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 $\mathrm{q}/\overline{\mathrm{q}}$ 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 $\mathrm{q}_i$ 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:

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the relative probability to pick a $\overline{\mathrm{q}}\overline{\mathrm{q}}$ diquark rather than a $\mathrm{q}$;

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the extra suppression associated with a diquark containing a strange quark (over and above the ordinary $\mathrm{s}/ \u$ suppression factor $\gamma_s$); and

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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).

The extra strange diquark suppression factor comes about since what appears in the exponent of the tunnelling formula is $m^2$ and not $m$, so that the diquark and the strange quark suppressions do not factorize.

Only two baryon multiplets are included, i.e. there are no $L = 1$ excited states. The two multiplets are:

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$S = J = 1/2$: the `octet' multiplet of SU(3);

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$S = J = 3/2$: the `decuplet' multiplet of SU(3).

In contrast to the meson case, different flavour combinations have different numbers of states available: for $\u\u\u$ only $\Delta^{++}$, whereas $\u\d\mathrm{s}$ may become either $\Lambda$, $\Sigma^0$ or $\Sigma^{*0}$.

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. $\c$ 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 $D$ $T_{D}$, the number of spin states $\sigma_{D}$ and the SU(6) factor for $D$ and a quark $\mathrm{q}$ $SU_{D,\mathrm{q}}$, the model prediction for the $(\mathrm{q}\rightarrow B + D)/(\mathrm{q}\rightarrow M + \mathrm{q}')$ ratio is

$$S_{\mathrm{q}}=\frac{P(\mathrm{q}\mathrm{q})}{P(\mathrm{q})}\sum_{D} T_{D} \sigma_{D} SU_{D,\mathrm{q}} ~.$$ (222)

(Neglecting this flavour dependence e.g. leads to an enhancement of the $\Omega^-$ relative to primary proton production with approximately a factor $1.2$, using JETSET 7.4 default values.) Since the chosen algorithm implies the normalization $\sum_{D} T_{D} \sigma_{D} = 1$ and $SU_{D,\mathrm{q}} \le 1$, the final diquark production rate is somewhat reduced from the $P(\mathrm{q}\mathrm{q})/P(\mathrm{q})$ input value.

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 $\mathrm{q}_i \overline{\mathrm{q}}_i$ pairs. The picture is the following. Assume that the original $\mathrm{q}$ is red $r$ and the $\overline{\mathrm{q}}$ is $\overline{r}$. Normally a new $\mathrm{q}_1 \overline{\mathrm{q}}_1$ pair produced in the field would also be $r \overline{r}$, so that the $\overline{\mathrm{q}}_1$ is pulled towards the $\mathrm{q}$ end and vice versa, and two separate colour-singlet systems $\mathrm{q}\overline{\mathrm{q}}_1$ and $\mathrm{q}_1 \overline{\mathrm{q}}$ are formed. Occasionally, the $\mathrm{q}_1 \overline{\mathrm{q}}_1$ pair may be e.g. $g \overline{g}$ ($g$ = green), in which case there is no net colour charge acting on either $\mathrm{q}_1$ or $\overline{\mathrm{q}}_1$. Therefore, the pair cannot gain energy from the field, and normally would exist only as a fluctuation. If $\mathrm{q}_1$ moves towards $\mathrm{q}$ and $\overline{\mathrm{q}}_1$ towards $\overline{\mathrm{q}}$, the net field remaining between $\mathrm{q}_1$ and $\overline{\mathrm{q}}_1$ is $\overline{b} b$ ($b$ = blue; $g + r = \overline{b}$ if only colour triplets are assumed). In this central field, an additional $\mathrm{q}_2 \overline{\mathrm{q}}_2$ pair can be created, where $\mathrm{q}_2$ now is pulled towards $\mathrm{q}\mathrm{q}_1$ and $\overline{\mathrm{q}}_2$ towards $\overline{\mathrm{q}}\overline{\mathrm{q}}_1$, with no net colour field between $\mathrm{q}_2$ and $\overline{\mathrm{q}}_2$. If this is all that happens, the baryon $B$ will be made up out of $\mathrm{q}_1$, $\mathrm{q}_2$ and some $\mathrm{q}_4$ produced between $\mathrm{q}$ and $\mathrm{q}_1$, and $\overline{B}$ of $\overline{\mathrm{q}}_1$, $\overline{\mathrm{q}}_2$ and some $\overline{\mathrm{q}}_5$, i.e. the $B$ and $\overline{B}$ will be nearest neighbours in rank and share two quark pairs. Specifically, $\mathrm{q}_1$ will gain energy from $\mathrm{q}_2$ in order to end up on mass shell, and the tunnelling formula for an effective $\mathrm{q}_1 \mathrm{q}_2$ diquark is recovered.

