In the following sections three realistic applications of WHIZARD
are presented: Higgs production in the four-fermion channels, W
production (four fermions), and strong WW scattering (six fermions).
With certain refinements and modifications, they could be used for
actual physical applications. However, here they are intended as a
form of tutorial, showing typical setups and common problems.
| 6.1 |
Higgs production at LEP |
|
Let us simulate the production of a 115 GeV Standard Model Higgs
in e+e- collisions at 209 GeV. With this mass, the Higgs
boson decays mainly into b_ b with a small fraction of
t+t- decays. Here, we only consider the signal together
with its irreducible background, contributing to the processes
where q=u,d,s,c and n=ne,nµ,nt. This list of
processes directly translates into the configuration file
whizard.prc:
# WHIZARD configuration file
# The selected model:
model SM
# Processes
# (Methods: chep=CompHEP, mad=MadGraph, omega=O'Mega)
# (Options: number=QCD order [Madgraph])
#
# Tag In Out Method Option
#=============================================================
# On-shell process:
zh e1,E1 Z,H chep
# Full four-fermion matrix elements (no QCD):
nnbb e1,E1 n1:n2:n3,N1:N2:N3,b,B omega
qqbb e1,E1 u:d:s:c,U:D:S:C,b,B omega
bbbb e1,E1 b,B,b,B omega
eebb e1,E1 e1,E1,b,B omega
mmbb e1,E1 e2,E2,b,B omega
qqtt e1,E1 u:d:s:c,U:D:S:C,e3,E3 omega
bbtt e1,E1 b,B,e3,E3 omega
# QCD contribution (gluon splitting):
uubb_qcd e1,E1 u,U,b,B mad 2
ddbb_qcd e1,E1 d,D,b,B mad 2
ssbb_qcd e1,E1 s,S,b,B mad 2
ccbb_qcd e1,E1 c,C,b,B mad 2
bbbb_qcd e1,E1 b,B,b,B mad 2
Here, we use flavor summation for the quark and missing-energy
channels. Since O'Mega can't yet handle QCD corrections where one
quark pair originates from gluon fragmentation, those are implemented
using MadGraph for the matrix elements. (We could also use
CompHEP for that purpose. For 2-> 4 processes, however,
CompHEP matrix elements typically need considerable more CPU time both
in compilation and execution.)
After this file has been saved in the conf subdirectory, we
should run configure in the directory where we had unpacked the
distribution, if not already done,
> ./configure
and make and install the executable
> make install
After this step is completed, we are ready for running the program.
| 6.1.1 |
The on-shell process |
|
All necessary files have now been installed in the results
subdirectory:
> cd results
> ls
Makefile whizard whizard.cut5 whizard.mdl
Makefile.in whizard.cut1 whizard.in whizard.prc
The file whizard.prc is a copy of the one we prepared above.
whizard.mdl contains all SM-specific definitions. Here, only
the vertex list contained within is used for phase space generation;
usually we don't need to modify it. The two cut configuration files
are empty. The file which is of interest for us now is
whizard.in which we have to edit:
&process_input
process_id = "zh"
sqrts = 209
/
&integration_input
stratified = F
/
&simulation_input /
&diagnostics_input /
¶meter_input
MH = 115
wH = 0.3228E-02
Mb = 2.9
Me = 0
Ms = 0
Mc = 0
/
&beam_input /
&beam_input /
As a warm-up, we examine the on-shell process e-e+-> ZH, labeled
zh. We insert the c.m. energy 209 GeV and the Higgs mass
115 GeV. The next two parameters are not relevant here, but will
be needed for the other processes: The Higgs width value wH
is the one returned by HDECAY [8]. The b mass Mb is
the running mass evaluated at a scale around mH. This is important
since the Hbb coupling is proportional to
mb.26 The setting stratified=F calls for
importance sampling instead of stratified sampling (see below). All
other fields are left empty.
Having saved whizard.in, the program can be started:
> ./whizard
(if you like, you can also type make run). Since this process
is trivial, the results are there immediately:
! WHIZARD 1.30 (Sep 20 2004)
! Reading process data from file whizard.in
! Wrote whizard.out
! Process zh:
! e a-e -> Z H
! 8 4 -> 1 2
! Reading vertices from file whizard.mdl ...
! Model file: 48 vertices found.
! Generating phase space channels for process zh...
! Phase space: 1 phase space channels generated.
! Scanning phase space channels for equivalences ...
! Phase space: 1 equivalence relations found.
! Wrote phase space configurations to file whizard.phx
! Created grids: 1 channels, 2 dimensions with 20 bins
!=============================================================================
! WHIZARD run for process zh:
!-----------------------------------------------------------------------------
! It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It]
!-----------------------------------------------------------------------------
! Reading cut configuration data from file whizard.cut1
! No cut data found for process zh
! Preparing (fixed weights): 1 sample of 5000 calls ...
1 5000 1.6114541E+02 1.16E-02 0.01 0.01* 99.44 0.00 1
!-----------------------------------------------------------------------------
! Adapting (variable wgts.): 3 samples of 5000 calls ...
2 5000 1.6114763E+02 1.16E-02 0.01 0.01 99.44
3 5000 1.6117303E+02 2.07E-02 0.01 0.01 98.28
4 5000 1.6118480E+02 3.57E-02 0.02 0.02 97.12
!-----------------------------------------------------------------------------
! Integrating (fixed wgts.): 3 samples of 5000 calls ...
