[1]Michael Joseph Riberdy

1]IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France
2]Department of Physics, The College of William & Mary, Williamsburg VA 23187,
US 3]National Centre for Nuclear Research (NCBJ), 02-093 Warsaw, Poland

# Combining lattice QCD and phenomenological inputs on generalised parton
distributions at moderate skewness

0000-0001-6105-5986<EMAIL_ADDRESS>Hervé Dutrieux  0000-0001-8334-4885
<EMAIL_ADDRESS>Cédric Mezrag  0000-0001-8678-4085<EMAIL_ADDRESS>Paweł Sznajder  0000-0002-2684-803X<EMAIL_ADDRESS>[  [  [

###### Abstract

We present a systematic study demonstrating the impact of lattice QCD data on
the extraction of generalised parton distributions (GPDs). For this purpose,
we use a previously developed modelling of GPDs based on machine learning
techniques fulfilling the theoretical requirements of polynomiality, a form of
positivity constraint and known reduction limits. A special care is given to
estimate the uncertainty stemming from the ill-posed character of the
connection between GPDs and the experimental processes usually considered to
constrain them, like deeply virtual Compton scattering (DVCS). Mock lattice
QCD data inputs are included in a Bayesian framework to the prior model which
is fitted to reproduce the most experimentally accessible information of a
phenomenological model by Goloskov and Kroll. We highlight the impact of the
precision, correlation and kinematic coverage of lattice data on GPD
extraction at moderate $\xi$ which has only been brushed in the literature so
far, paving the way for a joint extraction of GPDs.

###### keywords:

Generalized parton distributions, lattice QCD, pseudo-distributions, short-
distance factorization, skewness

##  1 Introduction

Generalised parton distributions (GPDs) were introduced in the 1990s [1, 2, 3]
and have been since then a very active topic for theoretical and experimental
studies (see _e.g._ the review papers [4, 5, 6] among others). This interest
is fuelled both by the interpretation of GPDs as 3D number densities of quarks
and gluons within the nucleon [7, 8] and by their connection to the energy
momentum tensor (EMT) [2, 9].

A recent study [10] pointed out the large magnitude of uncertainty underlying
the connection between GPDs and the exclusive processes usually considered to
constrain these objects, such as deeply virtual Compton scattering (DVCS) [11,
12], timelike Compton scattering (TCS) [13] or deeply virtual meson production
(DVMP) [14, 15]. More precisely, the coefficient functions of these processes
are not practically invertible, even when taking into account QCD corrections
on a large range of energy scale, and despite the constraints brought by
Lorentz covariance. The results of [10] were confirmed in the moderate $\xi$
range in [16]. In this region, which constitutes the bulk of the most precise
experimental data on DVCS, QCD evolution provides little help to alleviate the
so-called deconvolution problem. Recently, efforts were conducted to produce
GPD models giving a better account of the uncertainty associated to the ill-
defined extraction of GPDs from exclusive processes. The study [17] shows that
when all the theoretical properties of GPDs (including positivity) are taken
into account, the deconvolution uncertainties are mostly present in the so-
called Efremov-Radyushkin-Brodsky-Lepage (ERBL) kinematic region.

The ambiguities of the DVCS, TCS and DVMP processes have led to an on-going
considerable theoretical and experimental effort to characterise other
exclusive processes with richer kinematic structure. The best known case is
double DVCS (DDVCS) [18, 19] where both the incoming and outgoing photons are
deeply virtual. This additional kinematic degree of freedom allows, at leading
order and in a specific kinematic range, the extraction of the “deconvoluted”
GPDs. A new analysis performed in [20] suggests that DDVCS is measurable in
near-future experiments. Recently, other processes have been put forward such
as di-photon production [21, 22], photon meson pair production [23, 24], or
the more general single diffractive hard exclusive processes [25, 26].

In addition to the experimental inputs, a new source of information regarding
GPDs has emerged through lattice QCD simulations. Indeed, new formalisms
developed in [27, 28, 29, 30] have offered the possibility to connect non-
local spacelike Euclidean matrix elements computed on the lattice to lightlike
ones through perturbation theory. We focus here on two different formalisms.
First, the large-momentum effective theory [28], known as the “quasi-
distribution” formalism, has triggered a lot of interest, both in terms of
simulation and perturbative computation regarding the so-called matching
kernels. GPD matching kernels have been computed [31, 32] and the first
lattice QCD studies of GPDs have been performed [33].

The second formalism, and the one we will be focusing on through this paper,
is known as “short-distance factorisation” or “pseudo-distribution” formalism
[29]. Based on a Lorentz decomposition of the Euclidean matrix elements, it
allows the extraction of parton distributions in Ioffe time $\nu$, that is the
Fourier conjugate of the momentum fraction $x$. This method was first
introduced in [29], and steady progresses have been done since then, both on
the perturbative matching side [34, 35, 36], and on the lattice simulation one
(see for instance [37, 38]). The formalism has two advantages: working with
ratios of matrix elements greatly simplifies the renormalisation procedure and
allows an easier extrapolation to the continuum limit. It also presents
continuous matching kernels. However, this method - like any lattice-QCD
attempt to access lightcone distributions - comes with a difficulty in the
actual reconstruction of the momentum dependence of the $\overline{MS}$
distribution. As only a restricted range of Ioffe times can be probed
numerically with acceptable noise levels, the inversion of the Fourier
transform to reconstruct an $x$-dependent distribution is an ill-posed
inversion problem, also known as an imputation problem. Several attempts have
aleady been performed to try to handle this specific ill-posed problem [39,
40, 41].

The combination of phenomenological and non-local spacelike lattice inputs on
parton distributions has already been explored in some recent papers, for
instance on the proton PDF [42] and pion PDF [43]. For GPD studies, the
situation is made significantly more complicated due to the higher
dimensionality compared to ordinary PDFs. The recent works in [44, 45] have
associated lattice data with various phenomenological and experimental inputs,
where the authors have mostly considered GPD lattice data at vanishing
skewness. An additional difficulty in offering a framework for the joint study
of experimental and lattice inputs on GPDs is the number of parameters
involved. Indeed, for only two light quark flavours, [45] needs to model 20
different GPDs – two valence quark distributions, two sea quark, plus a gluon
distribution, repeated for the four helicity combinations $H$, $E$,
$\tilde{H}$ and $\tilde{E}$. This abundance of functions to extract forced the
authors to employ only basic modelling of the skewness relevant in the small
$\xi$ regime.

