Description of the method

The method consist to use a sample of events (weighted or unweighted events) and associate to those events a new weight corresponding to a new theoretical hypothesis. It corresponds to a multidimensional version of the unidimensional re-weighing method commonly used by experiments. Once computed, this weight can be propagate through the all simulation chain in order to avoid to have to perform the full-simulation on a huge number of sample. This methods works only if the original hypothesis and the new one are both significant in the same part of the phase-space.

We propose three types of reweightings. One for the Leading Order sample and two for the Next-to-Leading Order sample (called Kamikaze Reweighting and NLO Reweighting)

Leading Order

At Leading Order, the new weight is given by $$W_{new} = |M^{new}_h|2 /|M^{old}_h|2 * W_{old} $$ where h is the helicity associated to the events and $|M^{new/old}_h|2$ is the matrix element for that helicity. If the events is not associated to a specific helicity, then the sum over the helicity is used instead.

This method is fully LO accurate and do not present any bias. Note that the statistical fluctuation of the original sample can be enhanced by the reweighting. To get an idea of such propagation, one can use the naive formula of propagation of error:

$$\Delta\mathcal{O}_{new} = \bar R\cdot \Delta\mathcal{O}_{old} + \Delta R \cdot \mathcal{O}_{old} $$

where $\bar R$ is the avarage of the ratio of the matrix-element, $\Delta R$ the associated variance. $\mathcal{O}_{old/new}$ is the value of the observables under consideration for the associated hyppothesis and $\Delta\mathcal{O}_{old/new}$ the associated variance.

Kamikaze Reweighting

This correspond to a Leading Order type of reweighting. Both the soft and hard events are reweighted according to the associated tree-level matrix element related to the number of particles in the final state. i.e.,

$$W^{S_{new} = |M}{new}_{born}|^{2 /|M}{old}_{born}|^{2 * W}S_{old} $$ $$W^{H_{new} = |M}{new}_{real}|^{2 /|M}{old}_{real}|^{2 * W}H_{old} $$

For obvious reason, this method is, in general, not NLO accurate. This is available since 2.3.2

NLO reweighting:

We use the basis introduced in ​http://arxiv.org/pdf/1110.4738v1.pdf to decompose the matrix-element component independant of the scale and pdf variation:

$$d\sigma^{{H} = d\sigma}E - d\sigma^{{MC} $$ $$ d\sigma}{S} = d\sigma^{{MC} + \sum_{\alpha=S,C,SC} d\sigma}\alpha $$

Each of the $d\sigma^{\alpha$ can be written as}

$$ d\sigma^{\alpha=f_1(x_1,\mu_F)f_2(x_2,\mu_F) \left[\mathcal{W}}\alpha_0 + \mathcal{W}^{\alpha_F log\left(\mu_F/Q\right)}2 + \mathcal{W}^{\alpha_R log\left(\mu_R/Q\right)}2 \right] d\chi$$

Additionally, we keep track of which part of the $\mathcal{W}$ are proportional to the Born ($\mathcal{W}_B$), the finite piece of virtual ($\mathcal{W}_V$) and of the real ($\mathcal{W}_R$). The equations are available in the appendix of ​http://arxiv.org/pdf/1110.4738v1.pdf

In principle, the reweighting should be performed on each sub-part of the $\mathcal{W}$ according to the following formula

$$\mathcal{W}_B^{{new} = \frac{B}{new}}{B^{{old}} * \mathcal{W}_B}{old} $$ $$\mathcal{W}_V^{{new} = \frac{V}{new}}{V^{{old}} * \mathcal{W}_V}{old} $$ $$\mathcal{W}_R^{{new} = \frac{R}{new}}{R^{{old}} * \mathcal{W}_R}{old} $$

the final weight is then computed by recombining the weight according to the above formula.

However in MadGraph5_aMC@NLO, we use the virt-tricks method which avoid the computation of the virtual for some of the phase-space points. This speed optimisation method forbids the simple above reweighting since the generation will have $\mathcal{W}_V^{{old}=0$ even if $V_{old} \neq 0$. To avoid this problem, $\mathcal{W}_B$ is splitted in two piece :$\mathcal{W}_{BC}$, $\mathcal{W}_{BB}$. $\mathcal{W}_{BC}$ is the part proportional to the born and related to the one counter-term, while $\mathcal{W}_{BB}$ is for the other contribution (the born itself and the approximate virtual). The reweighting is then done as}

$$\mathcal{W}_{BB}^{{new} = \frac{(B}{new}+V^{new})}{(B{old}+V^{{old})} * \mathcal{W}_{BB}}{old} $$ $$\mathcal{W}_V^{{new} = \frac{(B}{new}+V^{new})}{(B{old}+V^{{old})} * \mathcal{W}_V}{old} $$ $$\mathcal{W}_{BC}^{{new} = \frac{B}{new}}{B^{{old}} * \mathcal{W}_{BC}}{old} $$ $$\mathcal{W}_R^{{new} = \frac{R}{new}}{R^{{old}} * \mathcal{W}_R}{old} $$

Such reweighting is fully NLO accurate. As in the LO case, the statistical uncertainty can be enhanced by the reweighting. Additionally the trick to support the virt-tricks adds an additional contribution to statistical uncertainty. This method will be released in a future version of MadGraph5_aMC@NLO and can currently be provided on request. Since this reweighting is based on a dedicated basis the NLO sample must be generated in a specific way to have the additional information in the leshouches event.

