Beam Remnants
 
  - Introduction
- Primordial kT
- Colour flow
- Further variables
Introduction
 
 
The BeamParticle class contains information on all partons 
extracted from a beam (so far). As each consecutive multiparton interaction 
defines its respective incoming parton to the hard scattering a 
new slot is added to the list. This information is modified when 
the backwards evolution of the spacelike shower defines a new 
initiator parton. It is used, both for the multiparton interactions 
and the spacelike showers, to define rescaled parton densities based 
on the x and flavours already extracted, and to distinguish 
between valence, sea and companion quarks. Once the perturbative 
evolution is finished, further beam remnants are added to obtain a 
consistent set of flavours. The current physics framework is further 
described in [Sjo04]. 
 
 
The introduction of rescattering 
in the multiparton interactions framework further complicates the 
processing of events. Specifically, when combined with showers, 
the momentum of an individual parton is no longer uniquely associated 
with one single subcollision. Nevertheless the parton is classified 
with one system, owing to the technical and administrative complications 
of more complete classifications. Therefore the addition of primordial 
kT to the subsystem initiator partons does not automatically 
guarantee overall pT conservation. Various tricks are used to 
minimize the mismatch, with a brute force shift of all parton 
pT's as a final step. 
 
 
Much of the above information is stored in a vector of 
ResolvedParton objects, which each contains flavour and 
momentum information, as well as valence/companion information and more. 
The BeamParticle method list() shows the 
contents of this vector, mainly for debug purposes. 
 
 
The BeamRemnants class takes over for the final step 
of adding primordial kT to the initiators and remnants, 
assigning the relative longitudinal momentum sharing among the 
remnants, and constructing the overall kinematics and colour flow. 
This step couples the two sides of an event, and could therefore 
not be covered in the BeamParticle class, which only 
considers one beam at a time. 
 
 
The methods of these classes are not intended for general use, 
and so are not described here. 
 
 
In addition to the parameters described on this page, note that the 
choice of parton densities is made 
in the Pythia class. Then pointers to the pdf's are handed 
on to BeamParticle at initialization, for all subsequent 
usage. 
 
 
Primordial kT
 
 
The primordial kT of initiators of hard-scattering subsystems 
are selected according to Gaussian distributions in p_x and 
p_y separately. The widths of these distributions are chosen 
to be dependent on the hard scale of the central process and on the mass 
of the whole subsystem defined by the two initiators: 
 
sigma = (sigma_soft * Q_half + sigma_hard * Q) / (Q_half + Q) 
  * m / (m + m_half * y_damp) 
 
Here Q is the hard-process renormalization scale for the 
hardest process and the pT scale for subsequent multiparton 
interactions, m the mass of the system, and 
sigma_soft, sigma_hard, Q_half, 
m_half and y_damp parameters defined below. 
Furthermore each separately defined beam remnant has a distribution 
of width sigma_remn, independently of kinematical variables. 
 
Note that, for external (LHE) events Q_half is treated as 
zero. This is so that LHE events with low-pT extra jets (e.g., in 
the context of POWHEG-style merging) are given the same primordial kT 
as their Born-level counterparts. 
 
flag   BeamRemnants:primordialKT   
 (default = on)
Allow or not selection of primordial kT according to the 
parameter values below. 
   
 
parm   BeamRemnants:primordialKTsoft   
 (default = 0.9; minimum = 0.)
The width sigma_soft in the above equation, assigned as a 
primordial kT to initiators in the soft-interaction 
limit. 
   
 
parm   BeamRemnants:primordialKThard   
 (default = 1.8; minimum = 0.)
The width sigma_hard in the above equation, assigned as a 
primordial kT to initiators in the hard-interaction limit. 
   
 
parm   BeamRemnants:halfScaleForKT   
 (default = 1.5; minimum = 0.)
The scale Q_half in the equation above, defining the 
half-way point between hard and soft interactions. For external (LHE) 
events, this parameter is treated as zero. 
   