In the present work, we develop a different strategy to combine phenomenology
and lattice data focusing on moderate skewness. In this domain, lattice
computations offer the perspective of a significant reduction of the
uncertainty associated to the deconvolution problem of the usually considered
exclusive processes. We use a previously developed GPD model presented in [17]
which offers significant flexibility precisely at moderate skewness. It is
illusory in the current state of experimental and lattice data to perform a
satisfactory flavour and helicity decomposition with this kind of flexible
model. Therefore, instead of adjusting it on actual experimental and lattice
results, we assume that the flavour and helicity decomposition has been
obtained already. On the experimental side, we use the phenomenological
information encoded in the Goloskokov and Kroll (GK) model [46, 47, 48],
whereas on the lattice side, we generate mock lattice data. Some tension
between lattice and experimental data is hinted at in [42] and [45], whereas
[43], using short-distance factorization, states that when taking into account
all sources of systematic uncertainty, lattice and experimental data are
generally compatible. We therefore use this assumption in the study, which
offers the possibility to merge phenomenology and lattice inputs thanks to a
Bayesian reweighting procedure. In short, we probe whether experimental data
which suffers from a deconvolution problem and lattice data which suffers from
an imputation one can be combined advantageously. For this first study, we
disregard the evolution properties of GPDs and thus the matching kernels
between pseudo-Ioffe distributions and GPDs. The impact of the perturbative
QCD corrections is left for future works.

In section 2, we briefly review key aspects of GPDs and their connection with
Ioffe-time distributions. We also revisit the main characteristics of the
flexible GPD model introduced in [17]. In section 3, we show how the model is
adjusted to a phenomenological parametrization so as to reproduce the main
experimental features of a GPD extraction. Section 4 is dedicated to the
generation of the set of mock lattice QCD data. We introduce in section 5 the
reweighting procedure to combine our mock lattice QCD data to the GPD model.
Finally, we discuss the results of the reweighting in section 6, and conclude.

##  2 Kinematics and GPD Modelling

###  2.1 Definition and properties of GPDs

GPDs can be formally defined as the Fourier transform of non-local matrix
elements evaluated on a lightlike distance $z^{-}$ [4]:

$\displaystyle p^{+}\int\frac{dz^{-}}{2\pi}e^{ixp^{+}z^{-}}\left\langle
p_{2}\bigg{|}\bar{q}\left(-\frac{z}{2}\right)\gamma^{+}q\left(\frac{z}{2}\right)\bigg{|}p_{1}\right\rangle_{z^{+}=|\vec{z}_{\perp}|=0}$
$\displaystyle=H^{q}(x,\xi,t)\bar{u}(p_{2})\gamma^{+}u(p_{1})$
$\displaystyle\hskip
20.0pt\quad+E^{q}(x,\xi,t)\bar{u}(p_{2})\frac{\sigma^{+\alpha}\Delta_{\alpha}}{m}u(p_{1}),$
(1)

where we have restricted here our definition to the unpolarized quark GPD in a
nucleon whose mass is labelled $m$. We also omitted the Wilson line in the
definition of the operator, that needs to be added when working in gauges
other than lightcone one. The average momentum of the incoming and outgoing
hadronic states $|p_{1}\rangle$ and $|p_{2}\rangle$ is defined as:

$p=\frac{p_{1}+p_{2}}{2},$ (2)

while $x$ is the usual lightcone momentum fraction defined as:

$x=\frac{k_{1}^{+}+k_{2}^{+}}{2p^{+}}$ (3)

with $k^{+}_{1,2}$ the incoming and outgoing momentum of the struck quark. The
total four-momentum transfer squared $t$ is given by:

$t=(p_{1}-p_{2})^{2}$ (4)

and the skewness $\xi$ is defined as:

$\xi=\frac{p_{1}^{+}-p_{2}^{+}}{2p^{+}}.$ (5)

GPDs have to obey several properties which play a crucial role in their
modelling. First, the forward limit of GPDs is given in terms of PDFs, such
that:

$H^{q}(x,0,0)=q(x)\Theta(x)-\bar{q}(-x)\Theta(-x)$ (6)

where $q(x)$ is the unpolarised quark PDF of flavour $q$ and $\Theta$ the
Heaviside step function. GPDs are also connected to electromagnetic form
factors (EFFs) through:

$\int_{-1}^{1}\textrm{d}xH^{q}(x,\xi,t)=F^{q}_{1}(t),$ (7)

where $F^{q}_{1}$ is the contribution of the quark flavour $q$ to the Dirac
EFF. GPDs have to obey the so-called polynomiality property, stating that
their Mellin moments are polynomials in $\xi$ of a given order [49, 50]:

$\displaystyle\int_{-1}^{1}\textrm{d}x\,x^{n}H^{q}(x,\xi,t)$
$\displaystyle=\sum_{i=0}^{\left[\frac{n}{2}\right]}\xi^{2i}A^{q}_{n,2i}(t)$
$\displaystyle+\textrm{mod}(n,2)C_{n}(t)\xi^{n+1},$ (8)

where $[\dots]$ is the floor function and $\textrm{mod}(n,2)$ is 0 for $n$
even, and 1 otherwise. The polynomiality property is understood as a
consequence of the Lorentz covariance of GPDs, and can be systematically
implemented thanks to the Radon transform [51, 52, 53] in the double
distribution formalism [50]. Finally, let us highlight that GPDs are also
constrained by the so-called positivity properties [54, 55, 56, 57], the most
classical one being given by:

$\displaystyle\left|H^{q}(x,\xi,t)\right|\leq\sqrt{\frac{q\left(\frac{x+\xi}{1+\xi}\right)q\left(\frac{x-\xi}{1-\xi}\right)}{1-\xi^{2}}}.$
(9)

Respecting all these constraints at once represents a challenge for the
phenomenology of GPDs.

GPDs can also be expressed as a function of the Ioffe time parameter
$\nu=p\cdot z$ [58], which is the Fourier conjugate of the average momentum
fraction $x$. The relation between $\hat{H}(\nu,\xi,t)$ and $H(x,\xi,t)$ is
thus given by [34]:

$\displaystyle\hat{H}^{q}(\nu,\xi,t)=\int_{-1}^{1}\textrm{d}xe^{ix\nu}H^{q}(x,\xi,t)$
(10)

In the following, we drop the “hat” on $\hat{H}(\nu)$ as the explicit
dependence in $\nu$ or $x$ is enough to indicate whether we work in Ioffe time
or momentum space.