5 15000 1.6115362E+02 1.08E-02 0.01 0.01 95.62 0.78 3
!-----------------------------------------------------------------------------
!
! Time estimate for generating 10000 unweighted events: 0h 00m 00s
!=============================================================================
! Total cross section summary (all processes):
!-----------------------------------------------------------------------------
! Process ID Integral[fb] Error[fb] Err[%] Eff[%] Chi2
!-----------------------------------------------------------------------------
zh 1.6115362E+02 1.08E-02 0.01 95.62 0.78
!-----------------------------------------------------------------------------
sum 1.6115362E+02 1.08E-02 0.01 95.62
!=============================================================================
! Wrote whizard.out
! Integration complete.
! No event generation requested
! WHIZARD run finished.
We read off the total cross section
s = 161.154 ± 0.011 fb
Clearly the grid adaptation was of no use at all, since the final
error is no smaller than the initial one. This would have been even
worse if we had not inserted stratified = F. However, for less
trivial processes stratified sampling usually gives better results,
and in the examples presented below we keep stratified = T.
If we were interested in a more accurate result, we could increase the
statistics like this
&integration_input
calls = 1 100000 1 100000 3 500000
...
(one initial run of 100,000 events, one run of 100,000 events for
adaptation, 3 runs of 500,000 events for integration) and would get
something like
s = 161.1410 ± 0.0002 fb.
This serves merely as an illustration: for this process the cross
section is easily obtained analytically, and the accuracy is limited
by the input parameters anyway.
| 6.1.2 |
The missing-energy channel |
|
The actual virtues of WHIZARD lie in calculating complete
multi-fermion processes. Requesting the process
|
e-e+ -> n |
|
b |
|
where n=ne,nµ,nt
|
nnbb,
&process_input
process_id = "nnbb"
sqrts = 209
/
and returning to the default integration setting
&integration_input /
we arrive at a result like this:
! WHIZARD 1.30 (Sep 20 2004)
! Reading process data from file whizard.in
! Wrote whizard.out
! Reading phase space configurations from file whizard.phx
! Process nnbb:
! e a-e -> nu_e a-nu_e b a-b
! 32 16 -> 1 2 4 8
! 12 phase space channels found for process nnbb
! Scanning phase space channels for equivalences ...
! Phase space: 12 equivalence relations found.
! Created grids: 12 channels, 8 dimensions with 20 bins
!=============================================================================
! WHIZARD run for process nnbb:
!-----------------------------------------------------------------------------
! It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It]
!-----------------------------------------------------------------------------
! Reading cut configuration data from file whizard.cut1
! No cut data found for process nnbb
! Preparing (fixed weights): 1 sample of 20000 calls ...
1 20000 1.0572597E+02 1.87E+00 1.77 2.50* 4.26 0.00 1
!-----------------------------------------------------------------------------
! Adapting (variable wgts.): 10 samples of 20000 calls ...
2 20000 1.0881157E+02 1.87E+00 1.72 2.43* 4.46
3 20000 1.0788791E+02 7.63E-01 0.71 1.00* 12.91
4 20000 1.0711324E+02 6.09E-01 0.57 0.80* 17.05
5 20000 1.0781600E+02 5.67E-01 0.53 0.74* 19.86
6 20000 1.0519942E+02 5.37E-01 0.51 0.72* 19.82
7 20000 1.0709944E+02 5.36E-01 0.50 0.71* 19.84
8 20000 1.0606741E+02 5.23E-01 0.49 0.70* 18.75
9 20000 1.0661251E+02 5.20E-01 0.49 0.69* 17.92
10 20000 1.0674035E+02 5.21E-01 0.49 0.69 18.98
11 20000 1.0691411E+02 5.16E-01 0.48 0.68* 19.06
!-----------------------------------------------------------------------------
! Integrating (fixed wgts.): 3 samples of 20000 calls ...
12 60000 1.0706821E+02 2.94E-01 0.27 0.67* 16.65 0.45 3
!-----------------------------------------------------------------------------
!
! Time estimate for generating 10000 unweighted events: 0h 00m 49s
!=============================================================================
! Total cross section summary (all processes):
!-----------------------------------------------------------------------------
! Process ID Integral[fb] Error[fb] Err[%] Eff[%] Chi2
!-----------------------------------------------------------------------------
nnbb 1.0706821E+02 2.94E-01 0.27 16.65 0.45
!-----------------------------------------------------------------------------
sum 1.0706821E+02 2.94E-01 0.27 16.65
!=============================================================================
! Wrote whizard.out
! Integration complete.
! No event generation requested
! WHIZARD run finished.
The cross section is27
s = 107.07 ± 0.29 fb
This calculation takes less than 4 minutes on a PC (Pentium IV
2.4 GHz, NAGWare Fortran compiler). Of course, if we
invest more CPU time, we can improve the accuracy, as explained in the
previous subsection.
In inspecting the output, the column tagged Err/Exp is of main
interest (read: relative error over expected relative error). This is
the estimated relative error multiplied by the square root of the
number of calls, a number which should be of the order one. Each time
this error improves, the entry is marked with a star. The best grid
so far (the last one with a star) will be used for integration, and
later for event generation. As evident from the column Eff[%],
the estimated reweighting efficiency improves as well. In the last
column the c2 value divided by the number of iterations is
shown. This value is meaningful only if there is more than one
iteration in the integration step. This is not the case here, so it
is zero in our example.
