Baryon production

The mechanism for meson production follows rather naturally from the simple picture of a meson as a short piece of string between two endpoints. There is no unique recipe to generalize this picture to baryons. The program actually contains three different scenarios: diquark, simple popcorn, and advanced popcorn. In the diquark model the baryon and antibaryon are always produced as nearest neighbours along the string, while mesons may (but need not) be produced in between in the popcorn scenarios. The simpler popcorn alternative admits at most one intermediate meson, while the advanced one allows many. Further differences may be found, but several aspects are also common between the three scenarios. Below they are therefore described in order of increasing sophistication. Finally the application of the models to baryon remnant fragmentation, where a diquark originally sits at one endpoint of the string, is discussed.

*Diquark picture*

Baryon production may, in its simplest form, be obtained by assuming that any flavour given above could represent either a quark or an antidiquark in a colour triplet state. Then the same basic machinery can be run through as above, supplemented with the probability to produce various diquark pairs. In principle, there is one parameter for each diquark, but if tunnelling is still assumed to give an effective description, mass relations can be used to reduce the effective number of parameters. There are three main ones appearing in the program:

- the relative probability to pick a diquark rather than a ;
- the extra suppression associated with a diquark containing a strange quark (over and above the ordinary suppression factor ); and
- the suppression of spin 1 diquarks relative to spin 0 ones (apart from the factor of 3 enhancement of the former based on counting the number of spin states).

Only two baryon multiplets are included, i.e. there are no excited states. The two multiplets are:

- : the `octet' multiplet of
**SU(3)**; - : the `decuplet' multiplet of
**SU(3)**.

An important constraint is that a baryon is a
symmetric state of three quarks, neglecting the colour degree of
freedom. When a diquark and a quark are joined to form a baryon,
the combination is therefore weighted with the probability that
they form a symmetric three-quark state. The program implementation
of this principle is to first select a diquark at random, with
the strangeness and spin 1 suppression factors above included,
but then to accept the selected diquark with a weight proportional to
the number of states available for the quark-diquark combination. This
means that, were it not for the tunnelling suppression factors, all
states in the **SU(6)** (flavour **SU(3)** times spin **SU(2)**)
56-multiplet would become equally populated. Of course also heavier
baryons may come from the fragmentation of e.g. quark jets, but
although the particle classification scheme used in the program is
**SU(10)**, i.e. with five flavours, all possible quark-diquark
combinations can be related to **SU(6)** by symmetry arguments.
As in the case for mesons, one could imagine an explicit further
suppression of the heavier spin 3/2 baryons.

In case of rejection, one again chooses between a diquark or a quark.
If choosing diquark, a new baryon is selected and tested, etc. (In
versions earlier than PYTHIA 6.106, the algorithm was instead to always
produce a new diquark if the previous one had been rejected. However,
the probability that a quark will produce a baryon and a antidiquark
is then flavour independent, which is not in agreement with the model.)
Calling the tunnelling factor for diquark , the number of
spin states and the **SU(6)** factor for and a
quark
, the model prediction for the
ratio is

(235) |

When a diquark has been fitted into a symmetrical three-particle state,
it should not suffer any further **SU(6)** suppressions. Thus the
accompanying antidiquark should `survive' with probability 1. When
producing a quark to go with a previously produced diquark, this is
achieved by testing the configuration against the proper **SU(6)**
factor, but in case of rejection keep the diquark and pick a new quark,
which then is tested, etc.

There is no obvious corresponding algorithm available when a quark from one
side and a diquark from the other are joined to form the last hadron of the
string. In this case the quark is a member of a pair, in which the antiquark
already has formed a specific hadron. Thus the quark flavour cannot be
reselected. One could consider the **SU(6)** rejection as a major joining
failure, and restart the fragmentation of the original string, but then
the already accepted diquark *does* suffer extra **SU(6)** suppression.
In the program the joining of a quark and a diquark is always accepted.

*Simple popcorn*

A more general framework for baryon production is the `popcorn' one [And85], in which diquarks as such are never produced, but rather baryons appear from the successive production of several pairs. The picture is the following. Assume that the original is red and the is . Normally a new pair produced in the field would also be , so that the is pulled towards the end and vice versa, and two separate colour-singlet systems and are formed. Occasionally, the pair may be e.g. ( = green), in which case there is no net colour field acting on either or . Therefore, the pair cannot gain energy from the field, and normally would exist only as a fluctuation. If moves towards and towards , the net field remaining between and is ( = blue; if only colour triplets are assumed). In this central field, an additional pair can be created, where now is pulled towards and towards , with no net colour field between and . If this is all that happens, the baryon will be made up out of , and some produced between and , and of , and some , i.e. the and will be nearest neighbours in rank and share two quark pairs. Specifically, will gain energy from in order to end up on mass shell, and the tunnelling formula for an effective diquark is recovered.