As we have mentioned before, non-local matrix elements computed on the lattice
are Euclidean, so the distance $z$ between the operators defining the GPD is
spacelike ($z^{2}<0$) contrary to the lightlike distance $z^{-}$ used in the
usual lightcone definition of parton distributions. As a result, Ioffe-time
pseudo-distributions computed on the lattice, once the continuum limit has
been appropriately taken, are functions $\mathfrak{M}(\nu,\xi,t,z^{2})$, which
can be matched perturbatively to the $\overline{MS}$ scheme to yield the
Ioffe-time distributions such as $H(\nu,\xi,t,\mu^{2})$ (see [34, 36]).
Matching from $z^{2}$-dependent pseudo-distributions to $\mu^{2}$-dependent
$\overline{MS}$ distributions is only a minor correction at small Ioffe time,
which we will neglect in this paper. This effect should however be properly
accounted for in further studies.

In the following, for simplicity we limit the scope of our study to the
singlet sea quark GPD $H^{q(+)}(x,\xi,t)=H^{q}(x,\xi,t)-H^{q}(-x,\xi,t)$.
Imposing this parity property means that we will only study the imaginary part
of $H(\nu)$. We also ignore the $t$-dependence of GPDs and the entanglement
between $t$ and $\xi$ which splits the kinematic domain between physical and
unphysical regions, to work with $t=0$ all along. This choice is made to
primarily focus on the deconvolution of the $x$ and $\xi$ dependence helped by
lattice data. The implications of the $t$ dependence are left for a future
work.

###  2.2 GPD Modelling with artificial neural networks

Our study is based on methods recently developed in [17], where the use of
artificial neural network (ANN) techniques in a direct modelling of GPDs was
proposed for the first time. This new way of modelling has been designed to
address the problem of model dependence and implementation of the theoretical
constraints one encounters in the GPD phenomenology, but also to facilitate a
future inclusion of lattice QCD information. To keep our article self-
consistent and self-contained, we remind now important details on ANN GPD
modelling.

Our GPD model based on ANNs significantly differs from a textbook
implementation of machine learning techniques (see e.g. Ref. [59]). The reason
for that is the desire to fulfil in the architecture of the neural network a
set of theory-driven constraints for a valid GPD model. These constraints are
among others linked to the parity properties of GPDs, the polynomiality
property (8) and known limits like (6). The positivity constraints (9) are not
guarantied at the level of the architecture of the network, but rather
enforced numerically during the training procedure.

The model proposed in Ref. [17] is built in the double distribution space
involving variables $\beta$ and $\alpha$. The relation with the GPD $H$ in
momentum space is given by the Radon transform:

$H(x,\xi,t)=\int\mathrm{d}\Omega\,F(\beta,\alpha,t)\,,$ (11)

where
$\mathrm{d}\Omega=\mathrm{d}\beta\,\mathrm{d}\alpha\,\delta(x-\beta-\alpha\xi)$
and $|\alpha|+|\beta|\leq 1$. In (11), $F(\beta,\alpha,t)$ is called a double
distribution and is the object we will model. The benefit of using the double
distribution space is an automatic fulfilment of the polynomiality property by
the resulting GPD $H(x,\xi,t)$.

To achieve a satisfactory flexibility and reproduction of known limits, our
double distribution model is composed of three parts:

$(1-x^{2})F_{C}(\beta,\alpha)+(x^{2}-\xi^{2})F_{S}(\beta,\alpha)+\xi
F_{D}(\beta,\alpha)\,.$ (12)

Let us address successively the role of each term in (12). The first one,

$F_{C}(\beta,\alpha)=f(\beta)h_{C}(\beta,\alpha)\frac{1}{1-\beta^{2}},$ (13)

ensures by design the proper reduction to the forward limit and has the
necessary flexibility to model the $x=\xi$ line, which is particularly
relevent for the current GPD phenomenology. The prefactor $(1-x^{2})$ of
$F_{C}(\beta,\alpha)$ in (12) combined with $1/(1-\beta^{2})$ in (13) was
introduced to facilitate the fulfilment of the positivity constraint (9). The
forward limit (unpolarised PDF for the GPD $H$) is denoted by $f(\beta)\equiv
H(\beta,0,0)$, while $h_{C}(\beta,\alpha)$ is a profile function generalising
that proposed by Radyushkin [54]. In this study it is given by:

$h_{C}(\beta,\alpha)=\frac{\mathrm{ANN}_{C}(|\beta|,\alpha)}{\displaystyle\int_{-1+|\beta|}^{1-|\beta|}\mathrm{d}\alpha\mathrm{ANN}_{C}(|\beta|,\alpha)}\,.$
(14)

Thanks to a special design of the activation function and the use of the
absolute value, the neural network $\mathrm{ANN}_{C}(|\beta|,\alpha)$ is even
w.r.t. both $\beta$ and $\alpha$ variables, and it vanishes at the edge of the
support region $|\beta|+|\alpha|=1$. The symmetry in $\beta$ is introduced to
keep the resulting GPD an odd function of $x$ (relevant for phenomenology of
DVCS and TCS), while the symmetry in $\alpha$ is mandatory, as it allows
fulfillment of the time reversal property, i.e. the invariance under
$\xi\leftrightarrow-\xi$ exchange. The normalisation by the integral over
$\alpha$, which can be done analytically, allows enforcement of the proper
forward limit, while the rest of the model is typically trained to reproduce
the diagonal $x=\xi$ probed by amplitudes of processes like DVCS, TCS and
DVMP.

