## Resonance Decays

The `ResonanceDecays` class performs the sequential decays of all resonances formed in the hard process. Note the important distinction between "resonances" and other "particles" made in PYTHIA.

- The list of resonances contains gamma^*/Z^0, W^+-, top, the Higgs, and essentially all new particles of Beyond-the-Standard-Model physics: further Higgs bosons, sfermions, gauginos, techniparticles, and so on. The partial widths to different decay channels are perturbatively calculable, given the parameters of the respective model, and branching ratios may be allowed to vary across a (reasonably broad) resonance peak. Usually resonances are short-lived, and therefore it makes sense to consider their decays immediately after the primary hard process has been set up. Furthermore, in several cases the decay angular distributions are encoded as part of the specific process, e.g. the W decays differently in f fbar → W^+-, f fbar → W^+ W^- and h^0 → W^+ W^- . All of these particles are (in PYTHIA) only produced as part of the hard process itself, i.e. they are not produced in showers or hadronization processes. Therefore the restriction to specific decay channels can be consistently taken into account as a corresponding reduction in the cross section of a process. Finally, note that all of these resonances have an on-shell mass above 20 GeV, with the exception of some hypothetical weakly interacting and stable particles such as the gravitino.
- The other particles include normal hadrons and the Standard-Model leptons, including the tau^+-. These can be produced in the normal hadronization and decay description, which involve unknown nonperturbative parameters and multistep chains that cannot be predicted beforehand: a hard process like g g → g g can develop a shower with a g → b bbar branching, where the b hadronizes to a B^0bar that oscillates to a B^0 that decays to a tau^+. Therefore any change of branching ratios - most of which are determined from data rather than from first principles anyway - will not be taken into account in the cross section of a process. Exceptions exist, but most particles in this class are made to decay isotropically. Finally, note that all of these particles have a mass below 20 GeV.

There is one ambiguous case in this classification, namely the photon. The gamma^*/Z^0 combination contains a low-mass peak when produced in a hard process. On the other hand, photons can participate in shower evolution, and therefore a photon originally assumed massless can be assigned an arbitrarily high mass when it is allowed to branch into a fermion pair. In some cases this could lead to double-counting, e.g. between processes such as f fbar → (gamma^*/Z^0) (gamma^*/Z^0), f fbar → (gamma^*/Z^0) gamma and f fbar → gamma gamma. Here it make sense to limit the lower mass allowed for the gamma^*/Z^0 combination, in `23:mMin`, to be the same as the upper limit allowed for an off-shell photon in the shower evolution, in `TimeShower:mMaxGamma`. By default this matching is done at 10 GeV.

In spite of the above-mentioned differences, the resonances and the other particles are all stored in one common particle data table, so as to offer a uniform interface to setting and getting properties such as name, mass, charge and decay modes, also for the particle properties in the event record. Some methods are specific to resonances, however, in particular for the calculation of partial widths and thereby of branching ratio. For resonances these can be calculated dynamically, set up at initialization for the nominal mass and then updated to the current mass when these are picked according to a Breit-Wigner resonance shape.

### Resonance Decays and Cross Sections

As already hinted above, you have the possibility to set the allowed decay channels of resonances, see Particle Data Scheme description. For instance, if you study the process q qbar → H^0 Z^0 you could specify that the Z^0 should decay only to lepton pairs, the H^0 only to W^+ W^-, the W^+ only to a muon and a neutrino, while the W^- can decay to anything. Unfortunately there are limits to the flexibility: you cannot set a resonance to have different properties in different places of a process, e.g. if instead H^0 → Z^0 Z^0 in the above process then the three Z^0's would all obey the same rules.

The restrictions on the allowed final states of a process is directly reflected in the cross section of it. That is, if some final states are excluded then the cross section is reduced accordingly. Such restrictions are built up recursively in cases of sequential decay chains. The restrictions are also reflected in the compositions of those events that actually do get to be generated. For instance, the relative rates of H^0 → W^+ W^- and H^0 → Z^0 Z^0 are shifted when the allowed sets of W^+- and Z^0 decay channels are changed.

We remind that only those particles that Pythia treat as resonances enjoy this property, and only those that are considered as part of the hard process and its associated resonance decays.

There is one key restriction on resonances:

`parm` ResonanceWidths:minWidth (`default = 1e-20`; `minimum = 1e-30`)
Minimal allowed width of a resonance, in GeV. If the width falls below this number the resonance is considered stable and will not be allowed to decay. This is mainly intended as a technical parameter, to avoid disasters in cases where no open decay channels exists at all. It could be used for real-life decisions as well, however, but then typically would have to be much bigger than the default value. Special caution would be needed if coloured resonance particles were made stable, since the program would not necessarily know how to hadronize them, and therefore fail at that stage.

