Chapter ‍8 Phase space parameterizations

• 8.1 General remarks
• 8.2 The flat method: rambo
• 8.3 The default method: wood
• 8.4 A new method: fast_wood
• 8.5 Phase space respecting restrictions on subdiagrams
• 8.6 Phase space for processes forbidden at tree level

8.1 General remarks

WHIZARD as a default performs an adaptive multi-channel Monte-Carlo 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 single-channel 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 single-channel, flat phase-space using RAMBO (on diet).

8.2 The flat method: rambo

The RAMBO algorithm produces a flat phase-space 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 final-state 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 final-state 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 fall-back option if the standard algorithm fails for physical reasons, see, e.g., Sec. ‍8.6.

8.3 The default method: wood

The 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 center-of-mass energy (or the mass of the decaying particle) and the masses of the final-state 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 phase-space forest is a collection of phase-space trees. A phase-space tree is a parameterization for a valid channel in the multi-channel 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 phase-space trees, collected in a phase-space 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,

  alias jet = u:d:s:c:b:U:D:S:C:B

all quarks including strange, charm and bottom have to be massless for the phase-space integration. WHIZARD can treat processes with subprocesses having final-state 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 phase-space parameterization, such that they can allow for different masses. E.g. in a 4-flavor scheme for massless u,d,s,c quarks one can write

  alias jet = u:d:s:c:U:D:S:C
  process eeqq = e1, E1 => (jet, jet) + (b, B)

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 high-energy 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 phase-space parameterizations in that way.

As a multi-particle 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 phase-space integration, following the assumption that they are related by some approximate symmetry. The VAMP adaptive multi-channel 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 phase-space setup, i.e. the detailed structure of trees and forests, are written by WHIZARD into a phase-space file that has the same name as the corresponding process (or process component) with the suffix .phs. For the wood phase-space method this file is written by a Fortran module which constructs a similar tree-like 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 phase-space generation has been reported to fail as WHIZARD could not find a valid phase-space channel. Such pathological cases cannot occur for the hard-coded model implementations inside WHIZARD. They can only happen if there are in principle two different Feynman diagrams contributing to the same phase-space 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 phase-space 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 phase-space channels according to a heuristic importance depending on the kinematic properties of the different phase-space channels in the groves. A phase-space (.phs) file looks typically like this:

 process sm_i1

! List of subprocesses with particle bincodes:
!  8  4      1   2
! e+ e- => mu+ mu-
!  8  4      1   2

   md5sum_process    = "1B3B7A30C24664A73D3D027382CFB4EF"
   md5sum_model_par  = "7656C90A0B2C4325AD911301DACF50EB"
   md5sum_phs_config = "6F72D447E8960F50FDE4AE590AD7044B"
   sqrts         =  1.000000000000E+02
   m_threshold_s =  5.000000000000E+01
   m_threshold_t =  1.000000000000E+02
   off_shell = 2
   t_channel = 6
   keep_nonresonant = T

 ! Multiplicity = 2, no resonances,  0 logs,  0 off-shell,  s-channel graph
 grove #1
 ! Channel #1
   tree  3

 ! Multiplicity = 1, 1 resonance,   0 logs,  0 off-shell,  s-channel graph
 grove #2
 ! Channel #2
   tree  3
   map   3 s_channel      23 ! Z

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 phase-space 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 center-of-mass 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 s-channel and t-channel-like 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 off-shell lines and of t-channel lines that WHIZARD will allow for finding valid phase-space channels, respectively. This neglects extremley multi-peripheral background-like diagram constellations which are very subdominamnt compared to resonant signal processes. The final flag specifies whether WHIZARD will keep non-resonant phase-space 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 phase-space channels which are connected by quasi-symmetries, together with the corresponding multiplicity of subchannels in that grove. In the phase-space file behind the multiplicity, WHIZARD denotes the number of (massive) resonances, logarithmcally enhanced kinematics (e.g. collinear regions), and number of off-shell lines, respectively. The final entry in the grove header notifies whether the diagrams in that grove have s-channel topologies, or count the number of corresponding t-channel lines.

