LHC EFT WG Note: SMEFT predictions, event reweighting, and simulation

Alberto Belvedere⋆, Saptaparna Bhattacharya† (ed.), Giacomo Boldrini⋄, Suman
Chatterjee‡, Alessandro Calandri∘, Sergio Sánchez Cruz§, Jennet Dickinson¶,
Franz J. Glessgen∘, Reza Goldouzian∥, Alexander Grohsjean⋆, Laurids Jeppe⋆,
Charlotte Knight⟂ (ed.), Olivier Mattelaer⋄, Kelci Mohrman×, Hannah Nelson∥,
Vasilije Perovic∘, Matteo Presilla∪ (ed.), Robert Schöfbeck‡ (ed.) and Nick
Smith¶

⋆ Deutsches Elektronen-Synchrotron (DESY), Hamburg, Germany

† Northwestern University, Evanston, Illinois, and Wayne State University,
Detroit, Michigan United States

⋄ Laboratoire Leprince-Ringuet (LLR), Ec̀ole Polytechnique, Palaiseau Cedex,
France

‡ Institute for High Energy Physics of the Austrian Academy of Sciences,
Vienna, Austria

∘ ETH Zürich, Zürich, Switzerland

§ CERN, Geneva, Switzerland

¶ Fermi National Accelerator Laboratory (FNAL), Batavia, Illinois, United
States

∥ University of Notre Dame, Notre Dame, Indiana, United States

⟂ Imperial College, London, United Kingdom

⋄ Catholic University of Louvain, Louvain, Belgium

× University of Florida, Gainesville , United States

∪ Institute for Experimental Particle Physics (ETP), Karlsruhe Institute of
Technology (KIT), Karlsruhe, Germany

## Abstract

This note gives an overview of the tools for predicting expectations in the
Standard Model effective field theory (SMEFT) at the tree level and one loop
available through event generators. Methods of event reweighting, the separate
simulation of squared matrix elements, and the simulation of the full SMEFT
process are compared in terms of statistical efficacy and potential biases.

###### Contents

  1. 1 Introduction and Motivation
  2. 2 Strategies for simulated predictions
  3. 3 Helicity aware and helicity ignorant reweighting
  4. 4 Best practices
  5. 5 Simulation studies of EFT prediction methods
    1. 5.1 Helicity and reweighting of predictions for the WZ process
    2. 5.2 Helicity and reweighting of predictions for the ZH process
    3. 5.3 The $\textrm{t}\overline{\textrm{t}}$ process
    4. 5.4 Studies in the $\textrm{t}\overline{\textrm{t}}Z$ process at NLO
    5. 5.5 Helicity aware and ignorant reweighting in the $\textrm{t}\overline{\textrm{t}}Z$ process
    6. 5.6 The VBF H process
    7. 5.7 Multiboson processes
      1. 5.7.1 Choice of dynamical scale
    8. 5.8 Post-generation reweighting
  6. 6 Summary

##  1 Introduction and Motivation

The Standard Model effective field theory (SMEFT) [1, 2, 3, 4, 5] provides a
low-energy parametrization of phenomena beyond the standard model (SM) in
terms of Wilson coefficients (WC). The WCs are the prefactors of symmetry-
preserving local field operators in the SMEFT Lagrangian whose measurement
allows for discriminating between different UV models.

The main organizing principle of the SMEFT operators is their mass dimension,
starting at six for phenomena relevant at the LHC. Accurate predictions for
high-dimensional SMEFT analyses require a versatile and robust toolkit whose
ranges of applicability and potential shortfalls must be understood in detail.
Earlier notes of the LHC EFT WG cover several important steps forward in this
regard. The relation between hypothetical high-scale physics and the SMEFT
operators can be obtained by matching the integrated effect of the high-scale
BSM phenomena to the SMEFT Wilson coefficients. Automated tools for this
matching are reviewed in Ref. [6]. A review of experimental SMEFT measurements
and observables is provided in Ref. [7]. Finally, strategies for treating
uncertainties related to the truncation of the EFT expansion at finite mass
dimension are discussed in Ref. [8]. In this work, we do not quote the range
of validity of the EFT expansion for the distributions used in the comparisons
because the consistency of the computational strategies is unaffected by the
validity of the expansion.

This note serves as a guide to obtaining SMEFT predictions from event
generators for usage in LHC data analyses. It assesses the quality of
reweighting- and sampling-based strategies for obtaining generator-level
predictions by comparing them to a reference strategy of “direct” simulation
at a specific fixed parameter point. It also aims to highlight best practices
and document common pitfalls but does not establish authoritative guidelines.

Section 2 discusses the different methodologies for obtaining simulated SMEFT
predictions in terms of the WCs. In Sec. 3, the role of the initial- and
final-state helicities is clarified. Best practices and common pitfalls are
summarized in Sec. 4. The main body of the work, a comparison of SMEFT
predictions obtained from different methods, is presented in Sec. 5. Section 6
gives a summary.

##  2 Strategies for simulated predictions

Our starting point is the SMEFT Lagrangian, extending the SM by a number
$M(d)$ of symmetry-preserving operators $\mathcal{O}$ with mass dimension
$d>4$,

$\mathcal{L}_{\textrm{SM-
EFT}}=\mathcal{L}^{(4)}_{\textrm{SM}}+\sum_{d>4}\sum_{a=1}^{M(d)}\frac{\theta_{a}\mathcal{O}_{(d)}^{a}}{\Lambda^{d-4}},$
(1)

where $\theta_{a}$ denotes the operators’ Wilson coefficients. Equation 1 is
designed to capture non-resonant phenomena beyond the SM (BSM) below a high
unknown new-physics scale. In practice, a dimensionful normalization scale
$\Lambda$ is introduced and often conventionally fixed to 1 TeV. Because a
generic SMEFT differential cross-section with single-operator insertions can
be expressed as

$\displaystyle\textrm{d}\sigma(\boldsymbol{\theta})$
$\displaystyle\propto\left|\mathcal{M}_{\textrm{BSM}}(\boldsymbol{z}_{p})\right|^{2}\textrm{d}{\boldsymbol{z}_{p}}=\left|\mathcal{M}_{\textrm{SM}}({\boldsymbol{z}_{p}})+\frac{1}{\Lambda^{2}}\sum_{a=1}^{M}\theta_{a}\mathcal{M}^{a}_{\textrm{EFT}}({\boldsymbol{z}_{p}})\right|^{2}\textrm{d}{\boldsymbol{z}_{p}},$
(2)

the SMEFT predictions for event rates with parton-level momenta
$\boldsymbol{z}_{p}$ can be accurately described by polynomials in the Wilson
coefficients. The matrix-elements (MEs) from the SM and SMEFT are denoted by
$\mathcal{M}_{\textrm{SM}}$ and $\mathcal{M}_{\textrm{EFT}}$, respectively.

In Eq. 2 and in the following, we collectively label observable features by
$\boldsymbol{x}$ and unobservable (latent) variables by $\boldsymbol{z}$. The
only exception is the Bjorken scaling variables where we do not break with the
convention and denote those by $x_{\textrm{Bjorken},1}$ and
$x_{\textrm{Bjorken},2}$ although these are part of $\boldsymbol{z}$. At the
parton level, $\boldsymbol{z}_{p}$ comprises the external partons’ four
momenta and, generically, the helicity configuration denoted by $h$. Whether
or not $h$ is considered a part of $\boldsymbol{z}_{p}$ is a matter of choice
with important practical implications for the reweighting-based strategies
which we discuss in Sec. 3. In the former case, we have

$\displaystyle\textrm{d}\boldsymbol{z}_{p}$
$\displaystyle=f_{1}(x_{\textrm{Bjorken},1},\mu_{F})f_{2}(x_{\textrm{Bjorken},2},\mu_{F})\textrm{d}\Omega_{\textrm{PS}}^{(h)}$
(3)

where $f_{i}(x_{\textrm{Bjorken},i},\mu_{F})$ is an estimate of the parton
distribution function for a factorization scale $\mu_{F}$. The per-helicity
kinematic phase space element of the external particles is denoted by
$\textrm{d}\Omega_{\textrm{PS}}^{(h)}$ and, consequently, includes the measure
over the Bjorken variables, such that

$\displaystyle\textrm{d}\sigma(\boldsymbol{\theta})$
$\displaystyle\propto\left|\mathcal{M}_{\textrm{BSM}}(\boldsymbol{z}_{p},h)\right|^{2}f_{1}(x_{\textrm{Bjorken},1},\mu_{F})f_{2}(x_{\textrm{Bjorken},2},\mu_{F})\textrm{d}\Omega_{\textrm{PS}}^{(h)}$
(4)

If unmeasurable, helicity is not part of the parton-level phase-space
definition, for example when this information is dropped by the matrix element
(ME) generator, the helicity dependence of the $|\mathcal{M}|^{2}$ terms is
summed, and we have

$\displaystyle\textrm{d}\sigma(\boldsymbol{\theta})$
$\displaystyle\propto\left(\sum_{h}\left|\mathcal{M}_{\textrm{BSM}}(\boldsymbol{z}_{p},h)\right|^{2}f_{1}(x_{\textrm{Bjorken},1},\mu_{F})f_{2}(x_{\textrm{Bjorken},2},\mu_{F})\right)\textrm{d}\Omega_{\textrm{PS}}$
(5)

with the important difference that $\textrm{d}\Omega_{\textrm{PS}}$ multiplies
a sum over $h$.

Either way, automated ME generators produce a numerical code for Eq. 2 which
can be efficiently re-evaluated for different $\boldsymbol{\theta}$ for given
$\boldsymbol{z}_{p}$. This computational efficiency is the basis for the
reweighting strategies discussed in this note.

In this note, we quantitatively compare three different strategies for
obtaining SMEFT predictions via event simulation using the SMEFT Lagrangian in
Eq. 1. All studies truncate the perturbative expansion at the leading order
(LO) or next-to-LO (NLO) in QCD. The simplest procedure chooses the desired
value of $\boldsymbol{\theta}$ and samples the SMEFT model at this parameter
point (“direct simulation”). While this approach is conceptually clean and
therefore our reference, it is not efficient enough for most practical
applications, because limits on Wilson coefficients require the comparison of
likelihoods for arbitrary $\boldsymbol{\theta}$, thus typically exceeding the
computing resources available for event simulation.