Technical details

Limitation

1. We do not perform any PDF and/or cut reweighting.
2. We do not allowed to change the functional form of alpha_S
3. In presence of decay chain the order of the particles in the events file is important. This is important if you want to use this tools with LHE events not produced by MadGraph5_aMC@NLO.

Installation

This module is built-in in MadGraph5_aMC@NLO.2.3.2. Since MadGraph5_aMC@NLO.2.3.2, this module relies on f2py to be installed. The easiest way to install f2py is to install numpy (if not already done).

Running the code

Running simultaneously with event generation

When running event generation at LO or NLO (either via ./bin/generate_events from the local directory or "launch" via the mg5 interface). You will be asked two questions. The phrasing/options of those two questions depends if you run at LO or NLO but both follow the same strategy. Here we will take the example of a NLO generation. In that case, the first question is:

The following switches determine which operations are executed:
 1 Perturbative order of the calculation:                               order=NLO
 2 Fixed order (no event generation and no MC@[N]LO matching):    fixed_order=OFF
 3 Shower the generated events:                                        shower=ON
 4 Decay particles with the MadSpin module:                           madspin=OFF
 5 Add weights to the events based on changing model parameters:     reweight=OFF
  Either type the switch number (1 to 5) to change its default setting,
  or set any switch explicitly (e.g. type 'order=LO' at the prompt)
  Type '0', 'auto', 'done' or just press enter when you are done.
 [0, 1, 2, 3, 4, 5, auto, done, order=LO, ... ][60s to answer]

As you can see, the question presents a series of switch which can take different value (in the example "NLO", "ON", "OFF"). In order to perform the reweighting, you need to put the reweight switch to "ON". Type

reweight=ON

You can also just type "5" but please avoid to use this mode in scripts. After pressing enter the question is reasked and you now should have:

The following switches determine which operations are executed:
 1 Perturbative order of the calculation:                               order=NLO
 2 Fixed order (no event generation and no MC@[N]LO matching):    fixed_order=OFF
 3 Shower the generated events:                                        shower=ON
 4 Decay particles with the MadSpin module:                           madspin=OFF
 5 Add weights to the events based on changing model parameters:     reweight=ON
  Either type the switch number (1 to 5) to change its default setting,
  or set any switch explicitly (e.g. type 'order=LO' at the prompt)
  Type '0', 'auto', 'done' or just press enter when you are done.
 [0, 1, 2, 3, 4, 5, auto, done, order=LO, ... ][60s to answer]

This allow you to change any other switch (note that "fixed_order" needs to stay on OFF). You can type enter when you want to pass to the next question:

Do you want to edit a card (press enter to bypass editing)?
  1 / param      : param_card.dat
  2 / run        : run_card.dat
  3 / reweight   : reweight_card.dat
  4 / shower     : shower_card.dat
 you can also
   - enter the path to a valid card or banner.
   - use the 'set' command to modify a parameter directly.
     The set option works only for param_card and run_card.
     Type 'help set' for more information on this command.
   - call an external program (ASperGE/MadWidth/...).
     Type 'help' for the list of available command
 [0, done, 1, param, 2, run, 3, reweight, 4, enter path, ... ][60s to answer]

For a NLO accurate reweighting (available since 2.4.0), type

set keep_rwgt_info True

This can also be done via the manual edition of the run_card (by typing 2). With this option on False (the default) the kamikaze reweighting will be performed.

Then type

3

to open an editor (in most system this use vi) where you can edit the content of the reweight_card. The format/options of that file are describe below. It allows you to specify which model/benchmark you want to use. When you are done, exit the file and press enter.

Running the code after the generation of events as been completed.

In order to run the reweighting on previously generated samples. You need to go to the associated process directory and run either ./bin/madevent or ./bin/aMC@NLOscript for respectively LO/NLO event generation. You can then type reweight RUN_NAME (RUN_NAME is typically run_01) and you will be asked the same questions as above.

Another options is to manually edit the Cards/reweight_card.dat file and then run one of the two following command:

./bin/madevent reweight RUN_NAME -f
./bin/aMC@NLO reweight RUN_NAME -f

The first sample is available in case of LO event generation, while the second is present in the case of NLO event generation.