Another example is shown here,

 ! Multiplicity = 3, no resonances,  2 logs,  0 off-shell,  1 t-channel line
 grove #1
 ! Channel #1
   tree  3 12
   map   3 infrared       22 ! A
   map  12 t_channel       2 ! u
 ! Channel #2
   tree  3 11
   map   3 infrared       22 ! A
   map  11 t_channel       2 ! u
 ! Channel #3
   tree  3 20
   map   3 infrared       22 ! A
   map  20 t_channel       2 ! u
 ! Channel #4
   tree  3 19
   map   3 infrared       22 ! A
   map  19 t_channel       2 ! u

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 phase-space setup that is generated and used by the wood phase-space method can be visualized using the SINDARIN option

  ?vis_channels = true

The wood phase-space method can be invoked with the SINDARIN command

  $phs_method = "wood"

Note that this line is unnecessary, as wood is the default phase-space method of WHIZARD.

8.4 A new method: fast_wood

This method (which is available from version 2.6.0 on) is an alternative implementation of the wood phase-space algorithm. It uses the recursive structures inside the O’Mega matrix element generator to generate all the structures needed for the different phase-space 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

  ?omega_write_phs_output = true
  $phs_method = "fast_wood"

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 phase-space 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 phase-space algorithm parses the tree-like represenation of the recursive set of one-particle off-shell 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 phase-space (.phs) file (cf. Sec. ‍8.3). In that way, the algorithm makes sure that only phase-space 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 phase-space setup (cf. next Sec. ‍8.5).

8.5 Phase space respecting restrictions on subdiagrams

The Fortran implementation of the wood phase-space 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 phase-space channels that might be absent when restrictions are imposed. This is not a principal problem, as in the adaptation of the phase-space 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 phase-space 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 phase-space method to make WHIZARD generation and integration of the phase space less cumbersome.

8.6 Phase space for processes forbidden at tree level

The phase-space generators wood and fast_wood are intended for tree-level processes with their typical patterns of singularities, which can be read off from Feynman graphs. They can and should be used for loop-induced or for externally provided matrix elements as long as WHIZARD does not provide a dedicated phase-space 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 hand-taylored matrix-element code that replaces the standard O’Mega code.

Currently, WHIZARD’s phase-space parameterization is nevertheless tied to the O’Mega generator, so for tree-level forbidden processes the phase-space construction process will fail.

There are two possible solutions for this problem:

1. It is possible to provide the phase-space parameterization information externally, by supplying an appropriately formatted .phs file, bypassing the automatic algorithm. Assuming that this phase-space file has been named my_phase_space.phs, the SINDARIN code should contain the following:

   ?rebuild_phase_space = false $phs_file = "my_phase_space.phs"

   Regarding the contents of this file, we recommend to generate an appropriate .phs for a similar setup, using the standard algorithm. The generated file can serve as a template, which can be adapted to the particular case.

   In detail, the .phs file consists of entries that specify the process, then a standard header which contains MD5 sums and such – these variables must be present but their values are irrelevant for the present case –, and finally at least one grove with tree entries that specify the parameterization. Individual parameterizations are built from the final-state and initial-state momenta (in this order) which we label in binary form as 1,2,4,8,…. The actual tree consists of iterative fusions of those external lines. Each fusion is indicated by the number that results from adding the binary codes of the external momenta that contribute to it.

   For instance, a valid phase-space tree for the process AA→ AA is given by the simple entry

   tree 3

   which indicates that the final-state momenta 1 and 2 are combined to a fusion 1+2=3. The setup is identical to a process such as e⁺e⁻→µ⁺µ⁻ below the Z threshold. Hence, we can take the .phs file for the latter process, replace the process tag, and use it as an external phase-space file.

2. For realistic applications of WHIZARD together with one-loop matrix-element providers, the actual number of final-state particles may be rather small, say 2,3,4. Furthermore, one-loop processes which are forbidden at tree level do not contain soft or collinear singularities. In this situation, the RAMBO phase-space integration method, cf. Sec. ‍8.2 is a viable alternative which does not suffer from the problem.