There are two main strategies for obtaining parametrized predictions. Firstly,
the SMEFT ME-squared terms in Eq. 2 can be expanded and the terms
corresponding to the same polynomial coefficient in $\boldsymbol{\theta}$ can
be sampled separately and independently (“separate simulation”). Events from
the resulting samples can then be weighted according to the desired value of
$\boldsymbol{\theta}$. If we denote the event sample at the SM by $S_{0}$, the
event samples obtained from the linear terms in Eq. 2 by $S_{a}$, etc., a
yield $\lambda_{\Delta\boldsymbol{z}}$ in a small phase space volume
$\Delta\boldsymbol{z}$ around the parton-level configuration
$\boldsymbol{z}_{p}$ is predicted to be

$\displaystyle\lambda_{\Delta\boldsymbol{z}}(\boldsymbol{\theta})$
$\displaystyle=\sum_{\boldsymbol{z}_{i}\in\Delta\boldsymbol{z}\,\cap\,S_{0}}w_{i,0}+\sum_{a=1}^{M}\theta_{a}\sum_{\boldsymbol{z}_{i}\in\Delta\boldsymbol{z}\,\cap\,S_{a}}w_{i,a}+\sum_{\begin{subarray}{c}a,b=1\\\
a\geq
b\end{subarray}}^{M}\theta_{a}\theta_{b}\sum_{\boldsymbol{z}_{i}\in\Delta\boldsymbol{z}\,\cap\,S_{ab}}w_{i,ab},$
(6)

where the constant weights $w_{i,0}$, $w_{i,a}$, and $w_{i,ab}$ are obtained
from the generator. The normalization can be chosen as

$\displaystyle\mathcal{L}\sigma(\boldsymbol{\theta})=\sum_{i\in
S_{0}}w_{i,0}+\sum_{a=1}^{M}\theta_{a}\sum_{i\in
S_{a}}w_{i,a}+\sum_{\begin{subarray}{c}a,b=1\\\ a\geq
b\end{subarray}}^{M}\theta_{a}\theta_{b}\sum_{i\in S_{ab}}w_{i,ab},$ (7)

where $\mathcal{L}$ is the integrated luminosity and
$\sigma(\boldsymbol{\theta})$ the inclusive cross-section.

Secondly, the per-event parton level configuration of an event from a sample
obtained with a specific SMEFT parameter reference point
$\boldsymbol{\theta}_{0}$, not necessarily the same as the SM at
$\boldsymbol{\theta}_{0}=\mathbf{0}$, can be used to re-evaluate Eq. 2 at
different values of $\boldsymbol{\theta}$. Because the differential cross
section is a quadratic function of the Wilson coefficients, a small number of
evaluations can be used to determine a polynomial that parametrizes the weight
of the event when computing the predicted yield as

$\displaystyle\lambda_{\Delta\boldsymbol{z}}(\boldsymbol{\theta})$
$\displaystyle=\sum_{\boldsymbol{z}_{i}\in\Delta\boldsymbol{z}}w_{i}(\boldsymbol{\theta})=\sum_{\boldsymbol{z}_{i}\in\Delta\boldsymbol{z}}\Bigg{(}w_{i,0}+\sum_{a=1}^{M}\theta_{a}w_{i,a}+\sum_{\begin{subarray}{c}a,b=1\\\
a\geq b\end{subarray}}^{M}\theta_{a}\theta_{b}w_{i,ab}\Bigg{)}$ (8)

To determine the per-event polynomial coefficients $w_{i,0}$, $w_{i,a}$, and
$w_{i,ab}$ from the event generator, a set of $k=1,....,K$ different SMEFT
base-points $\boldsymbol{\theta}^{(k)}$ is needed and $K$ must be at least
equal to the number of degrees of freedom, that is, $N=1+M+\frac{1}{2}M(M+1)$
at the quadratic order. If we let an index $n$ enumerate the constant term,
the $M$ linear terms and the $\frac{1}{2}M(M+1)$ quadratic terms, we can take
the constant factors from Eq. 8 to form the $K\times N$ matrix
$\Theta^{(k)}_{n}=\\{1,\theta^{(k)}_{a},\theta^{(k)}_{a}\theta^{(k)}_{b}\\}$.
For $K=N$, that is, if we have obtained just enough coefficients
$w_{i}(\boldsymbol{\theta}^{(k)})$ at the base points
$\boldsymbol{\theta}^{(k)}$, we can uniquely solve the linear set of equations

$\displaystyle
w_{i}(\boldsymbol{\theta}^{(k)})=\sum_{n}\Theta^{(k)}_{n}w_{i,n}$ (9)

for the polynomial coefficients $w_{i,n}$ of Eq. 8 in terms of the event
weights provided by the generator. Again, the index $n$ labels the constant
term, the $M$ linear terms, and the quadratic terms. For $K\geq N$, the
polynomial coefficients can be determined if the $K\times N$ matrix
$\Theta_{n}^{(k)}$ has full rank.

In the case of reweighting, it is an important practical difference whether
the generator computes the ME-squared separately for each helicity
configuration (helicity-aware, Eq. 4) or whether it first sums over helicities
(helicity-ignorant, Eq. 5). In the former case, the simulated helicity
contributions are accurately predicted at each stage. In the latter case, only
the sum of the EFT predictions over all helicity configurations is correct.
The advantage of helicity-ignorant reweighting is to avoid large weights in
the case a SMEFT operator introduces helicity configurations that are
suppressed in the SM. In both cases, the normalization of the reweighted
samples can be chosen at

$\displaystyle\mathcal{L}\sigma(\boldsymbol{\theta})=\sum_{i\in
S}w_{i}(\boldsymbol{\theta}).$ (10)

There also are important differences between the “reweighted simulation” in
Eq. 8 and the separate simulation in Eq. 6. Firstly, there is no stochastic
independence in the constant, linear, and quadratic terms when reweighting.
For each event, the probabilistic mass of its concrete parton-level
configuration, i.e., the weight of the event when computing yields, is known
in all SMEFT parameter space with potential benefits for machine-learning
applications [9, 10, 11, 12]. In contrast, the separate simulation of the
different ME-squared terms predicts the constant, linear, and quadratic terms
with uncorrelated statistical uncertainty, thereby potentially increasing the
CPU demand for a given requirement on statistical precision. Secondly, the
independent sampling does not depend on a reference point. In practice, event
reweighting can lead to large weights in those regions of phase space where
the parton-level differential cross sections greatly differ for
$\boldsymbol{\theta}$ and the reference $\boldsymbol{\theta}_{0}$. This can be
particularly acute when SMEFT operators introduce, e.g., helicity
configurations not present in the SM and helicity-aware reweighting is used.

Finally, let us clarify the relation to parametrized SMEFT predictions at
lower-level representations of the simulated data. Following the ME generators
providing the parton-level differential cross sections, a hierarchical
sequence of staged computer codes is used to simulate phenomena at lower
energy scales and using, typically, much higher-dimensional representations of
the events. The stages comprise the parton shower with ME-matching and merging
procedures, the hadronization of the shower algorithm’s output, the detector
interactions, and the event reconstruction. Many of these stages are at least
partially stochastic. Provided $S$ is sufficiently large for the statistical
uncertainty in $\lambda_{\Delta\boldsymbol{z}}$ to be acceptably small for any
$\Delta\boldsymbol{z}$ in the phase space covered by $S$, we use Eq. 6 or Eq.
8 to approximate the parton-level differential cross-section as

$\displaystyle\frac{1}{\mathcal{L}\sigma(\boldsymbol{\theta})}\frac{\lambda_{\Delta\boldsymbol{z}}(\boldsymbol{\theta})}{\Delta\boldsymbol{z}}\approx\frac{1}{\sigma(\boldsymbol{\theta})}\frac{\textrm{d}\sigma(\boldsymbol{z}_{p}|\boldsymbol{\theta})}{\textrm{d}\boldsymbol{z}_{p}}=p(\boldsymbol{z}_{p}|\boldsymbol{\theta})$
(11)

where the l.h.s. and r.h.s of the first equation each is a ratio of quadratic
polynomials. The last equality interprets the normalized differential cross
section as the parton-level probability density function. How do we transition
to other data representations, e.g., at the particle level or at the detector
level, commonly used in LHC data analyses? As illustrative examples, we first
define the particle-level $\boldsymbol{z}_{\textrm{ptl}}$ comprising stable
generated particles after hadronization and before interaction with the
detector material. Secondly, the detector level representation
$\boldsymbol{x}_{\textrm{det}}$ of the simulated processes shall consist of,
for example, jets, b-tagged jets, leptons, and other reconstructed high-level
objects. It is the simulated equivalent of the detector-level observation of
real data in a generic analysis. Equation 11 can then be used to express,
e.g., the detector-level cross-section as

$\displaystyle\frac{\textrm{d}\sigma(\boldsymbol{x}|\boldsymbol{\theta})}{\textrm{d}\boldsymbol{x}}$
$\displaystyle=\int\textrm{d}\boldsymbol{z}_{\textrm{ptl}}\int\textrm{d}\boldsymbol{z}_{p}\,p(\boldsymbol{x}|\boldsymbol{z}_{\textrm{ptl}})\,p(\boldsymbol{z}_{\textrm{ptl}}|\boldsymbol{z}_{p})\frac{\textrm{d}\sigma(\boldsymbol{z}_{p}|\boldsymbol{\theta})}{\textrm{d}\boldsymbol{z}_{p}}.$
(12)

The conditional distribution
$p(\boldsymbol{z}_{\textrm{ptl}}|\boldsymbol{z}_{p})$ is sampled by the shower
simulation, the hadronization model, and matching- and the merging procedures.
The conditional distribution $p(\boldsymbol{x}|\boldsymbol{z}_{\textrm{ptl}})$
governs the detector simulation and the event reconstruction. Both
distributions are intractable, i.e., they can be sampled for a fixed
conditional configuration but can not be evaluated as a function of the
condition for a fixed sampling instance. Nevertheless, it has been shown that
intractable factors cancel, provided that the SMEFT Wilson coefficients do not
modify these distributions [13, 14, 9, 15, 16]. Concretely, the probability to
obtain a certain observation $\boldsymbol{x}$ given a particle-level
configuration $\boldsymbol{z}_{\textrm{ptl}}$ shall not depend on the Wilson
coefficients and neither shall the probability to observe a certain particle-
level configuration when a parton-level event is given. In this case, dividing
both sides by the total cross section and using Eq. 11 then trivially re-
expresses the detector level probability density in terms of the parton-level
one. The conditional sequence relating the parton-level with the detector
level through the particle level could even be more refined, with more
intermediate integrations in Eq. 12, but as long as the SMEFT effects do not
affect anything other than $p(\boldsymbol{z}_{p}|\boldsymbol{\theta})$, it
follows that we can approximate any detector level yield
$\lambda_{\Delta\boldsymbol{x}}$ from separate simulation as

$\displaystyle\lambda_{\Delta\boldsymbol{x}}(\boldsymbol{\theta})$
$\displaystyle=\sum_{\boldsymbol{x}_{i}\in\Delta\boldsymbol{x}\,\cap\,S_{0}}w_{i,0}+\sum_{a=1}^{M}\theta_{a}\sum_{\boldsymbol{x}_{i}\in\Delta\boldsymbol{x}\,\cap\,S_{a}}w_{i,a}+\sum_{\begin{subarray}{c}a,b=1\\\
a\geq
b\end{subarray}}^{M}\theta_{a}\theta_{b}\sum_{\boldsymbol{x}_{i}\in\Delta\boldsymbol{x}\,\cap\,S_{ab}}w_{i,ab}$
(13)

using the same per-event weights as in Eq. 6. The corresponding prediction for
the case of event reweighting is