 
parm   BeamRemnants:halfMassForKT   
 (default = 1.; minimum = 0.)
The scale m_half in the equation above, defining the 
half-way point between low-mass and high-mass subsystems. 
(Kinematics construction can easily fail if a system is assigned 
a primordial kT value higher than its mass, so the 
mass-damping is intended to reduce some troubles later on.) 
   
 
parm   BeamRemnants:reducedKTatHighY   
 (default = 0.5; minimum = 0.; maximum = 1.)
For a system of mass m and energy E the 
damping factor y_damp above is defined as 
y_damp = pow( E/m, r_red), where r_red is the 
current parameter. The effect is to reduce the primordial kT 
of low-mass systems extra much if they are at large rapidities (recall 
that E/m = cosh(y) before kT is added). The reason 
for this damping is purely technical, and for reasonable values 
should not have dramatic consequences overall. 
   
 
parm   BeamRemnants:primordialKTremnant   
 (default = 0.4; minimum = 0.)
The width sigma_remn, assigned as a primordial kT 
to beam-remnant partons. 
   
 
 
A net kT imbalance is obtained from the vector sum of the 
primordial kT values of all initiators and all beam remnants. 
This quantity is compensated by a shift shared equally between 
all partons, except that the damping factor m / (m_half + m) 
is again used to suppress the role of small-mass systems. 
 
 
Note that the current sigma definition implies that 
<pT^2> = <p_x^2>+ <p_y^2> = 2 sigma^2. 
It thus cannot be compared directly with the sigma 
of nonperturbative hadronization, where each quark-antiquark 
breakup corresponds to <pT^2> = sigma^2 and only 
for hadrons it holds that <pT^2> = 2 sigma^2. 
The comparison is further complicated by the reduction of 
primordial kT values by the overall compensation mechanism. 
 
flag   BeamRemnants:rescatterRestoreY   
 (default = off)
Is only relevant when rescattering 
is switched on in the multiparton interactions scenario. For a normal 
interaction the rapidity and mass of a system is preserved when 
primordial kT is introduced, by appropriate modification of the 
incoming parton momenta. Kinematics construction is more complicated for 
a rescattering, and two options are offered. Differences between these 
can be used to explore systematic uncertainties in the rescattering 
framework.
 
The default behaviour is to keep the incoming rescattered parton as is, 
but to modify the unrescattered incoming parton so as to preserve the 
invariant mass of the system. Thereby the rapidity of the rescattering 
is modified.
 
The alternative is to retain the rapidity (and mass) of the rescattered 
system when primordial kT is introduced. This is made at the 
expense of a modified longitudinal momentum of the incoming rescattered 
parton, so that it does not agree with the momentum it ought to have had 
by the kinematics of the previous interaction.
 
For a double rescattering, when both incoming partons have already scattered, 
there is no obvious way to retain the invariant mass of the system in the 
first approach, so the second is always used. 
   
 
 
Colour flow
 
 
The colour in the separate subproccsses are tied together via the assignment 
of colour flow in the beam remnants. The assignment of colour flow is not 
known from first principles and therefore it is not an unambiguous procedure. 
Thus two different models have been implemented in Pythia. These 
will be referred to as new and old, based on the time of the implementation. 
 
 
The old model tries to reconstruct the colour flow in a way that a LO PS would 
produce the beam remnants. The starting point is the junction structure of the 
beam particle (if it is a baryon). The gluons are attached to a quark line and 
quark-antiquark pairs are added as if coming from a gluon splittings. Thus 
this model captures the qualitative behaviour that is expected from leading 
colour QCD. The model is described in more detail in [Sjo04]. 
 
 
The new model is built on the full SU(3) colour structure of QCD. The 
starting point is the scattered partons from the MPI. Each of these are 
initially assumed uncorrelated in colour space, allowing the total outgoing 
colour configuration to be calculated as an SU(3) product. Since the beam 
particle is a colour singlet, the beam remnant colour configuration has to be 
the inverse of the outgoing colour configuration. The minimum amount of gluons 
are added to the beam remnant in order to obtain this colour configuration. 
 