Content of the reweight_card

The cards is compose of two sections:

1. Options: those are options which change the behavior of the reweighting. Those lines need to be specified before the first 'launch' line to have effects.
   1. "change model <XXX>" performed the reweighting within a new model (you then need to profide a full param_card and not a difference)
   2. "change process <XXX>" change the process definition of the process.
   3. "change process <XXX> --add" add one process definition of the process to the new list.
   4. "change output <i>": Three options: 'default'(i.e. lhef version3 format), '2.0' (i.e. lhef version2 format, the main weight is replace), 'unweight' (a new unweighting is applied on the events sample.)
   5. "change helicity <True|False>": perform reweighting for the given helicity (True --default--) or do the sum over helicity.
   6. "change rwgt_dir <PATH>": change directory where the computation is performed. This can be use to avoid to recreate/recompile the fortran executable if pointing to a previously existing directory.
   7. change mode LO: For NLO sample, this flag force to use the kamikaze reweighting (available in 2.4.0)

2. benchmark definition: starting one benchmark starts with the line

   launch
   

   then you can have either a series of "set command" to specify how to edit the param_card from the original one. Are a couple of equivalent options:

   set mt 150
   set mass 6 150
   

   For a scan over parameter, you can use one of the following options (all are equivalent):

   set mt 6 scan:range(100,200,20)
   set mass 6 scan:[100,120,140,160,180]
   

   instead of using the "set command" you can also specify the path to a (valid) param_card

   PATH
   

Options of the code:

Note all the following options will be available in Madgraph5_aMC@NLO since version 2.3.2. They have to be included in the reweighting_card before the first launch command.

Input/Output format

1. the output format follows the Leshouches agreement version 3 [link arxiv]

as an example the header looks like:

<initrwgt>
<weightgroup type='mg_reweighting'>
<weight id='mg_reweight_1'>set param_card dim6 1 100.0
</weight>
<weight id='mg_reweight_2'>set param_card dim6 2 100.0
</weight>
<weight id='mg_reweight_3'>set param_card dim6 3 100.0
</weight>
</weightgroup>
</initrwgt>

and one associated events:

<event>
 8      0 +7.9887000e-06 1.24664300e+02 7.95774700e-02 1.23856500e-01
        1 -1    0    0  501    0 +0.0000000e+00 +0.0000000e+00 +1.3023196e+03 1.30231957e+03 0.00000000e+00 0.0000e+00 -1.0000e+00
       -2 -1    0    0    0  501 +0.0000000e+00 +0.0000000e+00 -1.4499581e+02 1.44995814e+02 0.00000000e+00 0.0000e+00 1.0000e+00
      -24  2    1    2    0    0 -1.2793809e+01 -8.3954553e+01 -1.1792566e+02 1.65987064e+02 8.02071978e+01 0.0000e+00 0.0000e+00
       23  2    1    2    0    0 +1.2793809e+01 +8.3954553e+01 +1.2752494e+03 1.28132832e+03 9.12640692e+01 0.0000e+00 0.0000e+00
       11  1    3    3    0    0 -1.2462673e+01 +1.3647422e+01 -2.6083861e+01 3.19677669e+01 0.00000000e+00 0.0000e+00 -1.0000e+00
      -12  1    3    3    0    0 -3.3113586e-01 -9.7601975e+01 -9.1841804e+01 1.34019297e+02 0.00000000e+00 0.0000e+00 1.0000e+00
        4  1    4    4  502    0 -1.8321803e+01 +9.0929609e+01 +9.3905973e+02 9.43629724e+02 0.00000000e+00 0.0000e+00 -1.0000e+00
       -4  1    4    4    0  502 +3.1115612e+01 -6.9750557e+00 +3.3618969e+02 3.37698598e+02 0.00000000e+00 0.0000e+00 1.0000e+00
<rwgt>
<wgt id='mg_reweight_1'> 4.55278761371e-06 </wgt>
<wgt id='mg_reweight_2'> 2.65941887458e-06 </wgt>
<wgt id='mg_reweight_3'> 8.68203803896e-06 </wgt>
</rwgt>
</event>

2. The reweight_card in that case was:

   launch
      set Dim6 1 100
      set Dim6 2 0
      set Dim6 3 0
      set Dim6 4 0
      set Dim6 5 0
   launch
      set Dim6 1 0
      set Dim6 2 100
      set Dim6 3 0
      set Dim6 4 0
      set Dim6 5 0
   launch
      set Dim6 1 0
      set Dim6 2 0
      set Dim6 3 100
      set Dim6 4 0
      set Dim6 5 0
   

Note:

1. The production events were in this case made via the SM where all those coefficient were in fact set to zero. Therefore all the line setting the line to zero were superfluous, and this is the reason why they didn't appear in the banner.
2. You can also specify a path to a param_card in the reweight_card. The content of the header being computed automatically with the difference of the two param_card.