$\displaystyle\lambda_{\Delta x}(\boldsymbol{\theta})$
$\displaystyle=\sum_{\boldsymbol{x}_{i}\in\Delta\boldsymbol{x}}\Bigg{(}w_{i,0}+\sum_{a=1}^{M}\theta_{a}w_{i,a}+\sum_{\begin{subarray}{c}a,b=1\\\
a\geq b\end{subarray}}^{M}\theta_{a}\theta_{b}w_{i,ab}\Bigg{)},$ (14)

again using the same weight functions as in Eq. 8. To the extent that the
intractable conditional likelihoods do not depend on the Wilson coefficients,
we can ignore the level we obtain the predictions for and simply accumulate
the event weight polynomials. In the case of event reweighting, we can,
furthermore, interpret the $w_{i}(\boldsymbol{\theta})$ as the total cross
section multiplied by the per-event likelihood of the joint observed and
generated features,

$w_{i}(\boldsymbol{\theta})=\sigma(\boldsymbol{\theta})\,p(\boldsymbol{x}_{i},\boldsymbol{z}_{\textrm{ptl},i},\boldsymbol{z}_{p,i}|\boldsymbol{\theta}),$
(15)

which agrees with the joint likelihood in Ref. [9] up to an overall cross-
section normalization. The conceptual simplification of the reweighting
strategy then appears in the ratio

$\displaystyle\frac{w_{i}(\boldsymbol{\theta})}{w_{i}(\textrm{SM})}=\frac{\sigma(\boldsymbol{\theta})}{\sigma(\textrm{SM})}\frac{p(\boldsymbol{x}_{i}|\boldsymbol{z}_{\textrm{ptl},i})p(\boldsymbol{z}_{\textrm{ptl},i}|\boldsymbol{z}_{\textrm{p},i})}{p(\boldsymbol{x}_{i}|\boldsymbol{z}_{\textrm{ptl},i})p(\boldsymbol{z}_{\textrm{ptl},i}|\boldsymbol{z}_{\textrm{p},i})}\frac{p(\boldsymbol{z}_{\textrm{p},i}|\boldsymbol{\theta})}{p(\boldsymbol{z}_{\textrm{p},i}|\textrm{SM})}=\frac{|\mathcal{M}(\boldsymbol{\theta})|^{2}(\boldsymbol{z}_{p,i})}{|\mathcal{M}(\textrm{SM})|^{2}(\boldsymbol{z}_{p,i})}$
(16)

via the cancellation of the extremely complicated, usually intractable,
conditional likelihood factors
$p(\boldsymbol{x}_{i}|\boldsymbol{z}_{\textrm{ptl},i})$ and
$p(\boldsymbol{z}_{\textrm{ptl},i}|\boldsymbol{z}_{\textrm{p},i})$. Once
$w_{i}(\textrm{SM})$ are known for an event sample, the easily calculable ME-
squared ratios are enough to obtain detector-level predictions for any values
of the Wilson coefficients.

##  3 Helicity aware and helicity ignorant reweighting

Any reweighting method consists of modifying the weight of a parton-level
event such that the resulting weighted event sample reproduces an alternative
scenario, leveraging the statistical power of a given event sample, possibly
removing the need of a dedicated shower- and detector simulation. At the LO,
matrix element generators customarily include the helicity configuration
associated to the events, even when using Eq. 5. For the nominal simulation,
for example at the SM parameter point, the Madgraph event generator [17]
selects helicity configurations randomly according to the probability

$\displaystyle p(h|\boldsymbol{z}_{p},\textrm{nom})$
$\displaystyle=\frac{\left|\mathcal{M}_{\textrm{nom}}(\boldsymbol{z}_{p},h)\right|^{2}}{\sum_{h}\left|\mathcal{M}_{\textrm{nom}}(\boldsymbol{z}_{p},h)\right|^{2}}$
(17)

where $\left|\mathcal{M}_{\textrm{nom}}(\boldsymbol{z}_{p},h)\right|^{2}$ is
the squared amplitude for a given helicity configuration $h$, comprising all
initial- and final-state particles. Helicity-aware re-weighting at LO to an
“alternative” parameter point is implemented by modifying the event weights by
a factor

$w_{\textrm{alt}}=w_{\textrm{nom}}\;\frac{\left|\mathcal{M}_{\textrm{alt}}(\boldsymbol{z}_{p},h)\right|^{2}}{\left|\mathcal{M}_{\textrm{nom}}(\boldsymbol{z}_{p},h)\right|^{2}},$
(18)

while the helicity ignorant reweighting amounts to

$w_{\textrm{alt}}=w_{\textrm{nom}}\;\frac{\sum_{h}\left|\mathcal{M}_{\textrm{alt}}(\boldsymbol{z}_{p},h)\right|^{2}}{\sum_{h}\left|\mathcal{M}_{\textrm{nom}}(\boldsymbol{z}_{p},h)\right|^{2}}.$
(19)

A few remarks are in order regarding the range of validity of these methods.
Firstly, even if the method is correct in the asymptotic limit of infinite
sample size, a real-world application is limited by the size of
$p(h|\boldsymbol{z}_{p},\textrm{alt})/p(h|\boldsymbol{z}_{p},\textrm{nom})$ as
a function of the parton-level momenta $\boldsymbol{z}_{p}$. If the alternate
scenario strongly differs in terms of helicity configurations or kinematic
dependence, this ratio can become very large. Because the statistical power of
the helicity-aware reweighted sample corresponds to the nominal sample, the
relative statistical uncertainty in the affected phase-space can grow
arbitrarily, sometimes entirely removing the feasibility of helicity-aware
reweighting. In practice, this is reflected by large event weights.

Secondly, we remark that the requirement of a similar phase-space density of
the alternate and the nominal hypothesis applies beyond SMEFT reweighting. For
example, reweighting can not be used for scanning mass values far outside of
the width of a resonance.

Finally, we remark that similar to the case of helicity, a choice is needed
for the reweighting according to the (leading) color assignment of an event,
which must be defined in the presence of a mixed perturbative expansion
(_e.g._ , the VBF process at LO with both QCD and QED amplitudes included). So
far, only color-ignorant re-weighting has been implemented which, in fact,
limits the applicability of re-weighting to BSM models that leave the relative
contributions of color assignments within a process unaffected by the Wilson
coefficients. Color-aware re-weighting is a theoretical possibility –even if
currently not implemented– with the same limitations and advantages as in the
case of helicity, as far as the hard-scatter parton level is concerned. The
color assignment, however, has a substantial and direct impact on the parton-
shower simulation, warranting careful and process-dependent validation in
case, e.g., four-fermion operators are used.

The particular case of mixed expansion not only creates an issue for the color
assignment at LO, but also in the handling of the re-weighting at NLO
accuracy. For technical reasons, in the presence of a mixed expansion, each
matrix-element is separated into terms with the same power of the coupling
constants. A correct re-weighting procedure then requires that each order is
re-weighted by the corresponding matrix element. Currently, this is not
implemented in MG5aMC re-weighting tool.

##  4 Best practices

We next address common challenges and pitfalls encountered during the studies
presented in Sec. 5. It should, therefore, be understood that the list is not
exhaustive.

Renormalization and factorization scales choice. It is customary to employ a
dynamic scale for various generated samples, often opting for the CKKW-L
clustering algorithm’s scale choice [18], where only clustering compatible
with the current integration channel is permitted. The event-specific nature
of this scale choice, dependent on both the channel of integration and the
event generation method, poses a potential issue for consistency tests
comparing direct simulation, separate simulation, and re-weighting. While this
is not an issue for the validity of the prediction, closure tests may only be
consistent up to scale variations when employing CKKW-L clustering.
Conversely, fixed scale choices determined solely by the event’s kinematics
remain unaffected. Unless noted otherwise, the scale choices in the closure
tests in Sec. 5, corresponding to $H_{T}/2$ by default, avoid this problem.

On the NLO SMEFT simulation. Another important aspect to consider concerns the
perturbative order of the MC simulation. While an NLO calculation generally
represents the best solution, NLO SMEFT simulations with MadGraph can involve
challenges with SMEFT operators involving electroweak vertices. These
complications stem from the fact that, at the time of writing, MadGraph SMEFT
calculations can account for NLO QCD effects but cannot account for QED loops.
For this reason, restricting the QED coupling order of NLO samples is
required. This restriction does not permit tree-level diagrams with a QED
order greater than or equal to two plus the lowest QED order tree-level
diagrams to enter; such tree-level contributions are not permitted because
they would enter with the same QED order as a QED loop added to the lowest QED
order diagrams. SMEFT couplings are assigned QED orders (which are somewhat
arbitrary and can differ between UFO models [19]), so tree-level diagrams
(with QED order larger than the QED order cutoff imposed for NLO calculations)
must be excluded in NLO calculations. However, LO calculations do not require
restrictions on the QED order, so the matched LO approach, where LO samples
with extra QCD emissions are taken as a proxy for partial NLO effects (see
below), does not involve this limitation.

When generating NLO SMEFT samples with operators and processes involving
electroweak vertices, it is advisable to also generate LO samples (without QED
order restrictions) to ensure that any SMEFT effects excluded at NLO by the
QED order restriction are fully understood.

SMEFT simulation with extra partons. When NLO samples are not available, it
can be beneficial to include an additional parton in the LO matrix element
calculation. Not only does this help to provide more accurate modeling of SM
kinematics, but it can also impact the SMEFT dependence of processes on
certain Wilson coefficients [19], though careful validation is important.
These effects are primarily due to the additional initial states that become
available with the inclusion of an additional parton, but other factors (such
as the topology of diagrams, energy scaling of vertices, and interference
effects) can also be relevant. Since it is difficult to predict a priori which
combinations of processes and operators will be strongly impacted by the
inclusion of the additional parton, it is beneficial to include the extra
parton whenever possible to avoid inadvertently neglecting relevant SMEFT
contributions. With the matched LO approach, the harder emission is handled
with MadGraph (with the extra parton explicitly included in the matrix element
calculation) while softer emission is handled by the parton shower; a matching
procedure (e.g. the $k_{T}$-jet version of the MLM matching scheme [20]) is
required to remove the overlap between the two regions. This matching
procedure involves choosing cutoff scales for the matrix element and parton
shower, and it is important to validate that the choices for the values of
these cutoff scales allow the matrix element generator and parton shower
simulation to smoothly fill the overlapping phase space. To perform such
validation, it is useful to study differential jet rate (DJR) distributions
[21, 22]; smooth transitions between the n and n+1 curves in a DJR
distribution is an indication that the chosen matching scales have allowed the
matrix element generator and parton shower simulation to be able to work
together to fill the overlapping space smoothly.