Let us now generate events. We define a nonzero luminosity
(100 fb-1, which is more than 100 times the LEP integrated
luminosity per year):
&process_input
process_id = "nnbb"
sqrts = 209
luminosity = 100
/
We do not want to repeat the adaptation and integration steps,
therefore we read in the previously adapted grids
&integration_input
read_grids = T
/
Now running WHIZARD will result in a repetition of the previous
result, before event generation is started:
! Reading analysis configuration data from file whizard.cut5
! No analysis data found for process nnbb
! Event sample corresponds to luminosity [fb-1] = 100.0
! Event sample corresponds to 64323 weighted events
! Generating 10707 unweighted events ...
! Event generation finished.
!=============================================================================
! Analysis results for process nnbb:
!-----------------------------------------------------------------------------
! It Events Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It]
!-----------------------------------------------------------------------------
13 10707 1.0706821E+02 1.03E+00 0.97 1.00 100.00
!-----------------------------------------------------------------------------
! Warning: Excess events: 2.2 ( 0.02% ) | Maximal weight: 1.01
! There were no errors and 2 warning(s).
! WHIZARD run finished.
The generation of 10,707 events takes one more minute in this case.
No further analysis has been requested, so we just get the total cross
section again. By definition, this is equal to the previously
calculated cross section, but the error now corresponds to the actual
event sample.
There are some excess events (weight greater than one), but the effect
of this excess being dropped is considerably less than the statistical
error of the event sample. If necessary, the excess could be removed
by setting safety_factor to a value greater than one (e.g.,
1.1), at the expense of extra CPU time since the reweighting
efficiency is reduced by this factor.
The event sample can be further analyzed: Let us plot the missing
invariant mass distribution and the dijet invariant mass distribution,
which should exhibit peaks at the Z and Higgs mass. To this end, we
tell WHIZARD to reread the generated event sample
&simulation_input
read_events = T
/
and make up an analysis configuration file whizard.cut5:
# cut/histogram configuration file
# e- e+ -> nu nubar b bbar
# 32 16 1 2 4 8
process nnbb, qqbb, bbbb, eebb, mmbb, qqtt, bbtt,
uubb_qcd, ddbb_qcd, ssbb_qcd, ccbb_qcd, bbbb_qcd
histogram M of 3 within 0 209 nbin 40
and
histogram M of 12 within 0 209 nbin 40
and
cut M of 12 within 114 116
histogram M of 3 within 0 209 nbin 40
This type of analysis can serve for all processes, therefore we have
added the other tags. The last histogram counts only those events
which have a bb_ invariant mass close to the Higgs mass. The
numbers result from adding up the binary codes of the external
particles, which are shown as a the comment in the file header.
Running WHIZARD this time takes almost no time since only the
previously generated files have to be read:
! Reading analysis configuration data from file whizard.cut5
! Found 3 analysis configuration datasets.
! Looking for raw event file whizard.evx ...
! Event sample corresponds to luminosity [fb-1] = 100.0
! Event sample corresponds to 64323 weighted events
! Reading 10707 unweighted events ...
! Event generation finished.
!=============================================================================
! Analysis results for process nnbb:
!-----------------------------------------------------------------------------
! It Events Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It]
!-----------------------------------------------------------------------------
13 10707 1.0706821E+02 1.03E+00 0.97 1.00 100.00
!-----------------------------------------------------------------------------
! Warning: Excess events: 2.2 ( 0.02% ) | Maximal weight: 1.01
!=============================================================================
! Analysis results for process nnbb:
!-----------------------------------------------------------------------------
! It Events Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It]
!-----------------------------------------------------------------------------
13 10707 1.0706821E+02 1.03E+00 0.97 1.00 100.00
!-----------------------------------------------------------------------------
! Warning: Excess events: 2.2 ( 0.02% ) | Maximal weight: 1.01
!=============================================================================
! Analysis results for process nnbb:
!
! Additional cuts:
! integration pass 5
cut M of 12 within 1.14000E+02 1.16000E+02
!-----------------------------------------------------------------------------
! It Events Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It]
!-----------------------------------------------------------------------------
13 2204 2.2039631E+01 4.69E-01 2.13 1.00 20.58
!-----------------------------------------------------------------------------
! Warning: Excess events: 0.4 ( 0.02% ) | Maximal weight: 1.02
! There were no errors and 6 warning(s).
! WHIZARD run finished.
From the last line we read off that the actual contribution of Higgs
production in this channel is 2204 events (or 20.58 %). Here
the efficiency is not the reweighting efficiency as before, but the
fraction of events remaining after cuts. The histograms can now be
found in the output file whizard.nnbb.dat:
!=============================================================================
! WHIZARD 1.30 (Sep 20 2004)
! Process nnbb:
! e a-e -> nu_e a-nu_e b a-b
! 32 16 -> 1 2 4 8
! Analysis results for process nnbb:
!
