Fragmentation
- Fragmentation functions
- Fragmentation pT
- Jet joining procedure
- Colour tracing
- Simplifying systems
- Ministrings
- Junction treatment
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.
Several methods are in place to allow the user to reweight generated
events in the parameters of the fragmentation model. This allows
the user to perform fast parameter variation or oversample rare
configurations, and reweight to the correct cross section. These
methods are documented on the separate
Hadronization Variations
page.
Note that the fragmentation in a
Hidden Valley scenario
is modelled in a similar spirit as for ordinary string fragmentation,
but with a separate set of parameters. Relevant parameters for it
are closely related to the scenario chosen, and therefore are
documented on the same 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 + 2StringPT: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.
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.
flag MiniStringFragmentation:tryAfterFailedFrag
(default = off)
If normal string fragmentation fails for a parton system, it is optionally
possible to attempt to fragment it as if it had been a ministring
(c.f. HadronLevel:mStringMin). This is mainly relevant for
heavy-ion collisions and for the QCD-based colour-reconnection model,
where many high-multiplicity events have a higher probability to have
failed fragmentation (hence throwing the whole event away may skew the
multiplicity distribution).
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:pNormJunction
(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. 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).
flag StringFragmentation:pearlFragmentation
(default = off)
Optionally allow to use a gluon approximation for pearl-on-a-string
junction topologies. Pearl-on-a-string is a junction topology such
that the junction gets stuck to a soft endpoint parton. In these cases
for light quarks, the propagation of the pT is similar to a
soft gluon kink, and can be fragmented by approximating the pearl as a
gluon.
parm StringFragmentation:pearlProbFactor
(default = 4; minimum = 1)
The gluon approximation for pearl-on-a-string cases is chosen
probabilistically. This parameter, f_pearl, modifies the
probability distribution used and has the general form of
prob(x) = 1 / (1 + (4Δv)fpearl).
Here Δv, which varies between 0 and 1/2, is a measure
of the total deceleration the pearl (ie the soft parton) experiences
over a time characteristic of hadronization (here given by the
pNormJunction parameter above).
flag StringFragmentation:strangeJunctions
(default = off)
Switching this parameter on allows strangeness enhancement for string breaks
next to a junction.
parm StringFragmentation:strangeJuncFactor
(default = 0.5; minimum = 0.; maximum = 1.0)
This parameter, s_J, scales how much strangeness enhancement
around a junction by modifying the probability
StringFlav:probStoUD,
P'(s:u/d) = P(s:u/d)(1 - s_J).
parm StringFragmentation:strangePearlFactor
(default = 0.5; minimum = 0.; maximum = 1.0)
Allow a different strangeness enhancement around a pearl junction.