Testing the Complex Mass Scheme NLO implementation
The complex mass scheme is the modern way of handling finite width effects of unstable particles in perturbation theory. In a nutshell, it consists in redefining the mass of unstable particles as complex with an imaginary part proportional to its width. The denominator of unstable propagators remains identical, but contrary to the naive implementation of the width, the complex mass will also appear in the numerator of fermion propagators and in all couplings proportional to the unstable particle mass. In doing so, the Lagrangian is simply continued to complex masses and the final result remains gauge invariant. The Complex Mass Scheme (CMS henceforth) can be activated by simply typing the following command in the interactive shell:
MG5_aMC> set complex_mass_scheme True
The CMS implementation is rather straightforward at leadingorder (LO) but it becomes more involved at nexttoleadingorder (NLO) because of mainly two points
 The widths must be LO accurate at least in the offshell region and NLO accurate in the onshell region.
 Because of the modified onshell renormalization condition in the CMS, the logarithms appearing in the UV wavefunction renormalization counterterms must be evaluated in the correct Riemann sheet.
The details of these issues will be discussed in a forthcoming publication; this wiki page is mainly to describe the various options to the command 'check cms' which automatically tests the consistency of the CMS implementation. The core idea of the test is to compare squared amplitudes in the CMS scheme () and in the case of widths set to zero () for a given kinematic configuration where all resonances are far offshell. The difference between these two amplitudes must be of higher order. More formally, this means that if we have , then we can write the following:
At LO, . The statement that the difference is of higher order is then equivalent to state that .
At NLO, this relation becomes
In order to check that and are indeed zero, the test proceeds by scaling down all relevant couplings and widths by the parameter and evaluate the expressions of for many progressively smaller values of , but always on the same offshell kinematic configuration. One can then plot the quantities and make sure that the asymptot for small values of lambda is the constant . Any divergent behavior would be a manifestation of the presence of the term which reveals an issue with the CMS implementation (most likely one of the two points mentioned above) which spoils the expected cancellation. Before we detail the option of this test, here is the expected output (generated automatically, incl. this plot) for the case of QCD and QED corrections to fully decayed top quark pair production:
In the upper inset, we clearly see the finite width effects for large values of . These become progressively smaller and indiscernible below . When dividing this difference by lambda, as done in the lower inset, we only see a mild deviation with respect to a constant. Changing the LO width used for the test by as little as 0.1 % already yields a larger than the residual one stemming from numerical inaccuracies. The figure on the right shows that incorrectly setting the analytical continuation of UV wavefunctions counterterms logarithms yields an asymptotic value of several thousands in the plot. This clearly establishes the sensitivity of the test towards any incorrect CMS implementation at NLO.
We now focus on the description of the command for this check, whose main syntax is
MG5_aMC> check cms [reuse] <process_definition> <options>
Minimal example check cms u d~ > e+ ve a [virt=QCD QED] Example with options check cms reuse u d~ > e+ ve a [virt=QCD QED] name=udx_epvea tweak=['default','allwidths>allwidths*0.99(widths_x_0.99)']
First, the 'reuse' suffix following 'cms' specifies that you want to reuse relevant information existing from previous runs. This includes potentially reusing the fortran output of the NLO matrix element if the same process was run before with the 'cms check' command. Also, if a name was given to this run (see option 'name' further) and the corresponding saved result python pickled file exists on disk, this run will be skipped and the result recycled from the pickle file. When 'reuse' is not specified, the test always restarts from scratch. In general, it is recommended to always use 'reuse'.
The <process_definition> which follows can be any process following the MG5 syntax. If this process is LO, the test will be adapted to test and the matrix element will be generated an evaluated dynamically directly in python. If the process definition is at NLO, then the test will test and the output will be done in fortran on disk, compiled and the corresponding standalone 'check' executable steered by MG5_aMC. The LO CMS test is mostly trivial, but it can be useful to investigate the expected sensitivity on the CMS implementation of the corresponding NLO process.
Finally, the following options are available and we detail their usage further below.
 Basic options
 name
 report
 tweak
 seed
 offshellness
 energy
 lambdaCMS
 show_plot
 lambda_plot_range
 recompute_width
 CTModeRun
 helicity
 reduction
 More technical options
 cms
 diff_lambda_power
 loop_filter
 resonances
 Special option
 analyze
Basic options
 name=auto

