Merging the VirtualMC branch to the main development branch (HEAD)
[u/mrichter/AliRoot.git] / ISAJET / doc / physics.doc
CommitLineData
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
8radiative corrections in the leading log approximation, independent
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 $2\to1$ $s$-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.