! Histograms:
!
histogram M of 3 within 0.00000E+00 2.09000E+02 nbin 40
2.61250000 0.00000000 0.00000000 0.00000000
7.83750000 2.00000000 1.41421356 0.00000000
13.0625000 5.00000000 2.23606798 0.00000000
18.2875000 3.00000000 1.73205081 0.00000000
23.5125000 7.00000000 2.64575131 0.00000000
28.7375000 12.0000000 3.46410162 0.00000000
33.9625000 14.0000000 3.74165739 0.00000000
39.1875000 13.0000000 3.60555128 0.00000000
44.4125000 22.0000000 4.69041576 0.131470099
49.6375000 31.0000000 5.56776436 0.00000000
54.8625000 42.0000000 6.48074070 0.242629656E-01
60.0875000 34.0000000 5.83095189 0.00000000
65.3125000 70.0000000 8.36660027 0.00000000
70.5375000 94.0000000 9.69535971 0.313546084
75.7625000 118.000000 10.8627805 0.238297301E-01
80.9875000 264.000000 16.2480768 0.00000000
86.2125000 1091.00000 33.0302891 0.425686281
...
In these histograms, the first column is the bin midpoint. The second
column is the number of entries, the third column the statistical
error on this number (N1/2). The fourth column shows the excess
events, which introduce an additional error. However, for each bin
this contribution is negligible compared to the statistical error
which is present anyway.
The actual histograms are shown in Fig. 15. (For better
illustration, we have changed the linear scale to a logarithmic one;
this is done in the file whizard-plots.tex by the manual
replacement of the lines
setup(linear,linear); graphrange (#0.00,#0), (#209.,??);
by
setup(linear,log); graphrange (#0.00,#.5), (#209.,#1e4);
and a subsequent rerun)
> make plots
Figure 15: Histograms for the process e-e+->nn_bb_ at
s1/2=209 GeV with mH=115 GeV. Top: Dijet
invariant mass distribution. Bottom: Missing invariant mass
distribution.
In the dijet invariant mass distribution the Z and Higgs peaks are
clearly visible. At low dijet invariant mass there is a tail of
continuum bb_ production. In the missing mass distribution the
peak is at the Z mass.
| 6.1.3 |
The four-jet channel |
|
There are two four-jet channels
If the light quark masses are set to zero (see the input file above),
all light quark channels can be treated in a single run. For unequal
masses, flavor summation is not possible, therefore the b channel
has to be treated separately. (We can't set the b mass to zero
since this would remove the b Yukawa coupling.)
Running the program as before for the process qqbb with the
given input and analysis configuration files takes about 10 minutes in
total. The cross section result is
s = 286.37 ± 0.76 fb
and 28638 events are generated in this run, 4681 of them within the
Higgs mass window.
To obtain a finite result, WHIZARD had to insert a cut of 10 GeV
on the dijet invariant mass of the light quarks. This cut is written
into the file whizard.qqbb.cut0:
! Automatically generated set of cuts
! Process qqbb:
! e a-e -> u a-u b a-b
! 32 16 -> 1 2 4 8
process qqbb
cut M of 3 within 1.00000E+01 1.00000E+99
The cut could be modified either by changing the default value in
whizard.in, e.g.
&integration_input
...
default_jet_cut = 5
/
or by providing a non-empty file whizard.cut1 of the same
format, with a different set of cuts. If an appropriate process entry
is found in this file, any default cuts are ignored.
The bb_bb_ channel is similar, although the presence of
identical particles in the final state makes phase space more
complicated. The adaptation again works well and we get
s = 44.51 ± 0.11 fb
(10)
Here, we have slightly cheated: The O'Mega matrix element which has
been integrated lacks the correct color factor for the interference
term (in the squared amplitude) where the b quark lines are crossed.
This is, however, a negligible effect: If we take MadGraph matrix
elements instead (which have exact color information), the result is
identical within the integration error.
| 6.1.4 |
The lepton channels |
|
In the µ+µ- channel adaptation takes somewhat longer due to
the small muon mass, if no cuts are provided. However, the 10
iterations of the adaptation step are sufficient, and in the final
integration step, the result
s = 52.90 ± 0.13 fb
is perfectly stable. With suitable cuts, we can set the muon mass to
zero, which as a side effect speeds up the calculation by a factor of
two.
A more difficult task is to get to a stable result for the cross
section of the process
without cuts, but with nonzero electron mass. The default choice for
the number of calls and iterations does not suffice here:
! WHIZARD run for process eebb:
!-----------------------------------------------------------------------------
! It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It]
!-----------------------------------------------------------------------------
! Reading cut configuration data from file whizard.cut1
! Replacing default cuts by user-defined cuts.
! Preparing (fixed weights): 1 sample of 20000 calls ...
1 20000 5.8506685E+02 2.01E+02 34.38 48.62* 0.20 0.00 1
!-----------------------------------------------------------------------------
! Adapting (variable wgts.): 10 samples of 20000 calls ...
2 20000 3.2960904E+02 4.39E+01 13.32 18.84* 0.31
3 20000 1.2921900E+03 4.96E+02 38.42 54.34 0.08
4 20000 2.4987669E+03 8.45E+02 33.83 47.85 0.10
5 20000 2.4882598E+03 4.07E+02 16.36 23.14 0.09
6 20000 3.4067505E+03 9.85E+02 28.92 40.89 0.06
7 20000 2.8709019E+03 3.28E+02 11.44 16.18* 0.11
8 20000 3.2983652E+03 3.47E+02 10.51 14.87* 0.18
9 20000 4.1803823E+04 2.89E+04 69.02 97.61 0.02
10 20000 3.8984197E+03 3.82E+02 9.81 13.87* 0.13
...
Apparently, convergence is not reached. One possibility to proceed is
to reuse these grids, but do further iterations:
&integration_input
calls = 1 20000 30 20000 3 20000
read_grids = T
/
For the first iterations the old results are copied. The additional
iterations now find convergence, although the final accuracy and
efficiency are not really satisfactory:
...