Part of the time, several $b \overline{b}$ colour pair productions may take place between the $\mathrm{q}_1$ and $\overline{\mathrm{q}}_1$, however. With two production vertices $\mathrm{q}_2 \overline{\mathrm{q}}_2$ and $\mathrm{q}_3 \overline{\mathrm{q}}_3$, a central meson $\overline{\mathrm{q}}_2 \mathrm{q}_3$ may be formed, surrounded by a baryon $\mathrm{q}_4 \mathrm{q}_1 \mathrm{q}_2$ and an antibaryon $\overline{\mathrm{q}}_3 \overline{\mathrm{q}}_1 \overline{\mathrm{q}}_5$. We call this a $BM\overline{B}$ configuration to distinguish it from the $\mathrm{q}_4 \mathrm{q}_1 \mathrm{q}_2$ + $\overline{\mathrm{q}}_2 \overline{\mathrm{q}}_1 \overline{\mathrm{q}}_5$ $B\overline{B}$ configuration above. For $BM\overline{B}$ the $B$ and $\overline{B}$ only share one quark-antiquark pair, as opposed to two for $B\overline{B}$ configurations. The relative probability for a $BM\overline{B}$ configuration is given by the uncertainty relation suppression for having the $\mathrm{q}_1$ and $\overline{\mathrm{q}}_1$ sufficiently far apart that a meson may be formed in between. The suppression of the $BM\overline{B}$ system is estimated by

$$\vert\Delta_F\vert^2\approx \exp(-2\mu_{\perp} M_{\perp}/\kappa)$$ (223)

where $\mu_{\perp}$ and $M_{\perp}$ is the transverse mass of $\mathrm{q}_1$ and the meson, respectively. Strictly speaking, also configurations like $BMM\overline{B}$, $BMMM\overline{B}$, etc. should be possible, but since the total invariant $M_{\perp}$ grows rapidly with the number of mesons, the probability for this is small in the simple model. Further, since larger masses corresponds to longer string pieces, the production of pseudoscalar mesons is favoured over that of vector ones. If only $B\overline{B}$ and $BM\overline{B}$ states are included, and if the probability for having a vector meson $M$ is not suppressed extra, two partly compensating errors are made (since a vector meson typically decays into two or more pseudoscalar ones).

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

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$\mathrm{q}_1 \to \mathrm{q}_2 + (\mathrm{q}_1 \overline{\mathrm{q}}_2)$ meson;

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$\mathrm{q}_1 \to \overline{\mathrm{q}}_2\overline{\mathrm{q}}_3 + (\mathrm{q}_1 \mathrm{q}_2 \mathrm{q}_3)$ baryon;

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$\mathrm{q}_1 \mathrm{q}_2 \to \overline{\mathrm{q}}_3 + (\mathrm{q}_1 \mathrm{q}_2 \mathrm{q}_3)$ baryon;

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$\mathrm{q}_1 \mathrm{q}_2 \to \mathrm{q}_1 \mathrm{q}_3 + (\mathrm{q}_2 \overline{\mathrm{q}}_3)$ meson;

with the constraint that the last process cannot be iterated to obtain several mesons in between the baryon and the antibaryon.