This will serve as the base_name for the fortran output folder for the loop matrix element and for the pickle files storing the results generated. The default 'auto' tries to smartly automatically assign one, but it is recommended to specify your own.
 Example: name=fully_decayed_ttx
 report=concise

By chosing 'full' instead of the default 'concise', MG5aMC will provide advanced information on the details (widths recomputed, offshellness considered, etc...) of the test performed. The two possible values are 'concise' and 'full'.
 Example: report=full
 tweak=default()

The tweak option is an important one as it lets you automatically performs changes to the computational setup which should deteriorate the quality of the CMS check (i.e. induce nonzero values of ) so that one can appreciate to which level of accuracy the current implementation is consistent. The keyword 'default' means no changes and would yield the blue curve in the figures below. What follows the tweak in parenthesis is its name which will be used to label the curve and to build the name of the pickle file storing this tweak. Possible tweak ares:
 <param_name>>expr(<param_name>): for instance 'WZ>1.1*WZ' would multiply the LOaccurate Zboson width by 10%. Notice that you can use the keyword 'allwidths' which will automatically apply the rule to all widths.
 seed<int>: for instance seed669 which specifies that for this tweak one must change the seed to value <int>
 [logplogmlog]>[logplogmlog]: for instance logp>logm which would change the matrix element fortran output so as to change the logarithms continued with '+2i' to '2i'
 Example: tweak=['default','allwidths>1.1*allwidths&seed333(Increased_widths_and_seed_333)','logp>logm&logm>logp(inverted_logs)']
 Example: tweak=alltweaks
 seed=666

Changes the seed for the generation of the offshell kinematic configuration, allowing for varying the config. The value '1' sets it so that it is different for every run.
 Example: seed=667
 offshellness=10.0

Sets what is the minimum requirement of offshellness for each resonance that the kinematic configuration must satisfay. The default offshellness of 10 is such that the momentum of each resonance must satisfy . The offshellness can be negative too, but always strictly larger than 1. Notice that when the offshellness required is negative, it is not guaranteed that MG5_aMC will find a valid kinematic configuration if external states are massive. Finally, notice that the phase kinematic configuration chosen satisfies extra requirement of isolations and hardness in terms of a minimal pt cut and R between all external legs.
 Example: offshellness=0.7
 energy=5000.0

Sets the target energy for the kinematic configuration to build. Notice that this energy will be automatically changed with a warning depending on its consistency with the offshellness required.
 Example: energy=2000.0
 lambdaCMS=(1.0e6,5)

Sets what values of the scaling parameter must be used for the test. This option can either be a tuple '(min_val, points_per_decade)', a float 'min_val' or a python list. The float 'min_val' is the minimal value of to probe and is the number points one should spread uniformly in each decade (i.e. interval []. Notice that the list must always contain the value 1.
 Example: lambdaCMS=(1.0e2,5)
 Example: lambdaCMS=[float('1.0e%d'%exp)\ for\ exp\ in\ range(8)]
 show_plot=True

Allow to turn off the matplotlib generation and only report the outcome of the numerical check. Turning it off also removes the progress_bar display during the check
 Example: show_plot=False
 lambda_plot_range=[1.0,1.0]]

Specifies the lower and upper bounds for the range of values to be plotted. A negative value means that it is automatically taken to the extremum of all the values used for producing the result of the cms check.
 Example: lambda_plot_range=[1e05,1e02]
 recompute_width=auto

Decides how to compute the leadingorder accurate width necessary for the test to pass. Four possible values 'never', 'first_time', 'always' or 'auto'. 'never' means that the width for will be taken from the value in the default param_card .dat and the widths for subsequent smaller values of will be computed via a simple scaling law. 'first_time' means that the widths will be computed (numerically or analytically if the model has a decay module) with MadWidth' for and scaled down for lower values. 'always' means that the widths will be recomputed for all values (this mode is only to be used for checking the width computation). Finally the default value 'auto' will be interpreted as 'never' for an LO test where LOaccurate width aren't necessary in the offshell region and 'first_time' if the check is NLO.
 Example: recompute_width=never
 CTModeRun=1