28 20000 7.9532372E+03 4.41E+02 5.55 7.85 0.31
29 20000 7.8702516E+03 2.87E+02 3.64 5.15 0.50
30 20000 7.5438112E+03 2.33E+02 3.09 4.37 1.04
31 20000 7.4196016E+03 1.62E+02 2.18 3.08* 1.08
!-----------------------------------------------------------------------------
! Integrating (fixed wgts.): 3 samples of 20000 calls ...
32 60000 7.6111773E+03 1.09E+02 1.43 3.50 0.60 1.57 3
!-----------------------------------------------------------------------------
To get a more precise result, we should also increase the number of
calls per iteration. This costs additional CPU time.
Fortunately, there is a shortcut. The problem of this calculation is
the fact that a large part of the cross section comes from the region
of very forward electrons. Furthermore, there is a piece which
corresponds to bb_g* production where the virtual photon
splits into the electron-positron pair. These small regions are
sampled only after several iterations have sufficiently adapted the
grids. The mappings built into WHIZARD assume the parameters
default_Q_cut and default_mass_cut as typical scales for
these subprocesses. The implicit assumption, which works in many
cases, is that this is not too far from the actual cuts. However, if
there are no cuts, the assumption is inadequate. We rather should set
those parameters to zero
&integration_input
default_Q_cut = 0
default_mass_cut = 0
/
(which actually replaces them by some finite small value), and start
again:
! WHIZARD run for process eebb:
!-----------------------------------------------------------------------------
! It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It]
!-----------------------------------------------------------------------------
! Reading cut configuration data from file whizard.cut1
! Replacing default cuts by user-defined cuts.
! Preparing (fixed weights): 1 sample of 20000 calls ...
1 20000 1.5053744E+04 6.09E+03 40.43 57.17* 0.20 0.00 1
!-----------------------------------------------------------------------------
! Adapting (variable wgts.): 10 samples of 20000 calls ...
2 20000 4.5557243E+03 1.46E+03 32.11 45.42* 0.16
3 20000 6.6003671E+03 3.98E+02 6.03 8.52* 0.47
4 20000 7.9219196E+03 5.13E+02 6.48 9.16 0.31
5 20000 7.5913298E+03 2.36E+02 3.11 4.39* 1.36
6 20000 7.4324783E+03 1.86E+02 2.51 3.55* 0.85
7 20000 7.4668913E+03 1.42E+02 1.91 2.70* 2.02
8 20000 7.1885920E+03 1.10E+02 1.53 2.16* 2.94
9 20000 7.5275492E+03 1.86E+02 2.47 3.49 0.89
10 20000 7.4377512E+03 1.38E+02 1.85 2.61 2.21
11 20000 7.6289727E+03 1.09E+02 1.43 2.02* 2.89
!-----------------------------------------------------------------------------
! Integrating (fixed wgts.): 3 samples of 20000 calls ...
12 60000 7.5581817E+03 6.53E+01 0.86 2.11 1.79 0.20 3
!-----------------------------------------------------------------------------
This time, the difficult regions are sampled from the beginning.
With standard cuts on the electron energies and angles, this process
his well-behaved. The extra effort is needed only if we are
particularly interested in the events with electron and positron going
in the very forward and backward regions, respectively.
The QCD background where one quark pair results from gluon splitting
is not included in the above. It has been listed separately in our
process set. Here, no flavor summation is (yet) possible, and we have
to calculate the individual quark flavors separately. Taking, e.g.,
the channel uubb_qcd
|
e-e+ -> u |
|
b |
|
(QCD contribution)
|
s = 51.66 ± 0.23 fb
with the default settings. The invariant mass distribution of the jet
pair can be analyzed as before (Fig. 16).
Figure 16: Invariant mass distribution of the light quark pair for the
process e-e+-> uu_ bb_, taking into account only the QCD
contribution.
Although the total cross section is quite large, the amount
contributing in the Higgs mass region is tiny (0.35 %), as the
analysis shows:
! Additional cuts:
! integration pass 5
cut M of 12 within 1.14000E+02 1.16000E+02
!-----------------------------------------------------------------------------
! It Events Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It]
!-----------------------------------------------------------------------------
13 18 1.8000350E-01 4.24E-02 23.57 1.00 0.35
| 6.2 |
6-fermion production: Higgs pairs |
|
A process which is of interest at a future Linear Collider is
Higgs pair production, which is sensitive to the Higgs trilinear
coupling. A process definition file whizard.prc for this
process could look like
# WHIZARD configuration file
model SM
#=============================================================
# On-shell process:
zhh e1,E1 Z,H,H chep
# Full six-fermion matrix elements (no QCD):
qqbbbb e1,E1 u:d:s:c,U:D:S:C,b,B,b,B omega
# QCD contribution (gluon splitting):
uubbbb_qcd1 e1,E1 u,U,b,B,b,B mad 2
uubbbb_qcd2 e1,E1 u,U,b,B,b,B mad 4
For simplicity, we consider only six-quark production where four
quarks are b quarks (the main decay channel of the Higgs pair), and
the remaining quark pair is light. This is supplemented by second-
and fourth-order QCD contributions calculated by MadGraph.
The input file whizard.in is prepared for the six-fermion
process at s1/2=500 GeV with default settings. We would
like to generate an event sample corresponding to 10 ab-1.