Part of the time, several
colour pair productions
may take place between the and
, however. With two
production vertices
and
, a central
meson
may be formed, surrounded by a baryon
and an antibaryon
.
We call this a
configuration to distinguish it from the
+
configuration
above. For
the and only share one
quark-antiquark pair, as opposed to two for
configurations. The relative probability for a
configuration is given by the uncertainty relation suppression for
having the and
sufficiently far apart that a meson
may be formed in between. The suppression of the
system is
estimated by

In total, the flavour iteration procedure therefore contains the following possible subprocesses (plus, of course, their charge conjugates):

- meson;
- baryon;
- baryon;
- meson;

When selecting flavours for
,
the quark coming from the accepted
is kept, and the other member
of
, as well as the spin of
, is chosen with weights
taking **SU(6)** symmetry into account. Thus the flavour of
is not influenced by **SU(6)** factors for
, but the flavour
of is.

Unfortunately, the resulting baryon production model has a fair
number of parameters, which would be given by the model only if
quark and diquark masses were known unambiguously.
We have already mentioned the
ratio and the
one; the latter has to be increased from 0.09 to 0.10 for the
popcorn model, since the total number of possible baryon
production configurations is lower in this case (the particle
produced between the and is constrained to be a
meson). With the improved **SU(6)** treatment introduced in PYTHIA
6.106, a rejected
may lead to the splitting
instead. This calls for an increase of the
input ratio by approximately 10%.
For the popcorn model, exactly the same parameters as
already found in the diquark model are needed to describe the
configurations. For
configurations, the square
root of a suppression factor should be applied if the factor is
relevant only for one of the and , e.g. if the
is formed with a spin 1 `diquark'
but the
with a spin 0 diquark
. Additional parameters
include the relative probability for
configurations,
which is assumed to be roughly 0.5 (with the remaining 0.5 being
), a suppression factor for having a strange meson
between the and (as opposed to having a lighter
nonstrange one) and a suppression factor for having a
pair (rather than a
one) shared between the and
of a
configuration. The default parameter
values are based on a combination of experimental observation
and internal model predictions.

In the diquark model, a diquark is expected to have exactly the same transverse momentum distribution as a quark. For configurations the situation is somewhat more unclear, but we have checked that various possibilities give very similar results. The option implemented in the program is to assume no transverse momentum at all for the pair shared by the and , with all other pairs having the standard Gaussian spectrum with local momentum conservation. This means that the and 's are uncorrelated in a configuration and (partially) anticorrelated in the configurations, with the same mean transverse momentum for primary baryons as for primary mesons.

*Advanced popcorn*

In [Edé97], a revised popcorn model is presented, where the
separate production of the quarks in an effective diquark is taken
more seriously. The production of a
pair which breaks the
string is in this model determined by eq. (), also
when ending up in a diquark. Furthermore, the popcorn model is
re-implemented in such a way that eq () could be used
explicitly in the Monte Carlo. The two parameters

Several new routines have been added, and the diquark code has been extended with information about the curtain quark flavour, i.e. the pair that is shared between the baryon and antibaryon, but this is not visible externally. Some parameters are no longer used, while others have to be given modified values. This is described in section .

*Baryon remnant fragmentation*

Occasionally, the endpoint of a string is not a single parton, but a diquark or antidiquark, e.g. when a quark has been kicked out of a proton beam particle. One could consider fairly complex schemes for the resulting fragmentation. One such [And81] was available in JETSET version 6 but is no longer found here. Instead the same basic scheme is used as for diquark pair production above. Thus a diquark endpoint is fragmented just as if produced in the field behind a matching flavour, i.e. either the two quarks of the diquark enter into the same leading baryon, or else a meson is first produced, containing one of the quarks, while the other is contained in the baryon produced in the next step.

Similarly, the revised algorithm for baron production can be applied to endpoint diquarks, though only with some care [Edé97]. The suppression factor for popcorn mesons is derived from the assumption of colour fluctuations violating energy conservation and thus being suppressed by the Heisenberg uncertainty principle. When splitting an original diquark into two more independent quarks, the same kind of energy shift does not obviously emerge. One could still expect large separations of the diquark constituents to be suppressed, but the shape of this suppression is beyond the scope of the model. For simplicity, the same kind of exponential suppression as in the "true popcorn" case is implemented in the program. However, there is little reason for the strength of the suppression to be exactly the same in the different situations. Thus the leading rank meson production in a diquark jet is governed by a new parameter, which is independent of the popcorn parameters and in eq. (). Furthermore, in the process (original diquark baryon+ ) the spin 3/2 suppression should not apply at full strength. This suppression factor stems from the normalization of the overlapping and wavefunctions in a newly produced pair, but in the process considered here, two out of three valence quarks already exist as an initial condition of the string.

The diquark picture above is not adequate when two or more quarks are kicked out of a proton. Below, in section , we will introduce string junctions to describe such topologies.