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:
  Enabled: 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 contrast, 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 is std::numeric_limits<double>::min()). 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 ¶

This sections documents 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 (VBs) is supported. Sherpa can simulate polarized cross sections of all possible polarization combinations 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 output. In addition to that, an additional weight is added describing the totaled interference between different polarizations of the current event. For massive VBs also all transversely polarized cross sections are calculated automatically. Sherpa supports two different definitions of transversely polarized cross sections, for details see section Transversely polarized cross sections. Beside this, user-specified cross sections can be produced as described in section Custom polarized cross sections. Weight names for automatically provided cross sections have the form PolWeight_ReferenceSystem.particle1.polarization1_particle2.polarization2... with + denoting right(plus)-, - left-handed (minus) and 0 longitudinal polarization. For a right(+)-handed \(\mathrm{W}^+\) boson and left(-)-handed \(\mathrm{W}^+\) boson in \(\mathrm{W}^+\mathrm{W}^+\) scattering, the weight name becomes PolWeight_ReferenceSystem.W+.+_W+.-. The sequence of the particles in the weight name corresponds to Sherpa’s internal particle ordering which can be obtained from the ordering in the process printed out when Sherpa starts running. The ReferenceSystem denotes the reference system which needs to be specified for an unambitious polarization definition (cf. section Reference system). The totaled interference contribution is called PolWeight_ReferenceSystem.int.

Polarized cross sections in SHERPA can currently be calculated at fixed leading order, LO+PS and in merged calculations. Furthermore, polarized NLO QCD corrections on the VB production part (not on the decays) can be simulated approximately by neglecting effects of virtual corrections as well as ultra-soft and ultra-collinear contributions below the parton shower IR cut-off on polarization fractions. This is currently only possible on particle level using SHERPA’s MC@NLO implementation for matching NLO hard matrix elements to the parton shower. Note that the resulting unpolarized prediction which is also used to compute the polarized cross sections from the polarization fractions contains all NLO QCD corrections.

More details about the definition of polarization for intermediate VBs and the implementation in Sherpa than covered by this manual entry can be found in [HSchonherrS24].

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 and decay of them. To neglect possible not-fully-resonant diagrams (i.e. diagrams where not each final state decay product particle comes from the decay of a resonant intermediate particle), for which this factorization and the definition of polarization for intermediate particles are 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 part of the process is specified in the Processes part of the run card whereas the possible decays are characterized in the Hard decays section. Details about PROCESSES and HARD_DECAYS definition are described in the corresponding sections of this manual. The following example shows PROCESSES and HARD_DECAYS definition of the same-sign \(\mathrm{W}^+ \mathrm{W}^+\) scattering with the \(\mathrm{W}^+\) boson decaying to electrons or muons.

PARTICLE_DATA:
  24:
    Width: 0
  23:
    Width: 0

WIDTH_SCHEME: Fixed

HARD_DECAYS:
  Enabled: true
  Spin_Correlations: 1  # can be omitted (default)
  Mass_Smearing: 1      # can be omitted (default)
  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. If VBs are considered as intermediate particles, the width of all VBs must be set to zero to preserve SU(2) Ward-Identities. This also holds true for processes where only one VB type participates as intermediate particle (e.g. same-sign \(\mathrm{W}^\pm \mathrm{W}^\pm\) scattering process).
For the calculation of polarized cross sections, spin correlations between production and decay of the intermediate particles must be enabled (which is the default).
Enabling the smearing of the mass of the intermediate VBs according to a Breit-Wigner distribution improves the applied spin-correlated narrow-width approximation by also retaining some of the off- shell effects, details cf. Hard decays section (corresponding setting can be omitted since it is the default).
WIDTH_SCHEME is set to Fixed to be consistent with setting all VB widths to zero.

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

HARD_DECAYS:
  Pol_Cross_Section:
    Enabled: true

The polarization vectors of massive VBs 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 VB polarization is not unambiguous. It can be specified by the following options described in the subsequent sections: Pol_Cross_Section:Spin_Basis and Pol_Cross_Section:Reference_System.

5.9.13.2. Spin basis ¶

For massive particles 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 VB polarization:

HARD_DECAYS:
  Pol_Cross_Section:
    Enabled: true
    Spin_Basis: Helicity

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 the described form, the following spinor gauge choice must be chosen:

COMIX_DEFAULT_GAUGE: 0

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/\(\sqrt{2}\))) will be 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 its polarization unambiguously. Sherpa supports the following options:

Reference_System: Lab (default): Particle polarization is defined in the laboratory frame.
Reference_System: COM: Particle polarization is defined in the center of mass system of all hard-decaying particles.
Reference_System: PPFr: Particle polarization is defined in the center of mass system of the two interacting partons.

