where the invariants are defined in the following way:
In this note we present status of work on possible implementation of
complete NLO corrections into public version of PHOTOS Monte Carlo for radiative
corrections in decays.
For that purpose a special correcting weight was introduced
which enables the implementation of - normally missing
in PHOTOS - residual non-leading terms. The solution, to be installed
into public version of PHOTOS, would nonetheless significantly
complicate program design. For the time being we leave it
as an exercise in domain of matching hard matrix elements with
parton shower-like algorithms in "toy environment" of QED.
The expression for the correcting weight is explicitly given
and results of numerical tests are shown as well,.
At present, PHOTOS Monte Carlo for radiative corrections in decays
is widely used in important applications such as W mass and couplings
measurements at Tevatron, or for measurement of quark mixing angles
parametrizing CKM matrix.
On the other hand, an algorithm was developed
and used in the first years of its existence as a secondary tool, of
unspecified quality, mainly for the simulation of radiative corrections
in decays of tau leptons [cpc1991]. Over the years, the situation has changed,
and already since [cpc1994] the effort on establishing its precision was started.
It was only in recent years, however, that the work was addressed with full energy. This point was addressed back in papers [Gizo], [Andodnow], where new comparisons with matrix elements for W and Higgs decay were revisited; at first order only.
In parallel, multiple-photon emission was introduced into PHOTOS [Golonka1], and the quality of the predictions were confronted with direct comparisons with matrix-element and high-precision Monte Carlos: KORALZ and KKMC, which were widely used at LEP, and WINHAC were documented. The agreement at the level of better than 0.1 % was found.
Ongoing effort in studying systematic errors of PHOTOS in decays like B to pi+ pi- should also be documented soon [Gizo1] (preliminary results are already available [PDF] [Gzipped PostScript]), as well as an the effort on predictions of PHOTOS downgraded to Low theorem approximation only [Philip].
Recently [Golonka2], some effort in documenting the organization of PHOTOS solution became available as well.
The dual approach, where on one side PHOTOS is based on complete matrix element calculations, and on the other hand some approximations were introduced to ease program design and to free the solution from burden of process-dependent weight is leading to some confusion [Chicago], [Caltex] [Baracchini].
That is why we decided to show that it is possible to install NLO correcting weights into PHOTOS and, at the same time, to have a prototype solution for matching exact matrix-element with parton-shower like algorithm for possible future applications in the domain of QCD. The possible relevance of this point may be found in ref. [cern-th/2003-102], for example.
Originally, PHOTOS was tailored on the first-order matrix-element
generator MUSTRAAL for the radiative corrections in fermion pair production
at LEP1 energies. For that purpose only the final-state radiation was kept,
and some approximations to enable modularity were introduced.
That is why we have started to turn back PHOTOS into complete O(alpha) algorithm
as of MUSTRAAL [MUSTRAAL]
and as it was in MUSTRAAL we concentrated on Z/gamma decay only.
The solution is not suitable for a general-purpose generator,
but may be useful for special applications, and is certainly interesting as an
exercises for matching parton showers with hard matrix-elements at NLO.
We checked that, on event-by-event basis, we were able to verify that angular parametrization of the phase space was identical, up to signs and/or pi/2. This was easy because our Born-level generator had beams of equal energies along the z-axis. In the case of MUSTRAAL, some of these variables were genuine internal variables used in generation; in the case of PHOTOS, those variables were reconstructed from four-vectors of Born-level events, which were fed into PHOTOS.
In the next step we could identify the approximation which was
used in PHOTOS. It washes-down to the approximation in formula
(3.5) of Comput.Phys.Commun.29:185-200,1983.
The formula has a form:
where the invariants are defined in the following way:
| s = 2 p+ • p- | s' = 2 q+ • q- |
| t = 2 p+ • q+ | t' = 2 p- • q- |
| u = 2 p+ • q- | u' = 2 p- • q+ |
xk = 
| k'± = q± • k |
,
In PHOTOS, instead of term in square brackets, only a single
Born expression was used for every singular term:
The terms 
are used to remove interference, which
will be re-installed later in PHOTOS (at this place it would be tailored
for Z decay only. In angular variables and at the peak of Z resonance
.
We have dropped the terms proportional to cos(
),
which are responsible for AFB (forward-backward asymmetry), because it is small.
The Altarelli-Parisi kernel is obtained from t2, u2, ... terms of formula 3.5, once
the average over photon direction is taken and non-leading terms are dropped.
Let us stress that, effectively, the same parametrization of phase-space was
used in MUSTRAAL and in PHOTOS. PHOTOS approximation could be explicitly written
as an approximation for the matrix element only, (phase space unchanged).
That is why it was straightforward to implement the correcting
weight for complete order-alpha into PHOTOS.
It is thus of no surprise that once the correcting weight was
introduced, the difference between PHOTOS and MUSTRAAL
(more precisely KORALZ, set with appropriate switches)
was consistent with zero, even for statistics of 100 M events.