Performing matching with SMEFT samples can introduce an additional
complication. Since SMEFT effects are included in the matrix element but not
in the parton shower, it is possible that the matching procedure could cause a
mismatch by removing SMEFT effects that the parton shower cannot replace. It
is expected that these effects are subdominant due to the additional momentum
dependence introduced by SMEFT operators, as argued in Ref. [23]. For this
reason, it is important to examine the DJR plots at various non-SM points
within the SMEFT space to ensure that there are no signs of mismatches between
the matrix element and parton shower contributions. These effects would be
most relevant to investigate for operators involving gluons or light quarks.
The generation and validation procedures for matched LO samples are described
in more detail in Ref. [19].

Table 1: The space of EFT operators under consideration in Sec. 5. They are categorized as operators affecting multiboson and top based processes, adopting thee Warsaw basis defined in Ref. [24]. Operator (CP even) | Multiboson | Studied in
---|---|---
${\epsilon}^{IJK}W^{I\nu}_{\mu}W^{J\rho}_{\nu}W^{K\mu}_{\rho}$ | ${{\mathcal{O}}_{W}}$ | Sec. 5.1
$\varphi^{\dagger}\tau^{I}\varphi W_{\mu\nu}^{I}B^{\mu\nu}$ | ${{\mathcal{O}}_{HWB}}$ | Sec. 5.2, 5.6
$\varphi^{\dagger}\varphi W_{\mu\nu}^{I}W^{I\mu\nu}$ | ${{\mathcal{O}}_{HW}}$ | Sec. 5.2, 5.6
$\varphi^{\dagger}\varphi B_{\mu\nu}B^{\mu\nu}$ | ${{\mathcal{O}}_{HB}}$ | Sec. 5.2, 5.6
$(\varphi^{\dagger}\varphi)\Box(\varphi^{\dagger}\varphi)$ | ${{\mathcal{O}}_{H\Box}}$ | Sec. 5.2, 5.6
$(\varphi^{\dagger}D_{\mu}\varphi)^{*}(\varphi^{\dagger}D_{\mu}\varphi)$ | ${{\mathcal{O}}_{HD}}$ | Sec. 5.2, 5.6
Operator (CP odd) | Multiboson | Section
${\epsilon}^{IJK}\tilde{W}^{I\nu}_{\mu}W^{J\rho}_{\nu}W^{K\mu}_{\rho}$ | ${{\mathcal{O}}_{\tilde{W}}}$ | Sec. 5.1
$\varphi^{\dagger}\varphi\tilde{W}_{\mu\nu}^{I}W^{I\mu\nu}$ | ${{\mathcal{O}}_{H\tilde{W}}}$ | Sec. 5.2, 5.6
$\varphi^{\dagger}\varphi\tilde{B}_{\mu\nu}B^{\mu\nu}$ | ${{\mathcal{O}}_{H\tilde{B}}}$ | Sec. 5.2, 5.6
$\varphi^{\dagger}\tau^{I}\varphi\tilde{W}_{\mu\nu}^{I}B^{\mu\nu}$ | ${{\mathcal{O}}_{H\tilde{W}B}}$ | Sec. 5.2, 5.6
Operator (CP even) | Vector boson and quark | Section
$i(\varphi^{\dagger}D_{\mu}\varphi-(D_{\mu}\varphi)^{\dagger}\varphi)(\bar{q}_{p}\gamma^{\mu}q_{r})$ | ${{\mathcal{O}}^{(1)}_{Hq}}$ | Sec. 5.2, 5.6
$i(\varphi^{\dagger}\tau^{I}D_{\mu}\varphi-(D_{\mu}\varphi)^{\dagger}\tau^{I}\varphi)(\bar{q}_{p}\tau^{I}\gamma^{\mu}q_{r})$ | ${{\mathcal{O}}^{(3)}_{Hq}}$ | Sec. 5.2, 5.6
$i(\varphi^{\dagger}D_{\mu}\varphi-(D_{\mu}\varphi)^{\dagger}\varphi)(\bar{u}_{p}\gamma^{\mu}u_{r})$ | ${{\mathcal{O}}_{Hu}}$ | Sec. 5.2, 5.6
$i(\varphi^{\dagger}D_{\mu}\varphi-(D_{\mu}\varphi)^{\dagger}\varphi)(\bar{d}_{p}\gamma^{\mu}d_{r})$ | ${{\mathcal{O}}_{Hd}}$ | Sec. 5.2, 5.6
$i(\tilde{\varphi}^{\dagger}D_{\mu}\varphi)(\bar{u}_{p}\gamma^{\mu}d_{r})$ | ${{\mathcal{O}}_{Hud}}$ | Sec. 5.2, 5.6
$(\bar{q}_{p}\sigma^{\mu\nu}u_{r})\tau^{I}\tilde{\varphi}W_{\mu\nu}^{I}$ | ${\pagecolor{orange!25}{{\mathcal{O}}_{uW}}}$ | Sec. 5.2, 5.6
$(\bar{q}_{p}\sigma^{\mu\nu}d_{r})\tau^{I}\tilde{\varphi}W_{\mu\nu}^{I}$ | ${{\mathcal{O}}_{dW}}$ | Sec. 5.2, 5.6
$(\bar{q}_{p}\sigma^{\mu\nu}u_{r})\tilde{\varphi}B_{\mu\nu}$ | ${{\mathcal{O}}_{uB}}$ | Sec. 5.2, 5.6
$(\bar{q}_{p}\sigma^{\mu\nu}d_{r})\tilde{\varphi}B_{\mu\nu}$ | ${{\mathcal{O}}_{dB}}$ | Sec. 5.2, 5.6
Operator (CP even) | Top quark | Section
$(\bar{q}_{p}\sigma^{\mu\nu}u_{r})\tilde{\varphi}B_{\mu\nu}$ | ${{\mathcal{O}}_{uB}}$ | Sec. 5.4
$(\bar{q}_{p}\sigma^{\mu\nu}u_{r})\tau^{I}\tilde{\varphi}W_{\mu\nu}^{I}$ | ${{\mathcal{O}}_{uW}}$ | Sec. 5.4
$(\bar{q}_{i}\sigma^{\mu\nu}T^{A}u_{j})\>\tilde{\varphi}G_{\mu\nu}^{A}$ | ${{{\mathcal{O}}^{(ij)}_{uG}}}$ | Sec. 5.3
$f^{ABC}G_{\mu}^{A\nu}G_{\nu}^{B\rho}G_{\rho}^{C\mu}$ | ${{\mathcal{O}}_{G}}$ | Sec. 5.3
$\ \varphi^{\dagger}\varphi G_{\mu\nu}^{A}G^{A\mu\nu}$ | ${{\mathcal{O}}_{\varphi G}}$ | Sec. 5.3
Table 2: The space of EFT operators under consideration in this document. They are categorized as operators affecting multiboson and top based processes. Operator (CP even) | Top quark (four fermion) | Section
---|---|---
$(\bar{q}_{i}\gamma^{\mu}q_{j})(\bar{q}_{k}\gamma_{\mu}q_{l})$ | ${{\mathcal{O}}^{1(ijkl)}_{qq}}$ | Sec. 5.3
$(\bar{q}_{i}\gamma^{\mu}\tau^{I}q_{j})(\bar{q}_{k}\gamma_{\mu}\tau^{I}q_{l})$ | ${{\mathcal{O}}^{3(ijkl)}_{qq}}$ | Sec. 5.3
$(\bar{q}_{i}\gamma^{\mu}q_{j})(\bar{u}_{k}\gamma_{\mu}u_{l})$ | ${{\mathcal{O}}^{1(ijkl)}_{qu}}$ | Sec. 5.3
$(\bar{q}_{i}\gamma^{\mu}T^{A}q_{j})(\bar{u}_{k}\gamma_{\mu}T^{A}u_{l})$ | ${{\mathcal{O}}^{8(ijkl)}_{qu}}$ | Sec. 5.3
$(\bar{q}_{i}\gamma^{\mu}q_{j})(\bar{d}_{k}\gamma_{\mu}d_{l})$ | ${{\mathcal{O}}^{1(ijkl)}_{qd}}$ | Sec. 5.3
$(\bar{q}_{i}\gamma^{\mu}T^{A}q_{j})(\bar{d}_{k}\gamma_{\mu}T^{A}d_{l})$ | ${{\mathcal{O}}^{8(ijkl)}_{qd}}$ | Sec. 5.3
$(\bar{u}_{i}\gamma^{\mu}u_{j})(\bar{u}_{k}\gamma_{\mu}u_{l})$ | ${{\mathcal{O}}^{(ijkl)}_{uu}}$ | Sec. 5.3
$(\bar{u}_{i}\gamma^{\mu}u_{j})(\bar{d}_{k}\gamma_{\mu}d_{l})$ | ${{\mathcal{O}}^{1(ijkl)}_{ud}}$ | Sec. 5.3
$(\bar{u}_{i}\gamma^{\mu}T^{A}u_{j})(\bar{d}_{k}\gamma_{\mu}T^{A}d_{l})$ | ${{\mathcal{O}}^{8(ijkl)}_{ud}}$ | Sec. 5.3
$(\bar{q}_{i}u_{j})\>\varepsilon\;(\bar{q}_{k}d_{l})$ | ${{{\mathcal{O}}^{1(ijkl)}_{quqd}}}$ | Sec. 5.3
$(\bar{q}_{i}T^{A}u_{j})\;\varepsilon\;(\bar{q}_{k}T^{A}d_{l})$ | ${{{\mathcal{O}}^{8(ijkl)}_{quqd}}}$ | Sec. 5.3

##  5 Simulation studies of EFT prediction methods

This section presents studies of the consistency of the different strategies
for obtaining SMEFT predictions. Unless noted otherwise, both linear and
quadratic terms are included. The vertical bars in the histogram correspond to
$\sqrt{\sum_{i}w_{i}^{2}}$ where the index $i$ extends over all events a bin
and the event’s weights are denoted by $w_{i}$.

###  5.1 Helicity and reweighting of predictions for the WZ process

Diboson production in proton-proton collision is extremely important to study
the electroweak sector of the SM due to its sensitivity to the self-
interaction of gauge bosons and to probe anomalous effects in trilinear gauge
couplings. The effects of dimension-6 SMEFT operators in the associated
production of a W and Z boson, referred to as WZ production, are studied in
this section. We restrict the study to the CP-even and -odd dimension-6
operators $\mathcal{O}_{\textrm{W}}$ and
$\mathcal{O}_{\widetilde{\textrm{W}}}$.

Those operators affect the triple gauge boson coupling, i.e., the interaction
vertex involving three electroweak vector bosons, as shown in Fig. 1.

Figure 1: Feynman diagram for WZ production where effects of dimension-6
operators affecting the triple gauge boson vertex are shown.

