TauSpinner算法：LHC τ-轻子自旋效应模拟工具详解
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TauSpinner算法文档详细介绍了在大型强子对撞机（LHC）上模拟$$ \uptau $$τ-轻子产生的自旋效应时使用的专用程序。在质子-质子（pp）碰撞中，τ-轻子的产生不仅有助于测量标准模型参数，还为新物理理论提供了探测手段。该程序的核心功能在于修改τ-轻子样本中的自旋特性，无论是通过改变生成矩阵元素还是考虑实际的自旋状态，都是在基于粒子运动学的基础上进行的。
程序主要关注τ-轻子本身的特性，包括τ-轻子本身（在W介导过程中还包括ντ中微子）以及可能伴随的硬喷射的四维矢量。此外，τ-轻子的衰变产物也被纳入考虑范围，因为这些都对自旋效应有直接影响。为了处理这些信息，程序不依赖于事件记录中的其他细节，只需对每个单独事件分配自旋或生产/衰减/矩阵元素权重，这样就不必重新生成整个事件，提高了效率。
自LEP 1时代以来，这种实验技术已显著发展并在LHC环境下得到了广泛应用。本论文不仅阐述了TauSpinner算法的物理学原理，还详述了其在实验和理论研究中的具体用途，以及如何确保计算的精确性和可靠性。作者来自雅盖隆大学和波兰科学院的核物理研究所，他们对算法进行了深入研究，并于2018年5月接收了论文，12月接受并通过审稿，最终于2019年1月在线发表。文章的版权归属作者，强调了这项工作对于粒子物理学领域的贡献和重要意义。
通过阅读这篇文献，研究者可以了解到如何有效地利用TauSpinner算法来分析LHC数据，以提升对τ-轻子物理的理解，同时也能发现可能的新物理现象线索。对于那些从事高能物理实验和理论分析的科学家来说，这是一个必不可少的工具。
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.
123
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