# 5.9. Hard decays¶

The handler for decays of particles produced in the hard scattering process (e.g. W, Z, top, Higgs) can be enabled and configured using the HARD_DECAYS collection of settings (and a small number of other other top-level settings). Which (anti)particles IDs should be treated as unstable is determined by the PARTICLE_DATA:<id>:Stable switch described in Models.

The syntax to configure HARD_DECAYS sub-settings is:

HARD_DECAYS:
<sub-setting>: <value>
# more sub-settings ...
Channels:
<channel id>:
<channel sub-setting>: <value>
# more sub-settings for <channel>
# more channels ...


The central setting to enable the hard decays is

HARD_DECAYS:
Enable: true


The channel ID codes are of the form a,b,c,..., where a is the PDG ID of the decaying particle and b,c,... are the decay products. The IDs for the decay channels can also be found in the decay table printed to screen during the run.

This decay module can also be used on top of NLO matrix elements, but it does not include any NLO corrections in the decay matrix elements themselves.

Note that the decay handler is an afterburner at the event generation level. It does not affect the calculation and integration of the hard scattering matrix elements. The cross section is thus unaffected during integration, and the branching ratios (if any decay channels have been disabled) are only taken into account for the event weights and cross section output at the end of event generation (if not disabled with the HARD_DECAYS:Apply_Branching_Ratios option, cf. below). Furthermore any cuts or scale definitions are not affected by decays and operate only on the inclusively produced particles before decays.

## 5.9.1. Status¶

This sub-setting to each channel defined in HARD_DECAYS:Channels allows to explicitly force or disable a decay channel. The status can take the following values:

Status: -1

Decay channel is disabled and does not contribute to total width.

Status: 0

Decay channel is disabled but contributes to total width.

Status: 1 (default)

Decay channel is enabled.

Status: 2

Decay channel is forced.

For example, to disable the hadronic decay channels of the W boson one would use:

HARD_DECAYS:
Channels:
24,2,-1:  { Status: 0 }
24,4,-3:  { Status: 0 }
-24,-2,1: { Status: 0 }
-24,-4,3: { Status: 0 }


In the same way, the bottom decay mode of the Higgs could be forced using:

25,5,-5:  { Status: 2 }


Note that the ordering of the decay products in <channel id> is important and has to be identical to the ordering in the decay table printed to screen. It is also possible to request multiple forced decay channels (Status: 2) for the same particle, all other channels will then automatically be disabled.

## 5.9.2. Width¶

This option allows to overwrite the calculated partial width (in GeV) of a given decay channel, and even to add new inactive channels which contribute to the total width. This is useful to adjust the branching ratios, which are used for the relative contributions of different channels and also influence the cross section during event generation, as well as the total width which is used for the lineshape of the resonance.

An example to set (/add) the partial widths of the H->ff, H->gg and H->yy channels can be seen in the following. The values have been taken from LHC Higgs WG):

PARTICLE_DATA:
25:
Mass: 125.09
Width: 0.0041

HARD_DECAYS:
Enabled: true
Channels:
25,5,-5:    { Width: 2.382E-03 }
25,15,-15:  { Width: 2.565E-04 }
25,13,-13:  { Width: 8.901E-07 }
25,4,-4:    { Width: 1.182E-04 }
25,3,-3:    { Width: 1E-06 }
25,21,21:   { Width: 3.354E-04 }
25,22,22:   { Width: 9.307E-06 }
25,23,22:   { Width: 6.318E-06 }


Another example, setting the leptonic and hadronic decay channels of W and Z bosons to the PDG values, would be specified as follows:

HARD_DECAYS:
Enabled: true
Channels:
24,2,-1:    { Width: 0.7041 }
24,4,-3:    { Width: 0.7041 }
24,12,-11:  { Width: 0.2256 }
24,14,-13:  { Width: 0.2256 }
24,16,-15:  { Width: 0.2256 }
-24,-2,1:   { Width: 0.7041 }
-24,-4,3:   { Width: 0.7041 }
-24,-12,11: { Width: 0.2256 }
-24,-14,13: { Width: 0.2256 }
-24,-16,15: { Width: 0.2256 }
23,1,-1:    { Width: 0.3828 }
23,2,-2:    { Width: 0.2980 }
23,3,-3:    { Width: 0.3828 }
23,4,-4:    { Width: 0.2980 }
23,5,-5:    { Width: 0.3828 }
23,11,-11:  { Width: 0.0840 }
23,12,-12:  { Width: 0.1663 }
23,13,-13:  { Width: 0.0840 }
23,14,-14:  { Width: 0.1663 }
23,15,-15:  { Width: 0.0840 }
23,16,-16:  { Width: 0.1663 }
6,24,5:     { Width: 1.32 }
-6,-24,-5:  { Width: 1.32 }