As the term $F_{C}(\beta,\alpha)$ was found to be tightly constrained by the
necessity of reproducing both $\xi=0$ and $x=\xi$ lines, an additional term
$F_{S}(\beta,\alpha)$ was introduced. This term explicitly vanishes on the
$\xi=0$ and $x=\xi$ lines, i.e. it does not contribute to the fit of
$F_{C}(\beta,\alpha)$ on the phenomenological inputs. Rather,
$(x^{2}-\xi^{2})F_{S}(\beta,\alpha)$ represents a contribution that is
entirely unconstrained by LO DVCS data and the knowledge of PDFs, and aims at
reproducing the deconvolution uncertainty of exclusive processes. Precisely,
it corresponds to a LO shadow distribution as defined and studied in [10].
$F_{S}$ is constructed in the following way:

$F_{S}(\beta,\alpha)=f(\beta)h_{S}(\beta,\alpha)\,.$ (15)

where:

$\displaystyle h_{S}(\beta,\alpha)/N_{S}$
$\displaystyle=\frac{\mathrm{ANN}_{S}(|\beta|,\alpha)}{\displaystyle\int_{-1+|\beta|}^{1-|\beta|}\mathrm{d}\alpha\mathrm{ANN}_{S}(|\beta|,\alpha)}$
$\displaystyle-\frac{\mathrm{ANN}_{S^{\prime}}(|\beta|,\alpha)}{\displaystyle\int_{-1+|\beta|}^{1-|\beta|}\mathrm{d}\alpha\mathrm{ANN}_{S^{\prime}}(|\beta|,\alpha)}\,.$
(16)

Since this contribution is not constrained in the fit, the major limit on its
size, aside from the maximal flexibility of the neural network, is really
imposed by the positivity constraint. During training, we seek to maximise the
$N_{S}$ normalisation factor in (2.2) within the limits allowed by positivity
so as to leverage the maximal flexibility. Writing the function
$h_{S}(\beta,\alpha)$ as the difference of two different profile functions
characterized by $\mathrm{ANN}_{S}(|\beta|,\alpha)$ and
$\mathrm{ANN}_{S^{\prime}}(|\beta|,\alpha)$ ensures that $F_{S}(\beta,\alpha)$
brings no contribution to the forward limit. The $f(\beta)$ factor helps to
enforce the positivity.

Finally, $F_{D}(\beta,\alpha)$ gives the additional flexibility necessary to
model the $D$-term, a degree of freedom of GPDs associated to the final terms
in $\xi^{n+1}$ in (8) and which plays a crucial role in the characterisation
of partonic matter [60, 61]. One has :

$F_{D}(\beta,\alpha)=\delta(\beta)D(\alpha)\,,$ (17)

and

$D(\alpha)=(1-\alpha^{2})\sum_{\begin{subarray}{c}i=1\\\
\mathrm{odd}\end{subarray}}^{N}d_{i}C_{i}^{\nicefrac{{3}}{{2}}}\left(\alpha\right)\,,$
(18)

where $d_{i}$ are coefficients of the expansion of $D$-term into Gegenbauer
polynomials, and where $N$ is an arbitrary truncation parameter.

##  3 Representation of experimental data and estimation of uncertainties

Let us now discuss how the model encodes a representation of experimental data
for processes like DVCS, TCS and DVMP which we will use as an input for the
lattice QCD impact study. We stress again that experimental data for the
aforementioned processes mostly probe GPDs at the $x=\xi$ line (with some
additional information on the $D$-term), and that at $t=0$, the forward limit,
i.e. the PDF, is very well known from a wealth of inclusive and semi-inclusive
processes.

We use a built-in PDF parameterisation proposed in [62] for $f(\beta)$
involved in (13) and (15). Free parameters of the $F_{C}(\beta,\alpha)$ term
are constrained by $200$ points of $H^{q}(x,\xi=x)$ generated with the GK
model [46, 47, 48] spanning over the range of
$-4<\log_{10}(x=\xi)<\log_{10}(0.95)$ at fixed $t=0$ and
$\mu^{2}=4~{}\mathrm{GeV}^{2}$. The term $F_{S}(\beta,\alpha)$ is only
constrained by the positivity requirement (9), giving rise to large
uncertainties when $x<\xi$. The details of the constraining procedure are
given in Ref. [17].

The uncertainty of the model is encoded in a collection of $101$ replicas.
This way is convenient for propagation of uncertainty to any related quantity
and for the use of Bayesian reweighting techniques. A single replica
represents the outcome of the independent fit to $200$ $x=\xi$ points
indicated in the previous paragraph, with a random choice of the initial
parameters (weights and biases of ANNs, and normalisation parameter $N_{S}$ in
(2.2)). To evaluate the mean and standard deviation of a quantity derived from
the GPD at a specific kinematic point – which can be the GPD itself, a 3D
number density, an observable, etc. – one may use any statistical estimator of
the mean and uncertainty of the population $X$ of $101$ values returned by
replicas. In this analysis we use respectively the median and the median
absolute deviation (MAD):

$\bar{\sigma}\equiv
1.4826\times\text{median}\bigg{(}|\text{median}(X)-X|\bigg{)},$ (19)

where the factor $1.4826$ is derived from the assumption of gaussianity.

We note that the population of $101$ values returned by replicas may contain
outliers, i.e. pathological values with respect to the other entries,
distorting the evaluation of mean and uncertainty. This may happen due to
instabilities in the numeric procedures, which cannot be entirely avoided due
to the complexity of GPD modelling and the constraining procedure. Many
methods for detecting and removing outliers were proposed in the literature,
like the $3\sigma$-method [63]. Alternatively, one can chose robust estimators
with respect to outliers, our motivation for using MAD.

Figure 1 shows $101$ replicas of $H(\nu,\xi,t=0)$ evaluated at three values of
$\xi$, together with its $1\bar{\sigma}$ band. For small values of $\xi$, the
replica bundle is extremely coherent, or auto-correlated, at small Ioffe time.
This is due to the fact that the positivity constraint limits considerably the
freedom of the model for $x>\xi$, on a region that is therefore all the more
extended that $\xi$ is small.

Figure 1: The set of GPD replicas $\textrm{Im}\,H(\nu,\xi,t=0)$ between Ioffe
times $\nu=0$ and $\nu=20$ at $\xi=0.1$ (top), $\xi=0.5$ (middle), $\xi=0.9$
(bottom) and their corresponding one standard deviation bands. Due to waning
support $x>\xi$ as $\xi$ increases, the replicas become less constrained by
positivity, oscillate more heavily, and decohere.

##  4 Generating mock lattice QCD data

After describing in the previous section the fit of our flexible GPD model
with phenomenological inputs, we turn to the question of the generation of
plausible mock lattice QCD data. In accordance with our choice of working in
the pseudo-distribution formalism, we will generate data at small Ioffe time
$\nu$. More precisely, for each value of $\xi$ that we will consider, we
generate the data in three independent batches, corresponding to the following
regions in $\nu$:

  1. 1.

