# Multiple-Parton Interactions model¶

The default underlying event model of Herwig++ is currently based on the eikonal model. It models the underlying event activity as additional semi-hard and soft partonic interactions. In doing so, it allows the description of minimum bias events as well as the underlying event in hard scattering processes.

## Main parameters of the model¶

The main parameters of the model are those controlling the matter distribution in the proton

\(\mu^2\), the inverse hadron radius squared which is set using:

set /Herwig/UnderlyingEvent/MPIHandler:InvRadius x

and

\(p_\perp^{\min}\), a transition scale between soft and semi-hard nteractions [1] which by default is now set using a power-law energy extrapolation

\(p_\perp^{\min}=p_{\perp,0}^{\min}\left(\frac{\sqrt{s}}{E_0}\right)^b\)

with parameters

\(p_{\perp,0}^{\min}\), the value of \(p_\perp^{\min}\) at the reference scale \(E_0\) set using:

set /Herwig/UnderlyingEvent/MPIHandler:pTmin0 x*GeV

\(b\) the power for power law extrapolation which is set using:

set /Herwig/UnderlyingEvent/MPIHandler:Power x

\(E_0\) the reference energy scale for power law energy extrapolation is set using:

set /Herwig/UnderlyingEvent/MPIHandler:ReferenceScale x*GeV

The default value is 7 TeV.

For colour reconnection, by default we use the so-called plain colour reconnection model with just one parameter:

\(p_{\rm reco}\) the colour reconnection probability in the plain colour reconnection model which is set using:

set /Herwig/Hadronization/ColourReconnector:ReconnectionProbability x

There is also more complicated statistical colour reconnection model available in Herwig++, see Colour reconnection models.

## Colour reconnection models¶

Currently there are two colour reconnection models implemented in Herwig: the plain model (default) and more complicated the statistical model,

`ColourReconnectionAlgorithm`

parameter selects the colour reconnection model to be used. Possible values are`Plain`

for the plain model (PCR), and`Statistical`

for the statistical model (SCR).

The PCR model was discussed above the more complex SCR has the following parameters:

\(c\), the initial temperature in the SCR model which is set using:

set /Herwig/Hadronization/ColourReconnector:InitialTemperature x

\(\alpha\), in the SCR model, the number of reconnection tries per temperature step is the number of clusters times this factor. It is set using:

set /Herwig/Hadronization/ColourReconnector:TriesPerStepFactor x

\(f\) the ratio of the temperatures in two successive temperature steps in the SCR model. It is set using:

set /Herwig/Hadronization/ColourReconnector:AnnealingFactor x

\(N_{\rm steps}\) the number of temperature steps in the SCR model. It is set using:

set /Herwig/Hadronization/ColourReconnector:AnnealingSteps x

## Energy extrapolation¶

As we mentioned above the only energy dependent parameter of the model is \(p_\perp^{\min}\). There are three options how to set \(p_\perp^{\min}\) at different energies steered by:

`EnergyExtrapolation`

parameter, which enables and selects the extrapolation of the ptmin cut-off. It is set using:set /Herwig/UnderlyingEvent/MPIHandler:EnergyExtrapolation x

x takes three possible values:

`Power`

: power-law (default), \(p_\perp^{\min}=p_{\perp,0}^{\min}\left(\frac{\sqrt{s}}{E_0}\right)^b\), see above how to set the parameters.`Log`

: logarithmic extrapolation, \(p_\perp^{\min}=A\log\left(\frac{\sqrt{s}}{B}\right)\), which is set using:set /Herwig/UnderlyingEvent/MPIHandler:EEparamA A set /Herwig/UnderlyingEvent/MPIHandler:EEparamB B

`No`

: no extrapolation, use \(p_\perp^{\min}\) as explicitly set using the interface:set /Herwig/UnderlyingEvent/MPIHandler:pTmin x*GeV

If no energy extrapolation is selected,

`MHatMin`

has to be set to exactly \(2p_\perp^{\min}\)!:set /Herwig/UnderlyingEvent/KtCut:MinKT x*GeV

[1] | If soft MPI model is switched off: ```
set /Herwig/UnderlyingEvent/MPIHandler:softInt No
``` the \(p_\perp^{\min}\) is not a transition scale but the minimum transverse momentum of additional semi-hard interactions. |