&process_input
process_id = "qqbbbb"
sqrts = 500
luminosity = 1000
/
&integration_input /
&simulation_input /
&diagnostics_input /
¶meter_input
mH = 115
wH = 0.3228E-02
mb = 2.9
me = 0
ms = 0
mc = 0
/
&beam_input /
&beam_input /
Without further considerations, we switch into the results
subdirectory and start integration and event generation
> ./whizard
After the initial message
! WHIZARD 1.30 (Sep 20 2004)
! Reading process data from file whizard.in
! Wrote whizard.out
! Process qqbbbb:
! e a-e -> u a-u b a-b b a-b
! 128 64 -> 1 2 4 8 16 32
! Warning: No color flow information available for process qqbbbb
! Reading vertices from file whizard.mdl ...
! Model file: 63 trilinear vertices found.
! Model file: 63 vertices usable for phase space setup.
! Generating phase space channels for process qqbbbb...
the program needs quite some time for generating the phase space
configuration (this would be faster without flavor summation).
Fortunately, the configuration is written to the file
whizard.phx and WHIZARD will reuse it when possible.
The absence of color flow information, indicated by the warning
message, is a problem of the O'Mega matrix element --- only the
leading color connection is evaluated. However, the resulting error
is typically negligible since the subleading terms are suppressed by
kinematical factors.
After this step is finished, a default cut for the light quark pair is
inserted and integration is started.
! Phase space: 408 phase space channels generated.
! Scanning phase space channels for equivalences ...
! Phase space: 1632 equivalence relations found.
! Note: The cross section may be infinite without cuts.
! Wrote default cut configuration file whizard.qqbbbb.cut0
! Wrote phase space configurations to file whizard.phx
! Created grids: 408 channels, 14 dimensions with 20 bins
!=============================================================================
! WHIZARD run for process qqbbbb:
!-----------------------------------------------------------------------------
! It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It]
!-----------------------------------------------------------------------------
! Reading cut configuration data from file whizard.cut1
! No cut data found for process qqbbbb
! Using default cuts.
cut M of 3 within 1.00000E+01 1.00000E+99
! Preparing (fixed weights): 1 sample of 100000 calls ...
The whole adaptation and integration run takes a considerable amount
of CPU time (about one day on an Alpha processor). If only one quark
flavor were considered, this could be reduced by a factor of four
(currently, WHIZARD/O'Mega does not take advantage of the fact that
some matrix elements with different flavor content are in fact
identical).
A usable result is already reached after about 10 iterations, with
considerable fluctuation around the optimal grid28:
1 100000 1.7245154E-01 9.02E-03 5.23 16.54* 0.73 0.00
!------------------------------------------------------------------------
! Adapting (var. weights): Generating 20 samples of 100000
events ...
2 100000 1.4965342E-01 2.15E-03 1.43 4.54* 1.97
3 100000 1.4929719E-01 1.29E-03 0.86 2.73* 2.16
4 100000 1.5426710E-01 5.29E-03 3.43 10.84 0.99
5 100000 1.5014838E-01 1.48E-03 0.99 3.12 1.93
6 100000 1.4741355E-01 1.11E-03 0.75 2.37* 3.02
7 100000 1.5119871E-01 1.33E-03 0.88 2.78 2.20
8 100000 1.4969993E-01 8.36E-04 0.56 1.77* 3.26
9 100000 1.5623658E-01 5.13E-03 3.28 10.38 1.17
10 100000 1.5145665E-01 7.81E-04 0.52 1.63* 3.55
11 100000 1.5170008E-01 1.05E-03 0.69 2.19 3.22
12 100000 1.5207701E-01 9.18E-04 0.60 1.91 3.35
13 100000 1.6122997E-01 1.03E-02 6.36 20.13 0.88
14 100000 1.5167050E-01 7.18E-04 0.47 1.50* 4.22
15 100000 1.5346776E-01 1.60E-03 1.04 3.30 2.59
16 100000 1.5190641E-01 9.46E-04 0.62 1.97 3.72
17 100000 3.9266569E-01 2.40E-01 61.19 193.49 0.11
18 100000 1.5427618E-01 1.94E-03 1.26 3.98 2.23
19 100000 1.5282396E-01 1.90E-03 1.24 3.94 2.57
20 100000 1.5299694E-01 1.11E-03 0.73 2.30 3.15
21 100000 1.6406596E-01 1.30E-02 7.90 24.98 0.68
!------------------------------------------------------------------------
! Integrating (fixed w.): Generating 1 sample of 100000
events ...
Nevertheless, the best grid obtained so far can safely be used for
event generation. The final estimate for the integral is
22 100000 1.5188597E-01 7.48E-04 0.49 1.56 4.19 0.00
!------------------------------------------------------------------------
!
! Time estimate for generating 10000 unweighted events: 2:24:24 hours
However, we don't need that many events: simulating 10 ab-1
now takes only half an hour:
! Event sample corresponds to luminosity [fb-1] = 0.1000E+05
!
! Generating 1519 unweighted events ...
!========================================================================
! Analysis results for the generated event sample:
!------------------------------------------------------------------------
! It Calls Integral[fb] Error[fb] Err[%] Err/Exp Eff[%] Chi2
!------------------------------------------------------------------------
23 1519 1.5188597E-01 3.90E-03 2.57 1.00 100.00
!------------------------------------------------------------------------
! Excess events: 6.8 (Error[%]: 0.45 )
! WHIZARD run finished.