For complete results of the comparison, see the
booklet
with test results for single-photon emission
(Note that until 13 of February 2006 a term (3/4 alpha/pi) was missing in
a form factor term of PHOTOS, causing discrepancy of 0.1% in branching ratios).
Note that these excellent results were obtained only because the complete
kinematics of PHOTOS, combined with Born-level generation,
and MUSTRAAL were the same. In particular, the beam
directions were along the z-axis,
that is why the angles 
or 
were matching the angle of Born level.
We leave the details of this discussion to the complete publication.
Encouraged by the results in O(alpha), we have turned to the case of multiple
PHOTON emission.
The details of the tests' definitions are explained in [GOLONKA1] or in the
toplevel webpage describing the tests of PHOTOS.
Let us at first look at the results for single-photon tests.
The improvement, with respect to the old booklet, is evident.
The results are presented in the
booklet with test results for multiple-photon emission.
and in another booklet in electron channel.
In this case, PHOTOS was compared with KKMC.
With respect to the
booklet
which compares standard exponentiated version of PHOTOS with exponentiated KKMC,
significant improvement was achieved.
Branching ratios now agree up to the level of 0.01% and
SDP improved from 0.2% to 0.003%
(for single-photon configuration).
Also, some of the plots for double-photon emissions improved as well.
In particular, the distribution of largest discrepancies before
(the invariant mass of mu+, mu- pair), improved from SDP=0.4% to
SDP=0.001%, once the NLO terms were switched on.
In contrary to this, the distribution of invariant mass of two photons
deteriorated (SDP changed from 0.27% to 0.96%), however
the "naked eye" comparison suggests that even in this case
improvement was achieved.
The development of the reduction procedure will continue.
We still have some trivial things to check.
(The results of KKMC - PHOTOS NLO comparison from
9 December 2005
with another type of the reduction procedure).
(The results for the electron channel are in
another booklet).
We produced comparisons with KKMC truncated to the first-order
matrix element as well. The agreement was by far worse.
We found it non-instructive, nevertheless we include these
results here.
At certain moment, it may be interesting to reproduce the
acoplanarity plots
using the NLO-corrected version of PHOTOS.
The assymetry in accoplanarity distribution is of NNLO nature.
It is thus of interest for studies of the limits of our reduction procedure.
We also observed that different options of NLO
term obtained from first-order expression lead to differences of
results of the same order as the difference between PHOTOS NLO and full KKMC O(alpha2).
We can provide appropriate booklets upon
request. We do not believe that the choice of
a better or worse option can be done without carefull analysis of
amplitudes or distributions, at order alpha2 at least.
The careful study, including study of gauge-invariant as in
ref. Eur. Phys. Jour. C44, vol. 4, pp.489-503, (2005)
(doi:10.1140/epjc/s2005-02381-y) is needed.
There, reduction procedure for initial state radiation is discussed only.
This interesting point does not seem to be of any value for QED bremsstrahlung simulations, but may be very important in attempts to extend the techniques of PHOTOS to QCD, especially in the domain of matching of parton showers with hard matrix elements and structure functions.
At the end, let us say that we still believe that the introduction of channel-dependent matrix-element weight into PHOTOS should be avoided, unless experimental precision will demand so. The necessary weight is not only process-dependent, but also requires that electroweak parameters and couplings must be included in its calculation; also, the effective beam orientation must be available for calculation of NLO (formerly: interference) weight. The special version of PHOTOS designed for our present tests requires that runs are performed either at the peak of the Z, or at low energies, where interference between Z and gamma s-channel exchanges is minimal. Thus, the Forward-Backward Asymmetry AFB is practically zero. For the case of other conditions, we will prepare an appropriate version of PHOTOS, if requested. At present, we found such - even though small - effort unjustified.
Finally, let us point out that even after NLO terms were included we still generate events ONLY with weight of = +1 (no negative weight at all).
In this on-line version of the note we documented the implementation of
complete NLO corrections into PHOTOS Monte Carlo for the case of Z
decays at the peak (where forward backward asymmetry is negligibly small).
Firstly, we showed that the parametrization of the phase-space in PHOTOS and in MUSTRAAL
Monte Carlo match with each other. All angular and energy variables agree and the difference
is up to signs and/or pi/2 terms, at most. Thanks to that we could explicitly write
the correcting weight, which removes the approximations in PHOTOS, using
matrix elements only. Once the correcting weight was introduced we got the agreement
between PHOTOS and MUSTRAAL-like option of KORALZ which is better than statistical error
of 100 M event runs.
In the second step we addressed the question of agreement between the multiple-photon option of PHOTOS (NLO weight included) and the KKMC Monte Carlo with second-order matrix-element and exponentiation. Excellent agreement was found for distributions defined with single hard photon in final states. For the two photon final states improvement was less significant. This is of no surprise because the correcting weight was calculated based on first-order matrix-element only, and refined study of the reduction procedures was not yet completed.
To conclude, we would like to stress the importance of fixed-order calculations in defining parton shower-like algorithms. This observation is true even for the simple, pedagogical example of QED final state bremsstrahlung alone.