alternative scale (i.e., not the coupling constant 353unification scale) for the RGE boundary conditions. 354 355 An alternative to the SUGRA model is the Gauge Mediated SUSY 356Breaking (GMSB) model of Dine, Nelson, and collaborators. In this model 357SUSY breaking is communicated through gauge interactions with messenger 358fields at a scale$M_m$small compared to the Planck scale and are 359proportional to gauge couplings times$\Lambda_m$. The messenger fields 360should form complete$SU(5)$representations to preserve the unification 361of the coupling constants. The parameters of the GMSB model, which are 362specified by the \verb|GMSB| keyword, are 363\begin{itemize} 364\item$\Lambda_m = F_m/M_m$: the scale of SUSY breaking, typically 36510--$100\,{\rm TeV}$; 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 369electroweak 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 372would have had if the only SUSY breaking scale were$F_m$. 373\end{itemize} 374In GMSB models the lightest SUSY particle is always the nearly massless 375gravitino$\tilde G$. The parameter$C_{\rm grav}$scales the gravitino 376mass and hence the lifetime of the next lightest SUSY particle to decay 377into it. The \verb|NOGRAV| keyword can be used to turn off gravitino 378decays. 379 380 A variety of non-minimal GMSB models can be generated using 381additional parameters set with the GMSB2 keyword. These additional 382parameters are 383\begin{itemize} 384\item$\slashchar{R}$, an extra factor multiplying the gaugino masses 385at 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 388shifts relative to the minimal model at the messenger scale. (These 389might 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 393gauge group messengers. They can be non-integer in general. 394\end{itemize} 395For discussions of these additional parameters, see S. Dimopoulos, S. 396Thomas, 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 401specifying a gravitino mass with \verb|MGVTNO|. The default is that such 402decays do not occur. 403 404Another alternative SUSY model choice allowed is 405anomaly-mediated SUSY breaking, developed by Randall and Sundrum. 406In this model, it is assumed that SUSY breaking takes place 407in other dimensions, and SUSY breaking is communicated to the visible sector 408via the superconformal anomaly. In this model, the lightest SUSY particle 409is usually the neutralino which is nearly pure wino-like. The chargino 410is nearly mass degenerate with the lightest neutralino. It can be 411very long lived, or decay into a very soft pion plus missing energy. 412The model incorporated in ISAJET, based on work by 413Ghergetta, Giudice and Wells (hep-ph/9904378), 414and 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. 416The parameter set is$m_0,\ m_{3/2},\ \tan\beta ,\ sign(\mu )$, and 417can be input via the$AMSB$keyword. Care should be taken with the chargino 418decay, since it may have macroscopic decay lengths, or even decay 419outside the detector. 420 421Since neutrinos seem to have mass, the effect of a massive right-handed 422neutrino has been included in ISAJET, when calculating the sparticle 423mass spectrum. If the keyword$SUGRHN$is used, then the user 424must input the 3rd generation neutrino mass (at scale$M_Z$) in units 425of GeV, and the intermediate scale right handed neutrino Majorana mass$M_N$, 426also 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 428Yukawa coupling is computed in the simple see-saw model, and 429renormalization group evolution includes these effects between 430$M_{GUT}$and$M_N$. Finally, to facilitate modeling of$SO(10)$431SUSY-GUT models, loop corrections to 3rd generation fermion masses have 432been included in the ISAJET SUSY models. 433 434 The ISASUSY program can also be used independently of the rest of 435ISAJET, either to produce a listing of decays or in conjunction with 436another event generator. Its physics assumptions are described in more 437detail in Section~\ref{SUSY}. The ISASUGRA program can also be used 438independently to solve the renormalization group equations with SUGRA, 439GMSB, or NUSUGRA boundary conditions and then to call ISASUSY to 440calculate the decay modes. 441 442 Generally the MSSM, SUGRA, or GMSB option should be used to study 443supersymmetry signatures; the SUGRA or GMSB parameter space is clearly 444more manageable. The more general option may be useful to study 445alternative SUSY models. It can also be used, e.g., to generate 446pointlike color-3 leptoquarks in technicolor models by selecting squark 447production and setting the gluino mass to be very large. The MSSM or 448SUGRA option may also be used with top pair production to simulate top 449decays to SUSY particles. 450 451\subsubsection{$e^+e^-$} An$e^+e^-$event generator is also included in 452ISAJET. The 453Standard Model processes included are$e^+e^-$annihilation through 454$\gamma$and$Z$to quarks and leptons, and production of$W^+W^-$and 455$Z^0Z^0$pairs. In contrast to WPAIR and HIGGS for the hadronic 456processes, the produced$W$'s and$Z$'s are treated as particles, so 457their spins are not properly taken into account in their decays. 458(Because the$W$'s and$Z$'s are treated as particles, their decay 459modes can be selected using \verb|FORCE| or \verb|FORCE1|, not 460\verb|WMODEi|. See Section [6] below.) Other Standard Model 461processes, including$e^+ e^- \to e^+ e^-$($t$-channel graph) and$e^+ e^-
462\to \gamma \gamma$, are not included. Once the primary reaction has been 463generated, QCD radiation and hadronization are done as for hadronic 464processes. 465 466The$e^+e^-$generator can be run assuming no initial state 467radiation (the default), or an initial state electron structure function 468can be used for bremsstrahlung or the combination bremsstrahlung/beamstrahlung 469effect. Bremsstrahlung is implemented using the Fadin-Kuraev 470$e^-$distribution function, and can be turned on using the \verb|EEBREM| 471command while stipulating the minimal and maximal subprocess energy. 472Beamstrahlung is implemented by invoking the \verb|EEBEAM| keyword. 473In this case, in addition the beamstrahlung parameter$\Upsilon$and 474longitudinal beam size$\sigma_z$(in mm) must be given. 475The definition for$\Upsilon$in terms of other beam parameters can be 476found in the article Phys. Rev. D49, 3209 (1994) by Chen, Barklow and Peskin. 477The bremsstrahlung structure function is then convoluted with the 478beamstrahlung distribution (as calculated by P. Chen) and a spline fit 479is created. Since the cross section can contain large spikes, event generation 480can be slow if a huge range of subprocess energy is selected for light 481particles; in these scenarios, \verb|NTRIES| must be increased well beyond 482the default value. 483 484$e^+e^-$annihilation to SUSY particles is included as well with 485complete lowest order diagrams, and cascade decays. The processes 486include 487\begin{eqnarray*} 488e^+ e^- &\to& \tilde q \tilde q \\ 489e^+ e^- &\to& \tilde\ell \tilde\ell \\ 490e^+ e^- &\to& \tilde W_i \tilde W_j \\ 491e^+ e^- &\to& \tilde Z_i \tilde Z_j \\ 492e^+ e^- &\to& H_L^0+Z^0,H_H^0+Z^0,H_A^0+H_L^0,H_A^0+H_H^0,H^++H^- 493\end{eqnarray*} 494Note that SUSY Higgs production via$WW$and$ZZ$fusion, which can 495dominate Higgs production processes at$\sqrt{s} > 500\,\GeV$, 496is not included. Spin correlations are neglected, although 4973-body sparticle decay matrix elements are included. 498 499$e^+e^-$cross sections with polarized beams are included for 500both Standard Model and SUSY processes. The keyword \verb|EPOL| is 501used to set$P_L(e^-)$and$P_L(e^+)$, where 502$$503P_L(e) = (n_L-n_R)/(n_L+n_R) 504$$ 505so that$-1 \le P_L \le +1$. Thus, setting \verb|EPOL| to$-.9,0$will 506yield a 95\% right polarized electron beam scattering on an unpolarized 507positron beam. 508 509\subsubsection{Technicolor} Production of a technirho of arbitrary 510mass and width decaying into$W^\pm Z^0$or$W^+ W^-$pairs. The cross 511section is based on an elastic resonance in the$WW$cross section 512with the effective$W$approximation plus a$W$mixing term taken from 513EHLQ. Additional technicolor processes may be added in the future. 