## Fragmentation

Fragmentation in PYTHIA is based on the Lund string model [And83, Sjo84]. Several different aspects are involved in the physics description, which here therefore is split accordingly. This also, at least partly, reflect the set of classes involved in the fragmentation machinery.

The variables collected here have a very wide span of usefulness. Some would be central in any hadronization tuning exercise, others should not be touched except by experts.

The fragmentation flavour-choice machinery is also used in a few other places of the program, notably particle decays, and is thus described on the separate Flavour Selection page.

### Fragmentation functions

The `StringZ` class handles the choice of longitudinal lightcone fraction z according to one of two possible shape sets.

The Lund symmetric fragmentation function [And83] is the only alternative for light quarks. It is of the form
f(z) = (1/z) * (1-z)^a * exp(-b m_T^2 / z)
with the two main free parameters a and b to be tuned to data. They are stored in

`parm` StringZ:aLund (`default = 0.68`; `minimum = 0.0`; `maximum = 2.0`)
The a parameter of the Lund symmetric fragmentation function.

`parm` StringZ:bLund (`default = 0.98`; `minimum = 0.2`; `maximum = 2.0`)
The b parameter of the Lund symmetric fragmentation function.

In the context of fits to experimental data, note that the a and b parameters typically exhibit a very high degree of correlation. An option for choosing an alternative parameterisation is therefore provided, whereby the user specifies the desired average value of the fragmentation function for primary rho mesons instead of the b parameter. The a parameter should still be given by `StringZ:aLund` as usual. The rho has been chosen as reference since its mass is near the average of the primary hadron production, while pions come to dominate only after secondary decays. This option can be enabled via the following flag:

`flag` StringZ:deriveBLund (`default = off`)
When set to `on`, the b parameter is treated as a derived quantity; i.e., the value of `StringZ:bLund` is ignored in favour of the `StringZ:avgZLund` parameter below. (The `StringZ:bLund` parameter is then also reset to the derived value so that it can be queried after initialisation, if desired.)

`parm` StringZ:avgZLund (`default = 0.55`; `minimum = 0.30`; `maximum = 0.70`)
When `StringZ:deriveBLund` is set to `on`, this parameter specifies the average of the fragmentation function for primary rho mesons, evaluated at mT^2 = mRho^2 + 2`StringPT:sigma`^2. The appropriate b-parameter value is computed automatically during initialisation and the `StringZ:bLund` parameter is updated accordingly. Note that the derived value is allowed to exceed the nominal limits given for `bLund` above. This is intended to allow fits to see the functional behaviour even outside the nominal limits.

In principle, each flavour can have a different a. Then, for going from an old flavour i to a new j one the shape is
f(z) = (1/z) * z^{a_i} * ((1-z)/z)^{a_j} * exp(-b * m_T^2 / z)
This is only implemented for s quarks and diquarks relative to normal quarks:

`parm` StringZ:aExtraSQuark (`default = 0.0`; `minimum = 0.0`; `maximum = 2.0`)
allows a larger a for s quarks, with total a = aLund + aExtraSQuark.

`parm` StringZ:aExtraDiquark (`default = 0.97`; `minimum = 0.0`; `maximum = 2.0`)
allows a larger a for diquarks, with total a = aLund + aExtraDiquark.

Finally, the Bowler modification [Bow81] introduces an extra factor
1/z^{r_Q * b * m_Q^2}
for heavy quarks. To keep some flexibility, a multiplicative factor r_Q is introduced, which ought to be unity (provided that quark masses were uniquely defined) but can be set in

`parm` StringZ:rFactC (`default = 1.32`; `minimum = 0.0`; `maximum = 2.0`)
r_c, i.e. the above parameter for c quarks.

`parm` StringZ:rFactB (`default = 0.855`; `minimum = 0.0`; `maximum = 2.0`)
r_b, i.e. the above parameter for b quarks.

`parm` StringZ:rFactH (`default = 1.0`; `minimum = 0.0`; `maximum = 2.0`)
r_h, i.e. the above parameter for heavier hypothetical quarks, or in general any new coloured particle long-lived enough to hadronize.

Within the string framework, the b parameter is universal, i.e. common for all flavours. Nevertheless, for fits to experimental data, better agreement can be obtained if both a_Q and b_Q can be set freely in a general expression
f(z) = 1/z^{1 + r_Q * b_Q * m_Q^2} * (1-z)^a_Q * exp(-b_Q m_T^2 / z)
The below switches and values can be used to achieve this. They should be used with caution and constitute clear deviations from the Lund philosophy.

