5.12. Hadronization

The hadronization setup covers the fragmentation of partons into primordial hadrons as well as the decays of unstable hadrons into stable final states.

5.12.1. Fragmentation

5.12.1.1. Fragmentation models

The FRAGMENTATION parameter sets the fragmentation module to be employed during event generation.

• The default is Ahadic, enabling Sherpa’s native hadronization model AHADIC++, based on the cluster fragmentation model introduced in [FW83], [Web84], [GM87], and [MW88] and implementing some modifications discussed in [WKS04].

• The hadronization can be disabled with the value None.

• To evaluate uncertainties stemming from the hadronization, Sherpa also provides an interface to the Lund string fragmentation in Pythia 8.3 [SAC+15] by using the setting Pythia8. In this case, the standard Pythia settings can be used to steer the behaviour of the Lund string, see [SAC+15]. They are specified in their usual form in Pythia in a dedicated settings block. Additionally a choice can be made to let Pythia directly handle hadron decays via the DECAYS setting (separate from the Model switch mentioned below) and whether Pythias or Sherpas default masses and widths should be used through the SHERPA_MASSES setting. By default the choice of generator for the masses and widths setting aligns with the decay setting.

SHERPA_LDADD: SherpaPythia
FRAGMENTATION: Pythia8
PYTHIA8:
  PARAMETERS:
    - StringZ:aLund: 0.68
    - StringZ:bLund: 0.98
      ...
  DECAYS: true
  SHERPA_MASSES: false

• Alternatively, Sherpa also provides an interface to Pythia 6.4 [SMS06] by using the setting Lund. In this case, the standard Pythia switches MSTJ, MSTP, MSTU, PARP, PARJ and PARU can be used to steer the behaviour of the Lund string, see [SMS06]. They can be specified as a 2xN matrix:

FRAGMENTATION: Lund
MSTJ:
- [<number1>, <value1>]
- [<number2>, <value2>]
  ...
MSTP:
- [<number1>, <value1>]
  ...

5.12.1.2. Hadron constituents

The constituent masses of the quarks and diquarks are given by

• M_UP_DOWN (0.3 GeV),

• M_STRANGE (0.4 GeV),

• M_CHARM (1.8 GeV), and

• M_BOTTOM (5.1 GeV).

The diquark masses are composed of the quark masses and some additional parameters,

with

• M_DIQUARK_OFFSET (0.3 GeV),

• M_BIND_0 (0.12 GeV), and

• M_BIND_1 (0.5 GeV).

Like all settings related to cluster fragmentation these are grouped under AHADIC.

AHADIC:
  - M_UP_DOWN: 0.3
    ...
  - M_DIQUARK_OFFSET: 0.3

5.12.1.3. Hadron multiplets

For the selection of hadrons emerging in such cluster transitions and decays, an overlap between the cluster flavour content and the flavour part of the hadronic wave function is formed. This may be further modified by production probabilities, organised by multiplet and given by the parameters

• MULTI_WEIGHT_R0L0_PSEUDOSCALARS (default 1.0),

• MULTI_WEIGHT_R0L0_VECTORS (default 1.0),

• MULTI_WEIGHT_R0L0_TENSORS2 (default 0.75),

• MULTI_WEIGHT_R0L1_SCALARS (default 0.0),

• MULTI_WEIGHT_R0L1_AXIALVECTORS (default 0.0),

• MULTI_WEIGHT_R0L2_VECTORS (default 0.0),

• MULTI_WEIGHT_R0L0_N_1/2 (default 1.0),

• MULTI_WEIGHT_R1L0_N_1/2 (default 0.0),

• MULTI_WEIGHT_R2L0_N_1/2 (default 0.0),

• MULTI_WEIGHT_R1_1L0_N_1/2 (default 0.0),

• MULTI_WEIGHT_R0L0_DELTA_3/2 (default 0.25),

In addition, there is a suppression factors applied to meson singlets,

• SINGLET_SUPPRESSION (default 1.0).