See also Use_HO_SM_Widths below for a global automatic switch to set these values.

## 5.9.3. Use_HO_SM_Widths¶

The partial decay widths (and thus BRs) calculated and used by the decay handler are only LO accurate. For SM setups, we provide pre-defined decay widths taking higher-order corrections into account. By default (HARD_DECAYS: { Use_HO_SM_Widths: true }) these will overwrite the LO widths with the values given in the Width example above.

## 5.9.4. Spin_Correlations¶

Spin correlations between the hard scattering process and the following decay processes are enabled by default. If you want to disable them, e.g. for spin correlation studies, you can specify the option Spin_Correlations: 0.

## 5.9.5. Store_Results¶

The decay table and partial widths are calculated on-the-fly during the initialization phase of Sherpa from the given model and its particles and interaction vertices. To store these results in the Results/Decays directory, one has to specify HARD_DECAYS: { Store_Results: 1 }. In case existing decay tables are to be read in the same configuration should be done. Please note, that Sherpa will delete decay channels present in the read in results but not in the present model with present parameters by default. To prevent Sherpa from updating the decay table files accordingly specify HARD_DECAYS: { Store_Results: 2 }.

## 5.9.6. Result_Directory¶

Specifies the name of the directory where the decay results are to be stored. Defaults to the value of the top-level setting RESULT_DIRECTORY.

## 5.9.7. Set_Widths¶

The decay handler computes LO partial and total decay widths and generates decays with corresponding branching fractions, independently from the particle widths specified by PARTICLE_DATA:<id>:Width. The latter are relevant only for the core process and should be set to zero for all unstable particles appearing in the core-process final state. This guarantees on-shellness and gauge invariance of the core process, and subsequent decays can be handled by the afterburner. In constrast, PARTICLE_DATA:<id>:Width should be set to the physical width when unstable particles appear (only) as intermediate states in the core process, i.e. when production and decay are handled as a full process or using Decay/DecayOS. In this case, the option HARD_DECAYS: { Set_Widths: true } permits to overwrite the PARTICLE_DATA:<id>:Width values of unstable particles by the LO widths computed by the decay handler.

## 5.9.8. Apply_Branching_Ratios¶

By default (HARD_DECAYS: { Apply_Branching_Ratios: true }), weights for events which involve a hard decay are multiplied with the corresponding branching ratios (if decay channels have been disabled). This also means that the total cross section at the end of the event generation run already includes the appropriate BR factors. If you want to disable that, e.g. because you want to multiply with your own modified BR, you can set the option {HARD_DECAYS: { Apply_Branching_Ratios: false }.

## 5.9.9. Mass_Smearing¶

With the default of HARD_DECAYS: { Mass_Smearing: 1 } the kinematic mass of the unstable propagator is distributed according to a Breit-Wigner shape a posteriori. All matrix elements are still calculated in the narrow-width approximation with onshell particles. Only the kinematics are affected. To keep all intermediate particles onshell {HARD_DECAYS: { Mass_Smearing: 0 }.

## 5.9.10. Resolve_Decays¶

There are different options how to decide when a 1->2 process should be replaced by the respective 1->3 processes built from its decaying daughter particles.

Resolve_Decays: Threshold

(default) Only when the sum of decay product masses exceeds the decayer mass.

Resolve_Decays: ByWidth

As soon as the sum of 1->3 partial widths exceeds the 1->2 partial width.

Resolve_Decays: None

No 1->3 decays are taken into account.