`flag` StringZ:useNonstandardC (`default = off`)
use the above nonstandard Lund ansatz for c quarks.

`flag` StringZ:useNonstandardB (`default = off`)
use the above nonstandard Lund ansatz for b quarks.

`flag` StringZ:useNonstandardH (`default = off`)
use the above nonstandard Lund ansatz for hypothetical heavier quarks.

`parm` StringZ:aNonstandardC (`default = 0.3`; `minimum = 0.0`; `maximum = 2.0`)
The a parameter in the nonstandard Lund ansatz for c quarks.

`parm` StringZ:aNonstandardB (`default = 0.3`; `minimum = 0.0`; `maximum = 2.0`)
The a parameter in the nonstandard Lund ansatz for b quarks.

`parm` StringZ:aNonstandardH (`default = 0.3`; `minimum = 0.0`; `maximum = 2.0`)
The a parameter in the nonstandard Lund ansatz for hypothetical heavier quarks.

`parm` StringZ:bNonstandardC (`default = 0.8`; `minimum = 0.2`; `maximum = 2.0`)
The b parameter in the nonstandard Lund ansatz for c quarks.

`parm` StringZ:bNonstandardB (`default = 0.8`; `minimum = 0.2`; `maximum = 2.0`)
The b parameter in the nonstandard Lund ansatz for b quarks.

`parm` StringZ:bNonstandardH (`default = 0.8`; `minimum = 0.2`; `maximum = 2.0`)
The b parameter in the nonstandard Lund ansatz for hypothetical heavier quarks.

As another nonstandard alternative, it is possible to switch over to the Peterson/SLAC formula [Pet83]
f(z) = 1 / ( z * (1 - 1/z - epsilon/(1-z))^2 )
for charm, bottom and heavier (defined as above) by the three flags

`flag` StringZ:usePetersonC (`default = off`)
use Peterson for c quarks.

`flag` StringZ:usePetersonB (`default = off`)
use Peterson for b quarks.

`flag` StringZ:usePetersonH (`default = off`)
use Peterson for hypothetical heavier quarks.

When switched on, the corresponding epsilon values are chosen to be

`parm` StringZ:epsilonC (`default = 0.05`; `minimum = 0.01`; `maximum = 0.25`)
epsilon_c, i.e. the above parameter for c quarks.

`parm` StringZ:epsilonB (`default = 0.005`; `minimum = 0.001`; `maximum = 0.025`)
epsilon_b, i.e. the above parameter for b quarks.

`parm` StringZ:epsilonH (`default = 0.005`; `minimum = 0.0001`; `maximum = 0.25`)
epsilon_h, i.e. the above parameter for hypothetical heavier quarks, normalized to the case where m_h = m_b. The actually used parameter is then epsilon = epsilon_h * (m_b^2 / m_h^2). This allows a sensible scaling to a particle with an unknown higher mass without the need for a user intervention.

### Fragmentation pT

The `StringPT` class handles the choice of fragmentation pT. At each string breaking the quark and antiquark of the pair are supposed to receive opposite and compensating pT kicks. How they are distributed depends on the following flag:

`flag` StringPT:thermalModel (`default = off`)
If switched off the quark pT is generated according to the traditional Gaussion distribution in p_x and p_y separately. If switched on, the new "thermal model" [Fis16] is instead used, wherein the quark pT is generated such that the resulting hadron receives a pT according to an exponential distribution. Also the hadronic composition is affected, see further below.

#### Gaussian Distribution

For `StringPT:thermalModel = off` the quarks receive pT kicks according to a Gaussian distribution in p_x and p_y separately. Call sigma_q the width of the p_x and p_y distributions separately, i.e.
d(Prob) = exp( -(p_x^2 + p_y^2) / 2 sigma_q^2).
Then the total squared width is
<pT^2> = <p_x^2> + <p_y^2> = 2 sigma_q^2 = sigma^2.
It is this latter number that is stored in

`parm` StringPT:sigma (`default = 0.335`; `minimum = 0.0`; `maximum = 1.0`)
the width sigma in the fragmentation process.

Since a normal hadron receives pT contributions for two string breakings, it has a <p_x^2>_had = <p_y^2>_had = sigma^2, and thus <pT^2>_had = 2 sigma^2.

Some studies on isolated particles at LEP has indicated the need for a slightly enhanced rate in the high-pT tail of the above distribution. This would have to be reviewed in the context of a complete retune of parton showers and hadronization, but for the moment we stay with the current recipe, to boost the above pT by a factor enhancedWidth for a small fraction enhancedFraction of the breakups, where

`parm` StringPT:enhancedFraction (`default = 0.01`; `minimum = 0.0`; `maximum = 1.`)
enhancedFraction,the fraction of string breaks with enhanced width.