514 515\subsubsection{Extra Dimensions} The possibility that there might be 516more than four space-time dimensions at a distance scale$R$much larger 517than$G_N^{1/2}$has recently attracted interest. In these theories, 518$$519G_N = {1 \over 8\pi R^\delta M_D^{2+\delta}}\,, 520$$ 521where$\delta$is the number of extra dimensions and$M_D$is the 522$4+\delta$Planck scale. Gravity deviates from the standard theory at a 523distance$R \sim 10^{22/\delta-19}\,{\rm m}$, so$\delta\ge2$is 524required. If$M_D$is of order$1\,{\rm TeV}$, then the usual heirarchy 525problem 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 529excitations with a mass splitting of order$1/R$. While any individual 530mode is suppressed by the four-dimensional Planck mass, the large number 531of modes produces a cross section suppressed only by$1/M_D^2$. The 532signature is an invisible massive graviton plus a jet, photon, or other 533Standard Model particle. The \verb|EXTRADIM| process implements this 534reaction using the cross sections of Giudice, Rattazzi, and Wells, 535hep-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 538off above the scale$M_D$; the model is not valid if the results depend 539on this flag. 540 541\subsection{Multiparton Hard Scattering} 542 543 All the processes listed in Section~\ref{hard} are either$2\to2$544processes like \verb|TWOJET| or$2\to1s$-channel resonance processes 545followed by a 2-body decay like \verb|DRELLYAN|. The QCD parton shower 546described in Section~\ref{qcdshower} below generates multi-parton final 547states starting from these, but it relies on an approximation which is 548valid only if the additional partons are collinear either with the 549initial or with the final primary ones. Since the QCD shower uses exact 550non-colliear kinematics, it in fact works pretty well in a larger region 551of phase space, but it is not exact. 552 553 Non-collinear multiparton final states are interesting both in 554their own right and as backgrounds for other signatures. Both the matrix 555elements and the phase space for multiparton processes are complicated; 556they have been incorporated into ISAJET for the first time in 557Version~7.45. To calculate the matrix elements we have used the MadGraph 558package 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 561KEK-91-11, that calculates the amplitude for any Feynman diagram in 562terms of spinnors, vertices, and propagators. The MadGraph code has been 563edited to incorporate summations over quark flavors. To do the phase 564space 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 566final state using the \verb|MTOT| keyword. Limits on the$p_T$and 567rapidity of each final parton can be set via the \verb|PT| and \verb|Y| 568keyworks, while limits on the mass of any pair of final partons can be 569set via the \verb|MIJTOT| keyword. These limits are sufficient to shield 570the infrared and collinear singularities and to render the result 571finite. However, the parton shower populates all regions of phase space, 572so careful thought is needed to combine the parton-shower based and 573multiparton based results. 574 575 While the multiparton formalism is rather general, it still takes 576a substantial amount of effort to implement any particular process. So 577far 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$582processes. The$Z$is defined to be jet 1; it is treated in the narrow 583resonance approximation and is decayed isotropically. The quarks, 584antiquarks, and gluons are defined to be jets 2 and 3 and are 585symmetrized in the usual way. 586 587\subsection{QCD Radiative Corrections\label{qcdshower}} 588 589 After the primary hard scattering is generated, QCD radiative 590corrections are added to allow the possibility of many jets. This is 591essential to get the correct event structure, especially at high 592energy. 