 
The above assumption of uncorrelated MPIs in colour space is a good 
assumption for a few well separated hard MPIs. However if the number of MPIs 
become large and ISR is included, such that the energy scale becomes lower 
(and thus distances becomes larger), the assumption loses its validity. This 
is due to saturation effects. The modelling of saturation is done in crude 
manner, as an exponential suppresion of high multiplet states. 
 
 
None of the models above can provide a full description of the colour 
flow in an event, however. Therefore additional colour reconfiguration 
is needed. This is referred to as colour reconnection. Several different 
models for colour reconnection are implemented, see 
Colour Reconection. 
 
mode   BeamRemnants:remnantMode   
 (default = 0; minimum = 0; maximum = 1)
Switch to choose between the two different colour models for the beam remnant. 
option  0 :  The old beam remnant model.    
option  1 :  The new beam remnant model.    
   
 
parm   BeamRemnants:saturation   
 (default = 5; minimum = 0.1; maximum = 100000)
Controls the suppresion due to saturation in the new model. The exact formula 
used is exp(-M / k), where M is the multiplet size and k is this 
parameter. Thus a small number will result in a large saturation. 
   
 
 
Further variables
 
 
mode   BeamRemnants:maxValQuark   
 (default = 5; minimum = 0; maximum = 5)
The maximum valence quark kind allowed in acceptable incoming beams, 
for which multiparton interactions are simulated. Default is that hadrons 
may contain all quarks except top, but e.g. a change to 3 would also 
exclude c and b hadrons. 
   
 
mode   BeamRemnants:companionPower   
 (default = 4; minimum = 0; maximum = 4)
When a sea quark has been found, a companion antisea quark ought to be 
nearby in x. The shape of this distribution can be derived 
from the gluon mother distribution convoluted with the 
g → q qbar splitting kernel. In practice, simple solutions 
are only feasible if the gluon shape is assumed to be of the form 
g(x) ~ (1 - x)^p / x, where p is an integer power, 
the parameter above. Allowed values correspond to the cases programmed. 
 
Since the whole framework is approximate anyway, this should be good 
enough. Note that companions typically are found at small Q^2, 
if at all, so the form is supposed to represent g(x) at small 
Q^2 scales, close to the lower cutoff for multiparton interactions. 
   
 
 
When assigning relative momentum fractions to beam-remnant partons, 
valence quarks are chosen according to a distribution like 
(1 - x)^power / sqrt(x). This power is given below 
for quarks in mesons, and separately for u and d 
quarks in the proton, based on the approximate shape of low-Q^2 
parton densities. The power for other baryons is derived from the 
proton ones, by an appropriate mixing. The x of a diquark 
is chosen as the sum of its two constituent x values, and can 
thus be above unity. (A common rescaling of all remnant partons and 
particles will fix that.) An additional enhancement of the diquark 
momentum is obtained by its x value being rescaled by the 
valenceDiqEnhance factor. 
 
parm   BeamRemnants:valencePowerMeson   
 (default = 0.8; minimum = 0.)
The abovementioned power for valence quarks in mesons. 
   
 
parm   BeamRemnants:valencePowerUinP   
 (default = 3.5; minimum = 0.)
The abovementioned power for valence u quarks in protons. 
   
 
parm   BeamRemnants:valencePowerDinP   
 (default = 2.0; minimum = 0.)
The abovementioned power for valence d quarks in protons. 
   
 
pvec   BeamRemnants:heavyQuarkEnhance   
 (default = {1.5,4.5,14.5}; minimum = 1.)
Enhancement factors for the momentum fraction assigned to heavy valence quarks 
(s, c, b respectively) relative to the normal u and 
d ones. The default values are based on the ratio of constituent 
masses, as can be motivated by an expectation of equal average velocity for 
all valence quarks. In low-mass  diffraction, its inverse is also used 
to provide the relative probability for each valence quark to be kicked 
backwards, out from the incoming hadron. (And thus the value itself 
the relative probability to remain in the beam direction.) This is 
consistent with the Additive Quark Model approach, where it is often 
assumed that each valence quark adds to the total cross section of 
its hadron inversely to its mass. Thus, in both cases, an above-unity 
value for a constituent quark flavour will tend to enhance its momentum 
fraction in the beam direction. Note that sea quarks are not affected by 
either of these two considerations. 
   