`parm` StringPT:enhancedWidth (`default = 2.0`; `minimum = 1.0`; `maximum = 10.0`)
enhancedWidth,the enhancement of the width in this fraction.

In the context of some toy studies [Fis16] the following three options have also been introduced, but are not part of any recommended framework.

`parm` StringPT:widthPreStrange (`default = 1.0`; `minimum = 1.0`; `maximum = 10.0`)
Prefactor multiplying the Gaussian width for strange quarks.

`parm` StringPT:widthPreDiquark (`default = 1.0`; `minimum = 1.0`; `maximum = 10.0`)
Prefactor multiplying the Gaussian width for diquarks. In case of diquarks with one or two strange quarks the prefactor is calculated by multiplying widthPreDiquark once or twice respectively with widthPreStrange.

`flag` StringPT:mT2suppression (`default = off`)
If switched on the flavour composition is chosen based on the hadronic transverse mass, mT^2_had, and not based on the quark masses. This implies a mass suppression factor exp(-m_had^2 / 2 sigma^2) .

#### Thermal Distribution

For `StringPT:thermalModel = on` the quark pT is generated such that the resulting hadron pT follows a thermal distribution
d(Prob) = exp( -pT_had/T) d^2pT_had
with temperature T, whose value is given by

`parm` StringPT:temperature (`default = 0.21`; `minimum = 0.1`; `maximum = 0.5`)
the temperature T in the fragmentation process.

`parm` StringPT:tempPreFactor (`default = 1.21`; `minimum = 1.0`; `maximum = 1.5`)
Temperature prefactor for strange quarks and diquarks. Default is determined to have the same average pT in u/d → s and s → u/d transistions.

#### Common setup for enhanced width

If strings are closely packed, e.g. as a consequence of MPIs, it is likely that they receive an increased string tension, which translates into a broader pT spectrum, see further [Fis16]. It also means an enhanced rate (or rather reduced suppression) of heavy-particle production relative to pions. This can be regulated by the flag and parameters below.

`flag` StringPT:closePacking (`default = off`)
If switched on then the two following parameters modify either `StringPT:sigma` or `StringPT:temperature`, respectively. Normally only one of the options below would be used, but technically both are allowed and then combine multiplicatively.

`parm` StringPT:expMPI (`default = 0.0`; `minimum = 0.0`; `maximum = 1.0`)
Exponent to the number of MPIs. The width/temparture will get the prefactor N(MPI)^expMPI.

`parm` StringPT:expNSP (`default = 0.13`; `minimum = 0.0`; `maximum = 1.0`)
Exponent for the number of effective nearby string pieces, calculated as
N(NSP) =1 + (Nstring-1)/(1+p2T had/ p2T 0) ,
where p2T had is the transverse momentum of the next produced hadron, estimated based on an educated guess of its momentum, and p2T 0 is the MPI regularization parameter `MultipartonInteractions:pT0Ref`. The width/temperature will get the prefactor N(NSP)^expNSP.

### Jet joining procedure

String fragmentation is carried out iteratively from both string ends inwards, which means that the two chains of hadrons have to be joined up somewhere in the middle of the event. This joining is described by parameters that in principle follows from the standard fragmentation parameters, but in a way too complicated to parametrize. The dependence is rather mild, however, so for a sensible range of variation the parameters in this section should not be touched.

`parm` StringFragmentation:stopMass (`default = 1.0`; `minimum = 0.0`; `maximum = 2.0`)
Is used to define a W_min = m_q1 + m_q2 + stopMass, where m_q1 and m_q2 are the masses of the two current endpoint quarks or diquarks.

`parm` StringFragmentation:stopNewFlav (`default = 2.0`; `minimum = 0.0`; `maximum = 2.0`)
Add to W_min an amount stopNewFlav * m_q_last, where q_last is the last q qbar pair produced between the final two hadrons.

`parm` StringFragmentation:stopSmear (`default = 0.2`; `minimum = 0.0`; `maximum = 0.5`)
The W_min above is then smeared uniformly in the range W_min_smeared = W_min * [ 1 - stopSmear, 1 + stopSmear ].

This W_min_smeared is then compared with the current remaining W_transverse to determine if there is energy left for further particle production. If not, i.e. if W_transverse < W_min_smeared, the final two particles are produced from what is currently left, if possible. (If not, the fragmentation process is started over.)