A detailed measurement of this process is performed by both the ATLAS and CMS
Collaborations using Run-2 LHC data [25, 26].

For the SM, the WZ process is generated at LO using Madgraph5 aMC@NLO v2.6.5
[17]. The NNPDF3.1 NNLO PDF set [27] is used. The renormalization and
factorization scales chosen are half of the sum of the transverse mass of
final state particles. The SMEFT effects are simulated at LO using the
SMEFTsim v3.0 [28] model with the topU3l flavor scheme. Event samples are
produced at both the SM point, i.e., setting all Wilson coefficients to zero
and with non-zero values of Wilson coefficients for the operators considered.
Several weights are stored for each event. Those are computed using
reweighting for matrix element method [29] following two approaches: helicity-
aware and -ignorant reweightings. Ten million events are generated for each of
the samples separately with helicity-aware and -ignorant reweightings. For the
generated samples, one million events are generated.

The WZ production at the SM is dominated by the helicity configuration where
both the bosons are transversely polarized with opposite helicities
($\pm$,$\mp$), whereas the SMEFT operators considered here affect the
configuration with both the bosons have the same transverse helicities
($\pm$,$\pm$) [30]. This is depicted in Fig. 2 as a function of W boson
p${}_{\textrm{T}}$ using the event samples produced for this study.

Figure 2: Helicity composition as a function of W boson pT at the SM point
(left) and a BSM point with both $c_{W}$ and $c_{\tilde{W}}$ set to 1 (right).
Here, L and T refer to longitudinal and transverse polarizations,
respectively, whereas OS and SS refer to the opposite- and same-sign
configurations, respectively. The helicity eigenstates are defined in the
laboratory reference frame.

Both the SMEFT operators modify the W boson p${}_{\textrm{T}}$ spectrum. So,
it is used as an observable to compare between samples where the prediction at
an EFT point is obtained using event weights and those produced directly at
that particular EFT point, referred to as reweighted and generated
predictions, respectively. The comparisons are shown in Fig. 3 for the case
where $c_{W}$ and $c_{\tilde{W}}$ have a value of 1. The top row of Fig. 3
shows the comparison of W boson p${}_{\textrm{T}}$ spectra, summed over all
possible helicity configurations, in reweighted and generated samples for two
choices of the reference point in reweighting: SM point and a BSM point, where
Wilson coefficients of both the operators are set to 0.5. For the SM reference
point, the helicity-ignorant reweighting can reproduce the W boson pT spectrum
predicted by the generated sample except at very high-pT values, where
statistics is small, but the helicity-aware reweighting fails. Both helicity-
aware and -ignorant reweightings model the generated W boson
p${}_{\textrm{T}}$ spectra very well once the BSM reference point is used in
reweighting. The bottom row of Fig. 3 shows the same comparison as the top row
but specifically for the helicity configuration affected by the SMEFT
operators. Here, it is evident that helicity-ignorant reweighting fails to
model the p${}_{\textrm{T}}$ spectra for individual helicity configurations
irrespective of the reference point chosen in reweighting, whereas the
helicity-aware reweighting with a BSM reference point can model the W boson
p${}_{\textrm{T}}$ spectrum for a specific helicity configuration, affected by
the SMEFT operators considered.

Figure 3: Comparison of W boson $p_{T}$ spectra between generated and
reweighted (both helicity-aware and -ignorant variants) at a BSM point
($c_{W}$, $c_{\tilde{W}}$ = 1, 1) for two reference points used in the
reweighting: the SM point (left) and a BSM point with both $c_{W}$ and
$c_{\tilde{W}}$ set to 0.5 (right). The upper row corresponds to the case
where all helicity configurations are summed, and the lower row corresponds to
only the same-sign transverse polarization configuration.

###  5.2 Helicity and reweighting of predictions for the ZH process

In this section, we study the modeling of SMEFT effects in Higgs production in
association with a W or a Z boson, referred to as VH production. This
particular Higgs production mode is extremely important to look for the
effects of new physics since its contribution becomes increasingly important
at high values of Higgs boson p${}_{\textrm{T}}$ [31]. The VH process has been
measured by both ATLAS and CMS Collaborations across different decay channels
[32, 33]. The VH production is affected by a number of SMEFT operators at
dimension 6 in the following ways:

  * •

modifying the vector boson coupling to quarks or giving rise to a four-point
interaction

  * •

modifying the Higgs boson interaction with $\mathup{{{W}}}$ or
$\mathup{{{Z}}}$ boson

For this study, we restrict ourselves to the ZH production only and focus on
one vector coupling operator ${{\mathcal{O}}^{(3)}_{Hq}}$ and two HVV
operators ${{\mathcal{O}}_{HW}}$ and ${{\mathcal{O}}_{H\tilde{W}}}$ that
affect both WH and ZH productions. The final state with the Z boson decaying
to the leptons and the Higgs boson decaying to a pair of b quarks, measured by
both the ATLAS and CMS Collaborations [34, 35, 36], is considered. The
${{\mathcal{O}}^{(3)}_{Hq}}$ operator mainly affects the helicity
configuration where the Z boson is longitudinally polarized, which is also
dominant in the SM. The HVV operators, on the other hand, also affect the
interference of scattering amplitudes with different helicities of Z boson
that get reflected in the distribution of angles constructed using Higgs boson
and leptons from Z boson [37] as depicted in Fig. 4.

Figure 4:  Decay planes and angles in the Z($\rightarrow$ l+l-
)H($\rightarrow$ b $\bar{\textrm{b}}$) production. The angle $\Theta$ and
$\phi$ are defined in the ZH rest frame, while $\theta$ is defined in the Z
boson rest frame.

The ZH production process followed by the leptonic decay of the Z boson and
the Higgs boson decay to a pair of bottom quarks is generated at LO with up to
one additional jet using Madgraph5 aMC@NLO v2.6.5 [17]. The NNPDF3.1 NNLO PDF
set [27] is used. The renormalization and factorization scales chosen are half
of the sum of the transverse mass of final state particles. The SMEFT effects
are simulated at LO using the SMEFTsim v3.0 [28] model with the topU3l flavor
scheme. For each event, several weights are stored, which are computed by ME
reweighting [29] following two approaches: helicity-aware and -ignorant
reweightings. The SM point, i.e. all Wilson coefficients set to 0, is used as
the reference point in reweighting. Separate samples are produced by turning
on one operator at a time with the following values: a) c${}_{Hq}^{(3)}$ =
0.1, b) cHW = 1, c) c${}_{H\tilde{W}}$ = 1; in each case, values of all other
Wilson coefficients are set to 0. These are referred to as the generated
samples. One million events are generated for each of the reweighted and
generated samples.

The particle-level Z boson p${}_{\textrm{T}}$ spectra predicted by reweighted
samples are compared to the one from the generated samples as shown in Fig. 5.
In this case, both helicity-aware and -ignorant variants of reweighting model
the effects of ${{\mathcal{O}}^{(3)}_{Hq}}$ and ${{\mathcal{O}}_{HW}}$ on Z
boson p${}_{\textrm{T}}$ well.

Figure 5: Comparison of Z boson p${}_{\textrm{T}}$ spectra (absolute cross
section per GeV on the y-axis) between generated and reweighted (with both
helicity-aware and -ignorant variants) ZH samples at c${}_{Hq}^{3}$=0.1 (left)
and cHW=1 (right). The helicity eigenstates are defined in the laboratory
reference frame.

Next, we compare the distributions of angle $\theta$ and $\phi$ depicted in
Fig. 4 between reweighted and generated samples in Fig. 6 for cHW and
c${}_{H\tilde{W}}$ separately. For both HVV operators, two variants of
reweighting model the angular distributions obtained from the generated
samples within statistical uncertainties. For the angle $\phi$, one can notice
that it follows a $\cos{2\phi}$ distribution for cHW=1, which is modified for
c${}_{H\tilde{W}}$=1 due to different terms in the interference between
scattering amplitudes being relevant for CP-even and -odd gauge coupling
operators, respectively. Therefore, the variable $\phi$ can be used to probe
the CP nature of Higgs to vector boson interaction.

Figure 6: Comparison of $\theta$ (top) and $\phi$ (bottom) distributions
(absolute cross section on the y-axis) between generated and reweighted (with
both helicity-aware and -ignorant variants) ZH samples at cHW=1 (left) and
c${}_{H\tilde{W}}$=1(right). The angle $\phi$ is defined in the ZH rest frame,
while $\theta$ is defined in the Z boson rest frame.

###  5.3 The $\textrm{t}\overline{\textrm{t}}$ process

In the SM, top-quark pair production
($\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$) in proton-proton
collisions proceeds predominantly via gluon-gluon fusion ($\sim 90\%$)
followed by quark-antiquark annihilation ($\sim 10\%$) at LO. At NLO in QCD
and beyond, channels with quark-gluon initial states appear. In this section,
we aim to include all operators that significantly impact the
$\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ processes. In Table
2, two-heavy-two-light four-quark operators are listed that can impact the
$\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ process. In
addition, the ${{{\mathcal{O}}^{(ij)}_{uG}}}$ operator from Table 1 affects
the $\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ rate
significantly. Therefore, both the real and imaginary parts of the WC for the
${{{\mathcal{O}}^{(ij)}_{uG}}}$ operator, ctGRe and ctGIm, are considered in
this study. In this work, we only include those operators of the Warsaw basis
that explicitly modify the couplings of the top quark with the other SM
fields. We therefore do not include the ${{\mathcal{O}}_{G}}$ and
${{\mathcal{O}}_{\varphi G}}$ operators that are well constrained by processes
that do not involve top quarks [38, 39].

The signal contribution is modeled at LO using the Madgraph5 aMC@NLO v2.6.5
event generator with the SMEFTsim model to incorporate the EFT effects. The
definitions of the operators associated with all of these WCs are provided in
[40]. As discussed in Sec. 1, the cross section (inclusive or differential)
depends quadratically on the WCs. We have parameterised each event weight as a
quadratic function of WCs, as described in Eq. 8, by including enough
reweighting points per event. The nominal
$\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ sample is generated
at a starting point in the WC parameter space far from the SM point. The
reference point of the nominal sample, called “LO (sample 1)”, is set to Pt1
as defined below. To simulate EFT impacts on the
$\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ production from
quark-gluon initial states and to predict distributions in the presence of
extra jets more accurately, we produce another sample that includes an
additional final-state parton in the matrix element generation. This sample is
also produced at reference point Pt1, and is called "LO+1 jet (sample 1)".
Both $\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ and
$\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ +1jet samples
include the dominant t $\rightarrow$ Wb decay followed with the leptonic
decays of W boson.

It is important to make sure that the generated sample is able to be
consistently reweighted to other points in EFT parameter space. In order to
check that, we have generated independent
$\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ samples at the
following points in EFT space,

  * •

SM point: all WCs set to zero.