$0.2\leq\nu\leq 2$, with data spaced regularly with interval $\Delta\nu=0.2$,

  2. 2.

$2.2\leq\nu\leq 4$, with interval $\Delta\nu=0.2$,

  3. 3.

$4.4\leq\nu\leq 6$, with interval $\Delta\nu=0.4$.

We introduce these three independent batches of lattice QCD data to loosely
mimic actual simulations where various sets of points with different
correlations arise from varying the boosted hadron momenta and the separation
between fields in the non-local operator.

We assume the correlation between different batches of data to be zero. On the
contrary, we consider the existence of correlations inside the batches,
characterized for commodity by a simple coefficient $0\leq c<1$, which is the
same between any two lattice data points in the same batch, and does not vary
from batch to batch. This choice of constant correlation coefficient is
obviously an oversimplification, but it already allows us to get a rough
estimate of the impact of correlations on the determination of GPDs.

We characterise how uncertainty grows from $\nu=0$ to $\nu=10$ by an
exponential slope parameter called $b$. Figure 2 presents the relative
uncertainty profiles which we will use in this study, given by

$\displaystyle
g(\nu;b,s=5\%,\nu_{\textrm{max}}=10)=\frac{s(b^{\nu}-b^{\nu_{\textrm{max}}})+1-b^{\nu}}{1-b^{\nu_{\textrm{max}}}}\,.$
(20)

As can be immediately checked, this formula guarantees that when
$\nu=\nu_{\textrm{max}}$, the relative uncertainty is 100%, and when $\nu=0$,
it saturates to $s$, here chosen as 5%. We choose the uncertainty to reach
100% at $\nu_{\textrm{max}}=10$ to replicate the common behaviour of lattice
data, which tend to exhibit uncertainty on the order of its central value
around this point in Ioffe time (see, for example, Fig. 9.a. of [38]). The
parameter $b$ controls the steepness of the uncertainty increase. For
$b\rightarrow 1$, uncertainty increases linearly with Ioffe time, while for
larger values of $b$, it starts with a plateau of good precision and then
degrades very quickly. The two cases $b=1.1$ and $b=2$ presented on figure 2
represent two archetypes of lattice uncertainties, corresponding respectively
to data of bad or good signal to noise ratio.

Figure 2: The relative uncertainty of the mock lattice data $g(\nu;b,0.05,10)$
is displayed as a function of the Ioffe time $\nu$ for $b=1.1$ (green) and for
$b=2$ (red).

We will therefore use as uncertainty of our mock lattice data

$\sigma^{\textrm{Latt}}(\nu)\equiv
g(\nu;b,s=5\%,\nu_{\textrm{max}}=10)\bar{\mu}(\nu)\,,$ (21)

where $\bar{\mu}(\nu)$ is the median of the output of our flexible GPD model
at $\nu$ and the relevant value of $\xi$. Each of the three batches of data
points contains a number of samples at Ioffe times $\nu_{i}$. We characterize
the correlation between these points inside each batch by a coefficient $c$
and express the associated covariance matrix

$\Omega_{i,i^{\prime}}^{\textrm{Latt}}\equiv\Big{(}\delta_{i,i^{\prime}}+(1-\delta_{i,i^{\prime}})c\Big{)}\sigma^{\textrm{Latt}}(\nu_{i})\sigma^{\textrm{Latt}}(\nu_{i}^{\prime}),$
(22)

We then draw the central values $\mu_{i}^{\textrm{Latt}}$ of our mock lattice
data points in a normal distribution centered on the median of the output of
the flexible GPD model at each $\nu_{i}$, with the covariance matrix
$\Omega_{i,i^{\prime}}^{\textrm{Latt}}$. This means that we choose our mock
lattice data to be globally compatible with the central value of our flexible
GPD model fitted on phenomenological inputs.

Figure 3 gives an example of mock lattice data set (orange points) superposed
to the replicas of our GPD model. The four panels demonstrate the effect of
various combinations of correlation coefficient $c$ and level of noise
parameter $b$. Increasing $c$ increases the degree to which one of the central
values influences the choice of the others within a given batch in $\nu$,
while increasing the parameter $b$ results in a data set much more
concentrated around the maximum likelihood of the GPD model.

Figure 3: The set of GPD replicas between Ioffe times $\nu=0$ and $\nu=6$ at
$\xi=0.1$ (green), their median (blue), the $1\sigma$ band (red) corresponding
to $b=1.1$ (top) and $b=2$ (bottom), and the corresponding generated mock
lattice data set (orange) determined by $c=0$ (left) and $c=0.5$ (right) are
shown.

##  5 Bayesian Reweighting

Now, we would like to assess the impact of the different sets of mock lattice
data generated in the previous section on the GPD model. For this purpose, for
each replica $R_{k}$, we introduce a goodness-of-fit estimator $\chi^{2}_{k}$:

$\displaystyle\chi^{2}_{k}$ $\displaystyle=$
$\displaystyle\sum_{\textrm{batches}}\sum_{i,i^{\prime}}(\mu_{i}^{\textrm{Latt}}-R_{k}(\nu_{i}))\Big{(}(\Omega^{\textrm{Latt}})^{-1}\Big{)}_{i,i^{\prime}}$
(23) $\displaystyle\times$
$\displaystyle(\mu_{i^{\prime}}^{\textrm{Latt}}-R_{k}(\nu_{i^{\prime}})),$

where $\mu_{i}^{\textrm{Latt}}$ is the central value of the lattice data
generated at point $\nu_{i}$. With this definition, $\chi^{2}_{k}$ estimates
the relative compatibility of a given replica $R_{k}$ with the mock data
within the uncertainty of the latter. Bayesian reweighting consists in
affecting to each replica $R_{k}$ a weight $\omega_{k}$ expressed from the
goodness-of-fit $\chi^{2}_{k}$ through [64]

$\omega_{k}=\frac{(\chi^{2}_{k})^{\frac{N-1}{2}}}{Z}e^{-\frac{\chi^{2}_{k}}{2}},$
(24)

where $Z$ is a normalization factor such that $\sum_{k}\omega_{k}=1$. One can
further define the effective fraction of replicas $R$ which are compatible
with the new data set as

$\tau\equiv\frac{\exp(\sum_{k}\omega_{k}\ln(\omega_{k}))}{K},$ (25)

where the exponentiated value is the Shannon entropy of the weight set.

In the following, we will generate several sets of mock lattice data at
various values of $\xi$ and reweight our GPD model altogether with these new
data at several $\xi$. We call this the “multikinematic” reweighting, as
opposed to the “monokinematic” reweighting where only one value of $\xi$ is
considered. As we do not model any correlation between mock lattice data at
different values of $\xi$, the total $\chi^{2}$ will then be the sum of the
$\chi^{2}$ evaluated at each $\xi$.

###  5.1 Reweighted statistics

Once weights $\omega_{k}$ have been attributed to the replicas, we need to
define the central value and standard deviation of a weighted distribution. At
each kinematic where we want to evaluate the weighted central value and
deviation, we first order the replicas monotoneously, and then define the
weighted median $\bar{\mu}_{\omega}$ as the element satisfying the condition

$\displaystyle\sum_{k=1}^{l-1}\omega_{k}\leq\frac{1}{2}\quad\textrm{and}\quad\sum_{k=l+1}^{k_{\textrm{max}}}\omega_{k^{\prime}}\leq\frac{1}{2}.$
(26)

The weighted median is the element which most accurately splits the weights
evenly. The weighted standard deviation estimator $\bar{\sigma}_{\omega}$ is
obtained by simply replacing the median by its weighted equivalent in (19).

###  5.2 Metrics of the impact of Bayesian reweighting

The effective fraction of replicas compatible with the data set $\tau$ defined
in (25) tells us how constraining the new data set is compared to the prior
model. It is mostly a tool to evaluate the statistical significance of the
reweighted distribution: if $\tau$ is too small, the weighted mean and
standard deviation should not be considered as statistically significant.
However, $\tau$ brings only indirect information on the reduction of
uncertainty. To characterise the latter, which is our main physical objective,
we introduce the ratio $\Sigma(y)$, such as

$\Sigma(y)\equiv\frac{\bar{\sigma}_{\omega}(y)}{\bar{\sigma}(y)}.$ (27)

At a given value of $y$ (which can represent any kinematic, in momentum space
or Ioffe time), it represents the fraction of uncertainty remaining after
reweighting. If $\Sigma(y)=1$, the uncertainty of the replica bundle at $y$
has not decreased via reweighting. If $\Sigma(y)<1$, some reduction of
uncertainty has occurred via reweighting at that point. We further define the
average retainment of uncertainty in Ioffe time as

$r_{\nu}=\frac{1}{\nu_{\max}-\nu_{\min}}\int_{\nu_{\min}}^{\nu_{\max}}d\nu\,\Sigma(\nu),$
(28)

and a corresponding ratio in momentum space where we adopt a logarithmic
scale,

$r_{\textrm{ln}x}=\frac{1}{\textrm{ln}(x_{\max}/x_{\min})}\int_{x_{\min}}^{x_{\max}}\frac{dx}{x}\Sigma(x).$
(29)

We calculate the uncertainty retainment values $r_{\nu}$ for the region
$0\leq\nu\leq 20$ and $r_{\textrm{ln}x}$ for the region $5\times 10^{-3}\leq
x\leq 1$. These two metrics, intended for $\nu$ and $x$ spaces respectively,
assign a global numerical value to the reduction of uncertainty following the
introduction of the mock lattice data, which will be convenient to compare
various scenarios.

One should note that although we generate mock lattice data in the range
$0\leq\nu\leq 6$, our uncertainty retainment $r_{\nu}$ metric extends up to
$\nu=20$. We do this to assess the ability for lattice data to decrease the
uncertainty of our GPD model even in Ioffe time regions where we do not expect
to be able to extract statistically significant lattice data. Indeed, [43]
concluded that the data at low values of Ioffe time, thanks to their smaller
uncertainties, represented in effect the bulk of the constraint even at larger
Ioffe times. We also do not include the entire region $0\leq x\leq 1$ in our
metric of uncertainty retainment $r_{\textrm{ln}x}$, integrating only from the
lower bound $x=5\times 10^{-3}$. We choose to cap the integration with this
lower bound given that $\Sigma(x)$ becomes relatively constant at lower $x$.
The inclusion of such behaviour in the integration region would completely
dominate the discrimination effects at large $x$.

##  6 Results of the reweighting

###  6.1 Monokinematic Reweighting

Let us now apply the tools of Bayesian reweighting using our GPD model fitted
on phenomenological inputs as a prior, and our mock data as the new
information. As a first example, we look at monokinematic reweighting, i.e. we
add mock data at a single value of $\xi$ and measure its impact on the GPD
extraction at this same value. We recall that, on average, the generated mock
lattice data becomes closer to the most likely output of the prior model as
$b$ increases. As $c$ increases, the mock lattice data remains more
consistently above or below the central value of the prior model.

The result for $b=2$ (high precision), $c=0$ (uncorrelated data) and $\xi=0.1$
is shown on figure 4, while figure 5 shows the result for $\xi=0.5$ and the
same other parameters. The right hand side plots show the effect of
reweighting in Ioffe time: we observe a large reduction of uncertainty, which
remains effective far outside of the range of the data (the light orange zone)
although the case $\xi=0.5$ shows more fluctuations linked to the lesser
coherence of the replica bundle. The average retainment of uncertainty in
Ioffe time is $r_{\nu}=0.16$ at $\xi=0.1$, and $r_{\nu}=0.25$ at $\xi=0.5$.

Figure 4: Upper plots: The set of GPD replicas at $\xi=0.1$ in momentum space
(green, left) and Ioffe times space (green, right), their central value (pink-
solid), one $\sigma$ band (purple-solid), and the reweighted central value
(blue-dashed) and reweighted one $\sigma$ band (red-dashed). The mock lattice
data was generated using $b=2$ (high precision) and $c=0$ (no correlation) at
$\xi=0.1$ (orange-dotted, right). Lower plots: The reweighted to initial
uncertainty ratio (purple-solid), the average uncertainty retainment in both
$x$ ($r_{\textrm{ln}x}=0.78$) and $\nu$ ($r_{\nu}=0.16$) (green-dashed), and
the range in which the lattice data was generated $\nu=0$ to $\nu=6$ (orange-
shade, right). The associated effective fraction of replicas retained after
reweighting is given by $\tau=0.3$. Figure 5: Same caption as figure 4 up to
the fact that the GPD is shown at $\xi=0.5$, with mock data added at $\xi=0.5$
and similarly $b=2$ (high precision) and $c=0$ (no correlation). The average
uncertainty retainments are $r_{\textrm{ln}x}=0.54$ and $r_{\nu}=0.25$,
$\tau=0.11$.