In spite of this technical parameter choice, it is possible to set a lifetime for a resonance, and thereby to obtain displaced vertices. If a resonance is allowed to decay it will do so, irrespective of the location of the decay vertex. This is unlike normal particle decays, where it is possible to define some region around the primary vertex within which all decays should happen, with particles leaving that region considered stable. The logic is that resonances as a rule are too short-lived for secondary vertices, so if you pick a scenario with a long-lived but unstable resonance it is because you want to study secondary vertices. How to interface those decays to a detector simulation program then is another story, to be solved separately. Do note that a special treatment is needed for coloured long-lived resonances, that form R-hadrons, and where charge and flavour may change between the production and decay vertices.

### Special properties and methods for resonances

The method `ParticleData::isResonance(id)` allows you to query whether a given particle species is considered a resonance or not. You can also change the default value of this flag in the normal way, e.g. `pythia.readString("id:isResonance = true")`.

Resonances come in two kinds.

- The standard built-in ones include Z^0, W^+-, t, h^0, and many more. These have explicit decay-widths formulae encoded, in classes derived from the `ResonanceWidths` base class. The formulae are used, e.g., to calculate all the partial widths as a function of the resonance masses you choose, and at initialization the existing total width values are overwritten. This is especially convenient for hypothetical states, like a Z', where the mass is not known and therefore routinely changed. Often the partial widths are associated with parameters that can be changed by the user, e.g. for MSSM Higgs states.
- If a resonance does not have a class of its own, with hardcoded equations for all relevant partial widths, then a simpler object will be created at initialization. This object will take the total width and branching ratios as is (with the optional variations explained in the next section). When you set a particle to be a resonance, and do not provide any class to go with its width calculations, this is where it will end up.

Sometimes experimentalists want to modify the physical width of a resonance, to understand how sensitive analyses are to this width, if at all. For the second, simpler kind of resonances, the `id:mWidth` can be changed right away, but for the first kind any change will be overwritten at initialization. To circumvent this problem, the `id:doForceWidth` flag can be changed from the default `off` to `on`. Then the width stored in `id:mWidth` is strictly used to describe the Breit-Wigner of the resonance. Partial widths are still recalculated to set the mass-dependent branching ratios, but then uniformly rescaled to the requested total width. The width can also run across the lineshape, so that it deviates from the nominal one in the wings of the Breit-Wigner.

For processes that contain interference terms between resonances, notably gamma^*/Z^0 or gamma^*/Z^0/Z'^0, it is not obvious how these contributions should be modified consistently. Therefore it is necessary to set `WeakZ0:gmZmode = 2` or `Zprime:gmZmode = 3`, to have a pure Z^0 or Z'^0, respectively, for width forcing to be allowed in these cases.

A warning is that the different processes have cross sections that rescale in different ways when the resonance width is varied. This depends on them not having been implemented in a guaranteed uniform way. To illustrate the point, consider the case of an s-channel resonance, where the cross section dependence on the width can be written as
sigmaHat(sHat) = constant * Gamma_in * Gamma_out / ((sHat - m^2)^2 + m^2 * Gamma^2)
Here the `doForceWidth = on` option ensures that the Gamma in the denominator is rescaled by some factor k relative to the natural width, but does not guarantee that Gamma_in and Gamma_out are rescaled as well. If all three are rescaled by the same factor k, as they should, then the integrated cross section also scales like k, assuming that the peak is reasonably narrow, so that the variation of PDF's across the Breit-Wigner can be neglected. This is the case for some processes. But in others either or both of the production and decay vertices can have been hardcoded, based on the coupling structure, and thus not scale with k. If only Gamma_out scales with k, say, the cross section remains (approximately) constant, and if neither scales the cross section will even go like 1/k. Such obvious normalization imperfections have to be corrected by hand.

Mainly for internal usage, the `ParticleData` contain some special methods that are only meaningful for resonances:

- `resInit(...)` to initialize a resonance, possibly including a recalculation of the nominal width to match the nominal mass;
- `resWidth(...)` to calculate the partial and total widths at the currently selected mass;
- `resWidthOpen(...)` to calculate the partial and total widths of those channels left open by user switches, at the currently selected mass;
- `resWidthStore(...)` to calculate the partial and total widths of those channels left open by user switches, at the currently selected mass, and store those as input for a subsequent selection of decay channel;
- `resOpenFrac(...)` to return the fraction of the total width that is open by the decay channel selection made by users (based on the choice of `onMode` for the various decay channels, recursively calculated for sequential decays);
- `resWidthRescaleFactor(...)` returns the factor by which the internally calculated PYTHIA width has to be rescaled to give the user-enforced width;
- `resWidthChan(...)` to return the width for one particular channel (currently only used for Higgs decays, to obtain instate coupling from outstate width).

These methods actually provide an interface to the classes derived from the `ResonanceWidths` base class, to describe various resonances.

### Modes for Matrix Element Processing

The `meMode()` value for a decay mode is used to specify nonisotropic decays or the conversion of a parton list into a set of hadrons in some channels of normal particles. For resonances it can also take a third function, namely to describe how the branching ratios and widths of a resonance should be rescaled as a function of the current mass of the decaying resonance. The rules are especially useful when new channels are added to an existing particle, or a completely new resonance added.

