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Eur. Phys. J. C (2019) 79 :91 Page 5 of 22 91
where
dσ
prod
=
flavours
dx
1
dx
2
f (x
1
,...)f (x
2
,...)
×d
parton level
prod
spin
|M
prod
parton level
|
2
wt
prod
d
τ
+
decay
= d
τ
+
spin
|M
τ
+
|
2
wt
τ
+
decay
,
d
τ
−
decay
= d
τ
−
spin
|M
τ
−
|
2
wt
τ
−
decay
. (14)
All four components of the overall weight (10)aregiven
separately to underline, wherever possible, their dependence
on all or only sub-set of phase space dimensions. Note that
for W
±
/H
±
→ τν
τ
we have not prepared so far any option
which allows to modify production processes.
3 WT calculations in TauSpinner
The formulae listed above are used in TauSpinner for
the calculation of distinct components of WT,seeEq.(10).
Let us start with the main weight of the spin effects, wt
spin
defined by Eq. (4). With this weight, for the samples where
spin effects of τ lepton production are absent, they can be
inserted into decay distributions. Alternatively, its inverse
can be used to remove spin effects from a sample where they
are taken into account. As a building block of this weight
introducing the dependence on the production process, the
matrix R
i, j
, is given by the formula (8). The other weights,
w
prod
and w
τ
±
decay
are defined by formulae (11, 12).
It is useful to introduce the notation P
z
τ
= R
t,z
= R
z,t
,
which represents the longitudinal polarization of the single
τ lepton (if one integrates out possible configurations of the
other τ lepton ). In usually sufficient approximation of helic-
ity states, that is when transverse momenta of τ decay prod-
ucts are neglected, as is the case of ultra-relativistic τ leptons,
the P
z
τ
is the only non-trivial (dynamic dependent) element
of matrix R. All others are equal to ±1, or can be set to 0.
We will return to the details later.
In the calculation of wt
spin
and w
prod
the physics uncer-
tainty depends on the accurateness of the factorization
assumptions for separation of patron level matrix elements
and PDFs and also on the choice of a particular PDFs
parametrization. Nothing in principle would change if instead
of 2 → 2 body production matrix element one would use the
the ones for 2 → 2 + n, where n denotes additional par-
tons/jets or other particles (of summed spin-states). In addi-
tion to the choice of PDFs (eventually also model of underly-
ing event interactions) for parton level matrix elements one
has to make a careful choice of how hard scattering kinemat-
ics is reconstructed from information available in the event.
Calculation of wt
τ
±
decay
involves modeling of τ lepton
decays only. This weight is useful in studies of the result-
ing systematic uncertainties. This part of the code is using
function of the Tauola library.
We will return to these points later, but one can keep it in
mind already now, while reading the following.
3.1 Kinematical frames
The components necessary for the WT weight are calculated
in different frames. This is a correct approach as long as
details of boosts and rotations connecting frames are metic-
ulously followed. Also spin states of the τ leptons may be
defined in different frames than four momenta of the hard
process. Bremsstrahlung photons and properties of the matrix
element in their presence, require a dedicated treatment.
The basic formula (1) can provide exact results in princi-
ple. In practice, we have to introduce approximations to adapt
to the fact that matrix elements are calculated from the parton
level amplitudes, and to the way how these amplitudes are
reconstructed by the TauSpinner algorithm where only
four momenta of outgoing τ leptons and partons can be used.
The definition of quantization frames is an essential element
of the arrangements, let us provide now the necessary details.
For calculating all components of the event weights (to
prepare kinematical configurations of events necessary for
that purpose) we define in total four frames (group of frames):
A : The basic frame (starting point) for kinematic trans-
formations and all other frame definitions.
The rest frame of the τ lepton pair is used (and not of τ
lepton pair with the final state bremsstrahlung photons
combined). Such choice is possible thanks to the prop-
erties of bremsstrahlung amplitudes. Emitted photons
do not carry out the spin. This non-trivial observation
is exploited also in Photos Monte Carlo phase space
parametrization, see Refs. [30,31]. The definition of
the rest frame of the τ lepton pairs is completed with
incoming partons set along the z-axis.
A
: The rest frame of the τ lepton pairs and the final state
bremsstrahlung photons combined. Incoming partons
are again set along the z-axis. This frame is used
for calculating the production weight wt
prod
and the
spin correlation matrix R
i, j
; if no bremsstrahlung
photons are present it is equivalent
3
to frame A. In
every case we reconstruct x
1
, x
2
, the arguments of
the PDF function, from the virtuality M and the lon-
gitudinal to the beam direction component ( p
L
)of
the intermediate state (sum of momenta of τ
+
τ
−
nγ)
3
In case when photons are present and for 2 → 4 matrix elements
modifications are needed to assure that on-mass shell kinematic config-
uration is passed to the routines calculating spin amplitudes.
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