On the other hand, the left hand side plots show a far less spectacular
reduction of uncertainty in momentum space. The situation is actually
inverted, with a larger reduction of uncertainty at $\xi=0.5$ compared to
$\xi=0.1$. Indeed, the retainment of uncertainty remains very large at
$r_{\textrm{ln}x}=0.78$ at $\xi=0.1$, whereas it is $r_{\textrm{ln}x}=0.54$ at
$\xi=0.5$. The fact that the uncertainty is less reduced in momentum space
compared to Ioffe time is an illustration of the imputation problem evoked in
the introduction. The reconstruction of the $x$-dependence from limited Ioffe
time data is an inverse problem which triggers a significant rise in
uncertainty because the mock data produces no control over the “high-
frequency” behaviour of the momentum distribution. Therefore, reweighting is
twice less efficient in momentum space compared to Ioffe time at $\xi=0.5$,
and five times less efficient at $\xi=0.1$. Why does it behave so poorly at
$\xi=0.1$ in momentum space? The answer comes from the extreme coherence of
the replicas of the GPD model in Ioffe time in the region where the
reweighting is performed. This means that the GPD model contains in the end
very little information in this region, such that the reweighting cannot
efficiently select features of the distribution that would make a significant
difference in momentum space.

We reiterate that the origin of the large coherence is the fact that
positivity constrains tightly the GPD on a large part of the $x$ dependence at
$\xi=0.1$ (namely for $x>0.1$). We add therefore lattice data in a region
where another theoretical constraint in momentum space has already
considerably reduced the freedom in modelling. Lattice data must be all the
better to bear any impact that the initial modelling uncertainties are small.
On the other hand, the model is much more flexible at $\xi=0.5$, allowing
lattice data a relatively better discriminating power in momentum space.

Figure 6: Effective fraction of replicas retained after reweighting $\tau$
(green curve), retainment of uncertainty in Ioffe time (blue curve) and in
momentum space (red curve) for the combinations of high and low noise (resp.
$b=1.1$ and $2$) and low and high correlation (resp. $c=0$ and $0.5$).

To compare the effect of reweighting at various values of $\xi$, we show in
figure 6 the effective fraction of surviving replicas $\tau$ and the
retainment of uncertainty in Ioffe time and momentum space as functions of
$\xi$. As the value of $\xi$ increases, the replica bundle decoheres and the
effective fraction of replicas $\tau$ drops quickly. For $\xi>0.7$, a full
refit seems necessary due to the low statistics. We observe that the reduction
of uncertainty is systematically better in Ioffe time compared to momentum
space, as expected due to the imputation problem. Using better and
uncorrelated data ($b=2,c=0$) produces generally a significant reduction of
uncertainty in Ioffe time compared to the other configurations, but that is
not reflected in momentum space. Overall, we observe in the case $b=1.1$ (low
precision) that uncertainty in momentum space decreases with larger values of
$\xi$. The fact that uncertainty is so erratic for $b=2$ has to do with the
small value of $\tau$, which makes results unreliable at large $\xi$, but
underlines how constraining the new data are compared to the prior model.

This highlights that with our modelling, it is in the large $\xi$ region,
where positivity does not provide significant constraints on the GPD, that we
would observe the largest effect due to the inclusion of lattice data.
However, we have worked here at $t=0$, where we could use the very well-known
unpolarised PDF as an input to our model, and used a simplified version of the
positivity constraint neglecting the role of the GPD $E$ for instance. In
general, the behaviour of GPDs at $\xi=0$ but $t\neq 0$ is one of the most
crucial aspects of GPD phenomenology and can be accessed readily on the
lattice. Although our current study invites to preferentially include data at
large $\xi$, in a more complete setting, the extraction of more precise
$t$-dependent PDFs plays a major role. We note that by studying both the limit
$\xi=0$ and large $\xi$, one handles two very interesting limits of GPDs: the
first is directly involved in the hadron tomography program or the spin sum
rule, whereas the latter probes a very different partonic dynamic in the
hadron, with a sensitivity to pairs of partons. The small but non-zero values
of $\xi$ can on the other hand be constrained rather efficiently by a
combination of experimental data, positivity constraints and arguments from
perturbation theory if working at a reasonably hard scale [65]. It remains
however interesting to verify whether lattice data at moderately small value
of $\xi$ are in good agreement with positivity constraints considering the
words of caution on relying excessively on these inequalities at low
renormalisation scale, as underlined for instance in [66].

Figure 7: Same caption as figure 4 up to the fact that the GPD is shown at
$\xi=0.5$, with mock data added at $\xi=0.1$, $b=1.1$ (low precision) and
$c=0.5$ (correlated data). The average uncertainty retainments are
$r_{\textrm{ln}x}=1.15$ and $r_{\nu}=0.93$, $\tau=0.83$.

###  6.2 Multikinematic Reweighting

We would like to explore one last aspect, namely how reweighting is propagated
to other values of $\xi$. So far, we have only studied the effect of
reweighting exactly at the same value of $\xi$ as the data we were simulating.
We show in figure 7 the result of a reweighting where the data is added at
$\xi=0.1$, but we observe the impact on the GPD at $\xi=0.5$, with $b=1.1$
(low precision) and $c=0.5$ (correlated data). With these large uncertainties,
the reweighting does not give any significant reduction of uncertainty at
$\xi=0.5$ in Ioffe time ($r_{\nu}=0.93$), and even an increase of uncertainty
in momentum space ($r_{\textrm{ln}x}=1.15$) by smearing the distribution.