For the latter, Sherpa also allows to redefine the mixing angles through parameters such as

• Mixing_0+ (default -14.1/180*M_PI),

• Mixing_1- (default 36.4/180*M_PI),

• Mixing_2+ (default 27.0/180*M_PI),

• Mixing_3- (default 0.5411),

• Mixing_4+ (default 0.6283),

And finally, some modifiers are applied to individual hadrons:

• ETA_MODIFIER (default 0.12),

• ETA_PRIME_MODIFIER (default 1.0),

5.12.1.4. Cluster transition to hadrons - flavour part

The phase space effects due to these masses govern to a large extent the flavour content of the non-perturbative gluon splittings at the end of the parton shower and in the decay of clusters. They are further modified by relative probabilities with respect to the production of up/down flavours through the parameters

• STRANGE_FRACTION (default 0.42),

• BARYON_FRACTION (default 1.0),

• CHARM_BARYON_MODIFIER (default 1.0),

• BEAUTY_BARYON_MODIFIER (default 1.0),

• P_{QS/P_{QQ}} (default 0.2),

• P_{SS/P_{QQ}} (default 0.04), and

• P_{QQ_1/P_{QQ_0}} (default 0.20).

The transition of clusters to hadrons is governed by the following considerations:

• Clusters can be interpreted as excited hadrons, with a continuous mass spectrum.

• When a cluster becomes sufficiently light such that its mass is below the largest mass of any hadron with the same flavour content, it must be re-interpreted as such a hadron. In this case it will be shifted on the corresponding hadron mass, and the recoil will be distributed to the “neighbouring” clusters or by emitting a soft photon. This comparison of masses clearly depends on the multiplets switched on in AHADIC++.

• In addition, clusters may becomes sufficiently light such that they should decay directly into two hadrons instead of two clusters. This decision is based on the heaviest hadrons accessible in a decay, modulated by another offset parameter,

  • DECAY_THRESHOLD (default 500 MeV).

• If both options, transition and decay, are available, there is a competition between

5.12.1.5. Cluster transition and decay weights

The probability for a cluster C to be transformed into a hadron H is given by a combination of weights, obtained from the overlap with the flavour part of the hadronic wave function, the relative weight of the corresponding multiplet and a kinematic weight taking into account the mass difference of cluster and hadron and the width of the latter.

For the direct decay of a cluster into two hadrons the overlaps with the wave functions of all hadrons, their respective multiplet suppression weights, the flavour weight for the creation of the new flavour q and a kinematical factor are relevant. Here, yet another tuning parameter enters,

• MASS_EXPONENT (default 4.0)

which partially compensates phase space effects favouring light hadrons,

5.12.1.6. Cluster decays - kinematics

Cluster decays are generated by firstly emitting a non-perturbative “gluon” from one of the quarks, using a transverse momentum distribution as in the non-perturbative gluon decays, see below, and by then splitting this gluon into a quark–antiquark of anti-diquark–diquark pair, again with the same kinematics. In the first of these splittings, the emission of the gluon, though, the energy distribution of the gluon is given by the quark splitting function, if this quark has been produced in the perturbative phase of the event. If, in contrast, the quark stems from a cluster decay, the energy of the gluon is selected according to a flat distribution.

In clusters decaying to hadrons, the transverse momentum is chosen according to a distribution given by an infrared-continued strong coupling and a term inversely proportional to the infrared-modified transverse momentum,

constrained to be below a maximal transverse momentum.

5.12.1.7. Splitting kinematics

In each splitting, the kinematics is given by the transverse momentum, the energy splitting parameter and the azimuthal angle. The latter, the azimuthal angle is always selected according to a flat distribution, while the energy splitting parameter will either be chosen according to the quark-to-gluon splitting function (if the quark is a leading quark, i.e. produced in the perturbative phase), to the gluon-to-quark splitting function, or according to a flat distribution. The transverse momentum is given by the same distribution as in the cluster decays to hadrons.

