## Timelike Showers

The PYTHIA algorithm for timelike final-state showers is based on the article [Sjo05], where a transverse-momentum-ordered evolution scheme is introduced, with the extension to fully interleaved evolution covered in [Cor10a]. This algorithm is influenced by the previous mass-ordered algorithm in PYTHIA [Ben87] and by the dipole-emission formulation in Ariadne [Gus86]. From the mass-ordered algorithm it inherits a merging procedure for first-order gluon-emission matrix elements in essentially all two-body decays in the standard model and its minimal supersymmetric extension [Nor01].

The normal user is not expected to call `TimeShower` directly, but only have it called from `Pythia`. Some of the parameters below, in particular `TimeShower:alphaSvalue`, would be of interest for a tuning exercise, however.

### Main variables

Often the maximum scale of the FSR shower evolution is understood from the context. For instance, in a resonace decay half the resonance mass sets an absolute upper limit. For a hard process in a hadronic collision the choice is not as unique. Here the factorization scale has been chosen as the maximum evolution scale. This would be the pT for a 2 -> 2 process, supplemented by mass terms for massive outgoing particles. For some special applications we do allow an alternative.

`mode` TimeShower:pTmaxMatch (`default = 1`; `minimum = 0`; `maximum = 2`)
Way in which the maximum shower evolution scale is set to match the scale of the hard process itself.
`option` 0 : (i) if the final state of the hard process (not counting subsequent resonance decays) contains at least one quark (u, d, s, c ,b), gluon or photon then pT_max is chosen to be the factorization scale for internal processes and the `scale` value for Les Houches input; (ii) if not, emissions are allowed to go all the way up to the kinematical limit (i.e. to half the dipole mass). This option agrees with the corresponding one for spacelike showers. There the reasoning is that in the former set of processes the ISR emission of yet another quark, gluon or photon could lead to doublecounting, while no such danger exists in the latter case. The argument is less compelling for timelike showers, but could be a reasonable starting point.
`option` 1 : always use the factorization scale for an internal process and the `scale` value for Les Houches input, i.e. the lower value. This should avoid doublecounting, but may leave out some emissions that ought to have been simulated. (Also known as wimpy showers.)
`option` 2 : always allow emissions up to the kinematical limit (i.e. to half the dipole mass). This will simulate all possible event topologies, but may lead to doublecounting. (Also known as power showers.)
Note: These options only apply to the hard interaction. Emissions off subsequent multiparton interactions are always constrainted to be below the factorization scale of the process itself. They also assume you use interleaved evolution, so that FSR is in direct competition with ISR for the hardest emission. If you already generated a number of ISR partons at low pT, it would not make sense to have a later FSR shower up to the kinematical for all of them.

`parm` TimeShower:pTmaxFudge (`default = 1.0`; `minimum = 0.25`; `maximum = 2.0`)
In cases where the above `pTmaxMatch` rules would imply that pT_max = pT_factorization, `pTmaxFudge` introduces a multiplicative factor f such that instead pT_max = f * pT_factorization. Only applies to the hardest interaction in an event, cf. below. It is strongly suggested that f = 1, but variations around this default can be useful to test this assumption.
Note:Scales for resonance decays are not affected, but can be set separately by user hooks.

`parm` TimeShower:pTmaxFudgeMPI (`default = 1.0`; `minimum = 0.25`; `maximum = 2.0`)
A multiplicative factor f such that pT_max = f * pT_factorization, as above, but here for the non-hardest interactions (when multiparton interactions are allowed).

`mode` TimeShower:pTdampMatch (`default = 0`; `minimum = 0`; `maximum = 2`)
These options only take effect when a process is allowed to radiate up to the kinematical limit by the above `pTmaxMatch` choice, and no matrix-element corrections are available. Then, in many processes, the fall-off in pT will be too slow by one factor of pT^2. That is, while showers have an approximate dpT^2/pT^2 shape, often it should become more like dpT^2/pT^4 at pT values above the scale of the hard process. This argument is more obvious for ISR, but is taken over unchanged for FSR to have a symmetric description.
`option` 0 : emissions go up to the kinematical limit, with no special dampening.
`option` 1 : emissions go up to the kinematical limit, but dampened by a factor k^2 Q^2_fac/(pT^2 + k^2 Q^2_fac), where Q_fac is the factorization scale and k is a multiplicative fudge factor stored in `pTdampFudge` below.
`option` 2 : emissions go up to the kinematical limit, but dampened by a factor k^2 Q^2_ren/(pT^2 + k^2 Q^2_ren), where Q_ren is the renormalization scale and k is a multiplicative fudge factor stored in `pTdampFudge` below.
Note: These options only apply to the hard interaction. Emissions off subsequent multiparton interactions are always constrainted to be below the factorization scale of the process itself.