3. The cross-section of the original file and those associated with the new hyppothesis are printed at the end of the script:

   INFO: Original cross-section: 0.80086112072 +- 0.0025669959099 pb 
   INFO: Computed cross-section: 
   INFO: 119 : 5.0238030968 
   INFO: 120 : 4.46724081967 
   INFO: 121 : 0.790019392142 
   

LO Validation

1. The comparison for the full cross-section are done like this:

./bin/madevent ./Cards/reweight_card.dat

p p > e+ e- cross-section

1. The reweight_card is the following:

   launch
    set aewm1 100
   launch 
    set aewm1 200
   launch 
    set aewm1 300
   

2. The associated cross-section are
   1. 1135.25 pb
   2. 1095.28 pb
   3. 1329.52 pb
3. The cross-section computed with MadEvent are
   1. 1130 +- 2.815 pb
   2. 1098 +- 2.478 pb
   3. 1336 +- 2.777 pb

EWDIM6 Validation

input

1. The model use for this validation is the EWDIM6 (See: ​http://arxiv.org/abs/arXiv:1205.4231). 10k events where generated with the standard model (cross-section: 0.8008 ± 0.0026 pb)
2. The reweight_card was:

   launch
      set Dim6 1 100
      set Dim6 2 0
      set Dim6 3 0
      set Dim6 4 0
      set Dim6 5 0
   launch
      set Dim6 1 10
      set Dim6 2 0
      set Dim6 3 0
      set Dim6 4 0
      set Dim6 5 0
   launch
      set Dim6 1 1
      set Dim6 2 0
      set Dim6 3 0
      set Dim6 4 0
      set Dim6 5 0
   launch
      set Dim6 1 0.1
      set Dim6 2 0
      set Dim6 3 0
      set Dim6 4 0
      set Dim6 5 0
   launch
      set Dim6 1 0.01
      set Dim6 2 0
      set Dim6 3 0
      set Dim6 4 0
      set Dim6 5 0
   

The same scan was done for the three first coupling (CWWW, CW, CB)

result:

1. For CWWW

Coupling value ($TeV{-2}$)	Reweight cross-section (pb)	MG5_aMC cross-section (pb)	Status
0.01	0.800810008029	0.7973 ± 0.0023	OK
0.1	0.800903791291	0.799 ± 0.0026	OK
1	0.802209013071	0.7987 ± 0.0025	OK
10	0.85200014698	0.8584 ± 0.00092	OK
100	5.0238030968	6.09 ± 0.0082	FAIL
100	5.04763	6.09 ± 0.0082	FAIL (done with a sample of 100k events)

The last entry fails since the expected distribution for such value of the coupling is too different from the distribution of the Standard Model. Such discrepancy are expected in this case. One hint is that the cross-section is an order of magnitude higher than the original one (Looking at the distribution confirm this). The inversed reweight (i.e. starting from the CWW=100 sample of events and reweight to find back the SM) is working properly. It returned: 0.803341120226

The various distribution for those generation are in attachment of this web page. The dashed blue curve is the one produced by reweighting. While the solid black is the curve generated by MadEvent. All sample are done with 100k events for the comparison of distributions.

2. For CW

Coupling value ($TeV{-2}$)	Reweight cross-section (pb)	MG5_aMC cross-section (pb)	Status
0.01	0.800798262059	0.7953 +- 0.002497	OK
0.1	0.801379445746	0.7988 ± 0.0023	OK
1	0.806872565125	0.8065 ± 0.0023	OK
10	0.889336417677	0.8832 ± 0.003	OK
100	4.46724081967	4.519 ± 0.015	FAIL
100	4.44273	4.519 ± 0.015	FAIL (done with a sample of 100k events)

Same comment as for the previous coupling.

3. For CB

Coupling value ($TeV{-2}$)	Reweight cross-section (pb)	MG5_aMC cross-section (pb)	Status
0.01	0.800798262059	0.7977 ± 0.0027	OK
0.1	0.800782626532	0.7985 ± 0.0024	OK
1	0.800626859275	0.7981 +- 0.002365	OK
10	0.799127987884	0.7971 ± 0.0024	OK
100	0.790019392142	0.7852 ± 0.0026	OK
100	0.786698206995	0.7852 ± 0.0026	OK (done with a sample of 100k events)

This operator has less impact on the cross-section/distributions, and therefore even a large value of the coupling is still working fine.

Note:

1. The cross-section obtained for 100k events sample is 0.7989 ± 0.00087
2. The statistical fluctuation of the original sample is reflected on the reweighing cross-section (as expected)