- 0 : channels for which hardcoded partial-width expressions are expected to exist in the derived class of the respective resonance. Should no such code exist then the partial width defaults to zero.
- 1 - 99 : same as 0, but normally not used for resonances.
- 100 : calculate the partial width of the channel from its stored branching ratio times the stored total width. This value remains unchanged when the resonance fluctuates in mass. Specifically there are no threshold corrections. That is, if the resonance fluctuates down in mass, to below the nominal threshold, it is assumed that one of the daughters could also fluctuate down to keep the channel open. (If not, there may be problems later on.)
- 101 : calculate the partial width of the channel from its stored branching ratio times the stored total width. Multiply by a step threshold, i.e. the channel is switched off when the sum of the daughter on-shell masses is above the current mother mass.
- 102 : calculate the partial width of the channel from its stored branching ratio times the stored total width. Multiply by a smooth threshold factor beta = sqrt( (1 - m_1^2/m_2 - m_2^2/m^2)^2 - 4 m_1^2 m_2^2/m^4) for two-body decays and sqrt(1 - Sum_i m_i / m) for multibody ones. The former correctly encodes the size of the phase space but misses out on any nontrivial matrix-element behaviour, while the latter obviously is a very crude simplification of the correct phase-space expression. Specifically, it is thereby assumed that the stored branching ratio and total width did not take into account such a factor.
- 103 : use the same kind of behaviour and threshold factor as for 102 above, but assume that such a threshold factor has been used when the default branching ratio and total width were calculated, so that one should additionally divide by the on-shell threshold factor. Specifically, this will give back the stored branching ratios for on-shell mass, unlike the 102 option. To avoid division by zero, or in general unreasonably big rescaling factors, a lower limit `minThreshold` (see below) on the value of the on-shell threshold factor is imposed. (In cases where a big rescaling is intentional, code 102 would be more appropriate.)

`parm` ResonanceWidths:minThreshold (`default = 0.1`; `minimum = 0.01`)
Used uniquely for `meMode = 103` to set the minimal value assumed for the threshold factor, sqrt( (1 - m_1^2/m_2 - m_2^2/m^2)^2 - 4 m_1^2 m_2^2/m^4) for two-body decays and sqrt(1 - Sum_i m_i / m) for multibody ones. Thus the inverse of this number sets an upper limit for how much the partial width of a channel can increase from the on-shell value to the value for asymptotically large resonance masses. Is mainly intended as a safety measure, to avoid unintentionally large rescalings.

All of these `meMode`'s may coexist for the same resonance. This would be the case e.g. if you want to add a few new channels to an already existing resonance, where the old partial widths come hardcoded while the new ones are read in from an external file. The typical example would be an MSSM Higgs sector, where partial widths to SM particles are already encoded, `meMode = 0`, while decay rates to sparticles are read in from some external calculation and maybe would be best approximated by using `meMode = 103`. Indeed the default particle table in PYTHIA uses 103 for all channels that are expected to be provided by external input.

Some further clarification may be useful. At initialization the existing total width and on-shell branching ratios will be updated. For channels with `meMode < 100` the originally stored branching ratios are irrelevant, since the existing code will anyway be used to calculate the partial widths from scratch. For channels with `meMode = 100` or bigger, instead the stored branching ratio is used together with the originally stored total width to define the correct on-shell partial width. The sum of partial widths then gives the new total width, and from there new branching ratios are defined.

In these operations the original sum of branching ratios need not be normalized to unity. For instance, you may at input have a stored total width of 1 GeV and a sum of branching ratios of 2. After initialization the width will then have been changed to 2 GeV and the sum of branching ratios rescaled to unity. This might happen e.g. if you add a few channels to an existing resonance, without changing the branching ratios of the existing channels or the total width of the resonance.

In order to simulate the Breit-Wigner shape correctly, it is important that all channels that contribute to the total width are included in the above operations. This must be kept separate from the issue of which channels you want to have switched on for a particular study, to be considered next.

In the event-generation process, when an off-shell resonance mass has been selected, the width and branching ratios are re-evaluated for this new mass. At this stage also the effects of restrictions on allowed decay modes are taken into account, as set by the `onMode` switch for each separate decay channel. Thus a channel may be on or off, with different choices of open channels between the particle and its antiparticle. In addition, even when a channel is on, the decay may be into another resonance with its selection of allowed channels. It is these kinds of restrictions that lead to the Gamma_out possibly being smaller than Gamma_tot. As a reminder, the Breit-Wigner for decays behaves like Gamma_out / ((s - m^2)^2 + s * Gamma_tot^2), where the width in the numerator is only to those channels being studied, but the one in the denominator to all channels of the particle. These ever-changing numbers are not directly visible to the user, but are only stored in a work area.