  * •

Dedicated point: All WCs set to zero except one which is set to -4,-2, 2, and
4 independently. The value of ctGRe is set to -0.4, -0.2, 0.2, 0.4 because of
its large effect on the cross section.

  * •

Pt1: WCs set to c${}_{\rm tG}^{Im}=0.7$, c${}_{\rm tG}^{Re}=0.7$, c${}_{\rm
Qj}^{38}=9.0$, c${}_{\rm Qj}^{18}=7.0$, c${}_{\rm Qu}^{8}=9.5$, c${}_{\rm
Qd}^{8}=12.0$, c${}_{\rm tj}^{8}=7.0$, c${}_{\rm tu}^{8}=9.0$, c${}_{\rm
td}^{8}=12.4$, c${}_{\rm Qj}^{31}=3.0$, c${}_{\rm Qj}^{11}=4.2$, c${}_{\rm
Qu}^{1}=5.5$, c${}_{\rm Qd}^{1}=7.0$, c${}_{\rm tj}^{1}=4.4$, c${}_{\rm
tu}^{1}=5.4$, c${}_{\rm td}^{1}=7.0$

  * •

Pt2: WCs set to c${}_{\rm tG}^{Im}=1.0$, c${}_{\rm tG}^{Re}=1.0$, c${}_{\rm
Qj}^{38}=3.0$, c${}_{\rm Qj}^{18}=3.0$, c${}_{\rm Qu}^{8}=3.0$, c${}_{\rm
Qd}^{8}=3.0$, c${}_{\rm tj}^{8}=3.0$, c${}_{\rm tu}^{8}=3.0$, c${}_{\rm
td}^{8}=3.0$, c${}_{\rm Qj}^{31}=3.0$, c${}_{\rm Qj}^{11}=3.0$, c${}_{\rm
Qu}^{1}=3.0$, c${}_{\rm Qd}^{1}=3.0$, c${}_{\rm tj}^{1}=3.0$, c${}_{\rm
tu}^{1}=3.0$, c${}_{\rm td}^{1}=3.0$.

The samples produced at Pt2 are called "LO (sample 2)" and "LO+1jet (sample
2)". In Fig. 7 and Fig. 8, relative SMEFT contributions to the
$\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ inclusive cross
section are shown for individual WCs. In each plot the quadratic function
extracted from LO (sample 1) and the LO (sample 2) are compared to the cross
section ratios calculated at the dedicated points. In general, there is good
agreement between the cross section ratio values predicted by reweighting and
those calculated at the dedicated points in EFT space.

In addition to the inclusive cross section estimation, LO+1jet (sample 1)
should be able to predict differential distributions of various kinematic
variables at different points in EFT space. In Fig. 9, distributions of top
quark $p_{T}$, leading lepton $p_{T}$, and $\Delta R$ between two leptons are
shown for $\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ events
with two leptons (electron or muon) with $p_{t}>20$ GeV and $|\eta|<2.5$, and
at least two jets with $p_{t}>20$ GeV and $|\eta|<2.5$. In the left column,
differential distributions are shown for LO+1jet (sample 1) and LO+1jet
(sample 2) reweighted to the SM point. In the right column LO+1jet (sample 1)
and LO+1jet (sample 2) are reweighted to Pt2, the starting point of the sample
2. These distributions show that our nominal sample is able to describe the
$\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ differential
distributions in different points in EFT space including the SM point.

Figure 7: Relative modification of the
$\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ total cross
section, $\sigma_{\text{SMEFT}}/\sigma_{\text{SM}}$, induced by the presence
of the SMEFT operators. Solid curves show the quadratic dependency of the
relative cross section to the WC values obtained from reweighting sample 1 and
sample 2 in EFT space. Blue points are obtained from dedicated simulations.

Figure 8: Relative modification of the
$\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ total cross
section, $\sigma_{\text{SMEFT}}/\sigma_{\text{SM}}$, induced by the presence
of the SMEFT operators. Solid curves show the quadratic dependency of the
relative cross section to the WC values obtained from reweighting sample 1 and
sample 2 in EFT space. Blue points are obtained from dedicated simulations.

Figure 9: Differential distributions for
$\mathup{{{t}}}$$\mathup{{\overline{{{\mathup{{{t}}}}}}}}$ +1jet process with
respect to the top quark average $p_{T}$ (top), leading lepton $p_{T}$
(middle) and $\Delta R(\text{leading lepton, sub leading lepton})$ (bottom).
Differential distributions obtained from reweighting both sample 1 and sample
2 to the SM point (left) and reweighting sample 1 to Pt2 (right).

###  5.4 Studies in the $\textrm{t}\overline{\textrm{t}}Z$ process at NLO

In this section, we analyze the EFT reweighting performance using as a
benchmark process the production of a top-antitop pair in association with a Z
boson emission, namely the $\textrm{t}\overline{\textrm{t}}\mathrm{Z}$
process. This process has been extensively studied both from ATLAS [41, 42]
and CMS [43, 44], showing a high sensitivity to possible EFT effects coming
from operators that modify the electroweak vertices. In this case, we focus on
the ${{\mathcal{O}}_{tZ}}$ operator in Table 1, which modifies the interaction
between the top quark and the Z boson.

To simulate the $\textrm{t}\overline{\textrm{t}}\mathrm{Z}$ process we use
Madgraph5 aMC@NLO v2.6.5, while the EFT effects are taken into account using
the SMEFTsim [28] and the SMEFT@NLO [45] models. The events are generated at
NLO in QCD when using the SMEFT@NLO model and at LO plus an additional parton
with the SMEFTsim model, while the parton shower is modeled using PYTHIA8[46].
For the generation of the LO samples, particular attention is dedicated to the
matching procedure as discussed in Sec.4. Both simulations are performed in
the 5 flavour scheme choosing as scale half of the sum of transverse masses.
We generate events assuming the SM and assuming $c_{\textrm{tZ}}=1$ in the
SMEFT@NLO convention. We then compare the definitions of the operator in the
two models to set the corresponding value of $c_{\textrm{tZ}}$ in the LO
simulation with the SMEFTsim model.

In Fig. 10 and Fig. 11, the distributions of the top quark and Z boson
transverse momentum for the SM and the different EFT prediction generation
strategies are shown. The left plot shows the distributions from the LO
simulation while in the right one, the NLO results are presented. It is not
possible to separate linear and quadratic EFT contributions in the NLO
simulation in MG v2, so only the reweighted simulation and the separate
simulation distributions are shown.

In the two central panels in Fig. 10 the ratio between the EFT distributions
and the SM is shown, highlighting an increase in sensitivity to the EFT
effects in the tail of the top quark transverse momentum distribution. In the
bottom panels, the ratio among the different EFT generation strategies is
presented, showing a good agreement along the whole spectrum.

Figure 10: Top $p_{T}$ distribution simulated at LO + 1 jet (left) and at NLO
(right). In black is depicted the SM distribution, while the other lines
represent the different available simulation methods to compute EFT
predictions. The first ratio plot highlights the sensitivity to the EFT
effects. While, in the second ratio plot, the agreement between the direct
simulation, the reweighted simulation, and the separate simulation (only for
LO + 1 jet) is shown.

The Z boson transverse momentum distribution, in Fig. 11, shows a higher
sensitivity to the EFT effect with respect to the top $p_{T}$ distribution.
However, the agreement between the reweighted simulation and the other two EFT
prediction strategies gets worse at high Z $p_{T}$ in the LO results. This
arises from employing the helicity-aware reweighting using the SM as a
reference point. In the NLO results, where only the helicity ignorant
reweighting is possible, there remains a good agreement between the reweighted
simulation and the direct simulation.

Figure 11: Z $p_{T}$ distribution simulated at LO + 1 jet (left) and at NLO
(right). In black is depicted the SM distribution, while the other lines
represent the different available simulation methods to compute EFT
predictions. The first ratio plot highlights the sensitivity to the EFT
effects. While, in the second ratio plot, the agreement between the direct
simulation, the reweighted simulation, and the separate simulation (only for
LO + 1 jet) is shown.

A downside of the reweighting simulation is the statistical fluctuations in
poorly populated phase space due to large event weights. This is evident in
Fig. 10 (left) and Fig. 11, where there are large statistical fluctuations in
the reweighted distribution at high Z and top transverse momentum.

###  5.5 Helicity aware and ignorant reweighting in the
$\textrm{t}\overline{\textrm{t}}Z$ process

In this section, we use the $\textrm{t}\overline{\textrm{t}}\mathrm{Z}$
process, which was already introduced in Sec. 5.4, to showcase the benefits
and potential limitations of the helicity-ignorant reweighting approach. To do
that, we generate $\textrm{t}\overline{\textrm{t}}\mathrm{Z}$ events using
Madgraph5 aMC@NLO v2.6.7 and simulate EFT from the $\mathcal{O}_{\textrm{tZ}}$
operator from Table 1.

The $\mathcal{O}_{\textrm{tZ}}$ operator, defined as
$-\sin\theta_{W}{{\mathcal{O}}_{uB}}+\cos\theta_{W}{{\mathcal{O}}_{uW}}$,
introduces $\textrm{t}\overline{\textrm{t}}\mathrm{Z}$ vertices with helicity
configurations that are not present in the SM and for this reason, the choice
between helicity-ignorant and aware reweighting is particularly relevant. This
is illustrated in Fig. 12, where we show the distribution of the spin of the
different partons involved in $\textrm{t}\overline{\textrm{t}}\mathrm{Z}$
events. We consider events generated assuming the SM and assuming
$c_{\textrm{tZ}}=5$. The figure also shows that certain helicity
configurations that arise naturally in the BSM scenario are not present or
suppressed in the SM. Because of this, it is not possible to use the helicity-
aware method to reweight SM samples so that they reproduce this specific BSM
scenario: the regions of the phase space spanned by these helicity
configurations will not be populated by SM samples.

Figure 12: Distribution of the different helicity configurations in
$\textrm{t}\overline{\textrm{t}}\mathrm{Z}$ generated at SM and
$c_{\mathrm{tZ}}$=5 in the partonic center-of-mass frame. Bins of the
distribution are labeled as
$(h_{\mathrm{Z}},h_{\mathrm{t}},h_{\bar{\mathrm{t}}},h_{\mathrm{g}_{1}},h_{\mathrm{g}_{2}})$,
where $h_{i}$ denotes the helicity of each particle. The distribution shows
that some helicity configurations are not present at the SM and, therefore,
helicity-aware reweighting cannot populate those regions of the phase space.