5.12.2. Hadron decays

The treatment of hadron and tau decays is steered by the parameters in a block named HADRON_DECAYS, e.g.

HADRON_DECAYS:
  Model: HADRONS++
  Max_Proper_Lifetime: 10.0
  QED_Corrections: 1

• Hadron properties like mass, width, and active can be set in full analogy to the settings for fundamental particles using PARTICLE_DATA, cf. Models.

• Max_Proper_Lifetime: [mm] (default: 10.0) Parameter for maximum proper lifetime (in mm) up to which hadrons are considered unstable. This will make long-living particles stable, even if they are set unstable by default or by the user. If you do not want to set this globally, set this to a value of -1 and steer the stability through PARTICLE_DATA:<id>:Stable, cf. Models.

• QED_Corrections: [0,1] (default: 1) Whether to dress hadron decays with QED corrections.

• Model: [HADRONS++, Lund, Off] (default: HADRONS++) It defaults to Hadrons to employ Sherpa’s built-in hadron decay module HADRONS++ described below. Alternatively one can select Lund (the default if FRAGMENTATION: Lund) to use the interface to Pythia6. Another option is to use the hadron decays from Pythia8 directly in the corresponding hadronisation interface, cf. Fragmentation above. To disable hadron decays completely, it can be disabled with the option Off.

HADRONS++ is the built-in module within the Sherpa framework which is responsible for treating hadron and tau decays. It contains decay tables with branching ratios for approximately 2500 decay channels, of which many have their kinematics modelled according to a matrix element with corresponding form factors. Especially decays of the tau lepton and heavy mesons have form factor models similar to dedicated codes like Tauola [JWDK93] and EvtGen [Lan01].

Its settings are also steered within the HADRON_DECAYS block as follows:

• Mass_Smearing: [0,1,2] (default: 1) Determines whether particles entering the hadron decay event phase should be put off-shell according to their mass distribution. It is taken care that no decay mode is suppressed by a potentially too low mass. HADRONS++ determines this dynamically from the chosen decay channel. Choosing option 2 instead of 1 will only set unstable (decayed) particles off-shell, but leave stable particles on-shell.

• Spin_Correlations: [0,1] (default: 0) A spin correlation algorithm is implemented and can be switched on with this setting. This might slow down event generation slightly.

• Channels: Many aspects of the decay tables and individual decay channels can be adjusted within this sub-block. The default settings of the Sherpa hadron decay data can be found in <prefix>/share/SHERPA-MC/Decaydata.yaml and can be overwritten individually in the run card, e.g. as follows:

  HADRON_DECAYS:
    Channels:
      111:
        22,22:
          BR: [0.98823, 0.00034]
          Origin: PDG2023
      15:
        16,-12,11:
          BR: [0.1782, 0.0004]
          Status: [1, 2, 1]
  

  The levels are structured first by decaying particle and then by decay products. For each decay channel the following settings are available:

  • BR: [<br>, <deltabr>] branching ratio and its uncertainty

  • Origin: <...> origin of BR for documentation purposes

  • Status: TODO

  • ME: lists the matrix elements used for the decay kinematics and the permutation that maps the external momenta of the decay into the internal convention in the ME implementation. Additionally, parameters for the ME calculation can be specified. Example:

    HADRON_DECAYS:
      Channels:
        521:
          321,11,-11:
            BR: [5.5e-07, 7e-08]
            Origin: PDG
            ME:
              - B_K_Semileptonic[0,1,2,3]:
                  Factor: [1.0, 0.0]
                  LD: 0
                  C1: -0.248
                  C2: 1.107
                  C3: 0.011
                  C4: -0.026
                  C5: 0.007
                  C6: -0.031
                  C7eff: -0.313
                  C9: 4.344
                  C10: -4.669
    

    If no ME information is specified, Sherpa will fall back to a generic matrix element based on the spins of the external particles.