`parm` TimeShower:pTdampFudge (`default = 1.0`; `minimum = 0.25`; `maximum = 4.0`)
In cases 1 and 2 above, where a dampening is imposed at around the factorization or renormalization scale, respectively, this allows the pT scale of dampening of radiation by a half to be shifted by this factor relative to the default Q_fac or Q_ren. This number ought to be in the neighbourhood of unity, but variations away from this value could do better in some processes.

The amount of QCD radiation in the shower is determined by

`parm` TimeShower:alphaSvalue (`default = 0.1383`; `minimum = 0.06`; `maximum = 0.25`)
The alpha_strong value at scale M_Z^2. The default value corresponds to a crude tuning to LEP data, to be improved.

The actual value is then regulated by the running to the scale pT^2, at which the shower evaluates alpha_strong

`mode` TimeShower:alphaSorder (`default = 1`; `minimum = 0`; `maximum = 2`)
Order at which alpha_strong runs,
`option` 0 : zeroth order, i.e. alpha_strong is kept fixed.
`option` 1 : first order, which is the normal value.
`option` 2 : second order. Since other parts of the code do not go to second order there is no strong reason to use this option, but there is also nothing wrong with it.

QED radiation is regulated by the alpha_electromagnetic value at the pT^2 scale of a branching.

`mode` TimeShower:alphaEMorder (`default = 1`; `minimum = -1`; `maximum = 1`)
The running of alpha_em.
`option` 1 : first-order running, constrained to agree with `StandardModel:alphaEMmZ` at the Z^0 mass.
`option` 0 : zeroth order, i.e. alpha_em is kept fixed at its value at vanishing momentum transfer.
`option` -1 : zeroth order, i.e. alpha_em is kept fixed, but at `StandardModel:alphaEMmZ`, i.e. its value at the Z^0 mass.

The rate of radiation if divergent in the pT -> 0 limit. Here, however, perturbation theory is expected to break down. Therefore an effective pT_min cutoff parameter is introduced, below which no emissions are allowed. The cutoff may be different for QCD and QED radiation off quarks, and is mainly a technical parameter for QED radiation off leptons.

`parm` TimeShower:pTmin (`default = 0.4`; `minimum = 0.1`; `maximum = 2.0`)
Parton shower cut-off pT for QCD emissions.

`parm` TimeShower:pTminChgQ (`default = 0.4`; `minimum = 0.1`; `maximum = 2.0`)
Parton shower cut-off pT for photon coupling to coloured particle.

`parm` TimeShower:pTminChgL (`default = 0.0005`; `minimum = 0.0001`; `maximum = 2.0`)
Parton shower cut-off pT for pure QED branchings. Assumed smaller than (or equal to) `pTminChgQ`.

Shower branchings gamma -> f fbar, where f is a quark or lepton, in part compete with the hard processes involving gamma^*/Z^0 production. In order to avoid overlap it makes sense to correlate the maximum gamma mass allowed in showers with the minumum gamma^*/Z^0 mass allowed in hard processes. In addition, the shower contribution only contains the pure gamma^* contribution, i.e. not the Z^0 part, so the mass spectrum above 50 GeV or so would not be well described.

`parm` TimeShower:mMaxGamma (`default = 10.0`; `minimum = 0.001`; `maximum = 50.0`)
Maximum invariant mass allowed for the created fermion pair in a gamma -> f fbar branching in the shower.

### Interleaved evolution

Multiparton interactions (MPI) and initial-state showers (ISR) are always interleaved, as follows. Starting from the hard interaction, the complete event is constructed by a set of steps. In each step the pT scale of the previous step is used as starting scale for a downwards evolution. The MPI and ISR components each make their respective Monte Carlo choices for the next lower pT value. The one with larger pT is allowed to carry out its proposed action, thereby modifying the conditions for the next steps. This is relevant since the two components compete for the energy contained in the beam remnants: both an interaction and an emission take avay some of the energy, leaving less for the future. The end result is a combined chain of decreasing pT values, where ones associated with new interactions and ones with new emissions are interleaved.

There is no corresponding requirement for final-state radiation (FSR) to be interleaved. Such an FSR emission does not compete directly for beam energy (but see below), and also can be viewed as occuring after the other two components in some kind of time sense. Interleaving is allowed, however, since it can be argued that a high-pT FSR occurs on shorter time scales than a low-pT MPI, say. Backwards evolution of ISR is also an example that physical time is not the only possible ordering principle, but that one can work with conditional probabilities: given the partonic picture at a specific pT resolution scale, what possibilities are open for a modified picture at a slightly lower pT scale, either by MPI, ISR or FSR? Complete interleaving of the three components also offers advantages if one aims at matching to higher-order matrix elements above some given scale.

`flag` TimeShower:interleave (`default = on`)
If on, final-state emissions are interleaved in the same decreasing-pT chain as multiparton interactions and initial-state emissions. If off, final-state emissions are only addressed after the multiparton interactions and initial-state radiation have been considered.

As an aside, it should be noted that such interleaving does not affect showering in resonance decays, such as a Z^0. These decays are only introduced after the production process has been considered in full, and the subsequent FSR is carried out inside the resonance, with preserved resonance mass.

One aspect of FSR for a hard process in hadron collisions is that often colour diples are formed between a scattered parton and a beam remnant, or rather the hole left behind by an incoming partons. If such holes are allowed as dipole ends and take the recoil when the scattered parton undergoes a branching then this translates into the need to take some amount of remnant energy also in the case of FSR, i.e. the roles of ISR and FSR are not completely decoupled. The energy taken away is bokkept by increasing the x value assigned to the incoming scattering parton, and a reweighting factor x_new f(x_new, pT^2) / x_old f(x_old, pT^2) in the emission probability ensures that not unphysically large x_new values are reached. Usually such x changes are small, and they can be viewed as a higher-order effect beyond the accuracy of the leading-log initial-state showers.

This choice is not unique, however. As an alternative, if nothing else useful for cross-checks, one could imagine that the FSR is completely decoupled from the ISR and beam remnants.

`flag` TimeShower:allowBeamRecoil (`default = on`)
If on, the final-state shower is allowed to borrow energy from the beam remnants as described above, thereby changing the mass of the scattering subsystem. If off, the partons in the scattering subsystem are constrained to borrow energy from each other, such that the total four-momentum of the system is preserved. This flag has no effect on resonance decays, where the shower always preserves the resonance mass, cf. the comment above about showers for resonances never being interleaved.

`flag` TimeShower:dampenBeamRecoil (`default = on`)
When beam recoil is allowed there is still some ambiguity how far into the beam end of the dipole that emission should be allowed. It is dampened in the beam region, but probably not enough. When on an additional suppression factor 4 pT2_hard / (4 pT2_hard + m2) is multiplied on to the emission probability. Here pT_hard is the transverse momentum of the radiating parton and m the off-shell mass it acquires by the branching, m2 = pT2/(z(1-z)). Note that m2 = 4 pT2_hard is the kinematical limit for a scattering at 90 degrees without beam recoil.

### Global recoil

The final-state algorithm is based on dipole-style recoils, where one single parton takes the full recoil of a branching. This is unlike the initial-state algorithm, where the complete already-existing final state shares the recoil of each new emission. As an alternative, also the final-state algorithm contains an option where the recoil is shared between all partons in the final state. Thus the radiation pattern is unrelated to colour correlations. This is especially convenient for some matching algorithms, like MC@NLO, where a full analytic knowledge of the shower radiation pattern is needed to avoid doublecountning. (The pT-ordered shower is described in [Sjo05], and the corrections for massive radiator and recoiler in [Nor01].)

Technically, the radiation pattern is most conveniently represented in the rest frame of the final state of the hard subprocess. Then, for each parton at a time, the rest of the final state can be viewed as a single effective parton. This "parton" has a fixed invariant mass during the emission process, and takes the recoil without any changed direction of motion. The momenta of the individual new recoilers are then obtained by a simple common boost of the original ones.

This alternative approach will miss out on the colour coherence phenomena. Specifically, with the whole subcollision mass as "dipole" mass, the phase space for subsequent emissions is larger than for the normal dipole algorithm. The phase space difference grows as more and more gluons are created, and thus leads to a way too steep multiplication of soft gluons. Therefore the main application is for the first one or few emissions of the shower, where a potential overestimate of the emission rate is to be corrected for anyway, by matching to the relevant matrix elements. Thereafter, subsequent emissions should be handled as before, i.e. with dipoles spanned between nearby partons. Furthermore, only the first (hardest) subcollision is handled with global recoils, since subsequent MPI's would not be subject to matrix element corrections anyway.

In order for the mid-shower switch from global to local recoils to work, colours are traced and bookkept just as for normal showers; it is only that this information is not used in those steps where a global recoil is requested. (Thus, e.g., a gluon is still bookkept as one colour and one anticolour dipole end, with half the charge each, but with global recoil those two ends radiate identically.)

`flag` TimeShower:globalRecoil (`default = off`)
Alternative approach as above, where all final-state particles share the recoil of an emission.
If off, then use the standard dipole-recoil approach.
If on, use the alternative global recoil, but only for the first interaction, and only while the number of particles in the final state is at most `TimeShower:nMaxGlobalRecoil` before the branching.

`mode` TimeShower:nMaxGlobalRecoil (`default = 2`; `minimum = 1`)
Represents the maximum number of particles in the final state for which the next final-state emission can be performed with the global recoil strategy. This number counts all particles, whether they are allowed to radiate or not, e.g. also Z^0. Also partons created by initial-state radiation emissions counts towards this sum, as part of the interleaved evolution. Without interleaved evolution this option would not make sense, since then a varying and large number of partons could already have been created by the initial-state radiation before the first final-state one, and then there is not likely to be any matrix elements available for matching.

The global-recoil machinery does not work well with rescattering in the MPI machinery, since then the recoiling system is not uniquely defined. `MultipartonInteractions:allowRescatter = off` by default, so this is not a main issue. If both options are switched on, rescattering will only be allowed to kick in after the global recoil has ceased to be active, i.e. once the `nMaxGlobalRecoil` limit has been exceeded. This should not be a major conflict, since rescattering is mainly of interest at later stages of the downwards pT evolution.

Further, it is strongly recommended to set `TimeShower:MEcorrections = off` (not default!), i.e. not to correct the emission probability to the internal matrix elements. The internal ME options do not cover any cases relevant for a multibody recoiler anyway, so no guarantees are given what prescription would come to be used. Instead, without ME corrections, a process-independent emission rate is obtained, and user hooks can provide the desired process-specific rejection factors.

### Radiation off octet onium states

In the current implementation, charmonium and bottomonium production can proceed either through colour singlet or colour octet mechanisms, both of them implemented in terms of 2 -> 2 hard processes such as g g -> (onium) g. In the former case the state does not radiate and the onium therefore is produced in isolation, up to normal underlying-event activity. In the latter case the situation is not so clear, but it is sensible to assume that a shower can evolve. (Assuming, of course, that the transverse momentum of the onium state is sufficiently high that radiation is of relevance.)

There could be two parts to such a shower. Firstly a gluon (or even a quark, though less likely) produced in a hard 2 -> 2 process can undergo showering into many gluons, whereof one branches into the heavy-quark pair. Secondly, once the pair has been produced, each quark can radiate further gluons. This latter kind of emission could easily break up a semibound quark pair, but might also create a new semibound state where before an unbound pair existed, and to some approximation these two effects should balance in the onium production rate. The showering "off an onium state" as implemented here therefore should not be viewed as an accurate description of the emission history step by step, but rather as an effective approach to ensure that the octet onium produced "in the hard process" is embedded in a realistic amount of jet activity. Of course both the isolated singlet and embedded octet are likely to be extremes, but hopefully the mix of the two will strike a reasonable balance. However, it is possible that some part of the octet production occurs in channels where it should not be accompanied by (hard) radiation. Therefore reducing the fraction of octet onium states allowed to radiate is a valid variation to explore uncertainties.