 
parm   BeamRemnants:valenceDiqEnhance   
 (default = 2.0; minimum = 0.5; maximum = 10.)
Enhancement factor for valence diquarks in baryons, relative to the 
simple sum of the two constituent quarks. 
   
 
parm   BeamRemnants:gluonPower   
 (default = 4.0; minimum = 0.)
The abovementioned power for gluons. 
   
 
parm   BeamRemnants:xGluonCutoff   
 (default = 1E-7; minimum = 1E-10; maximum = 1)
The gluon PDF is approximated with g(x) ~ (1 - x)^p / x, which 
integrates to infinity when integrated from 0 to 1. This cut-off is 
introduced as a minimum to avoid the problems with infinities. 
   
 
flag   BeamRemnants:allowJunction   
 (default = on)
The off option is intended for debug purposes only, as 
follows. When more than one valence quark is kicked out of a baryon 
beam, as part of the multiparton interactions scenario, the subsequent 
hadronization is described in terms of a junction string topology. 
This description involves a number of technical complications that 
may make the program more unstable. As an alternative, by switching 
this option off, junction configurations are rejected (which gives 
an error message that the remnant flavour setup failed), and the 
multiparton interactions and showers are redone until a 
junction-free topology is found. 
   
 
flag   BeamRemnants:beamJunction   
 (default = off)
This parameter is only relevant if the new colour reconnection scheme is used. 
(see  colour reconnection) 
This parameter tells whether to form a junction or a di-quark if more 
than two valence quarks are found in the beam remnants. If off a di-quark is 
formed and if on a junction will be formed. 
   
 
flag   BeamRemnants:allowBeamJunction   
 (default = on)
This parameter is only relevant if the new Beam remnant model is used. 
This parameter tells whether to allow the formation of junction structures 
in the colour configuration of the scattered partons. 
   
 
mode   BeamRemnants:unresolvedHadron   
 (default = 0; minimum = 0; maximum = 3)
Switch to to force either or both of the beam remnants to collapse to a 
single hadron, namely the original incoming one. Must only be used when this 
is physically meaningful, e.g. when a photon can be viewed as emitted from 
a proton that does not break up in the process. 
option  0 :  Both hadronic beams are resolved.    
option  1 :  Beam A is unresolved, beam B resolved.    
option  2 :  Beam A is resolved, beam B unresolved.    
option  3 :  Both hadronic beams are unresolved.    
   
 
 
Empirically it is known that the leading baryon in the fragmentation 
region is softer in PYTHIA than in data. The following settings 
allow the possibility to modify the leading-diquark fragmentation to 
(partly) address this issue. 
 
parm   BeamRemnants:dampPopcorn   
 (default = 1.; minimum = 0.; maximum = 1.)
Controls whether a beam remnant diquark can hadronize to a leading meson 
(by the popcorn mechanism) or not. If 1 then a remnant behaves just like 
in ordinary hadronization, while if 0 then a diquark will always produce 
a leading baryon. Intermediate values will interpolate between these two 
extremes. Has no influence on diquarks not coming from a beam remnant. 
   
 
flag   BeamRemnants:hardRemnantBaryon   
 (default = off)
Allows a remnant diquark to hadronize to a harder-than-normal baryon, 
assuming that forbidPopcorn is switched on. Then the next 
two parameters override the normal a and b of the 
Lund symmetric fragmentation function. 
   
 
parm   BeamRemnants:aRemnantBaryon   
 (default = 0.; minimum = 0.; maximum = 2.)
For the special hard remnant baryon handling above, this parameter is 
the power a in the (1-z)^a factor of the fragmentation 
function. 
   
 
parm   BeamRemnants:bRemnantBaryon   
 (default = 2.; minimum = 0.5; maximum = 5.)
For the special hard remnant baryon handling above, this parameter is 
the factor b in the exp(-b m_T^2/z factor of the 
fragmentation function.