When selecting flavours for $\mathrm{q}\mathrm{q}\rightarrow M+\mathrm{q}\mathrm{q}'$, the quark coming from the accepted $\mathrm{q}\mathrm{q}$ is kept, and the other member of $\mathrm{q}\mathrm{q}'$, as well as the spin of $\mathrm{q}\mathrm{q}'$, is chosen with weights taking SU(6) symmetry into account. Thus the flavour of $\mathrm{q}\mathrm{q}$ is not influenced by SU(6) factors for $\mathrm{q}\mathrm{q}'$, but the flavour of $M$ 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 $\mathrm{s}/ \u$ ratio and the $\mathrm{q}\mathrm{q}/ \mathrm{q}$ 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 $B$ and $\overline{B}$ is constrained to be a meson). With the improved SU(6) treatment introduced in PYTHIA 6.106, a rejected $\mathrm{q}\rightarrow B +\mathrm{q}\mathrm{q}$ may lead to the splitting $\mathrm{q}\rightarrow M+\mathrm{q}'$ instead. This calls for an increase of the $\mathrm{q}\mathrm{q}/ \mathrm{q}$ 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 $B\overline{B}$ configurations. For $BM\overline{B}$ configurations, the square root of a suppression factor should be applied if the factor is relevant only for one of the $B$ and $\overline{B}$, e.g. if the $B$ is formed with a spin 1 `diquark' $\mathrm{q}_1 \mathrm{q}_2$ but the $\overline{B}$ with a spin 0 diquark $\overline{\mathrm{q}}_1 \overline{\mathrm{q}}_3$. Additional parameters include the relative probability for $BM\overline{B}$ configurations, which is assumed to be roughly 0.5 (with the remaining 0.5 being $B\overline{B}$), a suppression factor for having a strange meson $M$ between the $B$ and $\overline{B}$ (as opposed to having a lighter nonstrange one) and a suppression factor for having a $\mathrm{s}\overline{\mathrm{s}}$ pair (rather than a $\u\overline{\mathrm{u}}$ one) shared between the $B$ and $\overline{B}$ of a $BM\overline{B}$ 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 $BM\overline{B}$ 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 $\mathrm{q}_1 \overline{\mathrm{q}}_1$ pair shared by the $B$ and $\overline{B}$, with all other pairs having the standard Gaussian spectrum with local momentum conservation. This means that the $B$ and $\overline{B}$ $p_{\perp}$:s are uncorrelated in a $BM\overline{B}$ configuration and (partially) anticorrelated in the $B\overline{B}$ 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 $\mathrm{q}\overline{\mathrm{q}}$ 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

$$\beta_{\mathrm{q}} \equiv 2\left\langle \mu_{\perp \mathrm{q... ...thrm{and}}~\Delta\beta \equiv \beta_{\mathrm{s}}-\beta_{\u },$$ (224)

then govern both the diquark and the intermediate meson production. In this algorithm, configurations like $BMM\overline{B}$ etc. are considered in a natural way. The more independent production of the diquark partons implies a moderate suppression of spin 1 diquarks. Instead the direct suppression of spin 3/2 baryons, in correspondence to the suppression of vector mesons relative to pseudo-scalar ones, is assumed to be important. Consequently, a suppression of $\Sigma$-states relative to $\Lambda^0$ is derived from the spin 3/2 suppression parameter.

Several new routines have been added, and the diquark code has been extended with information about the curtain quark flavour, i.e. the $\mathrm{q}\overline{\mathrm{q}}$ 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 $\mathrm{q}\mathrm{q}$ diquark endpoint is let to fragment just as would a $\mathrm{q}\mathrm{q}$ produced in the field behind a matching $\overline{\mathrm{q}}\overline{\mathrm{q}}$ 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 this must be made 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 $\beta$ parameter, which is independent of the popcorn parameters $\beta_{\mathrm{u}}$ and $\Delta \beta$ in eq. (). Furthermore, in the process (original diquark $\rightarrow$ baryon+ $\overline{\mathrm{q}}$) the spin 3/2 suppression should not apply at full strength. This suppression factor stems from the normalization of the overlapping $\mathrm{q}$ and $\overline{\mathrm{q}}$ wavefunctions in a newly produced $\mathrm{q}\overline{\mathrm{q}}$ pair, but in the process considered here, two out of three valence quarks already exist as an initial condition of the string.