Chapter 8 Phase space parameterizations
8.1 General remarksWHIZARD as a default performs an adaptive multichannel MonteCarlo integration. Besides its default phase space algorithm, wood, to be detailed in Sec. 8.3, WHIZARD contains a phase space method phs_none which is a dummy method that is intended for setups of processes where no phase space integration is needed, but the program flow needs a (dummy) integrator for internal consistency. Then, for testing purposes, there is a singlechannel phase space integrator, phs_single. From version 2.6.0 of WHIZARD on, there is also a second implementation of the wood phase space algorithm, called fast_wood, cf. Sec. 8.4, whose implementation differs technically and which therefore solves certain technical flaws of the wood implementation. Additionally, WHIZARD supports singlechannel, flat phasespace using RAMBO (on diet). 8.2 The flat method: ramboThe RAMBO algorithm produces a flat phasespace with constant volume for massless particles. RAMBO was originally published in [100]. We use the slim version, called RAMBO on diet, published in [98]. The overall weighting efficiency of the algorithm is unity for massless finalstate particles. For the massive case, the weighting efficiency of unity will decrease rendering the algorithm less efficient. But in most cases, the invariants are in regions of phase space where they are much larger than the masses of the finalstate particles. We provide the RAMBO mainly for cross checking our implementation and do not recommend it for real world application, even though it can be used as one. The RAMBO method becomes useful as a fallback option if the standard algorithm fails for physical reasons, see, e.g., Sec. 8.6. 8.3 The default method: woodThe wood algorithm classifies different phase space channels according to their importance for a full scattering or decay process following heuristic rules. For that purpose, WHIZARD investigates the kinematics of the different channels depending on the total centerofmass energy (or the mass of the decaying particle) and the masses of the finalstate particles. The wood phase space inherits its name from the naming schemes of structures of increasing complexities, namely trees, forests and groves. Simply stated, a phasespace forest is a collection of phasespace trees. A phasespace tree is a parameterization for a valid channel in the multichannel adaptive integration, and each variable in the a tree corresponds to an integration dimension, defined by an appropriate mapping of the (0,1) interval of the unit hypercube to the allowed range of the corresponding integration variable. The whole set of these phasespace trees, collected in a phasespace forest object hence contains all parameterizations of the phase space that WHIZARD will use for a single hard process. Note that processes might contain flavor sums of particles in the final state. As WHIZARD will use the same phase space parameterization for all channels for this set of subprocesses, all particles in those flavor sums have to have the same mass. E.g. in the definition of a "light" jet consisting of the first five quarks and antiquarks,
all quarks including strange, charm and bottom have to be massless for the phasespace integration. WHIZARD can treat processes with subprocesses having finalstate particles with different masses in an "additive" way, where each subprocess will become a distinct component of the whole process. Each process component will get its own phasespace parameterization, such that they can allow for different masses. E.g. in a 4flavor scheme for massless u,d,s,c quarks one can write
In that case, the parameterizations will be for massless final state quarks for the first subprocess, and for massive b quarks for the second subprocess. In general, for highenergy lepton colliders, the difference would not matter much, but performing the integration e.g. for √s = 11 GeV, the difference will be tremendous. WHIZARD avoids inconsistent phasespace parameterizations in that way. As a multiparticle process will contain hundred or thousands of different channels, the different integration channels (trees) are grouped into so called groves. All channels/trees in the same grove share a common weight for the phasespace integration, following the assumption that they are related by some approximate symmetry. The VAMP adaptive multichannel integrator (cf. Sec. 7.1) allows for equivalences between different integration channels. This means that trees/channels that are related by an exact symmetry are connected by an array of these equivalences. The phasespace setup, i.e. the detailed structure of trees and forests, are written by WHIZARD into a phasespace file that has the same name as the corresponding process (or process component) with the suffix .phs. For the wood phasespace method this file is written by a Fortran module which constructs a similar treelike structure as the directed acyclical graphs (DAGs) in the O’Mega matrix element generator but in a less efficient way. In some very rare cases with externally generated models (cf. Chapter 17) the phasespace generation has been reported to fail as WHIZARD could not find a valid phasespace channel. Such pathological cases cannot occur for the hardcoded model implementations inside WHIZARD. They can only happen if there are in principle two different Feynman diagrams contributing to the same phasespace channel and WHIZARD considers the second one as extremely subleading (and would hence drop it). If for some reason however the first Feynman diagram is then absent, no phasespace channel could be found. This problem cannot occur with the fast_wood implementation discussed in the next section, cf. 8.4. The wood algorithms orders the different groves of phasespace channels according to a heuristic importance depending on the kinematic properties of the different phasespace channels in the groves. A phasespace (.phs) file looks typically like this:
The first line contains the process name, followed by a list of subprocesses with the external particles and their binary codes. Then there are three lines of MD5 check sums, used for consistency checks. WHIZARD (unless told otherwise) will check for the existence of a phasespace file, and if the check sum matches, it will reuse the existing file and not generate it again. Next, there are several kinematic parameters, namely the centerofmass energy of the process, sqrts, and two mass thresholds, m_threshold_s and m_threshold_t. The latter two are kinematical thresholds, below which WHIZARD will consider schannel and tchannellike kinematic configurations as effectively massless, respectively. The default values shown in the example have turned out to be optimal values for Standard Model particles. The two integers off_shell and t_channel give the number of offshell lines and of tchannel lines that WHIZARD will allow for finding valid phasespace channels, respectively. This neglects extremley multiperipheral backgroundlike diagram constellations which are very subdominamnt compared to resonant signal processes. The final flag specifies whether WHIZARD will keep nonresonant phasespace channels (default), or whether it will focus only on resonant situations. After this header, there is a list of all groves, i.e. collections of phasespace channels which are connected by quasisymmetries, together with the corresponding multiplicity of subchannels in that grove. In the phasespace file behind the multiplicity, WHIZARD denotes the number of (massive) resonances, logarithmcally enhanced kinematics (e.g. collinear regions), and number of offshell lines, respectively. The final entry in the grove header notifies whether the diagrams in that grove have schannel topologies, or count the number of corresponding tchannel lines. Another example is shown here,
where WHIZARD notifies in different situations a photon exchange as infrared. So it detects a possible infrared singularity where a particle can become arbitrarily soft. Such a situation can tell the user that there might be a cut necessary in order to get a meaningful integration result. The phasespace setup that is generated and used by the wood phasespace method can be visualized using the SINDARIN option
The wood phasespace method can be invoked with the SINDARIN command
Note that this line is unnecessary, as wood is the default phasespace method of WHIZARD. 8.4 A new method: fast_woodThis method (which is available from version 2.6.0 on) is an alternative implementation of the wood phasespace algorithm. It uses the recursive structures inside the O’Mega matrix element generator to generate all the structures needed for the different phasespace channels. In that way, it can avoid some of the bottlenecks of the wood Fortran implementation of the algorithm. On the other hand, it is only available if the O’Mega matrix element generator has been enabled (which is the default for WHIZARD). The fast_wood method is then invoked via
The first option is necessary in order to tell O’Mega to write out the output needed for the fast_wood parser in order to generate the phasespace file. This is not enabled by default in order not to generate unnecessary files in case the default method wood is used. So the fast_wood implementation of the wood phasespace algorithm parses the treelike represenation of the recursive set of oneparticle offshell wave functions that make up the whole amplitude inside O’Mega in the form of a directed acyclical graph (DAG) in order to generate the phasespace (.phs) file (cf. Sec. 8.3). In that way, the algorithm makes sure that only phasespace channels are generated for which there are indeed (sub)amplitudes in the matrix elements, and this also allows to exclude vetoed channels due to restrictions imposed on the matrix elements from the phasespace setup (cf. next Sec. 8.5). 8.5 Phase space respecting restrictions on subdiagramsThe Fortran implementation of the wood phasespace does not know anything about possible restrictions that maybe imposed on the O’Mega matrix elements, cf. Sec. 5.4.3. Consequently, the wood phase space also generates phasespace channels that might be absent when restrictions are imposed. This is not a principal problem, as in the adaptation of the phasespace channels WHIZARD’s integrator VAMP will recognize that there is zero weight in that channel and will drop the channel (stop sampling in that channel) after some iterations. However, this is a waste of ressources as it is in principle known that this channel is absent. Using the fast_wood phasespace algorithm (cf. Sec. 8.4 will take restrictions into account, as O’Mega will not generate trees for channels that are removed with the restrictions command. So it advisable for the user in the case of very complicated processes with restrictions to use the fast_wood phasespace method to make WHIZARD generation and integration of the phase space less cumbersome. 8.6 Phase space for processes forbidden at tree levelThe phasespace generators wood and fast_wood are intended for treelevel processes with their typical patterns of singularities, which can be read off from Feynman graphs. They can and should be used for loopinduced or for externally provided matrix elements as long as WHIZARD does not provide a dedicated phasespace module. Some scattering processes do not occur at tree level but become allowed if loop effects are included in the calculation. A simple example is the elastic QED process
which is mediated by a fermion loop. Similarly, certain applications provide externally provided or handtaylored matrixelement code that replaces the standard O’Mega code. Currently, WHIZARD’s phasespace parameterization is nevertheless tied to the O’Mega generator, so for treelevel forbidden processes the phasespace construction process will fail. There are two possible solutions for this problem:
