SUBROUTINE QIHGEN - Main QCD-instanton hard process generator

For the generation of an event, this routine is called twice: in the first call, essentially a Monte Carlo weight corresponding to the instanton-induced total cross section is calculated. In the second call, the generation of the event is completed.

In the first call, with the variable GENEV set to .FALSE., the following steps are performed in QIHGEN:

• It gets the data on the incoming beams, does the initial state radiation (dummy routine EXFRAC), calculates derived parameters and checks limits by calling the subroutine QICALC.
• The HERWIG [1] identity of the current quark $q^{\prime\prime}$ and of the virtual quark $q^\prime$ are generated by calling QIHPAR.
• By calling QIHINS, the Bjorken variables $Q^{\prime 2}$ and $x^\prime$ of the instanton subprocess are generated and a Monte Carlo weight¹ $W_{Q^\prime x^\prime\sigma}$ corresponding to the integral
  

  $$\int\limits_{Q^{\prime 2}_{\rm min}}^{Q^{\prime 2}_{\rm max... ...\prime}\ \sigma^{(I)}_{q^\prime g}(x^\prime, Q^{\prime 2} )$$ (1)

  
  
  is calculated, where $\sigma^{(I)}_{q^\prime g}$ denotes the instanton-induced $q^\prime g$ total cross section from Ref. [2].
• The gluon multiplicity is generated by QIGMUL.
• The value of $z$ ($P$'s momentum fraction carried by the incoming gluon) is generated and the Monte Carlo weight $W_{z g}$ corresponding to the integral
  

  $$\int\limits_{{\rm max}\left(\frac{Q^{\prime 2}}{Sx^\prime ... ...min}}{x^\prime} \right)}^{z_{\rm max}} dz\,f_g (z, \mu_f ),$$ (2)

  
  
  where $f_g$ is the gluon distribution in the proton, is calculated. By default, $f_g$ is taken from Ref. [3] (set 1.1) and the factorization scale $\mu_f$ is identified with $Q^\prime_{\rm min}$.

  For default setting of the contrôl flags in QIINIT, the value of $z$ as well as the Monte Carlo weight $W_{z g}$ are generated as follows: Random points $y_z$, with
  

  $$y_z = \ln z\hspace{3ex} \Leftrightarrow\hspace{3ex} z = \exp y_z$$ (3)

  
  
  are generated uniformly in the interval between $\ln z_L$ and $\ln z_U$, where $z_L$ and $z_U$ are the lower and upper integration limits in Eq. (2), respectively. The Monte Carlo weight associated with Eq. (2) is then
  

  $$W_{zg} = (\ln z_U - \ln z_L )\,z\,f_g(z,\mu_f) .$$ (4)

  
  

• Next, the Bjorken variables $x_{\rm Bj}$ and $y_{\rm Bj}$ are generated analogously to the $z$ variable and the weight $W_{x_{\rm Bj}\,y_{\rm Bj}}$,
  

  $$W_{x_{\rm Bj}\,y_{\rm Bj}} = (\ln x_{{\rm Bj}\,U} - \ln x... ...j}\,L} )\, (\ln y_{{\rm Bj}\,U} - \ln y_{{\rm Bj}\,L} ) \, ,$$ (5)

  
  
  corresponding to the integral
  

  $$\int\limits_{x_{\rm Bj\,min}}^{x^\prime z - \frac{m_{q^{\p... ...rm Bj}}\ \theta (S x_{\rm Bj} y_{\rm Bj} - Q^2_{\rm min})\,$$ (6)

  
  
  is calculated. In Eq. (5), $\ln x_{{\rm Bj}\,U}$ and $\ln y_{{\rm Bj}\,U}$ denote the upper and $\ln x_{{\rm Bj}\,L}$ and $\ln y_{{\rm Bj}\,L}$ denote the lower integration limits in Eq. (6), respectively.

• The weight arising from the splitting of the photon into a quark-antiquark pair is inferred from QISPLT,
  

  $$W_{q^\prime} = \frac{1}{x^\prime z}\, P_{q^\prime}^{{ (I)}} (x^\prime ,Q^{\prime 2} ,z,Q^2=S x_{\rm Bj} y_{\rm Bj}).$$ (7)

  
  
  The factor $P_{q^\prime}^{{ (I)}}$ accounts for the flux of virtual (anti-)quarks $q^\prime$ in the $I$-background entering the $I$-induced $q^\prime p$-subprocess from the photon side [2,4].
• The weight from the flux of virtual photons is calculated,
  

  $$W_\gamma = \frac{1}{S x_{\rm Bj} y_{\rm Bj}} \left(1-y_{\rm Bj} +\frac{y_{\rm Bj}^2}{2}\right) .$$ (8)

  
  

• The remaining weight factors to yield finally a properly normalized cross section (in units of nb) are inferred from QIPVWT,
  

  $$W_n = 2\,\pi\,\alpha^2\, \sum\limits_{q^\prime} e_{q^\prime}^2\, x_{\rm Bj}\, x^\prime .$$ (9)

  
  
  Here $\alpha$ is the electro-magnetic fine-structure constant and $e_{q^\prime}$ is the fractional electro-magnetic charge of the virtual (anti-)quark $q^\prime$. The sum extends over all quarks and anti-quarks of all $n_f$ flavours.
• The total event weight (EVWGT) is built by multiplying the above weights,
  

  $$W_{eP}^{(I)} = W_{Q^\prime x^\prime\sigma}\, W_{z g}\,W_{x_{\rm Bj}\,y_{\rm Bj}}\,W_{q^\prime}\, W_\gamma\,W_n .$$ (10)

