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 DECAYMODEL 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.

FRAGMENTATION: Pythia8
PYTHIA8:
  PARAMETERS:
    - StringZ:aLund: 0.68
    - StringZ:bLund: 0.98
      ...
  DECAYS: false
  SHERPA_MASSES: true
  • 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 continous 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-iterpreted 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 paramter 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 inversemly 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 seleectred 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 pertrubative 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 specified by DECAYMODEL. Its allowed values are either the default choice Hadrons (the default if FRAGMENTATION: Ahadic), Lund (the default if FRAGMENTATION: Lund), or it can be disabled with the option Off.

HADRONS++ is the 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].

Some general switches which relate to hadron decays are

  • DECAYPATH The path to the parameter files for the hadron and tau decays (default: Decaydata/). It is important to note that the path has to be given relative to the current working directory. If it doesn’t exist, the default Decaydata directory (<prefix>/share/SHERPA-MC/Decaydata) will be used.

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

  • SOFT_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. While HADRONS++ determines this dynamically from the chosen decay channel, for Pythia as hadron decay handler its w-cut parameter is employed. Choosing option 2 instead of 1 will only set unstable (decayed) particles off-shell, but leave stable particles on-shell.

  • MAX_PROPER_LIFETIME = [mm] Parameter for maximum proper lifetime (in mm) up to which particles are considered unstable. If specified, this will make long-living particles stable, even if they are set unstable by default or by the user.

Many aspects of the above mentioned “Decaydata” can be adjusted. There exist three levels of data files, which are explained in the following sections. As with all other setup files, the user can either employ the default “Decaydata” in <prefix>/share/SHERPA-MC/Decaydata, or overwrite it (also selectively) by creating the appropriate files in the directory specified by DECAYPATH.

5.12.2.1. HadronDecays.dat

HadronDecays.dat consists of a table of particles that are to be decayed by HADRONS++. Note: Even if decay tables exist for the other particles, only those particles decay that are set unstable, either by default, or in the model/fragmentation settings. It has the following structure, where each line adds one decaying particle:

<kf-code>

<subdirectory>

<filename>.dat

decaying particle

path to decay table

decay table file

default names:

<particle>/

Decays.dat

It is possible to specify different decay tables for the particle (positive kf-code) and anti-particle (negative kf-code). If only one is specified, it will be used for both particle and anti-particle.

If more than one decay table is specified for the same kf-code, these tables will be used in the specified sequence during one event. The first matching particle appearing in the event is decayed according to the first table, and so on until the last table is reached, which will be used for the remaining particles of this kf-code.

Additionally, this file may contain the keyword CREATE_BOOKLET on a separate line, which will cause HADRONS++ to write a LaTeX document containing all decay tables.

5.12.2.2. Decay table files

The decay table contains information about outgoing particles for each channel, its branching ratio and eventually the name of the file that stores parameters for a specific channel. If the latter is not specified HADRONS++ will produce it and modify the decay table file accordingly.

Additionally to the branching ratio, one may specify the error associated with it, and its source. Every hadron is supposed to have its own decay table in its own subdirectory. The structure of a decay table is

{kf1,kf2,kf3,…}

BR(delta BR)[Origin]

<filename>.dat

outgoing particles

branching ratio

decay channel file

It should be stressed here that the branching ratio which is explicitly given for any individual channel in this file is always used regardless of any matrix-element value.

5.12.2.3. Decay channel files

A decay channel file contains various information about that specific decay channel. There are different sections, some of which are optional:

  • <Options>
        AlwaysIntegrate = 0
        CPAsymmetryC = 0.0
        CPAsymmetryS = 0.0
    </Options>
    
    • AlwaysIntegrate = [0,1] For each decay channel, one needs an integration result for unweighting the kinematics (see below). This result is stored in the decay channel file, such that the integration is not needed for each run. The AlwaysIntegrate option allows to bypass the stored integration result, and do the integration nonetheless (same effect as deleting the integration result).