In all cases, one can exclude the replacement of a particle below a given width threshold using Min_Prop_Width: (default 0.0). Both settings are sub-settings of HARD_DECAYS:

HARD_DECAYS:
Resolve_Decays: <mode>
Min_Prop_Width: <threshold>


## 5.9.11. Decay_Tau¶

By default, the tau lepton is decayed by the hadron decay module, Hadron decays, which includes not only the leptonic decay channels but also the hadronic modes. If Decay_Tau: true is specified, the tau lepton will be decayed in the hard decay handler, which only takes leptonic and partonic decay modes into account. Note, that in this case the tau needs to also be set massive:

PARTICLE_DATA:
15:
Massive: true
HARD_DECAYS:
Decay_Tau: true


## 5.9.12. Decay table integration settings¶

Three parameters can be used to steer the accuracy and time consumption of the calculation of the partial widths in the decay table: Int_Accuracy: 0.01 specifies a relative accuracy for the integration. The corresponding target reference is either the given total width of the decaying particle (Int_Target_Mode: 0, default) or the calculated partial decay width (Int_Target_Mode: 1). The option Int_NIter: 2500 can be used to change the number of points per integration iteration, and thus also the minimal number of points to be used in an integration. All decay table integration settings are sub-settings of HARD_DECAYS.

## 5.9.13. Simulation of polarized cross sections for intermediate particles¶

In this chapter it is described how Sherpa can be used to simulate polarized cross sections for unstable intermediate state particles. At the moment, only the simulation of polarized cross sections for massive vector bosons is supported. Sherpa can simulate all polarized cross sections in one simulation run. The polarized cross sections are handled during event generation and printed out as additional event weights similar to variation weights. By default, the cross sections for all polarization combinations of the intermediate particles are printed out. For massive vector bosons also all transverse weights are calculated automatically. Beside this, user-specified weights can be determined which is described in section Custom polarized cross sections. Weight names for automatically provided weights have the form PolWeight_ReferenceSystem.particle1.helicity1_particle2.helicity2... e.g. for two W+-bosons PolWeight.W+.+_W+.-. Ordering of the particles in the weight name corresponds to Sherpa’s internal particle ordering which can be read off from the ordering in the process printed out when Sherpa starts running. More details about the definition of polarization for intermediate vector bosons and the implementation can be found in .

### 5.9.13.1. General procedure¶

The definition of polarization for particles in intermediate states is only possible for processes which can be factorized into a production matrix element and decay matrix elements of them. To neglect possible non-resonant processes, for which this factorization is not possible, Sherpa applies an extended narrow-width approximation. All intermediate particles are considered as on shell but all spin correlations are preserved. The production process is specified in the Processes part whereas the possible decays are characterized in the Hard decays section. Details about PROCESSES and HARD_DECAYS definition are described in the corresponding chapters of this manual. The following example shows PROCESSES and HARD_DECAYS definition of the same-sign $$W^+ W^+$$-scattering with the W decaying to electrons or muons.

PARTICLE_DATA:
24:
Width: 0

HARD_DECAYS:
Enabled: true
Mass_Smearing: 1
Channels:
24,12,-11: {Status: 2}
24,14,-13: {Status: 2}

PROCESSES:
- 93 93 -> 24 24 93 93:
Order: {QCD: 0, EW: 4}


Things to notice:

• In PARTICLE_DATA the Width of the intermediate particles must be set to zero since they are handled as stable for the hard process matrix element calculation. The particles are then decayed by the internal (hard) decay module.

• Spin correlations must be enabled during the hard decays (which is the default).

The central setting to enable the calculation of polarized cross sections is:

HARD_DECAYS:
Pol_Cross_Section:
Enabled: true


The polarization vectors of massive vector bosons are implemented according to [Dit99], equation (3.19). Specifically, the polarization vectors are expressed in terms of Weyl spinors. For that, an arbitrary light-like vector needs to be chosen. The definition of vector boson polarization is not unambiguous. It can be specified by the options described in the following sections: Pol_Cross_Section:Spin_Basis and Pol_Cross_Section:Reference_System.