593 594 Consider the emission of one extra gluon from an initial or a 595final quark line, 596$$597q(p) \to q(p_1) + g(p_2) 598$$ 599From QCD perturbation theory, for small$p^2$the cross section is 600given by the lowest order cross section multiplied by a factor 601$$602\sigma = \sigma_0 \alpha_s(p^2)/(2\pi p^2) P(z) 603$$ 604where$z=p_1/p$and$P(z)$is an Altarelli-Parisi function. The same 605form holds for the other allowed branchings, 606\begin{eqnarray*} 607g(p) &\to& g(p_1) + g(p_2) \\ 608g(p) &\to& q(p_1) + \bar q(p_2) 609\end{eqnarray*} 610These factors represent the collinear singularities of perturbation 611theory, and they produce the leading log QCD scaling violations for the 612structure functions and the jet fragmentation functions. They also 613determine 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 617factors which dominate in the collinear limit but using exact, 618non-collinear kinematics. Thus higher order QCD is reduced to a 619classical cascade process, which is easy to implement in a Monte Carlo 620program. To avoid infrared and collinear singularities, each parton in 621the cascade is required to have a mass (spacelike or timelike) greater 622than some cutoff$t_c$. The assumption is that all physics at lower 623scales is incorporated in the nonperturbative model for hadronization. 624In ISAJET the cutoff is taken to be a rather large value, 625$(6\,\GeV)^2$, because independent fragmentation is used for the jet 626fragmentation; a low cutoff would give too many hadrons from 627overlapping partons. It turns out that the branching approximation not 628only incorporates the correct scaling violations and jet structure but 629also reproduces the exact three-jet cross section within factors of 630order 2 over all of phase space. 631 632 This approximation was introduced for final state radiation by 633Fox and Wolfram. The QCD cascade is determined by the probability for 634going from mass$t_0$to mass$t_1$emitting no resolvable radiation. 635For a resolution cutoff$z_c < z < 1-z_c$, this is given by a simple 636expression, 637$$638P(t_0,t_1)=\left(\alpha_s(t_0)/\alpha_s(t_1)\right)^{2\gamma(z_c)/b_0} 639$$ 640where 641$$642\gamma(z_c)=\int_{z_c}^{1-z_c} dz\,P(z),\qquad 643b_0=(33-2n_f)/(12\pi) 644$$ 645Clearly if$P(t_0,t_1)$is the integral probability, then$dP/dt_1$is 646the probability for the first radiation to occur at$t_1$. It is 647straightforward to generate this distribution and then iteratively to 648correct 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 651spacelike kinematics and of the structure functions. Sjostrand has 652shown how to do this by starting at the hard scattering and evolving 653backwards, forcing the ordering of the spacelike masses$t$. The 654probability that a given step does not radiate can be derived from the 655Altarelli-Parisi equations for the structure functions. It has a form 656somewhat similar to$P(t_0,t_1)$but involving a ratio of the structure 657functions for the new and old partons. It is possible to find a bound 658for this ratio in each case and so to generate a new$t$and$z$as for 659the final state. Then branchings for which the ratio is small are 660rejected in the usual Monte Carlo fashion. This ratio suppresses the 661radiation 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 663at small momentum transfer. 664 665 At low energies, the branching of an initial heavy quark into a 666gluon sometimes fails; these events are discarded and a warning is 667printed. 668 669 Since$t_c$is quite large, the radiation of soft gluons is cut 670off. To compensate for this, equal and opposite transverse boosts are 671made to the jet system and to the beam jets after fragmentation with a 672mean value 673$$674\langle p_t^2\rangle = (.1\,\GeV) \sqrt{Q^2} 675$$ 676The dependence on$Q^2$is the same as the cutoff used for DRELLYAN and 677the coefficient is adjusted to fit the$p_t$distribution for the$W$. 678 679 Radiation of gluons from gluinos and scalar quarks is also 680included in the same approximation, but the production of gluino or 681scalar quark pairs from gluons is ignored. Very little radiation is 682expected for heavy particles produced near threshold. 683 684 Radiation of photons,$W$'s, and$Z$'s from final state quarks is 685treated in the same approximation as QCD radiation except that the 686coupling constant is fixed. Initial state electroweak radiation is not 687included; it seems rather unimportant. The$W^+$'s,$W^-$'s and$Z$'s 688are decayed into the modes allowed by the \verb|WPMODE|, \verb|WMMODE|, 689and \verb|Z0MODE| commands respectively. {\it Warning:} The branching 690ratios implied by these commands are not included in the cross section 691because an arbitrary number of$W$'s and$Z$'s can in principle be 692radiated. 693 694\subsection{Jet Fragmentation:} 695 696 Quarks and gluons are fragmented into hadrons using the 697independent 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$$700u : d : s = .43 : .43 : .14 701$$ 702A meson$q \bar q_1$is formed carrying a fraction$z$of the momentum, 703$$704E' + p_z' = z (E + p_z) 705$$ 706and having a transverse momentum$p_t$with$\langle p_t \rangle =