Let us now add data for $\xi\in\\{0.1,0.2,0.3\\}$ while keeping $b=1.1$ and
$c=0.5$. The retainment of uncertainty at $\xi=0.5$ drops to $r_{\nu}=0.82$
and $r_{x}=1.0$ (figure 8). If we now add data for
$\xi\in\\{0.1,0.2,0.3,0.4,0.5\\}$, uncertainty retainment at $\xi=0.5$
tightens to $r_{\nu}=0.65$ and $r_{x}=0.75$ (figure 9). It is however not
better than a simple reweighting directly at $\xi=0.5$ which results in
$r_{\nu}=0.58$ and $r_{x}=0.64$ (see Table 1 for all results, including
various combinations of uncertainty and correlation that we have not mentioned
in the text or figures). This demonstrates that adding some data at a given
value of $\xi$ only produces a rather minimal effect on other values of higher
$\xi$ within our modelling of GPDs.

Figure 8: Same caption as figure 4 up to the fact that the GPD is shown at $\xi=0.5$, with mock data added at $\xi=\\{0.1,0.2,0.3\\}$, $b=1.1$ (low precision) and $c=0.5$ (correlated data). The average uncertainty retainments are $r_{\textrm{ln}x}=1.0$ and $r_{\nu}=0.82$, $\tau=0.77$. Figure 9: Same caption as figures 4 up to the fact that the GPD is shown at $\xi=0.5$, with mock data added at $\xi=\\{0.1,0.2,0.3,0.4,0.5\\}$, $b=1.1$ (low precision) and $c=0.5$ (correlated data). The average uncertainty retainments are $r_{\textrm{ln}x}=0.75$ and $r_{\nu}=0.65$, $\tau=0.57$. | Data |  |  |  |  |  | Results |
---|---|---|---|---|---|---|---|---
$\xi_{\text{Used}}$ | Precision | Correlation |  | $\xi_{\text{Shown}}$ |  | $\tau$ | $r_{\nu}$ | $r_{\textrm{ln}x}$
0.1 | Low | Low |  | 0.1 / 0.5 |  | 0.47 | 0.25 / 0.92 | 0.82 / 1.24
0.1 | Low | High |  | 0.1 / 0.5 |  | 0.83 | 0.85 / 0.93 | 1.02 / 1.15
0.1 | High | Low |  | 0.1 / 0.5 |  | 0.30 | 0.16 / 0.90 | 0.78 / 1.08
0.1 | High | High |  | 0.1 / 0.5 |  | 0.46 | 0.23 / 0.91 | 0.82 / 1.23
0.5 | Low | Low |  | 0.5 |  | 0.36 | 0.44 | 0.67
0.5 | Low | High |  | 0.5 |  | 0.52 | 0.58 | 0.64
0.5 | High | Low |  | 0.5 |  | 0.11 | 0.25 | 0.54
0.5 | High | High |  | 0.5 |  | 0.37 | 0.51 | 0.77
0.1 0.2 0.3 | Low | Low |  | 0.5 |  | 0.30 | 0.62 | 0.95
0.1 0.2 0.3 | Low | High |  | 0.5 |  | 0.77 | 0.82 | 1.00
0.1 0.2 0.3 | High | Low |  | 0.5 |  | 0.10 | 0.34 | 0.54
0.1 0.2 0.3 | High | High |  | 0.5 |  | 0.30 | 0.61 | 0.73
0.1 0.2 0.3 0.4 0.5 | Low | Low |  | 0.5 |  | 0.16 | 0.19 | 0.66
0.1 0.2 0.3 0.4 0.5 | Low | High |  | 0.5 |  | 0.57 | 0.65 | 0.75
0.1 0.2 0.3 0.4 0.5 | High | Low |  | 0.5 |  | 0.03 | 0.13 | 0.45
0.1 0.2 0.3 0.4 0.5 | High | High |  | 0.5 |  | 0.18 | 0.25 | 0.77
Table 1: Results as a function of the reweighting parameters. Low Correlation:
$c=0$, High Correlation: $c=0.5$, Low Precision: $b=1.1$, High Precision:
$b=2$, $r_{\textrm{ln}x}$: Average uncertainty retainment in $x$, $r_{\nu}$:
Average uncertainty retainment in $\nu$, $\tau$: Effective fraction of
replicas retained after reweighting.

##  7 Conclusion

We have presented a study of the impact of mock lattice QCD data at moderate
value of $\xi$ on a GPD model. The latter is based on machine learning
techniques and fitted to the forward limit and diagonal $x=\xi$ of the
phenomenological GK model, which represents the typical experimental
information available on GPDs. We further enforce a positivity constraint,
which considerably limits the freedom of the model in the region $x>\xi$. We
observe as a result that our model uncertainties are largely autocorrelated in
the small Ioffe-time region at small $\xi$, meaning that lattice data only
bring minimal additional reduction of uncertainty in momentum space. We
observe that the reduction of uncertainty in momentum space is systematically
inferior to that in Ioffe time space, as a consequence of the inverse problem
of relating the two representations of GPDs. We also observe that the addition
of data at some low values of $\xi$ impacts only minimally the GPD at higher
values of $\xi$. However, the latter point happens in a context where the
$t$-dependence is neglected. When restored, one recovers the $t$-dependent PDF
at zero skewness, a quantity which is not directly constrained by experimental
data. We thus expect that the impact of lattice QCD data in the low-$\xi$
region will increase with the value of $|t|$.

The Bayesian method employed here appears to be an adequate way to combine
both experimental and lattice knowledge on GPDs, when the lattice data is
globally in agreement with the prior model. However, our study highlighted the
impact of correlations within the lattice data on a potential joint
extraction. Real lattice data frequently present a very high degree of
correlation, along with systematic effects that are only starting to be under
control. This mandates a very careful treatment to avoid significant biases in
the assessment of uncertainty reduction, which represents a challenge for the
community.

Acknowledgments We thank Savvas Zafeiropoulos, Valerio Bertone, Hervé
Moutarde, Benoît Blossier and Kostas Orginos for stimulating discussions. This
work is supported in part in the framework of the GLUODYNAMICS project funded
by the ”P2IO LabEx (ANR-10-LABX-0038)” in the framework ”Investissements
d’Avenir” (ANR-11-IDEX-0003-01) managed by the Agence Nationale de la
Recherche (ANR), France. HD is supported in part by the U.S. DOE Grant #DE-
FG02-04ER41302 and the Gordon and Betty Moore Foundation. The work of P.S. was
supported by the Grant No. 2019/35/D/ST2/00272 of the National Science Centre,
Poland

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