  
  
  As is easily seen, the total event weight (10) is the relative contribution of the generated Bjorken variables, $(Q^{\prime 2},x^\prime ,z,x_{\rm Bj},y_{\rm Bj})$, to the total cross section of instanton-induced processes [2],
  

  $$\sigma_{eP}^{(I)}({\rm Cuts}) \simeq \frac{2\,\pi\,\alpha^2}{S}\,\sum\limits_{q^\prime} e_{q^\prime}^2... ...^\prime}\, \,\frac{\sigma^{(I)}_{q^\prime g}(x^\prime ,Q^{\prime 2})}{x^\prime}$$

  $$\times \int\limits_{{\rm max}\left(\frac{Q^{\prime 2}}{Sx^\prime ... ...rm Bj\,max}-\frac{Q^{\prime 2}}{Sx^\prime z}}} \frac{dx_{\rm Bj}}{x_{\rm Bj}}\,$$ (11)

  $$\times \int\limits_{{\rm max}\left( \frac{Q^{\prime 2}}{Sx^\prime... ..._{\rm min})\, \frac{1+(1-y_{\rm Bj})^2}{y_{\rm Bj}}\ P_{q^\prime}^{{ (I)}}\ .$$

  
  

This ends the first call of QIHGEN.

For standard settings of the contrôl flags in the HERWIG [1] routine HWIGIN (NOWGT=.TRUE.), the rejection method is applied to end up with unweighted events: The generated total event weight (10) is compared in the modified HERWIG routine HWEPRO with a random number $r\in (0,1)$ times the previously determined maximum weight $W_{\rm max}$ (see description of the subroutine QCLOOP). Only ``events'', i.e. generated values of $(Q^{\prime 2},x^\prime ,z,x_{\rm Bj},y_{\rm Bj})$, for which

$$W_{eP}^{(I)} > r\,\cdot W_{\rm max}$$ (12)

are accepted, that means for those events the variable GENEV is now set to .TRUE.. Furthermore, one associates to each accepted event an event weight equal to the total cross section (11). It is clear that the accepted events are then distributed according to the differential cross section

$$\frac{1}{\sigma_{eP}^{(I)}}\ \frac{d^5\sigma_{eP}^{(I)}} {dQ^{\prime 2}\,dx^\prime\, dz\, dx_{\rm Bj}\, dy_{\rm Bj}} \, ,$$ (13)

whose explicit expression can be easily read off from Eq. (11).

If the event is accepted, QIHGEN is called again, with the variable GENEV now set to .TRUE.. The following steps are then performed:

• The lepton beam is copied over.
• The 4-momentum of the incoming gluon, $g$, is generated by calling QIKPAR and put into the event record.
• The 4-momentum of the virtual photon, $q$, is generated by QIKGAM.
• The 4-momentum of the outgoing $e^\pm$, $e^\prime$, is calculated and stored.
• The 4-momentum of the virtual quark, $q^\prime$, and the outgoing current quark, $q^{\prime\prime}$, is generated by QIKGSP and these vectors are copied into the event record.
• The data for instanton generation routines are set up and the instanton subprocess momentum vector is calculated.
• The hard subprocess colour connections are inserted.
• The instanton decay products are generated by calling QISTID.

Table 1: Variables set in QIHGEN.

Name	Description
ENTOT	$eP$ CM energy $\sqrt{S}$
QISGAM	$eP$ CM energy squared $S$
IDN(1)	HERWIG identity (IDHW) of incoming $e$
IDN(2)	HERWIG identity (IDHW) of incoming $g$
EMSCA	Factorization scale $\mu_f=$ QIUPAR(18)
QIZPAR	Generated $z$
QIXONE	$x^\prime\,z$
QIYONE	$Q^{\prime 2}/(S x^\prime z )$
QIXBJG	Generated $x_{\rm Bj}$
QIYBJG	Generated $y_{\rm Bj}$
QIQ2GA	Bjorken variable $Q^2$ calculated via $Q^2=S x_{\rm Bj} y_{\rm Bj}$
QIELEN	Energy of incoming $e$, $e_t$
QIPREN	Energy of incoming $P$, $P_t$
CSFACT	Weight $W_{zg}\,W_{x_{\rm Bj}\,y_{\rm Bj}}$ corresponding to the product
	of the integrals (2) and (6)
WEIGHT	Weight $W_{q^\prime}\,W_\gamma\,W_n$ corresponding to the product
	of the integrals (7), (8) and (9)
EVWGT	Weight $W_{eP}^{(I)}$ (10) corresponding to the cross section (11)
QIMEWT	Internal storage of weight corresponding to Eq. (11) for analysis
QIPINC(1,3)	Particle data group identity (IDPDG) of virtual quark $q^\prime$
QIPINC(2,3)	Particle data group identity (IDPDG) of incoming gluon $g$
QIPINC(1,4)(=10)	IHEP pointer of current quark $q^{\prime\prime}$
QIPINC(2,4)(=6)	IHEP pointer of incoming $g$
QISHEP	CM energy squared of instanton subprocess, $W_i^2=(q^\prime + g)^2=s^\prime$