It is possible to obtain polarized cross sections for several different polarization definitions (differing in the reference systems chosen) with a single simulation run:

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

Additionally to the options explained above, any 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 VBs defined in different frames are distinguished by adding the corresponding reference frame keyword 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. Transversely polarized cross sections ¶

If some of the intermediate particles are VBs, also transversely polarized cross sections are output per default. Sherpa provides two different definitions of transversely polarized cross sections which can be selected by Transverse_Weights_Mode:

Transverse_Weights_Mode: 0: Transversely polarized cross sections result from adding the left(-)- and right(+)-handed polarized contribution (= incoherent definition). Transverse polarized particles are characterized by a small t in corresponding weight names.
Transverse_Weights_Mode: 1 (default): Transversely polarized cross sections result from adding the left(-)- and right(+)-handed polarized contribution as well as left-right interference terms (= coherent definition). Transverse polarized particles are characterized by a capital T in corresponding weight names. If this definition of the transverse polarized signals is chosen also a new interference weight is added containing the interference terms which are not in included in one of the transverse polarized weights. To distinguish it from the original interference weight, it is referred to as PolWeight_ReferenceSystem.coint.
Transverse_Weights_Mode: 2: Both, incoherently and coherently defined transverse polarized cross sections are simulated.

5.9.13.5. 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 specify a comma separated list of weight names from the automatically calculated cross sections. Corresponding cross sections are then added by Sherpa and printed out as additional event weights. On the other hand, partially unpolarized cross sections can be calculated. Those can be specified by the numbers of the particles which should be considered as unpolarized in the run card. Again, the numbering of the particles follows the Sherpa numbering scheme. Noteworthy, the partially unpolarized cross sections also contain contributions from terms describing the interference between different polarizations of the unpolarized intermediate particles. Therefore, an additional interference weight is added to the output describing the sum of the remaining interference contributions. If particles remaining polarized are massive VBs, Sherpa also output transversely polarized cross sections for the partially unpolarized cross sections.

Custom weights are generally specified by the option Weight in the run card. 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 \(\mathrm{W}^-\) boson is considered as unpolarized, Weight1 shows an example for how to specify polarized cross sections which should be added to a new weight. The result here would be the same as W+.t_W-.0.

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-.0, W+.-_W-.0
   Weight2: 3

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

The weight naming pattern is adjusted for custom polarized cross sections. The polarization of the unpolarized particles in the weight names is set to U and their spin labels are moved to the beginning of the weight name. If more than one intermediate particle is considered as unpolarized, the particle ordering among the unpolarized particles is preserved. The weight name is then prefixed by the name of the custom weight as specified in the run card. Thus, for the single-polarized cross section of a right-handed \(\mathrm{W}^+\) boson in opposite sign \(\mathrm{W}^+\mathrm{W}^-\) production with polarization defined in the laboratory frame (named Weight2 in the example run card above), the final weight name becomes PolWeight_Lab.Weight1_W-.U_W+.+. For custom cross sections specified by weight names (e.g. Weight1 in the example above), PolWeight_refsystem.Weightn is used instead to avoid long weight names. Hereby, Weightn corresponds to the corresponding setting in the run card.

The weight name syntax can also be used if a single or a sum of certain interference terms are of interest. Interference weight names have two instead of one polarization index per particle (first index stands for the polarization of the particle in the corresponding matrix element, the second index for its polarization in the complex conjugate matrix element). The example below is an excerpt of a run card for the simulation of polarized cross sections for single \(\mathrm{W}^+\) boson production in association with one jet at LO; Weight1 and Weight2 illustrate how single interference cross sections (Weight2) or a sum of selected ones (Weight1) can be printed out. Weight1 leads to the same result as W+.T (coherent transverse polarization definition) which is calculated automatically if Transverse_Weights_Mode: 1 or Transverse_Weights_Mode: 2.

HARD_DECAYS:
 Enabled: true
 Channels:
  24,12,-11: {Status: 2}
 Pol_Cross_Section:
   Enabled: true
   Weight1: W+.++, W+.+-, W+.-+, W+.--
   Weight2: W+.0+

PROCESSES:
 - 93 93 -> 24 93:
   Order: {QCD: 1, EW: 1}