If we wish to know how many of those events are originating from HH
pairs, we should set up an analysis configuration file
whizard.cut5 like this
! e- e+ -> q qbar b bbar b bbar
!128 64 1 2 4 8 16 32
process qqbbbb
cut M of 12 within 114 116
cut M of 48 within 114 116
and
cut M of 36 within 114 116
cut M of 24 within 114 116
and rerun the program, setting read_grids=T and
read_events=T. The result is
! Analysis results for the generated event sample:
!
! Additional cuts:
! integration level 5
cut M of 12 12 within 1.14000E+02 1.16000E+02
cut M of 48 48 within 1.14000E+02 1.16000E+02
!------------------------------------------------------------------------
! It Calls Integral[fb] Error[fb] Err[%] Err/Exp Eff[%] Chi2
!------------------------------------------------------------------------
23 150 1.4998614E-02 1.22E-03 8.16 1.00 9.87
!========================================================================
! Analysis results for the generated event sample:
!
! Additional cuts:
! integration level 5
cut M of 36 36 within 1.14000E+02 1.16000E+02
cut M of 24 24 within 1.14000E+02 1.16000E+02
!------------------------------------------------------------------------
! It Calls Integral[fb] Error[fb] Err[%] Err/Exp Eff[%] Chi2
!------------------------------------------------------------------------
23 199 1.9898162E-02 1.41E-03 7.09 1.00 13.10
! WHIZARD run finished.
The two event samples may be added (assuming that no events pass
both cuts simultaneously), to yield 349 ``signal'' events.
The stability of the result and the computing time can be improved by
reducing the number of phase space channels (see
Sec. 4.6.2).
| 6.3 |
Vector boson scattering: polarization and beamstrahlung |
|
In case no light Higgs boson exists, one will try to measure vector
boson scattering at high-energy colliders, e.g.
W+W- -> W+W-, ZZ
Such processes can be described in an effective-Lagrangian approach,
where higher-order corrections to the scattering amplitude are
described by new parameters a4,a5,.... WHIZARD
defines these anomalous couplings in the model SM_ac.mdl.
At an e+e- collider, WW scattering processes occur as a
subprocess of
This can be simulated in a single run, if we set up the process
configuration file whizard.prc as follows:
# WHIZARD configuration file
model SM_ac
# Tag In Out Method Option
#=================================================================
# On-shell process:
ww W+,W- W+,W- chep
zz W+,W- Z,Z chep
# Full six-fermion matrix elements (no QCD):
nnqqqq e1,E1 n1:n2:n3,N1:N2:N3,u:d,U:D,u:d,U:D omega
enqqqq e1,E1 e1,N1,u:d,U:D,u:d,U:D omega w:f
neqqqq e1,E1 n1,E1,u:d,U:D,u:d,U:D omega w:f
eeqqqq e1,E1 e1,E1,u:d,U:D,u:d,U:D omega w:f
# First neutrino generation only:
# WW and ZZ
nnuudd e1,E1 n1,N1,u,U,d,D omega
# WW only
nnucsd e1,E1 n1,N1,u,C,s,D omega
eeucsd e1,E1 e1,E1,u,C,s,D omega w:f
# ZZ only
nnuuss e1,E1 n1,N1,u,U,s,S omega
# WZ
enudss e1,E1 e1,N1,u,D,s,S omega w:f
# Second neutrino generation only:
# WW and ZZ
nnuudd2 e1,E1 n2,N2,u,U,d,D omega
# WW only
nnucsd2 e1,E1 n2,N2,u,C,s,D omega
# ZZ only
nnuuss2 e1,E1 n2,N2,u,U,s,S omega
This is actually an abridged version of the process file
whizard.prc.ww-strong that comes with the standard distribution.
In the chosen model, the Higgs boson is actually not absent, but its
mass is set to a very large value (resp. infinity) by default. Apart
from the signal process nnqqqq we have included some important
background processes, which must be considered if the final-state
electron is not observed. For these processes, gauge invariance is an
issue, and we must set the w:f (fudged-width) or w:c
(constant-width) option to obtain a consistent result.
The setup below is for the TESLA collider design. We may have
polarization and have to account for ISR and beamstrahlung. The input
file whizard.in specifies 80 % left-handed electron
polarization and 40 % right-handed positron polarization:
&process_input
process_id = "nnqqqq"
sqrts = 800
luminosity = 0
polarized_beams = T
structured_beams = T
/
&integration_input
calls = 1 100000 10 100000 5
default_Q_cut = 0
/
&simulation_input
/
&diagnostics_input
/
¶meter_input
ms = 0
mc = 0
a4 = 0
a5 = 0
/
&beam_input
particle_name = "e-"
polarization = 0.80 0
CIRCE_on = T
CIRCE_acc = 2
ISR_on = T
ISR_alpha = 0.0072993
ISR_m_in = 0.000511
/
&beam_input
particle_name = "e+"
polarization = 0 0.40
CIRCE_on = T
CIRCE_acc = 2
ISR_on = T
ISR_alpha = 0.0072993
ISR_m_in = 0.000511
/
Polarization is switched on by the entry polarized_beams in the
first block. Then, the polarization settings in the
beam_input blocks are respected. Similarly,
structured_beams switches on possible spectra and/or structure
function settings. Now, the beam particles are no longer
automatically defined by the process, but have to be explicitly
defined for each beam.
