Introduction

Numerical stability of the PHOTOS algorithm

Introduction

Recently introduced improvements in the PHOTOS algorithm, i.e. tripple-emmission and exponentiation put higher requirements on numerical stability of calculations.
Larger phase-space covered by radiation and multiple iterations of the single-emission kernel lead to rounding errors, which may have severe consequences.
In general, the smaller XPHCUT is, the larger the multiplicity of photon emission is, and the more important the energy-momentum conservation is.
The iterative nature of the PHOTOS algorithm leads to the accumulation of rounding errors (especially in routines that calculate boosts of light particles).

Special algorithms (described below) were thus employed to improve energy-momentum conservation (enforce double-precission accuracy).
Nevertheless, any modifications performed on the event record (particularly: its kinematics) need a lot of care - in general, some special-case solutions may be applied, but they may severely influence (damage) the outcome of PHOTOS in other cases!
Special treatment is needed i.e. for the cases of intermediate states of some widths, such as top-quarks, where particles' momenta are not on the mass shell because of physics, rather than rounding errors.

Currently our solution may not work in "every case".
Over time, we would like to improve PHOTOS operability, however to treat a large number of possible "special cases" may require a lot of effort. Therefore,

However, we want to stress that the STOP-warnings of PHOTOS are often the consequence of improperly-filled HEPEVT event record. The STOP messages may serve as a debugging tool in such cases.

Special routines for correction of kinematics

Since 1999, a subroutine PHCORK existed in PHOTOS, to correct improper kinematics of event record (strictly speaking: a part of the event record used by PHOTOS), as provided by host program. For example: host program is providing only single-precission event record. By default it is not active - a user may enable it (and select the mode of operation) by changing the parameter of PHCORK call in PHOINI subroutine. This point is "delicate" because we may encounter the cases where it is rather physisc, than rounding error, which affects the event record, for example due to the width of some intermediate states.

The following modes of operations of the PHCORK routune are currently implemented:

• PHCORK(1) [default]
  • Description:
    No kinematics correction
    
  • Design purpose:
    This is the default option.
    This option may be used for testing the quality of events produced by host event generator: PHOTOS usually stops and prints warning messages when the kinematics of the event record is not of sufficient precission.
    It often helps to debug other problems of HEPEVT filled by the host program.
    The typical case where PHOTOS will fail is in decays like D⁰ -> gamma e⁺ e^-, which should be coded as
    D⁰ -> gamma gamma^*, gamma^* -> e⁺ e^-.
    PHOTOS fails, in principle, because of rounding errors, but the real problem lays in improper presentation of the physics content for the event. This leads to exceptionally peaked phase-space distributions.
  • Limitations:
    In cases where the host generator produces events, which kinematics are not precise (e.g. some parts of the generator performs floating-point calculations in single-precission), boosting routines fail to calculate four-momentas of light particles and PHOTOS stops with a warning message. Also the cases of broad resonances may be treated in inappropriate way. In these situations, one of the options mentioned below should be activated.
• PHCORK(2)
  • Description:
    Corrects energies from masses for all decay products.
  • Design purpose:
    In boosting routines of PHOTOS, the lightest particles suffer the most from cumulative rounding errors. This routine corrects the energies from masses of the decay products, so that E²=m²+p².
    This correction is performed on a copy of the event record, before main PHOTOS routines are execured.
    This routine should be applied for light particles, i.e. leptons or pions, which in general should be on the mass shell.
  • Limitations:
    For resonances and other particles of certain width, this routine performs corrections that are unphysical: resonances are, in general, not on the mass shell because of their inherent physical nature, rather than numerical errors.
• PHCORK(3)
  • Description:
    Corrects masses from energies for all decay products.
  • Design purpose:
    As mentioned above, PHCORK(2) mode of kinematics correction is not applicable to resonances. On the contrary, this routine corrects masses from energies, again so as: m²=E²-p².
  • Limitations:
    Applying this routine to light ultra-relativistic particles, i.e. leptons, pions, often sets their mass to unphysical values, because of rounding errors.
• PHCORK(4)
  • Description:
    Corrects energy from mass for particles up to 0.4 GeV mass, for heavier ones corrects mass from energy. Applies corrections to decay products.
  • Design purpose:
    The two modes of operations mentioned above are applicable only to certain particles. This routine tries to make the two above modes applicable to any event. The "boundary" mass of 0.4 GeV has been chosen in such a way, to separate light particles, i.e. leptons, pions from resonances and massive, short-living particles of widths sizeable with respect to the mass.
  • Limitations:
    This routine and the routines above are often insufficient for PHOTOS working in exponentiated mode.
• PHCORK(5)
  • Description:
    Energies of decay products are corrected from their masses (as in PHCORK(2)), then the kinematics of decaying "mother" particle is re-constructed from four-momentas of the decay products to enforce momentum conservation, and its mass is set to put the particle on the mass shell.
  • Design purpose:
    This mode is dedicated to PHOTOS working in exponentiated mode. It is automatically activated for exponentiated mode (i.e. when IEXP=.TRUE. in PHOINI). In case of exponentiation, Energy-momentum conservation and phase-space volumes need to be calculated with far more superior precision that in the other options. Phase-space volumes are compared between iterations of the algorithm for single-photon emission.
    Additional loss of the precission of the order of E_typical/E_{gamma_min}. As the default value for this parameter is 10^-7, additonal care must be taken. That is why the energy-mometum conservation for decaying object needs to be corrected as well.
  • Limitations:
    This mode of operation may not be applicable to particles which are off-shell because of physical reasons, rather than numerical errors (i.e. heavy quarks or resonances). Possible solutions for these cases are currently being studied.

The following cases have been studied for the link to the web page of tests

• KK + PYTHIA: Z0->mu+ mu- at 2 TeV
  Exponentiated version of PHOTOS works in these regions,
  We have determined that the limits of instability regularization should be of order of 1.5D-8.
• PHOTOS + PYTHIA: Z0->e+ e- at LHC energies
  Exponentiated version of PHOTOS works in these regions,
  (notes concerning previous point apply).
• PHOTOS + PYTHIA: W+ -> mu+ nu_mu at LHC energies
  Exponentiated version of PHOTOS works in these regions,
  (notes concerning previous point apply).
• WINHAC + PHOTOS: W+ -> mu+ nu_mu at LHC energies
  Exponentiated version of PHOTOS works in these regions,
  We have determined that the limits of instability regularization should be of order of 2.5D-8.

The following cases have been identified

• PHOTOS + PYTHIA: tt~ production:
  under study now.
• PHOTOS + PYTHIA: meson decays in tt~ production
  
  • encountered instabilities in processes like D->gamma e+ e-:
    PHOTOS stops with "Too much bremsstrahlung" warning.
  • possible explanation: this is in fact a complex decay process:
    D -> gamma gamma*, gamma* -> e+ e-
    and the process is improperly encoded in the HEPEVT event record.
  • from technical point of view, it may be overcome by changing the limit in "Check on kinematical bounds" section of subroutine PHOENE, from -1.5D-8 to around -5E-3.
  • Possible impact to physical results: the source of the problem is of physical nature, rather than numerical; in cases like these, kinematics of the process after boost to decaying-particle rest-frame "gets crazy" because rounding errors become very important here; PHOTOS should not generate photons in events like that - it might result in having events with too much massive gamma*, etc.