### Colour tracing

`flag` StringFragmentation:TraceColours (`default = off`)
In some cases it is interesting to trace the primary hadrons back to the string pieces from which they were formed. If `StringFragmentation:TraceColours` is switched `on`, this is done by setting colour and anticolour indices for the primary hadrons to the indices of the string piece where the corresponding break-ups are assumed to have happened. To avoid the possible confusion of having colour indices on colour singlet particles, this flag is by default `off`.

### Simplifying systems

There are a few situations when it is meaningful to simplify the original task, one way or another.

`parm` HadronLevel:mStringMin (`default = 1.`; `minimum = 0.5`; `maximum = 1.5`)
Decides whether a partonic system should be considered as a normal string or a ministring, the latter only producing one or two primary hadrons. The system mass should be above mStringMin plus the sum of quark/diquark constituent masses for a normal string description, else the ministring scenario is used.

`parm` FragmentationSystems:mJoin (`default = 0.3`; `minimum = 0.2`; `maximum = 1.`)
When two colour-connected partons are very nearby, with at least one being a gluon, they can be joined into one, to avoid technical problems of very small string regions. The requirement for joining is that the invariant mass of the pair is below mJoin, where a gluon only counts with half its momentum, i.e. with its contribution to the string region under consideration. (Note that, for technical reasons, the 0.2 GeV lower limit is de facto hardcoded.)

`parm` FragmentationSystems:mJoinJunction (`default = 1.0`; `minimum = 0.5`; `maximum = 2.`)
When the invariant mass of two of the quarks in a three-quark junction string system becomes too small, the system is simplified to a quark-diquark simple string. The requirement for this simplification is that the diquark mass, minus the two quark masses, falls below mJoinJunction. Gluons on the string between the junction and the respective quark, if any, are counted as part of the quark four-momentum. Those on the two combined legs are clustered with the diquark when it is formed.

### Ministrings

The `MiniStringFragmentation` machinery is only used when a string system has so small invariant mass that normal string fragmentation is difficult/impossible. Instead one or two particles are produced, in the former case shuffling energy-momentum relative to another colour singlet system in the event, while preserving the invariant mass of that system. With one exception parameters are the same as defined for normal string fragmentation, to the extent that they are at all applicable in this case. A discussion of the relevant physics is found in [Nor00]. The current implementation does not completely abide to the scheme presented there, however, but has in part been simplified. (In part for greater clarity, in part since the class is not quite finished yet.)

`mode` MiniStringFragmentation:nTry (`default = 2`; `minimum = 1`; `maximum = 10`)
Whenever the machinery is called, first this many attempts are made to pick two hadrons that the system fragments to. If the hadrons are too massive the attempt will fail, but a new subsequent try could involve other flavour and hadrons and thus still succeed. After nTry attempts, instead an attempt is made to produce a single hadron from the system. Should also this fail, some further attempts at obtaining two hadrons will be made before eventually giving up.

### Junction treatment

A junction topology corresponds to an Y arrangement of strings i.e. where three string pieces have to be joined up in a junction. Such topologies can arise if several valence quarks are kicked out from a proton beam, or in baryon-number-violating SUSY decays. Special attention is necessary to handle the region just around the junction, where the baryon number topologically is located. The junction fragmentation scheme is described in [Sjo03]. The parameters in this section should not be touched except by experts.

`parm` StringFragmentation:eNormJunction (`default = 2.0`; `minimum = 0.5`; `maximum = 10`)
Used to find the effective rest frame of the junction, which is complicated when the three string legs may contain additional gluons between the junction and the endpoint. To this end, a pull is defined as a weighed sum of the momenta on each leg, where the weight is exp(- eSum / eNormJunction), with eSum the summed energy of all partons closer to the junction than the currently considered one (in the junction rest frame). Should in principle be (close to) sqrt((1 + a) / b), with a and b the parameters of the Lund symmetric fragmentation function.

`parm` StringFragmentation:eBothLeftJunction (`default = 1.0`; `minimum = 0.5`)
Retry (up to 10 times) when the first two considered strings in to a junction both have a remaining energy (in the junction rest frame) above this number.

`parm` StringFragmentation:eMaxLeftJunction (`default = 10.0`; `minimum = 0.`)
Retry (up to 10 times) when the first two considered strings in to a junction has a highest remaining energy (in the junction rest frame) above a random energy evenly distributed between eBothLeftJunction and eBothLeftJunction + eMaxLeftJunction (drawn anew for each test).

`parm` StringFragmentation:eMinLeftJunction (`default = 0.2`; `minimum = 0.`)
Retry (up to 10 times) when the invariant mass-squared of the final leg and the leftover momentum of the first two treated legs falls below eMinLeftJunction times the energy of the final leg (in the junction rest frame).