7070.35\,\GeV$. Baryons are included by generating a diquark with 708probability 0.10 instead of a quark; adjacent diquarks are not 709allowed, so no exotic mesons are formed. For light quarks$z$is 710generated with the splitting function 711$$712f(z) = 1-a + a(b+1)(1-z)^b, \qquad 713a = 0.96, b = 3 714$$ 715while for heavy quarks the Peterson form 716$$717f(z) = x (1-x)^2 / ( (1-x)^2 + \epsilon x )^2 718$$ 719is used with$\epsilon = .80 / m_c^2$for$c$and$\epsilon = .50 /
720m_q^2$for$q = b, t, y, x$. These values of$\epsilon$have been 721determined by fitting PEP, PETRA, and LEP data with ISAJET and should 722not be compared with values from other fits. Hadrons with longitudinal 723momentum less than zero are discarded. The procedure is then iterated 724for the new quark$q_1$until all the momentum is used. A gluon is 725fragmented like a randomly selected$u$,$d$, or$s$quark or 726antiquark. 727 728 In the fragmentation of gluinos and scalar quarks, supersymmetric 729hadrons are not distinguished from partons. This should not matter 730except possibly for very light masses. The Peterson form for$f(x)$is 731used with the same value of epsilon as for heavy quarks,$\epsilon =
7320.5 / m^2$. 733 734 Independent fragmentation correctly describes the fast hadrons in 735a jet, but it fails to conserve energy or flavor exactly. Energy 736conservation is imposed after the event is generated by boosting the 737hadrons to the appropriate rest frame, rescaling all of the 738three-momenta, and recalculating the energies. 739 740\subsection{Beam Jets} 741 742 There is now experimental evidence that beam jets are different in 743minimum bias events and in hard scattering events. ISAJET therefore uses 744similar a algorithm but different parameters in the two cases. 745 746 The standard models of particle production are based on pulling 747pairs of particles out of the vacuum by the QCD confining field, 748leading naturally to only short-range rapidity correlations and to 749essentially Poisson multiplicity fluctuations. The minimum bias data 750exhibit KNO scaling and long-range correlations. A natural explanation 751of this was given by the model of Abramovskii, Kanchelli and Gribov. 752In their model the basic amplitude is a single cut Pomeron with 753Poisson fluctuations around an average multiplicity$\langle n
754\rangle$, but unitarity then produces graphs giving$K$cut Pomerons 755with multiplicity$K\langle n \rangle$. 756 757 A simplified version of the AKG model is used in ISAJET. The 758number of cut Pomerons is chosen with a distribution adjusted to fit the 759data. For a minimum bias event this distribution is 760$$761P(K) = ( 1 + 4 K^2 ) \exp{-1.8 K} 762$$ 763while for hard scattering 764$$765P(1) \to 0.1 P(1),\quad P(2) \to 0.2 P(2),\quad P(3) \to 0.5 P(3) 766$$ 767For each side of each event an$x_0$for the leading baryon is selected 768with a distribution varying from flat for$K = 1$to like that for 769mesons for large K: 770$$771f(x) = N(K) (1- x_0)^c(K),\qquad c(K) = 1/K + ( 1 - 1/K ) b(s) 772$$ 773The$x_i$for the cut Pomerons are generated uniformly and then 774rescaled to$1-x_0$. Each cut Pomeron is then hadronized in its own 775center of mass using a modified independent fragmentation model with 776an energy dependent splitting function to reproduce the rise in 777$dN/dy$: 778$$779f(x) = 1 - a + a(b(s) + 1)^ b(s),\qquad 780b(s) = b_0 + b_1 \log(s) 781$$ 782The energy dependence is put into$f(x)$rather than$P(K)$because in 783the AKG scheme the single particle distribution comes only from the 784single chain. The probabilities for different flavors are taken to be 785$$786u : d : s = .46 : .46 : .08 787$$ 788to reproduce the experimental$K/\pi\$ ratio.