By default MadLoop will run in double precision, make stability tests and only if they are unsuccessful go to quadruple precision. Given the nature of the cancellation we are probing with this test, the plot can quickly become unstable for complicated processes and low values. It is then necessary to force MadLoop to perform the test in quadruple precision. This can be done by changing this option from its default 1 (which is where it performs stability tests) to 4 (forced quadruple precision).
 Example: CTModeRun=4
 helicity=1

The default value of 1 means that the test is performed with the matrix element summed over all helicity configurations. This can be slow when enforcing quadruple precision and looking at complicated processes. With this parameter set to 'helicity=<int>', the matrix elements will be evaluated on only the helicity configuration #<int> (check their order in the file HelConfigs.dat of the directory MadLoop5Resources output along with the process).
 Example: helicity=1
 reduction=1234

Specifies which reduction method to use and in which order (see comments on the option #MLReductionLib of the card MadLoopParams.dat for details.)
 Example: reduction=31
More technical options
 cms=QED&QCD,aewm1>10.0/lambdaCMS&as>0.1*lambdaCMS

This is one of the most complicated options and it contains two parts separated by a comma. The first part lists the coupling orders which take part in the expansion. The second part lists how to scale the corresponding parameter which drive the expansion. These must be external parameters present in the param_card. The value 'lambdaCMS' is a special tag which refers to the current value of the scaling parameter being considered. Notice that these replacement rules must be of the form <ext_param_name>>f(<ext_param_name>,lambdaCMS) where f is some function following python syntax. The default value start from fixed values 10.0 and 0.1 for , but it is possible to specify the original value of the parameter in the card like this 'aewm1>aewm1/lambdaCMS' in which case the base value of the external parameter aewm1 will be the one in the original param_card.dat. You should not need to change this default unless you are testing a new physics model with an extended gauge sector (in which case the modification could look like the example below).
 Example: cms=QED&QCD&NP,aewm1>10.0/lambdaCMS&as>0.1*lambdaCMS&newExpansionParameter>newExpansionParameter*lambdaCMS
 diff_lambda_power=1

This controls by which power to divide the difference term . The default is of course equal to one so as to test but at LO it is sometimes interesting to divide by so a to see if is vanishing as well. This is expected to be the case for all processes (i.e. the plot of has a constant asymptot as well, so that ). For such processes, the higher order contribution is zero by construction and the test is not sensitive to the CMS implementation
 Example: check cms u d~ > e+ ve diff_lambda_power=2
 loop_filter=None

Allows to specify a conditional expression to impose a requirement on the loop diagrams to be kept. This expression can only be an pyton expression of involving the following variables : 'n', the number of loop propagators; 'id' the loop diagram number as it can be read in the postscript generated with the command display diagrams; 'loop_pdgs' the list of absolute values of the PDG of the particles running in the loop; 'loop_masses' and 'struct_masses' the list of the parameter names of the masses running in the loop and the masses of the particles directly attached to the loop.
 Example: loop_filter='n>3'
 Example: loop_filter='n<4 and 6 in loop_pdgs and 3<=id<=7'
 resonances=1

Several kinematic configurations are constructed starting from each of the resonances detected in the process. Given that all resonances are anyway offshell, it is only necessary to run the test on one of these particular kinematic configurations. This option let you chose to instead run the test on the first 'n' such kinematic configurations. Alternatively, one can also specify the resonance(s) one wishes to run on, by specifying tuples '(resonance_PDG,(resonance_mother_numbers))'. Keep in mind however that no matter which kinematic configuration is picked, *all* resonances are tested at the same time, so this option is a bit superfluous as it is completely similar as only changing the seed. Finally, the keyword 'all' means that the checks is run on all kinematic configurations generated starting from each resonance detected.
 Example: resonances=all
 Example: resonances=3
 Example: resonances=(24,(3,4))
 Example: resonances=[(24,(3,4)),(23,(4,5))]
Special option
 analyze=None

This option must be used without any process definition. It is intended to specify the path of pickle files storing the results of previous results to be reanalyzed and replotted. A common usage of this option is:
 Example: check cms analyze=my_default_run.pkl,increased_widths.pkl(Increased_widths),logs_modified.pkl(Inverted_logs),seed_668.pkl(Different_seed)
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