If an octet onium state is chosen to radiate, the simulation of branchings is based on the assumption that the full radiation is provided by an incoherent sum of radiation off the quark and off the antiquark of the onium state. Thus the splitting kernel is taken to be the normal q -> q g one, multiplied by a factor of two. Obviously this is a simplification of a more complex picture, averaging over factors pulling in different directions. Firstly, radiation off a gluon ought to be enhanced by a factor 9/4 relative to a quark rather than the 2 now used, but this is a minor difference. Secondly, our use of the q -> q g branching kernel is roughly equivalent to always following the harder gluon in a g -> g g branching. This could give us a bias towards producing too hard onia. A soft gluon would have little phase space to branch into a heavy-quark pair however, so the bias may not be as big as it would seem at first glance. Thirdly, once the gluon has branched into a quark pair, each quark carries roughly only half of the onium energy. The maximum energy per emitted gluon should then be roughly half the onium energy rather than the full, as it is now. Thereby the energy of radiated gluons is exaggerated, i.e. onia become too soft. So the second and the third points tend to cancel each other.

Finally, note that the lower cutoff scale of the shower evolution depends on the onium mass rather than on the quark mass, as it should be. Gluons below the octet-onium scale should only be part of the octet-to-singlet transition.

`parm` TimeShower:octetOniumFraction (`default = 1.`; `minimum = 0.`; `maximum = 1.`)
Allow colour-octet charmonium and bottomonium states to radiate gluons. 0 means that no octet-onium states radiate, 1 that all do, with possibility to interpolate between these two extremes.

`parm` TimeShower:octetOniumColFac (`default = 2.`; `minimum = 0.`; `maximum = 4.`)
The colour factor used used in the splitting kernel for those octet onium states that are allowed to radiate, normalized to the q -> q g splitting kernel. Thus the default corresponds to twice the radiation off a quark. The physically preferred range would be between 1 and 9/4.

### Further variables

There are several possibilities you can use to switch on or off selected branching types in the shower, or in other respects simplify the shower. These should normally not be touched. Their main function is for cross-checks.

`flag` TimeShower:QCDshower (`default = on`)
Allow a QCD shower, i.e. branchings q -> q g, g -> g g and g -> q qbar; on/off = true/false.

`mode` TimeShower:nGluonToQuark (`default = 5`; `minimum = 0`; `maximum = 5`)
Number of allowed quark flavours in g -> q qbar branchings (phase space permitting). A change to 4 would exclude g -> b bbar, etc.

`flag` TimeShower:QEDshowerByQ (`default = on`)
Allow quarks to radiate photons, i.e. branchings q -> q gamma; on/off = true/false.

`flag` TimeShower:QEDshowerByL (`default = on`)
Allow leptons to radiate photons, i.e. branchings l -> l gamma; on/off = true/false.

`flag` TimeShower:QEDshowerByGamma (`default = on`)
Allow photons to branch into lepton or quark pairs, i.e. branchings gamma -> l+ l- and gamma -> q qbar; on/off = true/false.

`mode` TimeShower:nGammaToQuark (`default = 5`; `minimum = 0`; `maximum = 5`)
Number of allowed quark flavours in gamma -> q qbar branchings (phase space permitting). A change to 4 would exclude g -> b bbar, etc.

`mode` TimeShower:nGammaToLepton (`default = 3`; `minimum = 0`; `maximum = 3`)
Number of allowed lepton flavours in gamma -> l+ l- branchings (phase space permitting). A change to 2 would exclude gamma -> tau+ tau-, and a change to 1 also gamma -> mu+ mu-.

`flag` TimeShower:MEcorrections (`default = on`)
Use of matrix element corrections where available; on/off = true/false.

`flag` TimeShower:MEafterFirst (`default = on`)
Use of matrix element corrections also after the first emission, for dipole ends of the same system that did not yet radiate. Only has a meaning if `MEcorrections` above is switched on.

`flag` TimeShower:phiPolAsym (`default = on`)
Azimuthal asymmetry induced by gluon polarization; on/off = true/false.

`flag` TimeShower:recoilToColoured (`default = on`)
In the decays of coloured resonances, say t -> b W, it is not possible to set up dipoles with matched colours. Originally the b radiator therefore has W as recoiler, and that choice is unique. Once a gluon has been radiated, however, it is possible either to have the unmatched colour (inherited by the gluon) still recoiling against the W (`off`), or else let it recoil against the b also for this dipole (`on`). Before version 8.160 the former was the only possibility, which could give unphysical radiation patterns. It is kept as an option to check backwards compatibility. The same issue exists for QED radiation, but obviously is less significant. Consider the example W -> e nu, where originally the nu takes the recoil. In the old (`off`) scheme the nu would remain recoiler, while in the new (`on`) instead each newly emitted photon becomes the new recoiler.