A consequence of this is shown in Fig. 13 where the inclusive
$\textrm{t}\overline{\textrm{t}}\mathrm{Z}$ cross section dependence as a
function of $c_{\textrm{tZ}}$ is shown. This dependence is computed using
three independent samples with the same number of events. Two samples are
generated assuming the SM and a third one is generated at $c_{\textrm{tZ}}=5$.
The three samples are then weighted to $c_{\textrm{tZ}}=0,-1,1$ to obtain the
inclusive $\textrm{t}\overline{\textrm{t}}\mathrm{Z}$ cross section for those
values. From them, we interpolate the dependence using a second-order
polynomial, whose coefficients are shown in Fig. 13. We take the
$c_{\textrm{tZ}}=5$ as the reference value, which we expect to be more
accurate since that one is expected to populate the full kinematic phase
space. We use the helicity-aware and ignorant reweighting for each of the SM
samples.

The trend predicted by the helicity-ignorant reweighting approach agrees with
the one predicted by the reference value. In contrast, the helicity-aware
predicts a significantly smaller quadratic term than the reference value and
features larger statistical uncertainties than the other approaches. This is
due to two reasons. On the one hand, the smaller value predicted by the
helicity-aware approach is because it cannot populate phase space regions that
are forbidden in the SM but present when $c_{\textrm{tZ}}\neq 0$. On the other
hand, the larger uncertainties of this approach are due to a degradation of
the statistical power of the weighted sample, which is due to large event
weights arising from regions of the phase space that are suppressed but not
forbidden in the SM. This shows two possible advantages of helicity-ignorant
reweighting: it can be used to reweight SM samples to some BSM scenarios and,
in addition, can be used in general to increase the statistical power of
weighted samples.

Figure 13: Quadratic parametrization of the
$\textrm{t}\overline{\textrm{t}}\mathrm{Z}$ cross section as a function of
$c_{\mathrm{tZ}}$ obtained by reweighting samples generated at the SM and
$c_{\mathrm{tZ}}$=5 using the helicity-ignorant and aware methods.

Although the helicity-ignorant reweighting populates the kinematic phase space
more efficiently, it gives rise to reweighted samples that do not necessarily
reproduce the helicity configurations of the scenario that we are reweighting
to. The helicity of the produced partons is not directly measurable but it can
introduce correlations among the kinematics of the different final state
particles. In analyses where these correlations are relevant keeping track of
the helicity of the different particles is necessary.

To study this effect, we consider two different methods to model the decay of
the produced partons. In both scenarios, we produce samples
$c_{\textrm{tZ}}=5$ and reweight them to the SM and samples produced assuming
the SM. In one of the scenarios, we use Madspin to model the decay of the top
quarks and Z boson, and the weights are computed as a function of the top
quarks and Z boson, before their decay. In the other scenario, the decay is
modeled using Madgraph and the weights are computed as a function of the
particles produced in the decay of the top quarks and Z boson. We note that,
for this process, we only expect shortcomings due to the helicity-ignorant
reweighting in the earlier scenario. In the latter scenario, the reweighting
is performed as a function of the particles produced in the decay, and changes
in their helicity do not give rise to observable effects.

In Fig. 14 we show the $\Delta\phi$ distribution between the two leptons
produced in the Z boson decay. The plot shows that only the Madgraph decay
model can produce consistent results, while Madspin is introducing artificial
trends. These trends are present because Madspin computes and samples
implicitly over
$\mathcal{M}_{\mathrm{prod+decay}}/\mathcal{M}_{\mathrm{prod}}$, the ratio of
the production+decay over production matrix elements, which is computed at the
reference point. By default, Madspin does not recompute this ratio for
reweighted events, so it introduces artificial trends when reweighting to a
scenario for which
$\mathcal{M}_{\mathrm{prod+decay}}/\mathcal{M}_{\mathrm{prod}}$ is different
from the reference points.

Figure 14: Distribution of the $\Delta\phi$ between the two leptons produced
in the $\mathrm{Z}$ boson decay in samples where the top quarks and the
$\mathrm{Z}$ boson have been decayed using Madspin (left) and the Madgraph
decay syntax (right). Samples generated at the SM and $c_{\mathrm{tZ}}$=5 are
both reweighted to the SM.

###  5.6 The VBF H process

Vector boson fusion (VBF) is one of the dominant Higgs boson production
mechanisms at the LHC. This process has been measured by the ATLAS and CMS
experiments in a variety of Higgs boson decay channels [32, 33]. These
measurements rely on the presence of two forward (high $|\eta|$) quark-
initiated jets to distinguish them from other H production modes.

Operators, including CP-odd ones, at dimension 6 in standard model effective
field theory can modify the VBF process in a variety of ways by:

  1. 1.

altering only the total cross section,

  2. 2.

introducing anomalous couplings between the Higgs and vector bosons as shown
in Fig. 15a,

  3. 3.

introducing anomalous couplings between the quarks and vector bosons (Fig.
15b) or HVqq contact interactions (Fig. 15c).

Operators impacting Higgs boson decays are not considered.

Figure 15: Feynman diagrams representing VBF Higgs boson production with EFT
contributions to (a) the HVV interaction, (b) the Vqq interaction, and (c) a
VHqq contact interaction.

An SM VBF Higgs sample is generated at LO using Madgraph5 aMC@NLO v2.9.13 [17]
and the NNPDF3.1 PDF set [47]. The renormalization and factorization scales
are not kept fixed, and their values are the default in Madgraph5, namely the
transverse mass of the $2\rightarrow 2$ system resulting from $k_{\text{T}}$
clustering. The SMEFTsim framework [28] is used with the topU3l model and
$m_{W}$ input parameter scheme to provide per-event weights relative to the SM
accounting for EFT contributions (both linear and quadratic) from each of the
operators in Table 1. The EFT scale $\Lambda$ is chosen at 1 TeV, and Feynman
diagrams with single-operator insertions are considered. The reweighting
procedure used for this sample is helicity aware. Since the
${{\mathcal{O}}_{Hud}}$ operator induces a right-handed charged current, its
effects are not captured in the simulated sample.

Five EFT scenarios are selected as representative examples of the three
classes enumerated above $c_{H\Box}=1$, $c_{HW}=1$, $\tilde{c}_{HW}=1$,
$c_{Hq^{(1)}}=1$, and $c_{Hq^{(3)}}=1$. In each case, only the listed Wilson
coefficient is non-zero. In addition to the reweighted SM sample, each of
these five EFT points is simulated directly up to quadratic order. For each
EFT point, two additional samples are generated: one includes the linear term
only, and the other includes the quadratic term only. The sum of the standard
model, linear-only, and quadratic-only samples is expected to reproduce the
full sample. The cross section of each EFT sample is listed in Table 3. Good
agreement is observed between the various simulation methods.

Cross section [fb] | SM | $c_{H\Box}=1$ | $c_{HW}=1$ | $\tilde{c}_{HW}=1$ | $c_{Hj^{(1)}}=1$ | $c_{Hj^{(3)}}=1$
---|---|---|---|---|---|---
Reweighted | $-$ | 4.164 | 3.608 | 3.819 | 3.847 | 2.687
Direct sim. | 3.675 | 4.140 | 3.618 | 3.829 | 3.859 | 2.778
SM + Lin. + Quad. | $-$ | 4.135 | 3.617 | 3.824 | 3.865 | 2.804
Table 3: VBF cross section calculated by Madgraph5 from one million simulated
events.

Figure 16a shows the distribution of Higgs boson $p_{\textrm{T}}$ for the
$c_{H\Box}=1$ scenario for each of the simulated samples. The overall cross
section is enhanced with respect to the SM, as seen in Table 3. Note that the
upper limit of the $p_{\textrm{T}}$ distribution (700 GeV) may be beyond the
range of EFT validity. However, as this study is intended to validate the
performance of different simulation strategies, the range of EFT validity is
not considered. Figure 16b shows the angular separation $\Delta\eta$ between
the spectator quarks in the event, which is similar in shape to the SM
expectation.

Figure 16: Distribution of (a) the Higgs boson $p_{T}$ and (b) the angular
separation $\Delta\eta$ between the spectator quarks, for the $c_{H\Box}=1$
scenario. The SM expectation is shown in black. The prediction for
$c_{H\Box}=1$ obtained by reweighting the SM point is shown in red. The direct
simulation of $c_{H\Box}=1$ is shown in blue, and the sum of SM, linear-only,
and quadratic-only is shown in orange. The middle panel shows the ratio to the
SM, and the lower panel shows the ratio to the directly simulated sample.

Figure 17a shows the distribution of Higgs boson $p_{\textrm{T}}$ for the
$c_{HW}=1$ scenario. Compared to the SM expectation, a deficit is observed
below 50 GeV, and an enhancement is observed above about 150 GeV. Figure 17b
shows the azimuthal separation $\Delta\phi$ between the spectator quarks for
the $c_{HW}=1$ scenario. This distribution is strongly modified by the
presence of the EFT operator: while the SM expectation shows that large
angular separations are somewhat preferred, the distribution for $c_{HW}=1$ is
nearly flat in this variable as the ${{\mathcal{O}}_{HW}}$ operator affects
only the transverse amplitude $qV_{T}\rightarrow qH$ in VBF H production [48].

Figure 17: Distribution of (a) the Higgs boson $p_{T}$ and (b) the angular
separation $\Delta\phi$ between the spectator quarks, for the $c_{HW}=1$
scenario. The SM expectation is shown in black. The prediction for $c_{HW}=1$
obtained by reweighting the SM point is shown in red. The direct simulation of
$c_{HW}=1$ is shown in blue, and the sum of SM, linear-only, and quadratic-
only is shown in orange. The middle panel shows the ratio to the SM, and the
lower panel shows the ratio to the directly simulated sample.

Figure 18 shows the distribution of Higgs boson $p_{\textrm{T}}$ (a) and
azimuthal separation $\Delta\phi$ between the spectator quarks (b) for the
$\tilde{c}_{HW}=1$ scenario. Compared to the SM expectation, an enhancement in
the $p_{\textrm{T}}$ is observed above about 150 GeV. The $\Delta\phi$
distribution is also slightly modified, though the effect is far weaker than
that seen in Fig. 17b for the conjugate operator as the linear term, which
models the interference between the SM and the EFT scenario, is small.

Figure 18: Distribution of (a) the Higgs boson $p_{T}$ and (b) the angular
separation $\Delta\phi$ between the spectator quarks, for the
$\tilde{c}_{HW}=1$ scenario. The SM expectation is shown in black. The
prediction for $\tilde{c}_{HW}=1$ obtained by reweighting the SM point is
shown in red. The direct simulation of $\tilde{c}_{HW}=1$ is shown in blue,
and the sum of SM, linear-only, and quadratic-only is shown in orange. The
middle panel shows the ratio to the SM, and the lower panel shows the ratio to
the directly simulated sample.

Figure 19 shows the distribution of Higgs boson $p_{\textrm{T}}$ (a) and
angular separation $\Delta\eta$ between the spectator quarks (b) for the
$\tilde{c}_{Hj^{(1)}}=1$ scenario. Compared to the SM expectation, an
enhancement in the $p_{\textrm{T}}$ is observed above about 150 GeV. It should
be noted that the value of $\tilde{c}_{Hj^{(1)}}=1$ is large compared to
existing constraints [49], but the large deviation with respect to the SM at
high $p_{\textrm{T}}$ may fall outside the range of EFT validity. A small
enhancement with respect to the SM is also seen for $|\Delta\eta|\sim 0$.