### 5.9.13.2. Spin basis¶

For massive vector bosons the choice of a light-like vector for their description in the Weyl spinor formalism is not really arbitrary since it characterizes the spin axis chosen to define the polarization. By default, the reference vector is selected such that polarization vectors are expressed in the helicity basis since this is the common choice for vector boson polarization. The polarization vectors are then eigenvectors of the helicity operator and have the same form as in (3.15) in [Dit99] after transformation from spinor to vector representation. Sherpa provides several gauge choices for the Weyl spinors. To really get the polarization vectors in this form, the following spinor gauge choice must be chosen:

COMIX_DEFAULT_GAUGE: 0

HARD_DECAYS:
Pol_Cross_Section:
Enabled: true
Spin_Basis: Helicity


If Spin_Basis: ComixDefault is selected the COMIX default reference vector specified by COMIX_DEFAULT_GAUGE (default 1 which corresponds to (1.0, 0.0, 1/$$\sqrt{2}$$, 1/$$\sqrt{2}$$)) is used. Furthermore, it is possible to hand over any constant reference vector:

HARD_DECAYS:
Pol_Cross_Section:
Enabled: true
Spin_Basis: 1.0, 0.0, 1.0, 0.0


### 5.9.13.3. Reference system¶

The helicity of a massive particle is not Lorentz invariant. Therefore a reference system needs to be chosen to define vector boson polarization unambiguous. Sherpa supports the following options:

Reference_System: Lab (default)

Vector boson polarization is defined in the laboratory frame.

Reference_System: COM

Vector boson polarization is defined in the center of mass system of all hard-decaying particles.

Reference_System: PPFr

Vector boson polarization is defined in the center of mass system of the two interacting partons.

Reference_System: RestFrames

Vector boson polarization for each hard decaying particle is defined in its own center of mass system. If the helicity basis is selected as spin basis the spin axis lies along the flight direction of the hard decaying particle in the laboratory frame.

Sherpa allows the calculation of polarized cross sections for different polarization definitions specified by different reference systems in one simulation run:

HARD_DECAYS:
Pol_Cross_Section:
Enabled: true
Reference_System: [Lab, COM]


Additionally to the options explained above, each reference system defined by one or several hard process initial or final state particles can be used. This can be specified by the particle numbers of the desired particles according to the Sherpa numbering scheme. Distinct particle numbers should only be separated by a single white space, at least if more than one reference system is specified. The second reference frame in the following example is the parton-parton rest frame.

HARD_DECAYS:
Pol_Cross_Section:
Enabled: true
Reference_System: [Lab, 0 1]


In the Sherpa event output, polarized cross sections of vector bosons defined in different frames are distinguished by adding the reference frame to the weight names, e.g. PolWeight_Lab.W+.+_W+.-. For reference systems defined by particle numbers, refsystemn is added to avoid commas in weight names. n is the place in the reference system list specified in the YAML-File starting at 0. For the example above this means e.g. PolWeight_refsystem1.W+.+_W+.-.

### 5.9.13.4. Custom polarized cross sections¶

Sherpa provides the calculation of two different types of custom polarized cross sections. On the one hand, it is possible to hand over a comma separated list of weight names from the automatically calculated cross sections. These cross sections are then added by Sherpa and printed out as additional event weight. On the other hand, partially unpolarized cross sections can be calculated. These can be specified by the numbers of the particles which should be considered as unpolarized. Also here the numbering of the particles is according to the Sherpa numbering scheme. Custom weights are generally specified by Weight. By adding numbers to Weight e.g. Weight1, Weight2 … more than one custom cross section can be calculated. The number is limited to Weight10 by default but can be increased by using Number_Of_Custom_Weights. In the following example the W- is considered as unpolarized:

HARD_DECAYS:
Enabled: true
Mass_Smearing: 1
Channels:
24,12,-11: {Status: 2}
-24,-14,13: {Status: 2}
Pol_Cross_Section:
Enabled: true
Weight1: W+.+_W-.+, W+.-_W-.+
Weight2: 3

PROCESSES:
- 93 93 -> 24 -24 93 93:
Order: {QCD: 0, EW: 4}


In the case of partially unpolarized cross sections the helicity of the unpolarized particle is set to u in the weight names, e.g. PolWeight_Lab.W+.+_W-.u. For custom cross sections specified by weight names PolWeight_refsystem. Weightn is used instead to avoid long weight names. Hereby, Weightn corresponds to the corresponding setting in the YAML-File.