The beamstrahlung settings (CIRCE) are for the accelerator type 2
(TESLA), default parameterization version and revision numbers.
Concerning initial-state radiation (ISR), we should set the
electromagnetic coupling constant equal to the low-energy value of
1/137 since on-shell photons are radiated. The incoming mass must
be reset equal to 511 keV, since the physical electron mass
me has been set to zero.
In the parameter section, the two anomalous couplings a4 and
a5 are included. Here, we set them to zero which is also the
default value.
The signal process considered here is quite well-behaved, so, in the
integration section, we choose a number of iterations that is smaller
than the default for this class of processes, to limit execution time.
(The zero value set for the Q cut is unnecessary for the signal
process considered here and could be left out. It is useful for
integrating the background process e+e--> e+e- qq_ qq_.
With no Q cut on the final-state electron, we get the total cross
section. The parameter default_Q_cut has the side effect of
setting the scale for the integration over the transverse momentum.
If the parameter is zero, the electron mass is taken as setting the
scale, which is appropriate for the total cross section.) The actual
cuts for the signal and background processes may be taken as the
default ones, or we could write them explicitly in a file. The
default cuts require a minimum invariant mass of 10 GeV for each
quark pair.
The output shown below is for the complete process, including all
possible flavor combinations in the process labeled nnqqqq
|
e-e+ -> ne |
|
q |
|
q |
|
;
n = ne, nµ, nt;
q = u, d
|
! WHIZARD 1.30 (Sep 20 2004)
! Reading process data from file whizard.in
! Wrote whizard.out
! Process nnqqqq:
! e a-e -> nu_e a-nu_e u a-u u a-u
! 128 64 -> 1 2 4 8 16 32
! Warning: No color flow information available for process nnqqqq
! Active structure functions for beam 1:
! CIRCE: e -> e (generator)
! ISR: e -> e
! Active structure functions for beam 2:
! CIRCE: a-e -> a-e (generator)
! ISR: a-e -> a-e
! Warning: CIRCE: Beamstrahlung effect on polarization will be ignored.
! Warning: CIRCE: Beamstrahlung effect on polarization will be ignored.
! Warning: ISR: Effect on beam polarization will be ignored.
! Warning: ISR: Effect on beam polarization will be ignored.
! Reading vertices from file whizard.mdl ...
! Model file: 57 trilinear vertices found.
! Model file: 57 vertices usable for phase space setup.
! Generating phase space channels for process nnqqqq...
! Phase space: 419 phase space channels generated.
! Scanning phase space channels for equivalences ...
! Phase space: 2165 equivalence relations found.
! Note: The cross section may be infinite without cuts.
! Wrote default cut configuration file whizard.nnqqqq.cut0
! Wrote phase space configurations to file whizard.phx
! Created grids: 419 channels, 18 dimensions with 20 bins
!=============================================================================
! WHIZARD run for process nnqqqq:
!-----------------------------------------------------------------------------
! It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It]
!-----------------------------------------------------------------------------
! Reading cut configuration data from file whizard.cut1
! No cut data found for process nnqqqq
! Using default cuts.
cut M of 12 within 1.00000E+01 1.00000E+99
cut M of 20 within 1.00000E+01 1.00000E+99
cut M of 36 within 1.00000E+01 1.00000E+99
cut M of 24 within 1.00000E+01 1.00000E+99
cut M of 40 within 1.00000E+01 1.00000E+99
cut M of 48 within 1.00000E+01 1.00000E+99
! Preparing (fixed weights): 1 sample of 100000 calls ...
1 100000 6.3551041E+00 8.47E-01 13.33 42.14* 0.92 0.00 1
!-----------------------------------------------------------------------------
! Adapting (variable wgts.): 10 samples of 100000 calls ...
2 100000 6.2483097E+00 8.96E-01 14.34 45.34 0.90
3 100000 5.7224764E+00 1.37E-01 2.39 7.56* 0.74
4 100000 5.8265889E+00 9.12E-02 1.57 4.95* 1.07
5 100000 5.8355602E+00 7.12E-02 1.22 3.86* 1.55
6 100000 5.8333281E+00 5.98E-02 1.03 3.24* 1.72
7 100000 5.8430723E+00 6.24E-02 1.07 3.38 2.09
8 100000 5.7988184E+00 5.38E-02 0.93 2.93* 2.17
9 100000 5.7520261E+00 4.94E-02 0.86 2.71* 2.47
10 100000 5.8027805E+00 5.01E-02 0.86 2.73 2.32
11 100000 5.8475680E+00 5.04E-02 0.86 2.73 2.49
!-----------------------------------------------------------------------------
! Integrating (fixed wgts.): 5 samples of 100000 calls ...
12 500000 5.8180783E+00 2.56E-02 0.44 3.11 0.98 1.13 5
!-----------------------------------------------------------------------------
!
! Time estimate for generating 10000 unweighted events: 3h 55m 03s
!=============================================================================
! Total cross section summary (all processes):
!-----------------------------------------------------------------------------
! Process ID Integral[fb] Error[fb] Err[%] Eff[%] Chi2
!-----------------------------------------------------------------------------
nnqqqq 5.8180783E+00 2.56E-02 0.44 0.98 1.13
!-----------------------------------------------------------------------------
sum 5.8180783E+00 2.56E-02 0.44 0.00
!=============================================================================
! Wrote whizard.out
! Integration complete.
! No event generation requested
! There were no errors and 15 warning(s).
! WHIZARD run finished.