Figure 19: Distribution of (a) the Higgs boson $p_{T}$ and (b) the angular
separation $\Delta\eta$ between the spectator quarks, for the $c_{Hj^{(1)}}=1$
scenario. The SM expectation is shown in black. The prediction for
$c_{Hj^{(1)}}=1$ obtained by reweighting the SM point is shown in red. The
direct simulation of $c_{Hj^{(1)}}=1$ is shown in blue, and the sum of SM,
linear-only, and quadratic-only is shown in orange. The middle panel shows the
ratio to the SM, and the lower panel shows the ratio to the directly simulated
sample.

Figure 20 shows the distribution of Higgs boson $p_{\textrm{T}}$ (a) and
angular separation $\Delta\eta$ between the spectator quarks (b) for the
$\tilde{c}_{Hj^{(3)}}=1$ scenario. As for $\tilde{c}_{Hj^{(1)}}$, the value of
$\tilde{c}_{Hj^{(3)}}=1$ is large compared to existing constraints [49].
Compared to the SM expectation, an overall lower cross section and a deficit
in the range $100<p_{T}<300$ GeV is observed. A small deficit with respect to
the SM is also seen for $|\Delta\eta|\sim 0$.

Figure 20: Distribution of (a) the Higgs boson $p_{T}$ and (b) the angular
separation $\Delta\eta$ between the spectator quarks, for the $c_{Hj^{(3)}}=1$
scenario. The SM expectation is shown in black. The prediction for
$c_{Hj^{(3)}}=1$ obtained by reweighting the SM point is shown in red. The
direct simulation of $c_{Hj^{(3)}}=1$ is shown in blue, and the sum of SM,
linear-only, and quadratic-only is shown in orange. The middle panel shows the
ratio to the SM, and the lower panel shows the ratio to the directly simulated
sample.

The comparison of these simulated EFT samples indicates good agreement between
the predictions obtained by reweighting the SM prediction, directly
simulating, and combining separately generated SM, linear, and quadratic
components. In regions where the SM sample contains a limited number of
events, such as at the highest $p_{\textrm{T}}$ and small $|\Delta\eta|$, some
fluctuations and large uncertainties are observed in the reweighted spectrum.

###  5.7 Multiboson processes

####  5.7.1 Choice of dynamical scale

The dynamical scale, defined by certain functional forms, represents the
factorization ($\mu_{\textrm{F}}$) and renormalization ($\mu_{\textrm{R}}$)
scales. The following scale choices are implemented in Madgraph_aMC@NLO [50].

  * •

Option 1: transverse mass of the $2\rightarrow 2$ system resulting of a
$k_{\text{T}}$ clustering
$\frac{1}{2}\sum_{i=1}^{N}\sqrt{m_{i}^{2}+p_{T,i}^{2}}$.

  * •

Option 2: total transverse energy of the event $\sum_{i=1}^{N}\frac{E_{i}\cdot
p_{T,i}}{\sqrt{p_{x,i}^{2}+p_{y,i}^{2}+p_{z,i}^{2}}}$.

  * •

Option 3: sum of the transverse masses
$\sum_{i=1}^{N}\sqrt{m_{i}^{2}+p_{T,i}^{2}}$.

  * •

Option 4: half of the sum of the transverse masses
$\frac{1}{2}\sum_{i=1}^{N}\sqrt{m_{i}^{2}+p_{T,i}^{2}}$.

  * •

Option 5: partonic energy $\sqrt{\hat{s}}$.

The default scale choice is typically set to option 1 as defined above. This
choice is insufficient for the generation of dimension-8 operators as shown in
the left panel of Fig. 21. The dimension-8 operator under consideration is of
the form shown in Eq. 20. However, the inadequacy of the choice of the default
scale is independent of the exact choice of the operator.

${\cal O}_{T,0}=Tr[\hat{W}_{\mu\nu}\hat{W}^{\mu\nu}]\times
Tr[\hat{W}_{\alpha\beta}\hat{W}^{\alpha\beta}]$ (20)

Each process in Fig. 21 is generated separately representing the SM-only (as a
red-filled histogram), the interference between the SM and BSM components (as
a green-filled histogram), and the BSM-only (as a blue-filled histogram)
contribution. The exact syntax is given below (where the charge conjugate
process was not generated in the interest of keeping the computation time
low):

        - generate p p > w+ w+ w- T0=1 (for full generation, shown in black)
        - generate p p > w+ w+ w- (SM generation shown in red)
        - generate p p > w+ w+ w- T0^2==1
        (Interference between SM and BSM generation shown in green)
        - generate p p > w+ w+ w- T0^2==2 (BSM generation shown in blue)

The same syntax is used for both histograms on the left and the right. The
only difference is the choice of the dynamical scale, where the total
transverse energy in the event (Option 2 above) is used for the plot on the
right panel of Fig. 21. The total transverse energy in the event is a superior
choice for the generation of processes with the inclusion of dimension-8
operators in triboson processes in the bulk of the distribution (right panel
of Fig 21), while the tail of the distribution is better predicted by the
default choice of scale (left panel of Fig. 21). A possible reason for this
behavior is that a single scale choice is not valid for processes that
traverse a wide kinematic range. Therefore the a-priori expectation that the
same value of the scale will be sufficient for both the SM process and the BSM
process is flawed.

A similar effect is seen for vector boson scattering topologies as shown in
Fig. 22. However in this case the distribution for the SM process is shown,
where the SM distribution is obtained by reweighting a BSM distribution down
to the SM scenario. Other factors that could impact the process generation
non-negligibly such as the PDF are shown. In the ratio panel, the net effect
of the variation of these parameters can be seen, pointing to the need to
include these effects in analyses as possible sources of systematic
uncertainty.

Figure 21: Impact of the choice of scale in a triboson process. The SM, BSM,
and interference terms are generated separately and represented by the red,
blue, and green hatch-filled histograms. The full process generated with all
components of the generation is represented with a black hatch-filled
histogram. The syntax of the full process is analogous to the process
definition when the reweighting feature of Madgraph is used. Figure 22: Impact
of the choice of scale choice in a VBS process. The gray filled histogram is
produced for a direct generation of the SM term as a $2\rightarrow 6$ or p p >
e+ ve j j j j QCD=0 NP=0 SMHLOOP=0. The red, green, black, cyan and orange
open histograms show the same process for various generator parameters such as
the variation in the top width ($\Gamma_{t}$), different choice of the
functional form of the dynamical scale and various PDFs. The magenta points
show a further comparison with a $2\rightarrow 4$ or p p > v v j j QCD=0 NP=0
SMHLOOP=0 process [51].

###  5.8 Post-generation reweighting

Figure 23: Top quark $p_{T}$ (left) and $m_{t\overline{t}}$ (right) for SM
(black) and SMEFT with $c_{tG}=10$ for different production methods: Directly
generated (orange), reweighted from an SM sample (blue) and reweighted from a
combination of SM and other EFT samples (green). Both reweightings were
performed after the sample generation. The panels show the normalized
differential cross section (top), the ratio of two reweighting schemes to the
direct generation (center), and the ratio of EFT and SM (bottom).

The procedure of reweighting events is often done by the same application that
generated the events, and usually immediately after the generation, as part of
a single execution of the application. This is how all of the previous studies
shown in this report have been performed. This requires that the EFT model and
desired Wilson coefficient values (EFT points) are defined at the time of
sample generation. However, the need for a new model, or different EFT points,
might arise after generation, and re-generating a sample can come with
significant computing costs, especially when a full detector simulation is
involved.

Though it is less common, it is possible to reweight events post-generation,
thereby avoiding the significant computing costs associated with regeneration.
In practice, this can be achieved by generating an external matrix element
library from MadGraph 5. These libraries, which come in the format of a Python
module, are specific to a particular model and set of reweighting points. If
provided with the LHE-level information from the original generation, a new
weight for each reweighting point can be provided by the module.

An advantage of the post-generation approach is that any UFO model can be used
as long as the initial and final states of the process match that from the
original generation, and the phase space is adequately covered. This makes it
possible to reuse existing SM samples for an EFT analysis, leading to a
(possibly) quick interpretation of a SM measurement. Alternatively, an
existing EFT analysis could be easily reinterpreted with, for example, new
flavor assumptions. Furthermore, these advantages are not limited to samples
generated with MadGraph. As long as the LHE information is available, the
reweighting module can be applied to events from any generator.

An example of post-generation reweighting is shown in Fig. 23 for
$t\overline{t}$ production using the $c_{tG}$ Wilson coefficient in the
dim6top model. First, a SM $t\overline{t}$ sample is reweighted post-
generation to an EFT point with $c_{tG}=10$ (green), and compared to a direct
generation of the same point (orange).

Going further, it is also possible to combine multiple reference samples, each
reweighted independently to the same EFT point, into one large sample with
higher statistics. This can also be used if the different reference samples
cover different areas of the phase space (though care should be taken to
remove very large-weight events). The green line in Fig. 23 shows an example
where the reweighted SM $t\overline{t}$ sample was combined with reweighted
EFT samples with original Wilson coefficient values of $c_{tG}=1$ and
$c_{tG}=3$. The samples were combined by scaling the event weights of each
sample by $\sum w/\sum w^{2}$, where the sums are over an individual sample,
not the combination, and then concatenating the events. Given the reference
samples used, this approach leads to the lowest statistical uncertainty in the
inclusive sum of weights. In Fig. 23, one can see improved statistics in the
combined sample compared to the reweighted SM sample, especially in the
relevant high-energy tails of the distributions.

##  6 Summary

This note serves as a guide to strategies for obtaining simulation-based
predictions in the context of SMEFT. It does not aim at establishing
authoritative guidelines on how those predictions should be obtained but aims
at assessing the consistency and computational efficacy.

We provide the statistical interpretation of simulation-based and weight-based
strategies and discuss in some detail the intricacies related to the choice of
the role of the helicity configuration in the reweighting. Common pitfalls and
limitations of the methodology are discussed in Sec. 4 in the hope of alerting
the unaware user.

In the main body of the note, we compare predictions for various final states.
We study helicity-aware and ignorant reweighting in the WZ and ZH production
process. In the top quark sector, pair production and associate production of
a top quark pair and a Z boson are investigated. Good closure is observed in
almost all the cases and for most of the considered kinematic phase space. The
VBF production of an H boson and triboson processes complete the studies and
lay the ground for future EFT measurements at the LHC.

## Acknowledgements

This work was done on behalf of the LHC EFT WG and we would like to thank
members of the LHC EFT WG for stimulating discussions that led to this
document. The work of M. Presilla is supported by the Alexander von Humboldt-
Stiftung.

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