    One special type of ME used very often is Current_ME which corresponds to the contraction of two (V-A) currents that then have to be specified separately and can contain form factors etc. This structure allows to combine known currents flexibly without needing to implement a dedicated ME for each of these decays. Examples are semileptonic B/D-decays which can contain a leptonic current and a hadronic one or tau decays which can contain either two leptonic currents or also one hadronic one. Syntax example:

    HADRON_DECAYS:
      Channels:
        521:
          -423,12,-11:
            BR: [0.0558, 0.0022]
            Origin: PDG2022
            ME:
              - Current_ME:
                  J1:
                    Type: VA_F_F
                    Indices: [2,3]
                  J2:
                    Type: VA_P_V
                    Indices: [0,1]
                    FORM_FACTOR: 3
    
          -411,211,12,-11:
            BR: [0.0002, 0.0002]
            Origin: FS
            ME:
              - Current_ME:
                  J1:
                    Type: VA_F_F
                    Indices: [3,4]
                  J2:
                    Type: VA_B_DPi
                    Indices: [0,1,2]
                    Vxx: 0.04
    

  • PhaseSpace lists the phase-space mappings and optionally their (relative) weights. Example:

    PhaseSpace:
      - TwoResonances_a(1)(1260)+_2_rho(770)+_13:
          Weight: 0.5
      - TwoResonances_a(1)(1260)+_3_rho(770)+_12:
          Weight: 0.5
    

  • CPAsymmetryS: For CP violation in the interference between mixing and decay, cf. below.

  • CPAsymmetryC: For CP violation in the interference between mixing and decay, cf. below.

  • IntResults: This line stores the results from the phase space integration of the decay channel (width, MC uncertainty, maximum for unweighting). If they are missing, HADRONS++ integrates this channel during the initialization.

    Consequently, if some parameters are changed (also masses of incoming and outgoing particles) the maximum might change such that a new integration is needed in order to obtain correct kinematical distributions. In this case the IntResults line should be removed and replaced by the new one printed out to screen after integration.

• Constants Some globally used constants

• Aliases Create alias particles, e.g. to enforce specific decay chains. Example:

  HADRON_DECAYS:
    Aliases:
      999521: 521
  
    Channels:
      300553:
        999521,-999521:
          BR: 0.5
          [...]
        511,-511:
          BR: 0.5
          [...]
  
      999521:
        -423,12,-11:
          BR: [0.0558, 0.0022]
          Status: 2
          [...]
  

• Mixing: This block contains globally needed parameters for neutral meson mixing. Setting Mixing_<...> = 1 enables explicit mixing in the event record according to the time evolution of the flavour states. The Interference_X = 1 switch would enable rate asymmetries due to CP violation in the interference between mixing and decay (cf. CPAsymmetry settings below). By default, the mixing parameters are set to the following values:

  HADRON_DECAYS:
    Mixing:
      Mixing_D: 1
      Interference_D: 0
      x_D: 0.0032
      y_D: 0.0069
      qoverp2_D: 1.0
  
      Mixing_B: 1
      Interference_B: 0
      x_B: 0.770
      y_B: 0.0
      qoverp2_B: 1.0
  
      Mixing_B(s): 1
      Interference_B(s): 0
      x_B(s): 26.72
      y_B(s): 0.130
      qoverp2_B(s): 1.0
  

  If one wants to include time dependent CP asymmetries through interference between mixing and decay one can set the coefficients of the cos and sin terms respectively for each decay channel as described above (CPAsymmetryS/C). HADRONS++ will then respect these asymmetries between particle and anti-particle in the choice of decay channels.

• Partonics Some partonic decay tables (for c and b) that will be used to complement the decay table of hadrons if they don’t contain 100% BR and have spectators specified in their own setup like:

  521:
    Spectators: [ 2: { Weight: 1.0 } ]
  

• CreateBooklet: true to create a Latex booklet of all decay channels read in.