SUBROUTINE QISTID - Generate instanton decay products

The parton types, the momenta (in the $gq^\prime$ CMS) and the colour/flavour connections of the $n_g + 2\,n_f -1$ outgoing partons of the instanton-induced subprocess $g+q^\prime \Rightarrow n_g\,g's + (2\,n_f-1)\,q's$ are generated. This proceeds as follows:

• In the subroutine QIGLST, $n_f$ $[\,q \ldots \overline{q}\,]$ - ``strings'' of partons are set up, each beginning with a quark, followed by a random number of gluons and ending with an anti-quark of randomly chosen flavour. There are $n_g+1$ gluons in total and all $n_f$ flavours occur precisely once. All partons are marked as outgoing.

  A simple example for $n_f=3$ and $n_g=4$ outgoing gluons may provide some illustration:
  

  $$[\,d\; g\; g\;\overline{s}\,]\hspace{5ex} [\,u\;g\;\overline{d}\,] \hspace{5ex} [\,s\;g\;g\;\overline{u}\,]\, .$$ (1)

  
  

• One of the $n_g+1$ gluons in above strings is randomly chosen and marked as incoming. Furthermore, the anti-parton corresponding to the virtual quark $q^\prime$ is found in above strings and marked as incoming. All this is done in the subroutine QIGPAR.

  Example: Suppose we have the following instanton-induced subprocess,
  

  $$g+s\Rightarrow 4\,g's + d + \overline{d} + u + \overline{u} + s \, .$$ (2)

  
  
  Then the above procedure might lead to the following strings,
  

  $$[\,d\; g\; g\;\overline{\bf s}\,]\hspace{5ex} [\,u\;g\;\ove... ...e{d}\,] \hspace{5ex} [\,s\;g\;{\bf g}\;\overline{u}\,]\, ,$$ (3)

  
  
  where the boldface entries denote the partons marked as incoming.

• The appropriate mass $m_i$ is assigned to each outgoing parton in the subroutine QIPLST.

• The four-momenta $p_i$ of the $n=2n_f-1+n_g$ outgoing partons in the instanton subprocess CMS are generated uniformly in energy-weighted phase-space,
  

  $${\int \prod_{q=1}^{2n_f-1} \left\{ d^4p_q\,\delta \left( p_q^2-m_... ...k=1}^{n_g} \left\{ d^4p_k\,\delta \left( p_k^2-m_g^2\right) p^{t\,2}_k\right\}}$$

  $$\times \, \delta^{(3)}\left( \sum_{q=1}^{2n_f-1} \vec{p}_q+ \sum_... ...\delta \left( W_i - \sum_{q=1}^{2n_f-1} p_q^t - \sum_{k=1}^{n_g} p_k^t\right) ,$$

  
  
  corresponding to the leading-order matrix element [1] with different energy weights for gluons and quarks. This is done in the subroutine QIPSGN.

• The four-momenta and particle identities of all outgoing partons are stored into the PHEP common block of HERWIG [2]. The corresponding IHEP pointers start at IHEP=11 for the first outgoing parton from the instanton subprocess (a $d$ quark in our example (3) above) and are increased consecutively along the strings. This is done by calling the subroutine QIPSTO.

  For the example (3) above, the IHEP pointers would read,
  

  $$\begin{tabular}{llccccrclcccrclccccl} Strings: & $[$\ & $d$\ & $g... ... & 11 & 12 & 13 & & & & & 14 & 15 & 16 & & & & 17 & 18 & & 19 & . \end{tabular}$$ (5)

  
  

• The colour/flavour connections are set up in the subroutine QICCON. The colour flow is constructed by connecting the colour lines of neighbouring partons within each of the strings. The flavour flow is constructed by connecting the flavour lines of the quark at the beginning of a string with the flavour line of the anti-quark at the end of a string.

  Let us illustrate, within our example (2), how this is done in practice:

  The IHEP pointers of the current quark $q^{\prime\prime}$($=\overline{s}$ in our example (2)), IHEP=10, and of the incoming gluon, IHEP=6, set up in QIHGEN, are added to the list of IHEP pointers, such that we have, for our example (3),
  

  $$\begin{tabular}{llccccrclcccrclccccl} Strings: & $[$\ & $d$\ & $g... ... {\bf 10} & & & & 14 & 15 & 16 & & & & 17 & 18 & {\bf 6} & 19 & . \end{tabular}$$ (6)

  
  
  The corresponding HERWIG [2] mother/daughter pointers¹ specifying the colour/flavour flow are then:
  

  $$\begin{tabular}{\vert r\vert r\vert c\vert c\vert}\hline IHEP & D... ...in & 19 & 18 \\ 19 & $\overline{u}$\ out & 17 & 6 \\ \hline \end{tabular}\, .$$ (7)