    • CPAsymmetryC/CPAsymmetryS 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. HADRONS++ will then respect these asymmetries between particle and anti-particle in the choice of decay channels.

  • <Phasespace>
      1.0 MyIntegrator1
      0.5 MyIntegrator2
    </Phasespace>
    

    Specifies the phase-space mappings and their weight.

  • <ME>
      1.0 0.0 my_matrix_element[X,X,X,X,X,...]
      1.0 0.0 my_current1[X,X,...] my_current2[X,X,X,...]
    </ME>
    

    Specifies the matrix elements or currents used for the kinematics, their respective weights, and the order in which the particles (momenta) enter them. For more details, the reader is referred to [KLS].

  • <my_matrix_element[X,X,X,X,X,...]>
      parameter1 = value1
      parameter2 = value2
      ...
    </my_matrix_element[X,X,X,X,X,...]>
    

    Each matrix element or current may have an additional section where one can specify needed parameters, e.g. which form factor model to choose. Each parameter has to be specified on a new line as shown above. Available parameters are listed in [KLS]. Parameters not specified get a default value, which might not make sense in specific decay channels. One may also specify often needed parameters in HadronConstants.dat, but they will get overwritten by channel specific parameters, should these exist.

  • <Result>
      3.554e-11 6.956e-14 1.388e-09;
    </Result>
    

    These last three lines have quite an important meaning. If they are missing, HADRONS++ integrates this channel during the initialization and adds the result lines. If this section exists though, and AlwaysIntegrate is off (the default value, see above) then HADRONS++ reads in the maximum for the kinematics unweighting.

    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. There are two ways to enforce the integration: either by deleting the last three lines or by setting AlwaysIntegrate to 1. When a channel is re-integrated, HADRONS++ copies the old decay channel file into .<filename>.dat.old.

5.12.2.4. HadronConstants.dat

HadronConstants.dat may contain some globally needed parameters (e.g. for neutral meson mixing, see [KLS]) and also fall-back values for all matrix-element parameters which one specifies in decay channel files. Here, the Interference_X = 1 switch would enable rate asymmetries due to CP violation in the interference between mixing and decay (cf. Decay channel files), and setting Mixing_X = 1 enables explicit mixing in the event record according to the time evolution of the flavour states. By default, all mixing effects are turned off.

Mixing parameters with some example values

x_K = 0.946
y_K = -0.9965
qoverp2_K = 1.0
Interference_K = 0
Mixing_K = 0

x_D = 0.0
y_D = 0.0
qoverp2_D = 1.0
Interference_D = 0
Mixing_D = 0

x_B = 0.776
y_B = 0.0
qoverp2_B = 1.0
Interference_B = 1
Mixing_B = 0

x_B(s) = 30.0
y_B(s) = 0.155
qoverp2_B(s) = 1.0
Interference_B(s) = 0
Mixing_B(s) = 0

5.12.2.5. Further remarks

Spin correlations: a spin correlation algorithm is implemented. It can be switched on through the setting SOFT_SPIN_CORRELATIONS: 1.

If spin correlations for tau leptons produced in the hard scattering process are supposed to be taken into account, one needs to specify HARD_SPIN_CORRELATIONS: 1 as well. If using AMEGIC++ as ME generator, note that the Process libraries have to be re-created if this is changed.

Adding new channels: if new channels are added to HADRONS++ (choosing isotropic decay kinematics) a new decay table must be defined and the corresponding hadron must be added to HadronDecays.dat. The decay table merely needs to consist of the outgoing particles and branching ratios, i.e. the last column (the one with the decay channel file name) can safely be dropped. By running Sherpa it will automatically produce the decay channel files and write their names in the decay table.

Some details on tau decays: \(\tau\) decays are treated within the HADRONS++ framework, even though the \(\tau\) is not a hadron. As for many hadron decays, the hadronic tau decays have form factor models implemented, for details the reader is referred to [KLS].